Source-linked AI summary
Learning to Price and Stock Under Contextual and Censored Demand
Zean Han, Zezhen Ding, Jiheng Zhang
TL;DR
Existing joint pricing and inventory models often omit either contextual demand or censored lost-sales observations. This paper combines a linear contextual demand model with adaptive pricing and inventory control, achieving the stated regret rates with matching lower bounds and demonstrating effectiveness numerically.
Problem
Existing approaches often fail to account for both contextual information and censored demand observations in joint pricing and inventory control.
Method
The paper models demand as a linear combination of basis functions with unknown coefficients and uses an adaptive algorithm with exploration, demand estimation, and queue-based inventory control.
Results
The algorithm achieves O(K√T log T) regret under concave revenue and O(K^2/3 T^2/3 (log T)^1/2) in the general case, with matching lower bounds confirming minimax optimality.
Takeaways & Limitations
Context-aware pricing and inventory policies can be analyzed jointly despite censored demand, changing contexts, and stochastic inventory dynamics.
Takeaways & Limitations
The analysis assumes differentiable revenue and bounded noise with regularity conditions on the noise density.
Abstract
from arXiv · showhide
To make optimal joint pricing and inventory control decisions is a critical challenge for modern retailers. In practice, retailers face changing market conditions where demands are influenced by various contextual factors, while simultaneously dealing with the difficulty of lost sales that obscure true demand information. However, existing approaches often fail to account for both contextual information and censored demand observations. We address this gap by presenting a framework where we model demand as a linear combination of basis functions with unknown coefficients, allowing for adaptive pricing and inventory decisions that respond to changing contexts. We propose an efficient algorithm to achieve regret bound $\mathcal{O}(K\sqrt{T}\log T)$ under concave revenue conditions and $\mathcal{O}(K^{2/3}T^{2/3}(\log T)^{1/2})$ for the general case, with matching lower bounds confirming optimality. Extensive numerical experiments across diverse scenarios demonstrate our algorithm's effectiveness.
1 Introduction
The paper studies joint pricing and inventory control when demand depends on context and sales are censored by lost inventory. It introduces adaptive policies and regret guarantees for this setting.
- Existing models often overlook both censored demand and contextual information in joint pricing and inventory control.
- Demand is modeled with unknown coefficients over basis functions, while observed sales are censored when inventory is insufficient.
- Censored observations require unbiased demand estimation, while changing contexts require pricing and inventory decisions to adapt dynamically.
- The proposed framework makes both pricing and inventory policies responsive to changing contexts.
- The algorithm addresses censored demand, dynamic contexts, and stochastic inventory dynamics, with regret analysis under concave and general revenue conditions.
- Prior joint-control studies considered full information, data-driven learning, or censored demand, but existing models ignored contextual information.
2 Basic Setting
The model considers sequential pricing and order-up-to decisions under contextual demand, lost sales, inventory carryover, and unknown demand and noise distributions. It defines regret against a full-information myopic benchmark and imposes smoothness and bounded-noise assumptions.
- Each period begins with an observed context and inventory state, followed by a price and order-up-to decision before demand realizes.
- Demand equals an unknown context-price function plus independent mean-zero noise, and the firm learns using historical observations.
- Only sales min{D_t, y_t} are observed because demand exceeding available inventory is lost and unobservable.
- The objective is expected cumulative profit over T periods, with regret measured against the optimal price and inventory decisions.
- The analysis assumes differentiable revenue and bounded noise with regularity conditions on its density.
- Under the linear additive demand assumption, the optimal inventory level is y*(x, p) = λ(x, p) + z*(p).
3 Algorithm Design
The algorithm separates exploration from commitment to learn demand despite censoring, then uses estimated demand and a queue-based mechanism to manage context-driven inventory targets.
- During exploration, uniformly sampled prices and sufficiently high inventory improve demand observability by avoiding censoring.
