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Computers in Industry 31 (1996) 155-160

A genetic algorithm for job shop scheduling - A case study

N.S. Hemant Kumar, G. Srinivasan

Industri al Engineeri ng and Management Di vi sion. Deparrment of H umanit ies and Social Sciences, Indi an nstit ute of Technology, M adras

600 036, Indi a

Received 11 May 1995; revised 25 April 1996; accepted 21 May 1996

Abstract

The problem of scheduling n jobs on m machines with each job having a specific route has been one of considerable

research over the last several decades. Branch and Bound algorithms for determining the optimal makespan

have been

developed and tested on small sized problems and dispatching rule based heuristic algorithms to minimize specific

performance measures such as makespan, flowtime, tardiness, etc. are available to solve large sized problems. This paper

addresses the same problem faced by an organization and reports the solution of this problem using genetic algorithms (GA)

and a combination of dispatching rules. The proposed algorithm yields an improvement of about 30% in makespan over the

present system.

Keywords: Branch and bound algorithm; Genetic algorithm; Dispatching rules; Job shop scheduling

1 Introduction

Job shop scheduling deals with scheduling n given

jobs over given machines. Each job has a specific

route. The objective. in this paper is to minimize the

completion time of the last job on its last machine,

i.e., makespan. The problem is known to be hard

when there are man? than two jobs or two machines

[l]. Large sized problems can be solved using dis-

patching rules, which are used to choose a job to be

loaded on a machine from the queue [2].

The static job shop problem assumes that all jobs

are available at the beginning of the planning period

and that the processing times and set up times are

deterministic. In such cases, the set up times are

added to the proce,ssing times. The assumptions in

static job shop problems are discussed in Baker [2].

* Corresponding author.

Static job shop scheduling problems can be solved

using dispatching rules which are used to choose the

job to be loaded on a machine from a queue. Usu-

ally, non-delay schedules are evaluated where the

machine is not kept idle if there is at least one job in

the queue. Dispatching rules can be derived using

available information such as processing times and

due dates. A popular dispatching rule that uses pro-

cessing time information is the SPT (Shortest Pro-

cessing Time) rule where the job with the shortest

processing time is chosen from those in a queue. A

survey of dispatching rules can be found in Black-

stone et al. [3] where 34 dispatching rules are listed.

They also conclude that no single dispatching rule

has given consistently good results for different job

shop situations. There have also been many instances

where a combination of rules has been used success-

fully in job shop scheduling [3].

Search techniques such as Simulated Annealing

and Tabu Search have been used to solve the static

0166-3615/96/ 15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved.

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Hemant Kmur, G. Sriniuasm/ Computers in Industry 31 19961 155-160

job shop scheduling problem. Storer et al. [4] have

proposed algorithms using a set of dispatching rules

along with search heuristics such as Simulated An-

nealing, Tabu Search, etc. Van Laarhoven et al. [5]

have developed an algorithm using Simulated An-

nealing and have reported extensive computational

testing of their algorithms using standard problems

from literature.

Gen et al. [6] have developed a genetic algorithm

using mutation as the operator for minimizing

makespan in a static job shop. Biegal and Davem [7]

have illustrated the use of genetic algorithms to solve

single facility and general job scheduling problems

using crossover and mutation operators. Both the

above papers highlight the potential of genetic algo-

rithms in this field and illustrate the use of problems

of very small size. Genetic algorithms using partially

mapped crossover operators have also been used to

minimize flow time variance in single facility

scheduling problems [8] and in flow shop scheduling

191

In this paper, we use different dispatching rules

for different situations instead of using a single or a

combination rule consistently. The dispatching rule

is chosen randomly from a set of seven rules listed in

Section 3. We also develop a genetic algorithm that

progressively evaluates better solutions without get-

ting trapped in a local minimum. The genetic algo-

rithm includes the random dispatching rule approach

to choose jobs from the queue before every machine.

The genetic algorithm is applied to a real-life prob-

lem and results are presented.

The following section describes the problem situa-

tion while Section 3 explains in detail the advantages

of using different rules for different dispatching situ-

ations. The genetic algorithm is explained in Section

4 and the results obtained on applying the algorithm

to the case study are discussed in Section 5 followed

by the conclusions in Section 6.

2. The case problem

The job shop problem solved and reported in this

paper is extracted from an organization making a

product for defense purposes. There is a single final

product whose annual demand is known. There are

several assembly shops which feed sub assemblies to

Fig. 1. Schematic diagram of the production system.

the main assembly shop. There is a production shop

where components are manufactured and sent to the

assembly shops. A schematic description of the shop

is shown in Fig. 1.

