computer simulation approach for productive operators ... - Springer Link

Int J Adv Manuf Technol (2011) 56:329–343 DOI 10.1007/s00170-011-3186-9

ORIGINAL ARTICLE

A unique fuzzy multi-criteria decision making: computer simulation approach for productive operators’ assignment in cellular manufacturing systems with uncertainty and vagueness Ali Azadeh & Salman Nazari-Shirkouhi & Loghman Hatami-Shirkouhi & Ayyub Ansarinejad

Received: 24 August 2010 / Accepted: 17 January 2011 / Published online: 26 February 2011 # Springer-Verlag London Limited 2011

Abstract In today’s competitive world, manufacturing firms require to meet demand, increase quality, and decrease cost due to continuous changes in the market. Because of the importance of flexible manufacturing system, the optimum operator allocation problem in cellular manufacturing systems (CMSs) is a challenging issue. Hence, the aim of this paper is presenting a decision making approach based on Fuzzy Analytical Hierarchy Process (Fuzzy AHP), Technique for Order Performance by Similarity to Ideal Solution (TOPSIS), and computer simulation to determine the most efficient number of operators and the efficient measurement of operator assignment in CMS. Also, the proposed approach is performed by employing the number of operators, average lead time of demand, average waiting time of demand, number of completed parts, operator utilization, and average A. Azadeh (*) Department of Industrial Engineering, Center of Excellence for Intelligent Based Experimental Mechanics, University of Tehran, Tehran, Iran e-mail: [email protected] S. Nazari-Shirkouhi : A. Ansarinejad Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran S. Nazari-Shirkouhi e-mail: [email protected] A. Ansarinejad e-mail: [email protected] L. Hatami-Shirkouhi Islamic Azad University, Roudbar Branch, Roudbar, Iran e-mail: [email protected]

machine utilization as criteria for decision making. An actual case is considered and a computer simulation which considers various operators layout is developed with respect to the purpose of this study and 36 scenarios is produced. In order to find the best scenarios among 36 alternatives, a combined Fuzzy AHP and TOPSIS is employed. The Fuzzy AHP method is applied to determine the importance weight of the criteria. Finally, the TOPSIS method is utilized to rank and analyze scenarios. Also, a sensitivity analysis is carried out for validating the obtained results. Keywords Multi Criteria Decision Making (MCDM) . Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) . Computer simulation . Operator assignment . Cellular Manufacturing Systems (CMS) . Uncertainty

1 Introduction Cellular manufacturing systems (CMS) are usually designed as dual resource constraint (DRC) systems, where the number of operators is less than the total number of machines in the system [1]. The productive capacity of DRC systems is often specified by the combination of labor resources and machine. Jobs waiting to be processed may be delayed because of the non-availability of a machine or an operator or both. This subject creates the assignment of operators to machines as an important factor to determine the performance of CMSs and, therefore, the development of a multifunctional workforce a critical element in the design and operation of CM systems. Also, investigations indicate that the allocation of cross-trained operators is one

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of the most important decision points on CMS performance. The benefits of Cellular manufacturing in a series includes reduced setup and lead times, less work in process, less material handling, simplified production planning, and control. Black [2] discussed five different types of manufacturing systems including the CMS with pull systems and assembly cells. It is explained that such systems offer a high degree of flexibility if they are manned by workers who walk around cells and process parts on the machines. However, the insufficiency of traditional financial analysis and appraising measures lies on their non-stochastic nature. The conventional financial analysis methods do not appear to be suitable on their own for the evaluation of advanced manufacturing technologies investments due to the nonmonetary impacts posed by the manufacturing system [3]. Moreover, financial analysis such as net present value, return on investment, internal rate of return, etc., can lead to incorrect results in most real-world applications. Several studies have investigated operator allocation in CMS throughout the last decade. Russell et al. [4] examined labor resource in addition to the machine resource in CMS through a series of simulation experiments. Singh et al. [5] presented a bi-criterion framework for operator assignments in cellular manufacturing systems. Black and Schrorer [6] used a simulation model to study a U-shaped CMS composed of 13 stations that can be operated by a variable number of workers and great variations in processing times. Süer and Bera [7] used a two-phase hierarchical methodology based on mathematical models for generating alternative operator levels, finding the optimal operator and product assignments to the cells. Nakade and Ohno [8] considered an optimization problem for finding an allocation of operators at a U-shaped production line with multifunction operators to minimize the cycle time and number of workers which satisfies the demand. Ertay and Ruan [9] also proposed a framework based on data envelopment analysis (DEA) for the most suitable operator allocation in CMS. Their study concentrated on determining the most efficient number of operators and the efficient measurement of labor assignment in CMS when the demand rate and transfer batch size as a rate of the batch size change. Cesani and Steudel [10] studied the labor flexibility in CMSs characterized by intra-cell operator's mobility. Their special focus of the investigation was to explore the impact that using different labor allocation strategies have on system performance. Most researchers have proposed using MCDM technique both in the crisp and fuzzy environments. MCDM has also become a popular and common tool in the literature especially in problems with conflicting objectives [11]. Shih [12] evaluated the performances of alternatives by applying the group TOPSIS model in a case of robot selection. Wabalackis [13] developed a justified methodology based on the AHP to

Int J Adv Manuf Technol (2011) 56:329–343

evaluate many tangible and intangible benefits of an investment on flexible manufacturing system (FMS). Bayazit [14] applied the AHP approach for selecting a FMS. Karsak and Kuzgunkaya [15] presented a fuzzy approach with multiobjective programming for selection of FMS. Ayag and Ozdemir [16] presented AHP for machine tool selection problem due to the fact that it has been widely used in evaluating various types of MCDM problems in both academic research and practices. Also, Onut et al. [17] integrated TOPSIS and AHP method in fuzzy environment for the evaluation and selection of vertical CNC machining centers for a manufacturing company in Istanbul, Turkey. Azadeh and Anvari [18] presented a decision making model based on DEA, principle component analysis (PCA), and numerical taxonomy for optimization of operators’ allocation in CMS by computer simulation. Azadeh et al. [19] integrated fuzzy data envelopment analysis, fuzzy C-means and computer simulation for optimizing operator allocation in CMS. Azadeh et al. [1] presented a decision making approach based on hybrid Genetic Algorithm and TOPSIS for determining the most efficient number of operators and efficient measurement of operator assignment in CMS. Generally, after reviewing the literature, it could be stated that AHP and TOPSIS are not integrated in uncertain and vague environments. This study illustrates the use of MCDM methods to identify optimal scenario selection in a simulation model. It is further explained as how to identify optimum operator layout with respect to cellular condition. An integrated method is proposed by obtaining data from simulation and utilizing Fuzzy AHP and TOPSIS to solve this problem. The proposed method is applied to an operator allocation problem in a CMS. The case study considers a multilevel, multiproduct, and operator-constrained production environment. Its objectives are: (1) to increase average operators and machines utilization, (2) to decrease the lead time of the demands in CMS, (3) to increase annual production rate, (4) to decrease the average wait time of demands, and (5) to decrease the number of operators and also costs. The cells described in the case study are designed for flexibility not line balancing. The remaining parts of this paper are structured as follows: Section 2 describes the proposed methodology. In Section 3, an empirical illustration is presented and discussed. Section 4 discusses the results of proposed methodology. Also, a complete sensitivity analysis is presented. Finally, Section 5 presents concluding remarks.

