9. Decision-making in Complex Engineering...

School of Mechanical EngineeringAssociate Professor

Choi, Hae-Jin

9. Decision-making in Complex Engineering Design

SCHOOL OF MECHANICAL ENG.CHUNG-ANG UNIVERSITY-2-

Week 1: Decision Theory in Engineering Needs for decision-making in engineering Multiobjective optimization Goal Programming Compromise Decision Support Problem (cDSP)

Week 2: Decision-making under Uncertainty (I) Utility theory Utility based selection Decision Support Problem (usDSP)]

Week 3: Decision-making under Uncertainty (II) Robust design principles Robust Design Type I : Taguchi method Robust Design Type II: Robust Concept Exploration Method (RCEM) Robust Design Type III: RCEM - Error Margin Indices

Overview of Lectures


As engineers, we frequently think of ourselves as PROBLEM SOLVER. Being taught problem-solving skills as the major element of our

education, throughout our lives. (this is important)

The problem solving is NOT the principle activity of engineering; rather DECISION-MAKING

What is ENGINEERING?


Science (physical science) is the process of rationally and methodically seeking to understand nature, with the principal objective of developing a predictive or problem-solving capability.

Engineering involves the manipulation of nature to create systems from the benefit of at least some segment of mankind.

Science vs Engineering


Science vs Engineering

Load

Maximum StressHeightHeight

Given Height Maximum Stress??? Science

EngineeringHeight ??? Given MaximumAllowable Stress


Man cannot change nature; it is allowed only that man can manipulate nature

Something physical is created through engineering process

The process of creating something physical requires effective allocation of nature’s resources

Notion of Engineering Definition

Decision-making


A model is a necessary ingredient to make a decision

Why do we need “models”?

What are “models”?

System Modeling Why? What?

Load

HeightDeflection=F(Load, Height)

F: system model


The ability to make good engineering decisions lies in having large amount of good information.

Inexpensive way of trying many different decisions is essential.

Without models, we would have to build actual systems in order to gain decision information -> only few tests at most

With models, such as computer models, we might be able to test hundreds or even thousands of decision options.

Why do we need models?


A model is an abstraction of reality. (Hazelrigg)

A model is a simplified representation of something real.

Three basic classes of models:

Iconic models

Analog models

Symbolic models

What are models?


Scale representations, either large or small, of physical or real things

Examples: Wind tunnel models of airplanes, rocket, building and bridges Pilot plants; model trains and automobiles

Use when equations that adequately describe the behavior of the systems are not available.

Iconic Models


Models that use one property to represent another.

Examples Colors on a relief map Electrical circuit as an analog to heat transfer Voltage –Temperature gradient

Current – Heat flux

Resistance –Thermal resistivity

Analog Models


Are also called ‘Mathematical Models’ Use symbols to designate properties of the real system Are more abstract than other models covering a vast range Examples F=m*a; E=m*c2

Are often transformed to computer simulation Are used to examine system alternatives in a very

inexpensive way. -> big benefit for engineering (decision-making)

Symbolic Models


Compromise Decision-Making

portability

performance

heat vibration /noise

ergonomics

costsize

battery life

?

Multi-Objective Optimization Compromise Decision Support Problem


Selection Decision-Making

portability

performance

heat vibration /noise

ergonomicscost

size battery life

Multi-criteria Concept Evaluation

Selected Concept

Utility Theory Utility-based Selection Decision Support Problem


Optimization is derived from the Latin word“optimus”, the best.

Optimization characterizes the activities involvedto find “the best”.

People have been “optimizing” forever, but theroots for modern day optimization can be traced tothe SecondWorldWar.

Optimization


Note: There are MANY different optimization methods/algorithms

However, they are can be grouped by fundamental principles of: Model formulation or solution method/algorithm

Types of Optimization

Do not forget:

Optimization methods fall in the category of “decision support systems/methods”

Question: What are some other means for decision support?


Different types of optimization model formulations exist: Classical non-linear formulation Linear Programming formulation Baseline model formulation Goal Programming formulation Compromise Decision Support Problem formulation etc.

Basic classifications are: Constrained versus unconstrained Linear versus non-linear Single objective versus multi-objective

Another classification can be made by variables: continuous/discrete/mixed-integer

Problem Formulations


Decision-making is rather multi-objective by nature, so we will lookat some multi-objective

Some covered are: Baseline model Goal Programming (GP) model Compromise Decision Support Problem model Others exist

Single versus Multi-Objective


Multiobjective mathematical “programming” technique is Goal Programming(GP)

The term "goal programming" is used by its developers to indicate thesearch for an "optimal" program (i.e., a set of policies to be implemented)for a mathematical model that is composed solely of goals.

