Data envelopment analysis in cellular manufacturing systems considering worker assignment

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Int. J. Services and Operations Management, Vol. 18, No. 3, 2014

Data envelopment analysis in cellular manufacturing systems considering worker assignment Amin Aalaei* Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, P.O. Box:15875-4413, Hafez Ave, Tehran, Iran E-mail: [email protected] *Corresponding author

Mohammad Mahdi Paydar and Mohammad Saidi-Mehrabad School of Industrial Engineering, Iran University of Science and Technology, P.O. Box: 163-16765, Narmak, Tehran, Iran E-mail: [email protected] E-mail: [email protected] Abstract: In cellular manufacturing systems we are not only looking toward increasing production volumes, but are also looking to increasing productivity and efficiency of available resources. So in this article, we seek to increase production volumes and simultaneously increase productivity and efficiency of the resources. This paper develops an integer mathematical programming model to design the cellular manufacturing systems under data envelopment analysis. The aim of the proposed is to minimise cost of lost sales demand and intercellular costs. The data envelopment analysis (DEA) is performed to determine the most efficient alternative by employing the average machine utilisation, the average worker utilisation, and mean of product as the output variables and the number of machines, the number of workers, the number of parts and demand levels as the input variables. We are using the Tchebycheff norm method to rank best DMUs. Keywords: data envelopment analysis; DEA; cellular manufacturing; mathematical programming. Reference to this paper should be made as follows: Aalaei, A., Paydar, M.M. and Saidi-Mehrabad, M. (2014) ‘Data envelopment analysis in cellular manufacturing systems considering worker assignment’, Int. J. Services and Operations Management, Vol. 18, No. 3, pp.258–280. Biographical notes: Amin Aalaei received his BE in Industrial Engineering in 2007 and ME in Industrial Engineering in 2010 from Mazandaran University of Science and Technology, Babol, Iran. He began his PhD in 2011 from Amirkabir University of Technology, Tehran, Iran. His research interest includes cellular manufacturing systems, data envelopment analyses, scheduling, decision making theory and mathematical modelling. He has published more than 15 research papers in reputed international and national journals.

Copyright © 2014 Inderscience Enterprises Ltd.

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Mohammad Mahdi Paydar is a PhD candidate in Industrial Engineering at Iran University of Science and Technology. He received his MS and BS in Industrial Engineering from Mazandran University of Science and Technology. He has also been an invited University Lecturer in the past few years. His research interests are cellular manufacturing systems and modelling of manufacturing applications. He has published articles in some journals such as Computers and Operations Research, Computers & Industrial Engineering, Expert Systems with Applications, International Journal of Advanced Manufacturing Technology, Journal of Manufacturing Systems, International Journal of Production Research and 18 papers in international conferences. Mohammad Saidi-Mehrabad is a Professor of Industrial Engineering at Iran University of Science and Technology, Tehran, Iran. He received his BS in Mechanical Engineering from Oklahoma State University, USA, and MS and PhD, both in Industrial Engineering, from the Universities of Arkansas and West Virginia, USA, respectively. His research interests include: flexible manufacturing systems, dynamic cellular manufacturing, economic analysis, scheduling and operations research. He is the Editor-in-Chief of the Journal of Industrial Engineering.

1

Introduction

Group technology (GT) was first proposed by Mitrofanov (1966) which is a manufacturing philosophy in which similar parts are identified and grouped together to take advantages of their similarities in manufacturing and design. Cellular manufacturing system (CMS) is a successful application of GT concepts. The major advantages of CMS have been reported in the literature as reduction in setup time, reduction in throughput time, reduction in work-in-process inventories, reduction in material handling costs, better quality and production control, increment in flexibility, etc. (Heragu, 1994; Wemmerlov and Hyer, 1989). One of the key issues encountered in the implementation of a CMS is the cell formation problem (CFP). In the past several years, many solution methods have been developed for solving CFP by a binary machine-part incidence (two-dimensional) matrix. Some comprehensive summaries and taxonomies considering the CFP as a machine-part incidence matrix include Singh (1993), Offodile et al. (1994), Selim et al., (1998) and Mansouri et al. (2000). Moreover, recently some approaches that have been developed to the two-dimensional CFP are: genetic algorithms (Goncalves and Resende 2004; Mahdavi et al., 2009; Paydar and Saidi-Mehrabad, 2013 ), tabu search (Lozano et al., 1999; Wu et al., 2004), neural network (Soleymanpour et al., 2002; SudhakaraPandian and Mahapatra 2010), mathematical programming (Albadawi et al., 2005; Paydar et al., 2011; Ariafar et al., 2012; Soolaki and Izadi, 2013), simulated annealing (Wu et al., 2008; Pailla et al., 2010; Sangwan and Kodali, 2011) and similarity coefficients-based method (Yin and Yasuda 2006; Oliveira et al., 2008). Paydar et al. (2010) proposed a solution to solve the part-family and machine CFP considering the within-cell layout problem, simultaneously. The CMS is formulated as a

