Intuitionistic multiplicative analytic hierarchy process in group decision making

Accepted Manuscript Intuitionistic Multiplicative Analytic Hierarchy Process in Group Decision Making Peijia Ren, Zeshui Xu, Huchang Liao PII: DOI: Reference:

S0360-8352(16)30365-5 http://dx.doi.org/10.1016/j.cie.2016.09.025 CAIE 4481

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Please cite this article as: Ren, P., Xu, Z., Liao, H., Intuitionistic Multiplicative Analytic Hierarchy Process in Group Decision Making, Computers & Industrial Engineering (2016), doi: http://dx.doi.org/10.1016/j.cie.2016.09.025

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Intuitionistic Multiplicative Analytic Hierarchy Process in Group Decision Making Peijia Ren, Zeshui Xu*, Huchang Liao Business School, Sichuan University, Chengdu, Sichuan 610064, China

*

Corresponding

author.

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addresses:

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Intuitionistic Multiplicative Analytic Hierarchy Process in Group Decision Making ________________________________________________________________________________ Abstract Analytic Hierarchy Process (AHP), which analyzes complex decisions by organizing the problems into a multilayer hierarchic structure, is a simple yet popular decision technique used extensively in every decision field. But it is inadequate to handle the uncertain decision making problems. Taking advantage of intuitionistic multiplicative information in portraying the vagueness of problems with Saaty’s 1-9 scale, in this paper, we extend the intuitionistic multiplicative information into AHP to enhance the ability of AHP in tackling various decision making problems. We first verify that the intuitionistic multiplicative weighted geometric aggregation (IMWGA) operator has desirable characteristics to guarantee that the overall intuitionistic multiplicative preference relation (IMPR) is consistent when all individual IMPRs are consistent. Then, we provide a whole procedure of intuitionistic multiplicative analytic hierarchy process (IMAHP) for solving group decision making problems, including adjusting the individual IMPRs, aggregating the individual IMPRs and deriving the priorities from the overall IMPR. Finally, we present an example concerning the performance assessments of the hydropower stations to illustrate the effectiveness and applicability of the IMAHP. Keywords: Group decision making; Intuitionistic multiplicative information; Analytic hierarchy process; IMPR; IMAHP. ________________________________________________________________________________ 1

1. Introduction Decision theory, which aims to identify a desirable alternative according to the descriptive information of decision makers (DMs), has widespread use in every field of our modern life. Due to the incomplete information and the uncertainties of the problems, the DMs cannot accurately quantify the characteristics of alternatives. Till now, there are basically two types of preference relations which are used to express the DMs’ preferences on alternatives: fuzzy preference relations and multiplicative preference relations [1-4]. As for the multiplicative preference relations, Saaty [4] introduced the 1-9 scale to represent the characteristics of the objects. With this scale, the elements of the multiplicative preference relations are all in [1 9, 9] . However, Saaty’s 1-9 scale can only depict the affirmative preference information but ignores the negative and hesitative preference information over the objects. To overcome this drawback, Xia et al. [5] defined the intuitionistic multiplicative set (IMS), which includes a membership degree, a non-membership degree and a hesitancy degree, whose values vary between 1/9 and 9, to describe the DMs’ preferences more comprehensively. Analytic Hierarchy Process (AHP) introduced by Saaty [6,7], is one of the most significant methods to deal with the decision making problems. By organizing the objectives, the criteria (sub-criteria), and the alternatives into a multilayer hierarchical structure, AHP can analyze complex decision making problems efficiently. AHP obtains the priorities of each criterion and synthesizes the scores for each alternative on different criteria. The details of AHP involve the following steps: ⑴ Analyzing the problem and constructing the hierarchical structure; ⑵ Determining the multiplicative preference relations by pairwise comparisons of the criteria, and 2

provide the decision values of the alternatives with respect to each criterion; ⑶ Deriving the priorities of the criteria from the multiplicative preference relations; ⑷ Aggregating the comprehensive values for each alternative and ranking all alternatives. AHP is a simple yet popular decision making technique which has been widely used in the fields of business [8-11], industry [12-14], healthcare [15-17], and so on. Even though AHP has a strong ability in handling general decision making problems, it has also the limitations in applying the AHP to the uncertain decision making problems. The traditional AHP method, which is described by Saaty’s 1-9 scale, lacks the practicability in some cases due to the fact that the DMs may not provide the accurate numbers to represent their opinions with respect to the uncertainty and vagueness of the problems. To improve this situation, some scholars have combined the fuzzy set theory [18] and the AHP by introducing trapezoidal fuzzy numbers [19] and triangular fuzzy numbers [20] into the AHP. Later on, some studies have been conducted to derive the priorities [21-22] and developed the corresponding decision making methods [23-28] within the context of fuzzy AHP. The fuzzy AHP can only be used to solve the fuzzy decision making problems with the preference information of symmetrical distribution, however, we usually need to use the preference information with unbalanced distribution to deal with the decision making problems, just as the law of diminishing marginal utility in economics mentioned by Xia et al. [5]. Thus, in order to enhance the applicability of the AHP with Saaty’s 1-9 scale, it is urgent to do some work in extending the traditional AHP to the unbalanced distribution situation. As the IMS can describe the objects more comprehensively by providing the information of superiority, inferiority and hesitation, this paper aims to integrate the IMS into AHP so as to derive much more reasonable decision results in 3

practical decision making problems. Meanwhile, considering that a majority of decision processes require multiple stakeholders, the focus of our work is to tackle group decision making problems. Based on the above analysis, we organize the paper as follows: Section 2 reviews some fundamental knowledge about the intuitionistic multiplicative number (IMN) and the intuitionistic multiplicative preference relation (IMPR). Based on the consistency of the IMPR, Section 3 shows that the intuitionistic multiplicative weighted geometric aggregation (IMWGA) operator has desirable characteristics to guarantee that the overall IMPR is also consistent (or acceptably consistent) when all individual IMPRs are consistent (or acceptably consistent). Some properties of the IMWGA operator are also given in this section. Section 4 provides a whole procedure of IMAHP, including adjusting the individual IMPRs, aggregating the individual IMPRs and deriving the priorities from the overall IMPR. An example concerning the performance assessments of the hydropower stations is presented in Section 5 to illustrate the applicability of the IMAHP. Finally, some conclusions are listed in Section 6.

2. Preliminaries Here we review some elementary knowledge about intuitionistic multiplicative number (IMN) and intuitionistic multiplicative preference relation (IMPR).

2.1. IMNs As the basic component of IMPR, IMN [5] expressed as a = ( ra , s a ) is an effective tool to depict the superiority and the inferiority of an objective, where ra and s a , which both belong to

[1 9,9] ,

respectively indicate the membership degree and the non-membership degree. The 4

hesitation degree of a is defined by t a = 1 ( ras a ) . Some operational laws and the ranking method for IMNs can be given below: Definition 2.1 [5]. Let a = ( ra , s a ) , a1 = ( ra1 ,s a1 ) and a 2 = ( ra 2 , s a2 ) be three IMNs, and

l > 0 , then æ (1 + 2 r a1 )(1 + 2 r a2 ) - 1 2s a1s a 2 (1) a1 Å a 2 = ç , ç 2 (2 + s a1 )(2 + s a 2 ) - s a1s a2 è

ö ÷÷ ; ø

æ 2 ra1 r a2 (1 + 2s a1 )(1 + 2s a2 ) - 1 ö (2) a1 Ä a 2 = ç , ÷÷ ; ç (2 + ra )(2 + ra ) - ra ra 2 è 1 2 1 2 ø

æ (1 + 2ra )l - 1 ö 2s a l , (3) la = ç l l ÷; 2 (2 + s a ) - s a ø è æ 2ra l (1 + 2s a )l - 1 ö , (4) a = ç ÷. l l 2 è (2 + ra ) - ra ø l

Definition 2.2 [5]. The expressions s (a ) = ra s a and h (a ) = ra s a are respectively the score function and the accuracy function of an IMN a = ( ra , s a ) . Then for two IMNs a 1 and a 2 , we have: ⑴ If s (a1 ) > s (a 2 ) , then a 1 > a 2 ; ⑵ If s (a1 ) = s (a 2 ) , then (a) If h (a1 ) > h (a 2 ) , then a 1 > a 2 ; (b) If h (a1 ) = h (a 2 ) , then a 1 = a 2 .

