PARALLEL MACHINE EARLINESS-TARDINESS SCHEDULING
Comparison of Two Metaheuristic Approaches
Marcin Bazyluk, Leszek Koszalka
Chair of Computer Systems and Networks, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland
Keith J. Burnham
Control Theory and Applications Centre, Coventry University, Coventry, U.K.
Keywords:
Task scheduling, parallel machines, heuristics, genetic algorithms, tabu search.
Abstract:
This paper considers the problem of parallel machine scheduling with the earliness and tardiness penalties
(PMSP
E/T) in which a set of sequence-independent jobs is to be scheduled on a set of given machines to
minimize a sum of the weighted earliness and tardiness values. The weights and due dates of the jobs are
distinct positive numbers. The machines are diverse - each has a different execution speed of the respective
jobs, thus the problem becomes more complex. To handle this, it two heuristics are employed, namely: the
genetic algorithm with the MCUOX crossover operator and the tabu search. The performances of the both
approaches are evaluated and their dependency on the shape of the investigated instances examined. The
results indicate the significant predominance of the genetic approach for the larger-sized instances.
1 INTRODUCTION
The problem of job scheduling on parallel machines
(PMJS) has been considered recently in many scien-
tific and industrial communities. In the classical ex-
ample of this type of problem the collection of inde-
pendent jobs is investigated. Each of the tasks is to
be assigned to the single set of the available machines
which may beidentical or different. The problem con-
sidered here extends the PMJS with the weighted ear-
liness and tardiness penalties calculated for the exe-
cution of a job before or after its due date. A single
job consists of the following parameters, namely its
weight, also defined as the importance, and its due
date. Baker and Scudder (Baker and Scudder, 1990)
provide a survey on the job scheduling problem in-
volving a due date determination. In this paper, the
due date for the each job is considered as a specific
moment in the time. If a given job is accomplished
at its due date, the requirements are fulfilled and the
penalty is not subjected. The specification of a single
machine is implemented as a vector describingits pro-
cessing speed of the individual jobs. It is to be noted,
that the identical machines are characterized by the
identical vectors.
In the case, when the due dates of all of the jobs
are set to 0 (thus all the jobs require the fastest pos-
sible execution with no earliness penalty) and their
weights are equal, the problem considered is NP-hard
(Du and Leong, 1990). It is to be remarked, that when
the weights are arbitrary positive numbers, the task
becomes NP-hard in a strong sense. On the other
hand, the problem considered here may be thought of
as even more complex. Kanet examined the problem
of minimizing the total weighted earliness with the
tardiness on a single machine and common due dates
in (Kanet, 1981). He proposed the algorithm charac-
terized by a polynomial complexity. His work was
extended by Hall and Posner to the case of several
identical machines in (Hall and Posner, 1991). This
paper forms a further extension with the due dates al-
lowed to be different.
The example solution to the problem considered,
is a schedule of all jobs that are assigned to the sepa-
rate machines in a set of queues - one for each ma-
chine. The premise is to minimize the sum of the
penalties for all the jobs. The penalty for a single job
is calculated according to the Equation (1) where p
i
and w
i
are the penalty and the weight of the job i, re-
spectively. e
i
= max{0, d
i
c
i
} denoted the earliness
of the job i and t
i
= max{0, c
i
d
i
} is the tardiness
of the job i with d
i
being its corresponding due date
80
Bazyluk M., Koszalka L. and J. Burnham K. (2008).
PARALLEL MACHINE EARLINESS-TARDINESS SCHEDULING - Comparison of Two Metaheuristic Approaches.
In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - ICSO, pages 80-85
DOI: 10.5220/0001476500800085
Copyright
c
SciTePress
and c
i
its completion time (the time instant when the
processing of a job has been finished). The follow-
ing formula is introduced, see (2), accounting for the
total earliness and tardiness penalty (p), where I de-
notes the set of jobs.
p
i
= w
i
e
i
+ w
i
t
i
(1)
p =
iI
w
i
e
i
+
iI
w
i
t
i
(2)
2 PROBLEM FORMULATION
Let the problem of the scheduling of the N indepen-
dent lobs on a set of M available parallel machines
be considered. Each job may be completed by any
of the available machines, however the time required
differs, in general, depending on the machine chosen.
If the job has been accomplished too early or too late,
the penalty is calculated for the earliness or tardiness,
respectively, according to the difference between the
due date of the job and the real execution time. It
is noted, that the penalty is also proportional to the
weight of the particular job. By considering the pos-
sible decisions during the job scheduling procedure,
two optimization techniques have been developed to
minimize the total penalty for all the jobs. The mathe-
matical model presented here is a modification of that
proposed by Cao, Chen and Wan in (Cao et al., 2005).
