1. Multi-Attribute Utility Theory (MAUT)
Dr. Yan Liu
Department of Biomedical, Industrial & Human Factors Engineering
Wright State University

2. 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

3. 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 …

4. Automobile Example
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
Norushi
Standard Motors
Price ($k) 17 10 8
Life Spans (Years) 12 9 6
Worst
Best
Best
Worst

5. Life Span
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

6. 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

7. 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

8. Automobile Example (Cont.)
Utility Functions
Set UPrice(Standard) =UPrice(8000) = 1,
UPrice(Portalo) = UPrice(17000) = 0
ULife(Portalo) = ULife(12) = 1,
ULife(Standard) = ULife(6) = 0
: the worst value of attribute Xi ; : the best value of Xi
UPrice(Norushi) = UPrice(10000) = (10000 – 17000) / (8000 – 17000) = 0.78
ULife(Norushi) = ULife(9) = (9 – 6) / (12 – 6) = 0.5
Alternatives
Utilities
Portalo
Norushi
Standard Motors
UPrice
0.78 1
ULife
0.5

9. Weight Assessment 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

10. Weight Assessment (Cont.)
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) = ( )/( )= 0.933, ULife(7) = (7-6)/(12-6)=0.167
kPrice•1 + kLife•0 = kPrice• 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) •ULife(Portalo) = 0.286
U(Norushi) = 0.714•UPrice(Norushi) •ULife(Norushi) = 0.7
U(Standard) = 0.714•UPrice(Standard) •ULife(Standard) = 0.714

11. Indifference Curve
Alternatives falling on the same indifference curve have the same utility
The decision maker is indifferent among these alternatives
Utility
Life Span(Year)
0.714
0.7
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)

12. 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

13. 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

14. Attributes Swung from the Worst to best Consequence Sets to Compare
Automobile Example (Cont.)
Attributes Swung from the Worst to best
Consequence Sets to Compare
Rank
Rate
Weight
(Benchmark)
Life Span
Price
6 years, $17,000 3
12 years, $17,000 2 75
75/175=0.429
6 years, $8,000 1
100
100/175=0.571
KLife/kPrice = 75: 100 

15. 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.

16. Fundamental Objectives Hierarchy
Maximize Overall Satisfaction
A: Good conditions for staff
B: Easy access for clients
C: Suitability of space for center’s function
D: Administrative convenience
(0.43)
(0.24)
(0.19)
(0.14) A1
Office size
(0.39)
Closeness to clients’ homes
No. and suitability of counseling rooms
Adequacy of space B1
(0.50) C1
(0.52) D1
(0.64)
Convenience of commuting A2
(0.21)
Access to public transportation
Flexibility of space layout B2
(0.50) C2
No. and suitability of conference rooms D2
Office attractiveness
(0.32) A3
(0.14)
(0.36) A4
Office privacy
(0.14)
Suitability of reception and waiting area C3 A5
Parking space
(0.12)
(0.16)
Fundamental Objectives Hierarchy

17. Attributes
Sites 1 2 3 4 5 6 A A1 90 50 10 40 A2 30
100 A3 80 70 85 A4 35 A5 60 B B1 B2 95 C C1 C2 C3 D D1 D2
Ratings of Six Sites In terms of Attributes Corresponding to the Lowest-Level Fundamental Objectives

18. Attributes
Sites 1 2 3 4 5 6
A (0.43)
A1(0.39)
0.50
0.38
A2 (0.21)
0.47
0.26
0.05
A3 (0.14)
0.27
0.93
0.80
A4 (0.14)
0.25
0.31
A5 (0.12)
0.56
0.22
0.44
B (0.24)
B1(0.50)
0.33
B2 (0.50)
0.71
C (0.19)
C1 (0.52)
0.06
0.88
0.53
C2 (0.32)
0.63
C3 (0.16)
0.35
0.24
D (0.14)
D1 (0.64)
0.75
D2 (0.36)
0.42
0.11
Relative Weights of Attributes and Utilities of Six Sites In terms of Attributes

19. Calculate the overall utility using the additive utility function
Utility of site 1 w.r.t attribute A1
Expected utility of site 1 w.r.t attribute A
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∙ ∙ ∙ ∙ ∙0) ∙(0.5∙ ∙0.71) ∙(0.52∙ ∙ ∙0.47) ∙(0.64∙ ∙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∙ ∙ ∙ ∙ ∙0.56) ∙(0.5∙ ∙0.71) ∙(0.52∙ ∙ ∙0.35) ∙(0.64∙ ∙0.42)
=0.549

20. 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