The notation of extended intuitionistic fuzzy number is introduced and the representation of the weighted extended trapezoidal intuitionistic approximation of an intuitionistic fuzzy number is given in this paper. The problem related to aggregation is addressed about approximating the given intuitionistic fuzzy numbers or aggregating the given intuitionistic fuzzy numbers and then approximating the output of aggregation in two cases. Some examples are enumerated to show the benefits of the conclusions proposed in this paper and the conclusions are applied to multiple attribute decision making based on intuitionistic fuzzy information.
Since the fuzzy set theory was introduced by Zadeh [1], fuzzy set theory has been used widely in various fields. As a generalization of the concept of fuzzy set, intuitionistic fuzzy sets were introduced by Atanassov [2]. The problems of containing incomplete informations can be modelled more accurately using intuitionistic fuzzy sets than fuzzy sets. Intuitionistic fuzzy set has been successfully applied in various fields such as multi-criteria group decision [3], pattern recognition [4], information system problem with incomplete information [5], game theory [6] and so forth. It is easy to find that triangular and trapezoidal fuzzy numbers and triangular and trapezoidal intuitionistic fuzzy numbers are the most commonly used in applications. Because regular membership functions and nonmembership functions lead to calculations that are more simple and outputs that are more to handle. Those cause a need for approximations of fuzzy numbers and intuitionistic fuzzy numbers. Especially trapezoidal approximation of a fuzzy number has been recently elaborated by many researchers [7–9, 30]. The study of the approximation problem in the intuitionistic fuzzy case has also been investigated in the same way. Ban [11] discussed the nearest interval approximation of an intuitionistic fuzzy number. Ban and Coroianu [12] presented a method to obtain trapezoidal approximations of intuitionistic fuzzy numbers. Aggregation of intuitionistic fuzzy information has been studied by a few authors [23, 34], and aggregation operators are applied in multi-attribute group decision making. When lots of approximation and aggregation methods are given, one basic question should be considered. Should the approximation be performed before aggregation or after aggregation? This problem is answered by using the average as the aggregation operator in the fuzzy number case [16]. The same method is suitable to find the nearest trapezoidal intuitionistic fuzzy number of a given intuitionistic fuzzy number by using the well-known Karush-Kuhn-Tucker theorem in this paper. If the average as the aggregation operator, then a interesting result can be obtained. The solution of system of equations of the weighted extended trapezoidal intuitionistic approximation of given intuitionistic fuzzy numbers and the one of the weighted extended trapezoidal intuitionistic approximation of the output of aggregation are same. And two cases both with condition of preserving the expected interval and without one will be discussed below. The conclusions proposed in this paper can be used for aggregating intuitionistic fuzzy information and applied to multiple attribute decision making.
This paper is organised in the following manner. Some important definitions and propositions of intuitionistic fuzzy numbers are given in Section 2. The representation of weighted extended trapezoidal approximation of an intuitionistic fuzzy number is introduced. Main results on the approximation without restrictions on the expected interval and with restrictions on the expected interval are discussed in Sections 3 and 4, respectively. In Section 5, we apply the outputs to multiple attribute decision making based on intuitionistic fuzzy information. Finally, conclusions are given in Section 6.
Preliminaries
Intuitionistic fuzzy number
Definition 2.1. [2] Let X be a given nonempty set. An intuitionistic fuzzy set in X is an object A given by A = {〈x, μA (x), νA (x) 〉 : x ∈ X}, where μA : X → [0, 1] and νA : X → [0, 1] satisfy the condition 0 ≤ μA (x) + νA (x) ≤1, for every x ∈ X.
Definition 2.2. [10] An intuitionistic fuzzy set A = {〈x, μA (x), νA (x) 〉 : x ∈ R} such that μA and 1 - νA, where (1 - νA) (x) =1 - νA (x), ∀ x ∈ R, are fuzzy numbers is called an intuitionistic fuzzy number.
The set of all intuitionistic fuzzy numbers is denoted by IF (R). An intuitionistic fuzzy number A = 〈μA, νA〉, where μA and 1 - νA are trapezoidal fuzzy numbers, is called a trapezoidal intuitionistic fuzzy number. The set of all trapezoidal intuitionistic fuzzy numbers is denoted by IFT (R). An intuitionistic fuzzy number A = 〈μA, νA〉, where μA and 1 - νA are extended trapezoidal fuzzy numbers [17], is called an extended trapezoidal intuitionistic fuzzy number. The set of all extended trapezoidal intuitionistic fuzzy numbers is denoted by .
