"My garden" is a company which produces two kind of fertilisers and sells these: The first kind "Perfect Grass" and the second kind "Super Grass". In order to produce both kinds, you respectively need the 3 resources $M_1, M_2, M_3$ in different quantities. To produce $1\text{kg}$ "Perfect Grass", you need $2\text{kg}$ of $M_1$, $1\text{kg}$ of $M_2$ and $2\text{kg}$ of $M_3$. To produce $1\text{kg}$ "Super Grass", you need $3\text{kg}$ of $M_1$, $2\text{kg}$ of $M_2$ and $1\text{kg}$ of $M_3$. During the next week, the company can at most handle $1500\text{kg}$ of the resource $M_1$, $1200\text{kg}$ of $M_2$ and $1300\text{kg}$ of $M_3$. The company achieves a profit of $5 €$ per $\text{kg}$ of "Perfect Grass" and $8€$ per $1\text{kg}$ of "Super Grass". For the next week, the company has already received an order of $100\text{kg}$ of the kind "Perfect Grass" and affirmed the delivery.
How much kilogram of each kind of fertilisers should the company produce in the next week to achieve maximum profit? It's presumed that the company will sell the produced quantity entirely. Solve this task by using simplex algorithm.
Hint: Firstly solve this task without respecting the advance order and optimize afterwards, if necessary.
I only want know the correct inequalities / equalities. I can solve the rest of the task on my own. But if this part is wrong, everything will be wrong... So here is what I got from the text:
Perfect Grass: $x_1 = 2m_1+m_2+2m_3$
Super Grass: $x_2 = 3m_1+2m_2+m_3$
$m_1 \leq 1500, m_2 \leq 1200, m_3 \leq 1300$
The objective function is $P=5y_1+8y_2$
Is it alright like that?
In my view it is wrong. You have three resource constraints ($M_1, M_2, M_3$).
First I define the variables.
$x=$Amount of Perfect Grass in kg.
$y=$Amount of Super Grass in kg.
$m_i$=Amount of resource $i$ in kg.
$\color{olive}{\text{contraints}}$
$m_1=2x+3y\leq 1500 \qquad (M_1)$
$m_2=x+2y\leq 1200 \qquad (M_2)$
$m_3=2x+y\leq 1300\qquad (M_3)$
$x\geq 100$
This constraints are not equivalent to your constraints. Suppose the solution is $(x,y)=(500,100)$. Then $m_1=2\cdot 500+3\cdot 100=1300,m_2=700,m_3=1100$. This doesn´t satisfy for instance your equality for the amount of Perfect Grass:
$500\neq 2\cdot 1300+200+2\cdot 1100$
$\color{olive}{\text{obejective function}}$
$\texttt{max} \ \ 5x+8y$
$\color{olive}{\text{non-negativity condition}}$
$x,y\geq 0$