assume you own you own 1200 acres of land, and have $142,500 to invest
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
0 250 500 750 1,000 1,2500
250
500
750
X1 Number of Cattle
X2
Lo
ts o
f T
rees
0 250 500 750 1,000 1,2500
250
500
750
X1 Number of Cattle
X2
Lo
ts o
f T
rees
Z=$9,000
Z=$5,000
Objective Function Z = 9X1 + 20X2
0 250 500 750 1,000 1,2500
250
500
750
X1 Number of Cattle
X2
Lo
ts o
f T
rees
Z=$5,000
Z=$9,000
Z=$10,800
0 250 500 750 1,000 1,2500
250
500
750
X1 Number of Cattle
X2
Lo
ts o
f T
rees
Since each cattle requires 1 acre of land, and eachlot of 1000 trees requires 2 acres of land, and you have only 1200 acres,
1X1 + 2X2 =< 1200
Feasible Region
Infeasible Region
0 250 500 750 1,000 1,2500
250
500
750
X1 Number of Cattle
X2
Lo
ts o
f T
rees
Since each cattle costs $75 and eachlot of 1000 trees costs $300, and you have only $142,500,
75X1 + 300X2 =< 142500
Feasible Region
Infeasible Region
0
250
500
750
1,000
1,250
0
250
500
750
X1 Number of Cattle
X2
Lo
ts o
f T
rees
Infeasible Region
Feasible Region
Considering both constraints simultaneously
Feasible Region
Considering both constraints simultaneously
0
250
500
750
1,000
1,250
0
250
500
750
X1 Number of Cattle
X2
Lo
ts o
f T
rees
0
100
200
300
400
500
600
700
0 250 500 750 1000 1250 1500
X1 Number of Cattle
X2
Lot
s of
Tre
esConstraints and Ranging Analysis
Optimum – 500 cattle 350 lots of trees
Land Constraint
CapitalConstraint
300
320
340
360
380
450 470 490 510 530 550
X1 Number of Cattle
X2
Lot
s of
Tre
es“Zoom in” near optimum to see Results of Ranging Analysis better
Z=9X1 + 20X2 500 steer, 350 lots of trees =$11,500
300
320
340
360
380
450 470 490 510 530 550
X1 Number of Cattle
X2
Lot
s of
Tre
es“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
0
100
200
300
400
500
600
300 550 800 1050
X1 Number of Cattle
X2
Lot
s of
Tre
esChanges in “Technological Coefficients”
And Shadow Prices
Z=9X1 + 20X2 500 steer, 350 lots of trees =$11,500Z=9X1 + 20X2 700 steer, 300 lots of trees =$12,300$800 gain by adding 100 acres
Old land constraint
New land Constrain add 100 acres
Capital constraint
Formal Structure of LP problem
Maximize
Z = C1X1 + C2X2 + C3X3…
Such that,
A11X1 + A12X2 + A13X3 … <=B1A21X1 + A22X2 + A23X3 … <=B2A31X1 + A32X2 + A33X3 … <=B3A41X1 + A42X2 + A43X3 … <=B4
X1>=0X2>=0…
Ci = Objective function coefficients (cost coefficients)Aij = Technological coefficientsBi = Right hand side constraints
Integer Programming
0
1
2
3
4
5
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7
8
0 1 2 3 4
Number of Prey 1 eaten
Num
ber
of P
rey
2 ea
ten