material $3,21$ tons. Management wants to achieve the following $P_1$ priority level goals.

Goal $1$: Produce at least $30$ tons of fuel additive.

Goal $2$: Produce at least $15$ tons of solvent base.

Assume there are no other goals.

$($a$)$ Is it possible for management to achieve both $P_1$ level goals given the constraints on the amounts of each material available? If not, which constraint is the limiting factor?

Yes. It is possible to satisfy both $P_1$ level goals.

No$.$ There is an insufficient amount of Material $1.$

No$.$ There is an insufficient amount of Material $2.$

No$.$ There is an insufficient amount of Material $3.$

fuel additive produced, $x_2$ be the number of tons of solvent base produced, $d_{}$ be the deviation variable which exceeds the value of goal $i,$ and $d_{ni}$ be the deviation variable which is less than the value of goal $i,$ for $i=1,2.)$

Min

s$.$t$.$

Material $1$

Material $2$

Material $3$

Goal $1$

Goal $2$

$x_i,d_{ni},d_{}0,$ for $i=1,2$

$($c$)$ Use the graphical goal programming procedure to find the optimal solution for the model formulated in part $($b$).$

$(x_1,x_2)=(,)$

$($d$)$ If goal $1$ is twice as important as goal $2,$ what is the optimal product mix?