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Job shop scheduling with a combination of four buffering constraints

International Journal of Production Research

November 2017
56(9):1-20

DOI:10.1080/00207543.2017.1401240

Authors:

Shi-Qiang (Samuel) Liu

Fuzhou University

Erhan Kozan

Queensland University of Technology

Mahmoud Masoud

King Fahd University of Petroleum and Minerals

Yu Zhang

University of South Florida

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In this paper, a new scheduling problem is investigated in order to optimise a more generalised Job Shop Scheduling system with a Combination of four Buffering constraints (i.e. no-wait, no-buffer, limited-buffer and infinite-buffer) called CBJSS. In practice, the CBJSS is significant in modelling and analysing many real-world scheduling systems in chemical, food, manufacturing, railway, health care and aviation industries. Critical problem properties are thoroughly analysed in terms of the Gantt charts. Based on these properties, an applicable mixed integer programming model is formulated and an efficient heuristic algorithm is developed. Computational experiments show that the proposed heuristic algorithm is satisfactory for solving the CBJSS in real time.

Time elements of an operation in CBJSS.

…

Illustration of Property 1.

…

Illustration of Property 3.

…

Illustration of Property 4.

…

Illustration of Property 5.

…

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Job shop scheduling with a combination of four

buffering constraints

Shi Qiang Liu, Erhan Kozan, Mahmoud Masoud, Yu Zhang & Felix T.S. Chan

To cite this article: Shi Qiang Liu, Erhan Kozan, Mahmoud Masoud, Yu Zhang & Felix T.S. Chan

(2018) Job shop scheduling with a combination of four buffering constraints, International Journal of

Production Research, 56:9, 3274-3293, DOI: 10.1080/00207543.2017.1401240

To link to this article: https://doi.org/10.1080/00207543.2017.1401240

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Job shop scheduling with a combination of four buffering constraints

Shi Qiang Liu

, Erhan Kozan

, Mahmoud Masoud

, Yu Zhang

and Felix T.S. Chan

School of Economics and Management, Fuzhou University, Fuzhou, China;

School of Mathematical Sciences, Queensland

University of Technology, Brisbane, Australia;

Civil and Environmental Engineering, University of South Florida, Tampa, FL, USA;

Department of Industrial & Systems Engineering, Hong Kong Polytechnic University, Hong Kong

(Received 11 April 2017; accepted 19 October 2017)

In this paper, a new scheduling problem is investigated in order to optimise a more generalised Job Shop Scheduling

system with a Combination of four Buffering constraints (i.e. no-wait,no-buffer,limited-buffer and inﬁnite-buffer) called

CBJSS. In practice, the CBJSS is signiﬁcant in modelling and analysing many real-world scheduling systems in chemi-

cal, food, manufacturing, railway, health care and aviation industries. Critical problem properties are thoroughly analysed

in terms of the Gantt charts. Based on these properties, an applicable mixed integer programming model is formulated

and an efﬁcient heuristic algorithm is developed. Computational experiments show that the proposed heuristic algorithm

is satisfactory for solving the CBJSS in real time.

Keywords: job shop scheduling; buffer management; blocking; no-wait; mixed integer programming; constructive

algorithm; best-insertion-heuristic algorithm

1. Introduction

The classical job shop scheduling (JSS) problem is regarded as one of the most difﬁcult problems in combinatorial opti-

misation. An indication of its difﬁculty was given by the fact that a 10-job, 10-machine instance formulated for the ﬁrst

time (Fisher and Thompson 1963) was exactly solved after over 30 years by a branch-and-bound algorithm with more

than 5-h running time (Carlier and Pinson 1989). Because of its signiﬁcance from a theoretic view, the JSS is still an

important research topic till now (Amirghasemi and Zamani 2015; Peng, Lü, and Cheng 2015; Zhang and Chiong 2016;

Zhang et al., forthcoming).

However, the capacity of intermediate buffer storage in a classical job shop is assumed as inﬁnite, which implies that

speciﬁc buffering constraints (i.e. no-wait, no-buffer and limited-buffer) are neglected. In practice, this assumption

results in inapplicability for modelling many industrial systems. For example, in the food industry, the canning operation

must immediately follow the cooking operation to ensure freshness, which means that the no-wait constraint occurs in

this situation (Hall and Sriskandarajah 1996). For instance, the no-buffer ( blocking) constraint is required in chemical

industry, where partially processed chemical products sometimes have to be temporarily kept in the processing machine

because of high temperature or safety issues (Pacciarelli 2002). In manufacturing industry, the limited-buffer constraint

is critical to alleviate abrupt changes in fabrication lines, as intermediate buffers can accommodates the product parts

after a processing equipment unit or supply them to the next equipment unit among the contiguous process steps (Toba

2005; Chan and Choy 2011).

