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Page 1: Modelling the machine loading problem of FMSs and its solution using a tabu-search-based heuristic

This article was downloaded by: [Selcuk Universitesi]On: 21 December 2014, At: 22:10Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Computer IntegratedManufacturingPublication details, including instructions for authors and subscriptioninformation:http://www.tandfonline.com/loi/tcim20

Modelling the machine loading problem ofFMSs and its solution using a tabu-search-basedheuristicU. M. B. S. Sarma , Suman Kant , Rahul Rai & M. K. TiwariPublished online: 08 Nov 2010.

To cite this article: U. M. B. S. Sarma , Suman Kant , Rahul Rai & M. K. Tiwari (2002) Modelling the machineloading problem of FMSs and its solution using a tabu-search-based heuristic, International Journal of ComputerIntegrated Manufacturing, 15:4, 285-295, DOI: 10.1080/09511920110086926

To link to this article: http://dx.doi.org/10.1080/09511920110086926

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Page 2: Modelling the machine loading problem of FMSs and its solution using a tabu-search-based heuristic

Modelling the machine loading problem ofFMSs and its solution using a tabu-search-basedheuristic

U. M. B. S. SARMA, SUMAN KANT, RAHUL RAI and M. K. TIWARI

Abstract. This paper develops a modelling framework thataddresses the machine loading problem of FMSs. A generic0-1 mixed integer programming formulation that modelsthe loading problem, with the minimization of systemunbalance and maximization of throughput as the bi-criterion objective, has been proposed. The constraintsconsidered are availability of tool slots and time onmachines. Two heuristic methods are discussed to solvethe problem. In heuristic 1, sequence generation is carriedout using the fixed part sequencing rules. In heuristic 2,tabu-search-based methodology has been adopted toperturb the part sequences obtained from heuristic oneand then reallocation of operations on machines areperformed keeping in view the above objectives andconstraints. The proposed methodology has been testedon ten problems representing three types of FMSs: small,medium and large. Exhaustive computational experimentsreveal that the proposed methodology consistently offersgood results for all the test problems.

1. Introduction

A FMS consists of several numerically controlledworkstations that can machine a wide variety of parttypes. Parts are mounted on the pallets and automatictransportation devices are employed to transportthem. Machines are equipped with automatic toolchangers and processing can be performed onalternative machines if they are loaded with suitabletools. A supervisory computer is used to control thewhole system. There are two types of decision

problems associated with FMSs: Design problems (selec-tion of machines, robot layout decisions, AGVs, pathselection etc) and Operational problems (part typeselection, resource grouping, production ratio deter-mination, allocation of resources and loading). Themachine loading problem refers to selecting a subsetof parts from a part reservoir and assigning theiroperations to appropriate machines in the givenplanning horizon in order to meet certain specifiedobjectives while satisfying the system constraints(Stecke 1983).

These objectives are:

(1) Balancing the machine processing time.(2) Minimizing the number of movements.(3) Balancing the workloads per machine for a

system of groups of pooled machines of equalsizes.

(4) Unbalancing the load per machine for asystem of groups of pooled machines ofunequal sizes.

(5) Filling the tool magazines as densely as possible.(6) Maximizing the sum of operation priorities.

From the literature, it is evident that the majority ofthe performance measures of loading are quitestringent, and frequently involve multiple objectives(Stecke 1983, Kusiak 1985, Shanker and Srinivasulu1989).

Numerous solution methodologies have beendeveloped to solve the machine loading of FMSs.Mukhopadhyay and Tiwari (1995) solved the ma-chine loading problem using the principle ofconjoint measurement. Mukhopadhyay et al. (1991)prioritized the loading of machine tools and parts inrandom FMSs through eigenvalue analysis. The

INT. J. COMPUTER INTEGRATED MANUFACTURING, 2002, VOL. 15, NO. 4, 285–295

International Journal of Computer Integrated ManufacturingISSN 0951-192X print/ISSN 1362-3052 online # 2002 Taylor & Francis Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/09511920110086926

