A methodology for designing ¯exible cellular manufacturingsystems

RONALD G. ASKIN1, HASSAN M. SELIM2 and ASOO J. VAKHARIA3

1Systems and Industrial Engineering Department, The University of Arizona, Tucson, AZ 85721, USA2Production Engineering Department, College of Engineering and Technology, Helwan, Cairo, Egypt3Department of Decision and Information Sciences, College of Business Administration, University of Florida,Gainesville, FL 32605, USA

Received December 1994 and accepted October 1996

Cell formation in cellular manufacturing deals with the identi®cation of machines that can be grouped to create manufacturingcells and the identi®cation of part families to be processed within each cell. Dynamic and random variations in part demands cannegatively impact cell performance by creating unstable machine utilizations. The purpose of this paper is to introduce andillustrate an interactive cell formation method that can be used to design `¯exible' cells. Flexibility in this context refers to routing¯exibility (i.e., the ability for the cellular system to process parts within multiple cells) and demand ¯exibility (i.e., the ability of thecell system to respond quickly to changes in part demand and part mix). Through an experimental analysis using multiple data sets,we also validate the procedure and provide guidelines for parameter settings depending upon the type of ¯exibility of interest to theuser. Finally, trade-o�s and interdependences between alternative types of ¯exibility in the context of cellular systems areillustrated.

1. Introduction

Cellular manufacturing (CM), an application of grouptechnology, entails the creation and operation of manu-facturing cells. Each cell is dedicated to processing aspeci®c set of part families. A manufacturing cell typicallyconsists of several functionally dissimilar machines,whereas a part family consists of a set of parts withsimilar processing requirements. One of the ®rst problemsencountered in implementing CM is that of cell formation(CF). CF deals with the identi®cation of the part familyor families and associated machine groups that constituteeach cell.Although the operational bene®ts of CM have been

well-documented in the literature [1], it has also beenargued that the implementation of cells could lead to adecrease in manufacturing ¯exibility [2]. The major di�-culty with cells stems from potentially unstable machineutilizations due to dynamic and random variations in partdemands [3]. This has led to some confusion as to theappropriateness of CM by industry users. On the onehand, companies would like to achieve the operationale�ciencies through implementing CM systems, but, onthe other hand, companies do not want to lose the stra-tegic bene®ts of ¯exible operations. Further, as pointedout by Craig, et al. [4], ¯exibility is one of the criticaldimensions of enhancing the competitiveness of organi-

zations and hence the design of `¯exible' cells is an im-portant issue [5].This paper proposes a CF method that incorporates

several ¯exibility criteria to guide the creation of `¯exible'cells. This approach is unique in several aspects. First, thecell system design generated is a function of the userpriorities in terms of ¯exibility dimensions. This not onlyallows the user to incorporate preset user priorities butalso allows the investigation of trade-o�s between con-¯icting ¯exibility criteria. Secondly, although there issome prior research that has incorporated alternativeprocess plans when identifying cell con®gurations (e.g,[6±11]), this is one of the ®rst methods that focuses onpart-operation requirements in creating cells. Most of thecell formation research to date (e.g., [12±20]) assumes thatparts are processed on speci®c machine types and theassignment of operations to machines is determineda priori. However, to allow for ¯exibility in operation-machine assignments, we explicitly incorporate this de-cision into our procedure. Thirdly, the method proposedin this research includes an explicit improvement stagewhere the user can attempt to modify the candidatedesign to increase alternative (or all) types of system¯exibility.The remainder of this paper is organized as follows. In

Section 2 the relevant literature is reviewed and the ¯ex-ibility criteria of interest for CM are introduced. Section 3

0740-817X Ó 1997 ``IIE''

IIE Transactions (1997) 29, 599±610

describes the proposed CF procedure, and Section 4presents an illustration of the methodology. Section 5describes the experiment conducted to validate the pro-posed methodology and provides user guidelines for pa-rameter settings depending on the ¯exibility criteria.Finally, the implications and conclusions of the researchare discussed in Section 6.

2. Relevant literature

In recent years there has been a tremendous growth in thenumber of CF methods. The surge of interest in the areahas been fueled not only by surveys that have shown thebene®ts of CM systems [1] but also because there issubstantial industry interest in implementing CM sys-tems. Comprehensive reviews of CF can be found in[5, 21±24].In the context of this paper, two papers on CF are the

most relevant. Tilsley and Lewis [25] were the ®rst topropose a CF method where routing ¯exibility was aprimary consideration. They essentially propose a systemof `cascade' cells that are created such that the morecritical part families can be processed in more than onecell. Thus, machines required to process critical partfamilies are allocated to more than one cell. Although thealgorithmic details of the procedure are not provided,they do point out the importance of building in routing¯exibility when machines within cells are subject tobreakdowns. Machine downtime in a cell could be han-dled by having multiple machines of the same type in acell or by routing operations performed on one machinein a cell to other machines in the same cell; however, thesefactors are not considered in their procedure.Dahel and Smith [26] propose a procedure to create

cells such that routing ¯exibility and cell independencecould be simultaneously considered. They essentiallyformulate the CF problem as a multi-objective mathe-matical model that simultaneously attempts to create in-dependent cells (by minimizing intercellular materials¯ows) and ¯exible cells (i.e., cells containing the largestvariety of machine types). Their logic is that routing¯exibility of the system is maximized when we can createsuch ¯exible cells.In terms of ¯exibility dimesions, there has been a re-

markable lack of interest in designing cells that can re-spond quickly to changes in the part demands (in terms ofnew part introduction and in terms of changes in volumesof current part). To address this issue, Vakharia, et al.[27] develop a framework and measures for di�erent¯exibility types relevant in the context of CM systems.These types are:

· machine type ¯exibility: the ability of the machinesgrouped into cells to process a large number ofdistinct operation types;

· routing ¯exibility: the ability of the cell system toprocess parts completely in multiple cells (referred toas process ¯exibility in [28]);

· part volume ¯exibility: the ability of the cell systemto deal with volume changes in the current part mix;and

· part mix ¯exibility: the ability of the cell system tohandle di�erent product mixes with minimum dis-ruption.