The aggregation operators are useful to integrate IMNs in the decision making process. Here we 5

introduce a commonly used aggregation technique for IMNs, i.e., the intuitionistic multiplicative weighted operator. Moreover, it should be noted that the Å operation is associative, that is,

a Å ( b Å g ) = (a Å b ) Å g for any three IMNs a , b and g . Definition 2.3 [5]. Let a1 ,..., a n be a collection of IMNs, then an intuitionistic multiplicative weighted averaging (IMWA) operator is expressed as: n

IMWA (a1 , a 2 ,..., a n ) = Å (wia i ) i =1

(

æ n 1 + 2r Õ ai = ç i =1 ç 2 è where w = (w1 , w 2 ,..., w n )

T

)

wi

-1

2Õ i =1s awii n

,

Õ (2 + s ) - Õ wi

n

ai

i =1

is the weight vector of ai with wi Î [ 0,1] and

s wi i =1 ai n

å

n i= 1

ö ÷ ÷ ø

(2.1)

wi = 1 .

2.2. IMPRs IMPR is obtained by comparing objects in pairs to judge the preferred one, and the results of the pairwise comparisons are expressed by IMNs. More specifically, for a set of objects

( )

O={o1, o2 ,L, on} , Xia et al. [5] provided the IMPR A = aij

n´n

(

to compare one object with another,

which is measured by Saaty’s 1 9 - 9 scale, where aij = raij , s aij

)

is an IMN. raij and s a ij

respectively indicate the degree to which oi is preferred and not preferred to o j with the conditions raij = s aij , s aij = raij , raii = s aii = 1 , 0 < raij s aij £ 1 and 1 £ raij , s aij £ 9 . By the 9 expression t a = 1 ( ras a ) , the hesitancy degree that oi is preferred to o j is located in the interval [1, 81] . Generally, the consistency of IMPR is significant in the decision making process. One 6

definition about the consistency of IMPR was given as [29]:

(a )

Definition 2.4 [29]. Let A =

D = ( d ij )

n´n

ij n´n

(

)

( )

be an IMPR with aij = raij , s aij , C = cij

n´ n

and

be two MPRs, where ì ra ij , ï ï cij = í1, ï ïî1 s a ij ,

i< j i= j

and

i> j

ìs a ij , ï ï dij = í1, ï ïî1 raij ,

i< j i= j

(2.2)

i> j

then A is called to be consistent if the following formulas are satisfied:

cij = cik ckj

and

dij = dik d kj

(2.3)

for all i , j , k Î{1, 2,..., n} . For an inconsistent IMPR, its level of inconsistency is measured by the consistency index CI and the consistency ratio CR [6]: CI =

lmax - n and n -1

CR =

CI RI

(2.4)

where lmax is the largest eigenvalue of each MPR. The value of random index RI (defined by Saaty [6]) depends on the order of the MPR n (shown in Table 1).

Table 1. The values of RI

n RI

2

3

4

5

6

7

8

9

10

0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49

Generally, if CR < 0.1 , then the MPR is called to be acceptably consistent. 7

Definition 2.5 [29]. Let A , C and D be three IMPRs given in Definition 2.4. If C and D are acceptably consistent, then A is acceptably consistent.

3. Aggregation of the Consistent IMPRs with the IMWGA Operator This section tries to solve the problem that how to aggregate all individual IMPRs into an overall one by considering the consistency of the IMPRs in decision making process.

3.1. Description of the IMAHP Problems

IMAHP aims to obtain the priorities of multiple criteria and then rank all the alternatives from a finite set of the considered alternatives. In order to derive the priority weight of each criterion, the DMs need to compare the criteria in pairs. Considering the complexity and uncertainty of the decision making problem, we take the advantages of IMNs to express the comparison values by providing the membership degree, the non-membership degree and the hesitant degree, which reflects the DMs’ opinions about the criteria comprehensively. Suppose that a decision making problem has m alternatives xk for k = 1, 2,..., m and n criteria hi for i = 1, 2,..., n . There are s DMs et for t = 1, 2,..., s . The tth DM has weight g t for t = 1, 2,..., s . The weight vector is g = (g 1, g 2 ,..., g s )

å

s t= 1

T

with g t Î [ 0,1] for t = 1, 2,..., s and

g t = 1 . The DMs utilize the IMNs to express their results of pairwise comparisons. The DM

(

et provides his/her opinion as a ijt = r at , s at ij

ij

)

by comparing the criteria hi and h j , where rat ij

is the degree to which hi is prior to h j and s at ij is the degree to which hi is not prior to h j , with rat ij = s at ji , s at ij = rat ji , rat ii = s at ii = 1 , 0 < rat ij s at ij £ 1 and 1 £ rat ij , s at ij £ 9 . Thus, for tth 9 8

(

DM, his/her individual IMPR is At = (a ijt ) , where a ijt = rat ij , s at ij n ´n Obviously, in this problem, we have

s individual IMPRs

At = (a ijt )

)

n ´n

for all i, j = 1, 2,..., n . for t = 1, 2,..., s , which

can be shown as:

At = (aijt )

n´ n

h1 h = 2 M hn

h1

h2 L h n

æ a11t ç t ç a 21 ç M ç ç at è n1

a12t L a1tn

t a 22 L a 2t n M M M

t a nt 2 L a nn

ö ÷ ÷ , for t = 1, 2,..., s ÷ ÷ ÷ ø

(3.1)

After the DMs provide their individual IMPRs, we should check the consistency of all IMPRs, which is a significant property that all IMPRs must satisfy. The lack of consistency may lead to the incorrect priorities of criteria, which can impact the DMs’ judgments. If the individual IMPRs are inconsistent, we should adjust the inconsistent IMPRs by some rules or return the inconsistent IMPRs to the DMs to re-evaluate the IMPRs until they are all consistent. If all the individual IMPRs are consistent, then what we should notice is that how to aggregate all individual IMPRs into a collective one without losing the consistency property of the individual IMPRs. In what follows, we introduce an operator to aggregate the individual IMPRs into an overall one, which is also of the consistency property.

3.2. The Consistency of the Overall IMPR with the IMWGA operator

In Ref. [30], Xu proposed the intuitionistic fuzzy weighted geometric aggregation (IMWGA) operator to aggregate the individual IMPRs, which can be expressed as:

9

æ s IMWGAg (a1 , a 2 ,..., a s ) = ç Õ ra t g t , è t =1

(

where a t = ra t , s a t

)

s

Õs t =1

at

gt

ö ÷ ø

(3.2)

for t = 1, 2,..., s are a series of IMNs and g = (g 1 , g 2 ,..., g s )

T

is the weight

å

vector associated with the function IMWGA with g t Î [ 0,1] for t = 1, 2,..., s and

s t= 1

gt = 1.

Meanwhile, it was proven that the result derived by the IMWGA operator is also an IMN. Based on the IMWGA operator, we will discuss if it can maintain the consistency of IMPRs when we aggregate the individual IMPRs into a collective one.

Theorem 3.1. Suppose that the DMs et for t = 1, 2,..., s give their individual IMPRs as At = (a ijt )

n ´n

(

with a ijt = rat ij , s at ij

)

for t = 1, 2,..., s . The tth DM has weight g t for t = 1, 2,..., s .

The weight vector is g = (g 1, g 2 ,..., g s ) individual IMPRs A=

(a ) ij

n´ n

At =

(a )

t ij n ´n

T

with g t Î [ 0,1] for t = 1, 2,..., s and

for t = 1, 2,..., s

å

s t= 1

g t = 1 . If all

are consistent, then their overall IMPR

aggregated by the IMWGA operator is consistent.

Proof. By Definition 2.4, the individual IMPRs At = (a ijt ) for t = 1, 2,..., s can be divided into n ´n two kinds of MPRs C t = ( cijt ) for t = 1, 2,..., s and D t = ( d ijt ) for t = 1, 2,..., s . n ´n n´ n If the individual IMPRs are consistent, then cijt = cikt ckjt for t = 1, 2,..., s and dijt = dikt d kjt for

( )

t = 1, 2,..., s hold for all i, j, k Î {1, 2,..., n} , which means that the individual MPRs C t = cijt

( )

for t = 1, 2,..., s and Dt = d ijt

n´n

n´n

for t = 1, 2,..., s are all consistent. By using the weighted

( )

geometric mean (WGM) operator [31], the overall MPRs C = cij consistent, which can be expressed as:

10

n´n

( )

and D = d ij

n´n

are both

é ê ê ê ê1 C=ê ê ê ê ê ê1 ë

é ê ê ê ê1 D=ê ê ê ê ê ê1 ë

Õ(r )

...

1

...

s

1

t =1

Õ (s ) s

t =1

t =1

M

s

t a ni

t =1

Õ (r )

gt

Õ (s ) s

1

t an 2

t =1

Õ (s at 12 ) s

1

gt

Õ ( rat 2 t =1

)

M

L

1

...