In some industrial systems, no-wait,no-buffer, limited-buffer constraints must be considered in an integrated way

(Liu and Kozan 2009a). Investigation of a job shop system with combined buffering constraints is beneﬁcial to analyse

and optimise the operations in several industries. One implementation arises in healthcare industry, in which both outpa-

tients and inpatients are serviced in the hospital facilities. For example, the no-wait constraint incurs when an urgent sur-

gical case is treated for an acute outpatient; while the blocking constraint happens when an inpatient has to remain in a

ward bed until an operating theatre becomes available (Chien, Tseng, and Chen 2008; Pham and Klinkert 2008; Ruan

et al. 2016). Another important implementation occurs in railway industry due to the lack of crossing loops and the con-

sideration of train’s priority (D’Ariano, Pacciarelli, and Pranzo 2007, 2008; Burdett and Kozan 2009, 2010; Liu and

Kozan 2009b, 2011a, 2011b; Masoud, Kozan, and Kent 2011, 2015; Bürgy and Gröﬂin 2016; Masoud, Kent et al.

2016; Masoud, Kozan et al. 2016; Masoud et al. 2017). In a railway network, a tunnel section has to impose the no-wait

constraint for safety; a bridge section may require the no-buffer constraint due to absence of crossing loops; a loading

*Corresponding author. Email: f.chan@polyu.edu.hk

International Journal of Production Research, 2018

Vol. 56, No. 9, 3274–3293, https://doi.org/10.1080/00207543.2017.1401240

section with sidings must consider the limited-buffer constraint; and a depot section could allow the inﬁnite-buffer

constraint. One recent implementation of job shop scheduling with buffering constraints comes from aviation manage-

ment, which aims to optimise the take-off and landing operations at a busy European airport terminal with limited-

capacity infrastructure (Samà, D’Ariano et al., 2017).

In the literature about multi-stage scheduling with buffering constraints, the initial research efforts dealt with the

ﬂow shop scheduling (FSS) problem that is the simpliﬁed version of the JSS problem. To better understand the differ-

ence between the JSS and the FSS, the key characteristics of the classical FSS and JSS problems are stated as follows

(Liu and Ong 2002, 2004; Liu, Ong, and Ng 2005; Liu and Kozan 2012a; Bai et al. 2017; Rossit, Tohmé, and Frutos,

forthcoming). Given a set of jobs that have to be processed on a set of machines, each job consists of multiple opera-

tions. In the FSS, the operations of every job are required to be processed on such a set of machines in the same unidi-

rectional order. In comparison, in the JSS, each job has a prescribed processing order through the machines, but the

processing order for each job may be different. For the FSS with buffering constraints, the following papers published

in the leading journals are referred. Leisten (1990) presented the initial ideas of formulating the FSS problems with lim-

ited-buffer storage (including blocking, no-wait, limited-buffer constraints) but the proposed algorithm was ineffective.

Ronconi (2004) introduced two two-stage hybrid heuristic algorithms to solve the FSS with blocking constraints based

on the framework of the well-known NEH algorithm (Nawaz, Enscore, and Ham 1983). Grabowski and Pempera (2007)

developed a tabu search metaherusitc algorithm to solve the FSS with blocking constraints. Fink and Voß (2003) devel-

oped several metaheuristic algorithms for the FSS with now-wait constraints and evaluated the trade-off between running

time and solution quality for calibrating the algorithms. Qian et al. (2009) designed a hybrid differential evolution algo-

rithm to solve the FSS with limited-buffer constraints. Fu, Sivakumar, and Li (2012) developed a hybrid differential

evolution algorithm to solve the FSS with intermediate buffers and batch processors. Davendra and Bialic-Davendra

(2013) proposed a discrete self-organising migrating algorithm for solving the FSS with blocking in an efﬁcient way.

Ding et al. (2015) analysed the block properties of the FSS with blocking and developed an iterated greed algorithm to

solve the problem efﬁciently. Zhang et al. (2017) combined a greedy heuristic and a hybrid differential evolution algo-

rithm to solve the FSS with a batch processor and limited buffers. Han et al. (2016) proposed a so-called modiﬁed fruit

ﬂy optimisation algorithm to solve the FSS with blocking and demonstrated its efﬁciency based on benchmark

instances.

Compared to the literature of the classical JSS problem, the JSS problem with no-wait and no-buffer (blocking) con-

straints received much less attentions. This paper focuses on the JSS with various types of buffering constraints. For

ease of presentation, the JSS problems with a single buffering constraint (i.e. no-wait,blocking, limited-buffer) are,

respectively, called NWJSS,BJSS and LBJSS throughout this paper. An initial introduction of NWJSS and BJSS was

given by (Hall and Sriskandarajah 1996). Song and Lee (1998) developed a Petri-net time-marked graph to analyse the

deadlock detection properties of BJSS. Mati, Rezg, and Xie (2001) investigated an automated manufacturing system, in

which the deadlock-prone characteristics in a job shop were analysed based on an extended disjunctive graph. Mati,

