Progressively Interactive Evolutionary Multi-Objective Optimization ...

Table 6. Distance of obtained solution from the most preferred solution, function evaluations, and
the number of DM calls required by PI-NSGA-II for the five-objective modified DTLZ2 problem.
ds = 0.01 ds = 0.1
Minimum Median Maximum Minimum Median Maximum
Accuracy 0.0112 0.0329 0.1210 0.0395 0.0884 0.2777
Func. Evals. 20272 29298 37776 5083 6872 9919
# of DM Calls 51 69 96 9 12 17
6 Parametric Study
Besides the usual parameters associated with an evolutionary algorithm, such as population
size, crossover and mutation probabilities and indices, tournament size etc., in
the proposed PI-NSGA-II we have introduced a few additional parameters which may
effect the accuracy and number of DM calls. They are the number of generations between
DM calls (τ), termination parameter (ds), maximum archive size (|A| max ), KKT
error limit for terminating SQP algorithm in single-objective optimization used for the
termination check, and the parameter ρ used in the ASF function optimization. Of these
parameters, the first two have shown to have an effect on the chosen performance measures
— accuracy, the number of overall function evaluations, and the number of DM
calls.
A parametric study for ds has not been done in this section as results for two different
values of ds have already been presented in the previous section. The results show
an expected behavior, that is, a strict ds provides higher accuracy and requires a larger
number of DM calls and function evaluations, a relaxed ds provides lower accuracy and
requires less number of DM calls and function evaluations.
Thus, in this section, we study the effect of the parameter τ, while keeping ds =
0.01 and the other PI-NSGA-II parameters identical to that mentioned in the previous
section. Here, we use the two objective ZDT1 and three objective DTLZ2 test problems.
6.1 Effect of Frequency of DM Calls (τ )
We study the effect of τ by considering four different values: 2, 5, 10 and 20 generations.
The parameter ds is kept fixed at 0.01. To investigate the dependence of the
performance of the procedure on the initial population, in each case, we run PI-NSGA-
II from 21 different initial random populations and plot the best, median and worst
performance measures.
We plot three different performance measures — accuracy, number of DM calls and
number of function evaluations obtained for the modified ZDT1 problem in Figure 9.
It is interesting to note that all three median performance measures are best for τ =
5. A small value of τ means that DM calls are to be made more frequently. Clearly,
this results in higher number of DM calls, as evident from the figure. Frequent DM
calls result in more single-objective optimization runs for termination check, thereby
increasing the number of overall function evaluations. On the other hand, a large value
of τ captures too little information from the DM to focus the search near the most
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