The next two subsections introduce the nomenclature
and the definitions required, to formulate the problem.
2.1 Known Parameters
i, j = 1, 2, . . . , N, job indeces
m = 1, 2, . . . , M, machine index
z
im
processing time of the i-th job on the m-th
machine, z
im
= {1, 2, . . ., 10}
w
i
weight of the i-th job ,
w
i
= {1, 2, . . ., 10}
d
i
due date of the i-th job ,
d
i
= {1, 2. . ., 5}
c
i
execution time of the i-th job
e
i
earliness of jo i, e
i
= max{0, d
i
c
i
}
t
i
tardiness of jo i, t
i
= max{0, c
i
d
i
}
2.2 Decision Variables
x
ijm
=
1 if job j follows job i on machine m,
0 otherwise,
y
im
=
1 if job i is executed by machine m,
0 otherwise,
i = 0, 1, . .. , N, j = 1, . . . ,N,
j 6= i, m = 1, . . . , M
With the decision variables defined, the following
recursive formula (Equation (3)) is proposed for the
calculation of the moment of the job completion. The
job completion is defined as the sum of its process-
ing time together with the moment when the job that
is executed directly before the one being considered
here, is completed.
c
j
=
N
i=0
M
m=0
x
ijm
c
i
+ z
jm
(3)
2.3 Assumptions
Min T
c
=
N
i=1
w
i
(e
i
+ t
i
) (4)
N
i=1,i6= j
M
m=1
x
ijm
= 1 (5)
N
i=1,i6= j
x
ijm
= y
jm
, (6)
N
j=1, j6=i
x
ijm
y
im
(7)
c
j
+ A(1 x
ijm
) c
i
+ z
jm
(8)
In all the assumptions introduced, the parameters are
defined as follows:
i = 1, 2, . .. ,N, j = 1, 2, . .. , N, i 6= j,
m = 1, 2, . . . ,M, c
i
> 0, c
j
> 0
In the model presented, Equation (4) describes the ob-
jective function as the sum of the penalties for all the
jobs which are to be minimized. The following equa-
tions defines the restrictions to which the decision
variables and the parameters are subjected. Equation
(5) ensures that, first, every job is processed on a sin-
gle machine and, second, that its execution is not di-
vided into separate parts. Equation (6) imposes a re-
striction that the j-th job must immediately followany
other job on the m-th machine (or be left in the first
position in the case when i = 0). Thereby, the inter-
vals in the machine usage are not allowed. Equation
(7) states that, if the i-th job is processed on the m-th
machine, it will be immediately followed by at most
one other job on this machine. Therefore, the jobs
can be executed one at the each time instant. Equa-
tion (8) expresses the finite completion time of each
job, where the scalar A is a large positive number.
Due to the evident complexity of the problem con-
sidered, the investigations are restricted to the class of
heuristic algorithms. In Sections 3–4 two heuristic
based approaches are presented.
PARALLEL MACHINE EARLINESS-TARDINESS SCHEDULING - Comparison of Two Metaheuristic Approaches
81
START
Find the first solution.
LOOP
Find the best solution which is not tabu.
Update the global best solution.
Add a rule to the tabu list.
If the fixed number of the iterations are completed,
STOP;
else if the fixed number of the iterations without the
improvement regarding the the best global solution is
completed, STOP;
else repeat LOOP.
Figure 1: Tabu search algorithm.
3 TABU SEARCH ALGORITHM
APPROACH
The tabu search algorithm (TSA) was developed
mainly due to the work of the Polish mathematicians,
namely E. Nowicki and C. Smutnicki (Nowicki and
Smutnicki, 2005). It belongs to the class of local
search algorithms, where a concept of the local neigh-
bourhood of a given solution is considered. The local
neighbourhood is defined as a set of solutions that are
different from the basic solution with respect to a sin-
gle attribute. Many different types of neighbourhood
can be found in the literature, however the swap and
insert types, are considered in this paper.
TSA consists of the subsequent steps to determine
the best representative of the neighbour solutions set,
where a single step is conducted in each iteration. It
may occur that one of the neighbour solutions is su-
perior in respect to the basic solution (i.e. the solution
which the neighbourhood is considered at the given
iteration). In this case, the step is made towards the
direction of the newly found neighbourhood solution.
