3.4 Multi-Objective Optimization Formulation
3.4.2 Results and Discussion
3.4.2.1 Statistical Performance Analysis of EMO techniques . 49
The mean and standard deviation of IGD and HV indicators for 30 runs of EMO techniques are presented in Tables 3.16 and 3.17. Except for PAES, all EMO techniques show equivalent performance based on the mean values of indicators. Moreover, the standard deviation is quite small that suggests similar outcome of the non-dominated solutions from the EMO techniques.
3.4.2.2 Obtained Non-dominated Solutions
All EMO techniques showed equivalent performance on IGD and HV indicators, there- fore the solutions obtained from NSGA-II are considered in this section. The non- dominated solutions generated using model-1 are shown in Fig. 3.37. For discussion, these solutions are categorized into three groups. The first group of the non-dominated solutions is referred as the surface solutions (blue color symbols) in which most of the solutions are generated at lower P values. The second group of solutions (black color filled symbols) is referred as the knee-solutions that show a decent trade-off among the objectives. The third region is referred as the extension solutions in which the solu- tions do not show much trade-off between T andN, but objectiveP increases to higher value. Three extreme solutions are also shown in Fig. 3.37 in which solution ‘a1’ rep- resents minimum P, solution ‘z1’ represents minimum T, and solution ‘e1’ represents minimum N.
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50 Multi-Objective Optimization Formulations for Bulldozer and its Blade
Table 3.16: Mean and standard deviation of IGD indicator for five EMO techniques.
NSGAII SPEA2 PAES
Mean Std. dev. Mean Std. dev. Mean Std. dev.
Model-1 7.26e-04 6.0e-05 7.53e-04 5.0e-05 4.65e-03 6.7e-04 Model-2 7.50e-04 3.6e-05 7.55e-04 4.5e-05 5.02e-03 1.5e-03
SMPSO GDE3
Mean Std. dev. Mean Std. dev.
Model-1 8.12e-04 6.2e-05 8.63e-04 5.2e-05 Model-2 7.94e-04 4.3e-05 8.84e-04 3.4e-05
Table 3.17: Mean and standard deviation of HV indicator for five EMO techniques.
NSGAII SPEA2 PAES
Model-1 5.11e-01 2.1e-02 4.87e-01 2.2e-02 2.68e-01 7.5e-02 Model-2 4.86e-01 2.1e-02 4.69e-01 3.0e-02 2.38e-01 6.5e-02
SMPSO GDE3
Mean Std. dev. Mean Std. dev.
Model-1 5.76e-01 9.1e-02 5.17e-01 2.3e-02 Model-2 5.69e-01 1.1e-01 5.16e-01 2.6e-02
b1 e1
c1 z1
a1 d1
T
30000
10000 0
Extreme solutions Extension solutions Knee solutions
Surface solutions
0 20 40
60 80 100 200 150
50 0 P 100 N
20000
Figure 3.37: The approximate PO solutions from NSGA-II using model-1 are pre- sented. The solutions are categorized into three groups. Different colors are used for distinguishing the solutions lying in three groups. The unit of P and T are kN.m/s and s, respectively.
The scatter plots for three objectives are shown in Fig. 3.38 for model-1 solutions.
Since different colors for the symbols are used, three groups of the approximate PO solutions can be distinguished. The surface solutions are generated with lowerP values.
The same solutions show trade-off between N and T (refer Fig. 3.38c) in which these TH-2093_136103042
3.4 Multi-Objective Optimization Formulation 51
Extreme solutions Surface solutions
Knee solutions Extension solutions
e1 a1
z1
N 40
50 60 70
0 50 100 150 200
P 20
10 0 30
(a)P vs N
P
T
z1 a1
0 25000
0 50 100 150 200
15000 10000 5000 20000
(b)P vs T
a1
z1 e1 0 20000 25000
0 10 20 30 40 50 60 70
T
N 10000
5000 15000
(c)N vsT
Figure 3.38: The scatter plots for three objectives are presented in which the relation- ships among the objectives are shown for three groups of the approximate PO solutions using model-1.