Let A = 〈μA, νA〉 ∈ IF (R) and Aα = ((μA) α, (νA) α) is its α-cut [11], where (μA) α = [(μA) L (α), (μA) U (α)] and (νA) α = [(νA) L (α), (νA) U (α)]. With respect to the set (νA) α is the equalities (νA) L (α) = (1 - νA) L (1 - α) and (νA) U (α) = (1 - νA) U (1 - α).
Let A = 〈μA, νA〉 ∈ IFT (R), where μA = μA (t1, t2, t3, t4) and 1 - νA = (1 - νA) (s1, s2, s3, s4) . Let us adopt the following notation [16]:
Thus according to the above notations, μA (t1, t2, t3, t4) = μA (l, u ; x, y) [17] and
Some proposition and weighted extended trapezoidal intuitionistic approximations
Proposition 2.1.[17] For any l, u, x, y ∈ R such that x, y ≥ 0, x + y ≤ 2 (u - l), the family of intervals (1) defines a trapezoidal fuzzy number.
Then the following conclusion can be obtained immediately.
Proposition 2.2.Let A = 〈μA, νA〉 ∈ IF (R), where μA = μA (l, u ; x, y) and 1 - νA = (1 - νA) (l′, u′ ; x′, y′) . Then A = 〈μA, νA〉 is a trapezoidal intuitionistic fuzzy number iff x, y, x′, y′ ≥ 0, x + y ≤ 2 (u - l) and x′ + y′ ≤ 2 (u′ - l′).
Let λL (α) and λU (α) be non-negative functions on [0,1], called weighted functions, such that >0 and .
An extended trapezoidal fuzzy number is a structure given [16] by the set of all its α-cuts
where α ∈ [0, 1] and l, u, x, y ∈ R. Any trapezoidal fuzzy number μA (l, u ; x, y) can be represented as an extended trapezoidal fuzzy number [16], that is
then any trapezoidal intuitionistic fuzzy number A = 〈μA, νA〉 can be also represented as an extended trapezoidal intuitionistic fuzzy number, i.e. 〈μA, νA〉 ≡ , where is defined by (9) and is defined by
Similar to the proofs of Proposition 2.2 in [19], A new calculation formula of weighted distance can be obtained in the following.
Proposition 2.3.Let , , where , , , and . Then
Let A = 〈μA, νA〉 ∈ IF (R). The weighted extended trapezoidal intuitionistic approximation of A is the extended trapezoidal intuitionistic fuzzy number that minimizes the weighted distance , where . Let , where , and are determined by the following equalities [19]:
In the following, we will show that Te (A) is equal to the weighted extended trapezoidal intuitionistic approximation of A = 〈μA, νA〉. Similar to the proofs of Propositions 3.2 and 3.3 in [19], we may obtain the following Propositions 2.4 and 2.5.
Proposition 2.4.Let A = 〈μA, νA〉 ∈ IF (R) and, where and x′e, y′e) λ are defined as Equations (11)–(18). Thenholds for every .
Proposition 2.5.Let A = 〈μA, νA〉 ∈ IF (R) and , where and are defined as Equations (11)–(18). Then equals the weighted extended trapezoidal intuitionistic approximation of A = 〈μA, νA〉.
Because the weighted extended trapezoidal intuitionistic approximation Te (A) of the intuitionistic fuzzy number A can be calculated by Equations (11)–(18), Te (A) exists and is unique.
In the following sections we make use of the well-known Karush-Kuhn-Tucker theorem to prove the main results of this paper.
Theorem 2.1.[28] Let f, g1, ⋯, gm : Rn → R be convex and differentiable functions. Then solves the convex programming problemif and only if there exists ξi, i ∈ {1, . . ., m}, such that
;
;
ξi ≥ 0;
.
Weighted trapezoidal intuitionistic approximation and aggregation
Weighted trapezoidal intuitionistic approximation of an intuitionistic fuzzy number
Let A = 〈μA, νA〉 ∈ IF (R). TA = 〈μTA, νTA〉 is called weighted trapezoidal intuitionistic approximation of A if , where μTA = μTA (l, u ; x, y) and 1 - νTA = (1 - νTA) (l′, u′ ; x′, y′) . Let be weighted extended trapezoidal intuitionistic approximation of A, where and Now we want to find the TA.
