Some new operation rules and a new ranking method for interval-valued intuitionistic linguistic numbers

Abstract

An interval-valued intuitionistic linguistic set (IVILS) is introduced as an extension of both a linguistic term set and an interval-valued intuitionistic fuzzy set. However, the arithmetic operations and comparison methods of IVILS exist some limitations, which may result in information loss and distortion. To avoid this problem, the arithmetic operations are improved and a new comparison method based on a possibility degree is developed. Then, several arithmetic aggregation operators are proposed to aggregate interval-valued intuitionistic linguistic numbers. Moreover, a multi-criteria group decision-making (MCDM) approach based on the new aggregation operators is presented. Finally, the validity and effectiveness of the proposed method are confirmed by a comparable example.

Keywords

Interval-valued intuitionistic linguistic number operation rule possibility degree ranking method aggregation operator

1 Introduction

In 1986, Atanassov introduced the concept of intuitionistic fuzzy sets (IFSs) [1], which are a generalization of the fuzzy sets (FSs) introduced by Zadeh [2]. IFSs have been widely studied [3 –6]. Later, The notion of IFSs was further generalized by allowing the membership and non-membership functions to assume intervals, thereby the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) was introduced [7]. In recently years, IVIFSs have been widely applied to the multi-criteria decision-making (MCDM) [8 –11].

In a real-life MCDM situation, it is too complex or too ill-defined to be amenable for description in conventional quantitative expressions, so it is more suitable to provide assessment by means of linguistic values rather than numerical ones [12]. Several methods have been proposed to deal with linguistic information, these models are mainly classified into four types: (1) The methods based on transformation to the fuzzy numbers [13, 14]. (2) The methods based on symbols, which makes computations on the index of the linguistic terms [15, 16]. (3) The methods based on 2-tuple linguistic representation model [17, 18]. (4) The methods based on cloud model [19, 20].

Based on IFSs and linguistic term sets, Wang and Li [21] proposed the concept of intuitionistic linguistic sets (ILSs), which not only fit decision maker’s habit of language expression, but also consider the membership degree and non-membership degree. MCDM approach based on ILSs is an interesting research topic, which has been paid increasing attention. Liu [22] developed some generalized dependent aggregation operators with intuitionistic linguistic numbers (ILNs). Liu and Wang [23] introduced some intuitionistic linguistic power generalized aggregation operators. Afterwards, Wang et al. [24] further generalized the ILSs to develop the interval-valued intuitionistic linguistic sets (IVILSs), whose membership and non-membership degree are interval values. Interval-valued intuitionistic linguistic set theory has been mainly applied to the area of decision making [24, 25].

At present, Liu [27] proposed the interval-valued intuitionistic uncertain linguistic sets (IVIULSs). Wang et al. [25] and Liu [27] introduced some operational rules of IVILSs and IVIULSs. It is clear that the existing arithmetic operational rules had some limitations. For example, ${\tilde{α}}_{1} = 〈 h_{2}, [1, 1], [0, 0] 〉$ and ${\tilde{α}}_{2} = 〈 h_{4}, [0, 0], [1, 1] 〉$ are two interval-valued intuitionistic linguistic numbers. Based on the existing arithmetic operations of IVILS, it can be derived that ${\tilde{α}}_{1} + {\tilde{α}}_{2} = 〈 h_{6}, [1, 1], [0, 0] 〉$ , which is unreasonable. Because the reasonable result should be between [0, 0] and [1, 1]. So before developing several better arithmetic aggregation operators, some more reasonable arithmetic operational rules should be studied.

A ranking approach plays a very important role when applying the method based on the aggregation operators. In 1994, Chen and Tan [28] introduced a score function to rank the IFSs. Based on that, many scholars have improved the score function for intuitionistic fuzzy numbers [29, 30], interval-valued intuitionistic fuzzy numbers [31, 32] and interval-valued intuitionistic linguistic numbers [25, 27]. Although Liu and Wang [25, 27] introduced some new score functions for IVILNs, these functions have limitations and deficiencies. Therefore, searching for a more reasonable ranking approach is a significant work.

In this paper, we define some new arithmetic operational rules, a distance measure and develop some arithmetic aggregation operators with interval-valued intuitionistic linguistic numbers, then we propose a ranking method based on a possibility degree. The rest of this paper is organized as follows. In Section 2, we introduce some basic notions of IVILSs and related concepts. In Section 3, we propose some operators of IVILSs. Section 4 presents a method for multi-criteria decision making based on IVILSs. In Section 5, an illustrative example is provided, followed by a comparison analysis with other methods. At last, we conclude this paper.

2 IVILSs and related concepts

Let H = {h_i|i = 0, 1, …, 2t, t ∈ N^*} be a finite and linguistic term set, where h_i represents a possible value for a linguistic variable, and it should satisfy the following characteristics.

The set is ordered: h_i > h_j, if i > j.

There is the negation operator: h_i = neg (h_j), where i + j = 2t.

Max operator: max {h_i, h_j} = h_i, if i ≥ j.

Min operator: min {h_i, h_j} = h_j, if i ≥ j.

To preserve all the given information, we extend the discrete linguistic term set H to a continuous linguistic term set $\bar{H} = {h_{i} | i \in [0, l]}$ , in which h_i > h_j if i > j, and l (l > 2t) is a sufficiently large positive integer. If h_i ∈ H, then we call h_i the original additive linguistic term, otherwise, we call h_i the virtual additive linguistic term.

