Answer Questions $3$ and $4$ are based $on$ the following linear optimization problem.

Maximize $P_1+140P2+160P3+200P4$ Total profit

Subject $to$

$P_1+6P2+10P3+8P412000$

$P_1+3P2+2P3+6P43600$

$P3200$

Production budget constraint

And $P1,P2,P3,P40$

Minimum quantity needed for Product $3$ constraint

Non$-$negativity constraints

Where $P1,P2,P3$ and $P4$ represent the number $of$ units $of$ Product $1,$ Product $2,$ Product $3,$ and Product $4to$

$be$ manufactured. Constraints

$(a)$ Determine the optimal solution and the optimal value for this problem and interpret their meanings.

$(b)$ Determine the slack $(or$ surplus$)$ value for each constraint $in$ this problem and interpret its meaning.

$(a)$ What are the ranges $of$ optimality for the profit $of$ Product $1,$ Product $2,$ Product $3,$ and Product $4?$

$(b)$ Find the shadow prices $of$ the three constraints and interpret their meanings. What are the ranges $in$ which

each $of$ these shadow prices $is$ valid?

$(c)If$ the profit contribution $of$ Product $4$ changes from $$200$ per unit $to$ $$220$ per unit, what will $be$ the

optimal solution? What will $be$ the new total profit? $(Note$: Answer this question $by$ using the $sensitivity$

results given above. $Do$ not solve the problem again$).$

$(d)$ Which resource should $be$ obtained $in$ larger quantity $to$ increase the profit most? $(Note$: Answer this

question using the $sensitivity$ results given above. $Do$ not solve the problem again$).$

All questions attached $in$ order.