Flexible Job-shop Scheduling Problem with Sequence-dependent Setup

Times using Genetic Algorithm

Ameni Azzouz, Meriem Ennigrou and Lamjed Ben Said

Lab. SOIE., Strat

egies d’Optimisation et Informatique IntelligentE, ISG,

Institut Sup

erieur de Gestion, Universit

e de Tunis, Tunis, Tunisie

Keywords:

Job-shop Scheduling Problem, Flexible Manufacturing Systems, Sequence-dependent Setup Times, Genetic

Algorithms.

Abstract:

Job shop scheduling problems (JSSP) are among the most intensive combinatorial problems studied in liter-

ature. The ﬂexible job shop problem (FJSP) is a generalization of the classical JSSP where each operation

can be processed by more than one resource. The FJSP problems cover two difﬁculties, namely, machine as-

signment problem and operation sequencing problem. This paper investigates the ﬂexible job-shop scheduling

problem with sequence-dependent setup times to minimize two kinds of objectives function: makespan and

bi-criteria objective function. For that, we propose a genetic algorithm (GA) to solve this problem. To evaluate

the performance of our algorithm, we compare our results with other methods existing in literature. All the

results show the superiority of our GA against the available ones in terms of solution quality.

1 INTRODUCTION

Job-shop scheduling problem is one of the most im-

portant ﬁelds in manufacturing optimization where a

set of n jobs must be processed on a set of m speci-

ﬁed machines. Each job consists of a speciﬁc set of

operations, which have to be processed according to

a given order. This problem falls into the category of

NP-hard problems (Garey et al., 1976). The Flexible

Job Shop problem (FJSP), ﬁrst introduced by (Nui-

jten and Aarts, 1996), is a generalization of the above-

mentioned problem, where each operation can be pro-

cessed by a set of resources and has a processing time

depending on the resource used. Then, FJSP is more

difﬁcult than the classical JSP. Recently, many stud-

ies have been made to ﬁnd the near optimal solution

of FJSP using a varied range of tools and techniques

such as (Zhang et al., 2011; Azzouz et al., 2012; Zi-

aee, 2014; Turkyllmaz and Bulkan, 2014; Azzouz

et al., 2015). Most job-shop scheduling researches re-

ported in the literature ignore the setup times or con-

sider them as a part of the processing time. However,

in many real-life situations such as chemical, printing,

pharmaceutical and automobile manufacturing (Kim

and Bobrowski, 1994), the setup times are not only

often required between jobs but they are also strongly

dependent on job itself (sequence independent) and

the previous job that ran on the same machine (se-

quence dependent). Hence, reducing setup times is

an important task to improve shop performance. The

FJSP has been widely studied. However, few papers

have considered this problem with setup times. In this

paper, we propose a genetic algorithm for the FJSP

with sequence-dependent setup times (SDST). Then,

we show that our algorithm can be very effective with

respect to the state of the art. The remainder of this

paper is organized as follows. In section 2, we formu-

late the problem and we give an illustrative example.

Section 3 contains some related works of the problem

studied. Section 4 presents the proposed algorithm to

solve the SDST-FJSP. Section 5 describes the perfor-

mance of our algorithm on a set of benchmark prob-

lems and explains the most interesting results. Con-

clusions and some future works are presented in sec-

tion 6.

2 PROBLEM DEFINITION

The SDST-FJSP can be deﬁned as follows: this prob-

lem consists in performing n jobs on m machines. The

set machines is noted M, M = {M

, ..., M

}. Each

job i consists of a sequence of n

operations (routing).

Each routing has to be performed to complete a job.

The execution of each operation j of a job i (noted

i j

) requires one machine out of a set of given ma-

chines M

i, j

(i.e. M

i, j

is the set of machines available

Azzouz, A., Ennigrou, M. and Said, L.

Flexible Job-shop Scheduling Problem with Sequence-dependent Setup Times using Genetic Algorithm.

In Proceedings of the 18th International Conference on Enterprise Information Systems (ICEIS 2016) - Volume 2, pages 47-53

ISBN: 978-989-758-187-8

to execute O

i j

). The problem is to deﬁne a sequence

of operations together with assignment of start times

and machines for each operation. Assumptions con-

sidered in this paper are the following:

• jobs are independent of each other;

• machines are independent of each other;

• one machine can process at most one operation at

a time;

• no preemption is allowed;

• all jobs are available at time zero;

• Setup times are dependent on the sequence of

jobs. When one of the operations of a job t is

processed before one of those of job i (t 6= i) on

machine M

, the sequence dependent setup time

is S

t,i,k

> 0.

