MULTI-OBJECTIVE FLOW SHOP SCHEDULING USING METAHEURISTICS

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MULTI-OBJECTIVE FLOW SHOP SCHEDULING

USING METAHEURISTICS

Submitted in fulfillment

of the requirement for the award of the degree of

Doctor of Philosophy

in

Mechanical Engineering

by

Ashwani Kumar Dhingra

Registration No.

2K06-NITK-Ph D 1121M

Submitted to

Department of Mechanical Engineering

National Institute of Technology Kurukshetra

Kurukshetra 136 119 Haryana INDIA

Under the supervision of

Dr. Pankaj Chandna

Department of Mechanical Engineering National Institute of Technology Kurukshetra

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CERTIFICATE

This is to certify that the thesis entitled ‘MULTI-OBJECTIVE FLOW SHOP SCHEDULING USING METAHEURISTICS’ being submitted by Ashwani Kumar Dhingra, in fulfillment of the requirement for the award of degree of Doctor of Philosophy in Mechanical Engineering, has been carried out under my supervision and guidance.

The matter embodied in this thesis has not been submitted, in part or in full, to any other university or institute for the award of any degree, diploma or certificate.

(Dr. Pankaj Chandna) Associate Professor

Department of Mechanical Engineering National Institute of Technology Kurukshetra Kurukshetra 132 119 Haryana INDIA

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ABSTRACT

Production scheduling is generally considered to be the one of the most significant issue in the planning and operation of a manufacturing system. Better scheduling system has significant impact on cost reduction, increased productivity, customer satisfaction and overall competitive advantage. In addition, recent customer demand for high variety products has contributed to an increase in product complexity that further emphasizes the need for improved scheduling. Proficient scheduling leads to increase in capacity utilization efficiency and hence thereby reducing the time required to complete jobs and consequently increasing the profitability of an organization in present competitive environment. There are different systems of production scheduling including flow shop in which jobs are to be processed through series of machines for optimizing number of required performance measures.

Classical flow shop scheduling problems are mainly concerned with completion time related objectives, however, in modern manufacturing and operations management, on time delivery is a significant factor as for the reason of upward stress of competition on the markets. Industry has to offer a great variety of different and individual products while customers are expecting ordered goods to be delivered on time. Hence, there is a requirement of multi-objective scheduling system through which all the objectives can be achieved simultaneously. Also flow shop with sequence dependent setup times (SDST) are among the most difficult class of scheduling problems. Effective managing of sequence dependent setups is one of the critical factors to improve manufacturing system performance and hence it must be considered separately from the processing time when dealing with scheduling problems.

Hence, for the requirement of market competitiveness, effective utilization of resources for meeting goals of an industry or organization, multi-objective SDST flow shop scheduling has been considered in the present work in which objective is to minimize the weighted sum of total weighted squared tardiness, makespan, total weighted squared earliness and number of tardy jobs simultaneously. It is a known fact that flow shop scheduling problem with makespan or completion time based criteria is NP hard and even small size problems are difficult to solve. Multi-objective flow shop scheduling with sequence dependent set up time also becomes NP hard with greater complexity toward optimality in a reasonable time. Metaheuristics has become greater choice for solving NP

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hard problems because of their multi-solution and strong neighborhood search capabilities in a reasonable time. Genetic algorithm and simulated annealing are the main class among population and neighborhood search metaheuristics. Quality of these metaheuristics when applied to scheduling problems depends on the initial seed sequence as good initial solution might provide the better final solution. Therefore, in the present work, six modified NEH heuristic have been proposed along with some existing despatching rules for initial feasible sequence. Sequence obtained from the modified heuristics/despatching rules is combined with initial population of genetic algorithm and hence called as Hybrid Genetic Algorithms (HGA). Similarly, simulated annealing also starts from the initial feasible sequence for search towards optimality and called as Hybrid Simulated Annealing (HSA). Hence different HGAs and HSAs along with hybridization of Genetic Algorithm and Simulated Annealing (GASA) have been proposed for multi-objective SDST flow shop scheduling problem. As the solution quality of metaheuristics depends on stopping criteria of the algorithm, in the present work, stopping limit of all the metaheuristics have been considered as computational time limit which further depends upon on number of jobs and machines for fair comparison. The performance among different heuristics and metaheuristics based hybrid approaches have been compared with the help of a defined performance index upto 200 job and 20 machine benchmark problems of Taillard (1993).

Amongst different HGAs, HGA (NEH2), HGA (NEH4) and HGA (NEH5) provides better results and varies with the increase in number of machines especially for larger job size in SDST flow shop scheduling problems. Among three HGAs, It may be observed that HGA (NEH4) for 5 and 20 machine problems, HGA (NEH5) for 10 machine problems show improvement over other HGAs with all sets of weight values considered. However for 20 machine problems, HGA (NEH2) provides better results upto100 jobs.

Also, from the comparative analysis among HSAs, it has also been concluded that their performance varies with the size of machines and jobs as HSA (NEH6) for 5 machines, HSA (NEH4) for 10 and 20 machine problems shows better performance when compared with other HSAs especially for larger job size for all the considered sets of weight values.

Further, Proposed hybrid GASA also proves to be an effective approach for optimizing multi-objective SDST flow shop scheduling for any job and machine size as

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compared to GA and SA alone. The main advantage of GASA is that it provides a quick solution without any heuristics for initialization as required in HGA and HSA.

The performance of HGAs, HSAs and GASA have also been compared with same computational time limit based stopping criteria and it is shown that HGA (NEH2) for 5 & 20 machine and HSA (NEH5) for 10 machine problems confirm the superiority over others for all the considered sets of weight values. However for larger job size, HSA (NEH5), HGA (NEH5) and HGA (NEH4) provides improved results for 5, 10 and 20 machines problems respectively.

When applied to real life SDST flow shop scheduling problem in Tubular Component Division of Tube Investment of India verifies that proposed hybrid algorithm is a viable and effective approach for minimizing the multi-objective fitness function.

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ACKNOWLEDGEMENT

I wish to express my deep sense of gratitude to my erudite supervisor

Dr. Pankaj Chandna, Associate Professor, Mechanical Engineering Department, National Institute of Technology, Kurukshetra. I owe his overwhelming debt for helping me to plan and discuss the contents of this work. His ideas, stimulating comments, interpretations and suggestions increased my cognitive awareness and helped considerably in the fruition of my objective. I remain obliged to him for his help and able guidance through all stages of this thesis. His constant inspiration and encouragement to my efforts shall always be acknowledged. His observation and comments were highly enlightening.

