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European Journal of Operational Research 192 (2009) 414–428

Production, Manufacturing and Logistics

Integrated cellular manufacturing systems design withproduction planning and dynamic system reconfiguration

Steve Ah kioon a, Akif Asil Bulgak a, Tolga Bektas b,*

a Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Boulevard,

West Montreal (Quebec), Canada H3G 1M8b School of Management, University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom

Received 5 September 2006; accepted 14 September 2007Available online 25 September 2007

Abstract

This paper presents and analyzes a comprehensive model for the design of cellular manufacturing systems (CMS). A recurring themein research is a piecemeal approach when formulating CMS models. In this paper, the proposed model, to the best of the authors’ knowl-edge, is the most comprehensive one to date with a more integrated approach to CMS design, where production planning and systemreconfiguration decisions are incorporated. Such a CMS model has not been proposed before and it features the presence of alternateprocess routings, operation sequence, duplicate machines, machine capacity and lot splitting. The developed model is a mixed integernon-linear program. Linearization procedures are proposed to convert it into a linearized mixed integer programming formulation. Com-putational results are presented by solving some numerical examples, extracted from the existing literature, with the linearizedformulation.� 2007 Elsevier B.V. All rights reserved.

Keywords: Flexible manufacturing systems; Cellular manufacturing systems design; Production planning; Integer programming

1. Introduction

Cellular manufacturing (CM) is an application of group technology in manufacturing and involves the processing of acollection of similar parts (part families) on dedicated clusters of machines or manufacturing processes (cells). Withincreased global competition and shorter product life cycles, there has been a shift to demands for mid-volume andmid-variety product mixes. Job shops and flow lines cannot provide the efficiency and flexibility to adapt to such needs.Cellular manufacturing systems (CMS) have emerged to cope with such production requirements and have been imple-mented with favorable results (Wemmerlov and Hyer, 1989). Benefits include reduced set-up times, material handling,in-process inventory, better production efficiency and quality as well as market response time. It is important for companiesthat use CMS to invest sufficient time in the design and planning phase of any CMS implementation. The benefits of CMScan only accrue to the company if strategic decisions are based upon results obtained from models that accurately describeits structural and operational features.

We formulate our CMS model as a mathematical program and elaborate on its properties in Section 3, followed bysome preliminary computational work on small-scale examples. The linearization procedure and the resulting mixed inte-

0377-2217/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2007.09.023

* Corresponding author. Tel.: +44 (0) 23 8059 8969.E-mail address: T.Bektas@soton.ac.uk (T. Bektas).

S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428 415

ger linear problem are in Section 4. Section 5 shows the numerical examples used in the course of this study. Section 6presents the discussions. Conclusions and future research are in Section 7.

2. Literature review

Singh (1993) and Selim et al. (1998) have reviewed the many structured approaches to the Cell Formation Problem.Singh (1993) has classified the approaches to cell formation into the categories of coding and classification systems,machine-components group analysis methods, graph theoretic methods, neural networks and heuristics, fuzzy clusteringbased methods, similarity coefficient based mathematical models, knowledge and pattern recognition methods, mathemat-ical and heuristic methods. Selim et al. (1998) have categorized the approaches as cluster analysis, graph partitioning,descriptive procedures (part families identification, machine groups identification, part families/machine grouping), math-ematical programming (linear, integer and quadratic programming, goal and dynamic programming), artificial intelligenceapproaches. CM design is presently being researched with the emphasis on more integrated models and solution method-ologies (Mansouri et al., 2000; Selim et al., 1998; Singh, 1993). Some cell formation techniques have explicit or implicitobjectives, such as the minimization of intercellular movements that do not necessarily produce the best overall cell per-formance or satisfy application-specific objectives. Many design attributes have to be considered when designing CMS.

This paper aims at presenting an integrated CMS model that incorporates several important design attributes to bettermodel the realities of manufacturing. Our model considers the determination of the optimal cellular layout given differentknown demands of a mix of products within several planning horizons. Very few authors have considered production plan-ning and system reconfiguration simultaneously in CMS design models (Mungwattana, 2000; Song and Hitomi, 1996). Pro-duction planning brings certain advantages since inventory can be kept and used to satisfy known demands in futureperiods (Chen and Cao, 2004). With general-purpose machines being available, it is essential to consider alternate routingsfor each and every product. This means that the production of a certain part type can be split into two or more processroutings, where the same sequence of operations is carried out on different machine types. Mathematical programming isbeing extensively used to model CMS design problems that integrate other aspects of manufacturing. Existing research hasincorporated several manufacturing attributes, with the latter being assigned varying degrees of importance within the cor-responding model. To gain a better understanding of the research status quo, we present the most important manufactur-ing attributes in Table 1. We provide a more integrated approach in CMS design since our study encompasses more ofthese attributes within a single model.

Throughout the literature review and mathematical model development, emphasis has been put upon the works ofNsakanda et al. (2006), Chen and Cao (2004, 2005), Song and Hitomi (1996) and Mungwattana (2000) as they discussthe important issues of CMS design that are further addressed in this research. Nsakanda et al. (2006) consider inter-cellular, intracellular and outsourcing costs. The following are included in their work: part demands, machine capacitylimits, multiple process plans and alternative routings for each part type, the processing sequence of parts, the trade-offbetween intercellular and intracellular costs and the option of outsourcing. They do not take into account the presenceof multiple machines of the same type being available and production planning aspects. Using a genetic algorithm witha local optimizer (hybrid GA) to solve their model, their model tries to find the optimal assignment of machines tocells, the number of parts to be manufactured with each possible process plan and routing and the sizes of each cell,and the number of parts of each type to be outsourced. Chen and Cao (2004) have included the coordination of pro-duction planning within CMS design. An integrated approach was taken to study a production planning problem overa certain planning horizon in the CMS with fixed charge cost. The objective is to minimize the total cost includingintercellular material handling cost, fixed cell set-up cost, production set-up cost and product inventory cost in the sys-tem. By assuming that there are single process plans, limited machine capacities and multiple machines of the sametype, their model decides whether cells are to be formed and how many of these, which machine types and how manyto be allocated to the formed cells, the best time periods for part processing in a production planning horizon of multi-ple time periods. Song and Hitomi (1996) have also integrated production planning with a cellular manufacturing sys-tem. They formulate a mixed integer programming (MIP) model that allows inventory and layout adjusting (systemreconfiguration) for multiple periods (with varying dynamic deterministic product demand). The costs consideredinclude inventory holding, group setup, material handling (intercellular and intracellular) and layout adjustment.Machine capacity and demand satisfaction (taking into account inventory held) constraints are included and the pro-posed model is solved using the Benders decomposition approach. However, this model excludes multiple process plans,alternate routings and the outsourcing option. Mungwattana (2000) focused on dynamic and stochastic productionrequirements and routing flexibility. Two models working in a dynamic environment are developed: one where thedemands are uncertain (stochastic) and the other where the demands are deterministic. So, the model is compatible withdynamic and stochastic production requirements. This paves the way to various benefits that are exploited in the model,in particular the consideration of cell reconfiguration from period to period. The model also considers the machine allo-cation problem, machine operating and amortized costs together with machines that have multifunctional capabilities.

