Genetic Algorithm Combined with Tabu Search in a Holonic Multiagent

Model for Flexible Job Shop Scheduling Problem

Houssem Eddine Nouri

, Olfa Belkahla Driss

1,2

and Khaled Gh´edira

Strat´egies d’Optimisation et Informatique intelligentE, Higher Institute of Management of Tunis, Bardo, Tunis, Tunisia

Higher Business School of Tunis, University of Manouba, Manouba, Tunisia

Keywords:

Scheduling, Flexible Job Shop, Genetic Algorithm, Tabu search, Holonic Multiagent.

Abstract:

The Flexible Job Shop scheduling Problem (FJSP) is an extension of the classical Job Shop scheduling Prob-

lem (JSP) presenting an additional difﬁculty caused by the operation assignment problem on one machine out

of a set of alternative machines. The FJSP is an NP-hard problem composed by two complementary problems,

which are the assignment and the scheduling problems. In this paper, we propose a combination of a genetic

algorithm with a tabu search in a holonic multiagent model for the FJSP. In fact, ﬁrstly, a scheduler agent ap-

plies a genetic algorithm for a global exploration of the search space. Then, secondly, a local search technique

is used by a set of cluster agents to guide the research in promising regions of the search space and to improve

the quality of the ﬁnal population. To evaluate our approach, numerical tests are made based on two sets of

well known benchmark instances in the literature of the FJSP: Kacem and Brandimarte. The experimental

results show that our approach is efﬁcient in comparison to other approaches.

1 INTRODUCTION

Scheduling is a ﬁeld of investigationwhich has known

a signiﬁcant growth these last years. The scheduling

problems appear in all the economic areas, from com-

puter engineering to industrial production and manu-

facturing. The Job Shop scheduling Problem (JSP),

which is among the hardest combinatorial optimiza-

tion problems (Sonmez and Baykasoglu, 1998), is a

branch of the industrial production scheduling prob-

lems. The JSP is known as one of the most popu-

lar research topics in the literature due to its potential

to dramatically decrease costs and increase through-

put (Jones and Rabelo, 1998). The Flexible Job Shop

scheduling Problem (FJSP) is an extension of the clas-

sical JSP that allows to process operations on one ma-

chine out of a set of alternative machines. Hence,

the FJSP is more computationally difﬁcult than the

JSP. Furthermore the operation scheduling problem,

the FJSP presents an additional difﬁculty caused by

the operation assignment problem to a set of available

machines. This problem is known to be strongly NP-

Hard even if each job has at most three operations and

there are two machines (Garey et al., 1976).

To solve this problem, standard metaheuristic

methods are used for an approximate resolution and to

ﬁnd near-optimal solutions for the FJSP with accept-

able computational time. (Brandimarte, 1993) pro-

posed a hierarchical algorithm based on Tabu Search

metaheuristic for routing and scheduling with some

known dispatching rules to solve the FJSP. (Hurink

et al., 1994) developed a Tabu Search procedure for

the job shop problem with multi-purpose machines.

(Dauz`ere-P´er`es and Paulli, 1997) presented a new

neighborhood structure for the problem, and a list of

Tabu moves was used to prevent the local search from

cycling. (Mastrolilli and Gambardella, 2000) used

Tabu Search techniques and presented two neighbor-

hood functions allowing an approximate resolution

for the FJSP. (Bozejko et al., 2010a) presented a Tabu

Search approach based on a new golf neighborhood

for the FJSP, and in the same year, (Bozejko et al.,

2010b) proposed another new model of a distributed

Tabu Search algorithm for the FJSP, using a cluster ar-

chitecture consisting of nodes equipped with the GPU

units (multi-GPU) with distributed memory. A new

version of Tabu Search algorithm with a fast Pub-

lic Critical Block neighborhood structure (TSPCB)

was proposed by (Li et al., 2011) to solve the FJSP.

For the Genetic Algorithm, it was adopted by (Chen

et al., 1999), where their chromosome representation

of solutions for the problem was divided into two

parts. The ﬁrst part deﬁned the routing policy and

the second part took the sequence of operations on

573

Nouri H., Belkahla Driss O. and Ghédira K..

Genetic Algorithm Combined with Tabu Search in a Holonic Multiagent Model for Flexible Job Shop Scheduling Problem.

DOI: 10.5220/0005348105730584

In Proceedings of the 17th International Conference on Enterprise Information Systems (ICEIS-2015), pages 573-584

ISBN: 978-989-758-096-3

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

each machine. (Kacem et al., 2002a) used a Genetic

Algorithm with an approach of localization to solve

jointly the assignment and job shop scheduling prob-

lems with partial and total ﬂexibility, and a second

hybridization of this evolutionary algorithm with the

fuzzy logic was presented in (Kacem et al., 2002b).

