a filtered-beam-search-based heuristic algorithm for flexible job-shop scheduling problem

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A filtered-beam-search-based heuristicalgorithm for flexible job-shopscheduling problemWang Shi-Jin a , Zhou Bing-Hai a & Xi Li-Feng aa School of Mechanical Engineering , Shanghai JiaotongUniversity , Shanghai 200030, People's Republic of ChinaPublished online: 04 Apr 2008.

To cite this article: Wang Shi-Jin , Zhou Bing-Hai & Xi Li-Feng (2008) A filtered-beam-search-basedheuristic algorithm for flexible job-shop scheduling problem, International Journal of ProductionResearch, 46:11, 3027-3058, DOI: 10.1080/00207540600988105

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International Journal of Production Research,Vol. 46, No. 11, 1 June 2008, 3027–3058

A filtered-beam-search-based heuristic algorithm for flexible job-shop

scheduling problem

WANG SHI-JIN*, ZHOU BING-HAI and XI LI-FENG

School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200030,

People’s Republic of China

(Revision received June 2006)

Scheduling for the flexible job-shop is a very important issue in both fields ofcombinatorial optimization and production operations. However, due tocombination of the routing and sequencing problems, flexible job-shopscheduling problem (FJSP) presents additional difficulty than the classical job-shop scheduling problem and requires more effective algorithms. This paperdeveloped a filtered-beam-search-based heuristic algorithm (named as HFBS) tofind sub-optimal schedules within a reasonable computational time for the FJSPwith multiple objectives of minimising makespan, the total workload of machinesand the workload of the most loaded machine. The proposed algorithmincorporates dispatching rules based heuristics and explores intelligently thesearch space to avoid useless paths, which makes it possible to improve the searchspeed. Through computational experiments, the performance of the presentedalgorithm is evaluated and compared with those of existing literature and those ofcommonly used dispatching rules, and the results demonstrate that the proposedalgorithm is an effective and practical approach for the FJSP.

Keywords: Flexible job-shop scheduling; Filtered beam search; Heuristicalgorithm; Combinatorial optimization

1. Introduction

Scheduling is one of the most important issues in the planning and operation ofmanufacturing systems (Chen et al. 1999). But most scheduling problems associatedwith manufacturing are complex combinatorial optimization problems and verydifficult to solve. The classical job-shop scheduling problem (JSP) is such a problem,which deals with the sequencing operation of a set of jobs on a set of machines withthe objective to optimize some criterion or criteria. The key assumption of theproblem is that each operation of a job is to be processed only on one predeterminedmachine, i.e. predetermining machining routes without considering alternative route.It is well known that the classical JSP is NP-hard (Garey et al. 1976) and thuscomputationally intractable. Many approaches, especially artificial intelligence (AI)based meta-heuristics (e.g. simulated annealing (SA), tabu search (TS), beam search(BS), and genetic algorithm (GA) etc.) have been employed to solve this problem in

*Corresponding author. Email: [email protected]

International Journal of Production Research

ISSN 0020–7543 print/ISSN 1366–588X online � 2008 Taylor & Francis

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DOI: 10.1080/00207540600988105

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the last three decades. Readers are referred to Blazewicz et al. (1996), Jain andMeeran (1999) and Hart et al. (2005) for a relatively comprehensive overview.

Though the JSP has been well studied, its application to real-world scenarios isoften undermined by the assumption of the one-to-one mapping of operations tomachines (Ho and Tay 2004). In real manufacturing scenarios, due to constantdemands for a wide variety of products and faster production rates, flexibility isessential for manufacturing companies to survive in a highly competitive andchanging business environment (Sabuncuoglu 1998). In such cases, various modernmachine tools with considerable amount of overlapping capabilities are introducedinto job shops, which lead to machine route flexibility. Owing to its assumption ofpredetermining machining route, the JSP doesn’t reflect such requirements of jobshops (Baykasoglu 2002, Baykasoglu et al. 2004). Hence, an extension version of JSPrecently captured many researchers’ attention, which is referred as the flexible job-shop scheduling problem (FJSP). The FJSP extends the classical JSP by assumingthat a machine may be capable of performing more than one type of operation(Najib et al. 2002). The definition of the FJSP is to assign each operation to onemachine able to perform it (routing sub-problem) and to determine starting times foreach operation (sequencing sub-problem) in order to optimize one or several criteria.Thus, the FJSP presents two difficulties. The first one is how to assign each operationto a machine, and the second is how to compute start time of each operation on theassigned machine to optimize one or more given performance measures. So it is morecomplex than the already difficult classical JSP. However, due to its significanttheoretical and industrial importance, the FJSP has been explored to some extent inthe literature and still continues to attract researchers both in academia and industry.

Like the classical JSP, the FJSP is NP-hard as well (Dauzere-Peres and Paulli1997). The NP-hardness of an optimization problem suggests that it is not alwayspossible to find an optimal solution quickly (Mastrolilli and Gambardella 2000).Therefore, instead of investing the FJSP for an optimal solution with enormouscomputational efforts, researchers in recent years pay more attention to AI-basedmeta-heuristics (SA, GA, TS, BS etc.) for solving the problem. Brandimarte (1993)suggested a multiple start TS algorithm for the FJSP with makespan and weightedtotal tardiness minimisation. Dauzere-Peres and Paulli (1997) presented a tabusearch procedure based on a new defined neighbourhood structure. Mesghouni et al.(1998) put forward a genetic algorithm based model for FJSP. Utilising the Paretodominance, Hsu et al. (2002) developed a multi-objective evolutionary algorithm formulti-objective FJSP. Kacem et al. (2002a, b) proposed a genetic algorithmcontrolled by the assigned model, which is generated through the approach oflocalisation (AL). Baykasoglu et al. (2004) developed an approach for the FJSP,which made use of grammars, multiple objective TS and a dispatching rule-basedheuristics. These studies mentioned above mainly concentrate on TS-based andGA-based heuristics. Other various heuristics are also employed to solve thisproblem, including integrated greedy heuristic (e.g. Mati et al. 2001), modifiedsimulated annealing (e.g. Najib et al. 2002), particle swarm optimization andsimulated annealing (e.g. Xia and Wu 2005), multi-agent technique (e.g. Wu andWeng 2005). These existing approaches can be classified into two types: hierarchicalapproaches and integrated approaches (Najib et al. 2002, Xia and Wu 2005).In hierarchical approaches (e.g. Brandimarte 1993, Kacem et al. 2002a, b,Baykasoglu et al. 2004, Xia and Wu 2005), the assignment of operations to

3028 W. Shi-Jin et al.

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machines and the sequencing of operations on the machines are treated separately,whereas in integrated approaches (e.g. Dauzere-Peres and Paulli 1997, Mesghouniet al. 1998, Mati et al. 2001, Hsu et al. 2002, Najib et al. 2002, Wu and Weng 2005),the assignment and sequencing are not differentiated. Recently, some researcherssuggested that better scheduling decisions should integrate the assignment andsequencing together (Wu et al. 2002, Wu and Weng 2005).

Although various approaches mentioned above have been used for solvingthe FJSP, the problem has not been explored thoroughly compared with the classicalJSP. Therefore, there is still a need to develop effective and practical solutiontechniques for this complex problem (Baykasoglu et al. 2004). Filtered-beam-search-based heuristic algorithms can be considered as such techniques. As one of the mostcompetitive AI-based heuristic search methods, the beam search-based algorithm isfor efficient searching in the decision tree, particularly where the production space isvast (Ow and Morton 1988). Introduced firstly by Ow and Morton (1988), thefiltered beam search (FBS) approach, is an extension and improvement of BS. Thereare, in fact, many studies on the JSP and real-time flexible manufacturing system(FMS) scheduling problems using FBS-based heuristic algorithms, including De andLee (1990), Shih and Sekiguchi (1991), Sabuncuoglu and Karabuk (1998),Sabuncuoglu and Bayiz (1999), Sabuncuoglu and Kisilisik (2003). The highperformance and quick search speed of heuristic algorithms based on FBS forthe JSP and real-time FMS scheduling problems are demonstrated in the literature.Since the FJSP is an extension of the JSP and approximates a difficult subclassof scheduling problems encountered in the planning of FMS (Chen et al. 1999),FBS-based heuristic algorithms represent very promising techniques for the FJSPand deserve exploration. However, at present, few efforts employing heuristicalgorithms based on FBS to solve the FJSP are reported.

In this context, an integrated heuristic algorithm based on FBS (named as HFBS)is presented to solve the FJSP efficiently and practically, which combines the routingand sequencing sub-problems reasonably and simultaneously. Dispatching rules-based heuristics are incorporated into the proposed HFBS algorithm to improvethe search speed. The objectives considered are to minimize makespan, the totalworkload of machines, and the workload of the critical machines. The rest of thepaper is organised as follows. In section 2, the flexible job-shop problem isformulated. The proposed HFBS algorithm is developed in section 3, and thealgorithm is illustrated in detail through an example. In section 4, computationalexperiments are performed with the proposed algorithm, and results are evaluatedand compared with those of other methods. Finally, section 5 discusses conclusionsof this research.

