Closed-form solutions of consistency ratio in best worst method minmax optimization model: max of edge error matrix and minmax edge error determinant methods

Abstract

The Best Worst Method (BWM), a reduced version of the AHP, is a recent multi-criteria decision-making tool based on pairwise comparisons with reference to the best and worst criteria. Consistency Ratio (CR) measurement for the rating quality and prioritizations is still a controversial topic. Firstly, the computation for the current CR of BWM must rely on a software optimization solver to find the optimal values, and the solver may not always guarantee the exact optimal solutions, especially if the computational cost settings are not large enough for higher number of criteria. Secondly, much effort to evaluate optimization algorithms is needed to find the best solutions with the least computational resources due to diverse solvers possibly leading to different results with different performances. Thirdly, optimization programming code is not trivial to be implemented for general BWM users. To address these issues, this paper presents the closed-form solutions, Max of Edge Error Matrix (MEEM) (Eq. (44) of Theorem 4) and Minmax Edge Error Determinant (MEED) (Algorithm 1), to replace the BWM optimization models to directly calculate the CR values. Two simulations have been performed with a basic laptop using a single process. One simulation of twenty thousand random pairs of vectors took 26.34 h to perform to verify that the approximate results are higher than or very close to the exact closed-form values of both methods when high computational cost is allocated for the solver to increase the precision. Another simulation of one million random pairs of vectors only took 1.27 h to perform to verify that the MEED and MEEM methods always produce the same results for the number of criteria up to nine. The computational time for the exact results is dramatically reduced when the solver is not needed. The advantages of the proposed solutions include the following: the software to solve the optimization model to obtain CR is unnecessary, and the proposed calculation is extremely efficient to obtain the exact accuracy. The two-step optimization model can preserve the fixed Minmax Edge Error to find the weights which add up to one, which is the condition to determine if the model reaches exact optimal solutions. As the CR optimization model produces multiple versions of weights, which are recommended not to be used, the new method does not need to compute the unnecessary weight values to get the Minmax Edge Error. With the provision of equations leading to closed forms, users can understand the properties of CR in much clearer perspectives. Due to the computational efficiency and explainability, the proposed closed forms can replace the CR optimization model to compute CR efficiently and accurately for all diverse applications using BWM.