multiple colony ant algorithm for job-shop scheduling problem
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International Journal of ProductionResearchPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713696255
Multiple colony ant algorithm for job-shop schedulingproblemA. Udomsakdigool a; V. Kachitvichyanukul aa Industrial Engineering & Management, School of Advanced Technologies, AsianInstitute of Technology, Pathumthani 12120, Thailand
First Published: August 2008
To cite this Article: Udomsakdigool, A. and Kachitvichyanukul, V. (2008) 'Multiplecolony ant algorithm for job-shop scheduling problem', International Journal ofProduction Research, 46:15, 4155 — 4175
To link to this article: DOI: 10.1080/00207540600990432URL: http://dx.doi.org/10.1080/00207540600990432
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International Journal of Production Research,Vol. 46, No. 15, 1 August 2008, 4155–4175
Multiple colony ant algorithm for job-shop scheduling problem
A. UDOMSAKDIGOOL* and V. KACHITVICHYANUKUL
Industrial Engineering & Management, School of Advanced Technologies, Asian Institute of
Technology, PO Box 4, Klong Luang, Pathumthani 12120, Thailand
(Revision received November 2005)
Ant colony optimization (ACO) is a metaheuristic that takes inspiration from theforaging behaviour of a real ant colony to solve the optimization problem. Thispaper presents a multiple colony ant algorithm to solve the Job-shop SchedulingProblem with the objective that minimizes the makespan. In a multiple colonyant algorithm, ants cooperate to find good solutions by exchanging informationamong colonies which are stored in a master pheromone matrix that serves therole of global memory. The exploration of the search space in each colony isguided by different heuristic information. Several specific features are introducedin the algorithm in order to improve the efficiency of the search. Among othersis the local search method by which the ant can fine-tune their neighbourhoodsolutions. The proposed algorithm is tested over set of benchmark problemsand the computational results demonstrate that the multiple colony ant algorithmperforms well on the benchmark problems.
Keywords: Job-shop scheduling problem; Multiple colony ant algorithm;Metaheuristic; Local search
1. Introduction
The Job-shop Scheduling Problem (JSP) belongs to a class of problems that areknown to be strongly NP-hard problems. For every finite size problem instanceoptimal solutions can be obtained in bounded time via enumeration techniques suchas a branch-and-bound method and mathematical programming. However, whileconsiderable progress has been made, it was found that such algorithms may takea prohibitive amount of time, even for a moderate size problem. For pragmaticpurposes there is a need to look for more efficient approximate methods that can findgood solutions in an acceptable time.
In the past decade a new group of approximate algorithms called metaheuristicshave been widely studied to solve the JSP with the objective of minimizing themakespan. These approaches include genetic algorithms (Wang and Zheng 2001,Varela et al. 2003, Watanabe et al. 2005), a greedy randomized adaptivesearch procedure (Aiex et al. 2003), simulated annealing (Kolonko 1999,Aydin and Fogarty 2004), a tabu search (Nowicki and Smutinicki 1996, 2005,Pezzella and Merelli 2000) and the threshold accepting method
*Corresponding author. Email: [email protected]
International Journal of Production Research
ISSN 0020–7543 print/ISSN 1366–588X online � 2008 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/00207540600990432
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(Tarantilis and Kiranoudis 2002, Lee et al. 2004). A comprehensive survey of jobshop scheduling techniques can be founded in Jain and Meeran (1999).
In recent years a metaheuristic called ACO has been receiving extensive attentiondue to its successful applications for many combinatorial optimization problems.In the field of scheduling, ACO has been successfully applied to the single machineweighted tardiness problem (Gaene et al. 2002), the flow-shop scheduling problem(Shyu et al. 2004, Ying and Liao 2004), the open-shop scheduling problem(Blum 2005) and the resource constraint project scheduling (Merkle et al. 2002). Inthe industrial situation Gravel et al. (2002) proposed the ACO to solve the schedulingproblem in an aluminium casting centre. However, the application to the shop-scheduling problem, JSP has especially proven to be quite difficult. Colorni et al.(1994) were the first group of researchers who applied ACO to solve JSP and theiralgorithm was far from reaching a state-of-the-art performance. The earliestcompetitive ACO approach for solving the JSP was performed by Blum andSampels (2004) and their algorithm works well when applied to the open shopscheduling problem. In addition, there is no literature on applying a multiplecolony ant algorithm for JSP.
The present paper presents the development of a multiple colony ant algorithmfor solving the JSP. Each colony in the multiple colony ant algorithm is forcedto search in different regions of search space and to cooperate to find good solutionsby exchanging information among colonies. Several specific features are alsointroduced in the algorithm to improve the efficiency of the search. Amongthe features introduced are a local search in which the neighbourhood solutionsare defined using the method of Nowicki and Smutinicki (1996). The proposedalgorithm is investigated for its potential to solve the benchmark instances availablein the OR-Library (Beasley 2003).
The paper is organized as follows. In the next section a formal definition ofthe JSP in terms of a graph-based representation is presented to facilitate thedevelopment of ACO systems. The descriptions and special features are presented insection 3. The experimental evaluation to fine-tune the candidate list strategyand heuristic information, versions of ant algorithm, and choice of parameter settingare presented in section 4. The computational results on benchmark problemsare provided in section 5. Finally, the conclusion and discussion are presentedin section 6.