Authors: U. M. B. S. Sarma, Department of Mechanical Engineering, IndianInstitute of Technology (IIT), Guwahati, India; Suman Kant, Russian HostelCD-713/2, OHC, Ranchi-834 004, India; Rahul Rai and M. K. Tiwari,Department of Manufacturing Engineering, National Institute of Foundry andForge Technology (NIFFT), Hatia, Ranchi, India. E-mail: [email protected]

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Page 3: Modelling the machine loading problem of FMSs and its solution using a tabu-search-based heuristic

machine loading problem has been solved usingdifferent solution methodologies, which are given asfollows:

(1) Heuristic oriented (Stecke and Solberg 1981,Stecke 1983, Mukhopadhyay et al. 1992, Morinoand Ding 1993, Chen and Chung 1996, Tiwari etal. 1997).

(2) Simulation based (Jain et al. 1989, Sabuneuogloand Hommerzhein 1992, Basnet and Mize1993).

(3) Multi-criterion decision making (Kumar et al.1990, Chen and Askin 1990, Kim and Yano 1997,Sawik 1998).

(4) Mathematical programming (Stecke 1983, Las-kari et al. 1987, Shanker and Srinivasulu 1989,Sawik 1990, Liang and Dutta 1992, 1993,Guerrero et al. 1999).

Shanker and Tzen (1985) considered the bi-criterion objective for a loading problem that includesbalancing workloads and meeting due date of parttypes. Ammons et al. (1985) considered the bi-criterionobjective to be balancing workloads and minimizingworkstation visits to resolve the loading problem.Rajagopalan (1986) combined the loading problemwith other problems inherently found in the planningstage, such as job selection and production ratiodetermination, with the aim of achieving betterproduction schedules without too many iterations.Mukhopadhyay et al. (1992) and Tiwari et al. (1997)attempted the machine loading problem using heuristicapproaches with an objective of minimizing systemunbalance by maximizing throughput. Several objectivefunctions, such as maximization of workload balance onmachines, minimization of system unbalance, maximi-zation of system utilization, minimization of flow timeetc, have been considered in articles aiming at solvingthe machine loading problem in FMS.

The objective of this paper is to solve the machineloading problem, which includes a bi-criterion objectivefunction of minimizing system unbalance and maximiz-ing the throughput in the presence of availablemachine time and tool slots as constraints. Theseobjectives result in higher machine utilization, highersystem output, and hence result in limiting the parttardiness.

In this research, initially fixed part sequencingrules have been employed to solve the combined partsequencing and machine loading problem in whichoperation allocations on machines are carried out byobserving the maximum remaining processing timeon the machines. This solution approach is referredto as heuristic 1 in the article. A tabu-search-based

heuristic solution has been contemplated to generatea number of part sequences and a proposed newscheme for the neighbourhood generation whereeach unassigned part is replaced by an assigned onein a sequential manner. This aspect has been dealtwith by heuristic 2. To present the whole solutionmethodology adopted in this article in a simplifiedway, it can be said that heuristic 2 uses the output ofheuristic 1 (feasible part sequences that ensureminimizing of system unbalance and maximizing ofthroughput while satisfying the system’s technologicalconstraints).

In this research, the adoption of tabu search as apowerful optimization technique has been conceptua-lized effectively to carry out an iterative improvementthat begins with an initial feasible solution and attemptsto determine a better solution. The performance of aproposed heuristic is tested on ten problems represent-ing three types of FMSs (small, medium and large). Theproposed methodology has been demonstrated bysolving an illustrative example. An intractable problemthat has been encountered in machine loading hasbeen tackled nicely by this heuristic and its perfor-mance is found to be effective and reliable on a fairlylarge number of problems.

The paper is organized as follows. In the nextsection a brief description of the problem environmentand model formulation is given. Section 3 deals withsolution methodologies. An illustration of the proposedmethodology is presented in section 4. Computationalexperiments to evaluate the performance of thealgorithm are described in section 5. The paper isconcluded with final remarks in section 6.