The proposed genetic algorithm is tested using

data from the production shop. The shop has 80 jobs

and 59 machines. The maximum operations per job

is 37 while the minimum is two. The routes for the

jobs are such that a job may visit some machines

more than once. Wherever we have more than one

machine of the same type, we have considered a

common queue from which the jobs are loaded using

the randomly chosen dispatching rule. The compo-

nent routes are usually fixed but if there is a situation

where the required machine is busy while an identi-

cal machine is free, the jobs are loaded on the free

machine.

The annual demand of the final product is used to

compute the demand of each of the components.

Unit processing times are multiplied by a factor to

include lost time due to non-availability of machine

and operator. The batch size of each component is

equal to the monthly demand. The unit processing

time is multiplied with the batch size and the set up

times are added to result in the processing time of a

batch. Time to transport the items from one machine

to the next is also added to the processing times.

Orders for raw materials and purchased items are

usually placed in the beginning of the year. The

production rate is less during this period since orders

for most material would not have been received.

During busy times capacity cannot meet the demand

and a compromise between utilization (large batches)

and customer service (small batches) has to be made.

This results in a situation where the best schedule

has to be determined for production of components

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N.S. Hemant Kumar, G. Srinivasan/ Computers in Industry 31 1996) 155-160 157

in correct batch sizes to meet the requirements of the

subassembly shops in time and to ensure prompt

starting of the assembly of the final product, leading

to objectives of minnmizing tardiness and makespan.

The annual demand for the final product is found

to fluctuate and the production shop is required to

meet increased demand without a proportional in-

crease in the number of machines. The production

shop is usually behind schedule in such situations

which results in many tardy jobs. The shop also

faces the problem of rush orders from subassembly

shops which have to be met. The production shop

currently follows the Minimum Slack rule to choose

jobs from a queue to be loaded on a machine. Since

there are always a set of tardy jobs, there is a need to

find out whether this is because of an inefficient

dispatching rule, i.e., whether there are better dis-

patching rules that minimize makespan and tardiness.

This study focuses on the question of whether an-

other single dispatching rule or a combination of

rules can result in improvements in makespan or

tardiness which after discussions are identified as

important objectives to be considered.

3. Advantages in using different dispatching rules

for different situations

Let us consider the following seven popular dis-

patching rules from the literature [3]: SPT, TPT

(Total Processing Time), RPT (Remaining Process-

ing Time), DS (Dynamic Slack), EDD (Earliest Due

Date), RANDOM and FIFO (First-In-First-Out).

Each of the seven rules can be used individually to

yield seven different schedules from which the best

may be accepted.

Let us now consider the possibility of using a

randomly chosen ruble to choose jobs from the queue

whenever it is required. Let us consider a random

sequence of a fixed length (say 10) from numbers 1

to 7 given by [2 3 1 6 7 4 3 2 1 51.Here the number

represents the rule to be chosen. For example, num-

ber 1 indicates the SPT rule, 2 indicates the TPT rule

and so on.

In our scheduling procedure, we choose the dis-

patching rule according to the order in the random

sequence. The first dispatching rule chosen is rule 2

(TPT rule), the second is rule 3 (RPT rule) and S

on. In our example, we have created a sequence of

10 numbers. We actually need to have an infinitely

long string so that we have a rule every time a job is

to be chosen from the queue. However, if the string

length is sufficiently long, we can start from the

beginning once we have completed the string length.

In our example, if we need to choose a job to be

loaded for the eleventh time, we can use the TPT

rule (rule 2) again.

There is a possibility that a schedule generated

using this random sequence may be better than the

schedules generated using any of the seven used

individually in the schedule. The motivation to

choose different rules at different points comes from

the fact that the optimal solution can always be

looked upon as originating from such an infinitely

long string where the correct rules are in the required

order. It is also well known that no single rule has

consistently given the optimal solution for job shop

scheduling problems. It has been shown [lo] that

when more than one rule is used randomly, it is

possible with proper learning to obtain solutions with

better makespan than the solution obtained using the

poorest performing rule (among the rules considered).

A single randomly generated sequence may or

may not give better solutions when compared with

using the rules individually. If we generate many

sequences and use an algorithm to learn and progres-

sively improve the solutions, it is possible to obtain

better solutions than those obtained using each of the

rules individually.

4. A genetic algorithm GA) for job shop schedul-

ing

A genetic algorithm is a generic search procedure

for combinatorial optimization problems [ 111. It uses

the idea of survival of the fittest by progressively

accepting better solutions to the problems. The algo-

rithm starts with a population of parent chromo-

somes from which off springs are generated. These

are evaluated for the fitness function (objective) and

off springs with better fitness values are used for

further analysis.

