Heuristics for Scheduling Reentrant Flexible Job Shops with Sequence-dependent Setup Times and Limited Buffer Capacities

Transcription

1 Clemson University TigerPrints All Dissertations Dissertations Heuristics for Scheduling Reentrant Flexible Job Shops with Sequence-dependent Setup Times and Limited Buffer Capacities Jakrawarn Kunadilok Clemson University, Follow this and additional works at: Part of the Industrial Engineering Commons Recommended Citation Kunadilok, Jakrawarn, "Heuristics for Scheduling Reentrant Flexible Job Shops with Sequence-dependent Setup Times and Limited Buffer Capacities" (2007). All Dissertations This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact

2 HEURISTICS FOR SCHEDULING REENTRANT FLEXIBLE JOB SHOPS WITH SEQUENCE-DEPENDENT SETUP TIMES AND LIMITED BUFFER CAPACITIES A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Industrial Engineering by Jakrawarn Kunadilok August 2007 Accepted by: Dr. Mary Elizabeth Kurz, Committee Chair Dr. William G. Ferrell Dr. Byung Rae Cho Dr. Kevin M. Taaffe

3 ABSTRACT This research addresses the problem of scheduling complex job shops with minimizing total weighted tardiness as the objective. The complex job shop is characterized by reentrant job flow through a number of different work centers that contain one or more identical parallel machines. The processing restrictions for the complex job shop include possible non-zero ready times and sequence-dependent setup times. We categorize the complex job shop into two classes: the complex job shop with infinite or finite buffer capacities. We propose three solution methodologies for scheduling the problem with infinite buffer capacities. The first method is the use of mixed integer programming (MIP). The second method is an application of a genetic algorithm based on random keys encoding. The last method is developed based on tabu search. In total weighted tardiness problems the search spaces for local improvement approaches are large. A move search and a large step diversification are employed for developing the proposed random keys genetic algorithm (RKGA) and tabu search with large step diversification (TSLSD), respectively, in order to help improve performance. In the problem with finite buffer capacities, an MIP model is also developed mainly for comparison purposes in solving small-sized problems. We propose two versions of RKGA, called RKGA.b and RKGA.msb. We introduce buffer availability and job releasing scheme as the constraints for the flow of jobs throughout their assigned work centers. Then two schedule construction procedures are developed based on those constraints and the chromosome decoding for RKGA. The schedule construction ii

4 procedure for RKGA.b is designed without move search while RKGA.msb includes move search. A test-bed of problems has been developed for comparison of corresponding heuristic approaches. Computational results indicate that TSLSD and RKGA.msb are the most effective heuristic for scheduling the complex job shop with infinite and finite buffer capacities, respectively. iii

5 ACKNOWLEDGMENTS I would like to express my gratitude foremost to my advisor Dr. Mary E. Kurz for her guidance, encouragement, and extreme patience. Grateful acknowledgement is also due to my committee members Dr. William G. Ferrell, Dr. Byung Rae Cho, and Dr. Kevin M. Taaffe for their support and encouragement. I thank the Royal Thai Government for the financial support. I also thank for Dr. Banhan Lila and Dr. Ruephuwan Chantrasa who have given me valuable advices in both academic and general aspects since I entered the PhD program. Finally, I would like to thank my mother and brother who happily allow me to pursue my PhD at a time when they really need me to be close to them. It would have been impossible for me to complete this research without their love and encouragement. iv

6 TABLE OF CONTENTS TITLE PAGE...i ABSTRACT...ii ACKNOWLEDGMENTS...iv LIST OF TABLES...vii LIST OF FIGURES...ix CHAPTER 1. INTRODUCTION...1 Job Shops with Multiple Processing Restrictions...2 Research Objectives...4 Significant of Research...5 Organization of the Dissertation...6 Page 2. LITERATURE REVIEW AND BACKGROUND...7 Introduction...7 Job Shop Scheduling Problems with Infinite Buffer Capacities...7 Multi-stage Scheduling Problems with Buffer Constraints...37 Conclusion JOB SHOPS WITH INFINITE BUFFER CAPACITIES...41 Introduction...41 Mixed Integer Programming...43 Random Keys Genetic Algorithm...50 Tabu Search...65 Computational Results JOB SHOPS WITH FINITE BUFFER CAPACITIES Introduction Mixed Integer Programming v

7 Table of Contents (Continued) Page Random Keys Genetic Algorithm Computational Results CONCLUSIONS AND RECOMMENDATIONS Conclusions Recommendations for Manufacturing Application Future Research APPENDICES A: Appendix A Traveling Salesman Problem B: Appendix B Notation BIBLIOGRAPHY vi

8 LIST OF TABLES Table Page 2.1 Some Applications of SB, TS, and GA in Job Shop Scheduling Small-sized Problem Instances Data Generation Information MIP Solutions for FJc r j, s ijk, recrc Σw j T j Computational Results for Simple Heuristics Heuristic Comparison for Small-sized Problems with Known Optimal Solution Heuristic Comparison for Small-sized Problems with Unknown Optimal Solution Some Characteristics of Randomly Generated Problem Instances Initial Results in Evaluation of Simple Heuristics Comparison of Heuristics Using Large-sized Problems Chromosome Representation for Example Schedule Construction for Example Chromosome Representation for Example Schedule Construction for Example Example for SCP-b in Anti-deadlock Mode Small-sized Problem Instances Computational Results for FJc r j, s ijk,block Σw j T j Computational Results for FJc r j, s ijk,block recrc Σw j T j vii