Lahlou, and Dauzère-Pérès (2011) extended the BJSS problem by incorporating more features in a more ﬂexible manu-

facturing system. Mascis and Pacciarelli (2002) studied the BJSS and NWJSS problems by means of an alternative

graph, which is an extension of classical disjunctive graph. Pacciarelli (2002) further applied the alternative graph to

model a complex factory scheduling problem that incorporates the no-wait constraint and sequence-independent set-up

times. Schuster and Framinan (2003) developed a hybrid genetic algorithm and simulated annealing metaheuristic algo-

rithm for solving NWJSS. Hauptman and Jovan (2004) investigated a real-world process manufacturing system by trans-

forming it into a job shop with no-wait and no-buffer constraints. Brucker and Kampmeyer (2008) proposed a tabu

search metaheuristic for a cyclic BJSS problem by developing a recovering procedure in neighbourhood moves. Chien,

Tseng, and Chen (2008) modelled a patient scheduling problem as NWJSS and solved it by an evolutionary algorithm.

Gröﬂin and Klinkert (2009) developed a tabu search algorithm to solve BJSS with a special mechanism of satisfying the

blocking constraints based on an extended disjunctive graph. Gröﬂin, Pham, and Bürgy (2011) presented a local search

heuristic to solve the ﬂexible BJSS problem with the transfer and set-up times, based on the earlier work of (Gröﬂin and

Klinkert 2006, 2009) regarding the construction of a feasible neighbourhood structure. Santosa, Budiman, and Wiratno

(2011) developed a hybrid metaheuristic called CEGA (cross-entropy with genetic algorithm) to solve NWJSS. Samar-

ghandi and ElMekkawy (2013) developed a genetic algorithm to solve NWJSS with sequence-dependent set-up times

and single-server constraints. Pranzo and Pacciarelli (2016) developed an iterated greedy algorithm to solve to two vari-

ants of BJSS without/with swap allowed. Brucker et al. (2006) investigated there types of LBJSS by classifying buffers

into three categories: (i) machine-dependent output buffers; (ii) machine-dependent input buffers; (iii) job-dependent buf-

fers. Witt and Voß (2007) developed three heuristic algorithms to guarantee ﬁnding the high-quality LBJSS schedule and

indicated that the proposed approaches are functional to control the work-in-process operations that wait for processing

in the production system with limited intermediate storage. Zeng, Tang, and Yan (2014) extended the BJSS problem

International Journal of Production Research 3275

using a limited number of automated guided vehicles (AGV) to transfer jobs between machines. The so-called BJS-

AGV problem was solved by a two-stage heuristic algorithm based on the analysis of characteristics of this problem.

Louaqad and Kamach (2016) investigated the blocking and no-wait job shop scheduling problems in robotic cells and

developed a MILP model for solving this complex problem up to 10 jobs, 10 machines and 3 robots.

Based on the above literature review, it is noted that a combination of four buffering constraints in job shop environ-

ments was rarely found. To ﬁll this gap, a new scheduling problem called CBJSS (Job Shop Scheduling with a Combi-

nation of four Buffering constraints) is investigated in this study, which contributes to adding new knowledge in

scheduling theory. With a thorough analysis of problem properties, a novel heuristic algorithm that consists of several

interactive sub-algorithms, is developed to solve this complicated problem in a very efﬁcient way. The proposed CBJSS

model can be used as a fundamental tool to identify, analyse, conﬁgure and evaluate different types of scheduling sys-

tems that should consider various buffering requirements simultaneously. The proposed solution approach is useful for

many real-world applications in food, chemical, automation, railway, aviation and health care industries because diverse

buffering conditions occur frequently in these scheduling systems.

The remainder of this paper is organised as follows. In Section 2, the CBJSS problem is deﬁned and its problem

properties are analysed. In Section 3, a mixed integer programming model of CBJSS is formulated. An efﬁcient heuris-

tic algorithm is developed in Section 4. Extensive computational experiments are reported in Section 5. Contribution

and signiﬁcance of this study are concluded in the last section.

2. Deﬁnition and analysis

In a CBJSS system, there are nindependent and non-preemptive jobs that have to be processed on mmachines. The

objective is to minimise the makespan. Each job consists of at most moperations, each of which must be processed in a

given processing route, but this route may differ with jobs. Only one operation can be processed on one machine and

each machine can exactly process one operation at a time. Each machine should be associated with a speciﬁed buffering

constraint, due to technical, safety or service requirements. In comparison to the processing, blocking times or storing

times, the transferring times of jobs between machines (storage units) are negligible and thus omitted in this study. For

convenience, parameter b

∊{ɛ,∅,θ|τ

,∞}isdeﬁned to represent an intermediate buffering constraint associated with

Machine i.Ifb

=ɛ, this buffering constraint is deﬁned as ‘no-wait’, which implies that any job completed on Machine i

should be continuously processed without any delay. If b

=∅, this buffering constraint is deﬁned as ‘no-buffer’, which

means that there is no buffer storage to store any job after completed on Machine i.Ifb

