An α-cut intervals based fuzzy best-Worst method for Multi-Criteria Decision-Making
Harshit M Ratandhara, Mohit Kumar
TL;DR
This paper advances multi-criteria decision making by replacing the original fuzzy Best-Worst Method with an α-cut interval framework (α-FBWM) that optimizes the entire fuzzy judgment shape over a finite α-grid $F\subset[0,1]$. It defines a Degree of Approximation (DoA) to quantify accuracy and uses GMIR to defuzzify triangular fuzzy numbers, with interval weights derived from defuzzified α-cuts. A rigorously defined Consistency Index (CI) and Consistency Ratio (CR) assess weight-set accuracy, and the approach is demonstrated through numerical examples and a real-world Supply Chain 4.0 risk-ranking application. While solving the full non-linear, infinite-constraint problem is challenging, the finite-approximation strategy yields controllable accuracy and improved weight estimation over standard FBWM, offering a practical path for robust fuzzy-MCDM in complex domains.
Abstract
The Best-Worst Method (BWM) is a well-known Multi-Criteria Decision-Making (MCDM) method used to calculate criteria-weights in many real-life applications. It was observed that the decision judgments used to calculate weights in BWM may be imprecise due to human involvement. To incorporate this ambiguity into the weight calculation, Guo & Zhao proposed a model of BWM using fuzzy sets, known as Fuzzy BWM (FBWM). Although this model is known to have wide applicability, it has several limitations. One of the biggest limitations of this existing model is that the lower, modal and upper values of the fuzzy judgment are used in the weight calculation and the other values remain unused. To solve this limitation and optimize the entire shape, we propose a model of FBWM based on α-cut intervals. This helps in reducing information loss. It turns out that although it is possible to optimize the entire shape simultaneously, it is difficult to do so. Therefore, we approximate optimal weights using finite subset, say F, of [0, 1]. We then develop a technique to measure the Degree of Approximation (DoA) of a weight set and obtain a weight set with the desired DoA. For a given F, approximate weights are calculated using a minimization problem that has a non-linear nature and thus may lead to multiple weights. To solve this issue, we first compute the collection of all approximate weights of the criterion, which is an interval, and then adopt the center of this interval as the approximate weight of the criterion. To measure the accuracy of a weight set, we develop the concepts of Consistency Index (CI) and Consistency Ratio (CR) for the proposed model. Finally, we discuss some numerical examples and a real-world application of the proposed model in ranking of risk factors in supply chain 4.0 and compare the results with existing models.