Unique NSGA-II and MOPSO algorithms for improved dynamic cellular manufacturing systems considering human factors

Accepted Manuscript

Unique NSGA-II and MOPSO Algorithms for Improved Dynamic CMS by Considering Human Factors Azadeh , M. Ravanbakhsh , M. Rezaei-Malek , M. Sheikhalishahi , A. Taheri-Moghaddam PII: DOI: Reference:

S0307-904X(17)30122-1 10.1016/j.apm.2017.02.026 APM 11614

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

14 April 2016 3 February 2017 14 February 2017

Please cite this article as: Azadeh , M. Ravanbakhsh , M. Rezaei-Malek , M. Sheikhalishahi , A. Taheri-Moghaddam , Unique NSGA-II and MOPSO Algorithms for Improved Dynamic CMS by Considering Human Factors , Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.02.026

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Highlights A new multi-objective dynamic CMS by considering human factors. Human reliability and decision-making are considered Decision-making inconsistency is minimized over the planning horizon Workload of CMS is balanced by reliability and performance of individuals Two highly innovative meta-heuristic algorithms are developed to solve the problem

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Unique NSGA-II and MOPSO Algorithms for Improved Dynamic CMS by Considering Human Factors Azadeha,*, M. Ravanbakhsha, M. Rezaei-Maleka,b, M. Sheikhalishahia, A. Taheri-Moghaddama aSchool of Industrial Engineering, Center of Excellence for Intelligent Based Experimental

University of Tehran, Tehran, Iran bLCFC, Arts et Métiers ParisTech, 57078 Metz, France *Corresponding

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Mechanic and Department of Engineering Optimization Research, College of Engineering,

author. [email protected] or [email protected] Fax: +9821 88013102.

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Abstract This paper presents a new mathematical model for Multi objective Dynamic Cellular Manufacturing System (MDCMS) by considering human factors. Human factors are incorporated into the developed model in terms of human reliability and decision-making process. Three objective functions are simultaneously considered. The first objective minimizes total cost of MDCMS. The second objective function minimizes operators’ decision-making style inconsistency in the common manufacturing cells. The third objective function balances workload of cells with respect to operators’ efficiency which is calculated by human reliability analysis. Although various studies have been conducted in the field of MDCMS, human factors as one of the most important elements has not received enough attention. Due to NP-hardness of the MDCMS problem, two innovative metaheuristic algorithms (i.e. NSGA-II and MOPSO) are developed. The results of the algorithms are compared and analyzed using different criteria. Several test problems are applied to verify and validate the proposed model and solution methods. To the best of our knowledge, this is the first study that considers human reliability and decision-making styles for the large MDCMS in an actual production setting.

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Keyword: Dynamic cellular manufacturing system; Labor assignment; Decision-making style; Human reliability; Machine failure; Multi-objective optimization

1. Introduction Cellular Manufacturing System (CMS) is a significant application of Group Technology (GT) principles (Olumolade, 1996; Saad et al., 2002). The philosophy of GT divides parts into groups and machines into cells by considering similarities of parts regarding the processes of construction and design. CMS is the analysis of a set of similar parts within a group of machinery or production processes (Kooshan, 2012). Inefficiency of job shops and flow lines for quickly responding to radical fluctuations such as high demand changes, forced 2

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production companies to make use of a more efficient system, namely CMS. CMS simultaneously has the advantages of job shops and flow lines. In comparison to these traditional methods of workplace design, the advantages of CMS include reduction in setup time and cost, necessary tools, lead times, and work-in-process inventories. Moreover, CMS increases flexibility, improves promised delivery time with more reliability, and simplifies programming and the flow of parts and tools (Askin et al., 1999; Asokan et al., 2001). The CMS design includes four main steps in which Cell Formation Problem (CFP) is foremost among them. CFP categorizes machines and parts regarding to similarity among parts to take full efficiency and flexibility through standardization and common processing (Azadeh et al., 2015). Generally, to solve CFP, there are various techniques such as descriptive procedure, clustering analysis procedure, graph approach, artificial intelligence-based approach, and mathematical programming (Faber and Carter, 1986; Chen and Cheng, 1995; Naiv and Narendran, 1998; Albadawi et al., 2005; Safaei et al., 2008; Solimanpur and Foroughi, 2011; Javadi et al., 2013; Azadeh et al., 2015; Sakhaii et al., 2016; Rezaei-Malek et al., 2017). Previous studies in CMS have assumed that machines’ performance is %100 during programming period. Only few studies have regarded reliability while machinery is a key element in the production system and its failure brings about high costs. In this respect, Jabal-Ameli et al. (2008) used the ɛ-constraint method to minimize different costs of CMS by considering production line stoppage, maintenance costs and machinery replacement. In this paper, machineries’ reliability and lost time due to the machine failure have been incorporated into the model. Furthermore, as mentioned before, an important aspect of CMS is operator assignment to the cells. Since operators play an important role in operation of the machines, an appropriate allocation of them to the cells could increase the efficiency of a CMS. Manpower issues have not received enough attention in the CMS literature. In summary, contributions of this paper could be summarized as follows:  Presenting a new multi-objective dynamic mixed integer mathematical model for CMS design considering machine failure and Alternative Process Routes (APR),  Optimal assignment of manpower among cells with respect to two factors: (1) consistency of operators’ DMSs, (2) workload balance among cells by considering efficiency measures as human reliability concept,  Proposing two multi-objective evolutionary algorithms which are Non-Dominated Sorting Genetic Algorithm (NSGA-II) and Multi-Objective Particle Swarm Optimization (MOPSO);  Demonstrating the applicability of the proposed model through actual data set. Table (2) compares the proposed approach with previous studies in the CFWAP literature. {Please insert Table (2) about here} 3

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2. Literature review

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Various studies have been done in the field of CFP, but most of them tackled the problem in just one period or independent multiple periods (e.g. see Heragu and Chen (1998) and Zhao and Wu (2000)). Therefore, they assumed that product demand is determined and fixed in all the periods. However, because of increasing industrial competition in the recent years, the formation and demand level of products are changing in different periods of time (Aroombajestani et al., 2007). Thus, the formed configuration of cells in each time period is not necessarily optimized and cells reconfiguration is required at the end of each period (Aroombajestani et al., 2007). This type of CFP dealing with dynamic considerations is named Dynamic CFP (DCFP). The concept of dynamic manufacturing system in its current form of cellular reconfiguration with synchronic changes in part family and machinery group was first proposed by Rheult et al. (1996). One of the extensions for CFP is adding the problem of operator assignment (i.e. Cell Formation and Worker Assignment Problem; CFWAP) by incoporating related issues such as required skill (Süer and Tummaluri, 2008). Ignoring these issues may reduce profit and/or could increase failure of manufacturing systems. In the last decade, numerous studies have been dedicated to optimal manpower assignment to manufacturing cells in CMS. Bidanda et al. (2005) presented a large-scale analysis of manpower issues in CMS addressing a comprehensive review of the related literature. Cesani and Steudel (2005) presented a simulation model considering the worker flexibility in CMSs while operators can have mobility. Satoglu and Suresh (2009) developed a three-phase hybrid CMS model to decrease overall costs like hiring and firing costs of labor assignment and operators’ training cost. Mahdavi et al. (2010) presented an integer mathematical programming model for Dynamic CFWAP (DCFWAP) considering some constraints on the machine capacity, duplicate machines, and available time of workers in CMS. The objective function of the presented model was to minimize the backorder cost, holding cost, costs of intra- and inter-cell material handling, hiring and firing cost, cost of operators' salary and machinery reconfiguration cost. Rafiei and Ghodsi (2013) developed a bi-objective mathematical model for DCFP considering labor utilization. Their model considered the conventional assumptions of DCFWAP. The first objective function of this model minimized the machines purchase cost, relocation costs of machines, processing costs, inter- and intra-cell movement costs of parts, overtime costs of personnel, and labor shifting costs among cells. The second objective function maximized the labor utilization. Because the DCFWAP problem is categorized as a NP-hard problem, a hybrid algorithm of Genetic Algorithm (GA) and Ant Colony Optimization (ACO) was proposed. Bagheri and Bashiri (2014) developed a multi-objective mathematical model for DCFWAP and inter-cell layout problem. The objective functions of the proposed model were the minimization of inter- and intra-cell 4