(Jia et al., 2003) proposed a modiﬁed Genetic Algo-

rithm for the FJSP, where various scheduling objec-

tives can be achieved such as minimizing makespan,

cost and weighted multiple criteria. (Ho et al., 2007)

developed a new architecture named LEarnable Ge-

netic Architecture (LEGA) for learning and evolv-

ing solutions for the FJSP, allowing to provide an

integration between evolution and learning in an ef-

ﬁcient manner within a random search process. In

addition, other types of metaheuristics were devel-

oped in this last few years, such as (Yazdani et al.,

2010) implementing a Parallel Variable Neighbor-

hood Search (PVNS) algorithm to solve the FJSP us-

ing various neighborhood structures. A Knowledge-

Based Ant Colony Optimization (KBACO) algorithm

was presented by (Xing et al., 2010) for the FJSP.

Moreover, another branch of metaheuristic optimiza-

tion approaches appeared, by combining two or more

complementary approaches in order to create a supe-

rior solution procedure allowing to offer a more efﬁ-

cient resolution for complex problems. That is why,

(Glover et al., 1995) elaborated a study about the na-

ture of connectionsbetween the genetic algorithm and

tabu search metaheuristics, searching to show the ex-

isting opportunities for creating a hybrid approach

with these two standard methods to take advantage

of their complementary features and to solve difﬁcult

optimization problems. After this pertinent study, the

combination of these two metaheuristics has become

more well-known in the literature and is used to solve

many complex problems, such as the FJSP. An im-

proved genetic algorithm combined with local search

is proposed by (Zhang et al., 2008) to solve the FJSP.

In fact, this combination introduced a time-varying

the crossover probability for the genetic algorithm

and a time-varying maximum step size for the tabu

search, allowing to control the local search process

and the convergence to the global optimal solution.

After two years, (Zhang et al., 2010) presented a sec-

ond combined proposition of genetic algorithm with

tabu search to solve the multi-objective FJSP. They

adopted an additional external memory to save and

update the non-dominated solutions during the opti-

mization process. (Gao et al., 2008) adapted a hybrid

Genetic Algorithm (GA) and a Variable Neighbor-

hood Descent (VND) for FJSP. The GA used two vec-

tors to represent a solution and the disjunctive graph

to calculate it. Then, a VND was applied to improve

the GA ﬁnal individuals. (Thamilselvan and Balasub-

ramanie, 2009) developeda new algorithm integrating

Genetic Algorithm and Tabu Search methods to solve

the JSP. The proposed algorithm is based on the net-

work central-clients node architecture inspired from

(Andresen et al., 2002), where the genetic algorithm

is run on a central node to generate n initial solutions,

to be used by n client nodes as a starting solution

for the Tabu Search algorithm. (Zhang et al., 2014)

proposed a model of low-carbon scheduling in the

FJSP considering three factors, the makespan, the ma-

chine workload for production and the carbon emis-

sion for the environmental inﬂuence. A metaheuristic

hybridization algorithm was proposed combining the

original Non-dominated Sorting Genetic Algorithm II

(NSGA-II) with a Local Search algorithm based on a

neighborhood search technique. Moreover, the Par-

ticle Swarm Optimization was implemented by (Xia

and Wu, 2005) in a metaheuristic hybridization ap-

proach with the Simulated Annealing for the multi-

objective FJSP. A combined Particle Swarm Opti-

mization and a Tabu Search algorithm were proposed

by (Zhang et al., 2009) to solve the multi-objective

FJSP. (Moslehi and Mahnam, 2011) presented a meta-

heuristic approach based on a hybridization of the

Particle Swarm Optimization and Local Search algo-

rithm to solve the multi-objective FJSP. Furthermore,

a new heuristic was developed by (Ziaee, 2014) for

the FJSP. This heuristic is based on a constructivepro-

cedure considering simultaneously many factors hav-

ing a great effect on the solution quality. Furthermore,

distributed artiﬁcial intelligence techniques were used

for this problem, such as the multiagent model pro-

posed by (Ennigrou and Gh´edira, 2004) composed by

three classes of agents, job agents, resource agents

and an interface agent. This model is based on a lo-

cal search method which is the tabu search to solve

the FJSP. Also, this model was improved in (Enni-

grou and Gh´edira, 2008) where the optimization role

of the interface agent was distributed among the re-

source agents. Similarly, (Azzouz et al., 2012) de-

veloped a combination of a tabu search and a ge-

netic algorithm based on a multiagent system to solve

the FJSP. This approach used two classes of agents,

a ﬁrst class grouping resource agents responsible of

processing a local optimization technique based on

the tabu search of (Ennigrou and Gh´edira, 2008) and

a second class containing an interface agent responsi-

ble of processing a global optimization by a genetic

algorithm. (Henchiri and Ennigrou, 2013) proposed

a multiagent model based on a hybridization of two

metaheuristics, a local optimization process using the

tabu search to get a good exploitation of the good ar-

eas and a global optimization process integrating the

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Particle Swarm Optimization (PSO) to diversify the

search towards unexplored areas.

In this paper, we present a combination of a ge-

netic algorithm with a tabu search in a holonic multi-

agent model for the ﬂexible job shop scheduling prob-

lem. This new combined approach follows two prin-

cipal steps. In the ﬁrst step, a scheduler agent ap-

plies a genetic algorithm for a global exploration of

the search space. Then, in the second step, a local

search technique is used by a set of cluster agents to

improve the quality of the ﬁnal population. Numer-

ical tests were made to evaluate the performance of

our approach based on two data sets of (Kacem et al.,

2002b) and (Brandimarte, 1993) for the FJSP, where

the experimental results show its efﬁciency in com-

parison to other approaches.