Table 1CMS design attributes used in this research

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

a b a b a b a b c d e a b

Proposed model in paper · · · · · · · · · · · · · · · · ·Caux et al. (2000) · · · · · · ·Chen (2001) · · · ·Chen and Cao (2005) · · · · · ·Chen and Cao (2004) · · · · · · ·Defersha and Chen (2006) · · · · · · · · · · · · · ·Gupta et al. (1996) · · ·Jayaswal and Adil (2004) · · · · · ·Mungwattana (2000) · · · · · · · · · · · · ·Nsakanda et al. (2006) · · · · · · · ·Selim et al. (1998) · · · · · ·Solimanpur et al. (2004) · · · · · ·Song and Hitomi (1996) · · · · · · ·Spiliopoulos and Sofianopoulou (2003) · · · · ·Su and Hsu (1998) · · · · · · · ·Uddin and Shanker (2002) · · ·Vakharia and Chang (1997) · · ·Yin and Yasuda (2002) · · ·Zhao and Wu (2000) · · · · ·

Important design attributes for CMS design

1a. Intercellular materialhandling cost

1b. Intracellular materialhandling cost

4. Inventory holding in productionplanning

7a. Robust cell configuration7b. Agile cell configuration

10. Operation sequence

2. Part internal productioncost

5a. Stochastic demandrequirements5b. Deterministic demandrequirements

8a. Machines with multiple8b. Machine with limited capacities8c. Machine operating cost8d. Machine maintenance andoverhead cost8e. Machine relocation and cost copies

11. Lot splitting

3. Subcontracting cost 6. Multi-period planning 9. Alternate routingsa. Chosen from user-specifiedroutingsb. Chosen from all possible optionsbased on operation and machinetype

12. Cell size limits – upper andlower bound

416 S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428

However, the model ignores intracellular material handling, production planning (inventory holding) and outsourcing.A dynamic deterministic version of the model is solved via an optimal solution procedure. Defersha and Chen (2006)presented the latest model in CMS design that aims at incorporating various aspects of manufacturing in addition tothe cell formation problem. They take into consideration dynamic cell reconfiguration, lot splitting, operation sequence,multiple units of identical machines, machine capacity, workload balancing among cells, operation cost, outsourcing,setup cost, cell size limits and machine adjacency constraints. They have developed a non-linear mixed integer modeland have followed some linearization steps in order to obtain a mixed-integer linear problem and solve various scenar-ios. This is also a well-integrated model but it does not take into consideration certain issues that are addressed withinthis present paper: formation of compact cells by considering intra-cellular movement of parts, production planning(inventory holding), internal production cost.

In most of these papers, numerous assumptions have been made. The most recurrent and important one is that cell sizeis specified a priori. This assumption also holds in the model within this paper as it helps maintain mathematical tracta-bility. The number of cells in the system is user-specified in order to reduce the computational difficulty of assigning m

machines to k non-empty cells. If the number of cells is not pre-specified the problem is understood to be NP-complete(Garey and Johnson, 1979). However, when the number of cells is pre-specified, Venugopal and Narendran (1992) haveshown that the problem of machine allocation to cells becomes simpler.

The proposed model belongs to the class of NP-complete problems as it includes the machine-part grouping problem. Itis widely accepted in the literature that the whole problem of designing a CMS, taking into account the numerous phases

S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428 417

and criteria involved, belongs to the class of NP-complete problems (Ballakur, 1985; King and Nakornchai, 1982). Fur-thermore, the additional features of the proposed model increase its complexity and combinatorial nature. One phaseof the proposed model is to determine the number and types of machines to assign for each part type; then to determinethe process routings from multiple possible ones (machines can perform more than one operation and come in multiplecopies); Logendran et al. (1994) have shown that the problem involved in this phase is NP-hard. The consideration of cel-lular reconfiguration for a planning horizon is in itself a problem that belongs to the class of NP-complete problems. Chen(1998) has developed a model that considers system reconfiguration in terms of machine relocation, showing that solvingthe model is NP-complete. The proposed model in this paper is, therefore, NP-hard since it integrates the problem of cellformation, system reconfiguration along with the consideration of multiple process routings, production planning, machinecapacities and availabilities.

3. Formal description of the problem and of the mathematical model

3.1. Formal description of the problem

The integrated CMS model in this research consists of the classical cell formation problem, bridged with the machineallocation, part routing, production planning problems and system reconfiguration. The classical cell formation problem(CFP), see Fig. 1, is to group machines into cells, parts into part families and to assign part families to cells to form rel-atively independent cells. Within a manufacturing environment there is, on the one hand, machine types which have dif-ferent operational capabilities, limited capacities and possible multiple copies. On the other hand, there are different parttypes that each requires a certain sequence of operations and processing capacity to complete production. The overall strat-egy of forming this CMS environment is to group parts that require similar operations into the same cell whilst at the sametime assigning machines to these cells so that they are capable of performing these operations.

Fig. 1 shows a representation of a solution to the classical CFP where machines have been grouped into cells, e.g.machines 2 and 4 are in cell I, and where parts are grouped into part families in such a way that each of these part familiesare assigned to a cell containing the machines required for the operations. Part types 1 and 3 (belong to part family I) arethus assigned to machine cell I as they require machines 2 and 4 for processing, The ‘X’, termed an exceptional element,represents intercellular movement of part type 3 between cells I and II. This happens because machine cell I does not con-sist of machine type 1 required for an operation on part 3. So the latter has to be transferred to machine cell II as it containsa copy of machine type 1. Too many exceptional elements give rise to increased intercellular movement and relativelydependent cells, which can increase coordination effort between cells. It must be noted that this is a very simple situationas machine capacities, availabilities, multiple routings and other aspects of manufacturing are ignored. The ‘O’ represents avoid and this occurs because machine type 3 is only required to process parts 2 and 4 in cell II. Too many voids can lead tothe formation of large inefficient cells and can give rise to scheduling problems. A routing is defined as a sequence of specificmachines or work centres that a part type has to go through in order to complete all of its required operations. This impliesthat a part type can have alternate routings when machines have multiple capacities, capabilities and availabilities. There

PART

FAMILY I

PART

FAMILY II

Part

Type1 3 2 4 5

Machine

Type

1 1 2

1 1 4

MA

CH

INE

CE

LL

I

X 1 1 1 1

1 1 O 3

MA

CH

INE

CE

LL

II

Fig. 1. Classical cell formation problem.

