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category: literaturenote citekey: fazlollahtabarautonomousguidedvehicles2015 title: Autonomous Guided Vehicles authors: "Fazlollahtabar, Hamed; Saidi-Mehrabad, Mohammad" year: 2015 date: 2015-00-00 2015 doi: 10.1007/978-3-319-14747-5 url: "https://doi.org/10.1007/978-3-319-14747-5" zotero_key: TDHCPDHX zotero_storage: CJN2BSVB collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Fazlollahtabar, Hamed"

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Hamed Fazlollahtabar Mohammad Saidi-Mehrabad

Autonomous Guided Vehicles

Methods and Models for Optimal Path Planning

Studies in Systems, Decision and Control

Volume 20

Series editor

Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland e-mail: kacprzyk@ibspan.waw.pl

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Autonomous Guided Vehicles

Methods and Models for Optimal Path Planning

Hamed Fazlollahtabar Faculty of Industrial Engineering Iran University of Science and Technology Tehran Iran

Mohammad Saidi-Mehrabad Faculty of Industrial Engineering Center of Excellence for Advance Manufacturing Iran University of Science and Technology Tehran Iran

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-319-14746-8 ISBN 978-3-319-14747-5 (eBook) DOI 10.1007/978-3-319-14747-5

Library of Congress Control Number: 2014958756

Springer Cham Heidelberg New York Dordrecht London

c Springer International Publishing Switzerland 2015

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Preface

The Automated Guided Vehicle (AGV) is an automated guided cart that follows a guided path. This equipment is widely used in industrial fields and places of physical distribution. We have developed a new type of AGV that has additional functions such as following the motion of people and avoiding obstacles on the course it is traveling on. Therefore, it can work with service personnel even though there is no pre-determined guided path. In general, an AGV is a driverless transport system used for horizontal movement of materials. AGVs are especially used for the internal and external transport of materials. Moreover, vision or lasers can be used for determining the movement of an AGV.

Since their introduction in 1955, AGVs have found widespread industrial applications. AGVs are now found in all types of industries, with the only restrictions on their use mainly resulting from the dimensions of the goods to be transported or spatial considerations. Many applications of AGVs are technically feasible, but the purchase and implementation of such systems is usually based on economic considerations. The uses of AGVs can be divided into four main areas of application:

• 1) supply and disposal at storage and production areas;
• 2) production-integrated application of AGV trucks as assembly platforms;
• 3) retrieval, especially in wholesale trade; and
• 4) supply and disposal in special areas, such as hospitals and offices. In all of these settings.

AGVs have been found to reduce the damage to inventory, make production scheduling more flexible, and reduce staffing needs. But, as with any other major capital decision, implementation of these systems must be undertaken cautiously.

The AGVs were traditionally employed in manufacturing systems, but have recently extended their popularity to many other industrial applications, such as goods transportation in warehouses and container transshipment systems at container terminals. Industrial transportation systems using AGVs are used in warehouses and manufactures.

An AGV system is a fully automated industrial transport system that makes use of numerous AGVs. An AGV is a battery powered, computer controlled, VI Preface

unmanned vehicle that is capable of transporting goods in an industrial environment. The AGVs are restricted to follow pre-determined paths, on which they navigate by using sensors and stationary beacons. Transports, which consist of picking up a load on a certain location and transporting it to a certain destination, are generated by an external system and have to be assigned to the AGVs. Traditionally, one computer system (central server) is in charge of numerous complex and time-consuming tasks such as routing, collision avoidance or deadlock avoidance.

The AGVs themselves have little autonomy. Although this planning is efficient, it lacks flexibility. In highly dynamic systems, where the situation changes frequently, problems are experienced. A new and innovative architecture has been developed that additional qualities, like flexibility and openness, to cope with the highly dynamic environments.

AGV usage is growing. One reason is that as manufacturers strive to become more competitive, they are adopting flexible manufacturing systems (FMS). These systems integrate automated material handling systems, robots, numerically controlled machine tools, and automated inspection stations. Flexible manufacturing systems offer high capital utilization and reduced direct labor costs. They also reduce work-in-process inventories and make it possible to work with shorter lead times. Because the systems are flexible, they are more responsive to changes in production requirements. These systems offer high product quality and increased productivity.

In this book different models and methods for optimizing AGV's path are developed. The chapters are useful for academic courses in graduate studies. The models are classified into single and multiple AGVs. Also, models under uncertain conditions including fuzzy and statistical approaches are proposed.

The book chapters are organized as follows. In the first chapter an overview of the models and methods of scheduling and routing problems for AGVs is presented. In chapter 2, material flow optimization using AGV is discussed. In chapter 3, a manufacturing system equipped with AGV is modeled using a nonlinear stochastic mathematical programming. In chapter 4, reliability of AGVs is considered and optimized. In chapter 5, path planning under uncertainty is explained and modeled. In chapter 6, AGV routing under stochastic run time is modeled by cross entropy concept. In chapter 7, expert system is designed for AGV optimal path planning considering multiple criteria. In chapter 8, intelligent

Preface VII

agent is developed for path planning of AGV. In chapter 9, delay optimization for multiple AGVs is developed and modeled. In chapter 10, Markovian modeling is proposed for evaluation of multiple AGV system. In chapter 11, producer behavior in an AGV equipped manufacturing system is analyzed. In the last chapter risk evaluation for AGVs in a manufacturing system is described.

January 2015 Hamed Fazlollahtabar Tehran, Iran Mohammad Saidi-Mehrabad

Acknowledgments

We would like to express our gratitude to the many people who saw us through this book; to all those who provided support, talked things over, read, wrote, offered comments, allowed us to quote their remarks and assisted in the editing, proofreading and design.

We would like to thank Iran University of Science and Technology for enabling us to publish this book. Above all we want to thank our families, who supported and encouraged us in spite of all the time it took us away from them. It was a long and difficult journey for them.

We would like to thank Prof. Nezam Mahdavi-Amiri for helping us in the process of selection and editing. Thanks to our publisher who encouraged us.

Thanks to Prof. Abdolreza Sheikholeslami - without you this book would never find its way to the academician.

Last and not least: We beg forgiveness of all those who have been with us over the course of the years and whose names we have failed to mention.

H. Fazlollahtabar M. Saidi-Mehrabad

1.1	Summary	1
1.2	Scheduling Models	2
		4
		12
		13
		17
		17
		19
		20
		24
		25
		27
		27
		29
		30
		37
		38
		41
		41
		44
		45
		53
		54
		57
		57
	Statement of the Problem	59
	1.3 1.4 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 5.1 5.2	Routing Models Conclusions References Analytical Material Flow Model for AGV System Summary Statement of the Problem Mathematical Model Conclusions References Nonlinear Stochastic Model for AGV System Summary Statement of the Problem Mathematical Model Conclusions References Reliability Model for AGV Summary Statement of the Problem Mathematical Model Conclusions References Uncertain Optimal Path for AGV System Summary

	5.3	Mathematical Modeling	60
	5.4	Conclusions	75
		References	75
6		Cross Entropy Model for AGV Routing Time	79
	6.1	Summary	79
	6.2	Statement of the Problem	82
	6.3	Mathematical Model	83
	6.4	Conclusions	90
		References	90
7		Neuro-Fuzzy-Regression Expert System for AGV Optimal Path	93
	7.1	Summary	93
	7.2	Statement of the Problem	96
	7.3	Mathematical Model	98
	7.4	Conclusions	112
		References	112
8		Optimal Path for AGV System with Intelligent Agents	117
	8.1	Summary	117
	8.2	Statement of the Problem	120
	8.3	Mathematical Model	126
	8.4	Conclusions	130
		References	130
9		Earliness/Tardiness for a Multiple AGV System	133
	9.1	Summary	133
	9.2	Statement of the Problem	136
	9.3	Mathematical Model	138
	9.4	Conclusions References	144 144
		10 Markovian Model for Multiple AGV System	147
		10.1 Summary	147
		10.2 Statement of the Problem	151
		10.3 Mathematical Model	153
		10.4 Conclusions	166
		References	166

11 Producer's Behavior Analysis for AGV System
11.1 Summary
11.2 Statement of the Problem
11.3 Mathematical Model
11.4 Conclusions
References
12 Risk for Multiple AGV System
12.1 Summary
12.2 Statement of the Problem
12.3 Mathematical Model 12.4 Conclusions

Chapter 1 Models for AGVs' Scheduling and Routing

1.1 Summary

An automated guided vehicle (AGV) is a driverless material handling system used for horizontal movement of materials. AGVs were introduced in 1955 (Muller, 1983). The use of AGVs has grown enormously since their introduction. The number of areas of application and variation in types has increased significantly. AGVs can be used in inside and outside environments, such as manufacturing, distribution, transshipment and (external) transportation areas. At manufacturing areas, AGVs are used to transport all types of materials related to the manufacturing process. According to Gotting (2000) over 20,000 AGVs were used in industrial applications. The author states that the usage of AGVs will pay off for environments with repeating transportation patterns. Examples of these environments are distribution, transshipment and transportation systems. Warehouses and cross docking centers are examples of distribution areas. AGVs are used in these areas for the internal transport of, for example, pallets between the various departments, such as receiving, storage, sorting and shipment areas. At transshipment systems, such as container terminals, AGVs take care of the transport of products between the various modes of transport. Gotting (2000) presented an overview of available technology for automation in container terminals. Furthermore, navigation and vehicle guidance systems applicable in various indoor/outdoor environments are described. Haefner and Bieschke (1998) stated that AGV systems can provide benefits to both the port and its customers by executing transportation requests between vessels and inland transportation.

Namely, in non-automated terminals this transportation process is one of the least efficient and most costly processes. AGVs can also be used in the outdoor transportation process. An example of such a transportation system is an underground automated transportation system with AGVs travelling in tubes between companies and an airport (see Van der Heijden et al., 2002 a,b). In such systems, we notice a high traffic density and long tube driving times. It has even been studied if AGVs can be used as a communication system between work stations (see Maughan and Lewis, 2000). Clearly, the specifications of AGVs differ per environment. To transport a container, the capacity of an AGV should at

1

least be equal to 40 tons. Less capacity is required for the transport of pallets at warehouses. Furthermore, at container terminals self-lifting automated guided vehicles (ALVs) are used. For this type of AGV no other equipment is required to transfer a load to the vehicle. Vis and Harika (2004) and Yang et al. (2004) discussed this new type of AGVs in more detail.

In this chapter, we will discuss literature concerning the usage of AGVs in manufacturing and the new areas of application, namely distribution, transshipment and transportation systems. The most important differences between traditional and new areas of application are the number of AGVs used, the number of transportation requests, the occupancy degree of AGVs, the distances to be travelled and the number of pick-up and delivery points where transportation requests become available. At manufacturing systems, a small number of AGVs with relatively low occupancy degrees are used to transport a small number of requests over short distances between a few pick-up and delivery points. For continuous mass transport in these systems conveyors are used instead of large numbers of AGVs (see Gotting, 2000). In contrast to manufacturing systems, large numbers of AGVs (up to 400; Van der Heijden et al., 2002a) were used to execute a large number of repeating transportation tasks at container terminals and external transportation systems. Furthermore, operational conditions, such as weather conditions and spatial dimensions, found in outside environments (container terminals and external transportation systems) differ from the operational conditions in inside areas (manufacturing and distribution systems).

1.2 Scheduling Models

The literature discussed so far on scheduling of AGVs hardly considers (side) constraints such as capacity constraints of the machines where transportation jobs become available, schedules of other types of equipment and limited parking space for vehicles. These side constraints become more and more important in real life situations with large AGV systems. To take these constraints into account more attention should be paid to integrated scheduling of different types of material handling equipment, which also meet space and capacity requirements.

1.2.1 Artificial Intelligence

The increased use of FMS to efficiently provide customers with diversified products has created a significant set of operational challenges. Although extensive research has been conducted on design and operational problems of automated manufacturing systems, many problems remain unsolved. In particular, the scheduling task, the control problem during the operation, is of importance owing to the dynamic nature of the FMS such as flexible parts, tools and AGV routings. The FMS scheduling problem has been tackled by various traditional

optimization techniques. While these methods can give an optimal solution to small-scale problems, they are often inefficient when applied to larger-scale problems. Jerald et al. (2005) designed different scheduling mechanisms to generate optimum scheduling; these included non-traditional approaches such as genetic algorithm (GA), simulated annealing (SA) algorithm, memetic algorithm (MA) and particle swarm algorithm (PSA) by considering multiple objectives, i.e., minimizing the idle time of the machine and minimizing the total penalty cost for not meeting the deadline concurrently. The MA presented was essentially a GA with an element of SA. The results of the different optimization algorithms (MA, GA, SA, and PSA) were compared and conclusions were presented.

Solving multi-objective scientific and engineering problems is, generally, a very difficult goal. In these optimization problems, the objectives often conflict across a high-dimensional problem space and require extensive computational resources. Sarvana Sankar et al. (2006) developed a migration model of parallelization for a GA based multi-objective evolutionary algorithm (MOEA). The MOEA generated a near-optimal schedule by simultaneously achieving two contradicting objectives of an FMS. The parallel implementation of the migration model showed a speedup in computation time and needed less objective function evaluations when compared to a single-population algorithm. So, even for a single-processor computer, implementing the parallel algorithm in a serial manner (pseudo-parallel) delivers better results. Two versions of the migration model were constructed and the performance of two parallel GAs was compared for their effectiveness in bringing genetic diversity and minimizing the total number of functional evaluations.

1.2.2 Simulation

Gaur et al. (2003) studied the problem of scheduling an AGV in a flexible manufacturing system while minimizing completion times. A vehicle needs to visit each site only after its release time and before its due time. Sabuncuoglu (1998) used simulation to test various AGV scheduling rules.

Fazlollahtabar et al. (2012) concerned with applying tandem automated guided vehicle (TAGV) configurations as material handling devices and optimizing the production time considering the effective time parameters in a flexible automated manufacturing system (FAMS) using Monte Carlo simulation. Due to different configurations of TAGVs in an FAMS, the material handling activities are performed. With respect to various stochastic time parameters and the TAGV defects during material handling processes, sample data were collected and their corresponding probability distributions were fitted. Using the probability distributions, they modelled the TAGV material handling problem via Monte Carlo simulation. The effectiveness of the proposed model was illustrated in a case study.

1.3 Routing Models

If the dispatching decision is carried out, a route and schedule should be planned for the AGV to move the job from its origin to its destination within the AGV network. A route implies the path which should be taken by the AGV when making a pick-up or delivery. The related schedule gives arrival and departure times of the AGV at each part, pick-up and delivery point and intersection during the route to ensure collision free routing. The selection of a certain route and schedule is effective on the performance of the system. The longer it takes to transport a job, the fewer the jobs that can be handled within a certain time. Therefore, one of the objectives of the routing of AGVs is to minimize transportation times. Algorithms have to be developed to solve the routing problem. Two categories of algorithms can be distinguished, namely static and dynamic algorithms.

Analogies between these problems from transportation literature and routing and scheduling problems for AGVs in automated guided vehicle systems are clear. A number of loads at various locations have to be transported by vehicles at a certain start time or at a certain moment within a time window. However, the use of the described models from transportation literature is not always possible. These models do not take into account congestion in the system. Furthermore, most models are not developed to deal with real time response to dynamically changing transportation requests. Therefore, attention is paid in the literature to developing non-conflicting routes for AGVs. With a non-conflicting route, an AGV arrives as early as possible at the destination without conflicting with other AGVs.

In AGV routing with static algorithms the route from node i to node j is determined in advance and is always used if a load has to be transported from i to j. In this way, a simple assumption is to choose the route with the shortest distance from i to j. However, these static algorithms are not able to adapt to changes in the system and traffic conditions.

In dynamic routing, the routing decision is made based on real-time information and, as a result, various routes between i and j can be chosen. Static routing problems in AGV systems are related to vehicle routing problems (VRP) studied in transportation literature. In the vehicle routing problem a set of n clients with known demands need to be served by a fleet of m vehicles with limited capacity. The vehicles are all housed at one depot. The route of each vehicle starts and ends at this depot. m least costs (length) routes have to be planned such that each customer is served exactly once and that the total demand of the customers served by each vehicle does not exceed the capacity of each vehicle. The objective is to minimize the total distance of all m routes under previously mentioned conditions. This is an NP-hard problem to solve. The vehicle routing problem has been studied extensively in literature. Bodin et al. (1983), Laporte (1992) and Fisher (1995) provided an overview of literature in this area. A more recent paper observing this problem is from Kelly and Xu (1999). They proposed a set

1.3 Routing Models 5

partitioning based heuristic. In a systematic way fragments of routes are combined to obtain high-quality solutions. Vehicle scheduling problems can be seen as routing problems with additional constraints concerning times at which certain activities (e.g. delivery of a load) have to be executed. Vehicle activities have to be sequenced both in time and space. An overview of methods to solve vehicle scheduling problems has been given in Bodin et al. (1983).

Numerous studies on the vehicle routing problem with time windows have been executed. Branch and bound methods (Kolen et al., 1987), insertion heuristics (Solomon, 1987), extensions of vehicle routing problem heuristics (Solomon et al., 1988), Lagrangian relaxation (Fisher et al., 1997; Kohl and Madsen, 1997), constrained shortest path relaxation (Desrochers et al., 1992; Kohl et al., 1999) and set covering formulations (Bramel and Simchi-Levi, 1997) can be used to find solutions to the problem. Desrochers et al. (1988) provided an overview of solution methods. More recently, Cordeau et al. (2002) presented a survey of approximation and optimal approaches to solve the vehicle routing problem with time windows.

1.3.1 Artificial Intelligent

Owing to the lack of robust design and control algorithms, most current applications of AGV systems employ simple control methods despite the fact that the system is far from efficient. Petri nets have evolved into a powerful tool for modelling complex manufacturing systems. One of the advantages of the use of Petri nets is that analysis, simulation and on-line control can all be done on the same model once the model is built. By also using petri nets, conflict free routes can be determined (see Zeng et al., 1991). The purpose of Hsieh and Lin (1991) was to establish the research fundamentals in the field of the Petri-net modelling of an AGVS. The main contribution of the chapter was to define basic trafficcontrol nets which can be used directly to model an AGVS without too much thinking. Some basic AGV Petri-net control elements were described and illustrated in detail. Also, difficulties in the use of bidirectional flows were discussed, and it has been determined how Petri nets can be used to solve these problems.

The main problem for an AGV dispatching system is to assign vehicles to transport demands which optimize some predetermined objectives of a manufacturing shop. Singh and Tiwari (2004) presented a framework for an AGV dispatching system based on an object oriented approach using the unified modeling language (UML), and the development of a dispatching algorithm to facilitate a human controller to dispatch efficiently a fleet of AGVs in response to calls from any shop floor (or machine) operator. The main reason for their work was to model an AGV dispatching system as well as to develop a dispatching algorithm which can record details of the AGV position and movement and ensure their allocation of new orders. The provision of both immediate and pre-booked orders for an AGV was also incorporated in the proposed AGVs dispatching

system. The underlying AGV dispatching system and algorithm were capable of dispatching a vehicle automatically to handle a call at the required time. In order to overcome difficulties associated with tackling immediate orders, pre-booked orders, and processing of information related to AGVs, a comprehensive dispatching algorithm was developed which aims to minimize lateness, traveling time and distance of empty vehicles in a simulated jobshop scenario.

Kizil et al. (2006) evaluated the effects of various dispatching rules on the operation and performance of cellular manufacturing systems (CMS). When the study of a CMS considers the automated material handling, it is crucial to reduce the gridlock probability (i.e., the probability of an unsuccessful load transfer attempt occurring in the interface point between the intercell and intracell handling system). Preventing an unsuccessful load transfer is critical for the operation of the entire system as a blockage between the AGV and the overloaded cell results in a total system shutdown. The gridlock probability was influenced by the dispatching rule used to schedule the load transfers in the system. Therefore, in order to reduce this probability it was necessary to use a dispatching rule that will decrease the number of waiting loads in the transfer spurs. The main objective of the paper was to identify a dispatching rule that maintains the system operational at all times. A group of dispatching rules, including the first come first served, shortest imminent operation, longest imminent operation, most remaining operations, shortest processing time, shortest remaining process time, and a newly developed rule proposed by the authors, loads with the minimum number of processing first, were tested and evaluated with respect to whether the capacity of the transfer spurs of the cells was exceeded. The paper presented a simulation model of a cellular manufacturing system, which was used to further explore the effects of the dispatching rules on the system performance.

AGVs are the most flexible means to transport materials among workstations of a flexible manufacturing system. Complex issues associated with the design of AGV control of these systems are conflict-free shortest path, minimum time motion planning and deadlock avoidance. Srivastava et al. (2008) presented an intelligent agent-based framework to overcome the inefficacies associated with the aforementioned issues. Proposed approach described the operational control of AGVs by integrating different activities such as path generation, journey time enumeration, collision and deadlock identification, waiting node location and its time estimation, and decision making on the selection of the conflict-free shortest feasible path. It represented efficient algorithms and rules associated with each agent for finding the conflict-free minimum time motion planning of AGVs, which were needed to navigate unidirectional and bidirectional flow path network. A collaborative architecture of AGV agent and its different modules were also presented. Three complex experimental scenarios were simulated to test the robustness of the proposed approach. It was shown that the proposed agent-based controller was capable of generating optimal, collision- and deadlock-free path with less computational efforts.

The objective of Aized (2009) was to model and maximize performance of an integrated AGVS, which is embedded in a pull type multi-product, multi-stage and 1.3 Routing Models 7

multi-line FMS. The author examined the impact of guide-path flexibility on system performance through the development of three different guide-path configurations which range from dedicated to flexible relationships between AGVs and machine/assembly station resources. The system was modelled using coloured Petri net method (CPN) and the simulation results lead to identify the resource redundancy which can be rectified to achieve lower overall cost of the system through the development of flexible guide-path configurations. The study was extended to seek global near-optimal conditions for each guide-path configuration using response surface method, which yields improvements in system throughput and cycle time along with a decrease in the numbers of AGVs.

Material handling in manufacturing systems is becoming easier as the automated machine technology is improved. Nowadays, most of the research aims at increasing the flexibility and improving the performance of the AGV. Yahyaei et al. (2010) designed and made AGV in the Industrial Control Laboratory in Royce Lab at the University of Manchester Institute of Science and Technology. For controlling the navigation of the AGV, a newly developed controller integrated fuzzy logic with programmable logic controller was used. By using integrated fuzzy logic controller with programmable logic controller (IFLPLC), the flexibility of AGV was increased and they achieved great advantages. Since that AGV used programmable logic controller and fuzzy logic controllers together, it proved usefulness for factories which implement FMS. Online maintenance and sending the commands to other machines from AGV and so on were the advantages that can be used in FMS.

Fazlollahtabar and Mahdavi-Amiri (2012) proposed an approach for finding an optimal path in a flexible jobshop manufacturing system considering two criteria of time and cost. A network was configured in which the nodes are considered to be the shops with arcs representing the paths among the shops. An automated guided vehicle functioned as a material handling device through the manufacturing network. The expert system for cost estimation was based on fuzzy rule backpropagation network to configure the rules for estimating the cost under uncertainty. A multiple linear regression model was applied to analyze the rules and find the effective rules for cost estimation. The objective was to find a path minimizing an aggregate weighted unscaled time and cost criteria. A fuzzy dynamic programming approach was presented for computing a shortest path in the network. Then, a comprehensive economic and reliability analysis was worked out on the obtained paths to find the optimal producer's behavior.

Singh and Tiwari (2002) presented an intelligent agent framework to find a conflict-free shortest-time path for an AGV travelling in a bi- or unidirectional network.

Fazlollahtabar and Mahdavi-Amiri (2013) proposed an approach for finding an optimal path in a flexible jobshop manufacturing system considering two criteria of time and cost. With rise in demands, advancement in technology and increase in production capacity, the need for more shops persists. Therefore, a flexible jobshop system has more than one shop with the same duty. The difference among shops with the same duty is in their machines with various specifications.

A network was configured in which the nodes were considered to be the shops with arcs representing the paths among the shops. An AGV functioned as a material handling device through the manufacturing network. To account for uncertainty, the authors considered time to be a triangular fuzzy number and applied an expert system to infer cost. The objective was to find a path minimizing both the time and cost criteria, aggregately. Since time and cost have different scales, a normalization procedure was proposed to remove the scales. The model being biobjective, the analytical hierarchy process weighing method was applied to construct a single objective. Finally, a dynamic programming approach was presented for computing a shortest path in the network. The efficiency of the proposed approach was illustrated by a numerical example.

1.3.2 Simulation

Software aids for simulation are very important to practitioners of simulation. The widespread availability of inexpensive computing power now allows computer assistance in each stage of simulation activities such as input data analysis, modelling, programming, output analysis and so on. Therefore Ashayeri and Gelders (1987) described an interactive microcomputer GPSS simulation program generator for automated material handling systems. The program was written in Pascal and consisted of several modules to capture data, build the model, and generate the corresponding GPSS simulation program for automated guided vehicle systems as well as surge systems. The application of the program to a real life project was used to highlight practical advantages of the proposed approach.

Automated Guidance Vehicles' guidance techniques make use, in most vehicles, of a painted strip or a cable buried in the floor. Free ranging vehicles are now also available in limited types but at prohibitive cost. Katz and Bright (1992) presented a type of guidance technique whereby the vehicle follows a path created by the light emitted from suspended fluorescent lights. The method has been tested on a multi-directional vehicle. The results were encouraging when compared with other available guidance techniques. The possible integration of this guidance method with a navigation method could make AGVs more attractive to most of the manufacturing environments. The design, implementation and experimental results were outlined and described.

A simulation-based cost model was presented by Kasilingam and Gobal (1996) for determining the number of AGVs needed to meet the material handling requirements in a manufacturing system. The estimation of the number of vehicles was based on the sum of the idle-time costs of vehicles and machines, and the cost of waiting time of parts. While an increase in the number of vehicles reduces the waiting time of parts and the idle time of machines, it increases the idle time of vehicles. The application was illustrated using a hypothetical manufacturing system.

A TAGV system is obtained by partitioning all workstations into multiple zones and assigning a single vehicle to each zone. Kim et al. (2003) proposed an analytical model to design a tandem AGVS with multi-load AGVs. Using simulations, the performance of the proposed model was shown by comparing it with a conventional multi-load AGVS.

Kim and Jae (2003) presented an object-oriented simulation modelling environment, AgvTalk, to provide flexible modelling capabilities for the simulation of many alternative AGV systems. The hierarchical features and modularity of AgvTalk created possibilities for the extension and reuse of simulation object components. The detailed behaviour of each object in the AGV system can also be modelled easily and exactly in AgvTalk because there were no limiting modeling constructs. The modelling capabilities of AgvTalk were demonstrated by designing and simulating a conceptually different configuration AGV system, known as the tandem configuration. For the tandem and conventional AGV systems, the characteristics and design methodology in AgvTalk were described. Simulations between the two systems were compared using AgvTalk in a job shop environment.

Um et al. (2009) presented the simulation design and analysis of an FMS with an AGV system. To maximize the operating performance of FMS with AGVs, many parameters must be considered, including the number, velocity, and dispatching rule of AGV, part types, scheduling, and buffer sizes. Of the various critical factors, they considered the following three: (1) minimizing the congestion; (2) minimizing the vehicle utilization; and (3) maximizing the throughput. The authors considered the systematic analysis methods that combined a simulation-based analytic and optimization technique that was Multi-Objective Non-Linear Programming (MONLP) and Evolution Strategy (ES). MONLP determined the design parameters of the system through multi-factorial and regression analyses. ES was used to verify each parameter for simulationbased optimization. A validation test for the two methods was conducted. This method-based approach towards design yielded the correct experimental results, ensures confidence in the specification of design parameters and supports a robust framework for analysis.

Kim et al. (2009) proposed a simple blocking prevention method for a pathbased automated material handling system (AMHS) such as a semiconductor fabrication line. In their research, blocking means the situation in which a vehicle stands and waits without doing anything because the vehicle in front of it is in the process of loading or unloading. Since a typical bay type path-based AMHS has a single path in each bay and no sidetracks, the blocking issue is inevitable in a large complex system with many vehicles. The proposed method was based on the swapping of load assignments between retrieval vehicles on the same path. The simulation study on an example semiconductor line showed that the proposed method improves the AMHS productivity under various vehicle dispatching rules.

Guan and Dai (2009) developed a flexible, efficient, and deadlock-free dispatching method for automated guided vehicle systems. For this purpose, a deadlock-free multi-attribute dispatching method with dynamically adjustable weights (AWMA) was proposed. Traveling distance, input, and output buffer statuses were selected as dispatching attributes according to the efficiency and deadlock avoidance requirement. The weight for each attribute was dynamically adjusted according to the processing load and transportation load of the system. To ensure the system to be deadlock-free, a deadlock avoidance policy based on remaining capacity concept was introduced. It worked by temporarily forbidding critical tasks according to the system state, which will otherwise cause system deadlock. The AWMA method was formed by integrating the deadlock avoidance policy into the multi-attribute dispatching procedure. To validate the effectiveness of the proposed method, several simulation experiments were carried out to compare three commonly used dispatching methods with the proposed one under different system settings. The simulation results indicated that the deadlock avoidance policy can guarantee the system to be deadlock-free and that the proposed method was efficient.

In the majority of small and medium sized enterprises, the direct costs of material handling cannot be clearly measured. There are several reasons for this, including the large number of product types, complexity of their production cycle, and continuous change in markets. Therefore, production managers require flexible tools to create a suitable material handling system model which explicitly and rapidly calculates the indices required as these are traditionally neglected or laboriously approximated, (i.e., time and cost in material flow inside the factory, storage area requirements, and MH utilization percentage). Gamberi et al. (2009) proposed an integrated approach to analyzing and controlling material handling operations in an industrial manufacturing plant from a "full quantitative" point of view. The model presented united quite different fields of research into a unique methodology. The material handling model rapidly and automatically provided production managers with extensive and significant information. As a result, integrated layout flow analysis interrelated systematic layout planning with operational research algorithms and visual interactive simulation, using a complete software platform to implement them. This integrated layout flow analysis approach focused on determining the space requirement for manufacturing department buffers, the transportation system requirements, the performance indices, and the time and cost of material flows spent in the layout and in MH traffic jams.

Flexible material handling systems (FMHS) have been widely used to enhance productivity involved with product proliferation, and thus far, only fixed-track material handling systems such as Eton systems in the apparel industry are commonly used. Dai et al. (2009) explored the potential advantages of a FMHS using free-ranging automated-guided vehicles with a local positioning system for the apparel industry. First, the free-ranging FMHS (FRMHS) for the apparel industry has been designed. Then, through Monte Carlo simulation and analytical models, the performance in terms of manufacturing system effectiveness, workstation utilization, and the total transportation distance of the FRMHS were compared with those of the fixed-track system. Based on their analysis, the current proposed FRMHS can have significant advantages over the fixed-track system.

Kuttolamadom et al. (2010) dealt with the path tracking and stability of motion of automated guided vehicle systems and wheeled mobile robots. A two degreeof-freedom dynamic model was developed to represent the plane motion of the vehicle. This model along with the instantaneous posture errors (position and orientation errors) of the vehicle were used to formulate their path-tracking problem in state space format. Implementation of stability criterion and application of the actual physical limits of the pertinent parameters of the system were the strategies used to design the controller. It was shown that implementation of a control strategy based on the vehicles' position and orientation errors gives satisfactory results in vehicles' path tracking. The feasibility of the approach and the performance of the controlled system were demonstrated by using a prototype vehicle.

Joseph and Sridharan (2011) focused on a simulation-based experimental study of the interaction among routing flexibility, sequencing flexibility and part sequencing rules in a typical FMS. Two scenarios were considered for experimentation. Three routing flexibility levels, five sequencing flexibility levels and four scheduling rules for part sequencing decision were considered for detailed investigation. The performance of the FMS was evaluated using various measures related to flow time and tardiness of parts. The simulation results were subjected to statistical analysis. The analysis of results reveals that deterioration in system performance can be minimized substantially by incorporating either routing flexibility or sequencing flexibility or both. However, the benefits of either of these flexibilities diminish at higher flexibility levels. Part sequencing rules such as earliest due date and earliest operation due date provide better performance for all the measures at higher flexibility levels.

Routing flexibility is a major contributor of the flexibility of an FMS. Joseph and Sridharan (2011) focused on the evaluation of the routing flexibility of an FMS with the dynamic arrival of part types for processing in the system. A typical FMS configuration was chosen for detailed study and analysis. The system was set at five different levels of routing flexibility. Operations of part types can be processed on alternative machines depending upon the level of routing flexibility present in the system. Two cases have been considered with respect to the processing times of operations on alternative machines. A discrete-event simulation model has been developed to describe the operation of the chosen FMS. The performance of the system under various levels of routing flexibility was analyzed using measures such as mean flow time, mean tardiness, percentage of tardy parts, mean utilization of machines, mean utilization of automatic-guided vehicles, and mean queue length at machines. The routing flexibility for producing individual part types has been evaluated in terms of measures such as routing efficiency, routing versatility, routing variety and routing flexibility. The routing flexibility of the system has been evaluated using these measures. The flexibility levels were ranked based on the routing flexibility measure for the system. The ranking thus obtained has been validated with that derived using fuzzy logic approach.

Nowadays, so as to adapt to the global market, where competition is getting tougher, firms producing through the modern production approach need to bring not the only performance of the system designed both during the research and development phase and the production phase but also the performance of the product to be developed as well as the process to be improved to the highest level. The Taguchi method is an experimental design technique seeking to minimize the effect of uncontrollable factors, using orthogonal arrays. It can also be designed as a set of plans showing the way data are collected through experiments. Experiments are carried out using factors defined at different levels and a solution model generated in ARENA 3.0 program using SIMAN, which is a simulation language. Many experimental investigations reveal that the speed and capacity of automated-guided vehicle, the capacities of local depots, and the mean time between shipping from the main depot are the major influential parameters that affect the performance criteria of the storage system. For the evaluation of experiment results and effects of related factors, variance analysis and signal/noise ratio were used and the experiments were carried out in MINITAB15 according to Taguchi L16 scheme. The purpose of Subulan and Cakmakci (2012) was to prove that experimental design was an utilizable method not only for product development and process improvement but it can also be used effectively in the design of material handling–transfer systems and performance optimization of automation technologies, which were to be integrated to the firms.

Seifert et al. (1998) introduced a dynamic vehicle routing strategy based on hierarchical simulation. At each time a route decision has to be made, simulations were performed to evaluate a set of possible routes. The route with the smallest estimated travel time was chosen.

Except for finding conflict free routes, attention should be paid to the presence of interruptions in the system. Interruptions might occur due to, for example, vehicle breakdowns, objects on AGV paths and manual intervention. As a result of interruptions, AGVs may be blocked and routes cannot be finished. Therefore, if an AGV encounters an interruption it has to be rerouted in such a way that no conflicts with other AGVs occur. Narasimhan et al. (1999) used simulation to analyze rerouting of AGVs.

1.4 Conclusions

Summarizing, in the context of manufacturing areas, static and dynamic algorithms have been developed to solve the routing of vehicles. Network models, queuing networks, simulation and intelligent routing techniques are used to route AGVs conflict free through the network. The routing of AGVs through distribution, transshipment and transportation systems is hardly studied.

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Chapter 2

Analytical Material Flow Model for AGV System

2.1 Summary

Manufacturing automation has become increasingly important as the need to remain productive increases. In manufacturing of a product, many processes may be needed. For example, semiconductor manufacturing may include over 400 processing steps involving more than 100 different tools. Furthermore, the process route can include a high level of reentrance in which the same tool or tool types are used. An important aspect in manufacturing automation is material handling. To facilitate material handling, automated transport systems are employed.

A special attention has been given during the last few years to the problem of production control in manufacturing systems (Akella et al., 1990; Gershwin, 1989). Flow models are often used in the intermediate level to represent the inventory balance equation of the production system. The aim is to compute an average production rate for each product type in order to meet the demand with minimum surplus or backlogging costs. During the optimization procedure capacity changes due to machine failures must be taken into account. In (Gershwin, 1989 ), changes in the system's capacity due to machine failure are described as a function of the state of the system and since machine breakdowns cannot be predicted, capacity is considered as a stochastic set. This means that for the dynamic system representing the flow of parts, a sudden change in the system's state due to machine failure is transformed to a change (or a jump) for the capacity constraint.

Conventional automated transport systems are used in, for example, a semiconductor manufacturing plant typically comprises a plurality of bays. Each bay includes a plurality of tools which are used to process wafers. Transport vehicles, such as overhead transport (OHT) are provided for automatic transfer of wafers contained in a carrier. A stocker which temporarily stores carriers is provided for each bay. A transport system controller controls the movement of carriers within a bay or between bays. In recent years, the automated material handling system has rapidly developed as an efficient manufacturing system. The AGV (automated guided vehicle) system plays an especially significant role

and has become more widely used in modern manufacturing environments due to its flexibility and precision. With an AGV system, we can easily respond to changes in production volume, product mix, product routing and so on (Ho and Liao, 2009).

However, due to the continuously increasing size and complexity of the modern manufacturing system, controlling this system has become more difficult. Consequently, various types of research have been conducted to solve this problem. Bozer and Srinivasan (1992) defined the conventional AGV system and proposed tandem configuration based on the ''divide and conquer'' principle. Multiple vehicles can exist in a traditional AGV system and each vehicle can pickup, deliver and drop off a load at any workstation. A tandem AGV system is obtained by partitioning all workstations into multiple zones, assuming a single vehicle to each zone. As a result, any potential blocking, congestion or deadlock does not occur. And the performance of the tandem AGV system has been demonstrated (Bozer and Srinivassan, 1992; Laporte et al., 2006; Asef-Vaziri et al., 2001; Kaspi et al., 2002).

The emergence of high performance automated manufacturing systems (AMSs) has lead to the need for methods of modeling these types of system in order to maximize throughput, flexibility and competitiveness. AMSs belong to the domain of discrete event dynamic systems (DEDS) in which the evolution of the system depends on the complex interactions of various discrete events such as the arrival of raw materials, departure of finished goods, failure of equipment etc. The state of DEDS changes only at these discrete points in time. Over the last decade several models have been presented to describe DEDS and these can be grouped into two distinct area (Rajagopalan et al., 2004).

Qualitative models are concerned with the logical aspects of system evolution such as controllability, stability and the existence of deadlocks in system operation, etc. This category also includes Petri Nets, extended state machines and finitely recursive processes (Castillo et al., 2001).

Quantitative models are concerned with the quantitative system performance in terms of throughput and lead time. This category also includes discrete event simulation, min-max algebra, Markov Chains, stochastic Petri nets, queues, and queuing networks (Berman et al., 2009) . Quantitative models are a general term including performance modeling which is the area of interest to this article. Within the life cycle of an AMS various decisions are made concerning implementation, design and operation of the system. Typical decisions at the planning stage include number and type of machines, number of material handling devices, number of buffers, size of pallet pool and number of fixtures, best possible layout, tool storage capacity, evaluate candidate AMS configurations, part type selection, machine grouping, batching and balancing decisions, and scheduling policies (Huang et al., 2009)

During the operational phase of an AMS, performance modeling can be used to assist decisions about how to react in the event of a breakdown, removal or addition of resources and parts, optimal scheduling in the event of machine failure or sudden changes in the product or its demand and in the avoidance of unstable situations such as deadlocks (Demange et al., 2009).

Performance modeling is also used in the design stage of the system. It is used in decisions such as whether to use central versus local storage, push production versus pull production, shared versus distributed resources, the effect of flexibility, etc. Performance predictions obtained using faithful models, can be used to convince customers or investors and also give the designer another perspective on the design enabling better designs (Aized, 2009).

The performance of an AMS can be measured by a set of generic measures. These are manufacturing lead time, work in progress, throughput, machine utilization, capacity, flexibility, performance, and quality (Gen et al., 2009). Using performance measuring these values can be evaluated and used to compare AMS performances.

Performance modeling has become a very important part of automated manufacturing system design and is equally important for maintaining the system at its peak of ability. The manufacturing methods in use by companies has changed dramatically in recent years with the use of advanced robotics and computer control to optimize production, this has lead to reduced prices and higher quality of product. The production lines can only get better with more modeling and investment and this is best achieved with the use of performance modeling.

2.2 Statement of the Problem

Here, we consider a jobshop layout which employs an AGV for material handling. The AGV carries raw material, semi-produced and final products in batch sizes. Because of the increase in demands, advance in technology, and rise in the production capacity more shops than the existing shops are required. The new shops are associated with higher technology machines. Therefore, more than one shop with the same performance is evolved. The difference among shops with the same performance is machines with various specifications that effect the production time/ cost and productivity. As a result, the system could be a flexible jobshop model where multi shops of the same performance exist and each operation is possible to be processed on any type of machine. The sequences of jobs are specified and the jobs are independent.

