Three Genetic Algorithm Approaches to the Unrelated Parallel
Machine Scheduling Problem with Limited Human Resources
Fulvio Antonio Cappadonna
1
, Antonio Costa
2
and Sergio Fichera
2
1
Dipartimento di Ingegneria Elettrica, Elettronica ed Informatica, University of Catania, Viale A. Doria 6, Catania, Italy
2
Dipartimento di Ingegneria Industriale e Meccanica, University of Catania, Viale A. Doria 6, Catania, Italy
Keywords: Scheduling, Parallel Machines, Human Resources, Makespan, Genetic Algorithms.
Abstract: This paper addresses the unrelated parallel machine scheduling problem with limited and differently-skilled
human resources. Firstly, the formulation of a Mixed Integer Linear Programming (MILP) model for
solving the problem is provided. Then, three proper Genetic Algorithms (GAs) are presented, aiming to
cope with larger sized issues. Numerical experiments put in evidence how all GAs proposed are able to
approach the global optimum given by MILP model for small-sized instances. Moreover, a statistical
comparison among proposed meta-heuristics algorithms is performed with reference to larger problems.
1 INTRODUCTION
The parallel machine production system is a very
common environment that can be found in many
manufacturing situations. In the parallel machine
scheduling problem, a set of n jobs has to be
processed by only one out of m machines in parallel;
minimizing makespan in such a system is a NP-hard
problem as demonstrated by Garey and Johnson
(1979). In the more general case of unrelated parallel
machines, the processing time of each job depends
on the machine it is assigned to, as workstations are
supposed to be non-identical. The unrelated parallel
machine system has been widely addressed in
literature in the past few years; many techniques
have been proposed for the resolution of this
problem. In Kim et al. (2002) a Simulated Annealing
approach is presented for the unrelated parallel
machine problem with sequence dependent setup
times. Ghirardi and Potts (2005) developed a
Recovering Beam search Algorithm for minimizing
makespan in an unrelated parallel machine system
within a polynomial time, also in case of very large
instances. Recently, Vallada and Ruiz (2011)
developed a genetic algorithm for solving the
makespan minimization problem within an unrelated
parallel machine production system with sequence
dependent setup times.
Although in the last decades a number of studies
have been presented with reference to the unrelated
parallel machine issue, the effect of the human factor
on this scheduling problem has not been properly
addressed yet. Nevertheless, the impact of workforce
on the performance of production systems has been
widely discussed in the scheduling literature.
Norman et al. (2002) considered the problem of
assigning workers to manufacturing cells in order to
maximize the effectiveness of the organization. In
Celano et al. (2008) a first approach for solving the
scheduling problem of unrelated parallel
manufacturing cells with limited human resource is
given, through the development of an integrated
simulation framework that studies the effect brought
on system performances by the variation of workers
employed within the production shop.
In this paper, an unrelated parallel machine
problem with limited and differently-skilled human
resources is addressed with reference to the
makespan minimization objective. To this aim, a
Mixed-Integer Linear Programming Model (MILP)
and three different Genetic Algorithms (GAs) are
proposed.
The remainder of the paper is organized as
follows. In Section 2 the problem statement is
reported. In Section 3 the description of a MILP
model able to optimally solve small instances of the
aforementioned problem is given. Section 4
illustrates the genetic algorithms developed. In
Section 5 obtained results are discussed. Finally,
Section 6 concludes the paper.
170
Cappadonna F., Costa A. and Fichera S..
Three Genetic Algorithm Approaches to the Unrelated Parallel Machine Scheduling Problem with Limited Human Resources.
DOI: 10.5220/0004116501700175
In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 170-175
ISBN: 978-989-8565-33-4
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
2 PROBLEM STATEMENT
The proposed unrelated parallel machine problem
can be stated as follows. Let us consider a set N of n
jobs that has to be worked in a production stage
made of a set M of m parallel machines, aiming at
the minimization of the total completion time, i.e.,
makespan. Each job has to be processed by only one
machine before the exit from the system is allowed.
