Multi-product Dynamic Pricing with Limited Inventories Under Cascade Click Model PDF Download
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Author: Sajjad Najafi
Publisher:
ISBN:
Category :
Languages : en
Pages : 39
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Book Description
We consider a multi-product dynamic pricing problem with limited inventories under the so-called Cascade Click model, which is one of the most popular click models used in practice for analyzing customers' click-and-search behavior in large-scale web analytic applications. We present three fundamental results. First, despite the highly non-linear structure of the problem, we derive a sufficiently general characterization of the optimal pricing policy and show that it has a different structure than the optimal policy under the standard pricing model. Second, we show that the optimal expected total revenues under the Cascade Click model can be upper-bounded by the objective value of an approximate deterministic pricing problem, and that this deterministic problem can be efficiently solved. This result is reminiscent of the classic upper-bound result in the standard Revenue Management (RM) setting, whose importance and impact on RM research in the past two decades cannot be overstated. Third, we show that two heuristic policies that are known to have strong performance guarantees in the standard RM setting can be properly adapted to the setting with Cascade Click model and retain their strong performance guarantees.
Author: Sajjad Najafi
Publisher:
ISBN:
Category :
Languages : en
Pages : 39
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Book Description
We consider a multi-product dynamic pricing problem with limited inventories under the so-called Cascade Click model, which is one of the most popular click models used in practice for analyzing customers' click-and-search behavior in large-scale web analytic applications. We present three fundamental results. First, despite the highly non-linear structure of the problem, we derive a sufficiently general characterization of the optimal pricing policy and show that it has a different structure than the optimal policy under the standard pricing model. Second, we show that the optimal expected total revenues under the Cascade Click model can be upper-bounded by the objective value of an approximate deterministic pricing problem, and that this deterministic problem can be efficiently solved. This result is reminiscent of the classic upper-bound result in the standard Revenue Management (RM) setting, whose importance and impact on RM research in the past two decades cannot be overstated. Third, we show that two heuristic policies that are known to have strong performance guarantees in the standard RM setting can be properly adapted to the setting with Cascade Click model and retain their strong performance guarantees.
Author: Kevin D. Ferreira
Publisher:
ISBN: 9780494212776
Category :
Languages : en
Pages : 162
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Book Description
Three models are presented that dynamically price multiple product types. The first model prices the products over an infinite selling horizon, the second is a discrete-time model with a finite selling horizon, and finally we present a continuous-time finite horizon model. Numerical analysis exhibits that when the inventory is reduced to only one product type, then the pricing policies are non-increasing with respect to time and for higher inventories the discounting occurs earlier; however, it is evident that a product that has a higher expected rate of revenue associated with the subsequent price can be discounted at later times, and higher prices for that product can be used for higher levels of inventory. When pricing multiple products, it is observed that the decisions are not necessarily non-increasing with time and inventory, and that there are cases where higher overall inventory levels are discounted later than lower levels.
Author: Xiangyu Gao
Publisher:
ISBN:
Category :
Languages : en
Pages : 43
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Book Description
We consider a pricing problem for a set of products displayed on a list. We assume a general cascade click model, in which customers examine the products in a decreasing order of display, from the top to (potentially) the bottom of the list. At each step, customers can decide to either purchase the current product, forego the current product and continue examining the next product, or simply terminate the search without purchasing any product. The objective of the firm is to optimally price the products to maximize its expected total revenues. We first study the case where the firm knows all problem parameters and derive a relatively explicit expression for the optimal prices of the products, for some cases. This is useful for uncovering some interesting managerial insights regarding the properties of the optimal prices when customers behave in the manner prescribed by the general cascade click model. We then study the case where the parameters are unknown and need to be learned/estimated from the data. For this case, we develop an online algorithm that jointly learns the unknown parameters and optimizes the prices of the products. Moving beyond the base model where the only decision of the firm is the prices, we also consider several extensions to more complex settings that include filtering options (i.e., we allow customers to filter out some of the products using some filtering options) and display ranking decisions. We discuss how our algorithm in the base setting can be extended to these more general settings. Our numerical results highlight the value of properly taking into account customer search behavior when designing a learning algorithm.
Author: Mengzi Amy Guo
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
We consider an infinite-horizon multi-product dynamic pricing problem with reference effects in a monopolistic setting, where the reference price is an exponentially weighted average of historical prices. In each period, the demand follows the multinomial logit (MNL) model, where the utility depends on both the current price and the reference price, and consumers can have product-differentiated sensitivities to the price and the reference price. We conduct thorough analyses of the myopic pricing policy, including its solution, long-run convergence behavior, and performance guarantee compared to the long-term discounted revenue of the optimal pricing policy. Furthermore, we establish the structural properties of the optimal pricing policy. When consumers are loss-neutral towards all products, we explicitly characterize the optimal pricing policy if it converges to a steady state, and based on our characterization we show that the steady state price can be computed efficiently by a binary search. Interestingly, we find that such a convergence behavior of the optimal pricing policy heavily relies on the upper bound of the admissible price range, and a low price upper bound facilitates the policy to converge. In contrast, when consumers are gain-seeking towards all products, we prove that the optimal pricing policy admits no steady state regardless of the price range. Nevertheless, if consumers are only gain-seeking towards certain but not all products, the optimal pricing policy can potentially be convergent. In addition, our numerical experiments show that loss-aversion over all products does not rule out price fluctuations. This finding is at odds with the classic belief on loss-averse consumers and hence, highlights the significance of accounting for cross-product effects through the MNL demand.
