Online Joint Replenishment Problem with Arbitrary Holding and Backlog Costs

2025-07-21

AI Summary

This paper studies the online Joint Replenishment Problem (JRP) under a general setting where holding and stockout costs are arbitrary nonnegative functions. It presents the first deterministic online algorithm with a constant competitive ratio for this setting. To overcome the limitations of traditional deadline-based scheduling, the authors introduce a dynamic priority mechanism that selects subsets of pending requests based on real-time cost evolution. Theoretical analysis establishes competitive ratios of 4 for the single-item case and 16 for the multi-item case—breaking prior restrictions that required costs to be linear or convex. This resolves a long-standing open problem in online JRP. The result provides the first universally applicable, provably competitive algorithmic foundation for joint optimization of nonlinear inventory and delay costs in online supply chain planning.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In their seminal paper Moseley, Niaparast, and Ravi introduced the Joint Replenishment Problem (JRP) with holding and backlog costs that models the trade-off between ordering costs, holding costs, and backlog costs in supply chain planning systems. Their model generalized the classical the make-to-order version as well make-to-stock version. For the case where holding costs function of all items are the same and all backlog costs are the same, they provide a constant competitive algorithm, leaving designing a constant competitive algorithm for arbitrary functions open. Moreover, they noticed that their algorithm does not work for arbitrary (request dependent) holding costs and backlog costs functions. We resolve their open problem and design a constant competitive algorithm that works for arbitrary request dependent functions. Specifically, we establish a 4-competitive algorithm for the single-item case and a 16-competitive for the general (multi-item) version. The algorithm of Moseley, Niaparast, and Ravi is based on fixed priority on the requests to items, and request to an item are always served by order of deadlines. In contrast, we design an algorithm with dynamic priority over the requests such that instead of servicing a prefix by deadline of requests, we may need to service a general subset of the requests.

Problem

• Solves arbitrary holding and backlog costs in JRP
• Provides constant competitive algorithm for dynamic priorities
• Generalizes solution for single-item and multi-item cases

Innovation

• Dynamic priority algorithm for arbitrary costs
• 4-competitive single-item case solution
• 16-competitive multi-item version solution