2 The proposed methodology In this section, we describe our proposed methodology to assigning the optimal CMS scenarios by simulation,


Fuzzy AHP, and TOPSIS. In order to determine the optimal CMS scenarios by simulation, the following procedure is devised: 1. Define the problem and identify variables. 2. Simulate, verify, and validate the CMS being studied. 3. Determine scenarios as problem solutions. Data related to problem objectives would be extracted from simulation with respect to selected scenarios. 4. Apply the Fuzzy AHP method to determine the importance weight of the criteria to be used in fuzzy TOPSIS. 5. Utilize the TOPSIS method to rank and analysis of scenarios. 6. Make the decision and assign optimum labor. The proposed methodology to determine the optimal CMS scenarios is depicted in Fig. 1. The superiority and advantages of this integrated approach is shown through a comparative study composed by TOPSIS, DEA, PCA, and the recent studies in the field of operator allocation which is the case study of this paper. Table 1 shows the comparison of the proposed method with previous studies. Several important features are considered for comparison. These features comprise of multiple inputs, multiple outputs, having pairwise comparison between indicators or criteria, use of experts’ judgment in decision making, ranking and experimentation capability, treating uncertainty and ambiguities, multi products modeling, and flexibility on the basis of assigning weights. All methods

Define the problem and identify variables

Simulate, verify and validate the CMS being studied

Determine scenarios as problem solutions

Apply the Fuzzy AHP method (to determine the importance weight of the criteria to be used in fuzzy TOPSIS) Utilize the TOPSIS method to rank and analysis of scenarios

Make the decision and assign Optimum labor Fig. 1 Steps of the Fuzzy AHP-TOPSIS-Computer Simulation Approach in Uncertain and Vague Environment

331

can solve problems with multiple inputs and outputs, but the proposed approach can locate the best alternative by using use of experts’ judgment in decision making and pairwise comparison between indicators in fuzzy environment. Also, all of the methods have ranking capability. It is notable that the proposed methodology in this paper covers all of the abovementioned features for assigning the optimal CMS scenarios. According to Table 1, this paper presents a unique framework which has several advantages over the previous papers. Also, this paper presents a simple approach in the complex situations to overcome this problem in the fuzzy setting. 2.1 Fuzzy AHP Saaty [20] defined the AHP as a decision method that decomposes a complex multi-criteria decision problem into a hierarchy. Traditional methods of AHP can be useless when uncertainty is observed in the data of problems. To address such uncertainties, Zadeh [21] for the first time introduced and then used fuzzy sets theory. Because the real world is actually full of ambiguities or in one word is fuzzy, several researches have combined fuzzy theory with AHP. The first method of implementing Fuzzy AHP was proposed by Van Laarhoven and Pedrycz [22] in which triangular fuzzy numbers were compared according to their membership functions. Despite many weight calculation methods for TOPSIS (such as Entropy method, weighted least square method, and linear programming for multi-dimensions of analysis preference), use of Fuzzy AHP method has several benefits. Fuzzy AHP is based on a principle of pairwise comparison. It is a systematic approach to decision problems. Finally, it can be widely applied to decision problems in uncertain and vague environments. In this paper, triangular fuzzy numbers are used as the membership function, which is illustrated in Fig. 2. Triangular fuzzy numbers are used, since they help the decision maker make easier decisions [23]. Membership function of a triangular fuzzy number can be found in equation (1) and is usually notated by the triplet (l, m, u). 8 xl > > l x m > > < ml ux U ðxÞ ¼ ð1Þ m x u > > u m > > : 0 Otherwise The AHP method proposed by Saaty [20] uses pairwise comparisons shown in equation (2). Number aij shows the relative importance of criterion i (ci) in comparison with criterion j (cj) in Saaty’s scale [20].

332


Table 1 Comparison of the proposed method with previous studies Features\Models

Multiple inputs

Multiple outputs

TOPSIS Nakade and Ohno (1999) [8] PCA DEA Azadeh and Anvari [18] Azadeh et al. [1] Azadeh et al. [19] The Integrated Fuzzy AHPTOPSISSimulation

√ √

√ √

√ √

√ √ √

√ √ √

√ √ √

√

√ √ √

√ √ √

√ √ √

√ √ √

c1

2

c1 1

6 c2 6 1=a21 A ¼ aij ¼ . 6 . .. 6 4 .. cn 1=a1n

Pair-wise comparison between indicators

√

c2 a12

1 .. .

.. .

1=a2n

Use of experts’ judgment in decision making

√

cn 3 a1n a2n 7 7 7 .. 7 . 5

1 aji

2 ½1 = 9; 9

ð3Þ

lij ¼ min Bijk

ð4Þ

√ √

√

ð5Þ

1

ð6Þ

In which Bijk shows relative importance of criteria ci and cj given by expert k. The linguistic scale and corresponding triangular fuzzy numbers are illustrated in Table 2 based on Saaty’s scale [20]. The parameter θ is considered equal to 1 in this paper. The fuzzy matrix Ã in equation (7) will be used in the remaining steps of Fuzzy AHP. The number ãij is a triangular fuzzy number that represents the relative importance of criteria ci and cj based on experts’ judgments according to equations (3)–(6):

U ~f ( x )

C1

2

1 6 6 ~ ~ C2 6 1= a12 A ¼ ~aij ¼ . 6 .. 6 .. 4 . C1

Cn

U ~f ( x ) =

Flexibility on the basis of assigning weights

uij ¼ max Bijk

1

: 8i 6¼ j

x−l m−l

Multi products modeling

√

ð2Þ

To analyze the data and achieve the consensus of the experts, eigenvector method proposed by Buckley [24] is used here. As mentioned before, triangular fuzzy number can be represented by the triplet (l, m, u). Triangular fuzzy ~ number U ij is constructed as the following: ~ U ij ¼ lij ; mij ; uij : lij mij uij ; lij ; mij ; uij

U ~f ( x ) =

Treating uncertainty and ambiguities

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi n Y Bijk mij ¼ n

Where aij ¼ 1 : 8i ¼ j; aij ¼

Ranking and experimentation capability

1= ~a1n

C2 ~a

... ... .. .

