Integrating cell formation with cellular layout and operations scheduling

ORIGINAL ARTICLE

Integrating cell formation with cellular layout and operationsscheduling

Jamal Arkat & Mehdi Hosseinabadi Farahani &Leila Hosseini

Received: 31 July 2010 /Accepted: 26 October 2011 /Published online: 19 November 2011# Springer-Verlag London Limited 2011

Abstract Designing a cellular manufacturing (CM) systeminvolves three major decisions: cell formation (CF), cellularlayout (CL), and cellular scheduling (CS). The integrateddesign of CM systems is investigated in this paper byproposing two mathematical models. The first modelintegrates cellular layout problem with cell formationproblem to determine optimal cell configuration and thelayout of machines and cells in order to minimize the totalmovement costs. The second model takes also the cellularscheduling into consideration with the objective of mini-mizing the total completion time of parts. Two geneticalgorithms are developed to solve the real-sized problems.The proposed models are formulated as mixed integerlinear programming, and two numerical examples aresolved in order to investigate the effects of integration inthe CM systems design. The results show that consid-ering CF, CL, and CS decisions in a simultaneousmanner can significantly improve the performance of theCM systems.

Keywords Cellular manufacturing systems . Cellformation . Cellular layout . Operations scheduling . Geneticalgorithm

1 Introduction

Group Technology (GT) is a manufacturing philosophybased on the organizing and grouping of common tasks toimprove the productivity of the manufacturing system [1].One of the most important applications of GT is cellularmanufacturing that deals with the formation of manufac-turing cells in a manner that each part family is processedusing a machine cell [2]. Cellular manufacturing (CM) hasmany advantages including reduction in material handlingcosts, setup times, expediting costs, in-process inventories,part makespan, and improvement of human relations andoperator expertise [1]. Cell formation, cellular layout, andcell management are three major steps which should betaken into consideration in a successful design of a CMsystem [3]. Cell formation (CF) includes identifyingmachine cells and part families with the objective ofminimizing the number of or the total cost of intercellularmovement of parts. Designing a cellular layout includestwo stages: (1) specifying the layout of cells in the shopfloor (that is the intercell layout) and (2) determining thelayout of machines within each cell (i.e., intracell layout).The most prevalent objective for the cellular layout (CL)design is to minimize transportation cost [4]. Cell manage-ment deals with planning issues such as cellular scheduling(CS) which is the scheduling of part families and individualparts within each cell [3]. Different objective functions havebeen considered in the literature for cellular schedulingproblem. Most addressed objective functions are makespanminimization and total completion time minimization.

The interactions between the three major decisions playan important role in the CM system design. The cellularconfiguration (CF) depends directly on the transportationcost of parts which is directly determined according to thelayout of cells and machines. CL is affected by the number

J. Arkat (*) :M. Hosseinabadi Farahani : L. HosseiniDepartment of Industrial Engineering, University of Kurdistan,Pasdaran Boulevard,Sanandaj, Irane-mail: [email protected]

M. Hosseinabadi Farahanie-mail: [email protected]

L. Hosseinie-mail: [email protected]

Int J Adv Manuf Technol (2012) 61:637–647DOI 10.1007/s00170-011-3733-4

of exceptional elements and bottleneck machines which aredetermined in the CF. In addition, when the cell sizebecomes small, intercell moves increase, while large cellsizes complicate the cell management and scheduling.Transportation times are affected by the layout of cellsand layout of machines inside cells. For a detaileddescription of the interactions among CF, CL, and CS, thereader is referred to [4, 5].

Most studies have been focused on the CF problem asthe most important issue in the CM system design. Theseresearches can be categorized into two main groups basedon the used production data [6]. While the first categoryincludes methods that use machine-part incidence matrix asthe only input data, methods in the second category applyother manufacturing data such as production volumes,processing and setup times, operation sequence, machinecapabilities, alternative process routings, reliability ofmachines, and so on. Many exact and heuristic solutionprocedures have been developed to solve the CF problemincluding cluster analysis approaches, graph partitioningapproaches, branch and bound algorithms, and metaheur-istic algorithms. Detailed reviews of the various approachesto the CF problem are available in the literature [7–10].

Cellular layout design is the subject of some studies inthe CM system design. Most studies assume the first stage(CF) as a priori and solve the intercell and intracell layoutproblems. Chandrasekharan and Rajagopalan [11] haveproposed a multidimensional scaling algorithm for design-ing intercell layout by considering predetermined cellformation. CLASS is a simulated annealing-based methodpresented by Jajodia et al. [12] which determines theintercell and intracell layout simultaneously. Salum [13] hasproposed a two-phased procedure for the cellular manufac-turing layout problem through simulation. Urban et al. [14]have presented an integrated model to formulate themachine layout problem and the product assignmentproblem in which the manufacturing requirements dictatethe configuration of the manufacturing system.

