Exact penalty functions in the problem of choosing the optimal wholesale order in the face of rapid fluctuations in demand

The current article discusses a different situation in the market, when there is a rush demand for a new product followed by a sharp drop in demand. The trading company uses the following scheme for the wholesale order of goods. The ordered product is divided into two parts, and the first batch of goods arrives immediately, it is sold over a certain period of time $[0, T_1]$. The second batch of goods is delivered at the time $T$, but at the time interval $[0, T]$. This batch is sold at a discount and is completely sold out. Time $T$ corresponds to the end of the sale of the entire product. The time points $T_1, T$ are selected by the trading firm from the condition of maximizing income. The need to consider such a wholesale order scheme is related to the fact that, firstly, the warehouses of the trading firms have limited capacity and cannot accommodate all the ordered goods, and secondly, a manufacturer may not offer the entire ordered batch of goods, since not all goods can be produced at the initial (zero) point of time immediately after receiving the order. At the time $T_1$, the trading company completely sells the first batch of goods and receives financial resources, part of which is paid to the manufacturer. At the moment of time $T$, the complete sale of all purchased goods is completed. The choice of time points $T_1$ and $T$ allow to determine the volume of the first batch of ordered goods and the total volume of product ordered from the manufacturer. In the article, a mathematical model is proposed that makes it possible to choose the optimal ordering strategy for a trading company in the conditions of excessive growth of demand for the new product in time $\tau_1,\tau_2$ at some unknown point in time and $\tau_*\in [\tau_3,\tau_4]$, and the subsequent sharp drop in demand in the period of time $[\tau_3,~\tau_4]$ due to the saturation of the market with a new product. Four possible variants of optimization problems are considered. A method of exact penalty function is suggested, which allows one to find their solutions.