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Contents lists available at ScienceDirect
Journal ofManufacturing Systems
j ournal homepage : www.elsevier .com/ locate / jmansys
Multi-objective optimization ofparallel machine scheduling
integrated with multi-resources preventive maintenance planning
Shijin Wang , Ming Liu
Department ofManagement Science and Engineering, School of Economics& Management, Tongji University, Shanghai, PR China
a r t i c l e i n f o
Article history:
Received 10 March 2015
Received in revised form 28 May 2015
Accepted 11 July 2015
Available online xxx
Keywords:
Multi-objective optimization
Production scheduling
Preventive maintenance
NSGA-II
Multi-resource maintenance
a b s t r a c t
Many studies on the integration optimization ofproduction scheduling and preventive maintenance usu-
allyonly consider one resource, i.e., machine. However, in real-world manufacturing, multiple dependent
resources (e.g., human, tools and machines) are needed at the same time to avoid mismatch of multi-
resource usage, which makes it highly important tojointly schedule production and maintenance tasks of
multiple resources in order to improve system availability and system throughput simultaneously. Inthis
paper, a multi-objective parallel machine scheduling problem with two kinds ofresources (machines and
moulds) and with flexible preventive maintenance activities on resources are investigated. The objective
isto simultaneously minimize the makespan for the production aspect, the unavailability ofthe machine
system, and the unavailability ofthe mould system for the maintenance aspect. A multi-objective inte-
grated optimization method with NSGA-II adaption is proposed to solve this problem. The extensive
computational experiments are conducted. The results show that the integrated optimization method of
production scheduling and preventive maintenance outperforms the method with periodic preventive
maintenance for this problem, in terms ofmulti-objective metrics, and the results also show the effects
ofdifferent flexibilities ofresources for job processing.
2015 The Society ofManufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In the literature on production scheduling, most studies assume
that the resources (e.g., machines, tools, people) are always avail-
able. However, in the real-worldmanufacturingor service industry,
resource unavailability including breakdown, failure and inspec-
tion, often occurs, which interrupts the current production or
service. Hence, scheduling problems integrated with preventive
maintenance (PM) on resourceshave been received more andmore
attention [18].
However, most studies focus on single-resource (i.e., machine)
maintenance during production scheduling, which may not be
sufficient to improve production system reliability as a whole
[9], because in a real manufacturing or service system, pro-
duction or service usually involves several important resources
simultaneously [10,11]. For example, in a flexible manufacturing
system or cell, typically, machines and corresponding tools should
work together to process certain job [12]; in plastic production,
injection machines and matched injection moulds should work
together [9,10]; in real-world Dual Resource Constrained (DRC)
Corresponding author. Tel.: +86 15026613178.
E-mail address: [email protected](S. Wang).
manufacturing systems, both machine and human resources are
critical [13]. All these resources (machine, tool, mould) are subject
to deterioration and need PM activities to restore the working con-
ditions. For the human resource, leisure time or vacation are also
necessary. The introduction of the match requirements between
the resources, and the PM planning of multiple resources further
make the integrated optimization problem more complicated, but
more practical relevance.
In this paper, motivated by a problem from a car-component
manufacturing shop floor with more than 10 machines and mul-
tiple moulds, we investigate a multi-objective parallel machine
scheduling problem with two kinds of resources and with flexi-
ble preventive maintenance activities on resources. One resource
is machine, and another resource is mould (or tool, or people) that
is associated with the machine. There are m parallel machines and
moparallel moulds. Each jobcan only beperformed on onemachine
with one mould. Fullflexibility and partial flexibilityof resource eli-
gibility for job processing are considered. Both of these two kinds
of resources are subject to random failure with the time to fail-
ure (resp. time to repair) subject to exponential distributions. The
objective is to simultaneously minimize the makespan for the pro-
duction aspect, the unavailability of the machine system, and the
unavailability of the mould system for the maintenance aspect.
To the best of our knowledge, such a multi-objective scheduling
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problem with two kinds of resources and with flexible preventive
maintenance activities on resources has not been documented in
the literature. The application of the NSGA-II (Non-dominatedSort-
ing Genetic Algorithm version 2) for the problem is also realized.
The paper is organized as follows. A literature review is
presented in Section 2 and then Section 3 gives the detailed
description of the multi-objective integratedoptimization problem
of production scheduling with two kinds of resources and with PM
activities, the mathematical formulation is also given. In Section 4,
an adapted NSGA-II with implementation details is described. In
Section 5, the results of computational experiments with compar-
isons are reported. Finally, Section 6 concludes the paper andgives
future research.
2. Literature review
Recently, realizing the inherent conflicts between production
andmaintenance,more andmore researchesemphasize on produc-
tion scheduling integrated with maintenance planning. In general,
two types of maintenance activities are included in the integrated
problem: fixed and flexible. The former is performed periodically
with a fixed time interval, see [14] and [5] for a relative compre-
hensive overview. While in the latter type, maintenance intervals
orthe starting time of intervalsare supposedto beflexible andmust
be determined during the process of production scheduling.