The rest of the paper is organized as follows. In

section 2, we deﬁne the formulation of the FJSP with

its objective function and a simple problem instance.

Then, in section 3, we detail the proposed combined

approach with its holonic multiagent levels. The ex-

perimental and comparison results are provided in

section 4. Finally, section 5 rounds up the paper with

a conclusion.

2 PROBLEM FORMULATION

The ﬂexible job shop scheduling problem (FJSP)

could be formulated as follows. There is a set of n

jobs J = {J

,...,J

} to be processed on a set of m

machines M = {M

,...,M

}. Each job J

is formed

by a sequence of n

operations {O

i,1

i,2

,...,O

i,ni

}

to be performed successively according to the given

sequence. For each operation O

i, j

, there is a set of al-

ternative machines M(O

i, j

) capable of performing it.

The main objective of this problem is to ﬁnd a sched-

ule minimizing the end date of the last operation of

the jobs set which is the makespan. The makespan

is deﬁned by Cmax in equation (1), where C

is the

completion time of a job J

Cmax = max

1≤i≤n

) (1)

The FJSP scheduling problem is divided in two

sub-problems:

• The operations assignment sub-problem that as-

signs each operation to an appropriate machine.

• The operations sequencing sub-problem that de-

termines a sequence of operations on all the ma-

chines.

Furthermore, the adopted hypotheses in this prob-

lem are:

• All the machines are available at time zero;

• All jobs are ready for processing at time zero;

• The order of operations for each job is predeﬁned

and cannot be modiﬁed;

• There are no precedence constraints among oper-

ations of different jobs;

• The processing time of operations on each ma-

chine is deﬁned in advance;

• Each machine can process only one operation at a

time;

• Operations belonging to different jobs can be pro-

cessed in parallel;

• Each job could be processed more than once on

the same machine;

• The interruption during the process of an opera-

tion on a machine is negligible.

To explain the FJSP, a sample problem of three

jobs and ﬁve machines is shown in table 1, where the

numbers present the processing times and the tags “–

” mean that the operation cannot be executed on the

corresponding machine.

Table 1: A simple instance of the FJSP.

Job Operation M

1,1

2 9 4 5 1

1,2

– 6 – 4 –

2,1

1 – 5 – 6

2,2

3 8 6 – –

2,3

– 5 9 3 9

3,1

– 6 6 – –

3,2

3 – – 5 4

3 GENETIC ALGORITHM

COMBINED WITH TABU

SEARCH IN A HOLONIC

MULTIAGENT MODEL

(Glover et al., 1995) elaborated a study about the na-

ture of connections between the genetic algorithm and

tabu search metaheuristics, searching to show the ex-

isting opportunities for creating a hybrid approach

with these two standard methods to take advantage

of their complementary features and to solve difﬁ-

cult optimization problems. After this pertinent study,

the combination of these two metaheuristics has be-

come more well-known in the literature, which has

motivated many researchers to try the hybridization

of these two methods for the resolution of different

complex problems in several areas.

(Ferber, 1999) deﬁned a multiagent system as an

GeneticAlgorithmCombinedwithTabuSearchinaHolonicMultiagentModelforFlexibleJobShopSchedulingProblem

575

artiﬁcial system composed of a population of au-

tonomous agents, which cooperate with each other to

reach common objectives, while simultaneously each

agent pursues individual objectives. Furthermore, a

multiagent system is a computational system where

two or more agents interact (cooperate or compete, or

a combination of them) to achieve some individual or

collectivegoals. The achievementof these goals is be-

yond the individual capabilities and individual knowl-

edge of each agent (Botti and Giret, 2008).

(Koestler, 1967) gave the ﬁrst deﬁnition of the

term “holon” in the literature, by combining the two

Greek words “hol” meaning whole and “on” meaning

particle or part. He said that almost everything is both

a whole and a part at the same time. In fact, a holon

is recursively decomposed at a lower granularity level

into a community of other holons to produce a hol-

archy (Calabrese, 2011). Moreover, a holon may be

viewed as a sort of recursive agent, which is a super-

agent composed by a sub-agents set, where each sub-

agent has its own behavior as a complementary part

of the whole behaviour of the super-agent. Holons

are agents able to show an architectural recursiveness

(Giret and Botti, 2004).

In this work, we propose a combined metaheuris-

tic approach processing two general steps: a ﬁrst step

of global exploration using a genetic algorithm to ﬁnd

promising areas in the search space and a clustering

operator allowing to regroup them in a set of clusters.

In the second step, a tabu search algorithm is applied

to ﬁnd the best individual solution for each cluster.

The global process of the proposed approach is imple-

mented in a two hierarchical holonic levels adopted

by a recursive multiagent model, named Genetic Al-

gorithm combined with Tabu Search in a Holonic

Multiagent model (GATS+HM), see ﬁgure 1. The

ﬁrst holonic level is composed by a Scheduler Agent

which is the Master/Super-agent, preparing the best

promising regions of the search space, and the sec-

ond holonic level containing a set of Cluster Agents

which are the Workers/Sub-agents, guiding the search

to the global optimum solution of the problem. Each

holonic level of this model is responsible to process a

step of the hybrid metaheuristic algorithm and to co-

operate between them to attain the global solution of

the problem.

