Computational Intelligence - February 2017 - 35

fuzzy similarity matrix R can be obtained by normalizing the
traffic flow between cell i and cell j:

The fuzzy equivalence matrix R ) can be obtained by the transitive closure method [36] from the fuzzy similarity matrix R.
The element r )ij of the fuzzy matrix R ) can tell the degree to
which cell i and cell j belong to the same TA. The fuzzy clustering based initialization tends to divide cells among TAs such
that the traffic flow is maximal among the cells within the same
TA. If two adjacent cells such as cell i and cell j have no shared
boundary crossing, then R )ij should be reset to zero. In this way,
we can avoid the situation that two cells are assigned to the
same TA when they should not be. It works as follows:
Step 1: Generate a random sequence LN including the N
elements of {1, 2, ..., N};
Step 2: Find pairs of cells that should not be assigned into
the same TA according to Eq. (9), and store them in the matrix
D 2 # d . To be specific, for any j ! {1, 2, ..., d} cell D 1j and cell
D 2j cannot be assigned to the same TA. Here, d is the total
member of cells that cannot be assigned into the same TA;
Step 3: Calculate the equivalence matrix R * from the similarity matrix RNN, and let i = 1;
Step 4: If i > N, stop; or, for all j ! {1, 2, ..., d}, compare
the element of matrix R ) in row Li column D 1j with the element in row Li column D 2j , and the element with the smaller
value is reset to zero. Check the cells one by one, and assign the
cells having a relatively large but not zero value with cell Li
into the same TA as much as possible, i " i + 1.

constraints presented in Eqs. (7) and (8) are relevant to paging
capacity, and the constraints violation value of individual m will
be calculated by Vm = ( p m /p max) + (t m /t max) , where p m =
/ Hi =1 max (p )i -P BS, 0), t m =/ Hh=1 max `/ celli ! MMEh p i -T MME, 0 j,
H is the total number of TAs, and pmax, tmax is the maximal
value of pm and tm in the whole population. Vm = 0 means that
all the constraints are satisfied, while Vm ! 0 means that at least
one of the constraints is voilated.
If the newly generated solution violates the constraint presented by Eq. (9), a code repair process must be conducted.
The violation can be expressed as that cell i and cell j have the
same code but allow no user boundary crossing. This kind of
violation includes three situations:
1) Both cell i and cell j have the same code as their adjacent
cells. Then we split the TA into two new TAs according
to the fuzzy equivalence matrix. The cells having larger
traffic flow with cell i are divided into one TA, and the
cells having larger traffic flow with cell j into another TA.
2) Only one of them such as cell i has adjacent cells with a different code. Check all the adjacent TAs of cell i, and assign
cell i into the TA which has the largest traffic flow with it.
3) Both of them only have adjacent cells with different
codes. Calculate the sum of the traffic flow of the two
cells with their adjacent cells in the TA. The cell with the
larger traffic flow is kept in the TA while the cell with the
smaller traffic flow is assigned to a new TA using the
method described in 2).
According to the feasibility rules [34], feasible solutions are
firstly selected for the next generation population based on the
Tchebycheff method. If the number of feasible solutions produced is not enough for the next generation, infeasible solutions will be selected based on minimal constraint violations.

D. Crossover and Mutation

F. MOEA/D and M2M Decomposition Strategy

Good crossover and mutation operators can enhance the performance of EMO algorithms, especially in complex combinatorial
optimization problems. Both of them play an important role in
exploring the Pareto optimal solutions in the searching space.
The parental individuals for crossover are selected based on the
M2M framework to make best use of its advantages in local
search. Suppose one individual for crossover is x 1 , the other
individual x 2 is randomly selected with probability 0.7 from its
own subpopulation, otherwise from other subpopulations. The
crossover operation is then processed as follows: randomly generate a number r in [0, 1], if r > 0.5, an intermediate offspring is
generated from the two given solutions x 1 and x 2 by a singlepoint crossover; otherwise, the intermediate offspring is generated by multi-point crossover [37]. After crossover, the new
individual is modified by randomly mutating one code of the
representation, according to the mutation probability 1/N .

MOEA/D [28] is a decomposition based EMO algorithm, and
various decomposition methods, such as Tchebycheff, weighted
sum, and boundary intersection, can be used for decomposition. MOEA/D can decompose a multi-objective optimization
problem into a number of single-objective subproblems. In our
study, the Tchebycheff method is used for decomposition in
m
MOEA/D. Let w = (w 1, f, w m) $ 0 ( / i = 1 w i = 1 ) be a weight
vector, and the Tchebycheff method can be expressed as:

i!j
(m ij)
rij = * j =max
1, 2,..., N
1
i = j.
m ij

E. Constraint Handling and Repair Strategy

In this paper, every cell belongs to a single TA, and MMEs are
assigned based on the TA configuration. Therefore the constraints presented in Eqs. (4)-(6) are necessarily met. Those

minimize g te (x | w) = max {w i | fi (x) - z i |},
i = 1, f, m

(10)

where zi is the minimum value of the ith objective. Except for
a set of weight vectors, a niching parameter T that is used to
define the neighboring weight vectors for crossover and mutation also needs to be predefined in MOEA/D.
M2M [30] is a particular population decomposition framework for EMO algorithms. Unlike MOEA/D [28], M2M can
decompose a multi-objective optimization problem into a set
of multi-objective optimization subproblems. Each subproblem
in M2M has its own subpopulation, and these subproblems can
be solved collaboratively. The selection in each subpopulation is

FEbruary 2017 | IEEE ComputatIonal IntEllIgEnCE magazInE

Table of Contents for the Digital Edition of Computational Intelligence - February 2017