2. Problem definition and graph-based representation of problem
2.1 Problem definition
The classical job shop scheduling (Jain and Meeran 1999), denoted n/m/G/Cmax,consists of a set J of n jobs fJig
ni¼1 to be processed on a set M of m machines fMjg
mj¼1.
Oij is the operation of job Ji, which has to be processed on machine Mj for anuninterrupted processing period pij. Each job Ji consists of a chain of operationsfOijg
mj¼1, which represents the predetermined order of job Ji through the machines
(precedence constraint). Each machine can process at most one job, and each jobcan be processed by only one machine at a time (capacity constraint). In this paperno pre-emption is allowed. The duration in which all operations for all jobs are
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completed is referred to as the makespan, Cmax. The objective is to determine thestarting time, Sij, for each operation in order to minimize the makespan whilesatisfying all precedence and capacity constraints as given in equation (1):
minCmax ¼ minfmaxðSij þ pijÞ : 8Ji 2 J, 8Mj 2M g ð1Þ
Feasible schedule.
2.2 Graph-based representation
Every instance of the job-shop problem can be represented in an ACO approachvia a disjunction graph:
G ¼ ðO,C,DÞ,
where O is the set of all nodes (processing operations); C is a set of conjunctivedirected arcs; and D is a set of disjunctive undirected arcs, respectively.C corresponds to the precedence relationship between operations of a single job.Thus, operations belonging to the same job are connected in sequence. D representsthe machine constraint of operations belonging to different jobs. The operationsof jobs that are processed on the same machine are pair-wise connected in bothdirections. Two additional fictitious nodes, the source (the immediate job predecessorof the first operation of every job) and the sink (the immediate job successor of thelast operation of every job) of the zero processing time are added to the set. Thesenodes are the dummy operations of nest and food source, respectively. A path Pis defined as acyclic sequence of total operations that represent the possible solutionof the instance. The makespan of a schedule, Cmax(P), is equal to the longest pathfrom source to sink in path P. This path is called critical path C(P) and theoperations that it passes through are called critical operations.
An illustrative example job-shop problem is introduced here. The instanceconsists of nine operations that are partitioned into three jobs and have to beprocessed on three machines. Job J1 consist of operations {O1!O2!O3},J2 of {O4!O5!O6}, and J3 of {O7!O8!O9}. Machine M1 processes the setof operations {O1,O5,O9}, M2 of {O2,O4,O8}, and M3 of {O3,O6,O7}. Theprocessing times of the nine operations are 2, 3, 4, 3, 2, 2, 2, 3, and 2 units of time,respectively. The disjunctive graph of the exemplary instance is presented in figure 1,where dotted lines represent the conjunctive directed arcs of machines and the boldlines represent the undirected arcs of jobs. When the processing order is determined,the direction of arcs will be selected in such a way that path P is acyclic; the resultis a feasible solution. For example, the sequence of operations of P is to be{source!O1!O7!O2!O3!O4!O8!O5!O9!O6! sink}. The sameschedule is presented in figure 2 using the Gantt chart representation. As can beobserved, the makespan of the sample schedule is 13 units of time since the longestpath or critical path in the graph passes through {O1!O2!O4!O8!O9}.
2.3 ACO applied to JSP
The ACO algorithm is a kind of natural algorithm inspired by the foraging behaviorof real ants. This behavior is the basis for local interaction of the ant which leads
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to the emergence of shortest path. The main idea of ACO is to use a parameterizedprobabilistic model to construct solutions that are then used to update the modelparameter values with the aim of increasing the probability of constructing high-quality solutions (Blum and Sampels 2004). In every iteration the artificial ants builda solution by applying a probabilistic decision to move from the present state tothe adjacent state. The decision policy is a function of both prior informationrepresented by the problem specification and the local modifications in pheromonetrails induced by the past ants. After they complete their paths, the pheromonevalues are released depending on the quality of a solutions, the shorter the distancethe stronger the pheromone value. After the ants repeat this procedure for a certainnumber of iterations, a solution will emerge from the corporation of ants. That is,the path with the strongest pheromone value will become the dominant solution(for more detail, see Dorigo et al. 1996, 1999, 2000, Dorigo and Stutzle 2004).
To apply ACO for JSP, the arcs between nodes in figure 1 are weighted by a pairof numbers (�, �). The first is the pheromone trail level; the second is the visibilitythat is computed from the problem specific heuristic such as the shortest processing
O1 O5 O9
O7 O6O3
O4 O8O2
1086420 12 14
Makespan=13
Time (Unit)
Machine
1
3
2
J1 J2 J3
Thick boxes denote operations in the critical path
Figure 2. Gantt chart.
Disjunctive arc for machine M1Disjunctive arc for machine M2Disjunctive arc for machine M3Conjunctive arc of jobs
Source Sink
O1 O2 O3
O4 O5 O6
O7 O8 O9
Figure 1. Disjunctive graph of 3/3/G/Cmax problem.
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time or the shortest remaining time. All ants are initially at source operation andconstruct a sequence of operations step by step, selecting one operation from a set ofallowable operations by applying a probability transition rule. When all ants havecompleted their solutions, the processing order of operations that ate passed throughby the ants represents the solution and the longest path from source to sink inpath P is equal to the makespan. The pheromone-updating rule is then applied.The pheromone values are released depending on the quality of solutions; the shorterthe makespan the stronger the pheromone value. After the ants repeat this procedurefor certain number of iterations, a solution will emerge.
