Approach to multiple attribute decision making with interval-valued intuitionistic fuzzy information and its application

Abstract

In this paper, we investigate the multiple attribute decision making with interval-valued intuitionistic fuzzy numbers. Motivated by the ideal of dependent aggregation, we develop the dependent interval-valued intuitionistic fuzzy Einstein ordered weighted average (DIVIFEOWA) operator, in which the associated weights only depend on the aggregated interval-valued intuitionistic fuzzy arguments and can relieve the influence of unfair hesitant fuzzy arguments on the aggregated results by assigning low weights to those “false” and “biased” ones and then apply them to develop some approaches for multiple attribute decision making with interval-valued intuitionistic fuzzy numbers. Finally, an illustrative example for evaluating the computer network security is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Interval-valued intuitionistic fuzzy numbers operational laws dependent interval-valued intuitionistic fuzzy Einstein ordered weighted average (DIVIFEOWA) operator computer network security

1 Introduction

Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set [3] whose basic component is only a membership function. Later, Atanassov and Gargov [4, 5] further introduced the interval-valued intuitionistic fuzzy set (IVIFS). The fundamental characteristic of the IVIFS is that the values of its membership function and non-membership function are intervals. Xu [6] proposed some arithmetic averaging operators, such as, the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator, the interval-valued intuitionistic fuzzy ordered weighted averaging (IVIFOWA) operator and the interval-valued intuitionistic fuzzy hybrid aggregation (IVIFHA) operator. All the above operators are based on the algebraic operational laws of IVIFSs for carrying the combination process [7 –14] and are not consistent with the limiting case of ordinary fuzzy sets [15]. Recently, Wang and Liu [16] treated the intuitionistic fuzzy aggregation operators with the help of Einstein operations. Therefore, how to extend the Einstein operations to aggregate the interval-valued intuitionistic fuzzy information is a meaningful work, which is also the focus of this paper.

In order to do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to interval-valued intuitionistic fuzzy sets and some existing interval-valued intuitionistic fuzzy Einstein aggregating operators. In Section 3, we develop the dependent interval-valued intuitionistic fuzzy Einstein ordered weighted average (DIVIFEOWA) operator. In Section 4, we apply the dependent interval-valued intuitionistic fuzzy Einstein ordered weighted average (DIVIFEOWA) operator to deal with multiple attribute decision making with interval-valued intuitionistic fuzzy information. In Section 5, an illustrative example for evaluating the computer network security is pointed out. In Section 6 we conclude the paper and give some remarks.

2 Preliminaries

Atanassov and Gargov [4, 5] further introduced the interval-valued intuitionistic fuzzy set (IVIFS), which is a generalization of the IFS.

Definition 1. [4, 5] Let X be an universe of discourse, An IVIFS $\tilde{A}$ over X is an object having the form: $\tilde{A} = {〈 x, {\tilde{μ}}_{A} (x), {\tilde{ν}}_{A} (x) 〉 | x \in X}$ (1) where ${\tilde{μ}}_{A} (x) \subset [0, 1]$ and ${\tilde{ν}}_{A} (x) \subset [0, 1]$ are interval numbers, and $0 \leq sup ({\tilde{μ}}_{A} (x)) + sup ({\tilde{ν}}_{A} (x)) \leq 1$ , ∀ x ∈ X.

For convenience, let ${\tilde{μ}}_{A} (x) = [a, b]$ , ${\tilde{ν}}_{A} (x) = [c, d]$ , so $\tilde{A} = ([a, b], [c, d])$ .

Definition 2. Let $\tilde{a} = ([a, b], [c, d])$ be an interval-valued intuitionistic fuzzy number, the score function S of $\tilde{a}$ can be represented as follows: $S (\tilde{a}) = \frac{a - c + b - d + 2}{4}, S (\tilde{a}) \in [- 1, 1] .$ (2)

