Session 6a
Overview
Multiple Objective Optimization
•
•
•
•
•
Two Dimensions
More than Two Dimensions
Finance and HR Examples
Efficient Frontier
Pre-emptive Goal Programming
Intro to Decision Analysis
Decision Models -- Prof. Juran
2
Scenario Approach Revisited
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.
Decision Models -- Prof. Juran
3
Using the same notation as in the GMS case, 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. (In the GMS case, we
used thousands of dollars as the units.)
Each investment i has a percent return under each scenario j, which
we represent with the symbol rij.
Decision Models -- Prof. Juran
4
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
Decision Models -- Prof. Juran
S0

15.21  14.13
 0.076
14.13
5
The portfolio return under any scenario j is given by:
5
R j   rij x i
i 1
Decision Models -- Prof. Juran
6
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 
Decision Models -- Prof. Juran
 R
8
j 1

2


j
R Pj
7
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 
Decision Models -- Prof. Juran
j
j 1
j  R

2
8
8
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.
Decision Models -- Prof. Juran
(1)
(2)
(3)
9
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
Decision Models -- Prof. Juran
(2)
(3)
10
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
Decision Models -- Prof. Juran
11
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 “assume
non-negative” box in the Solver Options.
Decision Models -- Prof. Juran
12
Decision Models -- Prof. Juran
13
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
Decision Models -- 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
14
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%.
Decision Models -- Prof. Juran
15
2. Show how the optimal portfolio changes as the
required return varies.
Decision Models -- Prof. Juran
16
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
Decision Models -- 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
17
Optimal Portfolio
100%
90%
Lilly
Proportion of Portfolio
80%
70%
60%
Kellogg
50%
Ford
40%
Merck
30%
20%
HP
10%
0%
15%
20%
25%
30%
35%
Required Return
Decision Models -- Prof. Juran
18
3. Draw the efficient frontier for portfolios composed of
these five stocks.
Decision Models -- Prof. Juran
19
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)
Decision Models -- Prof. Juran
20
Repeat Part 2 with shorting allowed.
Decision Models -- Prof. Juran
21
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
Decision Models -- Prof. Juran
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
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
22
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)
Decision Models -- Prof. Juran
23
GMS Case Revisited
Assuming that Torelli's goal is to minimize the
standard deviation of the portfolio return, what is
the optimal portfolio that invests all $10 million?
Decision Models -- Prof. Juran
24
Formulation
Decision Variables
The decision variables are four amounts: x1, x2, x3, and x4, representing GMS
stock, Put Option A, Put Option B, and Put Option C, respectively.
Objective
Minimize Z =  R 
 R
7
j 1
j
 R
P
2
j
Constraints
4
x
i 1
i
 10 ,000
xi  0
for all investments i.
Decision Models -- Prof. Juran
25
Optimal Solution
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
GMS price
B
C
D
E
F
G
H
I
J
K
L
100
Scenarios for GMS stock in one month
Scenario GMS price Probability
1
150
0.05
2
130
0.10
3
110
0.20
4
100
0.30
5
90
0.20
6
80
0.10
7
70
0.05
Returns from one unit of each investment
GMS
Put A
Put B
Put C
50%
-100%
-100%
-100%
30%
-100%
-100%
-100%
10%
-100%
-100%
-100%
0%
-100%
-100%
-20%
-10%
-100%
56%
60%
-20%
355%
213%
140%
-30%
809%
369%
220%
Put options on GMS stock that expire in one month
Option
A
B
C
Strike price
90
100
110
Option price
$2.20
$6.40
$12.50
Investment decision (thousands of dollars spent on each investment)
GMS
Put A
Put B
Put C
8297
-8
-665
2376
Portfolio
Return
Sqdev
2446
5201674
786
385978
-873
1077810
198
1068
230
4237
225
3543
219
2912
Now the nonnegativity conditions for the
changing cells is removed, and the investor
sells short on the put A and B options. This
lowers the standard deviation of the portfolio
(and also increases its mean).
