Session 5b
Overview
Evolutionary Solver
(Genetic Algorithm)
• Advertising Example
• Product Design Example
– Conjoint Analysis
Decision Models -- Prof. Juran
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Nonlinear Problems
Some nonlinear problems can be formulated in a
linear fashion (i.e. some network problems).
Other nonlinear functions can be solved with our
basic methods (i.e. smooth, continuous functions
that are concave or convex, such as portfolio
variances).
However, there are many types of nonlinear
problems that pose significant difficulties.
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Nonlinear Problems
The linear solution to a nonlinear (say, integer)
problem may be infeasible.
The linear solution may be far away from the actual
optimal solution.
Some functions have many local minima (or
maxima), and Solver is not guaranteed to find the
global minimum (or maximum).
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3 Solvers
• Simplex LP Solver
• GRG Nonlinear Solver
• Evolutionary Solver
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Radio Advertising Example
Music radio WABC has commercials of the
following lengths (in seconds):
15, 15, 20, 25, 30, 35, 40, 57
The commercials must be assigned to 60-second
breaks. What is the fewest number of breaks that
are needed to air all of the commercials?
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Managerial Problem Definition
Decision Variables
Which commercials get assigned to which
programming breaks.
Objective
Minimize the total number of breaks.
Constraints
Every advertisement must be aired.
No break can be longer than 60 seconds.
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Formulation
Decision Variables
Define xi to be an integer variable identifying the
break to which commercial i is assigned. For example,
if commercial 1 is assigned to break 4, then x1 = 4.
It should be clear that we won’t need any more than
eight breaks, because there are only eight
commercials. These eight x variables are the decision
variables.
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Formulation
Define yj to be a binary variable, such that yj = 0 if no
commercials are assigned to break j, and yj = 1 if any
commercials are assigned to break j.
Define vij to be a binary variable such that vij = 1 if
commercial i is assigned to break j, and vij = 0
otherwise.
Define wi to be the duration of commercial i.
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Formulation
8
Note that the duration of break j is equal to
w v
i 1
i
ij
Let tj be the amount of “overtime” in break j. That is,
 8

t j  max  w i v ij  60 , 0 
 i 1

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Formulation
Objective
Our objective, then, is to:
8
Minimize Z =
y
j 1
8
j
 
 m t j
j 1
where m is a “large number”.
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Formulation
This is a good example of an advanced optimization
trick: taking a constraint and building it into the
objective function.
It doesn’t really matter what value we use for m, as
long as it is sufficiently large as to prevent any tj > 0.
As it happens, in this problem m = 100 works fine.
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Formulation
Constraints
For all xi, 1≤ xi ≤ 8.
For all xi, xi is an integer.
For all yj, yj is binary.
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Solution Methodology
E
D
C
B
A
60
1 Length of break (seconds)
2
Assignments of commercials to breaks
Lengths of commercials
3
Break
Commercial
Length
Commercial
4
1
1
15
1
5
1
2
15
2
6
1
3
20
3
7
1
4
25
4
8
1
5
30
5
9
1
6
35
6
10
1
7
40
7
11
1
8
57
8
12
13
=SUMIF($E$5:$E$12,A16,$B$5:$B$12)
Information on breaks
14
Amt over max
Used?
Seconds
Break
15
177
1
237
1
16
0
0
0
2
17
0
0
0
3
18
0
0
0
4
19
0
0
0
5
20
0
0
0
6
21
0
0
0
7
22
0
0
0
8
23
24
=IF(B23>0,1,0)
17701
Total "cost"
25
=MAX(B23-$B$1,0)
26
=SUM(C16:C23)+100*SUM(D16:D23)
27
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Solution Methodology
The objective function is in B25, including a penalty of 100
units per second if any breaks go over 60 seconds.
The decision variables (xi) are in the range E5:E12 (in the
spreadsheet shown, all commercials are assigned to break
1, so xi = 1 for all i).
The range B5:B12 contains the durations of each
commercial (wi), and the range B16:B23 uses the Excel
SUMIF function to calculate the duration of each
commercial break.
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Solution Methodology
The range C16:C23 keeps track of which breaks have
any assignments (the yi variables), while the range
D16:D23 keeps track of how much the breaks go over
the maximum limit (the tj variables). Recall that the yi
and tj variables are the basic ingredients of the
productive function.
Notice how the use of the IF function in C16:C23
precludes the need to have an explicit binary constraint
in Solver for the yi variables.
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Solution Methodology
The standard simplex algorithm (Solver’s default
method) won’t work on this problem. The GRG
Nonlinear algorithm will make an honest effort, but is
likely to give up without finding the optimal solution.
