A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system

Computers & Industrial Engineering xxx (2013) xxx–xxx

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A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system q Javad Rezaeian Zeidi a,⇑, Nikbakhsh Javadian a,1, Reza Tavakkoli-Moghaddam b,2, Fariborz Jolai b,2 a b

Department of Industrial Engineering, Faculty of Engineering, Mazandaran University of Science and Technology, P.O. Box 734, Babol, Iran Department of Industrial Engineering, Faculty of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 28 August 2010 Received in revised form 22 February 2012 Accepted 19 August 2013 Available online xxxx Keywords: Incremental cell formation Multi-objective optimization Genetic algorithm Neural network

a b s t r a c t One important issue related to the implementation of cellular manufacturing systems (CMSs) is to decide whether to convert an existing job shop into a CMS comprehensively in a single run, or in stages incrementally by forming cells one after the other, taking the advantage of the experiences of implementation. This paper presents a new multi-objective nonlinear programming model in a dynamic environment. Furthermore, a novel hybrid multi-objective approach based on the genetic algorithm and artificial neural network is proposed to solve the presented model. From the computational analyses, the proposed algorithm is found much more efficient than the fast non-dominated sorting genetic algorithm (NSGA-II) in generating Pareto optimal fronts. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The cellular manufacturing (CM) problem has captured a great deal of attention of many manufacturers and researchers. CM is the implementation of group technology (GT) and a manufacturing philosophy, in which similar parts are identified and grouped into part families; meanwhile, machines are grouped into machine cells to take advantage of their similarities in manufacturing and design (Chung, Wu, & Chang, 2011). CM, which is a flexible manufacturing system (FMS), can respond to the increasingly competitive environment facing manufacturers. Specially, manufacturers need to quickly improve their efficiency, response time and quality, but with a minimum of upfront investment of the capital and time. One of the implied assumptions in the modeling and development of CMSs is that the product mix remains stable over the time and a major deterrent to implement CMSs is changing the layout by entering new demands or variability of them. The variety and the uncertainty of demand, variety of characteristics of the product and manufacturing process are the reasons that motivated the request for more flexibility. In the recent decades, it has been tried to develop new layouts and new models of cell formation with more

q

This manuscript was processed by Area Editor Gursel A. Suer.

⇑ Corresponding author. Tel.: +98 1112291205; fax: +98 1112290118. 1 2

E-mail address: [email protected] (J.R. Zeidi). Tel.: +98 1112291205; fax: +98 1112290118. Tel.: +98 2161113358; fax: +98 2166409348.

flexibility (Hamedi, Esmaeilian, Ismail, & Ariffin, 2012). A number of researchers have suggested incremental cell formation in the literature. Cell formation is one of the important issues of CMSs, in which similar parts are grouped in a family known as part families and required machines to process parts are determined. Many models and solution approaches have been developed to deal with a cell formation problem (CFP), but virtually all of them look at CM in terms of the total number of products to be made and the total number of machines or machine types available (or needed), and then try to plan a conversion of the entire shop into cells, possibly keeping a remainder cell. In other words, planning the conversion of a job shop to CM is performed comprehensively (non-incremental) rather than incrementally (Marsh, Shafer, & Meredith, 1999). In situation of this kind, all machines belonging to shops will move to cells in a period totally, as shown in Fig. 1 (Rezaeian, Javadian, Tavakkoli-Moghaddam, & Jolai, 2011). Wemmerlov and Johnson (2000) in a survey carried out on 126 cells in 46 plants mentioned that academic (and some practitioner) writers on cell formation often seem to perceive the problem as one, in which multiple cells emerge from a single analysis of the factory. The reality is that most cells in industries are created and implemented sequentially over time. Incremental cell formation follows a sequential process of forming the cells proposed in the master plan. In this case, cells are implemented one-by one rather than all-at-once, in which a sample of this kind of cell formation is illustrated in Fig. 2. According to this arrangement,

0360-8352/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.08.015

Please cite this article in press as: Zeidi, J. R., et al. A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.015

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Fig. 1. Non-incremental cellular manufacturing system.