I am grateful to Dr. S.S. Rattan, Professor and Chairman, Department of Mechanical Engineering and Dr. K.S. Kasana, Professor and former Chairman, Department of Mechanical Engineering, National Institute of Technology, Kurukshetra to facilitate my work and for their support and encouragement during the course of my work.

I thank my Father, my wife Sunita Dhingra, sweet daughter Avnee and my other relatives for their support in all work that I choose to commence. Without their motivation I may never have reached this goal and I owe this to them.

I am also grateful to Dr. K.K Dubey, Mr. Bhim Singh and Mr Arun Gupta for their help in my thesis. I would like to thank the officials of the concerned industry specially Er. Gagan Chopra and Er. Ajay Kumar for their cooperation in data collection. Most important, I thank my friends, who for their large number cannot be named here, but who are loved and appreciated for seeing me through this thesis and a lot more. There have been numerous influences, big and small, that have helped me complete my thesis. I would like to thank them all.

Above all, I express my indebtedness to the ALMIGHTY for all his blessing and kindness.

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CHAPTER 4 PROPOSED HEURISTIC AND METAHEURISTICS

55 - 69

4.1 INTRODUCTION 55

4.2 MODIFIED NEH HEURISTIC 55

4.3 METAHEURISTICS 58

4.4 HYBRID GENETIC ALGORITHM (HGA) 59

4.4.1 Proposed HGA 60

4.4.2 Parameters of Genetic Algorithm 62

4.5 SIMULATED ANNEALING 64

4.5.1 Proposed Hybrid Simulated Annealing (HSA) 65 4.5.2 Parameters of Hybrid Simulated Annealing 66

4.6 HYBRIDIZATION OF GENETIC ALGORITHM

AND SIMULATED ANNEALING (GASA)

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CHAPTER 5 RESULTS AND DISCUSSIONS 70 - 95

5.1 HYBRID GENETIC ALGORITHM (HGA) 71

5.2 HYBRID SIMULATED ANNEALING (HSA) 81

5.3 HYBRID GENETIC ALGORITHM AND

SIMULATED ANNEALING (GASA)

90

5.4 COMPARATIVE ANALYSIS OF HGA, HSA AND

GASA

94

CHAPTER 6 CASE STUDY 96 - 105

6.1 INTRODUCTION 96

6.2 DATA COLLECTION 98

6.3 RESULTS AND DISCUSSIONS 101

CHAPTER 7 CONCLUSIONS AND FURTHER SCOPE OF WORK

106 – 108

7.1 CONCLUSIONS 106

7.2 LIMITATIONS AND FURTHER SCOPE OF THE WORK

107

REFERENCES 109 - 117

APPENDIX A LIST OF PUBLICATIONS 118

APPENDIX B HSA CODE (MATLAB) FOR MULTI-OBJECTIVE SDST FLOW SHOP SCHEDULING

119 - 131

APPENDIX C HGA CODE (MATLAB) FOR MULTI-OBJECTIVE SDST FLOW SHOP SCHEDULING

132 - 146

APPENDIX D GASA CODE (MATLAB) FOR

MULTI-OBJECTIVE SDST FLOW SHOP SCHEDULING

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NOTATIONS USED

n Number of jobs

m Number of machines

j Index for Jobs

i Index for machines

α Weight for total weighted squared tardiness

β Weight for makespan

γ Weight for total weighted squared earliness δ Weight for no. of tardy jobs

j

C Completion time of job ‘j’

j

d Due date of job ‘j’

j

E Earliness of job ‘j’

j

T Tardiness of job ‘j’

t

N _{Number of tardy jobs}

j

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x

LIST OF TABLES

Table No. Title Page No.

Table 4.1 Different despatching rules/heuristics for comparative analysis of proposed HGA and HSA.

61

Table 5.1 Parameters fixed for different HGAs. 72

Table 5.2 Mean RPD (%) of HGAs for 5, 10 and 20 machines. 80

Table 5.3 Parameters for different HSAs. 81

Table 5.4 Mean RPD for different HSAs. 89

Table 5.5 Parameters fixed for GASA. 91

Table 5.6 Mean RPD (%) for GASA, GA and SA. 94

Table 5.7 Comparison of average RPD (%) amongst HGA, HSA and GASA.

95

Table 6.1 Processing time of 15 jobs on 7 machines. 99

Table 6.2 Due date, weight and demand of jobs. 99

Table 6.3 Sequence dependent set up time on SPCNCM. 99

Table 6.4 Sequence dependent set up time on SPSM. 99

Table 6.5 Sequence dependent set up time on SPDM. 100

Table 6.6 Sequence dependent set up time on TM. 100

Table 6.7 Sequence dependent set up time on MCM. 100

Table 6.8 Sequence dependent set up time on HMM1. 101

Table 6.9 Sequence dependent set up time on HMM2. 101

Table 6.10 Completion time matrix for α = 0.2, β = 0.1, γ = 0.4, δ = 0.3 and

α = 0.1, β = 0.4, γ = 0.3, δ = 0.2.

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Table 6.11 Idle time matrix for α = 0.2, β = 0.1, γ = 0.4, δ = 0.3 and α = 0.1, β = 0.4, γ = 0.3, δ = 0.2.

102

Table 6.12 Completion time matrix for α = 0.4, β = 0.3, γ = 0.2, δ = 0.1 and

α = 0.25, β = 0.25, γ = 0.25, δ = 0.25.

103

Table 6.13 Idle time matrix for α = 0.4, β = 0.3, γ = 0.2, δ = 0.1 and α = 0.25, β = 0.25, γ = 0.25, δ = 0.25.

103

Table 6.14 Completion time matrix for α = 0.3, β = 0.2, γ = 0.1, δ = 0.4. 104

Table 6.15 Idle time matrix for α = 0.3, β = 0.2, γ = 0.1, δ = 0.4. 104

Table 6.16 Values of objective function for different weights for case study.

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LIST OF FIGURES

Figure No. Title Page No.

Figure 1.1 Information flow diagram in a manufacturing system. 5

Figure 1.2 Classification of scheduling problems based on requirement generations.

6

Figure 1.3 Classification of common search methodologies. 15

Figure 1.4 Classification of common metaheuristics. 15

Figure 5.1 RPD for different HGAs for 5 machine problems with α = 0.4, β = 0.3, γ = 0.2, δ = 0.1.

72

Figure 5.2 RPD for different HGAs for 10 machine problems with α

= 0.4, β = 0.3, γ = 0.2, δ = 0.1.