2. Problem definition

The flexible job-shop scheduling problem is normally described as follows. A set of njobs is to be processed on a set of m machines denoted by M. Each job j (1� j� n)consists of a sequence of nj operations. Each operation Oij (1� j� n, 1� i� nj) has tobe performed to complete one job. The process of each operation requires onemachine out of a set Mij of given machines, Mij �M. Thus, the FJSP is to determineboth an assignment and a sequence of the operations on the machines that optimize

A filtered-beam-search-based heuristic algorithm for FJSP 3029

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one or several criteria (Najib et al. 2002, Xia and Wu 2005). In this study, the

following three criteria are to be minimized:

F1 Makespan or maximal completion time of machines.F2 Total workload of the machines; which represents the total working

time of all machines.F3 Critical machine workload, i.e. the workload of the most loaded

machine.

The main reason for considering these three criteria is that although a minimum

makespan implies a high utilisation of the machines (Carlyle et al. 2001, Liu and

MacCarthy 1996) and is equivalent to a high throughput measure in some sense, its

application alone cannot reflect and guide the reduction of total workload and

workload balance of machines; because minimising the global makespan is not

necessarily aimed at these for modern machine tools with overlapping capacities.

While both workload-related criteria for modern machine tools and high utilisation

of the machines are very important in the practical sense of the FJSP, since

investment in and installation of modern machine tools is very capital intensive, and

their potential capacities are needed fully realised to ensure the system economical

justification and the high system flexibility.These three criteria are conflicting (Hsu et al. 2002, Ivanescu et al. 2002, Kacem

et al. 2002b). The explanation of this point is illustrated in the appendix. As in Xia

and Wu (2005), the objective of this paper is the weighted sum of the above three

objective functions. A range of weights is given beforehand from the perspective of

practical production: makespan objective is relatively more important than the other

two objectives, and the other two objectives are given the same preference of

importance. According to this decision-makers’ preference, the three weights are

generated randomly beforehand in a certain range. The best objective function value

obtained is finally selected.Assumptions considered in this paper are as follows:

1. The jobs are non-pre-emptive.2. The jobs are independent of each other and all jobs can be performed at

rj (release time).3. All machines are independent of each other and are available at t¼ 0.4. Set-up times of machines and move times between operations are neglected.5. Every machine is available every time, i.e. there are no stopping times or

breakdowns.

In the following, the mathematic model of FJSP is formulated. Firstly, the

detailed notation for the mathematical model is given below.Subscripts

j, l job, 1� j� n, 1� l� ni, u, v operation, 1� i, u� nj, 1� v� nlk, h machine, 1� k, h�m

Parameters

n number of jobsm number of machines


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nj number of operations of job jM the set of all machinesOij the operation i of job jMij the machine set that can perform Oij, Mij �M (if Mij¼M for each

operation, the problem is total flexibility problem, implies that eachoperation can be processed on any one machine ofM; ifMij �M, it ispartial flexibility problem.)

Mk a machinepijk the processing time if operation i of job j is to performed on machine

k, Mk2Mij

cijk the completion time of operation i of job j on machine k, Mk2Mij

cujh the completion time of operation u of job j on machine h, u 6¼ i,Mh2Muj

cvlk the completion time of operation v of job l on machine k, Mk2Mvl

L0,L a large enough positive numberwm the weight of F1

wTW the weight of F2

wMW the weight of F3

Decision variables

xijk ¼1, if operation i of job j is assigned on machine k

0, otherwise

�

yiujkh ¼1,

if operation i of job j processing on machine k

precedes operation u of job j on machine h

0, otherwise

8<:

y0

iujkh ¼1, if xijk þ xujh ¼ 2

0, otherwise

�

zijvlk ¼1,

if operation i of job j processing on machine k

precedes operation v of job l on machine k

0, otherwise

8<:

z0

ijvlk ¼1, if xijk þ xvlk¼ 2

0, otherwise

�

Thus, the mathematic model of FJSP is given as follows.

minðwmF1 þ wTWF2 þ wMWF3Þ

where,F1 ¼ maxðcijkÞ, i ¼ nj,8j

F2 ¼Xnji¼1

Xnj¼1

XMk2Mij

xijkpijk

F3 ¼ maxXnji¼1

Xnj¼1

xijkpijk

!, Mk 2Mij

wm þ wTW þ wMW ¼ 1


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Subject to XMk2Mij

xijk ¼ 1 ð1Þ

cujh � pujh þ Lð1� yiujkhÞ þ L0

ð1� y0

iujkhÞ � cijk ð2Þ

cvlk � cijk þ Lð1� zijvlkÞ þ L0

ð1� z0

ijvlkÞ � pvlk ð3Þ

cijk > 0 ð4Þ

pijk > 0 ð5Þ

The first constraint (1) ensures that a particular operation is performed on onlyone alternative machine. The second constraint (2) guarantees the right operationprecedence relation. The third constraint (3) avoids processing conflict on a samemachine. The fourth constraint (4) ensures non-negativity of operation completiontime. The fifth constraint (5) ensures non-negativity of operation processing time.

Once weights wm, wTW and wMW are determined, the above model is a mixedinteger linear programming problem. It allows obtaining the optimal solutionfor the problem. However, it is with an extremely high number of decisionvariables and constraints. Assuming that nj¼ J for all j, the number of decisionvariable is nmJþ 2n(J� 1)m2

þ n(n� 1)mJ2, and the number of constraints isnJþ n(J� 1)m2

þ n(n� 1)mJ2/2þ 2nJm, where, m is the number of machine, n isthe number of jobs. Let us consider a problem with 10 machines, 10 jobs each having3 operations (i.e. m¼ 10, n¼ 10 and nj¼ J¼ 3 for all j), the number of decisionvariables is 12 400 and the number of constraints is 6680. A problem of this size isdifficult to solve, even for some commercial standard packages. As a result, to obtainoptimal solutions for practical problems is unrealistic, especially for large sizeproblems. Hence, there is search for finding a better heuristic algorithm, which canprovide a solution nearer to optimal, if not optimal. In the next sections, a filtered-beam-search-based heuristic algorithm is given.

3. Filtered beam search

Filtered beam search is an extension of beam search. Beam search is an adaptation ofthe branch and bound method in which it only explores the promising nodes level bylevel without backtracking. The number of nodes explored at each level is called thebeamwidth. The node evaluation process at each level is a key issue in the beamsearch technique (Valente and Alves 2005). An evaluation function is usually usedto determine which nodes to continue the search on, which poses the problem offinding a good trade-off between quick, but poor, evaluation and morecomputationally demanding, but better, evaluation (Sabuncuoglu and Bayiz 1999).

To find a good trade-off mentioned above economically and quickly, filteredbeam search is introduced (Ow and Morton 1988). The filtered beam searchprocedure uses both crude and accurate evaluations in two phases: filtering phaseand beam selection phase. All nodes generated from a parent node are evaluatedcrudely by a computationally inexpensive filtering procedure (normally named as


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local evaluation function) in the first phase, leaving filterwidth filtered nodes forfurther accurate evaluation by a total cost evaluation function (normally named asglobal evaluation function) in the second phase. Those nodes found most promisingare added to a partial solution. This procedure is repeated on parallel beamwidthpaths until the complete schedules are reached. Hence, the number of solutions savedis equal to size of the beamwidth. Eventually, the solution with the best objectivefunction value is as the final schedule.

As shown in figure 1, the promising nodes (beam nodes) are determined byapplying local and global evaluation functions and proceed with the search throughthese selected nodes. After determining the beam nodes in level 1, the filtered beamsearch procedure is employed independently to generate one partial tree from each ofthem (in figure 1, there are two partial trees since beamwidth¼ 2). These partial treesare referred as beam paths. In each beam path, once a beam node is determined, nodesare generated in next level from this beam node by applying branching scheme. Thesegenerated nodes first subject to filtering phase, in which some of them are pruned bylocal evaluation and filterwidth (filterwidth¼ 2 in figure 1) filtered nodes are left forbeam selection phase. Through global evaluation function, one of the filtered nodeswith best global evaluation function value is determined as beam node for the level.The procedure is repeated in figure 1 and eventually it forms two beam paths.

4. Proposed heuristic algorithm

4.1 Representation scheme

A successful FBS-based heuristic algorithm for FJSP should solve four majorelements:

1. Search tree representation to define a solution space.2. Determination of beamwidth and filterwidth.

Root

Level 1

Level 2

Level 3

beamwidt h=2filterwidth=2

Beam nodes

Nodes left for global evaluation,but pruned by global evaluation

Nodes pruned by local evaluation

Figure 1. Representation of filtered beam search tree.


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3. Branching scheme.4. Local and global evaluation functions selection.

Among them, 3 and 4 are the more important elements.The specific definitions of these four elements are different for different

scheduling problems. For the defined problem in this paper, these four elements ofthe proposed algorithm are illustrated in detail in the following.

1. The solution space for the FJSP can be visualised as a search tree (see figure 1)with each path in the tree representing a potential solution, and the nodesrepresent a scheduling decision of an operation, including selecting a machinefrom alternative machines and determining the starting time on the selectedmachine. A set of sequential nodes corresponds to a partial schedule. A linebetween two nodes represents the decision to add an operation to the existingpartial schedule. Consequently, leaf nodes at the end of the tree correspond tocomplete schedules.