The current SDST-FJSP based on these assumptions

is aimed to minimize two kinds of objective functions:

• Minimize the makespan ( i.e. the time required to

complete all jobs)

• Minimize Aggregate objective function (AOF)

where AOF = αF1 + (1 − α)F2 and de-note the

weight given respectively to makespan (F1) and

mean tardiness (F2).

FJSP is classiﬁed as Total FJSP and Partial FJSP

(Kacem and Borne, 2002). In Total FJSP (T-FJSP),

each operation can be processes by all machines.

However, in Partial FJSP (P-FJSP), at least one oper-

ation may not be processed on all machines. Several

researches pointed out that the P-FJSP is more com-

plex as compared to T-FJSP on the same scale. In this

paper, we consider the P-FJSP.

Table 1: Processing times.

Job Operation M1 M2 M3

O11 4 - 5

O12 - 3 4

O13 6 5 -

O21 3 - 4

O22 4 5 -

O23 - 4 7

O31 5 3 -

O32 - 4 -

O33 4 5 3

Table 2: Sequence-dependent setup times.

Machine1 Machine2 Machine3

Job1 Job2 Job3 Job1 Job2 Job3 Job1 Job2 Job3

Dummy 3 2 1 2 4 1 4 3 2

Job1 0 1 3 0 2 4 0 2 3

Job2 1 0 2 4 0 3 2 0 4

Job3 1 3 0 2 1 0 3 2 0

Figure 1: Gantt Chart of solution.

To illustrate this problem, we consider an instance

with three jobs and three machines. In table1, we

show the processing time of each operation. The

symbol ”-” means that the machine can not execute

the corresponding operation. Table2 presents the se-

quence dependent setup times of each job. For in-

stance, the setup time for job3 after job2 on machine

is S

2,3,1

= 2 and the setup time for job2 after job1

on machine M

is S

1,2,3

= 2. Dummy job signiﬁes the

starting of a job on each machine. When one of the

operations of a jobi is the ﬁrst operation executed on

machine M

, the dummy job is D

i,k

. For example, if

3,1

is the ﬁrst operation executed on machine M

then, dummy job value would be D

3,1

= 1. In order

to explain better this problem, we represent a Gantt

Chart of solution on ﬁgure 1.

3 RELATED WORK

The majority of researches in scheduling problem

with SDST are restricted to the ﬂowshop problem.

Due to the complexity of this problem, most of the lit-

erature based on meta-heuristic methods like genetic

algorithms (Kaweegitbundit, 2011; Li and Zhang,

2012; Mirabi, 2014) tabu search (Varmazyar and

Salmasi, 2012; Santos et al., 2014) and ant colony op-

timization (Gajpal and Ziegler, 2006; Mirabi, 2011).

Despite the relevance of the job shop in real man-

ufacturing systems, not many papers consider both

sequence-dependent setup times and job shop envi-

ronments. Among these, (Cheung and Zhou, 2001)

propose a hybrid algorithm based on genetic algo-

rithm and heuristic rules to solve SDST-JSP with min-

imizing the makespan. For the same problem, (Zhou

et al., 2006) propose an immune algorithm which

certiﬁes the diversity of the antibody. (Moghaddas

and Houshmand, 2008) develop a mathematical and

heuristic model based on priority rules. (Naderi and

Ghomi, 2009) consider the job shop scheduling with

sequence-dependent setup times and preventive main-

tenance policies using four meta-heuristics based on

simulated annealing and genetic algorithms.

Considering the ﬂexibility constraints, ﬂexible job-

ICEIS 2016 - 18th International Conference on Enterprise Information Systems

shop problem presents additional difﬁculty than the

classical JSP and requires more effective algorithms.

