LINEAR PROGRAMMING PROJECT



LINEAR PROGRAMMING PROJECT

[pic] You are the new owner of a game shop in Queen Creek. The previous owner is now programming drones for the NSA to secretly monitor Apache Junction flash mobs. Your first duty as new owner and store manager is to create an advertising plan based on the budget available. You must figure out how many radio and TV ads to purchase.

• Radio ads cost $600 per airing.

• TV ads cost $1200 per airing.

• You total advertising budget is $9,000.

1. If we let x = radio ads and y = TV ads, write an inequality for our advertising budget.

2. The television station called to say that we are only allowed to purchase up to 6 TV ads on their Saturday morning gaming retrospective. Write an inequality for this constraint.

3. The radio station has informed us that their DJ threatened to quit if he had to listen to more than one add from a game shop per show. They are limiting us to 7 radio ads. Write an inequality for this constraint.

4. It is impossible to buy a negative number of TV ads. Write an inequality for this constraint.

5. It is impossible to buy a negative number of radio ads. Write an inequality for this constraint.

Graph this system of inequalities on the graph paper provided. This will help us determine our region of feasible solutions.

Name: ________________________ Period: ___ Date:________

Graph the system of inequalities you found on the previous page. When you’re finished. Clearly outline your region of feasible solutions (where all of the graphs overlap).

Determine the five points where the graphs intersect. These are called the vertices of the region of feasible solutions. Show work if necessary.

1. 2. 3.

4. 5.

Name: ________________________ Period: ___ Date:________

Radio Ads

The following data was collected in past years to try to determine how radio ads affect game sales. Plot the points and estimate the line of best fit.

|Number of radio|Increase in game sales|

|ads | |

|0 |0 |

|5 |725 |

|2 |250 |

|6 |900 |

|4 |450 |

|3 |400 |

|5 |750 |

|3 |600 |

|2 |350 |

|4 |575 |

|3 |450 |

|5 |700 |

|1 |150 |

|2 |325 |

|6 |950 |

Number of radio Ads

Use the points (1, 150) and (6, 900) to estimate the line of best fit.

1. Graph the line that goes through both points.

2. Find the slope of the line.

3. Write the equation of the line.

4. What does the slope tell you about how each radio ad affects sales?

For every radio ad, game sales increased by about ___________.

Name: ________________________ Period: ___ Date:________

TV Ads

The following data was collected in past years to try to determine how TV ads affect game sales. Plot the points and estimate the line of best fit.

|Number of TV ads |Increase in game |

| |sales |

|1 |100 |

|8 |725 |

|6 |590 |

|7 |725 |

|4 |375 |

|8 |800 |

|5 |440 |

|9 |900 |

|2 |150 |

|6 |630 |

|3 |300 |

|7 |640 |

|4 |410 |

|2 |275 |

|5 |560 |

Number of TV Ads

Use the points (1, 100) and (8, 800) to estimate the line of best fit.

1. Graph the line that goes through both points.

2. Find the slope of the line.

3. Write the equation of the line.

4. What does the slope tell you about how each TV ad affects sales?

For every TV ad, game sales increased by about ______.

Name: ________________________ Period: ___ Date:________

We now have all of the information we need to solve the linear program.

1. Write an objective function for CD sales. (Hint: Think about how each TV and radio ad affects sales.)

2. Substitute the coordinates of the vertices into the objective function.

| |(objective function) | |

|(x, y) | |f (x, y) |

|Vertex Point | |Total Sales |

| | | |

| | | |

| | | |

| | | |

| | | |

3. What is the maximum and where did it occur?

4. Knowing this information, how many TV and radio ads should you buy?

CONGRATULATIONS!!! You’re finally FINISHED! [pic]

Name: ________________________ Period: ___ Date:________

Extra Credit

Now that you have found your optimal solution, think about how a change in our allowable advertising values might affect the solution. This is called sensitivity analysis. Your job now is to find out how much a change in one of your constraint values will affect your final profit.

• For example, what if instead of being allowed to buy 7 radio commercials, you were only allowed to buy 5. What would be the new optimal solution? (show your work)

• If you were allowed to increase one constraint value in order to increase your profit, which one should you change? In other words, would you rather be allowed to buy one more TV ad or one more radio ad? Why?

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0 100 200 300 400 500 600 700 800 900 1000

Increase in game sales

0 1 2 3 4 5 6 7 8 9 10

0 100 200 300 400 500 600 700 800 900 1000

Increase in game sales

0 1 2 3 4 5 6 7 8 9 10

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