Complexity

Volume 2018 (2018), Article ID 6251384, 14 pages

https://doi.org/10.1155/2018/6251384

## Preference Attitude-Based Method for Ranking Intuitionistic Fuzzy Numbers and Its Application in Renewable Energy Selection

^{1}College of Computer and Information, Fujian Agriculture and Forestry University, Fuzhou 350002, China^{2}Institute of Big Data for Agriculture and Forestry, Fujian Agriculture and Forestry University, Fuzhou 350002, China^{3}School of Business, Central South University, Changsha 410083, China^{4}School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Riqing Chen

Received 19 September 2017; Accepted 16 January 2018; Published 22 February 2018

Academic Editor: Sigurdur F. Hafstein

Copyright © 2018 Jian Lin 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

Many applications of intuitionistic fuzzy sets depend on ranking or comparing intuitionistic fuzzy numbers. This paper presents a novel ranking method for intuitionistic fuzzy numbers based on the preference attitudinal accuracy and score functions. The proposed ranking method considers not only the preference attitude of decision maker, but also all the possible values in feasible domain. Some desirable properties of preference attitudinal accuracy and score functions are verified in detail. A total order on the set of intuitionistic fuzzy numbers is established by using the proposed two functions. The proposed ranking method is also applied to select renewable energy. The advantage and validity of the proposed method are shown by comparing with some representative ranking methods.

#### 1. Introduction

Atanassov [1] introduced the concept of intuitionistic fuzzy sets characterized by a membership function, a nonmembership function, and a hesitancy function. Due to the increasing complexity of real life problems, intuitionistic fuzzy set is very suitable for representing fuzzy information under complicated and uncertain settings as an extension of traditional fuzzy set. Intuitionistic fuzzy set theory has been deeply discussed by many scholars since the notations appearance and applied in various fields, such as decision-making [2–7], supplier selection [8–10], pattern recognition [11–14], medical diagnosis [15, 16], and artificial intelligence [17, 18].

Many applications of intuitionistic fuzzy sets depend on ranking or comparing intuitionistic fuzzy numbers. A number of researchers focus on the order relation of intuitionistic fuzzy numbers over the past two decades. Chen and Tan [19] proposed a score function to evaluate the score of intuitionistic fuzzy values. Through analysis on the limitation of Chen and Tan’s score function, Hong and Choi [20] improved their ranking method by adding an accuracy function. Xu [21] gave a kind of order relation to rank the intuitionistic fuzzy numbers by combining the score and accuracy function. Wang et al. [22] introduced a novel score function to measure the degree of suitability. Some desirable properties of the novel score function were discussed. Ye [23] developed an improved algorithm for score functions based on hesitancy degree. By using the intuitionistic fuzzy point operators, Liu and Wang [24] defined a series of new score functions for dealing with multicriteria decision-making problems. Jafarian and Rezvani [25] presented a method for mapping the intuitionistic fuzzy numbers into the crisp values and described the concept of spread value of intuitionistic fuzzy number. To analyze the fuzzy meaning of an intuitionistic fuzzy value, Yu et al. [26] formalized an intuitionistic fuzzy value as a fuzzy subset and determined the dominance relation between two intuitionistic fuzzy values. Guo [27] built a new ranking model based on the viewpoint of amount of information. A total order which extended the usual partial order was analyzed in deep. Zhang and Xu [28] used a special function to define the order of intuitionistic fuzzy numbers. Some good mathematical properties on algebraic intuitionistic fuzzy numbers were also given. Lakshmana et al. [29] derived a total order on the entire class of intuitionistic fuzzy number by applying upper lower dense sequence to the interval. Gupta et al. [30] utilized relative comparisons based on the advantage and disadvantage scores of intuitionistic fuzzy numbers. The relative comparison of intuitionistic fuzzy numbers took the membership, nonmembership, and hesitancy degree into account. Xing et al. [31] proposed an Euclidean distance-based approach to ranking intuitionistic fuzzy numbers. However, the above-mentioned methods for ranking intuitionistic fuzzy numbers do not consider the risk attitude of decision maker which is very flexible and useful in real-world applications. A valid ranking method should be established in accordance with the preference attitude of decision maker. To do this, Chen [32] conducted a comparative analysis of score functions for ranking intuitionistic fuzzy numbers. A parameterized score function was developed to represent a mixed result of positive and negative outcome expectations. Wang et al. [33] proposed a new score function based on relative entropy. The risk attitudes of decision makers were defined by risk preference indexes. Wan et al. [34] utilized the closeness degree to characterize the amount of information based on TOPSIS method. A novel risk attitudinal ranking measure is developed to rank the intuitionistic fuzzy numbers. Nevertheless, the above preference attitude-based ranking methods just focus on the extreme points of feasible domain; the other possible values in feasible domain are ignored. Therefore, these ranking methods may lose some valuable information, which can be useful in determining the order relation of intuitionistic fuzzy numbers. To overcome the shortages of existing ranking methods, this paper proposes a preference attitudinal method for ranking intuitionistic fuzzy numbers based on preference attitudinal accuracy and score functions. The proposed ranking method considers not only the preference attitude of decision maker, but also all the possible values in feasible domain. Some desirable properties of preference attitudinal accuracy and score functions are discussed. Moreover, the proposed method is applied to deal with renewable energy selection problem. The advantage and validity of the proposed method are shown in detail by comparing with some representative ranking methods.

The rest of this paper is structured as follows. Section 2 introduces some basic concepts on intuitionistic fuzzy set. In Section 3, the preference attitudinal accuracy and score functions of intuitionistic fuzzy numbers are proposed. Section 4 presents the order relation between intuitionistic fuzzy numbers. Numeral examples and comparison are shown in Section 5.

#### 2. Preliminaries

In the following, some basic concepts on intuitionistic fuzzy set are introduced to facilitate future discussions.

*Definition 1 (see [1]). *Let be a universe of discourse. An intuitionistic fuzzy set over is expressed aswhere and are the membership degree and nonmembership degree of to , respectively, such that , . The hesitation degree of to is denoted by .

Obviously, for each , we have . For simplicity, is called an intuitionistic fuzzy number (IFN), such that , , and . The set of all IFNs is denoted by .

Let and be two IFNs. Clearly, if and only if and . Based on the score function [19] and accuracy function [20], Xu [21] introduced the operational laws and ordering relation among intuitionistic fuzzy numbers as follows.

*Definition 2. *Let and be two IFNs; then (1),(2), .

*Definition 3. *Let be an IFN. The accuracy and score function of are, respectively, represented byIt is clear that and . From (2) and (3), we have . Thus, holds.

*Definition 4. *Let and be two IFNs. The ordering relation is established as follows:(1)If , then .(2)If , then(a)if , then ,(b)if , then .

#### 3. The Preference Attitudinal Accuracy and Score Functions of Intuitionistic Fuzzy Numbers

In this section, we will propose the definitions of preference attitudinal accuracy and score functions by adding an attitudinal parameter in double integral.

##### 3.1. The Preference Attitudinal Accuracy Function

*Definition 5. *Let be an IFN. The feasible domain of is represented bywhere . By considering the hesitation degree of , denotes the set of feasible intuitionistic fuzzy numbers with respect to . The feasible domain is illustrated in Figure 1.