1. GA Approaches to Multi-Objective Optimization
Scott Noble
Fred Iskander
18 March 2003

2. Multi-Objective Optimization Problems (MOPs)
Multiple, often competing objectives
In the case of a commensurable variable space, can often be reduced to a single objective function (or sequence thereof) and solved using standard methods
Some problems cannot be reduced and must be solved using pure MO techniques

3. Three General Approaches
Preemptive Optimization
sequential optimization of individual objectives (in order of priority)
Composite Objective Function
weighted sum of objectives
Purely Multi-Objective
Population-Based
Pareto-Based

4. Preemptive Optimization Steps
1. Prioritize objectives according to predefined criteria (problem-specific)
2. Optimize highest-priority objective function
3. Introduce new constraint based on optimum value just obtained
4. Repeat steps 2 & 3 for every other objective function, in succession

5. Composite Objective Functions
1. Assign weights to each function according to predefined criteria (problem-specific)
MAX and MIN objectives receive opposite signs
2. Sum weighted functions to create new composite function
3. Solve as a regular, single-objective optimization problem

6. Transformation Approaches
Advantages:
Easy to understand and formulate
Simple to solve (using standard techniques)
Disadvantages:
A priori prioritization/weighting can end up being arbitrary (due to insufficient understanding of problem): oversimplification
Not suited to certain types of MOPs

7. Pure MOPs: Population-Based Solutions
Allow for the investigation of tradeoffs between competing objectives
GAs are well suited to solving MOPs in their pure, native form
Such techniques are very often based on the concept of Pareto optimality

8. Pareto Optimality MOP  tradeoffs between competing objectives
Pareto approach  exploring the tradeoff surface, yielding a set of possible solutions
Also known as Edgeworth-Pareto optimality

9. Pareto Optimum: Definition
A candidate is Pareto optimal iff:
It is at least as good as all other candidates for all objectives, and
It is better than all other candidates for at least one objective.
We would say that this candidate dominates all other candidates.

10. Dominance: Definition
Given the vector of objective functions
we say that candidate dominates , (i.e ) if:
(assuming we are trying to minimize the objective functions).
(Coello Coello 2002)

11. Pareto Non-Dominance
With a Pareto set, we speak in terms of non-dominance.
There can be one dominant candidate at most. No accommodation for “ties.”
We can have one or more candidates if we define the set in terms of non-dominance.

12. Pareto Optimal Set
The Pareto optimal set P contains all candidates that are non-dominated. That is:
where F is the set of feasible candidate solutions
(Coello Coello 2002)

13. Examples
(Fonseca and Fleming 1993)

14. Examples Candidate f1 f2 f3 f4 1 (dominated by: 2,4,5) 5 6 3 10
3 (non-dominated) 2 11
4 (non-dominated)
5 (non-dominated) 9

15. Example: Pareto Ranking
(1)
(6)
(3)
(2)
(Fonseca and Fleming 1993)

16. Pareto Front
The Pareto Front is simply values of the optimality vector evaluated at all candidates in the Pareto Optimal Set

17. Pareto Front
(Tamaki et al. 1996)

18. Non-Pareto Selection VEGA (Parallel Selection) Tournament Selection
Vector Evaluated Genetic Algorithm
Next-generation sub-populations formed from separate objective functions
Tournament Selection
Pair wise comparison of individuals w.r.t. objective functions (pre-prioritized or random)
Random Objective Selection
Repetitive selection using a randomly selected objective function (predetermined probabilities)

19. Pareto-Based Selection
Pareto Ranking
Tournament Selection with Dominance
pair wise comparison against a comparison set based on dominance
Pareto Reservation (Elitism)
carry non-dominated candidates forward from previous generation
use additional selection method to regulate population size
Pareto-Optimal Selection

20. Diversity Lack of genetic diversity is an inherent issue with GAs
Fitness sharing encourages diversity by penalizing candidates from the same area of the solution or function space

21. Summary There are multiple approaches to MOPs.
GAs are well suited to exploring a multi-objective solution space.
They provide insight into the tradeoffs associated with MOPs, not necessarily a particular solution.

22. Further Reading
Coello Coello, C.A “Introduction to Evolutionary Multiobjective Optimization.”
Fonseca, C.M. and P.J. Fleming Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization. Genetic Algorithms: Proceedings of the Fifth International Conference. S. Forrest, ed. San Mateo, CA, July 1993.
Tamaki, H., H. Kita and S. Kobayashi Multi-Objective Optimization by Genetic Algorithms: A Review. Proceedings of the IEEE Conference on Evolutionary Computation, ICEC 1996, pp
Younes, A., H. Ghenniwa and S. Areibi An Adaptive Genetic Algorithm for Multi-Objective Flexible Manufacturing Systems. GECCO, New York, July 2002.