Multi-product Pricing Problem Research Articles

A Representative Consumer Model in Data-Driven Multiproduct Pricing Optimization

We develop a data-driven approach for the multiproduct pricing problem, using the theory of a representative consumer in discrete choice. We establish a set of mathematical relationships between product prices and demand for each product in the system, including that of the outside option. We provide identification conditions to recover the underlying representative consumer model and show that, with sufficient pricing experiments, the approach can identify the underlying demand model (more precisely, the associated perturbation function in the representative consumer model) accurately up to a constant shift and a given tolerance level. This holds even when the demand data obtained are a noisy realization of the theoretical demand. We use this approach to solve the multiproduct pricing problem using a (mixed integer) linear optimization method. Extensive tests using both synthetic and industry data clearly demonstrate the benefits of this approach, which addresses the issue of model misspecification in traditional pricing methods using discrete choice models and circumvents the computational issues associated with pricing methods that assume a known consumer valuation of each product. This paper was accepted by David Simchi-Levi, big data analytics.

Mixed-integer formulations for the Capacitated Rank Pricing Problem with envy

Pricing under a consumer choice model has been extensively studied in economics and revenue management. In this paper, we tackle a generalization of the Rank Pricing Problem (RPP), a multi-product pricing problem with unit-demand customers and a ranking-based consumer choice model. We generalize the RPP assuming that each product has a limited amount of copies for sale, and we call this extension the Capacitated Rank Pricing Problem (CRPP). We compare the envy-free allocation of the products (a fairness criterion requiring that customers receive their highest-ranked product given the pricing) with the envy version of the problem. Next, we focus on the CRPP with envy. We introduce two integer linear formulations for the CRPP and derive valid inequalities leveraging the structure of the problem. Afterwards, we develop separation procedures for the families of valid inequalities of greater size. The performance of the formulations and the resolution algorithms developed is tested by means of extensive computational experiments.

What Is the Impact of Nonrandomness on Random Choice Models?

Problem definition: This paper examines the impact of nonrandomness on random choice models and studies various operations problems under the new discrete choice models. Academic/practical relevance: The literature often assumes that the random utility components follow some independent and identically distributed distribution. This assumption is too restrictive in some real-world scenarios, because, for example, consumers may have known well about the attribute values for the product that they have repeatedly purchased. Methodology: We adopt the random utility maximization framework and characterize the choice probabilities when the utility of some alternative is deterministic. The log-likelihood function is jointly concave in the attribute coefficients under the linear utility-attribute assumption; an expectation-maximization algorithm is developed to overcome the missing data issue in estimation. Results: Surprisingly, if the utility of a particular product is deterministic, the assortment problem is still polynomial-time solvable, whereas if the utility of the no-purchase option is deterministic, the decision problem corresponding to the assortment optimization is NP-complete. We show that the price minus the reciprocal of price sensitivity is product invariant at optimality, which helps to simplify the multiproduct pricing problems. Managerial implications: Empirical study on real data shows that incorporating nonrandomness into random choice models can increase model fitting and prediction accuracy. Failure of accounting for the impact of nonrandomness may result in substantial losses.

Bounds and Heuristics for Multi-Product Personalized Pricing

We present tight bounds and heuristics for personalized, multi-product pricing problems. Under mild conditions we show that offering a non-personalize price in the direction of a positive vector (factor) has a tight profit guarantee relative to optimal personalized pricing. An optimal non-personalized price is the choice factor, when known. Using a factor vector with equal components results in uniform pricing and has exceedingly mild sufficient conditions for the bound to hold. A robust factor is presented that achieves the best possible performance guarantee. As an application, our model yields a tight lower-bound on the performance of linear pricing relative to personalized non-linear pricing, and suggests effective non-linear price heuristics relative to personalized solutions. Additionally, our model provides guarantees for simple strategies such as bundle-size pricing and component-pricing with respect to personalized mixed bundling policies. Heuristics to cluster customer types are also developed with the goal of improving performance by allowing each cluster to price along its own factor. Numerical results are presented for a variety of demand models, factors and clustering heuristics. In our experiments, economically motivated factors coupled with machine learning clustering heuristics performed best.

