### Table 3: Multi-objective optimisation algorithms based on simulated annealing. Dominance energy Volume energy

"... In PAGE 92: ...based or volume based) and whether the search is exploratory (computational temperature T gt; 0) or greedy (T = 0). Table3 summarises greedy and exploratory algorithms using dominance and volume energies, together with single solution and set states, which are described in this section; their performance on standard test problems is compared in section 4.4.... In PAGE 99: ... Results on MOSA and SAMOSA give a direct comparison of single solution states against set states, while dominance based and volume based energy measures are compared via the SAMOSA and VOLMOSA algorithms. As displayed in Table3 , the temperature zero versions of the algorithms are denoted by MOSA0 and SAMOSA0. Performance is evaluated on well-known test functions from the literature, namely the DTLZ test suite problems 1-6 [Deb et al.... ..."

### Table 4: Multi-objective search trace for the safety-critical example.

2000

"... In PAGE 22: ...Table 5: E ects of over-constraining in the control application. Table4 shows how the search progresses in the multi-objective optimization of the safety-critical application. In the table it can be seen that in the rst found solution, the load-balancing objective is estimated to have reached 92 percent of its optimal value while the lateness only has reached 10 percent.... In PAGE 23: ... It would be interesting to see if such reformulation is possible. As can be seen in Table4 , the multi-objective optimization strategy obtains an optimal value (20) of the communication. However, the estimated proximity of the communication in the optimal solution is only 66 percent.... ..."

Cited by 3

### Table 3. Density estimation techniques in multi-objective evolutionary algorithms and operators used in this study.

2001

"... In PAGE 11: ... Many advanced multi-objective evolutionary algorithms use some form of density dependent selection. Furthermore, nearly all techniques can be expressed in terms of density estimation, a classification is given in Table3 . We will make use of this as a further step towards a common framework of evolutionary multi-objective optimizers, and present the relevant enhancement of the unified model.... ..."

Cited by 20

### Table 3. Density estimation techniques in multi-objective evolutionary algorithms and operators used in this study.

2001

"... In PAGE 11: ... Many advanced multi-objective evolutionary algorithms use some form of density dependent selection. Furthermore, nearly all techniques can be expressed in terms of density estimation, a classification is given in Table3 . We will make use of this as a further step towards a common framework of evolutionary multi-objective optimizers, and present the relevant enhancement of the unified model.... ..."

Cited by 20

### Table 2. Results of the our proposed multi-objective approach after 1-hour runtime

2007

"... In PAGE 13: ...99 and the num ber of iterations within SA to be 1,000,000. Table2 lists the re- sults of using different evaluation functions on the obtained solutions. For the weighted-sum objective function, we use the sam e set of weight values as in formula (29), and list the num - ber of archived non-dom inated solutions (see colum n 2) and the best solution under this evaluation function (see colum n 3).... In PAGE 14: ...Table 2. Results of the our proposed multi-objective approach after 1-hour runtime A ccording to the results in Table2 , we can see that our proposed approach is very prom ising in solving the m ulti-objective nurse scheduling problem . In terms of the solution quality evaluated by the sam e objective function, our approach performs similar to the IP-based VNS, and significantly improve the best results of the hybrid genetic algorithm and the hybrid VNS by 25.... ..."

### Table 3. The average time for obtaining a solution for the multi-objective optimization problems by using TGP. The results are averaged over 30 independent runs.

"... In PAGE 13: ...igure 5. Diversity metric computed at every 10 generations. The results are averaged over 30 independent runs. Numerical values of the convergence and diversity metrics for the last generation are also given in section 9. 8 Running time Table3 is meant to show the efiectiveness and simplicity of the TGP algorithm by giving the time needed for solving these problems using a PIII Celeron computer at 850 MHz. Table 3 shows that TGP without archive is very fast.... In PAGE 13: ... 8 Running time Table 3 is meant to show the efiectiveness and simplicity of the TGP algorithm by giving the time needed for solving these problems using a PIII Celeron computer at 850 MHz. Table3 shows that TGP without archive is very fast. An average of 0.... ..."

### Table 2.1 2.3.2 Single- and Multi-Objective problems Quite often, the engineering problems require, that several contradicting criterions are satisfied simultaneously. Thus, it is suitable to refer to them as multi-objective problems. Example for such a problem in the Control Theory is the controller synthesis, where we want to have both minimal error between the set value and the system output, and control effort as small (economical) as possible.

### Table 4: Results for CVRP as a multi-objective problem.

"... In PAGE 24: ... This is the reason why we run 30 times our multi-objective GA (denoted as GAm), compute the average result and standard deviation for each instance, and compare them to the ones obtained with our single-objective GA. Results are shown in Table4 . For each method, the best result in all runs, the average of the best results, the standard deviation, and the relative difference between the best known result and the average of the best results are given.... ..."

### Table 4: Results of Multi-objective Experiments. Method Pareto

2007

"... In PAGE 6: ... Table 3: Results of Multimodal Experiments. Function SGA SGA with elite Tabu-GA Rastrigin 829/829/1 141/497/5 73/264/10 FMS- parameters -/-/0 126/282/2 143/608/10 Table4 shows the results of some multi-objective experiments, where Pareto is the number of Pareto optima and (n) is the tabu list size. While Ranking Selection GA gets six Pareto solutions on the frontier line of Pareto optima, Tabu-GA can get flexible and diverse Pareto solutions depending on a tabu list size.... ..."

### Table 2. Results comparison of game calculation and multi-objective optimization Objective function Design variable

2005