Joint Inventory Placement, Assortment Personalization, and Order Fulfillment for Substitutable Products

Abstract: Modern online retailers leverage networks of distributed warehouses to rapidly fulfill customer orders. We study a problem of jointly deciding (i) how to allocate inventories across warehouses in the network subject to warehouse capacity and product supply constraints, (ii) how to dynamically select personalized product assortments based on customer preferences and location, as well as real-time stock levels, and (iii) which warehouse to use to fulfill the product chosen by the customer to maximize expected profit. In our model, the firm chooses an inventory placement across warehouses at the start of the selling horizon. Customers of different types then arrive at discrete time periods over a finite time horizon according to a known distribution. When offered an assortment, each customer type chooses at most one product according to a discrete choice model. The firm then chooses a feasible warehouse from which to fulfill the chosen product, depleting its stock and earning a profit that depends on the product, customer type, and warehouse choice. We develop approximation algorithms and policies with provable theoretical guarantees and strong empirical performance. Under general probabilistic choice models, we establish asymptotically optimal approximation algorithms as warehouse capacities and product supplies scale. Under the multinomial logit (MNL) choice model, we obtain constant-factor approximation algorithms, with approximation factors ranging from 0.080 in the most general setting to 0.199 under stationarity and unlimited product supplies, which means that only warehouse capacity constraints are present. To the best of our knowledge, these results provide the first provably efficient algorithms for the joint inventory placement, assortment personalization, and order fulfillment problem.

• The paper develops joint fluid-relaxation algorithms for inventory placement, personalized assortment selection, and warehouse fulfillment, achieving square-root convergence under general choice models and constant-factor guarantees under MNL.
• The proposed policies address stationary and non-stationary demand, preserve expected profit when converted from inventory-agnostic to inventory-aware operation, and improve on prior single-warehouse cubic-root rates.
• Computational experiments show fluid-based methods reach up to 98.05% of the joint fluid upper bound at scale, demonstrating strong practical performance despite conservative worst-case guarantees and heuristic conversion steps.

Problem setting and motivation

This paper studies a joint optimization problem faced by online retailers operating networks of distributed warehouses. The decision-maker must simultaneously determine (i) an initial inventory placement of $n$ products across $L$ warehouses, subject to per-warehouse capacity constraints $K^\ell$ and per-product supply constraints $C_i$; (ii) a dynamic policy that selects personalized assortments for arriving customers based on customer type, time period, and remaining inventory; and (iii) a fulfillment rule assigning each chosen product to a warehouse. A defining feature is the triadic profit structure: the profit $r_{ij}^{\ell}$ from selling product $i$ to customer type $j$ from warehouse $\ell$ depends on all three components, capturing shipping costs and personalized pricing effects. Profits may be negative when shipping costs exceed revenue.

Customers arrive over a finite horizon of $T$ discrete periods according to known arrival probabilities $\lambda_{jt}$, choose at most one product from an offered assortment under a discrete choice model satisfying weak substitutability, and each sale depletes one unit of stock at the fulfilling warehouse. For any fixed placement, the optimal dynamic policy solves a high-dimensional dynamic program whose state space grows exponentially with products and warehouses, making exact solution intractable. The paper develops approximation algorithms with provable guarantees — to the authors' knowledge, the first for this joint problem with distributed warehouses and type- and location-dependent profits.

Main results

The guarantees are organized along two axes: stationary versus non-stationary arrivals, and general versus multinomial logit (MNL) choice models. Let $L$0, $L$1, and let WH denote settings with only capacity constraints and WH+SPL those with both capacity and supply constraints.

Choice model	Stationary WH	Stationary WH+SPL	Non-stationary WH	Non-stationary WH+SPL
General	$L$2	$L$3	$L$4	$L$5
MNL	0.199	0.158	0.101	0.080

Under general choice models, the algorithms are asymptotically optimal as capacities and supplies scale, achieving square-root convergence rates. This improves on the cubic-root rate of Bai et al. (2025), who studied only the single-warehouse case with product-only profits. Under MNL, constant-factor approximations range from 0.080 in the most general setting to $L$6 under stationarity and unlimited supplies. These constants are comparable to prior single-warehouse results but hold in the substantially richer multi-warehouse, triadic-profit setting.

Inventory-agnostic policies via fluid relaxations

The analytical core treats inventory placement as fixed and works with an inventory-agnostic relaxation: the firm may offer out-of-stock products or decline fulfillment, formalized by adding a virtual warehouse $L$7 with infinite stock and zero profit. A fluid linear program LP$L$8 upper-bounds the relaxed optimum $L$9 and is solvable in polynomial time via column generation under a standard assortment optimization oracle.

Stationary arrivals. The Simple Inventory-Agnostic Policy samples assortments proportionally to the fluid variables $K^\ell$0 and fulfills from warehouses proportionally to $K^\ell$1. Under stationarity, expected demand matches the fluid targets exactly, and an exact expression for expected profit decomposes into a potential profit term minus a fractional loss $K^\ell$2. Combining two tail bounds yields a $K^\ell$3-approximation relative to $K^\ell$4, where $K^\ell$5 is the smallest positive stocking quantity.

Non-stationary arrivals. The simple policy can perform arbitrarily badly here: the authors construct a two-period instance with skewed profits ($K^\ell$6) showing its ratio approaches $K^\ell$7, so uniform scaling achieves at best $K^\ell$8. They instead develop a selective scaling procedure: per product-warehouse pair, customer types are ordered by profit, and fulfillment probabilities of low-profit demands are suppressed via parameters $K^\ell$9 computed through a three-case construction involving carefully tuned constants ($C_i$0, $C_i$1) and auxiliary functions balancing profit thresholds against demand suppression. The resulting Advanced Policy guarantees $C_i$2 of the fluid value. A third, Modified Policy introduces customer-specific virtual capacities $C_i$3 and, using a variance-based concentration inequality, attains $C_i$4-approximation, driving the ratio to one as inventories scale.