Some researchers attempted to consider the flexible main-
tenance in production scheduling with different optimization
objectives in different manufacturing shop floors. Qi et al. [15]
studied the problem of simultaneouslyscheduling jobsand mainte-
nance tasks on a singlemachine to minimize thesum of completion
times. The problem is proven to be NP-hard, and heuristics and
a branch and bound (B & B) method are proposed. Cassady and
Kutanoglu [16,17] proposed an integrated model for a single
machine with time to failure subject to a Weibull distribution,
to minimize the total weighted tardiness of jobs and the total
weighted completion time, separately. Kubzin and Strusevich [2]
studied both a two-machine open shop problem and a two-machine flow shop problem with flexible maintenance activities
to minimize the makespan. Ruiz et al. [3] considered the integrated
scheduling problem with different preventive maintenance poli-
cies in regular flow shops to minimize makespan. Naderi et al.
[18] investigated a job shop scheduling problem with sequence-
dependent setup times and maintenance activities to minimize
the makespan. Sun and Li [19] studied the scheduling problems
with multiple maintenances on two identical parallel machines
to minimize the makespan and total completion time, separately.
Naderi et al. [20] investigated a flexible flow shop schedul-
ing problem with periodic preventive maintenance to minimize
makespan. Rustogi and Strusevich [21] presented polynomial-time
algorithms for single machine problems with generalized posi-
tional deterioration effects and imperfect machine maintenance tominimize the makespan. Dalfard and Mohammadi [22] discussed a
multi-objective flexible job shop scheduling problem (FJSP) with
maintenance, in which three objectives are equally treated and
weighted into one. Two meta-heuristic algorithms, a genetic algo-
rithm (GA) and a simulated annealing (SA) are proposed. Dong
[23] studied a parallel machine scheduling problem with flexi-
ble maintenance activity to minimize the total cost involved with
the completion time and the unavailable time. A B & B method is
proposed. Nouri et al. [24] investigated a non-permutation flow
shop scheduling problem with flexible maintenance activities to
minimize the sum of tardiness costs and maintenance costs. A SA
based heuristic is employed. Sarkar et al. [25] studied a job shop
scheduling problem with maintenance activities to minimize the
makespan. A hybrid evolutionary algorithm is developed.
While all these works provide a strong basis for further work,
it was observed that the integrated problems have been treated
as single-objective optimization problems. Since production and
maintenance must collaborate to achieve the common goal of
productivity maximization, both objectives of maintenance and
production are suggested to be considered with the same impor-
tance level [6]. Proper integrated scheduling can provide an
effective means to tradeoff between objectives related to the pro-
duction scheduling and maintenance aspects.
In the following, we review the researches of bi-objective or
multi-objectiveoptimization of productionscheduling and preven-
tive maintenance in different settings of machine environments.
For the single machine, Jin et al. [26] extended the model in [17]
to a multi-objective optimization problem to minimize the mainte-
nance cost,makespan,total weighted completion time of jobs,total
weighted tardiness and machine unavailability. A multi-objective
genetic algorithm (MOGA) is proposed.
For the parallel machine, Berrichi et al. [27] studied a schedul-
ing problem with PM activities to minimize the makespan and the
system unavailability simultaneously. Two multi-objective genetic
algorithms, NSGA-II and WSGA (Weighted Sum Genetic Algorithm)
are employed. In their later work, Berrichi et al. [6] proposed
a multi-objective ant colony optimization (MOACO) to solve the
same problem. The performance of the proposed MOACO is com-
pared with those of the well-known SPEA 2 (Strength Pareto
Evolutionary Algorithm version 2) and NSGA-II. In their further
work, Berrichi and Yalaoui [7] considered the similar problem in
which two objectives of the total tardiness and the unavailabil-
ity of the production system are included. A multi-objective ant
colony optimization approach is proposed. Moradi and Zandieh
[28] introduced a similarity-based subpopulation genetic algo-
rithm (SBSPGA) to solve the same problem. The performance of the
algorithm is compared with those of two other evolutionary algo-
rithms. Ben Ali et al. [29] studied a scheduling problem integrated
with preventive maintenance tasks that should be executed in a
tolerance interval. Two objectives: the makespan and the main-
tenance cost are to be minimized simultaneously. An MOGA is
proposed. Rebai et al. [30] considered a multi-objective parallelmachine schedulingproblem withthe requirement of maintenance
once on each machine, to minimize the total sum of the jobs
weighted completion times and the preventive maintenance cost.
A heuristic method with two-phases is proposed.
There arealso a handful of researchesin other complicated envi-
ronments. Moradi et al. [31] investigated a bi-objective FJSP with
PM activities to minimize the makespan and the system unavail-
ability simultaneously. Four multi-objective optimization methods
are usedto solve the problem.Li and Pan [32] proposed an effective
discrete chemical-reaction optimization (DCRO) algorithm to solve
the multi-objective FJSP with maintenance activity constraints.