=θ|τ

, this buffering constraint

is deﬁned as ‘limited-buffer’if 0 < τ

<n.Ifb

=∞, this buffering constraint is deﬁned as ‘inﬁnite-buffer’if τ

≥n. Note

that the limited-capacity buffers in our proposed CBJSS problem are ‘machine-dependent output buffering storage’,

which means that a storage unit may store a job just after its processing on the associated machine. In a sense, these

buffer storage units could be treated as ‘dummy parallel machines’but one main difference is that the utilisation of each

buffer storage unit depends on dynamic scheduling scenarios. For more discussions between machines and output buf-

fers, please refer to a book chapter by Brucker and Knust (2012). By extending three-tuple descriptor in scheduling the-

ory, the CBJSS problem is denoted as Jmðbi2e;;;hjsi;1fg;i¼1;2;...;mÞnjjCmax, where J

represents a job shop

with mmachines; nis the number of jobs; C

max

is the makespan; bi2e;;;hjsi;1

;i¼1;2;...;mdeﬁnes a ﬂexible

combination of four different buffering constraints associated with each machine. If the descriptor is

Jm n

Cmaxjbi¼;;8i¼1;2;...;m, then this problem (system) is regarded as the BJSS problem (system), which is an

extreme case of the CBJSS as all of buffering requirements associated with all machines are deﬁned as ‘no-buffer’.If

the descriptor is Jm n

Cmaxjbi¼e;8i¼1;2;...;m, then this problem (system) is treated as the NWJSS problem (sys-

tem), which is also extreme case of the CBJSS as all of buffering requirements associated with all machines are deﬁned

as ‘no-wait’.

To analyse, formulate and solve the CBJSS, the following notations are deﬁned.

Indices and parameters

nnumber of jobs

mnumber of machines

jjob index, j=1,2,…,n

Ja set of jobs

Job j;J

∊J

imachine index, i=1,2,…,m

Ma set of machines

3276 S.Q. Liu et al.

machine i;M

∊M

number of operations for job j

ooperation’s order index (o=1,2,…,π

) of job jin the given processing route

the oth operations of job j

oji

1, if O

requires M

in the given processing route; 0, otherwise

processing time of O

a buffering constraint associated with Machine i

1, if the buffering constraint of M

is no-wait (b

=ɛ); or 0, otherwise

1, if the buffering constraint of M

is no-buffer (b

=ϕ); or 0, otherwise

1, if the buffering constraint of M

is limited-buffer (b

=θ); or 0, otherwise

1, if the buffering constraint of M

is inﬁnite-buffer (b

=∞); or 0, otherwise

number of buffers associated with Machine iwhen b

=θ

kbuffer index, k¼1;2;...;si

Ha large constant positive value

Variables

yojo0j01, if O

proceeds Oo0j0; or 0, otherwise

wojo0j0k1, if O

proceeds Oo0j0in buffer k∊{1, 2, …,τ

}ofM

; or 0, otherwise

ojk

1, if O

is stored in buffer k∊{1, 2, …,τ

} after its completion on M

; or 0, otherwise

starting time of O

completion time of O

blocking time of O

departure time of O

ojk

storing time of O

in buffer k|k=1,2,…,τ

of M

ojk

leaving time of O

in buffer k|k=1,2,…,τ

ojk

max

maximum completion time or makespan

the current available time of buffer k|k∊{1, 2, …,τ

}ofM

when b

=θ

A

ithe earliest buffer available time of M

;A

i¼mink2f1;2;...;sigAkwhen b

=θ

the assigned buffer with the earliest buffer available time; k¼argmink2f1;2;...;sigAkwhen b

=θ

Figure 1is drawn to elucidate the time elements of an operation O

, including starting time, processing time, com-

pletion time, blocking time, departure time, storing time and leaving time. In Figure 1, the values of horizontal axis are

measured in timing units (e.g. minutes); the labels of vertical axis are described by each machine together with its

buffering constraint indicated by b

∊{ɛ,∅,θ|τ

,∞}.

In the following, critical properties of the CBJSS problem are thoroughly analysed.

Property 1: If b

=∅, then Boj ¼maxð0;Eoþ1;jCojÞ.

Analysis: Illustrated in Figure 2, blocking time B

of operation O

equals the gap between its completion time C

and its same-job successor’s starting time E

o+1,j

, if the buffering constraint associated with M

is no-buffer, i.e. b

=∅.

Due to the absence of buffer storage, M

has to be blocked until the job is able to be transferred when the downstream

machine becomes available.

Figure 1. Time elements of an operation in CBJSS.

International Journal of Production Research 3277

Property 2: If b

=θand Coj ¼Eoþ1;j, then B

= 0 and Sojk ¼0.