On Data-Driven Multi-Product Pricing

To handle optimization with only historical data, we present a novel learning framework combining parametric estimation and pricing optimization in the multi-product pricing problem. Motivated by the existence of errors, we first introduce the task-based learning with decision bias for handling estimation errors, which can lead to better decision making under demand parameter uncertainty. Then, we follow the idea of model-free learning, which can design better revenue estimators without knowing the parameter structure to handle model misspecification. Furthermore, to design a more robust estimator, we incorporate the boosting idea to combine a number of estimators for more robust pricing. We validate the superior performance of this framework with numerical studies.

Operations Management under Consumer Choice Models with Multiple Purchases

This paper investigates the effects of multiple purchases that arise in the retailing of consumer goods, in which the product choice and consumer surplus depend not only on what to purchase but also on how many units to purchase. We incorporate the multiple purchases into consumer choice behavior and study a series of associated operational problems. Most of the discrete choice models in the current literature often assume that a customer chooses exactly one unit of a product. The assumption of “one purchase” is too restrictive in some practical scenarios (e.g., consumer goods) because customers often purchase multiple units of a product. We take the widely-used multinomial logit (MNL) model as a showcase and incorporate the effects of multiple purchases into the classic discrete choice model. In the new choice framework, consumers first form a consideration set, then select one product from consideration set and determine the purchase quantity of the selected product. In the absence of fixed cost, we characterize the structure of the optimal policy for the assortment optimization problem; whereas in the presence of product-differentiated fixed costs, the assortment problem becomes NP-complete, so we propose an efficient heuristic. We further develop a polynomial time algorithm for the assortment problem with identical fixed cost for each product. For the joint assortment and pricing problem, we show it can be decoupled into multiple multi-product pricing problems with different assortment sizes, each of which can be transformed into a single-variable problem. For the price competition problem, we characterize the existence and uniqueness of the Nash equilibrium. We combine the alternating optimization algorithm with the expectation maximization algorithm to overcome the non-concavity and missing data issues in estimation. An empirical study on JD.com data shows that incorporating the effects of multiple purchases into discrete choice models can improve model fitting and prediction accuracy, while ignoring the effects of multiple purchases may lead substantial losses.

Scalable relaxation techniques to solve stochastic dynamic multi-product pricing problems with substitution effects

In many businesses, firms are selling different types of products, which share mutual substitution effects in demand. To compute effective pricing strategies is challenging as the sales probabilities of each of a firm’s products can also be affected by the prices of potential substitutes. In this paper, we analyze stochastic dynamic multi-product pricing models for the sale of perishable goods. To circumvent the limitations of time-consuming optimal solutions for highly complex models, we propose different relaxation techniques, which allow to reduce the size of critical model components, such as the state space, the action space, or the set of potential sales events. Our heuristics are able to decrease the size of those components by forming corresponding clusters and using subsets of representative elements. Using numerical examples, we verify that our heuristics make it possible to dramatically reduce the computation time while still obtaining close-to-optimal expected profits. Further, we show that our heuristics are (i) flexible, (ii) scalable, and (iii) can be arbitrarily combined in a mutually supportive way.

What is the Impact of Non-Randomness on Random Choice Models?

