Group decision making based on novel fuzzy modified TOPSIS method

Applied Mathematical Modelling 35 (2011) 4257–4269

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Group decision making based on novel fuzzy modified TOPSIS method Behnam Vahdani ⇑, S. Meysam Mousavi, Reza Tavakkoli-Moghaddam Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 29 July 2010 Received in revised form 16 February 2011 Accepted 23 February 2011 Available online 26 February 2011 Keywords: Multi-criteria analysis Modified TOPSIS Fuzzy sets Robot selection Rapid prototyping process selection

a b s t r a c t The aim of this paper is to present a novel fuzzy modified technique of order preference by a similarity to ideal solution (TOPSIS) method by a group of experts, which can select the best alternative by considering both conflicting quantitative and qualitative evaluation criteria in real-life applications. The proposed method satisfies the condition of being the closest to the fuzzy positive ideal solution and also being the farthest from the fuzzy negative ideal solution with multi-judges and multi-criteria. The performance rating values of alternatives versus conflicting criteria as well as the weights of criteria are described by linguistic variables and are transformed into triangular fuzzy numbers. Then a new collective index is introduced to discriminate among alternatives in the evaluation process with respect to subjective judgment and objective information. This paper shows that the proposed fuzzy modified TOPSIS method is a suitable decision making tool for the manufacturing decisions with two examples for the robot selection and rapid prototyping process selection. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Multi-criteria decision making (MCDM) is regarded as a main part of modern decision science and operational research, which contains multiple decision criteria and multiple decision alternatives. The objective of the MCDM is to find the most desirable alternative(s) from a set of available alternatives versus the selected criteria [1]. The MCDM methods are able to obtain solutions for a wide range of society, economics, engineering and management problems [1–4]. Technique for order preference by similarity to an ideal solution (TOPSIS) is one of the well-known classical MCDM methods proposed by Hwang and Yoon [5] for solving the decision making problems. It is based on the concepts that the chosen alternative should have the shortest distance from the positive ideal point and the farthest distance from the negative ideal point concurrently. In fact, the ideal point is the solution, in which the benefit criteria are maximized and the cost criteria are minimized. Consequently, the negative ideal point is the solution, in which the cost criteria are maximized and the benefit criteria are minimized [6–8]. The increasing complexity of the engineering and management environment leads to benefit from a group of experts or decision makers (DMs) to investigate all relevant aspects of decision making problems. In the recent decade, some studies focused on MCDM problems to provide reliable results and take into account the analysis of the DMs instead of the analysis of a single DM [1,4,7–9]. On the other hand, MCDM methods consist of the DMs’ subjective judgments and preferences including qualitative and/or quantitative criteria ratings as well as weights of criteria. These issues can be regarded as uncertain, imprecise and indefinite values that make the decision making process complicated in the real-life applications. To address the issues, the decision making process under a fuzzy environment is presented, where the information available is imprecise and uncertain to cope with such problems in the DMs’ assessments [8]. ⇑ Corresponding author. Tel./fax: +98 21 44649066. E-mail addresses: [email protected], [email protected] (B. Vahdani), [email protected] (S.M. Mousavi). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.02.040

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In this paper, the TOPSIS method by a group of DMs is considered for ranking and selection problems because of the merits and advantages provided as below [9,10]: A concept that is rational and readily comprehensible The computation process involved is simple and straightforward A sound logic that is capable of considering both the best and the worst alternative concurrently The ability to depict the best performance of the available alternative versus each assessment criterion in a simple mathematical form  Incorporating the relative importance described by objective and subjective information into the evaluation process  An easy implementation into a spreadsheet software    

Furthermore, both TOPSIS and fuzzy TOPSIS methods have been extensively applied to engineering and management fields over the last two decades. For instance, alternative-fuel buses selection [2], bridge scheme selection [11], evaluating conceptual bridge design [5], inter-company comparison [10], project risk assessment [3,8], rapid prototyping process selection [12], robot selection [13,14], supplier evaluation and selection [15], and temporary storage design [16]. In the last decade, numerous attempts have been made to propose the TOPSIS method for multi-criteria analysis along with uncertainty issues. Chen [17] developed the TOPSIS method for the group decision making under a fuzzy environment. The rating of each alternative and the weight of each criterion are provided by linguistic terms which can be expressed in triangular fuzzy numbers. Then a vertex method is presented to calculate the distance between two triangular fuzzy numbers. Chu and Lin [13] presented a fuzzy TOPSIS method for robot selection. The values of objective criteria are transformed into dimensionless indices to make sure compatibility between the values of objective criteria and the linguistic ratings of subjective criteria. Also, the membership function of each weighted rating is obtained by interval arithmetic of fuzzy numbers. Byun and Lee [12] introduced the selection of an optimal rapid prototyping system based on a decision support system and modified TOPSIS method. A modified TOPSIS is considered to assess both quantitative and qualitative data by using triangular fuzzy numbers and defuzzification centroid values. Chen et al. [15] presented a fuzzy approach based on the concept of the TOPSIS method for supplier evaluation and selection in supply chain management. Linguistic terms are applied to evaluate the ratings and weights for the factors and then they are expressed in trapezoidal or triangular fuzzy numbers. Rao and Padmanabhan [18] designed a methodology on the basis of digraph and matrix methods for the assessment of alternative industrial robots. In addition, they applied an MCDM method (TOPSIS) for the selection of a robot in an industrial application and compared the results. Rao [14] introduced multi-attributes decision making (MADM) methods in the manufacturing environment, such as evaluating flexible manufacturing system, selection of modern machining methods and robot selection. For this purpose, the TOPSIS and the modified TOPSIS methods along with other classical MADM methods are taken into account. In the traditional TOPSIS method, the normalized decision matrix is weighted by multiplying each column of the matrix by its relative importance of criterion. Then the overall performance of an alternative is determined by its Euclidean distance to the positive and negative ideal points; but, the distance depends on the relative importance of each criterion, and should be included in the measurement. In the modified TOPSIS method proposed by Deng et al. [10], weighted Euclidean distances are applied instead of constructing a weighted decision matrix. Hence, the positive and negative ideal points do not interrelate with the weighted decision matrix. Mahdavi et al. [19] proposed a model of the TOPSIS in a fuzzy environment for MCDM problems in order to obtain ideal solutions. They use a measurement of fuzzy distance value with a lower bound of alternatives by triangular fuzzy numbers. Chu and Lin [20] designed an interval arithmetic based fuzzy TOPSIS model, where ratings of alternatives versus criteria and weights of criteria are described by linguistic terms and represented by fuzzy numbers. Also, the membership function of each fuzzy weighted rating is obtained by interval arithmetic of fuzzy numbers. Wang and Lee [21] considered a fuzzy TOPSIS for evaluating alternatives, in which subjective weighting method and entropy-based objective weighting method are applied. Ebrahimnejad et al. [3] proposed an approach based on fuzzy TOPSIS and fuzzy LINMAP methods in order to rank high risks in build-operate-transfer projects. Their ranking process is performed on the basis of triangular fuzzy number and fuzzy Euclidean distance. Kumar and Garg [22] provided deterministic quantitative model according to distance based approach (DBA) method for the selection and ranking of robots. The presented model was illustrated with an application example in the robot selection. Chatterjee et al. [23] carried out an attempt to solve the robot selection problem with the application of two MADM methods, namely compromise ranking and outranking, and discussed their relative performances for a given industrial application. The review of the literature indicates that the fuzzy TOPSIS method under the group decision making process has received much less attention; there is a need for simple and logical mathematical tool to help the DMs in order to make a best decision. Hence, in this paper a new fuzzy modified TOPSIS method by a group of the DMs is proposed that considers both subjective judgment and objective information in real-life situations. The proposed method is based on concepts of positive ideal and negative ideal points for solving decision making problems with multi-judges and multi-criteria in a fuzzy environment. In this method, the performance rating values of each alternative under the selected criteria as well as the weights of criteria are linguistic variables expressed as triangular fuzzy numbers. Then a new collective index is introduced to discriminate among alternatives in the evaluation process by constructing ideal separation and anti-ideal separation matrixes simultaneously. Finally, for the purpose of proving the validity and applicability of the proposed method, two illustrative

