Group Decis Negot (2010) 19:57–76 DOI 10.1007/s10726-009-9164-z
A Deviation-Based Approach to Intuitionistic Fuzzy Multiple Attribute Group Decision Making Zeshui Xu
Published online: 29 May 2009 © Springer Science+Business Media B.V. 2009
Abstract The aim of this article is to investigate the approach to multiple attribute group decision making (MAGDM) with intuitionistic fuzzy information. We first introduce a deviation measure between two intuitionistic fuzzy numbers, and then utilize the intuitionistic fuzzy hybrid aggregation operator to aggregate all individual intuitionistic fuzzy decision matrices into a collective intuitionistic fuzzy decision matrix. Based on the deviation measure, we develop an optimization model by which a straightforward formula for deriving attribute weights can be obtained. Furthermore, based on the intuitionistic fuzzy weighted averaging operator and information theory, we utilize the score function and accuracy function to give an approach to ranking the given alternatives and then selecting the most desirable one(s). In addition, we extend the above results to MAGDM with interval-valued intuitionistic fuzzy information. Keywords Intuitionistic fuzzy set (IFS) · Multiple attribute group decision making (MAGDM) · Deviation measure · Intuitionistic fuzzy weighted averaging (IFWA) operator · Intuitionistic fuzzy ordered weighted averaging (IFOWA) operator · Intuitionistic fuzzy hybrid aggregation (IFHA) operator
1 Introduction Atanassov (1986, 1999) introduced the intuitionistic fuzzy set (IFS), which is more suitable for dealing with fuzziness and uncertainty than the ordinary fuzzy set developed by Zadeh (1965). In general, the IFS is a generalization of Zadeh’ fuzzy set. Gau and Buehrer (1993) gave the notion of vague set, but Bustince and Burillo (1996)
Z. Xu (B) Antai School of Economic and Management, Shanghai Jiaotong University, 200052 Shanghai, China e-mail:
[email protected]
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showed that it is an equivalent of the IFS. Some authors have given the applications of the IFS theory to decision making. For example, Szmidt and Kacprzyk (2004) proposed a similarity measure to assess the extent of agreement in a group of experts giving their opinions expressed by intuitionistic fuzzy preference relations. Szmidt and Kacprzyk (2002a) gave a method based on fuzzy linguistic majority Kacprzyk (1986) for group decision making with intuitionistic fuzzy preference relations. Atanassov et al. (2005) provided a tool to solve the MAGDM problems, in which the attribute weights are given as exact numerical values and the alternative scores (attribute values) corresponding to all attributes are expressed in intuitionistic fuzzy numbers (IFNs). Chen and Tan (1994) introduced a score function, and utilized it and the minimum and maximum operations to deal with multiple attribute decision making problems based on vague sets. Later, Hong and Choi (2000) defined an accuracy function to evaluate the accuracy degree of vague value, and used the minimum and maximum operations, score function and accuracy function, to develop another technique for handling multiple attribute decision making problems. The aforementioned techniques using the minimum and maximum operations to carry the combination process produce the loss of information, and hence lack precision in the final results. Li (2005), and Lin et al. (2007) proposed some methods to solve the single-person multiple attribute decision making problems with IFSs and partial weight information. However, in many MAGDM situations, such as group negotiations, high technology project investment of venture capital firms, medical diagnosis, personnal evaluations, strategic planning, supply chain management, and forecasting, etc. Xu (2004), the decision makers may provide their preferences for alternatives to a certain degree, but it is possible that they are not so sure about it Deschrijver and Kerre (2003); Xu (2007a), and the information about attribute weights may be completely unknown due to time pressure or because of the complexity and uncertainty of the problem. In such cases, the preference information provided by decision makers is very suitable to be expressed in IFNs, and the attribute weights can only be derived from the given intuitionistic fuzzy information. In this paper, we shall focus our attention on this issue. The remainder of the paper is organized as follows. In Sect. 2, we review some basic notions and aggregation operators. In Sect. 3, we introduce a deviation measure between two IFNs, and then utilize the intuitionistic fuzzy hybrid aggregation (IFHA) operator to aggregate all individual intuitionistic fuzzy decision matrices into a collective intuitionistic fuzzy decision matrix. Furthermore, we establish an optimization model to derive attribute weights and develop an approach to MAGDM with intuitionistic fuzzy information. Sect. 4 provides a numerical example to illustrate the developed approach, and finally, in Sect. 5, we conclude the paper.
2 Basic Notions and Aggregation Operators In this section, we review some basic notions and aggregation operators related to intuitionistic fuzzy sets (IFSs):
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Definition 2.1 Atanassov (1986). Let a set X = {x1 , x2 , . . . , xn } be fixed, an IFS A in X is an object of the following form: A = {< xi , t A (xi ), f A (xi ) > |xi ∈ X }
(1)
where the functions t A : X → [0, 1] and f A : X → [0, 1] determine the membership degree and non-membership degree of the element xi ∈ X , respectively, and for every xi ∈ X : 0 ≤ t A (xi ) + f A (xi ) ≤ 1
(2)
π A (xi ) = 1 − t A (xi ) − f A (xi ), for all xi ∈ X
(3)
For each IFS A in X , if
then π A (xi ) is called an indeterminacy degree of xi to A Atanassov (1986), or called a hesitancy degree of xi to A Szmidt and Kacprzyk (2002b). Especially, if π A (xi ) = 0, for all xi ∈ X , then the IFS A reduces to an ordinary fuzzy set. For convenience, we call α = (tα , f α ) an intuitionistic fuzzy number (IFN) Xu (2007b), where tα ∈ [0, 1],
f α ∈ [0, 1], tα + f α ≤ 1
(4)
and let be the set of all IFNs. Let α1 = tα1 , f α1 and α2 = tα2 , f α2 be two IFNs, then we call d (α1 , α2 ) =
1 |tα1 − tα2 | + | f α1 − f α2 | 2
(5)
the distance between α1 and α2 . By (5), we have: Theorem 2.1 Let αi (i = 1, 2, 3) be any three IFNs, then 1. 0 ≤ d (α1 , α2 ) ≤ 1, especially, d (α1 , α1 ) = 0; 2. d (α1 , α2 ) = d (α2 , α1 ); Proof See Appendix A. Xu (2007b) introduced some operations and relations related to IFNs as follows: Definition 2.2 Let α1 = tα1 , f α1 , α2 = tα2 , f α2 and α = (tα , f α ) be three IFNs, then 1. α1 ⊕ α2 = tα1 + tα2 − tα1 tα2 , f α1 f α2 ; 2. λα = 1 − (1 − tα )λ , f αλ , λ > 0. Theorem 2.2 Let α1 = tα1 , f α1 , α2 = tα2 , f α2 and α = (tα , f α ) be three IFNs, and let β = α1 ⊕ α2 and γ = λα (λ > 0), then both β and γ are also IFNs.
