Optimal Pricing Problem Research Articles

Research on Vertical Cooperation and Pricing Strategy of Electric Vehicle Supply Chain

To determine a vertical cooperation strategy and address the optimal pricing problem of the electric vehicle (EV) supply chain, a supply chain system consisting of two competing EV manufacturers (M1 and M2) and a battery supplier is studied. Firstly, three cooperation strategy models were constructed for the battery supplier and the EV manufacturers, namely: Strategy N (neither the battery supplier nor the two manufacturers cooperate with each other); Strategy I (M1 cooperates with the battery supplier); and Strategy II (M2 cooperates with the battery supplier). Then, the Stackelberg solution method was used to obtain the optimal equilibrium decisions under the three strategic models. Finally, the effect of the preference coefficient of consumers for leasing EVs per unit on the optimal equilibrium decision was analyzed. We found that: (1) The wholesale price of batteries provided by the battery supplier to M1 is always greater than to M2. (2) Strategies I and II prompt M1 and M2 to reduce the unit and fixed rental prices of EVs to some extent, while intensifying the competition between the two manufacturers in terms of EV lease prices. (3) When the consumer preference coefficient (θ) for leasing EVs per unit provided by manufacturer M1 is relatively small, the cooperation alliance S2 and the supply chain achieve the maximum profit under Strategy II; however, while θ is large, M1, cooperative alliance S1, and the entire supply chain could benefit the most under Strategy I.

Price Optimization for a Multistage Choice Model

Considering the real-world situations where a customer’s purchase choices in previous stages can influence the prices she encounters in subsequent stages, this research examines the multiproduct price optimization problem under a multistage choice model. Particularly, the seller commits to a multistage pricing policy and determines product prices based on the customer’s purchase history, and the customer makes purchase decisions such that the total expected utility is maximized. We show that the pricing problem has a unique optimal solution under some mild conditions and the optimal solution satisfies a modified equal adjusted markup property. Based on the property, the problem can be solved efficiently by reducing it to a single-dimensional search problem. Moreover, the optimal pricing policy has an important property, namely, the product with a higher adjusted markup in earlier stages should always lead to lower prices in subsequent stages. We also show that compared to customers who are myopic, the seller should offer higher first-stage prices and lower second-stage prices to forward-looking customers, which will lead to a higher profit. Numerical analyses are also conducted to demonstrate the above results.

Collaborative Vehicle-to-Grid Operations in Frequency Regulation Markets

Problem definition: We study the operations of electric vehicles (EVs) providing frequency regulation services to the electric grid in vehicle-to-grid (V2G) systems. In particular, individually owned EVs collaboratively bid in the regulation market, coordinated by a platform that operates the network of charging equipment. We study how the platform determines optimal pricing incentives for drivers to plug in their EVs, accounting for heterogeneous driving schedules. Methodology/results: We model the platform’s pricing optimization problem as a bilevel program: At the upper level, the platform determines hourly rebates for EV owners to plug in their EVs and capacity bids in the regulation market; at the lower level, individual travelers optimize their travel and charging schedules in response to pricing incentives. To account for uncertainties and heterogeneity in regulation market prices and travel patterns, we adopt distributionally robust optimization techniques to formulate the problem as a mixed-integer second-order cone program. We conduct a computational study based on the California Household Travel Survey data set and actual frequency regulation prices. Our results show that the ability to offer time-varying rebates and install workplace chargers can significantly improve the V2G platform’s expected profits. Managerial implications: As EV adoption progresses past the nascent stage, V2G business models become more viable. Successful implementation of V2G provides economic incentive for switching to EVs, potentially helps sustain adoption growth, and complements the growth of renewable power by helping stabilize the grid. Our findings shed light on the design of driver incentives for V2G systems. Funding: R. Tang acknowledges support from the National Natural Science Foundation of China [Grant 72201206]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2022.0133 .

Anticipated Wait and Its Effects on Consumer Choice, Pricing, and Assortment Management