Analysis: If the buffering constraint associated with M

is limited-buffer (i.e. b

=θ) and the completion time Coj of

operation O

is equal to its same-job successor’s starting time E

o+1,j

, then the blocking time B

is zero. In this case, its

storing time is also zero as none of buffers of M

are required.

Property 3: If b

=θand C

o+1,j

and Coj A

i, then B

= 0 and Sojk ¼Eoþ1;jCoj.

Analysis: As shown in Figure 3, if the buffering constraint associated with M

is limited-buffer (i.e. b

=θ) and its

completion time C

is less than its same-job successor’s starting time E

o+1,j

but greater than or equal to the earliest buf-

fer time A

iof M

, then blocking time B

of operation O

is zero because this job can be immediately transferred to an

available buffer (i.e. buffer k

) after its completion on M

. Moreover, the storing time Sojkequals the gap between its

same-job successor’s starting time E

o+1,j

and its completion time C

Property 4: If b

=θand C

o+1,j

and Coj\A

iand A

i\Eoþ1;j, then Boj ¼A

iCoj and Sojk¼Eoþ1;jA

Analysis: As illustrated in Figure 4, if the buffering constraint associated with M

is limited-buffer (i.e. b

=θ)and

its completion time C

is less than both its successor’s starting time E

o+1,j

and the earliest buffer time A

i, in addition,

the earliest buffer time A

iis less than its successor’s starting time E

o+1,j

, then B

equals the gap between its completion

time C

and the earliest buffer time A

i, implying that M

has to be blocked till the earliest availability of buffer k

After the blocking duration on M

, this job can be transferred to buffer k

with the earliest buffer time A

i. Thus, the stor-

ing time Sojkof O

is the gap between A

iand E

o+1,j

. The buffer k

becomes available when the next machine in the

processing route of this job becomes available.

Property 5: If b

=θand C

o+1,j

and Coj\A

iand A

iEoþ1;j, then B

o+1,j

−C

and Sojk¼0.

Analysis: As illustrated in Figure 5, if the buffering constraint associated with M

is limited-buffer (i.e. b

=θ) the

earliest buffer time A

iis greater than or equal to its completion time C

and its successor’s starting time E

o+1,j

, none

the buffers of M

are available to be used. In this scenario, the storing time Sojkof O

must be zero and its blocking

time B

equals the gap between C

and E

o+1,j

The following properties are analysed for the scenarios when the buffering constraint associated with M

is no-wait

(i.e. b

=ɛ).

Property 6: If b

=ɛ, then B

= 0 and C

o+1,j

Figure 2. Illustration of Property 1.

Figure 3. Illustration of Property 3.

3278 S.Q. Liu et al.

Figure 4. Illustration of Property 4.

Figure 5. Illustration of Property 5.

Figure 6. Illustration of Property 6.

Figure 7. Illustration of Property 7.

International Journal of Production Research 3279

Analysis: As shown in Figure 6, the blocking time of O

must be zero, if the buffering constraint associated with

is no-wait. This implies that this job must be immediately processed on the next machine without any delay due to

the requirement of no-wait constraint.

Property 7: In case b

=ɛand C

o+1,j

, the tune-up procedure should be applied to update E

o+1,j

−P

satisfy the no-wait constraint.

Analysis: As illustrated in Figure 7, the tune-up procedure (Liu and Kozan 2009b) may be applied to satisfy the no-

wait constraint in an iterated way. In this case, the starting time E

of O

is re-calculated and equals the gap between

its same-job successor’s starting time E

o+1,j

and its processing time P

Property 8: If Eo0j0[Eoj and Eo0j0\Doj



or ðEoj [Eo0j0and Eoj\Do0j0Þ, two conﬂicting scenarios may incur in the

solution procedure.

Analysis: As shown in Figure 8(a1), one conﬂicting scenario happens when the starting time of Oo0j0is greater than

the starting time of O

but less than the departure time of O

. On the other hand, as shown in Figure 8(b1), the starting

time of O

is greater than the starting time of Oo0j0but less than the departure time of Oo0j0.

The following additional notations are particularly deﬁned for Property 9 to be used only in the proposed heuristic

algorithm in Section 4(Figure 9).

Additional Parameters for Property 9

ready time of operation O

of job J

that is currently considered in the solution procedure

number of operations that have already been scheduled on M

at the current stage

x½ sequence index of the xth|x=1,…,n

operation already scheduled on M

Ox½;ithe xth operation already scheduled on M

Ex½;istarting time of Ox½;ialready scheduled on M

px½;iprocessing time of Ox½;ialready scheduled on M

Dx½;ideparture time of Ox½;ialready scheduled on M

Additional Variables for Property 9

E½k

oj the earliest starting time of O

with the best insertion position k

kthe best insertion position index to insert a new operation O

into the current sequence of operations already

scheduled on M

Property 9: If E1½;iRoj Poj , then E½1

oj ¼Roj and k¼1; else if E1½;iRoj \Poj and min

x2f2;...;nigðEx½;iRoj ;Ex½;i

Dx1½;iÞPoj, then k¼argmin

x22;...;ni

ðmaxðRoj;Dx1½;iÞÞ and E½k

oj ¼maxðRoj;Dk1½;iÞ; else if E1½;iRoj \Poj and

min

x2f2;...;nigðEx½;iRoj ;Ex½;iDx1½;iÞ\Poj, then k¼niþ1 and E½k

oj ¼maxðRoj;Dni

½;iÞ.

