Optimal Pricing Policy Research Articles

A Decentralized Production/Remanufacturing Inventory Model with Multiple Refurbished Components

This study aims to investigate the optimal inventory and pricing policies in a decentralized production/remanufacturing system with multiple refurbished components. A basic inventory problem with deterministic demand rate and return rate is considered. The manufacturer purchases recycled components from the collector, and sells remanufactured products as well as new products to customers. It is assumed that the customers tend to pay different prices for remanufactured products and new products. Both the manufacturer and the collector try to maximize their own profits. The equilibrium of the problem is formulated as a variational inequality. Numerical examples show that the manufacturer and the collector can change the retail prices and return-facilitating costs to influence the product demands and the quantity of returned end-of-use products, such that no surplus or shortage of any products and components will occur.

Secret versus public reserve price in an “outcry” English procurement auction: Experimental results

In a set of human experiments, we analyze the impact of the buyer׳s reserve price policy in an online open outcry reverse auction. We measure the impact of revealing or not the reserve price on the buyer׳s expected cost when suppliers may be risk-averse. Results confirm that the optimal public reserve price policy dominates for the buyer when the lowest cost supplier is weakly risk-averse but in contrast with theory, the secret reserve price policy does not dominate when he becomes very risk-averse. Analysis of strategic suppliers׳ bidding behavior shows, in conformity with theory, that they react to the secret reserve price and are more aggressive when they are more risk-averse. This explains why the optimal public reserve price does not perform better in this context but is equivalent to the secret one. However, strongly risk-averse bidders are less aggressive than predicted by theory. That is why the secret reserve price policy does not strictly dominate.

Pricing, inventory and production policies in a supply chain of pharmacological products with rework process: a game theoretic approach

In this paper, an economic production quantity model in a three levels supply chain including multiple non-competing suppliers, single manufacturer and multiple non-competing retailers for multiple products with rework process under integrated and non-integrated structures is developed. In this chain, suppliers ship raw material to the manufacturer and the manufacturer combines the fixed percentage of different kinds of raw material to produce the finished products and then delivers them to the retailers. The Stackelberg game is established among the members of the chain. Optimizing the total profit of the chain under both structures applying the optimal pricing, inventory and production policies is the purpose of the research. Eventually, a numerical example is presented for comparing the total profit of the integrated and non-integrated chains of proposed model and some sensitivity analyses are performed to study the effects of parameter changes on the decision variables and the chain profit.

Revenue management by sequential screening

Using a mechanism design approach, we consider a firm's optimal pricing policy when consumers are heterogeneous and learn their valuations at different times. We show that by offering a menu of advance-purchase contracts that differ in when, and for how much, the product can be returned, a firm can more easily price discriminate between privately-informed consumers. In particular, we show that screening on when the return option can be exercised increases firm profits, relative to screening on the size of the refund alone, only if the expected gains from trade are higher for consumers who learn later. We show that in some settings (mean-preserving spread) the firm can achieve the complete-information profits and analyze the optimal contract in other settings (first-order stochastic dominance) in which the first-best allocation is not always feasible.

The impact of China's carbon allowance allocation rules on the product prices and emission reduction behaviors of ETS-covered enterprises

It is an important task for China to allocate carbon emission allowance to realize its carbon reduction target and establish carbon trading market. China has designed several allocation rules within seven pilot regions. What influence those rules may cause is closely related with the enthusiasm of emission trading scheme (ETS) covered enterprises' participation in carbon market, and more importantly, with the mechanism design and sustainable development of carbon market. For this purpose, the multi-stage profit model is developed to analyze the ETS-covered enterprises' product prices and emission reduction behaviors under different allocation rules. The results show that, first, under the rules of grandfathering, self-declaration and auctioning, when deciding the optimal product price and optimal carbon emission reduction, those enterprises may focus on maximizing current stage profit; however, under the rule of benchmarking, those enterprises may care more about the impact of current decisions on the profit in next stage. Second, the optimal product price policy is positively correlated with the price of the same kind products, consumers' low-carbon awareness and government subsidy. Finally, along with the increase of carbon price, consumers' low-carbon awareness and government subsidy and the decrease of carbon emission cap, those enterprises tend to reduce carbon emissions.

Optimal dynamic pricing problem considering patient and impatient customers’ purchasing behaviour

We divide customers into two types according to their purchasing behaviour: one is impatient and the other is patient. An impatient customer will be immediately lost if his reservation price of the product is below the current price, while a patient customer will wait until the price drops below his reservation price and buy one item. The seller’s objective is to maximise his expected discounted overall revenue by dynamically pricing. We prove that the optimal pricing policy has a control limit structure, examine the impact of the proportion parameter on the seller’s expected discounted overall revenue and show that the limiting behaviour of the optimal pricing policy when there are little patient customers is the same as the optimal static pricing policy. We also show that the seller will make more profit if he partially knows the number of waiting customers by providing subscription service to customers. A set of numerical results illustrates the value of dynamic pricing and information with respect to problem parameters such as the arriving probability and the subscription probability.