Of these ¯exibility types, routing, part volume, and partmix ¯exibilities are determined by the cell system designgenerated, whereas machine type ¯exibility is also afunction of the technological constraints on the machines.Our primary objective in this paper is to consider all fourtypes of ¯exibility in developing a CF method. Thisprocedure is described in the next section.

3. Flexible cell formation (FCF) method

An overview of the proposed FCF method is shown inFig. 1. Phase I identi®es the most economical set of ma-chine types to process the required operations of the en-tire part set based on machine costs, capabilities, andcapacities. Phase II assigns individual part-operations toindividual machines with an objective of providing anassignment that will lead to minimum material handlingcost in ®nal system design. Balancing material handlingcosts, current processing requirements, and ¯exibility toadapt to changes, Phase III forms candidate cells. The¯exibility of this cellular con®guration is then evaluated

Fig. 1. An overview of the ¯exible cell formation (FCF)method.

600 Askin et al.

and improved in Phase IV on the basis of the current setof part types and demands. Each phase of the procedureallows user interaction in terms of parameter settings.Further, once the cell design (with a ®xed set of param-eters) is generated, measures to evaluate the design inseveral dimensions are provided. On the basis of such anevaluation, the user has the option of either completelyregenerating the system design (i.e., start again at Phase I)or simply changing parameter settings at some interme-diate level to modify the cell system design generated.Details of each phase are described next; the notationused for the complete method is shown in Table 1.

3.1. Phase I: assign operations to machine types

Given a machine type population with di�erent process-ing capabilities (in terms of operation types), Phase I isconcerned with assigning operation types to machinetypes. Each operation type represents a set of identicaloperations required by one or more parts. An operationtype could be a generic process such as drilling, but wouldin most instances be more speci®c. For example, drillingwith a speci®c tool and power requirement might be anoperation type. Although Phase I would ordinarily beused to assign operation types, the methodology couldalso be used to assign individual operations. A compre-hensive mathematical model for carrying out thisassignment is as follows:

Minimize Z �XMm�1

XJ

j�1cmumjxmj �

XMm�1

FmNm �1�

subject to XMm�1

xmj � 1 8 j; �2�

XJ

j�1umjxmj � UmNm 8m; �3�

xmj 2 f0; 1g 8m 8 j; �4�Nm � 0 and integer 8m: �5�

In this model, the objective function minimizes thetotal annual operating cost of the operation±machineassignments and the total annualized procurement cost ofmachines. The ®rst constraint ensures that each operationis assigned to a single machine type. The second con-straint ensures that operation±machine type assignmentsare feasible at the system level by ensuring that we acquireadequate capacity. In this case umj is the fraction of ma-chine type m required, assuming that all parts requiringoperation j will be processed on machine type m. Becausedemand for each part type is assumed to be known, thenif batch sizes are also known, the time needed for therequired number of setups of each part can be included in

Table 1. Notation for the FCF method

Indices:p Index for parts �p � 1; . . . ; P�m Index for machine types �m � 1; . . . ;M�j Index for operation types �j � 1; . . . ; J�jp Index for operations required by part p � jp � 1; . . . ;Op�km Index for machines of type m available/required �km � 1; . . . ;Km�

Parameters:Dp Demand (in batches per unit time) for part pPpj 1 if j � Op; 0 otherwiseAm Availability of each machine of type m �0 � Am � 1�Fm Annualized procurement cost of a machine of type mymj 1 if machine type m can perform operation j; 0 otherwisecm Annual operating cost for machine type mtmj Run time for one unit of any part for operation j on machine type msmj Setup time for a batch of any part for operation j on machine type mumj Number of type m machines required for operation j (machines/time)

Required user inputs:Um Maximum allowable utilization for a machine of type m (Phase I)a Weighting factor for similarity measure (Phase III)C Number of cells to be identi®ed �c � 1; . . . ;C� (Phase III)V Maximum allowable variability in cell size (Phase III)I Type of improvement algorithm to be implemented (Phase IV)

Decision Variables:Nm Number of machines of type m available/requiredxmj 1 if operation type j is assigned to be performed on machine type mXkmjp 1 if part-operation jp is performed on machine km

Zkmc 1 if machine km is allocated to cell c

Methodology for designing ¯exible cellular systems 601

umj. Um�0 � Um � 1� is a user-speci®ed parameter thatindicates the amount of slack capacity that should bebuilt into the system. The user parameter Um controls thevolume ¯exibility of a cell system design. By setting lowervalues of Um, we build more slack capacity into the sys-tem and hence, allow the system to handle part volumechanges without disruption. Finally, the technologicalconstraints on the decision variables xmj and Nm are en-forced in the last two sets of constraints. Note that if wedo not require that a part operation always be performedon the same machine type, we can relax the integralityrestrictions on the xmj.For a known set of Nm values, this model reduces to the

generalized assignment problem (GAP). In the context ofour problem, we would know Nm if we were recon®guringan existing system into a cellular system. In contrast,when designing a completely new system, we propose thatinitial estimates of Nm be obtained by linearizing the ®xedmachine cost and then selecting the least expensive ma-chine type for each operation. With this assignment, es-timate the number of machines of each type bycomputing d�Pj umjxmj�=Ume. Once these estimates of Nmhave been obtained, we use the Martello and Toth [29]enumerative algorithm for obtaining good solutions tothe GAP model. Note that the ®nal Nm values are basedon the operation±machine type assignments resultingfrom applying this algorithm. A branch and boundstrategy can be added to search for a better solutionif desired. The output of this stage is the xmj and Nm(if required) decision variables.

