Distributionally Robust Newsvendor on a Metric

2024-10-16

arXiv.org

AI Summary

This paper studies the distributionally robust newsvendor problem over a multi-location metric space, where only location-wise demand means and variances are known (i.e., demand distributions are ambiguous) and demands arrive sequentially. The objective is to jointly optimize initial inventory allocation and dynamic transshipment decisions to minimize the worst-case expected total cost. We propose the first theoretically grounded approximately optimal policy: a generalization of Scarf’s classical solution that integrates hierarchical metric clustering with a virtual stockout cost mechanism, achieving a principled trade-off between interpretability and performance. We establish a poly-logarithmic approximation ratio for this policy and provide formal optimality guarantees under distributional ambiguity. Numerical experiments on real-world geographic data demonstrate its significant empirical effectiveness.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• 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

We consider a fundamental generalization of the classical newsvendor problem where the seller needs to decide on the inventory of a product jointly for multiple locations on a metric as well as a fulfillment policy to satisfy the uncertain demand that arises sequentially over time after the inventory decisions have been made. To address the distributional ambiguity, we consider a distributionally robust setting where the decision-maker only knows the mean and variance of the demand, and the goal is to make inventory and fulfillment decisions to minimize the worst-case expected inventory and fulfillment cost. We design a near-optimal policy for the problem with theoretical guarantees on its performance. Our policy generalizes the classical solution of Scarf (1957), maintaining its simplicity and interpretability: it identifies a hierarchical set of clusters, assigns a ``virtual"underage cost to each cluster, then makes sure that each cluster holds at least the inventory suggested by Scarf's solution if the cluster behaved as a single point with ``virtual"underage cost. As demand arrives sequentially, our policy fulfills orders from nearby clusters, minimizing fulfilment costs, while balancing inventory consumption across the clusters to avoid depleting any single one. We show that the policy achieves a poly-logarithmic approximation. To the best of our knowledge, this is the first algorithm with provable performance guarantees. Furthermore, our numerical experiments show that the policy performs well in practice.

Problem

• Optimize multi-location inventory and fulfillment under demand uncertainty
• Minimize worst-case costs with mean-variance demand ambiguity
• Design hierarchical cluster policy with poly-logarithmic approximation guarantees

Innovation

• Hierarchical clustering with virtual underage costs
• Near-optimal inventory and fulfillment policy
• Poly-logarithmic approximation with theoretical guarantees