Fig. 2. Incremental cellular manufacturing system.

machines belonging to shops will move to cells in some periods gradually. Similar observations are found in Mahesh and Srinivasan (2002) and Johnson (1998). The following reasons for a company gradually implementing group technology (GT) cells is mentioned by Adil and Ghosh (2005). 1. To implement a pilot cell: A company may select a few strategically important parts/machines in a cell. This may provide the fruitful learning experience for the future cell implementation. 2. Economic reasons: Budget constraints may not permit a conversion of the whole plant into cells. 3. Master plan: Some companies follow a master plan where they gradually cellularize their plant. 4. Data not amenable to perfect grouping: There may be some bottleneck parts and bottleneck machines inherent that are not amenable to cellularization. 5. Service cell: Some machines serve multiple part families and hence may be dedicated to the shared service cell. The CFP, which groups machines into cells and parts into families, has received considerable attention in the literature. Mahesh and Srinivasan (2002) clustered a number of techniques and provided an overview of various algorithms that forms cells comprehensively (i.e., non-incrementally) in total. However, empirical findings on cell formation (Johnson, 1998; Mahesh & Srinivasan,

2002; Marsh et al., 1999; Wemmerlov & Johnson, 2000) suggest that cells are implemented incrementally. More recently, a few studies have developed methods for solving incremental CFPs. Shambu and Suresh (2000) simulated a shop floor under several degrees of cellularization for a wide variety of shop conditions. They measured the performance of hybrid cellular manufacturing and job shop in a dynamic environment. Balakrishnan and Cheng (2007) proposed a model which considers cell formation over a multi-period planning horizon with demand and resource uncertainties. In this study, CF has been done non-incrementally, in which at each period the cell configuration can be changed. However, planning, implementation or capital investment issues have not been addressed in their study. Despite many previous researches, Gravel, Price, and Gagne (2000) stated that in some cases the implementation of a CMS is unhelpful. They showed that in this situations hybrid cellular configuration would be beneficial. Adil and Ghosh (2005) developed a mathematical model and a heuristic approach based on a greedy random adaptive search procedure (GRASP) to address the incremental CFP. Venkumar and Noural Hag (2006) solved the fractional CFP (similar to incremental CFP) using the modified adaptive resonance theory network. Rezaeian et al. (2011) presented a new mathematical model for the incremental cell formation problem, and their presented model considered a single objective function. It is known that the CFP is one of the NP-hard combinational problems (Yasuda, Hu, & Yin, 2005). On the other hand, most

Please cite this article in press as: Zeidi, J. R., et al. A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.015

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researches consider a single criterion while designers in the real world consider optimizing more than one criterion, such as minimizing inter-cell movements and maximizing the machine utilization. Hence, the problem is very hard in terms of the mathematical point of view. Approximation algorithms (or heuristic) have been developed to obtain an optimum (or close to optimum) non-dominated solutions. Multi- objective evolutionary algorithms (MOEA) are efficient methods to find and evaluate the pareto-optimal set in difficult multi-objective optimization problems, such as vector evaluated genetic algorithm (VEGA) (Schaffer, 1984), non-dominated sorted genetic algorithm (NSGA) (Srinivas & Deb, 1994), strength pareto evolutionary algorithm (SPEA) (Zitzler & Thiele, 1999) and NSGA-II (Deb, Pratap, Agarwal, & Meyarivan, 2002). Konak, Coit, and Smith (2006) perused a review of the genetic algorithm (GA) developed for problems with multiple objectives. One of the major difficulties in applying evolutionary algorithms to real problems is the complexity of large number of evaluations of the objective functions necessary in order to obtain an acceptable solution, typically the order of several thousands. Often these are time-consuming evaluations obtained by solving numerical codes with expensive methods, such as finite-differences or finite-elements. Hence, the use of approximate models in evolutionary algorithms (EA) has received little attention, particularly for multi-objective optimization. Several approaches can be used to approximate objective functions, such as statistical methods or artificial neural networks (ANN) (Gaspar-Cunha & Vieira, 2005). Gaspar-Cunha and Vieira (2005) presented a further integration of the ANN in the MOEA by proposing two different hybrid approaches to estimate the functions or solutions used by a multiobjective evolutionary algorithm, namely the reduced pareto set genetic algorithm with elitism (RPSGAe) (Gaspar-Cunha, Oliveira, & Covas, 1997) and (Gaspar-Cunha & Covas, 2004). Pettersson, Chakraborti, and Saxe´ (2007) proposed a GA-based multi-objective optimization technique in the training process of a feed forward neural network. In a study by Hakimi-Asiabar, Ghodsypour, and Kerachian (2009), a new method SOM-based multi-objective GA (SBMOGA) is proposed to improve the genetic diversity. They used a self-organizing map network by a GA to improve the both local and global search. Jamali, Nariman-zadeh, Darvizeh, Masoumi, and Hamrang (2009) suggested a new method that is based on multi- objective EAs, known as non-dominated sorting genetic algorithm (NSGA-II), with a new diversity-preserving mechanism. It is applied for the Pareto optimal design of a group method of data handling GMDH-type neural networks. Almeida and Ludermir (2010) presented a method for optimizing the parameters and performance of ANNs through a search for weights, architectures (nodes and layers), transfer functions and training algorithm rates. The proposed method is based on fully connected supervised multi-layer Perceptrons composed of a combination of evolution strategies (ES), particle swarm optimization (PSO) and GA. In the area of CMSs, Guerrero, Lozano, Smith, Canca, and Kwok (2002) proposed a new self-organizing neural network, Soleymanpour, Vrat, and Shanker (2002) applied a transiently chaotic neural network, Peker and Kara (2004) also provided a fuzzy art neural network to design CMSs, and furthermore Venkumar and Noural Hag (2006) proposed self-organizing map network. Ateme-Nguema and Dao (2009) presented an application of modified Hopfield neural networks in order to solve CFP, namely the quantized and fluctuated Hopfield neural networks (QFHN). Sudhakara Pandian and Mahapatra (2009) proposed an adaptive resonance theory (ART) based algorithm to handle the cell formation problem with the operation time and operation sequence of the parts. This paper presents a new mathematical model and a hybrid approach based on GA and ANN in order to obtain the non-dominated solutions of a CMS, where cells are formed incrementally. The solutions are produced and arranged based on NSGA-II, and