Õ (s ) s

gt

t a1 n

t =1

1

...

t a

t =1

M

Õ ( rat n1 )

gt

Õ (r ) s

1

t =1

t an2

t =1

gt

ù ú ú ú gt ú ú ú ú ú ú ú û

gt

Õ (s ) s

gt

M s

n

M

t =1

s

t a

t =1

M

Õ (s )

gt

t a1 n

s

gt

t a2

ù ú ú ú gt ú ú ú ú ú ú ú û

Õ (r ) s

gt

t a12

M

M

L

1

n

According to Definition 2.4, these two consistent MPRs constitute an IMPR as: é (1,1) ê ê ê s s gt gt ö êæ t t r , s Õ Õ a a ç ÷ 21 21 ê t =1 ø G = ê è t =1 ê M ê ê s s gt gt ö t êæ r , s at n1 ÷ Õ Õ a n1 ç êëè t =1 t =1 ø

(

(

)

)

(

( )

æ s where a ij = ç Õ rat ij è t =1

(

gt

)

Õ (s ) t =1

(

)

s

,

æ s t ç Õ ra12 è t =1

t aij

gt

) , Õ (s ) s

gt

t =1

t a12

gt

ö æ s t ... ÷ ç Õ ra1n ø è t =1

(

(1, 1)

...

M æ s t ç Õ ra n 2 è t =1

(

) , Õ (s ) s

gt

t =1

t an 2

gt

æ s t ç Õ ra 2 n è t =1

(

) , Õ (s ) s

gt

t a1 n

t =1

)

gt

s

(

, Õ s at 2 n t =1

M

M

ö ÷ L ø

(1,1)

öù ÷ú øú ú g t öú ÷ú øú ú ú ú ú úû

gt

)

ö ÷. ø

On the other hand, by the IMWGA operator, the overall IMPR A = (a ij ) derived from n´ n individual IMPRs At = (a ijt ) for t = 1, 2,..., s can be aggregated as: n ´n

11

s

é (1,1) ê ê ê s s gt gt ö êæ t r , s at 21 ÷ Õ a 21 ê çè Õ t =1 ø A = ê t =1 ê M ê ê s s gt gt ö êæ rat n1 , Õ s at n1 ÷ Õ ç êëè t =1 t =1 ø

(

)

(

)

(

æ s t ç Õ ra12 è t =1

(

)

(

)

) , Õ (s ) s

gt

t =1

t a12

gt

(

) , Õ (s )

æ s t ç Õ ra 2 n è t =1

) , Õ (s )

(1,1)

...

M æ s t ç Õ ran 2 è t =1

(

) , Õ (s ) gt

s

t =1

t an2

gt

öù ÷ú øú ú gt ö ú ÷ú øú ú ú ú ú úû

ö æ s t ... ÷ ç Õ ra1 n ø è t =1

(

s

gt

t a1 n

t =1 s

gt

t a2 n

t =1

M

M

ö ÷ L ø

(1,1)

gt

which is the same as the matrix G . That is to say, the overall IMPR A = (a ij ) aggregated by the n´ n IMWGA operator can be divided into two MPRs C and D , which are both consistent. Then by Definition 2.4 in Ref. [29], A is consistent since the MPRs C and D are consistent, which completes the proof of Theorem 3.1. ■

The previous theorem guarantees the consistency of the overall IMPR when the individual IMPRs provided by the DMs are consistent. However, in some cases, it cannot ensure that all individual IMPRs are consistent in most cases. Therefore, the acceptable consistency is very important in handling the decision making process. Based on Definition 2.4, Jiang et al. [29]

( )

proposed that if MPRs C = cij

n´ n

( )

and D = d ij

n´n

are both acceptably consistent, then the

IMPR A = (aij )n´n is acceptably consistent. Furthermore, the relationship about the acceptable consistency between the individual IMPRs and the overall IMPR can be linked by the IMWGA operator. Theorem 3.2. Suppose that the DMs et for t = 1, 2,..., s give their individual IMPRs as

At = (a ijt )

n´n

(

with a ijt = rat ij , s at ij

)

for t = 1, 2,..., s . The tth DM has weight g t for t = 1, 2,..., s .

The weight vector is g = (g 1, g 2 ,..., g s )

( )

overall IMPR A = a ij

n´n

T

with g t Î [ 0,1] for t = 1, 2,..., s and

å

s t= 1

g t = 1 . The

aggregated by the IMWGA operator is acceptably consistent if all 12

individual IMPRs At = (a ijt ) for t = 1, 2,..., s are acceptably consistent. n ´n The proof of this theorem is trivial. ■ Theorem 3.1 and Theorem 3.2 manifest the desirable characteristics of the IMWGA operator in maintaining the consistency of the individual IMPRs. Moreover, we shall investigate some properties of the IMWGA operator in the following subsection.

3.3. The Properties of the IMWGA Operator Property 1. Let a t = ( rat , s a t ) for t = 1, 2,..., s be a collection of IMNs, and g = ( g 1 , g 2 ,..., g s )

T

be the associated weight vector, which satisfy g t Î [ 0,1] and

å

s t= 1

g t = 1 . If the IMNs

at = (rat ,sat ) for t = 1, 2,..., s are all equal, i.e., at = a = (r , s ) for t = 1, 2,..., s , then s æ s ö IMWGAg (a 1 , a 2 ,..., a s ) = ç Õ r ag tt , Õ s ag tt ÷ = ( r , s ) . t =1 è t =1 ø

(3.3)

Proof. This property is easy to be proven by

æ s IMWGAg (a1 , a 2 ,..., a s ) = ç Õ rag tt , è t =1

æ s gt gt ö s = Õ at ÷ çÕr , t =1 ø è t =1 s

s

Õs t =1

gt

ö ÷ ø

= ( r g1 r g 2 L r g s , s g1s g 2 Ls g s ) = ( r (g1 +g 2 +Lg s ) , s (g1 +g 2 +Lg s ) ) = ( r , s ) . ■ Property 2. Let at = (rat ,sat ) for t = 1, 2,..., s be a collection of IMNs, whose weights are

g = ( g 1 , g 2 ,..., g s )

T

with g t Î [ 0,1] and

å

s

g = 1 . Let l1 and l 2 be two real numbers. Then

t= 1 t

l1IMWGAg (a1 , a 2 ,..., a s ) Å l2 IMWGAg (a1 , a 2 ,...,a s ) = (l1 + l2 ) IMWGAg (a1, a 2 ,..., a s ) .

(3.4)

Proof. Notice that IMWGAg (a 1 , a 2 ,..., a s ) = ( r , s ) , where r and s are given in Eq. (3.2).

13

Thus, in order to prove this property, it is sufficient to prove l1 ( r , s ) Å l2 ( r , s ) = ( l1 + l2 ) ( r , s ) . In fact, by Definition 2.1, we have

l1 (r , s ) Å l2 (r , s ) æ (1 + 2 r )l1 - 1 2s l1 =ç , l ç 2 ( 2 + s ) 1 - s l1 è

ö æ (1 + 2 r )l2 - 1 2s l2 , ÷Åç l ÷ ç 2 ( 2 + s ) 2 - s l2 ø è

æ (1 + 2 r )l1 (1 + 2 r )l2 - 1 2s l1s l2 =ç , l l ç 2 ( 2 + s ) 1 ( 2 + s ) 2 - s l1s l2 è

ö ÷ ÷ ø

ö ÷ ÷ ø

æ (1 + 2 r )( l1 + l2 ) - 1 ö 2s ( l1 + l2 ) ÷ = ( l1 + l2 ) ( r , s ) . ■ =ç , ( l1 + l2 ) ( l1 +l2 ) ÷ ç 2 2 + s s ( ) è ø Property 3. Let at = (rat ,sat ) for t = 1, 2,..., s and bt = ( r bt , s bt ) for t = 1, 2,..., s be two collections of IMNs, whose weight vector is g = ( g 1 , g 2 ,..., g s )

T

with g t Î[ 0,1] and

å

s

g = 1,

t= 1 t

l is a real number. Then l IMWGAg (a 1 , a 2 ,..., a s ) Å l IMWGAg ( b 1 , b 2 ,..., b s ) = l éë IMWGAg (a 1 , a 2 ,..., a s ) Å IMWGAg ( b1 , b 2 ,..., b s )ùû .

