SOLUTION: Find the minimum and maximum values of z=10x+7​y, if​ possible, for the following set of constraints. x+y≤5 −x+y≤3 2x−y≤8

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Question 1102108: Find the minimum and maximum values of z=10x+7​y, if​ possible, for the following set of constraints.
x+y≤5
−x+y≤3
2x−y≤8
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


If the constraints also include x>=0 and y>=0, then the minimum value of the objective function is obviously at (0,0), where the value of the objective function is 0.

If x and/or y values less than 0 are allowed, then of course there is no minimum value of the objective function.

For the maximum value of the objective function subject to the given constraints, I solve the problem by putting the equations for the constraint boundary lines in slope-intercept form:
x+y=5 --> y = -x+5 --> slope -1
-x+y=3 --> y = x+3 --> slope 1
2x-y=8 --> y = 2x-8 --> slope 2

The standard method for finding the maximum and minimum values of the objective function is to graph the constraints to determine the feasibility region, and then evaluate the objective function at each vertex of the feasibility region.

If we do that, we find the vertices of the feasibility region at (0,0), (0,3), (1,4), (13/3,2/3), and (4,0). The objective function values at each vertex are
(0,0): 10(0)+7(0) = 0
(0,3): 10(0)+7(3) = 21
(1,4): 10(1)+7(4) = 38
(13/3,2/3): 10(13/3)+7(2/3) = 130/3+14/3 = 144/3 = 48
(4,0): 10(4)+7(0) = 40

The maximum value is 48, at (13/3,2/3).

A more efficient way to find the maximum value of the objective function is to find the slope of the objective function and look at the feasibility region.

10x+7y = z --> slope -10/7

A look at the feasibility region shows two places where a line with slope -10/7 will just touch a vertex without passing through the feasibility region -- at (0,0) and (13/3,2/3).

That means the maximum value of the objective function will be at (13/3,2/3) and the minimum value will be at (0,0).

By looking at the feasibility region and the slope of the objective function, we can determine where the maximum value if going to be obtained. That means we don't need to do the work to find all of the vertices of the feasibility region; we only need to find the coordinates of the vertex where we know the maximum value will be obtained.
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