Solving a Weighted Set Covering Problem for Improving Algorithms for Cutting Stock Problems with Setup Costs by Solution Merging

There are many different kinds of cutting stock problems (CSPs) occurring in practice and in theory having in common that they ask for a set of patterns, where each pattern is a collection of elements, to satisfy given element demands while minimizing the costs of the patterns. The classical CSP only considers fixed costs for each individual pattern, but in many practical applications an additional cost factor are setup costs arising whenever the machine has to be set up to cut a different pattern. Thus, finding a solution involving a small number of different types of patterns is often preferred. Most approaches to solve CSPs generate many candidate solutions yielding, if collected, a large and diverse set of patterns. We formalize an extension of the weighted set covering problem which exploits all these collected patterns by deriving an optimal combination of a subset of them resembling a feasible, possibly new incumbent solution. Solving this subproblem can be seen as a kind of solution merging. It can be applied either as a post-processing or as an intermediate step to also lead the pattern construction in a more promising direction. We investigate this extension specifically on K-staged two-dimensional CSPs with setup costs. The merging problem is defined as follows. Given is a set of elements E = {1, . . . , n} with a demand vector (di)ni=1 ∈ N and the set of collected patterns P . The actual structure of the patterns is not relevant here, but each pattern p ∈ P has an associated element vector (epi ) n i=1 ∈ N n indicating how often an element i ∈ E occurs in p. Every pattern p ∈ P has associated production costs cp and setup costs c S p. The goal is to find amounts a = (ap)p∈P ∈ N |P | such that