73

= 0.4, β = 0.3, γ = 0.2, δ = 0.1.

73

74

= 0.1, β = 0.4, γ = 0.3, δ = 0.2.

74

= 0.1, β = 0.4, γ = 0.3, δ = 0.2.

75

Figure 5.7 RPD for different HGAs for 5 machine problems with α = 0.2, β = 0.1, γ = 0.4, δ = 3.

75

= 0.2, β = 0.1, γ = 0.4, δ = 0.3.

76

= 0.2, β = 0.1, γ = 0.4, δ = 0.3.

76

77

= 0.3, β = 0.2, γ = 0.1, δ = 0.4.

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= 0.3, β = 0.2, γ = 0.1, δ = 0.4.

78

= 0.25, β = 0.25, γ = 0.25, δ = 0.25.

79

= 0.25, β = 0.25, γ = 0.25, δ = 0.25.

79

Figure 5.16 RPD for different HSAs for 5 machine problems with α = 0.4, β = 0.3, γ = 0.2, δ = 0.1.

82

Figure 5.17 RPD for different HSAs for 10 machine problems with α

= 0.4, β = 0.3, γ = 0.2, δ = 0.1.

82

= 0.4, β = 0.3, γ = 0.2, δ = 0.1.

83

= 0.1, β = 0.4, γ = 0.3, δ = 0.2.

84

= 0.1, β = 0.4, γ = 0.3, δ = 0.2.

84

85

= 0.2, β = 0.1, γ = 0.4, δ = 0.3.

85

= 0.2, β = 0.1, γ = 0.4, δ = 0.3.

86

= 0.3, β = 0.2, γ = 0.1, δ = 0.4.

87

= 0.3, β = 0.2, γ = 0.1, δ = 0.4.

87

Figure 5.28 RPD for different HSAs for 20 machine problems for α = 0.25, β = 0.25, γ = 0.25, δ = 0.25.

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= 0.25, β = 0.25, γ = 0.25, δ = 0.25.

88

= 0.25, β = 0.25, γ = 0.25, δ = 0.25.

89

Figure 5.31 RPD for GASA with α = 0.4, β = 0.3, γ = 0.2, δ = 0.1. 91

Figure 5.32 RPD for different GASA with α = 0.1, β = 0.4, γ = 0.3, δ

= 0.2.

92

= 0.3.

92

= 0.4.

93

Figure 5.35 RPD for GASA with α = 0.25, β = 0.25, γ = 0.25, δ = 0.25.

93

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CHAPTER 1

INTRODUCTION

1.1. SCHEDULING

Scheduling is the process of generating the schedule and schedule is a physical document and generally tells the happening of things and shows a plan for the timing of certain activities. Generally, scheduling problem can be approached in two steps; in the first step sequence is planned or decides how to choose the next task. In the second step, planning of start time and perhaps the completion time of each task is performed.

In a scheduling process, the type and amount of each resource should be known so that accomplishing of tasks can be feasibly determined. Boundary of scheduling problem can be efficiently determined if resources are specified. In addition, each task is described in terms of such information as its resource requirement, its duration, the earliest time at which it may start and the time at which it is due to complete. Any technological constraints (precedence restrictions) should also be described that exist among the tasks. Information about resources and tasks defines a scheduling problem and its solution is fairly a complex matter. Many of the early progress in the field of scheduling were aggravated by problems and hence it is usual to employ the vocabulary of manufacturing when describing scheduling problems. Now, although scheduling work is of considerable significant in many non manufacturing areas with still using terminology of manufacturing. Thus, resources are usually called machines and tasks are called jobs. Sometimes, jobs may consist of several elementary tasks called operations. The environment of the scheduling problem is called the job shop or simply the shop. Scheduling problems in an industry contain a set of tasks to be carried out with a set of limited resources available to perform these tasks. The general problem is to determine the timing of the tasks while recognizing the capability of the resources with given tasks and resources, together with some information about uncertainties. This problem usually arises within a decision making hierarchy in which scheduling follows some earlier, more basic decisions. In industries, analogous decisions are usually said to be part of the planning function. Among other things, the planning function might describe the design

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of a company’s products, the technology available for making and testing the required parts, and the volumes to be produced. In short, the planning function determines the resources available for production and the tasks to be scheduled.

The present industrial environment is characterized by markets facing competition from which customer requirements and expectations are becoming increasingly high in terms of quality, cost and delivery times. This evolution is made stronger by rapid development of new information and communication technologies which provide a direct connection between industries and customers. Hence, Industry performance is built on two dimensions:

• A technological dimension, whose goal is to develop intrinsic performance of marketed products in order to satisfy requirements of quality and lower cost of ownership for these products, technological innovation plays an important role and can be a differentiating element for market development and penetration. In this regard, it is known that rapid product technological growth and the personalization requirements for these products expected by markets often lead companies to forsake mass production and instead focus on small or medium-sized production runs, even on-demand manufacturing. This requires them to have flexible and progressive production systems, able to adapt to market demands and needs quickly and efficiently.

• An organizational dimension intended for performance development in terms of production cycle times, respect of expected delivery dates, inventory and work in process management, adaptation and reactivity to variations in commercial orders, etc. This dimension plays an increasingly important role as markets are increasingly volatile and progressive, and require shorter response times from companies.

Therefore, Industries must have powerful methods and tools at their disposal for production organization and control and that focused attention on satisfying customer needs under the best possible conditions.

To achieve these goals, an organization relies on the implementation of a number of functions together with scheduling which plays a very important role. Indeed, the scheduling function is intended for the organization of human and technological resource

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use in company workshops to directly satisfy customer’s requirements or demands issued from a production plan prepared by the company planning function. Considering market development and requirements, this function must organize the simultaneous completing of several jobs using flexible resources which are available in limited amounts, which becomes a complex problem to solve. In addition, it is this function which ultimately responsible for product manufacturing. Its efficiency and failures will therefore highly affect the company’s relationship with its customers. Within companies, this function has obviously always been present, but today it must face increasingly complex problems because of the large number of jobs that must be executed simultaneously with shorter manufacturing times.