Problem definition: We investigate the effects of waiting time, mainly due to production in a make-to-batch-order (MTBO) system, on consumer choice behavior, pricing, assortment, and model estimation. In an MTBO system, the seller/manufacturer first collects orders placed within a certain period of time into a batch and then starts the production process. After the production of all orders in a batch are complete, the products are then shipped and delivered to individual consumers. Because of batch production and shipping, the disutility of the waiting time exhibits negative externality. Methodology/results: We adopt the widely used multinomial logit (MNL) model as a starting point and incorporate the anticipated wait into consumers’ decision making. The derived model, referred to as the MNL with wait model, is a solution of the rational expectation equilibrium and is capable of capturing the effects of negative externality induced by anticipated wait that may change the substitution patterns dramatically. We characterize the multiproduct price optimization problem under the MNL with wait model by establishing a one-to-one mapping between the price vector and the choice probability vector. We find that firms tend to charge higher prices for time-consuming items and charge lower prices for time-saving items compared with the optimal prices under the standard MNL model. In addition to price competition, we also study the Cournot-type competition, in which the decision is the choice probability for each firm and establish the existence of a Nash equilibrium. For assortment optimization, we identify mild conditions under which the optimality of revenue-ordered assortments still holds. However, the assortment problem under the MNL with wait model is generally NP-hard, so we develop approximation algorithms with performance guarantees and provide an easy-to-compute tight upper bound. Moreover, we develop an efficient maximum likelihood-based algorithm for model calibration and further conduct numerical studies to showcase the importance of incorporating disutility due to wait in estimation, pricing, and assortment planning problems. Managerial implications: The MNL with wait model can increase prediction accuracy for consumers’ choice behavior especially when they are aware of the potential wait. Failure to take into account the effects of anticipated wait in firms’ decision making may lead to substantial losses. Funding: The research of C. Ke is supported in part by the National Natural Science Foundation of China [Grant 72101113]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2020.0346 .

Joint optimal pricing strategy of shared autonomous vehicles and road congestion pricing: A regional accessibility perspective

Current studies on shared autonomous vehicles (SAVs) and road congestion pricing (RCP) mainly focus on their independent effect on mobility, and rarely consider the joint impact of the two policies on accessibility. In light of this incentive, this paper establishes a bi-level programming model with multi-objective functions and multi-decision variables to solve the joint optimal pricing problem of SAVs and RCP considering accessibility. Because of the complex relationship between the decision variables and the objective functions, the proposed bi-level programming model belongs to a black-box problem. A machine learning algorithm based on multi-objective Bayesian optimization was developed to solve the proposed model, by taking Jiangyin City as an empirical case study. The results show that the combined implementation of SAVs and RCP plays an essential role in enhancing the performance of land-use and transportation systems. However, this advantageous outcome relies on an appropriately appropriate pricing strategy. Failing to account for the influence of the pricing strategy on implementation efficacy, SAVs and RCP could result in a 22.2 % and 5.0 % degradation in regional accessibility and total flow time, respectively. In contrast, the joint optimal pricing strategy of SAVs and RCP can not only mitigate traffic congestion and improve transportation network efficiency but also optimize urban spatial distribution and promote compact urban development, ultimately achieving a 14.0 % improvement in regional accessibility and 18.5 % reduction in total flow time. Sensitivity analysis reveals that with the increase in SAV market penetration rate, regional accessibility sees a steady rise. Conversely, the total flow time initially escalates before experiencing a decline. The method proposed in this paper serves not only in determining the joint optimal pricing strategy of SAVs and RCP to enhance accessibility, but also in analyzing and optimizing the accessibility impact of other land-use and transportation policies.

Distance traveled, transit design and optimal pricing

Should transit prices vary with distance? And if so, how? Two arguments that point in opposing directions have been presented in the literature: long distance users induce larger operators’ costs while short distance users induce larger in-vehicle times. Here we formulate a stylized model of a corridor where these and other elements are taken into account to approach the optimal strategic design (frequency, vehicle size, and route length) and its associated optimal pricing problem considering identical users that travel different distances. A crucial element is the consideration of the crowding caused by passengers entering the system at different points along the corridor, affecting the value of in-vehicle time which increases with the load factor through a crowding factor that, ultimately, is the single element that triggers optimal prices that increase with distance at a decreasing rate.

Prescriptive price optimization using optimal regression trees

This paper is concerned with prescriptive price optimization, which integrates machine learning models into price optimization to maximize future revenues or profits of multiple items. The prescriptive price optimization requires accurate demand forecasting models because the prediction accuracy of these models has a direct impact on price optimization aimed at increasing revenues and profits. The goal of this paper is to establish a novel framework of prescriptive price optimization using optimal regression trees, which can achieve high prediction accuracy without losing interpretability by means of mixed-integer optimization (MIO) techniques. We use the optimal regression trees for demand forecasting and then formulate the associated price optimization problem as a mixed-integer linear optimization (MILO) problem. We also develop a scalable heuristic algorithm based on the randomized coordinate ascent for efficient price optimization. Simulation results demonstrate the effectiveness of our method for price optimization and the computational efficiency of the heuristic algorithm.

