1 12-MultiobjectiveEAs

2 Multiobjective Evolutionary Algorithms

3 Multi-Objective Problems (MOPs)

4 An example: Buying a car


5 Two spaces

Decision (variable) space

Objective space


6 Comparing solutions

Optimization task:Minimize both f1 and f2

Then:a is better than ba is better than ca is worse than ea and d are incomparable

Objective space


7 Dominance relation


solutions dominated by x

solutions dominating x

8 Pareto optimality

Solution x is non-dominated among a set of solutions Q if no solution from Q dominates x

A set of non-dominated solutions from the entire feasible solution space is the Pareto-optimal set , its members Pareto-optimal solutions

Pareto-optimal front : an image of the Pareto-optimal set in the objective space

Illustration of the concepts

Illustration of the concepts

A practical example: The beam design problem

Minimize weight and deflection of a beam (Deb, 2001) :


Formal definition


m inimize

subject to


(beam weight)

(beam deflection)

(maximum stress)

Feasible solutions

Decision (variable) space


Goal: Finding non-dominated solutions


9 Goal of multiobjective optimizers


10 Single- vs. multiobjective optimization

Characteristic Singleobjective optimisation Multiobjective optimisation
Number of objectives one more than one
Spaces single two: decision (variable) space, objective space
Comparison of candidate solutions x is better than y x dominates y
Result one (or several equally good) solution(s) Pareto-optimal set
Algorithm goals convergence convergence, diversity

11 Two approaches to multiobjective optimization

Preference-based: traditional, using single objective optimization methods

Ideal: possible with novel multiobjective optimization techniques,enabling better insight into the problem

12 Preference-based approach

Given a multiobjective optimization problem,

use higher-level information on importance of objectives

to transform the problem into a singleobjective one,

and then solve it with a single objective optimization method

to obtain a particular trade-off solution.

13 An example approach: Weighted-sum

Modified problem:

H yperplanes in the objective space !


14 Ideal approach

Given a multiobjective optimization problem,

solve it with a multiobjective optimization method

to find multiple trade-off solutions,

and then use higher-level information

to obtain a particular trade-off solution.

15 Multiobjective optimization with evolutionary algorithms

Population-based method

Can return a set of trade-off solutions (approximation set) in a single run

Allows for the ideal approach to multiobjective optimization

16 EC approach: Advantages

Population-based nature of search means you can simultaneously search for set of points approximating Pareto front

Don’t have to make guesses about which combinations of weights might be useful

Makes no assumptions about shape of Pareto front - can be convex / discontinuous etc.

17 EC approach: Requirements

18 EC approach: Fitness Assignment

19 EC approach: Diversity maintenance

20 EC approach: Remembering Good Points

21 MOP - summary

MO problems occur very frequently

EAs are very good in solving MO problems

MOEAs are one of the most successful EC subareas