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Math Help - simplex method maximize objective function

  1. #1
    BenDunn
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    simplex method maximize objective function

    A farmer has 100 acres to plant in corn and beans. Let c represent the # of acres planted in corn and b = # of acres planted in beans. It costs $72 / acre to plant corn and $40 / acre to plant beans. If the farmer has $12,000 to spend planting crops, how many acres should he plant in each crop if corn has a profit of $24 / acre and beans $32 / acre?

    Write the objective function and the system of constraints.

    Show the initial tableau and solve.

    My thoughts... I guess we are trying to get maximum revenue. It seems to me since beans are cheaper to plant and have higher profit, that we are going to plant all beans.

    Objective function: z = 32b + 24c (this seems like the amount of revenue?)

    Constraints ? b + c <= 100

    40b + 72c <= 12000

    Intial Tableau

    Can anyone give me any feedback as to whether I am on the right track?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by BenDunn View Post
    A farmer has 100 acres to plant in corn and beans. Let c represent the # of acres planted in corn and b = # of acres planted in beans. It costs $72 / acre to plant corn and $40 / acre to plant beans. If the farmer has $12,000 to spend planting crops, how many acres should he plant in each crop if corn has a profit of $24 / acre and beans $32 / acre?

    Write the objective function and the system of constraints.

    Show the initial tableau and solve.

    My thoughts... I guess we are trying to get maximum revenue. It seems to me since beans are cheaper to plant and have higher profit, that we are going to plant all beans.

    Objective function: z = 32b + 24c (this seems like the amount of revenue?)

    Constraints ? b + c <= 100

    40b + 72c <= 12000

    Intial Tableau

    Can anyone give me any feedback as to whether I am on the right track?

    What you have is correct.

    To get the initial tableau you introduce initial slack variables e, f so we have:


    z - 32b - 24c = 0

    b + c +e = 100

    40b + 72c + f= 12000

    Then the initial tableau in matrix form is:

    \left[<br />
\begin{array}{ccccc}<br />
1&-32&-24&0&0\\<br />
0&1&1&1&0\\<br />
0&40&72&0&1<br />
\end{array}<br />
\right] <br />
\left[ \begin{array}{c}<br />
z\\b\\c\\e\\f<br />
\end{array}<br />
\right]<br />
=<br />
\left[ \begin{array}{c}<br />
0\\100\\1200<br />
\end{array}<br />
\right]<br />

    RonL
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