ANN is implemented to approximate the acceptable solutions because of complexity of these problems. The rest of the paper is organized as follows. Section 2 introduces the problem; this is done by giving the problem description, assumptions, notations and new mathematical model. The proposed hybrid approach is designed in Section 3. The analysis and discussion of this proposed approach is illustrated in Section 4. In Section 5, some experimentations and comparison are shown. Finally, Section 6 presents conclusions. 2. Problem formulation 2.1. Problem description This paper focuses on cell formation decisions incrementally. In the traditional cell formation approaches, designers tried to convert a job shop system to a CMS comprehensively while in reality it will be done incrementally. Hence, here a functional layout is considered at the beginning of the planning horizon being composed of multi periods. N parts are considered with each part visiting shops based on its requirements. In general, M machines are available in shops. The goal is to decide the number of cells formed in a period and the assignment of machines to cells such that the total cost and the total number of exceptional elements are minimized. The total cost consists of intra and inter-cell material handling, intra and inter-shop material handling, and material handling between cell and shop costs. An element is the exceptional element if it visits more than one cell in a period. Main assumptions of the presented model are as follows. 1. The demand for each part type in each period is known. 2. The number of cells formed in each period is limited. 3. Each cell consists of a minimum and maximum number of machines. 4. The unit cost of inter-cell movements, intra-cell movements, intra-shop movements and movements between cell and shop are known and constant over time. 5. The number of machines available is known and constant over time. 2.2. Mathematical description of the problem This section provides a brief description of the mathematical model. A set of N parts, P = {p1, p2, . . . , pn}, a set of M machines, M = {m1, m2, . . . , mM}, a set of S shops, S = {s1, s2, . . . , sS}, a set of C cells, C = {c1, c2, . . . , cmax} and a set of T periods, t = {1, 2, . . . , T} are considered. Let Dpt be the demand of part p in period t. The following parameters and decision variables are used throughout the paper: Parameters

a b d

c x Dpt LB UB Zpm = 1 Decision variables

intra-cell material handling cost inter-cell material handling cost intra-shop material handling cost cost of material handling between cell and shop inter-shop material handling cost demand for product p in period t minimum number of machines to be assigned to a cell maximum number of machines to be assigned to a cell if part p needs machine m; 0, otherwise

(continued on next page)

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Xcu = 1 Ymcu = 1

if cell c is formed in period u; 0, otherwise if machine m is assigned to cell c in period u; 0, otherwise if part p visits cell c in period u; 0, otherwise if part p visits a cell in period t; otherwise if machine m belongs to shop s in period t; 0, otherwise if part p visits a shop in period t; 0, otherwise if part p visits shop s in period t; 0, otherwise

Bpcu = 1 dpt = 1 Kmst = 1 kpt ¼ 1

1pst = 1

2.3. The problem formulation The objective functions and constraints can be formulated as follows: " #

Min z1 ¼

cmax T X t X X

P M X X Dpt  ðY mcu  Z pm Þ  Bpcu

t¼1 u¼1 c¼1

p¼1

a  X cu 

m¼1

" # cmax T X P t X X X þ b  Dpt  ðX cu  Bpcu Þ  dpt t¼1 p¼1

þ

u¼1

"

