using penalty function and tabu search to solve cell formation problems with fixed cell cost

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Computers & Operations Research 31 (2004) 21–37www.elsevier.com/locate/dsw

Using penalty function and Tabu search to solve cellformation problems with 'xed cell cost

Dong Cao, Mingyuan Chen∗

Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd.,West Montreal, Quebec, Canada H3G 1M8

Received 1 December 2001; received in revised form 1 June 2002

Abstract

In this paper, an integrated approach for manufacturing cell formation with 'xed charge cost is presented.The solution of the problem includes not only cell formation decisions but also cell set-up decisions. Amixed integer non-linear programming model is formulated to solve the problem. The NP-hardness of theproblem makes direct solution computationally prohibitive for real-life applications. A heursitic algorithm wasdeveloped to solve the problem e4ciently based on the features of the model and model duality analysis.Tabu search was used to 'nd the optimal or sub-optimal solutions of the problem. Numerical examples arepresented.

Scope and purpose

Today’s manufacturing industry is facing strong competition in providing high quality and low cost prod-ucts to ever demanding consumers. Cellular manufacturing is one of the widely used approaches to improvemanufacturing productivity to achieve this purpose. In cellular manufacturing, machine tools are grouped intodi7erent manufacturing cells for improved e4ciency and 8exibility. This research proposes a mathematicalprogramming model to form the cells so that material handling cost, machine tool cost and eventually the over-all production cost can be minimized. We also developed an e4cient solution method to solve the complicatedmathematical model.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Cell formation; Fixed charge cost; Integer programming; Tabu search

∗ Corresponding author. Tel.: +1-514-848-3131; fax: +1-514-848-3175.E-mail address: [email protected] (M. Chen).

0305-0548/04/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.PII: S0305-0548(02)00144-2

mailto:[email protected]

22 D. Cao, M. Chen / Computers & Operations Research 31 (2004) 21–37

1. Introduction

Cellular manufacturing (CM) is a widely studied approach for organizing machines and people intogroups to produce a variety of parts in part families. CM is also an e7ective approach to implement8exible manufacturing systems (FMS) and is normally associated with automated batch production[1]. Successful implementation of CM will result in reduced set-up times, reduced material 8owand in-process inventory, better system management, improved production e4ciency and productquality [2,3]. In the last 30 years, various CM problems have been studied by many researchersas summarized in [4,5]. More recently, CM research has been expanded to developing integratedmodels and methods as discussed in [6,7]. For example, Atmani et al. [8] presented a model forsimultaneous cell formation and operation allocation. Lockwood et al. [9] studied CM schedulingproblems. Production planning in CM systems was discussed in [10,11]. A variety of e7ective meth-ods were developed to investigate and solve CM problems. These include traditional mathematicalprogramming [12] and simulation studies [13,14]. Non-traditional methods such as neural networks,Tabu search, genetic search and searches using simulated annealing have also been used to searchfor near optimal solutions for various CM system problems. Recent work can be found in [15–18].Some of the methods were compared in [19] for their e7ectiveness and e4ciency in solving CMproblems. In the research presented in this paper, we also followed the integrated approach to studya cellular manufacturing problem with 'xed charge cost. In most existing cell formation models, thenumber of manufacturing cells has been given and the cells have been identi'ed. The solutions ofthe problems were then to allocate proper machines to the cells and to decide the part families tobe processed. Objectives of such models are generally to minimize certain criteria such as materialhandling cost or dis-similarities among part families. The integer programming model developed inthis paper considers a more general situation where the number of cells is to be determined by thesolution of the model. The model is similar to that of a 'xed charge problem [20] with decisionvariables for cell formation. The objective of the model is to minimize the total cost includinginter-cell material handling cost, 'xed cell set-up cost and 'xed machine and operating costs in thesystem. Detailed description of the problem and the development of the integer programming modelare given in the next section. Due to the NP-hardness of the problem we employed an embeddedoptimization technique to transform the original model to a parametric LP problem following theapproach developed in [21]. The parametric model was then transformed into a uni'ed computablepenalty problem. This solution procedure and model transformations are presented in Section 3.We used a Tabu search [22] procedure to 'nd optimal or near-optimal solutions of the transformedproblem. As shown in recent literature, Tabu search is an e7ective search process to quickly arrivein optimal or near-optimal solutions of large-scale combinatorial optimization problems. Details ofapplying the Tabu search process to solve the transformed cell formation model are presented anddiscussed in Section 4. We tested the developed methods using several numerical examples. Dataand computational results of two examples are presented in Section 5. In Section 6, we present thesummary, conclusions and our plans for future research in this area.

