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Preference Relations Based on Intuitionistic Multiplicative Information Meimei Xia, Zeshui Xu, Senior Member, IEEE, and Huchang Liao
Abstract—Preference relations are powerful techniques to express the preferences over alternatives (or criteria) and mainly fall into two categories: fuzzy preference relations (also called reciprocal preference relations) and multiplicative preference relations. For a pair of alternatives, a fuzzy preference relation only gives the degree that an alternative is prior to another; thus, the intuitionistic fuzzy preference relation is introduced by adding the degree that an alternative is not prior to another, which can describe the preferences over two alternatives more comprehensively. However, the intuitionistic fuzzy preference uses the symmetrical scale to express the decision makers’ preference relations, which are inconsistent with our intuition in some situations. If we use the unsymmetrical scale to express the preferences about two alternatives instead of the symmetrical scale in intuitionistic fuzzy preference relation, then a new concept is introduced, which we call the intuitionistic multiplicative preference relation reflecting our intuition more objectively. In this paper, we study the aggregation of intuitionistic multiplicative preference information, propose some aggregation techniques, investigate their properties, and apply them to decision making based on intuitionistic multiplicative preference relations.
TABLE I COMPARISON BETWEEN THE 0.1–0.9 SCALE AND THE 1–9 SCALE
Index Terms—Aggregation operator, decision making, intuitionistic fuzzy set, preference relation.
I. INTRODUCTION N a decision making problem, to give a ranking of n alternatives xi (i = 1, 2, . . . , n), preference relations are the most common techniques, which can be mainly classified into two forms: fuzzy preference relations [1] (also called reciprocal preference relations [2]) and multiplicative preference relations [3]. A fuzzy preference relation is defined as R = (rij )n ×n with the conditions rij + rj i = 1 and 0 ≤ rij ≤ 1, where rij indicates the degree that the alternative xi is preferred to xj . Concretely speaking, the case rij = 0.5 indicates that there is indifference between the alternatives xi and xj ; rij > 0.5 indicates that the alternative xi is preferred to xj ; especially, rij = 1 means
I
Manuscript received August 3, 2011; revised December 1, 2011 and March 1, 2012; accepted May 1, 2012. Date of publication June 5, 2012; date of current version January 30, 2013. The work was supported in part by the National Natural Science Foundation of China (Nos. 71071161 and 61273209) and the China Postdoctoral Science Foundation under Grant 2012M520311. M. Xia is with the School of Economics and Management, Tsinghua University, Beijing 100084, China (e-mail:
[email protected]). Z. Xu is with the Institute of Sciences, PLA University of Science and Technology, Nanjing 210007, China (e-mail:
[email protected]). H. Liao is with the Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TFUZZ.2012.2202907
that the alternative xi is absolutely preferred to xj ; rij < 0.5 indicates that the alternative xj is preferred to xi ; especially, rij = 0 means that the alternative xj is absolutely preferred to xi . It is noted that each element rij in a fuzzy preference relation R is expressed by using the 0.1–0.9 scale which assumes that the grades between “extremely not preferred” and “extremely preferred” are distributed uniformly and symmetrically (see Table I), but in real life, there exist the problems that need to assess their variables with the grades that are not uniformly and symmetrically distributed [4]–[10]. The unbalanced distribution may appear due to the nature of attributes of the problem; the law of diminishing marginal utility in economics is a good example. To invest the same resources to a company with bad performance and to a company with good performance, the former enhances more quickly than the latter. In other words, the gap between the grades expressing bad information should be smaller than the one between the grades expressing good information. Saaty’s 1–9 scale [3] (see Table I) is a useful tool to deal with such a situation, especially in expressing a multiplicative preference relation S = (sij )n ×n with the conditions sij sj i = 1 and 1/9 ≤ sij ≤ 9, where sij indicates the degree that the alternative xi is preferred to xj . Concretely speaking, sij = 1 indicates that there is indifference between the
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alternatives xi and xj ; sij > 1 indicates that the alternative xi is preferred to xj ; especially; sij = 9 means that the alternative xi is absolutely preferred to xj ; sij < 1 indicates that the alternative xj is preferred to xi ; especially, sij = 1/9 means that the alternative xj is absolutely preferred to xi . It is noted that the element sij in the multiplicative preference relation S is not symmetrically distributed around 1, which can express the decision makers’ preferences more objectively. In the past decades, a lot of research work has been done about fuzzy preference relations and multiplicative preference relations, such as the methods to obtain the priorities of preference relations [11]–[15], the consistency of a preference relation [16]–[22], the consensus of a group of preference relations [23]–[30], and so on. Due to the complexity and uncertainty involved in the real-world decision problems and incomplete information or knowledge, it is difficult to provide a precise preference relation. In these situations, the decision makers could be more comfortable providing interval values to express their preferences. An interval-valued fuzzy preference relation is given ˜ = (˜bij )n ×n , whose element ˜bij = [b− , b+ ] denotes the as B ij ij interval-valued preference degree or the intensity of the alterna+ − − tive xi over xj , and satisfies b+ ij + bj i = bij + bj i = 1 and 0 ≤ + b− ij , bij ≤ 1. Due to the fact that an interval-valued fuzzy num+ ber ˜bij = [b− ij , bij ] is equivalent to an intuitionistic fuzzy num+ ber βij = (μβ i j , vβ i j ) = (b− ij , 1 − bij ), then the interval fuzzy ˜ = (˜bij )n ×n can be equivalent to an intupreference relation B itionistic fuzzy preference relation [31] B = (βij )n ×n , where βij = (μβ i j , vβ i j ), μβ i j indicates the degree to which the alternative xi is preferred to xj , and vβ i j indicates the degree to which the alternative xi is not preferred to xj and both of them satisfy the conditions μβ i j = vβ j i , vβ i j = μβ j i , 0 ≤ μβ i j + vβ i j ≤ 1, and 0 ≤ μβ i j , vβ i j ≤ 1. It is noted that the intuitionistic fuzzy number βij = (μβ i j , vβ i j ) can describe the preference relation about the alternatives xi and xj more comprehensively than the + interval number [b− ij , bij ], because the intuitionistic fuzzy number is constructed by two functions: the membership function and the nonmembership function, while the interval number only provides a variation of the membership degree. On the other hand, an interval-valued multiplicative preference rela+ ˜ij = [a− tion is given as A˜ = (˜ aij )n ×n , whose element a ij , aij ] denotes the interval-valued preference degree or intensity of − − + the alternative xi over xj , satisfying a+ ij aj i = aij aj i = 1 and + 1/9 ≤ a− ij ≤ aij ≤ 9. Similarly, the interval-valued fuzzy num− ber a ˜ij = [aij , a+ ij ] can be written as αij = (ρα i j , σα i j ) = + − (aij , 1 aij ), which we call an intuitionistic multiplicative number (IMN), then A˜ = (˜ aij )n ×n can be written as A = (αij )n ×n which we call an intuitionistic multiplicative preference relation, where αij = (ρα i j , σα i j ), and ρα i j indicates the degree to which the alternative xi is preferred to xj , σα i j indicates the degree to which the alternative xi is not preferred to xj and both of them should satisfy the conditions ρα i j = σα j i , σα i j = ρα j i , ρα i j σα i j ≤ 1, and 1/9 ≤ ρα i j , σα i j ≤ 9. It is noted that the intuitionistic multiplicative number (IMN) αij = (ρα i j , σα i j ) can describe the preference information about the alterna-
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tives xi and xj better than the interval-valued fuzzy number + a ˜ij = [a− ij , aij ], for containing two parts: the membership function and the nonmembership function. More importantly, we can also find that the intuitionistic multiplicative preference relation is based on the unbalanced scale, the Saaty’s 1–9 scale, which is more reasonable and comprehensive and has many advantages in some situations. Therefore, it is sometimes more useful than the intuitionistic fuzzy preference relation. Up to now, lots of work has been done about interval fuzzy preference relations [32]–[34], interval multiplicative preference relations [35]–[39], and the intutionistic fuzzy preference relation [31], [40], [41]. However, nothing has been done about the intuitionistic multiplicative preference relation. In this paper, we introduce this new type of preference relation, give some basic operational laws, investigate its properties, and propose some operators to aggregate intuitionistic multiplicative information. Then, we apply these results to decision making based on intuitionistic multiplicative preference relations.