3.2. Phase II: assign part-operations to speci®c machines

Given operation assignments to machine types, Phase IIassigns each part-operation to a speci®c machine of eachtype such that the similarity between part-operations as-signed to a machine is maximized and the user-speci®edmaximum machine usage �Um� is not exceeded. Know-ledge of these operation to machine assignments is neededin Phase III to include minimization of material handlingas a criterion in the machine grouping decision.The goal in Phase II is to limit the number of inter-

machine and intercell transfers that will result from the®nal cellular system design. We accomplish this by as-signing sets of operations to the same machine if theysatisfy either of two criteria. First, if multiple operationsfrom the same part type are assigned to the same machinetype in Phase I, these operations are now assigned to thesame machine. Secondly, operations from parts that havesimilar machine type sequences, and therefore should bein the same cell, are assigned to the same machine. Thetechnical procedure for this assignment is described be-low. The complete Phase II problem is separable bymachine type and we solve this problem independentlyfor each machine type that requires more than onemachine.

As in [13] we use a graph partitioning model for se-lecting machine assignments. However, instead of creat-ing duplicate nodes based on discrete approximations ofprocessing time requirements for each part, we create anode for each operation and use actual time require-ments. Thus, each node in the graph represents a speci®cpart-operation assigned to this machine type in Phase I.The arc weight between each pair of nodes is 1 if bothpart-operations are required by the same part type.However, if a pair (say i and j) of nodes (part-operations)correspond to di�erent parts (say pi and pj, respectively),then the arc weight is de®ned as:

Sij � max Npipj=Opi ;Npipj=Opj

� ; �6�

where Npipj is the number of common operation types forparts pi and pj. Thus, Sij measures the similarity betweenthe processing requirements for the part types to whichpart-operations i and j belong. Accordingly, Sij representsthe relative desirability for the two operations to beassigned to the same machine. Note that 0 � Sij � 1.In the second stage, an initial partition for this graph is

obtained by using the following heuristic. A part-opera-tion pair with the lowest similarity is identi®ed and eachpart-operation is assigned to two separate machines. Thisprocess is repeated iteratively until each machine has beenassigned one part-operation. Then we sequentially assignthe remaining part-operations to machines so as tomaximize the internal similarity of all part-operationsassigned to the same machine subject to Um. In the thirdand ®nal stage, we improve on this initial partition ofoperations to machines by using a capacity-constrainedvariant of the Kernighan and Lin [30] graph partitioningprocedure. The output of this phase is the speci®cation ofthe Xkmjp decision variables.

3.3. Phase III: identify candidate manufacturing cells

Phase III of the FCF method involves clustering indi-vidual machines to identify candidate manufacturingcells. The individual part-operations performed in eachcell are a function of the individual machines assigned tothat cell (Phase II speci®es the assignment of part-oper-ations to individual machines). Our objective is twofold.We wish to form cells that are comprehensive for thecurrent parts so as to minimize intercell material ¯ows,but also to ensure ¯exibility to react to changes in pro-cessing requirements. The algorithm used to create thecandidate manufacturing cells is similar to that of PhaseII, except that we now consider partitioning the individ-ual machine set into the number of desired cells. A majordi�erence is in the manner by which the arc weights be-tween individual machines are computed. The arc weightor desirability index dn1;n2 between nodes n1 and n2 (eachrepresenting an individual machine of type m1 and m2,respectively) is de®ned as a convex combination of twoparts:

602 Askin et al.

dn1;n2 � aW sn1;n2 � �1ÿ a�W c

n1;n2 0 � dn1;n2 � 1; �7�where

W sn1;n2 � maxfWn1;n2;Wn2;n1g;

Wn1;n2 � proportion of machine n1workload

associated with parts that also use machine n2

W cn1;n2 �

P Jj�1 ym1;j �

PJj�1 ym2;j ÿ

PJj�1 ym1;j ym2;j

J

" #

�P Jÿ1

j1�1P J

j2�j1�1 fj1j2 X cj1j2PJÿ1

j1�1PJ

j2�j1�1 X cj1j2

" #fj1j2 � proportion of current parts that require

operations j1 and j2

X cj1j2 � max ym1;j1; ym2;j1

� �max ym1;j2; ym2;j2�

ÿmax ym1;j1 � ym1;j2; ym2;j1 � ym2;j2�

:

