Strong Bounds for Resource Constrained Project Scheduling: Preprocessing and Cutting Planes

TL;DR

This paper introduces a cutting plane algorithm and preprocessing techniques that significantly improve bounds for resource-constrained project scheduling problems, enabling optimal solutions for many previously unsolved instances.

Contribution

It presents a novel cutting plane method with new lifted inequalities and a preprocessing routine that enhance linear relaxation bounds for RCPSPs.

Findings

Improved linear relaxation bounds for RCPSPs.

Solved 754 previously open instances to optimality.

Enhanced solver performance with new inequalities.

Abstract

Resource Constrained Project Scheduling Problems (RCPSPs) without preemption are well-known NP-hard combinatorial optimization problems. A feasible RCPSP solution consists of a time-ordered schedule of jobs with corresponding execution modes, respecting precedence and resources constraints. In this paper, we propose a cutting plane algorithm to separate five different cut families, as well as a new preprocessing routine to strengthen resource-related constraints. New lifted versions of the well-known precedence and cover inequalities are employed. At each iteration, a dense conflict graph is built considering feasibility and optimality conditions to separate cliques, odd-holes and strengthened Chv\'atal-Gomory cuts. The proposed strategies considerably improve the linear relaxation bounds, allowing a state-of-the-art mixed-integer linear programming solver to find provably optimal solutions for 754 previously open instances of different variants of the RCPSPs, which was not possible using the original linear programming formulations.