Optimal Pricing Problem Research Articles

Multi-period price optimization problem for omnichannel retailers accounting for customer heterogeneity

In recent times many brick and mortar retailers have considered adding online channels to compete with e-tailers. These retailers have further enabled omnichannel features into their businesses considering customer's channel switching behavior. This article presents a price optimization problem for such a retailer who already had many offline stores but then added an online channel to improve its digital presence. The retailer has a distribution center that replenishes these stores periodically. Furthermore, the retailer uses either the distribution center or any of the offline stores to fulfill online orders. The paper takes an integrated approach to price optimization, inventory control, and e-fulfillment problem and developed a decision support model for such a retailer having multiple objectives of profit maximization and lost sales minimization. The lost sales minimization is achieved either by decreasing the probability of no-purchase option (by reducing the posted price) or by fulfilling online demands at relatively higher costs (it is not optimal to meet all the demand). The paper tests the efficacy of conventional Non-dominated sorting genetic algorithm-II (NSGA-II) and recently proposed direct Zigzag search in order to solve the resulting NP-hard discrete multi-objective optimization problem. We offer a new algorithm that exploits the neighborhood search feature of the Zigzag method to extend the NSGA-II front further. Furthermore, using realistic and real-world inspired data, we generated some useful insights for omnichannel retailers.

Product-Line Pricing Under Discrete Mixed Multinomial Logit Demand

We study a product-line price optimization problem with demand given by a discrete mixed multinomial logit (MMNL) model. The market is divided into a finite number of market segments, with product demand in each segment governed by the multinomial logit (MNL) model. We show that the concavity property with respect to the choice probability vector shown for the MNL model breaks down under MMNL even for entirely symmetric price sensitivities. In addition, the equal markup property identified for the MNL model no longer holds under MMNL, suggesting that heterogeneity in customer population justifies non-equal-markup pricing. In this paper, we characterize the profit function under MMNL as the sum of a set of quasiconcave functions and present efficient optimization algorithms. We demonstrate the application of our methods using data from Intel Corporation. Our results show that the optimal prices exploit segment differences through redistribution of sales and profit among customer segments. In addition, we use our methods to generate the profit-share efficient frontier, helping the firm to set prices that effectively balance these two measures. The tools developed in this research function in three important ways at Intel: (1) they provide a new alternative for market share prediction among Intel products for different customer segments, adding to the suite of independent demand forecasting tools; (2) they optimize product prices based on segment-specific customer preferences revealed through sales data, a capability that Intel’s current pricing tools include only heuristically; (3) they quantify the trade-off between profit and market share, adding a new decision support capability to the company. The online appendix is available at https://doi.org/10.1287/msom.2017.0675 .

Price and Assortment Optimization under Choice Models with Anticipated Wait

Customers often face multiple choices when purchasing a product or service. After making a choice, they sometimes have to wait for a while before receiving their purchased item due to the firm’s limited capacity to process orders. This paper incorporates the anticipated wait for receiving purchased products or services into customers’ choice behavior. The resulting choice model shares the same spirit of the rational expectation equilibrium, and captures the effects of negative externality caused by the anticipated wait, because all orders may be processed by a common facility. Our analysis shows that the anticipated wait may change the substitution patterns dramatically. We further investigate the effects of the anticipated wait on the decisions of capacity investment, product pricing and assortment planning. We establish the one-to-one mapping between the price vector and the choice probability vector, and show that the equivalent profit function of the choice probabilities is explicitly defined and more tractable. We characterize the multi-product price optimization problem under the MNL model with waiting. In addition to price competition, we also study the Cournot competition, in which the decision is the choice probability for each firm, and show that there exists a Nash equilibrium. For the assortment optimization, we identify the conditions under which the optimality of the revenue-ordered assortment still holds. Because the assortment problem with waiting is generally NP-hard, we develop efficient approximations with performance guarantee and also provide an easy-to-compute tight upper bound. The new model has the potential to increase prediction accuracy for customers’ choice behavior especially when customers faced with multiple choices are aware of the possible waiting for their purchased products. Failure to take into account the effects of the anticipated wait in customers’ purchase behavior may result in substantial losses to firms.

Capacitated assortment and price optimization under the multilevel nested logit model

We study the capacitated assortment and price optimization problem under the multilevel nested logit model, where the retailer needs to determine the revenue-maximizing subset of products and their prices within a capacity limit. Customers’ purchase behavior is modeled using the multilevel nested logit model, which includes the multinomial logit and the two-level nested logit models as special cases. We formulate this problem as a non-linear mixed integer program and develop an efficient algorithm that finds its ϵ-approximate solution.

