۱- Introduction

۲- Definitions and notations

۳- Review of some existing similarity measures between fuzzy numbers and its limitations

۴- Proposed similarity measure between GFNs with different left heights and right heights

۵- Comparative analysis

۶- Application of the proposed method in risk analysis

۷- Conclusions

References

Abstract

Similarity measure of fuzzy numbers plays an important role in the risk analysis problem. Generally, it is tool, which gives lingustic term to the risk obtained. In recent times, a vast numbers of literature are evident on application of similarity measure in risk analysis. It has been observed that the existing similarity measure on fuzzy numbers have numerous drawbacks and limitations. Hence, a robust method of similarity measure is necessary. With this point of view, a new method to measure the degree of similarity between fuzzy numbers has been proposed. The method has been discussed based on the concept of value, ambiguity, radius of gyration point, geometric distance and the height of fuzzy numbers. The concept of value and ambiguity have never been used in similarity measure of fuzzy numbers. However, the inclusion of these concepts value and ambiguity contributed in many ways in overcoming the limitations and drawbacks of the existing similarity measures. The out-performance of the proposed method is illustrated by comparing with existing methods of similarity measure. Further the proposed method is effectively applied in risk analysis of poultry farming.

Introduction

Fuzzy risk analysis has become very popular in recent times as the knowledge of expressing imprecise quantity in terms of fuzzy numbers has emerged. Most of the time similarity measure between fuzzy numbers is used in the risk analysis problem and other decision making problem. The similarity measures are defined on the different characteristic of the fuzzy number such as geometric distance, center of gravity (COG), area, radius of gyration (ROG) etc. Further, these measures are being generalized for use in different types of fuzzy numbers. It has been observed that the existing similarity measures on fuzzy numbers bear various limitations and drawbacks. A review of some of the existing methods to measure the degree of similarity reveals various limitations and drawbacks. Chen (1996) defined a similarity measure based on the geometric distance. This definition does not carry the information about the shape of the fuzzy numbers such as triangular, trapezoidal, etc. Hence, in many circumstances this method fails to give a proper degree of similarity between fuzzy numbers. Hsieh and Chen (1999) proposed a similarity measure between two fuzzy numbers using graded mean integration representation distance. This method has no contribution from heights and shapes of the fuzzy numbers. Hence, the method is confined to normal fuzzy numbers. As like Hsieh and Chen’s method Lee’s (2002) method is just confined to normal fuzzy numbers. As such, it is not going to give correct similarity between fuzzy numbers having different heights and shapes. So far, the information about the heights is missing in the similarity measures. Hence, Chen and Chen (2001) developed a similarity measure for generalized fuzzy number (GFN) using the concept of the COG. Although this method seems to outperform in many situations, yet drawbacks are obtained in some situations as discussed in the Section 3. Replacing Chen and Chen’s COG by ROG, Yong, Wenkang, Feng, and Qi (2004) proposed a new similarity measure and applied in pattern recognition problems. The method seems very promising. However, it fails to give proper similarity between crisp-valued fuzzy numbers. Wei and Chen (2009) proposed a measure based on the geometric distance and the perimeter of the fuzzy numbers. However, the method fails to give proper similarity between fuzzy numbers depicting similar shape located at different positions. Xu, Shang, Qian, and Shu (2010) again used the COG and the geometric distance in measuring the degree of similarity between GFNs. Although the method is based on GFNs yet it fails to measure similarity between fuzzy numbers depicting similar shape with different heights. Hejazi, Doostparast, and Hosseini (2011) used the concept of geometric distance, perimeter, area and height to discuss the degree of similarity. However, the drawbacks are pointed out by Patra and Mondal (2015).