Int. J. Agile Systems and Management, Vol. 3, Nos. 1/2, 2008 37

Copyright © 2008 Inderscience Enterprises Ltd.

Design of flexible manufacturing cell considering uncertain product mix requirement

Amit Rai Dixit* Department of Mechanical Engineering and Mining Machinery Engineering, Indian School of Mines University, Dhanbad, India E-mail: amitraidixit@gmail.com *Corresponding author

P.K. Mishra

Department of Mechanical Engineering, M.N.N.I.T., India E-mail: pkm@mnnit.ac.in

Abstract: In the real-world manufacturing, the requirement of number and the attribute of product may not be known exactly at the time of designing the manufacturing cell. The design of manufacturing cell requires to identify machine groups and part families in Cellular Manufacturing (CM) system. The success of a CA system is sensitive to fluctuations in the demand for products and the product mix. A large number of papers on manufacturing cell design have been published so far, but very few of them have considered random product mix constraint at the design stage. Little work has been reported which incorporated real-life production parameters like operation sequence, production volume, batch size, material handling capacity, processing time, setup time, machine capacity and cost factors. Considerations of these important parameters make the cell formation problem more complex, but realistic. This paper presents a new formulation of the part family/machine cell formation problem that addresses the dynamic nature of the production environment by considering a multiperiod forecast of product mix and demand during the formation of part families and machine cells. The computational procedure of the algorithm has been illustrated by an example. Numerical results indicate that the proposed methodology is flexible, efficient and may be effective even for industrial problems.

Keywords: cellular manufacturing systems; random product mix; intercell movement; operation sequence.

Reference to this paper should be made as follows: Dixit, A.R. and Mishra, P.K. (2008) ‘Design of flexible manufacturing cell considering uncertain product mix requirement’, Int. J. Agile Systems and Management, Vol. 3, Nos. 1/2, pp.37–60.

Biographical notes: Amit Rai Dixit is the Research Fellow in the Department of Mechanical Engineering at Motilal Nehru National Institute of Technology, India. He is also a Senior Lecturer in the Department of Mechanical Engineering and Mining Machinery Engineering at Indian School of Mines University, India. He received his BTech in Mechanical Engineering from BIET, India and his MTech in Production Engineering from MNNIT, India.

38 A.R. Dixit and P.K. Mishra

His research interests include group technology, advanced manufacturing technology, operations research and supply chain management. He is a Member of ASME and IAENG.

P.K. Mishra is a Professor in the Department of Mechanical Engineering at MNNIT, India. He received PhD from University of Roorkee, India. He received his BTech in Mechanical Engineering from MNREC, Allahabad and his MTech in Production Engineering from MNREC, Allahabad. His areas of research include FMS, CAPP, CAM and Genetic Algorithm. He is a Member of IE (India).

1 Introduction

Cellular Manufacturing (CM) has been proved as an attractive compromise between flow line production and job shop production as it incorporates the attributes of both of them. CM is basically based on the philosophy of Group Technology. Group Technology (GT) is an approach to manufacturing and engineering management that helps to manage diversity by capitalising on underlying similarities in products and activities. In the manufacturing context, GT has been defined as a manufacturing philosophy identifying similar parts and grouping them together into families to take advantage of their similarities in manufacturing and design. Grouping the production equipment into machine cells, where each cell specialises in the production of part families, is called as CM. So, CM is the application of the GT philosophy in manufacturing. CM is concerned with the creation and operation of manufacturing cells which are dedicated to the production of a set of part families. In order to introduce CM, it is necessary to identify parts and machine types to be used in the cellular configuration.

The first problem faced in implementing CM is cell formation. Cell formation deals with the identification of the family of parts and the group of machines to process these parts. The problem of cell formation is defined as:

“If the number, types, and capacities of production machines, the number and types of parts to be manufactured, and the routing plans and machine standards for each part are known, which machines and their associated parts should be grouped together to form cells?” (Wu and Salvendey, 1993).

In some cells, the definition of cell formation is expanded to allow choice of processing operations to achieve specific features. Since the last three decades, a considerable amount of researches have been directed to ease this type of problem. Burbidge (1971) developed an intuitive method, namely Production Flow Analysis (PFA) which is relatively easy to implement. PFA may be suitable for the small size problem, but it would definitely have difficulties coping with real life cell formation problems when the machine-part incidence matrix becomes more complex because of problem size. Many approaches have been developed to deal with the difficulties of intuitive method. These approaches are usually classified into part-oriented approaches (based on part characteristics) and process-oriented approaches (based on production methods). The part-oriented techniques usually employ some classification and coding system, and analyse parts for their similarities in design features and functionalities. However, these do not influence directly the configuration of manufacturing cells (Choobineh, 1988). The process-oriented approaches to the cell formation are based on manufacturing

Design of flexible manufacturing cell 39

data such as production methods, part routing information and process plans. The process-oriented approach is classified into four groups namely: Descriptive methods, Array-based methods, similarity coefficient methods and other analytical methods (Yasuda and Yin, 2001).