9 List of Tables (Continued) Table Page 4.9 Heuristic Comparison for Small-sized Problems with Known Optimal Solution Heuristic Comparison for Small-sized Problems with Unknown Optimal Solution Comparison of Heuristics Using Type I of 30 Large Problem Instances Comparison of Heuristics Using Type II of 30 Large Problem Instances viii

10 LIST OF FIGURES Figure Page 1.1 Paradigm for a Reentrant Flexible Job Shop Disjunctive Graph for a Classical Job Shop Problem Modified Disjunctive Graph for a Flexible Job Shop with Recirculation Pseudo-code for the RKGA Adapted from Norman and Bean (1999) Chromosome Representation Example Example of Parameterized Crossover with Crossover Probability = Pseudo-code for SCP Example of Problem Data and Chromosome Representation Gantt Chart Pseudo-code for Critical Path Determination One-Step Look-back Interchanges Example Insertion of Operation O km [x] Adapted from Nowicki and Smutnicki (1998) Infeasible Operation Sequence Created from Insertion Time Window for Operation Moving Operation within the Same Machine Direct Undesired Idle Time Created After Initial Moves Pseudo-code for Multi-step Look-back Move Pseudo-code for Move Evaluation...94 ix

11 List of Figures (Continued) Figure Page 4.1 Modified Disjunctive Graph for FJc r j, s ijk, block, recrc Σw j T j Partial Feasible Schedule in a Modified Disjunctive Graph A Feasible Schedule for Problem 630 with Finite Buffer Capacities Mixed Integer Programming Model Schedule Construction result in Case Schedule Construction result in Case Schedule Construction result in Case Schedule Construction Result in Case Pseudo-code for the Main Procedure for SCP-b Problem Instance Pseudo-code for SCP-b in Anti-deadlock Mode Pseudo-code for Move Search in Regular Mode of SCP-msb Pseudo-code for Move Search in Anti-deadlock Mode of SCP-b before Step Pseudo-code for Move Search in Anti-deadlock Mode of SCP-b before Step x

12 CHAPTER 1 INTRODUCTION Many modern manufacturing environments are complex job shops. In some production stages, identical parallel machines may be used to achieve an appropriate capacity. Modern technology machines allow different jobs to be processed on the same machine by changing machine tools. Setup activities of a new job then depend on the previous job on the machine. Some complicated products require multiple visits to the same processing station such as wafers in semiconductor manufacturing, print circuit boards, and painting and coating products. Scheduling these kinds of complex job shops is more difficult when dealing with the fact that all jobs are not ready to start their processing operation at the same time. Under unlimited storage buffer capacities conditions, a completed job is released immediately to the downstream machine. It may be processed on the machine instantly or wait in the machine s buffer if the machine is not available. Hopp and Spearman (2000) state that buffer capacities never become infinite in real production. When releasing jobs under limited buffer capacities conditions, the completed job must remain on the current machine, resulting in the machine being blocked until a buffer space or a machine at the downstream work center becomes available. The limited buffer capacities may result directly from limited working space or material handling capacities. In addition, buffers may be limited through management policy. Management may specify maximum buffer capacities at each work center in order to set a limit on the maximum amount of inventory in the production system. 1

13 In today s competitive business, manufacturers have to respond quickly to orders and meet shipping dates committed to the customers, as failure to do so may result in a significant loss of goodwill. This requires the ability to schedule production activities to satisfy on-time delivery to customers. There are several performance measures that can be used to evaluate due-date based scheduling in manufacturing. Total weighted tardiness is one due-date related scheduling objective. It incorporates differences in the importance of individual jobs that may lead to different penalties when the delivery is not on time. Because of the complication of these real world manufacturing environments at the present time, numerous job shop scheduling methodologies in academic research cannot be applied to real world scheduling problems. In scheduling a complex job shop problem, production planners may routinely apply several dispatching rules that tend to provide a relatively poor schedule causing a low level of customer satisfaction. Reentrant Flexible Job Shops with Multiple Processing restrictions This research considers reentrant flexible job shop with the processing restrictions including possibly non-zero ready times, sequence-dependent setup times, and possibly finite buffer capacities. A reentrant flexible job shop refers to the machine environment in which each job has to be processed on each one of a series of work centers, has its own predetermined route to follow, and may visit a work center more than once where some work centers may consist of identical parallel machines. Figure 1.1 shows a reentrant flexible job shop with g jobs and m k parallel machines at work center k where a buffer (with both infinite and finite capacities) for each work center is under consideration. 2

