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Thread: Solving Linear Program Geometrically

  1. #1
    Junior Member
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    Solving Linear Program Geometrically

    Maximize: $\displaystyle 2x-3y$

    Subject to:

    $\displaystyle y-x \leq 5$
    $\displaystyle x+y \leq 11$
    $\displaystyle x \leq 4$
    $\displaystyle x,y \geq 0$

    I had to miss class and this is what we learned....if anyone knows of a good website with how-to or example problems that would be greatly helpful.

    If not, a little help to start me off would be great and what I need to do.
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  2. #2
    Senior Member
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    A good way to visualize this problem is to note that the maximum value of $\displaystyle z = 2x - 3y$ cannot be attained at a point inside the polygonal domain; for at that point $\displaystyle 2x - 3y$ would have to curve back downward in all directions, and no plane can do this.

    The maximum value must therefore be attained at a point on the boundary of the domain. Now, as you move across the boundary lines, $\displaystyle z$ either increases or decreases (or stays the same) continually at the same rate due to its planar nature. Thus, we will certainly find the peak value of $\displaystyle z$ at a vertex of the domain, as we can move in the increasing direction of the boundary until we find it.

    The problem then becomes to find the intersections of the lines

    $\displaystyle
    \begin{array}{rcl}
    y-x & = & 5 \\
    x+y & = & 11 \\
    x & = & 4 \\
    x,y & = & 0
    \end{array}
    $

    that belong to the boundary of the domain (not all intersections will), and find the one at which $\displaystyle z$ is greatest.

    Hope this helps.
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  3. #3
    Senior Member DeMath's Avatar
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    Quote Originally Posted by toop View Post
    Maximize: $\displaystyle 2x-3y$

    Subject to:

    $\displaystyle y-x \leq 5$
    $\displaystyle x+y \leq 11$
    $\displaystyle x \leq 4$
    $\displaystyle x,y \geq 0$

    I had to miss class and this is what we learned....if anyone knows of a good website with how-to or example problems that would be greatly helpful.

    If not, a little help to start me off would be great and what I need to do.
    See also this graphical interpretation of your problem

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