Optimal instant discounts of multiple ride options at a ride-hailing aggregator

Recently, ride-hailing aggregators have emerged to help passengers find the best ride options. It aggregates the results from different service providers, allowing sorting and filtering all in a single app. It is not unusual for an aggregator to offer instant discounts on the prices returned from the service providers to increase its Gross Transaction Value (GTV) or Total Transaction Volume (TTV). In this research, we study this optimal instant discount problem, which can be viewed as a new variant of the classical multi-product price optimization problem. We first use the Nested Logit model to predict the probability a passenger would complete the trip with each ride option. We then formulate the instant discount problem as a nonlinear optimization model with a budget constraint and a group of discount bound constraints. To solve this model, we construct a surrogate relaxation formulation with strong duality. We develop a Lagrangian-dual-based approach to decompose this problem into a series of subproblems, and then design heuristic methods to give feasible solutions. For both cases when the GTV or the TTV is maximized, we quantify the optimality gap, give its asymptotic properties, and establish conditions under which it becomes zero and tight. Finally, we use real data from Meituan, a leading ride-hailing aggregator in China, to validate the proposed approach. Results show that compared to the baseline methods, we can improve the GTV by 1.293%, improve the TTV by 0.475%, and decrease the magnitude of the optimality gap.

On the continuity of outcomes in a monopoly market

A monopolist with a constant marginal cost faces an arbitrary nondecreasing and upper-semicontinuous demand function on R+ that takes a value in {0,1} outside of a fixed compact interval. This paper derives topological properties of outcomes induced by this monopolist’s optimal pricing problem. Specifically, the monopolist’s optimal profit is continuous in the marginal cost and the demand (under the weak-* topology); the optimal price and output correspondences are upper-hemicontinuous in the marginal cost and the demand; and the consumer surplus is upper (lower)-semicontinuous when the monopolist charges the lowest (highest) optimal price. These results further imply similar topological properties of outcomes in settings that feature either second-degree price discrimination or third-degree price discrimination.

Differential pricing in social networks with strategic consumers

This study analyzes a monopolistic seller's optimal differential pricing problem with strategic consumers connected in social networks. The consumers who purchase in the later period can get positive externalities from their friends who purchased in the early period but have to bear a utility discount for the delayed consumption. We first characterize consumers' strategic purchase decisions under general network structures. We then derive the optimal differential pricing strategies and demonstrate that different network structures lead to substantially different strategies. We find that when the intensity of the network externality effect is lower than a threshold and the influence matrix is symmetric, it is always optimal for the seller to conduct an increasing‐pricing strategy. However, when the network externality effect is strong, a decreasing‐pricing strategy may also be optimal. We further examine how the imbalance of influence, degree heterogeneity, and network topology impact the optimal pricing policy and profit. We find that when the intensity of network externality is relatively low, it is more profitable to sell products through many interconnected low‐influencer networks; however, when the network externality intensity is high, it is better to sell through a few high‐influencer networks. Finally, we show that the profit loss caused by uniform pricing strategies or by ignoring consumer network structures can be significant under certain conditions, thereby revealing the substantial value of differential pricing in social networks.

Pricing Decision and Cancellation Strategy Selection of ‘Hotel+OTA’ Based on Customer Cancellation Behavior

This paper is based on customer cancellation behavior and combines the Hotelling model framework to construct the demand for hotel direct sales channels and online travel agency (OTA) online channels. Then, the Stackelberg game model is used to analyze the optimal pricing problem between hotels and OTA with competitive relationships, and the optimal cancellation strategy for hotel direct channel is analyzed. We demonstrate that the hotel’s most profitable cancellation strategy depend on the loss cost of customer cancellation to the hotel，the cost of channel transfer for customers, and the OTA’s monopoly market segment. When the cost of loss to the hotel due to customer cancellation is relatively small, the hotel chooses a strategy where customers can cancel their reservations (i.e. Y strategy), and the pricing of hotel rooms increases as the cost of cancellation loss increases; otherwise, the hotel chooses a strategy where the customer cannot cancel the reservation (i.e. N strategy), and the pricing of hotel rooms decreases as the cost of cancellation loss increases. When the cost of channel transfer for customers is low, the hotel chooses N strategy; otherwise, the hotel chooses Y strategy. When the OTA’s monopoly market segment (i.e., the customers attracted solely to the OTA channel) is large, the hotel chooses N strategy; otherwise, the hotel chooses Y strategy. These findings can assist hotel managers in improving the design of their pricing and cancellation strategy.

