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Multi-Attribute Utility Theory (MAUT)

Dr. Yan Liu

Department of Biomedical, Industrial & Human Factors Engineering

Wright State University

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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

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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

…

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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.

  Alternatives

Attributes Portalo NorushiStandard Motors

Price ($k) 17 10 8

Life Spans (Years)

12 9 6

Worst Best

WorstBest

Automobile Example

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None of the cars is dominated

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

Start with the Standard Motor, the cheapest among the three alternativesPrefer 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

Portalo

Norushi

Standard

Life Span

Price

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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

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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

m

iiii

mmmm

xk

xkxkxkxxx

1

22211121

)(U

)(U)(U)(U),,,(U

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

)1( 21 mkkk

1),,,(U0 21 mxxx

1)(U0 ii x

ki – the weight of the ith attribute

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

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

Utility Functions

UPrice(Portalo) = UPrice(17000) = 0

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

  Alternatives

Utilities Portalo NorushiStandard Motors

UPrice 0 0.78 1ULife 1 0.5 0

)(U x

ii

i

xx

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

ix

ULife(Portalo) = ULife(12) = 1, ULife(Standard) = ULife(6) = 0

Automobile Example (Cont.)

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• Directly specify the ratio of the weights

Weight Assessment

e.g. kPrice= 2kLife

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

U(Portalo) = 2/3•UPrice(Portalo) + 1/3•ULife(Portalo) = 2/3(0) + 1/3(1) =1/3

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• Indirectly specify the tradeoffs between objectives

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

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

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

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

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)

Weight Assessment (Cont.)

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

kPrice + kLife = 1 (Eq. 2)

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

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

Portalo

Norushi

Standard

Price($K)

0.7140.7 0.286Life Span(Year)

Utility

8.6

7

Hypothetical car

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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:

|-|/

|-|/-+

-+

jjj

iii

xxk

xxkijM

kLife = 0.286, kPrice= 0.714

|k17$-k8$|/714.0|yr6-yr12|/286.0

=LPM = $0.6k per year = $600 per year

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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

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Attributes Swung from the Worst to best

Consequence Sets to Compare

Rank Rate Weight

(Benchmark)    

Life Span      

Price      

6 years, $17,000

12 years, $17,000

6 years, $8,000

3

1

2

0

100

75 75/175=0.429

100/175=0.571

Automobile Example (Cont.)

KLife/kPrice = 75: 100

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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.

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A: Good conditions for staff

B: Easy access for clients

C: Suitability of space for center’s function

D: Administrative convenience

Maximize Overall Satisfaction

Office size

Convenience of commuting

Office attractiveness

Office privacy

Parking space

Closeness to clients’ homesAccess to public transportation

No. and suitability of counseling rooms

Suitability of reception and waiting area

No. and suitability of conference rooms

Adequacy of space

Flexibility of space layout

(0.43) (0.24) (0.19) (0.14)(0.39)

(0.21)

(0.14)

(0.14)

(0.12)

(0.50)

(0.50)

(0.52)

(0.32)

(0.16)

(0.64)

(0.36)

Fundamental Objectives Hierarchy

A1

A2

A3

A4

A5

B1

B2

C1

C2

C3

D1

D2

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AttributesSites

1 2 3 4 5 6

A

A1 90 50 10 50 10 40A2 50 30 100 10 5 30A3 30 80 70 10 85 80A4 90 30 40 10 35 50A5 10 60 30 10 100 50

BB1 30 30 0 50 90 30B2 70 70 95 50 10 70

CC1 10 80 5 50 90 50C2 60 50 10 10 90 50C3 50 40 50 10 95 30

DD1 10 70 50 90 50 60D2 0 40 50 95 10 40

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

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AttributesSites

1 2 3 4 5 6

A (0.43)

A1(0.39) 1 0.50 0 0.50 0 0.38A2 (0.21) 0.47 0.26 1 0.05 0 0.26A3 (0.14) 0.27 0.93 0.80 0 1 0.93A4 (0.14) 1 0.25 0.38 0 0.31 0.50A5 (0.12) 0 0.56 0.22 0 1 0.44

B (0.24)B1(0.50) 0.33 0.33 0 0.56 1 0.33B2 (0.50) 0.71 0.71 1 0.47 0 0.71

C (0.19)C1 (0.52) 0.06 0.88 0 0.53 1 0.53C2 (0.32) 0.63 0.50 0 0 1 0.50C3 (0.16) 0.47 0.35 0.47 0 1 0.24

D (0.14)D1 (0.64) 0 0.75 0.50 1 0.50 0.63D2 (0.36) 0 0.42 0.53 1 0.11 0.42

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

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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

Utility of site 1 w.r.t attribute A1 Expected utility of site 1 w.r.t attribute A

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U(site 3) = 0.378

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