Stochastic Approach for Price Optimization Problems with Decision-dependent Uncertainty

TL;DR

This paper introduces a general stochastic optimization method for price problems where demand uncertainty depends on prices, using a stochastic gradient approach to improve revenue outcomes.

Contribution

It presents a novel non-convex stochastic optimization framework for decision-dependent demand uncertainty, with an unbiased gradient estimator and a practical solution method.

Findings

Proposed method achieves higher revenues than baseline approaches.

Effective in synthetic and real retail data experiments.

Demonstrates applicability to a broad class of decision-dependent uncertainty problems.

Abstract

Price determination is a central research topic of revenue management in marketing. The important aspect in pricing is controlling the stochastic behavior of demand, and the previous studies have tackled price optimization problems with uncertainties. However, many of those studies assumed that uncertainties are independent of decision variables (i.e., prices) and did not consider situations where demand uncertainty depends on price. Although some price optimization studies have dealt with decision-dependent uncertainty, they make application-specific assumptions in order to obtain an optimal solution or an approximation solution. To handle a wider range of applications with decision-dependent uncertainty, we propose a general non-convex stochastic optimization formulation. This approach aims to maximize the expectation of a revenue function with respect to a random variable representing demand under a decision-dependent distribution. We derived an unbiased stochastic gradient estimator by using a well-tuned variance reduction parameter and used it for a projected stochastic gradient descent method to find a stationary point of our problem. We conducted synthetic experiments and simulation experiments with real data on a retail service application. The results show that the proposed method outputs solutions with higher total revenues than baselines.