Progressively Interactive Evolutionary Multi-Objective Optimization ...

f 2
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
P4
P2
P1
1 2 3 4 5 6 7
f
1
Fig. 3. Cobb-Douglas value function for P 1 ≻ P 2 ≻ P 3 ≻ P 4 ≻ P 5.
The equation for the value function is V (f1, f2) = f 0.27
1
f 2
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
P4
P2
P1
P3
P5
f 0.73
2
1 2 3 4 5 6 7
f
1
Fig. 4. Cobb-Douglas value function for P 1 ≻ P 2 ≻ P 3 ≻ P 4 ≡ P 5.
The equation for the value function is V (f1, f2) = f 0.29
1
P3
P5
f 0.71
2
The determined value function is used to guide the search
of the EMO towards the region of interest. A local search
based termination criteria was also proposed in the study.
The termination condition is set-up based on the expected
progress which can be made with respect to the constructed
value function.
In this study we replace the previously used polynomial
value function with a generalized polynomial value function.
To begin with, a polynomial value function with p = 1 is
optimized. In case the optimization process is unsuccessful
with a negative ǫ, the value of p is incremented by 1. This
is done until a value function is found which is able to fit
the preference information. This ensured that any information
received by the decision maker always gets fitted with a value
function. This makes the algorithm more efficient eliminating
cases where a value function cannot be fitted to the DM
preferences.
In the previous study two, three and five objective unconstrained
test problems were successfully solved using the
algorithm. The algorithm was able to produce solutions close
f 2
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
P4
P2
P1
1 2 3 4 5 6 7
f
1
Fig. 5. Polynomial value function for P 1 ≻ P 2 ≻ P 3 ≻ P 4 ≻ P 5. The
equation for the value function is V (f1, f2) = f2(f1 + 0.54f2 − 0.54)
f 2
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
P4
P2
P1
1 2 3 4 5 6 7
f
1
Fig. 6. Polynomial value function for P 1 ≻ P 2 ≻ P 3 ≻ P 4 ≡ P 5. The
equation for the value function is V (f1, f2) = f2(f1 + 0.49f2 − 0.49)
to the most preferred point with a very high accuracy. A
decision maker (DM) was replaced with a DM emulated
value function which was used to provide preference information
during the progress of the algorithm. Though the
possibility of the decision maker’s indifference towards a pair
of solutions was discussed but the algorithm was evaluated
only with a DM emulated value function which provides
perfect ordering of the points. In the following part of this
paper we evaluate the efficacy of the algorithm on three
and five objective test problems with constraints. We also
evaluate, how does the efficiency of the algorithm change in
case the decision maker is unable to provide perfectly ordered
set of points i.e. he/she finds some of the pairs incomparable.
IV. RESULTS
In this section, we present the results of the PI-NSGA-
II-VF procedure on three, and five objective test problems.
DTLZ8 and DTLZ9 test problems are adapted to create
maximization problems. In all simulations, we have used the
following parameter values:
51
P3
P3
P5
P5