Total
10000
=
Budget
10000
Return from portfolio ($1000)
Mean
165
Stdev
718
Units of investments purchased (shares for GMS, number of puts for options)
GMS
Put A
Put B
Put C
82972
-3798
-103843
190058
Decision Models -- Prof. Juran
26
Efficient Frontier for GMS
GMS Risk vs. Return
$600
Efficient Frontier with Shorting
Expected Return (x 1000)
$500
$400
Efficient Frontier - No Shorting
$300
Minimum Risk with Shorting
$200
Minimum Risk - No Shorting
$100
GMS Stock Only
"One-for-One"
$$-
$200
$400
$600
$800
$1,000
$1,200
$1,400
$1,600
$1,800
$2,000
Std Dev of Return (x 1000)
Decision Models -- Prof. Juran
27
Parametric Approach Revisited
From Session 5a:
(a) Determine the minimum-variance portfolio that attains an
expected annual return of at least 0.12, with no shorting of stocks
allowed.
(b) Draw the efficient frontier for portfolios composed of these
three stocks.
(c) Determine the minimum-variance portfolio that attains an
expected annual return of at least 0.12, with no shorting of stocks
allowed.
Decision Models -- Prof. Juran
28
Formulation
Objective
Minimize Z =    12 x 12   22 x 22   32 x 32  2 x 1 x 2 COV1 , 2  2 x 2 x 3COV2 , 3  2 x 1 x 3COV1 , 3
Constraints
  x 1  1  x 2  2  x 3  3  0.12
3
x
i 1
i
 1.0
For all i, x i  0
Decision Models -- Prof. Juran
(1)
(2)
(3)
29
Optimal Solution
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Mean return
Variance of return
StDev of return
B
Stock 1
0.140
0.200
0.447
C
Stock 2
0.110
0.080
0.283
D
Stock 3
0.100
0.180
0.424
Stock 1
1.00
0.80
0.70
Stock 2
0.80
1.00
0.90
Stock 3
0.70
0.90
1.00
Stock 1
0.333
Stock 2
0.667
Stock 3
0.000
Actual
0.120
>=
Required
0.120
E
F
Correlations
Stock 1
Stock 2
Stock 3
G
H
I
J
Stock 1
0.2000
0.1012
0.1328
Stock 2
0.1012
0.0800
0.1080
Stock 3
0.1328
0.1080
0.1800
Covariances
Stock 1
Stock 2
Stock 3
Investment decision
Fractions to invest
Total
1
=
Required
1
Expected portfolio return
Portfolio variance
Portfolio stdev
0.103
0.321
Decision Models -- Prof. Juran
30
SolverTable
Decision Models -- Prof. Juran
31
Decision Models -- Prof. Juran
32
SolverTable Output
0.100
0.101
0.102
0.103
0.104
0.105
0.106
0.107
0.108
0.109
0.110
0.111
0.112
0.113
0.114
0.115
0.116
0.117
Stock 1 Stock 2 Stock 3
-0.244
1.861
-0.616
-0.244
1.861
-0.616
-0.244
1.861
-0.616
-0.244
1.861
-0.616
-0.244
1.861
-0.616
-0.244
1.861
-0.616
-0.244
1.861
-0.616
-0.244
1.861
-0.616
-0.244
1.861
-0.616
-0.240
1.859
-0.619
-0.212
1.849
-0.637
-0.185
1.840
-0.655
-0.158
1.830
-0.673
-0.130
1.821
-0.691
-0.103
1.811
-0.709
-0.075
1.802
-0.726
-0.048
1.792
-0.744
-0.021
1.783
-0.762
Decision Models -- Prof. Juran
Risk
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Return
A
B
C
D
E
F
3 Input (cell $D$18) values along side, output cell(s) along top
0.109
0.109
0.109
0.109
0.109
0.109
0.109
0.109
0.109
0.109
0.110
0.111
0.112
0.113
0.114
0.115
0.116
0.117
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.240
0.241
0.241
0.242
0.244
0.245
0.247
0.249
33
Efficient Frontier
Expected Return
16%
15%
14%
Stock 1
13%
12%
11%
Stock 2
10%
Stock 3
9%
8%
20%
25%
30%
35%
40%
45%
50%
Risk
Decision Models -- Prof. Juran
34
Parametric Approach, cont.
(c) Determine the minimum-variance portfolio that
attains an expected annual return of at least 0.12, with
shorting of stocks allowed.
All we need to do here is remove the non-negativity
constraint and re-run SolverTable.