Definition 1. Let X be the universe of discourse and h_θ(x) ∈ H. An interval-valued intuitionistic linguistic set $\tilde{A}$ in X is an object: $\tilde{A} = {〈 h_{θ_{(x)}}, {\tilde{μ}}_{\tilde{A}} (x), {\tilde{ν}}_{\tilde{A}} (x) 〉 | x \in X},$ where ${\tilde{μ}}_{\tilde{A}} (x) \subseteq [0, 1]$ and ${\tilde{ν}}_{\tilde{A}} (x) \subseteq [0, 1]$ areintervals, and for every x ∈ X, $0 \leq \sup {\tilde{μ}}_{\tilde{A}} (x) + \sup {\tilde{ν}}_{\tilde{A}} (x) \leq 1$ . The intervals ${\tilde{μ}}_{\tilde{A}} (x)$ and ${\tilde{ν}}_{\tilde{A}} (x)$ determine the membership degree and non-membership degree of the element x to the linguistic value h_θ(x), respectively, but for each x ∈ X, ${\tilde{μ}}_{\tilde{A}} (x)$ and ${\tilde{ν}}_{\tilde{A}} (x)$ are closed intervals rather than real numbers and their lower and upper boundaries are denoted by ${\tilde{μ}}_{\tilde{A}}^{L} (x)$ , ${\tilde{μ}}_{\tilde{A}}^{U} (x)$ , ${\tilde{ν}}_{\tilde{A}}^{L} (x)$ , ${\tilde{ν}}_{\tilde{A}}^{U} (x)$ , respectively. Therefore, another equivalent expression of an IVILS $\tilde{A}$ is $\tilde{A} = {〈 h_{θ_{(x)}}, [{\tilde{μ}}_{\tilde{A}}^{L} (x), {\tilde{μ}}_{\tilde{A}}^{U} (x)], [{\tilde{ν}}_{\tilde{A}}^{L} (x), {\tilde{ν}}_{\tilde{A}}^{U} (x)] 〉 | x \in X},$ where ${\tilde{μ}}_{\tilde{A}}^{U} (x) + {\tilde{ν}}_{\tilde{A}}^{U} (x) \leq 1$ , ${\tilde{μ}}_{\tilde{A}}^{L} (x) \geq 0$ and ${\tilde{ν}}_{\tilde{A}}^{L} (x) \geq 0$ .

For each element x ∈ X, the hesitation interval relative to $\tilde{A}$ can be calculated as $\begin{matrix} {\tilde{π}}_{\tilde{A}} (x) & = & [{\tilde{π}}_{\tilde{A}}^{L} (x), {\tilde{π}}_{\tilde{A}}^{U} (x)] \\ = & [1 - {\tilde{μ}}_{\tilde{A}}^{U} (x) - {\tilde{ν}}_{\tilde{A}}^{U} (x), 1 - {\tilde{μ}}_{\tilde{A}}^{L} (x) - {\tilde{ν}}_{\tilde{A}}^{L} (x)] . \end{matrix}$

For any given x, $〈 h_{θ_{(x)}}, {\tilde{μ}}_{\tilde{A}} (x), {\tilde{ν}}_{\tilde{A}} (x) 〉$ is referred to as an IVILN. For convenience, $〈 h_{θ_{(x)}}, {\tilde{μ}}_{\tilde{A}} (x), {\tilde{ν}}_{\tilde{A}} (x) 〉$ is often denoted by $\tilde{α} = 〈 h_{θ (\tilde{α})}, [a, b], [c, d] 〉$ , where [a, b] ⊆ [0, 1], [c, d] ⊆ [0, 1] and b + d ≤ 1.

2.1 Operational rules of IVILNs

In this section, we briefly review arithmetic operational rules of IVILNs and analyze the drawbacks of the existing operational rules. Then, we develop some novel operational rules of IVILNs.

Let ${\tilde{α}}_{1} = 〈 h_{θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}], [c_{1}, d_{1}] 〉$ and ${\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{2})}, [a_{2}, b_{2}], [c_{2}, d_{2}] 〉$ be two IVILNs and the exsiting operational rules of ${\tilde{α}}_{1}$ and ${\tilde{α}}_{2}$ are reviewed as follows:

Liu’s arithmetic operational rules [27]:

${\tilde{α}}_{1} + {\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{1}) + θ ({\tilde{α}}_{2})}, [1 - (1 - a_{1}) (1 - a_{2}), 1 - (1 - b_{1}) (1 - b_{2})], [c_{1} c_{2}, d_{1} d_{2}] 〉$ ,

$λ {\tilde{α}}_{1} = 〈 h_{λ \times θ ({\tilde{α}}_{1})}, [1 - (1 - a_{1})^{λ}, 1 - (1 - b_{1})^{λ}], [c_{1}^{λ}, d_{1}^{λ}] 〉, λ \geq 0$ .

Example 1. Let ${\tilde{α}}_{1} = 〈 h_{2}, [1, 1], [0, 0] 〉$ and ${\tilde{α}}_{2} = 〈 h_{4}, [0, 0], [1, 1] 〉$ be two IVILNs, by applying Liu’s arithmetic operational rules, we have ${\tilde{α}}_{1} + {\tilde{α}}_{2} = 〈 h_{6}, [1, 1], [0, 0] 〉$ .

However, the result is obviously unreasonable. Because when two IVILNs refer to different linguistic values, the equivalent membership degrees and non-membership degrees should have different meanings.

