A study of the Bienstock-Zuckerberg algorithm, Applications in Mining and Resource Constrained Project Scheduling

TL;DR

This paper explores the Bienstock-Zuckerberg algorithm's broader applicability to various scheduling problems, providing new proofs, computational techniques, and empirical evidence of its superior performance over traditional methods.

Contribution

It extends the BZ algorithm's application to a wider class of scheduling problems, offers a new proof of correctness, and demonstrates its computational advantages through experiments.

Findings

BZ algorithm outperforms Dantzig-Wolfe in tested problems

Speed-up techniques significantly reduce solution times

Computational experiments validate the algorithm's effectiveness

Abstract

We study a Lagrangian decomposition algorithm recently proposed by Dan Bienstock and Mark Zuckerberg for solving the LP relaxation of a class of open pit mine project scheduling problems. In this study we show that the Bienstock-Zuckerberg (BZ) algorithm can be used to solve LP relaxations corresponding to a much broader class of scheduling problems, including the well-known Resource Constrained Project Scheduling Problem (RCPSP), and multi-modal variants of the RCPSP that consider batch processing of jobs. We present a new, intuitive proof of correctness for the BZ algorithm that works by casting the BZ algorithm as a column generation algorithm. This analysis allows us to draw parallels with the well-known Dantzig-Wolfe (DW) algorithm. We discuss practical computational techniques for speeding up the performance of the BZ and DW algorithms on project scheduling problems. Finally, we present computational experiments independently testing the effectiveness of the BZ and DW algorithms on different sets of publicly available test instances. Our computational experiments confirm that the BZ algorithm significantly outperforms the DW algorithm for the problems considered. Our computational experiments also show that the proposed speed-up techniques can have a significant impact on solution time. We provide some insights on what might be explaining this significant difference in performance.