A New Similarity Measure of Interval-Valued Intuitionistic Fuzzy Sets Considering Its Hesitancy Degree and Applications in Expert Systems

Abstract

As an important content in fuzzy mathematics, similarity measure is used to measure the similarity degree between two fuzzy sets. Considering the existing similarity measures, most of them do not consider the hesitancy degree and some methods considering the hesitancy degree are based on the intuitionistic fuzzy sets, intuitionistic fuzzy values. It may cause some counterintuitive results in some cases. In order to make up for the drawback, we present a new approach to construct the similarity measure between two interval-valued intuitionistic fuzzy sets using the entropy measure and considering the hesitancy degree. In particular, the proposed measure was demonstrated to yield a similarity measure. Besides, some examples are given to prove the practicality and effectiveness of the new measure. We also apply the similarity measure to expert system to solve the problems on pattern recognition and the multicriteria group decision making. In these examples, we also compare it with existing methods such as other similarity measures and the ideal point method.

1. Introduction

In 1989, interval-valued intuitionistic fuzzy set (IVIFS), which is characterized by a membership degree range and a nonmembership degree range, was proposed by Atanassov and Gargov [1]. Due to the high complexity of the real world and the limitation of people’s judgment, the use of interval-valued intuitionistic fuzzy set has gained much attention [2–9]. As one of the most important applications, the expert system to solve problems under fuzzy environment is a hot research topic [10–15]. We give an example of the decision problem which is also an important aspect of the expert system. As the time pressure and lack of data, the decision information is often imprecise or uncertain. The experts may not express their preference over the alternatives considered precisely. So the evaluation can be given in the form of interval-valued intuitionistic fuzzy numbers [1]. Accordingly, IVIFS is a very suitable tool to describe the imprecise or uncertain decision information and deal with the uncertainty and vagueness in decision making.

As one of the important topics of fuzzy theory, the similarity measures are used to estimate the degree of similarity between two fuzzy sets. Functions expressing the degree of similarity of items or sets are used in many different fields, such as numerical taxonomy, ecology, information retrieval, and psychology and the similarity measure plays a very important role [16]. For example, many scholars take advantages of the similarity measures, especially the Jaccard, Dice, and cosine methods [17–19] to research the problems in information retrieval, citation analysis, and automatic classification. Therefore, the similarity measure has been investigated by many authors [20–22]. For the similarity measure to IVIFSs, Xu and Chen [23] generalized some formulas of similarity measures which are based on the distance measures for two IVIFSs. Besides, Wei et al. [24] gave a formula of a similarity measure of IVIFSs based on entropy theory [25]. Ye [16] presented a Dice similarity measure based on the reduct intuitionistic fuzzy sets of interval-valued intuitionistic fuzzy sets and applied it to multicriteria decision making problems. Singh [26] and Ye [21] also proposed a new cosine similarity measure for two IVIFSs, respectively. Xu and Yager [27] had studied the intuitionistic and interval-valued intuitionistic fuzzy preference relations and suggested some new similarity measures to evaluate the agreement within a group. However, many of these measures for two IVIFSs do not consider the degree of hesitancy and may lead to counterintuitive results in some cases. For example, for 𝐴,𝐵,𝐶∈  IVIFSs, we assume that the differences of membership and nonmembership degree in 𝐴,𝐵 and 𝐴,𝐶 are very close and the difference of hesitancy degree between 𝐵 and 𝐶 is very large. Then, it may be incapable of distinguishing which one between 𝐵 and 𝐶 is more similar to 𝐴 using the existing similarity measure. Besides, many other similarity measures for intuitionistic fuzzy values or intuitionistic fuzzy sets have been proposed. Xu [28] developed some similarity measures of intuitionistic fuzzy sets which consider the hesitancy degree and also define the notions of positive ideal intuitionistic fuzzy set and negative ideal intuitionistic fuzzy set. Later, Xia and Xu [29] proposed a series of similarity measures for intuitionistic fuzzy values (IFVs) based on the intuitionistic fuzzy operators and the similarity measures were also taking the hesitancy degree into consideration. In 2011, Xu and Xia [30] also studied the distance and similarity measures for hesitant fuzzy sets. The similarity measures considering the hesitancy degree outperform the existing methods and can deal with the group decision well. So we establish a new method of similarity measure by taking the hesitancy degree into consideration and using the entropy measure of IVIFSs.

The structure of this paper is organized as follows. Section 2 focuses on the concepts of the interval-valued intuitionistic fuzzy sets and analyses the drawbacks of some existing similarity measures. Section 3 presents a new similarity measure and illustrates its advantage. Section 4 describes the application to expert system based on the similarity measure of the IVIFSs and presents some illustrative examples. Section 5 concludes our work.

2. Basic Notions and the Drawback of Existing Similarity Measures

2.1. Preliminaries

Definition 1 (see [1]). Aninterval-valued intuitionistic fuzzy set in 𝑋 is an expression 𝐴 given by 𝐴=𝑥,𝑢_𝐴(𝑥),𝑣_𝐴(𝑥)∣𝑥∈𝑋, (1) where 𝑢_𝐴∶𝑋⟶[0,1],𝑣_𝐴∶𝑋⟶[0,1], (2) with the condition 0≤𝑢_𝐴(𝑥)+𝑣_𝐴(𝑥)≤1 for all 𝑥∈𝑋. The intervals 𝑢_𝐴(𝑥) and 𝑣_𝐴(𝑥) denote the degree of membership and nonmembership of 𝑥 to 𝐴, respectively.
If we let 𝑢_𝐴(𝑥)=[𝑢⁻_𝐴(𝑥),𝑢⁺_𝐴(𝑥)], 𝑣_𝐴(𝑥)=[𝑣⁻_𝐴(𝑥),𝑣⁺_𝐴(𝑥)], the interval-valued intuitionistic fuzzy set 𝐴 can be represented as follows: 𝐴=𝑥,𝑢⁻_𝐴(𝑥),𝑢⁺_𝐴(𝑥),𝑣⁻_𝐴(𝑥),𝑣⁺_𝐴(𝑥)∣𝑥∈𝑋, (3) where 0≤𝑢⁺_𝐴(𝑥)+𝑣⁺_𝐴(𝑥)≤1. The interval, 1−𝑢⁺_𝐴(𝑥)−𝑣⁺_𝐴(𝑥),1−𝑢⁻_𝐴(𝑥)−𝑣⁻_𝐴(𝑥), (4) is denoted by 𝜋_𝐴 and abbreviated by [𝜋⁻_𝐴,𝜋⁺_𝐴], which is called the intuitionistic index of element 𝑥 in 𝐴. We call it a hesitancy degree of 𝑥 to 𝐴.
For all 𝐴,𝐵∈IVFSs(𝑋), we have the following expression defined in [1]:(1)𝐴⊆𝐵 if and only if 𝑢⁻_𝐴(𝑥)≤𝑢⁻_𝐵(𝑥), 𝑢⁺_𝐴(𝑥)≤𝑢⁺_𝐵(𝑥), 𝑣⁻_𝐴(𝑥)≥𝑣⁻_𝐵(𝑥), 𝑣⁺_𝐴(𝑥)≥𝑣⁺_𝐵(𝑥) for 𝑥∈𝑋;(2)𝐴=𝐵 if and only if 𝐴⊆𝐵, 𝐴⊇𝐵;(3)𝐴^𝑐={(𝑥,[𝑣⁻_𝐴(𝑥),𝑣⁺_𝐴(𝑥)],[𝑢⁻_𝐴(𝑥),𝑢⁺_𝐴(𝑥)])∣𝑥∈𝑋}.

Definition 2 (see [24]). For 𝐴∈IVFSs(𝑋), an entropy measure on IVFSs(𝑋) can be given by 𝐸(𝐴)=1𝑛^𝑛_𝑗=1min𝑢⁻_𝐴𝑥_𝑗,𝑣⁻_𝐴𝑥_𝑗+min𝑢⁺_𝐴𝑥_𝑗,𝑣⁺_𝐴𝑥_𝑗+𝜋⁻_𝐴𝑥_𝑗+𝜋⁺_𝐴𝑥_𝑗×max𝑢⁻_𝐴𝑥_𝑗,𝑣⁻_𝐴𝑥_𝑗+max𝑢⁺_𝐴𝑥_𝑗,𝑣⁺_𝐴𝑥_𝑗+𝜋⁻_𝐴𝑥_𝑗+𝜋⁺_𝐴𝑥_𝑗⁻¹. (5) It also has the following properties.(1)𝐸(𝐴)=0 if and only if 𝐴 is the crispest.(2)𝐸(𝐴)=1 if and only if [𝑢⁻_𝐴(𝑥_𝑗),𝑢⁺_𝐴(𝑥_𝑗)]=[𝑣⁻_𝐴(𝑥_𝑗),𝑣⁺_𝐴(𝑥_𝑗)] for all 𝑥_𝑗∈𝑋.(3)𝐸(𝐴)=𝐸(𝐴^𝑐).(4)𝐸(𝐴)≤𝐸(𝐵) if 𝐴⊆𝐵 when 𝑢⁺_𝐵(𝑥_𝑗)≤𝑣⁺_𝐵(𝑥_𝑗) and 𝑢⁻_𝐵(𝑥_𝑗)≤𝑣⁻_𝐵(𝑥_𝑗), for 𝑥_𝑗∈𝑋, or 𝐴⊇𝐵 when 𝑢⁺_𝐵(𝑥_𝑗)≥𝑣⁺_𝐵(𝑥_𝑗) and 𝑢⁻_𝐵(𝑥_𝑗)≥𝑣⁻_𝐵(𝑥_𝑗) for 𝑥_𝑗∈𝑋.

Definition 3 (see [21]). Let 𝐴,𝐵∈IVFS(𝑋), 𝑆∶IVFS(𝑋)×IVFS(𝑋)→[0,1]; then, the degree of similarity between 𝐴 and 𝐵 is 𝑆(𝐴,𝐵), if it satisfies the following requirements:(1)0≤𝑆(𝐴,𝐵)≤1;(2)𝑆(𝐴,𝐵)=1 if and only if 𝐴=𝐵;(3)𝑆(𝐴,𝐵)=𝑆(𝐵,𝐴);(4)if 𝐴,𝐵,𝐶∈IVFS(𝑋),  𝐴⊆𝐵⊆𝐶, then 𝑆(𝐴,𝐶)≤𝑆(𝐴,𝐵), 𝑆(𝐴,𝐶)≤𝑆(𝐵,𝐶).