1.1.1 Significance of Scheduling

Scheduling is a decision making practice that is used on a regular basis in many manufacturing and services industries. Its aim is to optimize one or more objectives with the allocation of resources to tasks over given time periods. The resources and tasks in an organization can take a lot of different forms. The resources may be machines in a workshop, crews at a construction site, processing units in a computing environment, and runways at an airport and so on. The tasks may be operations in a production process, take-offs and landings at an airport, executions of computer programs, stages in a construction project, and so on. Each task can have a definite priority level, an earliest likely starting time and a due date. The objectives can also take many different forms and one objective may be the minimization of the completion time of the last job and another may be the minimization of the number of jobs completed after their respective due dates. Scheduling plays an important role in most manufacturing and service systems as well as in most information processing environments.

Scheduling derives its importance from the two different considerations:

• Ineffective scheduling results in deprived utilization of available resources. A noticeable symptom is the idleness of facilities, human resources and apparatus waiting for orders to be processed. As a result of this cost of production increases.

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• Poor scheduling normally create delays in the flow of some orders through the systems. Thus calls for advance measures that again increase cost.

1.1.2 Scheduling in a Manufacturing System

There are several functions that have to be performed to deal with operational problems of the manufacturing system during transformation of the raw material into the finished product. The functions that are generally followed in the management of the manufacturing system are as follows:

i. Aggregate production planning: This function determines the type and quantities of products to be produced in specific time periods.

ii. Process planning: It determines the production process and their routes through which the materials are transformed into finished products.

iii. Scheduling: This function determines an implementation of plan for the time schedule for every job contained in the process route adopted.

iv. Production plan implementation: It executes the actual production operations according to the time schedule.

v. Production control: This function monitors the production plan implementation function. Whenever actual production progresses and performance deviates from the production standards in plan and schedule, such deviations are measured and appropriate modifications are made. Since the focus of present work is on scheduling function, so this important function is discussed in detail in the present work.

Consider the manufacturing environment and the role of scheduling that orders that are released in a manufacturing setting have to be translated into jobs with associated due dates. These jobs often have to be processed on the machines in a given order or sequence. The processing of jobs may sometimes be delayed if certain machines are busy and preemptions may occur when high priority jobs arrive at machines that are busy. Unforeseen events on the shop floor, such as machine breakdowns or longer-than-expected processing times, also have to be taken into account, since they may have a major impact on the schedules.

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Pinedo (2005) demonstrated the information flow in a manufacturing system as shown in figure 1.1. In a manufacturing environment, the scheduling function has to interact with other decision making functions. One popular system that is widely used is the Material Requirements Planning (MRP) system. After a schedule has been generated it is necessary that all raw materials and resources are available at the specified times. The

Figure 1.1 Information flow diagram in a manufacturing system.

ready dates of all jobs have to be determined jointly by the production planning/scheduling system and the MRP system. MRP systems are normally fairly elaborate. Each job has a Bill of Materials (BOM) itemizing the parts required for production for which the MRP system keeps track of the inventory of each part. Furthermore, it determines the timing of the purchases of each one of the materials. In doing so, it uses techniques such as lot sizing and lots scheduling that are similar to those used in scheduling systems. There are many commercial MRP software packages available and, as a result, there are many manufacturing facilities with MRP systems. In the cases where the facility does not have a scheduling system, the MRP system may be used for production planning purposes. However, in complex settings it is not easy for an MRP system to do the detailed scheduling satisfactorily.

Shopfloor Scheduling and rescheduling Dispatching Shopfloor management Shop status Data collection status Job loading Schedule Detailed scheduling Shop orders, release dates Scheduling constraints Capacity status Material requirements, planning, capacity planning Production planning, master scheduling Quantities, due dates Material requirements Orders, demand forecasts Schedule performance

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6 1.1.3 Classification of Scheduling Problems

According to French (1982), the general scheduling problem is to find a sequence, in which the jobs (e.g., a basic task) pass between the resources (e.g., machines), which is a feasible schedule, and optimal with respect to some performance criterion. Graves (1981) introduced a functional classification scheme for scheduling problems. This scheme categorizes problems using the following dimensions:

a) Requirement generation

b) Processing complexity

c) Scheduling criteria

d) Parameter variability

e) Scheduling environment

(a) Based on requirements generation, it can be classified as an open shop or a closed shop. An open shop is "build to order" and no inventory is stocked and when orders are filled from existing inventory it is called closed shop. Closed shop can further classified into job shop and flow shop and detailed classification of scheduling problem is shown in figure 1.2.

Figure 1.2 Classification of scheduling problems based on requirement generations.

Job shop with duplicate machines Job shop Open shop Flow shop Hybrid Flow shop Hybrid Flow shop Single operation defined for each job

Identical routing defined for each job

Identical routing defined for each job

Identical routing per job & single resource per operation Permutation flow shop Single machine flow shop Specifically identical

routing defined for each job

Identical routing per job single resource per operation and “non passing” constraints

Single resource per operation

Single operation & single resource per job Specific routing

definition for each job

Single resource per operation

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(b) Processing complexity refers to the number of processing steps and workstations associated with the production process. This dimension can be decomposed further as follows:

(i) One stage, one processor

(ii) One stage multiple processors

(iii) Multistage flow shop

(iv) Multistage job shop

The one stage, one processor and one stage, multiple processors problems require one processing step that must be performed on a single resource or multiple resources respectively. In the multistage, flow shop problem each job consists of several tasks, which require processing by distinct resources; but there is a common route for all jobs. Finally, in the multistage, job shop situation, alternative resource sets and routes can be chosen, possibly for the same job, allowing the production of different part types.

(c) Scheduling criteria, states the desired objectives to be met. “They are numerous, complex, and often conflicting”. Some commonly used scheduling criteria include the following:

Minimize total tardiness, Minimize the number of late jobs, Maximize system/resource utilization, Minimize in-process inventory, Balance resource usage, Maximize production rate etc.

(d) The dimension ‘parameters variability’ indicates the degree of uncertainty of the various parameters of the scheduling problem. If the degree of uncertainty is insignificant (i.e. the uncertainty in the various quantities is several orders of magnitude less than the quantities themselves), the scheduling problem could be called deterministic. For example, the expected processing time is six hours, and the variance is one minute. Otherwise, the scheduling problem could be called stochastic.

(e) The dimension, scheduling environment, defined the scheduling problem as static or dynamic. Scheduling problems in which the number of jobs to be considered and their ready times are available are called static. On the other hand, scheduling problems in which the number of jobs and related characteristics change over time are called dynamic.