Optimal online markdown and markup pricing policies with demand uncertainty

In this paper, we study the dynamic pricing problem with demand uncertainty from the perspective of online algorithms and competitive analysis which eliminate the need for both the functional relationship between price and demand and the customer arrival rate. Assuming customer's reservation price falls in a closed interval, we prove the upper bound of any online policy for the HBL problem where the reservation price of a customer showing up later is always no more than the reservation price of a customer showing up earlier. And an optimal online markdown pricing policy whose competitive ratio matches the upper bound is proposed. For the symmetrical problem with LBH manner, we prove that there is no deterministic online policy whose competitive ratio is larger than the ratio of the lowest reservation price and the highest reservation price. For a special LBH problem with enough potential demand, we propose an optimal online markup policy dealing with not only how to price but also how much item to limit for each price.

Optimal Pricing and Sequencing in a Make-to-Order System With Batch Demand

In this paper we consider the problem of jointly managing production sequencing and pricing in a two product manufacturing system where both products (type-1 and type-2) are made using the same resource. Type-1 product is made-to-order to fulfill contractual demand arriving according to a compound Poisson process. Type-2 product is also made-to-order but to fulfill demand arising in the open market where demand arrives according to a Poisson process that is a function of a price posted by the manufacturer. The objective of the manufacturer is to make production sequencing and pricing decisions that maximize the total expected discounted profit earned over an infinite time horizon. We characterize the optimal sequencing policy using a monotone threshold curve and the optimal pricing policy using monotone switching curves. We also conduct a sensitivity analysis to learn how the manufacturer's profitability and the optimal production sequencing and pricing policies are impacted by various problem parameters.

Determining the retailer's replenishment policy considering multiple capacitated suppliers and price-sensitive demand

This article presents a mixed integer nonlinear programming model to find the optimal selling price and replenishment policy for a particular type of product in a supply chain defined by a single retailer and multiple potential suppliers. Each supplier offers all-unit quantity discounts as an incentive mechanism. Multiple orders are allowed to be submitted to the selected suppliers during a repeating order cycle. The demand rate is considered to be not constant but dependent upon the selling price. The model provides the optimal number of orders and corresponding order quantities for the selected suppliers, and the optimal demand rate and selling price that maximize the total profit per time unit under suppliers’ capacity and quality constraints. In addition, we provide sufficient conditions under which there exists an optimal solution where the retailer only orders from one supplier. We also apply the Karush–Kuhn–Tucker conditions to investigate the impact of supplier's capacity on the optimal sourcing strategy. The results show that, there may exist a range of capacity values for the dominating supplier, where the retailer's optimal sourcing strategy is to consider multiple suppliers without fully utilizing the dominating supplier's capacity. A numerical example is presented to illustrate the proposed model.

Optimal policy instruments for externality-producing durable goods under present bias

When consumers exhibit present bias, the standard solution to market failures caused by externalities—Pigouvian pricing—is suboptimal. I investigate policies aimed at externalities for present-biased consumers. Optimal policy includes an instrument to correct the externality and an instrument to correct the present bias. Either instrument can be an incentive-based policy (e.g. a tax on fuel economy) or a command-and-control policy (e.g. a fuel economy mandate). Under consumer heterogeneity, a command-and-control policy may dominate an incentive-based policy. Calibrated to the US automobile market, simulation results suggest that the second-best gasoline tax is 3–30% higher than marginal external damages. The optimal price policy includes a gasoline tax set about equal to marginal external damages and a fuel economy tax that increases the price of an average non-hybrid car by about $550–$2200 relative to the price of an average hybrid car.