2.2. The Existing Similarity Measures

Wei et al. [24] define a method to construct similarity measure of IVFSs as follows. Firstly, for 𝐴,𝐵∈IVFSs(𝑋), a new IVFS is defined, written by 𝑀(𝐴,𝐵). Consider 𝑀_𝐴𝐵1(𝑥)=1+min𝑢⁻_𝐴(𝑥)−𝑢⁻_𝐵(𝑥),𝑣⁻_𝐴(𝑥)−𝑣⁻_𝐵(𝑥)2,𝑀_𝐴𝐵2(𝑥)=1+min𝑢⁺_𝐴(𝑥)−𝑢⁺_𝐵(𝑥),𝑣⁺_𝐴(𝑥)−𝑣⁺_𝐵(𝑥)2,𝑀_𝐴𝐵3(𝑥)=1−max𝑢⁻_𝐴(𝑥)−𝑢⁻_𝐵(𝑥),𝑣⁻_𝐴(𝑥)−𝑣⁻_𝐵(𝑥)2,𝑀_𝐴𝐵4(𝑥)=1−max𝑢⁺_𝐴(𝑥)−𝑢⁺_𝐵(𝑥),𝑣⁺_𝐴(𝑥)−𝑣⁺_𝐵(𝑥)2. (6) Let 𝑢⁻_{𝑀()𝐴,𝐵}(𝑥)=min𝑀_𝐴𝐵1(𝑥),𝑀_𝐴𝐵2(𝑥,𝑢_{+𝑀()𝐴,𝐵})(𝑥)=max𝑀_𝐴𝐵1(𝑥),𝑀_𝐴𝐵2(𝑥,𝑣_{−𝑀()𝐴,𝐵})(𝑥)=min𝑀_𝐴𝐵3(𝑥),𝑀_𝐴𝐵4(𝑥,𝑣_{+𝑀()𝐴,𝐵})(𝑥)=max𝑀_𝐴𝐵3(𝑥),𝑀_𝐴𝐵4(𝑥). (7) Then we get 𝑀(𝐴,𝐵)=𝑢⁻_{𝑀(𝐴,𝐵)}(𝑥),𝑢⁺_{𝑀(𝐴,𝐵)}(𝑥,𝑣_{−𝑀()𝐴,𝐵})(𝑥),𝑣⁺_{𝑀()𝐴,𝐵}(𝑥)∣𝑥∈𝑋. (8) Then 𝑀(𝐴,𝐵)∈IVFSs(𝑋) and the degree of similarity between 𝐴 and 𝐵 is 𝑆₁(𝐴,𝐵)=𝐸(𝑀(𝐴,𝐵)).

Xu [31] also defines four different similarity measures of 𝐴 and 𝐵. We assume that 𝑋 is finite; that is, 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} and 𝑗={1,2,…,𝑛}. Let 𝐴=𝑥_𝑗,𝑢⁻_𝐴𝑥_𝑗,𝑢⁺_𝐴𝑥_𝑗,𝑣⁻_𝐴𝑥_𝑗,𝑣⁺_𝐴𝑥_𝑗∣𝑥_𝑗,𝑥∈𝑋𝐵=_𝑗,𝑢⁻_𝐵𝑥_𝑗,𝑢⁺_𝐵𝑥_𝑗,𝑣⁻_𝐵𝑥_𝑗,𝑣⁺_𝐵𝑥_𝑗∣𝑥_𝑗∈𝑋 (9) be two IVFSs. Let 𝑤={𝑤₁,𝑤₂,…,𝑤_𝑛} be the weight vector of elements 𝑥_𝑗,  𝑗=1,2,…,𝑛. The similarity measures of 𝐴 and 𝐵 are as follows: 𝑆₂()1𝐴,𝐵=1−4^𝑛_𝑗=1𝑤_𝑗𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗+𝑢⁺_𝐴𝑥_𝑗−𝑢⁺_𝐵𝑥_𝑗+𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗+𝑣⁺_𝐴𝑥_𝑗−𝑣⁺_𝐵𝑥_𝑗,𝑆₃()1𝐴,𝐵=1−4^𝑛_𝑗=1𝑤_𝑗𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗²+𝑢⁺_𝐴𝑥_𝑗−𝑢⁺_𝐵𝑥_𝑗²+𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗²+𝑣⁺_𝐴𝑥_𝑗−𝑣⁺_𝐵𝑥_𝑗^21/2,𝑆₄(𝐴,𝐵)=1−^𝑛_𝑗=1𝑤_𝑗×max𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗,𝑢⁺_𝐴𝑥_𝑗−𝑢⁺_𝐵𝑥_𝑗,𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗,𝑣⁺_𝐴𝑥_𝑗−𝑣⁺_𝐵𝑥_𝑗,𝑆₅(𝐴,𝐵)=1−^𝑛_𝑗=1𝑤_𝑗max𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗²,𝑢⁺_𝐴𝑥_𝑗−𝑢⁺_𝐵𝑥_𝑗²,𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗²,𝑣⁺_𝐴𝑥_𝑗−𝑣⁺_𝐵𝑥_𝑗^21/2. (10)(11)(12)(13)

2.3. The Drawback of Existing Similarity Measures

We give an example to calculate the similarity of two fuzzy sets using the above five different similarity measures. It proves that the five different similarity measures are not reliable in some cases.

Example 4. It is an example of group voting. We have three plans written in the form of IFSs. Let 𝐴={𝑥,0.5,0.5}, 𝐵={𝑥,0.3,0.7}, and 𝐶={𝑥,0.3,0.3} be three IFSs and denote the three plans. Plan 𝐴 denotes that the percentage of supporters or dissenters is fifty percent. As to plan 𝐵, thirty percent of the voters support the plan and seventy percent of the voters oppose it. For plan 𝐶, we can know that the percentage of supporters or dissenters is thirty percent while forty percent of the voters abstain from voting.
Then we can get the results of the similarity of 𝐴 and 𝐵 based on the above similarity measure. Consider 𝑀_𝐴𝐵1(𝑥)=𝑀_𝐴𝐵2(𝑥)=0.5∗1+min𝑢_𝐴(𝑥)−𝑢_𝐵(𝑥),𝑣_𝐴(𝑥)−𝑣_𝐵(𝑥𝑀)=0.6,_𝐴𝐵3(𝑥)=𝑀_𝐴𝐵4(𝑥)=0.5∗1−max𝑢_𝐴(𝑥)−𝑢_𝐵(𝑥),𝑣_𝐴(𝑥)−𝑣_𝐵(𝑥𝑀()={})=0.4,𝐴,𝐵𝑥,0.6,0.4,𝑆₁()(𝑀(=2𝐴,𝐵=𝐸𝐴,𝐵))3. (14) In the same way, we can get the similarity of 𝐴 and 𝐶; that is, 𝑆₁(𝐴,𝐶)=𝐸(𝑀(𝐴,𝐶))=2/3. Supposing 𝑤_𝑗=1, 𝑛=1, we can get the following results: 𝑆₂(𝐴,𝐵)=𝑆₂()[]𝑆𝐴,𝐶=1−0.25∗0.2+0.2+0.2+0.2=0.8,₃(𝐴,𝐵)=𝑆₃(𝐴,𝐶)=1−[]𝑆0.25∗0.04+0.04+0.04+0.04=0.8,₄(𝐴,𝐵)=𝑆₄(){}𝑆𝐴,𝐶=1−max0.2,0.2,0.2,0.2=0.8,₅(𝐴,𝐵)=𝑆₅(𝐴,𝐶)=1−max{0.04,0.04,0.04,0.04}=0.8. (15)
Obviously, we could not distinguish which one between 𝐵 and 𝐶 is more similar to 𝐴 using the five different similarity measures. However, if we make a deep analysis, we can know that 𝐴 is more similar to 𝐶 than to 𝐵. According to formula (4), the hesitancy degree of 𝐶 is twenty percent while the hesitancy degree of 𝐵 is zero. For plan 𝐶, the reason of abstention may be different. For example, someone may be absent for the vote. So, it is reasonable to assume that the twenty percent of abstention has a certain degree 𝑟(0≤𝑟≤1) to support the plan.
If 𝑟=0 or 𝑟=1, all of the abstainers oppose or support the plan 𝐶. We have the following results: {}{},𝑆𝐶=𝑥,0.3,0.7or𝐶=𝑥,0.7,0.3_𝑖(𝐴,𝐶)=𝑆_𝑖(𝐴,𝐵),𝑖=1,2,…,5. (16) If 𝑟=0.5, that is, half of the abstainers to plan 𝐶 are supporters or dissenters, the percentage of supporters or dissenters is fifty percent. We can get the result as follows: 𝐶={𝑥,0.5,0.5},𝑆_𝑖()𝑆𝐴,𝐶=1,_𝑖(𝐴,𝐶)>𝑆_𝑖(𝐴,𝐵),𝑖=1,2,…,5. (17)
But in general, both situations do not always happen, and it is just between them; that is, 0<𝑟<0.5,  0.5<𝑟<1. So we can get the result as follows: 𝑆(𝐴,𝐶)>𝑆(𝐴,𝐵).
According to our analysis, more reasonable result is that 𝐴 is more similar to 𝐶 than to 𝐵. It is different from the result calculated by the existing five similarity measures. Besides, we can also know that the five similarity measures do not take the hesitancy degree into consideration according to its definition. It is not reasonable to define the similarity measure without considering the hesitancy degree of the interval-valued intuitionistic fuzzy sets; otherwise, it may make mistakes in some cases. So we present a new method of similarity measure.

