Method to Determine the Criteria Weights

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Method to Determine the Criteria Weights

Reza Tavakkoli-Moghaddam¹⁽^&⁾, Hossein Gitinavard², Seyed Meysam Mousavi³, and Ali Siadat⁴

1 School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

[email protected]

2 School of Industrial Engineering,

Iran University of Science and Technology, Tehran, Iran [email protected]

3 Industrial Engineering Department, Faculty of Engineering, Shahed University, Tehran, Iran

[email protected]

4 LCFC, Arts et Métier Paris Tech, Centre de Metz, Metz, France [email protected]

Abstract. In a multi-criteria group decision analysis, numerous methods have been developed and proposed to determine the weight of each criterion; however, the group decision methods, except AHP, have rarely considered for obtaining the criteria weights. This study presents a new TOPSIS method based on interval-valued hesitant fuzzy information to compute the criteria weights. In this respect, the weight of each expert and the experts’ judgments about the criteria weights are considered in the proposed procedure. In addition, an application example about the location problem is provided to show the capability of the proposed weighting method. Finally, results of the proposed method are compared with some methods from the related literature in the presented illustrative example to show the validation of the proposed interval-valued hesitant fuzzy TOPSIS method.

Keywords: Criteria weights

Group decision making

Interval-valued hesitant fuzzy set

Utility degree

Individual regret

1 Introduction

In modern group decision analysis, multi-criteria group decision making (MCGDM) problems are important part of operations research, which can rank the candidate potential alternative regarding to experts’judgments. One of the main factors that can affect ranking results is the criteria weight. In some decision problems, the authors focused on determining the criteria weights based on subjective, objective and integrated methods.

The subjective methods compute the criteria weights based on preferences and judgments of experts, such as ranking ordering method of criteria [1–3], direct rating method [4, 5], Delphi method [6], eigenvector method [7], point allocation method

©Springer International Publishing Switzerland 2015

B. Kamiński et al. (Eds.): GDN 2015, LNBIP 218, pp. 157–169, 2015.

DOI: 10.1007/978-3-319-19515-5_13

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[8,9], linear programming of preference comparisons [10], and linear programming techniques for multidimensional analysis of preferences [11]. In addition, the objective methods calculate the criteria weights by utilizing the information of objective decision matrix, such as criteria’ importance through inter-criteria correlation method [12], entropy method [6,12], maximizing deviation method [13,14], and standard deviation method [12,15]. Thus, in integrated methods, the weight of each criterion is deter- mined based on considering the objective decision matrix and experts’ subjective judgments [16,17].

In real-world, the natures of the objects have been uncertain and imprecise, because the preferences and judgments of experts are hesitant or vague. Therefore, the criteria of group decision-making problems in an uncertain condition should be expressed by fuzzy values [18,19], such as interval values [20, 21], linguistic variables [21, 22], intuitionistic fuzzy values [23, 24], hesitant fuzzy sets [25,26], and interval-valued hesitant fuzzy sets [27,28].

Fuzzy sets theory and its extensions have widely utilized in imprecise situations for evaluating the problems in manyﬁelds, such as artiﬁcial intelligence [29], management [30], pattern recognition [31] and group decision making [32, 33]. Hence, in fuzzy group decision-making problems the criteria weights is an important issue to provide the best solution. Therefore, some researchers have studied on determining the criteria weights regarding to the uncertain environment. In this respect, Fan et al. [34] proposed an optimization model to determine the criteria weights by according to the experts’ fuzzy judgments and objective fuzzy decision matrices. Wang and Parkan [35] presented a general multi-attribute decision making framework by considering the objective information and subjective preferences to determine the criteria weights under fuzzy environment. Chen and Lee [36] proposed a fuzzy AHP method regarding to triangular fuzzy numbers for determining the criteria weights of professional conference organizer.

In some complex situations, the experts have defined their preferences and judgments by assigning some interval-values membership degrees for an object under a set to decrease the uncertainty risk and margin of errors. Therefore, the interval-valued hesitant fuzzy set (IVHFS), whichfirst introduced by Chen et al. [27] is a powerful tool to deal with these situations. Thus, each criterion can be defined based on IVHFS and expressed in terms of experts’preferences. In this case, Zhang et al. [37] proposed an objective weighting approach by utilizing the Shannon information entropy under a hesitant fuzzy environment. Xu and Zhang [38] developed an optimization model regarding to the maximizing deviation method to determine the criteria weights under hesitant fuzzy and interval-valued hesitant fuzzy-environments. They proposed a hybridized group decision making method under some steps to specify the criteria weighs which led to be more easy to use versus the optimization model. Beg and Rashid [39] proposed a method to aggregate the preferences expert’s judgments among the different criteria, in which the experts’opinions are expressed based on the hesitant fuzzy linguistic variables sets. Zhang et al. [40] constructed a hesitant fuzzy multiple attribute group decision making approach based on the distance measure to avoid the aggregation complexity of the hesitant fuzzy information. Feng et al. [41] utilized the TOPSIS (technique for order performance by similarity to ideal solution) method to solve the hesitant fuzzy multiple attribute decision making problems, in which the