418 S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428

may be numerous feasible configurations because of these alternate routings. For these possible configurations, we arelooking for the one that minimizes the total cost of material handling in terms of intercellular (from cell to cell) and intra-cellular (within the same cell) movement. Within the context of a multiple planning period, it is assumed that the demandsfor parts vary in a deterministic way. This allows the model to consider producing more in a period so that this inventorycan be used in future periods or to subcontract parts when internal production is not feasible either due to insufficientmachine capacity or uneconomical repercussions. Simultaneously, the CMS environment can respond by undergoingsystem reconfiguration where machines are relocated from one cell to another and/or where new routings are selectedfor the part types. The overall objective is to minimize the total cost of machine relocation, machine maintenance, machineoperation, outsourcing, inventory holding, internal part production cost, intercellular material and intracellular materialhandling. The notations used for the model are presented followed by the objective function, constraints and modelproperties.

Sets:

P ¼ f1; 2; 3; . . . ; Pg index set of part types

KðpÞ ¼ f1; 2; 3; . . . ;Kpg index set of operations indices for part type p

M ¼ f1; 2; 3; . . . ;Mg index set of machine types

Qðk; pÞ ¼ f1; 2; 3; . . . ;Qkpg index set of machine types that can perform operation k for part type p

C ¼ f1; 2; 3; . . . ;Cg index set of cells

T ¼ f1; 2; 3; . . . ; Tg index set of time periods

Model parameters:

dm relocation cost per machine type m per period 8m 2M

am maintenance and overhead costs per machine type m 8m 2M

em operating cost per unit time per machine type m 8m 2M

op outsourcing cost per part type p 8p 2P

Hp inventory holding cost per part type p per time period 8p 2P

bp production cost per part type p 8p 2P

IEp intercellular material handling cost per part type p 8p 2 P

IAp intracellular material handling cost per part type p 8p 2 P

Dp(t) demand for part type p at time period t 8p 2P; 8t 2T

Am(t) quantity of machine type m available at time period t 8m 2M ; 8t 2T

Tm capacity of one unit of machine type m during one period 8m 2M

BU upper cell size limit

BL lower cell size limit

ekpm processing time of operation k on machine m per part p 8k 2KðpÞ; 8p 2 P; 8m 2 Qðk; pÞ

Model decision variables

Xkpmc(t) number of parts of type p processed by operation k on machine type m in cell c at time t 8k 2KðpÞ; 8p 2P; 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T

Op(t) number of parts p to be outsourced at time t 8p 2 P; 8t 2T

Vp(t) quantity of inventory of part type p kept in period t and carried over to period t + 1 8p 2 P, 8t 2T

Nmc(t) number of machines of type m present at cell c at time t 8m 2M ; 8c 2 C; 8t 2T

NþmcðtÞ number of machines of type m added to cell c at time t 8m 2M ; 8c 2 C; 8t 2T

N�mcðtÞ number of machines of type m removed from cell c at time t 8m 2M ; 8c 2 C; 8t 2T

Zkpmc(t) 1 if operation k for part type p is carried out on machine type m in cell c at time t, and 0 otherwise8k 2KðpÞ; 8p 2P; 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T

3.2. The integer programming model

The developed CMS model is now formulated as a non-linear mixed integer program and is referred to as model F:

S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428 419

MinimizeXt2T

Xc2C

Xm2M

dm � NþmcðtÞ þ N�mcðtÞ� �

ð1:1Þ

þXt2T

Xc2C

Xm2M

am � NmcðtÞ ð1:2Þ

þXt2T

Xp2P

Xk2KðpÞ

Xm2Qðk;pÞ

Xc2C

em � ekpmX kpmcðtÞ ð1:3Þ

þXt2T

Xp2P

op � OpðtÞ ð1:4Þ

þXt2T

Xp2P

H p � V pðtÞ ð1:5Þ

þXt2T

Xp2P

Xk2KðpÞ

Xm2Qðk;pÞ

Xc2C

bp � X kpmcðtÞ ð1:6Þ

þXt2T

Xp2P

Xk2KðpÞnfKpg

Xc2C

IEp �X

m2Qðkþ1;pÞX kþ1;pmcðtÞ �

Xm2Qðk;pÞ

X kpmcðtÞ !�����

����� ð1:7Þ

þXt2T

Xp2P

Xk2KðpÞnfKpg

Xm2Qðk;pÞ

Xc2C

IAp � Zkþ1;pmcðtÞ �X

n2Qðk;pÞnfn¼mgZkpncðtÞ � X kpncðtÞ� � !

ð1:8Þ

subject to

X kpmcðtÞ 6 M � ZkpmcðtÞ 8k 2KðpÞ; 8p 2P; 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T ð2ÞXm2Qð1;pÞ

Xc2C

X k¼1;pmcðtÞ þ V pðt � 1Þ � V pðtÞ þ OpðtÞ ¼ DpðtÞ 8p 2P; 8t 2T ð3ÞXc2C

NmcðtÞ 6 AmðtÞ1 8m 2M ; 8t 2T ð4ÞXm2M

N mcðtÞ 6 BU 8c 2 C; 8t 2T ð5ÞXm2M

N mcðtÞP BL 8c 2 C; 8t 2T ð6Þ

NmcðtÞ ¼ Nmcðt � 1Þ þ NþmcðtÞ � N�mcðtÞ 8m 2M ; 8c 2 C; 8t 2T ð7ÞXp2P

Xk2KðpÞ

X kpmcðtÞ � ekpm 6 T m � NmcðtÞ 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T ð8ÞXc2C

Xm2Qðkþ1;pÞ

X kþ1;pmcðtÞ ¼Xc2C

Xm2Qðk;pÞ

X kpmcðtÞ 8k 2KðpÞ=fKpg; 8p 2 P; 8t 2T ð9Þ

X kpmcðtÞP 0 and integer 8k 2KðpÞ; 8p 2P; 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T ð10ÞOpðtÞP 0and integer 8p 2P; 8t 2T ð11ÞV pðtÞP 0 and integer 8p 2P; 8t 2T n fT g ð12ÞNmcðtÞP 0;NþmcðtÞP 0;N�mcðtÞP 0 and integer 8m 2M ; 8c 2 C; 8t 2T ð13ÞZkpmcðtÞ 2 f0; 1g 8k 2KðpÞ; 8p 2 P; 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T ð14Þ

The objective function consists of eight cost components. Term (1.1) is the machine relocation (reconfiguration) cost.Term (1.2) denotes the machine maintenance and overhead costs. Term (1.3) represents the machine operating cost. Term(1.4) represents the outsourcing cost. Term (1.5) corresponds to the inventory holding cost. Term (1.6) is the internal partproduction cost. Term (1.7) denotes the intercellular material handling cost. This cost is incurred whenever consecutiveoperations of the same part type are carried out in different cells. For instance, assume that operation k of part type p

is done on machine m in cell c at time t. If the next operation, k + 1, of part p is done on any machine but in another cell,then there is intercellular cost. The latter is directly proportional to the number of parts moved between these two cells. Inthis model, the unit intercellular is expressed only as a function of the part type being handled. For instance, larger parttypes are more expensive to handle but the cost of moving equal quantities of this part type from, say cell 1 to cell 2, is thesame as from cell 4 to cell 5. Term (1.8) is the intracellular material handling cost. This cost is sustained whenever consec-utive operations of the same part type are done in the same cell but on different machines. For instance, say that operationk of part type p is done on machine m in cell c at time t. If the next operation, k + 1, of part p is done on any other machine

but within the same cell, then there is intracellular cost. The latter is directly proportional to the number of parts moved

420 S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428

between the different machines in the same cell. Again, it is assumed that the unit intra-cellular cost only depends on thetype of part type being handled. Within a cell, the distances between any two machines are considered to be the same. So,larger part types are more expensive to handle but would fetch the same intra-cellular cost were they moved from machinetype 1 to type 2 or from machine type 4 to type 5 within the same cell.

