Pairwise comparison (PC) is a widely used scientific technique to compare criteria or alternatives in pairs in order to express the decision maker’s judgments without the need for a unique common measurement unit between criteria. The method constructs a PC matrix by requesting the assessments of the decision maker(s) in the judgment acquisition phase and calculates an inconsistency measure to determine whether the judgments are adequately consistent with each other before subsequent phases. Although the method requires the decision maker to make all judgments in a PC matrix, it does not force him/her to make a judgment for each element of the matrix. If any judgment in a PC matrix is absent, for this reason, the judgment acquisition phase yields an incomplete PC matrix rather than a complete one. Missing judgments are calculated by multiplication of the judgments made by the decision maker. If the judgements of the decision maker are transitive and well-proportioned, missing judgments will not disturb the consistency of the resulting PC matrix. In other words, consistency of a PC matrix relies on the judgments made by the decision maker. Since the current consistency analysis procedure is designed for complete PC matrices, the suitability for evaluating the inconsistency of incomplete PC matrices is questionable. Probability density functions of random PC matrices with altering numbers of missing judgments show distinct features, indicating an incomplete PC matrix and a complete PC matrix do not come from the same probability function, and their mean consistency index (RI) is different. Consequently, we propose an extended consistency analysis procedure to evaluate the consistency of incomplete PC matrices.
Consistency Analysis, Decision Support Systems, Pairwise Comparison, Random index
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