Weaknesses of Ant System for the Distributed Job Shop Scheduling

Problem

Imen Chaouch

1,2

, Olfa Belkahla Driss

1,3

and Khaled Ghedira

1,4

COSMOS Laboratory, Universit

e de la Manouba, La Manouba, Tunisia

Ecole Nationale des Sciences de l’Informatique, Universit

e de la Manouba, La Manouba, Tunisia

Ecole Sup

erieure de Commerce de Tunis, Universit

e de la Manouba, La Manouba, Tunisia

Institut Sup

erieur de Gestion de Tunis, Universit

e de Tunis, Tunis, Tunisia

Keywords:

Ant System, Distributed, Job Shop, Makespan, Scheduling, Multi-factory.

Abstract:

Globalization has opened up huge opportunities for the plant and industrial investors. The problem of single

plant is now more generalised, namely, multi factory problem. This paper deals with the problem of Distributed

Job shop Scheduling in multi-factories. The problem solving process consists of ﬁnding an effective way to

assign jobs to factories then, to generate a good operation schedule. To make this, an Ant System algorithm

is implemented. Several numerical experiments are conducted to evaluate the performance of the Ant System

algorithm applied to the Distributed Job shop Scheduling, and the results show the shortcoming of the standard

Ant System algorithm compared to developed algorithms in the literature.

1 INTRODUCTION

The manufacturing industry has undergone an impor-

tant evolution these recent years due to the trend of

globalisation. Owing to this evolution, there have

been signiﬁcant changes in the structure of produc-

tion plants. Industrial companies are increasingly

merging to distributed ones and thus, the structure

of their shops changes from simple conﬁgurations to

distributed ones. This system enables ﬁrms to in-

crease their competitiveness and responsiveness in the

global markets (Karimi and Davoudpour, 2017). Dis-

tributed workshops are becoming a popular thematic

to study in the ﬁeld of scheduling problems : Dis-

tributed Flow shop ( (Naderi and Ruiz, 2010), (Gao

and Chen, 2011), (Bargaoui et al., 2016), etc.), Dis-

tributed Job shop ( (Jia et al., 2007), (Naderi and

Azab, 2015), etc.).

In this paper, we focus on the Distributed Job shop

Scheduling Problem (DJSP), which can be considered

as an extension of the simple Job shop Scheduling

Problem (JSP). It can be treated as a set of f fac-

tories, which are geographically distributed in dif-

ferent areas. Each factory contains m machines on

which certain number of jobs have to be processed. A

generic representation can be seen in ﬁgure.1. Dis-

tributed Scheduling problems in multi-factory pro-

duction are much more complicated than classical

scheduling problems (Chung et al., 2009) since two

decisions have to be taken: allocating jobs to suitable

factories and sequencing the operations on machines

so that yield a feasible schedule aiming to minimize

one or more predeﬁned performance criteria.

In this work, we seek to minimize the maxi-

mum completion time (makespan, denoted as C

max

)

of DJSP, which is the maximum makespan among all

factories using an Ant System algorithm.

(Garey et al., 1976) proved that the JSP is strongly

NP-hard. Hence, the DJSP is ordinarily NP-hard and

the case of the simple JSP can be obtained when f =

In this paper, we focus our attention on the DJSP

in multi-factories assuming that all factories are iden-

tical due to the complexity of the problem in such a

system. The rest of the paper is organized as follows.

Section 2 gives the speciﬁcations of the DJSP. Then a

short literature review of the limited existing literature

on DJSP is provided. Section 3 proposes an effective

way to assign jobs to factories and sketches the pro-

posed Ant System algorithm. Section 4 conducts the

numerical experiments. Finally, Section 5 concludes

the paper and suggests few future research directions.

574

Chaouch, I., Driss, O. and Ghedira, K.

Weaknesses of Ant System for the Distributed Job Shop Scheduling Problem.

