Slides by
JOHN
LOUCKS
St. Edward’s
University
© 2008 Thomson South-Western. All Rights Reserved
Slide 1
Chapter 8
Nonlinear Optimization Models




Production Application
Constructing an Index Fund
Markowitz Portfolio Model
Forecasting Adoption of a New Product
© 2008 Thomson South-Western. All Rights Reserved
Slide 2
Introduction

Many business processes behave in a nonlinear
manner.
• The price of a bond is a nonlinear function of
interest rates.
• The price of a stock option is a nonlinear function
of the price of the underlying stock.
• The marginal cost of production often decreases
with the quantity produced.
• The quantity demanded for a product is often a
nonlinear function of the price.
© 2008 Thomson South-Western. All Rights Reserved
Slide 3
Introduction



A nonlinear optimization problem is any
optimization problem in which at least one term in
the objective function or a constraint is nonlinear.
Nonlinear terms include x13 , 1/x2 , 5x1x2 , and log x3 .
The nonlinear optimization problems presented on
the upcoming slides can be solved using computer
software such as LINGO and Excel Solver.
© 2008 Thomson South-Western. All Rights Reserved
Slide 4
Example: Production Application

Armstrong Bike Co.
Armstrong Bike Co. produces two new lightweight
bicycle frames, the Flyer and the Razor, that are made
from special aluminum and steel alloys. The cost to
produce a Flyer frame is $100, and the cost
to produce a Razor frame is $120.
We can not assume that Armstrong
will sell all the frames it can produce.
As the selling price of each frame
model – Flyer and Razor - increases, the quantity
demanded for each model goes down.
© 2008 Thomson South-Western. All Rights Reserved
Slide 5
Example: Production Application

Assume that the demand for Flyer frames F
and the demand for Razor frames R are given by:
F = 750 – 5PF
R = 400 – 2PR
where PF = the price of a Flyer frame
PR = the price of a Razor frame.

The profit contributions (revenue – cost) are:
PF F - 100F for Flyer frames
PR R - 120R for Razor frames
© 2008 Thomson South-Western. All Rights Reserved
Slide 6
Example: Production Application

Profit Contribution as a Function of Demand
• Solving F = 750 - 5PF for PF we get:
PF = 150 - 1/5 F
Substituting 150 - 1/5 F for PF in PF F - 100F we get:
PF F - 100F = F(150 - 1/5 F) - 100F =
50F - 1/5 F 2
• Solving R = 400 - 2PR for PR we get:
PR = 200 - 1/2 R
Substituting 200 - 1/2 R for PR in PR R - 120R we get:
PR R - 120R = R(200 - 1/2 R) - 120R =
© 2008 Thomson South-Western. All Rights Reserved
80R - 1/2 R2
Slide 7
Example: Production Application

Total Profit Contribution
Total Profit Contribution =
50F – 1/5 F2 + 80R – 1/2 R2
This function is an example of a quadratic function
because the nonlinear terms have a power of 2.
© 2008 Thomson South-Western. All Rights Reserved
Slide 8
Example: Production Application
A supplier can deliver a maximum of 500
pounds of the aluminum alloy and 420 pounds of the
steel alloy weekly. The number of pounds of each alloy
needed per frame is summarized below.
Aluminum Alloy
Flyer
2
Razor
4
Steel Alloy
3
2
How many Flyer and Razor frames should
Armstrong produce each week?
© 2008 Thomson South-Western. All Rights Reserved
Slide 9
Example: Production Application

Problem Formulation
Max 50F – 1/5 F2 + 80R – 1/2 R2 (Total Weekly Profit)
s.t.
2F + 4R < 500
3F + 2R < 420
F, R > 0
(Aluminum Available)
(Steel Available)
(Non-negativity)
© 2008 Thomson South-Western. All Rights Reserved
Slide 10
Example: Production Application

Total Profit Contribution
First, we will solve the unconstrained version of
this nonlinear program to find the values of F and R
that maximize the above total profit contribution
function (with the production constraints ignored).
© 2008 Thomson South-Western. All Rights Reserved
Slide 11
Example: Production Application

Optimal Solution for Unconstrained Problem
x2
250
3F + 2R < 420
200
Unconstrained
Optimum
(125, 80)
Profit = $6,325.00
150
100
50
Feasible
Region
50
2F + 4R < 500
100
150
200
250
© 2008 Thomson South-Western. All Rights Reserved
300
x1
Slide 12
Example: Production Application

Total Profit Contribution
Now we will solve the constrained version of this
nonlinear program to find the values of F and R that
maximize the total profit contribution function with
the production constraints enforced.
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Slide 13
Example: Production Application

