Multiple criteria group decision making with belief distributions and distributed preference relations

Accepted Manuscript

Multiple criteria group decision making with belief distributions and distributed preference relations Chao Fu , Wenjun Chang , Min Xue , Shanlin Yang PII: DOI: Reference:

S0377-2217(18)30692-1 https://doi.org/10.1016/j.ejor.2018.08.012 EOR 15305

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

3 February 2018 22 June 2018 10 August 2018

Please cite this article as: Chao Fu , Wenjun Chang , Min Xue , Shanlin Yang , Multiple criteria group decision making with belief distributions and distributed preference relations, European Journal of Operational Research (2018), doi: https://doi.org/10.1016/j.ejor.2018.08.012

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

Highlights  A group decision making method with two different modes of assessments is developed.  Belief distributions and distributed preference relations are unified.  Internal consistency and Pareto principle for unifying the two modes are proven.  The proposed method is used to solve the problem of selecting a key field.

AC

CE

PT

ED

M

AN US

CR IP T

 Internal consistency and Pareto principle are verified using the data in the problem.

1

ACCEPTED MANUSCRIPT

Multiple criteria group decision making with belief distributions and distributed preference relations Chao Fu1,2,*, Wenjun Chang1,2, Min Xue1,2, Shanlin Yang1,2 1

School of Management, Hefei University of Technology, Hefei, Box 270, Hefei 230009, Anhui, P.R. China 2

CR IP T

Key Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education, Hefei 230009, Anhui, P.R. China

Abstract

To solve multiple criteria group decision making (MCGDM) problems with belief

AN US

distributions (BDs) and distributed preference relations (DPRs), this paper proposes a new method. For unifying BDs and DPRs, the transformation from BDs into DPRs is developed. Two important properties of the transformation, which are the internal consistency and the Pareto principle of social choice theory, are theoretically proven on the condition that the evidential reasoning algorithm is used to combine DPRs. With a view to relieving the burden

M

on decision makers to provide complete DPR matrices, the process of generating solutions to

ED

the MCGDM problems from the DPRs between neighboring alternatives and belief matrices composed of BDs is presented through the consistency between the score intervals of the

PT

DPRs. The proposed method is used to select a key filed for an enterprise located in Hefei, Anhui Province, China. The selection is validated by analyzing actual situations to

CE

demonstrate the applicability and validity of the proposed method. The internal consistency and the Pareto principle of social choice theory are verified through the relevant results of

AC

conducting the selection and simulation experiments based on selected data and random parameters in the field selection problem. Keywords: Group decisions and negotiations; Belief distribution; Distributed preference relation; Internal consistency; Pareto principle *

Corresponding author. Tel: 0086-551-62904930; fax: 0086-551-62905263. E-mail address: [email protected] (C. Fu).

1. Introduction Emerging information technologies (ITs), such as cloud computing, big data, and Internet 2

ACCEPTED MANUSCRIPT

of things, have propelled the transformation and upgrading of economy and the rapid development of society. In the process, researchers have recognized many important issues. Among them, one issue is that the individual capabilities of perceiving and understanding real problems are limited, and thus group capabilities are preferred to analyze real problems and generate satisfactory or acceptable solutions. Another important issue is that increasing attention has been paid to sociological and ecological perspectives except for economic

CR IP T

perspective when real problems are analyzed in the current era. Over the past several years, many real situations have been managed by depending on group capabilities. For example, concerns about the environment and sustainability are addressed by ways that include evaluating the management plans of a region to protect the ecosystem of the region from

AN US

economic, ecological, and social perspectives (Zendehdel et al., 2010), evaluating and selecting e-waste recycling programs to protect the environment (Wibowo and Deng, 2015), selecting the recycling and reusing technologies of waste materials (Soltani et al., 2016), and evaluating alternative marine fuels from the sustainability dimensions of ecology, sociology,

M

and technology (Ren and Liang, 2017). To focus on emergencies, representative studies include selecting rescue plans after an earthquake (Xu et al., 2015) and selecting emergency

ED

technologies to manage water source pollution accidents (Qu et al., 2016). In addition, studies that focus on traditional problems integrated with emerging ITs have addressed the selection

PT

of IT outsourcing providers (Li and Wan, 2014) and the evaluation and selection of green suppliers in green supply chain management for an automobile enterprise (Qin et al., 2017).

CE

With the aim of handling real situations such as those listed above, a number of group decision making (GDM) methods have been developed. These methods have focused on the

AC

aggregation of decision makers’ preference information (Blagojevic et al., 2016; Bottero et al., 2018; Chen et al., 2016; Chiclana et al., 2007; Liu et al., 2016a; Liu, 2017; Merigó et al., 2016), the guaranteeing of both the consistency of individual preference information and group consensus (Altuzaarra et al., 2010; Dong et al., 2010; Meng and Chen, 2015; Wan et al., 2017; Wu and Xu, 2016; Zhang et al., 2015), group consensus convergence (Dong et al., 2015a; Fu and Yang, 2010,2012; Zhang, 2017) and large-scale GDM with more than 20 decision makers (Liu et al., 2014; Wu and Liu, 2016; Liu et al., 2016b). There are generally two modes of characterizing the preference information of decision makers about alternatives 3

ACCEPTED MANUSCRIPT

in these methods. One mode is to compare alternatives in pairs, such as multiplicative preference relations (Dong et al., 2010), fuzzy linguistic preference relations (Wu and Xu, 2016) and fuzzy preference relations (Zhang et al., 2015). The other mode is to evaluate alternatives solely, such as linguistic assessments (Dong et al., 2015a; Merigó et al., 2016), type-2 fuzzy assessments (Wu and Liu, 2016) and belief distributions (Fu and Yang, 2010). Although the sole application of the two modes is feasible in many situations, that is not

CR IP T

always the case. In some situations in which decision makers in GDM have different backgrounds, knowledge, experience, and cognitive habits, it is a flexible way for decision makers to provide preference information about alternatives in their preferred modes.

Many prior studies have been conducted to address situations in which decision makers

AN US

prefer different preference modes. As summarized by Chen et al. (2015), there are three types of methods to address GDM with the two modes of preference information, including indirect methods, direct methods, and optimization-based methods. With indirect methods, heterogeneous preference information is unified through transformation functions and then

M

aggregated into collective preference to generate a group solution (Chiclana et al., 1998, 2001; Dong et al., 2009; Herrera-Viedma et al., 2002). However, with direct methods, individual

ED

preference (or priority) vectors are derived from heterogeneous preference information and then aggregated into a collective preference vector to generate a group solution. Two

PT

important properties, including the internal consistency and the Pareto principle of social choice theory, are required to be satisfied (Dong and Zhang, 2014; Dong et al., 2015b).

CE

Finally, in optimization-based methods, heterogeneous preference information is aggregated in an optimization model to determine the collective preference vector and then generate a

AC

group solution (Fan et al., 2006; Ma et al., 2006; Zhang and Guo, 2014). The existing studies on direct and optimization-based methods indicate that heterogeneous preference information in these two types of methods is usually associated with individual and (or) collective preference vectors, in which each element represents the priority or the ranking value of an alternative (Chen et al., 2015). However, in some situations, such association may be difficult to construct. For example, when decision makers select belief distributions (Fu and Yang, 2010; Voola and Babu, 2017; Zhang et al., 2017) and distributed preference relations (Fu et al., 2016) to provide assessments, it is not easy to derive preference vectors from preference 4

ACCEPTED MANUSCRIPT

information. As a result, the indirect method seems to be a feasible way to handle GDM with belief distributions (BDs) and distributed preference relations (DPRs). Meanwhile, this presents a challenge because little attention has been paid to the indirect GDM method with BDs and DPRs and the transformation between BDs and DPRs. However, some existing indirect methods (e.g., Chiclana et al., 1998, 2001; Herrera-Viedma et al., 2002) may violate the internal consistency and the Pareto principle of social choice theory, as numerically

rational solution by using an indirect GDM method.

CR IP T

demonstrated by Dong and Zhang (2014). This results in another challenge of generating a

In this paper, we propose a new method to address these two challenges. After presenting the modeling of multiple criteria group decision making (MCGDM) problems with BDs and

AN US

DPRs, we describe the process of analyzing the MCGDM problems and identify why BDs are transformed into DPRs. With the aid of the utilities of the grades associated with BDs and the scores of the grades associated with DPRs (see Section 2.1), the transformation from BDs into DPRs is discussed in detail. Two important properties of the transformation, including the

M

internal consistency and the Pareto principle of social choice theory, are analyzed and theoretically proven. By using the transformation, the process of generating solutions to the

ED

MCGDM problems is presented.

The rest of this paper is organized as follows. Section 2 introduces the modeling and

PT

analysis of MCGDM problems with BDs and DPRs. Section 3 presents the proposed method. In Section 4, a problem of selecting a key field for an enterprise is investigated to demonstrate

CE

the applicability of the proposed method. The paper’s conclusions are presented in Section 5. 2. Modeling and Analysis of MCGDM Problems with BDs and DPRs

AC

In this section, we simply present the modeling of MCGDM problems with BDs and DPRs and the process of analyzing the problems to find group solutions. 2.1. The modeling of MCGDM problems with BDs and DPRs In a MCGDM problem, suppose that there are M alternatives al ( l = 1, …, M ), which are evaluated by T decision makers t j ( j = 1, …, T ) on L criteria ei ( i = 1, …, L ). The relative weights of the L criteria are denoted by w = ( w1 , w2 , …, wL ) such that 0  wi  1 and

i 1 wi  1 . L

By considering the different knowledge, experience

5

ACCEPTED MANUSCRIPT

and specialties of decision makers, the relative weights of the T decision makers on each criterion can be different. Under the conditions, the relative weights of the T decision makers on criterion ei are denoted by  (ei ) = ( 1 (ei ) , …,  T (ei ) ) such that

0   j (ei )  1 and

 j 1  j (ei )  1 . T

Due to the incomplete information and materials and the limited cognitive capabilities of

CR IP T

the decision makers, it is feasible and rational for the decision makers to provide uncertain assessments. By considering the different backgrounds, knowledge, experience, and cognitive habits of the decision makers, they can select BDs or DPRs to express their preference information, which are two types of representative uncertain expressions. Note that BDs are

AN US

used to evaluate alternatives solely and DPRs are used to compare alternatives in pairs. Without loss of generality, assume that decision makers t1 , …, tq prefer to use BDs to characterize preference information and decision makers tq 1 , …, tT prefer to use DPRs to characterize preference information. The two modes of preference information associated

M

with two types of decision makers are introduced in the following.

