Table of Contents
Chapter 8 (Nonlinear Programming)
The Challenges of Nonlinear Programming (Section 8.1)
NLP with Decreasing Marginal Returns: Wyndor (Section 8.2)
NLP with Decreasing Marginal Returns: Portfolio Selection (Section 8.2)
Separable Programming (Section 8.3)
Difficult Nonlinear Programming Problems (Section 8.4)
Evolutionary Solver and Genetic Algorithms (Section 8.5)
8.2–8.15
8.16–8.20
8.21–8.25
8.26–8.38
8.39–8.40
8.41–8.54
Nonlinear and Separable Programming (UW Lecture)
8.55–8.70
These slides are based upon a lecture from the MBA elective course “Modeling with Spreadsheets”
at the University of Washington (as taught by one of the authors).
Evolutionary Solver (UW Lecture)
8.71–8.82
These slides are based upon a lecture from the MBA elective course “Modeling with Spreadsheets”
at the University of Washington (as taught by one of the authors).
McGraw-Hill/Irwin
8.1
© The McGraw-Hill Companies, Inc., 2008
Examples of Linear and Nonlinear Formulas
Linear Formulas
Nonlinear Formulas
SUMPRODUCT(D4:D6, C4:C6)
[(D1 + D2) / D3] * C4
IF(D2 >= 2, 2*C3, 3*C4)
SUMIF(D1:D6, 4, C1:C6)
SUM(D4:D6)
2*C1 + 3*C4 + C6
C1 + C2 + C3
SUMPRODUCT(C4:C6, C1:C3)
[(C1 + C2) / C3] * D4
IF(C2 >= 2, 2*C3, 3*C4)
SUMIF(C1:C6, 4, D1:D6)
ROUND(C1)
MAX(C1, 0)
MIN(C1, C2)
ABS(C1)
SQRT(C1)
C1 * C2
C1 / C2
C1 ^2
Data cells are located in D1:D6 and changing cells are in C1:C6.
McGraw-Hill/Irwin
8.2
© The McGraw-Hill Companies, Inc., 2008
The Challenges of Nonlinear Programming
•
Nonlinear programming is used to model nonproportional relationships
between activity levels and the overall measure of performance, whereas linear
programming assumes a proportional relationship.
•
Constructing the nonlinear formula(s) needed for a nonlinear programming
model is considerably more difficult than developing the linear formulas used
in linear programming.
•
Solving a nonlinear programming model is often much more difficult (if it is
possible at all) than solving a linear programming model.
McGraw-Hill/Irwin
8.3
© The McGraw-Hill Companies, Inc., 2008
The Challenge of Nonproportional Relationships
•
Proportionality Assumption of Linear Programming: The contribution of
each activity to the value of the objective function is proportional to the level
of the activity. In other words, the term in the objective function involving this
activity consists of a coefficient times the decision variable.
•
Nonlinear programming problems arise when any activity has a
nonproportional relationship where the contribution of the activity to the
measure of performance is not proportional to the level of the activity.
McGraw-Hill/Irwin
8.4
© The McGraw-Hill Companies, Inc., 2008
Profit Graphs for Wyndor Glass Co.
(Proportional Relationship)
Weekly
Profit
($)
Weekly
Profit
($)
1200
3000
2500
900
2000
600
1500
1000
300
500
0
2
Production rate for doors
McGraw-Hill/Irwin
4
0
D
8.5
4
2
Production rate for windows
6
W
© The McGraw-Hill Companies, Inc., 2008
Profit Graphs with Nonproportional Relationships
Piecewise Linear with
Decreasing Marginal Returns
Decreasing Marginal Returns
Profit
Profit
Level of an activity
(a)
Profit
McGraw-Hill/Irwin
Level of an activity
(b)
8.6
Profit
© The McGraw-Hill Companies, Inc., 2008
Profit Graphs with Nonproportional Relationships
Level of an activity
(a)
Level of an activity
(b)
Decreasing Marginal Returns
Except for Discontinuities
Increasing Marginal Returns
Profit
Profit
Level of an activity
Level of an activity
(c)
McGraw-Hill/Irwin
(d)
8.7
© The McGraw-Hill Companies, Inc., 2008
Constructing a Nonlinear Formula
A
1
B
C
Constructing a Nonlinear Formula
2
3
4
5
6
7
8
Level of Activity
2
4
5
7
10
Profit
$16
$24
$28
$30
$33
Profit vs. Level of Activity
Profit
$35
$30
$25
$20
$15
$10
$5
$0
0
2
4
6
8
10
Level of Activity (x)
McGraw-Hill/Irwin
8.8
© The McGraw-Hill Companies, Inc., 2008
Format Trendline Dialogue Box
McGraw-Hill/Irwin
8.9
© The McGraw-Hill Companies, Inc., 2008
The Trendline (Quadratic Equation)
A
1
B
C
Constructing a Nonlinear Formula
2
3
4
5
6
7
8
Level of Activity
2
4
5
7
10
Profit
$16
$24
$28
$30
$33
Profit vs. Level of Activity
Profit
$35
$30
$25
$20
$15
$10
y = -0.3002x2 + 5.661x + 6.1477
$5
$0
0
2
4
6
8
10
Level of Activity (x)
McGraw-Hill/Irwin
8.10
© The McGraw-Hill Companies, Inc., 2008
Solving Nonlinear Programming Models
Consider the following model in algebraic form:
Maximize Profit = 0.5x5 – 6x4 + 24.5x3 – 39x2 + 20x
subject to
x≤5
x≥0
McGraw-Hill/Irwin
8.11
© The McGraw-Hill Companies, Inc., 2008
Solver Solution Starting with x = 0
A
1
B
C
D
E
A Simple NLP
2
3
4
x=
0.371
<=
Maximum
5
5
6
7