Definition 2. Let ${\tilde{α}}_{1} = 〈 h_{θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}], [c_{1}, d_{1}] 〉$ and ${\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{2})}, [a_{2}, b_{2}], [c_{2}, d_{2}] 〉$ be two IVILNs, then $\begin{matrix} (1) & {\tilde{α}}_{1} + {\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{1}) + θ ({\tilde{α}}_{2})}, \\ [\frac{θ ({\tilde{α}}_{1}) a_{1} + θ ({\tilde{α}}_{2}) a_{2}}{θ ({\tilde{α}}_{1}) + θ ({\tilde{α}}_{2})}, \frac{θ ({\tilde{α}}_{1}) b_{1} + θ ({\tilde{α}}_{2}) b_{2}}{θ ({\tilde{α}}_{1}) + θ ({\tilde{α}}_{2})}], \\ [\frac{θ ({\tilde{α}}_{1}) c_{1} + θ ({\tilde{α}}_{2}) c_{2}}{θ ({\tilde{α}}_{1}) + θ ({\tilde{α}}_{2})}, \frac{θ ({\tilde{α}}_{1}) d_{1} + θ ({\tilde{α}}_{2}) d_{2}}{θ ({\tilde{α}}_{1}) + θ ({\tilde{α}}_{2})}] 〉, \\ (2) & λ {\tilde{α}}_{1} = 〈 h_{λ \times θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}], [c_{1}, d_{1}] 〉, λ \geq 0 . \end{matrix}$

Obviously, with the above operations of IVILNs, we have the following theorem.

Theorem 1. Let ${\tilde{α}}_{i} = 〈 h_{θ ({\tilde{α}}_{i})}, [a_{i}, b_{i}], [c_{i}, d_{i}] 〉 (i = 1, 2, 3)$ be any three IVILNs, then

${\tilde{α}}_{1} + {\tilde{α}}_{2} = {\tilde{α}}_{2} + {\tilde{α}}_{1}$ ,

$({\tilde{α}}_{1} + {\tilde{α}}_{2}) + {\tilde{α}}_{3} = {\tilde{α}}_{1} + ({\tilde{α}}_{2} + {\tilde{α}}_{3})$ ,

$λ ({\tilde{α}}_{1} + {\tilde{α}}_{2}) = λ {\tilde{α}}_{1} + λ {\tilde{α}}_{2}$ , λ ≥ 0,

$λ_{1} {\tilde{α}}_{1} + λ_{2} {\tilde{α}}_{1} = (λ_{1} + λ_{2}) {\tilde{α}}_{1}$ , λ₁, λ₂ ≥ 0.

It is obvious that the above rules hold.

2.2 Comparison of IVILNs

In this section, we briefly review some score function and accuracy function of IVILNs and analyse the drawbacks of them. Then, we develop a novel comparison method of IVILNs.

As the IVIULS is an extension of the IVILS, Liu’s score function and accuracy function of IVIULS can be reduced the score function and accuracy function of IVILS [27]:

Definition 3. Let $\tilde{α} = 〈 h_{θ (\tilde{α})}, [a, b], [c, d] 〉$ be an IVILNs, an expected value of $\tilde{α}$ can be computed as $E_{L} (\tilde{α}) = \frac{h_{θ (α)}}{2} (\frac{a + b}{2} + 1 - \frac{c + d}{2}) .$ (1)

Definition 4. Let $\tilde{α} = 〈 h_{θ (\tilde{α})}, [a, b], [c, d] 〉$ be an IVILNs, an accuracy function of $\tilde{α}$ can be computed as $H_{L} (\tilde{α}) = h_{θ (\tilde{α})} \times (\frac{a + b}{2} + \frac{c + d}{2}) .$ (2)

Definition 5. If ${\tilde{α}}_{1} = 〈 h_{θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}], [c_{1}, d_{1}] 〉$ and ${\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{2})},$ [a₂, b₂], [c₂, d₂]〉 are two IVILNs, then

If $E_{L} ({\tilde{α}}_{1}) > E_{L} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} > {\tilde{α}}_{2}$ .

If $E_{L} ({\tilde{α}}_{1}) = E_{L} ({\tilde{α}}_{2})$ , then

If $H_{L} ({\tilde{α}}_{1}) > H_{L} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} > {\tilde{α}}_{2}$ .

If $H_{L} ({\tilde{α}}_{1}) = H_{L} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} = {\tilde{α}}_{2}$ .

Example 2. Assume that ${\tilde{α}}_{1} = 〈 h_{1}, [0.2, 0.8], [0, 0.2]$ and ${\tilde{α}}_{2} = 〈 h_{2}, [0, 0], [0, 0.6]$ , by applying Equations (1) and (2), we can obtain

$E_{L} ({\tilde{α}}_{1}) = h_{0.7}$ , $E_{L} ({\tilde{α}}_{2}) = h_{0.7}$ , $H_{L} ({\tilde{α}}_{1}) = h_{0.6}$ , $H_{L} ({\tilde{α}}_{2}) = h_{0.6}$ .

It can be obtained that ${\tilde{α}}_{1} = {\tilde{α}}_{2}$ , but the result is obviously wrong. Becuase we can see that ${\tilde{α}}_{1}$ is better than ${\tilde{α}}_{2}$ .

Wang’s score function and accuracy function [25]:

Definition 6. Let $\tilde{α} = 〈 h_{θ (\tilde{α})}, [a, b], [c, d] 〉$ be an IVILNs, a score function of $\tilde{α}$ can be computed as $S_{W} (\tilde{α}) = \frac{1}{2} (a - c + b - d) θ (\tilde{α}) .$ (3)

Definition 7. Let $\tilde{α} = 〈 h_{θ (\tilde{α})}, [a, b], [c, d] 〉$ be an IVILNs, an accuracy function of $\tilde{α}$ can be computed as $H_{W} (\tilde{α}) = \frac{1}{2} (a + b + c + d) θ (\tilde{α}) .$ (4)

Definition 8. If ${\tilde{α}}_{1} = 〈 h_{θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}], [c_{1}, d_{1}] 〉$ and ${\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{2})},$ [a₂, b₂], [c₂, d₂]〉 are two IVILNs,then

If $S_{W} ({\tilde{α}}_{1}) > S_{W} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} > {\tilde{α}}_{2}$ .