Definition 1. (Formal description of BD) (Yang and Xu, 2013) Suppose that decision maker

ED

t j ( j = 1, …, q ) evaluates alternative al on criterion ei using a set of grades denoted by

PT

 = { H1 , H 2 , …, H N } where the grades are arranged in ascending order, i.e., from worst to best. The evaluation is profiled by a BD represented by B j (ei (al )) = {( H n ,  nj,i (al ) ), n

CE

= 1, …, N ; (  , j ,i (al ) )}, where  nj,i (al ) such that  nj,i (al ) ≥ 0 and

n1 nj,i (al ) N

≤1

represents the belief degree assigned to grade H n and j ,i (al ) = 1  n1 nj,i (al )

AC

N

represents the degree of global ignorance (uncertainty). In Definition 1, if j ,i (al ) = 0, the BD is complete; otherwise, it is incomplete. The

degree of global ignorance can be assigned to any grade, and the allowance of global ignorance helps decision makers to provide flexible uncertain assessments. The utilities of grades u ( H n ) ( n = 1, …, N ) are used to reflect the difference among the grades, satisfying the constraint 0 = u ( H1 ) < u ( H 2 ) < … < u ( H N ) = 1.

6

ACCEPTED MANUSCRIPT

Definition 2. (Formal description of DPR) (Fu et al., 2016) Suppose that decision maker t j ( j = q  1 , …, T ) compares alternatives al and am on criterion ei using a set of grades denoted by  = { G1 , G2 , …, GX } where X is an odd number, G X 1 /2 stands for an indifferent grade between al and am , G X 1 /21 , ..., GX for the grades with increasing

CR IP T

preferred intensity of al over am , and G1 , ..., G X 1 /21 for the grades with decreasing non-preferred intensity of al over am . The comparison is profiled by a DPR represented by

D j (ei (alm )) ={( Gx , d xj,i (alm ) ), x = 1, ..., X ; (  , dj ,i (alm ) )}, where d xj,i (alm ) such that 0≤ d xj,i (alm ) ≤1, d xj,i (alm ) = d Xj  x 1,i (aml ) and

 x 1 d xj,i (alm ) ≤ X

1 represents the belief degree

assessed to grade Gx and dj ,i (alm ) = dj ,i (aml ) = 1   x 1 d xj,i (alm ) represents the degree

AN US

X

of global ignorance (uncertainty) between al and am .

In Definition 2, we have d(jX 1)/2,i (all ) = 1. Note that by using a DPR decision maker t j can express the preferred, non-preferred, indifferent, and uncertain degrees of al over am

M

simultaneously. Similar to the situation of BD, the degree of global ignorance in DPR can be

ED

assigned to any grade. The score values of grades Gx ( x = ( X  3) / 2 , …, X ) denoted by s(Gx ) are used to characterize the difference among the grades, satisfying 0 <

PT

s(G( X 3)/2 ) < ... < s(Gx ) = 1 and s(Gx )  s(GX  x 1 ) . We especially have s(G( X 1)/2 )  0 . An example is offered in Section A.1 of Appendix A in the supplementary material to

CE

demonstrate Definition 2. Next, we discuss how to analyze MCGDM problems with BDs and DPRs.

AC

2.2. Analysis of MCGDM problems with BDs and DPRs To address MCGDM problems with BDs and DPRs, it can be found from the above

contents that the relationship between the individual priority vector of alternatives and BD and the relationship between the priority vector and DPR may be difficult to construct. As a result, BD and DPR may not be used to directly generate the priority of each alternative or construct an optimization model to determine the priority of each alternative. This means that direct and optimization-based methods may not be applicable in this situation. The remaining indirect method seems to be a feasible way. 7

ACCEPTED MANUSCRIPT

With a view to solving MCGDM problems with BDs and DPRs, the two modes of assessments should be unified when the indirect method is preferred. There are two possible ways to unify BDs and DPRs. One is to transform BDs into DPRs, and the other is to transform DPRs into BDs. When the second way is adopted, there may be an essential question of how to guarantee that the resulting BDs are what is anticipated by the decision makers. A DPR may be transformed into different sets of BDs because the information carried

CR IP T

by the DPR is not sufficient to implement a one-to-one correspondence between the DPR and the transformed BDs. For this reason, it is questionable whether BDs transformed from DPRs and the solution generated from the BDs are satisfactory for the decision makers. Under the conditions, transforming BDs into DPRs is justified in a sense. The detailed transformation

AN US

process will be introduced in Section 3. The assessments of all the decision makers are unified as D j (ei (alm )) ( j = 1, …, T , i = 1, …, L , l , m = 1, …, M ) after the transformation.

To analyze the MCGDM problem with unified DPRs, the DPRs of the decision makers on

M

each criterion are combined by using the evidential reasoning (ER) algorithm (Yang and Xu, 2013) and  (ei ) to generate collective DPRs between alternatives on each criterion denoted

ED

by D(ei (alm )) = {( Gx , d x ,i (alm ) ), x = 1, ..., X ; (  , d,i (alm ) )}. The resulting D(ei (alm )) can be further combined by using the ER algorithm and w to generate collective

PT

DPRs between the alternatives denoted by D(alm ) = {( Gx , d x (alm ) ), x = 1, ..., X ; (  ,

CE

d (alm ) )}. The application of the ER algorithm requires the following assumption.

Assumption 1. For the MCGDM problem described in Section 2.1, the assessment of

AC

decision maker t j on criterion ei does not depend on the assessment of the decision maker on any other criterion ek , and it also does not depend on the assessments of any other decision maker tm on all criteria. As D(alm ) is a distribution, it is difficult to directly compare alternatives al and am by using D(alm ) . To facilitate the comparison between alternatives, D(alm ) is transformed into a score interval denoted by S (alm ) = [ S  (alm ) , S  (alm ) ] via S (Gx ) ( x = ( X  3) / 2 , …,

X ) because the degree of global ignorance d (alm ) in D(alm ) can be assigned to any

8

ACCEPTED MANUSCRIPT

grade. In detail, S (alm ) is calculated by

S  (alm ) =  x ( X 1)/21 d x (alm )  s(Gx ) -  x 1

d x (alm )  s(Gx ) - d (alm )  s(G1 ) and

(1)

S  (alm ) =  x ( X 1)/21 d x (alm )  s(Gx ) -  x 1

d x (alm )  s(Gx ) + d (alm )  s(GX ) .

(2)

( X 1)/2 1

X

( X 1)/2 1

X

It can be easily found from Eqs. (1)-(2) that S (alm )  [-1, 1] (Fu et al., 2016). From the condition d xj,i (alm ) = d Xj  x 1,i (aml ) shown in Definition 2, S (aml ) = [ S  (aml ) , S  (aml ) ]

CR IP T

can be determined using Eqs. (1)-(2), i.e.,

S  (aml ) = S  (alm ) and

(3)

S  (aml ) = S  (alm ) .

(4)

AN US

Until now, the comparison between alternatives al and am is transformed into the comparison between two intervals S (alm ) and S (aml ) . To carry out the comparison, the possibility degree of one interval being superior to the other developed by Zhang et al. (1999) is adopted, which is

 a  b ,

a

(5)



 b

PT

ED

M

1, a   b    2   2   2  (a  b )  (a  b )  (a  b ) , b  a   b      2  (a  a )  (b  b )  (a   b  )  (a   b  )     0.5  (a   a  )  (b   b  ) ,  a  b  a p ( a  b) =       a  b  0.5  b  b ,  a   b   b     a  a a a  a   b a  a  0.5   ,  b   a   a      b  b b  b   b   a  0,

CE

where a = [ a  , a  ] and b = [ b  , b  ] are two intervals. By using Eqs. (3)-(5), the possibility degree of S (alm ) being superior to S (aml ) is obtained as

AC

p(S (alm )  S (aml )) =

1,  2S  (alm ) 2 , 1    ( S (alm )  S  (alm )) 2  2S  (alm ) 2  ,  ( S  (alm )  S  (alm )) 2  0,

S  (alm )  S  (aml ) S  (aml )  S  (alm )  S  (aml )  S  (alm ) 

S (alm )  S



 aml   S

. 