Later, they proposed a novel discrete artificial bee colony (DABC)
algorithm for the same problem [33]. Xiong et al. [34] studied a
FJSP with random machine breakdowns, in terms of bi-objectivesof makespan and robustness. An evolutionary algorithm based
on the NSGA-II is proposed to solve the problem. Lei [35] stud-
ied an interval job shop scheduling problem with non-resumable
jobs and flexible maintenance to minimize interval makespan
and total interval tardiness. An effective multi-objective artificial
bee colony (MOABC) is proposed. Azadeh et al. [36] considered
a multi-objective open shop scheduling problem with preventive
maintenance and they applied the NSGA-II and a multi-objective
particle swarm optimization (MOPSO) to solve the problem.
From the above review, it was observed that the integrated
scheduling with PM activities has been conducted on one main
resource, i.e., machine. Till date, it is surprising that despite
the fruitful results of researches on multi-objective production
scheduling integrated with PM activities as mentioned above, only
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few researches are available on production scheduling integrated
with maintenanceon multiple resources. Forthe first time in thelit-
erature, Wong et al. [911] consider the joint production schedule
with the PM planning of injection machines and injection moulds
to minimize the makespan in the context the plastics production
systems. However, only one objective related to production aspect
is considered.
In general, if the production is continuously proceeded, the
availability or reliability of the machine will be decreased. In
the contrary, although a preventive maintenance consumes the
production time, it can increase or at least keep the availabil-
ity or reliability of the machine. To some extent, production and
maintenance objectives are conflicting. Hence, a bi-objective or
multi-objective of multi-resource production scheduling problem
with the PM planning is more practical relevant and thus more
important.
In this paper, motivated by a real-world case, and inspired from
the work of Wong et al. [9] and Berrichi et al. [6], we investigate
a multi-objective parallel machine scheduling problem with two
kinds of resources (machines and moulds) and with flexible PM
activities on resources. On the one hand, the work in this paper
extends the work of[9] into a multi-objective optimization, and on
the other hand, it extends the match relation between machines
and moulds into full and partial flexibility. In Wong et al. [9],
a mould for a job is specific beforehand while in this work full
and partial flexibility are allowable for machines and moulds. In
addition, in Berrichi et al. [6], the exponential distribution of main-
tenance feature of machines is assumed, this paper also employs
this assumption but extends the single resource into two kinds of
resources (machines and moulds), in which the match relationship
between machines and moulds further complicates the problem.To
the best of our knowledge, there is no such research documented
yet in the literature.
3. Problem formulation
3.1. Problem description
There are n jobs to be processed in a shop floor with m inde-
pendent parallel machines and mo moulds. Jobs are available at
the beginning of the production horizon and job preemption is not
allowed. Each job can only be performed on one machine with one
mould.
Let N= {1, . . ., n}, M= {1, . . ., m} and MO= {1, . . ., mo} be theset of jobs, machines and moulds, respectively. Each job i (iN)
can be processed on any alternative machine k (kM) with one
alternative mould l (lMO) with theprocessing timepikl.Ifeachjob
can be processed by any machine with any mould, we call this full
flexibility case. Forone job i, if alternative machines and alternative
moulds are only subset ofM (denoted by SubMi, SubMiM) and
subset ofMO (denoted bySubMOi, SubMOiMO), respectively, thenthis is the partial flexibility case.
With a particular mould loaded, a machine will perform as the
same processing time asothermachinesthatoperatewith thesame
mould, i.e.,pikl can be denoted aspil. But for keeping the machine
assignment information, we still usepikl instead ofpil in this paper.
Tables 1 and 2 show one example data, in which there are six jobs
that have to be processed in a shop floor with two machines and
two moulds. Each job has its batch size, and the total processing
time of the job equals the number of units multiplies by the unit
processing time. For example, job 1 can be processed on machine
1 or 2 with mould 1, orcan beprocessed on machine 1 with mould
2. If mould 1 is selected, the processing time is p1k1 = 220=40;
if mould 2 is selected, the processing time is p112 = 215=30. In
this paper, we assume that each job will be operated according
Table 1
Job data.
Job Units
1 2
2 1
3 3
4 1
5 2
6 3
to its batch size without splitting. The example is the case of full
flexibility of moulds and partial flexibility of machines.
For the production aspect, in order to obtain a schedule, we
should determine the assignment of jobs on machines, the assign-
ment of jobs on moulds, and the processing sequence of jobs on
each machine and on each mould.
Besides, in order to maintain its high availability, preventive
maintenance activities have to be conducted on each machine and
on each mould. The preventive maintenance of a machine intro-
duced by Berrichi et al. [6] is adapted in this paper. It is assumed
that the time to failure (resp. time to repair) of machine k (kM)
is a random variable with exponential probability distribution hav-
ing failure rateparameterk(resp. repair rateparameterk). Other
probability distributioncould also be considered. It is also assumed
that at time zero, the machine k is perfect and a PM restores the
machine back to an as good as new condition. By taking into
account these assumptions, from the initial instant, the point avail-
ability of machine k at time t is given by the following expression
[37,6], which represents the probability that the machine k is oper-
ating at time t:
Ak(t) =k
k + k+
kk + k
e(k+k)t (1)
IfTkis thecompletion time of themost recentPM activity on the
machine k, the expression of the availabilityAk(t) (t>Tk) isgivenby
the following expression [37,6], with the steady-state availability
k/(k +k) when t:
Ak(t) =k
k + k+
kk + k
e(k+k)(tTk) (2)
Since there arem independentparallelmachines,the machining
system availabilityAs(t) is given as follows:
As(t) = 1
mk=1
(1 Ak(t)) (3)
Hence, the unavailability of the machine system is
As(t) =
mk=1
(1 Ak(t)) (4)
For each machine, k
/(k
+k
) is the lower bound of availabil-
ity. we can get the upper bound of system unavailability As(t) =mk=1
(1 Ak(t)) mk=1k/(k + k). That is to say, 0 As(t) m
k=1k/(k + k).
