Approaches for Multiattribute Assessment Utility Independence Multiattribute Assessment Procedure Lecture 07 Multiattribute Utility Theory Jitesh H. Panchal ME 597: Decision Making for Engineering Systems Design Design Engineering Lab @ Purdue (DELP) School of Mechanical Engineering Purdue University, West Lafayette, IN http://engineering.purdue.edu/delp September 25, 2014 c Jitesh H. Panchal Lecture 07 1 / 32
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Lecture 07 Multiattribute Utility Theory · 2 u Y (y) is a conditional utility function on Y normalized by u Y (yo) = 0 and u Y (y 1) = 1 3 u Z (z) is a conditional utility function
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Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Lecture 07Multiattribute Utility Theory
Jitesh H. Panchal
ME 597: Decision Making for Engineering Systems Design
Design Engineering Lab @ Purdue (DELP)School of Mechanical Engineering
Purdue University, West Lafayette, INhttp://engineering.purdue.edu/delp
Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge, UK, CambridgeUniversity Press. Chapter 5.
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Conditional AssessmentsAssessing Utility Functions over “Value” FunctionsDirect Utility AssessmentQualitative Structuring of Preferences
Focus of this Lecture
Assume that the attributes for the problem are X1,X2, . . . ,Xn. If xi denotes aspecific level of Xi , the the goal is to find a utility function
u(x) = u(x1, x2, . . . , xn)
over the n attributes.
Property of the multiattribute utility function: Given two probabilitydistributions A and B over multi-attribute consequences x̃, probabilitydistribution A is at least as desirable as B if and only if
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Conditional AssessmentsAssessing Utility Functions over “Value” FunctionsDirect Utility AssessmentQualitative Structuring of Preferences
Conditional Unidimensional Utility Theory
Consider a two parameter scenario with consequence space defined by(xi ,wi). If the decision maker knew that wi were to prevail then we can carryout utility assessments for single parameter x .
x ∼ 〈x∗, πi(x), xo〉
πi(x) is the conditional utility function for x values given the state wi ,normalized by
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Conditional AssessmentsAssessing Utility Functions over “Value” FunctionsDirect Utility AssessmentQualitative Structuring of Preferences
Using Value Functions
A utility function is a value function, but a value function is not a utilityfunction!
1 First assign a value function for each point in the consequence space, x2 The utility function is monotonically increasing in V .3 Assess unidimensional utility functions for u[v(x)]
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Conditional AssessmentsAssessing Utility Functions over “Value” FunctionsDirect Utility AssessmentQualitative Structuring of Preferences
Direct Utility Assessment
Assigning utilities to possible consequences x1, x2, . . . , xR directly. Arbitrarilyset
u(xo) = 0 and u(x∗) = 1
where xo is the least preferred and x∗ is the most preferred.
If lottery 〈x∗, πr , xo〉 ∼ xr then
u(xr ) = πr
Shortcomings of the procedure:1 it fails to exploit the basic preference structure of the decision maker2 the requisite information is difficult to assess3 the result is difficult to work with in expected utility calculations and
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility FunctionMutual Utility IndependenceOne Utility-Independent AttributeGeneral Case - No Utility Independence
Utility Independence
Definition (Utility Independence)
Y is utility independent of Z when conditional preferences for lotteries on Ygiven z do not depend on the particular level of z
If Y is utility independent of Z , allconditional utility functions alonghorizontal cuts would be positive lineartransformations of each other.Therefore,
u(y , z) = g(z) + h(z)u(y , z′)
In other words, the conditional utilityfunction over Y given z does notstrategically depend on z.
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility FunctionMutual Utility IndependenceOne Utility-Independent AttributeGeneral Case - No Utility Independence
Benefits of Utility Independence in Utility Assessment
Y
Z
zo
z*
yo y*Y
Z
zo
z*
yo y*Y
Z
zo
z*
yo y*
Y
Z
zo
z*
yo y*Y
Z
zo
z*
yo y*Y
Z
zo
z*
yo y*
(a) (b) (c)
(d) (e) (f)
y1y2
z1
z2
z1
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Figure : 5.3 on page 228 (Keeney and Raiffa) (a) No independence condition holds, (b)Y is utility independent of Z , (c) Z is utility independent of Y , (d, e) Y ,Z are mutuallyutility independent, (f) Additivity assumption holds.
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility FunctionMutual Utility IndependenceOne Utility-Independent AttributeGeneral Case - No Utility Independence
Additive Independence
Definition (Additive Independence)
Attributes Y and Z are additive independent if the paired preferencecomparison of any two lotteries, defined by two joint probability distributionson Y × Z , depends only on their marginal probability distributions.
Alternate description for two attributes: For additive independence, thefollowing two lotteries must be equally preferable:
oru(y , z) = kY uY (y) + kZ uZ (z) + kYZ uY (y)uZ (z)
where1 u(y , z) is normalized by u(y0, z0) = 0 and u(y1, z1) = 1 for arbitrary y1
and z1 such that (y1, z0) � (y0, z0) and (y0, z1) � (y0, z0)
2 uY (y) is conditional utility on Y normalized by uY (y0) = 0 and uY (y1) = 13 uZ (z) is conditional utility on Z normalized by uZ (z0) = 0 and uZ (Z1) = 14 kY = u(y1, z0)
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility FunctionMutual Utility IndependenceOne Utility-Independent AttributeGeneral Case - No Utility Independence
The Multiplicative Representation
If two attributes are mutually utility independent, their utility function can berepresented by either a product form, when k 6= 0, or an additive form, whenk = 0.
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility FunctionMutual Utility IndependenceOne Utility-Independent AttributeGeneral Case - No Utility Independence
One Utility Independent Attribute
Assuming that Z is utility independent of Y , for any arbitrary y0,
u(y , z) = c1(y) + c2(y)u(y0, z), c2(y) > 0, ∀y
The two-attribute utility function can be specified by:1 three conditional utility functions, or2 two conditional utility functions and an isopreference curve, or3 one conditional utility function and two isopreference curve
Approaches for Multiattribute AssessmentUtility Independence
Multiattribute Assessment Procedure
Additive Independence and Additive Utility FunctionMutual Utility IndependenceOne Utility-Independent AttributeGeneral Case - No Utility Independence
General Case - No Utility Independence
Possibilities:1 Transformation of Y and Z to new attributes that might allow exploitation
of utility independence properties2 Direct assessment of u(y , z) for discrete points and then
interpolation/curve fitting.3 Application of independence results in subsets of the Y × Z space.4 Development of more complicated assumptions about the preference
structure that imply more general utility functions.
Assessment Procedure for Multiattribute Utility Functions
1 Introducing the terminology and ideas2 Identifying relevant independence assumptions3 Assessing conditional utility functions or isopreference curves4 Assessing the scaling constants5 Checking for consistency and reiterating
1 Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives:Preferences and Value Tradeoffs. Cambridge, UK, Cambridge UniversityPress. Chapter 5.
2 Clemen, R. T. (1996). Making Hard Decisions: An Introduction toDecision Analysis. Belmont, CA, Wadsworth Publishing Company.