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Robustness of Online Inventory Balancing to Inventory Shocks

Yiding Feng, Rad Niazadeh, Amin Saberi

TL;DR

This work designs a new family of optimal competitive algorithms calledatched Inventory Balancing (BIB), and bound the competitive ratio of BIB against optimal offline, which is asymptotically optimal and converges to (1-1/e) as initial inventories grow, in contrast to the original IB which no longer achieves the optimal ratio in this new model.

Abstract

In classic adversarial online resource allocation problems such as AdWords, customers arrive online while products are given offline with a fixed initial inventory. To ensure revenue guarantees under uncertainty, the decision maker must balance consumption across products. Based on this, the prevalent policy "inventory balancing (IB)" has proved to be optimal or near-optimal competitive in almost all classic settings. However, these models do not capture various forms of inventory shocks on the supply side, which play an important role in real-world online assortment and can significantly impact the revenue performance of the IB algorithm. Motivated by this paradigm, we introduce a variant of online assortment planning with inventory shocks. Our model considers adversarial exogenous shocks (where supply increases unpredictably) and allocation-coupled endogenous shocks (where an inventory reduction is triggered by the algorithms and re-adjusted after a usage duration), whose combination leads to non-monotonic inventory fluctuations. As our main result, we show the robustness of IB-type strategies against such shocks by designing a new family of optimal competitive algorithms called "Batched Inventory Balancing (BIB)." Using a novel randomized primal-dual method, we bound the competitive ratio of BIB against optimal offline. We show that with proper choice of a certain parameter, this competitive ratio is asymptotically optimal and converges to (1-1/e) as initial inventories grow, in contrast to the original IB which no longer achieves the optimal ratio in this new model. Moreover, we characterize BIB's competitive ratio parametric by its penalty function and show that it matches exactly the competitive ratio of IB without shocks. Our refined analysis reduces the dual construction to a combinatorial "interval assignment problem" whose algorithmic solution may be of independent interest.

Robustness of Online Inventory Balancing to Inventory Shocks

TL;DR

This work designs a new family of optimal competitive algorithms calledatched Inventory Balancing (BIB), and bound the competitive ratio of BIB against optimal offline, which is asymptotically optimal and converges to (1-1/e) as initial inventories grow, in contrast to the original IB which no longer achieves the optimal ratio in this new model.

Abstract

In classic adversarial online resource allocation problems such as AdWords, customers arrive online while products are given offline with a fixed initial inventory. To ensure revenue guarantees under uncertainty, the decision maker must balance consumption across products. Based on this, the prevalent policy "inventory balancing (IB)" has proved to be optimal or near-optimal competitive in almost all classic settings. However, these models do not capture various forms of inventory shocks on the supply side, which play an important role in real-world online assortment and can significantly impact the revenue performance of the IB algorithm. Motivated by this paradigm, we introduce a variant of online assortment planning with inventory shocks. Our model considers adversarial exogenous shocks (where supply increases unpredictably) and allocation-coupled endogenous shocks (where an inventory reduction is triggered by the algorithms and re-adjusted after a usage duration), whose combination leads to non-monotonic inventory fluctuations. As our main result, we show the robustness of IB-type strategies against such shocks by designing a new family of optimal competitive algorithms called "Batched Inventory Balancing (BIB)." Using a novel randomized primal-dual method, we bound the competitive ratio of BIB against optimal offline. We show that with proper choice of a certain parameter, this competitive ratio is asymptotically optimal and converges to (1-1/e) as initial inventories grow, in contrast to the original IB which no longer achieves the optimal ratio in this new model. Moreover, we characterize BIB's competitive ratio parametric by its penalty function and show that it matches exactly the competitive ratio of IB without shocks. Our refined analysis reduces the dual construction to a combinatorial "interval assignment problem" whose algorithmic solution may be of independent interest.

Paper Structure

This paper contains 22 sections, 3 theorems, 15 equations, 3 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotically Optimal Algorithm: Batched Inventory Balancing
  4. An Overview of the BIB Algorithm
  5. Main Result: Competitive Ratio of BIB
  6. Corollaries and Implications of Theorem \ref{['thm:concave competitive ratio']}
  7. Competitive Ratio Analysis: Proof of Theorem \ref{['thm:concave competitive ratio']}
  8. Batch-Specified Assortment and LP Revenue Benchmark
  9. Competitive Ratio Analysis of Batched Inventory Balancing
  10. Interval Assignment Problem
  11. Conclusion
  12. Further Related Work
  13. Discussion on Practical Relevance
  14. Numerical Experiments
  15. Numerical Performance under Stylized Instances
  16. ...and 7 more sections

Key Result

Theorem 1

For any monotone non-decreasing, concave and differentiable penalty function $\Psi$, and any batch-size scalar $\gamma\in[1, {c_{0}}]$, the competitive ratio of the Batched Inventory Balancing algorithm is at least $\Gamma(\Psi, \gamma) = \min\{\Gamma_1(\Psi,\gamma), \Gamma_2(\Psi,\gamma)\}$, wher and ${c_{0}} = \min_{i\in[n]}c_i$ is the smallest initial inventory.

Figures (3)

  • Figure EC.1: Box and whisker comparison of revenue under different policies (\ref{['apx:numerical random instance']}); Results are based on 20 iterations of Monte-Carlo simulation.
  • Figure EC.2: \ref{['example:failure of actual consumption level']}: A path of four overlapping intervals. Each interval overlaps only with its immediate neighbor. Both endpoints $(a_i, b_i)$ for each interval $i$ are marked in red. The bold number inside each interval indicates its IAP assignment ${\hat{I}}_i$, and the blue dashed arrows illustrate the predecessor links $p_j = i$ that define the partition satisfying \ref{['lem:IAP']}.
  • Figure EC.3: \ref{['example:failure of greedy']}: A complete binary-tree hierarchy of seven intervals. Each parent fully contains its two children. Both endpoints $(a_i, b_i)$ for each interval $i$ are marked in red. The bold number inside each interval indicates its IAP assignment ${\hat{I}}_i$, and the blue dashed arrows illustrate the predecessor links $p_j = i$ that define the partition satisfying \ref{['lem:IAP']}.

Theorems & Definitions (9)

  • Definition 1: Competitive ratio
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • Example EC.1: A Path of Overlapping Intervals
  • Example EC.2: A Binary-Tree Structure of Intervals