Figure 8. Illustration of Property 8.

3280 S.Q. Liu et al.

3. Mathematical formulation

Based on the analysis of problem properties, the CBJSS is mathematically formulated below.

3.1 CBJSS formulation

Minimise:

Cmax (1)

The objective function is to minimise the maximum completion time, i.e. makespan.

Subject to:

H2hoji ho0j0i



þMyojo0j0þEoj Eo0j0þPo0j0þBo0j0(2)

H2hoji ho0j0i



þM1yojo0j0



þEo0j0Eoj þPoj þBoj (3)

hoji þho0j0i1yojo0j0(4)

o¼1;2;...;pj1;o0¼1;2;...;pj01;j;j0¼1;2;...;njj6¼ j0;i¼1;2;...;m

Constraints (2–4) satisfy the exclusive precedence relationship of a pair of operations (i.e. O

and Oo0j0) that belong to

two jobs, respectively, (i.e. J

and Jj0).

i¼1

hoji Eoj þPoj



X

i¼1

hoþ1;j;iEoþ1;j(5)

o¼1;2;...;pj1;j¼1;2;...;n;

Figure 9. Illustration of Property 9.

International Journal of Production Research 3281

Constraints (5) requires that the completion time of an operation O

must be less than or equal to the starting time of

its same-job successor O

o+1,j

, whatever the buffering constraint is.

i¼1

aihoji Eoj þPoj



¼X

i¼1

hoþ1;j;iEoþ1;j(6)

i¼1

aixojiBoj ¼0 (7)

o¼1;2;...;pj1;j¼1;2;...;n;

Constraints (6–7) deﬁne the no-wait constraint of M

excluding the last machine in the processing route of each job. In

this case, the completion time of O

must be equal to the starting time of its same-job successor O

o+1,j

. Constraint (7)

requires that the blocking time of O

must be zero under the no-wait requirement.

i¼1

bihojiðEoj þPoj þBoj Þ¼X

i¼1

hoþ1;j;iEoþ1;j

o¼1;2;...;pj1;j¼1;2;...;n(8)

Constraint (8) deﬁnes the no-buffer constraint of M

excluding the last machine. In this case, after the completion of

may be blocked due to the absence of associated buffers. Thus, the blocking time (B

)ofO

may be non-zero.

i¼1

cihojiðEoj þPoj þBoj þSojk Þ¼X

i¼1

hoþ1;j;iEoþ1;j(9)

Doj ¼Eoj þPoj þBoj (10)

i¼1

cihojizojk ¼1 (11)

zojk þzo0j0k1wojo0j0k(12)

H2zojk zo0j0k



þMwojo0j0kþDoj Do0j0þSo0j0k(13)

H2zojk zo0j0k



þMð1wojo0j0kÞþDo0j0Doj þSojk (14)

o¼1;2;...;pj1;o0¼1;2;...;pj01;j;j0¼1;...;njj6¼ j0;k¼1;2;...;si;

Constraints (9–14) deﬁne the limited-buffer constraint of M

excluding the last machine. In this case, after the comple-

tion of O

, its leaving time in buffer kmust be equal to the starting time of its same-job successor O

o+1,j

as its storing

time may be non-zero. Moreover, only one buffer kðk21;2;...;siÞof M

can be used to store O

at a time. If both

operations O

and Oo0j0require the same buffer k, i.e. zojk ¼zo0j0k¼1, then the precedence relationship between them is

deﬁned. In a sense, Constraints (13–14) imply that the limited-capacity intermediate buffers (storing units) could be

regarded as the dummy machines that may not be used. If the storing time of O

is non-zero, it means that a storage

unit is used by O

i¼1

dihpj;j;iðEpj;jþPpj;jÞCmax (15)

i¼1

dihpj;j;iBpj;j¼0 (16)

j¼1;2;...;n

3282 S.Q. Liu et al.

Constraints (15–16) deﬁne the inﬁnite-buffer constraint of the last machine in the processing route of each job. In this

case, blocking time of the last operation must be zero, as the totally ﬁnished job (the ﬁnal product) is allowed to be

immediately removed from the production environment. In addition, it needs to satisfy maximum completion time con-

straints, i.e. the completion time of the last (π

th) operation Opj;jof Jjshould not exceed the makespan.

Eoj;Boj ;Coj;Doj;Sojk 0 (17)

yojo0j0;zojk ;wojko0j0k2f0;1g(18)

o¼1;2;...;pj;o0¼1;2;...;pj0;j;j0¼1;...;njj6¼ j0;k¼1;2;...;si;

Constraints (17–18) declare non-negativity and binary constraints, respectively.

