Reallocation Problems with Minimum Completion Time

TL;DR

This paper studies the complexity of reallocation scheduling problems involving fixed, non-fungible products, providing polynomial solutions for uniform cases, proving NP-completeness for more general cases, and proposing approximation algorithms under capacity augmentation.

Contribution

It offers a comprehensive complexity analysis of reallocation problems, introduces capacity augmentation techniques, and develops constant-factor approximation algorithms for large-capacity scenarios.

Findings

Polynomial-time solution for uniform product size and transition time.

NP-completeness of the problem in more general settings.

Constant-factor approximation algorithms for large warehouse capacities.

Abstract

Reallocation scheduling is one of the most fundamental problems in various areas such as supply chain management, logistics, and transportation science. In this paper, we introduce the reallocation problem that models the scheduling in which products are with fixed cost, non-fungible, and reallocated in parallel, and comprehensively study the complexity of the problem under various settings of the transition time, product size, and capacities. We show that the problem can be solved in polynomial time for a fundamental setting where the product size and transition time are both uniform. We also show that the feasibility of the problem is NP-complete even for little more general settings, which implies that no polynomial-time algorithm constructs a feasible schedule of the problem unless P$=$NP. We then consider the relaxation of the problem, which we call the capacity augmentation, and derive a reallocation schedule feasible with the augmentation such that the completion time is at most the optimal of the original problem. When the warehouse capacity is sufficiently large, we design constant-factor approximation algorithms under all the settings. We also show the relationship between the reallocation problem and the bin packing problem when the warehouse and carry-in capacities are sufficiently large.