II. INTUITIONISTIC MULTIPLICATIVE PREFERENCE RELATIONS In this section, we first introduce the concepts of intuitionistic fuzzy set and intuitionistic fuzzy preference relation, and based on this, we define the concepts of intuitionistic multiplicative set (IMS) and intuitionistic multiplicative preference relation. Let X = {x1 , x2 , . . . , xn } be a fixed set, a fuzzy set [42] defined on X can be given as C = {(xi , μC (xi ))|xi ∈ X}, where the value μC (xi )(0 ≤ μC (xi ) ≤ 1) is the membership degree of the element xi to X in C. It is noted that the fuzzy set only contains the membership information. To describe the uncertainty and vagueness in more detail, Atanassov [43] introduced the intuitionistic fuzzy set containing the membership, nonmembership, which can be given as B = {(xi , μB (xi ), vB (xi ))|xi ∈ X}, where the values μB (xi ) and vB (xi ) denote, respectively, the membership degree and the nonmembership degree of the element xi to X in B, with the conditions 0 ≤ μB (xi ), vB (xi ) ≤ 1, and μB (xi ) + vB (xi ) ≤ 1. For convenience, the pair (μB (xi ), vB (xi )) is called an intuitionistic fuzzy number (IFN) [44]. Supposing that there are n alternatives xi (i = 1, 2, . . . , n), the decision maker provides his/her preference over the alternatives xi and xj , denoted by IFN βij = (μβ i j , vβ i j ), where μβ i j indicates the intensity that the alternative xi is prior to xj ; vβ i j indicates the intensity that the alternative xi is not prior to xj , with the conditions μβ i j = vβ j i , vβ i j = μβ j i , 0 ≤ μβ i j + vβ i j ≤ 1, and 0 ≤ μβ i j , vβ i j ≤ 1. All the IFNs βij (i, j = 1, 2, . . . , n) construct the intuitionistic fuzzy preference relation [31] B = (βij )n ×n . For a collection of IFNs, Xu [44] proposed some aggregation operators, the basic one of which is the intuitionistic fuzzy weighted averaging (IFWA) operator:
XIA et al.: PREFERENCE RELATIONS BASED ON INTUITIONISTIC MULTIPLICATIVE INFORMATION
Definition 1: Let βi (i = 1, 2, . . . , n) be a collection of IFNs, an IFWA operator is defined as n n IFWA (β1 , β2 , . . . , βn ) = 1 − (1 − μβ i )w i , vβwii i=1
i=1
(1) where w = (w1 , w2 , . . . , wn )T isthe weight vector of βi (i = 1, 2, . . . , n) with wi ∈ [0, 1] and ni=1 wi = 1. Based on the IFWA operator, other aggregation operators have been developed. However, the IFWA operator has a disadvantage discussed as in the following example. Example 1: Let β1 = (1, 0), β2 = (0, 1), β3 = (0, 1), β4 = (0, 1), β5 = (0, 1), and β6 = (0, 1) be six IFNs, and let w = (1/6, 1/6, 1/6, 1/6, 1/6, 1/6)T be their weight vector, then by the IFWA operator, we have IFWA (β1 , β2 , . . . , β6 ) = (1, 0) which is somewhat inconsistent with our intuition in this special case. In what follows, we try to solve this issue by using other scales to express the intuitionistic fuzzy information. As discussed in the introduction, if the two parts of an intuitionistic fuzzy set, the membership function and the nonmembership function, are expressed by using the 1–9 scale instead of the 0.1–0.9 scale, a new concept will be given as follows. Definition 2: Let X be a fixed set, an IMS is defined as D = {< x, ρD (x), σD (x) > | x ∈ X}
(2)
which assigns to each element x a membership information ρD (x) and a nonmembership information σD (x), with the conditions 1/9 ≤ ρD (x), σD (x) ≤ 9, ρD (x) σD (x) ≤ 1, ∀x ∈ X. (3) For convenience, let the pair (ρD (x), σD (x)) be an IMN and M be the set of all IMNs. Definition 3: Let X = {x1 , x2 , . . . , xn } be n alternatives, then an intuitionistic multiplicative preference relation is expressed as A = (αij )n ×n , where αij = (ρα i j , σα i j ) is an IMN, and ρα i j indicates the degree to which the alternative xi is preferred to xj , σα i j indicates the degree to which the alternative xi is not preferred to xj and both of them should satisfy the conditions ρα i j = σα j i , σα i j = ρα j i , ρα i j σα i j ≤ 1, and 1/9 ≤ ρα i j , σα i j ≤ 9. It is noted that the fundamental elements of an intuitionistic multiplicative preference relation are the IMNs. To obtain the priority of an intuitionistic multiplicative preference relation, we have to aggregate the intuitionistic multiplicative preference information (or the IMNs) for each alternative and compare them. Therefore, in the following, we will mainly focus on the operation, comparison, and aggregation of the IMNs. From Definition 3, we can find that the bigger the degree to which the alternative xi is preferred to the alternative xj , i.e., the bigger the value of ρα i j , and the smaller the degree to which the alternative xi is not preferred to xj , i.e., the smaller
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the value of σα i j , then the higher priority of the alternative xi than that of xj . In the case where ρα i j and σα i j increase, we will know more and more information about the preference information about the alternatives xi and xj , especially, if the product of ρα i j and σα i j is equivalent to 1, then we can know all the preference information. In such a case, we assume that the more information we know, then the higher priority that the alternative xi has than that xj . Based on the previous analysis, to rank any two IMNs, we define the following comparison laws. Definition 4: For an IMN α = (ρα , σα ), we call s(α) = ρα /σα the score function of α, and h(α) = ρα σα the accuracy function of α. To compare two IMNs α1 and α2 , the following laws can be given: 1) If s(α1 ) > s(α2 ), then α1 > α2 ; 2) If s(α1 ) = s(α2 ), then If h(α1 ) > h(α2 ), then α1 > α2 ; If h(α1 ) = h(α2 ), then α1 = α2 . Definition 5: Let α1 and α2 be two IMNs, we denote the partial order as α1 ≥P α2 if and only if ρα 1 ≥ ρα 2 and σα 1 ≤ σα 2 . Especially, α1 = α2 if and only if ρα 1 = ρα 2 and σα 1 = σα 2 . The top and bottom elements are 9P = (9, 1/9) and 1/9P = (1/9, 9), respectively. According to Definition 5, if α1 ≥P α2 , then ρα 1 ≥ ρα 2 and σα 1 ≤ σα 2 , which indicates that s(α1 ) ≥ s(α2 ). If s(α1 ) > s(α2 ), then α1 > α2 ; if s(α1 ) = s(α2 ), then ρα 1 /σα 1 = ρα 2 /σα 2 , also since ρα 1 ≥ ρα 2 and σα 1 ≤ σα 2 , then ρα 1 = ρα 2 and σα 1 = σα 2 , which indicates that α1 = α2 . That is to say, if α1 ≥P α2 , then we have α1 ≥ α2 . Moreover, some operational laws can be defined as follows. Definition 6: Let α, α1 , and α2 be three IMNs, and λ > 0, then 1) αc = (σα , ρα ); 2) α1 ∧ α2 = (min(ρα 1 , ρα 2 ), max(σα 1 , σα 2 )); 3) α1 ∨ α2 = (max(ρ α 1 , ρα 2 ), min(σα 1 , σα 2 )); (1 + 2ρα 1 )(1 + 2ρα 2 ) − 1 4) α1 ⊕ α2 = 2 2σα 1 σα 2 ; (2 + σα 1 )(2 + σα 2 ) − σα 1 σα 2 2ρα 1 ρα 2 5) α1 ⊗ α2 = (2 + ρα 1 )(2 + ρα 2 ) − ρα 1 ρα 2 (1 + 2σα 1 )(1 + 2σα 2 ) − 1 ; 2 2σαλ (1 + 2ρα )λ − 1 , 6) λα = ; 2 (2 + σα )λ − σαλ (1 + 2σα )λ − 1 2ραλ λ , 7) α = . 2 (2 + ρα )λ − ραλ Theorem 1: Let α, α1 , and α2 be three IMNs and λ > 0, then 1) α1 ⊕ α2 = α2 ⊕ α1 ; 2) α1 ⊗ α2 = α2 ⊗ α1 ; 3) λ(α1 ⊕ α2 ) = λα1 ⊕ λα2 , λ > 0; 4) (α1 ⊗ α2 )λ = α1λ ⊗ α2λ , λ > 0;
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5) λ1 α ⊕ λ2 α = (λ1 + λ2 )α, λ1 ,λ2 > 0; 6) αλ1 ⊗ αλ2 = αλ1 + λ2 , λ1 , λ2 > 0. The proof of Theorem 1 is provided in the Appendix. Theorem 2: For three IMNs α, α1 , and α2 , the following are valid: 1) α1c ∨ α2c = (α1 ∧ α2 )c ; 2) α1c ∧ α2c = (α1 ∨ α2 )c ; 3) (αc )λ = (λα)c ; 4) λ(αc ) = (αλ )c ; 5) α1c ⊕ α2c = (α1 ⊗ α2 )c ; 6) α1c ⊗ α2c = (α1 ⊕ α2 )c . The proof of Theorem 2 is provided in the Appendix. III. INTUITIONISTIC MULTIPLICATIVE AGGREGATION OPERATORS To aggregate intuitionistic multiplicative information, we develop some aggregation operators for IMNs in this section. Definition 7: Let αi (i = 1, 2, . . . , n) be a collection of IMNs, an intuitionistic multiplicative weighted averaging (IMWA) operator is a mapping M n → M , such that n
IMWA (α1 , α2 , . . . , αn ) = ⊕ (wi αi )
(4)
i=1
where w = (w1 , w2 , . . . , wn )T isthe weight vector of αi (i = 1, 2, . . . , n) with wi ∈ [0, 1] and ni=1 wi = 1. Especially, if w = (1/n, 1/n, . . . , 1/n)T , then the IMWA operator reduces to the intuitionistic multiplicative averaging (IMA) operator: IMA (α1 , α2 , . . . , αn ) =
1 n ⊕ αi . n i=1
(5)
Theorem 3: Let αi (i = 1, 2, . . . , n) be a collection of IMNs, T and w = (w 1n, w2 , . . . , wn ) be their weight vector with wi ∈ [0, 1] and i=1 wi = 1, then IMWA (α1 , α2 , . . . , αn ) = n wi 2 ni=1 σαwii −1 i=1 (1 + 2ρα i ) n , n . wi wi − 2 i=1 (2 + σα i ) i=1 σα i (6) The proof of Theorem 3 is provided in the Appendix. Example 2: Let α1 = (1/3, 1/4), α2 = (2, 1/6), α3 = (4, 1/7), and α4 = (1/9, 5) be four IMNs, w = (0.1, 0.4, 0.2, 0.3)T be the weight vector of αi (i = 1, 2, 3, 4), then, IMWA (α1 , α2 , α3 , α4 ), as shown at the bottom of the page. IMWA (α1 , α2 , α3 , α4 ) =
If we use the corresponding IMNs of the IFNs in Example 1 to express the aggregation information, then we have α1 = (9, 1/9), α2 = (1/9, 9), α3 = (1/9, 9), α4 = (1/9, 9), α5 = (1/9, 9), and α6 = (1/9, 9); thus, by the IMWA operator, we get IMWA (α1 α2 , . . . , α6 ) = (0.4654, 2.1486) which is consistent with our intuition. Therefore, the IMNs can contain more original information in the aggregation process. We can conclude that the IMNs are prior to the IFNs in dealing with such a situation. Definition 8: Let αi (i = 1, 2, . . . , n) be a collection of IMNs, and let IMWG: M n → M , if n
IMWG (α1 , α2 , . . . , αn ) = ⊗ αiw i
then the IMWG function is called an intuitionistic multiplicative weighted geometric (IMWG) operator, where w = (w1 , w2 , . . . , wn )T isthe weight vector of αi (i = 1, 2, . . . , n) with wi ∈ [0, 1] and ni=1 wi = 1. In the case where w = (1/n, 1/n, . . . , 1/n)T , the IMWG operator reduces to the intuitionistic multiplicative geometric (IMG) operator: n
1/n
IMG (α1 , α2 , . . . , αn ) = ⊗ αi
.
