An improved fuzzy risk analysis based on a new similarity measures of generalized fuzzy numbers

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Abstract

This paper presents a novel method of fuzzy risk analysis based on a new similarity measure of generalized fuzzy numbers. This similarity measure considers many features of generalized fuzzy numbers such as the area, perimeter, height and geometric distance of these kinds of fuzzy numbers. Using some sets of generalized fuzzy numbers, we show the power of this similarity measurement method to overcome the drawbacks that other methods are suffering. Applying the proposed method, we present an improved fuzzy risk analysis method which develops the capability of fuzzy risk analysis methods to deal with sophisticated problems. In the proposed method we use new factors such as probability of failure detection and economic disbenefits of failure occurrence which have not been used in fuzzy risk analysis methods before.

Introduction

The task of handling risk analysis problem plays an important role in industries. Many different approaches in this field have been introduced. Wei and Chen (2009) proposed a fuzzy risk analysis method based on a new similarity measure between fuzzy numbers. Wang, Chin, Poon, and Yang (2009) presented a risk evolution in failure and effect analysis method using fuzzy weighted geometry mean. Lee and Chen (2008) presented a fuzzy risk analysis method based on fuzzy numbers with different shapes and different deviations. Wang & Chen (2008) presented a Fuzzy risk analysis based on ranking fuzzy numbers using a-cuts. Chen and Chen (2009) presented a fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads.

Chen (2007) presented a fuzzy risk analysis method based on similarity measures of fuzzy numbers. Chen & Chen proposed a new method for handling fuzzy risk analysis problems using ranking generalized fuzzy numbers. Chen & Chen (2003) presented a method for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers. Schmucker (1984) presented a method for fuzzy risk analysis based on fuzzy number arithmetic operations. Chen and Chen (2006) presented a method for ranking generalized fuzzy numbers for handling fuzzy risk analysis problems. Wang and Elhang (2006) presented the TOPSIS method based on alpha level sets and nonlinear programming (NLP) solution procedure. It is obvious that many facts can affect the result of a similarity measure, e.g., the shapes of fuzzy numbers, the positions of fuzzy numbers, …, etc. In recent years, some similarity measures between fuzzy numbers have been presented. (Chen, 1996, Hsieh and Chen, 1999, Lee, 1999). Wei & Chen (2008) presented a similarity measure between fuzzy numbers combining the concept of geometric distance, height and perimeters of generalized fuzzy numbers. Chen (1996) presented a similarity measure between fuzzy numbers for subjective mental workload assessment and fuzzy risk analysis. Chen & Chen (2001) combined the concept of the geometric distance and the center of gravity (COG) distance to propose a similarity measure between generalized fuzzy numbers. Hsieh and Chen (1999) presented a similarity measure between fuzzy numbers using the “graded mean integration representation distance”.

Most of the fuzzy risk analysis methods which have been introduced so far do not take some important aspects of risk analysis into consideration. In these approaches only two factors, probability of failure and severity of loss are taken into account and some influencing factors such as factor of failure detection, factor of maintainability, chance of failure and economic safety are overlooked.

In this paper we present a modified fuzzy risk analysis method based on a new similarity measure between fuzzy numbers. In this modified method some important factors like factor of failure detection, factor of maintainability, chance of failure and factor of economic reasonability are considered. Also the new similarity measure presented here, combines the concept of geometric distance, the area of generalized fuzzy number, the perimeter and the height of the generalized fuzzy number.

Section snippets

Preliminaries

In this section we briefly describe the basic concepts of generalized fuzzy numbers. Chen (1985) proposed the concept of generalized fuzzy numbers.

Let A be a generalized trapezoidal fuzzy number, A=(a1,a2,a3,a4,WA), as shown in Fig. 1, where a1, a2, a3, a4 are real values, WA denotes the height of the generalized fuzzy number A, and wA[0,1]. If −1   a1  a2  a3  a4  1, then A is called a standardized generalized fuzzy number. If WA=1, then A becomes a traditional fuzzy number and can be

A review of the existing similarity measures between fuzzy numbers

In this section, we review some of existing similarity measures between fuzzy numbers. These are from Hsieh and Chen, 1999, Lee, 1999, Chen and Chen, 2008, Wei and Chen, 2009.

Let A and B be two trapezoidal fuzzy numbers, where A=(a1,a2,a3,a4) and B=(b1,b2,b3,b4) as shown in Fig. 2.

Hsieh and Chen (1999) presented a similarity measure between fuzzy numbers using “graded mean integration representation distance”, where the degree of similarity S(A,B) between the fuzzy numbers A and B is

A new method for determining the degree of similarity between two generalized fuzzy numbers

Our proposed method is based on the model delineated by Wei & Chen (2008). In this method the geometric distance, the perimeter of the two fuzzy numbers and the area of the two fuzzy numbers are considered.

Let A and B be two generalized trapezoidal fuzzy numbers, where A=(a1,a2,a3,a4;wA) and B=(b1,b2,b3,b4;wB),0a1a2a3a41 and 0  b1  b2  b3  b4  1.

Now we define our measure of similarity that will be used in other sections of the paper.S(A,B)=1-i=14|ai-bi|4×min(P(A),P(B))max(P(A),P(B))

A comparison of similarity measures

In this section we make comparisons between the proposed similarity measure method with five other methods using 15 sets of fuzzy numbers which are shown in Fig. 3 and then we will analyze the quality of the results of each method in the following. These results are shown in Table 1. These sets of fuzzy numbers are from Wei & Chen (2009).

As it can see from results of comparisons between different similarity measures using different trapezoidal fuzzy numbers in Table 1, the drawbacks of these

Proposed fuzzy risk analysis method based on the modified similarity measure

In this section we propose our new fuzzy risk analysis method which considers new factors in fuzzy risk analysis.

Up to now, many different fuzzy risk analysis methods are delineated that each of them uses diverse factors to deal with such problems. In this paper we use Schmuker Risk Analysis Method (Schmucker, 1984) as a basis for our proposed method. Unlike nearly all fuzzy risk analysis methods which are proposed so far, our method considers some other influencing factors like Probability of

Conclusions

In this paper we proposed a modified approach to the task of fuzzy risk analysis method. To achieve this we delineated a new similarity measure between fuzzy numbers that combined different characteristics of generalized fuzzy numbers such as geometric distance, height, area and perimeter of these kinds of fuzzy numbers. Our main approach was to represent a method that can evaluate the risk of manufacturing components in a timely manner.

In this paper, we used linguistics variables that can be

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