Optimal Pricing of User-Initiated Data-Plan Sharing in a Roaming Market

A smartphone user’s personal hotspot (pH) allows him to share a cellular connection to another (e.g., a traveler) in the vicinity, but such sharing consumes the limited data quota in his two-part tariff plan and may lead to an overage charge. This paper studies how to motivate such pH-enabled data-plan sharing between local users and travelers in the ever-growing roaming markets, and proposes pricing incentive for a data-plan buyer to reward surrounding pH sellers (if any). The pricing scheme practically takes into account the information uncertainty at the traveler side, including the random mobility and the sharing cost distribution of selfish local users who potentially share their pHs. Though the pricing optimization problem is non-convex, we show that there always exists a unique optimal price to a tradeoff between the successful sharing opportunity and the sharing price. We further generalize the optimal pricing to the case of heterogeneous selling pHs who have diverse data usage behaviors in the sharing cost distributions, and we show such diversity may or may not benefit the traveler. Lacking selfish pHs’ information, the traveler’s expected cost is higher than that under the complete information, but the gap diminishes as the pHs’ spatial density increases. Finally, we analyze the challenging scenario that multiple travelers overlap for demanding data-plan sharing, by resorting to a near-optimal pricing scheme. We show that a traveler suffers as the travelers’ spatial density increases.

Optimal price decisions for joint ventures between port operators and shipping lines under the congestion effect

The congestion effect refers to the phenomenon that more customers choosing to use the same facility reduces the facility’s utility. This work addresses the optimal pricing problem for a firm operating a joint-venture terminal under the congestion effect. The firm is formed between a port terminal operator and a shipping line, thus being able to provide a bundle of ocean transportation and port terminal solutions to cargo suppliers. The objective is to determine the optimal prices charged to cargo suppliers to maximize the total profit of the firm. First, we develop a tractable flow-based optimization model that uses a fixed-point equation to capture the interaction between the congestion effect and cargo suppliers’ choice. Second, we characterize the optimal solution for a variety of cases, including the single origin-destination case, partially homogeneous case that includes the fully homogeneous case as a special case, and heterogeneous case. Third, we evaluate the profit loss incurred by ignoring the congestion effect with numerical studies. Moreover, we propose one-variable and two-variable search methods for the partially homogeneous and heterogeneous cases, respectively. We learn that the firm should quote the same price to all cargo suppliers under the fully homogeneous case. However, this is not necessarily optimal under the partially homogeneous or heterogeneous cases. The profit loss incurred by neglecting the congestion effect can be significant and increases as cargo suppliers become less tolerant of congestion.

Pricing When Customers Have Limited Attention

We study the optimal pricing problem of a monopolistic firm facing customers with limited attention and capability to process information about the value quality of a single offered product. We mod...

A user-centric economic-driven paradigm for rate allocation in non-orthogonal multiple access wireless systems

In this paper, a novel approach towards optimizing users’ rate allocation and price customization in a non-orthogonal multiple access (NOMA) wireless network under quality of service (QoS)-differentiated requested services is proposed and studied. A multi-service wireless system is considered, where each user’s QoS requirements are reflected through a utility function, alongside his willingness to pay for the corresponding service. Within this setting, in order to jointly allocate the customized price and rate, a two-variable optimization problem arises. Based on the principles of S-modular theory, the above two-variable (rate and price) optimization problem is modeled and solved as a distributed non-cooperative game. The existence and convergence to the Nash equilibrium point with reference to both user’s uplink transmission rate and price is proven. The proposed approach, allowing for decision-making at the user side, is well aligned with the self-optimization and self-adaptation objectives of future emerging 5G wireless networks. The performance evaluation of the devised framework is conducted via modeling and simulation under various scenarios, and the numerical results clearly demonstrate its superiority against other existing approaches.

