Section: New Results

Packing problems

In the two-dimensional guillotine cutting-stock problem, the objective is to minimize the number of large plates used to cut a list of small rectangles. We consider a variant of this problem, which arises in glass industry when different bills of order (or batches) are considered consecutively. For practical organisation reasons, leftovers are not reused, except the large one obtained in the last cutting pattern of a batch, which can be reused for the next batch. The problem can be decomposed into an independent problem for each batch. In [6] we focus on the one-batch problem, the objective of which is to minimize the total width of the cutting patterns used. We propose a diving heuristic based on column generation, in which the pricing problem is solved using dynamic programming (DP). This DP generates so-called non-proper columns, i.e. cutting patterns that cannot participate in a feasible integer solution of the problem. We show how to adapt the standard diving heuristic to this “non-proper” case while keeping its effectiveness. We also introduce the partial enumeration technique, which is designed to reduce the number of non-proper patterns in the solution space of the dynamic program. This technique strengthens the lower bounds obtained by column generation and improves the quality of the solutions found by the diving heuristic. Computational results are reported and compared on classical benchmarks from the literature as well as on new instances inspired from glass industry data. According to these results, variants of the proposed diving heuristic outperform constructive and evolutionary heuristics.instances than those previously managed in the literature.

The bin packing problem with generalized time lags (BPGL) consists of a set of items, each having a positive weight, and a set of precedence constraints with lags between pairs of items, allowing negative and non-negative lags. The items must be packed into the minimum possible number of bins with identical capacity, and the bins must be assigned to time periods satisfying the precedence constraints with lags on the items. In [18] we show a solution strategy using a generic branch-and-price algorithm (implemented in the software platform BaPCod) and applying some problem specific cuts. Our approach outperformed the compact Mixed Integer Programming (MIP) formulation solved by the MIP solver Cplex.