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weight information are completely known. The literature review shows that determining the criteria weights based on ranking methods and especially under the IVHF- environments is still an open problem. In this paper, a hybridized group decision making approach is proposed based on the TOPSIS method and preferences experts’ judgments about the criteria weights to determine the weight of each criterion under hesitant fuzzy environment. In addition, a group of experts is established to assess the problem based on the linguistic variables that indicate their subjective preferences. In sums, some merits and advantages of this study, which provide the proposed method to be more precise are expressed as follows: (1) Proposing a new TOPSIS method in an interval-valued hesitant fuzzy setting; (2) a group of experts is established to evaluate the problem by assigning their opinions by linguistic terms based on the interval-valued hesitant fuzzy information, which converted to interval-valued hesitant fuzzy elements;

and (3) proposing a new relative closeness index to obtain the criteria weights. In addition, the weight of experts is applied in procedure of the proposed method. The validation of the proposed approach is obtained by comparing with other weighting methods for determining criteria weights.

The rest of this paper organized as follows. In Sect.2, some methods to determine the criteria weights are explained and the interval-valued hesitant fuzzy TOPSIS (IVHF-TOPSIS) method for estimating the criteria weights are elaborated. In Sect.3, an illustrative example in the selection of the best site for building a new factory provided to show the implementation process of the proposed approach. Finally, in Sect.4, the paper is concluded.

2 Proposed Method

In this section, three techniques for computing the weight of criteria under the interval- valued hesitant fuzzy environment are extended to compare the computational results of criteria weights with the proposed approach.

2.1 Methods of Determining the Criteria Weights

In this subsection, these three approaches are considered to compute the criteria weights. Firstly, the criteria weights can be computed based on maximizing a deviation method introduced by Xu and Zhang [38] when the information completely unknown.

In this paper, the method is extended based on the interval-valued hesitant fuzzy Hamming distance measure to determine the optimal weight vector as follows; let h¼hh^L_ij;h^U_iji

mn is an interval-valued hesitant fuzzy element:

w_j¼

P

m i¼1

P

m r¼1

1 2l

P

l k¼1

h^{r k}_ij^{ð Þ}^L h^{r k}_rj^{ð Þ}^L

þh^{r k}_ij^{ð Þ}^U h^{r k}_rj^{ð Þ}^U

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pⁿ

j¼1

P^m

i¼1

P^m

r¼1 1 2l

P^l

k¼1

h^{r k}^{ð Þ}

L

ij h^{r k}^{ð Þ}

L

rj

þ h^{r k}^{ð Þ}

U

ij h^{r k}^{ð Þ}

U

rj

²

s ð1Þ

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w_j ¼ wj

Pⁿ

j¼1

wj

; j¼1;2;. . .;n ð2Þ

where theh^{r k}^{ð Þ}

L

ij ,h^{r k}^{ð Þ}

U

ij ,h^{r k}^{ð Þ}

L

rj and h^{r k}^{ð Þ}

U

rj are the kth largest value inh^L_ij,h^U_ij,h^L_rj and h^U_rj, respectively and also thew_j is the normalized criteria weight vector.

Secondly, the decision makers (DMs) specify the relative importance of criteria weights by linguistic variables that can be converted to the IVHFS and denoted bytj¼hl^L_j;l^U_j i

. In this respect, the ﬁnal aggregated criteria weights regarding to DMs’judgments are obtained by the HIVFG operator are as follows:

tj¼HIVFG ~h1;~h2;. . .;~hK

¼ ^K

k¼1 k^f_k~hk

¹_K

¼ [_~_c

12~h₁;~c₂2~h₂;...;~c_k2~h_k

Y^K

j¼1

k^L_kc^L_k

_K¹

; Y^K

j¼1

k^U_kc^U_k

_K¹

" #

( )

ð3Þ

where the weight of each DM is represented ask^f_k¼k^L_k;k^U_k

and is considered in the computational process of the criteria weights to decrease the errors.