(6)



(alm )  S (aml )

S  (alm )  S  (aml )

Details about the generation of p(S (alm )  S (aml )) shown in Eq. (6) are presented in Section A.1. From Eq. (6), it can be known that p(S (alm )  S (aml )) + p(S (aml )  S (alm )) =1. When p(S (alm )  S (aml ))  0.5 , we have p(S (aml )  S (alm ))  0.5 and can conclude that 9

ACCEPTED MANUSCRIPT

alternative al is preferred to am with the possibility degree p(S (alm )  S (aml )) . If p(S (alm )  S (aml ))  0.5 , a converse conclusion can be drawn. Note that al is completely

equal to am when S (alm ) = S (aml ) = [0, 0]. Meanwhile, there exists a relationship between the middle point of S (alm ) (or S (aml ) and p(S (alm )  S (aml )) ) (or p(S (aml )  S (alm )) ). Proposition 1. Suppose that S (alm ) = [ S  (alm ) , S  (alm ) ] is known. Then, we have

CR IP T

(1) p(S (alm )  S (aml ))  0.5 if (S  (alm )  S  (alm )) / 2  (S  (aml )  S  (aml )) / 2 and (2) p(S (alm )  S (aml ))  0.5 if (S  (alm )  S  (alm )) / 2  (S  (aml )  S  (aml )) / 2 .

This proposition is proven in Section A.2 of Appendix A, which offers the easy comparison between two alternatives based on their score intervals. A solution can be generated from

AN US

p(S (alm )  S (aml )) , which will be presented in Section 3.2. In the following, we present an

indirect method for solving MCGDM problems with BDs and DPRs. 3. Proposed Method

In this section, we present a method of solving MCGDM problems with BDs and DPRs,

M

which is an indirect method. The transformation from BDs into DPRs is introduced in the method. To guarantee the validity of the transformation in MCGDM, it is theoretically proven

ED

that the transformation satisfies the internal consistency and the Pareto principle of social choice theory, which are two important properties of GDM methods with different formats of

PT

preferences (Dong and Zhang, 2014). The whole process of generating a solution to the MCGDM problem is then presented, in which the transformation and the analysis of the

CE

MCGDM problem shown in Section 2.2 are included. 3.1. Transformation from BDs into DPRs

AC

As analyzed in Section 2.2, the indirect method is suitable to handle MCGDM problems with BDs and DPRs, and BDs should be transformed into DPRs to unify the assessments of different decision makers. This means that the assessments of decision makers t j ( j = 1, …,

q ) will be transformed into DPRs. In the following we discuss the transformation from BDs into DPRs and present its relevant properties. Without loss of generality, we take the transformation from B j (ei (al )) and B j (ei (am )) into D j (ei (alm )) as an example to present the transformation from BDs into DPRs. There 10

ACCEPTED MANUSCRIPT

are four possible situations that must be considered, which are (1) complete B j (ei (al )) and complete B j (ei (am )) , (2) complete B j (ei (al )) and incomplete B j (ei (am )) , (3) incomplete

B j (ei (al )) and complete B j (ei (am )) , and (4) incomplete B j (ei (al )) and incomplete B j (ei (am )) . As the first situation is a basic case and the second and third situations are two

(1) Complete B j (ei (al )) and complete B j (ei (am ))

CR IP T

special cases of the fourth situation, we discuss the first and fourth situations in the following.

As indicated in Definition 1, B j (ei (al )) and B j (ei (am )) are two distributions on a set of grades H n ( n = 1, …, N ). The comparison between B j (ei (al )) and B j (ei (am ))

AN US

depends on to which grades the belief degrees in them are assigned. For example, suppose that alternative al is evaluated as H n with the belief degree  nj,i (al ) and alternative am as H k with  kj,i (am ) . If n  k , alternative al is said to be preferred to am with the

M

belief degree nj,i (al )  kj,i (am ) . Conversely, n  k means that alternative al is said to be

ED

non-preferred to am with the belief degree nj,i (al )  kj,i (am ) . In particular, n  k means that alternative al is said to be equal to am with the belief degree nj,i (al )  kj,i (am ) . By

PT

following the comparison principle, the transformation from B j (ei (al )) and B j (ei (am ))

CE

into D j (ei (alm )) can be considered as the assignment of the belief degree nj,i (al )  kj,i (am ) to grade Gx in D j (ei (alm )) . The assignment of the belief degree nj,i (al )  kj,i (am ) to

AC

grade Gx is determined by the correlation of the difference between grades H n and H k in  with grade Gx . Meanwhile, the difference among grades H n in  and the difference among grades Gx in  are reflected by u ( H n ) and s(Gx ) . By considering these facts, the generalized difference between grades H n and H k in  is defined to carry out the assignment. Definition 3. Suppose that B j (ei (al )) = {( H n ,  nj,i (al ) ), n = 1, …, N }, B j (ei (am )) = {( H k ,  kj,i (am ) ), k = 1, …, N }, and utilities of grades u ( H n ) ( n = 1, …, N ) are 11

ACCEPTED MANUSCRIPT

provided. The generalized difference between grades H n and H k

is defined as

f (u( H n ), u( H k )) . Here, f ( y, z ) is a mapping 0, 1  0, 1   1, 1 , which satisfies

(1) f ( y, z)  0 if y  z ,

(7)

(2) f ( y, z)  0 if y  z ,

(8)

(3) f (1, 0)  1,

(9)

(4) f (0,1)  1,

CR IP T

(10)

(5) f ( y, y)  f ( z, z)  0 ,

(11)

(6) f ( y1 , z)  f ( y2 , z) if y1  y2 , and

(12)

(7) f ( y, z1 )  f ( y, z2 ) if z1  z2 .

(13)

In Definition 3, Eqs. (7)-(8) indicate two situations where al is said to be preferred and

AN US

non-preferred to am with the belief degree nj,i (al )  kj,i (am ) . Two special cases of Eqs. (7)-(8) are shown in Eqs. (9)-(10). Eq. (11) indicates the special situation where alternative al is said to be indifferent to am with the belief degree nj,i (al )  kj,i (am ) . Eqs. (12) and (13)

degree to which alternative

is preferred to am , and vice versa. By using

shown in Definition 3, the transformation from

ED

f (u( H n ), u( H k ))

al

M

indicate that the larger the generalized difference between grades H n and H k , the larger the

B j (ei (al )) and

PT

B j (ei (am )) into D j (ei (alm )) is defined. Definition 4. Suppose that B j (ei (al )) = {( H n ,  nj,i (al ) ), n = 1, …, N }, B j (ei (am )) =

CE

{( H k ,  kj,i (am ) ), k = 1, …, N }, utilities of grades u ( H n ) ( n = 1, …, N ) associated

AC

with B j (ei (al )) and B j (ei (am )) and scores of grades s(Gx ) ( x = 1, …, X ) associated with the objective D j (ei (alm )) = {( Gx , d xj,i (alm ) ), x = 1, …, X } are provided. Assume that

 s(Gx ), x  ( X  1) / 2,..., X s (Gx ) =   s(Gx ), x  1,...,( X  1) / 2  1

(14)

is used to reflect the preferred and non-preferred relationships between alternatives al and am . Then, the belief degree d xj,i (alm ) ( x = 1, …, X ) in the transformed D j (ei (alm )) is

12

ACCEPTED MANUSCRIPT

calculated using f (u( H n ), u( H k )) shown in Definition 3 as n,k d xj,i (alm ) = n1 k 1 d x,i (alm ) , N

N

(15)

where

unL,k , x =

f (u ( H n ), u (H k ))  s (Gx1 ) , s (Gx )  s (Gx 1 )

unR,k , x =

s (Gx 1 )  f (u ( H n ), u ( H k )) , s (Gx 1 )  s (Gx )

s (Gx 1 )  f (u ( H n ), u ( H k ))  s (Gx ) s (Gx )  f (u ( H n ), u ( H k ))  s (Gx 1 ) , others

(16)

(17)

CR IP T

  nj,i (al )   kj,i (am )  unL,k , x  j j R d xn,,ik (alm ) =   n,i (al )   k ,i (am )  un ,k , x  0 

(18)

and G0 and GX 1 are two virtual grades such that s (G0 )  s (G1 ) and s (GX 1 )  s (GX ) .

AN US

Example A.1 is offered in Section A.1 to demonstrate Definition 4. (2) Incomplete B j (ei (al )) and incomplete B j (ei (am ))

When B j (ei (al )) and B j (ei (am )) shown in Definition 4 become incomplete, i.e., j j j ,i (al )  0 and j ,i (am )  0 , the assignment of the belief degrees n,i (al )  ,i (am ) and

M

j ,i (al )  kj,i (am ) to d xj,i (alm ) ( x = 1, …, X ) in the transformed D j (ei (alm )) becomes

ED

the main difficulty to transform B j (ei (al )) and B j (ei (am )) into D j (ei (alm )) . Definition 4 shows that the assignment of the belief degree nj,i (al )  j ,i (am ) to d xj,i (alm ) is closely

PT

associated with the generalized difference between H n and  , i.e., f (u( H n ), u( )) .

