Advances in Fuzzy Systems

Volume 2018, Article ID 3637897, 10 pages

https://doi.org/10.1155/2018/3637897

## Distance Based Entropy Measure of Interval-Valued Intuitionistic Fuzzy Sets and Its Application in Multicriteria Decision Making

Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan

Correspondence should be addressed to Tabasam Rashid; moc.liamg@dihsar.masabat

Received 9 May 2018; Accepted 1 July 2018; Published 17 July 2018

Academic Editor: Katsuhiro Honda

Copyright © 2018 Tabasam Rashid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Fuzzy entropy means the measurement of fuzziness in a fuzzy set and therefore plays a vital role in solving the fuzzy multicriteria decision making (MCDM) and multicriteria group decision making (MCGDM) problems. In this study, the notion of the measure of distance based entropy for uncertain information in the context of interval-valued intuitionistic fuzzy set (IVIFS) is introduced. The arithmetic and geometric average operators are firstly used to aggregate the interval-valued intuitionistic fuzzy information provided by the decision makers (DMs) or experts corresponding to each alternative, and then the fuzzy entropy of each alternative is calculated based on proposed distance measure. Several numerical examples are solved to demonstrate the application to MCDM and MCGDM problems to show the effectiveness of the proposed approach.

#### 1. Introduction

In fuzzy set theory, the measurement of the degree of fuzziness in fuzzy sets and other extended higher order fuzzy sets is an important concept in dealing with real world problems. Zadeh [1] proposed the fuzzy entropy theory in 1965 which has been realized as a great achievement in various fields of MCDM. To measure of the amount of information in decision processes, De Luca and Termini [2] introduced their axiomatic construction of entropy of fuzzy sets by using the idea of Shannon’s probability entropy. Kaufmann [3] pointed out that the entropy of a fuzzy set can be found by calculating the distance between the fuzzy set and its nearest nonfuzzy set. However, Yager [4] constructed the entropy of a fuzzy set by using the distance of a fuzzy set to its complement. Based on the concept of entropy measure of a fuzzy set proposed by De Luca and Termini [2] and Zadeh [5], Loo [6] further extended the definition of entropy and developed new entropy measure of a fuzzy set. Liu [7] proposed the axiomatic definition of similarity, distance, and entropy measure of fuzzy set whereas Mi et al. [8] introduced a generalized axiomatic definition of entropy based on a distance in comparison with the axiomatic definition of Liu [7]. A great number of scholars have introduced various techniques of entropy measures for fuzzy set [9–11]. Szmidt [12] and Vlachos [13] introduced some entropy measures for intuitionistic fuzzy sets (IFSs) and discussed their applications in pattern recognition. Furthermore, Burillo and Bustince [14] proposed some entropy measures for interval-valued fuzzy sets (IVFSs) and IFSs while Zeng [15] and Zhang [16] proposed different entropy measures in the context of IVFSs. Liu [7], Zeng [17, 18], and Li [19] investigated some valuable relationships between similarity, distance, and entropy measures for fuzzy sets, IVFSs and IFSs. Farhadinia [20] came up with some entropy measures in the context of hesitant fuzzy sets and interval-valued fuzzy soft sets.

The IVIFS introduced by Atanassov [21] is a very useful generalization of IFS [22]. The IVIFS greatly helps in working under vagueness and uncertainty with a membership and nonmembership interval values of an element belonging to an IVIFS instead of real values in . In the uncertain environments, IVIFS has played a powerful role and received great consideration from the researchers. Most of the researchers got useful results in solving the MCDM [23–26] and MCGDM [27, 28] problems in the context of IVIFSs. In the past few years, the distance, similarity, inclusion, and information entropy measures for IVIFSs were very important topics. Therefore, there are a large number of researchers who have investigated their studies considering all the measures as mentioned above. Huang et al. [29] proposed an idea of information entropy for IVIFS and used it in uncertain system control and decision making. Furthermore, more work on entropy measures for IVIFSs has been proposed by many researchers in different viewpoints; for example, Liu [30], Zhang [31–33], Wei [34], and Ye [35] have investigated and contributed a lot of work on this topic and discussed their applications to solve different MCDM problems in real life. However, many entropy measures for IVIFS are very complicated and unacceptable in the intuitive sense. In order to overcome this weakness in the existing entropy measures, Zhang et al. [36] proposed the distance based information entropy measure for IFS and IVIFSs and then established some relationships between the entropy, similarity, and inclusion measures of IVIFSs.