Decision Models -- Prof. Juran
35
Efficient Frontier
Expected Return
16%
15%
14%
13%
Stock 1
12%
11%
Stock 2
10%
9%
8%
20%
Stock 3
25%
30%
35%
40%
45%
50%
Risk
Decision Models -- Prof. Juran
36
Preemptive Goal Programming:
Consulting Example
The Touche Young accounting firm must complete
three jobs during the next month. Job 1 will require
500 hours of work, job 2 will require 300 hours, and
job 3 will require 100 hours. At present the firm
consists of five partners, five senior employees, and
five junior employees, each of whom can work up to
40 hours per month.
Decision Models -- Prof. Juran
37
The dollar amount (per hour) that the company can
bill depends on the type of accountant assigned to
each job, as shown in the table below. (The "X"
indicates that a junior employee does not have
enough experience to work on job 1.)
Partner
Senior employee
Junior employee
Decision Models -- Prof. Juran
Job 1
160
120
X
Job 2
120
90
50
Job 3
110
70
40
38
All jobs must be completed.
Touche Young has also set the following goals, listed
in order of priority:
•Goal 1: Monthly billings should exceed $74,000.
•Goal 2: At most one partner should be hired.
•Goal 3: At most three senior employees should be hired.
•Goal 4: At most one junior employee should be hired.
Decision Models -- Prof. Juran
39
Managerial Formulation
Decision Variables
There are three types of decisions here.
First, we need to decide how many people to hire in each
of the three employment categories.
Second, we need to assign the available human resources
(which depend on the first set of decisions) to the three
jobs.
Finally, since it is not apparent that we will be able to
satisfy all of Touche Young’s goals, we need to decide
which goals not to meet and by how much.
Decision Models -- Prof. Juran
40
Objective
In the long run we want to minimize any negative
difference between actual results and each of the four
goals. Of course, our optimization methods require that we
only have one objective at a time, so we will use a variation
of goal programming to solve the problem four times.
The approach here will be to treat each of the goals as an
objective until it is shown to be attainable, after which we
will treat it as a constraint. For example, we will solve the
model with the goal of minimizing any shortfall in the
$74,000 revenue target. Once we find a solution that has no
shortfall, we will solve the problem again, with an added
constraint that the shortfall be zero.
Decision Models -- Prof. Juran
41
Constraints
The numbers of people hired must be integers.
(1)
Each project must receive its required number of man-hours.
(2)
Our model must take into account any difference between the
actual performance of the plan and the four targets.
(3)
We can’t assign people to jobs unless we hire them.
(4)
Decision Models -- Prof. Juran
42
Mathematical Formulation
Define tk to be the target amount for goal k. For example, the target for goal 1 (the
billing goal) is $74,000; therefore t1 = 74,000.
Define δk to be the “negative difference” between what we have achieved and the
target for “current” goal k. (δ is the Greek letter delta.) In the case of the billing
goal, the negative difference would be any amount less than 74,000. In the case of
goal 2, the negative difference would be any amount of new partners hired
greater than 1. If our current solution yields $60,000 in billings, then δ1 = 74,000 –
60,000 = 14,000.
Define vk to be the best value for goal k that has previously been achieved in our
model. For example, if the best solution we can find yields only $60,000 in
billings, then v1 = 14,000.
Define xi to be the number of new hires of type i.
Define Aij to be the number of man-hours of type i assigned to job j.
Define Rij to be the number of man-hours of type i required for job j.
Decision Models -- Prof. Juran
43
Decision Variables
xi (three decisions), Aij (nine decisions), δk (up to three
decisions)
Objective
Minimize Z =  k
Constraints
All xi are integers.
(1)
Aij = Rij for all i, j.
(2)
δ Goals ≠ k = vk for all goals < k
(3)
3
A
j 1
ij
 40 x i
for all i.