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multiple departures single destination multiple travelling salesman problem (MDmTSP) and a solution methodology based on simulated annealing is proposed to solve the formulated model. Pailla et al. (2010) developed two approaches to solve the CFP. Firstly, an evolutionary algorithm was introduced that improves the efficiency of the standard genetic algorithm by considering cooperation with a local search around some of the solutions it visits. Secondly, an approach based on simulated annealing was applied that utilises the same representation scheme of a feasible solution. Arkat et al. (2011) proposed a multi-objective programming model with the aim of minimising the number of exceptional elements (EEs) and the number of voids, simultaneously. They have also developed a bi-objective genetic algorithm for large-scale problems. Díaz et al. (2012) addressed a greedy randomised adaptive search procedure (GRASP) heuristic to achieve lower bounds for the optimal solution of the CFP. Their method contains of two phases. In the first phase an initial partition of machines into machine-cells or parts into part families is obtained, while in the second phase the assignment of parts to machine cell or machines to part-families is considered. Paydar and Saidi-Mehrabad (2013) presented a linear fractional programming model with the objective of maximising the grouping efficacy while the number of cells is unknown. Then, to solve the model for real-sized applications, a hybrid meta-heuristic algorithm in which genetic algorithm and variable neighbourhood search are combined. Using the grouping efficacy measure, they had also compared the performance of the proposed algorithm on a set of 35 test problems from the literature. One of the main points in CM is considering human issues since ignoring this factor can considerably reduce benefits of the utility of the cell manufacturing. In some of the previous research papers this issue is discussed. Nembhard (2001) addressed a greedy heuristic method based on individual learning rate for the improvement of productivity in organisations through targeted assignment of workers to tasks. Norman et al. (2002) proposed a mixed integer programming model for assigning workers to manufacturing cells in order to maximise the profit. Bidanda et al. (2005) presented an overview and evaluation of the diverse range of human issues involved in CM based on an extensive literature review. In Wirojanagud et al. (2007) proposed a workforce planning model that incorporates individual worker differences in ability to learn new skills and perform tasks. The model allows a number of different staffing decisions (i.e., hire and fire) which to minimise workforce related and missed production costs. Aryanezhad et al. (2008) presented a new model to deal with dynamic cell formation and worker assignment problem with considering part routing flexibility and machine flexibility and also promotion of workers from one skill level. Mahdavi et al. (2012) presented a new mathematical model for CFP based on a threedimensional machine-part-worker incidence matrix which demonstrates a cubic representation of assignment in CMS. Also, the new concept of EEs is discussed to show the interpretation of inter-cell movements of both workers and parts for processing on corresponding machines. The proposed method minimises total number of EEs and voids in a CMS. One important feature of the CFP is its efficiency measurement procedures. But, there are few papers on the efficiency measurement of the CFP. Ertay and Ruan (2005) proposed a decision making approach based on data envelopment analysis (DEA) for determining the most efficient number of operators and the efficient measurement of labour assignment in cell. They studied concentrates on efficiency measurement and the

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determination of the number of operators in cell when the demand rate and the transfer batch size as a rate of batch size change as the input variables and employing the average lead time, the average operator utilisation as the output variables. Both inputs and outputs are procured by means of simulation of CMS. In this paper we develop an integer mathematical programming to design the CMS, By means of considering several situation for each of input variables, the number of machines, the number of workers, the number of parts and demand levels, we get several different alternatives to decision maker. To determine the most efficient alternative, for each alternative we use the developed CMS model to gain the average machine utilisation, the average worker utilisation, and mean of product as the output variables of the alternative and then DEA performed to determine the most efficient scenario among all the scenarios that considered.

2

DEA methodology

DEA that has proposed by Charnes et al. (1978) is a mathematical programming technique that measures the relative efficiency of decision making units (DMUs) with multiple inputs and outputs but with no obvious production function to aggregate the data in its entirety. In most models of DEA (such as CCR), the best performers have efficiency score unity, and, from experience, we know that usually there are plural DMUs which have this ‘efficient status’. To discriminate between these efficient DMUs is an interesting research subject. Ranking DMUs is one of the main problems in DEA. There are some methods for ranking DMUs, see for example Adler et al. (2002). Recently, several authors have proposed some methods based on norms. Jahanshahloo et al. (2004) introduced L1-norm approach and Rezai Balf et al. (2011) presented ranking model L∞-norm (or Tchebycheff norm) in DEA. In this paper, we are use the ranking method based on the Tchebycheff norm proposed by Rezai Balf et al. (2011) that it seems to have superiority over other existing methods, because this method is able to remove the existing deficiencies in some methods, such as Andersen and Peterson (1993) that it is sometimes infeasible. The L∞-norm model always is feasible.

2.1 Background DEA 2.1.1 DEA model DEA is a mathematical model that measures the relative efficiency of DMUs with multiple inputs and outputs but with no obvious production function to aggregate the data in its entirety. By comparing n units with s outputs denoted by yrj = (r = 1, …, s) and m inputs denoted by xij = (i = 1, …, m) that all of them are non-negative and each DMU has at least one strictly positive input and output. The efficiency of a specific DMUP can be evaluated by the CCR model (Charnes et al., 1978), of DEA as follows:

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∑u y r

ro

r =1 m

Max

∑v x

i io

i =1 s

∑u y r

s.t.

∑v x

(1)

rj

r =1 m

≤ 0,

j = 1,..., n,

i ij

i =1

u r ≥ 0, v i ≥ 0,

r = 1,..., s, i = 1,..., m.

where ur(r = 1, …, s) and vi(i = 1, …, m), represent the output and input weights, respectively. Besides, the fractional programme is not used for actual computation of the efficiency scores due to its non-convex and nonlinear properties. Hence, by using Charnes and Cooper (1962) transformation, model (1) can be equivalently transformed into the linear programme below for solution: s

Max

∑u y r

ro

r =1 m

s.t.