Setup operations performed on a given workstation
by a single worker must precede each job processing
on the same workstation. A team W of w workers is
assumed to be committed to these operations being,
in general, w
m ; this means that operators
represent a critical resource. In addition, each
worker is featured by a certain skill level, on the
basis of which he is able to perform setup operations
slower or faster than his colleagues.
After setup operation, the job remains on a given
machine until its own processing has been
completed, as pre-emption is not allowed. Setup
times are assumed to be sequence-independent;
nevertheless, being the machines unrelated and the
workers differently-skilled, setup times depend both
on the worker selected for performing setup
operations and the machine actually processing the
job. In addition, processing times depend also on the
machines jobs are assigned to.
3 MILP MODEL
A first goal of the proposed research has consisted in
the development of a Mixed Integer Linear
Programming (MILP) model, aiming to both
optimally solve a set of small instances of the
problem in hand and validate performances of the
provided GAs. In the following, mathematical
formulation is reported.
Indices
,1,2,,
j
ln
jobs
1, 2, ,im
machines
1, 2, ,kw
workers
Parameters
ij
T
processing time of job j on machine i
ikj
S
setup time of job j performed by worker k on
machine i
M a big number
Binary variables
ikj
X
1 if job is processed on machine with
setup performed by worker
0 otherwise
ji
k
j
l
Q
auxiliary variable for either-or constraint
Continuous variables
j
CS
setup completion time of job j
j
C
completion time of job j
max
C
makespan
Model
minimize
max
C
subject to:
11
1
mw
ikj
ik
X
j
(1)
11
mw
j
ikj ikj
ik
CS X S
j
(2)
11
mw
j
jijikj
ik
CCS TX
j
(3)
11
11
(2 ( ) )
(2 ( ) 1 )
ww
j l ikj ikj ikj ikl jl
kk
ww
l j ikl ikl ikj ikl jl
kk
CS C X S M X X Q
CS C X S M X X Q
(4)
,,; ijl j l
11
11
(2 ( ) )
(2 ( ) 1 )
mm
j l ikj ikj ikj ikl jl
ii
mm
l j ikl ikl ikj ikl jl
ii
CS CS X S M X X Q
CS CS X S M X X Q
(5)
,,; kjlj l
max
j
CC j
(6)
0;1
ikj
X ,,ik j
(7)
0;1
jl
Q
,jl
(8)
Constraint (1) ensures that each job is assigned to
one and only one machine, and that its setup is
performed by one and only one worker. Constraint
(2) forces the setup completion time of each job to
be equal or greater than the actual setup time of the
job itself. Constraint (3) states that, after setup
completion, the processing time must elapse before
the job can be considered definitively completed.
Through the couple of constraints (4) is imposed
that, if two jobs are assigned to the same machine,
ThreeGeneticAlgorithmApproachestotheUnrelatedParallelMachineSchedulingProblemwithLimitedHuman
Resources
171
no overlap between that is allowed (i.e., one job
must be completed before setup of the other one is
started, or vice versa). Similarly, constraints (5)
avoids overlapping of setup operations performed by
the same worker (i.e., setup of one job must be
completed before setup of the other one is started, or
vice versa). Constraint (6) forces the makespan to be
equal or greater than the completion time of each
job. Finally, through constraints (8) and (9), the
binary variables are defined.
4 PROPOSED GENETIC
ALGORITHMS
The MILP model proposed is able to optimally solve
small instances of the problem at issue.
Nevertheless, it cannot be employed for larger
examples, because of the high computational burden
required. In order to solve a set of large-sized
instances of the aforementioned problem, three
different meta-heuristic procedures based on genetic
algorithms (GAs) have thus been developed. In the
following subsections, a detailed description of
proposed GAs is reported.
4.1 Permutation-based GA
A first approach towards the resolution of the
aforementioned problem by means of GAs consisted
in the development of a genetic algorithm equipped
with a permutation-based encoding scheme,
hereinafter PGA. In such procedure, each
chromosome directly describes the order in which
jobs have to be processed in the manufacturing
stage, while the assignment of jobs to machines and
workers is performed by the decoding procedure, on
the basis of a time-saving rule. Below, a detailed
description of such algorithm is provided.