By Proposition 2.4, we have
By Equations (9), (10) and Proposition 2.3, we can compute . Since is constant, to minimize is as to minimizing the following function
Taking into account Proposition 2.2, to find the minimum of Equation (19) can be reduced to the following problem.
Problem 3.1. Find (l, u, x, y, l′, u′, x′, y′) ∈ Ω1, which minimizes Equation (19), where
We will show Problem 3.1 has an unique solution in the following.
Proof. Let us define an inner product 〈 ·, · 〉 in R8 by . If (xi, yi, zi, oi, pi, qi, ri, si) ∈ R8, i ∈ {1, 2}, then
introduce a distance in R8. We know a finite inner produce space is complete and a complete inner produce space is usually called a Hilbert space. Hence (R8, 〈 ·, · 〉) is a Hilbert space. It is easy to verify that the set (20) is a closed convex subset of R8. Hence the Problem 3.1 has a unique solution, which is the projection of (le, ue, xe, ye, l′e, u′e, x′e, y′e) ∈ R8 onto Ω1 under D [33]. It is immediately obtained that the trapezoidal intuitionistic fuzzy number TA nearest (with respect to ) to a given intuitionistic fuzzy number A = 〈μA, νA〉 ∈ IF (R) exists and is unique. That is the weighted trapezoidal intuitionistic approximation TA of A = 〈μA, νA〉 exists and is unique. □
Weighted trapezoidal intuitionistic approximation of intuitionistic fuzzy numbers
Now we want to aggregate efficiently intuitionistic fuzzy numbers A1, A2, ⋯, An. T(A1,⋯,An) = 〈μT(A1,⋯,An), νT(A1,⋯,An)〉 is called weighted trapezoidal intuitionistic approximation of A1, A2, ⋯, An if ⋯, An), T(A1,⋯,An)), where μT(A1,⋯,An) = μT(A1,⋯,An) (l, u ; x, y), 1 - νT(A1,⋯,An) = (1 - νT(A1,⋯,An)) (l′, u′ ; x′, y′) . By Proposition 2.2, the T(A1,⋯,An) minimizes T(A1,⋯,An)), where (l, u, x, y, l′, u′, x′, y′) ∈ Ω1, and Ω1 is defined by (20).
By Proposition 2.4, to minimize is as to minimizing . is weighted extended trapezoidal intuitionistic approximation of the intuitionistic fuzzy number Ai, where and By Equations (9), (10) and Proposition 2.3, we have
So minimum of Equation (21) can be reduced to the following problem.
Problem 3.2. Find (l, u, x, y, l′, u′, x′, y′) ∈ Ω1, which minimizes Equation (21), where Ω1 is defined by (20).
According to the Karush-Kuhn-Tucker theorem, T(A1,⋯,An) = 〈μT(A1,⋯,An), νT(A1,⋯,An)〉 ∈ IFT (R) is a solution of Problem 3.2 if and only if there exist such that the following system of equations holds:
Although the solution of the system of equations may require complex calculations, the system of equations needn’t be solved. The clear way for getting the trapezoidal intuitionistic approximation is needed.
Weighted trapezoidal intuitionistic approximation of an aggregation
The arithmetic mean of given intuitionistic fuzzy numbers A1, A2, ⋯, An is denoted by , that is . is called weighted trapezoidal intuitionistic approximation of if, where and Taking into account Proposition 2.4, to minimize ) is as to minimizing where is weighted extended trapezoidal intuitionistic approximation of . is weighted extended trapezoidal intuitionistic approximation of Ai ∈ IF (R). From (11)–(18), we can compute le, ue, xe, ye, l′e, u′e, x′e and y′e . Hence we obtain where
By (9) and (10) and Proposition 2.3, we have
Taking into account Proposition 2.2, to find the minimum of Equation (50) can be reduced to the following problem.
Problem 3.3. Find , which minimizes Equation (50), where
According to the Karush-Kuhn-Tucker theorem, an trapezoidal intuitionistic fuzzy number is a solution of Problem 3.3 if and Only if there exist such that the following system of equations holds:
Conclusion and example
If we compare (22)–(47) and (52)–(77), we obtain the following result:
Theorem 3.1.The weighted trapezoidal intuitionistic approximation T(A1,⋯,An) of A1, A2, ⋯, An is the weighted trapezoidal intuitionistic approximation of .
