Asymmetric Fuzzy Preference Relations Based on the Generalized Sigmoid Scale and Their Application in Decision Making Involving Risk Appetites

Numerical preference relations constitute a useful decision-making technique and can be classified into two types on the basis of the 0.1-0.9 and 1-9 scales. However, these numerical scales cannot fully describe some desired properties, such as asymmetry, consistency, variability, and diminishing utility, in accordance with the preference relations. To address this issue, we first develop a generalized sigmoid function and define a continuous preference set, on the basis of which we propose the generalized sigmoid scale. Then, we introduce the optimal discrete fitting and risk preference selection approaches to solve the risk appetite parameters in the generalized sigmoid scale. Using these new methods, we further propose the asymmetric fuzzy preference relation (AFPR), examine its additive transitivity property and some weak transitivity properties, and design an approximate consistency test. The corresponding five-step modeling process under an asymmetric fuzzy preference environment is constructed, which can be applied in decision making involving different risk appetites. Finally, two examples are used to demonstrate the properties and advantages of these new methods. The first example is simple and shows the differences between the traditional fuzzy preference relations and the AFPRs, while the second is a practical case and is used to illustrate the feasibility and reasonability of the AFPRs.