Solution:

According to the solution of exercise 12, we have

The objective function for minimum cost is $\mathrm{Z}=400 \mathrm{x}+200 \mathrm{y}$

Subject to the constrains;

$5 x+2 y \geq 30 \ldots \ldots$ (i)

$2 x+y \leq 15 \ldots$ (ii)

$x \leq y \ldots$ (iii)

and $x \geq 0, y \geq 0$ (non-negative constraints)

Now, let’s construct a constrain table for the above Table for (i)

$$\begin{tabular}{|l|l|l|}
\hline x & 0 & 6 \\
\hline y & 15 & 0 \\
\hline Table for (ii) \\
\hline x & 0 & $7.5$ \\
\hline y & 15 & 0 \\
\hline Table & for (iii) \\
\hline x & 1 & 0 \\
\hline y & 1 & 0 \\
\hline
\end{tabular}$$

Next, solving (i) and (iii), we obtain

$x=30 / 7$ and $y=30 / 7$, so the corner point is A $(30 / 7,30 / 7)$

On solving (ii) and (iii), we obtain

$x=5$ and $y=5$, so the corner point is $B(5,5)$

Here, $\mathrm{ABC}$ is the shaded feasible region whose corner points are $\mathrm{A}(30 / 7,30 / 7), \mathrm{B}(5,5)$ and $\mathrm{C}(0,15)$ On evaluating the value of $Z$, we get

$$\begin{tabular}{|l|l|}
\hline Corner point & Value of $\mathrm{Z}=400 \mathrm{x}+200 \mathrm{y}$ \\
\hline $\mathrm{A}(30 / 7,30 / 7)$ & $\mathrm{Z}=400(30 / 7)+200(30 / 7)=18000 / 7=2571.4$ \\
\hline $\mathrm{B}(5,5)$ & $\mathrm{Z}=400(5)+200(5)=3000$ \\
\hline $\mathrm{C}(0,15)$ & $\mathrm{Z}=400(0)+200(15)=3000$ \\
\hline
\end{tabular}$$

It is seen from the table that the minimum value is $2571.4$

So, the required minimum cost is Rs $2571.4$ at $(30 / 7,30 / 7)$