Developers argue that any mathematical programming model may find anequivalent representation in GP.

“GP provides an alternative representation that often is more effective incapturing the nature of real world problems.”

Goal Programming (GP)


In Goal Programming a distinction is made between an objective and a goal:

Objective: In mathematical programming, an objective is a function that weseek to optimize, via changes in the problem variables.

The most common forms of objectives are those in which we seek tomaximize or minimize. For example,

Minimize Z = A(X)

Goal: In short, a goal is an objective with a “right hand side”.

This right hand side (T) is the target value or aspiration level associated withthe goal. For example,

A(X) T

Difference between Objectives and Goals


Solving multi-objective models is NOT standard practice(yet).

Often, a first step in solving these models is a modeltransformation into a model that CAN be solved using anexisting algorithm/solver.

Solving Multi-objective Models


Step 1 : Transform all objectives into goals by establishing associated aspirationlevels based on the belief that a real world decision maker can usually cite (initial)estimates of his or her aspiration levels. Hence,

where T rand Ts are the respective aspiration levels (targets).

Step 2 : Rank-order each goal according to its perceived importance. Hence, the setof hard goals (i.e., constraints in traditional math programming) is always assignedthe top priority or rank.

Step 3 : All the goals must be converted into equations through the addition ofdeviation variables

Transforming into a GP model

maximize Ar(X) becomes Ar(X) Tr for all r

minimize As(X) becomes As(X) Ts for all s.


In Goal Programming and other approaches (like compromise Decision SupportProblem) “deviation” variables are used to convert inequalities to equalities.

The deviation variable d is (then) defined as:

Deviation Variables - “Distance to target”

d = Ti - Ai(X)

• Note: Mathematically, the deviation variable d can be negative,positive, or zero.

• From a reality point of view, a deviation variable represents thedistance (deviation) between the aspiration level (target) and theactual attainment of the goal.


The deviation variable d can be replaced by two variables:

Two Deviation Variables instead of One

d = di- - di

+ where di- • di

+ = 0 and di-, di

+ 0

Ai(X) + di- - di

+ = Ti; i = 1,2, . . . , msubject to di

- • di+ = 0 and di

-, di+ 0

• Why? Many optimization algorithms do not “like” negative numbers and the preceding ensures that the deviation variables never take on negative values.

• The product constraint ensures that one of the deviation variableswill always be zero.

• The goal formulation (now) becomes:


Values of Deviation Variables

Note that a goal is always expressed as an equality:

Ai(X) + di- - di

+ = Ti; i = 1,2, . . . , m

And when considering this equality, the following will be true:

if Ai(X) < Ti is true, then (di- > 0 AND di

+ = 0) must be true;

if Ai(X) > Ti is true, then (di- = 0 AND di

+ > 0) must be true;

if Ai(X) = Ti is true, then (di- = 0 AND di

+ = 0) must be true.

When in doubt, just use a numerical example.


Again, note that a goal is always expressed as an equality.

“Desired” Values of Deviation Variables

To achieve a goal (i.e., reach the target), 3 situations are possible:

1. To satisfy Ai(X) Ti, we must ensure that the deviation variable di+ is zero.

- The deviation variable di- is a measure of how far the performance of the actual design is from the

goal.

2. To satisfy Ai(X) Ti, the deviation variable di- must be made equal to zero.

- In this case, the degree of “overachievement” is indicated by the positive deviation variable di+.

3. To satisfy Ai(X) = Ti, both deviation variables, di- and di

+ must be zero.

Question: How would this change if we only had a single di that can be positive or negative?

Ai(X) + di- - di

+ = Ti; i = 1,2, . . . , m

Thus, to achieve a target, we must minimize the unwanted deviation(s)!


Minimizing deviationsConsider the preceding three situations again.

To achieve a goal (i.e., reach the target), 3 situations are possible:

1. To achieve Ai(X) Ti, we must minimize ( di+ )

2. To achieve Ai(X) Ti, we must minimize ( di- )

3. To achieve Ai(X) = Ti, we must minimize (di- + di

+ ).

(How would this change if we only had a single di that can be positive or negative?_

Big Question: What if you have more than one goal?That is, how do you minimize multiple deviation variables?


Goals are not equally important to a decision maker. How do we represent our preferences?

Two approaches are: Assign weights and calculate the sum of the deviation variables

(‘distance to target’) multiplied by their individual weights.