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Proof See Appendix A. Theorem 2.3 Let α1 = tα1 , f α1 , α2 = tα2 , f α2 and α = (tα , f α ) be three IFNs, λ, λ1 , λ2 > 0, then 1. α1 ⊕ α2 = α2 ⊕ α1 ; 2. λ (α1 ⊕ α2 ) = λα1 ⊕ λα2 ; 3. λ1 α ⊕ λ2 α = (λ1 + λ2 ) α. Proof See Appendix A. Definition 2.4 Let α j = tα j , f α j ( j = 1, 2, . . . , n) be a collection of IFNs, and let IFWA: n → , if IFWAw (α1 , α2 , . . . , αn ) = w1 α1 ⊕ w2 α2 ⊕ · · · ⊕ wn αn
(6)
then IFWA is called an intuitionistic fuzzy weighted averaging (IFWA) operator of dimension n , where w = (w1 , w2 , . . . , wn )T is the weight vector of α j ( j = 1, 2, . . . , n), with w j ∈ [0, 1] and nj=1 w j = 1. The aggregated value by using the IFWA operator is also an IFN, and satisfies: ⎞ n n w j wj 1 − tα j , fα j ⎠ IFWAw (α1 , α2 , . . . , αn ) = ⎝1 − ⎛
j=1
(7)
j=1
Chen and Tan (1994) gave a score function to measure an IFN: Definition 2.5 Let α = (tα , f α ) be an IFN, then s(α) = tα − f α
(8)
is called a score of α, and s is called a score function, where s(α) ∈ [−1, 1]. The larger the score s(α), the greater the degree of deviation between tα and f α , which means that the value tα gets bigger and the value f α gets smaller, and thus the IFN α gets greater. Later, Hong and Choi (2000) further defined an accuracy function as follows: Definition 2.6 Let α = (tα , f α ) be an IFN, then h (α) = tα + f α
(9)
is called an accuracy degree of α, and h is called an accuracy function, where h(α) ∈ [0, 1]. The larger the value of h(α), the more the accuracy degree of α. Xu (2007b) developed a method for the comparison between two IFNs, which is based on the score function s and the accuracy function h:
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Definition 2.7 Let α1 = tα1 , f α1 and α2 = tα2 , f α2 be two IFNs, then • If s (α1 ) < s (α2 ), then α1 is smaller than α2 , denoted by α1 < α2 ; • If s (α1 ) = s (α2 ), then 1. If h(α1 ) = h(α2 ), then α1 and α2 represent the same information, denoted by α1 = α2 ; 2. If h(α1 ) < h(α2 ), then α1 is smaller than α2 , denoted by α1 < α2 . In Yager (1988), Yager introduced an ordered weighted averaging (OWA) operator which weights the ordered positions of the real-valued data instead of weighting the data themselves: Definition 2.8 An ordered weighted averaging (OWA) operator of dimension n is a T mapping OWA:R n → nR, that has an associated vector ω = (ω1 , ω2 , . . . , ωn ) such that ω j ∈ [0, 1] and j=1 ω j = 1. Furthermore OWA (b1 , b2 , . . . , bn ) =
n
ωjcj
(10)
j=1
where c j is the jth largest of bi (i = 1, 2, . . . , n), and R is the set of all real numbers. Clearly, the fundamental aspect of the OWA operator is the re-ordering step. Based on Definition 2.7, Xu (2007b) extended the OWA operator (10) to accommodate the situations where the input data are IFNs. Definition 2.9 Let α j = tα j , f α j ( j = 1, 2, . . . , n) be a collection of IFNs. An intuitionistic fuzzy OWA (IFOWA) operator of dimension n is a mapping IFOWA: T n → n, that has an associated vector ω = (ω1 , ω2 , . . . , ωn ) , such that ω j ∈ [0, 1] and j=1 ω j = 1. Furthermore, IFOWAω (α1 , α2 , . . . , αn ) = ω1 ασ (1) ⊕ ω2 ασ (2) ⊕ · · · ⊕ ωn ασ (n)
(11)
where (σ (1), σ (2), . . . , σ (n)) is a permutation of (1, 2, . . . , n), such that ασ ( j−1) ≥ ασ ( j) for all j. The aggregated value by using the IFOWA operator is also an IFN, and satisfies: ⎞ n n ω j ωj 1 − tασ ( j) , f ασ ( j) ⎠ IFOWAω (α1 , α2 , . . . , αn ) = ⎝1 − ⎛
j=1
(12)
j=1
The IFOWA operator has some desirable properties similar to those of the OWA operator, such as commutativity, idempotency, monotonicity, and boundedness Yager (1988). However, the IFWA operator weights only the IFNs, while the IFOWA operator weights only the ordered positions of the IFNs instead of weighting the IFNs themselves. To overcome this limitation, Xu (2007b) developed an intuitionistic fuzzy hybrid aggregation (IFHA) operator as follows:
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Definition 2.10 An intuitionistic fuzzy hybrid aggregation (IFHA) operator of dimenn → , which has an associated vector ω = (ω , ω , . . . , sion n is a mapping IFHA: 1 2 T ωn ) , with ω j ∈ [0, 1] and nj=1 ω j = 1 such that IFHAω,w (α1 , α2 , . . . , αn ) = ω1 α˙ σ (1) ⊕ ω2 α˙ σ (2) ⊕ · · · ⊕ ωn α˙ σ (n)
(13)
where α˙ σ ( j) is the jth largest of the weighted IFNs α˙ j α˙ j = nw j α j , j = 1, 2, . . . , n , w = (w1 , w2 , . . . , wn )T is the weight vector of α j ( j = 1, 2, . . . , n), with w j ∈ [0, 1] and nj=1 w j = 1, and n is the balancing coefficient, which plays a role of balance [in such a case, if the vector (w1 , w2 , . . . , wn )T approaches (1/n, 1/n, . . . , 1/n)T , then the vector (nw1 α1 , nw2 α2 , . . . , nwn αn )T approaches (α1 , α2 , . . . , αn )T ]. Especially, if ω = (1/n, 1/n, . . . , 1/n)T , then the IFHA operator reduces to the IFWA operator; if w = (1/n, 1/n, . . . , 1/n)T , then the IFHA operator reduces to the IFOWA operator. Let α˙ σ ( j) = tα˙ σ ( j) , f α˙ σ ( j) ( j = 1, 2, . . . , n; j = 1, 2, . . . , n), then ⎞ n n ω j ω IFHAω,w (α1 , α2 , . . . , αn ) = ⎝1 − 1 − tα˙ σ ( j) , f α˙ σj( j) ⎠ ⎛
j=1
(14)
j=1
and the aggregated value derived by using the IFHA operator is also an IFN. The IFHA operator weights all the given IFNs and their ordered positions, and thus generalizes both the IFWA and IFOWA operators. The IFHA operator is very suitable to be applied to MAGDM with intuitionistic fuzzy information, in the process of information aggregation, the IFHA not only can consider the weight of each decision maker, but also can relieve the influence of unfair data (too large or too small IFNs) on the decision result by assigning low weights to those unfair ones. Atanassov and Gargov (1989) introduced the notion of interval-valued intuitionistic fuzzy set (IVIFS), which is characterized by a membership function and a non-membership function whose values are intervals rather than exact numbers. An IVIFS A˜ over X is an object having the form:
A˜ = < xi , µ˜ A˜ (xi ) , v˜ A˜ (xi ) > |xi ∈ X
(15)
where µ˜ A˜ (xi ) ⊂ [0, 1] and v˜ A˜ (xi ) ⊂ [0, 1] are intervals, and for every xi ∈ X : sup µ˜ A˜ (xi ) + sup v˜ A˜ (xi ) ≤ 1
(16)
Xu and Chen (2007) called the pair µ˜ A˜ (xi ), v˜ A˜ (xi ) an interval-valued intuitionistic fuzzy number (IVIFN), and denoted an IVIFN by ([a, b], [c, d]), where [a, b] ⊂ ([0, 1], [c, d]) ⊂ [0, 1], b + d ≤ 1
(17)
˜ be the set of all IVIFNs. Then the above result of IFSs can be extended to and let IVIFNs Xu and Chen (2007).
