\section*{Task 1 - V11231}

In a large state-owned enterprise in the electrical industry, 12,000 units of a certain semi-finished product are annually procured from a supplier at a price of 1.00 DM per unit. The orders were previously placed twice a year, on January 1 and July 1.

The administrative costs for each order (issuing and sending the order, monitoring the delivery date, invoice review, booking, etc.) amount to 30.00 DM. The storage costs (space costs, administration, "shrinkage" due to spoilage and damage, etc.) amount to 20% of the value of the average inventory. The costs for ordering and storage were therefore:

2 orders ... 60.00 DM

Storage costs 20% of the average inventory (3000 units), i.e., 20% of 3000.00 DM, which is 600.00 DM

Total 660.00 DM

In a production meeting, it is suggested to reduce costs by ordering the quantity needed for each quarter (3000 units) four times a year.

a) How high are the costs according to this suggestion?

b) For how many orders do the lowest costs occur? How high are the costs in this case?

Hint: The term "Deutsche Mark" was only changed to "Mark der Deutschen Notenbank" (MDN) in 1964 in East Germany, and subsequently to "Mark" in 1968.
}

Let \( \mathrm{x} \) be the number of orders per year and the annual total costs, consisting of ordering costs and holding costs for this semi-finished product,

\[
K_{G}=K_{B}+K_{L}=30 x+\frac{1200}{x}
\]

then the costs for a)

\[
K_{G}=K_{B}+K_{L}=30 \cdot 4+\frac{1200}{4}=420
\]

and the lowest total costs for b)

\[
\begin{gathered}
K_{G}^{\prime}=30-\frac{1200}{x^{2}} \\
30 x^{2}-1200=0 \\
x=[\sqrt{40}]=6
\end{gathered}
\]

By substituting \( \mathrm{x}=6 \), one indeed obtains the lowest value of 380. With \( x=5, x=7 \), the costs start to rise again.