This, therefore, allows the algorithm to alleviate the
possibility to be trapped in local extremes of the ob-
jective function considered. To avoid repetitive steps
between the same solutions in the adjacent extremes,
the tabulist consisting of the assumed a priori number
of the rules regarding the quality of the solutions vis-
ited recently, is proposed. Figure 1 provides a more
formal description of the proposed algorithm.
There exists, however, an exception when the so-
lution found is superior to the best one to date. In this
case the algorithm steps towards its direction uncon-
ditionally and the tabu list is erased. It is remarked,
that a considerable amount of time is required to esti-
mate even a single solution in each iteration of the al-
gorithm, for the instances relatively large. This leads
START
Generate the first population.
LOOP
Find the best chromosome in the current population.
Update the global best chromosome.
Choose the representing chromosomes for the
crossover.
Create a new population.
Execute a swap mutation with the fixed probability.
Execute a bit mutation with the fixed probability.
If the fixed number of the iterations are completed,
STOP;
else if the fixed number of the iterations without the
improvement regarding the the best global solution is
completed, STOP;
else repeat LOOP.
Figure 2: Genetic algorithm.
to the conclusion that the quality of the first generated
solution from which the algorithm starts is essential.
In this paper, a modified algorithm is suggested, to
handle this task.
The swap move is carried out via the exchange
of the positions between two randomly chosen jobs
taken from the list assigned to a single machine. In-
sert one removes a random job from a one machine
and allocates it on the second machine, directly be-
fore randomly chosen job from its list.
4 GENETIC ALGORITHM
APPROACH
For more details on the subject of the genetic algo-
rithms (GAs), see (Schmitt, 2000) and (Davis, 1991).
Due to the fact that many different approaches have
been described in the literature for solving the prob-
lem considered, this section presents the exact param-
eters of the algorithm implemented.
The more formal definition of the algorithm pro-
posed is presented in figure 2. The crossover operator
is the basis of the genetic algorithms, hence, it is es-
sential to focus on its proper implementation. The is-
sue that was shown experimentally in (Bazyluk et al.,
2006) for a simpler type of job scheduling problem
is that the utilization of a popular PMX leads to an
impossibility to obtain satisfying results. Therefore,
the multi-component uniform order-based (MCUOX)
crossover operator proposed by Sivrikaya and Ulusoy
in (Sivrikaya-Serifogluand Ulusoy, 1999) was chosen
for the purpose of this paper.
It is noted, that in the scheme chosen a single gene
ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics
82
accommodates both the object and the associated se-
lection associated with this object. In the case of the
problem considered, this refers to the situation where
each gene corresponds to the job-machine pair. The
construction of a descendant from two parent chro-
mosomes is presented in Figure 3.
START
With the first position on the parents and the descen-
dant.
LOOP
Choose one parent randomly.
Find the first job corresponding to the chosen parent
which has not been assigned to the descendant.
If the machine assigned to the job is the same for both
parents, make the same selection for the job;
else choose one of the machines from the parents ran-
domly and assign the job to it.
Assign the job-machine pair to the rst empty position
of the descendant.
If all genes of the descendant chromosomes are set,
STOP;
else proceed to the next gene on the descendant and re-
peat LOOP.
Figure 3: MCUOX crossover algorithm.
The mutation implemented comprises two mecha-
nisms. The first, randomly selects two positions on a
chromosome and exchanges their contents. The sec-
ond, incorporates the reassignment of the machine to
a randomly chosen job in a chromosome. It is to be
noted, that this operation may lead to the same ma-
chine selection as before.
The selection of the chromosomes for the
crossover from a population builds on a rule that the
probability of choosing a chromosome to be a par-
ent in the next generation is proportional to its fitness.
The value of the fitness function is for a given chro-
mosome is defined by Equation (9) where K is the
size of the population and F
k
= CT
c
(k)
1
the fitness
function of the chromosome k that is inversely propor-
tional to the objective function with C being a fixed
constant.
Table 1: Experiment parameters.
tabu list size (TSA) 10
population size (GA) 25
swap mutation probability (GA) 0,1
bit mutation probability (GA) 0,05
crossover probability (GA) 0,9
P
k
=
F
k
K
k=1
F
k
(9)
The appropriate tuning of the parameters of the GA
is one of the crucial issues that influences signifi-
cantly its efficiency and effectiveness, see (Grefen-
stette, 1986) for more detailed discussion. For this
purpose, in the problem considered, a meta-genetic
algorithm with a regular PMX operator was addition-
ally implemented.