2. Problem environment

2.1. Problem description

Consider an FMS having four parts to be processedon two machines. To demonstrate the complexitiesassociated with the machine loading problem, con-sider each part having at least one essential and oneoptional operation that are to be processed on givenmachines. Therefore, for the above problem, the totalpossible number of part-operation-machine allocationsis (4!626262 = 192). To arrive at an optimal/nearoptimal solution for the above problem it is essentialto explore each allocation with respect to a givenobjective function by satisfying the constraints relatedto the availability of tool slots on machines. As the sizeof the problem grows, the number of possibleallocations to be explored is exponentially increased.The problem of searching optimal operation machine

U. M. B. S. Sarma et al.286

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allocation becomes more complicated when theflexibility related to routings, volumes, machines,toolings etc is also considered in the above problem.It is clear from the above facts that even a moderatesize shop floor problem will require a large computa-tional time and search space to explore all the optionsand to arrive at optimal/near optimal solutions. Aspointed out by Chen and Askin (1990), mathematical-programming-based solution methodologies to such acomputationally complex problem are unable todeliver optimal/near optimal solutions in a reasonabletime. Therefore, heuristic solutions were suggested toresolve these types of problems. Since the 1980s, a fewnovel intelligent heuristic techniques — namely TabuSearch (TS), Simulated Annealing (SA) and GeneticAlgorithm (GA) — have been extensively used byresearchers to solve computationally complex optimi-zation problems. Many successful applications of thesetechniques for obtaining optimal/near optimal solu-tions for the variety of problems are reported in theliterature. This research aims to find near optimalsolutions of part sequencing and machine loadingproblems using the tabu-search-based heuristic ap-proach. Details of the heuristic and implementation ofthe methodology will be discussed in the comingsubsections.

In order to test the performance of the proposedheuristic, authors have simulated ten test problemsrepresenting three types of FMSs, as suggested byBarash (1978) and Ito (1981). It is fairly logical toinclude five machines, to investigate the performanceof an FMS, from the publications record of the Japaneseproduction technological investigation society ‘Collec-tion of European and American FMS (1981)’, where

there is analysed data pertaining to 79 FMSs. It has beenfound that the maximum number of machines in FMSsis four to six machines. Keeping the above facts in mindand also the problem formulated by Shanker and Tzen(1985), the authors have considered the three types ofFMSs (the details are given in table 1). An illustrativeexample, consisting of seven jobs to be processed onfour machines, along with the batch size, operationnumber, machine number, unit processing time andtool slot required, is given in table 2. If the partsequencing problem is also combined with the opera-tion allocation then there exists (7!6144 = 725 760)total possible part operation machine allocations. Fordemonstrating the application of heuristics, the pro-blem given in table 2 is solved and analysed in detail.For the remainder of the nine problems, data related topart number, processing times, tool slot requirementetc, for small to large FMSs, are simulated and areavailable from the authors.

Modelling the machine loading problem 287

Table 1. Details of different FMS scenarios.

FMSType

Numberof

machines

Available timeon eachmachine

Number oftool slots on

each machine

FMS 1 4 480 min, 480 min,480 min, 480 min

5, 5, 5, 5

FMS 2 5 960 min, 960 min,960 min, 960 min,

960 min

10, 12, 10, 12and 10

FMS 3 6 960 min, 960 min,960 min, 960 min,960 min, 960 min

14, 14, 14, 14,14 and 16

Table 2. Detailed description of part types of test problem.

Partnumber

Batchsize

Operationnumber

Machinenumber

Unit processingtime Slots

Total processingtime

1 10 1 4 16 1 1602 4, 2, 3 7, 7, 7 1, 1, 1 70

2 13 1 1, 2, 3 25 1 3252 1, 2 17 1 2213 1 24 3 312

3 14 1 4, 1 26, 26 2, 2 3642 3 11 3 154

4 7 1 3 24 1 1682 4 19 1 133

5 9 1 1, 4 25 1 2252 4 25 1 2253 2 22 1 198

6 8 1 3 20 1 1607 9 1 2, 3 22, 22 2, 2 198

2 2 25 1 225

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2.2. Model formulation

2.2.1. Notationp part type p = 1,2,. . .,P,m machine m = 1,2,. . .,M,o operation o = 1,2,. . .,Op,um under-utilized time on machine m,om over-utilized time on machine m,op number of operations of part type p,T length of planning horizon (= 8 hours) in all