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Let α˜ 1 = ([a1 , b1 ], [c1 , d1 ]) and α˜ 2 = ([a2 , b2 ], [c2 , d2 ]) be two IVIFNs, then we call d (α˜ 1 , α˜ 2 ) =
1 (|a1 − a2 | + |b1 − b2 | + |c1 − c2 | + |d1 − d2 |) 4
(18)
the distance between α˜ 1 and α˜ 2 . By (18), we have: Theorem 2.4 Let α˜ i (i = 1, 2, 3) be any three IVIFNs, then 1. 0 ≤ d (α˜ 1 , α˜ 2 ) ≤ 1, especially, d (α˜ 1 , α˜ 1 ) = 0; 2. d (α˜ 1 , α˜ 2 ) = d (α˜ 2 , α˜ 1 ); 3. d (α˜ 1 , α˜ 3 ) ≤ d (α˜ 1 , α˜ 2 ) + d (α˜ 2 , α˜ 3 ). Proof See Appendix A. Let α˜ 1 = ([a1 , b1 ], [c1 , d1 ]) , α˜ 2 = ([a2 , b2 ], [c2 , d2 ]) and α˜ = ([a, b], [c, d]) be three IVIFNs, then 1. α˜ 1 ⊕ α˜ 2 = ([a1 + a2 − a1 a2 , b1 + b 2 − b1 b2 ] , [c1 c2 , d1 d2 ]) ; 2. λα˜ = 1 − (1 − a)λ , 1 − (1 − b)λ , cλ , d λ , λ > 0. Both α˜ 1 ⊕ α˜ 2 and λα˜ are also IVIFNs. Definition 2.11 Let α˜ j = [a j , b j ], [c j , d j ] ( j = 1, 2, . . . , n) be a collection of ˜ n → , ˜ if IVIFNs, and let IVIFWA: IVIFWAw (α˜ 1 , α˜ 2 , . . . , α˜ n ) = w1 α˜ 1 ⊕ w2 α˜ 2 ⊕ · · · ⊕ wn α˜ n
(19)
then IVIFWA is called an interval-valued intuitionistic fuzzy weighted averaging (IVT IFWA) operator of dimension n, where w = (w n1 , w2 , . . . , wn ) is the weight vector of α˜ j ( j = 1, 2, . . . , n), with w j ∈ [0, 1] and j=1 w j = 1. The aggregated value by using the IVIFWA operator is also an IVIFN, and satisfies: IVIFWAw (α˜ 1 , α˜ 2 , . . . , α˜ n ) ⎛⎡ ⎤ ⎡ ⎤⎞ n n n n w j w j w w ⎦, ⎣ = ⎝⎣1 − 1 − aj 1 − bj , 1− c j j, d j j ⎦⎠ j=1
j=1
j=1
j=1
(20) Let α˜ = ([a, b], [c, d]) be an IVIFN, then s(α) ˜ =
1 (a − c + b − d) 2
(21)
is a score of α, ˜ and s is a score function, where s(α) ˜ ∈ [−1, 1]. Moreover, h(α) ˜ =
1 (a + b + c + d) 2
(22)
is an accuracy degree of α, ˜ and h is an accuracy function, where h(α) ˜ ∈ [0, 1].
