\section*{Task 1 - 021221}

The founder of linear programming, Prof. Dr. L.W. Kantorovich, presents the following example:

In a company, the following are available for milling work:
a) 3 milling machines,
b) 3 milling machines with revolver head clamping device,
c) 1 automatic machine.

Two types of parts are to be produced in equal numbers. The production per working day for the machines listed above is as follows:
a) 10 pieces of type 1 or 20 pieces of type 2,
b) 20 pieces of type 1 or 30 pieces of type 2,
c) 30 pieces of type 1 or 80 pieces of type 2.

How many parts can be produced with these machines under the given conditions?
Let \(a\) denote the number of milling machines for the production of pieces of type 1, \(b\) the number of milling machines with revolver head clamping device for the production of pieces of type 1, and \(c\) the number of machines for the production of pieces of type 1.

Then, \(u=10 a+20 b+30 c\) pieces of type 1 and \(v=20(3-a)+30(3-b)+80(1-c)\) pieces of type 2 are produced per day.

Obviously, \(u \leq 120\), and \(v=0\) when \(u=120\).

For \(u=110\), \((a, b, c)=(2,3,1)\), i.e., \(v=20\).

For \(u=100\), \((a, b, c) \in\{(1,3,1),(3,2,1)\}\), i.e., \(v \in\{40,30\}\).

For \(u=90\), \((a, b, c) \in\{(0,3,1),(2,2,1),(3,3,0)\}\), i.e., \(v \in\{60,50,80\}\).

Therefore, a maximum of 80 pairs of pieces of both types can be produced.

Taken from [2]