Improved Design of CMS by Considering Operators ...

Improved Design of CMS by Considering Operators DecisionMaking Styles A. Azadeh1, M. Rezaei-Malek, F. Evazabadian and M. Sheikhalishahi School of Industrial Engineering and Center of Excellence for Intelligent Based Experimental Mechanic, College of Engineering, University of Tehran, Iran

Abstract Cell formation is one of the oldest problems in cellular manufacturing systems including assigning parts, machines, and operators to cells. Cell manufacturing contains a number of cells where each cell is responsible for processing the family of similar parts. Another important aspect of cell formation is worker assignment to cells. Since operators work together in long periods, it is suggested to consider operators’ personal characteristics to increase their satisfaction and the productivity of system. This paper considers decision-making styles of operators (as an index of operator’s personal characteristics) and presents a new mathematical programming model for clustering parts, machines and workers simultaneously. The model includes two objectives; (1) minimization of intracellular movements and cell establishment costs, (2) minimization of decisionmaking style inconsistency among operators in each cell. The paper applies εconstraint method for solving the problem and gathering non-dominated solutions as Pareto optimal solutions. Furthermore, this paper uses common weighted MCDA-DEA method to choose the best solution from the candidate Pareto optimal solutions that have been achieved by solving the mathematical model. A real case study is investigated to show the capability of the proposed model to design cellular manufacturing system in the assembly unit. The proposed design assists decision makers to develop cellular systems with more operators’ satisfaction and productivity. Keywords: Cell formation; Mathematical Modelling; Bi-objective Optimization; DecisionMaking Styles

1. Introduction Cellular Manufacturing is a model for workplace design, and an integral part of lean manufacturing systems (Heragu, 2006). Cellular Manufacturing System (CMS)

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Corresponding author. E-mail: [email protected]

1

is based upon the principles of group technology, so that it categorizes machines and parts with regard to similarity among parts (Cell Formation Problem; CFP) to take full efficiency and flexibility through standardization and common processing. Applying these similarities ensures an efficient production system, and would save time, money, and energy (Heragu, 2006). Mathematical programming approaches widely used in CFP because of its capability to model design requirements in mathematic way (Kusiak, 1987; Shtub, 1989; Chen, 1989; Rajamani et al., 1990; Adil et al., 1993; Szwarc et al., 1997; Albadawi et al., 2005; Solimanpur and Foroughi, 2011; Mahdavi et al., 2013; Javadi et al., 2013; Paydar et al., 2014). Kusiak (1987) developed a general p-median model by considering some alternative routings. Shtub (1989) formulated the problem as an assignment problem. Chen (1989) considered CFP and developed a model and solved it by Benders decomposition technique. Rajamani et al. (1990) added budget and machine capacity constraints to the problem. Adil et al. (1993) presented a mixed integer programming model for CFP considering investment and operational costs. Albadawi et al. (2005) proposed a two-phase approach. In the first phase, machine cells are determined by using factor analysis to the matrix of similarity coefficients. Then, in the second phase, an integer programming model is applied to assign parts to the determined machine cells. Solimanpur and Foroughi (2011) developed a mathematical model for CFP considering the issues such as machine requirement, sequence of operations, alternative processing routes, processing time, production volume, budget limitation, cost of machines, etc. Mahdavi et al. (2013) presented a new integrated mathematical model that consists of cell formation and cell layout simultaneously and considers forward and backtracking movements. Javadi et al. (2013) presented a mixed integer nonlinear programming model for cell formation and layout design in CMS. Their model determined optimal cell configuration and itra and intercell layout regarding to the important operational features such as routing flexibility, operation sequence, machine capacity, etc. Paydar et al. (2014) developed a mixed integer linear programming model for integrating procurement and production planning in supply chain and design of cell formation simultaneously. Furthermore, they considered uncertainties about some critical parameters such as customer demands and machine capacities. Mohammadi and Forghani (2014) proposed an integrated approach for designing CMS and its inter- and intra-cell layouts regarding to the various production factors such as part demands, operation sequences, alternative processing routings, processing times, capacity of machines, etc. In summary, in the CFP modelling there are many issues that have been considered such as intra and intercell movement, machine investment cost, size constraint for cells and part families, demand fluctuation, and sequence of operations. Another aspect of CFP is operator assignment to cells by considering special features for operators such as skill and working ability. In the recent decades, several studies have investigated operator allocation in CMS. Russell et al. (1991) 2

