An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment

Accepted Manuscript

An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment Jindong Qin , Xinwang Liu , Witold Pedrycz PII: DOI: Reference:

S0377-2217(16)30811-6 10.1016/j.ejor.2016.09.059 EOR 14020

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

23 January 2015 28 June 2016 4 September 2016

Please cite this article as: Jindong Qin , Xinwang Liu , Witold Pedrycz , An extended TODIM multicriteria group decision making method for green supplier selection in interval type-2 fuzzy environment, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.09.059

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

Highlights  Proposed an extended TODIM behavior decision method to green supplier selection.  Proposed a bounded rationality method to MCGDM under type-2 fuzzy environment.  A new parametric distance measure for interval type-2 fuzzy set is proposed.  Comparisons between IT2F-TODIM and IT2F-TOPSIS research are made.

1

ACCEPTED MANUSCRIPT

An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment Jindong Qina, Xinwang Liub, Witold Pedryczc,d,e a

School of Management, Wuhan University of Technology, Wuhan, 430070, Hubei, China School of Economics and Management, Southeast University, Nanjing, 211189, Jiangsu, China c Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 d System

CR IP T

b

Research Institute, Polish Academy of Sciences, Warsaw, Poland

e

Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

AC

CE

PT

ED

M

AN US

Abstract: With the increasing economic globalization and intensification of market competition, green supply chain management (GSCM), as a new management mode to pursue both economic benefits and the coordinated of environment sustainable development, has become highly relevant topic in modern enterprise production operation management. The green supplier evaluation and selection is the essential core of the GSCM, which can directly impact the manufacturer‘s performance. The green supplier selection can be regarded as a multiple criteria group decision making (MCGDM) problem that involves many conflict evaluation criteria, both being of qualitative and quantitative nature. Due to the increasing complexity and uncertainty of social economic environment, some evaluations of criteria are not adequately represented by numerical assessments and type-1 fuzzy sets (T1FSs). In addition, the decision makers (DMs) usually do not exhibit complete rationality under many practical decision situations. In this paper, we extend the TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) technique to solve MCGDM problems within the context of interval type-2 fuzzy sets (IT2FSs) and present its application to green supplier selection problem. First, we introduce a new distance based on the fuzzy logic and  -cut of the IT2FSs. Then, an extended novel TODIM method based on prospect theory to solve MCGDM problem under IT2FSs environment is developed. Finally, a green supplier selection example is provided to demonstrate the usefulness of the proposed method. Furthermore, a sensitivity analysis carried out with the aid of granular computing and the comparative analysis with TOPSIS technique is also performed. Key words: Multiple criteria analysis; Green supplier selection; TODIM method; Interval type-2 fuzzy sets; Group decision making. 1. Introduction In view of the shortage of resources and environmental pollution being on the rise today, facing the requirement of resource conservation and environment friendliness to trade off the economic benefits and the environment sustainable development becomes an important issue in modern enterprise production operation management. Green supply chain management (GSCM) (Vanchon 2007; Cabral et al. 2012), which has emerged as a new management mode takes the environment 

Corresponding author. Tel.:+862787859059; fax: +862787859039. Email address: [email protected] (J.Qin). 2

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

performance into consideration. The GSCM involves numerous key links, such as green product design, green supplier evaluation, green production, green packaging and transportation, green marketing and resource recycling. Due to the fact that the green supplier is located upstream of the entire supply chain, it acts on the cost savings and environmental protection that be able to pass through all aspects of the downstream supply chain. It can effectively improve the compatibility of the supply chain and the environment performance. The green supplier selection is the core part of the GSCM (Blome et al. 2013), which can directly impacts on the manufacturer‘s and environment protection performance. It can be regarded as a multiple criteria group decision making (MCGDM) problem that involves many conflicting evaluation criteria, including such as resource consumption, green image, green competencies, and product life cycle cost. In practice, for green supplier selection problems, most of the evaluation detailed information is not known and many factors are impacted by uncertainty. As a result, the ―traditional‖ type-1 fuzzy sets (T1FSs) might be insufficient to model practical situations because of the increasing complexity of the problem at hand. In such cases, type-2 fuzzy sets (T2FSs) could be considered as viable technique for handling higher uncertainty (Mendel and John 2001). T2FSs were first proposed by Zadeh (1975), and can be regarded as a useful extension of T1FSs. They are characterized by two membership functions: primary membership function (PMF) and secondary membership function (SMF). Interval type-2 fuzzy sets (IT2FSs) are the most widely used in type-2 fuzzy sets, given that their computational complexity is much lower than the general type-2 fuzzy sets (GT2FSs). Therefore, they are easy to use in real-world management application areas. The TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) method was developed by Gomes and Lima (1992), which is a discrete multi-criteria decision making (MCDM) method derived from the prospect theory (Kahneman and Tversky 1979) and comes now as one of the most well-known classical MCDM methods in modern behavior decision theory. Compared with other behavior decision methods, the main advantage of the TODIM method is that the decision maker‘s bounded rationality behavior character is taken into account. The main reason is that the method is able to capture the loss and gain under uncertainty from the view of reference point and the decision maker is more sensitive to the loss. Moreover, in complete rationality decision making, the decision maker pursuits utility maximization, while in TODIM method, the decision maker aims to value function maximization. Therefore, the TODIM method can be regarded as a useful bounded rationality behavioral decision making method. Gomes and Rangel (2009) presented an evaluation study of residential properties carried out together with real estate agents by using TODIM method of multi-criteria aiding. Gomes et al. (2009) studied the natural gas destination in Brazil with the aid of TODIM approach. Pereira et al. (2014) proposed a robustness analysis in a TODIM-based multi-criteria evaluation model of rental properties. Furthermore, the version of the TODIM method for solving sorting and classification problems is presented in Passos et al. (2014). The version of TODIM applicable when there are interactions between criteria is available in Gomes et al. (2014). Recently, a number of studies have extended the TODIM method to a variety of fuzzy environments (Lourenzutti and Krohling 2013; Fan et al. 2013; Krohling et al. 2013; Zhang and Xu 2014; Tseng et al. 2014; Liu and Teng 2014 a,b). However, little attention has been paid to the extension of the TODIM into high type fuzzy environment. As mentioned above, the green supplier selection process both involves the uncertainty information fusion and DM‘s behavior character, most of the evaluation is not known

3

ACCEPTED MANUSCRIPT

CR IP T

and many factors are impacted by uncertainty. As a result, the type-1 fuzzy sets and other extended versions might be insufficient to model practical situations because of the increasing complexity of the green supplier selection problem. In such cases, the IT2FSs could be as one of the most useful tool to handle this problem because it can easily express uncertainty. Meanwhile, the TODIM can sufficiently reflect the DM‘s bounded rationality character based on the prospect theory, and the TODIM method is able to test specific forms of the loss and gain functions (risks) under uncertainty. Therefore, it is justifiable to study the extended the TODIM method for green supplier selection within the context of IT2FSs. Motivated by this idea, we extend the TODIM method to accommodate interval type-2 fuzzy environment and further develop an interval type-2 fuzzy TODIM MCGDM method based on decision maker‘s bounded rationality behavior for handling real-life green supplier selection problems, which is the essential objective of this study. The paper is structured as follows. Section 2 reviews some related literature about green supplier evaluation and selection. Section 3 briefly introduces some basic concepts related to IT2FSs and TODIM method. Section 4 defines a new distance based on fuzzy logic and  -cut for

AN US

IT2FSs. Section 5 extends the TODIM to develop an MCGDM method under interval type-2 fuzzy environment. Section 6 provides a case study concerning green supplier selection example and covers a sensitivity and comparison analysis with the aid of granular computing to demonstrate the applicability and validity of the proposed methodology. Finally, some conclusions are presented in Section 7. 2. Literature review

AC

CE

PT

ED

M

Recently, a large number of methods for green supplier selection have been developed. Handfield et al. (2002) used AHP method to evaluate the environmental criteria to green supplier assessment. Lee et al. (2009) developed an extended fuzzy AHP decision model for green supplier selection, which took the environment issue into consideration, and applied this model to high-tech industry. Tsai and Huang (2009) developed a fuzzy goal programming method for green supplier selection optimization under activity-based costing and performance evaluation with a value-chain structure. Zhu et al. (2010) proposed a portfolio–based analytic method for green supplier management performance. Bai and Sarkis (2010) put forward an analytical evaluation based on rough set theory for green supplier selection. Kuo et al. (2010) integrated artificial neural network and MADA methods for green supplier selection. Fu et al. (2012) studied the evaluation of green supplier development programs at a telecommunications systems provider based on formalized grey-based DEMATEL methodology. Buyukozkan and Cifci (2012) proposed a novel hybrid MCDM approach based on fuzzy DEMATEL, TOPSIS and ANP to evaluate the green suppliers. Wan and Chan (2012) proposed a hierarchical fuzzy TOPSIS approach to improve the organization strategy in implementing green practices. Shen et al. (2013) proposed a fuzzy multiple criteria decision making for green supplier selection with linguistic preference. Hsu et al. (2013) proposed the DEMATEL decision model of carbon supplier selection. Kannan et al. (2014) proposed an approach to green supplier selection based on fuzzy axiomatic design, and presented a case study. Akman (2014) integrated the Fuzzy C Means (FCM) clustering technique and VIKOR method for evaluating green supplier selection. Dou et al. (2014) presented a grey analytical network process-based methodology to identify green development programs, which can directly impact the suppliers‘ performance. Dobos and Vörösmarty (2014)

4

ACCEPTED MANUSCRIPT

used data envelopment analysis (DEA) method with environmental, green issues. Blome et al. (2014) utilized the structure equation to analyze an impact of green procurement and green supplier development on supplier performance. Tsui et al. (2014) integrated the preference ranking organization method for enrichment evaluations (PROMETHEE) and the influential network relation map (INRM) for enhancing the reliability of green supplier selection in TFT-LCD industry. Hashemi et al. (2015) integrated green supplier selection approach with analytic network process and improved grey relational analysis. The overview of the previous literature on green supplier selection is provided in Table 1. Table 1 Summary of some of the relevant research on green supplier selection Method