(8)
i=1
Based on Definition 8 and Theorem 3, the following theorem can be obtained easily. Theorem 4: Let αi (i = 1, 2, . . . , n) be a collection of IMNs, T and w = (w 1n, w2 , . . . , wn ) be their weight vector with wi ∈ [0, 1] and i=1 wi = 1, then IMWG (α1 , α2 , . . . , αn ) =
2
n i= 1
n ×
n
ρw i i= 1 α i )w i −
(2 + ρα i
i= 1
n i= 1
(1 + 2σα i )w i − 1 2
ρwα ii
,
. (9)
If we use the IMWG operator instead of the IMWA operator in Example 2, then we have, IMWG(α1 , α2 , α3 , α4 ), as shown at the bottom of the next page. If we apply the generalized ordered weighted aggregation operator proposed by Yager [45] to the IMWA operator, then the following operator can be further given:
(1 + 2 × 1/3)0.1 × (1 + 2 × 2)0.4 × (1 + 2 × 4)0.2 × (1 + 2 × 1/9)0.3 − 1 , 2 (2 + 1/4)0.1
(7)
i=1
2 × (1/4)0.1 × (1/6)0.4 × (1/7)0.2 × 50.3 ×(2 + 1/6)0.4 × (2 + 1/7)0.2 × (2 + 5)0.3 − (1/4)0.1 × (1/6)0.4 × (1/7)0.2 × 50.3
= (1.1510, 0.3566) .
XIA et al.: PREFERENCE RELATIONS BASED ON INTUITIONISTIC MULTIPLICATIVE INFORMATION
If λ = 1, then the GIMWG operator becomes the IMWG operator; if w = (1/n, 1/n, . . . , 1/n)T , then the GIMWG operator reduces to the generalized intuitionistic multiplicative geometric (GIMG) operator: 1 n 1/n ⊗ (λαi ) . (14) GIMGλ (α1 , α2 , . . . , αn ) = λ i=1
Definition 9: For a collection of IMNs αi (i = 1, 2, . . . , n), a generalized intuitionistic multiplicative weighted averaging (GIMWA) operator is a mapping GIMWA: M n → M such that GIMWAλ (α1 , α2 , . . . , αn ) =
n ⊕ (wi αiλ )
1 /λ (10)
i=1
where λ > 0 and w = (w1 , w2 , . . . , wn )T isthe weight vector of αi (i = 1, 2, . . . , n), with wi ∈ [0, 1] and ni=1 wi = 1. Especially, if λ = 1, then the GIMWA operator reduces to the IMWA operator; if w = (1/n, 1/n, . . . , 1/n)T , then the GIMWA operator reduces to the generalized intuitionistic multiplicative averaging (GIMA) operator: GIMAλ (α1 , α2 , . . . , αn ) =
1 n λ ⊗ α n i=1 i
Similar to Theorem 5, we have the following theorem. Theorem 6: Let αi (i = 1, 2, . . . , n) be a collection of IMNs, . , wn )T be the weight vector of them, such that w = (w1 , w2 , . . wi ∈ [0, 1] and ni=1 wi = 1, then we have (15), shown at the bottom of the next page. Example 3: Let α1 = (2, 1/7) and α2 = (1/6, 3) be two IMNs and w = (0.3, 0.7)T be the weight vector of αi (i = 1, 2), then we can use the GIMWA and GIMWG operators to aggregate α1 and α2 . 1) If we use the GIMWA operator to aggregate the values given in Example 3, then we can get, GIMWA3 (α1 , α2 , . . . , αn ), shown at the bottom of the next page. 2) If we use the GIMWG operator to aggregate the values given in Example 3, then we can get, GIMWG3 (α1 , α2 , . . . , αn ), shown at the bottom of the next page. In fact, the correlations of these developed operators can be further discussed in the following. Theorem 7: Let αi (i = 1, 2, . . . , n) be a collection of IMNs T with the weight n vector w = (w1 , w2 , . . . , wn ) such that wi ∈ [0, 1] and i=1 wi = 1, λ > 0, then 1) IMWA(α1c , α2c , . . . , αnc ) = (IMWG(α1 , α2 , . . . , αn ))c ; 2) IMWG(α1c , α2c , . . . , αnc ) = (IMWA(α1 , α2 , . . . , αn ))c ; 3) GIMWAλ (α1c , α2c , . . . , αnc ) = (GIMWGλ (α1 , α2 , . . . , αn ))c ;
1 / λ .
(11)
Theorem 5: Let αi (i = 1, 2, . . . , n) be a collection of IMEs, . , wn )T be the weight vector of them, such that w = (w1 , w2 , . . wi ∈ [0, 1] and ni=1 wi = 1, then we have (12), shown at the bottom of the page. The proof of Theorem 5 is provided in the Appendix. If we give a further extension of the IMWG operator, then the following definition can be given. Definition 10: Let αi (i = 1, 2, . . . , n) be a collection of IMNs and let w = (w1 , w2 , . . . , wn )T be the weight vector of them, such that wi ∈ [0, 1] and ni=1 wi = 1. A generalized intuitionistic multiplicative weighted geometric (GIMWG) operator is a mapping M n → M , and 1 GIMWGλ (α1 , α2 , . . . , αn ) = λ
n
⊗ (λαi )w i
i=1
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, λ > 0. (13)
IMWG (α1 , α2 , α3 , α4 ) 2 × (1/3)0.1 × 20.4 × 40.2 × (1/9)0.3 = , (2 + 1/3)0.1 × (2 + 2)0.4 × (2 + 4)0.2 × (2 + 1/9)0.3 − (1/3)0.1 × 20.4 × 40.2 × (1/9)0.3 (1 + 2 × 1/2)0.1 × (1 + 2 × 1/7)0.4 × (1 + 2 × 1/6)0.2 × (1 + 2 × 8)0.3 − 1 2 = (0.6240, 0.7613).
GIMWAλ (α 1 , α 2 , . . . , α n ) ⎛ ⎜ ⎝
2
=⎜
n i= 1
n i= 1
(2 + ρ α i ) λ + 3ρ λ α
w i +3
i
3 + (1 + 2σ α i ) λ
w i +3
i= 1
n i= 1
n i= 1
2
n
(2 + ρ α i
) λ + 3ρ λ α
n i= 1
i
−
n i= 1
(2 + ρ α i
w i 1 λ n λ
(2 + ρ α i ) λ − ρ α
(1 + 2σ α i ) λ − 1
w i
−
i
i= 1
w i 1 λ n
3 + (1 + 2σ α i ) λ
−
w i
−
n i= 1
)λ − ρ λ α
w i 1 λ
i
(2 + ρ α i ) λ + 3ρ λ α
i
w i
−
n i= 1
(2 + ρ α i ) λ − ρ λ α
w i 1 λ
i
w i 1 λ ⎞ w i n 3 + (1 + 2σ α i ) λ − (1 + 2σ α i ) λ − 1 ⎟ i= 1 i= 1 ⎟ . (12) ⎠
w i 1 λ (1 + 2σ α i ) λ − 1
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4) GIMWGλ (α1c , α2c , . . . , αnc ) = (GIMWAλ (α1 , α2 , . . . , αn ))c . The proof of Theorem 7 is provided in the Appendix. Now, we discuss some desirable properties of the developed operators, and take the IMWA operator as an example. Property 1: Let αi (i = 1, 2, . . . , n) be a collection of IMNs, if all αi (i = 1, 2, . . . , n) are equal, i.e., αi = α = (ρα , σα ), for
all i, then n
IMWA (α1 , α2 , . . . , αn ) = IMWA (α, α, . . . , α) = ⊕ wi α = α i=1
(16) which is called the idempotency. ¯i = Property 2: Let αi = (ρα i , σα i ) (i = 1, 2, . . . , n) and α (ρα¯ i , σα¯ i ) (i = 1, 2, . . . , n) be two collections of IMNs, if ¯ i for all i, then αi ≤P α α1 , α ¯2 , . . . , α ¯n ) IMWA (α1 , α2 , . . . , αn ) ≤P IMWA (¯
λ ⎛ w n λ i 3 + (1 + 2ρ ) α ⎜ i i= 1 =⎜ ⎝
GIM WG (α 1 , α 2 , . . . , α n )
+3
n i= 1
2
2
n i= 1
(2 + σ α i ) λ + 3ρ λ αi
w i +3
(1 + 2ρ α i
n i= 1
n i= 1
n
)λ − 1
w i 1 λ n −
3 + (1 + 2ρ α i ) λ
(2 + σ α i ) λ + 3ρ λ αi
w i
w i
−
−
i= 1
n i= 1
n i= 1
(1 + 2ρ α i ) λ − 1
−
i= 1
w i
−
n i= 1
(1 + 2ρ α i
)λ − 1
w i 1 λ ,
w i 1 λ
(2 + σ α i ) λ − σ αλi
w i 1 λ n λ
(2 + σ α i ) λ − σ α i
3 + (1 + 2ρ α i
)λ
⎞
w i 1 λ
(2 + σ α i ) λ + 3ρ λ αi
w i
−
(17)
n i= 1
(2 + σ α i ) λ − σ αλi
⎟ ⎟ . ⎟ w i 1 λ ⎟ ⎠
i= 1
(15)
GIMWA3 (α1 , α2 , . . . , αn ) ⎛ 1/3 2 × ((2 + 2)3 + 3 × 23 )0.3 × ((2 + 1/6)3 + 3 × (1/6)3 )0.7 − ((2 + 2)3 − 23 )0.3 × ((2 + 1/6)3 − (1/6)3 )0.7 ⎜ ⎜ (((2 + 2)3 + 3 × 23 )0.3 × ((2 + 1/6)3 + 3 × (1/6)3 )0.7 + 3 × ((2 + 2)3 − 23 )0.3 × ((2 + 1/6)3 − (1/6)3 )0.7 )1/3 , =⎜ ⎜ 1/3 ⎝ − (((2 + 2)3 + 3 × 23 )0.3 × ((2 + 1/6)3 + 3 × (1/6)3 )0.7 − ((2 + 2)3 − 23 )0.3 × ((2 + 1/6)3 − (1/6)3 )0.7 )
((3 + (1 + 2 × 1/7))3 )0.3 × (3 + (1 + 2 × 3))3 )0.7 + 3 × (2 + 2 × 1/7)3 − 1)0.3 × (2 + 2 × 3)3 − 1)0.7 )1/3 −((3 + (1 + 2 × 1/7))3 )0.3 × (3 + (1 + 2 × 3))3 )0.7 − (2 + 2 × 1/7)3 − 1)0.3 × (2 + 2 × 3)3 − 1)0.7 )1/3 2 × ((3 + (1 + 2 × 1/7))3 )0.3 × (3 + (1 + 2 × 3))3 )0.7 − (2 + 2 × 1/7)3 − 1)0.3 × (2 + 2 × 3)3 − 1)0.7 )1/3
⎞ ⎟ ⎟ ⎟ ⎠
= (0.9774, 0.4913).