3. The New Similarity Measures between IVIFSs

3.1. New Similarity Measure

Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a finite universe of discourse. For 𝐴,𝐵∈IVFSs(𝑋), we define IVFSs, written by 𝑀(𝐴,𝐵): 𝑀_𝐴𝐵1𝑥_𝑗=2+𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗+𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗−𝜋⁺_𝐴𝑥_𝑗+𝜋⁺_𝐵𝑥_𝑗×4⁻¹,𝑀_𝐴𝐵2𝑥_𝑗=2+𝑢⁺_𝐴𝑥_𝑗−𝑢⁺_𝐵𝑥_𝑗+𝑣⁺_𝐴𝑥_𝑗−𝑣⁺_𝐵𝑥_𝑗−𝜋⁻_𝐴𝑥_𝑗+𝜋⁻_𝐵𝑥_𝑗×4⁻¹,𝑀_𝐴𝐵3𝑥_𝑗=2−𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗+𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗−𝜋⁺_𝐴𝑥_𝑗+𝜋⁺_𝐵𝑥_𝑗×4⁻¹,𝑀_𝐴𝐵4𝑥_𝑗=2−𝑢⁺_𝐴𝑥_𝑗−𝑢⁺_𝐵𝑥_𝑗+𝑣⁺_𝐴𝑥_𝑗−𝑣⁺_𝐵𝑥_𝑗−𝜋⁻_𝐴𝑥_𝑗+𝜋⁻_𝐵𝑥_𝑗×4⁻¹. (18) And define 𝑢⁻_{𝑀()𝐴,𝐵}𝑥_𝑗=min𝑀_𝐴𝐵1𝑥_𝑗,𝑀_𝐴𝐵2𝑥_𝑗,𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗=max𝑀_𝐴𝐵1𝑥_𝑗,𝑀_𝐴𝐵2𝑥_𝑗,𝑣⁻_{𝑀()𝐴,𝐵}𝑥_𝑗=min𝑀_𝐴𝐵3𝑥_𝑗,𝑀_𝐴𝐵4𝑥_𝑗,𝑣⁺_{𝑀()𝐴,𝐵}𝑥_𝑗=max𝑀_𝐴𝐵3𝑥_𝑗,𝑀_𝐴𝐵4𝑥_𝑗. (19) Then, we get 𝑀(𝐴,𝐵)=𝑥_𝑗,𝑢⁻_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑣⁻_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑣⁺_{𝑀()𝐴,𝐵}𝑥_𝑗∣𝑥_𝑗∈𝑋. (20)

Theorem 5. Suppose that 𝐸 is an entropy measure for 𝐼𝑉𝐹𝑆. The similarity measure on 𝐼𝑉𝐹𝑆(𝑋) is defined by 𝑆(𝐴,𝐵)=𝐸(𝑀(𝐴,𝐵)) for each pair of 𝐼𝑉𝐹𝑆𝑠𝐴 and 𝐵.

Proof. It is easy to prove that 𝐸(𝑀(𝐴,𝐵)) satisfies the conditions listed in Definition 3. (1)We can easily get 0≤𝐸(𝑀(𝐴,𝐵))≤1 since 0≤𝐸(𝐴)≤1 for all 𝐴∈IVIFS(𝑋) and 𝑀(𝐴,𝐵)∈IVIFS(𝑋).(2)With the definition of entropy and similarity measure, we can obtain 𝐸(𝑀(𝐴,𝐵))=1⟺𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗=𝑣⁺_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑢⁻_{𝑀()𝐴,𝐵}𝑥_𝑗=𝑣⁻_{𝑀()𝐴,𝐵}𝑥_𝑗,∀𝑥_𝑗∈𝑋⟺𝑢⁺_𝐴𝑥_𝑗=𝑢⁺_𝐵𝑥_𝑗,𝑣⁺_𝐴𝑥_𝑗=𝑣⁺_𝐵𝑥_𝑗,𝑢⁻_𝐴𝑥_𝑗=𝑢⁻_𝐵𝑥_𝑗,𝑣⁻_𝐴𝑥_𝑗=𝑣⁻_𝐵𝑥_𝑗,∀𝑥_𝑗∈𝑋⟺𝐴=𝐵. (21)(3)From the definition of 𝑀(𝐴,𝐵), we can easily have that 𝑀(𝐴,𝐵)=𝑀(𝐵,𝐴). So it is obvious that 𝐸(𝑀(𝐴,𝐵))=𝐸(𝑀(𝐵,𝐴)).(4)As 𝐴⊆𝐵⊆𝐶, that is, 𝑥_𝑗∈𝑋, 𝑣⁻_𝐴(𝑥_𝑗)≥𝑣⁻_𝐵(𝑥_𝑗)≥𝑣⁻_𝐶(𝑥_𝑗), 𝑣⁺_𝐴(𝑥_𝑗)≥𝑣⁺_𝐵(𝑥_𝑗)≥𝑣⁺_𝐶(𝑥_𝑗), 𝑢⁻_𝐴(𝑥_𝑗)≤𝑢⁻_𝐵(𝑥_𝑗)≤𝑢⁻_𝐶(𝑥_𝑗), 𝑢⁺_𝐴(𝑥_𝑗)≤𝑢⁺_𝐵(𝑥_𝑗)≤𝑢⁺_𝐶(𝑥_𝑗). We have 𝑀_𝐴𝐶1𝑥_𝑗=142+𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐶𝑥_𝑗+𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐶𝑥_𝑗−𝜋⁺_𝐴𝑥_𝑗+𝜋⁺_𝐶𝑥_𝑗=12𝑢⁻_𝐶𝑥_𝑗+𝑣⁻_𝐴𝑥_𝑗,𝑀_𝐴𝐵1𝑥_𝑗=142+𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗+𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗−𝜋⁺_𝐴𝑥_𝑗+𝜋⁺_𝐵𝑥_𝑗=12𝑢⁻_𝐵𝑥_𝑗+𝑣⁻_𝐴𝑥_𝑗. (22)So we can get 𝑀_𝐴𝐶1(𝑥_𝑗)≥𝑀_𝐴𝐵1(𝑥_𝑗) because of 𝑢⁻_𝐵(𝑥_𝑗)≤𝑢⁻_𝐶(𝑥_𝑗).
Thus in the same way, we can easily prove that 𝑀_𝐴𝐶1𝑥_𝑗≥𝑀_𝐴𝐵1𝑥_𝑗,𝑀_𝐴𝐶2𝑥_𝑗≥𝑀_𝐴𝐵2𝑥_𝑗,𝑀_𝐴𝐶3𝑥_𝑗≤𝑀_𝐴𝐵3𝑥_𝑗,𝑀_𝐴𝐶4𝑥_𝑗≤𝑀_𝐴𝐵4𝑥_𝑗. (23) We can also obtain 𝑢⁻_{𝑀()𝐴,𝐶}𝑥_𝑗≥𝑢⁻_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑢⁺_{𝑀()𝐴,𝐶}𝑥_𝑗≥𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑣⁻_{𝑀()𝐴,𝐶}𝑥_𝑗≤𝑣⁻_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑢⁺_{𝑀()𝐴,𝐶}𝑥_𝑗≤𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗. (24) Therefore 𝑀(𝐴,𝐵)⊆𝑀(𝐴,𝐶).
Besides, we have 𝑢⁻_{𝑀(𝐴,𝐵)}(𝑥_𝑗)≥𝑣⁻_{𝑀(𝐴,𝐵)}(𝑥_𝑗), 𝑢⁺_{𝑀(𝐴,𝐵)}(𝑥_𝑗)≥𝑣⁺_{𝑀(𝐴,𝐵)}(𝑥_𝑗) by the definition of 𝑀(𝐴,𝐵). We can get the result that 𝐸(𝑀(𝐴,𝐵))≥𝐸(𝑀(𝐴,𝐶)),𝑆(𝐴,𝐵)≥𝑆(𝐴,𝐶). (25) In the same way, we can also have 𝐸(𝑀(𝐵,𝐶))≥𝐸(𝑀(𝐴,𝐶)).

Remark 6. Let 𝐴={⟨𝑥,[0.1,0.2],[0.5,0.6]⟩} and 𝐵={⟨𝑥,[0.2,0.3],[0.5,0.6]⟩} be two IVIFSs. With (18), we can have 𝑀_𝐴𝐵1(𝑥)()𝑀=0.25∗2+0.1−0.7=0.35,_𝐴𝐵2(𝑥)()𝑀=0.25∗2+0.1−0.3=0.45,_𝐴𝐵3(𝑥)()𝑀=0.25∗2−0.1−0.7=0.3,_𝐴𝐵2(𝑥)()𝑢=0.25∗2−0.1−0.3=0.4._{−𝑀(𝐴,𝐵)}(𝑥)=min𝑀_𝐴𝐵1(𝑥),𝑀_𝐴𝐵2(𝑥𝑢)=0.35,_{+𝑀(𝐴,𝐵)}(𝑥)=max𝑀_𝐴𝐵1(𝑥),𝑀_𝐴𝐵2(𝑥𝑣)=0.45,_{−𝑀(𝐴,𝐵)}(𝑥)=min𝑀_𝐴𝐵3(𝑥),𝑀_𝐴𝐵4(𝑥𝑣)=0.3,_{+𝑀(𝐴,𝐵)}(𝑥)=max𝑀_𝐴𝐵3(𝑥),𝑀_𝐴𝐵4(𝑥)=0.4. (26) Hence we obtain that 𝑀(𝐴,𝐵)={⟨𝑥,[0.35,0.45],[0.3,0.4]⟩}.
We can get 𝐸(𝑀(𝐴,𝐵))=12/13, if we chose the entropy formula of IVIFSs defined in Definition 2.

Corollary 7. Let 𝐸 be the entropy measure defined in Definition 2; that is, 𝐴∈𝐼𝑉𝐼𝐹𝑆(𝑋), 𝐸(𝐴)=1𝑛^𝑛_𝑗=1min𝑢⁻_𝐴𝑥_𝑗,𝑣⁻_𝐴𝑥_𝑗+min𝑢⁺_𝐴𝑥_𝑗,𝑣⁺_𝐴𝑥_𝑗+𝜋⁻_𝐴𝑥_𝑗+𝜋⁺_𝐴𝑥_𝑗×max𝑢⁻_𝐴𝑥_𝑗,𝑣⁻_𝐴𝑥_𝑗+max𝑢⁺_𝐴𝑥_𝑗,𝑣⁺_𝐴𝑥_𝑗+𝜋⁻_𝐴𝑥_𝑗+𝜋⁺_𝐴𝑥_𝑗⁻¹. (27) Then the similarity measure is defined by 𝑆()=1𝐴,𝐵𝑛^𝑛_𝑗=14−𝑢⁻_𝑗+𝑢⁺_𝑗+𝑣⁻_𝑗+𝑣⁺_𝑗+𝜋⁻_𝑗+𝜋⁺_𝑗4+𝑢⁻_𝑗+𝑢⁺_𝑗+𝑣⁻_𝑗+𝑣⁺_𝑗+𝜋⁻_𝑗+𝜋⁺_𝑗, (28) for 𝐴,𝐵∈𝐼𝑉𝐼𝐹𝑆(𝑋), where 𝑢⁻_𝑗=𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗,𝑢⁺_𝑗=𝑢⁺_𝐴𝑥_𝑗−𝑢⁺_𝐵𝑥_𝑗,𝑣⁻_𝑗=𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗,𝑣⁺_𝑗=𝑣⁺_𝐴𝑥_𝑗−𝑣⁺_𝐵𝑥_𝑗,𝜋⁻_𝑗=𝜋⁻_𝐴𝑥_𝑗+𝜋⁻_𝐵𝑥_𝑗,𝜋⁺_𝑗=𝜋⁺_𝐴𝑥_𝑗+𝜋⁺_𝐵𝑥_𝑗. (29)
Let 𝑤={𝑤₁,𝑤₂,…,𝑤_𝑛} be the weight vector of elements 𝑥_𝑗(𝑗=1,2,…,𝑛), where 𝑤_𝑗≥0, ∑^𝑛_𝑗=1𝑤_𝑗=1.
Then we get the weighted similarity measure as follows: 𝑆(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗4−𝑢⁻_𝑗+𝑢⁺_𝑗+𝑣⁻_𝑗+𝑣⁺_𝑗+𝜋⁻_𝑗+𝜋⁺_𝑗4+𝑢⁻_𝑗+𝑢⁺_𝑗+𝑣⁻_𝑗+𝑣⁺_𝑗+𝜋⁻_𝑗+𝜋⁺_𝑗. (30)