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examples are provided in the manufacturing environment for problems of the robot selection and rapid prototyping process selection. The remaining of this paper is organized as follows: In Section 2, we briefly introduce the classical TOPSIS method. Section 3 reviews basic definitions of fuzzy sets and fuzzy numbers. Novel fuzzy modified TOPSIS method is proposed to solve MCDM problems in Section 4. Section 5 presents two illustrative examples in the manufacturing environment including two applications to select the robot and rapid prototyping process. The paper is concluded in Section 6.

2. TOPSIS method TOPSIS method is a technique to order preference by similarity to ideal solution proposed by Hwang and Yoon [24]. This method regards the principle that the chosen alternative should have the shortest distance from the ideal solution and the longest distance from the negative ideal solution [6,24]. In fact, the ideal solution is a solution that maximizes the benefit criteria and minimizes the cost criteria. If each criterion has a monotone increasing (or decreasing) effective function, the ideal solution, which consists of the best criterion values, and the negative ideal solution, which consists of the worst, is computed [12]. Suppose an MADM problem that has m alternatives, A1, . . ., Am and n decision criteria,C1, . . ., Cn. Each alternative is assessed with respect to the n criteria. All the performance ratings assigned to the alternatives with respect to each criterion form a decision matrix denoted by X ¼ ðxij Þmn . Let W = (w1, w2, . . . , wn) be the relative weight vector about the criteria, P satisfying nj¼1 wj ¼ 1. Then, the TOPSIS method can be summarized as follows: Step 1. Normalize the decision matrix X ¼ ðxij Þmn using the equation below.

xij rij ¼ qffiffiffiffiffiffiffiffiffiffi P ffi

m 2 i¼1 xij

; i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n;

ð1Þ

where rij is the normalized criteria rating. Step 2. Calculate the weighted normalized decision matrix V ¼ ðv ij Þmn: .

v ij ¼ wj rij ; i ¼ 1; 2; . . . ; m;

j ¼ 1; 2; . . . ; n;

ð2Þ Pn

where wj is the relative weight of the jth criterion, and j¼1 wj ¼ 1:. Step 3. Determine the ideal and negative-ideal solutions.

A ¼ v 1 ; . . . ; v m ¼ fðmaxj v ij j jXb Þ; ðminj v ij j jXc Þg;

ð3Þ

A ¼ v 1 ; . . . ; v m ¼ fðminj v ij j jXb Þ; ðmaxj v ij j jXc Þg;

ð4Þ

where Xb and Xc are the sets of benefit criteria and cost criteria, respectively. Step 4. Calculate the Euclidean distances of each alternative from the positive ideal solution and the negative ideal solution, respectively.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u n Di ¼ t ðv ij  v j Þ2 ;

i ¼ 1; 2; . . . ; m;

ð5Þ

j¼1

Di

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u n ¼ t ðv ij  v j Þ2 ;

i ¼ 1; 2; . . . ; m;

ð6Þ

j¼1

Step 5. Calculate the relative closeness of each alternative to the ideal solution. The relative closeness of the alternative Ai with respect to A⁄ is defined as

RC i ¼

Di

Di ; þ Di

i ¼ 1; 2; . . . ; m:

ð7Þ

Step 6. Rank the alternatives according to the relative closeness to the ideal solution. The bigger the RCi, the more desirable the alternative Ai will be. The best alternative is the one with the greatest relative closeness to the ideal solution. 3. Fuzzy numbers In this section, some basic definitions of fuzzy sets and fuzzy numbers are reviewed from Kaufmann and Gupta [25] and Ross [26]. Below, the basic definitions and notations of fuzzy sets and fuzzy numbers are presented which are applied throughout this paper until otherwise stated.

