New Crossover Operator in a Hybrid Genetic Algorithm for the Single

Machine Scheduling Problem with Sequence-dependent Setup Times

Aymen Sioud

, Marc Gravel

and Caroline Gagn´e

D´epartement d’Informatique et Math´ematique, Universit´e du Qu´ebec `a Chicoutimi,

555 Boulevard Universit´e, Chicoutimi, Canada

D´epartement des Sciences

Economiques et Gestion, Universit´e du Qu´ebec `a Chicoutimi,

555 Boulevard Universit´e, Chicoutimi, Canada

Keywords:

Genetic Algorithm, Hybrid Crossover, Constrained based Scheduling, Total Tardiness, Single Machine.

Abstract:

This paper presents a new crossover operator based on Constraint Based Scheduling (CBS) approach in a

Genetic Algorithm (GA) for solving a scheduling problem. The proposed hybrid crossover, noted as HCX, is

applied in Hybrid Genetic Algoritym (HGA) to a single machine scheduling problem with sequence-dependent

setup times for the objective of minimizing the total tardiness. Through numerical experiments we compare

the performance of the GA and the HGA approaches on different benchmarks from the literature. These results

indicate that the HGA is very competitive and generates solutions that approach those of the known reference

sets.

1 INTRODUCTION

In a survey of industrial projects, (Conner, 2009)

has pointed out, in more 250 projects, that 50%

of these projects contain sequence-dependent setup

times, and when these setup times are well applied,

92% of the order deadline could be met. Produc-

tion of good schedules often relies on management

of these setup times, and decision makers must there-

fore organize the job scheduling by trying to mini-

mize downtime while respecting the different dead-

lines (Pinedo, 2002; Allahverdi et al., 2008; Zhu and

Wilhelm, 2006).

The single machine problem has been used in

the literature to investigate scheduling issues relat-

ing to more complex scheduling problem. But, in

many industries, including the pharmaceutical indus-

try, metallurgical production, electronics and automo-

tive manufacturing, the production system can lim-

ited by a machine bottleneck, where scheduling might

be done by considering only this bottleneck machine

(Pinedo, 2002; Leung et al., 2004). This present pa-

per considers the single machine scheduling problem

with sequence dependent setup times with the objec-

tive to minimize total tardiness of the jobs (SMS-

DST). This problem, noted as 1|s

|ΣT

in accordance

with the notation of (Graham et al., 1979), is an NP-

hard problem (Du and Leung, 1990).

The 1|s

|ΣT

may be deﬁned as a set of n jobs

available for processing at time zero on a continu-

ously available machine. Each job j has a processing

time p

, a due date d

, and a setup time s

which is

incurred when job j immediately follows job i. It is

assumed that all the processing times, due dates and

setup times are non-negative integers. A sequence of

the jobs S = [q

, q

,..., q

n−1

, q

] is considered where

is the subscript of the j

job in the sequence. The

due date and the processing time of the j

job in se-

quence are denoted as d

and p

, respectively. Thus,

the completion time of the j

job in sequence will be

expressed as C

∑

k=1

k−1

+ p

) while the tar-

diness of the j

job in sequence will be expressed as

= max(0, C

−d

). The objective of the schedul-

ing problem studied is to minimize the total tardiness

of all the jobs which will be expressed as

∑

j=1

In this present paper, we propose a new crossover

operator called HCX in a genetic algorithm. The

HCX crossover integrates Constraint Based Schedul-

ing (CBS). This approach has become a widely used

form for modeling and solving scheduling problems

using the constraint programming approach. Compu-

tational testing is performed on a set of test problems

available from literature. We report on our experi-

mental results and conclude with some remarks and

future research directions. As a constraint program-

ming environment, we use the ILOG IBM CP envi-

144

Sioud A., Gravel M. and Gagné C..

New Crossover Operator in a Hybrid Genetic Algorithm for the Single Machine Scheduling Problem with Sequence-dependent Setup Times.

DOI: 10.5220/0004113101440151

In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 144-151

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

ronment using ILOG Solver and ILOG Scheduler via

the C++ APIs (ILOG, 2003b; ILOG, 2003a). The use

of this kind of platforms has been encouraged by the

steady improvement of general purpose solvers over

the past decade. Such solvers have become signiﬁ-

cantly more effective and robust (Yunes et al., 2010).

The body of this paper is organized into ﬁve

sections. Section 3 presents the used pure GA of

(Sioud et al., 2009), the CBS approach and the hy-

brid crossover HCX. The computational testing and

discussion are presented in Section 4. Finally, we

conclude with some remarks and future research di-

rections.