Decision Models -- Prof. Juran
(4)
44
A
B
C
D
E
F
G
1 Assigned Hours Job 1
Job 2
Job 3
Assigned
Available
2
Partners
0
0
0
0
<=
200
3
Seniors
0
0
0
0
<=
200
4
Juniors
0
0
0
0
<=
200
5
Assigned
0
0
0
6
=
=
= =SUM(D2:D4)
=SUM(B4:D4)
7
Needed
500
300
100
=B10+C10-D10
=SUMPRODUCT(J2:L4,B2:D4)
8
9
Goals
Actual Under
Over
Net
Goal
10
Billings
0
0
0
0
=
74000
11 Partners Hired
0
0
0
0
=
1
12 Seniors Hired
0
0
0
0
=
3
13 Juniors Hired
0
0
0
0
=
1
14
=C10
15
Deviation
Already attained
Priority
16
Billings
0
<=
0
1
17 Partners Hired
0
<=
0
2
18 Seniors Hired
0
<=
0
3
19 Juniors Hired
0
<=
0
4
=D11
20
Decision Models -- Prof. Juran
H
I
J
Billling Rate Job 1
Partners
160
=$J$11*L9 Seniors
120
Juniors
NA
Present staff
Partners
Seniors
Juniors
5
5
5
Hours/month
40
K
L
Job 2 Job 3
120
110
90
70
50
40
Added Total
0
5
0
5
0
5
=J7+K7
45
The tk targets are in cells G10:G13.
The δk “negative difference” variables will be in B16:B19. We use the
range C10:D13 to track all deviations (both positive and negative), and
then refer to the “undesirable” one in B16:B19. For example, it is
undesirable to have billings under 74,000, so B16 refers to C10. It is
undesirable for the number of new partners to be over 1, so B17 refers to
D11.
The vk “best achieved” variables will be in D16:D19.
The xi are in K7:K9.
The Aij assignments are in B2:D4.
The Rij requirements are in B7:D7.
Cells G2:G4 keep track of constraint (4).
We constrain B4 to be zero.
Decision Models -- Prof. Juran
46
First Iteration
At first we’ll ignore all of the goals except the billing target of $74,000.
Decision Variables
xi (three decisions, cells K7:K9), Aij (nine decisions, cells B2:D4)
Objective
Minimize Z = 1 (the shortfall, if any, between planned billings and $74,000)
Constraints
All xi are integers.
(1)
Aij = Rij for all i, j.
(2)
A11 = 0.
(4)
3
A
j 1
ij
 40xi for all i.
(5)
Note that constraint (3) doesn’t matter in this iteration.
Also note the balance equation constraint, forcing E10 = G10.
Decision Models -- Prof. Juran
47
Decision Models -- Prof. Juran
48
A
B
C
D
E
F
G
1 Assigned Hours Job 1
Job 2
Job 3
Assigned
Available
2
Partners
0
0
0
0
<=
320
3
Seniors
500
0
0
500
<=
520
4
Juniors
0
300
100
400
<=
480
5
Assigned
500
300
100
6
=
=
=
7
Needed
500
300
100
8
9
Goals
Actual Under
Over
Net
Goal
10
Billings
79000
0
5000
74000
=
74000
11 Partners Hired
3
0
0
3
=
1
12 Seniors Hired
8
0
0
8
=
3
13 Juniors Hired
7
0
0
7
=
1
14
15
Deviation
Already attained
Priority
16
Billings
0
<=
0
1
17 Partners Hired
0
<=
0
2
18 Seniors Hired
0
<=
0
3
19 Juniors Hired
0
<=
0
4
H
I
J
Billling Rate Job 1
Partners
160
Seniors
120
Juniors
NA
K
L
Job 2 Job 3
120
110
90
70
50
40
Present staff
Partners
Seniors
Juniors
5
5
5
Added Total
3
8
8
13
7
12
Hours/month
40
We have verified that it is feasible to have billings of $74,000.
Decision Models -- Prof. Juran
49
Second Iteration
Meeting the first goal will now be a constraint, and we’ll focus on the second
goal.
Decision Variables
xi (as before), Aij (as before), δk (for the billings goal)
Objective
Minimize Z =  2 (the number of partners hired more than the goal of 1)
Constraints
All xi are integers.
(1)
Aij = Rij for all i, j.
(2)
δ 1 = v1 (forcing the 74,000 billing goal to be met)
(3)
A11 = 0.
(4)
3
A
j 1
ij
 40 x i for all i.
(5)
Note that v1 = 0 from the previous iteration.
