16. Multi-Objective Optimization - CAUisdl.cau.ac.kr/education.data/DOEOPT/16.Multiobjective... ·...
Transcript of 16. Multi-Objective Optimization - CAUisdl.cau.ac.kr/education.data/DOEOPT/16.Multiobjective... ·...
Hae-Jin ChoiSchool of Mechanical Engineering,
Chung-Ang University
16. Multi-Objective Optimization
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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
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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
DOE and Optimization
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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
DOE and Optimization
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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)
DOE and Optimization
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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
DOE and Optimization
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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.
DOE and Optimization
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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:
DOE and Optimization
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Minimizing Deviations
To achieve a goal
To achieve Ai(x) = Ti, we must minimize (di- + di
+ ).
DOE and Optimization
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
+ )
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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.
DOE and Optimization
Utility Function Method
Utility
Attribute (objective value)0
1
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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)
DOE and Optimization
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
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Maximizing Overall Utility
To achieve a goal, we must maximize overall utility
DOE and Optimization
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
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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
DOE and Optimization
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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
DOE and Optimization
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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)
DOE and Optimization
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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?
DOE and Optimization
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“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)
DOE and Optimization
Conflicting Objectives
Note: We will use the terms “objective”, “goal”, and “criterion” interchangeably.
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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.
DOE and Optimization
Review: Methods for Trading Off Across Objectives
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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.
DOE and Optimization
Pareto Optimality Curve (Pareto Frontier)
f2 f2
f2 f2
f1 f1
f1 f1
Pareto
Maximizing both f1 and f2
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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)
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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
DOE and Optimization
Closure
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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
DOE and Optimization
Advanced Topics in DOE and Optimization