14 Jobs ; 1 ( WC route = 1->k->1->3->k ) 2 ( WC route = 2->3->1->2->k ) 3 g Work center 1 machine 1 Buffer : ; machine m 1 2 Work center 2 machine 1 Buffer : ; machine m Work center 3 machine 1 Buffer : ; machine m 3 1 Work center k machine 1 Buffer : ; machine m k 2 1 Figure 1.1 Paradigm for a Reentrant Flexible Job Shop Scheduling reentrant flexible job shops with the goal of minimizing the total weighted tardiness is one of the most complex scheduling problems. Single-machine job shop scheduling to minimize makespan has been proved to be NP-hard (Garey and Johnson 1979). There are several researchers who developed methods to solve this problem although they may not be applicable in practice, especially in modern manufacturing environments. In the last decade, processing restrictions, such as sequence-dependent setup times and recirculation, as well as due-date based objectives in job shop scheduling problems, have received more attention from many researchers. 3

15 However, job shop scheduling with limited buffer capacities that results in blocking seems to be absent in the open literature. As mentioned earlier, blocking can occur in the real world production environment. It causes the computation of the job completion times to be much more complex. Consequently, the scheduling problem is very difficult to comprehend. To specify assumptions for this study, we briefly describe characteristics of the problem in the following. 1. All data including processing times, setup times, ready times, and due dates are known deterministically. 2. Preemption is not allowed. 3. All machines are available continuously. 4. Setup for a job cannot begin until the job is available to the current work center and the desired machine in the work center is idle. 5. Each machine can process at most one job at a time, and each job can only be processed on one machine at a time. Research Objectives The objective in this research is to develop several procedures for scheduling complex job shops with the total weighted tardiness as the objective. In literature minimizing makespan is usually found in job shop scheduling problems because it concerns in reduction the completion time for only the last complete job by moving operations in the longest path in a given schedule (for instance, Nowicki and Smutnicki 1996). However, the total weighted tardiness is considered more relevance scheduling 4

16 objective as customer satisfaction in term of on-time delivery is an important factor in today s business. For method development perspective in job shop scheduling problems, minimizing total weighted tardiness is more difficult comparing to minimizing makespan. It takes every late job in a given schedule into consideration while only single job with the largest completion is only considered in the makespan problem. Therefore the solution space for total weighted tardiness being relatively large results in the difficulties in developing an effective scheduling method. The complex job shops, which refer to the reentrant flexible job shops with nonzero ready times and sequence-dependent setup times, are divided into two classes: the complex job shops with infinite and finite buffer capacities. The scheduling procedures developed include mixed integer programming model (MIP) and local search algorithms for each complex job shop class. These procedures are evaluated, in terms of solution quality and execution speed, based on the size of the complex job shops. In complex job shops with a small number of total operations for all jobs, local search algorithms are compared to the MIP when we are able to solve it optimally. In larger complex job shops, the performance of the local search algorithms is measured by the improvement in solution quality by comparison to several dispatching rules often used in industrial application. Significance of Research The significance of this research can be summarized as follow. 1. This research takes into consideration generalized job shop problems including identical parallel machines, sequence-dependent setup times, non-zero ready times, 5

17 recirculation, and possibly limited buffer capacities. These characteristics usually occur in real world production environments. Local search algorithms developed in this research for scheduling these problems can guarantee production of a feasible schedule. 2. MIP formulations for scheduling complex job shops with both infinite and finite capacities can provide the optimal schedule with lowest total weighted tardiness when dealing with small-sized problems that is specified by the total number of operations for all jobs. These formulations are of value due to the need to compare heuristic solutions with known optimal values. 3. A test-bed of problems has been developed and will be made available to the research community, facilitating further experimentation in this problem domain. Organization of the Dissertation The remaining parts of this dissertation are organized as follows. Chapter 2 presents the literature review of job shop scheduling. In next two chapters we discuss the design and implementation of MIP formulations and local search algorithms for scheduling the complex job shops, where the problems with infinite and finite buffer capacities are presented in chapters 3 and 4, respectively. Lastly, in chapter 5 we summarize the work that we have completed, make recommendations for the implementation of the most suitable algorithms for each complex job shop class, and suggest future research directions. 6

18 CHAPTER 2 LITERATURE REVIEW AND BACKGROUND Introduction Job shop scheduling problems have been widely presented in the open literature. A classical job shop, denoted Jm C max using the notation presented in Pinedo (2002), refers to a job shop with a single machine in each work center and a makespan (C max ) as the scheduling objective. The classical job shop problem has extensively received attention in theoretical research, despite its lack of realistic detail, due to its inherent difficulty. On the other hand, several researchers attempt to create a method which can be applied to job shop scheduling problems with various features occurring in real world manufacturing environments. These features are explained in chapter 1, and include flexible job shops (FJc), ready times (r j ), sequence-dependent setup times (s ij ), recirculation (recrc), finite buffer capacities (block), and due date-based objectives. In this chapter, we provide a literature review consisting of two sections. The first section focuses on job shop scheduling problems with infinite buffer capacities. The second section deals with buffer constraints in job shop scheduling problems as well as flow shop scheduling problems. Job Shop Scheduling Problems with Infinite Buffer Capacities Because the movement of jobs on machines is complex in job shops, a disjunctive graph representation of a classical job shop is presented at the beginning of this section. Next, branch and bound algorithms and mathematical programming formulations 7