examined labour resource in addition to machine resource in CMS through a number of simulation experiments. Black and Schrorer (1993) used a simulation model to study a U-shaped CMS including 13 stations that can be operated by a variable number of labour and processing time variations. Süer and Bera (1998) used a two-phase hierarchical methodology based on mathematical model to generate alternative operator levels and find the optimal assignment of operators and products to cells. Nakade and Ohno (1999) considered an optimization problem to assign multi-skill operators in a U-shaped production line while minimizing the cycle time and number of labourers which satisfies the demand. Ertay and Ruan (2005) proposed a framework based on Data Envelopment Analysis (DEA) to find the most suitable operator assignment in CMS. Their study concentrated on determining the most efficient numbers of workers and efficient labours assignment in CMS while the demand rate and transfer batch size are changing. Cesani and Steudel (2005) studied the worker flexibility in CMSs by intracell operator's mobility. Their special focus was to explore the impact of using different labour allocation strategies on system performance. Satoglu and Suresh (2009) proposed a goal programming model for hybrid CMS. Their study consists of three main phases and their aim was to minimize the total cost including training cost, hiring and firing cost over assignment of worker to cells. Mahdavi et al. (2010) proposed an integer mathematical model for CFP in dynamic environment by considering machine capacity, multi-period production planning, worker assignment, and available time of worker. The aim of their model was to minimize of the holding and backorder cost, inter and intracel;lular movement, hiring and firing and salary cost. Bashiri and Bagheri (2013) presented a new mathematical model based on extracted distances and also worker related issues, including salary, hiring, firing and cross-training. Bagheri and Bashiri (2014) developed a new mathematical model to solve the cell formation, operator assignment and intercell layout problems, simultaneously. The objectives of the proposed model were minimization of inter and intracell part trips, machine relocation cost and operator related issues. Motivation and significance Clustering of parts, machines, and operators is a complex problem in CMS (Heragu, 2006). According to the literature review, although there are several studies which incorporated operator assignment in CFP, in most of them operators have been considered as machines and humanitarian aspects are remained unattended. Since operators have to work with each other around 60 hours a week in common groups in manufacturing systems, it is highly appreciated to assign operators with compatible characteristics to the cells (Driver, 1979 and Driver et al., 1998). This prevents from possible conflicts between operators in workplace, and operators’ satisfaction will be increased (Driver, 1979 and Driver et al., 1998). Therefore,

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considering decision-making style as an index of operator’s personal characteristics can be an interesting area of study. In conclusion, there are lots of studies that coped with CFP, but regarding to our knowledge there is no study that handled the problem and operator assignment by considering humanitarian aspects. Therefore, this paper proposes a new mathematical model for CMS design problem by considering operators decision-making styles as an index of personal characteristics to consider the humanitarian aspect of the problem. 2. Decision-making styles Cellular manufacturing models are based upon the principles of group technology. Therefore, operators are categorized into different groups and they must work together. Personality characteristic plays an important role in interaction among people. In other words, if operators that work together have compatible characteristic, they can have effective interaction in workplace and their satisfaction level and work efficiency will be increased (Driver et al., 1998). One of indexes of operator’s personal characteristics is Decision-Making style (Driver et al., 1998). There are some categorizations of the decision-making styles (Driver, 1979; Driver et al., 1998; Tullier, 1999). However, the categorization presented by Driver et al. (1998) is in high agreement with the others. Driver et al. (1998) divided the decision-making styles into 5 categories named: Decisive, Flexible, Hierarchical, Integrative, and Systemic. The explanations of aforementioned decision-making styles are briefly provided as follows (Driver et al., 1998): 2.1. The decisive style Satisficing and unifocus are features of the decisive style. Decisives apply a minimum amount of information (satisficing) to immediately reach a clear solution (unifocus) for a problem. 2.2. The flexible style Satisficing and multifocus are features of the flexible style. Flexibles like decisives move fast. However, the emphasis is on adaptability here. When the solution appears not to be working, they will drop one tactic in favour of another (multifocus). 2.3. The hierarchic style Maximizing and unifocus are features of the hierarchic style. Hierarchic apply a lot of information (maximizing) to analyse a problem and accurately present a best solution.