12

1 .. . 1= ~a

2n

...

Cn ~a 3 1n 7 ~a 7 2n 7 .. 7 7 . 5

ð7Þ

1

There are different methods to defuzzify fuzzy numbers. In this paper, the method proposed by Liou and Wang [25] is used to defuzzify fuzzy matrix Ã into crisp matrix gα,β as shown in equations (8) and (9): ~ g a ¼ b:f l þ ð1 bÞ:f u ; 0 a; b 1

u−x u−m

α

a;b

ij

a

ij

a

ij

ð8Þ fα(l) =(m −l)α +l

fα (u) = u−(u−m)α

Fig. 2 Left and right representation of triangular fuzzy numbers

x

ga;b ~aij ¼ 1 = ga;b ~aji ; 0 a; b 1 : i > j

ð9Þ

Int J Adv Manuf Technol (2011) 56:329–343 Table 2 The linguistic scale and corresponding triangular fuzzy numbers

333

Fuzzy number

Linguistic scales

Scale of fuzzy number

1 ~ 3 ~ 5 ~ 7 ~ 9 ~~~~ 2;4;6;8 1=~ x

Equally important Weakly important Essentially important Very strongly important Absolutely important Intermediate values Between two adjacent judgments

(1, 1, 1) (3−θ, 3, 3+θ) (5−θ, 5, 5+θ) (7−θ, 7, 7+θ) (8, 9, 9) (x−θ, x, x+θ) (1/(x+θ), 1/x, 1/ (x−θ))

Because of presenting explicitly preferences (α) and risk tolerance (β) of the decision maker, decision makers can more thoroughly understand the risk they face in different circumstances [26]. The single pairwise comparison matrix is expressed in (10). 2

C1

C2

C1 a12 Þ C2 6 6 1=ga;b ðe e ¼ ga;b e aij ¼ . 6 ga;b A . 6 .. 4 .. Cn

1

a1n Þ 1=ga;b ðe

...

a12 Þ ga;b ðe

...

1 .. .

... .. .

a2n Þ 1=ga;b ðe

...

Cn 3 a1n Þ ga;b ðe a2n Þ 7 ga;b ðe 7 7 .. 7 5 .

The following characteristics of the TOPSIS method make it an appropriate approach which has good potential for Productive Operators’ Assignment CMSs problem: & &

1

ð10Þ To determine the consistency of the matrix, Saaty [20] suggests consistency index (C.I.) and consistency rate (C. R.). Values of C.I. and C.R. can be calculated from equations (11) and (12): lmax n C:I: ¼ n1

C:R: ¼

C:I: R:I:

&

ð11Þ

ð12Þ

Where 1max is the eigenvalue of the matrix gα,β.

&

2.2 TOPSIS TOPSIS, which is based on choosing the best alternative having the shortest distance to the positive ideal solution and the farthest distance from the negative-ideal solution, was first proposed by Hwang and Yoon [27]. The positive ideal solution is the solution that maximizes the benefit and also minimizes the total cost. On the contrary, the negative-ideal solution is the solution that minimizes the benefit and also maximizes the total cost. This approach has been widely employed for four reasons: (1) TOPSIS logic is rational and understandable; (2) the computation processes are straightforward; (3) the concept permits the pursuit of best alternatives for each criterion depicted in a simple mathematical form; and (4) the importance weights are incorporated into the comparison procedures [28,29].

& &

An unlimited range of cell properties and performance attributes can be included. In the context of operator assignment, the effect of each attribute cannot be considered alone and must always be seen as a trade-off with respect to other attributes. Any change in, for instance, the number of demand, lead time and operator utilization indices can change the decision priorities for other parameters. In light of this, the TOPSIS model seems to be a suitable method for multi-criteria operator assignment problems as it allows explicit tradeoffs and interactions among attributes. More precisely, changes in one attribute can be compensated for in a direct or opposite manner by other attributes. The output can be a preferential ranking of the alternatives (scenarios) with a numerical value that provides a better understanding of differences and similarities between alternatives, whereas other MADM techniques (such as the ELECTRE method) only determine the rank of each scenario. Pairwise comparisons, required by methods such as the AHP are avoided. This is particularly useful when dealing with a large number of alternatives; the methods are completely suitable for linking with computer databases dealing with scenario selection. It can include a set of weighting coefficients for different attributes which in this paper AHP in fuzzy environment is suggested. It is relatively simple and fast, with a systematic procedure.

The TOPSIS solution method consists of the following steps: 1. Normalize the decision matrix. The normalization of the decision matrix is carried of using the following transformation: rij nij ¼ sffiffiffiffiffiffiffiffiffiffi ; i ¼ 1; 2; :::; n; j ¼ 1; 2; :::; m; m P rij2 i¼1

ð13Þ

334


Where n is the number of scenarios (Alternatives), m is the number of criteria, and rij is an element of the decision matrix. 2. Multiply the columns of the normalized decision matrix by the associated weights obtained from Fuzzy AHP. The weighted and normalized decision matrix is obtained as:

Vij ¼ nij :Wj ; i ¼ 1; 2; :::; n; j ¼ 1; 2; :::; m;

ð14Þ

Where Wj represents the weight of the jth criteria obtained from Fuzzy AHP in this paper and V is weighted normalized decision matrix. 3. Determine the positive ideal and negative ideal solutions. The positive ideal and the negative ideal value sets are determined, respectively, as follows: Vþ ¼

ðmax Vij j 2 KÞ; min Vij jj 2 K 0 Þji ¼ 1; 2; :::; n; j ¼ 1; 2; :::; m i

i

ð15Þ

V ¼

ðmin Vij j 2 KÞ; max Vij jj 2 K 0 Þji ¼ 1; 2; :::; n; j ¼ 1; 2; :::; m i

i

ð16Þ Where K depends on the index set of benefit criteria and k 0 the index set of cost criteria. 4. Measure distances from the positive ideal and negative solutions. The two Euclidean distances for each alternative are, respectively, calculated as:

The higher the closeness shows the better the rank.

