A company produces three products, X$,$ Y$,$ and Z$.$ Each unit of Product X requires $2$ hours of labor, $1$ hour

of machine time, and $3$ hours of assembly time. Each unit of Product $Y$ requires $1$ hour of labor, $2$ hours of

machine time, and $2$ hours of assembly time. Each unit of Product $Z$ requires $3$ hours of labor, $2$ hours of

machine time, and $1$ hour of assembly time.

The company has the following resources available: $200$ hours of labor, $150$ hours of machine time, $120$

hours of assembly time

The profit per unit for each product is as follows: Product X: Rs$.30/-,$ Product Y: Rs$.25/-,$ Product Z: Rs$.$

$20/-.$ The company wants to maximize its total profit.

The final simplex table for the above problem is given below.

Note: $s1,s2$ and $s3$ relates to products $x,Y,$ and $Z$ respectively.

As per the above table;

$($i$)$ What is the optimal solution?

$($ii$)$ Interpret the shadow prices of labor hours, machine hours, and assembly hours.

$($iii$)$ Construct the dual for the problem.

$($iv$)$ Find the optimal values for the dual variables.

$($v$)$ Find the range within which the production of $x$ can be changed without affecting the optimal

solution.

$($vi$)$ Find the range within which the production of $Y$ can be changed without affecting the optimal

solution.

$($vii$)$ Find the range within which the production of $Z$ can be changed without affecting the optimal

solution.

$($viii$)$ Identify the range within which labor hours can be varied without altering the optimal

solution.

$($ix$)$ Identify the range within which machine hours can be varied without altering the optimal

solution.

$($x$)$ Identify the range within which assembly hours can be varied without altering the optimal

solution.