10. utility theory in complex engineering...
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10. Utility Theory in Complex Engineering Design
School of Mechanical EngineeringAssociate Professor
Choi, Hae-Jin
SCHOOL OF MECHANICALENG.
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Importance of properly formulated objective functions in decision-making
Need of rigorous mathematical framework within which we can examine the preferences of individuals
Decision-making under conditions of risk
Why Utility Theory??
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Notion of Utility Modeling of the preference of individual
If A is prefer to B then utility of A is greater than utility of B If A B, then UA > UB
If the person is indifferent between A and B, then utility of A is equal to utility of B
If A ~ B, then UA = UB
Ordinal utility: provides only preference ordering Cannot added or subtracted Cannot measure strength of preference
Cardinal utility : provides ordering and strength of utility Some experts assert that cardinal utilities do not exist. However, much of the following materials are based on cardinal
utilities.
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As the amount of a good consumed increases, the marginal utility of the good decreases.
The Law of Diminishing Marginal Utility
Saturation point
Slope =Marginal utility
Utility
Quantity consumed(System performance)
Utility Function
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Monotonicity
Monotonicity vs Non-monotonicity
utili
ty
system performance
Non-Monotonicity utili
ty
System performance
Target
Larger is betterOr Smaller is better
The nominal, the better
SCHOOL OF MECHANICALENG.
C A U-6-
Most goods can be described in terms of a set of descriptors, called attributes. E.g., attributes of an airplane are cost, speed, range, payload, takeoff
distance, landing distance, reliability, maintenance cost per flight hour, etc.
Objective function in multiobjective optimization is often a weighted sum of the multiattribute
Multiattribute Utility Functions
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Multiattribute Utility Functions
Combining multiattribute utility functions Linearly additive utility
Multiplicative utility
Log-linear utility
1
1
1
( )
( )
log ( )
where, is the utility of attribute occuring in quantity and is a weighting factor
n
s i i ii
n
s i iin
s i i ii
i
i i
u a u x
u u x
u a u x
u ix a
In designing a commercial jet, the objective may be Maximizing company profit Company profit = f(air plane design)
SCHOOL OF MECHANICALENG.
C A U-8-
Decision-making under Risk
Bernoulli paradox A fair coin is flipped until, on the nth flip, it lands heads. You
then win a prize of $2n. What would you pay to enter this game? The expected value of the prize, P, is given by
Von Neumann-Morgenstern Lotteries1 1 1
1(probability of n)(prize given n)= 2 12
nn
i i iP
1
1{ } ln 22
nn
iE u
Decreasing Marginal Value
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Risk aversion, neutral, and proverse
Von Neumann-Morgenstern Utility
Risk aversion
Risk neutral
Risk proverse
Wealth
Util
ity
U’’=0 : NeutralU’’<0 : AversionU’’>0 : Proverse
SCHOOL OF MECHANICALENG.
C A U-10-
With discrete events
Expected Utility
1{ } ( )
n
i ii
E u p u x
1
1n
ii
p
, where
and xi comprise the full set of possible outcomes
With continuous probability functionmax
min{ } ( ) ( )
x
i i ixE u u x p x dx
utility
xi
p(xi)
u(xi)
SCHOOL OF MECHANICALENG.
C A U-11-
Decision-making Under Variability
Machining accuracy=1/error
utilityPDF of machine A Sampling results
PDF of machine B Sampling results
Risk averter’s decision is ‘Machine A’Expected utility of machine A is greater that that of machine B
Risk taker’s decision is ‘Machine B’ Expected utility of machine B is greater that that of machine A
d
Error=abs(d-dactual)
SCHOOL OF MECHANICALENG.
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Utility based Selection Decision SupportTesting Snap-Fits on a Light Switch Cover Plate Assembly
Primary Goals for Rapid Prototyping (RP) in this example…
Functional Product Validation Determining closeness of fit/tolerance of the
two interfacing components Obtaining a basic feel for the product Visual and physical confirmation of 3D
interface integrity
Challenge:Resource Selection Choosing a suitable RP Material and Process
CombinationProducing a Rapid Prototype that closely
resembles the final production part Functional behavior Geometry
Detail Accuracy
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Class of manufacturing processes that build parts in additive manner.
