Question

Please show work using matrices, thank you!

A farmer has 400 acres of available land and $80,000 to spend He wants to plant the combination of crops which 'maximizes his profit. Complete parts a. through c below Profits and Constraints for Crops Potatoes Corn X2 $128 $40 Cabbage Total Number of Acres Cost (per acre Profit (peracre a. Give tne auai propiem X400 $320 $120 $224 S80,000 $60 2 subject to 320y2 120 128y20 2242 60 with y120, y2 20 b. The solution to the dual problem can be interpreted as shadow profits Use shadow profits to ostimate the far increases to $84 000 mer's profit if land is cut to 360 acres but capital The farmer's estimated profitis S c. Suppose the farmer has 440 acres of land but only $76.000 Find the optimum proft and the planting strategy that will produce this profit Planting acres of potatoes, acres of com, and acres of cabbage will produce an optimal profit of s

Answer 1

Dual of the LP

Minimize W = 400y1+80000y2

S.T.

y1+400y2>=180

y1+160y2>=60

y1+280y2>=90

y1,y2>=0

b) Sensitivity Report and shadow prices

. 2

The sensitivity report says the shadow price of constraint 1 is 0 and constraint 2 is 0.45

Making the change in land acres will not affect the optimal value but making one unit change in budget will change the optimal value by 0.45

So changing acres to 360 and budget to 84000

Change in profit = 0+0.45*4000=1800

New profit = 36000+1800=$37800

c) New optimal plan

The acre of land of potatoes = 190

Acre of land of corn = 0

Acre of land of cabbage =0

Total profit = 34200