To evaluate the performance of the proposed AMS, we assess the material flow between any two shops of different types. In the proposed model the aim is to optimize the material flow, i.e. finding a set of shops which minimize the material flow throughout the system. Here, flow is considered as the distance which the AGV moves to satisfy the production plan and demand. The proposed model is presented schematically in Figure 1.

Fig. 1 The Schematic presentation of the Proposed Model

2.3 Mathematical Model

In this section the mathematical model of the proposed flexible jobshop problem is represented. The indices, parameters, and decision variables are as follows:

Indices:

• m Index for shops, m=1,2, … , M.
• k Index for shops, k=1,2, … , K.
• n Index for shop type mth , n=1,2, … , N.
• h Index for shop type k th, h=1,2, … , H.
• i Index for products, i=1,2, … , I.
• p Index for job position, p=1,2, … , P.

Parameters:

$$C_{ipm_n}$$ Completion time of product $i^{th}$ in position $p^{th}$ in shop $n^{th}$ of type $m^{th}$

$$PT_{im_n}$$ Processing time of shop $n^{th}$ of type $m^{th}$ for product $i^{th}$

Transferring time from shop $$n^{th}$$ of type $m^{th}$ to shop $h^{th}$ of type

$$TT_{im_nk_h}$$ Transferring time from shop $n^{th}$ of type $m^{th}$ to shop $h^{th}$ of type $k^{th}$ for product $i^{th}$

$$V_{AGV}$$ Velocity of AGV

$$f_{im_nk_h}$$ Flow (distance) for product $i^{th}$ between shop $n^{th}$ of type $m^{th}$ and shop $h^{th}$ of type $k^{th}$

$$W_{im_n}$$ Waiting time for product $i^{th}$ in shop $n^{th}$ of type $m^{th}$

$$C.T_i$$ Cycle Time for product $i^{th}$

T Total working time in each day

$$z_{ipm} = \begin{cases} 0 & \text{o.w} \ 1 & \text{if shop } m^{th} \text{ is allocated for product } i^{th} \text{ in position } p^{th} \end{cases}$$

Decision variable:

$$z_{ipm_n} = \begin{cases} 0 & \text{o.w} \ 1 & \text{if shop } n^{th} \text{ of type } m^{th} \text{ is chosen for product } i^{th} \text{ in position } p^{th} \end{cases}$$ Objective Function:

$$Min \qquad \sum{h=1}^{H} \sum{k=1}^{K} \sum{n=1}^{N} \sum{\substack{m=1\ m \neq k}}^{M} \sum{p=1}^{P} \sum{i=1}^{I} z{ipm} z{i(p-1)kh} T T{im_n k_h} \tag{1}$$

S.T :

$$C_{ipma} = \sum{p=1}^{P} \sum{n=1}^{N} \sum{m=1}^{M} \left( z_{ipma} \left( P.T{ima} + W{ima} \right) \right) + \sum{n=1}^{H} \sum{k=1}^{K} \sum{n=1}^{N} \sum{m=1}^{M} \sum{p=1}^{P} z_{ipma} . z{i(p-1)ma} . T.T{im_a k_b}, \ \forall i, p, m, n,$$ (2)

$$C_{ipmn} - C{ip-1kh} \ge PT{imn} + W{imn} + TT{im_nk_h}, \quad \forall i, p, m, n, k, h,$$ (3)

$$\sum{n=1}^{N} z{ipmn} = z{ipm}, \qquad \forall i, p, m, \tag{4}$$

$$Z_{ipmn}.C{ipm_n} \le C.T_i, \qquad \forall i, p, m, n,$$ (5)

$$C_{ipm_n} \le T, \tag{6}$$

$$T.T_{m_n kh} = \frac{f{i m_n kh}}{V{AGV}}, \qquad \forall i, k, h, m, n,$$ (7)

$$z_{ipm_n} \in {0,1} \tag{8}$$

Equation (1), is the objective function of the proposed problem which minimizes the material flow. The output of the objective function is the types of the shops which minimize the total material flow. Equation (2), indicates the computation of completion time for each product. Equation (3), certifies that the differences between completion time of product i th in position p th in shop nth of type mth is larger than or equal to the addition of the processing time and waiting time of shop nth of type mth and the transferring time between shop nth of type mth and shop hth of type k th. Equation (4), warranties that if any shop in any position for any product is allocated then the corresponding shop type is also chosen. Equation (5), guarantees that if a shop is chosen then the corresponding completion time for each product is lower than or equal to the cycle time. Equation (6), certifies that the completion time for any product in any position in any shop is lower than or equal to the total working time in each day. Equation (7), indicates the transferring time between two shops is related directly to the flow of each product. Equation (8), presents the values of the decision variables.

After finding the shops which optimize the material flow, now we analyze the performance of the system considering these shops.

2.3.1 Regression Model for Performance Analysis

Here, we consider the output of the mathematical program in last section and analyze the performance of the obtained shops to satisfy the demands.We want to estimate the coefficients which imply the capability of the obtained shops to satisfy the demand of the manufacturing system. We should investigate how much a shop is significant on the demand of the system. One way to survey the effect of independent variables on dependent variable is multiple linear regression model. Assume that Dt is the demand in time t and Xj is the maximum production rate of shop j th. Therefore, we consider the following equation,

$$D{t} = \beta{0} + \beta{1}X{1} + ... + \beta{j}X{j} + \varepsilon_{j}, \qquad t = 1,...,T; j = 1,...,J.$$ (9)

2.3 Mathematical Model 23

where β0 is the intercept, β ^j s are the coefficients for *Xj*s and ε is the error term. The aim is first determining the value of the coefficients to see whether they are lower than 1 or not, and second identifying the β ^j s which are not important on demand (Dt). For the first objective, if the value of any coefficient is higher than 1, then it indicates that the corresponding shop can not satisfy the allocated demands with the current working time and lead to have extra working time. For the second objective, we apply two-sided hypothesis testing as follows;

$$\begin{split} H_0: \beta_j &= 0; \ H_1: \beta_j &\neq 0. \end{split} \tag{10}$$

The test of the null hypothesis H0 against the two sided alternative proceeds as in the three steps. The first is to compute the standard error of β ^j , ( )^j SE β . The standard error of β ^j is an estimator of ^{β j} σ , the standard deviation of the sampling distribution of β ^j .

$$\sigma{\beta{j}}^{2} = \frac{1}{m} \times \frac{\frac{1}{m-2} \times \sum{j=1}^{m} (r{j} - \bar{r})^{2} \varepsilon{j}^{2}}{\left[\frac{1}{m} \sum{j=1}^{m} (r_{j} - \bar{r})^{2}\right]^{2}},$$ (11)

$$SE(\betaj) = \sqrt{\sigma{\beta_j}^2} \ . \tag{12}$$

Although the formula for ^{2 β j σ} is complicated, in applications the standard error is computed by regression software. The second step is to compute the t-statistic,

$$t = \frac{\beta_j - 0}{SE(\beta_j)} \tag{13}$$

The third step is to compute the *p-*value, the probability of observing a value of β ^j at least as different from zero as the estimate actually computed ( act β ^j ), assuming that the null hypothesis is correct. Stated mathematically,

$$p - \text{value} = \Pr_{H_0} \left[ \left| \beta_j - 0 \right| > \left| \betaj^{\text{act}} - 0 \right| \right] \Rightarrow \Pr{H_0} \left[ \left| \frac{\beta_j - 0}{SE(\beta_j)} \right| > \left| \frac{\beta_j^{\text{act}} - 0}{SE(\betaj)} \right| \right] \Rightarrow \Pr{H_0} \left( \left| t \right| > \left| t^{\text{act}} \right| \right) (14)$$

where ⁰ Pr^H denotes the probability computed under the null hypothesis, the second equality follows by dividing by ( )^j SE β , and act t is the value of the t-statistic actually computed. Because β ^j is approximately normally distributed in large samples, under the null hypothesis the t-statistic is approximately distributed as a standard normal random variable, so in large samples,

$$p - \text{value} = \Pr(|\mathbf{Z}| > |t^{act}|) = 2\Phi(-|t^{act}|), \tag{15}$$

A small value of the *p-*value, say less than 5%, provides evidence against the null hypothesis in the sense that the chance of obtaining a value of β ^j by pure random variation from one sample to the next is less than 5% if, in fact, the null hypothesis is correct. If so, the null hypothesis is rejected at the 5% significance level. Simply we can reject the null hypothesis at the 5% significance level if, > 1.96 act t . Therefore, if the hypothesis is accepted then we find that the corresponding coefficient is not important and has no effect on the demand.

After describing the aspects of the proposed model for this type of flexible jobshop automated manufacturing system, a numerical illustration is provided in the next section.

2.4 Conclusions

In this chapter a new model in a flexible jobshop automated manufacturing systems has been proposed. The innovation is in the multi shops of the same performance, but different specification. The proposed flexible jobshop model is associated with an AGV for material handling. The mathematical model is identifying the optimal material flow amongst the shops with respect to job sequence, cycle time, and AGV capability constraints. Also a performance analysis has been worked out on the selected shops to consider their capability for satisfying demands. The numerical illustrations indicate the applicability and efficiency of the proposed approaches to investigate the proposed problem.

References 25

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Chapter 3

Nonlinear Stochastic Model for AGV System

3.1 Summary

Advanced automated manufacturing systems are widely used in industrial companies where productivity objectives have to be met. These systems often being costly, they must be designed to be as efficient as possible. Here, we focus on automated manufacturing systems in a job shop layout considering automated guided vehicle as a material handling resource. The key issue in manufacturing operations is how to produce high quality products at low costs in such a way that the diversified demand are met. Hence, modern manufacturing companies should become as responsive as possible in order to satisfy customer demands.

Material handling accounts for 30–75% of the total cost of a product, and efficient material handling can result in reducing the manufacturing system operations cost by 15–30% (Sule, 1994). These points underscore the importance of material handling costs reduction as a key element in improving the cost structure of a product. The determination of a material handling system involves both the selection of suitable material handling equipment and the assignment of material handling operations to each individual piece of equipment. Hence, material handling system selection can be defined as the selection of material handling equipment to perform material handling operations within a working area considering all aspects of the products to be handled (Rau and Liu, 2006).The material handling system plays a crucial role in automated manufacturing systems. When inadequately designed, the material handling system indeed can adversely affect the overall performance of the system and lead to substantial losses in productivity and competitiveness, and to unacceptably long lead times. Thus, to avoid such pitfalls, material handling system design must be integrated into the overall design of the manufacturing system centering on the selection of machines and the allocation of operations to the machines (Gholipour-Kanani at al., 2011).

Automated manufacturing systems (AMSs), equipped with several CNC machines and an AGV-based material handling system, are designed and implemented to secure automation and gain efficiency of production. To achieve these benefits, the planning in the AMS decision making process is critical because the planning decision has an influence on the subsequent decision processes such as scheduling, dispatching, etc. Planning in automated manufacturing systems can be characterized to be online and short-term in order to respond to frequently changing production orders. Given a production order, the manufacturing planning function is responsible for establishing a plan by decomposing the production task into a set of subtasks (Azadeh and Anvari, 2009).Subtasks are classified into two types for material processing and delivery. To pursue a global optimization of material processing and transportation cost, a manufacturing plan should consider the two costs simultaneously. Moreover, to obtain a better solution for subsequent decision making processes, not only total workloads are evenly distributed among the workstations, but workloads are well balanced among AGV's network segments.

The manufacturing industry is presently being much affected by the structural changes caused by the internal and external factors in an enterprise. The market conditions are becoming more dynamic, more global, and more customer-driven. The manufacturing performance is no longer driven merely by the product price; instead, other competitive factors such as automation, quality, delivery, and customer service have become equally important. The demand of the customer for tailored product has caused a shorter product life, reduced batch quantities and increased product varieties.

Manufacturing firms need to give prominence to issues such as reduction of manufacturing lead time and have flexibility to adapt to changes in the market. The improvement in productivity and reduction of costs of goods and services has become a key element for maintaining the market share.Operations management in an AMS is more complex than that of the conventional manufacturing systems. Managing an AMS demands for more decisions for its effective performance as compared to a transfer line or a job shop production system. The optimal selection of machines and tools and the assignment of part operations to the selected machines turn out to be difficult tasks for the production planner. This is due to the versatile machine tools capable of performing many different operations resulting in many alternative routes for a part type, and due to system capability for processing of the parts concurrently (Stecke, 1983).

Chan (1999) discussed the effects of universal loading station along with operational control rules. Rajagopalan (1986) presented a formulation and heuristic solution for the part grouping and tool loading in AMS. Ram et al. (1986) developed a model and a solution procedure for the machine loading and tool allocation problem in an AMS. Lashkari et al. (1987) extended the formulation of the operation allocation problem to include the aspects of refixturing and limited tool availability. Kusiak (1985), Stecke and Morin (1985), Singhal et al. (1987) and van Looveren et al. (1993) addressed the interrelationship of various decisions and their hierarchies in AMS. Kim and Yano (1993) presented a number of heuristic approaches for loading problems in automated manufacturing systems. Kim and Yano (1994) also presented a model for the loading problem in AMS with unequal workload targets across machine groups, and demonstrated how an existing branch-and-bound algorithm for the workload balancing objective may be used to solve the model. Stecke and Raman (1994) presented a queuing network

production planning model to determine the optimal machine workload assignments in an AMS. Liang (1994) proposed a two stage approach to the joint problem of part selection, machine loading, and machine speed selection problem in AMS. In the first stage, the mathematical model solves the part selection and machine loading problem, whereas in the second stage, it determines the optimal cutting speed for all job–tool–machine combinations.

A wide spectrum of multi-objective loading problems by combining two or more criteria has been addressed in literature (Swarnkar and Tiwari, 2004; Chan et al., 2004). However, considering the multi-objective problems, some of the objectives turn out to be contradictory in various situations, while for others, they may equally be applicable. When the machines are not pooled into groups, the loading objective used most commonly is balancing the workload on all the machines. It is established that this objective maximizes the expected production (Stecke and Morin, 1985). The mixed-integer programming (MIP) approach for solving the machine loading problem is shown to be computationally infeasible even for deterministic formulation.

Here, considering the stochastic nature of the proposed problem parameters, a stochastic programming approach is applied to optimize production time and cost in an automated manufacturing system.

3.2 Statement of the Problem

In our proposed model, it is assumed that a manufacturing system equipped with an AGV for material handling purposes is designed in a job shop layout. The AGV would receive the raw material and start visiting the shops to complete the production cycle. In each shop, one part of the product is processed and at the final shop the complete product is ready to be carried to the warehouse. The configuration of the proposed model is represented in Figure 1.

Fig. 1 Configuration of the proposed model

The AGV should wait until the processing of a job is finished in a shop and then move the semi-produced product to the next shop. Therefore, due to the processing conditions in a shop, the waiting time in a shop is not deterministic and is therefore supposed to be stochastic. While a job is processing on a machine, the machine may break down. Due to the stochastic nature of the failure, the breakdown cost is not known. Also, the semi-produced products carried by AGV may be damaged during traveling between shops, but the rate is not deterministic and thus is considered to be stochastic.

Considering these assumptions, we are confronted with a stochastic model. The distributions of the stochastic items are assumed to be normal or can be appropriately estimated by normal distributions. In the next section, the mathematical model of the proposed problem is given.

3.3 Mathematical Model

Note that some parameters are stochastic, based on the nature of the problem. Therefore, the stochastic parameters are presented using the corresponding expected value and variance. As a result, the following nonlinear deterministic mathematical model is configured. As mentioned before, the aim is to optimize production time and cost simultaneously. Hence, the model is multi-objective. The notations needed for the model are indicated below.

Indices:

i	Number of shops;	i=1,2,…,I
j	Type of machines;	j=1,2,…,J.

Notations:

Defect cost

Co Operational cost of AGV ⁱ Ct Tool cost in the *i*th shop

bj C Breakdown cost for *j*th machine

pj t Job processing time on *j*th machine

wij t Waiting time in *i*th shop for *j*th machine

tm Material handling time

tc Cycle time

β Upper limit for defect rate B1 Available budget for AGV B2 Available budget for tool

Di Demand for *i*th shop

Mi Number of machines in *i*th shop

N Number of jobs

E(.) Expected value of a parameter

Var(.) Variance of a parameter ZP Standard normal value for percentile P.

Decision variables:

Xij Number of produced products in *i*th shop by *j*th machine

Objective function 1 (cost minimization):

$$\begin{aligned} & Min \ (\sum{j=1}^{J} \sum{i=1}^{I} E(C{b{j}}) \times X{ij}) + Z{P} \sqrt{(\sum{j=1}^{J} \sum{i=1}^{I} Var(C{b{j}}) \times X{ij}^{2})} + (C{o} \times (\sum{j=1}^{J} \sum{i=1}^{I} C{t{i}} \times X{ij})) + \ & (E(C{d}) \times \sum{j=1}^{J} \sum{i=1}^{I} X{ij}) + Z{P} \sqrt{Var(C{d}) \times (\sum{j=1}^{J} \sum{i=1}^{I} X{ij}^{2})}, \end{aligned} \tag{1}$$

Objective function 2 (time minimization):

$$Min \ (tm \times (\sum{j=1}^{J} \sum{i=1}^{J} t{pj} \times X{ij})) + \sum{j=1}^{J} \sum{i=1}^{J} E(t{w{ij}}) \times X_{ij}) + Zp \sqrt{\sum{j=1}^{J} \sum{i=1}^{J} Var(t{w{ij}}) \times X{ij}^2)},$$ (2)

Constraints:

$$(N \times (\sum{j=1}^{J} \sum{i=1}^{I} t{p{j}} \times X{ij})) + (\sum{j=1}^{J} \sum{i=1}^{I} E(t{w{ij}}) \times X{ij}) + Z{p} \sqrt{\sum{j=1}^{J} \sum{i=1}^{I} Var(t{w{ij}}) \times X{i}^{2})} + (t{m} \times \sum{j=1}^{J} \sum{i=1}^{I} X{ij}) \leq (t{c} \times \sum{j=1}^{J} \sum{i=1}^{I} X{ij}),$$ (3)

$$Co \times \sum{j=1}^{J} \sum{i=1}^{I} X{ij} \le B_1,$$ (4)

$$(E(Cd) \times \sum{j=1}^{J} \sum{i=1}^{I} X{ij}) + Z_P \sqrt{Var(Cd) \times (\sum{j=1}^{J} \sum{i=1}^{I} X{ij}^2)} \le \beta,$$ (5)

$$\sum{j=1}^{J} \sum{i=1}^{I} \frac{C_{ti} \times X{ij}}{M_i} \le B_2,$$ (6)

$$\sum{j=1}^{J} X{ij} \ge D_i, i=1,2,...,I, \tag{7}$$

$$X_{ij} \ge 0, \text{integer}, \quad i=1,2,...,J.$$ (8)

The formula in (1) is the first objective function, the total cost of production, which is to be minimized. Formula (2) is the second objective function, the total time of manufacturing, which is also to be minimized. Inequality (3) confines the total production time to the cycle time. Inequality (4) considers the limitation of operational budget for AGVs. Inequality (5) represents a constraint for desirable defect rate. Inequality (6) indicates that the available budget for tools in the system is limited. The Inequalities (7) certify that the demands of the shops are covered. The constraints (8) ensure the nonnegativity and integrality of the variables.

Since the proposed model is nonlinear, we apply a linear approximation method for optimization.

3.3.1 Successive Linear Programming Approach

Successive Linear Programming (SLP) algorithms solve nonlinear optimization problems via a sequence of linear programs. They have been widely used, particularly in the oil and chemical industries, beginning with their introduction by Griffith and Stewart of Shell Development Company in 1961 (Griffith and Stewart, 1961). As the name implies, the method uses linear programming as a search technique. A starting point is selected as an initial estimate of the solution, and the nonlinear model and constraints are linearized at this point to give a linear problem which can be solved by the simplex method or its variations. The solution of the linear programming problem is used to construct a new estimate of the solution of the original problem, and the process is continued until a stopping criterion is met. As shown by Reklaitis et al. (1983), this procedure works, with no need for safeguards, when problems are mildly nonlinear. However, it is necessary to bound the steps taken in the iterations to insure that the model solution estimate improves, the values of the independent variables remain in the feasible region and the process converges to an optimal point. These safeguards are bounds on the independent variables specified in advance for solving the linear programming problem. The net result is that the bounds are additional constraints. If the bounds are set too small, then the process proceeds slowly toward a solution. If they are set too large, then infeasible solutions may be generated. Consequently, precautionary steps are incorporated into computer programs to expand the bounds when slow progress is detected and shrink them when the solution estimates do

not seem to be improving much in successive iterations, while being far from a stationary point. Now, the general nonlinear optimization problem can be written as:

Optimize: $$y(\mathbf{x})$$ , (9)

s.t. $$f_i(\mathbf{x}) < b_i$$ , $i = 1, 2, ..., m$ , $l_j \le x_j \le u_j$ , $j = 1, 2, ..., n$ ,

where upper and lower limits are shown specifically on the independent variables. In a general step k, the model y(x) and the constraints fi(x) are linearized at x k to give:

Optimize: $$z = \sum_{j=1}^{n} c_j \Delta x_j + y(x^k), \qquad (10)$$

s.t. $$\sum{j=1}^n a{ij} \Delta x_j + f_i(x^k) \le b_i , \qquad i=1,2,...,m,$$

$$l_j \le x_j^k + \Delta x_j \le u_j , \quad j=1,2,...,n,$$

where,

$$c_j = \frac{\partial y(x^k)}{\partial xj}, \qquad a{ij} = \frac{\partial f_i(x^k)}{\partial x_j}.$$

The problem (10) is now a linear programming one, but the values of Δxj can take on either positive or negative values. To have the linear program the standard nonnegativity imposition on the variables Δxj, a change of variables was made by Griffith and Stewart (1961) as follows:

$$\Delta x_j = \Delta x_j^+ - \Delta x_j^-, \tag{11}$$

where,

$$\Delta x_j^+ = \begin{cases} \Delta x_j, & \text{if } \Delta x_j \ge 0 \ 0, & \text{if } \Delta x_j < 0 \end{cases}$$ $$\Delta x_j^- = \begin{cases} -\Delta x_j, & \text{if } \Delta x_j \le 0 \ 0, & \text{if } \Delta x_j > 0. \end{cases}$$

Substituting (11) into (10), the linear programming problem has then the form:

Optimize: $$\sum{j=1}^{n} c{j} \Delta x{j}^{+} - \sum{j=1}^{n} c{j} \Delta x{j}^{-} + y(x^{k}),$$ s.t. $$\sum{j=1}^{n} a{ij} \Delta x{j}^{+} - \sum{j=1}^{n} a{ij} \Delta x{j}^{-} \leq b{i} - f{i}(x^{k}), \quad i=1,2,...,m,$$

$$\Delta x{j}^{+} - \Delta x{j}^{-} \leq (u{j} - x{j}^{k}),$$

$$\Delta x{j}^{+} - \Delta x{j}^{-} \geq (l{j} - x{j}^{k}), \qquad j=1,2,...,n,$$

$$\Delta x{j}^{+}, \Delta x{j}^{-} \geq 0, \qquad j=1,2,...,n.$$ (12)

The upper and lower limit bounds on the variables are specified by ( ^{k j j} u − x ) and ( ^{k j j} l − x ). The value of the next solution estimate for linearization is then defined to be k+^{1 j} x ^{=+ −} +Δ ^j −Δ ^j k ^j x x x , where the Δxj + and Δxj¯ are obtained as the solution of (12). The procedure is started by specifying a starting point x⁰ (*k=*0). We note that the values of the bounds uj and lj may affect the rate of convergence of the algorithm. To obtain an optimization algorithm using the SLP, we need some preliminaries.

Let * . and . be two arbitrary norms in ^{n ℜ} . It is well known that there are positive constants *c1*and c2 such that for all ⁿ x ∈ℜ ,

$$c1 |x|* \le |x| \le c2 |x|*. \tag{13}$$

Note that for every x ∈ X and r > 0 (r is called the trust region radius), the following inclusions hold:

$${h \mid x + h \in X, ||h|| \le c1 r} \subseteq {h \mid x + h \in X, ||h||* \le r} \subseteq {h \mid x + h \in X, ||h|| \le c_2 r}.$$ (14)

For x ∈ X and r > 0 , consider the following problem:

$$LP(x,r): \min \nabla^T f(x)h$$ s.t. $x+h \in X$ , (15) $$||h||_* \le r$$ ,

where,∇f (x) is a column vector composed of the partial derivations of f at x.

Clearly, when * . is a polyhedral norm and X is a polyhedral set, LP(x,r) is a linear programming problem. Denote H*(x,r) and v*(x,r) as the solution set and the optimal value of LP(x,r), respectively. The inclusions (14) imply that

$$v(x,c2r) \le v*(x,r) \le v(x,c_1r),$$ (16)

for every x ∈ X and r ≥ 0 .

Lemma 1. For every x ∈ X and r > 0 , we have:

$$x \in X{\text{stat}} \Leftrightarrow v*(x,r) = 0$$ ,

*where, X*stat is a stationary set.

In order to justify the technique being used to update the trust region radius, we need a simple technical result as given next.

Lemma 2. For every^δ > 0 and stat *x*∈ X \ X , there exist ^τ and ^ρ , with 0 < ^τ < ρ , such that the following properties hold:

$$(i) \forall r \in [\rho, \infty), \forall h \in H_*(x, r), f(x) - f(x + h) \le \frac{\delta}{2} r^2.$$

$$(ii) \forall r \in [0, \tau], \forall h \in H_*(x, r), f(x) - f(x+h) \ge \frac{\delta}{2} r^2$$

Proof let ^δ > 0 and stat *x*∈ X \ X .

Proof for (i). Define ( ) opt ¹ := 2 f (x) − f ^{− ρ δ} so that for allr ∈[ρ,∞),

$$f(x) - f(x+h) \le f(x) - f_{\text{opt}} = \frac{\delta}{2} \rho^2 \le \frac{\delta}{2} r^2.$$ (17)

Proof for (ii). Let ^{2 1 1} : 2 ( ) − a = c Lc + δ , so that there exists ^τ > 0 with a (x,c ) ≤ a ^{0 1 τ} . Take an arbitrary (0, ] ^{− r ∈ r} and ( , ) * ^{h ∈ H x r} . Then,

$$a_0(x, c_1 r) \le a_0(x, c_1 \tau) \le a_0$$ (18)

and

$$v(x, c_1 r) \le -\frac{1}{a_0(x, c_1 r)} (c_1 r)^2 \le -\frac{c_1^2}{a} r^2.$$ (19)

It follows from (16) and (18) that

$$\left\langle \nabla f(x), h \right\rangle = v_*(x, r) \le v(x, c_1 r) \le -\frac{c_1^2}{a} r^2. \tag{20}$$

Therefore, by (17),

$$f(x+h) - f(x) \le \langle \nabla f(x), h \rangle + \frac{L}{2} ||h||^2 \le -\frac{c_1^2}{\underline{a}} r^2 + \frac{Lc_2^2}{2} ||h||^2 \le -\frac{c_1^2}{\underline{a}} r^2 + \frac{Lc_2^2}{2} r^2 = -\frac{1}{2} \left( \frac{2c_1^2}{\underline{a}} - Lc_2^2 \right) r^2$$

$$= -\frac{\delta}{2} r^2.$$ (21)

We now state the main algorithm. The parameter k counts the number of iterations, while μk represents the number of subproblems solved at the k th iteration.

Algorithm SLP: Successive Linear Programming for Nonlinear Optimization

Parameters: , 0, 0 1. ^δ r* > < ^θ <

Input: Choose x ∈ X ¹ .

Step 0: (Initialization): Set k:=0, rk :=r*, *k*stop:= ∞ .

Step 1: Set k:=k+1, ^μ := 1, r:=rk-1. Solve the problem LP(x^k ,r) to

obtain a solution ( , ) ( , ) * h x r H x r ^{k k} ∈ . Set

$$v \coloneqq \begin{cases} 0, & \text{if } v*(x^k, r) = \left\langle \nabla f(x^k), h(x^k, r) \right\rangle = 0, \ -1, & \text{if } v_(x^k, r) \neq 0 \text{ and } f(x^k) - f(x^k, h*(x^k, r)) \geq \frac{\delta}{2} r^2, \ 1, & \text{if } v(x^k, r) \neq 0 \text{ and } f(x^k) - f(x^k, h_(x^k, r)) < \frac{\delta}{2} r^2. \end{cases}$$

If *v=*0 then set *kstop=k*, ^{k k} x = x stop and return.

Step 2: Set ^μ := ^μ + 1 and ^v r := r^θ . Solve the problem LP(x^k ,r) to obtain a solution ( , ) ( , ) * h x r H x r ^{k k} ∈ .

3.4 Conclusions 37

Step 3: If $$(v = -1 \text{ and } f(x^k) - f(x^k, h*(x^k, r)) < \frac{\delta}{2} r^2)$$ or $(v = 1 \text{ and } f(x^k) - f(x^k, h*(x^k, r)) \ge \frac{\delta}{2} r^2)$ then go to Step 4, else go to Step 2.

Step 4: If $$v=-1$$ then set $r_k=\theta r$ , else set $r_k=r$ . Let $\muk=\mu$ and $$x^{k+1}=x^k+h*(x^k,r_k) \text{ and go to Step 1}.$$

Note that is Step 1, the parameter v can take on one of three values. If *v=*0, then a stationary point has been found and the algorithm terminates. If *v=*1, then sufficient decrease was not achieved and thus the subproblem needs to be resolved with a smaller value of r. If v = −1 , then sufficient decrease was achieved, but the subproblem is solved again, this time with a larger value of r. Both of these resolves take place in Step 2. If the algorithm terminates after a finite number of iterations, then we can define the iterate stop ^k x as the approximate solution for the problem. We apply the proposed SLP algorithm to solve some numerical experiments.

3.4 Conclusions

A new model for automated manufacturing systems was proposed. Material handling was performed via AGV and the manufacturing system was considered to be a jobshop. The novelty of our approach is in our consideration of the stochastic nature of parameters and our use of stochastic programming. The mathematical model is a nonlinear bi-objective one, which considers both time and cost minimization. We first applied a successive linear programming technique for optimizing the nonlinear model. Numerical test results showed the linear optimization approach to be slow for large size problems, and thus presented a genetic algorithm (GA) for solving large size problems. We finally analyzed the differences of the solutions obtained by LINGO and GA using test of hypothesis. We concluded that there was a significant difference between LINGO's and GA's obtained solutions and that their performance was not the same due to variety in the problem size. For small and medium sized problems, LINGO was shown to be efficient enough to provide accurate solutions but for larger sized problems, we observed that the GA was successfully effective in finding good solutions.

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Chapter 4 Reliability Model for AGV

4.1 Summary

The Material Handling System (MHS) in a manufacturing setting plays an important role in the performance of the entire system. Inadequately designed MHSs can interfere with the overall performance of the manufacturing system and lead to substantial losses in productivity and competitiveness, and to unacceptably long lead times. Among the advanced technologies available for MHSs, Automated Guided Vehicles (AGVs) have found increasing applications because of their capability to transport a variety of part types from point to point without human intervention.

Today's automated MHSs are technologically advanced and increasingly complex. Uncertainty is an inevitable consequence of the complexities generated by technological advancements. Jain et al. (2013) show that most of the automated manufacturing studies have used single-item measures. They argue that singleitem measures are appropriate for relatively simple manufacturing systems. As the different dimensions of automated MHSs are complex in nature, single-item measures are inappropriate. Choosing adequate and relevant performance measures is critical in accurately analyzing MHSs (Beamon, 1998). A company also needs to address uncertainties in the manufacturing system to survive and compete in such an uncertain environment (Jain et al. 2013).

AGVs are the most flexible means for transporting pieces among workstations in an automated manufacturing system. An AGV is a driverless and programmed vehicle used to transfer the load from one part of a manufacturing facility to another part (Maniya and Bhatt, 2011). Maxwell and Muckstadt (1982) first recognized the importance of AGV-based MHS design. They developed an optimization model that minimized the total travel time and determined the maximum number of AGVs needed to efficiently transfer material from one shop to another. AGVs increase efficiency, reduce costs, and improve flexibility by automating an MHS. Time-based performance measures are often used to evaluate MHSs with AGVs. However, moving materials from one part of the manufacturing floor to another part utilize time and incur costs. In this study, we consider both time and cost measures in an optimization model and evaluate an MHS with AGVs.

The use of AGVs increases flexibility and has a significant impact on the overall performance and reliability of MHSs (Sarker and Gurav, 2005). As AGVs become larger and more complex, the traditional design requires more attention to issues such as control, cost, time, reliability, flexibility, etc. A number of different MHS design and evaluation methods (e.g., simulation, optimization, and the genetic algorithm) have been proposed in the literature. Simulation is an acceptable method for analyzing manufacturing systems. However, simulation is often challenging and time consuming (Law and Kelton, 2000; Kuo et al., 2007), particularly, when it is used for modeling complex manufacturing systems such as MHSs with AGVs.

The problem of scheduling AGVs in an automated MHS has been studied extensively. Abdelmaguid et al. (2004) addressed the problem of simultaneous scheduling of machines and AGVs with the objective of minimizing the makespan. This problem is composed of two interrelated decision problems: the scheduling of machines, and the scheduling of AGVs. They showed that each problem is an NP-complete problem and a simultaneous consideration of the two problems results in a more complicated NP-complete problem. They proposed a hybrid genetic-algorithm/heuristic coding scheme to solve the problem. Deroussi et al. (2008) also studied this problem and proposed a solution based on vehicles rather than machines. Each solution was evaluated using a discrete event approach. Gnanavel Babu et al. (2010) studied this problem further and proposed a meta-heuristic differential evolution algorithm for solving it. They introduced an iterative algorithm that anticipated the complete set of flow requirements for a given machine schedule and made vehicle assignments accordingly. Le-Anh and De Koster (2006) have compiled a comprehensive review of the AGV design and control models and methods in the literature.

Farling et al. (2008) used a simulation model to compare the performance of three AGV configurations under a variety of experimental conditions. They showed that system size, load/unload time, and machine failure rate factors have significant impacts on the operation and reliability of MHSs. Smith (1993) defined reliability as the probability that an item will perform a required function, under stated conditions, for a specific period of time. A reliability measure is a metric for quantifying this probability. A number of different reliability measures (i.e., availability, unavailability, failure rate, and mean time between failures) have been proposed in the literature. For degradable systems, such as MHSs, the performance of the system during a specific period of time can be described by different levels of performance as a function of machine failures (Beamon, 1998). Miriyala and Viswanadham (1989) developed several measures and algorithms for evaluating part-based reliability and system-based reliability for automated MHSs. Beamon (1995) proposed an analytical model for designing guide paths for automated MHSs as a function of reliability and quantified the reliability of the handling components.

In order to ensure an acceptable service level for each machine in each shop, we adopt and further extend the concept of reliability proposed by Ball and Lin (1993) in the model. We define reliability as the probability that the system is operational until time t. A failure is when a machine in a shop breaks down. 4.1 Summary 43

A desired level of reliability can be achieved by limiting the failure probabilities. This approach for handling reliability is called the chance constraints method and was initially proposed by Charnes and Cooper (1959) in the context of mathematical programming. The use of chance constraints in the vehicle routing problem was illustrated in Stewart and Golden (1983). Carbone (1974) used chance constraints for selecting multiple facilities under normally distributed demand. The model minimized an upper bound on the total demand-distance while ensuring that the constraints were satisfied with a specified chance or probability. Shiode and Drezner (2003) used a similar approach in a competitive location problem on a tree network.

In many real-life applications, the parameters in a manufacturing system may have varying values. This value variation may result from machine breakdowns, lack of training, unexpected delays, non-qualified operators, or complex tasks, among others (Özcan, 2010). Stochastic programming can provide an effective means for incorporating uncertainty in real-life MHSs (Birge and Louveaux, 1997). Stochastic programming has been used frequently in the design and control of MHSs with AGVs (Sayarshad and Tavakkoli-Moghaddam, 2010).

The presence of uncertainties in automated material handling and AGV systems has also motivated researchers to explore compensator design such as neural networks (Pamosoaji et al., 2013). Kuo et al. (2007) argues that simulation is very time consuming for large MHSs and the process of collecting adequate sample data places limitations on any analysis. They proposed to overcome this problem by developing a neural network simulation metamodel that required only a comparably small training data set. Artificial neural networks are composed of interconnected adaptive elements which are intended to respond to stimuli in a manner not unlike the human nervous system (Kohonen, 1988). McCulloch and Pitts (1943) introduced one of the first artificial neuron models in 1943. The main feature of their neuron model is that a weighted sum of input signals is compared to a threshold to determine the neuron output. Unlike biological networks, the parameters of their networks had to be designed, as no training method was available. However, the perceived connection between biology and digital computers generated a great deal of interest. In the late 1950s, Rosenblatt (1985) and several other researchers developed a class of neural networks called perceptrons. The neurons in these networks were similar to those of McCulloch and Pitts (1943). Rosenblatt's (1985) key contribution was the introduction of a learning rule for training perceptron networks to solve pattern recognition problems.

In this chapter, we consider a manufacturing system with the following physical characteristics: (1) the manufacturing system is a job shop, and (2) single-load AGVs perform the material handling job in the shop. We use a stochastic programming framework and propose a bi-objective optimization model (which has not been used in the MHS evaluation studies) to determine the optimal production time and cost in a manufacturing system with an automated MHS and AGVs. The contribution of this chapter is fivefold: we consider (1) a stochastic programming problem and decompose the optimization process into manageable steps and integrate the results to arrive at a solution consistent with organizational goals and objectives; (2) stochastic parameters in multi-objective optimization models in general and the proposed bi-objective optimization model in particular; (3) machine reliability as an important component in the proposed optimization model; (4) machine maintenance through the breakdown rate for a realistic representation of the system; and (5) the elements of uncertainty within the proposed structured framework by using a perceptron neural network to weigh the two objectives in the proposed bi-objective optimization model.

4.2 Statement of the Problem

In this problem, we consider a manufacturing system equipped with an AGV for material handling in a job shop environment. The AGV moves a part-type from one shop to another to complete a production cycle. In each shop, the part-type is partially processed. The part-types completed in the final shop are moved to the warehouse. For example, let us consider a manufacturer of cylinder block in the automotive industry. The manufacturer of cylinder block can install an AGV system to supply the lines with parts and to transfer cylinder blocks between five shops. Generally, the sequence of operations on a typical machining line for cylinder blocks involves five shops as shown in Figure 1:

This machining line for cylinder blocks is comprised of the following 16 processes performed in each of the five job shops:

• Shop 1: Assuring a uniform wall thickness for the cylinder bores qualifying (Process-1); rough mill pan and head faces (Process-2); and rough machining cylinder bores (Process 3)
• Shop 2: Milling bearing cap width and slots (Process 4) and finishing mill pan and bearing cap width (Process 5)
• Shop 3: Drilling oil holes compound angles (Process 6); drilling, reaming, tapping - Left and right, pan and head faces (Process 7); assembly of bearing caps (Process 8); finish front and rear end (Process 9); and drilling, reaming, tapping - end faces (Process 10)
• Shop 4: Line boring crankbore (Process 11); finish tappet bores (Process 12); assembly cam liners, finish line boring (Process 13); finish cylinder bores (Process 14); and finish mill/grind head face (Process 15)
• Shop 5: Hone and grade (Process 16)

The AGV waiting time in the job shop is not known precisely. It is also possible for a machine to breakdown while processing a job. The number of breakdowns (and the breakdown cost) is also not known precisely. The distributions for the stochastic parameters in the problem are assumed to be normal or can be estimated by the normal distribution. Additionally, the machines' reliability is also considered as a stochastic parameter following the exponential distribution. We present the mathematical details of the proposed model in the next section.

4.3 Mathematical Model 45

Fig. 1 Cylinder block machining line example

4.3 Mathematical Model

In this section, we propose a mathematical model for a simultaneous optimization of the production times and costs.

4.3.1 Reliability Component

We assume that the reliability of each machine-type is independent according to Exponential processes. Also, J is the total number of machine types (i.e., milling, drilling, turning, welding, etc.) We discuss the reliability based model as follows:

$$R(t){\text{system}} = \begin{cases} \left(1 - \prod{j=1}^{J} (1 - Rj(t))\right), & \text{when machines in each shop are in parallel} \ \left(\prod{j=1}^{J} R_j(t)\right), & \text{when machines in each shop are in series} \end{cases}$$ (1)

where R (t) ^j is the probability that the machine-type j works for a period of t time units.