2.1 Einstein operations of interval-valued intuitionistic fuzzy set

In this section, we shall introduce the Einstein operations on interval-valued intuitionistic fuzzy sets. the Einstein product (denoted by ${\tilde{a}}_{1} \otimes_{ɛ} {\tilde{a}}_{2}$ ) and Einstein sum (denoted by ${\tilde{a}}_{1} \oplus_{ɛ} {\tilde{a}}_{2}$ ) on two IVIFSs ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ , respectively, as follows [17]. $\begin{matrix} {\tilde{a}}_{1} \oplus_{ɛ} {\tilde{a}}_{2} ([\frac{a_{1} + a_{2}}{1 + a_{1} a_{2}}, \frac{b_{1} + b_{2}}{1 + b_{1} b_{2}}], \\ [\frac{c_{1} c_{2}}{1 + (1 - c_{1}) (1 - c_{2})}, \frac{d_{1} d_{2}}{1 + (1 - d_{1}) (1 - d_{2})}]); \\ λ {\tilde{a}}_{1} = ([\frac{{(1 + a_{1})}^{λ} - {(1 - a_{1})}^{λ}}{{(1 + a_{1})}^{λ} + {(1 - a_{1})}^{λ}}, \frac{{(1 + b_{1})}^{λ} - {(1 - b_{1})}^{λ}}{{(1 + b_{1})}^{λ} + {(1 - b_{1})}^{λ}}], \\ [\frac{2 c_{1}^{λ}}{(2 - c_{1})^{λ} + c_{1}^{λ}}, \frac{2 d_{1}^{λ}}{(2 - d_{1})^{λ} + d_{1}^{λ}}]), λ > 0 . \end{matrix}$

Furthermore, Wang and Liu [17] proposed the interval-valued intuitionistic fuzzy Einstein ordered weighted average (IVIFEOWA) operator [18].

Definition 3. Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ (j = 1, 2, ⋯ , n) be a collection of interval-valued intuitionistic fuzzy numbers. An interval-valued intuitionistic fuzzy Einstein ordered weighted average (IVIFEOWA) operator of dimension n is a mapping IVIFEOWA: Q ⁿ → Q, that has an associated vector w = (w₁, w₂,..., w_n)^T such that w_j >0 and $\sum_{j = 1}^{n} w_{j} = 1$ . Furthermore,

$\begin{matrix} {IVIFEOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {\oplus_{ɛ}}_{j = 1}^{n} (w_{j} {\tilde{a}}_{σ (j)}) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{w_{j}}}], \\ [\frac{2 \prod_{j = 1}^{n} c_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - c_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} c_{σ (j)}^{w_{j}}}, \\ \frac{2 \prod_{j = 1}^{n} d_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(2 - d_{σ (j)})}^{w_{j}} + \prod_{j = 1}^{n} d_{σ (j)}^{w_{j}}}]) \end{matrix}$ (3) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{α}}_{σ (j - 1)} \geq {\tilde{α}}_{σ (j)}$ for all j = 2, ⋯ , n.

3 Some dependent aggregation operators with interval-valued intuitionistic fuzzy information

Definition 4. Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ be a collection of interval-valued intuitionistic fuzzy numbers, the average value of the score function of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ is computed as $s (\tilde{a}) = \frac{\sum_{j = 1}^{n} s ({\tilde{a}}_{j})}{n}$ (4)

Definition 5. Let ${\tilde{a}}_{1} = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and ${\tilde{a}}_{2} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ be two interval-valued intuitionistic fuzzy numbers, then the Hamming distance between ${\tilde{a}}_{1} = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and ${\tilde{a}}_{2} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ is defined as follows [18]:

$\begin{matrix} d ({\tilde{a}}_{1}, {\tilde{a}}_{2}) & = & | a_{1} - a_{2} | + | b_{1} - b_{2} | + | c_{1} - c_{2} | \\ + | d_{1} - d_{2} | \end{matrix}$ (5)

Definition 6. Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ be be a collection of interval-valued intuitionistic fuzzy numbers, and $s (\tilde{a})$ the arithmetic mean scores of these interval-valued intuitionistic fuzzy numbers, then we define the standard deviation of these scores of these interval-valued intuitionistic fuzzy numbers as $σ = \sqrt{\frac{1}{n} \sum_{j = 1}^{n} d {(s ({\tilde{a}}_{j}), s (\tilde{a}))}^{2}}$ (6)

Definition 7. Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ be be a collection of interval-valued intuitionistic fuzzy numbers, then we call

$\begin{matrix} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) & = & 1 - \frac{d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}{\sum_{j = 1}^{n} d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}, \\ j = 1, 2, \dots, n . \end{matrix}$ (7)

the degree of similarity between the jth largest hesitant fuzzy values $s ({\tilde{a}}_{σ (j)})$ and the mean $s (\tilde{a})$ , where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that $s ({\tilde{a}}_{σ (j - 1)}) \geq s ({\tilde{a}}_{σ (j)})$ for all j = 2, ⋯ , n.

