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Applications of Linear Programming

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1 Applications of Linear Programming
Section 3-6 (a.k.a. STORY PROBLEMS)

2 Review Graph a system of inequalities.
Find the vertices of the feasible region. Determine which lines are intersecting. Solve that system of equations. Plug the vertices into the objective function to find the maximum and minimum values.

3 How will the applications be different?
Your problems will be in paragraph form. You will have to create your own system of inequalities. You also get to create your own function.

4 Not to worry, I have a plan!

5 Step 1 Define your variables.
Look at the last sentence in your problem. Write “x =“ and “y =“ as part of your work.

6 Step 2 Organize the given information in a chart.

7 Step 3 Write a system of inequalities to represent all of the limitations. Can the items in my problem be negative? Does the chart show me anything?

8 Step 4 Graph the system of inequalities and find the vertices of the feasible region.

9 Step 5 Write a function to be maximized or minimized.
Does the chart tell you anything? Often the function represents cost, revenue, or profit.

10 Step 6 Test your points and answer the question. SENTENCES!!

11 Sounds easy, ready to try?

12 The AC Telephone Company manufactures two styles of cordless telephones, deluxe and standard. Each deluxe phone nets the company $9 in profit and each standard phone nets $6. Machines A and B are used to make both styles of telephones. Each deluxe model requires 3 hours of machine A time and 1 hour on B. Each standard phone requires 2 hours on both machines. If the company has 12 hours available on machine A and 8 hours available on B, determine the mix of phones that will maximize the company’s profit.

13 “determine the mix of phones that will maximize the company’s profit”
Let x = # of deluxe phones. Let y = # of standard phones.

14 Make a chart. Profit Machine A Time Machine B # Deluxe x $9 3 1
# Standard y $6 2

15 System of Inequalities
Can I make negative phones? Does my table help? x ≥ 0 y ≥ 0 3x + 2y ≤ 12 x + 2y ≤ 8

16 Graph and find the vertices.

17 “maximize the company’s profit”
P(x, y) = 9x + 6y P(0, 0) = 9(0) + 6(0) = 0 P(0, 4) = 9(0) + 6(4) = 24 P(2, 3) = 9(2) + 6(3) = 36 P(4, 0) = 9(4) + 6(0) = 36

18 The company should either make 2 deluxe phones and 3 standard phones or 4 deluxe phones and 0 standard phones.

19 If you are the CEO, which option would you choose? Why?

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