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ARTICLE IN PRESSG Model

JMSY-414; No. of Pages11

Journal of Manufacturing Systems xxx (2015) xxxxxx

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

http://dx.doi.org/10.1016/j.jmsy.2015.07.002

0278-6125/ 2015 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
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2 S. Wang,M. Liu / Journal of ManufacturingSystems xxx(2015) xxxxxx

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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S. Wang,M. Liu / Journal of ManufacturingSystems xxx(2015) xxxxxx 3

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


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



0.724 0 0.071 0.366 44 29 2224 2490 571 539 432 495

Partial flexibility (A)vs specific


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



0.875 0 0.054 0.652 25 24 2133.5 2508 509 524 459 450

Partial flexibility (B)vs specific


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,


11/11

Please cite this article in press as: Wang S Liu M Multi-objective optimization of parallel machine scheduling integrated with multi-




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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multi-objective optimization of parallel machine scheduling integrated with multi-resources...

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