∑v x

i io

=1

i =1 s

m

∑u y − ∑v x r

r =1

i ij

rj

≤ 0,

j = 1,..., n,

(2)

i =1

u r ≥ 0, v i ≥ 0,

r = 1,..., s, i = 1,..., m.

The economic meaning of the above model can be interpreted as “DMUP searches for a set of input and output weights to maximize its efficiency as a whole and at the same time to make its each output being as efficient as possible to produce sufficient efficiency as an individual”.

2.1.2 L∞-norm in DEA By comparing n units with s outputs denoted by yrj, r = 1, …, s and m inputs denoted by xij, i = 1, …, m that all of them are non-negative and each DMU has at least one strictly positive input and output. The production possibility sets (PPS) is defined as: ⎧⎪ Tc = ⎨( X , Y ) | X ≥ ⎩⎪

n

∑ j =1

n

λj X j ,Y ≤

∑λ Y ,λ j

j =1

j

j

⎫⎪ ≥ 0, j = 1,..., n ⎬ ⎭⎪

(3)

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Rezai Balf et al. (2011) introduced ranking model L∞-norm in DEA. They assumed that the DMUo is extreme efficient. By omitting (Xo, Yo) from TC, they defined the PPS TC ′ as: ⎧⎪ Tc′ = ⎨( X , Y ) | X ≥ ⎩⎪

n



⎫⎪ λ j Y j , λ j ≥ 0, j = 1,..., n ⎬ j =1, j ≠ o ⎭⎪ n

λj X j ,Y ≤

j =1, j ≠ o



(4)

They consider the following model to obtain the ranking score of DMUo: ⎛⎧ ⎪ Min Φ oc ( X , Y ) = Max ⎜ ⎨ xio − ⎜ ⎜ ⎪⎩ ⎝

n ⎪⎫ ⎪⎧ λ j xij ⎬ , ⎨ yro − λ j yrj ⎪⎭i =1,..., m ⎪⎩ j =1, j ≠ o j =1, j ≠ o n





⎞ ⎪⎫ ⎟ ⎬ ⎟ ⎪⎭r =1,..., s ⎟ ⎠

n

s.t.



λ j xij ≥ xio ,

i = 1,..., m,

λ j yrj ≤ yro ,

r = 1,..., s,

(5)

j =1, j ≠ o n



j =1, j ≠ o

j = 1,..., n.

λ j ≥ 0,

where X = (x1, …, xm), Y = (y1, …, ys) and Λ = λ1, …, λo–1, λo+1, …, λn are the variables of the model (5) and Φ oc ( X , Y ) is a distance (Xo, Yo) from (X, Y) by using L∞-norm. It is obvious that the model (5) is nonlinear. In order to converting this model to a linear form, the set Tc′′ is define as: Tc′′ = Tc′

∩{( X , Y ) X ≥ X

o

, Y ≤ Yo } .

Therefore, by added the constrains X ≥ Xo and Y ≤ Yo to the model (5) they obtained the linear form as follows: Min φo n

s.t. φo ≥



λ j xij − xio ,

i = 1,..., m,

j =1, j ≠ o

(6)

n

φo ≥ yro −



r = 1,..., s,

λ j yrj ,

j =1, j ≠ o

λ j ≥ 0,

j = 1,..., n.

⎛⎧ n ⎞ n ⎪ ⎪⎫ ⎪⎧ ⎪⎫ ⎟. where φo = Max ⎜ ⎨ λ j xij − xio ⎬ , ⎨ yro − λ j yrj ⎬ ⎜⎜ ⎟ ⎪⎭i =1,..., m ⎪⎩ ⎪⎭r =1,..., s ⎟⎠ j =1, j ≠ o ⎝ ⎪⎩ j =1, j ≠ o





Theorem 1: Suppose (Xo, Yo) ∈ Tc is extreme efficient. For each ( X , Y ) ∈ Tc′ \ Tc′′ there

(

)

(

)

exists at least a member of Tc′′ , say X , Y , such that Φ oc X , Y ≤ Φ oc ( X , Y ) .

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Theorem 2: In any optimal solution the model (6), at least one of inputs (outputs) constraints is active. Theorem 3: The projected point of DMUo in model (6) lies on the efficient frontier. Theorem 4: Model (6) is always feasible and bounded.

3

Problem formulation

In this section, the mathematical model has been presented based on CMS with worker flexibility under following assumptions: •

the processing time for all operations of a part type on different machine types are known and deterministic



the demand for each part type is known and deterministic



the capacity of each machine type is known



the available time of each worker is known



the number of production for each part littler than the number of demand for each part.

