International Journal of Computational Intelligence Systems

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Volume 11, Issue 1, 2018, Pages 15 - 32

A Multiple Attribute Decision Making Approach Based on New Similarity Measures of Interval-valued Hesitant Fuzzy Sets

Authors
Yi Liu1, 2, 3, *, liuyiyl@126.com, Jun Liu3, J.liu@ulster.ac.uk, Zhiyong Hong4, hongmr@163.com
1Data Recovery Key Lab of Sichuan Province, Neijiang Normal University, Neijiang 641000, Sichuan, P.R. China
2School of Mathematics and Information Science, Neijiang Normal University, Neijiang 641000, Sichuan, P.R. China
3School of Computing and Mathematics, Ulster University, Jordanstown Campus, Northern Ireland BT 37 0TR, UK
4School of Computer Science, Wuyi University, Jiangmen 529020, Guangdong, P.R. China
Corresponding Author
Received 15 June 2017, Accepted 14 September 2017, Available Online 1 January 2018.
DOI
10.2991/ijcis.11.1.2How to use a DOI?
Keywords
Interval-valued hesitant fuzzy set; II-type interval-valued hesitant fuzzy distance; interval-valued hesitant fuzzy Lp distance; relative similarity measure; multiple attribute decision making
Abstract

Hesitant fuzzy sets, as an extension of fuzzy sets to deal with uncertainty, have attracted much attention since its introduction, in both theory and application aspects. The present work is focused on the interval-valued hesitant fuzzy sets (IVHFSs) to manage additional uncertainty. Now that distance and similarity as a kind of information measures are essential and important numerical indexes in fuzzy set theory and all their extensions, the present work aims at investigating distance and similarity measures in the IVHFSs and then employing them into multiple attribute decision making application. To begin with, II-type generalized interval-valued hesitant fuzzy distance is firstly introduced in the IVHFS, along with its properties and its relationships with the traditional Hamming-Distance and the Euclidean distance. Afterwards, another interval-valued hesitant fuzzy Lp distance based on Lp metric is proposed and its relationship with the Hausdorff distance is discussed. In addition, different from most of similarity measures with dependent on the corresponding distances, a new similarity measure based on set-theoretic approach for IVHFSs is introduced and its properties are discussed; especially, a relative similarity measure is proposed based on the positive ideal IVHFS and the negative ideal IVHFS. Finally, we describe how the IVHFS and its relative similarity measure can be applied to multiple attribute decision making. A numerical example is then provided to illustrate the effectiveness of the proposed method.

Copyright
© 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Fuzzy set, a generalization of classical set theory introduced by Zadeh23, has drawn the attention of many researchers who have extended the fuzzy sets to intuitionistic fuzzy sets (IFSs)2, interval-valued intuitionistic fuzzy sets (IVIFSs)1, interval-value Pythagorean fuzzy set15 and so on, which are also applied to some decision making11. In 2010, Torra20 introduced an important extension of fuzzy sets named the hesitant fuzzy sets (HFSs) which permit the membership degree of an element to a set to be represented as several possible values between 0 and 1, human beings hesitate among a set of membership degrees and they need to represent such a hesitation. Rodriguez17,18 recently provided a position and perspective analysis of HFSs in decision making, which gave a detailed review on HFS and its application in decision making, especially pointed out some important challenges.

In many real world decision making problems, due to insufficiency in available information, it may be difficult for decision makers to exactly quantify their opinions with a crisp number, instead they may prefer an interval number within [0, 1]. Accordingly, Chen5,6 introduced the concept of interval-valued hesitant fuzzy sets (IVHFSs), which represent the membership degrees of an element to a set with several possible interval values.

Information measures (specially, distance, similarity, and entropy) for IVHFSs have played the key roles in the development of the IVHFS theory and its applications. IVHFS distance and similarity measures indicate the distance and similarity degrees of two IVHFSs, and the entropy of an IVHFS describes the fuzziness degree of an IVHFS. Therefore, many researchers have investigated these three concepts for IVHFSs from different points of views. For example, Farhadinia7 investigated the relationship among the entropy, the similarity and the distance for IVHFSs and applied the similarity measures to two clustering algorithms with applications in data analysis and classification. He also investigated the approach for deriving the correlation coefficient of dual IVHFSs8 and proposed division and subtraction formulas for IVHFSs9. Gitinavard10 proposed a decision model based on IVHFSs using the distance measures and applied it to the industrial selection problems. Jin13 proposed a multiple attribute group decision making method on the basis of some information measures under the interval-valued hesitant fuzzy environment, where the axiomatic definitions of continuous entropy and continuous similarity measures on IVHFSs were introduced and the method was applied to emergency risk assessment. Quiros16 studied entropy measures under interval-valued hesitant fuzzy environment and built the entropy measure using three different measures: fuzziness, lack of knowledge and hesitance. Wei22 proposed a variety of distance measures for IVHFSs, based on which the corresponding similarity measures are derived, and some properties of these distance measures and similarity measures were investigated, but the general definition of distance measures and similarity measure are not given like the notions of fuzzy sets. Farhadini9 gave the axiom definition of distance measures of HFSs and IVHFSs, and all distance must be in [0, 1], however, there are many distance measures beyond [0, 1]. So we modify the axiom definitions of distance and similarity measures for IVHFSs and propose some new distance and similarity measures between IVHFSs, the new distance measures are more reasonable since they are very similar with the existing ones except for some minor modification of the expression. Wei22 introduced a generalized interval-valued hesitant Hamming-Hausdorff distance which is a generalization of interval-valued hesitant normal Hamming-Hausdorff distance, we will introduce new distance measure whose limit is the interval-valued hesitant normal Hamming-Hausdorff distance from the point of view of Lp metric. They are another new reasonable distance measures from different point of view and the main advantages of the new distance measures is that they do not need to be normalized and more reasonable since they are very similar with the existing ones except for some minor modification of the expression.

The IVHFSs method and its distance measures were applied to show potential evaluation of emerging technology commercialization with interval-valued hesitant fuzzy information.

As pointed out in17, one of the great challenges is to further develop foundation in the HFS theory, e.g., the notion of entropy, similarity and distance measures. Based on the existing work as reviewed above about the information measures for IVHFSs, in the present work, we continue investigating distance and similarity measures for IVHFSs from different points of views. The main contributions are summarized as below: (1) introduce a new generalized interval-valued hesitant fuzzy distance, named II-type generalized interval-valued hesitant fuzzy distance which is a generalization of the famous Haming-distance and the Euclidean distance; (2) introduce an interval-valued hesitant fuzzy Lp distance based on Lp metric, and shows that the Hausdorff distance of two IVHFSs is the limit of Lp distance when p → +∞; (3) propose a new similarity measure of IVHFSs based on the set theoretical view after analysing and summarizing the existing similarity measures which are dependent on the distance measures; (4) propose a new relative information measure, that is, a new relative similarity measure based on the positive ideal IVHFS and the negative ideal IVHFS; and 5) propose a multiple attributes decision making method under IVHFSs based on the new relative similarity measure.

