Non-parametric optimal service pricing: a simulation study

doi:10.1080/16843703.2016.1208487

Journal Article•DOI•

Non-parametric optimal service pricing: a simulation study

Asif Muzaffar¹, Shiming Deng², Ammar Rashid¹•Institutions (2)

University of Central Punjab¹, Huazhong University of Science and Technology²

03 Apr 2017-Quality Technology and Quantitative Management (Taylor & Francis)-Vol. 14, Iss: 2, pp 142-155

TL;DR: The main focus of this research is to provide some guidance for the selection of sample sizes based on the test significance and the measure of its power when actual mean and variance for revenue are unknown.

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Abstract: In this paper, we study a price discovery algorithm for searching the optimal price for a service with price-sensitive demand. The customer’s response to price is unknown and the customer arrival p...

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Journal Article•DOI•

Stochastic Estimation of the Maximum of a Regression Function

[...]

J. Kiefer, Jacob Wolfowitz

01 Sep 1952-Annals of Mathematical Statistics

TL;DR: In this article, the authors give a scheme whereby, starting from an arbitrary point, one obtains successively $x_2, x_3, \cdots$ such that the regression function converges to the unknown point in probability as n \rightarrow \infty.

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Abstract: Let $M(x)$ be a regression function which has a maximum at the unknown point $\theta. M(x)$ is itself unknown to the statistician who, however, can take observations at any level $x$. This paper gives a scheme whereby, starting from an arbitrary point $x_1$, one obtains successively $x_2, x_3, \cdots$ such that $x_n$ converges to $\theta$ in probability as $n \rightarrow \infty$.

...read moreread less

2,141 citations

"Non-parametric optimal service pric..." refers background in this paper

...Kiefer and Wolfowitz (1952) estimate the point of maximum when the function is unknown and stochastic approximation algorithm proposed uses simulation-based optimization....
[...]

Journal Article•DOI•

Optimal dynamic pricing of inventories with stochastic demand over finite horizons

[...]

Guillermo Gallego¹, Garrett van Ryzin¹•Institutions (1)

Columbia University¹

01 Aug 1994-Management Science

TL;DR: In this paper, the authors investigate the problem of dynamically pricing such inventories when demand is price sensitive and stochastic and the firm's objective is to maximize expected revenues, and obtain structural monotonicity results for the optimal intensity resp, price as a function of the stock level and the length of the horizon.

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Abstract: In many industries, managers face the problem of selling a given stock of items by a deadline We investigate the problem of dynamically pricing such inventories when demand is price sensitive and stochastic and the firm's objective is to maximize expected revenues Examples that fit this framework include retailers selling fashion and seasonal goods and the travel and leisure industry, which markets space such as seats on airline flights, cabins on vacation cruises, and rooms in hotels that become worthless if not sold by a specific time We formulate this problem using intensity control and obtain structural monotonicity results for the optimal intensity resp, price as a function of the stock level and the length of the horizon For a particular exponential family of demand functions, we find the optimal pricing policy in closed form For general demand functions, we find an upper bound on the expected revenue based on analyzing the deterministic version of the problem and use this bound to prove that simple, fixed price policies are asymptotically optimal as the volume of expected sales tends to infinity Finally, we extend our results to the case where demand is compound Poisson; only a finite number of prices is allowed; the demand rate is time varying; holding costs are incurred and cash flows are discounted; the initial stock is a decision variable; and reordering, overbooking, and random cancellations are allowed

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1,537 citations

"Non-parametric optimal service pric..." refers background in this paper

...These assumptions are standard in literature and such demand functions are called regular demand functions (Gallego & Ryzin, 1994)....
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Journal Article•DOI•

Dynamic Pricing for Non-Perishable Products with Demand Learning

[...]

Victor F. Araman, René Caldentey

01 Dec 2005-Social Science Research Network

TL;DR: The retailer's problem is formulated as a (Poisson) intensity control problem and the structural properties of an optimal solution are derived, and a simple and efficient approximated solution is suggested.