Recently, the studies on the similarity and entropy measures of different extensions of fuzzy sets are growing; for example, Pramanik et al. [37] discussed NS-cross entropy and designed a MAGDM strategy which is symmetric, free from undefined phenomena, and very useful for MCGDM problems. Dalapati et al. [38] further introduced IN-cross entropy and weighted IN-cross entropy and used them to design a MCGDM strategy. Pramanik et al. [39] discussed vector similarity measures of single-valued and interval neutrosophic sets by hybridizing the ideas of dice and cosine similarity measures and presented its applications in MCDM problems. Uluay et al. [40] proposed dice similarity, weighted dice similarity, hybrid vector similarity, and weighted hybrid vector similarity measures for bipolar neutrosophic sets and presented a MCDM strategy. Pramanik et al. [41] discussed cross entropy measures of bipolar neutrosophic sets and interval bipolar neutrosophic sets and designed MCDM strategies based on the cross entropy measures.

The distance between fuzzy sets and the entropy measure are very important to calculate the degree of fuzziness of a fuzzy set, and a lot of work as mentioned above has been contributed to the literature by various researchers. Therefore, motivated by the advantages of the entropy measures of different extensions of fuzzy sets, in this paper we proposes an entropy measure based on new distance measure for IVIFSs. In this approach, we first put forward a useful distance measure for IVIFSs which is completely different from the existing ones and then propose an entropy measure formula based on this distance measure. Furthermore, a useful comparison of the proposed entropy measure with the existing entropy measures formulae is performed to avoid any inconsistency in the proposed approach. In order to ensure the practicality and effectiveness of the proposed approach, we finally apply this approach to solve various MCDM and MCGDM problems using IVIFSs.

The remaining part of the paper can be summarized briefly as follows. Some basic concepts related to the work are presented in Section 2. A comparison of the performance of proposed entropy measure to those of all known entropy measures is established in Section 3. Different approaches are developed in Section 4 to handle MCDM and MCGDM problems using IVIFSs. Several numerical examples are provided to make out the practicality and effectiveness of the proposed approach in Section 5. We wind up the paper with some useful remarks in Section 6.

#### 2. Preliminary

In this section, basic definitions of IVIFS, the weighted arithmetic and geometric average operators, and some distance measures for IVIFSs are presented. The entropy measure for IVIFS based on different distance measures along with some basic properties is also discussed in this section. Furthermore, throughout the paper, the set of all closed subintervals of is denoted by [], the domain or universe of discourse is denoted by a set , the collection of all the crisp sets is denoted by , and the set of all IVIFSs in are denoted by .

*Definition 1 (see [14]). *Let , . For the comparison of two elements of , one defines

*Definition 2 (see [21]). *An IVIFS in is defined by a set , where are called the interval membership and nonmembership degrees of an element to , respectively, provided that for any .

*Definition 3. *For , where and , some basic operations and relations are defined as follows: (1);(2);(3);(4) and ;(5) and

*Definition 4 (see [27, 42]). *Let and . Then (1);(2);(3);(4)In the following, Xu [43] introduced two weighted aggregation operators related to IVIFSs as follows.

*Definition 5 (see [43]). *The weighted arithmetic average operator for , is defined by where and . The element is denoted as the weight of . In particular, is called the arithmetic average operator for IVIFSs if .

*Definition 6 (see [43]). *The weighted geometric average operator for , , is defined by where and . The element is denoted as the weight of . In particular, is called the geometric average operator for IVIFSs if .

*Definition 7. *The distance measure of IVIFSs on is a real valued function which satisfies the following properties: (1)If A is a crisp set, then ;(2);(3);(4)If , then and

*Definition 8. *For any two IVIFSs in , the following six distance measures were discussed in [36].