Objective Function Contour Lines
x2
250
200
$6,200.00
Contour
$6,325.00
150
$5,500.00
Contour
100
$6,075.47
Contour
50
50
100
150
200
250
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300
x1
Slide 14
Example: Production Application

Optimal Solution for Constrained Problem
x2
Constrained
Optimum
(92.45, 71.32)
Profit = $6,075.47
250
200
150
100
$6,075.47
Contour
50
50
100
150
200
250
© 2008 Thomson South-Western. All Rights Reserved
300
x1
Slide 15
Example: Production Application

Optimal Solution
•
•
•
•
Produce 92.45 Flyer frames per week.
Produce 71.32 Razor frames per week.
Profit per week is $6,075.47.
Use 470.2 pounds of aluminum alloy per week (of
the 500 pounds available per week).
• Use the entire 420 pounds of steel alloy available
per week.
© 2008 Thomson South-Western. All Rights Reserved
Slide 16
Local and Global Optima



A feasible solution is a local optimum if there are no
other feasible solutions with a better objective
function value in the immediate neighborhood.
• For a maximization problem the local optimum
corresponds to a local maximum.
• For a minimization problem the local optimum
corresponds to a local minimum.
A feasible solution is a global optimum if there are no
other feasible points with a better objective function
value in the feasible region.
Obviously, a global optimum is also a local optimum.
© 2008 Thomson South-Western. All Rights Reserved
Slide 17
Multiple Local Optima


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
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Nonlinear optimization problems can have multiple
local optimal solutions, in which case we want to find
the best local optimum.
Nonlinear problems with multiple local optima are
difficult to solve and pose a serious challenge for
optimization software.
In these cases, the software can get “stuck” and
terminate at a local optimum.
There can be a severe penalty for finding a local
optimum that is not a global optimum.
Developing algorithms capable of finding the global
optimum is currently a very active research area.
© 2008 Thomson South-Western. All Rights Reserved
Slide 18
Multiple Local Optima

2 ( - X 2 -( Y  1)2 )
Consider the function f ( X , Y )  3(1 - X ) e
- 10( X /5 - X - Y )e
3
-e




( - ( X  1)2 -Y 2 )
5
( - X 2 -Y 2 )
/3.
The shape of this function is shown on the next slide.
The hills and valleys in the graph show that this
function has several local maximums and local
minimums.
There are two local minimums, one of which is the
the global minimum.
There are three local maximums, one of which is the
global maximum.
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Slide 19
Multiple Local Optima
2 ( - X 2 -(Y 1)2 )
f (X , Y )  3(1 - X ) e
( - X 2 -Y 2 )
- 10( X /5 - X - Y )e
3
5
© 2008 Thomson South-Western. All Rights Reserved
( -( X 1)2 -Y 2 )
-e
/3
Slide 20
Single Local Optimum

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Consider the function f ( X , Y )  -X 2 - Y 2 .
The shape of this function is shown on the next slide.
A function that is bowl-shaped down is called a
concave function.
The maximum value for this particular function is 0
and the point (0, 0) gives the optimal value of 0.
Functions such as this one have a single local
maximum that is also a global maximum.
This type of nonlinear problem is relatively easy to
maximize.
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Slide 21
Single Local Optimum
Concave Function f (X ,Y )  -X 2 - Y 2
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Slide 22
Single Local Optimum

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
Consider the function f ( X , Y )  X 2  Y 2 .
The shape of this function is shown on the next slide.
A function that is bowl-shaped up is called a convex
function.
The minimum value for this particular function is 0
and the point (0, 0) gives the optimal value of 0.
Functions such as this one have a single local
minimum that is also a global minimum.
This type of nonlinear problem is relatively easy to
minimize.
© 2008 Thomson South-Western. All Rights Reserved
Slide 23
Single Local Optimum
Convex Function f ( X , Y )  X 2  Y 2
40
20
Z
4
2
0
-2
-4
-4
0
-2
X
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2
4
Y
Slide 24
Constructing an Index Fund

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
Index funds are a very popular investment vehicle in
the mutual fund industry.
Vanguard 500 Index Fund is the largest mutual fund
in the U.S. with over $70 billion in net assets in 2005.
An index fund is an example of passive asset
management.
The key idea behind an index fund is to construct a
portfolio of stocks, mutual funds, or other securities
that closely matches the performance of a broad
market index such as the S&P 500.
Behind the popularity of index funds is research that
basically says “you can’t beat the market.”
© 2008 Thomson South-Western. All Rights Reserved
Slide 25
Example: Constructing an Index Fund

Lymann Brothers Investments
Lymann Brothers has a substantial number of clients
who wish to own a mutual fund portfolio that
closely matches the performance of the
S&P 500 stock index.
A manager at Lymann Brothers has
selected five mutual funds (shown on the
next slide) that will be considered for inclusion
in the portfolio. The manager must decide what
percentage of the portfolio should be invested in each
mutual fund.
© 2008 Thomson South-Western. All Rights Reserved
Slide 26
Example: Constructing an Index Fund

Mutual Fund Performance in 4 Selected Years
Mutual Fund
Annual Returns (Planning Scenarios)
Year 1
Year 2
Year 3
Year 4
International Stock
Large-Cap Blend
Mid-Cap Blend
Small-Cap Blend
Intermediate Bond
25.64
15.31
18.74
14.19
7.88
27.62
18.77
18.43
12.37
9.45
5.80
11.06
6.28
-1.92
10.56
-3.13
4.75
-1.04
7.32
3.31
S&P 500
13.00
12.00
7.00
2.00
© 2008 Thomson South-Western. All Rights Reserved
Slide 27
Example: Constructing an Index Fund

Define the 9 Decision Variables
IS = proportion of portfolio invested in international stock
LC = proportion of portfolio invested in large-cap blend
MC = proportion of portfolio invested in mid-cap blend
SC = proportion of portfolio invested in small-cap blend
IB = proportion of portfolio invested in intermediate bond
R1 = portfolio return for scenario 1 (year 1)
R2 = portfolio return for scenario 2 (year 2)
R3 = portfolio return for scenario 3 (year 3)
R4 = portfolio return for scenario 4 (year 4)
© 2008 Thomson South-Western. All Rights Reserved
Slide 28
Example: Constructing an Index Fund

Define the Objective Function
Min (R1 – 13)2 + (R2 – 12)2 + (R3 – 7)2 + (R4 – 2)2

Define the 6 Constraints (including non-negativity)
25.64IS + 15.31LC + 18.74MC + 14.19SC + 7.88IB = R1
27.62IS + 18.77LC + 18.43MC + 12.37SC + 9.45IB = R2
5.80IS + 11.06LC + 6.28MC - 1.92SC + 10.56IB = R3
- 3.13IS + 4.75LC - 1.04MC + 7.32SC + 3.31IB = R4
IS + LC + MC + SC + IB = 1
IS, LC, MC, SC, IB > 0
© 2008 Thomson South-Western. All Rights Reserved
Slide 29
Example: Constructing an Index Fund

Optimal Solution for Lymann Brothers Example
•
•
•
•
•
•
•
•
•
R1 = 12.51
R2 = 12.90
R3 = 7.13
R4 = 2.51
IS = 0
LC = 0
MC = .332
SC = .161
IB = .507
(12.51% portfolio return for scenario 1)
(12.90% portfolio return for scenario 2)
( 7.13% portfolio return for scenario 3)
( 2.51% portfolio return for scenario 4)
( 0.0% of portfolio in international stock)
( 0.0% of portfolio in large-cap blend)
(33.2% of portfolio in mid-cap blend)
(16.1% of portfolio in small-cap blend)
(50.7% of portfolio in intermediate bond)
100.0% of portfolio
© 2008 Thomson South-Western. All Rights Reserved
Slide 30
Example: Constructing an Index Fund

Lymann Brothers Portfolio Return vs. S&P 500 Return
Scenario
1
2
3
4
Portfolio Return
S&P 500 Return
12.51
12.90
7.13
2.51
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13.00
12.00
7.00
2.00
Slide 31
Markowitz Portfolio Model




There is a key tradeoff in most portfolio optimization
models between risk and return.
The index fund model (Lymann Brothers example)
presented earlier managed the tradeoff passively.
The Markowitz mean-variance
portfolio model provides a very
convenient way for an investor to
actively trade-off risk versus return.
We will now demonstrate the
Markowitz portfolio model by
extending the Lymann Brothers example.
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Slide 32
Example: Markowitz Portfolio Model


In the Lymann Brothers example there were four
scenarios and the return under each scenario was
defined by the variables R1, R2, R3, and R4.
If ps is the probability of scenario s, and there are n
scenarios, then the expected return for the portfolio R
is
n
R   ps Rs
s1

If we assume that the four scenarios in the Lymann
Brothers model are equally likely, then
1
R   Rs or
s 1 4
4
1 4
R   Rs
4 s 1
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Slide 33
Example: Markowitz Portfolio Model


The measure of risk most often associated with the
Markowitz model is the variance of the portfolio.
For our example, the portfolio variance is
n
Var   ps ( Rs - R)2
s 1