2. The selection of beamwidth and filterwidth appears to be very problem specific(Coffin and Taylor 1996). In general, the determination of beamwidth andfilterwidth has a significant effect on the speed and the performance of theFBS based algorithm. The values of the filterwidth and beamwidth are usuallydetermined empirically (Sabuncuoglu and Karabuk 1998). Therefore,different combinations of beamwidth and filterwidth will be investigated tobalance the computational times and the solution quality.

3. The branching scheme adopted in this paper is a modification form of thenondelay procedure proposed by Sabuncuoglu and Bayiz (1999) and we nameit M_NONDELAY procedure. To describe the procedure some variables arefirstly defined.

Let PSl be a partial schedule containing l scheduled operations, Sl be the set ofschedulable operations at level l, corresponding to a given PSl, sij be the earliest timeat which the operation Oij2Sl could be started, and s(iþ 1)j¼ sijþ �ij, where� ij¼mink(pijk) 8 1� i� nj, 1� j� n. s1j¼ rj (rj is release time). Mij be the set ofmachines that can perform the operation Oij, and Tijk be the earliest time analternative machine can be ready to perform the operation Oij, Mk2Mij. Then theM_NONDELAY procedure is explained as follows:

Step 1: Determine T� ¼ minOij2Slfsijg, and let the operations Oij2Sl with sij¼T* be

a set �Sl.

Step 2: Select an operation Oij 2 �Sl, for each machine of Mij, generate a new nodethat corresponds to the partial schedule, in which operation Oij is added to PSl andstarted at time sijk¼Max(T�, Tijk).

Thus, at level l, each node for selection determines the assignment and start timesimultaneously. Such node contains information of operation assignment andsequencing.

4. The key point in utilising FBS-based heuristic algorithm is the selection oflocal and global evaluation. Typically, local evaluation is performed by acomputationally inexpensive dispatching rule. Global evaluation of a node isdetermined as the estimation of the upper/lower bound value for the solutionsthat can be generated if that node is added to the partial schedule. This is


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performed by generating a complete schedule from a given partial scheduling

by applying selected local and global evaluation function. It is, however,

very difficult to find a unique rule with high performance regardless

of production objective and manufacturing environment (Shih 1994).

In this paper, we investigate the following dispatching rules based heuristics

as local and global evaluation functions. Note that in the following, i� j� k

denotes the decision that the operation Oij is to be assigned on the

machine Mk.

M_SPT (modified shortest processing time):Favour the i� j� k with the shortest processing time (i.e. favour the i� j� k with

min(pijk)). If there exists more than one decision for selecting, the tie will be broken

by selecting the machine that has been used the least so far, which helps to balance

the distribution of workload among the machines (Tunali 1997). Then, ties are

broken by a random choice.M_EET (modified earliest ending time): Favour the i� j� k with the earliest

ending time (i.e. favour the i� j� k with min(sijkþ pijk)). Balance the loads on

machines for the tie breaking. Then, ties are broken by a random choice.M_LWR (modified least work remaining): Favour the i� j� k with the least

work remaining (i.e. favour the i� j� k with minðPnj

q¼i minðpqjkÞÞ). Balance the loads

on machines for the first tie breaking. A random choice is used as the second tie

breaking.M_PT/TOT (modified smallest ratio of processing time of an operation to the

total processing time of the job): Favour the i� j� k with the smallest ratio of

processing time of the operation to the total processing time of the job (i.e. favour the

i� j� k with minðpijk=Pnj

q¼1 minðpqjkÞÞ). Balance the loads on machines for the first

tie breaking. A random choice is used as the second tie breaking.

4.2 Complete HFBS algorithm

On the basis of the description mentioned above, the flow chart of the proposed

HFBS algorithm is shown in figure 2.The procedure form of the algorithm corresponding to the flow chart is given as

follows:

Step 1: Initialization:Let bn¼ 0, l¼ 0; input beamwidth b, filterwidth f; input total number of

operations To; input detailed information of n jobs, m machines; let partial schedule

sets PSl be empty.

Step 2: Determining beam nodes:

(i) By using the M_NONDELAY subroutine, generate nodes from the root node.

Check the total number of nodes N. Let l¼ lþ 1, update the PSl with generated

nodes.(ii) If N5b, then move down to one more level (i.e. l¼ lþ 1), generate new nodes

(update N) using M_NONDELAY subroutine with PSl as the partial schedule,

and update PSl. If N5b go to step 2 (ii); else go to step 2 (iii).


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l = l+1, Update PSl, Check thetotal number of nodes N

Initialization: bn = 0, l = 0; Input: b,f; get To, information of jobs, machines;

PSl is empty;

Generate nodes from theparent nodes by using

M_NONDELAY N>b?

Compute global evaluation function value, constructb beam nodes with the best value and sequence

sets: PSl (1) … PSl (b), l ′ = l+1

bn = bn+1, bn>b ?

l = l+1, l>To ?

Y

N

Generate nodes from the root node by usingM_NONDELAY subroutine

N

Generate Nbn,l number of nodes from the parent beam node byusing M_NONDELAY subroutine, compute local evaluationfunction value, select min(Nbn, l, f )number of nodes with the

best value

Compute global evaluation function value ofmin(Nbn, l, f ) number of nodes, select the best node,

update PSl (bn)

Formulate the complete bn-th schedule PS (bn)

Select schedule set or sets with the best objectivefunction value among b schedule sets

Y

N

Y

Start

l = l ′

Stop

Figure 2. The flow chart of the proposed algorithm.


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(iii) Compute the global evaluation function values for all the nodes and select thebest b number of nodes (initial beam nodes). Meanwhile, determine potentialsets PSl(1), . . . ,PSl(b).

Step 3: bn¼ bnþ 1; if bn4b, then go to step 4; otherwise, move on

(i) Form beam nodes of the lth level for each initial beam node:

(a) l¼ lþ 1, if l4To, then go to step 3 (ii); otherwise, move on.(b) Generate new nodes (the number of nodes is Nbn,l) from the beam node

according to the M_NONDELAY subroutine with PSl(bn) as the partialschedule represented by the beam node. Compute the local evaluationfunction values of all nodes generated and select min(Nbn, l, f) number ofnodes with the best values for further evaluation.

(c) Compute the global evaluation function values of min(Nbn,l, f) number ofnodes, select the node with the best value and add the node into the partialschedule PSl(bn), update the PSl(bn). Go to (a).

(ii) Formulate the bnth complete schedule PS(bn).

Step 4: Select the schedule set or schedule sets with the best objective functionvalues among the final b schedule sets; Stop.

4.3 An illustrative sample

In order to understand the developed algorithm more easily, a sample problem isgiven in the following. The problem is to execute three jobs (total six operations) ontwo machines according to the processing times pijk described in table 1. Theobjective function chosen is 0.4F1þ 0.3F2þ 0.3F3. Set b¼ 2 and f¼ 2. The M_SPT isselected as the local evaluation function and the global evaluation function is alsorepresented by the M_SPT rule.

The complete beam search tree of the problem is shown in figure 3, in whichF and G represent local and global evaluation function values, respectively. Asdescribed in M_NONDELAY subroutine,sijk refers to the start time of an operationif operation Oij is to be assigned on machine Mk, Mk2Mij. For example, in level twoof the beam path 1, if O11 is to be assigned on machine 1 (represented by 1-1-1), thestart time of the operation on the machine is s111¼ 1.

After initiation, nodes generated from the root are determined by using theM_NONDELAY subroutine. They are expressed by i� j� k and sijk. Six nodes aregenerated (i.e. 1-1-1 s111¼ 0, 1-1-2 s112¼ 0, 1-2-1 s121¼ 0, 1-2-2 s122¼ 0, 1-3-1 s131¼ 0,

Table 1. Processing time information.

M1 M2

J1 O11 3 8O21 5 2

J2 O12 1 4O22 7 3

J3 O13 7 2O23 2 6


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1-3-2 s132¼ 0). Since the number of nodes (N¼ 6) are greater than b (¼ 2), thestep 2(iii) in the proposed algorithm is executed to compute the global evaluationfunction value for each node and to determine the initial two beam nodes. The globalevaluation function value is the objective function value of the complete schedulegained by the M_SPT rule from the partial schedule represented by the node. Basedon the global evaluation function value, node 1-2-1 s121¼ 0 and node 1-3-2 s132¼ 0 areselected as the initial beam nodes.

Then we apply the algorithm to these two nodes separately and generate onepartial tree from each of them according to step 3, and finally form two parallelbeam paths. In level 2, 3, and 4 of beam path 1, since the global evaluation values ofnodes left for global evaluation are same, workload balance among machines is usedto break tie. For example, before determining the beam node in level 4 of the beampath 1, the workload on machine 1 is 4 and that on machine 2 is 2, therefore, node2-2-2 is selected as beam node in level 4 and its start time is 2. At the end of thealgorithm, two schedules are gained. In fact, these two schedules are the same.

