Improved Job sequencing Bounds from Decision Diagrams

TL;DR

This paper presents a novel method using decision diagram relaxations and Lagrangian duality to obtain tight bounds for complex job sequencing problems, improving solution quality and computational efficiency.

Contribution

It introduces a general relaxation technique for decision diagrams applicable to job sequencing and dynamic programming, with guidelines for identifying suitable problems.

Findings

Lagrangian relaxation yields very tight bounds for certain job sequencing classes.

Proves best known solutions for Biskup-Feldman instances are within 1% of optimal.

Sometimes finds optimal solutions for challenging instances.

Abstract

We introduce a general method for relaxing decision diagrams that allows one to bound job sequencing problems by solving a Lagrangian dual problem on a relaxed diagram. We also provide guidelines for identifying problems for which this approach can result in useful bounds. These same guidelines can be applied to bounding deterministic dynamic programming problems in general, since decision diagrams rely on DP formulations. Computational tests show that \mbox{Lagrangian} relaxation on a decision diagram can yield very tight bounds for certain classes of hard job sequencing problems. For example, it proves for the first time that the best known solutions for Biskup-Feldman instances are within a small fraction of 1% of the optimal value, and sometimes optimal.