GIMWG3 (α1 , α2 , . . . , αn ) ⎛ 1/3 2 × ((2 + 1/7)3 + 3 × (1/7)3 )0.3 × ((2 + 3)3 + 3 × 33 )0.7 − ((2 + 1/7)3 − (1/7)3 )0.3 × ((2 + 3)3 − 33 )0.7 ⎜ ⎜ ((2 + 1/7)3 + 3 × (1/7)3 )0.3 × ((2 + 3)3 + 3 × 33 )0.7 + 3 × ((2 + 1/7)3 − (1/7)3 )0.3 × ((2 + 3)3 − 33 )0.7 1/3 ⎜ , =⎜ ⎜ − (((2 + 1/7)3 + 3 × (1/7)3 )0.3 × ((2 + 3)3 + 3 × 33 )0.7 − ((2 + 1/7)3 − (1/7)3 )0.3 × ((2 + 3)3 − 33 )0.7 )1/3 ⎝
((3 + (1 + 2 × 2))3 )0.3 × (3 + (1 + 2 × 1/6))3 )0.7 + 3 × (2 + 2 × 2)3 − 1)0.3 × (2 + 2 × 1/6)3 − 1)0.7 )1/3 −((3 + (1 + 2 × 2))3 )0.3 × (3 + (1 + 2 × 1/6))3 )0.7 − (2 + 2 × 2)3 − 1)0.3 × (2 + 2 × 1/6)3 − 1)0.7 )1/3 2 × ((3 + (1 + 2 × 2))3 )0.3 × (3 + (1 + 2 × 1/6))3 )0.7 − (2 + 2 × 2)3 − 1)0.3 × (2 + 2 × 1/6)3 − 1)0.7 )1/3
= (0.2562, 2.2222).
⎞ ⎟ ⎟ ⎟ ⎠
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which is called the monotonicity. The proof of Property 2 is provided in the Appendix. Based on the monotonicity, the following property can be obtained. Property 3: Let αi = (ρα i , σα i )(i = 1, 2, . . . , n) be a collection of IMNs, and α− = (min{ρα i }, max{σα i }), α+ = (max{ρα i }, min{σα i }) i
i
i
i
(18) then α− ≤P IMWA (α1 , α2 , . . . , αn ) ≤P α+
(19)
which is called the boundedness. Property 4: Let αi = (ρα i , σα i )(i = 1, 2, . . . , n) be a collecT be the weight tion of IMNs, and let w = (w1 , w2 , . . . , wn ) n vector of them such that wi ∈ [0, 1] and i=1 wi = 1. If α ¯ = (ρα¯ , σα¯ ) is an IMN, then ¯ , α2 ⊕ α ¯ , . . . , αn ⊕ α) ¯ IMWA (α1 ⊕ α = IMWA (α1 , α2 , . . . , αn ) ⊕ α ¯.
(20)
The proof of Property 4 is provided in the Appendix. Property 5: Let αi = (ρα i , σα i )(i = 1, 2, . . . , n) be a collection of IMNs, and let w = (w1 , w2 , . . . , wn )T be the weight vector of them such that wi ∈ [0, 1] and ni=1 wi = 1. If r > 0, then IMWA (rα1 , rα2 , . . . , rαn ) = r IMWA (α1 , α2 , . . . , αn ). (21) The proof of Property 5 is provided in the Appendix. Property 6: Let αi = (ρα i , σα i )(i = 1, 2, . . . , n) be a collec. , wn )T be the weight tion of IMNs, and let w = (w1 , w2 , . . vector of them such that wi ∈ [0, 1] and ni=1 wi = 1. If r > 0, α ¯ = (ρα¯ , σα¯ ) is an IMN, then ¯ , rα2 ⊕ α ¯ , . . . , rαn ⊕ α ¯) IMWA (rα1 ⊕ α = r IMWA (α1 , α2 , . . . , αn ) ⊕ α ¯.
(22)
The proof of Property 6 is provided in the Appendix. ¯ i = (ρα¯ i , σα¯ i ) Property 7: Let αi = (ρα i , σα i ) and α (i = 1, 2, . . . , n) be two collections of IMNs and w = )T be the weight vector of them such that (w1 , w2 , . . . , wn wi ∈ [0, 1] and ni=1 wi = 1, then ¯ 1 , α2 ⊕ α ¯ 2 , . . . , αn ⊕ α ¯n ) IMWA (α1 ⊕ α = IMWA (α1 , α2 , . . . , αn ) ⊕ IMWA (¯ α1 , α ¯2 , . . . , α ¯ n ). (23) The proof of Property 7 is provided in the Appendix. IV. DECISION MAKING BASED ON INTUITIONISTIC MULTIPLICATIVE PREFERENCE RELATIONS Suppose that there are n alternatives xi (i = 1, 2, . . . , n), the decision maker provides his/her preference about the alternatives xi and xj , denoted by IMN αij = (ρα i j , σα i j ), where ρα i j indicates the intensity that the alternative xi is prior to xj ; σα i j indicates the intensity that the alternative xi is not prior to xj , with the conditions ρα i j = σα j i , σα i j = ρα j i , 1/9 ≤ ρα i j , σα i j ≤ 9, and ρα i j σα i j ≤ 1, all the IMNs αij (i,
Fig. 1.
Hierarchy structure.
j = 1, 2, . . ., n) construct the intuitionistic multiplicative preference relation A = (αij )n ×n . To get the best alternative, the following steps are given. Step 1: Suppose that the weight vector is w = (1/n, 1/n, . . . , 1/n)T , utilize the GIMWA or GIMWG operator to aggregate the preference values αij , and obtain the IMNs αi for the alternative xi . Step 2: Calculate the scores s(αi ) and the accuracy degrees h(αi ) of αi (i = 1, 2, . . . , m) by Definition 4. Step 3: Get the priority of the alternatives xi by ranking s(αi ) and h(αi )(i = 1, 2, . . . , m). In what follows, we use the example adapted from Gong et al. [41] to illustrate the developed method. Example 4: Benefit analysis and assessment of industry meteorological service is a new business of the China Meteorological Administration (CMA). In the assessment process, we need to know exactly the relationship between short-term change of meteorological conditions (temperature and precipitation) and the industrial economy; we then select the highly sensitive industries and thus need to evaluate the industry meteorological services. CMA has invited a decision maker group composed of meteorologists, industry experts, and economists to evaluate and compare the meteorological sensitivity of seven industries, including agricultural (AG), light industry (LI), heavy industry (HI), energy (ER), construction industry (CI), communications and transportation (CT), and commerce (CM), the results of which are the intuitionistic multiplicative preferences. Industries with the higher meteorological sensitivity selecting can be modeled as a hierarchical structure, as shown in Fig. 1. For convenience, let X = {x1 = AG, x2 = LI, x3 = HI, x4 = ER, x5 = CI, x6 = CT, x7 = CM} be the set of the seven industries. The intuitionistic multiplicative preference relation t )n ×n for seven industries with respect to temperAt = (αij ature and the intuitionistic multiplicative preference relation p )n ×n for seven industries with respect to precipitaAp = (αij tion are presented in Tables II and III, respectively. We utilize the GIMWA or GIMWG operator to aggregate the t and obtain the IMNs αit (i = 1, 2, . . . , 7) preference values αij for the alternatives (without loss of generality, let λ = 1) with
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TABLE II INTUITIONISTIC MULTIPLICATIVE DECISION MATRIX WITH RESPECT TO TEMPERATURE
TABLE III INTUITIONISTIC MULTIPLICATIVE DECISION MATRIX WITH RESPECT TO PRECIPITATION
Fig. 2. Scores of industries obtained by the GIMWA operator under temperature.
By Definition 4, we can calculate the scores of αip (i = 1, 2, . . . , 7) as follows: s(α1p ) = 10.6548, s(α2p ) = 1.5361, s(α3p ) = 0.2469 s(α4p ) = 0.2902, s(α5p ) = 0.5432, s(α6p ) = 1.6064 s(α7p ) = 5.3715.
respect to temperature:
Since
α1t = (1.6436, 0.3177), α2t = (0.7023, 0.7657)
s(α1p ) > s(α7p ) > s(α6p ) > s(α2p ) > s(α4p ) > s(α5p ) > s(α3p )
α3t = (0.3754, 1.8373), α4t = (0.3910, 1.5802)
then the ranking of the precipitation sensitivities of seven industries is
α5t = (0.5916, 0.9289), α6t = (1.3280, 0.5753) α7t = (3.1001, 0.1996). By Definition 4, we can calculate the scores of 1, 2, . . . , 7) as follows:
αit (i
AGp > CMp > CTp > LIp > ERp > CIp > HIp . =
s(α1t ) = 5.1738, s(α2t ) = 0.9171, s(α3t ) = 0.2043 s(α4t ) = 0.2474, s(α5t ) = 0.6369, s(α6t ) = 2.3084
The decision maker group views that the importance of temperature and the importance of precipitation are 0.6 and 0.4, respectively. Thus, we can get the scores of seven industries as s(αi ) = s(αit )0.6 s(αip )0.4 (i = 1, 2, . . . , 7): s(α1 ) = 6.9072, s(α2 ) = 1.1272, s(α3 ) = 0.2204
s(α7t ) = 15.5317.
s(α4 ) = 0.2637, s(α5 ) = 0.5976, s(α6 ) = 1.9968
Since
s(α7 ) = 10.1571.
s(α7t ) > s(α1t ) > s(α6t ) > s(α5t ) > s(α2t ) > s(α4t ) > s(α3t ) then the ranking of the temperature sensitivities of seven industries is CMt > AGt > CTt > LIt > CIt > ERt > HIt . Moreover, we utilize the GIMWA operator to aggregate the p (j = 1, 2, . . . , 7), and obtain the IMNs preference values αij p αi (i = 1, 2, . . . , 7) for the alternatives with respect to precipitation (without loss of generality, let λ = 1), for example α1p = (2.4785, 0.2326), α2p = (0.8104, 0.5276) α3p = (0.3715, 1.5046), α4p = (0.3370, 1.1610) α5p = (0.5200, 0.9573), α6p = (0.8698, 0.5415) α7p = (1.7641, 0.3284).
