Using max-continuous T fuzzy relation equations to determine the vector

Abstract

Solving fuzzy relation equations (FRQs) can be used to determine the weight vector in fuzzy synthetic evaluations, etc. However, the solutions of FRQs derived by real problems usually are infinite, so the vector cannot be determined. Many real problems need use FRQs to determine an objective and unknown vector, a particular solution. This paper discusses this problem for FRQs with max-continuous t-norm composition, which contain the two most frequently used types, max-min and max-product FRQs. It further analyzes the traits that FRQs have a unique solution, gives a sufficient condition that any vector can be determined by FRQs, and points out that it does for max-continuous T FRQs. Generally, when the FRQs cannot determine the vector, we need do some new tests and add those new equations into the system so as to the new system can determine it. But the new equations derived by blind testing often are redundant, or have little value, so the vector still cannot be determined, even if we add a lot. To solve this problem, the paper investigates how to extract valuable information from the old system and then to do new tests purposely so as to let the new system efficiently determine the vector. A procedure to realize it is thus given. Besides, three examples are also given to show the study content.