![](bg8.png)
nounced with the integration of local search proce-
dures. The introduction of the two intensification pro-
cedures improves essentially the **1 and the **5 in-
stances. Also, the optimal schedule is always reached
by GA
HY B
for the 608 instance. The GA
HY B
found
the optimal solution for all the instances at least one
time and this was not the case either for GA
PCX
or
GA
IP
.
The convergence of both GA and the GA
PCX
al-
gorithms are similar. Indeed, the average conver-
gence generation is equal to 1837 and 1845 genera-
tions for GA and GA
PCX
, respectively. Concerning
the GA
IP
algorithm, the average convergence gener-
ation is equal to 1325 generations. So, we can con-
clude that the two intensification procedures based
on the CBS approach are permitting a faster genetic
algorithm convergence than the PCX crossover but
achieving worse results. The GA
HY B
average conver-
gence generation is equal to 825 and compared to the
GA
PCX
, the introduction of the intensification proce-
dures speeds up the convergence of the solution with
reaching better results.
Exact methods are well known to be time ex-
pensive. The same applies to their hybridization of
them with metaheuristics. Indeed, times execution in-
creases significantly with such hybridization policies
due to some technicality during the exchange of infor-
mation between the two methods (Talbi, 2009; Talbi,
2002; Puchinger and Raidl, 2005; Jourdan et al.,
2009) and this is what has been observed here. How-
ever, in this paper, the solution quality is our main
concern. So, we concentrated our efforts on it.
5 CONCLUSIONS
In this paper, we describe the hybridization into a
Genetic Algorithm of both a crossover operator and
intensification process based on Constraint Based
Scheduling. The PCX crossover operator uses the
direct precedence constraints to improve the CBS
search and consequently the schedules quality. The
precedence constraints are built from the selected par-
ents information in the reproduction process.
The intensification procedures are based on two
different CBS approaches after fixing a jobs block :
the first minimizes the total tardiness which represents
the considered problem objective function while the
second minimizes the makespan which also enhances
the exploration process and is well adapted to some
instances. These three policies hybridization repre-
sent the main contribution of this paper.
Compared to a simple GA, the use of the PCX
crossover improves all the results but for some in-
stances the difference is still noticeable. The hybrid
algorithm which uses the PCX crossover and the in-
tensification process improves the results and speeds
up the convergence of the solution. These results sug-
gest that the latter model seems to outperform the sin-
gle GA, the genetic algorithm with the hybrid PCX
crossover and the genetic algorithm with the intensi-
fication process.
A possible area of research in the future would
be to improve the precedence constraints quality. In-
deed, it is possible to consider constraints related to
a jobs set or to intervals time and indirect constraint.
Another possible area for further research would be
to employ a chromosome representation based on the
start times of activities. Hence, it will be possible to
get more accurate combination of start times.
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