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An Optimistic-Robust Approach for Dynamic Positioning of Omnichannel Inventories

Pavithra Harsha, Shivaram Subramanian, Ali Koc, Mahesh Ramakrishna, Brian Quanz, Dhruv Shah, Chandra Narayanaswami

TL;DR

The paper presents BIO, a data-driven, distribution-free bimodal optimization framework for omnichannel inventory that blends worst-case robustness with optimistic demand to improve average profitability. By introducing an allied-adversary structure with an optimism parameter $\lambda$, BIO interpolates between pure RO and best-case scenarios, and proves a linear superposition property linking BIO-$\lambda$ to BIO-0 and BIO-1. It provides an exact MILP reformulation of the inner problem via RLT and solves large-scale instances with a Column-and-Cut Generation (CCG) method, validated on real data from a large retailer. Computational experiments show BIO can outperform pure RO by up to 27% in realized profitability and exceed baselines under imperfect distributional information by more than 10%, highlighting its practical value in cross-location, cross-channel inventory positioning. The work also offers extensions for inventory movement, information-edge exploitation, and closed-form solutions, making BIO a scalable, data-driven alternative to traditional RO for omnichannel operations.

Abstract

We introduce a new class of data-driven and distribution-free optimistic-robust bimodal inventory optimization (BIO) strategy to effectively allocate inventory across a retail chain to meet time-varying, uncertain omnichannel demand. The bimodal nature of BIO stems from its ability to balance downside risk, as in traditional Robust Optimization (RO), which focuses on worst-case adversarial demand, with upside potential to enhance average-case performance. This enables BIO to remain as resilient as RO while capturing benefits that would otherwise be lost due to endogenous outliers. Omnichannel inventory planning provides a suitable problem setting for analyzing the effectiveness of BIO's bimodal strategy in managing the tradeoff between lost sales at stores and cross-channel e-commerce fulfillment costs, factors that are inherently asymmetric due to channel-specific behaviors. We provide structural insights about the BIO solution and how it can be tuned to achieve a preferred tradeoff between robustness and the average-case performance. Using a real-world dataset from a large American omnichannel retail chain, a business value assessment during a peak period indicates that BIO outperforms pure RO by 27% in terms of realized average profitability and surpasses other competitive baselines under imperfect distributional information by over 10%. This demonstrates that BIO provides a novel, data-driven, and distribution-free alternative to traditional RO that achieves strong average performance while carefully balancing robustness.

An Optimistic-Robust Approach for Dynamic Positioning of Omnichannel Inventories

TL;DR

The paper presents BIO, a data-driven, distribution-free bimodal optimization framework for omnichannel inventory that blends worst-case robustness with optimistic demand to improve average profitability. By introducing an allied-adversary structure with an optimism parameter , BIO interpolates between pure RO and best-case scenarios, and proves a linear superposition property linking BIO- to BIO-0 and BIO-1. It provides an exact MILP reformulation of the inner problem via RLT and solves large-scale instances with a Column-and-Cut Generation (CCG) method, validated on real data from a large retailer. Computational experiments show BIO can outperform pure RO by up to 27% in realized profitability and exceed baselines under imperfect distributional information by more than 10%, highlighting its practical value in cross-location, cross-channel inventory positioning. The work also offers extensions for inventory movement, information-edge exploitation, and closed-form solutions, making BIO a scalable, data-driven alternative to traditional RO for omnichannel operations.

Abstract

We introduce a new class of data-driven and distribution-free optimistic-robust bimodal inventory optimization (BIO) strategy to effectively allocate inventory across a retail chain to meet time-varying, uncertain omnichannel demand. The bimodal nature of BIO stems from its ability to balance downside risk, as in traditional Robust Optimization (RO), which focuses on worst-case adversarial demand, with upside potential to enhance average-case performance. This enables BIO to remain as resilient as RO while capturing benefits that would otherwise be lost due to endogenous outliers. Omnichannel inventory planning provides a suitable problem setting for analyzing the effectiveness of BIO's bimodal strategy in managing the tradeoff between lost sales at stores and cross-channel e-commerce fulfillment costs, factors that are inherently asymmetric due to channel-specific behaviors. We provide structural insights about the BIO solution and how it can be tuned to achieve a preferred tradeoff between robustness and the average-case performance. Using a real-world dataset from a large American omnichannel retail chain, a business value assessment during a peak period indicates that BIO outperforms pure RO by 27% in terms of realized average profitability and surpasses other competitive baselines under imperfect distributional information by over 10%. This demonstrates that BIO provides a novel, data-driven, and distribution-free alternative to traditional RO that achieves strong average performance while carefully balancing robustness.
Paper Structure (20 sections, 8 theorems, 20 equations, 6 figures, 6 tables, 1 algorithm)

This paper contains 20 sections, 8 theorems, 20 equations, 6 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Problem setting, notation and assumptions
  4. Omnichannel robust inventory optimization model
  5. Static robust policy to omnichannel inventory management
  6. Optimistic-robust optimization: An ally-adversary model
  7. Column and Cut Generation (CCG) Method to solve BIO-$\lambda$
  8. Computational Experiments
  9. Inputs for inventory management
  10. Baseline Inventory Optimization Methods
  11. Experimental design and setup
  12. Batch Monte-Carlo simulations with optimizer executed just once:
  13. Business Value Assessment:
  14. Extension to incorporate the movement of pre-existing inventory across nodes
  15. Optimistic-robust modeling to exploit an information edge
  16. ...and 5 more sections

Key Result

Theorem 1

Given $\lambda$, there exists an $x^*_{BIO-\lambda}$ such that $x^*_{BIO-\lambda} = \lambda x^*_{BIO-1} + (1-\lambda)x^*_{BIO-0}$ and the corresponding optimal objective value $Z^*_{BIO-\lambda}$ satisfies $Z^*_{BIO-\lambda} = \lambda Z^*_{BIO-1} + (1-\lambda)Z^*_{BIO-0}$. In other words, for any gi

Figures (6)

  • Figure 1: Example of an omnichannel retailer.
  • Figure 2: The realized profitability box plots across different methods in case 2 of example 1.
  • Figure 3: Actual sales trends by channel (left) and the actual e-commerce sales fraction of the total sales (right) for select SKUs around sales peak.
  • Figure 4: Total inventory allocation across stores and DCs and resultant total profitability distribution in the Monte Carlo simulation. Initial Inventory is zero in (a-c), mean in (d-f), and excess in DC in (g-i). Price and back order penalty multipliers ($p,b$) to the weekly item price are: $p=1, b=0$ in (a,d,g), $p=1, b=1$ in (b,e,h) and $p=0, b=1$ in (c,f,i). Grey-shaded settings: perfect distributional info; others: imperfect or distribution-free.
  • Figure 5: KPIs of different replenishment policies using the transaction-level what-if simulator. Grey-shaded settings assume perfect distributional information; the rest reflect imperfect/distribution-free settings.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Corollary 1
  • Example 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Proposition 4
  • proof
  • proof
  • ...and 1 more