In fact, the choice of this new metaheuristic com-

bination is justiﬁed by that the standard metaheuristic

methods use generally the diversiﬁcation techniques

to generate and to improve many different solutions

distributed in the search space, or by using local

search techniques to generate a more improved set of

neighbourhood solutions from an initial solution. But

they did not guarantee to attain promising areas with

good ﬁtness convergingto the global optimum despite

the repetition of many iterations, that is why they need

to be more optimized. So, the novelty of our approach

is to launch a genetic algorithm based on a diversiﬁca-

tion technique to only explore the search space and to

select the best promising regions by the clustering op-

erator. Then, applying the intensiﬁcation technique of

the tabu search allowing to relaunch the search from

an elite solution of each cluster autonomously to at-

tain more dominant solutions of the search space.

The use of a multiagent system gives the opportu-

nity for distributed and parallel treatments which are

very complimentary for the second step of the pro-

posed approach. Indeed, our combined metaheuris-

tic approach follows the paradigm of “Master” and

“Workers” which are two recursive hierarchical lev-

els adaptable for a holonic multiagent model, where

the Scheduler Agent is the Master/Super-agent of its

society and the Cluster Agents are its Workers/Sub-

agents.

Figure 1: Genetic algorithm combined with tabu search in a

holonic multiagent model.

3.1 Scheduler Agent

The Scheduler Agent (SA) is responsible to process

the ﬁrst step of the hybrid algorithm by using a genetic

algorithm called NGA (Neighborhood-based Genetic

Algorithm) to identify areas with high average ﬁtness

in the search space. In fact, the goal of using the NGA

is only to explore the search space, but not to ﬁnd

the global solution of the problem. Then, a clustering

operator is integrated to divide the best identiﬁed ar-

eas by the NGA in the search space to different parts

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where each part is a cluster CL

∈ CL the set of clus-

ters, where CL = {CL

,CL

,...,CL

}. In addition,

this agent plays the role of an interface between the

user and the system (initial parameter inputs and ﬁnal

result outputs). According to the number of clusters N

obtained after the integration of the clustering opera-

tor, the SA creates N Cluster Agents (CAs) preparing

the passage to the next step of the global algorithm.

After that, the SA remains in a waiting state until the

reception of the best solutions found by the CA for

each cluster. Finally, it ﬁnishes the process by dis-

playing the ﬁnal solution of the problem.

3.1.1 Individual’s Solution Presentation

The ﬂexible job shop problem is composed by two

sub-problems: the machine assignment problem and

the operation scheduling problem, that is why the

chromosome representation is encoded in two parts:

Machine Assignment part (MA) and Operation Se-

quence part (OS).

Figure 2: The chromosome representation of a scheduling

solution.

The ﬁrst part MA is a vector V

with a length

L equal to the total number of operation and where

each index represents the selected machine to pro-

cess an operation indicated at position p, see ﬁg-

ure 2 (a). For example p = 2, V

(2) is the se-

lected machine M

for the operation O

1,2

. The sec-

ond part OS is a vector V

having the same length

of V

and where each index represents an operation

i, j

according to the predeﬁned operations of the

job set, see ﬁgure 2 (b). For example the opera-

tion sequence 1 − 2− 1 − 3 − 2 − 3− 2 can be trans-

lated to: (O

1,1

) → (O

2,1

) → (O

1,2

) →

3,1

) → (O

2,2

) → (O

3,2

) → (O

2,3

To convert the chromosome values to an active

schedule, we used the priority-based decoding of

(Gao et al., 2008). This method considers the idle

time which may exist between operations on a ma-

chine m, and which is caused by the precedence con-

straints of operations belonging to the same job i. Let

i, j

is the starting time of an operation O

i, j

(which

can only be started after processing its precedent op-

eration O

i,( j−1)

) with its completion time C

i, j

. In ad-

dition, we have an execution time interval [t

]

starts form t

and ends at t

on a machine m to al-

locate an operation O

i, j

. So, if j = 1, S

i, j

takes t

else if j ≥ 2, it takes max{t

i,( j−1)

}. In fact, the

availability of the time interval [t

] for an oper-

ation O

i, j

is validated by verifying if there is a sufﬁ-

cient time period to complete the execution time p

ijm

of this operation, see equation (2):

if j = 1,t

+ p

ijm

≤ t

(2)

if j ≥ 2, max{t

i,( j−1)

} + p

ijm

≤ t

The used priority-based decoding method allows

in each case to assign each operation to its reserved

machine following the presented execution order of

the operation sequence vector V

. Also, to schedule

an operation O

i, j

on a machine m, the ﬁxed idle time

intervals of the selected machine are veriﬁed to ﬁnd

an allowed available period to its execution. So, if a

period is found, the operation O

i, j

is executed there,

else it is moved to be executed at the end of the ma-

chine m.

