1. Assume you own you own 1200 acres of land,
and have $142,500 to invest.
Your land is most suited for tree farming
or raising cattle. You want to decide what is
the best mix of trees and cattle to maximize your income.
Cattle require 1 acre per animal, and cost $75
Trees require 1 acre per 500, and cost $300 per lot of 1000
Annual returns are $9/steer, and $10 per acre of trees

8. Constraints and Ranging Analysis
Land Constraint
Capital
Constraint
Optimum – 500 cattle
350 lots of trees

9. “Zoom in” near optimum to see
Results of Ranging Analysis better
Z=9X1 + 20X2 500 steer, 350 lots of trees =$11,500

10. “Zoom in” near optimum to see
Results of Ranging Analysis better
Z=9X1 + 20X2 500 steer, 350 lots of trees =$11,500
Z=5X1 + 20X2 500 steer, 350 lots of trees=$9,500

11. Changes in “Technological Coefficients”
And Shadow Prices
New land
Constrain add 100 acres
Capital
constraint
Old land
constraint
Z=9X1 + 20X2 500 steer, 350 lots of trees =$11,500
Z=9X1 + 20X2 700 steer, 300 lots of trees =$12,300
$800 gain by adding 100 acres

12. Formal Structure of LP problem
Maximize
Z = C1X1 + C2X2 + C3X3…
Such that,
A11X1 + A12X2 + A13X3 … <=B1
A21X1 + A22X2 + A23X3 … <=B2
A31X1 + A32X2 + A33X3 … <=B3
A41X1 + A42X2 + A43X3 … <=B4
X1>=0
X2>=0 …
Ci = Objective function coefficients
(cost coefficients)
Aij = Technological coefficients
Bi = Right hand side constraints

13. Integer Programming