McGraw-Hill/Irwin
Profit = 0.5x
=
5
4
-6x +24.5x
3
2
-39x +20x
$3.19
8.12
© The McGraw-Hill Companies, Inc., 2008
Solver Solution Starting with x = 3
A
1
B
C
D
E
A Simple NLP
2
3
4
x=
3.126
<=
Maximum
5
5
6
7
McGraw-Hill/Irwin
Profit = 0.5x
=
5
4
-6x +24.5x
3
2
-39x +20x
$6.13
8.13
© The McGraw-Hill Companies, Inc., 2008
Solver Solution Starting with x = 4.7
A
1
B
C
D
E
A Simple NLP
2
3
4
x=
5.000
<=
Maximum
5
5
6
7
McGraw-Hill/Irwin
Profit = 0.5x
=
5
4
-6x +24.5x
3
2
-39x +20x
$0.00
8.14
© The McGraw-Hill Companies, Inc., 2008
The Profit Graph
P ro fit ($)
x
McGraw-Hill/Irwin
8.15
© The McGraw-Hill Companies, Inc., 2008
Original Wyndor Glass Co. Spreadsheet
A
1
B
C
D
E
F
G
Wyndor Glass Co. Product-Mix Problem
2
3
4
Unit Profit
Doors
Windows
$300
$500
5
6
Hours Used Per Unit Produced
1
0
7
Plant 1
8
Plant 2
0
9
Plant 3
3
Hours
Hours
Used
Available
2
<=
4
2
12
<=
12
2
18
<=
18
10
11
12
Doors
Units Produced
McGraw-Hill/Irwin
Windows
2
6
8.16
Total Profit
$3,600
© The McGraw-Hill Companies, Inc., 2008
Quadratic (non-linear) programs
•
A quadratic program is an example of a non-linear program in which each
constraint is linear, but the objective function has the form:
f(x1, x2, …, xn) = i→nj→ncijx1xj + i→ndixi
Where cij and di are known constants. Example:
Quadratic program with
linear constraints, quadratic
objective function, n = 2
variables, c11 = 1, c12 = c21 = 0,
c22 = 1, and d1 = d2 = 0.
Introducti
minimize: z = x12 + x22
subject to: x1 – x2 = 3
x2 ≥ 3
McGraw-Hill/Irwin
Issue 8. Non-linear programming
on to
Wyndor Glass with Marketing Costs
•
Market research indicates that Wyndor could sell small numbers of doors and
windows with no advertising. However, extensive advertising would be
required to sell all that could be produced.
•
A curve-fitting procedure was used to estimate the weekly marketing costs
required to sustain a production rate of D doors and W windows:
– Marketing cost for doors = $25D2
– Marketing costs for windows = ($662/3)W2
•
The gross profit per door sold is about $375, and the gross profit per window
is about $700. Therefore, the net profits are as follows:
– Net profit for doors = $375D – $25D2
Net profit for windows = $700W – ($662/3)W2
•
Thus, the revised objective function is
Maximize Profit = $375D – 25D2 + $700W –($662/3)W2
Question: Considering the nonlinear marketing costs, how many doors and
windows should Wyndor produce?
McGraw-Hill/Irwin
8.18
© The McGraw-Hill Companies, Inc., 2008
Profit Graphs for Doors and Windows
Weekly
profit
($)
1,800
1,600
Weekly
profit
($)
1,400
1,200
1,200
1,000
1,000
800
800
600
600
400
400
200
200
0
2
4
D
0
Production rate for doors
McGraw-Hill/Irwin
2
4
6
W
Production rate for windows
8.19
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
A
1
B
C
D
E
F
G
H
Wyndor Problem With Nonlinear Marketing Costs
2
3
4
Unit Profit (Gross)
Doors
Windows
$375
$700
5
Hours
Hours
6
Used
Available
Hours Used Per Unit Produced
1
0
7
Plant 1
8
Plant 2
0
9
Plant 3
3
3.214
<=
4
2
8.357
<=
12
2
18
<=
18
10
11
12
Doors
Windows
Units Produced
3.214
4.179
Gross Profit from Sales
$4,130
Marketing Cost
$258
$1,164
Total Marketing Cost
$1,422
13
14
15
16
McGraw-Hill/Irwin
Total Profit
8.20
$2,708
© The McGraw-Hill Companies, Inc., 2008
Graphical Display of Nonlinear Formulation
W
Production rate for windows
6
5
(3
3
,4
14
5
) = optimal solution
28
4
Profit = $2,800
3
Feasible
region
Profit = $2,708
Profit = $2,600
Profit = $2,500
2
1
0
McGraw-Hill/Irwin
1
2
3
Production rate for doors
8.21
4
5
D
© The McGraw-Hill Companies, Inc., 2008
Portfolio Selection
•
It is now common practice for professional managers of large stock portfolios
to use computer models based on nonlinear programming to guide them.
•
Investors are concerned about both the expected return and the risk.
•
One way of formulating their approach is as a nonlinear version of a costbenefit trade-off problem:
– Minimize Risk
subject to Expected return ≥ Minimum acceptable level
•
Consider a portfolio with 3 stocks.
Question: What is the portfolio that will minimize the risk subject to
achieving at least an 18% expected return?
McGraw-Hill/Irwin
8.22
© The McGraw-Hill Companies, Inc., 2008
Data for Stocks
Stock
Expected
Return
Risk
(Standard
Deviation)
Pair
of
Stocks
Joint Risk
per Stock
(Covariance)
1
21%
25%
1 and 2
0.040
2
30
45
1 and 3
–0.005
3
8
5
2 and 3
–0.010
McGraw-Hill/Irwin
8.23
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Minimize Risk = (0.25S1)2+(0.45S2)2+(0.05S3)2+2(0.04)S1S2+2(–0.005)S1S3+2(–0.01)S2S3
subject to
(21%)S1 + (30%)S2 + (8%)S3 ≥ 18%
S1 + S2 + S3 = 100%
and
S1 ≥ 0, S2 ≥ 0, S3 ≥ 0.