T X S X P X

d  Dpt

t¼1 s¼1 p¼1

c¼1

M X

#

ðK mst  Z pm Þ  kpt þ

m¼1

T X P X

c

t¼1 p¼1

" # T X P S X X  kpt  dpt  Dpt þ x  Dpt fpst  kpt ; t¼1 p¼1

" # cmax T t X X X Min z2 ¼ Bpcu  1 t¼1

ð1  Bpcu Þ 

3. Hybrid approach for incremental cell formation problem

ð1Þ

s¼1

ð2Þ

u¼1 c¼1

s:t:X cu P X ðcþ1Þu M X

It is worth noting that the evaluation of fitness functions for complex functions is time consuming and expensive. Hence, the combination of the GA and ANN is performed in order to build a hybrid approach that is capable of seeking near-optimal or even optimal neural networks for a given problem. The overall schema

ð3Þ

Y mcu  Z pm ¼ 0

ð4Þ

m1 M X

The objective functions (1) and (2) represent the total cost and the total number of exceptional elements respectively. The total cost consists the costs of intra-cell material handling (first term in the objective function), inter-cell material handling (i.e., the second term), intra-shop material handling (i.e., the third term), material handling between cell and shop (i.e., the fourth term) and inter-shop material handling (i.e., the fifth term). The total number of exceptional elements includes between cells movements only. Eq. (3) ensures the order of CF in a period. Eqs. (4) and (5) show that part p visits cell c, when at least one of the required machines to process the part is allocated to the cell. Eq. (6) ensures that a machine can belong to a shop if it is in that shop in the preceding period. Eq. (7) represents that a machine can be allocated only to a cell or a shop in each period. Eqs. (8) and (9) show that part p visits shop s when at least one of the required machines to process the part is allocated to this shop. Eqs. (10) and (11) ensure that a part moves to the another cell (i.e., inter-cell movements) if the part visits more than one cell in a period. Eq. (12) ensures that each machine can be allocated to at most one cell in each period. Eqs. (13) and (14) ensure that a cell is formed in a period if at least one machine is allocated to the cell. Eqs. (15) and (16) show that part p visits shop s, when at least one of the required machines to process the part is allocated to the shop.

Y mcu  Z pm P Bpcu

Generate initial solutions randomly

ð5Þ

m¼1

ð1  K msu Þ  K msðuþ1Þ ¼ 0 K msu  K msu  K msðuþ1Þ 

ð6Þ

C max X

Y mcðuþ1Þ ¼ 0

ð7Þ

c¼1

ð1  kpt Þ 

S X M X

K mst  Z pm ¼ 0

Approximate the degree of infeasibility by ANN

ð8Þ

s¼1 m¼1 S X M X

K mst  Z pm P kpt

ð9Þ

s¼1 m¼1

ð1  dpt Þ 

C max X

Bpcu ¼ 0

If the degree of infeasibility > threshold value ( )

ð10Þ

No

c¼1 C max X

Bpcu P dpt

ð11Þ

c¼1

Feasible solutions

C max X

Y mcu 6 1

Yes

Infeasible solutions

ð12Þ

c¼1

ð1  X cu Þ 

M X Y mcu ¼ 0

ð13Þ Perform the GA operators

m¼1 M X

Y mcu P X cu

ð14Þ

m¼1 M X ð1  fpst Þ  K mst  Z pm ¼ 0 m1 M X K mst  Z pm P fpst 

ð15Þ ð16Þ

Implement the NSGA II Fig. 3. Diagram of the proposed algorithm.

m1

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of the hybridized approach depicted in Fig. 3 consists of the following steps. The proposed mathematical model presents a complex non-linear problem that solving such a problem optimally by exact algorithms is too consuming. Hence, GA as a random search algorithm can be efficient for this problem; however, the general GA generates a number of solutions, called population, in each iteration. It assigns a fitness value to each solution that is derived from the objective functions. The process of evaluation of fitness functions is very complex and time consuming for real problems such as incremental cell formation problem, which a large part of computational time is consumed for infeasible solutions. For this reason, here an ANN is designed to detect the feasible and infeasible solutions and measures the degree of infeasibility then the GA evaluate the fitness values for feasible solutions and assigns a penalty cost to infeasible solutions proportional to the corresponding degree of infeasibility. The steps of the proposed algorithm are as given below. 1. Set t = 0. 2. Initialization. Randomly generate the initial population of individuals (Pt). Each individual is presented by a chromosome, in which the structure of chromosome is formed as a matrix shown in Fig. 4. Each row represents a period and each column represents a machine. The value of each cell is an integer number between 0 and Cmax that demonstrates the status of machines in each period. Value 0 is equivalent to a shop and other values are equivalent to cells (see Fig. 5). 3. Evaluation. Approximate the degree of infeasibility (DI) of individuals by ANN. 4. Feasibility. The DI of each individual is compared with a threshold value (a). 5. Clustering. The population of individuals is clustered into two subsets of A with DI lower than a as feasible solutions and B with DI greater than a as infeasible solution. 6. Fitness value calculation. For individuals of set A, the fitness function is calculated exactly, and for individuals of set B proportionate to their DI a penalty cost is considered. 7. Pareto fronts determination. Sort the individuals in different fronts using Non-dominated sorting genetic algorithm (NSGA-II) (Deb et al., 2002). 8. Dummy fitness assignment. Assign an identical dummy fitness value rj to solutions of front j, in which the value of Pareto front is lower than other fronts. 9. Crowding distance measurement. Measure crowding distance cd(i) for each solution i of a front by relation (17).