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~ in a universe of discourse X is characterized by a membership function lAðxÞ ~ Definition. 3.1. A fuzzy set A which associates ~ with each element x in X a real number in the interval [0, 1]. The function value lAðxÞ is termed the grade of membership of x ~ in A. ~ ¼ ða1 ; a2 ; a3 Þ, the membership function of the fuzzy numDefinition. 3.2. The triangular fuzzy numbers can be denoted as A ~ is defined as follows: ber A

~ ¼ lAðxÞ

8 > > > <

0

x < a1 ;

ðx  a1 Þ=ða2  a1 Þ a1  x  a2 ;

> ða3  xÞ=ða3  a2 Þ a2  x  a3 ; > > : 0 x > a3 :

ð8Þ

Definition. 3.3. A non-fuzzy number r can be expressed as (r, r, r). The fuzzy sum  and fuzzy Subtraction  of any two triangular fuzzy numbers are also triangular fuzzy numbers; however, the multiplication of any two triangular fuzzy ~ ¼ ða1 ; a2 ; a3 Þ, numbers is only an approximate triangular fuzzy number. Given any two positive triangular fuzzy numbers, A ~ ~ ~ B ¼ ðb1 ; b2 ; b3 Þ and a positive real number r, some main operations of fuzzy numbers A and B can be expressed as follows:

~B ~ ¼ ða1 þ b1 ; a2 þ b2 ; a3 þ b3 Þ; A

ð9Þ

~B ~ ¼ ða1  b3 ; a2  b2 ; a3  b1 Þ; A

ð10Þ

~ r ¼ ða1 r; a2 r; a3 rÞ; A

ð11Þ

~ B ~ ¼ða A ~ 1 b1 ; a2 b2 ; a3 b3 Þ:

ð12Þ

4. Proposed novel fuzzy modified TOPSIS method The MADM problem including both crisp numbers and fuzzy numbers can be expressed in the matrix format for each DM as

2

x11

6 6 x21 6 6 6 6 .. 6 . 4 xm1



x1ðk1Þ

~x1k







x1ðk2Þ

~x2k



.. .

.. .

.. .

.. .

xmðk1Þ

~xmk

~x1n 3 7 ~x1n 7 7 7 7: .. 7 . 7 5

ð13Þ

~xmn

A set of K DMs is regarded for the above MADM problem. For a convenient calculation, it is assumed that xi1 ; . . . ; xiðk1Þ ði ¼ 1; . . . ; mÞ are crisp numbers and~ xik ; . . . ; ~ xin ði ¼ 1; . . . ; mÞ are fuzzy numbers for each DM. The linguistic variables ~ xij are triangular fuzzy numbers ðaij ; bij ; cij Þ. Furthermore, weights of criteria are linguistic variables which can be expressed in triangular fuzzy numbers. Therefore, the procedure of novel fuzzy modified TOPSIS method by a group of DMs can be described as follows. Step 1. Establish a group of k DMs. Step 2. Define and describe a set of relevant criteria. Step 3. Obtain the rating of a potential alternative versus each criterion for each DM. ~ j Þ. Step 4. Aggregate the ratings of alternatives versus each subjective criterion ð~ xij Þand fuzzy weights of selected criteriaðw Let the fuzzy ratings of all DMs be triangular fuzzy numbers ~ xijk ¼ ðaijk ; bijk ; cijk Þ, k = 1, 2, . . ., K. Then the aggregated fuzzy rating can be obtained as

~xij ¼ ðaij ; bij ; cij Þ; where

aij ¼

k 1X aijk ; K k¼1

bij ¼

k 1X bijk ; K k¼1

k ¼ 1; 2; . . . ; K;

ð14Þ

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cij ¼

k 1X cijk : K k¼1

~ jk ¼ ðwjk1 ; wjk2 ; wjk3 Þ; Let the fuzzy weights of selected criteria be triangular fuzzy numbers w gated fuzzy weight of each criterion can be obtained as

~ j ¼ ðwj1 ; wj2 ; wj3 Þ; w

k ¼ 1; 2; . . . ; K. Then the aggre-

k ¼ 1; 2; . . . ; K;

ð15Þ

where

wj1 ¼

k 1X wjk1 ; K k¼1

wj2 ¼

k 1X wjk2 ; K k¼1

wj3 ¼

k 1X wjk3 : K k¼1

Also, the ratings of alternatives versus each objective criterion are aggregated by the arithmetic mean. Step 5. Compute the normalized decision matrix. Vector normalization is applied to calculate rij and ~r ij .

xij rij ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm 2 ; i¼1 xij

i ¼ 1; . . . ; m;

! aij bij cij ; ; ; ej ej ej

~rij ¼

j ¼ 1; . . . ; k  1;

i ¼ 1; . . . ; m;

ð16Þ

j ¼ k; . . . ; n;

ð17Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffi u m uX  c2ij : ej ¼ t i¼1

~ ¼ ½v ~ ij mn . The fuzzy weighted normalized decision Step 6. Construct the fuzzy weighted normalized decision matrix, V ~j Þ, which is assigned by pairwise matrix is calculated by multiplying each column of the matrix by the fuzzy weight ðw comparisons of elements. Thus,

v~ ij ¼ w~ j rij ;

i ¼ 1; . . . ; m;

j ¼ 1; . . . k  1;

ð18Þ

v~ ij ¼ w~ j rij ;

i ¼ 1; . . . ; m;

j ¼ k; . . . n;

ð19Þ

Step 7. Defuzzify fuzzy numbers. Each fuzzy number is defuzzified by using the centroid method. For triangular fuzzy numbers, R v~ ¼ ða; b; cÞ, the defuzzification centroid value is obtained by

v ¼

v lðv Þdv 1 ¼ ða þ b þ cÞ: lðv Þdv 3

R

ð20Þ

~ ij ¼ ðaij ; bij ; cij Þ, contained in the weighted normalized decision matrix is Each centroid value for fuzzy numbers, v

v ij ¼

1 ðaij þ bij þ cij Þ; ¼ 1; . . . ; m; 3

j ¼ k; . . . ; n:

ð21Þ

Step 8. Determine the positive ideal and negative ideal solutions. The values for A⁄ and A are defined as

A ¼ ðv 1 ; v 2 ; . . . ; v k1 ; v~ k ; v~ kþ1 ; . . . ; v~ n Þ          ¼ max v ij j j  J ; min v ij j j  J ; max v ij j j  J ; min v ij j j  J j i ¼ 1; . . . ; m ;

ð22Þ

A ¼ ðv 1 ; v 2 ; . . . ; v k1 ; v~ k ; v~ kþ1 ; . . . ; v~ n Þ          ¼ min v ij j j  J ; max v ij j j  J ; min v ij j j  J ; max v ij j j  J j i ¼ 1; . . . ; m ;

ð23Þ

i

i

i

i

i

i

Where,

J ¼ fj ¼ 1; 2; . . . ; n j j associated with benefit attributeg; J_ ¼ fj ¼ 1; 2; . . . ; n j j associated with cost attributeg;

i

i

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For benefit criterion, the DM desires to have a maximum value among the alternatives. For cost criterion, however, the DM desires to have a minimum value among them. Obviously, A⁄indicates the most preferable alternative or the ideal solution. Similarly, A indicates the least preferable alternative or the anti-ideal solution. Step 9. Construct ideal separation matrix (D⁄) and anti-ideal separation matrix (D) which are defined as follow.

