Optimization I
Outline
• Basic Optimization: Linear
programming
– Graphical method
– Spreadsheet Method
• Extension: Nonlinear programming
– Portfolio optimization
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What is Optimization?
• A model with a “best” solution
• Strict mathematical definition of
“optimal”
• Usually unrealistic assumptions
• Useful for managerial intuition
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Elements of an Optimization Model
• Formulation
– Decision Variables
– Objective
– Constraints
• Solution
– Algorithm or Heuristic
• Interpretation
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Optimization Example:
Extreme Downhill Co.
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1. Managerial Problem Definition
Michele Taggart needs to decide how
many sets of skis and how many
snowboards to make this week.
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2. Formulation
a. Define the choices to be made by the manager
(decision variables).
b. Find a mathematical expression for the
manager's goal (objective function).
c. Find expressions for the things that restrict
the manager's range of choices (constraints).
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2a: Decision Variables
Variable Name
Skis
Snowboards
Symbol
X
Y
Operations Management -- Prof. Juran
Units
100s of pairs of skis
100s of snowboards
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2b: Objective Function
Find a mathematical
expression for the
manager's goal (objective
function).
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EDC makes $40 for every snowboard it
sells, and $60 for every pair of skis.
Michele wants to make sure she chooses
the right mix of the two products so as to
make the most money for her company.
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What Is the Objective?
Profit  6000X  4000Y
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2c: Constraints
Find expressions for the things
that restrict the manager's range
of choices (constraints).
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Molding Machine Constraint
The molding machine takes three hours to
make 100 pairs of skis, or it can make 100
snowboards in two hours, and the
molding machine is only running 115.5
hours every week.
The total number of hours spent molding
skis and snowboards cannot exceed 115.5.
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Molding Machine Constraint
3X  2Y  115.5
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Cutting Machine Constraint
Michele only gets to use the cutting
machine 51 hours per week. The cutting
machine can process 100 pairs of skis in
an hour, or it can do 100 snowboards in
three hours.
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Cutting Machine Constraint
1X  3Y  51
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Delivery Van Constraint
There isn't any point in making more
products in a week than can fit into the
van The van has a capacity of 48 cubic
meters. 100 snowboards take up one
cubic meter, and 100 sets of skis take up
two cubic meters.
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Delivery Van Constraint
2 X  1Y  48
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Demand Constraint
Michele has decided that she will
never make more than 1,600
snowboards per week, because she
won't be able to sell any more than
that.
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Demand Constraint
Y  16
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Non-negativity Constraints
Michele can't make a negative
number of either product.
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Non-negativity Constraints
X0
Y 0
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Solution Methodology
Use algebra to find the best solution.
(Simplex algorithm)
George B. Dantzig
1914 - 2005
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Point X Y
A
0 0
B
0 16
C
3 16
D 18.6 10.8
E 24 0
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Calculating Profits
Point X
Y
A
0
0
B
0
16
C
3
16
D
18.6 10.8
E
24
0
Objective function
Profit
6000(0)+4000(0) =$0.00
6000(0)+4000(16) =$64,000.00
6000(3)+4000(16) =$82,000.00
6000(18.6)+4000(10.8) =$154,800.00
6000(24)+4000(0) =$144,000.00
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The Optimal Solution
• Make 1,860 sets of skis and 1,080
snowboards.
• Earn $154,800 profit.