Thirdly, the above-mentioned methods can be hybridized, and thus the following relation for computing the criteria weights can be proposed by:

xj¼tj:wj ð4Þ

tj¼l^L_j þl^U_j

2 8j ð5Þ

x_j ¼ xj

Pⁿ

j¼1

xj

; j¼1;2; :. . .n ð6Þ

wherex_j is the normalized criteria weight vector.

2.2 TOPSIS Method with IVHFS

In this subsection, the proposed novel TOPSIS method is introduced based on the IVHFSs. In this respect, the DMs’ opinions about the relative importance of each criterion are considered in process of the proposed method. Therefore, the procedure of the proposed method is deﬁned based on the following steps:

Step 1. Construct an interval-valued hesitant fuzzy decision matrix (IVHF-decision matrix) for each criterion (Cj; 1,2,…,n) regarding to the possible alternatives (Ai; 1,2,

…,m) and opinions of each DM (k; 1,2,…,K).

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k₁ k₂ k_K

Gj¼ A₁

... A_m

l^L1_1j;l^U1_1j

h i

l^L2₁₂;l^U2₁₂

hl^Lk_1j;l^Uk_1ji

...

... .. . ... l^L1_mj;l^U1_mj

h i

l^L2_mj;l^U2_mj

h i

hl^Lk_mj;l^Uk_mji 0

B B B B B

@

1 C C C C C A

mk

8j ð7Þ

wherehl^Lk_mj;l^Uk_mji

is the interval-value membership degree for them-th alternative that expressed by thek-th expert to construct thej-th IVHF-decision matrix.

Step 2. Normalize each IVHF-decision matrix G^N_j ¼hg^Lk_ij ;g^Lk_iji

mk

based on the following relations [42]:

g^Lk_ij ¼ l^Lk_ij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P^m

i¼1

P^K

k¼1

½ðl^Lk_ijÞ²þ ðl^Uk_ij Þ²

s 8i;j;k ð8Þ

g^Uk_ij ¼ l^Uk_ij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P

m i¼1

P

K k¼1

½ðl^Lk_ij Þ²þ ðl^Uk_ij Þ²

s 8i;j;k ð9Þ

Step 3. Determine the interval-valued hesitant fuzzy positive ideal solution matrix (IVHF-PIS) and the interval-valued hesitant fuzzy negative ideal solution matrix (IVHF-NIS) by regarding to normalize IVHF-decision matrix as follows:

k1 k2 kK

A¼l^Lk_i ;l^Uk_i

mk¼

A1

... Am

l^L1₁ ;l^U1₁

l^L2₁ ;l^U2₁

l^LK₁ ;l^UK₁

...

... .. .

... l^L1_m ;l^U1_m

l^L2_m ;l^U2_m

l^LK_m ;l^UK_m 0

B B B

@

1 C C C A

mk

ð10Þ

k1 k2 kK

A ¼l_i^Lk;l_i^Uk

mk¼ A1

... Am

l₁^L1;l₁^U1

l₁^L2;l₁^U2

l₁^LK;l₁^UK ...

... .. .

... l_m^L1;l_m^U1

l_m^L2;l_m^U2

l_m^LK;l_m^UK 0

B B B

@

1 C C C A

mk

ð11Þ where the average of the group decision matrix is calculated by the following relations:

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l^Lk_i ¼1 n

Xⁿ

j¼1

l^Lk_ij 8i;k ð12Þ

l^Uk_i ¼1 n

Xⁿ

j¼1

l^Uk_ij 8i;k ð13Þ

l_i ^Lk¼min

j nl^Lk_ijo

8i;k ð14Þ

l_i ^Uk¼max

j nl^Uk_ij o

8i;k ð15Þ

Step 4. Compute the separation measure for each normalized IVHF-decision matrix from the IVHF-PIS matrix and the IVHF-NIS matrix by using the IVHF-Euclidean distance measure, which indicates byS_j and S_j , respectively.

S_j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2lx_i

X

m i¼1

X

K k¼1

X

l_xi

k¼1

l^{Lkr k}_i ^{ð Þ}ð Þxi A^{Lkr k}_i ^{ð Þ}ð Þxi

2

þl^{Ukr k}_i ^{ð Þ}ð Þxi A^{kUr k}_i ^{ð Þ}ð Þxi

2

v u u

t 8j

ð16Þ

S_j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2lxi

X^m

i¼1

X^K

k¼1

X

l_xi

k¼1

l^{Lkr k}_i ^{ð Þ}ð Þxi A_i^{Lkr k}^{ð Þ}ð Þxi

2

þl^{Ukr k}_i ^{ð Þ}ð Þxi A_i^{Ukr k}^{ð Þ}ð Þxi

2

v u u

t 8j:

ð17Þ Step 5. Specify the relative closenessC_j

for determining the most important criterion.

