0–1 reformulations of the multicommodity capacitated network design problem

Abstract

We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and the other based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.

Efficient heuristics for energy-aware routing in networks with bundled links

“…Reference [36] studies a 0-1 reformulation of MCND, and shows that extended linking inequalities, derived from variable disaggregation techniques, are equivalent to residual capacity inequalities. The authors of [36] provide a heuristic method that produces a lower bound for MCND with a value that is equivalent to one computed by a LP.…”

“…The authors of [36] provide a heuristic method that produces a lower bound for MCND with a value that is equivalent to one computed by a LP. Note that their method can be used to generate the initial routes for energy aware routing as an alternative to the LP approach used in [13] and shortest paths, as used in our algorithm, i.e., Step 1 in Figure 1, described in Section 4.…”

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). …”

Transforming Mathematical Models Using Declarative Reformulation Rules

“…A view based on complexity theory was proposed in [2], but since it requires a polynomial time mapping between the problems it already cuts off a number of well-known reformulation techniques where the mapping is pseudo-polynomial [6] or even exponential in theory [3,5,8], but quite effective in practice. Limited to MIP problems, general ideas based on variable redefinition were proposed by [13,14], without finding wide application due to its complexity, remaining unknown (or unused) by the "average" user.…”