3.1 Indices and their upper bounds P W M C i w m k

Number of part types Number of worker types Number of machine types Number of cells Index for part type (i = 1, 2, … P) Index for worker (w = 1, 2, … W) Index for machine type (m = 1, 2, … M) Index for cell (k = 1, 2, … C)

3.2 Input parameters rimw aim LMk LPk LWk RWw RMm timw Di

εi αi A

1 if machine type m is able to process part i with worker w; = 0 otherwise 1 if part i needs machine type m; = 0 otherwise Minimum size of cell k in terms of the number of machine types Minimum size of cell k in terms of the number of parts Minimum size of cell k in terms of the number of workers Available time for worker w Available time for machine m Processing time of part i on machine type m with worker w Demand of part i Unit cost of lost sales demand of part i Unit cost of inter-cell movement of part i An arbitrary big positive number

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3.3 Decision variables xmk

1 if machine type m is assigned for cell k; = 0 otherwise

yik

1 if part i is assigned to cell k; = 0 otherwise

zwk

1 if worker w is assigned for cell k; = 0 otherwise

dimwk

1 if part i is to be processed on machine type m with worker w in cell; = 0 otherwise

Pi

Number of part i to be produced

3.4 Mathematical formulation 3.4.1 Objective functions P

Min =

∑ ε [D − P ] i

i

(7.1)

i

i =1

⎡ C M W ⎤ ⎢ ⎥ ⎣⎡ yik xmk (1 − zwk ) dimwk ⎦⎤ ⎢ k =1 m =1 w=1 ⎥ ⎢ C M W ⎥ P + αi Pi ⎢ + ⎡⎣ 2 × xmk (1 − yik )(1 − zwk ) dimwk ⎤⎦ ⎥ ⎢ ⎥ i =1 ⎢ k =1 m =1 w=1 ⎥ ⎢ C M W ⎥ ⎡⎣ xmk (1 − yik ) zwk dimwk ⎤⎦ ⎢+ ⎥ ⎣⎢ k =1 m =1 w=1 ⎦⎥

∑∑∑



(7.2)

∑∑∑

(7.3)

∑∑∑

(7.4)

3.4.2 Contraints C

M

P

∑∑∑ d

imwk timw Pi

≤ RWw

∀ w;

(8)

∀ m, k ;

(9)

k =1 m =1 i =1 W

P

∑∑ d

imwk timw Pi

≤ RM m

w =1 i =1

Di ≥ Pi

∀ i;

(10)

dimwk ≤ rimw xmk

∀i, m, w, k ;

(11)

∀i, m;

(12)

=1

∀ i;

(13)

≥ LPK

∀ k;

(14)

C

W

∑∑ d

imwk

= aim

k =1 w =1 C

∑y

ik

k =1 P

∑y

ik

i =1

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∑x

=1

∀m;

(15)

mk

≥ LM k

∀k ;

(16)

wk

=1

∀w;

(17)

wk

≥ LWk

∀k ;

(18)

∀i, m, w, k ;

(19)

∀i;

(20)

mk

k =1 M

∑x m =1 C

∑z k =1 W

∑z w =1

xmk , yik , zwk , dimwk ∈ {0,1} ≥0

Pi

The objective function consists of several costs items as follows: •

(7.1) The cost of lost sales demand: The cost of shortage in delivery of all parts. This item is calculated the number of demand for each part, minus the number of production for each part, multiply by the unit cost of lost sales demand each part.



(7.2), (7.3), (7.4) Inter-cell movements cost: These terms are to minimise the total number of inter-cell movements in machine-part-worker incidence matrix. The numbers of inter-cell movements for parts are calculated based on the status of availability of corresponding machine and worker as shown in Table 1. If the corresponding machine and worker both are not in the cell, the number of inter-cell movements will take value 1 or 2 depending on the availability of machine and worker in one cell or at different cells, respectively. The equations (7.2)α(7.4) can be simplified as follows: P

C

M

W

∑∑∑∑ ⎡⎣ x

mk

( 2 − yik − zwk ) dimwk

i =1 k =1 m =1 w =1

Table 1 Case

⎤⎦

Status of exceptional elements Part

Machine

Worker

Inter-cell movements

1

3

3

3

0

2

3

3

×

1

3

×

3

×

2

4

×

3

3

1

Note: ‘3’ denotes included and ‘×’excluded.

To clarify calculation of inter-cell movements, this concept is discussed in Figures 1, 3 and 4. In Figure 1, part type 3 needs worker 5 to get processed on machine type 2. However, part type 3 and machine type 2 have been assigned to cell 3 while worker 5 is in cell 2. Thus, worker 5 has to come to cell 3 which implies one intercellular movement (case 2 of Table 1).

DEA in cellular manufacturing systems considering worker assignment Figure 1

Inter-cell movement for case 2 in Table 1

w3 p5

w4 p1 Cell 1

m3

m1

p2 w1 w5 m4

m5

Cell 2

m2

m1

w2 w6 p3 p 4

m4

m4

m1

m3

m4

m2

Figure 2

267

Cell 3

Inter-cell movement for case 3 in Table 1

m1

m1

p2 w5 w1 m4

m5

Cell 2

Cell 1

m2

m1

w2 w6 p3 p4

m4

w3 p5

w4 p1

m3

m3

m4

m2

Cell 3

In Figure 2, let us discuss case 3 of Table 1. In this figure, suppose machine type 2 and worker 1 are required to process part type 2. Furthermore, part type 2, machine type 2 and worker 1 have been assigned to cell 2, cells 1 and 3, and cell 2, respectively. Since part type 2 and worker 1 have to move to cell 3, the number of inter-cell movements will be 2. Figure 3

Inter-cell movement for case 4 in Table 1

m1 w3 p5

Cell 1

m1

p2 w1 w5 m4

m5

Cell 2

m2

m1

w2 w6 p3 p4

m4

w4 p1

m3

m3

m4

m2

Cell 3

Case 4 of Table 1 is demonstrated in Figure 3. In this figure, suppose machine type 5 and worker 1 are required to process part type 4. Furthermore, suppose part type 4 has been assigned to cell 3 while machine type 5 and worker 1 are in cell 2. Therefore, part type 4 has to move to cell 2 which results in one intercellular movement in this case.

