Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3080679, 9 pages

http://dx.doi.org/10.1155/2016/3080679

## Fuzzy Risk Analysis for a Production System Based on the Nagel Point of a Triangle

Department of Mathematics, Faculty of Science, Anadolu University, 26470 Eskisehir, Turkey

Received 17 December 2015; Accepted 9 March 2016

Academic Editor: Rosana Rodriguez-Lopez

Copyright © 2016 Handan Akyar. 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

Ordering and ranking fuzzy numbers and their comparisons play a significant role in decision-making problems such as social and economic systems, forecasting, optimization, and risk analysis problems. In this paper, a new method for ordering triangular fuzzy numbers using the Nagel point of a triangle is presented. With the aid of the proposed method, reasonable properties of ordering fuzzy numbers are verified. Certain comparative examples are given to illustrate the advantages of the new method. Many papers have been devoted to studies on fuzzy ranking methods, but some of these studies have certain shortcomings. The proposed method overcomes the drawbacks of the existing methods in the literature. The suggested method can order triangular fuzzy numbers as well as crisp numbers and fuzzy numbers with the same centroid point. An application to the fuzzy risk analysis problem is given, based on the suggested ordering approach.

#### 1. Introduction

Recently, ordering and ranking fuzzy numbers and their comparisons have become the main research interest in fuzzy decision analysis and real world problems (e.g., see [1–10] and references therein).

Prominent methods for the ordering and ranking of fuzzy numbers have been published since the method for ranking fuzzy numbers was initially proposed by Jain (e.g., see [2, 3, 10–25] and references therein). The most widely used approach for ordering and ranking triangular fuzzy numbers is based on the centroid of a triangle, proposed by Yager (see [12]). Yager’s method considers only the horizontal coordinate of the centroid for the ranking, which lessens its effectiveness. Murakami et al. improved Yager’s method using both coordinates of the centroid (see [10]). To overcome the shortcomings of the aforementioned centroid methods, S.-J. Chen and S.-M. Chen submitted a ranking method that uses the centroid point and the standard deviation of fuzzy numbers (see [3]). In [26], Nazirah and Daud wholly reviewed certain centroid index methods.

In [22], Lee and Chen propose a new method for fuzzy risk analysis based on fuzzy numbers with different shapes and deviations. In [14], Abbasbandy and Asady propose a sign distance method, where its drawbacks are treated in [15] by constructing a revised sign distance method. A new method is proposed by Ezzati et al. [20] that is a modification of Abbasbandy and Hajjari’s method (see [23]) which can also be used for ranking symmetric fuzzy numbers. In [19], Ezzati et al. revise this method to overcome certain shortcomings.

Another approach for the ranking and ordering of fuzzy numbers is given by Akyar et al., based on the incenter of a triangle, which is the center of an inscribed circle (see [2, 18]). However, it will be shown that this method has certain shortcomings, namely, the fact that the result of this method is sometimes inconsistent with human intuition. Düzce suggests a different method using the center of the nine-point circle of a triangle (see [21]). However, Düzce’s method also has certain drawbacks, namely, the fact that this method cannot rank fuzzy numbers with the same centroid.

Recently, an improved method, which considers the areas of the positive side, the areas of the negative side, and the spreads of generalized fuzzy numbers as the ranking factors for ranking fuzzy numbers, has been given by Jiang et al. [24]. Another approach for the ranking of generalized fuzzy numbers with different left and right heights is given by Chutia et al. using integral values [25]. This method considers nonnormal -norm trapezoidal fuzzy numbers as well as -norm generalized fuzzy numbers with different left and right heights.

In this paper, the incenter method will be revised using the Nagel point of a triangle to increase efficiency. The Nagel point is the point of intersection of the line segments from the vertices of the triangle to the points of tangency of the opposite excircles [27]. In order to save most of the information of triangular fuzzy numbers, we associate these fuzzy numbers with triplets instead of real numbers. We will show that, unlike previously mentioned methods, we can efficiently order triangular fuzzy numbers by the proposed method.

The rest of this paper is organized as follows. In Section 2, fuzzy numbers, their fundamental properties, basic definitions, and theorems regarding the Nagel point of a triangle are reviewed. Next, in Section 3, we present a new method for ordering triangular fuzzy numbers based on the Nagel point of a triangle. In Section 4, we investigate reasonable properties of ordering triangular fuzzy numbers by the proposed method. In Section 5, numerical examples are given. We compare the results of the proposed method with other existing methods. An application to the fuzzy risk analysis problem of the proposed method is presented in Section 6. Finally, a conclusion is given in Section 7.

#### 2. Preliminaries

##### 2.1. Fuzzy Numbers

Here, certain essential concepts of fuzzy numbers and their basic properties will be given. For further information see [28, 29].

A fuzzy set on a set is a function . Commonly, the symbol is used for the function , and it is said that the fuzzy set is characterized by its membership function which associates each with a real number . The degree to which belongs to is interpreted by the value of .

Let be a fuzzy set on . The support of is given as and the height of is defined as The fuzzy set is called a normal fuzzy set if .

Let be a fuzzy set on and . The -cut (-level set) of the fuzzy set is given by where denotes the closure of sets.

A fuzzy set on is called a convex fuzzy set if its -cuts are convex sets for all .

If is a fuzzy set in , then is called a fuzzy number if(1) is convex,(2) is normal,(3) is upper semicontinuous,(4)the support of is bounded.

Hereafter, lower case lettering such as will be used to denote fuzzy numbers.

Generally, certain special types of fuzzy numbers, such as triangular, trapezoidal, and -fuzzy numbers, are used for real life applications (e.g., see [1–10] and references therein).

Let be two continuous and increasing functions satisfying , . Let be real numbers. The fuzzy number is an -fuzzy number if

In particular, we get trapezoidal fuzzy numbers when the functions and are linear. Furthermore, if and then the fuzzy number is called a triangular fuzzy number. Then we can denote a triangular fuzzy number by a triplet and we can write .

Let and be fuzzy numbers. Then the sum of fuzzy numbers and is defined as In particular, if and are triangular fuzzy numbers, then one can verify that The multiplication of fuzzy numbers and is also defined as Although and are triangular fuzzy numbers, their multiplication is not a triangular fuzzy number. For convenience, we can express it as a triangular fuzzy number using two end points and one peak point. Therefore, for triangular fuzzy numbers and we define . Similarly, we define .

The set of all triangular fuzzy numbers will be denoted by . In this paper, we will only consider triangular fuzzy numbers.

##### 2.2. Nagel Point of a Triangle

The Nagel point of a triangle is defined as the point of intersection of the cevians , , and , where , , and are points where the excircles touch the sides (Figure 1). The existence of the Nagel point was proved in 1836, by German mathematician C. H. Nagel.