Review paper

Overview

AHP

Fuzzy MCDM model

TOPSIS

M

DEMATEL/ PROMETHEE

ED

Integrated hybrid methods

PT

Fuzzy optimization model

DEA

Fuzzy C means

Fuzzy logic

Fuzzy Inference

CE

Fuzzy clustering

AC

Uncertain theory

Other methodologies

Govindan et al. (2015) Noci (1997); Handfield et al. (2002); Lu et al. (2007); Chiou et al. (2008); Grisi et al.(2010); Lee et al. (2009) Buyukozkan and Cifci (2011); Hsu and Hu (2007, 2009) Awasthi et al. (2010); Wan and Chan (2012); Roshandel et al.(2013); Kannan et al. (2014) Hsu et al. (2013); Tsui et al. (2014) Yan (2009); Kuo et al. (2010); Wen and Chi (2010); Buyukozkan and Cifci (2012) Kumar and Jain (2010); Dobos and Vörösmarty (2014); Kuo and Lin (2011); Tsai and Huang (2009) Akman (2014); Keskin (2014) Humphreys et al. (2006) Kannan et al. (2014) Li and Zhao (2009); Fu et al. (2012); Hashemi et al. (2015); Dou et al. (2014) Bai and Sarkis (2010) Large and Thomsen (2011) ; Blome et al. (2014) Feyzioglu and Buyukozkan (2010) Hong-jun and Bin (2010)

AN US

ANP

Literature

CR IP T

Category

Grey relational analysis Rough set theory Structural equation model Choquet integral Factor Analysis

As shown in Table 1, we can see that the fuzzy sets theory has been extended to green supplier selection problems. However, little attention has been paid to higher type-2 fuzzy environment to handle multiple criteria green supplier selection. Therefore, it is beneficial to 5

ACCEPTED MANUSCRIPT

integrate the multiple criteria decision making method and its preference information within the interval type-2 fuzzy information. It can not only enhance the model ability of high-order uncertainties, but also address green supplier selection problems with imprecise and uncertain decision information. 3. Fundamentals of TODIM method and interval type-2 fuzzy set theory

CR IP T

3.1 TODIM decision model The TODIM method is a useful behavior decision technique based on prospect theory, which can capture the decision maker‘s bounded rationality in the process of making an actual decision. The basic idea behind this method is to determine the dominance degree of each alternative over the others by using the utility function coming from the prospect theory. First, let us consider a decision matrix R  (rij )mn , which is composed of alternatives and criteria described in the following way

where A1 , A2 ,

C2

Cn

A1 r11 A2 r21

r12 r22

r1n r2 n

Am rm1

rm 2

rmn

AN US

R

C1

, Am are m alternatives, C1 , C2 ,

, Cn are n criteria, rij is the rating of the

M

alternative Ai with respect to criterion C j , and   (1 , 2 , associated with the set of criteria C  {C1 , C2 ,

(1)

, n )T is the weight vector

, Cn } , which satisfies the following conditions

n

ED

 j [0,1] and  j 1 j  1 .

The TODIM method involves the following steps:

PT

Step 1.Calculate the relative weight  jr of the criterion C j to the reference criterion Cr

 jr 

j ( j  1, 2, r

AC

CE

expressed as follows:

, n)

(2)

where  j is the weight of the criterion C j and r  max{ j } . j

Step 2.Calculate the dominance degree of each alternative Ai over each alternative Ak with respect to criterion C j ( j  1, 2,

, n) using the following expression:

6

ACCEPTED MANUSCRIPT   (r  r ) jk ij kj  n   jk   j 1   j ( Ai , Ak )  0  n   jk (rkj  rij )  1  j 1   jk  

if rij  rkj  0 if rij  rkj  0

(3)

if rij  rkj  0

CR IP T

where the parameter  indicates the attenuation factor of the losses.  stands for the attenuation factor of the losses. Different choices of  lead to different shapes of the prospect theoretical value function in the negative quadrant. The range of the values of this parameter is   0 , if 0    1 , then the influence of loss will increase; if   1 , then the influence of loss will decrease.

AN US

Step 3. Calculate the overall dominance degree of each alternative Ai over each alternative Ak with respect to criterion C j in the following form:

n

 ( Ai , Ak )    j ( Ai , Ak )

(4)

j 1

Step 4. Calculate the global prospect value of the alternative Ai (i  1, 2,

M

, m) according to the

following expression:

ED

  ( A , A )  min   ( A , A )  (A )  max   ( A , A )  min   ( A , A ) m

k 1

i

m

i

k

k 1

i

i

m

k 1

i

k

k 1

i

i

(5)

k

PT

i

k

m

Step 5. Rank all the alternatives based on the global prospect values of alternatives. The higher the

 ( Ai ) is, the better alternative Ai is.

CE

value of

3.2 Interval type-2 fuzzy sets theory Definition 1 (Mendel and John 2002). Let X be the universe of discourse, a type-2 fuzzy set A can

AC

be represented by type-2 membership function  A ( x, u ) as follows:

A  (( x, u),  A ( x, u)) | x  X , u  J x  [0,1]

(6)

where J x denotes an interval in [0,1] . Moreover, the type-2 fuzzy set can also be expressed as the following form:

A

xX

where J x



uJ x

 A ( x, u) ( x, u)  

xX

(

uJ x

[0,1] is the primary membership at x , and u Jx

7

 A ( x, u) u) x A

(7)

( x, u ) u indicates the second

ACCEPTED MANUSCRIPT membership at x . For discrete space,

is replaced by the summation

.

Definition 2 (Mendel and John 2002). Let A be a type-2 fuzzy sets in the universe of discourse

X represented by a type-2 membership function  A ( x, u) . If all  A ( x, u)  1 , then A is called an interval type-2 fuzzy sets (IT2FSs). An interval type-2 fuzzy set can be regarded as a special case of the type-2 fuzzy sets, which is defined in the form: xX



uJ x

1 ( x, u )  

xX



uJ x

1u

x

(8)

CR IP T

A

It is obvious that the IT2FSs A defined in X is completely determined by the primary membership which is called the footprint of uncertainty (FOU). The FOU can be expressed as follows:

FOU( A) 

Jx 

xX

xX

( x, u) | u  J x  [0,1]

(9)

L

U

AN US

Because of the operations on IT2FSs are computationally demanding, so IT2FSs are usually considered in some simplified form. Here, we follow the results of Chen (2013), who adopted trapezoid interval type-2 fuzzy sets (TrIT2FSs) for solving MCGDM problems. Definition 3 (Chen 2013). Let A and A be two generalized trapezoidal fuzzy sets (TrFSs), where L

U

the height of a generalized fuzzy number is positioned in [0, 1]. Let hA and hA be the heights L

U

( AL , AU )

L

L

L

(a1L , a2L , a3L , a4L ; hAL ),(a1U , a2U , a3U , aU4 ; hUA )

ED

A L

M

of A and A , respectively. An IT2TrFN A in the universe of discourse X is defined as:

L

U

U

U

U

(10)

U

where a1 , a2 , a3 , a4 , hA , b1 , b2 , b3 , b4 , hA are all real numbers and satisfying the inequality

a2L

a3L U

a4L , b1U

b2U

b3U

b4U ,0

hAL

hUA

PT

a1L

1 . The upper membership function L

AC

CE

(UMF) A ( x) and the lower membership function (LMF) A ( x) are defined as:

U

A ( x)

( x a1U )hUA a2U a1U

a1U

x

a2U

hUA

a2U

x

a3U

(a4U x)hUA a4U a3U

a3U

x

a4U

0

otherwise

and

8

(11)

ACCEPTED MANUSCRIPT

AL ( x)

( x a1L )hAL a2L a1L

a1L

x

a2L

hAL

a2L

x

a3L

(a4L x)hAL a4L a3L

a3L

x

a4L

0

otherwise

( x)

CR IP T

A

AU

U A

h

hAL

a1U

a1L a2U a2L

AN US

AL

0

(12)

a3L

a3U a4L

a4U

x

M

Fig.1 An interval type-2 fuzzy sets A with geometrical interpretation Liu and Mendel (2011) proposed the analytic centroid computation method, which can be regarded as the continuous form of Karnik-Mendel (KM) algorithm, the centroid interval



ED

[C AL , CUA ] of A can be obtained by solving the following two equations: C AL

a1U

CA

PT

and

a1U

(CAL  x) AU ( x)dx   L (C AL  x) AL ( x)dx  C AL

C AR

aU 4

a1

CA

(14)

CE

 U (CAL  x) AL ( x)dx   R (CAL  x) AU ( x)dx  CAR

(13)

3.3 The ranking method for interval type-2 fuzzy sets

AC

Definition 4 (Qin and Liu 2015). Let A be an IT2FSs, then the single ranking of A can be defined using the following expressions:

a 4

a a h h R(1) ( A)    2 4  U 1

U 4

L A

U A

   

i 1

L i

 aiU 

8

1 4   R(2) ( A)   a1U  a4U   hAL hUA  4   8  aiL aiU i 1  

9

(15)

(16)

ACCEPTED MANUSCRIPT

 aU aU h L hU R(3) ( A)  16  U1 4 U  L A A U a a hA  hA 4  1

   

1  1 1    U  L ai  i 1  ai 4

(17)

where R(1) ( A), R(2) ( A) and R(3) ( A) are denoted three types of ranking functions, respectively. Definition 5 (Qin and Liu 2015). Let A be an IT2FSs. Then the combined ranking value of A is defined as:

where Ci 

q i 1

(18)

(i  1, 2,3) and multiplier q and orness level  are determined as follows:

2

q

CR IP T

R( A)  C1R1 ( A)  C2 R2 ( A)  C3 R3 ( A)

j

j 0

3

AN US

2 q 2   (2  i  1)q3i  0

(19)

i 2

and

 n  i  ln 2  R(i ) ( A) ln R(i ) ( A)  3   i 1 n  1    ln 2  R(i ) ( A) ln R(i ) ( A)  i 1 3

M





     

(20)

If R( A)  R( B) , then A

B;



If R( A)  R( B) , then A

B;



If R( A)  R( B) , then A

B.

