10. utility theory in complex engineering...

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


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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)


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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


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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)


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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)


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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


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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…



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


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…



Utility Functions for each AttributeID qualitative preference characteristicsID quantitative preference characteristicsFit a utility function Check for consistency


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Application…



Attribute Monotonicity

Tensile Strength Target Averse

Attitude Towards

Risk



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Application…



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


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…



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


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Application…




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


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Application…



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


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Application…



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


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Application…



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…



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


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.

10. utility theory in complex engineering...

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