Decision Models -- Prof. Juran
50
Decision Models -- Prof. Juran
51
A
B
C
D
E
F
G
1 Assigned Hours Job 1
Job 2
Job 3
Assigned
Available
2
Partners
0
0
0
0
<=
240
3
Seniors
500
0
0
500
<=
720
4
Juniors
0
300
100
400
<=
400
5
Assigned
500
300
100
6
=
=
=
7
Needed
500
300
100
8
9
Goals
Actual Under
Over
Net
Goal
10
Billings
79000
0
5000.000171
74000
=
74000
11 Partners Hired
1
0
0
1
=
1
12 Seniors Hired
13
0
0
13
=
3
13 Juniors Hired
5
0
0
5
=
1
14
15
Deviation
Already attained
Priority
16
Billings
0
<=
0
1
17 Partners Hired
0
<=
0
2
18 Seniors Hired
0
<=
0
3
19 Juniors Hired
0
<=
0
4
H
I
J
Billling Rate Job 1
Partners
160
Seniors
120
Juniors
NA
K
L
Job 2 Job 3
120
110
90
70
50
40
Present staff
Partners
Seniors
Juniors
5
5
5
Added Total
1
6
13
18
5
10
Hours/month
40
Now we know that it is feasible to achieve both of the first two goals.
Decision Models -- Prof. Juran
52
Third Iteration
Meeting both the first and second goals will now be constraints, and we’ll focus
on the third goal.
Decision Variables
xi (three decisions), Aij (nine decisions), δk (two decisions)
Objective
Minimize Z = 3 (the number of senior employees hired above the goal of 3)
Constraints
All xi are integers.
(1)
Aij = Rij for all i, j.
(2)
δ Goals  k = vk (for goals 1 and 2)
(3)
A11 = 0.
(4)
3
A
j 1
ij
 40xi for all i.
(5)
Note that v1 = v2 = 0 from the previous iteration.
Decision Models -- Prof. Juran
53
Decision Models -- Prof. Juran
54
A
B
C
D
E
F
G
1 Assigned Hours Job 1
Job 2
Job 3
Assigned
Available
2
Partners
195.25 6E-07
6.2152E-07
195.25
<=
200
3
Seniors
304.75
0
0
304.75
<=
320
4
Juniors
0
300
99.99999938
400
<=
400
5
Assigned
500
300
100
6
=
=
=
7
Needed
500
300
100
8
9
Goals
Actual Under
Over
Net
Goal
10
Billings
86810
0
12809.99833
74000
=
74000
11 Partners Hired
0
1
0
1
=
1
12 Seniors Hired
3
0
0
3
=
3
13 Juniors Hired
5
0
0
5
=
1
14
15
Deviation
Already attained
Priority
16
Billings
0
<=
0
1
17 Partners Hired
0
<=
0
2
18 Seniors Hired
0
<=
0
3
19 Juniors Hired
0
<=
0
4
H
I
J
Billling Rate Job 1
Partners
160
Seniors
120
Juniors
NA
K
L
Job 2 Job 3
120
110
90
70
50
40
Present staff
Partners
Seniors
Juniors
5
5
5
Added Total
0
5
3
8
5
10
Hours/month
40
All of the first three goals are feasible.
Decision Models -- Prof. Juran
55
Fourth Iteration
Meeting all of the first three goals will now be constraints, and we’ll focus on the
fourth goal.
Decision Variables
xi (three decisions), Aij (nine decisions), δk (for the first three goals)
Objective
Minimize Z = 4
(the number of new juniors hired above the goal of 1)
Constraints
All xi are integers.
(1)
Aij = Rij for all i, j.
(2)
δ Goals  k = vk for goals 1, 2, 3
(3)
A11 = 0.
(4)
3
A
j 1
ij
 40xi for all i.
(5)
Note that v1 = v2 = v3 = 0 from the previous iteration.
Decision Models -- Prof. Juran
56
Decision Models -- Prof. Juran
57
A
B
C
D
E
F
G
1 Assigned Hours Job 1
Job 2
Job 3
Assigned
Available
2
Partners
195
7
30
232
<=
240
3
Seniors
305
15
0
320
<=
320
4
Juniors
0
278
70
348
<=
360
5
Assigned
500
300
100
6
=
=
=
7
Needed
500
300
100
8
9
Goals
Actual Under
Over
Net
Goal
10
Billings
89990
0
15990
74000
=
74000
11 Partners Hired
1
0
0
1
=
1
12 Seniors Hired
3
0
0
3
=
3
13 Juniors Hired
4
0
3
1
=
1
14
15
Deviation
Already attained
Priority
16
Billings
0
<=
0
1
17 Partners Hired
0
<=
0
2
18 Seniors Hired
0
<=
0
3
19 Juniors Hired
3
<=
0
4
Decision Models -- Prof. Juran
H
I
J
K
L
Billling Rate Job 1 Job 2 Job 3
Partners
160
120
110
Seniors
120
90
70
Juniors
NA
50
40
Present staff
Partners
Seniors
Juniors
5
5
5
Hours/month
40
Added Total
1
6
3
8
4
9
58
Fifth Iteration
We can do one more round here. Given that we are going to hire 1 partner, 3
seniors, and 4 juniors, why not maximize the revenue from that combination of
workers?