CE



PT

ED

Qin and Liu (2015) presented a ranking order relation between two IT2FSs A, B as follows:

AC

4. New distance related to interval type-2 fuzzy sets In this section, we first propose a new ranking-based distance function of IT2FSs based on cut and the decomposition theorem, and then develop a new distance for IT2FSs associated with the ranking-based distance function. 4.1 The ranking-based distance function of IT2FSs Definition 6. Let A

( AL , AU ) be an IT2FSs, for any

[0,1] , the

cut of A is defined

as:

A ( x) where is a certain coefficient, 0

AL ( x)

(1

1. 10

) AU ( x)

(21)

ACCEPTED MANUSCRIPT

( AL , AU ) be an IT2FSs, for any

Definition 7. Let A

AL ( x)

1, x A ( x)

[0,1] , then A ( x) can be defined as:

AL ( x), x

, x

AU ( x)

(22)

AU ( x)

0, x

Based on Definitions 6 and 7, we provide a new rank function of IT2FSs, which is defined as follows:

CR IP T

(a1L , a2L , a3L , a4L ; hAL ),(a1U , a2U , a3U , a4U ; hUA ) be an IT2FSs defined in the

Definition 8. Let A

universe of discourse X . The rank-based distance function between A and 1 are defined as follows: b

min( A ( x),1 ( x))dx

1

Rd ( A,1)

a

1

d

b

0

A ( x)1 ( x)dx

AN US

a

where 1

(23)

(1,1,1,1;1),(1,1,1,1;1) .

Then based on Definitions 6 and 7, we can derive the following Theorem 1.

(a1L , a2L , a3L , a4L ; hAL ),(a1U , a2U , a3U , a4U ; hUA ) be an IT2FSs defined in the

Theorem 1. Let A

1 a4L

(a1L

L 4

(a

a1U

a4U

L 3

L 2

a

L 1

a

1 hU ( (a2L 2hAL hUA A

a4L )

ED

Rd ( A,1)

M

universe of discourse X . The rank-based distance function between A and 1 are defined as follows:

L A

U 4

a2U

a1U ) (24)

U 3

a

a

(a4L

a3L

a2L

(aU4

a3U

a4L

a3L )

hUA ( (a2L

a1L

a2U

a1U ) (a4L

a )) h (a

L 4

a1L

L 3

a )

PT

Proof. From Eqs.(11-12) and Definition 7, we have b

b

min( A ( x),1 ( x))dx

CE

a

L A

h

(a2L

AC

a4L

hAL (a4U

a1L

a2U

(a1L

a1U

a3U

a4L

A ( x)dx a

a1U ) (a4U a4U

a4L )

a4L )

L U A A

h h

a4L

U A

h

L A

h

a3L )

and

b

b

A ( x)1 ( x)dx a

A ( x)dx

1

a

Therefore, based on Eq.(23), we have

11

a1L )

(a1L

a3L

a1U )

a2L

a1L ))

ACCEPTED MANUSCRIPT

b

min( A ( x),1 ( x))dx

1

Rd ( A,1)

a b

1 0

d max( A ( x),1 ( x))dx

a

0

(a1L

L 4

L 3

(a 1 a4L (a4L

hAL

when 0

hUA

Rd ( A,1)

1 a4L (a4L

L 2

a

a

(a1L

a1U

a3L

hA

a1U

a2L

a4U

a4L )

L 1

L A

U 4

a )) h (a a4U

hAL hUA U 3

a

hUA ( (a2L L 4

a1L )) hAL (a4U

a3U

a4L

a2U

a1U ) d

L 3

a

a )

1 hU ( (a2L 2hAL hUA A

a4L )

a1L

a1L

a2U

a1U )

CR IP T

1 a4L

a3L )

1 , then the equation can be rewritten as follows: (a1L a3L

a1U a2L

a4U

a4L )

a1L )) hA (a4U

which completes of proof of Theorem 1.

1 h ( (a2L 2hA2 A a3U

a4L

a1L

a2U

a1U )

a3L )

AN US

1

M

Remark1. It is worth noting that the proposed ranking-distance function involves the parameter  . This parameter can be regarded as a measure reflecting the attitude character of the decision maker (DM). If the DM is optimistic, then we let   0 ; if the DM is neutral, then we let   0.5 ; and if the DM is pessimistic, we let   1 . The value of this parameter should reflect the DM‘s attitude preference in the certain decision making problem. For the sake of simplicity, we assume

ED

the DM‘s attitude preference is neutral, therefore, we let   0.5 in practical computation process. According to Definition 6 and Theorem 1, the ranking-based distance function of two IT2FSs

A and B can be determined by their corresponding ranking-based distance values Rd ( A,1) and

PT

Rd ( B,1) . The reason is that the ranking values Rd ( A,1) and Rd ( B,1) are real numbers. Thus, Rd ( B,1) , Rd ( A,1)

Rd ( B,1) , or

CE

one of the following three conditions must hold: Rd ( A,1)

Rd ( A,1)

Rd ( B,1) . This means that the proposed ranking-based distance function satisfies the

AC

laws of tracheotomy. For any two IT2FSs A and B , we have the following order relationship: Definition 9. Let A and B be two IT2FSs defined on the universe of discourse X , then the ranking of A and B by the rank-based distance Rd ( A,1) and Rd ( B,1) on X can be defined as follows: (1) If Rd ( A,1)

Rd ( B,1) , then A is superior to B , denoted by A

(2) If Rd ( A,1)

Rd ( B,1) , then A is indifferent to B , denoted by A

(3) If Rd ( A,1)

Rd ( B,1) , then A is inferior to B , denoted by A

Property 1. Let A

B;

B;

B.

(a1L , a2L , a3L , a4L ; hAL ),(a1U , a2U , a3U , a4U ; hUA ) be an IT2FSs defined on the 12

ACCEPTED MANUSCRIPT

universe of discourse X , then A

1 , if and only if Rd ( A,1)

0.

Proof. Based on Eq.(24), the conclusion is obvious. Based on the definition of the order relation for IT2FSs, we derive the following two theorems to demonstrate that the proposed ranking method satisfies the linear order and admissible order. Theorem 2. Let L be the set of all IT2FSs in X , and the order preference relation 



 is



a linear order and L, , 0,1 is a

CR IP T

set L is a binary relation. Then the order preference 

 on the

complete lattice with the smallest element 0  (0,0,0,0;0),(0,0,0,0;0)  and the largest element 1   (1,1,1,1;1),(1,1,1,1;1)  . Proof. First, we prove that the order

  is a partial order. Based on the set theory, we prove that

AN US

this order relation satisfies the properties of reflexivity, antisymmetricity and transitivity. 1. Reflexivity: For any IT2FSs A  L , we have Rd ( A,1)

Rd ( A,1) . Based on Eq.(24), it can

be easily shown that the ranking-based distance function is monotonically increasing with respect to its parameter. Therefore, we obtain that A

M

2. Antisymmetricity. For any two A, B  L , if A

Rd ( B,1) and Rd ( B,1)

ED

Rd ( A,1)

have

A.

B and B Rd ( A,1)

A , then based on Eq.(24), we Rd ( A,1)

Rd ( B,1) ,

then

A

B.

PT

according to the monotonicity of the ranking-based distance function, we can easily obtain

CE

3. Transitivity. For all IT2FSs A, B, C  L , if A have Rd ( A,1)

Rd ( B,1) and Rd ( B,1)

B and B

C , then based on Eq.(24), we

Rd (C,1) . Because of Rd ( A,1), Rd ( B,1), Rd (C,1)

AC

are all real numbers, then based on the transitivity on the line of real numbers R , we derive

Rd ( A,1)

Rd ( B,1)

Rd (C,1) . This implies that A C .

Based on the analysis shown above, we can prove that the order 

 is

a partial order

(Bustince, 2013). Theorem 3. Let  L,



be a partial order set, then 

 is a admissible order.

Proof. Based on the definition of admissible order (Bustince, 2013), we only need to prove the 

 satisfies the following two conditions: 13

ACCEPTED MANUSCRIPT

(1)

  is a linear order on L . According to Theorem 1, the conclusion is obvious.

(2) For any two IT2FSs A, B  L , A

B whenever Rd ( A,1)

Rd ( B,1) . Based on the principle

described in Definition 7, the conclusion is also obvious, so we omit the proof. which completes the proof of Theorem 3. 4.2 The distance measure of IT2FSs based on ranking-based distance function

CR IP T

Based on the proposed ranking-based distance function, we derive a new distance measure for IT2FSs, which is defined as follows: Definition 10. Let A and B be two IT2FSs. Then the distance between A and B is defined as:

d ( A, B)

Rd ( A,1)

Rd ( B,1)

(25)

Based on Eq. (25), it can be easily shown that the provided distance measure satisfies the properties of the metric space described in Theorem 4.

function d : L L (1) 0

L be two IT2FSs. Then the metric distance d in a set L is a real

AN US

Theorem 4. Let A, B,C

R , which satisfies the following three axioms:

d ( A, B) 1 .In particular, d ( A, B) d ( B, A) (Symmetry);

(3) d ( A, C )

d ( A, B)

A

B (Positivity);

M

(2) d ( A, B)

0

d ( B, C ) (Triangle inequality).