In the job shop scheduling problem, the fitness

function is the makespan of a given schedule which

has to be minimized.

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N.S. Hemant Kumar, G, Sriniuasan/ Computers in ndustry 31 (19 ) 155-160

The genetic algorithm developed for job shop

scheduling developed in this paper has an initial

population of 50 chromosomes. The fitness value for

all these chromosomes are computed and the mean

and standard deviation are calculated. The fitness

values of the chromosomes in the population are

assumed to be normally distributed and the probabil-

ity of acceptance being the difference of the cumula-

tive distribution function at the fitness value from 1.

A chromosome is picked randomly from the popula-

tion and is put into the mating pool if the acceptance

probability is more than a randomly generated num-

ber between 0 and 1. This process is continued till

the mating pool thus created has 100 chromosomes.

The next population is created from the mating

pool using operators such as single point crossover,

two point crossover, inversion and mutation. Single

point crossover chooses two chromosomes and one

position randomly. The values in the two chromo-

somes are interchanged from that position onwards.

For example, if two chromosomes [2 4 3 1 51 and [5

3 2 4 11 are chosen and the position is 3, the resultant

chromosomes are [2 4 2 4 11 and [5 3 3 1 51

If the same procedure is repeated with two posi-

tions and the values are changed in the interval, it is

called two point crossover. If the two positions are 1

and 4, the two point crossover results in new chro-

mosomes [5 3 2 4 51 and [2 4 3 1 11. If we pick one

chromosome randomly and choose two positions

randomly and invert the values between this posi-

tions, it becomes inversion. If we take our first

chromosome and the positions are 1 and 4, then

inversion results in the chromosome

[l

3 4 2 51.

Mutation is a random interchange of values in two

positions. Mutation for the previous situation will

result in [l 4 3 2 51.

In the genetic algorithm, the randomly generated

string of dispatching rules has a length of 10. This is

found to give better results than the solutions ob-

tained using the dispatching rules individually. This

is possible because the GA considers several such

random strings and also performs genetic operations

to change the order of appearance of the rules in

each string.

In our algorithm, mutation is used 2% of the

times, single point crossover 30% of the times, two

point crossover 40% of the times and inversion 28%

of the times. This setting is found to give good

results. In general, GA is found to give good results

when crossover operations are used more times than

mutation and inversion. Mutation is to be kept to a

minimum since it does the role of pairwise inter-

change in our algorithm.

From the mating pool an off spring is created

using these operators and accepted into the next

population if the fitness value is less than the mean

of the previous population. All populations are cre-

ated with 50 chromosomes. From the new popula-

tion, off springs are created and the procedure is

repeated till either 30 generations are completed or if

the population repeats again. The best solution which

is periodically updated is accepted from the algo-

rithm.

5. Results

The

solution using the proposed genetic algorithm

(GA) to minimize makespan is shown in Table 1.

Here, solutions using the seven individual dispatch-

ing rules are also shown for comparison. The due

dates are taken to be twice the processing times after

discussions with the management of the organiza-

tion.

Table 1 shows the makespan for completing a

single batch of 1000 items using the seven rules

individually and using the proposed GA. It can be

seen that there is an improvement of about 39% over

the EDD rule (poorest performing among the seven

rules considered) and an improvement of about 3%

over the best of the individual rules (SPT). There is

an improvement of about 37.5% over the result given

by the Dynamic Slack rule, which the organization

Table 1

Performance of various rules and the genetic algorithm

Rule Makespan CPU-time (seconds)

SPT 98003 3.32

TPT

131688 3.28

RPT 131688 3.34

DS 131688 3.68

EDD 133713 3.21

RS 120672 3.10

FIFO

111161 3.32

GA 95758 998.00

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Table 2

Batch sizes for producing 2000 items

Rule Makespan

Batch size (number)

SPT

1568048

TPT 2107010

RPT

2107010

DS 2107010

EDD

2139408

RS

1930752

FIFO 1778576

GA 1532128

143 (14)

154 (13)

154 (13)

154 (13)

118 (17)

125 (16)

134 (15)

143 (14)

currently follows. The CPU times for these are also

shown in Table 1. It is seen that the GA takes a

considerable amount of time as expected. However,

considering the improvement in the solutions and the

availability of conrputers, it may be worthwhile to

spend on increased computing time.

