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Math Help - How to Structure Linear Programming Problems

  1. #1
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    How to Structure Linear Programming Problems

    Hey guys Im taking an operations research class this semester. Take a look at this scenario:

    1. The Los Angeles Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for the year 2006 program and the allocation of the budget to different forms of advertising. The advertising alternatives include television, radio, and newspaper. The goal is to reach the largest possible audience. Audience estimates, costs, and maximum media usage limitations are as shown below:

    Constraint Television Radio Newspaper

    Audience per advertisement 100,000 18,000 40,000

    Cost per advertisement $2,000 $300 $600

    Maximum media usage 10 20 10
    (# of ads/spots)


    To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized. If the promotional budget is limited to $18,200, how many commercial messages should be run on each medium to maximize total audience contact? Structure this linear programming problem. (8 points)
    --------

    All I need to do is structure the problems so I just want to verify my models are correct and any help is appreciated.

    I just need to make sure I'm getting this down.

    1)
    Here set my variables x1 x2 and x3 to television, radio, and newspaper respectively.
    I set my objective function Z= 100000x1+18000x2+40000x3

    subject to constraints:

    (1) Budget constraint: 2000x1+300x2+600x3<=18200

    (2) Media Usage constraint: Im a bit confused over here. From the chart Im assuming that the max media usage for each alternative can be summed to be the "total advertisements authorized". If this is so then i need to make sure that a. The total ads do not exceed 50 b. radio ads are no more than 25, but then our max is 20 anyway c. at least 5 television ads should be used. Now Im thinking that these are to be seperate constraints:

    x2<=20
    x1>=5
    x1+x2+x3<=50

    Is this correct? Where am I going wrong if not? Am I misinterpreting the data?

    Thanks a lot
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  2. #2
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    Re: How to Structure Linear Programming Problems

    1. "set my variables x1 x2 and x3 to television, radio, and newspaper respectively. " Makes no sense. "television, radio, and newspaper" are NOT numbers. Did you mean that x1, x2, and x3 are the amounts of advertising money?

    2. Do you not understand what "50%" means?
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  3. #3
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    Re: How to Structure Linear Programming Problems

    Quote Originally Posted by HallsofIvy View Post
    1. "set my variables x1 x2 and x3 to television, radio, and newspaper respectively. " Makes no sense. "television, radio, and newspaper" are NOT numbers. Did you mean that x1, x2, and x3 are the amounts of advertising money?

    2. Do you not understand what "50%" means?
    1. I meant that they are the number of television, radio, and newspaper ads/spots. The problem asked me to maximize the audience contact.

    2. Yes I do. What I want to clarify is if the RHS of constraint should be 25 since thats 50% of the total or 20 bc for that specific medium the max is 20.
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  4. #4
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    Re: How to Structure Linear Programming Problems

    Heres the second problem

    2. The Monroe County Sheriff’s Department schedules police officers for 8-hour shifts. The beginning times for the shifts are 8:00 a.m., noon, 4:00 p.m., 8:00 p.m., midnight, and 4:00 a.m. An officer beginning a shift at one of these times works for the next eight hours. During normal weekday operations, the number of officers needed varies depending on the time of the day. The department staffing guidelines require the following minimum number of officers on duty:

    Time of Day Minimum Officers on Duty

    8:00 a.m. - noon 5
    Noon - 4:00 p.m. 6
    4:00 p.m. - 8:00 p.m. 10
    8:00 p.m. - midnight 7
    Midnight - 4:00 a.m. 4
    4:00 a.m. - 8:00 a.m. 6


    How many police officers should be scheduled to begin the 8-hour shifts at each of the six times in order to minimize the total number of officers required. Structure this linear programming problem.
    -------

    I think I did this right:

    decision variables:
    x1= Number of officers scheduled to come in on Shift 1
    x2= Number of officers scheduled to come in on Shift 2
    x3= Number of officers scheduled to come in on Shift 3
    x4= Number of officers scheduled to come in on Shift 4
    x5= Number of officers scheduled to come in on Shift 5
    x6= Number of officers scheduled to come in on Shift 6

    Obj. Function:
    MIN Z=x1+x2+x3+x4+x5+x6

    subject to:

    x6+ x1>=5
    x1+ x2>= 6
    x2+ x3>= 10
    x3+ x4>= 7
    x4+ x5>= 4
    x5+ x6>= 6
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