Modeling of the ordering process for piecewise-linear demand with saturation

This article discusses the behavior of trading companies when the demand function for a product is a piecewise linear function with some pre-defined level of saturation. As a result of special marketing research the demand function is determined. It is assumed that the following scheme is used. The ordered product is divided into two parts. The first batch will arrive immediately and will be sold within a certain period of time up to $T_1$. The second batch of goods will be delivered only at time $T$ and this batch will be sold with reduction in price as follows from deficit of the goods. The consideration of a such a scheme for the sale stems from the fact that, first, commercial enterprises have limited capacity and can not accommodate the entire volume of products ordered, and secondly, the manufacturer may not offer the entire ordered quantity at the initial time. An important feature of this problem is selection of parameters which are subsequently optimized. For trading firms the moment time $T_1$ has very high importance because trading companies have sold the first batch and received money. After this moment the trading firms will pay to the manufacturers, and manufacturers can to continue the activities. The time moment $ T_1$ is important for trading firms, as it will allow them to fulfill various financial obligations associated with the payment of loans and other expenses. Time $T$ is also important, so this will mean the successful completion in full of selling all these goods. Refs 16. Fig. 1.