the problems,tm tool slots capacity on machine m,Bp batch size of part type p, p = 1,2,. . .,P,Sm set of machines on which operation o of part

type p can be performed,tropm time remaining on machine m after allocation

of operation o of part type p,taopm time available on machine m before allocation

of operation o of part type p,tcopm time required by machine m for operation o of

part type p,Tropm tool slots remaining on machine m after

allocation of operation o of part type p.Taopm tool slots remaining on machine m before

allocation of operation o of part type p.Tcopm tool slots required by machine m for operation

of part type p.Yopm 0-1 variable,Yp 0-1 variable,SU system unbalance,SUmax maximum system unbalance (= 1920 min),SUmin minimum system unbalance (= 0 min),SUseq system unbalance corresponding to a particu-

lar part sequence,SUo system unbalance for initial feasible solution,TH throughput,THmax maximum throughput,THmin minimum throughput (= 0 units),THseq throughput corresponding to a particular part

sequence,THo throughput corresponding to part sequence of

initial feasible solution,UA unassigned part types for part sequence,UAo unassigned part types for part sequences of

initial feasible solution,F objective function,S part sequence,FC best objective function from the candidate

solution,F1 objective function of current solution,Fo objective function of initial feasible solution,F(S) objective function of part sequence S,Sb best part sequence encountered so far,So part sequence of initial feasible solution,

SC best part sequence in the candidate solution,S1 part sequence of current solution,A(S) aspiration level of part sequence encountered

so far,A(S1) aspiration level of current solution,T1 tabu list,i number of iterations,imax maximum number of iterations,N(S) neighbourhoods of sequence S.

2.2.2. Formulation of objective functions and constraints.The problem described above is formulated as the bi-criterion objective, which is a combination of twoobjective functions.

(1) The first objective function is to minimize thesystem unbalance equivalently, and to maximizethe system utilization. Thus, the first objectivefunction (F1) is

Maximize

M £ T ¡XM

m¡1

…Um ‡ Om†

M £ T:

(2) The second objective function is to maximize thethroughput or, equivalently, to maximize thesystem efficiency. Thus, the second objectivefunction (F2) is

Maximise

XP

pˆ1

Bp ¢ Yp

XP

pˆ1

Bp

:

Hence, overall objective function is to maximizeF

Maximize F ˆM £ T ¡

XM

mˆ1

…Um ‡ Om†

M £ T‡

XP

pˆ1

Bp ¢ Yp

XP

pˆ1

Bp

subject to the following constraints.

(1) System unbalance. The system unbalancedeals with the ideal time remained onmachines after allocation of all feasiblepart types: the constraint can be expressedas

XM

mˆ1

…Um ‡ Om† ¶ 0

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Page 6: Modelling the machine loading problem of FMSs and its solution using a tabu-search-based heuristic

(2) Tool slots. The tool slots constraint ensuresthat the number of slots required to performthe operations of the part types on amachine should always be less than or equalto the tool slots capacity of that machine.The constraint can be expressed as

XP

pˆ1

XO

oˆ1

TcopmYopm µ tm m ˆ 1; 2; . . . M :

(3) Unique part type routing. Although flexibilityexists in the selection of a machine foroptimal operations, once a machine isselected for it, the operation must becompleted on the same machine. Theconstraint can be expressed asX

k2Sm

YpoSm µ 1 p ˆ 1; 2 . . . P o ˆ 1; 2; . . . O :

(4) Non-splitting of part type. This implies that apart type undertaken for processing is to becompleted for all its operations beforeconsidering a new part type. The constraintcan be expressed as

XOp

oˆ1

XM

mˆ1

Yopm ˆ Yp ¢ Op p ˆ 1; 2; . . . P :

(5) Integrality of decision variable. Several decisionvariables possessing the characteristics of 0-1integers, are given as follows:

Yopm = 1 if operation O of part type p isassigned on machine m0 otherwise,

Yp = 1 if part type p is selected0 otherwise

3. Solution methodologies

In this article, attempts are made to address thecomplexities of the loading problem that has two stages,which are to be executed sequentially. Here, heuristic 1is proposed to deal with the operation allocation onmachines based on fixed part ordering rules, whereasvaried part sequences are generated by perturbing thepart sequence obtained through heuristic 1. The tabu-search-based heuristic (heuristic 2) is then used tofurther improve the solution. Section 3.1 deals withheuristic 1 whereas section 3.2 gives an overview of thetabu search. Finally, in section 3.3, heuristic 2 isexplained at length.