2. Model description

Consider a manufacturing system consisting of di7erent machines to process various types of parts.Each part-type may require some or all of the machines for processing. The manufacturing system

D. Cao, M. Chen / Computers & Operations Research 31 (2004) 21–37 23

follows a CM approach. The machines are formed in several manufacturing cells. The cell or cellsto be constructed will be determined from the problem solution, depending on the 'xed costs ofcell construction, cell capacities and other related factors. The types and units of machines to beallocated in the constructed cells are also to be decided from the problem solution. In this section,we present a non-linear mixed integer programming model to solve this cell formation problemwith 'xed cell set-up cost. The objective function of this model is to minimize inter-cell materialhandling cost, cell set-up cost and machine operating cost. In formulating the problem, we assumethat there is a single process plan for each part-type. Part processing cost, therefore, is not includedin the objective function. The model is formulated with the assumption that the distances betweenthe cells can be di7erent. This assumption is more general, resulting in non-linear material handlingcost in the objective function. Other cost and constraint terms in the model are linear. Minimizingthe given cost function is subject to known manufacturing process requirements. We assume thatmachine capacities are limited so that multiple units of machines may be required in a same cell.Again, this assumption is more realistic but makes the model more di4cult to solve. Before thedetails of the model are presented, we 'rst give the notations.

Known parameters and coe5cients

i—part-type index, i = 1; : : : ; I ,j—index of operations of part-type i; j = 1; : : : ; Ji,k—machine index, k = 1; : : : ; K ,c—cell index, c = 1; : : : ; C,Pi—amount of part type i to be processed,dlm—distance between cell l and cell m; l; m= 1; : : : ; C; l �= m,Sc—set-up and operating cost of cell c; c = 1; : : : ; C,Ti( j; k)—capacity (in number of machines) requirement to process operation j of part-type i bytype k machine, i = 1; : : : ; I; j = 1; : : : ; Ji,Wk—unit cost of type k machine, k = 1; : : : ; K ,Uc—number of machines allowed in cell c; c = 1; : : : ; C,Nk—number of type k machines available in the system k = 1; : : : ; K .

Decision variables

nkc—number of type k machines in cell c,

xi( j; k)c =

{1 if operation j of part-type i by type k machine will be processed in cell c;

0 otherwise;

yic =

{1 if part-type i will be processed in cell c;

0 otherwise;

zc =

{1 if cell c will be formed;

0 otherwise:


The parameter Ti( j; k) indicates the capacity required for type k machine to process operation j ofpart-type i. Subscripts j and k are related by given process plan and are not independent to eachother. Using the above de'ned variables and parameters, the mathematical model for cell formationwith 'xed charge cost can be expressed as below:

MP minM (x; y; z; n) =I∑i=1

C∑l=1

C∑m=1; �=l

12Rilmyilyim +

C∑c=1

Sczc +K∑k=1

Wk

C∑c=1

nkc (1)

s:t:C∑c=1

xi( j; k)c = 1; j = 1; : : : ; Ji; i = 1; : : : ; I; (2)

xi( j; k)c6yic; j = 1; : : : ; Ji; i = 1; : : : ; I; c = 1; : : : ; C; (3)

yic6 zc; i = 1; : : : ; I; c = 1; : : : ; C; (4)

I∑i=1

Ti( j; k)xi( j; k)c6 nkc; k = 1; : : : ; K; c = 1; : : : ; C; (5)

K∑k=1

nkc6Uczc; c = 1; : : : ; C; (6)

C∑c=1

nkc6Nk; k = 1; : : : ; K; (7)

xi( j; k)c; yic; zc = 0; 1; nkc¿ 0; are integer variables; ∀i; j; c: (8)