ED

5. An extended TODIM method for MCGDM problems based on interval type-2 fuzzy information

AC

CE

PT

In this study, we assume that the decision makers (DMs) expect to form linguistic terms (see Table 2) to assign linguistic value to express their decision preferences with trapezoid interval type-2 fuzzy information. Table 2 shows the linguistic term set L={ "Very Poor"(VP), "Poor"(P), "Medium Poor"(MP), "Medium"(M), "Medium Good"(MG), "Good"(G), "Very Good"(VG)} and their corresponding trapezoid interval type-2 fuzzy sets (TrIT2FSs), respectively, which are shown in Fig. 2. Table 2. Linguistic terms and their corresponding TrIT2FSs Linguistic terms

Trapezoid interval type-2 fuzzy sets

Very Poor(VP) Poor (P) Medium Poor (MP) Medium (M) Medium Good (MG) Good (G) Very Good (VG)

((0, 0, 0, 0.1;1), (0, 0, 0, 0.05;0.9)) ((0, 0.1, 0.1,0.3;1), (0.05, 0.1, 0.1,0.2;0.9)) ((0.1, 0.3, 0.3,0.5;1), (0.2, 0.3, 0.3,0.4;0.9)) ((0.3, 0.5, 0.5,0.7;1), (0.4, 0.5, 0.5,0.6;0.9)) ((0.5, 0.7, 0.7,0.9;1), (0.6, 0.7, 0.7,0.8;0.9)) ((0.7, 0.9, 0.9,1;1), (0.8, 0.9, 0.9,0.95;0.9)) ((0.9, 1, 1, 1;1), (0.95, 1, 1, 1;0.9))

14

ACCEPTED MANUSCRIPT

VP L

0

MP

0.1

0.2

M

0.3

0.4

MG

0.5

0.6

0.7

G

VG

CR IP T

1 0.9

0.8

0.9

1

Fig.2 Membership functions of IT2FSs linguistic terms

In addition, the complementary relations corresponding interval type-2 fuzzy sets in Table 3. Linguistic terms (L) Complentary terms (Lc)

VP VG

P G

AN US

Table 3. The complementary relations MP MG

M M

MG MP

G P

VG VP

5.1 The description of the MAGDM problems under interval type-2 fuzzy sets Let A  { A1 , A2 ,

, Cn } is a set of criteria,

, Dp } be a set of decision makers (DMs), p is the number of DMs, and

M

D  {D1 , D2 ,

, Am } be a set of alternatives, C  {C1 , C2 ,



p k 1

ED

  (1 , 2 , ,  p )T is a weight vector associated with them, which satisfies k [0,1] and k  1 . Let R( k )  (aij( k ) )mn be an interval type-2 fuzzy decision matrix, where aij( k )  L is

PT

the rating of alternative Ai  A with respect to criterion C j  C provided by decision maker

CE

Dk  D , which is shown in Table 4. Table 4 The interval type-2 fuzzy decision matrix R

(k )

C2

Cn

a

(k ) 12

a

a1(nk )

A2

(k ) a21

(k ) a22

a2( kn)

Am

am( k1)

am( k2)

(k ) amn

Alternative

AC

A1

C1 (k ) 11

5.2 The extended TODIM method for interval type-2 fuzzy information Step 1. Based on the principle of criteria category, the decision attribute can be divided into two sets: benefit criteria set F1 (the higher, the better) and cost criteria set F2 (the smaller the better), which satisfies F1

F2  C and F1

F2   , where  is an empty set. In general, the decision

15

ACCEPTED MANUSCRIPT

matrix has to be normalized before the actual decision process takes place unless all the criteria are formed having the same type. In this paper, we use the following formula to normalize the initial decision matrix R

(k )

(k )  aij aij   ( k ) c  (aij )

for benefit criteria C j  F1

(26)

for cost criteria C j  F2

where (aij ) is the complement of aij such that (aij )  L . Then we form the normalized (k ) c

(k ) c

(k )

 (aij( k ) )mn .

CR IP T

decision matrix R

c

Step 2. Aggregate the individual criteria weights into group criteria weights by using interval type-2 fuzzy weighted (IT2FWA) aggregation operator. Let the linguistic weight of the criteria

C j provided by DM Dk is represented by  (j k )  L , and its corresponding IT2FSs  j can be denoted as:

(k ) (k ) (k ) (k ) (k )  (j k )  ( (jk1) ,  (jk2) ,  (jk3) ,  (jk4) ; h( k ) ), ( j1 ,  j 2 ,  j 3 ,  j 4 ; h )   

AN US

j

j

(27)

Then, the aggregated weight of each criterion is calculated in the form: (2)  j  IT 2 FWA  (1) j , j ,

,  (j p ) 

p p p p  p  (k ) (k ) (k ) (k ) (k )    k  j1 ,  k  j 2 ,  k  j 3 ,  k  j 4 ;1   (1 h j ) k  k 1 k 1 k 1 k 1   k 1

ED

M

p p p p (k ) (k ) (k ) (k ) (k )   p k   k  j1 ,  k  j 2 ,  k  j 3 ,  k  j 4 ;1   (1 h j ) k  1 k  1 k  1 k  1 k 1 

(28)

  

Step 3. Calculate the centroid of aggregated weight  j by using the KM algorithm (Karnik and

PT

Mendel 2001).

AC

CE

Based on this algorithm, the centroid interval of

C(

j

)

 j is calculated as follows: b

x min

a

j

( x)dx

x

( x)dx j

(29)

b

[ a ,b ] a

( x)dx j

( x)dx j

b

x C(

j)

max

a

( x)dx j

x

( x)dx j

b

[ a ,b ] a

j

( x)dx

( x)dx j

According to Eqs.(29-30) , we obtain the centroid value of

C ( j ) 

j :

C ( j )  C ( j ) 2

Then the normalized weights of criteria C j ( j  1, 2, 16

(30)

, n) can be calculated by Eq.(32)

(31)

ACCEPTED MANUSCRIPT

j 

C ( j )

(32)

n

 C ( ) j

j 1

Step 4. Calculate the relative weight  jr of the criterion C j to the reference criterion Cr as:

 jr 

j ( j  1, 2, r

, n)

(33)

CR IP T

where  j is the weight value of the criterion C j and r  max{ j } . j

Step 5. Calculate the dominance degree of each alternative Ai over each alternative Ak with respect to criterion C j using the following expression:

(34)

ED

M

AN US

  d (r , r ) jk ij kj  if R (rij )  R (rkj )  0 n   jk   j 1   j ( Ai , Ak )  0 if R(rij )  R (rkj )  0  n   jk d (rkj , rij )  1  j 1  if R (rij )  R (rkj )  0  jk   where the paremeter  indicates the attenuation factor of the losses.

PT

Step 6. Calculate the overall dominance degree of each alternative Ai over each alternative Ak with respect to criterion C j in the following form: p

n

k 1

j 1

CE

 ( Ai , Ak )   k   j ( Ai , Ak )

(35)

AC

Step 7. Calculate the global prospect value of the alternative Ai according to the following expression:

  ( A , A )  min   ( A , A )  (A )  max   ( A , A )  min   ( A , A ) m

m

k 1

i

i

k

k 1

i

m

i

k 1

i

k

m

i

k

i

k 1

i

(36)

k

Step 8. Rank all the alternatives based on the global prospect values of alternatives. The bigger

 ( Ai ) is, the better alternative Ai is.

17

ACCEPTED MANUSCRIPT

6. An illustrative example

AN US

CR IP T

In this section, an illustrative example is provided to present the application of the proposed method for green supplier selection problem. 6.1 Problem description With the continuous development of economic globalization and environment protection, the green supply chain management has played an important role in marketing economic and become the most hotly research topic in modern management science, which directly impact on the manufactures‘ and environment performance. Green supplier evaluation and selection is one of the most important problems in green supply chain management. Consider a decision problem in an automobile manufacturing enterprise, which aims to search for the best supplier for purchasing the key components of its new automobile equipments. After preliminary screening, four potential automobile equipments suppliers (A1, A2, A3, A4) have been identified for further evaluation. Ten criteria to be considered in the evaluation process are: C1: Green product innovation; C2: Green image; C3: Use of environmentally friendly technology; C4: Resource consumption; C5: Green competencies; C6: Environment management; C7: Quality management; C8: Total product life cycle cost; C9: Pollution production; C10: Staff environmental training (See Table 5). Three decision makers D1, D2, D3 with different risk preferences (D1: Risk averse; D2: Risk neutral; D3: Risk appetite) are invited to carry out the evaluation and e= (0.2, 0.4, 0.4) be a set of weight vector of them. Three DMs can use the IT2FSs linguistic term to evaluate the importance of these ten criteria shown in Table 6. The decision matrices are listed in Tables 7-9.

Name

C1

Green product innovation

C2

Green image

Resource consumption

CE

C4

friendly technology

PT

C3

Use of environmentally

AC

C5

Green competencies

C6

Environment management

C7

Quality management

Definition

Green product innovation addresses environmental issues through

ED

Criteria

M

Table 5. Criteria for evaluating green supplier

product design and technique innovation

The ratio of green customers to total customers

The application of the environmental science to conserve the natural environment and resources, and to curb the negative impacts of human involvement.

Resource consumption in terms of raw material, energy and water during the measurement period Materials used in the supplied components that reduce the impact on natural resources ability to alter process and product for reducing the impact on natural resources Applying the management technique to seek to balance economic and supplier effective with the constration of environment Supply chain management activities and functions involved in determination of quality policy, quality planning and quality control Life cycle cost of product is the all life costing of product including

C8

Total product life cycle cost

from the selection, design, manufacture, test, use, maintenance, repair, to the waste of product.