Proof. We get the proof immediately substituting ξ1, ξ2, in (22)–(47) by . □
In order to illustrate the benefits of Theorem 3.1, we consider the case of n = 2 and n = 3.
Example 3.1. Let A1 = 〈μA1, νA1〉 ∈ IF (R) and A2 = 〈μA2, νA2〉 ∈ IF (R), where (1 - νA1) α = [- 1 + α3, 4 - 2α2] and (1 - νA2) α = [1 + α3, 3 - α2] (see Figs. 1 and 2). Then where and (see black solid line in Fig. 3). Let us consider λL (α) = λU (α) =1 . From step 1 of algorithm in [18], The weighted trapezoidal intuitionistic approximation of is the where and (see blue dashed line in Fig. 3).
Theorem 3.1 shows is also the weighted trapezoidal intuitionistic approximation of A1, A2 .
Intuitionistic fuzzy number A1 = 〈μA1, νA1〉.
Intuitionistic fuzzy number A2 = 〈μA2, νA2〉.
Intuitionistic fuzzy numbers and .
Example 3.2. Intuitionistic fuzzy numbers A1 = 〈μA1, νA1〉 and A2 = 〈μA2, νA2〉 are considered in Example 3.1. Let A3 = 〈μA3, νA3〉 ∈ IF (R), where and (see Fig. 4). Then where and (see black solid line in Fig. 5). We still consider λL (α) = λU (α) =1 . Using the same algorithm (step 1 of algorithm in [18]), we obtain the weighted trapezoidal intuitionistic approximation of , that is where and (see blue dashed line in Fig. 5).
By Theorem 3.1, is also the weighted trapezoidal intuitionistic approximation of A1, A2and A3.
Intuitionistic fuzzy number A3 = 〈μA3, νA3〉.
Intuitionistic fuzzy numbers and .
According to Theorem 3.1 and Equations (2), (5) in [36], the following consequence can be obtained immediately.
Corollary 3.1.Let Ai = 〈μAi, νAi〉 ∈ IFT (R), where The weighted trapezoidal intuitionistic approximation of A1, A2, ⋯, An is where and
Trapezoidal intuitionistic approximation preserving the weighted expected interval and aggregation
We will prove that the conclusion in Section 3 remains valid under the condition of preserving the expected interval in this section.
Weighted expected interval
Definition 4.1. The weighted expected interval of A = 〈μA, νA〉 ∈ IF (R) is defined as
where and
Definition 4.2. The weighted expected value of A = 〈μA, νA〉 ∈ IF (R) is defined as .
The weighted expected interval of a sequence of intuitionistic fuzzy numbers A1, A2, ⋯, An can be defined as
Let be the weighted extended trapezoidal intuitionistic approximation of the intuitionistic fuzzy number A, where xe, ye) λ and Straightforward calculations using (3), (4) and (78) show that
Thus combining (79) and (81) we get a result that
Weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of intuitionistic fuzzy numbers
Similar to the proof of subsection 3.1, we may prove that the weighted trapezoidal intuitionistic approximation preserving the weighted expected interval TA of A = 〈μA, νA〉 ∈ IF (R) exists and is unique.
Intuitionistic fuzzy numbers A1, A2, ⋯, An should be efficiently aggregation to such intuitionistic fuzzy number that the weighted expected interval of this aggregation would be equal to the weighted expected interval of A1, A2, ⋯, An. T(A1,⋯,An) = 〈μT(A1,⋯,An), νT(A1,⋯,An)〉 is called weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of A1, A2, ⋯, An if ⋯, An), B) ⋯, An), T(A1,⋯,An)) under condition EIw (T(A1,⋯,An)) = EIw (A1, ⋯, An), where μT(A1,⋯,An) = μT(A1,⋯,An) (l, u ; x, y) and 1 - νT(A1,⋯,An) = (1 - νT(A1,⋯,An)) (l′, u′ ; x′, y′) . From Equations (1), (2) and Definition 4.1, we get
By Equation (83), EIw (T(A1,⋯,An)) = EIw (A1, ⋯, An) can be rewritten as:
Hence to find the weighted trapezoidal intuitionistic approximation preserving the weighted expected interval T(A1,⋯,An) is as to solve Problem 3.2 under the conditions Equations (84) and (85). Then we have the following problems.