Noting that the chromosome ﬁtness is calculated

by Fitness(i) which is the ﬁtness function of each

chromosome i and Cmax(i) is its makespan value,

where i ∈ {1,...,P} and P is the total population size,

see equation (3).

Fitness(i) =

Cmax(i)

(3)

3.1.2 Population Initialization

The initial population is generated randomly follow-

ing a uniform law and based on a neighborhood pa-

rameter to make the individual solutions more diver-

siﬁed and distributed in the search space. In fact, each

new solution should have a predeﬁned distance with

all the other solutions to be considered as a new mem-

ber of the initial solution. The used method to deter-

minate the neighborhood parameter is inspired from

GeneticAlgorithmCombinedwithTabuSearchinaHolonicMultiagentModelforFlexibleJobShopSchedulingProblem

577

(Bozejko et al., 2010a), which is based on the per-

mutation level of operations to obtain the distance be-

tween two solutions. In fact, the dissimilarity distance

is calculated by verifying the difference between two

chromosomes in terms of the placement of each op-

eration O

i, j

on its alternative machine set in the ma-

chine assignment vector V

and its execution order in

the operation sequence vector V

. So, if there is a dif-

ference in the vector V

, the distance is incremented

by M(O

i, j

) (is the number of possible n placement

for each operation on its machine set, which is the

alternative machine number of each operation O

i, j

)

because it is in the order of O(n). Then, if there

is a difference in the vector V

, the distance is in-

cremented by 1 because it is in the order of O(1).

Let Chrom1(MA

,OS

) and Chrom2(MA

,OS

) two

chromosomes of two different scheduling solutions,

M(O

i, j

) the alternative number of machines of each

operation O

i, j

, L is the total number of operations of

all jobs and Dist is the dissimilarity distance. The

distance is calculated ﬁrstly by measuring the differ-

ence between the machine assignment vectors MA

and MA

which is in order of O(n), then by verify-

ing the execution order difference of the operation

sequence vectors OS

and OS

which is in order of

O(1), we give here how to proceed:

Noting that Distmax is the maximal dissimilar-

ity distance and it is calculated by equation (4), rep-

resenting 100% of difference between two chromo-

somes.

Distmax =

i,ni

∑

i,1

[M(Oi, j)] + L (4)

3.1.3 Selection Operator

The selection operator is used to select the best par-

ent individuals to prepare them to the crossover step.

This operator is based on a ﬁtness parameter allow-

ing to analyze the quality of each selected solution.

But progressively the ﬁtness values will be similar

for the most individuals. That is why, we integrate

the neighborhoodparameter, where we propose a new

combined parent selection operator named Fitness-

Neighborhood Selection Operator (FNSO) allowing

to add the dissimilarity distance criteria to the ﬁtness

parameter to select the best parents for the crossover

step. The FNSO chooses in each iteration two parent

individuals until engaging all the population to cre-

ate the next generation. The ﬁrst parent takes succes-

sively in each case a solution i, where i ∈ {1, . . . , P}

and P is the total population size. The second par-

ent obtains its solution j randomly by the roulette

wheel selection method based on the two Fitness and

Neighborhood parameters relative to the selected ﬁrst

parent, where j ∈ {1,...,P} \ {i} in the P population

and where j 6= i. In fact, to use this random method,

we should calculate the Fitness-Neighborhood total

FN for the population, see equation (5), the selec-

tion probability sp

for each individual I

, see equa-

tion (6), and the cumulative probability cp

, see equa-

tion (7). After that, a random number r will be gener-

ated from the uniform range [0,1]. If r ≤ cp

then the

second parent takes the ﬁrst individual I

, else it gets

the k

individual I

∈ {I

,...,I

} \ {I

} and where

k−1

< r ≤ cp

• The Fitness-Neighborhood total for the popula-

tion:

FN =

∑

k=1

[1/(Cmax[k] × Neighborhood[i][k])]

(5)

• The selection probability sp

for each individual

1/(Cmax[k] × Neighborhood[i][k])

(6)

• The cumulative probability cp

for each individ-

ual I

∑

h=1

(7)

=⇒ For equations (5), (6) and (7), k =

{1,2, . . . , P} \ { i}

3.1.4 Crossover Operator

The crossover operator has an important role in the

global process, allowing to combine in each case the

chromosomes of two parents in order to obtain new

individuals and to attain new better parts in the search

space. In this work, this operator is applied with

two different techniques successively for the parent’s

chromosome vectors MA and OS.

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Machine Vector Crossover. A uniform crossover

is used to generate in each case a mixed vector

between two machine vector parents, Parent1-MA1

and Parent2-MA2, allowing to obtain two new chil-

dren, Child1-MA1

′

and Child2-MA2

′

. This uniform

crossover is based on two assignment cases, if the

generated number is less than 0.5, the ﬁrst child gets

the current machine value of parent1 and the second

child takes the current machine value of parent2. Else,

the two children change their assignment direction,

ﬁrst child to parent2 and the second child to parent1.

Operation Vector Crossover. An improved prece-

dence preserving order-based on crossover (iPOX),

inspired from (Lee et al., 1998), is adapted for the

parent operation vector OS. This iPOX operator is ap-

plied following four steps, a ﬁrst step is selecting two

parent operation vectors (OS

and OS

) and generat-

ing randomly two job sub-sets Js

/Js

from all jobs.

