Maximizing Crop Profits with Land, Labor, and Capital Constraints, Study notes of Agricultural engineering

Download Maximizing Crop Profits with Land, Labor, and Capital Constraints and more Study notes Agricultural engineering in PDF only on Docsity! 1 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 = + + ≤ + ≤ 1. Labor 1 2 1 2 1 2 2 1 1 2 .3 .2 25 .3 25 .2 250 2 3 3 250 0 83.33 3 0 125 x x x x x x x x x x + = = − = − = ⇒ = = ⇒ = B 2. Land 1 2 1 2 2 1 1 2 100 100 0 100 0 100 x x x x x x x x + = = − = ⇒ = = ⇒ = 3. Objective 1 2 1 2 1 2 50 10 50 10 1 50 5 z x x x z x z x x = + = − = − 4. How does the objective function change as x1 and x2 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 1 250 2 100 3 3 1 50 3 3 50 50 100 50 50 x x x x x x − = − = = = ⇒ = − = AEB 6592 Lecture III Professor Charles B. Moss 2 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? 1. Reformulate the objective function 1 1 2 2 1 1 2 2 2 1 2 1 1 z z x z x z x z z x z z x x z z = + = − = − Assuming z2 is fixed at 10, what level of z1 would cause the optimum allocation to change? 2. Any increase in the value of z will result in no change in the optimal solution. 3. 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 10 2 3 30 2 15 x x z z z = = = Thus, the coefficient on cotton could decline to 15, a decrease of $35/acre before the optimal solution changes. 4. 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 2 2 2 2 2 2 50 3 3 100 33.33 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. AEB 6592 Lecture III Professor Charles B. Moss 5 D. Sensitivity Analysis–Over what ranges are the solution stable? 1. What if land were increased? What constraint would become binding? 12 12 10 20 8 24 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 x1=4 and x2=12 or 16 acres of land (an increase of 4 acres). The value of the objective function at this point is: ( ) ( )40 4 20 12 160 240 400 400 360 40 10 16 12 4 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 x1=8 and x2=0 (a decrease of 8 acres of land) ( ) ( )40 8 20 0 320 320 360 40 10 8 12 4 z ShadowValue = + = − − = = = − − AEB 6592 Lecture III Professor Charles B. Moss 6 2. What if the labor constraint were increased? What constraint would become binding? 12 12 10 20 8 24 Land Labor Capital Objective Oats Corn Increasing the labor would cause the Capital n Land intersection to become binding. Thus, the largest change in labor that would keep the constraint active is x1=8 and x2=4 ( ) ( ) ( ) ( ) : 6 8 2 4 48 8 56 40 8 20 4 320 80 400 400 360 40 5 56 48 8 NewLabor z ShadowValue + = + = = + = + = − = = = − The labor constraint can likewise be reduced until the point where the land is restricted without change in the active constraints. 3. Capital could be decreased until the land and labor intersection is binding at x1=6 and x2=6. E. Over what range of objective coefficients is the solution stable? 1 1 2 2 2 1 2 1 1 2 1 1 1 1 1 1 20 20 1 20, 20 40 20 20 1 60, 60 40 20 3 z z x z x z z x x z z Fixingz z z z z z z = + = − = − = − ⇒ = ∆ = − = − − = − ⇒ = ∆ = − = AEB 6592 Lecture III Professor Charles B. Moss 7 1 2 2 2 2 2 2 40 1 40, 40 20 20 40 1 40 , 13.33 20 6.67 40 3 3 Fixingz z z z z z z = = ⇒ = ∆ = − = = ⇒ = ∆ = − = −