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Optimization based approaches to product pricing
Abstract
─ We consider various axioms for customer behaviour using utility functions and so-called "reservation prices" and then based on these axioms, we discuss some mathematical models (employing integer programming, convex programming and classical nonlinear programming) for deciding on product prices to maximize the total profit (or perhaps another suitable objective function also involving minimization of risk). We also share some of our experiences from a recent collaborative research project involving a company in the tourism sector.
We propose improvements to some of the best heuristic algorithms for optimal product pricing problem originally designed by Dobson and Kalish in the late 1980s and in the early 1990s. Our improvements are based on a detailed study of a fundamental decoupling structure of the underlying mixed integer programming (MIP) problem and on incorporating more recent ideas, some from the theoretical computer science literature, for closely related problems. We provide very efficient implementations of the algorithms of Dobson and Kalish as well as our new algorithms. We show that our improvements lead to algorithms which generate solutions with better objective values and that are more robust in overall performance. Our computational experiments indicate that our implementation of Dobson–Kalish heuristics and our new algorithms can provide good solutions for the underlying MIP problems where the typical LP relaxations would have more than a trillion variables and more than three trillion constraints.
We consider the problem of determining a set of optimal tariffs for an agent in the network, who owns a subset of all the arcs, and who receives revenue by setting the tariffs on the arcs he owns. Multiple rational clients are active in the network, who route their demands on the least expensive paths from source to destination. The cost of a path is determined by fixed costs and tariffs on the arcs of the path. We introduce a remodeling of the network, using shortest paths. We develop three algorithms based on this shortest-path graph model: a combinatorial branch-and-bound algorithm, a path-oriented mixed integer program, and a known-arc-oriented mixed integer program. Combined with reduction methods, this remodeling enables us to solve the problem to optimality, for quite large instances. We provide computational results for the methods developed and compare them with the results of the arc-oriented mixed integer programming formulation of the problem, applied to the original network.
A significant application of conjoint analysis is in pricing decisions for new products. Conceptually, profit maximization is an important criterion for selecting the price of a product. However, the maximization of profit necessitates estimation of fixed and variable costs, which are difficult to estimate reliably for the large number of products available for evaluation in conjoint analysis. Consequently, users of conjoint models have begun to use a share simulation to screen a small set of attractive products. For each screened product, fixed and variable costs are estimated separately and used to simulate its profits at different price levels. The limitation of this approach is that the profit simulations are based on the assumption that the conjoint data, and hence the predicted profits, are error free. Also, though the purpose of examining alternative prices is to determine the best price at which to offer a new product, current conjoint simulators do not focus explicitly on optimal pricing decisions. The authors describe and illustrate a model for optimal pricing of screened products in conjoint analysis, incorporating the effect of measurement and estimation error on predicted profits.
The problem of maximizing the share of a new product introduced in a competitive market is shown to be NP-hard. A directed graph representation of the problem is used to construct shortest-path and dynamic-programming heuristics. Both heuristics are shown to have arbitrarily-bad worst-case bounds. Computational experience with real-sized problems is reported. Both heuristics identify near-optimal solutions for the simulated problems, the dynamic-programming heuristic performing better than the shortest-path heuristic.
This survey reviews the forty-year history of research on transportation revenue management (also known as yield management). We cover developments in forecasting, overbooking, seat inventory control, and pricing, as they relate to revenue management, and suggest future research directions. The survey includes a glossary of revenue management terminology and a bibliography of over 190 references. In the forty years since the first publication on overbooking control, passenger reservations systems have evolved from low level inventory control processes to major strategic information systems. Today, airlines and other transportation companies view revenue management systems and related information technologies as critical determinants of future success. Indeed, expectations of revenue gains that are possible with expanded revenue management capabilities are now driving the acquisition
The authors propose a probabilistic approach to optimally price a bundle of products or services that maximizes seller's profits. Their focus is on situations in which consumer decision making is on the basis of multiple criteria. For model development and empirical investigation they consider a season ticket bundle for a series of entertainment performances such as sports games and music/dance concerts. In this case, they assume consumer purchase decisions to be a function of two independent resource dimensions, namely, available time to attend performances and reservation price per performance. Using this information, the model suggests the optimal prices of the bundle and/or components (individual performances), and corresponding maximum profits under three alternative strategies: (a) pure components (each performance is priced and offered separately), (b) pure bundling (the performances are priced and offered only as a bundle), and (c) mixed bundling (both the bundle and the individual performances are priced and offered separately). They apply their model to price a planned series of music/dance performances. Results indicate that a mixed bundling strategy is more profitable than pure components or pure bundling strategies provided the relative prices of the bundle and components are carefully chosen. Limitations and possible extensions of the model are discussed.