3.1. Description of heuristic 1

The main objective of heuristic 1 is to minimize thesystem unbalance and thereby maximize the systemthroughput by satisfying the technological constraints.This heuristic works based on the fixed part orderingrule in the presence of optional operations for parttypes. Optional operations means an operation that canbe carried out on more than one machine. It is alwayspreferable to allocate the optional operations on thosemachines that have the maximum remaining proces-sing times. (The heuristic is explained via the flowchartshown in figure 1.)

3.2. Overview of tabu search technique

The tabu-search-based heuristic is regarded as a‘higher level’ iterative improvement/procedure that isused for solving various computationally complexproblems. Many successful implementations of tabusearch, for obtaining optimal or near optimal solutionof problems pertaining to process planning, scheduling,set partitioning and loading etc, are reported in theliterature (Glover 1990, Glover et al. 1993, Taillerd1990, Aljaber et al. 1997, McMullen 1998).

It begins with an initial feasible solution andattempts to find a better solution by investigationamong a large pool of neighbourhood solutions. Thismethod is characterized by its inherent simplicity, highadaptability, a short term memory process via a tabulist, to escape local optima and avoid cycling. Theprocess also allows backtracking to previous solutions,which may lead to a better solution, known asaspiration. These features of a tabu list and aspirationmake tabu search a powerful optimization tool forsolving the machine loading problem. If, for instance,the search is entrapped in a local optima (maximumobjective function in this case), then the best move ofthe next iteration will have a lower value of objectivefunction. In such a case, the previous solution shouldbe deleted from the search domain, so that thechances of the algorithm returning to the samesolution and identifying it as the best solution areminimized. An aspiration criterion is a checkingcondition for the acceptance of a solution. In general,the application of the tabu search technique can becharacterized by:

(1) Generation of an initial feasible solution,(2) Neighbourhood generation,(3) Tabu list size,(4) Aspiration level,(5) Stopping criterion.

Modelling the machine loading problem 289

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3.3. Description of heuristic 2

In heuristic 1, we considered the pre-determinedfixed part sequence and then addressed the operationallocations on machines to satisfy the given objectivesand constraints. Obviously, this attempt is aimed atsolving first the sequencing problem then the operationallocation. In order to ensure optimal/near optimalsolutions of machine loading problems of random FMS,these issues are need to be simultaneously addressed.Encompassing the above features is the main motivebehind the development of the tabu-search-basedheuristic, i.e. heuristic 2 (see figure 2). The tabu-search-based heuristic is discussed in detail in latersubsections.

3.3.1. Initial feasible solution. The initial feasible solu-tion is obtained from heuristic 1. In this research, the‘shortest processing time’ (SPT) based part sequencingrule has initially been considered to resolve the partordering/part input sequencing problems. According

to Shanker and Tzen (1985), SPT as a dispatching rulefor part types performs better on average for theloading problem of a random FMS as far as theobjective of balancing the workload is concerned. TheSPT as a part sequencing rule attempts to maximize thethroughput in comparison to LIFO, FIFO, LPT etc. Inlight of the above facts, the initial part sequencing ishere considered according to the SPT sequencing rule.

3.3.2. Generation of neighbourhood. Generation ofneighbourhood is carried out by the following steps.

Step 1. For a given part sequence enlist the numberof unassigned part types.

Step 2. Interchange the unassigned part typesamong the assigned types in the givensequence only.

This procedure is illustrated with an example.Let us assume that S is a feasible part sequence, and

UA is a set of unassigned part types.

U. M. B. S. Sarma et al.290

Figure 1. Flowchart representing heuristic 1.

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Consider a problem having seven jobs. Say, for aspecific case, with a given number of operations andother details, the part sequence and unassigned parttypes are

S = {6,1,4,7,3,5,2}UA = {2,5}

Then the neighbourhood generated is illustrated intable 3.