As stated earlier, the machine-types in each shop are parallel and the shops are organized in series. Therefore, the reliability of the system can be measured as follows:

$$\left(1 - \prod_{j=1}^{J} (1 - R_j(t))\right) \ge \alpha, \tag{2}$$

whereα is the lower bound for a desirable system reliability during the time period t. As previously assumed the reliability of each machine-type is independent and can be measured according to the following exponential distribution:

$$R{j}(t) = e^{\frac{-t}{\theta{j}}},\tag{3}$$

where^{θ j} is the exponential parameter for machine-type breakdown. Then,

$$\left(1 - \prod_{j=1}^{J} \left(1 - e^{\frac{-t}{\theta_j}}\right)\right) \ge \alpha \tag{4}$$

In order to obtain a higher level of reliability, more cost is incurred to the system. Hence, a cost function (C (t) ^j ) is defined to keep machine-type *j*reliable for the time period t. The following represents the cost for the entire system:

$$\sum_{j=1}^{J} C_j(t) \tag{5}$$

In order to validate the stated cost function, we consider the costs as losses and use the minimum expected loss (or minimum risk) associated with the system. We represent these losses with a quadratic loss function which is mathematically more tractable than other loss functions because of its symmetric property (i.e., an error above the target causes the same loss as the same magnitude of error below the target). If the target value of the pre-planned exponential parameter is f, then a quadratic loss (cost) function is

$$C{j}(t) = L \left| f - \theta{j} \right|^{2} \tag{6}$$

Where L is a constant and its value could be set to 1 if the constant makes no difference to a decision.

We replace our proposed stochastic parameters with a combination of the expected value and the variance for that parameter and the following nonlinear deterministic mathematical model is derived. As indicated earlier, our goal is to simultaneously optimize production time and cost in the following bi-objective model:

Indices: 
i Number of shops i=1,2,…,I
j Number of machines j=1,2,…,J
Parameters: 
Cd Stochastic cost per defective unit 
Co AGV operational cost per unit produced 
i Ct Tool cost for shop i
 pj
t Job processing time on machine j
 wij
t Stochastic waiting time of machine j in shopi
tm Material handling time 
tc Cycle time 
β Total daily cost of the defective items 
B1 Total AGV budget 
B2 Total tools budget 
B3 Total machines budget 
Di Total Demand for shop i
Mi Number of machines in shop i
N Number of jobs 
ZP Standard normal Z value for percentile P
Decision variables: 
Xij Number of units produced in shop i by machine j
θ j Exponential reliability function of machinej

Objective function 1 (cost minimization):

$$Min \left( \sum{j=1}^{J} \sum{i=1}^{J} E(C{j}(t)).X{ij} \right) + Z{p} \sqrt{\left( \sum{j=1}^{J} \sum{i=1}^{J} Var(C{j}(t)) X{ij}^{2} \right)} + \left( \left( \sum{i=1}^{J} C{i{i}} + C{o} \right). \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( E(C{d}). \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + Z{p} \sqrt{Var(C{d}). \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right)} \right) + \left( \left( \sum{i=1}^{J} C{i{i}} + C{o} \right). \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{J} X{ij} \right) + \left( \sum{j=1}^{J} \sum{$$

Objective function 2 (time minimization):

$$Min \left( \left( \sum{j=1}^{J} t{p{j}} + t{m} \right) \cdot \sum{j=1}^{J} \sum{i=1}^{I} X{ij} \right) + \left( \sum{j=1}^{J} \sum{i=1}^{I} E(t{w{ij}}) \cdot X{ij} \right) + Z{p} \sqrt{\left( \sum{j=1}^{J} \sum{i=1}^{I} Var(t{w{ij}}) \cdot X{ij}^{2} \right)}$$ (8)

Subject to:

$$\left(\left(N\sum{j=1}^{J}t{p{j}}\right)\cdot\left(\sum{j=1}^{J}\sum{i=1}^{I}X{ij}\right)\right) + \left(\sum{j=1}^{J}\sum{i=1}^{I}E(t{w{ij}}).X{ij}\right) + Z{p}\sqrt{\left(\sum{j=1}^{J}\sum{i=1}^{I}Var(t{w{ij}}).X{ij}^{2}\right)} + \left(t{m}\cdot\sum{j=1}^{J}\sum{i=1}^{I}X{ij}\right) \leq \left(t{c}\cdot\sum{j=1}^{J}\sum{i=1}^{I}X_{ij}\right)$$ (9)

$$Co.\sum{j=1}^{J}\sum{i=1}^{I}X{ij} \le B_1, \tag{10}$$

$$\left(E(Cd).\sum{j=1}^{J}\sum{i=1}^{I}X{ij}\right) + Z_P \sqrt{Var(Cd).\left(\sum{j=1}^{J}\sum{i=1}^{I}X{ij}^2\right)} \le \beta,$$ (11)

$$\left(1 - \prod_{j=1}^{J} \left(1 - e^{\frac{-t}{\theta_j}}\right)\right) \ge \alpha,\tag{12}$$

$$\sum{j=1}^{J} \sum{i=1}^{I} \frac{C_{ti} \cdot X{ij}}{M_i} \le B_2 , \tag{13}$$

$$\left[\sum{j=1}^{J} C{j}(t)\right] R{ij}(t){system} \leq B_{3}, \tag{14}$$

$$\sum{j=1}^{J} X{ij} \ge D_i, i=1,2,...,I,$$ (15)

$$X_{ij}, \theta_j \ge 0, i=1,2,...,I; j=1,2,...,J.$$ (16)

Note that, E() and Var() are the expected value and the variance of the stochastic parameters, respectively. Equation (7) is the first objective function intended to minimize the total cost of production. Equation (8) is the second objective function and is intended to minimize the total time of production. Equation (9) shows that the total production time is limited to the cycle time. Equation (10) indicates the limitation of the operational budget for the AGVs. Equation (11) represents a constraint for the acceptable defect rate. Equation (12) shows that the reliability of the system is restricted to a lowerbound α . Equation (13) indicates that the total available budgets for the required tools in each shop are limited to a pre-specified upper-bound value. Equation (14) indicates that the total available budget for the reliability of the machines is limited. Equation (15) indicates that the demand at each shop must be satisfied. Equation (16) enforces the non-negativity of the variables. Finally, ^{θ j} represents the stochastic breakdown of machine j. ^{θ j} is one of our decision variables, and our goal is to obtain the ^{θ j} values that can ensure no machine breakdown.

4.3.2 Perceptron Neural Network

In this study we apply perceptron to weight our time and cost objectives. To conceptualize a perceptron structure for our objectives consider a single-neuron perceptron with two-input and one neuron, as shown in Figure 2.

Fig. 2 Two input/single-output perceptron

In this figure, p1 and p2 are the inputs and w11 and w12 are their corresponding weights, respectively. Also, (^T ) wp is the weighted input corresponding to the target output. The output of this network is determined by:

$$a = \text{hardlim}(n) = \text{hardlim}(wp^{(T)} + b) = \text{hardlim}(w{11} \cdot p1 + w{12} \cdot p_2 + b)$$ . (17)

The decision boundary is determined by the input vectors for which the net input n is zero; that is,

$$n = wp^{(T)} + b = w{11} \cdot p1 + w{12} \cdot p_2 + b = 0.$$ (18)

The weights are the learning rates used in the proposed mathematical model. The summation of the learning weights are used as a single learning rate ξ . In order to configure our perceptron network, we consider the two factors of machine and operation as the inputs for time and the two factors of material and operator as the inputs for cost. The goal here is to find the final weights of the time and cost outputs. The perceptron network is run using the data observed in the manufacturing system and the weights of W1 (for time) and W2 (for cost) are obtained after the convergence. The perceptron computations are implemented in the MATLAB 7.0 package. These weights are used to unify the objective function. The weights are 0.2 for time and 0.1 for cost.

The mathematical model gives us the ^{θ j} s considering a confidence level as a reliability of the job-shop system. Now, we investigate the ^{θ j} s for the shops separately by data collection. The aim is to analyze the ^{θ j} s obtained from data collection which does not have any interactions with other shops, in comparison with the ^{θ j} s gained from the mathematical model in the last section. We collect data for a specific working time t=24 (minute). Then for t>24, our data are type I censored data. Assume 1: 2: : , , , n n rn xx x … are the r censored data in a specific shop, then the estimated θ (θ ˆ) using maximum likelihood estimation (MLE) is as follows:

Assume that X follows an exponential distribution,

$$X \approx EXP(\theta), \tag{23}$$

The cumulative distribution function is,

$$F(x) = 1 - e^{\frac{-x}{\theta}},\tag{24}$$

The probability density function is,

$$f(x) = \frac{1}{\theta} e^{\frac{-x}{\theta}},\tag{25}$$

Then for all xi:n , the ordered statistic is,

$$g(x{1:n}, x{2:n}, ..., x_{r:n}) = \frac{n!}{(n-r)!} \left( 1 - (1 - e^{\frac{-t0}{\theta}}) \right) \prod{i=1}^{r} \frac{1}{\theta} e^{\frac{-x_{i:n}}{\theta}}, \tag{26}$$

The likelihood function is,

$$L(\theta) = \frac{n!}{(n-r)!} \cdot \theta^{-r} \cdot e^{-\sum{i=1}^{r} x{i,n} + (n-r)t_0} \theta,$$ (27)

The logarithm of both sides of (27) gives,

$$\ln(L(\theta)) = \ln \frac{n!}{(n-r)!} - r \ln \theta - \sum{i=1}^{r} x{in} + (n-r)t_0,$$ (28)

The partial derivative of (28) with respect to θ is,

$$\frac{\partial \ln(L(\theta))}{\partial \theta} = -\frac{r}{\theta} + \frac{\sum{i=1}^{r} x{i:n} + (n-r)t_0}{\theta},$$ (29)

Then if we set (31) equal to zero, we have:

$$\hat{\theta}{MLE} = \frac{\sum{i=1}^{r} x_{i:n} + (n-r)t_0}{r} \,. \tag{30}$$

wheret0 is the end time of observation (t=24) and r is the number of failures. While setting the manufacturing system with the exact value of ^{θ j} obtained from the mathematical model is difficult due to mechanical specifications' changes during the manufacturing process, we propose a confidence interval for θ . Here,

using $$\frac{2r\hat{\theta}}{\theta} \approx \chi^2(2r)$$ as a pivot (2r is degree of freedom), we can set a

confidence interval for ^θ . To set a confidence interval using (2 ) ^{ˆ 2 2} r r ^{χ θ θ ≈} , we obtain:

$$P\left[\chi{1-\frac{\alpha}{2}}^{2}(2r) < \frac{2r\hat{\theta}}{\theta} < \chi{\frac{\alpha}{2}}^{2}(2r)\right] = 1 - \alpha, \tag{31}$$

By inverting (31) and multiplying all fractions by θ ˆ 2r we obtain:

$$P\left[\frac{2r\hat{\theta}}{\chi{\frac{\alpha}{2}}^{2}(2r)} < \theta < \frac{2r\hat{\theta}}{\chi{\frac{1-\alpha}{2}}^{2}(2r)}\right] = 1 - \alpha, \qquad (32)$$

where (34) is a two sided (1−^α )% confidence interval based on type I censored data. In our case, we collected data for a special working period. We were supposed to collect 30 observations but a failure occurred (type I censored data) in the 20th observation. Therefore, using (32) we can obtain the estimated^θ .

$$\hat{\theta}_{MLE} = \frac{235 + (30 - 20)10}{20} = 16.75.$$

Clearly the estimated θ is different from the one obtained from the mathematical model. Here, we configure the confidence interval for θ using the estimated value with a 0.95% confidence level (equation 33).

$$P\left[\frac{2\times20\times16.75}{\chi{0.025}^2(40)} < \theta < \frac{2\times20\times16.75}{\chi{0.975}^2(40)}\right] = 1 - 0.05; \quad \text{we} \quad \text{obtain}$$

[ ] 11.3,27.46 with a 95% confidence interval for ^θ .

The results show that the output produced in the integrated case was higher than the outputs produced by separate cost and time objective function considerations. In addition, the proposed model resulted in an increase in the availability of the manufacturing system through the improvements in the reliability of the machine types and the shops. These improvements help to maintain the system at a desirable level of reliability and to prevent sudden breakdowns.

4.4 Conclusions 53

4.4 Conclusions

AGV systems complement the operation in manufacturing systems by providing integrated automated material handling that capitalizes on the system's flexibility. Previous research considering AGVs systems has focused primarily on complex control strategies in MHSs. In this study, we focused on the time and cost measures in an optimization model used to evaluate an MHS with AGVs. The automated manufacturing system considered in this study has the following physical characteristics: (1) the manufacturing system is a job shop, and (2) single-load AGVs perform the material handling job in the shop.

We took into account the reliability of the manufacturing system because of the need for steadiness and stability in the system. Reliability was included in the model as a cost function. Furthermore, we considered bi-objective stochastic programming to optimize the time and cost objectives because of the uncertainties inherent in the optimization parameters in real-world problems. Finally, we used perceptron neural networks to transform the bi-objective optimization model into a single objective model.

The MHS proposed in this study could potentially be extended (or revised) to improve the effectiveness of a wide variety of decision tools in productivity improvement. For example, current trends in scheduling systems provide the production scheduler with powerful tools which can be used to optimize real-time workloads in various stages of production. The MHS could potentially be integrated within such a real-time scheduling system. The proposed method could also be potentially useful for general applications of business process improvement which strive to improve workflows within and between functional groups. These approaches often utilize systematic methods to process a large amount of imprecise and complex information to redesign critical business processes.

The limitations of this study and the futures research directions can be summarized as follows:

• Although we used optimization in this study, alternatively, a simulation method or a heuristic algorithm could be used in lieu of the proposed model. Buzacott and Yao (1986) presented a comprehensive review of the analytical models developed for the design and control of automated and flexible manufacturing systems. They advocated analytical methods over the simulation models because analytical methods provide a better insight into the system performance. Ho et al. (2000, p. 490), Lee et al. (2006, p. 1828), Kuo et al. (2007, p. 1002), and Crombecq et al. (2011, p. 683) have also confirmed that although computer simulations are often the first choice for modeling systems and even for optimization purposes, the simulation of complex

• systems with multiple input and output parameters can be both prohibitively expensive and time consuming. On the other hand, heuristics seek good feasible solutions to optimization problems where the complexity of the problem or the available time for its solution does not allow exact algorithms. Although the main measure of success for exact algorithms is time efficiency, we must often evaluate the quality of solutions when an exact optimum is not available (Gnanavel Babu et al. 2010).
• Although we did not experience this limitation in our study, the time complexity of the learning phase in some complex perceptron neural networks could be high depending on the heuristic used for calculating the weights and the halting condition. In such complex perceptron neural networks, a large number of passes may be required. Each pass involves computation of the outcome for every training set point followed by modification of the weights.
• Although in this study we usedthe time and cost measures in an optimization model, performance measures may be categorized on the basis of: time, cost, quality, and flexibility measures. Consideration of quality and flexibility is a natural extension of the model proposed in this study.
• Although in this study we considered single-load AGVs, researchers (e.g., Ho et al. 2004; Lee et al. 1996; Ozden, 1988) have already proven that multipleload AGV systems have many advantages over single-load ones. Another natural extension of this research is using multiple-load AGVs in the optimization model.

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Chapter 5

Uncertain Optimal Path for AGV System

5.1 Summary

Discrete event systems are characterized by changes in state over time, based on current state and state transition rules, where each state is separated from its neighbor by a step rather than a continuum of intermediate infinitesimal states. Examples of such systems are information systems, operating systems, networking protocols, banking systems, business processes and telecommunications systems, and flexible manufacturing systems. Traditional manufacturing has relied on dedicated mass-production systems to achieve high production volumes at low costs. As living standards improve and the demands for new consumer goods rise, manufacturing flexibility gains prominence as a strategic tool for rapidly changing markets. Flexibility, however, cannot be properly incorporated in the decisionmaking process if it is not well defined and measured in a quantitative manner. Today, manufacturing flexibility remains an elusive notion because of its inherent complexity and generality, in spite of a large body of published research work. There exist more than 50 definitions of (Sethi and Sethi, 1990) and six different approaches for obtaining a quantitative flexibility measure (Gupta and Goyal, 1989). Flexibility in its most rudimentary essence is the ability of a manufacturing system to respond to changes and uncertainties associated with the production process (Zelenovic, 1982; Buzacott, 1982; Gerwin, 1982). A comprehensive classification of eight flexibility types was proposed in Browne et al. (1984). Resource and system flexibilities were examined in Slack (1987), whereas global measures for flexible manufacturing systems (FMSs) were defined in Gupta and Buzacott (1989). Routing flexibility based on information theoretic concepts was examined in Yao and Pei (1990) and Kumar (1987). Flexibility measures for one machine, a group of machines, and whole industry were presented in Brill and Mandelbaum (1989), involving appropriate weights and machine efficiencies in carrying out sets of tasks.

Flexibility in its most rudimentary sense is the ability of a manufacturing system to respond to changes and uncertainties associated with the production process (Miettinen et al., 2010; Kumar and Sridharan, 2009; Das et al., 2009). In Barad and Sipper (1988), the period needed by a system to recover after a change was used as the central flexibility measure, whereas a stochastic dynamic programming model for its assessment was presented in Kulatilaka (1988). Artificial intelligence (AI) methods seem appropriate in most practical situations where numerical data are not readily available and linguistic variables are more amenable to handling imprecise knowledge (Dooner, 1991). The flexibility of competing systems can be ranked appropriately using an algorithmic approach (Abdel-Malek and Wolf, 1991) or a decision support system (Suresh, 1991) based on performance and economic criteria. Also, integer programming methods were proposed in Chandra and Tombak (1992), and a graphical representation method of production processes was presented in Kochikar and Narendran (1992).

Manufacturing flexibility is associated with uncertainty in all levels of a firm's operation, such as variation in the demand and characteristics of a product or unanticipated interruptions of the production process because of machine failures. In addition, human operators or managers use imprecise concepts and vague notions when they attempt to define or measure flexibility. Fuzzy set theory (Dubois and Prade, 1980; Zimmermann, 1991), and especially fuzzy logic, constitute natural frameworks for the representation and manipulation of uncertainty.Indeed, fuzzy set theory is an algebra of imprecise propositions and gradual statements such as "machine A is more flexible than machine B because it is more versatile." In previous treatments, uncertainty was handled by probability theory under the assumption that probabilities can be obtained precisely. Mandelbaum and Buzacott (1990), examining the meaning and use of flexibility in decision-making processes, admit that for real-world problems with increased complexity, the existing modeling methods are inadequate to represent reality. For context-dependent situations where conceptual imprecisions exist, however, as in the description of machine flexibility itself, fuzzy sets and fuzzy logic appear to be more appropriate for the definition and analysis of the problem.

Manufacturing costs for products are very crucial in decision making and strategic planning. And with respect to cost estimation, research and development departments in the past could only estimate the final product's total cost. Moreover, rules of thumb of the engineers are often applied as the cost estimation benchmarks, making the results controversial in terms of accuracy (Mostafaee et al., 2010; Eklin et al., 2009; Qian and Ben-Arieh, 2008; Wang, 2007; Verlinden et al., 2008; Wang, 2007).

Jobshop is a flexible, scalable and intelligent production planning and control system offering advanced functionality and value in key areas of manufacturing and assembly. The flexible jobshop problem (FJP) is an extension of the classical job shop problem allowing for an operation to be processed by any machine from a given set.

Automated guided vehicles (AGVs) reduce costs of manufacturing and increase efficiency in a manufacturing system. These trailers can be used to move raw materials in line to get them ready for manufacturing (Aized, 2009; Hsueh, 2010). To conceptualize an AGV, it is necessary to understand the fundamentals of flexible manufacturing systems (FMSs). Rather than using humans to perform repetitive tasks, a machine is used to perform the task. Each shop performs a specific task to assist in the manufacturing of a product. Although FMS is fast and efficient, but it is not cheap as it requires lots of expensive machines to operate. In the next section, we give a comprehensive description of our proposed problem of investigation.

5.2 Statement of the Problem

Consider a jobshop layout which applies an AGV for material handling. The AGV carries raw material, semi-produced and final products in batch sizes. Due to mounting demands, advancing technology, and rising production capacity, the need for increasingly more shops is mounting over time. The new shops are expected to have more advanced machines. Therefore, more than one shop with the same duty are evolved. The difference among shops having the same duty shows up in the shop's specifications that affect the production cost. As a result, the system would consider a flexible jobshop model where multi shops of the same duty exist and each operation can be processed on any type of machine in any shop. The sequences of jobs are specified and the jobs are assumed to be independent.

The structure of such a problem would configure a network. In this network, the nodes are the shops and the arcs are the flow paths of the AGV to each shop. Shops in each stage are of the same type but have different specifications such as different machine types and equipments, varied operator proficiencies, different rates of defect, etc.. Each flow path for the AGV is associated with a time parameter and also a cost parameter related to each shop. The aim is to find a path for the AGV minimizing an aggregate time and cost objective. Considering the variable status of the AGV flow among shops, the time of each flow path of the AGV is a triangular fuzzy number. In each shop, different machines and operators are working. Due to unpredictable events during working times a cost may incur. This cost is inferenced from an expert system via fuzzy logic "IF …. Then……." rules. Cost parameters of each shop are considered to be 3: (1) Equipment sensitivity, (2) operator proficiency, and (3) product specifications, each being specified by one of the three levels of low, moderate, and high. A configuration for the proposed problem is presented in Figure 1.

Fig. 1 A configuration of the proposed problem

As stated before, time is considered to be a triangular fuzzy number and cost is inferenced from an expert system. Our decision model is to consider both time and cost parameters, and thus an integration of the two parameters would be required. The integration is to have a weighted sum of cost and time as an arc length in the proposed network. Time is a triangular fuzzy number and cost is considered to be an indirect triangular fuzzy number. A description of fuzzy logic, numbers and systems are given in the next section.

5.3 Mathematical Modeling

5.3.1 Fuzzy Logic and Membership Functions

In Zadeh's words, who first introduced the notion of fuzzy logic (Zadeh, 1965), fuzzy logic is a tool for ''Computing with Words''. He stated that the main role of fuzzy logic was to serve as a methodology for computing with words when no other methodology could attain such purpose (Zadeh, 1996, 1999).

Fuzzy logic accommodates for the emulation of the human reasoning process and making decisions based on vague or imprecise data. Linguistic terms can better represent knowledge, experience, and subjective viewpoints of decision makers in more intuitive ways and natural language formats. Each linguistic term can be expressed by a fuzzy set. In fuzzy set theory, elements of a set are allowed to have membership values between zero and one. If we depict this membership value by α , then α can have any value in the interval [0,1]. Membership values for a fuzzy set are usually determined by a membership function. Triangular membership functions are common (Pedrycz, 1994).

Definition 1. A triangular fuzzy number a can be defined by a triplet ( , , ) a¹ a² a³ . Its conceptual schema and mathematical form are shown by:

$$\mu_{\bar{a}}(x) = \begin{cases} 0, & x \le a_1 \ \frac{x - a_1}{a_2 - a_1}, & a_1 < x \le a_2 \ \frac{a_3 - x}{a_3 - a_2}, & a_2 < x \le a_3 \ 0, & a_3 < x. \end{cases}$$ (1)

A triangular fuzzy number a in the universe of discourse X that conforms to this definition is shown in Figure 2.

Fig. 2 A triangular fuzzy number a

Definition 2. Assuming that both 123 a aaa =(, ,) and 123 b bb b ⁼( , ,) are triangular numbers, then the basic fuzzy operations are:

$$\begin{split} \tilde{a}\times\tilde{b} &= \left(a_1\times b_1, a_2\times b_2, a_3\times b_3\right) \text{ for multiplication,} \ \tilde{a}+\tilde{b} &= \left(a_1+b_1, a_2+b_2, a_3+b_3\right) \text{ for addition.} \end{split} \tag{2}$$

In Bastian (2000), different rule determination approaches have been described. Expert knowledge is the most popular and accepted method, in practice, while other common practices such as meta-rule techniques and fuzzy classifier systems which utilize genetic algorithms or artificial neural networks are applicable when expert knowledge is not available.

Roughly speaking, a fuzzy rule is the implication stated as an If-Then rule in which the premise and the conclusion are fuzzy sets. The basic components are the ''If'' part, usually referred to as the antecedent, and the ''Then'' part, usually referred to as the consequent (like If antecedent Then consequent). The antecedent can be composed of a single condition or a set of conditions combined by conjunction operators like ''AND'' and ''OR''.

Once the rules are determined, the next step is to determine the matching degree of the inputs with respect to the fuzzy rules to perform the inference process. If there are multiple inputs, a conjunction operator is used to combine the matching degree of the fuzzy inputs utilizing ''min'', ''max'', or ''product'' (Klir and Yuan, 1995; Zimmermann, 1996). The two most principal methods of fuzzy inference are ''clipping'' and ''scaling'' methods (Klir and Yuan, 1995; Zimmermann, 1996), both suppressing the membership functions for the consequent, depending upon the degree of matching. Once the final outputs of the fuzzy rules are obtained through fuzzy inference, the outputs are combined into one single aggregated output. Since there might be more than one rule with a matching degree greater than zero, then more than one rule may be configured (Klir and Yuan, 1995; Zimmermann, 1996).

5.3.2 Fuzzy Systems

The fuzzy logic and rule-based reasoning have found various applications in the control of industrial processes, modeling of complex systems and the development of fuzzy inference systems (Lin and Chen, 2004; Machacha and Bhattacharya, 2000). Due to their varied usage, fuzzy inference systems are also known as fuzzy expert systems, fuzzy rule-based systems, fuzzy associative memories or, in short, fuzzy systems. In general, a fuzzy system is composed of five major blocks:

• Fuzzifier which transforms the crisp inputs into degrees of match with linguistic values;
• dictionary which defines the membership functions of the fuzzy sets used in the fuzzy rules;
• rule base which contains fuzzy if-then rules and along with the dictionary, comprises the knowledge base of the fuzzy system;
• decision maker which performs the inference operations on the rules;
• defuzzifier which converts the fuzzy results of inference into crisp outputs.

Based on the different trends in forming the major blocks of a fuzzy system and various kinds of application to which the fuzzy systems are applied, different types of fuzzy systems have been introduced. Mamdani fuzzy system and TSK fuzzy system are two types being commonly used in the literature (Klir and Yuan, 1995; Zimmermann, 1996). Each one is being adopted in its own special domain because of its particular accommodation for the different ways of knowledge representation. However, other types of fuzzy systems such as Tsukamoto's (Tsukamoto, 1979), ANFIS (Jang, 1993; Jang and Sun, 1995), etc., have their own special applications. Next, we will give a brief discussion of the first two types.

• Mamdani Fuzzy System

Mamdani fuzzy system was proposed as a first attempt to control a steam engine and boiler combination by a set of linguistic control rules obtained from experienced human operators. Rules in Mamdani fuzzy systems are like these (Klir and Yuan, 1995; Zimmermann, 1996):

If $$x_1$$ is $A_1$ AND/OR $x_2$ is $A_2$ Then $y$ is $B_1$ ,

whereA1, A2 and B1 are fuzzy sets. The fuzzy set acquired from aggregation of rules' results will be defuzzified using defuzzification methods like centroid (center of gravity), max membership, mean-max, and weighted average. The centroid method is very popular, in which the ''center of mass'' of the result provides the crisp value. In this method, the defuzzified value of the fuzzy set A, d(A), is calculated by

$$d(A) = \frac{\int{X} x.\mu{A}(x)/x}{\int{X} \mu{A}(x)/x},$$ (3)

where (.) ^{μ A} is the membership function of the fuzzy set A (Klir and Yuan, 1995; Zimmermann, 1996).

For our problem in which various possible conditions of parameters are stated in forms of fuzzy sets, we utilize the Mamdani fuzzy system, because the fuzzy rules representing the expert knowledge in Mamdani fuzzy systems would consider fuzzy sets in their consequences, while in TSK fuzzy systems, the consequences are expressed as crisp functions. In general, designing a fuzzy system is composed of the following major steps (Klir and Yuan, 1995):

• Step 1. Identifying pertaining input and output variables. Besides, the meaningful linguistic states along with appropriate fuzzy sets for the variable ought to be selected.

• Step 2. Introducing a fuzzification method for input variables that expresses the associated measured uncertainty. The purpose of the fuzzification method is to interpret measurements of input variables which are expressed by real numbers.

• Step 3. Formulating pertaining knowledge in terms of fuzzy inference rules. There are two principal ways in which relevant inference rules can be determined. One is to elicit them from experienced humans and the other is to obtain them from empirical data by suitable learning methods, usually with the help of neural networks.

• Step 4. Combining measurements of input variables with relevant fuzzy rules to inference, regarding the output variables in which the purpose of inference engine will be obtained.

• Step 5. Ascertaining a suitable defuzzification method to convert the aggregated fuzzy set of implications into a real number.

5.3.3 Modeling the Proposed Fuzzy Problem

As stated before, time is a triangular fuzzy number, ensued from the experts' knowledge. Brainstorming and expert knowledge versus meta-rule techniques (neural networks, genetic algorithms, etc.) are two common approaches for defining fuzzy rules and membership functions. While available empirical data is requisite for using the second approach, due to unavailability of historical data for cost, we make use of the first approach to obtain the membership functions and fuzzy rules.

The input to our Mamdani type fuzzy system is composed of equipment sensitivity, operator proficiency, and product specifications. For any of the inputs, three linguistic terms of "low", "moderate", and "high" are defined. The output of the system is cost that is identified by any of the three linguistic terms, "low", "moderate", or "high". The maximum membership grade of linguistic term ''high'' is 30.

As a result, a triangular fuzzy number as the time and a numerical value as the cost are obtained. We intend to consider an integrated time-cost value as the value of each proposed arc in the network. Time and cost having different scales, it would not be possible to perform basic operations such as addition on their original forms. Thus, to remove the scales, we normalize the time and cost values throughout the network and then consider a weighing method to make them appropriate for basic operations. On the other hand, time is a triangular fuzzy number and cost is a crisp numerical value. But, as implied by Definition 2, we need to have triangular numbers for both parameters to perform the basic operations. For this, we consider cost as a trivial triangular fuzzy number and show it by a triplet (0,C,0), where C is the numerical value inferenced from the expert system for cost with the membership value equal to 1. The right and left sides of C are zero since their membership values are zero. In the next section, we present an approach for weighing the parameters.

5.3.4 Operations on the Parameters of the Problem: Normalization of the Parameters

As stated, to perform basic operations on the time and cost parameters with their different scales, we need to remove their scales. To do this, two normalization processes are proposed for time and cost, separately. In (4), (5) and (6) below, i is an index to show the node and j is an index to show the stage (shop type) in the proposed network, *rij*is the value of either time or cost parameter in each node, and *nij*is its corresponding normalized value. Considering time as a triangular fuzzy number, the normal value with a positive view is given by

$$n{ij} = \frac{r{ij} - r_j^{\min}}{r_j^{\max} - r_j^{\min}}, \qquad \forall i, j,$$ (4)

while the normal value with a negative view is given by,

$$n_{ij} = \frac{rj^{\text{max}} - r{ij}}{r_j^{\text{max}} - r_j^{\text{min}}}, \qquad \forall i, j,$$ (5)

where max ^j r and min ^j r are maximum and minimum values in each column of an assumed matrix of time or cost, respectively. Since time in our proposed model is a criterion implicating a negative aspect in decision making, then we choose (5) for normalizing time in our approach.

Assuming cost as a crisp value, we normalize the cost values as follows:

$$n{ij} = \frac{r{ij}}{\sqrt{\sum{k} r{kj}^2}}, \qquad \forall i, j,$$ (6)

To compute the minimum or maximum value in (4) and (5), comparisons are needed to be made. This means that it is necessary to have a method for ranking and comparing fuzzy numbers. An operator ≺ for ordering fuzzy numbers can be defined as follows (see Mahdavi et al., 2009):

$$\tilde{A} \leq \tilde{B} \iff (a_1 \leq b_1) \wedge (a_2 \leq b_2) \wedge (a_3 \leq b_3) \wedge (a_4 \leq b_4). \tag{7}$$

However, this relation is not a complete ordering, as fuzzy numbers A and B satisfying

$$\exists i, j \in {1, 2, 3, 4}) : (a_i < b_i) \land (a_i > b_i), \tag{8}$$

would not be comparable by ≺ .

The ranking or ordering methods for fuzzy quantities have been proposed by several authors. For summaries, see Bortolan and Degani (1985) and Delgado et al. (1988). Admittedly, none of these methods is commonly accepted. Various ranking functions may produce conflicting and controversial results in comparison of fuzzy numbers (these issues have been discussed in Wang and Kerre, 2001, through a specific example).

Here, we use a fuzzy ranking method recently adopted by Mahdavi et al. (2009). Consider the fuzzy min and max operations defined in analogy with the fuzzy addition as follows:

$$\begin{aligned} & MINV = \text{Min value}(\widetilde{\mathbf{a}}, \widetilde{\mathbf{b}}) = (\min(\mathbf{a}_1, b_1), \min(a_2, b_2), \min(a_3, b_3), \min(a_4, b_4)). \end{aligned} \ & MAXV = \text{Max value}(\widetilde{\mathbf{a}}, \widetilde{\mathbf{b}}) = (\max(\mathbf{a}_1, b_1), \max(a_2, b_2), \max(a_3, b_3), \max(a_4, b_4)). \end{aligned}$$

It is evident that, for non-comparable fuzzy numbers a ~ and b ^~ , the fuzzy min operation results in a fuzzy number different from both of them. For example, for ^{a =} (5,10,13,19) and b = (6,9,15,20) , we get from (9) a fuzzy MINV = (5,9,13,19) which differs from a and b . To alleviate this drawback, a method based on the distance between fuzzy numbers is proposed. We use the distance function introduced in Sadeghpour Gildeh and Gien (2001). The main advantages of this distance function, to be defined next, are the generality of its usage on various fuzzy numbers, and its reliability in distinguishing unequal fuzzy numbers.

Definition 3. The Dp q, -distance, indexed by parameters 1 < <∞ p and 0 1 < < q , between two fuzzy numbers a and b is a nonnegative function given by:

$$D_{p,q}(\widetilde{a},\widetilde{b}) = \begin{cases} \left[ (1-q) \int0^1 \left| a\alpha^- - b_\alpha^- \right|^p d\alpha + q \int0^1 \left| a\alpha^+ - b\alpha^+ \right|^p d\alpha \right]^{\frac{1}{p}}, & p < \infty, \ (1-q) \sup{0 \le \alpha \le 1} \left( \left| a\alpha^- - b\alpha^- \right| \right) + q \inf{0 \le \alpha \le 1} \left( \left| a\alpha^+ - b_\alpha^+ \right| \right), & p = \infty, \end{cases}$$

$$(10)$$

where ^{+ −} a^α a^α , and ^{+ −} b^α b^α , are the corresponding right and left ^α -cuts of a and ^b, respectively. The analytical properties of Dp q, depend on the first parameter ^p, while the second parameter q of Dp q, characterizes the subjective weight

attributed to the end points of the support; that is, ⁺ a^α and ⁻ a^α . of the fuzzy numbers. If there is no reason for distinguishing any side of the fuzzy numbers, then 1 2 p D is recommended. Having q close to 1 results in considering the right

side of the support of the fuzzy numbers more favorably. Since the significance of the end points of the support of the fuzzy numbers is assumed to be the same, then we consider 1 ^{2 q =} .

For triangular fuzzy numbers ( ) ¹²³ a aa a ⁼ , , and ( ) ¹²³ b bb b ⁼ , , , the above distance with p = 2 and 1 ^{2 q =} is then calculated as:

$$D{2,\frac{1}{2}}(\widetilde{a},\widetilde{b}) = \sqrt{\frac{1}{6} \left[ \sum{i=1}^{3} (b_i - a_i)^2 + (b_2 - a2)^2 + \sum{i=1}^{2} (b_i - ai)(b{i+1} - a_{i+1}) \right]}$$ (11)

We are now able to compute the distance between a b, and MINV using the proposed distance function (11); for a ~ and b ~ as given before, we get , ) 0.1667 ~( 1 2, ^{D a} MINV ⁼ and , ) 1.33 ^~ ( 2 1 2, D b MINV = . The number with

lowest distance is the minimum. Now, we are ready to propose the following algorithm for the comparison of fuzzy numbers.

Algorithm 1 (compare two triangular fuzzy numbers).

Input: Two fuzzy numbers ( ¹ ~ L and 2 ~ L ).

Output: Minimum or maximum between the two triangular fuzzy numbers ( min L or ~max ^L )

Step 1: Compute the minimum or maximum value ( MINV or MAXV ) by Equation (9).

Step 2: Find the distance of MINV or MAXV from ( ¹ ~ L and 2 ~ L ) using Equation (11).

Step 3: Determine min ^L or ~max ^L giving the smallest or largest distance.

5.3.5 Weighing the Parameters

To weigh the parameters, we take a multi criteria decision making approach. Multi-criteria decision making (MCDM), dealing primarily with problems of evaluation or selection (Keeney and Raiffa, 1976; Teng, 2002), is a rapidly developing area in operatios research and management science. The analytical hierarchy process (AHP), developed by Saaty (1980), is a technique of considering data or information for a decision in a systematic manner (Schniederjans and Garvin, 1997). AHP is mainly concerned with the way to solve decision problems with uncertainties in multiple criteria characterization. It is based on three principles: (1) constructing the hierarchy, (2) priority setting, and (3) logical consistency. We apply AHP to weigh the parameters.

5.3.6 Construction of the Hierarchy

A complicated decision problem, composed of various attributes of an objective, is structured and decomposed into sub-problems (sub-objectives, criteria, alternatives, etc.), within a hierarchy.

5.3.7 Priority Setting

The relative "priority" given to each element in the hierarchy is determined by pair-wise comparisons of the contributions of elements at a lower level in terms of the criteria (or elements) with a causal relationship. In AHP, multiple paired comparisons are based on a standardized comparison scale of nine levels (see Table 1 due to Saaty, 1980).

| | | | Table 1 Scale of relative importance | |--|--|--|--------------------------------------| |--|--|--|--------------------------------------|

Intensity of importance	Definition of importance
1	Equal
2	Weak
3	Moderate
4	Moderate plus
5	Strong
6	Strong plus
7	Very strong or demonstrated
8	Very, very strong
9	Extreme

Let { }ⁿ C c ,...,c = 1 be the set of criteria. The result of the pair-wise comparisons on n criteria can be summarized in an n × n evaluation matrix A in which every element aij is the quotient of weights of the criteria, as shown below:

$$A = (a_{ij}), i, j = 1, ..., n.$$ (12)

The relative priorities are given by the eigenvector (w) corresponding to the largest eigenvalue

$$(\lambda{\max})$$ as: $$Aw = \lambda{\max}w. \tag{13}$$

When pair-wise comparisons are completely consistent, the matrix A has rank 1 and λmax = n. In that case, weights can be obtained by normalizing any of the rows or columns of A.

The procedure described above is repeated for all subsystems in the hierarchy. In order to synthesize the various priority vectors, these vectors are weighed with the global priority of the parent criteria and synthesized. This process starts at the top of the hierarchy. As a result, the overall relative priorities to be given to the lowest level elements are obtained. These overall, relative priorities indicate the degree to which the alternatives contribute to the objective. These priorities represent a synthesis of the local priorities, and reflect an evaluation process that permits integration of the perspectives of the various stakeholders involved.

5.3.8 Consistency Check

A measure of consistency of the given pair-wise comparison is needed. The consistency is defined by the relation between the entries of A; that is, we say A is consistent if aik= aij · ajk, for all i,j,k. The consistency index (CI) is:

$$CI = \frac{(\lambda_{\text{max}} - n)}{(n - 1)}.\tag{14}$$

The final consistency ratio (CR), on the basis of which one can conclude whether the evaluations are sufficiently consistent, is calculated to be the ratio of the CI and the random consistency index (RI):

$$CR = \frac{CI}{RI}.\tag{15}$$

The value 0.1 is the accepted upper limit for CR. If the final consistency ratio exceeds this value, the evaluation procedure needs to be repeated to improve consistency. The measurement of consistency can be used to evaluate the consistency of decision makers as well as the consistency of all the hierarchies.

We are now ready to give an algorithm for computing parameter weights using the AHP. The following notations are used.

Notations and definitions:

n: Number of criteria.

i: Number of parameters.

p: Index for parameters, *p=1or 2.*

d: Index for criteria, 1 ≤ d ≤ D .