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Similar to Definition 2.7, in the following, we introduce a method for the comparison between two IVIFNs: Definition 2.12 Let α˜ 1 = ([a1 , b1 ], [c1 , d1 ]) and α˜ 2 = ([a2 , b2 ], [c2 , d2 ]) be two IVIFNs, then • If s (α˜ 1 ) < s (α˜ 2 ), then α˜ 1 is smaller than α˜ 2 , denoted by α˜ 1 < α˜ 2 ; • If s(α˜ 1 ) = s(α˜ 2 ), then 1. If h(α˜ 1 ) = h(α˜ 2 ), then α˜ 1 and α˜ 2 have no difference, denoted by α˜ 1 ∼ α˜ 2 ; 2. If h (α˜ 1 ) < h (α˜ 2 ), then α˜ 1 is smaller than α˜ 2 , denoted by α˜ 1 < α˜ 2 . Based on Definition 2.12, Xu and Chen (2007) developed the following two interval-valued intuitionistic fuzzy aggregation operators: Definition 2.13 Let α˜ j = [a j , b j ], [c j , d j ] ( j = 1, 2, . . . , n) be a collection of IVIFNs. An interval-valued intuitionistic fuzzy OWA (IVIFOWA) operator of dimen˜ that has an associated vector ω=(ω1 , ω2 , . . . , ˜ n → , sion n is a mapping IVIFOWA: T ωn ) , such that ω j ∈ [0, 1] and nj=1 ω j = 1. Furthermore, IVIFOWAω (α˜ 1 , α˜ 2 , . . . , α˜ n ) = ω1 α˜ σ (1) ⊕ ω1 α˜ σ (1) ⊕ · · · ⊕ ωn α˜ σ (n)
(23)
where (σ (1), σ (2), . . . , σ (n)) is a permutation of (1, 2, . . . , n), such that α˜ σ ( j−1) > If α˜ σ ( j−1) ∼ α˜ σ ( j) , then we replace α˜ σ ( j) or α˜ σ ( j−1) ∼ α˜ σ ( j) , for all j. Espeically, both α˜ σ ( j−1) and α˜ σ ( j) by their average α˜ σ ( j−1) ⊕ α˜ σ ( j) /2. The aggregated value by using the IVIFOWA operator is also an IVIFN, and satisfies: IVIFOWAω (α˜ 1 , α˜ 2 , . . . , α˜ n ) ⎛⎡ ⎤ ⎡ ⎤⎞ n n n n ω j ω j ω ω = ⎝⎣1 − , 1− cσ (j j) , dσ (j j) ⎦⎠ 1 − aσ ( j) 1 − bσ ( j) ⎦ , ⎣ j=1
j=1
j=1
j=1
(24) Definition 2.14 An interval-valued intuitionistic fuzzy hybrid aggregation (IVIFHA) ˜ n → , ˜ which has an associated operator of dimension n is a mapping IVIFHA: n T vector ω = (ω1 , ω2 , . . . , ωn ) , with ω j ∈ [0, 1] and j=1 ω j = 1, such that n
IVIFHAω,w (α˜ 1 , α˜ 2 , . . . , α˜ n ) = ⊕
j=1
ω j α˙˜ σ ( j)
(25)
where α˙˜ σ ( j) is the jth largest of the weighted IVIFNs α˙˜ j α˙˜ j =nw j α˜ j , j=1, 2, . . . , n , w = (w1 , w2 , . . . , wn )T is the weight vector of α˜ j ( j = 1, 2, . . . , n) , with w j ∈ [0, 1] and nj=1 w j = 1, and n is the balancing coefficient, which plays a role of balance. Especially, if ω = (1/n, 1/n, . . . , 1/n)T , then the IVIFHA operator reduces T , then the IVIFHA operator to the IVIFWA operator; if w = (1/n, 1/n, . . . , 1/n)
˙ reduces to the IVIFOWA operator. If let α˜ σ ( j) = a˙ σ ( j) , b˙σ ( j) , c˙σ ( j) , d˙σ ( j) ( j=1, 2, . . . , n), then
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IVIFHAω,w (α˜ 1 , α˜ 2 , . . . , α˜ n ) ⎛⎡ ⎤ ⎡ ⎤⎞ n n n n ω j ω j ωj ωj 1 − a˙ σ ( j) 1 − b˙σ ( j) ⎦ , ⎣ = ⎝⎣1 − , 1− c˙σ ( j) , d˙σ ( j) ⎦⎠ j=1
j=1
j=1
j=1
(26) and the aggregated value derived by using the IVIFHA operator is also an IVIFN. 3 An Optimization Model The MAGDM problem considered in this paper can be represented as follows: Let B = {B1 , B2 , . . . , Bn } be a finite set of alternatives, E = {e1 , e2 , . . . , el } T be the set of decision makers, and l ξ = (ξ1 , ξ2 , . . . , ξl ) be the weight vector of decision makers, ξk ∈ [0, 1] and k=1 ξk = 1. Each decision maker suggests a starting set of attributes, after that, the group discuss and analyse all these attributes, and then chose a feasible set of attributes from them, without loss of generality, let T ...,w G = {G 1 , G 2 , . . . , G m } be a feasible set of attributes, and w = (w1 , w2 , m ) m (k) be the weight vector of attributes, wi ∈ [0, 1] and i=1 wi = 1. Let R (k) = ri j m×n (k) (k) (k) be an intuitionistic fuzzy decision matrix, where ri j = ti j , f i j is an IFN, provided by the decision maker ek ∈ E for the alternative B j ∈ B with respect to the (k) attribute G i ∈ G, ti j indicates the degree that the alternative B j should satisfy the attribute G i expressed by the decision maker ek ∈ E, while f i(k) j indicates the degree that the alternative B j should not satisfy the attribute G i , expressed by the decision maker ek ∈ E, and (k)
ti j ∈ [0, 1],
(k)
(k)
(k)
f i j ∈ [0, 1], ti j + f i j ≤ 1, i = 1, 2, . . . , m; j = 1, 2, . . . , n (27)
To obtain the collective intuitionistic fuzzy decision information, we (14) to utilize (k) aggregate all individual intuitionistic fuzzy decision matrices R (k) = ri j (k = m×n
1, 2, . . . , l) into a collective intuitionistic fuzzy decision matrix R = (ri j )m×n , where ri j = IFHAω,ξ
(1) (2) (l) (σ (1)) (σ (1)) (σ (l)) ri j , ri j , . . . , ri j = ω1r˙i j ⊕ ω1r˙i j ⊕ · · · ⊕ ωl r˙i j ,
i = 1, 2, . . . , m; j = 1, 2, . . . , n
(28)
(k) (k) where r˙i(σj (k)) is the kth largest of the weighted IFNs r˙i(k) j r˙i j =lξk ri j , k = 1, 2, . . . , l . Based on the collective intuitionistic fuzzy decision matrix R = (ri j )m×n , we utilize the IFWA operator: r j = w1 r 1 j ⊕ w2 r 2 j ⊕ · · · ⊕ wm r m j ,
j = 1, 2, . . . , n
(29)
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to get the overall value of the alternative B j . The greater the value of r j , the better the alternative B j is. In the situations where the information about attribute weights is completely known, we develop the following approach to MAGDM with intuitionistic fuzzy information: (Approach I) Step 1. Utilize (28)to aggregate all individual intuitionistic fuzzy decision matri (k) (k = 1, 2, . . . , l) into a collective intuitionistic fuzzy ces R (k) = ri j m×n