5 NUMERICAL ANALYSIS
5.1 Generation of Test Instances
Both implemented algorithms were validated for the
same instances of the various sizes considered. Due
to the fact that no benchmark problems were found
in the literature, such as in particular the presented
shape, the test benchmarks were generated randomly.
After the tuning of the parameters the following val-
ues, presented in Table 1, were set.
5.2 Research Results
The implemented algorithms were evaluated on a
IBM-compatible machine, equipped with the Intel
Pentium M 740 1, 73GHz processor and 512 Mbytes
of RAM. The first experiment was conducted for a
set of 100 small instances comprising of 10 jobs and
3 machines. The evolution of the average objective
function, denoted (T
c
), for the both algorithms is de-
picted in Figure 4. It is observed, that TSA begins
with the solution that is nearly twice as large as T
c
,
however it relatively quickly overtakes GA (first in-
tersection of the graphs in Figure 4). This property
can be noted for all the instances consisting of the job
sizes 100 or less, in general. In the case of TSA, as
opposed to GA, it seems less difficult to determine
the optimal solution, but, on the other hand, signifi-
cantly more computation time is required. This is due
to the property of the slow movement across the ar-
guments area of the TSA based algorithm. It is to be
remarked, that the aforemention drawback of TSA is
not significant for the relatively small instances. It can
be expected that the GA would obtain the solution of
a similar quality as the TSA after a certain number
of iterations, which can be observed in Figure 5 (the
second intersection).
In the following investigation the impact is placed
to examine the two aforementioned intersection
PARALLEL MACHINE EARLINESS-TARDINESS SCHEDULING - Comparison of Two Metaheuristic Approaches
83
0 20 40 60 80 100
150
200
250
300
350
400
450
500
iteration
average Tc
tabu search
genetic algorithm
Figure 4: Objective function evolution.
points. In all the experiments conducted the three ar-
eas of the objective function evolution were observed:
1. predominance of GA in the first iterations (up to
the 1st intersection)
2. predominance of TSA for a given set of iterations
(up to the 2nd intersection)
3. predominance of GA again (after the 2nd intersec-
tion)
The purpose of the following investigation is to ana-
lyze the location of the both intersection points with
the accordance to the size of the particular instance.
Analogously to the results presented previously,
see Figures 4-5, the first intersection is observed rela-
tively quickly (it is a one of the first iterations) when
the instances smaller than 100 jobs and 10 machines
are considered. Therefore, the usage of the TSA has
proven to be a considerably better choice, when the
relatively short calculation time is to be expected. On
the other hand, in the cases when the location of the
optimum solution is of the prime interest, the GA al-
gorithm is the more appropriate choice. The radi-
cal changes are observed for the relatively large in-
stances of 100 jobs or more. In these cases, the TSA
is not able to move sufficiently fast across the argu-
ments area and, thus, the first intersection point, after
a larger number of iterations, is reached. The example
average objectivefunction evolution for 100 instances
of 200 jobs and 10 machines is presented in Table
2. It is noted, that with the growth of the instance
size, the corresponding time required for a comple-
tion of a single iteration increases, accordingly. Af-
ter 10 iterations it may be readily observed that the
TSA evolved from 59509 to 59137 with the decrease
of 0, 63%. On the other hand, the utilization of the GA
leads to the decrease from 55828 to 54381 at the ratio
of 2, 59%. The time required for carrying out the ex-
periment in the same configuration setup, for the time
0 50 100 150 200 250 300 350 400
150
200
250
300
350
400
450
500
iteration
average Tc
tabu search
genetic algorithm
Figure 5: Objective function evolution.
0
2
4
6
8
10
10
20
30
40
50
0
0.5
1
1.5
2
machines no.
jobs no.
F
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Figure 6: Effectiveness proportion of GA to TSA for differ-
ent instance sizes.
long enough to reach the first intersection point, was
approximately calculated to be few hours.
Table 2: Objective function evolution.
Iteration T
c
(TSA) T
c
(GA)
1 59509 55828
2 59470 55016
3 59396 54859
4 59346 54758
5 59315 54670
6 59271 54539
7 59243 54529
8 59210 54455
9 59174 54410
10 59137 54381
The further work will comprise the investigation of
the three-dimensional graphs to illustrate the compar-
ison results as a function of the both attributes of the
ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics
84
1
2
3
4
5
6
7
8
9
10
0
20
40
60
0
10
20
30
40
50
60
70
machines no.
jobs no.
iteration
0
10
20
30
40
50
60
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 25 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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