MADM method based on prospect theory and evidential reasoning approach with unknown attribute weights under intuitionistic fuzzy environment

Accepted Manuscript

MADM method based on prospect theory and evidential reasoning approach with unknown attribute weights under intuitionistic fuzzy environment Tiantian Bao , Xinlian Xie , Peiyin Long , Zhaokun Wei PII: DOI: Reference:

S0957-4174(17)30482-7 10.1016/j.eswa.2017.07.012 ESWA 11428

To appear in:

Expert Systems With Applications

Received date: Revised date: Accepted date:

27 February 2017 10 July 2017 11 July 2017

Please cite this article as: Tiantian Bao , Xinlian Xie , Peiyin Long , Zhaokun Wei , MADM method based on prospect theory and evidential reasoning approach with unknown attribute weights under intuitionistic fuzzy environment , Expert Systems With Applications (2017), doi: 10.1016/j.eswa.2017.07.012

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Highlights: The extended measures of intuitionistic fuzzy entropy and cross entropy are proposed. The proposed optimization models ensure the objectivity and reasonability of weights. The proposed approach can reduce the loss of decision making information.

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The decision results based on human bounded rationality can be more realistic.

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Title: MADM method based on prospect theory and evidential reasoning approach with unknown attribute weights under intuitionistic fuzzy environment Author names and affiliations: 1. Tiantian Bao

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E-mail address: [email protected] Full affiliation: Transportation Management College, Dalian Maritime University, Dalian 116026, China

2. Xinlian Xie*

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E-mail address: [email protected] Full affiliation: Transportation Management College, Dalian Maritime University, Dalian 116026, China

3. Peiyin Long E-mail address: [email protected]

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Full affiliation: Guizhou Highway Survey and Design Academy Co., Ltd., Guizhou 550081, China

4. Zhaokun Wei

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E-mail address: [email protected] Full affiliation: Transportation Management College, Dalian Maritime University, Dalian 116026, China

Corresponding author:

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Xinlian Xie

Tel.: +86 411 84729951

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E-mail address: [email protected] Complete postal address: Transportation Management College, Dalian Maritime University, Dalian 116026, China

Present/permanent address: Room 322, Transportation Management College, Dalian Maritime University, Dalian 116026, China

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We are very grateful again to the respected editor and the anonymous reviewers for their helpful comments. Based on the reviewers' comments, our manuscript is revised in which the modified parts are marked in yellow, and the Point-to-Point responses are completed as follows. If any explanation in the Point-to-Point responses is not clear, we would like to further illustrate and modify it.

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Abstract: This paper proposes an intuitionistic fuzzy decision method based on prospect theory and the evidential reasoning approach, aiming at analyzing multi-attribute decision making problems in which the criteria values are intuitionistic fuzzy numbers and the information of attributes weights is unknown. Firstly, the measures of entropy and cross entropy are defined for

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intuitionistic fuzzy sets by taking into consideration the preference of decision maker towards hesitancy degree. Secondly, combined with bounded rationality, the prospect decision matrix is calculated in the light of prospect theory and intuitionistic fuzzy distance. Thirdly, the correlational analyses are conducted between the attribute weights and three indicators which

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are entropy, cross entropy and prospect value, and optimization models for identifying attribute weights are built under the circumstances that the weights are incomplete and

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unknown. Finally, in order to avoid the loss of decision making information, the evidential reasoning approach is applied to the calculation of comprehensive prospective values for all

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alternatives. Following the value calculation, the ranking and the optimal alternative are determined based on the comprehensive prospective values. Illustrating examples demonstrate

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that the proposed method is reasonable and feasible.

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Keywords: Decision analysis; Multi-attribute decision making; Prospect theory; Evidential reasoning approach; Entropy

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1. Introduction Multi-attribute decision making (MADM) is of great importance in modern decision science, and widely adopted in management, economics and other fields. Generally, there are two key aspects in the process of MADM: (1) Collect information including all information contained in a decision matrix (Yang & Xu, 2002) and determine attribute weights;

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(2) Aggregate the information to evaluate decision alternatives.

For the first aspect, because of the complexity of the real world problems and the subjective nature of decision makers’ preferences and judgments, the information of MADM problems is often uncertain. One way of dealing with the subjective uncertainty is to use

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intuitionistic fuzzy numbers (IFNs) to express decision makers’ evaluations of decision alternatives. Intuitionistic fuzzy set (IFS) (Atanassov, 1986) is an extension of fuzzy set (Zadeh, 1965) and employed to deal with complex decision making problems (He & Teng, 2014; Park, Cho, & Kwun, 2013). For MADM problems under intuitionistic fuzzy (IF)

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environment, it is often difficult to give the complete weights objectively and accurately in the real world, so the estimation of attribute weights is an important research topic. Based on the

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related studies on MADM with unknown attribute weights, the solutions of determining attribute weights can be summed up in two ways. The one is to establish optimization models

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by maximizing similarity degree (Chen & Yang, 2011), satisfaction degree (Xu, 2012), distance from the anti-ideal alternative (Wang & Zhang, 2013) and collective overall rating

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(Xu, Wan, & Dong, 2016), or by minimizing distance from the ideal alternative (Wang & Zhang, 2013). The other way is to define IF entropy or cross entropy and adopt them to

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identify the weighting vector objectively (Li & Wu, 2016; Qi, Liang, & Zhang, 2015; Xia & Xu, 2012; Ye, 2013). However, these weighting methods suffer from some limitations, for example the IF entropy measure (Qi et al., 2015) will be invalid in some special situations. For the second aspect, the aggregation methods for MADM problems with IFSs have drawn the attention of many scholars. In general, the aggregation operators of IFSs are used to aggregate fuzzy decision information, such as IF weighted averaging operator (Qi et al., 2015), IF weighted geometric operator (Khaleie & Fasanghari, 2012), simple additive weighting operator (Chen & Yang, 2011; Z. Xu, 2012), symmetric IF weighted averaging operator and

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symmetric IF ordered weighted averaging operator (Xia & Xu, 2012). However, these operators still have some shortcomings. For example, the loss of information may be introduced when using the aforementioned operators to aggregate decision information, which will influence the ranking of alternatives. In order to overcome the shortcomings, we propose to introduce evidence theory into MADM problem analysis to aggregate the IF information about alternatives. In recent years, evidence theory has been developed and employed to solve

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MADM problems by a few researchers (Fu, Yang, & Yang, 2015; Liu, Liao, & Yang, 2015; Wang, Nie, Zhang, & Chen, 2013; Wang & Zhang, 2013; Wang, Zhu, Song, & Lei, 2016).

However, the vast majority of MADM approaches assume that the decision makers (DMs) are fully rational. In a real decision process, the behaviors and psychologies of DMs

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can have impacts on decision analysis. Prospect theory (Kahneman & Tversky, 1979) is regarded as one of the most well-known behavior decision theories, so there are increasing research interests in the MADM methods based on prospect theory to consider the DMs’ risk attitude and bounded rationality (Chen, Chin, Ding, & Li, 2016; Fan, Zhang, Chen, & Liu,

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2013; Hu & Yang, 2011; Peng, Liu, Liu, & Su, 2014). However, some gaps still exist in the literature on the representation and aggregation of complex information, such as the fuzzy

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decision information described by IFS in this context. The aim of this paper is to investigate the MADM problems in which attribute values are

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IFSs and the weight information is incomplete or unknown. Although plenty of methods and approaches have been developed to deal with the related MADM problems, there are several

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important issues:

(1) The IF entropy and cross entropy, common weighting methods, need to be extended

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to take into account the DMs’ preferences and address the “no definition” issues of the existing measures in some particular cases; (2) To decrease the loss of decision information, the existing IF aggregation operators need to be replaced by other approaches such as the evidential reasoning approach which is known for making the maximum use of all available while taking care not to distort uncertainty and unknown information (Yang & Xu, 2002); (3) It is necessary to consider the influence of DMs’ bounded rationality based on prospect theory for the MADM problems under IF environment, especially with partially

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and completely unknown weights. In the rest of this paper, Section 2 is devoted to provide an overview of the related work on MADM methods. Then, some basic concepts and definitions of IFSs, evidence theory and prospect theory are briefly reviewed in Section 3. In Section 4, the extended entropy and cross entropy measures for IFSs are defined to flexibly reflect the DMs’ preferences. Furthermore, an MADM method is proposed based on prospect theory and the evidential reasoning

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approach, in which some corresponding programming models are established to determine criteria’s weights in Section 5. Finally, examples and conclusion are given in Sections 6 and 7, respectively.

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2. Related work 2.1. MADM methods under intuitionistic fuzzy environment

Intuitionistic fuzzy set, which is characterized by a membership function and a non-membership function, is one of important tools to deal with the uncertainty and vagueness. In recent years, it has been widely applied in the MADM field. For example, Park

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et al. (2013) proposed a dynamic intuitionistic fuzzy MADM method based on the VIKOR method. He and Teng (2014) investigated dynamic hybrid MADM problems using IFS.

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Because the weight information plays an important role in the decision process, an increasing number of experts have investigated the weight determination in the MADM problems under

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IF environment. For example, Chen and Yang (2011) investigated some optimization models

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for determining the attribute weights of the multi-attribute group decision making (MAGDM) problems by maximizing the similarity degree. Xia and Xu (2012) developed two pairs of IF

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entropy and cross entropy for eliciting attribute weights of MADM problems. Khaleie and Fasanghari (2012) defined two entropy measures of IFS to calculate the weight vector of criteria. Xu (2012) established a multi-objective optimization model to determine the weight information. Wang and Zhang (2013) proposed a nonlinear programming model to obtain the optimal criteria’s weights based on the distances from ideal alternative and anti-ideal alternative. Ye (2010, 2013) determined weights of attributes and experts for MAGDM by using IF entropy. Qi et al. (2015) proposed a generalized cross entropy to identify the weights of attributes and experts objectively. Xu et al. (2016) solved the heterogeneous MAGDM

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problem based on TOPSIS and constructed a multiple objective IF programming model to obtain the optimal criteria’s weights. Li and Wu (2016) constructed a mathematical programming model based on IF cross entropy and grey correlation analysis to calculate attribute weights of MADM problems. 2.2. MADM methods based on evidence theory

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Evidence theory, also called D-S theory (Dempster, 1967), is one of the main methods in the uncertainty reasoning and of great advantage in processing uncertain multi-source information. Wang et al. (2016) studied the combination operation of unreliable evidence sources in the IF multi-criteria decision making (MCDM) framework based on D-S theory. In addition, Yang and Xu (2002) proposed the evidential reasoning approach (ERA) to aggregate

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multiple attributes based on the combination rule of evidence theory and belief decision matrix. Till now, the ERA has been successfully adopted to aggregate the uncertain and imprecise information of MADM problems. For example, Wang et al. developed a series of approaches based on the ERA for analyzing MADM problems under the IF environment

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(Wang et al., 2013; Wang & Zhang, 2013). Liu et al. (2015) employed the ERA to aggregate the uncertain information represented by the belief structure in the MAGDM problems. Fu et

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al. (2015) investigated the MAGDM problems with the belief structure and combined the group assessments by utilizing the evidential reasoning rule (Yang & Xu, 2013).

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2.3. MADM methods based on prospect theory Prospect theory, initiated by Kahneman and Tversky (1979), was proposed based on the

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assumption of bounded rationality. It explained the major violations of utility theory which assumed that all reasonable people would obey the axioms of the theory. Utility theory has

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dominated the analysis of decision making under risk (Kahneman & Tversky, 1979), but there have been some studies about MADM based on prospect theory in recent years. For example, Hu and Yang (2011) developed a dynamic stochastic MCDM method by combining cumulative prospect theory and set pair analysis. Fan et al. (2013) considered aspiration-levels of attributes based on prospect theory to analyze the MADM problem. Peng et al. (2014) proposed a method based on prospect theory to investigate the MADM problem with randomness uncertainty. Chen et al. (2016) studied the MCDM problem based on parametric

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estimation, score functions and prospect theory. 2.4. Argumentation-based approaches to MADM Argumentation is a reasoning model based on the construction and evaluation of interacting arguments which are intended to support, explain or attack statements (Amgoud & Prade, 2009). Argumentation theory was examined by Dung (1995), who suggested that an

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argumentation framework (AF) consisting of a set of arguments and the attack relation among them (Chai & Ngai, 2016). Since then, argumentation-based approaches have emerged for decision making. For instance, Amgoud and Prade (2009) proposed the first general and abstract argument-based framework, and applied it to the decision making under uncertainty and to MCDM. Bench-Capon et al. investigated the AFs (Atkinson et al., 2012) and used an

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argumentation-based frame to model decision making in economic experiments (Bench-Capon et al., 2012). Marey et al. (2015) studied agents’ uncertainty in argumentation-based negotiation and built a comprehensive framework for decision making. Samet et al. (2016) proposed an evidential argumentation framework which took into account

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imprecision and uncertainty based on evidence theory. Chai and Ngai (2016) developed a complex group argumentation (CGA) framework for complex group decision making. Ferretti

on dynamic AF.

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et al. (2017) proposed an abstract decision framework for single-agent decision making based

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In summary, the existing approaches have made significant contributions to solve the MADM problems. For the MADM problems with unknown weights under IF environment,

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the weight information can be determined based on the optimization models and IF entropy. However, there is still some room for improvement and developement through analysing and

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studying these weighting methods, such as resolving the invalidation problem of IF entropy and considering the DM’s preference and risk attitude. In order to make the minimum missing of all available in the aggregation of complex and uncertain information of MADM problems, the ERA will be introduced into the MADM approaches. The achivements based on evidence theory are still relatively limited, so further research is required to solve the MADM problems with IF decision information and unknown attribute weights. Finally, the DMs’ decision making behaviors and psychologies will influence the decision results of MADM problems,

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so the experts have coped with such issue through prospect theory. However, the representation and aggregation of prospect values in the prospect theory should be promoted and enhanced in the situation of MADM with uncertain information. Therefore, it is necessary to investigate the MADM problems with unknown weights based on IFS, evidence theory and prospect theory. Through the above discussion, the MADM methods have focused on looking for

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principles for comparing different alternatives which are supposed to be feasible. Namely, such methods aim at ranking a group of alternatives rather than understanding an alternative individually. Since each choice and decision has pros and cons of various strengths, adopting argumentation in the decision making is necessary in order to provide the reasons underlying

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the recommendation (Amgoud & Prade, 2009). However, some gaps still exist in the argumentation-based approaches, such as determining the unknown weights of attributes, which is the main topic discussed in the paper. The proposed method could be modified which based on argumentation theory to make and explain decisions in the future research.

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3. Preliminaries 3.1. Intuitionistic fuzzy set 1

(Atanassov,

1986).

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Definition



A  x,  A  x  , A  x  | x  X



Suppose

X

be

a

finite

universal

set,

is defined as an intuitionistic fuzzy set in X, where

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 A  0,1 and  A  0,1 , with the condition 0   A  x    A  x   1 , for all x  X .

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 A  x  and  A  x  stand for the degree of membership and non-membership of the element x to A, respectively, and  A  x   1   A  x    A  x  is the hesitancy degree. Then an

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intuitionistic fuzzy number can be denoted by a   a ,a  . Definition 2 (Atanassov, 1986). Let a   a ,a  , a1   1 ,1  and a2   2 ,2  be the IFNs, then

(1) a  a , a  ;

  

 ,1  1     ,

(2)  a  1  1  a  ,a ,   0 ; (3) a



 a



a

  0;

ACCEPTED MANUSCRIPT (4) a1  a2   1  2  12 ,12  . Definition 3 (Xu et al., 2016). For an IFN a   a ,a  , a score function is defined as

S a   a  a , and an accuracy function can be defined as H  a   a  a . For two IFNs

a1   1 ,1  and a2   2 ,2  , then (1) a1  a2 , if S a1   S a2  or S a1   S a2   H  a1   H  a2  ;

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(2) a1  a2 , if S a1   S a2  or S a1   S a2   H  a1   H  a2  ; (3) a1  a2 , if S a1   S a2   H  a1   H  a2  .

Definition 4 (Li, Wu, & Zhu, 2014). An intuitionistic fuzzy weighted averaging (IFWA)

operator of dimension n is a mapping, IFWA : n   , that has n IFNs ai and an associated

  1 , 2 , , n  , and

IFWA  a1 , a2 ,

, an   1a1  2a2 

3.2. Evidence theory

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weight vector

 n an .

Definition 5. Suppose the hypothesis space  be a finite set, 2 be a power set of  . If a mapping m : 2  0,1 satisfies m     0 and

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



A

m  A  1 , then m is the basic

probability mass of  , which is also called mass function. If m  A  0 , A is a focal

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element.

Definition 6 (Dempster, 1967). For two element B and C, the combination rule is defined as

m  A 



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follows,

B ,C  , B C  A

m1  B  m2  C 

,

(1)

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1 k where, k   B,C , B C  m1  B  m2  C  .

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3.3. Prospect theory Prospect theory assumes that DMs choose the optimal alternatives based on prospect

values which represent various consequences of risks. As an important part of prospect theory, the value function (Kahneman & Tversky, 1979) is a power function as follows:    x v  x         x 

x0 x0

,

(2)

where, x  0 means gain, while x  0 means loss;  and  are coefficients of risk attitude, defined by 0   ,   1 ; The higher values of  and  means that the DMs are

ACCEPTED MANUSCRIPT more prone to risk.  is the loss aversion coefficient, and when   1 , it means that the DMs are sensitive towards loss risk.

4. Extended cross entropy and entropy for intuitionistic fuzzy numbers 4.1. Extended cross entropy for measuring divergence between IFNs IF cross entropy is an important concept to measure the divergence of information in IF theory. The classical cross entropy measure was defined between two probability distributions

, pn  and Q  q1 , q2 ,

, qn  , as follows (Kullback & Leibler, 1951):

H  P, Q   i 1 pi ln pi qi , n

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P   p1 , p2 ,

(3)

which describes the degree of discrimination between distributions P and Q . However,

H  P, Q  is invalid when qi  0 and pi  0 . To resolve this issue, a modified cross

K  P, Q    i 1 pi ln n

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entropy of P from Q was defined as (Lin, 1991)

pi .  pi 2    qi 2 

(4)

For n  2 , assume that P   p,1  p and Q  q,1  q , then

2 1  p  2p .  1  p  ln pq 1  p   1  q 

(5)

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K  P, Q  =p ln

Shang and Jiang (1997) further extended the definition of cross entropy in Eq. (5) into

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the environment of fuzzy sets and defined the fuzzy cross entropy. Let fuzzy sets A and B be in the given universe X   x1 , x2 ,

, xn  , the fuzzy cross entropy is defined as

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  2 1   A  xi   2 A  xi  n   . (6) E  A, B    i 1   A  xi  ln  1   A  xi   ln  A  xi   B  xi  1   A  xi    1  B  xi     

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Moreover, Zhang and Jiang (2008) derived a cross entropy of vague sets. Assume that

A  xi     A  xi  ,1   A  xi  and B  xi    B  xi  ,1  B  xi  are two vague sets in the given

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universe X   x1 , x2 ,

, xn  , where  A  xi  and  B  xi  are the degrees of membership

of element xi to A and B respectively, while  A  xi  and  B  xi  are non-membership degrees of that element xi to A and B respectively. Then the vague cross entropy is defined as

D  A, B  



 A  xi   1   A  xi 

 ln

2   A  xi   1   A  xi  

  A  xi   1   A  xi      B  xi   1   B  xi   . (7) 2 1   A  xi    A  xi   1   A  xi    A  xi    ln 2 1   A  xi    A  xi    1   B  xi    B  xi   xi X xi X

2

ACCEPTED MANUSCRIPT Since the relationship between the membership and the non-membership degrees of a vague set is similar to that of an IFS, the IF cross entropy can be devised from Eq. (7) to measure the divergence degree of one IFN from another. For two IFNs     ,  and

    ,  , the cross entropy D  ,   can be defined as D  ,   

  1  

 ln

2    1   



2 1      1      ln 2 1       1      , (8)

m 

can be transformed as

2

  1   2



      2

Similarly, there is m 

     2    k  , k  1 2 .

  1   2

(9)

   k  , where k  1 2 . And their geometric

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where m 

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   1        1    2 1  m  2m  m  ln  1  m   ln m  m 1  m   1  m    1   2

explanations can be illustrated as shown in Fig. 1.

According to Fig. 1, it can be seen that the hesitancy degree



is divided into the

degrees of membership and non-membership equally ( k  1 2 ) in Eq. (9). However, for

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many real-world complex situations, DMs may prefer to distribute



to membership and

non-membership in different split. Therefore a preference factor k ( 0  k  1 ) is introduced

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into the definition of cross entropy for IFSs to fully consider the DMs’ preference.

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m 

  1   2



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

2

2



1  

0

m 



1

  1   2







2

2



1  

0

1

Fig. 1. The geometric explanation of

m

and

m .

Furthermore, the cross entropy measure of Eq. (8) will be undefined if

m  1 , i.e. it is invalid for  =  0,1

or

m  0

or

 = 1,0  . To address this problem, the IF cross

entropy in Eq. (8) can be corrected based on a measures of fuzzy entropy (Parkash, Sharma, & Mahajan, 2008), as follows:

ACCEPTED MANUSCRIPT

CE  ,   

 2 1  t 1  m   2 1  tm  1    1  t 1  m    ln 1  tm   ln  T   1  tm  1  tm     1  t 1  m    1  t 1  m    

(10) where, T  1  t  ln 1  t    2  t  ln  2  t   ln 2 ,





.

m    k  and m    k  .

When t  0 , T is an increasing function of t . Then a symmetric form of cross entropy for IFSs can be defined to satisfy the symmetry of cross entropy. Definition 7. For two IFNs     ,  and     ,  , the symmetric intuitionistic fuzzy

CR IP T

cross entropy PCE  ,   can be defined as,

1  CE  ,    CE   ,   2 , (11) 2 1  t 1  m   2 1  tm  1    1  t 1  m    ln  1  tm   ln 2T  1  tm   1  tm   1  t 1  m    1  t 1  m     2 1  t 1  m   2 1  tm   1  tm   ln 1  tm  1 tm  1  t 1  m   ln    1  t 1  m  1  t 1  m                 

AN US

PCE  ,   





where T  1  t  ln 1  t    2  t  ln  2  t   ln 2 , t  0 ; m    k  , m    k  , and k ( 0  k  1) is the preference factor.

PCE  ,   can be utilized to measure the difference between two IFNs  and  .

M

According to Shannon’s inequality (Lin, 1991), it is easy to prove that (Zhang & Jiang, 2008)

ED

PCE  ,    0 and PCE  ,    0 when    ,    . 4.2. Extended entropy for measuring fuzziness of IFNs

PT

Intuitionistic fuzzy entropy is a useful tool to measure the degree of fuzziness. The relationship between cross entropy and entropy, which are two significant concepts in IF

CE

theory, has been verified under IF environment, as follows. Theorem 1 (Vlachos & Sergiadis, 2007). Let



be an IFN, its entropy EIFS   and

  satisfy  ,    1 ,

AC

cross entropy DIFS  , 

EIFS     DIFS

where  is a normalization factor. Assume that

   in Definition 7, then there is

(12)

ACCEPTED MANUSCRIPT

 12 CE  ,   CE  , 



PCE  ,  

 2 1  t 1  m   2 1  tm   1  t 1  m    ln  1  tm   ln 1  tm  1  tm      1  t 1  m    1  t 1  m   . (13)  2 1  t 1  m   2 1  tm  1  tm   ln 1  tm  1 tm   1  t 1  m   ln 1  t 1  m   1 t 1  m               2 1  t 1  m   2 1  tm  1    1  t 1  m    ln   1  tm   ln 2T  2  t  t  2k  1   2  t  t 1  2k    2 1  t 1  m    2 1  tm  1  tm   ln 2  t  t  2k  1   1  t 1  m   ln 2 t  t 1  2k       1 2T

CR IP T



In the light of Theorem 1 and Eq. (13), an extended entropy of IFS can be stated to measure fuzziness degree.

Definition 8. Let     ,  be an IFN, the IF entropy PE   can be defined as

1 2T





M

0  k  1.

AN US

 2 1  t 1  ma   2 1  tma   1  t 1  ma    ln   1  tma   ln 2  t  t  2k  1   2  t  t 1  2k     , (14)  2 1  t 1  m    b   2 1  tmb   1  t 1  mb    ln  1  tmb   ln  2  t  t  2k  1   2  t  t 1  2k     where, T  1  t  ln 1  t    2  t  ln  2  t   ln 2 , t  0 ; ma    k  , mb    k , PE    1 

Theorem 2. PE   satisfies the axiomatic conditions:

ED

(P1) PE    0 , if and only if  =  0,1 or  = 1, 0  ;

   ;

PT

(P2) PE    1 , if and only if (P3) PE    PE    , if

       or       ;

 

CE

(P4) PE    PE  .

AC

Therefore, PE   is an intuitionistic fuzzy entropy measure.

Proof.



Because



of



the

symmetry

of

cross

entropy

PCE  ,   , there is

 



PCE  ,   PCE  ,  . Then it is easy to prove PE    PE 

according to Theorem

1. Let H  1  tma  ln 1  tma   1  t 1  ma  ln 1  t 1  ma   1  tmb  ln 1  tmb   1  t 1  mb  ln 1  t 1  mb   2

 1  tma   1  tmb  1  tma   1  tmb  1  t 1  ma    1  t 1  mb   1  t 1  ma   1  t 1  mb   , ln  ln   2 2 2 2  

where

ACCEPTED MANUSCRIPT ma    k  , mb    k ,when 0  k  1, t  0 . Then, there is PE    1 

H , where T  1  t  ln 1  t    2  t   ln  2  t   ln 2  . T

f  x   1  tx  ln 1  tx   1  t 1  x   ln 1  t 1  x   when 0  x  1 , then

1  tx f  x   t ln 1  t 1  x  '

and

concave-up function of

t2 t2 f  x    0 . Thus 1  t 1  x  1  tx "

f  x

is

a

x , and H is convex. So H reaches its minimum at

CR IP T

Let

ma  mb  0 and maximum at ma  mb  1. Then H increases as ma  mb increases, and PE   decreases as decreases as   

ma  mb

increases. Since

increases.

ma  mb     , PE  

AN US

Therefore, PE   attains its minimum value when PE    0 (or when  =  0,1 or  = 1, 0  ) and its maximum value when PE    1 (or when

  

), as

PE    0 if and only if  =  0,1 or  = 1,0  , and PE    1 if and only if

   .

       or       , then        .Therefore,

M

If

ED

PE    PE    can be determined. 4.3. Applications

PT

In order to demonstrate the reasonability of the extended cross entropy and entropy for IFSs, the examples shown in document (Vlachos & Sergiadis, 2007) are taken.

CE

Example 1. Consider a pattern recognition problem. Suppose that there are three known patterns P1, P2 and P3, which have classifications C1, C2 and C3 respectively and represented

AC

by the following IFSs in X = {x1, x2, x3}.

P1 

 x ,1,0

P2 

 x ,0.8,0.1

x1 , x2 ,1,0 x2 , x3 ,0.9,0 x3 ;

P3 

 x ,0.6,0.2

x1 , x2 ,0.8,0 x2 , x3 ,1,0 x3 .

1

1

1



x1 , x2 ,0.8,0 x2 , x3 ,0.7,0.1 x3 ;



Given an unknown pattern Q 



 x ,0.5,0.3 1



x1 , x2 ,0.6,0.2 x2 , x3 ,0.8,0.1 x3 .

Based on the symmetric IF cross entropy (see Definition 7), the discrimination information

ACCEPTED MANUSCRIPT between Q and Pk (k =1, 2, 3) can be calculated by

DI  Q, Pk    x X PCE  Q  xi  , Pk  xi   .

(15)

i

Let t=1 and k=0.6, then Table 1 presents the values of DI  Q, Pk  between Q and Pk. According to the principle of minimum discrimination information, it is obvious that Q can be classified to the class C3. This result is in agreement with the one obtained in document

Table 1 The discrimination information between Q and Pk (k =1, 2, 3).

Q

P1

P2

P3

0.1913

0.1504

0.072

CR IP T

(Vlachos & Sergiadis, 2007).

AN US

Example 2. Consider a medical diagnosis problem. Suppose that there are four patients denoted by P = {Al, Bob, Joe, Ted}, five diagnoses represented by D = {Viral fever, Malaria, Typhoid, Stomach problem, Chest problem}, and five symptoms expressed by S = {Temperature, Headache, Stomach pain, Cough, Chest pain}. The doctors give the degrees of

M

membership and non-membership of each symptom for each diagnose and each patient, which are shown in Table 2 and Table 3 respectively.

ED

Let t=1 and k=0.6, then the discrimination information between each patient’s symptoms and the possible diagnoses can be obtained by utilizing the symmetric IF cross entropy and

PT

shown in Table 4. Similarly, according to the principle of minimum discrimination information, it can be concluded that Al and Ted suffer from viral fever, while Bob and Joe

CE

face stomach problem and typhoid respectively. Then the results obtained by using the proposed IF cross entropy are also accordant with the ones in document (Vlachos & Sergiadis,

AC

2007). Therefore, the extended IF cross entropy can be used to describe the discrimination information between IFSs and give reasonable results. Table 2

Symptoms characteristic for the diagnoses considered. Temperature

Headache

Stomach pain

Cough

Chest pain

Viral fever

(0.4, 0)

(0.3, 0.5)

(0.1, 0.7)

(0.4, 0.3)

(0.1, 0.7)

Malaria

(0.7, 0)

(0.2, 0.6)

(0, 0.9)

(0.7, 0)

(0.1, 0.8)

Typhoid

(0.3, 0.3)

(0.6, 0.1)

(0.2, 0.7)

(0.2, 0.6)

(0.1, 0.9)

ACCEPTED MANUSCRIPT

Stomach problem

(0.1, 0.7)

(0.2, 0.4)

(0.8, 0)

(0.2, 0.7)

(0.2, 0.7)

Chest problem

(0.1, 0.8)

(0, 0.8)

(0.2, 0.8)

(0.2, 0.8)

(0.8, 0.1)

Table 3 Symptoms characteristic for the patients. Headache

Stomach pain

Cough

Chest pain

Al

(0.8, 0.1)

(0.6, 0.1)

(0.2, 0.8)

(0.6, 0.1)

(0.1, 0.6)

Bob

(0, 0.8)

(0.4, 0.4)

(0.6, 0.1)

(0.1, 0.7)

(0.1, 0.8)

Joe

(0.8, 0.1)

(0.8, 0.1)

(0, 0.6)

(0.2, 0.7)

(0, 0.5)

Ted

(0.6, 0.1)

(0.5, 0.4)

(0.3, 0.4)

(0.7, 0.2)

(0.3, 0.4)

CR IP T

Temperature

Table 4

The discrimination information between each patient’s symptoms and the possible diagnoses. Malaria

Typhoid

Al

0.1822

0.2545

0.348

Bob

0.8566

1.5586

0.5234

Joe

0.3103

0.7233

0.154

Ted

0.186

0.3618

0.4889

Stomach problem

Chest problem

AN US

Viral fever

1.3009

1.5814

0.0487

0.9791

1.0411

1.3436

0.812

1.1041

M

5. MADM method based on prospect theory and evidential reasoning approach For a multi-attribute decision making problem, assume that there are m alternatives

, m and n attributes represented by C  c j | j  1, 2, , n .

ED

denoted by A  ai | i  1, 2,





n j 1

, n , satisfying 0   j  1 and

PT

The attribute weighting information is W   j | j  1, 2,

 j =1 . An intuitionistic fuzzy decision matrix is expressed by D   dij  mn and

CE

dij   ij ,ij  , in which each element is provided by the DMs for an alternative ai in

AC

relation to an attribute c j on the fuzzy concept “excellence”. 5.1. Establishment of prospect decision matrix The decision reference point is the key concept in prospect theory which concerns the

disparity between the outcome and expectation rather than the outcome alone. The decision



o o reference point vector can be denoted by O  o j  and o j   j , j 1n



for n attributes. The

DMs can determine the reference points by practice. In general, it can be taken as

o j   0.5,0.5 based on the definition and score function of IFS. Based on the value function in prospect theory (see Eq. (2)), IF prospect matrix V  vij  can be derived. mn

ACCEPTED MANUSCRIPT Definition 9. For IF decision matrix D   dij 

mn

and decision reference point vector

O  o j  , IF prospect matrix V  vij  mn can be calculated as, 1n





 D d ,o  ,d  o IFS  ij j ij j  , vij     DIFS  dij , o j  , dij  o j 



(16)



where vij is IF prospect value, which is the element of matrix V and denoted by

  ,  . v ij

v ij

which can be stated as (Li et al., 2014),







min( ij ,  oj )





CR IP T

dij  o j or dij  o j are determined based on Definition 3. DIFS  dij , o j  is an IF distance,



DIFS  dij , o j   1  max L  dij , o j  , H  dij , o j  , min L  dij , o j  , H  dij , o j  ,





min(1  ij ,1   oj )

max(1  ij ,1   oj )

.

PT

ED

M

max( ij ,  oj )

, H dij , o j 

AN US



where, L dij , o j 

(17)

CE

Fig. 2. Intuitionistic fuzzy prospect values with the reference point o j   0.5,0.5 .



Obviously, the IF distance DIFS dij , o j



is also an IFN, denoted by



DIFS ij



,ijDIFS .

AC

Then based on Definition 2, we can employ Fig.2 to depict the relationship between the IF



decision information dij  ij ,ij





v v and the IF prospect value vij = ij ,ij



when

o j   0.5,0.5 , where the values ijv and ijv are denoted by the colors of each point

  ,  ij

ij

ij =0.5

on the simplex. It is obvious that ij reaches the minimum value (i.e. ij =0 ) at v

v

and ij reaches the minimum value (i.e. ij =0 ) at v

v

ij =0.5 .

5.2. Determination of attribute weights In practice, the weights of attributes may be partially or even fully unknown. The

ACCEPTED MANUSCRIPT

information of incomplete weights can be denoted as a constraint set  and divided into the following types: (1) A weak ranking:

   i

j

| i  j ;

      ; (3) A ranking of multiples:      ; i

j

i

i

(4) A ranking of differences: (5) An interval form:

i

   i

j

j

 k  l | j  k  l ;

i  i  i   i  , where i

and

i

CR IP T

(2) A strict ranking:

are nonnegative constants.

Firstly, to determine the attribute weights, the IF cross entropy (see Definition 7) is utilized to calculate the deviation degree of alternative

ai

DDij 

1 m PCE  vij , vkj  .  m  1 k 1,k i

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denoted as

in relation to attribute c j ,

(18)

Thus, the comprehensive cross entropy of attribute c j can be determined by

DD j  i 1 DDij . DD j reflects the overall divergence of m alternatives under attribute c j .

M

m

The smaller value of DD j means the smaller deviation of m alternatives.

ED

Secondly, the fuzziness of decision making information should also be considered. According to Definition 8, the fuzziness degree for attribute c j can be formulated by IF entropy as follows: m

PT

FD j  i 1 PE  vij  .

(19)

CE

Obviously, smaller value of FD j means the lower degree of fuzziness in the

information. Therefore, the decision making information is more useful and the attribute is

AC

more important. Overall, DD j and FD j depict the information of attribute c j from aspects of deviation and fuzziness degree respectively. The weighting function E j is defined by the combination of DD j and FD j . Large value of E j results in more importance of attribute c j .

E j  DD j  1  FD j 





m  1 m .   i 1  PCE v , v  1  PE v      ij kj ij   m  1 k 1,k i 

(20)

ACCEPTED MANUSCRIPT We can build the following programming model (M-1) to obtain the optimal weight vector with incomplete weighting information, denoted as

 j  .

(M-1)



n n m  1 m max E W    j 1 E j   j   j 1  i 1  PCE  vij , vkj   1  PE  vij    j W  m  1 k 1,k i

s.t. j 1  j =1,  j  0, j  1, 2, n

   

j

,n

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j  Finally, according to prospect theory, the DMs would like to select the alternative with a larger prospect value, denoted by



v   j . Therefore, the following programming

n

j 1 ij

model (M-2) can be constructed to maximize the prospect value. (M-2)

max V W    j 1  i 1 vij   j m

 j W

s.t. j 1  j =1,  j  0, j  1, 2, n

,n

j 

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n

Generally, the solutions to (M-1) and (M-2) are different. To combine (M-1) and (M-2)

M

for determination of the attribute weights, programming model (M-3) is established by integrating (M-1) and (M-2) for a solution. (M-3)



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n n m  1 m max E W    j 1 E j   j   j 1  i 1  PCE  vij , vkj   1  PE  vij   k 1, k  i  j W  m 1

   

j

max V W    j 1  i 1 vij   j  j W

m

PT

n

s.t. j 1  j =1,  j  0, j  1, 2, n

CE

j 

,n

AC

To solve the above model (M-3), it is firstly transformed into model (M-4) based on the

linear equal-weighted summation method. (M-4)

max f  E W   V W    j 1 E j   j   j 1  i 1 vij   j n

n

m

 j W

s.t. j 1  j =1,  j  0, j  1, 2, n

,n

j  Since the prospect value vij is an IFN, model (M-4) is transformed into mathematical programming models (M-5) to (M-7) for calculating the optimal solution [17].

ACCEPTED MANUSCRIPT (M-5)

max f 0   j 1 E j   j   j 1  i 1 vijM   j n

n

m

 j W

s.t. j 1  j =1,  j  0, j  1, 2, n

,n

j  M ij

where, v

 1     v ij

v ij



.

2.

(M-6)

f1   j 1 E j   j   j 1  i 1 vij   j = j 1  j  E j   j 1  j  v j

 j , v j , j 1, n

n

s.t. j 1  j =1,  j  0, j  1, 2, n



m

n

n

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n

max

,n

ijv  v j   i 1 1  ijv 

m

m

i 1

(M-7)

max

 , j , v j , j 1, n

f2  

s.t. j 1  j =1,  j  0, j  1, 2, n



E   j   j 1 v j   j  d *  f 0* j 1 j

n

n

m i 1

ijv 

j 



 2

m

 ijv  v j   i 1 1  ijv   m

i 1



 2

m

i 1

 ijv

M



,n

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j 

where f 0 and f1 are the optimal solution of (M-5) and (M-6) respectively, d  f1  f 0 . *

*

*

*

*

ED

By solving model (M-7), we can obtain the optimal solutions, denoted by  * ,  j , v j , *



, n under incomplete weights  .

PT

* * and gain the optimal weight vector W   j | j  1, 2,

*

The constraint set  is conducive to guaranteeing the reasonability of attribute weights. For

CE

example, suppose that there is an attribute irrelevant to an MADM problem and yet having high IF cross entropy and low IF entropy after attribute selection. Then reasonable weights can be determined by setting proper constraints in the models. However, the information of

AC

weights may be completely unknown in practice. In this case, another measure combining entropy, cross entropy and prospect value, as similar to model (M-3), is established to obtain the weights. Assume that the integrated value of attribute c j is denoted by CFj based on the concepts of entropy, cross entropy and prospect value, as follows:

CFj  E j   i 1 vijM m

. (21) m  1 m m  v v   i 1  PCE v , v  1  PE v  1     2         ij kj ij ij ij  i 1  m  1 k 1,k i 



Then the weight of attribute c j can be calculated as



ACCEPTED MANUSCRIPT



 j  CFj where



n j 1

n j 1

CFj ,

(22)

 j =1,  j  0, j  1, 2, , n .

5.3. Aggregation for alternatives on attributes In this paper, prospect value vij can be considered as the performance for alternative ai in relation to attribute c j . For prospect matrix V  vij  , a grade set can be defined as mn

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H  H h | h  1, 2 , where H1    v , v   1,0  when the performance is exactly the same as DMs expected, and H 2    v , v    0,1 when the performance is worst. Using the set of grades, the prospect value of alternative



ai

on attribute c j can be expressed as follows:



S  c j  ai     H h , h,ij  ， h  1, 2 , v

belonging to grade If



2 h 1

v

Hh

 h,ij denotes the prospect belief degree of alternative ai

on attribute c j .

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where 1,ij  ij ,  2,ij  ij , and

(23)

 h,ij  1 , i.e. ijv ijv  1 and  ijv  1  ijv  ijv  0 , there is no hesitancy on

ai

on attribute c j . If



2

h 1

h,ij  1 , there exists uncertainty in

M

the judgement of alternative

S  c j  ai   . Let H H   0,0  , which represents the prospect grade of hesitancy. Then we can get

ED

 H ,ij , the belief degree of alternative ai belonging to the grade of hesitancy as  H ,ij  1   h1 h,ij . When the hesitancy uncertainty in an intuitionistic fuzzy environment is

PT

2

expressed as a belief structure shown in Eq. (23), the ERA can then be applied for aggregating the attribute values of the alternatives based on the combination rule (see Eq. (1)).

CE

Let mh ,ij be a basic probability mass assigned to grade

Hh

on the jth attribute c j

AC

for an alternative ai . Let mH ,ij be the remaining probability mass which is unassigned to any individual grade on the jth attribute c j . They can be calculated as

mh,ij   j h,ij ,

(24)

mH ,ij  1   h1 mh,ij  1   j  h1 h,ij . 2

2

(25)

Let m H ,ij  1   j and m H ,ij   j  H ,ij , then the probability mass mh ,iJ  j  and the remaining probability mass mH ,iJ  j  , which represent the mass to grade to neither grade on the first j aggregated attributes for alternative

ai

Hh

and the mass

respectively, can be

ACCEPTED MANUSCRIPT calculated as follows:

mh,iJ  j 1  K J  j 1 mh,iJ  j  mh,i j 1  mH ,iJ  j  mh ,i  j 1  mh ,iJ  j  mH ,i  j 1  ,

(26)

mH ,iJ  j 1  K J  j 1 mH ,iJ  j  mH ,i j 1  mH ,iJ  j  mH ,i  j 1  mH ,iJ  j  mH ,i  j 1  ,

(27)

mH ,iJ  j 1  K J  j 1 mH ,iJ  j  mH ,i j 1 ,

(28)

mH ,iJ  j 1  mH ,iJ  j 1  mH ,iJ  j 1c ,

(29)

where K J  j 1 is a normalization factor, as follows: 1





 1  m1,iJ  j  m2,i  j 1  m2,iJ  j  m1,i  j 1

1

.

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2 2 K J  j 1  1   h 1 t 1,t  h mh,iJ  j  mt ,i j 1   

After the above aggregation process is completed for all the n attributes, the belief degree to which alternative ai belongs to grade H h is combined as,





h,i  mh,iJ  n 1  mH ,iJ  n ， h  1, 2 ,

Thus

the

1  m

H ,iJ  n 

comprehensive



is the belief degree representing aggregated hesitancy.

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where  H ,i  mH ,iJ  n

(30)

prospect

value

of

alternative

ai

is

stated

as

V  ai    iv ,iv  , where iv  1,i and iv   2,i .

M

5.4. Ranking the alternatives

After aggregation, we can obtain a set of comprehensive prospect values for alternative



ED

ai  A , denoted by V  ai , iv ,iv , i  1, 2,



m . Using Definition 3, the ranking and

prioritization of alternatives can be determined through comparing the prospect values in V .