Rank order goal deviations in priority levels, often referred to as apreemptive formulation. The preemptive formulation does not excludethe assignment of weights.

Two Approaches to Prioritizing Objectives


Assigning weights, or weighted sum approach, is one of the most common ways of converting multi-objective/multi-goal problems into a single objective problem.

Min z = (w1d1- + w2d2

+ + ….) = (widi- + wkdk

+ )

The weights (w) can be any value, in principle. In case the sum of the weights equals 1, then we speak of an

archimedean formulation.

However, assigning weights without thought can cause problems.

Weighted Sum Approach


In Rank Ordering, you prioritize one goal/objective above each other without giving explicit mathematical weights. Basically, in words, Goal A has to be achieved before Goal B. I should not even

think about Goal B yet if Goal A has not been achieved yet.

One mathematical construct that is used in rank ordered formulations is the Lexicographic Minimum.

The concept of a lexicographic minimum is used in several multi-objective formulations Goal Programming Compromise DSP

Rank Ordering


31

Compromise Decision Support ProblemMulti-Objective Decision Support:

Compromise DSP…Traditional Single-Objective

Optimization…

Given n, number of decision variablesp, number of equality constraintsq, number of inequality constraintsf(x), an objective functiongi(x), constraint functions

Find xSubject to

g(x)=0 i=1,...,pg(x)<0 i=p+1,...,p+q

Optimizef(x)

Given n, number of decision variablesp, number of equality constraintsq, number of inequality constraintsm, number of system goalsgi(x), constraint functions

Find x (system variables)di

- ,di+ (deviation variables)

Satisfy System constraints:

g(x)=0 i=1,...,pg(x)<0 i=p+1,...,p+q

System goals:Ai(x)/Gi + di

- - di+ = 1

Bounds:Xi

min < Xi < Ximax

di- ,di

+ > 0 and di- . di

+ = 0 Minimize

Z = [f1(di-,di

+), …, fk (di-,di

+)] preemptiveZ = Wi (di

- + di+) Archimedean

Constraints from Math. Programming

Goals and Deviation Variables from Goal Programming


It is important to note that differences in formulation CAN cause differences in results.

The most influential factors are the choices of: Objectives versus goals Goal Priorities Constraints versus goals (constraints are higher priority) Goal targets

The Effect of Selecting a Formulation


“The typical role of a design engineer is to resolve conflicting objectivesand arrive at a design that represents an acceptable or desired balance of all objectives.” (Mattson & Messac 2002)

Classical examples of conflicting objectives: Truss Design: Weight versus Strength

Flywheel design: Kinetic Energy stored versus Weight

Finite Element Meshes: Aspect Ratio versus Distortion Parameter

Standard problem definition (Textbook’s notation):

Minimize f = [ f1(x), f2(x), … , fm(x) ],

where each fi is an objective function

Subject to x Ω (constraints on space of design variables)

Pareto Optimality


1. Weighting of objectives (Archimedean)

minimize f = w1f1(x) + w2f2(x)+ … ; subject to x Ω; where wi > 0 and Σ wi = 1.

2. Lexicographic minimum: preemptive ranking of objectives

3. A slight twist: Picking one objective as primary, transforming remaining objectives into constraints

minimize f1(x);

subject to f2(x) c2, f3(x) c3, … and fm(x) cm where ci is a limit

x Ω

These all provide point solutions (x*) based on an assignment of preferences among objectives.

Methods for Trading Off Across Objectives


Thus far in class, preferences, weights, & limits were all chosen by “engineering judgment” — trial and error, experience, etc. Varying weights & preferences to explore goal tradeoffs is manually intensive.

How can we visualize a global picture of the tradeoffs in optimum solutions over a wide range of weights?

Answer: Transform graphical solutions from “design (variable) space” to “criterion space” (also called “objective space”).

The Need Globally Viewing Tradeoffs in Optimality

Ωx1

x2 f1

f2

design space

f1

f2

Ω'

criterionspace


This part of the boundary is called the Pareto Curve (or Pareto Frontier) Or, the “functionally efficient” solution set There are Pareto curves in both the design

variable space and the criterion space. Pareto curves contain Pareto points (solutions) Bold lines in the pictures (right) represent

Pareto curves when maximizing objectives.

The Pareto Optimality Curve

f2 f2

f2 f2

f1 f1

f1 f1

Pareto

Maximization Problem

In criterion space, we can identify a special “trade-off curve” on the boundary where: Changing the weights in an Archimedean

(weighted) objective function traces out the curve’s path.

No point is “better” than any other point on the line with respect to both objectives.

No improvements can be made in any objective without trading off (worsening) the other.