cdðiÞ ¼ dij þ dik

ð17Þ

10. Domain of neighborhood. Whereas each solution as i has a maximum acceptable neighbor distance (MAND), the eligible neighbor set (ENS) is defined for each solution that has cubic distance to solution i lower than MAND. The total neighbor distance (TND) of solution i and the average distance of solutions of a front to solution i are determined by relations (18) and (19).

M1

M2

M2

M4

T1

0

0

1

0

T2

0

2

1

0

T3

0

2

1

2

Fig. 4. Chromosome structure.

f2

X dij Y j

dik i

k f1

Fig. 5. Crowding distance and the domain of approach of NSGA-II.

X

TNDðiÞ ¼

cdðjÞ

ð18Þ

j2ENSðiÞ

TNDðiÞ ¼

TNDðiÞ jENSj

ð19Þ

11. Fitness value assignment. Assign a fitness value to solution i of front j by relation (20).

f v ði; jÞ ¼ P

TNDðiÞ  i TNDðiÞ  r j

ð20Þ

12. Probability of selection. Calculate the percentage of roulette wheel based on the fv(i, j) of solution i of front j and the probability of selection of this solution PV(i, j) corresponds to its percentage.

f v ði; jÞ PVði; jÞ ¼ P P j i f v ði; jÞ

ð21Þ

13. Selection. Initially, define the probability of front j (PSELj) and the expected number of selected solutions of front j (NSSFj) by relations (22) and (23).

PSELj ¼

X PVði; jÞ

ð22Þ

j

NSSF j ¼ minðPSELj  POPS; NSF j Þ

ð23Þ

where NSFj is the number of solutions of front j and POPS is the population size. From first front NSSFj number of solutions is selected, if the number of solutions (NSFj) on this front is greater than NSSFj then of available solutions based on their probability of selection, required number of solutions will be selected. If the number of solutions (NSFj) is lower than NSSFj then the solutions of next front will be added to selected solutions. This process is repeated for all fronts and the selected solutions will moved to mating pool. If the size of mating pool is lower than POPS then the solutions with best fitness value are selected and PV*POPS number of this solution in mating pool is repeated. Delete this solution and repeat this manner while the mating pool is filled. 1. Generation. Set t = t + 1, by using the GA operators as follows, the new generation will be produced. 14.1 Crossover operator. Here, the matrixes of the two parents are cut in two sections at some random point and are recombined into one new solution. The crossover operator is given in Fig. 6. 14.2 Mutation operator. The mutation operator changes the value of a column between 0 and Cmax randomly. For example, Fig. 7 shows this operator.

Please cite this article in press as: Zeidi, J. R., et al. A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.015

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Fig. 6. Crossover operator.

Fig. 7. Mutation operator.

C11 W111 C12 M1

T1

Feasible

C1max

M2

DI

Ti Cij

Mk

Infeasible

Tt

Ct1 Ct2 Wktmax

Ctmax Input Layer

First hidden Layer

Second hidden Layer

Output Layer

Fig. 8. A multilayer perceptron.

