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arxiv: 2604.04740 · v1 · submitted 2026-04-06 · 🧮 math.OC

Recognition: no theorem link

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

Hyunwoo Lee , Taesu Cheong
Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:42 UTC · model grok-4.3

classification 🧮 math.OC
keywords strip packingmultiple stripsheterogeneous costsinteger programmingBenders decompositionnormal-position formulationexact methodspacking problem
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The pith

Normal-position formulation with Benders decomposition solves generalized multiple strip packing with heterogeneous per-unit-area costs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the generalized multiple strip packing problem where rectangles of fixed sizes must be placed without overlap into several open strips that have different widths and different per-unit-area costs. It unifies prior separate objectives such as total area used, total height across identical strips, and makespan under this single cost model. Two integer programming formulations are given, one using big-M constraints and one using normal-position variables, and a Benders decomposition algorithm named BendM is developed for the normal-position version. Tests on 180 instances built from standard benchmarks show the approach handles three cost structures effectively. Readers care because the work supplies exact methods for packing decisions that arise when material or space costs differ across available strips.

Core claim

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 unifies several objectives studied separately in the literature. We propose two exact integer programming formulations: 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 Bend

What carries the argument

The normal-position formulation, which encodes non-overlapping placements via discrete relative positioning variables extended to multiple heterogeneous strips, together with the BendM Benders decomposition algorithm that separates the model across strips.

If this is right

  • The heterogeneous cost objective unifies total area minimization, total height for identical strips, and makespan into one framework.
  • BendM solves the normal-position model effectively across three cost structures on 180 instances derived from standard strip-packing benchmarks.
  • The normal-position formulation successfully extends prior single-strip positioning methods to the case of multiple strips with differing widths and costs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition structure could be adapted to related problems such as bin packing or cutting stock with resource-specific costs.
  • The exact methods provide a reference point for developing scalable heuristics or hybrid approaches on instances larger than the current 180.
  • Industries that assign different costs to strips or lanes of varying width can directly apply the unified model to minimize total expenditure.

Load-bearing premise

The big-M and normal-position integer programs correctly encode non-overlapping placement and the heterogeneous per-unit-area cost objective without hidden modeling errors.

What would settle it

A concrete benchmark instance in which an optimal solution returned by BendM contains overlapping rectangles or yields a total cost different from the sum of each strip's used area times its specific per-unit cost.

Figures

Figures reproduced from arXiv: 2604.04740 by Hyunwoo Lee, Taesu Cheong.

Figure 1
Figure 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 × IQR and outliers are shown as circles. 7 Conclusion We introduced the cost-weighted area objective P i CiWiHi for the generalized multiple strip packing… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the Generalized Multiple Strip Packing Problem (GMSPP) with heterogeneous per-unit-area costs for packing rectangular items without overlap into multiple open-ended strips of varying widths. It proposes two exact integer programming formulations (a big-M formulation adapted from prior work and a normal-position formulation extending single-strip models) and develops a Benders decomposition algorithm called BendM for the normal-position model. Computational experiments on 180 instances derived from standard strip-packing benchmarks demonstrate the effectiveness of the approaches across three cost structures.

Significance. If the formulations correctly encode non-overlap and the heterogeneous cost objective, and if BendM performs as reported, the work unifies previously separate objectives (total area, total height, makespan) under a single framework and provides practical exact methods for a novel variant. The extension of normal-position constraints and Benders cuts to multiple heterogeneous strips, together with results on a sizable benchmark set, strengthens the literature on exact packing algorithms in operations research.

minor comments (3)
  1. [Abstract] Abstract: the claim that the cost-weighted area objective 'unifies several objectives studied separately' lists total area, total height for identical strips, and makespan but does not explicitly map each to the new objective; adding one sentence with the mappings would improve immediate clarity.
  2. [§3] §3 (normal-position formulation): the extension of single-strip normal-position constraints to multiple strips via item-to-strip assignment variables is described, but the interaction between the per-strip height variables and the heterogeneous cost coefficients in the objective is not illustrated with a small numerical example; such an example would aid verification of the linear objective encoding.
  3. [Computational experiments] Computational experiments section: results are presented with standard metrics across the 180 instances and three cost structures, yet no summary table reports the fraction of instances solved to optimality or the average optimality gap per formulation and cost structure; adding this would strengthen the effectiveness claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly highlights the novelty of the heterogeneous per-unit-area cost objective in the GMSPP, the two IP formulations, and the BendM Benders algorithm. We appreciate the recommendation for minor revision and the recognition that the work unifies several packing objectives under one framework.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core contributions are two explicit integer programming formulations (big-M and normal-position) presented as direct, standard extensions of prior single-strip models to the multi-strip heterogeneous-cost setting, plus a Benders decomposition algorithm (BendM) derived from the normal-position model. The new cost-weighted area objective is introduced by definition as a linear combination of per-strip used heights and strip-specific costs; no equations reduce this objective or the non-overlap constraints to fitted parameters or self-referential inputs. Computational results on 180 benchmark-derived instances report exact solving performance without any predictive claims that loop back to the model parameters. No uniqueness theorems, ansatzes, or load-bearing self-citations are invoked to force the formulations or algorithm. The derivation chain is therefore self-contained and externally verifiable through the stated constraints and standard Benders machinery.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard integer-programming modeling assumptions and Benders decomposition theory; no new free parameters, axioms beyond standard math, or invented entities are introduced.

axioms (2)
  • standard math Standard integer programming assumptions for non-overlapping rectangle placement
    Invoked when defining the big-M and normal-position formulations.
  • standard math Benders decomposition converges for the chosen master and subproblem structure
    Used to justify the BendM algorithm.

pith-pipeline@v0.9.0 · 5450 in / 1193 out tokens · 50118 ms · 2026-05-10T19:42:30.833541+00:00 · methodology

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Reference graph

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