Most of the suggested algorithms/models consider binary machine-part incidence matrix A = [aij], with

1 if part type requires machine type

0 otherwiseij

i ja

⎧= ⎨⎩

The binary part-machine matrix only represents the occurrence of an operation on a machine, but not the actual intercell movements of parts. Most of the cell formation methods based on binary part-machine matrix do not consider the following:

• Operation sequence of parts: the sequence of operation is an important manufacturing attribute in the design of CM systems. The operation sequence may be defined as an ordering of the machines on which the part is sequentially processed. The sequence of operation has an impact on the flow of material in the system. An intermediate operation of a component to be performed outside its cell requires two intercell transfers while the first or last operation requires only one such transfer (Choobineh, 1988). Vakharia and Wemmerlov (1990), Logendran (1991), Wu and Salvendy (1993), Yin and Yasuda (2002), Defersha and Chen (2006) and Dixit and Mishra (2007) considered operation sequence matrix in the design of manufacturing cell. Sarker and Xu (1998) presented a brief review of cell formation based on the operation sequences.

• Random product mix: product mix refers to a set of part types to be manufactured. In practice, the product requirement is uncertain at the design stage of the manufacturing system.

• Production volume: the production volume is the total number of different parts to be manufactured in the given period. The merits of incorporating production volume were depicted by Gupta and Seifoddini (1990).

• Number of intercell movements: an ideal clustering consists of mutually exclusive attributes between two clusters of data. However, it is rare to achieve such perfect clustering results in reality. Clustering results often contain exceptional elements in the case of binary data, an exceptional element indicates a clustering discrepancy between data and attributes in terms of unity elements. A 1 outside the diagonal block is called an exceptional element and it indicates only one intercell movement. Parts are generally processed in batches with unequal volumes. In fact, the volume of intercell movement depends upon the batch size and capacity of material handling device. If the batch size is large and the capacity of material handling system is limited, then the volume of intercell movement will be large.

• Machine capacity: the basic requirement, in the design of a CM system, is to have the adequate capacity to process all the parts. The machine capacity has an impact on the workload induced by the part. In binary part-machine matrix, the 0s inside the diagonal block are referred as VOIDS. A void indicates that a machine assigned to a cell is not required for the processing of a part in the cell. The corresponding machine is said to be underutilised. But the actual utilisation of machine depends upon the machine capacity and the workload imposed by the parts on it.

40 A.R. Dixit and P.K. Mishra

• Processing and setup times: the processing and setup time required by the part on a machine is another important parameter. It influences workload and machine utilisation.

• Material handling capacity: the material handling capacity is considered as the number of parts transported to the machines/cells. So, it depends upon the load carrying capacity and the pallet size.

In the recent years, some of the researchers considered different production parameters in cell formation problem (Adil et al., 1996; Askin and Subramaniam, 1987; Choobineh, 1988; Dixit and Mishra, 2004; Harhalakis et al., 1990a, b; Kumar and Vannelli, 1987; Kusiak and Chow, 1987; Wu and Salvendy, 1993).

But, little work has been reported in the design of CM systems under uncertain product mix environment. However, it has gained interest from researchers in recent years.

2 Past review

Sankaran and Kasilingam (1993) developed a mixed integer programming model for different size of cells within a single layout. In this model, the intracell transfer costs of parts produced in a cell are based on the size of the cell that is defined in terms of the number of machines in the cell. The model only considers a single period. This model was tested on a small problem with very limited set of data consisting of three cost scenarios. Each of the three scenarios required over 6500 sec of CPU time on a mainframe computer to find the model’s optimal cost and a solution. For more large sized problems, the computational time would be prohibitive, so, a heuristic procedure is proposed. The heuristic procedure starts out with all parts assigned to a single cell. The procedure then attempts to create cells for individual parts so that intracell transfer costs are reduced. This procedure is tested on the example problem and very quickly generates the optimal solution for each of the three cost scenarios.

Harhalakis et al. (1994) proposed a two-stage solution methodology for the robust cellular manufacturing design. They focused on product demand variations over a system design horizon which was divided into elementary time periods. The objective was to obtain a cellular design with the minimum expected intercell material handling cost over the entire design horizon. In the first stage, a production volume for each product was determined and corresponding cell configuration was obtained by using a heuristic method in the second stage. The first stage began with mapping the forecast of product demand to a set of feasible production volumes with given resource capacity constraints. If sufficient capacity existed to produce all products at their demand level, product demand gave feasible production volumes. Otherwise, the projected demands were used in a linear program that gave a set of feasible production volumes such that profit was maximised. Given several product mixes, a procedure for calculating the joint probabilities for every feasible production mix was presented. The joint probabilities were used to evaluate the mean production volume for each product. The heuristic method proposed by Harhalakis et al. (1990a) was used to obtain a near-optimal cell formation in the second stage. This paper assumed that the product mix for each period was the same; no new products were introduced nor old products discontinued.

Design of flexible manufacturing cell 41

Product demand in each period was assumed to be normally distributed, where the mean and standard deviation were time-invariant. Additional machines of same kind were not considered.