C9

Pollution production

C10

Staff environmental training

Average volume of air emission pollutant, waste water, solid wastes and harmful materials releases per day during measurement period Staff training on environmental targets

18

ACCEPTED MANUSCRIPT

Table 6. Linguistic variables for relative importance weights of criteria Trapezoid interval type-2 fuzzy sets

Very Low (VL) Low (L) Medium Low (ML) Medium (M) Medium High (MH) High (H) Very High (VH)

((0, 0, 0, 1;1), (0, 0, 0, 0. 5;0.9)) ((0, 1, 1, 3;1), (0. 5, 1, 1, 2;0.9)) ((1, 3, 3, 5;1), (2, 3, 3, 4;0.9)) ((3, 5, 5, 7;1), (4, 5, 5, 6;0.9)) ((5, 7, 07, 9;1), (6, 7, 7, 8;0.9)) ((7, 9, 9,10;1), (8, 9, 9, 9.5;0.9)) ((9, 10, 10, 10;1), (9.5, 10, 10, 10;0.9))

CR IP T

Linguistic variable

Table 7 The decision matrix R1 C2

C3

C4

C5

C6

C7

C8

C9

C10

VP P VP VG

MP MG MP MP

M M MG G

G VG M P

VG P G M

M P MP VP

P MG P VP

VP G VP MP

MP M G P

P VP M VP

AN US

A1 A2 A3 A4

C1

Table 8 The decision matrix R2 C2

C3

C4

P VP MP VP

G VP VP MP

MG G VG M

VG MG MP G

C5

C6

C7

C8

C9

C10

VP VG MP VG

P P VG M

G MP G P

MG VP MP VP

M MG VG MP

P MP P P

M

A1 A2 A3 A4

C1

MP VP MP VP

VP MP VP VP

C3

C4

C5

C6

C7

C8

C9

C10

VG M VG G

MP G MP MG

MP VG MP VG

VG M VG P

G P G MP

MP VP MP VP

VG MP VG MG

P P P MP

PT

C2

CE

A1 A2 A3 A4

C1

ED

Table 9 The decision matrix R3

AC

The weights of these ten criteria are derived from three DMs based on Table 2 and are presented in Table 10.

D1 D2 D3

Table 10 The weight preference matrix by DMs

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

VL VH H

ML L VH

MH M MH

H MH H

VH ML ML

ML VL VL

L M H

VL MH VL

VH H L

H L VH

6.2 The evaluation steps (1) Since C1,C2,C3,C5,C6,C7,C10 are benefit criteria, C4,C8,C9 are cost criteria, so we fist normalize the decision matrices based on Eq.(26) and Table 3, the normalized decision matrices

R1 , R2 , R3 are listed in Tables 11-13. 19

ACCEPTED MANUSCRIPT

Table 11 The decision matrix R1

A1 A2 A3 A4

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

VG G VG VP

MP MG MP MP

M M MG G

P VP M G

VG P G M

M P MP VP

P MG P VP

VG P VG MG

MG M P G

P VP M VP

A1 A2 A3 A4

CR IP T

Table 12 The decision matrix R2 C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

G VG MG VG

G VP VP MP

MG G VG M

VP MP MG P

VP VG MP VG

P P VG M

G MP G P

MP VG MG VG

M MP VP MG

P MP P P

A1 A2 A3 A4

C1

C2

C3

C4

MG VG MG VG

VP MP VP VP

VG M VG G

MG P MG MP

AN US

Table 13 The decision matrix R3 C5

C6

C7

C8

C9

C10

MP VG MP VG

VG M VG P

G P G MP

MG VG MG VG

VP MG VP MP

P P P MP

ED

M

(2) Aggregating the individual criteria weights into group criteria weights by using interval type-2 aggregation operator (Wang et al. 2012). Based on IT2FWA operator described as Eq.(28), we can obtain the group attributes weights. The results are shown as follows:

1  (3.1, 4.7,6.4,7.8;1),(3.7,5.2,7.4,8.1;0.9) , 2  (1.1, 2.7,3.6, 4.2;1),(2.7,3.8, 4.6,6.1;0.9)

PT

3  (2.1,3.7,5.2,6.4;1),(4.3,5.4,6.7,7.5;0.9) , 4  (0.9,1.4, 2.8,3.2;1),(1.7,3.1, 4.5,5.9;0.9)

CE

5  (5.3,6.4,7.1,8.2;1),(6.1,7.4,8.5,9.2;0.9) , 6  (3.7, 4.8,6.1,6.9;1),(4.2,5.4,6.8,7.7;0.9)

AC

7  (4.2,5.3,6.5,7.2;1),(5.4,6.3,7.1,8.1;0.9) , 8  (2.5,3.9, 4.6,5.4;1),(3.4, 4.9,6.1,7.3;0.9) 9  (0.9,1.3, 2.2,3.5;1),(1.2, 2.4,3.7, 4.9;0.9) , 10  (5.1,6.8,7.7,8.4;1),(6.5,7.8,8.6,9.5;0.9)

(3) Calculate the centroid of aggregated weight  j and obtain the relative weight  jk . According to Eqs.(29-33), we can obtain the result as follows:

 jk  (0.12,0.08,0.17,0.11,0.05,0.14,0.21,0.04,0.03,0.05)T (4) Calculate the dominance of each alternative Ai over each alternative Ak based on the decision

20

ACCEPTED MANUSCRIPT

maker D p with respect to criterion C j . Based on Eq.(34), and we set

  1 (  i 1 ei i ) , 3

where (1 , 2 , 3 )  (2,1,0.5) , the obtained results are listed in Tables 14-23 (See Appendix). (5) Calculate the global dominance of each alternative Ai over each alternative Ak based on the decision maker D p with respect to criterion C j . Based on Eq.(35), we can obtain the related results

DM1

DM2

The global dominance matrix

The global dominance matrix

 ( Ai , Aj )

 ( Ai , Aj )

1

DM3

The global dominance matrix

 3 ( Ai , Aj )

2

A1

A2

A3

A4

A1

A2

A3

A4

A1

A2

A3

A4

0

-0.873

-0.723

0.025

0

-1.096

-0.867

0.047

0

-0.737

-0.927

0.145

-0.824

0

-0.612

-0.372

-0.981

0

-0.932

-0.129

-0.83

0

-0.798

-0.988

-1.132

-0.937

0

0.164

-1.121

-0.097

-0.109

0.437

0

-2.402

AN US

A1 A2 A3 A4

CR IP T

listed in Table 24. Table 24 The global dominance matrix of each alternative over others with respect to criterion

-1.057

0

0.658

-0.11

-1.135

0

-0.929

-2.253

-1.746

0

-1.02

-0.497

-1.132

0

(6) Calculate the overall dominance of alternative Ai .

Ai

A1 0.7356

 (Ai)

M

According to Eq.(39), the final result is shown in Table 25. Table 25 The ranking of alternatives A2 1

A3 0.8237

A4 0.2345

ED

(7) Rank all the alternatives Ai (i  1, 2,3, 4) and select the best one(s) in accordance with the

PT

value  ( Ai ) .

CE

According to Table 25, it is obvious that

 ( A2 )   ( A3 )   ( A1 )   ( A4 )

AC

Therefore, we have

where the symbol ―

A2

A3

A1

A4

‖means ―superior to‖. Thus, the best supplier is A2 .

6.3 Sensitivity analysis In order to reflect the influence of different values of parameter  on the produced results, we use different values  and assess the obtained ranking of the alternatives. The corresponding results are shown in Table 26. Table 26 Ranking orders of alternatives with different  Different values of 

The ranking of alternatives A1

A2

A3

21

A4

Ranking orders of alternatives

ACCEPTED MANUSCRIPT  1  2  3  4

0.7356 0.8342 0.6379 0.5926

1 1 1 1

0.8237 0.9107 0.7842 0.6531

A2 A2 A2 A2

0.2345 0.1772 0.2423 0.1352

A3 A3 A3 A3

A1 A1 A1 A1

A4 A4 A4 A4

In order to see the influence of change the value of  , we provide a radar diagram based on Table 26 to show the result of the sensitivity analysis, which is shown in Fig.3.

1 0.8 0.6 0.4 0.2 0

2

3

Supplier Supplier Supplier Supplier

A1 A2 A3 A4

AN US

4

CR IP T

1

Fig.3 The radar plot showing the result of the sensitivity analysis

M

From Table 26, it is apparent that the ranking orders obtained by different values of  from 1 to 4 are the same in this example. This means the ranking results are not sensitive to the values

ED

of  . In other words, in spite of decision process involving various values of the attenuation parameter  , the final ranking results remain consistent. In addition, it is worth pointing out that the above sensitivity analysis is based on e . In what follows, we carry out the sensitivity analysis of the decision makers‘ weighting vector e . We

PT

assume  =1 , and the sensitivity analysis is performed by modifying the weighting vector e , and recalculating the ranking orders of alternatives for different values of e .