3. Description of a multiple colony ant algorithm
There are two main components in an ant algorithm: the construction of a solutionand the update of the pheromone. In the construction step an ant selects oneoperation from a set of operations that can be scheduled to construct a feasiblesolution. The strategy for selecting an operation from a set of operations by an antis called a candidate list strategy. The candidate list strategy plays an important rolein reducing the number of schedules in the region that contains an optimal solution.After the candidate operations are determined, an ant will select the operationby applying a probability transitional rule guided by the pheromone trail andheuristic information. The question is which candidate list strategy and heuristicinformation should be used to guide the ant in the search space toward regions thatcontain high-quality solutions. Therefore, it is necessary to fine-tune the candidatelist strategy and heuristic information before they are used in the final versionof algorithm. After all the ants construct the solutions, the local improvement isapplied. In the pheromone-updating step each colony has its own pheromone matrixcalled a local pheromone matrix. On top of that there is a master pheromone matrixthat collects the information from all the colonies. The ant performs both the localpheromone matrix updating and the master pheromone updating. The briefdescription is shown in figure 3; the details are described in the next subsection.The information exchange in multiple colony ant algorithm is illustrated in figure 4.
Input: A problem instance of JSPInitialize pheromone value and parameter valueswhile (termination condition not met ) dofor colony l = 1 to n do
for ant k = 1 to m do Construct solutionApply local improvement
next antUpdate pheromone in colony (Local pheromone matrix)next colonyUpdate pheromone among colonies (Master pheromone matrix)
if the restart condition is reachedApply restart process
end ifend while
Output: Best solution
Figure 3. Multiple colony ant algorithm.
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3.1 Initialize pheromone and parameter setting
Unlike the strategies used by previous researches (Colorni et al. 1994, Blum 2005),
where each for path the pheromone value �ij is initialized with a constant, in the
proposed method the pheromone value on each path is initialized with the value
drawn from a random number in the interval (0.1, 0.5). The reason behind that
is to enforce the diversification at the start of the algorithm. The lower bound of
pheromone value is set to a small positive constant (0.001) to prevent the algorithm
from prematurely converging to a solution. To select the candidate list strategy and
heuristic information the parameters that control the search are first set according to
values suggested by Dorigo and Stutzle (2004), as shown in subsection 4.1. After the
effective candidate list strategy and heuristic information are selected, various
combinations of parameters values are studied in subsection 4.2. The appropriate
parameter values were used in the final version of the ant algorithm.
3.2 Construct solution
At each step of a solution construction an ant selects one operation from a set of
candidate operations: S0t, which is a subset of the operations; and St, which can be
scheduled at a construction step by applying a probability transitional rule. The
probability for each operation in the set S0t to be chosen depends on the two-step
random proportional probability transition rule model that was first described
by Dorigo and Gambarella (1997) as follows. While building the solution the
selection of an operation by an ant is guided by the pheromone information and
heuristic information. In iteration t the kth ant selects an operation by taking
a random number q. If q� q0, the operation is chosen according to equation (2).
Otherwise, an operation is selected according to equation (3):
j kðtÞ ¼arg max
o2S0iðiÞ �
kioðtÞ
� ���kio� ��n o
if q � q0
J otherwise,
(ð2Þ
Best solutionBest solution Master pheromone trail
Global memory(Master pheromone matrix)
Local memory(Local pheromone matrix)
Ant process:1. Reception of master and local pheromone trail2. Solution construction3. Pheromone updating - update local pheromone trail - send the best solution to update the master pheromone trail
Local memory(Local pheromone matrix)
Ant process:1. Reception of master and local pheromone trail2. Solution construction3. Pheromone updating - update local pheromone trail - send the best solution to update the master pheromone trail
Colony nColony 1 . . .
Figure 4. Information exchange in multiple colony ant algorithm.
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where J is an operation selected from the random proportional transition ruledefined as equation (3):
pkioðtÞ ¼
�kioðtÞ� ��
�kio� ��P
o02S0tðiÞ
�kio0ðtÞ
� ���kio0
� ��0 otherwise,
8><>: ð3Þ
where i is the operation at the present step; o is the operation an ant selects forthe next step; o0 is the operation in a candidate list; pio is the probability of selectingan operation; o is for the next step; �io is the pheromone trail between operationi and o; �io is the heuristic information between operation i and o; and S0tðiÞ is the setof operations in a candidate list of operation i. Also, � and � are the parametersthat determine the relative important of the pheromone trail and the heuristicinformation (�40, and �40); q is the random number uniformly distributed in [0, 1];and q0 2 ð0, 1Þ is the parameter that determines the relative importance betweenexploitation and exploration.
The candidate list strategy and the heuristic information used by the ant are firstevaluated as presented in section 4 and the most effective of them are used in the finalversion of the algorithm.
In a multiple colony ant algorithm each colony of ants might have differentcandidate list strategies and heuristic information to guide their search. When anant constructs a solution it uses information from both the local pheromone matrixand the master pheromone matrix. The strength of pheromone trail betweenoperation i and o of ant k in iteration t is computed as follows:
�kioðtÞ ¼ ð1� wÞ�kl,ioðtÞ þ w�km,ioðtÞ, ð4Þ
where w is the important weight of master pheromone trail; �l,io is the pheromonetrail between operation i and o from the local pheromone matrix; and �m,io is thepheromone trail between operation i and o from the master pheromone matrix.