Rpd : The weight of pth item with respect to dth criterion.

wd : The weight of dth criterion.

Algorithm 2: PWAHP (compute parameter weights using the AHP).

Step 1: Define the decision problem and the goal.

Step 2: Structure the hierarchy from the top through the intermediate to the lowest level.

Step 3: Construct the parameter-criteria matrix using steps 4 to 8 using the AHP. {Steps 4 to 6 are performed for all levels in the hierarchy.}

Step 4: Construct pair-wise comparison matrices for each of the lower levels for each element in the level immediately above by using a relative scale measurement. The decision maker has the option of expressing his or her intensity of preference on a nine-point scale. If two criteria are of equal importance, a value of 1 is set for the corresponding component in the comparison matrix, while a 9 indicates an absolute importance of one criterion over the other (Table 1 shows the measurement scale defined by Saaty, 1980).

Step 5: Compute the largest eigenvalue by the relative weights of the criteria and the sum taken over all weighted eigenvector entries corresponding to those in the next lower level of the hierarchy.

Analyze pair-wise comparison data using the eigenvalue technique. Using these pair-wise comparisons, estimate the parameters. The eigenvector of the largest eigenvalue of matrix A constitutes the estimation of relative importance of the attributes.

Step 6:Construct the consistency check and perform consequence weights analysis as follows:

$$A = (a{ij}) = \begin{bmatrix} 1 & w{1} \ w{2} & \cdots & w{1} \ w{2} \ w{1} & 1 & \cdots & w{2} \ w{1} & \vdots & \vdots & \ddots \ w{n} \ w{1} & w{n} \ w{2} & & 1 \end{bmatrix}.$$

Note that if the matrix A is consistent (that is, aik= aij · ajk, for all i jk n , , 1, 2, ..., = ), then we have (the weights are already known),

$$a_{ij} = \frac{w_i}{w_j}, \qquad i, j = 1, 2, ..., n.$$

If the pair-wise comparisons do not include any inconsistencies, then λmax= n.

The more consistent the comparisons are, the closer the value of computed λmax is to n. Set the consistency index (CI), which measures the inconsistencies of pairwise comparisons, to be:

$$CI = \frac{\left(\lambda_{\max} - n\right)}{\left(n - 1\right)},\,$$

and let the consistency ratio (CR) be:

$$CR = 100 \left( \frac{CI}{RI} \right),$$

wheren is the number of columns in A and RI is the random index, being the average of the CI obtained from a large number of randomly generated matrices.

Note that RI depends on the order of the matrix, and a CR value of 10% or less is considered acceptable (Saaty, 1980).

Step 7: Form the parameter-criteria matrix as specified in Table 2:

Table 2 The parameter-criteria matrix.

	C1	C2	…	Cd
parameter 1	R11	R12	…	R1d
parameter 2	R21	R22	…	R2d

| Step 8: As a result, configure the pair-wise comparison for criteria-criteria matrix | | | | |--------------------------------------------------------------------------------------|--|--|--| | as in Table 3: | | | |

	C1	C2	…	Cd	w d
Criteria 1	1	a12	…	a1d	w1
Criteria 2	1/a12	1	…	a2d	w2
#	#	#	#	#	#
Criteria d	1/a1d	1/a2d	…	1	wd

Table 3 The criteria-criteria pair-wise comparison matrix.

The wd are gained by a normalization process. The wd are the weights for criteria.

Step 9: Compute the overall weights for the parameters, using tables 2 and 3, as follows:

$$\psi$$ = Total weight for parameter $1 = R_{11} \times w1 + R{12} \times w2 + ... + R{1d} \times wd$ ,
$\psi$ ' = Total weight for parameter $2 = R{21} \times w1 + R{22} \times w2 + ... + R{2d} \times w_d$ , (16)

where ψ '= 1 −ψ . Note that due to the significance and sensitivity of different shops of the same type in any stage the weighing process is performed for each arc. Here, we obtain the weights for the parameters of each arc in the proposed network. If we consider ⁿ C and ⁿ T as normalized cost and time, respectively, then the total weighted normalized value of each arc is determined as follows:

$$P = (\psi \times T^n) + (\psi \times C^n). \tag{17}$$

In the next section, we propose an approach to identify the shortest path in the network using the total weighted normalized value for each arc. Before that, consider a ranking method for triangular fuzzy numbers.

5.3.9 Ranking of Triangular Fuzzy Numbers

Several fuzzy ranking methods have been proposed (Bortalan and Degani, 1985; Luis and Antonio, 1989; Kim and Park, 1990). Since the graded mean integration representation method (Chen and Hseih, 2000) not only alleviates some drawbacks of the existing methods, but also possesses the advantages of being easily implementable and quite effective in problem solving (see Lee et al., 2007), we will use it to transform the total weighted normalized value of each arc in our proposed network. If a ⁼ ( , , ) a¹ a² a³ is a triangular fuzzy number, then the graded mean integration ( ) ~ R(a ) is defined to be:

$$R(\tilde{a}) = \frac{a_1 + 4a_2 + a_3}{6}. (18)$$

We apply (18) to transform the fuzzy numbers to crisp values and use them to find the optimal path.

5.3.10 Shortest Path in a Network

Let G = (V,A) be a graph, where V = { } 1,...,N is the set of nodes, and A ⊂V ×V is the set of arcs. We write (i, j) ∈ A , if there exists an arc from node *i*∈V to node j ∈V . Furthermore, let ≥ 0 ij t denote the distance (or travel time, or any other measure of cost) from i to j. If (i, j) ∉ A ,then set tij = +∞. Note that the travel time from node i to node j is assumed to be stationary; i.e., independent of the actual arrival time at node i. Let *fij*denote the length of the shortest-path from i to j in the graph.

It is a well-known principle that every additive deterministic dynamic programming formulation can be equivalently viewed as the problem of finding the shortest path in a directed network, where the states, decisions, and decision costs of the former correspond to the nodes, arcs, and arc lengths of the latter (Dreyfus and Law, 1977). It is perhaps for this reason that the task of efficiently computing shortest paths is found prominence in the mathematical programming literature. Next, we describe a dynamic programming approach for computing the optimal path.

Dynamic programming (DP) was introduced by Bellman (1957). Toth (1980) presented the early DP-based approaches and reported numerical experiments with a limited success. Hybrid methods, combining dynamic programming and implicit enumeration, were developed later. The first approach was developed by Plateau and Elkihel (1985). A recent approach, the so-called combo algorithm, is able to solve very large instances of up to 10000 variables within less than one second, with basically no difference in the required solution times for ''easy'' and ''hard'' instances (Martello et al., 1999). Marsten and Morin (1978) proposed the first hybrid method, which combined heuristic algorithms, dynamic programming and branch-and-bound approaches. More sophisticated methods can be found in Ibaraki (1987).

Dynamic programming is a technique to tackle multistage decision processes. A given problem is subdivided into smaller subproblems, which are sequentially solved until the initial problem is solved by the aggregation of the subproblem solutions. In each stage, a set of states is defined. The states would describe all possible conditions of the process in the current decision stage, which corresponds to every feasible partial solution. The set of all possible states is known as the state space. The states of a stage u can be transformed to states of a stage u+1, using a transition. A transition indicates the decisions adopted in a stage, and a sequence of transitions taken to reach a state starting from another state is known as a policy. Dynamic programming approaches can be seen as transformations of the original problem to one associated with the exploration of a multistage graph G(S,T), where the vertices in S correspond to the state space and the arcs in T correspond to the set of transitions, leading to an optimal policy.

The basis of dynamic programming can be traced to the optimality principle of Bellman (2003). The optimality principle states that an optimal policy should be constituted by optimal policies from every state of the decision chain to the final state.

Here, we make use of a dynamic programming approach for our proposed network to identify the optimal manufacturing path. This model helps the manufacturing system to determine the more profitable shops. The advantages of such a model are simplicity, the ability to determine the exact optimal value, and implementability on sophisticated networks. The backward dynamic model would be defined as:

Indices:

s Number of stages; s= 0,1,2,…, n. i' Start node number; i'=1,2, …, I; i'=0 (for the start node). i End node number; i=1,2, …, I.

Notations:

(i') ^ϕs The minimum value of moving from node i' in stage s to an end node i in stage s-1.

Pi'i Numerical value of an arc between node i' to node i.

Objective function:

$$\varphi{s}(i') = \underset{i \text{ in layer } s+1}{\min} \left{ \varphi{s+1}(i) + P_{i'i} \right}, \qquad \forall i' \text{ in stage } s, (s = 0,1,2,...,n).$$

$$\varphi_{s}(i) = (0),$$

$$\varphi^{*} = \varphi_{0}(0).$$

$$(19)$$

Note that ϕ * identifies the optimal path.

5.4 Conclusions 75

5.4 Conclusions

We proposed an approach for finding an optimal path in a flexible jobshop manufacturing system considering two criteria of time and cost. The proposed flexible jobshop system has more than one shop with the same duty. The difference among shops with the same duty is in their machines with various specifications. The shops configure a network in which they are considered as nodes and the paths among them are considered as network arcs. An Automated Guided Vehicle (AGV) functions as a material handling device through the manufacturing network. Time is considered to be a triangular fuzzy number and cost is inferred from an expert system considering three parameters of equipment sensitivity, operator proficiency, and product specification via linguistic variables. As unique contributions of the work, the objective was to find a path which minimizes both the time and cost criteria, aggregately. Since time and cost had different scales, a normalization process was used to remove the scales and because the model was bi-objective, the AHP weighing method was applied to gain a single objective. A dynamic programming approach was used to compute a shortest path in the proposed network. Finally, the proposed approach was illustrated by a numerical example. Various applicability and advantages of the proposed model were discussed. The limitations of the research were the work load of the AGVs and the defect rate of the guide path. As future research, other multi-objective optimization approaches could be tested and compared with the proposed method.

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Chapter 6

Cross Entropy Model for AGV Routing Time

6.1 Summary

Today, in order to survive in the rapid challenging environment of the modern manufacturing era, manufacturers are forced to adopt new technologies, especially for products that are made in small batch production. Automated Guided Vehicles (AGV) has been applied for the automated manufacturing system. In industrial application, manufacturing factory is brought the mobile vehicle to incorporate working with other machine in order to being the automated manufacturing system (Arai et al., 2002).Advanced automated manufacturing systems are widely used in industrial companies where productivity objectives have to be met. These systems often being costly, they must be designed to be as efficient as possible. Here, an automated manufacturing system in a job shop layout considering AGV as a material handling resource is focused. The key issue in manufacturing operations is how to produce high quality products at low costs in such a way that the diversified demand is met. Hence, modern manufacturing companies should become as responsive as possible in order to satisfy customer demands. Material handling accounts for 30–75% of the total cost of a product, and efficient material handling can result in reducing the manufacturing system operations cost by 15– 30% (Sule, 1994). These points underscore the importance of material handling costs reduction as a key element in improving the cost structure of a product. The determination of a material handling system involves both the selection of suitable material handling equipment and the assignment of material handling operations to each individual piece of equipment. Hence, material handling system selection can be defined as the selection of material handling equipment to perform material handling operations within a working area considering all aspects of the products to be handled.The material handling system plays a crucial role in automated manufacturing systems. When inadequately designed, the material handling system indeed can adversely affect the overall performance of the system and lead to substantial losses in productivity and competitiveness, and to unacceptably long lead times. Thus, to avoid such pitfalls, material handling system design must be integrated into the overall design of the manufacturing system centering on the selection of machines and the allocation of operations to the machines (Butdee and Suebsomran, 2006; Shirazi et al., 2010).Automated guided vehicle is an intelligent machine that has 'intelligence' to determine its motion status according to the environmental conditions systems. AGVs are advanced material handling devices extensively used in automated manufacturing systems (AMS) to transport materials among workstations (Vis, 2006).

Automated manufacturing systems (AMSs), equipped with several CNC machines and an AGV-based material handling system, are designed and implemented to secure automation and gain efficiency of production. To achieve these benefits, the planning in the AMS decision making process is critical because the planning decision has an influence on the subsequent decision processes such as scheduling, dispatching, etc. Subtasks are classified into two types for material processing and delivery. To pursue a global optimization of material processing and transportation cost, a manufacturing plan should consider the two costs simultaneously. Moreover, to obtain a better solution for subsequent decision making processes, not only total workloads are evenly distributed among the workstations, but workloads are well balanced among AGV's network segments. Many factories were adopted it into assembly line or production line such as automobile, food processing, wood working, port container terminal, and other factories.

The cross-entropy (CE) method is a general Monte Carlo approach to combinatorial and continuous multi-extremal optimization(De Boer et al., 2005). The method originated from the field of rare event simulation (Rubinstein, 1997), where very small probabilities need to be accurately estimated, for example in network reliability analysis, queuing models, or performance analysis of telecommunication systems. The CE method can be applied to static and noisy combinatorial optimization problems such as the traveling salesman problem, the quadratic assignment problem, DNA sequence alignment, the max-cut problem and the buffer allocation problem, as well as continuous global optimization problems with many local extrema (Rubinstein, 1999).

In a nutshell the CE method consists of two phases:

• Generate a random data sample (trajectories, vectors, etc.) according to a specified mechanism.
• Update the parameters of the random mechanism based on the data to produce a "better" sample in the next iteration.

Here, we aim to obtain the total material handling time for an AGV while the AGV activity times are stochastic. Due to importance of material handling in manufacturing systems specially AMS, estimating the material handling time is substantial for planning purposes to cover demands considering the capability and availability of the machines and equipment.

Automated Guided Vehicle has firstly developed and conducted as a research by (Butdee and Suebsomran, 2006 and 2007; Butdee et al., 2006) in the attempt to be used at Jumbo Truck Manufacturing in Thailand. An AGV need to sense its environment to be able to plan its operations and then act based on this plan. Thus, an AGV system possesses more flexibility and capacity then other

6.1 Summary 81

conventional material-handling systems and plays an important role in AMS (Reveliotis, 2000).

An automated guided vehicle system enables flexible material routing and dispatching, and is especially suited for flexible manufacturing environments in which product mix and priorities may continuously vary (Dubois, 1983). An AGV based material handling system is designed and implemented to gain production the flexibility and efficiency (Co and Wysk, 1986; Suri and Hildebrant, 1984; Dallery and David, 1984). There are four basic AGV material flow configurations: conventional network, unidirectional loop network, tandem configuration and segmented loop topology (Asef-Vaziri and Laporte, 2005). For tandem layout we can point to a recent work by Shirazi et al., (2010) where they illustrated a nonlinear multi-objective problem for minimizing the material flow intra and interloops and minimization of maximum amount of inter cell flow, considering the limitation of tandem automated guided vehicle (TAGV) work-loading.

The use of AGV has increased considerably since their introduction in the 1950s, along with the number of AGV application areas and types (Vis, 2006). As the complexity and size of AGVS applications increase, evaluating performance becomes more difficult. The last decade has also seen great progress in control methodologies of autonomous multi-robot systems (Buzacott, 1999). The objective of optimization model is the minimization of the total distance traveled by vehicles to transport the material handling system. The route planning of AGVs in FMS was proposed by (Co and Wysk, 1986; Suri and Hildebrant, 1984; Dallery and David, 1984). Authors in (Co and Wysk, 1986) presented the new approach for dynamic route planning and scheduling problem of AGVs. They applied the search algorithm and some heuristic rules to solve the route assignment in dynamic situations. Suri and Hildebrant (1984) also proposed the path planning strategy of AGV to navigation, collision avoidance and docking to the target. The path planning was implemented on-board computer in order to avoid the wireguided path. Not only the AGV was moved along the path with collision avoidance, but also it should be navigated with no deadlock condition as done by Srinivasan et al. (1994).

For an AMS, it is desirable to be able to increase or decrease the product with the rise and fall of demand. Therefore, Fazlollahtabar et al. (2010) proposed a flexible jobshop automated manufacturing system to optimize the material flow. The flexibility was on the multishops of the same type and also multiple products that can be produced. An AGV was applied for material handling. The objective was to optimize the material flow regarding the demand fluctuations and machine specifications.

Application of cross entropy (CE) method to rare event simulation has been studied to obtain an accurate estimation of very small probabilities. CE mainly uses various importance sampling schemes to estimate rare event probabilities in Markovian systems such as Markovian reliability models and Jackson networks.Kaynar and Ridder (2010), present a general state-dependent importance sampling method which partitions the state space and applies the cross entropy method to each partition. They investigated two versions of their proposed algorithm and employed them in several examples of reliability and queuing models.

6.2 Statement of the Problem

We consider a jobshop manufacturing system make use of AGV for material handling purpose as shown in Figure 1. Since uncertainty is elusive in automated manufacturing systems, the activities' times are stochastic and follow statistical distributions. The statistical distribution of each AGV activity is considered for obtaining the total material handling time. The activities that the AGVs perform in material handling process are such as direct movement, waiting, pickup/delivery, breakdown, etc. These activities are accompanied with time span. These time spans follow statistical distributions due to uncertainty and stochastic nature of automated equipment such as AGV. Therefore, the collected data (activity times) are considered by goodness of fit technique to obtain the probability distribution function.

Fig. 1 A configuration of AGV-based jobshop manufacturing system

The material handling begins with taking the raw materials from the depot and moving to the shops considering the production plan and job sequences to produce the final product. Products are in batches. Since, material handling activities for each shop are effective on the total material handling time, all the shops' activities should be considered, i.e., the total AGV material handling time is an aggregation of all shops' activities.

The aim here is to obtain the AGV total stochastic material handling time. While the times are considered to follow statistical distributions, we need to 6.3 Mathematical Model 83

propose special techniques to obtain the total AGV material handling time. Our proposed approach is called heuristic statistical method (HSM). The flowchart of the proposed methodology to consider AGV stochastic activity times to obtain total material handling time is shown in Figure 2. To analyze the system, the data are collected for each activity. Then, the collected data are tested via goodness of fit technique to clarify the probability distribution function (pdf). If the goodness of fit technique does not provide a known pdf, then cross entropy (to be discussed later) approach is used. If the pdfs are known and same, i.e., all activity times follow exponential distribution functions, the HSM is employed to find total material handling time. Again, if the pdfs are not same, then cross entropy approach is used. Both HSM and cross entropy approach are verified using simulation study.

6.3 Mathematical Model

To model the problem, we consider a simple case in which an AGV is assumed to do the material handling with two activities of direct movement and waiting showing by X and Y, respectively. The probability distribution for each activity is exponential as follows:

$$X \approx EXP(\lambda),$$ (1)

and the corresponding probability density function is,

$$f(X) = \lambda e^{-\lambda x},\tag{2}$$

Also,

$$Y \approx EXP(\lambda),$$ (3)

$$f(Y) = \lambda e^{-\lambda y} \,. \tag{4}$$

To obtain the total handling time, we need to investigate the joint distribution function of X and Y. Since they are times, then they are independent. Thus, the joint density function of independent variables is equal to the products of their density functions, as follows:

$$f(X,Y) = f(X) \times f(Y), \tag{5}$$

and thus we want to obtain,

$$f(X < z, Y < w) = ? \tag{6}$$

This problem is easily solved using the distribution function integral of each variable due to independent variables. But we are looking for a density function to illustrate the total handling time of the AGV. Therefore, we apply cumulative distribution function technique.

Fig. 2 Flowchart of the proposed methodology

Here, we want to know what the distribution function of W = X +Y is, i.e., what the cumulative distribution function of X and Y is. The space of this cumulative function is shown in Figure 3.

Fig. 3 The space representation of the cumulative function

We have,

$$f(X,Y) = f(X) \times f(Y) = \lambda e^{-\lambda x} \times \lambda e^{-\lambda y}, \tag{7}$$

$$F_W(a) = P(W < a) = P(X + Y < a),$$ (8)

$$\int{0}^{a} \int{0}^{-\lambda x} \lambda e^{-\lambda x} \times \lambda e^{-\lambda y} dy dx = \int{0}^{a} \lambda e^{-\lambda x} \left(1 - e^{-\lambda(a - x)}\right) dx = \int{0}^{a} \lambda e^{-\lambda x} - \int_{0}^{a} \lambda e^{-\lambda a} dx = 1 - e^{-\lambda a} - \lambda a e^{-\lambda a}$$ (9)

Now, we set partial derivative to obtain the summation density function,

$$f{W}(a) = \frac{dF{W}(a)}{da} = \lambda e^{-\lambda a} - \left(\lambda e^{-\lambda a} - \lambda^{2} a e^{-\lambda a}\right) = \lambda^{2} a e^{-\lambda a} = \frac{\lambda e^{-\lambda a} \left(\lambda a\right)^{1}}{1!}, \quad a > 0. \quad (10)$$

Therefore, the obtained density function for W follows Erlang probability distribution with λ and ^α = 2 .

Note that the Erlang probability density function is,

$$f(x;\alpha,\lambda) = \frac{\lambda^{\alpha} x^{\alpha-1} e^{-\lambda x}}{(\alpha-1)}, \quad x,\lambda \ge 0.$$ (11)

As a result, if X1, X2,…, Xn are independent exponential random variables with λ , then ∑= = i Y Xi 1 follows Erlang distribution with λ and ^α = n.

To minimize the total AGV handling time, we have to investigate the minimum cumulative distribution function. Therefore, in an easy case, we consider, ( ) X ≈ EXP ^λ1 , ( ) Y ≈ EXP ^λ2 and we are looking for W = min( ) X,Y . We apply the cumulative distribution function, as follows:

$$F{W}(a) = 1 - P(W > a) = 1 - P(\min(X, Y) > a) = 1 - P(X > a, Y > a) = 1 - (P(X > a) \times P(Y > a)) = 1 - (e^{-a\lambda{i}} \times e^{-a\lambda{i}}) = 1 - (e^{-a(\lambda{i} + \lambda_{i})a}).$$ (12)

Here, we set the derivative of the cumulative distribution function to obtain the density function, as below:

$$f_W(a) = \frac{dF_W(a)}{da} = (\lambda_1 + \lambda_2)e^{-(\lambda_1 + \lambda_2)a}, \quad a \ge 0.$$ (13)

As a result, if X1, X2,…, Xn are independent exponential random variables with ^{λ λ λn} , ,..., ^{1 2} , then ( ) W X X Xn min , ,..., = ^{1 2} follows exponential distribution with ^λ = ^λ + ^λ + + ^λn ... ^{1 2} .

This result is reliable when all the activities' probability distributions are exponential, but if the distributions are different, then we have to explore another approach.

To break through the model when the pdfs are different or even when the pdfs are not known, a cross entropy algorithm is proposed and developed in next section.

6.3.1 Cross Entropy (CE)

Due to the drawback of our heuristic proposed approach to aggregate stochastic AGV activity times when the probability distribution functions are different, we employ cross entropy to break through the problem.

Let X { } X Xn ,..., = ¹ be a random vector taking values in some space ^χ . Let ( ) f (.;v) be a family of probability density functions (pdfs) on χ , with respect to some base measure μ , where v is a real-valued parameter (vector). Thus,

$$EH(X) = \int_{\mathcal{X}} H(X) f(x; v) \mu(dx), \qquad (14)$$

for any function H. Note that E stands for expected value. For simplicityμ(dx)is considered to be dx. Let S be some real function on ^χ .

Suppose we are interested in the probability that S(X) is greater than or equal to some real number ξ - we consider it as AGV activity time- under f(.;u). This probability can be expressed as

$$l = P_u(S(X) \ge \xi) = Eu I{{S(X) \ge \xi}}, \tag{15}$$

Where I is an indicator random variable. A straightforward way to estimate l is to use Monte Carlo simulation. Draw a random sample X Xn ,..., ¹ from f(.;u); then

$$\frac{1}{N} \sum{i=1}^{N} I{\left{S(X_i \ge \xi)\right}},\tag{16}$$

is an unbiased estimator of l. This way is a time consuming simulation effort. An alternative way is based on importance sampling (Smith et al., 1997; Srinivasan, 2002). Take a random sample X Xn ,..., ¹ from an importance sampling density ^g on χ , and estimate l using the likelihood ratio estimator.

$$\hat{l} = \frac{1}{N} \sum{i=1}^{N} I{{S(X_i) \ge \xi}} \frac{f(X_i; u)}{g(X_i)}.$$ (17)

The best way to estimate l is to use change of measure with density

$$g^{*}(x) = \frac{I_{{S(x) \ge \xi}} f(x; u)}{l}.$$ (18)

Therefore, we obtain from (17)

$$I_{{S(X_i) \ge \xi}} \frac{f(X_i; u)}{g^*(X_i)} = l, \qquad \text{for all } i.$$ (19)

l is a constant value with zero variance. Here, the problem is that g* depends on the unknown parameter l. it is convenient to choose a g in the family of densities { } f (.;v) . Now, we need to choose the reference parameter v such that the distance between the density g* in (18) and f(.;v) is minimal. A usual measure of distance between two densities, say g and h, is the Kullback-Leibler distance (Kullback and Leibler, 1951; Kullback, 1959) or the cross entropy between g and h. The Kullback-Leibler distance is defined as,

$$D = (g, h) = E_g \ln \frac{g(X)}{h(X)} = \int g(x) \ln g(x) dx - \int g(x) \ln h(x) dx.$$ (20)

Minimizing the Kullback-Leibler distance between g* in (18) and f(.;v) is equivalent to choosing v such that ∫ ^{∗ − g} (x)ln ^f (x, ^v)dx is minimized, which is solving the following maximization problem,

$$\max_{v} \int g^{*}(x) \ln f(x, v) dx, \qquad (21)$$

Substituting g* from (18) into (21) we obtain the maximization program

$$\max{v} \int \frac{I{{S(x) \ge \xi}} f(x; u)}{l} \ln f(x, v) dx, \qquad (22)$$

which is,

$$\max{v} D(x) = \max{v} E{u} I{{S(x) \ge \xi}} \ln f(x, v).$$ (23)

Again, using importance sampling with a change of measuref(.;w) we can rewrite (23) as

$$\max{v} D(v) = \max{v} E{w} I{{S(X) \ge \xi}} W(X; u, w) \ln f(x, v),$$ (24)

for any reference parameter w, where,

$$W(x;u,w) = \frac{f(x,u)}{f(x,w)},\tag{25}$$

Is the likelihood ratio at x, between f(.;u) and f(.;w). The optimal solution of (24) can be written as:

$$v^* = \arg\max_{v} Ew I{{S(X) \ge \xi}} W(X; u, w) \ln f(X; v),$$ (26)

We can solve v* by solving the following stochastic program

$$\max{v} \hat{\mathbf{D}}(v) = \max{v} \frac{1}{N} \sum{i=1}^{N} I{{S(X_i) \ge \xi}} W(X_i; u, w) \ln f(X_i; v).$$ (27)

where X Xn ,..., ¹ is a random sample from f(.;w).

6.3 Mathematical Model 89

this way we obtain an estimated probability for stochastic AGV activity time. While these probabilities are real values, we can obtain the aggregated value by simple adding all activities' times. Here, all stochastic times from various probability distribution functions can be aggregated by CE algorithm to compute the total AGV material handling time. A substantial advantage of the proposed CE algorithm is to be capable for data collections without any known probability distribution function. The following algorithm is composed to facilitate the CE computations:

Algorithm 1. Cross entropy method for aggregating various probability distribution functions

Step 1. Define v = u 0ˆ . Set t=1 (iteration)

Step 2. Generate a sample X Xn ,..., ¹ from density f(.;vt-1) and compute the sample (1-q)-quantileξ ^t ˆ of the performances according to ^t ( ) ⎡ ⎤ ^{q N S} (1 ) ^{ˆ = − ξ} , provided ξ ^t ˆ is less than ξ . Otherwise set ξ^t = ξ ˆ .

Note that,

$$P{\nu{t-1}}(S(X) \ge \xi_t) \ge q,$$

$$P{\nu{t-1}}(S(X) \le \xi_t) \ge 1 - q.$$

Step 3. Use the same sample X Xn ,..., ¹ to solve the stochastic program

$$\max{v} \hat{D}(v) = \max{v} \frac{1}{N} \sum{i=1}^{N} I{{S(X_i) \geq \frac{\hat{e}}{\xi_t}}} W(Xi; u, \hat{v}{t-1}) \ln f(X_i; v).$$

Denote the solution by *^t v*ˆ .

Step 4.If ξ^t < ξ ˆ , set t=t+1 and reiterate from Step 2. Else proceed with Step 5.

Step 5. Estimate probability l using likelihood ratio estimate

$$\hat{l} = \frac{1}{N} \sum{i=1}^{N} I{{S(X_i) \ge \xi}} W(X_i; u, \hat{v}_T).$$

where T denotes the final number of iterations.

Next section presents verification and validation of both HSM and CE approach for an AGV-based jobshop manufacturing system. The most significant advantage of our proposed heuristic method and CE is to be helpful when the data are few.

6.4 Conclusions

Here, a heuristic statistical technique to compute total stochastic material handling time in an automated guided vehicle (AGV) equipped jobshop manufacturing system was proposed. With respect to stochastic times of AGVs material handling process, each activity was associated with a probability distribution. Using the probability distributions, we modeled the AGV material handling problem using a heuristic statistical method (HSM) when the activities' probability distribution functions are same. To model the problem when the activities' probability distribution functions are different, a cross entropy algorithm is developed. The problem was investigated and implemented in two cases with same probability distributions and different ones. The effectiveness of the proposed model was validated in an example. An experimental test problem was worked out for each of the cases. For the second case, to verify the obtained probability distribution of the total AGV handling time, we applied chi-square and Kolmogorov-Smirnov goodness-of-fit tests. The simulation was also implemented by several softwares to analyze their behaviors for the proposed problem ensuing to better performance of ARENA. The computations validated by simulation illustrated the efficiency of the HSM approach in the case while all distributions were same and the CE method for the case having different probability distributions of activities.

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Chapter 7

Neuro-Fuzzy-Regression Expert System for AGV Optimal Path

7.1 Summary

According to Turban and Aronson (1998), a decision support system (DSS) is a computer-based information system combining models and data in an attempt to solve non-structured problems with extensive user involvement. They advocated that an expert system (ES) was a computer system that applied reasoning methodologies on knowledge to render advice or recommendations much like a human expert. When expert system technology was first applied to decisionmaking problems, it fell short in several respects. Early expert systems were rulebased and thus were not capable of handling the classical DSS functions being more computational than logical. Recently, artificial intelligence researchers have noted the necessity for using statistical techniques to build intelligent decision support systems (Nolan, 1998; Weiss and Kulikowski, 1991). Examples of such statistical techniques include fuzzy logic, neural networks, rule induction and various Bayesian techniques. Turban and Aronson (1998) believe that although uncertainty is widespread in the real world, but practical treatment in artificial intelligence is very limited.

Cost estimation plays a very important role in design and production stages as well as a fairly important role in company business decision-making. Effective cost estimation can contribute to an optimal design leading to customers' satisfaction due to cost reduction, improved quality and timely delivery (Wu and Zhou, 2005).

Manufacturing costs in products are very crucial in decision making and strategic planning. With respect to cost estimation, research and development departments were mainly concerned with the estimate of the total cost for the finished product (Vis, 2006). Moreover, engineers often applied rules of thumb as the cost estimation benchmarks, making the results controversial in terms of accuracy (Khayat et al., 2006). Although calculation by cost model has the advantage of timeliness, only representative values exclusive of indirect tasks cost and raw materials cost are calculated resulting in inadequate estimation accuracy. Niazi et al. (2006) pointed out that back propagation network (BPN) could be

applied for training to deduce unprecedented problems by accumulated knowledge and information. Specially, it can find out solutions in uncertain circumstances and has satisfactory results in dealing with nonlinear problems. Therefore, BPN is one of the most popular neural network models being applied and it properly fits the nature of product cost estimation. McKim (1993) proposed a discussion on applying BPN in cost estimation projects. Finally, by integrating a user interface, the demand of a fast response cost estimation model was met at the initial product development stage.

Cavalieria et al. (2004) on the spare parts cost estimation in auto industry pointed out that, at the final stage of the product life cycle, most product costs continuously rise because they are determined at the product concept design stage. Yang and Lin (1997), Rehman and Guenov (1998) and Gayretli and Abdalla (1999) also pointed out time and again that the product design initial stage determined 70–80% of the total product cost.

Cost affects sales price, sale volume and profit most directly. In addition, Vrbsky (1997) pointed out that the research and development design cost often accounted for almost 50% of the product development cost and was an important factor determining whether the product total cost would be reasonable or not. For the same reason, accurately estimated design cost cannot only help reduce the capital-losing risks in the product research and development process but also can shorten the product development time period. Such methods can really reduce the expense for product development (Ghasemzadeh et al., 2009).

The automotive industry operates in a very competitive market which requires controlling product costs, improving the product quality and shortening the development lead time. Cost estimation requires quality data and information. Roy et al. (2011) presented the various data and information requirements for detailed cost estimating in automotive industry. The research project has identified the common cost estimation process model within the identified industry sector. The study identified the types of data and information requirements for cost estimating. It has also constructed the relevant data infrastructure as the basis for a Web Portal, which was the physical presentation of the information infrastructure. Multiple sources of data collection techniques were employed to identify the types of data needed for detailed cost estimates within a manufacturing company.

Quintana and Ciurana (2011) proposed a machine tool selection problem consisting of selecting the most suitable machine to satisfy manufacturing company requirements. The main goal of the work was to develop a cost estimation support tool for vertical high speed machining centers based on final part and productivity requirements of the company linked with machine tool characteristics available in the catalogues in order to apply the cost model and to calculate machine tool cost estimations. The cost model presented was based on multiple regression analyses and provided reasonably accurate market cost predictions. Applying the proposed cost model will help the user to determine the approximate market cost of the machine and can be especially interesting for decision makers in the preliminary stages of a selection process because it avoids long and costly studies.

7.1 Summary 95

Flexible material handling systems (MHS) have been widely used to enhance productivity involved with product proliferation, and thus far, only fixed-track MHSs such as Eton systems in the apparel industry are commonly used. Dai and Lee (2012) explored the economic feasibility of a flexible MHS using free-ranging automated guided vehicles (AGV) with a local positioning system (LPS) for the apparel industry. A component-based and modified activity-based costing methodology was proposed to estimate the additional cost of adopting flexible MHSs, and then the internal rate of return (IIR) and payback periods were applied to evaluate the project economic performance.

Accurate cost estimation plays a significant role in industrial product development and production. Deng and Yeh (2011) applied least squares support vector machines (LS-SVM) method solving the problem of estimating the manufacturing cost for airframe structural projects. They evaluated the estimation performance using back-propagation neural networks and statistical regression analysis.

Product cost estimation varies widely ranging from standard spare parts manufacturing cost estimation to the cost analysis of the optimized technology and marketing fees of highly customized assembled products, with appropriate product estimation models available from the product concept design stage to the product design cycle's final stages. Zhang et al. (1996) categorized cost estimation techniques into traditional detailed breakdown, simplified-breakdown, grouptechnology-based, regression-based and activity-based cost approaches. Ben-Arieh and Qian (2003) divided cost estimation models into intuitive, analogical, parametric and analytical approaches. Shehab and Abdalla (2001) proposed intuitive, parametric, variant-based and generative cost estimating approaches. Cavalieria et al. (2004) provided three cost analyses of analogy-based, parametric and engineering approaches. Niazi et al. (2006), on the basis of the integrated cost estimation approaches, categorized the qualitative and quantitative cost estimation techniques along with key advantages and limitations of each cost estimation technique.

Here, we employ soft computing techniques for configuring an expert system in the presence of uncertainty. The purpose of our study is to design a cost estimation model for an AGV based automated manufacturing system. We choose three main factors accompanied by their sub-factors and then propose a neural network based approach to analyze the effects of the sub-factors on the main factors. Due to uncertainty of cost elements in manufacturing environment, we apply fuzzy logic to determine an appropriate range for each of the linguistic variables. Then, using the linguistic variables, we extract the possible fuzzy rules and using a multiple linear regression analysis we investigate the significant rules and eliminate inappropriate ones. A sensitivity analysis is considered along with the regression method. We then use a dynamic programming approach to find an optimal path for the proposed manufacturing system.

7.2 Statement of the Problem

Consider a jobshop layout which applies an AGV for material handling. The AGV carries raw material, semi-produced and finished products in batch sizes. Due to mounting demands, advancing technology, and rising production capacity, the need for increasingly more shops is mounting over time. The new shops are expected to have more advanced machines. Therefore, more than one shop with the same duty are evolved. The difference among shops having the same duty shows up in the shop's specificities that affect the production cost. As a result, the system would consider a flexible jobshop model where multi shops of the same duty exist and each operation can be processed on any type of machine in any shop. The sequences of jobs are specified and the jobs are assumed to be independent.

The structure of such a problem would configure a network. In this network, the nodes are the shops and the arcs are the flow paths of the AGV to each shop. Shops in each stage are of the same type but have different specifications such as different machine types and equipments, varied operator proficiencies, different rates of defect, etc.. Each flow path for the AGV is associated with a cost parameter related to each shop. The aim is to find a path for the AGV minimizing the cost objective. In each shop, different machines and operators are working. Due to unpredictable events during working times a cost may incur. This cost is inferred from an expert system via fuzzy logic "IF …. Then……." rules. Cost parameters of each shop are considered to be 3 main factors: (1) Equipment sensitivity, (2) operator proficiency, and (3) product specifications, each being specified by one of the three levels of low, moderate, and high. A configuration for the proposed problem is presented in Figure 1.

Fig. 1 A configuration of the proposed problem.

As stated before, cost is inferred from an expert system. A flowchart for the proposed expert system is presented in Figure 2. In our expert system, a back propagation neural network is considered to estimate the cost factors (outputs) using their corresponding sub-factors (inputs). The cost factors and their related sub-factors are shown in Figure 3. Using the existing data, we train the network and then by the resulting pattern we can apply the model to obtain the output with respect to the proposed sub-factors. The numerical results would show a lower bound and an upper bound for each cost factor. We utilize these bounds as our cost factors ranges for the fuzzy rule base. Considering the uncertainty in the cost factors due to dynamic changes in neural network inputs, we specify the cost factors in three levels of low, moderate, and high. The problem is to specify the range of these levels, i.e., the numerical range of low, moderate or high for each of the cost factors. This process is repeated for each AGV movement between any two shops for determining the arcs cost values.

Fig. 2 A proposed expert system.

Fig. 3 The cost factors and their related sub-factors.

7.3 Mathematical Model

We apply an inductive reasoning technique to obtain the appropriate range for the levels. This method is based on an ideal scheme describing the input and output relationships for a well-established data base. This method is called entropy. A key goal of entropy minimization analysis is to determine the quantity of information in a given data set. The entropy of a probability distribution is a measure of the uncertainty of the distribution. This information measure estimates the uncertain range of data using a predetermined inappropriate range to start the process. The higher the prior estimate of the probability for an outcome to occur, the lower will be the information gained by observing it to occur. The entropy on a set of possible outcomes of a trial where exactly one outcome is possible is defined by the sum of probabilities. In other words, the entropy is the expected value of information. For a simple one-dimensional (one uncertain variable) case, let us assume that the probability of the i th sample *wi*to be true is ( ) wi p . If we actually observe the sample wi in the future and discover that it is true, then we gain the following information, I(wi):

$$I(w_i) = -k \ln p(w_i), \tag{1}$$

wherek is a normalizing parameter. If we discover that it is false, we still gain this information:

$$I(\overline{w}_i) = -k \ln[1 - p(w_i)]. \tag{2}$$

Thus, the entropy of the inner product of all the samples (N) is:

$$S = -k \sum_{i=1}^{N} [p_i \ln p_i + (1 - p_i) \ln(1 - p_i)],$$ (3)

where*pi*is the probability of the i th sample to be true. Note that S ≥ 0 , because ln x ≤ 0 , for 0 ≤ x ≤ 1 .

The entropy of a rule should be minimized. Minimum entropy (S) is associated with all the pi being as close to ones or zeros as possible, which in turn implies that they have a very high probability of either happening or not happening, respectively. Note in equation (3) that if *pi=*1 then *S=*0.

This result makes sense, since pi is the probability measure of whether a value belongs to a partition or not. The precedence for partitioning the data is as follows.

First, we assume that we are seeking a threshold value for a sample in the range between [*x*1, x] and [x, *x*2]. We denote the first region p and the second region q. by moving an imaginary threshold value x between *x*1 and *x*2, we calculate entropy for each value of x.