4.1.1 Encoding/Decoding Scheme
In PGA, each solution is represented by a
permutation
of n elements, where n is the number
of jobs to be scheduled in the manufacturing stage.
More in detail, let
(l) be the job on the l-th position
of the considered permutation (l=1,2,…,n) to be
scheduled on an unrelated parallel machine
production system with
m machines and w workers
(
w m ); S
ik
(l)
denotes the time required by worker k
(
k=1,2,…,w) to perform setup of job
(l) on machine
i (i=1,2,…m), while T
i
(l)
indicates processing time
of job
(l) on machine i. The decoding procedure
considers jobs in the order they appear in the
permutation and assigns them to the couple
machine-worker that can complete them earlier than
any other. Thus, indicating with
TM
i
the time at wich
machine i is ready to accept a new job, and with TW
k
the time at which worker k is ready to start a new
setup operation after all jobs preceding
(l) in the
permutation have been scheduled, the completion
time
C
(l)
is calculated as follows:
C
(l)
=
()
,
min
ik l
ik
E
(9)
where E
ik
(l)
indicates the estimated completion time
of job
(l) if processed on machine i with setup
performed by worker
k, calculated according to the
following formula:
() () ()
max ;
ik l i k ik l i l
ETMTWST
(10)
Then, denoting with
i* and k* respectively, the
machine and the worker to which job
(l) is assigned
(i.e., those minimizing E
ik
(l)
), quantities TM
i*
and
TW
k*
are updated as follows:
TM
i*
= C
(
l
)
(11)
TW
k*
=C
(l)
- T
i*
(l)
(12)
Lastly, after the aforementioned procedure has been
performed for all jobs in the permutation, the
makespan is calculated according to the following
formula:
max ( )
max
l
l
CC
(13)
4.1.2 Selection, Crossover and Mutation
Operators
With regards to selection mechanism, the well-
known roulette-wheel scheme (Michalewicz, 1994)
has been adopted, assigning to each solution a
probability of being selected inversely proportional
to makespan value. A position-based crossover
(Syswerda, 1991) has been employed for generating
new offspring from a couple of selected parents.
With reference to mutation procedure, a simple swap
operator (Oliver, 1987) has been chosen. The
algorithm has also been equipped with an elitist
procedure which copies the best two individuals of
each generation into the new population. Finally, a
total number of makespan evaluations has been set
as stopping criterion for the algorithm.
4.2 Multi-encoding GA
The proposed PGA allows considerably limited
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
172
computational burdens, because of the very simple
encoding exploited. Nevertheless such algorithm
moves over a space which cannot embrace the
entirety of solutions, as the decoding procedure may
define, for each permutation, one and one only
scheme regarding assignment of jobs to machines
and workers. A second approach towards the
resolution of the proposed problem by means of
GAs, consisted thus in the development of a genetic
algorithm, hereinafter MGA, equipped with a multi-
encoding scheme able to describe a wider solution
space compared to PGA. In such procedure, two
arrays for driving the assignment of jobs to
machines and workers, respectively, are added to job
permutation in the chromosome structure. Below, a
detailed description of such algorithm is provided.
4.2.1 Encoding/Decoding Scheme
In order to illustrate the encoding procedure
exploited by MGA, let us use the same nomenclature
defined for PGA. Thus, assuming to have
n jobs to
be scheduled on an unrelated parallel machine
system with
m workstations and w workers (w m
),
each chromosome is represented by the following
substrings:
a permutation
’
of n elements;
an array
’’ of n integers ranging from 0 to m,
driving the assignment of jobs to machines;
an array
’’’ of n integers ranging from 0 to w,
driving the assignment of jobs to workers.
In order to introduce the decoding procedure, let
’(l) be the job on the l-th position of the
permutation
’ (l=1,2,…,n); i
indicates the element
at position
’(l) of array
’’; k
indicates the element
at position
’(l) of array
’’’. S
ik
'(l)
denotes the time
required by worker
k (k=1,2,…,w) to perform setup
of job
’(l) on machine i (i=1,2,…m) while T
i
'(l)
indicates processing time of job
’(l) on machine i.
