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Improved Online Algorithms for Inventory Management Problems with Holding and Delay Costs: Riding the Wave Makes Things Simpler, Stronger, & More General

David Shmoys, Varun Suriyanarayana, Seeun William Umboh

TL;DR

The paper tackles online inventory management with both holding and delay costs, focusing on the Joint Replenishment Problem (JRP) and its single-item lot-sizing special case. It develops a primal-dual online framework with a wavefront dual and a premature-service mechanism that ranks future demands by when their delay cost would equal the current holding cost, enabling coordinated replenishment decisions. The authors achieve a 5-competitive online algorithm for JRP with arbitrary monotone holding-delay costs and, for the single-item problem, raise the competitive ratio to $1+ ext{phi}$ under general monotone costs, significantly improving prior results. They further extend ideas to non-monotone costs, showing hardness connections to set cover and providing a nuanced analysis that blends online dual fitting with phase-initiating and premature servicing concepts, yielding insights with practical impact for inventory coordination under time-sensitive costs.

Abstract

The Joint Replenishment Problem (JRP) is a classical inventory management problem, that aims to model the trade-off between coordinating orders for multiple commodities (and their cost) with holding costs incurred by meeting demand in advance. Moseley, Niaparast and Ravi introduced a natural online generalization of the JRP in which inventory corresponding to demands may be replenished late, for a delay cost, or early, for a holding cost. They established that when the holding and delay costs are monotone and uniform across demands, there is a 30-competitive algorithm that employs a greedy strategy and a dual-fitting based analysis. We develop a 5-competitive algorithm that handles arbitrary monotone demand-specific holding and delay cost functions, thus simultaneously improving upon the competitive ratio and relaxing the uniformity assumption. Our primal-dual algorithm is in the spirit of the work Buchbinder, Kimbrel, Levi, Makarychev, and Sviridenko, which maintains a wavefront dual solution to decide when to place an order and which items to order. The main twist is in deciding which requests to serve early. In contrast to the work of Moseley et al., which ranks early requests in ascending order of desired service time and serves them until their total holding cost matches the ordering cost incurred for that item, we extend to the non-uniform case by instead ranking in ascending order of when the delay cost of a demand would reach its current holding cost. An important special case of the JRP is the single-item lot-sizing problem. Here, Moseley et al. gave a 3-competitive algorithm when the holding and delay costs are uniform across demands. We provide a new algorithm for which the competitive ratio is $φ+1 \approx 2.681$, where $φ$ is the golden ratio, which again holds for arbitrary monotone holding-delay costs.

Improved Online Algorithms for Inventory Management Problems with Holding and Delay Costs: Riding the Wave Makes Things Simpler, Stronger, & More General

TL;DR

The paper tackles online inventory management with both holding and delay costs, focusing on the Joint Replenishment Problem (JRP) and its single-item lot-sizing special case. It develops a primal-dual online framework with a wavefront dual and a premature-service mechanism that ranks future demands by when their delay cost would equal the current holding cost, enabling coordinated replenishment decisions. The authors achieve a 5-competitive online algorithm for JRP with arbitrary monotone holding-delay costs and, for the single-item problem, raise the competitive ratio to under general monotone costs, significantly improving prior results. They further extend ideas to non-monotone costs, showing hardness connections to set cover and providing a nuanced analysis that blends online dual fitting with phase-initiating and premature servicing concepts, yielding insights with practical impact for inventory coordination under time-sensitive costs.

Abstract

The Joint Replenishment Problem (JRP) is a classical inventory management problem, that aims to model the trade-off between coordinating orders for multiple commodities (and their cost) with holding costs incurred by meeting demand in advance. Moseley, Niaparast and Ravi introduced a natural online generalization of the JRP in which inventory corresponding to demands may be replenished late, for a delay cost, or early, for a holding cost. They established that when the holding and delay costs are monotone and uniform across demands, there is a 30-competitive algorithm that employs a greedy strategy and a dual-fitting based analysis. We develop a 5-competitive algorithm that handles arbitrary monotone demand-specific holding and delay cost functions, thus simultaneously improving upon the competitive ratio and relaxing the uniformity assumption. Our primal-dual algorithm is in the spirit of the work Buchbinder, Kimbrel, Levi, Makarychev, and Sviridenko, which maintains a wavefront dual solution to decide when to place an order and which items to order. The main twist is in deciding which requests to serve early. In contrast to the work of Moseley et al., which ranks early requests in ascending order of desired service time and serves them until their total holding cost matches the ordering cost incurred for that item, we extend to the non-uniform case by instead ranking in ascending order of when the delay cost of a demand would reach its current holding cost. An important special case of the JRP is the single-item lot-sizing problem. Here, Moseley et al. gave a 3-competitive algorithm when the holding and delay costs are uniform across demands. We provide a new algorithm for which the competitive ratio is , where is the golden ratio, which again holds for arbitrary monotone holding-delay costs.
Paper Structure (20 sections, 23 theorems, 6 equations, 1 figure, 3 algorithms)

This paper contains 20 sections, 23 theorems, 6 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Joint Replenishment Problem with Holding and Delay Costs (JRP):
  4. The Single-Item Lot-Sizing with Holding and Delay Costs:
  5. The Single-Item Lot-Sizing Problem
  6. An Offline Exact Optimization Algorithm
  7. Description of primal solution:
  8. Online Algorithm
  9. High-Level Description:
  10. The dual solution and replenishment orders:
  11. Identifying which demands to serve when an order is placed:
  12. Extending to the Joint Replenishment Problem
  13. Algorithm Description
  14. Dual solution and placing replenishment orders:
  15. Identifying which item types to serve when an order is placed:
  16. ...and 5 more sections

Key Result

Lemma 3.1

For every demand $t$, it holds that $I^d_t \cap O \neq \emptyset$.

Figures (1)

  • Figure 3.1: When placing an order at time $s$, we rank demands with desired service time $>s$ in ascending order of when their delay cost equals their current holding cost. For each demand $t_i$, this happens at time $g_{t_i}$. When considering demands in ascending order of their $g_t$ value, we consider demand $t_2$ first because $g_{t_2}<g_{t_1}<g_{t_{3}}$. $H^{t_2}_s=0.75K<K$, so we can serve $t_2$. However, if we also tried to serve $t_1$, the total holding cost would be $1.25K>K$, therefore, we serve $t_2$ but not $t_1$ at time $s$. The algorithm will not even consider serving $t_3$ at time $s$ eventhough $H^{t_2}_s+H^{t_3}_s=0.9K<K$ because $g_{t_3}>g_{t_1}$ and $t_1$ will not be served at time $s$.

Theorems & Definitions (45)

  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Lemma 3.3
  • Proof 3
  • Lemma 3.4
  • Proof 4
  • Theorem 3.5
  • Proof 5
  • ...and 35 more