In real-life situations, the interval-valued intuitionistic fuzzy numbers ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ usually take the form of a collection of n preference values provided by n different individuals. Some individuals may assign unduly high or unduly low preference values to their preferred or repugnant objects. In such a case, we shall assign very low weights to these “false” or “biased” opinions, that is to say, the closer a preference value (argument) is to the mid one(s), the more the weight. As a result, based on (7), we define the IVIFEOWA operator weights as $\begin{matrix} w_{j} = \frac{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}{\sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}, j = 1, 2, \dots, n \end{matrix}$ (8) Obviously, w_j ≥ 0, j = 1, 2, ⋯ , n and $\sum_{j = 1}^{n} w_{j} = 1$ .

Especially, if $s ({\tilde{a}}_{j}) = s (\tilde{a})$ , for all i, j = 1, 2, ⋯ , n, then by (8), we have w_j = 1/n, for all j = 1, 2, ⋯ , n.

By (3), we have

$\begin{matrix} IVIFEOWA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \oplus_{j = 1}^{n} (\frac{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}{\sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} {\tilde{a}}_{σ (j)}) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} + \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} + \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}], \\ [\frac{2 \prod_{j = 1}^{n} c_{σ (j)}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(2 - c_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} + \prod_{j = 1}^{n} c_{σ (j)}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}, \\ \frac{2 \prod_{j = 1}^{n} d_{σ (j)}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(2 - d_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} + \prod_{j = 1}^{n} d_{σ (j)}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}]) \end{matrix}$ (9)

Since $\sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) {\tilde{a}}_{σ (j)} = \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}),$ $s (\tilde{a})) {\tilde{a}}_{j}$ and $\sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) = \sum_{j = 1}^{n} (s ({\tilde{a}}_{j}), s (\tilde{a}))$ then we replace (9) by

$\begin{matrix} IVIFEOWA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \oplus_{j = 1}^{n} {\tilde{a}}_{j} (sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + a_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} - \prod_{j = 1}^{n} {(1 - a_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + a_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} + \prod_{j = 1}^{n} {(1 - a_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + b_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} - \prod_{j = 1}^{n} {(1 - b_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + b_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} + \prod_{j = 1}^{n} {(1 - b_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}], \\ [\frac{2 \prod_{j = 1}^{n} c_{j}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(2 - c_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} + \prod_{j = 1}^{n} c_{j}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}, \\ \frac{2 \prod_{j = 1}^{n} d_{j}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(2 - d_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} + \prod_{j = 1}^{n} d_{j}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}]) \end{matrix}$ (10)

We call (10) a dependent interval-valued intuitionistic fuzzy Einstein ordered weighted average (DIVIFEOWA) operator.

Similar to Xu [19], we have the following result:

Theorem 1. Let ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ be be a collection of interval-valued intuitionistic fuzzy numbers, and let $s (\tilde{a})$ the average value of the score function of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that $s ({\tilde{a}}_{σ (j - 1)}) \geq s ({\tilde{a}}_{σ (j)})$ for all j = 2, ⋯ , n. If $sim (s ({\tilde{a}}_{i}), s (\tilde{a})) \geq sim (s ({\tilde{a}}_{j}), s (\tilde{a}))$ , then w_i ≥ w_j.

The normal distribution is one of the most commonly observed and is the starting point for modeling many natural process, it is usually found in events that are the aggregation of many smaller, but independent random events. Xu [19] introduced a normal distribution method to determine the weight of DOUWA operator. Motivated by the idea, we shall give another method for deriving the DIVIFEOWA weights: $w_{j} = \frac{1}{\sqrt{2 π} σ} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}, j = 2, \dots, n .$ (11)

where $s (\tilde{a})$ and σ are the arithmetic mean and the standard deviation of these hesitant fuzzy arguments variables ${\tilde{a}}_{σ (j)} (j = 1, 2, \dots, n)$ , respectively, (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that $s ({\tilde{a}}_{σ (j - 1)}) \geq s ({\tilde{a}}_{σ (j)})$ for all j = 2, ⋯ , n.

