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Optimization in Engineering Mechanics 5 – Multiobjective Optimization. Definition. Multiple objective functions. The problem is then: In general, no solution vector X exists that minimizes all the k objective functions simultaneously. New concept: Pareto optimal solution.

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definition
Definition

Multiple objective functions. The problem is then:

In general, no solution vector X exists that minimizes all the k objective functions simultaneously. New concept: Pareto optimal solution.

Multiobjective optimization grew out from three areas: economic equilibrium and wellfare theories, game theory and mathematics.

pareto optimality
Pareto Optimality

All x from P to Q are Pareto Optimal

A feasible solution X* is called a Pareto Optimal , if there exist no other feasible solution Y* such that fi(Y*)≤ fi(X*) for i=1,2,..,k with fj(Y*)< fj(X*) for at least one j.

A feasible vector X* is called a Pareto optimal if no other feasible solution Y* that would reduce some objective function without causing a simultaneous increase in at least one other objective function.

articulation of preferences
Articulation of Preferences

The preference of the decision-maker is its opinion concerning points in the design space.

Some methods require an a priori definition of the preference to obtain the optimal solution.

Some find the Pareto set and then, using an a-posteriori definition of the preference, find the optimal.

Finally, some methods find an optimal solution with no articulation of the preferences.

methods with a priori articulation of preferences i
Methods with a priori articulation of preferences (I)

Utility function method: the importance of each objective function compared to others is defined. Then a total utility function is defined:

The optimum solution X* si found to maximize the global utility U, subject to the constraints.

The most typical form of the utility function is:

methods with a priori articulation of preferences ii
Methods with a priori articulation of preferences (II)

Global Criterion Method: a global criterion is defined, such as, the sum of the squares of the relative deviations of the individual objective functions from the feasible ideal solution:

Usually p=2

methods with a priori articulation of preferences iii
Methods with a priori articulation of preferences (III)

s.t.:

solution

s.t.:

solution

solution

Final solution

s.t.:

Lexicographic method: the objectives are ranked in order of importance by the designer. The optimum solution is then found by minimizing the objective functions starting with the most important and proceeding according to the order of importance of the objectives. If f1(X) and fk(X) denote the most and least important objective functions. The first problem is formulated as:

final remarks
Final remarks

MATLAB Goal Attainment Method: fgoalattain

Multiobjective optimization problems are typical from economics but also engineering: minimize cost and weight, cost and efficiency,…

Genetic Algorithms are specially well suited for multiobjective