3 jv 11  v 1 j jv 1ðk1Þ  v k1 j jv 1k  v k j jv 1n  v n j 6 jv 21  v  j jv 2ðk1Þ  v  j jv 2k  v  j jv 2n  v  j 7 6 1 n 7 k1 k  7; D ¼ ½dij  ¼ 6 .. .. .. .. .. .. 7 6 5 4 . . . . . . jv m1  v 1 j jv mðk1Þ  v k1 j jv mk  v k j jv mn  v n j

ð24Þ

3 jv 11  v 1 j jv 1ðk1Þ  v k1 j jv 1k  v k j jv 1n  v n j 7 6  6 jv 21  v 1 j jv 2ðk1Þ  v k1 j jv 2k  v k j jv 2n  v n j 7 7 6   D ¼ ½dij  ¼ 6 7: .. .. .. .. .. .. 7 6 . . . . . . 5 4     jv m1  v 1 j jv mðk1Þ  v k1 j jv mk  v k j jv mn  v n j

ð25Þ

2

2

Step 10. Calculate collective index (CI). The CI is calculated by

0

11 k  K X dij A HiðD ; D Þ ¼ @ þ Z ij0  dij i¼1 



8i ¼ 1; 2; . . . ; m;

ð26Þ

ðAÞ

P   0 where the first summation ð A Þ refers to all j for which dij > 0 while ðZ ij0 Þ refers to all j for which dij ¼ 0. Moreover, Z ij0 can    max1 w j j d 0   for which dij0 > 0 and wj for dij0 ¼ 0. be calculated such that Z ij0 ¼ maxj0 dij0 ij





Ii ðD ; D Þ ¼

K X l¼1

!m1  dij

0 þ@

11 k K X 1A þ Q ij0  d l¼1ðAÞ ij

8i ¼ 1; 2; . . . ; m;

ð27Þ

P   0 where the second summation ð A Þ refers to all j for which dij > 0 while ðQ ij0 Þ refers to all j for which dij ¼ 0 and   maxj wj    Q ij0 ¼ minj0 dij0 for which dij > 0 and wj for dij ¼ 0. The collective index is calculated as follows:

CI ¼ H i þ Ii :

ð28Þ

Step 11. Rank the preference order. The best satisfactory alternative can be determined according to preference rank order of H i and Ii . Minimum values of the CI indicate the better performance for the alternative i. 5. Application of the proposed method in solving problems 5.1. Illustrative example 1 for the robot selection In this section, the illustrative example from Liang and Wang [27] is presented to illustrate the validity and applicability of the proposed novel fuzzy modified TOPSIS method for the robot selection problem. Assume that a manufacturing company requires a robot to perform a material-handling task. A committee of four DMs (DM1, DM2, DM3 and DM4) is formed to assess and select the most suitable robot. In this example, it is assumed that degrees of the importance for four DMs are equal. Three robots (R1, R2 and R3) are chosen, and six criteria (Ci = 1, 2, ... , 6)are selected for further assessment (Steps 1 and 2). Table 1 Linguistic variables for rating the importance of selected criteria. Linguistic variables

Triangular fuzzy numbers

Very high (VH) high (H) Medium high (MH) Medium (M) Medium low (ML) low (L) Very low (VL)

(0.7, 0.9, 1) (0.6, 0.7, 0.8) (0.4, 0.5, 0.6) (0.1, 0.3, 0.5) (0, 0.2, 0.3) (0, 0.1, 0.2) (0, 0, 0.2)

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The importance weights of the five criteria are described by using the following linguistic terms: Very low (VL), low (L), medium low (ML), medium (M), medium high (MH), high (H) and very high (VH), which are defined in Table 1. The performance ratings of the alternatives with respect to criteria are characterized by the following linguistic terms: Very poor (VP), poor (P), medium poor (MP), fair (F), medium good (MG), good (G) and very good (VG), which are defined in Table 2. The selected subjective and objective criteria are provided for the robot selection in Table 3. The weights of these six criteria are obtained by four DMs according to linguistic terms described in Table 1 and are given in Table 4. Then the ratings of alternatives with respect to three subjective criteria are represented by four DMs according to linguistic terms in Table 2 and are shown in Table 5. Also, the data of objective criteria is provided in Table 6 (Step 3). The ratings of three robots (alternatives) obtained by all four DMs versus each subjective criterion are aggregated by Eq. (14) and given in Table 7. Also, the relative importance of selected criteria provided by all four DMs is aggregated by Eq. (15) and their calculations are shown in Table 8 (Step 4). The normalized decision making matrix and the weighted normalized decision making matrix are constructed. The respective results are presented in Tables 9 and 10, respectively (Steps 5 and 6). The weighted normalized decision matrix is defuzzified by Eq. (21) and is illustrated in Table 11 (Step 7). Consequently, positive ideal and negative ideal solutions are determined (Step 8). Ideal separation matrix (D⁄) and anti-ideal separation matrix (D) are constructed by using Eqs. (24) and (25), and are given below (Step 9).

2

0:048 0:103 0:033

6  D ¼ ½dij  ¼ 4 0:107 0

0

0:01

0:01

0

0:01

0

0:141

0

0:045

0

3

7 0:004 0:051 0:028 5;

and

2 

D ¼

 ½dij 

6 ¼4

0:059 0

0

0

0

0:051

0:103 0:023 0:006

0:107 0:093 0:033

0:01

0

0

3

7 0:113 5:

0:006 0:141

Table 2 Linguistic variables for the rating of alternatives. Linguistic variables

Triangular fuzzy numbers

Very good (VG) Good (G) Medium good (MG) Fair (F) Medium poor (MP) Poor (P) Very poor (VP)

(9, 10, 10) (7, 9, 10) (5, 7, 9) (3, 5, 7) (1, 3, 5) (0, 1, 3) (0, 0, 1)

Table 3 Selected criteria for the robot selection problem. Subjective criteria

Objective criteria

Man–machine interface (C1) Programming flexibility (C2) Vendor’s service contract (C3)

