A company producing soft drinks has a contrast which requires a minimum of 80 units of chemical $A$ and 60 , units of chemical $B$ to go in each bottle of the drink. The chemical are available in a prepared mix from two different suppliers. Supplier $X$ has a mix of 4 units of $A$ and 2 units of $B$ that costs Rs.10, and the supplier $Y$ has a mix of 1 unit of $A$ and 1 unit of $B$ that costs \mp4. How many mixes from $X$ and $Y$ should the company purchase to honor the contract requirement and yet minimize the cost?

Home » A company producing soft drinks has a contrast which requires a minimum of 80 units of chemical and 60 , units of chemical to go in each bottle of the drink. The chemical are available in a prepared mix from two different suppliers. Supplier has a mix of 4 units of and 2 units of that costs Rs.10, and the supplier has a mix of 1 unit of and 1 unit of that costs \mp4. How many mixes from and should the company purchase to honor the contract requirement and yet minimize the cost?

A company producing soft drinks has a contrast which requires a minimum of 80 units of chemical $A$ and 60 , units of chemical $B$ to go in each bottle of the drink. The chemical are available in a prepared mix from two different suppliers. Supplier $X$ has a mix of 4 units of $A$ and 2 units of $B$ that costs Rs.10, and the supplier $Y$ has a mix of 1 unit of $A$ and 1 unit of $B$ that costs \mp4. How many mixes from $X$ and $Y$ should the company purchase to honor the contract requirement and yet minimize the cost?

A company producing soft drinks has a contrast which requires a minimum of 80 units of chemical $A$ and 60 , units of chemical $B$ to go in each bottle of the drink. The chemical are available in a prepared mix from two different suppliers. Supplier $X$ has a mix of 4 units of $A$ and 2 units of $B$ that costs Rs.10, and the supplier $Y$ has a mix of 1 unit of $A$ and 1 unit of $B$ that costs \mp4. How many mixes from $X$ and $Y$ should the company purchase to honor the contract requirement and yet minimize the cost?

Let $x$ and $y$ be number of mixes from suppliers $X$ and $Y$ .
$\therefore$ According to the question,
$4 x+y \geq 80,2 x+y \geq 60, x \geq 0, y \geq 0$
Minimize $Z=10 x+4 y$
The feasible region determined by $4 x+y \geq 80,2 x+y \geq 60, x \geq 0, y \geq 0$ is given by

The feasible region is unbounded .The corner points of feasible region are $A(0,80), B(10,40), C(30,0)$ .
The value of $Z$ at corner points are

$\begin{tabular}{|l|l|l|} \hline Corner Point & $\mathrm{Z}=10 \mathrm{x}+4 \mathrm{y}$ & \\ \hline $\mathrm{A}(0,80)$ & 320 & \\ \hline $\mathrm{B}(10,40)$ & 260 & Minimum \\ \hline $\mathrm{C}(30,0)$ & 300 & \\ \hline \end{tabular}$

The minimum value of $Z$ is 260 at point $(10,40)$ .
Hence, the company should buy 10 mixes from supplier $\mathrm{X}$ and 40 mixes from supplier $\mathrm{Y}$ to minimize the cost.

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