3.3.3. Tabu list size. The size of tabu list assumed inthis article is three. A lower value of tabu list size willresult in insufficient memory for matching of thecurrent best candidate solution with those of previousiterations. A higher tabu list size did not improve the

quality of the result, hence the tabu list size of three. Inthe beginning, the tabu list is a null set, which meansthat the tabu list does not have any sequence. In theproposed heuristic, revision of the tabu list is carriedout after every iteration. After the first iteration, thebest candidate solution is included in the tabu list at thefirst place from the left-hand side. Thereafter, for eachsuccessive iteration, the best solution of the previousiteration is shifted one place towards the right-handside of the list and the first place is occupied by thecurrent best solution of the iteration just carried out.

3.3.4. Aspiration level. The aspiration criterion is achecking condition for the acceptance of a solution. Inthis paper, the best optimal value (maximum value) ofthe objective function is taken out as the aspiration level

Modelling the machine loading problem 291

Figure 2. Flowchart illustrating heuristic 2 (Tabu Search based heuristic).

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to check the aspiration level criterion. As the objectivefunction is a function of throughput and systemutilization, its maximization is chosen as the aspirationcriterion. Hence, if the objective function value of thecurrent solution is greater than the aspiration level, thetabu status of the solution is replaced and this solutionis considered as the current best solution.

3.3.5. Stopping criterion. The iteration procedure goeson until one of the following conditions is satisfied.

(1) The total number of iterations completed sincethe start of algorithm have been elapsed. As aresult of exhaustive computational experiments,it has been found that optimal/near optimalsolutions are achieved well within the fourthiteration. Therefore the maximum number ofiterations considered in this article is four only.

(2) A certain number of iterations since the lastcurrent best sequence was found has occurred.

4. Illustrative example

For numerical illustration of the proposed algo-rithm, the following are the several iterations requiredfor solving the problem given in the table 2. Nine moreproblems have been solved to assess the computationalperformance of the proposed heuristic.

Consider an initial feasible solution

So= {6,1,4,7,3,5,2}SUo= 288THo= 48Fo= 725UAo= {2,5}

The above results are obtained after applyingheuristic 1.

Tabu list = {f,f,f}i = 0Sb = SC= SO= {6,1,4,7,3,5,2}Fb = FC= FO= 0.7250

First iterationPlace S0 in the tabu list.Tabu list = {S0,f,f}The neighbourhood of S0 is generated as follows.

Candidate solution Systemunbalance

Throughput Value of objectivefunction

Part typesunassigned

S1= {2,1,4,7,3,5,6} 108 39 0.7155 3,5,6

S2= {6,2,4,7,3,5,1} 178 37 0.6848 3,5,7

S3= {6,1,2,7,3,5,4} 299 40 0.6851 3,5,9

S4= {6,1,4,2,3,5,7} 371 38 0.6408 3,5,7

S5= {6,1,4,7,2,5,3} 158 43 0.7276 2,3

S6= {5,1,4,7,3,6,2} 158 43 0.7276 2,3

S7= {6,5,4,7,3,1,2} 158 43 0.7276 2,3

S8= {5,1,4,7,3,6,2} 158 43 0.7276 2,3

S9= {6,1,4,5,3,7,2} 63 48 0.7835 2,7

S10= {6,1,4,7,5,3,2} 158 43 0.7276 2,3

The best candidate solution (having maximum value ofobjective function) from the generated neighbourhoodis with sequence S9. Since S9 is not in the tabu list, themove from S0 to S9 is therefore accepted.

Second iterationPlace S9 in the tabu list.Tabu list = {S9, S0, f}The neighbourhood generated is as follows.