Quantity-Based RM.- Single-Resource Capacity Control.- Network Capacity Control.- Overbooking.- Price-based RM.- Dynamic Pricing.- Auctions.- Common Elements.- Customer-Behavior and Market-Response Models.- The Economics of RM.- Estimation and Forecasting.- Industry Profiles.- Implementation.
We consider revenue management models for pricing a product line with several customer segments, working under the assumption that every customer's product choice is determined entirely by their reservation price. We model the customer choice behavior by several prob- abilistic choice models and formulate the problems as mixed-integer programming problems. We study special properties of these formulations and compare the resulting optimal prices of the difierent probabilistic choice models. We also explore some heuristics and valid inequali- ties to improve the running time of the mixed-integer programming problems. We illustrate the computational results of our models on real and generated customer data taken from a company in the tourism sector.
Firms should be able to apply the time-based philosophy of revenue management to their sales forces. To do so requires a revision in the way most sales divisions traditionally have viewed salesperson time. Hence, a different type of proposed measure, revenue per available salesperson hour, is proposed to better integrate the value of the salesperson''s time as a factor in sales potential and revenue calculation. This article seeks to (1) foster a positive perception of revenue management as a viable sales approach, (2) establish a framework for such a strategy, and (3) set a useful road map for facilitating execution.
The multinomial logit model is a standard approach for determining the probability of purchase in product line problems. When the purchase probabilities are multiplied by product contribution margins, the resulting profit function is generally nonconcave. Because of this, standard nonlinear search procedures may terminate at a local optimum which is far from the global optimum. We present a simple procedure designed to alleviate this problem. The key idea of this procedure is to find a "path" of prices from the global optimum of a related, but concave logit profit function, to the global optimum of the true (but nonconcave) logit profit function.
Designing and pricing a product-line is the very essence of every business. In recent years quantitative methods to assist managers in this task have been gaining in popularity. Conjoint analysis is already widely used to measure preferences for different product profiles, and build market simulation models. In the last few years several papers have been published that suggest how to optimally choose a product-line based on such data.
We formalize this problem as a mathematical program where the objective of the firm is either profit or total welfare. Unlike alternative published approaches, we introduce fixed and variable costs for each product profile. The number of products to be introduced is endogenously determined on the basis of their desirability, fixed and variable costs, and in the case of profits, their cannibalization effect on other products. While the problem is difficult (NP-complete), we show that the maximum welfare problem is equivalent to the uncapacitated plant location problem, which can be solved very efficiently using the greedy interchange heuristic. Based on past published experience with this problem, and on simulations we perform, we show that optimal or near optimal solutions are obtained in seconds for large problems. We develop a new greedy heuristic for the profit problem, and its application to simulated problems shows that it too runs quickly, and with better performance than various alternatives and previously published heuristics. We also show how the methodology can be applied, taking existing products of both the firm and the competition into account.
Bundle pricing is a widespread phenomenon. However, despite its importance as a pricing tool, surprisingly little is known about how to find optimal bundle prices. Most discussions in the literature are restricted to only two components, and even in this case no algorithm is given for setting prices. Here we show that the single firm bundle pricing problem is naturally viewed as a disjunctive program which is formulated as a mixed integer linear program. Multiple components, and a variety of cost and reservation price conditions are investigated with this approach. Several new economic insights on the role and effectiveness of bundling are presented. An added benefit of the solution to the bundle pricing model is the selection of products which compose the firm's product line. Computational testing is done on problems with up to 21 components (over one million potential product bundles), and data collection issues are addressed.
In this marketing-oriented era where manufacturers maximize profits through customer satisfaction, there is an increasing
need to design a product line rather than a single product. By offering a product line, the manufacturer can customize his
or her products to the needs of a variety of segments in order to maximize profits by satisfying more customers than a single
product would. When the amount of data on customer preferences or possible product configurations is large and no analytical
relations can be established, the problem of an optimal product line design becomes very difficult and there are no traditional
methods to solve it. In this paper, we show that the usage of genetic algorithms, a mathematical heuristics mimicking the
process of biological evolution, can solve efficiently the problem. Special domain operators were developed to help the genetic
algorithm mitigate cannibalization and enhance the algorithm’s local search abilities. Using manufacturer’s profits as the
criteria for fitness in evaluating chromosomes, the usage of domain specific operators was found to be highly beneficial with
better final results. Also, we have hybridized the genetic algorithm with a linear programming postprocessing step to fine
tune the prices of products in the product line. Attacking the core difficulty of cannibalization in the algorithm, the operators
introduced in this work are unique.