This is because of our use of MAX, IF, and SUMIF
functions, resulting in discontinuities in our productive
function and constraints as functions of the decision
variables.
However, the Evolutionary Solver, a genetic algorithm,
can do a good job with a problem like this.
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Solution Methodology
The Evolutionary Solver operates in a completely
different way from the other types. Instead of searching
in a structured way guaranteed to reach the optimal
solution, genetic algorithms operate somewhat like
biological evolutionary processes, with some degree of
randomness in the steps taken from one solution to the
next.
In a finite period of time, the Evolutionary Solver is not
guaranteed to find the optimal solution, but it will find
very good solutions and try to improve upon them.
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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
25
A
Length of break (seconds)
B
60
Lengths of commercials
Commercial
1
2
3
4
5
6
7
8
Length
15
15
20
25
30
35
40
57
Information on breaks
Break
1
2
3
4
5
6
7
8
Seconds
0
0
57
60
60
60
0
0
Total "cost"
4
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C
D
E
Assignments of commercials to breaks
Commercial
Break
1
5
2
5
3
4
4
6
5
5
6
6
7
4
8
3
Used?
0
0
1
1
1
1
0
0
Amt over max
0
0
0
0
0
0
0
0
20
Conclusions
The solution indicates that commercials 1, 2, and 5 should
go in one break, 3 and 7 should go in another, 4 and 6
should go in another, and 8 should go by itself.
A reasonably bright person could solve this problem in
their head, of course. The trick here was to set it up so that
a computer could solve it, providing a method for the
solution of much larger problems with the same basic
structure.
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Product Design Example
Conjoint Analysis is a multivariate technique used
specifically to understand how respondents
develop preferences for products or services.
It is based on the simple premise that consumers
evaluate the value or utility of a product (real or
hypothetical) by combining the utility provided by
each attribute characterizing the product.
-- Prof. Pradeep Chintagunta, Univ. of Chicago
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Conjoint Analysis
Conjoint Analysis is a decompositional method.
Respondents provide overall evaluations of products
that are presented to them as combinations of
attributes. These evaluations are then used to infer
the utilities of the individual attributes comprising
the products.
In many situations, this is preferable to asking
respondents how important certain attributes are, or
to rate how well a product performs on each of a
number of attributes.
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Conjoint Analysis
After determining the contribution of each attribute to the
consumer’s overall evaluation, one could
1.Define the product with the optimal combination of features
2.Predict market shares of different products with different
sets of features
3.Isolate groups of customers who place differing importances
on different features
4.Identify marketing opportunities by exploring the market
potential for feature combinations not currently available
5.Show the relative contributions of each attribute and each
level to the overall evaluation of the product
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Product Design Example
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Product Design Example
Assume that a consumer's purchase decision on an
electric razor is based on four attributes, each of
which can be set at one of three levels (1, 2, or 3).
Using conjoint analysis, our analysts have divided
the market into five segments (labeled as customers
1, 2, 3, 4, and 5) and have determined the "partworth" that each customer gives to each level of
each attribute.
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We assume here that all customers within a particular
segment view electric razors more or less the same in
terms of which levels of which attributes constitute an
attractive product.
We also assume that customers in a segment conduct a
sort of mathematical analysis (perhaps unconsciously) in
which they weigh the various attributes of a product to
come up with an overall value with respect to competing
products.
Conjoint analysis usually assumes the customer buys the
product yielding the highest total part-worth.
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Part-Worths
Segment 1 Segment 2 Segment 3 Segment 4 Segment 5
Levels
4
4
1
1
1
1 (Slim)
Attribute 1
4
3
4
1
1
2 (Wide)
(Handle)
3
3
3
4
2
3 (Long)
4
2
3
1
4
1 (Rubber)
Attribute 2
3
1
4
4
1
2 (Textured Plastic)
(Grip)
1
1
3
1
4
3 (Smooth Plastic)
1
3
3
1
4
1 (Blue)
Attribute 3
1
3
4
4
1
2 (Black)
(Color)
4
4
2
2
3
3 (Red)
4
4
3
4
4
1 (Aloe)
Attribute 4
3
2
2
2
2
2 (Lube)
(Strip)
4
2
2
2
2
3 (None)
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For example, consider Segment 1 and these two products:
Attribute 1 (Handle)
Attribute 2 (Grip)
Attribute 3 (Color)
Attribute 4 (Strip)
Razor A
1 (Slim)
1 (Rubber)
1 (Blue)
1 (Aloe)
Razor B
2 (Wide)
2 (Textured Plastic)
2 (Black)
2 (Lube)
We assume that customers in Segment 1 will not buy
Product B, because they value Product A at 1 + 4 + 4 + 4 =
13 and Product B at 1 + 1 + 1 + 2 = 5.
