Some methods to derive the priority weights from the best-worst method matrix and weight efficiency test in view of incomplete pairwise comparison matrix

The Best-Worst Method (BWM) has been recently proposed to derive the weights of criteria using two vectors of the pairwise comparison. For BWM, the best criteria and the worst criteria of alternatives are first determined by the decision-maker (DM). Then, the DM gives his best-to-others vector (BV) and the others-to-worst vector (WV). In this paper, we show that the BV and WV can intrinsically be formulated as an incomplete reciprocal matrix, we call it BWM matrix. Thus, to derive the weights for BWM can be transformed to derive the weights from an incomplete reciprocal preference relation. In this view, we present several models to derive priority weights from a BWM matrix. Especially, we also show that the initial BWM model is a special case of our proposed method, the concept of efficiency is extended to the incomplete reciprocal preference relation. Furthermore, these methods are extended to derive the priority weights for group decision making problems. Additionally, some inconsistency indices are introduced to measure the inconsistency degree of a BWM matrix. Finally, one example is illustrated to derive the optimal weights from a BWM matrix and another example is illustrated to show the efficiency of the weight vectors, respectively. Monte Carlo simulations and comparative analyses are carried out to show the effectiveness of the proposed priority methods.