Candidate solution Systemunbalance

Throughput Value of objectivefunction

Part typesunassigned

S11= {2,1,4,5,3,7,6} 13 44 0.7716 5,6,7

S12= {6,2,4,5,3,7,1} 83 42 0.7408 1,5,7

S13= {6,1,2,5,3,7,4} 24 40 0.7437 3,4,7

S14= {6,1,4,2,3,7,5} 371 38 0.6408 1,3,7

S15= {6,1,4,5,2,7,3} 158 43 0.7276 1,3,7

S16= {7,1,4,5,3,6,2} 158 43 0.7276 1,3,7

S17= {6,7,4,5,3,1,2} 158 43 0.7276 1,3,7

S18= {6,1,7,5,3,4,2} 158 43 0.7276 1,3,7

S19= {6,1,4,7,3,5,2} 288 48 0.7250 2,5

S20= {6,1,4,7,5,3,2} 158 43 0.7276 2,3

The best candidate solution from the neighbourhood isfound in sequence S11. Since S11 is not in the tabu list,the move from S9 to S11 is therefore considered.

U. M. B. S. Sarma et al.292

Table 3. Methodology of neighbourhood generation.

Part type sequence (initial) Unassigned part types Sequence of neighbourhood Explanation

(6, 1, 4, 7, 3, 5, 2) (2, 5) (2, 1, 4, 7, 3, 5, 6) Replace job 6 with job 2(6, 1, 4, 7, 3, 5, 2) (2, 5) (6, 2, 4, 7, 3, 5, 1) Replace job 1 with job 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Third iterationPlace S11 in the tabu list.Tabu list = {S11, S9, S0}The neighbourhood generated is as follows.

Candidate solution Systemunbalance

Throughput Value of objectivefunction

Part typesunassigned

S21 = {5,1,4,2,3,7,6} 233 40 0.6919 2,6,7

S22 = {2,5,4,1,3,7,6} 113 24 0.6518 1,3,6

S23 = {2,1,5,4,3,7,6} 24 40 0.7437 3,4,7

S24 = {2,1,4,3,5,7,6} 13 44 0.7716 5,6,7

S25 = {7,1,4,5,3,2,6} 158 43 0.7276 2,3

S26 = {2,7,4,5,3,1,6} 108 39 0.7156 3,5,6

S27 = {2,1,7,5,3,4,6} 108 39 0.7156 3,5,6

S28 = {2,1,4,5,7,3,6} 108 39 0.7156 3,5,6

S29 = {6,1,4,5,3,7,2} 63 48 0.7835 2,7

S30 = {2,6,4,5,3,7,1} 83 45 0.7408 1,5,7

S31 = {2,1,6,5,3,7,4} 24 40 0.7437 3,4,7

S32 = {2,1,4,5,6,7,3} 371 38 0.6440 3,5,7

The best candidate solution from the above generatedneighbourhood is with sequence S29. Since S9= S29, themove from S11 to S29 is rejected, and there is also noother sequence in the above neighbourhood with a bestcandidate solution. From the above results, it is notpossible to consider any other move. The stoppingcriteria mentioned in subsection 3.3.5 (ii) are satisfied.Sb = S9= S29 is found to be the best solution.

5. Computational experience

The performance of the heuristics is tested accord-ing to an objective function, which is the combinationof the minimization of system unbalance and max-imization of throughput. Shanker and Srinivasulu(1989), Mukhopadhyay et al. (1992), Tiwari et al.(1997) have solved similar types of machine loadingproblems. As a first step, the SPT-based part sequencingrule has been adopted by them, and then the heuristicprocedure was implemented to address the operationallocations on machines. Attempts are made in thisarticle to do away with the part sequencing rules, whichwere extensively referred to by the earlier researchers.

The authors have found that the proposed tabu-search-based heuristic solution methodology is char-acterized by its inherent simplicity and high adapt-ability and short-term memory process. Whileperforming the computational experiments with theproposed tabu-search-based heuristic, the followingparameters were chosen to solve the underlinedloading problems. These are: size of tabu list = 3,maximum iteration = 4 and perturbation policy, inwhich ‘each assigned part is replaced by an unassigned

one in a sequential manner’ is adopted to generate theneighbourhood solution.

In this research, exhaustive experiments have beencarried out to assess the solution quality by changingthe size of the tabu list and also the maximum numberof iterations. It is observed that the solution gets frozenat the earlier stages, even if the number of maximumiterations is increased. Also, the average computationaltime remains fairly unchanged as the size of the tabu listgrows. In this article, the proposed heuristic attempts tooptimize an objective function, which is a combinationof system unbalance and throughput. It is worthobserving that the majority of earlier researcherstreated system unbalance and throughput separatelyas their objective functions.