The rest of the paper is organized as follows. In Section 2, we review some definitions on IVHFSs, some existing distance and similarity measures of IVHFSs, which are used in the analysis throughout this paper. Section 3 is devoted to the main results concerning the distances of IVHFSs: 1) II-type generalized interval-valued hesitant fuzzy distance; 2) the interval-valued hesitant fuzzy Lp distance based on Lp metric. Section 4 is focused on the new similarity measures base on set theory after analysing the existing similarity measures and proposing a generic definition of a family of similarity measures, which are dependent on the distance measures. In Section 5, we propose the concept of relative similarity measure of IVHFSs and construct multiple attribute decision-making approach based on the relative similarity measure. Consequently, a practical example is provided in Section 6 to illustrate this method and compare the proposed methods with the existing one in Wei 2013. This paper is concluded in Section 7.

2. Preliminaries

Necessary basic concepts to understand the definition of distance and similarity measures proposed in this work are given in this section. It has been split into two subsections. Basic concepts about the interval-valued hesitant fuzzy set are explained in the first subsection. In the second subsection, the axiomatic definitions of distance and similarity measures for IVHFSs are given, which establish foundations for the follow-up development of this work in other sections.

2.1. Interval-Valued Hesitant Fuzzy Set

Definition 1.20

Let X = {x1, x2, ⋯ , xn} be a reference set. A hesitant fuzzy set (HFS) F on X is defined in terms of a function hF(x) that returns a subset of [0, 1] when it is applied to X, i.e., F = {⟨x, hF(x)|x ∈ X⟩} where hF(x) is a set of some different values in [0, 1], representing the possible membership degrees of the element x ∈ X to F. hF(x) is called a hesitant fuzzy element (HFE), a basic unit of HFS.

Definition 2.5,6

Let X = {x1, x2, ⋯ , xn} be a reference set. An interval-valued hesitant fuzzy set (IVHFS) on X is

F˜={〈x,hF˜(x)〉|x∈X},
where hF˜(x) denotes all possible interval-valued membership degrees of the element x ∈ X to the set F˜. For convenience, we call hF˜(x) an interval- valued hesitant fuzzy element (IVHFE), which reads
hF˜(x)={r˜|r˜∈hF˜(x)}.

Here r˜ is an interval number. An IVHFE is the basic unit of an IVHFS, and it can be considered as a special case of the IVHFS. The relationship between IVHFE and IVHFS is similar to that between interval-valued fuzzy number and interval-valued fuzzy set.

Example 1.

If X = {x1, x2, x3} is a reference set, hF(x1) ={0.1, 0.2}, hF(x2) = {0.4, 0.5, 0.7}, and hF(x3) = {0.2} are the possible membership degrees of xi(i = 1, 2, 3) to the set F, then F can be regarded as a HFS and is presented as follows:

F={〈x1,{0.1,0.2}〉,〈x2,{0.4,0.5,0.7}〉,〈x3,{0.2}〉}.

For the reference set X = {x1, x2, x3}, if hF˜(x1)={[0.1,0.2]}, hF˜(x2)={[0.5,0.7],[0.3,0.4]}, hF˜(x3)={[0.2,0.5]} are the possible membership degrees of xi(i = 1, 2, 3) to the set F˜. then F˜ can be regarded as an IVHFS and is presented as follows:

F˜={〈x1,{[0.1,0.2]}〉,〈x2,{[0.5,0.7],[0.3,0.4]}〉,〈x3,{[0.2,0.5]}〉}.

Hereafter, for the notation convenience, a real number a can be regarded as an interval [a, a], so it is also regarded as a special IVHFS, that is, ⟨x, {[a, a]}⟩.

Remark 1.

From Definition 2.2, it should be noted that HFSs permit the membership of an element to be a set of several possible values. All these possible values are crisp real numbers that belong to [0, 1]. However, IVHFSs permit the membership degrees of certain elements to a set with several possible interval values. Therefore, IVHFSs are more reasonable for experts to assign their evaluation values in some practical problems. Whist we can see that both HFSs and IVFSs are all particular cases of IVHFSs.

We use ℐ 𝒱 ℋ ℱ 𝒮 (X) to denote the class of all IVHFSs of a reference set X. In addition, for an IVHFE hF˜(xi), it is necessary to arrange the intervals in an increasing order. To achieve the goal, the score function and accuracy function, given in16, are employed to compare two intervals and described as follows:

Let X = [x1, x2], the score function of X defined as S(X) = x2 − x1 and accuracy function of X defined as H(X) = x1 + x2. Let A = [a1, b1] and B = [a2, b2] be two intervals, then

A⩽B⇔{H(A)⩽H(B)orH(A)=H(B) and S(A)⩽S(B).

Let hF˜σ(j)(xi) stand for the jth smallest interval in the hF˜(xi), where

hF˜σ(j)(xi)=[hF˜σ(j)L(xi),hF˜σ(j)U(xi)].hF˜σ(j)L(xi)=infj(hF˜σ(j)(xi)),hF˜σ(j)U(xi)=supj(hF˜σ(j)(xi)),1⩽i⩽n,1⩽j⩽n(hF˜(xi)).

Here n(hF˜(xi)) denotes the number of intervals in the hF˜(xi).

Example 2.

Let X = {x1, x2} be a reference set. Define an IVHFS F˜ on X as follows:

F˜={〈x1,{[05,0.6],[0.2,0.3],[0.4,0.6]}〉,〈x2,{[0.3,0.5],[0.4,0.6]}〉}.

Then hF˜σ(1)(x1)=[0.2,0.3], hF˜σ(2)(x1)=[0.4,0.6], hF˜σ(3)(x1)=[0.5,0.6], hF˜σ(1)(x2)=[0.3,0.5], hF˜σ(2)(x2)=[0.4,0.6].

It follows from Example 1 and Example 2 that the number of intervals in different IVHFEs may be different. We use n(hF˜(x)) to stand for the number of intervals in the IVHFE hF˜(x). For any two IVHFSs Ã, B˜on the same reference set X, set nx(A˜,B˜)=max{n(hA˜(x)),n(hB˜(x))} for some x ∈ X. In order to compare two IVHFSs conveniently and correctly, the two IVHFEs hÃ(x) and hB˜(x) should have the same number of intervals, that is, nx(A˜,B˜). Therefore, if there are less intervals in hÃ(x) than in hB˜(x), an extension of hÃ(x) should be considered optimistically by repeating its maximum interval until it has the same number of the intervals as that in hB˜(x), that is nx(A˜,B˜), in short denoted as nx ( of course, an extension of hÃ(x) is also considered pessimistically by repeating its minimal interval until it has the same number of the intervals as that in hB˜(x). Although the results may be different if we extend the shorter one by adding different values, this is reasonable because the decision makers risk preferences can directly influence the final decision. In this paper, we assume that the decision makers are all optimistic (other situations can be studied similarly)). In the subsequent section, without loss of generality, assume that the IVHFEs involved in distance or similarity measures have the same number of intervals.

Example 3.

Let X = {x1, x2} be a reference set. Define two IVHFSs F˜ and G˜ on X as follows:

F˜={〈x1,{[0.5,0.6],[0.2,0.3],[0.4,0.6]}〉,〈x2,{[0.3,0.5],[0.4,0.6]}〉};G˜={〈x1,{[0.4,0.5],[0.5,0.6]}〉,〈x2,{[0.3,0.5],[0.4,0.6],[0.6,0.7]}〉}.

Then, to keep the same number of intervals, after the extensions on both F˜ and G˜ respectively, these two IVHFSs can be re-expressed respectively as follows:

F˜={〈x1,{[0.2,0.3],[0.4,0.6],[0.5,0.6]}〉,〈x2,{[0.3,0.5],[0.4,0.6],[0.4,0.6]}〉};G˜={〈x1,{[0.4,0.5],[0.5,0.6],[0.5,0.6]}〉,〈x2,{[0.3,0.5],[0.4,0.6],[0.6,0.7]}〉}.