Some researchers have studied the joint problem of CFand CL. Arvindh and Irani [5] have investigated the effectsof four subproblems in CM system design (i.e., CF,machine duplication, intracell layout, and intercell layout)on each other. They have proposed an iterative solutionprocedure for solving these subproblems simultaneously.Akturk [15] has presented a mathematical model fordetermining CF and intracell layout with the objective ofminimizing material handling cost. He has also proposed adissimilarity measure between parts based on operationsequences. Chiang and Lee [16] have investigated the jointproblem of CF and intercell layout considering linear layoutfor machine cells with the objective of minimizing intercellflow cost. They have used a simulated annealing approachincorporated with dynamic programming algorithm to solve

the problem. Mahdavi et al. [17] have proposed a heuristicalgorithm based on flow matrix to determine the formationof cells and the layout of machines inside cells in asimultaneous manner. Ahi et al. [18] have developed a two-stage solution procedure to solve the CF and CL problemssimultaneously. In the first stage, an initial solution is foundusing the technique for order preference by similarity to theideal solution, and in the second stage, this solution isimproved. Wu et al. [19, 20] have proposed a mathematicalmodel and developed a genetic algorithm to solve the cellformation problem and cellular layout simultaneously.

On the other hand, the CS decision in the CM systemshas been studied by some researchers. Sridhar andRajendran [21] have proposed a hybrid simulated annealingalgorithm to solve the scheduling problem in the CMsystems. Considering the formation of cells as a priori, thealgorithm tries to obtain a sequence of operations minimiz-ing the total flow time in a flow line cell. Atmani et al. [22]have introduced a mathematical programming model tosolve the cell formation problem and operation allocationproblem simultaneously. The objective of the model is tominimize the sum of the operation, refixturing, andtransportation costs. They have also considered multipleprocess plans for each part in which each operation of a parttype can be performed on more than one machine. SVSalgorithm is a two-stage heuristic algorithm proposed bySolimanpur et al. [23] to minimize the makespan in a CMsystem. The first and the second stages (namely, intracel-lular scheduling and intercellular scheduling) determine thesequence of operations and the sequence of cells, respec-tively. Franca et al. [24] have used evolutionary algorithmsto solve the problem of scheduling part families and jobsinside each part family in a flow shop manufacturing cellby considering sequence-dependent family setup times.Reddy and Narendran [25] have proposed a number ofheuristics to schedule jobs within a part family byidentifying subfamilies. The objective is to improve theutilization of machines within a cell and to reduce thetardiness and the number of tardy jobs.

It is obvious from the literature that most studies in thefield of CM systems have attempted to solve one of thethree mentioned decisions (CF, CL, and CS), and some ofthem have considered two or three decisions sequentially.These approaches yield to solutions that are probablyefficient to one of the three problems, but do not providesatisfactory solutions to the overall system. This occursbecause the three mentioned decisions influence each other.In other words, the three mentioned decisions are subpro-blems of the CM system design problem. As describedbefore, these subproblems are interrelated, and the inter-actions among them are significant. Solving a subproblemwithout considering these interactions is not guaranteed toresult in a solution satisfying two other solutions. The

638 Int J Adv Manuf Technol (2012) 61:637–647

importance of integrating cell formation with cellular layouthas been emphasized by Logendran [26] and Arvindh andIrani [5]. Furthermore, Wu et al. [3] have shown that CFand CS are interrelated, and the solution obtained for thescheduling problem is directly affected by the formation ofcells. The only mathematical model that integrates cellformation with intracell and intercell layout has beenproposed by Wu et al. [19, 20], but their model andsolution algorithm does not handle the cellular layout in acorrect manner, and cells may overlap each other. It isnoteworthy that intercell layout is just attained by deter-mining non-overlapping cells. If cells overlap each other,such as what occurs in designing virtual cells, the intercelllayout would be meaningless. Wu et al. [3] have alsoemployed the mathematical model proposed in [19, 20] todevelop a new model which integrates three mentioneddecisions. As the previous model, this model has also aninaccuracy in determining the layout of cells. We discussthis issue in detail in the next section.

In this study, two approaches, namely sequential andconcurrent, are investigated in the CM system design. Inthe sequential approach, a novel mathematical model isproposed to integrate the CF problem with the CL problemwhich is able to form rectangular non-overlapping cells andto determine the exact location of machines in the shopfloor. Considering the results obtained by solving thismodel (cellular configuration and layout), the CS problembecomes a job shop scheduling problem with transportationtimes. The second proposed model is based on theconcurrent design of the CM system and integrates the CSproblem with CF and CL problems.