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Spreadsheet Optimization
A
1 objective:
2
3 decision variables:
4
5
6 constraints:
7
8
9
10
11
12
B
(in 100s)
C
6000
D
4000
E
F
G
H
I
$ 10,000 = profit
=SUMPRODUCT(C1:D1,C4:D4)
J
skis snowboards
1
1
Molding
Cutting
Van
Demand
Nonnegativity (skis)
Nonnegativity (snowboards)
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3
1
2
0
1
0
2
3
1
1
0
1
5
4
3
1
1
1
=SUMPRODUCT($C$4:$D$4,C7:D7)
<=
115.5
<=
51
<=
48
<=
16
>=
0
>=
0
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A
1 objective:
2
3 decision variables:
4
5
6 constraints:
7
8
9
10
11
12
B
(in 100s)
Molding
Cutting
Van
Demand
Nonnegativity (skis)
Nonnegativity (snowboards)
Operations Management -- Prof. Juran
C
6000
D
4000
E
F
$ 154,800 = profit
G
77.4
51
48
10.8
18.6
10.8
115.5
51
48
16
0
0
skis snowboards
18.6
10.8
3
1
2
0
1
0
2
3
1
1
0
1
<=
<=
<=
<=
>=
>=
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A
B
C
14 Objective Cell (Max)
15
Cell
Name
16
$E$1 objective:
17
18
19 Variable Cells
20
Cell
Name
21
$C$4 skis
22
$D$4 snowboards
23
24
25 Constraints
26
Cell
Name
27
$E$11 Nonnegativity (skis)
28
$E$12 Nonnegativity (snowboards)
29
$E$7 Molding
30
$E$8 Cutting
31
$E$9 Van
32
$E$10 Demand
Operations Management -- Prof. Juran
D
E
Original Value
$
154,800
Final Value
$
154,800
Original Value
18.6
10.8
Cell Value
18.6
10.8
77.4
51
48
10.8
F
G
Final Value
Integer
18.6 Contin
10.8 Contin
Formula
$E$11>=$G$11
$E$12>=$G$12
$E$7<=$G$7
$E$8<=$G$8
$E$9<=$G$9
$E$10<=$G$10
Status
Slack
Not Binding
18.6
Not Binding
10.8
Not Binding
38.1
Binding
0
Binding
0
Not Binding
5.2
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A
B
C
6 Variable Cells
7
8
Cell
Name
9
$C$4 skis
10
$D$4 snowboards
11
12 Constraints
13
14
Cell
Name
15
$E$11 Nonnegativity (skis)
16
$E$12 Nonnegativity (snowboards)
17
$E$7 Molding
18
$E$8 Cutting
19
$E$9 Van
20
$E$10 Demand
D
E
F
G
H
Final Reduced Objective
Value
Cost
Coefficient
18.6
0
6000
10.8
0
4000
Allowable
Allowable
Increase
Decrease
2000 4666.666667
14000
1000
Final Shadow Constraint
Value
Price
R.H. Side
18.6
0
0
10.8
0
0
77.4
0
115.5
51
400
51
48
2800
48
10.8
0
16
Allowable
Increase
18.6
10.8
1E+30
13
27.2
1E+30
Allowable
Decrease
1E+30
1E+30
38.1
27
26
5.2
Most important number: Shadow Price
The change in the objective function that would result from
a one-unit increase in the right-hand side of a constraint
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Nonlinear Example:
Scenario Approach to Portfolio Optimization
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
Ford
14.13
15.21
19.33
28.13
37.69
50.50
43.63
26.63
28.13
Lilly Kellogg Merck HP
14.47 8.09
5.02 42.38
16.50 10.00
5.22 33.88
27.88 17.38
7.61 36.75
37.13 25.88 13.76 41.88
39.00 26.19 17.61 58.25
42.75 32.13 19.25 47.25
68.50 33.81 25.83 31.88
73.25 37.94 29.96 57.00
83.50 65.38 55.50 69.88
Use the scenario approach to determine the minimumrisk portfolio of these stocks that yields an expected
return of at least 22%, without shorting.
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The percent return on the portfolio is represented by the random
variable R.
5
R   ri xi
i 1
In this model, xi is the proportion of the portfolio (i.e. a number
between zero and one) allocated to investment i.
Each investment i has a percent return under each scenario j, which
we represent with the symbol rij.
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We calculate the percent return on each of the stocks in each year:
Year
1984
1985
1986
1987
1988
1989
1990
1991
Ford
0.076
0.271
0.455
0.340
0.340
-0.136
-0.390
0.056
Lilly Kellogg Merck HP
0.140 0.236
0.040 -0.201
0.690 0.738
0.458 0.085
0.332 0.489
0.808 0.140
0.050 0.012
0.280 0.391
0.096 0.227
0.093 -0.189
0.602 0.052
0.342 -0.325
0.069 0.122
0.160 0.788
0.140 0.723
0.852 0.226
For example, Ford went from $14.31 to $15.21 in 1984, so the return on Ford stock
in 1984 was:
r1 j 
S1 j  S0
S0
Operations Management -- Prof. Juran

15.21  14.13
 0.076
14.13
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The portfolio return under any scenario j is given by:
5
R j   rij x i
i 1
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Let Pj represent the probability of scenario j occurring.
The expected value of R is given by:
8
 R   R j Pj
j 1
The standard deviation of R is given by:
R 
 R
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j 1

2


j
R Pj
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In this model, each scenario is considered to
have an equal probability of occurring, so we
can simplify the two expressions:
8
R 
R
j 1
8
 R
8
R 
Operations Management -- Prof. Juran
j
j 1
j  R

2
8
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Managerial Formulation
Decision Variables
We need to determine the proportion of our portfolio to invest
in each of the five stocks.
Objective
Minimize risk.
Constraints
All of the money must be invested.
The expected return must be at least 22%.
No shorting.
