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Dimensional Analysis. I know my equations are right

  1. Nov 26, 2011 #1
    1. The problem statement, all variables and given/known data

    A birchwood table company has an individual who does all its finishing
    work and it wishes to use him in this capacity at least 36 hours each
    week. By union contract, the assembly area can be used at most 48 hours
    each week. The company has three models of birch tables, T1, T2 and T3.
    T1 requires 1 hour for assembly, 2 hours for finishing, and 9 board feet of
    birch. T2 requires 1 hour for assembly, 1 hour for finishing and 9 board
    feet of birch. T3 requires 2 hours for assembly, 1 hour for finishing and 3
    board feet of birch. Write a LOP that will compute how many of each model
    should be made in order to minimize the board feet of birchwood used.


    2. What I want to do with this problem

    Here is the thing, I wrote out the equations, but my variables don't mean a thing. I tried to make some sense out of it

    Here is the equations

    [tex]1x_1 + 1x_2 +2x_3 \leq 48[/tex]
    [tex]2x_1 + 1x_2 +1x_3 \geq 36[/tex]
    [tex]P = 9x_1 + 9x_2 + 3x_3[/tex]

    [tex]x_1, x_2, x_3 \geq 0[/tex]

    For instance the first equation is

    [tex]1x_1 + 1x_2 +2x_3 \leq 48[/tex]

    Right hand side is hours, so I expect the units on [tex]1x_1 + 1x_2 +2x_3[/tex] cancel out so that it gives me hours too

    Look at the coefficients of [tex]1x_1 + 1x_2 +2x_3[/tex]

    The "1" in front of [tex]x_1[/tex] represents "hour for assembly" or "hour/assembly". So to make things work out, [tex]x_1[/tex] has units "assembly for T_1[/tex]"

    But that doesn't work for the second equation because I will need [tex]x_1[/tex] to have units "finishing for T_1[/tex]"
     
    Last edited: Nov 26, 2011
  2. jcsd
  3. Nov 26, 2011 #2

    I like Serena

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    x1 is the number of tables of type T1 that is produced per week.

    For assembly, the 1 in front of x1, is the hours of assembly/table of type T1.
     
  4. Nov 26, 2011 #3
    I don't see how those units could cancel out...

    x1 would change for finishing work, but I need a consistent unit
     
  5. Nov 26, 2011 #4

    Ray Vickson

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    If x1 is the number of pieces of T1 to produce, x1 is a dimensional number. The number of assembly hours per piece of T1 is 1, so x1 units need 1*x1 hours. Note that 1*x1 is dimensionless; the "hours" occurs outside the expression, because we take 1 as the number of hours, not a time of 1 hour. Similarly, the number of hours we have available is 48; the '48' is dimensionless. I avoided saying the available time is 48 hours, in favor of saying the number of hours available is 48. See the difference?

    RGV
     
  6. Nov 26, 2011 #5

    I like Serena

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    How would x1 change?

    For finishing work we have:
    x1 is still the "number of tables of type T1"
    The 1 in front of x1 is the "hours of finishing/table of type T1".

    The "number of tables of type T1" cancels out, and the result is "hours of finishing".
    The total "hours of finishing" is supposed to exceed 36 "hours of finishing".
     
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