3.3 Local improvement
Local improvement plays an important role for improving the solution especially ina metaheuristic method. The local improvement procedure explores the best solutionfrom a certain neighbourhood of a given schedule and keeps it as the solution. In theant algorithm local improvement is performed after the ants complete their solutions.The local improvement method used in this paper is adapted from the methodproposed by Nowicki and Smutinicki (1996). The detail of the neighbourhoodstructure of this method is summarized as follows.
A neighbourhood schedule is defined using the so-called blocks of operations ona critical path. The neighbourhood schedule is defined by moving the operation nearthe border line of blocks on a single critical path, C(P) in schedule, P. The blocks,B1, . . . ,Br in C(P) are defined. The first two (and the last two) operations in everyblocks B2, . . . ,Br�1, each of which contains at least two operations that are swapped.In the first block the last two operations are swapped and in the last block the firsttwo operations are swapped. After the swap the makespan will be recalculate and thebest makespan is kept as solution. If the swap does not improve the objective the
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original solution is kept. From figure 2 there is only one block where two
neighbourhood solutions can be defined from swapping: {O2!O4} and {O4!O8}.
3.4 Pheromone updating
In ant algorithm with a single colony the pheromone-updating rule is performed only
on the local pheromone matrix. In a multiple colony ant algorithm a two-step
pheromone-updating rule is performed. That is, after each colony updates its localpheromone matrix, the master pheromone matrix is updated.
3.4.1 Local pheromone matrix updating. In each colony after all ants complete theirsolution the best solution found since the start of algorithm, Ss, of each colony
is used to update its local pheromone matrix. The rule of pheromone updating
is defined as equation (4):
�ioðtþ 1Þ ¼ ð1� �Þ�ioðtÞ þ ���ioðtÞ, ð5Þ
where
��ioðtÞ ¼1 if ði, oÞ 2 tour of best ant0 otherwise
�
where � 2 ½0, 1Þ is the pheromone evaporating parameter. The minimum pheromone
value is set to 0.001. When applying the pheromone updating, the pheromone value
that is less than this number is set back to 0.001.
3.4.2 Master pheromone matrix updating. The information exchange amongcolonies is performed after all colonies finish their solution. The best Ss amongcolonies is used to update the master pheromone matrix following equation (5).
3.5 Restart
When the system is stuck in an area of the search space, a restart processis performed. All pheromone values in every path are re-initialized and the algorithm
is started again. The reason for restart is to diversify to find a new, possibly better,
solution in another search space.
4. Computational experiences
The experiments were conducted in two parts. The first part aims to find effective
combinations of a candidate list strategy and heuristic information in the
construction step. The second part is carried out to find appropriate parametervalues used in the final version of the ant algorithm.
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4.1 Evaluation of candidate list strategy and heuristic information
The experiments were conducted with the aim to select the robust candidate liststrategy and heuristic information for a wide range of instances. In these experimentsthe parameters that control the search are set to �¼ 1, �¼ 2, �¼ 0.1, and q0¼ 0.5.The number of ants is set to 0.1 times the number of total operations, O. Thealgorithm terminates when the total number of iterations reaches 1000. The detailof the candidate list strategy and heuristic information are described in the followingsubsection.
4.1.1 Candidate list strategy. In order to cut down the number of schedules to beconsidered, the candidate operation strategies for the ants are restricted to onlyactive and non-delay schedules which are known to contain the optimal solution.Three strategies to define a set of candidate operations are described as follows:
. Active: the earliest possible completion times of all operations in St arecalculated. One of the machines, M�, with minimal completion time, t�,is chosen and a set of S 0t is defined as a subset of all operations that need to beprocessed on machine M� and whose earliest possible starting time is lessthan t�.
. Non-delay: the earliest possible starting time among all operations in St isdetermined. Then the S0t is defined as a subset of all operations that can startat time t.
. Random: ant randomly selects between active and non-delay.
4.1.2 Heuristic information. The priority dispatching rules are used as heuristicinformation for an ant to guide the search. The dispatching rules tested are listed intable 1. To translate the dispatching rule into heuristic information, every operationin S0t is normalized as shown using an example of shortest processing time (SPT).Every operation in S0t, the heuristic information of operation, �0 is computed as:
�0 1=ptðoÞP
o02S0t1=ptðo0Þ
ð6Þ
where pt is the processing time of the operation.
Table 1. Dispatching rules used as heuristic information.
Rule Description
EST Earliest starting timeEFT Earliest finishing timeSPT Shortest processing timeLPT Longest processing timeLWR Least work remaining in the jobMWR Most work remaining in the jobSMT Shortest value obtained by multiplying process time with total process timeLMT Largest value obtained by multiplying process time with total process timeSDT Shortest value obtained by dividing process time with total process timeLDT Largest value obtained by multiplying process time with total process time
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4.1.3 Experimental evaluation of candidate list strategy and heuristic information.