Since s(α7 ) > s(α1 ) > s(α6 ) > s(α2 ) > s(α5 ) > s(α4 ) > s(α3 ) the ranking of the meteorological sensitivities of seven industries is CM > AG > CT > LI > CI > ER > HI. It is noted that we let λ = 1 in the aforementioned analysis. In fact, the parameter λ can be assigned different values according to the decision maker’s preferences. To investigate the variation of the ranking of seven industries with respect to the value of the parameter λ, we assign λ the values between 0 and 9, and calculate the scores of these seven industries; more details can be found in Figs. 2–7.
XIA et al.: PREFERENCE RELATIONS BASED ON INTUITIONISTIC MULTIPLICATIVE INFORMATION
Fig. 3.
Fig. 5.
More details in Fig. 4.
Fig. 6.
Scores of industries obtained by the GIMWA operator.
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More details in Fig. 2.
Fig. 4. Scores of industries obtained by the GIMWA operator under precipitation.
Fig. 2 illustrates the scores of industries with respect to temperature obtained by using the GIMWA operator; we can find that scores of each industries with respect to temperature increase as the values of λ change from 0 to 9. The scores of industries HI and ER are very closer, to give a clear comparison, Fig. 3 is given, from which we can find that 1) If 0 < λ ≤ 3.011, then we have CM > AG > CT > LI > CI > ER > HI. 2) If 3.011 ≤ λ ≤ 9, then we have CM > AG > CT > LI > CI > HI > ER.
Figs. 4 and 5 illustrate the scores of industries with respect to precipitation obtained by using the GIMWA operator; it is noted that the scores increase as the values of λ increase from 0 to 9 and 1) If 0 < λ ≤ 1.654, then we have AG > CM > CT > LI > CI > ER > HI. 2) If 1.654 < λ ≤ 1.7504, then we have AG > CM > LI > CT > CI > HI > ER. 3) If 1.7504 < λ ≤ 3.5487, then we have AG > CM > LI > CT > CI > HI > ER. 4) If 3.5487 ≤ λ ≤ 4.62, then we have AG > CM > LI > CT > HI > CI > ER.
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Fig. 7.
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More details in Fig. 6.
Fig. 9.
More details in Fig. 8.
We can find that contrary to the GIMWA operator, the scores obtained by the GIMWG operator decrease with the increase of the parameter λ. In addition, by comparing Figs. 6 and 8, Figs. 7 and 9, we can find that when 0 < λ ≤ 9, most of the scores obtained by the GIMWA operator are much bigger than the ones obtained by the GIMWG operator, as the value of the parameter becomes bigger, and as the deviation becomes bigger.
V. CONCLUSION
Fig. 8.
Scores of industries obtained by GIMWG operator.
5) If 4.62 ≤ λ ≤ 9, then we have AG > CM > LI > CT > HI > ER > CI. Figs. 6 and 7 illustrate the scores of industries obtained by using the GIMWA operator, and it is noted that scores increase as the values of λ increase from 0 to 9, and 1) If 0 < λ ≤ 2.111, then we have CM > AG > CT > LI > CI > ER > HI. 2) If 2.111 < λ ≤ 9, then we have CM > AG > CT > LI > CI > ER > HI > ER. If we use the GIMWG operator to aggregate the intuitionistic multiplicative preference information, then the scores of industries are given in Figs. 8 and 9, from which the ranking is CM > AG > CT > LI > CI > ER > HI.
In this paper, we have investigated multiplicative preference relations under intuitionistic environments. Based on the concept of interval multiplicative preference relation, we have introduced the concept of intuitionistic multiplicative preference relation whose basic units are the IMNs, in which the membership information and the nonmembership information of an element to a set represented by Saaty’s 1–9 scale instead of the 0.1–0.9 scale in the intuitionistic fuzzy set. To aggregate the intuitionistic multiplicative preference information, several aggregation operators have been developed, such as the GIMWA operator and the GIMWG operator. The illustrative example has shown that most of the scores obtained by the former aggregation operator increase as the value of the parameter increases, while the scores obtained by the latter decrease as the value of the parameter decreases. In addition, the parameter can be assigned different values according to the decision makers’ preferences and the actual situations. In the future, we will apply the classical ordered weighted averaging [46] operation to the proposed aggregation operators and investigate the methods [47] to determine the weight vectors of these operators under intuitionistic multiplicative environments.
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APPENDIX Proof of Theorem 1: For three IMNs α, α1 , and α2 , we have (1 + 2ρα 1 )(1 + 2ρα 2 ) − 1 2σα 1 σα 2 1) α1 ⊕ α2 = , 2 (2 + σα 1 )(2 + σα 2 ) − σα 1 σα 2 (1 + 2ρα 2 )(1 + 2ρα 1 ) − 1 2σα 2 σα 1 , = 2 (2 + σα 2 )(2 + σα 1 ) − σα 2 σα 1 = α2 ⊕ α1 2)
α1 ⊗ α2 = =
2ρα 1 ρα 2 (1 + 2σα 1 )(1 + 2σα 2 ) − 1 , (2 + ρα 1 )(2 + ρα 2 ) − ρα 1 ρα 2 2 2ρα 2 ρα 1 (1 + 2σα 2 )(1 + 2σα 1 ) − 1 , (2 + ρα 2 )(2 + ρα 1 ) − ρα 2 ρα 1 2
= α2 ⊗ α1
3)
(1 + 2ρα 1 )(1 + 2ρα 2 ) − 1 2σα 1 σα 2 , λ(α1 ⊕ α2 ) = λ 2 (2 + σα 1 )(2 + σα 2 ) − σα 1 σα 2 ⎛ λ λ
(1+2ρ α 1 )(1+2ρ α 2 )−1 2σ α 1 σ α 2 2 1 + 2 − 1 2 (2+σ α 1 )(2+σ α 2 )−σ α 1 σ α 2 ⎜ , =⎝ λ λ 2 2σ α 1 σ α 2 2σ α 1 σ α 2 2 + (2+σ α )(2+σ − (2+σ α 1 )(2+σ α 2 )−σ α 1 σ α 2 α 2 )−σ α 1 σ α 2 1 2(σα 1 σα 2 )λ ((1 + 2ρα 1 )(1 + 2ρα 2 ))λ − 1 , = 2 ((2 + σα 1 )(2 + σα 2 ))λ − (σα 1 σα 2 )λ 2σαλ1 2σαλ2 (1 + 2ρα 1 )λ − 1 (1 + 2ρα 2 )λ − 1 , , ⊕ λα1 ⊕ λα2 = 2 2 (2 + σα 1 )λ − σαλ1 (2 + σα 2 )λ − σαλ2 ⎛ (1+2ρ α 1 ) λ −1 (1+2ρ α 2 ) λ −1 1 + 2 1 + 2 −1 2 2 ⎜ ⎜ =⎝ 2
2 2+ =
4)
2σ αλ1
(2+σ α 1 ) λ −σ αλ1
2σ αλ1
⎞
2σ αλ2
(2+σ α 1 ) λ −σ αλ1 (2+σ α 2 ) λ −σ αλ2
2+
2σ αλ2