An entropy with each value of x in the region *x*1 and *x*2 is:

$$S(x) = p(x)S_p(x) + q(x)S_q(x),$$ (4)

where,

$$S{p}(x) = -[p{1}(x)\ln p{1}(x) + p{2}(x)\ln p_{2}(x)], \tag{5}$$

$$S{q}(x) = -[q{1}(x)\ln q{1}(x) + q{2}(x)\ln q_{2}(x)], \tag{6}$$

with pk(x) and qk(x) as conditional probabilities that the class k sample is in the region [*x*1, *x1+x*] and [*x1+x*, *x*2], respectively, p(x) and q(x) are probabilities that all samples are in the region [*x*1, *x1+x*] and [*x1+x*, *x*2], respectively. Also,

$$p(x) + q(x) = 1. (7)$$

A value of x that gives the minimum entropy is the optimum threshold value. We calculate entropy estimates pk(x), qk(x), p(x), and q(x) as follows:

$$p_k(x) = \frac{n_k(x) + 1}{n(x) + 1},$$ (8)

$$q{k}(x) = \frac{N{k}(x) + 1}{N(x) + 1},$$ (9)

$$p(x) = \frac{n(x)}{n},\tag{10}$$

$$q(x) = 1 - p(x),$$ (11)

where nk(x) is the number of class k samples located in [*x*1, *x1+x*], n(x) is the total number of samples located in [*x*1, *x1+x*], Nk(x) is the number of class k samples located in [*x1+x*, *x*2], N(x) is the total number of samples located in [*x1+x*, *x*2], and n is the total number of samples in [*x*1, *x*2]. While moving x in the region [*x*1, *x*2], we calculate the values of entropy for each position of x.

This procedure finds the region for the levels of the cost factors. After each update in input data the ranges are specified using an entropy technique. After determining the regions of the cost factors, we then compose the fuzzy rules which clarify the total cost for each shop. To find the effective rules, we configure a multiple linear regression model using the previous data and identify the regression coefficients. Since we have three cost factors each of which contains three levels of low, moderate and high, then twenty seven (3*3*3) possible rules exist. Thus, using the hypothesis testing, the effective rules are identified. By these rules we estimate the cost of each shop (arc length in the network). Consequently, a dynamic program is applied to find the optimal path in the proposed jobshop automated manufacturing network.

7.3.1 Artificial Neural Network

Neural networks are being widely used in many fields of study. This could be attributed to the fact that these networks attempt to model the capabilities of human brains. Since the last decade, neural networks have been used as a theoretically sound alternative to traditional statistical models. Although neural networks (NNs) originated in mathematical neurobiology, the rather simplified practical models currently in use have moved steadily towards the field of statistics. A number of researchers have illustrated the connection of neural networks to traditional statistical models. For example, Gallinari et al. (1991) presented analytical results establishing a link between discriminant analysis and 7.3 Mathematical Model 101

multilayer perceptrons (MLP) used for classification problems. Cheng and Titterington (1994) made a detailed analysis and comparison of various neural network models with traditional statistical models. They showed strong associations of the feed-forward neural networks with discriminant analysis and regression models, and unsupervised networks such as self-organizing neural networks with clustering. Neural networks are being used in areas of prediction and classification, areas where regression models and related statistical techniques have traditionally been used. Ripley (1994) discusses the statistical aspects of neural networks and classifies neural networks as one of a class of flexible nonlinear regression models. Sarle (1994) translates neural network terminologies into statistical ones and shows the relationship between neural networks and statistical models such as generalized linear models, projection pursuit and cluster analysis. He explains that neural networks and statistical approaches are not competing methodologies for data analysis and there is a considerable overlap between the two. Warner and Misra (1996) present a comparison between regression analysis and neural network computation in terms of notation and implementation. They also discuss when it would be advantageous to use a neural network model in place of a parametric regression model, as well as some of the difficulties in implementation. Schumacher et al. (1996) and Vach et al. (1996) present a comparison between feed-forward neural networks and the logistic regression. The conceptual similarities and discrepancies between the two methods are also analyzed.

Some of the commonly used traditional statistical techniques applied for prediction and classification are multiple regression, discriminant analysis, and logistic regression. These methods are being widely used in various applications due to their established methodologies. However, neural networks are being used as alternatives to these traditional techniques and gaining much popularity in recent years. This has led to a number of studies comparing the traditional statistical techniques with neural networks in a variety of applications.

Artificial neural networks have been applied successfully to many manufacturing and engineering areas. Zhengrong et al. (1996) used quadratic regression to assess the results of neural network for improving the efficiency of fermentation process development. The results show that different sizes of neural nets within a certain range give an equally good prediction by using the ''stopping training" technique, while quadratic regressions are sensitive to the size of the data sets. Smith and Mason (1997) mentioned that regression and neural network modeling methods have become two competing empirical model-building methods. They compared the predictive capabilities of NNs and regression methods in manufacturing cost estimation problems. This study examines the performance, stability and ease of cost estimation modeling using regression versus neural networks to develop cost estimating relationships (CERs). Results show that neural networks have advantages when dealing with data not adhering to the generally chosen low order polynomial forms, or data for which there is little a priori knowledge of the appropriate CER to select for regression modeling. However, in cases where an appropriate CER can be identified, regression models

have significant advantages in terms of accuracy, variability, model creation and model examination. Both simulated and actual data sets were used for comparison purposes. Researchers have also discussed some important issues other than model accuracy to be considered when using regression versus neural network to estimate cost functions. Finnie et al. (1997) used neural network models to estimate software development efforts. Improving the estimation techniques available to project managers would facilitate more effective control of time and budgets in software development. They compared three estimation techniques using function points as an estimate of system size. The models considered were based on regression analysis, artificial neural networks and case-based reasoning. Although regression models performed poorly on the data set of 299 software projects, both artificial neural networks and case-based reasoning appeared to be useful for software development effort estimation models. Also, Heiat (2002) compared artificial neural network and regression to estimate the amount of effort required for developing an information system, an important project management concern. The results of the study indicated that for one experiment (when only third generation language data set was considered), regression analysis and neural networks performance were nearly identical and for a second experiment (when a combined third generation and fourth generation language data set as used), the neural network produced improved performance over conventional regression analysis in terms of mean absolute percentage error (MAPE). They also stated that autocorrelation could have been a possible reason for the large MAPE values and not its non-normality as in both cases the data sets were not normally distributed. Lee and Um (2000) used multiple regression analysis and artificial neural network methods for predicting the relationship between welding parameters and geometry of the welding area in arc welding. It was found that the error rate predicted by the artificial neural network was smaller than that predicted by the multiple regression analysis. Shuhui et al. (2001) compared regression and neural networks to predict the power produced by wind farms and found that neural networks performed better than regression models. Applicability of NN for cost estimation in building construction was tested by Setyawati et al. (2002). The authors concluded that neural networks outperformed linear regression models given the same training data and the same variables. Krishnaswamy and Krishnan (2002) used regression analysis and neural network in nozzle wear rate prediction and found a statistical significant difference in the prediction of validation data. Feng and Wang (2002) compared nonlinear regression and neural network models in developing empirical models for estimating the digitizing accuracy of a given coordinatemeasuring machine. It was shown that both prediction models appear to provide a satisfactory prediction with the regression model providing a slightly better performance in both model construction and model verification. Feng and Wang (2002) in a different article applied two competing data mining techniques, nonlinear regression analysis and computational neural networks in developing the empirical models for surface roughness prediction. It was demonstrated by hypothesis testing of relative errors and absolute errors for the process of modeling construction, model validation, and combined data modeling that no

7.3 Mathematical Model 103

statistically significant differences were found between the performances of the two methods. Kim et al. (2004) applied three techniques, namely multiple regression analysis, neural network and case based reasoning for estimating construction costs of Korean residential buildings. Authors concluded that although the best NN model performed better than the other models for accuracy of estimation, the training procedure to train the best NN model was slow because of the trial and error process. In addition to the accuracy of estimation results, the time and accuracy tradeoffs and also the clarity of explanation should be considered in a cost estimation model. Yesilnacar and Topal (2005) used a neural network and compared it with logistic regression for landslide susceptibility mapping in a study for Hendek region in Turkey. The landslide susceptibility map produced using the neural network method predicts higher percentages of landslides, specially in high and very high zones than the logistic regression method. Few studies in the project management literature concentrated on the critical success factors that affect project success or failure. A comparison between neural network and linear regression analysis was used for identifying critical managerial factors affecting the success of high-tech defense projects by Dvir et al. (2006). The study showed that neural networks have significantly better explanatory and prediction powers, and they enable the exploration of relationships among the data that are difficult to arrive at by traditional statistical methods. Pendharkar (2006) investigated the factors influencing the objectoriented (OO) component size and source code documentation.

Using empirical data, researcher compared the performance of nonlinear artificial neural network (ANN) forecasting models and linear regression models. Experimental results confirmed the superior performance of ANN over the linear multiple regression model when variable returns to scale economies exist between multiple inputs and multiple outputs.

7.3.2 The Backpropagation Neural Network

The backpropagation algorithm trains a given feed-forward multilayer neural network for a given set of input patterns with known classifications. When each entry of the sample set is presented to the network, the network examines its output response to the sample input pattern. The output response is then compared to the known and desired output and the error value is calculated. Based on the error, the connection weights are adjusted. The backpropagation algorithm is based on Widrow-Hoff delta learning rule in which the weight adjustment is done through mean square error of the output response to the sample input (Abdi et al., 1996).

Algorithm 1. General steps of backpropagation

• 1. Propagate inputs forward in the usual way, i.e., all outputs are computed using sigmoid thresholding of the inner product of the corresponding weight and input vectors. All outputs at stage n are connected to all the inputs at stage n+1
• 2. Propagate the errors backwards by apportioning them to each unit according to the amount of the error the unit is responsible for.

We now discuss how to develop the stochastic backpropagation algorithm for the general case. The derivation is simple, but unfortunately the book-keeping is a little messy. The following notations and definitions are needed:

j x ^G : input vector for unit j (xji = *i*th input to the *j*th unit)

wj ^G : weight vector for unit j (wji = weight on xji)

^{j j j z w x G G =} . : the weighted sum of inputs for unit ^j

oj : output of unit j ( ( ) ^{j j} o = ^σ z )

tj : target for unit j

Downstream(j) : set of units whose immediate inputs include the output of j

Output : Set of output units in the final layer.

Since we update after each training example, we can simplify the notation somewhat by assuming that the training set consists of exactly one example and so the error can simply be denoted by E.

We want to calculate wji E ∂ ∂ corresponding to each input weight wji of each

output unit j. Note first that since zj is a function of wji regardless of where in the network unit j is located,

$$\frac{\partial E}{\partial w_{ji}} = \frac{\partial E}{\partial z_j} \cdot \frac{\partial zj}{\partial w{ji}} = \frac{\partial E}{\partial zj} . x{ji}, \tag{12}$$

Furthermore, j z E ∂ ^∂ is the same regardless of which input weight of unit j we are

trying to update. So, we denote this quantity by ^{δ j} .

Consider the case when j is an output unit. We know that

$$E = \frac{1}{2} \sum_{k \in \text{Outputs}} (t_k - \sigma(z_k))^2$$ (13)

Since the outputs of all units k ≠ j are independent of wji, we can then drop the summation and consider just the contribution to E by j and we call it ^{δ j} :

$$\delta{j} = \frac{\partial E}{\partial z{j}} = \frac{\partial}{\partial z{j}} \frac{1}{2} (t{j} - o{j})^{2} = -(t{j} - o{j}) \frac{\partial o{j}}{\partial z{j}} = -(t{j} - o{j}) \frac{\partial}{\partial z{j}} \sigma(z_{j})$$

$$= -(t{j} - o{j})(1 - \sigma(z{j}))\sigma(z{j}) = -(t{j} - o{j})(1 - o{j})o{j}.$$ (14)

Thus,

$$\Delta w{ji} = -\eta \frac{\partial E}{\partial w{ji}} = \eta \deltaj x{ji}. \tag{15}$$

Now, consider the case when j is a hidden unit. Like before, we make the following two important observations:

• 1. For each unit k downstream from j, zk is a function of zj.
• 2. The contribution to error by all unitsl ≠ j , in the same layer as j, is independent of wji.

We want to calculate wji E ∂ ∂ for each input weight wji for each hidden unit j.

Note that wji influences just zj which influences oj which influences zk, ∀k ∈Downstream( j), each of which influences E. So, we can write,

$$\frac{\partial E}{\partial w{ji}} = \sum{k \in Downstream(j)} \frac{\partial E}{\partial z_k} \cdot \frac{\partial z_k}{\partial o_j} \cdot \frac{\partial o_j}{\partial z_j} \cdot \frac{\partial zj}{\partial w{ji}} = \sum_{k \in Downstream(j)} \frac{\partial E}{\partial z_k} \cdot \frac{\partial z_k}{\partial o_j} \cdot \frac{\partial o_j}{\partial zj} \cdot x{ji}. \quad (16)$$

Again, note that all the terms except xji in (16) are the same regardless of which input weight of unit j we are trying to update. Like before, we denote this common

quantity by $$\delta_j$$ . Also, note that $\frac{\partial E}{\partial z_k} \delta_k$ , $\frac{\partial z_k}{\partial oj} w{kj}$ and $\frac{\partial o_j}{\partial z_j} = o_j (1 - o_j)$ .

Substituting them in (14),

$$\delta{j} = \sum{k \in Downstream(j)} \frac{\partial E}{\partial z{k}} \cdot \frac{\partial z{k}}{\partial o{j}} \cdot \frac{\partial o{j}}{\partial z{j}} = \sum{k \in Downstream(j)} \delta{k} \cdot w{kj} \cdot o{j} (1 - o{j}), \tag{17}$$

we obtain:

$$\delta{k} = o{j} (1 - o{j}) \sum{k \in Downstream(j)} \delta{k} . w{kj}.$$

$$\tag{18}$$

We are now in a position to state the backpropagation algorithm formally.

Algorithm 2. Formal statement of stochastic backpropagation

(Training examples, η, ni, nh, no)

Each training example is of the form ^{x t G G} , , where x G is the input vector and t G is the target vector, ηis the learning rate (e.g., 0.05), ni, nh and no are the number of input, hidden and output nodes, respectively. Input from unit i to unit j is denoted by xji and its weight is denoted by wji. Create a feed-forward network with ni inputs, nh hidden units, and no output units.

Initialize all the weights to small random values (e.g., between -0.05 and 0.05).

While termination condition is not met Do

For each training example ^{x t G G} , ,

• 1. Input the instance x G and compute the output ou of every unit.
• 2. For each output unit k, calculate

$$\delta_k = o_k (1 - o_k)(t_k - o_k). \tag{19}$$

1. For each hidden unit h, calculate

$$\delta_h = o_h (1 - oh) \sum{k \in Downstream(h)} \deltak . w{kh} . \tag{20}$$

1. Update each network weight wji as follows:

$$w{ji} \leftarrow w{ji} + \Delta w_{ji} \,. \tag{21}$$

where,

$$\Delta w_{ji} = \eta \deltaj x{ji} \,. \tag{22}$$

7.3.3 Fuzzy Logic

Fuzzy Logic (FL) is a problem-solving control system methodology that lends itself to implementation of systems ranging from simple, small, embedded microcontrollers to large, networked, multi-channel PC or workstation-based data acquisition and control. It can be implemented in hardware, software, or a combination of both. FL provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. The FL approach to control problems mimics how a person would make decisions, only much faster.

FL incorporates a simple, rule-based IF X AND Y THEN Z approach to solving a control problem rather than attempting to model a system mathematically. The FL model is empirically-based, relying on the operators' experience rather than their technical understanding of the system.

FL requires some numerical parameters in order to operate such as what is considered to be significant error and significant rate-of-change-of-error, but exact values of these numbers are usually not critical unless very responsive performance is required in which case empirical tuning would determine them.

7.3.4 Mamdani Fuzzy System

Mamdani fuzzy system was proposed as a first attempt to control a steam engine and boiler combination by a set of linguistic control rules obtained from experienced human operators. Rules in Mamdani fuzzy systems are like these (Klir and Yuan, 1995; Zimmermann, 1996):

If $$x_1$$ is $A_1$ AND/OR $x_2$ is $A_2$ Then $y$ is $B_1$ ,

where*A*1, *A*2 and *B*1 are fuzzy sets. The fuzzy set acquired from aggregation of rules' results are defuzzified using defuzzification methods like centroid (center of gravity), max membership, mean-max, and weighted average. The centroid method is very popular, in which the ''center of mass'' of the result provides the crisp value. In this method, the defuzzified value of the fuzzy set A, d(A), is calculated by:

$$d(A) = \frac{\int_{X} x.\muA(x)/x}{\int{X} \mu_A(x)/x},$$ (23)

where (.) ^{μ A} is the membership function of the fuzzy set A (Klir and Yuan, 1995; Zimmermann, 1996).

For our problem in which various possible conditions of parameters are stated in forms of fuzzy sets, we utilize the Mamdani fuzzy system, because in this system the fuzzy rules representing the expert knowledge would consider fuzzy sets as their consequences. In general, designing a fuzzy system is composed of the following major steps (Klir and Yuan, 1995).

Algorithm 3: General steps for the design of a fuzzy logic system

• Step 1. Identify pertaining input and output variables. Besides, select the meaningful linguistic states along with appropriate fuzzy sets for the variable.
• Step 2. Employ a fuzzification method for input variables that expresses the associated measured uncertainty. The purpose of the fuzzification method is to interpret measurements of input variables which are expressed by real numbers.
• Step 3. Formulate pertaining knowledge in terms of fuzzy inference rules. There are two principal ways in which relevant inference rules can be determined. One is to elicit them from experienced humans and the other is to obtain them from empirical data by suitable learning methods, usually with the help of neural networks.
• Step 4. Combine measurements of input variables with relevant fuzzy rules to inference, considering the output variables in a way to admit the purpose of the inference engine.
• Step 5. Ascertain a suitable defuzzification method to convert the aggregated fuzzy set of implications into a real number.

7.3.5 Multiple Linear Regression

As we stated before, some rules affect the cost of the system. We should investigate whether a rule is significant on cost or not. One way to survey the effect of independent variables on dependent variable is to use a multiple linear regression model. Therefore, we consider the following equation,

$$u_i = \beta_0 + \beta_1 r_1 + \dots + \beta_j r_j + \varepsilon_j,$$ $i = 1, \dots, n, j = 1, \dots, m,$ (24)

where β0 is the intercept, the β ^j are the coefficients of the *rj*parameters and the ^{j ε} are the error terms. The aim is to identify the β ^j not being important on total cost of the proposed system. Here, we apply two-sided hypothesis testing as follows:

$$H_0: \beta_j = 0,$$

$$H_1: \beta_j \neq 0.$$ (25)

The null hypothesis indicates whether a rule coefficient is zero or not. If the hypothesis is accepted (a certain rule coefficient is zero) then we can omit the corresponding rule from cost estimation process. The test of the null hypothesis H0 against the two sided alternative proceeds in three steps. The first is to compute the standard error of β ^j , ( )^j SE β . The standard error of β ^j is an estimator of β j ^σ , the standard deviation of the sampling distribution of β ^j :

$$\sigma{\beta{j}}^{2} = \frac{1}{m} \times \frac{\frac{1}{m-2} \times \sum{j=1}^{m} (r{j} - \bar{r})^{2} \varepsilon{j}^{2}}{\left[\frac{1}{m} \sum{j=1}^{m} (r_{j} - \bar{r})^{2}\right]^{2}},$$ (26)

$$SE(\betaj) = \sqrt{\sigma{\beta_j}^2} \ . \tag{27}$$

Although the formula for ^{2 β j σ} is complicated, but in practice the standard error is computed by a regression software. The second step is to compute the t-statistic:

$$t = \frac{\beta_j - 0}{SE(\beta_j)}. (28)$$

The third step is to compute the *p-*value, the probability of observing a value of β ^j at least as different from zero as the estimate actually computed ( act β ^j ), assuming that the null hypothesis is correct. Stated mathematically,

$$p - \text{value} = \Pr_{H_o} \left[ \left| \beta_j - 0 \right| > \left| \betaj^{\text{act}} - 0 \right| \right] \Rightarrow \Pr{H_o} \left[ \left| \frac{\beta_j - 0}{SE(\beta_j)} \right| > \left| \frac{\beta_j^{\text{act}} - 0}{SE(\betaj)} \right| \right] \Rightarrow \Pr{H_o} \left( |t| > |t|^{\text{act}} \right), \quad (29)$$

where ⁰ Pr^H denotes the probability computed under the null hypothesis, the second equality follows by dividing into ( )^j SE β , and act t is the value of the t-statistic actually computed. Because β ^j is approximately normally distributed in large samples, under the null hypothesis the t-statistic is approximately distributed as a standard normal random variable, and so for large samples,

$$p - \text{value} = \Pr(\left|Z\right| > \left|t^{act}\right|) = 2\Phi(-\left|t^{act}\right|). \tag{30}$$

A small value of the *p-*value, say less than 5%, provides evidence against the null hypothesis in the sense that the chance of obtaining a value of β ^j by pure random variation from one sample to the next is less than 5% if, in fact, the null hypothesis is correct. If so, the null hypothesis is rejected at the 5% significance level. Simply, we can reject the null hypothesis at the 5% significance level if > 1.96 act t . Therefore, if the hypothesis is accepted, then we ensue that the corresponding rule is not important and is not influential on the cost. The estimation of the coefficients is performed using a regression software (EViews 3.0).

7.3.6 Shortest Path in a Network

Let G = (V,A) be a graph, where V = { } 1,...,N is the set of nodes, and A ⊆V ×V is the set of arcs. We write (i, j) ∈ A , if there exists an arc from node *i*∈V to node j ∈V . Furthermore, let ≥ 0 ij t denote the distance (or travel time, or any other measure of cost) from i to j. If (i, j) ∉ A ,then set tij = +∞ . Note that the travel time from node i to node j is assumed to be stationary; i.e., independent of the actual arrival time at node i. Let *fij*denote the length of the shortest-path from i to j in the graph.

It is a well-known principle that every additive deterministic dynamic programming formulation can be equivalently viewed as the problem of finding the shortest path in a directed network, where the states, decisions, and decision costs of the former correspond to the nodes, arcs, and arc lengths of the latter (Dreyfus and Law, 1977). It is perhaps for this reason that the task of efficiently computing shortest paths is found prominence in the mathematical programming literature. Next, we describe a dynamic programming approach for computing the optimal path.

7.3.7 Dynamic Programming

Dynamic programming (DP) was introduced by Bellman (1957). Toth (1980) presented the early DP-based approaches and reported numerical experiments with a limited success. Hybrid methods, combining dynamic programming and implicit enumeration, were developed later. The first approach was developed by Plateau and Elkihel (1985). A recent approach, the so-called combo algorithm, is able to solve very large instances of up to 10000 variables within less than one second, with basically no difference in the required solution times for ''easy'' and ''hard'' instances (Martello et al., 1999). Marsten and Morin (1978) proposed the first hybrid method, which combined heuristic algorithms, dynamic programming and branch-and-bound approaches. More sophisticated methods can be found in Ibaraki (1987).

Dynamic programming is a technique to tackle multistage decision processes. A given problem is subdivided into smaller subproblems, which are sequentially solved until the initial problem is solved by the aggregation of the subproblem solutions. In each stage, a set of states is defined. The states would describe all possible conditions of the process in the current decision stage, which corresponds to every feasible partial solution. The set of all possible states is known as the state space. The states of a stage u can be transformed to states of a stage u+1, using a transition. A transition indicates the decisions adopted in a stage, and a sequence of transitions taken to reach a state starting from another state is known as a policy. Dynamic programming approaches can be seen as transformations of the original problem to one associated with the exploration of a multistage graph G(S,T), where the vertices in S correspond to the state space and the arcs in T correspond to the set of transitions, leading to an optimal policy.

The basis of dynamic programming can be traced to the optimality principle of Bellman (2003). The optimality principle states that an optimal policy should be constituted by optimal policies from every state of the decision chain to the final state.

Here, we make use of a dynamic programming approach for our proposed network to identify the optimal path in the manufacturing system. The advantages of such a model are simplicity, the ability to determine the exact optimal value, and implementability on sophisticated networks. The backward dynamic model would be defined as:

Indices:

s Number of stages; s= 0,1,2,…, n. i' Start node number; i'=1,2, …, I; i'=0 (for the start node). i End node number; i=1,2, …, I.

Notations:

(i') ^ϕs The minimum value of moving from node i' in stage s to an end node i in stage *s-*1.

Pi'i Numerical value of an arc from node i' to node i.

Optimal policy:

$$\varphi{s}(i') = \underset{i \text{ in layer } s+1}{Min} { \varphi{s+1}(i) + P_{i'i} }, \qquad \forall i' \text{ in stage } s (s = 0,1,2,...,n).$$

$$\varphi_{s}(i) = (0),$$

$$\varphi^{*} = \varphi_{0}(0).$$ (31)

Note that ϕ * identifies the optimal path.

7.4 Conclusions

We proposed an approach for finding an optimal path in a flexible jobshop manufacturing system. The proposed flexible jobshop system has more than one shop with the same duty. The difference among shops with the same duty is in their machines with various specifications. The shops configure a network in which they are considered as nodes and the paths among them are considered as network arcs. An Automated Guided Vehicle (AGV) functions as a material handling device through the manufacturing network. Cost of traveling from one node to another is uncertain and needs to be approximated. Cost is inferenced from an expert system considering three parameters of equipment sensitivity, operator proficiency, and product specification via linguistic variables. This cost is associated with some main and sub factors. To approximate the cost, a back propagation neural network was applied to estimate the sub-factors with the corresponding main factors and gain a bound for the main factors. These bounds were processed by the entropy technique to obtain regions for the linguistic variables. After determining the regions for the linguistic variables, we composed the rules and verified their significance using regression analysis. The purified rules were used to infer the cost of the system. The objective was to find a path minimizing the cost in the manufacturing network. A dynamic programming approach was used to compute an optimal path in the proposed network. Finally, the proposed approach was illustrated by a numerical example.

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Chapter 8

Optimal Path for AGV System with Intelligent Agents

8.1 Summary

An AGV is a material handling equipment that travels on a network of guide paths. The guide path is composed of aisle segments on which the vehicles are assumed to travel at a constant speed. The vehicles can travel forward or backward. As many vehicles travel on the guide path simultaneously, collisions must be avoided. AGV systems are implemented in various industrial contexts: container terminals, part transportation in heavy industry, flexible manufacturing systems.

The most elementary version of the vehicle routing problem is the capacitated vehicle routing problem (CVRP). The CVRP is described as follows: n customers must be served from a unique depot. Each customer asks for a quantity qi of goods (i = 1,..., n) and a vehicle of capacity Q is available to deliver goods. Since the vehicle capacity is limited, the vehicle has to periodically return to the depot for reloading. In the CVRP, it is not possible to split customer delivery. Therefore, a CVRP solution is a collection of tours where each customer is visited only once and the total tour demand is at most Q.

From a graph theoretical point of view the CVRP may be stated as follows: Let G = (C,L) be a complete graph with node set C = (co, c1, c2,..., cn) and arc set L = (ci, cj): *ci , cj*∈C, i < j. In this graph model, co is the depot and the other nodes are the customers to be served. Each node is associated with a fixed quantity qi of goods to be delivered (a quantity qo = 0 is associated to the depot co). To each arc (ci, cj) is associated a value tij representing the travel time between *ci*and cj. The goal is to find a set of tours of minimum total travel time. Each tour starts from and terminates at the depot co, each node ci(i = 1,..., n) must be visited exactly once, and the quantity of goods to be delivered on a route should never exceed the vehicle capacity Q.

An important extension of the CVRP is the vehicle routing problem with time windows (VRPTW). In addition to the mentioned CVRP features, this problem includes, for the depot and for each customer ci (i = 0,..., n) a time window [bi, ei] during which the customer has to be served (with b0 the earliest start time and e0 the latest return time of each vehicle to the depot). The tours are performed by a fleet of l identical vehicles. The additional constraints are that the service beginning time at each node ci (i = 1,..., n) must be greater than or equal to bi, the beginning of the time window, and the arrival time at each node ci must be lower than or equal to ei, the end of the time window. In case the arrival time is less than bi, the vehicle has to wait till the beginning of the time window before starting servicing the customer. In the literature the fleet size l is often a variable and a very common objective is to minimize l. Usually, two different solutions with the same number of vehicles are ranked by alternative objectives such as the total traveling time or total delivery time (including waiting and service times).

For a general review on AGV problems, the reader is referred to (Co and Tanchoco, 1991; King and Wilson, 1991; Ganesharajah et al., 1998). For a recent review on AGVs scheduling and routing problems and issues, the reader is referred to the survey of Qiu et al. (2002). These authors identified three types of algorithms for AGVs problems:

• (1) for general path topology,
• (2) for path optimization and
• (3) for specific path topologies.

Methods of the first type can be divided in three categories:

(1a) static methods, where an entire path remains occupied until a vehicle completes its route; (1b) time-window based methods, where a path segment may be used by different vehicles during different time-windows;

and (1c) dynamic methods, where the utilization of any segment of path is dynamically determined during routing rather than before as with categories (1a) and (1b). The method presented in this article belongs to the third category (1c) and addresses the conflict free routing problem with an optimization approach.

A number of authors have addressed the conflict free routing problem with a static transportation requests set, i.e., with all requests known a priori. Lee et al. (1998) present a two-staged traffic control scheme to solve a conflict free routing problem. Their heuristic method consists of generating off-line k-shortest paths in the first stage before the on-line traffic controller picks a conflict free shortest path whenever a dispatch command for an AGV is issued (second stage). Rajotia et al. (1998) propose a semi-dynamic time window constrained routing strategy. They use the notions of reserved and free time windows to manage the motion of vehicles. Krishnamurthy et al. (1993) propose an optimization approach. Their objective is to minimize the make span. They assume that the assignment of tasks to AGVs is given and they solve the routing problem by column generation (Ho and Wu, 2002). Their method generates very good solutions in spite of the fact that it is not optimal (column generation is performed at the root node of the

8.1 Summary 119

search tree only). Oboth et al. (1999) present a heuristic method to solve the dispatching and routing problems but not simultaneously. Scheduling is performed first and a sequential path generation heuristic (SPG) is used to generate conflict free routes. The SPG is inspired from Krishnamurthy et al. (1993) static version of the AGV routing problem and applied to a dynamic environment while relaxing some of the limiting assumptions like equal and constant speeds of AGVs. When conflict is encountered, no feedback is sent to the scheduling module. The AGV being routed has to be delayed if an alternate route cannot be generated (Lin et al., 2002).

The authors use rules for positioning idle AGVs instead of letting the system manage them. Langevin et al. (1996) propose a dynamic programming based method to solve exactly instances with two vehicles. They solve the combined problem of dispatching and conflict free routing. Desaulniers et al. (2003) propose an exact method that enables to solve instances with up to four vehicles. Their approach combines a greedy search heuristic (to find a feasible solution and set bound on delays), column generation and a branch and cut procedure. Their method presents however some limits since its efficiency depends highly on the performance of the starting heuristic. If no feasible solution is found by the search heuristic, then no optimal solution can be found. The search heuristic performs poorly when the level of congestion increases.

The problem considered in this chapter is a generalization of the Vehicle Routing Problem (VRP) and the Pickup and Delivery Problem (PDP) see Cordeau et al. (2004), Mitrovi´c-Mini´c (1998) and secondary literature given there. The most widely studied vehicle routing problems are the capacitated VRP and the Vehicle Routing Problem with Time Windows (VRPTW) which are surveyed by Laporte and Semet (2002) and Cordeau et al. (2002). Efficient methods for handling complex side constraints in insertion methods are presented in Campbell and Savelsbergh (2004). Comprehensive surveys on construction methods, neighborhood search methods, and meta-heuristics for the VRPTW are given by Bräysy and Gendreau (2005a) and Bräysy and Gendreau (2005b). Variable Neighborhood Search (VNS) is a meta-heuristic based on the idea of systematically changing the neighborhood structure during the search, see Mladenovi ´c and Hansen (1997) and Hansen and Mladenovi´c (2003). VNS systematically exploits the following observations:

• a) a local optimum with respect to one neighborhood structure is not necessary so for another;
• b) a global optimum is a local optimum with respect to all possible neighborhood structures;
• c) for many problems local optima with respect to one or several neighborhoods are relatively close to each other.

A recent example of a VNS algorithm for vehicle routing problems is the algorithm for the multi-depot VRPTW presented by Polacek et al. (2004). Large Neighborhood Search (LNS) has been introduced for the VRPTW by Shaw (1997) and can be interpreted as a special case of VNS. Kilby et al. (2000) have shown that LNS is well suited for rich VRP.

In many cases it is assumed that transportation requests are accepted before planning begins and tours are generated assuming that all accepted transportation requests must be served (Lin et al., 2004). Work regarding load acceptance issues for the Traveling Salesman Problem (TSP) has been surveyed by Feillet et al. (2005), but only few attempts have been made to tackle extensions of this problem, for example, by Schönberger et al. (2002). VRP with multiple pickup and delivery locations have been studied by Savelsbergh and Sol (1995), Savelsbergh and Sol (1998) and Hasle (2003). A comprehensive discussion of dynamic vehicle routing can be found in Psaraftis (1988) and Psaraftis (1995). Dynamic real-life problems often require rich models, in most of the literature on dynamic routing problems; however, some simplifying assumptions are made. For example, in the dynamic full-truckload PDP, which recently has received increasing attention, see Fleischmann et al. (2004), Yang et al. (2004), and Powell et al. (2000), each vehicle can only carry one transportation request at a time and cannot load further shipments until all currently loaded shipments are unloaded. The only work known to the authors regarding rich VRP in a dynamic context is presented by Savelsbergh and Sol (1998). A column generation approach is used to solve the General Pickup and Delivery Problem (GPDP) presented by Savelsbergh and Sol (1995).

8.2 Statement of the Problem

In a material requirement planning (MRP) for a product some shops are existed to provide the needed materials. According to the bill of material, an AGV is applied to provide the materials for the final assembly in the master shop. Considering the hierarchical structure of the BOM, shops are in levels. For each level some shops are existed being differentiated by the cost, time, and AGV capability. In this way from each shop of a level to the next shop from another level some paths are considered. As it is stated, the difference between paths is in cost, time, and AGV capability. Hence, the AGV should decide to select the path to minimize time, cost, and AGV capability, concurrently. The materials are assembled from the lowest level of the BOM and lead to the final product. Each shop in each level can satisfy a corresponding level of the proposed BOM. The proposed BOM is illustrated in Figure 1.

Fig. 1 The Proposed BOM

Based on the hierarchical structure of the BOM, the shops are in a level structure each of which provides the materials in the certain level of the BOM. The AGV would begin to assemble the materials from shops in varied levels to reach the final product in the last stage in the main shop. The model is looking for a path that fulfills the BOM and also is optimal considering the decision parameters. A configuration of the shops in the Material Requirement Planning is presented in Figure 2.

Fig. 2 A Configuration of the Shops in the Material Requirement Planning

AGV needs to decide which route to select to satisfy the objectives of the problem. In the stated condition, AGV should consider all time, cost, and AGV capability to select a route i.e. the lowest cost, shortest time, and the lowest occupied AGV capability must be regarded. To achieve this goal, mathematical programming approach is applied for optimization. To find out the cost, time, and capability parameters intelligent agents are proposed. Intelligent agents help AGV to evaluate all paths.

In artificial intelligence, an intelligent agent (IA) is an autonomous entity which observes and acts upon an environment (i.e. it is an agent) and directs its activity towards achieving goals (i.e. it is rational). Intelligent agents may also learn or use knowledge to achieve their goals. They may be very simple or very complex: a reflex machine such as a thermostat is an intelligent agent, as is a human being, as is a community of human beings working together towards a goal.

Intelligent agents are often described schematically as an abstract functional system similar to a computer program. For this reason, intelligent agents are sometimes called abstract intelligent agents (AIA) to distinguish them from their real world implementations as computer systems, biological systems, or organizations. Some definitions of intelligent agents emphasize their autonomy, and so prefer the term autonomous intelligent agents. Still others considered goaldirected behavior as the essence of intelligent and so prefer a term borrowed from economics, "rational agent".

Intelligent agents in artificial intelligence are closely related to agents in economics, and versions of the intelligent agent paradigm are studied in cognitive science, ethics, the philosophy of practical reason, as well as in many interdisciplinary socio-cognitive modeling and computer social simulations.

Fig. 3 A configuration of the intelligent agent and information transfer process

Intelligent agents are also closely related to software agents (an autonomous software program that carries out tasks on behalf of users). In computer science, the term intelligent agent may be used to refer to a software agent that has some intelligence, regardless if it is not a rational agent.

In this chapter, we propose three intelligent agents for time, cost and capability, respectively. The core of these agents is optimization. The results of all agents are transferred to a computer data base which conducts the AGV through guide path in the shop floor. A configuration of the intelligent agent and information transfer process is shown in Figure 3.

The AGV collects the information and process them in the separate intelligent agents. The results are sent to the data base and view by the controller. The controller issue required orders to the AGV.

To estimate the time of traveling from one shop (node) to another one in our proposed network, a mathematical optimization is worked out by an intelligent agent. While the velocity (v) of AGV is known, the time is computed as follows:

$$d = v.t, (1)$$

Where d is the movement distance, v is AGV velocity and t is the time of movement. Due to a Cartesian coordinates structure as shown in Figure 4.

Fig. 4 Cartesian coordinates for AGV movement

to compute the distance we use,

$$d = \sqrt{(x - x_0)^2 + (y - y_0)^2},$$ (2)

wherex is the horizontal movement and y is the vertical movement. As a result, we have,

$$t = \frac{d}{v} \,. \tag{3}$$

Therefore, time is computed and transferred to the central computer.

Cost is integrated with the time computed from last section. Our proposed intelligent agent use the following formulae to estimate the cost of moving from one shop to another one,

$$cost = (variable cost \times d) + fix cost,$$ (4)

Variable cost depends on the distance that the AGV move (such as, defect, amortization, etc.). But the fix cost is independent of the distance and include set up cost, controller salary, etc.

Here, we want to estimate the AGV capability during the assembly process. Some parameters affect the capability of an AGV. We should investigate whether a parameter is significant on capability or not. One way to survey the effect of independent variables on dependent variable is multiple linear regression model. Therefore, we consider the following equation which is computed in our proposed intelligent agent for capability,

$$u_i = \beta_0 + \beta_1 r_1 + ... + \beta_j r_j + \varepsilon_j,$$ $i = 1,...,n; j = 1,...,m.$ (5)

where β0 is the intercept, β ^j s are the coefficients for *rj*parameters and ε is the error term. The aim is identifying the β ^j s which are not important on capability (ui). Here, we apply two-sided hypothesis testing as follows;

$$\begin{split} H_0: \beta_j &= 0; \ H_1: \beta_j &\neq 0. \end{split} \tag{6}$$

The test of the null hypothesis H0 against the two sided alternative proceeds as in the three steps. The first is to compute the standard error of β ^j , ( )^j SE β . The standard error of β ^j is an estimator of ^{β j σ} , the standard deviation of the sampling distribution of β ^j .

$$\sigma{\beta{j}}^{2} = \frac{1}{m} \times \frac{\frac{1}{m-2} \times \sum{j=1}^{m} (r{j} - \bar{r})^{2} \varepsilon{j}^{2}}{\left[\frac{1}{m} \sum{j=1}^{m} (r_{j} - \bar{r})^{2}\right]^{2}},$$ (7)

$$SE(\betaj) = \sqrt{\sigma{\beta_j}^2}$$ (8)

Although the formula for ^{2 β j σ} is complicated, in applications the standard error is computed by regression software. The second step is to compute the t-statistic,

$$t = \frac{\beta_j - 0}{SE(\beta_j)} \tag{9}$$

The third step is to compute the *p-*value, the probability of observing a value of β ^j at least as different from zero as the estimate actually computed ( act β ^j ), assuming that the null hypothesis is correct. Stated mathematically,

$$p - \text{value} = \Pr_{H_0} \left[ \left| \beta_j - 0 \right| > \left| \betaj^{\text{act}} - 0 \right| \right] \Rightarrow \Pr{H_0} \left| \left| \frac{\beta_j - 0}{SE(\beta_j)} \right| > \left| \frac{\beta_j^{\text{act}} - 0}{SE(\betaj)} \right| \right| \Rightarrow \Pr{H_0} \left( |t| > |t^{\text{act}}| \right)$$ (10)

Where 0 Pr^H denotes the probability computed under the null hypothesis, the second equality follows by dividing by ( )^j SE β , and act t is the value of the t-statistic actually computed. Because β ^j is approximately normally distributed in large samples, under the null hypothesis the t-statistic is approximately distributed as a standard normal random variable, so in large samples,

$$p - \text{value} = \Pr(|Z| > |t^{act}|) = 2\Phi(-|t^{act}|)$$ (11)

A small value of the *p-*value, say less than 5%, provides evidence against the null hypothesis in the sense that the chance of obtaining a value of β ^j by pure random variation from one sample to the next is less than 5% if, in fact, the null hypothesis is correct. If so, the null hypothesis is rejected at the 5% significance level. Simply we can reject the null hypothesis at the 5% significance level if, > 1.96 act t . Therefore, if the hypothesis is accepted then we ensue that the corresponded parameter is not important and has no effect on the capability.