Two main purposes are outlined in this research: (1) topropose appropriate mathematical models and to provide amathematical framework for integrating three stages(CF, CL, and CS) in the CM system design and (2) toinvestigate the effects of considering CS in designing aCM system.

The remainder of this paper is organized as follows: InSection 2, an integrated model of CF and CL problems ispresented via a mixed integer linear programming. InSection 3, a mathematical programming is proposed forintegrating CS, CF, and CL. Solution procedures aredescribed in Section 4. Computational results are providedin Section 5, and the conclusion is given in Section 6.

2 Integrating cell formation with cellular layout

The first model proposed in this paper is developed to solvethe cell formation, intracell layout, and intercell layoutconcurrently. The aim of this model is to determine theformation of cells and the exact location of machines inthe shop floor with the objective of minimizing total

transportation cost of parts. Total transportation cost isdefined as the sum of intracellular and intercellulartransportation costs. In order to evaluate the transporta-tion cost of parts in a correct manner, the sequence ofoperations is taken into consideration. The importanceof considering sequence of operations in evaluating theintercell and intracell moves has been emphasized byLogendran [26].

It is evident that in an accurate intercell layout, cellsshould not be overlapped. This is the case that has not beenrespected in the models proposed by Wu et al. [3, 19, 20].In their models, a number of positions are considered in theshop floor to which machines are assigned. Each positionaccepts only one machine, and each machine is assigned toonly one cell. Each machine can be assigned to any of thecells without considering its position in the shop floor, andthere is no restriction to prevent the cells from beingoverlapped. Hence, it is possible to assign two machines toa cell while another machine located between them hasbeen assigned to a different cell. The model presented inthis paper prevents cells from being overlapped byimposing some restrictions on assigning machines to cells.When two machines are allocated to a cell, any othermachine which is located between these two machinesshould also be assigned to the same cell. When we say thata machine is located between two other machines, we meanthat the horizontal and vertical coordinates of its center arebetween the corresponding coordinates of the centers ofthose two machines. Figure 1 shows an example of threemachines. It can be seen that x1≤x2≤x3 and y1≤y2≤y3where xi and yi are horizontal and vertical coordinates ofthe center of machine i, respectively. When machines 1and 3 are allocated to a cell, a restriction is needed toimpose that machine 2 is also assigned to the same cell.This restriction is essential to form rectangular non-overlapping cells.

The layout of machines inside each cell should be in theform of one of the common configurations for themanufacturing systems such as a single row (flow line),multi-rows (job shop), or loop layout [4]. In the proposedmodel, the job shop configuration is considered for the

Machine 1

Machine 3

Machine 2

x1 x2

y1

y2

y3

x3

Fig. 1 Illustrative example for forming non-overlapping cells

Int J Adv Manuf Technol (2012) 61:637–647 639

intracell layout. It is assumed that cells are rectangles, andthere is no clearance between machines. Furthermore,machines are squares of equal area, and hence, they areconsidered to have a unit dimension [27].

2.1 Mathematical model

The following notations and definitions are considered inthe mathematical models:

i Index for machines i=1,…, mj Index for parts j=1,…, nk Index for cells k=1,…, Co Index for operations o=1,…, Nj

m Number of machinesn Number of partsC Number of cells to be performedR Maximum number of machines allowed in each

cellH Horizontal dimension of the floor planV Vertical dimension of the floor planNj Number of operations of part jAj Intracellular transfer unit cost for part jEj Intercellular transfer unit cost for part jA

0j Intracellular transfer unit time for part j

E0j Intercellular transfer unit time for part j

tjo Processing time of operation o of part jgjo Completion time of operation o of part jg(j) Completion time of part jxi Horizontal distance between center of machine i

and vertical reference lineyi Vertical distance between center of machine i and

horizontal reference lineCj oo0ð Þ Intracellular or intercellular transportation cost for

part j between operations o and o0

Tj oo0ð Þ Intracellular or intercellular transportation time for

part j between operations o and o0

Si Set of operations to be performed on machine iUjoi 1, if operation o of part j should be processed on

machine i, and 0 otherwisewoo0 j 1, if operation o of part j precedes operation o

0,

and 0 otherwisezik 1, if machine i is assigned to cell k, and 0

otherwisepiuv 1, if xi<xv<xu or xu<xv<xi, and 0 otherwiseqiuv 1, if yi<yv<yu or yu<yv<yi, and 0 otherwiseM A large positive number

The proposed model is presented as follows:

MinX

nj¼1

XNj�1o¼1 Cj o;oþ1ð Þ ð1Þ

Subject to

XCk¼1

zik ¼ 1 i ¼ 1; . . . ;m ð2Þ

Xmi¼1

zik � R k ¼ 1; . . . ;C ð3Þ

Xmi¼1

zik � 1 k ¼ 1; . . . ;C ð4Þ

xi � xuj j þ yi � yuj j � 1 i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m

ð5Þ

xv � xið Þ xu � xvð Þ < Mpiuv 8i; u; v ð6Þ

� xv � xið Þ xu � xvð Þ � M 1� piuvð Þ 8i; u; v ð7Þ

yv � yið Þ yu � yvð Þ < Mqiuv 8i; u; v ð8Þ

� yv � yið Þ yu � yvð Þ � M 1� qiuvð Þ 8i; u; v ð9Þ

zik þ zuk þ piuv þ qiuv � 3 � zvk 8i; u; v ð10Þ

zik þ zuk þ piuv þ quiv � 3 � zvk 8i; u; v ð11Þ

zik þ zuk þ puiv þ qiuv � 3 � zvk 8i; u; v ð12Þ

zik þ zuk þ puiv þ quiv � 3 � zvk 8i; u; v ð13Þ

Cj oo0ð Þ ¼ UjoiUjo0uzikzuk xi � xuj j þ yi � yuj jð ÞAj

þ UjoiUjo0uzik 1� zukð Þ xi � xuj j þ yi � yuj jð ÞEj

o ¼ 1; . . . ;Nj � 1 o0 ¼ oþ 1

ð14Þ

xi � H i ¼ 1; 2; . . . ;m ð15Þ

yi � V i ¼ 1; 2; . . . ;m ð16Þ


zik ; piuv; qiuv;Ujoi;woo0 j ¼ 0 or 1 ð17Þ

xi; yi ¼ integer i ¼ 1; . . . ;m ð18Þ

Equation 1 is the objective function which minimizesthe total transportation cost of parts. Equation 2 ensuresthat each machine is assigned to only one cell. Equation 3indicates the capacity constraint in cells. Equation 4imposes that at least one machine is assigned to each cell.Equation 5 is to prevent machines from overlapping eachother. Constraints 6 through 13 prevent cells from beingoverlapped and force cells to be rectangular. Equation 14calculates intercellular and intracellular transportationcost for part j between operations o and o0. Constraints15 and 16 force machines to be located within theboundaries of the floor plan. Equations 17 and 18 specifythat the decision variables are binary and integervariables.

Note that when the horizontal coordinate of the center ofmachine v is between the horizontal coordinates of thecenters of machines i and u (i.e., xi≤xv≤xu or xu≤xv≤xi),constraints 6 and 7 impose that piuv is equal to 1. Similarly,according to Eqs. 8 and 9, when the vertical coordinate ofthe center of machine v is between the vertical coordinatesof the centers of machines i and u (i.e., yi≤yv≤yu or yu≤yv≤yi) qiuv is equal to 1. Now, with respect to Eq. 10, if piuv=1,qiuv=1, and machines i and u are assigned to cell k, thenmachine v should also be located in cell k. Hence, machinesare located in distinct rectangular areas which form non-overlapping cells.

It is noteworthy that after determining machine cells,part families can be determined easily by assigning eachpart to a cell in which the part has the most operationsto be performed. However, when the objective is tominimize transportation cost of parts, the objective isnot affected by the part assignments. Every part has adistinct rout and is transported based on its routregardless of part family to which it has been assigned.The transportation cost of a part depends on the formationof cells, layout of cells in the shop floor, layout of machinesinside cells, and transportation unit cost. Therefore,transportation cost is independent of determined partfamilies.

2.2 Linearization

In this section, the linearization procedure and the mixedinteger linear programming formulation of the model arepresented. The nonlinearity of the model results from theabsolute terms in Eqs. 5 and 14 and the product of decisionvariables in Eqs. 6–9 and Eq. 14. Absolute terms in Eq. 5

can be eliminated by replacing this constraint with thefollowing four equations:

xi � xu þMriu þMsiu � 1 i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m

ð19Þ

� xi � xuð Þ þM 1� riuð Þ þMsiu � 1

i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m

ð20Þ

yi � yu þMriu þM 1� siuð Þ � 1

i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m

ð21Þ

� yi � yuð Þ þMriu þM 1� siuð Þ � 1

i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m

ð22Þwhere riu and siu are binary variables. At least one of thefollowing equations holds for machines i and u:

xi � xu � 1 i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m ð23Þ

� xi � xuð Þ � 1 i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m ð24Þ

yi � yu � 1 i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m ð25Þ

� yi � yuð Þ � 1 i ¼ 1; . . . ;m� 1 u ¼ iþ 1; . . . ;m ð26ÞTherefore, the horizontal or vertical coordinates of the

center of machine i have a difference of at least one unitwith horizontal or vertical coordinates of the center ofmachine u, respectively. In view of the fact that machineshave a unit dimension, Eqs. 19–22 prevent machines fromoverlapping.

