Approximation of intuitionistic fuzzy numbers by trapezoidal intuitionistic fuzzy numbers

Abstract

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.

Keywords

Intuitionistic fuzzy number trapezoidal intuitionistic fuzzy number weighted trapezoidal intuitionistic approximation aggregation expected interval

1 Introduction

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.

2 Preliminaries

2.1 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, μ_A (x), ν_A (x) 〉 : x ∈ X}, where μ_A : X → [0, 1] and ν_A : X → [0, 1] satisfy the condition 0 ≤ μ_A (x) + ν_A (x) ≤1, for every x ∈ X.

Definition 2.2. [10] An intuitionistic fuzzy set A = {〈x, μ_A (x), ν_A (x) 〉 : x ∈ R} such that μ_A and 1 - ν_A, where (1 - ν_A) (x) =1 - ν_A (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 = 〈μ_A, ν_A〉, where μ_A and 1 - ν_A are trapezoidal fuzzy numbers, is called a trapezoidal intuitionistic fuzzy number. The set of all trapezoidal intuitionistic fuzzy numbers is denoted by IF^T (R). An intuitionistic fuzzy number A = 〈μ_A, ν_A〉, where μ_A and 1 - ν_A 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 $\tilde{{IF}^{T}} (R)$ .

Let A = 〈μ_A, ν_A〉 ∈ IF (R) and A_α = ((μ_A) _α, (ν_A) _α) is its α-cut [11], where (μ_A) _α = [(μ_A) _L (α), (μ_A) _U (α)] and (ν_A) _α = [(ν_A) _L (α), (ν_A) _U (α)]. With respect to the set (ν_A) _α is the equalities (ν_A) _L (α) = (1 - ν_A) _L (1 - α) and (ν_A) _U (α) = (1 - ν_A) _U (1 - α).

Let A = 〈μ_A, ν_A〉 ∈ IF^T (R), where μ_A = μ_A (t₁, t₂, t₃, t₄) and 1 - ν_A = (1 - ν_A) (s₁, s₂, s₃, s₄) . Let us adopt the following notation [16]: $\begin{matrix} l = \frac{t_{1} + t_{2}}{2}, u = \frac{t_{3} + t_{4}}{2}, \\ x = t_{2} - t_{1}, y = t_{4} - t_{3}, \\ l^{'} = \frac{s_{1} + s_{2}}{2}, u^{'} = \frac{s_{3} + s_{4}}{2}, \\ x^{'} = s_{2} - s_{1}, y^{'} = s_{4} - s_{3} . \end{matrix}$

Thus according to the above notations, μ_A (t₁, t₂, t₃, t₄) = μ_A (l, u ; x, y) [17] and $(μ_{A})_{α} = [l + x (α - \frac{1}{2}), u - y (α - \frac{1}{2})] .$ (1)

Analogously, (1 - ν_A) (s₁, s₂, s₃, s₄) = (1 - ν_A) (l′, u′ ; x′, y′) and $(1 - ν_{A})_{α} = [l^{'} + x^{'} (α - \frac{1}{2}), u^{'} - y^{'} (α - \frac{1}{2})] .$ (2)

2.2 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 = 〈μ_A, ν_A〉 ∈ IF (R), where μ_A = μ_A (l, u ; x, y) and 1 - ν_A = (1 - ν_A) (l′, u′ ; x′, y′) . Then A = 〈μ_A, ν_A〉 is a trapezoidal intuitionistic fuzzy number iff x, y, x′, y′ ≥ 0, x + y ≤ 2 (u - l) and x′ + y′ ≤ 2 (u′ - l′).

Let λ_L (α) and λ_U (α) be non-negative functions on [0,1], called weighted functions, such that $\int_{0}^{1} λ_{L} (α) d α$ >0 and $\int_{0}^{1} λ_{U} (α) d α > 0$ .

Let us introduce the following notations [16]: $a = \int_{0}^{1} λ_{L} (α) d α,$ (3) $b = \int_{0}^{1} λ_{U} (α) d α,$ (4) $c = \int_{0}^{1} (α - w_{L})^{2} λ_{L} (α) d α,$ (5) $d = \int_{0}^{1} (α - w_{U})^{2} λ_{U} (α) d α,$ (6) $w_{L} = a^{- 1} \int_{0}^{1} α λ_{L} (α) d α,$ (7) $w_{U} = b^{- 1} \int_{0}^{1} α λ_{U} (α) d α,$ (8)

An extended trapezoidal fuzzy number ${\tilde{μ}}_{A} (l, u; x, y)_{λ}$ is a structure given [16] by the set of all its α-cuts $\begin{matrix} ({\tilde{μ}}_{A})_{α} = [l + x (α - w_{L}), u - y (α - w_{U})], \end{matrix}$ where α ∈ [0, 1] and l, u, x, y ∈ R. Any trapezoidal fuzzy number μ_A (l, u ; x, y) can be represented as an extended trapezoidal fuzzy number [16], that is

$\begin{matrix} μ_{A} (l, u; x, y) \\ \equiv {\tilde{μ}}_{A} {(l + x (w_{L} - \frac{1}{2}), u - y (w_{U} - \frac{1}{2}), x, y)}_{λ}, \end{matrix}$ (9) then any trapezoidal intuitionistic fuzzy number A = 〈μ_A, ν_A〉 can be also represented as an extended trapezoidal intuitionistic fuzzy number, i.e. 〈μ_A, ν_A〉 ≡ $〈 {\tilde{μ}}_{A},$ ${\tilde{ν}}_{A} 〉$ , where ${\tilde{μ}}_{A}$ is defined by (9) and $1 - {\tilde{ν}}_{A}$ is defined by

$\begin{matrix} 1 - {\tilde{ν}}_{A} \\ = (1 - {\tilde{ν}}_{A}) (l^{'} + x^{'} (w_{L} - \frac{1}{2}), \\ {u^{'} - y^{'} (w_{U} - \frac{1}{2}), x^{'}, y^{'})}_{λ} . \end{matrix}$ (10)

Similar to the proofs of Proposition 2.2 in [19], A new calculation formula of weighted distance ${\tilde{d}}_{λ} (\tilde{A}, \tilde{B})$ can be obtained in the following.

Proposition 2.3. Let $\tilde{A} = 〈 {\tilde{μ}}_{A}, {\tilde{ν}}_{A} 〉$ , $\tilde{B} = 〈 {\tilde{μ}}_{B}, {\tilde{ν}}_{B} 〉 \in \tilde{{IF}^{T}} (R)$ , where ${\tilde{μ}}_{A} = {\tilde{μ}}_{A} (l_{1}, u_{1}; x_{1}, y_{1})_{λ}$ , $1 - {\tilde{ν}}_{A} = (1 - {\tilde{ν}}_{A}) (l_{1}^{'}, u_{1}^{'}; x_{1}^{'}, y_{1}^{'})_{λ}$ , ${\tilde{μ}}_{B} = {\tilde{μ}}_{B} (l_{2}, u_{2}; x_{2}, y_{2})_{λ}$ , and $1 - {\tilde{ν}}_{B}$ $= 1 - {\tilde{ν}}_{B} (l_{2}^{'}, u_{2}^{'}; x_{2}^{'}, y_{2}^{'})_{λ}$ . Then $\begin{matrix} {\tilde{d}}_{λ}^{2} (\tilde{A}, \tilde{B}) \\ = \frac{1}{2} a (l_{1} - l_{2})^{2} + \frac{1}{2} b (u_{1} - u_{2})^{2} + \frac{1}{2} c (x_{1} - x_{2})^{2} \\ + \frac{1}{2} d (y_{1} - y_{2})^{2} + \frac{1}{2} a (l_{1}^{'} - l_{2}^{'})^{2} + \frac{1}{2} b (u_{1}^{'} - u_{2}^{'})^{2} \\ + \frac{1}{2} c (x_{1}^{'} - x_{2}^{'})^{2} + \frac{1}{2} d (y_{1}^{'} - y_{2}^{'})^{2} . \end{matrix}$