decision matrix R = (ri j )m×n . Step 2. Utilize (29) to get the overall values r j ( j = 1, 2, . . . , n) of the alternatives B j ( j = 1, 2, . . . , n). Step 3. Utilize (8) to calculate the score s(r j ) of the overall value r j of the alternative Bj. Step 4. Utilize the scores s(r j )( j = 1, 2, . . . , n) to rank the alternatives B j ( j = 1, 2, . . . , n), and then to select the most desirable one(s) [if there is no difference between two scores s(ri ) and s(r j ), then we need to calculate the accuracy degrees h(ri ) and h(r j ) of the overall values ri and r j by using (9), respectively. After that, we rank the alternatives Bi and B j in accordance with the accuracy degrees h(ri ) and h(r j )]. If the information about the attribute weights is completely unknown in the considered problem, then we need to determine the attribute weights in advance. According to the information theory, if all alternatives have similar attribute values with respect to an attribute, then a small weight should be assigned to the attribute, this is due to that such attribute does not help in differentiating alternatives Zeleny (1982). By (5), we introduce the deviation between the alternative B j and the other alternatives with respect to the attribute G i : di j (w) =
d ri j , rik wi , i = 1, 2, . . . , m; j = 1, 2, . . . , n
(30)
k= j
and let n di (w) = d ri j , rik wi , i = 1, 2, . . . , m
(31)
j=1 k= j
denote the sum of all the deviations di j (w) ( j = 1, 2, . . . , n). Then we construct the deviation function: d(w) =
m i=1
di (w) =
m n i=1 j=1
di j (w) =
m n
d(ri j , rik )wi
(32)
i=1 j=1 k= j
Obviously, a reasonable vector of attribute weights w = (w1 , w2 , . . . , wm )T should be determined so as to maximize d(w), and thus, we establish the following optimization model:
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Maximized(w) =
m n
67
d(ri j , rik )wi
i=1 j=1 k= j
Subject to:
m
wi2 = 1, wi ≥ 0, i = 1, 2, . . . , m
i=1
To solve this model, we construct the Lagrange function: m 1 2 L(w, ζ ) = d(w) + ζ wi − 1 2
(33)
i=1
where ζ is the Lagrange multiplier. Differentiating (33) with respect to wi (i = 1, 2, . . . , m) and ζ , and setting these partial derivatives equal to zero, the following set of equations is obtained: ∂ L(w,ζ
) ∂wi ∂ L(w,ζ ) ∂ζ
= nj=1 k= j d ri j , rik + ζ wi = 0 m = i=1 wi2 − 1 = 0
(34)
∗ T , where By solving (34), we get the optimal solution w∗ = w1∗ , w2∗ , . . . , wm n wi∗
=
j=1
m n i=1
k= j
j=1
d ri j , rik
, i = 1, 2, . . . , m 2 k= j d ri j , rik
(35)
Obviously, wi∗ ≥ 0, for all i. Normalizing (35), we get the normalized attribute weights: n
wi =
j=1 k= j d ri j , rik , m n i=1 j=1 k= j d ri j , rik
i = 1, 2, . . . , m
(36)
m In such case, we have wi ∈ [0, 1] and i=1 wi = 1. If all the decision makers agree with the attribute weights derived from (36), then we utilize Approach I to rank and select the given alternatives; otherwise, we ask the decision makers to revise the attribute weights derived from (36), and get T (k) (k) (k) , k = 1, 2, . . . , l w (k) = w1 , w2 , . . . , wm
(37)
Then we aggregate all the weight vectors w(k) (k = 1, 2, . . . , l) into the collective one w¯ = (w¯ 1 , w¯ 2 , . . . , w¯ m )T , where w¯ i =
l
(k)
ξk wi , i = 1, 2, . . . , m
(38)
k=1
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Based on (37) and (38), we calculate the deviation between w (k) and w: ¯ m (k) |wi − w¯ i | d w (k) , w¯ =
(39)
i=1
¯ ≤ α, then we call that w (k) and w¯ are of acceptable deviation, where α is If d(w (k) , w) the threshold of acceptable deviation, which can be predefined by the decision makers in practical applications; otherwise, we shall return the weight vector w(k) together with w¯ to the decision maker ek for revaluation. We repeat this process until w (k) and w¯ are of acceptable deviation, or the process will stop as the repetition times reach the maximum number predefined by the decision makers, and then we utilize Approach I to rank and select the given alternatives. In what follows, we extend the above results to MAGDM with interval-valued intuitionistic fuzzy information: (k) (k) ˜ be an interval-valued intuitionistic fuzzy decision matrix, Let R = r˜i j m×n (k) (k) (k) is an IVIFN which is provided by the decision maker where r˜i j = t˜i j , f˜i j (k) ek ∈ E for the alternative B j ∈ B with respect to the attribute G i ∈ G, t˜i j indicates the degree range that the alternative B j should satisfy the attribute G i , expressed by (k) the decision maker ek ∈ E, while f˜i j indicates the degree range that the alternative B j should not satisfy the attribute G i , expressed by the decision maker ek ∈ E, and (k) (k) (k) (k) t˜i j ⊂ [0, 1], f˜i j ⊂ [0, 1], sup t˜i j + sup f˜i j ≤ 1, i = 1, 2, . . . , m, j = 1, 2, . . . , n
If the information about attribute weights is completely known, then we develop the following approach to group decision making: (Approach II) Step 1. Utilize the IVIFHA operator (26) to aggregate all individual interval-valued intuitionistic fuzzy decision matrices R˜ (k) = r˜i(k) = 1, 2, . . . , l) (k j m×n
into a collective interval-valued intuitionistic fuzzy decision matrix R˜ = (˜ri j )m×n . Step 2. Utilize r˜ j = w1r˜1 j ⊕ w2 r˜2 j ⊕ · · · ⊕ wm r˜m j ,
j = 1, 2, . . . , n
(40)
to get the overall values of the alternatives B j ( j = 1, 2, . . . , n). Step 3. Utilize (21) to calculate the score s(˜r j ) of the overall value r˜ j of the alternative Bj. Step 4. Utilize the scores s(˜r j )( j = 1, 2, . . . , n) to rank all the alternatives B j ( j = 1, 2, . . . , n). If there is no difference between two scores s(˜ri ) and s(˜r j ), then we need to calculate the accuracy degrees h(˜ri ) and h(˜r j ) of the overall values r˜i and r˜ j by using (22), and then rank the alternatives Bi and B j in accordance with the accuracy degrees h(˜ri ) and h(˜r j ).
