Maximize f=7x+5y subject to the following constraints and and find the shadow price for Constraint 4 :

Question

Constraint 1:  3x+4y<=24

Constraint 2:  2x+y<=10

Constraint 3:  x =>1

Constraint 4:  y<=5

x and y are non-negative 

Answer 1

Graph the constraints (think of them as equalities to graph the boundary lines but consider the inequality when deciding which side of the line to "count as good" or "color":

Find the coordinates of the 5 intersection points

(1, 0) and (1. 5) and (5, 0) are easy

To find the intersection of constraint 1 and constraint 4, use 3x + 4y = 24 but let y = 5 so 3x + 4(5) = 24 meaning x = 4/3 so the point is (4/3, 5)

To find the intersection of constraint 1 and constraint 2, solve 3x + 4y = 24 and 2x + y = 10 simultaneously. You can use a variety of methods but one way is to make the second equation y = 10 - 2x and plug "10 - 2x" into 3x + 4y = 24 in place of "y": 3x + 4(10 - 2x) = 24 meaning x = 3.2 then by plugging x = 3,2 into y = 10 - 2x, we get y = 3.6 so the point is (3.2, 3.6)

Now build a table that includes all these points and the value of "f" (which is 7x + 5y) at each intersection point:

x y f

------------------

1 0 7

1 5 32

5 0 35

4/3 5 103/3

3.2 3.6 40.4

Since we are trying to maximize "f", look for the largest value which is 40.4 occurring at x = 3.2 and y = 3.6.

To find the shadow price of constraint 4, increase constraint 4 by 1 so y < 6 and redo the table. You find out that a new intersection point is found at the intersection of y = 6 and constraint 2 and it is (2, 6) and that point gives an "f" value of 44. Subtract the new maximum from the old maximum (so 44 - 40.4) to get a shadow price of 3.6