Optimal prices

Danzon and Liu22 show that the short-term effect of RP is to produce a kink in the demand curve at the point corresponding to the RP, assuming that all doctors have perfect information on prices. The kinked demand model put forward by these authors to explain the behaviour of prices subject to RP predicts that it will never be optimal to fix a price below RP, the optimal pricing response being EFP = RP (see box above). 

Utilization-optimal prices for these businesses can be systematically identified using simulation-based optimization of prices with contributions from Kurus (2006). 

Anjos MF, Cheng RCH, Currie CSM (2005) Optimal pricing policies for perishable products. European Journal of Operational Research 166 (1) 246-254... 

Product/market strategy Within each market segment, what price level gives your product the optimal price/benefit position relative to competitors ... 

The grading of timber should be viewed as part of a marketing strategy, designed to ensure that timber buyers obtain the quality of timber appropriate for their needs and timber sellers receive an optimal price for their product. Unfortunately, grading suffers from conflicting objectives and can be best described as an attempt to bring some order out of what would otherwise be a chaotic situation. 

In many real-hfe situations, retailers deal with perishable products such as fresh fruits, food-stuffs, and vegetables. The inventory of these products is depleted not only by demand but also by deterioration. Yang [7] has developed the model to determine an optimal pricing and an ordering policy for deteriorating items with quantity discount, which is offered by the vendor. His model assumed that the... 

Pasternack,B. (1985). Optimal pricing and returns policies for perishable commodities. Marketing Science, 4, 166-176. 

The second model which must be parameterized is the time-series forecast model. This is equivalent to determining the location (placement) of the demand curve for each customer segment for each product. The placement of the demand curve is of critical importance, since it represents the potential size of the market and whether or not supply constraints will bind. If demand from a market segment is large relative to supply, then the optimal prices are adjusted to reflect the opportunity cost associated with limited supply. 

In addition, further research should be done on the impact of price discrimination on the inventory in a supply chain. For example, how will downstream companies respond to different price discrimination policies Is the seller better off implementing a static pricing policy, or should he dynamically determine the optimal prices ... 

In fact, the software does not incorporate any constraints, with the possible exception of restrictions imposed on the winning prices. For example, the supplier may incorporate a constraint requiring the price(s) or win probability to be above/below a certain threshold level. While supply constraints are not explicity represented in NTP, the effects of the prospective supply constraints can be incorporated as opportunity costs and factored into the optimal price determination process. 

Gumani, H., K. Karlapalem. 2001. Optimal Pricing Strategies for Internet-Based Software Dissemination. Journal of the Operational Research Society 52(1), 64-70. 

Lieber, Z., A. Bamea. 1977. Dynamic Optimal Pricing to Deter Entry under Constrained Supply. Operations Research 25(4), 696-705. 

Mills [107] concentrates on showing the effect of uncertainty on a monopolist s short-run pricing policy under the assumption of additive demand. In his model, the demand is specified as D p, s) = y p) -f- e,where y p) is a decreasing function of price p and e is a random variable defined within some range. In particular, he shows that the optimal price under stochastic demand is always no greater than the optimal price under the assumption of deterministic demand, called the riskless price. Lau and Lau [90] and Polatoglu [122] both study different cases of demand process for linear demand case where y p) — a bp, where a, 6 > 0. 

On the other hand, Karlin and Carr [76] used the following multiplicative case where the demand D p, s) = y p)e. They show that the optimal price under stochastic demand is always no smaller than the riskless price, which is opposite of the corresponding relationship found to be true by Mills [107] for the additive demand case. 

Petruzzi and Dada [116] provide a unified framework to reconcile this apparent contradiction by introducing the notion of a base price and demonstrating that the optimal price can be interpreted as the base price plus a premium. 

The (p,p) policy outlined in [151], states that, for each period t, if inventory is above yt no production should be made, and the price should be set to the point pt (/) on the optimal price trajectory based on a current inventory level of I. If inventory is below yu production should be made to bring the inventory level to yu and the price should be set to Pt yt)-... 

Since the conditions in general are difficult to verify, Thowsen discusses special cases under which the (y,p) policy is optimal for the problem considered. For instance, under a linear expected demand curve, linear stockout costs, convex holding costs, and a demand distribution that is a PF2 distribution, the iViP) policy is optimal. Furthermore, if excess demand is backlogged, the demand curve is concave and the revenue is collected a fixed number of periods after the time orders are placed, then no assumptions are needed on the cost and demand distribution for optimality of the critical number policy, and for this case the decision on price and quantity decisions can be made separately. Thowsen also shows that if negative demand is disallowed, the optimal price will be a decreasing function of increasing initial inventory. 

Using the structure of the problem, Federgruen and Heching show that expected profit-to-go is concave, and that the optimal price is nonincreasing as a function of initial inventory. Further, similar to Thowsen, they characterize the... 

In Biller et al [18], the authors analyze a pricing and production problem where (in extensions), multiple products may share limited production capacity. When the demand for products is independent and revenue curves are concave, the authors show that an application of the greedy algorithm provides the optimal pricing and production decisions. 

Work on problems with multiple classes owes much to a few seminal papers that studied optimal pricing of homogeneous customers arriving to a service facility, see for example [44], [104], and [144]. The focus is generally on determining a price that balances the delay cost to the customer and the utilization of the server. The long-term service rate is also considered in many of these papers, which is effectively a capacity decision for the facility. 

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