Problem definition. This paper examines the impact of non-randomness on random choice models, and then studies various operations problems under the new discrete choice models. Academic/Practical Relevance. The literature often assumes that the random utility components follow some i.i.d. distribution. This assumption may be too restrictive in some real-world scenarios, because for example some consumers may know well about attribute values for some product (which could also be the no-purchase option) that they have repeatedly purchased. Methodology. We adopt the random utility maximization framework, and characterize the choice probabilities when the utility of some alternative is deterministic. The log-likelihood function is still jointly concave; the EM algorithm is developed to overcome the missing data issue. Results. Surprisingly, if the utility of a particular product is deterministic, the assortment problem is polynomial-time solvable; whereas if the utility of the no-purchase option is deterministic, the assortment problem is NP-hard. We show that the prices are product-invariant at optimality and use this result to simplify the multi-product pricing problems. Managerial Implications. Empirical study on real data shows that incorporating non-randomness into random choice models can increase model fitting and prediction accuracy. Failure of accounting for the impact of non-randomness on random choice models may result in substantial losses.

Optimal Pricing Under Diffusion-Choice Models

We develop a solution approach to the centralized pricing problem of a firm managing multiple substitutable products. Demand of these products undergoes a diffusion process, and customers choose among the products, with the choice probability of each product given by the logit model. We examine the firm’s optimal pricing problem when product demand can be described by such “diffusion-choice” models. In particular, we focus on two models with proven merits and study a generalized version of the two models. To our knowledge, our work is the first to study the multiproduct pricing problem under the integrated diffusion-choice models, which are of both theoretical appeal and practical advantage. We establish uniqueness of the optimal solution and propose an efficient solution approach in addition to characterizing the optimal prices and their time trend. We show that the price trend can be attributed to diffusion dynamics as well as intertemporal changes in product costs and customer price sensitivity, all of which are integrated into a unified framework in this paper. Our model applies to both simultaneous and sequential product introductions and adapts to stochastic demand.

Discrete Choice Models with Different Levels of Utility Uncertainty

In this paper, we relax the restriction on the identical distribution for the random utility parts under discrete choice models. The derived new choice model can allow more flexible substitution pattern, and has the potential to describe choice behavior more accurately. If an alternative's nominal utility is relatively high, its choice probability is higher when an individual uses its mean of utility in her choice process, whereas the choice probabilities for other alternatives are lower. We show that in the pricing problem the optimal prices are product-invariant for products with the same levels of utility uncertainty and use this result to simplify the multi-product pricing problem. We also characterize the oligopolistic problems for competition in price and choice probability respectively, and provide efficient algorithms to compute the Nash equilibrium. The assortment problem is generally NP-hard, so we develop a fully polynomial-time approximation scheme that can find an arbitrarily near-optimal solution in a timely manner. Surprisingly, if the utility of a product of the focal firm rather than the outside option is deterministic, the revenue-ordered assortment is optimal for the assortment problem. To implement the newly proposed choice model with different levels of utility uncertainty, we develop an efficient estimation algorithm with estimated product attractiveness in closed form. Several extensions are also considered, including relaxing the restriction under the multi-stage nested logit model.

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

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.

Technical Note—Multiproduct Pricing Under the Generalized Extreme Value Models with Homogeneous Price Sensitivity Parameters

We consider unconstrained and constrained multiproduct pricing problems when customers choose according to an arbitrary generalized extreme value (GEV) model and the products have the same price sensitivity parameter. In the unconstrained problem, there is a unit cost associated with the sale of each product. The goal is to choose the prices for the products to maximize the expected profit obtained from each customer. We show that the optimal prices of the different products have a constant markup over their unit costs. We provide an explicit formula for the optimal markup in terms of the Lambert-W function. In the constrained problem, motivated by the applications with inventory considerations, the expected sales of the products are constrained to lie in a convex set. The goal is to choose the prices for the products to maximize the expected revenue obtained from each customer, while making sure that the constraints for the expected sales are satisfied. If we formulate the constrained problem by using the prices of the products as the decision variables, then we end up with a nonconvex program. We give an equivalent market-share-based formulation, where the purchase probabilities of the products are the decision variables. We show that the market-share-based formulation is a convex program, the gradient of its objective function can be computed efficiently, and we can recover the optimal prices for the products by using the optimal purchase probabilities from the market-share-based formulation. Our results for both unconstrained and constrained problems hold for any arbitrary GEV model. The e-companion is available at https://doi.org/10.1287/opre.2018.1740 .