This research also considers questions pertaining tooverloading of machines. In case the overloading ofmachines is permitted, the optimum value of systemunbalance should be maintained within a thresholdlimit. Such an effort will lead to maintaining flexibilityof the system. In this context it is important to mentionthat earlier researchers, such as Mukhopadhyay et al.(1992) considered unutilized machines and over-utilized machine time in their system unbalancewhereas, Shanker and Srinivasulu (1989) have consid-ered only unutilized time in their system unbalance.Mukhopadhyay et al. (1992) have taken the positivevalue of the sum of the remaining machine time afterallocating the parts in sequence. They considered thenext part from the set of unassigned parts only whenthe modular value of system unbalance starts increasingwith regards to the previous values, even though theactual system unbalance is not positive. In this research,all these questions have been examined carefully andoverloading of machines is taken into considerationwhile determining the system unbalance. In order tomaintain the system flexibility, the proposed heuristictakes into account the allocation of parts from the set ofunassigned parts until a positive value of systemunbalance is found.

Some of the notable features of the proposedheuristics are:

(1) Various combinations of part sequences are tobe evolved by perturbing initial sequences.

(2) A new sequence is generated and the corre-sponding operation allocation is made until theoptimal/near optimal results are achieved.

(3) The effectiveness of the algorithm has beenjudged by the following performance measures,namely system utilization and system efficiency,

where system utilization …%†ˆSUmax¡SUseq

SUmax¡SUmin£100;

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Page 11: Modelling the machine loading problem of FMSs and its solution using a tabu-search-based heuristic

where system efficiency …%†ˆTHmax¡THseq

THmax¡THmin£100:

The efficient performance of the proposed heuristiccan be verified from table 4, where it has been appliedto solve the loading problem of small, medium andlarge FMSs. The proposed tabu search heuristic hasbeen coded in C language and the program was run onIBM PC with Pentium CPU at 133 MHz. The results andanalysis of ten problems consisting of small, mediumand large FMSs are available from M. K. Tiwari uponrequest.

6. Conclusions

This research is mainly focused on developing aconsistent and efficient heuristic for solving themachine loading problem in FMSs. For solving theFMS loading problem, previous studies have consideredpart sequencing and operation allocation problemsseparately. The objectives of the loading problemconsidered in this research are minimization of systemunbalance and maximization of throughput, whereasthe system constraints are maximum available time andtool slots on each machine.

Exhaustive computations have been carried out toassess the effectiveness of the proposed tabu-search-based heuristic. The efficacy of the proposed heuristichas been tested using SPT as a fixed part sequencingrule and the results obtained have been evaluated interms of system efficiency and system unbalance. Fromthe above discussions, it has been observed thatperformance of the proposed heuristics offers consis-tently better results for most of the problems related todifferent types of FMSs.

The application of the proposed heuristics isrestricted to certain cases where there is a fairly largenumber of fixture, pallets and AGVs available on the

shop floor. Further research can be carried out byconsidering a few more objective functions, namelyminimization of path movements, tool changeovers, setup changeovers etc.

Acknowledgements

M. K. Tiwari wishes to offer his most sincere thanks toProfessor S. T. Newman and the anonymous refereesfor their constructive comments, which led to theconsiderable improvement in the quality of the article.The authors also wish to acknowledge the assistanceprovided by prominent researchers such as Professor K.E. Stecke (Michigan, USA), Professor S. P. Dutta(Windsor, Canada), Professor Y. Narahari (IISc, Banga-lore, India), Professor H. Ohta (Osaka, Japan),Professor T. J. Sawik (Cracow, Poland), ProfessorVenkat Allada (University of Missouri-Rolla, USA),Professor H. C. Zhang, (University of Texas Tech.,USA) in the form of sending their valuable reprints. Wealso express our sincere thanks to Mr S. Kameshwaran(PhD candidate at IISc, Bangalore, India), Mr DebjitRoy (Graduate Student at NIFFT, India), and Dr RaviShankar (IIT, Delhi) for fruitful discussions andcomputational support related to this research.

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