3.5 Description of constraints Constraints (8) and (9) ensure that the available time for workers and capacity of machines are not exceeded. Constraint (10) the number of production for each part littler than the number of demand for each part. Constraint (11) ensures that when machine type m is not in cell k, then dimwk = 0. Equation (12) implies that only one worker is allotted for

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processing each part type on each machine type in one cell. This model is flexible for doing same job with different workers. This means that if one part type is required to be processed by one machine type; more than one worker would be able to service this machine type. Equation (13) ensures that each part type is assigned to only one cell. Constraint (14) forces the lower bound for the number of parts to be allocated to each cell. Equation (15) guarantees that each machine type is assigned to only one cell. Constraint (16) prevents from assigning less than LMk machines to cell k. Equation (17) guarantees that each worker will be assigned to only one cell. Constraint (18) ensures that at least LWk workers will be assigned to cell k in each period.

3.6 Linearisation of the proposed model In this section, an attempt is made to linearise the objective function of the mathematical model proposed in Section 3.4.

Procedure The linearisation procedure that we propose here consists of two steps that are given by the two lemmas stated below. The nonlinear terms in the objective function and constraints (8), (9) are multiplication of binary and integer variables which can be linearised using the following auxiliary integer variables Eimwk, Fimwk, Simwk, and Gimwk. Each lemma for linearisation is followed by a proof that illustrates the meaning of each auxiliary (linearisation) variable and the expressions where they are used. Lemma 1: The nonlinear terms in the objective function and constraints (8) and (9) of the mathematical model can be linearised with Eimwk = Pi .dimwk, and Fimwk = ymk .Eimwk and Simwk = zwk .Eimwk under the following sets of constraints: Eimwk ≤ Pi + A (1 − dimwk )

∀i, m, w, k ;

(21.1)

Eimwk ≥ Pi − A (1 − d imwk )

∀i, m, w, k ;

(21.2)

Eimwk ≤ A.dimwk

∀i, m, w, k ;

(21.3)

Fimwk ≤ Eimwk i + A (1 − ymk )

∀i, m, w, k ;

(21.4)

Fimwk ≥ Eimwk i − A (1 − ymk )

∀i, m, w, k ;

(21.5)

Fimwk ≤ A. ymk

∀i, m, w, k ;

(21.6)

Simwk ≤ Eimwk i + A (1 − zwk )

∀i, m, w, k ;

(21.7)

Simwk ≥ Eimwk i − A (1 − z wk )

∀i, m, w, k ;

(21.8)

Simwk ≤ A.zwk

∀i, m, w, k ;

(21.9)

and

and

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

DEA in cellular manufacturing systems considering worker assignment 1

269

∀i, m, w, k;

dimwk . Pi = Pi.

Such a situation arises when dimwk = 1 so, constraints (21.1) and (21.2) implies Eimwk ≤ Pi and Eimwk ≥ Pi and ensures that Eimwk = Pi. 2

dimwk. Pi = 0. Such a situation arises under one of the following three sub-cases: a

dimwk = 1 and Pi = 0.

∀i, m, w, k;

b

dimwk = 0 and Pi > 0.

∀i, m, w, k;

c

dimwk = 0 and Pi = 0.

∀i, m, w, k;

In all of the three sub-cases given above, Eimwk takes the value of 0, because in these cases, constraint (21.3) implies Eimwk ≤ 0 and ensures that Eimwk = 0. Because Eimwk has not a strictly positive cost coefficient, the minimising objective function doesn’t ensures that Eimwk = 0. Thus, constraint (21.3) should be added to the mathematical model. The performance of constraints (21.4)–(21.9) is similar to constraints’ (21.1) and (21.3). Lemma 2: The nonlinear terms in the objective function can be linearised with Gimwk = xik .Eimwk, under the following set of constraints: Gimwk ≥ Eimwk − A (1 − xik )

∀i, m, w, k ;

(22)

Proof: Consider the following two cases: 1

xik . Eimwk = 0. Such a situation arises under one of the following three sub-cases: a

xik = 1 and Eimwk = 0.

∀i, m, w, k;

b

xik = 0 and Eimwk > 0.

∀i, m, w, k;

c

xik = 0 and Eimwk = 0.

∀i, m, w, k;

In all of the three sub-cases given above, the value of Gimwk = 0, because in these cases, constraint (22) implies Gimwk ≥ 0 or –∞ and since Gimwk has a strictly positive cost coefficient, the minimising objective function ensures that Gimwk = 0. 2

∀i, j;

xik . Eimwk = Eimwk > 0.

Such a situation arises when xik = 1 and Eimwk > 0 so, constraint (22) implies Gimwk ≥ Eimwk and since Gimwk has a strictly positive cost coefficient, the minimising objective function ensures that Gimwk = Eimwk.

