INCOMPLETE PAIRWISE COMPARISON MATRICES AND OPTIMIZATION TECHNIQUES

Pairwise comparison matrices (PCMs) play a key role in multi-criteria decision making, especially in the analytic hierarchy process. It could be necessary for an expert to compare alternatives based on various criteria. However, for a variety of reasons, such as lack of time or insufficient knowledge, it may happen that the expert cannot provide judgments on all pairs of alternatives. In this case, an incomplete pairwise comparison matrix is formed. In the first research part, an optimization algorithm is proposed for the optimal completion of an incomplete PCM. It is intended to numerically minimize a constrained eigenvalue problem, in which the objective function is difficult to write explicitly in terms of variables. Numerical simulations are carried out to examine the performance of the algorithm. The simulation results show that the proposed algorithm is capable of solving the minimization of the constrained eigenvalue problem. In the second part, a comparative analysis of eleven completion methods is studied. The similarity of the eleven completion methods is analyzed on the basis of numerical simulations and hierarchical clustering. Numerical simulations are performed for PCMs of different orders considering various numbers of missing comparisons. The results suggest the existence of a cluster of five extremely similar methods, and a method significantly dissimilar from all the others. In the third part, the filling in patterns (arrangements of known comparisons) of incomplete PCMs based on their graph representation are investigated under given conditions: regularity, diameter and number of vertices, but without prior information. Regular and quasi-regular graphs with minimal diameter are proposed. Finally, the simulation results indicate that the proposed graphs indeed provide better weight vectors than alternative graphs with the same number of comparisons. This research problem’s contributions include a list of (quasi-)regular graphs with diameters of 2 and 3, and vertices from 5 up to 24.