Dynamic Joint Assortment and Pricing Optimization With Demand Learning

Problem definition: We consider a joint assortment optimization and pricing problem where customers arrive sequentially and make purchasing decisions following the multinomial logit (MNL) choice model. Not knowing the customer choice parameters a priori and subjecting to a display capacity constraint, we dynamically determine the subset of products for display and the selling prices to maximize the expected total revenue over a selling horizon. Academic/practical relevance: Assortment and pricing are important decisions for firms such as online retailers and have received enormous attention in the operations literature. In this paper we present the first learning algorithm for the dynamic joint assortment optimization and pricing problem (based on the MNL choice model) when the firm has limited prior knowledge about customer demand. Methodology: We design a learning algorithm that balances the trade-off between demand learning and revenue extraction, and evaluate the performance of the algorithm using Bayesian regret. This algorithm uses the method of random sampling to simultaneously learn the demand and maximize the revenue on the fly. Results: An instance independent upper bound for the Bayesian regret of the algorithm is obtained and numerical results show that it performs very well. Managerial implications: Our work is the first one to develop effective learning algorithm for joint assortment and pricing optimization problem when customer demand information is not known a priori. The algorithm concurrently learns customer demand while making adaptive assortment and pricing decisions, and it is an effective approach for revenue maximization.

Pricing for sales and per-use rental services with vertical differentiation

An increasing number of firms are simultaneously offering consumers products for sale and for per-use rental services. Typically, the products offered in per-use rental services are different from those offered for sale. For example, automobiles offered in Daimler’s Car2Go program are “Smart” cars, which have a smaller size than the typical automobiles sold to consumers. Such a difference, which is objectively measurable and captured by quality, is called a vertical differentiation. This study investigates the optimal pricing problem for per-use rental services and sales, and reveals how per-use rental services interplay with sales when vertical differentiation exists. The optimal solution is unique and determined by a model that uses a marginal renter and a marginal buyer as decision variables rather than the prices of per-use rental services and sales. The vertical differentiation significantly influences a firm’s potential profitability. When the vertical differentiation falls into a certain interval, the potential profitability from per-use rental services is higher than that from sales, which explains why the per-use rental business model is increasing in popularity. Finally, the results show that offering products with a relatively high (low) quality in per-use rental services (sales) is highly profitable for product categories with a strong pooling effect or when there are high firm-side benefits from ownership. This finding enriches the existing literature by providing novel perspectives on ownership and the pooling effect, and these insights are illustrated through a numerical example.

Optimal worst-case pricing for a logit demand model with network effects

We consider optimal pricing problems for a product that experiences network effects. Given a price, the sales quantity of the product arises as an equilibrium, which may not be unique. In contrast to previous studies that take a best-case view when there are multiple equilibrium sales quantities, we maximize the seller’s revenue assuming that the worst-case equilibrium quantity will arise in response to a chosen price. We compare the best- and worst-case solutions, and provide asymptotic analysis of revenues.

Optimal dynamic pricing for smart grid having mixed customers with and without smart meters

This paper investigates an optimal day-ahead dynamic pricing problem in an electricity market with one electricity retailer and multiple customers. The main objective of this paper is to support the retailer to make the best day-ahead dynamic pricing decision, which maximizes its profit under the realistic assumption that mixed types of customers coexist in the electricity market where some customers have installed smart meters with the embedded home energy management system in their home whereas other customers have not installed smart meters. To this end, we propose a hybrid demand modelling framework which firstly uses an optimal energy management algorithm with bill minimization to model the behavior of customers with smart meters and secondly use a data-driven demand modelling method to model the behavior of customers without smart meters. Such a hybrid demand model can not only schedule usages of home appliances to the interests of customers with smart meters but also be used to understand electricity usage behaviors of customers without smart meters. Based on the established hybrid demand model for all customers, a profit maximization algorithm is developed to achieve optimal prices for the retailer under relevant market constraints. Under the condition of no growth of the revenue (i.e. no increase of total bill from all customers), simulation results indicate our optimization algorithm can improve the profit for around 5% on average.

Price Optimization Under the Finite-Mixture Logit Model

We consider price optimization under the finite-mixture logit model. This model assumes that customers belong to one of a number of customer segments, where each customer segment chooses according to a multinomial logit model with segment-specific parameters. We reformulate the corresponding price optimization problem and develop a novel characterization. Leveraging this new characterization, we construct an algorithm that obtains prices at which the revenue is guaranteed to be at least (1-epsilon) times the maximum attainable revenue for any prespecified epsilon>0. Existing global optimization methods require exponential time in the number of products to obtain such a result, which practically means that the prices of only a handful of products can be optimized. The running time of our algorithm, however, is exponential in the number of customer segments and only polynomial in the number of products. This is of great practical value, since in applications the number of products can be very large, while it is has been found in various contexts that a low number of segments is sufficient to capture customer heterogeneity appropriately. The results of our numerical study show that our algorithm runs fast on a broad range of problem instances.

