Centroid defuzzification and the maximizing set and minimizing set ranking based on alpha level sets

Centroid defuzzification and the maximizing set and minimizing set methods are two commonly used approaches to ranking fuzzy numbers and often require membership functions to be known. In this paper, the two methods are reinvestigated when explicit membership functions are not known but alpha level sets are available. Two analytical formulas are derived under the assumption that the exact membership functions can be approximated by using piecewise linear functions based on alpha level sets. The derived analytical formulas are of significant importance and provide very useful decision supports for a wide variety of applications of the two methods in industrial engineering and other areas. Numerical examples are offered to test the derived formulas and illustrate their computational processes and applications in risk assessment of a software development.