The objective function, Eq. (1), is to minimize inter-cell material handling cost, 'xed cell cost andmachine cost. The 'rst term of this cost function is the inter-cell material handling cost to moveparts between cells where Rilm = Pi × dlm. The second term is the 'xed charge cost for setting upmanufacturing cells. The third term is the cost of purchasing and operating the required machinesin the system. This model is more general than other similar models where the number of cellsin the system were pre-determined. The 'rst constraint in the model, Eq. (2), enforces that anyoperation of a part will be processed in only one of the cells. Eq. (3) presents the relationshipbetween operations and parts. Eq. (4) ensures that if a part is processed in a certain cell, thenthat cell must be constructed. Eq. (5) presents machine requirements for each machine type basedon the capacity requirement by each operation of the parts. Eq. (6) gives the maximum numberof machines to be placed in a cell should it be built. Eq. (7) enforces that the total number ofunits for each machine type will not exceed the available number of machines. Eq. (8) is integerrequirement for the decision variables. Model MP is a mixed integer programming model similar to'xed charge problem models [20] with additional decision variables for cell formation. Solving suchmodels for real life problems by a brute-force algorithm such as branch and bound method may becomputationally prohibitive due to NP-hardness [20]. Computational di4culties are mainly caused bylarge number of integer variables and complicating constraint functions in Eqs. (5)–(7). All of themare related to integer decision variables nkc. Unfortunately, in solving the above discussed problem,nkc cannot be approximated by continuous variables. The numbers of di7erent types of machines in


each cell are relatively small. Rounding o7 continuous variables cannot yield meaningful approximatesolutions to the original problem. In the next section, we present a heuristic algorithm to search foroptimal solution of model MP.

3. Solution method

In this section, we present a functional variation formulation for model MP to reduce its com-plexity. We then apply a Tabu search procedure to search for optimal or near-optimal solutions ofthe problem.

3.1. Functional variation and linear programming variation

Let (x; y; z) = {xi( j; k)c; yic; zc} be a partial solution to problem MP and consider the followingsub-problem SP:

SP min SM (x; y; z) =K∑k=1

Wk

C∑c=1

nkc

s:t: (5)–(7) and nkc¿ 0; are integer variables; ∀k; c:We observe that solving MP is equivalent to solving the following problem:

FP min M (x; y; z) =I∑i=1

C∑l=1

C∑m=1; �=l

12Rilmyilyim +

C∑c=1

Sczc + SM (x; y; z)

s:t: (2)–(4) and xi( j; k)c; yic; zc = 0; 1;

where SM (x; y; z) is the objective function value of sub-problem SP for a given solution (x; y; z).Obviously, the objective function value SM (x; y; z) is an implicit function of (x; y; z). Notice thatif (x; y; z) is infeasible to MP, then the corresponding sub-problem SP has no feasible solution. Forsub-problem SP, we have the following result.

For a given solution (x; y; z) to problem SM, solving sub-problem SP is equivalent to solving thefollowing modi'ed sub-problem:

SP′ min SM (x; y; z) =K∑k=1

Wk

C∑c=1

nkc

s:t:

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉6 nkc; k = 1; : : : ; K; c = 1; : : : ; C;

K∑k=1

nkc6Uczc; c = 1; : : : ; C;


C∑c=1

nkc6Nk; k = 1; : : : ; K;

nkc¿ 0; are integer variables; ∀k; c;where in the 'rst constraint of SP′; �∗� is the smallest integer greater than or equal to the realnumber ∗. This conclusion can be veri'ed directly by noticing the integral requirement of nkc.Sub-problem SP′ is an integer programming problem. However, SP′ can be solved as a linear pro-gramming problem as shown below.

Theorem. Sub-problem SP′ is a transportation problem with integral demand and supply. Integersolutions can be obtained when it is solved as a linear programming problem.

Proof. Let n′kc = nkc − �∑Ii=1 Ti( j; k)xi( j; k)c�. Since �∑I

i=1 Ti( j; k)xi( j; k)c� are integers, n′kc¿ 0, ∀k; c,are also integers. Problem SP′ can then be transformed into:

SP′′ min SM (x; y; z) =K∑k=1

Wk

C∑c=1

n′kc +K∑k=1

Wk

C∑c=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉

s:t:K∑k=1

n′kc6Uczc −K∑k=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉; c = 1; : : : ; C;

C∑c=1

n′kc6N ′k = Nk −

C∑c=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉; k = 1; : : : ; K;

n′kc¿ 0; are integer variables; ∀k; c:It is a standard transportation problem with integer requirements on supply and demand. Optimalsolutions will all be integers if the problem is solved as a linear programming problem [23].

We now conclude that solving the original problem MP is equivalent to solving problem FP withthe implicit function SM (x; y; z) de'ned by sub-problem SP′′, where SP′′ can be solved as a linearprogramming problem.

