Assessing the quality of convex approximations using sampling
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چکیده
We consider two types of convex approximations of two-stage totally unimodular integer recourse models. Although worst-case error bounds are available for these approximations, their actual performance has not yet been investigated, mainly because this requires solving the original recourse model. In this chapter we assess the quality of the approximating solutions using Monte Carlo sampling, or more specifically, using the so-called multiple replications procedure. Based on numerical experiments for an integer newsvendor problem and a fleet allocation and routing problem, we conclude that the actual performance is much better than the error bounds suggest, especially if the variability of the random parameters in the model is medium to large. In case this variability is small, the performance of the approximations is not so good. However, these are precisely the cases for which sampling methods, with modest sample sizes, may perform best. In this sense, the convex approximations and sampling methods can be considered as complementary solution methods. Finally, for a fleet allocation and routing problem, we derive a new error bound dealing with deterministic second-stage side constraints and relatively complete recourse. This chapter is submitted for publication as [58].
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