Definition 3.7

Let X be a reference set, F˜ and G˜ be two IVHFSs on X. Then, one kind of ordering for IVHFSs, i.e., the component-wise ordering of IVHFSs, is defined as follows:

F˜⩽G˜⇔hF˜σ(j)L(xi)⩽hG˜σ(j)L(xi),hF˜σ(j)U(xi)⩽hG˜σ(j)U(xi),
where xi ∈ X, 1 ⩽ i ⩽ n, 1 ⩽ j ⩽ nxi.

2.2. Axiomatic Definitions of Distance and Similarity Measures for IVHFSs

Distance and similarity measures are the fundamental and important issues of theory of fuzzy sets. The axiomatic definitions of distance and similarity measure between two IVHFSs is introduced in7 and reviewed as below in Def. 4 and Def. 5 respectively.

Definition 4.7

Let F˜ and G˜ be IVHFSs on the reference set X = { x1, x2, ⋯ , xn} and d a real function: ℐ 𝒱 ℋ ℱ 𝒮 × ℐ 𝒱 ℋ ℱ 𝒮 → [0, 1]. d is a distance measure between F˜ and G˜ if d satisfies the following properties:

  • (D1) 0⩽d(F˜,G˜)⩽1;

  • (D2) d(F˜,G˜)=0⇔F˜=G˜;

  • (D3) d(F˜,G˜)=d(G˜,F˜);

  • (D4) d(F˜,F˜C)=1 if and only if F˜={〈x,{[0,0]}〉} or F˜={〈x,{[1,1]}〉};

  • (D5) let E˜ be an IVHFS and E˜⩽F˜⩽G˜. Then d(E˜,F˜)⩽d(E˜,G˜) and d(F˜,G˜)⩽d(E˜,G˜). Here in (D4), F˜C={〈x,∪r˜∈hM˜(x){r˜C}}〉|x∈X} denotes the complement of F˜, and r˜C is the complement of an interval r˜.

Moreover, various distance measures of two IVHFSs F˜ , G˜ are proposed in7,21. For example, given two IVHFSs F˜, G˜, hF˜σ(j)(xi)=[hF˜σ(j)L(xi),hF˜σ(j)U(xi)] and hG˜σ(j)(xi)=[hG˜σ(j)L(xi),hG˜σ(j)U(xi)] denote the jth smallest intervals in hF˜(xi) and hG˜(xi), respectively.

The interval-valued hesitant fuzzy Hamming distance is defined in7 as follows:

d1(F˜,G˜)=1n∑i=1n[12nxi∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|)].

The interval-valued hesitant fuzzy Euclidean distance is defined in21 as follows:

d2(F˜,G˜)=1n∑i=1n[12nxi∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|2+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|2)].

The generalized interval-valued hesitant fuzzy distance is defined in21 as follows:

d3(F˜,G˜)={1n∑i=1n[12nxi∑j=1nxi(|hF˜ σ(j)L(xi)−hG˜ σ(j)L(xi)|p+|hF˜ σ(j)U(xi)−hG˜ σ(j)U(xi)|p)]}1p,p>0.

We can see from the above distances that di(F˜,G˜)(i=1,2,3) all satisfy 0⩽di(F˜,G˜)⩽1, sometimes some defined operators satisfy (D2) − (D5), but did not satisfy (D1). For example,

d1'(F˜,G˜)=1n∑i=1n[1nxi∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|)].

Obviously, d1'(F˜,G˜)⩽2, which shows that d1'(F˜,G˜) does not satisfy the condition (D1). In order to distinguish it from Def. 6 introduced in the subsequent section, d(F˜,G˜) in Def. 4 is also called an interval-valued hesitant fuzzy normalized distance between two IVHFSs F˜ and G˜.

Therefore, di(F˜,G˜)(i=1,2,3) are called interval-valued hesitant fuzzy normalized Hamming distance, interval-valued hesitant fuzzy normalized Euclidean distance and generalized interval-valued hesitant fuzzy normalized distance, respectively.

Definition 5.7.

Let F˜ and G˜ be IVHFSs on the reference set X = {x1, x2, ⋯ , xn} and s a real function: ℐ 𝒱 ℋ ℱ 𝒮 × ℐ 𝒱 ℋ ℱ 𝒮 → [0, 1]. s is called a similarity measure between F˜ and G˜ if s satisfies the following properties:

  • (S1) 0⩽s(F˜,G˜)⩽1;

  • (S2) s(F˜,G˜)=1⇔F˜=G˜;

  • (S3) s(F˜,G˜)=s(G˜,F˜);

  • (S4) if F˜={〈x,{[0,0]}〉} or F˜={〈x,{[1,1]}〉}, then s(F˜,F˜C)=0;

  • (S5) let Ẽ be an IVHFS and E˜⩽F˜⩽G˜. Then s(E˜,F˜)⩾s(E˜,G˜) and s(F˜,G˜)⩾s(E˜,G˜). Here, F˜C={〈x,∪r˜∈hM˜(x){r˜C}}〉|x∈X}, as the same as defined above.

3. New Distance Measures for IVHFSs

In this section, we will introduce two new distance measures between two IVHFSs, that is, II-type generalized interval-valued hesitant fuzzy distance and the interval-valued hesitant fuzzy Lp distance based on Lp metric, along with their properties. They are another new reasonable distance measures from different point of view and the main advantages of the new distance measures is that they do not need to be normalized and more reasonable since they are very similar with the existing ones except for some minor modification of the expression. If not otherwise specified, in the subsequent sections, we always assume that F˜, G˜ are two IVHFSs and hF˜σ(j)(xi)=[hF˜σ(j)L(xi),hF˜σ(j)U(xi)] and hG˜σ(j)(xi)=[hG˜σ(j)L(xi),hG˜σ(j)U(xi)] denote the jth smallest intervals in hF˜(xi) and hG˜(xi), respectively.

3.1. II-type generalized interval-valued hesitant fuzzy distance

We have listed three kinds of important distances in Section 2. Obviously, if p = 1, p = 2 in Eq.(3), then d3(F˜,G˜) will be degenerated to d1(F˜,G˜) and d2(F˜,G˜) respectively, which reflects the connection between these two sets of distances. From d1(F˜,G˜), assume

di=12nxi∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|).

It is actually the distance between the ith IVHFEs of F˜ and G˜. However, d1(F˜,G˜) is the distances between all IVHFEs of F˜ and G˜. From this point of view, we can modify d3(F˜,G˜) and redefine another generalized measure d4(F˜,G˜) between F˜ and G˜ as follows:

d4(F˜,G˜)=1n∑i=1n[12nxi∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p)]1p,p>0.

If p = 1 in Eq. (5), Eq. (5) is also degenerated to Eq. (1). If p = 2 in Eq. (5), Eq. (5) will not be degenerated to Eq. (2), instead, it is degenerated to another measure d5(F˜,G˜) between F˜ and G˜ as follows:

d5(F˜,G˜)=1n∑i=1n12nxi∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|2+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|2).

Of course, it is natural to consider the rationality of Eq. (5) in terms of distance measure. In other words, we need to check if d4(F˜,G˜) satisfies (D1)–(D5) in Def. 4 or not. The answer is positive and stated in the following theorem.