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parts trips, machines relocation costs, and operators related issues costs including hiring, firing and salary that all are transformed to one objective by using the LP-metric approach. Decision-Making Style (DMS) is a psychological personality characteristic (Driver et al., 1998). DMS is the way individuals make decisions. Therefore, it can be considered as a human psychological factor. This factor plays an important role in manufacturing systems which rely heavily on human resources. In this regard, in the manufacturing systems that personnel interact with each other for doing their jobs, it has already justified that the consistency of DMSs would affect job satisfaction and productivity (Azadeh et al., 2015; Rezaei-Malek et al., 2017). As a good example, in CMSs, operators working in common manufacturing cells are in high degree of interaction. There are various classifications for DMSs. One of the most acceptable of them is Driver’s classification that classifies DMSs into five categories (Driver et al., 1998): Decisive, Hierarchic, Flexible, Integrative, and Systemic. People who have decisive style use information as much as necessary and they will consider only one option for decision making. Speed and efficiency are crucial to such individuals. Individuals with hierarchic style fully investigate a subject from every aspect, so this leads to the best single solution. People who are associated with flexible style use information to the extent which it is necessary and if one method does not lead to a decision they will look for another option. People who have integrative style comprehensively investigate an issue and then they present some options for decision-making, however, they are not decisive. Finally, people with systemic style alternatively use the integrative style and hierarchic one and assess the options by considering one or several criteria. This style is very ordered and rigorous (Driver et al., 1998). Because workers cooperate with each other in long periods of

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time (about 40 hours per week) in the same production cells, considering consistency of personal characteristics (i.e. DMSs) among them could improve their job satisfaction and productivity of the manufacturing system (Rezaei-Malek et al. 2017). Azadeh et al. (2015) determined a compatibility degree of DMSs, specifically in the manufacturing areas based on Driver et al. (1998) (see Table 1). This paper

{Please insert Table (1) about here}

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applies these values for assigning operators with more compatible DMSs to common cells.

Azadeh et al. (2015) developed a novel bi-objective mathematical model for CFWAP which considers the workers' DMS. The objectives of the model were; (1) minimization of inter-cell material movement and cells establishment costs, and (2) minimization of inconsistency of DMSs among assigned operators to same cells. Rezaei-Malek et al. (2017) extended the work of Azadeh et al. (2015). They considered DMS consistency between the skilled operators and the assigned machines in all the manufacturing cells. Furthermore, Rezaei-Malek et al. (2017) presented a hybrid algorithm of multi-objective fireworks and 5

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NSGA-II for the large-sized test problems. The proposed hybrid algorithm outperforms NSGA-II and multi-objective algorithm. Their model lacks the consideration of operators and machines reliability in dynamic environment. Nowadays, due to the increased flexibility and multi-functioning of machines, there are Alternative Process Routes (APR) for each part to be proceeded (Shu-Hsing et al., 2010). Increased flexibility in production routes can increase flexibility of the CMS design (Kusiak, 1987; Adil et al., 1996). Adil et al. (1996) considered APR assumption and proposed a nonlinear integer programming model to identify groups of parts and machines cells. Aryanezhad et al. (2009) considered APR feature and the promotion capability for workers from one skill level to another. Ghotboddinia et al. (2011) proposed a dynamic model for CFWAP and simultaneously considered two objective functions. The first one minimizes the total costs including fixed and variable expenses of machinery operations, inter- and intracell replacements, purchase and sale during different time periods. The second objective function maximizes manpower productivity ratio in different cells. In CFWAPs, the most significant point in optimal assignment of operators is the proportion of workload of each cell to the number of assigned individuals to the cell. But, it should be noticed that individuals’ abilities are not equal, so the reliability and efficiency of operator should be considered. Nobody can deny the role of individuals in designing, manufacturing, and utilizing industrial processes. For instance, the statistics indicate that more than 90 percent of industrial accidents are directly or indirectly caused by workers’ errors (Khakestani, 2010). Human reliability as a part of human factors and ergonomics is involved in the diverse fields such as production, transportation, medicine, etc. Indeed, individual performance may be influenced by factors including health, age, intellectual aptitude, sentiments, inclination to some common mistakes, cognitive prejudices, etc. For the first time, Williams (1958) pointed out that for increasing system reliability, human reliability should be well considered. Human Reliability Analysis (HRA) is rooted in human performance studies. Early HRA studies are done in empirical psychology and behavioral sciences that have created structural blocks regarding contemporary analysis and qualifying techniques. These blocks include classifications of behaviors, work analysis techniques and psychometrics techniques. Common methods for estimating human reliability are: Confusion Matrix, Expert Estimation, Task Analysis Linked Evaluation Technique, Human Error Rate Prediction (THERP), and Simulation of Personnel Performance. Efficiency can be assessed using individuals’ reliability. Givi et al. (2015) presented a new mathematical model estimating the human error rate in an assembly station under the impact of learning–forgetting and fatigue–recovery. This model is a HRA model that can dynamically measure the human error rate and reliability. Khakestani (2010) used influential factors on HRA questionnaire that includes questions about risk rates, physical and intellectual health, communication, inherent psychological traits, stress causes, skill level, anthropometry, and human error. Afterward, by using the collected data, they determined 6

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3. Problem description and formulation

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input and output variables of the Neural-Fuzzy inference system. After finishing network training, they calculated each operator’s efficiency using ANFIS. In this paper, we apply the proposed approach by Khakestani (2010) to calculate the efficiency score of each operator based on his/her reliability. In other words, the efficiency of the operators is calculated as an index of their reliability.

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In this section, a multi-objective mathematical model for DCFWAP is presented considering some real-world assumptions. The proposed model allocates human resource with respect to mutual consistency of individuals’ DMS as well as grouping of parts and machines. Also, the workload of different cells is balanced by considering individuals’ efficiency. Objective function (1) minimizes various costs of DCFWAP including fixed and variable costs of machinery operations, inter- and intra-cell movement of parts, machinery purchase and sale within the planning horizon, and manpower movement among cells. Objective function (2) maximizes mutual DMS compatibility of individuals assigned to each cell. Finally, objective function (3) balances utilization of employees in different cells by considering their efficiency. The problem is formulated regarding the following assumptions:

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3.1. Assumptions  Each part needs several operations to be processed and each operation must be done based on its respective order.  Each part can be processed in several routes; however, only one of them should be selected.  Fixed costs for each machine and cost of manpower during regular time and overtime are known.  Machines are multi-functional.  The required time for machinery operations, manpower, and setup are known.  Capacity of each machine for each period in the regular time is fixed.  Purchasing cost and marginal revenue of selling each machine are fixed.  Size of each batch for every intra- and inter-cell movement is fixed, and it is assumed that the costs of intra- and inter-cell movement are the same and equal.  Demand for each part is predetermined and it may change in each period.  Backorder is not allowed.  Costs of inter- and intra-cell manpower transfer are fixed and determined.  In each period, the maximum number of cells that can be formed and their respective capacity are known.  Number of operators is fixed for all periods and hiring or firing is not allowed.