A second step is allowing to copy any element in

/OS

that belong to Js

/Js

into child individual

′

/OS

′

and retain them in the same position. Then

the third step deletes the elements that are already in

the sub-set Js

/Js

from OS

/OS

. Finally, ﬁll orderly

the empty position in OS

′

/OS

′

with the reminder ele-

ments of OS2/OS1 in the fourth step, see the example

in the ﬁgure 3.

Figure 3: An iPOX crossover example.

3.1.5 Mutation Operator

The mutation operator is integrated to promote the

children generation diversity. In fact, this operator is

applied on the chromosome of the new children gen-

erated by the crossover operation. Also, each part of a

child chromosome MA and OS has separately its own

mutation technique.

Machine Vector Mutation. This ﬁrst operator uses

a random selection of an index from the machine vec-

tor MA. Then, it replaces the machine number in the

selected index by another belonging to the same alter-

native machine set, see ﬁgure 4.

Figure 4: A machine vector mutation example.

Operation Vector Mutation. This second operator

selects randomly two indexes index1 and index2 from

the operation vector OS. Next, it changes the position

of the job number in the index1 to the second index2

and inversely.

3.1.6 Replacement Operator

The replacement operator has an important role to

prepare the remaining surviving population to be con-

sidered for the next iterations. This operator replaces

in each case a parent by one of its children which has

the best ﬁtness in its current family.

Figure 5: The ﬁnal population transformation by applying

the clustering operator.

3.1.7 Clustering Operator

By ﬁnishing the last iteration of the genetic algorithm,

the Scheduler Agent applies a clustering operator us-

ing the hierarchical clustering algorithm of (Johnson,

GeneticAlgorithmCombinedwithTabuSearchinaHolonicMultiagentModelforFlexibleJobShopSchedulingProblem

579

1967) to divide the ﬁnal population into N Clusters,

see ﬁgure 5, to be treated by the Cluster Agents in

the second step of the global process. The clustering

operator is based on the neighbourhood parameter

which is the dissimilarity distance between individu-

als. The clustering operator starts by assigning each

individual Indiv(i) to a cluster CL

, so if we have P

individuals, we have now P clusters containing just

one individual in each of them. For each case, we

ﬁxe an individual Indiv(i) and we verify successively

for each next individual Indiv( j) from the remaining

population (where i and j ∈ {1,...,P},i 6= j) if the

dissimilarity distance Dist between Indiv(i) and

Indiv( j) is less than or equal to a ﬁxed threshold

Dist fix (representing a percentage of difference X%

relatively to Distmax, see equation (8)) and where

Cluster(Indiv(i)) 6= Cluster(Indiv( j)). If it is the

case, Merge(Cluster(Indiv(i)),Cluster(Indiv( j))),

else continue the search for new combination with

the remaining individuals. The stopping condition is

by browsing all the population individuals, where we

obtained at the end N Clusters.

Dist fix = Distmax× X% (8)

Figure 6: Distribution of the Cluster Agents in the different

clusters of the search space.

3.2 Cluster Agents

Each Cluster Agent CA

is responsible to apply suc-

cessively to each cluster CL

a local search technique

which is the Tabu Search algorithm to guide the re-

search in promising regions of the search space and to

improve the quality of the ﬁnal population of the ge-

netic algorithm. In fact, this local search is executed

simultaneously by the set of the CAs agents, where

each CA starts the research from an elite solution of

its cluster searching to attain new more dominant in-

dividual solutions separately in its assigned cluster

, see ﬁgure 6. The used Tabu Search algorithm is

based on an intensiﬁcation technique allowing to start

the research from an elite solution in a cluster CL

promising part in the search space) in order to collect

new scheduling sequence minimizing the makespan.

Let E the elite solution of a cluster CL

, E

′

∈ N(E) is

a neighbor of the elite solution E, each CL

plays the

role of the tabu list with a dynamic length and Cmax

is the makespan of the obtained solution. So, this al-

gorithm applies in each case two steps, a move and

insert ﬁrst step inspired from (Mastrolilli and Gam-

bardella, 2000) to generate new scheduling combina-

tion and a second step to save the best found solution

in the tabu list (which is CL

) after each generation

case. The stopping condition is by attaining the max-

imum allowed number of neighbors for a solution E

without improvement. We give here how to proceed:

By ﬁnishing this local search step, the CAs agents

terminate the process by sending their last best solu-

tions to the SA agent.

4 EXPERIMENTAL RESULTS

4.1 Experimental Setup

The proposed GATS+HM is implemented in java lan-

guage on a 2.10 GHz Intel Core 2 Duo processor and 3

Gb of RAM memory, where we use the IntegratedDe-

velopment Environment (IDE) eclipse to code the al-

gorithm and the multiagent platform Jade (Bellifem-

ine et al., 1999) to create the different agents of our

holonic model. To evaluate its efﬁciency, numerical

tests are made based on two sets of well known bench-

mark instances in the literature of the FJSP:

ICEIS2015-17thInternationalConferenceonEnterpriseInformationSystems

580

Table 2: Results of the Kacem instances (part 1).