Proof. Similar to the proof of Property 2, we need to prove

l ( ra , s a ) Å l ( rb , s b ) = l éë( ra , s a ) Å ( rb , s b )ùû . Since l ( ra , s a ) Å l ( r b , s b ) l ö æ (1 + 2ra )l - 1 ö æ (1 + 2r b ) - 1 2s b l 2s a l ç ÷ =ç , ÷ Å , l l l ÷ l ÷ ç ç 2 2 2 + s s ( ) 2 + s s ( b) b ø a a ø è è

æ (1 + 2 r )l (1 + 2 r )l - 1 2s a ls b l a b =ç , l l ç 2 ( 2 + s a ) ( 2 + s b ) - s a ls b l è 14

ö ÷ ÷ ø

(3.5)

and l éë( ra , s a ) Å ( r b , s b )ùû æ (1 + 2 ra ) (1 + 2 r b ) - 1 2s a s b =lç , ç 2 ( 2 + s a ) ( 2 + s b ) - s as b è

ö ÷ ÷ ø

l æ é(1 + 2 r ) 1 + 2 r ù l - 1 ö 2 (s a s b ) a ( b )û ë ç ÷ = , l l ÷ çç 2 é( 2 + s a ) ( 2 + s b ) ù - (s a s b ) ÷ ë û è ø

then from the above equations, we can easily obtain l ( ra ,sa ) Ål ( rb ,sb ) = l éë( ra ,sa ) Å( rb ,sb )ùû . ■ Property 4. Let at = (rat ,sat ) for t = 1, 2,..., s be a collection of IMNs, whose weight vector is

g = ( g 1 , g 2 ,..., g s )

T

with g t Î [ 0,1] and

å

s

g = 1 . l1 and l2 are two real numbers. Then

t= 1 t

l1

l2

éë IMWGAg (a1,a2 ,...,as ) ùû Ä éë IMWGAg (a1 ,a2 ,..., as )ùû = IMWGAg (a1 ,a2 ,...,as ) Proof. To prove this property, we need to verify the equality

( r ,s )

l1

Ä ( r ,s )

l2

(l1 +l2 )

.

(3.6)

= ( r ,s ) 1

l + l2

. In

fact,

( r ,s )

l1

Ä ( r ,s )

æ 2r l1 =ç , ç ( 2+r ) l1 - r l1 è

l2

(1 + 2s ) 2

l +l æ 2r ( 1 2 ) =ç , ç ( 2+r )( l1 +l2 ) - r ( l1 + l2 ) è

l1

-1 ö æ 2r l2 ÷ Äç , ÷ ç ( 2+r )l2 - r l2 ø è

(1 + 2s )(

l1 + l2 )

2

(1 + 2s )

l2

2

-1 ö ÷ ÷ ø

-1ö l +l ÷ = ( r,s ) 1 2 . ■ ÷ ø

Property 5. Let at = ( rat , s at ) for t = 1, 2,..., s and b t = ( r b t , s b t ) for t = 1, 2,..., s be two collections of IMNs, whose weight vector is g = ( g 1 , g 2 ,..., g s )

T

l is a real number. Then

15

with g t Î [ 0,1] and

å

s t= 1

g t = 1.

l

éë IMWGAg (a 1 , a 2 ,..., a s ) ùû Ä éë IMWGAg ( b 1 , b 2 ,..., b s )ùû

l

l

= éë IMWGAg (a 1 , a 2 ,..., a s ) Ä IMWGAg ( b1 , b 2 ,..., b s )ùû .

(3.7)

Proof. Similarly, this property holds if the following expression holds:

( ra , s a )

l

Ä ( r b , s b ) = éë( ra , s a ) Ä ( r b , s b )ùû . l

l

In fact,

( ra , s a ) Ä ( r b , s b ) = ç l

l

æ

2ra l

ç ( 2+r )l - r l a a è

,

(1 + 2s a )

l

2

2r b l - 1 ö æç ÷Ä , ÷ ç 2+r l - r l ) ø è( b b

(1 + 2s a ) (1 + 2s b )

æ 2ra l r b l =ç , ç ( 2+r )l ( 2+r )l - r l r l a b a b è

l

l

2

Property 6. (Monotonicity) Let a t = ( rat , s a t )

(1 + 2s ) b

2

T

-1 ö ÷ ÷ ø

-1 ö l ÷ = é( ra , s a ) Ä ( r b , s b ) ù . ■ ë û ÷ ø

for t = 1, 2,..., s

t = 1, 2,..., s be two collections of IMNs, g = ( g 1 , g 2 ,..., g s )

l

bt = ( r bt , s bt ) for

and

be the weight vector of IMNs with

g t Î [ 0,1] . If ra t £ r bt and s at ³ s bt for all t = 1, 2,..., s , then IMWGAg (a1 , a 2 ,..., a s ) £ IMWGAg ( b1 , b 2 ,..., b s ) .

(3.8)

Proof. Since s æ s IMWGAg (a1 , a 2 ,..., a s ) = ç Õ rag tt , Õs ag tt t =1 è t =1

and ra t £ r bt , s at ³ s bt for all t , then

s

ö æ s g t s gt ö IMWGA b , b ,..., b = , ( ) g 1 2 s ÷ ç Õ r b t , Õs b t ÷ t =1 ø è t =1 ø

Õr t =1

gt at

s

£ Õ r bg tt and t =1

s

Õs t= 1

gt at

s

³ Õ s bg tt . For the IMNs t= 1

æ s gt s gt ö æ s gt s gt ö r , s and ç Õ at Õ at ÷ ç Õ r bt , Õ s bt ÷ , according to the score function of IMNs (see Definition t =1 t= 1 è t =1 ø è t= 1 ø 2.2) [5], we can obtain 16

s æ s ö s s ç Õ rag tt , Õs ag tt ÷ = Õ rag tt t =1 è t =1 ø t =1

æ s gt s gt ö s gt gt s £ s Õ at ç Õ r bt , Õs bt ÷ = Õ r bt t =1 t =1 è t =1 ø t =1 s

s

Õs t =1

gt bt

Thus IMWGAg (a1 , a 2 ,..., a s ) £ IMWGAg ( b1 , b 2 ,..., b s ) . ■ Property 7. (Boundary) Let at = (rat ,sat ) for t = 1, 2,..., s be a collection of IMNs, a - =

( min ( r ) ,max (s )) at

at

(

( )

( )) .

and a + = max rat , min s a t

a t with g t Î [ 0,1] and

å

s t=1

g = ( g 1 , g 2 ,..., g s )

T

is the weight vector of

g t = 1 , then

a - £ IMWGAg (a1 ,a 2 ,..., a s ) £ a + .

(3.9)

Proof. In accordance with Property 1 and Property 6, it follows s æ s ö IMWGAg (a1 ,a 2 ,...,a s ) = ç Õ rag tt , Õ s ag tt ÷ t =1 è t =1 ø

æ s ³ ç Õ min rat è t =1

(

) , Õ ( max s ) s

gt

gt

at

t =1

( ( )

( )) = a

ö ÷ = min rat , max s at ø

-

s æ s ö IMWGAg (a1 ,a 2 ,...,a s ) = ç Õ rag tt , Õ s ag tt ÷ t =1 è t =1 ø

æ s £ ç Õ max rat è t =1

(

)

gt

Õ ( min s ) s

,

gt

at

t =1

(

( )) = a

ö ÷ = max rat , min s at ø

( )

+

.■

4. The Decision Making Process of IMAHP 4.1. The Adjustments of the Individual Inconsistent IMPRs

For the problems described as before, the individual IMPRs should be firstly provided by the DMs to compare the criteria in pairs. However, the IMPRs are rarely consistent, which may cause 17

unreasonable results. We shall first check the IMPRs’ consistency, which is a very significant property. However, the IMPRs cannot always be perfectly consistent, and thus, the acceptable consistency is proposed. That is to say, if the IMPRs are checked to be acceptably consistent, then the results derived from the IMPRs are considered to be reliable. This subsection aims to check the consistency of the individual IMPRs and adjust any inconsistent IMPR into a consistent one. The process can be shown as follows:

( )

Step 1. Divide each of the individual IMPRs At = a ijt

( )

MPRs C t = cijt

n´ n

( )

for t = 1, 2,..., s and D t = d ijt

n´ n

n´n

for t = 1, 2,..., s into two kinds of

for t = 1, 2,..., s in accordance with

(

Definition 2.4, where n is the order of the IMPR, and a ijt = rat ij , s at ij

)

for i, j = 1, 2,..., n ;

t = 1, 2,..., s are all IMNs. Step 2. Calculate the consistency index CI and the consistency ratio CR [6] for each of the

( )

MPRs C t = cijt

n´n

( )

for t = 1, 2,..., s and Dt = d ijt

n´n

for t = 1, 2,..., s by Eq. (2.4).