Thirty different combinations of candidate list strategy and heuristic information(table 2) are tested on the 30 benchmark problems from OR-Library, which areFT06, FT10, FT20, LA01–LA03, LA06–LA08, LA11–LA13, LA16–LA18,LA21–LA24, LA26–LA29, LA36–LA39, and YN1–YN3. The number of jobsand machines (n�m) of these problems range from 6� 6 to 20� 20. All versionsof algorithm tested at this stage do not include the local improvement and restartprocess because the objective is to test for significant effects of the candidate liststrategy and heuristic information on the solution quality. Each problem instancewas repeated for ten trials in order to ensure that the random generator did not bearany influence on the quality of results obtained.
The solutions obtained from each algorithm are transformed into rank numberand the analysis of variance (ANOVA) is used to test the significant effects ofcandidate list strategy and heuristic information type on the quality of a solution.In the analyses the problems are categorized into two groups. The first group isa small size problem, which n�m is less than or equal to 10� 10 and the secondis a large size problem, which n�m is more than 10� 10. The analyses of varianceare shown in table 3(a) and (b).
The results indicate that in both small size and large size problems the typesof candidate list strategy and heuristic information significantly affect the qualityof the solution. The effect of heuristic information on solution quality is moresignificant than that of the candidate list strategy. Duncan’s multiple range test(Montgomery 2001) is used to test for the homogenous subset of the combinationsof candidate list strategy and heuristic information that provides the best solutionquality. The analyses are shown in tables 4 and 5.
It can be summarized that in a small size problem the non-delay algorithms withmost work remaining (MWR), non-delay with earliest finish time (EFT), non-delaywith earliest start time (EST), random with EFT, and random with EST are in thegroup that provides the best solution quality. For a large size problem the non-delayalgorithm with MWR, non-delay with EFT, and non-delay with EST are in thegroup that provides the best solution quality.
Table 2. Different versions of algorithms.
No.
Candidatelist
strategyHeuristic
information No.
Candidatelist
strategyHeuristic
information No.
Candidatelist
strategyHeuristic
information
1 N EST 11 A EST 21 R EST2 N EFT 12 A EFT 22 R EFT3 N SPT 13 A SPT 23 R SPT4 N LPT 14 A LPT 24 R LPT5 N LWR 15 A LWR 25 R LWR6 N MWR 16 A MWR 26 R MWR7 N SMT 17 A SMT 27 R SMT8 N LMT 18 A LMT 28 R LMT9 N SDT 19 A SDT 29 R SDT10 N LDT 20 A LDT 30 R LDT
N: Non-delay, A: active, R: random between non-delay and active.
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4.1.4 Versions of ant algorithm. Two versions of the algorithm, a single colony antalgorithm and a multiple colony ant algorithm, are studied in order to investigatetheir performance. First, the performance of single colony ant algorithm, SAnt,with the best combination of candidate list strategy and heuristic information wasinvestigated. The multiple colony ant algorithm, MAnt, is then investigated andcompared with the single colony ant algorithm. In the experiment a non-delayalgorithm with MWR is used in a single colony ant algorithm. For a multiplecolony ant algorithm three colonies are used with all three different combinationsof a candidate list strategy and heuristic information. In other words, one colonywith a non-delay algorithm with MWR, one colony with non-delay with EFT, andone colony with non-delay with EST.
Two versions of each algorithm are tested: one with a local search and restartprocess and one without.
4.2 Choice of parameter setting
To select the appropriate parameter values, ad-hoc experiments are conducted.Several values of parameters are tested on a small size problem (FT06, LA01, LA06,and LA16) and on a large size problem (LA21, LA26, LA36, YN1) with the objectiveto find the guide number for a wide range of problem instance. The values ofeach parameter tested are: �2f1,3,5,7,10g, �2f1,3,5,7,10g, �2 f0:1,0:3,0:5,0:7,0:9g,q0 2 f0:1, 0:3, 0:5, 0:7, 0:9g, and n 2 f0:1 � O, 0:3 � O, 0:5 � O, 1 � O, 1:5 � Og.Duncan’s multiple range test is again used to test the homogenous subset of eachparameter that provides the best quality of a solution. The analyses are separatedinto a small size and a large size problem. The results are summarized as follows:
. �: for a small size problem, two levels of �, 1 and 3, are in the group thatprovides the best quality of a solution, and in the large size problem all levelsof � are in the best group, as shown in table 6.
Table 3. The analysis of variance for testing the significant effect of candidate list strategyand heuristic information on the quality of solution.
SourceType III sumof squares df Mean square F Sig.
(a) Small size problemCAN 3552.49 2 1776.24 86.81 0.00HEU 19 707.23 9 2189.69 107.01 0.00CAN �HEU 1827.00 18 101.50 4.96 0.00Error 8593.76 420 20.46Total 141 793.00 450
(b) Large size problemCAN 9230.00 2 4615.00 519.17 0.000HEU 18 588.03 9 2065.33 232.34 0.000CAN �HEU 2158.53 18 119.91 13.49 0.000Error 3733.43 420 8.88Total 141 822.50 450
R squared¼ 0.745 (Adjusted R squared¼ 0.727).CAN: Candidate list strategy, HEU: Heuristic information.R squared¼ 0.889 (Adjusted R squared¼ 0.882).
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. �: for a small size problem, all levels of � are in the best group that providesthe best quality of a solution, and in the large size problem three levels of�, 5, 7 and 10, are in the best group, as displayed in table 7.