CE

Because j ,i (am ) can be assigned to any grade involved in  or any combination of grades involved in  , u ( ) may be any value limited to [0, 1]. To determine the

AC

representative value of u ( ) , we compare two same intervals [0, 1] and [0, 1] by using Eq. (5) and then obtain that p([0, 1]  [0, 1])  0.5 . Meanwhile, it can also be found from Eq. (5) that p([0, 1]  [0.5, 0.5])  0.5 . By comparing the two possibility degrees, the representative value of u ( ) can be set as 0.5. By using the representative value of u ( ) , the transformation from B j (ei (al )) and B j (ei (am )) into D j (ei (alm )) is defined below. Definition 5. Suppose that B j (ei (al )) = {( H n ,  nj,i (al ) ), n = 1, …, N ; (  , j ,i (al ) )},

B j (ei (am )) = {( H k ,  kj,i (am ) ), k = 1, …, N ; (  , j ,i (am ) )}, utilities of grades 13

ACCEPTED MANUSCRIPT

u ( H n ) ( n = 1, …, N ) associated with B j (ei (al )) and B j (ei (am )) and scores of grades

s(Gx ) ( x = 1, …, X ) associated with the transformed D j (ei (alm )) = {( Gx , d xj,i (alm ) ), x = 1, …, X ; (  , dj ,i (alm ) )} are provided. Assume that

 s(Gx ), x  ( X  1) / 2,..., X s (Gx ) =   s(Gx ), x  1,...,(X  1) / 2 1

(19)

CR IP T

is used to reflect the preferred and non-preferred relationships between alternatives al and am . Then, the belief degree d xj,i (alm ) ( x = 1, …, X ) in the transformed D j (ei (alm )) is

calculated by using f (u( H n ), u( H k )) shown in Definition 3 as

n1 k 1 d xn,,ik (alm ) N

N

+

n1 d xn,,i (alm )

dj ,i (alm ) = ,i (al )  ,i (am ) , j

j

where

+

k 1 d x,i,k (alm ) N

and

s (Gx 1 )  f (u ( H n ), u ( H k ))  s (Gx ) s (Gx )  f (u ( H n ), u ( H k ))  s (Gx 1 ) , others

M

  nj,i (al )   kj,i (am )  unL,k , x  j j R d xn,,ik (alm ) =   n,i (al )   k ,i (am )  un ,k , x  0 

N

AN US

d xj,i (alm ) =

(20)

(21)

(22)

f (u ( H n ), u (H k ))  s (Gx1 ) , s (Gx )  s (Gx 1 )

(23)

unR,k , x =

s (Gx 1 )  f (u ( H n ), u ( H k )) , s (Gx 1 )  s (Gx )

(24)

ED

unL,k , x =

s (Gx 1 )  f (u ( H n ),0.5)  s (Gx ) s (Gx )  f (u ( H n ),0.5)  s (Gx 1 ) , others

CE

PT

  nj,i (al )  j ,i (am )  unL,, x  j j R d xn,,i (alm ) =   n,i (al )  ,i (am )  un,, x  0 

(25)

f (u ( H n ),0.5)  s (Gx 1 ) , s (Gx )  s (Gx 1 )

(26)

unR,, x =

s (Gx 1 )  f (u (H n ), 0.5) , s (Gx 1 )  s (Gx )

(27)

AC

unL,, x =

  nj,i (al )  j ,i (am )  unL,, x  j j R d x,i,k (alm ) =   n,i (al )  ,i (am )  un,, x  0 

s (Gx 1 )  f (0.5, u ( H k ))  s (Gx ) s (Gx )  f (0.5, u ( H k ))  s (Gx 1 ) , others

(28)

uL ,k , x =

f (0.5,u (H k ))  s (Gx 1 ) , s (Gx )  s (Gx 1 )

(29)

uR ,k , x =

s (Gx 1 )  f (u (H n ), 0.5) , s (Gx 1 )  s (Gx )

(30)

14

ACCEPTED MANUSCRIPT

and G0 and GX 1 are two virtual grades such that s (G0 )  s (G1 ) and s (GX 1 )  s (GX ) . Example A.2 is offered in Section A.1 to demonstrate Definition 5. Because the situation of incomplete B j (ei (al )) and incomplete B j (ei (am )) can cover the other three situations, we focus on this situation to discuss two important properties of the transformation from

B j (ei (al )) and B j (ei (am )) into D j (ei (alm )) , which are the internal consistency (Chiclana

CR IP T

et al., 2001) and the Pareto principle of social choice theory (Arrow, 1963). The relevant concepts of the two properties are recalled in Definitions 6 and 7.

Definition 6. (Chiclana et al., 2001) The rankings of alternatives derived from the transformed preference information are the same as those from the original preference

AN US

information.

Definition 7. (Arrow, 1963) Given two alternatives al and am , if all decision makers prefer alternative al to am , then alternative al should be the collective choice. To facilitate the discussion of the two properties, the minimum and maximum expected

M

utilities of B j (ei (al )) are calculated as (Fu and Yang, 2010)

u j  (ei (al )) = n1 nj,i (al )  u( H n ) + j ,i (al )  u( H1 ) and

(31)

u j  (ei (al )) = n1 nj,i (am )  u( H n ) + j ,i (al )  u( H N ) .

(32)

N

ED

N

PT

The utility interval of B j (ei (al )) is then obtained as u j (ei (al )) = [ u j  (ei (al )) , u j  (ei (al )) ]. By using Definition 5, the score interval of the transformed D j (ei (alm )) can be calculated,

CE

which is shown in the following proposition.

AC

Proposition 2. Suppose that D j (ei (alm )) is derived from B j (ei (al )) and B j (ei (am )) in accordance with Definition 5. Then, with the aid of Eqs. (1)-(2) the score interval of

D j (ei (alm )) is obtained as S j (ei (alm )) = [ S j  (ei (alm )) , S j  (ei (alm )) ], satisfying



j j S j  (ei (alm )) = n1 k 1 n,i (al )  k ,i (am )  f (u( H n ), u( H k )) N

N

n1 nj,i (al )  j ,i (am )  f (u(H n ),0.5)  N

k 1 j ,i (al )  kj,i (am )  f (0.5, u( H k ))  N

15



+

+ - j ,i (al )  j ,i (am )

(33)

ACCEPTED MANUSCRIPT



j j S j  (ei (alm )) = n1 k 1 n,i (al )  k ,i (am )  f (u( H n ), u( H k )) N

N



+

n1 nj,i (al )  j ,i (am )  f (u(H n ),0.5) 

+

k 1 j ,i (al )  kj,i (am )  f (0.5, u( H k )) 

+ j ,i (al )  j ,i (am ) .

N

N

(34)

This proposition is proven in Section A.2. With the utility intervals of B j (ei (al )) and

CR IP T

B j (ei (am )) calculated using Eqs. (31)-(32) and the score interval of the transformed D j (ei (alm )) calculated using Eqs. (33)-(34), the internal consistency described in Definition 6 is redefined to focus on the transformation from B j (ei (al )) and B j (ei (am )) into

AN US

D j (ei (alm )) .

Definition 8. Suppose that the utility intervals of B j (ei (al )) and B j (ei (am )) , i.e.,

u j (ei (al )) = [ u j  (ei (al )) , u j  (ei (al )) ] and u j (ei (am )) = [ u j  (ei (am )) , u j  (ei (am )) ] are obtained using Eqs. (31)-(32), and the score interval of D j (ei (alm )) transformed from

M

B j (ei (al )) and B j (ei (am )) , i.e., S j (ei (alm )) = [ S j  (ei (alm )) , S j  (ei (alm )) ] is obtained

ED

using Eqs. (33)-(34). Then, the internal consistency between the two BDs and the transformed DPR is said to be satisfied when

PT

(1) p(u j (ei (al ))  u j (ei (am ))) ≥ 0.5 deduces p(S j (ei (alm ))  S j (ei (aml ))) ≥ 0.5 and

CE

(2) p(u j (ei (al ))  u j (ei (am ))) < 0.5 deduces p(S j (ei (alm ))  S j (ei (aml ))) < 0.5. Until now, we do not provide a specific function f ( y, z ) satisfying the conditions

AC

specified in Definition 3. This is because an abstract function f ( y, z ) does not influence the conclusions shown in Definitions 4 and 5 and Proposition 2. However, this is not the case when the internal consistency must be satisfied. For this reason, we provide a specific function f ( y, z ) = y  z and prove that the internal consistency shown in Definition 8 is satisfied when such a function is applied in Definition 5 and Proposition 2. Proposition 3. Given a function f ( y, z ) = y  z satisfying the conditions specified in Definition 3, suppose that the function is used to transform B j (ei (al )) and B j (ei (am )) into

16

ACCEPTED MANUSCRIPT

D j (ei (alm )) in accordance with Definition 5 and calculate the score interval of the transformed D j (ei (alm )) in accordance with Proposition 2. Thus, the two conclusions shown in Definition 8 still hold. This proposition is proven in Section A.2. As to the second property, i.e., the Pareto principle of social choice theory, it is closely associated with the method of combining

CR IP T

individual preference information. As presented in Section 2.2, D j (ei (alm )) ( j = 1, …, T ) is combined by using the ER algorithm (Yang and Xu, 2013) and  (ei ) . To facilitate understanding the combination of D j (ei (alm )) and discussing the Pareto principle, the combination process by using the ER algorithm is simply presented.

k1 =



X x 1

ˆ e(2) mˆ xe,(2) i (alm )  m,i (alm )



1

AN US

e (2) ˆ xe,(2) ˆ e(2) = k1  m d xe,(2) i (alm ) ( x = 1, …, X ), d ,i (alm ) = k1  m,i (alm ) i (alm )

(35)

(36)

2 1 1 1 2 2 mˆ xe,(2) i (alm ) = [(1   (ei ))   (ei )d x ,i (alm )  (1   (ei ))   (ei ) d x ,i ( alm )] +

M

[1 (ei )d 1x,i (alm )   2 (ei )d x2,i (alm )  1 (ei )d x1,i (alm )   2 (ei )d2 ,i (alm ) +

ED

1 (ei )d1 ,i (alm )   2 (ei )d x2,i (alm )]

(37)

2 1 1 1 2 2 mˆ e(2) ,i (alm ) = [(1   (ei ))   (ei )d,i (alm )  (1   (ei ))   (ei ) d,i ( alm )] +

PT

[1 (ei )d1 ,i (alm )   2 (ei )d2 ,i (alm )]

(38)

CE

The combination process presented in Eqs. (35)-(38) is an iterative process. The final combination result is D(ei (alm )) = {( Gx , d x ,i (alm ) = d xe,(iT ) (alm ) ), x = 1, …, X ; (  ,

AC

d,i (alm ) = de(,Ti ) (alm ) )}. Such a combination process can make the Pareto principle

satisfied.

