Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $$63,$ $$95,$ and $$135,$ respectively. The production requirements per unit are as follows:

Number of

Fans Number of

Cooling Coils Manufacturing

Time $($hours$)$

Economy $118$

Standard $1212$

Deluxe $1414$

For the coming production period, the company has $260$ fan motors, $300$ cooling coils, and $2,600$ hours of manufacturing time available. How many economy models $($E$),$ standard models $($S$),$ and deluxe models $($D$)$ should the company produce in order to maximize profit? The linear programming model for the problem is as follows:

Max $63$E $+95$S $+135$D

s$.$t$.$

$1$E $+1$S $+1$D $<=260$ Fan motors

$1$E $+2$S $+4$D $<=300$ Cooling coils

$8$E $+12$S $+14$D $<=2,600$ Manufacturing time

E$,$ S$,$ D $>=0$

The computer solution is shown below.

Optimal Objective Value $=17660\mathrm{.}00000$

Variable Value Reduced Cost

E $220\mathrm{.}000000\mathrm{.}00000$

S $40\mathrm{.}000000\mathrm{.}00000$

D $0\mathrm{.}0000024\mathrm{.}00000$

Constraint Slack$/$Surplus Dual Value

$10\mathrm{.}0000031\mathrm{.}00000$

$20\mathrm{.}0000032\mathrm{.}00000$

$3360\mathrm{.}000000\mathrm{.}00000$

Variable Objective

Coefficient Allowable

Increase Allowable

Decrease

E $63\mathrm{.}0000012\mathrm{.}0000015\mathrm{.}50000$

S $95\mathrm{.}0000031\mathrm{.}000008\mathrm{.}00000$

D $135\mathrm{.}0000024\mathrm{.}00000$ Infinite

Constraint RHS

Value Allowable

Increase Allowable

Decrease

$1260\mathrm{.}0000040\mathrm{.}00000110\mathrm{.}00000$

$2300\mathrm{.}0000090\mathrm{.}0000040\mathrm{.}00000$

$32600\mathrm{.}00000$ Infinite $360\mathrm{.}00000$

$($a$)$

Identify the range of optimality for each objective function coefficient. $($If there is no upper or lower limit$,$ enter NO LIMIT.$)$

E

to

S

to

D

to

$($b$)$

Suppose the profit for the economy model is increased by $$6$ per unit, the profit for the standard model is decreased by $$2$ per unit, and the profit for the deluxe model is increased by $$4$ per unit. What will the new optimal solution be$?$

E

units

S

units

D

units

profit $

$($c$)$

Identify the range of feasibility for the right$-$hand$-$side values. $($If there is no upper or lower limit$,$ enter NO LIMIT.$)$

constraint $1$

to

constraint $2$

to

constraint $3$

to

$($d$)$

If the number of fan motors available for production is increased by $60,$ will the dual value for that constraint change? Explain.

Yes, the dual value will change because $60$ is greater than the allowable increase of $12.$

Yes, the dual value will change because $60$ is greater than the allowable increase of $40.$

No$,$ the dual value will not change because there is no upper limit to how much the constraint can increase.

No$,$ the dual value will not change because $60$ is less than the allowable increase of $300.$