Instance

Problem

n× m

AL+CGA (2002) LEGA (2007) KBACO (2010)

MOPSO+LS

(2011)

Best

Dev

(%)

Best

Dev

(%)

Best

Dev

(%)

Best

Dev

(%)

case 1 4×5 16 13,250 11 0 11 0 16 31,25

case 2 8×8 15 6,666 N/A – 14 0 14 0

case 3 10×7 15 26,666 11 0 11 0 15 26,666

case 4 10×10 7 0 7 0 7 0 7 0

case 5 15×10 23 52,173 12 8,333 11 0 11 0

Table 3: Results of the Kacem instances (part 2).

Instance

Problem

n× m

Hybrid NSGA-II

(2014) Heuristic (2014) GATS+HM

Best

Dev

(%) Best

Dev

(%) Best

Avg

Cmax

Avg CPU

(in seconds)

case 1 4×5 11 0 11 0 11 11,00 0,05

case 2 8×8 15 6,666 15 6,666 14 14,20 0,36

case 3 10×7 N/A – 13 15,384 11 11,40 0,72

case 4 10×10 7 0 7 0 7 7,60 1,51

case 5 15×10 11 0 12 8,333 11 11,60 29,71

• Kacem Data (Kacem et al., 2002b): The data set

consists of 5 problems considering a number of

jobs ranging from 4 to 15 with a number of op-

erations for each job ranging from 2 to 4, and a

number of operations for all jobs ranges from 12

to 56, which will be processed on a number of

machines ranging from 5 to 10.

• Brandimarte Data (Brandimarte, 1993): The data

set consists of 10 problems considering a number

of jobs ranging from 10 to 20 with a number of

operations for each job ranging from 5 to 15, and

a number of operations for all jobs ranges from

55 to 240, which will be processed on a number

of machines ranging from 4 to 15.

Due to the non-deterministic nature of the pro-

posed algorithm, we run it ﬁve independent times for

each case of the two data instances in order to ob-

tain signiﬁcant results. The computational results are

presented by four metrics such as the best makespan

(Best), the average of makespan (Avg Cmax), the av-

erage of CPU time in seconds (Avg CPU), and the

standard deviation of makespan (Dev %) which is cal-

culated by equation (9). The Mko is the makespan

obtained by Our algorithm and Mkc is the makespan

of an algorithm that we chose to Compare to.

Dev = [(Mkc− Mko)/Mkc] × 100% (9)

The used parameter settings for our algorithm are

adjusted experimentally and presented as follow:

• Crossover probability 1.0

• Mutation probability 1.0

• Maximum number of iterations 1000

• The population size ranged from 15 to 300 de-

pending on the complexity of the problem.

4.2 Experimental Comparisons

To show the efﬁciency of our GATS+HM algorithm,

we compare its obtained results from the two previ-

ously cited data sets with other well known algorithms

in the literature of the FJSP. The chosen algorithms

are the TS of (Brandimarte, 1993), the AL+CGA of

(Kacem et al., 2002b), the LEGA of (Ho et al., 2007),

the MATSLO+ of (Ennigrou and Gh´edira, 2008), the

KBACO of (Xing et al., 2010), the TS3 of (Bozejko

et al., 2010a), the MOPSO+LS of (Moslehi and Mah-

nam, 2011), the MATSPSO of (Henchiri and Enni-

grou, 2013), the Hybrid NSGA-II of (Zhang et al.,

2014) and the Heuristic of (Ziaee, 2014). The differ-

ent comparative results are displayed in the tables 2,

3, 4 and 5, where the ﬁrst column takes the name of

each instance, the second column gives the size each

instance, with n the number of jobs and m the num-

ber of machines (n × m), and the remaining columns

detail the experimental results of the different chosen

approaches in terms of the best Cmax (Best) and the

standard deviation (Dev %). The bold values in the

tables signify the best obtained results and the N/A

means that the result is not available.

By analyzing the table 2 and table 3, it can be

GeneticAlgorithmCombinedwithTabuSearchinaHolonicMultiagentModelforFlexibleJobShopSchedulingProblem

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Table 4: Results of the Brandimarte instances (part 1).

Instance

Problem

n× m

TS (1993) LEGA (2007)

MATSLO+

(2008)

KBACO (2010)

Best

Dev

(%)

Best

Dev

(%)

Best

Dev

(%)

Best

Dev

(%)

MK01 10×6 42 4,761 40 0 40 0 39 -2,564

MK02 10×6 32 15,625 29 6,896 32 15,625 29 6,896

MK03 15×8 211 3,317 N/A – 207 1,449 204 0

MK04 15×8 81 20,987 67 4,477 67 4,477 65 1,538

MK05 15×4 186 6,989 176 1,704 188 7,978 173 0

MK06 10×15 86 24,418 67 2,985 85 23,529 67 2,985

MK07 20×5 157 8,280 147 2,040 154 6,493 144 0

MK08 20×10 523 0 523 0 523 0 523 0

MK09 20×10 369 15,718 320 2,812 437 28,832 311 0

MK10 20×15 296 25 229 3,056 380 41,578 229 3,056

Table 5: Results of the Brandimarte instances (part 2).