DOI: 10.5220/0006332405740581

In Proceedings of the 19th International Conference on Enterprise Information Systems (ICEIS 2017) - Volume 1, pages 574-581

ISBN: 978-989-758-247-9

2 PROBLEM STATEMENT AND

STATE-OF-THE-ART

Scheduling problems have become a popular issue for

researchers and industrialists in the last three decades,

particularly the JSP since it is one of the most difﬁcult

tasks. (Colorni et al., 1994), (Dell’Amico and Tru-

bian, 1993), (Adams et al., 1988) and (Davis, 1985)

are pioneer researches in the literature that dealt with

the JSP. Recently, the JSP has evolved from the simple

conﬁguration with one factory to the Distributed one

and becomes increasingly, one of the most important

issues to raise.

The DJSP can be stated as follows: a set J =

{

... j

}

of independent jobs, each of which consists of an or-

dered set of operations. Each operation must be exe-

cuted on a speciﬁc machine from a set M =

{

...i

}

of machines geographically distributed on f identi-

cal factories. The main objective of the problem is

to ﬁnd an optimal scheduling minimizing a speci-

ﬁed criterion which is generally time related such as

makespan, maximum tardiness or total tardiness. In

our case, we are aiming to minimize the maximum

completion time (makespan) among all factories.

There are various constraints on both jobs and ma-

chines. The DJSP entails the following assumptions:

• All jobs are independent and available for pro-

cessing at time 0 and all machines are continu-

ously available.

• Once a job is assigned to a factory it cannot be

transferred to another factory as the remaining op-

erations must be completed in the same plant.

• All factories are able to process all jobs.

• There are no precedence constraints among the

operations of different jobs.

• Each operation needs to be processed during an

uninterrupted time of a ﬁxed processing period

and a given machine.

• A job can be processed by at most one machine at

a time and a machine can process at most one job

at a time.

• It is assumed that a job does not visit the same

machine twice and neither the release times nor

due dates are speciﬁed.

• Setup times of machines and transit times between

operations are negligible.

The following example will make the idea clear

about the representation of the problem. Consider a

DJSP with f = 2, n = 6 and m = 2. The processing

time matrix is shown in table 1. A feasible Gantt chart

of this problem is shown in ﬁgure 2. Makespan in fac-

tory 1 is 17 and makespan in factory 2 is 15, leading to

the conclusion that the makespan of this DJSP is equal

to the maximum makespan between the two factories,

which is 17.

Table 1: Processing time matrix.

Job Machine Processing Route

1 2

1 4 7 {1,2}

2 6 6 {2,1}

3 5 6 {2,1}

4 6 4 {1,2}

5 3 5 {2,1}

6 4 4 {2,1}

Researchers are beginning to study the DJSP re-

cently. We can ﬁnd Jia et al. (Jia et al., 2002) which

have studied the DJSP and proposed a Genetic Al-

gorithm (GA) approach in order to facilitate collabo-

ration between geographically distributed plants. In

their next paper, to solve the same problem in a multi-

factory network, authors in (Jia et al., 2003) presented

a Modiﬁed Genetic Algorithm (MGA) in which two-

step encoding method was used to encode the factory

candidates and to affect jobs and operations. Later

(Jia et al., 2007), they reﬁned their previous approach

and proposed a GA integrated with Gantt Chart (GC)

to derive the factory combination and schedule.

Recently, the problem of DJS have been math-

ematically formulated by Naderi and Azab (Naderi

and Azab, 2014) with two different Mixed Inte-

ger Linear Programming models (MILP). In addi-

tion, three well-known heuristics were ﬁrst adapted

to the problem; these are Shortest Processing Time

ﬁrst (SPT), Longest Processing Time ﬁrst (LPT) and

Longest Remaining Processing Time (LRPT). Then,

three Greedy Heuristics have been deployed (GH1,

GH2 and GH3). In their next paper (Naderi and Azab,

2015), authors have differently treated the problem.

Different forms of a developed simulated annealing

have been designed and implemented and to further

improve the algorithm, two additional mechanisms of

local search and restart phase were designed. The al-

gorithm has been hybridized as well with a greedy

heuristic.