CE

Table 27 Ranking orders of alternatives with different e

Cases

AC

1 2 3 4

Different values of e

e1

e2

e3

0.2 0.1 0.3 0.4

0.4 0.45 0.5 0.25

0.4 0.45 0.2 0.35

Ranking orders of alternatives A2 A2 A3 A2

A3 A3 A2 A1

A1 A1 A1 A3

A4 A4 A4 A4

As shown in Table 27, depending on the different value of e , the ranking orders of alternatives may be slightly different, and the results may lead to different decisions. So, when unknown attribute weights, it is significant important to choose an appropriate method for determining them. From Tables 26 and 27, the final decision result depends on two factors: parameter  and the decision makers‘ weighting vector e . How to match this two parameters are important for this problem. In what follows, we use the optimal allocation information granularity computing 22

ACCEPTED MANUSCRIPT

CR IP T

method based on PSO algorithm which was proposed by Pedrycz and Song (2011) to conduct the sensitivity analysis. First, based on the experts input who provide the following reciprocal linguistic preference relations using the set of five linguistic terms, the progression of the optimization is quantified in terms of the fitness function obtained in successive generations labels. The PSO algorithm returns the optimal cutoff points equal to 0.22, 0.26, 0.32, 0.38, 0.49, and 0.5062, for the linguistic term set S1, and 0.08, 0.52, 0.67, and 0.72, for the linguistic term set S2, respectively. The parameters of the PSO were set up as follows: the number of particles is 100, the number of generations is set to 300, while c1=c2=2. Then, based on aggregation phase and the exploitation phase, the reciprocal collective preference relation with the higher performance index Q is given below and the progression of the optimization is quantified in terms of the performance index fitness obtained in successive generations, see Fig.4.

max Q   ln  n   n i i 1   n ni ei    i 1 n 1 

AN US

(37)

Ranking preference fitness Termination generation＝ 300 0.8

0.6

M

0.5

0.4

0.3

ED

Performance index fitness

0.7

0.2

0

50

100

150 Generation

200

250

300

PT

0.1

Fig.4. The preference index Q in successive PSO generations

From Fig.4, we observe that when we set  =1 and e  (0.25,0.4,0.35) in this example,

CE

T

AC

the performance converges quickly with the drop in the values of the fitness function occurring in the first generations. This result verifies that in this example the  =1 is robust. In such cases, we can match the optimal value of   1 and its corresponding decision makers‘ risk weight vector by solving the inverse problem of Eq.(37) with the aid of the PSO algorithm. 6.4 Comparative analysis In order to verify the validity of our method, we complete a comparative analysis of the proposed method with the interval type-2 fuzzy TOPSIS (IT2F-TOPSIS) method, which was proposed by Chen and Lee (2011). The result is shown as follows: Determine the positive ideal solution (PIS) A (NIS) A

(

1

,

2

,

,

m

) , where

23

(

1

,

2

,

,

m

) and the negative solution

ACCEPTED MANUSCRIPT

max Rank ( ij ) , if fi

F1

min Rank ( ij ) , if fi

F2

min Rank ( ij ) , if fi

F1

max Rank ( ij ) , if fi

F2

1 j n i

(38)

1 j n

and 1 j n i

(39)

CR IP T

1 j n

where F1 is denoted as a benefit type criteria set and F2 is denoted as a cost type criteria set, which satisfies F1

F2  C (C is criteria set) and F1

F2   .

Calculate the distance d ( Aj ) between each alternative A j and the positive ideal solution

AN US

A , which comes as follows: m

d ( Aj )

( Rank (vij )

i

)2

(40)

( Rank (vij )

i

)2

(41)

i 1 m

d ( Aj )

i 1

M

Calculate the relative degree of closeness C ( Aj ) of A j with respect to the positive ideal

ED

solution A , expressed as:

d ( Aj ) d ( Aj )

(42)

d ( Aj )

PT

C ( Aj )

Using Eqs.(40-42), the corresponding distance measures d i and d i and the closeness degree

CE

Ci are produced. The results are shown in Table 28. Table 28 The closeness coefficients of alternatives along with final ranking

AC

Alternative A1 A2 A3 A4

d i

d i

Ci

Ranking order

0.523 0.543 0.523 0.639

0.776 0.695 0.776 0.527

0.497 0.561 0.597 0.452

3 2 1 4

From Table 28, it is apparent that the ranking orders obtained by these two methods are slightly different. By using the proposed IT2F-TODIM method, the best alternative is A2 , while using the IT2F-TOPSIS method the best alternative is A3 . The main reason is that the proposed

24

ACCEPTED MANUSCRIPT

method considers the decision maker‘s bounded rationality behavior in the MCGDM problems, while the IT2F-TOPSIS assume the DM is complete rationality, it fails to consider the DM‘s bounded rationality psychological behavior, so it produces more suitable decision result according to the decision maker‘s actual needs and behavior preference. Therefore, the ranking result obtained by the proposed method is more accurate and reasonable. 7. Conclusions and future works

AN US

CR IP T

The green supplier selection problem is one of the most important issues in the green supply chain management, which directly impacts the manufactures‘ performance. From this perspective, development and extension of a new green supplier selection decision making method is of substantial significance. Although many fuzzy MCDM methods have been used to green supplier selection problem, those methods can not consider the decision maker‘s bounded rationality behavior. and Furthermore, they cannot solve fuzzy group decision making problems. In this work we have focused on the group decision making under interval type-2 fuzzy environment for a green supplier selection by using the classical behavior decision TODIM method. TODIM method is a useful behavior decision making technique to solve the MCGDM problems, especially in the situations where the DMs‘ bounded rationality is taken into consideration. We have extended the novel TODIM method to handle MCGDM problems under interval type-2 fuzzy environment, in which all the decision information and criteria weights information provided by DMs are represented by IT2FSs. Firstly, we have defined a new distance and the weighted distances based on the fuzzy logic and  -cuts between IT2FSs. Then, we

AC

CE

PT

ED

M

develop a TODIM-based MCGDM method within the context of IT2FSs, in this method, we can use the multiple criteria utility function based on prospect theory to calculate the dominance degree and obtain the best alternative(s) for the group. Finally, we use a real green supplier selection example to illustrate the proposed method, and make a sensitivity analysis with granular computing technique and carry out a comparison analysis with the IT2F-TOPSIS method, the result shows that the proposed method is more reasonable and flexible than IT2F-TOPSIS method and easy to be applied and spread. In future research, we will integrate the TODIM method with other classical decision methods, such as AHP, DEMATEL, PROMETHEE etc., and further consider the interaction among the criteria by using the Choquet integral. Furthermore, we will continue our studies with anticipation that the method could be found applicable to other similar supplier selection problems, such as low carbon supplier selection, strategic supplier selection, sustainable supplier selection, and others.

Acknowledgments The work is supported by the National Natural Science Foundation of China (NSFC) under Projects 71171048 and 71371049, Ph.D. Program Foundation of Chinese Ministry of Education 20120092110038, and the Scientific Research and Innovation Project for College Graduates of Jiangsu Province CXZZ13_0138, and the Scientific Research Foundation of Graduate School of Southeast University YBJJ1454.

Appendix Table 14 The dominance matrix of each alternative over others with respect to criterion C1 25

ACCEPTED MANUSCRIPT

DM1

DM2

The dominance matrix

A1 A2 A3 A4

11

DM3

12

The dominance matrix

The dominance matrix

13

A1

A2

A3

A4

A1

A2

A3

A4

A1

A2

A3

A4

0

0.243

0.175

-0.247

0

0.379

0.577

0.246

0

0.621

-0.324

0.483

-0.654

0

0.429

0.321

-0.525

0

0.638

-0.743

-0.264

0

0.731

0.825

-0.732

-0.478

0

-0.432

-0.412

-0.238

0

0.195

0.775

-0.307

0

0.267

0.692

-0.545

0.563

0

-0.694

-0.325

-0.779

0

-0.637

-0.245

-0.725

0

DM1

DM2

The dominance matrix

DM3

The dominance matrix

22

The dominance matrix

A1

A2

A3

A4

A1

A2

A3

A4

0

-0.552

-0.668

-0.847

0

-0.723

0.154

0.129

0.127

0

0.835

0.423

0.128

0

0.147

0.152

0.235

-0.121

0

0.632

-0.844

-0.675

0

0.009

-0.327

-0.298

0

-0.793

-0.773

-0.817

23

A1

A2

A3

A4

0

0.037

0.245

0.473

-0.824

0

0.197

-0.363

0.194

-0.724

-0.554

0

0.217

0

-0.513

0.617

-0.698

0

AN US

A1 A2 A3 A4

21

CR IP T

Table 15 The dominance matrix of each alternative over others with respect to criterion C2

Table 16 The dominance matrix of each alternative over others with respect to criterion C3 DM2

The dominance matrix

The dominance matrix

A1

The dominance matrix

33

A2

A3

A2

A3

A4

A1

A2

A3

A4

0

-0.062

0.111

0.203

0

0.237

-0.139

0.317

0

0.328

0.117

0.237

0.854

0

0.273

0.172

-0.731

0

0.256

-0.645

-0.527

0

0.239

-0.158

-0.783

-0.636

0

0.369

0.813

-0.672

0

0.192

-0.774

0.747

0

0.473

-0.664

-0.745

-0.552

0

-0.625

0.272

-0.723

0

-0.692

0.772

-0.425

0

ED

A4

32

DM3

A1

PT

A1 A2 A3 A4

31

M

DM1

CE

Table 17 The dominance matrix of each alternative over others with respect to criterion C4 DM1

DM2

41

The dominance matrix

42

The dominance matrix

43

AC

The dominance matrix

DM3

A1

A2

A3

A4

A1

A2

A3

A4

A1

A2

A3

A4

0

0.147

-0.258

0.426

0

0.428

0.355

0.215

0

-0.363

0.242

0.176

-0.773

0

0.713

0.245

-0.515

0

-0.176

0.284

0.624

0

0.246

0.125

0.654

-0.119

0

-0.554

-0.598

0.812

0

-0.353

-0.715

-0.698

0

0.332

-0.538

-0.715

0.442

0

-0.725

-0.624

0.577

0

-0.823

-0.825

0.663

0

A1 A2 A3 A4

Table 18 The dominance matrix of each alternative over others with respect to criterion C5 DM1

DM2

26

DM3

ACCEPTED MANUSCRIPT

The dominance matrix

A1 A2 A3 A4

51

52

The dominance matrix

The dominance matrix

53

A1

A2

A3

A4

A1

A2

A3

A4

A1

A2

A3

A4

0

0.194

0.234

0.274

0

0.028

-0.437

0.336

0

-0.134

-0.325

0.233

-0.551

0

0.166

0.232

-0.428

0

-0.215

0.117

0.698

0

-0.613

0.342

-0.669

-0.457

0

0.165

0.379

0.639

0

-0.432

0.495

0.249

0

0.253

-0.784

-0.656

-0.476

0

-0.547

-0.812

0.557

0

-0.577

-0.617

-0.713

0

DM1

DM2

The dominance matrix

DM3

62

The dominance matrix

The dominance matrix

A1

A2

A3

A4

A1

A2

A3

A4

0

-0.145

0.333

-0.541

0

0.103

0.239

0.787

0

0.124

-0.625

-0.882

0

-0.565

-0.738

0

0.332

-0.671

0.432

0.321

-0.666

0

0.535

63

A1

A2

A3

A4

-0.444

0

-0.115

0.247

-0.327

-0.624

0.354

0.77

0

-0.254

0.109

0.317

0

-0.138

0.64

0.723

0

0.277

-0.633

0.728

0

0.59

-0.679

-0.657

0

AN US

A1 A2 A3 A4

61

CR IP T

Table 19 The dominance matrix of each alternative over others with respect to criterion C6