Wicks (1995) addressed the dynamic nature of production environment by considering a multiperiod forecast of the product mix and resource availability during the formation of part families and machine cells. The objectives considered were the minimisation of intercell material handling cost, the minimisation of investment in additional machines and the minimisation of the cost of system reconfiguration over the planning horizon. A mixed-integer formulation of the multiperiod Part Families and Machine Cells (PF/MC) formation problem was developed. The multiperiod PF/MC formation procedure belongs to machine-grouping solution strategy where machine cells are formed first, followed by the assignment of parts to the machine cells. The assignment of machines to cells over the planning horizon is made via a genetic algorithm. A heuristic for assigning parts to cells is also embedded in the algorithm.

Song and Hitomi (1996) developed a methodology to integrate production planning and cellular layout via a long-run planning horizon. The problem was formulated as a mixed-integer problem. It contains two types of integer programming problems: determining the production quantity for each product and the timing of adjusting the cellular layout in a finite planning horizon period with dynamic demand. The objective of the model was to minimise the sum of inventory-holding cost, group-setup cost, material handling cost and layout-adjusting cost. The Benders decomposition method was employed to solve the problem. The periodic fluctuating demand was absorbed by adjusting both layout configuration and inventory level. The demand for each part family in each period was assumed to be known and constant.

Chen (1998) developed a mixed integer programming model to minimise intercell material handling and machine investment costs as well as cell reconfiguration cost in a dynamic CM environment with anticipated changes of demand or production process for multiple time periods. The problem was decomposed into simpler cell formation subproblems by removing the system reconfiguration cost from the objective function and the corresponding coupling constraints from the model. Thus, the decomposed subproblems correspond to different time periods. The commercial optimisation software packages were used to solve binary-integer programming problems optimally. The model only examines reconfiguration of cells in terms of the machine types included in each cell and does not include capacity constraints to determine how many units of each machine type are needed in each cell during each period.

Wicks and Reasor (1999) developed an integer model for multiperiod cell formation. Their model considered intercell transfer cost, machine equipment cost and machine relocation cost. The model requires a minimum number of machines and parts to be assigned to each cell (the number of cells must also be specified). Production costs based on the size of the cell are not considered. The authors also propose a genetic algorithm to generate solutions to the model.

Mungwattana (2000) developed a solution approach for designing CM systems that addresses the dynamic and stochastic production requirements. A simulated annealing-based heuristic was developed to obtain good solutions within reasonable amounts of time. The developed heuristic was evaluated in two ways. First, different CM design problems were generated and solved using the heuristic. Then, solutions obtained from the heuristic were compared with lower bounds of solutions obtained from the optimal solution procedure (CPLEX software). The lower bounds were used instead of

42 A.R. Dixit and P.K. Mishra

optimal solutions because of the computational time required to obtain optimal solutions. The objectives considered were the minimisation of intercell material handling cost, machine investment cost, operating cost and the cost of system reconfiguration over the planning horizon.

Schaller (2007) proposes an integer model that considers part reallocation or equipment reallocation between cells as alternative for the design of a CM system to handle long-term demand change. The objective of the model was to minimise the sum of amortised machine cost, cost of reallocating equipment and the cost of producing parts. A problem specific heuristic (called CB procedure) and metaheuristic (tabu search) was employed to obtain the acceptable solution. The proposed model does not consider important parameters like amount of intercell movement of parts, operational sequence of parts, batch size, etc.

This paper describes a solution methodology for the problem of manufacturing cell formation in uncertain product mix environment. The goal of the multiperiod formulation is to obtain a cellular design that continues to perform well with respect to the design objectives as the product mix/volume changes with time.

This paper is organised as follows: notations and definitions are explained in Section 3. The problem formulation is presented in Section 4. The solution methodology is presented in Section 5. Computational analysis and results are presented in Section 6. Conclusion is presented in Section 7.

3 Notations and definitions

i Part type index

j Machine type index

k Cell type index

n Operation type index

m Number of machines M = (m1, m2,… ,mj,…,mm)

p Number of parts P = (p1, p2,…,pi,…,pp)

c Number of cells C = (c1, c2,…,ck,…,cc)

kjh Number of identical machines of machine type j for the product mix h

ir Maximum number of operations for components i

pvih Total production volume of part type i for the product mix h

ihbs Batch size of part type i for the product mix h

MHCi Capacity of material handling device for part i

ijt Processing time of part type i on machine type j

Design of flexible manufacturing cell 43

ijst Setup time of part type i on machine type j

jT Capacity of a machine type j

jO Operation cost per hour of machine type j

jP Procurement cost of machine type j

iIT Intercell movement cost per batch of part type i

jUMC Non-utilisation penalty cost of machine type j

hjK Number of machine type j which are not utilised for the product mix ‘h’

UB Upper bound on cell size (maximum number of machines in a cell)

m(k) Number of machines in cell type k

ijw proportionate workload induced by part type i on machine type j

[ ]( )× + ×=

pv st pv / bsij ih ij ih ihij

i

tw

T (1)

wckj Workload on machine type j in cell type k

=∀

= ∑1,

wcp

k ij jk ik j

ik j

X Y w (2)

jk

j kX

1 if machine type is in cell type

0 otherwise

⎧= ⎨⎩

ik

i kY

1 if part type is assigned in cell type

0 otherwise

⎧= ⎨⎩

st, , M achine type ' ' is required for part type ' ' for operationr sOP a a r s=

( ) Total number of operations required by part type ' 'O p p

( )χ⎧ +

= ⎨⎩

th1 if and ( 1) operation is performed on machine type ' ' and ' ';,

0 otherwise

k k a ba b

abΨ total movement between two distinct cells ca and cb.