Purchase cost (C4) Load capacity (C5) Positioning accuracy (C6)

Table 4 Weights of selected criteria. Criteria

C1 C2 C3 C4 C5 C6

Decision makers DM1

DM2

DM3

DM4

H VH M M VH VH

VH H L M VH H

VH VH M M H H

H M L L VH H

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Table 5 Ratings of robots under subjective criteria. Criteria

Robots

Decision makers DM1

DM2

DM3

DM4

C1

R1 R2 R3

F F G

F G F

G F VG

VG F G

C2

R1 R2 R3

G VG G

P G F

G VG VG

F F G

C3

R1 R2 R3

F G G

F F G

G VG G

F G VG

Table 6 Ratings of robots under objective criteria. Robots

Purchase cost($  1000), C4

Load capacity, C5

Positioning accuracy (±)in, C6

R1 R2 R3

73 70 68

50 45 45.5

0.12 0.16 0.17

Table 7 Aggregated fuzzy ratings of the alternatives under subjective criteria by four DMs. Criteria

Robots

Decision makers

Aggregated ratings

DM1

DM2

DM3

DM4

C1

R1 R2 R3

(3, 5, 7) (3, 5, 7) (7, 9, 10)

(3, 5, 7) (7, 9, 10) (3, 5, 7)

(7, 9, 10) (3, 5, 7) (9, 10, 10)

(9, 10, 10) (3, 5, 7) (7, 9, 10)

(5.5, 7.25, 8.5) (4, 6, 7.75) (6.5, 8.25, 9.25)

C2

R1 R2 R3

(7, 9, 10) (9, 10, 10) (7, 9, 10)

(1, 3, 5) (7, 9, 10) (3, 5, 7)

(7, 9, 10) (9, 10, 10) (9, 10, 10)

(3, 5, 7) (3, 5, 7) (7, 9, 10)

(4.25, 6, 7.5) (7, 8.5, 9.25) (6.5, 8.25, 9.25)

C3

R1 R2 R3

(3, 5, 7) (7, 9, 10) (7, 9, 10)

(3, 5, 7) (3, 5, 7) (7, 9, 10)

(7, 9, 10) (9, 10, 10) (7, 9, 10)

(3, 5, 7) (7, 9, 10) (9, 10, 10)

(4, 6, 7.75) (6.5, 8.25, 9.25) (7.5, 9.25, 10)

Table 8 Aggregation of the relative importance of each selected criteria by four DMs. Criteria

C1 C2 C3 C4 C5 C6

Decision makers DM1

DM2

DM3

DM4

(0.6, 0.7, 0.8) (0.7, 0.9, 1) (0.1, 0.3, 0.5) (0.1, 0.3, 0.5) (0.7, 0.9, 1) (0.7, 0.9, 1)

(0.7, 0.9, 1) (0.6, 0.7, 0.8) (0, 0.1, 0.2) (0.1, 0.3, 0.5) (0.7, 0.9, 1) (0.6, 0.7, 0.8)

(0.7, 0.9, 1) (0.7, 0.9, 1) (0.1, 0.3, 0.5) (0.1, 0.3, 0.5) (0.6, 0.7, 0.8) (0.6, 0.7, 0.8)

(0.6, 0.7, 0.8) (0.1, 0.3, 0.5) (0, 0.1, 0.2) (0, 0.1, 0.2) (0.7, 0.9, 1) (0.6, 0.7, 0.8)

Aggregated weights

Defuzzify fuzzy weights

(0.65, 0.8, 0.9) (0.525, 0.7, 0.825) (0.025, 0.2.0.35) (0.075, 0.25, 0.425) (0.675, 0.85, 0.95) (0.625, 0.75, 0.85)

0.784 0.684 0.192 0.25 0.825 0.742

Table 9 Normalized decision matrix. Robots

C1

C2

C3

C4

C5

C6

R1 R2 R3

(0.372, 0.491, 0.575) (0.271, 0.406, 0.525) (0.440, 0.558, 0.626)

(0.281, 0.397, 0.497) (0.464, 0.563, 0.613) (0.431, 0.547, 0.613)

(0.255, 0.382, 0.494) (0.414, 0.526, 0.590) (0.478, 0.590, 0.638)

0.598 0.574 0.557

0.615 0.554 0.560

0.457 0.609 0.647

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B. Vahdani et al. / Applied Mathematical Modelling 35 (2011) 4257–4269 Table 10 Weighted normalized decision matrix. Robots

C1

C2

C3

C4

C5

C6

R1 R2 R3

(0.241, 0.392, 0.517) (0.176, 0.324, 0.472) (0.286, 0.446, 0.563)

(0.147, 0.277, 0.41) (0.243, 0.394, 0.505) (0.226, 0.382, 0.505)

(0.006, 0.076, 0.172) (0.011, 0.105, 0.206) (0.011, 0.118, 0.223)

(0.044, 0.149, 0.254) (0.043, 0.143, 0.243) (0.041, 0.139, 0.236)

(0.415, 0.522, 0.584) (0.373, 0.471, 0.526) (0.378, 0.476, 0.532)

(0.285, 0.342, 0.388) (0.381, 0.456, 0.517) (0.404, 0.485, 0.549)

Table 11 Defuzzified weighted normalized decision matrix. Robots

C1

C2

C3

C4

C5

C6

R1 R2 R3 Positive ideal solutions (R⁄) Negative ideal solutions (R)

0.383 0.324 0.431 0.431 0.324

0.278 0.381 0.371 0.381 0.278

0.084 0.107 0.117 0.117 0.084

0.149 0.143 0.139 0.139 0.149

0.507 0.456 0.462 0.507 0.456

0.338 0.451 0.479 0.479 0.338

Table 12 The values H i ,Ii and CI by the proposed novel fuzzy modified TOPSIS. Robots

Hi

Ii

CI ¼ H i þ Ii

Final ranking

R1 R2 R3

(0.901 + 0.756)=1.657 (1.077 + 0.6116) = 1.688 1.403

(0.833 + 6.046 + 0.109) = 6.958 (0.764 + 3.888 + 0.0146) = 4.667 (0.616 + 2.616) = 3.232

