On Branch-and-Price for Project Scheduling

TL;DR

The paper investigates why branch-and-price (BP) methods using Dantzig-Wolfe decomposition often underperform on classical resource-constrained project scheduling problems (RCPSP). Through theoretical analysis and computational experiments, it identifies degeneracy, weak relaxation strength, and NP-hard pricing as key structural obstacles, arguing that BP rarely outperforms solving compact formulations directly. It presents a general Dantzig-Wolfe reformulation for multi-project, multi-mode RCPSP with resource-capacity decisions and analyzes the master and pricing problems across variants, showing limited or marginal bound improvements except in special cases (e.g., dedicated resources or preemptive scheduling). The findings suggest redirecting research toward alternative formulations with robust pricing problems or effective stabilization, and toward BP approaches tailored to specific, harder variants where NP-hard pricing can unlock stronger bounds.

Abstract

Integer programs for resource-constrained project scheduling problems are notoriously hard to solve due to their weak linear relaxations. Several papers have proposed reformulating project scheduling problems via Dantzig-Wolfe decomposition to strengthen their linear relaxation and decompose large problem instances. The reformulation gives rise to a master problem that has a large number of variables. Therefore, the master problem is solved by a column generation procedure embedded in a branching framework, also known as branch-and-price. While branch-and-price has been successfully applied to many problem classes, it turns out to be ineffective for most project scheduling problems. This paper identifies drivers of the ineffectiveness by analyzing the structure of the reformulated problem and the strength of different branching schemes. Our analysis shows that the reformulated problem has an unfavorable structure for column generation: It is highly degenerate, slowing down the convergence of column generation, and for many project scheduling problems, it yields the same or only slightly stronger linear relaxations as classical formulations at the expense of large increases in runtime. Our computational experiments complement our theoretical findings.

Figures (1)

• Figure 1: Convergence for solving the linear relaxation of the master problem via column generation (logarithmic axes).

Theorems & Definitions (9)

• Proposition 1
• Theorem 1: desrosiers2024, Chapter 4.1
• Definition 1: guignard2003lagrangean, integrality property
• Theorem 2
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6