The very basic model to determine the optimal size of the delivery lot, is a model providing the so called Harris - Wilson formula.
\(\lambda\) - annual consumption of the goods in physical units; \(c_N\) - unit price of the good; \(c_S\) - the annual storage costs of one piece of good (\(c_s = c_N r\) - \(r\) is interest rate); \(c_D\) - the costs connected with one delivery of the goods;
\(Q\) is a seize of one delivery lot; \(Q/2\) average level of the stock;
\(\lambda/Q\) - number of deliveries within the one year period;
\[C(Q) = \frac{Q}{2}c_S + \frac{\lambda}{Q}c_N\] To find the optimal value of \(Q\), we diferenciate the above given function to get
\[\frac{dC}{dQ}= \frac{c_S}{2} - \frac{\lambda c_N}{Q^2}\] If researching for the first order condition of the function extrema, we get
\[\frac{c_S}{2} - \frac{\lambda c_N}{Q^2} = 0 \quad \leftrightarrow \quad Q^* = \sqrt{\frac{2c_N}{c_S}}\] which is an optimal size of the delivery lot.