M.H. van der Vlerk

Convex Approximations for Totally Unimodular Integer Recourse Models: A Uniform Error Bound

We consider a class of convex approximations for totally unimodular (TU) integer recourse models and derive a uniform error bound by exploiting properties of the total variation of the probability density functions involved. For simple integer recourse models this error bound is tight and improves the existing one by a factor 2, whereas for TU integer recourse models this is the first nontrivial error bound available. The bound ensures that the performance of the approximations is good as long as the total variations of the densities of all random variables in the model are small enough.

Convex hull approximation of TU integer recourse models:Counterexamples, sufficient conditions, and special cases

We consider a convex approximation for integer recourse models. In particular, we show that the claim of Van der Vlerk (2004) that this approximation yields the convex hull of totally unimodular (TU) integer recourse models is incorrect. We discuss counterexamples, indicate which step of its proof does not hold in general, and identify a class of random variables for which the claim in Van der Vlerk (2004) is not true. At the same time, we derive additional assumptions under which the claim does hold. In particular, if the random variables in the model are independently and uniformly distributed, then these assumptions are satisfied.

A two-stage model for a day-ahead paratransit planning problem

Abstract We consider a dynamic planning problem for the transport of elderly and disabled people. In particular, we focus on a decision to take one day ahead: which requests should be served with own vehicles, and which requests should be subcontracted to taxis? We call this problem the Day-ahead Paratransit Planning problem (DaPP). The model developed for DaPP is a non-standard two-stage integer recourse model from stochastic programming. Both stages consist of two parts: requests are first clustered into routes, and these routes are in turn assigned to vehicles (own vehicles and taxis). To solve this model, a genetic algorithm approach is used. Computational results are presented for randomly generated data sets.