Now consider Eqs. 6 and 7. These constraints ensure thatpiuv is equal to 1 when terms (xv−xi) and (xu−xv) have thesame signs (either both are positive or both are negative)and 0 otherwise. By defining a binary variable diu which isequal to 1 if (xi−xu) is positive and 0 otherwise, Eqs. 6 and 7can be transformed into the following linear equations:

xv � xi < Mdvi 8i; v ð27Þ

� xv � xið Þ � M 1� dvið Þ 8i; v ð28Þ


xu � xv < Mduv 8u; v ð29Þ

� xu � xvð Þ � M 1� duvð Þ 8u; v ð30Þ

dvi þ duv � 1 � piuv 8i; u; v ð31Þ

�dvi � duv þ 1 � piuv 8i; u; v ð32Þ

dvi � duv þ 1 � piuv 8i; u; v ð33Þ

�dvi þ duv þ 1 � piuv 8i; u; v ð34Þ

Equations 31–34 impose that piuv equals 1 when dvi andduv are either both 1 or both 0 and otherwise equals 0.Equations 8 and 9 can be linearized in a similar way as wasdone with Eqs. 6 and 7.

At last, Eq. 14 is replaced with the following equation:

Cj oo0ð Þ ¼ UjoiUjo0uFiukAj þ UjoiUjo0uF0iukEj

o ¼ 1; . . . ;Nj � 1 o0 ¼ oþ 1

ð35Þ

and additional constraints are added to the original model asfollows:

zik þ zuk � 2 � M liuk � 1ð Þ 8i; u; k ð36Þ

zik þ zuk � 1 � Mliuk 8i; u; k ð37Þ

Fiuk � xi � xuð Þ þ yi � yuð Þ þM liuk � 1ð Þ 8i; u; k ð38Þ

Fiuk � xi � xuð Þ � yi � yuð Þ þM liuk � 1ð Þ 8i; u; k ð39Þ

Fiuk � � xi � xuð Þ þ yi � yuð Þ þM liuk � 1ð Þ 8i; u; k ð40Þ

Fiuk � � xi � xuð Þ � yi � yuð Þ þM liuk � 1ð Þ 8i; u; k ð41Þ

where liuk is a binary variable which equals 1 if zik and zukboth are equal to 1 and otherwise equals 0.When liuk equals 1,Eqs. 38–41 ensure that Fiuk � xi � xuj j þ yi � yuj j. SinceFiuk has a positive coefficient Aj, the minimizing objectivefunction guarantees that Fiuk equals 0 when wiuk=0. Similarequations are added to the original model for variable F

0iuk .

3 Integrating cell formation and cellular layoutwith cellular scheduling

The model proposed in this section is an extension of theprevious model which formulates the cell formation,intracellular layout, and intercellular layout and cellularscheduling simultaneously. Hence, all the assumptions ofthe previous model are retained.

Some regular objective functions for the scheduling problemare total weighted completion time, makespan, the maximumlateness, the number of tardy jobs, the total tardiness, and thetotal weighted tardiness. The objective function of the proposedmodel is to minimize the total completion time. Minimizing thetotal completion time yields to increasing throughputs, reduc-ing work-in-process inventory and minimizing job averagespent time in the production system [28].

The objective function of the previous model is replacedwith the following:

MinXnj¼1

gðjÞ ð42Þ

Equations 2–13 remain unchanged, and Eq. 14 issubstituted with the following equations:

Tj oo0ð Þ ¼ UjoiUjo0uzikzuk xi � xuj j þ yi � yuj jð ÞA0j

þ UjoiUjo0uzik 1� zukð Þ xi � xuj j þ yi � yuj jð Þ0jo ¼ 1; . . . ;Nj � 1 o 0 ¼ oþ 1

ð43Þ

gjo0 � gjo � tjo0 þ Tj oo0ð Þ

j ¼ 1; . . . ; n o ¼ 1; . . . ;Nj � 1 o0 ¼ oþ 1

ð44Þ

gjo0 � gj 0o þM 1� woo0j

� � � tjo0

for all o; o0

h i2 Si; i ¼ 1; . . . ;m

ð45Þ

gjo0 � gj 0o þMwoo0 j � tj 0o for all o; o0

h i2 Si; i ¼ 1; . . . ;m

ð46Þ

gjo � tjo 8j; o ð47Þ

gðjÞ � gjo 8j; o ð48Þ


Equation 42 minimizes the total completion time ofparts. Equation 43 calculates intercellular and intracellulartransportation time for part j between operations o and o0.Equation 44 imposes that each part is processed on machinesaccording to the defined precedence of operations in itsprocess routing. Constraints 45 and 46 impose that at most,one part is being processed by each machine at a time.Constraint 47 ensures that completion time of operation o ofpart j is greater than or equal to its processing time, andconstraint 48 calculates the completion time of part j.

