Page 74 - 35Linear Algebra
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74 The Simplex Method
of linear programming problems, the optimal answer must lie at a vertex of
the feasible region. Rather than prove this, lets look at a plot of the linear
function s(x, y) = 5x + 10y.
Example 37 (The sugar function)
Plotting the sugar function requires three dimensions:
The plot of a linear function of two variables is a plane through the origin.
Restricting the variables to the feasible region gives some lamina in 3-space.
Since the function we want to optimize is linear (and assumedly non-zero), if
we pick a point in the middle of this lamina, we can always increase/decrease
the function by moving out to an edge and, in turn, along that edge to a
corner. Applying this to the above picture, we see that Pablo’s best option
is 110 grams of sugar a week, in the form of 8 apples and 7 oranges.
It is worthwhile to contrast the optimization problem for a linear function
with the non-linear case you may have seen in calculus courses:
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