Theorem 1.

d4(F˜,G˜) is an interval-valued hesitant fuzzy normalized distance between two IVHFSs, F˜ and G˜.

Proof.

Obviously, it is straightforward to see that d4(F˜,G˜) satisfies (D1), (D2), (D3) and (D4) in Definition 4. Now, we consider the axiom (D5). Let E˜⩽F˜⩽G˜. Then hE˜σ(j)L(xi)⩽hF˜σ(j)L(xi)⩽hG˜σ(j)L(xi), hE˜σ(j)U(xi)⩽hF˜σ(j)U(xi)⩽hG˜σ(j)U(xi). Therefore, for any p > 0, we have

|hE˜σ(j)L(xi)−hF˜σ(j)L(xi)|p⩽hE˜σ(j)L(xi)−hG˜σ(j)L(xi)|p,|hE˜σ(j)U(xi)−hF˜σ(j)U(xi)|p⩽hE˜σ(j)U(xi)−hG˜σ(j)U(xi)|p.

It follows that

|hE˜σ(j)L(xi)−hF˜σ(j)L(xi)|p+|hE˜σ(j)U(xi)−hG˜σ(j)U(xi)|p⩽|hE˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hE˜σ(j)U(xi)−hG˜σ(j)U(xi)|p.

Hence,

12nxi∑j=1nxi(|hE˜σ(j)L(xi)−hF˜σ(j)L(xi)|p+|hE˜σ(j)U(xi)−hF˜σ(j)U(xi)|p)⩽12nxi∑j=1nxi(|hE˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hE˜σ(j)U(xi)−hG˜σ(j)U(xi)|p).
and
[12nxi ∑j=1nxi(|hE˜ σ(j)L(xi)−hF˜ σ(j)L(xi)|p+|hE˜ σ(j)U(xi)−hF˜ σ(j)U(xi)|p)]1p ≤[12nxi∑j=1nxi(|hE˜ σ(j)L(xi)−hG˜ σ(j)L(xi)|p+|hE˜ σ(j)U(xi)−hG˜ σ(j)U(xi)|p)]1p.

Therefore,

1n∑i=1n[12nxi ∑j=1nxi(|hE˜ σ(j)L(xi)−hF˜ σ(j)L(xi)|p+|hE˜ σ(j)U(xi)−hF˜ σ(j)U(xi)|p)]1p⩽1n∑i=1n[12nxi∑j=1nxi(|hE˜ σ(j)L(xi)−hG˜ σ(j)L(xi)|p+|hE˜ σ(j)U(xi)−hG˜ σ(j)U(xi)|p)]1p.

That is, d4(E˜,F˜)⩽d4(E˜,G˜). Similarly, we can prove that d4(F˜,G˜)⩽d4(E˜,G˜). Thus d4(E˜,F˜) satisfies the axiom (D5). The completes the proof.

Remark 2.

d5(F˜,G˜) is the special case of d4(F˜,G˜) when p = 2. d4(F˜,G˜) is called II-type generalized interval-valued hesitant fuzzy normalized distance measure and d5(F˜,G˜) is called II-type interval-valued hesitant fuzzy normalized Euclidean distance measure corresponding to d2(F˜,G˜). Accordingly, the interval-valued hesitant fuzzy normalized Hamming distance d1(F˜,G˜) is a special case of II-type generalized interval-valued hesitant fuzzy normalized distance measures when p = 1.

Based on Eq. (5), we can further define another measure d4'(F˜,G˜) between F˜ and G˜ as follows:

d4'(F˜,G˜)=∑i=1n[12nxi ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p)]1p,p>0.

Obviously, d4'(F˜,G˜) is not normalized and does not satisfy (D1). So it is not a distance measure as defined in Def. 4. (D1) is a rather restricted property and if it was modified, (D4) would need to be modified accordingly.

In the following, we will introduce another new distance measure which generalise (D1), i.e., the distance measure is defined in the ℝ+, instead of [0, 1], in order to cover more general cases, accordingly, (D4) is also modified.

Definition 6.

Let F˜ and G˜ be two IVHFSs on the reference set X = {x1, x2, ⋯ , xn} and d a real function : ℐ 𝒱 ℋ ℱ 𝒮 × ℐ 𝒱 ℋ ℱ 𝒮 → ℝ*. d is a distance measure between F˜ and G˜ if d satisfies the following properties:

  • (D1′) 0⩽d(F˜,G˜);

  • (D2′) d(F˜,G˜)=0⇔F˜=G˜;

  • (D3′) d(F˜,G˜)=d(G˜,F˜);

  • (D4′) d(F˜,F˜C)=maxA˜,B˜∈ℐ𝒱ℋℱ𝒮(X)d(A˜,B˜) if and only if F˜={〈x,{[0,0]}〉} or F˜={〈x,{[1,1]}〉};

  • (D5′) let E˜ be an IVHFS and E˜⩽F˜⩽G˜. Then d(E˜,F˜)⩽d(E˜,G˜) and d(F˜,G˜)⩽d(E˜,G˜). where, F˜C={〈x,∪r˜∈hF˜(x){r˜C}}〉|x∈X}.

Corresponding to Def. 4, d(F˜,G˜) in Definition 6 is called an interval-valued hesitant fuzzy distance between F˜ and G˜.

Remark 3.

According to definitions, we can see that the normalized distance measure defined in Def. 4 is a special case of the distance measures defined in Def. 6. According to Def. 6, similar to the proof of Theorem 1, we have

Theorem 2.

d4'(F˜,G˜) is an interval-valued hesitant fuzzy distance between F˜ and G˜.

Remark 4.

The normalized distance measure can be obtained by a distance measure, i.e., if d is a distance measure and dmax is the maximal distance of any two interval-valued hesitant fuzzy sets, then ddmax is a normalized distance measure.

Remark 5.

It is easy to prove that (D5′ ) is equivalent to

  • (D5″): Let D˜, E˜, F˜, G˜ be four IVHFSs and D˜⩽E˜⩽F˜⩽G˜. Then d(D˜,G˜)⩾d(E˜,F˜).

3.2. Interval-valued hesitant fuzzy Lp distance based on Lp metric

In12, Huang studied the similarity of intuitionistic fuzzy sets based on Lp metric. The reader can refer to12 about the Lp metric concept. Huang extend the normal Lp metric concept onto the one on the interval, this basic idea as follows:

Let ℜ be a Euclidean space and I = {[a, b]| a, b ∈ ℜ, a ≤ b} be the set of closed intervals in ℜ. For any two interval x = [x1, y1], y = [x2, y2] ∈ I, the distance between two intervals x, y is defined as follows:

dp(x,y)=(|x1−x2|p+|y1−y2|p)12,p⩾1.

This corresponds to representing an interval [a, b] ∈ I as a point (a, b), where the lower bounds of the intervals are represented in the x-axis, and the upper bounds in the y-axis, and then computing the Lp distance between the points (x1, x2) and (y1, y2). Therefore, the distance dp is a suitable extension of the Lp metric on the interval.

Hamming distance and Hausdorff distance are two important distance. Wei22 introduced a generalized interval-valued hesitant Hamming-Hausdorff distance which is a generalization of interval-valued hesitant normal Hamming-Hausdorff distance. Since the IVHFEs of an IVHFS are sets of some intervals and the distance of two IVHFSs are closely related the distances of IVFHEs, we will introduce new distance measure named interval-valued hesitant fuzzy Lp distance whose limit is the interval-valued hesitant normal Hamming-Hausdorff distance based on Lp metric idea. This new measure between F˜ and G˜, denoted as dp(F˜,G˜), is defined as follows:

dp(F˜,G˜)=1n∑i=1n( ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p)]1p,p⩾1.

