Computing interval weights for incomplete pairwise-comparison matrices of large dimension - a weak consistency based approach

Multiple-criteria decision making and evaluation problems dealing with a large number of objects are very demanding, particularly when the use of pairwise-comparison (PC) techniques is required. A major drawback arises when it is not possible to obtain all the PCs, due to time or cost limitations, or to split the given problem into smaller subproblems. In such cases, two tools are needed to find acceptable weights of objects: an efficient method for partially filling a pairwise-comparison matrix (PCM) and a suitable method for deriving weights from this incomplete PCM. This paper presents a novel interactive algorithm for large-dimensional problems guided by two main ideas: the sequential optimal choice of the PCs to be performed and the concept of weak consistency. The proposed solution significantly reduces the number of needed PCs by adding information implied by the weak consistency after the input of each PC (providing sets of feasible values for all missing PCs). Interval weights of objects are computed from the resulting incomplete weakly consistent PCM adapting the methodology for calculating fuzzy weights from fuzzy PCMs. The computed weight intervals, thus, cover all possible weakly consistent completions of the incomplete PCM. The algorithm works both with Saaty's PCMs and fuzzy preference relations. The performance of the algorithm is illustrated by a numerical example and a real-life case study. The performed simulation demonstrates that the proposed algorithm is capable of reducing the number of PCs required in PCMs of dimension 15 and greater by more than 60% on average.