Question 514025
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Maximize *[tex \Large P(x,y)\ =\ 10x\ +\ 20y]


Subject to:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ 3y\ \leq\ 30]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ +\ y\ \geq\ 12]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ \geq\ 0]


Note:  No need for constraining *[tex \Large x] non-negative since the first two constraints cover that possibility.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ {(x,y)\,|\, x,\,y\ \in\ \mathbb{Z}]


Note:  Constrain to the integers because you can't make and sell a fractional part of a box in this scenario.


Step 1:  Graph the constraint inequalities.


The triangular area where the solution sets overlap is the area of feasibility.  The optimum point must be one of the vertices of the feasibility polygon.  The integer constraint means that the optimum solution will be an ordered pair of integers that is closest to the ordered pair representing the vertex of the feasibility polygon.  In the case of this problem, all three vertices have integer coordinates.


Determine the ordered pairs representing the three vertices of the feasibility triangle and test each one in the Profit function.  The one that gives the biggest profit value is the optimum solution.


Full solution with graphs available.  Write for quote.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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