Comparison, ranking and determination of the ranks of fuzzy numbers based on a satisfaction function

Abstract

Fuzzy numbers have been widely used in many application areas because they are proper to represent uncertain and vague values. In those areas, comparing fuzzy numbers and some operations related with this issue are important. The operations on crisp numbers are well established and their results are clear, but those on fuzzy numbers are not so easy to define. Moreover, their results could also be vague because of the inherent vagueness of fuzzy numbers. This thesis focuses on how to manage the vagueness of fuzzy numbers in comparing, ranking and determining the rank of fuzzy numbers, and how to represent the vagueness in the results of these operations. For these purposes, two comparison measures, two ranking methods and one method for determining the rank of fuzzy numbers are proposed. In addition to these, the ranking method is used in two applications in order to show the usefulness of the proposed method. A comparison measure is extended from an existing measure called the satisfaction function on a discrete domain(SFD). The SFD was defined to estimate the satisfaction degree of an arithmetic relation between fuzzy numbers on a discrete domain. It is extended to be applicable to fuzzy numbers on a continuous domain, so it is called the satisfaction function on a continuous domain(SFC). The SFC evaluates the degree to which a comparison is satisfied in consideration of the vagueness of fuzzy numbers, and represents the comparison result with a real number in [0,1]. The SFC is extended to the fuzzy satisfaction function(FSF) that represents the degree to which a comparison is satisfied by a fuzzy set on [0,1]. Since it is extended from the SFC, its output includes that of the SFC and it also considers vagueness of fuzzy numbers for comparison. Two ranking methods are developed which use the SFC or the FSF as evaluation measures. Since fuzzy numbers are vague, evaluations can be easily affected by users```` viewpoints. Thus, both methods provide users with ...