We have an export order for 1500 units of instrument Type I and 800 units of instrument Type II. The instruments can be manufactured in plants $A$ and $B$. Plant $A$ can produce 30 units of Type I or 20 units of Type II per day,

How should we divide the production between the two plants so that the shipment can be completed in the shortest possible time?
I. solution. It is more advantageous if plant $B$ starts with the production of type II, because its daily production is twice as high as that of $A$, while in the production of type I it is only $5 / 3$ times higher. Therefore, we assign plant $A$ to start with the production of type I.

To complete the ordered quantity, $A$ (from type I) needs 50 days, and $B$ (from type II) needs 20 days. Therefore, starting from the 21st day, $B$ will also produce type I. By then, $A$ will have produced 600 units of type I. The remaining $900$ units will take $900: 80=11.25$ days to complete, because it is obviously the fastest way to complete them if neither stops before the other, and thus $80$ units are produced daily in the two plants together. During this period, $B$ will produce $562.5$ units of type I, and $A$ will produce an additional $337.5$ units.

According to this, the production plan is as follows: plant $A$ will produce $937.5$ units of type I, and plant $B$ will produce $562.5$ units of type I and $800$ units of type II, with both plants working for $31$ and $1 / 4$ days.

However, since production cannot be divided into half units, the fractional number of units will be rounded to the nearest whole number, one down and one up. It is more advantageous to assign $937$ units to $A$ and $563$ units to $B$, because this way the work will be completed faster, although only by minutes.

If the packaging unit is larger than the unit, further rounding may occur, but this will not significantly extend the more than one-month production time.

Note. We could also start our reasoning as follows: It is more advantageous to assign the production of type I to $A$, because the daily quantity of type I is $50 \%$ higher than that of type II, while in plant $B$ it is only $25 \%$ higher. This also leads to the conclusion that plant $B$ should start with the production of type II.

II. solution. Let plant $A$ produce type I for $x$ days and type II for $y$ days. During this time, it will produce $30 x$ and $20 y$ units, respectively, and plant $B$ will need to produce $1500-30 x$ and $800-20 y$ units. For this, it will need $(1500-30 x) / 50=30-0.6 x$ and $(800-20 y) / 40=20-0.5 y$ days, respectively, totaling $50-0.6 x-0.5 y$ days of work.

In the most favorable case, the working time of the two plants is equal; otherwise, the total time can be reduced by having the plant that finishes earlier take over part of the remaining work of the other. Thus,

$$
x+y=50-0.6 x-0.5 y, \quad \text { i.e., } \quad 1.6 x+1.5 y=50
$$

Here, neither $x$ nor $y$ is negative, and we are looking for when the total production time $t=x+y$ is the smallest. Writing our equation in the form $1.6 t=50+0.1 y$, it is clear that this happens when $y$ is as small as possible, i.e., 0. In this case, $x=t=\frac{50}{1.6}=31 \frac{1}{4}$ days. During this time, plant $A$ will produce $30 x=937 \frac{1}{2}$ units of type I. Plant $B$ will need $562 \frac{1}{2}: 50=11 \frac{1}{4}$ days to produce the remaining $562 \frac{1}{2}$ units of type I, and $800 / 40=20$ days to produce $800$ units of type II, thus a total of $31 \frac{1}{4}$ days. - In practice, of course, the results need to be rounded appropriately.

László Csirmaz (Budapest, J. Hunyadi Secondary School, 8th grade)

Note. We can also start from the production plan of plant $B$: produce type I for $u$ days and type II for $v$ days, from which $8 u+9 v=270$ follows. However, now $v \leq 20$, otherwise it would produce more than the order, so

$$
t=u+v=\frac{270-v}{8}
$$

its minimum value is obtained by taking the largest $v$, which is $(270-20): 8=31$ and $1 / 4$ days.