Table 20 The dominance matrix of each alternative over others with respect to criterion C7 DM1

DM2

72

The dominance matrix

The dominance matrix

73

A1

A2

A3

A4

A1

A2

A3

A4

0

-0.325

0.611

0.304

0

-0.117

0.229

0.245

-0.332

0.525

0

-0.266

0.353

0.823

0

0.114

-0.307

0

-0.565

-0.321

0.625

0

0.255

-0.697

-0.665

0

0.223

0.324

0

-0.538

-0.533

-0.623

0

-0.645

0.603

-0.621

0

A2

A3

A4

0

-0.311

0.245

-0.461

0.573

0

0.921

-0.429

-0.102

0.531

0.653

ED

A1

PT

A1 A2 A3 A4

71

M

The dominance matrix

DM3

Table 21 The dominance matrix of each alternative over others with respect to criterion C8

CE

DM1

The dominance matrix

81

DM3

The dominance matrix

82

The dominance matrix

83

A1

A2

A3

A4

A1

A2

A3

A4

A1

A2

A3

A4

0

0.621

-0.104

0.233

0

0.433

0.192

-0.155

0

0.324

-0.217

0.089

-0.238

0

-0.141

0.521

-0.523

0

-0.209

0.333

-0.449

0

0.187

-0.253

0.607

0.772

0

0.227

-0.818

0.652

0

-0.415

0.633

-0.712

0

-0.301

-0.595

0.623

-0.673

0

0.799

-0.557

0.546

0

-0.087

0.632

0.519

0

AC A1 A2 A3 A4

DM2

Table 22 The dominance matrix of each alternative over others with respect to criterion C9 DM1 The dominance matrix

DM2

91

The dominance matrix

27

DM3

92

The dominance matrix

93

ACCEPTED MANUSCRIPT

A1 A2 A3 A4

A1

A2

A3

A4

A1

A2

A3

A4

A1

A2

A3

A4

0

0.201

-0.312

0.225

0

0.337

-0.105

0.247

0

0.431

-0.332

0.254

-0.777

0

0.119

-0.154

-0.552

0

0.343

-0.111

0.437

0

0.117

-0.315

0.635

-0.737

0

0.247

0.838

-0.633

0

0.292

0.703

-0.73

0

0.662

-0.724

0.692

-0.363

0

-0.645

0.827

-0.744

0

0.461

0.635

-0.312

0

Table 23 The dominance matrix of each alternative over others with respect to criterion C10 DM2

The dominance matrix

A2

A3

The dominance matrix

A4

102

The dominance matrix

A1

A2

A3

A4

103

A1

A2

A3

A4

0

-0.112

-0.117

0.232

0

0.221

-0.132

-0.121

0

0.224

-0.135

0.243

0.808

0

0.145

-0.414

-0.723

0

0.133

0.204

-0.32

0

0.722

-0.165

0.774

-0.745

0

0.362

0.812

-0.647

0

-0.196

0.73

-0.188

0

-0.337

-0.693

0.517

-0.339

0

0.878

-0.704

0.549

0

-0.47

0.726

0.433

0

AN US

A1 A2 A3 A4

A1

101

DM3

CR IP T

DM1

References

Akman, Gülşen. 2015. "Evaluating suppliers to include green supplier development programs via Fuzzy c-means and VIKOR methods." Computers & Industrial Engineering 86:69-82. Gomes, Luiz F.A.M. and Rangel, Luis A.D. 2009. Än application of the TODIM method to the multicriteria rental evaluation of residential properties". European Journal of Operational

M

research 193 (1): 204-211.

Awasthi, Anjali, Satyaveer S Chauhan, and SK Goyal. 2010. "A fuzzy multicriteria approach for evaluating environmental performance of suppliers." International Journal of Production

ED

Economics 126 (2):370-378.

Büyüközkan, Gülçin. 2012. "An integrated fuzzy multi-criteria group decision-making approach for green supplier evaluation." International Journal of Production Research 50 (11):2892-2909.

PT

Büyüközkan, Gülçin, and Gizem Çifçi. 2012. "A novel hybrid MCDM approach based on fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS to evaluate green suppliers." Expert Systems with Applications 39 (3):3000-3011.

CE

Bai, Chunguang, and Joseph Sarkis. 2010a. "Green supplier development: analytical evaluation using rough set theory." Journal of Cleaner Production 18 (12):1200-1210.

Bai, Chunguang, and Joseph Sarkis. 2010b. "Integrating sustainability into supplier selection with grey

AC

system and rough set methodologies." International Journal of Production Economics 124 (1):252-264.

Bali, Ozkan, Erkan Kose, and Serkan Gumus. 2013. "Green supplier selection based on IFS and GRA." Grey Systems: Theory and Application 3 (2):158-176.

Blome, Constantin, Daniel Hollos, and Antony Paulraj. 2014. "Green procurement and green supplier development: antecedents and effects on supplier performance." International Journal of Production Research 52 (1):32-49. Bustince, H., Fernández, J., Kolesárová, A., & Mesiar, R. (2013). Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets and Systems, 220, 69-77 Cabral, Izunildo, Antonio Grilo, and Virgílio Cruz-Machado. 2012. "A decision-making model for Lean, Agile, Resilient and Green supply chain management." International Journal of Production 28

ACCEPTED MANUSCRIPT

Research 50 (17):4830-4845. Chai, Junyi, James NK Liu, and Eric WT Ngai. 2013. "Application of decision-making techniques in supplier selection: A systematic review of literature." Expert Systems with Applications 40 (10):3872-3885. Chen, Shyi-Ming, and Li-Wei Lee. 2010. "Fuzzy multiple attributes group decision-making based on the interval type-2 TOPSIS method." Expert Systems with Applications 37 (4):2790-2798. Chen, Ting-Yu. 2013. "A linear assignment method for multiple-criteria decision analysis with interval type-2 fuzzy sets." Applied Soft Computing 13 (5):2735-2748. Cifci, Gizem, and Gülçin Büyüközkan. 2011. "A fuzzy MCDM approach to evaluate green suppliers."

CR IP T

International Journal of Computational Intelligence Systems 4 (5):894-909.

Dai, Jing, and Jennifer Blackhurst. 2012. "A four-phase AHP–QFD approach for supplier assessment: a sustainability perspective." International Journal of Production Research 50 (19):5474-5490. Datta, Saurav, Chitrasen Samantra, Siba Sankar Mahapatra, Sabyasachi Banerjee, and Asish Bandyopadhyay. 2012. "Green supplier evaluation and selection using VIKOR method embedded in fuzzy expert system with interval–valued fuzzy numbers." International Journal

AN US

of Procurement Management 5 (5):647-678.

Dobos, Imre, and Gyöngyi Vörösmarty. 2014. "Green supplier selection and evaluation using DEA-type

composite

indicators."

157:273-278.

International

Journal

of

Production

Economics

Dou, Yijie, Qinghua Zhu, and Joseph Sarkis. 2014. "Evaluating green supplier development programs with a grey-analytical network process-based methodology." European Journal of Operational Research 233 (2):420-431.

M

Fan, Zhi-Ping, Xiao Zhang, Fa-Dong Chen, and Yang Liu. 2013. "Extended TODIM method for hybrid multiple attribute decision making problems." Knowledge-Based Systems 42:40-48. Fu, Xiaoyong, Qinghua Zhu, and Joseph Sarkis. 2012. "Evaluating green supplier development

ED

programs at a telecommunications systems provider." International Journal of Production Economics 140 (1):357-367.

Genovese, Andrea, SC Lenny Koh, Giuseppe Bruno, and Paolo Bruno. 2010. Green supplier selection:

PT

A literature review and a critical perspective. Paper presented at the Supply Chain Management and Information Systems (SCMIS), 2010 8th International Conference on.

CE

Gomes, LFAM, and MMPP Lima. 1992. "TODIM: Basics and application to multicriteria ranking of projects with environmental impacts." Foundations of Computing and Decision Sciences 16 (4):113-127.

AC

Gomes, Luiz Flavio Autran Monteiro, Maria Augusta Soares Machado, Francisco Ferreira da Costa, and Luis Alberto Duncan Rangel. 2013. "Criteria interactions in multiple criteria decision aiding: A Choquet formulation for the TODIM method." Procedia Computer Science 17:324-331.

Gomes, Luiz Flavio Autran Monteiro, Maria Augusta Soares Machado, and Luis Alberto Duncan Rangel. 2013. "Behavioral multi-criteria decision analysis: the TODIM method with criteria interactions." Annals of Operations Research 211 (1):531-548. Gomes, Luiz Flavio Autran Monteiro, Luis Alberto Duncan Rangel, and Francisco José Coelho Maranhão. 2009. "Multicriteria analysis of natural gas destination in Brazil: An application of the TODIM method." Mathematical and Computer Modelling 50 (1):92-100. Govindan, Kannan, Sivakumar Rajendran, Joseph Sarkis, and P Murugesan. 2013. "Multi criteria

29

ACCEPTED MANUSCRIPT

decision making approaches for green supplier evaluation and selection: a literature review." Journal of Cleaner Production. Govindan, Kannan, Joseph Sarkis, Charbel José Chiappetta Jabbour, Qinghua Zhu, and Yong Geng. 2014. "Eco-efficiency based green supply chain management: Current status and opportunities." European Journal of Operational Research 233 (2):293-298. Govindan, K., Soleimani, H., & Kannan, D. (2015). Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future. European Journal of Operational Research, 240(3), 603-626. Handfield, Robert, Steven V Walton, Robert Sroufe, and Steven A Melnyk. 2002. "Applying

CR IP T

environmental criteria to supplier assessment: A study in the application of the Analytical Hierarchy Process." European Journal of Operational Research 141 (1):70-87.