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8 1.2 FLOW SHOP PRODUCTION SYSTEM

A general job shop problem suppose having n jobs {j1, j2, j3 --- jn} to be processed through m machine {m1, m2, m3 --- mm}. Technological constraints demand that each job should be processed through the machine in a particular order and gives an important special case named as flow shop. Thus in case of flow shop jobs pass between the machine in the same order i.e. if j1 must be processed on m1 before machine m2 then the same the true for all jobs. This technological constraint therefore gives the form like:

Job Processing order

j1 m1 m2 m3 --- mm

j2 m1 m2 m3 --- mm

- ---

jn m1 m2 m3 --- mm

For a general job shop problem defined above the number of possible sequences are (n!) m, where n is number of jobs and m is the number of machines. With the above technological constraints in case of flow shop number of different sequence reduces to (n!). This reduced number is quite large for even moderate size problems and recognized to be NP hard (Garey et al., 1976; Gonzalez and Sahni, 1978; Pinedo, 2005 and several others). There has been an attempt to solve this problem with typical objective function being the minimization of average flow time, minimizing the time required to complete all the jobs or makespan, minimizing average lateness values or tardiness, minimizing maximum tardiness, and minimizing the number of tardy jobs.

1.3 SET UP TIME

In actual practice, many problems are encountered in a real world whenever some time is spent in bringing a given facility to a desired state for processing the job. The time spent is called set up time and its magnitude depends upon the job just completed and job waiting to be processed and hence occupied a substantial percentage of the available production time on the manufacturing equipment. Some example of the above situation

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includes container manufacturing industry: where the machines are to be adjusted whenever the dimensions of the containers are changed; the paint industry: in which parts of different colors are produced on the same piece or equipment and printing press: where printing in different colors is to be done on the same machine; a paper bag industry: in which a machine is to be switched over from one type of bag to another, a set-up time is incurred. The length of the set-up time on the machine depends on the similarities between the consecutive orders (the number of colors in common, the differences in bag sizes, etc.).

Flow shop scheduling research has been explored in several general scheduling review works, such as Allahverdi et al. (1999), Graham et al. (1979). Some reviews, such as Dudek, Panwalkar, and Smith (1991) focused entirely on flow shop scheduling. Still other reviews focus on one particular aspect of flow shop scheduling. For example, Dannenbring (1977) and Park, Pegden, and Enscore (1984) focused on flow shop scheduling heuristics, whereas Hall and Sriskandarajah (1996) focused on flow shop scheduling problems with blocking and no-wait in process.

The above-mentioned focused flow shop scheduling assume that the setup times, defined as the time required to shift from one job to another on a given machine, are either included in the processing times or are negligible, and hence are ignored. However, in general, the breakdown structure of various operations of a job on a machine given by Allahverdi et al. (2008) may be as follows:

i. Setup time that is independent of the job to be processed. This operation consists of activities such as fetching the required jigs and fixtures and setting them up on the machine.

ii. Setup time that is dependent on the sequence of job to be processed. This operation includes the time required to set the job in the jigs and fixtures and to adjust the tools as per job sequence and called as sequence dependent set up time.

iii. Processing time of the job being processed.

iv. Removal time that is dependent on the job that has been processed. This operation includes activities such as disengaging the tools from the job, and releasing the job from the jigs and fixtures.

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v. Removal time that is independent of the job that has been processed. This operation includes activities such as dismounting the jigs, the fixtures and/or tools, inspecting/sharpening of the tools, and cleaning the machine and the adjacent area.

Combination of (i, ii) and (iv, v) called as separable set up and separable removal times and (iii) denotes processing time. An operation is separable from processing if it is not part of the processing operation and hence it is obvious that separable setup/removal times should be considered explicitly when they are not negligible in the scheduling problem.

1.3.1 Sequence Dependent Set up Time (SDST)

There are situations in which it is simply not acceptable to assume that the time required to set up machine for the next job is independent of the job was the immediate predecessor on the machine. Infact, the variation of setup time with sequence provides the dominant criterion for evaluating schedule. These situations are often found in process industries and are frequently associated with the problem of lot sizing. For example the scheduling problem in the group technology environment for each family of parts a long set up time is required to initiate the process of family parts after which a short changeover time is required which depend upon the sequence of jobs preceding a particular family part (job being processed). In such cases it is advantages to consider these changeover times explicitly in the identification of an optimal schedule.

It has been reported in the literature that sequence dependent set up time (SDST) is one of the most recurrent additional complications in the scheduling problem. Very limited number of research on flow shop scheduling under SDST environment for due date related performance measure have been done. As setup times are defined to be the work to arrange the resources, process, or bench for tasks which includes obtaining tools, positioning work in process material, cleaning up, adjusting and returning tools and inspecting material in manufacturing system. Kim and Bobrowski (1994) conducted a simulation study to demonstrate the impact of sequence dependent setup times on shop performance and concluded that setup time must be considered separately in any scheduling strategy. The single machine scheduling problem with sequence-dependent

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setup times is known to be NP-hard (Pinedo, 2005). Setup times and lot size are key factors that affect shop performance (Krazewski et al., 1997). Wortman (1992) stated that if sequence dependent setup times have not been addressed adequately than greater competitiveness has been hindered. Hwang and Sun (1998) considered ‘n’ jobs on two machines problem in which the setup time for a job in the first machine was dependent not on the immediately preceding job but on the job which was two steps prior to it. The impact of sequence dependent setup times has been investigated by researchers for real-world job shop environment in glass industry (Chevalier et al., 1996), metallurgical industry (Narasimhan and Mangiameli, 1987) , paper & textile industry (Sherali et al., 1990), chemical industry (Fortemps et al., 1996) and aerospace industry (Li, 1997).

In a survey of industrial schedulers, 70% of the schedulers reported that they had to deal with sequence dependent setups (Luh et al. 1998). Industrial evidence and academic research suggest that effective managing of sequence-dependent setups is one of the critical factors to improve manufacturing system performance. Pinedo (2005) cited machine setup time as a significant factor when production scheduling is devised in all flow patterns, and it may easily consume more than 20% of available machine capacity if not well handled. Also, the completion times of production and machine setups are influenced by the product mix and production sequence. Scheduling problems with sequence dependent setup times are among the most difficult classes of scheduling problems. A single-machine sequence-dependent setup scheduling problem is NP-hard (Pinedo 2005) and even for a small system, the complexity of this problem is beyond the reach of existing theories (Luh et al. 1998).

Gendreau et al. (2001) concluded that setup times have to be considered separately in the textile industry as the fabric type, when, transformed on a machine, the wrap chain must be replaced and the time it takes depends on the current and the preceding fabric types. Yi and Wang (2003) addressed the significance of setup times in stamping plants that are used by most automakers. In such plants, the setup time between manufacturing parts involves the shifting of heavy dies. Andre’s et al. (2005) addressed the importance of set up time and considered the dilemma of product grouping in the tile industry with sequence dependent setup times and concluded that minimisation of the changeover (setup) time results in reduction of the production time. Recent survey of flow shop scheduling problems including set up times has been done by Ali Allahverdi et al.