3 Empirical illustration Manned cells are a very flexible system that can adapt to changes in the customer’s demand or changes quite easily and rapidly in the product design. The cells described in this study are designed for flexibility, not line balancing. The walking multi-functional operators permit rapid rebalancing in a U-shaped. The considered cell has eight stations and can be operated by one or more operators, depending on the required output for the cell. The times for the operations at the stations do not have to be balanced. The balance is achieved by having the operators walk from station to station. The sum of operation times for each operator is approximately equal. In other words, any division of stations that achieves balance between the operators is acceptable. Operators perform scenario movements in cell. Once a production batch size arrives to the cell, it is divided to transfer batch sizes. Transfer batch size is the transfer quantity of intra-cell movements of parts. The ability to quickly rebalance the cell to obtain changes in the output of the cell can be demonstrated by the developed simulation model. The existing manned cell example for a case model is presented in Fig. 3 [19]. The proposed integrated approach is applied to an actual CMS. Each step of the integrated approach is applied and discussed for the CMS. The Fuzzy AHP and TOPSIS methods and simulation process focus on formulating and optimizing operator allocation of the CMS.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u n Siþ ¼ t Vij Vjþ ; i ¼ 1; 2; :::; n; j ¼ 1; 2; :::; m j¼1

ð17Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u n Vij Vj ; i ¼ 1; 2; :::; n; j ¼ 1; 2; :::; m Si ¼ t j¼1

ð18Þ

5. Calculate the relative closeness to the positive ideal » solution (Ci ). The relative closeness to the positive ideal solution can be defined as:

D: Decoupler Station: Operator Operator movement when out of work

S » » Ci ¼ þ i ; i ¼ 1; 2; :::; n; 0 Ci 1 Si þ Si

ð19Þ

Operator movement with parts Fig. 3 The existing manned cell example for the case model


3.1 Defining the problem and identifying variables The system being studied is a U-shaped CMS with eight machines. The walking multifunctional operators in the Ushaped CMS permit rapid rebalancing and can be operated by one or more operators. In this cell, operators are used for transferring parts between machines, as well as performing loading/unloading activities. Lead time of demands, waiting times of demands, and the number of the operators (in a working day) are defined as the index set of cost criteria and operator/machine utilization and annual number of completed parts are defined as index set of benefit criteria [30]. 3.2 Data generation by simulation A flexible simulation model is built by Visual SLAM [31] which incorporates all scenarios for quick responses and results (Fig. 4). The results of the simulation experiments are used to compare the efficiency of the alternatives. Furthermore, the ability to quickly rebalance the cell to obtain changes in output of the cell can be demonstrated by the simulation model. The eight operators are defined by the RESOURCE BLOCKS OP1 to OP8.

Fig. 4 Visual SLAM model for operator assignment in CMS

335

They are modeled in a flexible manner and are used upon the request by the scenario. A scenario may use only two while the other scenario may use four. The eight machines are defined by the RESOURCE BLOCKS M1 to M8. The task performed (by operators on machines) is defined by Task=LTRIB [1]. Initially, it is set to 1 and the integrated simulation environment of Visual SLAM makes sure that all eight tasks are performed in the production line. This is done by identifying a condition in Activity number 11 and the following ASSIGN node of the network which increments the Task by 1. Simulation model of operators’ allocation in CMS is shown in Fig. 4. The type of demands (three types) and products (two types) are specified by two distinct arrays at the beginning of the network after the CREATE node. Furthermore, ATRIB [2] is set to demandkind and demandkind=DPROB (4, 5) and ATRIB [3] is set to parttype and parttype=DPROB (7, 8). Moreover, the DPROB (4, 5) selects the type of demand based on the assigned probabilities. In addition, DPROB (7, 8) selects the type of products and their assigned processing times by the assigned probabilities for product 1 and 2, respectively. Azadeh et al. [1] defines the number of jobs performed by the operators. This is done by incrementing it by one in an ASSIGN node right after

336


ACTIVITY number 11. It is further accumulated and computed in COLCT node with the label of “total performed work.” The cycle time (or time in system) for each demand is stored in another COLCT node with the command of TNOW-ATRIB [1]. This COLCT node is labeled as demandkind. The classification and sorting of demand types are achieved by a BATCH node before this

COLCT node. The setup time between tasks is shown in ACTIVITY number 9 with UNFRM (0.2, 0.5). This would also contain the setup time for each product. The process time for each product is exponentially distributed with expon ([array, parttype, task]). However, when all eight tasks on machines are performed by the operators, it is shown as expon ([array, parttype, 8]). This is shown in

Table 3 Simulation results for the case model (decision matrix) C1 Number of operators

C2 Average lead time of demand

C3 Average waiting time of demand

C4 Number of completed parts

C5 Operator utilization

C6 Average machine utilization

Sen1-1 Sen1-2 Sen1-3 Sen2-1 Sen2-2 Sen2-3 Sen3-1 Sen3-2 Sen3-3 Sen4-1 Sen4-2 Sen4-3

8 16 24 7 14 21 6 12 18 5 10 15

303.41 251.75 213.07 269.95 232.12 220.35 267.90 231.43 213.70 305.55 245.62 227.37

136.35 110.21 118.69 118.80 88.63 95.50 101.12 86.08 97.71 239.57 105.61 82.83

675 815 740 585 655 870 595 740 690 565 885 655

20.19 7.65 10.80 15.77 10.20 4.03 28.90 28.38 10.00 50.85 16.71 6.81

43.59 16.08 22.68 28.55 18.54 7.06 43.13 43.91 15.86 65.20 21.98 8.96

Sen5-1 Sen5-2 Sen5-3 Sen6-1 Sen6-2 Sen6-3 Sen7-1 Sen7-2 Sen7-3 Sen8-1 Sen8-2 Sen8-3 Sen9-1 Sen9-2 Sen9-3 Sen10-1 Sen10-2 Sen10-3