2 LITERATURE REVIEW

Different approaches have been proposed by a num-

ber of researchers to solve the 1|s

|ΣT

problem.

(Rubin and Ragatz, 1995) proposed a Branch and

Bound approach, which quickly showed its limita-

tions. It could optimally solve only small instances

of benchmark ﬁles of 15, 25, 35 and 45 jobs pro-

posed by these authors. (Bigras et al., 2008) have op-

timally solved all instances proposed by (Rubin and

Ragatz, 1995) using a Branch and Bound approach

with linear programming relaxation bounds. They

also demonstrated and used the problem’s similarity

with the time-dependent traveling salesman problem

(TSP). This Branch and Bound approach solved some

of these instances in more than 7 days. For their

part, (Sioud et al., 2010b) introduce a constraint based

programming approach proposing an ILOG API C++

model. This exact approach also shown its limits and

fail to resolve small 25 job instances in less than 8

hours. Because this problem is NP-hard, many re-

searchers used a wide variety of metaheuristics to

solve this problem, such as a genetic algorithm(Sioud

et al., 2009; Franca et al., 2001), a memetic algo-

rithm (Rubin and Ragatz, 1995; Franca et al., 2001;

Armentano and Mazzini, 2000), a simulated anneal-

ing (Tan and Narasimhan, 1997), a GRASP (Gupta

and Smith, 2006) and an ant colonies optimization

(ACO) (Gagn´e et al., 2002; Liao and Juan, 2007).

Heuristics such as Random Start Pairwise Interchange

(RSPI) (Rubinand Ragatz, 1995)and Apparent Tardi-

ness Cost with Setups (ATCS) (Lee et al., 1997) have

also been proposed for solving this problem. More

robust methods such the hybrid metaheuristics have

also been used. Indeed, (Gagn´e et al., 2005) intro-

duce a Tabu/VNS which propose a version of the tabu

metaheuristic that incorporates variable neighbour-

hood search. (Sioud et al., 2010a) integrate a CBS

approach in a crossover operator using direct prece-

dence constraints to enhance the CBS search. The

CBS approach has also been integrated in the inten-

siﬁcation search space process using additional con-

straints. (Sioud et al., 2012) introduce for their part

a hybrid genetic algorithm which is based on the in-

tegration between a genetic algorithm and concepts

from constraint programming, multi-objective evolu-

tionary algorithms and ant colony optimization.

Concerning the pure genetics algorithms (GA),

only (Sioud et al., 2009) succeeded in proposing an

efﬁcient GA, suggesting that this metaheuristic is not

well suited to deal with the speciﬁcities of this prob-

lem. Indeed, the authors have proposed a GA in-

tegrating the RMPX crossover operator which takes

greater account of the relative and absolute position

of a job. Indeed, (Rubin and Ragatz, 1995; Armen-

tano and Mazzini, 2000; Tan and Narasimhan, 1997)

have shown the importance of relative and absolute

order positions for solving the 1|s

|ΣT

problem. The

proposed GA outdoes the performance of all the GAs

found in the literature but is still less efﬁcient than the

Tabu/VNS of (Gagn´e et al., 2005) and the hybrid ge-

netic algorithm of (Sioud et al., 2012) whichrepresent

the best approaches found in the literature.

3 THE HYBRID GENETIC

ALGORITHM

Several researchers have used metaheuristics varia-

tions and hybridizations to improve the effectiveness

of these methods (Talbi, 2009; Puchinger and Raidl,

2005). In general, hybridization combines two or

more methods in a single algorithm to solve combina-

torial optimization problems. (Puchinger and Raidl,

2005) divide hybrid methods into two categories :

collaborative and integrative hybridization. The algo-

rithms that exchange information in a sequential, par-

allel or interlaced way fall into the category of collab-

orativehybridization. We talk aboutan integrative hy-

bridization when a technique is an embedded compo-

nent of another technique. In this paper, we introduce

a collaborative hybridization which incorporates CBS

approach sequentially in a crossover operator into a

genetic algorithm. We introduce ﬁrst the used pure

algorithm genetic, then we describe the HCX hybrid

crossover.