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2.4. The integrative style Maximizing and multifocus are features of the integrative style. Integratives like hierarchics apply lots of information to analyse a problem and his/her tendency is to explore a problem for many perspectives to reach a variety of alternatives for coping with the problem. 2.5. The systemic style Systemic has two-stage decision process. First, he/she investigates a problem as an integrative (applying lots of information, considering the situation from different perspectives, presenting various alternatives). Then, he/she shifts into a more hierarchic mode and analysing the alternatives regarding to one or more criteria. Driver et al. (1998) investigated the interaction of persons with the different decision-making styles. They addressed, based on specific features of different styles, there are different situations about interaction of styles. For instance, the interaction of systemic style with decisive is terrible. This style does not permit to systemic to clearly explain his/her opinion. Driver et al. (1998) determined compatible decision-making styles especially in production areas (see Table 1).

Systemic

Integrative

Hierarchic

Flexible

Decision making styles

Decisive

Table 1. Consistency of decision-making styles

Decisive C NCNI NCNI NCNI SI Flexible NCNI SC I C I Hierarchical NCNI I C C C Integrative I C NCNI SC C Systemic SI SI I NCNI C SC; Strongly Consistent, C; Consistent, NCNI; Neither Consistent Nor Inconsistent, I; Inconsistent, SI; Strongly Inconsistent

According to the literature, while operator assignment in CFP has been considered from different points of view, the role of operators’ decision-making styles in CFPs remained unattended. Therefore, in this paper a new mathematical model is proposed considering operators’ decision-making style index for the first time to reach more efficiency and operators’ satisfaction in production areas. 3. Problem description and formulation The problem that this paper deals with is the traditional CFP with workers assignment to cells with some new considerations. In the traditional problem, there are certain parts, machines and operators which should be classified under similarity among parts as different cells. 5

The problem of this paper has different assumptions that can be broken into two categories; traditional and specific assumptions as follows: Traditional assumptions The traditional assumptions are same as assumptions of previous studies that have been considered before;  Each part needs specific operations to be done by certain machines (e.g., considered at Kusiak, (1987)),  Each operator has specific ability to work with certain machines (e.g., considered at Satoglu and Suresh, (2009)),  Each machine needs a certain amount area to place in cells (e.g., considered at Mahdavi et al., (2013)). Specific assumptions The specific assumptions are related to the newly applied concept of this paper (operator's decision-making style) and the real case study of that. Related to the decision-making style concept (Driver et al., 1998):  Each operator has certain decision-making style,  There are different levels of decision-making style inconsistency among operators that directly work together (e.g., work at same cells). Related to the case study:  Maximum allowable number of cells are certain and area capacity of each of them are known and identical (e.g., considered at Mahdavi et al., 2010; Mahdavi et al., (2013)); in the case study of this paper, the constructed assembly unit has formed based on CMS and the physical specifications of cells (e.g., number and area capacity) are certain, so the design group just need to redesign cells in terms of combination of parts, machines, and operators,  Establishing cell l has a certain fixed cost Cl (this cost consists of maintenance, equipment, charge of land). This cost difference among cells is related to physical condition of each cell and the different requirements to activate that (e.g., considered at Javadi et al., (2013)),  The only noticeable cost of the considered CMS is intracellular movement cost. The other costs are minor in comparison with that. Moreover, in the proposed case study, intracellular movement cost between two cells only depends on number of movements. This assumption is not far from the truth, since the cost difference between various scenarios due to distance is negligible (Adenso-Diaz et al., 2001). The problem is to assign parts, machines and operators to cells to minimize cost of travelling between cells, and to minimize inconsistencies of decisionmaking styles in cells.

