A Study on the Block Relocation Problem: Lower Bound Derivations and Strong Formulations

TL;DR

This paper advances understanding of the Block Relocation Problem by reviewing variants, deriving stronger lower bounds, and developing robust MIP formulations for exact solutions, with significant computational improvements.

Contribution

It introduces a comprehensive classification, a general lower bound framework, and a new strong MIP formulation for multiple BRP variants, enhancing solution quality and computational efficiency.

Findings

New lower bound dominates existing bounds

MIP formulations outperform previous methods

Computational results show improved solution times

Abstract

The block relocation problem (BRP) is a fundamental operational issue in modern warehouse and yard management, which, however, is very challenging to solve. In this paper, to advance our understanding on this problem and to provide a substantial assistance to practice, we (i) introduce a classification scheme and present a rather comprehensive review on all 16 BRP variants; (ii) develop a general framework to derive lower bounds on the number of necessary relocations and demonstrate its connection to existing lower bounds of the unrestricted BRP variants; (iii) propose and employ a couple of new critical substructures concepts to analyze the BRP and obtain a lower bound that dominates all existing ones; (iv) build a new and strong mixed integer programming (MIP) formulation that is adaptable to compute 8 BRP variants, and design a novel MIP-formulation-based iterative procedure to compute exact BRP solutions; (v) extend the MIP formulation to address four typical industrial considerations. Computational results on standard test instances show that the new lower bound is significantly stronger, and our new MIP computational methods have superior performances over a state-of-the-art formulation.