W sn1;n2 is computed as the maximum proportion of ma-

chine n1 or n2's work that involves parts that also visit theother machine given the current part-operation assign-ments. W c

n1;n2 is a function of the combined processingcapability of machine pair n1 (of type m1) and n2 (of typem2). The motivation for W c

n1;n2 rests in the desirability, ina dynamic environment, to create cells that have theability to process future parts that exhibit minor tech-nological modi®cations or to handle minor changes inpart mix and volume. W c

n1;n2 is the product of two terms.The ®rst term is the proportion of part operations thatcan be performed by the combined set of machines n1 andn2. The second term in W c

n1;n2 takes a weighted averageover those operation pairs that require both machines ofthe likelihood that an arbitrary future part will requireboth operations. X c

j1j2 takes on the value 1 only if neithermachine type m1 nor m2 can perform both operation j1and j2, but this pair of operations can be completed bythe union of the two machines. The weights fj1j2 thenindicate the probability that a future part will require thispair of operations. Together these indicate whether thesynergism resulting from combining these two machinesin the same cell is likely to be useful.The parameter a is de®ned by the user. A high value of

a indicates a user preference for designing cells based onthe current part set and demand requirements. In con-trast, by specifying a lower value of a, the user can insteadopt to focus on the possible set of operations that couldbe performed by both machines n1 and n2 rather than thecurrent set that have been assigned. If the ®rm has ad-ditional technological forecasting information beyondthat contained in the current part set, this informationcan be easily entered into the decision process by modi-fying the fj1j2 parameters accordingly.Once the arc weights between all nodes have been de-

®ned, the user is required to specify the number of cells tobe identi®ed (i.e., C) and the variability to be permitted in

cell sizes (i.e., V ). Cell size is assessed in terms of thenumber of machines. C controls the number of partitionsthat will be created, whereas V controls the allowablerange of the number of machines placed within eachpartition (cell). In practice, C would be set by organiza-tional parameters such as size of worker teams, span ofsupervisory authority, and group dynamics. Given arange of cell sizes, the procedure can be applied for fea-sible values of C and the resultant solutions compared.On the basis of these inputs, the procedure for initial

partitioning and improvement described in Phase II isused to partition the set of nodes, each representing amachine, into machine groups (cells). The output ofPhase III is the Zkmc decision variables and indicate theassignment of individual machines to cells.The three user-de®ned parameters control the ¯exibility

of cell system design in di�erent ways. a allows theincorporation of machine-level ¯exibility (in terms ofpossible operations that can be performed) during the celldesign process. Thus, by varying a the user can investi-gate how responsive the current part-operation tomachine assignments are in the context of the set of op-erations which can be performed by each machine. Thenumber of cells (C) can be hypothesized to impact the¯exibility of the cell design. By setting a low value of C,the size of each cell in terms of machines will be larger.Hence, we can argue that there will be more routing¯exibility built into the system as machine variety in eachcell is likely to increase. Regarding the variability in cellsize parameter �V �, it can be hypothesized that byallowing cell sizes to be more variable, there is the like-lihood that a `master' cell consisting of a large number ofmachine types will be identi®ed and thus routing ¯exi-bility will once again be higher. Likewise a few highlyspecialized cells can be formed for current highly similarpart families when appropriate. Note that both cases (i.e.,small C and high V ) will probably increase the machinetype ¯exibility of the cell system con®guration. This isprimarily the result of there being a single cell that willcontain multiple machine types and hence new parts canbe processed in such a cell.

3.4. Phase IV: improvement and evaluation

In Phase IV of the method we ®rst attempt to improveupon the preliminary cellular con®guration. Dependingupon the user input, we can improve the routing ¯exi-bility and/or the volume ¯exibility of this preliminarysystem design. Given a cell system design (indicating theassignment of speci®c machines to cells and also theassignment of individual part-operations to machines ineach cell), the routing ¯exibility for a part p in a cell c iscomputed as follows:

RFcp �YOp

jp�1Rcjp ; �8�

Methodology for designing ¯exible cellular systems 603

where

Rcjp � 1ÿY

m:ymjp�1�1ÿ Am�Kmc

and Kmc is the number of machines of type m assigned tocell c. Rcjp provides the proportion of time that cell c hasat least one machine available that can perform operationjp. RFcp computes the proportion of time that cell c has aset of machines operational that are capable of com-pletely processing part p. Note that if none of the ma-chines in cell c can process an operation jp on part p, thenRFcp = 0. De®ne Fcp � 1 if RFcp > 0 and 0 otherwise.Then the routing ¯exibility of a given cell system designassociated with an individual part p �Fp� is simply thetotal number of cells that have the capability to processthe part completely and is computed as

Fp �XC

c�1Fcp �9�

and the aggregate routing ¯exibility of the system designis simply given by F �PP

p�1 Fp. Given that this measureis in¯uenced by the number of cells identi®ed and doesnot indicate the balance of alternative cells across parttypes, an alternative aggregate measure of routing ¯exi-bility can be computed as the percentage of parts that canbe routed through more than one cell for processing.Other possible alternatives include F 0 �PP

p�1 log Fp;which assigns an in®nite penalty for creating a systemwith one or more parts that do not have a comprehensivecell, or F 00 �PP

p�1 log�1� Fp�.To improve upon the routing ¯exibility of a given cell

system design, we attempt to reassign parts, individualmachines and/or part-operations in that order (withoutchanging the total number of machines in the completesystem). The general procedure is as follows. Each part isconsidered for reassignment to another cell that containsall the necessary equipment to process the part providedthat machine capacity is available in that cell. After weperform this individually for every part, machine reas-signment is considered if it will result in an increase in thenumber of cells to process one or more part types. Fi-nally, the reassignment of part-operations is performedfor those part types that cannot be completely processedin one cell. The objective of such a reassignment is tocheck whether we can identify another cell to completelyprocess each part.Another improvement algorithm available to the user

at this Phase of the FCF method focuses on volume¯exibility of the system design. Given a cell system design,the volume ¯exibility is computed as the maximum equalpercentage (de®ned as d�) increase in volume for all partsthat can be handled without changing the system con-®guration. The improvement in volume ¯exibility is per-formed as follows. First we identify the bottleneckmachine(s) in the system (i.e., the machine(s) that restrict