Partial Trade Credit Policy of Retailer in Economic Order Quantity Models for Deteriorating Items with Expiration Dates and Price Sensitive Demand

In a supplier-retailer-customer supply chain, a credit-worthy retailer frequently receives a permissible delay on the entire purchase amount without collateral deposits from his/her supplier (i.e., an up-stream full trade credit). By contrast, a retailer usually requests his/her credit-risk customers to pay a fraction of the purchase amount at the time of placing an order, and then grants a permissible delay on the remaining balance (i.e., a down-stream partial trade credit). Also, in selecting an item for use, the selling price of that item is one of the decisive factors to the customers. It is well known that the higher selling price of item decreases the demand rate of that item where the lesser price has the reverse effect. Hence, the demand rate of an item is dependent on the selling price of that item. In addition, many products such as fruits, vegetables, high-tech products, pharmaceuticals, and volatile liquids not only deteriorate continuously due to evaporation, obsolescence and spoilage but also have their expiration dates. However, only a few researchers take the expiration date of a deteriorating item into consideration. This paper proposes an economic order quantity model to allow for: (a) the strategy that supplier offers retailer a full trade credit policy whereas the retailer offers their customers a partial trade credit policy, (b) selling price dependent demand rate, (c) a profit maximization objective and (d) deteriorating items not only deteriorate continuously but also have their expiration dates. For the objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. An efficient algorithm is designed to determine the optimal pricing and inventory policies for the retailer. Finally, numerical examples are presented to illustrate the proposed model and the effect of key parameters on optimal solution is examined.

Dynamic pricing with time-dependent elasticities

Many stochastic dynamic sales applications are characterized by time-dependent price elasticities of demand. However, in general, such problems cannot be solved analytically. To determine smart pricing heuristics for general time-dependent dynamic pricing models, we solve a general class of deterministic dynamic pricing problems for perishable and durable goods. The continuous time model has several time-dependent parameters, for example, discount rate, marginal unit costs and price elasticity. We show how to derive the value function and optimal pricing policies. On the basis of the feedback solution to the deterministic model, we propose a method for constructing heuristics to be applied to general stochastic models. For the case of isoelastic demand, we analytically verify the excellent performance of this approach for both small and large inventory levels.

Dealing with Dynamic Agency

I study the implications of agency frictions for the pricing policy of institutional market makers. In a setting where a market maker cannot observe the actions of an employed trader, I derive the optimal compensation structure and pricing policy. The theory demonstrates that incentive contracting and the price for immediacy are inherently linked. When the trader’s compensation is optimally deferred according to order flow, market making efficiency is improved and the quoted spreads are minimized. In other words, optimizing trader compensation leads to a liquidity gain.

Optimal two-level piecewise-constant price discrimination for a storable perishable product

Price differentiation over time is an additional policy that firms might consider when determining prices for perishable products. The common policy of a fixed price regardless of freshness might result in leaving some expired inventory unsold. Price differentiation can impact the demand for perishable products, which declines as the expiration date approaches. We develop an optimisation model with the goal of evaluating the monetary effectiveness of the strategy of simultaneously combining price discrimination across heterogeneous consumers with price differentiation over time for perishable inventory under separable multiplicative demand factors of price and time. Necessary optimality equations are derived, and their solutions are proved to constitute a unique global optimal solution. It is proved that an optimal pricing policy is to implement price discrimination with respect to consumers’ sensitivity to freshness, while dynamically changing the price over time, starting with a lower price at the early stages of the product’s shelf life and increasing it at a later stage. The monetary benefit that the retailer and consumer can derive from the suggested pricing policy is evaluated by comparing the model to other models in which price discrimination or dynamic pricing are not implemented. A numerical example that illustrates the significance is introduced. From the analysis of a numerical illustration of the model, it is concluded that a dynamic price discrimination policy can be approximated by an identical-to-all dynamic pricing policy in order to maximise the retailer’s profit and thus, mitigate the retailer’s risk from failing in the process of implementing price discrimination.

Pricing and replenishment policies in dual-channel supply chain under continuous unit cost decrease

This paper explores pricing and replenishment policies for a high-tech product in a dual-channel supply chain that consists of a brick-and-mortar channel and an internet channel. The unit cost of the product decreases over its short life cycle. Assuming the manufacturer as the Stackelberg leader, the optimal pricing and replenishment policy is analysed mathematically. It is found that there is a severe price competition between the retail and online channel, and product compatibility has a significant impact on the pricing policy. In particular (i) customers’ higher retail channel preference above a threshold leads to non-coexistence of dual-channel, (ii) the dual-channel is non-profitable for product compatibility outside an interval and (iii) higher or lower retail price in comparison to online price is dependent on product compatibility. Also, the retailer’s higher setup cost may lead to non-existence of online channel. Finally, a profit sharing mechanism through wholesale price adjustment resolves channel conflict. A numerical example is illustrated to justify our proposed model.