( ) ( )( )O pP

ab p i p ip i

A p OP OP

a b m

1

, , 11 1

,

,

χ−

+= =

Ψ = ×

∀ ∈

∑ ∑ (3a)

44 A.R. Dixit and P.K. Mishra

4 Problem formulation

This paper addresses two problems, firstly, the machine/part grouping problem considering various production parameters and secondly, the identification of suitable cell configuration under dynamic product mix scenarios.

4.1 Machine/part grouping problem The most fundamental objective of cell formation is to achieve cell independency which in turn significantly simplifies shopfloor control. Further, the potential to increase the accountability, responsibility and autonomy of the workers is enhanced. It also reduces material handling that results in less damage to work-in-process (Shafer and Rogers, 1993). To achieve cell independency, intercell movements must be minimised.

In past few years, some researchers have incorporated operation sequence in the formulation of cell formation problem. But very few of them have considered the non-consecutive operations on the same machines. In practical situations, same machine is used more than once in a part routing, and if such a part has to move outside its assigned cell, the implication on material handling are significant (Harahalakis et al., 1990a). But Harahalakis et al. (1990a) did not incorporate the non-consecutive operation consideration due to difficulties encountered in proposed matrix formulation. The non-consecutive operations on the same machine are incorporated in the proposed formulation.

The mathematical formulation for the design of manufacturing cell is developed such that initially machine cells are formed and then the parts are assigned to the appropriate cells. However, it could be more complicated to model and would result in a large mathematical model, which requires a substantial amount of time to solve. The mathematical formulation is given below:

−

= = +

ψ⎡ ⎤= ⎢ ⎥+⎣ ⎦∑ ∑

1

11 1

Minimise ( ) ( )

m mab

a b a

Fm a m b

(3)

Subject to constraint:

<( ) UBm k (4)

=

= =∑1

1 for 1,2,3,..., c

jkk

X j m (5)

=

≥ =∑1

1 for 1,2,3..., m

jkj

X k c (6)

=

≥ =∑1

1 for 1,2,3..., p

iki

Y k c (7)

The objective function (3) minimises the total intercell movement of parts in the system. Constraint (4) ensures that the merging cells/groups satisfy cell size. Constraints (5) ensures that each part can only be assigned to one cell. Constraint (6) and (7) ensures that each cell must contain at least one machine and one part.

Design of flexible manufacturing cell 45

4.1.1 Performance measure

Group Technology Efficiency has been used to measure the performance of the proposed algorithm to group machines into cells and parts into their respective families.

GTη ,Group Technology Efficiency (Harhalakis et al. 1990a), is defined as the ratio of the

difference between the maximum number of intercell movements possible and the number of intercell movements actually required by the system to the maximum number of intercell movements possible. Mathematically GTη can be defined as:

( ) /GT I U Iη = − (8)

( )=

= −∑1

pv 1p

i ii

I r (9)

λ−

= =

=∑ ∑1

1 1

pvirp

i iri r

U (10)

where

0 if operations , +1 are performed

in the same cell

1 otherwiseir

r r

λ⎧⎪= ⎨⎪⎩

4.2 Economic justification of cell configuration

The cell configuration is designed for each product mix and the effect of the uncertainty due to occurrence of the other product mix is analysed by the cost function ( , )h sZ .

It consists of operating cost of machines, investment cost of machines, intercell movement cost of parts and non-utilisation penalty cost of machine. If the CMS is designed according to a specific product mix scenario and some other product mix demand arises, then the following possibilities may arise

1 All machine types are being utilised.

2 Only few machine types are being utilised and the rest remain idle for the given time period.

3 All machine types are utilised, but the number of same type of machines required may be less. Therefore, few identical machines remain idle for the given period of time.

4 All machine types are utilised, but the number of same type of machines required may be more due to high production requirement. Therefore, more identical machines have to be procured to meet the production requirement within production schedule.

If the machines remain idle in a particular production period, it indicates the poor cell configuration planning. Hence, a penalty cost is introduced which is assumed to be 0.25 times the machine investment cost.

46 A.R. Dixit and P.K. Mishra

( )h sZ , = Cost of cell configuration, designed for sth product mix, due to the occurrence

of hth product mix.

( )

( )λ

= = =

−

= = =

⎛ ⎞= + + ×⎜ ⎟

⎝ ⎠

⎛ ⎞+ × + × ∀⎜ ⎟

⎝ ⎠

∑∑ ∑

∑ ∑ ∑

,1 1 1

1

1 1 1

st pv pv

pvUMC ,

bs

i

p m mij ih

ih ij j jh jhh si j jih

rpmh ih

j j i irj i rih

Z t O K Pbs

K IT h s

(11)

The cost function (11) represents the total sum of operating cost of machines, procurement cost of machines, non-utilisation penalty cost for machine type j and cost of intercell movement of parts (in batches) for a given product mix h.