It is easy to prove that dp(F˜,G˜) is a distance measure in terms of Definition 3.

Wei21 introduced the interval-valued hesitant fuzzy normalized Hamming-Hausdorff distance as below:

d7(F˜,G˜)=1n∑i=1n(maxj=1,2,…,nxi{|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|,|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|}).

In the following, we propose and prove an interesting result:

limp→+∞dp(F˜,G˜)=d7(F˜,G˜).

To prove the interesting result Eq. (9), the following lemma is necessary.

Lemma 1.12

Let a, b be two non-negative real numbers. We have

limp→+∞(ap+bp)1p=max{a,b},p⩾1.

The following Theorem 3 can be easily obtained from Lemma 1.

Theorem 3.

limp→+∞dp(F˜,G˜)=d7(F˜,G˜).

Proof.

By the properties of a limit and Lemma 1, we have

limp→+∞dp(F˜,G˜)=limp→+∞(1n∑i=1n( ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p))1p)=1n∑i=1nlimp→+∞( ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p))1p=1n∑i=1n(maxj=1,2,…,nxi{|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|,|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|})=d7(F˜,G˜).

In the multiple attribute decision making problems, when we select different distance functions, it may lead to different results. Usually, the weight of each element xi ∈ X should be taken into account. Hereafter, assume that the weight of the element xi ∈ X is ωi (i = 1, 2, ⋯ , n) with ωi ∈ [0, 1] and ∑i=1nωi=1. Several weighted distance measures for the IVHFSs have been presented in21. Among all weighted distance measures for the IVHFSs, the most popular one is the generalized interval-valued hesitant fuzzy weighted Hamming distance. For two IVHFSs F˜ and G˜, it is defined as below:

d8(F˜,G˜)=[∑i=1nωi(12nxi ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p)]1p,p>0.

Here, we define a generalized interval-valued hesitant fuzzy weighted Lp distance as below:

d9(F˜,G˜)=∑i=1nωi( ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p))1p,p⩾1.

It is also easy to prove that d9(F˜,G˜) is a distance measure in terms of Definition 6.

Obviously, if each element has the same importance, i.e., ωi=1n, then Eq. (10) and Eq. (11) will be degenerated to Eq. (3) and Eq. (7) respectively.

4. New Similarity Measures for IVHFSs

In this section, we focus on the new similarity measures in IVHFSs. The existing similarity measures of IVHFSs are analysed and then a new kind of new and more general similarity of IVHFSs are given; whist a new similarity measure, which is not dependent on the distance measure, but based on the set-theoretic approach, is also introduced in this section.

4.1. Analysis on the existing similarity measures for IVHFSs

Similarity measures are generally used for determining the degree of similarity between two objects. Similarity measure between two fuzzy sets is mostly dependent on their corresponding distance measures, so the former can normally be derived from the latter. This is also true for IVHFSs as detailed in21. In addition, in most of cases, the relation between the similarity measure s and the distance measure d defined in the interval [0, 1]for IVHFSs can be naturally expressed as:

s=1−d.

However, there are also some other ways or formulas to derive the similarity measure from the distance measure for IVHFSs. For example, a new similarity measure is defined with the following theorem.

Theorem 4.7

Let F˜ and G˜ be two IVHFSs, Z : [0, 1] → [0, 1] a strictly monotone decreasing real function, and d a distance measure between F˜ and G˜. Then,

sd(F˜,G˜)=Z(d(F˜,G˜))−Z(1)Z(0)−Z(1)
is a similarity measure between F˜ and G˜ based on the corresponding distance d.

As indicated in7, the distance d is a normalized distance in terms if Definition 4, so 0⩽d(F˜,G˜)⩽1. In Section 3, we have the modified normalized distance with the update of the axiom (D(1)):0⩽d(F˜,G˜)⩽1 to (D1′):0⩽d(F˜,G˜)⩽dmax, where dmax ∈ ℝ+ is the maximal distance. Hence we can give an extension of Theorem 4.

Theorem 5.

Let F˜ and G˜ be two IVHFSs, Z : [0, 1] → [0, 1] a strictly monotone decreasing real function, d a distance measure between F˜ and G˜, and dmax the maximal distance. Then,

sd'(F˜,G˜)=Z(d(F˜,G˜))−Z(dmax)Z(0)−Z(dmax)
is a similarity measure between F˜ and G˜ based on the corresponding distance d.

Proof.

It only needs to prove that the measure sd'(F˜,G˜) is a similarity measure in terms of Definition 7. Since Z is a monotone decreasing function and 0⩽d(F˜,G˜)⩽dmax, we have

Z(dmax))⩽Z(d(F˜,G˜)⩽Z(0).

It follows that

Z(d(F˜,G˜))−Z(dmax)⩽Z(0)−Z(dmax).

Therefore,

0⩽Z(d(F˜,G˜))−Z(dmax)Z(0)−Z(dmax)⩽1.

Hence (S1) holds. Similarly, we can prove (S5) holds according to (D5) in Definition 6, the details are omitted. For (S2)-(S4) are straightforward.

By Theorem 4 and Theorem 5, if we choose the monotone decreasing function Z(x) = e−x, or 1 − x, or 1−x1+x, or others), then different similarity measures between F˜ and G˜ will be obtained, and they are closely related to the distance. It is also consistent with most similarity measures corresponding to distant. Hence, Theorem 4 and Theorem 5 present two ways to construct the similarity measures for IVHFSs based on some distance measures. For example, if we choose the function Z(x) = 1 − x in Theorem 5, then we have sd'(F˜,G˜)=1−d(F˜,G˜)dmax. If the distance d is normalized into [0, 1], then we have sd(F˜,G˜)=1−d(F˜,G˜), which is consistent with Eq. (12). This means both sd and sd' are more general definitions of similarity measures than the most common case as in Eq. (12). Based on this idea and different distance measures for IVHFSs defined in Section 2, we can construct the following new similarity measures accordingly:

• Similarity measure for IVHFSs based on II-type generalized interval-valued hesitant fuzzy distance measures:

s1(F˜,G˜)=1−1n∑i=1n[12nxi ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p)]1p,p>0.

If we take the weight of each element x ∈ X into account in s1(F˜,G˜) and s2(F˜,G˜), where ω = (ω1, ω2, ⋯ , ωn), ωi ∈ [0,1] and ∑i=1nωi=1. Then we obtain

• Similarity measure with weight ω for IVHFSs based on II-type generalized interval-valued hesitant fuzzy distance measures

s1'(F˜,G˜)=1−∑i=1nωi[12nxi ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p)]1p,p>0.

• Similarity measure for IVHFSs based on interval-valued hesitant fuzzy Lp distance

s2(F˜,G˜)=1−1n∑i=1n( ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p))1p,p⩾1.

• Similarity measure with weight ω for IVHFSs based on interval-valued hesitant fuzzy Lp distance:

s2'(F˜,G˜)=1−∑i=1nωi( ∑j=1nxi(|hF˜σ(j)L(xi)−hG˜σ(j)L(xi)|p+|hF˜σ(j)U(xi)−hG˜σ(j)U(xi)|p))1p,p⩾1.

