A general model of decision-theoretic three-way approximations of fuzzy sets based on a heuristic algorithm

A model of decision-theoretic three-way approximations of fuzzy sets that can calculate a pair of interpretable thresholds (α, β) by combining shadowed sets with three-way decisions was established by Deng. In that model, the significant contribution is that the value 0.5, which denotes a membership grade with the highest uncertainty, is used to replace the uncertain region with the interval [0,1] in shadowed sets. From the principle of the minimum decision cost, although the lowest overall cost can be achieved precisely in some data distributions based on the value 0.5, there is a difference between the obtained overall cost and the least cost in some cases. Therefore, in this paper, based on Deng’s model, the concept of a general three-way approximation of a fuzzy set is proposed to replace 0.5 with a variable value c(0 < c < 1). Then, the loss function composed of the elevation and reduction operations in shadowed sets is established. In addition, the relationship between the required thresholds (α, β) with the different significance and variable value c is discussed. To optimize the loss function, particle swarm optimization (PSO) as a heuristic algorithm is introduced to search for the optimal value c by minimizing the total cost. Finally, the experimental results indicate that the proposed model, which is an extension of Deng’s model, can provide richer insight into data analysis. These conclusions further enrich shadowed sets and three-way decisions to simplify complex problems by a few membership grades.