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(1)
(2)
(3)
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Mathematical Formulation
Decision Variables
x1, x2, x3, x4, and x5 (corresponding to Ford, Lilly, Kellogg, Merck, and HP).
 R
8
Objective
Minimize Z =  R 
Constraints
j 1
i 1
 R

2
8
5
x
j
i
 1.0
(1)
8
R 
R
j 1
8
j
 0.22
For all i, xi ≥ 0
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(2)
(3)
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A
1
2 Exp Return
3 StDev
4
5 req. return
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
B
C
=
=
0.127
0.265
=
0.220
D
E
Total
1
F
Ford
1.000
G
Lilly
0.000
Ford
14.13
15.21
19.33
28.13
37.69
50.50
43.63
26.63
28.13
Lilly
14.47
16.50
27.88
37.13
39.00
42.75
68.50
73.25
83.50
Historical data on returns
Year
Ford
1984
0.076
1985
0.271
1986
0.455
1987
0.340
1988
0.340
1989
-0.136
1990
-0.390
1991
0.056
mean
0.127
stdevp
0.265
Lilly
0.140
0.690
0.332
0.050
0.096
0.602
0.069
0.140
0.265
0.235
H
Kellogg
0.000
I
Merck
0.000
J
HP
0.000
Kellogg
8.09
10.00
17.38
25.88
26.19
32.13
33.81
37.94
65.38
Merck
5.02
5.22
7.61
13.76
17.61
19.25
25.83
29.96
55.50
HP
42.38
33.88
36.75
41.88
58.25
47.25
31.88
57.00
69.88
Kellogg
0.236
0.738
0.489
0.012
0.227
0.052
0.122
0.723
0.325
0.271
Merck
0.040
0.458
0.808
0.280
0.093
0.342
0.160
0.852
0.379
0.290
HP
-0.201
0.085
0.140
0.391
-0.189
-0.325
0.788
0.226
0.114
0.341
=SUM(G2:K2)
=AVERAGE(B19:B26)
Historical data
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
=SQRT(AVERAGE(C19:C26))
=SUMPRODUCT($F$2:$J$2,F19:J19)
return
0.076
0.271
0.455
0.340
0.340
-0.136
-0.390
0.056
deviation^2
0.003
0.021
0.108
0.045
0.045
0.069
0.267
0.005
=(B19-$C$2)^2
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The decision variables are in F2:J2.
The objective function is in C3.
Cell E2 keeps track of constraint (1).
Cells C2 and C5 keep track of constraint (2).
Constraint (3) can be handled by checking the
“Unconstrained Variables Non-negative” box.
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A
1
2 Exp Return
3 StDev
4
5 req. return
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
B
C
=
=
0.220
0.128
=
0.220
return
0.042
0.450
0.366
0.205
0.076
0.194
0.175
0.253
deviation^2
0.032
0.053
0.021
0.000
0.021
0.001
0.002
0.001
Operations Management -- Prof. Juran
G
F
Ford
0.173
Lilly
0.426
Ford
14.13
15.21
19.33
28.13
37.69
50.50
43.63
26.63
28.13
Lilly
14.47
16.50
27.88
37.13
39.00
42.75
68.50
73.25
83.50
Historical data on returns
Ford
Year
0.076
1984
0.271
1985
0.455
1986
0.340
1987
0.340
1988
-0.136
1989
-0.390
1990
0.056
1991
0.127
mean
0.265
stdevp
Lilly
0.140
0.690
0.332
0.050
0.096
0.602
0.069
0.140
0.265
0.235
D
E
Total
1
Historical data
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
J
I
Merck
0.105
HP
0.241
Kellogg
8.09
10.00
17.38
25.88
26.19
32.13
33.81
37.94
65.38
Merck
5.02
5.22
7.61
13.76
17.61
19.25
25.83
29.96
55.50
HP
42.38
33.88
36.75
41.88
58.25
47.25
31.88
57.00
69.88
Kellogg
0.236
0.738
0.489
0.012
0.227
0.052
0.122
0.723
0.325
0.271
Merck
0.040
0.458
0.808
0.280
0.093
0.342
0.160
0.852
0.379
0.290
HP
-0.201
0.085
0.140
0.391
-0.189
-0.325
0.788
0.226
0.114
0.341
H
Kellogg
0.054
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Conclusions
Invest 17.3% in Ford, 42.6% in Lilly, 5.4% in
Kellogg, 10.5% in Merck, and 24.1% in HP.
The expected return will be 22%, and the standard
deviation will be 12.8%.
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2. Show how the optimal portfolio changes as the
required return varies.
