Lexicographical minimization of routing hops in telecommunication networks

Consider a telecommunications network with given link capacities and a set of commodities with known demands that must be routed over the network. We aim to determine a single routing path for each commodity such that the whole set of paths does not violate the link capacities and the number of routing hops is minimized in a lexicographical sense, i.e., minimizing the number of paths with the worst number of hops; then, among all such solutions, minimizing the number of paths with the second worst number of hops; and so on. We present two approaches for solving this problem. The first approach is iterative where a sequence of hop constrained problems is solved and the optimal solution value of each iteration defines a new constraint added to the problems of subsequent iterations. The second approach is based on defining a single integer programming model for the whole problem. In this approach, we consider appropriate cost parameters associated with the number of hops of each routing path such that the cost of a path with h hops is higher than the cost of all paths with h-1 hops. In both cases, we propose multi-commodity flow and hop-indexed models and compare them both in terms of linear programming relaxation values and in terms of efficiency