( )

Step 3. If CR £ 0.1 , then the MPR is acceptably consistent. If C t = cijt

n´n

( )

and Dt = d ijt

n´n

for t Î {1, 2,..., s} are both acceptably consistent, then terminate the checking process and utilize

( )

the correspondingly IMPR directly; if CR > 0.1 for either or both C t = cijt

n´n

( )

and D t = d ijt

n´ n

for t Î {1, 2,..., s} , then go to the next step. Step 4. Utilize the row geometric mean method (RGMM) [32] to obtain the weights of the

( )

inconsistent MPR C t = cijt

n´n

t Î {1, 2,..., s} , which are given by 1n

æ n ö t i = ç Õ cij ÷ è j =1 ø

1n

æ n ö ç Õ cij ÷ å i =1 è j =1 ø n

18

(4.1)

By the definition of the consistent MPR, we can obtain the corresponding perfect consistent

( )

MPR C% t = c%ijt

c%ijt =

n´n

( )

t Î {1, 2,..., s} for any inconsistent MPR C t = cijt

n´n

t Î {1, 2,..., s} , where

ti for i, j = 1, 2,..., n ; t Î {1, 2,..., s} . In the same way, we can get the perfect consistent MPR tj

( )

D% t = d%ijt

n ´n

( )

t Î {1, 2,..., s} . Thus, a perfect consistent IMPR A% t = a%ijt

( )

constituted by C% t = c%ijt

n´n

( )

and D% t = d%ijt

n´ n

n´n

( t Î {1, 2,..., s} ) can be

for t Î {1, 2,..., s} .

( )

Step 5. Calculate the differences between the elements of the original IMPR At = a ijt

( )

t Î {1, 2,..., s} and those of the perfect IMPR A% t = a%ijt

n´n

t Î {1, 2,..., s} by

Ltij = lg r ijt - lg r% ijt + lg s ijt - lg s% ijt

We need to find the maximum value of Ltij for i = 1,..., n - 1 ; corresponding element

(r

t i* j*

, s it* j* ) and

( r%

t i* j*

n´n

(4.2)

j = i + 1,..., n and the

, s% it* j* ) , and then let

rit* j* ( k ) = rit* j* ( k - 1) g r%it* j* and d it* j* ( k ) =

d it* j* ( k - 1) g d%it* j*

(4.3)

where k ( k ³ 1) is the number of adjustment, rit* j* ( 0 ) = rit* j* and d it* j* ( 0 ) = d it* j* . By Eq. (4.3),

rit* j* and s it* j* get closer gradually to r%it* j* and s% it* j* respectively. According to the upper triangle of the IMPR, we can obtain the lower triangle matrix correspondingly. Then a new IMPR

A ( k ) ( k ³ 1) can be constructed. Step 6. Check the consistency of the new IMPR A ( k ) through Step 2. By the rule in Step 3, if the new IMPR is acceptably consistent, then we can use it directly; otherwise, readjust the new IMPR by Step 5. Repeat the steps until the IMPR is acceptably consistent. Let t for 1, 2,..., s , then

( )

all inconsistent IMPRs can be adjusted to the acceptably consistent IMPRs At = aijt 19

n´n

for

t = 1, 2,..., s .

4.2. The Priority Method of the Overall IMPRs with Acceptable Consistency After all the individual IMPRs are checked or adjusted to be of acceptable consistency, we can use the IMWGA operator to aggregate all these IMPRs into a collective one. According to Theorem

( )

3.1 and Theorem 3.2, the overall IMPR A = aij

( )

case, how to obtain the priorities from A = aij

n´n

n´n

is of acceptable consistency as well. In this

is a crucial part in IMAHP. Below we propose

a method to calculate the priorities. Let w = ( w1 , w2 ,..., wn )

T

( )

be the priorities derived from the overall IMPR A = aij

(

value of wi w j must be able to reflect the element a ij = ra ij , s a ij

)

n´n

, then the

for all i, j = 1, 2,..., n . As

mentioned before, raij and s a ij indicate the degree to which hi is preferred and not preferred to

h j respectively. For raij s aij £ 1 , we know that the degree to hi being preferred to h j is at least raij , and the degree to hi being not preferred to h j is at least s a ij . According to the relationship between the interval and the IMN, we can get the degree to hi being preferred to h j is at least ri f and no more than 1 s i f , expressed as

[ rif ,1 s if ] . Thus, we have

n n é n ù a i f = Õ a ij = êÕ raij , Õ 1 s a ij ú , for all i = 1, 2,..., n j =1 j =1 ë j =1 û

The vector a i f = (a1f , a 2 f ,..., a n f )

T

(4.4)

indicates the degree to hi being preferred to other criteria

n é n ù with ai f = êÕ raij , Õ1 saij ú for all i = 1,2,..., n . Let j= 1 ë j= 1 û

[ wi , wi ]

for i = 1, 2,..., n be a sequence of

intervals, which are respectively equal to a i f for i = 1, 2,..., n . Then we calculate the possibility degree [33] by: 20

æ ö ïü wk - wi ïì p ( wi ³ wk ) = max í1 - max ç ,0 ÷ , 0 ý è wi - wi + wk - wk ø þï îï

(4.5)

where h k is the criterion in the problem. Let pik = p ( wi ³ wk ) , then a possibility degree matrix

P = ( pik )n´n can be constructed, where pik ³ 0 , pik + pki = 1 and pii = 0.5 . The priority vector of P is able to reflect the importance of the criteria, which can be got by [34]:

wi =

æ n 1 n ö ç å pik + - 1 ÷ n ( n - 1) è j =1 2 ø

(4.6)

Thus, the priorities of criteria in IMAHP are obtained as w = (w1 , w2 ,..., wn ) . T

4.3. Procedure of the IMAHP In the following, we conduct the whole process of IMAHP to solve the considered decision making problems. Step 1. Identify the alternatives xi for i = 1, 2,..., n and the criteria h j for j = 1, 2,..., m of the considered decision making problem, and then construct the hierarchical structure of the problem. There are s DMs, and the tth DM et has a weight g t for t = 1, 2,..., s . The weight vector is g = (g 1 , g 2 ,..., g s )

T

with g t Î [ 0,1] for t = 1, 2,..., s and

å

s t= 1

g t = 1 , are invited to make

pairwise comparisons of the criteria and provide a series of individual IMPRs (shown as Eq. (3.1)). Meanwhile, the DMs also give their assessments over the alternatives with respect to each criterion, and thus construct some decision matrices. Step 2. Check the consistency of each individual IMPRs by Eq. (2.4). If all IMPRs are acceptably consistent, then go to step 4; otherwise, go to the next step. 21

Step 3. Find the corresponding perfect consistent IMPRs of the inconsistent IMPRs and adjust these inconsistent IMPRs by Step 5 and Step 6 in Section 4.1 until they are all acceptably consistent. Step 4. Aggregate all acceptably consistent IMPRs into the collective one by Eq. (3.2). Step 5. Derive the priorities of the criteria by Eq. (4.4), Eq. (4.5) and Eq. (4.6). Step 6. Combine the evaluation values of the alternatives given by the DMs into an overall decision matrix by Eq. (2.1). Step 7. Aggregate the comprehensive values of each alternative in accordance with the priorities of the criteria and the overall decision values by Eq. (2.1), and then calculate the scores of each alternative and rank them. The bigger the score value, the better the alternative.

The whole process of the IMAHP can be briefly listed by the following diagram:

[INSERT FIGURE 1 HERE] Some discussions are provided to illustrate the methodology: (1) By the asymmetric 1 - 9 scale, the IMPRs are firstly constructed to comprehensively depict the DMs’ preferences from the superior and inferior aspects. The IMPRs have significance in practical decision making problems. In general, it is too ideal to let the DMs provide the consistent IMPRs, but yet the lack of consistency probably leads the unreasonable decision results. Thus, a technique is introduced to check the consistency of IMPRs, then an adjustment process is given to repair and improve the consistency of inconsistent IMPRs. 22

(2) For group decision making problems, it is natural to aggregate all individual IMPRs into an overall IMPR so as to get the comprehensive decision results. Moreover, based on the acceptably consistent IMPRs obtained before, the aggregation operator is required to guarantee that the overall IMPR is also acceptably consistent. We have proven that the IMWGA operator has this kind of desirable characteristics. (3) We give a method to derive the priorities of criteria from the overall IMPR. Later on, by combining the information from the decision matrices and the priorities of criteria, we can get the score of each alternative and rank alternatives.