In model F, constraint (2) shows that the number of parts produced internally can be positive only if Zkpmc(t) = 1, thatis, it has been decided that part p would be produced internally by operation k on machine m in cell c. Constraint (3) showsthat part demand can be satisfied in period t through internal production and/or outsourcing and/or inventory carried overfrom the previous period. It is not necessary to formulate constraint (3), more specifically the term representing internalpart production, based on all of the operations for part type p. It is sufficient to formulate the demand constraint (3) basedonly on the first operation (Kp = 1) of each part type, since the material conservation flow constraint (9) ensures that allconsecutive operations of a part type consist of equal production quantities, thus always resulting in fully processed inter-nally produced parts. Therefore, in the proposed model, any inventory carried over to future periods consist of parts thathave been fully processed. Constraint (4) implies that the total number of machines of each type assigned to cells is less thanor equal to the number of machines of the same type that are available during the period considered. The cell size is user-defined through constraints (5) and (6), where the cell size lies within lower and upper bounds. Constraint (7) accounts formachine relocation, where the number of machines of a type assigned to a cell during a period is equal to the number ofmachines in the previous period, plus the number of machines added and minus the number of machines removed. Con-straint (8) shows the how machine capacity constraints are respected. Constraint (9) is for material flow conservation. Con-straints (10)–(14) are the logical binary and non-negativity integer requirements on the decision variables.

3.3. Properties of the model

The model takes a holistic approach in CMS design in that it integrates several aspects of manufacturing which can begrouped into the key categories of production planning, subcontracting, dynamic system reconfiguration and cell forma-tion. These are further detailed in the following.

Production planning and subcontracting. The model recognizes the fact that, under a multiple period planning horizon,the system can be set to various levels of internal production, inventory holding and outsourcing during each time period.Depending on the demand and total cost of meeting that demand, the system could produce some surplus parts in a timeperiod which can be used to supply part or all of the demand for the same part in future planning periods. Furthermore,due to limited machine capacities, outsourcing can be used to procure some of the required parts to meet the marketdemand. It is assumed that there is no initial inventory and that no inventory is left over at the end of the whole planninghorizon.

Dynamic system reconfiguration. As a result of dynamic deterministic demands within the planning horizon, a CMS con-figuration for a period might not be optimal or even feasible for other planning periods. Therefore, the model presents tothe user the best configuration within each planning period in terms of the types and number of machines assigned to cells,part types assigned to cells and part routings. Allowing system reconfiguration enhances the flexibility of CMS to respondto variations in part mix demands.

Cell formation. Part routings: a routing is defined in terms of the possible machines that can be used to perform therequired operations for a part. With multi-functional machines and multiple copies of each machine type allowed in thesystem, the presence of alternate routings is important since this gives more flexibility in deciding upon CMS configura-tions. Within this research, we have developed a model in such a way that the system decides on the best routes insteadof the user specifying pre-determined routes. In most literature, routes are pre-specified relying upon the operationsrequired by a part and machine capabilities. We have designed our model in such a way that all possible routes can co-exist and that more than one route can be chosen to make a part. Operation sequence: this directly influences the routesthat are to be assigned to a part type. Moreover, operation sequence is important in the calculation of intercellular andintracellular material handling cost since it gives a more accurate count of the number of times that a part either has tomove between cells or between different machines within the same cell. Lot splitting: this allows more than one machineto simultaneously perform the same required operation on a part within a specified route. This means that a part operationcan be split onto two machines within the same cell or even in different cells. Lot splitting is useful since it improvesmachine utilization, reduces material handling and movement and can enable better process routings to be found bythe model.

3.4. Preliminary computational experience

Four small-scale problems were solved in order to validate the proposed mathematical formulation that represents theintegrated CM model of this research. The models were solved using Extended Lingo (LINDO Systems Inc., 2005), withthe results shown in Table 2. The total number of variables and constraints can be calculated as follows:

Table 2Results of the implementation of model F to four small-scale scenarios in terms of solution time and solution status

Number ofpart types

Number oftime periods

Number ofm/c types

Numberof cells

Number ofoperations perpart

Number ofvariables

Number ofconstraints

Time taken to reachsolution (seconds)

Solutionstatus

1 2 3 2 2 88 52 4 Localoptimum

1 5 3 2 2 220 130 1290 Localoptimum

2 5 3 2 2–3 410 345 1955 Localoptimum

3 5 3 3 2–3 795 645 N/A N/A

S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428 421

• Total number of variables = 2PT + 3MCT + 2KpPMCT, where Kp is the number of operations for part type p.• Total number of constraints = 2MCT + 2CT + MT + PT + (2Kp � 1)PMCT.

For the first three scenarios, local optimum solutions were obtained after the shown computational time. No solutionwas obtained for scenario four (three part types), even after 50 hours (481,332 iterations). The figures given in Table 2imply that it is not possible to guarantee getting a global optimum solution when solving model F with Extended Lingo.Furthermore, as the size of the problems grows, it will be more arduous to solve this model. We, therefore, propose in thenext section, some linearization steps to transform this mixed integer non-linear model in to a mixed integer linear pro-gramming formulation.

4. Linearization of the model

In this section, we present the linearization procedure and the mixed integer linear programming formulation for theproblem.

4.1. Procedure

The linearization procedure that we propose here consists of two steps that are given by the two propositions statedbelow. The non-linearity results from terms (1.7) and (1.8), therefore, these two terms will be linearized using the followingauxiliary continuous variables YPkpmc(t), YMkpmc(t), Zkpmnc(t) and Wkpmnc(t). Each proposition for linearization is followedby a proof and a numerical example that illustrates the meaning of each auxiliary (linearization) variable and the expres-sions where they are used.