Similarly, for the mould system, we also assume that the time
to failure (resp. time to repair) of mould l (lMO) is a randomvari-
able with exponential probability distribution having failure rate
Table 2
Unit processing time data.
Unit processing time
Mould M1 M2
1 20 20
2 15
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parameter ml
(resp. repair rate parameterml ). It is also assumed
that at time zero, themould l is perfect anda PM restores themould
back to an as good as new condition. Since moulds are indepen-
dent and parallel, the mould system unavailability Asmo(t) is alsoas
Asmo(t) =
mol=1
(1 AMl(t)) (5)
whereAMl(t) is the availability of mould l at time t.
Similarly, 0 Asmo(t) mol=1ml /(m
l + m
l ).
In this paper, we assume that the maintenance features of
moulds are independent of those of machines. It is also assumed
that the number of repair or maintenance staffs is sufficient to
ensure that repair and maintenance times of machines and moulds
are independent.
The integrated model considers three objectives simultane-
ously:the minimization of makespan forthe production aspect, the
minimum unavailability of the machine system, and the minimum
unavailability of the mould system for the maintenance aspect. We
assume that the maintenance activity is not preemptive by jobs
processing and vice versa.
To obtain a feasible schedule, there are five decisions should betaken jointly: the assignment of jobs on machines, the assignment
of jobson moulds,the processingsequenceof jobs,the maintenance
decision on machines after each job and the maintenance decision
on moulds after each job.
Let Ci be the completion time of job i and Cmax the completion
time of the last job (i.e., makespan). Then we have
Cmax = max{Ci}, i = 1,2, . . . , n (6)
Let D1 = {0, t1, t2, . . ., tr1, Cmax}, where ti (i= 1 , 2, . . ., r1) are thestarting times of all PM actions on all machines, and r1 is the total
number of PM actions on machines. r1 is not determined before-
hand andneededto be determined with the production scheduling
together. Since the unavailability is an increasing function in each
interval [ti, ti+1], i= 0, 1, . . ., r1, with t0 = 0 and tr1+1 =Cmax, and as
assumed that a machine becomes as good as new at the end of
each PM action, the system unavailability is only computed at the
times t1, t2, . . ., tr1+1[6]. Then, the highest value found is taken asthe unavailability of the machine system.
Similarly, let D2 = {0, 1, 2, . . ., r2, Cmax} are the starting timesof all PM actions on all moulds, and r2 is the total number of PM
actions on moulds. The unavailability of the mould system can also
be determined by comparing the system unavailability at inter-
val [i, i+1], i= 0, 1, . . ., r2, with 0 = 0 and r2+1 =Cmax. In thispaper, we assume that the processing times of a PM action on a
machine, PTk, and ona mould,PTml , is the mean time of the preven-
tivemaintenance activity,i.e.,PTk = 1/k, kMandPTml = 1/
ml , l
MO, respectively.
Then, the three objective functions to be optimized are as fol-lows:
f1 = min{Cmax} (7)
f2 = min
maxtD1As(t)
(8)
f3 = min
maxD2Asmo()
(9)
3.2. Mathematical model of the problem
First, some indices, parameters and variables are defined.
Indices:
i,j: job indices;
k,w: machine indices;
l, o: mould index.
Parameters and variables:
L: a sufficiently large integer;
xik: 1 if job i is assigned on machine k; 0, otherwise.
yil
: 1 if job i is assigned on mould l; 0, otherwise.
uijk: 1 i f job i is sequenced before jobj on machine k; 0, otherwise.
vijl: 1 if job i is sequenced before jobj on mould l; 0, otherwise.zxik: 1 if a PM activity is executed after job i on machine k; 0,
otherwise.
zyil: 1 if a PM activity is executed after job i on mould l; 0, other-
wise.
