On Branch-and-Price for Project Scheduling

TL;DR

This paper investigates the limitations of branch-and-price methods for resource-constrained project scheduling, revealing structural issues that hinder their effectiveness and comparing their performance to classical formulations.

Contribution

It identifies structural drivers behind the ineffectiveness of branch-and-price in project scheduling and analyzes how reformulations impact convergence and computational efficiency.

Findings

Reformulated problems are highly degenerate, slowing column generation.

Branch-and-price often offers minimal relaxation improvements over classical methods.

Structural issues limit the practical effectiveness of branch-and-price for project scheduling.

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.