Proposition 1. The non-linear component of the objective function (1.7) in problem F can be linearized by the following

transformation jXk+1,pmc(t) � Xkpmc(t)j = YPkpmc(t) + YMkpmc(t), under the following set of constraints:

X kþ1;pmcðtÞ � X kpmcðtÞ ¼ YP kpmcðtÞ � YMkpmcðtÞ 8k 2KðpÞ=fKpg; 8p 2P; 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T ð15Þ

Proof. Consider the following three cases:

(i) Xk+1,pmc(t) > Xkpmc(t). By (15), YPkpmc(t) � YMkpmc(t) > 0. Since this is a minimization problem and the objectivefunction cost coefficients are strictly positive, YMkpmc(t) = 0 and YPkpmc(t) = Xk+1,pmc(t) � Xkpmc(t) will hold in theoptimal solution.

(ii) Xk+1,pmc(t) < Xkpmc(t). By (15), YPkpmc(t) � YMkpmc(t) < 0. In this case, again with the coefficients of YPkpmc(t) andYMkpmc(t) being strictly positive, the objective function will ensure that YPkpmc(t) = 0 and thusYMkpmc(t) = Xkpmc(t) � Xk+1,pmc(t) will hold in the optimal solution.

(iii) Xk+1,pmc(t) = Xkpmc(t). By (15), YPkpmc(t) � YMkpmc(t) = 0. In this case, both YPkpmc(t) = 0 and YMkpmc(t) = 0, willhold in the optimal solution since their coefficients in the objective function are strictly positive. h

YPkpmc(t) and YMkpmc(t) are both non-negative continuous variables that are used to linearize an expression of the typegiven in term (1.7), one where the absolute term creates non-linearity. When considered together in the expression YPkpmc-(t) + YMkpmc(t), under constraint (15), they give the magnitude of the quantity of parts involved in intercellular movement.This indicates the quantity of parts moved from operation k (performed on machine type m in cell c at time t) to anothercell where the next operation k + 1 in the process routing takes place (performed on any machine type m capable of doingoperation k + 1).

Table 3Examples to the linearization stated in Proposition 1

Example 1 Example 2

Xkpmc(t) 5 10Xk+1,pmc(t) 10 5Xk+1,pmc(t) � Xkpmc(t) 5 �5jX kþ1;pmcðtÞ � X kpmcðtÞj 5 5YPkpmc(t) + YMkpmc(t) 5 5YPkpmc(t) � YMkpmc(t) 5 �5YPkpmc(t) 5 0YMkpmc(t) 0 5

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We illustrate how the linearization in Proposition 1 works through the two numerical examples shown in Table 3.Xkpmc(t) and Xk+1,pmc(t) can take any positive general integer value. The non-negative variables (YPkpmc(t) and YMkpmc(t))take values in such a way that the computations for intercellular cost using the linear terms generate the same final resultsas would be obtained by the original variables and non-linear terms.

In Table 3, consider example 1 where Xkpmc(t) = 5 and Xk+1,pmc(t) = 10. This means that five units of part p were pro-cessed by operation k on machine type m in cell c at time t. The next operation for p, that is k + 1, involves the processingof 10 units on the same machine m in the same cell c at time t. This implies that five additional parts p have been transferredfrom another cell (where they underwent operation k) to destination cell c for the current operation k + 1. The termjXk+1,pmc(t) � Xkpmc(t)j gives the magnitude of the quantity of parts transferred from different cells between two successiveoperations of the same part p in the same period t. Thus YPkpmc(t) and YMkpmc(t) take non-negative values in such a waythat YPkpmc(t) + YMkpmc(t) also gives the correct magnitude for the same intercellular material flow; in this case five unitsof parts p. Upon considering example 2 in Table 3, it is seen that Xkpmc(t) = 10 and Xk+1,pmc(t) = 5. Out of the 10 units ofpart p processed by operation k on machine type m in cell c, only 5 undergo the next operation k + 1 within the samecell. Thus, the other five units of part p have been transferred to another cell to undergo the next operationk + 1. jXk+1,pmc(t) � Xkpmc(t)j = 5 gives the correct magnitude for the intercellular flow involved, as does YPkpmc(t) +YMkpmc(t) = 5, where both YPkpmc(t) and YMkpmc(t) have taken the required non-negative values.

Proposition 2. The non-linear intracellular cost terms in the objective function (1.8) of problem F can be linearized withWkpmnc(t) = Zkpmnc(t) Æ Xkpnc(t), where Zkpmnc(t) = Zk+1,pmc(t) Æ Zkpnc(t), under the following sets of constraints:

ZkpmncðtÞP Zkþ1;pmcðtÞ þ ZkpncðtÞ � 1 8k 2KðpÞ=fKpg; 8p 2P; 8m 2 Qðk; pÞ; 8n 2 Qðk; pÞ=fn ¼ mg; 8c2 C; 8t 2T ð16Þ

W kpmncðtÞ 6 M � ZkpmncðtÞ 8k 2KðpÞ; 8p 2P; 8m 2 Qðk; pÞ; 8n 2 Qðk; pÞ=fn ¼ mg; 8c 2 C; 8t 2T ð17Þ

where M is a large positive value.

Proof. This can be shown for each of the two possible cases that can arise.

(i) Zk+1,pmc(t) Æ Zkpnc(t) = 0. Such a situation arises under one of the following three sub-cases:

(a) Zk+1,pmc(t) = 1 and Zkpnc(t) = 0. 8k 2KðpÞ=fKpg; 8p 2P; 8m 2 Qðk; pÞ; 8n 2 Qðk; pÞ=fn ¼ mg; 8c 2 C;

8t 2T(b) Zkþ1;pmcðtÞ ¼ 0 and ZkpncðtÞ ¼ 1:8k 2KðpÞ=fKpg; 8p 2P; 8m 2 Qðk;pÞ; 8n 2 Qðk;pÞ=fn¼ mg; 8c 2C; 8t 2T(c) Zk+1,pmc(t) = 0 and Zkpnc(t) = 0. 8k 2KðpÞ=fKpg,8p 2 P, 8m 2 Qðk; pÞ; 8n 2 Qðk; pÞ=fn ¼ mg; 8c 2 C;

8t 2T

In all of the three sub-cases given above, the value of the non-linear term (1.8) is given by Zk+1,pmc(t) Æ Zkpnc(t) Æ Xkpnc(t) = 0, under any value of Xkpnc(t). In this case, constraint (16) implies Zkpmnc(t) P � 1 and since Zkpmnc(t)has a strictly positive cost coefficient, the minimizing objective function ensures that Zkpmnc(t) = 0. Thus, under con-straint (17), Wkpmnc(t) = 0. Hence, the value of the non-linear term (1.8) and Wkpmnc(t) have equivalent values for thiscase.

(ii) Zkþ1;pmcðtÞ � ZkpncðtÞ ¼ 1: 8k 2KðpÞ=fKpg; 8p 2P; 8m 2 Qðk; pÞ; 8n 2 Qðk; pÞ=fn ¼ mg; 8c 2 C; 8t 2T.