For a given probability ( sΡ ) of the occurrence of the product mix(s), the total

expected cost of cell configuration is given as:

( )γ= =

=∑∑1 1

S S

s s (h,s )s h

E P Z (12)

The cell configuration with minimum ( )sE γ is selected for the design of the CM system.

5 Solution methodology

The design of CM is combinatorial complex. The number of ways in which m machines may be assigned to exactly k cells is given by the Stirling number of the second kind (Venugopal and Narendran, 1992a).

( )( )

!

11

k

j

s

k

j

mK

j

jk

k∑

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

So, there will be 34105 distinct partitions of 10 machines into 4 cells, but this number increases to 1,12,59,66,000 approximately; if 19 machines are to be partitioned into 4 cells. However, the number of cells is usually not known in advance. But the maximum and minimum limit of the number of cell can be 1 and m, respectively. The total number of ways in which machine-cell assignments may be made explodes to

( )( )

∑∑

∑=

=

−

=⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

=m

k

k

j

mK

j

jkm

k

k

k

j

s1

1

1 !

1

Therefore, this class of problem is NP-complete. Many approaches have been proposed by different researchers to solve the problem. Heuristic approaches are also used to obtain good solutions within acceptable amount of time. Numerous papers can be found in the literature for cell formation using heuristics (Adenso-Diaz et al., 2005; Baykasoglu

Design of flexible manufacturing cell 47

et al., 2001; Burke and Kamal, 1992; Cao and Chen, 2004; Carpenter and Grossberg, 1987; Chen et al., 1995; Dixit and Mishra, 2004; Dobando et al., 2002; Gupta et al., 1995; Harhalakis et al., 1990a, b; Lozano et al., 2001; Muruganandam et al., 2005; Onwubolu and Mutinigi, 2001; Peker and Kara, 2004; Sofianopoulou, 1997; Solimanpur et al., 2004; Su and Hsu, 1998; Venugopal and Narenderan, 1992a,b; Xambre and Vilarinho, 2003; Yasuda et al., 2005; Zolfagari and Liang, 2003).

In real-world manufacturing, production requirements may not be known exactly at the time of designing CM Systems. It is likely that a set of possible production requirements (scenarios) with certain probabilities may be given at the time of design. Therefore, uncertainty in production requirements needs to be incorporated at the design stage of the CM System. Dealing with uncertain production requirements in the design of CM System has not been extensively investigated. Seifoddini (1990) considered the uncertainty when designing CM System. The algorithm proposed by Seifoddini chooses a cell configuration from a set of cell configurations generated from different product mixes. The chosen configuration is the one that has the lowest expected intercell material handling cost. The primary drawback in his algorithm is that only the optimal designs of each product mix are considered. It is possible that a system design exists with a lower expected intercell material handling cost over all product mixes, although, it is not optimal with respect to any individual product mix. Other important parameters like machine requirement, utilisation level of machines (under utilised/over utilised) etc have not been considered. In this Paper, a procedure to obtain CM design solutions for a single-period planning considering uncertainty is presented. Such a procedure can be extended for multiperiod planning. The following assumptions have been considered:

1 there exists a finite number of possible product mixes (scenarios) which can occur

2 each product mix is represented by a unique set of part types and their demands

3 each product mix has a known probability of occurrence.

We have also applied the heuristic based approach for manufacturing cell design under uncertain production requirement

5.1 Phase-I (cell configuration for given product mix)

In phase-I, initial basic feasible solution is obtained. The machine cells and part families are identified at this stage. Initially, a symmetric matrix with [m×m] entries is constructed using equation (3a). An entry abΨ in the matrix indicates the number of

immediate movements between distinct machines ma and mb. Then, the machine-pair having maximum normalised intercell flow is grouped into a machine cell and the matrix is again revised. The procedure of iteration is continued until the upper bound condition on cell size is not violated. Once the machine cell is constructed, the parts are assigned optimally to the cells. Later, a local refinement procedure is applied to find the better solution.

The general procedure of the proposed heuristic algorithm is presented as follows.

48 A.R. Dixit and P.K. Mishra

5.1.1 Machine cell formation algorithm:

Step 1 Initially assign each machine type to a cell. Thus, at the beginning of the algorithm, the number of cells is the same as the number of machine types. (Number of cells = Number of machines).

Step 2 The total immediate movement of parts (Ψab) is calculated between all pairs of cells between the cells, ca and cb :

( ) ( )( )−

+= =

Ψ = × χ

∀ ∈

∑ ∑1

, , 11 1

,

,

O pP

ab p i p ip i

A p OP OP

a b m

A symmetric matrix of size (m × m) is constructed from the above formulation.

Step 3 The normalised intercell flow abT , is calculated between all pair of cells, ca

and cb.

Τ ψ=

+( ) ( )ab

ab m a m b

Where m(a) and m(b) are the number of machines in cell ca and cb, respectively.

Step 4 The pair of cells that corresponds to the maximum normalised intercell flow is identified.

If Tie occurred (more than one cell-pair has same value of normalised intercell flow), then the cell-pair having less number of machines is selected.

If the selected cell pairs satisfy the cell size constraint ( ( )m k UB< ), both cells are merged to form a single cell.