Cj¼

d S _j ;S d S _j ;S

þd S _j;S 8j ð18Þ

S¼1 n

Xⁿ

j¼1

S_j ð19Þ

S ¼1 n

Xⁿ

j¼1

S_j ð20Þ

where S the average of S_jðj¼1;2; ::;nÞ, and also the average ofS_j ðj¼1;2; ::;nÞ represented asS . The DM often speciﬁes their opinion by linguistic variables, which applied in our proposed method and established a new hybrid approach. In this respect, the hybrid relative closenessC_j^h

is deﬁned as follows:

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C_j^h¼ -j:d S _j ;S d S _j ;S

þd S _j;S 8j ð21Þ

-j¼HIVFG ~h1;~h2;. . .;~hK

¼ ^K

k¼1 k^f_k~hk

_K¹

¼ [_~_c

12~h1;~c22~h2;...;~ck2~hk

Y

K j¼1

k^L_kc^L_k _K¹

; Y

K j¼1

k^U_kc^U_k

_K¹

" #

( )

ð22Þ

-j¼l^L_j þl^U_j

2 8j ð23Þ

where theﬁnal weight of each DM is indicated bykk¼k^L_k;k^U_k

that is computed by:

k^L_k¼ P^m

i

Pⁿ

j

l^Lk_ij

P

K k

P

m i

P

n j

l^Lk_ij

ð24Þ

k^U_k ¼ P^m

i

Pⁿ

j

l^Uk_ij

P

K k

P

m i

P

n j

l^Uk_ij

ð25Þ

k^f_k¼ P^m

i

Pⁿ

j

l^Uk_ij þl^Lk_ij

P

K k

P

m i

P

n j

l^Uk_ij þl^Lk_ij

ð26Þ

where, theﬁnal DMs’weightsk^f_k is P

K k¼1

k^f_k¼1.

Step 6. Estimate the weight of each criterionw_jaccording to the relative closeness.

wj¼ C_j Pⁿ

j¼1

C_j

8j ð27Þ

3 Illustrative Example

In this section, an illustrative example about the location problem from the literature [28]

is considered to show the capability of the proposed method for determining the weight of each criterion. In this respect, for showing the veriﬁcation of the proposed method,

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the application example is solved under three decision approaches, and then we compare them with our decision method. The application example is about the best site selection for building a new factory that is established by four DMs (k = 1, 2, …, 4), three potential alternative (A1,A₂,A₃) and the following six criteria:C₁: Climate, Condition;

C₂: Regional demand; C₃: Expansion possibility; C₄: Transportation availability;

C₅: Labor force; andC₆: Investment cost.

As represented in Tables 1 and 2, the relative importance of each hesitant fuzzy linguistic term for rating the importance of each criteria and rating of potential alternatives are deﬁned, respectively. In addition, the evaluation of alternatives expressed by DMs’opinions with hesitant fuzzy linguistic terms that shown in Table3. Similarly, the DMs’ judgments about relative importance of each criterion are expressed as hesitant fuzzy linguistic terms in Table4. Then, these linguistic variables are converted to interval-valued hesitant fuzzy elements (IVHFEs).

The IVHF-decision matrix is normalized by regarding Eqs. (8) and (9); then, the IVHF-PIS matrix and the IVHF-NIS matrix are constructed by considering the relations Eqs. (10)−(15). In addition, the separation measure for each criterion is calculated by applying Eqs. (16) and (17). Then, the relative closeness is speciﬁed by Eqs. (18)−(20).

Also, in hybrid decision approach, the DMs’judgments about the relative importance of each criterion are considered and computed by Eqs. (21)−(26). The Eq. (27) is utilized to compute the weight of each criterion in the proposed approach and in the

Table 1. Hesitant fuzzy variables for rating the importance of criteria and the DMs Hesitant fuzzy linguistic variables IVHFE

Very important (VI) [0.90,0.90]

Important (I) [0.75, 0.80]

Medium (M) [0.50, 0.55]

Unimportant (UI) [0.35, 0.40]

Very unimportant (VUI) [0.10,0.10]

Table 2. Hesitant variables for the rating of possible alternatives Hesitant fuzzy linguistic variables IVHFE

Extremely good (EG) [1.00,1.00]

Very very good (VVG) [0.90,0.90]

Very good (VG) [0.80, 0.90]

Good (G) [0.70, 0.80]

Medium good (MG) [0.60, 0.70]

Fair (F) [0.50, 0.60]

Medium bad (MB) [0.40, 0.50]

Bad (B) [0.25, 0.40]

Very bad (VB) [0.10, 0.25]

Very very bad (VVB) [0.10,0.10]

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proposed hybrid approach. The above-mentioned results have shown in Tables5and6.