While we determined the effective parameters on capability, we can find the regression value for capability omitting the noneffective parameters.

8.3 Mathematical Model

Considering the network nature of the problem and the objective which is identifying shortest path or route, mathematical programming approach is applied. The network of the proposed problem is represented in Figure 5.

Fig. 5 The Network of the Problem

All nodes indicate the shops except node 0 which shows the AGV. Arcs are identified with three criteria, cost, time, and AGV capability which are consequenced from the intelligent agents. Number of nodes are N and depends on the number of the shops that provide materials. Nodes are in levels as stated before. Mathematical programming approach in details is as follows.

Indices:

i	Start node	i=0,1,2,,N
i'	End node	i'=1,2,…,N
j	Level number	j=0,1,2,J

Notations:

Cii' Cost of moving from node i to node i'

Tii' Time duration of moving from node i to node i'

(i) ^{ϕ j} The minimum value of moving from the start node i to the end nod i'

in layer j

N Number of nodes

Maximum capability of an AGV δ

τ Maximum time duration in hand to satisfy the demands

B Maximum budget in hand

Pii' Capability of an AGV for moving from node i to node i'

$$\xi_{ii'} = \begin{cases} 1 & \text{if path } ii' \text{ is selected} \ 0 & \text{otherwise} \end{cases}$$

Objective Function:

$$\varphi{j}(i') = \underset{i \neq i'}{Min} \left{ \varphi{j-1}(i) + C_{ii'} \right} \quad \text{and}$$ (12)

$$\underset{i\neq i'}{Min} \left{ \varphi{j-1}(i) + T{ii'} \right} \text{and}$$ (13)

$$\min{i \neq i} \left{ \varphi{j-1}(i) + P_{ii'} \right} \tag{14}$$

Subject to:

$$\varphi_0(0) = 0 \tag{15}$$

$$\sum{i}^{N} \sum{i}^{N} \xi{ii'} \times C{ii'} \le B \tag{16}$$

$$\sum{i'}^{N} \sum{i}^{N} \xi{ii'} \times T{ii'} \le \tau \tag{17}$$

$$\sum{i'}^{N} \sum{i}^{N} \xi{ii'} \times P{ii'} \le \delta \tag{18}$$

$$T{ii'}, C{ii'}, P_{ii'} \ge 0$$ (19)

Equations (12), (13), and (14) are the objective functions which aim to find the shortest path based on cost, time, and AGV capability, respectively. Equation (15) is the primary condition for starting the travel on nodes. Equation (16) guarantees that our travels won't exceed our accessible budget. Equation (17) represents that the time needed to satisfy the demands is confined. Equation (18) guarantees the capability of AGV during the travels. Equation (19) is the sign relation.

As it is cleared, the objective functions cause the decision maker not to be able to decide at once about time and cost and capability simultaneously. Hence, we do some changes on objective functions to mix them in to a single function that take into account all time and cost and capability as follows.

8.3.1 Normalization Process

To transform the above three objective functions (12, 13 and 14) the normalization is required for making it possible to obtain time, cost and AGV capability, simultaneously. The normalization process is as follows:

$$T{ii'}^{n} = \frac{T{ii'}}{\sqrt{\sum{i'=1}^{N} \sum{i=0}^{N} T_{ii'}^{2}}}$$ (20)

where *ⁿ Tii*′ is the normalized value of time duration for each arc.

The same procedure is existed for costs:

$$C{ii'}^{n} = \frac{C{ii'}}{\sqrt{\sum{i'=1}^{N} \sum{i=0}^{N} C_{ii'}^{2}}}$$ (21)

where *ⁿ Cii*′ is the normalized value of cost for each arc.

The same procedure is existed for AGV capability:

$$P{ii}^{n} = \frac{P{ii'}}{\sqrt{\sum{i'=1}^{N} \sum{i=0}^{N} P_{ii'}^{2}}}$$ (22)

By the means of normalization the objective functions could be transformed to a single one as follows:

$$\varphi{j}(i') = Min\left{\varphi{j-1}(i) + T{ii'}^{n} + C{ii'}^{n} + P_{ii'}^{n}\right}$$ (23)

Equation (23) does time-cost-AGV capability optimization at the same time and its result is the optimum combination of time, cost, and AGV capability which ensue to the optimal path or route. Also, note that their importance weights are considered to be equal. But, if a decision maker tends to consider different importance for time, cost and capability, it is easily performed using common weighing approaches in the literature.

Algorithm for Solving

The following heuristic algorithm could be applied for solving the proposed problem in some stages.

Step0- Identify number of nodes (factories)

Step1- Identify Tii' ,Cii' and, Pii'

Step2- Normalizing process

Step3- Selecting the best combination of time, cost, and capability

Step4- Satisfying the constraints

Step5- Finding the optimal path (route)

Step6- Termination

The flow chart of the heuristic algorithm is illustrated in Figure 6.

Fig. 6 The Flowchart of the Solving Algorithm

Regarding to the flowchart, in termination stage a path with all optimal time, cost, and AGV capability is identified at the same time.

8.4 Conclusions

The obtained results extracted from the proposed methodology in a material handling system are helpful in providing better managerial insights for decision making in AGV planning and guide path maintenance. While a path contains several arcs in the proposed material handling network determining the arcs especially the most used ones helps fortifying them and prohibit halt in the system due to AGV or path break downs. Another managerial aspect is to make real time decisions underlying the proposed agents as decision supports. While demand and material supplies are dynamic therefore AGV planning should also be dynamic. Therefore the proposed agents help to make decisions at any time with respect to any demand fulfilled and material provision. We have considered an extension of the classical vehicle routing problem in which three dimensional packing constraints were introduced i.e. time, cost, and AGV capability. This problem featured three classical combinatorial optimization problems. Intelligent agents were designed for the triple criteria which used optimization tools to estimate the values and report them to the computer data base. We have applied a mathematical programming approach for optimizing the problem. By a normalization process all time, cost, and AGV capability were included as the arc's value of the proposed network. The effectiveness of the proposed algorithm was validated using two illustrative examples.

References

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Chapter 9

Earliness/Tardiness for a Multiple AGV System

9.1 Summary

Flexible manufacturing systems (FMSs), container terminals, warehousing systems, and service industries including hospital transportations are employing automated guided vehicle systems (AGVs) for the material handling to maintain flexibility and efficiency of production and distribution. For the efficient operation, it is requested to realize the synchronized operations for the simultaneous scheduling of production systems and transportation systems. The main issue treated in this chapter is the simultaneous optimization problems for penalized earliness and tardiness for the AGVs in the manufacturing system. The production scheduling problems asks an optimal production sequence and starting time of operations for jobs at machines for multi-stages with respect to a specified technical precedence relation. The vehicle management problems are classified into:

• (1) dispatching, which is to assign tasks to vehicles;
• (2) routing, which is to select specific paths taken by vehicles;
• (3) scheduling, which is to determine the arrival and departure times.

Unlike the classical vehicle routing problem (VRP) formulation, conflict-free constraints should be considered for the routing of AGVs for semiconductor fabrication. The interaction between production and transportation control is discussed by Mantel and Landerweerd (1995). In the flowshop production systems, the production and transportation schedules are usually controlled by a pull type of policy in case of fork lifts or conveyor systems. However, for FMSs environment with AGV systems, the optimal machine schedules highly depend on the selection of dispatching and routing because it is extremely difficult to predict the transportation time when the conflicts and interferences between vehicles cannot be neglected.

Automated guided vehicle (AGV) is a material handling equipment traveling on a network of guide paths. The FMS is a configuration of various shops, also called working stations, each with a specific function such as milling, washing, or assembly. Each shop is connected to the guide path network by a pick-up/delivery (P/D) station where pallets are transferred from/to the AGVs. Pallets of products are moved between the shops by the AGVs. The guide path is composed of aisle segments on which the vehicles are assumed to travel at a constant speed. The vehicles can travel forward or backward. As many vehicles travel on the guide path simultaneously, collisions must be avoided. AGV systems are implemented in various industrial contexts: container terminals, part transportation in heavy industry, flexible manufacturing systems. For a general review on AGV problems, the reader is referred to (Co and Tanchoco, 1991; King and Wilson, 1991; Ganesharajah et al., 1998). For a recent review on AGVs scheduling and routing problems and issues, the reader is referred to the survey of Qiu et al. (2002). These authors identified three types of algorithms for AGVs problems:

• (1) for general path topology,
• (2) for path optimization,
• (3) for specific path topologies.

Methods of the first type can be divided in three categories:

• (1a) static methods, where an entire path remains occupied until a vehicle completes its route;
• (1b) time-window based methods, where a path segment may be used by different vehicles during different time-windows;
• (1c) dynamic methods, where the utilization of any segment of path is dynamically determined during routing rather than before as with categories (1a) and (1b).

This chapter addresses a penalized earliness and tardiness scheduling problem for AGVs in a manufacturing system.

Scheduling problems arise in areas as diverse as production planning, personnel planning, product configuration, and transportation. An overview of the wide range of constraints in scheduling, together with the most powerful propagation algorithms for these constraints are given (Baptiste et al., 2001; Baptiste et al., 1995).

Production scheduling, dispatching, routing and scheduling decisions for AGVs can be made simultaneously or separately. Most of the literature treats one or two of the problems at the same time. An extensive review has been addressed by Vis (2006) for operational control of AGVs. A widely used technique for dispatching is the simulation. The heuristic rules are used in on-line control systems. For routing and scheduling of AGVs, several techniques have been used to maximize the total system performance taking in to account for deadlock or conflicts for AGVs. Kim and Tanchoco (1991) studied the problem of finding conflict-free routes in a bi-directional network. The algorithm is based on the shortest path methods through the concept of time-window graph. Petri net is used to analyze deadlock and conflict-free conditions (Dotoli and Fanti, 2004; Wu and Zhou, 2005). Singh and Tiwari (2002) presented an intelligent agent framework to find a conflict-free shortest-time path. Nishi et al. (2005) provided a mathematical model for routing problem. Lagrangian decomposition technique was used solve the problem. Ghasemzadeh et al. (2009) presented a conflict-free scheduling and

9.1 Summary 135

routing in mesh topologies. It can generate the shortest path for scheduling predicting conflicts and select another path in the case of failure.

The literature discussed above on scheduling of AGVs hardly considers the capacity constraints of the machines where transportation jobs become available and sequencing of operations at the machines. The simultaneous production scheduling and transportation routing problem is one of the difficult joint problems. The problems for AGVs have been studied mostly in operations research and/or FMS literature. A common approach for FMS scheduling is based on the discrete event simulation with dispatching rules (Tunali, 1998).

Lacomme et al. (2005) introduced a branch and bound algorithm coupled with discrete event simulation. Blazewicz et al. (1994) addressed the two steps algorithm for integrating machine scheduling and the conflict-free routing problems. In their approach, the production scheduling and routing problems are solved separately. Bilge and Ulusoy (1995) developed a time-window approach to solve the simultaneous scheduling of machines and material handling in FMSs. They formulated the problem as a mixed integer programming problem. Ulusoy et al. (1997) and Jerald et al. (2006) dealt with the application of the genetic algorithm on the problem. Khayat et al. (2006) studied an integrated method with mixed integer linear programming (MILP) and constrained programming. Never the less in their model, vehicles can always select a shortest path from a machine station to another machine station without the consideration of conflict and collision on the detailed routing for vehicles. Corre´a et al. (2007) proposed an integrated scheduling of dispatching and vehicle routing with the consideration of conflict-free path selection, but it does not take into account the scheduling of machines and vehicles simultaneously.

In the above literature, it is extremely difficult to consider production scheduling and conflict-free routing because the number of decision variables is significantly increased. Therefore, the conventional decomposition algorithm is not sufficient to solve the problem efficiently. The integration of cut generation with various decomposition methods is widely studied recently (2006). The logicbased Benders decomposition method was introduced by Hooker (2003). The advantage of the logic-based Benders is that it permits to combine MILP and the constraint programming approach. Similar idea was applied to solve the simultaneous planning and scheduling problems (2006).

A number of authors have addressed the conflict free routing problem with a static transportation requests set, i.e., with all requests known a priori. Lee et al. (1998) present a two-staged traffic control scheme to solve a conflict free routing problem. Their heuristic method consists of generating off-line k-shortest paths in the first stage before the on-line traffic controller picks a conflict free shortest path whenever a dispatch command for an AGV is issued (second stage). Rajotia et al. (1998) propose a semi-dynamic time window constrained routing strategy. They use the notions of reserved and free time windows to manage the motion of vehicles. Krishnamurthy et al. (1993) propose an optimization approach. Their objective is to minimize the makespan. They assume that the assignment of tasks to AGVs is given and they solve the routing problem by column generation. Their method generates very good solutions in spite of the fact that it is not optimal (column generation is performed at the root node of the search tree only).

Oboth et al. (1999) present a heuristic method to solve the dispatching and routing problems but not simultaneously. Scheduling is performed first and a sequential path generation heuristic (SPG) is used to generate conflict free routes. The SPG is inspired from Krishnamurthy et al. (1993) static version of the AGV routing problem and applied to a dynamic environment while relaxing some of the limiting assumptions like equal and constant speeds of AGVs. When conflict is encountered, no feedback is sent to the scheduling module. The AGV being routed has to be delayed if an alternate route cannot be generated. The authors use rules for positioning idle AGVs instead of letting the system manage them.

Langevin et al. (1996) propose a dynamic programming based method to solve exactly instances with two vehicles. They solve the combined problem of dispatching and conflict free routing. Desaulniers et al. (2003) propose an exact method that enables to solve instances with up to four vehicles. Their approach combines a greedy search heuristic (to find a feasible solution and set bound on delays), column generation and a branch and cut procedure. Their method presents however some limits since its efficiency depends highly on the performance of the starting heuristic. If no feasible solution is found by the search heuristic, then no optimal solution can be found. The search heuristic performs poorly when the level of congestion increases.

9.2 Statement of the Problem

Consider a jobshop manufacturing system with multiple AGVs performing material handling. There is a number of AGVs pre-specified for material handling. The AGVs guide paths may be occupied in the time that an AGV is sent to do the material handling. Therefore, finding the free path to fulfill the function is important. The manufacturing process plan of all jobs processing time is cleared. If an AGV arrives early, it should waits until the part processing is finished. The waiting time is related to the distance the AGV moves and the due date of jobs in shops. The overall problem is to determine the manufacturing schedule and routing for AGVs to minimize the total penalized earliness/tardiness and AGVs' waiting times at the shops in jobshop configuration. The following assumptions are considered for modeling the proposed problem.

• The number of jobs, processing time for each job and the number of AGVs are given.
• The started job cannot be interrupted once the processing of that is started at a shop.
• Each AGV can transport a single load at a time. A transportation task is the set of a starting node and a destination node.

The overall decision variables for the problem consist of the allocation of AGVs to the shops for material handling and the conflict-free routing for vehicles.

To estimate the time of traveling from one shop (node) to another in our proposed network, a mathematical relation is employed. While the velocity (v) of AGV is known, the time is computed as follows:

$$d = vt, (1)$$

where d is the movement distance, v is AGV velocity and t is the time of movement, due to a Cartesian coordinates structure as shown in Figure 1.

Fig. 1 Cartesian coordinates for AGV movement

to compute the distance, we use rectilinear formulae,

$$d = |x - x_0| + |y - y_0|, (2)$$

where x is the horizontal movement and y is the vertical movement. As a result, we have,

$$t = \frac{d}{v} \,. \tag{3}$$

Therefore, AGV movement time is computed.

9.3 Mathematical Model

Here, we formulate the proposed problem, mathematically. The indices, parameters and decision variable are introduced below.

Indices:

i Index for jobs; i = 1,2,..., m j Index for AGVs; j = 1,2,..., n k,k' Index for shops; k = 1,2,..., o, k '= 2,3,..., o

Parameters:

^α weight for total earliness; ^α ≥ 0 β weight for total tardiness; β ≥ 0

pik processing time of job i being processed in shop k

duik due date of job i being processed in shop k dkk' distance between any two shops k and k'

V velocity of AGV (same for all AGVs)

tjkk' Movement time for AGV j between nodes (shops) k and k'

Eik earliness of job i being processed in shop k TAik tardiness of job i being processed in shop k

Cik completion time for job i being processed in shop k

Qik allocation of job i to shop k; 1, if job i is allocated to shop k, 0, otherwise

Decision variables:

$$X_{jik} = \begin{cases} 1 & \text{if AGV } j \text{ is scheduled on job } i \text{ in shop } k \ 0 & \text{otherwise,} \end{cases}$$

$$Y_{jik} = \begin{cases} 1 & \text{if a path between shops } k \text{ and } k' \text{ is busy by AGV } j \ 0 & \text{otherwise,} \end{cases}$$

fori = 1,..., m , j = 1,..., n, k = 1,..., o. Objective function:

k

i

= = = =

1 1 1 1

k

i

$$Min z = \sum{i=0}^{\infty} \sum{k=0}^{\infty} \alpha \cdot E{ik} + \sum{k=0}^{\infty} \sum{k=0}^{\infty} \beta \cdot TA{ik}$$ (4)

where,

$$\alpha = \sum{k=1}^{o} \sum{i=1}^{m} |p{ik} - du{ik}|^{2},$$

$$i = 1, ..., m, \quad k = 1, ..., o.$$

$$\beta = \sum{k=1}^{o} \sum{i=1}^{m} |p{ik} - C{ik}|^{2},$$ (5)

The weights for total earliness and tardiness are loss functions of processing times, due dates and completion times. Note that, α and β are computed based on quadratic loss function with a positive squared subtraction of processing times from due dates and completion times for earliness and tardiness, respectively.

Constraints:

$$\sum{j=1}^{n} X{jik} = 1, i = 1, ...., m, k = 1, ...., o, (6)$$

$$\sum{i=1}^{m} \sum{k=1}^{o} X_{jik} = 1, j = 1, ..., n, (7)$$

$$X{jik}, Y{jkk'} = 1,$$ $i = 1, ..., m, j = 1, ..., n, k = 1, ..., o,$ (8)

$$\sum{j=1}^{n} Y{jkk'} = 1, k = 1, ..., o, k' = k+1, (9)$$

$$\sum{j=1}^{n} Y{jkk'} \cdot Q_{ik} = 1, i = 1, ..., m, k = 1, ..., o, k' = k+1, (10)$$

$$\sum{j=1}^{n} X{jik}.Q_{ik} = 1, i = 1,...,m, k = 1,...,o, (11)$$

$$p{ik} = \sum{j=1}^{n} X{jik} t{jik'},$$ $i = 1,...,m, k = 1,...,o,$ (12)

$$C{ik} \ge C{i,k-1} + p_{ik}, \qquad i = 1, ..., m, \quad k = 1, ..., o,$$ (13)

$$TA{ik} = \max \left{ t{jkk'}, C{ik} - du{ik} \right}, \quad i = 1, ..., m, \quad k = 1, ..., o,$$ (14)

$$E{ik} = \max{t{jkk'}, du{ik} - C{ik}}, \quad i = 1, ..., m, \quad k = 1, ..., o,$$ (15)

$$X{jik}, Y{jik'} \in {0,1}, \quad i = 1, ..., m, \quad k = 1, ..., o, \quad j = 1, ..., n.$$ (16)

Constraints (6) specify that exactly one AGV is scheduled at the kth shop. Constraints (7) define each AGV to be scheduled only once. Constraints (8) emphasize that when an AGV is scheduled for a job in a shop then the path from that shop to the next immediate shop is busy. Constraints (9) imply that only one AGV is allocated to a shop. Constraints (10) show that for a job-shop allocation only one AGV occupies the path between any two nodes, too. Constraints (11) indicate that for a job-shop allocation only one AGV is scheduled. Relations (12) compute the processing times of jobs in shops. Relations (13) show an inequality between the completion times of job in a previous shop and the processing time of the job in next shop. Relations (14) present the tardiness. Relations (15) present the earliness. Relations (16) indicate that all variables are binary.

Since formulae (14) and (15) are nonlinear, we linearize them as follows:

$$E{ik} \ge t{jkk'}, \quad i = 1, ..., m, \quad k = 1, ..., o,$$ (17)

$$E{ik} \ge du{ik} - C_{ik}, \qquad i = 1, ..., m, \quad k = 1, ..., o,$$ (18)

$$TA{ik} \ge t{jkk'}, \qquad i = 1, ..., m, \quad k = 1, ..., o,$$ (19)

$$TA{ik} \ge C{ik} - du_{ik}, \qquad i = 1, ..., m, \quad k = 1, ..., o.$$ (20)

Clearly, the equation (8) is nonlinear due to product of two binary decision variables leading to inefficiency of the outputs and spending longer time for convergence purposes. So, we linearize it by substituting the binary variable ijkl ^ε as follows,

$$\varepsilon{ijkk'} = X{jik} Y_{jkk'}, \qquad \forall i, j, k, k'. \tag{21}$$

Meanwhile, the following linear inequalities should be added to the mathematical model,

$$\varepsilon{ijkk'} - X{jik} - Y_{jkk'} + 1.5 \ge 0, \qquad \forall i, j, k, k', \tag{22}$$

$$1.5\varepsilon{ijkk'} - X{jik} - Y_{jkk'} \le 0, \qquad \forall i, j, k, k'. \tag{23}$$

Therefore, the final linearized mathematical model is given below:

Eq. (4)

Eq. (5)

s.t.

Eq. (6) and (7)

Eq. (21)-(23)

Eq. (9)-(13) Eq. (17)-(20) Eq. (16).

Next, we test the applicability of the proposed mathematical model in the numerical results.

Here, we work out on the computational aspects of the proposed mathematical model. The number of variables and constraints in the linearized model are presented parametrically in tables 1 and 2, respectively, based on the variable indices.

Table 1 The number of variables in the linearized model

Variable	Count
X jik	n×m×o
Yjkk'	n×o×(o-1)
	Sum= (n×m×o) + (n×m×(o-1))

Table 2 The number of constraints in the linearized model

Con.	Count	Con.	Count
(6)	m×o	(11)	i×o
(7)	n	(12)	i×o
(21-23)	3(n×m×o×(o-1))	(13)	i×o
(9)	o×(o-1)	(17-20)	4(i×o)
(10)	m×o×(o-1)	(16)	m×o×n

Sum= (m×o) +( *n)+3(n×m×o×(o-1))+(o×(o-1))+(m×o×(o-1))+*7(i×o)+( m×o×n)

According to tables 1 and 2, when the number of jobs, shops and AGVs increase, the search space increases drastically. The solution should provide the optimal paths for each AGV and prohibit the conflict among AGVs minimizing the total earliness and tardiness.

AGVs begin their movement from a corresponding shop according to the process plan and go on through other shops to complete the pre-defined job sequence and deliver the finished product to the depot. These movements of AGVs configure a network as shown in Figure 2. The AGVs movements are based on the due dates of the jobs and distances between any two shops. Thus, to prevent the AGVs conflicts the solution algorithm should consider an allocation between jobs and AGVs so that no conflict occurs.

At each iteration, the arcs (paths between any two shops) in the network are divided into two sets; the arcs occupied by an AGV (A); the free arcs (B). A network structure (A, B) is optimal if an AGV serves as a handling device for a job

is one and at the same time the AGV waiting time to move to next job is minimized. With those conditions, the current solution is optimal. Otherwise, there are arcs in the network that violate the optimal conditions. An arc is a violated arc if it belongs to B and has lower waiting time than an ark in A. This Algorithm maintains a feasible arc at each iteration and successfully goes toward the optimality conditions until it becomes optimal. (k,k')∈A

Fig. 2 A network configuration of the problem

Heuristic Search Algorithm

To create the initial or Basic Feasible Solution (BFS) in Step 0, an artificial node 0 and artificial arcs are appended to the network. The node '0' will be the source of the network and the artificial arcs connect the nodes to the source. The set B consists of the main arcs in the network, and the set A is empty.

Selection of an earliness/tardiness scheme is an important decision in Step 1. During this step the waiting time of the AGVs on non-basic arcs are recalculated. If there is at least one that violates its optimality condition it is a candidate to enter the basis.

In Step 1, appending the entering arc (k, k'), an arc with violation, to the network forms a unique cycle, W, with the arcs of the basis.

In order to eliminate this cycle (Step 2), one of its arcs must leave the basis. The cycle is eliminated when we have augmented flow by a minimum amount of earliness or tardiness to force the flow in one or more arcs of the cycle to the next shop. By augmenting the flow in an empty arc, the objective value of the solution is improved. The first task in determining the leaving arc is the identification of all arcs of the cycle. The flow (AGV path) change is determined by the equation ^θ = min{Yjkk' for all(k, k')∈W}. The leaving arc is selected based on cycle W.

The substitution of entering for the leaving arc and the reconstruction of new network is called a pivot (Step 3). After pivoting to change the basis, the earliness/tardiness for each arc (k,k')∉ A is calculated. If the earliness/tardiness for all (k,k')∈{ } A,B satisfies the optimality condition then the current basic feasible solution is optimal. Otherwise, an arc (k, k') where there is a violation should be chosen and operations of the algorithm should be repeated. The steps of the proposed search algorithm are specified in Figure 3.

Fig. 3 The proposed search algorithm

Architecture of Solution Approach

Here, the architecture and operations of the proposed solution approach is described, briefly. At the start of process, the Job Generator generates a few jobs for the AGVs. These jobs will be appended to the remaining jobs, which is empty at the beginning. The remaining jobs are used by the heuristic search algorithm and the output of this method is an individual job for every AGV. When the time is running, the travelled and waited times of every AGV should be updated. At the same time, if the AGV picks up the job, the assigned job for the AGV will be deleted and removed from the list of remaining jobs. If the job should be delivered to the shop, it could not be removed until meeting time between the preceding and proceeding shops. The Job Generator has to generate a few new jobs, when it finds out any idle AGV. The main architecture of the proposed solution approach is shown in Figure 4.

Fig. 4 Block diagram for the proposed solution approach

9.4 Conclusions

In this chapter, the automated guided vehicle scheduling problem was formulated as a special case of the earliness/tardiness minimization. In the proposed problem the number of jobs, processing time for each job and the number of AGVs were assumed to be given. Also, the started job couldn't be interrupted once the processing of that was started at a shop, and each AGV could transport a single load at a time. Besides scheduling, the model was able to fulfill the conflict free routs for the AGVs. Due to nonlinear equations, a linearization process performed to make the mathematical model smooth for the solution approach. The number of parameters and variables and therefore the search space increased when the number of jobs, shops and AGVs increase. The solution should provide the optimal paths for each AGV and prohibit the conflict among AGVs minimizing the total earliness and tardiness. Then, a heuristic search algorithm and a solution methodology based on network concepts were developed for tackling the problem. Our computational results implied that the algorithms were efficient to find the optimal solution for large scale problems in negligible time. The heuristic search algorithm accompanied with the proposed solution architecture was a complete algorithm and it was efficient for dealing with the problems without any certain limits in size.

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Chapter 10

Markovian Model for Multiple AGV System

10.1 Summary

Traditional manufacturing has relied on dedicated mass-production systems to achieve high production volumes at low costs. As living standards improve and the demands for new consumer goods rise, manufacturing flexibility gains prominence as a strategic tool for rapidly changing markets. Flexibility, however, cannot be properly incorporated in the decision-making process if it is not well defined and measured in a quantitative manner. Flexibility in its most rudimentary sense is the ability of a manufacturing system to respond to changes and uncertainties associated with the production process (Miettinenet al., 2010; Kumar and Sridharan, 2009; Das et al., 2009). A comprehensive classification of eight flexibility types was proposed in Browne et al. (1984).

Flexible manufacturing systems (FMS) are crucial for modern manufacturing to enhance productivity involved with high product proliferation (Paraschidiset al., 1994). As one of the critical components of the FMS, the flexible material handling system (MHS) plays a strategic role in the implementation of the FMS (Beamon, 1998). According to Tompkins et al. (2002), about 20–50% of the total production cost is spent on material handling. This makes the subject of material handling increasingly important. In addition, all the complexity of manufacturing is passed on to the MHS. Therefore, the flexible MHS has been vital for improving the FMS to fulfill the requirements of high product proliferation.

Automated manufacturing systems (AMS), which are equipped with several CNC machines and AGV-based material handling system are designed and implemented to gain the automation and efficiency of production. To make use of all features of AMS, the planning in the AMS decision making process is critical because the planning decision has influence on the subsequent decision processes such as scheduling, dispatching, etc. The planning in automated manufacturing systems can be characterized as being online and short-term nature to respond to frequently changing production order. Given a production order, manufacturing planning function is responsible to establish a plan by decomposing the production task into a set of subtasks. An analysis of AMS dealing with changing demand can be found in (Terkaj et al., 2009). An extensive review of the loading problem for

an FMS can be found in (Grieco et al., 2001). An early stochastic programming approach to address the short-term production planning for an FMS can be found in (Terkaj and Tolio, 2006).

Automated Guided Vehicle System (AGVS) becomes popular in many industrial fields because of its flexibility, reliability, safety, and contribution to the increase of productivity and to the improvement of housekeeping. But, the performance of the material handling system is significantly influenced by several operating policies. One of the important operating policies is the positioning strategy of idle vehicles on the guide path (Egbehi, 1993; Kim, 1995).

In most manufacturing systems, decision making is worked out at several stages of design, planning and operation. The role of performance modeling is significant in advanced manufacturing systems from economic viewpoints. However, events such as machine breakdown, changes in part type and volume, tool replacement, raw material and other short interruptions are effective on the desired performance of a manufacturing system. This problem is critical due to its impacts on the capacity of the system (Stoop and Wiers, 1996). Researches on the automated manufacturing systems imply that the machine failure is the major problem in analyzing system performance in comparison with other factors like raw material, equipment, software and workers (Sanchez, 1994). Therefore, reliability considerations should be taken into account for manufacturing system analysis. Researchers who studied this problem include (Hilderbrant, 1980; Kimemia, 1982; Liberopulos, 1993; Viswanadham and Narahari, 1992; Vinod, 1983; Vinod and Solberg, 1984; Choiand Lee, 1998).

Since the manufacturing systems experience different failure states, therefore considering these states in modeling a reliability problem is of importance. The best way for considering system states in modeling is to employ Markovian property. Reibman (1990) stated the problem in estimating the probability of failure in different state is vital for reliability computations.

The increasing demand for the reliability assessment in manufacturing systems under several random parameters has been investigated by several approaches facilitating the computations of probability estimations. According to the following brief literature review, studies to compare Markovian and neural networks are few. Especially, modeling the reliability of an advanced manufacturing system considering AGVs is also rare.

The improvement of safety in the process industries is related to assessment and reduction of risk in a cost-effective manner. Kančev and Čepin(2011) addressed the trade-off between risk and cost related to standby safety systems. An age-dependent unavailability model that integrated the effects of the test and maintenance (T&M) activities as well as component ageing was developed and represented the basis for calculating risk. The repair "same-as-new" process was considered regarding the T&M activities. Costs were expressed as a function of the selected risk measure. The time-averaged function of the selected risk measure was obtained from probabilistic safety assessment, i.e. the fault tree analysis. This function was further extended with inclusion of additional parameters related to T&M activities as well as ageing parameters related to component ageing. In that

10.1 Summary 149

sense, a new model of system unavailability, incorporating component ageing and T&M costs, was presented. The testing strategy was also addressed. Sequential and staggered testings were compared. The developed approach was applied on a standard safety system in nuclear power plant although the method was applicable to standby safety systems that were tested and maintained in other industries as well.

The problem of selecting a suitable maintenance policy for repairable systems and for a finite time period was presented by Marquez and Heguedas (2002). Since the late seventies, examples of models assessing corrective and preventive maintenance policies over an equipment life cycle exist in the literature. However, there are not too many contributions regarding real implementation of these models in the industry, considering realistic timeframes and for repairable systems. Modeling this problem requires normally the representation of different corrective and/or preventive actions that could take place at different moments, driving the equipment to different states with different hazard rates. An approach to pattern the system under finite periods of time has been the utilization of semi-Markovian probabilistic models, allowing later a maintenance policy optimization using dynamic programming. These models are very flexible to represent a given system, but they are also complex and therefore very difficult to handle when the number of the system possible states increases. Marquez and Heguedas (2002) explored the trade-off between flexibility and complexity of these models, and presented a comparison in terms of model data requirements versus potential benefits obtained with the model.

In generalized renewal process (GRP) reliability analysis for repairable systems, Monte Carlo (MC) simulation method instead of numerical method is often used to estimate model parameters because of the complexity and the difficulty of developing a mathematically tractable probabilistic model. Wang and Yang (2012) proposed a nonlinear programming approach to estimate restoration factor for the Kijima type GRP model I, as well as the model II based on the conditional Weibull distribution for repairable systems, using negative loglikelihood as an objective function and adding inequality constraints to model parameters. The method minimized the negative log-likelihood directly, and avoided solving the complex system of equations. Three real and different types of field failure data sets with time truncation for NC machine tools were analyzed by the proposed numerical method. The sampling formulas of failure times for the GRP models I and II were derived and the effectiveness of the proposed method was validated with MC simulation method.

Ke et al. (2013) considered a multi-repairmen problem comprising of M operating machines with W warm standbys (spares). Both operating and warm standby machines were subject to failures. With a coverage probability c, a failed unit was immediately detected and attended by one of R repairmen if available. If the failed unit was not detected with probability 1−c, the system entered an unsafe state and must be cleared by a reboot action. The repairmen were also subject to failures which result in service (repair) interruptions. The failed repairman resumed service after a random period of time. In addition, the repair rate depended on number of failed machines. The entire system was modeled as a finite-state Markov chain and its steady state distribution was obtained by a recursive matrix approach. The major performance measures were evaluated based on this distribution. Under a cost structure, the authors proposed to use the Quasi-Newton method and probabilistic global search Lausanne method to search for the global optimal system parameters.

Nowadays VoIP has become an evolutionary technology in telecommunications. Hence it is very important to study and enhance its dependability attributes. An analytical dependability model for VoIP was proposed by Gupta and Dharmaraja (2011). The study was focused on analyzing the combined effects of resource degradation and security breaches on the Quality of Service (QoS) of VoIP, to enhance its overall dependability. As a preventive maintenance policy to prevent or postpone software failures which cause resource degradation, software rejuvenation was adopted. The dependability model was analyzed using semi-Markov process which captures the effects of non-Markovian nature of the time spent at various states of the system. The steady-state as well as the time-dependent analysis of the dependability model was presented.

Zhou et al. (2012) presented a maintenance optimization method for a multistate series-parallel system considering economic dependence and state-dependent inspection intervals. The objective function considered in the chapter was the average revenue per unit time calculated based on the semi-regenerative theory and the universal generating function (UGF). A new algorithm using the stochastic ordering was also developed in the chapter to reduce the search space of maintenance strategies and to enhance the efficiency of optimization algorithms. A numerical simulation was presented in the study to evaluate the efficiency of the proposed maintenance strategy and optimization algorithms.

Reliability assessment for hard disk drives (HDDs) is important yet difficult for manufacturers. Motivated by the fact that the particle accumulation in the HDDs, which accounts for most HDD catastrophic failures, is contributed from the internal and external sources, a counting process with two arrival sources was proposed by Ye et al. (2013) to model the particle cumulative process in HDDs. The model successfully explained the collapse of traditional ALT approaches for accelerated life test data. Parameter estimation and hypothesis tests for the model were developed and illustrated with real data from a HDD test. A simulation study was conducted to examine the accuracy of large sample normal approximations that were used to test existence of the internal and external sources.

An R out of N repairable system consisting of N independent components is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R − 1. A failed component is sent to a repair facility having several repairmen. Life times of working components are i.i.d random variables having an exponential distribution. Repair times are i.i.d random variables having a phase type distribution. Both cold and warm stand-by systems are considered. Barron et al. (2006) presented an algorithm deriving recursively in the number of repairmen the generator of the Markov process that governs the process. Then they derived formulas for the point

availability, the limiting availability, the distribution of the down time and the up time. Numerical examples were given for various repair time distributions. The numerical examples showed that the availability is not very sensitive to the repair time distribution while the mean up time and the mean down time might be very sensitive to the repair time distributions.

According to the brief reviewed literature, studies to compare Markovian and neural networks are few. Especially, modeling the reliability of an advanced manufacturing system considering AGVs is also rare.

10.2 Statement of the Problem

Here, a jobshop manufacturing system having multiple AGVs for material handling purpose is considered. In each shop several machines perform the part processing according to a process plan. To transfer the parts among different shops AGVs are employed. The reliability of the whole manufacturing system is concerned with the reliability of the machines in shops and the reliability of the AGVs. The failure of the machines and AGVs could be considered in different states. The failure causes for machines are:

• Amateur operator
• Equipment deficiency
• Inappropriate part specifications

Also, the failures of AGVs are due to:

• Carrier overload
• Guide path fracture

Using Markovian property, we can configure the transition diagram and the corresponding matrix.

The result of the Markovian process is the failure probability for machines and AGVs. These probabilities are applied in reliability computations. For reliability, first we conceptualize different scenarios exist in the proposed manufacturing system. The shops are in parallel since the parts are disseminated through the system according to the process plan. The sequence of machines in a shop may be important or not, i.e., the part processing in a shop should be performed sequentially on the machines or the sequence is not important and parallel machining is possible. Therefore, two separate cases of series and parallel should be modeled. AGVs are in series since if one AGV break down then the whole system should wait until the AGV is repaired or taken out of the system.

The aim of the decision maker is to maximize the performance of the whole system. To achieve the aim, two objectives namely maximizing the total reliability of machines in shops in the whole jobshop system and the maximizing the total reliability of the AGVs, should be investigated. Also, for the economic viewpoint of the system performance the third objective is to minimize the total repair cost in the system. As a unit (machine or AGV) in the system is broken down, the repair should be performed on it for preparing it to function.

The aims of conducting this study are:

• Developing a reliability assessment methodology for AGV based manufacturing systems
• Analyzing and including fault sources in machine-AGV state modeling in manufacturing systems
• Markovian modeling for reliability assessment of a machine-AGV manufacturing system
• Comparing Markovian reliability assessment with the neural network method

It is necessary to incorporate reliability into the model to ensure the level of service for each machine in each shop and the AGVs. For modeling reliability, the approach of Ball and Lin (1993) is adopted and further extended.

The reliability is defined as the probability that the system works until time t. If a machine in a shop is broken down, it can be regarded as a failure. A desired level of reliability can be achieved by limiting the failure probabilities. This approach for handling reliability is called the method of chance constraints in the context of mathematical programming. The use of chance constraints in vehicle routing problem was illustrated in Stewart and Golden (1983). Carbone (1974) used chance constraints for selecting multiple facilities under normally distributed demand. The model minimized an upper bound on the total demand-weighted distance while ensuring that constraint was satisfied with specified chance or probability. Shiode and Drezner (2003) used a similar approach in a competitive location problem on a tree network.

It is assumed that the reliability of each machine type and the AGV are independently according to Exponential processes. Also, J is total types of machines, i.e., drilling machines, turning machines, bending machines show three machine types. We discuss the reliability based model as follows:

R (t) ^j : The probability that machine type ^j th works until time t.

$$R(t){\textit{system}} = \begin{cases} \left(1 - \prod{j=1}^{J} (1 - Rj(t))\right), & \text{when machines in each shop are in parallel case} \ \left(\prod{j=1}^{J} R_j(t)\right), & \text{when machines in each shop are in series case} \end{cases}$$ (1)

In our proposed problem, AGVs are series and the machine types in each shop may be in parallel or series cases and the shops are parallel, i.e., a composite system is configured. Therefore, the reliability of the system is as follows:

$$\left(1 - \prod_{j=1}^{J} (1 - R_j(t))\right) \ge \alpha, \tag{2}$$

owhere α is the lower bound for a desirable reliability of the system until time t. As previously assumed, the reliability of each machine type and AGV are independently according to Exponential distribution:

$$R{j}(t) = e^{\frac{-t}{\theta{j}}},\tag{3}$$

Where ^{θ j} is the exponential parameter for machine type or AGV breakdown. Then,

$$\left(1 - \prod_{j=1}^{J} \left(1 - e^{\frac{-t}{\theta_j}}\right)\right) \ge \alpha . \tag{4}$$

It is obvious that to obtain a higher level of reliability, more cost is incurred to the system. Hence, a cost function (C (t) ^j ) is defined to keep machine type j th reliable until time t. For the whole system we have:

$$\sum_{j=1}^{J} C_j(t) \,. \tag{5}$$

10.3 Mathematical Model

In this section, we construct the proposed failure state diagrams and matrices for machines and AGVs' using Markov system, separately. A Markov system is a system that can be in one of several (numbered) states, and can pass from one state to another each time step according to fixed probabilities. If a Markov system is in state i, there is a fixed probability, pij, of it going into state j the next time step, and pij is called a transition probability. A Markov system can be illustrated by means of a state transition diagram, which is a diagram showing all the states and transition probabilities. The entries in each row add up to 1.