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Attribute 1
(Handle)
Attribute 2
(Grip)
Attribute 3
(Color)
Attribute 4
(Strip)
Razor A
Levels
1 (Slim)
2 (Wide)
3 (Long)
1 (Rubber)
2 (Textured Plastic)
3 (Smooth Plastic)
1 (Blue)
2 (Black)
3 (Red)
1 (Aloe)
2 (Lube)
3 (None)
Total
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Seg. 1
1
1
2
4
1
4
4
1
3
4
2
2
13
Razor B
Levels
1 (Slim)
2 (Wide)
3 (Long)
1 (Rubber)
2 (Textured Plastic)
3 (Smooth Plastic)
1 (Blue)
2 (Black)
3 (Red)
1 (Aloe)
2 (Lube)
3 (None)
Seg. 1
1
1
2
4
1
4
4
1
3
4
2
2
5
30
Currently there is a single product in the market that sets
all four attributes equal to 1 (call it Razor 0). We want to
introduce two new types of electric razors, and capture
as much of the market as possible.
We want to design a two-product line that maximizes
the number of market segments that will buy one of our
two products. Assume that in the case of a tie, the
consumer does not purchase our product.
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Managerial Formulation
Decision Variables
Which levels of each attribute to design into each of our
two products.
Objective
Maximize the number of customer segments who will
buy one of our products.
Constraints
There are only four attributes, each of which must be
assigned to one of three existing levels for each product.
(In other words, no product can have more or less than
one level per attribute.)
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Formulation: Preliminaries
There are five customer segments, and we will index them
from 1 to 5 with the subscript letter i.
There are three products (the existing Razor 0, plus our
Razors 1 and 2 to be designed), and we will index them from
1 to 3 with the subscript letter j.
There are four product attributes, and we will index them
from 1 to 4 with the subscript letter k.
There are three possible levels for each attribute, and we will
index them from 1 to 3 with the subscript letter l.
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Symbol
Variable Description
x ij
A binary variable; 1 if segmenti will buy product j,
0 otherwise.
v ij
The total “value” that segmenti places on productj.
For our two products (j = 1, 2), x ij  1 if
v ij > v i products
not j
a jkl
A binary variable; 1 if productj has level l of
attribute k, 0 otherwise.
There are 12 of these per product; 36 total in this
problem.
b ikl
The “value” placed by segmenti on level l of
attribute k.
There are 12 of these per segment; 60 total for this
problem (as shown in the table onslide 27).
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Example: Using the example on slide 28, consider Segment
1’s evaluation of Razors A and B.
For Razor A:
v1A
4
3
  a Akl b 1 kl
k 1 l 1
 a A , 1 , 1 b 1 , 1 , 1  a A ,1 , 2 b 1 , 1 , 2  a A ,1 , 3 b 1 , 1 , 3  a A , 2 ,1 b 1 , 2 , 1  a A , 2 , 2 b 1 , 2 , 2  a A , 2 , 3 b 1 , 2 , 3  ...  a A , 4 , 3 b 1 , 4 , 3
 1 1   0 1  0 2   1 4   0 1  0 4   ...  0 2 
 100 400 400 400
 13
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For Razor B:
v1B
4
3
  a Bkl b 1 kl
k 1 l 1
 a B , 1 , 1 b 1 , 1 , 1  a B , 1 , 2 b 1 , 1 , 2  a B , 1 , 3 b 1 , 1 , 3  a B , 2 , 1 b 1 , 2 , 1  a B , 2 , 2 b 1 , 2 , 2  a B , 2 , 3 b 1 , 2 , 3  ...  a B , 4 , 3 b 1 , 4 , 3
 0 1   1 1   0 2   0 4   1 1   0 4   ...  0 2 
 010010010020
5
Since v 1 A > v 1 B , x 1 A  1 , and x 1 B  0 .
In English, customer segment 1 will buy Razor A and not buy
Razor B.
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Formulation
Decision Variables
We need to set 12 a jkl variables (attribute levels)
for each of our two products.
Objective
Maximize Z =
5
2
 x
i 1 j 1
ij
Constraints
3
For each attribute k in product j,
a
l 1
jkl
1
All a jkl are binary.
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Solution Methodology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
A
Razor 1 Attributes
B
C
D
E
F
G
H
J
K
L
Number buying ours
=SUM(M6:Q6)
2
Level
2
2
2
2
Part-worths for segments
1 2 3 =VLOOKUP($B5,$K$16:$Q$19,E$21+2)
4 5
Wide
1 1 4 3 4
Textured Plastic
1 4 4 1 3
Maximums
Black
1 4 4 3 1
Buy ours?