Of course, we can also choose other types of monotone decreasing functions to construct other different similarity measures for IVHFSs. Many similarity measures can be obtained by the relation function: sd(F˜,G˜)=1−d(F˜,G˜) or sd'(F˜,G˜)=1−d(F˜,G˜)dmax, will not be listed one by one here.

4.2. New similarity measures for IVHFSs based on set-theoretic approach

It follows from the above analysis on the existing similarity measures that most of similarity measures are dependent on the corresponding distance measures. In this section, we will introduce a new similarity measure for IVHFSs, which is not dependent on the distance measure, but based on the set-theoretic approach.

Theorem 6.

Let F˜, G˜ be two IVHFSs. Then

s3(F˜,G˜)=1n∑i=1nmin{∑j=1nximin{hF˜σ(j)L(xi),hG˜σ(j)L(xi)}∑j=1nximax{hF˜σ(j)L(xi),hG˜σ(j)L(xi)},∑j=1nximin{(hF˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}∑j=1nximax{(hF˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}}
is a similarity measure between F˜ and G˜.

Proof.

It needs to prove that s3(F˜,G˜) satisfies (S1)-(S5) in Definition 5. Obviously, s3(F˜,G˜) satisfies (S1), (S3) and (S4). Now, we need to prove s3(F˜,G˜) satisfies (S2) and (S5).

For (S2), s3(F˜,G˜)=1, that is,

1n∑i=1nmin{∑j=1nximin{hF˜σ(j)L(xi),hG˜σ(j)L(xi)}∑j=1nximax{hF˜σ(j)L(xi),hG˜σ(j)L(xi)},∑j=1nximin{(hF˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}∑j=1nximax{(hF˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}}=1

if and only if

min{hF˜σ(j)L(xi),hG˜σ(j)L(xi)}=max{hF˜σ(j)L(xi),hG˜σ(j)L(xi)}
and
min{(hF˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}=max{(hF˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}
if and only if
hF˜σ(j)L(xi)=hG˜σ(j)L(xi) and hF˜σ(j)U(xi)=hG˜σ(j)U(xi)
if and only if F˜=G˜.

For (S5). Let E˜, F˜, and G˜ be three IVHFSs and E˜⩽F˜⩽G˜.

s3(E˜,F˜)=1n∑i=1nmin{∑j=1nximin{(hE˜σ(j)L(xi),hF˜σ(j)L(xi)}∑j=1nximax{(hE˜σ(j)L(xi),hF˜σ(j)L(xi)},∑j=1nximin{hE˜σ(j)U(xi))c,(hF˜σ(j)U(xi))c}∑j=1nximax{hE˜σ(j)U(xi))c,(hF˜σ(j)U(xi))c}}=1n∑i=1nmin{∑j=1nxihE˜σ(j)L(xi)∑j=1nxihE˜σ(j)L(xi),∑j=1nxi(hF˜σ(j)U(xi))c∑j=1nxi(hF˜σ(j)U(xi))c}⩾1n∑i=1nmin{∑j=1nxihE˜σ(j)L(xi)∑j=1nxihE˜σ(j)L(xi),∑j=1nxi(hG˜σ(j)U(xi))c∑j=1nxi(hG˜σ(j)U(xi))c}=1n∑i=1nmin{∑j=1nximin{(hE˜σ(j)L(xi),hG˜σ(j)L(xi)}∑j=1nximax{(hE˜σ(j)L(xi),hG˜σ(j)L(xi)},∑j=1nximin{hE˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}∑j=1nximax{hE˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}}=s3(E˜,G˜)

Similarly, we have s3(E˜,G˜)⩽s3(F˜,G˜). Therefore, s3 is a similarity measure for IVHFSs according to Definition 5.

In fact, we can also propose another new measure between F˜ and G˜ as follows:

s4(F˜,G˜)=1n∑i=1nmin{∑j=1nximin{hF˜σ(j)L(xi),hG˜σ(j)U(xi)}∑j=1nximax{hF˜σ(j)L(xi),hG˜σ(j)U(xi)},∑j=1nximin{(hF˜σ(j)U(xi))c,(hG˜σ(j)L(xi))c}∑j=1nximax{(hF˜σ(j)U(xi))c,(hG˜σ(j)L(xi))c}}.

Similar to the proof of Theorem 6, we have

Theorem 7.

s4(F˜,G˜) is a similarity measure between F˜ and G˜.

By comparing s3(F˜,G˜) and s4(F˜,G˜), we can find that they have a dual relationship. In this sense, a compromise between s3(F˜,G˜) and s4(F˜,G˜) is meaningful so that we can have a new measure as a weighted average of s3(F˜,G˜) and s4(F˜,G˜) as follows:

s5(F˜,G˜)=ωs3(F˜,G˜)+(1−ω)s4(F˜,G˜),0⩽ω⩽1.

It follows from Theorem 6 and Theorem 7 that

Theorem 8.

s5(F˜,G˜) is a similarity measure between F˜ and G˜.

The proof of Theorem 8 is easy from Theorem 7 and Theorem 6, so we omit its proof.

If we take the weight of each element x ∈ X into account in s3(F˜,G˜) and s4(F˜,G˜), then we obtain

s3'(F˜,G˜)=∑i=1nωi[min{∑j=1nximin{hF˜σ(j)L(xi),hG˜σ(j)U(xi)}∑j=1nximax{hF˜σ(j)L(xi),hG˜σ(j)U(xi)},∑j=1nximin{(hF˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}∑j=1nximax{(hF˜σ(j)U(xi))c,(hG˜σ(j)U(xi))c}}
and
s4'(F˜,G˜)=∑i=1nωi[min{∑j=1nximin{hF˜σ(j)L(xi),hG˜σ(j)U(xi)}∑j=1nximax{hF˜σ(j)L(xi),hG˜σ(j)U(xi)},∑j=1nximin{(hF˜σ(j)U(xi))c,(hG˜σ(j)L(xi))c}∑j=1nximax{(hF˜σ(j)U(xi))c,(hG˜σ(j)L(xi))c}}],
where ωi ∈ [0, 1] and ∑i=1nωi=1. Specially, if ωi=1n(i=1,2,…,n), then Eq. (20) and Eq. (21) will be degenerated to Eq. (17) and Eq. (18), respectively.

5. A Multiple Attribute Decision Making Approach Based on Interval-Valued Hesitant Fuzzy Information

In this section, we shall utilize the interval-valued hesitant fuzzy similarity measures to multiple attribute decision making with interval-valued hesitant fuzzy information, especially, a relative similarity measure is proposed in order to fit into the multiple attribute decision making approach.