Pricing Game Models of Hybrid Channel Supply Chain: A Strategic Consumer Behavior Perspective

The current sales model combining online and offline channels meets the diverse requirements of consumers. However, consumers’ inter-channel switching behavior and strategic behavior also pose significant challenges to pricing decisions in the hybrid channel. Using game theory and consumer utility theory, a retailer-driven pricing model is developed to study the optimal pricing problem for each channel in a mixed-channel supply chain considering the characteristics of channel competition and the waiting behavior of strategic consumers. Study results show there is a negative correlation between the proportion of strategic consumers and the optimal pricing and profit of each channel, and as the proportion of strategic consumers rises, the optimal pricing and profit of manufacturers and retailers all trend downward. Incorporating strategic consumers into the pricing model will assist the supply chain in elucidating the behavior of consumer heterogeneity during various decision-making periods and in making reasonable pricing decisions. Effective guiding strategies, such as pre-discount and purchase restrictions, can reduce the profit loss caused by strategic consumer behavior. The optimal combination of pre-announcement discount and strategic consumer ratio can generate the greatest profit for retailers and the supply chain.

A differential game model research on dynamic pricing and coordination of fresh agricultural products supply chain based on freshness

For the problem of optimal dynamic pricing and coordination of fresh agricultural products supply chain, the differential equations related to freshness and market needs are constructed, and the optimal control theory is used to solve the optimal strategy under centralized and decentralized decision-making, i.e., optimal selling price and optimal preservation input. The equilibrium results under the two scenarios are also compared. The comparison finds that the decentralized scenario leads to low overall supply chain profits, based on which a two-part pricing contract is proposed to coordinate the supply chain operating companies. The results show that freshness effectiveness directly impacts pricing, and the centralized scenario does not necessarily lead to high prices for fresh produce due to the ‘double marginal effect’. the two pricing contracts can successfully coordinate the supply chain. The freshness effectiveness increases the supply chain coordination ability of fresh produce suppliers, while operational inefficiencies, on the contrary, decrease the coordination ability.

Joint location and pricing optimization of self-service in urban logistics considering customers’ choice behavior

Self-services by widely-deployed stations allow customers to pick up their packages themselves and have great potential to reduce last-mile delivery cost by aggregating spatial demands. An efficient self-service design for the locations of self-service stations and the service price will help to shift customers from the home-delivery service to the cost-effective self-service, thus reducing the vehicle routing cost of the dedicated fleet. This study proposes a comprehensive modeling and optimization framework for the joint self-service station location and service price (SLSP) optimization problem considering customers’ choice behavior and vehicle routing in order to maximize the profit of the logistics service providers in the last-mile delivery system. Logit equilibrium and Wardrop equilibrium conditions are employed to characterize the delivery service choice and route choice behavior of customers. A convex optimization model is developed to capture the distribution of customers opting for home-delivery service and self-services at various locations, who aim to maximize their expected utilities. We then formulate the SLSP problem as a mathematical program with complementarity constraints. An active set algorithm is proposed to solve the model effectively. Numerical experiments based on the Nguyen-Dupuis and Sioux Falls networks are conducted to demonstrate the proposed framework and derive managerial insights.

Quantum game perspective on green product optimal pricing under emission reduction cooperation of dual-channel supply chain

PurposeFrom the quantum game perspective, this paper aims to study a green product optimal pricing problem of the dual-channel supply chain under the cooperation of the retailer and manufacturer to reduce carbon emissions.Design/methodology/approachThe decentralized and centralized decision-making optimal prices and profits are obtained by establishing the classical and quantum game models. Then the classical game and quantum game are compared.FindingsWhen the quantum entanglement is greater than 0, the selling prices of the quantum model are higher than the classical model. Through theoretical research and numerical analysis results, centralized decision-making is more economical and efficient than decentralized decision-making. Publicity and education on carbon emission reduction for consumers will help consumers accept carbon emission reduction products with slightly higher prices. When the emission reduction increases too fast, the cost of emission reduction will form a significant burden and affect the profits of manufacturers and supply chain systems.Originality/valueFrom the perspective of the quantum game, the author explores the optimal prices of green product and compares them with the classical game.

