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

TL;DR

The paper addresses a novel generalization of bin packing, the BPPS, where items are partitioned into classes and packing any item from a class in a bin incurs a setup weight and a setup cost, in addition to a fixed bin cost. It develops a natural ILP formulation and analyzes its LP relaxation, revealing that the baseline bound can be arbitrarily weak; to strengthen it, the authors introduce Minimum Class Inequalities (MCIs) and the Minimum Bins Inequality (MBI), proving a worst-case 1/2 bound on the strengthened LP. They further reduce model size with an instance-dependent upper bound on the number of bins, leading to the ILP variants ${\textnormal{ILP}^\dagger_\textnormal{N}}$, ${\textnormal{ILP}^\ddag_\textnormal{N}}$, and ${\textnormal{ILP}^{\star}_\textnormal{N}}$, and demonstrate substantial computational gains across 480 benchmark instances. The empirical results show that MCIs and MBIs markedly improve LP bounds and that the upper-bound-driven reduction significantly speeds up solving BPPS instances, highlighting practical applicability to production and logistics settings with setup considerations.

Abstract

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.

Figures (6)

• Figure 1: The figure considers a BPPS instance with bin capacity $d = 6$, number of items $n = 8$, and number of classes $m = 2$. The item weights are $w_1 = w_2 = w_3 = w_4 = 3$ and $w_5 = w_6 = w_7 = w_8 = 1$, while the setup weights and costs are $s_1 = s_2 = 1$, $f_1 = 2$ and $f_2 = 3$. The items are partitioned into the classes ${\mathcal{I}}_1 = \{1,2,3,4\}$ and ${\mathcal{I}}_2 = \{5,6,7,8\}$. The figure shows two optimal BPPS solutions for this instance for two different values of the bin cost ${r}$. Used bins are shown along the horizontal axis, while the vertical axis represents the bin capacity $d$. For each bin, the figure displays the total weight of the packed items and the setup weight associated with their classes. The height of each item and setup block corresponds to its weight, visually illustrating bin utilization. The remaining empty space in each bin is shown in gray. In part (a), with ${r} = 10$, the optimal solution consists of four used bins: $\{\{1,5\}, \{2,6\}, \{3,7\}, \{4,8\}\}$, yielding a total minimum cost of ${\psi} = 60$. In part (b), with ${r} = 1$, the optimal solution consists of five used bins: $\{\{1\}, \{2\}, \{3\}, \{4\}, \{5,6,7,8\}\}$, with a total minimum cost of ${\psi} = 16$.
• Figure 2: The left part of the figure illustrates an optimal solution of ${\textnormal{LP}_\textnormal{N}}$ for the worst-case instance used in the proof of Proposition \ref{['prop:ratio']} with a given value of $n$. Each bin is filled up to the value $\frac{2n-1}{n}$. The blocks represent the fraction of the item weights and of the class setup weight assigned to each bin. The right part of the figure illustrates an optimal BPPS solution to the same instance, where each bin is completely filled by the weight of one item plus the setup weight of the class.
• Figure 3: The left part of the figure illustrates an optimal solution of ${\textnormal{LP}^\dag_\textnormal{N}}$ for the worst-case instance used in the proof of Proposition \ref{['prop:worstcase']} with a given value of $\vartheta$. Each bin is filled up to the value $\vartheta + \frac{\gamma_1}{n}$. The blocks represent the fraction of the item weights and of the class setup weight assigned to each bin. The right part of the figure illustrates an optimal BPPS solution to the same instance, where at most one item can be packed in each bin, since two items together with the setup weight do not fit.
• Figure 4: Box plot of the solution times of the different variants of the ILP formulation for the BPPS, for the instances solved to optimality by ${\textnormal{ILP}^{\star}_\textnormal{N}}$. The vertical axis is on a logarithmic scale. Time limit: 1800 seconds.
• Figure 5: Box plot of the optimality gap of the different variants of the ILP formulation for the BPPS for the instances not solved to optimality within the time limit by ${\textnormal{ILP}^{\star}_\textnormal{N}}$. Time limit of 1800 seconds.

Theorems & Definitions (22)

• Proposition 1
• Proposition 2
• proof
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• proof
• Proposition 7
• Proposition 8