Exact solution of multicommodity network optimization problems with general step cost functions

Abstract

We describe an exact solution procedure, based on the use of standard LP software, for multicommodity network optimization problems with general discontinuous step-increasing cost functions. This class of problems includes the so-called single-facility and multiple-facility capacitated network loading problems as special cases. The proposed procedure may be viewed as a specialization of the well-known BENDERS partitioning procedure, leading to iteratively solving an integer 0 -1 linear programming relaxed subproblem which is progressively augmented through constraint generation. We propose an improved implementation of the constraint generation principle where, at each step, several (O(N )) new constraints are included into the current problem, thanks to which the total number of iterations is greatly reduced (never exceeding 15 in all the test problems treated). We report on systematic computational experiments for networks up to 20 nodes, 37 links and cost functions with an average six steps per link.

An improved Benders decomposition applied to a multi-layer network design problem

“…Gabrel et al introduce a constraint-generation procedure based on a Benders decomposition for capacitated network design problems [12]. Then, Knippel and Lardeux extend this work to multi-layered networks [16] and multi-period time scheduling [13].…”

A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems

“…Specific applications include the network loading problem [3,13,15,19], the facility location problem with multiple capacity options [17,18], and the merge-in-transit problem [7].…”

Models and algorithms for network design problems

“…Benders decomposition is one of the most common decomposition for (ND) (Gabrel et al, 1999;Costa, 2005), maybe because it is very easy to implement. However, it su ers from two drawbacks.…”

“…They solve the problem using two di erent heuristics. Gabrel et al (1999) introduce a constraint generation procedure based on a Benders decomposition for capacitated network design problems. Knippel and Lardeux (2007) and Ge ard et al (2007) extend this work to multi-layered networks and multi-period time scheduling, respectively.…”