Let A = 〈μ_A, ν_A〉 ∈ IF (R). The weighted extended trapezoidal intuitionistic approximation of A is the extended trapezoidal intuitionistic fuzzy number that minimizes the weighted distance ${\tilde{d}}_{λ} (A, \tilde{X})$ , where $\tilde{X} = 〈 {\tilde{μ}}_{X}, {\tilde{ν}}_{X} 〉 \in \tilde{{IF}^{T}} (R)$ . Let $T_{e} (A) = 〈 {\tilde{μ}}_{T_{e}}, {\tilde{ν}}_{T_{e}} 〉 \in \tilde{{IF}^{T}} (R)$ , where ${\tilde{μ}}_{T_{e}} = {\tilde{μ}}_{T_{e}} (l^{e}, u^{e}; x^{e}, y^{e})_{λ}$ , and $1 - {\tilde{ν}}_{T_{e}} = (1 - {\tilde{ν}}_{T_{e}}) (l^{' e}, u^{' e}; x^{' e}, y^{' e})_{λ}$ are determined by the following equalities [19]: $l^{e} = a^{- 1} \int_{0}^{1} μ_{A_{L}} (α) λ_{L} (α) d α,$ (11) $u^{e} = b^{- 1} \int_{0}^{1} μ_{A_{U}} (α) λ_{U} (α) d α,$ (12) $x^{e} = c^{- 1} \int_{0}^{1} μ_{A_{L}} (α) (α - w_{L}) λ_{L} (α) d α,$ (13) $y^{e} = - d^{- 1} \int_{0}^{1} μ_{A_{U}} (α) (α - w_{U}) λ_{U} (α) d α,$ (14) $l^{' e} = a^{- 1} \int_{0}^{1} (1 - ν_{A})_{L} (α) λ_{L} (α) d α,$ (15) $u^{' e} = b^{- 1} \int_{0}^{1} (1 - ν_{A})_{U} (α) λ_{U} (α) d α,$ (16) $x^{' e} = c^{- 1} \int_{0}^{1} (1 - ν_{A})_{L} (α) (α - w_{L}) λ_{L} (α) d α,$ (17) $y^{' e} = - d^{- 1} \int_{0}^{1} (1 - ν_{A})_{U} (α) (α - w_{U}) λ_{U} (α) d α .$ (18)

In the following, we will show that T_e (A) is equal to the weighted extended trapezoidal intuitionistic approximation of A = 〈μ_A, ν_A〉. 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 = 〈μ_A, ν_A〉 ∈ IF (R) and $T_{e} (A) = 〈 {\tilde{μ}}_{T_{e}}, {\tilde{ν}}_{T_{e}} 〉 \in \tilde{{IF}^{T}} (R)$ , where ${\tilde{μ}}_{T_{e}} = {\tilde{μ}}_{T_{e}} (l^{e}, u^{e}; x^{e}, y^{e})_{λ}$ and $1 - {\tilde{ν}}_{T_{e}} = (1 - {\tilde{ν}}_{T_{e}}) (l^{' e}, u^{' e};$ x^′e, y^′e) _λ are defined as Equations (11)–(18). Then ${\tilde{d}}_{λ}^{2} (A, \tilde{X}) = {\tilde{d}}_{λ}^{2} (A, T_{e} (A)) + {\tilde{d_{λ}}}^{2} (T_{e} (A), \tilde{X})$ holds for every $\tilde{X} = 〈 μ_{\tilde{X}}, ν_{\tilde{X}} 〉 \in \tilde{{IF}^{T}} (R)$ .

Proposition 2.5. Let A = 〈μ_A, ν_A〉 ∈ IF (R) and $T_{e} (A) = 〈 {\tilde{μ}}_{T_{e}}, {\tilde{ν}}_{T_{e}} 〉 \in \tilde{{IF}^{T}} (R)$ , where ${\tilde{μ}}_{T_{e}} = {\tilde{μ}}_{T_{e}} (l^{e}, u^{e}; x^{e}, y^{e})_{λ}$ and $1 - {\tilde{ν}}_{T_{e}} = (1 - {\tilde{ν}}_{T_{e}}) (l^{' e}, u^{' e}; x^{' e}, y^{' e})_{λ}$ are defined as Equations (11)–(18). Then $T_{e} (A) = 〈 {\tilde{μ}}_{T_{e}}, {\tilde{ν}}_{T_{e}} 〉$ equals the weighted extended trapezoidal intuitionistic approximation of A = 〈μ_A, ν_A〉.

Because the weighted extended trapezoidal intuitionistic approximation T_e (A) of the intuitionistic fuzzy number A can be calculated by Equations (11)–(18), T_e (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, ⋯, g_m : Rⁿ → R be convex and differentiable functions. Then $\bar{x}$ solves the convex programming problem ${\begin{matrix} minf (x) \\ s . t . g_{i} (x) \leq h_{i}, i \in {1, . . ., m} \end{matrix}$ if and only if there exists ξ_i, i ∈ {1, . . ., m}, such that

$\nabla f (\bar{x}) + \sum_{i = 1}^{m} ξ_{i} \nabla g_{i} (\bar{x}) = 0$ ;

$g_{i} (\bar{x}) - h_{i} \leq 0$ ;

ξ_i ≥ 0;

$ξ_{i} (h_{i} - g_{i} (\bar{x})) = 0$ .

3 Weighted trapezoidal intuitionistic approximation and aggregation

3.1 Weighted trapezoidal intuitionistic approximation of an intuitionistic fuzzy number

Let A = 〈μ_A, ν_A〉 ∈ IF (R). T_A = 〈μ_{T
_A}, ν_{T
_A}〉 is called weighted trapezoidal intuitionistic approximation of A if $min_{B \in {IF}^{T} (R)} {\tilde{d}}_{λ}^{2} (A, B) = {\tilde{d}}_{λ}^{2} (A, T_{A})$ , where μ_{T
_A} = μ_{T
_A} (l, u ; x, y) and 1 - ν_{T
_A} = (1 - ν_{T
_A}) (l′, u′ ; x′, y′) . Let $T_{e} (A) = 〈 {\tilde{μ}}_{T_{e}}, {\tilde{ν}}_{T_{e}} 〉$ be weighted extended trapezoidal intuitionistic approximation of A, where ${\tilde{μ}}_{T_{e}} = {\tilde{μ}}_{T_{e}} (l^{e}, u^{e}; x^{e}, y^{e})_{λ}$ and $1 - {\tilde{ν}}_{T_{e}} = (1 - {\tilde{ν}}_{T_{e}}) (l^{' e}, u^{' e}; x^{' e}, y^{' e})_{λ} .$ Now we want to find the T_A.

By Proposition 2.4, we have ${\tilde{d}}_{λ}^{2} (A, T_{A}) = {\tilde{d}}_{λ}^{2} (A, T_{e} (A)) + {\tilde{d}}_{λ}^{2} (T_{e} (A), T_{A}) .$

By Equations (9), (10) and Proposition 2.3, we can compute ${\tilde{d}}_{λ}^{2} (T_{e} (A), T_{A})$ . Since ${\tilde{d}}_{λ}^{2} (A, T_{e} (A))$ is constant, to minimize ${\tilde{d}}_{λ}^{2} (A, T_{A})$ is as to minimizing the following function

$\begin{matrix} f (l, u, x, y, l^{'}, u^{'}, x^{'}, y^{'}) = {\tilde{d}}_{λ}^{2} (T_{e} (A), T_{A}) \\ = \frac{1}{2} a {(l^{e} - l - x (w_{L} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} b {(u^{e} - u + y (w_{U} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} c (x^{e} - x)^{2} + \frac{1}{2} d (y^{e} - y)^{2} \\ + \frac{1}{2} a {(l^{' e} - l^{'} - x^{'} (w_{L} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} b {(u^{' e} - u^{'} + y^{'} (w_{U} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} c (x^{' e} - x^{'})^{2} + \frac{1}{2} d (y^{' e} - y^{'})^{2} . \end{matrix}$ (19)

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

$\begin{matrix} Ω_{1} & = & {(l, u, x, y, l^{'}, u^{'}, x^{'}, y^{'}) \in R^{8} | x, y, x^{'}, y^{'} \geq 0, \\ x + y \leq 2 (u - l), x^{'} + y^{'} \leq 2 (u^{'} - l^{'})} . \end{matrix}$ (20)

We will show Problem 3.1 has an unique solution in the following.

