Production, Manufacturing and LogisticsDynamic pricing with real-time demand learning
Introduction
Dynamic pricing is a business strategy that adjusts the product price in a timely fashion in order to allocate the right service to the right customer at the right time. The rationale of dynamic pricing can be understood with an example of an airline company. When an airline sells seats in the same class, it offers different fares depending on time to departure and current seat inventory. The airline has the incentive to promote sale when the departure time is approaching with a lot of vacancies on hand, because each empty seat is worth nothing after the airplane takes off. On the other hand, the airline still wants to reserve a certain number of seats for possible last-minute travelers who are willing to pay substantially more in price. As a consequence, airfare often fluctuates in its selling horizon.
Products such as airline seats are called perishable products, which have three major characteristics: (1) the quantity is fixed and reordering is not possible; (2) there is a deadline for sale; and (3) the marginal cost of selling one more item is little, so most part of revenue goes directly to profit. Because of these characteristics, perishable products are particularly suitable for dynamic pricing. Besides being used extensively in the airline industry, dynamic pricing can also be found in other travel industries––such as hotel rooms [4], rental cars, and cruise lines [11]––to incorporate seasonal fluctuation in demand. Interested readers are referred to survey papers, such as [20] and [17], for an overview of dynamic pricing and its role in revenue management.
In general, there are two major sources of randomness in demand: customer arrival rate and customer reservation price distribution. Most existing literature concerning dynamic pricing assumes that both customer arrival rate and customer reservation price distribution are well known before the sale begins. In many service industries, however, whereas the seller can use historical data to estimate the customer reservation price to a good extent, it is rather difficult to accurately forecast the customer arrival rate before the sale begins. For example in the travel industry, the demand rates for air travel services and for hotel rooms on different days may be different if an event––such as a commencement, a trade show, or a conference––takes place at the destination city. For another example in the entertainment industry, when a pop singer goes on an international tour, it is relatively easy to know how much a loyal fan is willing to pay for a ticket, but it is rather difficult to know how many fans there are in each city and how many of those fans will be aware of the event. In these cases, if the seller roughly estimates the customer arrival rate and dynamically sets the product price based on this rough estimation, he faces a significant risk. If the true customer arrival rate is much lower than the estimated rate, the seller will end up with many unsold items at the end. On the other hand, if the true arrival rate is much higher, the seller will be out of stock quickly and loses the opportunity to take advantage of the excess demand. The dynamic pricing literature does not adequately address this risk.
In this paper, we present a dynamic pricing model where customers arrive in accordance with a conditional Poisson process, whose rate is not known to the seller in advance. Instead, through preliminary pre-sale market research, the seller obtains a prior distribution of the customer arrival rate. As the sale moves forward, the seller uses real-time sales data from the realized demand to fine-tune the arrival rate estimation, and then uses the fine-tuned arrival rate estimation to better understand the demand curve in the future. Consequently, the seller updates the future demand distribution in real time, and then dynamically sets the product price to maximize the expected total revenue.
In recent years, the problem of dynamic pricing has drawn much attention. Most research on dynamic pricing assumes that the customers arrive according to a stochastic process that has independent increments; that is, the numbers of customers in disjoint time intervals are independent random variables. With this assumption, knowing the number of customers that have shown up so far provides no additional information about how many more customers will show up later on, so learning is not possible. For example, in a continuous-time setting, a common assumption is that customers arrive according to a Poisson process with a given intensity function [6], [8], [9]. In a discrete time setting, time is divided into small intervals such that in each time interval there is a small probability a customer will arrive, independent of everything else [13], [19]. With the assumption that the demand process has independent increments, the problem is often formulated as a Markov decision process. In most cases, it can be shown that the optimal product price increases in the remaining time and decreases in the current inventory level. However, because the optimal policy is difficult to derive, most research focuses on developing heuristic policies.
Learning models have been studied in the operations management literature to better forecast the future demand curve. Most work assumes that price is exogenous, while the firm decides how much inventory to replenish in each time period [2], [12], [16]. Learning models that incorporate both price and replenishment decisions include [3] and [18]. In both papers, the demand curve in each time period is a deterministic and identical function of the price, while the parameters of the function are unknown to the decision maker. Based on the realized demand in early periods, the decision maker learns about the demand curve in order to set a proper price later on. Burnetas and Smith [5] considered a similar problem except that the demands in different periods are independent and identically distributed random variables. They developed a policy with which the realized profit converges with probability one to the optimal value under complete information. These learning models are different from our model because the seller learns from repetition of identical experiments (same flight number through different days), and in our model the seller learns throughout the sales horizon of a single event.
The rest of this paper is organized as follows. In Section 2, we introduce a dynamic pricing model where customers arrive according to a conditional Poisson process. We show how the seller can improve the estimation on the customer arrival rate from the real-time sales data as the sale moves forward. Motivated by these preliminary results, we consider a surrogate dynamic pricing model and derive its optimal policy in Section 3. Then in Section 4, we use the results from this surrogate model to develop the variable-rate policy for the original problem described in Section 2. The numerical experiments show that this variable-rate policy is not only nearly optimal, but also robust even when the pre-sale estimation on the customer arrival rate is relatively poor. In Section 5, we extend the results to four settings that are often encountered in practice: (1) lost customers are not observable; (2) the customer arrival process is non-stationary; (3) each customer can request more than one item; and (4) the allowable price set is discrete. Finally we conclude the paper and discuss future research directions in Section 6.
Section snippets
The model and preliminaries
Consider a dynamic pricing model where a seller sells a given stock of identical items over a finite time horizon [0, T]. Customers arrive according to a conditional Poisson process with an unknown rate Λ. Upon arrival, a customer will purchase one item if the posted product price is lower than her reservation price, or leaves empty-handed otherwise. We assume the reservation prices of all customers are independent and identically distributed with a continuous cumulative distribution function F.
A surrogate model
It is difficult, if not impossible, to derive the optimal policy for the continuous-time dynamic pricing problem in the previous section, because the distribution of the customer arrival rate depends on both time elapsed and the number of customers that have shown up. In Section 3.1, we consider a surrogate dynamic pricing problem, which is motivated by the observation that whenever the seller needs to set the product price for an arriving customer, the number of future customers follows a
Dynamic pricing with real-time demand learning
In this section, we return to the model in Section 2, where the seller updates the demand distribution in real time. In Section 4.1, we present a dynamic pricing policy such that the seller sets the product price based on the updated demand distribution. We then present numerical examples to demonstrate this policy’s efficiency in Section 4.2 and its robustness in Section 4.3.
Extensions
We next consider several extensions to the basic model. The first extension deals with the situation when the seller can track only the number of items sold but not the number of customers. The second extension considers the case when the customer arrival rate is time dependent. For these two extensions, we modify the VR policy so that the seller can quote the price from the same three-dimensional table discussed in Section 4.1. The third extension allows each customer to buy multiple items,
Conclusions
In this paper we present a dynamic pricing model where the seller needs to sell a given stock of identical items by a deadline. Unlike traditional dynamic pricing models where the seller knows the customer arrival rate, a key assumption in our model is that the seller can only estimate the arrival rate. As the sale moves forward, the seller collects the sales data in real time to fine-tune the customer arrival rate estimation. He then uses this fine-tuned arrival rate estimation to better
Acknowledgements
The author thanks anonymous referees for careful reviews and helpful comments. This material is based upon work supported by the National Science Foundation under Grant No. 0223314. Most of the work was done when the author was in the Grado Department of Industrial and Systems Engineering at Virginia Tech.
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