In this paper, we provide a heuristic procedure, that performs well from a global optimality point of view, for an important and difficult class of bilevel programs. The algorithm relies on an interior point approach that can be interpreted as a combination of smoothing and implicit programming techniques. Although the algorithm cannot guarantee global optimality, very good solutions can be obtained through the use of a suitable set of parameters. The algorithm has been tested on large-scale instances of a network pricing problem, an application that fits our modeling framework. Preliminary results show that on hard instances, our approach constitutes an alternative to solvers based on mixed 0–1 programming formulations.
Developed by General Motors (GM), the Auto Choice Advisor website ( http://www.autochoiceadvisor.com ) recommends vehicles to consumers based on their requirements and budget constraints. Through the website, GM has access to large quantities of data that reflect consumer preferences. Motivated by the availability of such data, we formulate a nonparametric approach to multiproduct pricing.
We consider a class of models of consumer purchasing behavior, each of which relates observed data on a consumer’s requirements and budget constraint to subsequent purchasing tendencies. To price products, we aim at optimizing prices with respect to a sample of consumer data. We offer a bound on the sample size required for the resulting prices to be near-optimal with respect to the true distribution of consumers. The bound exhibits a dependence of O(n log n) on the number n of products being priced, showing that—in terms of sample complexity—the approach is scalable to large numbers of products.
With regards to computational complexity, we establish that computing optimal prices with respect to a sample of consumer data is NP-complete in the strong sense. However, when prices are constrained by a price ladder—an ordering of prices defined prior to price determination—the problem becomes one of maximizing a supermodular function with real-valued variables. It is not yet known whether this problem is NP-hard. We provide a heuristic for our price-ladder-constrained problem, together with encouraging computational results.
Finally, we apply our approach to a data set from the Auto Choice Advisor website. Our analysis provides insights into the current pricing policy at GM and suggests enhancements that may lead to a more effective pricing strategy.
We consider the problem of maximizing the revenue raised from tolls set
on the arcs of a transportation network, under the constraint that users
are assigned to toll-compatible shortest paths. We first prove that this
problem is strongly NP-hard. We then provide a polynomial time algorithm
with a worst-case precision guarantee of , where
denotes the number of toll arcs. Finally we show that the
approximation is tight with respect to a natural relaxation by
constructing a family of instances for which the relaxation gap is
reached.
Product line selection and pricing decisions are critical to the profitability of many firms, particularly in today’s competitive business environment in which providers of goods and services are offering a broad array of products to satisfy customer needs.We address the problem of selecting a set of products to offer and their prices when customers select among the offered products according to a share-of-surplus choice model. A customer’s surplus is defined as the difference between his utility (willingness to pay) and the price of the product. Under the share-of-surplus model, the fraction of a customer segment that selects a product is defined as the ratio of the segment’s surplus from this particular product to the segment’s total surplus across all offered products with positive surplus for that segment.We develop a heuristic procedure for this non-concave, mixed-integer optimization problem. The procedure utilizes simulated annealing to handle the binary product selection variables, and a steepest-ascent-style procedure that relies on certain structural properties of the objective function to handle the non-concave, continuous portion of the problem involving the prices. We also develop a variant of our procedure to handle uncertainty in customer utilities. In computational studies, our basic procedures perform extremely well, producing solutions whose objective values are within about 5% of those obtained via enumerative methods. Our procedure to handle uncertain utilities also performs well, producing solutions with expected profit values that are roughly 10% higher than the corresponding expected profits from solutions obtained under the assumption of deterministic utilities.
Maximum Utility Product Pricing Models and Algorithms Based on Reservation Prices Revenue Management Under a General Discrete Choice Model of Consumer Behavior
- R Shioda
- L Tunçel
- T Myklebust
- Canada
- G J Van Ryzin
- K T Talluri
Shioda, R., Tunçel, L., Myklebust, T. (2007), " Maximum Utility Product Pricing
Models and Algorithms Based on Reservation Prices ", Dept. of Combinatorics and
Optimization, University of Waterloo, Research Report CORR 2007-08, Canada.
van Ryzin, G. J., Talluri, K. T. (2004), " Revenue Management Under a General
Discrete Choice Model of Consumer Behavior ", Management Science, Vol. 50, No. 1,
pp. 15-33.