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A B C D E F
Stock 1 Stock 2
Expected Profit (on $50,000) $12,500 $20,000
Risk (Stand. Dev. for $50,000) $5,000 $30,000
Stock 1 Stock 2 Total
Portfolio (% of $50,000) 13.3% 86.7% 100% <= 100%
$6,667 $43,333
Minimum
Expected
Portfolio Profit
Expected Profit $19,000 >= $19,000
Risk (Variance) $676,444,444
Risk (Stand. Dev.) $26,009
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A B C D E F G H I
Portfolio Selection Problem (Nonlinear Programming)
Stock 1 Stock 2 Stock 3 Stock 4
Expected Return 21% 30% 8% 17%
Risk (Stand. Dev.) 25% 45% 5% 18%
Joint Risk (Covar.) Stock 1 Stock 2 Stock 3 Stock 4
Stock 1 0.040 -0.005 -0.015
Stock 2 -0.010 -0.025
Stock 3 0.003
Stock 4
Stock 1 Stock 2 Stock 3 Stock 4
Total
Portfolio 24.5% 12.4% 17.6% 45.5% 100% =100%
Minimum
Expected
Portfolio Return
Expected Return 18.0% >= 18.0%
Risk (Variance) 0.0095
Risk (Stand. Dev.) 9.8%
Minimum Risk Expected
Expected Return Stock 1 Stock 2 Stock 3 Stock 4 (St. Dev.) Return
24.5% 12.4% 17.6% 45.5% 9.8% 18.0%
8% 7.5% 3.9% 85.6% 3.1% 3.8% 10.1%
10% 7.5% 3.9% 85.6% 3.1% 3.8% 10.1%
12% 11.6% 5.9% 69.2% 13.3% 4.4% 12.0%
14% 15.9% 8.1% 52.0% 24.0% 5.9% 14.0%
16% 20.2% 10.2% 34.8% 34.7% 7.7% 16.0%
18% 24.5% 12.4% 17.6% 45.5% 9.8% 18.0%
20% 28.8% 14.5% 0.5% 56.2% 11.9% 20.0%
22% 24.5% 30.9% 0.0% 44.6% 16.0% 22.0%
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A B C D E F G H I
24% 19.9% 47.7% 0.0% 32.4% 22.3% 24.0%
26% 15.2% 64.5% 0.0% 20.2% 29.6% 26.0%
28% 10.6% 81.3% 0.0% 8.0% 37.2% 28.0%
30% 0.0% 100.0% 0.0% 0.0% 45.0% 30.0%
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A B C D E F
Unit Profit = a(Rate) + b(Rate)2, where
Product 1 Product 2
a$200 $300
b -$100 -$100
Product 1 Product 2 Total Total Profit
Production Rate 0.75 1.25 2$313
<=
2
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A B C D E F
Unit Profit Power Saws Power Drills
Regular Time $150 $100
Overtime $50 $75
Total
Used Available
Power Supplies 1 1 10,000 <= 10,000
Gear Assemblies 2 1 13,000 <= 15,000
Power Saws Power Drills Power Saws Power Drills
Regular Time 3,000 5,000 <= 3,000 5,000
Overtime 0 2,000 <= 2,000 3,000
Total 3,000 7,000
Total Profit
$1,100,000
Units Produced
Maximum
Used per Unit Produced
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A B C D E F
Unit Profit Doors Windows
Gross Profit ($hundred) 4 6
Marketing Cost ($hundred)
(Doors)32(Windows)2
Total Resource
Used Available
Resource 1 1 3 5.6547 <= 8
Resource 2 5 2 8.7735 <= 14
Total Profit
Doors Windows ($hundreds)
Production Rate 1.155 1.500 7.58
Used Per Unit Produced
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18
A B C D E F
Unit Profit ($hundred) Doors Windows
First 3 4
Second -3 0
Third -19 -4
Total Resource
Used Available
Resource 1 1 3 4 <= 8
Resource 2 5 2 7 <= 14
Power Saws Power Drills Doors Windows
First 1 1 <= 1 1
Second 0 0 <= 1 1
Third 0 0 <= 1 1
Total 1 1
Total Profit
($hundred)
7
Units Produced
Maximum
Used per Unit Produced
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A B C D E F G H
Unit Profit Product 1 Product 2 Product 3
First Group $360 $240 $450
Second Group $30 $120 $300
Third Group –$90 $180
Total Resource
Resource Used per Unit Produced Used Available
Resource 1 1 1 1 60 <= 60
Resource 2 3 2 0 85 <= 200
Resource 3 1 0 2 65 <= 70
Units Produced Maximum
Product 1 Product 2 Product 3 Product 1 Product 2 Product 3
First Group 15 20 10 <= 15 20 10
Second Group 0 0 5 <= 30 20 5
Third Group 0 10
Total 15 20 25 Total Profit
$18,000
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A B C D E F
Unit Profit ($millions) Campaign 1 Campaign 2
0 to 2 3 5
2 to 4 -1 1
4 to 5 -4 -2
Total Resource
Used Available
Resource 1 4 1 12 <= 20
Resource 2 1 4 18 <= 20
Campaign 1 Campaign 2 Campaign 1 Campaign 2
0 to 2 2 2 <= 2 2
2 to 4 0 2 <= 2 2
4 to 5 0 0 <= 1 1
Total 2 4
Total Profit
($millions)
18
Maximum
Resource Used per Unit
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A B C D E F G
Unit Profit ($millions) Campaign 1 Campaign 2
0 to 2 3 4 0 to 3
2 to 2.5 0.5 0.5 3 to 3.5
2.5 to 3 –0.5 -0.5 3.5 to 4
3 to 5 -3 -2 4 to 5
Total Resource
Used Available
Resource 1 4 1 13.5 <= 20
Resource 2 1 4 16.5 <= 20
Campaign 1 Campaign 2 Campaign 1 Campaign 2
0 to 2 2 3 <= 2 3 0 to 3
2 to 2.5 0.5 0.5 <= 0.5 0.5 3 to 3.5
2.5 to 3 0 0 <= 0.5 0.5 3.5 to 4
3 to 5 0 0 <= 2 1 4 to 5
Total 2.5 3.5
Total Profit
($millions)
18.5
Maximum
Resource Used per Unit
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A B C D E F
Unit Profit Campaign 1 Campaign 2
Net Profit
3X1 – (X1 – 1)23X2 – (x2-2)2
Total Resource
Used Available
Resource 1 4 1 13.5 <= 20
Resource 2 1 4 16.5 <= 20
Total Profit
Campaign 1 Campaign 2 ($millions)
Level 2.5 3.5 13.50
Used Per Unit Produced