Question

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 1 1 8
Standard 1 2 12
Deluxe 1 4 14
For the coming production period, the company has 240 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 63E + 95S + 135D
s.t.
1E + 1S + 1D ≤ 240 Fan motors
1E + 2S + 4D ≤ 300 Cooling coils
8E + 12S + 14D ≤ 2,600 Manufacturing time
E, S, D ≥ 0
The computer solution is shown below.
Optimal Objective Value = 17040.00000
Variable Value Reduced Cost
E 180.00000 0.00000
S 60.00000 0.00000
D 0.00000 −24.00000
Constraint Slack/Surplus Dual Value
1 0.00000 31.00000
2 0.00000 32.00000
3 440.00000 0.00000
Variable Objective
Coefficient Allowable
Increase Allowable
Decrease
E 63.00000 12.00000 15.50000
S 95.00000 31.00000 8.00000
D 135.00000 24.00000 Infinite
Constraint RHS
Value Allowable
Increase Allowable
Decrease
1 240.00000 60.00000 90.00000
2 300.00000 110.00000 60.00000
3 2600.00000 Infinite 440.00000
(a)
Identify the range of optimality for each objective function coefficient. (If there is no upper or lower limit, enter NO LIMIT.)
E

Correct: Your answer is correct.
to

Correct: Your answer is correct.
S

Correct: Your answer is correct.
to

Correct: Your answer is correct.
D

Correct: Your answer is correct.
to

Correct: Your answer is correct.
(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
220
units
S
40
units
D
0

Correct: Your answer is correct.
units
profit $
16440

Incorrect: Your answer is incorrect.
(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

Correct: Your answer is correct.
to

Correct: Your answer is correct.
constraint 2

Incorrect: Your answer is incorrect.
to

Incorrect: Your answer is incorrect.
constraint 3

Incorrect: Your answer is incorrect.
to

Correct: Your answer is correct.
(d)
If the number of fan motors available for production is increased by 90, will the dual value for that constraint change? Explain.
Yes, the dual value will change because 90 is greater than the allowable increase of 12.
Yes, the dual value will change because 90 is greater than the allowable increase of 60.
No, the dual value will not change because 90 is less than the allowable increase of 300.
No, the dual value will not change because there is no upper limit to how much the constraint can increase.

Answer 1

For Doggies Paradise Inc., the profit-maximizing quantity is four units, where marginal revenue equals marginal cost. Producing four units results in the highest profit, with a total profit of $4 after subtracting the total costs from the total revenue at this quantity level.

Step-by-step explanation:

To find the profit maximizing quantity for Doggies Paradise Inc., we first need to calculate the total revenue (TR), marginal revenue (MR), total cost (TC), and marginal cost (MC) for each output level from one to five units. Given the fixed cost of $100, price per unit of $72, and variable costs at different output levels, we can formulate a table.

Here's the calculation in tabular form:

Quantity
TR ($72 * Quantity)
MR (Change in TR/Change in Quantity)
TC (Fixed + Variable Cost)
MC (Change in TC/Change in Quantity)


1
$72
$72
$164 ($100 + $64)
$64


2
$144
$72
$184 ($100 + $84)
$20 ($84 - $64)


3
$216
$72
$214 ($100 + $114)
$30 ($114 - $84)


4
$288
$72
$284 ($100 + $184)
$70 ($184 - $114)


5
$360
$72
$370 ($100 + $270)
$86 ($270 - $184)

Next, by plotting TR and TC curves, we can observe the point at which TR exceeds TC, indicating profit. Similarly, plotting MR and MC helps us identify the level at which MR equals MC, which marks the profit maximizing quantity. By analyzing the table, we can see that up to the fourth unit, the marginal revenue exceeds marginal cost. However, at the fifth unit, the marginal cost of $86 is higher than the marginal revenue of $72, hence producing the fifth unit would result in a loss.

Therefore, the profit-maximizing quantity is at the output level where marginal revenue equals marginal cost, which is the fourth unit. Producing four units yields the highest profit before costs exceed revenues.

As referenced in Step 5, after determining the profit-maximizing output level, we look at the profit amount which would be the total revenue minus total cost at the fourth unit ($288 - $284 = $4).