3.1 Genetic Algorithm

Genetic algorithms are methods based upon bio-

logical mechanisms such as the genetic inheritance

laws of Mendel and the natural selection concept of

NewCrossoverOperatorinaHybridGeneticAlgorithmfortheSingleMachineSchedulingProblemwith

Sequence-dependentSetupTimes

145

Charles Darwin, where the best adapted species sur-

vive. The basic concepts of GA have been described

by the investigation carried out by Holland (1992)

who explainedhow to add intelligence into a program

computing with the crossover exchange of genetic

material and transfer as a source of genetic diversity.

In a GA, a population of individuals or chromosomes

incurs a sequence of transformations by means of ge-

netic operators to form a new population. Two main

operators are used for this purpose : crossover and

mutation. Crossover creates new individuals by com-

bining parts of two individuals and mutation creates

new individuals by a small change in a single individ-

ual.

Based on the GA proposed by (Sioud et al., 2009),

we redeﬁne a simple genetic algorithm. A solution is

coded as a permutation of the considered jobs. The

populationsize is set to n to ﬁt with the considered in-

stance size. The initial population is randomly gener-

ated for60% and also for 20% using a pseudo-random

heuristic which favors setup times and promotes a rel-

ative order for the jobs. The last 20% is generated

using a pseudo-random heuristic which depends on

due dates and promotes an absolute order for the jobs.

A binary tournament selects the chromosomes for the

crossover. The proposed GA uses the OX crossover

(Michalewicz, 1996) to generate 30% of offspringand

the RMPX crossover (Sioud et al., 2009) to generate

the rest of the children population. Both of the OX

and RMPX crossovermaintain both of the relativeand

the absolute order positions, but the RMPX crossover

seems to give better results. The RMPX crossover can

be described in the following steps : (i) two parents

P1 and P2 are considered and two distinct crossover

points C1 and C2 are selected randomly, as shown in

Figure 1; (ii) an insertion point p

ns is then randomly

chosen in the offspring E as p

ns = random (n – ( C2 –

C1)); (iii) the part [C1, C2] of P1, shaded in Figure 1,

is inserted in the offspring E from p

ns. The insertion

is to be done from the position 2 showing in Figure 1;

and (iv) the rest of the offspring E is completed from

P2 in order of appearance since its ﬁrst position.

ins

Figure 1: Illustration of RMPX.

The crossover probability pc is set to 0.8, i.e.

therefore n*0.8 offspring are generated at each gener-

ation in which a mutation is applied with a probability

pm equal to 0.3. The mutation consists of exchanging

the position of two distinct jobs which are randomly

chosen. The replacement is elitist and the duplicates

individuals in the population were replaced by chro-

mosomes which are generated by one of the pseudo-

random heuristics used in the initialization phase. The

stop criterion is set to 3000 generations.

3.2 The Constraint based Scheduling

Constraint solving methods such as domain reduc-

tion and constraint propagation have proved to be

well suited for a wide range of industrial applica-

tions (Baptiste et al., 2001). These methods are

increasingly combined with classical solving tech-

niques from operations research, such as linear, in-

teger, and mixed integer programming (Talbi, 2002),

to yield powerful tools for constraint-based schedul-

ing by adopting them. In the recent years, the CBS

has become a widely used form for modeling and

solving scheduling problems using the constraint pro-

gramming approach (Baptiste et al., 2001; Allahverdi

et al., 2008). A scheduling problem is the process of

allocating tasks to resources over time with the goal

of optimizing one or more objectives (Pinedo, 2002).

A scheduling problem can be efﬁciently encoded like

a constraint satisfaction problem (CSP).

The activities, the resources and the constraints,

which can be temporal or resource related, are the

basis for modeling a scheduling problem in a CBS

problem. Based on representations and techniques

of constraint programming, various types of variables

and constraints have been developed speciﬁcally for

scheduling problems. Indeed, the domain variables

may include intervals domains where each value rep-

resents an interval (processing or early start time for

example) and variable resources for many classes of

resources. Similarly, various research techniques and

constraints propagation have been adapted for this

kind of problem.

In Constraint Based Scheduling, the single ma-

chine problem with setup dependent times can be efﬁ-

ciently encoded in terms of variables and constraints

in the following way. Let M be the single resource.

We associate an activity A

for each job j. For each

activity A

four variables are introduced, start(A

end(A

), proc(A

) and dep(A

). They represent the

start time, the end time, the processing time and the

departure time of the activity A

, respectively. The

departure time represents the needed setup time of an

activity when the latter starts the schedule.