First, we configure the machines' state diagram. As stated before, the machines may be broken down in three states, namely, (A) amateur operator, (B) equipment deficiency, and (C) inappropriate part specifications. Note that, the states refer to the break down states causes, i.e., the machine is working or it is broken down due to the failure states such as amateur operator, equipment deficiency, and inappropriate part specifications. In another word, since we are modeling the reliability of the system considering different state changes, it is common in Markovian computations to monitor the state transition while all are the causes of breakdown. The state transition diagram for machines is shown in Figure 1.

Fig. 1 The state transition diagram for machines

As a result the corresponding transition matrix Pij is,

$$P_{ij} = \begin{bmatrix} 1 - \alpha - \varepsilon & \alpha & \varepsilon \ \beta & 1 - \beta - \gamma & \gamma \ v & \delta & 1 - \delta - \nu \end{bmatrix}.$$ (6)

where ^α , β , γ , δ ,^ε , and ν are the transition probability from the three states given in Figure 1. Using the probability transition matrix and the limiting probability we obtain each state's occurrence probability as follows.

$$\pi{j} = \sum{i=1}^{3} \pi{i} p{ij}, \text{ for } j=1,2,3.$$ (7)

$$\sum{j=1}^{3} \pi{j} = 1. \tag{8}$$

Using these probabilities, we can compute the reliability of each state helping us to assess the total reliability of the system.

We also can compute the long run probability for each state using steady state distribution given below.

$$\begin{bmatrix} A & B & C \end{bmatrix} \begin{bmatrix} 1 - \alpha - \varepsilon & \alpha & \varepsilon \ \beta & 1 - \beta - \gamma & \gamma \ v & \delta & 1 - \delta - v \end{bmatrix} = \begin{bmatrix} A & B & C \end{bmatrix}, \tag{9}$$

having A+B+C=1.

The same computations exist for AGVs different failure state, while we stated 2 states, i.e., we have two state probability and a 2× 2 transition matrix.

Now, for reliability we have,

$$R(t)=1-F(t), (10)$$

where, F(t) is the failure probability computed above as states' probabilities. Note that, we can compute the reliability in two cases, first for current state, and second for steady state. The numerical comparison of the two could be interesting.

Having the current state of the system by Markovian model and by the means of neural network, we can compute the steady state probabilities. Next, we review the artificial neural network and the backpropagation neural network for our proposed work. The reason is to find the difference in accuracy of the two methods and determine the most effective one. It is obvious that the neural network can be more efficient due to using past data in training stage.

The aim to compute the steady state probability and reliability is to obtain an estimation of the system availability for long run planning horizon. Therefore, it is significant for a decision maker to determine steady state reliability using the corresponding probability, accurately.

Artificial Neural Network

Neural networks are being widely used in many fields of study. This could be attributed to the fact that these networks attempt to model the capabilities of human brains. Since the last decade, neural networks have been used as a theoretically sound alternative to traditional statistical models. Although neural networks (NNs) originated in mathematical neurobiology, the rather simplified practical models currently in use have moved steadily towards the field of statistics. A number of researchers have illustrated the connection of neural networks to traditional statistical models. For example, Gallinari et al. (1991) presented analytical results establishing a link between discriminant analysis and multilayer perceptrons (MLP) used for classification problems. Cheng and Titterington (1994) made a detailed analysis and comparison of various neural network models with traditional statistical models. They showed strong associations of the feed-forward neural networks with discriminant analysis and regression models, and unsupervised networks such as self-organizing neural networks with clustering. Neural networks are being used in areas of prediction and classification, areas where regression models and related statistical techniques have traditionally been used. Ripley (1994) discusses the statistical aspects of neural networks and classifies neural networks as one of a class of flexible nonlinear regression models. Warner and Misra (1996) presented a comparison between regression analysis and neural network computation in terms of notation and implementation. They also discuss when it would be advantageous to use a neural network model in place of a parametric regression model, as well as some of the difficulties in implementation. Vach et al. (1996) presented a comparison

between feed-forward neural networks and the logistic regression. The conceptual similarities and discrepancies between the two methods are also analyzed.

Artificial neural networks have been applied successfully to many manufacturing and engineering areas. Zhengrong et al. (1996) used quadratic regression to assess the results of neural network for improving the efficiency of fermentation process development. The results show that different sizes of neural nets within a certain range give an equally good prediction by using the ''stopping training" technique, while quadratic regressions are sensitive to the size of the data sets. Smith and Mason (1997) mentioned that regression and neural network modeling methods have become two competing empirical model-building methods. They compared the predictive capabilities of NNs and regression methods in manufacturing cost estimation problems.

The Backpropagation Neural Network

The backpropagation algorithm trains a given feed-forward multilayer neural network for a given set of input patterns with known classifications. When each entry of the sample set is presented to the network, the network examines its output response to the sample input pattern. The output response is then compared to the known and desired output and the error value is calculated. Based on the error, the connection weights are adjusted. The backpropagation algorithm is based on Widrow-Hoff delta learning rule in which the weight adjustment is done through mean square error of the output response to the sample input. The general steps of backpropagation are given below.

• 1. Propagate inputs forward in the usual way, i.e., all outputs are computed using sigmoid thresholding of the inner product of the corresponding weight and input vectors. All outputs at stage n are connected to all the inputs at stage n+1
• 2. Propagate the errors backwards by apportioning them to each unit according to the amount of the error the unit is responsible for.

We now discuss how to develop the stochastic backpropagation algorithm for the general case. The following notations and definitions are needed:

j x
G : input vector for unit j (xji = ith input to the jth unit) 
wj
 G : weight vector for unit j (wji = weight on xji) 
j j j z w x G G = . : the weighted sum of inputs for unit j
oj : output of unit j ( ( ) j j o = σ z ) 
tj : target for unit j
Downstream(j) : set of units whose immediate inputs include the output of j
Output : Set of output units in the final layer.

Since we update after each training example, we can simplify the notation somewhat by assuming that the training set consists of exactly one example and so the error can simply be denoted by E.

We want to calculate wji E ∂ ∂ corresponding to each input weight wji of each

output unit j. Note first that since zj is a function of wji regardless of where in the network unit j is located,

$$\frac{\partial E}{\partial w{ji}} = \frac{\partial E}{\partial z{j}} \cdot \frac{\partial z{j}}{\partial w{ji}} = \frac{\partial E}{\partial z{j}} \cdot x{ji}, \tag{11}$$

Furthermore, j z E ∂ ^∂ is the same regardless of which input weight of unit j we are

trying to update. So, we denote this quantity by ^{j δ} .

Consider the case when j is an output unit. We know that

$$E = \frac{1}{2} \sum_{k \in \text{Outputs}} (t_k - \sigma(z_k))^2$$ (12)

Since the outputs of all units k ≠ j are independent of wji, we can then drop the summation and consider just the contribution to E by j and we call it ^{j δ} :

$$\delta{j} = \frac{\partial E}{\partial z{j}} = \frac{\partial}{\partial z{j}} \frac{1}{2} (t{j} - o{j})^{2} = -(t{j} - o{j}) \frac{\partial o{j}}{\partial z{j}} = -(t{j} - o{j}) \frac{\partial}{\partial z{j}} \sigma(z_{j})$$

$$= -(t{j} - o{j}) (1 - \sigma(z{j})) \sigma(z{j}) = -(t{j} - o{j}) (1 - o{j}) o{j}.$$ (13)

Thus,

$$\Delta w{ji} = -\eta \frac{\partial E}{\partial w{ji}} = \eta \deltaj x{ji}. \tag{14}$$

Now, consider the case when j is a hidden unit. Like before, we make the following two important observations:

• 1. For each unit k downstream from j, zk is a function of zj.
• 2. The contribution to error by all unitsl ≠ j , in the same layer as j, is independent of wji.

We want to calculate wji E ∂ ∂ for each input weight wji for each hidden unit j. Note

that wji influences just zj which influences oj which influences zk, ∀k ∈Downstream( j), each of which influences E. So, we can write,

$$\frac{\partial E}{\partial w{ji}} = \sum{k \in Downstream(j)} \frac{\partial E}{\partial z_k} \cdot \frac{\partial z_k}{\partial z_j} \cdot \frac{\partial o_j}{\partial z_j} \cdot \frac{\partial zj}{\partial w{ji}} = \sum_{k \in Downstream(j)} \frac{\partial E}{\partial z_k} \cdot \frac{\partial z_k}{\partial o_j} \cdot \frac{\partial o_j}{\partial zj} \cdot x{ji}$$ (15)

Again, note that all the terms except xji in (15) are the same regardless of which input weight of unit j we are trying to update. Like before, we denote this common quantity by ^{j δ} . Also, note that ^k k z ^{E δ} ∂ ^∂ , kj j ^k w o z ∂ ∂ and (1 ) ^{j j} j ^j o o z ^{o = − ∂ ∂} . Substituting them in (13),

$$\delta{j} = \sum{k \in Downstream(j)} \frac{\partial E}{\partial z{k}} \cdot \frac{\partial z{k}}{\partial o{j}} \cdot \frac{\partial o{j}}{\partial z{j}} = \sum{k \in Downstream(j)} \delta{k} \cdot w{kj} \cdot o{j} (1 - o{j}),$$ (16)

we obtain:

$$\delta_k = o_j (1 - oj) \sum{k \in Downstream(j)} \deltak . w{kj}. \tag{17}$$

Fig. 2 A configuration of the proposed neural network

To adapt the backpropagation algorithm on our proposed model, consider the failure causes for machines and AGVs as inputs and the current state failure probability of machines and AGVs as outputs. We train the network collecting data in different time periods and compute the importance weight for each input resulting the corresponding output. A configuration of the proposed neural network is shown in Figure 2.

We are now in a position to state the backpropagation algorithm formally.

Algorithm 1: Formal statement of stochastic backpropagation.

(Training examples, η, ni, nh, no)

Each training example is of the form ^{x t G G} , , where x G is the input vector and t G is the target vector, ηis the learning rate (e.g., 0.05), ni, nh and no are the number of input, hidden and output nodes, respectively. Input from unit i to unit j is denoted by xji and its weight is denoted by wji. Create a feed-forward network with ni inputs, nh hidden units, and no output units.

Initialize all the weights to small random values (e.g., between -0.05 and 0.05).

While termination condition is not met Do

For each training example ^{x t G G} , ,

• 1. Input the instance x G and compute the output ou of every unit.
• 2. For each output unit k, calculate

$$\delta_k = o_k (1 - o_k)(t_k - o_k). \tag{18}$$

1. For each hidden unit h, calculate

$$\delta_h = o_h (1 - oh) \sum{k \in Downstream(h)} \deltak . w{kh} . \tag{19}$$

1. Update each network weight wji as follows:

$$w{ji} \leftarrow w{ji} + \Delta w_{ji} \,. \tag{20}$$

where,

$$\Delta w_{ji} = \eta \deltaj x{ji} \,. \tag{21}$$

This way, we can compare the performance of backpropagation neural network and limiting distribution model for computing the steady state probabilities using the current state probabilities.

Mathematical Optimization

Here, the mathematical optimization model is given. As stated before we aim to maximize the total reliability of machines in shops in the whole jobshop system and maximize the total reliability of the AGVs. Since the reliability model is stochastic, one may think about simulation study. But considering multipleobjectives and especially including cost factors in the form of a composite mathematical function is difficult and requires tiring and complicated simulation efforts. Also, as we will present further, we considered several 0/1 integer variables which are easier to be modelled mathematically.

Maximizing Total Reliability of Machines

The following mathematical notations are employed to model this maximization problem:

Mathematical notations:

$$k & \text{Index for machines,} & k=1,...,K. \ l & \text{Index for jobs,} & l=1,...,L. \ m & \text{Index for shops,} & m=1,...,M. \ Rk & \text{Reliability of machine } k. \ \tau{kl} = \begin{cases} 1 & \text{if machine } k \text{ process job } l. \ 0 & \text{otherwise.} \end{cases}$$

$$\phi_{km} = \begin{cases} 1 & \text{if machine } k \text{ in shop } m \text{ is chosen.} \ 0 & \text{otherwise.} \end{cases}$$

The mathematical model:

$$Max \sum{m} \sum{k} (Rk \cdot \varphi{km}), \qquad (22)$$

s.t.

$$\sum{k} \varphi{km} \tau_{kl} = 1, \qquad \forall l, m, \tag{23}$$

$$\varphi_{km} \in {0,1}, \ \forall k, m. \tag{24}$$

Maximizing Total Reliability of AGVs

The following mathematical notations are employed to model this maximization problem:

Mathematical notations:

n Index for AGVs, $$n=1,...,N$$ .
$R_n$ Reliability of AGV $n$ .

$$\varsigma_{nm} = \begin{cases} 1 & \text{if AGV } n \text{ can service shop } m. \ 0 & \text{otherwise.} \end{cases}$$

$$\zeta_n = \begin{cases} 1 & \text{if AGV } n \text{ is chosen.} \ 0 & \text{otherwise.} \end{cases}$$

The mathematical model:

$$Max \sum{n} R{n} \cdot \zeta_{n} , \qquad (25)$$

s.t.

$$\sum{n} \zeta{n} \zeta_{nm} = 1, \qquad \forall m, \tag{26}$$

$$\zeta_n \in {0,1}, \ \forall n.$$ (27)

As a result a multi-objective mathematical model is configured as follows:

$$Max \sum{m} \sum{k} (Rk \cdot \varphi{km}), \qquad (28)$$

$$Max \sum{n} R{n} \zeta_{n}, \qquad (29)$$

s.t.

$$\sum{k} \varphi{km} \tau_{kl} = 1, \qquad \forall l, m, \tag{30}$$

$$\sum{n} \zeta{n} \zeta_{nm} = 1, \qquad \forall m, \tag{31}$$

$$\sum{k} \varphi{km} = 1, \quad \forall m, \tag{32}$$

$$\zeta_n \in {0,1}, \ \forall n. \tag{33}$$

$$\varphi_{km} \in {0,1}, \ \forall k, m. \tag{34}$$

Next, an approach to optimize the proposed multi-objective model is given. We use objectives weighing method to integrated and optimize the model.

Analytic Hierarchy Process (AHP) for Multi-objective Optimization

To weight the objectives, we take a multi-criteria decision-making approach. Multi-criteria decision-making (MCDM), dealing primarily with problems of evaluation or selection, is a rapidly developing area in operations research and management science. AHP is a technique of considering data or information for a decision in a systematic manner. It is mainly concerned with a way of solving decision problems with uncertainties in multiple criteria characterization. It is based on three principles: constructing the hierarchy, priority setting, and logical consistency. We apply AHP to weight the objectives.

Construction of the Hierarchy

A complicated decision problem, composed of various attributes of an objective, is structured and decomposed into sub-problems (sub-objectives, criteria, alternatives, etc.), within a hierarchy.

Priority Setting

The relative "priority" given to each element in the hierarchy is determined by pairwise comparisons of the contributions of elements at a lower level in terms of the criteria (or elements) with a causal relationship. In AHP, multiple paired comparisons are based on a standardized comparison scale of nine levels (see table 1).

| | Table 1 Scale of relative importance | |--|--------------------------------------| | | |

| Intensity of importance | Definition of importance | | | |-------------------------|-----------------------------|--|--| | 1 | Equal | | | | 2 | Weak | | | | 3 | Moderate | | | | 4 | Moderate plus | | | | 5 | Strong | | | | 6 | Strong plus | | | | 7 | Very strong or demonstrated | | | | 8 | Very, very strong | | | | 9 | Extreme | | |

Let { }ⁿ C c ,...,c = 1 be the set of criteria. The result of the pair-wise comparisons on n criteria can be summarized in an n × n evaluation matrix A in which every element aij is the quotient of weights of the criteria, as shown below:

$$A = (a_{ij}), i, j = 1, ..., n.$$ (35)

The relative priorities are given by the eigenvector (w) corresponding to the largest eigenvalue

$$(\lambda{\max})$$ as:
$Aw = \lambda{\max}w$ . (36)

When pair-wise comparisons are completely consistent, the matrix A has rank 1 and λmax = n. In that case, weights can be obtained by normalizing any of the rows or columns of A.

The procedure described above is repeated for all subsystems in the hierarchy. In order to synthesize the various priority vectors, these vectors are weighted with the global priority of the parent criteria and synthesized. This process starts at the top of the hierarchy. As a result, the overall relative priorities to be given to the lowest level elements are obtained. These overall, relative priorities indicate the degree to which the alternatives contribute to the objective. These priorities represent a synthesis of the local priorities, and reflect an evaluation process that permits integration of the perspectives of the various stakeholders involved.

Consistency Check

A measure of consistency of the given pair-wise comparison is needed. The consistency is defined by the relation between the entries of A; that is, we say A is consistent if aik= aij · ajk, for all i,j,k. The consistency index (CI) is:

$$CI = \frac{(\lambda_{\text{max}} - n)}{(n - 1)}.\tag{37}$$

The final consistency ratio (CR), on the basis of which one can conclude whether the evaluations are sufficiently consistent, is calculated to be the ratio of the CI and the random consistency index (RI):

$$CR = \frac{CI}{RI}. (38)$$

The value 0.1 is the accepted upper limit for CR. If the final consistency ratio exceeds this value, the evaluation procedure needs to be repeated to improve consistency. The measurement of consistency can be used to evaluate the consistency of decision-makers as well as the consistency of all the hierarchies.

We are now ready to give an algorithm for computing objective weights using the AHP. The following notations and definitions are used.

n: number of criteria

i: number of objectives

p: index for objectives, *p=*1or 2

d: index for criteria, 1≤ d ≤ D

Rpd : the weight of pth item with respect to dth criterion

wd : the weight of dth criterion

Algorithm 2: OWAHP (compute objective weights using the AHP)

Step 1: Define the decision problem and the goal.

Step 2: Structure the hierarchy from the top through the intermediate to the lowest level.

Step 3: Construct the objective-criteria matrix using steps 4 to 8 using the AHP. (Steps 4 to 6 are performed for all levels in the hierarchy.)

Step 4: Construct pair-wise comparison matrices for each of the lower levels for each element in the level immediately above by using a relative scale measurement. The decision-maker has the option of expressing his or her intensity of preference on a nine-point scale. If two criteria are of equal importance, a value of 1 is set for the corresponding component in the comparison matrix, while a 9 indicates an absolute importance of one criterion over the other (table 1 shows the measurement scale).

Step 5: Compute the largest eigenvalue by the relative weights of the criteria and the sum taken over all weighted eigenvector entries corresponding to those in the next lower level of the hierarchy.

Analyze pair-wise comparison data using the eigenvalue technique. Using these pair-wise comparisons, estimate the objectives. The eigenvector of the largest eigenvalue of matrix A constitutes the estimation of relative importance of the attributes.

Step 6:Construct the consistency check and perform consequence weights analysis as follows:

$$A = (a{ij}) = \begin{bmatrix} 1 & w{1} / & \dots & w{1} / & \ w{2} / & 1 & \dots & w{2} / & \ w{1} & \vdots & \ddots & \ddots & \ w{n} / & w{n} / & w_{2} & & 1 \end{bmatrix}.$$

Note that if the matrix A is consistent (that is, aik= aij · ajk, for all i jk n , , 1, 2, ..., = ), then we have (the weights are already known),

$$a_{ij} = \frac{w_i}{w_j}, \quad i, j = 1, 2, ..., n.$$

(39)

If the pair-wise comparisons do not include any inconsistencies, then λmax= n. The more consistent the comparisons are, the closer the value of computed λmax

is to n. Set the consistency index (CI), which measures the inconsistencies of pairwise comparisons, to be:

$$CI = \frac{\left(\lambda_{\max} - n\right)}{\left(n - 1\right)},\,$$

and let the consistency ratio (CR) be:

$$CR = 100 \left( \frac{CI}{RI} \right),$$

where n is the number of columns in A and RI is the random index, being the average of the CI obtained from a large number of randomly generated matrices.

Note that RI depends on the order of the matrix, and a CR value of 10% or less is considered acceptable.

Step 7: Form the objective-criteria matrix as specified in table 2:

Table 2 The objective-criteria matrix

	C1	C2	…	Cd
objective 1	R11	R12	…	R1d
objective 2	R21	R22	…	R2d

Step 8: As a result, configure the pair-wise comparison for criteria-criteria matrix as in table 3:

Table 3 The criteria-criteria pair-wise comparison matrix

	C1	C2	…	Cd	w d
Criteria 1	1	a12	…	a1d	w1
Criteria 2	1/a12	1	…	a2d	w2
#	#	#	#	#	#
Criteria d	1/a1d	1/a2d	…	1	wd

The wd are gained by a normalization process. The wd are the weights for criteria.

Step 9: Compute the overall weights for the objectives, using tables 2 and 3, as follows:

$$\psi$$ = Total weight for objective $1 = R_{11} \times w1 + R{12} \times w2 + ... + R{1d} \times wd$ ,
$\psi'$ = Total weight for objective $2 = R{21} \times w1 + R{22} \times w2 + ... + R{2d} \times w_d$ , (40)

where ψ ψ+ =' 1. Thus the integrated objective function is formed as:

$$Max\left(\psi.\sum{m}\sum{k}\left(R{k}.\varphi{km}\right)+\psi'.\sum{n}\left(R{n}.\zeta_{n}\right)\right).$$

10.4 Conclusions

We proposed a Markovian model for flexible manufacturing systems (FMSs). The model considered two features of automated flexible manufacturing systems equipped with automated guided vehicle (AGV) namely, the reliability of machines and the reliability of AGVs in a multiple AGV jobshop manufacturing system. We made use of current state transition matrix for the failure of the machines and AGVs in different states. Therefore, a Markovian model was proposed for reliability assessment. Also, for steady state probability computations, the limiting theorem was compared with adapted backpropagation neural network showing neural network's effectiveness. Using the reliabilities, we worked out an optimization mathematical model. The optimization objectives in the proposed model were maximizing the total reliability of machines in shops in the whole jobshop system and maximizing the total reliability of the AGVs. The computational results illustrated the applicability of our proposed model. A strategic viewpoint of such computations was to enable the management to control the failures of AGVs and Machines to satisfy the optimization purposes.

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Chapter 11

Producer's Behavior Analysis for AGV System

11.1 Summary

Traditional manufacturing has relied on dedicated mass-production systems to achieve high production volumes at low costs. As living standards improve and the demands for new consumer goods rise, manufacturing flexibility gains prominence as a strategic tool for rapidly changing markets. Flexibility, however, cannot be properly incorporated in the decision-making process if it is not well defined and measured in a quantitative manner. Flexibility in its most rudimentary sense is the ability of a manufacturing system to respond to changes and uncertainties associated with the production process (Miettinen et al., 2010; Kumar and Sridharan, 2009; Das et al., 2009). A comprehensive classification of eight flexibility types was proposed in Browne et al. (1984). Resource and system flexibilities were examined in Slack (1987), whereas global measures for flexible manufacturing systems (FMSs) were defined in Gupta and Buzacott (1989). Routing flexibility based on information theoretic concepts was examined by Yao and Pei (1990) and Kumar (1987). Flexibility measures for one machine, a group of machines, and the whole industry were presented in Brill and Mandelbaum (1989), involving appropriate weights and machine efficiencies in carrying out sets of tasks.

Artificial Intelligence (AI) methods seem appropriate in most practical situations where numerical data are not readily available and linguistic variables are more amenable to handling imprecise knowledge (Gen et al., 2009; Shin et al., 2009). The flexibility of competing systems can be ranked appropriately using an algorithmic approach (Abdel-Malek and Wolf, 1991) or a decision support system (Suresh, 1991) based on performance and economic criteria. According to Turban and Aronson (1998), a Decision Support System (DSS) is a computer-based information system that combines models and data in an attempt to solve nonstructured problems with extensive user involvement. They believe that an Expert System (ES) is a computer system that applies reasoning methodologies on knowledge to render advice or recommendations much like a human expert. When expert systems technology was first applied to decision- making problems, it fell short in several respects. Early expert systems were rule-based. They were not

capable of handling the classical DSS functions being more computational than logical. Recently, artificial intelligence researchers have seen the necessity of using statistical techniques to build intelligent decision support systems (Nolan, 1998; Weiss and Kulikowski, 1991). Examples of such statistical techniques include fuzzy logic, neural networks, rule induction and various Bayesian techniques. Turban and Aronson (1998) believed that although uncertainty was widespread in the real world, its treatment in the practical world of artificial intelligence was very limited.

Manufacturing costs for products are very crucial in decision making and strategic planning. And with respect to cost estimation, research and development departments in the past could only estimate the final product's total cost. Moreover, rules of thumb of the engineers are often applied as the cost estimation benchmarks, making the results controversial in terms of accuracy (Mostafaee et al., 2010; Eklin et al., 2009; Qian and Ben-Arieh, 2008; Wang, 2007; Verlinden et al., 2008; Wang, 2007). Although calculation by cost model has the advantage of being timely, only representative values exclusive of indirect tasks cost and raw materials cost are calculated resulting in an inadequate accuracy of the estimation. Niazi et al. (2006) pointed out that Backpropagation Network (BPN) could be applied for training to deduce unprecedented problems by accumulated knowledge and information. Specially, it could find out solutions in uncertain circumstances and have satisfying results in dealing with non-linear problems. Therefore, BPN is the most popular neural network models being applied and it fits the nature of product cost estimation the best. McKim (1993) discussed applying BPN in cost estimation projects. Finally, by integrating a user interface, the demand of a fast response cost estimation model was met at initial product development stage in his work. Zhang et al. (1996) categorized cost estimation techniques into traditional detailed breakdown, simplified-breakdown, group-technology-based, regressionbased and activity-based cost approaches. Ben- Arieh and Qian (2003) divided cost estimation models into intuitive, analogical, parametric and analytical approaches. Shehab and Abdalla (2001) proposed intuitive, parametric, variantbased and generative cost estimating approaches. Cavalieria et al. (2004) proved three cost analyses of analogy-based, parametric and engineering approaches. Niazi et al. (2006) discussed, on the basis of the integrated cost estimation approaches, qualitative and quantitative cost estimation techniques pointing out the key advantages and limitations of each cost estimation technique.

Jobshop is a flexible, scalable and intelligent production planning and control system offering advanced functionality and value in key areas of manufacturing and assembly. The Flexible Jobshop Problem (FJP) is an extension of the classical jobshop problem allowing for an operation to be processed by any machine in a given set. Automated Guided Vehicles (AGVs) reduce the cost of manufacturing and increase efficiency in a manufacturing system. These trailers can be used to move raw materials in line to get them ready for manufacturing (Aized, 2009; Hsueh, 2010). To conceptualize an AGV, it is necessary to understand the fundamentals of Flexible Manufacturing Systems (FMSs).

Here, soft computing techniques are considered for configuring an expert system in the presence of uncertainty. The purpose of this study is to design a model for an AGV based automated manufacturing system. Three main factors accompanied by their sub-factors are chosen and then propose a neural network based approach to analyze the effects of the sub-factors on the main factors. Due to uncertainty of cost elements in the manufacturing environment, fuzzy logic is applied to determine an appropriate range for each of the linguistic variables. Then, using the linguistic variables, the possible fuzzy rules are extracted and using a multiple linear regression model the significant rules are investigated and eliminate inappropriate rules. A sensitivity analysis is associated with the regression method. Then, considering time as a triangular fuzzy number, an aggregation process to combine both the obtained cost and the time parameters is proposed. Then a dynamic program is proposed for finding an optimal path in the proposed manufacturing system.

11.2 Statement of the Problem

Consider a jobshop layout which applies an AGV for material handling. The AGV carries raw material, semi-produced and final products in batch sizes. Due to mounting demands, advancing technology, and rising production capacity, the need for increasingly more shops is mounting over time. The new shops are expected to have more advanced machines. Therefore, more than one shop with the same duty are evolved. The difference among shops having the same duty shows up in the shop's specifications that affect the production cost. As a result, the system would consider a flexible jobshop model where multi shops of the same duty exist and each operation can be processed on any type of machine in any shop. The sequences of jobs are specified and the jobs are assumed to be independent.

The structure of such a problem would configure a network. In this network, the nodes are the shops and the arcs are the flow paths of the AGV to each shop. Shops in each stage are of the same type but have different specifications such as different machine types and equipments, varied operator proficiencies, different rates of defect, etc.. Each flow path for the AGV is associated with a time parameter and also a cost parameter related to each shop. The aim is to find a path for the AGV minimizing an aggregate time and cost objective. Considering the variability of the AGV flow among shops, the time of each flow path of the AGV is a triangular fuzzy number. In each shop, different machines and operators are working. Due to unpredictable events during working times a cost may incur. This cost is inferred from an expert system via a fuzzy neural network. Cost parameters for each shop are considered to be 3: (1) Equipment sensitivity, (2) operator proficiency, and (3) product specifications, each being specified by one of the three levels of low, moderate, and high. A configuration for the proposed problem is presented in Figure 1.

Fig. 1 A configuration of the proposed problem

As stated before, time is considered to be a triangular fuzzy number and cost is inferred from an expert system. In our expert system a backpropagation neural network is considered to estimate the cost factors (outputs) using their corresponding sub-factors (inputs). The backpropagation algorithm trains a given feed-forward multilayer neural network for a given set of input patterns with known classifications. The backpropagation algorithm is based on Widrow-Hoff delta learning rule in which the weight adjustment is done through mean square error of the output response to the sample input (Abdi et al., 1996).

The cost factors and their related sub-factors are shown in Figure 2.

Fig. 2 The cost factors and their related sub-factors.

Using the existing data, the network is trained and then by the resulting pattern one can apply the model to obtain the output with respect to the proposed subfactors. The numerical results would show a lower bound and an upper bound for each cost factor. These bounds are utilized as our cost factors ranges for the fuzzy rule base. Considering the uncertainty in the cost factors due to dynamic changes in neural network inputs, the cost factors in three levels of low, moderate, and high is specified. The problem is to specify the range of these levels; that is, the numerical range of low, moderate or high for each of the cost factors.

11.3 Mathematical Model

An inductive reasoning technique to obtain the appropriate range for the levels is applied. This method is based on an ideal scheme describing the input and output relationships for a well-established data base. This method is called entropy. A key goal of entropy minimization analysis is to determine the quantity of information in a given data set. The entropy of a probability distribution is a measure of the uncertainty of the distribution. This information measure estimates the uncertain range of data using a predetermined inappropriate range to start the process. The higher the prior estimate of the probability for an outcome to occur, the lower will be the information gained by observing it to occur. The entropy on a set of possible outcomes of a trial where exactly one outcome is possible is defined by the sum of probabilities. In other words, the entropy is the expected value of information. For a simple one-dimensional (one uncertain variable) case, let us assume that the probability of the i th sample *wi*to be true is ( ) p wi . If the sample wi is actually observed in the future and discover that it is true, then the following information, I(wi) is gained:

$$I(w_i) = -k \ln p(w_i), \qquad (1)$$

where k is a normalizing parameter. If it is discovered to be false, we still gain some information:

$$I(\overline{w}_i) = -k \ln[1 - p(w_i)]. \tag{2}$$

Thus, the entropy of the inner product of all the samples (N) is:

$$S = -k \sum_{i=1}^{N} [p_i \ln p_i + (1 - p_i) \ln(1 - p_i)],$$ (3)

where *pi*is the probability of the i th sample to be true. Note that S ≥ 0 , because ln x ≤ 0 , for 0 ≤ x ≤ 1 .

The entropy of a rule should be minimized. Minimum of entropy S is associated with all the pi being as close to one or zero as possible, which in turn implies that they have a very high probability of either happening or not happening, respectively. Note in equation (3) that if *pi=*1, then *S=*0.

This result makes sense, since pi is the probability measure of whether a value belongs to a partition or not. The precedence for partitioning the data is explained next.

First, seeking a threshold value for a sample in the ranges [*x*1, x] and [x, *x*2] are assumed. It is denoted that the first region p and the second region q. by moving an imaginary threshold value x between *x*1 and *x*2, entropy is calculated for each value of x.

An entropy with each value of x in the region containing *x*1 and *x*2 is:

$$S(x) = p(x)S_p(x) + q(x)S_q(x),$$ (4)

where,

$$S{p}(x) = -[p{1}(x)\ln p{1}(x) + p{2}(x)\ln p_{2}(x)], \tag{5}$$

$$S{q}(x) = -[q{1}(x)\ln q{1}(x) + q{2}(x)\ln q_{2}(x)], \tag{6}$$

with pk(x) and qk(x) as conditional probabilities that the class k sample is in the regions [*x*1, *x1+x*] and [*x1+x*, *x*2], respectively, p(x) and q(x) are probabilities that all samples are in the regions [*x*1, *x1+x*] and [*x1+x*, *x*2], respectively. Also,

$$p(x) + q(x) = 1. (7)$$

A value of x that gives the minimum entropy is the optimum threshold value. The entropy estimates pk(x), qk(x), p(x), and q(x) are calculated as follows:

$$p_k(x) = \frac{n_k(x) + 1}{n(x) + 1},$$ (8)

$$q{k}(x) = \frac{N{k}(x) + 1}{N(x) + 1},$$ (9)

$$p(x) = \frac{n(x)}{n},\tag{10}$$

$$q(x) = 1 - p(x),$$ (11)

where nk(x) is the number of class k samples located in [*x*1, *x1+x*], n(x) is the total number of samples located in [*x*1, *x1+x*], Nk(x) is the number of class k samples located in [*x1+x*, *x*2], N(x) is the total number of samples located in [*x1+x*, *x*2], and n is the total number of samples in [*x*1, *x*2]. While moving x in the region [*x*1, *x*2], the values of entropy are calculated for each position of x.

This procedure finds the region for the levels of the cost factors. After each update in input data, the ranges are specified using an entropy technique. After determining the regions of the cost factors, the fuzzy rules are then composed which clarify the total cost for each shop. To find the effective rules, a multiple linear regression model is configured using the previous data and identify the regression coefficients. Since we have three cost factors each with three levels of low, moderate and high, then twenty seven (3*3*3) possible rules exist. Thus, using the test of hypothesis, the effective rules are identified.

As stated before, some rules affect the cost of the system. Whether a rule is significant for cost or not should be investigated. One way to survey the effect of independent variables on dependent variable is multiple linear regression model. Therefore, the following equation is considered,

$$u_i = \beta_0 + \beta_1 r_1 + ... + \beta_m r_m + \varepsilon_i,$$ $i = 1,...,n,$ (12)

whereui is the total cost in the i th period, β0 is the intercept, the β ^j are the coefficients for the rj and ^{i ε} is the error term (note that the β ^j , 0 ≤ j ≤ m , depend on the period i, but for simplicity the index i is omitted here). The aim is to identify the β ^j which are not important for the total cost of the proposed system. Here, a two-sided hypothesis testing is applied as follows;

$$\begin{aligned} \boldsymbol{H}_0: \boldsymbol{\beta}_j &= 0; \ \boldsymbol{H}_1: \boldsymbol{\beta}_j &\neq 0. \end{aligned} \tag{13}$$

The test of the null hypothesis H0 against the two sided alternative proceeds in three steps. The first is to compute the standard error of β ^j , ( )^j SE β . The second step is to compute the t-statistic,

$$t = \frac{\beta_j - 0}{SE(\beta_j)},\tag{14}$$

and the third step is to compute the *p-*value, the probability of observing a value of β ^j at least as different from zero as the estimate actually computed ( act β ^j ), assuming that the null hypothesis is correct.

A small value of the *p-*value, say less than 5%, provides evidence against the null hypothesis in the sense that the chance of obtaining a value of β ^j by pure random variation from one sample to the next is less than 5% if, in fact, the null hypothesis is correct. If so, the null hypothesis is rejected at the 5% significance level. Simply, one can reject the null hypothesis at the 5% significance level if

1.96 act t . Therefore, if the hypothesis is accepted, then it is ensued that the corresponding rule is not important and thus is not effective on the cost. The estimation of the coefficients is performed using a regression software.

By these rules the cost of each shop is estimated (arc length in the network). Consequently, a dynamic program is applied to find the optimal path in the proposed jobshop automated manufacturing network. A flowchart for the proposed expert system is presented in Figure 3.

Fig. 3 A proposed expert system

Our decision model is to consider both time and cost parameters, and thus an integration of the two parameters would be required. The integration is to have a weighted sum of cost and time as an arc length in the proposed network. Time is a triangular fuzzy number and cost is considered to be an indirect triangular fuzzy number.

As stated before, time is a triangular fuzzy number, ensued from the experts' knowledge. Brainstorming and expert knowledge versus meta-rule techniques (neural networks, genetic algorithms, etc.) are two common approaches for defining fuzzy rules and membership functions. While available empirical data is requisite for using the second approach, due to unavailability of historical data for cost, we make use of the first approach to obtain the membership functions and fuzzy rules.

In Zadeh's words (Zadeh, 1965), fuzzy logic is a tool for ''Computing with Words''. He stated that the main role of fuzzy logic was to serve as a methodology for computing with words when no other methodology could attain such purpose (Zadeh, 1996, 1999). Triangular membership functions are common (Pedrycz, 1994). Mamdani fuzzy system is being commonly used in the literature (Klir and Yuan, 1995; Zimmermann, 1996). The fuzzy set acquired from aggregation of rules' results will be defuzzified using defuzzification methods like centroid (center of gravity), max membership, mean-max, and weighted average. The centroid method is very popular, in which the ''center of mass'' of the result provides the crisp value. For our problem in which various possible conditions of parameters are stated in forms of fuzzy sets, the Mamdani fuzzy system is utilized, because the fuzzy rules representing the expert knowledge in Mamdani fuzzy systems would consider fuzzy sets.

The input to our Mamdani type fuzzy system (Babuska, 1998) is composed of equipment sensitivity, operator proficiency, and product specifications. For any of the inputs, three linguistic terms of "low", "moderate", and "high" are defined. The output of the system is cost being identified by any of the three linguistic terms, "low", "moderate", or "high". The maximum membership grade of linguistic term ''high'' is 30.

As a result, a triangular fuzzy number as the time and a numerical value as the cost are obtained. We intend to consider an integrated time-cost value as the value of each proposed arc in the network. Time and cost having different scales, it would not be possible to perform basic operations such as addition on their original forms. Thus, to remove the scales, the time and cost values are normalized throughout the network.

As stated, to perform basic operations on the time and cost parameters with their different scales, we need to remove their scales. To do this, two normalization processes are proposed for time and cost, separately. In (15), (16) and (17) below, i is an index to show the node and j is an index to show the stage (shop's type) in the proposed network, *rij*is the value of either time or cost parameter in each node, and *nij*is its corresponding normalized value. Considering time as a triangular fuzzy number, the normal value with a positive view is given by

$$n{ij} = \frac{r{ij} - r_j^{\min}}{r_j^{\max} - r_j^{\min}}, \qquad \forall i, j,$$ (15)

while the normal value with a negative view is given by,

$$n_{ij} = \frac{rj^{\text{max}} - r{ij}}{r_j^{\text{max}} - r_j^{\text{min}}}, \qquad \forall i, j,$$ (16)

where max ^j r and min ^j r are maximum and minimum values in each column of an assumed matrix of time or cost, respectively. Since time in our proposed model is a criterion implicating a negative aspect in decision making, then (16) is chosen for normalizing time in our approach.

Assuming cost as a crisp value, the cost values are normalized as follows:

$$n{ij} = \frac{r{ij}}{\sqrt{\sum{k} r{kj}^2}}, \qquad \forall i, j,$$ (17)

To compute the minimum or maximum value for (15) and (16), comparisons are needed to be made. This means that it is necessary to have a method for ranking and comparing fuzzy numbers. To compare fuzzy numbers, the efficient approach developed by Mahdavi et al. (2009) based on the distance between fuzzy numbers is used. They used the distance function introduced by Sadeghpour Gildeh and Gien (2001). The main advantages of this distance function are the generality of its usage on various fuzzy numbers, and its reliability in distinguishing unequal fuzzy numbers.