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3.1. Sets I J K L

The set of parts, The set of machines, The set of operators, The set of cells.

3.2. Parameters aij bjk skr M Cl CM vj vv

1; if part i needs machine j for processing, 0; otherwise, 1; if operator k has the skill of working with machine j, 0; otherwise, decision-making style inconsistency of two operators k and r (k≠r), big number, cost of establishing cell l, cost of movement between two cells, amount of needed area for machine j, available area of each cell.

3.3. Variables xil yjl zkl fl

1; if part i is assigned to cell l, 0; otherwise, 1; if machine j is assigned to cell l, 0; otherwise, 1; if operator k is assigned to cell l, 0; otherwise, 1; if cell l is established, 0; otherwise.

3.4. Mathematical model min 𝐶𝑀 ∑ ∑ 𝑎𝑖𝑗 (1 − ∑ 𝑥𝑖𝑙 𝑦𝑗𝑙 ) + ∑ 𝐶𝑙 𝑓𝑙 𝑖∈𝐼 𝑗∈𝐽

𝑙∈𝐿

(1)

𝑙∈𝐿

min ∑ ∑ ∑ 𝑠𝑘𝑟 𝑧𝑘𝑙 𝑧𝑟𝑙 𝑘 ≠ 𝑟

(2)

𝑘∈𝐾 𝑟∈𝐾 𝑙∈𝐿

s.t. ∑ 𝑏𝑗𝑘 𝑧𝑘𝑙 ≥ 𝑦𝑗𝑙 ∀𝑗 ∈ 𝐽, 𝑙 ∈ 𝐿

(3)

𝑘∈𝐾

∑ 𝑣𝑗 𝑦𝑗𝑙 ≤ 𝑣𝑣

∀𝑙 ∈ 𝐿

(4)

𝑗∈𝐽

∑ 𝑦𝑗𝑙 ≤ 𝑀 ∑ 𝑥𝑖𝑙 𝑗∈𝐽

∀𝑙 ∈ 𝐿

(5)

∀𝑙 ∈ 𝐿

(6)

𝑖∈𝐼

∑ 𝑧𝑘𝑙 ≤ 𝑀 ∑ 𝑦𝑗𝑙 𝑘∈𝐾

𝑗∈𝐽

∑ 𝑥𝑖𝑙 ≤ 𝑀𝑓𝑙 ∀𝑙 ∈ 𝐿

(7)

𝑖∈𝐼

∑ 𝑥𝑖𝑙 = 1 ∀𝑖 ∈ 𝐼 𝑙∈𝐿

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(8)

∑ 𝑦𝑗𝑙 = 1 ∀𝑗 ∈ 𝐽

(9)

𝑙∈𝐿

∑ 𝑧𝑘𝑙 = 1 ∀𝑘 ∈ 𝐾

(10)

𝑙∈𝐿

(11)

𝑥, 𝑦, 𝑧, 𝑓 ∈ {0,1}

Equation (1) minimizes the cost of intracellular movement and cell establishment. Equation (2) minimizes summation of decision-making style inconsistency in the cells. Constraint (3) assures there is at least one operator (with appropriate skill) for an assigned machine in a cell. Since the cells are considered identical in this work, area constraint is shown by Equation (4). Equations (5) and (6) are feasibility constraints. Equation (5) assures that a machine is assigned to a cell if a part is allocated to the cell. In similar way, Equation (6) guarantees that an operator is assigned to a cell if a machine is allocated to the cell. Equation (7) assures that a cell is established if at least one part is assigned to it. Equations (8), (9) and (10) show that each part, machine and operator are assigned to one cell. Constraint (11) shows variable types of the model. Since the proposed model is non-linear, it takes long time to solve large size problems. Thus, the model is linearized by rewriting Equations (1) and (2). For this purpose two new variables are defined as follows: xyijl 1; if part i and machine j are assigned to cell l, 0; otherwise, ukrl 1; if operators k and r are assigned to cell l, 0; otherwise. By these new variables, Equations (12-17) are added:

𝑢𝑘𝑟𝑙

𝑥𝑦𝑖𝑗𝑙 ≥ 𝑥𝑖𝑙 + 𝑦𝑗𝑙 − 1 ∀𝑖 ∈ 𝐼. 𝑗 ∈ 𝐽. 𝑙 ∈ 𝐿 𝑥𝑦𝑖𝑗𝑙 ≤ 𝑥𝑖𝑙 ∀𝑖 ∈ 𝐼. 𝑗 ∈ 𝐽. 𝑙 ∈ 𝐿 𝑥𝑦𝑖𝑗𝑙 ≤ 𝑦𝑗𝑙 ∀𝑖 ∈ 𝐼. 𝑗 ∈ 𝐽. 𝑙 ∈ 𝐿 ≥ 𝑧𝑘𝑙 + 𝑧𝑟𝑙 − 1 ∀𝑘 ∈ 𝐾. 𝑟 ∈ 𝐾. 𝑙 ∈ 𝐿. 𝑟 ≠ 𝑘 𝑢𝑘𝑟𝑙 ≤ 𝑧𝑘𝑙 ∀𝑘 ∈ 𝐾. 𝑟 ∈ 𝐾. 𝑙 ∈ 𝐿 𝑢𝑘𝑟𝑙 ≤ 𝑧𝑟𝑙 ∀𝑘 ∈ 𝐾. 𝑟 ∈ 𝐾. 𝑙 ∈ 𝐿

(12) (13) (14) (15) (16) (17)

By adding these constraints, Equations (1) and (2) are rewritten as the follows: min 𝐶𝑀 ∑ ∑ 𝑎𝑖𝑗 (1 − ∑ 𝑥𝑦𝑖𝑗𝑙 ) + 𝐶 ∑ 𝑓𝑙 𝑖

𝑗

𝑙

min ∑ ∑ ∑ 𝑠𝑘𝑟 𝑢𝑘𝑟𝑙 𝑘

𝑟

(19)

𝑙

(20)

𝑙

4. Methodology In this paper decision-making styles are defined and the interactions between each two styles are cleared. For this purpose 5 decision making styles are considered 8

according to Drive et al. (1998), and the inconsistency between each two styles is defined (see Table 1). For quantification of Table 1, numbers {1, 3, 5, 7, and 9} are applied. For Strongly inconsistent styles as worst case number 9 is considered and for two persons with strongly consistent decision-making styles the inconsistency number is 1 (see Table 2).

Decision making styles

Decisive

Flexible

Hierarchic

Integrative

Systemic

Table 2. Quantified consistency of decision-making styles

Decisive Flexible Hierarchic Integrative Systemic

3 5 5 7 9

5 1 7 3 9

5 7 3 5 7

5 3 3 1 5

9 7 3 3 3

4.1. The ε-constraint method In the proposed model two objectives have to be optimized simultaneously. Moreover, because of existence of numerous binary variables, the presented model is categorized as a non-convex problem and the feasible region of the problem is not continuous (Mishra, 2011). Hence, ε-constraint method that is capable to deal with the non-convex problems is applied to determine Pareto optimal solutions. In addition, the following three reasons motivate this research to apply the εconstraint method (Haimes et al., 1971):  Producing efficient solutions with adjusted varieties and number by controlling grid points for each objective function;  Being consistent with different scales of objective functions;  Producing unsupported efficient solution in integer programming problems. In the ε-constraint method introduced by Haimes et al. (1971), one of the objective functions is chosen for optimization and the other objectives are converted into constraints. These constraints have upper bound (εk). Assume the following model: Max (f1(x),f2(x),…,fp(x)) s.t. xϵS,