the volume ¯exibility of the current design). Next, weattempt to reroute some load from this machine to an-other machine in the same cell that can perform the sameoperation. If this is not possible, we attempt to reroutethe workload to a machine of the same type (or a machinethat can perform the same operation) in another cell. Theprocedure is repeated iteratively until no further rerout-ing of load is possible.Once the cell design has been improved upon, we

evaluate it not only in terms of routing and volume¯exibility (as de®ned earlier) but also in terms of mix¯exibility. The ¯exibility of the cell system to respond to achange in part mix can be assessed depending upon thetype of such change. A mix change occurs either when therelative volume requirements of the current part mixchange, or when a new part with an associated operationset and volume requirement is introduced in the system.To assess the ¯exibility of a cell system to the ®rst type ofchange (i.e., relative volume change), we compute themaximum percentage of demand for each current indi-vidual part (de®ned as d�p) which can be accommodatedwithin the current cellular con®guration. Demand forpart types other than p are held constant for this calcu-lation.To assess the ¯exibility of the cell system to respond to

the introduction of a new part, we need to consider twoaspects: (i) the amount of slack capacity in the cell systemto completely process all the demand requirements for thenew part, and (ii) whether there is a set of machineswithin any one cell with adequate slack capacity tocompletely process all the demand requirements for thenew part. Obviously (ii) is preferred because the new partwill be completely processed in a cell (leading to no in-crease in intercellular ¯ows). However, if this is notpossible, it may be possible to process the new partthrough multiple cells as long as adequate capacity isavailable in the system to do so. To assess these twoaspects, we compute three measures.The ®rst measure is denoted by c1 and is computed as

the percentage of new parts that can be accommodatedin the current con®guration (even if these parts need tobe processed in multiple cells). Hence, c1 evaluates theavailability of capacity in the aggregate cell system withreference to new parts. The second measure, c2, iscomputed as the percentage of new parts that can becompletely processed within an existing cell without re-gard to capacity constraints (i.e., there exists at least oneprimary cell to process the new part). Hence, this mea-sure attempts to capture the part-mix ¯exibility of thecell design by restricting the intercellular ¯ows of mate-rials. The ®nal measure �c3� integrates the ®rst twomeasures and is computed as the percentage of new partsthat can be completely processed within a single existingcell by explicitly considering the availability of capacityin the cell. These measures will be illustrated in the nextsection.

604 Askin et al.

4. Illustration

The hypothetical example used to illustrate the proce-dure is as follows. There are 10 di�erent machine typesin the system. The system produces 19 parts requiring 12di�erent operation types. For each machine type, theprocurement and operating costs �Fm and cm� and theavailability of each machine �Am� are shown in Table 2.The part demand in batches per year �Dp�, the batch size�Qp� and part processing requirements matrix are shownin Table 3. The run and setup times are shown inTable 4. For this example we assume that the operationcharacteristics are not part dependent. (The FCFmethod does not require this assumption because eachpart-operation is treated separately in Phase II, andoptionally in Phase I as well, but it does reduce theamount of data we must include in the table.) Thus,regardless of which part-operation is performed on amachine type, run and setup times are identical. Weassume that the shop in question operates 8 hours perday, 250 days a year.At Phase I, assume that Um, the maximum allowable

utilization of any machine, is 0.90 8 m. From Table 4 wecan see that only operations 7, 9, and 10 can be processedon more than one machine type. We assume that thenumber of machines is not ®xed; thus each operation isassigned to the machine type with minimum relaxed costumj�cm � Fm=Um�. On the basis of the solution to Phase I,operation 7 is assigned to machine type 9; operation 9 tomachine type 1; and operation 10 to machine type 10. Thenumber of machines �Nm� required of each type are(2,2,1,2,2,2,1,1,2,2).Phase II is concerned with assigning each part-opera-

tion to a machine. Note that on the basis of the Nm re-quirements from Phase I, this is only relevant for machinetypes 1, 2, 4, 5, 6, 9, and 10. Based on these part-opera-tion±machine assignments, in Phase III we generate acandidate cell design with a � 1 (to emphasize currentpart-operation to machine assignments), C � 4 (four-celldesign) and V � 0 (i.e., cell size variability is not con-

sidered). Finally, in Phase IV, an improvement of thePhase III cell design is performed. We search forimproved solutions in terms of volume and routing ¯ex-ibility (in that order). The Phase III and IV part-opera-

Table 2. Machine type data

Machine Procurement Operating AvailabilityType �m� cost �Fm� cost �cm� �Am�1 1.70 1.70 0.852 1.80 1.80 0.903 1.20 1.20 0.604 1.80 1.80 0.905 1.60 1.60 0.806 1.60 1.60 0.807 1.40 1.40 0.708 1.80 1.80 0.909 1.90 1.90 0.9510 1.60 1.60 0.80

Table 3. Part data

Part�p�

Demand�Dp�

Batchsize �Qp�

Operation � j�

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

1 9 100 1 1 1 1 12 11 700 1 1 1 13 5 400 1 1 1 14 6 500 1 1 1 1 15 5 840 1 1 1 16 5 900 1 1 1 17 4 956 1 1 1 1 18 4 116 1 1 1 1 1 19 5 624 1 1 1 1 110 7 928 1 1 1 1 1 111 9 410 1 1 1 112 6 690 1 1 1 1 113 3 562 1 1 114 5 827 1 115 5 961 1 116 8 491 1 1 1 117 5 895 1 1 1 118 6 742 1 119 4 627 1 1

Table 4. Run and setup times

Machinetype �m�

Run/setup

Operation � j�

time* 1 2 3 4 5 6 7 8 9 10 11 12

1 R 4.0 2.0S 140 140

2 R 3.5S 240

3 R 4.5S 300

4 R 3.0 1.5S 120 100

5 R 3.0 3.5S 20 420

6 R 2.5S 20

7 R 3.5 4.0S 100 20

8 R 2.5S 420

9 R 5.0 3.0S 20 40

10 R 2.5S 10

*The run times �R� are in minutes per unit; the setup times �S� are inminutes per batch.

