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machines no.
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Figure 7: Location of first intersection point for different
instance sizes.
instance size i.e. the number of the jobs and the num-
ber of the machines. Figure 6 illustrates the propor-
tion, denoted (F), of the objective function value ob-
tained via the GA and the TSA. The experiment was
carried out for 100 instances of the each size taken
from the set of 1 to 10 machines and 10 to 50 jobs.
Both algorithms were investigated for 25 iterations.
The value of F > 1 on the graph corresponds to the
situations where the efficiency of GA was superior to
TSA, F < 1 otherwise.
Figure 7 presents the average number of iterations,
when the first intersection was observed. The first in-
tersection point can be interpreted as a point starting
from which the GA is superior over the TSA.
6 CONCLUSIONS
In the paper the problem of the parallel machine job
scheduling with the weighted earliness and tardiness
has been addressed. Two heuristic algorithms, that
proven to be efficient, have been proposed and nu-
merically validated. The investigation of the sensitiv-
ity of both approaches as a function of the size of the
problem instance, has been carried out. The results
obtained suggest that the TSA is appropriate to han-
dle the relatively small and medium instances. On
the other hand, utilization of the GA coupled with
MCUOX crossover operator, becomes more benefi-
cial with the increase of the problem instances. An
important property was observed, namely a signifi-
cant deterioration of the efficiency of the TSA for in-
stances containing 2−5 machines in comparison with
other values was noted, see Figures 6-7. Considering
these figures from the instance of (1 machine, 10 jobs)
to (10 machines, 50 jobs) a constant improvement of
the GA in comparison with the TSA up to the point of
its predominance, can be seen. It has been shown ex-
perimentally that the point is located in the vicinity of
the instance size of 100 and 200 jobs, see (Figure 5).
Furthermore, the execution time of GA with MCUOX
increases with a decreasing rate as a function of the
increasing problem size whilst the increasing rate is
observed for the TSA.
The problem extension could consist of the se-
quential dependency of jobs and the possibility of in-
troducing the idle time intervals between subsequent
execution of jobs. Further improvement of the pro-
posed heuristics can be achieved by considering a hy-
brid algorithm that inherits the advantages of both ap-
proaches i.e. the GA and the TSA.
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