Instance

Problem

n× m

TS3 (2010) MATSPSO (2013) Heuristic (2014) GATS+HM

Best

Dev

(%)

Best

Dev

(%)

Best

Dev

(%)

Best

Avg

Cmax

Avg CPU (in

seconds)

MK01 10×6 40 0 39

2,564

42 4,761 40 40,80 0,93

MK02 10×6 29 6,896 27 0 28 3,571 27 27,80 1,18

MK03 15×8 204 0 207 1,449 204 0 204 204,00 1,55

MK04 15×8 65 1,538 65 1,538 75 14,666 64 65,60 4,36

MK05 15×4 173 0 174 0,574 179 3,351 173 174,80 8,02

MK06 10×15 68 4,411 72 9,722 69 5,797 65 67,00 110,01

MK07 20×5 144 0 154 6,493 149 3,355 144 144,00 19,73

MK08 20×10 523 0 523 0 555 5,765 523 523,00 11,50

MK09 20×10 326 4,601 340 8,529 342 9,064 311 311,80 79,68

MK10 20×15 227 2,202 299 25,752 242 8,264 222 224,80 185,64

seen that our algorithm GATS+HM is the best one

which solves the ﬁves instances of Kacem. In fact,

the GATS+HM outperforms the AL+CGA in four out

of ﬁve instances, the LEGA, the MOPSO+LS and the

Hybrid NSGA-II in two out of ﬁve instances, and

the Heuristic in three out of ﬁve instances. Also,

our algorithm attains the same results obtained by

the KBACO, but the efﬁciency of GATS+HM can

be noted by the average of the CPU time in sec-

onds which is very acceptable for a real manufactur-

ing case.

From table 4 and table 5, the comparison re-

sults show that the GATS+HM obtains nine out of

ten best results for the Brandimarte instances. In-

deed, our algorithm outperforms the TS in nine out

of ten instances. Moreover, our GATS+HM outper-

forms the LEGA and the MATSLO+ in eight out of

ten instances. For the comparison with the KBACO,

the GATS+HM obtains the best solutions for the

MK02, MK04, MK06 and MK10 instances, but it gets

slightly worse result for the MK01 instance. Also, our

hybrid approach outperformsthe TS3 in ﬁveout of ten

instances. Furthermore, the MATSPSO attained the

best result for the MK01 instance, but our algorithm

obtains a set of solutions better than it for the remain-

ing instances. In addition, our algorithm outperforms

the Heuristic in all the Brandimarte instances. By

solving this second data set, our GATS+HM attains

the same results obtained by some approaches such

as the MK01 for LEGA, MATSLO+ and TS3; the

MK02 for MATSPSO; the MK03 for KBACO, TS3

and Heuristic; the MK05 and MK07 for KBACO and

TS3; the MK09 for KBACO; the MK08 for all the

algorithms only it is not the case for the Heuristic.

By analyzing the computational time in seconds

and the comparison results of our algorithm in term

of makespan, we can distinguish the efﬁciency of the

new proposed GATS+HM relatively to the literature

of the FJSP. This efﬁciency is explained by the ﬂex-

ible selection of the promising parts of the search

space by the clustering operator after the genetic al-

gorithm process and by applying the intensiﬁcation

technique of the tabu search allowing to start from an

elite solution to attain new more dominant solutions.

ICEIS2015-17thInternationalConferenceonEnterpriseInformationSystems

582

5 CONCLUSIONS

In this paper, we present a new combination of a ge-

netic algorithm with a tabu search in a holonic multi-

agent model, called GATS+HM, for the ﬂexible job

shop scheduling problem (FJSP). In this approach,

a Neighborhood-based Genetic Algorithm is adapted

by a Scheduler Agent (SA) for a global exploration

of the search space. Then, a local search technique is

applied by a set of Cluster Agents (CAs) to guide the

research in promising regions of the search space and

to improve the quality of the ﬁnal population. Numer-

ical tests are made to measure the performance of the

proposed approach using two well known data sets

from the literature of the FJSP. The experimental re-

sults show that the proposed approach is efﬁcient in

comparison to others approaches. In the future work,

improvements will be done on the tabu search of each

Cluster Agent. In fact, if a Cluster Agent CA

ﬁnds

a new dominant solution E

′

and it does not attain the

allowed number of neighbors from its current solu-

tion E, it does not stop, but it creates in each case a

new Sub-Cluster Agent SCA

i, j

allowing to relaunch

the search from the new obtained solution E

′

. Then

it continues to generate the remaining allowed neigh-

bors from its solution E searching other dominant so-

lutions. In other words, each Cluster Agent distributes

the new found solutions E

′

from its elite solution E

on a new sub-set of Sub-Cluster Agents SCA

i, j

launch from them their new search processes, which

enhances more the search in the selected promising

areas and shows more the efﬁcient use of the notion of

the holonic agents (recursive agents) in our proposed

solution for the FJSP problem. Also, we will search

to treat other extensions of the FJSP, such as by inte-

grating new transportation resources constraints in the

shop process. So, we will make improvements to our

approach to adapt it to this new transformation and

study its effects on the makespan.

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