Mathematical models and exact algorithms for the Colored Bin Packing Problem

Abstract

This paper focuses on exact approaches for the Colored Bin Packing Problem (CBPP), a generalization of the classical one-dimensional Bin Packing Problem in which each item has, in addition to its length, a color, and no two items of the same color can appear consecutively in the same bin. To simplify modeling, we present a characterization of any feasible packing of this problem in a way that does not depend on its ordering. This allows us to describe the problem with a simple mathematical model. Furthermore, we present four exact algorithms for the CBPP. First, we propose a generalization of Valério de Carvalho’s arc flow formulation for the CBPP using a graph with multiple layers, each representing a color. Second, we present an improved arc flow formulation that uses a more compact graph and has the same linear relaxation bound as the first formulation. And finally, we design two exponential set-partition models based on reductions to a generalized vehicle routing problem, which are solved by a branch-cut-and-price algorithm through VRPSolver. To compare the proposed algorithms, a varied benchmark set with 574 instances of the CBPP is presented. Experimental results show that our best model, the improved arc flow formulation, was able to solve instances of up to 500 items and 37 colors. The set-partition models are also shown to exceed their arc flow counterparts in instances with a very small number of colors.