Assume A = {A1, A2, ⋯ , Am} is a discrete set of alternatives, X = {x1, x2, ⋯ , xn} is a set of attributes, and ω = {ω1, ω2, ⋯ , ωn} is the weight vector of the attribute xj (j = 1, 2, ⋯ , n), where ωj ∈ [0, 1] and ∑j=1n=1. Assume that the characteristics for alternatives Ai are presented by the IVHFS as follows:

Ai={〈xj,hAi(xj)〉|xj∈X},i=1,2,…,m,
where hAi(xj) indicates the degree that the alternative Ai satisfies the attribute xj. This also means that, if the decision makers provide several interval-valued values for the alternative Ai under the attribute xj on condition of anonymity, these values can be considered as an interval-valued hesitant fuzzy element hAi(xj). Before the decision making method is introduced, we firstly introduce the concepts of the positive ideal IVHFS, the negative ideal IVHFS and the relative similarity measure for IVHFS as follows:

Definition 7.14

Let Ai(i = 1, 2, ⋯ , m) be IVHFSs on the reference set X. The positive ideal IVHFS A+ and the negative ideal IVHFS A− are defined as follows, respectively:

A+={〈xj,hA+(xj)〉|xj∈X},A−={〈xj,hA−(xj)〉|xj∈X},
where,
hA+(xj)={maxi{hAiσ(k)(xj)}|k=1,2,…,nxi, i=1,2,…,m}hA−(xj)={mini{hAiσ(k)(xj)}|k=1,2,…,nxi, i=1,2,…,m}

Note that both the positive ideal IVHFS and the negative ideal IVHFS are special case of an IVHFS with only one IVHFE in each IVHFS.

In order to choose the desired alternative in multiple attribute decision making problem, based on some aforementioned similarity measures for IVHFSs in Section 4, we can compute the similarity degree between the positive ideal IVHFS A+ and the alternative Ai, as well as the similarity degree between the negative ideal IVHFS A− and the alternative Ai, denoted by s(A+, Ai) and s(A−, Ai), respectively. Intuitively, the larger the similarity s(A+, Ai), the better the alternative; while the smaller the s(A−, Ai), the better the alternative. Motivated by the well-known TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution method4, we take both s(A+, Ai) and s(A−, Ai) into consideration. This leads naturally to the concept of relative similarity measure.

Definition 8.

Let A = {A1, A2, ⋯, Am} be a discrete set of alternatives. The relative similarity measures rsi corresponding to the alternative Ai are defined as:

rsi=s(A+,Ai)s(A+,Ai)+s(A−,Ai),i=1,2,…,m.

Based on the above analysis and concepts, we propose the following procedure of a multiple attribute decision making:

Step 1. Construct the corresponding interval-valued hesitant fuzzy sets according to the characteristics of the alternatives:

Ai={〈xj,hAi(xj)〉|xj∈X},i=1,2,…,m.

Step 2. Construct the notions of the positive ideal IVHFS A+ and the negative ideal IVHFS A according to Def.7.

Step 3. Calculate similarity measure s(A+, Ai) and s(A−, Ai) according to some similarity measures given in the Section 4. We will choose any weighted similarity measures when the attributes with weight information. Otherwise, we can choose other similarity measures without weight.

Step 4. Calculate the relative similarity measure rsi corresponding to the alternative Ai according to Eq.(23).

Step 5. Obtain the priority of the alternative Ai(i = 1, 2, ⋯ , m) by ranking the rsi(i = 1, 2, ⋯ , m).

Step 6. End of the steps.

6. An Example Illustration

We choose a practical problem as an illustrative example, which have been discussed in22, in order to illustrate the effeteness of the proposed method compared with the the one in22.

Assume that there is an investment company, which wants to invest a sum of money in the best option. There is a panel with five possible alternatives Ai(i = 1, 2, ⋯ , 5) to invest money: (1) A1 is a car company; (2) A2 is a food company; (3) A3 is a computer company; (4) A4 is an cosmetic company; (5) A5 is a TV company. The investment company must take a decision according to the following attributes xi(i = 1, 2, 3, 4): (1) x1 is the risk analysis; (2) x2 is the growth analysis; (3) x3 is the social-political impact analysis; (4) x4 is the environment impact analysis. In order to avoid influence each other, the decision makers are required to evaluate the five companies Ai(i = 1, 2, ⋯ , 5) under the above four attributes X = (x1, x2, x3, x4) on condition of anonymity. And suppose that the weight vector of attributes is (0.4, 0.2, 0.3, 0.1). The decision matrix is presented in Table 1 below, which shows all interval-valued hesitant fuzzy information.

x1 x2 x3 x4
A1 {[0.2, 0.3], [0.3, 0.4]} {[0.2, 0.5]} {[0.7, 0.8], [0.8, 0.9]} {[0.4, 0.5]}
A2 {[0.2, 0.3], [0.4, 0.5], [0.5, 0.6]} {[0.3, 0.4], [0.6, 0.7]} {[0.3, 0.4]} {[0.5, 0.6], [0.8, 0.9]}
A3 {[0.5, 0.7]} {[0.2, 0.3], [0.4, 0.5]} {[0.8, 0.9], [0.9, 1.0]} {[0.3, 0.5]}
A4 {[0.3, 0.4], [0.7, 0.8]} {[0.1, 0.3]} {[0.6, 0.7], [0.8, 0.9]} {[0.5, 0.7]}
A5 {[0.2, 0.3]} {[0.4, 0.6]} {[0.2, 0.3], [0.6, 0.7]} {[0.6, 0.7]}
Table 1:

Interval-valued hesitant fuzzy decision matrix

Step 1. Construct the corresponding interval-valued hesitant fuzzy sets according to the characteristics of the alternatives as follows based on Table 1:

A1={〈x1,{[0.2,0.3],[0.3,0.4]}〉,〈x2,{[0.2,0.5]}〉,〈x3,{[0.7,0.8],[0.8,0.9]}〉,〈x4,{[0.4, 0.5]}〉}A2={〈x1,{[0.2,0.3],[0.4,0.5],[0.5,0.6]}〉,〈x2,{[0.3,0.4],[0.6,0.7]}〉,〈x3,{[0.3,0.4]}〉,〈x4,{[0.5,0.6],[0.8,0.9]}〉},A3={〈x1,{[0.5,0.7]}〉,〈x2,{[0.2,0.3],[0.4,0.5]}〉,〈x3,{[0.8,0.9],[0.9,1.0]}〉,〈x4,{[0.3,0.5]}〉},A4={〈x1,{[0.3,0.4],[0.7,0.8]}〉,〈x2,{[0.1,0.3]}〉,〈x3,{[0.6,0.7],[0.8,0.9]}〉,〈x4,{[0.5,0.7]}〉},A1={〈x1,{[0.2,0.3]}〉,〈x2,{[0.4,0.6]}〉,〈x3,{[0.2,0.3],[0.6,0.7]}〉,〈x4,{[0.6,0.7]}〉}.

Step 2. Calculate the positive ideal IVHFS and the negative ideal IVHFS as follows, respectively:

A+={〈x1,[0.7,0.8]}〉,〈x2,{[0.6,0.7]}〉,〈x3,{[0.9,1.0]}〉,〈x4,{[0.8,0.9]}〉},A−={〈x1,{[0.2,0.3]}〉,〈x2,{[0.1,0.3]}〉,〈x3,[0.2,0.3]}〉,〈x4,{[0.3,0.5]}〉}.

For Step 3, we can utilize different similarity measures provided in Section 4 to determine the priority of the alternatives.

(Case 1) Utilizing the similarity measure s1' with the weight vector ω for IVHFSs based on II-type generalized interval-valued hesitant fuzzy distance measures, i.e., Eq. (14).

Step 3. By (14), we calculate s1'(A+,Ai) and s1'(A−,Ai)(i=1,2,3,4,5) and assume that p = 2.

s1'(A+,A1)=0.6682,s1'(A+,A2)=0.6140,s1'(A+,A3)=0.8070,s1'(A+,A4)=0.7037,s1'(A+,A5)=0.5868;s1'(A−,A1)=0.7556,s1'(A−,A2)=0.7850,s1'(A−,A3)=0.6256,s1'(A−,A4)=0.6999,s1'(A−,A5)=0.8296.