All additional equations in the second model are linearexcept Eq. 43 which can be linearized in a similar way aswas done with Eq. 14. Hence, this model can also betransformed to a mixed integer linear programming model.

4 Genetic algorithm approach

The proposed models involve complex shapes of search space.General search methods such as genetic algorithm (GA) areapplicable for efficient exploration when solution space iscomplicated. Introduced by Holland [29], GAs are searchalgorithms based on the principle of “survival of the fittest” innatural evolution. GA starts with an initial set of solutions andevolves through generations by the use of operators forselection, crossover, and mutation. GAs have been appliedwidely for combinatorial optimization problems to find optimalor suboptimal solutions. In this paper, two GAs are developed:a GA called GA_CFCL for integrating cell formation withcellular layout and a GA called GA_CFCLCS for integratingcell formation and cellular layout with cellular scheduling. Themain concepts of the proposed GAs are discussed below.

4.1 Representation

The first stage of the GA process involves encoding asolution into the format of a chromosome string. In

GA_CFCL, the length of chromosome is equal to 3m (mis the number of machines). The first m genes are used toencode the CF results, and the remaining genes arededicated to the CL results. The horizontal distancebetween the centers of machines and the vertical referenceline is represented by the second m genes, and the verticaldistance between the centers of machines and the horizontalreference line is depicted by the third m genes. InGA_CFCLCS, each chromosome consists of two sections.The first section of the chromosome is as used forGA_CFCL. The second section has a length of n (n is thenumber of parts) and is a permutation of all jobsrepresenting the scheduling sequence for parts.

4.2 Initialization and fitness evaluation

In the proposed algorithms, the initial populations aregenerated randomly in a manner that each generatedchromosome is feasible. A chromosome is feasible if eachcell has at least one and at most the predetermined numberof machines (R) assigned to it; no two machines or twocells overlap, and the sequence of parts is as a permutation.If no machine is assigned to a cell, one of the machinesfrom a randomly selected cell having more than onemachine is chosen randomly and assigned to the cell. If acell has more than R machines, one of its machines isselected randomly, removed from the cell, and assigned to arandomly selected cell having less than R machines. Thisprocedure is started all over again until the number ofmachines assigned to the cell is equal to R. If two machinesoverlap, one of them is selected randomly and locatedrandomly to an unoccupied area in the shop floor. In orderto prevent cells from being overlapped, for any couple ofmachines assigned to the same cell, if there exists anothermachine located between these two machines, this machineis also assigned to that cell.

P4P3P2P1P4P3P2P1M1 1 2 1 0 M1 3 8 10 0M2 2 1 2 1 M2 11 10 7 15 M3 0 3 3 2 M3 0 6 30 19

(b)(a)

Fig. 2 Data related to the first example: a process routings of partsand b processing time of operations

Machine 3 Machine 2 Machine 1

Cell 2 Cell 1


Cell 1 Cell 2 (b) (a)

Fig. 3 Optimal results obtainedfrom two approaches: a sequen-tial and b concurrent for thefirst example

P1 P2 P3 P4 P1 P2 P3 P4

M1 1 1 3 0 M1 24 4 2 0

M2 3 0 1 0 M2 10 0 7 0

M3 2 3 0 2 M3 10 13 0 11

M4 0 2 2 1 M4 0 20 16 16

(b)(a)

Fig. 4 Data related to the second example: a process routings of partsand b processing time of operations


Fitness function f(x)is calculated for individual x asfollows:

f ðxÞ ¼ gmax � gðxÞ

In GA_CFCL, g(x) is the total transportation cost, and inGA_CFCLCS, g(x) is the total completion time for parts;gmax is the maximum amount of g(x) in the currentpopulation.

4.3 Reproduction

Roulette wheel approach is used for selection phase inthe proposed algorithms. This approach is a fitness propor-tional selection in which a new population is selected withrespect to the probability distribution based on the fitnessvalues.

