Mixed-Integer Reaggregated Hull Reformulation of Special Structured Generalized Linear Disjunctive Programs

TL;DR

This paper introduces a novel mixed-integer reformulation approach for generalized disjunctive programming, combining convex hull characterization with time-slot reaggregation to produce compact, strong formulations for scheduling and packing problems.

Contribution

It develops a unified reformulation methodology for linear disjunctive sets, leading to more efficient and tighter mixed-integer models for specific structured problems.

Findings

New formulations for single-unit scheduling with strong theoretical guarantees.

Symmetry-breaking formulations for strip-packing improve computational efficiency.

Reformulations yield compact models with strong continuous relaxations.

Abstract

Generalized Disjunctive Programming (GDP) provides a powerful framework for combining algebraic constraints with logical disjunctions. To solve these problems, mixed-integer reformulations are required, but traditional reformulation schemes, such as Big-M and Hull, either yield a weak continuous relaxation or result in a bloated model size. Castro and Grossmann showed that scheduling problems can be formulated as GDP by modeling task orderings as disjunctions with algebraic timing constraints. Moreover, in their work, a particular representation of the single-unit scheduling problem, namely using a time-slot concept, can be reformulated as a tight yet compact mixed-integer linear program with notable computational performance. Based on that observation, and focusing on the case where the constraints in disjunctions are linear and share the same coefficients, we connect the characterization of the convex hull of these disjunctive sets by Jeroslow and Blair with Castro and Grossmann's time-slot reaggregation strategy to derive a unified reformulation methodology. We test this reformulation in two problems, single-unit scheduling and two-dimensional strip-packing. We derive new formulations of the general precedence concept of single-unit scheduling and symmetry-breaking formulations of the strip-packing problem, yielding mixed-integer programs with strong theoretical guarantees, particularly compact formulations in terms of continuous variables, and efficient computational performance when solving them with commercial mixed-integer solvers for these problems.