(2+σ α 2 ) λ −σ αλ2
−
2σ αλ1
2σαλ1 σαλ2 (1 + 2ρα 1 )λ (1 + 2ρα 2 )λ − 1 , 2 (2 + σα 1 )λ (2 + σα 2 )λ − σαλ1 σαλ2
2σ αλ2
(2+σ α 1 ) λ −σ αλ1 (2+σ α 2 ) λ −σ αλ2
⎟ ⎟ ⎟ ⎠
λ 2ρα 1 ρα 2 (1 + 2σα 1 )(1 + 2σα 2 ) − 1 , (2 + ρα 1 )(2 + ρα 2 ) − ρα 1 ρα 2 2 ⎛ λ 2ρα 1 ρα 2 2 ⎜ (2 + ρα 1 )(2 + ρα 2 ) − ρα 1 ρα 2 ⎜ = ⎜ λ λ ⎝ 2ρα 1 ρα 2 2ρα 1 ρα 2 2+ − (2 + ρα 1 )(2 + ρα 2 ) − ρα 1 ρα 2 (2 + ρα 1 )(2 + ρα 2 ) − ρα 1 ρα 2
(α1 ⊗ α2 )λ =
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1+2
(1+2σ α 1 )(1+2σ α 2 )−1 2
2
λ
⎞ − 1⎟ ⎠
((1 + 2σα 1 )(1 + 2σα 2 ))λ − 1 2(ρα 1 ρα 2 )λ , = 2 ((2 + ρα 1 )(2 + ρα 2 ))λ − (ρα 1 ρα 2 )λ 2ρλ 2ρλ (1 + 2σα 1 )λ − 1 (1 + 2σα 2 )λ − 1 α1 α2 λ λ α1 ⊗ α2 = , , ⊗ 2 2 (2 + ρα 1 )λ − ραλ1 (2 + ρα 2 )λ − ραλ2 ⎛ 2ρ λ 2ρ λ α1 α2 ⎜ 2 λ λ ⎜ (2+ρ α 1 ) λ −ρ λ (2+ρ α 2 ) −ρ α 2 α1 =⎜ ⎜ 2ραλ1 2ρλ 2ρλ 2ρλ ⎝ α2 α1 α2 2+ 2+ − (2 + ρα 1 )λ − ραλ1 (2 + ρα 2 )λ − ραλ2 (2 + ρα 1 )λ − ραλ1 (2 + ρα 2 )λ − ραλ2 ⎞ (1+2σ α 1 ) λ −1 (1+2σ α 2 ) λ −1 1+2 1 + 2 − 1 2 2 ⎟ ⎟ ⎠ 2
=
2ραλ1 ρλ α2
(1 + 2σα 1 )λ (1 + 2σα 2 )λ − 1 , 2 (2 + ρα 1 )λ (2 + ρα 2 )λ − ραλ1 ρλ α2
5)
2σαλ1 2σαλ2 (1 + 2ρα )λ1 − 1 (1 + 2ρα )λ2 − 1 λ1 α ⊕ λ2 α = , , ⊕ 2 2 (2 + σα )λ1 − σαλ1 (2 + σα )λ2 − σαλ2 ⎛ (1+2ρ α ) λ1 −1 (1+2ρ α ) λ2 −1 1 + 2 −1 1 + 2 2 2 ⎜ ⎜ =⎝ 2 λ1 λ 2σ α 2 2 λ λ (2+σ α ) λ1 −σ α 1 (2+σ α ) λ2 −σ α 2 λ λ λ 2σ α 1 2σ α 2 2σ α 1 2+ 2 + − λ λ λ (2+σ α ) λ1 −σ α 1 (2+σ α ) λ2 −σ α 2 (2+σ α ) λ1 −σ α 1
⎞
2σ α
=
(1 + 2ρα )λ1 (1 + 2ρα )λ2 − 1 2σαλ1 σαλ2 , 2 (2 + σα )λ1 (2 + σα )λ2 − σαλ1 σαλ2
λ2 λ (2+σ α ) λ2 −σ α 2 2σ α
⎟ ⎟ ⎟ ⎠
= (λ1 + λ2 )α
6)
α λ1 ⊗ α λ 2 =
⎛
1 2ρλ (1 + 2σα )λ1 − 1 α , 1 2 (2 + ρα )λ1 − ρλ α
⊗
2 2ρλ (1 + 2σα )λ2 − 1 α , 2 2 (2 + ρα )λ2 − ρλ α
2 λ1 2ρλ α λ 2 (2+ρ α ) λ1 −ρ α 1 (2 + ρα )λ2 − ρλ α 1 λ 2ραλ1 2ρλ 2ρ α 2 α 2+ − λ λ 2 1 (2+ρ α ) λ2 −ρ α (2 + ρα )λ1 − ρα 1 (2 + ρα )λ1 − ρλ α ⎞ λ1 λ2 1 + 2 (1+2σ α2 ) −1 1 + 2 (1+2σ α2 ) −1 − 1 ⎟ ⎟ ⎠ 2
⎜ ⎜ =⎜ ⎜ ⎝ 2+
2
2ρ α
λ2 λ (2+ρ α ) λ2 −ρ α 2 2ρ α
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=
125
1 λ2 2ρλ (1 + 2σα )λ1 (1 + 2σα )λ2 − 1 α ρα , 1 λ2 2 (2 + ρα )λ1 (2 + ρα )λ2 − ρλ α ρα
= αλ1 + λ2 . Proof of Theorem 2: For three IMNs α, α1 , and α2 , we have
1)
α1c ∨ α2c = (max(σα 1 , σα 2 ), min(ρα 1 , ρα 2 )) = (min(ρα 1 , ρα 2 ), max(σα 1 , σα 2 ))c = (α1 ∧ α2 )c
2)
α1c ∧ α2c = (min(σα 1 , σα 2 ), max(ρα 1 , ρα 2 )) = (max(ρα 1 , ρα 2 ), min(σα 1 , σα 2 ))c = (α1 ∨ α2 )c
3)
(αc )λ =
2σαλ (1 + 2ρα )λ − 1 , 2 (2 + σα )λ − σ λ
=
α
4)
λα = c
2ραλ (1 + 2σα )λ − 1 , 2 (2 + ρα )λ − ραλ
5)
α1c
⊕
α2c
= =
=
(1 + 2ρα )λ − 1 2σαλ , 2 (2 + σα )λ − σαλ
(1 + 2σα )λ − 1 2ρλ α , 2 (2 + ρα )λ − ρλ
c
c
= (λα)c
= (αλ )c
α
(1 + 2σα 1 )(1 + 2σα 2 ) − 1 2ρα 1 ρα 2 , 2 (2 + ρα 1 )(2 + ρα 2 ) − ρα 1 ρα 2 2ρα 1 ρα 2 (1 + 2σα 1 )(1 + 2σα 2 ) − 1 , (2 + ρα 1 )(2 + ρα 2 ) − ρα 1 ρα 2 2
c
= (α1 ⊗ α2 )c 6)
α1c
⊗
α2c
= =
2σα 1 σα 2 (1 + 2ρα 1 )(1 + 2ρα 2 ) − 1 , (2 + σα 1 )(2 + σα 2 ) − σα 1 σα 2 2 (1 + 2ρα 1 )(1 + 2ρα 2 ) − 1 2σα 1 σα 2 , 2 (2 + σα 1 )(2 + σα 2 ) − σα 1 σα 2
c
= (α1 ⊕ α2 )c . Proof of Theorem 3: We prove the theorem by using mathematical induction on n. We first prove (6) holds for n = 2. Since w1 α1 = w2 α2 =
2σαw11 (1 + 2ρα 1 )w 1 − 1 , 2 (2 + σα 1 )w 1 − σαw11 2σαw22 (1 + 2ρα 2 )w 2 − 1 , 2 (2 + σα 2 )w 2 − σαw22
we have IMWA (α1 , α2 ) = w1 α1 ⊕ w2 α2 ⎛ (1+2ρ α 1 ) w 1 −1 (1+2ρ α 2 ) w 2 −1 1+2 1 + 2 −1 2 2 =⎝ 2
(24) (25)
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013
⎞ 2σαw11 2σαw22 2 ⎟ (2 + σα 1 )w 1 − σαw11 (2 + σα 2 )w 2 − σαw22 ⎟ w1 w1 w2 w2 ⎠ 2σ 2σα 1 2σ 2σ α1 α2 2+ 2 + (2+σ )αw22 −σ w 2 − w1 w1 w2 w w w 1 1 2 α α 2 (2 + σα 1 ) − σα 1 (2 + σα 1 ) − σα 1 (2 + σα 2 ) − σα 2 2 w1 w2 w1 w2 2σα 1 σα 2 (1 + 2ρα 1 ) (1 + 2ρα 2 ) − 1 = , . (26) 2 (2 + σα 1 )w 1 (2 + σα 2 )w 2 − σαw11 σαw22 If (6) holds for n = k, that is ⎛
k
wi
(1 + 2ρα i ) ⎜ ⎜ IMWA (α1 , α2 , . . . , αk ) = ⎜ i=1 2 ⎝
−1
k
2 ,
i=1 k
(2 + σα i
⎞ σαwii
)w i
−
i=1
k
σαwii
⎟ ⎟ ⎟ ⎠
(27)
i=1
then, when n = k + 1, by the operational laws in Definition 6, we have IMWA (α1 , α2 , . . . , αk +1 ) ⎞ ⎛⎛ k (1+2ρ α i ) w i −1 wk+1 ⎟ ⎜⎜ −1 ⎟ 1 + 2 (1+2ρ α k + 1 ) ⎜ ⎜1 + 2 i = 1 −1 2 2 ⎝ ⎠ ⎜ ⎜ ⎜ =⎜ ⎜ 2 ⎜ ⎜ ⎜ ⎝
2
2 k ⎛ ⎜ ⎜2 + k ⎝ ⎛
k
2
w
σ α ii
i= 1
k
(2+σ α i ) w i −
i= 1
k
⎞
w
σ α ii
⎞
⎟ ⎟ 2+ ⎠
w
σ α ii
i= 1
(2+σ α i ) w i −
i= 1
w
k
w 2σ α 1k + 1 (2+σ α k + 1 ) w k + 1
wi
wi
(1 + 2ρα i ) ⎜ ⎜ i=1 =⎜ 2 ⎝
⎛
w −σ α kk ++ 11
⎜ −⎜ k ⎝
−1
2 ,
2 , k +1 i=1
2
k
w
σ α ii
i= 1
(2+σ α i ) w i −
i= 1
wk + 1
−1
w
(2+σ α k + 1 ) w k + 1 −σ α kk ++ 11
w σ α ii
i= 1
k +1 i=1
k i=1
k
(2 + σα i )w i −
k +1
σαwii
k
w
σ α ii
i= 1
k
w
(2+σ α k + 1 ) w k + 1 −σ α kk ++ 11
⎟ ⎟ ⎟ ⎟ ⎟ ⎞⎟ ⎟ ⎟ ⎟⎟ ⎟⎟ ⎠⎠
⎞
w
σαwii σα 1k + 1
⎟ ⎟ ⎟ ⎠
i=1
⎞
σαwii
w
2σ α 1k + 1
w σαwii σα 1k + 1
(2 + σα i )w i (2 + σα k + 1 )w k + 1 −
i=1 k +1
2σ α 1k + 1
i= 1
(1 + 2ρα i ) (1 + 2ρα k + 1 ) ⎜ ⎜ = ⎜ i=1 2 ⎝ ⎛
k
⎟ ⎟ ⎟ ⎠
i=1
(28) that is, (6) holds for n = k + 1. Therefore, (6) holds for all n, which completes the proof of the theorem. Proof of Theorem 5: Since 2ρλ (1 + 2σα i )λ − 1 αi λ αi = , 2 (2 + ρα i )λ − ραλ i
(29)
XIA et al.: PREFERENCE RELATIONS BASED ON INTUITIONISTIC MULTIPLICATIVE INFORMATION
127
then n ⊕ (wi αiλ )
i=1
⎛
n
⎜ ⎜ i=1 =⎜ ⎝
1+2
−1
(2+ρ α i ) λ −ρ λ αi
2 ,
2
⎛ n
⎜ i=1 =⎜ ⎝
w i
2ρ λ αi
n
2
n i=1
w i
−
n i=1
2+
(1+2σ α i ) λ −1 2
(1+2σ α i ) λ −1 2
(2 + ρα i )λ − ραλi
(2 + ρα i )λ − ραλi
i=1
i=1
(2 + ρα i )λ + 3ραλi
n
w i
n
−
⎞
w i
⎟ ⎟ w i ⎟ ⎠ (1+2σ α i ) λ −1 2
i=1
w i
w i
2
, n
n
w i (1 + 2σα i )λ − 1
⎞
⎟ w i w i ⎟ n ⎠ (1 + 2σα i )λ + 3 (1 + 2σα i )λ − 1 − i=1
i=1
i=1
(30)
and
GIMWAλ (α 1 , α 2 , . . . , α n ) = ⎛
⊕ (w i α λ i ) n
1 λ
i= 1
⎛ n
w i ⎞1 λ
w n λ i
(2 + ρ α i ) λ + 3 ρ α − (2 + ρ α i ) λ −ρ λ αi ⎜ i ⎜i= 1 ⎟ ⎜ i= 1 2⎝