Proof. We have 𝑢⁻_{𝑀(𝐴,𝐵)}(𝑥)≥𝑣⁻_{𝑀(𝐴,𝐵)}(𝑥), 𝑢⁺_{𝑀(𝐴,𝐵)}(𝑥)≥𝑣⁺_{𝑀(𝐴,𝐵)}(𝑥), with the definition of 𝑀(𝐴,𝐵). Then we can obtain 𝐸(𝑀(=1𝐴,𝐵))𝑛^𝑛_𝑗=1min𝑢⁻_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑣⁻_{𝑀()𝐴,𝐵}𝑥_𝑗+min𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑣⁺_{𝑀()𝐴,𝐵}𝑥_𝑗+𝜋⁻_{𝑀()𝐴,𝐵}𝑥_𝑗+𝜋⁺_{𝑀()𝐴,𝐵}𝑥_𝑗×max𝑢⁻_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑣⁻_{𝑀()𝐴,𝐵}𝑥_𝑗+max𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗,𝑣⁺_{𝑀()𝐴,𝐵}𝑥_𝑗+𝜋⁻_{𝑀()𝐴,𝐵}𝑥_𝑗+𝜋⁺_{𝑀()𝐴,𝐵}𝑥_𝑗⁻¹=1𝑛^𝑛_𝑗=1𝑣⁻_{𝑀()𝐴,𝐵}𝑥_𝑗+𝑣⁺_{𝑀()𝐴,𝐵}𝑥_𝑗+𝜋⁻_{𝑀()𝐴,𝐵}𝑥_𝑗+𝜋⁺_{𝑀()𝐴,𝐵}𝑥_𝑗×𝑢⁻_{𝑀()𝐴,𝐵}𝑥_𝑗+𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗+𝜋⁻_{𝑀()𝐴,𝐵}𝑥_𝑗+𝜋⁺_{𝑀()𝐴,𝐵}𝑥_𝑗⁻¹=1𝑛^𝑛_𝑗=12−𝑢⁻_{𝑀()𝐴,𝐵}𝑥_𝑗−𝑢⁺_{𝑀()𝐴,𝐵}𝑥_𝑗2−𝑣⁻_{𝑀()𝐴,𝐵}𝑥_𝑗−𝑣⁺_{𝑀()𝐴,𝐵}𝑥_𝑗=1𝑛^𝑛_𝑗=12−𝑀_𝐴𝐵1𝑥_𝑗−𝑀_𝐴𝐵2𝑥_𝑗2−𝑀_𝐴𝐵3(𝑥)−𝑀_𝐴𝐵4(𝑥). (31) If we let 𝑢⁻_𝑗=𝑢⁻_𝐴𝑥_𝑗−𝑢⁻_𝐵𝑥_𝑗,𝑢⁺_𝑗=𝑢⁺_𝐴𝑥_𝑗−𝑢⁺_𝐵𝑥_𝑗,𝑣⁻_𝑗=𝑣⁻_𝐴𝑥_𝑗−𝑣⁻_𝐵𝑥_𝑗,𝑣⁺_𝑗=𝑣⁺_𝐴𝑥_𝑗−𝑣⁺_𝐵𝑥_𝑗,𝜋⁻_𝑗=𝜋⁻_𝐴𝑥_𝑗+𝜋⁻_𝐵𝑥_𝑗,𝜋⁺_𝑗=𝜋⁺_𝐴𝑥_𝑗+𝜋⁺_𝐵𝑥_𝑗, (32) then 𝐸(𝑀(=1𝐴,𝐵))𝑛^𝑛_𝑗=14−𝑢⁻_𝑗+𝑢⁺_𝑗+𝑣⁻_𝑗+𝑣⁺_𝑗+𝜋⁻_𝑗+𝜋⁺_𝑗4+𝑢⁻_𝑗+𝑢⁺_𝑗+𝑣⁻_𝑗+𝑣⁺_𝑗+𝜋⁻_𝑗+𝜋⁺_𝑗. (33)
Now we let 𝑤={𝑤₁,𝑤₂,…,𝑤_𝑛} be the weight vector of elements 𝑥_𝑗(𝑗=1,2,…,𝑛); the weighted similarity measure can be written by the form 𝐸(𝑀(𝐴,𝐵))=^𝑛_𝑗=1𝑤_𝑗4−𝑢⁻_𝑗+𝑢⁺_𝑗+𝑣⁻_𝑗+𝑣⁺_𝑗+𝜋⁻_𝑗+𝜋⁺_𝑗4+𝑢⁻_𝑗+𝑢⁺_𝑗+𝑣⁻_𝑗+𝑣⁺_𝑗+𝜋⁻_𝑗+𝜋⁺_𝑗. (34)
The function 𝑆 defined by (30) is a similarity measure on IVIFS(𝑋), from Theorem 5.

3.2. Comparison with Some Existing Similarity Measures

Example 8. We test the new similarity measure by Example 4 given in Section 2.3. Consider 𝐴={𝑥,0.5,0.5}, 𝐵={𝑥,0.3,0.7}, and 𝐶={𝑥,0.3,0.3}. From (18), we can have 𝑀_𝐴𝐵1=𝑀_𝐴𝐵2=0.25∗2+𝑢_𝐴(𝑥)−𝑢_𝐵(𝑥)+𝑣_𝐴(𝑥)−𝑣_𝐵(𝑥−𝜋_𝐴)(𝑥)+𝜋_𝐵(𝑥)𝑀=0.6,_𝐴𝐵3=𝑀_𝐴𝐵4=0.25∗2−𝑢_𝐴(𝑥)−𝑢_𝐵(𝑥)+𝑣_𝐴(𝑥)−𝑣_𝐵(𝑥−𝜋_𝐴)(𝑥)+𝜋_𝐵(𝑥)=0.4. (35)
Then we can get 𝑀(𝐴,𝐵)={𝑥,0.6,0.4}. The similarity of 𝐴 and 𝐵 is 𝑆()(𝑀(=2𝐴,𝐵=𝐸𝐴,𝐵))3, (36) by (30). In the same way, we can easily get 𝑀()={}()(𝑀(=5𝐴,𝐶𝑥,0.5,0.3,𝑆𝐴,𝐶=𝐸𝐴,𝐶))7. (37)
So we can get the right answer that 𝐴 is more similar to 𝐶 than to 𝐵; that is, 𝑆(𝐴,𝐵)<𝑆(𝐴,𝐶). While using the similarity measures 𝑆₁, 𝑆₂, 𝑆₃, 𝑆₄, and 𝑆₅, we can get the result 𝑆(𝐴,𝐵)=𝑆(𝐴,𝐶) from Section 2.3. It shows that the new similarity measure is much more reasonable than the 𝑆₁, 𝑆₂, 𝑆₃, 𝑆₄, 𝑆₅.
In [32], Hung and Yang proposed three similarity measures between IFSs. Let 𝐴,𝐵∈IFSs(𝑋) and 𝑑(𝐴,𝐵) is the distance between two IFSs. Consider 𝑆₆()(),𝑆𝐴,𝐵=1−𝑑𝐴,𝐵₇()=()𝐴,𝐵1−𝑑𝐴,𝐵(),𝑆1+𝑑𝐴,𝐵₈()=𝑒𝐴,𝐵^{−𝑑(𝐴,𝐵)}−𝑒⁻¹1−𝑒⁻¹, (38) where 𝑑(𝐴,𝐵)=(1/𝑛)∑^𝑛_𝑗=1max{|𝑢_𝐴(𝑥_𝑗)−𝑢_𝐵(𝑥_𝑗)|,|𝑣_𝐴(𝑥_𝑗)−𝑣_𝐵(𝑥_𝑗)|}.

The following example shows some drawbacks of formulas (38).

Example 9. Let 𝐴={𝑥,0.5,0.5}, 𝐵={𝑥,0.4,0.6}, and 𝐶={𝑥,0.4,0.5} be three IFSs.
Intuitively, one can see that 𝐴 is more similar to 𝐶 than to 𝐵. However, using the similarity measures given by (38), we have 𝑑()|,|𝑑()|,|𝐴,𝐵=max{|0.5−0.40.5−0.6|}=0.1,𝐴,𝐶=max{|0.5−0.40.5−0.5|}=0.1. (39)
So 𝑆₆(𝐴,𝐵)=𝑆₆(𝐴,𝐶), 𝑆₇(𝐴,𝐵)=𝑆₇(𝐴,𝐶), 𝑆₈(𝐴,𝐵)=𝑆₈(𝐴,𝐶), because of 𝑑(𝐴,𝐵)=𝑑(𝐴,𝐶). Now we can calculate the similarity measures 𝑆(𝐴,𝐵) and 𝑆(𝐴,𝐶) by formula (30). We can get 𝑀_𝐴𝐵1=𝑀_𝐴𝐵2()𝑀=0.25∗2+0.1+0.1−0=0.55,_𝐴𝐵3=𝑀_𝐴𝐵4()𝑀()={},𝑆()(𝑀(=9=0.25∗2−0.1−0.1−0=0.45,𝐴,𝐵𝑥,0.55,0.45𝐴,𝐵=𝐸𝐴,𝐵)),𝑀11_𝐴𝐶1=𝑀_𝐴𝐶2()𝑀=0.25∗2+0+0.1−0.1=0.5,_𝐴𝐶3=𝑀_𝐴𝐶4()𝑀()={},𝑆()(𝑀(==0.25∗2−0−0.1−0.1=0.45,𝐴,𝐶𝑥,0.5,0.45𝐴,𝐶=𝐸𝐴,𝐶))10,𝑆()(),11𝐴,𝐶>𝑆𝐴,𝐵 (40) which means that 𝐴 is more similar to 𝐶 than to 𝐵, consistent with the intuition. Therefore, the new similarity measure (30) is proved to be more reasonable than 𝑆₆,𝑆₇,𝑆₈ in some cases.
Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} and 𝐴,𝐵∈IFSs(𝑋). Chen [33] presented the following similarity measures between IFSs𝐴 and 𝐵: 𝑆₉()∑𝐴,𝐵=1−^𝑛𝑗=1𝑢_𝐴𝑥_𝑗−𝑣_𝐴𝑥_𝑗−𝑢_𝐵𝑥_𝑗−𝑣_𝐵𝑥_𝑗,𝑆2𝑛₁₀(𝐴,𝐵)=1−^𝑞∑^𝑛_𝑗=1𝜓_𝐴𝑥_𝑗−𝜓_𝐵𝑥_𝑗^𝑞𝑛, (41) where 1≤𝑞≤∞, and, for each 𝑗, 𝜓_𝐴𝑥_𝑗=0.5𝑢_𝐴𝑥_𝑗−𝑣_𝐴𝑥_𝑗,𝜓+1_𝐵𝑥_𝑗=0.5𝑢_𝐵𝑥_𝑗−𝑣_𝐵𝑥_𝑗+1. (42)

The following example indicates that the above similarity measures 𝑆₉, 𝑆₁₀ are also not reasonable in some cases.

