Elsevier

Applied Soft Computing

Volume 61, December 2017, Pages 935-946
Applied Soft Computing

Full length article
Correlation coefficients of hesitant multiplicative sets and their applications in decision making and clustering analysis

https://doi.org/10.1016/j.asoc.2017.08.011Get rights and content

Highlights

  • Some correlation coefficient formulas are proposed to measure the relationship between two HMSs.

  • Improve the addition and multiplication operations of Hesitant multiplicative elements to avoid the increasing of values in the derived HME.

  • Demonstrate two practical examples to validate the validity and usability of the correlation coefficients.

Abstract

Hesitant multiplicative set (HMS) can reflect the preferences of decision makers objectively for the reason that it uses the unsymmetrical scale to express the preferences about two alternatives. Considering that there is still no research on correlation coefficients between HMSs, in this paper, we propose some correlation coefficient formulas to measure the relationship between two HMSs. Firstly, we improve the addition and multiplication operations of Hesitant multiplicative elements (HMEs) to avoid the increasing of values in the derived HME. Then a series of new concepts related to the HME and the HMS are defined. Then, some correlation coefficient formulas between two HMSs are defined, and the weighted forms of the correlations and correlation coefficients between HMSs are developed. Finally, we provide two practical examples (online car selling, and star-classification of cities’ happiness by clustering analysis) to validate the validity and usability of the correlation coefficients.

Introduction

Decision making is one of the common human activities existing in politics, economy, technology and so on. It is necessary to take effective way to rank the alternatives and then select the best one [1], [2]. Since it was first introduced by Zadeh [3], the fuzzy set theory has been successfully used for handling fuzzy decision making problems [4], [5], [6]. With the development of society, practical decision making problems become increasingly complex. Besides, the fuzzy set theory has been extended to accommodate different fuzzy environments, and a series of generalized types of fuzzy sets have been put forward, such as type-2 fuzzy set [7], interval type-2 fuzzy set [8], intuitionistic fuzzy set [9], [10], [11], interval-valued intuitionistic fuzzy set [12], and hesitant fuzzy linguistic term set [13], [14], [15].

Torra [16] noticed that when the decision makers make decisions, they are usually hesitant and irresolute, which makes it hard to reach a final agreement. To cope with this problem, Torra [16] introduced the concept of hesitant fuzzy set (HFS) to permit the membership having different possible values. The HFS allows the membership grade of an element to a given set represented by several possible values between 0 and 1. When dealing with the complex decision making problems under uncertainty issues, the hesitate fuzzy theory is more flexible than other extended fuzzy set theories [17], [18]. So the HFS has attracted a lot of attention and turns out to be a powerful tool to handle uncertainty and vagueness in the decision making process. The hesitant fuzzy information is based on the 0–1 scale, which is distributed symmetrically. However, in reality, the information is usually unsymmetrically distributed. Under this circumstance, it is more suitable to use Saaty’s unsymmetrical 1–9 scale instead of the symmetrical 0–1 scale [19]. The law of diminishing marginal utility in economics is a good example to justify the usefulness of the unsymmetrical scale. When increasing the same amount of investments, a company with bad performance gains more utility than that with good performance. That is to say, the gap between the grades expressing good information should be bigger than that between the grades expressing bad information [20], [21]. So the hesitation multiplicative set (HMS) was proposed [22]. Similar to hesitant fuzzy information, the hesitant multiplicative information is also a set of possible values, indicating the preferences of the candidates [23]. But unlike hesitant fuzzy information, the hesitant multiplicative information uses the unsymmetrical 1–9 scale to express the decision makers’ preferences. As the hesitant multiplicative information can reflect the preferences of the decision makers more objectively, decision-making based on hesitant multiplicative information has broad application prospects.

Correlation is an indicator which measures how well two variables move together in a linear fashion. It has been widely used in many fields, such as artificial intelligence [24], [25], biomedical [26], earthquake engineering [27], digital image process [28], [29], pattern recognition [30], and decision analysis [31], [32]. The correlation coefficient has been extended into different fuzzy circumstances. Different forms of fuzzy correlations and correlation coefficients have been proposed, such as fuzzy correlations and correlation coefficients [33], [34], [35], and hesitant fuzzy correlations and correlation coefficients [36], [37].

As presented above, the HMS is more objectively in expressing vagueness and uncertainty. However, the addition and multiplication operations of HMEs have some drawbacks, that is, both the addition and multiplication operations of HMEs can increase the number of values in the derived HME and they are also complicated in the calculation process. In addition, there is still no investigation on the correlation coefficients between HMSs. Based on these reasons, in this paper, we firstly improve the addition and multiplication operations of HMEs. Then we propose the concepts of mean and variance of the HMS as well as some correlations and correlation coefficients for HMSs. Afterwards, we provide two practical applications to validate the validity and usability of correlation coefficients formulas for HMSs.

The remainder of this paper is organized as follows: Section 2 reviews the concepts of HFS and HMS, and then some improvements of the operations on HMSs are proposed. In Section 3, some types of correlations and correlation coefficients between HMSs are proposed. In Section 4, two numerical and practical application examples are provided to support our studies. Finally, the paper ends with some concluding remarks in Section 5.

Section snippets

Preliminaries

In this section, let us first review several concepts related to HFSs and HMSs.

Several correlation coefficients for HMSs

In the following, we first define the concept of correlation coefficients for HMSs, and then give several correlation coefficients and discuss their properties.

Correlation coefficient is a basic tool to represent the relationship between elements with uncertain information. Xu and Xia [36] presented several correlation coefficients of HFEs on the assumption that the two compared HFEs have the same length. Furthermore, Chen et al. [37] introduced a correlation coefficient of HFSs, assuming that

Applications of the correlation coefficient between HMSs

In this section, two examples are provided to show the practical applications of the correlation coefficients between HMSs in different fields.

Conclusion and further study

In practical applications, the HMS is an efficient tool to represent uncertain information, which can reflect the preferences of the decision makers more objectively. As the research on HMS is still relatively slow, and there is still no work on the correlation coefficients between HMSs, in this paper, we have proposed some correlation coefficients to measure the relationship between two HMSs. Firstly, in order to solve the problem that the existing addition and multiplication operations of

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