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(2008) and concluded that including sequence dependent set up time in flow shop scheduling problem has to obtain considerable savings.

1.4 PERFORMANCE MEASURES

It is not easy to state objectives in scheduling as they are numerous, complex and often conflicting. A measure of performance is said to be regular if it is a non-decreasing function of job completion times and the scheduling objective is to minimize the performance measure. A large number of scheduling problems have been studied with regular performance measures. The most widely considered regular performance measures are makespan, total flow time, total tardiness, maximum tardiness and number of tardy jobs etc. Makespan and total flow time are related to maximizing system utilization and work in process inventory while the remaining measures are related to job due dates. Scheduling with makespan criteria is very important in order to increase the productivity and maximum utilization of resources. Since, in modern manufacturing and operations management, on time delivery is a critical factor towards realizing customer satisfaction. As lack of success in meeting due dates can result in the loss of customer and market competitiveness. Hence, scheduling problems with due date related objectives have attracted increasing attention from managers and researchers. In today’s competitiveness environment, cost of production must be reduced in order to survive in this dynamic environment which has been done by effective utilisation of all the resources and production in shorter time to increase the productivity also simultaneously considering due dates of the job. As minimisation of makespan with not meeting the due date is of no use for an industry since there is loss of market competitiveness and customer.

Traditionally it has been most difficult to find optimal solutions for tardiness based objectives and focus has shifted from regular to non regular measures with the advent of the just in time philosophy. A non-regular performance measure is usually not a monotone function of the job completion times. An example of such a measure is job earliness, wherein jobs are penalized if they are completed earlier than their due-dates. The non-regular characteristic of the performance measure led researchers to develop

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completely new methodologies for scheduling problems, as the earlier ones were no longer applicable.

Many real world scheduling problems are multi-objective by nature, i.e. several objectives should be achieved simultaneously. Classical flow shop scheduling problems are mainly concerned with completion time related objectives (e.g. flow time and makespan) and aims to decrease production time and increase productivity and capacity utilization. In modern manufacturing and operations management, on time delivery is a significant factor as for the reason of upward stress of competition on the markets. Industry has to offer a great variety of different and individual products while customers are expecting ordered goods to be delivered on time. Most of the research reported in the literature is focused on the single objective case of shop scheduling problems, in which the makespan should be minimized. Some researchers have investigated multi-objective perspective of scheduling problems but the amount of literature in this area is still scarce compared to the single objective case. In this dynamic and conflicting environment, industries have to achieve number of performance measures for survival and hence scheduling system with multi-objective performance measures have given due attention since 1980. Various researchers have considered multi-objective nature of scheduling problem but restricted to two or three criteria including regular and non regular performance measures.

1.5 COMBINATORIAL OPTIMIZATION TECHNIQUES

It is known that the decision making associated with the scheduling problem belongs to the category of combinatorial optimization problems. The range of techniques that have been applied to tackle combinatorial optimization problems can be classified in two general category, firstly, the exact methods and secondly the approximate (heuristic) methods. Exact methods seek to solve a problem to guaranteed optimality but their execution on large real world problems usually requires too much computation time. Consequently, resolution by exact methods is not realistic for large problems, justifying the use of powerful heuristic and metaheuristics methods. For practical use heuristic methods seek to find high quality solutions (not necessarily optimal) within reasonable computation times (Poole et al., 1998). Metaheuristics are a class of heuristic techniques

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that have been successfully applied to solve a wide range of combinatorial optimization problems over the years (Voss et al., 1999; Osman and Laporte, 1996). The most prominent and successful meta-heuristics are the following:-

• Standard local search: Starting from a solution created at random or by some problem specific heuristic, standard local search tries to improve on it by iteratively deriving a similar solution in the neighborhood of the so-far best solution. Responsible for finding a neighboring solution is a move-operator, which must be carefully defined according to the problem. The major disadvantage of standard local search is its high probability of getting trapped at a poor local optimum. A simple improvement is iterated local search. It applies standard local search multiple times from different starting solutions and returns the best local optimum identified.

• Simulated annealing: Another way for enabling local search to escape from local optima and approach new areas of attraction in the search space is to sometimes also accept worse neighboring solutions. Simulated annealing does this in a probabilistic way. At the beginning of the optimization, worse solutions are accepted with a relatively high Probability, and this probability is reduced over time in order to achieve convergence.

• Tabu search: This strategy extends local search by the introduction of memory. Stagnation at a local optimum is avoided by maintaining a data structure, called history, in which the last created solutions or alternatively the last moves (i.e., changes from one candidate solution to the next) are stored. These solutions, respectively moves, are forbidden (tabu) in the next iteration, and the algorithm is forced to approach unexplored areas of the search space

• Evolutionary algorithms (EAs): They are a broader class of meta-heuristics based on the common idea of adopting principles from natural evolution in simplified ways.

However detailed classification of different methods has been shown in figure 1.2 and 1.3.

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Figure 1.3 Classification of common search methodologies.

Figure 1.3 Classification of common metaheuristics.

Optimization

Continuous Combinatorial

Linear Quadratic Approximate

Method Nonlinear

Local Method

Classical Method Metaheuristic

Population Based Exact Method Global Method Heuristic Neighborhood Based Metaheuristics Neighborhood-based algorithms Evolutionary programming Tabu search Simulated annealing Population-based algorithm Swarm Intelligence Evolutionary computation Ant colony optimization Particle swarm optimization Genetic algorithm Genetic programming Differential Evolution Evolutionary strategies

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In the present work, heuristics and metaheuristics based hybrid approaches have been proposed for multi-objective performance measures in a flow shop scheduling problems with sequence dependent set up time. The regular and irregular performance measures have been considered for this dynamic and competitive environment. This work will also benefit the development of optimization techniques that can be applied to other combinatorial optimization problems. Some work has been reported in literature on exact and approximation optimization methods for flow shop scheduling problems but restricted to the single or bi-objectives performance measures including total flow time, makespan, total weighted tardiness, number of tardy jobs etc. Further, present work demonstrates the suitability of applying metaheuristic techniques for multi-objective SDST flow shop scheduling problems.