3.2. Dual sub-problem and penalty formulation

In the above discussion, we have not explicitly addressed the infeasibility issue in solving sub-problem SP′′ for a given partial solution (x; y; z). Now consider SPD′′, the dual problem of SP′′:

SPD′′ max SDM (x; y; z) =C∑c=1

�c

(Uczc −

K∑k=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉)

+K∑k=1

�k

(Nk −

C∑c=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉)


+K∑k=1

Wk

C∑c=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉

s:t: �c + �k6Wk; c = 1; :::C; k = 1; : : : ; K;

�c; �k6 0:

From LP duality theory, we have

• For any k; c, if

Uczc −K∑k=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉¿ 0 and N ′

k¿ 0;

then

SDM (x; y; z) = SM (x; y; z) =K∑k=1

Wk

C∑c=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉: (9)

• For any k; c; if

Uczc −K∑k=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉¡ 0 or N ′

k ¡ 0

then

SDM (x; y; z) = +∞: (10)

If the original cell formation problem MP has feasible solutions, then a su4ciently large lower bound−� can be imposed on the dual variables �c and �k in SPD′′. The objective function of sub-problemSDM (x; y; z;�) can be expressed by

• If Uczc −∑K

k=1

⌈∑Ii=1 Ti( j; k)xi( j; k)c

⌉¿ 0 and N ′

k¿ 0, then

SDM (x; y; z;�) = SM (x; y; z) =K∑k=1

Wk

C∑c=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉: (11)

• If Uczc −∑K

k=1

⌈∑Ii=1 Ti( j; k)xi( j; k)c

⌉¡ 0 or N ′

k ¡ 0, then

SDM (x; y; z;�) =−�∑k⊂K−

N ′k −�

∑c⊂C−

(Uczc −

K∑k=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉)

+K∑k=1

Wk

C∑c=1

⌈I∑i=1

Ti( j; k)xi( j; k)c

⌉; (12)

where in Eq. (12) C− and K− are the index subsets whose values are negative. The relationshipgiven in Eqs. (11) and (12) indicates that penalty will be imposed if (x; y; z) is not feasible. Thefollowing lemma gives the relationship between problem FP and the penalty formulation.


Lemma. If problem FP has feasible solutions, there exists a positive number �∗, such that for�¿�∗, solving FP is equivalent to solving:

FPD min M (x; y; z) =I∑i=1

C∑l=1

C∑m=1;�=l

12Rilmyilyim +

C∑c=1

Sczc + SDM (x; y; z;�)

s:t: (2); (3); (4) and xi( j; k)c; yic; zc = 0; 1: (13)

In Eq. (13), SDM (x; y; z;�) is de:ned by Eqs. (11) and (12).

Proof. The proof of this lemma is based on the fact that the admissible solution region consists oflimited number of alternatives. The penalty used in such a bounded admissible region must be anexact penalty, i.e., there exists a positive number �∗ such that for �¿�∗, problems FP and FPDhave identical solutions [21].

3.3. A uni:ed computable penalty problem

Problem FPD provides a computable formulation for the cell formation problem MP discussedin Section 2. Before the detailed solution procedure is presented, we can remove the dependentdecision variables in the problem formulation to make the search procedure more e4cient. A carefulobservation on model MP reveals that among the sets of decision variables (x; y; z), only x areindependent. Since Rilm and Sc in MP are all non-negative, (y; z) are uniquely determined by x asfollows:

TY (X ) : yic =max(xi( j; k)c;∀(j; k)); (14)

TZ(YX ) : zc =max(yic;∀i): (15)

The above equations state that if operation j of part i is to be processed in cell c, then part i mustbe processed in cell c, that is, yic=1; otherwise, yic=0; Similarly, if part i is to be processed in cellc, then cell c must be composed, that is, zc = 1; otherwise, zc = 0. FPD can hence be reduced to

FPD′ min F(x; y(x); z(x)) =I∑i=1

C∑l=1

C∑m=1;�=l

12Rilmyilyim +

C∑c=1

Sczc + SDM (x; y; z;�)

s:t:C∑c=1

xi( j; k)c = 1; j = 1; : : : ; Ji; i = 1; : : : ; I;

where yic and zc are de'ned in Eqs. (14) and (15), respectively. Due to the simpler structure ofFPD′, we use a Tabu search procedure to search for its optimal or near optimal solutions. Detailsof the Tabu search process are presented next.

4. Tabu search for cell formation

Since FPD′ is a reformulation of the original problem FPD, it does not change the NP-hard natureof the problem. Traditional optimization method cannot generate optimal solutions e4ciently for


larger size problems. In this section, we present a short-term Tabu search method to solve problemFPD′ for 'nding the optimal or near optimal solutions of FPD′, leading to solutions of the originalcell formation problem.

