A mathematical programming model for manufacturing cell formation to develop multiple configurations

Journal of Manufacturing Systems 33 (2014) 149–158

Contents lists available at ScienceDirect

Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys

Technical paper

A mathematical programming model for manufacturing cell formation to develop multiple configurations C.R. Shiyas, V. Madhusudanan Pillai ∗ Department of Mechanical Engineering, National Institute of Technology Calicut, NIT Campus (P.O), Calicut 673601, Kerala, India

a r t i c l e

i n f o

Article history: Received 3 April 2013 Received in revised form 14 July 2013 Accepted 2 October 2013 Available online 1 November 2013 Keywords: Manufacturing cells Heterogeneity of cells Mathematical model Genetic algorithm Grouping efficacy

a b s t r a c t This paper presents and analyses a mathematical model for the design of manufacturing cells which considers two conflicting objectives such as the heterogeneity of cells and the intercell moves. A genetic algorithm (GA) based solution methodology is developed for the model which is also solved using an optimization package. The model is suitable for getting multiple potential solutions in a structured way for the cell formation problem by making a trade-off between the two objectives, instead of reaching at a single negotiating solution. This model provides the decision maker the flexibility of choosing a suitable cell design from different alternatives by considering the practical constraints. A part assignment heuristic is also developed by which part-families can be identified and is integrated with the GA based solution procedure. A comparison of the proposed method is made with other seven methods using 36 problems from the literature. Grouping efficacy is the basis for comparison and it is found to give reasonably good results. © 2013 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction

2. Literature review and problem description

Cellular manufacturing (CM) has the flexibility of job shops and high production rate of flow lines. It is a group technology (GT) concept and is one of the best methods for achieving flexibility without compromising productivity. GT is a theory of management, based on the principle that a group of things which are having similarities can be done in a similar fashion so as to ensure saving in time, effort and cost. The goal of cellular manufacturing is to have the flexibility to produce a high variety of medium demand products, while maintaining the high productivity of large scale production, by grouping parts and machines according to the similarity between them. Design of a cellular manufacturing system (CMS), generally, involves grouping of parts with similar design features/processing requirements into part-families and the associated machines into machine cells [1]. CMS has reduced cycle time compared to job shops, and increased flexibility and greater job satisfaction compared to flow shops. CMS has been proven to be better than the other traditional layout types given that the demand and the part mix remain constant over the planning horizon.

The most widely used methods for CMS design in the literature are based on (i) array management, (ii) hierarchical clustering, (iii) non-hierarchical clustering, (iv) mathematical models, and (v) heuristic techniques. All these methods generally use the part machine incidence matrix as the input. In 1980 King [2] introduced an array based clustering method, namely, rank order clustering (ROC) which can identify part-families and machine groups simultaneously by rearranging the 1’s in a part machine incidence matrix to form a block diagonal structure. Later, a modified version called MODROC was developed by Chandrasekharan and Rajagopalan [3]. Other array-based clustering methods include bond energy algorithm (BEA) of McCormick et al. [4], and the direct clustering algorithm of Chan and Milner [5]. The main drawback of array based clustering methods is that the quality of solution depends on the initial configuration of part machine incidence matrix [6]. In hierarchical clustering method, the input data set is described in terms of a similarity or a distance function and produces a hierarchy of clusters [7]. Non-hierarchical methods are used in ZODIAC [8] and GRAPHICS [9]. While CASE [10] describes a CMS design using a similarity coefficient based non-hierarchical clustering for sequence data. Better solution is obtained in the non-hierarchical methods compared to the earlier methods. The CMS design problems are combinatorial optimization problems and many mathematical programming approaches are available for better solution compared to the other methods. Kusiak and Chow [11] developed a p-median model for CMS design, where

∗ Corresponding author. Tel.: +91 495 2287804/9895367804; fax: +91 495 2287250. E-mail addresses: [email protected] (C.R. Shiyas), [email protected] (V. Madhusudanan Pillai).

0278-6125/$ – see front matter © 2013 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jmsy.2013.10.002

150

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158

the objective is to minimize the sum of distances between each product/machine pair. Another p-median model based on a new similarity coefficient for CMS design is developed by Youkyung and Currie [12]. A two-stage procedure for CMS design was developed by Choobineh [13] which identifies product families initially and then uses an integer programming model for machine cell formation. In their model, the objective is to minimize the production cost and the machine acquisition cost. Srinivasan et al. [14] proposed an assignment model for the part-families and machine grouping problem. They formulated the CMS design problem as an assignment problem and proposed an algorithm for forming manufacturing cells after part-family formation. Adil et al. [15] formulated an assignment allocation algorithm (AAA), a non-linear mathematical model, to identify the part-families and machine groups simultaneously. The objective of the model is to explicitly minimize the weighted sum of the exceptional elements and voids, so that multiple solutions are obtained by varying the weights. Their model may lead to formation of machine cells without any machines, but with part assignment. Mahdavi et al. [16] developed a mathematical model to solve cell formation problem and the objective is to minimize the number of voids and exceptional elements in a three dimensional (cubic) machine-part-worker incidence matrix. This model is able to capture the capability of workers in doing different jobs and such models are rarely seen in the literature. The early literature in CMS considered single objective functions but there are a certain recent research papers that consider multi-objective mathematical model. Yasuda et al. [17] proposed a multi-objective model for CMS design which can provide a variety of solutions. But, Yin et al. [18] proved that the formulation of Yasuda et al. [17] can be considered as an algorithm for cell formation only. Pitombeira Neto and Goncalves Filho [19] presented a Pareto-optimal multi-objective model based on GA and simulation for solving the manufacturing cell formation problem. The solution fitness analysis is carried out by means of simulation and finally it results in a number of alternative solutions. The bi-objective model proposed by Arkat et al. [20] optimizes voids and exceptional elements in a system. Their solution procedure consists of an ε constrained method based on GA which is a step by step procedure and it uses new constraints in every step. This solution procedure leads to a single solution. Rabbani et al. [21] suggested a bi-objective model for cell formation under stochastic production situations and developed a two-phase fuzzy linear programming approach as solution methodology. They compared their approach with other two methods to prove the effectiveness. Raflei and Ghodsi [22] modelled a dynamic cell formation problem as a bi-objective problem and a hybrid ant colony optimization-genetic algorithm solution methodology is developed. Other cell formation methods are mainly based on heuristic approaches such as genetic algorithms (GA), simulated annealing (SA), tabu search, bacteria foraging algorithm (BFA). They are used either directly or as a solution method for mathematical models. These approaches are generally applied when pure mathematical modelling is not possible or the models require long computational time for a solution under available methods, especially for large size problems. Chan et al. [23], Rogers and Kulkarni [24], Leepost [25], Mak et al. [26], Wu et al. [27], Onwubolu and Mutingi [28], Tariq et al. [29], Joines et al. [30], and Deljoo et al. [31] used GA based solution methodology in part-family identification and cell formation problems. Wide use of GA, in solving cell formation models, is due to the combinatorial nature of the problem and the possibility of easy representation of solutions in the form of chromosomes. Arkat et al. [32], Zolfaghari and Liang [33], Adil et al. [15] and Chen et al. [34], Ariafar and Ismail [35] used simulated annealing for solving their cell design models while, Zolfaghari and Liang [36] used hybrid of tabu search and simulated annealing for