G=9.2

G=8.8 G=8.8

G=8.8 G = 8.8

G=9.2 G=8.8

G=8.8

G=15.9 G=8.8 G=12.5 G=14.5 G=8.81-1-1

1-1-1 1-1-2

1-1-2 2-2-1 2-2-2

1-3-1 1-3-2 1-1-1

1-1-1 2-2-1 2-2-2

1-1-2 1-2-1 1-2-2

1-1-2 1-2-1 1-2-2 1-3-1 1-3-2

F=3

G=8.8

G=8.8

G=8.8

G=8.8G=12.5

G=8.8

1-1-1

2-2-1

2-1-1

2-1-1

2-1-2

2-1-2

2-2-2 2-3-1

2-1-1

2-1-1

G=8.8 G=8.8

G=12.5 G=8.8

G=8.82-3-1

2-3-2

2-1-2

2-1-2

F=3

F=3 F=2

F=2 F=2 F=2

F=6F=7

F=5 F=2

F=8

F=8 F=7

2-2-1 2-3-1

2-3-1

F=7

F=5

F=3

F=7 F=2 F=8 F=1 F=4F=3

F=3

Rootl=0

l=1

l=2

l=3

l=4

l=5

l=6

s111 = 0 s121 = 0s112 = 0

s111 = 1

s111 = 1

s221 = 4

s211 = 4 s212 = 5

s212 = 5s211 = 6

s222 = 2 s231 = 4

s231 = 4

s232 = 2

2-3-2F=6

s232 = 5

s112 = 2 s221 = 1

s221 = 4

s231 = 4s211 = 4

s211 = 6

s212 = 5

s212 = 5

s222 = 2

G=8.8 G=8.82-2-2 2-3-2

2-3-2

F=3 F=2 F=6

F=6

s222 = 2 s231 = 4 s232 = 2

s232 = 5

s112 = 0 s132 = 0 s111 = 0

s111 = 1

s112 = 2

1-1-2F=8 F=3F=7

s112 = 2 s221 = 1 s222 = 2

s121 = 0 s122 = 2s131 = 1

s122 = 0 s131 = 0 s132 = 0

Beam path 2Beam path 1

b = 2f = 2

Figure 3. Filtered-beam search tree for the illustrative sample.


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5. Computational experiments

To demonstrate the practicability and efficiency of the proposed algorithm for theFJSP, different numerical simulations are tested and evaluated. The algorithm wasrun on a personal computer with an Intel Pentium IV, 512 MB RAM, on MicrosoftWindows 2000 Professional. The codes are written in the VCþþ 6.0 language.

5.1 Simulation 1: different combinations of local/global evaluation functions

Since the quality of the proposed algorithm depends on the quality of local andglobal evaluation functions, an analysis must be carried out to evaluate the effects ofthe proposed different local and global evaluation functions (i.e. M_SPT, M_EET,M_LWR, M_PT/TOT) under the same beamwidth and filterwidth. In this paper, theperformances of four representative different combinations of local and globalevaluation functions are measured. They are (M_SPT, M_SPT), (M_EET, M_EET),(M_LWR, M_LWR) and (M_PT/TOT, M_PT/TOT), where, the former one in theblank represents the local evaluation function, and the latter represents the globalevaluation function. The weights wm, wTW and wMW are generated randomly toaggregate a set of different objective functions. Representative results aresummarised under an objective function 0.4�F1þ 0.3�F2þ 0.3�F3 with b¼ 5,f¼ 2, 3, . . . , 8.

The data sets used for the simulation come from Brandimarte (1993). The dataare randomly generated using a uniform distribution between given limits. Theyconsist of 10 problems (i.e. Mk01,Mk02, . . . ,Mk10) where the number of jobsranges from 10 to 20, the number of machines ranges from 4 to 15, operations foreach job range from 5 to 15 and the maximum number of equivalent machines peroperation ranges from 3 to 6 (Mastrolilli and Gambardella 2000). The averagenumber of equivalent machines per operation ranges from 1.5 to 3.5 (namely, Mk01is 2, Mk02 is 3.5, Mk03 is 3, Mk04 is 2, Mk05 is 1.5, Mk06 is 3, Mk07 is 3, Mk08 is1.5, Mk09 is 3, Mk10 is 3.). These 10 problems are all partial flexibility problems.In this simulation, these partial flexibility problems are converted into total flexibilityones by substituting infinite processing time for each forbidden state according to themethod provided by Kacem et al. (2002a).

The simulation results are summarised in table 2. Where, F represents theobjective function value, T (s) represents the CPU seconds in our computer.

The results indicate that under the same b and f, the combination (M_SPT,M_SPT) runs the fastest, whereas the combination (M_LWR, M_LWR) runs theslowest. It also can be observed that as f increases, the computational time increases,which can be explained by more nodes entering for evaluation in the filtering phase.Results also show that for the same b and f, the combination (M_EET, M_EET)yields the best objective function value (except the Mk09 problem), while (M_LWR,M_LWR) yields the worst value. When f¼ 5, a relative best objective function valueis found in this trial range for the first time. After f¼ 5, performance of the algorithmalmost remain the same.

The values of three criteria corresponding to the best objective function value aregiven in the table. For example, the obtained best objective function value ofproblem Mk07 is 312.6, the corresponding criteria are F1¼ 162, F2¼ 664, F3¼ 162.


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Table

2.

Sim

ulationresultswithrespectto

differentcombinationsoflocalandglobalevaluationfunctions

f¼2

f¼3

f¼4

f¼5

f¼6

f¼7

f¼8

b¼5

FT(s)

FT

(s)

FT(s)

FT(s)

FT(s)

FT(s)

FT(s)

Mk01(10jobs6machines

55operations)

F1¼45F2¼160F3¼45

(M_SPT,M_SPT)

84.1

6.2

82.2

8.3

81.4

9.7

81.1

10.6

80.3

11.2

80.3

11.4

80.3

11.6

(M_EET,M_EET)

83.7

6.5

81.1

8.5

80.1

10.1

79.5

11.1

79.5

11.6

79.5

11.9

79.5

12.1

(M_LWR,M_LWR)

84.6

7.0

83.1

9.2

82.0

10.6

81.2

11.4

81.2

11.9

81.2

12.2

81.2

12.4

(M_PT/TOT,M_PT/TOT)

89.8

6.6

81.3

8.8

80.2

10.3

80.2

11.1

80.2

11.6

80.2

12.0

80.2

12.2


58operations)

F1¼33F2¼146F3¼31

(M_SPT,M_SPT)

69.4

9.3

68.6

12.4

68.3

15.9

67.8

18.3

67.4

20.5

67.4

22.6

67.4

24.2

(M_EET,M_EET)

69.7

9.7

67.7

13.1

67.4

16.2

66.3

19.3

66.3

22.2

66.3

24.9

66.3

26.2

(M_LWR,M_LWR)

72.1

11.1

68.9

14.8

68.1

18.3

67.8

21.9

67.8

24.6

67.8

26.9

67.8

28.9

(M_PT/TOT,M_PT/TOT)

70.6

10.4

68.4

13.9

66.8

16.9

66.8

20.0

66.8

22.8

66.8

24.8

66.8

26.3


150operations)

F1¼213F2¼875F3¼213

(M_SPT,M_SPT)

427.9

103.3

420.7

188.2

418.4

262.7

414.9

319.5

414.9

343.3

414.9

365

414.9

394

(M_EET,M_EET)

423.6

104.9

419.2

198.2

417.3

281.3

411.6

345.7

411.6

383.7

411.6

410.3

411.6

435.1

(M_LWR,M_LWR)

426.4

129.2

423.1

215.1

420.5

324.6

416.8

381.6

416.8

409.5

416.8

438.2

416.8

466.7

(M_PT/TOT,M_PT/TOT)

433

127.4

419.5

210.6

417.7

311.9

411.9

346.5

411.9

378.0

411.9

400.3

411.9

439.5

Mk04(15jobs,8machines,90operations)

F1¼69F2¼356F3¼67

(M_SPT,M_SPT)

170.9

16.5

162.6

36.8

160.8

43.1

158.0

52.4

158.0

55.4

158.0

58.9

158.0

60.2

(M_EET,M_EET)

160.9

28.4

156.3

37.0

155.8

44.6

154.5

53.0

154.5

56.9

154.5

60.0

154.5

62.9

(M_LWR,M_LWR)

172.0

30.3

168.1

39.6

165.0

46.4

159.7

55.6

159.7

59.1

159.7

62.6

159.7

65.1

(M_PT/TOT,M_PT/TOT)

168.2

30.0

161.2

39.3

158.0

45.8

155.2

54.2

155.2

57.4

155.2

60.3

155.2

62.5


F1¼179F2¼682F3¼178

(M_SPT,M_SPT)

354.2

61.7

344.9

85.7

342.3

108.7

339.6

122.8

337.2

144.3

337.2

156.2

337.2

167.5

(M_EET,M_EET)

346.8

62.3

341.3

86.7

335.6

109.6

334.3

128.8

333.5

148.7

329.6

159.6

329.6

168.7

(M_LWR,M_LWR)

356.7

72.3

345.1

103.0

344.7

126.2

343.1

145.5

341.0

165.6

341.0

182.1

341.0

190.5

(M_PT/TOT,M_PT/TOT)

347.4

66.0

342.3

92.6

341.3

121.6

337.4

134.8

335.2

157.3

334.3

169.7

334.3

181.2


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F1¼92F2¼340F3¼90

(M_SPT,M_SPT)

172.6

176.4

169.8

291.0

168.3

365.8

166.6

440.5

166.6

464.4

166.6

531.8

166.6

562.3

(M_EET,M_EET)

170.6

176.1

168.6

293.1

166.8

388.9

165.8

473.1

165.8

491.8

165.8

562.3

165.8

596.3

(M_LWR,M_LWR)