19 developed in order to determine an optimal solution are reviewed. After that, heuristic approaches including constructive heuristics and neighborhood search heuristics are discussed. Disjunctive Graph Pinedo (2002) describes a graph used to represent job shop scheduling problems called a disjunctive graph. The disjunctive graph is introduced by Roy and Sussman in In general, the graph consists of sets of conjunctive arcs, disjunctive arcs, and nodes, including a dummy source node and a dummy sink node (Figure 2.1). All arcs are directed. Node (i, j) represents the operation of processing job j on machine i. U 0 0 2,2 0 p 22 p 13 1,1 1,3 p 11 p 13 p 11 p 13 p 23 p 21 2,1 p 32 p 12 p 22 p 12 p 42 1,2 4,2 3,2 p p p 12 p 23 p 42 p 43 2,3 p 21 p 21 p 23 3,1 4,3 p 31 p 43 p 31 V job machine sequence processing time p 11 p 21 p p 22 p 12 p 42 p p 13 p 23 p 43 Figure 2.1 Disjunctive Graph for a Classical Job Shop Problem Conjunctive arcs indicate the precedence constraints between operations of the same job. Two disjunctive arcs, which go in opposite directions, connect two operations that belong to two different jobs processed on the same machine. The length of any arcs leading from node (i, j) indicates the processing time of job j on machine i. Determination of an optimal solution for minimum makespan is achieved by finding a set of disjunctive arcs that minimizes the length of the longest path from the source node (U) to the sink 8

20 node (V). This longest path consists of a set of operations of which the first starts at time 0 and the last finishes at the time of the makespan. Each operation on this path is immediately followed by either the next operation on the same machine or the next operation of the same job on another machine. Note that the selected disjunctive arcs must satisfy a feasible schedule. Pinedo also describes a disjunctive programming formulation for a classical job shop originally presented by Roy and Sussman in The formulation is most often used because it is closely related to the disjunctive graph representation of the job shop. However, the formulation cannot be solved by using typical mathematical programming solution techniques, as a result of disjunctive constraints. Consequently, branch and bound methods are required to obtain optimal solutions. Optimization Techniques Developing an exact method for determining an optimal schedule is one of the current research areas in job shop scheduling, although most of the job shop problems, except J2 C max, are NP-hard (Garey and Johnson 1979). In scheduling simple job shop problems with a single machine at each work center, optimization techniques are capable of finding an optimal solution for scheduling a small number of jobs and machines within a reasonable amount of time. Moreover there are methods that can be applied to these optimization techniques for improving solution speed and solving larger problem sizes. In scheduling more complex job shop problems, optimization techniques may only be able to solve a very small problem. However, developing an optimization technique may still lead to the creation of an approximation algorithm for solving larger problem sizes. A 9

21 number of researchers develop an optimization technique and then use it in a preliminary test for their proposed heuristic approaches for large-sized problems. For example, Chen and Luh (2003) propose a mixed integer program for flexible job shop scheduling. The objective of their scheduling problem is to minimize total weighted tardiness and earliness cost. Then they present an alternative framework to the Lagrangian relaxation approach in which the operation precedence constraints are relaxed. To determine a good operation schedule, the relaxed problem is decomposed into single-stage machine subproblems which are solved using fast heuristic algorithms. Numerical results show that this approach outperforms dispatching rules. In this section, we provide a review of two broad optimization techniques in job shop scheduling research. These techniques include branch and bound techniques and mathematical programming formulations. The branch and bound techniques, one of the most efficient optimization techniques for job shop problems, have often been applied to classical job shop problems. In general the principle of the branch and bound technique is the implicit enumeration of a minimization problem such that branches, which do not contribute to optimal solution, are detected as early as possible. A branch of the enumeration tree defines a subset of the set of all feasible solutions of the original problem. For each subset the objective value of its best solution is estimated by a lower bound. In a level of the enumeration tree a lower bound associated with each branch is compared to the best known upper bound. In case the lower bound exceeds the upper bound, the corresponding branch can be dropped from the next level of the tree. 10

22 Otherwise, the search is continued from the most promising subset which is divided into smaller subsets through the ending level of the tree. The enumeration tree generation and the lower bound determination are the vital parts of the application of the branch and bound algorithms for job shop scheduling problems. Blazewicz et al. (1996) provide an excellent survey of branch and bound algorithms for a classical job shop problem. The authors report that the enumeration tree can be created based on the generation of active schedules using Giffer and Thomson s algorithm or based on disjunctive arcs. The use of Lagrangian relaxation as well as minimization of the maximum lateness (L max ) for a one machine subproblem with release dates and due dates can be used to obtain the lower bound for makespan. For instance, Pinedo (2002) presents a branch and bound technique for this problem. A branching tree is created based on the generation of active schedules introduced by Giffer and Thomson (1960). In the first generation, called level 1, an initial lower bound for the makespan is calculated after adding disjunctive arcs from a selected operation to other operations. Then each job s release dates and due dates for each machine are identified based on a disjunctive graph. The optimal maximum lateness for each machine is found with respect to the current set of jobs assigned to each machine. The lower bound of each node in the branching tree is the sum of the initial lower bound and the highest optimal L max. After examining every node in this level of the branching tree, the search is continued. This approach has found the optimal solution of several 100 operations classical job shop problems, e.g. Fisher and Thomson s benchmark problems. However, with larger problems, e.g. 250 operations, the computation time is prohibitive. 11