Consider that w_j ≥ 0, j = 1, 2, ⋯ , n and $\sum_{j = 1}^{n} w_{j} = 1$ , then by (11), we have

$\begin{matrix} w_{j} & = & \frac{\frac{1}{\sqrt{2 π} σ} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}{\sum_{j = 1}^{n} \frac{1}{\sqrt{2 π} σ} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} \\ = & \frac{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}{\sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}, j = 1, 2, \dots, n . \end{matrix}$ (12) then by (3), we have

$\begin{matrix} DIVIFEOWA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \oplus_{j = 1}^{n} (\frac{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}{\sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} {\tilde{a}}_{σ (j)}) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} - \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + a_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} + \prod_{j = 1}^{n} {(1 - a_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} + \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}], \\ [\frac{2 \prod_{j = 1}^{n} c_{σ (j)}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(2 - c_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} + \prod_{j = 1}^{n} c_{σ (j)}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}, \\ \frac{2 \prod_{j = 1}^{n} d_{σ (j)}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(2 - d_{σ (j)})}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} + \prod_{j = 1}^{n} d_{σ (j)}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}]) \end{matrix}$ (13)

Since $\begin{matrix} \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} {\tilde{a}}_{σ (j)} & = & \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} {\tilde{a}}_{j}, \\ \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} & = & \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} \end{matrix}$ then, (13) can be rewritten as $\begin{matrix} DIVIFEOWA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \oplus_{j = 1}^{n} (\frac{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}{\sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} {\tilde{a}}_{j}) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + a_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}} - \prod_{j = 1}^{n} {(1 - a_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + a_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}} + \prod_{j = 1}^{n} {(1 - a_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + b_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}} - \prod_{j = 1}^{n} {(1 - b_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + b_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}} + \prod_{j = 1}^{n} {(1 - b_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}}}], \\ [\frac{2 \prod_{j = 1}^{n} c_{j}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(2 - c_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}} + \prod_{j = 1}^{n} c_{j}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}}}, \\ \frac{2 \prod_{j = 1}^{n} d_{j}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(2 - d_{j})}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}} + \prod_{j = 1}^{n} d_{j}^{e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s ({\tilde{a}}_{j}), s (\tilde{a}))))}^{2}}{2 σ^{2}}}}}]) \end{matrix}$ (14)

Obviously, (14) is also a neat and dependent interval-valued intuitionistic fuzzy Einstein ordered weighted average (DIVIFEOWA) operator.

From (8) and (11), we know that all the associated weights of the DIVIFEOWA operator only depend on the aggregated interval-valued intuitionistic fuzzy numbers, and can relieve the influence of unfair arguments on the aggregated results by assigning low weights to those “false” and “biased” ones, and thus make the aggregated results more reasonable in the practical applications.

4 An approach to multiple attribute decision making with interval-valued intuitionistic fuzzy information

In this section, we shall apply DIVIFEOWA operator to the multiple attribute decision making problems

with interval-valued intuitionistic fuzzy numbers. Let A ={ A ₁, A ₂, ⋯ , A _m } be a discrete set of alternatives, and G ={ G ₁, G ₂, ⋯ , G _n } be the set of attributes. The information about attribute weights is completely known. Suppose that $\tilde{R} = {({\tilde{r}}_{ij})}_{m \times n} = {([a_{ij}, b_{ij}], [c_{ij}, d_{ij}])}_{m \times n}$ is the interval-valued intuitionistic fuzzy decision matrix, [a _ij, b _ij] ⊂ [0, 1], [c _ij, d _ij] ⊂ [0, 1], b _ij + d _ij ≤ 1, i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n.

In the following, we apply the DIVIFEOWA operator to multiple attribute decision making with interval-valued intuitionistic fuzzy information. The method involves the following steps:

Step 1. Utilize the decision information given in matrix $\tilde{R}$ , and the DIVIFEOWA operator

$\begin{matrix} {\tilde{r}}_{i} = ([a_{i}, b_{i}], [c_{i}, d_{i}]) = DIVIFEOWA ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) = \oplus_{j = 1}^{n} (\frac{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}{\sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} {\tilde{a}}_{ij}) \\ = ([\frac{\prod_{j = 1}^{n} {(1 + a_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} - \prod_{j = 1}^{n} {(1 - a_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(1 + a_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} {(1 - a_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + b_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} - \prod_{j = 1}^{n} {(1 - b_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(1 + b_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} {(1 - b_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}], \\ [\frac{2 \prod_{j = 1}^{n} c_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(2 - c_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} c_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}, \\ \frac{2 \prod_{j = 1}^{n} d_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(2 - d_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} d_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}]) \end{matrix}$

Step 2. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ of the collective overall values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ to rank all the alternatives A _i (i = 1, 2, ⋯ , m) and then to select the best one(s).

Step 3. Rank all the alternatives A _i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with $S ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ .

Step 4. End.