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

146

A setup time s

is introduced and it is incurred

when job j immediately follows job i. In our case,

the setup times are activity related and not resource-

related. For this purpose, we assign a type to each ac-

tivity and a lattice to the unary machine. Then, when

we calculate the objective function,it is possible to as-

sociate the transition times between two distinct types

of activities. The tardiness criterion is represented by

an additional variable Tard. Its value is determined by

Tard =

∑

max(end(A

) − d

, 0).

(ILOG, 2003a) provides several predeﬁned search

algorithms named as goals and activity selectors. We

used the IloSetTimesForward algorithm with the Ilo-

SelFirstActMinEndMin activity selector. The IloS-

etTimesForward algorithm schedules activities on a

single machine forward initializing the start time of

the unscheduled activities. The activity selector de-

ﬁnes the heuristic scheduling variables representing

start times, which chooses the next activity to sched-

ule. The IloSelFirstActMinEndMin tries ﬁrst the ac-

tivity with the smallest start time and in case of equal-

ity the activity with the smallest end time. For his

part, (ILOG, 2003b) provides four strategies to ex-

plore the search tree : the default Depth-First Search

(DFS), the Slice-Based Search (SBS) (Beck and Per-

ron, 2000), Interleaved Depth-First Search (IDFS)

(Meseguer, 1997) and the Depth-Bounded Discrep-

ancy Search (DDS) (Walsh, 1997). In this pa-

per, we use the CBS model introduced by (Sioud

et al., 2010b) where the IloSetTimesForward algo-

rithm with the IloSelFirstActMinEndMin activity se-

lector are used driven by the Depth-Bounded Discrep-

ancy Search (DDS) (Walsh, 1997) algorithm search

engine.

3.3 The Hybrid Crossover

When we handle a basic single machine model, there

is no precedence constraint between activities as is

the case in a ﬂow-shop or job-shop where addingcon-

straints improves the CBS approach. The main idea of

integrating the CBS in a crossover is to provide to this

latter precedence constraints between activities when

generating offspring. In this work, we consider all

the precedence constraints during the introduced new

crossover HCX while (Sioud et al., 2010a)considered

only the direct constraints between two jobs. Also,

(Sioud et al., 2010a) extracted these direct constraints

from two selected parents while the HCX crossover

extract this information from the whole population.

Thereby, using more precedence constraints the HCX

crossover extract more information and use it to gen-

erate better offspring taking greater account of the rel-

ative position of a jobs.

The introduced new crossover HCX can be de-

scribed in the following steps : (i) from a population

at time t we build a precedence job matrix noted as

MPREC; (ii) for each job j in MPREC we calculate

the number of time that j precede the other jobs; (iii)

using a roulette wheel based on the sum calculated at

the second step we choose nbr

job

jobs; (iv) using a

threshold t

job

we extract some precedence constraints

from the x considered jobs in the third step; and (v)

from a random chosen individual in the actual popu-

lation, the CBS approach tries to solve the problem

while adding the precedence constraints built in the

previous step and an upper bound consisting of the

objective function value of the considered individual.

The upper bound is added to discard faster bad solu-

tions when branching during the solver process. As a

reminder, the ILOG Solver uses a Branch and Bound

approach to solve a problem (ILOG, 2003b). The

HCX crossover will be done under probability p

HCX

To better understand how the HCX crossover

works, consider the 5-jobs example in Figure 2. The

Figure (a) represents a population at time t with ﬁve

individuals. At the HCX crossover ﬁrst step the

MPREC matrix is build fromthis population as shown

in Figure(b). So, we can remark, for example, that the

job 2 precedes ﬁve times the job 5 and the job 4 pre-

cedes only once the job 3. Then, from the MPREC

matrix and for each job, we calculate the sum of the

precedence constraints. As shown in Figure (c), the

job 2 have 14 precedence constraints while job 5 have

only 5. After, at the third step, using a roulette wheel

we choose nbr

job

jobs. We consider in this example

that we choose the two jobs 2 and 4. We also consider

a threshold t

job

equals to 2. Next, in the fourth step,

we keep onlytwo precedenceconstraints withoutcon-

sidering mutuals constraints. So, the constraints ”2

before 4” and ”4 before 2” will be remove. These two

precedence constraints are those with the greatest val-

ues. In the example in Figure (d), we keep the two

precedence constraints ”2 before 5” and ”2 before 1”

for the job 2 shaded in gray. Concerning the job 4,

we keep the two precedence constraint ”4 before 1”

and ”4 before 5” also shaded in gray. In the case that

we have more than two equal values, we keep ran-

domly only two constraints. Finally, using the CBS

approachfrom a randomly chosen individual from the

population we try to solve the problem while adding

objective function value of the considered individual

and the four precedence constraints : ”2 before 5”, ”2

before 1”, ”4 before 1” and ”4 before 5”.

