| Meal 1 | Meal 2 | Meal 3 | |
|---|---|---|---|
| Milk (%) | 10 | A | B |
| Meat (%) | 10 | 12 | A |
| Veggies (%) | C | 4 | 12 |
These are milk, meat and veggies rates of meals. A, B and C values will be given as the input. In order to make the daily requirement of milk, meat and veggies at least 100%, in order to keep the total amount of meals(Meal 1, Meal 2 and Meal 3) taken to a minimum, how many units these meals should be added to the unit diet program.
For example: These are given: A = 15, B = 20 and C = 20;
I created this matrix: \begin{bmatrix}10&15&20\\10&12&15\\20&4&12\end{bmatrix}
When I solve with this equation with Gauss elimination method, I found these values:
M1 = 6.74 M2 = 10.87 M3 = -6.52
However, quantity of Meal 3 cannot be equal to a negative number.
I mean, these should be $$ M1 \geq 0 , M2 \geq 0 \ and \ M3 \geq 0 $$
Daily requirement is eg. total percentage of milk should be at least 100%. Also, total percentage of meat should be at least 100% and total percentage of veggies should be at least 100%. It means that $$10x + 15y + 20z \geq 100 $$ $$10x + 12y + 15z \geq 100 $$ $$20x + 4y + 12z \geq 100 $$
How can I find the solution set of the smallest numbers of meals that satisfy these constraints?