Proof. Let us define an inner product 〈 ·, · 〉 in R⁸ by $〈 (x_{1}, y_{1}, z_{1}, o_{1}, p_{1}, q_{1}, r_{1}, s_{1}), (x_{2}, y_{2}, z_{2}, o_{2}, p_{2}, q_{2}, r_{2}, s_{2}) 〉 = \frac{1}{2} ({ax}_{1} x_{2} + {by}_{1} y_{2} + {cz}_{1} z_{2} + {do}_{1} o_{2} + {ap}_{1} p_{2} + {bq}_{1} q_{2} + {cr}_{1} r_{2} + {ds}_{1} s_{2})$ . If (x_i, y_i, z_i, o_i, p_i, q_i, r_i, s_i) ∈ R⁸, i ∈ {1, 2}, then $\begin{matrix} D^{2} ((x_{1}, y_{1}, z_{1}, o_{1}, p_{1}, q_{1}, r_{1}, s_{1}), \\ (x_{2}, y_{2}, z_{2}, o_{2}, p_{2}, q_{2}, r_{2}, s_{2})) \\ = 〈 (x_{1} - x_{2}, y_{1} - y_{2}, z_{1} - z_{2}, o_{1} - o_{2}, \\ p_{1} - p_{2}, q_{1} - q_{2}, r_{1} - r_{2}, s_{1} - s_{2}), \\ (x_{1} - x_{2}, y_{1} - y_{2}, z_{1} - z_{2}, o_{1} - o_{2}, \\ p_{1} - p_{2}, q_{1} - q_{2}, r_{1} - r_{2}, s_{1} - s_{2}) 〉 \\ = \frac{1}{2} (a (x_{1} - x_{2})^{2} + b (y_{1} - y_{2})^{2} + c (z_{1} - z_{2})^{2} \\ + d (o_{1} - o_{2})^{2} + a (p_{1} - p_{2})^{2} + b (q_{1} - q_{2})^{2} \\ + c (r_{1} - r_{2})^{2} + d (s_{1} - s_{2})^{2}) \end{matrix}$ 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 (l^e, u^e, x^e, y^e, l^′e, u^′e, x^′e, y^′e) ∈ R⁸ onto Ω₁ under D [33]. It is immediately obtained that the trapezoidal intuitionistic fuzzy number T_A nearest (with respect to ${\tilde{d}}_{λ}$ ) to a given intuitionistic fuzzy number A = 〈μ_A, ν_A〉 ∈ IF (R) exists and is unique. That is the weighted trapezoidal intuitionistic approximation T_A of A = 〈μ_A, ν_A〉 exists and is unique. □

3.2 Weighted trapezoidal intuitionistic approximation of intuitionistic fuzzy numbers

Now we want to aggregate efficiently intuitionistic fuzzy numbers A₁, A₂, ⋯, A_n. T_{(A₁,⋯,A_n)} = 〈μ_{T
_{(A₁,⋯,A_n)}}, ν_{T
_{(A₁,⋯,A_n)}}〉 is called weighted trapezoidal intuitionistic approximation of A₁, A₂, ⋯, A_n if $min_{B \in {IF}^{T} (R)} {\tilde{d}}_{λ}^{2} ((A_{1}, A_{2},$ $\dots, A_{n}), B) = {\tilde{d}}_{λ}^{2} ((A_{1}, A_{2},$ ⋯, A_n), T_{(A₁,⋯,A_n)}), where μ_{T
_{(A₁,⋯,A_n)}} = μ_{T
_{(A₁,⋯,A_n)}} (l, u ; x, y), 1 - ν_{T
_{(A₁,⋯,A_n)}} = (1 - ν_{T
_{(A₁,⋯,A_n)}}) (l′, u′ ; x′, y′) . By Proposition 2.2, the T_{(A₁,⋯,A_n)} minimizes $\sum_{i = 1}^{n} d_{λ}^{2} (A_{i},$ T_{(A₁,⋯,A_n)}), where (l, u, x, y, l′, u′, x′, y′) ∈ Ω₁, and Ω₁ is defined by (20).

By Proposition 2.4, to minimize ${\tilde{d}}_{λ}^{2} (A_{i}, T_{(A_{1}, \dots, A_{n})})$ is as to minimizing ${\tilde{d}}_{λ}^{2} (T_{e} (A_{i}), T_{(A_{1}, \dots, A_{n})})$ . $T_{e} (A_{i}) = {\tilde{T}}_{i} 〈 {\tilde{μ}}_{{\tilde{T}}_{i}},$ ${\tilde{ν}}_{{\tilde{T}}_{i}} 〉$ is weighted extended trapezoidal intuitionistic approximation of the intuitionistic fuzzy number A_i, where ${\tilde{μ}}_{{\tilde{T}}_{i}} = {\tilde{μ}}_{{\tilde{T}}_{i}} (l_{i}^{e}, u_{i}^{e}; x_{i}^{e}, y_{i}^{e})_{λ}$ and $1 - {\tilde{ν}}_{{\tilde{T}}_{i}} = (1 - {\tilde{ν}}_{{\tilde{T}}_{i}}) (l_{i}^{' e}, u_{i}^{' e};$ $x_{i}^{' e}, y_{i}^{' e})_{λ} .$ By Equations (9), (10) and Proposition 2.3, we have

$\begin{matrix} \sum_{i = 1}^{n} {\tilde{d}}_{λ}^{2} (T_{e} (A_{i}), T_{(A_{1}, \dots, A_{n})}) \\ = \frac{1}{2} a \sum_{i = 1}^{n} {(l - l_{i}^{e} + x (w_{L} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} b \sum_{i = 1}^{n} {(u - u_{i}^{e} - y (w_{U} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} c \sum_{i = 1}^{n} (x - x_{i}^{e})^{2} + \frac{1}{2} d \sum_{i = 1}^{n} (y - y_{i}^{e})^{2} \\ + \frac{1}{2} a \sum_{i = 1}^{n} {(l^{'} - l_{i}^{' e} + x^{'} (w_{L} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} b \sum_{i = 1}^{n} {(u^{'} - u_{i}^{' e} - y^{'} (w_{U} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} c \sum_{i = 1}^{n} (x^{'} - x_{i}^{' e})^{2} + \frac{1}{2} d \sum_{i = 1}^{n} (y^{'} - y_{i}^{' e})^{2} . \end{matrix}$ (21)

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,⋯,A_n)} = 〈μ_{T
_{(A₁,⋯,A_n)}}, ν_{T
_{(A₁,⋯,A_n)}}〉 ∈ IF^T (R) is a solution of Problem 3.2 if and only if there exist $ξ_{1}, ξ_{2}, ξ_{3}, ξ_{1}^{'}, ξ_{2}^{'},$ $ξ_{3}^{'}$ such that the following system of equations holds: $anl + an (w_{L} - \frac{1}{2}) x - a \sum_{i = 1}^{n} l_{i}^{e} + 2 ξ_{3} = 0,$ (22) $bnu + bn (\frac{1}{2} - w_{U}) y - b \sum_{i = 1}^{n} u_{i}^{e} - 2 ξ_{3} = 0,$ (23) $\begin{matrix} an (w_{L} - \frac{1}{2}) l + (an {(w_{L} - \frac{1}{2})}^{2} + cn) x \\ - a (w_{L} - \frac{1}{2}) \sum_{i = 1}^{n} l_{i}^{e} - c \sum_{i = 1}^{n} x_{i}^{e} - ξ_{1} + ξ_{3} = 0, \end{matrix}$ (24) $\begin{matrix} bn (\frac{1}{2} - w_{U}) u + (bn {(\frac{1}{2} - w_{U})}^{2} + dn) y \\ - b (\frac{1}{2} - w_{U}) \sum_{i = 1}^{n} u_{i}^{e} - d \sum_{i = 1}^{n} y_{i}^{e} - ξ_{2} + ξ_{3} = 0, \end{matrix}$ (25) ${anl}^{'} + an (w_{L} - \frac{1}{2}) x^{'} - a \sum_{i = 1}^{n} l_{i}^{' e} + 2 ξ_{3}^{'} = 0,$ (26) ${bnu}^{'} + bn (\frac{1}{2} - w_{U}) y^{'} - b \sum_{i = 1}^{n} u_{i}^{' e} - 2 ξ_{3}^{'} = 0,$ (27) $\begin{matrix} an (w_{L} - \frac{1}{2}) l^{'} + (an {(w_{L} - \frac{1}{2})}^{2} + cn) x^{'} \\ - a (w_{L} - \frac{1}{2}) \sum_{i = 1}^{n} l_{i}^{' e} - c \sum_{i = 1}^{n} x_{i}^{' e} - ξ_{1}^{'} + ξ_{3}^{'} = 0, \end{matrix}$ (28) $\begin{matrix} bn (\frac{1}{2} - w_{U}) u^{'} + (bn {(\frac{1}{2} - w_{U})}^{2} + dn) y^{'} \\ - b (\frac{1}{2} - w_{U}) \sum_{i = 1}^{n} u_{i}^{' e} - d \sum_{i = 1}^{n} y_{i}^{' e} - ξ_{2}^{'} + ξ_{3}^{'} = 0, \end{matrix}$ (29) $x \geq 0,$ (30) $y \geq 0,$ (31) $x^{'} \geq 0,$ (32) $y^{'} \geq 0,$ (33) $x + y \leq 2 (u - l),$ (34) $x^{'} + y^{'} \leq 2 (u^{'} - l^{'}),$ (35) $ξ_{1} \geq 0,$ (36) $ξ_{2} \geq 0,$ (37) $ξ_{3} \geq 0,$ (38) $ξ_{1}^{'} \geq 0,$ (39) $ξ_{2}^{'} \geq 0,$ (40) $ξ_{3}^{'} \geq 0,$ (41) $ξ_{1} x = 0,$ (42) $ξ_{2} y = 0,$ (43) $ξ_{3} (x + y - 2 u + 2 l) = 0,$ (44) $ξ_{1}^{'} x^{'} = 0,$ (45) $ξ_{2}^{'} y^{'} = 0,$ (46) $ξ_{3}^{'} (x^{'} + y^{'} - 2 u^{'} + 2 l^{'}) = 0 .$ (47)

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.