Proposed metaheuristic has also been applied for scheduling the jobs in a flow shop manufacturing system with sequence dependent set up time. Output in the form of completion time and idle time of individual job and machine can be obtained with corresponding optimal sequence. Proposed metaheuristic is practical for scheduling any number of jobs on m-machines in SDST flow shop environment for meeting the said multi-objectives for achieving organizational and individual goals.

1.7 CONTRIBUTIONS OF THE PRESENT WORK

The contributions of this thesis are summarized as follows:

i. A description and formulation of the multi-objective SDST flow shop scheduling problem with minimizing the weighted sum of total weighted squared tardiness, makespan, total weighted squared earliness and number of tardy jobs. Experimental study has been conducted on the benchmarks problem derived by Taillard (1993).

ii. Modification of NEH heuristic that can be applied to proposed multi-objective performance measures for obtaining initial seed sequence in SDST flow shop scheduling as original NEH has been limited to makespan criteria.

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iii. Suitability of proposed heuristics when combined with initial population of genetic algorithm as population search techniques called as hybrid genetic algorithm to solve the multi-objective SDST flow shop scheduling problem has been explored. Already available heuristics and despatching rule based hybrid genetic algorithm has also been developed for comparative analysis with proposed heuristic based hybrid genetic algorithm. It is shown that proposed HGA can produce solutions of better quality than those generated by other dispatching rules and heuristics based HGA when compared with same computational time limit as stopping criteria.

iv. Regarding to local search techniques, proposed heuristic based hybrid simulated annealing has also been developed along with other available heuristics and dispatching rule based HSA for comparative analysis. Further, it is also shown that proposed HSA can produce solutions of better quality than those generated by other dispatching rules and heuristics based HSA when compared with same computational time limit as stopping criteria.

v. Among the modern meta-heuristics methods simulated annealing and genetic algorithms represent powerful combinatorial optimization methods with complementary strengths and weaknesses. Hence, borrowing the respective advantages of the two paradigms, an effective combination of GA and SA called hybrid GASA has also been proposed for multi-objective flow shop scheduling problems including sequence dependent set up time (SDST) with same computational time limit as stopping criteria. It has been shown that hybridization of genetic algorithm with simulated annealing (GASA) proves to be effective approach for minimizing any subset of performance measures including multi-objectives.

vi. Finally, the application of the proposed hybrid meta-heuristic has been carried out in Tubular Component Division (TCD) of the Tube Products of India (TPI), which is one of the largest integrated manufacturers of quality automobile components in India. There are 15 jobs to be processed on 7 machines resulting into 15x7 SDST flow shop scheduling problem. The completion time and idle time of each job on each machine for the optimal sequence with corresponding value of all the said

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objectives for any subsets of weight values for the considered multi-objective performance measures is being obtained.

1.8 ORGANIZATION OF THE THESIS

The remainder of this thesis has been organized as follows. In the 2nd chapter, review of the published research on the subject of flow shop scheduling is presented together with an account and brief description of a range of heuristics/dispatching rules and meta-heuristic approaches proposed in the literature.

Chapter 3 presents the problem statement along with assumptions made. It is devoted to formulation of multi-objective model of flow shop scheduling problem for minimizing the weighted sum of total weighted squared tardiness, makespan, total weighted squared earliness and number of tardy jobs under sequence dependent set up time.

In Chapter 4, metaheuristics and heuristics based hybrid metaheuristics approaches have been proposed. It covers the heuristics/dispatching rules based hybrid genetic algorithm (HGA) and hybrid simulated annealing (HSA) and also metaheuristics based hybrid genetic algorithm and simulated annealing (GASA) for multi-objective SDST flow shop scheduling problem.

Chapter 5 shows the results and discussions for the proposed HGA, HSA and GASA for the multi-objective SDST flow shop scheduling. Computational analysis upto 200 jobs and 20 machines have been shown with six modified NEH and four dispatching rules based hybrid GA and SA. Also comparative analyses amongst different HGA and HSA have been shown with the help of a defined performance index. Further, results obtained by GASA have also been shown for comparison with genetic algorithm (GA) and simulated annealing (SA) alone followed by discussions.

Chapter 6 covers the industrial application of the proposed hybrid meta-heuristic to 15x7 SDST flow shop scheduling problem in Tubular Component Division (TCD) of the Tube Products of India (TPI), which is one of the largest integrated manufacturers of quality automobile components in India. The problem has been solved by considering the

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different weight values for the multi-objective function and significant improvement in meeting the said multi-objectives has been reported.

Finally, conclusions and some suggestions for further work in this area are presented in chapter 7.

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CHAPTER 2

LITERATURE REVIEW

2.1 HISTORICAL BACKGROUND

Scheduling theory has been developed to solve problems occurring in the production facilities in which set of jobs to be executed on the set of machines, such that all side-constraints are met. Obviously, this should be done in such a way that the resulting solution, which is called a schedule that minimizes the given objective function. Since the first scheduling paper appeared in 1954, many variants of the basic scheduling problem have been formulated by differentiating between machine environments, side-constraints and objective functions. Until the late 1980s, however, it was common practice that in the objective function only one performance criteria was taken into account. In practice, however, quality is a multidimensional notion. A company, for instance, judges a production scheme on the basis of a number of criteria, for example, work-in-process inventories and observance of due dates. If only one criterion is taken into account, then the outcome is likely to be unbalanced, no matter what criterion is considered. If everything is set on keeping work-in-process inventories low, then some products are likely to be completed far beyond their due dates. Whereas, if the main goal is to keep the customers satisfied by observing due dates, then the work-in process inventories are likely to be large. In order to reach an acceptable compromise, quality of solution has to be measured on all important criteria. This notice has led to the development of the area of multi-objective scheduling.

Further, an appealing and pragmatic variation of the flow shop problem is when jobs have sequence dependent setup times on machines, that is, there is a significant setup time on each machine before processing the next job that depends on the job currently being processed. When these setup times are specifically considered the problem is usually referred to as SDST flow shop. This scheduling problem was shown to be NP-complete by Gupta (1986) so it is unlikely that a polynomial algorithm exists for solving such NP complete problem.

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Nagar et al. (1995) have surveyed papers in the bi-criteria and multi-criteria scheduling upto 1994 for regular performance measures. From the survey it was found that only 32 papers covering the bi-criteria and multi-criteria scheduling problems. And out of 32, 25 papers considered bi-criteria and only 7 papers relates to multi-criteria. Also majority of papers are on single machine problems with only four papers covering multi-machines and the techniques employed was only the exact techniques. Since 1994, also very limited research has been done on the multi-objective performance measures but none has been considered said multi-objective simultaneously in SDST flow shop scheduling.