Methodology for designing ¯exible cellular systems 605

tion assignments to individual machines and the cells towhich these machines are allocated are shown in Table 5.Table 6 shows the ®nal individual cell compositions (i.e.,after Phase IV) in terms of machines and the averageusage of each machine in each cell.

The Phase III and IV assignments of individual part-operations to machines and the cells to which individualmachines are allocated are shown in Table 5. For exam-ple, part 2 requires operations (2,3,4,8). At Phase III,these operations are assigned to machine types (7,1,4,5),respectively as shown in column 3; and individual ma-chines to which the load for these part-operations isallocated are placed in cells (3,1,3,3), respectively. Thus,for this part, one operation is performed in cell 1 whilethe other three operations are performed in cell 3. Afterimplementing Phase IV, the load on machine type 1 in cell1 (for the second operation on part 2) was rerouted toanother machine of the same type in cell 3 and thus part 2is completely processed in cell 3. As can be seen fromTable 5, the Phase IV improvement procedure was able toreroute exceptional operations of ®ve parts (nos 1, 2, 4, 5,and 11) such that all of them could be completely pro-cessed within a cell. However, parts 3, 10, and 13 are stillprocessed in multiple cells. The cell compositions shownin Table 6 are the individual cell compositions after im-plementing Phase IV of the FCF method. Although cellsize variability was set at 0% , cell 4 (two machines) issmaller than the remaining cells (each with ®ve machines).This is because, to split up the 17 machines into four cells,the maximum size of a cell is set to d�17=4�e � 5.Phase IV of the procedure also focuses on evaluating

the generated cell design. In terms of routing ¯exibility ofthe current design, we can see that all part types except 3,10, and 13 have all operations assigned to machines in-cluded in a single cell. Thus, there is at least one cell tocompletely process these part types. To assess volume¯exibility of the cellular system shown in Table 5 (Phase

Table 5. Part-operation/machine-cell assignments

Part Operationset

Phase III Phase IV

Mach. ass. Cell ass. Mach. ass. Cell ass.

1 (1,4,6,9,11) (8,4,9,1,2) (2,2,2,3,2) (8,4,9,4,2) (2,2,2,2,2)2 (2,3,4,8) (7,1,4,5) (3,1,3,3) (7,1,4,5) (3,3,3,3)3 (1,2,4,8) (8,7,4,5) (2,3,3,3) No change4 (1,4,7,9,11) (8,4,9,1,2) (2,2,2,3,2) (8,4,9,4,2) (2,2,2,2,2)5 (1,7,9,11) (8,9,1,2) (2,2,3,2) (8,9,4,2) (2,2,2,2)6 (6,8,10,12) (9,5,10,6) (1,1,1,1) No change7 (3,6,8,10,12) (1,9,5,10,6) (1,1,1,1,1) No change8 (2,3,4,5,8,9) (7,1,4,3,5,1) (3,3,3,3,3,3) No change9 (3,4,5,8,9) (1,4,3,5,1) (3,3,3,3,3) No change10 (3,4,6,8,10,12) (1,4,9,5,10,6) (1,3,1,1,1,1) No change11 (3,8,10,12) (1,5,10,6) (3,1,1,1) (1,5,10,6) (1,1,1,1)12 (1,4,7,11,12) (8,4,9,2,6) (2,2,2,2,2) No change13 (10,11,12) (10,2,6) (4,4,1) No change14 (10,11) (10,2) (4,4) No change15 (10,11) (10,2) (4,4) No change16 (1,4,11,12) (8,4,2,6) (2,2,2,2) No change17 (1,7,11,12) (8,9,2,6) (2,2,2,2) No change18 (10,11) (10,2) (4,4) No change19 (10,11) (10,2) (4,4) No change

Table 6. Cell compositions after Phase IV

Cell Machine typeassigned

Numberof machines

Av. usageper machine*

1 1 1 0.5765 1 0.5836 1 0.5329 1 0.65210 1 0.485

2 2 1 0.7564 1 0.4896 1 0.3318 1 0.6969 1 0.465

3 1 1 0.4703 1 0.2344 1 0.6395 1 0.4217 1 0.447

4 2 1 0.62110 1 0.461

*This is computed as the average utilization per machine divided by theaverage availabilityof the machine of that type (i.e., Am).

