A picture fuzzy evaluation framework based on a novel approach incorporating bidirectional projection measures and the VIKOR method

The complexity of multi-criteria decision-making (MCDM) makes it challenging for decision-makers (DMs) to reach an optimal decision. Therefore, this study sought to develop a novel picture fuzzy MCDM framework with partially known criterion weight information to enable DMs to select the most appropriate alternatives. As such, this study made several key contributions. Firstly, this study defined a novel picture fuzzy bidirectional projection measure that simultaneously accounted for distance and direction, thus overcoming the inherent drawbacks of existing picture fuzzy projection measures. Secondly, both objectivity and subjectivity were taken into account in the decision-making process, thus avoiding the limitations of considering only a single subjective weight or an objective weight. Additionally, based on the proposed bidirectional projection measure, Jaynes’s maximum entropy principle, and minimal deviations, a nonlinear programming model was constructed to determine the comprehensive weight vector of the criteria. Thirdly, the existing VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method always utilizes distance to measure the closeness degrees to the ideal solutions while ignoring the influence of the direction factor. Therefore, an improved picture fuzzy VIKOR method incorporating bidirectional projection was proposed to conduct selection process. Finally, a picture fuzzy evaluation framework was constructed by coupling bidirectional projection measures and the VIKOR method. The applicability of the proposed MCDM framework was then evaluated by solving a supplier selection problem, after which sensitivity and comparison analyses were performed to illustrate. its robustness and reliability.