Various mechanisms exist to contextualize complex scheduling problems with respect to existing literature. Problem classification is an important prerequisite to the selection of a suitable solution strategy since information regarding problem complexity and existing algorithms provide useful points of departure for new algorithm development. Flow shop scheduling problem is a known to be NP-hard and only exhaustive search guarantees the optimal solutions. But these can become prohibitively expensive to compute even for small problems (Pinedo, 2005). Several methods have been used to solve combinatorial optimization problem but each having its own limitation and advantages. In this section, existing combinatorial optimization techniques to solve the flow-shop scheduling problem has been explored. Classification of common combinatorial optimization can be broadly classified into two, firstly, the exact methods and secondly approximate methods. Hence, some of the important literature on single and multi-objective flow shop scheduling also been presented in respect of techniques ranging from the traditional exact methods to modern metaheuristic methods.

2.2 EXACT TECHNIQUES

Several exact techniques have been used for regular or non-regular performance measures for flow shop scheduling. Yet theoretical analysis is far from satisfaction due to its complexity and there exists neither a universally accepted technique nor an analytical algorithm for solving such NP Hard problems. Traditional optimization techniques have been successfully applied to flow shop scheduling problems with simple settings (Pinedo, 2005) but have drawback of long computational time as being exact techniques. Due to

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the complexity of flow shop scheduling problems, using exact methods to solve them is impracticable for instances of more than a few jobs and/or machines.

One of the exact techniques to obtain an optimal solution is the straightforward enumeration where every possible solution is explored in order to find the optimal solution. However, due the computational complexity, it is not practical even for problems of moderate sizes. In the case of implicit enumeration, some of the solutions can be eliminated due to feasibility issues and hence this reduces the number of solutions to be evaluated marginally. Another exact technique is the branch and bound technique where the known upper bounds or lower bounds for the solution are used to restrict the search space. The efficiency of these algorithms is dependent on the quality of the lower bounds.

The first branch and bound algorithms developed by Ignall and Schrage (1965) for the permutation flow-shop problem with makespan minimization. Lockett and Muhlemann (1972) proposed branch and bound algorithm for scheduling jobs with sequence dependent setup times on a single processor to minimize the total number of tool changes. The algorithm was computationally restrictive, suitable for only small sized problems.

Gupta (1982) presented a mathematical model based on the branch and bound technique to solve static scheduling problems involving n jobs and m machines for minimize the cost of setting up the machines as objective. He assumed that sets up times were sequence dependent and not included in processing times. The algorithm was only used to solve small size problems only. It provided an insight into the structure of optimal solutions, which could be used to devise good heuristic rules, because heuristic rules were generally more appropriate to solver large size scheduling problems where computational efforts increase rapidly with problem size for optimizing techniques.

Gupta (1986) proposed a branch and bound algorithm to minimize setup cost in ‘n’ jobs and ‘m’ machines flow shop with sequence dependent setup times and concluded that this algorithm has been restricted to small problems and however heuristic rules were the preferred techniques for large size scheduling problems with computational efforts increase rapidly with problem size.

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Sen et al. (1989) proposed a branch and bound algorithm for the two machine flow shop problem. The authors presented a comparison against the earliest due date (EDD), shortest processing time (SPT) and minimum slack (SLACK) dispatching rules. The experiments were carried out using a set of 640 problems where the processing times for both machines are randomly generated from a uniform distribution over the values 1 and 10. The due dates are also randomly generated from a uniform distribution between P (1 − T − R/2) and P (1 − T + R/2) where T and R are two parameters called tardiness factor and due date range. The P is commonly a lower bound on the makespan but in this case is the sum of the processing times in machine two plus the smallest processing time in machine one. Several problems were proposed where T = {0.25, 0.5, 0.75, 1}, R = {0.25, 0.5, 0.75, 1} and n= {6, 8, 10, 12}. Therefore, 64 combinations are generated each of which is repeated 10 times. The results show that the SPT rule performs very well for large T values and the number of nodes processed by the branch and bound to find an optimal solution tends to increase with increasing values of T and R.

Another branch and bound algorithm for the two-machine case is that of Kim (1993). The authors proposed a lower bound based on the sum of two lower bounds computed from the set of jobs in the partial sequence and the set of jobs not included in it, respectively. Dominance rules to prune sequences are also presented. A total of 240 problems were randomly generated to test the performance of the method. Processing times were uniformly distributed between 1 and 30 and due dates of the jobs were computed using the method previously commented where P in this case is the sum of the processing times of all operations divided by two. Several combinations of the parameters T, R and the number of jobs were considered where T = {0.1, 0.2, 0.3, 0.4, 0.5}, R= {0.8, 1, 1.2, 1.4, 1.6, 1.8} and n = {10, 11, 12, 13, 14, 15}.

The author compares the proposed branch and bound using the lower bounds and dominance rules presented in the paper against the branch and bound proposed by Sen et al. (1989) and a similar branch and bound but adding the dominance rule presented in Sen et al. (1989). The results show that the said branch and bound was outperformed by the other two proposed algorithms which showed a very similar performance.

Kim (1995) proposed a branch and bound to compute a lower bound with objective of total tardiness objective in flow shop scheduling. The depth first rule is used and the lower bound is computed for each node and a procedure to check the existence of

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dominated sequences is applied in the root node to reduce the size of the problem. The algorithm was tested using 480 randomly generated instances where the processing time of the jobs are uniformly distributed between 1 and 30 and the due dates were generated randomly. The results show that the proposed algorithm solved all 20 instances considered of each group until n = 13 jobs and m = 8 machines. It was able to solve also all the instances of the group n = 14 and m = 4 machines. Regarding the special sets of instances, the branch and bound proposed was not able to solve all the 20 problems within the CPU limit time set to 3600 s. In order to test the results, the algorithm is compared against the branch and bound for the two-machine case presented in Sen et al. (1989) since the authors did not find any other work to solve the m-machine problem optimally. For this case, 120 problems were generated with n = {10, 11, 12, 13, 14, 15} jobs and 20 instances for each n value. The proposed branch and bound was able to solve optimally all the problems generated while the branch and bound proposed in Sen et al. (1989) did not solve some of the 20 problems where the number of jobs is greater or equal to 13.

Carlier et al. (1996) propose two branch and bound algorithms for the per

MULTI-OBJECTIVE FLOW SHOP SCHEDULING USING METAHEURISTICS