The time taken to produce 2000 items in batches

of 1 to 20 are found. From this, the best batch size

that minimizes makespan and the corresponding

number of batches are found. These are shown in

Table 2 for the seven dispatching rules used individ-

ually and for the proposed genetic algorithm. The

batch size and the number of batches (shown in

brackets) to complete 2000 items of the final product

are shown.

The same problem is solved for multiple objec-

tives of minimizing makespan, minimizing tardiness

and minimizing the number of tardy jobs. Here, the

fitness function is taken to be the product of the

values of the three objectives. The results are shown

in Table 3. For the seven individual rules, the three

objectives are evaluated and reported. Two solutions

Table 3

Solutions using various rules and the genetic algorithm for multi-

ple objectives

_ _

Rule Makespar

Tardiness Number of tardy jobs

SPT 98003 1040947 63

TPT 131688

889999 63

RPT 131688 846120

58

DS 131688

889999 63

EDD 133713

954005 66

RANDOM

120672

1523430

68

FIFO 111161

1573501 71

GA 125842

888340 58

GA 120856

872315 61

are reported using the genetic algorithm. Both of

these have the same product of the values of the

three objectives.

If we consider only minimizing the number of

tardy jobs, one of the GA solutions has 58 jobs as

against the value of 63 given by the Dynamic Slack

rule which the organization is currently following.

The RPT rule results in the same number of tardy

jobs but has an increased makespan compared to

both the GA solutions. The only instance where the

proposed algorithm performs inferior to a single

dispatching rule is when the tardiness values are

compared to the solution using RPT. However, the

GA solution has a better makespan than the RPT

solution and the product of the values of the three

performance measures is lower for the solution using

the proposed algorithm than for the solution using

the RPT rule. If the genetic algorithm is used only to

minimize tardiness, it is possible to obtain better

values of minimum tardiness than what is obtained

in the solution for the multiple objective case. The

individual dispatching rules are unable to provide

good solutions when all the three objectives are

combined because they are invariably aimed at opti-

mizing a single objective, while the GA which at-

tempts to optimize the three objectives together pro-

vides better solutions.

6. Conclusions

In this paper we addressed the important problem

of job shop scheduling for a real-life situation using

genetic algorithms. The approach uses different dis-

patching rules from a set of seven rules, for different

situations and improves the solution using a genetic

algorithm.

Results indicate that for the problem situation

considered, the concept of using different dispatch-

ing rules for different situations in a genetic algo-

rithm yields better results than using a single dis-

patching rule. The proposed algorithm yields an

improvement of about 37.5% in makespan over the

existing rule and an improvement of 3% over the

best among the seven rules. The superior results

obtained using the algorithm have been communi-

cated to the organization and the algorithm is being

considered for implementation.

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Hemant Kumar, G. Sriniuasan/Computers in Industry 31 1996) 155-160

A very obvious conclusion is that by changing the

current dispatching rule from slack to SPT, the orga-

nization can obtain significant improvement in

makespan. Though the proposed algorithm is little

complicated for the factory managers compared to

using simple dispatching rules, the superiority of the

results and the use of recent sophisticated tools such

as genetic algorithms are expected to facilitate im-

plementation of the results of the proposed aIgo-

rithm.

The proposed algorithm uses seven popular dis-

patching rules and chooses from these. It may be

possible to get better results by increasing the set of

dispatching rules and by including combination rules

such as slack/NOP, SPT/RPT, etc. Further research

can include considering a combination of Simulated

Annealing and dispatching rules which can be com-

pared to the proposed genetic algorithm and dis-

patching rules. Further research can concentrate on

using consistently different dispatching rules for dif-

ferent machines and improve the solutions using a

genetic algorithm.

Acknowledgements

The authors are grateful to the reviewers for their

comments which have improved the presentation and

contents of the paper considerably.

References

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N.S. Hemant Kumar is currently a

Doctoral Student in Management at the

University of Texas at Austin. He has a

Bachelors degree in Aerospace Engi-

neering and a Masters degree in Mat-

agement from the Indian Institute of

Technology, Madras, India. His research

interests are in the areas of scheduling

and management of manufacturing sys-

tems.

G. Srinivasan is an Assistant Professor

of Industrial Engineering and Manage-

ment at the Indian Institute of Technol-

ogy, Madras, India. He has a Bachelors

degree in Mechanical Engineering from

Anna University, Madras, India fol-

lowed by a Masters degree and a Ph.D.

from the Indian Institute of Technology,

Madras. He has six years of teaching

and research experience in operations

management, operations research and

cellular manufacturing. He has pub-

lished in the International Journal of Production Research and

Computers and Industrial Engineering.

He is also a referee for

research in cellular manufacturing for these journals.