. �: for a small size problem, all levels of � are in the best group that providesthe best quality of a solution, and in the large size problem three levels of�, 0.1, 0.3 and 0.5, are in the best group, as illustrated in table 8.
. q0: for a small size problem, all levels of q0 except 0.9 are in the bestgroup that provides the best quality of a solution, and in the large sizeproblem three levels of q0, 0.1, 0.3 and 0.5, are in the best group, as shownin table 9.
. Number of ants: it is observed that in all problems when the number of antsis more than the number of total operations the solutions are slightly betteror stable.
Table 4. The Duncan’s multiple range test of homogenous subset of candidate list strategyand heuristic information of small problem size.
DuncanaSubset
CAN&HEU N 1 2 3 4 5 6 7 8 9 10 11
6 15 3.872 15 4.331 15 4.87 4.8722 15 4.93 4.9321 15 7.30 7.30 7.3026 15 8.03 8.03 8.0316 15 8.13 8.13 8.1311 15 8.30 8.30 8.3012 15 8.63 8.6327 15 10.20 10.20 10.207 15 10.70 10.70 10.7029 15 11.30 11.303 15 12.3323 15 13.109 15 16.4728 15 17.23 17.2324 15 17.33 17.3325 15 18.77 18.7730 15 19.33 19.33 19.3317 15 19.47 19.47 19.478 15 20.47 20.47 20.474 15 22.57 22.57 22.5713 15 22.73 22.73 22.7320 15 23.00 23.00 23.0019 15 23.60 23.6010 15 24.00 24.005 15 24.2718 15 24.6714 15 25.6715 15 29.40
aSample size¼ 15.
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From the statistical analyses the robust parameter values are set for all problemswhich are �¼ 1, �¼ 5, �¼ 0.1, q0¼ 0.5, and the number of ants is set to the numberof total operations. When the Ss solution is not improved after 50 iterations, thealgorithm is restarted. The algorithm is stopped when it is run for 1000 iteration.
The parameter values used in SAnt and MAnt are the same including the samenumber of ants. For multiple colony ant algorithm there is a question about how todistribute ants among three colonies with the same number of ants used in a singlecolony. The primary experiments are conducted. Two sets of parameters are testedon FT06, LA01, LA06, LA16, LA21, LA26, LA36, and YN1. Experiments arecarrying for two cases: a case of three colonies with the same number of ants whichequal O/3; and a second case with the number of ants in three colonies which equal0.5 �O, 0.25 �O, and 0.25 �O, respectively. In both cases, the combinations ofcandidate list strategy and heuristic information used in each colony are non-delaywith MWR, non-delay with EFT, and non-delay with EST, respectively.
Table 5. The Duncan’s multiple range test of homogenous subset of candidate list strategyand heuristic information of large problem size.
DuncanaSubset
CAN&HEU N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
6 15 1.40
2 15 2.20
1 15 3.00 3.00
7 15 5.07 5.07
26 15 5.80
16 15 6.97 6.97
3 15 7.00 7.00
11 15 8.47 8.47
21 15 9.80
9 15 9.80
12 15 9.83
22 15 10.53
8 15 15.87
27 15 16.23 16.23
4 15 16.43 16.43
10 15 16.83 16.83
23 15 17.20 17.20
29 15 17.20 17.20
5 15 18.37 18.37
28 15 19.67 19.67
24 15 21.00
30 15 21.67 21.67
17 15 23.33 23.33
25 15 23.90 23.90 23.90
18 15 24.00 24.00
14 15 24.40 24.40 24.40
13 15 26.20 26.20 26.20
20 15 26.43 26.43
19 15 26.73
15 15 29.67
aSample size¼ 15.
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The important weight of the master pheromone trail is 0.7 in both cases. The resultsindicate that the latter set of parameter values yielded better solutions, as shownin table 10. Therefore, in the final version of a multiple colony ant algorithmthe number of ants in three colonies are set to 0.5 �O, 0.25 �O, and 0.25 �O,respectively. The important weight of the master pheromone trail is 0.7.
5. Experimental results
The ant algorithm was tested on benchmark problems, which are available from theORLIB. The algorithm was coded in C and run on a Pentium IV 2.4-GHz computer
Table 7. The Duncan’s multiple range test of homogenous subset of �.
� Subset
Duncana N 1 2
(a) Small size problem3.00 4 2.12505.00 4 2.87501.00 4 3.500010.00 4 3.50007.00 4 4.0000
(b) Large size problem5.00 4 1.75007.00 4 2.000010.00 4 2.00003.00 4 4.00001.00 4 5.0000
aSample size¼ 4.
Table 6. The Duncan’s multiple range test of homogenous subset of �.
� Subset
Duncana N 1 2
(a) Small size problem1.00 4 1.12503.00 4 1.87505.00 4 3.75007.00 4 4.000010.00 4 4.2500
(b) Large size problem1.00 4 2.25003.00 4 2.25007.00 4 2.500010.00 4 3.75005.00 4 4.2500
aSample size¼ 4.
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1-GB RAM Window platform. To evaluate the algorithm, each of the problem
instances was repeated for ten trials. The best solution is obtained from ten trails
of this tested algorithm. The percentage deviation from the optimal solution is
calculated as:
Percentage deviation from the optimal solution
¼best solution� optimal solution
optimal solution
� �� 100:
The CPU time is in seconds. The final results are listed in table 11.