174.6

201.8

170.1

311.3

168.9

403.0

166.8

488.1

166.8

508.7

166.8

586.8

166.8

624.0

(M_PT/TOT,M_PT/TOT)

171.2

190.5

169.0

306.5

168.1

384.2

166.3

464.9

166.3

487.2

166.3

578.2

166.3

616.4


F1¼162F2¼664F3¼162

(M_SPT,M_SPT)

331.6

45.2

326.2

63.9

319.3

82.0

319.3

96.2

315.0

109.2

315.0

124.0

315.0

135.6

(M_EET,M_EET)

345.1

46.3

323.6

67.6

322.7

88.8

312.6

109.7

312.6

128.6

312.6

147.8

312.6

162.7

(M_LWR,M_LWR)

332.9

51.6

328.9

74.3

323.3

98.3

322.1

109.3

318.6

140.5

317.7

159.2

317.7

171.1

(M_PT/TOT,M_PT/TOT)

330.6

52.5

315.6

75.3

315.6

95.6

314.7

115.3

314.7

134.2

314.7

150.4

314.7

163.5


F1¼524F2¼2547F3¼524

(M_SPT,M_SPT)

1148.2

350.1

1143.7

519.0

1137.5

716.2

1137.5

864.3

1133.6

938.2

1133.6

1110.5

1133.6

1280.8

(M_EET,M_EET)

1146.8

355.9

1136.9

534.3

1135.4

714.6

1135.4

886.4

1130.9

932.3

1130.9

1111.2

1130.9

1201.4

(M_LWR,M_LWR)

1160.8

420.3

1145.3

660.8

1142.3

842.5

1139.5

931.6

1137.5

1000.4

1137.5

1323.6

1137.5

1336.3

(M_PT/TOT,M_PT/TOT)

1144.6

378.3

1139.3

542.3

1136.9

734.6

1136.9

864.1

1133.1

912.4

1133.1

1121.3

1133.1

1225.6


240operations)

F1¼396F2¼2257F3¼368

(M_SPT,M_SPT)

990.4

483.7

979.2

843.8

974.2

1325.5

968.6

1613.4

961.1

1858.5

949.6

2028.2

949.6

2222.6

(M_EET,M_EET)

977.6

504.8

964.1

855.5

962.3

1375.6

950.2

1623.8

947.0

1862.1

947.0

2052.8

947.0

2264.1

(M_LWR,M_LWR)

1001.8

617.2

988.2

1002.4

980.8

1565.4

973.1

1884.1

965.8

2051.3

950.9

2216.7

950.9

2486.4

(M_PT/TOT,M_PT/TOT)

992.1

538.1

967.4

802.3

965.9

1391.2

964.1

1723.3

945.9

1923.7

945.9

2186.5

945.9

2364.9


240operations)

F1¼294F2¼1866F3¼282

(M_SPT,M_SPT)

797.5

585.6

789.1

1211.2

785.6

1466.7

772.6

1972.5

767.7

2471.5

767.7

2809.1

767.7

3029.5

(M_EET,M_EET)

796.5

596.8

787.0

1232.3

779.4

1474.2

770.3

1984.6

762.0

2582.1

762.0

2912.2

762.0

3131.2

(M_LWR,M_LWR)

799.8

705.7

790.8

1331.7

786.3

1694.3

773.4

2103.3

768.2

2792.6

768.2

3421.6

768.2

3342.4

(M_PT/TOT,M_PT/TOT)

797.4

619.4

787.8

1239.5

779.9

1583.5

771.4

2091.4

763.8

2686.2

763.8

3019.5

763.8

3234.6


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For clarity, figure 4 depicts the experimental results for the four differentcombinations of local and global evaluation functions when b¼ 5, f¼ 2� 8. Thevertical axis shows average objective function value of the 10 problems. Thehorizontal axis measures the filterwidth-value. Each curve represents the averageobjective function value of the 10 problems under a particular combination of localand global evaluation function. As shown in the figure, the combination (M_EET,M_EET) leads to the best solution.

5.2 Simulation 2: FJSP with total flexibility

Three representative problem instances (denoted by problem n�m/N operations)based on practical data have been selected to compute. These problems are all fromKacem et al. (2002b), including the small size problem, problem 4� 5/12 operations;medium size problem, problem 10� 10/30 operations, and relatively large sizeproblem, problem 15� 10/56 operations. These problems are all total flexibility.In the following, the proposed algorithm is used to test the three problems, andresults are summarised to compare with those of other algorithms.

In order to balance the computational time and quality of solution, we set b¼ 5and f¼ 7 for the former two problems and b¼ 7 and f¼ 7 for the problem 15� 10/56operations. According to the results of section 5.1, in the following M_EET isselected as both local and global evaluation functions. The weights wm, wTM andwMW are generated randomly to aggregate a set of different objective functions. Finalfeasible solution is the best one of the solutions generated under the set of objectivefunctions.

5.2.1 Problem 43 5/12 operations. This problem is same as the instance 1 ofKacem et al. (2002b), which is a relatively simple instance of total flexibility. The

432

436

440

444

448

452

456

460

2 3 4 5 6 7 8

filterwidth

Ave

rage

F v

alue

(M_SPT, M_SPT)

(M_EET, M_EET)

(M_LWR, M_LWR)

(M_PT/TOT, M_PT/TOT)

Figure 4. Results comparison for four combinations of local and global evaluationfunctions.


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release time of jobs are set as r1¼ 3, r2¼ 5, r3¼ 1, and r4¼ 6. The obtained results bythe proposed algorithm are given in figure 5 in the form of a Gantt chart. Numbers(in the form of [job-operation]) inside the blocks are the operations associated withthe jobs, for example, 4-2 represents the operation 2 of job 4. The item ‘CPU time(millisecond)’ is the computational time in the form of millisecond (only about0.6 second in our computer for this problem). The item ‘Objective Function’indicates the objective function the algorithm is adopted and the values of thefunctions are also given.

The obtained solution by our proposed algorithm is summarised in table 3 incomparison with those of Kacem et al. (2002b), in which WT represents totalworkload of all machines, i.e. the value of F2, and Max(Wk) represents the workloadof the most loaded machine, i.e. the value of F3. The column labelled ‘ALþCGA’refers to the algorithm proposed by Kacem et al. (2002a, b). For clarity comparison,only two solutions of Kacem et al. (2002b) are given in table 3 (there are total of foursolutions in their pareto-optimality set). The results in table 3 show that the solutionof the proposed HFBS algorithm is better than those of ALþCGA for this problem.

5.2.2 Problem 103 10/30 operations. This problem is same as the instance 3 ofKacem et al. (2002b) with rj¼ 0, 8j2 n. It is a middle size problem with totalflexibility. It is also tested by PSOþSA in Xia and Wu (2005). The scheduling result

Figure 5. Solution of the problem 4� 5/12 operations with the proposed algorithm.


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is shown in figure 6 in the form of a Gantt chart. The computational time isabout 4.2 seconds in our computer. Due to the lack of computational time of otheralgorithms, it is impossible to compare the computational time between the proposed

algorithm and other previous methods.The comparison of the proposed algorithm with other algorithms is shown

in table 4. The column labeled ‘TD’ refers to the temporal decomposition proposedby F. Chetouane (Kacem et al. 2002a) and ‘Classic GA’ represents the classical


Table 3. Comparison of results on the problem 4� 5/12operations.

ALþCGA HFBS

Makespan 16 16 16WT 35 34 32Max(Wk) 9 10 9


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genetic algorithm, ‘AL’ and ‘ALþCGA’ are two algorithms proposed by Kacem

et al. (2002a, b), ‘PSOþ SA’ refers to the approach proposed by Xia and Wu (2005).

The result of the proposed HFBS algorithm is better than those of temporaldecomposition, Classical GA, AL and PSOþSA for this problem.

5.2.3 Problem 153 10/56 operations. The problem is a relatively large-scaleinstance, which has 15 jobs with 56 operations that need to be processed on

10 machines with total flexibility. The problem is same as the instance 4 of Kacemet al. (2002b) with given release times: r1¼ 5, r2¼ 3, r3¼ 6, r4¼ 4, r5¼ 9, r6¼ 7,

r7¼ 1, r8¼ 2, r9¼ 8, r10¼ 0, r11¼ 14, r12¼ 13, r13¼ 11, r14¼ 12, and r15¼ 5.The corresponding Gantt chart representations are shown in figure 7. The

computational time is about 55 seconds in our computer.The comparison of our proposed algorithm with ALþCGA (Kacem et al.

2002b) is shown in table 5. The results in table 5 show that the solution of HFBS is

better than one of pareto-optimality solutions obtained by ALþCGA for thisproblem.

5.2.4 Discussion. The above simulation results of these instances show that theproposed HFBS algorithm performs better in terms of objective function values.

This can be explained as follows:

. The proposed FBS based algorithm is an integrated approach based on the

built-in branching scheme and evaluation functions. Instead of dictatingspecific machines at the first stage like hierarchical approaches, the proposed

HFBS approach provides late commitment on selection of machines, thus

increases the degree of scheduling freedom, and thereby the potential foroptimization and updating.

Table 5. Comparison of results on the problem 15� 10/56operations.

ALþCGA HFBS

Makespan 23 23WT 95 93Max(Wk) 11 10

Table 4. Comparison of results on the problem 10� 10/30 operations.