Example 10. Let 𝐴={𝑥,0.5,0.5}, 𝐵={𝑥,0.2,0.2}, and 𝐶={𝑥,0.49,0.51} be three IFSs. It is obvious that 𝐴 is more similar to 𝐶 than to 𝐵.
We calculate the degree of similarity between 𝐴 and 𝐵, as well as 𝐴 and 𝐶 using the new similarity measure given by (30). Then 𝑆(𝐴,𝐵)=5/6, 𝑆(𝐴,𝐶)=99/101, which show that 𝐴 is more similar to 𝐶 than to 𝐵 and is consistent with intuition. But if we use formula (41) to calculate the similarity measure, then 𝑆₉()|(𝑆𝐴,𝐵=1−0.5∗0.5−0.5−0.2−0.2)|=1,₉()|(𝜓𝐴,𝐶=1−0.5∗0.5−0.5−0.49−0.51)|=0.99,_𝐴𝑥_𝑗=0.5,𝜓_𝐵𝑥_𝑗=0.5,𝜓_𝐶𝑥_𝑗𝑆=0.49,₁₀(𝐴,𝐵)=1−^𝑞|0.5−0.5|^𝑞1𝑆=1,₁₀(𝐴,𝐶)=1−^𝑞|0.5−0.49|^𝑞1𝑆=0.99,₉(𝐴,𝐵)>𝑆₉(𝐴,𝐶),𝑆₁₀(𝐴,𝐵)>𝑆₁₀(𝐴,𝐶), (43) which is not reasonable. So the new similarity measure is illustrated to be more effective than formula (41).
Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} and 𝐴,𝐵∈IFSs(𝑋). Xu [28] gave many similarity measures based on the distance measure. The weight of each element is 𝑤_𝑗, 𝑗=1,2,…,𝑛. Consider 𝑆₁₁()1𝐴,𝐵=1−2^𝑛_𝑗=1𝑤_𝑗𝑢_𝐴𝑥_𝑗−𝑢_𝐵𝑥_𝑗^𝛼+𝑣_𝐴𝑥_𝑗−𝑣_𝐵𝑥_𝑗^𝛼+𝜋_𝐴𝑥_𝑗−𝜋_𝐵𝑥_𝑗^𝛼1/𝛼,𝑆₁₂(𝐴,𝐵)=1−^𝑛_𝑗=1𝑤_𝑗𝑢_𝐴𝑥_𝑗−𝑢_𝐵𝑥_𝑗^𝛼+𝑣_𝐴𝑥_𝑗−𝑣_𝐵𝑥_𝑗^𝛼+𝜋_𝐴𝑥_𝑗−𝜋_𝐵𝑥_𝑗^𝛼×^𝑛_𝑗=1𝑤_𝑗𝑢_𝐴𝑥_𝑗+𝑢_𝐵𝑥_𝑗^𝛼+𝑣_𝐴𝑥_𝑗+𝑣_𝐵𝑥_𝑗^𝛼+𝜋_𝐴𝑥_𝑗+𝜋_𝐵𝑥_𝑗^{𝛼−11/𝛼},𝑆₁₃()=𝐴,𝐵_𝑛𝑗=1𝑤_𝑗min𝑢_𝐴𝑥_𝑗,𝑢_𝐵𝑥_𝑗+min𝑣_𝐴𝑥_𝑗,𝑣_𝐵𝑥_𝑗+min𝜋_𝐴𝑥_𝑗,𝜋_𝐵𝑥_𝑗×^𝑛_𝑗=1𝑤_𝑗max𝑢_𝐴𝑥_𝑗,𝑢_𝐵𝑥_𝑗+max𝑣_𝐴𝑥_𝑗,𝑣_𝐵𝑥_𝑗+max𝜋_𝐴𝑥_𝑗,𝜋_𝐵𝑥_𝑗⁻¹,𝑆₁₄()=𝐴,𝐵_𝑛𝑗=1𝑤_𝑗𝑢_𝐴𝑥_𝑗∗𝑢_𝐵𝑥_𝑗+𝑣_𝐴𝑥_𝑗∗𝑣_𝐵𝑥_𝑗+𝜋_𝐴𝑥_𝑗∗𝜋_𝐵𝑥_𝑗×max^𝑛_𝑗=1𝑤_𝑗𝑢²_𝐴𝑥_𝑗+𝑣²_𝐴𝑥_𝑗+𝜋²_𝐴𝑥_𝑗,_𝑛𝑗=1𝑤_𝑗𝑢²_𝐵𝑥_𝑗+𝑣²_𝐵𝑥_𝑗+𝜋²_𝐵𝑥_𝑗⁻¹. (44)

Example 11. Let 𝐴={𝑥,0.5,0.5}, 𝐵={𝑥,0.3,0.7}, and 𝐶={𝑥,0.3,0.4} be three IFSs.
For IFSs𝐴,𝐵, and 𝐶, we have the following results using the similarity measures 𝑆₁₁, 𝑆₁₂, 𝑆₁₃, 𝑆₁₄: 𝑆₁₁(𝐴,𝐵)=0.8,𝑆₁₁()𝑆𝐴,𝐶=0.735,₁₂(𝐴,𝐵)=0.804,𝑆₁₂()𝑆𝐴,𝐶=0.698,₁₂(𝐴,𝐵)=0.804,𝑆₁₂()𝑆𝐴,𝐶=0.698,₁₄(𝐴,𝐵)=0.862,𝑆₁₄(𝐴,𝐶)=0.7. (45) It indicates that 𝐴 is more similar to 𝐵 than to 𝐶. As the membership of 𝐵 and 𝐶 is the same, we only analyse the change of the membership. Comparing 𝐴 with 𝐵, the change degree of nonmembership is from 0.5 to 0.7. While comparing 𝐶 with 𝐴, the change degree of nonmembership is from 0.4 to 0.5. Hence, intuitively, 𝐴 is more similar to 𝐶 than to 𝐵 and the results of 𝑆₁₁, 𝑆₁₂, 𝑆₁₃, 𝑆₁₄ are not reasonable. Now we calculate the similarity measures 𝑆(𝐴,𝐵) and 𝑆(𝐴,𝐶) by formula (30). Then 𝑆(𝐴,𝐵)=2/3 and 𝑆(𝐴,𝐶)=19/27, which indicated that 𝐴 is more similar to 𝐶 than to 𝐵 and is consistent with intuition.
Later, Xia and Xu [29] also proposed the following similarity measures. Xu and Yager [27] named 𝛼_𝑟=(𝑢_𝛼𝑟,𝑣_𝛼𝑟) an intuitionistic fuzzy value (IFV). Let 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛) and 𝛽=(𝛽₁,𝛽₂,…,𝛽_𝑛) be two collections of IFVs and 𝜅,𝜆∈[0,1], 𝑡_𝑎𝑗=𝑢_𝛼𝑗−𝑢_𝛽𝑗, 𝑡_𝑏𝑗=𝑣_𝛼𝑗−𝑣_𝛽𝑗, 𝑡_𝑐𝑗=𝜋_𝛼𝑗−𝜋_𝛽𝑗=−𝑡_𝑎𝑗−𝑡_𝑏𝑗, 𝑗=(1,2,…,𝑛). Then the author defined the similarity measures as 𝑆^𝜂_𝐷𝜅𝛼,𝛽=1−^𝑛_𝑗=1𝑤_𝑗(1−𝜅)𝑡_𝑎𝑗−𝜅𝑡_𝑏𝑗^𝜂1/𝜂. (46) For 𝜅+𝜆≤1, 𝑆^𝜂_{𝐹𝜅,𝜆}1𝛼,𝛽=1−2^𝑛_𝑗=1𝑤_𝑗(1−𝜅)𝑡_𝑎𝑗−𝜅𝑡_𝑏𝑗^𝜂+(1−𝜆)𝑡_𝑎𝑗−𝜆𝑡_𝑏𝑗^𝜂+(1−𝜅−𝜆)𝑡_𝑎𝑗+𝑡_𝑏𝑗^𝜂1/𝜂,𝑆^𝜂_{𝐺𝜅,𝜆}1𝛼,𝛽=1−2^𝑛_𝑗=1𝑤_𝑗𝜅𝑡_𝑎𝑗^𝜂+𝜆𝑡_𝑏𝑗^𝜂+𝜅𝑡_𝑎𝑗+𝜆𝑡_𝑏𝑗^𝜂1/𝜂,𝑆^𝜂_{𝐻𝜅,𝜆}1𝛼,𝛽=1−2^𝑛_𝑗=1𝑤_𝑗𝜅𝑡_𝑎𝑗^𝜂+𝜆𝑡_𝑎𝑗−(1−𝜆)𝑡_𝑏𝑗^𝜂+(𝜅−𝜆)𝑡_𝑎𝑗+(1−𝜆)𝑡_𝑏𝑗^𝜂1/𝜂,𝑆^𝜂_{𝐻∗𝜅,𝜆}1𝛼,𝛽=1−2^𝑛_𝑗=1𝑤_𝑗𝜅𝑡_𝑎𝑗^𝜂+𝜅𝜆𝑡_𝑎𝑗−(1−𝜆)𝑡_𝑏𝑗^𝜂+(1−𝜆)𝜅𝑡_𝑎𝑗+𝑡_𝑏𝑗^𝜂1/𝜂,𝑆^𝜂_{𝐽𝜅,𝜆}1𝛼,𝛽=1−2^𝑛_𝑗=1𝑤_𝑗(1−𝜅)𝑡_𝑎𝑗−𝜅𝑡_𝑏𝑗^𝜂+𝜆𝑡_𝑏𝑗+(1−𝜅)𝑡_𝑎𝑗+(𝜆−𝜅)𝑡_𝑏𝑗^𝜂1/𝜂,𝑆^𝜂_{𝐽∗𝜅,𝜆}1𝛼,𝛽=1−2^𝑛_𝑗=1𝑤_𝑗(1−𝜅)𝑡_𝑎𝑗−𝜅𝜆𝑡_𝑏𝑗^𝜂+𝜆𝑡_𝑏𝑗+(1−𝜅)𝑡_𝑎𝑗+𝜆𝑡_𝑏𝑗^𝜂1/𝜂. (47) For 𝜅+𝜆≤1, 𝑆^𝜂_{𝑃𝜅,𝜆}1𝛼,𝛽=1−2^𝑛_𝑗=1𝑤_𝑗max𝜅,𝑢_𝛼𝑗−max𝜅,𝑢_𝛽𝑗^𝜂+min𝜆,𝑣_𝛼𝑗−min𝜆,𝑣_𝛽𝑗^𝜂+max𝜅,𝑢_𝛼𝑗−max𝜅,𝑢_𝛽𝑗+min𝜆,𝑣_𝛼𝑗−min𝜆,𝑣_𝛽𝑗^𝜂1/𝜂. (48) For 𝜅+𝜆≤1, 𝑆^𝜂_{𝑄𝜅,𝜆}1𝛼,𝛽=1−2^𝑛_𝑗=1𝑤_𝑗min𝜅,𝑢_𝛼𝑗−min𝜅,𝑢_𝛽𝑗^𝜂+max𝜆,𝑣_𝛼𝑗−max𝜆,𝑣_𝛽𝑗^𝜂+min𝜅,𝑢_𝛼𝑗−min𝜅,𝑢_𝛽𝑗+max𝜆,𝑣_𝛼𝑗−max𝜆,𝑣_𝛽𝑗^𝜂1/𝜂. (49)

Example 12. Let 𝐴={𝑥,0.3,0.3}, 𝐵={𝑥,0.4,0.4}, and 𝐶={𝑥,0.4,0.5} be three IFVs. It is clear to see that 𝐴 is much more similar to 𝐵 than to 𝐶. However, using the similarity measure (46), we can get 𝑆⁵_𝐷0.2(𝐴,𝐵)=0.94, 𝑆⁵_𝐷0.2(𝐴,𝐶)=0.96. Hence, 𝑆⁵_𝐷0.2(𝐴,𝐵)<𝑆⁵_𝐷0.2(𝐴,𝐶), which is not reasonable. If we use the new similarity measure (30), we have 𝑆(𝐴,𝐵)=0.8571, 𝑆(𝐴,𝐶)=0.7857. Therefore, the new similarity measure is illustrated to be more reasonable than 𝑆^𝜂_𝐷𝜅.

Example 13. Let 𝐴={𝑥,0.5,0.5}, 𝐵={𝑥,0.2,0.4}, and 𝐶={𝑥,0.2,0.65} be three IFVs. We can get the following results using formula (47). We adopt the values of 𝜂,𝜅,𝜆 from [31] 𝑆⁵_𝐹0.2,0.1(𝐴,𝐵)=0.7203,𝑆⁵_𝐹0.2,0.1()𝑆𝐴,𝐶=0.7219,_5𝐺0.7,0.8(𝐴,𝐵)=0.7381,𝑆⁵_𝐺0.7,0.8()𝑆𝐴,𝐶=0.8145,_5𝐻0.8,0.3(𝐴,𝐵)=0.7691,𝑆⁵_𝐻0.8,0.3()𝑆𝐴,𝐶=0.778,_{5𝐻∗0.9,0.2}(𝐴,𝐵)=0.7158,𝑆⁵_𝐻^∗_0.9,0.2()𝑆𝐴,𝐶=0.7597,_5𝐽0.1,0.6(𝐴,𝐵)=0.704,𝑆⁵_𝐽0.1,0.6()𝑆𝐴,𝐶=0.7447,_{5𝐽∗0.2,0.7}(𝐴,𝐵)=0.7301,𝑆⁵_𝐽^∗_0.2,0.7(𝐴,𝐶)=0.769. (50) For similarity measures (48)-(49), the results are varying when the values of 𝜅,𝜆 are different. For example, let 𝜅=0; we can get the following: 𝑆^𝜂_{𝑃𝜅,𝜆}(𝐴,𝐵)≤𝑆^𝜂_{𝑃𝜅,𝜆}()𝑆𝐴,𝐶,if𝜆≤0.6,_{𝜂𝑃𝜅,𝜆}(𝐴,𝐵)>𝑆^𝜂_{𝑃𝜅,𝜆}()𝑆𝐴,𝐶,if𝜆>0.6,_{𝜂𝑄𝜅,𝜆}(𝐴,𝐵)≥𝑆^𝜂_{𝑄𝜅,𝜆}()𝑆𝐴,𝐶,if𝜆≤0.65,_{𝜂𝑄𝜅,𝜆}(𝐴,𝐵)=𝑆^𝜂_{𝑄𝜅,𝜆}(𝐴,𝐶),if𝜆>0.65. (51)
If we use formula (30) proposed in this paper, we can get 𝑆(𝐴,𝐵)=0.7143 and 𝑆(𝐴,𝐶)=0.6642. Furthermore, the change of the membership from 𝐴 to 𝐵 and 𝐴 to 𝐶 is the same. But the change degree of nonmembership is different. Comparing 𝐴 with 𝐵, the change is from 0.5 to 0.4 which is smaller than that comparing 𝐴 with 𝐶. Accordingly, 𝐴 is much more similar to 𝐵 than to 𝐶 and our similarity measure is more effective in some cases.

4. Application to Expert System

IVIFS is a very suitable tool to the expert system to process the imperfect information. We apply the new similarity measure defined by (30) to some applications.

4.1. Pattern Recognition

Step 1. Suppose that there exist 𝑚 patterns which are represented by IVIFSs for a pattern recognition problem 𝐴_𝑖=𝑥_𝑗,𝑢⁻_𝐴𝑖𝑥_𝑗,𝑢⁺_𝐴𝑖𝑥_𝑗,𝑣⁻_𝐴𝑖𝑥_𝑗,𝑣⁺_𝐴𝑖𝑥_𝑗∣𝑥_𝑗∈𝑋, (52) for 𝑖=1,2,…,𝑚, in the feature space 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛}, and suppose that there is a sample to be recognized which is represented by IVIFSs 𝐵=𝑥_𝑗,𝑢⁻_𝐵𝑥_𝑗,𝑢⁺_𝐵𝑥_𝑗,𝑣⁻_𝐵𝑥_𝑗,𝑣⁺_𝐵𝑥_𝑗∣𝑥_𝑗∈𝑋. (53)

Step 2. Calculate the similarity degree 𝑆(𝐴_𝑖,𝐵) between 𝐴_𝑖 and 𝐵 by formula (30).

Step 3. Select the largest 𝑆(𝐴_𝑘,𝐵), from 𝑆(𝐴_𝑖,𝐵),𝑖=1,2,…,𝑚. According to the principle of maximum similarity degree between IVIFSs, the sample 𝐵 belongs to the pattern 𝐴_𝑘.

Example 14 14 (see [32]). Assume that there are three patterns denoted by IFSs in 𝑋={𝑥₁,𝑥₂,𝑥₃}𝐴₁=𝑥₁,0.1,0.1,𝑥₂,0.5,0.1,𝑥₃,𝐴,0.1,0.9₂=𝑥₁,0.5,0.5,𝑥₂,0.7,0.3,𝑥₃,𝐴,0.0,0.8₃=𝑥₁,0.7,0.2,𝑥₂,0.1,0.8,𝑥₃,0.4,0.4. (54)
Assume that a sample 𝐵={(𝑥₁,0.4,0.4),(𝑥₂,0.6,0.2),(𝑥₃,0.0,0.8)} is given and let the weight vector 𝑤=(1/3,1/3,1/3)^𝑇 and 𝑞=2. According to formula (30) and the similarity measures 𝑆₉,𝑆₁₀ proposed by Chen [33], the following results can be obtained: 𝑆𝐴₁,𝐵=0.7857,𝑆𝐴₂𝑆𝐴,𝐵=0.8889,₃𝑆,𝐵=0.5097,₉𝐴₁,𝐵=1,𝑆₉𝐴₂𝑆,𝐵=1,₉𝐴₃𝑆,𝐵=0.6,₁₀𝐴₁,𝐵=1,𝑆₁₀𝐴₂𝑆,𝐵=1,₁₀𝐴₃,𝐵=0.5817. (55)
We can see that the similarity measures 𝑆₉,𝑆₁₀ cannot recognize which pattern 𝐵 belongs to. Based on the principle of maximum degree of similarity, it is obvious that the sample 𝐵 belongs to 𝐴₂ using our similarity measure and the results are the same with some other similarity measures [32].

Example 15 15 (see [34]). We discuss the medical diagnosis problem. Let us consider a set of diagnoses 𝐴={𝐴₁(Viralfever),𝐴₂(Malaria),𝐴₃(Typhoid)} as well as a set of symptoms 𝑋={𝑥₁(Temprature),𝑥₂(Headache),𝑥₃(Cough)}. Assume that a patient can be represented as follows: 𝐵=𝑥₁,[0.6,0.8],[0.1,0.2],𝑥₂,[],[],𝑥0.3,0.70.2,0.3₃,[0.6,0.8],[0.1,0.2]. (56) Let the weight vector 𝑤=(1/3,1/3,1/3)^𝑇 and each diagnosis can be denoted by IVIFSs as follows: 𝐴₁=𝑥₁,[0.4,0.5],[0.3,0.4],𝑥₂,[],[],𝑥0.4,0.60.2,0.4₃,[],[],𝐴0.4,0.80.1,0.2₂=𝑥₁,[0.3,0.6],[0.3,0.4],𝑥₂,[],[],𝑥0.5,0.60.3,0.4₃,[],[],𝐴0.4,0.50.1,0.3₃=𝑥₁,[0.7,0.8],[0.1,0.2],𝑥₂,[],[],𝑥0.6,0.70.1,0.3₃,[0.3,0.4],[0.1,0.2]. (57) Our goal is to categorize the patient 𝐵 in one of the classes 𝐴₁,𝐴₂,𝐴₃. Then, we can have the following results based on formula (30) and (10)–(13): 𝑆𝐴₁,𝐵=0.8366,𝑆𝐴₂𝑆𝐴,𝐵=0.7586,₃𝑆,𝐵=0.8577.₂𝐴₁,𝐵=0.8834,𝑆𝐴₂𝑆𝐴,𝐵=0.8334,₃𝑆,𝐵=0.9.₃𝐴₁,𝐵=0.517,𝑆₃𝐴₂𝑆,𝐵=0.4084,₃𝐴₃𝑆,𝐵=0.8292.₄𝐴₁,𝐵=0.8,𝑆₄𝐴₂𝑆,𝐵=0.7334,₄𝐴₃𝑆,𝐵=0.7334.₅𝐴₁,𝐵=0.784,𝑆₅𝐴₂𝑆,𝐵=0.7292,₅𝐴₃,𝐵=0.7056. (58)
Based on the recognition principle of the maximum similarity degree between the IVIFSs, we can get result that the similarity degree between 𝐴₃ and 𝐵 is the largest using formula (30), (10), and (11). Consider 𝑆𝐴₃,𝐵=max𝑆𝐴₁,𝐵,𝑆𝐴₂,𝐵,𝑆𝐴₃,𝑆,𝐵₂𝐴₃,𝐵=max𝑆₂𝐴₁,𝐵,𝑆₂𝐴₂,𝐵,𝑆₂𝐴₃,𝑆,𝐵₃𝐴₃,𝐵=max𝑆₃𝐴₁,𝐵,𝑆₃𝐴₂,𝐵,𝑆₃𝐴₃,𝐵. (59)
It is the right answer that 𝐵 belongs to 𝐴₃, which is the same as in [34]. But the similarity measures 𝑆₄,𝑆₅ would have the wrong answer. Thus, we can assign the patient to diagnosis 𝐴₃ according to the recognition of principle. So we can diagnose that the illness of the patient is typhoid.
The application of pattern recognition in medical diagnosis problem can be very significant. We can take advantage of the similarity measure to help us to diagnose the patient based on the database of the symptoms of illnesses. So it can help the patients much faster to know their illness, as well as alleviating the burden of doctors.

Example 16 (see [24]). Assume that there are four classes of building materials 𝐴={𝐴₁(poor),𝐴₂(average),𝐴₃(good),𝐴₄(excellent)} that are written in the form of IVIFSs 𝐴_𝑖={(𝑥_𝑗,[𝑢⁻_𝐴𝑖(𝑥_𝑗),𝑢⁺_𝐴𝑖(𝑥_𝑗)],[𝑣⁻_𝐴𝑖(𝑥_𝑗),𝑣⁺_𝐴𝑖(𝑥_𝑗)])∣𝑥_𝑗∈𝑋}(𝑖=1,2,3,4) in the future space 𝑋={𝑥₁,𝑥₂,…,𝑥₁₂} that denote twelve different indicators. And the weight vector 𝑤 is as follows: ()𝑤=0.1,0.05,0.08,0.06,0.03,0.07,0.09,0.12,0.15,0.07,0.13,0.05^𝑇. (60)
And there are an unknown building material 𝐵 and four classes of building materials represented by IVIFSs with respect to all indicators as follows: 𝐴₁=𝑥₁,[0.1,0.2],[0.5,0.6],𝑥₂,[],[],𝑥0.1,0.20.7,0.8₃,[0.5,0.6],[0.3,0.4],𝑥₄,[],[],𝑥0.8,0.90.0,0.1₅,[0.4,0.5],[0.3,0.4],𝑥₆,[],[],𝑥0.0,0.10.8,0.9₇,[0.3,0.4],[0.5,0.6],𝑥₈,[],[],𝑥1.0,1.00.0,0.0₉,[0.2,0.3],[0.6,0.7],𝑥₁₀,[],[],𝑥0.4,0.50.4,0.5₁₁,[0.7,0.8],[0.1,0.2],𝑥₁₂,[],[],𝐴0.4,0.50.4,0.5₂=𝑥₁,[0.5,0.6],[0.3,0.4],𝑥₂,[],[],𝑥0.6,0.70.1,0.2₃,[1.0,1.0],[0.0,0.0],𝑥₄,[],[],𝑥0.1,0.20.6,0.7₅,[0.0,0.1],[0.8,0.9],𝑥₆,[],[],𝑥0.7,0.80.1,0.2₇,[0.5,0.6],[0.3,0.4],𝑥₈,[],[],𝑥0.6,0.70.2,0.3₉,[1.0,1.0],[0.0,0.0],𝑥₁₀,[],[],𝑥0.1,0.20.7,0.8₁₁,[0.0,0.1],[0.8,0.9],𝑥₁₂,[],[],𝐴0.7,0.80.1,0.2₃=𝑥₁,[0.4,0.5],[0.3,0.4],𝑥₂,[],[],𝑥0.6,0.70.2,0.3₃,[0.9,1.0],[0.0,0.0],𝑥₄,[],[],𝑥0.0,0.10.8,0.9₅,[0.0,0.1],[0.8,0.9],𝑥₆,[],[,𝑥0.6,0.70.2,0.3]₇,[0.1,0.2],[0.7,0.8],𝑥₈,[],[],𝑥0.2,0.30.6,0.7₉,[0.5,0.6],[0.2,0.4],𝑥₁₀,[],[],𝑥1.0,1.00.0,0.0₁₁,[0.3,0.4],[0.4,0.5],𝑥₁₂,[],[,𝐴0.0,0.10.8,0.9]₄=𝑥₁,[1.0,1.0],[0.0,0.0],𝑥₂,[],[],𝑥1.0,1.00.0,0.0₃,[0.8,0.9],[0.0,0.1],𝑥₄,[],[],𝑥0.7,0.80.1,0.2₅,[0.0,0.1],[0.7,0.9],𝑥₆,[],[,𝑥0.0,0.10.8,0.9]₇,[0.1,0.2],[0.7,0.8],𝑥₈,[],[],𝑥0.1,0.20.7,0.8₉,[0.4,0.5],[0.3,0.4],𝑥₁₀,[],[],𝑥1.0,1.00.0,0.0₁₁,[0.3,0.4],[0.4,0.5],𝑥₁₂,[,𝑥0.0,0.1][0.8,0.9]𝐵=₁,[0.9,1.0],[0.0,0.0],𝑥₂,[],[],𝑥0.9,1.00.0,0.0₃,[0.7,0.8],[0.1,0.2],𝑥₄,[],[],𝑥0.6,0.70.1,0.2₅,[0.0,0.1],[0.8,0.9],𝑥₆,[],[],𝑥0.1,0.20.7,0.8₇,[0.1,0.2],[0.7,0.8],𝑥₈,[],[],𝑥0.1,0.20.7,0.8₉,[0.4,0.5],[0.3,0.4],𝑥₁₀,[],[],𝑥1.0,1.00.0,0.0₁₁,[0.3,0.4],[0.4,0.5],𝑥₁₂,[0.0,0.1],[0.7,0.9]. (61)
We have to recognize which class the unknown pattern 𝐵 belongs to. According to the recognition principle of maximum similarity degree between IVIFSs, the process of recognizing 𝐴_𝑗 to 𝐵 is derived based on 𝑗=argmax{𝑆(𝐴_𝑖,𝐵)}. Using the new similarity measure proposed in this paper and 𝑆₁, we can obtain 𝑆𝐴₁,𝐵=0.454,𝑆𝐴₂𝑆𝐴,𝐵=0.4157,₃,𝐵=0.7456,𝑆𝐴₄𝑆,𝐵=0.9594.₁𝐴₁,𝐵=0.4811,𝑆₁𝐴₂𝑆,𝐵=0.4272,₁𝐴₃,𝐵=0.7616,𝑆₁𝐴₄,𝐵=0.9563. (62)
It is obvious that the similarity degrees 𝑆(𝐴₄,𝐵), 𝑆₁(𝐴₄,𝐵) are the largest one in the four different similarity methods and so 𝐵 belongs to 𝐴₄. So we can know that the quality of the building material is excellent. It can be very useful and easy for us to detect the quality of building material with the help of the similarity measure.

4.2. Multicriteria Group Decision Making

For a decision making problem, let 𝐴={𝑎₁,𝑎₂,…,𝑎_𝑚} be a set of alternatives, let 𝐶={𝑐₁,𝑐₂,…,𝑐_𝑛} be a set of criteria, let 𝑊={𝑤₁,𝑤₂,…,𝑤_𝑛} be a set of criteria weights, and let 𝐷𝑊={𝑑𝑤₁,𝑑𝑤₂,…,𝑑𝑤_𝑙} be a set of decision maker’s weights. We also have the condition that ∑^𝑛_𝑖=1𝑤_𝑖=1,𝑤_𝑖∈[0,1] and ∑^𝑙_𝑖=1𝑑𝑤_𝑖=1,𝑑𝑤_𝑖∈[0,1].

The decision making procedure designed to find the best alternative is given by the following steps.

Step 1. The evaluation of the alternative 𝑎_𝑖 with respect to the criterion 𝑐_𝑗 is an intuitionistic fuzzy number represented by 𝑎_𝑖𝑗=⟨𝑐_𝑗,[𝑢⁻_𝑖𝑗,𝑢⁺_𝑖𝑗],[𝑣⁻_𝑖𝑗,𝑣⁺_𝑖𝑗]⟩. In this case, the alternative 𝑎_𝑖 is presented by the following IVIFS: 𝑎_𝑖=𝑐_𝑗,𝑢⁻_𝑖𝑗,𝑢⁺_𝑖𝑗,𝑣⁻_𝑖𝑗,𝑣⁺_𝑖𝑗∣𝑐_𝑗∈𝐶, (63) where 0≤𝑢⁺_𝑖𝑗+𝑣⁺_𝑖𝑗≤1, 0≤𝑢⁻_𝑖𝑗≤𝑢⁺_𝑖𝑗≤1, 0≤𝑣⁻_𝑖𝑗≤𝑣⁺_𝑖𝑗≤1, 𝑗=1,2,…,𝑛, 𝑖=1,2,…,𝑚.