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If the information about the attribute weights is completely unknown in the considered problem, then similar to (36), we have n
wi =
j=1 k= j d r˜i j , r˜ik , m n i=1 j=1 k= j d r˜i j , r˜ik
i = 1, 2, . . . , m
(41)
which is used to derive the attribute weights. If all the decision makers agree with the attribute weights derived from (41), then we utilize Approach II to rank and select the given alternatives; otherwise, we proceed with the interaction process presented earlier. 4 Numerical Example In this section, a MAGDM problem involves the prioritization of a set of information technology improvement projects [adapted from Ngwenyama and Bryson (1999)] is used to illustrate the developed approach. The information management steering committee of Midwest American Manufacturing Corp., which comprises (1) the Chief Executive Officer, (2) the Chief Information Officer, and (3) the Chief Operating Officer, must prioritize for development and implementation a set of eight information technology improvement projects B j ( j = 1, 2, . . . , 8), which have been proposed by area managers. The committee is concerned that the projects are prioritized from highest to lowest potential contribution to the firm’s strategic goal of gaining competitive advantage in the industry. Suppose that there are four members ek (k = 1, 2, 3, 4)in the committee, whose weight vector is ξ = (0.4, 0.2, 0.1, 0.3)T . In assessing the potential contribution of each project, each member needs to suggest a starting set of attributes, and then the group discuss and analyse all the given attributes, from which a feasible set of six attributes are chosen: (1) G 1 —productivity, (2) G 2 —technological innovation capability, G 3 —marketing capability, G 4 —differentiation, G 5 —management, and G 6 —risk avoidance. The following is the list of proposed information systems projects: (1) B1 —quality management information, (2) B2 —inventory control, (3) B3 —customer order tracking, (4) B4 —materials purchasing management, (5) B5 —fleet management, (6) B6 —design change management, (7) B7 —employee skills tracking, and (8) B8 —budget analysis. These four members represent the characteristics of the projects B j ( j = 1, 2, . . . , 8) by the IFNs r˜i j (i = 1, 2, . . . , 6; j = 1, 2, . . . , 8) with respect , 6), listed in Tables 1, 2, 3,4 (i.e., intuitionistic fuzzy to the factors G i (i = 1, 2, ... (k = 1, 2, 3, 4)). decision matrices R (k) = ri(k) j 6×8
Now we use the developed approach to rank the projects: Firstly, we give the associated vector ω = (0.155, 0.345, 0.345, 0.155)T of the IFHA operator by using the normal distribution based method Xu (2005) which can relieve the influence of unfair data on the decision results by weighting these data with small values, and utilize (28) to aggregate the individual intuitionistic fuzzy decision (k) (k) = ri j (k = 1, 2, 3, 4) into the collective intuitionistic fuzzy matrices R 6×8
decision matrix R = (ri j )6×8 (see Table 5).
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Table 1 Intuitionisitc fuzzy decision matrix R (1)
G1 G2 G3 G4 G5 G6
B1
B2
B3
B4
B5
B6
B7
B8
(0.5, 0.3) (0.6, 0.2) (0.3, 0.4) (0.8, 0.1) (0.6, 0.2) (0.9, 0.1)
(0.4, 0.2) (0.5, 0.4) (0.8, 0.1) (0.7, 0.2) (0.6, 0.3) (0.4, 0.1)
(0.7, 0.1) (0.2, 0.7) (0.5, 0.2) (0.8, 0.2) (0.5, 0.5) (0.9, 0.1)
(0.2, 0.3) (0.5, 0.3) (0.7, 0.1) (0.2, 0.4) (0.4, 0.6) (0.8, 0.1)
(0.6, 0.1) (0.4, 0.2) (0.1, 0.6) (0.5, 0.1) (0.6, 0.3) (0.7, 0.2)
(0.3, 0.5) (0.7, 0.1) (0.6, 0.2) (0.4, 0.4) (0.4, 0.1) (0.3, 0.3)
(0.5, 0.2) (0.6, 0.4) (0.4, 0.5) (0.7, 0.1) (0.3, 0.4) (0.5, 0.2)
(0.3, 0.1) (0.5, 0.3) (0.7, 0.2) (0.4, 0.6) (0.1, 0.3) (0.6, 0.1)
Table 2 Intuitionisitc fuzzy decision matrix R (2)
G1 G2 G3 G4 G5 G6
B1
B2
B3
B4
B5
B6
B7
B8
(0.6, 0.2) (0.7, 0.1) (0.5, 0.4) (0.8, 0.2) (0.7, 0.1) (0.8, 0.1)
(0.5, 0.4) (0.3, 0.3) (0.8, 0.1) (0.6, 0.1) (0.7, 0.2) (0.5, 0.1)
(0.7, 0.2) (0.4, 0.4) (0.5, 0.2) (0.6, 0.2) (0.4, 0.5) (0.8, 0.2)
(0.4, 0.5) (0.8, 0.1) (0.5, 0.3) (0.3, 0.5) (0.5, 0.4) (0.7, 0.1)
(0.5, 0.1) (0.7, 0.2) (0.4, 0.5) (0.5, 0.2) (0.5, 0.3) (0.6, 0.1)
(0.4, 0.2) (0.6, 0.3) (0.5, 0.1) (0.5, 0.4) (0.5, 0.1) (0.4, 0.3)
(0.4, 0.2) (0.5, 0.4) (0.5, 0.3) (0.6, 0.2) (0.4, 0.5) (0.6, 0.1)
(0.3, 0.3) (0.6, 0.2) (0.6, 0.2) (0.5, 0.4) (0.3, 0.2) (0.6, 0.3)
Table 3 Intuitionisitc fuzzy decision matrix R (3)
G1 G2 G3 G4 G5 G6
B1
B2
B3
B4
B5
B6
B7
B8
(0.6, 0.1) (0.4, 0.5) (0.7, 0.1) (0.9, 0.1) (0.8, 0.2) (0.7, 0.1)
(0.5, 0.2) (0.6, 0.3) (0.7, 0.2) (0.8, 0.2) (0.7, 0.1) (0.6, 0.3)
(0.4, 0.4) (0.7, 0.1) (0.6, 0.3) (0.7, 0.3) (0.4, 0.4) (0.8, 0.2)
(0.6, 0.3) (0.7, 0.2) (0.5, 0.4) (0.4, 0.6) (0.5, 0.2) (0.6, 0.3)
(0.8, 0.1) (0.4, 0.4) (0.3, 0.5) (0.4, 0.5) (0.3, 0.4) (0.8, 0.1)
(0.4, 0.5) (0.8, 0.1) (0.5, 0.2) (0.5, 0.4) (0.4, 0.2) (0.5, 0.1)
(0.6, 0.2) (0.5, 0.3) (0.7, 0.1) (0.4, 0.2) (0.6, 0.1) (0.4, 0.2)
(0.4, 0.1) (0.6, 0.3) (0.7, 0.2) (0.5, 0.3) (0.3, 0.1) (0.7, 0.3)
Table 4 Intuitionisitc fuzzy decision matrix R (4)
G1 G2 G3 G4 G5 G6
B1
B2
B3
B4
B5
B6
B7
B8
(0.7, 0.2) (0.5, 0.3) (0.6, 0.4) (0.7, 0.1) (0.6, 0.2) (0.8, 0.2)
(0.6, 0.3) (0.4, 0.4) (0.5, 0.2) (0.6, 0.1) (0.7, 0.3) (0.4, 0.1)
(0.5, 0.1) (0.8, 0.1) (0.3, 0.6) (0.6, 0.2) (0.5, 0.4) (0.7, 0.2)
(0.4, 0.5) (0.7, 0.2) (0.5, 0.3) (0.3, 0.5) (0.4, 0.2) (0.8, 0.1)
(0.4, 0.4) (0.6, 0.2) (0.5, 0.1) (0.6, 0.3) (0.4, 0.3) (0.7, 0.1)
(0.7, 0.1) (0.4, 0.5) (0.6, 0.2) (0.5, 0.4) (0.3, 0.1) (0.4, 0.2)