3.1. The neural network for fitness function approximation The artificial neural network (ANN) implemented by a multilayer preceptron is a flexible scheme capable of approximating an arbitrary complex function. It consists of an input layer one or more hidden layers and an output layer. Backpropagation, which have been successfully used in modeling, classification, forecasting, design, control, and pattern recognition, is one of the most popular algorithms for training multilayer perceptrons (Yan, Chen, & Chang, 2009). The ANN consists of an input layer, two hidden layers and output layer as shown in Fig. 8. The input layer represents the available machines in shops and the output layer the approximation of objective functions that include M units and 1 unit, respectively. All the nodes in the input

layer are connected to every node in the first hidden layer. The first hidden layer is clustered into T sections, where every section includes Cmax units. The jth node of ith section of the first hidden layer represents the cell number j at period i. The weight Wijk associated with unit i of the input layer and unit k of section j of the hidden layer represents the dependency of machine i to cell k in period j (see Fig. 9). Let the vector Wjk = [wijk, . . . , wmjk] denotes the input and W 0jkl represents the output of the unit k of section j of first hidden layer. The output of first hidden layer units is a function of activation function (ujk) that is defined by:

W 0jkl

¼ ujk ðW jk Þ ¼

8 <1 :

if LB 6

X

wijk 6 UB

i

1 otherwise

Please cite this article in press as: Zeidi, J. R., et al. A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.015

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Section j

Wij

Mi

Cjk

Wmjk X3

Mm

X2

Fig. 9. Inputs to a unit of the first hidden layer.

X1 Fig. 11. Clustered area by the ANN.

Section j Cj1 W j1j‘ Cj2

‘ W j2j

W ''j

W‘kjk= ∞ Tj

Ckj

‘ W jmaxj

'’'

W k=∞ ‘ W kjk < ∞ Tk

'’'

Wk <∞

Feasible o.f Infeasible

Cjmax

First hidden layer

2 W 00j

3

w0j1j

6 w0 6 j2j 6 ¼6 6: 6: 4

w0jmaxj

7 7 7 7 7 7 5

  are obtained by: The outputs of the second hidden layer W 000 j

W 000 j

¼

8 > <1 > :

if

Max X W 0jkj 6 C max k¼1

1 otherwise

The outputs of the second hidden layer are connected to output layer as the inputs of output layer. Finally, the output of the output layer is calculated by:

o:f ¼



feasible

if W 000 for j ¼ 1; . . . ; T j 6 1

Output layer

Fig. 12. Feasible and infeasible units.

Fig. 10. Inputs to a unit of the second hidden layer.

The second hidden layer parallel to first hidden layer sections includes T units. All nodes of the same section of the first hidden layer are connected to a related node of the second hidden layer. For example, the kth unit of section j of the first hidden layer will be connected to jth unit of the second hidden layer. The input vectors to second hidden layer W 0jkj form a new weight between the first and second layer units connection is called W 00j , where W 00j is a Cmax-by-M matrix as shown in Fig. 10.

Second hidden layer

3.2. Training The main objective of the proposed ANN is to cluster solution area into the feasible and infeasible area. For this reason, some solutions with the known objective function values are used to train the network. In this study, some solutions indicate the obvious layout of machines, for example, all machines remain unchanged from initial layout (job shop) and hence calculation of the exact objective function value for such a solution is not complex. The proposed neural network weights assign a solution to the incremental cell formation problem, and the feasibility of the solution is determined. This assignment is done in the input layer by binary values for related weights Wijk. The training set (i.e., known solutions) is used to adjust weights with back propagation. For each solution (i.e., input weights), the artificial neural network (ANN) calculates and assigns the objective function value to the feasible or infeasible area. In this algorithm, the training is performed by minimizing the error function as follows.

E¼

N X

2 ðo:fn  o:~f n Þ

ð24Þ

n¼1

where o.fn are the true value and o:~f n the output produced by the ANN. For feasible solutions, value 1 and infeasible value 1 are assigned to the output of ANN. By minimizing Eq. (24), the genetic algorithm operators are used based on steps of Section 3 for updating the weights of the input layer (see Fig. 13).

infeasible otherwise

In general, the proposed ANN in the output layer clusters the solution area into the feasible and infeasible areas as shown in Fig. 11. Moreover, from every unit of hidden layers, a connection is designed to the output layer if the output weight of the unit is set to 1. To clarify this subject, suppose that the output weight of the unit j of section k of first hidden layer has a value 1. Hence, this solution is infeasible and without continuing the process, the processing is finished (see Fig. 12).

