Minimizing the maximum network flow: models and algorithms with resource synergy considerations

In this paper, we model and solve the network interdiction problem of minimizing the maximum flow through a network from a given source node to a terminus node, while incorporating different forms of superadditive synergy effects of the resources applied to the arcs in the network. Within this context, we examine linear, concave, and convex–concave synergy relationships, illustrate their relative effect on optimal solution characteristics, and accordingly develop and test effective solution procedures for the underlying problems. For a concave synergy relationship, which yields a convex programme, we propose an inner-linearization procedure that significantly outperforms the competitive commercial solver SBB by improving the quality of solutions found by the latter by 6.2% (within a time limit of 1800 CPU s), while saving 84.5% of the required computational effort. For general non-concave synergy relationships, we develop an outer-approximation-based heuristic that achieves solutions of objective value 0.20% better than the commercial global optimization software BARON, with a 99.3% reduction in computational effort for the subset of test problems for which BARON could identify a feasible solution within the set time limit.