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### Multi-Attribute Utility Theory (MAUT)

Dr. Yan Liu

Department of Biomedical, Industrial & Human Factors Engineering

Wright State University

Introduction

- Conflicting Objectives and Tradeoffs in Decision Problems
- e.g. higher returns vs. lower risks in investment, better performance vs. lower price of computer
- Objectives with Incomparable Attribute Scales
- “Attribute” refers to the quantity used to measure the accomplishment of an objective
- e.g. maximize profits vs. minimize impacts on environments
- Multi-Attribute Decision Making (MADM)
- A study of methods and procedures that handle multiple attributes
- Usages
- Identify a single most preferred alternative
- Rank alternatives
- Shortlist a limited number of alternatives for subsequent detailed appraisal
- Distinguish acceptable from unacceptable possibilities

Introduction (Cont.)

- Types of MADM Techniques
- Multiattribute scoring model (in Chapter 4)
- Covert attribute scales to comparable scales
- Assign weights to these attributes and then calculate the weighted average of each consequence set as an overall score
- Compare alternatives using the overall score
- Multi-Attribute Utility Theory (MAUT)
- Use utility functions to convert numerical attribute scales to utility unit scales
- Assign weights to these attributes and then calculate the weighted average of each consequence set as an overall utility score
- Compare alternatives using the overall utility score
- …

You want to buy a car with a long expected life span and a low price. You have narrowed down your choices to three alternatives: the Portalo (a relatively expensive sedan with a reputation for longevity) , the Norushi (renowned for its reliability), and the Standard Motors car (a relatively inexpensive domestic automobile). You have done some research and evaluated these three cars on both attributes, as follows.

Worst

Best

Best

Worst

Portalo

None of the cars is dominated

How much are you willing to pay to increase the life span of your car? (subjective judgment)

Norushi

Standard

Price

- Start with the Standard Motor, the cheapest among the three alternatives
- Prefer Norushi to Standard if you are willing to pay $2k or more to increase the life span of your car by 3 years
- Prefer Portalo to Norushi if you are willing to pay extra $7k or more for an additional 3 years

Trading Off Conflicting Objectives

- Need Systematic Techniques to Handle Any Decision Situation Efficiently
- Three or more objectives
- Objectives with incomparable attribute scales
- Issues to be addressed
- Construct a quantitative model of preferences to compare alternatives
- Numerical weight must be assessed for each attribute

Additive Utility Function

- A Simplified Utility Model
- Ignores interactions among attributes
- For a consequence set that has values x1, x2, …, xm on the attributes of m objectives, its overall utility is computed as

Ui(xi) – the utility function of the ith attribute

ki– the weight of the ith attribute

: the worst value of attribute Xi; : the best value of Xi

Automobile Example (Cont.)

Utility Functions

UPrice(Portalo) = UPrice(17000) = 0

Set UPrice(Standard) =UPrice(8000) = 1,

ULife(Standard) = ULife(6) = 0

ULife(Portalo) = ULife(12) = 1,

UPrice(Norushi) = UPrice(10000) = (10000 – 17000) / (8000 – 17000) = 0.78

ULife(Norushi) = ULife(9) = (9 – 6) / (12 – 6) = 0.5

- Directly specify the ratio of the weights

e.g. kPrice= 2kLife

Because kPrice+ kLife =1, then kPrice=2/3 and kLife = 1/3

U(Portalo) = 2/3•UPrice(Portalo) + 1/3•ULife(Portalo)

= 2/3(0) + 1/3(1) =1/3

U(Norushi) = 2/3•UPrice(Norushi) + 1/3•ULife(Norushi)

= 2/3(0.78) + 1/3(0.5) =0.69

U(Standard) = 2/3•UPrice(Standard) + 1/3•ULife(Standard)

= 2/3(1) + 1/3(0) =2/3

- Indirectly specify the tradeoffs between objectives

e.g. You are willing to pay up to $600 for an extra year of life span

Suppose taking the Standard Motors as the base case. You are indifferent between paying $8000 for 6 years of life span and paying $8,600 for 7 years of life span

U($8,000, 6 Years) = U($8,600, 7 Years)

kPrice•UPrice(8000) + kLife•ULife(6) = kPrice•UPrice(8600) + KLife•ULife(7)

UPrice(8600) = (8600-17000)/(8000-17000)= 0.933, ULife(7) = (7-6)/(12-6)=0.167

kPrice•1 + kLife•0 = kPrice•0.933 + kLife•0.167 0.067kPrice= 0.167kLife (Eq. 1)

kPrice + kLife = 1 (Eq. 2)

Solve Eqs (1) and (2) kPrice= 0.714, kLife = 0.286

U(Portalo) = 0.714•UPrice(Portalo) + 0.286•ULife(Portalo) = 0.286

U(Norushi) = 0.714•UPrice(Norushi) + 0.286•ULife(Norushi) = 0.7

U(Standard) = 0.714•UPrice(Standard) + 0.286•ULife(Standard) = 0.714

Indifference Curve

- Alternatives falling on the same indifference curve have the same utility
- The decision maker is indifferent among these alternatives

