Symmetry breaking generalized disjunctive formulation for the strip packing problem

The two-dimensional strip packing is a special case of “Cutting and Packing” problems that arises in many applications. The problem seeks to pack a given set of rectangles into a strip of given width in order to minimize its length. Some applications in which the strip packing problem arises are: Cutting pieces from wooden boards, cutting pieces from glass or steel sheets, optimal layout of industrial facilities, etc. Lodi et al.[1] present a comprehensive survey of application and methods. Following the typology proposed by Wascher et al.[2] for cutting and packing problems, the two-dimensional strip packing is classified as twodimensional open dimension problem (2-ODP). Although there exist several heuristic and meta heuristic algorithms for solving the problem[3], exact algorithms have been proposed for the solution of the two-dimensional strip packing. Martello et al.[4] present a branch and bound algorithm. The lower bound of the nodes in this method is obtained considering the geometry of the problem. Alvarez-Valdes et al.[5] improve the branch and bound method of Martello et al. by obtaining stronger lower bounds and deriving new dominance conditions. In addition to specialized algorithms, the two-dimensional strip packing problem has been formulated as a mixed-integer linear program (MILP). Westerlund et al.[6] present an MILP model for the N-dimensional allocation, which includes the strip packing problem. Castro and Oliveira[7] present two different MILP formulations for the problem, based on scheduling models. One formulation follows a discrete-space approach, and it is based on the Resource-Task Network process representation. The other formulation uses a