Optimal Product Pricing Research Articles

Contributors

Contributors

TECHNICAL NOTE—Optimal Product Pricing and Component Production Quantities for an Assembly System Under Supply Uncertainty

We consider an assemble-to-order system where one of the components faces uncertainty in the supply process in which the actual available quantity is equal to some random fraction of the production quantity. Demand is assumed to be price-dependent. We analyze how the supply uncertainty of one component affects the product pricing and production quantities of all the components under the assembly structure. We show that it is profitable for the firm to assemble the product only if the product price exceeds a certain threshold. This price threshold increases as the unit cost of each component or the degree of variability of the supply reliability distribution increases, but it is independent of the underlying demand function and demand distribution. Also, the optimal product price decreases as supply uncertainty decreases. We further show that under deterministic demand, the components can be managed independently such that the production quantity or unit cost of the component with supply uncertainty does not affect the optimal production quantity of the other components, as long as it is profitable to assemble the product. However, when demand is stochastic, the optimal production quantity of each component depends on the supply reliability distribution as well as the unit costs of the other components. For a fixed product price, the optimal production quantities of the components are smaller when the unit product price is low, and they are higher when the unit product price is high as compared to the case with no supply uncertainty.

The Value of Flexibility in Make-to-Order Systems: The Effect of Demand Correlation

We consider a manufacturer of mass-customized modular products who orders components under demand uncertainty, and sets prices, produces to order, and trades excess components in a secondary market after this uncertainty is resolved. The sequence of events reflects, in a parsimonious fashion, the considerable reduction in demand uncertainty between the procurement stage and the selling season, typical of industries with long supply lead times and short product life cycles. We prove that, in contrast to conventional wisdom, the value of production flexibility and expected profit increase with demand correlation if, and only if, commonality between the corresponding products does not exceed a threshold. We also prove that the value of flexibility and expected profit may each increase or decrease with demand variability, depending on demand correlations and component commonalities across the entire product line. Finally, we prove that when demand shocks are independent, the optimal product prices are positively correlated if, and only if, the degree of commonality between the corresponding products exceeds a threshold.

Optimal price and pro rata decisions for combined warranty policies with different repair options

A repairable product under a non-renewing combined warranty policy that is subject to a displaced log-linear demand function of the product's price and pro rata period length is considered. Expressions for the manufacturer's long-run average profit per unit time under replacement, minimal and general repair options are obtained. In addition, expressions for the stationary points and second-order conditions of the profit function are presented. Numerical illustrations that demonstrate optimal product pricing, pro rata length determination, and repair option selection to maximize the manufacturers, profit are given.

Optimal reliability, warranty and price for new products

The success of a new product depends on both engineering decisions (product reliability) and marketing decisions (price, warranty). A higher reliability results in a higher manufacturing cost and higher sale price. Consumers are willing to pay a higher price only if they can be assured about product reliability. Product warranty is one such tool to signal reliability with a longer warranty period indicating better reliability. Better warranty terms result in increased sales and also higher expected warranty servicing costs. Warranty costs are reduced by improvements in product reliability. Learning effects result in the unit manufacturing cost decreasing with total sales volume and this in turn impacts on the sale price. As such, reliability, price and warranty decisions need to be considered jointly. The paper develops a model to determine the optimal product reliability, price and warranty strategy that achieve the biggest total integrated profit for a general repairable product sold under a free replacement-repair warranty strategy in a market and looks at two scenarios for the pricing and warranty of the product. The model assumes that the sale rate increases as the warranty period increases and decreases as the price increases. The maximum principle method is used to obtain optimal solutions for dynamic price and warranty situations. Finally, numerical examples are given to illustrate the proposed model.

Optimal Control of High-Volume Assemble-to-Order Systems with Delay Constraints

We consider an assemble-to-order system with a high volume of prospective customers arriving per unit time. A companion paper established that with optimal product prices, component production capacity, and sequencing of orders for assembly, the system can be approximated by a diffusion process with dimension equal to the number of components (rather than the number of components plus the number of products). This state space collapse allows us to incorporate product-specific delay constraints and dynamic control of component inventory. First, we choose product prices, component production capacities, and propose a discrete review policy for expediting components and sequencing orders for assembly that satisfies delay constraints perfectly, and is asymptotically optimal when expediting is very costly. We also propose a simpler version of this policy, based on the backlog of demand for each component, that asymptotically complies with delay guarantees and achieves the same limiting expected discounted profit. Second, we numerically solve an approximating diffusion control problem for systems with both expediting and salvaging of components, and highlight useful structural properties.

Optimal pricing and ordering policies for perishable commodities

In the pricing problem considered, a seller must determine the price for several units of a perishable or seasonal product to be sold for a limited period of time. The list price should be posted in advance of the sale and is not negotiable with potential buyers. Any product not sold by the end of the sales period will be disposed at a lower price. Assuming that the customer's demand is represented as a negative binomial distribution, we determine the optimal product price based on the demand rate, buyers' preferences, and length of the sales period. Because the seller's average revenue decreases as the number of items for sale increases, we also consider the optimal-order quantity that maximizes the seller's total expected profit. For the case where the seller can divide the sales period into several short periods, we finally propose a multi-period pricing model.

Note—A Numerical Approach to Deriving Long-Run Equilibrium Solutions in Spatial Positioning Models

Choi, DeSarbo and Harker (1990) have recently proposed a numerical methodology for optimal product positioning and pricing under the assumption that the incumbents react only with price changes in the short run. The current note extends this methodology to include incumbents' long-run strategy of competitive product repositioning, in addition to price reactions. This note also introduces the notion of product relocation cost to the equilibrium analysis, and applies the concept of (full and partial) adjustment processes toward an equilibrium to model various strategic behavior of firms. This methodology is illustrated using an existing set of data collected for a major telecommunication equipment manufacturer.

A controlled birth and death process model of optimal product pricing under stochastically changing demand

We consider the problem of product pricing when the firm's market share is changing stochastically according to a birth and death process. The current market share together with the price prevailing determine the current rate of profit made as well as the birth and death rates. The optimal pricing policy must balance the immediate advantage of setting a high price in terms of increased current profit against the disadvantage in terms of a possible erosion of the future market share. We formulate a continuous-time Markov decision model and analyse it using a recent technique developed by Lippman [6] for optimization of exponential queueing systems. The optimal pricing policy is characterized as having a sort of monotonicity property. We also analyse the dependence of the optimal policy on the problem parameters and indicate further extensions of the model.

A controlled birth and death process model of optimal product pricing under stochastically changing demand

We consider the problem of product pricing when the firm's market share is changing stochastically according to a birth and death process. The current market share together with the price prevailing determine the current rate of profit made as well as the birth and death rates. The optimal pricing policy must balance the immediate advantage of setting a high price in terms of increased current profit against the disadvantage in terms of a possible erosion of the future market share. We formulate a continuous-time Markov decision model and analyse it using a recent technique developed by Lippman [6] for optimization of exponential queueing systems. The optimal pricing policy is characterized as having a sort of monotonicity property. We also analyse the dependence of the optimal policy on the problem parameters and indicate further extensions of the model.