developing optimization algorithms for CMS design. They conducted a comparative study using GA, simulated annealing and tabu search in solving cell formation problems and their study indicate that simulated annealing outperforms both GA and tabu search mostly for large size problems. These studies also reveal that GA seems slightly better than that of the tabu search method for the comprehensive grouping problems which involve machine/part types, processing times, lot sizes, and machine capacities, etc. Islier [37] used an ant system algorithm for a general cell formation problem which is found to be better than GA, tabu and SA when an equal number of solution alternatives are concerned. Fuzzy programming approach [38] and algorithms based on neural networks [39–41] are also used in recent researches to obtain good configuration for traditional cell formation problems. The bacteria foraging algorithm (BFA) is a new computation technique developed from the foraging behaviour of Escherichia coli (E. coli) bacteria. BFA is found to be an efficient optimization tool used to solve a large class of problems by exploring all regions of the state space and exponentially make use of potential areas through chemotaxis, swarming, reproduction, and elimination–dispersal operations of E. coli bacteria [42]. This technique is used in many scientific applications which require optimization. Nouri et al. [42] presented the first application of BFA in the design of CMS. In their work, an algorithm has been developed based on BFA to solve the CMS design problem, while taking into consideration the minimization of number of voids in the cells and the number of inter-cell travels based on operational sequences of the parts visiting the machines. CMS design models considering operation time [43], and cell load variation [44] are also available based on BFA. Other newly developed approaches of CMS design include a new branch and bound algorithm for the design of CMS [45], firefly inspired algorithm [46], etc. Generally, solution procedure for all the cell formation methods converges to a single configuration so that the decision maker has no choice other than selecting it. But in real situations, implementation of such a configuration is compromising as the solution may be difficult to implement due to some practical conditions which are not quantified in the model. Rarely, all the practical conditions are modelled into the mathematical model. So developing alternative configurations has specific advantages in such conditions and the proposed model is able to provide multiple configurations. AAA [15] gives the possibility of different solutions, but it may lead to formation of cells with part assignment for which there are no machines assigned. The algorithm for multi-objective optimization by Dimopoulos [47] also gives an opportunity of multiple solutions, but it lacks the presence of a complete mathematical formulation for forming machine cells. In their model part assignment is not discussed and as a result the quality of solutions cannot be analyzed in a traditional way using grouping efficiency or grouping efficacy, etc. The model of Pitombeira Neto and Goncalves Filho [19] provides alternative configuration; however it is different from the proposed model when we consider the nature of input parameters and objectives. Usually, models in the literature receive number of cells as input [11,14,19,24,26,28–32,37,38]. In the proposed model, number of cells is automatically evolved and can be interpreted from the value of the decision variables. So, the main advantages of the proposed model are (i) the number of cells is not an input, (ii) multiple configurations are obtained by solving the model, and (iii) a comparison with other models based on the quality of solutions is possible unlike other multi-objective models. The present paper focuses on the fundamental cell formation problem and a mathematical (non-linear programming) model has been developed for that situation. Moreover, a software package (LINGO 11.0) is used for solving the proposed model of small size problems. A GA based solution methodology is also developed since the computational

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158

time required is very high with the software package, particularly for large problems. The basic aims of this paper are (i) to develop a concept that will provide a measure called heterogeneity in terms of diversity of machine types in a cell with respect to the visiting parts and use it to develop a mathematical formulation for assigning machine types into different cells, (ii) to identify a large number of potentially feasible configurations and (iii) to compare the solution quality of the proposed method with the solution quality of other methods. The entire problem can be summarized as follows: For a given number of parts and machines, grouping of machines into cells and parts into families needs to be recognized from the part machine incidence matrix. For assigning machine types into cells, two conflicting objectives such as intercell moves and the weighted heterogeneity of cells are considered. The two conflicting objectives are combined into a single objective formulation using a weight factor. For different values of the weight factor different solutions are possible. The part assignment is made using a customized rule after the machine assignment to cells. The remaining part of this paper is organized as follows. The proposed model and the heterogeneity concepts are discussed in Section 3. The solution methodology for the mathematical formulation is discussed in Section 4. An illustrative example demonstrating the proposed model is given in Section 5. Comparison of solution quality of various methods with proposed model is provided in Section 6 and conclusion is given in Section 7.

3. The proposed model A manufacturing cell is a group of heterogeneous machines [19,48]. Minimization of intercell moves without a limitation in the number of cells will lead to a single cell for the production system. Such a cell has the maximum number of heterogeneous machine types. To have the advantages of cellular manufacturing system, there should be more than one cells or the cell should be small in size. That is, the number of types of machines and parts assigned to a cell should be at a manageable level. This is achieved in this paper by a measure called heterogeneity. Heterogeneity is a concept which is concerned with the machine types required for the processing of parts visiting a cell. It is similar to the concept of heterogeneity of operations assigned in a cell for the design of dedicated cells [49]. Every part visiting a cell contributes to the heterogeneity of the cell. The heterogeneity due to a part visiting a cell is the number of machine types present in a cell which are not required for the part to process. The heterogeneity of a cell is the sum of the heterogeneity of parts visiting the cell. The heterogeneity of a CMS is the sum of the heterogeneity of all cells. The sum of heterogeneity of all cells formed is one of the objectives, which is to be minimized in this model. If all the machine types are assigned to a single cell, the heterogeneity is maximum, and it will be minimum (zero) if a cell is dedicated for each type of machine. Operators will be gaining a lot of experience at minimum heterogeneity (zero) and confusions are less in the operation of the cell at this level. But, the maximum intercell moves will be created at minimum heterogeneity. A large number of intercell moves cannot be tolerated and hence minimization of intercell moves is the second objective considered in this model. The new cellular design approach proposed in this paper aims at minimizing the total intercell moves and the heterogeneity which are two conflicting objectives. The trade-off between these two functions will be taken care of by a single objective function which is the sum of these two functions. Grouping efficacy [50] is, generally, used for measuring the goodness of a solution. If equal importance is given for both objectives of the objective function, the solution may not be good. Also, we may be interested in alternative solutions. To get a better solution as per the goodness measure and to have alternative solutions,

