## Section: Application Domains

### Packing and Covering Problems

Realopt team has a strong experience on exact methods for cutting and packing problems. These problems occur in logistics (loading trucks), industry (wood or steel cutting), computer science (parallel processor scheduling).

We developed a branch-and-price algorithm for the Bin Packing Problem with Conflicts which improves on other approaches available in the literature [74]. The algorithm uses our methodological advances like the generic branching rule for the branch-and-price and the column based heuristic. One of the ingredients which contributes to the success of our method are fast algorithms we developed for solving the subproblem which is the Knapsack Problem with Conflicts. Two variants of the subproblem have been considered: with interval and arbitrary conflict graphs.

We also developed a branch-and-price algorithm for a variant of the bin-packing problem where the items are fragile. In [33] we studied empirically different branching schemes and different algorithms for solving the subproblems.

We studied a variant of the knapsack problem encountered in inventory routing problem [62]: we faced a multiple-class integer knapsack problem with setups [61] (items are partitioned into classes whose use implies a setup cost and associated capacity consumption). We showed the extent to which classical results for the knapsack problem can be generalized to this variant with setups and we developed a specialized branch-and-bound algorithm.

We studied the orthogonal knapsack problem, with the help of graph
theory [56], [55],
[58], [57]. Fekete and Schepers proposed to model
multi-dimensional orthogonal placement problems by using an efficient representation of all
geometrically symmetric solutions by a so called *packing class* involving one *interval graph* for each dimension. Though Fekete & Schepers' framework is very efficient,
we have however identified several weaknesses in their algorithms: the most obvious one is
that they do not take advantage of the different possibilities to represent interval
graphs. We propose to represent these graphs by matrices with consecutive ones on each
row. We proposed a branch-and-bound algorithm for the 2D knapsack problem that uses our 2D
packing feasibility check. We are currently developing exact optimization tools for
glass-cutting problems in a collaboration with Saint-Gobain
[38]. This 2D-3stage-Guillotine cut problems are very hard to solve
given the scale of the instance we have to deal with. Moreover one has to issue cutting
patterns that avoid the defaults that are present in the glass sheet that are used as raw
material. There are extra sequencing constraints regarding the production that make the
problem even more complex.

We have also organized a European challenge on packing with society Renault. This challenge was about loading trucks under practical constraints.