3.3 Weighted trapezoidal intuitionistic approximation of an aggregation

The arithmetic mean of given intuitionistic fuzzy numbers A₁, A₂, ⋯, A_n is denoted by $\bar{A}$ , that is $\bar{A} = \frac{1}{n} (A_{1} + A_{2} + \dots + A_{n})$ . $T_{\bar{A}} = 〈 μ_{T_{\bar{A}}}, ν_{T_{\bar{A}}} 〉$ is called weighted trapezoidal intuitionistic approximation of $\bar{A}$ if $min_{B \in {IF}^{T} (R)}$ ${\tilde{d}}_{λ}^{2} (\bar{A}, B) = {\tilde{d}}_{λ}^{2} (\bar{A}, T_{\bar{A}})$ , where $μ_{T_{\bar{A}}} = μ_{T_{\bar{A}}} (\bar{l}, \bar{u}; \bar{x}, \bar{y})$ and $1 - ν_{T_{\bar{A}}} = (1 - ν_{T_{\bar{A}}}) ({\bar{l}}^{'}, {\bar{u}}^{'};$ ${\bar{x}}^{'}, {\bar{y}}^{'}) .$ Taking into account Proposition 2.4, to minimize ${\tilde{d}}_{λ}^{2} (\bar{A}, T_{\bar{A}})$ ) is as to minimizing ${\tilde{d}}_{λ}^{2} (T_{e} (\bar{A}), T_{\bar{A}}),$ where $T_{e} (\bar{A}) = \tilde{T} 〈 {\tilde{μ}}_{\tilde{T}},$ ${\tilde{ν}}_{\tilde{T}} 〉$ is weighted extended trapezoidal intuitionistic approximation of $\bar{A} \in IF (R)$ . $T_{e} (A_{i}) = {\tilde{T}}_{i} 〈 {\tilde{μ}}_{{\tilde{T}}_{i}},$ ${\tilde{ν}}_{{\tilde{T}}_{i}} 〉$ is weighted extended trapezoidal intuitionistic approximation of A_i ∈ IF (R). From (11)–(18), we can compute l^e, u^e, x^e, y^e, l^′e, u^′e, x^′e and y^′e . Hence we obtain $T_{e} (\bar{A}) = \tilde{T} 〈 {\tilde{μ}}_{\tilde{T}},$ ${\tilde{ν}}_{\tilde{T}} 〉,$ where $\begin{matrix} {\tilde{μ}}_{\tilde{T}} = {\tilde{μ}}_{\tilde{T}} {(\frac{1}{n} \sum_{i = 1}^{n} l_{i}^{e}, \frac{1}{n} \sum_{i = 1}^{n} u_{i}^{e}; \frac{1}{n} \sum_{i = 1}^{n} x_{i}^{e}, \frac{1}{n} \sum_{i = 1}^{n} y_{i}^{e})}_{λ} \end{matrix}$ (48) $\begin{matrix} 1 - {\tilde{ν}}_{\tilde{T}} = (1 - {\tilde{ν}}_{\tilde{T}}) (\frac{1}{n} \sum_{i = 1}^{n} l_{i}^{' e}, \frac{1}{n} \sum_{i = 1}^{n} u_{i}^{' e}; \\ {\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{' e}, \frac{1}{n} \sum_{i = 1}^{n} y_{i}^{' e})}_{λ} . \end{matrix}$ (49)

By (9) and (10) and Proposition 2.3, we have

$\begin{matrix} f (\bar{l}, \bar{u}, \bar{x}, \bar{y}, {\bar{l}}^{'}, {\bar{u}}^{'}, {\bar{x}}^{'}, {\bar{y}}^{'}) = {\tilde{d}}_{λ}^{2} (T_{e} (\bar{A}), T_{\bar{A}}) \\ = \frac{1}{2} a {(\bar{l} - \frac{1}{n} \sum_{i = 1}^{n} l_{i}^{e} + \bar{x} (w_{L} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} b {(\bar{u} - \frac{1}{n} \sum_{i = 1}^{n} u_{i}^{e} - \bar{y} (w_{U} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} c (\bar{x} - \frac{1}{n} \sum_{i = 1}^{n} x_{i}^{e})^{2} + \frac{1}{2} d {(\bar{y} - \frac{1}{n} \sum_{i = 1}^{n} y_{i}^{e})}^{2} \\ + \frac{1}{2} a {({\bar{l}}^{'} - \frac{1}{n} \sum_{i = 1}^{n} l_{i}^{' e} + {\bar{x}}^{'} (w_{L} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} b {({\bar{u}}^{'} - \frac{1}{n} \sum_{i = 1}^{n} u_{i}^{' e} - {\bar{y}}^{'} (w_{U} - \frac{1}{2}))}^{2} \\ + \frac{1}{2} c {({\bar{x}}^{'} - \frac{1}{n} \sum_{i = 1}^{n} x_{i}^{' e})}^{2} \\ + \frac{1}{2} d {({\bar{y}}^{'} - \frac{1}{n} \sum_{i = 1}^{n} y_{i}^{' e})}^{2} . \end{matrix}$ (50)

Taking into account Proposition 2.2, to find the minimum of Equation (50) can be reduced to the following problem.

Problem 3.3. Find $(\bar{l}, \bar{u}, \bar{x}, \bar{y}, {\bar{l}}^{'}, {\bar{u}}^{'}, {\bar{x}}^{'}, {\bar{y}}^{'}) \in Ω_{2}$ , which minimizes Equation (50), where

$\begin{matrix} Ω_{2} & = & {(\bar{l}, \bar{u}, \bar{x}, \bar{y}, {\bar{l}}^{'}, {\bar{u}}^{'}, {\bar{x}}^{'}, {\bar{y}}^{'}) \in R^{8} | \bar{x}, \bar{y}, {\bar{x}}^{'}, {\bar{y}}^{'} \geq 0, \\ \bar{x} + \bar{y} \leq 2 (\bar{u} - \bar{l}), {\bar{x}}^{'} + {\bar{y}}^{'} \leq 2 ({\bar{u}}^{'} - {\bar{l}}^{'})} . \end{matrix}$ (51)

According to the Karush-Kuhn-Tucker theorem, an trapezoidal intuitionistic fuzzy number $T_{\bar{A}} = 〈 μ_{T_{\bar{A}}},$ $ν_{T_{\bar{A}}} 〉$ is a solution of Problem 3.3 if and Only if there exist $η_{1}, η_{2}, η_{3}, η_{1}^{'}, η_{2}^{'}, η_{3}^{'}$ such that the following system of equations holds: $a \bar{l} + a (w_{L} - \frac{1}{2}) \bar{x} - \frac{a}{n} \sum_{i = 1}^{n} l_{i}^{e} + 2 η_{3} = 0,$ (52) $b \bar{u} + b (\frac{1}{2} - w_{U}) \bar{y} - \frac{b}{n} \sum_{i = 1}^{n} u_{i}^{e} - 2 η_{3} = 0,$ (53) $\begin{matrix} a (w_{L} - \frac{1}{2}) \bar{l} + a {(w_{L} - \frac{1}{2})}^{2} \bar{x} - \frac{c}{n} \sum_{i = 1}^{n} x_{i}^{e} \\ - \frac{a}{n} (w_{L} - \frac{1}{2}) \sum_{i = 1}^{n} l_{i}^{e} + c \bar{x} - η_{1} + η_{3} = 0, \end{matrix}$ (54) $\begin{matrix} b (\frac{1}{2} - w_{U}) \bar{u} + b {(\frac{1}{2} - w_{U})}^{2} \bar{y} - \frac{d}{n} \sum_{i = 1}^{n} y_{i}^{e} \\ - \frac{b}{n} (\frac{1}{2} - w_{U}) \sum_{i = 1}^{n} u_{i}^{e} + d \bar{y} - η_{2} + η_{3} = 0, \end{matrix}$ (55) $a {\bar{l}}^{'} + a (w_{L} - \frac{1}{2}) {\bar{x}}^{'} - \frac{a}{n} \sum_{i = 1}^{n} l_{i}^{' e} + 2 η_{3}^{'} = 0,$ (56) $b {\bar{u}}^{'} + b (\frac{1}{2} - w_{U}) {\bar{y}}^{'} - \frac{b}{n} \sum_{i = 1}^{n} u_{i}^{' e} - 2 η_{3}^{'} = 0,$ (57) $\begin{matrix} a (w_{L} - \frac{1}{2}) {\bar{l}}^{'} + (a {(w_{L} - \frac{1}{2})}^{2} + c) {\bar{x}}^{'} \\ - \frac{a}{n} (w_{L} - \frac{1}{2}) \sum_{i = 1}^{n} l_{i}^{' e} - \frac{c}{n} \sum_{i = 1}^{n} x_{i}^{' e} - η_{1}^{'} \\ + η_{3}^{'} = 0, \end{matrix}$ (58) $\begin{matrix} b (\frac{1}{2} - w_{U}) {\bar{u}}^{'} + (b {(\frac{1}{2} - w_{U})}^{2} + d) {\bar{y}}^{'} \\ - \frac{b}{n} (\frac{1}{2} - w_{U}) \sum_{i = 1}^{n} u_{i}^{' e} - \frac{d}{n} \sum_{i = 1}^{n} y_{i}^{' e} - η_{2}^{'} \\ + η_{3}^{'} = 0, \end{matrix}$ (59) $\bar{x} \geq 0,$ (60) $\bar{y} \geq 0,$ (61) ${\bar{x}}^{'} \geq 0,$ (62) ${\bar{y}}^{'} \geq 0,$ (63) $\bar{x} + \bar{y} \leq 2 (\bar{u} - \bar{l}),$ (64) ${\bar{x}}^{'} + {\bar{y}}^{'} \leq 2 ({\bar{u}}^{'} - {\bar{l}}^{'}),$ (65) $η_{1} \geq 0,$ (66) $η_{2} \geq 0,$ (67) $η_{3} \geq 0,$ (68) $η_{1}^{'} \geq 0,$ (69) $η_{2}^{'} \geq 0,$ (70) $η_{3}^{'} \geq 0,$ (71) $η_{1} \bar{x} = 0,$ (72) $η_{2} \bar{y} = 0,$ (73) $η_{3} (\bar{x} + \bar{y} - 2 \bar{u} + 2 \bar{l}) = 0,$ (74) $η_{1}^{'} {\bar{x}}^{'} = 0,$ (75) $η_{2}^{'} {\bar{y}}^{'} = 0,$ (76) $η_{3}^{'} ({\bar{x}}^{'} + {\bar{y}}^{'} - 2 {\bar{u}}^{'} + 2 {\bar{l}}^{'}) = 0 .$ (77)

3.4 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,⋯,A_n)} of A₁, A₂, ⋯, A_n is the weighted trapezoidal intuitionistic approximation $T_{\bar{A}}$ of $\bar{A}$ .

Proof. We get the proof immediately substituting ξ₁, ξ₂, $ξ_{3}, ξ_{1}^{'}, ξ_{2}^{'}, ξ_{3}^{'}$ in (22)–(47) by $n η_{1}, n η_{2}, n η_{3}, n η_{1}^{'},$ $n η_{2}^{'}, n η_{3}^{'}$ . □

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 = 〈μ_{A
₁}, ν_{A
₁}〉 ∈ IF (R) and A₂ = 〈μ_{A
₂}, ν_{A
₂}〉 ∈ IF (R), where $(μ_{A_{1}})_{α} = [- 1 + α^{2}, 3 - \sqrt{α}],$ $(μ_{A_{2}})_{α} = [1 + \sqrt{α}, 3 - \sqrt{α}],$ (1 - ν_{A
₁}) _α = [- 1 + α³, 4 - 2α²] and (1 - ν_{A
₂}) _α = [1 + α³, 3 - α²] (see Figs. 1 and 2). Then $\bar{A} =$ $\frac{1}{2} (A_{1} + A_{2}) = 〈 μ_{\bar{A}}, ν_{\bar{A}} 〉,$ where ${(μ_{\bar{A}})}_{α} = [\frac{1}{2} α^{2}$ $+ \frac{1}{2} \sqrt{α}, 3 - \sqrt{α}]$ and ${(1 - ν_{\bar{A}})}_{α} = [α^{3}, \frac{7}{2} - \frac{3}{2} α^{2}]$ (see black solid line in Fig. 3). Let us consider λ_L (α) = λ_U (α) =1 . From step 1 of algorithm in [18], The weighted trapezoidal intuitionistic approximation of $\bar{A}$ is the $T_{\bar{A}} = 〈 μ_{T_{\bar{A}}}, ν_{T_{\bar{A}}} 〉,$ where $μ_{T_{\bar{A}}}$ $= μ_{T_{\bar{A}}} (\frac{1}{20}, \frac{19}{20}, \frac{29}{15}, \frac{41}{15})$ and $1 - ν_{T_{\bar{A}}} = (1 - ν_{T_{\bar{A}}}) (- \frac{1}{5}, \frac{7}{10}, \frac{9}{4}, \frac{15}{4})$ (see blue dashed line in Fig. 3).

Theorem 3.1 shows $T_{\bar{A}}$ is also the weighted trapezoidal intuitionistic approximation of A₁, A₂ .

Fig.1

Intuitionistic fuzzy number A₁ = 〈μ_A
₁, ν_A
₁〉.

Fig.2

Intuitionistic fuzzy number A₂ = 〈μ_A
₂, ν_A
₂〉.

Fig.3

Intuitionistic fuzzy numbers $\bar{A} = 〈 μ_{\bar{A}}, ν_{\bar{A}} 〉$ and $T_{\bar{A}} = 〈 μ_{T_{\bar{A}}}, ν_{T_{\bar{A}}} 〉$ .

Example 3.2. Intuitionistic fuzzy numbers A₁ = 〈μ_{A
₁}, ν_{A
₁}〉 and A₂ = 〈μ_{A
₂}, ν_{A
₂}〉 are considered in Example 3.1. Let A₃ = 〈μ_{A
₃}, ν_{A
₃}〉 ∈ IF (R), where $(μ_{A_{3}})_{α} = [- \frac{1}{3} + α^{2}, 4 - α^{2}],$ and $(1 - ν_{A_{3}})_{α} = [- \frac{1}{2} + α^{3}, 6 - 3 α^{2}]$ (see Fig. 4). Then $\bar{A} = \frac{1}{3} (A_{1} + A_{2} + A_{3}) = 〈 μ_{\bar{A}}, ν_{\bar{A}} 〉,$ where ${(μ_{\bar{A}})}_{α} = [- \frac{1}{9} + \frac{2}{3} α^{2}$ $+ \frac{1}{3} \sqrt{α}, \frac{10}{3} - \frac{2}{3} \sqrt{α} - \frac{1}{3} α^{2}]$ and ${(1 - ν_{\bar{A}})}_{α} = [- \frac{1}{6} +$ $α^{3}, \frac{13}{3} - 2 α^{2}]$ (see black solid line in Fig. 5). We still consider λ_L (α) = λ_U (α) =1 . Using the same algorithm (step 1 of algorithm in [18]), we obtain the weighted trapezoidal intuitionistic approximation of $\bar{A}$ , that is $T_{\bar{A}} = 〈 μ_{T_{\bar{A}}}, ν_{T_{\bar{A}}} 〉,$ where $μ_{T_{\bar{A}}}$ $= μ_{T_{\bar{A}}} (- \frac{2}{15}, \frac{4}{5}, \frac{211}{90}, \frac{289}{90})$ and $1 - ν_{T_{\bar{A}}} =$ $(1 - ν_{T_{\bar{A}}}) (- \frac{11}{30}, \frac{8}{15}, \frac{8}{3}, \frac{14}{3})$ (see blue dashed line in Fig. 5).

By Theorem 3.1, $T_{\bar{A}}$ is also the weighted trapezoidal intuitionistic approximation of A₁, A₂and A₃.

Fig.4

Intuitionistic fuzzy number A₃ = 〈μ_A
₃, ν_A
₃〉.

Fig.5

Intuitionistic fuzzy numbers $\bar{A} = 〈 μ_{\bar{A}}, ν_{\bar{A}} 〉$ and $T_{\bar{A}} = 〈 μ_{T_{\bar{A}}}, ν_{T_{\bar{A}}} 〉$ .

According to Theorem 3.1 and Equations (2), (5) in [36], the following consequence can be obtained immediately.

Corollary 3.1. Let A_i = 〈μ_{A
_i}, ν_{A
_i}〉 ∈ IF^T (R), where $μ_{A_{i}} = μ_{A_{i}} (t_{1}^{i}, t_{2}^{i}, t_{3}^{i}, t_{4}^{i}), 1 - ν_{A_{i}} = (1 - ν_{A_{i}}) (s_{1}^{i}, s_{2}^{i}, s_{3}^{i}, s_{4}^{i}), i \in {1, \dots, n} .$ The weighted trapezoidal intuitionistic approximation of A₁, A₂, ⋯, A_n is $\bar{A} = 〈 μ_{\bar{A}}, ν_{\bar{A}} 〉,$ where $μ_{\bar{A}} = μ_{\bar{A}} (\frac{1}{n} \sum_{i = 1}^{n} t_{1}^{i}, \frac{1}{n} \sum_{i = 1}^{n} t_{2}^{i}, \frac{1}{n} \sum_{i = 1}^{n} t_{3}^{i}, \frac{1}{n} \sum_{i = 1}^{n} t_{4}^{i})$ and $1 - ν_{\bar{A}} = (1 - ν_{\bar{A}}) (\frac{1}{n} \sum_{i = 1}^{n} s_{1}^{i}, \frac{1}{n} \sum_{i = 1}^{n} s_{2}^{i}, \frac{1}{n} \sum_{i = 1}^{n} s_{3}^{i}, \frac{1}{n} \sum_{i = 1}^{n} s_{4}^{i}) .$

4 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.

4.1 Weighted expected interval

Definition 4.1. The weighted expected interval of A = 〈μ_A, ν_A〉 ∈ IF (R) is defined as ${EI}^{w} (A) = [E_{*} (A), E^{*} (A)],$ (78) where $E_{*} (A) = \frac{1}{2 a} \int_{0}^{1} (μ_{A_{L}} (α) + (1 - ν_{A})_{L} (α)) λ_{L} (α) d α$ and $E^{*} (A) = \frac{1}{2 b} \int_{0}^{1} (μ_{A_{U}} (α) + (1 - ν_{A})_{U} (α)) λ_{U} (α) d α .$

Definition 4.2. The weighted expected value of A = 〈μ_A, ν_A〉 ∈ IF (R) is defined as ${EV}^{w} (A) = \frac{E_{*} (A) + E^{*} (A)}{2}$ .

From Definition 4.1, Equations (11), (12), (15) and (16), we get ${EI}^{w} (A) = [\frac{1}{2} (l^{e} + l^{' e}), \frac{1}{2} (u^{e} + u^{' e})] .$ (79)

The weighted expected interval of a sequence of intuitionistic fuzzy numbers A₁, A₂, ⋯, A_n can be defined as

$\begin{matrix} {EI}^{w} (A_{1}, \dots, A_{n}) \\ = \frac{1}{n} ({EI}^{w} (A_{1}) + \dots + {EI}^{w} (A_{n})) \\ = [\frac{1}{2 n} \sum_{i = 1}^{n} (l_{i}^{e} + l_{i}^{' e}), \frac{1}{2 n} \sum_{i = 1}^{n} (u_{i}^{e} + u_{i}^{' e})] \end{matrix}$ (80)

Let $T_{e} (A) = 〈 {\tilde{μ}}_{{\tilde{T}}_{e}}, {\tilde{ν}}_{{\tilde{T}}_{e}} 〉$ be the weighted extended trapezoidal intuitionistic approximation of the intuitionistic fuzzy number A, where ${\tilde{μ}}_{T_{e}} = {\tilde{μ}}_{T_{e}} (l^{e}, u^{e};$ x^e, y^e) _λ and $1 - {\tilde{ν}}_{T_{e}} = (1 - {\tilde{ν}}_{T_{e}}) (l^{' e}, u^{' e}; x^{' e}, y^{' e})_{λ} .$ Straightforward calculations using (3), (4) and (78) show that ${EI}^{w} (T_{e} (A)) = [\frac{1}{2} (l^{e} + l^{' e}), \frac{1}{2} (u^{e} + u^{' e})] .$ (81)

Thus combining (79) and (81) we get a result that ${EI}^{w} (T_{e} (A)) = {EI}^{w} (A) .$ (82)

4.2 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 T_A of A = 〈μ_A, ν_A〉 ∈ IF (R) exists and is unique.

Intuitionistic fuzzy numbers A₁, A₂, ⋯, A_n 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, ⋯, A_n. T_{(A₁,⋯,A_n)} = 〈μ_{T
_{(A₁,⋯,A_n)}}, ν_{T
_{(A₁,⋯,A_n)}}〉 is called weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of A₁, A₂, ⋯, A_n if $min_{B \in {IF}^{T} (R)} {\tilde{d}}_{λ}^{2} ((A_{1}, A_{2},$ ⋯, A_n), B) $= {\tilde{d}}_{λ}^{2} ((A_{1}, A_{2},$ ⋯, A_n), T_{(A₁,⋯,A_n)}) under condition EI^w (T_{(A₁,⋯,A_n)}) = EI^w (A₁, ⋯, A_n), where μ_{T
_{(A₁,⋯,A_n)}} = μ_{T
_{(A₁,⋯,A_n)}} (l, u ; x, y) and 1 - ν_{T
_{(A₁,⋯,A_n)}} = (1 - ν_{T
_{(A₁,⋯,A_n)}}) (l′, u′ ; x′, y′) . From Equations (1), (2) and Definition 4.1, we get

$\begin{matrix} {EI}^{w} (T_{(A_{1}, \dots, A_{n})}) \\ = [\frac{1}{2} (l + l^{'} + x (w_{L} - \frac{1}{2}) + x^{'} (w_{L} - \frac{1}{2})), \\ \frac{1}{2} (u + u^{'} - y (w_{U} - \frac{1}{2}) - y^{'} (w_{U} - \frac{1}{2}))] \end{matrix}$ (83)

By Equation (83), EI^w (T_{(A₁,⋯,A_n)}) = EI^w (A₁, ⋯, A_n) can be rewritten as: $\begin{matrix} l + l^{'} + x (w_{L} - \frac{1}{2}) + x^{'} (w_{L} - \frac{1}{2}) \\ = \frac{1}{n} \sum_{i = 1}^{n} (l_{i}^{e} + l_{i}^{' e}) \end{matrix}$ (84) $\begin{matrix} u + u^{'} - y (w_{U} - \frac{1}{2}) - y^{'} (w_{U} - \frac{1}{2}) \\ = \frac{1}{n} \sum_{i = 1}^{n} (u_{i}^{e} + u_{i}^{' e}) \end{matrix}$ (85)

Hence to find the weighted trapezoidal intuitionistic approximation preserving the weighted expected interval T_{(A₁,⋯,A_n)} 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,⋯,A_n)} = 〈μ_{T
_{(A₁,⋯,A_n)}}, ν_{T
_{(A₁,⋯,A_n)}}〉 ∈ IF^T (R) is a solution of Problem 4.1 if and only if there exist $ς_{1}, ς_{2}, ς_{3}, ς_{4}, ς_{1}^{'},$ $ς_{2}^{'},$ $ς_{3}^{'}, ς_{4}^{'}$ such that the following system of equations holds: $\begin{matrix} anl + an (w_{L} - \frac{1}{2}) x - a \sum_{i = 1}^{n} l_{i}^{e} + 2 ς_{3} + ς_{4} = 0, \end{matrix}$ (86) $\begin{matrix} bnu + bn (\frac{1}{2} - w_{U}) y - b \sum_{i = 1}^{n} u_{i}^{e} - 2 ς_{3} + ς_{4}^{'} = 0, \end{matrix}$ (87) $\begin{matrix} an (w_{L} - \frac{1}{2}) l + (an {(w_{L} - \frac{1}{2})}^{2} + cn) x \\ - a (w_{L} - \frac{1}{2}) \sum_{i = 1}^{n} l_{i}^{e} - c \sum_{i = 1}^{n} x_{i}^{e} - ς_{1} + ς_{3} \\ + ς_{4} (w_{L} - \frac{1}{2}) = 0, \end{matrix}$ (88) $\begin{matrix} bn (\frac{1}{2} - w_{U}) u + (bn {(\frac{1}{2} - w_{U})}^{2} + dn) y \\ - b (\frac{1}{2} - w_{U}) \sum_{i = 1}^{n} u_{i}^{e} - d \sum_{i = 1}^{n} y_{i}^{e} - ς_{2} + ς_{3} \\ - ς_{4}^{'} (w_{U} - \frac{1}{2}) = 0, \end{matrix}$ (89) $\begin{matrix} {anl}^{'} + an (w_{L} - \frac{1}{2}) x^{'} - a \sum_{i = 1}^{n} l_{i}^{' e} + 2 ς_{3}^{'} + ς_{4} = 0, \end{matrix}$ (90) $\begin{matrix} {bnu}^{'} + bn (\frac{1}{2} - w_{U}) y^{'} - b \sum_{i = 1}^{n} u_{i}^{' e} - 2 ς_{3}^{'} + ς_{4}^{'} = 0, \end{matrix}$ (91) $\begin{matrix} an (w_{L} - \frac{1}{2}) l^{'} + (an {(w_{L} - \frac{1}{2})}^{2} + cn) x^{'} \\ - a (w_{L} - \frac{1}{2}) \sum_{i = 1}^{n} l_{i}^{' e} - c \sum_{i = 1}^{n} x_{i}^{' e} - ς_{1}^{'} + ς_{3}^{'} \\ + ς_{4} (w_{L} - \frac{1}{2}) = 0, \end{matrix}$ (92) $\begin{matrix} bn (\frac{1}{2} - w_{U}) u^{'} + (bn {(\frac{1}{2} - w_{U})}^{2} + dn) y^{'} \\ - b (\frac{1}{2} - w_{U}) \sum_{i = 1}^{n} u_{i}^{' e} - d \sum_{i = 1}^{n} y_{i}^{' e} - ς_{2}^{'} + ς_{3}^{'} \\ - ς_{4}^{'} (w_{U} - \frac{1}{2}) = 0, \end{matrix}$ (93) $x \geq 0,$ (94) $y \geq 0,$ (95) $x^{'} \geq 0,$ (96) $y^{'} \geq 0,$ (97) $x + y \leq 2 (u - l),$ (98) $x^{'} + y^{'} \leq 2 (u^{'} - l^{'}),$ (99) $\begin{matrix} l + l^{'} + x (w_{L} - \frac{1}{2}) + x^{'} (w_{L} - \frac{1}{2}) \\ = \frac{1}{n} \sum_{i = 1}^{n} (l_{i}^{e} + l_{i}^{' e}) \end{matrix}$ (100) $\begin{matrix} u + u^{'} - y (w_{U} - \frac{1}{2}) - y^{'} (w_{U} - \frac{1}{2}) \\ = \frac{1}{n} \sum_{i = 1}^{n} (u_{i}^{e} + u_{i}^{' e}) \end{matrix}$ (101) $ς_{1} \geq 0,$ (102) $ς_{2} \geq 0,$ (103) $ς_{3} \geq 0,$ (104) $ς_{4} \geq 0,$ (105) $ς_{1}^{'} \geq 0,$ (106) $ς_{2}^{'} \geq 0,$ (107) $ς_{3}^{'} \geq 0,$ (108) $ς_{4}^{'} \geq 0,$ (109) $ς_{1} x = 0,$ (110) $ς_{2} y = 0,$ (111) $ς_{3} (x + y - 2 u + 2 l) = 0,$ (112) $\begin{matrix} ς_{4} (l + l^{'} + x (w_{L} - \frac{1}{2}) + x^{'} (w_{L} - \frac{1}{2}) \\ - \frac{1}{n} \sum_{i = 1}^{n} (l_{i}^{e} + l_{i}^{' e})) = 0 \end{matrix}$ (113) $ς_{1}^{'} x^{'} = 0,$ (114) $ς_{2}^{'} y^{'} = 0,$ (115) $ς_{3}^{'} (x^{'} + y^{'} - 2 u^{'} + 2 l^{'}) = 0 .$ (116) $\begin{matrix} ς_{4}^{'} (u + u^{'} - y (w_{U} - \frac{1}{2}) - y^{'} (w_{U} - \frac{1}{2}) \\ - \frac{1}{n} \sum_{i = 1}^{n} (u_{i}^{e} + u_{i}^{' e})) = 0 \end{matrix}$ (117)

4.3 Weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of an aggregation

$T_{\bar{A}} = 〈 μ_{T_{\bar{A}}}, ν_{T_{\bar{A}}} 〉$ is called weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of $\bar{A}$ if $min_{B \in {IF}^{T} (R)} {\tilde{d}}_{λ}^{2} (\bar{A}, B) = {\tilde{d}}_{λ}^{2} (\bar{A}, T_{\bar{A}})$ under condition ${EI}^{w} (T_{\bar{A}}) = {EI}^{w} (\bar{A})$ , where $μ_{T_{\bar{A}}} = μ_{T_{\bar{A}}} (\bar{l},$ $\bar{u}; \bar{x}, \bar{y})$ and $1 - ν_{\bar{A}} = (1 - ν_{\bar{A}}) ({\bar{l}}^{'}, {\bar{u}}^{'}; {\bar{x}}^{'}, {\bar{y}}^{'}) .$ By Equation (82), we have ${EI}^{w} (T_{e} (\bar{A})) = {EI}^{w} (T_{\bar{A}})$ . From Equations (1), (2), (9), (10), (48), (49), (79) and (81), the condition ${EI}^{w} (T_{\bar{A}}) = {EI}^{w} (\bar{A})$ can be rewritten as $\begin{matrix} \bar{l} + {\bar{l}}^{'} + \bar{x} (w_{L} - \frac{1}{2}) + {\bar{x}}^{'} (w_{L} - \frac{1}{2}) \\ = \frac{1}{n} \sum_{i = 1}^{n} (l_{i}^{e} + l_{i}^{' e}) \end{matrix}$ (118) $\begin{matrix} \bar{u} + {\bar{u}}^{'} - \bar{y} (w_{U} - \frac{1}{2}) - {\bar{y}}^{'} (w_{U} - \frac{1}{2}) \\ = \frac{1}{n} \sum_{i = 1}^{n} (u_{i}^{e} + u_{i}^{' e}) \end{matrix}$ (119)

Hence to find the weighted trapezoidal intuitionistic approximation preserving the weighted expected interval $T_{\bar{A}}$ is as to solve Problem 3.3 under the conditions Equations (118) and (119). Then we have the following problems.

Problem 4.2. Find $(\bar{l}, \bar{u}, \bar{x}, \bar{y}, {\bar{l}}^{'}, {\bar{u}}^{'}, {\bar{x}}^{'}, {\bar{y}}^{'}) \in Ω_{2}$ , which minimizes Equation (50) under the conditions Equations (118) and (119), where Ω₂ is defined by (51).