Cik: job is completion time on machine k, equal to0 if job i is not
assigned to machine k;
Ci: job is completion time;
Then, the problem is subject to the following non-linear math-
ematical formulation constraints:
m
k=1
xik= 1, i N (10)
mol=1
yil = 1, i N (11)
(uijk + ujik) xik xjk = 1, i, j N, i /= j,k M (12)
(vijl + vjil) yil yjl = 1, i, j N, i /= j,l MO (13)
Cik L xik, i N, k M (14)
Ci = maxkM
{Cik}, i N (15)
Ci +max{PTk zxik, PTml zyil} +pjkl xjk yjl Cj
+ L(1 uijk) i, j N, j /= i, k M, l MO (16)
tw = minj{Ci zxik Cj xjw zxjw zxjw PTw > 0}
i, j N, i /= j,k,w M (17)
to = minj{Ci zxil Cj yjo zyjo zyjo PT
mo > 0}
i, j N, i /=j,l, o MO (18)
xik, yil, zxik, zyil {0,1}, i N,k M, l MO (19)
uijk, vijl {0,1}, i, j N, i /=j, k M, l MO (20)
In this model, Eqs. (10) and (11) ensure that one job can be
only assigned on one machine and on one mould, respectively.Eqs. (12) and (13) ensure that job i is sequenced before job j, or
job j is sequenced before job i, on the same machine k and on the
same mould l, respectively. Inequality (14) defines job is comple-
tion time on machine k (i.e., the completion time on the assigned
mould). Constraint (15) is used to determine the completion time
of job i, which can be used to calculate the objective f1 via Eqs.
(6) and (7). Constraint (16) ensures that no two jobs i and j on
the same machine and on the same mould can overlap in time.
Constraint (17) obtains the time interval from the most recent PM
activity on the machinew Muntil the completion time of job i on
the machinek. Onlywhenzxik = 1, Ci zxik Cj xjw zxjw zxjw PTwwill be calculated and the minimum value is recorded ifCi zxik
Cj xjw zxjw zxjw PTw > 0 for the different jobj. The obtained tw
is t Tw on the machine w,w M in Eq. (2), which can be used
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Fig. 2. A schedule result of an example chromosome.
the alternative mould (say l) with the earliest completion
time, ECTli =min(CTMl +pikl), where CTMl is the available
time of alternative moulds before processing the job [i],
andpiklis the processing times of the job [i] on the alter-
native moulds. In the full flexibility case, l= 1 , 2, . . ., mo,while in the partial flexibility case, l is from one subset of
alternative moulds for this job (i.e., l SubMO[i] MO). If
there is a tie, themouldwith thesmaller index is selected.
(2) Then, w e select t he machine, s ay i, with the earliest com-
pletion time CTk before processing the job [i], since the
processing times on alternative machines aresame.In the
full flexibilitycase, k=1,2, . . .,m, while in thepartialflexi-
bility case,k is from thesubset of alternative machines for
this job (i.e., k SubM[i] M). Then, we can determine the
starttimeofthecurrentjob[i],S[i] =max(CTk,CTMl), which
is the maximum value of available times of the selected
machine and the selected mould.
(3) Then the job is labelled as assigned one, and its comple-
tion time C[i]can be calculated. The machine age after the
recent PM activity and the mould age after the recent PM
activity can be updated. Then according to the mainte-
nance strategy of machines and moulds, and the genes in
part B and C, the maintenance decision could be made.The process is repeated until all jobs are processed. Once
a schedule is obtained, three objectives f1, f2, and f3 can
be calculated.
According to the problem data in Tables 1 and 2, and the chro-
mosome in Fig.1, wecan followthe greedyruleto obtaina schedule,
which is shown in Fig. 2. In this example, k =0.1, k = 0.05, k= 1,2, and m
l = 0.2, m
l = 0.04, l = 1,2. The processing times of a PM
action on machines and moulds is the mean time of PM activity,
which are 10 and 5 time units, respectively. In the figure, the num-
ber in the job operation box is the job index. The implementation
process of the greedy rule is explained as follows:
(1) According to the chromosome, the first unassigned job isjob 2. Since at the beginning, min(CTMl +pikl) =min(0+20,
0 + 1 5), hence, mould 2 is selected to process job 2.
Also, we can easily determine machine 1 for job pro-
cessing. C[1] = S[1] +p2k2 =15. Then, job 2 is labelled as an
assigned job, and its starting time and completion time
are S2 = 0 and C2 = 15. Correspondingly, the completion
times of machine 1 and mould 2 are updated CT1 =15 and
CTM2 = 15. According to the chromosome, after this job,
there is a PM action on the mould assigned. After execut-
ing the PM action, the earliest available time of mould 2
is CTM2 =20.
(2) Then, the first unassigned job is 3.
min(CTMl +pikl) =min(0+60, 20+45)= 60. Hence, mould
1 is selected. The machine available time CT1 = 1 5 and
Fig. 3. An example of crossover operation.
CT2 =0, hence, machine 2 is selected. The starting time
of the job is S3 =max(CT2, CTM1) = 0 , and C3 =60. Cor-
respondingly, the completion times of machine 2 and
mould 1 are updated CT2 = 6 0 and CTM1 = 60. According
to the chromosome, after this job, there is a PM action on
the machine assigned. Then, theCT2 =60+10=70.
(3) Next, the first unassigned job is job 5.
min(CTMl +pikl) =min(60 +40, 20+ 30) = 50, hence, mould
2 is selected. Then, only machine 1 can be selected. The
starting time of the job is S5 =max(CT1, CTM2)= 20. And
C5 = 2 0 + 215= 50. Correspondingly, the completion
times of machine 1 and mould 2 are updated CT1 =50 and
CTM2 =50. According to the chromosome, after this job,
there is a PM action on the mould assigned. Then, after
the PM activity,CTM2 =50+5=55.
