A parallel branch-and-bound-and-prune algorithm for irregular strip packing with discrete rotations

Abstract

The irregular strip-packing problem consists of the computation of a non-overlapping placement of a set of polygons onto a rectangular strip of fixed width and the minimal length possible. Recent performance gains of the Mixed-Integer Linear Programming (MILP) solvers have encouraged the proposal of exact optimization models for nesting. The Dotted-Board (DB) MILP model solves the discrete version of the nesting problem by constraining the positions of the polygons to be on a grid of fixed points. However, its number of non-overlapping constraints grows exponentially with the number of dots and types of polygons, which encouraged the proposal of a reformulation called the DB Clique Covering (DB-CC) that sets the current state-of-the-art by significantly reducing the constraints required. However, DB-CC requires a significant preprocessing time to compute edge and vertex clique coverings. Moreover, current knowledge of the stable set polytope suggests that achieving a tighter formulation is unlikely. Thus, our hypothesis is that an ad-hoc exact algorithm requiring no preprocessing might be a better option to solve the DB model than the costly Branch-and-Cut approach. This work proposes an exact branch-and-bound-and-prune algorithm to solve the DB model from the conflict inverse graph based on ad-hoc data structures, bounding, and forward-checking for pruning the search space. We introduce two 0-1 ILP DB reformulations with discrete rotations and a new lower-bound algorithm as by-products. Our experiments show that DB-PB significantly reduces the resolution time compared to our replication of the DB-CC model. Seventeen open instances are solved up to optimality.