An efficient MILP-based algorithm for the qualitative flexible multi-criteria method under incomplete or conflicting weights

This study first proposes a mixed-integer linear programming model for the qualitative flexible multi-criteria method (QUALIFLEX) within an interval type-2 fuzzy environment. This extends an efficient QUALIFLEX method that already exists in the literature. The computational complexity of QUALIFLEX grows exponentially with an increase in the number of alternatives, and the extended model efficiently solves a multi-criteria decision problem and determines the best permutation regardless of the number of alternatives. A new QUALIFLEX algorithm is also developed to handle imprecise and conflicting preference structures for criteria weights. This algorithm includes both a single-objective and a bi-objective model to address incomplete and conflicting weight information, respectively, and these models are subsequently linearized. The newly developed algorithm solves the models only once to produce the best permutation and the corresponding weights, rather than requiring the solution ofm!nonlinear models as in previous studies. The implications of the proposed extended and developed algorithms are illustrated using numerical examples, and their performance is analyzed against existing methods across a set of 30 problems with varying numbers of alternatives. The formulated model achieves similar results to the previous version with a limited number of alternatives using only one model-solving attempt and demonstrates superior performance in terms of computation time for problems with a larger number of alternatives.