4 8 12 6 12 18 4 8 12 3 6 9 3 6 9 3 6 9

281.89 255.15 230.73 284.90 247.50 225.47 333.03 261.38 242.44 329.63 258.76 246.91 389.24 343.72 305.07 337.32 299.19 263.20

115.80 101.38 85.77 87.99 116.91 103.73 129.15 97.55 102.74 100.47 107.42 201.60 152.93 165.53 152.91 114.61 238.46 204.81

515 710 635 615 740 645 545 840 800 570 665 685 445 740 670 530 850 720

38.55 24.38 7.99 21.73 19.98 11.48 35.89 20.51 13.99 40.65 23.60 20.85 60.25 47.70 25.35 53.45 32.20 23.45

42.56 28.86 8.61 34.14 31.90 18.24 42.05 23.40 15.99 36.83 20.99 18.43 59.75 46.86 25.00 46.60 28.54 21.23

Sen11-1 Sen11-2 Sen11-3 sen12-1 sen12-2 sen12-3

3 6 9 2 4 6

363.18 276.19 261.57 361.91 329.36 291.53

196.82 110.25 107.71 185.12 248.24 106.71

535 615 680 505 875 660

60.20 29.15 23.25 64.88 74.33 31.13

55.70 26.61 21.95 49.63 54.90 22.15

Scenarios


337

ACTIVITY number 8 and by the two branching conditions after the FREE node which is right after ACTIVITY number 11. The proposed model considers different demand levels and part types. System performance is monitored for different workforce levels and shifts by means of simulation. In simulation experiments, each of the received demand’s parts has a special type and level. There are two part types and three demand levels. The processing times of jobs for each machine activity are related to the part type. The objective of the scenarios is to reduce the number of operators in the cell and also to observe how the operations are redistributed among the operators. 3.3 Determining scenarios Twelve scenarios (alternatives) are considered for the purpose of this study as follows: 1. Eight operators with one operator dedicated to one machine each 2. Seven operators with one operator dedicated to two machines and six operators dedicated to one machine each 3. Six operators with two operators dedicated to two machines each and four operators dedicated to one machine each 4. Five operators (one by one operator to two by two machines, one operator for each of the others) 5. Four operators (one by one operator to two by two machines) 6. Six operators with one operator dedicated to three machines and five operators dedicated to one machine each 7. Four operators with two operator dedicated to three machines each and two operators dedicated to one machine each 8. Three operators with two operators dedicated to three machines each and one operator dedicated to two machines

9. Five operators with one operator dedicated to four machines and four operator dedicated to one machine each 10. Three operators with one operator dedicated to four machines and two operators dedicated two machines each 11. Three operators with one operator dedicated to four machines, one operator dedicated to three machines and one operator dedicated to one machine 12. Two operators with two operators dedicated to four machines each As mentioned, outputs collected from simulation model are the average lead time of demand, the average of waiting time of demand, average operator and machine utilization, and number of completed parts per annum. The results of the simulation experiment are used to compare the efficiency of the alternatives. Each labor assignment scenario considers three shifts with one, two, or three shifts per day. Moreover, a flexible simulation model is built by Visual SLAM (See Fig. 4) which incorporates all 36 scenarios for quick response and results. Furthermore, the ability to quickly rebalance the cell to obtain changes in output of cell can be demonstrated by simulation model. Hence, in the developed model, different demand levels and part types have taken into consideration. The Table 3 shows Output of the simulation for considered actual case in this paper. 3.4 Applying fuzzy AHP After running the simulation model for each scenario, now Fuzzy AHP method is applied to determine the importance weight of the criteria to be used in fuzzy TOPSIS. According to equations (2)–(6), based on Table 4, the fuzzy decision matrix for the required criteria to assign optimum operator in CMS is attained from a verbal questionnaire filled by eight different experts and then converted to fuzzy numbers based on Saaty’s scale [20]. In this paper, α and β are considered equal to 0.5.

Table 4 Aggregated fuzzy pair-wise comparison of criteria

C1 C2 C3 C4 C5 C6

C1

C2

C3

C4

C5

C6

(1, 1, 1)

(2, 3.448, 7) (1, 1, 1)

(0.333, 0.843, 2) (0.333, 0.653, 1)

(2, 4.888, 9) (0.125, 1.235, 2)

(0.125, 0.484, 1) (0.167, 0.563, 2)

(0.897, 1.328, 2) (0.111, 0.654, 1)

(1, 1, 1)

(2, 5.873, 8) (1, 1, 1)

(0.111, 0.834, 2) (0.111, 0.483, 1) (1, 1, 1)

(0.25, 1.675, 3) (0.125, 0.247, 1) (0.5, 1.235, 2) (1, 1, 1)

338


Table 5 Final defuzzified matrix of criteria

C1 C2 C3 C4 C5 C6

C1

C2

C3

C4

C5

C6

1 0.2516 0.9951 0.1925 1.9117 0.7204

3.9738 1 1.5153 0.8706 1.2145 1.6545

1.0049 0.6599 1 0.1839 1.0554 0.606

5.1938 1.1486 5.4366 1 1.9275 2.4716

0.5231 0.8234 0.9475 0.5188 1 0.8049

1.3882 0.6044 1.6502 0.4046 1.2424 1

Selecting α=0.5 indicates that environmental uncertainty is steady; additionally β = 0.5 indicates that a future attitude would be fair. After building the fuzzy matrix, the matrix of Table 4 should be defuzzified. The matrix can be defuzzified based on equation (8) and (9) as an example: fo:5 ðl12 Þ ¼ ð3:448 2Þ 0:5 þ 2 ¼ 2:724 fo:5 ðu12 Þ ¼ 7 ð7 3:448Þ 0:5 ¼ 5:224

Table 6 Weighted normalized decision matrix

Scenarios

C1

C2

C3

C4

C5

C6

Sen1-1 Sen1-2 Sen1-3 Sen2-1 Sen2-2

0.030825 0.061649 0.092474 0.026971 0.053943

0.018497 0.015348 0.01299 0.016457 0.014151

0.03674 0.029697 0.031982 0.032011 0.023882

0.010754577 0.012985156 0.011790203 0.009320633 0.010435923

0.02148 0.008139 0.01149 0.016777 0.010852

0.032018 0.011811 0.016659 0.020971 0.013618


0.080914 0.023118 0.046237 0.069355 0.019265 0.038531 0.057796 0.015412 0.030825 0.046237 0.023118 0.046237 0.069355 0.015412 0.030825 0.046237 0.011559 0.023118