4. Solution approach

Solving CBJSS is a challenge due to the following difﬁculties: (i) the potential conﬂicts may be arisen due to the no-

buffer constraint, especially when two operations on a blocked machine are not allowed to be swapped; (ii) the no-wait

constraint is so restrictive that the starting times of the same-job predecessors of an operation may need to be reversely

tuned up in an iterated way; (iii) the limited-buffer constraint requires the determination of the earliest available buffer

in a dynamic way. Based on Properties 1–9, an innovative heuristic algorithm (called the CBJSS-CA-BIH algorithm) is

developed to obtain a high-quality CBJSS solution in an efﬁcient way. In the framework of best-insertion-heuristic

(BIH), a complicated constructive algorithm (called the CBJSS-CA algorithm) is developed and embedded inside the

CBJSS-CA-BIH algorithm.

A small-size example is given to illustrate the CBJSS-CA-BIH heuristic algorithm. In this example, it is assumed that

the current partial sequence of scheduled jobs is Π={J

→J

}; the list of unscheduled jobs is U={J

}. By

CBJSS-CA algorithm

Step 1: For each job J

in sequence, get the number of operations for J

in an alternative: π

Step 2: Initialise the order index of the current operation of J

:o←1.

Step 3: While (o≤π

)

3.1 Get the machine that is assigned to process operation O

3.2 Determine the earliest starting time of O

with the best insertion position k, by inserting

into the current sequence of operations that have been already scheduled on M

based on Property 9: E½k

oj and k.

3.3 If o> 1, determine the blocking time B

o-1,j

and the storing time S

o-1,j

of O

o-1,j

(i.e. the same-job predecessor of O

)

in terms of the given buffering constraint (i.e. deﬁned by the value of b

i-1

)ofM

i-1

3.3.1 If bi1¼;(i.e. ‘no-buffer’), then apply Property 1 to set Bo1;j maxð0;Eoj Co1;jÞ;S

o-1,j

←0.

3.3.2 Else if bi1¼h(i.e. ‘limited-buffer’), then apply Properties 2–5 in terms of the following four scenarios.

3.3.2.1 If C

o-1,j

, then set B

o-1,j

←0; S

o-1,j

←0.

3.3.2.2 Else if Co1;j\Eoj and Co1;jA

i1, then set B

o-1,j

←0; So1;jk Eoj Co1;j;So1;j So1;jk.

3.3.3.3 Else if Co1;j\Eoj and Co1;j\A

i1and A

i1\Eoj, then set Bo1;j A

i1Co1;j;So1;jk Eoj A

i1;

So1;j So1;jk.

3.3.3.4 Else if Co1;j\Eoj and Co1;j\A

i1and A

i1Eoj, then set B

o-1,j

←E

−C

o-1,j

←0.

3.3.3 Else if bi1¼e(i.e. ‘no-wait’), then apply Properties 6–7 to set

o-1,j

←max(E

o-1,j

−P

o-1,j

); C

o-1,j

←E

o-1,j

←0; S

o-1,j

←0.

3.4 Set D

o-1,j

←C

o-1,j

←D

o-1,j

3.5 Justify the conﬂicting status based on Property 8.

3.6 If non-conﬂicting, then set o←o+ 1 and go to Step 3.

3.7 Else, apply the following to eliminate the conﬂict.

3.7.1 Get the number of scheduled operations on M

i-1

3.7.2 For xfrom 1 to n

i-1

If Ex½;i1Eo1;jand Ex½;i1\Do1;j



or ðEo1;jEx½;i1and Eo1;j\Dx½;i1Þ, then update the ready time of O

o-1,j

Ro1;j maxðEo1;j;Dx½;i1Þand break.

3.7.3 Reset the starting time E

o-1,j

based on the updated R

o-1,j

3.7.4 Apply Steps 3.3–3.6 to re-determine time information of same-job predecessors (i.e. from o−2to1)

of O

o-1,j

in a reverse order until a new non-conﬂicting operation at Step 3.6 is found.

3.8 If o≤π

, go to Step 3; else, go to Step 1 for the next job in sequence.

International Journal of Production Research 3283

inserting each unscheduled job in Uinto the current partial sequence of scheduled jobs Π, a set of alternatives (i.e. a set

of new sequences) Ais obtained as: A=[{J

→J

}, { J

→J

}, { J

→J

}, {

→J

}, …,{J

→J

}], as analysed in detail as below. In Step 4.1.1, the number of alter-

natives in the set Ais obtained as 8, that is, nA¼nuðnsþ1Þ¼23þ1ðÞ¼8. For each alterative in A, the pro-

posed CBJSS-CA construction algorithm is implemented to construct the feasible CBJSS schedule in Step 4.1.2. The

best alternative with the minimum makespan is selected in Step 4.2 to update the set Pin Step 4.3 and the set Uin

Step 4.4 for the next iteration.