TD Classic GA AL ALþCGA PSOþ SA HFBS

Makespan 15 7 8 7 7 7WT 59 53 46 45 44 42Max(Wk) 16 7 6 5 6 6


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. As far as the solution stability is concerned, once the beamwidth and thefilterwidth are determined, the obtained results of the proposed algorithm arestable. This helps decrease the possibility of loss of good sub-optimalsolutions and improve the computational efficiency since it need not runmore times to compensate the computational randomness.

5.3 Simulation 3: comparison with dispatching rules

Dispatching rules are frequently used to solve scheduling in practice. To furtherinvestigate the proposed HFBS algorithm, its performance is compared with those ofthe following five modified dispatching rules, which are based on the proposedM_NONDELAY procedure.

SPT: In the earliest schedulable set of operations (deduced from theM_NONDELAY procedure), favour the i� j� k with min(pijk).

LPT (longest processing time): In the earliest schedulable set of operations, favourthe i� j� k with the longest processing time (i.e. favour the i� j� k with max(pijk)).

LWR: In the earliest schedulable set of operations, favour the i� j� k with theminð

Pnjq¼i minðpqjkÞÞ.

MWR (most work remaining): In the earliest schedulable set of operations,favour the i� j� k with the most work remaining (i.e. favour the i� j� k withmaxð

Pnjq¼i minðpqjkÞÞ).



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EET: In the earliest schedulable set of operations, favour the i� j� k withmin(sijkþ pijk).

The tests are performed using random instances that are generated according tothe scheme suggested by Scrith et al. (2004). Each instance can be characterised by thefollowing parameters: number of jobs n, number of machinesm, number of operationsper job nj, processing time of operation pijk, set of machines that can process theoperation Mij and number of alternative machines for each operation Aij (i.e. Aij iscardinality of set Mij). Table 6 shows the simulation parameters used to generate theinstances, where U[a, b] denotes a discrete uniform distribution between a and b.

The parameter Aij determines the flexibility level of the problem (partial or totalflexibility). Two levels are considered: 0.5 (medium partial flexibility) and 1 (totalflexibility) of the total number of the machines. The processing times pijk aregenerated in the following way. A given operation Oij has Aij alternative machines:the first machine Mij(1) is chosen from U[1, m] and the corresponding processingtime pijk(1) is selected from U[1,15]. The remaining processing times of this operationon the alternative machines depend on pijk(1) and are chosen from U[pijk(1),min(2� pijk(1),15)]. This helps avoid large discrepancies among the processing timesof an operation on alternative machines. The generated testing instances are denotedby dimension (n�m). Twelve different dimensions are designed, and for eachdimension, two different flexibility levels are considered. For each dimension andeach flexibility level, a group of 10 instances are generated.

The objective function chosen is F¼ 0.4F1þ 0.3F2þ 0.3F3, b¼ 3, f¼ 3, andM_EET is selected both as local and global evaluation functions. Simulation resultsare shown in table 7. In which, the HFBS is compared with the dispatching rules onthe basis of the average objective function value �F for each dimension. In the table,G represents the group number of instances, �FD denotes the �F value obtained bydispatching rules (i.e. D denotes SPT, LPT, LWR, MWR and EET), �FHFBS denotesthe �F value obtained by the proposed HFBS algorithm. P.I. refers to the percentageimprovement for the �F of the HFBS compared with the dispatching rules, which isdefined as:

P:I: ¼�FD � �FHFBS

�FD

� 100% ð6Þ

The results show that the average objective function values obtained by theHFBS algorithm are clearly better than those of the dispatching rules regardless ofthe partial or total flexibility. The resulting �F-values of the HFBS are lower than

Table 6. Simulation parameters.

Parameter Values

n 5, 10, 20, 30, 50m 6, 8, 12nj U[m/2, m]Aij 0.5�m, mpijk U[1, 15] for pijk(1) on Mij(1)

U[pijk(1), min(2� pijk(1), 15)] for Mij(2), . . . ,Mij(Aij)


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Table

7.

Thesimulationresultswithcomparisonin

theaverageobjectivefunctionvalues

Aij¼0.5�m

Aij¼m

G(n�m)

HFBSvs.

SPT

HFBSvs.

LPT

HFBSvs.

LWR

HFBSvs.

MWR

HFBSvs.

EET

HFBSvs.

SPT

HFBSvs.

LPT

HFBSvs.

LWR

HFBSvs.

MWR

HFBSvs.

EET

15�6

� FD

70.08

110.08

72.7

108.7

66.25

64.66

114.09

64.9

115.9

61.85

� FHFBS

62.92

55.79

P.I.(%

)10.22

42.84

13.45

42.12

5.03

13.72

51.10

14.04

51.86

9.80

t4.36

18.65

5.34

15.73

3.61

6.13

16.10

6.42

17.8

4.48

210�6

� FD

126.72

200.36

134.73

200.16

124.31

119.32

208.14

120.28

200.87

114.4

� FHFBS

117.67

107.37

P.I.(%

)7.14

41.27

12.66

41.21

5.34

10.02

48.41

10.73

46.55

6.15

t8.83

23.02

8.93

23.85

4.72

31.02

27.90

16.50

11.59

6.78

330�6

� FD

340.58

523.44

354.55

514.98

340.29

337.87

590.66

337.55

589.81

327.74

� FHFBS

324.68

321.6

P.I.(%

)4.67

37.97

8.42

36.95

4.59

4.82

45.55

4.73

45.47

1.87

t7.49

25.04

10.59

28.01

4.21

5.85

32.00

6.30

32.72

6.06

45�8

� FD

88.1

149.6

90.77

150.67

85.19

86.44

149.74

86.32

148.9

81.13

� FHFBS

80.8

77.91

P.I.(%

)8.29

45.99

10.98

46.37

5.15

9.87

47.97

9.74

47.68

3.97

t9.41

17.05

5.24

18.23

6.24

5.22

14.81

5.09

15.48

5.11

510�8

� FD

147.68

240.9

154.22

235.93

145.42

155.76

284.27

156.48

284.67

149.27

� FHFBS

139.41

144.06

P.I.(%

)5.60

42.13

9.60

40.91

4.13

7.51

49.32

7.94

49.39

3.49

t6.67

20.41

4.74

23.14

7.44

5.75

21.99

6.47

21.99

5.98

620�8

� FD

295.56

487.49

311.01

467.69

296.68

291.26

512.84

292.06

511.31

273.72

� FHFBS

286.69

267.04

P.I.(%

)3.00

41.19

7.82

38.70

3.37

8.32

47.93

8.57

47.77

2.44

t3.89

25.56

6.92

22.32

6.96

6.47

35.22

6.85

38.11

7.63


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750�8

� FD

704.95

1130.65

735.08

1138.5

698.03

676.94

1221.45

676.9

1220.02

663.06

� FHFBS

686.86

649.98

P.I.(%

)2.57

39.25

6.56

39.67

1.60

3.98

46.79

3.98

46.72

1.97

t5.65

43.06

8.18

40.03

6.44

4.91

39.13

4.99

39.02

4.63

85�12

� FD

120.33

210.23

121.98

209.78

116.6

130.06

226.47

130.78

227.57

122.34

� FHFBS

114.14

117.85

P.I.(%

)5.14

45.71

6.43

45.59

2.11

9.39

47.96

9.89

48.21

3.67

t7.87

23.36

7.25

29.24

3.69

11.36

41.55

8.54

40.16

9.80

910�12

� FD

209.49

362.71

215.17

363.55

207.77

206.56

375.23

206.4

372.3

193.06

� FHFBS

199.73

188.71

P.I.(%

)4.66

44.93

7.18

45.06

3.87

8.64

49.71

8.57

49.31

2.25

t6.78

43.53

7.76

26.81

10.93

7.36

39.67

7.54

36.32

7.99

10

20�12

� FD

393.85

693.13

405.77

683.51

403.24

385.48

700.5

387

699.86

378.94

� FHFBS

391.92

367.75

P.I.(%

)0.49

43.46

3.41

42.66

2.81

4.60

47.50

4.97

47.45

2.95

t1.05

85.05

6.69

56.68

5.74

8.09

47.80

7.81

50.17

7.38

11

30�12

� FD

577.91

994.23

589.57

994.42

591.08

567.18

1039.74

571.2

1038.44

560.58

� FHFBS

567.93

542.27

P.I.(%

)1.73

42.88

3.67

42.89

3.92

4.39

47.85

5.06

47.78

3.27

t5.33

53.15

6.40

67.43

11.64

12.70

57.64

9.98

62.04

14.83

12

50�12

� FD

960.86

1598.42

974.18

1590.76

966.94

956.02

1627.96

970.22

1594.98

966.54

� FHFBS

939.60

916.40

P.I.(%

)2.29

42.17

3.96

41.56

3.29

4.01

45.48

4.84

44.95

4.58

t7.42

58.57

8.83

74.27

29.22

15.34

37.43

15.97

29.37

22.84


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competitor methods in the range of 1.60� 9.80% for HFBS versus EET,

0.49� 13.72% for HFBS versus SPT, 3.41� 14.04% for HFBS versus LWR,

37.97� 51.10% for HFBS versus LPT and 36.95� 51.86% for HFBS versus MWR.The results also indicate that among the rules, the LPT and MWR rules are worst.