Table 8. The Duncan’s multiple range test of homogenous subset of �.
� Subset
Duncana N 1 2 3
(a) Small size problem0.30 4 2.50000.50 4 2.75000.90 4 2.87500.10 4 3.37500.70 4 3.5000
(b) Large size problem0.10 4 1.75000.30 4 2.00000.50 4 2.7500 2.75000.70 4 4.0000 4.00000.90 4 4.5000
aSample size¼ 4.
Table 9. The Duncan’s multiple range test of homogenous subset of q0.
q0 Subset
Duncana N 1 2
(a) Small size problem0.50 4 2.00000.10 4 2.2500 2.25000.30 4 2.7500 2.75000.70 4 3.7500 3.75000.90 4 4.2500
(b) Large size problem0.10 4 1.75000.30 4 2.00000.50 4 2.00000.70 4 4.25000.90 4 4.7500
aSample size¼ 4
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Table 11 presents the results of SAnt and MAnt with and without local searchand restart process. The aim is to investigate the performance of the algorithm whena multiple colony strategy is added while the numbers of ants remain the same and toinvestigate the performance of algorithm when local search and restart process areapplied.
The results in table 11 illustrate that there are 12 instances: FT06, LA01,LA05–LA14; and that both algorithms yielded the optimal solution without usinglocal search and restart process. The MAnt algorithm can achieve better solutionsover those found by the SAnt algorithm. In addition, the solution obtained from theMAnt algorithm converged quicker than the solution obtained from SAnt, asillustrated in figure 5.
When the local search and restart process are included, it is found that the bestsolution obtained from SAnt and MAnt are both improved. For all the probleminstances the MAnt algorithm yields a more significant improvement on the bestsolution than the SAnt algorithm. In addition, the MAnt algorithm reached theoptimal solution for two additional problem instances, LA04 and LA18.
6. Conclusions and discussions
In this paper, an ACO algorithm is proposed to solve the JSP with the objective ofminimizing the makespan. To adapt the ACO in the JSP, the scheduling problem istransformed into the graph-based model. The candidate list strategy and heuristicinformation used in the construction step are tested. The statistical analyses showthat in a small size problem the non-delay algorithm with MWR, non-delay withEFT, non-delay with EST, random with EFT, and random with EST are in thegroup that provides the best solution quality. For a large size problem, the non-delayalgorithm with MWR, non-delay with EFT and non-delay with EST are in the groupthat provides the best solution quality.
The performances of MAnt and SAnt are compared. In the construction step,the non-delay algorithm with MWR is used in SAnt and the non-delay with MWR,
Table 10. Results of testing two set of number of ants in colony on benchmark problems.
Number of ants in colony, n
O/3 0.50 �O, 0.25 �O, 0.25 �O
Instance n�m Optimal Best %D Time Best %D Time
FT06 6� 6 55 55 0.00 3 55 0.00 3LA01 10� 5 666 666 0.00 8 666 0.00 8LA06 15� 5 926 926 0.00 22 926 0.00 22LA16 10� 10 945 1008 6.67 27 982 3.92 28LA21 15� 10 1046 1167 11.57 75 1142 9.17 74LA26 20� 10 1218 1328 9.03 156 1313 7.80 157LA36 15� 15 1268 1372 8.20 155 1369 7.97 164YN1 20� 20 888a 1067 20.16 565 1006 13.28 568
aUpper bound.
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Table
11.
Resultsofbenchmark
problems.
Withoutlocalsearchandrestart
Withlocalsearchandrestart
SAnt
MAnt
SAnt
MAnt
Blum
and
Sampel
(2004)
Instance
n�m
Optimal
Best
%D
Tim
eBest
%D
Tim
eBest
%D
Tim
eBest
%D
Tim
eBesta
FT06
6�6
55
55
0.00
355
0.00
355
0.00
41
55
0.00
21
FT10
10�10
930
968
4.09
28
968
4.09
29
946
1.72
66
944
1.51
70
FT20
20�5
1165
1247
7.04
46
1227
5.32
46
1178
1.12
85
1178
1.12
82
LA01
10�5
666
666
0.00
8666
0.00
8666
0.00
13
666
0.00
13
LA02
10�5
655
716
9.31
8666
1.68
8666
1.68
12
658
0.46
14
LA03
10�5
597
632
5.86
8622
4.19
8619
3.69
14
603
1.01
13
LA04
10�5
590
632
7.12
7598
1.36
8612
3.73
12
590
0.00
12
LA05
10�5
593
593
0.00
8593
0.00
8593
0.00
13
593
0.00
14
LA06
15�5
926
926
0.00
21
926
0.00
22
926
0.00
34
926
0.00
35
LA07
15�5
890
890
0.00
22
890
0.00
22
890
0.00
34
890
0.00
36
LA08
15�5
863
863
0.00
21
863
0.00
22
863
0.00
35
863
0.00
36
LA09
15�5
951
951
0.00
22
951
0.00
22
951
0.00
36
951
0.00
35
LA10
15�5
958
958
0.00
20
958
0.00
21
958
0.00
34
958
0.00
37
LA11
20�5
1222
1222
0.00
46
1222
0.00
46
1222
0.00
82
1222
0.00
80
LA12
20�5
1039
1039
0.00
44
1039
0.00
45
1039
0.00
85
1039
0.00
82
LA13
20�5
1150
1150
0.00
47
1150
0.00
47
1150
0.00
87
1150
0.00
82
LA14
20�5
1292
1292
0.00
45
1292
0.00
45
1292
0.00
92
1292
0.00
88
LA15
20�5
1207
1253
3.81
45
1250
3.56
47
1240
2.73
83
1240
2.73
85
LA16
10�10
945
985
4.23
26
982
3.92
28
979
3.60
51
977
3.39
52
LA17
10�10
784
811
3.44
26
793
1.15
30
793
1.15
60
793
1.15
54
LA18
10�10
848
872
2.83
27
866
2.12
30
866
2.12
54
848
0.00
54
LA19
10�10
842
895
6.29
27
880
4.51
29
874
3.80
55
860
2.14
56
LA20
10�10
902
964
6.87
27
922
2.22
29
932
3.33
62
925
2.55
55
LA21
15�10
1046
1177
12.52
72
1142
9.17
74
1139
8.89
325
1063
1.62
390
1047
LA22
15�10
927
1008
8.74
70
1005
8.41
75
960
3.56
324
954
2.91
409
(continued)
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Table
11.