Proposition 4. Suppose that D j (ei (alm )) ( j = 1, …, T ) is combined to generate D(ei (alm )) by using the ER algorithm and  (ei ) , and the score intervals of D j (ei (alm ))

and D(ei (alm )) are calculated as S j (ei (alm )) = [ S j  (ei (alm )) , S j  (ei (alm )) ] and S (ei (aml )) = [ S  (ei (alm )) , S  (ei (alm )) ] by using Eqs. (1) and (2). On the assumption that 17

ACCEPTED MANUSCRIPT

p(S j (ei (alm ))  S j (ei (aml ))) > 0.5 ( j = 1, …, T ), it is certain to hold that p(S (ei (alm ))  S (ei (aml ))) > 0.5.

This proposition is proven in Section A.2. Next, we discuss the generation of solutions to MCGDM problems with BDs and DPRs by using the transformation from BDs into DPRs presented in this section.

CR IP T

3.2. Generation of a solution After B j (ei (al )) ( j = 1, …, q ) is transformed into D j (ei (alm )) by using the technique presented in Section 3.1, the complete DPR matrices of T decision makers can be used to generate a solution to the MCGDM problem in theory. However, there are really two

AN US

difficulties. One is that providing DPRs for M  (M  1)  L / 2 pairs of alternatives is a burden to decision maker t j ( j = q  1 , …, T ), as indicated by Fu et al. (2016). The other is how to guarantee the transitivity of the possibility matrix derived from the final score matrix (Abel et al., 2018), which directly influences the rationality of the solution. In the

M

following, we discuss how to overcome the two difficulties and then generate a rational solution.

ED

To overcome the first difficulty, only (M  1)  L pairs of neighboring alternatives are required to be provided, i.e., D j (ei (al l 1 )) ( j = q  1 , …, T , l = 1, …, M  1 ). By

PT

using the ER algorithm,  (ei ) , and wi ( i = 1, …, L ), D j (ei (al l 1 )) is combined to

CE

generate D(ei (al l 1 )) and further D(al l 1 ) , which is used to generate S (ei (al l 1 )) = [ S  (ei (al l 1 )) , S  (ei (al l 1 )) ] and further S (al l 1 ) = [ S  (al l 1 ) , S  (al l 1 ) ] via Eqs.

AC

(1)-(2) and S (Gx ) ( x = ( X  3) / 2 , …, X ). The score intervals of other pairs of alternatives can be derived from S (al l 1 ) by following the additive consistency developed by Fu et al. (2016). As a result, the transitivity of the possibility matrix derived from the resulting score matrix is guaranteed, which means that the second difficulty is overcome. The adopted additive consistency is presented. Definition 9. (Fu et al., 2016) Suppose that S = ( S (alm ) = [ S  (alm ) , S  (alm ) ]) M M represents a score matrix. Then, S is said to be consistent if 18

ACCEPTED MANUSCRIPT

(1) S  (aln ) =  g (| S  (alm ) |, | S  (amn ) | ) , S  (alm ) < 0 and S  (amn ) < 0, l , m , n  {1, …, M },

(39)

(2) S  (aln ) = g (S  (alm ), S  (amn )) , S  (alm ) > 0 and S  (amn ) > 0, l , m , n  {1, …, M },

(40)

(3) S  (aln ) = S  (alm ) + S  (amn ) , S  (alm ) · S  (amn ) < 0, l , m , n  {1, …, M },

CR IP T

(41)

(4) S  (aln ) =  g (| S  (alm ) |, | S  (amn ) | ) , S  (alm ) < 0 and S  (amn ) < 0, l , m , n  {1, …, M },

(42)

AN US

(5) S  (aln ) = g (| S  (alm ) |, | S  (amn ) | ) , S  (alm ) > 0 and S  (amn ) > 0, l , m , n  {1, …, M } and

(43)

(6) S  (aln ) = S  (alm ) + S  (amn ) , S  (alm ) · S  (amn ) < 0, l , m , n  {1, …, M }. (44)

M

A two-variable function g ( y, z ) = y  z  (1  b)  yz)) / (1  b  yz) with a parameter b  (, 1) is adopted by Fu et al. (2016). The relevant properties of the function g ( y, z )

ED

can be found in (Fu et al., 2016) and thus omitted here. On the assumption that S (al (l 1) ) = [ S  (al (l 1) ) , S  (al (l 1) ) ] is obtained, DPRs between another pair of nonadjacent alternatives

PT

must be provided to determine the parameter b of the function g ( y, z ) . This is because the

CE

additive consistency shown in Definition 9 is based on the preference information of decision makers, as indicated by Fu et al. (2016). By following the idea of Fu et al. (2016), an

AC

appropriate pair of nonadjacent alternatives denoted by al* (l*  2) is determined by N l* = max { Nl  Nl , l = 1, ..., M  2 } where N l = |{ ei | S  (ei ( al l 1 )) · S  (ei (al 1l  2 )) > 0, i = 1, …, L }|, N l = |{ ei | S  (ei (al l 1 )) · S  (ei (al 1l  2 )) > 0, i = 1, …, L }|, and |·| denotes the cardinality of a set. Focusing on al* (l*  2) , decision makers t j ( j = q  1 , …,

T ) then provide D j (ei (al* (l*  2) ) ( i = 1, …, L ), which is combined with the transformed

D j (ei (al* (l*  2) ) ( j = 1, …, q ) by using the ER algorithm and  (ei ) to generate 19

ACCEPTED MANUSCRIPT

D(ei (al* (l*  2) ) . By using Eqs. (1)-(2) and S (Gx ) ( x = ( X  3) / 2 , …, X ), S (ei (al* (l*  2) ) = [ S  (ei (al* (l*  2) ) , S  (ei (al* (l*  2) ) ] is derived from D(ei (al* (l*  2) ) . An optimization model constructed by Fu et al. (2016) is then used to determine the parameter b of the function g ( y, z ) , where Si12 , Si23 , and Si13 represent S (ei (al* (l* 1) ) , S (ei (a(l* 1)(l*  2) ) , and

MIN F =

i 1,S

12  23 Si 0 i

( g * (Si13 )  g (| Si12 |,| Si23 |))2 +

i 1,S

12  23 Si 0 i

( g * (Si13 )  g (| Si12 |,| Si23 |))2

L

L

CR IP T

S (ei (al* (l*  2) ) for simplicity.

(45)

| Si12 |  | Si23 | (1  b) | Si12  Si23 | , 1  b | Si12  Si23 |

(46)

g (| Si12 |,| Si23 |) =

| Si12 |  | Si23 |  (1  b ) | Si12  S i23 | , 1  b | Si12  Si23 |

(47)

AN US

g (| Si12 |,| Si23 |) =

s.t.

(48)

| S 13 | , if Si13  max{ Si12  , Si23  }>0  g * ( Si13 ) =  i 12 , 23   g (| Si |,| Si |), otherwise

(49)

M

| S 13 | ,if  Si13  max{  Si12 ,  Si23  }>0  g * ( Si13 ) =  i 12 , 23   g (| Si |,| Si |),otherwise

  b 1.

(50)

ED

When the parameter b of the function g ( y, z ) is determined, the score intervals of the pairs of nonadjacent alternatives are derived from S (al (l 1) ) using Definition 9. The score

PT

matrix S = ( S (alm ) = [ S  (alm ) , S  (alm ) ]) M M is then formed. Through Proposition 1, the

CE

score matrix can be used to construct a preferring matrix P  ( plm )M M , in which

1, ( S  (alm )  S  (alm )) / 2  ( S  (aml )  S  (aml )) / 2 . plm =      0, ( S (alm )  S (alm )) / 2  ( S (aml )  S (aml )) / 2

(51)

AC

From the matrix, the number of alternatives non-preferred to alternative al is obtained as NP(al ) =  m1,ml plm . M

(52)

The ranking of alternative al is derived from NP(al ) , which is r (al ) = M  NP(al ) .

(53)

From p(S (alm )  S (aml )) ( l , m = 1, …, M ) obtained using Eq. (6) and r (al ) ( l = 1, …, M ), a ranking order of the M alternatives with possibility degrees is generated, which is considered as a solution to the MCGDM problem with BDs and DPRs. Note that

20

ACCEPTED MANUSCRIPT because the consistency of the score matrix S is guaranteed using Definition 9, the problem of comparison cycle (Yang et al., 2016) can be avoided, which guarantees the validity of the generated solution. Meanwhile, the solution generated based on the transformation from BDs into DPRs is generally acceptable to all decision makers, even when most decision makers provide BDs. If this is not the case, DPRs must be transformed into BDs to generate a solution that involves the preference information of each decision maker. Such a transformation is difficult, as shown in Section 2.2.