Author: Dimitris Bertsimas
Publisher:
ISBN:
Category :
Languages : en
Pages : 20
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Book Description
A periodical multi-product pricing and inventory control problem with applications to production planning and airline revenue management is studied. The objective function of the single-period model is shown to be convex for certain types of demand distributions, thus tractable for large instances. A heuristic is proposed to solve the more complex multi-period problem, which is an interesting combination of linear and dynamic programming. Numerical experiments and theoretical bounds on the optimal expected revenue suggest that the extent to which a dynamic policy based on a stochastic model will outperform a simple static policy based on a deterministic model depends on the level of demand variability as measured by the coefficient of variation.
Author: Xing Hu
Publisher:
ISBN:
Category :
Languages : en
Pages : 32
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Book Description
Many online retail channels face high rates of product returns. This poses a new challenge to the sellers' dynamic pricing problem when some returns in good condition can be resold in the selling season. To study the impact of product returns and guide sellers to adjust pricing policies, we build a product returns model by augmenting the classic monopolist's dynamic pricing framework. We show that the return dynamics can complicate the problem by making it generally not Markovian. We address the technical challenges both analytically and numerically. Our analysis finds that ignoring returns leads to over-pricing and can cause significant revenue loss when the demand is high, initial inventory is moderate, product return speed is high, and intuitively when return probability is high. The analysis yields easy-to-implement heuristic policies that have good and robust performance relative to the theoretical benchmarks. We obtain many important findings for managers. For example, restocking product returns can be highly profitable even when the restocking cost is considerably high. Gaining visibility to customers' product return decisions, albeit helps forecasting returns and gauge total sellable inventory level, often provides small revenue benefits once the seller properly adjusts its dynamic pricing.
Author: 劉心鈺
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Author: Sajjad Najafi
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
I study the dynamic pricing problem of a firm selling limited inventory of multiple differentiated products over a finite planning horizon, where the firm wishes to maximize the expected revenue. I formulate the firm's optimization problem as a Markov decision process and investigate the pricing problem in the presence of a variety of operational settings. First, I integrate consumer's sequential search behavior into the pricing problem. The consumer inspects products one at a time by incurring non-zero search cost, and makes decision by comparing the utility of the best product so far versus the reservation utility, a threshold at which the consumer is indifferent between continuation and stopping of the sequential search. The firm aims at maximizing the expected revenue by offering the products in the right sequence and at the right prices. I analytically derive the optimal prices in each period. I show that under some condition it is optimal to present products in the descending order of quality. Second, I address a problem in which the firm is subject to a set of sales volume constraints required to be satisfied at different time points along the sales horizon. Due to stochastic nature of sales, I incorporate a risk measure that allows the firm to manage the total sales while the expected revenue is maximized. I formulate the problem as a chance-constrained dynamic programming and show that the Karush-Kuhn-Tucker conditions are not only necessary but also sufficient for the optimal price. Third, I assimilate consumer's consideration sets to the dynamic pricing problem. When to make a purchase decision, consumers use a two-stage decision-making process, i.e., consumers constitute a consideration set including a subset of the available products using a screening rule (e.g., brands, quality, and budget), and they only evaluate the products in the consideration set using a utility comparison process and opt for the product with the maximum utility. I show that the first-order condition is sufficient for the optimal price of products if consumers apply a quality-based screening rule.
Author: Ioannis Stamatopoulos
Publisher:
ISBN:
Category :
Languages : en
Pages : 46
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Book Description
We study a model in which a monopoly firm designs the quality profile of its inventory and then dynamically updates its pricing menu for a finite selling horizon to maximize revenue. In a counterfactual scenario, a social planner goes through the same process to maximize total welfare. We show that in both scenarios the problem of dynamically pricing heterogeneous-quality (vertically differentiated) inventories is equivalent to that of dynamically pricing homogeneous-quality inventories, in the sense that a solution to one implies a solution to the other. Moreover, we prove a strong scarcity result, which suggests that the sale of a product drives up the prices on all remaining products, whether of higher or lower quality. We then consider product line design under a production technology that utilizes costly and potentially limited resources. We show that with unlimited (but costly) resources, the revenue maximizer under-supplies quality to all products compared to the social planner. With limited resources, we show that the revenue maximizer exhibits elitism: he over-allocates (under-allocates) resources on the production of high-quality (low-quality) products. However, as the volume of expected consumer arrivals increases to infinity, both the revenue maximizer and the welfare maximizer allocate resources equally across products.
Author: Rainer Schlosser
Publisher:
ISBN:
Category :
Languages : en
Pages : 28
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Book Description
We analyze stochastic dynamic multi-product pricing models for durable goods and consider a single advertising channel to promote all types of products. We include general adoption effects, unit costs as well as inventory holding costs. In case of isoelastic, exponential and linear demand, we derive solution formulas for the expected profit, the optimal feedback prices for all types of products, and the optimal advertising rate. In order to evaluate optimally controlled sales processes over time, we use efficient simulation techniques. Moreover, for the case of exponential demand, we demonstrate how to include risk aversion in the model.
