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# Quantitative Methods Optimal Solution a-j

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Quality Air conditioning manufactures three hoe 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
Economy 1 1 8 hours
Standard 1 2 12 hours
Deluxe 1 4 14 hours

For the coming production period, the company has 200 fan motors, 320 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 63E + 95S + 135D
s.t.
1E + 1S + 1D < 200 Fan motors
1E + 2S + 4D < 320 Cooling coils
8E + 12 S + 14 D < 2400 Manufacturing Time
E, S, D, > 0

Use the 100% rule for objective function coefficients and right hand side ranges where appropriate. Do not run the changed model. Assume that any changes given in a of the problem are the only changes being made in the model

a) What is the optimal solution? What values of the decision variables lead to it? (3 points) I believe this is the right answer for this first question, the rest I am not sure of.

E = 80, S = 120, D = 0. Profit= \$16,440

b) Suppose the coefficient of E is changed to 67. Will the optimal solution change? Why or why not?(3 points)

c) Suppose the coefficient of D is changed to 167 . Will the optimal solution change? Why or why not? (3 points)

d) Suppose the coefficient of E is changed to 70.5, that of S to 90.5 and that of D to 130. Will the optimal solution change? Why or why not? (6 points)

e) Suppose the coefficient of E is changed to 60.25, that of S to 97 and that of D to 142. Will the optimal solution change? Why or why not? (6 points)

f) Which resource is not completely used? (3 points)

g) Suppose the number of available fan motors is increased to 225. By how much will Quality Airâ??s profit increase? (5 points)

h) Suppose the number of available cooling coils is increased to 375. By how much will Quality Airâ??s profits increase? (5 points)

i) Suppose the number of available fan motors is changed to 215, that of cooling coils to 310 and the number of available manufacturing hours to 2350. Will the dual prices of those constraints change? Why or why not? (7 points)

j) Suppose number of available fan motors is changed to 185, the number of cooling coils to 345 and the number of available manufacturing hours to 2275. Will the dual prices of those constraints change? Why or why not? (7 points)

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#### Solution Preview

I am going to solve these problems using the SOLVER function in Excel.

Refer to the excel worksheet for the optimal amount for each of the two objectives. If you are not familiar with linear programming with excel, refer to this tutorial here http://it.usu.edu/plugins/work/sitemaps/107/files/Linear_Programming_Using_Excel.pdf. I use the exact same setup as the tutorial so it should be very easy to follow.

1) In the first sheet in the excel spreadsheet (named 1), we solve the first problem. The solution is set up in the same way as the tutorial above. We get

E = 80, S = 120, D = 0. Profit= \$16,440

2) In the second sheet (named 2), we change the coefficient of E to 67. After the change, we still have the same solution as before, namely, E = 80, S = 120, D = 0. The new profit is \$16,760.

3) In the third sheet, we ...

#### Solution Summary

Quantitative Methods Optimal Solution a-j

\$2.19