If $S_{W} ({\tilde{α}}_{1}) = S_{W} ({\tilde{α}}_{2})$ , then

If $H_{W} ({\tilde{α}}_{1}) > H_{W} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} > {\tilde{α}}_{2}$ .

If $H_{W} ({\tilde{α}}_{1}) = H_{W} ({\tilde{α}}_{2})$ , then ${\tilde{α}}_{1} = {\tilde{α}}_{2}$ .

Example 3. Assume that ${\tilde{α}}_{1} = 〈 h_{1}, [0.3, 0.5], [0, 0.2]$ and ${\tilde{α}}_{2} = 〈 h_{2}, [0.1, 0.3], [0, 0.1]$ , by applying Equations (3) and (4), we can obtain

$S_{W} ({\tilde{α}}_{1}) = 0.3$ , $S_{W} ({\tilde{α}}_{2}) = 0.3$ , $H_{L} ({\tilde{α}}_{1}) = 0.5$ , $H_{L} ({\tilde{α}}_{2}) = 0.5$ .

It can be obtained that ${\tilde{α}}_{1} = {\tilde{α}}_{2}$ , so we cannot get which one is better.

Therefore, the above methods have some limitations and drawbacks.

In recent years, the method of possibility degree has been widely applied by many researches [36, 37]. Here, a ranking method based on a possibility degree is as follows.

Let $\tilde{α} = 〈 h_{θ (\tilde{α})}, [a, b], [c, d] 〉$ be an IVILN. According to the definition of the IVILN, the lower boundary of membership degree to $\tilde{α}$ is denoted by ${\tilde{μ}}^{-} = [a, b]$ and the upper boundary of membership degree to $\tilde{α}$ is denoted by ${\tilde{μ}}^{+} = [1 - d, 1 - c]$ . Then the lower boundary of certainty degree of $\tilde{α}$ is computed by $θ (\tilde{α}) {\tilde{μ}}^{-}$ . Similarily, the upper boundary of certainty degree of $\tilde{α}$ is computed by $θ (\tilde{α}) {\tilde{μ}}^{+}$ .

Definition 9. Given two IVILNs ${\tilde{α}}_{1} = 〈 h_{θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}],$ [c₁, d₁]〉 and ${\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{2})}, [a_{2}, b_{2}], [c_{2}, d_{2}] 〉$ , the possibility degree for the comparison between lower boundaries of membership degrees of two IVILNs ${\tilde{α}}_{1}$ and ${\tilde{α}}_{2}$ can be represented as follows:

$\begin{matrix} p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) \\ = \min {\max {\frac{θ ({\tilde{α}}_{1}) b_{1} - θ ({\tilde{α}}_{2}) a_{2}}{θ ({\tilde{α}}_{1}) (b_{1} - a_{1}) + θ ({\tilde{α}}_{2}) (b_{2} - a_{2})}, \\ 0}, 1} . \end{matrix}$ (5)

If a₁ = b₁ and a₂ = b₂,

if $θ ({\tilde{α}}_{1}) a_{1} > θ ({\tilde{α}}_{2}) a_{2}$ , then $p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 1$ ,

if $θ ({\tilde{α}}_{1}) a_{1} = θ ({\tilde{α}}_{2}) a_{2}$ , then $p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = \frac{1}{2}$ ,

if $θ ({\tilde{α}}_{1}) a_{1} < θ ({\tilde{α}}_{2}) a_{2}$ , then $p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 0$ .

Theorem 2. Given two IVILNs ${\tilde{α}}_{1} = 〈 h_{θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}],$ [c₁, d₁]〉 and ${\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{2})},$ [a₂, b₂], [c₂, d₂]〉, then it can be easily proved that the followings are true.

$0 \leq p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) \leq 1$ ,

$p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 1$ if and only if $θ ({\tilde{α}}_{1}) b_{1} \geq θ ({\tilde{α}}_{2}) a_{2}$ ,

$p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 0$ if and only if $θ ({\tilde{α}}_{1}) b_{1} \leq θ ({\tilde{α}}_{2}) a_{2}$ ,

$p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) + p^{-} ({\tilde{α}}_{2} \geq {\tilde{α}}_{1}) = 1$ , especially $p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{1}) = 0.5$ ,

$p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) \geq 0.5$ if and only if $θ ({\tilde{α}}_{1}) a_{1} + θ ({\tilde{α}}_{1}) b_{1} \geq θ ({\tilde{α}}_{2}) a_{2} + θ ({\tilde{α}}_{2}) b_{2}$ . Especially, $p^{-} ({\tilde{α}}_{1}$ $\geq {\tilde{α}}_{2}) = 0.5$ if and only if $θ ({\tilde{α}}_{1}) a_{1} + θ ({\tilde{α}}_{1}) b_{1} = θ ({\tilde{α}}_{2}) a_{2} + θ ({\tilde{α}}_{2}) b_{2}$ .