(21)

Where x is the vector of decision variables, f1(x),f2(x),…,fp(x) are p objective functions and S is the feasible region. As mentioned above, in the ε-constraint method, one of the objective functions is optimized while the other objective 9

functions are considered as constraints, as shown below (Chankong and Haimes, 1983; Cohon, 1978): Max f1(x) s.t. f2(x) ≥ ε2, f3(x) ≥ ε3, ... fp(x) ≥ εp,

(22)

By parametrical variation in the RHSs of the constrained objective functions (εi) the efficient solution of the problem is obtained (Mavrotas, 2009). 4.2. Common weighted MCDA-DEA method The ε-constraint method determines the Pareto optimal solution so that there would be certain non-dominated solutions which have not been preferred to each other. A common weighted Multi Criteria Decision Analysis (MCDA)-DEA method is used for choosing the optimal solutions. DEA is used for performance evaluation of decision-making units (DMUs). DMUs convert multiple inputs into multiple outputs. There is a great application of DEA in evaluating performance of DMUs in many applicable industries such as hospitals, US Air Force, courts, universities, etc (Zhou et al., 2007). The DEA method calculates a measure of the relative efficiency of each DMU. This is carried out by comparing each DMU to all of the remaining ones. The problem of evaluating each DMU is formulated as a linear programming model. Evaluating the performance of n different DMUs consists of the solution of n different linear programming problems (Sofianopoulou, 2006). One of the most basic DEA models, appropriately named as the CCR model, is developed by Charnes et al. (1978). Indeed, there are n DMUs and related numerical data for each of the m inputs and s outputs for all DMUs. The fractional mathematical programming problem that is solved to gain values for the input weights (vi; i=1,…,m) and the output weights (ur; r=1,…,s) variables is as follows (See Model (23)): Max ℎ(𝑢. 𝑣) =

∑𝑟 𝑢𝑟 𝑦𝑟𝑗0 ⁄∑ 𝑣 𝑥 𝑖 𝑖 𝑖𝑗0

(23)

s.t.

∑𝑟 𝑢𝑟 𝑦𝑟𝑗 ⁄∑ 𝑣 𝑥 ≤ 1 𝑖 𝑖 𝑖𝑗 𝑢𝑟 . 𝑣𝑗 ≥ 0 Index j0 is related to the DMU being evaluated. The objective function of the model maximizes the ratio of output to input of the DMU under evaluation by 10

calculating the proper weights vi and ur. The constraint of the model assures that this ratio does not exceed 1 for each DMU. This infers that the objective function value will place between 0 and 1; the latter value means that the DMU under examination is efficient (Sofianopoulou, 2006). For considering composite indicators (CIs), Zhou et al. (2007) presented a mathematical programming approach based on CCR model constructing CIs. A CI is a mathematical aggregation of a set of individual indicators that measure multidimensional concepts but usually have no common units of measurement (Nardo et al., 2014). Hatefi and Torabi (2010) extended the model proposed by Zhou et al. (2007). Their model used common weight approach to construct CIs which have more discriminating power than the best weights that were used in Zhou et al. (2007). Since the model of present paper deals with CIs (cost sub-indicator; as objective function #1, and inconsistency sub-indicator; as objective function #2), we apply the common weighted MCDA-DEA method proposed by Hatefi and Torabi (2010) that is as follows (See Model (24)): min 𝑀

(24)

s.t.