To further analyze the results, the 10 pair-wise comparisons in each dimension

are subjected to paired-sample t-tests on the difference between �F, like the data

processing by Tai and Boucher (2002). The test statistic for a two-tail test with nine

degrees of freedom and a 99% confidence interval is 3.25. The t-value of the

10-paired trials for each dimension is given in the table. Besides 1.05 (located in thecell of group 10, Aij¼ 0.5�m and HFBS versus SPT), all other t-values are larger

than 3.25. This indicates that the results could not reasonably have occurred by

chance since they are statistically significant above the 99% confidence level. This

further demonstrates that the HFBS algorithm performs much better than thedispatching rules in terms of the average objective function value.

It is important to note that, although HFBS outperforms the dispatching rules in

accuracy, its computation time is larger than that of dispatching rules. For example,

the average CPU seconds of 10 instances with largest problem size (i.e. 50� 15 and

Aij¼m) are around 2552.12 and 30.95 seconds for HFBS and the rules, respectively.Table 8 shows the percentage improvement for average values of the three

individual objective functions of the HFBS compared with those of the dispatching

rules. In the table, the values in the first row of each dimension are the corresponding

average objective function values obtained by the HFBS algorithm. I-HFBSD is the

percentage improvement of the HFBS, which is defined as:

I-HFBSD ¼�fD � �fHFBS

fD� 100%, ð7Þ

where, D refers to certain dispatching rule, fD is the average value of one

individual objective function obtained by certain dispatching rule, and fHFBS is the

corresponding average value obtained by the HFBS algorithm.The results indicate that the HFBS are clearly better than the LPT, MWR and

EET rules. Although the total workload obtained by the SPT and LWR rules are

better than that of the HFBS (�0.33��6.34% for I-HFBSSPT and �0.33��4.67%

I-HFBSLWR), the makespan and the maximum workload of the SPT and LWR are

much worse (15.78� 32.55% for I-HFBSSPT and 17.63� 36.51% for I-HFBSLWR inmakespan, 16.46� 47.20% for I-HFBSSPT and 16.37� 47.20% for I-HFBSLWR in

the maximum workload).In table 8, the total workload objective function value of the HFBS is larger than

those of the SPT and LWR. This is mainly that the dispatching rules have the

property of inherent selection inclination, e.g. the SPT always selects the decisionwith the shortest processing time. Therefore, it tends to obtain the lower bound of

the total workload of an instance. LRW is similar to the SPT. To demonstrate this,

three instances mentioned in section 5.2 are tested by the SPT and LRW rules, and

the results with comparison are shown in table 9. The results demonstrate that the

proposed SPT and LWR rule can reach the lower-bound value of the total workloadWT. This also explains the reason that the values of WT of the HFBS are worse than

those of the SPT and LWR rules.


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Table 8. The percentage improvement results in three individual objective function values ofHFBS compared with dispatching rules.

Aij¼ 0.5�m Aij¼m

G (n�m) Makespan Max(WK) WT Makespan Max(WK) WT

1 5� 6 HFBS 37.3 27.3 132.7 30.8 22.4 122.5I-HFBSSPT(%) 25.10 31.06 (�4.0) 29.84 40.43 (�2.6)I-HFBSLPT(%) 52.24 54.73 34.47 62.12 66.01 40.53I-HFBSLWR(%) 31.05 33.58 (�2.55) 30.79 40.43 (�2.6)I-HFBSMWR(%) 50.33 54.35 34.44 63.11 66.52 41.13I-HFBSEET(%) 10.12 12.50 1.19 13.48 21.68 5.84


3 30� 6 HFBS 135.5 132.8 768.8 134.7 130.3 762.1I-HFBSSPT(%) 19.73 16.79 (�2.40) 19.44 19.12 (�2.68)I-HFBSLPT(%) 48.96 47.92 32.31 53.65 52.96 41.57I-HFBSLWR(%) 25.51 16.37 1.50 19.05 19.12 (�2.68)I-HFBSMWR(%) 46.55 47.22 31.78 53.10 53.08 41.62I-HFBSEET(%) 9.67 8.41 2.60 4.60 4.33 0.77



6 20� 8 HFBS 104.5 95.3 721 92.5 87.6 679.2I-HFBSSPT(%) 19.18 21.04 (�4.18) 32.04 33.13 (�3.16)I-HFBSLPT(%) 53.16 56.88 35.42 59.78 60.31 42.54I-HFBSLWR(%) 30.89 26.35 (�2.17) 33.02 33.13 (�3.16)I-HFBSMWR(%) 47.03 49.36 34.91 58.93 60.33 42.60I-HFBSEET(%) 9.13 7.21 1.62 8.51 6.41 0.70

7 50� 8 HFBS 232 224.3 1755.9 219.6 214.3 1659.5I-HFBSSPT(%) 17.85 18.85 (�3.48) 17.91 18.39 (�1.36)I-HFBSLPT(%) 48.21 47.57 36.00 55.53 55.73 43.34I-HFBSLWR(%) 23.86 23.21 (�0.23) 17.88 18.39 (�1.36)I-HFBSMWR(%) 49.75 51.11 35.46 55.13 55.74 43.36I-HFBSEET(%) 3.65 3.82 0.94 3.17 2.81 1.65

8 5� 12 HFBS 66.2 28.2 264 71.5 26.4 271.1I-HFBSSPT(%) 15.78 24.80 (�2.01) 15.88 47.20 (�0.33)I-HFBSLPT(%) 52.10 58.95 41.05 51.16 65.49 43.89

(continued)


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6. Conclusions

In this paper, a new filtered-beam-search-based heuristic algorithm (HFBS) isdeveloped to solve the flexible job-shop scheduling problem with multiple objectiveswith respect to minimisation of makespan, the total workload of machines and theworkload of the most loaded machine. In the proposed algorithm, a modifiedbranching scheme (M_NONDELAY) is designed to integrate the two sub-problemsof the FJSP simultaneously. Different dispatching rule-based heuristics areincorporated into the proposed algorithm as local and global evaluation functions.Through simulations, the performance of the proposed algorithm is evaluated andcompared with those of other approaches. The results show that the proposedalgorithm can obtain very satisfactory solutions in short computational time forthe FJSP.

The FJSP does not have a rich literature and still needs further exploration. Theproposed HFBS algorithm in this paper can provide an effective and simple

Table 8. Continued.

Aij¼ 0.5�m Aij¼m

G (n�m) Makespan Max(WK) WT Makespan Max(WK) WT

I-HFBSLWR(%) 18.27 25.20 (�1.19) 17.63 47.20 (�0.33)I-HFBSMWR(%) 50.04 61.63 41.22 51.43 65.54 44.18I-HFBSEET(%) 1.34 19.43 0.08 0.28 35.29 0.11



11 30� 12 HFBS 145.5 134.1 1565 137 129.3 1495.6I-HFBSSPT(%) 22.48 19.02 (�3.61) 26.23 22.94 (�1.38)I-HFBSLPT(%) 55.83 54.85 39.29 60.77 59.40 44.23I-HFBSLWR(%) 25.77 21.99 (�2.15) 28.65 25.17 (�1.38)I-HFBSMWR(%) 55.92 55.55 39.17 60.45 59.43 44.21I-HFBSEET(%) 10.30 7.71 2.72 10.63 7.44 1.90

12 50� 12 HFBS 227.1 218.6 2610.6 212.9 204 2566.8I-HFBSSPT(%) 24.35 21.82 (�3.37) 25.82 26.25 (�1.74)I-HFBSLPT(%) 53.24 52.45 39.41 57.69 57.50 42.34I-HFBSLWR(%) 24.12 22.95 (�1.25) 26.51 25.63 (�0.68)I-HFBSMWR(%) 51.60 51.56 39.04 56.35 56.12 42.11I-HFBSEET(%) 9.70 6.42 2.21 14.33 11.80 2.73


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Table

9.

ResultscomparisonoftheSPT

andLWR

ruleswiththelower-boundvalues

forthreebenchmark

instances.

Problem

4�5/12operations

Problem

10�10/30operations

Problem

15�10/56operations

Lower-boundvalues

(Kacem

etal.2002b)

SPT/LWR

Lower-boundvalues

(Kacem

etal.2002b)

SPT/LWR

Lower-boundvalues

(Kacem

etal.2002b)

SPT/LWR

Makespan

16

23

714

23

27

WT

32�

32�

41�

41�

91�

91�

Max(W

k)

718

513

10

22

*Represents

thelower

boundvalue.


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approach for this complex problem. Due to its effectiveness and simplicity, theproposed algorithm is promising for practical complex scheduling problemsencountered in flexible shop floors if further modification (e.g. if other additionalresources can be incorporated into the algorithm, and dynamic flexible job-shopscheduling can be realised).

Acknowledgements

This work was supported by the National Science Foundation of China under Grant60574054 and the Program for New Century Excellent Talents in the University ofChina (NCET 2006). The authors would like to express sincere appreciation to theanonymous referees for their detailed and helpful comments to improve the qualityof the paper. Appreciation is also expressed to Dr Yu Jianbo for his invaluableadvice and encouragement.

Appendix

Ivanescu et al. (2002) mentioned that the makespan of a job set is influenced by itsworkload. More specifically, the workload on the bottleneck resource puts a lowerbound to the makespan. Logically, we borrow the definitions of objective conflictdefined by Brizuela and Aceves (2003).