In the CBJSS-CA-BIH algorithm, the number of total alternatives is computed as:

fnðÞ¼ X

ns¼n1

ns¼1

nns

ðÞnsþ1ðÞ¼n1ðÞ

ns¼n1

ns¼1

nsþX

ns¼n1

ns¼1

nX

ns¼n1

ns¼1

¼n1ðÞn1ðÞn

2þnn1ðÞ

n1ðÞnð2n1Þ

6¼1

6n3þ1

2n22

3n:

Computational complexity indicates how much effort is needed to apply an algorithm, estimated by the rate growth

function (i.e. fnðÞ) for solution space in terms of problem size. According to this analysis, the asymptotic complexity of

the proposed CBJSS-CA-BIH heuristic algorithm depends mainly on the ﬁrst term (1

6n3) due to the cubic growth of the

ﬁrst term as the other two terms can be disregarded for the large n. Hence, the computational complexity of the pro-

posed CBJSS-CA-BIH heuristic algorithm is O(n

5. Computational results

To evaluate the performance of the CBJSS-CA-BIH algorithm, a collection of benchmark CBJSS instances are estab-

lished based on Lawrence’sJSS data from OR-Library (Beasley 1990). In the literature, the Lawrence’sJSS instances

(40 instances in total) are well known as the difﬁcult benchmarks. To differentiate the instance types between JSS and

CBJSS, the original Lawrence’s data are named JSS-LA instances. By keeping the same processing times and the

CBJSS-CA-BIH algorithm

Step 1: Set J

which is the initial list of sorted jobs by sorting the jobs in the non-increasing order of the largest sum of the

processing times on machines, each of which the buffering constraint is no-wait.

Step 2: Initialise Π, which is the partial sequence of scheduled jobs containing the ﬁrst sorted job in J

Step 3: Initialise U=J

−Π, which is the list of sorted jobs that are unscheduled.

Step 4: While Uis not empty,

4.1 For each job J

∊Uin order:

4.1.1 Construct a set of alternatives A¼[

nuðnsþ1Þ

ða¼1ÞP0

aðJjÞ;each of which is a new sequence obtained by adding an

unscheduled job J

∊Uinto the current partial sequence Π. The number of alternatives equals: n

×(n

+ 1) that is all

possible combinations of n

unscheduled jobs with n

+ 1 insertion positions, where n

is the number of unscheduled jobs in U

and n

is the number of currently scheduled jobs in Π.

4.1.2 For each alternative in A:

4.1.2.1 Apply the CBJSS-CA algorithm to construct the feasible CBJSS schedule.

4.1.2.2 According to the constructed schedule, get and save the makespan value for each alternative.

4.2 Among all alternatives in A, determine the best alternative that leads to the minimum makespan and the best inserted job

denoted as Jj.

4.3 Update Πthat is the best alternative determined in Step 4.2.

4.4 Remove Jjfrom U:U UfJjg.

J={J

}P={J

→J

}U={J

}

↓↓↓↓P0

1(J

)={J

→J

}P0

5(J

)={J

→J

}

→J

2(J

)={J

→J

}P0

6(J

)={J

→J

}

n=5,n

=3,n

=n−n

=2 P0

3(J

)={J

→J

}P0

7(J

)={J

→J

}

|A|=n

×(n

+1)=8 P0

4(J

)={J

→J

}P0

8(J

)={J

→J

}

3284 S.Q. Liu et al.

processing routes of jobs in Lawrence’s data as well as adding the speciﬁed buffering constraints of machines, the

40 established benchmark data for CBJSS are called CBJSS-LA (i.e. CBJSS_LA01 to CBJSS-LA40) instances, which are

deﬁned in the following way.

•b

=ɛ;b

=∅;b

=θ|1; b

=θ|2; b

=θ|3 for 5-machine instances (i.e. from CBJSS-LA01 to CBJSS-LA15).

•b

=ɛ;b

=∅;b

=θ|1; b

=θ|2; b

=θ|3; b

=ɛ;b

=∅;b

=θ|1; b

=θ|2; b

=θ|3 for 10-machine instances

(i.e. from CBJSS-LA16 to CBJSS-LA35).

•b

=ɛ;b

=∅;b

=θ|1; b

=θ|2; b

=θ|3; b

=ɛ;b

=∅;b

=θ|1; b

=θ|2; b

=θ|3; b

=ɛ;b

=∅;

=θ|1; b

=θ|2; b

=θ|3 for 15-machine instances (i.e. from CBJSS-LA36 to CBJSS-LA40).

For example, the 10-job, 5-machine CBJSS_LA01 instance is denoted as J5ðb1¼e;b2¼;;b3¼hjs3¼

1;b4¼hjs4¼2;b5¼hjs5¼3Þj10jCmax and the buffering constraints of this instance are explained below: