Interval programming method for hesitant fuzzy ... - Semantic Scholar

Computers & Industrial Engineering 75 (2014) 217–229

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Interval programming method for hesitant fuzzy multi-attribute group decision making with incomplete preference over alternatives q Xiaolu Zhang a, Zeshui Xu b,⇑ a b

School of Economics and Management, Southeast University, Nanjing 211189, China Business School, Sichuan University, Chengdu, Sichuan 610065, China

a r t i c l e

i n f o

Article history: Received 29 August 2013 Received in revised form 23 February 2014 Accepted 7 July 2014 Available online 15 July 2014 Keywords: Hesitant fuzzy set Multi-attribute group decision making Incomplete preference Interval programming

a b s t r a c t The aim of this study is to employ the main structure of LINMAP (LINear programming technique for Multidimensional Analysis of Preference) to propose an interval programming method for solving multi-attribute group decision making (MAGDM) problems in which the ratings of alternatives are taken as hesitant fuzzy elements (HFEs) and all pair-wise comparison judgments over alternatives are represented by interval numbers. The contribution of this study is fivefold: (1) we define the new consistency and inconsistency indices; (2) we construct an interval programming model to determine the hesitant fuzzy positive ideal solution and the optimal weights of attributes, and at the same time present a decision algorithm; (3) we discuss several special cases of the proposed model in detail; (4) we show that compared with intuitionistic fuzzy LINMAP method (Li et al., 2010), the proposed approach reveals more useful information including the interval preference information, and does not need to transform HFEs into intuitionistic fuzzy numbers but directly deals with MAGDM problems and thus obtains better final decision results; and (5) we demonstrate the applicability and implementation process of the proposed approach by using an energy project selection example. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Multi-attribute decision making (MADM) is to find an optimal alternative that has the highest degree of satisfaction from a set of feasible alternatives which are characterized in terms of multiple attributes. MADM problems arise in many real world situations. Considering a classical MADM problem, let M = {1, 2, . . . ,m}, N = {1, 2, . . . ,n}, denote the alternative set by A = {A1, A2, . . . ,Am} and the attribute set by C = {C1,C2, . . . ,Cn}, and let xij(i 2 M, j 2 N) be the ratings of the alternative Ai with respect to the attribute Cj and x = (w1, w2, . . . ,wn)T be the weight vector of all attributes. Thus the MADM problem is commonly expressed in following decision table (Table 1). To effectively handle this sort of problems, many corresponding MADM methods have been developed, for instance, the TOPSIS (technique for order preference by similarity to ideal solution) (Hwang and Yoon, 1981; Shidpour et al., 2013), the LINMAP (linear programming technique for multidimensional analysis of preference) (Srinivasan and Shocker, 1973), the VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method q

This manuscript was processed by Area Editor Imed Kacem.

⇑ Corresponding author.

E-mail addresses: [email protected] (X. Zhang), [email protected] (Z. Xu). http://dx.doi.org/10.1016/j.cie.2014.07.002 0360-8352/Ó 2014 Elsevier Ltd. All rights reserved.

(Opricovic and Tzeng, 2002) and the ELECTRE (ELimination Et Choix Traduisant la REalité) method (Roy, 1968; Zandi and Roghanian, 2013), etc. Among these MADM approaches, the LINMAP method proposed by Srinivasan and Shocker (1973) proves to be a practical and useful approach for determining the weights of attributes and the ideal solution. In the classical LINMAP approach, the decision maker (DM) cannot only provide the ratings of alternatives with respect to each attribute but also simultaneously give the incomplete preference relations on pair-wise comparisons of alternatives. The underlying logic of LINMAP is to define the consistency and inconsistency indices based on pair-wise comparisons of alternatives. The consistency index is to measure the degree of consistency between the given preference relations over alternatives by the DM in advance and the derived preference relations by analyzing data in decision table, whereas the inconsistency index is to measure the degree of inconsistency between the given preference relations and the derived preference relations. Based on the consistency and inconsistency indices, a crisp LINMAP model is constructed to derive the ideal solution and the attribute weights. Thus the best compromise alternative has the shortest distance to the ideal solution from the set of feasible alternatives. All decision information in the classical LINMAP are known precisely or given as crisp (no-fuzzy) values. In other words, the

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X. Zhang, Z. Xu / Computers & Industrial Engineering 75 (2014) 217–229

Table 1 The MADM problem with decision table format. Alternatives

A1 A2 ... Am

Attributes C1

C2

...

Cn

x11 x21 ... xm1

x12 x22 ... xm2

... ... ... ...

x1n x2n ... xmn

classical LINMAP method cannot be used to directly handle the MADM problem under fuzzy contexts because of the uncertainty of attribute values or preference information over alternatives. To this end, some researchers have extended the LINMAP method for solving MADM problems within a variety of different fuzzy environments. For example, considering the uncertainty of attribute values, Li and Yang (2004), Li and Sun (2007) and Xia et al. (2006) respectively proposed fuzzy LINMAP methods for solving MADM problems under fuzzy environments. Li (2008) and Li et al. (2010) respectively extended the LINMAP methods for solving MADM and multi-attribute group decision making (MAGDM) problems with intuitionistic fuzzy number (IFN) (Xu, 2007). Wang and Li (2012) further extended the LINMAP method for solving MAGDM problems with interval-valued IFNs. Combining with comparison possibilities and hybrid averaging operations, Chen (2013) also proposed an interval-valued intuitionistic fuzzy LINMAP method for solving MAGDM problems. On the other hand, considering the uncertainty of preference information over alternatives, Sadi-Nezhad and Akhtari (2008) developed a possibility LINMAP for solving the MAGDM problems with triangular fuzzy truth degree. Li and Wan (2013, 2014) proposed a LINMAP-based fuzzy linear programming technique for solving heterogeneous MADM problems with trapezoidal fuzzy truth degree. Wan and Li (2013) presented a generalization of LINMAP method for solving heterogeneous MADM problems with intuitionistic fuzzy truth degree. They also pointed out that their method is the generalization of the other extended LINMAP methods. It is easy to see that the LINMAP has been extended into various forms for solving MADM or MAGDM problems under a variety of different environments. Nevertheless, these LINMAP-based methods are not applicable to the MAGDM problem with incomplete weight information under hesitant fuzzy scenario. In fact, there always exists a certain situation in real life in which the ratings of alternatives provided by the DMs are taken as hesitant fuzzy elements (HFEs). For example, to get a reasonable decision result, a decision organization included several DMs are authorized to estimate the degree that an alternative should satisfy an attribute. Suppose that there are three cases: some DMs provide 0.3, some provide 0.5, and the others provide 0.6, and meanwhile these three parts cannot persuade with each other. In this situation the DMs usually provide their preferences with a HFE {0.3,0.5,0.6}. It is noted that the HFE {0.3,0.5,0.6} can describe the above situation more objectively than the crisp number 0.3 (or 0.5, or 0.6), or the interval number [0.3,0.6], or the IFN (0.3,0.4), because the degrees that the alternative should satisfy the attribute are not the convex of 0.3 and 0.6, or the interval between 0.3 and 0.6, but just three possible values (Xia et al., 2013). Considering the fact that the use of hesitant fuzzy assessments makes the DMs’ judgments more reliable and informative in decision making, some scholars recently discussed the HFS’s basic operators and their properties (Torra, 2010), aggregating operators (Wei, 2012; Xia and Xu, 2011; Xia et al., 2013; Zhang, 2013; Zhu et al., 2012), correlation coefficient (Chen et al., 2013), and their distance and similarity measures (Farhadinia, 2013a; Peng et al., 2013; Xu and Xia, 2011a, 2011b). These research works have made great contributions to enrich hesitant fuzzy theory and have been

applied to various fields, such as cluster analysis (Zhang and Xu, 2012, 2013), and mainly in decision making fields (Farhadinia, 2013b; Peng et al., 2013; Xu and Xia, 2011a, 2011b; Xu and Zhang, 2013; Zhang and Wei, 2013; Zhang, 2013). For example, Farhadinia (2013b) proposed a HFE ranking-based approach to solve MADM problems under hesitant fuzzy context. Xu and Zhang (2013) put forward a hybrid approach combining TOPSIS and the maximizing deviation to handle MADM problems in which the evaluation information is expressed by HFEs and the information about attribute weights is incomplete. Zhang and Wei (2013) extended the VIKOR method and the TOPSIS method to develop a new method for solving MADM problems with HFEs. Zhang (2013) developed a method based on hesitant fuzzy power aggregation operators for the MAGDM problem with HFEs. In this study we extend the classic LINMAP to put forward an interval programming technique for solving MAGDM problems in which the ratings of alternatives with respect to each attribute are represented by HFEs and all pair-wise comparison judgments over alternatives are taken as interval numbers. To do so, we organize the paper as follows: In Section 2, we review some concepts related to HFSs, introduce the interval preference relations and describe a class of optimization problems with interval coefficients. In Section 3, we describe the MAGDM problem with HFEs and put forward an interval programming technique for solving it. Meanwhile we also discuss some special cases of the proposed model in detail and present a decision algorithm. Section 4 employs an energy project selection problem to demonstrate the applicability and implementation process of the proposed methodology and the study finishes with some concluding remarks in Section 5. 2. Some basic concepts In what follows, we introduce some basic concepts and terminologies which will be used in the next sections. Readers familiar with these topics are encouraged to proceed directly to Section 3. 2.1. The basic concepts related to hesitant fuzzy sets Hesitant fuzzy sets (HFSs) were first introduced by Torra (2010), which permit the membership degree of an element to a set to be represented as several possible values between 0 and 1. HFSs are very useful in coping with the situations where people have hesitancy in providing their assessments. Definition 1. (Torra, 2010). Let X be a reference set, a HFS A on X is defined in terms of a function hA(x) when applied to X returns a subset of [0,1]. To be easily understood, Xia and Xu (2011) expressed the HFS A by a mathematical symbol:

A ¼ f< x; hA ðxÞ > jx 2 Xg

ð1Þ

where hA(x) is a set of some different values in [0,1], representing the possible membership degrees of the element x 2 X to A. For convenience, Xia and Xu (2011) called hA(x) a HFE denoted by h. The operations of HFEs are also defined as follows: let k > 0, given three HFEs represented by h, h1 and h2, respectively, we have (Torra, 2010; Xia and Xu, 2011): h1 [ h2 ¼ [c1 2h1 ;c2 2h2 maxfc1 ; c2 g; h1 \ h2 ¼ [c1 2h1 ;c2 2h2 minfc1 ; c2 g; kh ¼ [c2h f1 ð1 cÞk g k

c

h ¼ [c2h fck g; h ¼ [c2h f1 cg; h1 h2 ¼ [c1 2h1 ;c2 2h2 fc1 þ c2 c1 c2 g; h1 h2 ¼ [c1 2h1 ;c2 2h2 fc1 c2 g:

Torra (2010) also showed that the envelop of a HFE is an IFN expressed in the following definition:

219


Definition 2. (Torra, 2010). Given a HFE h, we define the IFN Aenv(h) as the envelope of h, where Aenv(h) can be represented as þ ðh ; 1 h Þ, with h = min{cjc 2 h} and h+ = max{cjc 2 h}. It is noted that the number of values for different HFEs may be different, and the values are usually out of order, then we should arrange them in any order for convenience. Suppose that we arrange the values of a HFE h in a decreasing order, and let cl be the lth largest value in h. Let h1 and h2 be two HFEs and ‘ = max{l(h1),l(h2)}, where l(h1) and l(h2) are respectively the numbers of values in the HFEs h1 and h2. In order to more accurately calculate the distance between two HFEs, we should extend the shorter one until both of them have the same length when we compare them with l(h1) – l(h2). Usually, the value which we need to add into the shorter HFE mainly depends on the DM’s risk preference. To this end, Xu and Zhang (2013) developed a method of extension with a parameter g which can reveal the DM’s risk preference to identify the adding value. Definition 3. (Xu and Zhang, 2013). Denote a HFE by h = {cljl = 1, 2, . . . , l(h)} and stipulate that h+ and h are the maximum and minimum values in the HFE h, respectively; then we call ¼ ghþ þ ð1 gÞh an adding value, where g (0 6 g 6 1) is the h parameter determined by the DM according his/her risk preference.

Remark 1. In general, we can add different values to the HFE according the DM’s risk preference. If g = 1, then the adding value ¼ hþ , which indicates that the DM’s risk preference is riskh ¼ h , which means that the DM’s risk prefseeking; if g = 0, then h ¼ ðhþ þ h Þ=2, which erence is risk-averse; while if g ¼ 12, then h reflects the DM’s risk preference being risk-neutral. In this study, we assume that the DMs are all risk-averse. Based on the above operational laws and the principle of extension, Xu and Xia (2011b) defined the hesitant fuzzy Euclidean distance of HFEs as follows:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ‘ 2 u1 X l l dðh1 ; h2 Þ ¼ t h1 h2 ‘ l¼1

ð2Þ l

l

Table 2 Linguistic meanings of interval preferences. Linguistic meanings

Interval numbers

Strong preferable Preferable Almost preferable Moderate preferable Scarcely preferable

[0.9,1.0] [0.8,0.9] [0.7,0.8] [0.6,0.7] [0.5,0.6]

In real-life decision process, the interval number usually represents a certain kind of meanings. In other words, there exist some corresponding relations between interval numbers and linguistic meanings. These relations can be described in Table 2. For example, ~ p ðo; rÞ ¼ ½0:9; 1:0 expresses the meanings that the DM Dp strongly C ~ p ðo; rÞ ¼ ½0:6; 0:7 prefers the alternative Ao to Ar; whereas C expresses the meanings that Dp moderately prefers the alternative Ao to Ar. 2.3. Optimization problems with interval-objective functions ~ ¼ ½aL ; aU ¼ fajaL 6 a 6 aU ; a 2 Rg be an interval number, Let a ~ where aL and aU are left and right limits of the interval number a ~ ¼ ½aL ; aU on the real number set R, respectively. If aL = aU, then a ~Þ ¼ reduces to a real number aða ¼ aL ¼ aU Þ. Given that mða ~Þ ¼ ðaU aL Þ=2 are respectively the midpoint ðaL þ aU Þ=2 and nða ~, thus the interval a ~ can be alternatively repreand half-width of a ~ ¼ ðmða ~Þ; nða ~ÞÞ. sented as a ~ be two interval numbers represented by a ~ and b ~ ¼ ½aL ; aU Let a ~ ¼ ½bL ; bU , respectively, some operations on them can be and b described as (Tong, 1994): ~ ¼ ½aL þ bL ; aU þ bU ; ~þb (1) a L U ~ ¼ ½ba ; ba ðb > 0Þ ; (2) ba 0 ðb ¼ 0Þ ~ if and only if aL 6 bL and aU 6 bU. ~ 6 b, (3) a Now we consider the following minimization problem with interval-objective functions:

where h1 and h2 are two HFEs; h1 and h2 are the lth largest values in h1 and h2, respectively, which will be used thereafter.

~g min fa ~ 2 X1 s:t: a

2.2. Interval-valued preference relations

which is equivalent to the bi-objective mathematical programming problem as follows (Ishibuchi and Tanaka, 1990; Lai et al., 2002):

One of the most disadvantages in multidimensional analysis of preference is to consider pair-wise comparison between alternatives with the crisp degree of truth 0 or 1 (Sadi-Nezhad and Akhtari, 2008). Because in practical decision problems DMs are not sure enough in all pair-wise comparisons over alternative and usually express their opinions with a degree of insurance. As a result, using interval numbers instead of numerical values is more realistic. In this paper the DM’s preferences can be considered as interval numbers given through pair-wise comparisons between alternatives. Definition 4. For the DM Dp and each pair of alternatives Ao and Ar, if the DM Dp prefers the alternative Ao to Ar with the degree of truth ~ p ðo; rÞ, we could define Xp as: C

Xp ¼ fðo; rÞjAo C~ p ðo;rÞ Ar ; ðo; r 2 MÞg

ð3Þ

where Xp is a set of ordered pairs (o,r) provided by the DM Dp, and the degree of truth is expressed as an interval number denoted by ~ p ðo; rÞ ¼ ½C por ; C por ð0 6 C por 6 C por 6 1Þ. C

~Þg min faU ; mða ~ s:t: a 2 X1

ð4Þ

ð5Þ

~ should where X1 is a set constraint conditions which the variable a satisfy according to requirements in real situations. The maximization problem with interval-objective function is described as:

~g max fa ~ s:t: a 2 X2

ð6Þ

which is also equivalent to the bi-objective mathematical programming problems as (Ishibuchi and Tanaka, 1990; Lai et al., 2002):

~Þg max faL ; mða ~ 2 X2 s:t: a

ð7Þ

~ should where X2 is a set constraint conditions which the variable a satisfy.

220


3. Interval programming method for MAGDM problems with HFEs In this section, we first describe MAGDM problems with HFEs. Then we use the main structure of LINMAP to put forward an interval programming technique for solving this sort of MAGDM problems. Afterwards, we discuss some special cases of the proposed model in detail. At length, a decision algorithm is presented. 3.1. The description of MAGDM problems MAGDM usually refers to the process that multiple DMs make evaluations with their respective knowledge, experience and preference for a set of alternatives over multiple attributes, to rank all the alternatives or give evaluation information of each alternative, and then decision results from each DM are aggregated to form an overall ranking result for these alternatives. Let F = {1, 2, . . . ,f} and assume that the DMs Dp(p 2 F) have to rank the alternatives Ai (i 2 M) or to choose one from them on basis of the attributes Cj (j 2 N). Denote a group of DMs by D = {D1, D2, . . . ,Df}, the alternative set by A = {A1, A2, . . . ,Am} and the attribute set by C = {C1, p C2, . . . ,Cn}. Let Hp ¼ hij be the hesitant fuzzy decision matrix, mn

p

where hij is the rating of the alternative Ai with respect to the attribute Cj given by Dp, which is taken as a HFE. Thus the MAGDM problem with HFEs can be concisely expressed in the matrix format as below:

It is easy to find that there exist three major difficulties and challenges for solving such a MAGDM problem. The first is how to effectively take into account the DMs’ hesitation when they provide the assessment values of alternatives with respect to each attribute, and simultaneously consider the DM’s interval preference about pair-wise comparisons of alternatives. The second is how to define the new consistency and inconsistency indices. The third is how to construct the appropriate mathematical programming models and further develop an effective method to solve them. Bearing this fact in mind, we next develop an interval programming approach for solving the aforementioned MAGDM problem. 3.2. The proposed approach Our proposed approach starts with the definitions of new consistency and inconsistency indices. Then an interval programming model is constructed to derive the optimal weight vector and the hesitant fuzzy positive ideal solution (HFPIS). Afterwards, the distances of each alternative from the HFPIS can be calculated and each DM’s ranking for alternatives is also obtained. At length, the collective alternative ranking is obtained using social choice functions such as the Borda’s function and Copeland’s function (Hwang and Yoon, 1981). The details of the approach are elaborated as follows: þ þ þ Denote the unknown HFPIS by Aþ ¼ h1 ; h2 ; . . . ; hn , where þ þ 1 þ 2 þ ‘ þ hj ðj 2 NÞ is expressed as a HFE hj ¼ ððhj Þ ; ðhj Þ ; . . . ; ðhj Þ Þ. Then, using Eq. (2) the square of the weighted Euclidean distance between each alternative Api and the HFPIS A+ can be calculated as:

Spi Denote the attributes weighting vector by x = (w1, w2, . . . ,wn)T, where wj is the relative weight of the attribute Cj, satisfying the norP malization condition: nj¼1 wj ¼ 1 and wj P 0(j 2 N). Usually, there are both benefit attributes (the larger the attributes values the better) and cost attributes (the smaller the attributes values the better) in the MAGDM problems. The dimensions and measurements of attributes functions or values are often different since the types of attributes are different. To ensure compatibility between the values of all attributes, all attributes must be converted into a compatible scale (or dimensionless indices), i.e., normalization. Consequently, to eliminate the effect of different physical dimensions and measurements on the final decision, in this paper, we transform the attributes values of the cost type into the attributes values of the benefit type by normalizing the hesitant fuzzy decision matrix p Hp ¼ hij to yield a corresponding normalized hesitant fuzzy mn decision matrix Gp ¼ g pij , using the method of Zhu et al. (2012): mn

8 p < hij ; for benefit attribute C j g pij ¼ p c ; i 2 M; j 2 N; p 2 F : h ; for cost attribute C j ij

ð8Þ

S p c p p c where ðhij Þ is the complement of hij , such that ðhij Þ ¼ c2hp f1 cg. ij Moreover, the DMs in some special situations may not only provide the ratings of alternatives with respect to each attribute but also give the incomplete preference information on pair-wise comparisons of alternatives. As mentioned previously, this study assumes that the DM Dp gives incomplete preference information between alternatives by a set of ordered pairs Xp ¼ fðo; rÞj ~ p ðo; rÞ. The Ao C~ p ðo;rÞ Ar ; ðo; r 2 MÞg with the interval truth degrees C h i ~ p ðo; rÞ is an interval number represented by C ~ p ðo; rÞ ¼ C por ; C por , C satisfying 0 6 C por 6 C por 6 1ðo; r 2 M; p 2 FÞ.

! n n ‘ 2 X p þ 2 X 1X p þ l l ; ¼ wj d hij ; hj ¼ wj hij hj ‘ l¼1 j¼1 j¼1

i2M ð9Þ

As mentioned previously, the DMs in this situation do not only provide the ratings of alternatives with respect to each attribute but also give the incomplete preference relations between alterna n o tives by a set of ordered pairs Xp ¼ ðo; rÞAo C~ p ðo;rÞ Ar ; ðo; r 2 MÞ ~ p ðo; rÞðo; r 2 MÞ. Thus the square with interval fuzzy truth degrees C of the weighted Euclidean distance between each pair alternatives (o,r) 2 Xp and the HFPIS A+ can be calculated as:

Spo ¼

! n n ‘ 2 p þ 2 X X 1X p þ l l ; d hoj ; hj wj ¼ wj hoj hj ‘ l¼1 j¼1 j¼1

i2M ð10Þ

and

Spr

! n n ‘ 2 p þ 2 X X 1X p þ l l ¼ d hrj ; hj wj ¼ wj hrj hj ; ‘ l¼1 j¼1 j¼1

i2M ð11Þ

Let t pro ¼ Spr Spo , we can obtain: ! ! n ‘ n ‘ p þ 2 p þ 2 X X 1X 1X l l l l wj hrj hj wj hoj hj ‘ l¼1 ‘ l¼1 j¼1 j¼1 2 2 X n X ‘ n X ‘ X p p wj 2wj l þ l p l p l l hoj hrj hj hoj hrj ¼ ‘ ‘ j¼1 l¼1 j¼1 k¼1

t pro ¼

ð12Þ

It is easy to see that for each pair alternatives (o,r) 2 Xp, if Spr P Spo (i.e., t pro P 0), then the alternative Ao is closer to the HFPIS A+ than the alternative Ar. Thus the ranking order of the alternatives Ao and Ar determined by Spo and Spr based on (x,A+) is consistent with the preference (o,r) 2 Xp given by the DM Dp. Conversely,

221


if Spr < Spo , then the ranking order of the alternatives Ao and Ar determined by Spo and Spr based on (x,A+) is inconsistent with the preference relation (o,r) 2 Xp given by the DM Dp. In the following, we define an inconsistency index Spr Spo to measure the degree of inconsistency between the ranking order of the alternatives Ao and Ar (determined by Spo and Spr ) and the preferences relation (o,r) 2 Xp (given by the DM Dp) as:

p Sr Spo ¼

(

p 0 Sr P Spo p ~ p ðo; rÞ Sp Sp Sr < Spo C o r

Spr Spo

n o ~ p ðo; rÞ Sp Sp ¼ max 0; C o r

n X p X o ~ p ðo; rÞ Sp Sp Sr Spo ¼ max 0; C o r ðo;rÞ2Xp

ð14Þ

ð15Þ

ðo;rÞ2Xp

which is called the inconsistency index of the DM Dp. Thus, the collective inconsistency index of all DMs is defined as: ~¼ B

f f f n o X X X p X X ~ p ðo; rÞ Sp Sp ~p ¼ Sr Spo ¼ max 0; C B o r p¼1 ðo;rÞ2Xp

p¼1

ð16Þ

p¼1 ðo;rÞ2Xp

þ In a similar way, a consistency index Spr Spo is defined as:

p þ Sr Spo ¼

(

p ~ p ðo; rÞ Sp Sp Sr P Spo C p r p o 0 S r < So

ð17Þ

which can measure the consistency between the ranking order of the alternatives Ao and Ar (determined by Spo and Spr ) and the preferences relation (o,r) 2 Xp. Similarly, the consistency index in Eq. (17) can be also rewritten as:

Spr Spo

þ

n o ~ p ðo; rÞ Sp Sp ¼ max 0; C r o

ð18Þ

Hence, the collective consistency index of all DMs is defined as: ~¼ G

f X p¼1

~p ¼ G

f X X

Spr Spo

p¼1 ðo;rÞ2Xp

þ

¼

f X X

n o ~ p ðo; rÞ Sp Sp max 0; C r o

Spo

þ

~ p ðo; rÞ Sp Sp Spr Spo ¼ C r o

ð21Þ

Combining with Eqs. (9), (16), (19) and (21), we get:

~ B ~¼ G

f X X

Spr Spo

þ

f X X ~ p ðo;rÞ t p Spr Spo ¼ C ro

p¼1 ðo;rÞ2Xp

p¼1 ðo;rÞ2Xp

ð22Þ Using Eqs. (16) and (22), the mathematical programming model (20) can be rewritten as:

Let

~p ¼ B

Spr

ð13Þ

For convenience, we rewrite the inconsistency index as:

It can be easily derived from Eqs. (13) and (17) that

ð19Þ

p¼1 ðo;rÞ2Xp

min

8 f > ~ p ðo; rÞ t p ea~ > C > ro > > > p¼1 ðo;rÞ2Xp > > < þ þ þ 2 ‘ s:t: 1 P h1j P hj P P hj P 0; > > > > n > X > > > wj ¼ 1; j 2 N > : 1 P wj P 0;

ð23Þ j2N

j¼11

For each pair alternatives (o,r) 2 Xp, let kpor ¼ max 0; Spo Spr , p p p p p p then kor P So Sr (i.e., t ro þ kor P 0) and kor P 0. Thus n o ~ p ðo; rÞ Sp Sp ~ p ðo; rÞ kp . can be rewritten as C max 0; C o r or Then, the above mathematical programming model (23) can be converted into the following programming model (24) as:

min

8 f > ~ p ðo; rÞ t p ea~ > C > ro > > p > p¼1 ðo;rÞ2 X > > > > > > tpro þ kpor P 0 ððo; rÞ 2 Xp ; p 2 FÞ > < p p s:t: kor P 0 ððo; rÞ 2 X ; p 2 FÞ þ þ > > > 1 P h1 P h2 P P h‘ þ P 0; > > j j j > > > > n > X > > > wj ¼ 1; j 2 N > 1 P wj P 0; :

ð24Þ

j2N

j¼11

Remark 2. It is noticed that both the collective consistency index ~ and inconsistency index B ~ are interval numbers because the DM’s G preferences on pair-wise comparisons of alternatives are interval numbers. In real-life decision process, the smaller the collective inconsistency index is, the better the decision result is. In general, the col~ should be no bigger than the lective inconsistency index B ~ In this sense, a mathematical procollective consistency index G. gramming model is constructed to determine the weighting vector x and the HFPIS A+ as follows:

~ minfBg 8 ~ ~ ea~ GB > > > þ þ þ > > < 1 P h1 P h2 P P h‘ P 0; j 2 N j j j s:t: n > X > > > wj ¼ 1; j 2 N > : 1 P wj P 0;

ð20Þ

j¼1

~ which intends to minimize the collective inconsistency index B ~ is under the condition in which the collective consistency index G ~ by ea~ . Here, no smaller than the collective inconsistency index B ea~ ¼ ½a; a is an arbitrary positive interval number given by the DMs in advance.

Because the objective function and first constraint condition in model (24) are interval numbers, the model (24) can be further rewritten as the following model:

82 39 f f < X = X X X min 4 C por kpor ; C por kpor 5 : p¼1 ; p¼1 ðo;rÞ2Xp ðo;rÞ2Xp 3 82 f f X X p > > 4X X p p > > C t ; C or t pro 5 P ½a; a or ro > > > p¼1 ðo;rÞ2Xp p¼1 ðo;rÞ2Xp > > > > > tp þ kp P 0 ððo; rÞ 2 Xp ; p 2 FÞ > > or < ro p p s:t: kor P 0 ððo; rÞ 2 X ; p 2 FÞ > þ > þ þ > 1 2 ‘ > > P hj P P hj P 0; j 2 N 1 P hj > > > > > n X > > > 1 P w P 0; > wj ¼ 1; j 2 N j :

ð25Þ

j¼11

Obviously, the model (25) is a standard linear interval programming model, which may be solved by the existing methods (Ishibuchi and Tanaka, 1990; Lai et al., 2002; Sengupta and Pal, 2000). Here, the model (25) is solved in the sense of the model (5). Thus the model (25) is transformed to the bi-objective linear programming model (26) as:

222

min


8 f X X p > > > C or tpro P a > > > p > p¼1 ðo;rÞ2X > > > > > f >X X p > > > C or tpro P a > > > p > p¼1 ðo;rÞ2X > > < p s:t: t pro þ kpor P 0 ððo; rÞ 2 X ; p 2 FÞ > > > > kpor P 0 ððo; rÞ 2 Xp ; p 2 FÞ > > > > þ þ þ > > 1 2 ‘ > > 1 P hj P hj P P hj P 0; > > > > > > n X > > > > 1 P wj P 0; wj ¼ 1; j 2 N > :

using Eq. (9). Afterwards, we can obtain each DM’s ranking for the alternatives Ai(i 2 M) according to the increasing orders of the distances Spi ði 2 MÞ, respectively. At length, the collective alternative ranking is obtained using social choice functions such as the Borda’s function and Copeland’s function (Hwang and Yoon, 1981).

ð26Þ

3.3. The proposed algorithm Based on the above models and analysis, a practical decision algorithm is presented for solving MAGDM problems where the weights of attributes are partial known or completely unknown, the attribute values take the form of HFEs, and the DMs express their preferences on pair-wise comparison between alternatives with interval numbers. The proposed algorithm involves the following steps:

j2N

j¼1

Using the weight average approach, the model (26) can be transformed into the parametric linear programming model (27): 9 8 f f = < X X X X p p p p p min h C or kor þ ð1 hÞ C or þ C or =2 kor ; : p p p¼1 p¼1 ðo;rÞ2X

ðo;rÞ2X

8 f > X X p > > > C or tpro P a > > > p > p¼1 ðo;rÞ2 X > > > > > f >X X p > > > C or tpro P a > > > > p¼1 ðo;rÞ2Xp > > < p s:t: tpro þ kpor P 0 ððo;rÞ 2 X ;p 2 FÞ > > > p p > kor P 0 ððo; rÞ 2 X ;p 2 FÞ > > > > þ þ þ > > 1 2 ‘ > > P hj P P hj P 0; j 2 N > 1 P hj > > > > > n X > > > > wj ¼ 1; j 2 N > 1 P wj P 0; : j¼1

ð27Þ

Step 1. Form a group of DMs and identify all alternatives and attributes. Denote the sets of alternatives and attributes by A = {A1, A2, . . . ,Am} and C = {C1, C2, . . . , Cn}, respectively. p Step 2. Determine the performance ratings hij ði 2 M; j 2 N; p 2 FÞ for all alternatives with respect to each attribute by the DM Dp using the HFEs, and then construct the hesitant fuzzy decision matrices Hp(p 2 F). Step 3. Express pair-wise comparison judgments between alternatives by a set of ordered pairs Xp ¼ fðo; rÞjAo C~ p ðo;rÞ Ar ; ~ p ðo; rÞ for all ðo; r 2 MÞg with the interval truth degrees C the DMs Dp(p 2 F). Step 4. Define the collective consistency index and collective inconsistency index using Eqs. (16) and (19), respectively. Step 5. Construct the interval mathematical programming model by using Eq. (25), and transform such a model into the linear programming model (27). Step 6. Solve the linear programming model (27) using the Maltab and Lingo soft package with the given parameter h(0 6 h 6 1) by the DMs in advance, and then determine the optimal weight vector x and the HFPIS A+. Step 7. Calculate the distances Spi ði 2 M; p 2 FÞ of each alternative Ai(i 2 M) to the HFPIS A+ using Eq. (9). Step 8. Rank the alternatives Ai(i 2 M) for each DM Dp(p 2 F) according to the increasing orders of the distances Spi ði 2 MÞ, respectively. Step 9. Rank the alternatives using social choice functions such as the Borda’s function and Copeland’s function (Hwang and Yoon, 1981), and then determine the optimal alternative from the alternatives Ai(i 2 M).

where 0 6 h 6 1 is a parameter which is determined by the DMs in advance. Apparently, in model (27) t pro is the linear function of x and A+, P f p and there exist variables that need to be p¼1 jX j þ n þ n‘ Pf p p p determined, including p¼1 jX j variables kor P 0ððo; rÞ 2 X Þ; n weights of attributes wj(j 2 N), and n‘ variables of þ l hj ðl 2 L; j 2 NÞ, and at the same time there have P 2 fp¼1 jXp j þ n þ n‘ þ 1 inequalities (excluding the nonnegative

3.4. Other generalizations of the proposed model

constraints for the variables and the weighs). It is clear that the linear programming model (27) can be easily solved by the Simplex method and needs very low time cost relative to the non-linear programming model. In order to determine objectively these vari P ables, the number 2 fp¼1 jXp j þ n þ n‘ þ 1 of inequalities should Pf p not be very small. In general, the larger p¼1 jX j (i.e., pair-wise comparison between alternatives) is, the more precise and reliable the weighting vector and the HFPIS are to be determined. But if the number of conflicting preference relations in Xp(p 2 F) is very large, the model (27) may become infeasible. Thus, collecting the preference information over alternatives is an important issue for the sake of effectively implementing the LINMAP procedure (Chen, 2013). By solving the model (27), we can determine the weight vector x and the HFPIS A+. Then, we calculate the distances Spi ði 2 M; p 2 FÞ of each alternatives Ai(i 2 M) to the HFPIS A+

It is worth pointing out that the MAGDM problem mentioned in Section 3.1 is a general and complex decision problem, in real life we may encounter some special cases of this problem, such as, in this MAGDM problem the weights of attributes are completely known or partial known, the HFPIS is known in advance, or there may have just one DM (i.e., the MADM problem), etc. Can the proposed approach (i.e., the model (27)) be generalized to deal with these special cases of MAGDM problems under hesitant fuzzy environments? In this part, we try to answer this question. Case 1: The MAGDM problem mentioned above only includes the single DM, namely, the MAGDM problem has been reduced to the MADM problem, which is similar to the LINMAP-based approaches (Li, 2008; Wan and Li, 2013; Xia et al., 2006). In this case, p = 1, the model (27) can be converted into the following model:

223


( min h

X

C or kor þ ð1 hÞ

ðo;rÞ2X

X

)

Case 3: The weights of attributes are completely known in advance. In this case, the variables in the model (27) are only the HFPIS, and thus the model (27) reduces to the following model (30). We just need to derive the HFPIS by solving the model (30).

ððC or þ C or Þ=2Þ kor

ðo;rÞ2X

8 X C or t or P a > > > > ðo;rÞ2X > > > X > > > > C or t or P a > > > ðo;rÞ2X > > > > > < t ro þ kor P 0 ðo; rÞ 2 X s:t: > > > kor P 0 ðo; rÞ 2 X > þ þ þ > > 2 ‘ > > 1 P h1j P hj P P hj P 0; > > > > > > n > X > > > 1 P wj P 0; wj ¼ 1; j 2 N > :

8
> t pro P ea~ > > > p > p¼1 ðo;rÞ2 X > > > > p p > > t pro þ kor P 0 ððo; rÞ 2 X ; p 2 FÞ > < p p s:t: kor P 0 ððo; rÞ 2 X ; p 2 FÞ > þ þ þ > > 1 2 ‘ > P hj P P hj P 0; 1 P hj > > > > > > n X > > > > 1 P wj P 0; wj ¼ 1; j 2 N :

f X X

ð29Þ

j2N

j¼1

where ea~ is not an interval number but an arbitrary positive real number given by the DMs in advance. Remark 3. As mentioned previously, in practical decision problems DMs are not sure enough in pair-wise comparisons over alternative and usually express their opinions with a degree of insurance. As a result, the DM’s pair-wise comparison judgments over alternatives may be represented by interval numbers, triangular fuzzy numbers, trapezoidal fuzzy numbers or HFEs, etc. It is noticed that the different forms of DM’s preferences on pair-wise comparisons need different LINMAP-based models to solve them. In this paper, an interval programming model is proposed to solve MAGDM problems in which the ratings of alternatives are taken as HFEs and all pair-wise comparison judgments over alternatives are represented by interval numbers. While if the DM’s pair-wise comparison judgments are represented by triangular fuzzy numbers, trapezoidal fuzzy numbers or HFEs, both the collective ~ and inconsistency index B ~ are triangular fuzzy consistency index G numbers, trapezoidal fuzzy numbers or HFEs, respectively. Thus the proposed model cannot solve them. In fact, Sadi-Nezhad and Akhtari (2008) have developed a possibility LINMAP for solving the MAGDM problems with triangular fuzzy truth degree. Li and Wan (2013, 2014) have also proposed a LINMAP-based fuzzy linear programming technique for solving heterogeneous MADM problems with trapezoidal fuzzy truth degree. But for the case that the DM’s pair-wise comparison judgments are represented by HFEs, there is no method to solve it, which may sever as a suggestion for further research.

C por

kpor

ðo;rÞ2Xp

9 f = X X p p p þ ð1 hÞ C or þ C or =2 kor ; p p¼1 ðo;rÞ2X

8 f X X p > > > C or t pro P a > > > p > p¼1 ðo;rÞ2 X > > > > f > > >X X p < C or t pro P a p

p¼1 ðo;rÞ2X > > > > t pro þ kpor P 0 ððo; rÞ 2 Xp ; p 2 FÞ > > > > > kpor P 0 ððo; rÞ 2 Xp ; p 2 FÞ > > > > > : 1 P h1 þ P h2 þ P P h‘ þ P 0;j 2 N j

j

j

ð30Þ

Case 4: The HFPIS is already known in advance. In this case, the variables need to be determined are just the weights of attributes, thus the model (27) reduces to the following model (31) as: 8
> > C por t pro P a > > > > p¼1 ðo;rÞ2Xp > > > > f > > p¼1 ðo;rÞ2Xp > > > > > t pro þ kpor P 0 ððo; rÞ 2 Xp ; p 2 FÞ > > > p p > > > kor P 0 ððo; rÞ 2 X ; p 2 FÞ > Pn : 1 P wj P 0; j¼1 wj ¼ 1; j 2 N

ðo;rÞ2X

ð31Þ

Case 5: The weights of attributes are partially known. It is worth to notice that the weights of attributes aforementioned are completely unknown or known. However, there are some real-life situations that the information about the weights of all attributes is not completely unknown but partially known. For convenience, let Z be a set of the known weight information, according to the Refs. Kim and Ahn (1999), Park (2004), Park and Kim (1997)), Z can be constructed by the following forms: (1). A weak ranking form: {wi P wj}(i – j); (2). A strict ranking form: {wi wj P ai}(ai > 0); (3). A ranking of differences form: {wi wi wj P wk wl}(i – j – k – l); (4). A ranking with multiples form: {wi P aiwj}(i – j,0 6 ai 6 1); (5). An interval form: {ai 6 wi 6 ai + ei}(0 6 ai 6 ai + ei 6 1). In this case, the weights of attributes are partially known, thus the model (27) is transformed into the model (32): min

s:t:

8
> > > C or tpro P a > > > p¼1 ðo;rÞ2Xp > > > > > f >X X p > > > C or tpro P a >
> tpro þ kpor P 0 ððo; rÞ 2 Xp ; p 2 FÞ > > > > p > ððo; rÞ 2 Xp ; p 2 FÞ k > or P 0 > > þ þ > þ > 1 2 ‘ > > P hj P P hj P 0; 1 P hj > > > : x2Z

j2N ð32Þ

It is apparent from the above analysis that the model (27) is extremely flexible, according to the different actual needs, this model can

224


reduce to various special models for solving different decision making problems under different circumstances. 4. An energy project selection problem and comparison analysis of obtained results In this section, an energy project selection problem adopted from Xu and Xia (2011a) is used to demonstrate the applicability and the implementation process of the proposed method. The comparison analysis of computational results is also conducted to show its superiority. 4.1. An energy project selection problem and the analysis process Energy is an indispensable factor for the social and economic development. The correct energy policy affects economic development and environment, and thus the most appropriate energy policy selection is very important. Suppose that there are five alternatives (energy projects) Ai(i = 1, 2, 3, 4, 5) to be invested, and four attributes to be considered: C1: technological; C2: environmental; C3: socio-political; C4: economic (more details about them can be found in Kahraman and Kaya (2010)). Three decision organizations Dp (p = 1, 2, 3) are invited to evaluate the performances of the five alternatives. For an alternative under an attribute, all DMs of the decision organization provide their evaluated values and some of these values may be repeated. But a value repeated more times does not indicate that it is more importance than other values repeated less times. For example, the value repeated one time may be provided by a DM who is an expert at this area, and the value repeated twice may be provided by two DMs who are not familiar with this area. In such cases, the value repeated one time may be more important than the one repeated twice. To get a more reasonable result, it is better that the DMs give their evaluations anonymously. We only collect all the possible values for an alternative under an attribute, and each value provided only means that it is a possible value, but its importance is

unknown. Thus the times that the values repeated are unimportant, and it is reasonable to allow these values repeated many times appear only once. The HFE is just a tool to deal with such cases, and all possible evaluations for an alternative under the attributes can be considered as a HFE. The results evaluated by three decision organizations are contained in hesitant fuzzy decision matrices, shown in Tables 3–5. Meanwhile, the decision organizations Dp(p = 1, 2, 3) also provide their comparison preference (incomplete preference) information between alternatives as follows: X1 ¼ fhðA1 ; A2 Þ; ½0:7; 0:8i; hðA3 ; A4 Þ;½0:8; 0:9i;hðA5 ; A3 Þ; ½0:8;0:9ig; X2 ¼ fhðA1 ; A3 Þ; ½0:6; 0:7i; hðA3 ; A4 Þ;½0:8; 0:9i;hðA4 ; A2 Þ; ½0:7;0:8i; hðA5 ;A3 Þ; ½0:7; 0:8ig; X3 ¼ fhðA2 ; A1 Þ; ½0:7; 0:8i; hðA3 ; A1 Þ;½0:5; 0:6i;hðA4 ; A5 Þ; ½0:9;1:0ig:

where (A1,A3) in X2 means that the decision organization D2 moderately prefers A1 to A3, and the others have the similar meanings. In the following, we employ the proposed method in previous section to solve the above energy project selection problem. It is easy to see from the Tables 3–5 that the numbers of values in different HFEs are different. In order to more accurately calculate the distance between two HFEs, we should extend the shorter one until both of them have the same length when we compare them. According to the regulations mentioned above, we consider that the DMs are risk-averse in the example, and change the hesitant fuzzy data by adding the minimal values as listed in Tables 6–8. Then, we proceed to solve the above problem for selecting the most appropriate energy policy, which involves the following two cases: Case 1: The information about the attribute weights is completely unknown. For convenience, let þ ~þ ; h ~þ ; h ~þ ; h ~ þ ¼ ðhh11 ; h12 ; h13 ; h14 ; h15 i; hh21 ;h22 ;h23 ;h24 ;h25 i; h ¼ h 1 2 3 4 hh31 ; h32 ; h33 ; h34 ; h35 i; hh41 ; h42 ; h43 ; h44 ; h45 iÞ:

Table 3 ~ 1. Hesitant fuzzy decision matrix D

A1 A2 A3 A4 A5

C1

C2

C3

C4

{0.5,0.4,0.3} {0.5,0.3} {0.7,0.6} {0.8,0.7,0.4,0.3} {0.9,0.7,0.6,0.3,0.1}

{0.9,0.8,0.7,0.1} {0.9,0.7,0.6,0.5,0.2} {0.9,0.6} {0.7,0.4,0.2} {0.8,0.7,0.6,0.4}

{0.5,0.4,0.2} {0.8,0.6,0.5,0.1} {0.7,0.5,0.3} {0.8,0.1} {0.9,0.8,0.7}

{0.9,0.6,0.5,0.3} {0.7,0.3,0.4} {0.6,0.4} {0.9,0.8,0.6} {0.9,0.7,0.6,0.3}


A1 A2 A3 A4 A5

C1

C2

C3

C4

{0.6,0.4} {0.5,0.4,0.3} {0.7,0.6,0.5,0.4,0.2} {0.8,0.7,0.3} {0.9,0.7,0.6}

{0.9,0.8,0.7} {0.9,0.7,0.6,0.5,0.1} {0.9,0.6} {0.7,0.4,0.2} {0.8,0.7,0.6,0.4}

{0.5,0.4,0.2} {0.8,0.6,0.5,0.4} {0.7,0.5,0.3} {0.8,0.6,0.4} {0.9,0.8,0.7,0.6,0.5}

{0.9,0.6,0.5,0.3} {0.7,0.3,0.4} {0.6,0.5,0.4,0.3} {0.9,0.8,0.6,0.5,0.3} {0.9,0.8, 0.3}


A1 A2 A3 A4 A5

C1

C2

C3

C4

{0.7,0.6,0.5,0.4,0.3} {0.5,0.3} {0.7,0.6,0.5} {0.8,0.7,0.4,0.3} {0.9,0.7}

{0.8,0.7} {0.9,0.7,0.6,0.5} {0.9,0.6} {0.7,0.4,0.2} {0.8,0.7,0.6,0.5,0.4}

{0.5,0.4,0.3,0.2} {0.8,0.6,0.5} {0.7,0.5,0.3} {0.8,0.7,0.2,0.1} {0.9,0.8,0.7}

{0.9,0.6,0.5} {0.7,0.6,0.5,0.4} {0.8,0.7,0.6,0.5,0.4} {0.9,0.8,0.6} {0.9,0.7,0.6,0.3}

225


Then, we utilize the model (27) to construct the corresponding linear programming model as follows:

( min

s:t:

At length, on the basis of the Borda’s function (Hwang and Yoon, 1981), the Borda’s scores of each alternatives Ai (i = 1, 2, 3, 4, 5) can

ð0:05h þ 0:75Þ k112 þ ð0:05h þ 0:85Þ k134 þ ð0:05h þ 0:85Þ k153 þ ð0:05h þ 0:65Þ k213 þ ð0:05h þ 0:85Þ k234

)

þð0:05h þ 0:75Þ k242 þ ð0:05h þ 0:75Þ k253 þ ð0:05h þ 0:75Þ k321 þ ð0:05h þ 0:55Þ k331 þ ð0:05h þ 0:95Þ k345 8 0:014w1 þ 0:002w2 0:148w3 þ 0:136w4 þ 0:04w3 ð3h31 þ 2h32 þ 3h33 h34 h35 Þ 0:04w1 h12 > > > > 0:04w ðh þ h 4h h Þ 0:08w ðh þ h þ h Þ þ k1 P 0 > 2 22 23 24 25 4 41 42 43 > 12 > > > > 0:092w þ 0:296w þ 0:066w 0:306w 0:04w ð2h h h11 þ 3h14 þ 3h15 Þ þ 0:04w4 ð3h41 þ 4h42 > 1 2 3 4 1 13 12 > > > > þ2h þ 2h þ 2h Þ 0:04w ð4h h þ 2h þ 2h þ 2h Þ 0:08w2 ðh12 þ h22 þ 2h23 þ 2h24 þ 2h25 Þ þ k134 P 0 > 43 44 45 3 32 31 33 34 35 > > > > 0:382w3 0:088w2 0:034w1 þ 0:168w4 þ 0:04w4 ð3h41 þ 3h42 þ 2h43 h44 h45 Þ þ 0:04w2 ðh12 h22 > > > > 1 > þ2h > 24 þ 2h25 Þ 0:04w3 ð2h31 þ 3h32 þ 4h33 þ 4h34 þ 4h35 Þ 0:04w1 ð2h11 þ h12 3h14 5h15 Þ þ k53 P 0 > > > > > 0:134w2 0:06w1 0:096w3 þ 0:13w4 þ 0:04w1 ðh11 þ 2h12 þ h13 2h15 Þ 0:04w2 ð2h22 þ h23 þ h24 þ h25 Þ > > > > > w4 ð0:12h41 þ 0:04h42 þ 0:04h43 Þ þ 0:04w3 ð2h31 þ h32 þ h33 þ h34 þ h35 Þ þ k213 P 0 > > > > > > > 0:296w2 0:02w1 0:094w3 0:24w4 þ 0:04w4 ð3h41 þ 3h42 þ 2h43 þ 2h44 Þ þ 0:04w1 ðh11 þ h12 > 2 > > > > 2h13 h14 þ h15 Þ 0:08w2 ðh12 þ h22 þ 2h23 þ 2h24 þ 2h25 Þ þ 0:04w3 ðh31 þ h32 þ h33 þ h34 þ h35 Þ þ k34 P 0 > > > 0:144w1 0:23w2 0:018w3 þ 0:246w4 0:04w4 ð2h41 þ 4h42 þ 3h43 þ 2h44 Þ þ 0:04w3 h33 > > > > 2 > þ0:04w > 2 ð2h12 þ 3h22 þ 4h23 þ 3h24 h25 Þ 0:12w1 ðh11 þ h12 Þ þ k42 P 0 > > > > > 0:216w1 0:088w2 þ 0:308w3 þ 0:154w4 þ 0:04w2 ðh12 h22 þ 2h24 þ 2h25 Þ 0:04w3 ð2h31 þ 3h32 > > > > > þ4h33 þ 3h34 þ 2h35 Þ 0:04w4 ð3h41 þ 3h42 h43 Þ 0:04w1 ð2h11 þ h12 þ h13 þ 2h14 þ 4h15 Þ þ k253 P 0 > > > < 0:234w 0:088w 0:148w 0:1w þ 0:04w ð2h þ 3h þ 2h þ h Þ þ 0:04w ðh h 3

2

1

4

1

11

12

13

14

2

23

12

> þ2h24 þ 2h25 Þ 0:04w3 ð3h31 þ 2h32 þ 2h33 þ 3h34 þ 3h35 Þ þ 0:04w4 ð2h41 þ 2h44 þ 2h45 Þ þ k321 P 0 > > > > > 0:05w1 0:07w2 þ 0:086w3 0:004w4 þ 0:04w2 ðh22 h21 þ h23 þ h24 þ h25 Þ w1 ð0:04h14 þ 0:08h15 Þ > > > > > þ0:04w4 ðh41 h42 h43 þ h45 Þ 0:04w3 ð2h31 þ h32 þ h34 þ h35 Þ þ k331 P 0 > > > > > > 0:138w4 0:208w2 0:448w3 0:058w1 0:04w4 ðh42 þ 3h44 þ 3h45 Þ þ 0:04w1 ðh11 þ 2h13 2h15 Þ > > > > > þ0:04w2 ðh12 þ 3h22 þ 4h23 þ 2h24 þ 2h25 Þ þ 0:04w3 ðh31 þ 7h32 þ 6h33 þ 6h34 þ 6h35 Þ þ k345 P 0 > > > > > 0:0258w2 0:145w1 0:1558w3 0:2352w4 þ 0:028w1 h11 0:028w1 h12 þ 0:004w1 h13 þ 0:028w2 h12 þ 0:092w1 h14 > > > > > þ0:204w1 h15 þ 0:06w2 h22 þ 0:004w2 h23 0:212w2 h24 þ 0:012w2 h25 þ 0:016w3 h31 þ 0:028w3 h32 0:024w3 h33 > > > > > þ0:16w3 h34 þ 0:128w3 h35 þ 0:112w4 h41 þ 0:236w4 h42 þ 0:108w4 h43 0:028w4 h44 0:044w4 h45 P a > > > > > 0:0236w2 0:1352w1 0:1422w3 0:2191w4 þ 0:026w1 h11 0:026w1 h12 þ 0:004w1 h13 þ 0:026w2 h12 > > > > > þ0:086w1 h14 þ 0:19w1 h15 þ 0:056w2 h22 þ 0:004w2 h23 0:198w2 h24 þ 0:012w2 h25 þ 0:014w3 h31 þ 0:024w3 h32 > > > > > 0:024w3 h33 þ 0:148w3 h34 þ 0:118w3 h35 þ 0:104w4 h41 þ 0:22w4 h42 þ 0:1w4 h43 0:026w4 h44 0:04w4 h45 P a > > > > 1 1 1 2 2 2 2 3 3 3 > > k12 ; k34 ; k53 ; k13 ; k34 ; k42 ; k53 ; k21 ; k31 ; k45 P 0 > > > > > 1 P h11 P h12 P h13 P h14 P h15 P 0; 1 P h21 P h22 P h23 P h24 P h25 P 0 > > > > > 1 P h31 P h32 P h33 P h34 P h35 P 0; 1 P h41 P h42 P h43 P h44 P h45 P 0 > > : 1 P wj P 0 ðj ¼ 1; 2; 3; 4Þ; w1 þ w2 þ w3 þ w4 ¼ 1

¼ ½0:1; 0:1), the optimal Let h = 0.5 and ea~ ¼ 0:1 (i.e., ea~ ¼ ½a; a weight vector x and the HFPIS h+ can be obtained by solving the above linear programming model:

x ¼ ð0:2729; 0:1098; 0:1063; 0:5110ÞT ; þ

h ¼ ðh0:8498; 0:8498; 0:8498; 0:0; 0:0i; h1:0; 1:0; 1:0; 1:0; 0i; h1:0; 1:0; 1:0; 1:0; 1:0i; h1:0; 1:0; 1:0; 1:0; 1:0iÞ: Based on the obtained weight vector and the HFPIS, the distances Spi ði ¼ 1; 2; 3; 4; 5; p ¼ 1; 2; 3Þ of the alternatives Ai (i = 1, 2, 3, 4, 5) to the HFPIS h+ can be calculated by using the Eq. (9), and the results are listed in Table 9. According to the increasing orders of the distances Spi ði ¼ 1; 2; 3; 4; 5Þ, the ranking orders of the alternatives Ai (i = 1, 2, 3, 4, 5) for the decision organizations Dp(p = 1, 2, 3) can be obtained, which are also listed in Table 9.

be obtained, which are shown in Table 10. Therefore, the ranking order of the alternatives (energy projects) Ai (i = 1, 2, 3, 4, 5) can be easily obtained from Table 10 as A2 A4 A1 A3 A5, and the most desirable alternative is obtained as the energy project A5. Case 2: The information about the attribute weights is partly known and the known weight information is given as: Z ¼ f0:15 6 w1 6 0:2; 0:16 6 w2 6 0:18; 0:3 6 w3 6 0:35; 0:3 6 w4 P 6 0:45; 4j¼1 wj ¼ 1g. Analogously, denote the HFPIS h+ as þ

h ¼ ðhh11 ; h12 ; h13 ; h14 ; h15 i; hh21 ; h22 ; h23 ; h24 ; h25 i; hh31 ; h32 ; h33 ; h34 ; h35 i; hh41 ; h42 ; h43 ; h44 ; h45 iÞ: Then, we utilize the model (32) to construct the corresponding linear programming model as follows: ¼ ½0:1; 0:1, we can obtain the optiTaking h = 0.5 and ea~ ¼ ½a; a mal weight vector x and the HFPIS h+ by solving the above linear programming model as:

226


x ¼ ð0:15; 0:16; 0:3; 0:39ÞT þ

h ¼ ðh0:9086; 0:9086; 0:9086; 0:0123; 0:0123i; h1:0; 0:4716; 0:4716;0:4716; 0i; h1:0; 1:0; 0:2836; 0:2836;0:2836i; h1:0; 1:0;1:0;1:0;1:0iÞ:

( min

4.2. Comparison analysis with the intuitionistic fuzzy LINMAP method As aforementioned in the introduction, many researchers have extended the LINMAP approach into various forms for solving MADM or MAGDM problems under a variety of different environ-

ð0:05h þ 0:75Þ k112 þ ð0:05h þ 0:85Þ k134 þ ð0:05h þ 0:85Þ k153 þ ð0:05h þ 0:65Þ k213 þ ð0:05h þ 0:85Þ k234

)

þð0:05h þ 0:75Þ k242 þ ð0:05h þ 0:75Þ k253 þ ð0:05h þ 0:75Þ k321 þ ð0:05h þ 0:55Þ k331 þ ð0:05h þ 0:95Þ k345 8 0:014w1 þ 0:002w2 0:148w3 þ 0:136w4 þ 0:04w3 ð3h31 þ 2h32 þ 3h33 h34 h35 Þ 0:04w1 h12 > > > > 0:04w ðh þ h 4h h Þ 0:08w ðh þ h þ h Þ þ k1 P 0 > 2 22 23 24 25 4 41 42 43 > 12 > > > > 0:092w þ 0:296w þ 0:066w 0:306w 0:04w ð2h h h11 þ 3h14 þ 3h15 Þ þ 0:04w4 ð3h41 þ 4h42 þ 2h43 > 1 2 3 4 1 13 12 > > > þ2h þ 2h Þ 0:04w ð4h h þ 2h þ 2h þ 2h Þ 0:08w ðh þ h þ 2h þ 2h þ 2h Þ þ k1 P 0 > > 44 45 3 32 31 33 34 35 2 12 22 23 24 25 34 > > > > 0:382w 0:088w 0:034w þ 0:168w þ 0:04w ð3h þ 3h þ 2h h h Þ þ 0:04w ðh h > 3 2 1 4 4 41 42 43 44 45 2 12 22 > > > > > þ2h24 þ 2h25 Þ 0:04w3 ð2h31 þ 3h32 þ 4h33 þ 4h34 þ 4h35 Þ 0:04w1 ð2h11 þ h12 3h14 5h15 Þ þ k153 P 0 > > > > > 0:134w2 0:06w1 0:096w3 þ 0:13w4 þ 0:04w1 ðh11 þ 2h12 þ h13 2h15 Þ 0:04w2 ð2h22 þ h23 þ h24 þ h25 Þ > > > 2 > > w 4 ð0:12h41 þ 0:04h42 þ 0:04h43 Þ þ 0:04w3 ð2h31 þ h32 þ h33 þ h34 þ h35 Þ þ k13 P 0 > > > > > > > 0:296w2 0:02w1 0:094w3 0:24w4 þ 0:04w4 ð3h41 þ 3h42 þ 2h43 þ 2h44 Þ þ 0:04w1 ðh11 þ h12 2h13 > > 2 > > > h14 þ h15 Þ 0:08w2 ðh12 þ h22 þ 2h23 þ 2h24 þ 2h25 Þ þ 0:04w3 ðh31 þ h32 þ h33 þ h34 þ h35 Þ þ k34 P 0 > > > > > 0:144w1 0:23w2 0:018w3 þ 0:246w4 0:04w4 ð2h41 þ 4h42 þ 3h43 þ 2h44 Þ þ 0:04w3 h33 > > 2 > > > þ0:04w2 ð2h12 þ 3h22 þ 4h23 þ 3h24 h25 Þ 0:12w1 ðh11 þ h12 Þ þ k42 P 0 > > > > 0:216w1 0:088w2 þ 0:308w3 þ 0:154w4 þ 0:04w2 ðh12 h22 þ 2h24 þ 2h25 Þ 0:04w3 ð2h31 þ 3h32 þ 4h33 > > > 2 > > þ3h 34 þ 2h35 Þ 0:04w4 ð3h41 þ 3h42 h43 Þ 0:04w1 ð2h11 þ h12 þ h13 þ 2h14 þ 4h15 Þ þ k53 P 0 > > > < 0:234w 0:088w 0:148w 0:1w þ 0:04w ð2h þ 3h þ 2h þ h Þ þ 0:04w ðh h 3 2 1 4 1 11 12 13 14 2 23 12 s:t: 3 > þ2h þ 2h Þ 0:04w ð3h þ 2h þ 2h þ 3h þ 3h Þ þ 0:04w ð2h þ 2h þ 2h Þ þ k P0 24 25 3 31 32 33 34 35 4 41 44 45 > 21 > > > > 0:05w 0:07w þ 0:086w 0:004w þ 0:04w ðh h þ h þ h þ h Þ w ð0:04h þ 0:08h > 1 2 3 4 2 22 21 23 24 25 1 14 15 Þ > > > > þ0:04w4 ðh41 h42 h43 þ h45 Þ 0:04w3 ð2h31 þ h32 þ h34 þ h35 Þ þ k331 P 0 > > > > > 0:138w4 0:208w2 0:448w3 0:058w1 0:04w4 ðh42 þ 3h44 þ 3h45 Þ þ 0:04w1 ðh11 þ 2h13 2h15 Þ > > > > > þ0:04w2 ðh12 þ 3h22 þ 4h23 þ 2h24 þ 2h25 Þ þ 0:04w3 ðh31 þ 7h32 þ 6h33 þ 6h34 þ 6h35 Þ þ k3 P 0 > 45 > > > > 0:0258w2 0:145w1 0:1558w3 0:2352w4 þ 0:028w1 h11 0:028w1 h12 þ 0:004w1 h13 þ 0:028w2 h12 þ 0:092w1 h14 > > > > > > þ0:204w1 h15 þ 0:06w2 h22 þ 0:004w2 h23 0:212w2 h24 þ 0:012w2 h25 þ 0:016w3 h31 þ 0:028w3 h32 0:024w3 h33 > > > > > þ0:16w3 h34 þ 0:128w3 h35 þ 0:112w4 h41 þ 0:236w4 h42 þ 0:108w4 h43 0:028w4 h44 0:044w4 h45 P a > > > > > 0:0236w2 0:1352w1 0:1422w3 0:2191w4 þ 0:026w1 h11 0:026w1 h12 þ 0:004w1 h13 þ 0:026w2 h12 > > > > > þ0:086w1 h14 þ 0:19w1 h15 þ 0:056w2 h22 þ 0:004w2 h23 0:198w2 h24 þ 0:012w2 h25 þ 0:014w3 h31 þ 0:024w3 h32 > > > > > 0:024w3 h33 þ 0:148w3 h34 þ 0:118w3 h35 þ 0:104w4 h41 þ 0:22w4 h42 þ 0:1w4 h43 0:026w4 h44 0:04w4 h45 P a > > > > > k1 ; k1 ; k1 ; k2 ; k2 ; k2 ; k2 ; k3 ; k3 ; k3 P 0 > > 12 34 53 13 34 42 53 21 31 45 > > > > 1 P h11 P h12 P h13 P h14 P h15 P 0; 1 P h21 P h22 P h23 P h24 P h25 P 0 > > > > > 1 P h31 P h32 P h33 P h34 P h35 P 0; 1 P h41 P h42 P h43 P h44 P h45 P 0 > > : 0:15 6 w1 6 0:2; 0:16 6 w2 6 0:18; 0:3 6 w3 6 0:35; 0:3 6 w4 6 0:45; w1 þ w2 þ w3 þ w4 ¼ 1

Afterwards, the distances Spi ði ¼ 1; 2; 3; 4; 5; p ¼ 1; 2; 3Þ of the alternatives Ai (i = 1, 2, 3, 4, 5) to the HFPIS h+ can be derived by using the Eq. (9), and the results are listed in Table 11. Furthermore, the ranking orders of the alternatives Ai (i = 1, 2, 3, 4, 5) for each decision organization Dp(p = 1, 2, 3) can be obtained according to the increasing orders of the distances Spi ði ¼ 1; 2; . . . ; 5Þ, which are also listed in Table 11. Finally, the Borda’s scores of all alternatives Ai(i = 1, 2, 3, 4, 5) can be obtained by using the Borda’s function (Hwang and Yoon, 1981), and the results are listed in Table 12. Thus the ranking order of the alternatives (energy projects) Ai(i = 1, 2, 3, 4, 5) can be easily obtained as A2 A5 A1 A3 A4, and the best alternative is the energy project A4.

ments. Here, we make a comparison with intuitionistic fuzzy LINMAP (IF-LINMAP) method proposed in Ref. Li et al. (2010), which is the closest to the proposed approach. In order to compare with the IF-LINMAP method, we consider the HFEs’ envelopes, i.e., intuitionistic fuzzy data, and apply the IF-LINMAP method (Li et al., 2010) to solve the energy project selection problem under intuitionistic fuzzy environment. Because the envelope of the HFE h is the IFN Aenv(h), we can transform hesitant fuzzy data in the energy project selection problem into intuitionistic fuzzy data as depicted in Tables 13–15. Moreover, the IF-LINMAP method just considers pair-wise comparison between alternatives with the crisp degree of truth 0 or 1, ~ p ðo; rÞ in the above here we also assume that the interval numbers C

227

X. Zhang, Z. Xu / Computers & Industrial Engineering 75 (2014) 217–229 Table 6 ~1. Hesitant fuzzy decision matrix D

A1 A2 A3 A4 A5

C1

C2

C3

C4

{0.5,0.4,0.3,0.3,0.3} {0.5,0.3,0.3,0.3,0.3} {0.7,0.6,0.6,0.6,0.6} {0.8,0.7,0.4,0.3,0.3} {0.9,0.7,0.6,0.3,0.1}

{0.9,0.8,0.7,0.1,0.1} {0.9,0.7,0.6,0.5,0.2} {0.9,0.6,0.6,0.6,0.6} {0.7,0.4,0.2,0.2,0.2} {0.8,0.7,0.6,0.4,0.4}

{0.5,0.4,0.2,0.2,0.2} {0.8,0.6,0.5,0.1,0.1} {0.7,0.5,0.3,0.3,0.3} {0.8,0.1,0.1,0.1,0.1} {0.9,0.8,0.7,0.7,0.7}

{0.9,0.6,0.5,0.3,0.3} {0.7,0.4,0.3,0.3,0.3} {0.6,0.4,0.4,0.4,0.4} {0.9,0.8,0.6,0.6,0.6} {0.9,0.7,0.6,0.3,0.3}

C1

C2

C3

C4

{0.6,0.4,0.4,0.4,0.4} {0.5,0.4,0.3,0.3,0.3} {0.7,0.6,0.5,0.4,0.2} {0.8,0.7,0.3,0.3,0.3} {0.9,0.7,0.6,0.6,0.6}

{0.9,0.8,0.7,0.7,0.7} {0.9,0.7,0.6,0.5,0.1} {0.9,0.6,0.6,0.6,0.6} {0.7,0.4,0.2,0.2,0.2} {0.8,0.7,0.6,0.4,0.4}

{0.5,0.4,0.2,0.2,0.2} {0.8,0.6,0.5,0.4,0.4} {0.7,0.5,0.3,0.3,0.3} {0.8,0.6,0.4,0.4,0.4} {0.9,0.8,0.7,0.6,0.5}

{0.9,0.6,0.5,0.3,0.3} {0.7,0.4,0.3,0.3,0.3} {0.6,0.5,0.4,0.3,0.3} {0.9,0.8,0.6,0.5,0.3} {0.9,0.8,0.3,0.3,0.3}

C1

C2

C3

C4

{0.7,0.6,0.5,0.4,0.3} {0.5,0.3,0.3,0.3,0.3} {0.7,0.6,0.5,0.5,0.5} {0.8,0.7,0.4,0.3,0.3} {0.9,0.7,0.7,0.7,0.7}

{0.8,0.7,0.7,0.7,0.7} {0.9,0.7,0.6,0.5,0.5} {0.9,0.6,0.6,0.6,0.6} {0.7,0.4,0.2,0.2,0.2} {0.8,0.7,0.6,0.5,0.4}

{0.5,0.4,0.3,0.2,0.2} {0.8,0.6,0.5,0.5,0.5} {0.7,0.5,0.3,0.3,0.3} {0.8,0.7,0.2,0.1,0.1} {0.9,0.8,0.7,0.7,0.7}

{0.9,0.6,0.5,0.5,0.5} {0.7,0.6,0.5,0.4,0.4} {0.8,0.7,0.6,0.5,0.4} {0.9,0.8,0.6,0.6,0.6} {0.9,0.7,0.6,0.3,0.3}

Table 7 ~2. Hesitant fuzzy decision matrix D

A1 A2 A3 A4 A5

Table 8 ~3. Hesitant fuzzy decision matrix D

A1 A2 A3 A4 A5

Table 9 Distance between each alternative and HFPIS, and the ranking.