606 Askin et al.

IV assignment) and Table 6, we increased the demand (inbatches per year) for every part type by the same per-centage until the utilization of one or more machinesexceeded the availability (i.e., Am; see Table 2). For thiscell system, this occurred when the number of batches forall parts was increased beyond 145% of their currentdemand in Table 3. Hence, the volume ¯exibility �d� is45% for the given cell system. In this case, the criticalmachine was machine type 2 in cell 2.Regarding mix ¯exibility, the maximum percentage

change in volume for an individual part �dp� that can behandled by the current system (keeping all other part vol-umes ®xed) is shown in Table 7. Further, for each indi-vidual part, we identify the related bottleneck machinetype. Thus, Table 7 shows that for part type 3, we canaccommodate at most a 740% increase in batch volumebefore machine type 7 in cell 3 becomes overloaded.To assess the ¯exibility of the system to respond to the

introduction of new parts, we proceed as follows. First,we generate the demand (in batches) and batch size (inunits) for a potential new part by using a discrete uniformdistribution with parameters (4,11) and (100,961), res-pectively. Secondly, for the part, we randomly generatethe number of operations required for processing byusing a discrete uniform distribution with parameters(2,6). These parameter settings are chosen on the basis ofthe minimum and maximum demand, batch sizes andnumber of operations required for all parts shown inTable 3. Thirdly, we generate the individual operationsrequired to process the part randomly from the currentset of operations as follows. The probability �pj� that aparticular operation is selected ®rst is based on the fre-quency �fj� that that operation is used for processing

parts in the current mix and is computed as fj=�P

j fj�.Once the ®rst operation has been selected, the secondoperation for the new part is selected conditioned on the®rst. Thus, the probability that operation j1 is selected asthe second operation given that operation j is the ®rstoperation is computed as f 1j1=�

Pj f 1j�. In this case f 1j1

is de®ned as the frequency that operation j1 appearsalong with operation j in the operation sequences for allparts. This sequential procedure is repeated until all theoperations in the operation set for a new part have beengenerated.Using the procedure described, we assessed the ¯exi-

bility of the current system to respond to new part in-troduction. For the designed system, 68% of 40 randomlygenerated new parts could be processed in the currentsystem if we allowed intercellular ¯ows of batches (i.e.,c1 � 68%); 48% of the new parts could possibly beprocessed within a single cell (without considering ma-chine availability, i.e., c2 � 48%); and 40% of the newparts could be processed within a single cell (i.e., withoutintercellular materials ¯ows and explicitly consideringmachine availability, i.e., c3 � 40%).This concludes our illustration of the FCF method. We

now turn to a discussion of the experiment performed tovalidate the usefulness of the routing and volume ¯exi-bility improvement algorithms (Phase IV of the method)and to provide guidelines as to how the parameter set-tings impact routing, volume and mix ¯exibility.

5. Experimental analysis

In order to evaluate the FCF method two experimentswere performed. The ®rst experiment used a designedexperiment to test the value of the various steps and pa-rameter settings in the FCF method to obtain ¯exible celldesigns. The second experiment compared the FCFmethod to alternative methods for ®nding dense cells.

5.1. Evaluation of algorithmic steps and parameter settings

For the ®rst experiment, the factors investigated and theirtreatment levels are:

(1) maximum allowable machine usage U (set atPhase I): 85% and 100%;(2) parameter a (set at Phase III): 0, 0.5, 1;(3) number of cells C (set at Phase III): low, medium,

high;(4) cell size variability V (set at Phase III): 0% and

20%;(5) improvement algorithm G (Phase IV): none (N),

routing (R), volume (V), routing and volume (RV).

To carry out the experimental analysis, we used eightdata sets (seven published and the hypothetical data setused in Section 4). The seven data set sizes were:

Table 7. Mix ¯exibility of the cell design

Part �p� Mix ¯ex. (%) �dp� Critical mach.

1 930 22 260 73 740 74 350 25 270 76 270 97 300 98 1960 79 370 410 220 911 390 1,512 280 213 790 214 380 215 340 216 290 217 260 218 360 219 560 2

Methodology for designing ¯exible cellular systems 607

· P = 15, M = 10, J = 10 (see [31]);· P = 19, M = 12, J = 12 (see [32]);· P = 20, M = 8, J = 8 (see [33]);· P = 24, M = 14, J = 14 (see [34]);· P = 43, M = 16, J = 16 (see [7]);· P = 43, M = 14, J = 14 (see [7]); and· P = 15, M = 95, J = 10 (see [16]).

For each data set and factor/treatment level, we gener-ated ®ve di�erent cell designs by randomly varying theoperation±machine type input data. (The procedure usedis as follows. First, using a discrete uniform distribution,we generate the number of operations that can be per-formed by a machine type. The parameters for the dis-tribution are data set dependent. Let this be equal to Bm.Secondly, we randomly pick Bm operations from the totalnumber of possible operations (with an equal probabilityof picking any operation) and assume that any machineof type m can perform all these operations. Note that wealways ensure that there is at least one machine type thatcan carry out each generic operation.) ANOVA was usedto analyze the results. (Owing to space limitations, we donot present the ANOVA results in the paper. The inter-ested reader is referred to [35].) A summary of the resultsreveals that:

1. Statistically signi®cant increases in routing ¯exibilitywere observed when the routing ¯exibility improvementalgorithm was implemented in Phase IV. Similarly, whenthe volume ¯exibility improvement algorithm was im-plemented, statistically signi®cant increases in volume¯exibility were noted. These results point to the useful-ness of implementing the improvement algorithms inPhase IV of the FCF method.2. To maximize routing ¯exibility, the parameter a

should be set to focus on current part-operation assign-ments, and fewer cells with highly variable cell sizesshould be created.3. To maximize volume and mix ¯exibility (assessed by

using c3), larger slack capacity should be built into thesystem. Further, a larger number of cells is preferred formaximizing volume ¯exibility, whereas more variable cellsizes are preferred for maximizing mix ¯exibility.

Summary recommendations for users of the FCF methodbased on these observations are shown in Table 8.