151

a weight parameter is introduced. This weight is given to the heterogeneity term of the objective function. That is, the heterogeneity part of the objective function is the product of weight assigned and the heterogeneity. This weight is represented by  which can have value greater than or equal to zero. 3.1. The weight parameter sigma () The importance of the heterogeneity function in the objective function is regulated by the weight parameter . Depending on the value assigned to this parameter, the solutions obtained will vary in terms of the number of cells. A zero value of the parameter  will indicate that heterogeneity is not considered and in such a case only the intercell moves are to be minimized. This gives rise to a solution with single cell in which all machine types are assigned. This solution will continue to be the optimum one for small values of  and it will continue until a particular value for ; at further high value the model will establish more than one cell. At the other extreme, a very high value for  reduces the importance of material handling, giving priority to the internal homogeneity of the cells; resulting in the establishment of cells that are exclusively devoted to each machine type. There is a threshold value of  above which, the number of cells obtained will be equal to the number of machine types. In this model, increasing the value of  (results in increasing the number of cells) will produce a drop in the average number of types of machine assigned in each cell, and the average number of parts processed per cell. Thus for different value of , there is a possibility to get a different configuration. The mathematical formulation for identifying machine cells is given below. Notations i Index of machine types, i = 1, 2, . . ., M Index of cells k l Index of parts Parameter for regulating the importance of heterogeneity  of the cells formed 1, if part type l requires operation in machine of type i vil 0, otherwise Pl Heterogeneity due to part l Decision variables 1, if machine type i is assigned to cell k xik 0, otherwise 1, if part l visits cell k Slk 0, otherwise

 Min Z =





Pl +

l

   l

 Slk − 1

(1)

k

Subject to M 

xik = 1 ∀i

(2)

k=1 M 

xik vil ≤ M × Slk

∀l, k

(3)

i=1

Pl =

  Slk

k

xik (1 − vil ) ∀l

(4)

i

xik, Slk ∈ {0, 1}

(5)

152

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158

Table 1 Comparison of GA and LINGO run time. Problem no.

1 2 3 4 5 6 7

Size (Machine types × Parts)

4×4 4×5 5×3 5×7 7×5 7 × 12 8 × 20

solutions [44]. In GA, a solution is generally termed as a chromosome or string. Computational time (s) GA

LINGO

2 5 3 5 5 6 7

2 7 3 59 245 2750 NA

Note: NA indicates that solution is not available due to extensive runtime using LINGO. (Problem 7 is not solved even after 5 days of continuous running.)

The objective function is given in Eq. (1). The first term in this function is the weighted heterogeneity of the system. The second term is concerned with intercell moves. It is assumed that the number of intercell moves for a part is equal to the number of cells visited minus one. Constraints are numbered as (2), (3), (4) and (5). Constraint (2) assigns a machine type to a cell. Constraint (3) defines Slk which is used in the objective function for calculating intercell moves. Constraint (4) calculates Pl which is the heterogeneity of part l. Binary restrictions are given as constraint (5). 4. Solution methodology A small example problem (5 machine types × 3 parts) is taken initially and the solution for the model is obtained using LINGO which has the ability to solve linear and non-linear models. An algorithm is also developed based on genetic algorithm (GA) for solving the above formulation for any size satisfying all the constraints. LINGO and GA are giving same set of solutions for the tested problems which are given in Table 1. The experimentation with LINGO and GA reveals the efficiency of solving complex problems using GA compared to software package LINGO. The computational time required for solving seven problems of different size for a particular value of  using GA and LINGO is shown in Table 1. Table 1 shows that LINGO cannot be used for finding the solution for large size problems due to high computational time. For small size problems, both LINGO and GA are giving solution almost at the same time. But, increasing the number of machines and parts results in a large increase in computational time when LINGO is used. For ill-structured problems LINGO is taking high computational time even for problems of small sizes when compared to well-structured input matrix of the same size. Hence computational time using LINGO is not only depending on the problem size but also on the nature of the problem. On the other hand, GA based procedure could give solution to the problems of different sizes and nature within a reasonable time. The proposed GA gives machine cells for the suggested mathematical model. A solution for a CMS design problem contains part-families and machine cells. A heuristic is proposed to generate part-families. Finally, the goodness of a solution is to be evaluated. An algorithm is developed by integrating all these aspects and they are described in this section. 4.1. Genetic algorithm GA is one of the evolutionary search methods that can provide optimal or near optimal solutions for the combinatorial optimization problems. Compared to other stochastic search methods, GA searches the feasibility space by setting a population of feasible solutions simultaneously in order to find optimal or near-optimal

4.1.1. Chromosome representation The representation of chromosome plays a key role in the genetic algorithm. Here, we use integer numbers to represent a chromosome. Each gene represents a cell number and the position of a gene in a chromosome represents a machine type. The length of a chromosome represents the number of machine types. For example, a chromosome (string) ‘3 1 2 3 1 2’ represents a three cell solution with the following machine types in each cell: Cell 1: Machine types 2, 5 Cell 2: Machine types 3, 6 Cell 3: Machine types 1, 4 4.1.2. Initialization of population The initialization process is executed with a randomly generated solution space. The population size remains constant from generation to generation and has a considerable effect on the performance of genetic algorithm [51]. In this paper, the population size is taken as 2.5 times the length of a chromosome [52]. 4.1.3. Fitness function Before performing the crossover operation to create offsprings, each solution in the population should be evaluated with respect to its design objectives to find out its potential to appear in the next generation. Genetic algorithm works with maximization function and hence it is essential to convert the objective function which is to be minimized into a fitness function. A transformation used here for such a conversion is Fi =

1 (1 + Ti )

where Fi , fitness value of string i; Ti , objective function value of string i 4.1.4. Genetic operators To create the next generation, a new set of chromosomes called offsprings are produced by the execution of genetic operators such as reproduction, crossover and mutation. Crossover acts as the main operator for change while mutation acts as a secondary operator. The purpose of the reproduction operator is to pick parents for the next generation of solutions. In this paper, a method called remainder stochastic sampling without replacement policy [52,53] is used for reproduction. The expected number of copies, e(i), for the population member, i, is given by, e(i) = Pi × popsize where Pi =

F

i

Fi

Pi is the probability of selection of string i, Fi the fitness of string i and popsize is the population size. In the remainder stochastic sampling without replacement policy each member of the population is copied into the mating pool according to the integer portion of the expected number of copies. The fractional portions are normalized to the sum of one and treated as probabilities during the selection process to fill out the rest of the mating pool. A uniform number between zero and one is generated and the corresponding solution receives additional copy in the mating pool. The probability of the solution receiving another copy is set equal to zero. This process continues, with each population member receiving at most one additional copy, until the mating