Lube
2 2 2 2 3
=SUM(E4:E7)
5 11 14 9 11
Level
3
3
3
3
Part-worths for segments
1 2 3 4 5
Long
2 4 3 3 3
Smooth Plastic
4 1 3 1 1
Red
3 2 2 4 4
None
2 2 2 2 4
11 9 10 10 12
Attribute Level
1
1
1
1
Part-worths for segments
1 2 3 4 5
Slim
1 1 1 4 4
Rubber
4 1 3 2 4
Blue
4 1 3 3 1
Aloe
4 4 3 4 4
13 7 10 13 13
Attribute
Handle
Grip
Color
Strip
Totals
Razor 2 Attributes
Attribute
Handle
Grip
Color
Strip
Totals
Razor 0 Attributes
Handle
Grip
Color
Strip
Totals
I
M
13
0
Attribute Level
Handle
1 Slim
2 Wide
3 Long
N
O
P
=MAX(D26,D8,D17)
11
14
13
=IF(D26<M5,1,0)
1
1
0
Q
13
0
Sgmt 1 Sgmt 2 Sgmt 3 Sgmt 4 Sgmt 5
1
1
1
4
4
1
1
4
3
4
2
4
3
3
3
Grip
1 Rubber
2 Textured Plastic
3 Smooth Plastic
4
1
4
1
4
1
3
4
3
2
1
1
4
3
1
1 Blue
2 Black
3 Red
4
1
3
1
4
2
3
4
2
3
3
4
1
1
4
1 Aloe
2 Lube
3 None
4
2
2
4
2
2
3
2
2
4
2
2
4
3
4
Color
Decision Models -- Prof. Juran
Strip
38
Solution Methodology
We have used an elegant Excel trick to make the model more manageable,
mainly through the use of VLOOKUP functions in the ranges D4:H7, D13:H16,
and D22:H25. This lets us reduce the complete 5 x 12 matrix of b ikl “part-worth”
values (see slide 27) down to a 5 x 4 matrix relevant to the attribute levels
actually used by a given product.
This lets us collapse the 24a jkl decision variables down to 8 changing cells
(B4:B7 and B13:B16). Instead of being binary, these are now integers between 1
and 3.
The use of MAX and IF functions makes this a good problem for the
Evolutionary Solver.
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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
25
26
A
Razor 1 Attributes
B
C
D
E
F
G
H
Level
1
1
3
1
Part-worths for segments
1 2 3 4 5
Slim
1 1 1 4 4
Rubber
4 1 3 2 4
Red
3 2 2 4 4
Aloe
4 4 3 4 4
12 8 9 14 16
Level
3
1
1
1
Part-worths for segments
1 2 3 4 5
Long
2 4 3 3 3
Rubber
4 1 3 2 4
Blue
4 1 3 3 1
Aloe
4 4 3 4 4
14 10 12 12 12
Attribute Level
1
1
1
1
Part-worths for segments
1 2 3 4 5
Slim
1 1 1 4 4
Rubber
4 1 3 2 4
Blue
4 1 3 3 1
Aloe
4 4 3 4 4
13 7 10 13 13
Attribute
Handle
Grip
Color
Strip
Totals
Razor 2 Attributes
Attribute
Handle
Grip
Color
Strip
Totals
Razor 0 Attributes
Handle
Grip
Color
Strip
Totals
I
J
K
L
Number buying ours
5
M
N
O
P
Q
Maximums
Buy ours?
14
1
10
1
12
1
14
1
16
1
Attribute Level
Handle
1 Slim
2 Wide
3 Long
Sgmt 1 Sgmt 2 Sgmt 3 Sgmt 4 Sgmt 5
1
1
1
4
4
1
1
4
3
4
2
4
3
3
3
Grip
1 Rubber
2 Textured Plastic
3 Smooth Plastic
4
1
4
1
4
1
3
4
3
2
1
1
4
3
1
1 Blue
2 Black
3 Red
4
1
3
1
4
2
3
4
2
3
3
4
1
1
4
1 Aloe
2 Lube
3 None
4
2
2
4
2
2
3
2
2
4
2
2
4
3
4
Color
Decision Models -- Prof. Juran
Strip
41
Conclusions
It turns out that there is a line of two products that can
capture all five segments!
Razor 1, with attribute levels (1, 1, 3, 1), captures
segments 4 and 5.
Razor 2, with attribute levels (3, 1, 1, 1), captures
segments 1, 2, and 3.
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Summary
Evolutionary Solver
(Genetic Algorithm)
• Advertising Example
– Integer and Binary tricks
– Moving Constraints into the productive Function
– MAX, IF, SUMIF
• Product Design Example
– Conjoint Analysis
– VLOOKUP, MAX, IF
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