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Abstract: A retailer is endowed with a finite inventory of a non-perishable product. Demand for this product is driven by a price-sensitive Poisson process that depends on an unknown parameter, theta; a proxy for the market size. If theta is high then the retailer can take advantage of a large market charging premium prices, but if theta is small then price markdowns can be applied to encourage sales. The retailer has a prior belief on the value of theta which he updates as time and available information (prices and sales) evolve. We also assume that the retailer faces an opportunity cost when selling this non-perishable product. This opportunity cost is given by the long-term average discounted profits that the retailer can make if he switches and starts selling a different assortment of products. The retailer's objective is to maximize the discounted long-term average profits of his operation using dynamic pricing policies. We consider two cases. In the first case, the retailer is constrained to sell the entire initial stock of the non-perishable product before a different assortment is considered. In the second case, the retailer is able to stop selling the non-perishable product at any time to switch to a different menu of products. In both cases, the retailer's pricing policy trades-off immediate revenues and future profits based on active demand learning. We formulate the retailer's problem as a (Poisson) intensity control problem and derive structural properties of an optimal solution which we use to propose a simple approximated solution. This solution combines a pricing policy and a stopping rule (if stopping is an option) depending on the inventory position and the retailer's belief about the value of theta. We use numerical computations, together with asymptotic analysis, to evaluate the performance of our proposed solution.