NewCrossoverOperatorinaHybridGeneticAlgorithmfortheSingleMachineSchedulingProblemwith

Sequence-dependentSetupTimes

147

j1 j2 j3 j4 j5

3 - 3 3 5j2

- 2 1 2 2j1

4 2 - 4 3j3

3 0 2 2 -j5

3 2 1 - 3j4

j1 j2 j3 j4 j5

3 - 3 3 5j2

- 2 1 2 2j1

4 2 - 4 3j3

3 0 2 2 -j5

3 2 1 - 3j4

Total

j1 j2 j3 j4 j5

3 - 3 - 5j2

- 2 1 2 2j1

4 2 - 4 3j3

3 0 2 2 -j5

3 - 1 - 3j4

Total

(a) (b)

(d)(c)

MPREC matrix

2 3 4 5 1

3 4 1 2 5

2 5 3 1 4

4 3 2 5 1

1 2 5 3 4

Population P(t)

Figure 2: Illustration of HCX.

4 COMPUTATIONAL RESULTS

AND DISCUSSION

The benchmark problem set consists of eight in-

stances, each with a number of jobs of 15, 25, 35

and 45 jobs, and it is taken from the work of (Ragatz,

1993). These instances are available on the Internet at

https://www.msu.edu/˜rubin/ﬁles/c&ordata.zip. The

job processing times are normally distributed with a

mean of 100 time units and the setup times are also

uniformly distributed with a mean of 9.5 time units.

Each instance has three factors which have both high

and low levels. These factors are due date range, pro-

cessing time variance and tardiness factor. The tar-

diness factor determines the expected proportion of

jobs that will be tardy in a random sequence. All the

experiments were run on an Itanium with a 1.4 GHz

processor and 4 GB RAM. Each instance was exe-

cuted 5 times and the results presented represent the

average deviation with the optimal results of (Bigras

et al., 2008). All the algorithmsare coded in C++ lan-

guage under the ILOG IBM CP constraint environ-

ment using ILOG Solver and Scheduler via the C++

API (ILOG, 2003b; ILOG, 2003a).

Table 1 summarize the comparison of different al-

gorithms approaches. In this table, PRB denotes the

instance names and OPT the optimal solution found

by the B&B of (Bigras et al., 2008). These authors

have not given information about the execution time

of their approach. They only said that some instances

have been resolved after more than seven days. The

GA column shows the results average deviation to the

optimal solution of the genetic algorithm described in

the section 3.1 which gives the best results among all

pure genetic algorithms in the literature without an

intensiﬁcation process (Sioud et al., 2009). The GA

average CPU time is equal to 13.4 seconds for the 32

instances. The GA generally obtained fairly good re-

sults only for the instances 601, 605, 701 and 705.

The CBS column shows the deviations of the CBS

approach minimizing the total tardiness proposed by

(Sioud et al., 2010b) and used in the HCX crossover.

For this approach, the execution time is limited to 60

minutes. It can be noticed that the CBS approach re-

sults deteriorate with increasing the instances size and

especially for the **4, **5 and **8 instances. The

PCX

column shows the average deviation of the ge-

netic algorithm in which the hybrid crossover opera-

tor PCX of (Sioud et al., 2010a) is integrated. This

crossover used only direct constraints between two

jobs from two selected parents. The CBS approach

execution time in this hybrid crossover is limited to

15 seconds. The GA

PCX

average time execution is

equal to 15.2 minutes for the 32 instances.

The GA

HCX

column shows the average deviation

of the genetic algorithm in which the hybrid crossover

operator HCX presented in this paper is integrated.

The probability p

HCX

is equal to 0.05 and the CBS ap-

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

148

Table 1: Comparison of different algorithms.