Problem 4.1. Find (l, u, x, y, l′, u′, x′, y′) ∈ Ω1, which minimizes Equation (21) under the conditions Equations (84) and (85), where Ω1 is defined by (20).
According to the Karush-Kuhn-Tucker theorem, T(A1,⋯,An) = 〈μT(A1,⋯,An), νT(A1,⋯,An)〉 ∈ IFT (R) is a solution of Problem 4.1 if and only if there exist such that the following system of equations holds:
Weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of an aggregation
is called weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of if under condition , where and By Equation (82), we have . From Equations (1), (2), (9), (10), (48), (49), (79) and (81), the condition can be rewritten as
Hence to find the weighted trapezoidal intuitionistic approximation preserving the weighted expected interval is as to solve Problem 3.3 under the conditions Equations (118) and (119). Then we have the following problems.
According to the Karush-Kuhn-Tucker theorem, an trapezoidal intuitionistic fuzzy number is a solution of Problem 4.2 if and only if there exist such that the following system of equations holds:
Conclusion
If we compare (86)–(117) and (120)–(151), we obtain the following result:
Theorem 4.1.The weighted trapezoidal intuitionistic approximation preserving the weighted expected interval T(A1,⋯,An) of A1, A2, ⋯, An is the weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of .
Proof. We get the proof immediately substituting ς1, ς2, in (86)–(117) by nτ1, nτ2, nτ3, . □
Theorem 4.1 shows that there is no difference whether the approximation preserving the weighted expected interval is performed before or after aggregation with respect to the operator of average. We obtain the following result which is shown in Theorem 4.1.
Corollary 4.1.Let Ai = 〈μAi, νAi〉 ∈ IFT (R), where i ∈ {1, ⋯, n} . The weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of A1, A2, ⋯, An is where and
Proof. Corollary 3.1 has shown that the weighted trapezoidal intuitionistic approximation of A1, A2, ⋯, An is Hence we only prove that the weighted expected interval of A1, A2, ⋯, An is equivalent to the weighted expected interval of .
Since the weighted expected interval introduced in Definition 4.1 is linear, from (80) we obtain
Application
In the following, we apply the Theorem 3.1 to multiple attribute decision making based on intuitionistic fuzzy information, and we use the example in [34].
Let A = {A1, A2, ⋯, An} be a set of alternatives, and let G = {G1, G2, ⋯, Gm} be a set of attributes. We denote Aij = 〈μij, νij〉, where μij indicates the degree that the alternative Ai satisfies the attribute Gj, νij indicates the degree that the alternative Ai does not satisfy the attribute Gj, μij ∈ [0, 1], νij ∈ [0, 1], μij + νij ≤ 1, 1 ≤ i ≤ n, 1 ≤ j ≤ m.
In order to get the best alternatives, we can make use of the output of the Theorem 3.1:
The T(Ai1,Ai2,⋯,Aim) and are approximation operators. The can be calculated by algorithms in [18]. If Ai1, ⋯, Aim are trapezoidal intuitionistic fuzzy numbers, then is their arithmetic mean. If Ai1, ⋯, Aim are general intuitionistic fuzzy numbers, then is the weighted extended trapezoidal intuitionistic approximation of their arithmetic mean.
Then we calculate the scores [35] (i = 1, 2, ⋯, n) of the overall intuitionistic fuzzy numbers , and make use of the scores to rank the alternatives Ai, and then to select the best one.
Conclusion
In this paper, we introduced some notions about extended trapezoidal intuitionistic fuzzy numbers and weighted expected interval of intuitionistic fuzzy numbers. The representation of the weighted extended trapezoidal intuitionistic approximation of an intuitionistic fuzzy number is given. We obtain a conclusion the approximations preserving the weighted expected interval or without constraint are the same if we choose the average as the aggregation operator in Sections 3 and 4. The conclusion of Theorem 3.1 is applied to multiple attribute decision making based on intuitionistic fuzzy information. Choosing particular weighted functions we can get familiar distance of intuitionistic fuzzy numbers. Theorems 3.1 and 4.1 can be generalized into the case of approximations using other distances. Furthermore the conclusions in Sections 3 and 4 can be probably extended to the case of approximation under other additional constraints. In addition, the presentation of approximating intuitionistic fuzzy numbers by trapezoidal intuitionistic fuzzy numbers with conditions will be studied in the near future.
Footnotes
Acknowledgment
This work is supported by the National Natural Science Foundation of China (61374118).
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