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Machinery breakdowns follow Poisson distribution, where the amount of λ for each machine is known.

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e   x x!

3.2. c m p h j l

Indices Index of manufacturing cells (c = 1, …, C), Index of machine types (m = 1, …, M), Index of part types (p = 1, …, P), Index of time periods (h = 1, …, H), Index of required operations for processing of part p (j = 1, …, OP), Index of operators (l = 1, …, L).

P

Number of part types,

OP

Number of the required operations for part p,

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Number of machine types,

C L Dph

Number of cells,

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Parameters Number of periods,

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Allowable quantity of machines in each cell,

 pinter

Inter-cell movement cost per batch p (USD),

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Intra-cell movement cost per batch p (USD), Cost of purchasing machine type m (USD),

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Total number of labors, Demand for part p in period h, 1; if part p is planned to be produced in period h, 0; otherwise,

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3.3.

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Mean time to repair for each machine is specified. DMS of each operator is determined by using the Driver’s approach. Each operator has enough skill to work with every type of machines.

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x  

Revenue from selling machine type m (USD),

m

Fixed cost of machine type m in each period (USD),

h

Fixed cost of inter-cell labor movement in period h (USD),

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Relocation cost of machine type m (USD),

Tmh tipm

Time capacity of machine type m in period h in regular time (h), Required processing time to perform operation j of part type p on machine type m (h), 8

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ajpm BRm MTBFm MTTRm WT Efl

Required time for operator to perform operation j of part type p on machine type m (h), 1; if operation j of part p can be done on machine type m, 0; otherwise, Cost of breakdown of machine type m (USD), Mean time between failures on machine type m (h), Mean time to repair machine type m (h), Available time per worker (h), Efficiency of operator l,

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Variable cost of machine type m for each unit time in regular time (h),

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Batch size for movement of part type p, DMS inconsistency of two operators l and  .

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3.4. Decision variables x jpmch 1; if operation j of part type p is done on machine type m in cell c in period h, 0; otherwise,

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1; if operator l is assigned to cell c in period h, 0; otherwise,

Nmch

Number of machine type m allocated to cell c in period h,

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Number of machine type m added into cell c in period h,

Number of machine type m removed from cell c in period h,

 I mh

Number of machine type m sold in period h,

 I mh

Number of machine type m purchased in period h,

OPh CPch

Utilization percentage of operators in period h, Utilization percentage of operators in cell c in period h.

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3.5. Mathematical model Based on the above-mentioned definitions, the following multi-objective nonlinear mathematical model is presented. C

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  Min Z 1    N mch  m   I mh m    I mh m m 1 c 1 h 1

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   m D ph t jpm x jpmch

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M 1 H P inter  D ph  OP 1 C M  x  x jpmch    p B    j 1 pmch  2 h 1 p 1 m 1  p  j 1 c 1 m 1

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M M  1 H P intra  D ph  OP 1 C  M  x  x  x        j 1 pmch   p jpmch  j 1 pmch x jpmch  2 h 1 p 1 m 1 m 1  B p  j 1 c 1  m 1 

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1 H C L  h  c h 1l   chl 2 h 1 c 1 l 1

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M H C P OP D 1 H M C ph t jpm x jpmch BR m    m  k mch  k mch     2 h 1 m 1 c 1 MTBFm m 1 h 1 c 1 p 1 j 1

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Min Z 2     chl  ch  l  h 1 c 1 l 1   l 1

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Min Z 3   CPch OPh h 1 c 1

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 a

jpm

x jpmch ph

 j , p, h

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x jpmch  a jpm ph

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ph

p 1 j 1

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c 1 l 1 C

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 L

 chl  1

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chl

h, m

 h ,c , m

(4) (5) (6)

 h ,c , m

(7)

(9)

h

(10)

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 M P OP    D ph t jpm x jPmch  m 1 p 1 j 1  CPch   L    chl WT   l 1 Ef l  

 h ,c , m

(3)

(8)

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 h ,c

   N mch T mh 

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 N mc  h 1  I  mh  I  mh

N mc  h 1  k  mch  k  mch  N mch

N

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 MTTR m t jpm x jPmch 1   MTBFm

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s.t.

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(11)

c , h

(12)

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 C M p OP    D ph t jpm x jPmch  c 1 m 1 p 1 j 1  OPh   C L   chl  WT   c 1 l 1 Ef l  

h

(13)

x jPmch , chl 0,1 , k  mch , k  mch , I  mh , I  mh , N mch  N + mCh

, OPh , CPCh  0

(14)

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Objective function (1) minimizes total cost of DCFWAP in all the time periods consisting of the fixed cost of allocating machines to the cells (term 1), purchasing cost of required machines (term 2), revenue of selling the second-hand machines (term 3), variable cost of machines for processing the parts (term 4), cost of inter- and intra-cell movement of parts (terms 5 and 6, respectively), cost of inter-cell movement of manpower including the training cost for improving the workers' skills (term 7), reconfiguration cost of machines (term 8), and finally cost of machine failure which is calculated based on workload of each machine, failure cost, and the average time span between each two failures (term 9). Objective function (2) minimizes total DMS inconsistency among operators in common cells. Objective function (3) minimizes total difference between utilization percentage of all operators in CMS and utilization percentage of operators in each cell. In fact, this objective function maximizes the workload homogeneity of operators in different cells. Equation (4) guarantees that each operation on a part is processed by only one machine in a manufacturing cell. Constraint (5) does not allow variable x to be equal to one if its related parameter a is zero. Constraint (6) guarantees that total operating time of each machine does not exceed the available time. Equation (7) shows the quantity of machines in different periods regarding the purchased and sold machines. Equation (8) balances the quantity of machines in each cell regarding the replacement of machines in different periods. Constraint (9) specifies capacity limitation of each cell for number of machines. Equation (10) shows total number of operators. Equation (11) assures that every operator in each period is allocated only to one cell. Equation (12) computes the percentage of operator’s utilization in each period in each cell. Equation (13) calculates operators’ utilization in each period. Equation (14) identifies type of variables. 3.6. Linearization of the proposed model The proposed model is nonlinear and in this section it is tried to make it as linear as possible. Equation (1) is a nonlinear integer equation because of the absolute terms. To transform this function into linear one, six positive variables ( 1 2 1 2 1 2 z jpch , z jpch , y jpmch , y jpmch ,w chl ,w chl ) are defined and the objective function is rewritten as follows: 11

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  Min Z 1   N mch  m  I mh m   I mh m h 1 m 1 c 1

h 1 m 1

h 1 m 1

H C P OP M  BR m    D ph t jpm x jpmch   m   MTBFm  h 1 c 1 p 1 j 1 m 1 

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1 H P inter  D ph  OP 1 C 1 2 p      z jpch  z jpch  2 h 1 p 1  B P  j 1 c 1

1 H C L h w ch1 w ch2   2 h 1 c 1 l 1

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1 H M C    m  k mch  k mch   2 h 1 m 1 c 1

s.t. M

M

m 1

m 1

z 1jpch  z 2jpch  x  j 1 pmch  x jpmch y 1jpmch  y 2jpmch  x  j 1 pmch  x jpmch

 j , p , m ,c , h  j , p , m ,c , h

c , h , l

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1 2 w chl w chl   c  h 1l  chl