0.013433 0.016332 0.014109 0.013028 0.018627 0.014974 0.013861 0.017185 0.015555 0.014066 0.017369 0.015089 0.013745 0.020303 0.015935 0.01478 0.020095 0.015775

0.025733 0.027247 0.023195 0.026328 0.064553 0.028457 0.022319 0.031203 0.027317 0.023111 0.023709 0.031502 0.027951 0.0348 0.026285 0.027684 0.027072 0.028945

0.013861455 0.009479961 0.011790203 0.010993568 0.009001979 0.014100446 0.010435923 0.008205344 0.011312222 0.010117269 0.009798615 0.011790203 0.010276596 0.008683325 0.013383474 0.012746165 0.009081643 0.01059525

0.004287 0.030746 0.030193 0.010639 0.054098 0.017777 0.007245 0.041012 0.025937 0.0085 0.023118 0.021256 0.012213 0.038183 0.02182 0.014884 0.043247 0.025108

0.005186 0.03168 0.032253 0.01165 0.047891 0.016145 0.006581 0.031261 0.021198 0.006324 0.025077 0.023431 0.013398 0.030887 0.017188 0.011745 0.027053 0.015418

Sen8-3 Sen9-1 Sen9-2 Sen9-3 Sen10-1 Sen10-2 Sen10-3 Sen11-1 Sen11-2 Sen11-3 Sen12-1 Sen12-2 Sen12-3

0.034678 0.011559 0.023118 0.034678 0.011559 0.023118 0.034678 0.011559 0.023118 0.034678 0.007706 0.015412 0.023118

0.015053 0.02373 0.020954 0.018598 0.020564 0.01824 0.016046 0.022141 0.016838 0.015946 0.022063 0.020079 0.017773

0.054322 0.041208 0.044603 0.041202 0.030882 0.064254 0.055187 0.053034 0.029707 0.029023 0.049882 0.06689 0.028754

0.010913904 0.007090055 0.011790203 0.010674914 0.008444335 0.013542801 0.011471549 0.008523998 0.009798615 0.010834241 0.008046017 0.013941118 0.010515586

0.022182 0.064099 0.050747 0.026969 0.056864 0.034257 0.024948 0.064045 0.031012 0.024735 0.069024 0.079078 0.033119

0.013537 0.043888 0.03442 0.018363 0.034229 0.020963 0.015594 0.040913 0.019546 0.016123 0.036455 0.040325 0.01627

Int J Adv Manuf Technol (2011) 56:329–343 Table 7 Positive Ideal solutions and negative ideal solutions

Vþ V

339 C1

C2

C3

C4

C5

C6

0.007706 0.092474

0.01299 0.02373

0.022319 0.06689

0.014100446 0.007090055

0.079078 0.004287

0.047891 0.005186

go:5;0:5 ða12 Þ ¼ ½0:5 2:724 þ ð1 0:5Þ 5:224 ¼ 3:974

Final results of the proposed methodology for an actual case are shown in Table 8. According to the obtained

And finally: go:5;0:5 ða21 Þ ¼ 1 = go:5;0:5 ða12 Þ ¼

4 Results and discussion

1 ¼ 0:2516 3:974

The final defuzzified matrix is shown in Table 5. Therefore, the importance weight of the criteria (W) to be used in TOPSIS will be: Wj ¼ ½0:2421; 0:103; 0:2249; 0:0653; 0:2128; 0:1519 As a result of the abovementioned calculations, the weights of six considered criteria for assigning the optimum operator in CMS, i.e., number of operators, average lead time of demand, average waiting time of demand, number of completed parts, operator utilization, and average machine utilization are 0.2421, 0.103, 0.2249, 0.0653, 0.2128, and 0.1519. Then, C.I. and C. R. can be calculated using equations (11) and (12) and are 0.0710 and 0.0568. The results show that the matrix shown in Table 4 for the optimum operator assignment in CMS is consistent. 3.5 Implementation of TOPSIS After calculating the weight of criteria using Fuzzy AHP, here, TOPSIS method can be applied for the rank and analysis of scenarios. For the first step of this method, the decision matrix as mentioned before in Table 3, representing the performance values of each alternative with respect to each criterion is computed by simulation model. Afterwards, these performance values are normalized by equation (13). The normalized matrix is multiplied by the criteria weights calculated by Fuzzy AHP method which is shown in Table 6 and is called weighted decision matrix. Using equations (15)–(17), the ideal and negative ideal solutions according to the weighted decision matrix can be obtained which is shown in Table 7. The positive ideal solution consists of taking the best values of alternatives and with the similar principle; the negative ideal solution consists of taking the worst values of alternatives. To measure distances from the ideal and negative solutions equations (17) and (18) can be used.

Table 8 Overall Scores and ranking of alternatives »

Scenarios

Siþ

Si

Ci


0.065981 0.096464 0.11326 0.071457 0.089356 0.113087 0.053788 0.064236 0.099115 0.050982 0.075877 0.0969 0.043774 0.064111 0.090616 0.062683 0.074299 0.09751

0.075934 0.049972 0.039237 0.077266 0.059594 0.04449 0.088567 0.074602 0.048903 0.098486 0.069389 0.057518 0.096357 0.07859 0.064622 0.086445 0.064092 0.047862

0.535066 0.341256 0.257295 0.519531 0.400092 0.282337 0.622157 0.537331 0.330388 0.658911 0.477666 0.372483 0.68762 0.550731 0.416276 0.579669 0.463121 0.329239

18 32 36 21 30 35 11 17 33 10 24 31 7 16 29 14 25 34


0.047535 0.069141 0.083342 0.042793 0.065336 0.078635 0.027866 0.042273 0.068662 0.029286 0.068978 0.076139 0.036712 0.058645 0.068951 0.033325 0.046405 0.058539

0.093765 0.077467 0.062788 0.100726 0.082847 0.063073 0.110824 0.091377 0.068757 0.107067 0.077691 0.063959 0.107662 0.084663 0.073393 0.112459 0.113257 0.08524

0.663588 0.528394 0.429672 0.701829 0.559086 0.445091 0.799075 0.683703 0.500345 0.785221 0.529703 0.456529 0.745715 0.590778 0.515606 0.771409 0.709353 0.592852