(4) Similarly, following the procedure mentioned above, the
remaining three jobs can be assigned, and the schedulingresults are shown in Fig. 2.
The makespan of the schedule is 125. D1 = {0, t1, t2, . . ., tr1,Cmax}= {0, 60, 100, 125}, andD2 = {0, 1, 2, . . ., r2, Cmax}= {0, 15,
50, 110, 125}.
Then we have
maxtD1
{As(t)} = max{A1(60)A2(60), A1(100)A2(30), A1(15)
A2(55)} = max{0.111,0.110,0.099} = 0.111And we also have
maxD2
{Asmo()} = max{AM1(15) AM2(15), AM1(50)
AM2(30), AM1(110)AM2(55), AM1(10) AM2(70)} =
max{0.026,0.028,0.028,0.025} = 0.028Hence, the objective values of the schedule associated with the
example chromosome can be represented by a three-tuple {125,
0.111, 0.028}.
4.3.2. Crossover
Single point crossover is considered, in which one crossover
point is selected, say 1 rn (r is an integer), till this point the
genes in part A is copied from the first parent, then the genes in
part A of the second parent is scanned and if the number is not yet
in the offspring it is added sequentially. The corresponding genes
in part B and part C are also exchanged. Suppose we have two par-
ent chromosomes for the example problem mentioned above, and
the crossover point r= 3, then two offsprings can be generated, as
shown in Fig. 3.
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Fig. 4. An example of mutation operation.
To generate the offsprings, two parents are first selected ran-
domly. Then a random value between 0 and 1 is generated, if this
value is less than the crossover probability pc, two offsprings are
generated with the proposed single point crossover operator, oth-
erwise, these two selected parents are used as offsprings.
4.3.3. Mutation
The mutation operator with swap and flit combination is
employed. First, two random numbers r1 and r2 (1 r1, r2 n, r1,
r2 are integers) are generated, the genes on the positions r1and r2in part A are swapped, and the corresponding genes on positions
r1 and r2 in part B and part C are also swapped. Then, a randomnumber rBC, 1 rBCn (rBCis an integer) is generated, and if the
genes on position rBC in part B and part C is 1, they are flitted into
0 and vice versa. An example of the mutation operator is as shown
in Fig. 4, in which r1 = 2, r2 = 5 and rBC= 4.It is possible that rBCmay
equal to r1or r2.
To execute mutation operator or not is dependent on the muta-
tion probabilitypm. For everychromosome obtainedafter crossover
operation, a random number between 0 and 1 is generated and
if the number is less than pm, the chromosome will be mutated,
otherwise, no mutation is executed on this chromosome.
5. Computational experiments
The NSGA-II algorithm has been implemented in MATLAB 7.1.The results described in the following have been obtained on a
personal computer with an Intel 2.10GHz CPU and 1.96GB RAM.
5.1. Comparisons with the periodic PMplanning
5.1.1. The setting of periodic PMplanning
For each machine, the mean time to failure (MTTF) is set as
the cycle of periodic PM planning. The MTTF can be determined
as follows [37]
TFk = MTTFk =
0
e(k)tdt=1
k(21)
where k is the failure rate of a machine. TFk is used as thefixed interval for PM planning. Since the job processing is non-preemptive, there may be conflicts between the job processing and
PM activities if the fixed interval is strictly followed. To avoid the
conflict, the PM activity is executed only when the job processing
is finished and when the time since the last PM activity reaches
or exceeds TFk. The time of a PM activity is 1/k. We also use the
similar periodic PM planning for each mould and the PM execution
interval is TFml , which is obtained using the failure rate of a mouldml
in Eq. (21).
To realize the periodic PM, we can update the genes in part B
(resp., part C) as 1 when the time since the last PM activity reaches
or exceeds TFk(resp., TFml ). That means, unlike in the original inte-
gratedmethod,thegenesinpartBandpartCofchromosomesinthe
periodic PM planning are constrained by TFkand TFm
l
, respectively.
5.1.2. Data generation
We generate 6 n-jobs, m-machines and mo-moulds problems
and each test problem is denoted by the triple (n, m, mo). The test
problems are (20, 2, 2), (20, 2, 4), (20, 4, 2), (50, 4, 4), (50, 4, 6) and
(50, 6,4). Thefull flexibility of machines andmoulds areconsidered,
i.e., each job can be processed by a combination of any machine
and any mould. The processing time of jobs on moulds is randomly
generated between 10 and 50 units of time (As assumed above, the
processing times of jobs with the same mould but on alternative
machines are same). The batch size of jobs is randomly generated
between 1 and 5 units. The failure rate of machines and moulds
is randomly selected from a set {0.0025, 0.002, 0.001} (i.e., MTTF
is selected from {1000, 500, 400}, which is larger than at least a
batch with the maximum processing times 505=250), and the
repairrate of machines and moulds is randomly selected from a set
{0.025, 0.05, 0.1}, which means the PM activity time is taken from
{10, 20, 40} time units. The reason of setting repair rate like this is
that usually the repair rate is larger than the failure rate such that
the inherent availability (an equipment design parameter) [37] as
shown in Eq. (22) is not too small:
Akinh =k
k + k(22)
5.1.3. Measure metrics
The following two metrics are used to compare the quality of
two non-dominated fronts A and B obtained by the integration
method and the method with the periodic PM planning. The first
one isCmetric, which is proposed in [41] and is a relative measure
which allows clearly differentiating two fronts. The value ofC(A, B)
represents the percentage of solutions in B dominated by at least
one solution ofA [6]. The computing formula is
C(A, B) =|{b B|a A : aPb}|
|B|(23)
where a Pb means that a dominates b. The closer the value ofC(A,
B) to 1 is, the better the front A compared to B is. Since this mea-
sure is not symmetrical, i.e., C(B,A) /= 1C(A, B), it is necessary tocalculate C(B,A) andA is better than B ifC(A, B) >C(B,A).