(0.5, 0.1) (0.4, 0.6) (0.6, 0.4) (0.6, 0.2) (0.5, 0.3) (0.5, 0.1)
(0.4, 0.3) (0.7, 0.1) (0.5, 0.2) (0.5, 0.4) (0.2, 0.1) (0.5, 0.2)
Secondly, we utilize (36) to derive the weight vector of attributes: w = (0.10, 0.15, 0.16, 0.23, 0.19, 0.17)T
(42)
The decision makers e1 and e2 agree with the above weight vector of attributes, for convenience of describation, here, we denote w(1) = w (2) = w. However, the decision makers e3 and e4 disagree with the weight vector w, and revise it as follows:
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Table 5 Collective intuitionisitc fuzzy decision matrix R
G1 G2 G3 G4 G5 G6
B1
B2
B3
B4
B5
B6
B7
B8
(0.60, 0.21) (0.58, 0.17) (0.47, 0.33) (0.77, 0.12) (0.65, 0.17) (0.83, 0.13)
(0.52, 0.29) (0.42, 0.36) (0.70, 0.14) (0.64, 0.12) (0.68, 0.25) (0.44, 0.11)
(0.61, 0.13) (0.48, 0.34) (0.43, 0.32) (0.67, 0.21) (0.47, 0.45) (0.80, 0.17)
(0.37, 0.43) (0.68, 0.18) (0.56, 0.23) (0.28, 0.48) (0.46, 0.39) (0.77, 0.11)
(0.52, 0.18) (0.58, 0.21) (0.30, 0.39) (0.54, 0.24) (0.48, 0.31) (0.68, 0.12)
(0.45, 0.31) (0.58, 0.26) (0.57, 0.17) (0.48, 0.37) (0.39, 0.10) (0.38, 0.24)
(0.48, 0.15) (0.49, 0.46) (0.51, 0.36) (0.62, 0.17) (0.40, 0.32) (0.52, 0.12)
(0.35, 0.21) (0.60, 0.20) (0.60, 0.20) (0.48, 0.46) (0.20, 0.19) (0.57, 0.19)
w (3) = (0.20, 0.10, 0.10, 0.25, 0.15, 0.20)T w (4) = (0.15, 0.20, 0.15, 0.20, 0.20, 0.10)T
(43) (44)
Then by (38), we aggregate all the weight vectors w(k) (k = 1, 2, 3, 4) into the collective one: w¯ = 0.4w (1) + 0.2w (2) + 0.1w (3) + 0.3w (4) = (0.125, 0.160, 0.151, 0.223, 0.189, 0.152)T
(45)
By (39), we calculate the deviation between w (k) and w: ¯ d w (1) , w¯ = d w (2) , w¯ = 0.070, d w (3) , w¯ = 0.300, d w (4) , w¯ = 0.152
(46)
Suppose that the threshold α = 0.2, then d w (3) , w¯ > 0.2, in this case, we need to return the weight vector w (3) together with w¯ to the decision maker e3 for revaluation, and assume that the revaluated weight vector is: w˙ (3) = (0.15, 0.13, 0.12, 0.22, 0.18, 0.20)T
(47)
Then, by (38), we aggregate all the weight vectors w(k) (k = 1, 2, 4) and w˙ (3) into the collective one: w˙¯ = (0.120, 0.163, 0.153, 0.220, 0.192, 0.152)T
(48)
By (39), we calculate the deviations: d w (1) , w˙¯ = d w (2) , w¯ = 0.066, d w˙ (3) , w¯ = 0.156, d w (4) , w˙¯ = 0.150 (49) thus these deviations are acceptable.
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Thirdly, we utilize (29) and (48) to get the overall values r j ( j = 1, 2, . . . , 8) of the alternatives B j ( j = 1, 2, . . . , 8): r1 = (0.681, 0.172), r2 = (0.590, 0.186), r3 = (0.600, 0.256), r4 = (0.542, 0.277) r5 = (0.529, 0.231), r6 = (0.480, 0.221), r7 = (0.513, 0.237), r8 = (0.481, 0.237) Fourthly, we utilize (8) to calculate the score s(r j ) of the overall value r j of the alternative B j : s(r1 ) = 0.509, s(r2 ) = 0.404, s(r3 ) = 0.344, s(r4 ) = 0.265 s(r5 ) = 0.298, s(r6 ) = 0.259, s(r7 ) = 0.276, s(r8 ) = 0.244 Since s(r1 ) > s(r2 ) > s(r3 ) > s(r5 ) > s(r7 ) > s(r4 ) > s(r6 ) > s(r8 ) then B1 B2 B3 B5 B7 B4 B6 B8 and thus, the project B1 has the highest potential contribution to the firm’s strategic goal of gaining competitive advantage in the industry. 5 Conclusions In this article, we have utilized the deviation measure and information theory to develop an approach to MAGDM with intuitionistic fuzzy information. The approach utilizes the IFHA operator to aggregate all individual intuitionistic fuzzy decision matrices into a collective intuitionistic fuzzy decision matrix, which can consider all the given IFNs and their ordered positions. In the process of information aggregation, the IFHA operator not only can reflect the importance of each decision maker, but also can relieve the influence of unfair data (too large or too small IFNs) on the decision result by assigning low weights to those unfair ones. According to the information theory, a small weight should be assigned to the attribute with similar attribute values corresponding to an alternative, we have established an optimization model which integrates all the given IFNs and produces no loss of information. By solving the model, a straightforward formula has been obtained for determining the attribute weights. After doing so, two cases have been discussed: (1) If all the decision makers agree with the attribute weights derived from the model, then we utilize the IFWA operator to derive the overall values of alternatives, and then utilize the score function and accuracy function to rank the given alternatives; (2) If some decision makers disagree with the derived attribute weights, then the decision makers need to revise these weights, and thus an interactive mechanism has been established to reach group consensus so as to find a final compromise solution, which can sufficiently reflect the decision makers’ subjective desirability, the rest decision steps are the same as the case 1. In addition, we have
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extended the above results to the interval-valued intuitionistic fuzzy environments, in which the attribute values are expressed in IVIFNs, and the information about attribute weights is completely known or completely unknown. Acknowledgments The work was supported by the National Science Fund for Distinguished Young Scholars of China (no. 70625005). The author is very grateful to the anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper.