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158

pool is full. In this research the size of the mating pool is equal to the size of the population size. The crossover operator creates children by exchanging information enclosed in the parent solutions such that the children contain features from each parent. In this paper, two-point crossover method is used. Mutation is required because reproduction and crossover effectively search and recombine chromosomes, but occasionally they may lose some potentially valuable genetic material. It prevents the value of any parameter from remaining intact forever. 4.1.5. Selecting new population of solutions After the genetic operators have been applied, new strings are produced and these strings are called children or offsprings. The poorly performing offsprings are replaced in the new generation with a replacement policy [52]. The offsprings are evaluated using the fitness function. The objective of the replacement policy is to create a generation of solutions that, on an average, are superior to the previous generation. 4.1.6. Termination criteria The genetic algorithm continues to generate new generations until a criterion for termination is met. In this paper, two termination criteria are used together: (i) If there is no improvement found in the best solution for a pre-specified number of generations, the algorithm stops or (ii) It may run for a particular number of generations. 4.2. Part assignment heuristic A part may be visiting several cells and the mathematical model does not assign it to any cell. Identification of a part with a family of parts is required for proper planning and control of production operation of the part. Part assignment to cells is carried out after machine types are assigned to cells based on the mathematical model. In the proposed model, part assignment does not have any influence on the objective function and also on the different optimum configurations obtained based on the different values of weight parameter sigma (). The quality of solutions can be analyzed in a traditional manner, only if, the part assignment is made. The part assignment heuristic is developed with a view to reduce the number of exceptional elements and voids. The heuristic is as follows. • Assign a part to a cell in which it has maximum number of operations. • If a tie exists, the part is assigned to a cell where the number of operations (machine types) not required by this particular part is minimum. • And again if there is a tie, assign to any one of the cells in which the previous tie is satisfied. 4.3. Quality of solutions Even though, the decision of selecting a particular configuration can be made based on the values of the two terms in the objective function, it is possible to evaluate the configurations obtained, in the traditional manner using any of the grouping efficiency measures. Some of such measures are grouping efficiency [54], grouping efficacy [50], and doubly weighted grouping efficiency [55]. 4.3.1. Measure of performance For measuring the quality of a part-family/cell formed, several researchers have developed different efficiency measures. The quality of the final matrix obtained after block diagonalisation is

153

measured. The grouping efficacy proposed by Kumar and Chandrasekharan [50] is the most widely used and popular one among these efficiency measures. We have also used the same for finding the goodness of the solution. The grouping efficacy is defined as GE =

(m − e) (m + v)

where GE, grouping efficacy; m, total number of 1’s in the input matrix; e, total number of 1’s outside the diagonal blocks; v, total number of 0’s inside the diagonal blocks. 4.4. The algorithm The algorithm for determining machine cells and part-families is given below. This algorithm uses GA to generate solution for the mathematical model, part assignment heuristic to develop the part families, and grouping efficacy to evaluate the solution (cells and part-families) obtained through GA and part assignment heuristic. Step 1: Input the part-machine incidence matrix, the genetic algorithm parameters for the model and the sigma value. Step 2: Set number of cells equal to maximum number of possible cells which is equal to number of machine types. Step 3: Initialize the population. Step 4: Calculate objective function value for each member of the population and convert it into corresponding fitness value. Step 5: Reproduce strings using remainder stochastic sampling without replacement policy. Step 6: Do crossover using two-point crossover operator. Step 7: Perform Mutation. Step 8: Carryout the replacement strategy and evaluate the new generation. Step 9: Go to step 10 if the termination criteria is met, else go to step 5 Step 10: Select the best chromosome which is the final solution of the suggested mathematical model. Step 11: Assign parts to the cells (solution) based on the part assignment heuristic. Step 12: Calculate grouping efficacy value for the solution. Run the algorithm for different values of sigma for getting different solutions. The process is detailed below. Here sigma is an input parameter and for the same problem each value of sigma gives a solution. For a range of values of sigma the solution may be the same. For example: Case 1. Sigma = 0.1–0.5 (range) The result may be the same configuration with two cells for all values in this range of sigma. Case 2. Sigma = 0.6–0.8 (range) The result may be a different configuration compared to case 1and with two cells for all values in this range of sigma. Case 3. Sigma = 0.9–1.3 (range) The result may be the same configuration with three cells for all values in this range of sigma. Similarly, we can experiment with different range of values of sigma for getting different configurations. The selection of a particular solution is based on the practical considerations and the designer’s choice for value of the two functions included in the objective function, the grouping efficacy and the number of cells.

154

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158

Table 2 Input matrix. Parts 1 0 1 0 0 0 0 0

0 0 0 0 1 1 0 0

Machine type

1 1 0 1 0 0 1 1

0 1 0 1 0 0 1 1

0 0 0 0 1 1 0 0

0 1 0 1 1 0 1 1

0 1 0 1 0 0 1 1

1 0 1 0 0 0 0 0

1 0 1 0 0 1 0 0

0 0 0 1 1 1 0 0

The algorithm is tested with 36 problems selected from the literature and the results are analyzed. For a null value of the parameter , only intercell moves are considered for the design of CMS. The solution for this value of  is a single cell which contains all machine types. This solution will continue to be the optimum one for small values of  until reaching a particular value, above which the model will try to establish more than one cell. This process is repeated with increased value for  until a new solution is obtained. Like this different solutions can be obtained. A very high value of  means the importance for the intercell moves is less and the priority for the internal homogeneity of the cells is more. Such a cell configuration has cells which are exclusively devoted to each machine type. Beyond a particular value of , the number of cells obtained will be equal to the number of machine types. An example problem from Chandrasekharan and Rajagopalan [49] has been taken to illustrate the detailed working of the algorithm. The incidence matrix of the problem is provided in Table 2. The parameters used in the GA, which are fixed for all problems, are given below. Population size = 2.5 times the length of chromosome. Maximum generations = 3000. Probability of crossover = 0.85. Probability of mutation = 0.1.

Objective function value

The pre-specified number of iterations can have value in the range of 20–150. The value to be selected depends on the problem size. For small size problems the value can be chosen from the lower side of the range. Fig. 1 shows the convergence of GA at a particular value of sigma (0.05). The GA is run for the problem with sigma value ranging from 0 to 10. For a zero value of sigma a single cell configuration is obtained. At sigma value of 10 the configuration has the number of cells equal to 8 which is the number of machine types used. Between these two conditions a wide range of solutions are obtained and these solutions provide a choice for the designer to select from different alternatives. Fig. 2 shows the number of cells formed against various sigma values and it can be noted that beyond a threshold value of it (here 10) the number of cells remains the same which is equal to the number of machine types. Table 3 gives the value of heterogeneity and intercell moves when GA is used for solving the problem. This table also provides

9 8 7 6 5 4 3 2 1 0

0 0 0 0 1 1 1 0

1 0 1 0 0 0 0 0

1 1 1 0 0 0 0 0

0 0 0 0 1 1 0 0

1 0 1 0 0 0 0 0

1 0 1 0 1 0 0 0

0 1 0 1 0 0 1 1

1 0 1 0 0 0 0 0

0 1 0 1 0 1 1 1

Series1

0

0.01

0.02

0.05

0.2

2

5

7

10

12

20

Zigma

Fig. 2. Number of cells formed for various values of sigma.

