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

2026-02-12

AI Summary

This work addresses the online joint replenishment problem (JRP) under non-uniform and monotone holding and delay costs. The authors propose an online algorithm grounded in the primal-dual framework, which dynamically determines order timing and item bundling by maintaining a wavefront dual solution. Requests are prioritized for early service based on the time when their delay cost equals the current holding cost. The algorithm achieves a competitive ratio of 5 under arbitrary monotone demand-dependent costs, breaking through the previous barrier of a 30-competitive ratio and eliminating the restrictive assumption of uniform costs. In the single-item case, it attains the optimal competitive ratio of φ+1 ≈ 2.681, significantly advancing both the theoretical limits and practical applicability of online algorithms for JRP.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Algorithm Configuration
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: The sharing economy
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $\phi +1 \approx 2.681$, where $\phi$ is the golden ratio, which again holds for arbitrary monotone holding-delay costs.

Problem

• Joint Replenishment Problem
• Inventory Management
• Holding Cost
• Delay Cost
• Online Algorithms

Innovation

• Online Algorithms
• Joint Replenishment Problem
• Primal-Dual Method
• Holding and Delay Costs
• Competitive Ratio