Inverted prioritization in AHP: Identifying the comparison matrix for given vector of weights with genetic algorithm

In Analytic Hierarchy Process (AHP) it is required to compute so called priority vector, i.e. weights of the decision elements involved in pair wise comparisons and creation of the reciprocal judgment matrix. Standard AHP derives this vector as the principal eigenvector of a matrix, and method used for this is known as EVM (Eigenvector Method). However, it is well known that deriving priority vector is underdetermined problem with practically indefinite set of possible solutions (matrices). In fact, the matrix space is enlarging dramatically with raising the number of decision elements and order of the related matrix. This is mainly due to inherent inconsistencies of the decision maker's pair wise comparisons and limitations imposed by using certain ratio scale. In this paper, inversion problem is stated and solved with the use of genetic algorithm (GA) as an efficient stochastic search engine: for given priority vector identify most probable reciprocal matrix from which the vector could be derived. Rationale is to try to recognize possible decision maker's behavior during pair wise comparisons of the decision elements. Solving mechanism is based on original Saaty's scale [1/9,1/8, ..., 1/2, 1, 2, ..., 9] with fitness function defined as closeness of any individual priority vector, derived from generated matrix, to the given priority vector. All standard parameters in genetic algorithm are involved: tournament, single point crossover, mutation, elitism and re-initialization of population to preserve convergence. An example application of proposed approach is given, followed by brief discussion.