Decision Variables
xi (three decisions), Aij (nine decisions), δk (for the 2nd, 3rd, and 4th goals)
Objective
Maximize Z = Revenue
Constraints
All xi are integers.
(1)
Aij = Rij for all i, j.
(2)
δk = vk for goals 2, 3, 4
(3)
A11 = 0.
(4)
3
A
j 1
ij
 40 x i for all i.
(5)
Note that v2 = v3 = 0, and v4 = 3, from the previous iteration.
Decision Models -- Prof. Juran
59
Decision Models -- Prof. Juran
60
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
A
Assigned Hours
Partners
Seniors
Juniors
Assigned
C
Job 2
0
60
240
300
=
300
D
Job 3
0
0
100
100
=
100
E
Assigned
240
320
340
Needed
B
Job 1
240
260
0
500
=
500
Goals
Billings
Partners Hired
Seniors Hired
Juniors Hired
Actual
91000
1
3
4
Under
0
0
0
0
Over
0
0
0
3
Net
91000
1
3
1
<=
<=
<=
<=
Already attained
0
0
0
0
Deviation
Billings
0
Partners Hired
0
Seniors Hired
0
Juniors Hired
3
F
<=
<=
<=
=
=
=
=
G
Available
240
320
360
Goal
74000
1
3
1
H
I
J
K
L
Billling Rate Job 1 Job 2 Job 3
Partners
160
120
110
Seniors
120
90
70
Juniors
NA
50
40
Present staff
Partners
Seniors
Juniors
5
5
5
Hours/month
40
Added Total
1
6
3
8
4
9
Priority
1
2
3
4
There are two constraints that don’t show in the window:
$E$2:$E$4<=$G$2:$G$4, and $K$7:$K$9 = integer.
Decision Models -- Prof. Juran
61
Conclusions
It would appear that it is infeasible to reach all four of the
goals.
However, the first three can be met while delivering all of
the jobs as required.
The only goal that can’t be attained is the limit of hiring
only one new junior-level employee. The best solution we
found requires us to hire four new employees at this level.
It is possible to have revenues of $91,000 under this hiring
plan.
Decision Models -- Prof. Juran
62
Decision Analysis is a tool for studying situations in which there are several
decision alternatives and a set of uncertain future events.
Decision Alternatives are elements of a set of possible choices, represented by
d1, d2, d3, … di
States of Nature are elements of a set of N random future events, represented by
s1 , s 2 , s 3 , … sN
A Payoff Table lists outcomes associated with some combination of decision
alternative and state of nature. The payoff for decision alternative i under state of
nature j is symbolized by vij.
Decision analysis is a natural extension of our previous work with conditional
probability and systems of probabilities and payoffs; the only new element here
is the opportunity for a decision maker to make choices at certain discrete points
in time.
Decision Models -- Prof. Juran
63
A buyer for a large department store chain must place orders with an athletic
shoe manufacturer 6 months prior to the time the shoes will be sold in the
department stores. In particular, the buyer must decide on November 1 how
many pairs of the manufacturer's newest model of tennis shoes to order for sale
during the upcoming summer season.
Assume that each pair of this new brand of tennis shoes costs the department
store chain $45 per pair. Furthermore, assume that each pair of these shoes can
then be sold to the chain's customers for $70 per pair. Any pairs of these shoes
remaining unsold at the end of the summer season will be sold in a closeout sale
next fall for $35 each. Finally, assume that the department store chain must
purchase these tennis shoes from the manufacturer in lots of 100 pairs.
Decision Models -- Prof. Juran
64
The probability distribution of consumer demand for these tennis shoes (in
hundreds of pairs) during the upcoming summer season has been assessed by
market research specialists and is provided in the table below.