- Changing contexts and perishability can make ideal inventory targets infeasible, so a queue-based approach controls deviations from recommended targets.
- The estimated revenue function determines the price, after which the corresponding target inventory is computed.
- The algorithm uses exploration and commitment phases to separate learning from subsequent policy execution.
- Least-squares estimation exploits the linear basis-function structure to estimate demand during exploration.
4 Regret Analysis
The regret analysis establishes rates for the proposed contextual pricing-and-inventory algorithm under strongly concave and general revenue conditions. It combines exploration-based demand estimation with pricing optimization and queueing-based control of inventory mismatches.
- Strongly Concave Revenue Function: O(K√T log T) regret is achieved under strongly concave revenue assumptions.The bound is stated for the algorithm under the strongly concave setting.
- Non-concave Revenue Function: O(K^2/3T^2/3(log T)^1/2) regret is achieved without strong concavity.The analysis also provides a lower bound for m-th differentiable revenue functions and identifies minimax-optimal rates in the corresponding setting.
- Algorithmic Framework: The algorithm explores with uniformly sampled prices and high inventory, then estimates demand and residual distributions before committing to optimized contextual decisions.During commitment, it maximizes an estimated revenue function and sets inventory using the estimated contextual demand target, adjusted for current inventory.
- Assumptions: The analysis assumes interior full-information optimal prices, strong concavity of G in price, and twice-continuous differentiability of the revenue function.These assumptions support unique pricing decisions and the optimization arguments used in the regret analysis.
- Regret Decomposition: The regret proof decomposes total loss into exploration regret, exploitation loss from price and recommended-inventory errors, and implementation mismatch.The recommended inventory is estimated from the contextual demand model, while the implemented level accounts for the current inventory state.
- Inventory Feasibility: A queueing-based stochastic recursion bounds deviations between implemented and recommended inventory levels under changing contexts.The queue interpretation links inventory excess to waiting time and yields yt − ˆyt ≤ Bt with high probability.
5 Numerical Experiments
The numerical study evaluates relative regret across time horizons, cost parameters, and basis-function dimensions. Relative regret decreases as T grows, while increasing K raises regret but variance remains controlled.
- Experimental setup: The experiments use a linear demand model with normalized parameters, uniform noise, bounded prices, and 50 independent runs.Context vectors are normalized standard-normal draws, and performance is evaluated using relative regret across multiple horizons.
- Time-horizon results: Relative regret decreases as the time horizon T increases across different h and b settings.The reported trend indicates improved decisions as more data become available.
- Time-horizon results: The log-log results show a linear trend between relative regret and T across parameter settings.This pattern supports a power-law relationship and robustness across the tested settings.
- Dimension K: Relative regret increases consistently with K, indicating that higher-dimensional models are harder to learn under a fixed horizon.The study varies K from 4 to 12 while fixing T = 1000, h = 1, and b = 1.
- Dimension K: Standard deviations remain moderate across K, indicating controlled variance and stable performance despite increased model complexity.Figure 3 reports relative regret with standard deviation error bars.
6 Conclusion
The paper addresses joint pricing and inventory control with contextual information and censored demand, proposing adaptive policies and an algorithm with optimal regret guarantees. It also identifies broader modeling assumptions as a direction for future work.
- Conclusion: The paper addresses the gap in models that fail to jointly account for contextual factors and censored demand from lost sales.The setting is joint pricing and inventory control under a linear additive demand model.
- Conclusion: The proposed framework adapts both pricing and inventory policies to changing contexts.The algorithm also handles censored demand and stochastic inventory dynamics.
- Conclusion: The algorithm achieves O(K√T log T) regret under concave revenue and O(K^2/3 T^2/3 (log T)^1/2) in the general case, with matching lower bounds.The paper reports minimax optimality for these regret guarantees.
- Future work: Future work should examine alternative modeling assumptions beyond the linear additive framework and develop UCB-style analyses for censored demand.These directions target broader assumptions and principled exploration under partial observability.