Perform sensitivity analysis on the following linear programming with the optimal solution x$*=($x$1$ x$3$ x$5)$T $=(111)$T

see screenshot for given minimization

$($a$)$ We want to determine the range of values of the cost vector c $($components$)$ for which the optimal basis

and solution remain optimal.

$($b$)$ Find the range of b for the current optimality.

$($c$)$ Suppose that added is the following constraint: $2$x$1+4$x$2+3$x$37.$ Then what will happen?

$($d$)$ If a new constraint is added, then the objective function value cannot get better; either remains the

same or get worse. Is it true?

$($e$)$ Suppose that added is the following: x$7$ with c$7=2$ and a$7=(123)$T$.$ What will happen?

$($f$)$ Suppose that we have new cost vector components: c$1=2+,$ c$2=32.$ What is the range of

such that the previous optimal solution remains so$?$