McGraw-Hill/Irwin
8.24
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Model
A
1
B
C
D
E
F
G
H
=
100%
Portfolio Selection Problem (Nonlinear Programming)
2
3
4
Stock 1
Stock 2
Stock 3
Expected Return
21%
30%
8%
Risk (Stand. Dev.)
25%
45%
5%
Joint Risk (Covar.)
Stock 1
Stock 2
Stock 3
0.040
-0.005
5
6
7
8
9
Stock 1
10
Stock 2
11
Stock 3
-0.010
12
13
14
Portfolio
Stock 1
Stock 2
Stock 3
Total
40.2%
21.7%
38.1%
100%
15
16
Minimum
17
Expected
18
19
Portfolio
Expected Return
18.0%
Risk (Variance)
0.0238
Risk (Stand. Dev.)
15.4%
Return
>=
18.0%
20
21
22
23
McGraw-Hill/Irwin
8.25
© The McGraw-Hill Companies, Inc., 2008
Portfolio selection spreadsheet model
McGraw-Hill/Irwin
Introducti
on to
Issue 8. Non-linear programming
Using Solver Table to Examine Trade-Offs
Between Expected Return and Risk
B
C
D
E
25
26
F
Risk
Min Return
27
G
Expected
Stock 1
Stock 2
Stock 3
(St. Dev.)
Return
40.2%
21.7%
38.1%
15.4%
18.0%
28
8%
7.1%
3.7%
89.1%
3.9%
9.7%
29
10%
8.1%
4.3%
87.6%
3.9%
10.0%
30
12%
16.2%
8.6%
75.2%
5.6%
12.0%
31
14%
24.2%
13.0%
62.8%
8.6%
14.0%
32
16%
32.2%
17.3%
50.5%
12.0%
16.0%
33
18%
40.2%
21.7%
38.1%
15.4%
18.0%
34
20%
48.2%
26.1%
25.7%
18.9%
20.0%
35
22%
56.2%
30.4%
13.4%
22.5%
22.0%
36
24%
64.2%
34.8%
1.0%
26.1%
24.0%
37
26%
44.4%
55.6%
0.0%
30.8%
26.0%
38
28%
22.2%
77.8%
0.0%
37.3%
28.0%
39
30%
0.0%
100.0%
0.0%
45.0%
30.0%
8.27
© The McGraw-Hill Companies, Inc., 2008
McGraw-Hill/Irwin
Wyndor Glass When Overtime is Needed
•
Wyndor Glass has accepted a special order for hand-crafted goods to be made
in plants 1 and 2 throughout the next four months.
•
Filling this order will require borrowing certain employees from the work
crews of regular products.
•
The remaining workers will need to work overtime to utilize the full
production capacity of each plant’s machinery for the regular products.
•
The original constraints of Hours Used ≤ Hours Available are still valid.
However, the objective function will need to be modified because of the
additional cost of using overtime work.
•
In particular, because of the additional cost, the profit per unit will be reduced
for those units that require overtime.
Question: Considering overtime costs, how many doors and windows should
Wyndor produce?
McGraw-Hill/Irwin
8.28
© The McGraw-Hill Companies, Inc., 2008
Data for Wyndor When Overtime is Needed
Maximum Weekly Production
Product
Regular
Time
Overtime
Doors
3
Windows
3
Profit per Unit Produced
Total
Regular
Time
Overtime
1
4
$300
$200
3
6
500
100
(and 3D + 2W ≤ 18)
McGraw-Hill/Irwin
8.29
© The McGraw-Hill Companies, Inc., 2008
Profit Graphs for Doors and Windows
Weekly
profit
($)
1,800
1,500
Weekly
profit
($)
1,100
900
0
3
Production rate for doors
McGraw-Hill/Irwin
4
D
0
8.30
3
Production rate for windows
6
W
© The McGraw-Hill Companies, Inc., 2008
The Separable Programming Technique
•
For each activity that violates the proportionality assumption, separate its
profit graph into parts, with a line segment in each part.
•
Then, instead of using a single decision variable to represent the level of each
such activity, introduce a separate new decision variable for each line segment
on that activity’s profit graph.
•
Since the proportionality assumption holds for these new decision variables,
formulate a linear programming model in terms of these variables.
•
For the Wyndor problem, these new decision variables are
– DR = Number of doors produced per week on regular time
– DO = Number of doors produced per week on overtime
– WR = Number of windows produced per week on regular time
WO = Number of windows produced per week on overtime
McGraw-Hill/Irwin
8.31
© The McGraw-Hill Companies, Inc., 2008
Separable Programming Spreadsheet Model
A
1
B
C
D
E
F
G
Wyndor Problem with Overtime (Separable Programming)
2
3
Unit Profit
Doors
Windows
4
Regular
$300
$500
5
Overtime
$200
$100
6
7
Hours Used Per Unit Produced
1
0
8
Plant 1
9
Plant 2
0
10
Plant 3
3
Hours
Hours
Used
Available
4
<=
4
2
6
<=
12
2
18
<=
18
11
12
Units Produced
13
Doors
Maximum
Doors
Windows
Windows
14
Regular
3
3
<=
3
3
15
Overtime
1
0
<=
1
3
16
Total Produced
4
3
17
18
McGraw-Hill/Irwin
Total Profit
$2,600
8.32
© The McGraw-Hill Companies, Inc., 2008
Separable programming spreadsheet model
McGraw-Hill/Irwin
Introducti
on to
Issue 8. Non-linear programming
Separable Programming with Smooth Profit Graphs
Profit
Profit graph
Approximation
Level of activity
McGraw-Hill/Irwin
8.34
© The McGraw-Hill Companies, Inc., 2008
Advantages of Separable Programming
•
The Excel Solver can readily solve nonlinear problems that have decreasing
marginal returns, with the advantage that no approximation is needed.