Weaknesses of Ant System for the Distributed Job Shop Scheduling Problem

575

Figure 1: An outline of a typical Distributed Scheduling problem. f factories with m machines, on which jobs have to be

processed.

Figure 2: Gantt Chart of the Distributed Job shop Schedul-

ing Problem.

3 THE PROPOSED ANT SYSTEM

ALGORITHM

3.1 Job-factory Assignment

A crucial step in solving the DJS problem is the

allocation of jobs to suitable factory. The objective

is to partition jobs into factories so as to equilibrate

the workload in different factories. In our approach,

we use the job-facility assignment rule introduced in

(Naderi and Azab, 2014). As ﬁrst step, the workload

on each machine is separately calculated using the

following rule (1), which is deﬁned for each job j on

each machine i as follows:

workload( j, i) =

∑

k∈R

j,i

j,k

+ p

j,i

∀

i, j

(1)

Where R

j,i

is the set of all machines preceding

machine i in the processing of job j and P

j,i

is the

processing time of job j on machine i.

The Workload of each operation is calculated and

regarding the total workloads, the jobs are ranked in

descending order, from highest workload to the low-

est ones. Suppose that we have f factories. The n ﬁrst

jobs are assigned to factories

{

1...n

}

, respectively.

The workload of machines on different factories be-

comes equal to those of the assigned jobs and the

maximum workload in the f factories is determined.

To assign the next job, the maximum workload is

calculated if the job is assigned to a factory. All the

possibilities should be enumerated and the workload

is calculated at each time. Then, the job is assigned

to the factory with minimum of the maximum work-

load. The procedure repeats for subsequent jobs until

all jobs are assigned. To better illustrate this concept,

it is applied to the previous example with 2 factories,

6 jobs and 2 machines (Table 1). The Workload of

each operation is shown in Table 2.

Table 2: The workload of the example.

Job Machine Total Rank

1 2

1 4 11 15 4

2 12 6 18 1

3 11 6 17 2

4 6 10 16 3

5 8 5 13 5

6 8 4 12 6

Regarding the total workloads, the jobs are ranked

ICEIS 2017 - 19th International Conference on Enterprise Information Systems

576

(2, 3, 4, 1, 5, 6). The two ﬁrst jobs (jobs 2 and 3) are

assigned to factories 1 and 2, respectively. The work-

load of machines 1 and 2 on the ﬁrst factory becomes

12 and 6, respectively. The workload of machines 1

and 2 on the second factory becomes 11 and 6, re-

spectively. Thus, the maximum workload in factories

1 and 2 are 12 and 11, respectively. To assign the next

job (job 4), the maximum workload is calculated. If

job 4 is assigned to factory 1, the workloads become

18 and 16 while if it is assigned to factory 2, we have

workloads of 17 and 16. The maximum workload of

factories 1 and 2 are 18 and 17, respectively. There-

fore, job 4 is assigned to factory 2. The procedure

repeats for subsequent jobs.

This method proved to be efﬁcient to well equilibrate

workloads in different factories.

3.2 Ant System Applied to the DJSP

Once all jobs are affected to their corresponding fac-

tory, they need to be sequenced. To do this, Ant Sys-

tem algorithm is applied. It is important to mention

that this is the ﬁrst time that the Ant System algo-

rithm is applied to DJSP since the thematic areas is

recent. As the name suggests, ant algorithms have

been inspired by the behavior of real ant colonies, in

particular, by their collective foraging behaviour. The

ﬁrst Ant Colony System (ACS) was introduced by

Dorigo’s Ph.D. (Dorigo, 1992), which is called Ant

System (AS). The basic idea in AS is to imitate the

cooperative behavior of real ants to solve optimiza-

tion problems (Talbi, 2009). In the nature, ants are

able to ﬁnd the shortest path between a food source

and their nest according to their collective behaviour.