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solutions are generated over a wide range of N and T objectives. The second group is made of the knee-solutions, which is always important for a practitioner or decision maker. It is because the solutions show a decent trade-off among the posed objectives in this region. The extension solutions are generated with lower N and T values (refer Figs. 3.38a and 3.38b). However, higher P values can be observed.
The approximate PO solutions obtained using model-2 are shown in Fig. 3.39. The solutions again can be categorized into three groups. The surface solutions have lowerP values and wider ranges ofT andN. The knee-solutions show a decent trade-off among the objectives. The extension solutions are evolved with larger P values but with less N and T ranges. In the figure, solution ‘a2’ represents minimum P value, solution
‘e2’ represents minimum N value, and solution ‘z2’ represents minimum T value. The scatter plots are also shown in Fig. 3.40 for model-2 in which same observation can be made as observed with model-1 solutions.
b2 z2e2
a2 d2 c2
N 30000
10000 0
Surface solutions
Extreme solutions
80 100 40 60
0 20
200
100 50 0 150 P
T
Extension solutions Knee solutions
20000
Figure 3.39: The approximate PO solutions from NSGA-II using model-2 are pre- sented. The solutions are categorized into three groups. Different colors are used for distinguishing the solutions lying in three groups.
The non-dominated solutions are also generated using the ǫ−constraint method.
The starting solutions for fmincon solver are marked in Fig. 3.37 for which values of ǫN and ǫT are set from NSGA-II solutions. Table 3.18 presents NSGA-II solutions and the improved solutions using theǫ−constraint method. It can be observed that P values have marginally improved from its original values.
For model-2, the starting solutions for thefmincon solver are marked in Fig. 3.39.
The solutions obtained from NSGA-II andǫ−constraint method are presented in Table TH-2093_136103042
3.4 Multi-Objective Optimization Formulation 53
Knee solutions Extension solutions Extreme solutions Surface solutions
z2 e2
a2
0 40 50 60 70
0 50 100 150 200
N
P 20
10 30
(a)P vs N
T
P
z2 a2
0 25000
0 50 100 150 200
15000 10000 5000 20000
(b)P vs T
z2
a2
0 e2 20000 25000
0 10 20 30 40 50 60 70
T
N 10000
5000 15000
(c)N vsT
Figure 3.40: The scatter plots for three objectives are presented in which the relation- ships among the objectives are shown for three groups of the approximate PO solutions using model-2.
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54 Multi-Objective Optimization Formulations for Bulldozer and its Blade
Table 3.18: The solutions obtained from NSGA-II andǫ−constraint method considering model-1 are presented.
Solutions NSGA-II ǫ− constraint
(P, N, T) (P, ǫN, ǫT) a1 (21.936, 67, 19999.920) (21.935, 67, 19999.920) b1 (33.623, 27, 2849.314) (33.232, 27, 2849.314) c1 (50.859, 11, 2907.186) (49.605, 11, 2907.186) d1 (83.584, 9, 362.356) (80.299, 9, 362.356) z1 (193.231, 11, 134.563) (192.767, 11, 134.563)
Table 3.19: The solutions obtained from NSGA-II andǫ−constraint method considering model-2 are presented.
Solutions NSGA-II ǫ− constraint
(P, N, T) (P, ǫN, ǫT) a2 (28.635, 67, 19968.430) (28.635, 67, 19968.430) b2 (44.686, 26 881.653) (44.654, 26, 881.653) c2 (57.017, 12, 2536.945) (56.378, 12, 2536.945) d2 (97.844, 12, 295.281) (95.735, 12, 295.281) z2 (193.280, 12, 146.261) (193.276, 12, 146.261)
3.19. Again, marginal difference can be seen inP values among the solutions obtained from both techniques.