Incentivizing Users for Balancing Bike Sharing Systems

Bike sharing systems have been recently adopted by a growing number of cities as a new means of transportation offering citizens a flexible, fast and green alternative for mobility. Users can pick up or drop off the bicycles at a station of their choice without prior notice or time planning. This increased flexibility comes with the challenge of unpredictable and fluctuating demand as well as irregular flow patterns of the bikes. As a result, these systems can incur imbalance problems such as the unavailability of bikes or parking docks at stations. In this light, operators deploy fleets of vehicles which re-distribute the bikes in order to guarantee a desirable service level. Can we engage the users themselves to solve the imbalance problem in bike sharing systems? In this paper, we address this question and present a crowdsourcing mechanism that incentivizes the users in the bike repositioning process by providing them with alternate choices to pick or return bikes in exchange for monetary incentives. We design the complete architecture of the incentives system which employs optimal pricing policies using the approach of regret minimization in online learning. We investigate the incentive compatibility of our mechanism and extensively evaluate it through simulations based on data collected via a survey study. Finally, we deployed the proposed system through a smartphone app among users of a large scale bike sharing system operated by a public transport company, and we provide results from this experimental deployment. To our knowledge, this is the first dynamic incentives system for bikes re-distribution ever deployed in a real-world bike sharing system.

Cooperative optimal pricing for stochastic access control in overlaid radio access networks

As radio access technology has been greatly developed nowadays, dynamic spectrum allocation (DSA), with which secondary users (SUs) are able to access the spectrum allocated to primary users becomes an important method for service providers to efficiently utilize limited wireless resources. Meanwhile, call admission control of SUs in an environment with multiple possible heterogenous wireless networks supporting DSA is also evolving. In this paper, we propose a pricing-based resource allocation scheme for multiple cooperative providers. According to this scheme, SUs are charged based on current system state by service provider to join one of several radio access networks (RANs). We integrate stochastic call admission control with dynamic pricing and formulate the problem of maximizing the expectation of system revenue in the long run as an infinite horizon average reward problem which can be solved by stochastic dynamic programming. Characteristics of the optimal pricing policy are analyzed based on series of computation results under various system conditions. Simulations carried out for a small network model show that a maximum profit can be achieved under our scheme and this maximum value varies with primary user arriving rates, primary user blocking punishments and channel capacities. Using our scheme, the system income could be improved by nearly 60 % in particular cases. Also, if each RAN's probability of admitting a SU is carefully chosen, the system profit can be maximized and primary user blocking count minimized.

Optimal piecewise-constant price under heterogeneous sensitivity to product freshness

When perishable products are priced uniformly, regardless of the amount of time remaining until expiration, consumers may gravitate towards fresher products, leaving some inventory unsold. This research considers dynamic pricing policies as well as replenishment policies in the context of perishable products with a fixed shelf life. Consumers are assumed to be heterogeneous in their sensitivity to freshness, i.e. their willingness to pay more for fresher products. We develop a model for identifying an optimal (profit-maximising) dynamic pricing policy and for evaluating the extent to which both the retailer and the consumer benefit from the implementation of a dynamic pricing policy as opposed to a static policy. The model assumes that the retailer is able to utilise knowledge regarding the heterogeneous characteristics of incoming customers (e.g. the retailer can gather specific information about customers’ historical purchases). Unexpectedly, it is proven that in an optimal pricing policy, the retailer should assign a lower price to fresher products and then raise the price as the products approach expiration. A numerical illustration shows that profits are strongly influenced by the volatility of consumer sensitivity to freshness; specifically, this variable has the potential to reduce optimal profits by up to 8%.

Dynamic Inventory and Pricing Policy in a Periodic-Review Inventory System with Finite Ordering Capacity and Price Adjustment Cost

We consider a dynamic inventory control and pricing optimization problem in a periodic-review inventory system with price adjustment cost. Each order occurs with a fixed ordering cost; the ordering quantity is capacitated. We consider a sequential decision problem, where the firm first chooses the ordering quantity and then the sale price to maximize the expected total discounted profit over the sale horizon. We show that the optimal inventory control is partially characterized by as, s′, ppolicy in four regions, and the optimal pricing policy is dependent on the inventory level after the replenishment decision. We present some numerical examples to explore the effects of various parameters on the optimal pricing and replenishment policy.

Supply-based dynamic Ramsey pricing: Avoiding water shortages

In many countries, current water-pricing policies are dictated by the sole objective of financial breaking even. This results in large withdrawals, which are not sustainable in the long run, hence not optimal. In this paper, we derive the optimal dynamic pricing policy, which targets efficient distribution while breaking even through a rebate scheme. Using data from Turkey, we estimate the demand for water by user groups. We carry out simulations to compare the effects of the current and optimal pricing policies on the frequency and severity of shortages. We find that, under the policy of break-even prices, the supplier runs into a shortage every 8 years. In contrast, if the prices were to set optimally, shortages would be practically nonexistent over the next century.