8.615 6.355 4.635

3 2 1

Finally, the values of the H i , Ii and CI are computed by using Eqs. (26)–(28) and are shown in Table 12 (Steps 10 and 11). According to Table 12, the ranking order of three potential robots is R3, R2 and R1 for the robot selection problem. Hence, the best alternative versus six selected criteria is robot 3. 5.2. Illustrative example 2 for the rapid prototyping process selection To further demonstrate the novel fuzzy modified TOPSIS method, another illustrative example from Byun and Lee [12] is presented for the rapid prototyping process selection in the manufacturing environment. Assume that a manufacturing company aims to select an appropriate rapid prototyping process. This selection can be obtained according to various criteria. Dimensional accuracy, surface roughness, part cost, build time and material properties (tensile strength and elongation) are regarded as the main criteria for evaluating rapid prototyping parts. A group of four DMs (DM1, DM2, DM3 and DM4) is arranged to evaluate and select the appropriate rapid prototyping process. In this example, it is assumed that the degrees of importance for all four DMs are equal. Six rapid prototyping processes (i.e., SLA3500, SLS2500, FDM8000, LOM1015, Quadra, Z402) as potential alternatives ðRPP i ; i ¼ 1; 2; . . . ; 6Þ are provided, and six criteria ðC i ; i ¼ 1; 2; . . . ; 6Þ are identified for their evaluations (Steps 1 and 2). The selected subjective and objective criteria are provided for the rapid prototyping process selection in Table 13. The weights of these six criteria are obtained by four DMs according to linguistic terms described in Table 1 and are given in Table 14. Then the ratings of alternatives with respect to three subjective criteria are represented by four DMs according to linguistic terms in Table 2 and are shown in Table 15. Also, the data of objective criteria is provided in Table 16 (Step 3). The ratings of six rapid prototyping processes (alternatives) provided by all four DMs with respect to each criterion are aggregated by Eq. (14) and given in Table 17. Also, the relative importance of selected criteria obtained by all four DMs is aggregated by Eq. (15) and their results are illustrated in Table 18 (Step 4). The normalized decision making matrix and the weighted normalized decision making matrix are constructed. The respective results are presented in Tables 19 and 20, respectively (Steps 5 and 6).

Table 13 Selected criteria for the rapid prototyping process selection problem. Objective

Subjective

Accuracy C1 Roughness C2 Strength C3 Elongation C4

Part cost C5 Build time C6

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B. Vahdani et al. / Applied Mathematical Modelling 35 (2011) 4257–4269 Table 14 Weights of selected criteria. Criteria

Decision makers

C1 C2 C3 C4 C5 C6

DM1

DM2

DM3

DM4

MH M L ML VH VH

M MH ML L VH VH

MH M L ML H VH

M MH ML L VH H

Table 15 Ratings of rapid prototyping processes under subjective criteria. Criteria

Rapid prototyping processes

Decision makers DM1

DM2

DM3

DM4

C5

RPP1 RPP2 RPP3 RPP4 RPP5 RPP6

G G MG F G VG

G MG MG F G G

G G F MP G VG

MG G MG F MG VG

C6

RPP1 RPP2 RPP3 RPP4 RPP5 RPP6

MG F G F MP MP

MG MG MG F F MP

F MG G MP F MP

MG MG G F F P

Table 16 Ratings of rapid prototyping processes under objective criteria. Rapid prototyping processes

Accuracy C1

Roughness C2

Strength C3

Elongation C4

RPP1 RPP2 RPP3 RPP4 RPP5 RPP6

120 150 125 185 95 600

6.5 12.5 21 20 3.5 15.5

65 40 30 25 30 5

5 8.5 10 10 6 1

Table 17 Aggregated fuzzy ratings of the alternatives under subjective criteria by four DMs. Criteria

Rapid prototyping processes

Decision makers

Average ratings

DM1

DM2

DM3

DM4

C5

RPP1 RPP2 RPP3 RPP4 RPP5

(7, 9, 10) (7, 9, 10) (5, 7, 9) (3, 5, 7) (7, 9, 10)

(7, 9, 10) (5, 7, 9) (5, 7, 9) (3, 5, 7) (7, 9, 10)

(7, 9, 10) (7, 9, 10) (3, 5, 7) (1, 3, 5) (7, 9, 10)

(5, 7, 9) (7, 9, 10) (5, 7, 9) (3, 5, 7) (5, 7, 9)

(6.5, 8.5, 9.75) (6.5, 8.5, 9.75) (4.5, 6.5, 8.5) (2.5, 4.5, 6.5) (6.5, 8.5, 9.75)

C6

RPP6 RPP1 RPP2 RPP3 RPP4 RPP5 RPP6

(7, 9, 10) (5, 7, 9) (3, 5, 7) (7, 9, 10) (3, 5, 7) (1, 3, 5) (1, 3, 5)

(7, 9, 10) (5, 7, 9) (5, 7, 9) (5, 7, 9) (3, 5, 7) (3, 5, 7) (1, 3, 5)

(9, 10, 10) (3, 5, 7) (5, 7, 9) (7, 9, 10) (1, 3, 5) (3, 5, 7) (1, 3, 5)

(9, 10, 10) (5, 7, 9) (5, 7, 9) (7, 9, 10) (3, 5, 7) (3, 5, 7) (0, 1, 3)

(8.5, 9.75, 10) (4.5, 6.5, 8.5) (4.5, 6.5, 8.5) (6.5, 8.5, 9.75) (2.5, 4.5, 6.5) (2.5, 4.5, 6.5) (0.75, 2.5, 4.5)

The weighted normalized decision matrix is defuzzified by Eq. (21) and is given in Table 21 (Step 7). Consequently, positive ideal and negative ideal solutions are determined (Step 8).

