A negative-cycle algorithm for computing all the supported efficient solutions in multi-objective integer network flow problems

This paper presents a new algorithm for identifying all the supported nondominated vectors (or outcomes) in the objective space, as well as the corresponding efficient solutions in the decision space, for the multi-objective integer network flow problem. Identifying the set of supported non-dominated vectors is of the utmost importance for obtaining a first approximation of the whole set of non-dominated vectors. This approximation is crucial, for example, in the two-phase methods that first compute the supported non-dominated vectors and then the unsupported non-dominated ones. Our approach is based on the negative-cycle algorithm used in single objective minimum cost flow problems, applied to a sequence of parametric problems. The proposed approach uses the connectedness property of the set of supported non-dominated vectors/efficient solutions to find all the integer solutions in the maximal nondominated/efficient facets.