Such a situation arises when Zk+1,pmc(t) = Zkpnc(t) = 1 so that the non-linear term (1.8) can take any non-negative valuesince Zk+1,pmc(t) Æ Zkpnc(t) Æ Xkpnc(t) = Xkpnc(t). In this case, constraint (16) implies Zkpmnc(t) P 1 and since Zkpmnc(t) has astrictly positive cost coefficient, the minimizing objective function ensures that Zkpmnc(t) = 1. Then under constraint (17), it

Table 4Examples to the linearization stated in Proposition 2

Zkpnc(t) Zk+1,pmc(t) Zkpmnc(t) Xkpnc(t) Wkpmnc(t) Zk+1,pmc(t) Æ Zkpnc(t) Æ Xkpnc(t)

Example 1 1 1 1 20 20 20Example 2 1 0 0 Any value 0 0Example 3 0 1 0 Any value 0 0Example 4 0 0 0 Any value 0 0

S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428 423

follows that Wkpmnc(t) 6M, showing that Wkpmnc(t) can also take any non-negative value. Therefore, the non-linear term(1.8) and Wkpmnc(t) are equivalent for this case. h

Zkpmnc(t) and Wkpmnc(t) are non-negative continuous variables used to linearize the term (1.8), where the non-linearity iscaused by the multiplication of binary variables Zkpnc(t) and Zk+1,pmc(t) as well as integer variables Xkpnc(t). After linear-ization, the linearization variable Zkpmnc(t) shows whether there is intercellular movement. When operation k of part p attime t is done on machine type n in cell c and this is followed by the next operation, k + 1, being done on any machine typewithin the same cell, there is intracellular movement involved so that Zkpmnc(t) > 0. Otherwise, whenZkpmnc(t) = 0, there isno intracellular movement as the consecutive operations are done in different cells. Wkpmnc(t) gives the magnitude of thenumber of parts involved in the intracellular movement.

Proposition 2 is illustrated through the four numerical examples shown in Table 4, where the possible values for Zkpnc(t),Zk+1,pmc(t) and Xkpnc(t) are analyzed. Wkpmnc(t) can, therefore, be calculated and compared with the non-linear term (1.8).

In example 1 of Table 4, we are dealing with two successive operations on part p. Zkpnc(t) = 1 shows that operation k isperformed on machine type n in cell c whilst Zk+1,pmc(t) = 1 shows that the next operation (for the same part type p) is donewithin the same cell c on another machine m. Therefore, Zkpmnc(t) = 1 implies that there is intracellular movement, that istransfer of parts between different machines in the same cell c. Wkpmnc(t) = 20 shows that 20 units of part p are involved inthe intracellular transfer between machines m and n in cell c, when the operation sequence is k and k + 1 for part type p intime period t. Compared with Zk+1,pmc(t) Æ Zkpnc(t) Æ Xkpnc(t) = 20, which is the number of parts involved in the intracellulartransfer, it is seen that the linear term, Wkpmnc(t), gives the same correct value of 20. In examples 2, 3 and 4, Zkpmnc(t) = 0indicates that the consecutive operations k and k + 1 are performed in different cells, so that there is no intracellular move-ment. Therefore, Wkpmnc(t) = 0, under any value of Xkpnc(t). It is observed that Zk+1,pmc(t) Æ Zkpnc(t) Æ Xkpnc(t) = 0, whichalso indicates that there is no intracellular material flow, showing that the non-linear term (1.8) and Wkpmnc(t) areequivalent.

4.2. The linearized mixed integer programming formulation

We now present the transformed mixed integer linear programming formulation, now referred to as model F2:

Minimize ð1:1Þ–ð1:6ÞþXt2T

Xp2P

Xk2KðpÞ

Xc2C

Xm2Qðk;pÞ

IEp � ðYP kpmcðtÞ þ YMkpmcðtÞÞ ð18Þ

þXt2T

Xp2P

Xk2KðpÞ

Xm2Qðk;pÞ

Xc2C

Xn2Qðk;pÞnfn¼mg

IAp � W kpmncðtÞ ð19Þ

subject to (2)–(14)

X kþ1;pmcðtÞ � X kpmcðtÞ ¼ YP kpmcðtÞ � YMkpmcðtÞ 8k 2KðpÞ=fKpg; 8p 2P; 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T

ZkpmncðtÞP Zkþ1;pmcðtÞ þ ZkpncðtÞ � 1 8k 2KðpÞ=fKpg; 8p 2P; 8m 2 Qðk;pÞ; 8n 2 Qðk;pÞ=fn¼ mg; 8c 2C; 8t 2T

W kpmncðtÞ 6 M � ZkpmncðtÞ 8k 2KðpÞ; 8p 2P; 8m 2 Qðk; pÞ; 8n 2 Qðk; pÞ=fn ¼ mg; 8c 2 C; 8t 2T

YP kpmcðtÞ; YMkpmcðtÞP 0 8k 2KðpÞ; 8p 2 P; 8m 2 Qðk; pÞ; 8c 2 C; 8t 2T

ZkpmncðtÞP 0 8k 2KðpÞ; 8p 2P; 8m 2 Qðk; pÞ; 8n 2 Qðk; pÞ=fn ¼ mg; 8c 2 C; 8t 2T

W kpmncðtÞP 0 8k 2KðpÞ; 8p 2P; 8m 2 Qðk; pÞ; 8n 2 Qðk; pÞ=fn ¼ mg; 8c 2 C; 8t 2T

For this linearized mixed integer mathematical formulation, the number of constraints can be calculated as follows:PT + MT + 2CT(1 + M) + PMCT(2MKp + Kp �M � 1). The total number of variables can be calculated as follows:2PT + 3MCT + 4KpPMCT + 2KpPMCT(M � 1). The total number of constraints and variables are both used to rankthe problems in terms of size.

424 S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428

5. Numerical examples

The small-scale models used in Section 3.4 for the preliminary computational experience on the non-linear model F werenow tested using the linearized model F2. In addition, data sets of larger sizes have been used from Chen and Cao (2004),Defersha and Chen (2006), Nsakanda et al. (2006), Mungwattana (2000). All the computational experiments were per-formed on a 3 GHz Pentium 4 workstation running Linux. The models were solved using CPLEX 9.0.1 (ILOG Inc.,2006). Since the models developed and used by these authors are different from our proposed model, we added some addi-tional cost parameters for the features not addressed in their data sets and model (e.g. inventory holding cost). Theunknown cost parameters, which proved more difficult to get, were extracted by cross-referencing between the data setscontaining them and then incorporated within the other data sets that are missing this information. Therefore, all ofthe data sets used in each solved numerical example contain values within the same range in terms of unit costs. To betterillustrate this, such ‘grafting’ enabled numerical values of certain parameters from some problems (e.g. unit inventory costsof problems 5–7) to be exported to other ones (problems 1–4 and 8–11). To ensure that the data are as close as possible toreality, real-life raw data were used, collected from a company running in a CMS environment. Emphasis was put on thenumber of part types, machine types, operations and number of cells. The summarized data from the surveyed companyare shown in Table 5.