In recent decades, many attempts have been made to

ﬁnd the near optimal solution of SDST-FJSP using

a varied range of tools and techniques. (Imanipour,

2006) was the ﬁrst one who investigates the SDST-

FJSP. The author modeled the problem as a non lin-

ear mixed integer programming model and proposes

a tabu search for the same problem. (Saidi-Mehrabad

and Fattahi, 2007) presented a Tabu Search for solv-

ing the SDST-FJSP to minimize makespan. They as-

sumed in their research that each operation can be per-

formed by two machine alternatives. They compared

their obtained results with the results of the lingo soft-

ware. (Bagheri and Zandieh, 2011) propose a variable

neighborhood search (VNS) based on integrated ap-

proach to minimize an aggregate objective function

(AOF) where AOF = αF1 + (1 − α)F2 and α de-

note the weight given respectively to makespan (F1)

and mean tardiness (F2). To evaluate this model,

the authors generate randomly 20 problem instances

under four different classes. Using the same AOF,

(Sadrzadeh, 2013) present an artiﬁcial immune sys-

tem algorithm (AIS) and a particle swarm optimiza-

tion algorithm (PSO) and prove that both algorithms

works better than VNS of (Bagheri and Zandieh,

2011). (Mousakhani, 2013) formulate the SDST-

FJSP as a mixed integer linear programming model to

minimize total tardiness and present a meta-heuristic

based on iterated local search for the same problem.

(Oddi et al., 2011) considers the SDST-FJSP to mini-

mize the makespan using the iterative ﬂattering search

(IFS) and propose a new benchmark which is denoted

SDST-HUdata. It consists of 20 instances produced

as an extension of the existing well-known bench-

marks of FJSP of (Hurink et al., 1994). (Gonz

alez

and Varela, 2013) develop memetic algorithm to min-

imize the makespan which the tabu search was ap-

plied to every chromosome generated by the genetic

algorithm. In order to evaluate their model, they used

the same benchmark as in (Oddi et al., 2011) and

prove that the memetic algorithm has obtained a bet-

ter result than the IFS. Recently, (Rossi, 2014) inves-

tigate the SDST-FJSP with transportation times us-

ing ant-colony algorithm with reinforced pheromone.

The most recent comprehensive survey of schedul-

ing problem with setup times is given by (Allahverdi,

2015).

4 GENETIC ALGORITHM FOR

SDST-FJSP

Since the discovery of the genetic algorithms by

(Holland, 1975), they have been recognized as a pow-

erful methods for solving combinatorial optimization

problems such as scheduling problems. In our algo-

rithm, we generate the initial population according

several dispatching rules. After evaluating each solu-

tion in the population, if the stop criterion is not met,

there are two choices. According to the probability

crossover

and P

mutation

, the current individual executes

crossover operator or the mutation one respectively.

The stop criterion is that a certain number of iteration

is reached or the best solution has not been improved

for a certain number of iteration. Next, we present the

details of the implementation of our GA components.

4.1 Encoding Problem

For solving SDST-FJSP by GA, the ﬁrst step is to

represent a solution of a problem as a chromosome.

We try to design an efﬁcient coding of the individu-

als which respects the most important constraints of

our problem in order to increase the number of fea-

sible solutions produced after genetic recombination.

Then, our chromosome is designed as a binary matrix,

where:

• The rows correspond to all operations of jobs.

Furthermore, the order in which they appear in the

chromosome describes the sequence of operations

present in the solution.

• The columns correspond to all machines.

Moreover, in our representation, we present a con-

straint described as follows:

∑

k=1

i jk

= 1 (1)

i jk

= 1 when O

i j

is assigned to resource M

i jk

= 0 otherwise.

The sum of each row must be equal to one, to

guarantee that each operation is assigned to only one

machine. The order, in which the operations appear

in this representation, is found according to the start

times of the operations. When we have more than one

operation executed in the same time, we choose the

one which has the smallest number of jobs.

Figure2 illustrate our encoding scheme and repre-

sent the solution ﬁgured in the Gantt chart in Figure1.

4.2 The Initial Population

Initial population plays a signiﬁcant role in genetic

algorithms in order to get good result. Generally,

the initial population is generated randomly in order

to maintain the diversity of solutions. Therefore, to

increase the diversity of the ﬁrst generation and to

Flexible Job-shop Scheduling Problem with Sequence-dependent Setup Times using Genetic Algorithm







M1 M2 M3

0 0 1

1 0 0

0 1 0

0 0 1

1 0 0

0 0 1

0 1 0

1 0 0







Figure 2: A sample chromosome encoding by our represen-

tation.

maintain a certain quality, we propose an improved

function of initial population generation inspired from

(Pezzella et al., 2008) which is based on three tradi-

tional dispatching rules as following:

• 20% using shortest processing time (SPT): Jobs

are scheduled with this rule by sequencing them in

ascending order of job processing times per pro-

cess.

