Coherent weights for pairwise comparison matrices and a mixed-integer linear programming problem

Pairwise comparison matrices (PCMs) have been a long standing technique for comparing alternatives/criteria and their role has been pivotal in the development of modern decision making methods. In order to obtain general results, suitable for several kinds of PCMs proposed in the literature, we focus on PCMs defined over a general unifying framework, that is an Abelian linearly ordered group. The paper deals with a crucial step in multi-criteria decision analysis, that is to obtain coherent weights for alternatives/criteria that are compared by means of a PCM. Firstly, we provide a condition ensuring coherent weights. Then, we provide and solve a mixed-integer linear programming problem in order to obtain the closest PCM, to a given PCM, having coherent weights. Isomorphisms and the mixed-integer linear programming problem allow us to solve an infinity of optimization problems, among them optimization problems concerning additive, multiplicative and fuzzy PCMs.