Definition 10. Given two IVILNs ${\tilde{α}}_{1} = 〈 h_{θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}],$ [c₁, d₁]〉 and ${\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{2})}, [a_{2}, b_{2}], [c_{2}, d_{2}] 〉$ , the possibility degree for the comparison between upper boundaries of membership degrees of two IVILNs ${\tilde{α}}_{1}$ and ${\tilde{α}}_{2}$ can be represented as follows:

$\begin{matrix} p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) \\ = \min {\max {\frac{θ ({\tilde{α}}_{1}) (1 - c_{1}) - θ ({\tilde{α}}_{2}) (1 - d_{2})}{θ ({\tilde{α}}_{1}) (d_{1} - c_{1}) + θ ({\tilde{α}}_{2}) (d_{2} - c_{2})}, \\ 0}, 1} . \end{matrix}$ (6)

If c₁ = d₁ and c₂ = d₂,

if ${\tilde{α}}_{1} (1 - d_{1}) > {\tilde{α}}_{2} (1 - d_{2})$ , then $p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 1$ ,

if ${\tilde{α}}_{1} (1 - d_{1}) = {\tilde{α}}_{2} (1 - d_{2})$ , then $p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = \frac{1}{2}$ ,

if ${\tilde{α}}_{1} (1 - d_{1}) < {\tilde{α}}_{2} (1 - d_{2})$ , then $p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 0$ .

Theorem 3. Given two IVILNs ${\tilde{α}}_{1} = 〈 h_{θ ({\tilde{α}}_{1})}, [a_{1}, b_{1}],$ [c₁, d₁]〉 and ${\tilde{α}}_{2} = 〈 h_{θ ({\tilde{α}}_{2})},$ [a₂, b₂], [c₂, d₂]〉, then it can be easily proved that the followings are true.

$0 \leq p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) \leq 1$ ,

$p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 1$ if and only if $θ ({\tilde{α}}_{1}) (1 - c_{1}) \geq θ ({\tilde{α}}_{2}) (1 - d_{2})$ ,

$p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 0$ if and only if $θ ({\tilde{α}}_{1}) (1 - c_{1}) \leq θ ({\tilde{α}}_{2}) (1 - d_{2})$ ,

$p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) + p^{+} ({\tilde{α}}_{2} \geq {\tilde{α}}_{1}) = 1$ , especially $p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{1}) = 0.5$ ,

$p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) \geq 0.5$ if and only if $θ ({\tilde{α}}_{1}) (1 - d_{1}) + θ ({\tilde{α}}_{1}) (1 - c_{1}) \geq θ ({\tilde{α}}_{2}) (1 - d_{2}) + θ ({\tilde{α}}_{2}) (1 - c_{2})$ . Especially, $p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = 0.5$ if and only if $θ ({\tilde{α}}_{1}) (1 - d_{1}) + θ ({\tilde{α}}_{1}) (1 - c_{1}) = θ ({\tilde{α}}_{2}) (1 - d_{2}) + θ ({\tilde{α}}_{2}) (1 - c_{2})$ .

The possibility degree for the comparison between two IVILNs ${\tilde{α}}_{1}$ and ${\tilde{α}}_{2}$ can be represented as follows: $p ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) = \frac{p^{-} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2}) + p^{+} ({\tilde{α}}_{1} \geq {\tilde{α}}_{2})}{2} .$ (7)

Assume that there are n IVILNs ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ , each IVILN ${\tilde{α}}_{i}$ is compared to all IVILNs by using Equation (7), then a complementary matrix can be expressed as follows: $P = (p_{ij})_{n \times n},$

where p_ij ≥ 0, p_ij + p_ji = 1 and $p_{ii} = \frac{1}{2}$ .

According to [37], we have the following theorem.

Theorem 4. Let P = (p_ij) _n×n be a fuzzy complementary judgment matrice, and its priority vector ξ = (ξ₁, ξ₂, …, ξ_n) can be obtained as follows: $ξ_{i} = \frac{1}{n (n - 1)} (\sum_{j = 1}^{n} p_{ij} + \frac{n}{2} - 1) .$ (8)

Example 4. Assume that ${\tilde{α}}_{1} = 〈 h_{1}, [0.2, 0.8], [0, 0.2]$ and ${\tilde{α}}_{2} = 〈 h_{2}, [0, 0], [0, 0.6]$ , by applying Equations (7) and (8), we can obtain ξ₁ = 0.536, ξ₂ = 0.464, so ${\tilde{α}}_{1} ≻ {\tilde{α}}_{2}$ .

Example 5. Assume that ${\tilde{α}}_{1} = 〈 h_{1}, [0.3, 0.5], [0, 0.2]$ and ${\tilde{α}}_{2} = 〈 h_{2}, [0.1, 0.3], [0, 0.1]$ , by applying Equations (7) and (8), we can obtain ξ₁ = 0.375, ξ₂ = 0.625, so ${\tilde{α}}_{2} ≻ {\tilde{α}}_{1}$ .

3 Aggregation operators of IVILNs

In this section, we introduce some aggregation operators of IVILNs.

Definition 11. Let ${\tilde{α}}_{i} (i = 1, 2, \dots, n)$ be a collection of IVILNs. The weighted arithmetic averaging(IVIL-WAA) operator is defined as $IVIL - WAA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = \sum_{i = 1}^{n} ω_{i} {\tilde{α}}_{i},$ (9) where Ω = (ω₁, ω₂, …, ω_n) is the weight vector of $({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n})$ with ω_i ∈ [0, 1] (i = 1, 2, …, n) and $\sum_{i = 1}^{n} ω_{i} = 1$ . Especially, if $ω_{i} = \frac{1}{n} (i = 1, 2, \dots, n)$ , then the IVIL-WAA operator is reduced to an arithmetic averaging (IVIL-AA) operator for IVILNs.