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6. Illustrative example

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6.1. Case illustration

In order to illustrate the decision process and prove the validity of the proposed method,

a real-life case is utilized. As the competition of international shipping market becomes

AC

increasingly fierce, it is necessary to assess and rank shipping enterprise competitiveness for determining the status of an enterprise in the international shipping market. Assume that there are five liner shipping companies, named ai  i  1, 2,3, 4,5 . According to the characters of operation management in shipping industry, a set of five attributes is constructed as: return on assets ( c1 ), asset-liability ratio ( c2 ), size of fleet ( c3 ), compatibility between organizational structure and strategic environment ( c4 ), and safety ( c5 ). The decision matrix D   dij 

55

and the reference point vector O  o j 

15

can be

ACCEPTED MANUSCRIPT

collected and organized in the format of IFNs. For example, in term of return on assets ( c1 ), a manager or DM may state that he is 75% satisfied with company a1 and 10% unsatisfied with it. In the statement, the percentage values of 75 and 10 are referred to as the degrees of membership and non-membership of company a1 on attribute c1 . Then the assessment can be expressed as d11   0.75,0.1 . In addition, if the manager or DM assumes that the degrees of membership and non-membership of an expected alternative are both 0.5 for

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attribute c1 , the reference point value on attribute c1 can be determined by o1   0.5,0.5 . For comparison in the rest of this section, the data in document (Wang & Zhang, 2013) are applied to evaluation of shipping enterprise competitiveness, which is shown in Table 5.

O   0.5,0.5

that

the

reference

 0.5,0.5  0.5,0.5  0.5,0.5  0.5,0.5

point

vector

is

, because the ideal and anti-ideal

AN US

Suppose

alternatives in the article (Wang & Zhang, 2013) were denoted with (1,0) and (0,1) respectively. Table 5 The information of the decision matrix. c2

M

c1

c3

c4

c5

(0.75, 0.1)

(0.8, 0.15)

(0.4, 0.45)

(0.6, 0.15)

(0.55, 0.45)

a2

(0.6, 0.25)

(0.68, 0.2)

(0.75, 0.05)

(0.4, 0.4)

(0.7, 0.15)

a3

(0.8, 0.2)

(0.45, 0.5)

(0.6, 0.3)

(0.6, 0.3)

(0.65, 0.2)

a4

(0.7, 0.25)

(0.78, 0.2)

(0.85, 0.05)

(0.6, 0.3)

(0.8, 0.15)

a5

(1, 0)

(0.85, 0.1)

(0.9, 0.05)

(0.7, 0.2)

(0.8, 0.15)

PT

ED

a1

CE

Case I: Suppose that this is the situation in which the attribute weights are partially

known. The manager or DM can only give the following weight information:

0.2  1  0.3 ,

0.15  2  0.25 ,

0.1  3  0.3 ,

AC

1  3  2  5  4 ,

0.1  4  0.2 , 0.1  5  0.25 . Then the calculation steps are as follows. Step I-1. In Eq. (16), let  = =0.88 and  =2.25 based on prospect theory (Kahneman

& Tversky, 1979). Then the prospect matrix V  vij  can be worked out as follows: 55

ACCEPTED MANUSCRIPT   0.38, 0.51    0.21, 0.62  V   0.42, 0.58    0.33, 0.62   0.54, 0.46  

 0.42, 0.54   0.31, 0.58   0.73, 0   0.41, 0.58   0.46, 0.51

 0.54, 0.25   0.38, 0.48   0.21, 0.67   0.46, 0.48   0.49, 0.48

 0.21, 0.54   0.54, 0.41  0.21, 0.67   0.21, 0.67   0.33, 0.58 

 0.12, 0.88   0.33, 0.54    0.28, 0.58  .   0.42, 0.54   0.42, 0.54 

Step I-2. Suppose that t=1 and k=0.6 in Eq. (11) and (14), model (M-3) can be shown as follows:  j W



5 5 5 1 5 E W    j 1 E j   j   j 1  i 1   k 1,k i PCE  vij , vkj   1  PE  vij  3

max V W    j 1  i 1 vij   j 5

5

j

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 j W

 5  j =1,  j  0, j  1, 2, , n  j 1  1  3  2  5  4  0.2  1  0.3  s.t.  0.15  2  0.25  0.1  3  0.3   0.1  4  0.2  0.1  5  0.25 

   

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max

M

Models (M-5) to (M-7) can be programmed and solved in MATLAB. The attribute weighing vector obtained is W  (1 , 2 , 3 , 4 , 5 )T  (0.25,0.25,0.25,0.1,0.15)T .

ED

Step I-3. Calculate the comprehensive prospect value of each alternative using the ERA algorithm outlined in Section 5.3, then V  a1    0.371,0.539 , V  a2    0.316,0.582  ,

PT

V  a3    0.407,0.487  , V  a4    0.368,0.595 , and V  a5    0.471,0.507  . Step I-4. Obtain the scores of all alternatives based on Definition 3: S a1   0.168 ,

CE

S a2   0.266 , S a3   0.08 , S a4   0.227 , and S a5   0.036 . Then the

a3

a1

a4

a2 .

AC

ranking order can be gained in accordance with the scores: a5

According to the decision results, it is obvious that shipping company a5 is most

competitive in the comparison group while the least competitive company is a2 . The

comprehensive prospect value of company a5 is V  a5    0.471,0.507  , which means that the manager is 47.1% satisfied with company a5 and 50.7% unsatisfied with it after integrating five attributes and considering his bounded rationality. It is implied that the decision result has about 2.2% uncertainty for company a5 , due to the hesitancy degree of

ACCEPTED MANUSCRIPT

decision information to company a5 on each attribute (see Table 5). Case II: Suppose that this is the situation in which the weights are completely unknown. Then the calculation steps are as follows. Step II-1. As with the Step I-1 of Case I, the prospect matrix V is determined based on prospect theory in which  = =0.88 and  =2.25 . Step II-2. Suppose that t=1 and k=0.6 in Eq. (11) and (14), attribute weights are obtained

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by using Eq. (22).

CF  (2.42,3.64,2.83,2.56,2.66)T .

W  (1 , 2 , 3 , 4 , 5 )T  (0.17,0.26, 0.2,0.18,0.19)T .

Step II-3. Similar with the Step I-3 of Case I, the comprehensive prospect values of are

calculated

V  a1    0.331, 0.58 ,

as

V  a2    0.344,0.563 ,

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alternatives

V  a3    0.397,0.485  , V  a4    0.354,0.605 , and V  a5    0.446,0.525 . Step II-4: Using Definition 3, the values of score function are S a1   0.249 ,

S a2   0.219 , S a3   0.088 , S a4   0.251 , and S a5   0.079 . Then the

a3

a2

M

ranking order of all alternatives is a5

a1

a4 .

Therefore, the best alternative is also company a5 , while company a4 is most

ED

unsatisfactory. Similarly, the comprehensive prospect values of alternatives indicate that there

is still uncertain information for each alternative when the uncertainties exist in decision

PT

matrix.

6.2. Comparison with other methods

CE

In order to illustrate the effectiveness and performance of the proposed MADM method,

some comparisons between the proposed method and the approaches of articles (Chen & Yang,

AC

2011; Qi et al., 2015; Wang & Zhang, 2013) are conducted based on the Case I and II. Wang and Zhang (2013) developed an intuitionistic fuzzy MCDM method based on the ERA and established a nonlinear programming model to obtain the attributes’ weights, called as Wang’s method. Since the weighting information is incompletely certain in the Wang’s method, this approach is used to compare with the proposed method only for Case I to ensure the comparability of these two methods. In addition, some researchers (Chen & Yang, 2011; Qi et al., 2015) investigated the intuitionistic fuzzy MADM problems with unknown weights based

ACCEPTED MANUSCRIPT

on the IFWA operator which is one of the representative aggregation methods. When adopting the IFWA operator to aggregate the decision information of Case I and II, it would have different decision results based on the decision matrix (see Table 5) or the prospect matrix (see Step I-1). So two kinds of methods are derived from decision matrix and prospect matrix based on IFWA operator, named as decision IFWA-based method and prospect IFWA-based method respectively. In the decision IFWA-based method, attribute weights, denoted by  j , are firstly

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D

determined based on only entropy and cross entropy of the values in the decision matrix D . When the information of weights is partially known, we can substitute decision values d ij for prospect values vij in the model (M-1) and obtain the optimal weights by the simplex method. In another condition when the criteria’s weights  j

are unknown completely, the

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D

parameters CFj of Eq.(22) need to be replaced by the weighting functions of decision D

IFWA-based method E j

which are defined by substituting d ij for vij in the Eq.(20).

