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

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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

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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 …

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4 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 AttributesPortaloNorushi Standard Motors Price ($k)17108 Life Spans (Years) 1296 Worst Best Worst Best Automobile Example

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5 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 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 Portalo Norushi Standard Life Span Price

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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

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7 Additive Utility Function A Simplified Utility Model Ignores interactions among attributes For a consequence set that has values x 1, x 2, …, x m on the attributes of m objectives, its overall utility is computed as U i (x i ) – the utility function of the ith attribute k i – the weight of the ith attribute

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8 U Price (Norushi) = U Price (10000) = (10000 – 17000) / (8000 – 17000) = 0.78 Set U Price (Standard) =U Price (8000) = 1, Utility Functions U Price (Portalo) = U Price (17000) = 0 U Life (Norushi) = U Life (9) = (9 – 6) / (12 – 6) = 0.5 Alternatives UtilitiesPortaloNorushi Standard Motors U Price 00.781 U Life 10.50 : the worst value of attribute X i ; : the best value of X i U Life (Portalo) = U Life (12) = 1, U Life (Standard) = U Life (6) = 0 Automobile Example (Cont.)

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9 Directly specify the ratio of the weights Weight Assessment e.g. k Price = 2k Life Because k Price + k Life =1, then k Price =2/3 and k Life = 1/3 U(Norushi) = 2/3U Price (Norushi) + 1/3U Life (Norushi) = 2/3(0.78) + 1/3(0.5) =0.69 U(Standard) = 2/3U Price (Standard) + 1/3U Life (Standard) = 2/3(1) + 1/3(0) =2/3 U(Portalo) = 2/3U Price (Portalo) + 1/3U Life (Portalo) = 2/3(0) + 1/3(1) =1/3

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10 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) = 0.714U Price (Norushi) + 0.286U Life (Norushi) = 0.7 U(Standard) = 0.714U Price (Standard) + 0.286U Life (Standard) = 0.714 U(Portalo) = 0.714U Price (Portalo) + 0.286U Life (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) k Price U Price (8000) + k Life U Life (6) = k Price U Price (8600) + K Life U Life (7) U Price (8600) = (8600-17000)/(8000-17000)= 0.933, U Life (7) = (7-6)/(12-6)=0.167 k Price 1 + k Life 0 = k Price 0.933 + k Life 0.167 0.067k Price = 0.167k Life (Eq. 1) Weight Assessment (Cont.) Solve Eqs (1) and (2) k Price = 0.714, k Life = 0.286 k Price + k Life = 1 (Eq. 2)

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11 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) 0.714 0.7 0.286 Life Span(Year) Utility 8.6 7 Hypothetical car

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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 x i and x j, M ij, is: k Life = 0.286, k Price = 0.714 = $0.6k per year = $600 per year

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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

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14 Attributes Swung from the Worst to best Consequence Sets to Compare RankRateWeight (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.) K Life /k Price = 75: 100

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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.

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16 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’ homes Access 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) (0.12) (0.50) (0.52) (0.32) (0.16) (0.64) (0.36) Fundamental Objectives Hierarchy A1A1 A2A2 A3A3 A4A4 A5A5 B1B1 B2B2 C1C1 C2C2 C3C3 D1D1 D2D2

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17 Attributes Sites 123456 A A1905010501040 A2503010010530 A3308070108580 A4903040103550 A51060301010050 B B130 0509030 B270 95501070 C C110805509050 C2605010 9050 C3504050109530 D D1107050905060 D204050951040 Ratings of Six Sites In terms of Attributes Corresponding to the Lowest-Level Fundamental Objectives

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18 Attributes Sites 123456 A (0.43) A1(0.39)10.500 00.38 A2 (0.21)0.470.2610.0500.26 A3 (0.14)0.270.930.80010.93 A4 (0.14)10.250.3800.310.50 A5 (0.12)00.560.22010.44 B (0.24) B1(0.50)0.33 00.5610.33 B2 (0.50)0.71 10.4700.71 C (0.19) C1 (0.52)0.060.8800.531 C2 (0.32)0.630.50001 C3 (0.16)0.470.350.47010.24 D (0.14) D1 (0.64)00.750.501 0.63 D2 (0.36)00.420.5310.110.42 Relative Weights of Attributes and Utilities of Six Sites In terms of Attributes

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19 Calculate the overall utility using the additive utility function U(site 1) = k A ∙(k A1 ∙U 1, A1 + k A2 ∙U 1, A2 +k A3 ∙U 1, A3 +k A4 ∙U 1,,A4 + k A5 ∙U 1, A5 ) + K B ∙(K B1 ∙U 1, B1 +k B2 ∙U 1,B2 ) + k C ∙(k C1 ∙U 1, C1 + k C2 ∙U 1, C2 +k C3 ∙U 1, C3 ) + k D ∙(k D1 ∙U 1, D1 + k D2 ∙U 1, 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) = k A ∙(k A1 ∙U 2, A1 + k A2 ∙U 2, A2 +k A3 ∙U 2, A3 +k A4 ∙U 2,,A4 + k A5 ∙U 2, A5 ) + K B ∙(K B1 ∙U 2, B1 +k B2 ∙U 2,B2 ) + k C ∙(k C1 ∙U 2, C1 + k C2 ∙U 2, C2 +k C3 ∙U 2, C3 ) + k D ∙(k D1 ∙U 2, D1 + k D2 ∙U 2, 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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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

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