23 Besides being used for Jm C max, a branch and bound technique, developed by Single and Pinedo (1998), is applied to solve a single machine job shop problem with total weighted tardiness minimization as the objective. The branching tree is generated based on disjunctive arcs. For problems with more than 10 jobs and 10 machines, the authors select the most critical machine (the machine most likely to be the bottleneck machine) to be scheduled next in the order that generated the best solution in the SB- TWT heuristic developed by Single and Pinedo (1999). The arcs are inserted until the most critical machine has all of its job scheduled. Then the arc insertion scheme is applied to create the complete schedule of the remaining unscheduled machines. To find lower bounds, the authors use two different methods: the arc fixing lower bound modified from Carlier and Pinson (1989), and the multi-machine lower bound modified from Lageweg et al. (1977). This branch and bound algorithm can determine optimal solutions of 10 jobs-10 machines instances, which are generated by adding weights and due dates for every job to several well-known benchmark problems for Jm C max. The authors conclude that the solution time depends on the tightness of due dates and the job routing. The branch and bound algorithms seem to be the most efficient optimization techniques for a small-sized problem of a single machine job shop. However, their applications require a certain level of programmer skill and are limited based on the difficulty in building the branching tree and finding the lower bound in more complex job shops. There have been major contributions of integer programming formulations in job shop scheduling literature, as indicated in Blazewicz et al. (1991). This survey is the first one that attempts to compile a large number of mathematical programming formulations 12

24 for scheduling into a single paper to ease the task of model building and testing the formulations. Pan (1997) evaluates several MIP formulations for a classical job shop problem. The author concludes that the formulation developed by Manne (1960) is the best formulation in terms of model size. Recently, Pan and Chen (2005) propose four extended mixed integer programming formulations for a reentrant job shop to minimize makespan based on the formulation developed by Manne (1960). Layer division procedures are developed and incorporated in the models in order to improve the solution speed. Computational experiments were designed for solving 3 jobs with 5 and 10 machines instances, and 5 jobs with 3 and 5 machines instances. The average number of operations to be scheduled is up to 59 operations including reentrant operations. The results showed that the proposed formulations were able to find optimal solutions for almost all instances within 6 minutes. Kim and Egbelu (1998) propose an integer programming model with a preprocessing step for a job shop scheduling problem with multiple process plans per job to minimize makespan. The formulation is developed based on disjunctive constraint integer programming. The experimental results show that CPU times in solving small problems (up to 5 jobs, 4 machines, and 5 process plans per job) decrease dramatically when using the preprocessing method. Low and Wu (2001) develop an integer programming model for scheduling a flexible manufacturing system (FMS) with sequence-dependent setup times to minimize total tardiness. A disjunctive graph is used to represent the problem and helps in formulating the integer programming. Linearization of constraints has been applied to the original formulation which contains non-linear 13

25 equations. Although computational results are not shown in the paper, the authors report that the proposed formulation can be used to solve problems containing only a few jobs. The literature indicates that, while exact optimal solution techniques maybe desirable from a theoretical perspective, they are limited in their ability to solve moderate or larger problems. Approximation Algorithms Approximation algorithms, usually called heuristic approaches, are used for scheduling large scale job shop problems without guarantee of optimality. They are classified into two categories: constructive methods and iterative methods. The constructive methods build a schedule by adding an unscheduled operation into a partial schedule until the schedule is completed. These methods consist of priority rules, insertion algorithms, and shifting bottleneck heuristics. The priority rules known as dispatching rules are probably the heuristics that are most often used in practice. A number of dispatching rules in a job shop environment are presented by Morton and Pentico (1993). The performance of several dispatching rules is evaluated by Chang et al. (1996). An application of the insertion algorithm for job shop problems is developed by Werner and Winkler (1995). Dispatching rules and insertion algorithms can easily be applied to scheduling all types of job shop problems with a very small computing time. However, the solution quality is not superior when comparing to the solutions obtained from more sophisticated heuristics such as shifting bottleneck heuristics and iterative methods. 14

26 The iterative methods, often referred to as local search, start with a completed schedule and then try to obtain a better schedule by manipulating the current schedule. There are numerous algorithms in the iterative methods that are developed for scheduling several classes of job shop. Blazewicz et al. (1996) survey several local search approaches such as simulated annealing, tabu search, and genetic algorithms for scheduling job shops with makespan minimization. Also, Jain and Meeran (1999) review several iterative heuristics for scheduling a classical job shop problem, e.g. genetic algorithms, genetic local search, reinsertion algorithms, simulated annealing, tabu search. In this job shop class (Jm C max ), promising iterative-based heuristics, including simulated annealing, tabu search, and genetic algorithms, have achieved remarkable success. They provide better solutions compared to most constructive methods, including shifting bottleneck based heuristics. Simulated annealing and tabu search are similar in regard to the general idea of the search. They define how to obtain a new solution from a given schedule by moving one operation at a time based on the same neighborhood structures. The difference is in the mechanism for selection of candidate operations from the neighborhoods; the mechanism in simulated annealing is probabilistic while being deterministic in tabu search. In contrast, genetic algorithms commonly produce a new solution by combining one parent representing a completed schedule of every operation with another parent, in the so-called crossover operation. Within the iterative class of approaches, a sophisticated tabu search called i- TSAB developed by Nowicki and Smutnicki (2005) dominates other iterative-based heuristics in terms of solution quality while the computing time is relatively fast. Since 15

27 tabu search uses knowledge of a critical path in Jm C max to define a small neighborhood that occurs in this specific job shop environment, it remains unclear which local search is the most suitable heuristic to deal with the more complex job shop. The remainder of this section reviews the literature focusing on the use of the shifting bottleneck procedure, genetic algorithms, and tabu search. Shifting Bottleneck Procedure Adams et al. (1988) originally developed a shifting bottleneck (SB) procedure for the classical job shop problem, Jm C max. This SB procedure dominated other heuristics at that time in both solution quality and time. The idea of the SB procedure is to schedule each machine in a job shop optimally under the condition that disjunctive arc directions in each optimal schedule of every single machine concur with an optimal job shop schedule. The general steps of SB consist of subproblem identification and optimization, bottleneck machine determination, sequencing the bottleneck machine, and reoptimization of each scheduled machine. These steps are repeated until all the machines are scheduled. The subproblem identification and optimization intends to find an optimal sequence of jobs in each unscheduled machine. Each subproblem consists of a number of operations that are subject to release dates and due dates that are determined by sequences of operations on other machines. The sequence of operations on a given machine is determined by minimizing the maximum lateness in the associated subproblem. Then the bottleneck machine is the machine which has the highest maximum lateness from previous step. Next, disjunctive arcs in regard to the optimal sequence of jobs in the bottleneck machine are inserted into the disjunctive graph that 16

28 represents the job shop problem. In the reoptimization step, for each scheduled machine, a job sequence in a particular machine is redefined by finding a new optimal solution of this machine subproblem, while keeping job sequences fixed in the remaining machines. This reoptimization method is repeated for all scheduled machines. Although solution quality and computational time of the original SB are exceptional, the difficulty in performing reoptimization is a drawback of this procedure. Also the original SB procedure may sometimes yield infeasible solution resulting from cycle generation in the reoptimization step. To address this problem, Balas and Vazacopoulos (1995) and Dauzere-Peres and Lasserre (1993) suggest that delay precedence constraints must be involved in the subproblem identification and optimization to prevent cyclic schedules. They also propose algorithms for solving the subproblems with delay precedence constraints. There are numerous researchers trying to improve the performance of the original SB procedure. Besides the surveys given by Blazewicz et al. (1996), and Jain and Meeran (1999), some research in the SB-based procedure for Jm C max are Uzsoy and Wang (2000), and Wenqi and Aihua (2004). Uzsoy and Wang (2000) modify a SB procedure using a decomposition method to study the performance of their procedure when bottleneck machines are well-defined. A branch and bound algorithm is applied for rescheduling bottleneck machines. Computational results indicate that their proposed SB is a powerful heuristic for job shop scheduling problems and is better suited to job shops with well-defined bottleneck machines. Wenqi and Aihua (2004) propose two shifting bottleneck heuristics called the improved shifting bottleneck (ISB) and the refined version of ISB (ISBB) Several 17

29 existing SB-based heuristics for Jm C max are examined and compared to their proposed heuristics on a number of well-known benchmark instances. The results show that no individual heuristic dominates others when considering both solution time and quality. However, the benchmark heuristics did not include Balas et al. (1998) which provide the best solution comparing to other SB procedures; this may be due to the use of a local search in the reoptimization step in the Balas approach, meaning that the Balas approach may have not been considered a pure shifting bottleneck heuristic. The SB procedure is applied to many generalizations of Jm C max. The most significant research is developed by Pinedo and Singer (1999) who propose a shifting bottleneck heuristic for job shop scheduling to minimize total weighted tardiness (TWT). The classical disjunctive graph is modified with a minor change in the sink node for representing this problem. The subproblem identification and optimization is the approximation of a minimum total weighted tardiness for single machine scheduling with release time and delay precedent constraints. The proposed heuristic, called SB-TWT, is used to solve several 10 job-10 machine problem instances which are modified from well-known benchmark problems for Jm C max. Computational results show that the SB- TWT solutions are close to the optimal solutions obtained from a branch and bound algorithm (Single and Pinedo 1998). With minor changes, this heuristic approach can be used in a flexible job shop problem. Aspects of the modification approach are described by Pinedo (2002). Mason et al. (2002) proposed a modified shifting bottleneck called MSB for minimizing total weighted tardiness in complex job shops. They define the complex job 18