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30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
A
Required Return
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0.110
0.120
0.130
0.140
0.150
0.160
0.170
0.180
0.190
0.200
0.210
0.220
0.230
0.240
B
Risk
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.115
0.116
0.119
0.123
0.128
0.133
0.139
Operations Management -- Prof. Juran
C
Return
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.179
0.180
0.190
0.200
0.210
0.220
0.230
0.240
D
Ford
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.289
0.285
0.249
0.224
0.198
0.173
0.148
0.122
E
Lilly
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.407
0.413
0.430
0.429
0.428
0.426
0.425
0.424
F
Kellogg
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.029
0.038
0.046
0.054
0.063
0.071
G
Merck
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.007
0.039
0.072
0.105
0.138
0.171
H
HP
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.304
0.302
0.286
0.271
0.256
0.241
0.226
0.211
65
©The McGraw-Hill Companies, Inc., 2004
Optimal Portfolio
100%
90%
Lilly
Proportion of Portfolio
80%
70%
60%
Kellogg
50%
Ford
40%
Merck
30%
20%
10%
0%
15%
HP
20%
25%
30%
35%
Required Return
Operations Management -- Prof. Juran
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3. Draw the efficient frontier for portfolios composed of
these five stocks.
Operations Management -- Prof. Juran
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©The McGraw-Hill Companies, Inc., 2004
Efficient Frontier
50%
45%
40%
Merck
Expected Return
35%
Kellogg
30%
25%
Lilly
20%
Ford
15%
HP
10%
5%
0%
0%
5%
10%
15%
20%
25%
30%
35%
40%
Risk (Standard Deviation)
Operations Management -- Prof. Juran
68
©The McGraw-Hill Companies, Inc., 2004
Repeat Part 2 with shorting allowed.
Operations Management -- Prof. Juran
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©The McGraw-Hill Companies, Inc., 2004
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
A
Required Return
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
0.110
0.120
0.130
0.140
0.150
0.160
0.170
0.180
0.190
0.200
0.210
0.220
0.230
0.240
B
Risk
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.114
0.115
0.116
0.119
0.123
0.128
0.133
0.139
C
Return
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.166
0.170
0.180
0.190
0.200
0.210
0.220
0.230
0.240
Operations Management -- Prof. Juran
D
Ford
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.311
0.300
0.274
0.249
0.224
0.198
0.173
0.148
0.122
E
Lilly
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.432
0.431
0.430
0.429
0.428
0.426
0.425
0.424
F
Kellogg
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.013
0.021
0.029
0.038
0.046
0.054
0.063
0.071
G
Merck
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.074
-0.059
-0.026
0.007
0.039
0.072
0.105
0.138
0.171
H
HP
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.322
0.315
0.300
0.286
0.271
0.256
0.241
0.226
0.211
70
©The McGraw-Hill Companies, Inc., 2004
Efficient Frontier
50%
45%
40%
Merck
35%
Expected Return
Kellogg
30%
25%
Lilly
20%
Ford
15%
HP
10%
5%
0%
0%
5%
10%
15%
20%
25%
30%
35%
40%
Risk (Standard Deviation)
Operations Management -- Prof. Juran
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©The McGraw-Hill Companies, Inc., 2004
Juran’s Lazy Portfolio
•Invest in Vanguard mutual funds under
university retirement plan
•No shorting
•Max 8 mutual funds
•Rebalance once per year
•Tools used:
• Excel Solver
• Basic Stats (mean, stdev, correl, beta, crude version of CAPM)
Decision Models -- Prof. Juran
72
©The McGraw-Hill Companies, Inc., 2004
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
Decision Models -- Prof. Juran
DJ
7.8%
3.9%
-14.4%
31.5%
15.1%
10.4%
15.3%
8.6%
-41.5%
45.0%
17.8%
-8.5%
5.6%
6.0%
6.6%
S&P
-10.1%
-13.0%
-23.4%
26.4%
9.0%
3.0%
13.6%
3.5%
-38.5%
23.5%
12.8%
0.0%
13.4%
29.6%
11.4%
73
©The McGraw-Hill Companies, Inc., 2004
$1 Invested 12/31/1999
$2.50
DJ
$2.00
S&P
$1.50
$1.00
$0.50
$1999
2001
2003
2005
Mean
StDev
DJ
7.3%
19.6%
Correl
Beta
0.817
0.853
Decision Models -- Prof. Juran
2007
2009
2011
2013
S&P
4.1%
18.8%
74
©The McGraw-Hill Companies, Inc., 2004
Summary
• Basic Optimization: Linear
programming
– Graphical method
– Spreadsheet Method
• Extension: Nonlinear programming
– Portfolio optimization
Operations Management -- Prof. Juran
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©The McGraw-Hill Companies, Inc., 2004