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3 FINE-TUNING OF THE SA
Fine-tuning of parameters is the process of adjusting
the parameters of an algorithm to solve a problem.
Several works like (Barr et al., 1995), (Fink and S.,
2002) have addressed the importance of performing
a calibration of the parameters in heuristic and meta-
heuristic approaches. Related works about method-
ologies that can be followed to perform a fine-tuning
are shown in (D´ıaz and Laguna, 2006), (Fidanova
et al., 2009), (Taguchi, 1994).
A Covering Array(CA) (Lopez-Escogido et al.,
2008) is an array M of size N× k consisting of N vec-
tors of length k with entries from 0, 1, . .., v− 1 (v is
the size of the alphabet) such that every one of the v
t
possible vectors of size t occurs at least once in every
possible selection of t elements from the vectors. The
parameter t is referred to as the covering strength.
This process is based in a set of instances β of a
problem P and in a Covering Array CA(N;t, k, v) to
study the effect of the interaction between parameters
in the solution of P using an algorithm A .
In the CA, the value k is the number of param-
eters of the algorithm A and the problem P object
of the fine-tuning. The value of the alphabet v will
correspond to the cardinality of the set of values for
the parameters considered in the tuning process. The
level of interaction between parameters is initialized
to t = 2. During the process, the performance ϕ
t
of
the algorithm solving the problem P with a level of
interaction t is measured. If the performance is im-
proved, the values of the parameters of P and A are
adjusted in order to find better solutions. If the perfor-
mance ϕ
t
is no longer improved, the interaction level
t is increased by one only if ϕ
t
− ϕ
t−1
> ε. ε is a
threshold selected so that the fine-tuning process halts
in a suitable amount of time. Every time that the level
of interaction t is increased then a new CA(N;t, k, v)
must be constructed.
The next section presents the experiments done to
tune the values of the parameters of the Simulated An-
nealing to solve the BC problem using the methodol-
ogy described in this section.
4 EXPERIMENTAL RESULTS
The methodology described in Section 3 was used to
fine-tune the parameters of the Simulated Annealing
in order to solve the BC problem.
The initial values selected for the parameters dur-
ing the tuning process are shown in Table 2. The first
column shows the parameter, the last 3 columns are
the values selected for that parameter.
Table 2: Values for the parameters of the Simulated Anneal-
ing.
Type of Parameter Parameter 1
st
value (0) 2
nd
value (1) 3
rd
value (2)
Problem R R
a
R
b
R
c
Problem χ χ
a
χ
b
χ
c
Problem Π Π
1
Π
2
Π
3
Algorithm T
0
1.0 0.5 0.25
Algorithm α 0.99 0.90 0.85
Algorithm T
f
0.001 0.000001 0.000000001
Algorithm I 10000 100000 1000000
Algorithm L 500 800 1000
Algorithm Γ 5 10 15
Problem N N
1
N
2
N
3
Table 3: Summary of the results obtained during the fine-
tuning process using a CA(14;2, 10, 3).
Num. Avg. Avg. Avg. Avg.
Conf. Sol. Opt. Sol. Time Eval. Opt. Eval.
0002010112 14.00 0.00 37.29 0.00 20129.03 0.00
0101221200 12.00 6.00 32.26 0.90 97483.81 4666.67
0120112011 22.00 0.00 49.03 0.00 127741.88 0.00
0210002222 18.00 0.00 53.68 21.94 1331612.75 0.00
1001101122 14.00 0.00 38.84 0.00 91354.82 0.00
1011212100 12.00 2.00 32.77 0.58 47380.64 4800.00
1021020221 18.00 0.00 46.45 0.00 19354.84 0.00
1112000020 12.00 4.00 32.90 0.00 18322.58 4250.00
1210120210 12.00 2.00 37.16 0.00 18903.23 3000.00
2000021002 14.00 0.00 33.29 1.94 193548.45 0.00
2120200102 14.00 0.00 38.45 0.00 20129.03 0.00
2202122101 28.00 0.00 72.00 3.87 284903.31 0.00
2211201011 30.00 0.00 112.52 0.00 37741.95 0.00
2222211220 12.00 2.00 37.03 0.00 80903.23 45000.00
Table 3, in the first column, shows the first
CA(14;2, 10, 3) used for the tuning process. The al-
phabet for this CA is shown in the Table 2. Each digit
from left to right correspond to a parameter shown in
Table 2. The values 0, 1 and 2 correspond to values in
the columns 2, 3 and 4, in Table 2, respectively.
The BC problem was solved using the SA and the
CA shown in the first column of Table 3. The BC
problem was solved 31 times by each configuration.
Table 3 summarizes the results obtained during the
fine-tuning process using a level of interaction of two
t = 2. Column 2 shows the best solution achieved
by each configuration. Column 3 shows the number
of times that the optimal solution was reached. Col-
umn 4 presents the average cost of a solution using a
specific configuration. Column 5 presents the average
time measured in seconds that the SA spent in finding
the optimal solution. Column 6 is the average number
of times that the evaluation function was computed
during the SA. Column 7 shows the average number
of times that the SA needs to compute the evaluation
function in order to find the optimal solution.
The results presented in Table 3 show a low qual-
ity in the solution achieved by the SA using the com-
bination of the values described by CA(14;2, 10, 3).
The best configuration just achieved the optimum
value 19% of the times. The solutions of the SA aver-
aged a distance of 168% over the optimal solution, in
the best case.
In order to improve the performance of the SA
when solving the BC, the level of interaction between
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