SAA-regularized methods for multiproduct price optimization under the pure characteristics demand model

Utility-based choice models are often used to determine a consumer’s purchase decision among a list of available products; to provide an estimate of product demands; and, when data on purchase decisions or market shares are available, to infer consumers’ preferences over observed product characteristics. These models also serve as a building block in modeling firms’ pricing and assortment optimization problems. We consider a firm’s multiproduct pricing problem, in which product demands are determined by a pure characteristics model. A sample average approximation (SAA) method is used to approximate the expected market share of products and the firm profit. We propose an SAA-regularized method for the multiproduct price optimization problem. We present convergence analysis and numerical examples to show the efficiency and the effectiveness of the proposed method.

'Marginal Estimation+Price Optimization' with Robust Choice Models

We develop an estimation and optimization framework for the multi-product pricing problem by exploiting the properties of the marginal distributions of the random utilities in discrete choice models. We provide a novel interpretation of the marginal distribution choice model from a robust optimization viewpoint where the modeler evaluates the choice probabilities assuming a worst-case realization of the utilities of the products in a budgeted uncertainty set. Moreover, we show that the multi-product pricing problem under this robust choice model is a convex optimization problem when the marginal distributions are log-concave. Multinomial Logit and the Nested Logit models are special cases of this model.We use this approach to address partially the problem of model identification in choice modeling. Instead of presupposing the structural forms of the consumer’s utility functions, we use experimentations to guide us to the appropriate marginal distribution models for the pricing problems. More specifically, we develop a data-driven linear program to estimate the marginal distributions from a set of price experiments and to find the optimal prices. Mixed Integer Linear Programming models can be used to find the prices when side constraints are present. Tests using both simulated data and real data from the automobile industry illustrates the benefits of the “marginal estimation price optimization” approach.

Stochastic Bundling

Intertemporal price discrimination is widespread. The internet has made stochastic delivery possible. We study the multiproduct monopoly pricing problem and show these sales strategies are isomorphic. We derive a sufficient condition for these advanced selling strategies to be profitable: the appropriate cross-partial derivative of the profit function must be negative. Hence we show that Stokey's (QJE 1979) celebrated no discrimination across time result does not extend to multiple goods in many classes of cases, including normal distributions of valuations. We show that profits can be increased by dynamic pricing for the cross-sell. Further, welfare can also be increased with the introduction of stochastic delivery or intertemporal price discrimination.

Customer Data-Oriented Multi-Product Pricing Model

Through interaction with online consumers, e-commerce websites can gather data reflecting consumer preferences. The available data on consumer preferences together with sophisticated analytical tools enables companies’ increases in profit through optimization of prices. We consider a class of models of consumer purchasing behavior, each of which affected the data on a consumer’s requirements and budget constraint to subsequent purchasing tendencies. We can study for the multi-product pricing problem based on the max budgets of consumers, and a customer data-oriented multi-product pricing model is presented. Proposed models were solved through genetic algorithms. Optimal solution of the given examples shows that this model is effective for enterprises.

Mixed Bundling of Two Independently Valued Goods

This paper develops analytical results and insights for the mixed bundling problem of pricing a product line consisting of two component goods (with valuations distributed uniformly and independently in [0, a1] and [0, a2], respectively, with a1≤ a2) and a bundle of the two goods. This setting has previously been considered analytically intractable. By deriving component good prices as exact algebraic functions of the optimal bundle price, I reduce the multiproduct pricing problem to a univariate nonlinear optimization problem in the bundle price. When the two component goods are sufficiently asymmetric in valuations, the optimal solution is partial mixed bundling (one component good is not sold separately), for which I derive exact conditions and optimal prices. An exact analytical solution is also given for mixed bundling of two information goods (component goods with zero marginal costs). For the general case, I derive a closed-form approximation of the optimal bundle price: pB= 0.5724w1+ 0.5695w2+ 0.3516a1+ 0.4889a2+ 0.0054(a2/a1)w1−0.0201(a2/a1)w2. The approximation is highly accurate (the resulting profit is within a percent of the exact optimal profit), efficient (instant solutions for specific problem instances), and useful in other applications where mixed bundling is a subproblem. I demonstrate this by applying the solution to the mixed bundle problem in a vertical channel where a retailer must bundle and price component goods from multiple independent manufacturers.This paper was accepted by Lorin Hitt, information systems.

