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Graphical Solution and Sensitivity (Again)
Lecture III
I. Problem from Last Time
A.
1 2 1 2 1 2
max 50 10
.. .3 .2 25: 100:
z x x s t x x Labor x x Land
- Labor 1 2 1 2 1 2
2 1 1 2
x x x x x x
x x x x
B
- Land 1 2 1 2 2 1 1 2
x x x x x x x x
- Objective 1 2 1 2 1 2
z x x x z x x z x
- How does the objective function change as x 1 and x 2 change? 1 2 1 2
z Ax Bx dz Adx Bdx
B. Where do the labor and land constraints intersect? 2 2
2 2 2 1
x x
x x x x
Professor Charles B. Moss
C. What is the maximum allowable profit per acre of cotton such that the current solution remains optimal? Similarly, what is the minimum return on cotton for which this solution remains optimal?
- Reformulate the objective function 1 1 2 2 1 1 2 2 1 2 2 1 1
z z x z x z x z z x x z^ z x z z
Assuming z 2 is fixed at 10, what level of z 1 would cause the optimum allocation to change?
- Any increase in the value of z will result in no change in the optimal solution.
- The range over which the solution is stable in the other direction is bounded by the labor constraint. Specifically, if the objective function becomes more steep than the labor constraint, then the optimal solution will change. To solve for the coefficient at which the solution changes 1 1 1 1 1
x x z z z
Thus, the coefficient on cotton could decline to 15, a decrease of $35/acre before the optimal solution changes.
- Similarly for the coefficient on corn, the coefficient on corn could be decreased indefinitely without changing the optimal solution. However, if the coefficient were increased sufficiently corn would enter the solution (^22 )
2 2
z (^) x x
z z
B
Thus, if the rate of return per acre of corn were increased to $33.33/acre, then corn would enter the optimal solution.
II. New Problem A. The firm’s problem is how to organize production {choose levels of crop plantings} to maximize profits or net returns over variable costs given 12 acres of land, 48 hours of labor, and $360 of operating capital.
Professor Charles B. Moss
b. Capital n Labor 2 2
2 2 1
x x
x x x
c. Capital n Land 2 2
2 2 1
x x
x x x
C. Optimal combination of crops 1 2 1 2 1 2
z x x x z x x z x
This implies an optimal solution at the corner of the Land n Labor constraints: 1 1 1 2 3
Land
Labor
Capital
Objective
Oats
Corn
Professor Charles B. Moss
D. Sensitivity Analysis–Over what ranges are the solution stable?
- What if land were increased? What constraint would become binding?
Land
Labor
Capital
Objective
Oats
Corn
If land were sufficiently increased, then the combination of labor and capital will eventually become constraining. The constrained optimal at that point involves x 1 =4 and x 2 =12 or 16 acres of land (an increase of 4 acres). The value of the objective function at this point is:
z ShadowValue
On the down side, land can be decreased until the labor constraint is no longer binding before the solution changes. At that point x 1 =8 and x 2 =0 (a decrease of 8 acres of land)
z ShadowValue
= −^ = − =
Professor Charles B. Moss
1 (^22 )
(^22 )
Fixingz z (^) z z
z (^) z z