TM
i
and TW
k
denote times at which machine i and
worker
k, respectively, are ready to start a new setup
operation after all jobs preceding
’(l) in the
permutation have been scheduled.
The decoding procedure considers jobs in the
order they appear in permutation
’ and uses
information from arrays
’’ and
’’’ to perform the
assignment of jobs to machines and workers; if no
information is given by one or both arrays (i.e. if
i
=0 and/or k
=0), the same time-saving rule of
PGA is used. Hence, completion time
C
’(l)
is
calculated as follows:
C
’(l)
=
'( )
'( )
'( )
'( )
,
if 0 and 0
min if 0 and 0
min if 0 and 0
min if 0 and 0
ik l
ik l
k
ik l
i
ik l
ik
Eik
Eik
Eik
Eik
(14)
where E
ik
’(l)
indicates the estimated completion time
of job
’(l) if processed on machine i with setup
performed by worker k, calculated according to the
following formula:
'( ) '( ) '( )
max ;
ik l i k ik l i l
ETMTWST
(15)
According to such decoding procedure, the job is
assigned to machine
i
if
0i
, and to worker k
if
0k
; if not obtainable from arrays
’’ and
’’’,
machine and worker for processing job
’(l) are
chosen as those minimizing the estimated
completion time according to formula (14). Thus,
denoting with
i* and k* respectively, the machine
and the worker to which job
’(l) is assigned,
quantities
TM
i*
and TW
k*
are updated as follows:
TM
i*
= C
'
(
l
)
(16)
TW
k*
=C
'(l)
- T
i*
'(l)
(17)
Lastly, after the aforementioned procedure has been
performed for all jobs, the makespan is calculated
according to the following formula:
max '( )
max
l
l
CC
(18)
4.2.2 Selection, Crossover and Mutation
Operators
The same roulette-wheel mechanism exploited in
PGA has been adopted for selecting chromosomes,
assigning to each solution a probability of being
selected inversely proportional to makespan value.
Crossover procedure has been performed by
separately managing the mating between the three
parts of the parent structures (i.e., permutation
substrings, machine assignment arrays, worker
assignment arrays), with three distinct probabilities
p
cross
’, p
cross
’’, p
cross
’’’. Crossover between
permutation substrings of two parents has been
executed through a position-based operator as in
PGA. With reference to arrays driving the
assignment of jobs to machines and workers, a
simple uniform crossover operator (Syswerda, 1989)
has been employed. Similarly to crossover, mutation
procedure has been performed by separately
managing the three parts of the chromosome, using
ThreeGeneticAlgorithmApproachestotheUnrelatedParallelMachineSchedulingProblemwithLimitedHuman
Resources
173
three distinct probabilities p
mut
’, p
mut
’’, p
mut
’’’.
Mutation of the permutational substring of
chromosomes has been performed through the same
swap operator exploited in PGA. With reference to
assignment arrays, a simple uniform mutation
operator (Michalewicz, 1994) has been adopted. The
elitist procedure employed in PGA has been used as
well. Lastly, the same criterion of PGA has been
chosen for stopping the algorithm, i.e. the total
number of makespan evaluations.
4.3 Hybrid GA
The last approach for solving the proposed problem
through the employment of proper GAs consisted in
the development of a hybrid genetic algorithm,
hereinafter HGA, combining both the
aforementioned meta-heuristics. In such technique, a
first optimization phase is performed by PGA; then,
after a proper encoding conversion procedure is
executed, MGA is launched to complete the second
part of the algorithm. Through this method, the
space of solutions is quickly probed into as first, by
means of the “smart encoding” adopted by PGA;
then, a refined research is executed by MGA,
equipped with a more accurate encoding scheme.
The encoding conversion procedure occurs when a
fixed percentage of the total number of makespan
evaluations has been reached by PGA. It operates by
adding two assignment arrays to all chromosomes of
the last population obtained.