•
However, the separable programming approach also has certain advantages:
– Converting the problem into a linear programming problem tends to make it
quicker to solve, which can be very helpful for large problems.
– A linear programming formulation makes available Solver’s Sensitivity Report.
– Separable programming only requires estimating the profit from each activity at a
few points. Therefore, it is not necessary to use a curve fitting method to estimate
the formula for the profit graph.
McGraw-Hill/Irwin
8.35
© The McGraw-Hill Companies, Inc., 2008
Wyndor Problem with Both Overtime Costs and
Nonlinear Marketing Costs
•
The previous spreadsheet model does not include nonlinear marketing costs.
•
Recall that the curve-fitting procedure was used to estimate the weekly
marketing costs required to sustain a production rate of D doors and W
windows:
– Marketing cost for doors = $25D2
– Marketing costs for windows = ($662/3)W2
Question: Considering both overtime costs and nonlinear marketing costs,
how many doors and windows should Wyndor produce?
McGraw-Hill/Irwin
8.36
© The McGraw-Hill Companies, Inc., 2008
Data for Wyndor with Overtime Costs and
Nonlinear Marketing Costs
Maximum Weekly Production
Regular
Time
Overtime
Doors
3
Windows
3
Product
McGraw-Hill/Irwin
Gross Unit Profit
Total
Regular
Time
Overtime
Marketing
Costs
1
4
$375
$275
$25D2
3
6
700
300
662/3W2
8.37
© The McGraw-Hill Companies, Inc., 2008
Weekly Profit from Producing Doors
D
Gross
Profit
Marketing
Costs
Profit
Incremental
Profit
0
$0
$0
$0
—
1
375
25
350
350
2
750
100
650
300
3
1,125
225
900
250
4
1,400
400
1,000
100
McGraw-Hill/Irwin
8.38
© The McGraw-Hill Companies, Inc., 2008
Weekly Profit from Producing Windows
W
Gross
Profit
Marketing
Costs
Profit
Incremental
Profit
0
$0
$0
$0
—
1
700
662/3
6331/3
6331/3
2
1,400
2662/3
1,1331/3
500
3
2,100
600
1,500
3662/3
4
2,400
1,0662/3
1,3331/3
–1662/3
5
2,700
1,6662/3
1,0331/3
–300
6
3,000
2,400
600
–4331/3
McGraw-Hill/Irwin
8.39
© The McGraw-Hill Companies, Inc., 2008
Separable Programming Spreadsheet Model
A
1
B
C
D
E
F
G
Wyndor with Overtime and Marketing Costs (Separable)
2
3
Unit Profit
Doors
Windows
4
Regular (0-1)
$350.00
$633.33
5
Regular (1-2)
$300.00
$500.00
6
Regular (2-3)
$250.00
$367.67
7
Overtime
$100.00
-$300.00
8
9
10
Hours Used Per Unit Produced
Hours
Hours
Used
Available
11
Plant 1
1
0
4
<=
4
12
Plant 2
0
2
6
<=
12
13
Plant 3
3
2
18
<=
18
14
15
Units Produced
16
Doors
Windows
Doors
Maximum
Windows
17
Regular (0-1)
1
1
<=
1
1
18
Regular (1-2)
1
1
<=
1
1
19
Regular (2-3)
1
1
<=
1
1
20
Overtime
1
0
<=
1
3
21
Total Produced
4
3
22
23
McGraw-Hill/Irwin
Total Profit
$2,501
8.40
© The McGraw-Hill Companies, Inc., 2008
Separable Programming Spreadsheet Model
McGraw-Hill/Irwin
Introducti
on to
Issue 8. Non-linear programming
Nonlinear Programming Spreadsheet Model
A
1
B
C
D
E
F
G
H
Wyndor With Overtime and Marketing Costs (Nonlinear Programming)
2
3
Unit Profit (Gross)
Doors
Windows
4
Regular
$375
$700
5
Overtime
$275
$300
6
7
8
Hours Used Per Unit Produced
1
0
Hours
Hours
Used
Available
9
Plant 1
4
<=
4
10
Plant 2
0
2
6
<=
12
11
Plant 3
3
2
18
<=
18
12
13
14
Units Produced
Doors
Maximum
Doors
Windows
Windows
15
Regular
3
3
<=
3
3
16
Overtime
1
0
<=
1
3
17
Total Produced
4
3
18
19
Marketing Cost
$400
$600
20
McGraw-Hill/Irwin
Gross Profit from Sales
$3,500
Total Marketing Cost
$1,000
Total Profit
8.42
$2,500
© The McGraw-Hill Companies, Inc., 2008
Nonlinear Programming Spreadsheet Model
McGraw-Hill/Irwin
Introducti
on to
Issue 8. Non-linear programming
Difficult Nonlinear Programming Problems
•
Even if a model has a nonlinear objective function, so long as the model has
certain properties (e.g., linear constraints, decreasing marginal returns), the
Solver can easily find an optimal solution.
•
In some cases separable programming can be used to model a nonlinear
problem in such a way that linear programming can be used.
•
However, if a problem has increasing marginal returns, or nonlinear functions
in the constraints, or disconnected profit graphs, finding a solution is often
much more difficult.
– such problems may have many local optima
– Solver can get stuck at local optima, rather than finding the global optimum
•
One approach with such problems is to solve the problem many times, each
time starting with a different initial solution.
– Solver Table can be used to do this process more systematically when there are
only one or two variables.
McGraw-Hill/Irwin
8.44
© The McGraw-Hill Companies, Inc., 2008
Using Solver Table to Try Different Starting Points
A
1
B
C
D
E
F
G
H
I
Using Solver Table to Try Different Starting Points
2
3
4
Maximum
x=
0.371
<=
Starting
5
Point
Solution
x *
Profit
0.371
$3.19
0
0.371
$3.19
8
1
0.371
$3.19
9
2
3.126
$6.13
10
3
3.126
$6.13
11
4
3.126
$6.13
12
5
5.000
$0.00
5
6
7
x
Profit = 0.5x
=
McGraw-Hill/Irwin
5
4
-6x +24.5x
3
2
-39x +20x
$3.19
8.45
© The McGraw-Hill Companies, Inc., 2008
Evolutionary Solver and Genetic Algorithms
•
Evolutionary Solver uses an entirely different approach than the standard
Solver to search for an optimal solution for a model.