9 20 28 6 15 27 1 8 23 2 19 26 4 13 22 3 5 12

Ranking

340


results, scenario 9-1 (one operator to four machines and one operator for each of others with one shifts per day) is the most efficient. The second best scenario is the scenario 10-1 which is similar to scenario 9-1 except in assigning one by one operator to two by two machines. The third best scenario is 12-1 with two operators (one by one operator to four by four machines with one shift per day). Also, a sensitivity analysis is done for validating the obtained results. For the sensitivity analysis, the calculated

weights using Fuzzy AHP are replaced for two criteria while the other weights are constant. The aim is to » determine Ci values for alternatives in each action to observe new ranking. The various names are specified for each action. For example, C12 means the weights of Criteria 1 (C1) and criteria 2 (C2) are replaced while the other weights of criteria are constant and C24 means the weights of Criteria 2 (C2) and criteria 4 (C4) are replaced while the other weights of criteria are constant. Fourteen mutual

»

Table 9 Results of sensitivity analysis for Ci C12

C14

C16

C24

C26

C31

C35

C43

C45

C51

C52

C56

C63

C64

S1-1 S1-2

0.453 0.351

0.437 0.358

0.504 0.313

0.535 0.342

0.530 0.353

0.528 0.350

0.524 0.328

0.521 0.294

0.663 0.435

0.495 0.322

0.632 0.413

0.593 0.359

0.537 0.289

0.528 0.360

S1-3 S2-1 S2-2 S2-3 S3-1 S3-2 S3-3 S4-1 S4-2 S4-3 S5-1 S5-2 S5-3 S6-1 S6-2 S6-3 S7-1 S7-2

0.353 0.427 0.399 0.346 0.551 0.546 0.378 0.589 0.427 0.380 0.606 0.488 0.389 0.497 0.441 0.375 0.560 0.459

0.351 0.389 0.380 0.358 0.520 0.539 0.363 0.566 0.429 0.362 0.562 0.468 0.364 0.470 0.431 0.358 0.531 0.455

0.295 0.442 0.363 0.282 0.580 0.550 0.321 0.657 0.409 0.332 0.631 0.483 0.350 0.515 0.436 0.324 0.604 0.449

0.255 0.518 0.399 0.283 0.621 0.537 0.328 0.658 0.478 0.371 0.685 0.550 0.415 0.579 0.463 0.327 0.663 0.529

0.265 0.528 0.412 0.297 0.623 0.535 0.344 0.646 0.491 0.389 0.691 0.561 0.434 0.585 0.468 0.341 0.664 0.542

0.277 0.512 0.408 0.301 0.617 0.544 0.345 0.639 0.476 0.382 0.680 0.547 0.420 0.575 0.465 0.343 0.654 0.525

0.246 0.507 0.386 0.268 0.610 0.526 0.317 0.667 0.465 0.358 0.679 0.539 0.402 0.566 0.452 0.316 0.655 0.516

0.185 0.491 0.330 0.212 0.591 0.496 0.246 0.736 0.455 0.291 0.657 0.521 0.354 0.543 0.433 0.245 0.643 0.504

0.306 0.643 0.491 0.359 0.731 0.616 0.399 0.644 0.596 0.456 0.746 0.656 0.512 0.705 0.558 0.392 0.733 0.647

0.260 0.475 0.376 0.277 0.584 0.520 0.320 0.647 0.443 0.352 0.654 0.515 0.386 0.538 0.437 0.319 0.628 0.491

0.310 0.625 0.482 0.343 0.719 0.604 0.398 0.653 0.566 0.449 0.757 0.636 0.502 0.682 0.541 0.394 0.724 0.617

0.274 0.556 0.421 0.291 0.669 0.577 0.344 0.679 0.499 0.384 0.715 0.574 0.428 0.620 0.495 0.344 0.694 0.548

0.214 0.493 0.346 0.210 0.610 0.525 0.269 0.730 0.436 0.306 0.674 0.516 0.356 0.552 0.440 0.274 0.654 0.486

0.263 0.528 0.414 0.305 0.619 0.531 0.344 0.635 0.500 0.392 0.685 0.566 0.439 0.586 0.469 0.340 0.662 0.551

S7-3 S8-1 S8-2 S8-3 S9-1 S9-2 S9-3 S10-1 S10-2 S10-3 S11-1 S11-2 S11-3 S12-1 S12-2 S12-3

0.400 0.606 0.469 0.347 0.699 0.606 0.413 0.702 0.408 0.357 0.656 0.505 0.455 0.681 0.650 0.506

0.394 0.581 0.438 0.303 0.679 0.615 0.396 0.675 0.405 0.328 0.642 0.474 0.433 0.661 0.660 0.484

0.367 0.625 0.458 0.343 0.785 0.652 0.421 0.736 0.439 0.362 0.723 0.501 0.437 0.732 0.692 0.492

0.430 0.701 0.558 0.444 0.799 0.686 0.500 0.784 0.531 0.456 0.746 0.590 0.515 0.771 0.711 0.593

0.445 0.710 0.577 0.460 0.781 0.679 0.511 0.785 0.537 0.469 0.735 0.605 0.531 0.766 0.703 0.612

0.432 0.695 0.552 0.430 0.788 0.673 0.492 0.779 0.509 0.441 0.728 0.584 0.513 0.755 0.688 0.586

0.417 0.694 0.548 0.440 0.802 0.684 0.493 0.782 0.531 0.453 0.754 0.581 0.505 0.780 0.725 0.584

0.392 0.672 0.533 0.463 0.775 0.708 0.493 0.750 0.589 0.480 0.789 0.563 0.483 0.796 0.863 0.567

0.531 0.757 0.660 0.517 0.746 0.709 0.576 0.779 0.580 0.524 0.707 0.675 0.606 0.718 0.661 0.668

0.400 0.670 0.519 0.408 0.790 0.667 0.467 0.767 0.498 0.422 0.738 0.554 0.483 0.766 0.712 0.559

0.510 0.749 0.644 0.508 0.757 0.688 0.560 0.789 0.562 0.512 0.712 0.665 0.591 0.729 0.658 0.654

0.444 0.712 0.570 0.452 0.807 0.695 0.512 0.787 0.533 0.463 0.746 0.602 0.527 0.757 0.693 0.591

0.379 0.673 0.515 0.429 0.827 0.696 0.476 0.771 0.538 0.445 0.792 0.553 0.472 0.800 0.780 0.546