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B. Vahdani et al. / Applied Mathematical Modelling 35 (2011) 4257–4269 Table 18 Aggregation of the relative importance of each selected criteria by four DMs. Criteria

C1 C2 C3 C4 C5 C6

Decision makers DM1

DM2

DM3

DM4

(0.4, 0.5, 0.6) (0.1, 0.3, 0.5) (0, 0.1, 0.2) (0, 0.2, 0.3) (0.7, 0.9, 1) (0.7, 0.9, 1)

(0.1, 0.3, 0.5) (0.4, 0.5, 0.6) (0, 0.2, 0.3) (0, 0.1, 0.2) (0.7, 0.9, 1) (0.7, 0.9, 1)

(0.4, 0.5, 0.6) (0.1, 0.3, 0.5) (0, 0.1, 0.2) (0, 0.2, 0.3) (0.6, 0.70.8) (0.7, 0.9, 1)

(0.1, 0.3, 0.5) (0.4, 0.5, 0.6) (0, 0.2, 0.3) (0, 0.1, 0.2) (0.7, 0.9, 1) (0.6, 0.70.8)

Average weights

Defuzzify fuzzy weights

(0.25, 0.4, 0.55) (0.25, 0.4, 0.55) (0, 0.15, 0.25) (0, 0.15, 0.25) (0.675, 0.85, 0.95) (0.675, 0.85, 0.95)

0.4 0.4 0.13 0.13 0.825 0.825

Table 19 Normalized decision matrix. Rapid prototyping processes

C1

C2

C3

C4

C5

C6

RPP1 RPP2 RPP3 RPP4 RPP5 RPP6

0.17 0.22 0.185 0.27 0.14 0.88

0.18 0.34 0.58 0.55 0.09 0.43

0.71 0.43 0.32 0.27 0.32 0.05

0.27 0.46 0.54 0.54 0.32 0.05

(0.29, 0.38, 0.43) (0.29, 0.38, 0.43) (0.2, 0.29, 0.38) (0.11, 0.2, 0.29) (0.29, 0.38, 0.43) (0.38, 0.43, 0.44)

(0.24, 0.35, 0.45) (0.24, 0.35, 0.45) (0.35, 0.45, 0.52) (0.13, 0.24, 0.35) (0.13, 0.24, 0.35) (0.04, 0.13, 0.24)

Table 20 Weighted normalized decision matrix. Rapid prototyping processes

C1

C2

C3

C4

C5

C6

RPP1 RPP2 RPP3 RPP4 RPP5 RPP6

(0.042, 0.068, 0.093) (0.055, 0.088, 0.12) (0.046, 0.074, 0.101) (0.067, 0.108, 0.14) (0.035, 0.056, 0.077) (0.22, 0.35, 0.48)

(0.045, 0.072, 0.099) (0.085, 0.13, 0.18) (0.14, 0.23, 0.32) (0.13, 0.22, 0.3) (0.022, 0.036, 0.049) (0.107, 0.17, 0.23)

(0, 0.106, 0.177) (0, 0.064, 0.107) (0, 0.048, 0.08) (0, 0.04, 0.067) (0, 0.048, 0.08) (0, 0.007, 0.012)

(0, 0.04, 0.067) (0, 0.069, 0.115) (0, 0.081, 0.13) (0, 0.081, 0.13) (0, 0.048, 0.08) (0, 0.007, 0.012)

(0.19, 0.32, 0.41) (0.19, 0.32, 0.41) (0.13, 0.24, 0.36) (0.074, 0.17, 0.27) (0.19, 0.32, 0.40) (0.25, 0.36, 0.41)

(0.16, 0.29, 0.42) (0.16, 0.29, 0.42) (0.23, 0.38, 0.49) (0.087, 0.2, 0.33) (0.087, 0.2, 0.33) (0.027, 0.11, 0.228)

Table 21 Defuzzified weighted normalized decision matrix. Rapid prototyping processes

C1

C2

C3

C4

C5

C6

RPP1 RPP2 RPP3 RPP4 RPP5 RPP6

0.067 0.087 0.073 0.105 0.056 0.35

0.072 0.131 0.23 0.216 0.035 0.169

0.094 0.057 0.042 0.035 0.042 0.006

0.035 0.061 0.07 0.07 0.042 0.006

0.306 0.306 0.243 0.171 0.303 0.34

0.29 0.29 0.366 0.205 0.205 0.121

Positive ideal solutions (R⁄)

0.056

0.035

0.094

0.07

0.171

0.121

Negative ideal solutions (R)

0.35

0.23

0.006

0.006

0.34

0.366

The ideal separation matrix (D⁄) and anti-ideal separation matrix (D–) are constructed by using Eqs. (24) and (25), and are provided below (Step 9). and

2

0:011 0:037

0

0:035 0:135 0:169

3

6 0:031 0:096 0:037 0:009 0:135 0:169 7 7 6 7 6 6 0:017 0:195 0:052 0 0:072 0:245 7   7; D ¼ ½dij  ¼ 6 6 0:049 0:181 0:059 0 0 0:084 7 7 6 7 6 4 0 0 0:052 0:028 0:132 0:084 5 0:294 0:134 0:088 0:694 0:169

0

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Table 22 The values H i , Ii and CI by the proposed novel fuzzy modified TOPSIS. Rapid prototyping processes

Hi

Ii

CI ¼ H i þ Ii

Final ranking

RPP1 RPP2 RPP3 RPP4 RPP5 RPP6

1.4032 1.4192 (1.2244 + 1.5615) = 2.7859 1.5821 1.3594 (1.4821 + 2.5957) = 4.0778

(0.8536 + 2.1482) = 3.001 (0.8839 + 2.1333) = 3.0172 (0.9134 + 2.7515 + 0.0644) = 3.7293 (0.8484 + 2.2725) = 3.1209 (0.8163 + 2.1447) = 2.961 (1.0550 + 4.5249 + 0.0995) = 5.6794

4.4042 4.4364 6.5152 4.703 4.3204 9.7572

2 3 5 4 1 6

and

2

0:283 0:158 0:088 0:029 0:034 0:076

3

6 0:263 0:099 0:051 0:055 0:034 0:076 7 7 6 7 6 7 6 0:277 0 0:036 0:064 0:097 0  7 D ¼ ½dij  ¼ 6 6 0:245 0:014 0:029 0:064 0:169 0:161 7: 7 6 7 6 4 0:294 0:195 0:036 0:036 0:037 0:161 5 0