•
The philosophy of Evolutionary Solver is based on genetics, evolution and the
survival of the fittest. Hence, this type of algorithm is sometimes called a
genetic algorithm.
•
The standard Solver starts with a single solution, and then moves in directions
that will improve this solution. Evolutionary Solver begins by randomly
generating a whole population of solutions.
•
After generating the population, Evolutionary Solver creates a new generation
by pairing off solutions in the population to create “offspring”, combining
some elements from each parent.
McGraw-Hill/Irwin
8.46
© The McGraw-Hill Companies, Inc., 2008
Evolutionary Solver and Genetic Algorithms
•
Among solutions in the population, some will be good (or “fit”) and some will
be bad (or “unfit”), as measured by evaluating the objective function.
Borrowing from the principles of evolution and survival of the fittest, the “fit”
members are allowed to reproduce more frequently than the unfit members.
•
Another key feature is mutation. Like gene mutation in biology, Evolutionary
Solver will occasionally make a random change in a member of the
population. This helps the algorithm get unstuck if it is getting trapped near a
local optimum.
•
Evolutionary Solver keeps creating new generations of solutions until there
have been no improvements for several consecutive generations.
McGraw-Hill/Irwin
8.47
© The McGraw-Hill Companies, Inc., 2008
Selecting a Portfolio to Beat the Market
•
A common goal of portfolio managers is to beat the market.
•
If we assume that past performance is somewhat of an indicator of the future,
then picking a portfolio that beat the market most often in the past might yield
a portfolio that will more than likely beat the market in the future.
•
Consider a portfolio of five large stocks traded on the New York Stock
Exchange (NYSE):
–
–
–
–
–
Disney (DIS)
Boeing (BA)
General Electric (GE)
Procter & Gamble (PG)
McDonald’s (MCD)
Question: What mix of these five stocks will yield a portfolio that is likely to
beat the market in the future?
McGraw-Hill/Irwin
8.48
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Model
A
1
B
C
D
E
F
G
H
I
J
K
Beating the Market (Evolutionary Solver)
2
Beat
M arket
3
Quarter
Year
DIS
BA
GE
PG
M CD
Return
M arket?
(NYSE)
4
5
6
7
8
9
10
11
12
13
Q4
Q3
Q2
Q1
Q4
Q3
Q2
Q1
Q4
Q3
2005
2005
2005
2005
2004
2004
2004
2004
2003
2003
0.38%
-4.17%
-12.35%
3.34%
24.37%
-11.52%
2.00%
7.12%
16.79%
2.08%
3.77%
3.34%
13.36%
13.45%
0.67%
1.45%
24.99%
-2.17%
23.28%
0.55%
4.85%
-2.21%
-3.31%
-0.57%
9.32%
4.26%
6.80%
-0.89%
4.61%
4.60%
-2.15%
13.29%
0.04%
-3.33%
2.26%
-0.13%
4.31%
5.49%
8.12%
4.64%
2.74%
20.71%
-10.88%
-2.90%
16.50%
7.84%
-9.02%
15.07%
7.13%
6.70%
1.92%
6.19%
-2.63%
2.00%
10.62%
0.38%
5.81%
4.93%
11.99%
3.71%
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
No
Yes
1.59%
5.75%
0.70%
-1.14%
10.35%
-0.50%
0.06%
2.09%
14.53%
2.52%
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Q2
Q1
Q4
Q3
Q2
Q1
Q4
Q3
Q2
Q1
Q4
Q3
Q2
Q1
2003
2003
2002
2002
2002
2002
2001
2001
2001
2001
2000
2000
2000
2000
16.09%
4.36%
9.02%
-19.86%
-18.15%
11.42%
12.44%
-35.55%
1.03%
-1.20%
-23.81%
-1.45%
-5.91%
41.04%
37.76%
-23.61%
-2.84%
-23.80%
-6.39%
24.95%
16.34%
-39.57%
0.08%
-15.34%
2.55%
54.71%
11.00%
-8.47%
13.17%
5.59%
-0.49%
-14.56%
-21.84%
-6.24%
8.23%
-23.50%
17.06%
-12.37%
-16.81%
9.62%
2.16%
0.87%
0.60%
4.11%
-3.42%
0.61%
0.32%
14.39%
9.29%
14.73%
2.54%
-19.82%
17.65%
18.77%
0.48%
-48.07%
52.55%
-10.05%
-7.70%
-37.92%
2.51%
4.85%
-1.51%
0.28%
1.91%
-21.90%
13.37%
-8.36%
-11.87%
-7.29%
24.03%
-3.92%
-1.09%
-19.11%
-8.71%
9.87%
8.96%
-16.72%
4.52%
-14.13%
-1.41%
14.66%
-0.83%
-4.38%
Yes
Yes
No
No
Yes
Yes
Yes
No
Yes
No
No
Yes
No
No
16.38%
-5.40%
6.16%
-16.44%
-11.22%
1.80%
8.45%
-12.53%
4.38%
-9.32%
-0.93%
3.13%
-0.74%
-0.40%
0%
<=
20.0%
<=
100%
0%
<=
20.0%
<=
100%
0%
<=
20.0%
<=
100%
0%
<=
20.0%
<=
100%
0%
<=
20.0%
<=
100%
Sum
100%
=
100%
Portfolio
Num ber of Quarters
Beating the M arket
36
McGraw-Hill/Irwin
15
8.49
© The McGraw-Hill Companies, Inc., 2008
Premium Solver Dialogue Box
McGraw-Hill/Irwin
8.50
© The McGraw-Hill Companies, Inc., 2008
Solver Options Dialogue Box
McGraw-Hill/Irwin
8.51
© The McGraw-Hill Companies, Inc., 2008
Limit Options Dialogue Box
McGraw-Hill/Irwin
8.52
© The McGraw-Hill Companies, Inc., 2008
Evolutionary Solver Spreadsheet Solution
A
1
B
C
D
E
F
G
H
I
J
K
Beat
M arket
Beating the Market (Evolutionary Solver)
2
3
Quarter
Year
DIS
BA
GE
PG
M CD
Return
M arket?
