Multi Product Dynamic Pricing With Limited Inventories Under Cascade Click Model

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Multi-product Dynamic Pricing with Limited Inventories Under Cascade Click Model

Author : Sajjad Najafi
Publisher :
ISBN 13 :
Total Pages : 39 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Multi-product Dynamic Pricing with Limited Inventories Under Cascade Click Model by : Sajjad Najafi

Download or read book Multi-product Dynamic Pricing with Limited Inventories Under Cascade Click Model written by Sajjad Najafi and published by . This book was released on 2019 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Dynamic Pricing of Multiple Products with Segmented Customer Behaviour

Author : Kevin D. Ferreira
Publisher :
ISBN 13 : 9780494212776
Total Pages : 162 pages
Book Rating : 4.2/5 (127 download)

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Book Synopsis Dynamic Pricing of Multiple Products with Segmented Customer Behaviour by : Kevin D. Ferreira

Download or read book Dynamic Pricing of Multiple Products with Segmented Customer Behaviour written by Kevin D. Ferreira and published by . This book was released on 2006 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Multi-Product Dynamic Pricing with Reference Effects Under Logit Demand

Author : Mengzi Amy Guo
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)

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Book Synopsis Multi-Product Dynamic Pricing with Reference Effects Under Logit Demand by : Mengzi Amy Guo

Download or read book Multi-Product Dynamic Pricing with Reference Effects Under Logit Demand written by Mengzi Amy Guo and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Joint Learning and Optimization for Multi-product Pricing Under a General Cascade Click Model

Author : Xiangyu Gao
Publisher :
ISBN 13 :
Total Pages : 43 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Joint Learning and Optimization for Multi-product Pricing Under a General Cascade Click Model by : Xiangyu Gao

Download or read book Joint Learning and Optimization for Multi-product Pricing Under a General Cascade Click Model written by Xiangyu Gao and published by . This book was released on 2020 with total page 43 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Practical Process for Estimating the Coefficients of Demand Models for Multi-product Dynamic Pricing

Author : 劉心鈺
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (143 download)

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Book Synopsis Practical Process for Estimating the Coefficients of Demand Models for Multi-product Dynamic Pricing by : 劉心鈺

Download or read book Practical Process for Estimating the Coefficients of Demand Models for Multi-product Dynamic Pricing written by 劉心鈺 and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Dynamic Pricing and Inventory Control for Multiple Products

Author : Dimitris Bertsimas
Publisher :
ISBN 13 :
Total Pages : 20 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Dynamic Pricing and Inventory Control for Multiple Products by : Dimitris Bertsimas

Download or read book Dynamic Pricing and Inventory Control for Multiple Products written by Dimitris Bertsimas and published by . This book was released on 2014 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Dynamic Pricing and Inventory Control with Learning

Author : Nicholas C. Petruzzi
Publisher :
ISBN 13 :
Total Pages : 50 pages
Book Rating : 4.:/5 (368 download)

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Book Synopsis Dynamic Pricing and Inventory Control with Learning by : Nicholas C. Petruzzi

Download or read book Dynamic Pricing and Inventory Control with Learning written by Nicholas C. Petruzzi and published by . This book was released on 1996 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Stochastic Dynamic Multi-Product Pricing with Dynamic Advertising and Adoption Effects

Author : Rainer Schlosser
Publisher :
ISBN 13 :
Total Pages : 28 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Stochastic Dynamic Multi-Product Pricing with Dynamic Advertising and Adoption Effects by : Rainer Schlosser

Download or read book Stochastic Dynamic Multi-Product Pricing with Dynamic Advertising and Adoption Effects written by Rainer Schlosser and published by . This book was released on 2015 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Inventory-Based Dynamic Pricing with Costly Price Adjustment

Author : Wen Chen
Publisher :
ISBN 13 :
Total Pages : 33 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Inventory-Based Dynamic Pricing with Costly Price Adjustment by : Wen Chen