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 D  OP 1 C  M  1 H P    pintra  ph      y 1jpmch  y 2jpmch    z 1jpch  z 2jpch   2 h 1 p 1   B P  j 1 c 1  m 1

z 1jpch , z 2jpch , y 1jpmch , y 2jpmch ,w ch1 , w ch2  0

 j , p , m ,c , h

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(16) (17) (18) (19)

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3.7. Validation and Verification In this section, a small-size test problem is applied to validate the proposed model. Table (3) demonstrates parts route and demands for each part during each period and batches’ size. Table (4) and (5) show model parameters’ value and machines’ information, respectively. Table (6) illustrates labor information and inconsistency between them. {Please insert Table (3) about here} {Please insert Table (4) about here} {Please insert Table (5) about here} {Please insert Table (6) about here}

The normalized optimal value of each objective function is obtained by Gams 22.1 on PC ASUS Core i7, 2.2 GHz with 4GB RAM. Fig. 1 depicts the behavior of objective functions. According to the results, the objective functions do not have consistent behavior and they must be considered separately. {Please insert Fig. 1 about here} 12

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4. Methodology It should be mentioned, in this paper a questionnaire developed by Driver et al (1998) is used to assess the operators’ DMSs. Each style has a different consistency and compatibility with other styles and the quantification method proposed by Azadeh et al (2015) is applied (see Table (7)). Due to the rarity of the systemic style, only four basic styles are considered as follows:

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{Please insert Table (7) about here}

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To determine the human efficiency values, the questionnaire proposed by Khakestani (2010) is used. Because of the Np-hardness of the large-size problems, two meta-heuristic algorithms including Multi Objective Particle Swarm Optimization (MOPSO) and NonDominated Sorting Genetic Algorithm (NSGA-II) are developed to solve the proposed model. Non-dominated solution concept is described as follows:

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4.1. Non-dominated solution It is possible to simply determine the best solution for a single-objective problem, where a solution that has the lowest value (in a minimization form) would be an optimal solution. However, in real-world applications, decision-makers often deal with multi-objective problems where there is no single optimum solution for all the objectives (e.g. see Azadeh et al. (2015b), Salehi Sadghiani et al. (2015), Azadeh et al. (2016a), Azadeh et al. (2016b), Rezaei-Malek et al. (2014), Zahiri et al. (2014a,b), Firoozi et al. (2013), and Salehi et al. (2013)). In such problems, it is possible to obtain a set of non-dominated solutions where each solution is not better than the others. In other words, a feasible solution x is called non-dominated if there is not another feasible solution better than x in one or more objective functions without worsening the others (Ehrgott, 2005). The basic of solutions’ ranking in evolutionary multi-objective algorithms is based on the non-dominated pareto fronts. When the ranking of a non-dominated solution set is 1, it is called as Front #1. When the solutions of Front #1 are eliminated, the remaining non-dominated solutions are called Front #2 (their ranking is equal to 2). Similarly, when the solutions of Front #2 are eliminated (as well as Front #1), the remaining non-dominated solutions are called Front #3 (their ranks are equal to 3). This process should be repeated until the ranking of all the solutions are determined. 4.2. NSGA-II The NSGA-II algorithm is an evolutionary algorithm that works like the function of genes. In this algorithm, based on the legacy of the algorithm, the past data are extracted and used in the search process (Rezaei-Malek et al., 2016a; Rezaei-Malek et al., 2016b). A chromosome is a string of numbers which is called gene. In each iteration, new chromosomes (offspring) 13

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are generated by the algorithm’s operators including the crossover and mutation. Then, the offspring are evaluated and more qualified chromosomes are chosen by a selection procedure such that the selected chromosomes would have a population size same as the initial population and they would be transferred to the next generation. In this process, the algorithm converges to the best chromosome which represents the optimum or suboptimal solution. This paper uses another criterion (i.e., Crowding-Distance (CD)) for sorting the solutions with the same rank. In addition to convergence of the solutions to a real Pareto front, they should be uniformly distributed across the front. The crowdingdistance indicates the absolute distance among the normalized objective functions of the adjacent elements, and as much as its value is higher, the solution is more desirable. The structure of NSGA-II is depicted in Fig. 2. Single-cut crossover is applied and each set of variables is crossed with the same type variables of the parent’s gens. It is notable that one of the parents of the crossover operator is chosen from the Non-Dominated Solutions (NDSs) set and the other parent is chosen from the last population. The selection probability of each NDS is based on its CD value and so a greater CD leads to the higher selection chance. Equation (20) calculates CD, where xi-1, xi and xi+1 are consecutive members of NDS. f1(xi), f2(xi) and f3(xi) are the values of the first, second and third objective functions of the i-th point of NDS. f1max, f2max and f3max are the maximum values of the first, second and third objective functions on the Pareto front, respectively. f1min, f2min and f3min are the minimum values of the first, second and third objective functions on the Pareto front, respectively. The selection probability of the other parent (which is chosen from the population) is equal to the crossover rate (input parameter of NSGA-II) for all the members of population. f (x 1 )  f 1 (x i 1 ) f (x 1 )  f 2 (x i 1 ) f (x 1 )  f 3 (x i 1 ) CD i  1 i max  2 i max  3 i max (20) min min f1 f1 f 2 f 2 f 3  f 3min

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Exploitation (i.e. mutation) is only considered for the NDS set. The experiments showed that if neighborhood search is only established for the NDS set, computational time will be decreased without any change in the quality of solutions. Therefore, we muted only the members of the NDS set and a high-quality solution has the chance to be considered in the next population or even in the next NDS set. Probability of mutation for each chromosome (from the NDS set) is equal to the mutation rate, which is an input parameter of the NSGA-II algorithm. Next generation is selected from the previous generation and the new offspring which are generated by the crossover and mutation operators. The selection probability is based on CD which can be calculated by Equation (20). In other words, a chromosome with the greater CD has a higher chance to be selected for the next generation. This approach provides more variability for the generations production. Besides, if the number of NDSs be more than the archive size, NDSs with lower CD values will be eliminated by the algorithm. Optimum values of algorithm’s parameters are calculated by TAGUCHI method, which is a

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common approach for design of experiments. Table (9) presents the input parameters of the proposed NSGA-II. {Please insert Figure (2) about here}

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4.3. MOPSO Particle swarm optimization is one of the most well-known meta-heuristic algorithms which is modeled based on birds’ social behavior. In this algorithm, the initial birds (i.e., solutions) are generated randomly and then an initial velocity is assigned to each of them. According to the bird’s current velocity and its distance from the best position in the personal memory and its distance from the best position founded by the leader birds (best global founded solutions), a new velocity and position are calculated for the bird by Equation (21) and (22), respectively. If the new position dominates the best personal memory, it will replace the old personal memory (see Equations (23) and (24)), otherwise one of them will be selected randomly. Notations of the algorithm are as follows: xi indicates the current position of the particle, vi denotes the velocity of the particle, pi represents the best personal memory, gi indicates the best position found by the leader particles, r1 and r2 are random numbers between 0 and 1,  is the weight of inertia and, c1 and c2 are acceleration values.  , c1 and c2 represent the weighted-impact of the current velocity, the best position in the personal memory and the best global position on the new velocity, respectively. The proposed MOPSO algorithm assumes the NDS set as the leader particles. As it can be inferred from Equation (21), only one leader should be selected for each particle (gi) while the NDS set has several members. Leaders are selected from the NDS set according to their CD index which is calculated by Equation (20) similar to the NSGA-II algorithm. In other words, selection probability of each NDS, as the leader of each particle, is calculated regarding the CD index (NDS with greater CD has higher chance to be selected as the leader). The algorithm eliminates the NDSs with the lowest CD values from the NDS set if the number of NDSs exceeds the size of archive. The structure of the proposed MOPSO algorithm is shown in Fig. 3. (21)

x i (t )  x i (t  1) V i (t )

