linear programming problem by the method of corners.

Question

Maximize  P = 3x+2y

Subject to: x+y<6

                 x<3

              x>0 and y>0

Answer 1

First you need to graph the constraint area defined by the four inequalities:

x + y < 6

or y < -x +6

which defines an area below the line of slope -1 and y intercept 6

 

The other three inequalities are easy to graph:

x < 3

x > 0

y > 0

 

Now, we need to find the corner points of this constraint area. After you draw the graph, it's easy to see that the area has four corners: (0, 0), (3, 0), (0, 6), (3, 3)

 

Now, according to method of corners, the max value of a linear equation such as P = 3x+2y will occur at one of the corners, so we need to evaluate 3x+2y at the four corners.

 

3x+2y at (0, 0) = 0

3x+2y at (3, 0) = 3*3 + 0 = 9

3x+2y at (0, 6) = 3*0 + 2*6 = 12

3x+2y at (3, 3) = 3*3 + 2*3 = 15

 

So the max value of 3x+2y is 15

Answer 2

The first thing to do is to find the corners of the "feasible region" -- that is, the region that satisfied all the constraints.

 

You can do this by considering each pair of constraints as a pair of equations in two variables (x and y) and solving each pair to find the intersecting points.  All of the corners will be at the intersection of two of the constraints.  But not all of the intersections will be corners -- because some will be outside of the feasible region.  That is, some of the intersections will violate one of the constraints.

 

Because there are 4 constraints, that will mean that there are 6 different pairs of equations to solve.  It's not a lot of work, but I'm not going to work all of that through here.

 

It may also work to graph all the constraints and just eyeball where the feasible region is and where its corners are.  In this case, the constraints are simple enough that this works.  If you do this, you will see that the corners of the feasible region are at the following points: (0,0), (3.0), (3,3) and (0,6)

 

Now find the value of the objective function P at each of these points:

 

(0,0) -- P = 3*0 + 2*0 = 0

(3.0) -- P = 3*3 + 2*0 = 9

(3,3) -- P = 3*3 + 2*3 = 15

(0,6) -- P = 3*0 + 2*6 = 12

 

The corner with the maximum value of P is (3,3).

 

HOWEVER...  Notice that the constraints are all stated as pure inequalities.  Perhaps you wrote out the problem incorrectly and meant to have them be ≤ and ≥?  If so then the maximum value of the objective function is 15, at the point (3,3). 

 

But if you wrote the problem correctly, then that means that the corners are not actually in the feasible region.  So the best you can say is that P < 15 but there is no specific point at which P takes on a maximum value.