Download or read book Inventory-Based Dynamic Pricing with Costly Price Adjustment written by Wen Chen and published by . This book was released on 2014 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study an average-cost stochastic inventory control problem in which the firm can replenish inventory and adjust price at anytime. We establish the optimality to change the price from low to high in each replenishment cycle as inventory is depleted. With costly price adjustment, scale economies of inventory replenishment are reflected in the cycle time instead of lot size -- An increased fixed ordering cost leads to an extended replenishment cycle but does not necessarily increase the order quantity. A reduced marginal cost of ordering calls for an increased order quantity, as well as speeding up product selling within a cycle. We derive useful properties of the profit function that allows for reducing computational complexity of the problem. For systems requiring short replenishment cycles, the optimal solution can be easily computed by applying these properties. For systems requiring long replenishment cycles, we further consider a relaxed problem that is computational tractable. Under this relaxation, the sum of fixed ordering cost and price adjustment cost is equal to (greater than, less than) the total inventory holding cost within a replenishment cycle when the inventory holding cost is linear (convex, concave) in the stock level. Moreover, under the optimal solution, the time-average profit is the same across all price segments when the inventory holding cost is accounted properly. Through a numerical study, we demonstrate that inventory-based dynamic pricing can lead to significant profit improvement compared with static pricing and limited price adjustment can yield a benefit that is close to unlimited price adjustment. To be able to enjoy the benefit of dynamic pricing, however, it is important to appropriately choose inventory levels at which the price is revised.

Dynamic Pricing with Time-Dependent Elasticities

Author : Rainer Schlosser
Publisher :
ISBN 13 :
Total Pages : 33 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Dynamic Pricing with Time-Dependent Elasticities by : Rainer Schlosser

Download or read book Dynamic Pricing with Time-Dependent Elasticities written by Rainer Schlosser and published by . This book was released on 2015 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many stochastic dynamic sales applications are characterized by time-dependent price elasticities of demand. However, in general, such problems cannot be solved analytically. To determine smart pricing heuristics for general time-dependent dynamic pricing models, we solve a general class of deterministic dynamic pricing problems for perishable and durable goods. The continuous time model has several time-dependent parameters, e.g., discount rate, marginal unit costs, and price elasticity. We show how to derive the value function and optimal pricing policies. On the basis of the feedback solution to the deterministic model, we propose a method for constructing heuristics to be applied to general stochastic models. For the case of isoelastic demand, we analytically verify the excellent performance of this approach for both small and large inventory levels.

Dynamic Pricing of Limited Inventories with Product Returns

Author : Xing Hu
Publisher :
ISBN 13 :
Total Pages : 32 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Dynamic Pricing of Limited Inventories with Product Returns by : Xing Hu

Download or read book Dynamic Pricing of Limited Inventories with Product Returns written by Xing Hu and published by . This book was released on 2018 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Demand Shaping Through Bundling and Product Configuration

Author : Jing-Sheng Jeannette Song
Publisher :
ISBN 13 :
Total Pages : 50 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Demand Shaping Through Bundling and Product Configuration by : Jing-Sheng Jeannette Song

Download or read book Demand Shaping Through Bundling and Product Configuration written by Jing-Sheng Jeannette Song and published by . This book was released on 2018 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a dynamic, multi-item model to analyze the optimal joint inventory, pricing, and bundling decisions for a firm over a finite horizon. We develop a novel demand model that transfers the discrete bundling decision and the corresponding pricing decision into a market share decision. We show that the optimal policy is dictated by a no-order set. For items in this set, we do not place replenishment orders, because these items are overstocked. The rest of the policy parameters -- the order-up-to-levels for the items that we do order, the bundling and pricing decisions, and the bundle assembly quantity -- all depend on the overstock levels. Exploring the optimal policy features, we devise a branching algorithm that significantly simplifies the computation of the optimal policy. We also characterize how the optimal bundling decision depends on item complementarity, cost structure, inventory status, demand uncertainty, and supply responsiveness.

Essays in Dynamic Pricing of Multiple Substitutable Products

Author : Sajjad Najafi
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (133 download)

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Book Synopsis Essays in Dynamic Pricing of Multiple Substitutable Products by : Sajjad Najafi

Download or read book Essays in Dynamic Pricing of Multiple Substitutable Products written by Sajjad Najafi and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Dynamic Pricing with Reference Price Effects

Author :
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (931 download)

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Book Synopsis Dynamic Pricing with Reference Price Effects by :

Download or read book Dynamic Pricing with Reference Price Effects written by and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Tractable Multi-product Pricing Under Discrete Choice Models

Author : Philipp Wilhelm Keller
Publisher :
ISBN 13 :
Total Pages : 204 pages
Book Rating : 4.:/5 (864 download)