(22)

pi (t )  pi (t 1) if f (x i (t ))  f ( pi (t 1))

(23)

pi (t )  x i (t ) if f (x i (t ))  f ( pi (t  1))

(24)

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V i (t )   (t  1)V i (t )  c1r1 ( pi (t 1)  x i (t 1))  c 2r2 ( g i (t 1)  x i (t 1))

{Please insert Fig. 3 about here}

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Design of experiments by the TAGUCHI in MINITAB is applied to find the optimum values of input parameters of the MOPSO and NSGA-II algorithms (see Table (8)). {Please insert Table (8) about here}

k 1

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4.4. Evaluation of the meta-heuristic algorithms In this section, performance of the proposed algorithms (i.e., NSGA-II, MOPSO, WeightedSum) are evaluated. Five different common indexes are employed for comparing the performance of the proposed algorithms: 1) The number of Pareto optimal solutions which indicates the Pareto optimal variability. The larger value of this index results in better performance and more options for decision-makers. 2) The CPU time which shows the calculation time for solving a problem and the lower value of this index indicates better performance. 3) The diversity of the solutions (D) which shows the diversity of the Pareto solutions on the Pareto front and the larger values of this index are preferable. 4) The spacing metric of the Pareto optimal solutions (S) that is defined for measuring uniformly distribution of the Pareto solutions. Decision-makers always prefer solutions which are uniformly distributed, because they have more options to choose. For lower values of this index the Pareto solutions are distributed uniformly in comparison to the larger values. Thus, the lower values are preferable. 5) The quality of the Pareto solutions (Q(A,B)). This index is the most important index for comparing the performance of multi-objective algorithms. The quality of solutions of an algorithm cannot be calculated individually. In other words, the quality of two algorithms are defined in relation to each other. The D index is calculated by Equation (25) where i and j are solution indexes and k is objective function index. 3 (25) D   max f k (x i )  f k (x j )

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The spacing metric (S) implies how close the Pareto optimal solutions are. Equation (26) calculates this metric where n is the number of Pareto optimal solutions, di (which is calculated by Equation (27)) denotes the distance to the i-th closest optimal solution. (26) 1 n 2 S  ( d  d )  i n  1 i 1

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f 1 (x i )  f 1 (x j )  f 2 (x i )  f 2 (x j )  f 3 (x i )  f 3 (x j )

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For calculating the quality of solutions some explanations must be provided. Assume that A and B are two sets of solutions which are intended to compare with each other. Firstly, C(A,B) and C(B,A) should be calculated (see Equation (28)). Q (A, B) ratio is defined by Equation (29), in which 0≤Q(A,B)≤1 and Q(A,B)+Q(B,A)=1. If Q (A, B) > Q (B, A), it implies that solution A is better than solution B.

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(28)

Number of solutions B dominated by solutions A Number of solutions B C (A , B ) Q A , B   C (A , B )  C (B , A ) C (A , B ) 

(29)

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The algorithms are evaluated through ten different randomly generated numerical test problems with different sizes. Replication is considered for eliminating stochastic nature of the meta-heuristic algorithms. In other words, each test problem is solved several times by each algorithm to increase the evaluation accuracy. Table (9) presents the size of each test problem as well as the solving replications. {Please insert Table (9) about here}

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To show the applicability of the developed methods, Weighted-Sum Method (WSM) is coded in GAMS to solve the test problems. GAMS is unable to solve medium and large-size test problems and only the results of small-size test problems are compared with the proposed algorithms. The test problems are solved by the developed meta-heuristics and compared with WSM according to the explained metrics. Mean and standard deviation of the mentioned metrics for solving each test problem are reported in Tables 10 to 12.

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{Please insert Table (10) about here} {Please insert Table (11) about here} {Please insert Table (12) about here} {Please insert Table (13) about here} {Please insert Table (14) about here}

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Statistical tests are implemented to guarantee that there are statistical evidences that the meta-heuristic algorithms can achieve good Pareto solutions in reasonable time. 100 hypothesis tests (Montgomery and Ruger, 2014) are considered for the metrics and different problem sizes. 99% confidence level (i.e., type II error α=1%) is assumed for all the tests. The null hypotheses and the P-Values are represented in Table (15). As Table (15) shows, the proposed NSGA-II algorithm performs better than MOPSO and the weightedsum method in some cases such as: Number of the Pareto solutions, Diversity of the solutions and Quality of the solutions. The proposed MOPSO algorithm performs better in the CPU time. The weighted-sum method performs better in uniformly distributions of the Pareto solutions and the quality of the Pareto solutions. In some cases, all the algorithms perform well, so checking the weaknesses can be useful. Although the weighted-sum performs the best by the S and Q metrics, but it is unable to solve medium and large-size test problems. It takes too much CPU time, and the number of the Pareto solutions as well as the diversity of them are not acceptable. Although MOPSO performs the best by the CPU time metric, but it is defeated by other algorithms in other metrics. NSGA-II is the only 17

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algorithm that is capable of solving the test problems with different sizes in a reasonable time with high-quality. The statistical tests can approve this claim, which are briefly reported in Table (15). {Please insert Table (15) about here}

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The results of the comparison imply:  NSGA-II presents a larger archive of non-dominated solutions.  The required time to achieve optimum solutions in the MOPSO is less than NSGA-II.  NSGA-II has a greater average value of D in comparison to MOPSO. This represents a greater uniformity of MOPSO Pareto solutions in comparison to NSGA-II.  Value of S obtained from the weighted-sum Pareto solutions is better than MOPSO and NSGA-II algorithms.  The quality of non-dominated solutions of NSGA-II is better than MOPSO, and the weighted-sum method (for large-size problems).

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Regarding the obtained results, it is highly recommended to use NSGA-II for large-size problems. Although it takes a bit more time in comparison with MOPSO, the quality of its solutions is better. Noteworthy, the presented approach has been applied in a real CMS in Tehran, Iran. IMEN Compressed Air Industries company assemblies the reciprocating compressors type PC250. IMEN mentioned that by considering the operators’ efficiency and DMS, the job satisfaction level of operators has been increased. The company said that the number of complaint reports which they usually receive every month, has been substantially decreased because the operators are more satisfied with their coworkers in the common cells and the workload is fairly divided between different cells.

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5. Sensitivity analysis

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In this section, a sensitivity analysis is done on the fifth test problem. First, optimum value of the first objective function (i.e., total cost) is determined by using GA, PSO and GAMS regardless of other objective functions. Then, the third objective function is calculated by applying the obtained solution. Optimum value of the third objective function (i.e., workload homogeneity) is specified by using GA, PSO and GAMS, and the first objective function is calculated by applying the obtained solution. The results are given in Table (16). {Please insert Table (16) about here}

Regardless of other objective functions, optimum value of the second objective function (i.e., operators’ inconsistency) is determined by using GA, PSO and GAMS. Then, by using the optimal solution for the second objective function, the 8th term of the cost objective function is calculated. Since the second objective function depends only on the allocation of manpower, so it just affects the 8th term of the cost objective function. In other words, it has no influence on other cost components. Moreover, optimum value of the 8th term of the first 18

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objective function is found, then the second objective function is determined by using the obtained optimal solution and the results are given in Table (17). {Please insert Table (17) about here}

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In Table (16), the optimal value of total cost and the optimal value of balancing objective function are calculated. In addition, in Table (17), the minimum inconsistency and minimum costs resulting from labor transfer during time periods are presented. Now, the relationship between the first and third objective functions, and the relationship between the second objective function and the cost of labor transfer during the planning horizon are analyzed. In other words, the impacts of minimizing the inconsistency and balancing the workload of manpower on the costs of cellular manufacturing have been analyzed by using the Pareto frontier obtained by NSGA-II and MOPSO (see Figs. 4 and 5). The figures imply that the reduction of inconsistency and workload deviation would increase total cost.

