C. Main Loop
Initially, a random parent population is created. The pop-
ulation is sorted based on the nondomination. Each solution is
assigned a fitness (or rank) equal to its nondomination level (1
is the best level, 2 is the next-best level, and so on). Thus, mini-
mization of fitness is assumed. At first, the usual binary tourna-
ment selection, recombination, and mutation operators are used
to create a offspring population of size . Since elitism
is introduced by comparing current population with previously
found best nondominated solutions, the procedure is different
after the initial generation. We first describe the th generation
of the proposed algorithm as shown at the bottom of the page.
The step-by-step procedure shows that NSGA-II algorithm is
simple and straightforward. First, a combined population
is formed. The population is of size . Then, the
population is sorted according to nondomination. Since all
previous and current population members are included in ,
elitism is ensured. Now, solutions belonging to the best non-
dominated set are of best solutions in the combined popu-
lation and must be emphasized more than any other solution in
the combined population. If the size of is smaller then ,
we definitely choose all members of the set for the new pop-
ulation . The remaining members of the population
are chosen from subsequent nondominated fronts in the order of
their ranking. Thus, solutions from the set are chosen next,
followed by solutions fromthe set , and so on. This procedure
is continued until no more sets can be accommodated. Say that
the set is the last nondominated set beyond which no other
set can be accommodated. In general, the count of solutions in
all sets from to would be larger than the population size.
To choose exactly population members, we sort the solutions
of the last front using the crowded-comparison operator
in descending order and choose the best solutions needed to fill
all population slots. The NSGA-II procedure is also shown in
Fig. 2. The new population of size is now used for se-
lection, crossover, andmutation to create a newpopulation
of size . It is important to note that we use a binary tournament
selection operator but the selection criterion is now based on the
crowded-comparison operator . Since this operator requires
both the rank and crowded distance of each solution in the pop-
ulation, we calculate these quantities while forming the popula-
tion , as shown in the above algorithm.
Consider the complexity of one iteration of the entire algo-
rithm. The basic operations and their worst-case complexities
are as follows:
1) nondominated sorting is ;
2) crowding-distance assignment is ;
3) sorting on is .
The overall complexity of the algorithm is , which is
governed by the nondominated sorting part of the algorithm. If
C. Main Loop
Initially, a random parent population is created. The pop-
ulation is sorted based on the nondomination. Each solution is
assigned a fitness (or rank) equal to its nondomination level (1
is the best level, 2 is the next-best level, and so on). Thus, mini-
mization of fitness is assumed. At first, the usual binary tourna-
ment selection, recombination, and mutation operators are used
to create a offspring population of size . Since elitism
is introduced by comparing current population with previously
found best nondominated solutions, the procedure is different
after the initial generation. We first describe the th generation
of the proposed algorithm as shown at the bottom of the page.
The step-by-step procedure shows that NSGA-II algorithm is
simple and straightforward. First, a combined population
is formed. The population is of size . Then, the
population is sorted according to nondomination. Since all
previous and current population members are included in ,
elitism is ensured. Now, solutions belonging to the best non-
dominated set are of best solutions in the combined popu-
lation and must be emphasized more than any other solution in
the combined population. If the size of is smaller then ,
we definitely choose all members of the set for the new pop-
ulation . The remaining members of the population
are chosen from subsequent nondominated fronts in the order of
their ranking. Thus, solutions from the set are chosen next,
followed by solutions fromthe set , and so on. This procedure
is continued until no more sets can be accommodated. Say that
the set is the last nondominated set beyond which no other
set can be accommodated. In general, the count of solutions in
all sets from to would be larger than the population size.
To choose exactly population members, we sort the solutions
of the last front using the crowded-comparison operator
in descending order and choose the best solutions needed to fill
all population slots. The NSGA-II procedure is also shown in
Fig. 2. The new population of size is now used for se-
lection, crossover, andmutation to create a newpopulation
of size . It is important to note that we use a binary tournament
selection operator but the selection criterion is now based on the
crowded-comparison operator . Since this operator requires
both the rank and crowded distance of each solution in the pop-
ulation, we calculate these quantities while forming the popula-
tion , as shown in the above algorithm.
Consider the complexity of one iteration of the entire algo-
rithm. The basic operations and their worst-case complexities
are as follows:
1) nondominated sorting is ;
2) crowding-distance assignment is ;
3) sorting on is .
The overall complexity of the algorithm is , which is
governed by the nondominated sorting part of the algorithm. If
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C. Main Loop
Initially, a random parent population is created. The pop-
ulation is sorted based on the nondomination. Each solution is
assigned a fitness (or rank) equal to its nondomination level (1
is the best level, 2 is the next-best level, and so on). Thus, mini-
mization of fitness is assumed. At first, the usual binary tourna-
ment selection, recombination, and mutation operators are used
to create a offspring population of size . Since elitism
is introduced by comparing current population with previously
found best nondominated solutions, the procedure is different
after the initial generation. We first describe the th generation
of the proposed algorithm as shown at the bottom of the page.
The step-by-step procedure shows that NSGA-II algorithm is
simple and straightforward. First, a combined population
is formed. The population is of size . Then, the
population is sorted according to nondomination. Since all
previous and current population members are included in ,
elitism is ensured. Now, solutions belonging to the best non-
dominated set are of best solutions in the combined popu-
lation and must be emphasized more than any other solution in
the combined population. If the size of is smaller then ,
we definitely choose all members of the set for the new pop-
ulation . The remaining members of the population
are chosen from subsequent nondominated fronts in the order of
their ranking. Thus, solutions from the set are chosen next,
followed by solutions fromthe set , and so on. This procedure
is continued until no more sets can be accommodated. Say that
the set is the last nondominated set beyond which no other
set can be accommodated. In general, the count of solutions in
all sets from to would be larger than the population size.
To choose exactly population members, we sort the solutions
of the last front using the crowded-comparison operator
in descending order and choose the best solutions needed to fill
all population slots. The NSGA-II procedure is also shown in
Fig. 2. The new population of size is now used for se-
lection, crossover, andmutation to create a newpopulation
of size . It is important to note that we use a binary tournament
selection operator but the selection criterion is now based on the
crowded-comparison operator . Since this operator requires
both the rank and crowded distance of each solution in the pop-
ulation, we calculate these quantities while forming the popula-
tion , as shown in the above algorithm.
Consider the complexity of one iteration of the entire algo-
rithm. The basic operations and their worst-case complexities
are as follows:
1) nondominated sorting is ;
2) crowding-distance assignment is ;
3) sorting on is .
The overall complexity of the algorithm is , which is
governed by the nondominated sorting part of the algorithm. If
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