5．Case Study China possesses rich water resources which has been gradually exploited since the founding of New China in 1949. To build a resource-conserving society, strengthen the management of water resources, and promote the construction of ecological civilization, the hydropower stations are very significant in the economic development of the future. A fact that can be observed is that the hydropower resource in China has great potentialities. The exploitation of the hydropower stations is greatly developed in our daily life, which plays an important role in improving the condition with water and electricity in production and life. What’s more, the construction and operation of the hydropower stations can promote the local economy. There are plenty of water resources in the southwest of China, which are the poorer areas of the country. Thus, constructing the hydropower stations and operating them well are the good ways to strengthen the economic development of these poverty-stricken areas. 23

Dadu River is the largest tributary of Minjiang River in the Yangtze River Basin, which is originated from southeast of Guoluo Mountain, Qinghai Province. The length of Dadu River is 1062 km, where its length in Sichuan Province is 852 km. Its natural fall is 4175 m, and the volume of its runoff is 4.7 ´1010 cubic meters per year. Due to the rich water resources in the Dadu River valley, a majority of hydropower stations are built in large-scale. However, lots of them are constructed in poor areas, such as Luding, Liangshan, Yaan, etc. Thus, assessing the performances of some large hydropower stations in these areas has great benefits in analyzing the economic structure and improving the economy of the poor areas. In this study, we select a part of hydropower stations constructed in 2001-2010 in Dadu River valley. According to the different location areas of the selected hydropower stations, we classify these hydropower stations into three parts: l

Luding hydropower stations (LDHS), which is located in Ganzi. Its hydropower capacity is 92 GW and the static investment is 8.663 ´1010 yuan. Its construction began in 2009.

l

Pubugou-Shenxigou hydropower stations and Longshitou hydropower stations (PLHS), which are located in Yaan. Their hydropower capacity is 460 GW and the static investment is 2.539 ´1011 yuan. The first part of the construction began in 2001, while the other part began in 2005.

l

Shawan hydropower stations (SWHS), which is located in Liangshan. The hydropower capacity is 60 GW and the static investment is 1.308 ´1010 yuan. Its construction began in 2005.

We assess the performances of these three hydropower stations in terms of the promotions on 24

local economy. More specifically, we invite the DMs to assess the performance of the selected hydropower stations by comparing the local economy before and after the constructions of them. In order to reflect the local economy visually and comprehensively, we choose the following criteria in the assessment process: (1) The effect on the local Gross Domestic Product per person; (2) The effect on the local industrial structure; (3) The effect on the local Engel coefficient of rural residents; (4) The effect on the ratio of the fiscal revenue to fiscal expenditure. Now, we interpret the above criteria. As we all know, when considering the economic development of an area, the Gross Domestic Product and population of this area are very important factors and cannot be ignored. Combining these two pivotal factors, we choose the Gross Domestic Product per person as the first criterion. In addition, the industrial structure of the area is also crucial. Usually, a developed area must have a reasonable industrial structure. The three-industry composition should be harmonious, which means the primary industry supplies the materials for the residents’ everyday living, the secondary industry leads to economic growth, and the tertiary industry goes well. Considering the areas, i.e., Luding, Yaan and Liangshan, are poor, the Engel coefficient of rural residents are significant to judge the living standard of the poor residents in these three areas. Moreover, the ratio of the fiscal revenue to fiscal expenditure is an effective index to reflect the fiscal capacity of an area. The lower the ratio is, the weaker the fiscal capacity is. According to these criteria, we collect the economic statistics from 2000-2013 of the areas Ganzi, Yaan and Liangshan, which are listed in Table 1.

25

[INSERT TABLE 1 HERE] Three DMs are invited to participate in the assessment process. The tth DM has a weight g t for t = 1, 2,3 , and the weight vector is g = (g 1 , g 2 , g 3 ) = ( 0.3, 0.4,0.3 ) . We denote the four criteria T

T

mentioned above as h k for k = 1, 2,3, 4 . For this uncertain assessment problem, the DMs cannot give their accurate assessment values directly. Moreover, real numbers fail to depict the complexity of the problem. In such a situation, IMNs are effective to handle this issue, which not only can describe the uncertainty of the problem, but also can provide more information on both the positive aspect and the negative aspect. Thus, combining the hydropower capacity, the static investment, the year of construction and the economic statistics given in Table 1, the DMs utilize IMNs to express pairwise comparisons of the four criteria and give the attribute values of the selected hydropower stations with respect to each criterion. Thus, we can obtain three individual IMPRs At = (a ijt ) 4 ´4 for t = 1, 2,3 and three individual decision matrices G t = ( g ijt ) for t = 1, 2,3 , which are shown 3´ 4 in the following:

The individual IMPR A1 é(1,1) ê ê(1/4， 3) A1 = ê ê ê( 3,1/5 ) ê êë( 2,1/3 )

The individual IMPR A2

2)ù ( 3,1/4 ) (1/5,3 ) (1/3，

(1,1)

(1/6,1)

(1,1/6)

(1,1)

( 3,1/3) (1/5,3)

é(1,1) ê ê(1， 1) A2 = ê ê ê( 2,1/5 ) ê êë( 2,1/3 )

ú (1/3,3 ) úú ú (3,1/5 ) ú ú (1,1) úû

26

(1,1)

2 )ù (1/5,2 ) (1/3，

(1,1)

(1/5,3 ) (1/5, 2 ) úú

( 3,1/5 ) (1,1)

ú ú

(3,1/4 ) ú

( 2,1/5) (1/4,3) (1,1)

ú úû

3

The decision matrix G

The individual IMPR A é(1,1) ê ê(1/3,3 ) A3 = ê ê ê( 2,1/5) ê êë(1,1/3)

1) ù ( 3,1/3) (1/5, 2 ) (1/3，

(1,1)

(1/5, 2 )

( 2,1/5) (1,1) ( 3,1/5 ) (1/5,3 )

The decision matrix G h1 é( 5,1 / 7 ) LDHS ê G 2 = PLHS ê( 3,1 / 3 ) ê SWHS ê ë( 5,1 / 7 )

h1

ú (1/5,3 )úú ú ( 3,1/5 )ú ú (1,1) úû

é( 5,1 / 7 ) LDHS ê G1 = PLHS ê( 3,1 / 5 ) ê SWHS ê ë( 6,1 / 7 )

2

h2

h2

h4

h1

( 5,1 / 8 ) (3,1 / 5 ) (1,1 / 2 ) ù ( 2,1 / 5 ) ( 5,1 / 6 ) ( 3,1 / 5 ) (3,1 / 4 )

é( 6,1 / 7 ) LDHS ê G 3 = PLHS ê( 5,1 / 6 ) ê SWHS ê ë( 5,1 / 6 )

ú ( 2,1 / 4 )úú ú ( 2,1 / 3 )û

h3

h4

( 6,1 / 8 ) (3,1 / 4 ) ( 2,1 / 3 ) ù ú

( 3,1 / 5 ) ( 5,1 / 5 ) (1,1 / 3)úú ( 3,1 / 5 ) ( 3,1 / 5 ) (1,1)

The decision matrix G h3

1

ú û

3

h2

h3

h4

( 6,1 / 6 ) (3,1 / 5 ) ( 2,1 / 3 )ù ú

( 5,1 / 8 ) (5,1 / 7 ) (1,1 / 2 ) úú ( 6,1 / 8 ) (3,1 / 5 ) (1,1)

ú û

The structure of this assessment process can be illustrated in Fig. 2: [INSERT FIGURE 2 HERE] Based on the information of the selected hydropower stations and the local economic statistics, we can handle this assessment problem by the IMAHP method.

(1)

Check the consistency of the individual IMPRs and adjust the inconsistent one(s) Here we illustrate the process of checking the IMPR A 1 given by the first DM as an example.

Firstly, by Definition 2.4, we can obtain the matrices C 1 and D 1 . Their consistency ratios are

CRC1 = 0.0603 and CRD1 = 0.4089 . These ratios indicate that the IMPR A 1 is inconsistent, and thus needs to be adjusted. By RGMM, the weight vectors of the matrices C1 and D1 are 27

t C1 = ( 0.1231,0.0679,0.5668,0.2422 )

T

and t D1 = ( 0.2497,0.4200,0.1147,0.2156 ) , respectively. Thus, T

we can get the corresponding perfect consistent IMPR A% 1 of A 1 : é (1,1) ê ê( 0.5946,1.8128 ) A% 1 = ê ê ê( 2.1779, 0.2171) ê êë(1.1583, 0.5081)

(1.8128, 0.5946 ) ( 0.2171, 2.1779 ) ( 0.5081,1.1583) ù (1,1)

( 0.1198, 3.6628 )

( 3.6628, 0.1198 )

(1,1)

(1.9480, 0.2803 ) ( 0.5318, 2.3403 )

ú 0.2803,1.9480 ( )úú . ú ( 2.3402, 0.5318) ú ú (1,1) úû

Then, we use Eq. (4.3) and Eq. (4.4) to adjust the IMPR A 1 . By using Matlab software package, we can get the acceptably consistent IMPR: é (1,1) ê ê( 0.3856， 2.3321) 1 A =ê ê ê( 3.0000,0.2000 ) ê êë(1.5220,0.4116 )

The matrices

( 2.3321, 0.3856 ) ( 0.2000,3.0000 ) ( 0.4116,1.5220 )ù (1,1)

( 0.1301, 2.6477 )

( 2.6477, 0.1301)