During their move, they lay down a chemical trail

(pheromone) on the ground, which guides other ants

towards best path. In DJSP, the aim is to ﬁnd the best

path giving the minimum makespan among all pos-

sible paths. The ﬂowchart for the AS and the EAS

algorithms can be deﬁned in Figure 3.

The algorithm below presents the generic ant al-

gorithm. The metaheuristic consists of a parameter

initialization step and mainly two algorithmic proce-

dures which are repeated until a stopping criteria is

reached, a maximum number of iterations in our case.

Constructing Solutions by Ants

The ﬁrst procedure consists of a probabilistic con-

struction of solutions by all the ants according to the

State Transition Rule (2). The probability for an ant

to choose its next node is directed by both the amount

of pheromone on the route and heuristic distance from

its current location to the next one.

Figure 3: AS chart.

Algorithm 1: Ant System pseudocode.

Begin

Set parameters, initialize pheromone trails

While Stopping criterion not satisﬁed do

Position each ant in a starting node

Repeat

For each ant do

Choose next node by applying the state transition rule

End for

Until every ant has built a solution

Update best solution

Apply Global updating of pheromone

End While

End.

p(i, j)(t) =

[τ

i, j

(t)]

i, j

∑

j∈AllowedNodes

[τ

i, j

(t)]

i, j

(2)

with

• τ

i, j

quantity on pheromone between the node

and

node

• d

i, j

heuristic distance between node

and node

In our case, d

i, j

is the processing time of the oper-

ation.

Weaknesses of Ant System for the Distributed Job Shop Scheduling Problem

577

• p

i, j

probability to branch from node

to node

• The parameters α and β tune the relative impor-

tance in probability of the amount of pheromone

versus the heuristic distance.

Artiﬁcial ants can be considered as stochastic greedy

procedures that construct a solution in a probabilistic

manner by adding solution components to partial ones

until a complete solution is derived (Talbi, 2009).

In the general AS, the set of next operations for an

ant in node

to visit is all those not visited. Which is

not the case in the DJSP, choosing the next operation

should respect the operation precedence constraints.

Therefore, for each transition from a node

to node

the ant has to build its Allowed List containing the

operation that can visit.

Updating Pheromone

Once all ants generate a solution, the second pro-

cedure is applied which consist of updating the

pheromone trail using an updating rule (3).

i, j

(t + n) = ρ × τ

i, j

(t) + ρ × ∆τ

i, j

(t + n) (3)

∆τ

i, j

(t + n) =

BestC

max

ant

(4)

where:

ρ : evaporation coefﬁcient ∈ [0, 1]

Q : Constant

The pheromone updating rule is applied in two

phases:

• An evaporation phase where a fraction of the

pheromone evaporates and decreases automati-

cally, so as to diversify the search procedure into

larger solution spaces.

• a reinforcement phase where each ant deposits an

amount of pheromone which is proportional to the

generated solutions

4 NUMERICAL EXPERIMENTS

Experimental results are conducted over two phases.

First, the developed AS algorithm is tested on the

classic FT03 (Fisher and Thompson, 1963) with f =

1, leading us to the case of the simple Job shop Prob-

lem. This phase shows that our solution is not only

relevant for the DJSP but also for the simple case

when f = 1, leading to the simple JSP. Then, some

experiments were conducted on well-known bench-

marks with different level of f proposed by (Fisher

and Thompson, 1963) and (Lawrence, 1984). In the

second phase, 80 instances of Taillard benchmark for

job shop (Taillard, 1993) are tested with different lev-

els of f ( f = 2, 3, 4, 5), summing up 320 instances and

compared with other algorithms in the literature. The

results described in the following sections have been

obtained on a personal computer with 3.4 GHz Intel

Core i7 and 8 GB of RAM memory.

4.1 Small Instances

4.1.1 Traditional Scheduling Problem

If there is one factory, the DJSP becomes a traditional

JSP. Our proposed algorithm is able to deal with the

traditional JSP because each job has the information

of selected factory ID, and this ID is always ﬁxed and

set to 1. In other words, the total number of factories

is 1, that means that all jobs are affected to the same

factory and the job-factory assignment procedure is

omitted.