4.1. General procedure

Tabu search is a meta-heuristic developed by Glover [24–26] to 'nd optimal or near-optimalsolutions for various combinatorial optimization problems. Tabu search has been successfully usedto solve many manufacturing system analysis problems including cell formation problems [18]. Oneof the di4culties in solving such problems is the existence of local optima. In short-term Tabusearch, the search process attempts to avoid local optimum by further exploring the neighborhoodof any local optimal solution. It does not stop at a local solution xl but moves to the best possiblesolution xp in the neighborhood. If the neighborhood is too large to be explored e4ciently, asub-neighborhood, will be searched. The neighborhood or a sub-neighborhood of a local solutioncontains the search history. Certain attributes of recently visited solutions are classi'ed as “Tabu”.Future moves containing these elements are excluded. The Tabu mechanism prevents moves leadingback to these solutions. On the other hand, Tabu active status (Tabu list) of a move may bemodi'ed if certain aspiration criterion is satis'ed. This should lead the search process to reach a“good” solution. Before the step-by-step procedure of the short-term Tabu search is given, severalconcepts related to the search process are presented and explained below.

• Feasible solution: A feasible solution x = (xi( j; k)c), and hence the corresponding (yic) and (zc),satisfying all constraints of problem FPD′, where yic and zc are functions of xi( j; k)c as de'ned inEqs. (14) and (15), respectively.

• Swap operation: A swap operation is to change the value of a single component xi( j; k)c of asolution x. xi( j; k)c = 1 indicates that operation j of part i is executed in cell c. A swap operationis to make xi( j; k)c = 0 and xi( j; k)c′ = 1; c′ �= c, with all other decision variables unchanged. Letthe new solution generated after the swap operation be x′. The move from x to x′, expressed byx→ x′, is called a swap move. In solving our cell formation problem, such moves correspond tothe cell changes that operation j of part i is executed.

• Solution neighborhood: A neighborhood V (x) of a solution x= xi( j; k)c is de'ned by the ensembleof all swap moves.

• Aspiration criterion: The objective function value F(x) in model FPD′ is used to measure theaspiration level of the Tabu search.

• Stopping criterion: The Tabu search procedure stops if a predetermined number of iterationshas been reached; or the solution has not been improved after a certain number of consecutiveiterations [26]. These numbers are normally decided before the search started with reference togiven Tabu list size and problem size.

In the Tabu search process to solve our cell formation problem, the entire neighborhood of a localsolution will be searched. It is based on the consideration that the whole neighborhood of a solutionconsists of manageable number of feasible solutions. For example, if the cell formation problemhas 20 part types each having 've operations to be processed in four possible cells, then the wholeneighborhood of any solution contains 300 solutions. With today’s common computing platforms


such as a PC, it will take insigni'cant computing time and computer memory to complete the wholeneighborhood search. With the above introduced concepts and explanations, the short-term Tabusearch procedure can be described as follows [27]:

1. Initialization: Select an initial solution x1 ∈X, where X is the feasible solution region. Initializethe best value F∗ of F and the corresponding solution x∗ by letting:F(x1) → F∗x1 → x∗Tabu list TL is set empty.

2. Iterative search: Let x& denote the current solution, for step & = 1; 2; : : : : QF is used to storethe best accessible value of F found in exploring V (x&), the neighborhood of x&. Let Qx be thesolution in V (x&) for which F( Qx) = QF . QF is initially set to ∞.For all x∈V (x&), if F ¡ QF and if the move (x& → x) is not Tabu, or the move is Tabu butcorresponding F passes the aspiration criterion, then letF(x) → QF and x→ Qx.Let Qx→ x&+1.If QF ¡F∗, then Qx→ x∗ and QF → F∗.A move is established: x& → x&+1, with the modi'cation of the associated Tabu list information.

3. End: If the stopping criterion is reached, stop.

The above algorithm was coded in FORTRAN-77 and executed on a PC computer with PentiumIII-600 processor. Several numerical examples were developed to test this algorithm. Details of tworepresentative example problems are presented next.