Typically layer-by-layer.
Stereolithography, Selective Laser Sintering, Fused Deposition Modeling.
Range of processes and materials is constantly expanding.
Few limitations on shape.
Not just parts, but tools and patterns too.
Additive Fabrication (Rapid Prototyping)
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Layered Manufacturing Example
http://mecadserv1.technion.ac.il/public_html/images/face_low.gif
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Stereolithography
SLA-3500
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Selective Laser Sintering (SLS)
Tightly compacted powder is selectively melted by laser to form a layer of the object
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Fused Deposition Modeling (FDM)
A plastic filament is unwound from a coil and supplies material to an extrusion nozzle
http://home.att.net/~castleisland/fdm_int.htm
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Available RP Resources
RP Materials
RP Machines
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Motivation…
RP Resource Selection -A selection problem characterized by …
Large number of alternatives and attributes Measurement of attributes on different scales Uncertainty with respect to attribute values Potentially conflicting objectives Tradeoffs among attributes
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Need…Method of Decision Support for Making Selection Decisions in
Engineering Design… Provides structure Mathematically rigorous Records viewpoints factoring into decisions Accurately reflects, rather than imposes, designer preferences Preference consistent Capable of modeling preferences quantitatively Allow for the simultaneous consideration of large numbers of
objectives Explicitly takes into consideration factors of risk and uncertainty Provides for post-solution sensitivity analysis
SCHOOL OF MECHANICALENG.
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Synthesis…
Utility-Based Selection Decision Support Problem (u-sDSP)
Word FormulationGiven alternativesIdentify attributes and associated uncertaintiesAssess Decision Maker utilities w.r.t. attributes
and attribute combinationsEvaluate alternatives using utility functionsRank alternatives based on expected utility
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Application…
Resource Selection for Product Validation
GivenIdentifyAssessEvaluateRank
AlternativesMaterialsSOMOS 7110SOMOS 8120SL 7510P400TJ 65ProcessesSLA 250SLA 3500FDM 1650ACTUA 2100
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AttributesTensile StrengthYoung’s ModulusElongation at BreakFlexural StrengthFlexural ModulusHardnessImpact StrengthDensityHeat Deflection Temp.ResistanceDurabilityFunctionalityDetail CapabilityAccuracyBuild TimeCost
Application…
Resource Selection for Product Validation
GivenIdentifyAssessEvaluateRank
AttributesTensile StrengthYoung’s ModulusElongation at BreakFlexural StrengthFlexural ModulusHardnessImpact StrengthDensityHeat Deflection Temp.ResistanceDurabilityFunctionalityDetail CapabilityAccuracyBuild TimeCost
ProvideAcronymsScalesRanges
Attribute Acronym
Tensile Strength (TS)
Scale Lower Un-acceptable Ideal Upper Un-
acceptable
Ratio 50 65 75
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Levels and or probability distributions for each attribute for each alternativeTensile StrengthYoung’s ModulusElongation at BreakFlexural StrengthFlexural ModulusHardnessImpact StrengthDensityHeat Deflection Temp.ResistanceDurabilityFunctionalityDetail CapabilityAccuracyBuild TimeCost
Application…Resource Selection for Product Validation
GivenIdentifyAssessEvaluateRank
Process Material Type of Distribution
Lower Bound/ Mean
Upper Bound/ Variance
SLA250 DSM7110 Uniform 44 69SLA3500 SL7510 Uniform 42.3 55.46SLA3500 DSM8120 Uniform 23 29FDM1650 P400 Uniform 31 37MJM2100 TJ75 Uniform 9 11
Alternatives Tensile Strength
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
Utility Functions for each AttributeID qualitative preference characteristicsID quantitative preference characteristicsFit a utility function Check for consistency
SCHOOL OF MECHANICALENG.