For our example, the four planning scenarios are
equally likely. Thus,
1
Var   ( Rs - R)2 or
s 1 4
4
1 4
Var   ( Rs - R)2
4 s 1
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Slide 34
Example: Markowitz Portfolio Model



The portfolio variance is the average of the sum of
the squares of the deviations from the mean value
under each scenario.
The larger the variance value, the more widely
dispersed the scenario returns are about the average
return value.
If the portfolio variance were equal to zero, then
every scenario return Ri would be equal.
© 2008 Thomson South-Western. All Rights Reserved
Slide 35
Example: Markowitz Portfolio Model


There are two basic ways to formulate the Markowitz
model:
• (1) Minimize the variance of the portfolio subject
to constraints on the expected return, and
• (2) Maximize the expected return of the portfolio
subject to a constraint on risk.
We will now demonstrate the first (1) formulation,
assuming that Lymann Brothers’ client requires the
expected portfolio return to be at least 9 percent.
© 2008 Thomson South-Western. All Rights Reserved
Slide 36
Example: Markowitz Portfolio Model

Define the Objective Function
Minimize the portfolio variance:
1 4
Min  ( Rs - R)2
4 s 1

Define the Constraints
Define the return for each scenario:
25.64IS + 15.31LC + 18.74MC + 14.19SC + 7.88IB = R1
27.62IS + 18.77LC + 18.43MC + 12.37SC + 9.45IB = R2
5.80IS + 11.06LC + 6.28MC - 1.92SC + 10.56IB = R3
- 3.13IS + 4.75LC - 1.04MC + 7.32SC + 3.31IB = R4
© 2008 Thomson South-Western. All Rights Reserved
Slide 37
Example: Markowitz Portfolio Model

Define the Constraints (continued)
All the money must be invested in the portfolio:
IS + LC + MC + SC + IB = 1
Define the expected return for the portfolio:
1 4
Rs = R

4 s 1
The portfolio return must be at least 9 percent:
R  9
Non-negativity:
IS, LC, MC, SC, IB > 0
© 2008 Thomson South-Western. All Rights Reserved
Slide 38
Example: Markowitz Portfolio Model

Optimal Solution
•
•
•
•
•
•
•
•
•
•
R1 = 10.63
R2 = 12.20
R3 = 8.93
R4 = 4.24
Rbar = 9.00
IS =
0
LC = .251
MC = 0
SC = .141
IB = .608
(10.63% portfolio return for scenario 1)
(12.20% portfolio return for scenario 2)
( 8.93% portfolio return for scenario 3)
( 4.24% portfolio return for scenario 4)
( 9.00% expected portfolio return)
( 0.0% of portfolio in international stock)
(25.1% of portfolio in large-cap blend)
( 0.0% of portfolio in mid-cap blend)
(14.1% of portfolio in small-cap blend)
(60.8% of portfolio in intermediate bond)
100.0% of portfolio
© 2008 Thomson South-Western. All Rights Reserved
Slide 39
Forecasting Adoption of a New Product



Forecasting new adoptions (purchases) after a product
introduction is an important marketing problem.
We introduce here a forecasting model developed by
Frank Bass.
Nonlinear programming is used to estimate the
parameters of the Bass forecasting model.
© 2008 Thomson South-Western. All Rights Reserved
Slide 40
Forecasting Adoption of a New Product

The Bass model has three parameters that must be
estimated.
• m is the number of people estimated to eventually
adopt a new product
• q is the coefficient of imitation which measures the
likelihood of adoption due to a potential adopter
influenced by someone who has already adopted
the product
• p is the coefficient of imitation which measures the
likelihood of adoption assuming no influence from
someone who has already adopted the product.
© 2008 Thomson South-Western. All Rights Reserved
Slide 41
Forecasting Adoption of a New Product

Developing the Forecasting Model
• Ft , the forecast of the number of new adopters
during time period t , is
Ft = (likelihood of a new adoption in time period t)
x (number of potential adopters remaining at
the end of time period t – 1)
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Slide 42
Forecasting Adoption of a New Product

Developing the Forecasting Model
• Essentially, the likelihood of a new adoption is the
likelihood of adoption due to innovation plus the
likelihood of adoption due to imitation.
• Let Ct - 1 denote the number of people who have
adopted the product up to time t - 1.
• Hence, Ct - 1 /m is the fraction of the number of
people estimated to adopt the product by time t – 1.
• The likelihood of adoption due to imitation is
q(Ct - 1 /m).
• The likelihood of adoption due to innovation and
imitation is p + q(Ct - 1 /m).
© 2008 Thomson South-Western. All Rights Reserved
Slide 43
Forecasting Adoption of a New Product