Consider , where . (1)Normalized Euclidean distance(2)Normalized hamming distance(3)Normalized hamming distance measure induced by Hausdorff metric(4)Normalized distance measure induced by Hausdorff metric(5)Fifth distance measure(6)Generalized distance measure, for any ,

*Definition 9 (see [31]). *The entropy measure of IVIFSs is a function which satisfies the following four properties. (1) if is a crisp set;(2);(3), if is less fuzzy than ; i.e., , for ; , for ; , for ; , for ;(4).

*Definition 10 (see [36]). *The entropy measure of IVIFS on can be defined as a function which fulfills the properties as follows: (1), i.e., if is a crisp set;(2);(3)If , then ;(4).

*Remark 11. *The properties as discussed above reveal the following important points: (1)a crisp set is always nonfuzzy; i.e., an IVIFS has no vagueness when it degenerates to a crisp set;(2)the fuzziest IVIFS is ;(3)the IVIFS is more fuzzier when it is closer to ;(4)the IVIFS and its complement have the same fuzziness.

Zhang [36] introduced the following useful method to obtain an entropy measure of IVIFSs, which possesses all the above-mentioned four properties.

*Definition 12 (see [36]). *Suppose the six distance measures between IVIFSs are denoted by as mentioned before. Then, for any , , are known as the entropy measures of IVIFSs where are the distance measures as discussed in Definition 8.

#### 3. Comparison of Entropy Measures for IVIFSs

Liu [7] developed few distance measures of fuzzy sets and discussed some of their desired properties. Zeng [17] and Grzegorzewski [44] developed a number of normalized distance measures for IVFSs. Park [45] investigated various distances over IVIFSs in . Motivated by the above work, we can set our investigation over another useful distance measure of IVIFSs as follows.

*Definition 13. *For , where and , one defines The proposed distance measure of IVIFSs and is defined by .

*Remark 14. *It is easy to show that the new proposed distance satisfies all the four properties discussed in Definition 7.

Some important entropy measures of IVIFSs have been introduced in the literature by various researchers to handle MCDM problems of real-life applications [33–35]. Zhang [31] extended the intuitionistic fuzzy entropy axioms presented in [12] and introduced a new definition of entropy for IVIFSs. Zhang [36] further extended the entropy concept for fuzzy set [2, 7] to introduce another useful distance based entropy measure for IVIFS. Now, we introduce another entropy measure of IVIFSs based on a proposed distance measure as follows.

*Definition 15. *Let be the proposed distance measure (Definition 13) of IVIFSs. Then, is defined as an entropy measure of IVIFSs for any based on the distance measure .

Theorem 16. *Let be the proposed distance measure (Definition 13) between two IVIFSs. Then, is the entropy measure of IVIFS for any .*

*Proof. *We will try to prove that fulfills all the properties discussed in Definition 10.

(1) For or , i.e., if is a crisp set, then, from Definition 13 we have .

Hence, .

(2) (by Definition 13).

(3) ; then it immediately follows that .

(4) . Therefore, it can be easily observed that . Hence .

Zhang [36] compared the performances of the proposed entropies , and to those of [30], [31], [34], [33], and [35]. Now we shall also compare the performance of proposed entropy measure to those of all entropy measures as mentioned above with the help of the following example.

*Example 17. *We will consider here an example, as adapted by Hung and Yang in [46]. Suppose that . For any , define the IVIFS as follows:The IVIFS was defined on the universal set in [46] as follows:and, similarly, we can determine the following IVIFSs:In order to express the characterization of these sets in linguistic variables, De [47] considered as “LARGE” in . By using the above-mentioned operations,

can be considered as “more or less LARGE”;

can be considered as “very LARGE”;

can be considered as “quite very LARGE”;

can be considered as “very very LARGE”.

Based on the mathematical operations, the entropy measures of IVIFSs as discussed above should fulfill the condition as mentioned in [46]; i.e., . Now we try to compare the performance of proposed entropy measures by using the IVIFSs as mentioned above to those of , , and . The entropy values by using all the corresponding distance measures can be seen in Table 1.