Each one of the numerical examples used is solved as an integrated model and the solving ability of CPLEX is beingtested as the problem size increases (number of variables and constraints). We compare the computational time takenand optimality gap (difference between current feasible solution and best bound on optimal solution) with respect tothe various problem sizes.

The results are presented in Table 6, where elapsed time and optimality gap are shown for each problem instance. Alsoshown are the number of time periods for which the design is performed, the number of cells c used in each problem and thenumber of operations required for each part type.

It is evident that the required computational times (and number of iterations) increase as the problem sizes are increased.All of the small-scale problems (problems 1–5) were successfully solved in less than 5 seconds with the highest number ofvariables and constraints encountered being 2685 and 2580, respectively. The medium-sized problems (problems 6–8)required more computational times but were solved within 2051 seconds (34 minutes). The largest number of variablesand constraints in these medium-sized problems were 53,445 and 40,941, respectively. Problem 9 is considered to be amedium to large-scale problem and it could be solved within 6556 seconds (1.82 hours) with an optimality gap of

Table 5Summary of design data collected from company

Number of products (part types p) Number of operations per product type Number of cells within company Number of machine types

12 5–9 5 42

Table 6Summary of computational results from selected data sets

Problemscenario

Source Numberof parttypes

Number oftimeperiods

Numberof m/ctypes

Number ofoperationsper part

Numberof cells

Numberofvariables

Number ofconstraints

Timeelapsed(seconds)

Optimalitygap (%)

1 Small-scale 1 2 3 2 2 232 340 0.00 0.002 Small-scale 1 5 3 2 2 580 400 0.00 0.003 Small-scale 2 5 3 2–3 2 1310 915 0.01 0.004 Small-scale 3 5 3 2–3 3 2685 2580 0.10 0.005 Chen and Cao

(2004)4 3 4 2–3 2 2496 1764 4.25 0.01

6 Chen and Cao(2004)

5 6 5 2–4 4 20,580 12,588 46.93 0.01

7 Mungwattana(2000)

11 2 10 2–4 3 35,864 33,234 114 0.00

8 Nsakanda et al.(2006)

15 1 15 2–3 3 53,445 40,941 2051 0.01

9 Defersha and Chen(2006)

25 2 10 4–9 5 319,400 277,320 6556 0.01

10 Nsakanda et al.(2006)

40 1 20 2–9 5 242,230 209,160 12,952 0.04

11 Nsakanda et al.(2006)

20 1 20 2–9 5 462,340 409,250 11,040 38.41

S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428 425

0.01%. Therefore, the small to medium-scale problems (1–9) were solved within reasonable computational times. Problems10 and 11 are considered to be large-scale ones with the highest number of variables and constraints being 462,340 and409,250, respectively. Both of these two problems scenarios proved to be too difficult for the simplex-based branch andcut algorithm of CPLEX to solve since no optimal solution was obtained after 3 hours of computation. In fact, CPLEXstopped solving both problems (10 and 11) even before the time limit due to insufficient memory. After 12,952 seconds(3.5 hours) of computational effort, problem 10 had not yielded any optimal solution but had produced a feasible solutionwhich had an optimality gap of 0.04% (difference between current feasible solution and best bound on optimal solution).After 11,040 seconds (3.07 hours), CPLEX could not find an optimal solution for problem 11. The feasible solutionobtained for the latter at that point had a considerable optimality gap (38.41%). This clearly indicates that the branchand cut algorithm of CPLEX is unable to produce good quality solutions in reasonable computational times for large-scaleproblems of the CMS model, even when using the linear model F2.

6. Discussion of results

The largest problem that was solved to optimality was the one selected from Defersha and Chen (2006). We select thisnumerical example, which represents the largest model solved in Defersha and Chen (2006), to illustrate the various fea-tures of the solution obtained for our developed CMS model. Defersha and Chen (2006) used Extended Lingo (LINDOSystems Inc., 2005) to solve the problem. Using this problem instance and a linearized version of their model, an optimalsolution was obtained. The objective function values obtained in this paper and in Defersha and Chen (2006) cannot becompared because of the different objective costs involved. Thus, a comparison of these different objective costs wouldnot yield any meaningful information. The computational times required could also not be compared because these werenot addressed in Defersha and Chen (2006). However, a meaningful comparison can be made in terms of the CM decisionaspects provided by the two different solutions. Our approach is comparably more general since we also considered thefollowing aspects in addition to those considered by Defersha and Chen (2006): production planning in terms of inventoryholding, the outsourcing option, internal part production cost and intra-cellular material handling cost.

The solution obtained with the model in this paper on problem 9 is detailed out in the rest of this section and simulta-neously compared with that obtained in the model from Defersha and Chen (2006). Table 7 shows the part-machine cellallocation for period 1 using our model.

The cells are formed as shown in Table 7, with elements (exceptional elements) that lie outside of the diagonal blockrepresenting movement of parts from cell to cell (intercellular). In the above configuration, parts can be produced in multi-ple cells and/or different machines (lot splitting). This is shown through the sample in Table 8, demonstrating the selectedroutings (visited machines and cells) for part type P25 in period 1. P25 is processed in cells 1, 2 and 3 using machines M7,M8 and M9.

Part P25 has to undergo six operations in order to complete production. The first operation processes 1275 parts onmachine M7, cell C1. There is lot splitting for the second operation as 275 parts are again processed on machine M7, cellC1, whilst 1000 parts are transferred to cell C2 on machine M7. The third operation when 1225 parts are transferred tomachine M8 in cell 3 and the remaining 50 parts on the same machine type M8 but in cell 4. The entire of the fourth oper-ation takes place on machine M8 in cell 3, after the 50 parts of P25 from cell 4 are transferred to M8 in cell 3. The fifthoperation consists of sending 675 parts back to cell 1 on machine M8 and 600 parts back to cell 2 on machine M8. The finaloperation takes place in cell 2 on machine type M9, when all of the 1275 parts are sent there.

It has to be noted that although there is no demand for part P25 in period 1 the system produces 1275 parts whichwill be kept as inventory. This inventory will be used to satisfy part of the demand occurring in period 2. The system hasutilized any surplus machine capacities to produce this inventory, expecting machine capacities in period 2 to be insuf-ficient to meet the whole demand and outsourcing to be economically unprofitable. System reconfiguration also takesplace as a result of the changes in demand for the parts. The solution obtained from Defersha and Chen (2006) doesnot involve any internal production for part P25 in period 1; hence no comparison can be made in terms of the differentroutings that were chosen.