We apply here our algorithm on the instance FT03

(3 jobs, 3 machines) proposed by (Fisher and Thomp-

son, 1963) to measure the effectiveness of the pro-

posed algorithm for the traditional JSP (table 3) As

seen in ﬁgure 4, makespan obtained by the proposed

AS algorithm is 12 units which is the optimal value

obtained in the literature.

Table 3: Processing time matrix of the FT03.

Job Machine Processing Route

1 2 3

1 3 5 2 {2,3,1}

2 2 4 1 {1,2,3}

3 1 3 4 {1,3,2}

Figure 4: The resultant schedule of the encoded solution of

FT03 with f=1.

4.1.2 Distributed Scheduling Problem

To test the performance of the AS applied to the

DJSP on small instances, well-known benchmarks are

used with different level of f ((Fisher and Thomp-

son, 1963) and (Lawrence, 1984) ). Table 4 sum-

marizes the results obtained by our AS applied to

small instances. Unfortunately, there is no studies

ICEIS 2017 - 19th International Conference on Enterprise Information Systems

578

in the literature using (Fisher and Thompson, 1963)

and (Lawrence, 1984) for the DJSP. For this purpose,

we have been content to cite our obtained results for

small instances without comparison with previous re-

searches. As we can see from table 4, AS is performs

well when the number of factories increases.

Table 4: Obtained C

max

of small instances using AS for

DJSP.

n m f = 2 f = 3 f = 4 f = 5

ft03 3 3 10 10 - -

ft06 6 6 48 47 47 47

ft10 10 10 1053 839 727 655

ft20 20 5 1254 986 789 667

la01 10 5 688 446 459 413

la02 10 5 628 456 394 394

la03 10 5 516 380 356 356

la04 10 5 527 397 421 370

la05 10 5 550 380 380 380

4.2 Large Instances

To test the performance of the AS applied to the DJSP

on large instances, we use those of Taillard bench-

mark for job shops (Taillard, 1993). This bench-

mark includes 8 combinations for n and m, and 10

instances for each combination. It sums up to 80

instances. Each instance is solved by different lev-

els of f ( f = 2, 3, 4, 5); thus, there are 320 instances.

First, the AS parameters are tuning and the best re-

sults are obtained with the parameters initialized as

α = 0.2, β = 0.8, ρ = 0.7, the number of iterations is

ﬁxed at 2000. Table 5 shows the average RPD ob-

tained for the proposed AS applied to large instances.

For each instance, these results are calculated with the

maximum makespan among all factories. The perfor-

mance measure used in this research is Relative Per-

centage Deviation (RPD). It can be calculated as fol-

lows:

RPD =

Alg −Min

Min

× 100 (5)

where

• Alg is the makespan obtained by any of the algo-

rithm

• Min is the lowest makespan obtained for a given

instance.

Results are compared with six available algorithms,

GH3 proposed in (Naderi and Azab, 2014), SA, HAS,

Table 5: The RPD of the AS on large instances with differ-

ent level of f .

Instances 2 f 3 f 4 f 5 f Average

n m

15 15 75 39 4.3 -4.4 28

20 15 119 45 19 13 49

20 145 79 35 18 69

30 15 147 71 28 10 64

20 198 112 61 33 101

50 15 163 92 48 16 79

20 278 127 92 59 139

100 20 418 149 96 52 178

Table 6: The RPD of the AS compared with other algo-

rithms.

n m Algorithms

AS HSA SA GSA GA GH3

15 15 28 0.42 0.34 0.79 2.35 5.37

20 15 49 0.76 0.92 2.06 3.85 9.69

20 69 0.2 0.31 1.30 6.36 8.79

30 15 64 0.4 1.35 2.65 6.07 11.73

20 101 0.14 0.95 2.00 9.09 13.97

50 15 79 1.68 3.42 3.45 10.17 20.46

20 139 0.12 3.19 2.44 12.81 19.40

100 20 178 0.00 6.89 8.52 20.77 61.09

Average 88.46 0.46 2.17 2.90 8.93 18.81

GSA in (Naderi and Azab, 2015) and genetic algo-

rithm (GA) in (Jia et al., 2007). We should mention

that the RPD values of the SA, HAS, GSA and GA

are calculated by taken the best value given by the

algorithms as optimal value. In their paper, (Naderi

and Azab, 2015) didn’t mention the value of the ob-

tained makespan, that is why we consider the optimal

of (Taillard, 1993). Table 6 shows the results averaged

by the combinations of (n, m).