Efficient formulations for pricing under attraction demand models

We propose a modeling and optimization framework to cast a broad range of fundamental multi-product pricing problems as tractable convex optimization problems. We consider a retailer offering an assortment of differentiated substitutable products to a population of customers that are price-sensitive. The retailer selects prices to maximize profits, subject to constraints on sales arising from inventory and capacity availability, market share goals, bounds on allowable prices and other considerations. Consumers' response to price changes is represented by attraction demand models, which subsume the well known multinomial logit (MNL) and multiplicative competitive interaction demand models. Our approach transforms seemingly non-convex pricing problems (both in the objective function and constraints) into convex optimization problems that can be solved efficiently with commercial software. We establish a condition which ensures that the resulting problem is convex, prove that it can be solved in polynomial time under MNL demand, and show computationally that our new formulations reduce the solution time from days to seconds. We also propose an approximation of demand models with multiple overlapping customer segments, and show that it falls within the class of demand models we are able to solve. Such mixed demand models are highly desirable in practice, but yield a pricing problem which appears computationally challenging to solve exactly.

Multiproduct Price Optimization and Competition Under the Nested Logit Model with Product-Differentiated Price Sensitivities

We study firms that sell multiple substitutable products and customers whose purchase behavior follows a nested logit model, of which the multinomial logit model is a special case. Customers make purchasing decisions sequentially under the nested logit model: they first select a nest of products and subsequently purchase one within the selected nest. We consider the multiproduct pricing problem under the general nested logit model with product-differentiated price sensitivities and arbitrary nest coefficients. We show that the adjusted markup, defined as price minus cost minus the reciprocal of price sensitivity, is constant for all the products within a nest at optimality. This reduces the problem's dimension to a single variable per nest. We also show that the adjusted nest-level markup is nest invariant for all the nests, which further reduces the problem to maximizing a single-variable unimodal function under mild conditions. We also use this result to simplify the oligopolistic multiproduct price competition and characterize the Nash equilibrium. We also consider more general attraction functions that include the linear utility and the multiplicative competitive interaction models as special cases, and we show that similar techniques can be used to significantly simplify the corresponding pricing problems.

Optimal Solutions for Mixed Bundling of Two Independently-Valued Goods

This paper develops analytical results and insights for the mixed bundling problem of pricing a product line consisting of two component goods and a bundle of the two goods. Consumer valuations for the two goods follow a symmetric bivariate uniform distribution, a setting which has previously been considered analytically intractable. Prior analytical work has demonstrated the superiority of mixed bundling, but failed to provide an explicit specification of the optimal solution or prices. Alternately, researchers have employed numerical solution techniques to solve instances of the mixed bundling problem. I reduce the multi-product pricing problem to a univariate nonlinear optimization problem in the bundle price with a global maximum that can be identified uniquely. I derive tight lower and upper bound functions for the optimal bundle price. The average of these two functions can be used as an alternate, and extremely simple, analytical expression for the optimal bundle price. Another alternative formula for the optimal bundle price is obtained by applying statistical estimation over solutions for a large space of problem instances. Both approximations are highly accurate, producing a pro t that is within half percent of the exact optimal pro t. The optimal bundle price has an elegant analytical expression (a linear function of the marginal costs of the two goods) and prices for component goods are an algebraic function of the bundle price. With a bivariate distribution in the unit square and constant unit marginal costs w1 and w2, the optimal bundle price can be written as 0:86032 0:55929(w1 w2)/2.