Step 4. Calculate the relative similarity measure rsi corresponding to the alternative Ai using Eq. (23)

rs1=0.4693,rs2=0.4389,rs3=0.5633,rs4=0.5014,rs5=0.4121.

Step 5. Obtain the priority of the alternative Ai(i = 1, 2, ⋯ , 5) by ranking the rsi(i = 1, 2, ⋯ , 5).

A3≻A4≻A1≻A2≻A5.

(Case 2) Utilizing the similarity measure s2' with the weight vector ω for IVHFSs based on the interval-valued hesitant fuzzy Lp distance.

Step 3. Calculate s2'(A+,Ai) and s2'(A−,Ai) (i = 1, 2, 3, 4, 5) and assume that p = 2.

s2'(A+,A1)=0.3969,s2'(A+,A2)=0.2695,s2'(A+,A3)=0.6776,s2'(A+,A4)=0.4754,s2'(A+,A5)=0.3210;s2'(A−,A1)=0.5573,s2'(A−,A2)=0.5501,s2'(A−,A3)=0.3340,s2'(A−,A4)=0.4116,s2'(A−,A5)=0.7094.

Step 4. Calculate the relative similarity measure rsi corresponding to the alternative Ai using Eq. (23)

rs1=0.4159,rs2=0.3288,rs3=0.6698,rs4=0.5360,rs5=0.3115.

Step 5. Obtain the priority of the alternative Ai(i = 1, 2, ⋯ , 5) by ranking the rsi(i = 1, 2, ⋯ , 5).

A3≻A4≻A1≻A2≻A5.

(Case 3) Utilizing the similarity measure s 3 with the weight vector for IVHFSs based on the set-theoretic approach.

Step 3. Calculate s3'(A+,Ai) and s3'(A−,Ai) (i = 1, 2, 3, 4, 5).

s3'A+,A1)=0.2097,s3'(A+,A2)=0.2789,s3'(A+,A3)=0.3866,s3'A+,A4)=0.2667,s3'(A+,A5)=0.2810;s3'(A−,A1)=0.5593,s3'(A−,A2)=0.5126,s3'(A−,A3)=0.3481,s3'(A−,A4)=0.5057,s3'(A−,A5)=0.65.

Step 4. Calculate the relative similarity measure rsi corresponding to the alternative Ai using Eq. (23)

rs1=0.2727,rs2=0.3523,rs3=0.5262,rs4=0.3453,rs5=0.3018.

Step 5. Obtain the priority of the alternative Ai(i = 1, 2, ⋯ , 5) by ranking the rsi(i = 1, 2, ⋯ , 5).

A3≻A2≻A4≻A5≻A1.

According to the above numerical results for three cases utilizing different similarity measures under interval-valued hesitant fuzzy environment, the ordering of the alternatives are given in Eq. (24), Eq. (25), Eq. (26) respectively. The ≻ in Eqs. (24)–(26) means ”preferred to”. We can see from the results, depending on the similarity measures used, the ordering of the alternatives is slightly different. Therefore, depending on the similarity measure used, the result will lead to different ranking of other alternatives, but the best desirable alternative is the same. From Eq. (24) to Eq. (26), we know that alternative A3 is the best one obtained by using all three similarity measures proposed in Section 4.

In addition, in22, Wei. et al. applied an interval-valued hesitant fuzzy choquet ordered averaging (IVHFCOA) operator and a hesitant interval-valued fuzzy choquet ordered geometric (IVHFCOG) operator to the same practical example with interval-valued hesitant fuzzy information. As the results of aggregation are some intervals, so, a score function of IVFHNs was employed in order to obtain the ordering of the alternatives. The result is the same as the one obtained using the proposed method based on the relative similarity measures for IVHFSs. This shows the proposed method is rational and practically feasible.

7. Conclusions

IVHFSs, which is a generalization of the HFSs, have been used widely in decision problems17,19. IVHFS permits the memberships of an element to a given set having a few different interval values in [0, 1] rather than precise single numbers, so it can be considered as a useful tool to express uncertain information in the human decision making process. In this paper, our aim mainly focused on the following two aspects:

  1. (1)

    New information measure A generalization of popular Haming-distance (that is, a II-type generalized interval-valued hesitant fuzzy distance) and an interval-valued hesitant fuzzy Lp distance were introduced based on some existing distance measures for IVHFSs and Lp metric, the relation between interval-valued hesitant fuzzy Lp distance and interval-valued hesitant fuzzy Hausdorff distance was also discussed. About the similarity measure, we analysed the existing similarity measures, a family of similarity measures for IVHFSs were obtained and a new similarity measure, which is not dependent on the distance measure in IVHFSs from the view of set theory, was proposed.

  2. (2)

    Application of relative similarity measure We proposed the concept of relative similarity measure of IVHFSs based on the new similarity measure in (1) and constructed a multiple attribute decision making approach under interval-valued hesitant fuzzy environment and based on the relative similarity measure. Consequently, a practical example was provided to illustrate this method and compare it with the existing one in22.

In the future, we shall continue working applying the interval-valued hesitant fuzzy multiple attribute decision making approach to other problem domains, and also focus on other information measures on hesitant fuzzy set and its application in the real world.

Acknowledgments

The authors are grateful to the anonymous reviewers for their constructive comments and based on which the presentation of this paper has been greatly improved. This work is partially supported by the National Natural Science Foundation of P.R.China (Grant no. 61673320, 61305074); Chinese Scholarship Council of the Ministry of Education ([2016]5112); the Application Basic Research Plan Project of Sichuan Province (No.2015JY0120); the Scientific Research Project of Department of Education of Sichuan Province (15TD0027, 14ZA0245, 15ZB0270); Natural Science Foundation of Guangdong Province (2016A030310003).

References

6.N Chen and ZS Xu, Properties of interval-valued hesitant fuzzy sets, Journal of Intelligent and Fuzzy Systems, Vol. 27, 2014, pp. 143-158.
9.B Farhadinia, Study on division and subtraction operations for hesitant fuzzy sets, interval-valued hesitant fuzzy sets and typical dual hesitant fuzzy sets, Journal of Intelligent Fuzzy Systems, Vol. 28, 2015, pp. 1393-1402.
11.Y Liu, Y Qin, and Y Han, Multiple criteria decision making with probabilities in interval- valued Pythagorean fuzzy setting, International Journal of Fuzzy System.
21.GW Wei, R Lin, and HJ Wang, Distance and similarity measures for hesitant interval-valued fuzzy sets, Journal of Intelligent Fuzzy Systems, Vol. 27, 2014, pp. 19-36.
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Journal
International Journal of Computational Intelligence Systems
Volume-Issue
11 - 1
Pages
15 - 32
Publication Date
2018/01/01
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.11.1.2How to use a DOI?
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risenwbib
TY  - JOUR
AU  - Yi Liu
AU  - Jun Liu
AU  - Zhiyong Hong
PY  - 2018
DA  - 2018/01/01
TI  - A Multiple Attribute Decision Making Approach Based on New Similarity Measures of Interval-valued Hesitant Fuzzy Sets
JO  - International Journal of Computational Intelligence Systems
SP  - 15
EP  - 32
VL  - 11
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.11.1.2
DO  - 10.2991/ijcis.11.1.2
ID  - Liu2018
ER  -