Computational Intelligence - February 2016 - 64

Table 2 Distribution of QCs' cycle time [50].
PeRCeNTaGe

15%

20%

19%

10%

CYCLE TIME (SEC)

30-40

40-50

50-60

60-70

70-80

70-90

90-120

120-150

150-180

C. Graph Model of the FSP

We follow the approach in Subsec. II-A
to create a graph model for the FSP. To
do so, first we calculate all the possible
pickup times for each container, which
are the nodes in the graph, by dividing
the time window by a discretisation unit
d. To create the test cases we consider d
to be 60 seconds. Then, among all the
nodes, we identify those that are compatible with each other. To do so, we calculate the travel time of vehicles
between PDPs (QCs and SCs) based on
the distances between PDPs and the
speeds of vehicles, which are set to be 4
m/s and 2 m/s for the empty and loaded
vehicles respectively, based on common
industrial specifications. Given the
pickup and travel times, now we can calculate all the compatible nodes of a
given node using the procedure in II-A2.
By connecting all the compatible nodes
a graph model of the FSP can be created.

the parameters of FSEA, which are 15,
20, and 5 for the popSize, c, and b
respectively, where popSize is the size of
the population, and c and b are the
parameters of the adaptive learning
method (see Subsec. IV-E). FSEA uses
the rank selection. The local search and
mutation operators are applied to all individuals in each generation. FSEA stops
when one of the following criteria is met
first: FSEA reaches the global optima
found by CPLEX (if available) or FSEA
reaches its 2000th generation. In the latter case, the first time that FSEA finds the
best solution will be its processing time.
For CPLEX, the best estimate search for
node selection and the strong branching
for variable selection were considered.
The relative gap tolerance was set to
0.01% of the optimal value. If CPLEX is
out-of-memory due to gap tolerance, the
gap tolerance will be increased to 0.1%.
C. Performance Measures

VII. Experimental Results
in Static Environments
A. IBM ILOG CPLEX Optimizer as a
Benchmark

To the best of our knowledge, there has
been no existing EA research for the
FSP in ports. As a result, we considered
the integer programming (IP) model in
[3], which was solved using the CPLEX
solver from IBM, to be the benchmark.
CPLEX is the commercial, state-of-theart solver and mathematically it is proved
that the IP model in CPLEX can reach
the global optima given unlimited time
and resources. We developed the source
code for the IP model in [3] and then
solved it by CPLEX to find global
optima. We then applied FSEA to all the
test cases to find its optimal solutions and
compared these to CPLEX.
B. Parameter Settings of
FSEA and CPLEX

We conducted a series of experimental
studies to determine the best values for

We ran FSEA for 30 times and compared the results to that of CPLEX.
There are four possible comparison outcomes. First, FSEA can find the same
optimal fleet size as CPLEX in all runs.
Second, FSEA can only find the same
optimal fleet size as CPLEX in fewer
than 30 runs. Third, CPLEX runs out of
memory but it provides an integer lower
bound with no proof of optimality.
Fourth, CPLEX runs out of memory
and cannot find any lower bounds. In
the first case, the time to reach the global
optimum is used to compare the two
algorithms, using the Mann-Whitney
statistical test with a significance level of
95%. In the second case, we consider
CPLEX outperforms FSEA. In the third
case, the algorithm with the lower objective value is considered to be better. In
the case of equal objective values, the
Mann-Whitney statistical test identifies
the superior algorithm based on the processing time. In the fourth case, since
CPLEX cannot solve the problem, obviously FSEA outperforms CPLEX.

IEEE ComputatIonal IntEllIgEnCE magazInE | FEbruary 2016

D. Experimental Results

We coded FSEA in C++. All the experiments were conducted on a Core 2
Duo CPU 2.98 GHz with 3 GB RAM.
We created 165 test cases using the settings given in Sec.VI4.
Experimental results are summarized
into Tables 3 and 4. Table 3 shows that
FSEA significantly outperforms CPLEX
in 128 out of 165 cases. CPLEX outperforms FSEA in 31 out of 165 cases. In 6
out of 165 cases, no algorithm outperforms the other algorithm. CPLEX can
only solve 56 cases (the smaller-scale
ones) and it cannot solve 109 larger cases
due to out-of-memory issues, whereas
FSEA is able to solve all cases. This table
also shows the effectiveness of the FSEA
in terms of identifying the global optima.
In 50 out of 56 cases, FSEA finds the
global optima as found by CPLEX.
Table 4 shows the performance of
the two algorithms with respect to the
problems' size. Generally, it can be seen
that CPLEX cannot be used when the
sizes of the problems increase. For
example, with 100 containers and buffer
sizes $ 5, CPLEX runs out of memory.
Worse, when the number of containers
increases to 200 or 300, CPLEX can
only solve the problems if the size of
buffer is small, i.e. # 3 or 2, respectively.
In contrast, FSEA can solve all largerscale problems in a reasonable time.
Based on the results it can be seen that
FSEA not only has a reliable performance
regarding finding global optima, but it
also is able to solve the larger-scale problems where CPLEX fails5. FSEA was also
tested on large-scale cases where there
were 400-3000 containers. The results6
show that FSEA managed to find optimal
solutions in a reasonable length of time.
4

Detailed experimental results are shown in [http://
www.staff.ljmu.ac.uk/enrtngu1/Papers/detailed_
results.png].
5
Readers are referred to Sec.VII-F for a detailed analysis on the FSEA's operators and the impact of the buffer on the optimal fleet size.
6
Available in http://www.staff.ljmu.ac.uk/enrtngu1/
Papers/CIM_large_scales.pdf.

http://http:// http://www.staff.ljmu.ac.uk/enrtngu1/Papers/detailed_ http://www.staff.ljmu.ac.uk/enrtngu1/

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