(NYSE)
4
5
6
7
8
9
10
11
12
13
Q4
Q3
Q2
Q1
Q4
Q3
Q2
Q1
Q4
Q3
2005
2005
2005
2005
2004
2004
2004
2004
2003
2003
0.38%
-4.17%
-12.35%
3.34%
24.37%
-11.52%
2.00%
7.12%
16.79%
2.08%
3.77%
3.34%
13.36%
13.45%
0.67%
1.45%
24.99%
-2.17%
23.28%
0.55%
4.85%
-2.21%
-3.31%
-0.57%
9.32%
4.26%
6.80%
-0.89%
4.61%
4.60%
-2.15%
13.29%
0.04%
-3.33%
2.26%
-0.13%
4.31%
5.49%
8.12%
4.64%
2.74%
20.71%
-10.88%
-2.90%
16.50%
7.84%
-9.02%
15.07%
7.13%
6.70%
1.78%
7.26%
0.88%
1.80%
5.96%
2.05%
8.46%
3.13%
11.11%
3.79%
Yes
Yes
Yes
Yes
No
Yes
Yes
Yes
No
Yes
1.59%
5.75%
0.70%
-1.14%
10.35%
-0.50%
0.06%
2.09%
14.53%
2.52%
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Q2
Q1
Q4
Q3
Q2
Q1
Q4
Q3
Q2
Q1
Q4
Q3
Q2
Q1
2003
2003
2002
2002
2002
2002
2001
2001
2001
2001
2000
2000
2000
2000
16.09%
4.36%
9.02%
-19.86%
-18.15%
11.42%
12.44%
-35.55%
1.03%
-1.20%
-23.81%
-1.45%
-5.91%
41.04%
37.76%
-23.61%
-2.84%
-23.80%
-6.39%
24.95%
16.34%
-39.57%
0.08%
-15.34%
2.55%
54.71%
11.00%
-8.47%
13.17%
5.59%
-0.49%
-14.56%
-21.84%
-6.24%
8.23%
-23.50%
17.06%
-12.37%
-16.81%
9.62%
2.16%
0.87%
0.60%
4.11%
-3.42%
0.61%
0.32%
14.39%
9.29%
14.73%
2.54%
-19.82%
17.65%
18.77%
0.48%
-48.07%
52.55%
-10.05%
-7.70%
-37.92%
2.51%
4.85%
-1.51%
0.28%
1.91%
-21.90%
13.37%
-8.36%
-11.87%
-7.29%
19.73%
-4.12%
-2.71%
-14.53%
-7.27%
10.51%
9.57%
-11.69%
5.54%
-16.55%
3.44%
21.62%
1.89%
-18.41%
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
Yes
No
16.38%
-5.40%
6.16%
-16.44%
-11.22%
1.80%
8.45%
-12.53%
4.38%
-9.32%
-0.93%
3.13%
-0.74%
-0.40%
0%
<=
2.6%
<=
100%
0%
<=
25.0%
<=
100%
0%
<=
25.7%
<=
100%
0%
<=
34.8%
<=
100%
0%
<=
12.0%
<=
100%
Sum
100%
=
100%
Portfolio
Num ber of Quarters
Beating the M arket
36
19
McGraw-Hill/Irwin
8.53
© The McGraw-Hill Companies, Inc., 2008
Advantages and Disadvantages of Evolutionary Solver
•
Evolutionary Solver has two significant advantages over the standard Solver
for solving difficult nonlinear programming problems:
– The complexity of the objective function does not matter. As long as the function
can be evaluated for a given candidate solution (to determine the level of fitness), it
does not matter if the function has kinks, discontinuities, or many local optima.
– By evaluating whole populations of candidate solutions, Evolutionary Solver keeps
from getting trapped at a local optimum. Even if the whole population evolves
toward a locally optimal solution, mutation allows the possibility of getting
unstuck.
•
However, Evolutionary Solver is not a panacea.
– It can take much longer that standard Solver to find a final solution.
– Evolutionary Solver does not perform well on models that have many constraints.
– Evolutionary Solver is a random process. Running it again on the same model
usually will yield a different solution.
– The best solution found is typically not optimal (although it may be very close).
McGraw-Hill/Irwin
8.54
© The McGraw-Hill Companies, Inc., 2008
Problems That Solver will Solver Correctly
•
A maximization problem with linear constraints and a concave objective
function.
Line joining any two points
is on or below the curve
A Concave Function
McGraw-Hill/Irwin
8.55
© The McGraw-Hill Companies, Inc., 2008
Problems That Solver will Solver Correctly
•
A minimization problem with linear constraints and a convex objective
function.
Line joining any two points
is on or above the curve
A Convex Function
McGraw-Hill/Irwin
8.56
© The McGraw-Hill Companies, Inc., 2008
Quality Furniture Corporation
•
The Quality Furniture Corporation manufactures two products: benches and
tables.
•
They employ three carpenters. During the next week, 120 hours of labor are
available at regular wages ($8 per hour).
•
Up to 30 hours of overtime can be used at a wage rate of $12 per hour.
•
Up to 30 hours of weekend time can be utilized at a wage rate of $16 per hour.
•
540 pounds of wood is available at a cost of $2 per pound.
•
Each bench requires 3 labor hours and 12 pounds of wood. Each table requires
6 labor hours and 38 pounds of wood.