• 20% using longest processing time (LPT): Jobs

are scheduled with this rule by sequencing them

in descending order of job processing times per

process.

• 20% using heuristic rules based on local search

algorithm.

• The remaining with random solution.

4.3 Selection

The selection phase aims to choose the chromosomes

for reproduction to create the next generation. In this

study, we adopt our selection operator (Azzouz et al.,

2015). Four individuals are randomly chosen from

parent population and the ﬁtness of each of them are

compared in order to select the best and the worst so-

lution for reproduction.

4.4 Crossover Operator

The goal of the crossover is to obtain better chromo-

somes to improve the result by exchanging informa-

tion contained in the current good ones. As in (Az-

zouz et al., 2015), we have adopted the crossover op-

erator ”order1” (Davis, 1985).

We adapted this crossover to our own coding de-

scribed earlier. The idea of this operator is as follows:

We randomly select two positions XP1 and XP2 in

Parent1. The middle part is copied to the offspring1.

The rest is ﬁlled from the parent2 starting with po-

sition XP2 + 1 and jumping elements that are already

present in the offspring 1. The same steps are repeated

Figure 3: Crossover operator.

for the second offspring by starting with the Parent2.

To more explain the crossover operator, we present

an example in Figure3. Note that in this example, we

take XP1 = 3 and XP2 =7.

4.5 Mutation

Mutation operator is used also to get a new individual

having only one value different from an already exist-

ing one. In our work, we adopt intelligent mutation

proposed by (Pezzella et al., 2008) in which we se-

lect an operation on the machine with the maximum

workload (i.e. the amount of work that a machine pro-

duces in a speciﬁed time period), and assign it to the

machine with the minimum workload if possible.

5 RESULTS AND EXPERIMENTS

This section evaluates the performance of our pro-

posed genetic algorithm for two kinds of objective

functions: makespan and Aggregate Objective Func-

tion (AOF).

For that, we compare our GA against the avail-

able algorithms in the literature including variable

neighourbood search (VNS) proposed by (Bagheri

and Zandieh, 2011), an adapted tabu search (TS) pro-

posed by (Ennigrou and Ghedira, 2008), Artiﬁcial Im-

mune System (AIS) and Particle swarm Optimization

ICEIS 2016 - 18th International Conference on Enterprise Information Systems

(PSO) from (Sadrzadeh, 2013). Our proposed algo-

rithm has been implemented using JAVA and run on

PC with core2Duo, 2,6GHZ and 2GB RAM. In our

experiment, we tested different values for our GA pa-

rameters, and computational experience proves that

the following values are more effective:

• Population size: 150

• Crossover probability: 0.8

• Mutation probability: 0.2

• Number of iterations (stopping condition): 150

• Number of iterations with no improvement (stop-

ping condition): 30.

For the makespan objective function, we consider the

same benchmark as in (Oddi et al., 2011; Gonz

alez

and Varela, 2013) which is denoted SDST-HUdata. It

consists of 20 instances derived from the ﬁrst 20 in-

stances of the FJSP benchmark proposed in (Hurink

et al., 1994). Each instance was created by adding

to the original instance one setup time matrixS

t,k

for

each machine k. The same setup time matrix was

added for each machine in all benchmark instances.

Each matrix has size nn, and the value S

t,i,k

indicates

the setup time needed to reconﬁgure the machine k

when switches from job t to job i. These setup times

are sequence dependent and they fulﬁll the triangle

inequality. The non-deterministic nature of our algo-

rithm makes it necessary to carry out multiple runs

on the same instance in order to obtain meaningful

results.

After ten runs of each generated instance by the

above-mentioned algorithms, the best solutions ob-

tained for each instance (which is namedSol

min

) are

calculated. We use the relative percentage deviation

(RPD) measure to compare the performance of

algorithms. RPD is obtained as follows:

RPD =

Sol

algo

−Sol

min

Sol

min

× 100 (2)

where SOL

algo

is the makespan of each algorithm.

Table3 show the performance of the proposed GA

compared with others algorithms. The instance names

are listed in the ﬁrst column, the second column show

the size (n × m) of each instance. The third, fourth

and ﬁfth columns report the obtained results of VNS

algorithm (Bagheri and Zandieh, 2011), TS algorithm

(Ennigrou and Ghedira, 2008) and our GA.