3.6.1 The linearised model The new version of the first, second and third terms of objective function based on new variables, the linear mathematical model becomes as follows: P

Max = Eq.(5.1) −

C

M

W

∑∑∑∑ α [ 2 × G i

imwk

− Fimwk − Simwk ]

i =1 k =1 m =1 w =1

Eimwk , Fimwk , Gimwk , Simwk ≥ 0

∀i, m, w, k ;

Subject to constraints (10)–(23) and new version of constraints (8) and (9):

(23)

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M

P

∑∑∑ E

imwk timw

≤ RWw

∀ w;

(24)

∀ m, k ;

(25)

k =1 m =1 i =1 W

P

∑∑ E

imwk timw

≤ RM m

w =1 i =1

4

Using DEA in the CMS model

4.1 Choice the inputs and outputs for DEA model In this paper, the DEA is applied to the problem of comparing and evaluating the alternative rescores assignment in a CMS environment. In general, in a number of preceding DEA evaluation models, the criteria that are to be minimised are viewed as inputs, and the criteria to be maximised are considered as outputs (Doyle and Green, 1993). In other words, usually the DEA assumes that outputs are rising and more of an output is better than less of the output. Ertay and Ruan (2005) did not examine the concept of cellular manufacturing to design the whole structure of the cells to adapt to changes in the customer’s demand, but they only studied on the design of a flexible cell. They took advantage of the cross-efficiency evaluation to determine the best labour assignment in cell when the demand rate and the transfer batch size as a rate of batch size change as the input variables and employing the average lead time, the average operator utilisation as the output variables. In this article we are looking to increase production volumes, but given the limited resources available in production quantities of all the market demands are impossible. Moreover, the concept of cellular manufacturing to design the whole structure of the cells to adapt to changes in the customer’s demand, So that, there are the various alternatives for supply of machines and workers with different capabilities and many different products and production values. In other words, we seek to increase production volumes and simultaneously increase productivity and efficiency are the resources. In the simulation studies, different scenarios are compared in terms of the average machine utilisation, the average worker utilisation, and mean of product selected as performance measures. Increasing the resources available (machines, workers), increase the variety and quantities of products will follow. System performance is monitored for different demand levels, machine assignment alternatives, worker assignment alternatives and part assignment alternatives by means of simulation. This study will assist the mathematical model when a decision related to interchange of the machinery and workforce in a manufacturing cell is required as soon as the demand is changed. The objective of the alternatives consisted of reducing the cost of lost sales demand and Intercell movements cost. •

Choice 4 machine among 5 machine type that have different availability level the decision maker have 5 alternatives to choice machines 1, 2, 3, 4, 5, the alternatives are 1, 2, 3, 4 (alternative A), 1, 2, 3, 5 (B), 1, 2, 4, 5 (C), 1, 3, 4, 5 (D), 2, 3, 4, 5 (E).



Use 4 or 3 worker in the cell that the workers number 1 and number 2 have the same level and the workers number 3 and number 4 have the same level, since the decision

DEA in cellular manufacturing systems considering worker assignment

271

maker have 3 alternative for choose the workers, all the 4 workers (F), worker 1, worker 2 and worker 3 (G), worker 2, worker 3 and worker 4 (H). •

The number of parts type for produce is 4 or 5. The production value (backorder cost) all of the parts are the same level so decision maker have 2 alternatives to choice parts.



Numbers of demand for each part are 300 and 350, and decision maker have 2 alternatives to choice volume of the parts.

It has been considered 60 simulation experiments in this study (5 × 3 × 2 × 2 = 60; machine assignment alternatives × worker assignment alternatives × part assignment alternatives × demand levels). To illustrate the capability of the proposed model an alternative have been solved by branch and bound (B&B) method under Lingo 9.0 software package. In all alternatives we consider two cells with different machines, parts and workers. The dataset related to the all alternatives are shown in Tables 2 and 3. Table 2 indicates machines requirement of parts. For example, part type 3 requires machine types 2 and 4. Table 3 indicates capabilities of workers in working with different machines. For example, worker 3 is able to work with machine types 2 and 4. The available time of worker in each period is 20 hours and the available time of machine in each period is 20 hours. Also the processing time is presented in Table 4. Moreover, the unit cost of lost sales demand of each part types are 1. Also, the minimum size of each cell in terms of the number of machines, parts and workers has been considered to value one. Table 2

The input data of machine-part incidence matrix

Parts

Machines 1

2

3

4

5

1

1

1

1

1

1

2

1

0

1

0

1

3

0

1

0

1

0

4

0

1

0

1

0

5

1

0

1

0

1

Table 3

The input data of machine-worker incidence matrix

Machines

Workers 1

2

3

4

1

1

0

0

1

2

0

1

1

0

3

1

0

0

1

4

0

1

1

0

5

1

0

0

1

M5

M4

.01

.03

M3

M2

.02

M1

.01

.04

.01

.04

W3

Part 1

W2

.01

.03

.02

W4

.01

.03

.02

W1

W3

Part 2 W2

.01

.03

.02

W4

W1

.01

.04 .01

.04

W3

Part 3 W2

W4

W1

.01

.04

.01

.04

W3

Part 4 W2

W4

.01

.03

.02

W1

W3

Part 5 W2

.01

.03

.02

W4

Table 4

W1

272 A. Aalaei et al.

The processing time (hrs.)

2

3, 4

Worker

1, 2

1, 3, 4

2, 3, 4, 5

Cell 2

14

4

3

2

1

Machine

Notes: the worker movement between cells b the part movement between cells

a

Volume of product

1

Machine

Cell 1

Part

584

Inter-cell movements

1

2

50

3

Part 1

4 2

1

2

316

3

Part 2

a

2

4

1

100

2

2 1

b

3

Workers

Part 3

4

1

350

2

2 1

3 b

Part 4

4

2

1

350

2

3

Part 5

2

4

a

Table 5

The cost of lost sales demand

DEA in cellular manufacturing systems considering worker assignment The result of alternative 1

273

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A. Aalaei et al.