𝑀 ≥ 𝑑𝑖 . 𝑖 = 1. 2. … . 𝑚 ∑ 𝑤𝑗 𝐼𝑖𝑗 + 𝑑𝑖 = 1. 𝑖 = 1. 2. … . 𝑚 𝑗

𝑤𝑗 ≥ 𝜀𝑗 . 𝑑𝑖 ≥ 0. 𝑖 = 1. 2. … . 𝑚. 𝑗 = 1. 2. … . 𝑛 In this model the weights (wj) and deviations (di) are decision variables, and Iij denotes the value of sub-indicator j with respect to DMUi. Finally, the DMUi with highest efficiency score (1- di) is the best solution among the existent DMUs (Hatefi and Torabi, 2010). 5. Numerical Experiment: Actual Case Study As mentioned before, in this paper 5 decision-making styles are considered. These decision-making styles are shown in Table 1. For quantification of Table 1, numbers from 1 to 9 are applied. For worst case number 9 is considered, and for two persons with same decision-making style the inconsistency is defined as zero (see Table 2). The required data of the Case study are gathered from an assembly unit at IMEN Compressed Air Company which assembles air compressor block in Iran. Assembly unit consists of 15 parts, 7 machines and 12 operators. Each part needs certain machining operations, which are shown in Table 3.

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Table 3. Part-machine matrix Machine part

1

2

3

4

5

6

7

1

0

1

1

1

1

1

0

2

1

0

0

0

1

1

1

3

1

1

1

1

1

1

0

4

1

0

0

0

0

0

0

5

0

1

1

1

0

1

0

6

0

0

0

0

0

1

0

7

1

1

1

1

0

1

1

8

0

0

0

1

0

1

1

9

1

1

0

1

1

1

1

10

1

1

0

1

1

0

0

11

1

0

0

1

0

0

1

12

0

0

0

1

1

0

1

13

1

1

0

0

0

0

0

14

1

0

0

1

0

0

1

15

1

0

0

0

1

0

1

Part machine matrix shows the machines which should be worked on each part. For example a31=1, which is highlighted, shows that part 3 needs machines 1 to be produced. Each operator has special skills and is capable of working with just some of the machines (Table 4). Table 4. Machine-operator matrix

Machine

Operator bjk

1

2

3

4

5

6

7

8

9

10

11

12

1

0

0

0

1

1

0

0

1

0

1

1

1

2

1

1

1

1

0

0

1

0

1

1

0

0

3

1

1

1

0

1

0

0

0

1

0

1

1

4

0

1

0

0

1

0

0

0

1

1

1

0

5

1

1

1

0

0

0

0

1

0

1

0

1

6

1

1

1

1

1

1

1

1

0

1

0

0

7

1

1

0

0

1

0

0

0

1

0

1

1

Machine-operator matrix demonstrates operators’ abilities to work with different machines. For example b21 shows that machine operator 1 have the ability to work on machine 2. Also, each operator has a decision-making style and there is inconsistency between operators that is shown in Table 5. Decisionmaking style of operators is inferred from questionnaires by psychologist in human resource department of IMEN Company. For more information about the structure and questions of questionnaire, and its analysis method, the readers can refer to Driver et al. (1998).

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Table 5. Decision-making style inconsistency of operators skr

1

2

3

4

5

6

7

8

9

10

11

12

1

8

5

8

9

3

4

7

7

6

5

3

5

2

5

8

5

7

6

6

0

1

9

6

1

8

3

8

7

4

4

8

3

6

4

2

2

5

4

4

5

3

2

9

9

7

5

2

7

7

5

5

5

6

9

2

7

5

3

3

7

0

5

6

5

6

5

6

6

4

1

5

2

2

0

5

5

4

7

5

5

8

8

7

8

3

0

7

3

1

7

8

2

4

1

0

3

6

9

7

1

0

7

5

9

5

6

5

4

2

8

6

0

2

4

9

5

10

4

1

5

6

4

7

4

8

4

1

4

8

11

2

7

2

4

1

3

1

8

1

1

2

6

12

3

1

5

5

5

4

4

3

8

3

5

7

The maximum allowable number of the cells is 5. The cost of establishing cell l has shown in Table 6. Table 6. Establishing cost of cell l Cell