•
Completed benches sell for $80 each, and tables sell for $200 each.
Question: How many benches and how many tables should be produced?
McGraw-Hill/Irwin
8.57
© The McGraw-Hill Companies, Inc., 2008
Outdoor Furniture Labor Costs
Labor Cost
$1600
$16/ hr Weekend
$1320
$12/ hr Overtime
$960
$8/ hr Regular
30
60
90
120
150
180
Labor Hours
McGraw-Hill/Irwin
8.58
© The McGraw-Hill Companies, Inc., 2008
Nonlinear Programming Spreadsheet
A
1
B
C
D
E
F
G
Quality Furniture Corporation (Nonlinear)
2
3
4
Revenue/Unit
Benches
$85
Tables
$200
5
6
7
8
Labor
Wood
Usage per Unit Produced
3
6
12
38
Total Used
135
540
<=
<=
Available
180
540
9
Wood Cost/lb.
$2
Regular
Overtime
Sunday
Labor Cost
(per hour)
$8
$12
$16
Hours
Available
120
30
30
Production
Benches
45
Tables
0
23
Revenue
Wood Cost
Labor Cost
$3,825.00
$1,080.00
$1,140.00
24
Profit
$1,605.00
10
11
12
13
14
15
16
17
18
19
20
21
22
McGraw-Hill/Irwin
8.59
© The McGraw-Hill Companies, Inc., 2008
Outdoor Furniture Labor Costs
Labor Cost
$1600
$16/ hr Weekend
$1320
$12/ hr Overtime
$960
$8/ hr Regular
30
60
90
120
150
180
Labor Hours
McGraw-Hill/Irwin
8.60
© The McGraw-Hill Companies, Inc., 2008
Separable Programming Spreadsheet
A
1
B
C
F
E
D
G
Quality Furniture Corporation (Separable)
2
3
4
Revenue/Unit
Benches
$85
Tables
$200
5
6
7
8
Labor
Wood
Usage per Unit Produced
6
3
38
12
Total Used
135
540
<=
<=
Available
135
540
9
10
Wood Cost/lb.
$2
11
Regular
Overtime
Sunday
Labor Cost
(per hour)
$8
$12
$16
Hours
Available
120
30
30
Production
Benches
45
Tables
0
Regular Hours
Overtime Hours
Weekend Hours
Labor Used
120
15
0
<=
<=
<=
28
Revenue
Wood Cost
Labor Cost
$3,825.00
$1,080.00
$1,140.00
29
Profit
$1,605.00
12
13
14
15
16
17
18
19
20
21
22
23
24
120
30
30
25
26
27
McGraw-Hill/Irwin
8.61
© The McGraw-Hill Companies, Inc., 2008
Advertising Example
A
1
B
C
Advertising Example (Nonlinear)
2
3
4
5
6
7
8
Parameters:
Unit Variable Cost
Unit Price
Salesforce Salary
Fixed Overhead
Seasonality
$48
$65
$9,000
$23,000
1.2
9
10
11
Decision Variable:
Advertising
$122,949
12
Quarter
13
14
Q1
14994
Units Sold
15
16
17
18
Sales Revenue
Cost of Sales
Gross Margin
$974,610
$719,712
$254,898
Total Fixed Costs
$154,949
19
20
21
22
Profit
Sales  (35)  (Seasonality Factor)
McGraw-Hill/Irwin
$99,949

8.62
Advertising
+
Sales Force
2
© The McGraw-Hill Companies, Inc., 2008
The Sales Function
Sales  (35)  (Seasonality Factor)

Advertising
+
Sales Force
2
20000
18000
16000
Sales Level
14000
12000
10000
8000
6000
4000
2000
0
0
50000
100000
150000
200000
Advertising
McGraw-Hill/Irwin
8.63
© The McGraw-Hill Companies, Inc., 2008
Approximating a Nonlinear Function
A
1
B
C
D
Approximating the Nonlinear Sales Function
2
3
4
Seasonality =
Sales Force =
1.2
9000
5
6
7
8
9
10
11
McGraw-Hill/Irwin
Advertising Level
$0
$50,000
$100,000
$150,000
$200,000
Sales Level
2,817
9,805
13,577
16,509
18,993
8.64
Slope
0.1398
0.0754
0.0586
0.0497
© The McGraw-Hill Companies, Inc., 2008
Advertising Example Using Separable Programming
A
1
B
C
D
E
F
G
Advertising Example (Separable)
2
3
4
5
6
7
8
Parameters:
Unit Variable Cost
Unit Price
Salesforce Salary
Fixed Overhead
Seasonality
$48
$65
$9,000
$23,000
1.2
9
Units Sold per
Advertising Dollar
0.1398
0.0754
0.0586
0.0497
10
11
12
13
14
15
Advertising ($0-$50,000)
Advertising ($50,000-$100,000)
Advertising ($100,000-$150,000)
Advertising ($150,000-)
16
Quarter
17
18
Units Sold
$50,000
$50,000
$0
$0
$100,000
<=
<=
<=
$50,000
$50,000
$50,000
Total Advertising
Q1
13577
19
20
21
22
Sales Revenue
Cost of Sales
Gross Margin
$882,505
$651,696
$230,809
Total Fixed Costs
$132,000
23
24
25
26
Profit
McGraw-Hill/Irwin
$98,809
8.65
© The McGraw-Hill Companies, Inc., 2008
Evolutionary Solver
•
The standard Solver has difficulty with problems that are
–
–
–
–
•
highly nonlinear
are not smooth (have “kinks” in the objective)
have discontinuities (the objective jumps in value)
have many local optima (many hills and valleys)
Excel functions like IF, MAX, ABS, ROUND, etc., tend to cause one or more
of these problems.
McGraw-Hill/Irwin
8.66
© The McGraw-Hill Companies, Inc., 2008
Premium Solver
•
Included on the textbook CD is the “Premium Solver”. After installing, a new
button (“Premium”) is added to Solver.