the final solution as a chromosome for various values of sigma. It is clear that heterogeneity and intercell moves are conflicting objectives and hence both cannot be minimized simultaneously. The variation of one with respect to the other is shown in Fig. 3. Now, the designer has to decide the trade-off point based on these two design criteria. The generated solutions cover the entire tradeoff range of the two objectives considered. This is possible as there is no constraint like the number of cells which is usually an input in many of the models. Generally, the literature provides models which can suggest a best part-family/cell formation under a given number of cells. The proposed model is able to provide the decision maker with a wide range of solutions instead of a single one. Table 3 also shows the grouping efficacy for the different configurations obtained by the selection of different values for sigma. The best value of grouping efficacy is for solution number 4 and this solution configuration is the final configuration obtained for almost all the cell formation design models of the literature. This configuration is given in Table 4. But, the advantage of the proposed model is that, this is not the only solution obtained. The proposed model provides several solutions and hence the designer has the opportunity to choose the most fitting and practical solution for the real world problem. Based on the heterogeneity and intercell moves, other solution options can also be considered along with grouping efficacy to arrive at an appropriate decision. The physical size of a cell is a determining factor in the design of CMS. The number of workstations in a cell should not normally exceed six, otherwise the workers may be overwhelmed by different kind of tasks [56]. Moreover, large sized cells lack co-ordination and team spirit among workers. The heterogeneity variable is an indicator of the variety of tasks and hence, the value of heterogeneity can directly be taken as a criterion for selection of a particular

25 120

20

100

15

Series1

10 5 0

Heterogeniety

• • • •

Number of cells

5. Illustrative example

1 0 1 0 0 0 1 0

80 Series1

60 40 20

1

5

10

20

25

50

Number of Generations

Fig. 1. Convergence of GA.

100

120

0 0

5

11

9

15

20

26

32

41

Intercell move

Fig. 3. Variation of heterogeneity v/s intercell moves.

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158

155

Table 3 Configurations obtained for different values of sigma. Sl. no.

Sigma

Number of cells

1 2 3 4 5 6 7 8 9

0 0.01 0.02 0.05 0.2 2 5 7 10

1 2 3 3 4 5 6 7 8

Heterogeneity 99 49 29 17 9 5 2 1 0

Intercell moves

Chromosome

Grouping efficacy

0 5 11 9 15 20 26 32 41

11111111 12122222 12133332 12123322 12124423 12123542 51523641 41472356 32145786

0.3812 0.5847 0.6197 0.8524 0.7541 0.6721 0.5738 0.4754 0.3279

Table 4 Cell configuration (3 cells) corresponding to maximum grouping efficacy. 1 1 0 0 0 0 0 0

1 1 0 0 0 0 0 0

1 1 0 0 0 0 0 1

1 1 0 0 1 0 0 0

1 1 0 0 0 0 0 0

1 1 1 0 0 0 0 0

1 1 0 0 0 0 0 0

1 1 0 0 0 0 1 0

1 1 0 0 0 0 0 0

1 0 1 1 1 1 0 0

0 0 1 1 1 1 0 0

configuration. The proposed model helps in choosing appropriate cell size based on the heterogeneity. The best solution for the illustrative problem (based on grouping efficacy) has 3 cells. However, if the designer wants less number of machine types per cell then a four or a five cell solution will be suitable. Such a choice will improve the overall performance of the system including worker efficiency and team work. On the other hand, if the decision maker wants less number of cells and less intercell moves, he/she can go for a solution with two cells. If more than one solution is available with two cells, grouping efficacy can be used as a secondary criterion for the selection among alternatives. Table 5 gives a two-cell solution for the illustrative example using the proposed model. So the decision can be made based on the two objective function criteria along with the value of grouping efficacy. In real situations, this characteristic of the model will be very useful. Real situations are, generally, constrained by some limitations and it is often difficult to quantify. Certain such limitations are:

0 0 1 1 1 1 1 0

0 0 1 1 1 1 0 0

0 0 1 1 1 1 0 0

0 0 1 1 1 1 0 1

0 0 0 0 0 0 1 1

0 0 0 0 0 0 1 1

0 0 0 1 0 0 1 1

0 0 0 0 1 0 1 1

0 0 0 0 0 0 1 1

6. Comparative analysis To know the solution quality of the proposed model, 36 problems from the literature are solved using the proposed method. The selection of these 36 problems is mainly based on the availability of grouping efficacy values for these problems under seven different models from the literature. All the seven models did not solve all the problems but one of them has solved all the problems. The structure of the problems is also different in nature with respect to the easiness of grouping the data. For example, problem numbers from 23 to 28 are of the same size but with different grouping capabilities. Problem 23 is an ideal problem where perfect grouping is possible while problem 28 is very ill-structured such that it is difficult to group. The ill-structured nature of the problem increases from 23 to 28 and is reflected in the grouping efficacy values. The size of the problems is usually specified in terms of number of machine types and number of parts (i.e., machine types × parts). The size of problems considered here varies from 5 × 7 to 40 × 100. All the problems have been solved using the proposed method in a similar pattern as in the illustrative example. The solution is compared with the previously reported results based on the grouping efficacy. The solution with maximum value of grouping efficacy of the proposed model is taken for comparison. We compare the grouping efficacy obtained by our algorithm with the grouping efficacy obtained by the following seven methods and the selection of these methods is based on the available data from the literature. Among these methods, the algorithm of Tariq et al. [29] has produced the best results.

• Specific shape of a shop floor is such that dividing it into certain number of cells proposed by the solution of a model may not be possible. • Limited area and the input and output areas of the plant will make difficult to divide it into required number of divisions (cells) as per the solution of a model. • A good solution determined based on efficacy/efficiency may not be permitting uniform distribution of workload, etc. That is, practical considerations in real situations will make it difficult to implement the best solution and will force to choose a solution which is different from the best. Alternative configurations based on the same objectives from the present model provide a great advantage in such conditions.