In this paper, the possible routings for part p are defined in terms of both the sequence of operations required and themachines that have the operational capabilities. Thus, the co-existence of multiple possible routings for a part is allowed.Table 9 shows the routings chosen for part P25 during period 2 using our model. The processing of parts occurs onmachine types M7, M8 and M9 and is spread in cells C1, C2, C4 and C5. The routings obtained by Defersha and Chen(2006) are also shown in Table 9 since the production of part P25 occurs entirely in period 2. It is observed that productionalso occurs on the machine types M7, M8 and M9 since the latter are the only ones capable of performing the requiredoperations for P25. However, their model dictates that production for P25 occurs wholly in cell C3. For instance, operation1 involves machine M7 in cell C3 in Defersha and Chen (2006) whereas in our model, operation 1 also done on machineM7, is split in two different cells, C2 and C5. It has to be also noted that the quantity of P25 processed in period 2 using ourmodel is 3225 compared to 4500 in Defersha and Chen (2006).

Table 7Part-machine cell allocation for period t = 1

Cell Quantity of machines P7 P23 P17 P19 P18 P25 P1 P15 P8 P9 P16

C1 M1 = 5 · · · · ·M2 = 4 · ·M3 = 4 · ·M4 = 2 · · ·M5 = 3 · ·M7 = 4 ·M8 = 1M9 = 2

C2 M3 = 1 · ·M5 = 1 ·M6 = 3 ·M7 = 1 ·M8 = 2M9 = 2M2 = 1 ·

C3 M3 = 5 · ·M4 = 2 · ·M5 = 3M6 = 4 ·M8 = 1 ·

C4 M1 = 5 ·M2 = 2 ·M4 = 2 ·M5 = 3M6 = 3 ·M8 = 3 · ·M9 = 1 ·

C5 M2 = 2 ·M4 = 2 ·M7 = 5 · ·M8 = 1 ·

Table 8Routings for part type P25 in period 1

1 2 3 4 5 6

Operations and routings obtained with our developed CMS model

1275 M7/C1 275 M7/C1 1225 M8/C3 1275 M8/C3 675 M8/C1 1275 M9/C21000 M7/C2 50 M8/C4 600 M8/C2

Operations and routings obtained by Defersha and Chen (2006)

N/A N/A N/A N/A N/A N/A

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Table 10 shows how part demands are met for part types P25 and P16 through internal production, inventory holdingand outsourcing during the two planning periods. The solution obtained by Defersha and Chen (2006) is different in thattheir CM system does not produce any part type P25 in the first time period. Rather, the demand for part P25 is satisfiedwholly through internal production in period 2. Within our model, since the option of keeping inventory is considered, thesystem can leverage the excess capacity of capable machines to start production of P25 in period 1.Although the outsourc-ing option is present within the model from Defersha and Chen (2006), it is given a large value in the numerical exampleswith a view to favoring cell formation only. Therefore, no outsourcing information was obtained from Defersha and Chen(2006). Since their model does not consider production planning, there is no inventory held. By simultaneously consideringall of the three ways of satisfying the demand for part P25, the model in this paper presents the possible production plangiven in Table 10, showing a higher flexibility in meeting the part demand. This can provide tangible advantages if there areunexpected machine breakdowns or reduced machine capacities.

P16 is also only required to satisfy demand in period 2, with part of it being produced internally during period 1, accord-ing to our model. The demand for P16 is satisfied by outsourcing 5783 parts in period 2 and using the 217 parts kept in

Table 9Comparison of routings for part type P25 in period 2

1 2 3 4 5 6

Operations and routings obtained with our developed CMS model

1000 M7/C2 450 M7/C1 1550 M8/C1 3225 M8/C4 3225 M8/C4 725 M9/C12225 M7/C5 2775 M7/C5 1675 M8/C2 2500 M9/C4

Operations and routings obtained by Defersha and Chen (2006)

4500 M7/C3 4500 M7/C3 4500 M8/C3 4500 M8/C3 4500 M8/C3 4500 M9/C3

Table 10Comparison of production plans for P25 and P16

Time period/parts Our developed model Defersha and Chen (2006)

P25 P16 P25 P16

T1Internal production 1275 217 0 0Outsourced 0 0 0 0Inventory 1275 217 0 0Demand 0 0 0 0

T2Internal production 3225 0 4500 6000Outsourced 0 5783 0 0Inventory 1275 0 0 0Demand 4500 6000 4500 6000

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inventory from period 1, thereby supplying 6000 parts for P16. However, unlike part P25, part P16 has to be outsourced inthe second period to satisfy the demand. This can be due to insufficient machine availabilities and capacities for therequired operations. The solution for P16 in this paper is different from that in Defersha and Chen (2006) since in the latterP16 is only produced internally in period 2. Therefore, it can be inferred again that the simultaneous consideration of inter-nal production, outsourcing and inventory holding present an increased flexibility in the production plan.

Since the model in this paper also considers the minimization of the amount of intracellular cost involved in consecutiveoperations, the cellular layout is more compact. In fact, a trade-off is made between intercellular and intracellular move-ment by simultaneously minimizing both of these costs within the objective function. This is important since, on the onehand, high intra-cellular costs imply that the cells are quite large, reducing the effectiveness of the CM system. On the otherhand, high amounts of intercellular movement increase the dependence of cells on one another, increasing the coordinationeffort required between cells and adversely affecting the benefits of CM systems. In many of the models, the trade-offbetween intracellular and intercellular material handling is not sufficiently addressed as intracellular movement is ignored.This type of trade-off is only considered in Nsakanda et al. (2006) but the model in the latter does not consider multipleperiods, inventory holding, internal production cost, lot splitting, machine maintenance and overhead costs, multi-periodplanning and dynamic cell reconfiguration.

7. Conclusions and future research directions

We have developed a CMS model that integrates production planning, dynamic system reconfiguration, multiple rout-ings and several other attributes. With this mixed integer non-linear model, some random small-scale problems were solvedup to a certain problem size. Motivated by the computational difficulty of the non-linear formulation, some linearizationtechniques have been proposed to transform the model into a mixed integer linear programming formulation. The small-scale models were again solved using the new linearized model, resulting in better computational times. For instance, thelargest small problem was successfully solved in less than a second. Some more experiments were performed with this lin-earized model using problem data sets from the literature to find the size limits of solving the linearized model usingCPLEX. These were medium-sized problems which could be solved within up to 2 hours, the largest one being 25 parttypes by 10 machines. Larger sized problems (e.g. 20 part types · 20 machine types) prove to be more difficult to solve usingthe proposed approach. The solutions from this integrated approach have shown that additional CM structural and oper-ational design decisions that were not considered in previous research can be addressed with the proposed model. The nextstep in research is the investigation of the use of meta-heuristics, especially Tabu Search, Simulated Annealing and GeneticAlgorithm, to solve problems of larger scale for this integrated CMS model.

428 S. Ah kioon et al. / European Journal of Operational Research 192 (2009) 414–428

Acknowledgements

This research was in part supported by grants from the Natural Sciences and Engineering Council (NSERC). Theauthors also acknowledge the useful comments and suggestions of the two anonymous referees.

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