5 NUMERICAL EXAMPLES AND
COMPUTATIONAL RESULTS
In order to assess the performances of proposed GAs
in solving the unrelated parallel machine problem
with limited and differently-skilled human
resources, a comparison between the proposed meta-
heuristics and the MILP model developed has been
performed on a benchmark of small-sized test cases.
A total of 8 classes of problems have been generated
by combining the following factors:
number of jobs (
n): 2 levels (8, 10);
number of machines (
m): 2 levels (4, 5);
number of workers (
w): 2 levels (2, 3).
For each class, 10 instances have been generated
letting vary, with uniform distribution, processing
times in the range [1, 99] and setup times in the
range [1, 49]. Thus, a total of 80 problems has been
created. For each problem, the global optimum has
been found through the resolution of the MILP
model executed on a IBM ILOG CPLEX®
Vers.12.2 (64 bit) platform. Then, the whole set of
instances has been solved by the proposed GAs, with
all parameters tuned after a proper calibration phase
and termination criterion set at 10,000 makespan
evaluations. The Relative Percentage Deviation
(
RPD) from the global optimum has been computed
for each problem, according to the following
expression:
GA BEST
100
BEST
s
ol sol
sol
RPD
(19)
where BEST
sol
is the global optimum obtained
through the resolution of the mathematical
programming model, and GA
sol
is the best solution
provided by a given genetic algorithm after the
stopping criterion is reached. Table 1 shows average
RPDs obtained, grouping results by number
n of
jobs. Results show how all proposed GAs are able to
closely approach the global optimum with a limited
computational burden, as the amount of time
required by all meta-heuristics for solving a given
problem is, on, average, lower than 4 seconds.
Table 1: Average performances of GAs on small test
cases.
Number of jobs
(n)
Average RPD
PGA MGA HGA
8
4.368 2.532 3.676
10
3.883 3.416 3.204
Average
4.126
2.974 3.440
After having validated the performances of
proposed GAs, a wider set of large-size instances
has been created in order to carry out a comparison
among the three methods proposed. To this end, 36
new classes of problems have been generated by
combining the following factors:
number of jobs (
n): 4 levels (20, 40, 60, 100);
number of machines (
m): 3 levels (10, 15, 20);
number of workers (
w): 3 levels (5, 8, 10).
For each class, 10 problems have been generated
letting processing time vary in the range [1, 99] and
setup times in the range [1, 49]. Thus, a total of 360
problems has been created. All problems have been
solved five times by each GA. The performance
index chosen was the same RPD reported in
equation (19), considering as BEST
sol
the best
solution obtained by GAs for a given problem;
results obtained are reported in Table 2.
In order to infer some conclusion over the
statistical significance of differences between
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
174
performances of meta-heuristics proposed, an
analysis of variance (ANOVA) (Mongomery, 2007)
has been performed. Figure 1 shows the LSD
intervals (=0.05) regarding RPDs obtained by each
algorithm. It can be seen how HGA outperforms the
other meta-heuristics on large size instances, with a
statistically significant difference.
Table 2: Average performances of GAs on large test cases.
Number of jobs
(n)
Average RPD
PGA MGA HGA
20
2.015 2.395 1.980
40
3.952 3.914 3.534
60
3.141 4.000 2.534
100
2.809 2.372 2.217
Average
2.979
3.170 2.566
Figure 1: LSD plot for proposed GAs.
6 CONCLUSIONS
In this paper, the unrelated parallel machines
scheduling problem with limited and differently-
skilled human resources has been addressed with
regards to the makespan minimization objective. As
first, a MILP model has been developed, in order to
assess the performances of three Genetic Algorithms
(GAs) properly developed for the problem at issue.
Then, these latter procedures have been tested on a
wider set of large-sized instances, in order to carry
out a comparison among them. Results obtained
show how an hybrid approach, which combines two
GAs exploiting different encodings, outperforms the
single-encoding algorithms from which it is derived.
Statistical analysis confirms the significance of
difference between performances obtained, thus
giving evidence of the effectiveness of the proposed
hybrid procedure.
Further research should involve the consideration
of sequence-dependent setup times of jobs to be
scheduled in the manufacturing system.
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