Theorem 5. Let ${\tilde{α}}_{i} (i = 1, 2, \dots, n)$ be a collection of IVILNs, then, the result aggregated by IVIL-WAA operator is still an IVILN, and

$\begin{matrix} IVIL - WAA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = 〈 \sum_{i = 1}^{n} ω_{i} h_{θ ({\tilde{α}}_{i})}, \\ [\frac{\sum_{i = 1}^{n} ω_{i} θ ({\tilde{α}}_{i}) μ^{L} ({\tilde{α}}_{i})}{\sum_{i = 1}^{n} ω_{i} θ ({\tilde{α}}_{i})}, \frac{\sum_{i = 1}^{n} ω_{i} θ ({\tilde{α}}_{i}) μ^{U} ({\tilde{α}}_{i})}{\sum_{i = 1}^{n} ω_{i} θ ({\tilde{α}}_{i})}], \\ [\frac{\sum_{i = 1}^{n} ω_{i} θ ({\tilde{α}}_{i}) ν^{L} ({\tilde{α}}_{i})}{\sum_{i = 1}^{n} ω_{i} θ ({\tilde{α}}_{i})}, \frac{\sum_{i = 1}^{n} ω_{i} θ ({\tilde{α}}_{i}) ν^{U} ({\tilde{α}}_{i})}{\sum_{i = 1}^{n} ω_{i} θ ({\tilde{α}}_{i})}] 〉 . \end{matrix}$ (10)

The theorem can be easily proven by using mathematical induction method, and it is omitted here.

Definition 12. Let ${\tilde{α}}_{i} (i = 1, 2, \dots, n)$ be a collection of IVILNs. The ordered weighted averaging (IVIL-OWA) operator is defined as $IVIL - OWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = \sum_{i = 1}^{n} w_{i} {\tilde{α}}_{φ_{i}},$ (11) where W = (w₁, w₂, …, w_n) is the associated weight vector of $({\tilde{α}}_{φ_{1}}, {\tilde{α}}_{φ_{2}}, \dots, {\tilde{α}}_{φ_{n}})$ with w_i ∈ [0, 1] (i = 1, 2, …, n) and $\sum_{i = 1}^{n} w_{i} = 1$ , and (φ₁, φ₂, …, φ_n) is a permutation of (1, 2, …, n), such that ${\tilde{α}}_{φ_{i}} \geq {\tilde{α}}_{φ_{i + 1}}$ for all i = 1, 2, …, n - 1. w_i is determined based on the fuzzy linguistic quantitative rules [38]: $φ_{i} = Q (\frac{i}{n}) - Q (\frac{i - 1}{n}), i = 1, 2, \dots, n,$ (12)

And the regular unimodal quantifier is as follows: $Q = {\begin{matrix} 0, & r < δ_{1} \\ \frac{r - δ_{1}}{δ_{2} - δ_{1}}, & δ_{1} \leq r \leq δ_{2} \\ 1, & r > δ_{2} \end{matrix}$ (13)

Here, we utilize the principle of antonym pairs most, at least half, as many as possible, where the parameters (δ₁, δ₂) are equal to (0.3, 0.8), (0, 0.5) and (0.5, 1), respectively.

Theorem 6. Let ${\tilde{α}}_{i} (i = 1, 2, \dots, n)$ be a collection of IVILNs, then the result aggregated by IVIL - OWA operator is also an IVILN, and

$\begin{matrix} IVIL - OWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = 〈 \sum_{i = 1}^{n} w_{i} h_{θ ({\tilde{α}}_{φ_{i}})}, \\ [\frac{\sum_{i = 1}^{n} w_{i} θ ({\tilde{α}}_{φ_{i}}) μ^{L} ({\tilde{α}}_{φ_{i}})}{\sum_{i = 1}^{n} w_{i} θ ({\tilde{α}}_{φ_{i}})}, \frac{\sum_{i = 1}^{n} w_{i} θ ({\tilde{α}}_{φ_{i}}) μ^{U} ({\tilde{α}}_{φ_{i}})}{\sum_{i = 1}^{n} w_{i} θ ({\tilde{α}}_{φ_{i}})}], \\ [\frac{\sum_{i = 1}^{n} w_{i} θ ({\tilde{α}}_{φ_{i}}) ν^{L} ({\tilde{α}}_{φ_{i}})}{\sum_{i = 1}^{n} w_{i} θ ({\tilde{α}}_{φ_{i}})}, \frac{\sum_{i = 1}^{n} w_{i} θ ({\tilde{α}}_{φ_{i}}) ν^{U} ({\tilde{α}}_{φ_{i}})}{\sum_{i = 1}^{n} w_{i} θ ({\tilde{α}}_{φ_{i}})}] 〉 . \end{matrix}$ (14)

Definition 13. Let ${\tilde{α}}_{i} (i = 1, 2, \dots, n)$ be a collection of IVILNs. The hybrid arithmetic (IVIL-HA) operator is defined as $IVIL - HA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = \sum_{i = 1}^{n} w_{i} {\tilde{β}}_{σ_{i}},$ (15) where W = (w₁, w₂, …, w_n) is the associatedweight vector of $({\tilde{β}}_{σ_{1}}, {\tilde{β}}_{σ_{2}}, \dots, {\tilde{β}}_{σ_{n}})$ with w_i ∈ [0, 1] (i = 1, 2, …, n) and $\sum_{i = 1}^{n} w_{i} = 1$ . And ${\tilde{β}}_{σ_{j}}$ is jth largest one of all values $(n ω_{1} {\tilde{α}}_{1}, n ω_{2} {\tilde{α}}_{2}, \dots, n ω_{n} {\tilde{α}}_{n})$ , Ω = (ω₁, ω₂, …, ω_n) is the weight of $({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n})$ with w_i ∈ [0, 1] (i = 1, 2, …, n), $\sum_{i = 1}^{n} w_{i} = 1$ , and n is the balancing coefficient.

