Maximin approach in estimating of the goods order volume under condition of falling demand

The paper considers the problem of choosing the optimal strategy of behaviour trading company in the maxim in approach. Suppose that as a result of marketing research, it was determined that in some unknown moment of time there would be a noticeable change demand, the time points are known. It is assumed that the trading company uses the following wholesale order schemegoods. All ordered goods are divided into two parts, with the first batch goods arrive immediately, and it must be sold within a certain period $[0, T_{1}]$. The second batch of goods is shipped at time $T$, however, the interval of time $[T_{1}, T]$ the second batch of goods are sold at a discount and a full not implemented. The times $T_{1}$ and $T$ are was chosen by the trading company from the condition of maximizing income. The need to consider such a scheme wholesale order due to the fact that, firstly, the warehouses of a trading company have limited volume and cannot accommodate the entire volume of goods ordered and secondly, the manufacturer cannot offer all the ordered batch goods, since not all goods can be produced in the initial (zero) point in time when order is made. At time $T_{1}$ the trading company will fully sell the first batch of goods and receive funds, part of which she will pay to the manufacturer company. At time $T$ there is a completion of the full realization of all purchased goods. Selection of moments time $T_{1}$ and $T$ allows for trade company to determine the volume of the first batch of ordered goods and the total amount of all goods ordered from the manufacturing company. In work, a mathematical model is suggested that allows making the best choice order strategy for a trading company under the condition of falling demand at some unknown time from the interval. It is assumed that the interval is known. The minimum function with compelled constraints is investigated. Under some assumptions on the objective function and multi-valued mapping, the form of the derivative in directions is found.