$(20$ points$)$ Consider the following resource$-$allocation problem and accompanying

optimal tableau $($x$5,$ x$6,$ and x$7$ are the respective slack variables$).$

maximize $15$x$1+8$x$2+10$ x$312$x$4($Profit $$)$

subject to x$1+2$x$2+$ x$420($Resource $1)$

x$1+$ x$2+$ x$3+$ x$454($Resource $2)$

$2$x$1+$ x$3+$ x$436($Resource $3)$

x$1,$ x$2,$ x$3,$ x$40.$

z x $1$ x $2$ x $3$ x $4$ x $5$ x $6$ x $7$ RHS

$1-900-26-40-10-440$

$01/2101/21/20010$

$0-3/200-1/2-1/21-18$

$0201100136$

$(5$ points each$)$

a$)$Write the dual problem and specify the optimal dual solution from the foregoing tableau.

b$)$ What are the shadow prices of the resources? If you were to choose between increasing the amount of resources $1,2,$ or $3,$ which would you choose to increase and why?

c$)$ Suppose that the coefficient of x$4$ in the objective function changes from $(12)$ to $20.$ Using sensitivity analysis, find the new optimal solution.

d$)$ Suppose that the available amount of resource $1$ changes from $20$ to $40.$ Using sensitivity analysis, find the new optimal solution.