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{Please insert Fig. 4 about here} {Please insert Fig. 5 about here}

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Different scenarios are generated by changing the input parameters. These scenarios are employed to investigate the contributions of this paper including the introduction of DMS, labor reliability, MTBFm, MTTRm, and BRm into the DCFWAP problem. In each scenario, one of the parameters is changed. Table (18) illustrates different scenarios. In the first three scenarios individuals’ DMS are changed. In the next three scenarios, failure parameters are changed, and finally in the last three scenarios operators’ reliability are changed. The results are shown on the Pareto frontier and are compared with each other in Fig. 6.

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The results show that employing labor with integrative DMS would reduce inconsistency between workers because of their high consistency with other styles. Increasing the cost of damage or the frequency of system downtime would increase the system costs (almost 25%). Furthermore, by employing operators with various reliabilities, workload balance between different cells have been decreased.

6. Conclusion In this paper, a new mathematical model has been proposed for dynamic cell formation problem with worker assignment. Regarding the CMS literature, the proposed model has several novelties including the consideration of: (1) human resource reliability, (2) operators’ decision-making style assigned to the common cells and, (3) machines’ 19

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reliability. The problem is multi-objective and two meta-heuristic algorithms (i.e., MOPSO and NSGA-II) have been developed to solve it. To evaluate the validity of the newly added parameters, various test problems have been generated and analyzed. The performances of the presented algorithms have been compared by using several criteria (e.g., covered solution, spacing metric, CPU Time, number of Pareto solutions, and covered space). The results showed that it is better to use NSGA-II for the large-size problems. Although it takes a bit more time in comparison with MOPSO, the quality of its solution is better. The presented approach has been applied in a real CMS in Tehran, Iran. According to the results, the job satisfaction level of operators has been increased after incorporation of the operators’ efficiency and decision-making style. This claim is based on the decrease in the number of complaints per month. For future works, due to the abilities of individuals to work with different machines, different reliability values can be considered for operators. Moreover, decision-making style of an operator may be changed according to their assigned workload.

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Safaei, N., Saidi-Mehrabad, M., & Jabal-Ameli, M.S. (2008). A hybrid simulated annealing for solving an extended model of dynamic cellular manufacturing system. European Journal of Operational Research, 185(2), 563–592. Sakhaii, M., Tavakkoli-Moghaddam, R., Bagheri, M., & Vatani, B. (2016). A robust optimization approach for an integrated dynamic cellular manufacturing system and production planning with unreliable machines. Applied Mathematical Modelling, 40(1), 169–191. Salehi, N., Khayatian, A., Firoozi, M., & Siadat, A. (2013). Developing a multi-objective optimization model for locating backup facilities in multi-branch companies under spatial risks. The Proceedings of 2013 International Conference on Industrial Engineering and Systems Management (IESM), October 28-30, Rabat, Morocco. Salehi Sadghiani, N., Torabi, S.A., & Sahebjamnia, N. (2015). Retail supply chain network design under operational and disruption risks. Transportation Research Part E, 75, 95–114. Satoglu, S. I., & Suresh, N. (2009). A goal-programming approach for design of hybrid cellular manufacturing systems in dual resource constrained environments. Computers and Industrial Engineering, 56(2), 560–575. Shu-Hsing, C., Tai-His, W., & Chin-Chih, C. (2010). An efficient Tabu search algorithm to the cell formation problem with alternative routings and machine reliability considerations. Computers and Industrial Engineering, 60(1), 7–15. Solimanpur, M., & Foroughi, A. (2011). A new approach to the cell formation problem with alternative processing routes and operation sequence. International Journal of Production Research, 49(19), 5833–5849. Williams, H.L., 1958. Reliability evaluation of the human component in man- machine systems. Electrical Manufacturing. April 1958: 78-82. Zahiri, B., Tavakkoli-Moghaddam, R., Mohammadi, M., & Jula, P. (2014a). Multi-objective design of an organ transplant network under uncertainty. Transportation Research Part E., 72, 101-124. Zahiri, B., Tavakkoli-Moghaddam, R., & Pishvaee, M. S. (2014b). A robust possibilistic programming approach to multi-period location–allocation of organ transplant centers under uncertainty. Computers & Industrial Engineering, 74, 139–148. Zhalechian, M., Tavakkoli-Moghaddam, R., Zahiri, B., Mohammadi, M. (2016). Sustainable design of a closed-loop location-routing-inventory supply chain network under mixed uncertainty. Transportation Research Part E: Logistics and Transportation Review, 89, 182-214. Zhao, C., & Wu, Z. (2000). A genetic algorithm for manufacturing cell-formation whit multi routes and multiple objective. International Journal of Production Research, 38(2), 385-395.

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Olumolade, M. O. & Norrie, D. H. (1996). Reactive scheduling system for cellular manufacturing with failure-prone machines. International Journal of Computer Integrated Manufacturing 9(2), 131-144. Saad, S. M., Baykasoglu, A. & Gindy, N. (2002) A new integrated system for loading and scheduling in cellular manufacturing. International Journal of Computer Integrated Manufacturing 15(1), 37–49.

24

1.2 1 0.8 0.6 0.4 0.2 0

OF1

OF1 OF2 OF3

OF2

OF3

0.62

1

0.81

0.86

0.43

1

1

0.83

0.18

CR IP T

Normalized values

ACCEPTED MANUSCRIPT

Fig. 1. Behavior of the objective functions

start

Rate the new generation and send the rank number1 solutions to the pareto archive

AN US

Generate the initial population

Stop criteria reached?

Rating populations based on rank and crowding distance

YES

NO

Generate a new population of offspring by crossover and mutation operators

M

end

Combination and competitiveness of offspring with the parents

Create a new generation

PT

start

ED

Fig. 2. Flow chart of NSGA-II

CE

Generate the initial population

AC

Determine the velocity of each particle

Stop criteria reached?

