Consider the following resource allocation problem and the accompa$-$ nying optimal tableau, where x$4,$x$5$ and x$6$ are the slack variables for constraints $1$ through $3,$ respectively.

max z $=10$x$1+7$x$2+6$x$3$

S$.$t$.3$x$1+2$x$2+$ x$336$ x$1+$ x$2+2$x$3322$x$1+$ x$2+$ x$322$

x$1,$ x$2,$ x$30$

$($Profit $$)($Resource $1)($Resource$2)($Resource$3)$

z x$1$ x$2$ x$3$ x$4$ x$5$ x$6$ RHS z $1300105146$

x$2011010-114$ x$50-20011-32$ x$30101-1028$

$($a$)$Suppose that the profit of producing one unit of Product $3$ goes down from $$6$ to $$3.$ Use sensitivity analysis to find a new pro$-$ duction plan for the company.

$($b$)$ Suppose that the available amount of Resource $1$ increases from $36$ units to $41$ units, and the available amount of Resource $3$ decreases from $22$ units to $20$ units. How should the production plan be changed?

$($c$)$Due to technology improvement, the amount of Resource $1$ needed to produce one unit of Product $1$ is reduced from $3$ units to $2$ units. How should the production plan be changed?