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Problem $3-12($Algorithmic$)$

Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $$69,$ $$93,$ and $$131,$ 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 $200$ fan motors, $380$ cooling coils, and $2400$ 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 $69$E $+93$S $+131$D

s$.$t$.$

$1$E $+1$S $+1$D Fan motors

$1$E $+2$S $+4$D Cooling coils

$8$E $+12$S $+14$D Manufacturing time

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

The computer solution is shown in the figure below.

Optimal Objective Value $=18120\mathrm{.}00000$

Variable Value Reduced Cost

E $20\mathrm{.}000000\mathrm{.}00000$

S $180\mathrm{.}000000\mathrm{.}00000$

D $0\mathrm{.}00000-10\mathrm{.}00000$

Constraint Slack$/$Surplus Dual Value

$10\mathrm{.}0000045\mathrm{.}00000$

$20\mathrm{.}0000024\mathrm{.}00000$

$380\mathrm{.}000000\mathrm{.}00000$

Variable Objective

Coefficient Allowable

Increase Allowable

Decrease

E $69\mathrm{.}000005\mathrm{.}0000022\mathrm{.}50000$

S $93\mathrm{.}0000045\mathrm{.}000003\mathrm{.}33333$

D $131\mathrm{.}0000010\mathrm{.}00000$ Infinite

Constraint RHS

Value Allowable

Increase Allowable

Decrease

$1200\mathrm{.}0000020\mathrm{.}0000010\mathrm{.}00000$

$2380\mathrm{.}0000020\mathrm{.}00000180\mathrm{.}00000$

$32400\mathrm{.}00000$ Infinite $80\mathrm{.}00000$

$$

What is the optimal solution, and what is the value of the objective function? If required, round your answers to the nearest whole number. If your answer is zero, enter $"0".$