Planning Inventory Up Front Can Reduce the Need for Dynamic Product Selection

A retailer must decide both which products to carry and how many units of each to stock. These decisions interact: selling out of one item can change demand for the others. In their November 2026 arXiv preprint, Jacob Feldman, Heng Zhang and Yukai Huang develop algorithms for this joint planning problem and prove conditional guarantees on expected revenue. The paper presents theoretical results, not a retail field trial.

Two ways to offer products

The authors study a selling horizon with a fixed capacity measured in total units stocked. Inventory is chosen beforehand, cannot be replenished, and falls by one unit with each purchase. Customers arrive sequentially, choosing a product or making no purchase.

In the show-all setting, every available product must be offered. The retailer's only decision is the initial inventory; the displayed selection changes as products sell out.

In personalization, the retailer observes each customer's type and may offer a subset of available products. The benchmark permits these choices to depend on remaining inventory and the entire purchase history.

Both settings assume substitutable choice: removing products cannot reduce the purchase probability of an item that remains available. This allows several choice models, including multinomial logit and ranking-based models, but does not cover every possible pattern of customer behavior.

What the guarantees mean

The framework relies on a separate algorithm for a static planning problem. For show-all, that problem selects a limited number of products to maximize expected revenue from a customer, without tracking inventory depletion. Let its approximation factor be alpha: it achieves at least that fraction of the static optimum. The framework requires this capability for modified product rewards, not just the original revenues.

The authors prove that their inventory algorithm achieves at least 0.486 times alpha of optimal expected show-all revenue when the customer count is known in advance or follows a Poisson distribution. For an increasing-failure-rate, or IFR, customer count, the guarantee is 0.4328 times alpha.

Here IFR restricts how the customer-count distribution can behave: conditional on reaching a given customer, the probability that this is the last customer does not decrease as the count increases. It does not mean that customers arrive at an increasing rate over clock time.

These are worst-case bounds on expected revenue, averaging over customer uncertainty and the algorithm's randomized inventory choice. They are neither predictions of typical performance nor promises for each selling season. Capacity, however, is respected in every inventory realization.

Turning an optimistic plan into whole units

The method begins with a fluid relaxation, an optimistic mathematical model that accounts for expected sales without requiring a feasible whole-unit inventory plan. Its value upper-bounds the achievable optimum.

Because this relaxation has two resource constraints, the framework can work with a solution involving at most two assortments. For each, it constructs a fractional inventory allocation combining a baseline amount per product with an amount related to expected demand.

The analysis bounds sales by hypothetical demand capped at available inventory. Substitutability supports this comparison: while an item remains available, removing sold-out alternatives cannot make it less likely to be purchased.

Finally, dependent rounding converts fractional quantities into whole units while maintaining capacity and preserving an analytical revenue lower bound in expectation. It does not claim to preserve actual revenue exactly.

Personalized offers without repeated optimization

For personalization, the static subproblem is different. It jointly selects a shared product catalog and one assortment for each customer type. Its approximation factor—not the show-all static factor—determines the resulting revenue guarantee.

The implemented policy fixes those type-specific assortments before selling starts. Each customer sees every still-available product in their predetermined assortment. Stock is shared across types, and the policy neither introduces previously excluded products nor reoptimizes offers after stockouts.

Nevertheless, the authors obtain the same numerical factors against the optimal fully adaptive personalized policy. For multinomial logit choice, the cited static customization algorithm yields guarantees approaching 0.243 for deterministic or Poisson counts and 0.2164 for IFR counts. The paper explicitly notes that these do not improve the cited specialized multinomial-logit guarantees.

Where the conclusions stop

The assumptions include known choice probabilities and arrival parameters, independent purchase decisions conditional on offers and types, and observable types for personalization. With random customer counts, types must be independent and identically distributed, independently of that count; deterministic counts allow period-dependent type probabilities.

The authors also construct examples limiting comparison with the fluid benchmark: optimal show-all revenue can approach one-half of that bound under deterministic counts, or about 0.4328567 under IFR counts. These limits concern the relaxation's optimism, not impossibility results for algorithms measured against the true stochastic optimum.