The Bin Packing Problem with Setups: Formulation, Structural Properties and Computational Insights

We introduce and study a novel generalization of the classical Bin Packing Problem (BPP), called the Bin Packing Problem with Setups (BPPS). In this problem, which has many practical applications in production planning and logistics, the items are partitioned into classes and, whenever an item from a given class is packed into a bin, a setup weight and cost are incurred. We present a natural Integer Linear Programming (ILP) formulation for the BPPS and analyze the structural properties of its Linear Programming relaxation. We show that the lower bound provided by the relaxation can be arbitrarily poor in the worst case. We introduce the Minimum Classes Inequalities (MCIs), which strengthen the relaxation and restore a worst-case performance guarantee of 1/2, matching that of the classical BPP. In addition, we derive the Minimum Bins Inequality (MBI) to further reinforce the relaxation, together with an upper bound on the number of bins in any optimal BPPS solution, which leads to a significant reduction in the number of variables and constraints of the ILP formulation. Finally, we establish a comprehensive benchmark of 480 BPPS instances and conduct extensive computational experiments. The results show that the integration of MCIs, the MBI, and the upper bound on the number of bins substantially improves the performance of the ILP formulation in terms of solution time and number of instances solved to optimality.