Tables 5 show the results of alternative 1. It indicates the assignment of parts, machines and workers in cells. For instance, workers 3 and 4 are assigned in cell 1, and worker 1 and 2 is assigned in cell 2. Also machine type 2 is assigned in cell 1 and machines 1, 3 and 4 are assigned in cell 2. Moreover, it shows the allotment of worker for each part, in cell for work on corresponding machine. For instance, part 3 shall process with machine 2 (see Table 2) and workers 2 and 3 capability of working to this machine (see Table 3) which this operation is executed by worker 3 in cell 1 (see Table 5). The volume of products and objective function value including the cost of lost sales demand and number of inter-cell movements has been indicated in Table 5. As can be seen, the demand of part 2 in is 350 but this part is 316 produced. This means, the 34 volume of demand of part 2 is which causes the cost of lost sales demand. These alternatives inputs data of the mathematical model are indicated in the Table 6. Steps of the proposed methodology are presented in Figure 4. Figure 4

Steps of the proposed methodology

Data collection

Input data Alternative for resources assignment and demand Generation output data by mathematical programming

DEA for final resource assignment and demand

Final resource assignment and demand Source: Ertay and Ruan (2005)

For each alternative we use the developed CMS and calculate average machine utilisation (ATUM) and average worker utilisation (ATUW) and mean of product from the solutions and set them as the alternative results data of the mathematical model, Table 7 shows the results of 32 alternatives.

4.2 The most efficient alternative We used the model (2) for 60 inputs and outputs that showed in the Tables 6 and 7, The DEA is applied to the dataset of 60 DMUs. The efficiency scores obtained using DEA are listed in Table 8. The DEA results denote that 9 cases of 60 DMUs are relatively efficient; however, a ranking cannot be obtained for these DMUs. Since the efficiencies evaluate 9

DEA in cellular manufacturing systems considering worker assignment

275

of the 60 DMUs as efficient and cannot discriminate among them any further, a ranking method is needed. We are use the L∞-norm model (6) to rank these 9 alternatives. The results are shown in Table 9. According to the L∞-norm method in Table 9, DMU24 is the most efficient alternative, whereas DMU48 is the second most efficient followed by DMU23, DMU47 and others. Table 6

The inputs of the DEA model

Demand Number level of parts

Number Number Number Number Demand Number of of of of DMU DMU level of parts workers machines workers machines

350

4

3(F)

4(C)

31

350

5

4

4(A)

1

300

4

3(F)

4(C)

32

300

5

4

4(A)

2

350

5

3(G)

4(C)

33

350

4

4

4(A)

3

300

5

3(G)

4(C)

34

300

4

4

4(A)

4

350

4

3(G)

4(C)

35

350

5

3(F)

4(A)

5

300

4

3(G)

4(C)

36

300

5

3(F)

4(A)

6

350

5

4

4(D)

37

350

4

3(F)

4(A)

7

300

5

4

4(D)

38

300

4

3(F)

4(A)

8

350

4

4

4(D)

39

350

5

3(G)

4(A)

9

300

4

4

4(D)

40

300

5

3(G)

4(A)

10

350

5

3(F)

4(D)

41

350

4

3(G)

4(A)

11

300

5

3(F)

4(D)

42

300

4

3(G)

4(A)

12

350

4

3(F)

4(D)

43

350

5

4

4(B)

13

300

4

3(F)

4(D)

44

300

5

4

4(B)

14

350

5

3(G)

4(D)

45

350

4

4

4(B)

15

300

5

3(G)

4(D)

46

300

4

4

4(B)

16

350

4

3(G)

4(D)

47

350

5

3(F)

4(B)

17

300

4

3(G)

4(D)

48

300

5

3(F)

4(B)

18

350

5

4

4(E)

49

350

4

3(F)

4(B)

19

300

5

4

4(E)

50

300

4

3(F)

4(B)

20

350

4

4

4(E)

51

350

5

3(G)

4(B)

21

300

4

4

4(E)

52

300

5

3(G)

4(B)

22

350

5

3(F)

4(E)

53

350

4

3(G)

4(B)

23

300

5

3(F)

4(E)

54

300

4

3(G)

4(B)

24

350

4

3(F)

4(E)

55

350

5

4

4(C)

25

300

4

3(F)

4(E)

56

300

5

4

4(C)

26

350

5

3(G)

4(E)

57

350

4

4

4(C)

27

300

5

3(G)

4(E)

58

300

4

4

4(C)

28

350

4

3(G)

4(E)

59

350

5

3(F)

4(C)

29

300

4

3(G)

4(E)

60

300

5

3(F)

4(C)

30

276 Table 7 ATUW (%)

A. Aalaei et al. The outputs of the DEA model ATUM (%)

Mean of product (%)

DMU

ATUW (%)

ATUM (%)

Mean of product (%)