Developing the Forecasting Model
• The number of potential adopters remaining at the
end of time period t – 1 is m - Ct - 1 .
• Hence, the complete forecast model is given by
Ft = (p + q(Ct - 1 /m)) (m - Ct - 1)
© 2008 Thomson South-Western. All Rights Reserved
Slide 44
Forecasting Adoption of a New Product

Nonlinear Optimization Problem Formulation
N
Min  Et2
t 1
Ft = (p + q(Ct - 1 /m)) (m - Ct - 1),
Et = Ft - St , t = 1, …., N
t = 1, …., N
where N = number of time periods of data available
Et = forecast error for time period t
St = actual number of adopters (or a multiple of
that number such as sales) in time period t
© 2008 Thomson South-Western. All Rights Reserved
Slide 45
Example: Forecasting New-Product Adoption

Maid For You
Maid For You is a residential cleaning
service firm that has been quite successful
developing a client base in the Chicago area.
The firm plans to expand to other major
metropolitan areas during the next few years.
Maid For You would like to use its Chicago
subscription data (on the next slide) to develop a model
for forecasting service subscriptions in regions where it
might expand. The first step is to estimate values for p
(coefficient of innovation) and q (coefficient of imitation).
© 2008 Thomson South-Western. All Rights Reserved
Slide 46
Example: Forecasting New-Product Adoption

Subscribers and Cumulative Subscribers (1000s)
Month
Subscribers St
Cum. Subscribers Ct
1
2
3
4
5
6
7
8
9
0.53
2.94
3.60
4.85
3.44
2.76
1.82
0.93
0.61
0.53
3.47
7.07
11.92
15.36
18.12
19.94
20.87
21.48
© 2008 Thomson South-Western. All Rights Reserved
Slide 47
Example: Forecasting New-Product Adoption

Define the Objective Function
Minimize the sum of the squared forecast errors:
Min E12  E22  E32  E42  E52  E62  E72  E82  E92
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Slide 48
Example: Forecasting New-Product Adoption

Define the Constraints
Define the forecast for each time period:
1)
2)
3)
4)
5)
6)
7)
8)
9)
F1 = pm
F2 = (p + q( 0.53/m)) (m – 0.53)
F3 = (p + q( 3.47/m)) (m – 3.47)
F4 = (p + q( 7.07/m)) (m – 7.07)
F5 = (p + q(11.92/m)) (m – 11.92)
F6 = (p + q(15.36/m)) (m – 15.36)
F7 = (p + q(18.12/m)) (m – 18.12)
F8 = (p + q(19.94/m)) (m – 19.94)
F9 = (p + q(20.87/m)) (m – 20.87)
© 2008 Thomson South-Western. All Rights Reserved
Slide 49
Example: Forecasting New-Product Adoption

Define the Constraints (continued)
Define the forecast error for each time period:
10)
11)
12)
13)
14)
15)
16)
17)
18)
E1 = F1 – 0.53
E2 = F2 – 2.94
E3 = F3 – 3.60
E4 = F4 – 4.85
E5 = F5 – 3.44
E6 = F6 – 2.76
E7 = F7 – 1.82
E8 = F8 – 0.93
E9 = F9 – 0.61
© 2008 Thomson South-Western. All Rights Reserved
Slide 50
Example: Forecasting New-Product Adoption

Optimal Forecast Parameter Values
Parameter
p
q
m
Value
0.08
0.62
21.26
The value of the imitation parameter q = .62 is
substantially larger than the value of the innovation
parameter p = .08. Subscriptions gain momentum
over time due mainly to very favorable word-ofmouth.
© 2008 Thomson South-Western. All Rights Reserved
Slide 51
Example: Forecasting New-Product Adoption

Optimal Solution
Month Forecast Subscribers Error
1
1.77
0.53
1.24
2
2.05
2.94
-0.89
3
3.29
3.60
-0.31
4
4.12
4.85
-0.73
5
4.03
3.44
0.59
6
3.14
2.76
0.38
7
1.93
1.82
0.11
8
0.88
0.93
-0.05
9
0.27
0.61
-0.34
© 2008 Thomson South-Western. All Rights Reserved
Slide 52
Example: Forecasting New-Product Adoption

Subscribers versus Forecasts
Subscribers
Subscribers (1000s)
5
Forecast
4
3
2
1
1
2
3
4
5
6
7
© 2008 Thomson South-Western. All Rights Reserved
8
9
Month
Slide 53
End of Chapter 8
© 2008 Thomson South-Western. All Rights Reserved
Slide 54