5. Numerical examples

The above developed Tabu search method has been applied to solve more than 20 cell formationsample problems of di7erent sizes. Since the behavior of the algorithm in solving those problemsare similar, we present two most illustrative examples in this section. The 'rst one is of smaller sizeso that we are able to compare the solution from the developed algorithm and the optimal solutiongenerated by LINDO, an integer programming software [28]. The second example is much larger thanthe 'rst one. The heuristic algorithm is able to converge very quickly to 'nd a heuristic solution. It isimpossible to 'nd the optimal solution for problem of this size using integer programming softwaresuch as LINDO within tolerable computing time. The heuristic algorithm found sub-optimal solutionswith insigni'cant time of computation. These two examples show that the developed heuristic methodis able to generate optimal or near optimal solutions for small and large problems and is applicablein solving real-life industrial problems. In solving the two example problems, the initial solutionswere infeasible solutions to the original problem. The algorithm is able to 'nd feasible solutionsafter several iterations.

5.1. Example 1

The 'rst example considers 've machine types allocated to a maximum of four possible cells.Multiple units of machines can be allocated to the cells. These machines will process 've di7erent


Table 1Machine part operation data (Example 1)

Machine Unit machine Max Part numbernumber cost units

1 2 3 4 5

1 10 40 4.0 4.4 3.32 20 60 4.3 3.32 4.4 2.53 30 50 5.5 3.3 3.3 2.54 40 40 5.55 50 60 8.9 4.45 4.4

Table 2Cell data (Example 1)

Cell number 1 2 3 4

1 0.0 1.0 2.0 3.02 1.0 0.0 1.0 4.03 2.0 1.0 0.0 5.04 3.0 4.0 5.0 0.0

Cell capacity 30 40 30 30Fixed cell cost 28 32 22 35

types of parts with 2–4 operations each part. Data of machine cost, maximum number of machinesavailable and machine-part operations are given in Table 1. Table 1, for example, shows that theunit cost of type 1 machine is 10. The maximum number of type 1 machine which can be placedin the system is 40. Processing part type 1 requires 4.0, 4.3 and 5.5 units of machine types 1, 2and 3, respectively, in terms of machine capacities. Similar data for processing other parts for thisexample are also shown in Table 1. Table 2 presents the distances among the four possible cellsas well as maximum number of machines allowed for each cell and corresponding 'xed cell cost.It is assumed that the material handling amount for each part type is 1 unit, or the di7erences ofmaterial handling are re8ected in the di7erent cell distances.

The Tabu search process starts with the penalty ratio R=5000 and Tabu list size LT =6. The initialsolution was to construct Cell 2 only with all machines allocated to this cell. This initial solution maynot be feasible to the original problem but it is a feasible solution of the reduced problem FPD′. TheTabu search process terminates if solutions are not improved after 30 consecutive iterations. Detailsof the best solution found from this search process are summarized in Table 3. Objective functionvalues corresponding to the iterations are given in Table 4. This set of data in Table 4 are alsoplotted and shown in Figs. 1 and 2. Fig. 1 shows the general trend of the objective function whileFig. 2 presents the detailed moves of the function after 'rst few iterations with big drops. In order todetermine the quality of the heuristic solution, we used LINDO software to 'nd the optimal solution


Table 3Best cell part and machine allocation (Example 1)

Cell to set-up Cell 2 Cell 3Parts to be processed 3, 4, 5 1, 2Machine types/units 1/8, 2/7, 3/6, 4/6, 5/9 1/4, 2/8, 3/9, 5/9

Table 4Objective function (F) value evolution (Example 1)

Iteration 1 2 3 4 5 6 7 8 9 10F 132032 87065 57066 32097 12067 2067 2106 2097 2097 2097

Iteration 11 12 13 14 15 16 17 18 19 20F 2116 2096 2097 2106 2106 2105 2076 2065 2064 2085

Iteration 21 22 23 24 25 26 27 28 29 30F 2066 2105 2108 2109 2102 2103 2121 2118 2098 2098

Iteration 31 32 33 34 35 36 37 38 39 40F 2097 2106 2106 2096 2114 2066 2096 2097 2108 2108

Iteration 41 42 43 44 45 46 47 48 49 50F 2101 2121 2098 2096 2095 2104 2103 2064 2085 2066

0

20000

40000

60000

80000

100000

120000

140000

1 9 13 17 21 25 29 29 29 29

Iteration5

Fig. 1. General trend of the objective function (Example 1).

of this example problem. The nonlinear integer programming problem, after linearization, has 184integer variables and 245 constraint functions. It took about 30 min to determine the optimal solutionon the same PC computer. The Tabu search based heuristic was able to 'nd the same solution inless than 5 s.


2060

2070

2080

2090

2100

2110

2120

2130

0 10 20 30 40 50 60

Iteration

Fig. 2. Detailed moves at later iterations (Example 1).