Appendix A Proof of Theorem 2.1 Let αi = (tαi , f αi )(i = 1, 2, 3), then by (5), we have 1 |tα1 − tα2 | + | f α1 2 1 d (α1 , α1 ) = |tα1 − tα1 | + | f α1 2 1 |tα1 − tα2 | + | f α1 d(α1 , α2 ) = 2 1 d(α1 , α2 ) = |tα1 − tα2 | + | f α1 2 = d(α2 , α1 )
d (α1 , α2 ) =
1 (50) − f α2 | ≥ (0 + 0) = 0 2 1 − f α1 | = (0 + 0) = 0 (51) 2 1 (52) − f α2 | ≤ (1 + 1) = 1 2 1 |tα2 − tα1 | + | f α2 − f α1 | − f α2 | = 2 (53)
and thus, (1) and (2) hold, which completes the proof of Theorem 2.1. Proof of Theorem 2.2 Xu (2007b). Since αi = tαi , f αi (i = 1, 2) are two IFNs, then tα1 ∈ [0, 1], f α1 ∈ [0, 1], tα2 ∈ [0, 1], f α2 ∈ [0, 1]tα1 + f α1 ≤ 1, tα2 + f α2 ≤ 1
(54)
thus tα1 + tα2 − tα1 tα2 = tα1 (1 − tα2 ) + tα2 ≥ tα2 ≥ 0, f α1 f α2 ≥ 0 (55) tα1 + tα2 − tα1 tα2 + f α1 f α2 ≤ tα1 + tα2 − tα1 tα2 + (1 − tα1 )(1 − tα2 ) = 1 (56) and hence, β is an IFN. Since α = (tα , f α ) is an IFN, then 1 − (1 − tα )λ ≥ 0, λ
1 − (1 − tα ) +
f αλ
f αλ ≥ 0
(57)
≤ 1 − (1 − tα )λ + (1 − tα )λ = 1
(58)
and thus, γ is also an IFN. This completes the proof of Theorem 2.2. Proof of Theorem 2.3 Xu (2007b). Since α1 = (tα1 , f α1 ), α2 = (tα2 , f α2 ) and α = (tα , f α ) be three IFNs, λ, λ1 , λ2 > 0, then
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α1 ⊕ α2 = tα1 + tα2 − tα1 tα2 , f α1 f α2 = tα2 + tα1 − tα2 tα1 , f α2 f α1 =α ⊕α (59) 2 1 λ λ , f α1 f α2 λ (α1 ⊕ α2 ) = 1 − 1 − tα1 + tα2 − tα1 tα2 λ λ λ = 1 − 1 − tα1 1 − tα2 , f α1 f α2 λ λ λ λ = 2 − 1 − tα1 − 1 − tα2 − 1 − 1 − tα1 − 1 − tα2 λ λ λ , f α1 f α2 1 − tα2 + 1 − tα1 λ λ = 1 − 1 − tα1 + 1 − 1 − tα2 λ λ λ 1 − 1 − tα2 , 1 − f α1 f α2 − 1 − 1 − tα1 = λα1 ⊕ λα2 λ1 α ⊕ λ2 α = 2 − (1 − tα )λ1
(60)
− (1 − tα )λ2 − 1 − (1 − tα )λ1 1 − (1 − tα )λ2 , f αλ1 f αλ2 = 1 − (1 − tα )λ1 (1 − tα )λ2 , ( f α )λ1 +λ2 = 1 − (1 − tα )λ1 +λ2 , ( f α )λ1 +λ2 = (λ1 + λ2 ) α (61)
and thus, (1–3) hold, which completes the proof of Theorem 2.3.
Proof of Theorem 2.4. Let α˜ i = ([ai , bi ], [ci , di ]) (i = 1, 2, 3) be three IVIFNs, then by (18), we have
1 (|a1 − a2 | + |b1 − b2 | + |c1 − c2 | + |d1 − d2 |) 4 1 ≥ (0 + 0 + 0 + 0) = 0 4 1 d (α˜ 1 , α˜ 2 ) = (|a1 − a2 | + |b1 − b2 | + |c1 − c2 | + |d1 − d2 |) 4 1 ≤ (1 + 1 + 1 + 1) = 1 4 1 d(α˜ 1 , α˜ 1 ) = (|a1 − a1 | + |b1 − b1 | + |c1 − c1 | + |d1 − d1 |) 4 1 = (0 + 0 + 0 + 0) = 0 4 d (α˜ 1 , α˜ 2 ) =
i.e., 0 ≤ d(α˜ 1 , α˜ 2 ) ≤ 1 and d(α˜ 1 , α˜ 1 ) = 0. Also since
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(62)
(63)
(64)
A Deviation-Based Approach to Intuitionistic Fuzzy MAGDM
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1 (|a1 − a2 | + |b1 − b2 | + |c1 − c2 | + |d1 − d2 |) 4 1 = (|a2 − a1 | + |b2 − b1 | + |c2 − c1 | + |d2 − d1 |) 4 = d(α˜ 2 , α˜ 1 ) 1 d(α˜ 1 , α˜ 3 ) = (|a1 − a3 | + |b1 − b3 | + |c1 − c3 | + |d1 − d3 |) 4 1 = (|a1 − a2 + a2 − a3 | + |b1 − b2 + b2 − b3 | 4 +|c1 − c2 + c2 − c3 | + |d1 − d2 + d2 − d3 |) 1 ≤ (|a1 − a2 | + |a2 − a3 | + |b1 − b2 | + |b2 − b3 | 4 +|c1 − c2 | + |c2 − c3 | + |d1 − d2 | + |d2 − d3 |) = d(α˜ 1 , α˜ 2 ) + d(α˜ 2 , α˜ 3 )
d(α˜ 1 , α˜ 2 ) =
(65)
(66)
then (1–3) hold. This completes the proof of Theorem 2.4.
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