McGraw-Hill/Irwin
8.67
© The McGraw-Hill Companies, Inc., 2008
Premium Solver
•
Clicking on the “Premium” button switches to Premium Solver, which gives
the option of three different solvers.
– Standard GRG Nonlinear is equivalent to the regular Solver without choosing
“Assume Linear Model”.
– Standard Simplex LP is equivalent to the regular Solver with choosing “Assume
Linear Model”.
– Standard Evolutionary uses a genetic evolutionary algorithm that is only available
with Premium Solver.
McGraw-Hill/Irwin
8.68
© The McGraw-Hill Companies, Inc., 2008
How Genetic Algorithms (Evolutionary Solver) Work
Genetic algorithms (such as Evolutionary Solver) use principles from the theory of
evolution.
•
The Population: a large set of random solutions is generated.
•
Level of fitness: each member of the population (solution) is evaluated to
determine its level of “fitness” (value of objective).
•
Evolution: a new generation (set of solutions) is created as follows:
– Reproduction: pairs reproduce and create new solutions that share some properties
of each.
– Survival of the Fittest: more “fit” solutions reproduce more frequently, less “fit”
solutions are allowed to die out.
– Mutation: occasionaly random “mutations” are introduced.
McGraw-Hill/Irwin
8.69
© The McGraw-Hill Companies, Inc., 2008
Inventory Ordering Policy with Quantity Discounts
•
Consider a manufacturer that orders a given part from a supplier.
•
They require 10,000 parts per year.
•
There is a cost associated with each order (due to processing, receiving costs,
fixed shipping costs, etc.) of $20.
•
The cost of holding a part in inventory is estimated at $4 per year.
•
The supplier of this part offers quantity discounts on the purchasing cost of
this part according to the following schedule.
McGraw-Hill/Irwin
Order Quantity
Purchase Price (per unit)
1–99
$10.00
100–499
9.80
500–999
9.70
1,000–9,999
9.60
10,000+
9.50
8.70
© The McGraw-Hill Companies, Inc., 2008
Total Annual Cost
•
Recall from Operations Management class that the total annual cost (including
puchasing, ordering, and holding costs) is
Total Annual Cost = Dp + (Q / 2)H + (D / Q)S
where
D = Annual demand
p = purchase cost
Q = order size
H = annual holding cost per unit
S = ordering cost
McGraw-Hill/Irwin
8.71
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Model
A
1
B
C
D
E
F
G
H
I
Ordering Policy with Quantity Discounts
2
Min Order
Quantity
1
Price
$10.00
8
100
500
1,000
$9.80
$9.70
$9.60
9
10,000
$9.50
3
4
5
6
7
1
Order Quantity
<=
1000
<=
20,000
Annual Cost
Purchasing
Ordering
Holding
Total
$192,000
$400
$2,000
$194,400
10
11
12
13
Annual Demand
Ordering Cost
Annual Holding Cost
McGraw-Hill/Irwin
20,000
$20
$4
Price
8.72
$9.60
© The McGraw-Hill Companies, Inc., 2008
Attempts with Standard Solver
Various “solutions” provided by the standard Solver, depending on the starting
point:
Starting Point (Q)
Solution (Q*)
Cost
1
1,000
$194,400
200
500
195,800
400
447
197,789
600
500
195,800
1,200
1,000
194,400
11,000
10,000
210,040
12,000
1,000
194,400
McGraw-Hill/Irwin
8.73
© The McGraw-Hill Companies, Inc., 2008
Cost Function with Quantity Discounts
Cos t
$230,000
$225,000
$220,000
$215,000
$210,000
$205,000
$200,000
$195,000
$190,000
10
100
447
500 1000
10,000
Order Quantity (logarithmic scale)
McGraw-Hill/Irwin
8.74
© The McGraw-Hill Companies, Inc., 2008
Solving with the Evolutionary Solver
A
1
B
C
D
E
F
G
H
I
Ordering Policy with Quantity Discounts
2
Min Order
Quantity
1
Price
$10.00
1
8
100
500
1,000
$9.80
$9.70
$9.60
<=
1000
<=
9
10,000
$9.50
3
4
5
6
7
Order Quantity
20,000
Annual Cost
Purchasing
Ordering
Holding
Total
$192,000
$400
$2,000
$194,400
10
11
12
13
Annual Demand
Ordering Cost
Annual Holding Cost
McGraw-Hill/Irwin
20,000
$20
$4
Price
8.75
$9.60
© The McGraw-Hill Companies, Inc., 2008
Tips on Using Evolutionary Solver
•
Bounding all of the variables greatly aids the Evolutionary Solver by
decreasing the search space.
•
The limit options should be increased (Max Time, Max Subproblems, and Max
Feasible Sols) for challenging problems. Setting Tolerance to 0.0005 and Max
Time Without Improvements to 30 will ensure the algorithm will stop if the
Target Cell value has improved less than 0.05% in the last 30 seconds.
•
Experiment with different populations sizes and mutation rates to see what
works well. I have found that higher than default mutation rates can be helpful
in problems with lots of local optima.
•
The Evolutionary Solver can take a very long time, but it will usually find a
good solution.
McGraw-Hill/Irwin
8.76
© The McGraw-Hill Companies, Inc., 2008
Tips on Using Evolutionary Solver
•
There is no guarantee that Evolutionary Solver will find the best solution.
•
The Evolutionary Solver performs well even with nasty objective functions,
but is not very efficient at handling constraints.
•
Much of the solution process is driven by random numbers that direct the
search. Thus, two people running Evolutionary Solver on the same model may
get different results.
•
Once Evolutionary Solver has found a good solution, you can use GRG
Nonlinear Solver (the nonlinear algorithm that is included with the Premium
Solver software) to try to find a slightly better solution.
McGraw-Hill/Irwin
8.77
© The McGraw-Hill Companies, Inc., 2008