30 shop as a flexible job shop with sequence-dependent setup times, recirculation, batching, and possibly non-zero ready times. This type of production environment can be found in wafer fabrication facilities in semiconductor industries. In the proposed heuristic, a disjunctive graph is used to represent the problem. The apparent tardiness cost with setups (ATCS) index for parallel machines scheduling problem, developed by Lee and Pinedo (1997), is modified to accommodate batching problems. The modified ATCS, called BATCS, is used in the subproblem solution procedures. The re-optimization step, considering the newly added disjunctive arcs for the selected machine, is the optional step in MSB. To assess the ability of their proposed models, a benchmark problem is used to compare MSB with a number of existing dispatching rules. The results show that the MSB needs the re-optimization step in order to obtain a better solution but the computational time will increase about 50%. Moreover, only 4 of 10 instances without sequence-dependent setup times indicate that MSB is better than the best dispatching rule in term of solution quality. 6 of 10 instances with sequence-dependent setup times indicate that MSB is better than the best dispatching rule. However, since the benchmark examples contain only two product types, it is unclear how well MSB will perform in real-world problems with more than two product types. Mason and Oey (2003) report that cyclic paths in a disjunctive graph may occur when scheduling complex job shops, resulting in a feasible schedule not being generated during the execution of the MSB heuristic (Mason et al. 2002). To eliminate the cyclic paths, they modify the disjunctive graph representing complex job shop problems and identify all possible causes of infeasible schedule generation. Then, the cycle elimination 19

31 procedure (CEP) is developed to tackle this problem. The mechanism of CEP is spotting batching nodes that cause the cyclic schedule, and then removing them. Their proposed procedure is examined in practical wafer fabrication. The results indicate that solution times increase about 10% with the satisfied level of solution quality. However, only two product types and scheduling for one shift are executed, so utilization of the proposed MSB with CEP is still unclear. Genetic Algorithm Genetic algorithms (GAs) are motivated by the theory of evolution from the work by Holland (1975). They seek to imitate the biological phenomenon of evolutionary reproduction. In the problem domain of scheduling, genetic algorithms generally view schedules as individuals (chromosomes) which are members of a population. Each chromosome is characterized by its fitness which is measured through the surrogate of the scheduling objective function. In the search process, genetic algorithms work on populations iteratively from one generation to the next by the application of genetic operators. The most basic GA that uses a binary string solution representation and simple crossover, e.g. one point crossover, cannot be applied to scheduling problems. Davis (1985) initially explores the application of a GA to a classical job shop problem (Jm C max ). He proposes an order-based representation of the Traveling Salesman Problem (TSP). Each chromosome consists of m sub-chromosomes that define a job s preference list in each of the m machines. In decoding he proposes a procedure for mapping from chromosomes to solutions that always creates feasible schedules. Since simple crossover cannot be used in the TSP-based problems, Davis proposed order crossover (OX) in his 20

32 GA that is able to generate a feasible child given any two parents in the evolutionary process. The same idea is subsequently modified by Croce et al. (1995). Also Falkenauer and Bouffoix (1991) adapt this GA in scheduling job shop problems with release times and due dates (Jm r j, d j C max ). However, their works, including the original GA for Jm C max by Davis, seem complicated and not effective due to the difficulty of several specialized operators to be used to insure the feasibility of generated solutions. Cheng et al. (1996) give a tutorial survey of several genetic algorithm applications in job shop scheduling. The focus of their survey is on types of job shop scheduling representation and encoding problems for GA. Cheng et al. (1999) continue their survey and note that the operation sequences on each machine and the precedence constraints among operations for a job result in infeasible solutions after application of some genetic operators. They discuss how to encode a solution of the problem into a chromosome and to ensure that the chromosome will correspond to a feasible solution. Moreover, the authors suggest that incorporating other heuristic methods may enhance the performance of genetic search. Numerous researchers have been applied genetic algorithms in Jm C max as studied by Blazewicz et al. (1996), Jain and Meelan (1999), and Ponnambalam et al. (2001). Among those that do not incorporate other local search heuristics, a GA developed by Bierwirth (1995) seems to provide better quality of solution. He represents an individual as a string having length equal to the number of operations in the job shop. An entry in this string is job identification. For instance, there are three jobs A, B, C having 3, 4, 3 operations, respectively, then a randomly generated string may be (B, A, B, 21

33 B, C, A, C, C, B, A). In this string, entries 2, 6, and 10 correspond to the first, second, and third operation of job A. This representation is called a permutation representation. Now a simple crossover operation can be used to combine two parents. Vasquez et al. (2000) give a comparison of four different genetic algorithms across a set of benchmark problems. The four algorithms are developed by Vasquez and Whitley (2000), Hart and Ross (1998), Lin et al. (1997), and Bierwirth (1995). In Vasquez and Whitley s GA, the chromosome represents a permutation of the jobs for each machine, and each operation also contains its start time. Order-based operators are applied, and the Giffler and Thomson algorithm and non-delay algorithms used to generate schedules. A local search is applied to all offspring using the Grabowski critical neighborhood with a steepest descent technique to improve generating active schedules. A neighborhood is defined as repositioning of an operation to either the front or the rear of a critical block. Lin et al. s GA use the operation and starting time based representation. The time-horizon-crossover exchanges operation sequences from parents such that temporal relations among operations are preserved. The mutation operator selects two operations on the critical path and reverses these operations. In Hart and Ross s GA, each gene in a chromosome of length (number of jobs x number of machines) represent a pair of values, (method, heuristic), where method denotes a choice of two algorithms that should be used at each iteration of a scheduling algorithm to calculate a set of schedulable operations, and heuristic denotes the priority dispatching rule, which is simply a heuristic for selecting an operation to schedule when a machine become idle. 22

34 The computation results of this study show that no algorithm dominates others in all criteria including best and average makespan, variation of makespan, and solution time. Several solution improvement techniques have been applied in the attempt to reach optimal solutions of well-known benchmark problems. For example, Goncalves et al. (2005) develop a GA using random keys representation integrated with a parameterized active schedule builder and local search for fitness function improvement. Their concept design seems to achieve a remarkable results but the GA cannot reach optimal solutions in some hard benchmark problems with 15 jobs and 15 machines. Moreover, it takes greater than 30 minutes in solving time for each of those problems, whereas other local search methods provide better solutions within 1 minute. In general, it appears that tabu search may be better suited to the problem than GAs. A GA integrated with a simulated annealing algorithm by Wang and Zheng (2002), two hybrid GAs called single GA and parallel GA by Park et al. (2003), a GA with niching methods for finding multiple solutions by Perez (2003), Local search GAs by Ombuki and Ventresca (2004), and a GA with modified crossover operator by Watanabe et al. (2005) are the other GAs that fail to attack the well-known benchmark problems compared to a powerful tabu search such as fast tabu search developed by Nowicki and Smutnicki (1996). This is because tabu search (or other local search algorithms utilizing a neighborhood structure) exploits some specific properties in Jm C max problems to create a small neighborhood structure from the longest path (which is C max ) and search for a new solution within this structure without checking if the new solution is feasible. However, the advantage of the GA is that they require less knowledge about the 23

35 structural properties of a problem which leads to the ability to apply GA in complex job shop problems. Norman and Bean (1999) develop a genetic algorithm based on random keys representation, elitist reproduction, Bernoulli crossover, and immigration type mutation. The GA, called a random keys genetic algorithm (RKGA) is designed for solving complex scheduling problems. In this article the authors apply RKGA for scheduling identical parallel machines problems with non-zero ready times, sequence-dependent setup times, machine downtimes, and tool availability with the goal of minimizing the total tardiness. Their computation results show that RKGA is the best heuristic comparing to other existing heuristics. Norman and Bean (1997) also utilize the RKGA in scheduling Jm C max. A delay factor encoding is added into the original random keysbased representation in order to reduce undesirable idle time on a machine that may occur during decoding a chromosome to a feasible solution. With the delay factor, move search performs a look-ahead search to compensate for undesired idle times that may occur in job shop scheduling. Even though the RKGA is unable to provide the better solution compared to the fast tabu search (by Nowicki and Smutnicki), the solutions are close and statistically proved to be better than other benchmark heuristics such as the SB by Adam et al. Moreover, the authors conclude that the RKGA is able to be applied to scheduling a generalized job shop that may contain parallel machines, sequence-dependent setups, tool constraints, precedence, and alternative routings. Cheung and Zhou (2001) use an integration of a genetic algorithm and heuristic rules for job shop scheduling problems with sequence-dependent setup times to minimize 24

36 makespan (Jm s ij C max ). In the algorithm, only the first operations on each machine are obtained by GA while the rest of the operations on each machine are scheduled using a heuristic rule implemented by a simulator. The Shortest Processing Time (SPT) rule is used as the heuristic for specifying the processing priority of each job. The computational results indicate that the GA is better in solution quality but running time is much greater than a solution from a most work remained being scheduled first (MWKR) heuristic developed by Choi and Kokmaz (1997). Zhang and Yan (2005) develop a GA for a job shop scheduling problems with batching and setup times to minimize makespan (Jm s j, batch C max ). The authors motivate batching as being required when setup time is longer than processing time, as in a punching workshop in automobile manufacturing industries. A multi-section coding strategy is applied for chromosome representation. Three independent crossover operators are also developed in their hybrid GA. The results of comparing the GA to priority dispatching rules show that the GA outperforms other heuristics. Chen et al. (1999) develop a GA for a flexible job shop scheduling problem to minimize makespan (FJc C max ). In the representation, encoding each individual requires two chromosomes. The first chromosome defines the routing policy and the second chromosome defines the sequence of the operations on each identical parallel machine. In reproduction, single-point and two-point crossovers are applied in the first chromosome. For the second chromosome, two crossovers called order-preserving singlepoint crossover and order-preserving two-point crossover are developed. The GA is tested in a SPARC-workstation on three instances. The results show that the instances 25