CR IP T

4. Case Study

In this section, the problem of selecting a key field for an enterprise located in Hefei, Anhui Province, China, is solved to demonstrate the applicability of the proposed method. To facilitate the handling of the problem, a solution system is developed in the MATLAB

AN US

environment. 4.1. Description of the problem of selecting a key field

The rapid development of society and economy in recent years has significantly changed people’s life modes and quality requirements. In this situation speed, convenience, reliability, security, and enjoyment have become people’s basic requirements. With the aim of satisfying

M

such requirements from some special perspective, the voice industry, as an important strategic

ED

and leading industry, has greatly developed and its relevant products have been widely applied in various fields. The development of the voice industry has been further propelled by

PT

the development and application of emerging ITs such as cloud computing, big data, and Internet of things. With a rising technological tide that requires voice products in different

CE

fields, many well-known IT enterprises such as Microsoft, Qualcomm, Google, LG Electronics, and Philips have invested R&D resources in the voice industry and intend to

AC

achieve high market share. The R&D of famous enterprises provides technological support for the healthy and sustainable development of the voice industry while it promotes the formation of technological barriers in this industry. Under these conditions, opportunities and challenges coexist for Chinese enterprises in the voice industry. There is a huge market of voice products in various fields in China, such as mobile terminal, automotive electronics, culture and education, and intelligent television. The market provides an excellent external environment for the development of Chinese enterprises in the voice industry. In this environment, Chinese enterprises have rapidly developed and accumulated 21

ACCEPTED MANUSCRIPT

abundant knowledge of key technologies of voice products, which plays an important role in the successful application of the products in different fields. However, the successful application in different fields also requires the accumulation of knowledge and experience regarding the fields except for key technologies. Due to limited resources, Chinese enterprises generally do not invest sufficient R&D resources in multiple fields to achieve high market share in the fields simultaneously. As a result, enterprises face the challenge of selecting an

CR IP T

appropriate field to invest sufficient R&D resources to achieve their anticipated goals in the market.

As a famous enterprise in the voice industry of China, an enterprise located in Hefei, Anhui Province, China, has rapidly developed in the past years and accumulated abundant

AN US

knowledge of key technologies of voice products. The products of the enterprise currently cover several fields, including automotive electronics, intelligent television, mobile terminal, intelligent customer service, culture and education, and mobile internet. The enterprise has achieved relatively high market share in the six fields. With the increasing competency of the

M

market and the upgrading and regeneration of voice products, the enterprise realizes that it is very difficult to achieve what is anticipated in all fields due to limited resources. It plans to

ED

place an emphasis on one of the six fields to achieve the highest market share and continuously maintain competitive edge in the field. This also drives the development of the

PT

enterprise in the remaining fields with the improvement of its brand value, industry reputation, customer loyalty, and social benefits.

CE

To address the problem of selecting a key field, a vice general manager of the enterprise acts as the facilitator. He invites six decision makers to carry out the selection process. They

AC

are the director of financial department ( t1 ), the director of the department of technology and quality ( t2 ), the deputy director of R&D department ( t3 ), the director of marketing department ( t4 ), the director of president office ( t5 ), and an expert from the cooperation university ( t6 ). The six decision makers are required to evaluate the six fields mentioned above to select the most appropriate one as the key field, which are automotive electronics ( F1 ), intelligent television ( F2 ), mobile terminal ( F3 ), intelligent customer service ( F4 ), culture and education ( F5 ), and mobile internet ( F6 ). The evaluation is conducted from seven

22

ACCEPTED MANUSCRIPT

perspectives (or criteria), which are market potential ( e1 ), existing foundations ( e2 ), profitability ( e3 ), market competency ( e4 ), technological requirements ( e5 ), service requirements ( e6 ), and social benefits ( e7 ). The method employed in (Ölçer and Odabaşi, 2005) is adopted by the facilitator to determine the relative weights of the six decision makers

on the seven criteria, which are presented in Table 1. Table 1 Relative weights of the six decision makers

 2 (ei )

 3 (ei )

 4 (ei )

e1

0.13

0.11

0.13

0.26

e2

0.16

0.16

0.14

0.19

e3

0.26

0.13

0.15

0.21

e4

0.12

0.14

e5

0.08

0.26

e6

0.1

0.21

e7

0.11

0.13

 5 (ei )

 6 (ei )

0.21

0.16

0.27

0.08

0.15

0.1

CR IP T

1 (ei )

AN US

Criteria

0.16

0.23

0.19

0.16

0.24

0.11

0.13

0.18

0.19

0.24

0.14

0.12

0.21

0.13

0.16

0.26

M

The six decision makers can select BDs or DPRs to evaluate the six fields. By considering

ED

their respective knowledge, experience and cognitive habits, the first three decision makers select BDs, and the remaining three decision makers select DPRs. The first three decision

PT

makers solely evaluate the six fields by using the set of grades:  = { H n , n = 1, …, 6} = {very poor, poor, average, good, very good, excellent} while the remaining three decision

CE

makers compare the six fields in pairs by using the set of grades:  = { Gx , x = 1, …, 9} = {absolutely weaker, much weaker, moderately weaker, marginally weaker, indifferent,

AC

marginally stronger, moderately stronger, much stronger, absolutely stronger}. Each decision maker provides the assessment of each field on one criterion without the consideration of his assessment on any other criterion and the assessments of any other decision maker on the seven criteria. Under the conditions, Assumption 1 is satisfied, and the ER algorithm (Yang and Xu, 2013) can be applied to aggregate the assessments of decision makers. After a discussion between the facilitator and the six decision makers about the seven criteria and the two sets of grades for sole and pairwise evaluation, criterion weights are specified as w = (0.18, 0.16, 0.2, 0.16, 0.12, 0.1, 0.08) by using the method employed in 23

ACCEPTED MANUSCRIPT

(Ölçer and Odabaşi, 2005) and the utilities of grades H n ( n = 1, …, 6) and the scores of grades Gx ( x = 6, …, 9) are set as u ( H n ) ( n = 1, ..., 6) = (0, 0.2, 0.4, 0.6, 0.8, 1) and s(Gx ) ( x = 6, …, 9) = (0.25, 0.5, 0.75, 1) by using the probability assignment approach

(Winston, 2011). 4.2. Generation of solution to the problem of selecting a key field The first three decision makers provide their assessments by using H n ( n = 1, …, 6), as

CR IP T

presented in Tables B.1-B.3 of Section B.1 in Appendix B of the supplementary material, while the remaining three decision makers provide their assessments by using Gx ( x = 1, …, 9), as presented in Tables B.4-B.6 of Section B.1. To relieve the burden on the decision makers to provide DPRs between any pairs of fields and guarantee the transitivity of the

AN US

possibility matrix derived from the final score matrix, only 5·7 pairs of neighboring fields are compared by the decision makers t j ( j = 4, 5, 6), which can be seen in Tables B.4-B.6. Under the conditions, B j (ei ( Fl )) and B j (ei ( Fl 1 )) ( j = 1, 2, 3, l = 1, …, 5) are required

M

to be transformed into D j (ei ( Fl l 1 )) to solve the problem of selecting a key field. To make the transformation satisfy the internal consistency shown in Proposition 3 and the Pareto

ED

principle of social choice theory shown in Proposition 4, the function f ( y, z ) = y  z is selected. By using the selected function, the transformed D j (ei ( Fl l 1 )) is obtained in

PT

accordance with Definition 5, which is presented in Tables B.8-B.10 of Section B.2 in Appendix B. By using the ER algorithm,  (ei ) shown in Table 1, and wi ( i = 1, …, 7),

CE

D j (ei ( Fl l 1 )) ( j = 1, …, 6) is combined to generate D(ei ( Fl l 1 )) and further D( Fl l 1 ) ,

AC

which are presented in Tables B.11-B.12 of Section B.2. Through Eqs. (1)-(2), S ( Fl l 1 ) ( l = 1, …, 5) is derived from D( Fl l 1 ) , from which S ( Fl 1l ) is deduced by using Eqs. (3)-(4), as presented in Table 2. Table 2 Score intervals of any pairs of fields F1 F1

—

F2

F3

F4

F5

F6

[-0.1492,

[-0.192,

[0.0681,

[0.0621,

[-0.0404,

-0.0663]

-0.0312]

0.3079]

0.3624]

0.3305]

24

ACCEPTED MANUSCRIPT

—

[-0.045,

[0.2151,

[0.2091,

[0.1066,

0.035]

0.3685]

0.42]

0.3882]

[0.2601,

[0.2805,

[0.178,

0.3391]

0.3797]

0.3478]

[-0.006,

[-0.1083,

0.0638]

0.0319]

0.1492]

[0.0312,

[-0.035,

0.192]

0.045]

[-0.3079,

[-0.3685,

[-0.3391,

-0.0681]

-0.2151]

-0.2601]

[-0.3624,

[-0.42,

[-0.3797,

[-0.0638,

-0.0621]

-0.2091]

-0.2805]

0.006]

[-0.3305,

[-0.3882,

[-0.3478,

[-0.0319,

0.0404]

-0.1066]

-0.178]

0.1083]

—

F3

—

F4

F5

F6

[-0.1025,

CR IP T

[0.0663, F2

—

-0.0319]

[0.0319,

—

0.1025]

AN US

To achieve the score intervals of any pairs of nonadjacent fields from S ( Fl l 1 ) and S ( Fl 1l ) in accordance with Definition 9, the parameter b of the function g ( y, z ) must

be determined. For this purpose, S (ei ( Fl l 1 )) is obtained from D(ei ( Fl l 1 )) by using Eqs.

M

(1)-(2), as presented in Table B.13 of Section B.2. From Table B.13 it can be found that N l

ED

( l = 1, …, 4) = (2, 3, 4, 3) and N l ( l = 1, …, 4) = (4, 4, 5, 2), which determines that N 3* = max { Nl  Nl , l = 1, ..., 4}. According to this, the decision makers t j ( j = 4, 5, 6)

PT

provide D j (ei ( F35 )) ( j = 4, 5, 6), which is presented in Table B.7 of Section B.1.