DMU

66.3

51.4

83.1

31

72

72.9

66.5

1

75.8

53.6

91.6

32

72

72.9

73.3

2

74.1

37

56

33

72

72.9

83.1

3

66.3

49.2

58.3

34

60

68.5

91.6

4

68.5

38.5

78.5

35

75.8

56.9

51.4

5

63.6

47

83.3

36

75.8

56.9

60

6

59.2

59.2

78

37

75.8

56.9

64.2

7

57

58.5

77.3

38

70

58.5

75

8

58.5

58.5

97.4

39

77.4

58.5

60.8

9

55.8

52.9

100

40

83.6

62.5

66.6

10

44.1

33.6

58.8

41

88.5

66.3

76

11

77

58.5

84.3

42

83.6

62.9

83.3

12

44.1

33.6

73.7

43

59.2

59.2

66.2

13

43.6

26.3

77.6

44

55.8

55.8

73.3

14

83.6

47

78

45

74.1

74.1

83.2

15

74.1

59.2

84.3

46

70

70

91.6

16

78.5

58.5

97.4

47

66.3

49.2

47.6

17

70

52.9

100

48

66.3

49.2

55.5

18

64.1

54.1

66.5

49

66.3

49.2

59.4

19

64.1

64.1

73.2

50

66.3

49.2

69.4

20

64.1

64.1

83.2

51

99.2

74.1

67.7

21

58.5

61.4

91.6

52

93.6

70

73.3

22

74.1

55.8

56.5

53

99.2

74.1

83.2

23

75.8

56.3

66.6

54

93.6

70

91.6

24

75.8

56.3

71.4

55

44.3

53.6

68.5

25

75.8

51.4

83.3

56

65.8

65.8

73.3

26

77

58.5

60.8

57

57

57

85.7

27

73.6

55.8

66.6

58

56.3

64.1

91.6

28

77

54.1

76.8

59

74.1

56.3

66.5

29

73.6

55.8

83.3

60

71.4

49.2

73.3

30

DEA in cellular manufacturing systems considering worker assignment Table 8

277

Efficiency scores that are obtained by DEA

Efficiency score

DMU

Efficiency score

DMU

Efficiency score

DMU

Efficiency score

DMU

0.8999

46

0.8591

31

1.0000

16

0.9838

1

1.0000

47

0.9493

32

0.6683

17

0.9836

2

1.0000

48

0.7470

33

0.7083

18

0.9891

3

0.8650

49

0.7083

34

0.6847

19

0.9906

4

0.8702

50

0.8217

35

0.7445

20

0.7679

5

0.9128

51

0.8481

36

1.0000

21

0.8129

6

0.9466

52

0.8489

37

0.9446

22

0.7692

7

0.7530

53

0.8403

38

1.0000

23

0.8357

8

0.7771

54

0.9874

39

1.0000

24

0.7895

9

0.7978

55

1.0000

40

0.7587

25

0.8932

10

0.8813

56

0.5973

41

0.8879

26

0.9014

11

0.7894

57

0.8972

42

0.8862

27

0.9051

12

0.7698

58

0.7370

43

1.0000

28

0.7989

13

0.8339

59

0.7760

44

0.7738

29

0.7988

14

0.8772

60

0.8767

45

0.7902

30

1.0000

15

Table 9 DMU

The Tchebycheff values and ranking efficient DMUs DEA Tch. norm Value

Rank

15

1.8855e-009

5

16

8.0071e-010

7

21

1.4388e-009

6

Table 10

Tch. norm

DMU

Value

Rank

23

0.2295

3

24

6.7066

1

28

3.4715e-017

9

Cell 1 Cell 2 Machine w1 1, 2

Machine 1, 3, 4 Worker

Value

Rank

40

7.2731e-015

8

47

0.0131

4

48

1.9761

2

The result of alternative 24 Part 1

Part

Tch. norm

DMU

1

3, 4

1

2

2

2, 3

3

1

4

1

w2

Part 2 w3

1

w1

w2

Part 3 w3

w1

w2

Part 4 w3

w1

w2 w3

1 a

2

2

Volume of product

1

2

1 200

300

300

300

a

Note: the part movement between cells

As can be seen in Table 9, Alternative 24 has the best performance and thus the result of this alternative shows in Table 10. It indicates the assignment of parts, machines and workers in cells. For instance, part type 1 is processed on machine type 2 with worker 3 in cell 2. Then part type 1 is transferred from cell 1 to cell 2 and 200 inter-cell movements for this part type is accrued. Inter-cell movements in CMS resulted in a reduction in resource productivity and performance. Furthermore, Table 10 shows that

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only one mode inter-cell movement occurs, so the productivity and efficiency from other sources are more alternatives.

5

Conclusions

In recent years, many studies have been done on the role and importance of CMSs in production volumes. However, in these studies, less attention was paid on the productivity and effectiveness of using resources. In this study, we show that only looking for more production is not enough in the competitive market, but the productivity and efficiency of available resources are particularly important. Because of the limited resources, not all the factories and industrial enterprises are able to compete in the market by producing different types of products. Therefore, this paper presents a new approach to optimise productions considering both ability of competing in the market and appropriate usage of available resources. Furthermore, in this paper we proposed the mathematical model in CMSs considering production volumes and worker assignment. Moreover, the new concept of inter-cell movements is discussed to show the interpretation of inter-cell movements of both workers and parts for processing on corresponding machines. The proposed approach minimises cost of lost sales demand and intercellular cost in a CMS. The DEA approach performed for determining the most efficient alternative among 60 alternatives that considered. As a result of the application of classic DEA model, nine alternatives are determined as relatively efficient. To increase discriminating power among alternatives and ranking, the Tchebycheff-norm ranking method was employed. Future work could be focused on refining this problem by considering the multi horizon planning and uncertainty in the demand where a robust optimisation model of this problem could be formulated and solved to find optimal solution in a more practical situation.

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