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229 citations

"Non-parametric optimal service pric..." refers background in this paper

...For extensive research related to pricing and/or inventory decisions in Bayesian framework, see Araman and Caldentey (2009), etc. and the reference therein....
[...]
...For extensive research related to pricing and/or inventory decisions in Bayesian framework, see Araman and Caldentey (2009), etc. and the reference therein. There is plethora of literature on non-parametric approach in dynamic pricing. Ryzin and Mc Gill (2000) analyse adaptive algorithms for determining airline seat protection levels. In their model, the distributions of customers’ requests for each class are non-parametric. Besbes and Zeevi (2006) introduce a blind non-parametric approach for searching near optimal price setting under unknown demand pattern. Their proposed algorithm is asymptotically optimal. The decision-maker observes demand over time but is otherwise blind to the demand function which maps prices. Structural non-parametric assumptions are made for demand function and demand rate at any instant is estimated by multivariate Poisson process and intensity is determined by price vectors which denotes the set of feasible prices. The authors use non-parametric price algorithm which experiments with several feasible prices and selects price that is expected to optimize revenue based on the demand observed in the previous step. The advantage is the use of multiple products which are linked with various resource constraints contrary to previous approaches which were restricted to single product problem. Rusmevichientong, Roy, and Glynn (2006) formulate a multiproduct single resource pricing problem as a static optimization problem in which a car manufacturer seeks fixed prices that maximizes revenues based on historical preference data. Lim and Shanthikumar (2007) use a max–min approach to a single-product revenue management problem where the uncertainty arises at the level of the point process distribution characterizing the customers’ requests. Other min–max approaches to modeling uncertainty include minimax regret (see Eren, Maglaras, and Ryzin (2006) and Ball and Queyranne (2006) for references in the revenue management literature)....
[...]
...For extensive research related to pricing and/or inventory decisions in Bayesian framework, see Araman and Caldentey (2009), etc. and the reference therein. There is plethora of literature on non-parametric approach in dynamic pricing. Ryzin and Mc Gill (2000) analyse adaptive algorithms for determining airline seat protection levels. In their model, the distributions of customers’ requests for each class are non-parametric. Besbes and Zeevi (2006) introduce a blind non-parametric approach for searching near optimal price setting under unknown demand pattern. Their proposed algorithm is asymptotically optimal. The decision-maker observes demand over time but is otherwise blind to the demand function which maps prices. Structural non-parametric assumptions are made for demand function and demand rate at any instant is estimated by multivariate Poisson process and intensity is determined by price vectors which denotes the set of feasible prices. The authors use non-parametric price algorithm which experiments with several feasible prices and selects price that is expected to optimize revenue based on the demand observed in the previous step. The advantage is the use of multiple products which are linked with various resource constraints contrary to previous approaches which were restricted to single product problem. Rusmevichientong, Roy, and Glynn (2006) formulate a multiproduct single resource pricing problem as a static optimization problem in which a car manufacturer seeks fixed prices that maximizes revenues based on historical preference data....
[...]
...For extensive research related to pricing and/or inventory decisions in Bayesian framework, see Araman and Caldentey (2009), etc. and the reference therein. There is plethora of literature on non-parametric approach in dynamic pricing. Ryzin and Mc Gill (2000) analyse adaptive algorithms for determining airline seat protection levels. In their model, the distributions of customers’ requests for each class are non-parametric. Besbes and Zeevi (2006) introduce a blind non-parametric approach for searching near optimal price setting under unknown demand pattern. Their proposed algorithm is asymptotically optimal. The decision-maker observes demand over time but is otherwise blind to the demand function which maps prices. Structural non-parametric assumptions are made for demand function and demand rate at any instant is estimated by multivariate Poisson process and intensity is determined by price vectors which denotes the set of feasible prices. The authors use non-parametric price algorithm which experiments with several feasible prices and selects price that is expected to optimize revenue based on the demand observed in the previous step. The advantage is the use of multiple products which are linked with various resource constraints contrary to previous approaches which were restricted to single product problem. Rusmevichientong, Roy, and Glynn (2006) formulate a multiproduct single resource pricing problem as a static optimization problem in which a car manufacturer seeks fixed prices that maximizes revenues based on historical preference data. Lim and Shanthikumar (2007) use a max–min approach to a single-product revenue management problem where the uncertainty arises at the level of the point process distribution characterizing the customers’ requests. Other min–max approaches to modeling uncertainty include minimax regret (see Eren, Maglaras, and Ryzin (2006) and Ball and Queyranne (2006) for references in the revenue management literature). Kiefer and Wolfowitz (1952) estimate the point of maximum when the function is unknown and stochastic approximation algorithm proposed uses simulation-based optimization. The proposed Kiefer–Wolfowitz (KW) algorithm uses sequential procedure for stochastic approximation depending on four conditions. The algorithm does not require existence of derivative but optimality of tuning sequence in some reasonable sense is required for the convergence of the algorithm. Though sequence converges stochastically but the proposed algorithm says nothing about the optimal values of the sequence parameters and the stopping rule. Broadie, Deniz, and Assaf (2009) derive general bound on the mean squared error and most importantly the optimal choice of tuning sequences....
[...]
...For extensive research related to pricing and/or inventory decisions in Bayesian framework, see Araman and Caldentey (2009), etc. and the reference therein. There is plethora of literature on non-parametric approach in dynamic pricing. Ryzin and Mc Gill (2000) analyse adaptive algorithms for determining airline seat protection levels. In their model, the distributions of customers’ requests for each class are non-parametric. Besbes and Zeevi (2006) introduce a blind non-parametric approach for searching near optimal price setting under unknown demand pattern. Their proposed algorithm is asymptotically optimal. The decision-maker observes demand over time but is otherwise blind to the demand function which maps prices. Structural non-parametric assumptions are made for demand function and demand rate at any instant is estimated by multivariate Poisson process and intensity is determined by price vectors which denotes the set of feasible prices. The authors use non-parametric price algorithm which experiments with several feasible prices and selects price that is expected to optimize revenue based on the demand observed in the previous step. The advantage is the use of multiple products which are linked with various resource constraints contrary to previous approaches which were restricted to single product problem. Rusmevichientong, Roy, and Glynn (2006) formulate a multiproduct single resource pricing problem as a static optimization problem in which a car manufacturer seeks fixed prices that maximizes revenues based on historical preference data. Lim and Shanthikumar (2007) use a max–min approach to a single-product revenue management problem where the uncertainty arises at the level of the point process distribution characterizing the customers’ requests. Other min–max approaches to modeling uncertainty include minimax regret (see Eren, Maglaras, and Ryzin (2006) and Ball and Queyranne (2006) for references in the revenue management literature). Kiefer and Wolfowitz (1952) estimate the point of maximum when the function is unknown and stochastic approximation algorithm proposed uses simulation-based optimization. The proposed Kiefer–Wolfowitz (KW) algorithm uses sequential procedure for stochastic approximation depending on four conditions. The algorithm does not require existence of derivative but optimality of tuning sequence in some reasonable sense is required for the convergence of the algorithm. Though sequence converges stochastically but the proposed algorithm says nothing about the optimal values of the sequence parameters and the stopping rule. Broadie, Deniz, and Assaf (2009) derive general bound on the mean squared error and most importantly the optimal choice of tuning sequences. They present an adaptive version of the KW algorithm, i.e. scale and shifted KW (SSKW) which improved finite time behaviour dramatically. The key idea is to dynamically shift and scale tuning sequences to get better results for the unknown function and noise level. Although problem of oscillation and noisy gradient never affects the asymptotic convergence, SSKW algorithm greatly improves the rate of convergence for finite time than truncated KW algorithm. Xu and Yu-Hong (2008) give a stochastic approximation frame algorithm which uses the noisy evaluation of the negative gradient direction as the iterative direction....
[...]