5 Numerical example

This section presents a numerical example to evaluate the computer network security with uncertain linguistic information to illustrate the method proposed in this paper. There are five possible computer network systems A _i (i = 1, 2, 3, 4, 5) for four attributes G _j (j = 1, 2, 3, 4). The four attributes include the tactics (G ₁), technology and economy (G ₂), logistics (G ₃) and strategy (G ₄), respectively. The five possible computer network systems A _i (i = 1, 2, 3, 4, 5) are to be evaluated using the interval-valued intuitionistic fuzzy information by the decision maker under the above four attributes, as listed in the following matrix.

$\begin{matrix} \tilde{R} = [\begin{matrix} ([0.4, 0.5], [0.3, 0.4]) & ([0.4, 0.6], [0.2, 0.4]) \\ ([0.5, 0.6], [0.2, 0.3]) & ([0.6, 0.7], [0.2, 0.3]) \\ ([0.3, 0.5], [0.3, 0.4]) & ([0.1, 0.3], [0.5, 0.6]) \\ ([0.2, 0.5], [0.3, 0.4]) & ([0.4, 0.7], [0.1, 0.2]) \\ ([0.3, 0.4], [0.1, 0.3]) & ([0.7, 0.8], [0.1, 0.2]) \end{matrix} \\ \begin{matrix} ([0.3, 0.4], [0.4, 0.5]) & ([0.5, 0.6], [0.1, 0.3]) \\ ([0.5, 0.6], [0.3, 0.4]) & ([0.4, 0.7], [0.1, 0.2]) \\ ([0.2, 0.5], [0.4, 0.5]) & ([0.2, 0.3], [0.4, 0.6]) \\ ([0.4, 0.5], [0.3, 0.5]) & ([0.5, 0.8], [0.1, 0.2]) \\ ([0.5, 0.6], [0.2, 0.4]) & ([0.6, 0.7], [0.1, 0.2]) \end{matrix}] \end{matrix}$

In the following, we apply the DIVIFEOWA operator to multiple attribute decision making for evaluating the computer network security with interval-valued intuitionistic fuzzy information. The method involves the following steps:

Step 1. If we apply the DIVIFEOWA operator to decision making with interval-valued intuitionistic fuzzy information, we get the weight of DIVIFEOWA operator is: $w = (0.18, 0 . 32, 0 . 26, 0 . 24) .$

Step 2. Utilize the decision information given in matrix $\tilde{R}$ , and the DIVIFEOWA operator, we obtain the overall preference values ${\tilde{r}}_{i}$ of the computer network systems A _i (i = 1, 2, ⋯ , 5). $\begin{matrix} {\tilde{r}}_{1} = ([0.4025, 0.5346], [0.2495, 0.3968]) \\ {\tilde{r}}_{2} = ([0.4254, 0.5469], [0.1947, 0.2962]) \\ {\tilde{r}}_{3} = ([0.2323, 0.4337], [0.3627, 0.4710]) \\ {\tilde{r}}_{4} = ([0.3026, 0.5984], [0.1997, 0.3136]) \\ {\tilde{r}}_{5} = ([0.4879, 0.5982], [0.1042, 0.2627]) \end{matrix}$ Step 3. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, 5)$ of the overall interval-valued intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ $\begin{matrix} S ({\tilde{r}}_{1}) = 0.1454, S ({\tilde{r}}_{2}) = 0.2407 \\ S ({\tilde{r}}_{3}) = - 0.0839, S ({\tilde{r}}_{4}) = 0.1938 \\ S ({\tilde{r}}_{5}) = 0.3596 \end{matrix}$ Step 4. Rank all the computer network systems A _i (i = 1, 2, 3, 4, 5) in accordance with the scores $S ({\tilde{r}}_{i})$ of the overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ : A ₅ ≻ A ₂ ≻ A ₄ ≻ A ₁ ≻ A ₃, and thus the most desirable computer network systems is A ₅.

6 Conclusion

In this paper, we investigate the multiple attribute decision making with interval-valued intuitionistic fuzzy numbers. Motivated by the ideal of dependent aggregation, we develop the dependent interval-valued intuitionistic fuzzy Einstein ordered weighted average (DIVIFEOWA), in which the associated weights only depend on the aggregated interval-valued intuitionistic fuzzy arguments and can relieve the influence of unfair hesitant fuzzy arguments on the aggregated results by assigning low weights to those “false” and “biased” ones and then apply them to develop some approaches for multiple attribute decision making with interval-valued intuitionistic fuzzy numbers. Finally, an illustrative example for evaluating the computer network security is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Footnotes

Acknowledgments

The work was supported by the Science Development Program of Shunde, Foshan, Guangdong in 2012 (20120202088) and the Research Project of Education Science in Guangdong province during the Twelfth Five-year Guideline in 2012 (2012JK303).

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