Definition 1: The Ideal Point or Ideal Vector is defined as the point Z� composed ofthe best attainable objective values. This is, Z�j ¼ min or maxffjðxÞjx 2 Ag, A is theset (without constraints) of all possible x, j¼ 1, 2 . . . q, where q41 is the number ofobjective functions, x is one feasible schedule, fj(x) is the value of j th objectivefunction.

Definition 2: Two objectives are in conflict if the Euclidean distance from the idealpoint to the set of best values of feasible solutions is different from zero.

Definition 3: Three or more objectives are in conflict if they are in conflict pairwise.To show that the objectives are in conflict, we are going to construct a counter-exampleinstance where we can enumerate all solutions and see that there is not a single solutionwith all its objective values being better than or equal to the respective objective valuesof the other solutions. This implies that the distance from the set of solutions to the idealpoint is positive.

Table 10 presents an instance of the FJSP with three machines and two jobs eachhaving two operations. The processing times on alternative machines are given in thetable. All solutions for this problem are enumerated in table 11. Note that, for each

Table 10. Problem data of a simple illustrative FJSP.

Job Operation M1 M2 M3

Job 1 O11 3 1 2O12 2 4 3

Job 2 O21 4 2.5 6O22 3 2.5 3


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Table 11. The possible schedule results for the illustrative problem.

Assignment MakespanTotal workloadof machines

The workload ofmost loaded machines

O11M1, O12M1, O21M1, O22M1 12.0 12.0 12.0O11M1, O12M1, O21M1, O22M2 9.0 11.5 9.0O11M1, O12M1, O21M1, O22M3 9.0 12.0 9.0O11M1, O12M1, O21M2, O22M1 8.0 10.5 8.0O11M1, O12M1, O21M2, O22M2 5.0 10.0 5.0O11M1, O12M1, O21M2, O22M3 5.5 10.5 5.0O11M1, O12M1, O21M3, O22M1 9.0 14.0 8.0O11M1, O12M1, O21M3, O22M2 8.5 13.5 6.0O11M1, O12M1, O21M3, O22M3 9.0 14.0 9.0O11M1, O12M2, O21M1, O22M1 10.0 14.0 10.0O11M1, O12M2, O21M1, O22M2 9.5 13.5 7.0O11M1, O12M2, O21M1, O22M3 10.0 14.0 7.0O11M1, O12M2, O21M2, O22M1 7.0 12.5 6.5O11M1, O12M2, O21M2, O22M2 9.0 12.0 9.0O11M1, O12M2, O21M2, O22M3 7.0 12.5 6.5O11M1, O12M2, O21M3, O22M1 9.0 16.0 6.0O11M1, O12M2, O21M3, O22M2 9.5 15.5 6.5O11M1, O12M2, O21M3, O22M3 9.0 16.0 9.0O11M1, O12M3, O21M1, O22M1 10.0 13.0 10.0O11M1, O12M3, O21M1, O22M2 9.5 12.5 7.0O11M1, O12M3, O21M1, O22M3 10.0 13.0 7.0O11M1, O12M3, O21M2, O22M1 6.0 11.5 6.0O11M1, O12M3, O21M2, O22M2 6.0 11.0 5.0O11M1, O12M3, O21M2, O22M3 8.5 11.5 6.0O11M1, O12M3, O21M3, O22M1 9.0 15.0 9.0O11M1, O12M3, O21M3, O22M2 9.0 14.5 9.0O11M1, O12M3, O21M3, O22M3 12.0 15.0 12.0O11M2, O12M1, O21M1, O22M1 9.0 10.0 9.0O11M2, O12M1, O21M1, O22M2 6.5 9.5 6.0O11M2, O12M1, O21M1, O22M3 7.0 10.0 6.0O11M2, O12M1, O21M2, O22M1 6.5 8.5 5.0O11M2, O12M1, O21M2, O22M2 6.0 8.0 6.0O11M2, O12M1, O21M2, O22M3 5.5 8.5 3.5

O11M2, O12M1, O21M3, O22M1 9.0 12.0 6.0O11M2, O12M1, O21M3, O22M2 8.5 11.5 6.0O11M2, O12M1, O21M3, O22M3 9.0 12.0 9.0O11M2, O12M2, O21M1, O22M1 7.0 12.0 7.0O11M2, O12M2, O21M1, O22M2 7.5 11.5 7.5O11M2, O12M2, O21M1, O22M3 7.0 12.0 5.0O11M2, O12M2, O21M2, O22M1 7.5 10.5 7.5O11M2, O12M2, O21M2, O22M2 10.0 10.0 10.0O11M2, O12M2, O21M2, O22M3 7.5 10.5 7.5O11M2, O12M2, O21M3, O22M1 9.0 14.0 6.0O11M2, O12M2, O21M3, O22M2 8.5 13.5 7.5O11M2, O12M2, O21M3, O22M3 9.0 14.0 9.0O11M2, O12M3, O21M1, O22M1 7.0 11.0 7.0O11M2, O12M3, O21M1, O22M2 6.5 10.5 4.0O11M2, O12M3, O21M1, O22M3 7.0 11.0 6.0O11M2, O12M3, O21M2, O22M1 6.5 9.5 3.5

O11M2, O12M3, O21M2, O22M2 6.0 9.0 6.0O11M2, O12M3, O21M2, O22M3 7.0 9.5 6.0O11M2, O12M3, O21M3, O22M1 9.0 13.0 9.0

(continued)


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assignment of operations on machines, there is more than one solution due to

different sequences for the same assignment. In table 11, only the best solution is

given for the same assignment. For example, as for the assignment (O11M3, O12M1,

O21M1, O22M1), under the precedence constraints and resource constraints, there

are three possible sequence, i.e. O11M3 jO12M1 jO21M1 jO22M1, O11M3 j

O21M1 jO12M1 jO22M1, O11M3 jO21M1 jO22M1 jO12M1, the corresponding

objectives of three solutions are {11, 11, 9}, {9, 11, 9} and {9, 11, 9}. For brevity,

only {9, 11, 9} is given in table 11. From the table 11, it can be seen that the

assignments (O11M1,O12M1,O21M2,O22M2), (O11M3,O12M1,O21M2,O22M2) and

(O11M3,O12M3,O21M2,O22M2) have the best makespan value (the value is 5,

highlighted in bold), the assignment (O11M2,O12M1,O21M2,O22M2) has the

best value of total workload of machines (the value is 8), and the assignments

(O11M2,O12M1,O21M2,O22M3) and (O11M2,O12M3,O21M2,O22M1) have the best

value of the workload of most loaded machines (the value is 3.5). Therefore, the ideal

point Z* for this instance is given by Z*¼ {5, 8, 3.5}. The Euclidean distance from the

set of values in table 11 to the ideal point is positive. This implies that these three

performance measures are conflicting.

Table 11. Continued.

Assignment MakespanTotal workloadof machines

The workload ofmost loaded machines

O11M2, O12M3, O21M3, O22M2 9.0 12.5 9.0O11M2, O12M3, O21M3, O22M3 12.0 13.0 12.0O11M3, O12M1, O21M1, O22M1 9.0 11.0 9.0O11M3, O12M1, O21M1, O22M2 6.5 10.5 6.0O11M3, O12M1, O21M1, O22M3 7.0 11.0 6.0O11M3, O12M1, O21M2, O22M1 7.0 9.5 5.0O11M3, O12M1, O21M2, O22M2 5.0 9.0 5.0O11M3, O12M1, O21M2, O22M3 5.5 9.5 5.0O11M3, O12M1, O21M3, O22M1 11.0 13.0 8.0O11M3, O12M1, O21M3, O22M2 10.5 12.5 8.0O11M3, O12M1, O21M3, O22M3 11.0 13.0 11.0O11M3, O12M2, O21M1, O22M1 7.0 13.0 7.0O11M3, O12M2, O21M1, O22M2 8.5 12.5 6.5O11M3, O12M2, O21M1, O22M3 7.0 13.0 5.0O11M3, O12M2, O21M2, O22M1 6.5 11.5 6.5O11M3, O12M2, O21M2, O22M2 9.0 11.0 9.0O11M3, O12M2, O21M2, O22M3 6.5 11.5 6.5O11M3, O12M2, O21M3, O22M1 11.0 15.0 8.0O11M3, O12M2, O21M3, O22M2 10.5 14.5 8.0O11M3, O12M2, O21M3, O22M3 11.0 15.0 11.0O11M3, O12M3, O21M1, O22M1 7.0 12.0 7.0O11M3, O12M3, O21M1, O22M2 6.5 11.5 6.5O11M3, O12M3, O21M1, O22M3 8.0 12.0 8.0O11M3, O12M3, O21M2, O22M1 5.5 10.5 5.0O11M3, O12M3, O21M2, O22M2 5.0 10.0 5.0O11M3, O12M3, O21M2, O22M3 8.0 10.5 8.0O11M3, O12M3, O21M3, O22M1 11.0 14.0 11.0O11M3, O12M3, O21M3, O22M2 11.0 13.5 11.0O11M3, O12M3, O21M3, O22M3 14.0 14.0 14.0


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a filtered-beam-search-based heuristic algorithm for flexible job-shop scheduling problem

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