At the end of this step, the total intercell flow is reduced by the intercell flow value between the two cells being merged. It is emphasized that this is the maximum possible intercell flow value that could have been reduced by merging any pair of cells at this iteration. Hence, the values of normalised intercell flow have to be revised after iteration. The number of cells has been reduced by one unit and the number of machines in the newly constituted cell has been increased.

Step 5 If the maximum normalised intercell flow value is greater than zero, continue merging cells through step 4 until

1 the intercell flow between all the pairs of cells becomes zero,

2 it is impossible to further merge any cells without violating the cell size constraint.

5.1.2 Part allocation algorithm

Step 1 The parts are assigned to the cell having MAXIMUM number of machines required by the particular part.

If TIE occurred (more then one cell has equal number of machines required by the particular part).

Design of flexible manufacturing cell 49

• If operations are in sequence in TIE cells, then the part will be assigned to the cell having minimum number of total machines. If ties persist (total number of machines is also equal in TIE cells), part will be assigned to the cell having MINIMUM operation sequence.

• If operations are not in a sequence in one of the TIE cells, then the part will be assigned to the cell having operations in sequence.

• If operations are not in sequence in all the TIE cells, the part will be assigned to the cell having minimum number of machines. If tie persists (number of machines is same), part will be assigned to the cell having intermediate operation number.

Step 2 A refinement step is performed after all the parts have been assigned to different cells. Since the cell formation heuristic is a greedy algorithm, that is once machines have been merged together into cell, they cannot be removed from this cell, a refinement operation is performed to improve the solution.

This is performed by identifying the exceptional elements (bottleneck machines, bottleneck parts) and their respective cells.

Step 3 Identify the bottleneck machine which is more involved for processing bottleneck parts as compared to their regular operations for the part families within the cell.

Step 4 If these bottleneck parts belong to a single cell, shift the bottleneck machine from its parent cell to the cell having bottleneck parts.

If TIES occurred

• Number of exceptional elements are equal to the number of operations within the parent cell of the machine and if (number of parts in parent cell > number of parts in cell having bottleneck parts), ‘Shift the machine to the cell having bottleneck parts’.

Repeat the step for all the bottleneck machines.

Step 5 Apply only step 1 once.

Step 6 Stop.

5.2 Phase-II (selection of cell configuration)

Phase-II is applied to identify the overloaded machines, so that the requirement of the number of identical machines can be found to select the most economical cell configuration. The procedure is as follows:

Step 1 Evaluate the load induced by parts on machines ijw . Also, calculate the

cumulative load on machines. 1P ii j

jw=

∀∑

Step 2 Identify the overloaded machines and calculate the number of identical machines required to balance the load.

50 A.R. Dixit and P.K. Mishra

Step 3 If 11 2P ii j

jw=

∀< < ⇒∑ two identical machines of machine type j are required

( jK 2= ). Similarly if the above limit is 2–3, then three identical machines of

machine type j are required ( jK 3= ).

Step 4 Determine the cell configuration for each product mix (by applying Phase-I).

Step 5 Evaluate the cost of the cell configuration for each product mix.

Step 6 For each cell configuration, assign parts of other product mix to the machine cells.

Step 7 Evaluate the cost of cell configurations for all such assignment of different possible product mixes.

Step 8 Determine the total expected cost of cell configurations.

Step 9 Identify the minimum total expected cost of cell configuration.

Step 10 Select the cell configuration/s with minimum total expected cost.

Step 11 Stop.

6 Computational analysis and results

The purpose of this section is to provide some computational analysis and results that can be widely used to benchmark the effectiveness of heuristic approaches, which may be proposed in future research in this field, rather than addressing the algorithmic aspect. In order to validate the proposed heuristic, initial part-machine sequence matrix is taken from the published research papers. Due to non-availability of few production data, some of the input parameters were randomly generated. The values in the surveyed publications are used as a guideline for determining the range of the values of parameters. Table 1 summarises the value for parameters.

The algorithm has been implemented in script programming in MATLAB 7.0 and the experiments have been run on a Pentium IV, with 2.40 GHz and 512 MB RAM. The example data set contains 20 machines and 20 parts (Harhalakis et al., 1990a). The predicted demand of the parts, batch size, machine capacity and material handling device capacity has been generated as per the guideline of Table 1. The operation sequence has been the same as suggested by Harhalakis et al. (1990a). Table 2 shows the attributes of machine type, which includes procurement cost of the machine and the operating cost of machine (per hour). Machine capacity is considered to be fixed to 2000 hr. It is assumed that machine operates 8 hr/day, 5 days/week for 50 weeks (nearly one year). Table 3 shows the attributes of part type such as operation sequence, product mix demand, batch size, pallet size and intercell movement cost per batch. A zero demand of a part indicates that the part is not manufactured in the given period. Five product mixes have been considered. Each product mix is assumed to have equal probability of occurrence.

Design of flexible manufacturing cell 51

Table1 Values of parameters

Parameters Values Remark

Machine part sequence matrix

Taken from published papers

In case of 0–1 matrix, the entry 1 is replaced by non-repetitive discrete uniform distributed number

Part demand U(100 – 1000) Uniformly distributed random no.