Other weighting techniques for determining the relative importance of criteria are extended and illustrated in Subsect. 2.1. In this respect, each extended technique in our application example is applied and for showing the low difference between them and proposed approaches, the mean value and the variance of criteria weights are provided.

Utilizing different techniques commonly leads to different results. It is unsuitable to say which method is powerful and capable because every method has various results underlying assertion or theory. However, the extended TOPSIS for the criteria weights is more capable for compromise of nearby to the ideal and farther from the negative ideal. Since the criteria weights of the proposed approaches are generated from the DMs’ judgments,“biased”or “false”judgments lead to a low weight [43].

Table 3. Linguistic evaluations by the decision makers Main

criteria

Alternatives k1 k2 k3 k4

C₁ A₁ MG MG G VG

A2 VG G G G

A3 F MG F MG

C₂ A₁ G VG VG MG

A2 F MG F F

A3 VG VG VG G

C3 A1 F MG MG MG

A2 VG G VG VG

A₃ MG MG G MG

C4 A1 VG VG G G

A2 G G G MG

A₃ G G VG F

C5 A1 MG G G VG

A2 MG MG VG G

A3 VG G G G

C6 A1 VVG VG VVG EG

A₂ F B B MG

A3 VVG VVG VG VVG

Table 4. Linguistic evaluations for weights of criteria assigned by the decision makers k₁ k₂ k₃ k₄

C1 I VI M I C2 M I M I C₃ I M I VI C4 I UI M M C5 M I I M C₆ M M I I

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In this regard, the proposed approach versus the classical/modern methods is more precise, because of two main features as considering the IVHFSs and the DMs’ opinions about the criteria weights. The IVHFSs aid to DMs by assigning some interval-values membership degrees for an element under a set to margin of errors. In addition, the preferences DMs’ judgments about the relative signiﬁcance of the criteria are provided in procedure of the proposed method to decrease the errors.

4 Conclusions and Future Direction

In group decision making problems, determining the relative importance of each criterion has been very important issue. In this regard, a novel TOPSIS method has been proposed by utilizing the IVHFS regarding to the experts weights and their opinions about the criteria weights. In the proposed approach, the preferences experts’ judgments have been expressed by linguistic variables which transformed to interval-valued hesitant fuzzy element. Hence, an illustrative example about the location problem has been considered to illustrate the steps of the proposed decision method. Finally, the

Table 5. Computational results ofSj*

, Sj−andCj

Sj*

Sj− Cj

S1*

0.111190 S1− 0.200000 C1 0.034653 S2*

0.098327 S2− 0.191599 C2 0.514718 S₃^* 0.119171 S₃⁻ 0.208143 C₃ 0.38869 S4*

0.073633 S4− 0.186870 C4 0.285241 S₅^* 0.060673 S₅⁻ 0.182734 C₅ 0.273389 S6*

0.173451 S6− 0.229551 C6 0.306194 S 0.1060746 S 0.199816

Table 6. Final criteria weight by the proposed method and hybridized method and comparative analysis

wj Proposed approach

Proposed hybridized approach

Maximizing deviation method

Linguistic variables

Hybridized

maximizing deviation method and linguistic variables

w1 0.019221 0.021825 0.146788 0.182418 0.163515

w₂ 0.285496 0.283351 0.183486 0.159441 0.178648

w3 0.215593 0.244808 0.146788 0.182418 0.163515

w4 0.158213 0.130957 0.091743 0.132973 0.074495

w₅ 0.151639 0.150499 0.091743 0.159441 0.089324

w6 0.169835 0.168558 0.339449 0.159441 0.330500

X 0.166667 0.166666667 0.166666667 0.1626893 0.16666667 r² 0.0077124 0.008448048 0.008430828 0.00033864 0.00829643

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results have been compared with several approaches implemented in a practical example to show the validation of the proposed approach. For future direction, the proposed method can be enhanced by considering the hierarchical structure of the criteria.

Acknowledgments. This work has been supported ﬁnancially by the Center for International Scientiﬁc Studies & Collaboration (CISSC) and the French Embassy in Tehran as well as the Partenariats Hubert Curien (PHC) program in France. Additionally, the authors would like thank anonymous reviewers for their valuable comments.

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