Answer to Question #111647 in Operations Research for Prosper Mawuli

Question

1. A store sells two types of toys, A and B. The store owner pays $8 and $14 for
each one unit of toy A and B respectively. One unit of toys A yields a profit of
$2 while a unit of toys B yields a profit of $3. The store owner estimates that
no more than 2000 toys will be sold every month and he does not plan to
invest more than $20,000 in inventory of these toys. How many units of each
type of toys should be stocked in order to maximize his monthly total profit
profit?

Answer 1

Buy 56$ 7 toys type A. Then we will earn 2*7=14 $. If we buy type B, we will earn 3*4=12$. from this it seems that it is profitable to sell toys A. But here we must take into account that no more than 2000 toys of two types will be sold in total, and not only the costs are important, but also the individual profitability. I.e. we have the following system of inequations:

$\begin{Bmatrix} 8A+14B≤20000\\ A+B≤2000 \end{Bmatrix} \iff\begin{Bmatrix} 8(A+B)+6B≤20000 \\ A+B≤2000 \end{Bmatrix}$

(let me remind you that we strive to maximize 2A+3B)

2A+3B=2*(A+B)+B≤2*2000+B=4000+B

This means that we want to buy type b toys.

This means that we can already write the system as

$\begin{Bmatrix} 16000+6B≤20000 \\ A+B=2000 \end{Bmatrix}\iff\begin{Bmatrix} 3B≤2000 \\ A+B=2000 \end{Bmatrix}$

A=1334

B=666

2A+3B=2668+1998=4666