PRB OPT GA CBS

PCX

HCX

401 90 0.0 0.0 0.0 0.0

402 0 0.0 0.0 0.0 0.0

403 3418 0.5 0.0 0.0 0.0

404 1067 0.0 0.0 0.0 0.0

405 0 0.0 0.0 0.0 0.0

406 0 0.0 0.0 0.0 0.0

407 1861 0.0 0.0 0.0 0.0

408 5660 0.2 0.9 0.0 0.0

501 261 0.5 0.4 0.0 0.0

502 0 0.0 0.0 0.0 0.0

503 3497 0.2 2.5 0.0 0.0

504 0 0.0 0.0 0.0 0.0

505 0 0.0 0.0 0.0 0.0

506 0 0.0 0.0 0.0 0.0

507 7225 0.7 1.8 0.0 0.0

508 1915 0.0 35.8 0.0 0.0

601 12 169.4 41.7 6.7 3.9

602 0 0.0 0.0 0.0 0.0

603 17587 1.8 6.5 0.8 0.2

604 19092 1.8 21.1 1.1 0.6

605 228 13.0 122.4 2.6 1.2

606 0 0.0 0.0 0.0 0.0

607 12969 1.6 17.7 0.7 0.3

608 4732 1.7 156.6 0.7 0.0

701 97 30.7 20.6 6.8 2.9

702 0 0.0 0.0 0.0 0.0

703 26506 1.9 2.8 1.2 0.7

704 15206 3.4 94.8 1.6 0.8

705 200 33.7 72.5 6.1 3.1

706 0 0.0 0.0 0.0 0.0

707 23789 2.2 20.4 1.0 0.4

708 22807 2.8 50.0 1.5 1.0

Execution Time

(min)

- 0.225 60 15.2 14.5

proach execution time is limited to 15 seconds. The

two parameters nbr

job

and t

job

are ﬁxed to 0.15 * n

and 0.2 * n where n is the jobs considered number.

These two parameters were adjusted following empir-

ical tests on different instances. The ﬁrst observation

is that the GA

HCX

algorithm is always optimal for 15

and 25 jobs instances. It should be noted that the in-

tegration of the HCX crossover improves all of the

GA results and especially for the instances **1 and

**5 where the deviation became less than 4%. For

example, the deviation was reduced from 169.4% to

3.9% for the 601 instance. Using the precedence con-

straints allows the HCX crossover to enhance both

the GA exploration and the CBS search; and conse-

quently reaching better schedules. So, this new hy-

brid crossover can reach better space solution using

more precedence constraints.

The GA

HCX

average time execution is equal to

14.5 minutes for the 32 instances. This hybrid algo-

rithm improves all the results found by the GA

PCX

These improvements are more pronounced with the

integration of all the precedence constraints. This in-

tegration improves essentially the **1 and the **5 in-

stances. Also, the optimal schedule is always reached

by GA

HCX

for the 608 instance. The GA

HCX

found

the optimal solution for all the instances at least one

NewCrossoverOperatorinaHybridGeneticAlgorithmfortheSingleMachineSchedulingProblemwith

Sequence-dependentSetupTimes

149

time and this was not the case either for GA

PCX

The convergence of both GA and the GA

PCX

al-

gorithms are similar. Indeed, the average convergence

generation is equal to 1837 and 1845 generations for

GA and GA

PCX

, respectively. The GA

HCX

average

convergence generation is equal to 1358 and com-

pared to the GA

PCX

, the integration of the precedence

constraints speeds up the convergence of the solution

with reaching better results.

Exact methods are well known to be time ex-

pensive. The same applies to the hybridization of

them with metaheuristics. Indeed, execution times in-

creases signiﬁcantly with such hybridization policies

due to some technicality during the exchange of infor-

mation between the two methods (Talbi, 2009; Talbi,

2002; Puchinger and Raidl, 2005; Jourdan et al.,

2009) and this is what has been observed here. How-

ever, in this paper, the solution quality is our main

concern. So, we concentrated our efforts on it.

5 CONCLUSIONS

In this paper, we introduce a hybrid crossover into a

Genetic Algorithm to solve the sequence-dependent

setup times single machine problem with the objec-

tive of minimizing the total tardiness. The proposed

hybrid crossover extracts precedence constraints from

the population. These constraints improve the CBS

search and consequently the schedules quality.

Compared to a simple GA, the use of the HCX

crossover improves all the results but for some in-

stances the difference is still noticeable. Also, the

results of this crossover outdoes those of a hybrid

crossover taken from literature. Indeed, using the di-

rect and indirect precedence constraints from the pop-

ulation improves the results and speeds up the conver-

gence of the solution

Our results encourage us to use such hybridization

for other scheduling problems in particular and other

optimization problems in general. It is in this direc-

tion that our work is directed in the future. Also, to

making a self-adaptive method, we will work on re-

ﬁning the individual selection process for the hybdrid

HCX crossover and its two parameters : nbr

job

and

job

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