Then, a weighing method is considered to make the time and cost appropriate for basic operations.

To weigh the parameters, a multi criteria decision making approach is taken. Multi-Criteria Decision Making (MCDM), dealing primarily with problems of evaluation or selection (Keeney and Raiffa, 1976; Teng, 2002), is a rapidly developing area in operations research and management science. The Analytical Hierarchy Process (AHP), developed by Saaty (1980), is a technique of considering data or information for a decision in a systematic manner (Schniederjans and Garvin, 1997). AHP is mainly concerned with the way to solve decision problems with uncertainties in multiple criteria characterization. It is based on three principles: (1) constructing the hierarchy, (2) priority setting, and (3) logical consistency. AHP is applied to weigh the parameters. In AHP, multiple paired comparisons are based on a standardized comparison scale of nine levels (Saaty, 1980).

Here, the weights for the parameters of each arc in the proposed network are obtained. If ⁿ C and ⁿ T are considered as normalized cost and time, respectively, then the total weighted normalized value of each arc is determined as follows:

$$P = (\psi \times T^n) + (\psi \times C^n). \tag{18}$$

On the other hand, time is a triangular fuzzy number and cost is a crisp numerical value. But, it is needed to have triangular numbers for both parameters to perform the basic operations. For this, cost is considered as a trivial triangular fuzzy number and show it by a triplet (0,C,0), where C is the numerical value inferenced from the expert system for cost with the membership value equal to 1.

Several fuzzy ranking methods have been proposed (Bortalan and Degani, 1985; Luis and Antonio, 1989; Kim and Park, 1990). Since the graded mean integration representation method (Chen and Hseih, 2000) not only alleviates some drawbacks of the existing methods, but also possesses the advantages of being easily implementable and quite effective in problem solving (see Lee et al., 2007), we will use it to transform the total weighted normalized value of each arc in our proposed network. If a ⁼ ( , , ) a¹ a² a³ is a triangular fuzzy number, then the graded mean integration ( ) ~ R(a ) is defined to be:

$$R(\tilde{a}) = \frac{a_1 + 4a_2 + a_3}{6}. (19)$$

Equation (19) is applied to transform the fuzzy numbers to crisp values and use them to find the optimal path.

Shortest Path in a Network

Let G = (V,A) be a graph, where V = { } 1,...,N is the set of nodes, and *A*⊂V ×V is the set of arcs. We write (i, j) ∈ A , if there exists an arc from node *i*∈V to node j ∈V . Furthermore, let ≥ 0 ij t denote the distance (or travel time, or any other measure of cost) from i to j. If (i, j) ∉ A ,then set tij = +∞. Note that the travel time from node i to node j is assumed to be stationary; i.e., independent of the actual arrival time at node i. Let *fij*denote the length of the shortest-path from i to j in the graph. Next, a fuzzy dynamic programming approach for computing the optimal path is described.

An Algorithm for Computing a Fuzzy Shortest Path

The following dynamic programming algorithm is for computing the shortest path in a network. The algorithm is based on Floyd's dynamic programming method to find a shortest path, if it exists, between every pair of nodes i and j in the network (see Floyd, 1962, and also Dreyfus and Law, 1977). The following optimal value function f (i, j) ^k and the corresponding labeling function P (i, j) ^k are used:

f (i, j) ^k : length of the shortest path from node i to node j when the path is considered to use only the nodes from the set nodes { } 1,...,k .

P (i, j) ^k : the last intermediate node on the shortest path from node i to node ^j using { } 1,...,k as intermediate nodes.

The dynamic updating for the optimal path length and its corresponding labeling are:

$$fk(i,j) = \min{f{k-1}(i,j), f{k-1}(i,k) + f{k-1}(k,j)},\$$

$$Pk(i,j) = \begin{cases} P{k-1}(i,j) & \text{if } k \text{ is not on the shortest path from } i \text{ to } j \text{ using } {1,...,k} \ P_{k-1}(k,j) & \text{otherwise} \end{cases}$$

We are now ready to give steps of the algorithm.

Algorithm 1:A dynamic programming method for computing a shortest path in a fuzzy network with triangular fuzzy arc lengths.

Give G = (V , A) , where V is the set of nodes with V = N , and A is the set of arcs.

Step 0: Let k=0 and ^k dij ^{f i j ~} ( , ) ^{~ =} , for all (i, ^j) ^{∈ A} , ⁼ ( , ) ^{= ∞ ~} f i j ^k , for all (i, j) ∉ A . If an arc exists from node i to node *j*then let P i j i ^k ( , ) = .

Step 1: Let k = k +1.

Do the following steps for i = 1,2,3,..., N, j = 1,2,3,..., N,i ≠ j.

• 1.1 Compute the value of ( , ) min , ( , ) ( , ) ^{1 1 1} f i j f i j f i k f k j ^k = ^{k− k−} + ^k− , (for comparison of fuzzy numbers, where the Dp,q approach is used for computing the distances).
• 1.2 If node k is not on the shortest path using { } 1,2,...,k as intermediate nodes, then let ( , ) ( , ) ¹ P i j P i j ^k = ^k− else let ( , ) ( , ) ¹ P i j P k j ^k = ^k− .

Step 2: If k < N then go toStep 1.

Step 4: Obtain the shortest path using ^P (i, ^j) ^k . If ⁼ ( , ) ^{= ∞ ~} f i j ^N , then there is no path between i and j. the shortest path from node i to j, if it exists, is identified backwards and read by the nodes: j, P i j k ^N ( , ) = followed by P i k P i l i ^N ( , ),..., ^N ( , ) = , where l is the node immediately after i in the path.

Sensitivity Analysis

Analyzing Producer's Behavior

Here, a comprehensive analysis is worked out for the obtained paths in last section. We obtained a path and its corresponding value. These values are considered as costs for each path. The aim is to find the quantity of products to be carried by AGV to maximize the profit of a producer. Obviously, profit is the difference between total revenue and total expenses in a perfect competitive market. The notations of profit maximization model are as follows:

Index:

j Counter for paths; j=1,2,…,m.

Parameters:

pj Price of product in the j th path.

qj Quantity of product in the j th path.

rj Numerical value for path j.

xj Quantity of input material in path j.

Fj Product transformation function of q qm ,..., ¹ .

Then, the mathematical model for profit is:

$$Max \quad \pi = \sum{j=1}^{m} p{j} q{j} - \sum{j=1}^{m} r{j} x{j} \Rightarrow \pi = \sum{j=1}^{m} p{j} q{j} - \sum{j=1}^{m} r{j} F{i} (q{1}, ..., q{m}). \quad (20)$$

Obviously, the maximization function can be turned into minimization as follows:

$$Min - \pi = \sum_{i=1}^{m} r_i F_i(q_1, ..., qm) - \sum{i=1}^{m} p_i q_i , \qquad (21)$$

where the xi are functions of the quantities of the outputs qj ( ( ,..., ) ⁱ Fi q¹ qm x = ). It means that, considering the inputs given to the manufacturing system, some outputs are produced. The relationships among inputs and outputs are stated as mathematical functions. The model is an unconstrained nonlinear problem. To solve the profit maximization problem for a producer desiring to maximize the profit using the above model, the partial derivatives are set equal to zero (first order conditions):

$$\frac{\partial(-\pi)}{\partial qj} = \sum{i=1}^m r_i \frac{\partial F_i}{\partial q_j} - pj = \sum{i=1}^m r_i f_i - p_j = 0,$$ (22)

where fi is the marginal cost, j i q ^{x f ∂ ∂ =} , and the marginal productivity is

$$MP = \frac{1}{f_i} = \frac{1}{\frac{\partial x_i}{\partial q_j}}.$$

To investigate whether the obtained quantity of product is optimal or not, the second order conditions are checked. The second order conditions for the maximization of profit require that the relevant Hessian of π (or −^π ) be negative (or positive) semi-definite (if the Hessian of π (or −^π ) is not negative (or positive) semi-definite, then we are sure that the obtained path is not a maximizer). Here, the Hessian of ( −^π ) is:

$$\left[\nabla^2(-\pi)\right]_{ij} = -\frac{\partial^2 \pi}{\partial q_i \partial q_j},\tag{23}$$

To be positive semi-definite, the eigenvalues of ( ) ² ∇ −^π , the λⁱ , must satisfy:

$$\lambda_i \ge 0, \qquad \forall i,$$ (24)

If the solution point is so that ( ) ² ∇ −π is positive definite, that is,

$$\lambda_i > 0, \qquad \forall i \,, \tag{25}$$

then the point is a local maximizer (or minimizer) of π (or −^π ).

This way, an application of the obtained optimal paths have been seen. The optimization scheme helps the producer to handle economic batch sizes for the AGV to attain more profit considering the specified prices and costs.

Path Reliability

It is necessary to incorporate reliability into the model to ensure the level of service for every AGV in every path. For modeling reliability, the approach of Ball and Lin (1993) is adopted for extension.

The reliability is defined as the probability that the system functions until time t. If an AGV in a path is broken down, it can be regarded as a failure. A desired

level of reliability can be achieved by limiting the failure probabilities. This approach for handling reliability is called the method of chance constraints and was initially suggested by Charnes and Cooper (1959) in the context of mathematical programming. The use of chance constraints in vehicle routing problem was illustrated by Stewart and Golden (1983). Carbone (1974) used chance constraints for selecting multiple facilities under normally distributed demands. The model minimized an upper bound on the total demand-weighted distance while ensuring that constraints were satisfied with specified chance or probability. Shiode and Drezner (2003) used a similar approach for a competitive location problem on a tree network.

It is assumed that the reliability of moving an AGV on each arc is independently exponentially distributed. We now discuss the reliability based model. Let R (t) ^j be the probability of an AGV functioning on arc j until time t. Since the arcs composing a path are in series, then the corresponding reliability is computes to be

$$R(t){system} = \prod{j=1}^{m} R_j(t).$$ (26)

Therefore, the reliability of the system is determined to satisfy

$$\left(\prod{j=1}^{m} R{j}(t)\right) \geq \alpha, \tag{27}$$

where α is the lower bound for a desirable reliability of the system until time t. As previously assumed, the reliability of each arc is independently exponentially distributed:

$$R{j}(t) = e^{\frac{-t}{\theta{j}}}. (28)$$

Therefore, we need to have

$$\left(\prod{j=1}^{m} e^{\frac{-t}{\theta{j}}}\right) \ge \alpha. \tag{29}$$

Using the inequality (29), the ^{θ j} is obtained considering a confidence level. Now, we investigate the ^{θ j} for arcs separately by data collection. The aim is to analyze the ^{θ j} obtained from data collection not having any interaction with other arcs, as compared to the ^{θ j} obtained by (29). Data in a specific working time t=24 are collected. Then, for t>24 our data are censored (type I censored data). Assume x1:n, x2:n, … , *xr:n*are the r censored data in a specific path. Then, θ ˆ, the estimated θ , is obtained by maximum likelihood estimation (MLE) as follows.

$$\hat{\theta}{MLE} = \frac{\sum{i=1}^{r} x_{i:n} + (n-r)t_0}{r},$$ (30)

where t0 is the end time of observation (t=24) and r is the number of failures.

Here, using (2 ) ^{ˆ 2 2} r r ^{χ θ θ ≈} as a pivot (2r is the degrees of freedom), a confidence interval for θ can be set.

To set a confidence interval using (2 ) ^{ˆ 2 2} r r ^{χ θ θ ≈} ,

$$P\left[\chi{1-\frac{\alpha}{2}}^{2}(2r) < \frac{2r\hat{\theta}}{\theta} < \chi{\frac{\alpha}{2}}^{2}(2r)\right] = 1 - \alpha.$$ (31)

Inverting (31) and multiplying θ ˆ 2r into all fractions,

$$P\left[\frac{2r\hat{\theta}}{\chi{\frac{\alpha}{2}}^{2}(2r)} < \theta < \frac{2r\hat{\theta}}{\chi{\frac{1-\alpha}{2}}^{2}(2r)}\right] = 1 - \alpha, \qquad (32)$$

where (32) is a two sided (1−^α )% confidence interval based on type I censored data.

11.4 Conclusions

An approach for finding an optimal path in a flexible jobshop manufacturing system considering two criteria of time and cost is proposed. The proposed flexible jobshop system has more than one shop with the same duty. The difference among shops with the same duty is in their machines with various specifications. The shops configure a network in which they are considered as

References 185

nodes and the paths among them are considered as network arcs. An Automated Guided Vehicle (AGV) functions as a material handling device through the manufacturing network. Time is considered to be a triangular fuzzy number and cost is inferred from an expert system considering three parameters of equipment sensitivity, operator proficiency, and product specification via linguistic variables. This cost is associated with some main factors and some sub-factors. To approximate the cost, a backpropagation neural network was applied to estimate the sub-factors with the corresponding main factors and compute a bound for the main factors. These bounds were processed by the entropy technique to obtain regions for the linguistic variables. After determining the regions for the linguistic variables, we composed the rules and verified their significance using regression analysis. The purified rules were used to infer the cost of the system. The aim was to find a path which minimized an aggregate weighted unscaled time and cost objective. Since time and cost had different scales, a normalization process was used to remove the scales and because the model was bi-objective, the AHP weighing method was applied to gain a single objective. A fuzzy dynamic programming approach was used to compute a shortest path in the proposed network. Also, some sensitivity testing on the obtained shortest paths are performed. Finally, the proposed approach was fully illustrated by a numerical example.

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Chapter 12

Risk for Multiple AGV System

12.1 Summary

A standard formula for the quantitative definition of risk is, Risk (loss) (loss) = × P L , where risk is the function of the probability (P) of loss and the significance of its consequences (L) (Manuj and Mentzer, 2008). Hetland (2003) and Diekmann et al. (1988), on the other hand, view risk as the implication of an uncertain phenomenon. Waters (2007) explains the difference: risk occurs because there is uncertainty about the future, which means that unexpected events may occur. Knight's (1921) distinction between certainty, risk and uncertainty is probably the best known and most used typology of uncertainty for risk management. In his definition of risk Knight coined the terms (quantitative) ''measurable'' uncertainty and (nonquantitative) ''un-measurable'' uncertainty when there is only partial knowledge of outcomes in the form of beliefs and opinions (Vilko and Hallikas, 2012).

The researchers' role in the discussions was to present the data and guide the discussion in a holistic direction. Consensus was finally reached about the risks, their categorization and impact. The risk drivers found from the manufacturing system were classified by source. We found that an adaptation of Manuj and Mentzer's (2008) risk source classification with its wide perspective on manufacturing risk management provided a solid framework for our case; the classification is both qualitative and quantitative, taking into account both the direct and indirect impacts, and this facilitates in depth understanding of the risk sources without losing the holistic view. The risks were categorized as follows: Supply Risks, Security Risks, Operational Risks, Macro Risks, Policy Risks and Environmental Risks. No risk sources fitting the Demand Risks, Competitive Risks and Resource Risks classification were identified in the group discussions (Pons, 2010). Having identified the risks and made their semi-quantitative assessment, the expert panel turned to the delay impact, which was modeled in the form of triangular distributions representing the minimum, the most likely and the highest impact. Risk effects can be categorized in three different types: timebased, finance-based and quality-based. Time-based effects either delay or disrupt the material flow of the manufacturing system. In this case the disruption was identified as a breakdown in the chain such that the goods do not reach their destination by the time they are expected by the final customer. There are no clear time limitations on the delay due to the fact that it had significantly different consequences in different stages of the system. Manufacturing transportation could be delayed for times without serious consequences. The risk impact in this case varied highly depending on the goods. Risk likelihood and impact were evaluated on a scale of 0,1,3,9, where 0 implies zero likelihood or no impact and 9 denotes a very high likelihood or impact. The scale is adapted from the Quality Function Deployment design method (see e.g., Akao, 2004), which is commonly used in the context of new product/ service development.

Manufacturing risk assessment is a supporting tool for the contractor and program office decision-making process. It seeks to estimate the probabilities of success or failure associated with the manufacturing alternatives available (Nau et al., 2012). These risk assessments may reflect alternative manufacturing approaches to a given design or may be part of the evaluation of design alternatives, each of which has an associated manufacturing approach.Risk management is an overarching process that begins during the earliest stages of a program and continues throughout its entire life cycle. Risk encompasses the following steps:

• Risk identification;
• Risk analysis;
• Risk mitigation planning;
• Risk mitigation plan implementation; and
• Risk tracking.

Assessing manufacturing risks is a requirement, and it is required as early as pre-Milestone A where the Analysis of Alternatives (AoA) is required to assess the "manufacturing feasibility" of the proposed approach.

As a system progresses through its definition, design, development, testing and fielding, more information becomes available concerning the system's risk. If the risk management process is conducted continuously, then new information will lead to identifying and analyzing new risk root causes, and identifying and implementing mitigation plans for them. It will also lead to re-analyzing previously identified risk root causes, and re-evaluating and adjusting mitigation plans already in place. This continuous activity allows the PM to focus valuable program resources where they can be most effective, and shift resources as new future root causes are discovered and others are re-evaluated.

Iterative Systems Engineering process is the perfect vehicle for helping manufacturing managers to identify risk early through technical reviews and audits and to support the development of plans and mitigations to reduce those risks.

Critical success factors refer to identifying the factors that must be successfully mastered to execute a successful risk management program. Some examples of risk management critical success factors include:

12.1 Summary 191

Clearly define and establish feasible, stable, and well-understood user requirements;

• Establish a close partnership with users, industry, and other key stakeholders;
• Comprehensively plan, formally document, and continuously apply the risk management process, and ensure it is integral to all program processes;
• Use continuous, event-driven technical reviews as part of the risk management process; and
• Clearly define criteria for assessing the effectiveness of implemented risk mitigation actions.

Risk is time phased and should be tied to appropriate maturity models such as the Technology Readiness Level (TRL), Manufacturing Readiness Level (MRL) and Sustainment Readiness Model (SRM) that are considered best practices. Other chapters will discuss these models. These models provide for an assessment of a technology, manufacturing process, logistics/sustainment considerations of a component, subsystem, or weapon system. These models have been structured to:

• Define the current level of maturity;
• Identify maturity shortfalls and associated cost and risk; and
• Provide a basis for investments to mature the component, subsystem, or weapon system and thereby manage risk.

The risks were analyzed in terms of their effects on the manufacturing system. Some of the investigated organizations had severe problems informing a holistic and clear view, and clear over estimations as well as underestimations were evident in their assessment of the risk impact beyond their own functions. Expert panel discussions were held in order to verify the risk values (Nau et al., 2011). The first of these took place during the interview process in order to discuss the preliminary findings from the interviews conducted thus far and to explore the different viewpoints. The expert panel consisted of logistics field researchers and port operations experts in the case manufacturing system.

Here, a manufacturing network is proposed and the defects are determined in order to analyze risk and compute loss. The aim is to obtain paths with less loss and more reliability. With respect to the multi stage decision making process of the proposed network, we develop a dynamic program being a useful tool for multi stage decision making. To counteract the dynamism of digital data in different time periods, a Bayesian approach is employed to determine the loss function of moving through the stages of the proposed network.

The main contribution of this work is to propose an integrated methodology under uncertainty for risk evaluation in very high automated technology of material handling in manufacturing. Since, in past works risk evaluation for high technology systems neglected to consider uncertainty of dynamic changes in automated manufacturing system. Also, considering risk for AGV systems is novel.

12.2 Statement of the Problem

Several AGVs are considered for material handling purpose in a manufacturing system consisting of workstations to process parts to form products. To conceptualize a network, the workstations are proposed to be nodes and the AGV guide paths are the arcs. Due to demands sent to the AGVs, they carry the parts on the arcs thus AGVs may have conflict. Also, due to high technology employed in AGV configurations defects are inevitable.

Suppose that there are several kinds of parts that should be transported by AGVs from node A to node B. The distance between A and B is t and the velocities of the AGVs are given by 1 .Thus, it takes time t to transport the part from A to B. During the operation of an AGV, certain fatalrisks which may cause the failure of the transportation (e.g., defect in automation) arrive at the vehicle according to ahomogeneous Poisson process with intensityλ . A fatalrisk, independently of all others, causes the failure of the transportation with probability p. When more than one vehicle operates at the same time, given afatalrisk, the failures between vehicles are assumed to be independent. We have m kinds of parts (part1, part2, . . . , partm) that should be transported and m identical AGVs (AGV1, AGV2, . . . , AGVn). We define that the 'mission' is completed when all the m kinds of parts are safely transported. A configuration of the proposed manufacturing network is depicted in Figure 1.

Fig. 1 A configuration of the proposed manufacturing network

As shown in Figure 2, several AGVs carry parts amongst the work stations on the guide paths to process the manufacturing plan and satisfy product demands. Turning points are mounted as guide paths division centers to prohibit AGVs' conflicts during movements. The intersection of guide paths is determined as the turning point. In these points an AGV is directed according to the process plan sent from the control unit and with respect to work stations' demands. These points are the state variables for the proposed network to be handled by a dynamic program.

As a result, for an economic decision making, loss of the incurred risks can be considered. The loss here is composed of the risk of defect when an AGV moves in the manufacturing network. A loss concept is utilized to consider all aspects of a reliable manufacturing network.

We consider loss as a function of defect elements, that is, , where X1,…,Xp are the elements of defects, i.e., part m transported by AGV n. Clearly, defect data for AGVs are recovered in any time periods causing uncertainty of data. Therefore, the proposed loss function should be estimated. For loss function estimation, a Bayesian approach is utilized. To find Loss ( ,... ) X¹ X ^p ≡ f

Fig. 2 A flowchart of the proposed methodology

an optimal path in the manufacturing network having less risk, and therefore less loss, dynamic programming is applied. The reason is that the proposed network is composed of nodes as state variable and part processing in different work stations as stages of the dynamic program. A flowchart of the proposed methodology is shown in Figure 2.

12.3 Mathematical Model

The model applied in the present work belongs to the applied measurement approach (AMA) and it is called the loss distribution approach (LDA). It is characterized by the categorization of the losses in terms of 'Frequency' (the number of loss events during a certain time period) and 'Severity' (the impact of the event in terms of economic loss) (Valle and Giudici, 2006). Formally, for each intersection r (where r = 1, ... , R) and for a given number of AGV t, the total operational loss could be defined as the sum of a random number *nt*of losses:

$$L{rt} = X{r1} + X_{r2} + \cdots + X_r \, \boldsymbol{n}_t \,, \tag{1}$$

where Lrt denotes the total operational loss, *Xr1,* ... , Xr nt denote individual loss severities and *nt*denotes the frequency, for t =1, ... , T, T representing the number of AGVs. Note that, for each intersection and for each AGV, the total loss can be expressed as

$$L{l}=s{i}\cdot n_{l}, \tag{2}$$

where *nt*is the frequency, defined as before, and st(commonly referred to as the severity) is the mean loss for that AGV. The LDA assumes that, for each AGV:

• (1) the individual losses {Xrq}, where q = 1, ... , nt, are independent and identically distributed random variables;
• (2) the distribution of the frequency *nt*is independent of the distribution of the severities {Xrq}, for q = 1,… , nt. This implies that *nt*is independent of st;
• (3) besides, Lrt, for t = 1, ... ,T, are independent and identically distributed random variables.

For a given intersection, we construct a discrete probability density of the number of loss events *nt*for the AGV and *ntcontinuous probability densities of the loss severities (Alexander, 2003). Now, if we express the likelihood function for each intersection in a general way, we obtain the following equation, indicating the severity distribution with ( ) ξ^q f x* and the frequency distribution with f ( ) nt ^ψ , where ξ denotes the parameter vector of the severity distribution and ψ denotes the parameter vector of the frequency distribution, we obtain the following form:

$$L(\mathbf{x}, \mathbf{n}|\boldsymbol{\psi}, \boldsymbol{\xi}) = \prod{t=1}^{T} \left[ \prod{q=1}^{n_t} f(x_j | \boldsymbol{\xi}) \right] f(n_t | \boldsymbol{\psi}). \tag{3}$$

Different functional forms for the frequency and severity distributions exist. Frequency represents a discrete phenomenon. Since we want to determine the probability that a certain number of loss events occur in a predetermined time horizon, the most suitable probability distributions are the Poisson and the Negative Binomial. Severity, instead, is a continuous phenomenon and we chose to describe it by the Exponential, the Gamma and the Pareto distributions. For more details about these distributions see Johnson et al. (1994).

In the proposed problem the number of defects occurs for an AGV is pointed by frequency and the effect of loss is implied by severity. Our first problem is to estimate, on the basis of data, the parameters of the frequency and severity distributions, denoted by ^ψ and ξ , respectively, in equation (3). The classical approach suggests the employment of the method of moments or the method of maximum likelihood, as described, for example, by Gourieroux and Monfort (1995). An alternative approach is the Bayesian method, which allows the combination of quantitative data, coming from the time series of operational losses, and prior information, represented by expert opinions. See Berger (1985), Bernardo and Smith (1994) or Robert (1994) for an introduction to these methods. In this chapter we estimate the parameters with the maximum likelihood method and with the Bayesian method, making some comparisons between the classical and the Bayesian approaches. In particular, we choose the maximum likelihood method because of its good asymptotic properties, since the MLE converges almost surely to the true value of the parameter, under fairly general conditions (see Lehmann and Casella, 1998). Once the parameters have been estimated, the marginals are defined and the operational loss distribution has to be identified.

Bayesian Approach

Suppose a continuous probability distribution with probability density function (pdf) ƒΘ is assigned to an uncertain quantity Θ. In the conventional language of mathematical probability theory Θ would be a "random variable" (Lehmann and Casella, 1998). The probability that the event B will be the outcome of an experiment depends on Θ; it is P(B | Θ). As a function of Θ this is the likelihood function:

$$L(\theta) = P(B|\Theta = \theta). \tag{4}$$

Then the posterior probability distribution of Θ, i.e. the conditional probability distribution of Θ given the observed data B, has probability density function

$$f{\Theta}(\theta|B) = \text{constant} f{\Theta}(\theta) L(B|\theta),$$ (5)

where the "constant" is a normalizing constant so chosen as to make the integral of the function equal to 1, so that it is indeed a probability density function. This is the form of Bayes' theorem actually considered by Thomas Bayes (Berger, 1985). More generally still, the new data B may be the value of an observed continuously distributed random variable X. The probability that it has any particular value is therefore 0. In such a case, the likelihood function is the value of a probability density function of X given Θ, rather than a probability of B given Θ:

$$L(\theta) = f_X(x|\Theta = \theta). \tag{6}$$

Bayes's rule provides the framework for combining prior information with sample data. In this reference, we apply Bayes's rule for combining prior information on the assumed distribution's parameter(s) θ with sample data in order to make inferences based on the model. The prior knowledge about the parameter(s) is expressed in terms of a pdf f(θ), called the prior distribution. The posterior distribution of θ given the sample data, using Bayes rule, provides the updated information about the parameters θ. This is expressed with the following posterior pdf:

$$f(\theta|\text{Data}) = \frac{L(\text{Data}|\theta)\varphi(\theta)}{\int L(\text{Data}|\theta)\varphi(\theta)d\theta}.$$ (7)

where,

θ is a vector of the parameters of the chosen distribution,

ε is the range of θ,

L(Data|θ) is the likelihood function based on the chosen distribution and data,

f(θ) is the prior distribution for each of the parameters.

The integral in equation (7) is often referred to as the marginal probability and can be interpreted as the probability of obtaining the sample data given a prior distribution and it's a constant number. Generally, the integral in equation (7) does not have a closed form solution and numerical methods are needed for its solution.

As can be seen from equation (7), there is a significant difference between classical and Bayesian statistics. First, the idea of prior information does not exist in classical statistics. All inferences in classical statistics are based on the sample data. On the other hand, in the Bayesian framework, prior information constitutes the basis of the theory. Another difference is in the overall approach of making inferences and their interpretation. For example, in Bayesian analysis the parameters of the distribution to be "fitted" are the random variables. In reality, there is no distribution fitted to the data in the Bayesian case.

In order to obtain an estimate of the random variables, a probability needs to be specified or we can use the expected value of the posterior distribution. In order to demonstrate the procedure of obtaining results from the posterior distribution, we will rewrite equation (7) for a single parameter *θ*1:

$$f(\theta_1 | \text{Data}) = \frac{L(\text{Data} | \theta_1) \varphi(\theta_1)}{\int L(\text{Data} \theta_1) \varphi(\theta_1) d\theta_1}.$$ (8)

The expected value (or mean value) of the parameter *θ*1 can be obtained as follows:

$$E(\theta1) = \int{\varepsilon} \theta_1 \cdot f(\theta_1 | \text{Data}) d\theta_1.$$ (9)

Now, using the estimated loss function, we obtain the arc values being applied in dynamic programming with respect to state and stage concepts to find the optimal path.

A Dynamic Program

The output of the current state is ( ,..., ) F w¹ wl , being the obtained loss from the Bayesian method. While the current state is an n-dimensional vector having several nodes (work stations), we consider a state function^{δ l}(p), l =1,...,n . Also, a policy function is proposed to consider the AGV movement policy, and hence assuming an m-dimensional vector of policies ( m ≤ n ), we have βl(p), l =1,...,m, where βl( ) 0,1 p ∈{ } , i.e., policy movement occurs (=1) or does not occur (=0) for an AGV movement. Here, our problem becomes n dynamical equations of the form

$$\delta{l}(p+1) = \delta{l}(p) + f{l, p}(\delta{1}(p), ..., \delta{n}(p), \beta{1}(p), ..., \beta_{m}(p)), \quad l = 1, ..., n, p = 0, ..., N - 1.$$ (10)

Therefore, a dynamic programming approach can be used to model our multidimensional problem. Let ( ) ( ),..., ( ) ¹ S p p ^{p n δ δ} bethe minimum loss of the remaining nodes an AGV visits if we start stage p in state ( ),..., ( ) ^{δ 1} p ^{δ n} p .

Then, for p = 0,..., N − 1, the strategy function is:

$$S{p}(\delta{1}(p),...,\delta{n}(p)) = \min{\beta{1}(p),...,\beta{n}(p)} \left[ S{p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1}(p),f{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)),...,\delta{n}(p) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{1}(p),...,\delta{n}(p),\beta{1}(p),...,\beta{m}(p)) + S{p+1}(\delta{1,p}(\delta{$$

with the boundary condition,

$$S_N(\delta_1(N),...,\delta_n(N)) = h(\delta_1(N),...,\delta_n(N)).$$ (12)

This way, the state and the policy direction for the AGV is at hand. Solving (11) for the appropriate indices, we obtain the state and the policy direction in each time period.

Considering the boundary condition, we can compute *h*OPT (the optimum states, i.e., the optimal arcs leading to minimum loss of AGV movement) for given policy functions. Due to nonlinear format and operations on mathematical functions, it is easy to see that exact dynamic program (DP) requires O(n( * δ ) ^m)) calculations for the multi-dimensional dynamic program where { } * min ,..., ^{1 l δ} δδ = . Space requirements and value function computations become impractical for even moderate m. Thus, exact DP is not a practical methodology particularly for large-scale optimization problems. The basic idea behind approximate DP is to approximate the optimal value function ( ) 1( ),..., ( ) p n Sp p ^{δ δ} and to construct a suboptimal solution with respect to ( ) 1( ),..., ( ) ⁿ hp p ^{δ δ} . We introduce a heuristic algorithm to approximate the optimal value function in the following sections.

A Heuristic Solution Approach

This section introduces our proposed Adaptive Dynamic Program Heuristic (ADPH) approach for the multi-dimensional problem. The basic idea of ADPH is estimating the optimal value function by the solution value of a suboptimal methodology to the corresponding subproblem and constructing a solution through boundary condition equation. This suboptimal methodology is called the heuristic approach (Bertsimasand Demir, 2002).

Let ( ) ( ) ^l h p ^δ be a heuristic for the subproblem ( ) 1( ),..., ( ) ⁿ hp p ^{δ δ} . Let ( ) ^{l δ} p be the corresponding heuristic solution and ( ) ( ) p l S p ^δ be the heuristic value, i.e., an estimate of the optimal value ( ) 1( ),..., ( ) p n Sp p ^{δ δ} . The ADPH algorithm starts by applying ( ) ( ) ^l h p ^δ to the problem ( ) 1( ),..., ( ) ⁿ hp p ^{δ δ} and getting ( ) ^{l δ} p . If the solution ( ) ^{l δ} p is optimal, the algorithm terminates with an optimal solution. If the problem is infeasible, the algorithm terminates without a solution. Otherwise, the algorithm sets the variables best-solution BEST ( ) ^{l δ δ} = p and best-solution-value ( ) p l S p = ^δ (Step 2 in Figure 4).

The algorithm proceeds by applying reduced loss fixing as described below to fix some the βl(p), l =1,...,m variables to the corresponding values (0 or 1) in an optimal solution to the problem ( ) 1( ),..., ( ) ⁿ hp p ^{δ δ} _ (Step 3 in Figure 4). Reduced loss fixing might effectively reduce the number of variables in advance. Let M denote the set of indices of the fixable variables by the reduced loss criterion. We denote by x ^M the corresponding fixed values, that is,

$$x_l^M = 0$$ or 1, for all $l \in M$ .

estimated ones.

The algorithm iteratively sets the variables to 0 or 1 as described in Step 4 of Figure 4. At each iteration, we update the best-solution, BEST ( ) ^{l δ δ} = p and bestsolutionvalue, ( ) p l S p = ^δ . We check if early termination is possible (true) or not (false). We also allow fixing a set of variables, which we call lag-variablefixing. At the final step, we set the first variable to the optimal solution of the reduced problem (Step 5 of Figure 4) and the algorithm returns BEST ( ) ^{l δ δ} = p and ( ) p l S p = ^δ . We present the details of ADPH algorithm in Figure 3.

Variable Assignment: To calculate ^p S for a variable *l M*∉ , we employ an estimated value, Mlp (, ) , instead of optimal ones, Mlp (, ). This is the basic idea in ADPH. We denote by U(l,p) an upper bound to the problem ( ) ( ) ^l h p ^δ . In our computations, we set the upper bound U(l,p) to *h*OPT. Let ^ε denote the percentage deviation of ( 1, 1( ( ))) ^{l δ} β lp p − + with respect to the upper bound ^U(l-1,(p+1) ( ) ^l β p ) for ( ) ^l β p =0, 1 and also let ^ε εε * min , = { 0 1}. We estimate the optimal values, M(l−1,(p+1) ( ) ^l β p ), by ( 1,( 1) ( )) (1 *) ( 1,( 1) ( )) ^M l l l p p Ul p p − + =− − + βε ^β for ( ) ^l β p =0, 1. By the definition of ^ε *, the optimal values are approximated such that the percentage deviations of the estimate values are the same with respect to the associated upper bounds. Finally, we set Sp replacing the optimal values by the

1: Initialization: 
 l = n, p =0, Sp=0 
 early−termination = false 
2: Apply ( ) ( ) l h p δ and get ( ) ( ) l h p δ , ( ) p l S p = δ
Problem infeasible → exit, the problem is infeasible 
( ) l δ p optimal → exit, an optimal solution available 
BEST δ ← ( ) l δ p , p S ← ( ) l δ p
3: Apply reduced loss fixing and get M, xM
4: While (l > 2)and (early−termination = false)do 
 if (l ∈M)then 
 Sp← M
             l x
 else 
 Calculate Sp by calling variable assignment 
 Update best-solution and best-solution-value and get BEST δ , p S
 Check early termination criterion and get the status (true or false) 
 Fix variables by lag variable fixing and get M, SM
 p←p+1 
 l←l−1 
5: Set Sp to the optimal solution of ( ) ( ) l h p δ
6: Output: BEST δ and p S

Fig. 3 The ADPH algorithm for the ( ) 1( ),..., ( ) ⁿ hp p ^{δ δ}

Update Best-Known Solution and Value: Once we set Sp, if necessary, we update the best-solution, BEST ^δ , and best-solution-value, Sp as follows. Let p=p+1( ^{n j} p j l S = ∑ ). Because of the construction of Sp j for j in [l,p], ( ( )( 1, ), ,..., ) c ln l pp ^{δ δ} = − pl pS S is a feasible solution to the problem ( ) ( ) ^l h p ^δ , where the value of the solution ^{c δ} equals ( 1, ) ⁿ c j p p j l S hl p S = = − ∑ . We update BEST ^δ to ^{c δ} and Sp to ^{c p} S , respectively, if ^{c p} S is larger than Sp.

Early Termination: The ADPH algorithm can be terminated early while setting Sp if we determine that we find an optimal solution to the problem ( ) ( ) ^l h p ^δ ,

where p=p+1( ^{n j} p j l S = ∑ ). We apply the following tests to determine whether we

have an optimal solution or not to the problem ( ) ( ) ^l h p ^δ :

(a) We check if δ( 1, ) l p − and ( 1, ) ^{p δ} l pS − + are optimal solutions to the subproblems hl p ( ) ^δ( 1, ) − and ( ) ( 1, ) ^p h l pS ^δ − + , respectively.

We conclude that an optimal solution exists if hl p ( ) δ( 1, ) − = U(l-1, p) and ( ) ( 1, ) ^p h l pS ^δ − + = U(l-1,p+Sp);

• (b) We conclude that δ δ ( , ) ( 1, ),0 lp l p = − ( )is an optimal solution if hl p ( ) δ( 1, ) − is greater than U(l-1,p+Sp) and hl p ( ) δ( 1, ) − = U(l-1, p);
• (c) Similarly, we conclude that ( , ) ( 1, ),1 ( ) ^{p δ δ} lp l p S = −+ is an optimal solution if ( ) ( 1, ) ^p h l pS ^δ − + is greater than U(l-1,p) and ( ) ( 1, ) ^p h l pS ^δ − + = U(l-1,p+Sp).

Lag Variable Fixing: To allow the ADPH to generate solutions in significantly shorter times (possibly of lower quality), we allow fixing a set of variables Sp for j in [l-n, l-1], and *j M*∉ . We call the method lag-variable-fixing and those variables as lag-fixable variables. We denote by { } lag, : [ , 1 and ^l M j j l nl j M = ∈− − ∈ . Once the *l*th variable is assigned to 0 or 1, the variables ^{j M} S are set to ( 1, ) ^{j p} Sl p − or ( 1, ) ^j p p Sl pS − + , respectively, for all lag,*^l j M∈ . We update the set M by including those indices in lag,^l M* . *Sp ^j*are assigned to SM j for all lag,*^l j M*∈ in Step 4 of Figure 4. Lag-size parameter q specifies the number of variables to be fixed lagging from the *l*th variable while setting Sp through the relationship, q=[l/lag-time], where lag-time is a user-specified parameter.

12.4 Conclusions

We proposed a new automated manufacturing network considering the existing data on states and stages aiming to find the arcs with more convergence for value added purposes. The arcs were the guide paths that AGVs move through to carry parts. Regarding the multi stage decision making process of the AGVs, we develop a multi-dimensional dynamic program as a useful tool for multi stage decision making. To counteract the dynamism of the digital data in different time periods, two concepts of state and policy direction are introduced to determine the loss of moving through the stages of the proposed manufacturing network. Using this function, a dynamic model was proposed to consider the AGV's movement state and policy direction in each time period. A dynamic program was proposed to minimize loss of the remaining nodes assuming that the analysis started at an appropriate stage of the problem. Since the space requirements and value function computations become impractical for even moderate size, we approximated the optimal value function developing a heuristic algorithm. A comprehensive heuristic solution methodology was developed to approximate the proposed dynamic program due to nonlinearity and computational difficulties. The solution methodology was verified in comparison with some famous methods in the literature. In summary, the computational evidence suggests that the ADPH approach for the ( ) ( ) ^l h p ^δ seems an attractive alternative to existing methodologies as it produces near optimal solutions fast.

The advantages of the proposed methodology are:

• Controlling uncertainty and dynamism of future analysis for AGV movement
• Breakthrough the traditional obstacles of loss function estimation by Bayesian analysis using prior and posterior distributions
• Usefulness of the collected data (past data) for future analysis of AGV risk assessment
• Maximizing the expected benefit form an AGV movement process
• Competitive advantage of the management equipped with the proposed methodology
• Reducing the risk exist in the AGV based transportation

To sum up, the proposed methodology with the above capabilities is a significant tool for the automated manufacturing network since the current market is very competitive and absorbing large amount of markets requires subtle analysis and decision making. Also, the obtained results show the more reliablepaths for AGV movements. This is useful for redesigning a routing structure for optimization of extension of shop purposes. Another element of the results obtained is the conflict free notion in the AGV movement on guide paths being very important in designing an AGV system. Clearly, due to dynamism the model is capable to determine low risk conflict free AGV routings.

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