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Book Synopsis Tractable Multi-product Pricing Under Discrete Choice Models by : Philipp Wilhelm Keller

Download or read book Tractable Multi-product Pricing Under Discrete Choice Models written by Philipp Wilhelm Keller and published by . This book was released on 2013 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider a retailer offering an assortment of differentiated substitutable products to price-sensitive customers. Prices are chosen to maximize profit, subject to inventory/ capacity constraints, as well as more general constraints. The profit is not even a quasi-concave function of the prices under the basic multinomial logit (MNL) demand model. Linear constraints can induce a non-convex feasible region. Nevertheless, we show how to efficiently solve the pricing problem under three important, more general families of demand models. Generalized attraction (GA) models broaden the range of nonlinear responses to changes in price. We propose a reformulation of the pricing problem over demands (instead of prices) which is convex. We show that the constrained problem under MNL models can be solved in a polynomial number of Newton iterations. In experiments, our reformulation is solved in seconds rather than days by commercial software. For nested-logit (NL) demand models, we show that the profit is concave in the demands (market shares) when all the price-sensitivity parameters are sufficiently close. The closed-form expressions for the Hessian of the profit that we derive can be used with general-purpose nonlinear solvers. For the special (unconstrained) case already considered in the literature, we devise an algorithm that requires no assumptions on the problem parameters. The class of generalized extreme value (GEV) models includes the NL as well as the cross-nested logit (CNL) model. There is generally no closed form expression for the profit in terms of the demands. We nevertheless how the gradient and Hessian can be computed for use with general-purpose solvers. We show that the objective of a transformed problem is nearly concave when all the price sensitivities are close. For the unconstrained case, we develop a simple and surprisingly efficient first-order method. Our experiments suggest that it always finds a global optimum, for any model parameters. We apply the method to mixed logit (MMNL) models, by showing that they can be approximated with CNL models. With an appropriate sequence of parameter scalings, we conjecture that the solution found is also globally optimal.

Design and Dynamic Pricing of Vertically Differentiated Inventories

Author : Ioannis Stamatopoulos
Publisher :
ISBN 13 :
Total Pages : 46 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Design and Dynamic Pricing of Vertically Differentiated Inventories by : Ioannis Stamatopoulos

Download or read book Design and Dynamic Pricing of Vertically Differentiated Inventories written by Ioannis Stamatopoulos and published by . This book was released on 2018 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Dynamic Pricing and Inventory Control with Fixed Ordering Cost and Incomplete Demand Information

Author : Boxiao Chen
Publisher :
ISBN 13 :
Total Pages : 42 pages
Book Rating : 4.:/5 (129 download)

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Book Synopsis Dynamic Pricing and Inventory Control with Fixed Ordering Cost and Incomplete Demand Information by : Boxiao Chen

Download or read book Dynamic Pricing and Inventory Control with Fixed Ordering Cost and Incomplete Demand Information written by Boxiao Chen and published by . This book was released on 2020 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the periodic review dynamic pricing and inventory control problem with fixed ordering cost. Demand is random and price dependent, and unsatisfied demand is backlogged. With complete demand information, the celebrated (s,S,p) policy is proved to be optimal, where s and S are the reorder point and order-up-to level for ordering strategy, and p, a function of on-hand inventory level, characterizes the pricing strategy. In this paper, we consider incomplete demand information and develop online learning algorithms whose average profit approaches that of the optimal (s,S,p) with a tight O ̃(√T) regret rate. A number of salient features differentiate our work from the existing online learning researches in the OM literature. First, computing the optimal (s,S,p) policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in operations management that only require solving single-period optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profit-to-go function. The necessity of analyzing action-dependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult than existing bandit learning algorithms. Second, the pricing function p is of infinite dimension, and approaching it is much more challenging than approaching a finite number of parameters as seen in existing researches. The demand-price relationship is estimated based on upper confidence bound, but the confidence interval cannot be explicitly calculated due to the complexity of the DP recursion. Finally, due to the multi-period nature of (s,S,p) policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy p, which is unknown to the learner a priori. In this paper, the demand randomness is approximated by an empirical distribution constructed using dependent samples, and a novel Wasserstein metric based argument is employed to prove convergence of the empirical distribution.