Hashemi, S. H., Karimi, A., & Tavana, M. (2015). An integrated green supplier selection approach with analytic network process and improved Grey relational analysis. International Journal of Production Economics, 159, 178-191.

Ho, William, Xiaowei Xu, and Prasanta K Dey. 2010. "Multi-criteria decision making approaches for

AN US

supplier evaluation and selection: A literature review." European Journal of Operational Research 202 (1):16-24.

Hsu, Chia-Wei, Tsai-Chi Kuo, Sheng-Hung Chen, and Allen H Hu. 2013. "Using DEMATEL to develop a carbon management model of supplier selection in green supply chain management." Journal of Cleaner Production 56:164-172.

Igarashi, Mieko, Luitzen de Boer, and Annik Magerholm Fet. 2013. "What is required for greener supplier selection? A literature review and conceptual model development." Journal of

M

Purchasing and Supply Management 19 (4):247-263.

Kannan, Devika, Kannan Govindan, and Sivakumar Rajendran. 2014. "Fuzzy Axiomatic Design Production.

ED

approach based green supplier selection: a case study from Singapore." Journal of Cleaner Kannan, Devika, Ana Beatriz Lopes de Sousa Jabbour, and Charbel José Chiappetta Jabbour. 2014. "Selecting green suppliers based on GSCM practices: Using fuzzy TOPSIS applied to a

PT

Brazilian electronics company." European Journal of Operational Research 233 (2):432-447. Karnik, Nilesh N, and Jerry M Mendel. 2001. "Centroid of a type-2 fuzzy set." Information sciences

CE

132 (1):195-220.

Krohling, Renato A, André GC Pacheco, and André LT Siviero. 2013. "IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making." Knowledge-Based Systems 53:142-146.

AC

Kuo, RJ, and YJ Lin. 2012. "Supplier selection using analytic network process and data envelopment analysis." International Journal of Production Research 50 (11):2852-2863.

Kuo, RJ, YC Wang, and FC Tien. 2010. "Integration of artificial neural network and MADA methods for green supplier selection." Journal of Cleaner Production 18 (12):1161-1170.

Labib, Ashraf W. 2011. "A supplier selection model: a comparison of fuzzy logic and the analytic hierarchy process." International Journal of Production Research 49 (21):6287-6299. Lee, Amy HI, He-Yau Kang, Chang-Fu Hsu, and Hsiao-Chu Hung. 2009. "A green supplier selection model for high-tech industry." Expert Systems with Applications 36 (4):7917-7927. Liu, Peide, and Fei Teng. 2014. "An extended TODIM method for multiple attribute group decision-making based on 2‐dimension uncertain linguistic Variable." Complexity. Liu, Xinwang, and Jerry M Mendel. 2011. "Connect Karnik-Mendel algorithms to root-finding for

30

ACCEPTED MANUSCRIPT

computing the centroid of an interval type-2 fuzzy set." Fuzzy Systems, IEEE Transactions on 19 (4):652-665. Lourenzutti, Rodolfo, and Renato A Krohling. 2014. "The Hellinger distance in Multicriteria Decision Making: An illustration to the TOPSIS and TODIM methods." Expert Systems with Applications 41 (9):4414-4421. Mendel, Jerry M, and RI Bob John. 2002. "Type-2 fuzzy sets made simple." Fuzzy Systems, IEEE Transactions on 10 (2):117-127. N Karnik, Nilesh, and Jerry M Mendel. 2001. "Operations on type-2 fuzzy sets." Fuzzy sets and systems 122 (2):327-348.

CR IP T

Noci, Giuliano. 1997. "Designing ‗green‘vendor rating systems for the assessment of a supplier's environmental performance." European Journal of Purchasing & Supply Management 3 (2):103-114.

Passos, Aderson Campos, Marcello Goulart Teixeira, Katia Cristina Garcia, Anelise Menezes Cardoso, and Luiz Flavio Autran Monteiro Gomes. 2014. "Using the TODIM-FSE method as a decision-making support methodology for oil spill response." Computers & Operations

AN US

Research 42:40-48.

Pedrycz, W., & Song, M. (2011). Analytic hierarchy process (AHP) in group decision making and its optimization with an allocation of information granularity. Fuzzy Systems, IEEE Transactions on, 19(3), 527-539.

Pereira, Javier, Luiz Flavio Autran Monteiro Gomes, and Fernando Paredes. 2013. "Robustness analysis in a TODIM-based multicriteria evaluation model of rental properties." Technological and Economic Development of Economy 19 (sup1):S176-S190.

M

Qin, Jindong, and Xinwang Liu. 2014. "Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment." Information sciences 297:293-315. Sahu, Nitin Kumar, Saurav Datta, and Siba Sankar Mahapatra. 2012. "Establishing green supplier (3):395-418.

ED

appraisement platform using grey concepts." Grey Systems: Theory and Application 2 Sarkis, Joseph, Qinghua Zhu, and Kee-hung Lai. 2011. "An organizational theoretic review of green

PT

supply chain management literature." International Journal of Production Economics 130 (1):1-15.

CE

Shaik, Mohammed, and Walid Abdul-Kader. 2011. "Green supplier selection generic framework: a multi-attribute utility theory approach." International Journal of Sustainable Engineering 4 (01):37-56.

AC

Shen, Lixin, Laya Olfat, Kannan Govindan, Roohollah Khodaverdi, and Ali Diabat. 2013. "A fuzzy multi criteria approach for evaluating green supplier's performance in green supply chain with linguistic preferences." Resources, Conservation and Recycling 74:170-179.

Singh, P. (2014). Some new distance measures for type-2 fuzzy sets and distance measure based ranking for group decision making problems. Frontiers of Computer Science, 8(5), 741-752.

Thongchattu, Chakthong, and S Siripokapirom. 2010. Green supplier selection consensus by neural network. Paper presented at the Mechanical and Electronics Engineering (ICMEE), 2010 2nd International Conference on. Tsai, W-H, and Shih-Jieh Hung. 2009. "A fuzzy goal programming approach for green supply chain optimisation under activity-based costing and performance evaluation with a value-chain structure." International Journal of Production Research 47 (18):4991-5017.

31

ACCEPTED MANUSCRIPT

Tseng, Ming-Lang. 2011. "Green supply chain management with linguistic preferences and incomplete information." Applied Soft Computing 11 (8):4894-4903. Tseng, Ming-Lang, Yuan-Hsu Lin, Kimhua Tan, Rong-Hui Chen, and Yuan-Ho Chen. 2014. "Using TODIM to evaluate green supply chain practices under uncertainty." Applied Mathematical Modelling 38 (11):2983-2995. Tsui, Che-Wei, Gwo-Hshiung Tzeng, and Ue-Pyng Wen. 2014. "A hybrid MCDM approach for improving the performance of green suppliers in the TFT-LCD industry." International Journal of Production Research (ahead-of-print):1-19. Vachon, Stephan. 2007. "Green supply chain practices and the selection of environmental

CR IP T

technologies." International Journal of Production Research 45 (18-19):4357-4379.

Verma, Rohit, and Madeleine E Pullman. 1998. "An analysis of the supplier selection process." Omega 26 (6):739-750.

Wang, Weize, Xinwang Liu, and Yong Qin. 2012. "Multi-attribute group decision making models under interval type-2 fuzzy environment." Knowledge-Based Systems 30:121-128.

Wang, Xiaojun, and Hing Kai Chan. 2013. "A hierarchical fuzzy TOPSIS approach to assess

AN US

improvement areas when implementing green supply chain initiatives." International Journal of Production Research 51 (10):3117-3130.

Wang, Xiaojun, Hing Kai Chan, and Dong Li. 2015. "A case study of an integrated fuzzy methodology for green product development." European Journal of Operational Research 241 (1):212-223. Wang, Ying-Ming, and Kwai-Sang Chin. 2011. "Fuzzy analytic hierarchy process: A logarithmic fuzzy preference programming methodology." International Journal of Approximate Reasoning 52 (4):541-553.

M

Weber, Charles A, John R Current, and WC Benton. 1991. "Vendor selection criteria and methods." European Journal of Operational Research 50 (1):2-18. Xiao, Zhi, Weijie Chen, and Lingling Li. 2012. "An integrated FCM and fuzzy soft set for supplier (4):1444-1454.

ED

selection problem based on risk evaluation." Applied Mathematical Modelling 36 Yeh, Wei-Chang, and Mei-Chi Chuang. 2011. "Using multi-objective genetic algorithm for partner

PT

selection in green supply chain problems." Expert Systems with Applications 38 (4):4244-4253.

CE

Zadeh, Lotfi A. 1975. "The concept of a linguistic variable and its application to approximate reasoning—I." Information sciences 8 (3):199-249.

Zhang, Xiaolu, and Zeshui Xu. 2014. "The TODIM analysis approach based on novel measured

AC

functions under hesitant fuzzy environment." Knowledge-Based Systems 61:48-58.

Zhu, Qinghua, Yijie Dou, and Joseph Sarkis. 2010. "A portfolio-based analysis for green supplier management using the analytical network process." Supply Chain Management: An International Journal 15 (4):306-319.

Zhu, Qinghua, and Joseph Sarkis. 2007. "The moderating effects of institutional pressures on emergent green supply chain practices and performance." International Journal of Production Research 45 (18-19):4333-4355. Zhu, Qinghua, Joseph Sarkis, James J Cordeiro, and Kee-Hung Lai. 2008. "Firm-level correlates of emergent green supply chain management practices in the Chinese context." Omega 36 (4):577-591.

32