In an economy, according to the feeding program, each animal must receive at least 45 dkg of nutrient $A$, at least 60 dkg of nutrient $B$, and at least 5 dkg of nutrient $C$ daily. Water consumption is indifferent. The farm has two types of feed. The first feed contains $10 \mathrm{dkg} A, 10 \mathrm{dkg} B$ nutrients, and $80 \mathrm{dkg}$ water per kilogram, while the second feed contains $10 \mathrm{dkg} A, 20 \mathrm{dkg} B, 5 \mathrm{dkg} C$ nutrients, and $65 \mathrm{dkg}$ water per kilogram.

a) Under what program will the cost be minimal if the price of the first feed is $30 \mathrm{Ft} / \mathrm{q}$, and the price of the second feed is $120 \mathrm{Ft} / \mathrm{q}$?

b) Under what program will the feeding loss be minimal if the loss for the first feed is 10%, and for the second feed is 20%?

c) Is there a program under which both requirements can be met simultaneously, and what would an animal receive in this case? For practice 802 and problem 1208, see the introductory article of this issue.
A pig consumes $x$ kilograms of feed I and $y$ kilograms of feed II daily. The composition of these feeds in terms of nutrients $A, B, C$ and the animal's requirements are such that the following conditions must be met:

$$
10 x+10 y \geqq 45, \quad 10 x+20 y \geqq 60, \quad 5 y \geqq 5 .
$$

By simplification and supplementation, the problem can be described by the following system of inequalities:

$$
x+y \geqq 4.5, \quad x+2 y \geqq 6, \quad y \geqq 1, \quad x \geqq 0 .
$$

In the Cartesian coordinate system, all pairs of numbers $x, y$ that need to be considered are represented by the hatched region $H$ and its boundary points. The region $H$ is bounded by segments of the lines $x+y=4.5$ and $x+2 y=6$, and rays of the lines $y=1$ and $x=0$. The vertices of the region are $C_{1}(0 ; 4.5)$, $C_{2}(3 ; 1.5)$, and $C_{3}(4 ; 1)$. The boundary points belong to the region $H$. (The region $H$ is unbounded, extending to infinity, but the "distant" points are not relevant for feeding, as they indicate overfeeding of the animals.)

![](https://cdn.mathpix.com/cropped/2024_05_02_4a2485b4b77c59f73c4ag-1.jpg?height=794&width=938&top_left_y=736&top_left_x=580)

a) The total cost of feed for one pig is $K=30 x+120 y$ fillers, and the points $(x, y)$ that give the same $K$ value lie on parallel lines. For example, for 180 fillers, we can get 6 kg of feed I or 1.5 kg of feed II, so the direction of these lines is given by the line connecting the points $(6 ; 0)$ and $(0 ; 1.5)$. This line has no points in common with $H$. Setting the ruler to this triangle and then sliding it parallel away from $O$ - which represents higher $K$ values - the ruler first reaches the point $C_{3}$ from $H$, so the program with the lowest cost is: 4 kg of feed I and 1 kg of feed II per pig per day. In this case, $K_{\min }=2.40 \text{ Ft}$. (Here, we did not consider the possibility of loss mentioned in b).)

b) The feeding loss can be understood in two ways: by mass or by monetary value.

$b_{1}$) The amount of feed that goes to waste is $V_{t}=10 x+20 y$ decagrams. By drawing the lines corresponding to this and sliding the ruler, we find that the minimum $V_{t}$ is given by the points on the segment $C_{2} C_{3}$. Any point $(x, y)$ on this segment, when used to mix the feed, results in 60 decagrams of feed not being used as planned per pig per day.

$\left.b_{2}\right)$ The value of 10 decagrams of feed I that goes to waste is 3 fillers. The same economic indicator for feed II is 24 fillers. Thus, $V_{p}=3 x+24 y$ fillers, and using a similar procedure, $V_{p, \text{ min }}=36$ fillers per pig per day, with the program $x=4 \text{ kg}, y=1 \text{ kg}$ corresponding to point $C_{3}$.

c) There is a program that satisfies both a) and b), and in both variants of b), the program $x=4, y=1$ is the only one that meets the criteria.

It is also worth noting that we did not consider the impact of losses on the determination of the feed amounts.

Judit Kolcza (Győr, Kazinczy F. Secondary School III. grade) Tamás Rétházi (Miskolc, Bláthy O. Secondary School II. grade) [^0]

[^0]: ${ }^{1} 1$ Based on the article by Scharnitzky Viktor-Surányi János: "On Linear Programming" - K. M. L. 25 (1962/11) 97-104.