Step 2. We define an ideal IVIFS for each criterion in the ideal alternative 𝑎^∗ as 𝑎_𝑖𝑗=⟨𝑐_𝑗,[1,1],[0,0]⟩ for “excellence.” Then, by considering criteria weights and applying formula (3), we can gain the weighted similarity measure between the ideal alternative 𝑎^∗ and alternative 𝑎_𝑖(𝑖=1,2,…,𝑚) for each decision maker: 𝑊𝑆_𝑘𝑎_𝑖,𝑎^∗=^𝑛_𝑗=12−min𝜇⁻_𝑖,𝑣⁻_𝑖−min𝜇⁺_𝑖,𝑣⁺_𝑖2+max𝜇⁻_𝑖,𝑣⁻_𝑖+max𝜇⁺_𝑖,𝑣⁺_𝑖. (64)

Step 3. Calculate the last value of alternatives considering each decision maker’s evaluation 𝑊𝑆𝑎_𝑖,𝑎^∗=^𝑙_𝑘=1𝑊𝑆_𝑘𝑎_𝑖,𝑎^∗. (65)

Step 4. Determine the order of alternatives. The larger the value of 𝑊𝑆(𝑎_𝑖,𝑎^∗) is, the better the alternative is.

We compare it with the ideal point method [35] based on interval-valued intuitionistic fuzzy sets used in the multicriteria group decision making. As the ideal point method [35] fits multiperiod and one decision maker, we should improve the method to solve our decision problem which is just one period and multidecision maker. The procedure of ideal point method is as follows.

Step 1. Utilize the UDIFWA operator: 𝑟_𝑖𝑗=([𝑢⁻_𝑖𝑗,𝑢⁺_𝑖𝑗],[𝑣⁻_𝑖𝑗,𝑣⁺_𝑖𝑗],[1−𝑢⁺_𝑖𝑗−𝑣⁺_𝑖𝑗,1−𝑢⁻_𝑖𝑗−𝑣⁻_𝑖𝑗]).

Step 2. Define 𝑎⁺ and 𝑎⁻ as the uncertain intuitionistic fuzzy ideal solution and the uncertain intuitionistic fuzzy negative ideal solution, respectively.

Step 3. Calculate the distance between the alternative 𝑎_𝑖 and 𝑎⁺, 𝑎⁻: 𝑑𝑟_𝑖,𝑎⁺=^𝑙_𝑘=1𝑑𝑤_𝑘^𝑚_𝑗=1𝑤_𝑗𝑑𝑟_𝑖𝑗,𝑎⁺,𝑑𝑟_𝑖,𝑎⁻=^𝑙_𝑘=1𝑑𝑤_𝑘^𝑚_𝑗=1𝑤_𝑗𝑑𝑟_𝑖𝑗,𝑎⁻. (66)

Step 4. Calculate the closeness coefficient of each alternative: 𝑐𝑎_𝑖=𝑑𝑟_𝑖,𝑎⁻𝑑𝑟_𝑖,𝑎⁻+𝑑𝑟_𝑖,𝑎⁺. (67)

Step 5. Choose the alternative 𝑎_𝑖 which has the largest values of closeness coefficient. The greater the value (𝑎_𝑖), the better the alternative 𝑎_𝑖.

Example 17 (see [17]). We consider an investment company, which wants to invest money in the best option. There is a panel with four possible alternatives to invest the money: (1)  𝑎₁ is a car company; (2)  𝑎₂ is a food company; (3)  𝑎₃ is a computer company; (4)  𝑎₄ is an arms company. The investment company must consider the following three criteria in order to make decision: (1)  𝑐₁ is the risk analysis; (2)  𝑐₂ is the growth analysis; (3)  𝑐₃ is the environmental impact analysis. The criteria are independent and the criteria weights comprise a vector 𝑊=[0.5,0.3,0.2]^′. The four possible alternatives 𝑎_𝑖(𝑖=1,2,3,4) are to be evaluated using the interval-valued intuitionistic fuzzy information by the decision maker under the three criteria (see Table 1).
Then we give the ideal IVIFS for each criterion in the ideal alternative 𝑎^∗ as 𝑎^∗={(𝑐₁,[1,1],[0,0]),(𝑐₂,[1,1],[0,0]),(𝑐₃,[1,1],[0,0])}. With formula (30), we can get the weighted similarity degree between the ideal alternative 𝑎^∗ and alternative 𝑎_𝑖(𝑖=1,2,3,4) as follows: 𝑊𝑆𝑎₁,𝑎^∗=0.4418,𝑊𝑆𝑎₂,𝑎^∗𝑎=0.6040,𝑊𝑆₃,𝑎^∗=0.4997,𝑊𝑆𝑎₄,𝑎^∗=0.6613. (68) Besides, 𝑘=1; that is, there is only one decision maker, so we can see that 𝑊𝑆𝑎₄,𝑎^∗>𝑊𝑆𝑎₂,𝑎^∗>𝑊𝑆𝑎₃,𝑎^∗>𝑊𝑆𝑎₁,𝑎^∗. (69) The order of the alternatives is 𝑎₄>𝑎₂>𝑎₃>𝑎₁. So we can get that the best alternative is 𝑎₄.
Using the ideal point method, we can obtain 𝑟₁=],[],[,],[],[,],[],[,𝑟{([0.4,0.50.3,0.40.1,0.3])([0.4,0.60.2,0.40,0.4])([0.1,0.30.5,0.60.1,0.4])}₂=],[],[,],[],[,],[],[,𝑟{([0.6,0.70.2,0.30,0.2])([0.6,0.70.2,0.30,0.2])([0.4,0.70.1,0.20.1,0.4])}₃=],[],[,],[],[,],[],[,𝑟{([0.3,0.60.3,0.40,0.4])([0.5,0.60.3,0.40,0.2])([0.5,0.60.1,0.30.1,0.4])}₄=],[],[,],[],[,],[],[,𝑎{([0.7,0.80.1,0.20,0.2])([0.6,0.70.1,0.30,0.3])([0.3,0.40.1,0.20.4,0.6])}₊=],[],[,],[],[,],[],[,𝑎{([1,10,00,0])([1,10,00,0])([1,10,00,0])}₋=],[],[,],[],[,],[],[.{([0,01,10,0])([0,01,10,0])([0,01,10,0])} (70) The results of the ideal point method are shown in Table 2. We can know that the results from the ideal point method are also the same as our measures.

Table 1 

The evaluation of the decision maker.

Table 2 

The results of ideal point method.

Example 18 (see [8]). In order to strengthen academic education and promote the quality of teaching, the school of management of a Chinese university wants to introduce oversea outstanding professor. The panel of the decision makers consist of the university president 𝑑𝑤₁, dean of the management school 𝑑𝑤₂, and human resource officer 𝑑𝑤₃. Besides, the weight of the decision maker is 𝐷𝑊={0.5,0.3,0.2}. They make strict evaluation for five candidates 𝑎_𝑖(𝑖=1,2,3,4,5) from four aspects, namely, morality 𝑐₁, research capability 𝑐₂, teaching skills 𝑐₃, and education background 𝑐₄. The weights of the evaluation criteria comprise a vector 𝑊=[0.4,0.3,0.2,0.1]^′. The evaluations of each decision maker to the candidates construct the following interval-valued intuitionistic fuzzy information as present in Tables 3, 4, and 5.

Table 3 

The evaluation of the university president.

Table 4 

The evaluation of the dean of the management school.

Table 5 

The evaluation of the human resource officer.

Step 1. In order to calculate the similarity degree between the ideal alternative and the alternative 𝑎_𝑖(𝑖=1,2,3,4), we have the ideal alternative 𝑎^∗ as follows: 𝑎^∗=𝑐₁,[1,1],[0,0],𝑐₂,[],[],𝑐1,10,0₃,[1,1],[0,0],𝑐₄,[1,1],[0,0]. (71) It is done by formula (64). The result is shown in Table 6.

Table 6 

The similarity degree results.

Step 2. The last value of each alternative can be calculated by formula (65). Consider 𝑊𝑆𝑎₁,𝑎^∗=0.5244,𝑊𝑆𝑎₂,𝑎^∗𝑎=0.5603,𝑊𝑆₃,𝑎^∗=0.4662,𝑊𝑆𝑎₄,𝑎^∗𝑎=0.5490,𝑊𝑆₅,𝑎^∗=0.5361. (72)

Step 3. The ranking order of all alternatives is shown as follows: 𝑎₂>𝑎₄>𝑎₅>𝑎₁>𝑎₃. (73) Thus, the best alternative is 𝑎₂.

In the same way, we can also get the result based on the ideal point measure: 𝑎⁺=],[],[,],[],[,],[],[,],[],[,𝑎{([1,10,00,0])([1,10,00,0])([1,10,00,0])([1,10,00,0])}₋=],[],[,],[],[,],[],[,],[],[.{([0,01,10,0])([0,01,10,0])([0,01,10,0])([0,01,10,0])} (74) Table 7 describes the results of the ideal point method. We can get the same ranking order and the same best alternative from the two methods. We can conclude that the ideal point method and our similarity measure method have the same effect used to solve the multicriteria group decision making problem. The difference is that the method proposed in [35] uses the distance measure and closeness coefficient measure while our method takes advantage of the similarity measure.

Table 7 

The results of ideal point method.

5. Conclusion

In this paper, we have presented a new similarity measure for interval-valued intuitionistic fuzzy sets with considering the hesitancy degree and using the entropy measure of IVIFSs. In fact, the entropy measure has been used in constructing the similarity measure, such as Wei et al. [24]. However, these similarity measures including many other definition methods [16–18, 33] may make mistakes, especially when the differences of membership and nonmembership between the candidate fuzzy sets and the target one are close and the difference of hesitancy degree among the candidate fuzzy sets is large. In order to make up for the flaws, we take the influence of the hesitancy degree into account and give a new method to construct the similarity measure by entropy measure, by which the proposed measure is demonstrated to yield a similarity measure. Then, the efficiency of proposed similarity measure is demonstrated by the comparative analysis with the other existing similarity measures. Finally, we also apply the new similarity measure to expert system to solve the pattern recognition problem and the multicriteria group decision making problem in which we compare our method with the ideal point method [35]. Several examples are given to illustrate the practicality and effectiveness of the applications. Besides, the similarity measure between interval-valued intuitionistic fuzzy sets can be applied to many different fields.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This study was partly funded by National Natural Science Foundation of China (71271070) and China Scholarship Council (201306120159).