1. Performance Evaluation of Multiobjective Evolutionary Algorithms METRICS

2. MOEA Testing and Analysis
Specifically, they suggest that a well-designed experiment follows the following steps:
1. Define experimental goals;
2. Choose measures of performance - metrics;
3. Design and execute the experiment;
4. Analyze data and draw conclusions;
5. Report experimental results.

3. Choose measures of performance - metrics;
Every algorithm can maintain a group of nondominated individuals at the end of the run.
Sometimes the result from one algorithm fully dominates the other, which is the simplest condition. But generally, some results from one algorithm dominate some from another algorithm, and vice versa.
Another reason for the special consideration on the performance evaluation is that we are interested in not only the convergence to PF* but also the distribution of the individuals along PF*.
Adequately evaluating convergence and distribution is still an open problem in the field of MOEAs.
Benchmark problem design is also an interesting field because we want to conveniently generate problems with different shapes of PF*, different convergent difficulties, different dimensions, etc.

4. Performance Indices
After determining which benchmark MOPs to optimize, we need to make a careful decision on how to evaluate the performance of different MOEAs.
Those criteria are performance indices (PI).

5. Three are normally the issues to take into consideration to design a good metric in the given domain (Zitzler, 2000):
1. Minimize the distance of the Pareto front produced by our algorithm with respect to the true Pareto front (assuming we know its location).
2. Maximize the spread of solutions found, so that we can have a distribution of vectors as smooth and uniform as possible.
3. Maximize the amount of elements of the Pareto optimal set found.

7. The Need for Quality MeasuresB A B
Is A better than B?
independent of user preferences
Yes (strictly) No
dependent on user preferences
How much? In what aspects?
Ideal: quality measures allow to make both type of statements

8. Independent of User Preferences
Pareto set approximation (algorithm outcome) = set of incomparable solutions
weakly dominates = not worse in all objectives sets not equal
dominates = better in at least one objective
strictly dominates = better in all objectives
is incomparable to = neither set weakly better A B C D B A A C D C B C

9. Dependent on User Preferences
Goal: Quality measures compare two Pareto set approximations A and B.
application of quality measures
hypervolume
distance
diversity
spread
cardinality
6 5
A B A B
comparison and interpretation of quality values
“A better”

10. 1-Cardinality-based Performance Indices
The research produced in the last few years has included a wide variety of metrics that assess the performance of an MOEA in one of the three aspects. Some examples are the following:
1-Cardinality-based Performance Indices
Number of obtained solutions
Error Ratio
Coverage

11. The number of obtained solutions |Sj| (Cardinality)
K. Deb: Multi-Objective Optimization Using Evolutionary Algorithms, Wiley, Chichester, U.K., 2001.
Let S be the union of the J solution sets.
Let Sj be a solution set (j=1, 2, 3, …, J). For comparing J solution sets (S1, S2, …, SJ), we use the number of obtained solution |Sj| f1
: reference solution r
: current solution x
|Sj| f2

12. Error Ratio (ER):
This metric was proposed by Van Veldhuizen to indicate the percentage of solutions (from the non-dominated vectors found so far) that are not members of the true Pareto-optimal set:
where n is the number of solutions in the current set of non-dominated
set; ei = 0 if solution i is a member of the Pareto-optimal
set, and ei = 1 otherwise. It should then be clear that ER = 0 indicates
an ideal behavior, since it would mean that all the solutions generated by
our MOEA belong to the Pareto-optimal set of the problem. This metric
addresses the third issue from the list previously provided.

13. ER=2/3

14. In 1999, Zitzler suggested a binary PI called coverage (C)
C(S1, S2) is the percent of the individuals in S2 who are weakly dominated by S1.
The larger C(S1, S2) is, the better S1 outperforms S2 in C.

15. 2-Distance-based Performance Indices
Distance-based PI evaluate the performance of the solutions according to the distance to PF*.
Generational Distance (GD):
In 1999, Van Veldhuizen suggested a unary PI called generational distance (GD).
First we need to define the minimum distance from S to PF* as

16. The GD measure can be written as follows:
where S* is a reference solution set for evaluation the solution set Sj.
dxr is the distance between a current solution x and a reference solution r.
The GD measure is not the average distance from each solution in Sj to its nearest reference solution in S*.
It is referred to as the generation distance.
While the generation distance can only evaluate the convergency of the solution set Sj to S*, GD(Sj) can evaluate the distribution of Sj as well as the proximity of Sj to S*. f1
: reference solution r
: current solution x
dr-1, x-1
drx f2

17. It should be clear that a value of GD = 0 indicates that all the elements generated are in the Pareto-optimal set. Therefore, any other value will indicate how “far” we are from the global Pareto front of our problem.
This metric addresses the first issue from the list previously provided.

18. Metrics for Diversity
Spacing (SP): Here, one desires to measure the spread (distribution) of vectors throughout the non-dominated vectors found so far. Since the “beginning” and “end” of the current Pareto front found are known, a suitably defined metric judges how well the solutions in this front are distributed. Schott proposed such a metric measuring the range (distance) variance of neighboring vectors in the non-dominated vectors
found so far. This metric is defined as:

20. Hyperarea and Ratio (HA,HR): The hyperarea (hypervolume) and hyperarea ratio metric which are Pareto compliant relate to the area of coverage of PFknown with respect to the objective space for a two objective MOP.
This equates to the summation of all the rectangular areas, bounded by some reference point and (f1(x), f2(x)).

21. Mathematically, this is described in equation
Also proposed is a hyperarea ratio metric defined as:
where HA1 is the PFknown hyperarea and HA2 is the hyperarea of PFtrue.

22. Hypervolume in 2D Objective 2 A B C D E 5 4 3 2 1 Objective 1 1 2 3 4
Hypervolume {A, B, C, D, E} = 11 5 4 3 2
Reference point 1
Objective 1 1 2 3 4 5

23. PFtrue ’s H = 16+6+4+3 = 29 units2, and PFknown ’s H = 20+6+7.5 = 33.5
HR = 33.5/29 = 1.155