A continuous approximation method for dynamic pricing problem under costly price modifications

Bullwhip effect under substitute products

Pricing and inventory replenishment are important operations decisions for firms such as retailers. To make these decisions effectively, a firm needs to know the demand distribution and its dependency on selling price, which is usually estimated using sales data at various testing price levels. Although more testing prices can lead to a better estimation of the demand–price relationship, frequent price changes are costly and come with adverse effect such as customers’ negative perception. In this article, data-driven algorithms are developed that learn the demand structure with constraints on the number of price changes. These algorithms are shown to converge to the optimal clairvoyant solution, and the convergence rates are the best possible in terms of profit loss.

Dynamic pricing for perishable products with hybrid uncertainty in demand

Multi-sector specific factors model with two mobile factors

Is Ignorance Bliss? A Note on Dynamic Pricing with Demand Learning

We study a multi-period stochastic inventory system with backlogs. Demand in each period is random and price sensitive, but the firm has little or no prior knowledge about the demand distribution and how each customer responds to the selling price, so the firm has to learn the demand process when making periodic pricing and inventory replenishment decisions to maximize its expected total profit. We consider the scenario where the firm is faced with the business constraint that prevents it from conducting extensive price exploration, and develop parametric data-driven algorithms for pricing and inventory decisions. We measure the performances of the algorithms by regret, which is the profit loss compared with a clairvoyant who has complete information about the demand distribution. We analyze the cases where the number of price changes is restricted to a given number or a small number relative to the planning horizon, and show that the regrets for the corresponding learning algorithms converge at the best possible rates in the sense that they reach the theoretical lower bounds. Numerical results indicate that these algorithms empirically perform very well. Supplementary materials are available for this article. Go to the publisher’s online edition of IISE Transaction, for datasets, additional tables, detailed proofs, etc.

Technology products often experience a life‐cycle demand pattern that resembles a diffusion process, with weak demand in the beginning and the end of the life cycle and high demand intensity in between. The customer price‐sensitivity also changes over the life cycle of the product. We study the prespecified pricing decision for a product that exhibits such demand characteristics. In particular, we determine the optimal set of discrete prices and the times to switch from one price to another, when a limited number of price changes are allowed. Our study shows that the optimal prices and switching times show interesting patterns that depend on the product's demand pattern and the change in the customers' price sensitivity over the life cycle of the product. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012

In this paper, we consider a problem of the dynamic pricing and inventory control for non-instantaneous deteriorating items with uncertain demand, in which the demand is price-sensitive and governed by a diffusion process. Shortages and remains are permitted, and the backlogging rate is variable and dependent on the waiting time for the next replenishment. In order to maximize the expected total profit, the problem of dynamic pricing and inventory control is described as a stochastic optimal control problem. Based on the dynamic programming principle, the stochastic control model is transformed into a Hamilton-Jacobi-Bellman (HJB) equation. Then, an exact expression for the optimal dynamic pricing strategy is obtained via solving the HJB equation. Moreover, the optimal initial inventory level, the optimal selling pricing, the optimal replenishment cycle and the optimal expected total profit are achieved when the replenishment cycle starts at time 0. Finally, some numerical simulations are presented to demonstrate the analytical results, and the sensitivities analysis on system parameters are carried out to provide some suggestions for managers.

Dynamic pricing, which aims to dynamically adjust the charging price in a timely fashion to unlock the flexibility of electric vehicle (EV) customers, has been extensively studied with the rapid development of charging technologies. Many existing works on dynamic pricing focus on maximizing the social welfare of charging service providers and EV customers. Cases of high-dimensional charging environments, which are often encountered with the rapid growth of EV market penetration, have been rarely considered to date. This article proposes a new dynamic pricing framework for EV charging stations that can offer multiple charging options to customers over a finite-time horizon. The charging price can be dynamically adjusted to maximize the quality of service (QoS) with a differentiated service requirement level (SRL) whenever the arrival rates and queuing system capacities of the charging systems are given at the end of a time period. The dynamic pricing problem is formulated as a finite-discrete horizon Markov decision process (MDP) with a mixed state space. A customized deep reinforcement learning (DRL) approach is employed to solve the examined EV dynamic pricing problem. The simulation results demonstrate the effectiveness of the proposed method.

Distributed dynamic pricing of multiple perishable products using multi-agent reinforcement learning

We consider a dynamic pricing problem in a double-lane system consisting of one general purpose lane and one wireless charging lane (WCL). The electricity price is dynamically adjusted to affect the lane-choice behaviors of incoming electric vehicles (EVs), thereby regulating the traffic assignment between the two lanes with both traffic operation efficiency and charging service efficiency considered in the control objective. We first establish an agent-based dynamic double-lane traffic system model, whereby each EV acts as an agent with distinct behavioral and operational characteristics. Then, a deep Q-learning algorithm is proposed to derive the optimal pricing decisions. A regression tree (CART) algorithm is also designed for benchmarking. The simulation results reveal that the deep Q-learning algorithm demonstrates superior capability in optimizing dynamic pricing strategies compared to CART by more effectively leveraging system dynamics and future traffic demand information, and both outperform the static pricing strategy. This study serves as a pioneering work to explore dynamic pricing issues for WCLs.

In the modern travel market, online travel agencies (OTAs) compete intensively for online users who are valuable resources that support the OTAs’ long-term development. To expand their pool of online users, a few OTAs use aggressive pricing strategies (e.g., providing extremely low price for newly enlisted customers). However, such approach can create other management issues (e.g., financial burden). Noting that pricing strategies for sizing the pool of online users are less addressed by current literature, this paper considers a dynamic pricing problem in which an OTA actively optimizes entrance prices for newly enlisted customers to encourage them to join the pool. The essence of this problem is how to achieve an adequately sized pool supporting long-term profit through dynamic pricing. By analysing this problem, several findings on the interplay between price choosing and pool sizing are obtained. First, we proved the monotonicity of the optimal entrance price relative to the pool size, which demonstrates the positive effect of pool size on pricing decisions. Second, we showed that the optimal entrance price would be lower than operational cost, resulting in negative profits from newly enlisted customers when the pool size is smaller than a breakeven point. Third, we characterised the steady-state pool size and revealed the eventual convergence of the pool size and the pricing decisions. This result implies that the OTA has two pricing strategies, namely, penetration and constant.

This paper considers a dynamic pricing problem over a finite horizon where demand for a product is a time-varying linear function of price. It is assumed that at the start of the horizon there is a fixed amount of the product available. The decision problem is to determine the optimal price at each time period in order to maximize the total revenue generated from the sale of the product. In order to obtain structural results we formulate the decision problem as an optimal control problem and solve it using Pontryagin's principle. For those problems which are not easily solvable when formulated as an optimal control problem, we present a simple convergent algorithm based on Pontryagin's principle that involves solving a sequence of very small quadratic programming (QP) problems. We also consider the case where the initial inventory of the product is a decision variable. We then analyse the two-product version of the problem where the linear demand functions are defined in the sense of Bertrand and we again solve the problem using Pontryagin's principle. A special case of the optimal control problem is solved by transforming it into a linear complementarity problem. For the two-product problem we again present a simple algorithm that involves solving a sequence of small QP problems and also consider the case where the initial inventory levels are decision variables. Copyright © 2006 John Wiley & Sons, Ltd.

Dynamic pricing problems with elastic demand

Welfare implications of piracy with dynamic pricing and heterogeneous consumers