ZODIAC (Chandrasekharan and Rajagopalan) [8]; GRAFICS (Srinivasan and Narendran) [9]; MST – Clustering algorithm (Srinivasan) [57];

Table 5 Two-cell solution. 1 1 0 0 0 0 0 0

1 1 0 0 0 0 0 0

1 1 0 0 0 1 0 0

1 1 0 0 0 0 1 0

1 1 0 0 0 0 0 0

1 1 1 0 0 0 0 0

1 1 0 0 0 0 0 0

1 1 0 0 1 0 0 0

1 1 0 0 0 0 0 0

0 0 0 0 1 1 0 0

1 0 1 1 0 0 1 1

0 0 1 1 0 0 1 1

0 0 0 0 1 1 0 0

0 0 1 1 1 0 1 1

0 0 1 1 0 0 1 1

0 0 0 1 1 1 0 0

0 0 0 0 1 1 1 0

0 0 0 0 1 1 0 0

0 0 1 1 0 0 1 1

0 0 1 1 0 1 1 1

156

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158

Table 6 Comparison of solution of various methods. Problem no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Size (M × P) 5×7 5 × 18 6×8 7 × 11 7 × 11 8 × 12 8 × 20 8 × 20 10 × 10 10 × 15 12 × 15 10 × 20 14 × 23 14 × 24 16 × 24 16 × 30 16 × 43 18 × 24 20 × 20 20 × 23 20 × 35 20 × 35 24 × 40 24 × 40 24 × 40 24 × 40 24 × 40 24 × 40 27 × 27 28 × 46 30 × 41 30 × 50 30 × 50 30 × 90 37 × 53 40 × 100

ZODIAC

GRAPHICS

MST

GATSP

GP

73.68 77.36 76.92 39.13 70.37 68.3 58.33 85.24 70.59 92.00 – – 64.36 65.55 32.09 67.83 53.76 41.84 21.63 38.66 75.14 51.13 100 85.11 73.51 20.42 18.23 17.61 52.14 33.01 30.46 46.06 21.11 32.73 52.21 83.66

73.68 – – 53.12 – 68.3 58.13 85.24 70.59 92.00 – – 64.36 65.55 45.52 67.83 54.39 48.91 38.26 49.36 75.14 – 100 85.11 73.51 43.27 44.51 41.67 41.37 32.86 55.43 56.32 47.96 39.41 52.21 83.92

– – – – – – 58.72

– 77.36 76.92 46.88 70.37 – 58.33

–

70.59 92.00 – – 64.36

70.59 92.00 – – – 67.44 – – 53.89 – 37.12 46.62 75.28 55.14 100 85.11 73.03 49.37 44.67 42.50 – – 53.80 56.61 45.93 – – 84.03

48.70 67.83 54.44 44.20 – 43.01 75.14 – 100 85.11 73.51 51.81 44.72 44.17 51 40 55.29 58.70 46.30 40.05 – 83.92

– – – 58.72 85.24

– – – – – – – 49 – 100 – 73.51 – – – – – – – – – – 84.03

EA

HA

73.68 79.59 76.92 53.13 70.37 68.3 58.72 85.24 70.59 92.00 86.67 – 60.86 69.33 52.58 67.83 54.86 54.46 42.96 49.65 76.22 58.07 100 85.11 73.51 51.97 47.06 44.87 54.27 44.62 58.48 59.66 50.51 42.64 56.42 84.03

73.68 79.59 76.92 53.13 70.37 68.3 58.72 85.24 70.59 92.00 86.67 100 70.83 70.51 51.96 67.83 54.86 54.95 43.45 49.65 76.14 58.38 100 85.11 73.51 52.5 46.84 44.85 54.31 46.43 60.74 59.66 50.51 44.67 59.60 84.03

Proposed Model 73.68 79.59 76.92 58.62 70.83 70.45 58.72 85.24 70.59 92.00 86.67 100 72.85 69.33 45.1 68.31 55.11 55.91 42.31 48.87 77.16 58.89 100 85.11 73.51 50.00 47.76 45.21 51.56 46.64 61.04 59.76 50.56 45.53 55.98 84.03

Data set for problem numbers 1–6, 9–11, 13–16, 18–26, 28, 29–35 is taken from the paper of Goncalves and Resende [6] and these problems are from research articles published during 1972–1992. The origin of other data set is as follows: Problem-7 – Chandrasekharan and Rajagopalan [3]. Problem-8 – Chandrasekharan and Rajagopalan [54]. Problem-12 – Tariq et al. [29]. Problem-17 – King [2]. Problem-27 – Chandrasekharan and Rajagopalan [60]. Problem-36 – Chandrasekharan and Rajagopalan [8]. Note: Results of improved and those equal to previously known better solution using the proposed method is shown in bold.

GATSP – Genetic algorithm (Cheng et al.) [58]; GP – Genetic programming (Dimopoulos and Mort) [59]; EA – Evolutionary algorithm (Goncalves and Resende) [6]; HA – Hybrid algorithm (Tariq et al.) [29]. The grouping efficacy for the above approaches (and problems) is taken from Goncalves and Resende [6] and Tariq et al. [29]. The problem size and source along with the results are presented in Table 6. (The source of data is provided at the bottom of this table.) The proposed method provides better grouping efficacy for 16 (around 44%) problems compared to all the previously reported results while 7 (around19%) problems have shown reduced efficacy compared to some of the previously reported methods. For the remaining problems, we got the results same as the previously reported best values. We can see that better results are obtained for problems irrespective of the problem sizes. In case of ill-structured input matrix the proposed algorithm seems to reach at a good solution which is better than the other methods. For the ideal problems given in the literature where grouping efficacy of one is possible, the proposed model gives the perfect solution for almost all values of sigma. In the case of ideal problems where perfect grouping is possible and problems which are very close to the ideal problems where grouping is possible with very

less number of exceptional elements and voids, almost all methods are giving the same result. So, the efficiency of a method in cell design problems is the ability to provide good solutions for data having machines and parts which are difficult to group. Table 6 shows that there are six problems (Problem numbers 19, 20, 27, 28, 30 and 34) for which the best grouping efficacy (GE) obtained from other methods for comparison is less than 50%. These problems can be considered as very ill-structured and difficult to group. Among these problems, the proposed algorithm is performing better than other methods for four problems. These four problems are large size problems. This analysis shows the ability of the proposed method in handling problems with ill-structured data. The reduced efficacy value for a few problems may be because of the part assignment rule. A better part assignment rule may improve the solution. It can also be noted from Table 6 that the grouping efficacy (GE) values obtained for the proposed model are the same as that of the GE obtained for many other methods when the problem size is small (5 machine types × 7 parts to 10 machine types × 20 parts). This shows that many methods are efficient, if the input problem size is small. From Table 6 it may be noted that majority of the situations where better GE values are obtained with the proposed model compared to other models is for large size problems (24 machine types × 40 parts to 30 machines types × 90 parts). This shows the

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158 Table 7 Machine-part incidence matrix for problem number 4. Parts

