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$3($Profit $$)$

S$.$t$.3$x$1+2$x$2+$ x$336($Resource $1)$

x$1+$ x$2+2$x$332($Resource $2)$

$2$x$1+$ x$2+$ x$322($Resource $3)$

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

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 improvement of technology, the amount of Resource $1$

needed in order to produce one unit of Product $1$ is reduced from

$3$ units to $2$ units. How should the production plan be changed?