Global measure‐based feasibility modeling for continuous packing problems

Abstract

Abstract Continuous packing problems are traditionally modeled using systems of local non‐overlap and containment constraints whose number and structure depend on object interactions and geometry. While geometrically exact, such formulations often lead to large and structurally complex models that complicate feasibility analysis and optimization. This paper proposes a global, measure‐based formulation of feasibility for two‐dimensional packing problems. Feasibility is characterized through a single scalar quantity, called the configuration area, defined as the Lebesgue measure of the intersection between the container and the union of all placed objects. We establish a formal equivalence between classical geometric feasibility and a simple configuration‐area identity under mild regularity assumptions. We further investigate the smoothness structure of the configuration‐area functional and establish an explicit gradient formula, showing that the number of nonsmoothness sources does not grow with the number of objects, in contrast to classical φ‐function formulations. This result provides a rigorous theoretical justification for replacing systems of local constraints with a single global condition that is independent of the number of objects and agnostic to their shape. We further establish a quantitative bound showing that an approximate solution of the global condition, satisfied only up to a tolerance , certifies that every individual pairwise overlap and containment violation does not exceed . A numerical example illustrates the applicability of the proposed approach.

The authors' abstract, as published at the source. International Transactions in Operational Research, 2026 · DOI ↗