Exact approaches to the single‐source network loading problem

Abstract

This article considers the network design problem that searches for a minimum-cost way of installing capacities on the edges of a network to simultaneously route a flow from a given access point to a subset of nodes representing customers with positive demands. We first consider compact and exponential-sized Mixed Integer Programming (MIP) formulations of the problem and provide their theoretical and computational comparison. We also consider a stronger disaggregated flow formulation. To solve the problem in practice, we project out the flow variables and generate Benders cuts within a branch-and-cut framework. To the best of our knowledge, the combination of Benders approach and this specific disaggregation has not been considered so far. In an extensive computational study, we compare the performance of compact MIP models against a textbook implementation and several normalization variants of Benders decomposition. We introduce a set of 32 realworld instances and use these, together with 64 other instances from the literature, to test our approaches. The results show that our branch-and-cut approach outperforms the best performing compact formulation leading to the best exact algorithm today for solving the considered dataset.

Location Science

“…One way to formulate the problem is to explicitly introduce continuous variables g ij for .i; j / 2 A S , representing the amount of flow routed on arc .i; j / to satisfy the demands of the facilities. Although this formulation is compact, it is attractive from a computational point of view to project out flow variables and replace constraints (20.15) and (20.16) by the capacitated cut set inequalities (Ljubić et al 2012):…”

Models and algorithms for network design problems

“…The latter rewrite the problem with a very large number of binary variables in order to obtain a tight formulation (Frangioni and Gendron, 2009;Ljubic et al, 2009). …”

“…Our main achievement is a careful implementation of the Benders decomposition within a branchand-cut algorithm, obtaining signi cantly better results than previous works on the problems. Some recent works of Fischetti et al (2010) and Ljubic et al (2009), among others, have highlighted the importance of the normalization constraint in the separation problem. Herein, we investigate another aspect of the algorithm, namely, when to generate cuts.…”

“…Since neither of the objective functions of (ML) and (HOP) have costs associated with routing variables, we focus on the generation of feasibility cuts in what follows. In particular, we choose violated extreme rays by solving SP(y) augmented by a normalization constraint (see Fischetti et al (2010); Ljubic et al (2009) for testing of di erent normalization constraints). If its solution is strictly less than 0, we add a feasibility cut; otherwise no cut is added.…”

An exact approach for the multicommodity network optimization problem with a step cost function

“…The exact separation of these inequalities is investigated in [21]. Ljubić et al [20] investigate a single-source variant. They introduce several mixed-integer models that they solve to optimality using a Benders-based approach combined with a specific disaggregation technique.…”