3.3. Comparison of computational times of the traditional GA and the hybrid GA Computational time reduction is one of the main advantages of the hybrid GA. The traditional GA evaluates the fitness value of all population solutions, which consumes too much CPU time for complex problems, such as the incremental cell formation problem. Hence, the proposed hybrid GA clusters the generated

Please cite this article in press as: Zeidi, J. R., et al. A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.015

J.R. Zeidi et al. / Computers & Industrial Engineering xxx (2013) xxx–xxx

Percentage of accuracy

8

Crossover Wijk (t+1)

Wijk (t) Mutation

Fig. 13. GA operators for training the ANN.

population into feasible and infeasible solutions by ANN. Then, it evaluates the fitness value only for feasible solutions and considers a penalty for infeasible solutions. It seems that the proposed algorithm increases the CPU time for detection feasible solutions by ANN. Besides, the hybrid GA reduces the CPU time by reduction of the number of solutions that have to be evaluated. The tradeoff between this increment and reduction of the CPU time is shown in Table 1. For this reason, some problems of different sizes (i.e., small, medium and large) are solved by the traditional GA and the hybrid GA. Each problem consists of POPS number of solutions and is generated in one iteration. Table 1 shows that the CPU time of problems with small sizes solved by the traditional GA are lower than the hybrid GA and it decreases with the growing problem sizes so that the CPU time of the traditional GA is lower than the hybrid GA for large-sized problems. As the real problems are in large sizes, then the hybrid GA is efficient.

NSGAII ANN

Test number Fig. 14. Estimated accuracy of NSGA-II and ANN algorithms.

Percentage of accuracy

NSGAII ANN

Test number Fig. 15. Average required time for producing feasible solutions.

4. Discussion and analysis of algorithms’ performance In this section, to clarity an additional analysis of some problems is discussed, in which their objective functions are estimated by the NSGA-II and the proposed algorithm. The accuracy of estimations and computational times are factors that are used to measure the efficiency of algorithms. The accuracy factor is a tool for identifying the feasible and infeasible solutions. For this reason, all constraints are written in lingo, and the exact feasibility of them is determined. Each test problem is generated many times by the NSGA-II and ANN. The results are shown in Fig. 14. It shows that in all cases the ANN is performed more accurate recognition of the feasibility. The average required computational time that algorithms need to produce a feasible solution is calculated based on the following equation:

C:T ðComputational timeÞ ¼

NFS RT

diversification of solutions and domain of points of each solution as the performance measures are used to compare the NSGA-II and ANN algorithms. The NSGA-II and proposed ANN algorithms are coded in MATLAB 7.5 on a common PC. Commonly, many researchers consider a form of distance between points as a performance measure that the crowding distance proposed by Favuzza, Ippolito, and Riva Sanseverino (2006) is one of them. Here, three performance measures are used to evaluate the procedures. The first performance measure is the number of non-dominated solutions produced by each procedure on each data set. The second performance measure is the diversification of solutions solved by each procedure on each data set. The last performance measure is the domain of points of each solution. The diversification of solutions is achieved by considering the crowding distance proposed by Favuzza et al. (2006) and the domain of solution (DOS) using the following equations:

X CDi NNS i

The number of the feasible solution produced at a given number of iterations is called the NFS, and the total time taken for a number of the given generation is called RT. Fig. 15 shows a comparison of the computational time between algorithms. It is clear that the average computational times required for producing feasible solutions of the proposed ANN is shorter than the NSGA-II.

Diversification ¼

5. Computational experiments

The structures of experiments are selected to represent a wide variety of problems (small, medium and large scale sizes). To show the efficiency of the proposed algorithm, the solutions of a test problem by the proposed ANN and NSGA-II algorithms under equal condition are illustrated in Fig. 16.

In this section, 26 numerical examples are tested that their data are produced randomly. Each test problem is solved ten times and the average of the number of non-dominated solutions (NNS), the

DOS ¼ X þ Y where CDi is the crowding distance of point i as shown in Fig. 5

CDi ¼ dij þ dik

Table 1 Comparison of CPU time (S = Small, M = Medium, L = Large). Problem size

S

S

S

S

S

M

M

M

M

M

L

L

L

L

L

CPU time by GA CPU time by hybrid GA Percentage of difference

0.192 0.284 32.39

0.292 0.418 30.14

0.136 0.399 65.91

0.276 0.409 32.52

0.732 1.36 46.18

1.3 1.7 23.5

1.32 1.76 25

1.42 1.93 26

1.87 1.88 0

6.82 6.59 3.5

56.58 62.26 5.33

68.37 64.33 6.28

218.03 214.29 2

527 493 7

713 692 4

Please cite this article in press as: Zeidi, J. R., et al. A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.015

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J.R. Zeidi et al. / Computers & Industrial Engineering xxx (2013) xxx–xxx

Z2

Z2

×10

×10 6

6

z1

×

×

(a)

z1

(b)

Fig. 16. Pareto front of (a) NSGA-II and (b) proposed algorithm. Table 2 Data of experiments (S = Small, M = Medium, L = Large). Test number

