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Exact Methods for the Generalized Multiple Strip Packing Problem with Heterogeneous Costs

Hyunwoo Lee, Taesu Cheong

Abstract

We study the Generalized Multiple Strip Packing Problem (GMSPP) with heterogeneous per-unit-area costs, in which rectangular items of fixed dimensions must be packed without overlap into multiple open-ended strips of different widths, each incurring a cost proportional to the area used. This cost-weighted area objective is introduced here for the first time and unifies several objectives studied separately in the literature, including total area, total height for identical strips, and makespan. We propose two exact integer programming formulations for this problem: a big-M formulation adapted from recent work, and a normal-position formulation extending an earlier single-strip approach to multiple heterogeneous strips. For the normal-position formulation, we develop an exact Benders decomposition algorithm, called BendM (Benders' Method for Multiple strips). Comprehensive computational experiments on 180 instances derived from standard strip-packing benchmarks compare both formulations and demonstrate the effectiveness of BendM across three cost structures.

Exact Methods for the Generalized Multiple Strip Packing Problem with Heterogeneous Costs

Abstract

We study the Generalized Multiple Strip Packing Problem (GMSPP) with heterogeneous per-unit-area costs, in which rectangular items of fixed dimensions must be packed without overlap into multiple open-ended strips of different widths, each incurring a cost proportional to the area used. This cost-weighted area objective is introduced here for the first time and unifies several objectives studied separately in the literature, including total area, total height for identical strips, and makespan. We propose two exact integer programming formulations for this problem: a big-M formulation adapted from recent work, and a normal-position formulation extending an earlier single-strip approach to multiple heterogeneous strips. For the normal-position formulation, we develop an exact Benders decomposition algorithm, called BendM (Benders' Method for Multiple strips). Comprehensive computational experiments on 180 instances derived from standard strip-packing benchmarks compare both formulations and demonstrate the effectiveness of BendM across three cost structures.

Paper Structure

This paper contains 21 sections, 2 theorems, 10 equations, 1 figure, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Big-$M$ Formulation
  4. Normal-Position Formulation
  5. Normal positions
  6. Compact formulation
  7. Benders' Decomposition
  8. Master problem
  9. $y$-check subproblem
  10. Cut generation
  11. Stage 1: Standard Benders' cut.
  12. Stage 2: Combinatorial Benders' cut.
  13. Stage 3: Lifted combinatorial Benders' cut.
  14. Overall algorithm
  15. Computational Experiments
  16. ...and 6 more sections

Key Result

Proposition 1

$z^*_{\emph{LP-PC}} \le z^*$. That is, the LP relaxation of (P$\,|\,$cont$\,|\,C_{\max}$) provides a valid lower bound for both the normal-position formulation eq:gmspp0 and the big-$M$ formulation eq:bigm. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Distribution of objective value, lower bound, and optimality gap across 180 instances for the three exact methods. Boxes span the interquartile range (IQR) with the median shown as a horizontal line; red diamonds indicate the mean. Whiskers extend to $1.5\times\text{IQR}$ and outliers are shown as circles.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Remark 1: Cross-formulation bounding
  • Remark 2: $H_i$-aware cuts for the area-cost objective
  • Proposition 2: Joint validity of lifted cut
  • proof