Update p(i) and g(i)

YES

end Determined g(i)

NO Calculate the new velocity of each particle

Calculate the new position of the particles

Fig. 3. Flow chart of MOPSO 25

ACCEPTED MANUSCRIPT

NSGA-II

CR IP T

MOPSO

Fig. 4. MOPSO and NSGA-II Pareto front (OF3-OF1)

NSGA-II

AN US

MOPSO

Scenario1 Scenario2

Scenario4

Scenario7

Scenario5

Scenario8

Scenario6

Scenario9

CE

PT

Scenario3

ED

M

Fig. 5. MOPSO and NSGA-II Pareto front (OF2-OF1 (8th term))

AC

Fig. 6. Pareto front for the different scenarios (NSGA-II)

26

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Table 1 Consistency degree of decision-making styles (Azadeh et al., 2015). DMS Decisive Flexible Hierarchic Integrative Systemic

Decisive CO NCNI NCNI IN SI

Flexible NCNI SCO IN CO SI

Hierarchic NCNI IN CO NCNI IN

Integrative NCNI CO CO SCO NCNI

Systemic SI IN CO CO CO

CR IP T

DMS: Decision-making style, SCO: Strongly consistent, CO: Consistent, NCNI: Neither consistent nor inconsistent, IN: Inconsistent, SI: Strongly inconsistent

Table 2 Literature review of the human resources allocation in CMS

* Multiobjective

Operators Reliability

AN US

Skill

DMS

APR

Solution method

√

-

√

-

-

√

Hierarchical method

√

-

-

-

√

-

-

-

Goal programming

√

√

√

-

√

-

-

-

Lingo 8.0

√

√

√

√

-

-

-

√

Benders’ decomposition

√

√

√

√

-

-

-

√

Ant colony algorithm

√

√

√

-

√

-

-

-

LP-metric approach

√

-

-

-

-

-

NSGA-II and ε-constraint

ED

M

-

√

AC This paper

Operator workload balance

√

-

CE

Aryanezhad et al. (2009) Satoglu and Suresh (2009) Mahdavi et al. (2010) Ghotbodinia et al. (2010) Rafiei and Ghodsi (2013) Bagheri and Bashiri (2013) Aghajani et al. (2014) Azadeh et al. (2015) RezaeiMalek et al. (2016)

Dynamic

PT

Author(s)

Movement cost intra- Intercell cell

√

-

-

-

√

-

√

-

ε-constraint

√

-

-

-

√

-

√

-

MOFA* and NSGA-II

√

√

√

√

-

√

√

√

NSGA-II and MOPSO

firework algorithm

27

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Table 3 Part information of the test problem P1 1 .08 .09

M1 M2 M3 M4 Dph

2

3

.90

.61 .66

P2 1 .64

2 .11

2 .65 .45

.17 .57

.54

.97

Period1 Period2 Bp

P3 1 .48

3

.07

200 500 8

3 .46

P4 1 .48

3 .40

.36

.27

0 450 12

2 .82 .98

.56

.61

CR IP T

Route

0 0 10

650 500 16

Table 4 General information of the test problem Value

Parameter

Value

Parameter

H

2

M

4

h

500

OP

3

p

4

t ' jpm

~ t jpm /10

 Pintra

~  Pinter /10

C

3

WT

UB

3

 Pinter

100

100

12000

10000

M2

700

14000

M3

700

15000

M4

700

14000

Parameter

BR m  $ 



M m 1

a jpm

MTBFm

MTTR m

120

300

60

3

850

70

11500

140

150

70

1.5

510

50

10500

150

350

75

3.5

400

85

12300

140

200

70

2

370

90

CE DMS

L1 Flexible

L2 Hierarchic

L3 Flexible

L4 Decisive

L5 Hierarchic

L6 Integrative

Ef

0.38

0.65

0.85

0.62

0.82

0.73

inconsistency L1

_

7

1

5

7

3

L2 L3 L4 L5 L6

7 1 5 7 3

_ 7 5 3 5

7 _ 5 7 3

5 5 _ 5 7

3 7 5 _ 5

3 3 5 3 _

AC

2

m  $

Table 6 Labor information of the test problem

l

Value

 m $

PT

700

ED

M

Table 5 Machine information of the test problem  m $ T mh  $  h m  $  m  $  M1

Value

AN US

Parameter

28

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Table 7 Quantified consistency of DMS (adopted from Azadeh et al. (2015)) Decisive 3 5 5 7

Flexible 5 1 7 3

Hierarchic 5 7 3 5

Table 8 NSGA-II and MOPSO input parameters

MOPSO

Population

Number of generation

Crossover rate

100

50

0.7

Population

Number of generation

C1

C2



120

50

2.5

2.5

0.8

Table 9 Considered sizes for test problems

|J|×|P|×|M|×|C|×|H||×|L|

1 2 3

2 × 3 × 4 × 2× 2× 4 3 × 4 × 4 × 3× 2× 6 3 × 5 × 5 × 3× 2× 8

4

3 × 6 × 6 × 3× 2× 10

0.5

100

Mutation rate

Size of archive

0.3

100

Number of replications (Solving the same problem) 30 30

3 × 8 × 6 × 3× 3× 10

6

3 × 12 × 8 × 3× 3× 14

ED

5 7

3 × 16 × 10× 3× 3× 16

8

3 × 20× 12 × 4× 3× 16

4 × 22 × 16 × 4× 4× 18

PT

9

CE

Table 10 Number of Pareto solutions Sample size (replication Test of solving NSGAProblem the same II problem) 1 30 67.80 2 30 38.83 3 20 39.55 4 20 66.75 5 10 76.30 6 10 93.60

AC

Size of archive

M

Test problem number

10

Mutation rate

AN US

NSGA-II

4 × 24 × 18 × 4× 4× 20

Mean

Integrative 5 3 3 1

CR IP T

DMS Decisive Flexible Hierarchic Integrative

20 20 10 10 5 5 5 5

Standard deviation

MOPSO

WeightedSum

NSGAII

MOPSO

WeightedSum

44.27 29.07 43.85 51.40 62.20 74.80

7 11 13 8 12 Out-of

2.75 0.27 0.27 2.78 0.82 4.00

0.10 1.65 4.08 4.41 1.29 4.29

0.00 0.00 0.00 0.00 0.00 Out-of

29

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7 8 9 10

5 5 5 5

100 100 100 100

88.20 100 100 100

memory

0.02 0.00 0.00 0.00

2.05 0.04 0.00 0.00

memory

Table 11 CPU time (Seconds)

MOPSO

33.33 170.43 184.22 398.39 856.73 1104.32 1381.85 1943.86 2200.08 3759.37

21.06 104.47 115.14 246.23 521.97 693.52 5994.75 1199.84 1345.31 2906.90

WeightedSum

NSGAII

12.37 252.38 723.38 2529.03 8079.15

0.06 2.57 12.85 28.53 36.50 108.99 83.73 42.50 70.22 145.52

CR IP T

30 30 20 20 10 10 5 5 5 5

NSGA-II

Standard deviation MOPSO

0.99 9.02 5.99 2.50 4.22 26.33 306.44 81.78 396.70 218.34

AN US

1 2 3 4 5 6 7 8 9 10

Mean

Out-of memory

M

Test Problem

Sample size (replication of solving the same problem)

WeightedSum

1.19 1.69 5.73 4.40 7.84

Out-of memory

AC

CE

PT

ED

Table 12 Diversity of the solutions (D) Sample Mean Standard deviation size Test (replication WeightedWeightedProblems of solving NSGA-II MOPSO NSGA-II MOPSO Sum Sum the same problem) 1 30 866.69 639.61 11.49 7.41 0.00 2660.24 2 30 839.10 1101.60 46.88 4.43 0.00 4899.27 3 20 606.78 127.46 0.00 95739.09 26968.13 2391.17 4 20 1643.02 1849.55 0.00 3.91E+05 2.29E+05 1.05E+04 5 10 6.87E+05 8.70E+05 6.01E+04 58.75 5829.72 0.00 6 10 2.76E+06 3.45E+06 7111.49 7676.86 7 5 7351.67 84220.95 1.18E+07 1.09E+07 Out-of Out-of 8 5 257549.44 327163.16 4.19E+07 3.57E+07 memory memory 9 5 1345477.83 236310.33 2.79E+08 2.07E+08 10 5 5.30E+08 5.73E+08 1912703.76 5408625.18

