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<p>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 (e.g., transition time), non-fungible, and reallocated among warehouses 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<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="453_2025_1320_Article_IEq1.gif" Format="GIF" Height="6" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(=\)</EquationSource> <EquationSource Format="MATHML"><math> <mo>=</mo> </math></EquationSource> </InlineEquation>NP. We then consider to solve the problem by relaxing capacity constraints, 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. We also show the relationship between the reallocation problem and the bin packing problem when the warehouse and carry-in capacities are sufficiently large.</p>

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Reallocation Problems with Minimum Completion Time

  • 期刊论文

  • 发表日期：

    2025年5月16日

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  • Toshimasa Ishii,
  • Jun Kawahara,
  • Kazuhisa Makino,
  • Hirotaka Ono

摘要

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 (e.g., transition time), non-fungible, and reallocated among warehouses 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 to solve the problem by relaxing capacity constraints, 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. We also show the relationship between the reallocation problem and the bin packing problem when the warehouse and carry-in capacities are sufficiently large.