Inverse Problems in AHP

Abstract

Two classes of inverse problems in the process of the AHP are studied. The first class, which we call “inverse hierarchy problem”, is the inverse of the forward problem of obtaining the overall priority weight vector for the alternative set from the local priority weight data set, such as the weight vectors for criteria viewed from the goal and for alternatives viewed from each criterion. The second class, which we call “inverse eigenvector problem”, is the inverse of the forward problem of obtaining the eigenvector from the pairwise comparison matrix, where complete information of pairwise comparison data is assumed. For the inverse hierarchy problem, two types of the problems are mathematically formulated as constrained least p-th norm problems. The inverse hierarchy problem of type 1 is to estimate, from the given objective overall priority weight vector for the alternative set, the local priority weight vector for the criteria. The inverse hierarchy problem of type 2 is to estimate, from the given objective overall priority weight vector for the alternative set, some of the local priority weight vectors for the alternatives viewed from each criterion. For the inverse eigenvector problem, we show that, given a priority weight vector x and an eigenvalueλ(or Consistency Index CI=(λ-N)/(N-1)), the problem of finding an N×N matrix A which satisfies Ax=λx can be equivalently transformed into the problem of finding an N×N matrix E which satisfies E1=λ1, where eij=aij(xj/xi), x>0, and 1 is the all-1 column vector. For N=3, the reciprocity assumed and λ given, the error matrix E is determined uniquely in the sense that a quadratic equation has a unique pair of solutions. On the basis of this result for N=3, the inverse eigenvector problem “E1=λ1” is analyzed for cases of N=3m, 3m+1, and 3m+2, respectively, and the error matrix E is shown to be expressed explicitly with N(N-3)/2 free independent parameters.