Machine type

1 2 3 4 5 6 7

1

2

3

4

5

6

7

8

9

10

11

0 1 0 1 0 1 0

1 0 0 0 0 0 0

1 0 0 1 0 0 1

0 0 0 0 0 1 1

0 1 0 0 1 0 0

0 0 0 1 0 0 1

1 0 0 0 0 0 1

0 0 0 0 1 1 0

0 0 0 0 0 1 1

0 0 1 0 0 1 0

0 1 1 0 0 0 0

2

7

5

8

10

11

1

3

4

6

9

1 0 0 0 0 0 0

1 0 0 0 0 0 1

0 1 1 0 0 0 0

0 0 1 0 0 1 0

0 0 0 1 0 1 0

0 1 0 1 0 0 0

0 1 0 0 1 1 0

1 0 0 0 1 0 1

0 0 0 0 1 0 1

0 0 0 0 0 1 1

0 0 0 0 0 1 1

Table 8 Result of problem number 4. Parts

Machine type

1 2 5 3 4 6 7

capability of the proposed model in solving large size problems compared to other methods. The performance of the proposed model is analyzed using the result obtained for a small problem of 7 × 11 (problem number 4), which has given improved solution compared to other methods. Table 7 shows the input matrix and Table 8 shows the solution for this problem. A better solution to a small problem like this also is an indication of the power and efficiency of the proposed model. The comparison reveals that in addition to the advantage of a wide range of feasible solutions, grouping efficacy wise also the proposed method performs better than the previous approaches. 7. Conclusion The approach proposed in this paper gives a mathematical model for machine cell formation. The model is first solved using the optimization software package, LINGO, and then a GA based solution methodology is developed for solving problems of any size saving computational time. An algorithm is proposed which uses GA for forming machine cells, a part assignment heuristic for part family formation, and grouping efficacy to evaluate the goodness of the solution. Manufacturing cell is a group of heterogeneous machines. But, there is no quantitative measure available to measure the degree of heterogeneity of machine types in a cell. This study developed a measure called heterogeneity and used it in a mathematical programming model for CMS design. The heterogeneity is a direct indicator of the variety of machine types present in a cell and the physical size of the cell and hence it can be effectively used by the designer to take a decision. The model uses two conflicting objectives, namely, heterogeneity and intercell moves for identifying machine groups. The importance of heterogeneity in the objective function is managed using a weighting factor, . By changing this factor, the designer can generate alternative solutions in a structured manner. In the case of a perfectly groupable data, for any value of  the algorithm shows a tendency to converge into the best solution where all grouping efficiency measures will give its maximum value. On the other hand, for an input data which posses a number of grouping options and is ill-structured, the algorithm provides a wide range of alternative solutions based on the value of sigma. The advantage

157

of this model is the possibility of a large number of potential solutions instead of a single compromising solution. This may help in selecting appropriate solution suitable for a real case. For example, it may be possible to divide the layout into 4 cells only. But the best solution for the problem may be 5 cells based on some performance measure and normally, efficient algorithms converge to this solution. In such a situation, a 4-cell solution with best suitable grouping can be obtained using the proposed model. The trade-off between the two objectives of the objective function, and the priorities and criteria in the mind of the designer play an important role in selecting a particular configuration. The quality of solutions for comparison is analyzed using the grouping efficacy. Out of the 36 problems tested, 44% of problems have improved efficacy and for 36% of problems the efficacy is same as the best reported in the literature. So, the proposed algorithm is performing better than the previously reported methods/algorithms. Also, it has been noted from the comparative analysis that the model is performing well for problems with illstructured data and large size problems when compared to other CMS design methods discussed. Thus the proposed model is giving a comprehensive framework for the identification of machine cells and part families in a better way leaving so much of options to the designer for decision making. While the proposed algorithm is proven to be efficient compared to the other methods of CMS design, a better result may be possible with a different part assignment rule, because it has a very high impact on the grouping efficacy values. Further, the model may be properly modified to take ordinal and ratio level data input which are more realistic, and multiple configuration model in such situations will be more useful. Alternative routes, cell load variation and dynamic part demand can also be included in the design by making appropriate modifications in the model. Heterogeneity of cells which is a new parameter that can be used in the above said possibilities may lead to interesting and encouraging results in CMS design. References [1] Wu X, Chu CH, Wang Y, Yan W. A genetic algorithm for cellular manufacturing design and layout. Eur J Oper Res 2007;181:156–67. [2] King JR. Machine-component grouping in production flow analysis: an approach using rank order clustering algorithm. Int J Prod Res 1980;18(2):213–32. [3] Chandrasekharan MP, Rajagopalan R. MODROC: an extension of rank order clustering for group technology. Int J Prod Res 1986;24(5):1221–33. [4] McCormick WT, Schweitzer PJ, White TW. Problem decomposition and data reorganization by a clustering technique. Oper Res 1972;20(5):993–1009. [5] Chan H, Milner D. Direct clustering algorithm for group formation in cellular manufacturing. J Manuf Syst 1982;1(1):65–7. [6] Goncalves J, Resende MGC. An evolutionary algorithm for manufacturing cell formation. Comput Ind Eng 2004;47:247–73. [7] Seifoddini H. Single linkage versus average linkage clustering in machine cells formation applications. Comput Ind Eng 1989;16(3):419–26. [8] Chandrasekharan MP, Rajagopalan R. ZODIAC—an algorithm for concurrent formation of part families and machine cells. Int J Prod Res 1987;25(6):835–50. [9] Srinivasan G, Narendran TT. GRAFICS-a nonhierarchical clustering-algorithm for group technology. Int J Prod Res 1991;29(3):463–78. [10] Nair J, Narendran TT. CASE: a clustering algorithm for cell formation with sequence data. Int J Prod Res 1998;36(1):157–79. [11] Kusiak, Chow W. Efficient solving of the group technology problem. J Manuf Syst 1987;6(2):117–24. [12] Youkyung W, Currie KR. An effective p-median model considering production factors in machine cell/part family formation. J Manuf Syst 2006;25(1):58–64. [13] Choobineh F. A framework for the design of cellular manufacturing systems. Int J Prod Res 1988;26(7):1161–72. [14] Srinivasan G, Narendran TT, Mahadevan B. An assignment model for the partfamilies problem in group technology. Int J Prod Res 1990;28(l):145–52. [15] Adil GK, Rajamani D, Strong D. Assignment allocation and simulated annealing algorithms for cell formation. IIE Trans 1997;29(1):53–67. [16] Mahdavi, Aalaei A, Paydar MM, Solimanpur M. A new mathematical model for integrating all incidence matrices in multi-dimensional cellular manufacturing system. J Manuf Syst 2012;32(2):214–23. [17] Yasuda K, Hu L, Yin Y. A grouping genetic algorithm for the multi-objective cell formation problem. Int J Prod Res 2005;43(4):829–53.

