Lagrangean‐based decomposition algorithms for multicommodity network design problems with penalized constraints

Abstract

This article discusses problems in the context of multicommodity network design where additional constraints (such as capacity), rather than being imposed in a strict manner, are allowed to be violated at the expense of additional penalty costs. Such penalized cost structures allow these constraints to be treated as utilization targets and provide a better modelling framework in terms of strategic or tactical level planning of network design, especially in freight transportation systems. However, due to the penalized costs, these problems are generally in the form of a nonlinear integer multicommodity network design problem. This article presents two algorithms based on Lagrangean relaxation and decomposition for the solution of such problems. The first relies upon dualizing the capacity constraints that results in a flow decomposition, and the second is through relaxing flow constraints that results in an arc decomposition. It is shown that nonlinearities in the decomposed substructures can be handled in a very efficient manner. Arc decomposition is shown, through computational experiments, to have better convergence properties. Through the proposed algorithms, reasonably good solutions can be obtained for these problems where publicly available state-of-the-art nonlinear optimization codes fail to identify feasible solutions.

Multicommodity Network Flows With Nonconvex Arc Costs

“…A few authors have considered the case where some existing topology with already installed link capacities do exist and we want to expand these capacities to face an increasing traffic or to reduce congestion at minimal cost (see [13,43,78]). Of course that situation includes the general case but we focus here on the situation where the initial fixed cost is zero.…”

A Comparison of Models for Uncertain Network Design

“…To the best of our knowledge, we follow this approach for the first time for network design problems, by comparing which uncertainty set actually fits real-world data. We compare two robust optimization approaches for a network capacity expansion model with outsourceable demand (see, e.g., Bertsekas (1998); Bektas ¸et al (2010)). In this setting, we need to invest into the network infrastructure now, so that each commodity can be routed to satisfy its uncertain demand later.…”