## Presentation on theme: "Session 5b"— Presentation transcript

Slide1

Session 5bSlide2

Decision Models -- Prof. Juran

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Overview

Evolutionary Solver

(Genetic Algorithm)

Advertising Example

Product Design Example

Conjoint AnalysisSlide3

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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.Slide4

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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).Slide5

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3 Solvers

Simplex LP Solver

GRG Nonlinear Solver

Evolutionary SolverSlide6

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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?Slide7

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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.Slide8

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Formulation

Decision Variables

Define

x

i

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

x

1

= 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. Slide9

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Define

y

j

to be a binary variable, such that

y

j

= 0 if no commercials are assigned to break

j

, and

y

j

= 1 if any commercials are assigned to break

j

.

Define

v

ij

to be a binary variable such that

v

ij

= 1 if commercial

i

is assigned to break

j

, and

v

ij

= 0 otherwise.

Define

w

i to be the duration of commercial i.

FormulationSlide10

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Formulation

Note that the duration of break

j

is equal to

Let

t

j

be the amount of “overtime” in break

j

. That is,Slide11

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Formulation

Objective

Our objective, then, is to:

Minimize Z =

where

m

is a “large number”. Slide12

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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

t

j

> 0. As it happens, in this problem

m

= 100 works fine.

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Formulation

Constraints

For all

x

i

, 1≤

x

i

≤ 8.

For all

x

i

,

x

i

is an integer.

For all

y

j

,

y

j

is binary.Slide14

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Solution MethodologySlide15

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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 (

x

i

) are in the range E5:E12 (in the spreadsheet shown, all commercials are assigned to break 1, so

x

i

= 1 for all

i

).

The range B5:B12 contains the durations of each commercial (

w

i

), and the range B16:B23 uses the Excel SUMIF function to calculate the duration of each commercial break.Slide16

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Solution Methodology

The range C16:C23 keeps track of which breaks have any assignments (the

y

i

variables), while the range D16:D23 keeps track of how much the breaks go over the maximum limit (the

t

j

variables). Recall that the

y

i

and

t

j

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

y

i

variables.Slide17

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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.Slide18

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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. Slide20

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Optimal SolutionSlide21

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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.Slide22

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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 ChicagoSlide23

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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.Slide24

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Conjoint Analysis

After determining the contribution of each attribute to the consumer’s overall evaluation, one could

Define the product with the optimal combination of features

Predict market shares of different products with different sets of features

Isolate groups of customers who place differing importances on different features

Identify marketing opportunities by exploring the market potential for feature combinations not currently available

Show the relative contributions of each attribute and each level to the overall evaluation of the productSlide25

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Product Design ExampleSlide26

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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 "part-worth" that each customer gives to each level of each attribute. Slide27

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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. Slide28

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For example, consider Segment 1 and these two products:

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.Slide30

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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.Slide32

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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.)Slide33

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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

.Slide34

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Symbol

Variable Description

ij

x

A binary variable; 1 if segment

i

will buy product

j

,

0 otherwise.

ij

v

The total “value” that segment

i

places on product

j

.

For our two products (

j

= 1, 2),

1

=

ij

x

if

j

i

ij

v

v

not

products

>

jkl

a

A binary variable; 1 if product

j

has level

l

of

attribute

k

, 0 otherwise.

There are 1

2 of these per product; 36 total in this

problem.

ikl

b

The “value” placed by segment

i

on level

l

of

attribute

k

.

There are 12 of these per segment; 60 total for this

problem (as shown in the table on

slide 27

).

Slide35

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Example:

Using the example on slide 28, consider Segment 1’s evaluation of Razors A and B.

For Razor A:

Slide36

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For Razor B:

Since , , and .

In English, customer segment 1 will buy Razor A and not buy Razor B.Slide37

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FormulationSlide38

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Solution MethodologySlide39

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Solution MethodologySlide40

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Optimal SolutionSlide42

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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.Slide43

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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