Joint placement algorithms

For the full problem, both the stocking plan and the dynamic policy derive from a single joint fluid relaxation that combines LP$C_i$5 with relaxed capacity and supply constraints — departing from the conventional sequential two-stage approach. The stocking plan scales and rounds up the fluid solution, $C_i$6, with $C_i$7 chosen so the gap between capacity and expected demand supports the concentration inequality. This absolute-deviation analysis (rather than bounding consumption as a fraction of stock) yields the square-root rates in the table. Feasibility of the rounded plans under both constraint families is verified explicitly.

MNL-specific constant factors

Under MNL, the exponential fluid program collapses to a compact LP via duality arguments. Because the fluid optimum itself is not DR-submodular even in single-warehouse settings, the authors construct a surrogate $C_i$8 that caps each customer type's sales and each product's attraction weight at half their natural maxima. Two properties drive the algorithm: $C_i$9 is a $r_{ij}^{\ell}$0-approximation of LP$r_{ij}^{\ell}$1, and it is monotone DR-submodular — proved through a dual analysis establishing weak-DR submodularity of a per-type dual function via monotonicity of greedy knapsack allocations. Discretizing into unit inventory elements, the placement problem becomes monotone submodular maximization over one or two partition matroids, admitting $r_{ij}^{\ell}$2 or $r_{ij}^{\ell}$3 approximations respectively. Multiplying the three factors (submodular optimization, surrogate approximation, policy guarantee) produces the four MNL constants in Theorem 4.4.

Conversion to inventory-aware policies

A universal conversion maps any inventory-agnostic policy to an inventory-aware one that never offers unavailable products and always fulfills offered choices, preserving expected profit exactly. The key tool, adapted from Feng et al., states that under weak substitutability, for any assortment distribution and any available set, there exists a polynomially samplable distribution over subsets of the available set preserving every available product's marginal choice probability. Since these marginals match the fluid flow-balance targets, expected profit is unchanged; a coupling argument proves distributional equivalence of cumulative sales. Under MNL, the conversion admits a closed-form sampling distribution with compensating normalizer factors, eliminating the need to solve a linear program at each step. All guarantees therefore transfer verbatim to feasible policies for the original problem.

Computational experiments

Experiments use instances with $r_{ij}^{\ell}$4 products, $r_{ij}^{\ell}$5 customer types located on a map, $r_{ij}^{\ell}$6 warehouses, Euclidean shipping costs, MNL preferences with heterogeneous consideration sets, and non-stationary arrivals favoring "pickier" customers later in the horizon. Five algorithms are compared against a myopic static benchmark, with performance reported as a percentage of the joint fluid upper bound over 36 parameter configurations at horizons $r_{ij}^{\ell}$7 and $r_{ij}^{\ell}$8.

Key findings:

• Fluid-based rounding methods dominate. RDU and RDWH outperform the benchmark on average at both horizons (e.g., RDWH averages 92.74% at $r_{ij}^{\ell}$9 and 98.05% at $i$0). RSS, closest to the asymptotic theory, improves by 8 percentage points (88.31% to 96.37%) as the horizon grows while the benchmark stays flat, consistent with the theoretical convergence.
• The DR-submodular approach (DGSS) is less robust empirically, exhibiting large instance-to-instance variation (as low as ~61% on some instances) despite strong average performance (~88.84% at $i$1), attributed to variation in the submodular optimization step.
• Conversion heuristics matter. The provably correct subsampling is slightly conservative; the unproven pre-select-and-drop heuristic (RDWH) performs best at scale, while drop-if-unavailable oversells from distant warehouses at large $i$2.
• Artificial demand reduction hurts in practice: although needed for worst-case non-stationary guarantees, unscaled fluid solutions perform better empirically.

The synthetic data construction favors the benchmark, since supply levels are calibrated from the same myopic assortments the benchmark offers, making the fluid-based methods' performance particularly notable.

Limitations and open questions

Several caveats bear directly on the results. The asymptotic guarantees require minimum capacities and supplies to grow with $i$3, $i$4, and $i$5; for small instances only the modest MNL constants apply, and the general-choice-model results provide no constant-factor guarantee at all. The MNL constants themselves are conservative products of three multiplicative losses, and the authors identify improving them — ideally without artificial demand reduction — as open. Whether a constant-factor approximation exists for general choice models under the assortment oracle assumption remains unresolved, as do hardness bounds conditioned on P ≠ NP. Extensions to reusable resources and to settings with uncertain or misidentified customer types are left unaddressed. On the empirical side, the dual-greedy stocking heuristic lacks a theoretical guarantee, and the best-performing conversion heuristics (RDU, RDWH) carry no proven performance bounds.

Conclusion

This paper provides the first approximation algorithms with performance guarantees for jointly optimizing inventory placement, assortment personalization, and order fulfillment across warehouse networks with triadic profit heterogeneity. Its main technical contributions are a selective, profit-based demand-suppression mechanism for non-stationary arrivals, a joint fluid-based construction yielding square-root asymptotic convergence under general choice models, a DR-submodular surrogate enabling constant-factor MNL guarantees, and an exact profit-preserving conversion to inventory-aware policies. Computational experiments show the fluid-based methods achieve near-optimal profits (up to 98% of the fluid bound) and improve systematically with scale, indicating that the theoretical framework translates effectively into practice.