158

C.R. Shiyas, V. Madhusudanan Pillai / Journal of Manufacturing Systems 33 (2014) 149–158

[18] Yin Y, Xu C, Hu L. Some insight into Yasuda et al’s. “A grouping genetic algorithm for the multi-objective cell formation problem”. Int J Prod Res 2009;47(7):2009–10. [19] Pitombeira Neto AR, Goncalves Filho EV. A simulation-based evolutionary multiobjective approach to manufacturing cell formation. Comput Ind Eng 2010;59(1):64–74. [20] Arkat J, Hosseini L, Farahani MH. Minimization of exceptional elements and voids in the cell formation problem using a multi-objective genetic algorithm. Exp Syst Appl 2011;38(8):9597–602. [21] Rabbani M, Jolai F, Manavizadeh N, Radmehr N, Javadi B. Solving a biobjective cell formation problem with stochastic production quantities by a two-phase fuzzy linear programming approach. Int J Adv Manuf Technol 2012;58(5–8):709–22. [22] Raflei H, Ghodsi R. A bi-objective mathematical model toward dynamic cell formation considering labor utilization. Appl Math Modell 2013;37(4):2308–16. [23] Chan FTS, Lau KW, Chan LY, Lo VHY. Cell formation problem with consideration of both intracellular and intercellular movements. Int J Prod Res 2008;46(10):2589–620. [24] Rogers DS, Kulkarni SS. Optimal bivariate clustering and a genetic algorithm with an application in cellular manufacturing. Eur J Oper Res 2005;160:423–44. [25] Lee-Post. Part family identification using a simple genetic algorithm. Int J Prod Res 2000;38(4):793–810. [26] Mak KL, Wong YS, Wang XX. An adaptive genetic algorithm for manufacturing cell formation. Int J Adv Manuf Technol 2000;16:491–7. [27] Wu X, Chu CH, Wang Y, Yan W. Concurrent design of cellular manufacturing systems: a genetic algorithm approach. Int J Prod Res 2006;44(6):1217–41. [28] Onwubolu GC, Mutingi M. A genetic algorithm approach to cellular manufacturing systems. Comput Ind Eng 2001;39:125–44. [29] Tariq, Hussain I, Ghafoor A. A hybrid genetic algorithm for machine-part grouping. Comput Ind Eng 2009;56:347–56. [30] Joines JA, Culbreth CT, King RE. Manufacturing cell design: an integer programming model employing genetic algorithms. IIE Trans 1996;28:69–85. [31] Deljoo V, Mirzapour Al-e-hashem SMJ, Deljoo F, Aryanezhad MB. Using genetic algorithm to solve dynamic cell formation problem. Appl Math Modell 2010;34:1078–92. [32] Arkat J, Saidi M, Abbasi B. Applying simulated annealing to cellular manufacturing system design. Int J Adv Manuf Technol 2007;32:531–6. [33] Zolfaghari S, Liang M. Machine cell/part family formation considering processing times and machine capacities: a simulated annealing approach. Comput Ind Eng 1998;34:813–23. [34] Chen CL, Cotruvo NA, Baek W. A simulated annealing solution to the cell formation problem. Int J Prod Res 1995;33:2601–14. [35] Ariafar S, Ismail N. An improved algorithm for layout design in cellular manufacturing systems. J Manuf Syst 2009;28(4):132–9. [36] Zolfaghari S, Liang M. Comparative study of simulated annealing, genetic algorithms and tabu search for solving binary and comprehensive machinegrouping problems. Int J Prod Res 2002;40(9):2141–58. [37] Islier AA. Group technology by an ant system algorithm. Int J Prod Res 2005;43(5):913–32. [38] Shanker R, Vrat P. Some design issues in cellular manufacturing using the fuzzy programming approach. Int J Prod Res 1999;37(11):2545–63. [39] Pandian RS, Mahapatra SS. Manufacturing cell formation with production data using neural networks. Comput Ind Eng 2008;56:1340–7.

[40] Won Y, Currie KR. Fuzzy ART/RRR-RSS: a two-phase neural network algorithm for part-machine grouping in cellular manufacturing. Int J Prod Res 2007;45(9):2073–104. [41] Rao HA, Gu P. Expert self-organizing neural network for the design of cellular manufacturing systems. J Manuf Syst 1994;13(5):346–58. [42] Nouri H, Tang SH, Hang Tuah BT, Anuar MK. BASE: a bacteria foraging algorithm for cell formation with sequence data. J Manuf Syst 2010;29(2/3): 102–10. [43] Nouri H, Hong TS. A bacteria foraging algorithm based cell formation considering operation time. J Manuf Syst 2012;31(3):326–36. [44] Nouri H, Hong TS. Development of bacteria foraging optimization algorithm for cell formation in cellular manufacturing system considering cell load variations. J Manuf Syst 2013;32(1):20–31. [45] Arkat J, Abdollahzadeh H, Ghahve H. A new branch and bound algorithm for cell formation problem. Appl Math Modell 2012;36(10):5091–100. [46] Sayadi MK, Hafezalkotob A, Naini SGJ. Firefly-inspired algorithm for discrete optimization problems: an application to manufacturing cell formation. J Manuf Syst 2013;32(1):78–84. [47] Dimopoulos C. Multi-objective optimization of manufacturing cell design. Int J Prod Res 2006;44(22):4855–75. [48] Irani SA. Handbook of cellular manufacturing systems. New York: John Wiley & Sons, Inc.; 1999. [49] Adenzo-Diaz B, Lozano S. A model for the design of dedicated manufacturing cells. Int J Prod Res 2008;46(2):301–19. [50] Kumar KR, Chandrasekharan MP. Grouping efficacy: a quantitative criterion for goodness of block diagonal forms of binary matrices in group technology. Int J Prod Res 1990;28(2):233–43. [51] Gupta Y, Gupta M, Kumar A, Sundaram C. A genetic algorithm-based approach to cell composition and layout design problems. Int J Prod Res 1996;34(2):447–82. [52] Madhusudanan Pillai V, Subbarao K. A robust cellular manufacturing system design for dynamic part population using a genetic algorithm. Int J Prod Res 2008;46(18):5191–210. [53] Wicks EM, Reasor RJ. Designing cellular manufacturing systems with dynamic part populations. IIE Trans 1999;3:11–20. [54] Chandrasekharan MP, Rajagopalan R. An ideal seed non-hierarchical clustering algorithm for cellular manufacturing. Int J Prod Res 1986;24(2):451–64. [55] Sarker BR. Measures of grouping efficiency in cellular manufacturing systems. Eur J Oper Res 2001;130(3):588–611. [56] Nicholas JM. Competitive manufacturing management. New Delhi: Tata McGraw-Hill Publishing Company Ltd.; 2001. [57] Srinivasan G. A clustering algorithm for machine cell formation in group technology using minimum spanning trees. Int J Prod Res 1994;32: 2149–58. [58] Cheng CH, Gupta YP, Lee WH, Wong KF. A TSP-based heuristic for forming machine groups and part families. Int J Prod Res 1998;36(5):1325–37. [59] Dimopoulos C, Mort N. A hierarchical clustering methodology based on genetic programming for the solution of simple cell-formation problems. Int J Prod Res 2001;39(1):1–19. [60] Chandrasekharan MP, Rajagopalan R. GROUPABILITY: analysis of the properties of binary data matrices for group technology. Int J Prod Res 1989;27(6):1035–52.