The obtained results show that the proposed GA per-

forms better than the others algorithms in 16 in-

stances. Only in instance La02, La04 and La08 VNS

has gained better results. However, in instance La15,

TS obtained the better results. The proposed GA out-

performs the others algorithms with average RPD of

1.40 while the worst performing algorithm is TS with

average RPD of 4.15. Moreover, we notice that GA

obtained the best average RPD of 0.71 in the largest

number of machine against 4.58 and 6.48 for VNS

and TS respectively. To further evaluate the per-

formance of our algorithm, we study the interaction

between the performance of the algorithm and the

problem size in ﬁgure4. We remark that our algo-

rithm keeps its robust performance in different prob-

lem sizes.

Table 3: Summary of results in the SDST-FJSP to minimize

the makespan: SDST-HU data benchmark.

Instance Size n × m VNS TS GA

La01

10 × 5

2.54 7.36 0.00

La02 1.58 8.53 2.57

La03 2.66 5.33 0.13

La04 3.50 3.66 4.07

La05 1.11 3.58 0.16

La06

15 × 5

4.45 2.79 2.04

La07 2.92 2.99 1.97

La08 2.37 2.41 2.86

La09 3.82 3.21 0.41

La10 1.84 4.02 0.00

La11

20 × 5

3.32 2.08 1.61

La12 5.33 2.71 2.38

La13 3.30 3.14 3.02

La14 1.70 1.76 1.26

La15 2.37 1.18 1.97

La16

10 × 10

2.37 4.04 0.00

La17 4.34 2.39 0.07

La18 1.86 5.81 0.91

La19 6.37 6.67 0.11

La20 4.84 9.36 2.48

Average 3.12 4.15 1.40

Figure 4: The average RPD of the algorithms versus the

number of jobs.

Furthermore, for the AOF, we consider artiﬁcial

benchmarks according to the function proposed by

(Bagheri and Zandieh, 2011). We propose four

classes of instances. These classes are different in

number of jobs, n, number of operations for each jobi,

, and number of machines, m, that are denoted as

(n × n

× m). The generated instances have partial

ﬂexibility and the number of available machines for

each operation (AMO) is generated randomly accord-

Flexible Job-shop Scheduling Problem with Sequence-dependent Setup Times using Genetic Algorithm

Table 4: The characteristics of the instances.

n × n

× m AMO Processing SDST Dummy

time Jobs

Class1 10 × 5 × 5 U(1,5)

U(20,100) U(20,60) U(20,40)

Class2 10 × 5 × 8 U(1,8)

Class3 10 × 10 × 5 U(1,5)

Class4 15 × 10 × 10 U(1,10)

ing to uniform distribution. Table 4 summarizes the

characteristics of the artiﬁcial benchmarks used in this

paper. In order to introduce due dates, we consider the

same formula as in (Bagheri and Zandieh, 2011).

Overall, compared to VNS, AIS and PSO, our GA

has a superiority result to minimize the AOF for all α

values. Moreover, from the results shown in ﬁgure 5,6

Figure 5: The average RPD of the algorithms of each type

of problem class for α = 0.25.

Figure 6: The average RPD of the algorithms of each type

of problem class for α = 0.5.

Figure 7: The average RPD of the algorithms of each type

of problem class for α = 0.75.

and 7, we remark that GA is more effective with α =

0.75 then α = 0.25. Otherwise, our algorithms have

the best results with makespan against mean tardiness

objective function.

6 CONCLUSION

In this paper, we focus on solving the ﬂexible job shop

scheduling problem where sequence dependent setup

times are also taken into account. We have proposed

genetic algorithm to minimize two kinds of objective

functions: makespan and aggregate objectives func-

tion. For that, we tested GA on two kinds of bench-

mark. Results showed that the present GA is better

than other algorithms. In future works, it will be inter-

esting to investigate the dynamic scheduling problem

to closely reﬂect the real ﬂexible job shop schedul-

ing environment. For the same reason, we will con-

sider the multi-criteria scheduling problem and the

scheduling problems with learning effects consider-

ations.

ACKNOWLEDGEMENTS

The authors would like to say thanks to Miguel A.

Gonz

alez for providing us with the SDST-FJSP in-

stances.

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Flexible Job-shop Scheduling Problem with Sequence-dependent Setup Times using Genetic Algorithm