The second one is the D1Rmeasure [42,6], which can be used to
evaluate the distribution of frontA and the distance ofA to a refer-
ence front (i.e., the Pareto-optimal front or a near Pareto-optimal
front). Let S* be the reference solution set. If the Pareto front is not
known, the two (or more if more than two methods used) fronts
are combined and all the non-dominated solutions are selected to
form the set S*. The D1R measure of a frontA is given by
D1R(A) =1
|S|
YS
min{dXY: X A} (24)
where dXYis the distance between a solution F(X) and a reference
solution F(Y) inp-dimensional normalized objective space (in thiswork,p= 3)
dXY=
(f
1(X) f
1(Y))
2+ .+ (fp (X) f
1
(Y))2
(25)
wherefi ( ) is the ith objective that is normalized using the refer-
ence solution set S*. That is,
fi ( ) =
fi( ) fmini (S)
fmaxi (S) fmini (S
) (26)
where fmini
(S) and fmaxi
(S) represent the minimum and maxi-
mum ith objective value in S*. The smaller the value ofD1R(A) is,
the better the frontA is.
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Table 11
Thepartial flexibility mould case forthe problem 60915.
Job Alternative moulds Job Alternative moulds
1 [1, 14, 13] 31 [14, 12, 7, 4, 8]
2 [4, 9, 8, 7, 2, 5, 11, 10, 12, 6, 3, 14, 15] 32 [5, 6, 2, 11, 12]
3 [12, 4, 9, 13, 10, 11, 1, 5, 15] 33 [11, 6, 15, 1, 13]
4 [6, 11, 9, 8, 10, 13, 7, 12, 3, 1, 5, 14] 34 [3, 2, 14, 5, 10, 9, 4, 11, 1, 7, 6, 8, 15]
5 [6, 10, 5, 15, 11, 8, 4, 3, 14, 9, 12, 1, 7, 2, 13] 35 [6, 1, 5]
6 [6, 2, 4, 10, 8, 3, 7, 15, 13] 36 [8, 6, 2, 13, 5, 3, 9, 11]
7 [10, 7, 6, 14, 9, 8, 4, 11, 13] 37 [14, 11, 7]8 [13, 15, 9, 1] 38 [15]
9 [1, 6, 14] 39 [9, 10, 3, 13, 6, 11]
10 [9, 14, 11, 8, 1] 40 [2, 13, 1, 8, 11, 3, 14, 6, 9]
11 [3, 11, 10] 41 [6, 4, 13, 2, 15, 12, 9, 14, 3]
12 [9, 12, 5, 15] 42 [10, 11, 6, 15, 12, 2, 14, 13, 8, 1, 3, 5]
13 [5, 14, 11, 13, 15, 4, 7] 43 [1, 5, 10, 6, 14, 9, 2, 8, 4, 13, 7]
14 [14, 11, 12, 4, 10, 8, 3] 44 [12, 3, 15, 14, 4, 8, 1, 6, 11, 13, 7, 5, 9, 2, 10]
15 [14, 5, 3, 11, 15, 8, 4, 9, 1, 2, 13, 6, 12] 45 [15, 6]
16 [7, 9, 2, 6, 13, 3, 5, 12, 14, 1, 8] 46 [11, 5, 10, 3, 14, 4, 9, 8, 7]
17 [1, 7, 13] 47 [7, 6, 10, 3, 12, 11, 4, 5, 15, 14, 8, 13]
18 [8, 11, 3, 7, 13, 5, 10, 6, 1, 2, 4, 15, 9, 14] 48 [8, 10, 13, 6, 12, 3, 11, 1, 7, 2, 4, 9, 15, 5]
19 [15, 4, 8, 2, 10, 5, 1, 13, 7] 49 [6, 10]
20 [9] 50 [12, 1, 3, 9, 4, 15, 13, 10, 7, 6, 8, 14, 5, 2, 11]
21 [8, 1, 9, 4, 13, 12, 10, 6, 3, 7, 5, 15, 11] 51 [8, 13, 7, 6, 15, 3, 11, 14, 9, 10, 2, 4, 1, 12, 5]
22 [15, 5, 8, 3, 13, 4, 11, 14] 52 [10]
23 [8, 7, 9, 3, 12, 11, 10, 15, 13, 14, 1, 4, 6] 53 [5, 3, 1]
24 [8, 14, 4, 2] 54 [10]
25 [2, 14, 12, 1, 13, 5, 7, 11] 55 [9, 6, 15, 13, 2, 7, 5, 10]
26 [15, 4, 7, 9, 6, 8, 3, 1, 14, 13, 10, 12, 11, 5] 56 [5, 8, 9, 7, 1]
27 [15, 14, 9, 6, 8, 10, 13, 1] 57 [3, 2, 5, 10, 14, 6, 4, 9, 1, 12, 11, 13, 15, 7]
28 [5, 8, 11, 1, 3, 2, 12] 58 [2, 8, 11]
29 [6, 11, 1, 8, 10, 3, 12, 7] 59 [2, 7, 14, 15, 4, 13, 1, 5]
30 [2, 14] 60 [12, 15, 1, 3, 10]
Table 12
Thecomparisonsresults of full flexibility, partial flexibility and specific mould case.