(1,1)

( 3.0000, 0.3333 ) ( 0.3261, 2.6497 )

ú ( 0.3333,3.0000 )úú . ( 2.6497, 0.3261) úú ú (1,1) úû

C 1 and D1 can be determined by the IMPR A 1 and their consistency ratios are

CRC1 = 0.0150 and CRD1 = 0.0708 , respectively. Similarly, we can get the acceptably consistent IMPRs for the other IMPRs: é (1,1) ê ê(1.0000， 1.0000 ) 2 A =ê ê ê( 2.0000,0.2000 ) ê êë( 2.0000,0.3333)

(1.0000,1.0000 ) ( 0.2000, 2.0000 ) ( 0.3333, 2.0000 )ù (1,1)

( 0.2000,3.0000 )

( 3.0000, 0.2000 )

(1,1)

( 2.0000, 0.2000 ) ( 0.2500, 3.0000 )

28

ú (0.2000, 2.0000 )úú , (3.0000,0.2500 )úú ú (1,1) úû

é (1,1) ( 2.4298, 0.8952 ) ê ê( 0.8952， 2.4298 ) (1,1) 3 A =ê ê ê( 2.0000, 0.2000 ) ( 2.9516, 0.1520 ) ê êë(1.0000,0.3333 ) ( 3.0000,0.2000 )

( 0.2000, 2.0000 ) ( 0.3333,1.0000 ) ù ( 0.1502,2.9516 ) (1,1) ( 0.2885, 2.4298 )

ú (0.2000, 3.0000 )úú . ( 2.4298,0.2885 ) úú ú (1,1) úû

Suppose that the matrices At for t = 1, 2,3 are accepted by the stakeholders. (2)

Aggregate the acceptably consistent IMPRs into an overall one We use the IMWAG operator to aggregate the three acceptably consistent IMPRs and obtain: é (1,1) ê ê( 0.7268， 1.6827 ) A=ê ê ê( 2.2587,0.3991) ê ëê(1.4967,0.3551)

(1.6827,0.7268 ) (0.2000,1.1320 ) ( 0.3551,1.4967 ) ù (1,1)

( 0.1613, 2.8756 )

( 2.8756,0.1619 )

(1,1)

( 2.5508, 0.2331) ( 0.2826,2.7132 )

ú ( 0.2331, 2.5508) úú . ú ( 2.7132,0.2826 )ú ú (1,1) ûú

Since all individual IMPRs are acceptably consistent, then by Theorem 3.2, the overall IMPR

A is also acceptably consistent. (3)

Derive the weights of each criterion By the overall IMPR A , we can get the interval degree to which hi is preferred to h j . Then

by Eq. (4.4), we can calculate the degrees to which hi is preferred to other criteria:

a 4 f = ( [ 0.1195, 0.8121] , [ 0.0273, 0.0810 ] , [17.6226, 54.7625] , [1.0789,5.0835] ) . T

Afterwards, the associated possibility degree matrix P can be calculated:

29

1 0.5347 0.7958 ù é 0.5 ê 0 0.5 0 0 úú ê . P= ê0.4626 1 0.5 1 ú ê ú 0 0.5 û ë0.2042 1 By Eq. (4.6), the priorities of the criteria are obtained as:

w = ( 0.3194, 0.1250, 0.3302, 0.2254 ) . T

(4)

Aggregate the individual matrices into an overall decision matrix We use the IMWA operator to aggregate the individual matrices into an overall decision matrix:

é( 5.7827, 0.0103 ) ê G = ê( 4.0083， 0.0204 ) ê ê ë( 5.7827, 0.0108 ) (5)

( 6.0799, 0.0208 ) (3.5000, 0.0222 ) ( 2.0380, 0.0405)ù 0.0293) (5.5000, 0.0152 ) ( 3.5035， 0.0193 ) ( 4.2143, 0.0156 ) (3.5000，

ú (1.8401, 0.0405) úú . (1.8401, 0.0782 ) úû

Calculate the comprehensive values of the hydropower stations and rank them. By the IMWA operator, the comparison results of the three hydropower stations are calculated

as shown in Table 2. [INSERT TABLE 2 HERE] The priorities of the criteria imply that the adjusting effect on local industrial structure is the most important factor to impact local economy. According to the economic statistics in Table 1, the secondary industry was greatly promoted in these three areas, and thus can quickly enhance the economic conditions. However, the tertiary industry in Yaan and Liangshan cannot catch up with the development of the secondary industry, which led to the industrial structure a little bit unreasonable. In addition, the local Gross Domestic Product per person index and the ratio of the fiscal revenue to 30

fiscal expenditure in these three areas grew steadily. As for the index of the local Engel coefficient of rural residents, Yaan performed best in reducing it, which means that it obtained the maximum promotion of the living standard for poor residents. According to the comprehensive assessments on these hydropower stations, the Luding hydropower station is of the best performance. The performance assessments on some hydropower stations are of strong practical significance in manifesting the improvement of local economy. For the assessment technique of the uncertain group decision making problem, after establishing the hierarchical structure of the assessment problem, we use the IMAHP to handle it. It provides an appropriate tool to depict the uncertain assessment problems.

6．Conclusions Although AHP is a simple yet popular decision technique used in every decision fields, it is also inadequate to tackle uncertain problems. Considering IMPR is an effective tool to portray the vagueness of the problems by Saaty’s 1-9 scale, this paper has developed the IMAHP in group decision making. The contributions of the paper can be summarized as follows: (1) The paper has verified that the IMWGA operator is of great characteristics in remaining the consistency of the IMPRs. (2) To adjust the inconsistent IMPR(s) into an acceptably consistent one(s), this paper has proposed an iterative process to repair the maximum inconformity gradually, which makes less changes of the DMs’ original opinions. (3) Based on the consistency guarantee of the overall IMPR, this paper has provided a brief 31

and intuitive method to derive the priorities of criteria. In addition, the paper has presented the performance assessments of the hydropower stations in the southwest of China to demonstrate the effectiveness and applicability of the IMAHP. Moreover, the priorities can be derived from the IMPR by various methods from different visual angles, which we will discuss in detail in further research.

Acknowledgements The work was supported by the National Natural Science Foundation of China (No. 71571123, No. 71501135), the Program for Changjiang Scholars at Sichuan University (No. YJ201430) and the Young scholars high level academic team construction project at Sichuan University (skgt201501).

References [1] S. A. Orlovsky, “Decision-making with a fuzzy preference relation,” Fuzzy Sets and Systems, vol. 1, pp. 155-167, 1978. [2] F. Herrera, E. Herrera-Viedma, F. Chiclana, “Multiperson decision-making based on multiplicative preference relations,” European Journal of Operational Research, vol. 129, pp. 372-385, 2001. [3] Z. S. Xu, “A survey of preference relations,” International Journal of General Systems, vol. 36, pp. 179-203, 2007. [4] T. L. Saaty, The Analytic Hierarchy Process, New York: McGraw-Hill, 1980. [5] M. M. Xia, Z. S. Xu, H. C. Liao, “Preference relations based on intuitionistic multiplicative information,” IEEE Transactions on Fuzzy Systems, vol. 21, pp. 113-133, 2013. [6] T. L. Saaty, “A scaling method for priorities in a hierarchical structure,” Journal of Mathematical Psychology, vol. 12, pp. 234-281, 1977. 32

[7] T. L. Saaty, “An exposition of the AHP in reply to the paper “Remarks on the analytic hierarchy process”,” Management Science, vol. 36, pp. 259-268, 1990. [8] A. Arbel, Y. E. Orgler, “An application of the AHP to bank strategic planning: The mergers and acquisitions process,” European Journal of Operational Research, vol. 48, pp. 27-37, 1990. [9] G. N. Angelou, A. A. Economides, “A compound real option and AHP methodology for evaluating ICT business alternatives,” Telematics and Informatics, vol. 26, pp. 353-374, 2009. [10] M. K. Chen, S. C. Wang, “The use of a hybrid fuzzy-Delphi-AHP approach to develop global business intelligence for information service firms,” Expert Systems with Applications, vol. 37, pp. 7349-7407, 2010. [11] H. Smyth, L. Lecoeuvre, “Differences in decision-making criteria towards the return on marketing investment: A project business perspective,” International Journal of Project Management, vol. 33, pp. 29-40, 2015. [12] C. L. Yang, S. P. Chuang, R. H. Huang, “Manufacturing evaluation system based on AHP/ANP approach for wafer fabricating industry,” Expert Systems with Applications, vol. 36, pp. 11369-11377, 2009. [13] M. K. Chen, S. C. Wang, “The critical factors of success for information service industry in developing international market: Using analytic hierarchy process (AHP) approach,” Expert Systems with Applications, vol. 37, pp. 694-704, 2010. [14] F. M. Al-Oqla, S. M. Sapuan, M. R. Ishak, A. A. Nuraini, “Predicting the potential of agro waste fibers for sustainable automotive industry using a decision making model,” Computers and Electronics in Agriculture, vol. 113, pp. 116-127, 2015. [15] M. Danner, M. J. Hummel, F. Volz, M. V. Jeannette G, B. Wiegard, C. M. Dintsios, H. Bastian, A. Gerber, M. J. IJzerman, “Integrating patients' views into health technology assessment: Analytic hierarchy process (AHP) as a method to elicit patient preferences,” International Journal of Technology Assessment in Health Care, vol. 27, pp. 369-375, 2011. [16] S. Ajami, S. Ketabi, “Applying fuzzy AHP to evaluate the sustainability of knowledge-based virtual communities in healthcare industry,” Journal of Medical Systems, vol. 36, pp. 1165-1171, 2012. [17] S. M. Case, J. O'Leary, N. Kim, M. Tinetti，T. Fried, “Older adults’ recognition of trade-offs in 33