As we can see from table 6, AS provides worst re-

sults among the tested algorithms with average RPD

of 88, 46.

The reason for these poor results may be explained

by the fact that there is no local search included to

improve ant’s solutions or to guide the ants when ex-

ploring their space.

Figure 5 shows the average RPD of the tested al-

Weaknesses of Ant System for the Distributed Job Shop Scheduling Problem

579

gorithms versus the number of jobs. It is interesting

to see that the Average RPD for almost all algorithms

increases with the number of jobs being increased.

Figure 5: The average RPD of the tested algorithms versus

the number of jobs.

Some initial results for small instances were en-

couraging and have shown the viability of the ap-

proach. However, for larger instances AS gives a very

poor solution quality compared to state-of-the-art al-

gorithms. This is due to randomized character of the

AS algorithm which makes probabilistic decisions in

the construction of the solution. In the literature, re-

searchers presented various improvements over AS. A

ﬁrst improvement on the initial form of AS, is the Eli-

tist Strategy for Ant System, introduced in (Dorigo,

1992) and (Dorigo et al., 1996) where only the best-

so-far solution is used to update the pheromone trails

and the idea is to enhance the promising search space.

Other improvements were presented such as rank-

based Ant System (AS

rank

) (Bullnheimer et al., 1997),

it can be considered as an extension of the elitist

strategy. Ants are sorted according to the quality of

their solutions and the amount of pheromone that each

ant deposits on the trails decreases according to its

rank. Meanwhile, the best-so-far ant still deposits

pheromone at each iteration.

MAX – MIN Ant System (MMAS) (St

utzle,

1998; St

utzle and Hoos, 1997; St

utzle and Hoos,

2000) is another improvements of AS. MMAS intro-

duces upper and lower bounds to the values of the

pheromone trails, as well as a different initialization

of their values. All trails are maintained inside an in-

terval bounded [τ

min

, τ

max

] in order to avoid stagnation

caused by exploring best solutions.

There are others extensions of AS, for exam-

ple Ant Colony System (ACS) by Dorigo and

Gambardella (Dorigo and Gambardella, 1997a;

Dorigo and Gambardella, 1997b; Gambardella and

Dorigo, 1996),Approximate Non-deterministic Tree

Search (ANTS) by Maniezzo (Maniezzo, 1999) and

population-based ACO (P-ACO) by Guntsch and

Middendorf (Cagnoni et al., 2002)

It will be interesting to apply different AS forms

in order to see the inﬂuence of the improvements on

DJSP.

5 CONCLUSION AND

PERSPECTIVES

In this work, we have applied the Ant System algo-

rithm for the ﬁrst time to solve the Distributed Job

shop Scheduling Problem with makespan minimiza-

tion criterion. The algorithm is compared with other

algorithms in literature. Despite of the fact that AS

succeeded in solving small instances of DJSP, results

are very poor for large instances and show that the

standard AS is not competitive at all comparing to

other methods.

For future work, it will be interesting to in-

vestigate on the different variations of Ant System,

namely Elitist ant system (EAS), Max-Min Ant Sys-

tem (MMAS), etc. and maybe the integration of local

search leads to a possible improvement of the results.

Also, we can study the problem aiming to optimize

other objectives. And ﬁnally, we can consider the case

of a DJSP with non identical factories. Because some-

where, considering that the factories are the same, can

be an idealization to the real problem, since it is rarely

the case.

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