Expert: Paul Klarreich - 10/17/2007

Question
Could you please take the time to help me solve this problem: Using Geometric Approach, maximize P = 4 x + 6 y, subject to the constraints x > 0, y > 0, 2 x + 4 y < 24, and 4 x + 2 y < 24. Shade the region with feasible solutions, and identify, if possible, the optimum feasible solution (the point which makes P maximum).

Answer
Questioner:   John
Category:  Advanced Math
Private:  No
 
Subject:  Geometric Approach
Question:  Could you please take the time to help me solve this problem: Using Geometric Approach, maximize P = 4 x + 6 y, subject to the constraints x > 0, y > 0, 2 x + 4 y < 24, and 4 x + 2 y < 24. Shade the region with feasible solutions, and identify, if possible, the optimum feasible solution (the point which makes P maximum).
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Hi, John,

This is a standard Linear Programming exercise.  Alas, I can't show you a graph through the crude interface at this site, but I can outline the process for you.

Part I - Plotting graphs of straight lines.

1.  x > 0 means the part to the right of the y-axis.
2.  y > 0 means the part to the right of the x-axis.

That means we are limited to the first quadrant on your graph paper.  (You got some, didn't you?)

3.  2x + 4y < 24

First draw the graph of  2x + 4y EQUALS 24, which is the same as

x + 2y = 12

Fastest way:  The x-intercept is at  x=12 -- plot (12,0)  and the y-intercept is at  y=6 -- plot (0,6).

Draw the line between (12,0) and (0,6) and shade, very lightly, with diagonal slanted lines, the triangular area to its left and below.

4.  4x + 2y < 24.  Same stuff:

2x + y = 12  gives intercepts of  (6,0) and (0,12).  Draw the line, shade the triangle lightly with diagonal lines slanted THE OTHER WAY from the first one.

Now there is a region that is shaded TWICE.  That's your region to shade a bit darker.

....................

Now the maximization stuff.  Your shaded region has four corners:  (0,0), (6,0), (0,6) AND  (4,4).  You get that last one by solving the two simultaneous EQUATIONS that we wrote above.  If you did a really good job of plotting, you will see that point.

Now you just compute P at all four points:

P(x,y) = 4x + 6y

P(0,0) = 4(0) + 6(0) = 0

P(6,0) = 4(6) + 6(0) = 24

P(0,6) = 4(0) + 6(6) = 36

P(4,4) = 4(4) + 6(4) = 40  << that's your max.