30

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Table 13 Uniformly distribution of the Pareto solutions (S) Mean Sample size Test (replication of Problem solving the NSGA-II MOPSO same problem) 37.53 98.29 777.37 2964.26 9333.08 2.34E+04 8.71E+04 3.52E+05 1.77E+06 6.89E+06

25.04 53.75 355.64 2275.81 8305.95 2.52E+04 7.62E+04 2.77E+05 1.89E+06 4.33E+06

10.72 27.67 92.72 275.14 1524.14

Out-of memory

Table 14 Quality of pareto optimal solutions

MOPSO

0.22 0.79 6.70 3.98 62.53 204.95 436.66 2405.77 16932.34 52885.73

0.15 0.40 0.50 12.94 35.21 121.71 480.32 1437.80 18712.21 29913.20

2

AC

4 5 6 7 8 9 10

CE

3

30 30 20 20 10 10 5 5 5 5

Q(NSGA-II, WeightedSum)

ED

1

Q(NSGAII, MOPSO) 0.4884 0.5555 0.609 0.494 0.7015 0.6623 0.5866 0.864 0.827 0.7799

PT

Test Problem

M

Mean Sample size (replication of solving the same problem)

Q(MOPSO, WeightedSum)

0 0.4828 0.4869 0.5109 0.5572

0 0.449 0.4754 0.4907 0.5014

Out-of memory

Out-of memory

31

WeightedSum

NSGA-II

0.00 0.00 0.00 0.00 0.00

CR IP T

30 30 20 20 10 10 5 5 5 5

WeightedSum

AN US

1 2 3 4 5 6 7 8 9 10

Standard deviation

Out-of memory

Standard deviation

Q(NSGAII, MOPSO) 0.02 0.02 0.05 0.13 0.10 0.18 0.04 0.16 0.04 0.20

Q(NSGA-II, WeightedSum)

Q(MOPSO, WeightedSum)

0.06 0.10 0.12 0.10 0.04

0.11 0.18 0.12 0.04 0.17

Out-of memory

Out-of memory

ACCEPTED MANUSCRIPT

Table 15 P-Values of the hypothesis tests

3

4

5

7

Number of pareto solutions

0.000

0.000

1.000

0.000

0.000

CPU Time

1.000

1.000

1.000

1.000

1.000

Diversity (D)

0.000

0.000

0.000

0.000

1.000

Standard deviation (S)

1.000

1.000

1.000

1.000

1.000

Quality of solutions

1.000

0.000

0.000

0.685

0.000

Number of pareto solutions

0.000

0.000

0.000

0.000

0.000

100%

CPU Time

1.000

0.000

0.000

0.000

0.000

80%

Diversity (D)

0.000

0.000

0.000

0.000

0.000

Standard deviation (S)

1.000

1.000

1.000

1.000

1.000

0%

Quality of solutions

1.000

0.906

0.753

0.247

0.000

20%

Number of pareto solutions

0.000

CPU Time

1.000

9

0.000

0.000

0.000

0.000

0.000

90%

1.000

0.000

1.000

0.995

1.000

10%

1.000

0.000

0.000

0.000

1.000

70%

0.000

1.000

1.000

0.000

1.000

20%

0.000

0.000

0.000

0.000

0.000

80%

AN US

M

8

CR IP T

2

Out-of memory

100%

0.000

0.000

0.000

0.000

100%

0.000

0.000

0.000

0.000

80% Out-of memory

80%

0.000

1.000

0.000

0.000

0.000

Standard deviation (S)

1.000

1.000

1.000

1.000

1.000

0%

Quality of solutions

1.000

0.984

0.899

0.925

0.486

0%

PT

Diversity (D)

6

10

1

ED

Weighted sum performs better than MOPSO

Weighted sum performs better than NSGA-II

MOPSO performs better than NSGAII

Null Test Problem hypothesis Index (H0)

Percentage of P-Values which are less than 1% (The null hypothesis is rejected at 99% confidence level)

CE

Table 16 Optimum solution when one of the objectives (OF1 or OF3) is considered Solution method

Machine constant

Machine variable

Purchasing and sale

Inter-cell movement

Intra-cell movement

GA

Value of OF1 277005

3600

22089

124600

57900

PSO

.93

282032

6220

24048

157100

GAMS

.98

260186

3473

21845

GA

.003

394256

5430

PSO

.006

437093

GAMS

.001

380362

AC

Minimized objective function

OF1

Value of OF3 1.34

OF3

Labor transfer

Reconfiguration

Machine failure

5080

Over time working 9437

40000

757

13542

20300

8760

5454

45000

827

14323

141300

34700

6480

7465

30000

567

14356

37165

185300

64600

3975

7561

70000

489

19736

7394

34805

238400

74600

8360

8649

45000

948

18937

6548

31450

195200

54720

5482

8654

60000

654

17654

32

ACCEPTED MANUSCRIPT

Table 17. Optimum solution when one of the objectives (OF1; 8th term or OF2) is considered Value of Labor transfer (OF1) 0

Value of OF2 453

PSO

0

436

GAMS

0

462

GA

85000

246

PSO GAMS

50000 75000

251 242

OF2

Table 18. Different scenarios 3

4

5

6

7

8

9

10

MTBFm

DMS

D .3 8

H .6 5

D .8 5

F .6 2

H .7 3

I .5 6

F .8 2

I .7 3

H .5 2

D .8 5

U(300,900)

Reliability

DMS

I

I

I

I

I

I

I

I

H

H

Reliability

.3 8

.6 5

.8 5

.6 2

.7 3

.5 6

.8 2

.7 3

.5 2

.8 5

DMS

F

F

F

F

I

I

I

I

D

D

.3 8

.6 5

.8 5

.6 2

.7 3

.5 6

.8 2

.7 3

.5 2

.8 5

F

H

F

D

F

D

H

I

D

D

.3 8

.6 5

.8 5

.6 2

.7 3

.5 6

.8 2

.7 3

.5 2

.8 5

H

F

D

F

D

H

I

D

D

.6 5

.8 5

.6 2

.7 3

.5 6

.8 2

.7 3

.5 2

.8 5

Reliability DMS

4

5

Reliability DMS

F

Reliability

.3 8

H

F

D

F

D

H

I

D

I

.3 8

.6 5

.8 5

.6 2

.7 3

.5 6

.8 2

.7 3

.5 2

.8 5

Reliability DMS

F

H

F

D

F

D

H

I

D

I

Reliability

.2

.2

.8

.5

.7

.5 6

.7

1

6

1

DMS

F

H

F

D

F

D

H

I

D

I

Reliability

.3

.9

1

.4

.6

.8

.5

.4

.7

.3

DMS

F

H

F

D

F

D

H

I

D

I

Reliability

1

1

1

1

1

1

1

1

1

1

H: Hierarchic

,

AC

7

F

CE

DMS 6

8

9

D: Decisive

,

F: Flexible

,

33

I: Integrative

MTTR m

BR m

U(50,90)

U(100,400 )

U(300,900)

U(50,90)

U(100,400 )

U(300,900)

U(50,90)

U(100,400 )

U(200,800)

U(40,100)

U(100,500 )

U(100,300)

U(20,40)

U(200,800 )

U(200,1000 )

U(20,3 0)

U(100,400 )

U(300,900)

U(50,90)

U(100,400 )

U(300,900)

U(50,90)

U(100,400 )

U(300,900)

U(50,90)

U(100,400 )

AN US

3

2

M

2

1

ED

1

Characteristi c

PT

labors Scenario s

CR IP T

Minimized objective function

OF1; the 8th term

Solution method GA