4.4 Crossover

In GA_CFCL, one-point crossover is used to combine thefeatures of two chromosomes and to generate offspring. InGA_CFCLCS, partially mapped crossover (PMX) is incor-porated with one-point crossover. At first, a cross-point israndomly selected over the length of the chromosome; if thepoint is in the first section, then the one-point crossover isexecuted; otherwise, PMX is applied for the second section.

4.5 Mutation

Mutation as a background operator produces randomchanges in the gene’s values of various chromosomes. InGA_CFCL, a gene is randomly selected, and it is assigneda new value randomly selected from the correspondingrange of values. In GA_CFCLCS, the mutation operator isthe same as GA_CFCL except that if the selected gene is inthe second section of the chromosome, another gene israndomly selected, and then the values of two genes areexchanged.

5 Computational results

Two experiments are conducted in this section. The firstexperiment deals with global optimization experiences andis performed to investigate the effects of integrating CSwith CF and CL. The second one is conducted to evaluatethe performance of the proposed algorithms.

5.1 Global optimization experiences

In order to validate the proposed models and to show theeffect of integrating CS with CF and CL, the models havebeen examined for some problems. Two small-sizedexamples are solved optimally by a branch and bound


Cell 1

Cell 2 (a) (b)

Machine 3


Cell 1

Cell 2

Machine 1

Fig. 5 Optimal results obtainedfrom two approaches: a sequen-tial and b concurrent for thesecond example

Table 1 Comparison betweenGA_CFCL and B&B

aOptimal solution

No. No. of machines No. of parts No. of cells B&B GA_CFCL

Solution t (s) Best Average t (s)

1 3 4 2 29a 4 29 29 0.734

2 3 5 2 86a 6 86 86 0.824

3 4 5 2 72a 83 72 72 1.047

4 4 6 2 192a 804 192 192 1.248

5 5 8 2 61a 16,154 61 61 5.161

6 6 8 2 225 43,200 199 202.0 11.462

7 8 10 3 521 43,200 329 338.2 22.191

8 10 12 3 1,284 43,200 801 865.2 45.492

9 15 25 4 – 43,200 1,012 1,048.7 68.244

10 20 40 6 – 43,200 1,706 1,764.5 82.103


(B&B) method using LINGO 8.0 software on a 2.00-GHzpersonal computer with 2.00 GB of RAM.

A random generation procedure is used to producenumerical examples. The parameters that need to begenerated are process routing for each part and processingtime for each operation. The number of machines in eachpart process routing is considered to be an integer numberrandomly selected between 2 and [m/2]+1. Machines areselected randomly for each process routing and arranged ina random manner. The processing times of operations arerandom integer numbers between 2 and 30.

Two approaches are used to solve each example. Thefirst approach is a sequential procedure and consists of twoseparated phases. In the first phase, the first proposedmodel is solved to find the optimal cell formation andcellular layout for each numerical example. The secondphase is to determine the optimal solution of the schedulingproblem for the solution obtained in the first phase. In theother hand, the second approach is based on the concurrentdesign of the CM system. In this approach, the secondproposed model is optimally solved to find the solution ofthe integrated problem with the objective of minimizingtotal completion time of parts.

The first example consists of four parts to be processedon three machines. Machines should be assigned to two

cells, and maximum number of machines allowed in eachcell is two. The data related to the first example are shownin Fig. 2 in which rows and columns of matrices areconsidered as machines and parts respectively. Processroutings of parts are shown in Fig. 2a, and processing timesof operations are depicted in Fig. 2b. The layout ofmachines and cells for the optimal solution of the firstmodel (i.e., without considering CS) is depicted in Fig. 3a.As can be seen, machines 1 and 2 are assigned to cell 2, andmachine 3 is located in cell 1. The optimal value of totalcompletion time for this solution is 264 units of time.

By considering cellular scheduling and data provided inFig. 2, the obtained optimal solution of the second model (i.e.,integrated model) is presented by Fig. 3b in which machine 1is assigned to cell 1, and machines 2 and 3 are allocated to cell2. The optimal value of total completion time for this solutionis 244 units of time. As shown in Fig. 3, consideringscheduling has a significant effect in designing a CM system.In this example, the layout of machines is not affected byconsidering scheduling, but formation of cells is changed toobtain a better solution with an improvement of more than7.5% in terms of the total completion time.

The second example consists of four parts and fourmachines. Process routings of parts and processing times ofoperations are shown in Fig. 4. The optimal solutions thatresulted from the sequential and concurrent approaches areshown in Fig. 5a and b, respectively. The total completiontimes for these two solutions are 238 and 226 units of time,respectively. As can be seen, formation of cells is the samein both solutions, but concurrent design of the CM systemyields to a different layout and an improvement of morethan 5% in terms of the total completion time.

