10. Utility Theory in Complex Engineering Design. School of Mechanical Engineering Associate Professor Choi, Hae-Jin

Transcription

1 10. Utility Theory in Complex Engineering Design School of Mechanical Engineering Associate Professor Choi, Hae-Jin

2 Why Utility Theory?? 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 -2-

3 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. -3-

4 The Law of Diminishing Marginal Utility As the amount of a good consumed increases, the marginal utility of the good decreases. Utility Function Utility Slope = Marginal utility Saturation point -4- Quantity consumed (System performance)

5 Monotonicity vs Non-monotonicity Monotonicity utility Larger is better Or Smaller is better system performance Non-Monotonicity utility The nominal, the better Target System performance -5-

6 Multiattribute Utility Functions 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 -6-

7 Multiattribute Utility Functions In designing a commercial jet, the objective may be Maximizing company profit Company profit = f(air plane design) Combining multiattribute utility functions n u au ( x ) Linearly additive utility Multiplicative utility Log-linear utility u s i i i i1 n u ( x ) s i i i1 n u a log u ( x ) s i i i i1 where, ui is the utility of attribute i occuring in quantity x and a is a weighting factor i i -7-

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 $2 n. What would you pay to enter this game? The expected value of the prize, P, is given by 1 n P (probability of n)(prize given n)= 2 1 n i1 i12 i1 Von Neumann-Morgenstern Lotteries 1 Eu {} ln2 n i1 2 n Decreasing Marginal Value -8-

9 Von Neumann-Morgenstern Utility Risk aversion, neutral, and proverse Risk neutral Utility Risk aversion U =0 : Neutral U <0 : Aversion U >0 : Proverse Risk proverse Wealth -9-

10 Expected Utility With discrete events n Eu {} pux i ( i) i1, where n i1 p i 1 and xi comprise the full set of possible outcomes With continuous probability function x max E{ u} u( x ) p( x ) dx x min i i i utility u(xi) p(xi) xi -10-

11 Decision-making Under Variability Error=abs(d-dactual) utility PDF of machine A Sampling results PDF of machine B Sampling results Machining accuracy =1/error d 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 -11-

12 Utility based Selection Decision Support Testing 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 Combination Producing a Rapid Prototype that closely resembles the final production part Functional behavior Geometry Detail Accuracy -12-

13 Additive Fabrication (Rapid Prototyping) 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. -13-

14 Layered Manufacturing Example

15 Stereolithography SLA

16 Selective Laser Sintering (SLS) Tightly compacted powder is selectively melted by laser to form a layer of the object -16-

17 Fused Deposition Modeling (FDM) A plastic filament is unwound from a coil and supplies material to an extrusion nozzle -17-

18 Available RP Resources RP Machines RP Materials

19 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 -19-

20 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

21 Synthesis Utility-Based Selection Decision Support Problem (u-sdsp) Word Formulation Given alternatives Identify attributes and associated uncertainties Assess Decision Maker utilities w.r.t. attributes and attribute combinations Evaluate alternatives using utility functions Rank alternatives based on expected utility -21-

22 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Alternatives Materials SOMOS 7110 SOMOS 8120 SL 7510 P400 TJ 65 Processes SLA 250 SLA 3500 FDM 1650 ACTUA

23 Application Resource Selection for Product Validation Given Attributes Provide Identify Tensile Strength Acronyms Assess Young s Modulus Scales Evaluate Elongation at Break Ranges Rank Flexural Strength Flexural Modulus Hardness Attribute Acronym Scale Lower Unacceptable Ideal Upper Unacceptable Impact Strength Tensile Strength (TS) Ratio Density Heat Deflection Temp. Resistance Durability Functionality Detail Capability Accuracy Build Time Cost -23-

24 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Levels and or probability distributions for each attribute for each alternative Tensile Strength Young s Modulus Alternatives Tensile Strength Elongation at Break Type of Lower Bound/ Upper Bound/ Process Material Distribution Mean Variance Flexural Strength SLA250 DSM7110 Uniform Flexural Modulus SLA3500 SL7510 Uniform Hardness SLA3500 DSM8120 Uniform FDM1650 P400 Uniform Impact Strength MJM2100 TJ75 Uniform 9 11 Density Heat Deflection Temp. Resistance Durability Functionality Detail Capability Accuracy Build Time Cost -24-

25 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Utility Functions for each Attribute ID qualitative preference characteristics ID quantitative preference characteristics Fit a utility function Check for consistency -25-

26 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Utility Functions for each Attribute ID qualitative preference characteristics ID quantitative preference characteristics Fit a utility function Check for consistency Attribute Tensile Strength Monotonicity Target Attitude Towards Risk Averse -26-

27 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Utility Functions for each Attribute ID qualitative preference characteristics ID quantitative preference characteristics Fit a utility function Check for consistency Left Hand Side Utility Right Hand Side Utility Keeney, R.L. and Raiffa, H. (19976). Decisions with Multiple Objectives: Preferences and Value Tradeoffs, New York: John Wiley and Sons -27-

28 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Utility Functions for each Attribute ID qualitative preference characteristics ID quantitative preference characteristics Fit a utility function Check for consistency Utility Normalized Functions for Non-Monotonic Attributes (i.e., Tensile Strength) Left Hand Side Ux ( ) exp( x) Right Hand Side Ux ( ) exp( x) Lower Unacceptable Ideal Upper Unacceptable Tensile Strength -28-

29 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Utility Functions for each Attribute ID qualitative preference characteristics ID quantitative preference characteristics Fit a utility function Check for consistency Option A Certainty Equivalent Option B Lottery p=0.2 TS 1 TS 0.6 p=0.8 TS 0.5 E u TS Eu 0.2 u TS 0.8 u TS

30 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Multi-Attribute Utility Function ID relevant independence assumptions and corresponding functional form of the multi-attribute utility function Assess scaling constants for the multi-attribute utility function Check multi-attribute utility function for consistency -30-

31 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Multi-Attribute Utility Function ID relevant independence assumptions and corresponding functional form of the multi-attribute utility function Assess 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. n U ku i i( Ai) i1-31-

32 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Multi-Attribute Utility Function ID relevant independence assumptions and corresponding functional form of the multi-attribute utility function Assess scaling constants for the multi-attribute utility function Check multi-attribute utility function for consistency Attribute k-values Tensile Strength Young's Modulus Flexural Strength Flexural Modulus Detail Capability Accuracy

33 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Multi-Attribute Utility Function Each alternative Alternatives Expected Utility Process Material SLA250 DSM SLA3500 SL SLA3500 DSM FDM1650 P400 0 MJM2100 TJ

34 Application Resource Selection for Product Validation Given Identify Assess Evaluate Rank Each Alternative Expected Utility Alternatives Expected Utility Process Material SLA250 DSM SLA3500 SL SLA3500 DSM FDM1650 P400 0 MJM2100 TJ75 0 Suggestion: Use DSM 7110 resin on the SLA 250 Stereo- lithography machine -34-

35 References Utility-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-based selection decision support problem, Concurrent Engineering Research and Applications, Vol. 13, No. 1, pp