0.453 0.713 0.584 0.465 0.771 0.683 0.517 0.781 0.546 0.475 0.730 0.610 0.537 0.761 0.705 0.622


341

weight replacements are done for the sensitivity analysis in » this paper. New Ci values of the various scenarios (Alternatives) are shown in Table 9 and new ranking according to new obtained results are shown in Table 10. The results of the sensitivity analysis can be observed graphically from Fig. 5. It is obvious when the weights of criteria are » replaced mutually; Ci values and ranking of the various scenarios change. For example, while the weights of

»

criteria 1 and 2 are replaced, Ci value of scenario 9-1 from 0.799075 to 0.699 changes and likewise ranking is 1. Scenario 1-3 has the worst ranking in the most replacements. Although there are some deviations on ranking of alternatives in replacements, scenario 9-1 is the best alternative in the most replacements. This analysis shows that the obtained results of the proposed methodology are reliable and help the decision maker assign the optimum operator in CMS.

Table 10 The results of Ranking by sensitivity analysis The main ranking

C12

C14

C16

C24

C26

C31

C35

C43

C45

C51

C52

C56

C63

C64


18 32 36 21 30 35 11 17 33 10 24 31 7 16 29 14 25 34

20 34 33 23 27 36 11 12 30 9 22 29 6 16 28 15 21 31

19 33 34 26 27 31 12 10 29 8 22 30 9 16 28 15 21 32

14 34 35 20 27 36 11 12 33 6 25 31 8 17 29 13 23 32

18 32 36 21 30 35 11 17 33 10 24 31 8 16 29 14 25 34

21 32 36 22 30 35 11 19 33 10 24 31 7 16 29 14 26 34

18 32 36 21 30 35 11 17 33 10 24 31 7 16 29 14 25 34

19 32 36 21 30 35 11 18 33 9 24 31 8 16 29 14 26 34

17 31 36 22 30 35 11 20 33 6 26 32 9 18 29 15 27 34

13 32 36 19 30 35 6 20 33 18 22 31 4 16 29 10 25 34

19 32 36 22 30 35 11 15 33 9 24 31 8 17 29 14 25 34

17 32 36 18 30 35 7 20 33 14 22 31 3 16 29 10 25 34

14 32 36 19 30 35 11 16 34 10 24 31 5 17 29 12 25 33

16 32 35 20 30 36 11 17 34 6 26 31 8 18 29 13 25 33

22 32 36 21 30 35 12 20 33 10 24 31 7 16 29 14 26 34


9 20 28 6 15 27 1 8 23 2 19 26 4 13 22 3 5 12

10 18 26 7 17 35 2 8 24 1 25 32 4 14 19 3 5 13

11 17 25 7 18 36 1 6 24 2 23 35 5 14 20 3 4 13

10 19 26 9 18 30 1 7 24 2 21 28 4 15 22 3 5 16

9 20 28 6 15 27 1 7 23 2 19 26 4 13 22 3 5 12

9 17 28 5 15 27 2 8 23 1 18 25 4 13 20 3 6 12

9 19 27 5 15 28 1 8 23 2 22 26 4 13 20 3 6 12

10 20 28 6 15 27 1 7 23 2 17 25 4 13 22 3 5 12

10 19 28 8 16 25 4 7 21 5 12 24 3 14 23 2 1 13

5 17 26 2 15 28 3 8 24 1 23 27 9 11 21 7 14 12

10 20 28 6 16 27 1 7 23 2 18 26 4 13 21 3 5 12

6 19 27 4 15 28 2 9 24 1 23 26 8 11 21 5 12 13

8 20 28 6 18 27 1 7 23 2 21 26 4 13 22 3 9 15

10 21 28 9 19 27 1 7 22 5 15 24 3 12 23 2 4 14

9 17 28 5 15 27 2 8 23 1 18 25 4 13 19 3 6 11

342 Fig. 5 The changes of ranking by sensitivity analysis

Int J Adv Manuf Technol (2011) 56:329–343 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 The main ranking

C12

C14

C16

C24

5 Conclusion In this paper, for the first time, an integrated Fuzzy MCDM approach based on Fuzzy AHP and TOPSIS and simulation for assigning the optimal operator in CMS is proposed for those who work in the field of manufacturing system. Using fuzzy theory for AHP to determine the weight of criteria of TOPSIS can reduce ambiguities and uncertainties that are inherent for making a decision about the most efficient number of operators and the efficient measurement of operator assignment in CMS. The proposed methodology uses triangular fuzzy numbers for AHP, computer simulation for values of the criteria used to TOPSIS. Using linguistic variables and experts’ judgments makes the decision making process more realistic and reliable. An actual case was considered and a computer simulation which considers various operators layout was developed with respect to the purpose of this study. In order to find the best scenarios among 36 alternatives, a combination of Fuzzy AHP and TOPSIS was employed. The average lead time of demands, the average of waiting time of demand and number of the operators as the index set of cost criteria, and the average utilization of operator/ machine and the numbers of completed parts as the index set of benefit criteria in TOPSIS were considered. The values of the criteria were obtained by means of computer simulation. Also, a sensitivity analysis was performed to validate and explain the obtained results from the proposed approach. For the extension of this work, other decision making and ranking methods for selecting the appropriate scenario for the operator assignment in CMS can be employed. Also, the comparison of the results of these methods with the current methods is suggested. Additionally, the various weight calculation methods for TOPSIS such as Entropy method, weighted least square

C26

C31

C35

C43

C45

C51

C52

C56

C63

C64

S1-1 S1-2 S1-3 S2-1 S2-2 S2-3 S3-1 S3-2 S3-3 S4-1 S4-2 S4-3 S5-1 S5-2 S5-3 S6-1 S6-2 S6-3 S7-1 S7-2 S7-3 S8-1 S8-2 S8-3 S9-1 S9-2 S9-3 S10-1 S10-2 S10-3 S11-1 S11-2 S11-3 S12-1 S12-2 S12-3

method, linear programming for multi dimensions of analysis preference and PROMETHEE method can be applied. These methods have been extended in the fuzzy environment and can be used for the comparison of the results. As another direction for future research, more criteria and scenarios can be considered. Acknowledgement The authors are grateful for the valuable comments and suggestion from the respected reviewers. Their valuable comments and suggestions have enhanced the strength and significance of our paper.

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