Proportional picture fuzzy sets and their AHP extension: Application to waste disposal site selection

Picture fuzzy sets are an extension of intuitionistic fuzzy sets preferable when indecision degree is another parameter to define a membership of an element to a set additional to membership and non-membership degrees. A problem with picture fuzzy sets or another extension is that requiring decimal numbers for membership, non-membership and indecision degrees of an element in picture fuzzy sets from the experts is not sufficiently applicable. Instead, requiring proportional relations between membership and non-membership degrees is more applicable than the former. Our methodology is to develop some proportion-based models for picture fuzzy sets including their arithmetic operations and aggregation operators. Proportional picture fuzzy (PPF) sets need only the proportional relations between the parameters of the picture fuzzy set. Their contribution is that these models will ease the use of picture fuzzy sets with more correct data better-representing experts’ judgments. Besides, vague definition of the proportions is incorporated into the PPF models. We also develop PPF AHP method and use it for the comparison of alternative waste disposal site alternatives. The application, sensitivity and comparative analyses result in that proportional picture fuzzy sets are easily applicable to any problem and produce valid outcomes.