CAB CBA D1R(A) D1R(B) nnd(A) nnd(B) min {f1}A min {f1}B min {f2}A min {f2}B min {f3}A min {f3}B
3035
Full flexibility (A)vs specific
mould assignment (B)
1 0 0 0.387 100 36 2072 2160.7 400 440 270 364
Partial flexibility (A)vs specific
mould assignment (B)
1 0 0 0.271 36 25 2082 2160.7 385 440 270 364
Full flexibility (A)vs partialflexibility (B)
0.680 0.170 0.010 0.171 100 25 2072 2082 400 385 270 270
40610
Full flexibility (A)vs specific
mould assignment (B)
0.724 0 0.071 0.366 44 29 2224 2490 571 539 432 495
Partial flexibility (A)vs specific
mould assignment (B)
0.655 0 0.146 0.263 21 29 2211 2490 578 539 444 495
Full flexibility (A)vs partial
flexibility (B)
0.5714 0.023 0.023 0.117 44 21 2224 2211 571 578 432 444
60915
Full flexibility (A)vs specific
mould assignment (B)
0.875 0 0.054 0.652 25 24 2133.5 2508 509 524 459 450
Partial flexibility (B)vs specific
mould assignment (B)
1 0 0 0.678 40 24 2208.5 2508 486 524 440 450
Full flexibility (A)vs partial
flexibility (B)
0.525 0 0.097 0.081 25 40 2133.5 2208.5 509 486 459 440
solutions obtained in the specific mould case are dominated by at
least one solution of the full flexibility case. The next two columns
represent the D1R values. The results of the first line show that
D1R(A) = 0 and D1R(B) = 0.387, which means that the solutions sets
obtained in the full flexibility case are more distributed and bet-
ter approximate the reference front than the ones obtained in the
specific mould assignment case. The next two columns shows the
number of non-dominated solutions for the full flexibility case and
the specific mould case. The next six columns show the minimum
objective value of three objectives in the two cases, respectively.
The results of the first line show that for the full flexibility case,
min{f1}= 2072, min{f2}= 400, min{f3}= 2 70, and for the specific
mould case,min{f1}= 2160.6,min{f
2}= 440,min{f
3}= 364. Itshows
that all three objectives of the specific mould case are larger than
those of the full flexibility case.
The meaning of each column is same. The results show that in
terms ofCmetric and D1R values, for almost all different cases of
the threeexamples, the fullflexibility canget better non-dominated
solutions compared with the partial flexibility case, which in turn
can get better ones than those of the specific mould case.
6. Conclusions
In this paper, a multi-objective scheduling problem with two
kinds of resources (machines and moulds) and with flexible pre-
ventive maintenance activities on resources has been investigated,
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in whichmultiple parallel machines andparallelmouldsare consid-
ered together.Each jobcan only be performedon onemachine with
one mould. The time to failure of each machine and each mould is
subject to the exponential distribution. The full or partial flexibility
of machines and moulds for job processing are further consid-
ered.A multi-objective meta-heuristic method has been developed
based on the NSGA-II algorithm, which integrates the production
scheduling and the PM planning on machines and moulds simulta-
neously. The results show that the integrated method outperforms
the method with the periodic PM planning, in terms of multi-
objectivemetrics. Theresultsalsoshow theeffectsof thepartialand
full flexibility of resources. The case of full flexibility of machines
and moulds is better in general.
The main limit of the paper is that the random repair time is not
included in the makespan and the time length of a PM activity is
fixed for a resource. Future work can be done by minimizing the
expected makespan value by considering the random repair times
and random time length of PM activities. As the proposed model
only considers the parallel machine manufacturing environment,
further work can also be done for considering other manufacturing
environments, like a hybridsystem with serialand parallel devices.
Moreover, different failure probability distributions besides the
exponential distribution on machines or moulds could be inves-
tigated. Exploring other multi-objective meta-heuristic methods
could also be an interesting field for further research.
Acknowledgements
This work was supported by the National Science Foundation
of China [grant numbers 71101106, 71171149, 71428002]. It was
also supported by the Fundamental Research Funds for the Central
Universities.
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