Background

Pricing has been recognized as a significant tool used in the profit maximization of firms. Whether it be applied to areas in Revenue Management or Supply Chain Management, it is used in the daily operations of industries to manipulate demand, and to regulate the production and distribution of goods and services. However, many decision-making models use either incomplete demand functions which are defined only on a restricted domain, or functions that do not reflect market reality.

A demand function (DF) is an approximation of the demand-price relation in the real market. There are many ways to define a demand function. In microeconomics, a DF can be derived from the first-order conditions of the maximization of a utility function. That is, consumers choose to spend their money so as to maximize their utility subject to budget constraints. The reader can refer to http://en.wikipedia.org/wiki/Utility_maximization_problem for more details. Though theoretically justified, utility maximization (UM) may not be practical enough to model the real market. It can generate a DF only when a closed-form solution can be obtained. Thus in the literature, we have only seen a small special class of DFs derived from UM. They include Linear DFs and the Cobb-Douglas DFs.

Another way of determining a DF is through data-fitting. That is, some simple and generic functions can be used to fit the data observed from the markets. For example, linear functions are simple and thus commonly used in practice. However, as linear functions inevitably become negative at sufficiently high prices, they are not able to fit demand data corresponding to these high prices. Other commonly considered functions include Logit and CES functions. Though they can be used to approximate demands on all non-negative prices, they may not reflect real market behavior. For example, it follows from these functions that the demand for a product vanishes only when its price reaches an infinitely high value.

To correct the above-mentioned problems, Soon, Zhao and Zhang proposed the following approach to generate demand functions: a decision maker can first construct a reasonable function (linear or non-linear) in a region around the average prices of the products (say, in a set Omega), and then extend to outside Omega via the solution of a Complementarity problem, generating a Complementarity-Constrained Demand Function (CCDF) as a result. In that same work, a pricing model incorporating a piecewise defined CCDF was introduced. Due to the constraints governing the CCDF, the pricing model took the form of a mathematical program with equilibrium constraints (MPEC). See http://journals.cambridge.org/repo_A61WjS2A for more details.

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