Experimental evidence suggests that at high speeds the stressesare high near the hub of the rotating disk. For this reason, to getthe stresses within safe limits, it is advisable to have more massnear the hub.

The design criterion is to locate the points, P2 to P4 such that thekinetic energy is maximized and the mass of the rotating disk isminimized.

Rotating Disk (Flywheel) Example

w(r)

r

w1(r)w2(r)

w3(r)w4(r)

P1 = 0.10 P5 = 1.0

U

(L

+ U

)/2.0

L

P2 P3

P4

Z


“Baseline” Model for Flywheel

are the radial stress, tangential stress and yield stress

GivenThe relevant information for the disk:

Angular velocity of the disk = user input (rad/sec)Lower limit of thickness L = 0.01 (m)Upper limit of thickness U = 0.10 (m)Location of the hub P1 = 0.05 (m)Location of the rime P5 = 0.5 (m)Slope of the linear portion ’ = 0.9Density of the material of disk =7830 (kg/m3)Yield stress of the material of disk YS =1.48E9 (N/m2)

Relevant equations for the physics of the problem.Find

System variablesThey determine the profile of the rotating disk,

P2, P3, and P4.Satisfy

System constraintsThe stress constraints,

R(r) y,T(r) y,where R , T , and y respectively.

The constraints on the geometry of the rotating disk,P1 P2,P2 P3,P3 P4,P4 P5.

MaximizeThe kinetic energy (K) of the rotating disk is to be maximized.

MinimizeThe weight (M) of the rotating disk is to be minimized.


The traditional single-objective model is exercised in three ways,

one with kinetic energy as objective function and mass of the disk as constraint,

the other with mass of the disk as objective function and kinetic energy as constraint, and

as a weighted sum of the two objectives.

The compromise DSP template is exercised in three ways,

The deviation function is modeled in the preemptive form with the achievementof the kinetic energy goal as first priority.

the deviation function is modeled in the preemptive form with the achievement of the weightgoal as first priority.

the deviation function is formulated for the Archimedean form giving a weight of 0.6 for theachievement of the aspiration level of the kinetic energy of the disk and a weight of 0.4 for theweight of the disk.

Different Design Scenarios


Different single objective functions

MinimizeThe kinetic energy of the disk is to be maximized and its weight (M) is to be minimized

f ( P2, P3, P4) = 0.6(-K )+ 0.4M,where f is the objective function.

Weighted sum approach:

MinimizeThe kinetic energy (K) of the rotating disk is to be maximized,

where f is the objective function.-K(P2, P3, P4) = -K

MinimizeThe weight (M) of the rotating disk is to be minimized,

where f is the objective function.W(P2, P3, P4) = M


Satisfy (continued from the previous baseline model)System GoalsK(P2, P3, P4) /Gkinetic_energy + d1

- - d1+ = 1

W(P2, P3, P4) /Gweight + d2- - d2

+ = 1di

- , di+ > 0 and di

-·di+ = 0 where i=1,2

MinimizeZ = [g1(d1

-), g2( d2+)] Preemptive

Z = W1·d1- + W2·d2

+ Archimedean

Compromise Decision Support Problem


By way of illustrating the "power" of a preemptive formulation, acomparison of the results obtained is made for Scenario I:

First priority: maximize the kinetic energy of the disk

Second priority: minimize the mass.

• The aspiration levels for these objectives are set at 1000 MJ and 800 kgsrespectively

Differences in Results

500

600

700

800

900

1000

1100

1200

1300

KIN

ETIC

EN

ERG

Y (M

J)

500 1000 1500 2000 2500ROTATION SPEED (rad/s)

Compromise DSPSingle-objective approach

ASPIRED KINETIC ENERGY = 1000 MJASPIRED WEIGHT = 800 KGS

Note that compromiseDSP solution "sticks"to 1000 MJ whiletrying to minimizeweight.

Question: At what speeddo you expect aminimum weight?


ReferencesDecision-based Design Theory:Kemper E. Lewis, W. C., and Linda C. Schmidt, 2006, Decision Making in

Engineering Design, ASME, New York, NY.Hazelrigg, G. A., 1996, Systems Engineering: An Approach to Information-based

Design, Prentice-Hall, Upper Saddle River, NJ.

The Decision Support Problems:Mistree, F., Hughes, O. F. and Bras, B. A., "The Compromise Decision Support

Problem and the Adaptive Linear Programming Algorithm" in StructuralOptimization: Status and Promise, pages 247-286, (M. P. Kamat, Ed.),Washington, D.C.: AIAA, (1993).

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