1

2

3

4

5

Cost

20

21

22

23

24

Establishment cost shows the penalty for establishing extra cell. It can be seen from Table 6 that by increasing number of cells, establishment cost is increased. 6. Numerical Results In order to show the applicability and usefulness of the presented model and the solution method, ε-constraint is coded in general algebraic modelling system (GAMS) and integrated to the bi-objective mixed integer model to solve the proposed case study. GAMS23.5.2/CPLEX solver is capable of solving mixedinteger programming (MIP) models. A PC with 1.73 GHz seven processors and WINDOWS7 operating system is used as a technical platform. Pareto frontier is shown in Fig 1. The non-dominated solutions are as follows: YN={(z1,z2)|(420,203),(440,191),(480,151),(500,108),(565,107),(682,90),(701,90), (823,81)}

13

Z2

210 190 170 150 130 110 90 70

Z1

400

500

600

700

800

900

Figure 1. Non-dominated solutions As mentioned before, in this paper a common weighted MCDA-DEA method proposed by Hatefi and Torabi (2010), is applied. The non-dominated solutions considered as DMUs for selecting the preferred solution. The objective functions are considered as sub-indicators that their values (Iij) are shown in Table 7. Table 7. Value of sub-indicators. Sub-indicators

Non-dominated solutions/DMUs

Objective 1#

Objective #2

1 2 3 4 5 6 7 8

420 440 480 500 565 682 701 823

203 191 151 108 107 90 90 81

Deviation values (di) for each DMU are shown in Table 8. Efficiency scores would be obtained easily by subtracting deviation values from 1 (1-di). Thus the final ranking would be as 8>7>6>1>2>3>5>4. Table 8. Deviation values. DMU

1

2

3

4

5

6

7

8

Deviation value

0.136

0.143

0.19

0.264

0.201

0.121

0.102

0

The best solution for our problem achieved by the eighth scenario that the value for its cost objective function is 823 and the value for decision-making style inconsistency is 81. According to the results, the common weighted MCDA-DEA method selects the preferred Pareto frontier solution based on minimizing inconsistency goal that has important role in a production unit. To show the capability of the presented model and importance of applied decision-making style index, a sensitivity analysis is done. To do so, 7 scenarios are defined based on 14

different combinations of operators’ decision-making style. An integrated code of the ε-constraint and the common weighted MCDA-DEA method are applied to find the preferred solution for each scenario. Table 9 shows the designed scenarios, and also the preferred results for each scenario. Table 9. Different scenarios

Flexible

Hierarchical

Integrative

Systemic

Objectives

Decisive

Decision-Making style

z1

Scenario 1

12

-

-

-

-

608

90

Scenario 2

6

-

-

-

6

592

30

Scenario 3

3

3

-

3

3

629

57

Scenario 4

2

2

3

3

2

420

102

Scenario 5

-

12

-

-

-

825

22

Scenario 6

-

4

4

4

-

849

34

Scenario 7

4

2

1

1

4

445

92

Scenarios

z2

According to the results, it is obvious that by increasing one objective, the other objective is decreased. Regarding to the results, the preferred scenarios are scenarios number 2 and 5. The decision maker may select each of these scenarios as the preferred one. This would help the manager to employ personnel with compatible personality. 7. Conclusion and Future Research In this paper, CFP is considered by adding worker assignment in the cells. The workers had different decision-making styles and there was inconsistency between each two different decision-making styles. First, a bi-objective mathematical model is formulated to cluster parts, machines and workers simultaneously including two objectives; (1) minimization cost of intracellular movements and cell establishment, (2) minimization of decision-making style inconsistency. To find Pareto frontier of the model, the ε-constraint method is applied. For selecting the preferred solution of the problem, the common weighted MCDA-DEA method is used. A sensitivity analysis is also done to show the impact of different combination of decision-making styles on the preferred solution. According to the results the proposed model assigns workers to the cells so that decision-making style inconsistency value is minimized, while cost of movements and cell establishment is also considered. The proposed model of this study would help the decision makers to employ personnel with compatible personality, or to assign the current staff to the cells to minimize possible inconformity.

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