Normal wiggly probabilistic hesitant fuzzy-based TODIM approach for optimal solid waste disposal method selection.

The normal wiggly probabilistic hesitant fuzzy set (NWPHFS) enhances the conventional probabilistic hesitant fuzzy set (PHFS) by capturing not only explicit probabilistic information but also critical underlying details that may be hidden in the original inputs provided by decision-makers (DMs). This paper introduces a novel extension of the Tomada de Decisão Interativa Multicritério (TODIM) method, called the normal wiggly probabilistic hesitant fuzzy TODIM (NWPHFT) method based on the proposed distance measures of NWPHFSs. Initially, two novel basic operations over NWPHFSs-the subtraction and division operations-are defined. Additionally, several distance measures specific to normal wiggly probabilistic hesitant fuzzy sets are developed, and their properties are thoroughly examined. Furthermore, for scenarios where the weights of criteria are partially or completely unknown, two optimization models are established to determine these weights using the maximizing deviation approach and the Lagrange function technique, respectively. Next, the traditional TODIM approach is extended to develop the NWPHFT for addressing MCDM problems by utilizing the proposed distance measures and criteria weight determination models. The proposed method is then applied to a problem related to selecting solid waste disposal methods to demonstrate its practical applicability. Finally, comprehensive sensitivity analyses and comparisons are conducted to illustrate the stability and effectiveness of the proposed approach.