Secondly, we can calculate the comprehensive decision values of alternatives, denoted by

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DIFWA  ai  , using IFWA operator (see Definition 4) to aggregate the decision values d ij and the attribute weights  j . Finally, according to the score function and accuracy function (see D

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Definition 3), the corresponding values SIFWA  ai  and H IFWA  ai  can be gained to rank all alternatives.

D

D

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In the prospect IFWA-based method, the IFWA operator is utilized to combine the information of prospect matrix V (see Step I-1) and the attribute weights

 j ( see Step I-2

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and Step II-2) and obtain the comprehensive prospect values VIFWA  ai  . Then the ranking of alternatives is also determined based on the values of score function and accuracy function,

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denoted by SIFWA  ai  and H IFWA  ai  respectively. V

V

For Case I and II, assume that t=1 and k=0.6 in Eq. (11) and (14), we can obtain different

decision analysis outcomes using Wang’s method, the decision IFWA-based method and the prospect IFWA-based method (see Table 6). Then Fig. 3 displays the different values of score function and accuracy function using various methods including the proposed method and the above three approaches. Finally, the ranking orders of all alternatives determined by applying these four methods are displayed in Fig.4.

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Table 6 The decision making results using three methods for Case I and II. Wang’s method for Case I

Decision IFWA-based method

Prospect IFWA-based method

Case I

Case II

Case I

Case II

(0.655, 0.225)

(0.654, 0.206)

(0.652, 0.219)

(0.388, 0.474)

(0.358, 0.504)

a2

(0.681, 0.157)

(0.664, 0.147)

(0.669, 0.147)

(0.335, 0.537)

(0.36, 0.523)

a3

(0.673, 0.26)

(0.568, 0.283)

(0.652, 0.28)

(0.446, 0)

(0.439, 0)

a4

(0.797, 0.14)

(0.774, 0.143)

(0.777, 0.144)

(0.387, 0.565)

(0.376, 0.572)

a5

(0.913, 0.047)

(1, 0)

(1, 0)

(0.473, 0.5)

(0.453, 0.512)

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a1

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Fig. 3. The comparisons of decision making results based on score function and accuracy function.

Fig. 4. The rankings of alternatives for Case I and II.

Three points can be made from the comparisons. First, these methods produce different decision outcomes according to Table 6, Step I-3 and Step II-3. It is worth noting that some non-membership degrees are zero based on the

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decision IFWA-based method and the prospect IFWA-based method. By analyzing the noticeable phenomenon, this is due to a few zero values which exist in the decision matrix and prospect matrix when the IFWA operator is used to aggregate the decision information. This phenomenon means that the manager or DM is not against a certain alternative. Since the non-membership degrees of the decision matrix and prospect matrix are not all zero, which means there is still opposition in the decision process, it is obvious that the two methods

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based on IFWA operator may distort the real decision information.

Second, as demonstrated by the accuracy function of Fig. 3, we can conclude that V H  ai   H IFWA  ai  . Larger accuracy can be construed as an advantage because H  ai 

and H IFWA  ai  measure the degrees of certainty in the decision outcomes. The conclusion V

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V H  ai   H IFWA  ai  shows that the information synthesis process using the ERA is able to

minimize the loss of information and reduce the uncertainty in the decision outcomes. Third, it can be seen from Fig. 4 that the ranking orders by utilizing these methods are apparently variant for Case I and II. One of the reasons for the different ranking sequences is

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that prospect matrix according to prospect theory can take account of decision makers’ risk attitude and bounded rationality. In conjunction with the score function of Fig. 3, the best

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alternatives obtained by employing the proposed method and the prospect IFWA-based method are a5 and a3 respectively, but the ranking sequences of other alternatives based

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on the two methods are both a1

a4

a2 . So the decision results can be influenced by

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bounded rationality behaviors of decision makers and the methods of aggregating uncertain information.

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In addition, the input data used in these methods are same in order to guarantee their

comparability. Because prospect theory and evidence theory are introduced into the proposed method, it may be more complex in formula expression than other approaches. But the comparison results can demonstrate the performance and strengths of the proposed MADM method based on prospect theory and evidence theory. In another example, this method has been utilized to successfully solve a performance evaluation problem of shipping enterprises which is much more complex than the example used in the paper and consists of 18 attributes and 10 alternatives. Thus the availability of the proposed method can be illustrated for the

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complex decision problems which have a large amount of input data. 6.3. Sensitivity analyses In order to investigate the influence of factor k and t in Eq. (11) and (14) on the weighing

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vectors, the sensitivity analyses are conducted for Case I and II (see Fig.5).

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Fig. 5. Sensitivity analyses of factor k and t for Case I and II.

As can be seen from Fig.5, the attribute weights can be clearly divided into two kinds of

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outcomes for Case I with the variation of parameters k and t, which shows that the values of k and t have limited influences on attribute weights in Case I. The reason may be that the effect on attribute weights from changes of k and t values is constrained by the weight restrictions

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 . For Case II, attribute weights vary when t  1 and different values of factor k are used, while the changes of factor t do not much affect the results of weights when k  0.6 . It is

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illustrated that the attribute weights change with factor k more sensitively than with factor t

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when the information of weights is completely unknown.

7. Conclusion In this paper, based on prospect theory and the evidential reasoning approach, a new

method is proposed to handle multi-attribute decision making problems, where the information of attribute weights is partially or completely unknown and attribute values of alternatives are expressed by IFNs. As discussed in the paper and illustrated by the examples, four advantages of this method have been shown in details. First, the measures of the intuitionistic fuzzy entropy and cross entropy are defined to reflect decision makers’

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preference and overcome the invalidity of the existing measures in some special cases. Second, a few optimization models are proposed to determine the information of weights which is unknown partially or completely based on cross entropy, entropy and prospect value. Since the models combine the divergence and fuzziness of decision information and risk attitude of decision makers, the weights of attributes can be obtained objectively and reasonably by utilizing the optimization models. Third, the proposed method, which adopts

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the evidential reasoning approach to aggregate the values of the attributes and weights, can reduce the loss of decision making information. Finally, the ranking and prioritizing of alternatives based on prospect theory can reflect the human bounded rationality and produce a more realistic decision making result. Overall, the proposed method is suitable for

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intuitionistic fuzzy MADM problems with uncertain information and is of significant practical value.

As for future work, we are interested in estimating the degrees of membership and non-membership for IFS. It is challenging that decision maker declares directly the decision

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information described by IFS in the practical and complex situations which have a considerable higher number of attributes and alternatives. Moreover, some concepts should be

Acknowledgments

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further introduced, such as group decision and argumentation.

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Special thanks go to Professor Dong-Ling Xu for her contribution in enhancing the quality of this paper. The authors are also grateful to the anonymous reviewers and editor for

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their constructive suggestions and comments that have helped to improve the quality of this paper. This work is supported by the National Key Research and Development Program of

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China [2017YFC0805309] and the Fundamental Research Funds for the Central Universities [3132016358].

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