D1 D2 D3

Table 11 Distance between each alternative and HFPIS, and the ranking.

A1

A2

A3

A4

A5

Ranking

0.3250 0.2830 0.2472

0.3511 0.3395 0.2472

0.2905 0.3363 0.2287

0.2904 0.2963 0.2715

0.2904 0.2835 0.2411

A2 A1 A3 A4 A5 A2 A3 A4 A5 A1 A4 A1 A2 A5 A3

Table 10 Borda’s scores of alternatives with respect to each DM. Energy projects

A1 A2 A3 A4 A5

Decision organizations

D1 D2 D3

D2

D3

2 1 3 4 4

5 1 2 3 4

2 2 5 1 4

A2

A3

A4

A5

Ranking

0.2964 0.2638 0.2290

0.3634 0.3168 0.2290

0.2626 0.2935 0.2144

0.2624 0.2473 0.2691

0.2625 0.2731 0.2327

A2 A1 A3 A5 A4 A2 A3 A5 A1 A4 A4 A5 A1 A2 A3

Table 12 Borda’s scores of alternatives with respect to each DM. Borda’s scores

D1

A1

Energy projects

9 4 10 8 12

example reduces to the real number 0 or 1, i.e., the decision organizations Dp(p = 1, 2, 3) provide their comparison preference relations between alternatives as: X1 ¼ fðA1 ; A2 Þ; ðA3 ; A4 Þ; ðA5 ;A3 Þg; X2 ¼ fðA1 ;A3 Þ; ðA3 ; A4 Þ; ðA4 ; A2 Þ;ðA5 ; A3 Þg; X3 ¼ fðA2 ; A1 Þ; ðA3 ; A1 Þ; ðA4 ;A5 Þg;

where (A1,A2) in X1 means that the decision organization D1 prefers A1 to A2, and the others have the similar meanings. Then, we employ the IF-LINMAP method (Li et al., 2010) to construct the corresponding programming model and obtain its optimal solution as (where ea~ ¼ 0:1):

A1 A2 A3 A4 A5


Borda’s scores

E1

E2

E3

2 1 3 5 4

4 1 2 5 3

3 3 5 1 2

9 5 10 11 9

Using Eqs. (6), (61) and (62) in Ref. Li et al. (2010), the intuitionistic fuzzy positive ideal solution (IFPIS) A+ can be calculated as follows:

Aþ ¼ ðh1:0; 0:0i; h1:0; 0:0i; h0:8315; 0:1674i; h0:5296; 0:1312iÞ Therefore, the squares of the distances of the alternatives Ai (i = 1, 2, 3, 4, 5) from the IFPIS A+ can be calculated using Eq. (6) in Ref. Li et al. (2010) as: 1

1

1

1

1

d1 ¼ 0:0944; d2 ¼ 0:0786; d3 ¼ 0:0719; d4 ¼ 0:0604; d5 ¼ 0:0717; 2

2

2

2

2

d1 ¼ 0:086; d2 ¼ 0:05; d3 ¼ 0:086; d4 ¼ 0:0807; d5 ¼ 0:071;

and

x ¼ ð0:017; 0:013; 0:089; 0:881Þ; u ¼ ð0:017; 0:01; 0:074; 0:4682Þ;

v ¼ ð0; 0; 0:0149; 0:116Þ:

3

3

3

3

3

d1 ¼ 0:0761; d2 ¼ 0:0383; d3 ¼ 0:0661; d4 ¼ 0:0604; d5 ¼ 0:0604:

228


Table 13 ~ 1. Intuitionistic fuzzy decision matrix D

A1 A2 A3 A4 A5

Table 16 Borda’s scores of alternatives with respect to each DM.

C1

C2

C3

C4

(0.3,0.5) (0.3,0.5) (0.6,0.3) (0.3,0.2) (0.1,0.1)

(0.1,0.1) (0.2,0.1) (0.6,0.1) (0.2,0.3) (0.4,0.2)

(0.2,0.5) (0.1,0.2) (0.3,0.3) (0.1,0.2) (0.7,0.1)

(0.3,0.1) (0.4,0.3) (0.4,0.4) (0.6,0.1) (0.3,0.1)


A1 A2 A3 A4 A5

C1

C2

C3

C4

(0.4,0.4) (0.3,0.5) (0.2,0.3) (0.3,0.2) (0.6,0.1)

(0.7,0.1) (0.1,0.1) (0.6,0.1) (0.2,0.3) (0.4,0.2)

(0.2,0.5) (0.4,0.2) (0.3,0.3) (0.4,0.2) (0.5,0.1)

(0.3,0.1) (0.4,0.3) (0.3,0.4) (0.3,0.1) (0.3,0.1)


A1 A2 A3 A4 A5

C1

C2

C3

C4

(0.3,0.3) (0.3,0.5) (0.5,0.3) (0.3,0.2) (0.7,0.1)

(0.7,0.2) (0.5,0.1) (0.6,0.1) (0.2,0.3) (0.4,0.2)

(0.2,0.5) (0.5,0.2) (0.3,0.3) (0.1,0.2) (0.7,0.1)

(0.5,0.1) (0.4,0.3) (0.4,0.2) (0.6,0.1) (0.3,0.1)

Comparing these distances, the ranking orders of the alternatives Ai (i = 1, 2, 3, 4, 5) for the decision organizations DP(p = 1, 2, 3) are generated as:

A1 A2 A3 A5 A4 ;

A1 A3 A4 A5 A2 ; and A1 A3

A5 A4 A2 : Similarly, using the Borda’s function (Hwang and Yoon, 1981), the Borda’s scores of the alternatives Ai(i = 1, 2, 3, 4, 5) can be obtained as in Table 16: Then the ranking order of the alternatives (energy projects) Ai (i = 1, 2, 3, 4, 5) can be easily obtained from Table 16 as follows:

A1 A3 A5 A4 A2 : Thus the best alternative is the energy project A2. It is easy to see that the ranking order of the alternatives obtained by the IF-LINMAP (Li et al., 2010) is remarkably different from that obtained by the proposed approach. The main reason is that the proposed method considers the hesitant fuzzy information which is represented by several possible values, not by a margin of error (as in IFNs), while if adopting the IF-LINMAP method, it needs to transform HFEs into IFNs, which gives rise to a difference in the accuracy of data in the two types. Moreover, the proposed approach considered the situation that the set of all pair-wise comparison judgments from DMs are not crisp but interval numbers, while the IF-LINMAP method proposed in Ref. Li et al. (2010) only considered pair-wise comparison judgments between alternatives with the crisp degree of truth 0 or 1, which is one of the most disadvantages in multidimensional analysis of preference (Sadi-Nezhad and Akhtari, 2008). Thus it is not hard to see that our methodology has some desirable advantages over the IF-LINMAP approach (Li et al., 2010) as follows:

Energy projects

A1 A2 A3 A4 A5


Borda’s scores

E1

E2

E3

5 4 3 1 2

5 1 5 3 2

5 1 4 3 3

15 6 12 7 7

in MAGDM problems, does not need to transform HFEs into IFNs but directly handle such problems, and thus obtains better final decision results. (2) The proposed method can integrate more useful information into decision process by considering the situation that all pair-wise comparison judgments from DMs are interval numbers, and thus is capable of better modeling the real-life decision situations. In particular, when we meet some decision problems in which the decision data is represented by several possible values, the proposed approach demonstrates its great superiority in handling this sort of decision problems. 5. Conclusions LINMAP is a practical and useful approach for solving the reallife MAGDM problems, but it cannot be used to directly handle the MAGDM problems with fuzzy information. Considering the fact that the HFE, characterized by a membership function represented by a set of possible values, is a new effective tool to express human’s hesitancy in daily life, in this paper we have utilized the main structure of LINMAP to put forward an interval programming technique for solving MAGDM problems in which the ratings of alternatives provided by DMs are taken as HFEs, and all pair-wise comparison judgments are represented by interval numbers. The most important advantage of the proposed approach is its ability to sufficiently consider the DMs’ hesitancy in expressing their assessment information by using HFEs, and at the same time can take into account the uncertainty of preference information over alternatives by using interval numbers. Compared with the IF-LINMAP method (Li et al., 2010), the proposed technique integrates more useful information into the decision process, and does not need to transform HFEs into IFNs but directly deals with MAGDM problems with hesitant fuzzy information, and thus obtains better final decision results. In addition, the proposed method can be further extended to the MAGDM problems under interval-valued hesitant fuzzy environment, and can also be expected to be applicable to other similar decision making problems, such as performance evaluation, supply chain management, risk investment, and so on. Acknowledgements The authors are very grateful to the anonymous reviewers, the area editor Imed Kacem, and the Editor-in-Chief Mohamed I Dessouky for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China (No. 61273209), the Fundamental Research Funds for the Central Universities (No. CXZZ13_0139) and the Scientific Research Foundation of Graduate School of Southeast University (No. YBJJ1339). References

(1) The proposed method, by extending the LINMAP to take into account the hesitant fuzzy assessments which are wellsuited to handle the ambiguity and impreciseness inherent

Chen, N., Xu, Z. S., & Xia, M. M. (2013). Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Applied Mathematical Modelling, 37, 2197–2211.

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