Finite-horizon joint inventory-pricing optimization with non-concave demand and lost sales

In this paper, we study a periodic-review joint inventory and pricing optimization problem in a stochastic price-sensitive demand framework with lost sales. We contribute to the existing literature from three aspects: generalizing the form of demand functions, developing the technique of proving concavity preservation, and extending the scope of optimality conditions. Specifically, we set the demand in our model as a general function. In order to obtain the optimality condition under general demands, we first introduce the concept of Price Elasticity of the Slope (PES) and transform the original PES condition between sales and profit in literature into a PES condition between sales and demand. Then, we obtain the lower bound of the PES difference between sales and profit by utilizing our transformed condition. With the non-negativity of the lower bound, we obtain two groups of easy-to-verify conditions which consist of the mean demand and the probability distribution of the demand. Under specific demand forms, the obtained conditions are reduced to constraints on the distribution of price-free uncertainty and the deterministic functions of price, respectively. Compared with the existing literature, our study therefore extends the scope of optimality conditions and includes non-concave demand functions when adapting a base-stock list-price (BSLP) policy in a joint inventory-pricing problem. Our conditions ensure that more types of products (e.g., the products of which the price elasticity of demand is constant) meet the optimality of the BSLP strategy, and more retailers can make pricing and replenishment decisions based on this strategy.

Optimum positioning of network platforms on a plane

The paper presents a study of equilibrium in a two-sided market for network platforms with cross externalities between buyers and sellers. The proposed model is a generalization of Armstrong's (2006) monopoly model for the case of a duopoly in a two-sided market for network platforms located on a plane. The paper solves the problem of optimal pricing and investigates the question of the optimal location of platforms in the market, provided that the heterogeneous utility of agents of both groups (buyers and sellers) is formed taking into account the Hotelling specification with the Manhattan metric.

An exact algorithm for the static pricing problem under discrete mixed logit demand

Price differentiation is a common strategy in many markets. In this paper, we study a static multiproduct price optimization problem with demand given by a discrete mixed multinomial logit model. By considering a mixed logit model that includes customer specific variables and parameters in the utility specification, our pricing problem reflects well the discrete choice models used in practice. To solve this pricing problem, we design an efficient iterative optimization algorithm that asymptotically converges to the optimal solution. To this end, a linear optimization (LO) problem is formulated, based on the trust-region approach, to find a “good” feasible solution and approximate the problem from below. A convex optimization problem is designed using a convexification technique to approximate the optimization problem from above. Then, using a branching method, we tighten the optimality gap. The effectiveness of our algorithm is illustrated on several cases, and compared against solvers and existing state-of-the-art methods in the literature.

Distributed Data Offloading in Ultra-Dense LEO Satellite Networks: A Stackelberg Mean-Field Game Approach

The low earth orbit (LEO) satellite network is regarded as a promising technology for delivering seamless service to remote areas such as rural areas. In this paper, we consider a dynamic data offloading problem in ultra-dense LEO satellites networks with large-scale ground users, where each ground user makes the distributed offloading decision based on its state information, the influence from other ground users, and the fees paid to satellites. To investigate the interaction problem between ground users and satellites, we formulate the problem as a Stackelberg game. Specifically, the satellites serve as the leaders, who decide the data service price at each time slot. The ground users, on the contrary, are the followers who decide the power control based on their states, the influence from other ground users and the fees paid to satellites. Since the influence is difficult to estimate due to the large number of ground users, we employ the mean field game algorithm to transform the influence from others and satellites into the mean field term, and reformulate the optimization problem as a Stackelberg mean field game (SMFG). Each ground user makes the data offloading decision by learning the future impact of the whole network distributively. For ground users, we solve the power control optimization problem by utilizing the G-prox primal-dual hybrid gradient (PDHG) algorithm, where the Fokker-Planck-Kolmogorov (FPK) equation is converted into a linear form via the Taylor expansion. For satellites, we address the service pricing optimization problem by using the adjoint algorithm. Finally, the numerical results demonstrate the effectiveness of the proposed algorithm.

Online Pricing Incentive to Sample Fresh Information

Today mobile users such as drivers are invited by content providers (e.g., Tripadvisor) to sample fresh information of diverse paths to control the age of information (AoI). However, selfish drivers prefer to travel through the shortest path instead of the others with extra costs in time and gas. To motivate drivers to route and sample diverse paths, this paper is the first to propose online pricing for a provider to economically reward drivers for diverse routing and control the actual AoI dynamics over time and spatial path domains. This online pricing optimization problem should be solved without knowing drivers' costs and even arrivals, and is intractable due to the curse of dimensionality in both time and space. If there is only one non-shortest path, we leverage the Markov decision process (MDP) techniques to analyze the problem. Accordingly, we design a linear-time algorithm for returning optimal online pricing, where a higher pricing reward is needed for a larger AoI. If there are a number of non-shortest paths, we prove that pricing one path at a time is optimal, yet it is not optimal to choose the path with the largest current AoI. Then we propose a new backward-clustered computation method and develop an approximation algorithm to alternate different paths to price over time. Perhaps surprisingly, our analysis of approximation ratio suggests that our algorithm's performance approaches closer to optimum given more paths.