Journal Article•DOI•

Revenue Management Saves National Car Rental

[...]

M. K. Geraghty, Ernest Johnson

01 Feb 1997-Interfaces

TL;DR: In the early 1990s, National Car Rental faced liquidation, with the loss of 7,500 jobs, unless it could show a profit in the short term.

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Abstract: In 1993, National Car Rental faced liquidation. General Motors Corporation (National's parent) took a $744 million charge against earnings related to its ownership of National Car Rental Systems. National faced liquidation, with the loss of 7,500 jobs, unless it could show a profit in the short term. National initiated a comprehensive revenue management program whose core is a suite of analytic models developed to manage capacity, pricing, and reservation. As it improved management of these functions, National dramatically increased its revenue. The initial implementation in July 1993 produced immediate results and returned National Car Rental to profitability. In July 1994, National implemented a state-of-the-art revenue management system, improving revenues by $56 million in the first year. In April 1995, General Motors sold National Car Rental Systems for an estimated $1.2 billion.

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214 citations

"Non-parametric optimal service pric..." refers background in this paper

...Geraghty and Johnson (1997) mention National car rentals increased revenue by $56 million in the first year of use....
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Journal Article•DOI•

Dynamic Pricing with a Prior on Market Response

[...]

Vivek F. Farias¹, Benjamin Van Roy²•Institutions (2)

Massachusetts Institute of Technology¹, Stanford University²

01 Jan 2010-Operations Research

TL;DR: Computer results demonstrate that decay balancing offers significant revenue gains over recently studied certainty equivalent and greedy heuristics, and establish that changes in inventory and uncertainty in the arrival rate bear appropriate directional impacts on decay balancing prices in contrast to these alternatives.

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Abstract: We study a problem of dynamic pricing faced by a vendor with limited inventory, uncertain about demand, and aiming to maximize expected discounted revenue over an infinite time horizon. The vendor learns from purchase data, so his strategy must take into account the impact of price on both revenue and future observations. We focus on a model in which customers arrive according to a Poisson process of uncertain rate, each with an independent, identically distributed reservation price. Upon arrival, a customer purchases a unit of inventory if and only if his reservation price equals or exceeds the vendor's prevailing price. We propose a simple heuristic approach to pricing in this context, which we refer to as decay balancing. Computational results demonstrate that decay balancing offers significant revenue gains over recently studied certainty equivalent and greedy heuristics. We also establish that changes in inventory and uncertainty in the arrival rate bear appropriate directional impacts on decay balancing prices in contrast to these alternatives, and we derive worst-case bounds on performance loss. We extend the three aforementioned heuristics to address a model involving multiple customer segments and stores, and provide experimental results demonstrating similar relative merits in this context.

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205 citations

"Non-parametric optimal service pric..." refers background in this paper

...Farias and Roy (2009) analyse dynamic pricing with a prior on market response with maximizing expected discounted revenue over infinite horizon of time....
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Non-parametric optimal service pricing: a simulation study

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