Table 5Machine part operation data (Example 2)

Mach Mach Max Part numbernum cost units

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 25 60 4.0 4.4 3.3 0.5 0.72 20 80 4.3 3.1 4.4 2.5 0.43 30 60 5.5 1.3 3.3 2.5 3.0 0.6 1.04 40 60 2.5 2.0 2.0 0.3 0.5 0.45 50 80 4.1 4.5 4.4 2.1 1.0 0.8 0.36 23 60 2.2 1.2 3.0 1.5 1.2 3.0 2.57 31 80 1.1 3.0 2.0 2.0 3.08 42 60 1.5 1.0 1.2 2.0 2.0 2.5

5.2. Example 2

In this larger example problem, we consider four possible cells and eight di7erent types of ma-chines to process 15 part types. Each part has 2–5 operations. Detailed data of this example problemare given in Tables 5 and 6. The search process starts with the penalty ratio R=50; 000. Initial so-lution was also to construct Cell 2 only. The search procedure stops when it reaches 600 iterations.Best solutions corresponding to di7erent Tabu list sizes are presented in Table 7. Details of thebest solution corresponding to TL=15 are shown in Table 8. The changes of the objective functionvalue at each iteration with TL = 15 are presented in Table 9. It shows that the 'rst 13 iterationsare outside the admissible region of the original problem. Starting from iteration 14, all solutionsare admissible. The solution was found at iteration 229 with objective function value F =3725. Forthis example problem, we were not able to verify the global optimality of the solution by LINDOsince the required amount of computation exceeds our computing capacity.


Table 6Cell data (Example 2)

Cell number 1 2 3 4

1 0.0 1.0 2.0 3.02 1.0 0.0 1.0 2.03 2.0 1.0 0.0 1.54 3.0 2.0 1.5 0.0

Cell capacity 20 60 40 60Fixed cell cost 20 40 50 90

Table 7Best objective function value vs. di7erent Tabu list lengths (Example 2)

TL 5 10 15 20 30 40 50 60F∗ 3729 3727 3725 3728 3755 3732 3755 3755

Table 8Best cell part and machine allocation (Example 2, TL= 15)

Cell to set-up Cell 1 Cell 2 Cell 3

Parts to be processed 7, 9, 13 2, 4, 5, 6, 8, 11,12, 14, 15 1, 3, 10Machine types/units 3=3; 5=6; 6=3; 7=1; 8=3 1=4; 2=6; 3=5; 4=8; 5=10; 6=9; 7=8; 8=8 1=9; 2=9; 3=10; 5=5; 7=2

5.3. Remarks

• Tabu search is a meta-heuristic method. The quality of the best solution found by our methoddepends on several factors. We found that the initial solution of the problem a7ects the 'nalresults signi'cantly. The sizes of the Tabu list LT can make large di7erences in the 'nal solutionreached.

• There are no general guidelines to determine the optimal size of the Tabu list. In practice,the search process may be constructed with di7erent Tabu list sizes. From the computation ofthe example problems in this paper, we observed that good Tabu list sizes are proportional to theproblem sizes. This is consistent with the recommendations suggested in [26].

6. Summary and conclusions

In this paper, the problem of manufacturing cell formation with 'xed charge cost was discussed.The problem was formulated as a mixed integer programming model. The objective of the modelis to 'nd an optimal machine-cell allocation by minimizing the summation of inter-cell materialhandling cost, cell construction cost and machine related costs. The formulation integrates both cell


Table 9Objective function (F) value evolution (Example 2, TL= 15)

1 2 3 4 5 6 7 8 9 10

2,453,650 2,153,670 1,903,680 1,703,680 1,503,680 1,303,750 1,053,730 853,757 653,808 503,808

