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Math Help - System of equations with constraints (allocation of cost)

  1. #1
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    System of equations with constraints (allocation of cost)

    I have a real life business math problem. I need to allocate pay for multiple people across multiple locations with the following constraints.

    Let a = Hours to allocate to Place01 for Sue
    Let b = Hours to allocate to Place02 for Sue
    Let c = Hours to allocate to Place01 for Bob
    Let d = Hours to allocate to Place02 for Bob

    Bob's Hourly rate = 500 dollars/hour
    Sue's Hourly rate = 1000 dollars/hour
    Bob's Hours worked = 70 Hours
    Sue's Hours worked= 90 Hours

    The following must be true in any reasonable solution:
    a>0
    b>0
    c>0
    d>0

    1000*a + 500*c = $50,000 where 1000 represents Sue's hourly rate and 500 represents Bob's hourly rate.
    1000*b + 500*d = $75,000 where 1000 represents Sue's hourly rate and 500 represents Bob's hourly rate.
    c+d = 70 hours
    a+b = 90 hours

    The Problem

    I need to distribute the 160 total hours as evenly as possible across Place01 and Place02 for each Person.

    Thus, in math terms I would like to minimize the absolute difference between a and b
    I would also like to minimize the absolute difference between c and d.

    e.g. I want to select the solution from the universe of possible solutions such that:

    |a-b| is at a minimum and |c-d| is also minimized

    Notice that the total dollars allocated $125,000 = total dollars paid = $125,000

    Normally, to do this I would just split the hours for each person between each place. But as noted above, I now have the constraint that Place01 must get exactly $50,000 allocated and place two must get $75,000.

    How can I solve this in the best way possible that distributes the hours as evenly as possible for each employee across the two Places? Bear in mind that the solution will need to be scalable in the sense that I actually have about 100 places and as many people. Rates and hours for each person are known in advance and are constant. I simplified the example for the purpose of illustrating the problem at hand.

    Thanks in advance!
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: System of equations with constraints (allocation of cost)

    Check if you like this combo:

    a=45
    b=45
    c=35
    d=35

    Updated to minimize |a-b| and |c-d|.
    Last edited by MaxJasper; September 30th 2012 at 05:11 PM.
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    Re: System of equations with constraints (allocation of cost)

    MaxJasper, Thanks for the quick response.

    I do like combo. What technique did you use to the solve the problem and how do I know that the differences are minimized? Also, how can I do it if I have more than two people and more than two places to allocate cost?

    Edit: MaxJasper. Your modified answer indeed minimizes the difference between both (a and b) and (c and d). However, that change made it yield the wrong allocation for Place01 and Place02 if a and c are equal. Specifically Place01 now gets $62,500 allocated yet it must get $50,000 exactly.

    The answer you posted previously worked...I just didn't know if that was yielded the smallest difference between (a and b) and (c and d) subject to the requirement that Place01 get $50k and Place02 get $75k.
    Last edited by fusion1920; September 30th 2012 at 06:11 PM.
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    Senior Member MaxJasper's Avatar
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    Re: System of equations with constraints (allocation of cost)

    Updated values were for total $125000. Here are values to minimize |a-b|+|c-d| and to fulfill other constrains:

    a=65/2
    b=115/2
    c=35
    d=35

    Sorry couldn't get closer a & b values!
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  5. #5
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    Re: System of equations with constraints (allocation of cost)

    No problem on not being able to get closer. I am more interest in the specific technique you are using to arrive at your answer. Can you point me in the right direction for how to solve this problem in a general case?
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