CE

Meanwhile, B j (ei ( F3 )) and B j (ei ( F5 )) ( j = 1, 2, 3) are transformed into D j (ei ( F35 )) by using Definition 5. After D j (ei ( F35 )) ( j = 1, …, 6) is combined using the ER algorithm

AC

and  (ei ) , D(ei ( F35 )) is generated and used to calculate S (ei ( F35 )) in accordance with Eqs. (1)-(2), as presented in Table B.13. By solving the optimization model shown in Eqs. (45)-(50) with S (ei ( F34 )) , S (ei ( F45 )) , and S (ei ( F35 )) , the parameter b is obtained as -0.8265 with the objective of the model F = 0.0079. With the aid of the resulting parameter

b , the score intervals of the pairs of nonadjacent fields are derived from S ( Fl l 1 ) and S ( Fl 1l ) through Definition 9, as presented in Table 2.

By following Proposition 1, a preferring matrix P  ( plm )66 is generated from the score 25

ACCEPTED MANUSCRIPT

matrix in Table 2 using Eq. (51), as presented in Table B.14 of Section B.2. Through Eqs. (52)-(53), it is obtained that NP( Fl ) ( l = 1, …, 6) = (3, 4, 5, 1, 0, 2) and r ( Fl ) ( l = 1, …, 6) = (3, 2, 1, 5, 6, 4). A ranking order of the six fields is generated from r ( Fl ) , which is F3 F2

F1

F6

F4

F5 . The possibility degrees between the neighboring

fields in the ranking order are further calculated using Eq. (6) and the score intervals in Table 2, which are 0.6167, 1, 0.9763, 0.8964, and 0.9851. As a whole, a ranking order of the six 1

F2

0.9763

0.8964

0.9851

CR IP T

0.6167

fields with possibility degrees is obtained as F3

F1

F6

F4

F5 . This indicates that Mobile terminal ( F3 ) should be selected as the key field by

considering the opinions of the six decision makers.

The generated solution shows that mobile terminal ( F3 ), intelligent television ( F2 ), and

AN US

automotive electronics ( F1 ) are more appropriate than the other three fields to become the key field of the enterprise. For this reason, we take F3 , F2 and F1 as examples to analyze their current situations in China to validate the solution. First, in accordance with the statistical data published by Ministry of Industry and Information Technology of the People’s Republic

M

of China, approximately 1.4 and 1.5 billion of smartphones, which are characterized as a representative class of mobile terminal, are produced in China in 2015 and 2016, respectively.

ED

The continuous increase in smartphone production may be due to the continuous increase in new customers and old customers’ pursuit of new functionality and experience generated by

PT

the latest smartphones. The applications of voice techniques in smartphones such as input method and voice broadcast are usually considered as the rigid demands of people. As a result,

CE

they contribute to the rapid development of smartphones. Second, intelligent television shows great market potential in China, a populous country in which people are accustomed to

AC

owning televisions. The application of voice techniques in intelligent television facilitates the interaction between television and people. However, it has been difficult for existing intelligent televisions to obtain a balance between functionality and cost to satisfy customers’ value requirements. Third, automotive electronics has become a key incentive to propel the development of the automobile industry within the past 10 years. The applications of voice techniques in automobiles, such as navigation and information query, have shown great potential. However, these applications have not become the rigid demands of people in

26

ACCEPTED MANUSCRIPT

accordance with the current development of automobile industry. The above qualitative analyses of the three fields demonstrate the rationality of selecting mobile terminal ( F3 ) as the key field of the enterprise to some extent. Next, the internal consistency and the Pareto principle of social choice theory will be verified using the data in the problem of selecting a key field. 4.3. Verification of internal consistency and Pareto principle

CR IP T

In the above process of generating the solution to the problem of selecting a key field, two properties of the transformation shown in Definition 5, which are the internal consistency and the Pareto principle of social choice theory, are not discussed. Although the two properties are theoretically proven in Propositions 3 and 4, it is meaningful to use the data in the problem of

AN US

selecting a key field to verify them, which is presented as follows.

To verify the internal consistency shown in Proposition 3, p(u j (ei ( Fl ))  u j (ei ( Fl 1 ))) ( j = 1, 2, 3, l = 1, …, 5) and p(S j (ei ( Fl (l 1) ))  S j (ei ( F(l 1)l ))) must be obtained. To generate

M

p(u j (ei ( Fl ))  u j (ei ( Fl 1 ))) , u j (ei ( Fl )) ( j = 1, 2, 3, l = 1, …, 6) is derived from B j (ei ( Fl )) shown in Tables B.1-B.3 by using Eqs. (31)-(32), as presented in Tables

ED

B.15-B.17 of Section B.3 in Appendix B. Through Eq. (5), p(u j (ei ( Fl ))  u j (ei ( Fl 1 ))) ( j

PT

= 1, 2, 3, l = 1, …, 5) is derived from u j (ei ( Fl )) , as presented in Tables B.18-B.20 of Section B.3. To generate p(S j (ei ( Fl (l 1) ))  S j (ei ( F(l 1)l ))) , S j (ei ( Fl (l 1) )) is obtained from

CE

the transformed D j (ei ( Fl (l 1) )) ( j = 1, 2, 3, l = 1, …, 5) shown in Tables B.8-B.10 by

AC

using Eqs. (1)-(2), as presented in Tables B.21-B.23 of Section B.3. From the resulting S j (ei ( Fl (l 1) )) , p(S j (ei ( Fl (l 1) ))  S j (ei ( F(l 1)l ))) ( j = 1, 2, 3, l = 1, …, 5) is obtained using

Eqs. (3)-(4) and (6), as presented in Tables B.24-B.26 of Section B.3. By comparing the possibility degrees shown in Tables B.18-B.20 and B.24-B.26, respectively, the internal consistency described in Proposition 3 is verified. To verify the Pareto principle of social choice theory described in Proposition 4, except

p(u j (ei ( Fl ))  u j (ei ( Fl 1 ))) ( j = 1, 2, 3, l = 1, …, 5), p(S j (ei ( Fl (l 1) ))  S j (ei ( F(l 1)l )))

27

ACCEPTED MANUSCRIPT

( j = 4, 5, 6, l = 1, …, 5) and p(S (ei ( Fl (l 1) ))  S (ei ( F(l 1)l ))) must be obtained. To generate p(S j (ei ( Fl (l 1) ))  S j (ei ( F(l 1)l ))) ( j = 4, 5, 6, l = 1, …, 5), D j (ei ( Fl (l 1) )) shown in Tables B.4-B.6 is used to obtain S j (ei ( Fl (l 1) )) according to Eqs. (1)-(2), as presented in Tables B.27-B.29 of Section B.3. The resulting S j (ei ( Fl (l 1) )) is then used to

CR IP T

generate p(S j (ei ( Fl (l 1) ))  S j (ei ( F(l 1)l ))) ( j = 4, 5, 6, l = 1, …, 5) in accordance with Eqs. (3)-(4) and (6), as presented in Tables B.30-B.32 of Section B.3. Meanwhile,

p(S (ei ( Fl (l 1) ))  S (ei ( F(l 1)l ))) is derived from S (ei ( Fl (l 1) )) shown in Table B.13 using Eqs. (3)-(4) and (6), as presented in Table B.33 of Section B.3. By comparing the possibility

AN US

degrees shown in Tables B.24-B.26 and B.30-B.32 and those shown in Table B.33, the Pareto principle of social choice theory in Proposition 4 is verified.

The above verification is conducted by using the given u ( H n ) , s(Gx ) , and  (ei ) . In the situation where u ( H n ) ,  (ei ) , and (or) s(Gx ) can be variables, the verification of the two

the two properties still hold.

ED

5. Conclusions

M

properties is discussed in Section B.4 of Appendix B. It can be found from the discussion that

When a group of decision makers face a common decision problem, they usually seek to

PT

provide preference information with the different modes that are consistent with their knowledge, experience, and cognitive habits as they are allowed. To facilitate decision makers

CE

in flexibly providing preference information, two modes of preference information including the sole evaluation of alternatives and the pairwise comparison between alternatives are

AC

generally adopted in the existing studies. These studies mainly focus on GDM problems with the two modes of preference information in a fuzzy context; however, they pay little attention to MCGDM problems with BDs and DPRs. To address MCGDM problems with BDs and DPRs, we propose a new method, which is an indirect method. In this method, we develop a transformation from BDs into DPRs when the BDs to be transformed are complete or some or all of them are incomplete. To support the transformation, an abstract two-variable function is defined with the relevant properties that should be satisfied. A specific function is then selected for the transformation to satisfy the 28

ACCEPTED MANUSCRIPT

internal consistency and the Pareto principle of social choice theory, which is theoretically proven. Based on the transformation with the selected function, the process of generating solutions to MCGDM problems with BDs and DPRs is discussed. In the process, only the DPRs between neighboring alternatives are required to be provided to relieve the burden on decision makers who select to provide DPRs. While the consistency of a score matrix shown in Definition 9 and the optimization model shown in Eqs. (45)-(50) are used to generate the

CR IP T

complete and consistent score matrix of alternatives, from which a solution is generated using Eqs. (6) and (51)-(53). The problem of selecting a key field for an enterprise located in Hefei, Anhui Province, China, is solved to demonstrate the applicability of the proposed method.

In the proposed method, criterion weights and the relative weights of decision makers are

AN US

subjectively determined. More complex situations where interval-valued BDs and DPRs are offered for different types of reasons such as limitations of knowledge, information, or data are not addressed. Meanwhile, group consensus is not the focus of this paper. In the next step, we will propose new methods to learn criterion weights and the relative weights of decision

M

makers from historical decision data and analyze MCGDM problems with interval-valued BDs and DPRs as well as group consensus requirements. The internal consistency and the

Acknowledgements

ED

Pareto principle of social choice theory are also required to be satisfied.

PT

This research is supported by the National Natural Science Foundation of China (Grant Nos. 71622003, 71571060, 71690235, 71690230, and 71521001).