Processing time U(0.5–2) min Discrete uniform distribution

Setup time U(0.05–0.5) min Discrete uniform distribution

Machine capacity 2000 hr/period Fixed (8 hr/day, 5 days/week for 50 weeks i.e. 1 year)

Material handling device capacity

U(100–500) Discrete uniform distribution

Batch size U(50–100) Discrete uniform distribution

Investment cost U(10,000–50,000) Discrete uniform distribution

Operating cost/unit machining time

N(mean–50, SD–20) Normal distribution

Intercell cost U(20, 50)/batch Discrete uniform distribution

Non-utilisation penalty cost of machine

0.25 × investment cost Discrete uniform distribution

Table 2 Machine type attributes (Input)

Machine type Investments cost Operating cost

1 11,087.62 48.07

2 41,746.68 42.55

3 49,969.23 50.56

4 14.409.54 51.29

5 34,904.05 44.87

6 15,302.87 55.33

7 22.401.19 55.32

8 15.391.51 49.83

9 18.933.05 51.46

10 25,861.87 50.78

11 15,405.74 49.17

12 19,642.35 53.25

13 47,100.65 47.37

14 25,644.03 59.76

15 30,450.51 49.39

16 13,715.84 50.51

17 10,867.96 54.77

18 16,381.39 50.27

19 43,780.64 49.57

20 45,166.14 46.28

52 A.R. Dixit and P.K. Mishra

Table 3 Part type attributes (Input)

Product demand Part type

Operation sequence

PM 1 PM 2 PM 3 PM 4 PM 5

Batch size

Intercell movement

cost

1 M2-M3-M1-M4-M5 956 0 855 547 816 98 256.10

2 M3-M2-M1 309 418 0 910 962 67 497.39

3 M1-M3-M2 0 832 714 840 571 86 413.61

4 M3-M1-M4-M2 538 109 442 681 893 98 461.41

5 M1-M3-M4-M2 903 226 849 837 256 80 343.89

6 M5-M1-M2-M3-M4 786 0 0 695 982 95 348.80

7 M1-M2-M3 0 279 739 408 345 88 286.26

8 M5-M3-M4-M2-M1 117 0 487 0 328 69 218.28

9 M4-M2-M3-M5-M1 0 345 375 0 889 87 278.74

10 M3-M1-M2 501 279 0 581 764 59 255.88

11 M3-M1-M2 654 0 275 0 223 98 475.13

12 M5-M3-M1-M4-M2 813 773 0 379 111 60 236.98

13 M1-M2-M3-M4 930 501 373 855 905 89 204.03

14 M3-M4-M1-M2 0 939 588 0 280 81 310.91

15 M1-M2-M3-M4 0 520 236 434 369 59 409.59

16 M3-M2-M1-M4 466 477 0 733 696 52 466.80

17 M2-M1-M3 942 0 441 592 356 65 378.13

18 M1-M4-M2-M3 0 573 875 501 523 99 247.01

19 M2-M1-M4-M3 470 0 869 726 159 98 295.01

20 M3-M2-M4-M1 0 705 635 660 990 62 270.02

Note: PM = Product Mix.

In the first product mix, thirteen part types, 1, 2, 4, 5, 6, 8, 10, 11, 12, 13, 16, 17 and 19, are produced. Twenty machine types are needed in this period. Two units of machine type 1, 3, 5, 6, 7, 9, 11, 12, 14 and 15 and one unit of machine type 2, 4, 8, 10, 13, 16, 17, 18, 19 and 20 are used to meet the production requirement.

Cell 1 consists of machine types 1, 5, 9, 12 and 18, and part types 1, 12 and 17 are produced in this cell. Cell 2 consists of machine types 2, 3, 10 and 11, and part types 2, 4, 11 and 19 are produced in this cell. . Cell 3 consists of machine types 4, 6, 7, 13 and 15, and part types 5, 8, 13 and 16 are produced in this cell. Cell 4 consists of machine types 14, 16 and 17, and part type 6 is produced in this cell. Cell 5 consists of machine types 8, 19 and 20, and part type 10 is produced in this cell. The total cost of the cell configuration in this period includes:

1 The procurement cost of $7,36,260.

2 The operating cost of $21,37,000.

3 The intercell movement cost of $24,285.

Design of flexible manufacturing cell 53

The value of group technology efficiency is 0.7436. The total cost of cell configuration is $28,97,545. Table 4 shows the machine/part grouping for other product mixes. The number of machines of each type required to meet the product demand for different product mix scenarios are given in Table 5. Table 6 shows the total cost and group technology efficiency of cell configuration, design by the proposed algorithm.