According to the Karush-Kuhn-Tucker theorem, an trapezoidal intuitionistic fuzzy number $T_{\bar{A}} = 〈 μ_{T_{\bar{A}}},$ $ν_{T_{\bar{A}}} 〉$ is a solution of Problem 4.2 if and only if there exist $τ_{1}, τ_{2}, τ_{3}, τ_{4}, τ_{1}^{'}, τ_{2}^{'}, τ_{3}^{'}, τ_{4}^{'}$ such that the following system of equations holds: $\begin{matrix} a \bar{l} + a (w_{L} - \frac{1}{2}) \bar{x} - \frac{a}{n} \sum_{i = 1}^{n} l_{i}^{e} + 2 τ_{3} + τ_{4} = 0, \end{matrix}$ (120) $\begin{matrix} b \bar{u} + b (\frac{1}{2} - w_{U}) \bar{y} - \frac{b}{n} \sum_{i = 1}^{n} u_{i}^{e} - 2 τ_{3} + τ_{4}' = 0, \end{matrix}$ (121) $\begin{matrix} a (w_{L} - \frac{1}{2}) \bar{l} + (a {(w_{L} - \frac{1}{2})}^{2} + c) \bar{x} \\ - \frac{a}{n} (w_{L} - \frac{1}{2}) \sum_{i = 1}^{n} l_{i}^{e} - \frac{c}{n} \sum_{i = 1}^{n} x_{i}^{e} - τ_{1} \\ + τ_{3} + τ_{4} (w_{L} - \frac{1}{2}) = 0, \end{matrix}$ (122) $\begin{matrix} b (\frac{1}{2} - w_{U}) \bar{u} + (b {(\frac{1}{2} - w_{U})}^{2} + d) \bar{y} \\ - \frac{b}{n} (\frac{1}{2} - w_{U}) \sum_{i = 1}^{n} u_{i}^{e} - \frac{d}{n} \sum_{i = 1}^{n} y_{i}^{e} - τ_{2} \\ + τ_{3} - τ_{4}^{'} (w_{U} - \frac{1}{2}) = 0, \end{matrix}$ (123) $\begin{matrix} a {\bar{l}}^{'} + a (w_{L} - \frac{1}{2}) {\bar{x}}^{'} - \frac{a}{n} \sum_{i = 1}^{n} l_{i}^{' e} + 2 τ_{3}^{'} + τ_{4} = 0, \end{matrix}$ (124) $\begin{matrix} b {\bar{u}}^{'} + b (\frac{1}{2} - w_{U}) {\bar{y}}^{'} - \frac{b}{n} \sum_{i = 1}^{n} u_{i}^{' e} - 2 τ_{3}^{'} + τ_{4}^{'} = 0, \end{matrix}$ (125) $\begin{matrix} a (w_{L} - \frac{1}{2}) {\bar{l}}^{'} + (a {(w_{L} - \frac{1}{2})}^{2} + c) {\bar{x}}^{'} \\ - \frac{a}{n} (w_{L} - \frac{1}{2}) \sum_{i = 1}^{n} l_{i}^{' e} - \frac{c}{n} \sum_{i = 1}^{n} x_{i}^{' e} - τ_{1}^{'} \\ + τ_{3}^{'} + τ_{4} (w_{L} - \frac{1}{2}) = 0, \end{matrix}$ (126) $\begin{matrix} b (\frac{1}{2} - w_{U}) {\bar{u}}^{'} + (b {(\frac{1}{2} - w_{U})}^{2} + d) {\bar{y}}^{'} \\ - \frac{b}{n} (\frac{1}{2} - w_{U}) \sum_{i = 1}^{n} u_{i}^{' e} - \frac{d}{n} \sum_{i = 1}^{n} y_{i}^{' e} - τ_{2}^{'} \\ + τ_{3}^{'} - τ_{4}^{'} (w_{U} - \frac{1}{2}) = 0, \end{matrix}$ (127) $\bar{x} \geq 0,$ (128) $\bar{y} \geq 0,$ (129) ${\bar{x}}^{'} \geq 0,$ (130) ${\bar{y}}^{'} \geq 0,$ (131) $\bar{x} + \bar{y} \leq 2 (\bar{u} - \bar{l}),$ (132) ${\bar{x}}^{'} + {\bar{y}}^{'} \leq 2 ({\bar{u}}^{'} - {\bar{l}}^{'}),$ (133) $\begin{matrix} \bar{l} + {\bar{l}}^{'} + \bar{x} (w_{L} - \frac{1}{2}) + {\bar{x}}^{'} (w_{L} - \frac{1}{2}) \\ = \frac{1}{n} \sum_{i = 1}^{n} (l_{i}^{e} + l_{i}^{' e}) \end{matrix}$ (134) $\begin{matrix} \bar{u} + {\bar{u}}^{'} - \bar{y} (w_{U} - \frac{1}{2}) - {\bar{y}}^{'} (w_{U} - \frac{1}{2}) \\ = \frac{1}{n} \sum_{i = 1}^{n} (u_{i}^{e} + u_{i}^{' e}) \end{matrix}$ (135) $τ_{1} \geq 0,$ (136) $τ_{2} \geq 0,$ (137) $τ_{3} \geq 0,$ (138) $τ_{4} \geq 0,$ (139) $τ_{1}^{'} \geq 0,$ (140) $τ_{2}^{'} \geq 0,$ (141) $τ_{3}^{'} \geq 0,$ (142) $τ_{4}^{'} \geq 0,$ (143) $τ_{1} \bar{x} = 0,$ (144) $τ_{2} \bar{y} = 0,$ (145) $τ_{3} (\bar{x} + \bar{y} - 2 \bar{u} + 2 \bar{l}) = 0,$ (146) $\begin{matrix} τ_{4} (\bar{l} + {\bar{l}}^{'} + \bar{x} (w_{L} - \frac{1}{2}) + {\bar{x}}^{'} (w_{L} - \frac{1}{2}) \\ - \frac{1}{n} \sum_{i = 1}^{n} (l_{i}^{e} + l_{i}^{' e})) = 0 \end{matrix}$ (147) $τ_{1}^{'} {\bar{x}}^{'} = 0,$ (148) $τ_{2}^{'} {\bar{y}}^{'} = 0,$ (149) $τ_{3}^{'} ({\bar{x}}^{'} + {\bar{y}}^{'} - 2 {\bar{u}}^{'} + 2 {\bar{l}}^{'}) = 0 .$ (150) $\begin{matrix} ς_{4}^{'} (\bar{u} + {\bar{u}}^{'} - \bar{y} (w_{U} - \frac{1}{2}) - {\bar{y}}^{'} (w_{U} - \frac{1}{2}) \\ - \frac{1}{n} \sum_{i = 1}^{n} (u_{i}^{e} + u_{i}^{' e})) = 0 \end{matrix}$ (151)

4.4 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,⋯,A_n)} of A₁, A₂, ⋯, A_n is the weighted trapezoidal intuitionistic approximation preserving the weighted expected interval $T_{\bar{A}}$ of $\bar{A}$ .

Proof. We get the proof immediately substituting ς₁, ς₂, $ς_{3}, ς_{4}, ς_{1}^{'}, ς_{2}^{'}, ς_{3}^{'}, ς_{4}^{'}$ in (86)–(117) by nτ₁, nτ₂, nτ₃, $n τ_{4}, n τ_{1}^{'},$ $n τ_{2}^{'}, n τ_{3}^{'}, n τ_{4}^{'}$ . □

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 A_i = 〈μ_{A
_i}, ν_{A
_i}〉 ∈ IF^T (R), where $μ_{A_{i}} = μ_{A_{i}} (t_{1}^{i}, t_{2}^{i}, t_{3}^{i}, t_{4}^{i}), 1 - ν_{A_{i}} = (1 - ν_{A_{i}}) (s_{1}^{i}, s_{2}^{i}, s_{3}^{i},$ $s_{4}^{i}),$ i ∈ {1, ⋯, n} . The weighted trapezoidal intuitionistic approximation preserving the weighted expected interval of A₁, A₂, ⋯, A_n is $\bar{A} = 〈 μ_{\bar{A}}, ν_{\bar{A}} 〉,$ where $μ_{\bar{A}} = μ_{\bar{A}} (\frac{1}{n} \sum_{i = 1}^{n} t_{1}^{i}, \frac{1}{n} \sum_{i = 1}^{n} t_{2}^{i}, \frac{1}{n} \sum_{i = 1}^{n} t_{3}^{i}, \frac{1}{n} \sum_{i = 1}^{n} t_{4}^{i})$ and $1 - ν_{\bar{A}} = (1 - ν_{\bar{A}}) (\frac{1}{n} \sum_{i = 1}^{n} s_{1}^{i}, \frac{1}{n} \sum_{i = 1}^{n} s_{2}^{i}, \frac{1}{n} \sum_{i = 1}^{n} s_{3}^{i},$ $\frac{1}{n} \sum_{i = 1}^{n} s_{4}^{i}) .$

Proof. Corollary 3.1 has shown that the weighted trapezoidal intuitionistic approximation of A₁, A₂, ⋯, A_n is $\bar{A} = 〈 μ_{\bar{A}}, ν_{\bar{A}} 〉 .$ Hence we only prove that the weighted expected interval of A₁, A₂, ⋯, A_n is equivalent to the weighted expected interval of $\bar{A}$ .

Since the weighted expected interval introduced in Definition 4.1 is linear, from (80) we obtain $\begin{matrix} {EI}^{w} (A_{1} + \dots + A_{n}) \\ = {EI}^{w} (\frac{1}{n} (A_{1} + \dots + A_{n})) = {EI}^{w} (\bar{A}) . □ \end{matrix}$

5 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, ⋯, A_n} be a set of alternatives, and let G = {G₁, G₂, ⋯, G_m} be a set of attributes. We denote A_ij = 〈μ_ij, ν_ij〉, where μ_ij indicates the degree that the alternative A_i satisfies the attribute G_j, ν_ij indicates the degree that the alternative A_i does not satisfy the attribute G_j, μ_ij ∈ [0, 1], ν_ij ∈ [0, 1], μ_ij + ν_ij ≤ 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:

$\begin{matrix} {\tilde{A}}_{i} & = & T_{(A_{i 1}, A_{i 2}, \dots, A_{im})} \\ = & T_{\bar{A} = \frac{1}{m} \sum_{j = 1}^{m} A_{ij}} = 〈 {\tilde{μ}}_{i}, {\tilde{ν}}_{i} 〉 \end{matrix}$ (152)

The T_{(A_i1,A_i2,⋯,A_im)} and $T_{\bar{A} = \frac{1}{m} \sum_{j = 1}^{m} A_{ij}}$ are approximation operators. The $T_{\bar{A} = \frac{1}{m} \sum_{j = 1}^{m} A_{ij}}$ can be calculated by algorithms in [18]. If A_i1, ⋯, A_im are trapezoidal intuitionistic fuzzy numbers, then ${\tilde{A}}_{i}$ is their arithmetic mean. If A_i1, ⋯, A_im are general intuitionistic fuzzy numbers, then ${\tilde{A}}_{i}$ is the weighted extended trapezoidal intuitionistic approximation of their arithmetic mean.

Then we calculate the scores [35] $S ({\tilde{A}}_{i}) = {\tilde{μ}}_{i} - {\tilde{ν}}_{i}$ (i = 1, 2, ⋯, n) of the overall intuitionistic fuzzy numbers ${\tilde{A}}_{i}$ , and make use of the scores $S ({\tilde{A}}_{i})$ to rank the alternatives A_i, and then to select the best one.

6 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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