Life Span(Year)

0.7

0.714

0.286

Portalo

Hypothetical car

Norushi

7

Standard

Price($K)

8.6

Indifference Curves of the Automobile Example (Trade $600 for an additional year of life span)

Assessing Weights Indirectly

- Pricing Out
- Determine the marginal rate of substitution between one particular attribute (usually monetary) and any other attribute
- Marginal rate of substitution is the rate at which one attribute can be used to replace another (the slope of the indifference curves in additive utility function)
- e.g. One year of life span of a car is worth $600
- Appropriate for additive utility function

In an additive utility function, marginal rate of substitution between attributes xi and xj, Mij, is:

kLife = 0.286, kPrice= 0.714

= $0.6k per year = $600 per year

Assessing Weights Indirectly (Cont.)

- Swing Weighting
- Can be used virtually in any weight-assessment situation
- Requires a thought process of comparing individual attributes directly by imaging hypothetical outcomes
- Step One: Create a table in which the first row indicates the worst possible consequence set (with the worst level on each attribute), and each of the succeeding rows “swings” one of the attributes from the worst to best
- Step Two: Rank the consequence sets created in the above table
- Step Three: Assign a rating score to each consequence set
- Step Four: Calculate the weights from the rating scores

KLife/kPrice = 75: 100

Automobile Example (Cont.)

3

0

6 years, $17,000

12 years, $17,000

75

75/175=0.429

2

1

100

6 years, $8,000

100/175=0.571

Drug Counseling Center Choice

The drug-free center is a private nonprofit contract center that provides counseling for clients sent to it by the city courts as a condition of their parole. It has just lost its lease and must relocate.

The director of the center has screened the spaces to which it might move. After the prescreening, 6 sites are chosen for serious evaluation. The director must, of course, satisfy the sponsor, the Probation Department, and the courts that the new location is appropriate and must take the needs and wishes of both employees and clients into account. But as a first cut, the director wishes simply to evaluate the sites on the basis of values and judgments of importance that make sense internally to the center.

After consulting the members of the center staff, the director constructs a fundamental objective hierarchy that expresses the value-relevant objectives and attributes for comparing alternative center locations. Since the purpose of the evaluation to compare quality, cost is omitted.

A: Good conditions for staff

B: Easy access for clients

C: Suitability of space for center’s function

D: Administrative convenience

(0.24)

(0.43)

(0.19)

(0.14)

A1

Closeness to clients’ homes

Office size

Adequacy of space

No. and suitability of counseling rooms

(0.39)

(0.50)

B1

(0.64)

C1

(0.52)

D1

Convenience of commuting

A2

(0.21)

Access to public transportation

Flexibility of space layout

B2

D2

(0.50)

C2

No. and suitability of conference rooms

(0.32)

Office attractiveness

(0.14)

A3

(0.36)

A4

Office privacy

(0.14)

Suitability of reception and waiting area

C3

(0.16)

A5

Parking space

(0.12)

Fundamental Objectives Hierarchy

Ratings of Six Sites In terms of Attributes Corresponding to the Lowest-Level Fundamental Objectives

Relative Weights of Attributes and Utilities of Six Sites In terms of Attributes

Utility of site 1 w.r.t attribute A1

Expected utility of site 1 w.r.t attribute A

- Calculate the overall utility using the additive utility function

U(site 1) = kA∙(kA1∙U1, A1+ kA2∙U1, A2+kA3∙U1, A3+kA4∙U1,,A4 + kA5∙U1, A5) + KB∙(KB1∙U1, B1 +kB2∙U1,B2) + kC∙(kC1∙U1, C1+ kC2∙U1, C2+kC3∙U1, C3) + kD∙(kD1∙U1, D1+ kD2∙U1, D2)

= 0.43∙(0.39∙1+ 0.21∙0.47+0.14∙0.27+0.14∙1 + 0.12∙0) + 0.24∙(0.5∙0.33 +0.5∙0.71) + 0.19∙(0.52∙0.06+ 0.32∙0.63+0.16∙0.47) + 0.14∙(0.64∙0+ 0.36∙0)

=0.470

U(site 2) = kA∙(kA1∙U2, A1+ kA2∙U2, A2+kA3∙U2, A3+kA4∙U2,,A4 + kA5∙U2, A5) + KB∙(KB1∙U2, B1 +kB2∙U2,B2) + kC∙(kC1∙U2, C1+ kC2∙U2, C2+kC3∙U2, C3) + kD∙(kD1∙U2, D1+ kD2∙U2, D2)

= 0.43∙(0.39∙0.5+ 0.21∙0.26+0.14∙0.93+0.14∙0.25 + 0.12∙0.56) + 0.24∙(0.5∙0.33 +0.5∙0.71) + 0.19∙(0.52∙0.88+ 0.32∙0.5+0.16∙0.35) + 0.14∙(0.64∙0.75+ 0.36∙0.42)

=0.549

U(Site 4) = 0.404

U(Site 5) = 0.491

U(Site 6) = 0.488

In conclusion, because U(Site 2) is the highest, site 2 should be chosen

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