Continued.
Withoutlocalsearchandrestart
Withlocalsearchandrestart
SAnt
MAnt
SAnt
MAnt
Blum
and
Sampel
(2004)
Instance
n�m
Optimal
Best
%D
Tim
eBest
%D
Tim
eBest
%D
Tim
eBest
%D
Tim
eBesta
LA23
15�10
1032
1070
3.68
75
1074
4.07
76
1055
2.23
350
1055
2.23
384
LA24
15�10
935
1014
8.44
72
976
4.38
74
976
4.38
335
954
2.03
385
944
LA25
15�10
977
1028
5.22
72
1017
4.09
75
1014
3.78
330
1003
2.66
382
977
LA26
20�10
1218
1316
8.05
156
1313
7.8
157
1315
7.96
3020
1308
7.39
3155
LA27
20�10
1235
1300
5.26
156
1291
4.53
162
1292
4.61
3025
1269
2.75
3125
1243
LA29
20�10
1152
1336
15.97
158
1320
14.5
165
1239
7.55
3051
1162
0.86
3200
1168
LA30
20�10
1355
1455
7.38
160
1458
7.6
162
1418
4.65
3306
1411
4.13
3204
LA36
15�15
1268
1375
8.44
155
1369
7.97
164
1366
7.73
3004
1334
5.21
3096
LA37
15�15
1397
1463
4.72
155
1464
4.8
165
1461
4.58
3006
1457
4.29
2928
LA38
15�15
1196
1298
8.52
158
1296
8.36
168
1283
7.27
3024
1224
2.34
3062
1227
LA40
15�15
1222
1356
10.96
162
1325
8.42
170
1274
4.25
3044
1269
3.84
3045
1228
n:Number
ofjobs,m:number
ofmachines,Best:thebestmakespanfoundover
10runs,%
D:deviationofbestmakespanover
optimalsolution,Tim
e:theCPU
timein
seconds.
athebestsolutionof20trials,algorithm
term
inateswhen
restarts5–10times.
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the non-delay with EFT and the non-delay with EST are used in MAnt.The performances of MAnt and SAnt are compared with and without local search
and restart process. The experimental results show that MAnt can improve the best
solution in more instances over SAnt. The information exchange in MAnt influencesthe optimization behaviour that it allows colonies to benefit from the lesson learned
by the other colonies and converges quickly.The solutions obtained by SAnt and MAnt are both improved when local
improvement and restart process are included. The optimal solutions are obtained
in two more problem instances by MAnt. For other instances there is small deviation
from an optimal solution with less than a 5% deviation, except for problems LA26and LA36.
The result is compared with those published recently by Blum and Sampels
(2004); the proposed algorithm yields better solution for problems LA29 and LA38.However, the comparison can only be done qualitatively since only some problems
are reported by Blum and Sampels and the terminating criteria are different. In Blum
and Sampels the algorithm terminates when it restarts five to ten times, and thesolution is the best of 20 trials. The terminating criterion used in this research is
a 1000 iteration and the solution is best of ten trials.There are many possibilities to improve the algorithm further. First, the values
of parameters that control the search affect the quality of solutions. The parametervalues from the experiments can be used as initial values. The optimal values that
provide the highest solution quality should be fine-tuned for each instance. Forexample, a GA can be applied to fine-tune the parameter values. The chromosome
in each colony is encoded by a set of parameters. Colonies are in competition to
make it to the next generation. The fitness of each individual is evaluated by runningthe ant algorithm. When an algorithm runs until it reaches the terminate criteria,
the chromosome will converge to the best fitness set of parameter values. Second,
a faster local search method such as that by Nowicki and Smutinicki (2005) canbe substituted to improve further the speed of the algorithm. Finally, in a multiple
colony ant algorithm the strategy of exchange information should be examined.
For example, how many colonies should be included and which kind of informationshould be exchanged.
600
800
1000
Iteration
Mak
espa
n (u
nit o
f tim
e)
1 201 401 601 801
1200
SAnt
MAnt
Figure 5. Convergence of a solution of LA20.
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