Consumer Demand
1
2
3
4
5
6
Decision Models -- Prof. Juran
Probability
0.05
0.15
0.25
0.30
0.15
0.10
65
Basically, the contribution is given by the following formula, in which profit is
symbolized by the Greek letter π, purchase quantities (in hundreds) are
represented by p and demand states (in hundreds) are represented by d.
 ij
= Revenue from Regular Price + Revenue from Closeout Price – Cost to Purchase


 


 $70 * 100 * min p i , d j  $35 * 100 * max 0 , p i  d j  p i * 100 * 45 
For example, if we purchase 400 shoes and demand is 200, then p = 4 and d = 2.
 ij  $70 * 100 * min p i , d j   $35 * 100 * max 0 , p i  d j   p i * 100 * 45

 

 $70 * 100 * 2   $35 * 100 * 2   4 * 100 * 45
 $14 ,000  $7 ,000  $18 ,000
 $3 ,000
Decision Models -- Prof. Juran
66
1
2
3
4
A
$45
$70
$35
B
C
D
E
Cost of each pair of new tennis shoes
Selling price of each pair of shoes
Closeout sale price of each leftover pair
5
F
G
H
Payoff Table
Consumer Demand
6
7 Purchased
1
2
3
4
5
6
8
1
$2,500
$2,500
$2,500 $2,500
$2,500
$2,500
9
2
$1,500
$5,000
$5,000 $5,000
$5,000
$5,000
10
3
$500
$4,000
$7,500 $7,500
$7,500
$7,500
11
4
($500)
$3,000
$6,500 $10,000 $10,000 $10,000
12
5
($1,500) $2,000
$5,500 $9,000 $12,500 $12,500
13
6
($2,500) $1,000
$4,500 $8,000 $11,500 $15,000
14 Probability
0.05
0.15
0.25
0.3
0.15
0.1
15
=($A$2*100*MIN(G$7,$A13))+($A$3*100*MAX(0,($A13-G$7)))-($A13*$A$1*100)
16
Decision Models -- Prof. Juran
67
For each possible strategy, we can calculate the expected revenue and the
standard deviation of revenue.
A
5
B
C
D
E
F
G
H
I
J
K
Payoff Table
Consumer Demand
6
7 Purchased
1
2
3
4
5
6
Expected Revenue Std Dev of Revenue Variance
8
1
$2,500
$2,500
$2,500
$2,500
$2,500
$2,500
$2,500
$0
0
9
2
$1,500
$5,000
$5,000
$5,000
$5,000
$5,000
$4,825
$184
33993.75
10
3
$500
$4,000
$7,500
$7,500
$7,500
$7,500
$6,625
$625
390468.8
11
4
($500)
$3,000
$6,500
$10,000 $10,000 $10,000
$7,550
$1,197
1432025
12
5
($1,500) $2,000
$5,500
$9,000
$12,500 $12,500
$7,425
$1,467
2153244
13
6
($2,500) $1,000
$4,500
$8,000
$11,500 $15,000
$6,775
$1,613
2602819
14 Probability
0.05
0.15
0.25
0.3
0.15
0.1
15
=SUMPRODUCT(B13:G13,$B$14:$G$14) =SQRT(J13)
16
17 =(($B$14*(B13-H13))^2)+(($C$14*(C13-H13))^2)+(($D$14*(D13-H13))^2)+(($E$14*(E13-H13))^2)+(($F$14*(F13-H13))^2)+(($G$14*(G13-H13))^2)
18
The variance formula looks ugly, but it works.
Decision Models -- Prof. Juran
68
Risk Profile
$8,000
Expected Revenue
$7,000
Buy 400 Pairs
Buy 500 Pairs
Buy 600 Pairs
Buy 300 Pairs
$6,000
$5,000
Buy 200 Pairs
$4,000
$3,000
Buy 100 Pairs
$2,000
$1,000
$0
$0
$200
$400
$600
$800
$1,000
$1,200
$1,400
$1,600
$1,800
Std. Deviation of Revenue
Decision Models -- Prof. Juran
69
Summary
Multiple Objective Optimization
•
•
•
•
•
Two Dimensions
More than Two Dimensions
Finance and HR Examples
Efficient Frontier
Pre-emptive Goal Programming
Intro to Decision Analysis
Decision Models -- Prof. Juran
70