⎜ w ⎠ n i ⎜ 2 (2 + ρ α i ) λ −ρ λ αi ⎜ ⎜ i= 1 , =⎜ w i w i ⎞ 1 λ ⎛ w i w i ⎞ 1 λ ⎜⎛ n n n n ⎜ λ λ λ λ λ λ λ λ (2 + ρ α i ) + 3 ρ α − (2 + ρ α i ) −ρ α (2 + ρ α i ) + 3 ρ α − (2 + ρ α i ) −ρ α ⎜ i i i i ⎟ ⎜i= 1 ⎟ ⎜ ⎜2 + i = 1 i= 1 i= 1 −
w i w ⎠ ⎝ ⎠ n n ⎝⎝ i λ λ 2 (2 + ρ α i ) λ −ρ λ 2 (2 + ρ ) − ρ αi α α i
i
i= 1
⎛ ⎜ ⎝1 + 2 n
2
n
(1 + 2 σ α i ) λ −1
⎞1 λ
w i
⎟ w i w i ⎠ n λ λ (1 + 2 σ α i ) + 3 − (1 + 2 σ α i ) −1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
− 1⎟
i= 1
i= 1
i= 1
2
⎛ ⎜ ⎜ =⎜ ⎜ n ⎝
2 (2 + ρα i )λ + 3ρλ αi
n i= 1
(2 + ρα i )λ + 3ρλ αi
i= 1
w i +3
i= 1
n
n
(2 + ρα i )λ − ρλ αi
i= 1
3 + (1 + 2σ α i )λ
w i +3
n
(1 + 2σ α i )λ − 1
i= 1
w i
−
w i 1 λ
w i 1 λ
n i= 1
−
2
n
(2 + ρα i )λ + 3ρλ αi
i= 1
3 + (1 + 2σ α i )λ
w i
−
−
−
n
n
n
(2 + ρα i )λ − ρλ αi
, w i 1 λ
i= 1
3 + (1 + 2σ α i )λ
i= 1 n
w i
i= 1
i= 1
(2 + ρα i )λ − ρλ αi
w i 1 λ
(1 + 2σ α i )λ − 1
w i
−
w i 1 λ
n i= 1
⎞ w i 1 λ ⎟ (1 + 2σ α i )λ − 1 ⎟ ⎟. ⎟ ⎠
i= 1
(31)
128
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013
Proof of Theorem 7: ⎛ n
1)
⎜ i=1 n IMWA (α1c , α2c , . . . , αnc ) = ⊕ (wi αic ) = ⎜ ⎝
(1 + 2σα i )w i − 1
n
⎛ 2)
n
= ⊗
i=1
=
⎜ ⎜ =⎜ k ⎝
(αic )w i
n
⊕ wi αi
3) GIMWAλ ⎛ ⎜ ⎜ =⎜ ⎜ n ⎝
n
=
⊕
i= 1
(2 + σ α i )λ + 3σ αλi
w i
n
+3
3 + (1 + 2ρα i )λ
w i +3
n
1
λ
i=1
n
⊗ (λα i )w i
c
i= 1
k
σαwii
(2 + σα i )w i −
k
(2 + σ α i )λ − σ αλi
(1 + 2ρα i )λ − 1
n
⎞ wi
(1 + 2ρα i )
i=1
,
2
σαwii
− 1⎟ ⎟ ⎟ ⎠
i=1
w i
−
w i 1 λ
w i 1 λ
n
(2 + σ α i )λ − σ αλi
3 + (1 + 2ρα i )λ
w i
−
n
(2 + σ α i )λ + 3σ αλi
w i
−
i= 1
−
n
n
−
i= 1
w i 1 λ
i= 1
n
(2 + σ α i )λ − σ αλi
w i 1 λ
i= 1
3 + (1 + 2ρα i )λ
i= 1
2
2
i= 1
i=1
= (IMWA (α1 , α2 , . . . , αn ))c
(2 + σ α i )λ + 3σ αλi
i= 1
n
i ρw αi
⎟ ⎟ ⎠
1 λ
i= 1
i= 1
=
λ
n
2
i= 1
w i (α ci )
k
i=1
c
i=1
(α c1 , α c2 , . . . , α cn )
(2 + ρα i )w i −
n
= (IMWG (α1 , α2 , . . . , αn ))c
i=1
IMWG (α1c , α2c , . . . , αnc )
⎞ i ρw αi
i=1
c
⊗ αiw i
=
n i=1
, n
2
i=1
2
(1 + 2ρα i )λ − 1
w i
−
n
i= 1 w i 1 λ
⎞ w i 1 λ ⎟ (1 + 2ρα i )λ − 1 ⎟ ⎟ ⎟ ⎠
i= 1
= (GIMWG λ (α 1 , α 2 , . . . , α n ))c
n 1 4) GIMWG λ (α c1 , α c2 , . . . , α cn ) = ⊗ (λα ci )w i λ i= 1 ⎛ w i w i w i 1 λ w i 1 λ n n n n ⎜ 3 + (1 + 2σ α i )λ +3 (1 + 2σ α i )λ − 1 − 3 + (1 + 2σ α i )λ − (1 + 2σ α i )λ − 1 ⎜ i= 1 i= 1 i= 1 i= 1 ⎜ = ⎜ n w i w i 1 λ n ⎝ λ λ 2
3 + (1 + 2σ α i )
−
i= 1
2
(2 + ρα i )λ + 3ρλ αi
i= 1
3
=
n
n
(2 + ρα i )λ − ρλ αi
i= 1
⊕ wi αλ i n
i= 1
1 λ c
w i +
n
(1 + 2σ α i ) − 1
i= 1
(2 + ρα i )λ + 3ρλ αi
i= 1
= (GIMWAλ (α 1 , α 2 , . . . , α n ))c .
w i
−
w i 1 λ
n
(2 + ρα i )λ − ρλ αi
w i 1 λ
⎞
i= 1
−
n i= 1
(2 + ρα i )λ + 3ρλ αi
w i
−
n i= 1
(2 + ρα i )λ − ρλ αi
⎟ ⎟ ⎟ ⎟ λ 1 w i ⎠
XIA et al.: PREFERENCE RELATIONS BASED ON INTUITIONISTIC MULTIPLICATIVE INFORMATION
129
Proof of Property 2: For two collections of IMNs αi = (ρα i , σα i ) and α ¯ i = (ρα¯ i , σα¯ i ) (i = 1, 2, . . . , n), if αi ≤P α ¯ i for all i, that is ρα i ≤ ρα¯ i and σα i ≥ σα¯ i for all i, we have k i=1
2
2 k i=1
k
k
(1 + 2ρα i )w i − 1
(2 + σα i
≤
k
2
−
i=1
(32)
2
σαwii
)w i
(1 + 2ρα¯ i )w i − 1
i=1
k
i=1
≥ σαwii
i=1
k
σαw¯ ii
(2 + σα¯ i
)w i
(33)
k
−
i=1
σαw¯ ii
i=1
α1 , α ¯2 , . . . , α ¯ n ). then IMWA(α1 , α2 , . . . , αn ) ≤P IMWA(¯ Proof of Property 4: Since ¯= αi ⊕ α
(1 + 2ρα i )(1 + 2ρα¯ ) − 1 2σα i σα¯ , 2 (2 + σα i )(2 + σα¯ ) − σα i σα¯
(34)
then we have IMWA (α1 ⊕ α ¯ , α2 ⊕ α, ¯ . . . , αn ⊕ α ¯) ⎛
n (1+2ρ α i )(1+2ρ α¯ )−1 w i 1+2 −1 2 ⎜ i=1 , =⎜ n ⎝ 2 ⎛ n
⎜ i=1 =⎜ ⎝
2
i=1
2+
i=1
((1 + 2ρα i )(1 + 2ρα¯ ))w i − 1 2
2σ α i σ α¯ (2+σ α i )(2+σ α¯ )−σ α i σ α¯
2σ α i σ α¯ (2+σ α i )(2+σ α¯ )−σ α i σ α¯ n
2 , n
n
w i
−
n
i=1
⎞
w i
2σ α i σ α¯ (2+σ α i )(2+σ α¯ )−σ α i σ α¯
⎟ w i ⎟ ⎠
⎞
(σα i σα¯ )w i
i=1
((2 + σα i )(2 + σα¯ ))w i −
i=1
n
(σα i σα¯ )w i
⎟ ⎟ ⎠
(35)
i=1
and IMWA (α1 , α1 , . . . , αn ) ⊕ α ¯ ⎛⎛ n ⎜⎜ ⎜ ⎝1 + 2 i = 1 ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝ ⎛ n
⎜ i=1 =⎜ ⎝
(1+2ρ α i ) w i −1 2
⎞
2
⎟ ⎠ (1 + 2ρα¯ ) − 1
2 n , ⎛
2
⎜ ⎝2 + n
2
n
w σ α ii
(2+σ α i ) w i −
i= 1
(1 + 2ρα i )w i (1 + 2ρα¯ ) − 1 2
n
2 , n
i=1
n
w
σ α ii
i= 1
σαwii σα¯
(2 + σα i )w i (2 + σα¯ ) −
i=1
(2+σ α i
n
σαwii σα¯
)w i
⎞
⎞
w
σ α ii
i= 1
i= 1
i= 1
n
−
n i= 1
w σ α ii
σα¯
⎛
⎟ ⎜ ⎠ (2 + σα¯ ) − ⎝ n i= 1
⎞ ⎟ ⎟. ⎠
2
n
w
σ α ii
i= 1
(2+σ α i ) w i −
n
w
σ α ii
⎟ ⎟ ⎟ ⎟ ⎟ ⎞⎟ ⎟ ⎟ ⎟⎟ σα¯ ⎠ ⎠
i= 1
(36)
i=1
Proof of Property 5: According to Definition 6, we have rαi =
2σαr i (1 + 2ρα i )r − 1 , 2 (2 + σα i )r − σαr i
(37)
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013
then
⎛ n
⎜ i=1 IMWA (rα1 , rα2 , . . . , rαn ) = ⎜ ⎝ ⎛ n
⎜ i=1 =⎜ ⎝
(1+2ρ α i ) r −1 2
1+2
w i
−1 , n
2
i=1
(1 + 2ρα i )r w i − 1
2+
2
w i
−
σαr wi i
i=1
n
(2 + σα i )r w i −
i=1
⎞
w i
n
i=1
⎞
n
2
2σ αr i (2+σ α i ) r −σ αr i
2σ αr i (2+σ α i ) r −σ αr i
i=1
, n
n
2
2σ αr i (2+σ α i ) r −σ αr i
⎟ w i ⎟ ⎠
⎟ ⎟ ⎠
σαr wi i
(38)
i=1
and ⎛⎛
n
⎜⎜ ⎜ ⎝1 + 2 i = 1 ⎜ ⎜ ⎜ r IMWA (α1 , α2 , . . . , αn ) = ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ n
⎜ i=1 =⎜ ⎝
(1+2ρ α i ) w i −1 2
⎛
⎞r
⎜ 2⎝ n
⎟ ⎠ −1 , ⎛
2
⎜ ⎝2 + n
2
n
i= 1
(1 + 2ρα i )r w i − 1
2 , n
2
n i=1
(2 + σα i
−
i=1
⎞r
n
w
σ α ii
i= 1
n
n
w σ α ii
⎞
⎟ ⎠
i= 1
⎛
⎟ ⎜ ⎠ −⎝ n
2
n
w
σ α ii
i= 1
(2+σ α i ) w i −
i= 1
⎞
σαr wi i
)r w i
(2+σ α i ) w i −
w σ α ii
(2+σ α i ) w i −
⎞r
w
σ α ii
i= 1
i= 1
i= 1
n
2
n i= 1
⎟ ⎟. ⎠
σαr wi i
(39)
i=1
Proof of Property 6: According to Definition 6, we have 2σαr i (1 + 2ρα i )r − 1 rαi = , 2 (2 + σα i )r − σαr i and
⎛
(40)
⎞ 2σ r 2 (2+σ α α) ri −σ r σα¯ αi i ⎠ , ¯=⎝ rαi ⊕ α 2σ r 2σ r 2 2 + (2+σ α α) ri −σ r (2 + σα¯ ) − (2+σ α α) ri −σ r σα¯ αi αi i i r r 2σ σ (1 + 2ρα i ) (1 + 2ρα¯ ) − 1 ¯ αi α = , . 2 (2 + σα i )r (2 + σα¯ ) − σαr i σα¯ 1+2
(1+2ρ α i ) r −1 2
w
σ α ii
