Melting Probability Measure With OWA Operator to Generate Fuzzy Measure: The Crescent Method

Given probability information, i.e., a probability measure m with a random variable x on the outcome space N, the expected value of that random variable is commonly used as some valuable evaluation result for further decision making. However, there is no guarantee that the given probability information will be convincing to every decision maker. This is possible because decision makers may question the reliability of that provided probability information and can also be because decision makers often have their own different optimistic/pessimistic preferences. Often, such optimistic/pessimistic preferences can be easily embodied and expressed by some ordered weighted average (OWA) weight functions w. This study first compares and analyzes some simpler methods to melt the given OWA weight functions w with the given probability measure m to generate a new probability measure, pointing out their respective advantages and shortcomings. Then, this study proposes the melting axioms, which will both conform to our intuition and have mathematical reasonability. As the main finding of this study, we then propose the Crescent Method, which will effectively melt the given OWA weight function w with the given probability measure m to generate a final resulted fuzzy measure. Based on that melted fuzzy measure, we perform the Choquet integral of x as the more convincing evaluation result to decision makers with preference w. The study also proposes several interesting mathematical results such as the orness of resulted fuzzy measure will always be equal to the orness of the given OWA weight function w.