Optimal pricing on an age-specific inventory system for perishable items

With aging over time, perishable items with fixed shelf lives, can be differentiated by their ages. The age which reflects the quality level of the item can directly affect consumers’ willingness to pay, and then affect the consumer demand. Therefore, we consider that consumers’ sensitivity to item’s age is heterogeneous, which can induce price differentiation to balance consumers’ purchasing preferences. Taking the age heterogeneity into account, we model an age-specific inventory system to explore the optimal pricing problem for perishable items, which in fact, is an optimal control problem for distributed parameter system. We describe the evolution of the stock by a first-order partial differential equation of both time and age. Through applying the distributed parameter optimal control theory, we explicitly derive the optimal pricing strategy which prices items differentially based on their different ages. Numerical examples show that, for the different aged items at the same time, it is beneficial for the firm to discount the items with their ages, while for the same aged items, taking the penetration pricing policy over time is a better choice.

Optimal pricing for selling to a static multi-period newsvendor

This paper considers a multi-period supply chain model in which a supplier sells to a multi-period newsvendor. Such a problem is relevant in industries with long production lead times. We study the optimal pricing problem for the supplier. We derive procedures for solving the optimal prices and show that the optimal pricing sequence is decreasing in time. We also show that the optimal prices are increasing in the backorder cost when the cumulative demand functions have increasing generalized failure rates.

Multiproduct price optimization under the multilevel nested logit model

We study the multiproduct price optimization problem under the multilevel nested logit model, which includes the multinomial logit and the two-level nested logit models as special cases. When the price sensitivities are identical within each primary nest, that is, within each nest at level 1, we prove that the profit function is concave with respect to the market share variables. We proceed to show that the markup, defined as price minus cost, is constant across products within each primary nest, and that the adjusted markup, defined as price minus cost minus the reciprocal of the product between the scale parameter of the root nest and the price-sensitivity parameter of the primary nest, is constant across primary nests at optimality. This allows us to reduce this multidimensional pricing problem to an equivalent single-variable maximization problem involving a unimodal function. Based on these findings, we investigate the oligopolistic game and characterize the Nash equilibrium. We also develop a dimension reduction technique which can simplify price optimization problems with flexible price-sensitivity structures.

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.

Optimal Pricing for Submodular Valuations with Bounded Curvature

The optimal pricing problem is a fundamental problem that arises in combinatorial auctions. Suppose that there is one seller who has indivisible items and multiple buyers who want to purchase a combination of the items. The seller wants to sell his items for the highest possible prices, and each buyer wants to maximize his utility (i.e., valuation minus payment) as long as his payment does not exceed his budget. The optimal pricing problem seeks a price of each item and an assignment of items to buyers such that every buyer achieves the maximum utility under the prices. The goal of the problem is to maximize the total payment from buyers. In this paper, we consider the case that the valuations are submodular. We show that the problem is computationally hard even if there exists only one buyer. Then we propose approximation algorithms for the unlimited budget case. We also extend the algorithm for the limited budget case when there exists one buyer and multiple buyers collaborate with each other.

Capacitated assortment and price optimization for customers with disjoint consideration sets

We study the capacitated assortment and price optimization problem for customers with disjoint consideration sets. The objective is to find the revenue maximizing set of products and their prices subject to a capacity constraint on the total display space of the offered products. We formulate the problem as a mathematical program and demonstrate its NP-hardness. We propose a fully polynomial-time approximation solution scheme and show that when the weights of the products are identical, our approach yields the optimal solution.

Exact Optimal Pricing under the Multinomial Logit Choice Model with Network Effects

We obtain exact semi-explicit solutions to the optimal pricing problem for the multinomial logit (MNL) consumer choice model with network effects, through a novel conditioning argument and the use of the Lambert W function. Then we manage to characterize the exact optimal solution in the presence of negative, positive and weak or positive and strong network effects, and also in the setting of heterogeneous market parameters. Furthermore, we provide a semi-explicit expression of equilibrium sales quantities in terms of equilibrium prices, based on which we establish sufficient conditions for the uniqueness of market equilibria. In addition, we provide an exact semi-explicit expression for the steady-state choice probability, based on which we discuss the uniqueness of the steady-state choice probability.