⎟ ⎟ ⎟ ⎟ ⎟ ⎞r ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠
(1 + 2ρα¯ ) − 1
(41)
Then, we have ¯ , rα2 ⊕ α ¯ , . . . , rαn ⊕ α ¯) IMWA (rα1 ⊕ α ⎛
n (1+2ρ α i ) r (1+2ρ α¯ )−1 w i 1+2 −1 2 ⎜ i=1 =⎜ , n ⎝ 2 ⎛ n
⎜ i=1 =⎜ ⎝
i=1
2+
i=1
((1 + 2ρα i )r (1 + 2ρα¯ ))w i − 1 2
n i=1
⎛ n
((2 + σα i )r (2 + σα¯ ))w i −
⎜ i=1 r IMWA (α1 , α2 , . . . , αn ) ⊕ α ¯=⎜ ⎝
2
2 , n i=1
w i
σ α¯
n i=1
(1 + 2ρα i )r w i − 1
2σ αr i σ α¯ (2+σ α¯ )−σ αr i σ α¯
n i=1
−
n i=1
(σαr i σα¯ )w i
2σ αr i σ α¯ (2+σ α i ) r (2+σ α¯ )−σ αr i
⎟ w i ⎟ ⎠
n i=1
σαr wi i
σ α¯
⎞ ⎟ ⎟ ⎠
⎞ σαr wi i
(2 + σα i )r w i −
⎞
w i
)r
(σαr i σα¯ )w i
i=1
and
(2+σ α i
2σ αr i σ α¯ (2+σ α i ) r (2+σ α˙ )−σ αr i
2 , n
n
2
⎟ ⎟⊕α ⎠ ¯
(42)
XIA et al.: PREFERENCE RELATIONS BASED ON INTUITIONISTIC MULTIPLICATIVE INFORMATION
⎛⎛
n
⎜⎜ ⎜ ⎝1 + 2 i = 1 ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝ ⎛ n
⎜ i=1 =⎜ ⎝
(1+2ρ α i ) r w i −1 2
131
⎞
2
⎟ ⎠ (1 + 2ρα¯ ) − 1
2 n , ⎛
2
2
⎜ ⎝2 + n
i= 1
, n
2
rwi
σα i
n
(2 + σα i )r w i (2 + σα¯ ) −
i=1
−
rw σα i i
i= 1
σα¯
⎛
σαr wi i σα¯
2
n
rwi
σα i
i= 1
(2+σ α i ) r w i −
i= 1
⎞
σαr wi i σα¯
i=1
n
⎟ ⎜ ⎠ (2 + σα¯ ) − ⎝ n
i= 1
n
2
n
(2+σ α i ) r w i −
)r w i
⎞
rw σα i i
i= 1
(1 + 2ρα i )r w i (1 + 2ρα¯ ) − 1
(2+σ α i
⎞
rwi
σα i
i= 1
i= 1
n
n
n
rwi
⎟ ⎟ ⎟ ⎟ ⎟ ⎞⎟ ⎟ ⎟ ⎟⎟ σα¯ ⎠ ⎠
σα i
i= 1
⎟ ⎟. ⎠
(43)
i=1
Proof of Property 7: According to Definition 6, we have (1 + 2ρα i )(1 + 2ρα¯ i ) − 1 2σα i σα¯ i αi ⊕ α , ¯i = 2 (2 + σα i )(2 + σα¯ i ) − σα i σα¯ i
(44)
and ¯ 1 , α2 ⊕ α ¯ 2 , . . . , αn ⊕ α ¯n ) IMWA (α1 ⊕ α ⎛ w i n (1+2ρ α i )(1+2ρ α¯ i )−1 1+2 −1 2 ⎜ i=1 , =⎜ n ⎝ 2 ⎛ n
⎜ i=1 =⎜ ⎝
i=1
2σ α i σ α¯ i (2+σ α i )(2+σ α¯ i )−σ α i σ α¯ i
2σ α i σ α¯ i (2+σ α i )(2+σ α¯ i )−σ α i σ α¯ i
2+
i=1
((1 + 2ρα i )(1 + 2ρα¯ i ))w i − 1
2
w i
−
n
i=1
⎞
w i
2σ α i σ α¯ i (2+σ α i )(2+σ α¯ i )−σ α i σ α¯ i
⎟ w i ⎟ ⎠
⎞
n
(σα i σα¯ i )w i
i=1
, n
2
n
2
((2 + σα i )(2 + σα¯ i ))w i −
i=1
n
⎟ ⎟ ⎠
(σα i σα¯ i )w i
(45)
i=1
and IMWA (α1 , α2 , . . . , αn ) ⊕ IMWA (¯ α1 , α ¯2 , . . . , α ¯n ) ⎛⎛ ⎞⎛ n n ⎜⎜ ⎜ ⎝1 + 2 i = 1 ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝
(1+2ρ α i ) w i −1 2
⎟⎜ i= 1 ⎠ ⎝1 + 2
(1+2ρ α¯ i ) w i −1 2
2 n ⎛
⎛ n
⎜ i=1 =⎜ ⎝ ⎛ n
⎜ i=1 =⎜ ⎝
⎟ ⎠−1
2
2
⎜ ⎝2 + n
⎞
2
n
⎞⎛
w σ α ii
i= 1
(2+σ α i ) w i −
i= 1
n
w
σ α ii
n
(1 + 2ρα i )w i
w
σ α ii
i= 1
(2+σ α i
i= 1
⎟⎜ ⎠ ⎝2 + n
i= 1
n
2
)w i n
−
2 n
(2+σ α˙ i ) w i −
n
w
σ α¯ i
2
(2 + σα i )w i
2
i=1
n
n
w
σ α ii
n
n
σαwii
2
i= 1
n
(2+σ α i ) w i −
n i=1
n
((2 + σα i )(2 + σα¯ i ))w i −
i=1
(σα i σα¯ i )w i
n i=1
w
σ α¯ i i
(2+σ α˙ i ) w i −
i= 1
σαwii
i=1
n
n
n i= 1
σαw¯ ii
(σα i σα¯ i )w i
i=1
w
σ α ii
i= 1
(2 + σα¯ i )w i −
i=1
2 , n
2
w σ α¯ i i
i= 1
i=1
i=1
n i= 1
⎟ ⎜ ⎠−⎝ n
2
((1 + 2ρα i )(1 + 2ρα¯ i ))w i − 1
⎛
i= 1
, n
−
)w i
i
(1 + 2ρα¯ i )w i − 1
i=1
i
(2+σ α˙ i
i= 1
w σ α¯ i i
⎞
w
σ α¯ i
i= 1
⎞
i= 1
i= 1
i= 1
n
w σ α ii
n
σαw¯ ii
⎞
n
w
σ α¯ i
⎟ ⎟ ⎟ ⎟ ⎟ ⎞⎟ ⎟ ⎟ ⎟⎟ ⎠⎠
i
i= 1
⎟ ⎟ ⎠
⎞
⎟ ⎟. ⎠
(46)
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper.
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XIA et al.: PREFERENCE RELATIONS BASED ON INTUITIONISTIC MULTIPLICATIVE INFORMATION
Meimei Xia received the B.S. degree in information management and information systems, and the M.S. degree in operational research from Qufu Normal University, RiZhao, China, in 2006 and 2009, respectively, and the Ph.D. degree in management science and engineering from Southeast University, Nanjing, China, in 2012. She is currently a Postdoctoral Researcher with the School of Economics and Management, Tsinghua University, Beijing, China. Her current research interests include fuzzy information fusion, aggregation operators, and group decision making.
Zeshui Xu (M’08–SM’09) received the Ph.D. degree in management science and engineering from Southeast University, Nanjing, China, in 2003. From April 2003 to May 2005, he was a Postdoctoral Researcher with the School of Economics and Management, Southeast University. From October 2005 to December 2007, he was a Postdoctoral Researcher with the School of Economics and Management, Tsinghua University, Beijing, China. He is currently a Chair Professor with the Institute of Sciences, PLA University of Science and Technology, Nanjing. He is a member of the Editorial Boards of Information: An International Journal, the International Journal of Applied Management Science, the International Journal of Data Analysis Techniques and Strategies, System Engineering—Theory and Practice, Fuzzy Systems and Mathematics, the Journal of Systems Engineering, and the Chinese Journal of Management Science. He has authored the following books: Uncertain Multiple Attribute Decision Making: Methods and Applications (Tsinghua Univ. Press, 2004), Intuitionistic Fuzzy Information: Aggregation Theory and Applications (Science Press, 2008), and Decision Making with Linguistic Information: Theory and Methods (Science Press, 2008). He has contributed more than 300 journal articles to professional journals. His current research interests include information fusion, group decision making, computing with words, and aggregation operators.
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Huchang Liao received the B.S. degree in information and computing science from Guizhou Minzu University, Guiyang, China, in 2009 and the M.S. degree in management science and engineering from the Kunming University for Science and Technology, Kunming, China, in 2011. He is currently working toward the Ph.D. degree with the Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China. His current research interests include fuzzy information fusion, aggregation operators, and group decision making.