Theorem 7. Let ${\tilde{α}}_{i} (i = 1, 2, \dots, n)$ be a collection of IVILNs, then the result aggregated by IVIL - HA operator is also an IVILN, and

$\begin{matrix} IVIL - HA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = 〈 \sum_{i = 1}^{n} w_{i} h_{θ ({\tilde{β}}_{φ_{i}})}, \\ [\frac{\sum_{i = 1}^{n} w_{i} θ ({\tilde{β}}_{φ_{i}}) μ^{L} ({\tilde{β}}_{φ_{i}})}{\sum_{i = 1}^{n} w_{i} θ ({\tilde{β}}_{φ_{i}})}, \frac{\sum_{i = 1}^{n} w_{i} θ ({\tilde{β}}_{φ_{i}}) μ^{U} ({\tilde{β}}_{φ_{i}})}{\sum_{i = 1}^{n} w_{i} θ ({\tilde{β}}_{φ_{i}})}], \\ [\frac{\sum_{i = 1}^{n} w_{i} θ ({\tilde{β}}_{φ_{i}}) ν^{L} ({\tilde{β}}_{φ_{i}})}{\sum_{i = 1}^{n} w_{i} θ ({\tilde{β}}_{φ_{i}})}, \frac{\sum_{i = 1}^{n} w_{i} θ ({\tilde{β}}_{φ_{i}}) ν^{U} ({\tilde{β}}_{φ_{i}})}{\sum_{i = 1}^{n} w_{i} θ ({\tilde{β}}_{φ_{i}})}] 〉 . \end{matrix}$ (16)

Especially, if $Ω = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})^{T}$ , then the IVIL-HA operator is reduced to the IVIL-OWA operator. If $W = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})^{T}$ , then the IVIL-HA operator is reduced to the IVIL-WAA operator.

4 MCDM method based on the IVILNs

Assume that there are n alternatives X = {x₁, x₂, …, x_n} and m criteria C = {c₁, c₂, …, c_m}. Let Ω = (ω₁, ω₂, …, ω_m) be the weight vector of C, where ω_j ∈ [0, 1] and $\sum_{j = 1}^{m} ω_{j} = 1$ . Suppose there is a group consisting of t decision makers E = {e₁, e₂, …, e_g} whose corresponding weight vector is λ = (λ₁, λ₂, …, λ_g), where λ_k ∈ [0, 1] and $\sum_{k = 1}^{t} λ_{k} = 1$ . Based on a linguistic term set H = {h_i|i = 0, 1, …, 2t, t ∈ N^*}, the evaluation matrix given by e_k is denoted as $R = (r_{ij}^{k})_{n \times m}$ , where $r_{ij}^{k}$ takes the form of the IVILN for alternative x_i with respect to criterion c_j. Here, a method based on the IVIL-HA and IVIL-WAA operator is developed here.

Step 1. Group aggregation.

Use the IVIL-HA operator

$\begin{matrix} r_{ij} & = & IVIL - HA (r_{ij}^{1}, r_{ij}^{2}, \dots, r_{ij}^{g}) = \sum_{k = 1}^{g} w_{k} {\tilde{β}}_{ij}^{φ_{k}} \\ i = 1, 2, \dots, n, j = 1, 2, \dots, m . \end{matrix}$ (17) to aggregate all decision matrices R^k (k = 1, 2, …, g) into a collective decision matrix R = (r_ij) _n×m, where W = (w₁, w₂, …, w_k) is the associated weight vector with IVIL-HA operator, w_k ∈ [0, 1] and $\sum_{k = 1}^{g} w_{k} = 1$ . ${\tilde{β}}_{ij}^{φ_{k}}$ is kth largest of $(g λ_{1} r_{ij}^{1}, g λ_{2} r_{ij}^{2}, \dots, g λ_{g} r_{ij}^{g})$ , where λ = (λ₁, λ₂, …, λ_g) is the weight vector of (r_i1, r_i2, …, r_im), λ_k ∈ [0, 1] and $\sum_{k = 1}^{g} λ_{k} = 1$ , and g is the balancing coeficient.

Step 2. Aggregate the criteria values of each alternative.

Use the IVIL-WAA operator

$\begin{matrix} r_{i} & = & IVIL - WAA (r_{i 1}, r_{i 2}, \dots, r_{im}) = \sum_{j = 1}^{m} ω_{j} r_{ij}, \\ i = 1, 2, \dots, n \end{matrix}$ (18) to obtain the collective overall evaluation value r_i of the alternative x_i, where Ω = (ω₁, ω₂, …, ω_m) is the weight vector with IVIL-WAA operator, ω_j ∈ [0, 1] and $\sum_{j =}^{m} ω_{j} = 1$ .

Step 3. Rank the alternatives.

Use Equations (5)–(7) to derive the possibility degree matrix of all alternatives, then rank all alternatives according to their priority vector by applying Equation (8).

Step 4. End.

5 Illustrative example

In this section, we use an example to demonstrate the application of the proposed methods in this paper, then conduct a comparison analysis to validate the feasibility of the proposed method.

Suppose that four command and control systems x₁, x₂, x₃, x₄ are to be evaluated by three decision makers under three criteria of system availability c₁, information accuracy c₂ as well as picture completeness c₃, and the evaluation values are in the form of IVILNs with the linguistic term set H = {h₀ = extremely poor, h₁ = very poor, h₂ = poor, h₃ = fair, h₄ = good, h₅ = very good, h₆ = extremely good}. Suppose that the weight vector of decision makers and criteria are λ = (0.3, 0.4, 0.3) and Ω = (0.3727, 0.3500, 0.2773), respectively. The decision matrices are listed in Tables 1–3.