Table 4 Cell configuration for each product mix scenarios

Product Mix Cell configuration

Number of cells Machine grouping Part family

1, 5, 9, 12, 18 1, 12, 17

2, 3, 10, 11 2, 4, 11, 19

4, 6, 7, 13, 15 5, 8, 13, 16

14, 16, 17 6

PM1 5

8, 19, 20 10

1, 2, 10 4, 14, 20

3, 9, 11, 12, 18 2, 9, 12

4, 6, 7, 15 5, 13, 16

8, 19, 20 3, 10, 18

PM2 5

5, 13, 14, 16, 17 7, 15

1, 9, 10, 12, 18 1, 9, 14, 17, 20

2, 3, 11 4, 11, 19

4, 6, 7, 15 5, 8, 13

5, 13, 14, 16, 17 7, 15

PM3 5

8, 19, 20 3, 18

1, 9, 12, 18 1, 12, 17, 20

2, 3, 11 2, 4, 19

4, 6, 7, 15 5, 13, 16

5, 13, 14, 16, 17 6, 7, 15

PM4 5

8, 10, 19, 20 3, 10, 18

1, 9, 10, 12, 18 1, 9, 12, 14, 17, 20

2, 3, 11 2, 4, 11, 19

4, 6, 7, 15 5, 8, 13, 16

5, 13, 14, 16, 17 6, 7, 15

PM5 5

8, 19, 20 3, 10, 18

54 A.R. Dixit and P.K. Mishra

Table 5 Number of machines required to meet different product mix demands

Design of flexible manufacturing cell 55

Table 6 Cell configuration cost for different Product Mix

Product Mix

Cell configuration

Investment cost

Operating cost

Intercell movement

cost

Total cost

Group technology efficiency

PM1 CC1 7,36,260 21,37,000 24,285 28,97,545 0.7436

PM2 CC2 6,90,180 17,21,800 33,462 24,45,442 0.675

PM3 CC3 8,21,790 21,51,100 28,108 30,00,998 0.6591

PM4 CC4 8,54,310 25,30,400 28,217 34,12,927 0.7826

PM5 CC5 9,95,310 27,95,500 38,391 38,29,201 0.678

If the second product mix (PM2) demand occurs, then the parts will be assigned to the machine groups, arranged for first product mix (PM1). Three units of machine type 1, two units of machine type 8, 10, 12, 19 and 20 and one unit of machine type 2, 3, 4, 5, 6, 7, 9, 11, 13, 14, 15, 16, 17 and 18 are used to meet the production requirement. One more unit of machine type 1, 8, 10, 19 and 20 is required to be procured in order to meet the production schedule. One unit of machine type 3, 5, 6, 7, 9, 11 and 15 will remain idle for the given product mix scenario. Cell 1 consists of machine types 1, 5, 9, 12 and 18, and part types 9, 12 and 20 are produced in this cell. Cell 2 consists of machine types 2, 3, 10 and 11, and part types 2, 4 and 14 are produced in this cell. . Cell 3 consists of machine types 4, 6, 7, 13 and 15, and part types 5, 13 and 16 are produced in this cell. Cell 4 consists of machine types 14, 16 and 17 and part types 7 and 15 are produced in this cell. Cell 5 consists of machine types 8, 19 and 20, and part types 3, 10 and 18 are produced in this cell. The total cost of the cell configuration in this period includes:

1 Machine investment cost of $8,77,550.

2 Operating cost of $17,21,800.

3 Intercell movement cost of $32,109.

4 Non-productive penalty cost of $46,842.

The total cost of cell configuration is $26,78,301. Similarly, for other product mixes, the total cost of cell configuration (designed for particular product mix scenario) is given in Table 7.

The minimum total expected cost is $26,60,568 for the cell configuration designed for the product mix scenario 2(PM2). Hence, this configuration must be selected for the design of CM system. The execution time of the algorithm to obtain the result is found to be 0.4976 sec.

The algorithm has also been applied to the problems reported in different research papers and execution time of the proposed algorithm has been shown in Table 8. The execution time ranges from 0.2176 to 0.4926 sec.

56 A.R. Dixit and P.K. Mishra

Table 7 Payoff Matrix for different cell configuration cost

Design of flexible manufacturing cell 57

Table 8 Execution time of proposed algorithm for other published problems

No. Test Instances Source Size Number of product mixes

Execution time

(seconds)

1 Lee and Chen (1997) 15 × 9 3 0.1490

2 Jayakrishana and Narender (1998) 40 × 25 2 0.2176

3 Harhalakis et al. (1994) 20 × 17 3 0.2498

4 Venugopal and Narendran (1992a) 30 × 15 4 0.3476

5 Venugopal and Narendran (1992b) 30 × 15 3 0.2592

6 Venugopal and Narendran (1992a) 19 × 12 3 0.2467

7 Harhalakis et al. (1990a) 20 × 20 5 0.4976

8 Chandrasekharan and Rajagopalan (1989)

24 × 40 3 0.3109

7 Conclusion

In this paper, an efficient algorithm for a probabilistic machine cell formation model to deal with the uncertainty of the product mix for a single period has been proposed. While most of the existing methods of grouping are solely based on binary machine-part incidence matrix and a few have used either the operations sequence or combination of processing time, setup time, part demand, machine capacity, material handling capacity and lot size, simultaneous consideration of all these factors makes the cell formation problem complex, but more realistic. This paper also addressed the machine-part grouping problem considering all forementioned production parameters simultaneously. It is evident from the illustrative example that in uncertain product mix environment, the cell configuration with lowest total expected cost should be implemented.

The algorithm was applied to the numerical problems reported in different research papers and computational experience has been reported. The results obtained suggested that the algorithm is efficient and provides better solutions. The proposed heuristic approach is also capable of solving the industrial problems. This approach incorporates many of the real-life production parameters. Hence, it is bound to provide amicable solutions to the designers.

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