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INVALS: A Forward Looking Inventory Allocation System

Shiv Krishna Jaiswal, Karthik S. Gurumoorthy, Etika Agarwal, Shantala Manchenahally

TL;DR

INVALS addresses the challenge of allocating warehouse inventory to multiple stores by planning beyond immediate needs to exploit forward demand, thereby enhancing labor and trailer utilization while curbing stock-outs and excess stock. The authors formulate a MILP-style objective that combines item-store allocation, store trailer priority, and a min-capacity penalty, and solve it using a three-layer approach: incremental trailer assignment guided by submodularity, transformation of the remaining LP to a capacity-constrained OT problem, and double entropic regularization-based drainage (DRM) to obtain efficient solutions. The method hinges on the submodular structure of the trailer assignment objective and a careful reduction to OT, enabling fast, scalable computation on large retailer data with thousands of items and hundreds of stores. Empirical results show substantial improvements in labor and trailer utilization, a left-skew in days-of-supply distributions for pull-forward items, and around 90% agreement with globally optimal MILP solutions across replenishment cycles, underscoring practical applicability and potential for broader multi-echelon extension.

Abstract

We design an Inventory Allocation System (INVALS) that, for each item-store combination, plans the quantity to be allocated from a warehouse that replenishes multiple stores using trailers, while respecting typical operational constraints. We formulate a linear objective function which, when maximized, determines the allocation plan by considering not only the immediate store needs, but also its future (forward) expected demand. This forward-looking allocation significantly improves the utilization of labor and trailers in the warehouse. To reduce overstocking, we adapt from our objective to prioritize allocating those items in excess which are sold faster at the stores, keeping the days of supply (DOS) to a minimum. For the proposed formulation, which is an instance of Mixed Integer Linear Programming (MILP), we present a scalable algorithm using the concepts of submodularity and optimal transport theory by: (i) sequentially adding trailers to stores based on maximum incremental gain, (ii) transforming the resultant linear program (LP) instance to an instance of capacity constrained optimal transport (COT), solvable using double entropic regularization and incurring the same computational complexity as the Sinkhorn algorithm. Compared against the planning engine that only allocates for immediate store needs, INVALS increases labor utilization by 34.70% and item occupancy in trailers by 37.08% on average. The DOS distribution is also skewed to the left, indicating that higher-demand items are allocated in excess, reducing the days they are stocked. We empirically observed that for ~90% of the replenishment cycles, the allocation results of INVALS are identical to the globally optimal MILP solution.

INVALS: A Forward Looking Inventory Allocation System

TL;DR

INVALS addresses the challenge of allocating warehouse inventory to multiple stores by planning beyond immediate needs to exploit forward demand, thereby enhancing labor and trailer utilization while curbing stock-outs and excess stock. The authors formulate a MILP-style objective that combines item-store allocation, store trailer priority, and a min-capacity penalty, and solve it using a three-layer approach: incremental trailer assignment guided by submodularity, transformation of the remaining LP to a capacity-constrained OT problem, and double entropic regularization-based drainage (DRM) to obtain efficient solutions. The method hinges on the submodular structure of the trailer assignment objective and a careful reduction to OT, enabling fast, scalable computation on large retailer data with thousands of items and hundreds of stores. Empirical results show substantial improvements in labor and trailer utilization, a left-skew in days-of-supply distributions for pull-forward items, and around 90% agreement with globally optimal MILP solutions across replenishment cycles, underscoring practical applicability and potential for broader multi-echelon extension.

Abstract

We design an Inventory Allocation System (INVALS) that, for each item-store combination, plans the quantity to be allocated from a warehouse that replenishes multiple stores using trailers, while respecting typical operational constraints. We formulate a linear objective function which, when maximized, determines the allocation plan by considering not only the immediate store needs, but also its future (forward) expected demand. This forward-looking allocation significantly improves the utilization of labor and trailers in the warehouse. To reduce overstocking, we adapt from our objective to prioritize allocating those items in excess which are sold faster at the stores, keeping the days of supply (DOS) to a minimum. For the proposed formulation, which is an instance of Mixed Integer Linear Programming (MILP), we present a scalable algorithm using the concepts of submodularity and optimal transport theory by: (i) sequentially adding trailers to stores based on maximum incremental gain, (ii) transforming the resultant linear program (LP) instance to an instance of capacity constrained optimal transport (COT), solvable using double entropic regularization and incurring the same computational complexity as the Sinkhorn algorithm. Compared against the planning engine that only allocates for immediate store needs, INVALS increases labor utilization by 34.70% and item occupancy in trailers by 37.08% on average. The DOS distribution is also skewed to the left, indicating that higher-demand items are allocated in excess, reducing the days they are stocked. We empirically observed that for ~90% of the replenishment cycles, the allocation results of INVALS are identical to the globally optimal MILP solution.
Paper Structure (28 sections, 3 theorems, 18 equations, 4 figures)

This paper contains 28 sections, 3 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Replenishment Systems and Related Work
  4. Existing Replenishment System
  5. INVALS
  6. Mathematical Formulation
  7. Notations
  8. Utility Function
  9. Constraints
  10. Proposed Solution
  11. Incremental Trailer Assignment
  12. Batching
  13. Transformation to Optimal Transport Problem
  14. Experiments
  15. Set-up
  16. ...and 13 more sections

Key Result

Theorem 4.1

Let $P_{ij}^t = \alpha[t] q_{ij}^t$ represent the profit for unit allocation of item $i$ to $j^{th}$ on day $t$. Denote $M_j=Mx_j$ as the maximum possible allocation to store $j$ for the current number of assigned trailers $x_j$. By adding pseudo items and pseudo stores to produce the super-sets $\t subject to the constraints: satisfying $K = \sum_{i \in \tilde{\mathcal{I}}} S_i = \sum_{j \in \ti

Figures (4)

  • Figure 1: Disallowing PF in exp. A results in poor trailer and labour utilization, and disabling penalty on LMTC leads to poor trailer utilization in exp. B. Changing the optimization solver for exp. D between Da--Dd has almost no effect on the trailer and labour utilization, highlighting the quality of our approximate but fast solution approach.
  • Figure 2: With $q_{ij}^t=1$ in exp. C, the DOS distribution is further to the right of the time axis compared to exps. Da--Dd where it is inversely proportional to store-item DOS. Reduced DOS will decrease the inventory holding costs at the stores and increase savings. Note that the four optimization algorithms in exps. Da-Dd yield almost identical results.
  • Figure 3: Execution runtime of exps. Db, Dc and Dd for the largest LP instances of problems \ref{['problem:linear_program']} and \ref{['problem:COT']}. DRM in exp. Dd takes $61.16\%$ less time on average than GLOP in exp. Dc for the COT instance in \ref{['problem:COT']}.
  • Figure 4: Distribution of normalised $g_{\mathcal{X}}(k) - g_{\mathcal{Y}}(k)$

Theorems & Definitions (7)

  • Theorem 4.1
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Definition C.1: Incremental Gain
  • Definition C.2: Modularity, Submodularity, Non-negativity