0:061

0

0

0

0:245

Finally, the values of the H i , Ii and CI are computed by using Eqs. (26)–(28) and are shown in Table 22 (Steps 10 and 11). According to Table 22, the ranking order of six potential rapid prototyping processes is RPP5, RPP1, RPP2, RPP4, RPP3 and RPP6 for the problem of rapid prototyping process selection. Hence, the best alternative versus six selected criteria is rapid prototyping process 5. 6. Conclusion Fuzzy multi-criteria analysis under the group decision making process provides an effective framework for ranking and selecting the potential alternatives in terms of their overall performance with respect to conflicting criteria. In this paper, a novel fuzzy modified TOPSIS method was presented that could reflect both subjective judgment and objective information in real-life decision making. The proposed method was on the basis of the concepts of positive ideal and negative ideal points for solving decision making problems with multi-judges and multi-criteria in a fuzzy environment. In the presented method, the performance rating values of alternatives with respect to the selected criteria as well as the relative importance of the criteria were expressed in linguistic terms; then, these linguistic terms were converted into triangular fuzzy numbers. The new collective index was developed to discriminate among alternatives in the assessment and selection process by constructing ideal separation and anti-ideal separation matrixes concurrently. Finally, for the purpose of proving the validity and suitability of the proposed method, two illustrative examples were presented for the robot selection and rapid prototyping process selection in the manufacturing environment. Although the presented fuzzy modified TOPSIS method was applied for the manufacturing decisions, it can be utilized for making a best decision in any other areas of engineering and management problems. Acknowledgement The authors are grateful for the partially financial support from the University of Tehran under the research Grant No. 8106043/1/16. References [1] Z. Yue, A method for group decision-making based on determining weights of decision makers using TOPSIS, Appl. Math. Model. 35 (2011) 1926–1936. [2] B. Vahdani, M. Zandieh, R. Tavakkoli-Moghaddam, Two novel FMCDM methods for alternative-fuel buses selection, Appl. Math. Model., doi:10.1016/ j.apm.2010.09.018 [3] S. Ebrahimnejad, S.M. Mousavi, H. Seyrafianpour, Risk identification and assessment for build-operate-transfer projects: a fuzzy multi attribute decision making model, Expert Syst. Appl. 37 (2010) 575–586. [4] S.M.H. Mojtahedi, S.M. Mousavi, A. Makui, Project risk identification and assessment simultaneously using multi-attribute group decision making technique, Saf. Sci. 48 (4) (2010) 499–507. [5] H. Malekly, S.M. Mousavi, H. Hashemi, A fuzzy integrated methodology for evaluating conceptual bridge design, Expert Syst. Appl. 37 (2010) 4910– 4920. [6] S.J. Chen, C.L. Hwang, Fuzzy Multiple Attribute Decision Making Methods and Applications, Springer, Berlin, 1992. [7] M.-S. Kuo, G.-H. Tzeng, W.-C. Huang, Group decision-making based on concepts of ideal and anti-ideal points in a fuzzy environment, Math. Comput. Model. 45 (2007) 324–339. [8] A. Makui, S.M.H. Mojtahedi, S.M. Mousavi, Project risk identification and analysis based on group decision making methodology in a fuzzy environment, Int. J. Manage. Sci. Eng. Manage. 5 (2) (2010) 108–118. [9] H.S. Shih, H.J. Syur, E.S. Lee, An extension of TOPSIS for group decision making, Math. Comput. Model. 45 (2007) 801–813. [10] H. Deng, C.-H. Yeh, R.J. Willis, Inter-company comparison using modified TOPSIS with objective weights, Comput. Oper. Res. 27 (2000) 963–973.

B. Vahdani et al. / Applied Mathematical Modelling 35 (2011) 4257–4269

4269

[11] S.M. Mousavi, H. Malekly, H. Hashemi, S.M.H. Mojtahedi, A two phase fuzzy decision making methodology for bridge scheme selection, in: Proc. of the IEEE International Conference on Industrial Engineering and Engineering Management, M. Helander, M. Xie, R. Jiao (Eds.), Singapore, 8-11 December, pp. 415–419, 2008. [12] H.S. Byun, K.H. Lee, A decision support system for the selection of a rapid prototyping process using the modified TOPSIS method, Int. J. Adv. Manuf. Technol. 26 (2005) 1338–1347. [13] T.-C. Chu, Y.-C. Lin, A fuzzy TOPSIS method for robot selection, Int. J. Adv. Manuf. Technol. 21 (2003) 284–290. [14] R.V. Rao, Decision Making in the Manufacturing Environment Using Graph Theory and Fuzzy Multiple Attribute Decision Making Methods, SpringerVerlag, 2007. [15] C.-T. Chen, C.-T. Lin, S.-F. Huang, A fuzzy approach for supplier evaluation and selection in supply chain management, Int. J. Prod. Econ. 102 (2006) 289–301. [16] M. Heydar, R. Tavakkoli-Moghaddam, S.M. Mousavi, S.M.H. Mojtahedi, Fuzzy multi-criteria decision making method for temporary storage design in industrial plants, in: Proc. of the IEEE International Conference on Industrial Engineering and Engineering Management, M. Helander, M. Xie, R. Jiao (Eds.), Singapore, 8–11 December, pp. 1154–1158, 2008. [17] C.-T. Chen, Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets Syst. 114 (2000) 1–9. [18] R.V. Rao, K.K. Padmanabhan, Selection, identification and comparison of industrial robots using digraph and matrix methods, Robot. Comput. Int. Manuf. 22 (2006) 373–383. [19] I. Mahdavi, N. Mahdavi-Amiri, A. Heidarzade, R. Nourifar, Designing a model of fuzzy TOPSIS in multiple criteria decision making, Appl. Math. Comput. 206 (2008) 607–617. [20] T.-C. Chu, Y.-C. Lin, An interval arithmetic based fuzzy TOPSIS model, Expert Syst. Appl. 36 (2009) 10870–10876. [21] T.-C. Wang, H.-D. Lee, Developing a fuzzy TOPSIS approach based on subjective weights and objective weights, Expert Syst. Appl. 36 (2009) 8980–8985. [22] R. Kumar, R.K. Garg, Optimal selection of robots by using distance based approach method, Robot. Comput. Int. Manuf. 26 (2010) 500–506. [23] P. Chatterjee, V. Manikrao Athawale, S. Chakraborty, Selection of industrial robots using compromise ranking and outranking methods, Robot. Comput. Int. Manuf. 26 (2010) 483–489. [24] C.L. Hwang, K. Yoon, Multiple Attributes Decision Making Methods and Applications, Springer, Berlin Heidelberg, 1981. [25] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1991. [26] T.J. Ross, Fuzzy Logic with Engineering Applications, Second ed., John Wiley & Sons Ltd,, 2004. [27] G.H. Liang, M.-J.J. Wang, A fuzzy multi-criteria decision making approach for robot selection, Robot. Comput. Int. Manuf. 10 (1993) 267–274.