Question

Consider the following linear program
Max 3A + 2B
St
1A + 1B <= 10
3A + 1B <= 24
1A + 2B <= 16
A, B >= 0
The value of the optimal solution is 27. Suppose that the right hand side for constraint one is increased from 10 to 11.

a) Use the graphical solution procedure to find the new optimal solution.
b) Use the solution to part A to determine the shadow price for constraint 1
C) The sensitivity report for the linear programming problem one provides the following right hand range information

Constraint      Constraint      Allowable    Allowable
                        R H Side.        Increase.     Decrease
1.                        10,000.          1,200.           2,000
2.                        24,000.         6.000.          6,000
3.                        16,000.          Infinite.        3,000

What does the right hand side range information for constraint one tell you about the shadow price for constraint one

D) The shadow price for constraint 2 is 0.5 using this shadow price and right hand side range information and parts c what conclusion can you draw about the effect of changes to the right hand side of constraint 2

Answer 1

a)

Since the objective function line will pass through point C at last in the bounded solution. The value will lie at point C such that at the intersection point of lines x1 + x2 = 11 and 3x1 + x2 = 24.

Solving for the intersection point of these two lines,

x1 = 6.5 and x2 = 4.5

objection function value is;

z = 3(6.5) + 2(4.5) = 28.5

b)

Shadow price for constraint 1 = new objective function value - old objective function value

Shadow price for constraint 1 = 28.5 - 27 = 1.5

c)

The allowable increase for the constraint 1 = 1.2 and the allowable decrease for the constraint 1 = 2

Within this allowable range, the optimal solution will not change such that we can change the right-side value within this range and the objection function value will increase or decrease by shadow price multiplied by the change in the right-hand side value.

d)

Constraint 2:

Right-hand side value = 24

Allowable increase = 6

Allowable decrease = 6

Shadow price = 0.5

For 6 unit increase in the RH side value, the objective function value will increase by 0.5*6 =3

For 6 unit decrease in the RH side value, the objective function value will decrease by 0.5*6 =3