The robust machine availability problem – bin packing under uncertainty

Abstract

We define and solve the robust machine availability problem in a parallel machine environment, which aims to minimize the number of identical machines required while completing all the jobs before a given deadline. The deterministic version of this problem essentially coincides with the bin packing problem. Our formulation preserves a user-defined robustness level regarding possible deviations in the job durations. For better computational performance, a branch-andprice procedure is proposed based on a set covering reformulation. We use zero-suppressed binary decision diagrams for solving the pricing problem, which enable us to manage the difficulty entailed by the robustness considerations as well as by extra constraints imposed by branching decisions. Computational results are reported that show the effectiveness of a pricing solver with zero-suppressed binary decision diagrams compared with a mixed integer programming solver.

Solving robust bin-packing problems with a branch-and-price approach

“…In the column "# INSTANCE ", the number of tested instances for the corresponding set of instances is given. In the columns "# OPT ", "# OPT Vance", "# OPT SCIP", "# OPT Belov", and "# OPT Song", the number of instances solved to optimality in less than one minute per instance, by the proposed branch-and-price algorithm, and by the ones by Vance et al (1994) , by Gamrath et al (2016) , by Belov and Scheithauer (2006) , and by Song et al (2018) , are provided, respectively. For "# OPT Vance", "# OPT SCIP", and "# OPT Belov", the reported results come from the study by Delorme et al (2016) , in which a benchmark of several algorithms for BP was performed on the same computer, an Intel Xeon CPU at 3.1 GHz, with 8 GB of RAM, faster than the Intel Core i7-6700HQ CPU at 2.6 GHz with 4 GB of RAM that we used.…”

“…We first consider two papers introducing column-generation based algorithms to tackle variants of BP under uncertainty, then the literature about branch-and-price algorithms designed to solve the set-cover formulation of BP provided in Section 3.2 . Song et al (2018) address the robust machine availability problem in a parallel machine environment, which in its deterministic version coincides with BP . Using the methodology proposed by Bertsimas and Sim (2004) , they introduce a robust version with budgeted uncertainty, where at most jobs can deviate from their nominal processing times, which are assumed to belong to symmetric and bounded intervals.…”

“…, where v RMP is the linear relaxation value of the current restricted master problem and v PS is the optimal solution value to the current pricing subproblem. This bound is also used by Vance et al (1994) and Song et al (2018) . It was shown to be effective in reducing tailing off effects.…”

“…Another case of BP problems under uncertainty is the one dimensional cutting stock problem, as in Alem, Munari, Arenales, and Ferreira (2010) , where the demand or item sizes can be uncertain. A robust machine availability problem in a parallel machine environment with uncertain task durations in Song, Kowalczyk, and Leus (2018) is another example. The latter two articles are discussed in more detail in Section 3 , dedicated to the literature review.…”

Approximating Robust Bin Packing with Budgeted Uncertainty

“…Interestingly, these positive results do not extend to most scheduling problems (because they involve nonlinearities) and to the bin packing problem (because it involves a non-constant numbers of robust constraints). While in previous papers [13,12] (with authors in common) we provided approximability results on robust scheduling, no such results have yet been proposed for the bin packing problem, the only previous work focusing on numerical algorithms [27]. The purpose of this paper is to fill these gaps, as we present constant-factor approximation algorithms the bin packing problem, both for U Ω and U Γ .…”