5.2 Evaluation of the proposed GAs

The proposed algorithms have been coded in C# and run ona 2.00-GHz PC with 2.00 GB of RAM. Some parameters

14 12 8

5 1 15 20 2

11 13 4 6

7 16 17 19 18

9 10 3

Fig. 6 The obtained layout for problem instance 10

Table 2 Comparison betweenGA_CFCLCS and B&B No. No. of

machinesNo. ofparts

No. ofcells

B&B GA_CFCLCS

Solution t (s) Best Average t (s)

1 3 5 2 306* 700 306 306 2.724

2 4 5 2 469* 18,358 469 469 4.047

3 4 6 2 436* 32,112 436 436 4.381

4 5 7 2 573 43,200 543 543.8 7.976

5 5 8 2 578 43,200 552 555.6 10.602

6 6 8 2 624 43,200 561 568 18.245

7 8 10 3 822 43,200 764 771.1 31.469

8 10 12 3 1,804 43,200 1,162 1,173.0 47.712

9 15 25 4 – 43,200 1,214 1,238.7 72.283

10 20 40 6 – 43,200 1,922 1,986.1 101.249


need to be set while running the algorithms includingpopulation size, crossover rate, mutation rate, and numberof generations. The following parameters have been utilizedfor the proposed genetic algorithms: for GA_CFCL,population size 100, crossover rate 0.9, mutation rate 0.1,and number of generations 200: for GA_CFCLCS, popu-lation size 100, crossover rate 0.8, mutation rate 0.2, andnumber of generations 250.

In order to evaluate the performance of each of theproposed algorithms, ten instances have been generatedaccording to the random generation procedure described inSect. 5.1. The algorithms have been run ten replicates foreach test problem, and the best as well as average of theobtained solutions have been provided. For all test prob-lems, the proposed mixed integer linear programmingmodel has also been solved by a B&B method usingLINGO software. In large-scaled problems, the LINGOprogram has been interrupted after 12 h, and the bestsolution has been reported. Table 1 illustrates the informa-tion about test problems and the solutions found byGA_CFCL and B&B.

One of the advantages of the proposed approach is that theresulting layout is very compatible with the real-worldsituations. Exact location of machines in the cells and exactlocation of cells in the shop floor are determined. Therefore,the obtained layout can be easily implemented in a manufac-turing system. For example, resulting layout for probleminstance 10 of Table 1 has been depicted in Fig. 6. It isobvious that some modifications are needed before execu-tion. In the obtained layout, machines are squares of equalarea, and there is no clearance between machines. However,these assumptions are very common in layout problems anddo not pose any difficulty in implementation [27].

For five test problems that the optimal solution hasbeen found by B&B, GA_CFCL also finds the optimalsolution in a short time. For problems 6 to 8, the bestsolution found by GA_CFCL is better than the solutionfound using B&B in a much less time. For these probleminstances, the solution found by B&B provides an upperbound on the objective value. In order to have anestimation of the efficiency of the proposed GA, the solutionobtained by GA is compared with this upper bound. Forproblems 9 and 10, B&B finds no feasible solution evenafter 12 h.

The information about the random generated test prob-lems and the solutions found by GA_CFCLCS and B&Bare shown in Table 2. As can be seen from Table 2,GA_CFCLCS finds optimal solutions for problems 1 to 3which have been solved optimally by the B&B method. Forproblems 4 to 8, the best solution found by GA_CFCLCS isbetter than the solution found using B&B in much lesstime. No feasible solution is found by B&B for problems8 and 10.

6 Conclusion

In this paper, integrated design of CM systems has beeninvestigated. In order to show the effects of consideringcellular scheduling in the CM system design, twoapproaches have been investigated. The first approach is asequential one which consists of two phases. First, cellformation and cellular layout problems are solved simulta-neously. Then, by considering the solution found in the firstphase, the cellular scheduling problem is solved as a jobshop scheduling problem with transportation times. In thesecond approach, cellular scheduling is also integrated withcell formation and cellular layout. The appropriate mathe-matical models for the abovementioned approaches havebeen presented. Proposed mathematical models are able toform rectangular non-overlapping cells and to determine theexact location of machines in the shop floor. Thelinearization procedure and the resulted mixed integerlinear programming formulations of the models have beenpresented. Two genetic algorithms have also been devel-oped to solve the real-sized problems. In order to verify theperformance of the models and to examine the effects ofintegrated design of the CM system, two numericalexamples have been solved under two approaches, namelyconcurrent and sequential. The obtained results show thatconsidering cellular scheduling in CM system design canaffect both cell formation and cellular layout and canimprove the quality of the obtained solution.

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