11 12 13 14 15 16 17 18 19 20353,809 203,810 53,811 3762 3759 3758 3758 3757 3757 375621 22 23 24 25 26 27 28 29 303755 3755 3755 3754 3755 3755 3731 3731 3730 373031 32 33 34 35 36 37 38 39 403730 3731 3731 3731 3732 3732 3732 3731 3732 373141 42 43 44 45 46 47 48 49 503732 3732 3731 3731 3731 3731 3731 3730 3730 372951 52 53 54 55 56 57 58 59 603729 3729 3730 3730 3729 3730 3730 3731 3730 373161 62 63 64 65 66 67 68 69 703730 3730 3731 3730 3731 3731 3730 3729 3728 372771 72 73 74 75 76 77 78 79 803728 3728 3728 3728 3728 3728 3729 3728 3728 372981 82 83 84 85 86 87 88 89 903729 3729 3730 3730 3731 3732 3733 3734 3734 373391 92 93 94 95 96 97 98 99 1003733 3732 3732 3732 3732 3732 3731 3731 3732 3731101 102 103 104 105 106 107 108 109 1103731 3731 3730 3729 3729 3728 3728 3729 3729 3730111 112 113 114 115 116 117 118 119 1203730 3729 3729 3729 3729 3730 3730 3731 3732 3733121 122 123 124 125 126 127 128 129 1303732 3733 3733 3733 3733 3733 3732 3732 3732 3733131 132 133 134 135 136 137 138 139 1403733 3732 3732 3732 3732 3731 3731 3731 3732 3732141 142 143 144 145 146 147 148 149 1503730 3730 3730 3729 3728 3728 3729 3728 3728 3728151 152 153 154 155 156 157 158 159 1603728 3729 3729 3730 3730 3730 3730 3730 3730 3730161 162 163 164 165 166 167 168 169 1703731 3731 3731 3731 3730 3730 3731 3732 3731 3730171 172 173 174 175 176 177 178 179 1803730 3730 3730 3731 3731 3731 3731 3731 3731 3731181 182 183 184 185 186 187 188 189 1903731 3732 3732 3733 3733 3732 3732 3732 3732 3731191 192 193 194 195 196 197 198 199 2003730 3731 3730 3729 3730 3731 3730 3731 3731 3731201 202 203 204 205 206 207 208 209 2103732 3733 3734 3734 3733 3733 3733 3731 3730 3730211 212 213 214 215 216 217 218 219 2203730 3730 3730 3730 3730 3731 3731 3731 3730 3730221 222 223 224 225 226 227 228 229∗ 2303730 3729 3729 3729 3728 3728 3727 3726 3725∗ 3726


design and cell operation aspects in a single mathematical programming model. The complicatedfeatures of the NP-hard model make it impossible to solve within tolerable computational time onregular computing platforms. In order to obtain optimal or near-optimal solutions of the problem, anembedded optimization procedure was developed to transform the original mixed integer program-ming model into a pure binary problem. We then applied Tabu search method to 'nd optimal ornear optimal solution of the reduced problem. The developed search method is e4cient and can beapplied to solve real-world practical problems. One of the advantages of the developed approach isthat the search process does not need to start with a feasible solution of the original problem. The'nal solution found by the search process will be feasible even the initial solution was an infeasiblesolution of the original problem. Numerical example problems tested in this research showed thatthe developed search method can arrive at optimal solutions or near optimal solutions in most cases.

The model and algorithm developed in this paper can be further extended without much technicaldi4culty. For example, job scheduling requirement may be included in the model as side constraints[29]. They can be treated using penalty functions. Production planning in cellular manufacturingsystems may also be considered in developing a more comprehensive integrated model. The currentand extended models may be solved by developing deeper search procedures and more powerfulcomputational algorithms. For example, better lower bounds could be generated by Lagrangian re-laxation for deeper Tabu search as suggested in [22]. In addition, adaptive Tabu list size [30] couldbe used, instead of 'xed Tabu list size. The above-mentioned model and algorithm modi'cationswill be further explored in our future research in this area.

Acknowledgements

This research was supported by Concordia University Engineering and Computer Science ResearchSupport Fund and by Research Grant #OGP0121863 from Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada. The authors would like to thank the two anonymous referees for theirthorough review of this paper and their valuable comments and suggestions.

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Dr. Dong Cao is currently with the Department of Mechanical and Industrial Engineering at Concordia University,Montreal, Canada. He received his Master and Ph.D. degrees from the Center for Systems Engineering and AppliedMechanics at the University of Louvain, Louvain-la-Neuve, Belgium. His past and present research interests lie in the'elds of multi-attribute decision making theory, hierarchical production planning, and solution methodologies based onmeta-heuristics for manufacturing systems.Dr. Mingyuan Chen is an Associate Professor of Industrial Engineering in the Department of Mechanical and Industrial

Engineering, Concordia University, Montreal, Canada. He received his Ph.D. in Industrial Engineering from the Univer-sity of Manitoba, Canada. He has M.E. degree in Industrial and Management Engineering and B.E. degree in AppliedMathematics, both from Beijing University of Aeronautics and Astronautics, China. His research interests include appliedoperations research, manufacturing systems analysis and network analysis.

using penalty function and tabu search to solve cell formation problems with fixed cell cost

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