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
Attribute Monotonicity
Tensile Strength Target Averse
Attitude Towards
Risk
Utility Functions for each AttributeID qualitative preference characteristicsID quantitative preference characteristicsFit a utility function Check for consistency
SCHOOL OF MECHANICALENG.
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
0 0.25 0.5 0.75 0.75 0.5 0.25 0
50.00 51.75 54.43 58.88 65.00 68.86 71.32 73.40 75.00
Left Hand Side Utility Right Hand Side Utility1
Utility Functions for each AttributeID qualitative preference characteristicsID quantitative preference characteristicsFit a utility function Check for consistency
Keeney, R.L. and Raiffa, H. (19976). Decisions with Multiple Objectives: Preferences and Value Tradeoffs, New York: John Wiley and Sons
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
Utility Functions for each AttributeID qualitative preference characteristicsID quantitative preference characteristicsFit a utility functionCheck for consistency
Lower Unacceptable Ideal Upper Unacceptable
Util
ity
Tensile Strength
Normalized Functions for Non-Monotonic Attributes (i.e., Tensile Strength)
Left Hand Side ( ) 1.022 exp( 2.402 )U x x
Right Hand Side ( ) 2.018 exp(0.6934 )U x x
SCHOOL OF MECHANICALENG.
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
Utility Functions for each AttributeID qualitative preference characteristicsID quantitative preference characteristicsFit a utility function Check for consistency
TS0.6
TS1
TS0.5
p=0.2
p=0.8
Option A Certainty Equivalent
Option B Lottery
0.60.6E u TS
1 0.50.2 0.8 0.6E u u TS u TS
SCHOOL OF MECHANICALENG.
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
Multi-Attribute Utility FunctionID relevant independence assumptions and corresponding functionalform of the multi-attribute utility functionAssess scaling constants for the multi-attribute utility functionCheck multi-attribute utility function for consistency
SCHOOL OF MECHANICALENG.
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
Multi-Attribute Utility FunctionID relevant independence assumptions and corresponding functionalform of the multi-attribute utility functionAssess scaling constants for the multi-attribute utility function
Since both additive and mutual utility independence have been verified for the decision-maker in this example, the multi-attribute utility function may take an additive form.
1( )
n
i i ii
U k u A
SCHOOL OF MECHANICALENG.
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
Multi-Attribute Utility FunctionID relevant independence assumptions and corresponding functionalform of the multi-attribute utility functionAssess scaling constants for the multi-attribute utility functionCheck multi-attribute utility function for consistency
Tensile Strength 0.193741Young's Modulus 0.186446Flexural Strength 0.189041Flexural Modulus 0.192265Detail Capability 0.156366
Accuracy 0.082142
Attribute k-Values
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Application…
GivenIdentifyAssessEvaluateRank
Resource Selection for Product Validation
Process Material
SLA250 DSM7110 0.62214SLA3500 SL7510 0.44195SLA3500 DSM8120 0FDM1650 P400 0MJM2100 TJ75 0
AlternativesExpected Utility
Multi-Attribute Utility FunctionEach alternative
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Application…Resource Selection for Product Validation
Process Material
SLA250 DSM7110 0.62214SLA3500 SL7510 0.44195SLA3500 DSM8120 0FDM1650 P400 0MJM2100 TJ75 0
AlternativesExpected Utility
GivenIdentifyAssessEvaluateRank
Each AlternativeExpected Utility
Suggestion: Use DSM 7110 resin on the SLA 250 Stereo- lithography machine
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ReferencesUtility-Based Selection:
Fernandez, M. G., C. Conner Seepersad, D. W. Rosen, J. K. Allen and F. Mistree,2005, “Decision support in concurrent engineering - The utility-basedselection decision support problem,” Concurrent Engineering Research andApplications, Vol. 13, No. 1, pp. 13-27.