An incomplete design in the Analytic Hierarchy Process

In the 1970’s, Thomas Saaty developed the Analytic Hierarchy Process (AHP) which is a decision analysis tool. For each level of the hierarchy, pairwise comparisons are made by each judge. However, a problem with the method of paired comparisons is the large number of pairwise comparisons involved. In general, there have been a lot of papers dealing with reducing the number of pairs to be compared since L.L. Thurstone developed paired comparison scales 60 years ago. But as far as AHP is concerned, there are only six other authors, E.N. Weiss and V.R. Rao, P.T. Harker, Iwaro Takahashi, and E. Takeda and P.L. Yu, who tried to deal with incomplete design for the AHP. The present approach is different from theirs. The authors divide the objects of a level into several subsets such that all these subsets have one common object as a standard one. Then pairwise comparisons are performed for each subset and a weight vector is found by solving the corresponding eigenvector problem. Finally, a weight vector for the objects of the level of the hierarchy is derived by using the common object and weight vector of each subset. The present incomplete design is based on an important property of the comparison matrix, which is Theorem 1. A concrete example, which is given by Saaty from the wealth of nations, is chosen to compare the results of the incomplete design with the result of the complete design and the two results are surprisingly close to each other for many different incomplete designs.