Ranking fuzzy numbers based on epsilon-deviation degree
Graphical abstract
Introduction
Zadeh [1] introduced the fuzzy set theory as a great tool, mathematically representing uncertainty and vagueness in order to efficiently deal with knowledge associated with imprecision. A fuzzy set can greatly reduce uncertainty and has been applied to problems in a variety of fields such as supply chain management, underground mining, weaponry, and so forth [2], [3], [4], [5], [6], [7], [8], [9].
One of the most useful applications of the fuzzy set theory is in decision making, a cognitive process. The fuzzy set theory describes the approximate information of values and preferences of the decision maker that involve uncertainty, thus generating a decision by ranking fuzzy numbers representing the imprecise numerical measurement of alternatives. However, selecting an optimal alternative among a set of possibilities under a fuzzy environment is complex and challenging.
The literature over the past few decades has proposed numerous methods for ranking fuzzy numbers. Some of them are legendary for ranking fuzzy numbers, such as maximizing sets and minimizing sets, centroid points, and distance minimization. Jain [10] proposed the first ranking method using the maximizing set to order fuzzy numbers for selecting an optimal alternative. Yager [11] presented the centroid-index method, Dubios and Prade [12] used the maximizing set to order fuzzy numbers, and Chen [13] proposed the maximizing and minimizing set approach. Chu and Tsao [14] ranked fuzzy numbers with an area between the centroid point and original point. Wang and Yang [15] presented the centroid of fuzzy numbers. Abbasbandy and Asady [16] suggested sign distance, while Asady and Zendehnam [17] proposed the distance minimization method. Other methods have offered revisions to achieve more completeness [18], [19], but most of them are unable to give satisfied ranking results for all situations due to the complexity. Many researchers have recently employed maximizing set and minimizing set and the concept of a centroid point as the basis for comparing and ranking fuzzy numbers [20], [21], [22], [23], [24]. These methods have mainly concern the correlation between L-R areas and centroid point of a fuzzy number. Wang et al. [20] defined the L-R deviation degree of a fuzzy number and came up with the ranking rule, in which the larger the left deviation degree and the smaller the right deviation degree are, the larger the fuzzy number is. Asady [21] and Nejad Mashinchi [22] redefined the L-R deviation degree of a fuzzy number to overcome the shortcomings of Wang et al. [20]. However, most deviation degree approaches still display the same limitations due to the neglected decision maker's attitude, the incoherent transfer coefficient formula, and the unreliable ranking index computation.
To eliminate all these aforementioned drawbacks, this study introduces a new epsilon-deviation degree approach on the basis of maximizing set and minimizing set and centroid point and considers the decision maker's attitude for ranking fuzzy numbers. The rest of this paper is organized as follows. Section 2 briefly reviews the concept of fuzzy numbers, previous approaches and presents the shortcomings through several counter-examples. Section 3 shows the proposed new approach with two innovative indices for ranking fuzzy numbers. The proposed approach is compared with existing approaches. Results and discussions are presented in Section 4. Finally, conclusions are drawn in Section 5, in which the contributions of this research are presented and the major findings are highlighted.
Section snippets
Concept of fuzzy numbers
This section defines the concept of fuzzy numbers [12].
Definition 1: A real fuzzy number is described as any fuzzy subset of the real line R with membership function uA which is given by:where is a constant.
Note that and are strictly monotonic and continuous, mapping from R to closed interval Since and are strictly monotonic and continuous, their inverse functions
The proposed epsilon-deviation degree method
To overcome the weaknesses of the existing approaches based on deviation degree, this study introduces an epsilon-deviation degree method for ranking fuzzy numbers The proposed ranking index integrates an epsilon-transfer coefficient, L-R areas of a fuzzy number and decision maker's attitude.
Definition 3: The L-R areas of a generalized fuzzy number are defined by the following equations:
where
Results
In this section, the proposed method is used to solve the three counter-examples discussed in Section 2.3 to demonstrate its reliability. Two additional numerical examples are also presented to show the superiority of the proposed approach. Let ε1 = 0.01, ε2 = 0.01, ρ = 1 in all the examples.
Example 1
Set 1: Examine the three fuzzy numbers A1, A2 and A3 shown in Fig. 1. By using the new ranking index value, the ranking orders of the fuzzy numbers and their images are and ,
Conclusion
This study proposes a new epsilon-deviation degree approach for ranking fuzzy numbers. The proposed method not only considers the left and right side areas of a fuzzy number and the expectation value of centroid, but also the index of optimism. The new epsilon-transfer coefficient shows strong discrimination to overcome the identical centroid points of fuzzy numbers, which are the shortcomings in earlier studies [20], [21]. In comparison to other approaches, what is especially noteworthy is
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2018, Applied Soft Computing JournalCitation Excerpt :It has wide-ranging application in various fields include transportation [7–10], risk assessment [11,12], the wine industry [13], agriculture [14], renewable energy [15]. Ever since Jain [16] proposed the first ranking method for fuzzy numbers by using maximizing sets, many ranking methods for fuzzy numbers have been proposed [3,17–32]. In these approaches, ranking indices, which are associated with transforming fuzzy numbers in real numbers, are mainly generated from areas, a centroid point, median, distance, etc.
Ranking of fuzzy numbers by using value and angle in the epsilon-deviation degree method
2017, Applied Soft Computing JournalCitation Excerpt :Following are some of the shortcomings and limitations of some existing methods of ranking fuzzy numbers. Consider the fuzzy numbers A = (0.1, 0.2, 0.2, 0.6;1.0), B = (0.25, 0.275, 0.275, 0.3;1.0) and C = (0.2, 0.3, 0.3, 0.4;1.0) shown in Fig. 6 and partly derived from Ref. [43]. Ranking of these fuzzy numbers and their images by various methods are displayed in Table. 6.