healthcare decision making,” Journal of the American Geriatrics Society, vol. 63, pp. 1658-1662, 2015. [18] L. A. Zadeh, “Fuzzy sets,” Information Control, vol. 8, pp. 338-353, 1965. [19] J. J. Buckley, “Fuzzy hierarchical analysis,” Fuzzy Sets and Systems, vol. 17, pp. 233-247, 1985. [20] P. J. M. Van Laarhoven, W. Pedrycz, “A fuzzy extension of Saaty’s priority theory,” Fuzzy Sets and Systems, vol. 11, pp. 229-241, 1983. [21] C. G. E. Boender, J. G. de Graan, F. A. Lootsma, “Multicriteria decision analysis with fuzzy pairwise comparison,” Fuzzy Sets and Systems, vol. 29, pp. 133-143, 1989. [22] Y. C. Chou, C. C. Sun, H. Y. Yen, “Evaluating the criteria for human resource for science and technology (HRST) based on an integrated fuzzy AHP and fuzzy DEMATEL approach,” Applied Soft Computing, vol. 12, pp. 64-71, 2012. [23] K. J. Zhu, Y. Jing, D. Y. Chang, “A discussion on extent analysis method and applications of fuzzy AHP,” European Journal of Operational Research, vol. 116, pp. 450-456, 1999. [24] M. G. Gao, S. Y. Li, H. Y. Zhang, “Approaches to group decision making with incomplete information based on power geometric operators and triangular fuzzy AHP,” Expert Systems with Applications, vol. 42, pp. 7846-7857, 2015. [25] H. Zhang, A. Bouras, Y. Ouzrout, A. Sekhari, “Fuzzy multi-criteria lifecycle system maturity decision making based on an integrated Fuzzy AHP and VIKOR methodology,” Conference: Computational Science and Technology, pp. 1 – 6, 2014. [26] R. K. Jaiswal, N. C. Ghosh, A. K Lohani, T. Thomas, “Fuzzy AHP Based Multi Crteria Decision Support for Watershed Prioritization,” Water Resources Management, vol. 29, pp. 4205-4227, 2015. [27] G. Fan, “A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers,” Expert Systems with Applications, vol. 44, pp. 289-303, 2016. [28] S. H. Zyoud, L. G. Kaufmann, H. Shaheen, S. Samhan, D. Fuchs-Hanusch, “A framework for water loss management in developing countries under fuzzy environment: Integration of Fuzzy AHP with Fuzzy TOPSIS,” Expert Systems with Applications, vol. 61, pp. 86-105, 2016. 34

[29] Y. Jiang, Z. S. Xu, X. H. Yu, “Group decision making based on incomplete intuitionistic multiplicative preference relations,” Information Science, vol. 295, pp. 33-52, 2015. [30] Z. S. Xu, “Priority weight intervals derived from intuitionistic multiplicative preference relations,” IEEE Transactions on Fuzzy Systems, vol. 21, pp. 642-654, 2013. [31] J. Aczél，C. Alsina, “On synthesis of judgments,” Socio-Economic Planning Sciences, vol. 20, pp. 333-339, 1986. [32] J. Aguarón, J. M. M. Jiménez, “The geometric consistency index: Approximated thresholds,” European Journal of Operational Research, vol. 147, pp. 137-145, 2003. [33] Z. S. Xu, J. Chen, “Some models for deriving the priority weights from interval fuzzy preference relations,” European Journal of Operational Research, vol. 184, pp. 266-280, 2008. [34] Z. S. Xu, Uncertain Multi-Attribute Decision Making: Methods and Applications, Berlin: Springer-Verlag, 2015.

35

Fig 1. Procedure of the IMAHP

1

Fig 2. The performance assessments of the hydropower stations

2

Table 1. The economic data from 2000-2013 of the areas Ganzi, Yaan and Liangshan The Gross

The industrial structure

The Engel

The ratio of the

Domestic

coefficient of

fiscal revenue to

Product per

rural residents

fiscal expenditure

person (Yuan) Ganzi 2000

2797

7.27: 6.98: 10.43

0.7083

0.0712

2001

3092

7.69: 7.69: 12.19

0.6421

0.0555

2002

3492

8.50: 9.81: 13.02

0.6264

0.0650

2003

3904

9.20: 11.75: 14.28

0.7553

0.0629

2004

4615

10.45: 15.46: 16.19

0.7435

0.0617

2005

5439

11.08: 16.70: 22.27

0.7548

0.0783

2006

6469

12.80: 22.31: 25.06

0.7115

0.1054

2007

8204

20.18: 29.01: 29.16

0.6908

0.1295

2008

9640

23.66: 34.64: 35.71

0.6879

0.1492

2009

10324

25.44: 34.96: 42.75

0.6629

0.1268

2010

11659

28.77: 44.92: 49.14

0.6516

0.1235

2011

13889

37.70: 57.72: 56.80

0.6406

0.1196

2012

15753

43.09: 68.12: 63.81

0.6163

0.0979

2013

17809

47.32: 80.78: 73.12

0.6037

0.0804

2000

4949

17.82: 35.42: 20.7

0.5637

0.3503

2001

5386

19: 39.03: 23.02

0.5871

0.2860

2002

5895

21.32: 42.63: 25.57

0.5437

0.2055

2003

6564

21.84: 49.98: 28.53

0.5317

0.2430

2004

8014

26.75: 61.48: 33.36

0.5142

0.2090

2005

8310

28.35: 56.86: 41.26

0.5083

0.2408

2006

9680

34.30: 66.42: 46.99

0.5315

0.2356

2007

11473

39.26: 83.62: 53.47

0.5313

0.2598

2008

14051

46.08: 107.02: 60.13

0.4961

0.1295

2009

15710

44.80: 124.50: 70.31

0.4503

0.1329

2010

18881

49.97: 157.83: 78.74

0.4588

0.2020

2011

23153

56.87: 200.38: 92.88

0.4750

0.2672

2012

26157

60.39: 233.56: 104.10

0.4620

0.3248

2013

27317

63.25: 240.23: 114.49

0.3714

0.0879

2000

3630

56.44: 41.96: 46.15

0.6897

0.3525

2001

3969

59.74: 47.82: 52.84

0.6933

0.2597

2002

4360

62.67: 56.33: 58.98

0.6599

0.2340

2003

4900

66.63: 68.27: 67.45

0.6727

0.2516

2004

5803

83.36: 89: 78.19

0.6886

0.2178

2005

6934

92.07: 108.17: 99.98

0.6064

0.2462

2006

8255

91.63: 128.07: 107.06

0.6077

0.2875

Yaan

Liangshan

2007

10048

13.03: 18.64: 13.38

0.5995

0.3127

2008

12896

157.63: 246.12: 157.33

0.6206

0.3191

2009

14306

157.46: 256.08: 213.57

0.5875

0.3050

2010

17560

172.06: 371.05: 241.08

0.5629

0.3288

2011

22044

194.56: 523.57: 282.00

0.5689

0.3200

2012

24668

218.80: 587.87: 316.00

0.5448

0.3330

2013

26556

233.89: 642.21: 338.30

0.5626

0.3304

Table 2. The comparison results of three hydropower stations Hydropower stations

LDHS

PLHS

SWHS

Comprehensive values The score values Ranking orders

(3.9379,0.0197) 199.8185 1

(3.7108,0.0226) 164.4542 3

(3.6795,0.0213) 172.6762 2

Grapgical Abstract

Highlights

1、 Extends the intuitionistic multiplicative information into AHP. 2、 Proposes an operator to guarantee the consistency of the overall IMPR. 3、 Provides an adjustment process to repair and improves the consistency of inconsistent IMPR.

1