5.1 Procedures of MCDM method based on the IVIL-HA and IVIL-WAA operator

Step 1. Group aggregation.

Use the IVIL-HA operator $\begin{matrix} r_{ij} & = & IVIL - HA (r_{ij}^{1}, r_{ij}^{2}, \dots, r_{ij}^{3}) = \sum_{k = 1}^{3} w_{k} {\tilde{β}}_{ij}^{φ_{k}} \\ i = 1, 2, 3, 4, 5, j = 1, 2, 3 . \end{matrix}$ to aggregate all decision matrices R^k (k = 1, 2, 3) into a collective decision matrix R = (r_ij) _n×m, where W = (0.2429, 0.5142, 0.2429) and λ = (0.3, 0.4, 0.3), and $β_{ij}^{φ_{k}}$ is kth largest of $(3 λ_{1} r_{ij}^{1}, 3 λ_{1} r_{ij}^{2}, 3 λ_{1} r_{ij}^{3})$ . The collective decision matrix R is shown in Table 4.

Step 2. Aggregate the criteria values of each alternative.

Use the IVIL-WAA operator $\begin{matrix} r_{i} & = & IVIL - WAA (r_{i 1}, r_{i 2}, r_{i 3}) = \sum_{j = 1}^{3} ω_{j} r_{ij}, \\ i = 1, 2, 3, 4, 5 \end{matrix}$ to obtain the collective overall evaluation value r_i of the alternative x_i, where Ω = (0.3727, 0.3500, 0.2772) is the weight vector with IVIL-WAA operator. $\begin{matrix} r_{1} & = & 〈 h_{4.3}, [0.58, 0.73], [0.09, 0.22] 〉, \\ r_{2} & = & 〈 h_{3.9}, [0.66, 0.81], [0.04, 0.15] 〉, \\ r_{3} & = & 〈 h_{4.4}, [0.59, 0.72], [0.08, 0.23] 〉, \\ r_{4} & = & 〈 h_{4.1}, [0.50, 0.66], [0.10, 0.30] 〉, \\ r_{5} & = & 〈 h_{4.2}, [0.62, 0.78], [0.05, 0.20] 〉 . \end{matrix}$

Step 3. Rank the alternatives.

Use Equations (5)–(7) to derive the following possibility degree matrix of all alternatives: $p = [\begin{matrix} 0.5 & 0.51 & 0.39 & 0.79 & 0.44 \\ 0.49 & 0.5 & 0.37 & 0.81 & 0.43 \\ 0.61 & 0.63 & 0.5 & 0.9 & 0.55 \\ 0.21 & 0.19 & 0.1 & 0.5 & 0.16 \\ 0.56 & 0.57 & 0.45 & 0.84 & 0.5 \end{matrix}]$

By applying Equation (8), we have ξ₁ = 0.211, ξ₂ = 0.191, ξ₃ = 0.238, ξ₄ = 0.146, ξ₅ = 0.214.

Then, $x_{3} ≻ x_{5} ≻ x_{1} ≻ x_{2} ≻ x_{4} .$

5.2 Comparison analysis and discussion

In order to validate the feasibility of the proposed methods, another two methods based on aggregation operators in [27] and [25] are used to solve the same illustrative example.

(1) Applying the MCDM method based on the IVIULWGA operator and IVIULHG operator [27], the final ranking is: $x_{2} ≻ x_{3} ≻ x_{1} ≻ x_{5} ≻ x_{4} .$

(2) In [25] a method using geometric aggregation operators was developed, and the ranking of all alternatives is: $x_{3} ≻ x_{2} ≻ x_{1} ≻ x_{5} ≻ x_{4} .$

The comparison of three methods is shown in Table 5.

Obviously, the difference among the rankings is the order of x₂, x₃ and x₅. The main reasons are listed as follows.

The arithmetic operations of IVILNs defined in this paper are more reasonable and reliable, because the membership and non-membership degree are closely combined with the linguistic value of the original IVILNs. Compared with the operations in this paper, the operations in [27] and [25] have some limitations and may easily result in information loss and distortion.

The ranking methods in [27] and [25] are substantially the same. However, according to the analysis above, the ranking method exists some limitations and drawbacks. By contrast, the proposed comparison approach makes use of all information of IVILN including membership degree and non-membership degree as well as hesitation degree. Therefore the results derived by the proposed methods are more reliable.

6 Conclusion

In this paper, some new arithmetic operations, a new comparison method of IVILNs based on a possibility degree and some new arithmetic aggregation operators of IVILNs are defined. Then, a MCDM approach for IVILNs is introduced. This paper makes three contributions with respect to the existing studies. Firstly, the improvement of arithmetic operations of IVILNs makes great sense, which considers that membership and non-membership degree are closely combined with the linguistic value of the original IVILNs. Secondly, a comparison method based on a possibility degree is developed, which can make use of all information of IVILNs and avoid information loss. Lastly, a MCDM approach with IVIL information based on aggregation operators is introduced, which can provide solutions for group decision-making problems.

In future study, the related concepts of IVILSs including distance measures, similarity measures will be explored. In particular, a clustering model for IVILNs will be studied.

Footnotes

Acknowledgments

The authors are very grateful to the editor and anonymous reviewers for their insightful and constructive comments and suggestions, which have been very helpful in improving this paper. This work was supported by the National Natural Science Foundation of China (Nos. 709210014, 71171202, 71210003).

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