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

Problem size

S S S S S S M M M M M M M M M L L L L L L L L L L L

NSGA-II

Proposed ANN algorithm

DOS

Diversification

NNS

DOS

Diversification

NNS

22 13 8.5 8.5 16.5 4.2 8.25 9 7.65 27 13 17 11.5 6.5 12.5 14.5 12.5 14.5 12.7 8 11.5 12 11.5 13 15 22

25 14 8.5 15 25 6.2 12 12.67 9.75 42.3 21.7 26 18.7 7 17.7 23 18 31 19.4 17.3 19.7 21.2 26.3 33.4 22.8 39.6

3 3 3 6 7 7 5 5 7 7 6 10 6 3 3 5 4 8 11 6 9 7 8 8 6 7

13.5 14 9 19.5 18.5 4 8.54 7.25 7.75 36 12.5 19.4 14 6.8 16.8 18 15 20.17 16.7 16 10 13.5 14 10 27 25

21 18.5 11 32 27.3 6.5 9 11.5 8.76 54.6 23.4 31 24.5 8.2 23.2 29 27 32.6 23.2 47 43.6 45 33.7 31.5 52.3 48

8 4 4 6 8 8 8 13 11 13 13 11 9 6 4 7 4 14 12 21 20 15 10 13 9 8

As shown in Fig. 16, the crowding distance between points produced by NSGA-II maybe greater than the proposed algorithm, but it should be considered the domain of the proposed algorithm is larger and the number of non-dominated points is more than NSGA-II. The analysis of remaining experiments is shown in Table 2. From Table 2, it is seen that there is significant difference between the proposed ANN and NSGA-II algorithms; the proposed algorithm produces more non-dominated solutions than the NSGA-II, except two cases (i.e., test numbers 4 and 17) that the number of non-dominated solution for both algorithms are equal. Also, other performance measures for the proposed algorithm work better than the NSGA-II. In other words, the proposed ANN produces more solutions in all cases in a wide range of different values, which provides a higher degree of diversification of solutions. The difference between these algorithms for test number

16 is shown in Fig. 17. For this case, NSGA-II generates a lower number of solutions than ANN and the solutions are centralized in a bounded area tightly, while the ANN covers a wide range. Clearly, ANN provides the decision maker to choice the best decision from a lot of diverse solutions. Table 3 clarifies that the proposed ANN performs better than the NSGA-II in most cases (i.e., 92%, 77% and 81%). In some cases (i.e., 23% and 19%), it seems that the NSGA-II performs better than the proposed algorithm. This idea is very weak because in such cases, some performance measures of the NSGA-II performs better than the proposed algorithm, firstly the average of differences are inconsiderable (i.e., 2.575, 3.21) and secondly this cases performs very weak in other performance measures. For example, in Table 1, test number 8 has a domain and diversification value of 9 and 12.67 respectively that both of them are weakly better than the corresponding value for the proposed algorithm. However, the

Please cite this article in press as: Zeidi, J. R., et al. A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system. Computers & Industrial Engineering (2013), http://dx.doi.org/10.1016/j.cie.2013.08.015

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J.R. Zeidi et al. / Computers & Industrial Engineering xxx (2013) xxx–xxx

Z2

Z2

z1

z1

(a)

(b) Fig. 17. Comparison of (a) ANN and (b) NSGA-II.

Table 3 Analysis of experiments. Percentage of cases

Number of non-dominated solutions Average of differences Domain of solutions Average of differences Diversification Average of differences

Proposed algorithm > NSGA-II

Proposed algorithm = NSGA-II

Proposed algorithm < NSGA-II

92 3.81 77 3.8 81 9.58

8 0 0 0 0 0

0 0 23 2.575 19 2.81

proposed algorithm is strongly better in the number of the nondominated solution measure than the NSGA-II.

6. Conclusion Over the last three decades, many researches have developed a number of techniques for cell formation problems; however, most of them do not consider planning, implementation, and capital investment issues. They have focused only on forming manufacturing cells comprehensively or non-incrementally. This paper has presented a new multi-objective mathematical model that try to plan conversion of job shops to cellular manufacturing systems with minimization of the total material handling cost and total exceptional elements. A novel approach for implementing multiobjective optimization within the artificial neural network (ANN) has been constructed. This approach has performed a population based, such as NSGA-II, with two significant attributes. Initially, it is trained based on the ANN and produced feasible solutions at each generation. The next issue was the computational time to generate the population, which is lower than NSGA-II. Many test problems have been designed to evaluate the proposed ANN algorithm with three performance measures. The comparison results have demonstrated the effectiveness of the proposed algorithm.

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