CE

References

Abel, E., Mikhailov, L., & Keane, J. (2018). Inconsistency reduction in decision making via

AC

multi-objective optimisation. European Journal of Operational Research, 267(1), 212-226.

Altuzarra, A., Moreno-Jiménez, J. M., & Salvador, M. (2010). Consensus building in AHP-group decision making: a Bayesian approach. Operations Research, 58(6), 1755-1773. Arrow, K. J. (1963). Social choice and individual values (second edition). New York: Wiley. Blagojevic, B., Srdjevic, B., Srdjevic, Z., & Zoranovic, T. (2016). Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm. 29

ACCEPTED MANUSCRIPT

Information Sciences, 330, 260-273. Bottero, M., Ferretti, V., Figueira, J. R., Greco, S., & Roy, B. (2018). On the Choquet multiple criteria preference aggregation model: theoretical and practical insights from a real-world application. European Journal of Operational Research, 271(1), 120-140. Chen, S. M., Cheng, S. H., & Tsai, W. H. (2016). Multiple attribute group decision making based on interval-valued intuitionistic fuzzy aggregation operators and transformation

CR IP T

techniques of interval-valued intuitionistic fuzzy values. Information Sciences, 367-368, 418-442.

Chen, X., Zhang, H. J., & Dong, Y. C. (2015). The fusion process with heterogeneous preference structures in group decision making: A survey. Information Fusion, 24, 72-83.

AN US

Chiclana, F., Herrera, F., & Herrera-Viedma, E. (1998). Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets and Systems, 97, 33-48.

Chiclana, F., Herrera, F., & Herrera-Viedma, E. (2001). Integrating multiplicative preference

M

relations in a multipurpose decision-making based on fuzzy preference relations. Fuzzy Sets and Systems, 122, 277-291.

ED

Chiclana, F., Herrera-Viedma, E., Herrera, F., & Alonso, S. (2007). Some induced ordered weighted averaging operators and their use for solving group decision-making problems

383-399.

PT

based on fuzzy preference relations. European Journal of Operational Research, 182(1),

CE

Dong, Y. C., Chen, X., & Herrera, F. (2015a). Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making.

AC

Information Sciences, 297, 95-117. Dong, Y. C., Luo, N., & Liang, H. M. (2015b). Consensus building in multiperson decision making with heterogeneous preference structures: A perspective based on prospect theory. Applied Soft Computing, 35, 898-910. Dong, Y. C., Xu, Y. F., & Yu, S. (2009). Linguistic multiperson decision making based on the use of multiple preference relations. Fuzzy Sets and Systems, 160, 603-623. Dong, Y. C., & Zhang, H. J. (2014). Multiperson decision making with different preference representation

structures:

A direct

consensus 30

framework

and

its

properties.

ACCEPTED MANUSCRIPT

Knowledge-Based Systems, 58, 45-57. Dong, Y. C., Zhang, G. Q., Hong, W. C., & Xu Y. F. (2010). Consensus models for AHP group decision making under row geometric mean prioritization method. Decision Support Systems, 49(3), 281-289. Fan, Z. P., Ma, J., Jiang, Y. P., Sun, Y. H., & Ma, L. (2006). A goal programming approach to group decision making based on multiplicative preference relations and fuzzy preference

CR IP T

relations. European Journal of Operational Research, 174(1), 311-321.

Fu, C., Xu, D. L., & Yang, S. L. (2016). Distributed preference relations for multiple attribute decision analysis. Journal of the Operational Research Society, 67(3), 457-473.

Fu, C., & Yang, S. L. (2010). The group consensus based evidential reasoning approach for

AN US

multiple attributive group decision analysis. European Journal of Operational Research, 206(3), 601-608.

Fu, C., & Yang, S. L. (2012). An evidential reasoning based consensus model for multiple attribute group decision analysis problems with interval-valued group consensus

M

requirements. European Journal of Operational Research, 223(1), 167-176. Herrera-Viedma, E., Herrera, F., & Chiclana, F. (2002). A consensus model for multiperson

ED

decision making with different preference structures. IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, 32(3), 394-402.

PT

Liu, P. D. (2017). Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators. Computers & Industrial

CE

Engineering, 108, 199-212. Li, D. F., & Wan, S. P. (2014). A fuzzy inhomogeneous multiattribute group decision making

AC

approach to solve outsourcing provider selection problems. Knowledge-Based Systems, 67, 71-89.

Liu, J. P., Chen, H. Y., Xu, Q., Zhou, L. Q., & Tao, Z. F. (2016a). Generalized ordered modular averaging operator and its application to group decision making. Fuzzy Sets and Systems, 299, 1-25. Liu, Y., Fan, Z. P., & Zhang, X. (2016b). A method for large group decision-making based on evaluation information provided by participators from multiple groups. Information Fusion, 29, 132-141. 31

ACCEPTED MANUSCRIPT

Liu, B. S., Shen, Y. H., Chen, X. H., Sun, H., & Chen, Y. (2014). A complex multi-attribute large-group PLS decision-making method in the interval-valued intuitionistic fuzzy environment. Applied Mathematical Modelling, 38(17-18), 4512-4527. Ma, J., Fan Z. P., Jiang, Y. P., & Mao, J. Y. (2006). An optimization approach to multiperson decision making based on different formats of preference information. IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, 36(5), 876-889.

CR IP T

Meng, F. Y., & Chen, X. H. (2015). An approach to incomplete multiplicative preference relations and its application in group decision making. Information Sciences, 309, 119-137.

Merigó, J. M., Palacios-Marqués, D., & Zeng, S. Z. (2016). Subjective and objective

AN US

information in linguistic multi-criteria group decision making. European Journal of Operational Research, 248(2), 522-531.

Ölçer, A. İ., & Odabaşi, A. Y. (2005). A new fuzzy multiple attributive group decision making methodology and its application to propulsion/manoeuvring system selection problem.

M

European Journal of Operational Research, 166(1), 93-114.

Qin, J. D., Liu, X. W., & Pedrycz, W. (2017). An extended TODIM multi-criteria group

ED

decision making method for green supplier selection in interval type-2 fuzzy environment. European Journal of Operational Research, 258(2), 626-638.

PT

Qu, J. H., Meng, X. L., & You, H. (2016). Multi-stage ranking of emergency technology alternatives for water source pollution accidents using a fuzzy group decision making tool.

CE

Journal of Hazardous Materials, 310, 68-81. Ren, J. Z., & Liang, H. W. (2017). Measuring the sustainability of marine fuels: A fuzzy

AC

group multi-criteria decision making approach. Transportation Research Part D, 54, 12-29.

Soltani, A., Sadiq, R., & Hewage, K. (2016). Selecting sustainable waste-to-energy technologies for municipal solid waste treatment: a game theory approach for group decision-making. Journal of Cleaner Production, 113, 388-399. Voola, P., & Babu, V. (2017). Study of aggregation algorithms for aggregating imprecise software requirements’ priorities. European Journal of Operational Research, 267(1), 212-226. 32

ACCEPTED MANUSCRIPT

Wan S. P., Wang, F., & Dong, J. Y. (2017). Additive consistent interval-valued Atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making. European Journal of Operational Research, 263(2), 571-582. Wibowo, S., & Deng, H. P. (2015). Multi-criteria group decision making for evaluating the performance of e-waste recycling programs under uncertainty. Waste Management, 40, 127-135.

CR IP T

Winston, W. (2011). Operations research: Applications and algorithms. Beijing: Tsinghua University Press.

Wu, T., & Liu, X. W. (2016). An interval type-2 fuzzy clustering solution for large-scale multiple-criteria group decision-making problems. Knowledge-Based Systems, 114,

AN US

118-127.

Wu, Z. B., & Xu, J. P. (2016). Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations. Omega, 65, 28-40. Xu, X. H., Zhong, X. Y., Chen, X. H., & Zhou, Y. J. (2015). A dynamical consensus method

M

based on exit-delegation mechanism for large group emergency decision making. Knowledge-Based Systems, 86, 237-249.

ED

Yang, W. E., Ma, C. Q., Han, Z. Q., & Chen, W. J. (2016). Checking and adjusting order-consistency of linguistic pairwise comparison matrices for getting transitive

PT

preference relations. OR Spectrum, 38, 769-787. Yang, J. B., & Xu, D. L. (2013). Evidential reasoning rule for evidence combination. Artificial

CE

Intelligence, 205, 1-29.

Zendehdel, K., Rademaker, M., De Baets, B., & Van Huylenbroeck, G. (2010). Environmental

AC

decision making with conflicting social groups: A case study of the Lar rangeland in Iran. Journal of Arid Environments, 74(3), 394-402.

Zhang, Z. M. (2017). Hesitant fuzzy multi-criteria group decision making with unknown weight information. International Journal of Fuzzy Systems, 19(3), 615-636. Zhang, Q., Fan, Z. P., & Pan, D. H. (1999). A ranking approach for interval numbers in uncertain multiple attribute decision making problems. Systems Engineering - Theory & Practice, 5, 129-133 (in Chinese). Zhang, Z., & Guo, C. H. (2014). An approach to group decision making with heterogeneous 33

ACCEPTED MANUSCRIPT

incomplete uncertain preference relations. Computers & Industrial Engineering, 71, 27-36. Zhang, Z. M., Wang, C., & Tian, X. D. (2015). A decision support model for group decision making with hesitant fuzzy preference relations. Knowledge-Based Systems, 86, 77-101. Zhang, M. J., Wang, Y. M., Li, L. H., & Chen, S. Q. (2017). A general evidential reasoning

AC

CE

PT

ED

M

AN US

Journal of Operational Research, 257(3), 1005-1015.

CR IP T

algorithm for multi-attribute decision analysis under interval uncertainty. European

34