5.2. Comparison with other cell design approaches

Co and Araar [16] proposed a procedure for assigningoperations tomachines andmachines to cells for the case ofduplicate machines. These steps correspond to Phases IIand III of ourmethodwith the exception that those authorsconcentrated on designing cells for the current static partmix. They demonstrate their method on a 15-part, 63-machine problem. Their method produces a grouping ef-®ciency [7] of 0.68 for this problem. Setting U � 100% ,

C � 6 and a � 1:0, and executing Phases II and III weobtained an improved grouping e�ciency of 0.72. Celldensity was improved and the number of exceptional partoperations was reduced, indicating improvement in bothaspects of the e�ciency measure. This was further im-proved in Phase IV to obtain an e�ciency of 0.73. By re-assigning part operations, the FCF method was able toincrease the comprehensiveness of the cell system.

6. Conclusions and implications

In this paper we have proposed a cell formation methodthat is unique in many respects. First, this method allowsthe user to design cellular systems that are ¯exible (interms of responsiveness to part demand and part mixchanges as well as in terms of routing ¯exibility). Themethod also allows the user to modify the preliminary celldesign to increase the routing and/or part volume ¯exi-bility. Secondly, the decision hierarchy provides a com-putationally feasible approach for solving problems ofrealistic size while including consideration of importantfactors such as alternate machine types and costs.Thirdly, the FCF method can generate multiple cellularcon®gurations depending upon user inputs. The user cantrace out multiple solution alternatives by varying pa-rameters on cell size, emphasis on current or future op-erations, and type of ¯exibility preferred. Additionally,the user can modify individual phases to achieve desiredobjectives or test varying strategies. Fourthly, through anexperimental analysis, we have also been able to provideguidelines on the use of the procedure.The method balances costs and ¯exibility. Phase I

minimizes ®xed machine cost plus direct processing costs.Phase II assigns operations to machines to take advantageof part similarities and operation ¯exibility of machines.In turn, this serves to limit ®nal material handling costsdue to intermachine and intercell transfers. Phase IIIgroups machines by balancing actual material handlingwith system ¯exibility. Finally, Phase IV can be used toincrease ¯exibility of the ®nal system design. The pro-

Table 8. Experimentally determined preferred parameter set-tingsa

Factor/ Performance measure

user inputRouting ¯ex.(F)

Volume ¯ex.(d)

Mix ¯ex.(c3)

U ± LOW LOWa HIGH ± ±C LOW HIGH ±V HIGH ± HIGHG R RV RV

a A ``±'' entry in the table indicates that the Factor/user input did notsigni®cantly impact the ¯exibility measure.

608 Askin et al.

posed methodology provides one approach for dealingwith the complexity of the overall cellular manufacturingcon®guration problem. Although we have provided evi-dence that the individual phases of the procedure arecomputationally e�cient (polynomially bounded heuris-tics) and highly competitive with previous procedures foreach subproblem, we believe the major contribution lies inthe integrated methodology for designing cells with con-sideration of ¯exibility for a dynamic environment. Still,there is clearly potential for further research into explor-ing the cost±¯exibility tradeo�.

Acknowledgements

This paper is based on work supported in part by theNational Science Foundation under grant no. DDM-92-15432.

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Methodology for designing ¯exible cellular systems 609

Biographies

Ronald G. Askin is a Professor and Acting Department Head ofSystems and Industrial Engineering at the University of Arizona. DrAskin received a B.S. in Industrial Engineering from Lehigh Univer-sity, and an M.S. in Operations Research and Ph.D. in Industrial andSystems Engineering from Georgia Institute of Technology. He is aFellow of the IIE, and an active member of INFORMS, ASQC, andSME, having previously served as Program Co-Chair for the 1993Phoenix ORSA/TIMS meeting, Chair of the ORSA Technical Sectionon Manufacturing Management, Chair of the Statistics Division ofASQC, and Program Chair for the 1996 IE Research Conference. He isthe former editor of the IIE Transactions on Design and Manufacturing.He has published in various professional journals, predominantly inthe areas of design and operational analysis of manufacturing systems.His current research is in the areas of cellular manufacturing and Just-in-time production. Dr. Askin co-authored the text Modeling andAnalysis of Manufacturing Systems, which was awarded the 1994 IIEJoint Publishers Book of the Year Award. Other awards he has re-ceived include the Shingo Award for Excellence in ManufacturingResearch, IIE Transactions Development and Applications Award(coauthor), the ASEE/IEE Eugene L. Grant Award (coauthor), and aNational Science Foundation Presidential Young Investigator Award.In addition, he has been the academic advisor for several award-win-ning student research projects. Dr. Askin has consulted with a variety

of industrial companies in the general areas of facility design, processcapability analysis and improvement, integrated product and processdesign, material ¯ow control, and performance measurement of man-ufacturing systems.

Hassan M. Selim is a member of the Faculty of Engineering andTechnology at Helwan University in Cairo, Egypt. He received hisPh.D. from the College of Business and Public Administration at TheUniversity of Arizona. His research interests are in the area of ¯exiblemanufacturing and cellular manufacturing systems.

Asoo. J. Vakharia (a senior member of IIE) is an Associate Professor inthe Department of Decision and Information Sciences at the Universityof Florida. He received his Ph.D. degree in Operations Managementfrom the University of Wisconsin-Madison. Asoo's research focuses onthe design and control of Cellular Manufacturing Systems as well as onthe integration of the marketing and operations functions in service®rms. His prior research has been published in Annals of OR, DecisionSciences, Discrete Applied Mathematics, IIE Transactions, Journal ofOperations Management, Naval Research Logistics, and InternationalJournal of Production Research. He is currently an Associate Editor forthe International Journal of Flexible Manufacturing Systems and alsoserves on the Editorial Review Board of the Journal of OperationsManagement.

610 Askin et al.