16. Multi-Objective Optimization - CAUisdl.cau.ac.kr/education.data/DOEOPT/16.Multiobjective... · 2018-11-05 · SCHOOL OF MECHANICAL ENG.-2-CHUNG-ANG UNIVERSITY Often, a first step

Hae-Jin ChoiSchool of Mechanical Engineering,

Chung-Ang University

16. Multi-Objective Optimization

SCHOOL OF MECHANICAL ENG.

CHUNG-ANG UNIVERSITY-1-

Given

Problem definition

Find

x

Subject to

gi(x) ≤ 0; i=1 to n

Minimize

f1(x), f2(x), f3(x),…, fm(x)

DOE and Optimization

Multi-Objective Optimization

Issues to consider….

Formulation of multi-

objective problem

Prioritization of multiple

objectives

Management of Conflicting

objectives



Often, a first step in solving these models is a model transformation into a model

that CAN be solved using an existing algorithm/solver.

There are different ways to formulate a multi-objective optimization model

Some covered are:

Goal Programming (GP) method

Utility function method

Others exist

Different formulations




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




Another multiobjective mathematical “programming” technique is GoalProgramming (GP)

The term "goal programming" is used by its developers to indicate the searchfor an "optimal" program (i.e., a set of policies to be implemented) for amathematical 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 to maximizeor 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 with thegoal. For example,

A(X) T

Difference between Objectives and Goals




In Goal Programming, “deviation” variables are used to convert inequalities toequalities.

The deviation variable d is (then) defined as:

Deviation Variables - “Distance to target”

d = T i - A i(X)

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

• From a reality point of view, a deviation variable represents the distance(deviation) between the aspiration level (target) and the actual attainment of thegoal.




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, . . . , m

subject 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 variables will always be zero.

• The goal formulation (now) becomes:




Minimizing Deviations

To achieve a goal

To achieve Ai(x) = Ti, we must minimize (di- + di

+ ).


Given

Problem definition

Find

x

Subject to

gi(x) ≤ 0; i=1 to n

Minimize

f1(x), f2(x), f3(x),…, fm(x)

Given

Problem definition

Find

x

Subject to

gi(x) ≤ 0; i=1 to n

Ai(x) + di- - di

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

di- • di

+ = 0 and di-, di

+ 0

Minimize

z=∑ (di- + di

+ )



Utility is a measure of satisfaction, referring to the total satisfaction received

by a consumer from consuming a good or service

Utility of twenty sandwiches is not twenty times the utility of one sandwich, by

the law of diminishing returns.


Utility Function Method

Utility

Attribute (objective value)0

1



A function Ui(fi) is defined for each objective depending on the importance of fi

compared to the other objective functions. Then a total or overall utility U is

defined as

The solution is then found by maximizing the overall utility U (objective function)


Additive Von Neumann–Morgenstern Utility

Utility

f1(Engine power)0

1

Utility

f2(Fuel consumption)0

1

1

( )k

i i

i

U U f



Maximizing Overall Utility

To achieve a goal, we must maximize overall utility


Given

Problem definition

Find

x

Subject to

gi(x) ≤ 0; i=1 to n

Minimize

f1(x), f2(x), f3(x),…, fm(x)

Given

Problem definition

Find

x

Subject to

gi(x) ≤ 0; i=1 to n

Maximize

U=∑ Ui (fi) ; i=1 to m



Goals are not equally important to a decision maker.

How do we represent our preferences?

Two approaches are:

1) Assign weights and calculate the sum of the deviation variables (‘distance to target’) multipliedby their individual weights.

2) Rank order goal deviations in priority levels, often referred to as a preemptive formulation.The preemptive formulation does not exclude the assignment of weights.

Note: Other techniques exist, but right now we focus on the above two.

Prioritization of Multiple Goals




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.

In goal programming formulation, the objective function is Minimize 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. Can you name some?

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 including goal formulation

Lexicographic Minimum (Rank Ordering)




Lexicographic Minimum - Definition

LEXICOGRAPHIC MINIMUM Given an ordered array f(i) = (f1, f2, ... , fn)

of nonnegative elements fk’s, the solution given by f(1) is preferred to f(2) iff

fk(1) < fk

(2)

and all higher order elements (f1, …, fk-1) are equal. If no other solution is

preferred to f(1), then f(1) is the lexicographic minimum.

Examples?




“The typical role of a design engineer is to resolve conflicting objectives and 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)


Conflicting Objectives

Note: We will use the terms “objective”, “goal”, and “criterion” interchangeably.



So far, we have employed different techniques to achieve multi-objective optimization:

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

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


Review: Methods for Trading Off Across Objectives



In criterion (objective) space, we can identify a special “trade-off curve” on the boundary where: 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.

Changing the weights in an Archimedean (weighted) objective

function traces out the curve’s path.

This part of the boundary is called the Pareto

Curve (or Pareto Frontier) 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.


Pareto Optimality Curve (Pareto Frontier)

f2 f2

f2 f2

f1 f1

f1 f1

Pareto

Maximizing both f1 and f2


CHUNG-ANG UNIVERSITY-19-DOE and Optimization

Example of Pareto Frontier

EMI feasible

DET feasible

Pareto frontier

Design of biosensor

Two conflicting objectives

1. Mass sensitivity:

Degree of sensitiveness to small bio-

molecules (Larger the better)

2. Motional resistance:

Degree of signal quality from the sensor

(Smaller the better)



We have dealt with two different subjects – DOE and Optimization

Those two topics are, however, closely related.

Optimization formulation of a system requires models or functions in the

objective function and constraint conditions.

DOE may provide the mathematical models or functions of the system

optimization

Therefore, we may conclude that

DOE is for characterization of the system

Optimization is for decision-making in designing the system

RSM in DOE includes some of optimization techniques


Closure



Some advanced topics in DOE

Blocking and Confounding

Metamodeling techniques other than response surface model

Kriging,

Artificial Neural Network,

Partial Least Square (Projection to Latent Structure),

etc.

Uncertainty and reliability issues in optimization

Bayesian Approach

Fuzzy Set Theory

Utility Theory


Advanced Topics in DOE and Optimization

16. Multi-Objective Optimization - CAUisdl.cau.ac.kr/education.data/DOEOPT/16.Multiobjective... · 2018-11-05 · SCHOOL OF MECHANICAL ENG.-2-CHUNG-ANG UNIVERSITY Often, a first step

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