DBS-21-02｢Logic-based Benders decomposition method forthe seru scheduling problem withsequence-dependent setup time and
DeJong’s learning effecgt｣
Zhe Zhanga, Xiaoling Songa, Huijung Huanga, Xiaoyang Zhoub, Yong YincaSchool Economics & Management, Nanjing University of Science and Technology, Nanjing 210094, P. R. CHINA
bSchool of Management, Xi’an Jiaotong University, Xi’an 710049, P. R. CHINA
cGraduate School of Business, Doshisha University, Karasuma-Imadegawa, Kamigyo-ku, Kyoto, 602-8580, Japan

June, 2021

Abstract
This paper concentrates on the scheduling problem in seru production system (SPS), where seru is a
successful new-type production mode arising from the Japanese labor-intensive electronic assembly
industry. Motivated by the practical situations, the sequence-dependent setup time and DeJong’s
learning effect are considered in seru scheduling problems, and the objective is to minimize the makespan. The seru scheduling problem is formulated as a mixed integer programming (MIP), and thenreformulated to a set partitioning master problem and some independent sub-problems by employing
the logic-based Benders decomposition (LBBD) method. Subsequently, the set partitioning masterproblem is used to assign jobs to serus of SPS, and the subproblems are applied to find the optimalschedules in each seru given the assignment of the master problem. Finally, computational studies
are made, and results indicate that the LBBD method is able to return high-quality schedules forsolving seru scheduling problems.

Keywords: scheduling, seru production system, decomposition, sequence-dependent setup time, learning effectDBS Discussion Paper Series supported by the OMRON Foundation.DBS

Manuscript

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Click here to view linked References

Logic-based Benders decomposition method for the seru scheduling problem
with sequence-dependent setup time and DeJong’s learning effectZhe Zhanga , Xiaoling Songa , Huijung Huanga , Xiaoyang Zhoub , Yong Yinca School Economics & Management, Nanjing University of Science and Technology, Nanjing 210094, P. R. CHINA
b School of Management, Xi’an Jiaotong University, Xi’an 710049, P. R. CHINA

c Graduate School of Business, Doshisha University, Karasuma-Imadegawa, Kamigyo-ku, Kyoto, 602-8580, Japan

Abstract
This paper concentrates on the scheduling problem in seru production system (SPS), where seru is a successful newtype production mode arising from the Japanese labor-intensive electronic assembly industry. Motivated by the practical situations, the sequence-dependent setup time and DeJong’s learning effect are considered in seru schedulingproblems, and the objective is to minimize the makespan. The seru scheduling problem is formulated as a mixedinteger programming (MIP), and then reformulated to a set partitioning master problem and some independent subproblems by employing the logic-based Benders decomposition (LBBD) method. Subsequently, the set partitioning
master problem is used to assign jobs to serus of SPS, and the subproblems are applied to find the optimal schedules
in each seru given the assignment of the master problem. Finally, computational studies are made, and results indicatethat the LBBD method is able to return high-quality schedules for solving seru scheduling problems.
Keywords: scheduling, seru production system, decomposition, sequence-dependent setup time, learning effect

1. Introduction
Along with the high-speed development of information technology, product life cycles decrease and production
demands are fluctuating. More manufacturing companies recognize that the fast response is another dimension ofdemand in production practice apart from the product volume and product variety (Yin et al., 2018 [63]). In thissituation, traditional assembly lines, including Toyota production system (TPS) and lean, can not cope with the volatile
market with short product life cycles, uncertain product types, and fluctuating production volumes because they are
fit for a stable market and can not make speedy and timely adjustments. Accordingly, seru seisan (“seru” means
cell, and “seisan” means production in Japanese), which is a new production organization deriving from the Japaneseelectronic industry practice, is applied to offset the influence of fluctuant demands, and it could achieve efficiency,
flexibility and fast response, simultaneously (Stecke et al., 2012 [50]). Seru production system (SPS) is reconfigured
from the traditional assembly line, and it is composed of one or more workers and some simple and cheap equipment
to assemble products. Fig. 1 is an example of converting an assembly line into SPS contains three parallel serus. In
seru 1, a partially cross-trained worker 1 handles tasks 1 to 3, and a partially cross-trained worker 2 handles tasks 4to 5; in seru 2, completely cross-trained workers 3 and 4 handle all tasks from 1 to 5 without disruption, and they
move repeatedly from the entrance to the exit of seru 2; in seru 3, a single completely cross-trained worker 5 handles
all tasks from 1 to 5, and this worker is equipped with not only technical but also managerial skills. Compared to the
traditional production mode, serus in SPS can be constructed, modified, dismantled, and reconstructed frequently ina short time to accommodate the volatile market requirement, so SPS is very flexible. The comparison between SPS
and other production systems, such as Ford, TPS, and cellular manufacturing (CM), are presented in Liu et al., (2014)[32]), Yin et al. (2018) [63], and Yu and Tang (2019) [69].
In fact, since SPS combines advantages of Toyota’s lean philosophy and Sony’s one person production organization, it has brought tremendous benefits to its users and is denoted as “the next generation of lean” in Japanese
Email address: zhangzhe@njust.edu.cn. (Zhe Zhang)
Preprint submitted to European Journal of Operational Research

June 6, 2021

Traditional assembly line
task 1

task 2

task 3task 4task 5worker 1

worker 2

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inoutSeru production systemtask 2

task 1inworker 1

task 3seru 1

worker 2
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task 1

task 3inseru 2

worker 4
worker 3

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task 1inseru 3worker 5

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task 4task 5out

Figure 1: An example of an assembly line to be decomposed into SPS

production practice (Shinobu, 2003 [49]; Yin et al., 2007 [62]). Many global electronics giants, such as Sony, Canon,
Panasonic, Samsung, and LG, have adopted the seru production mode already (Yin et al., 2018 [63]). As it turns
out, SPS can reduce space requirements, workforce quantity, lead time, setup time, work-in-process (WIP) inventory,
finished product inventory, and cost (Stecke et al., 2012 [50]). Although so many benefits have been obtained, SPS
is still largely unknown outside Japan and the research on seru is few due to its short history. Fortunately, because
SPS can achieve high efficiency, high flexibility and fast response simultaneously in practice, it has attracted vastattention from some leading scholars and practitioners in operations management (OM) recently. Hopp and Spearman(2020) [24] provided a useful construct for lean training and implementation by describing four “Lenses of Lean”,
and showed that SPS can elevate both efficiency and responsiveness. Lewis (2019) [27] summarized an overview of
the current and classic OM research and indicated that seru production mode was one feasible solution to cope withmarket requirements for smaller volumes and higher variety. Yin et al. (2018) [63] analyzed the demand drivers for
production system evolution from Industry 2.0 through Industry 4.0 by employing supply-demand relationships, andpointed out that SPS may be the potential smart manufacturing system in the Internet of Things (IoT) age. Trevilleet al. (2017) mentioned that some Japanese electronics enterprises could achieve a fast response to market demands
using SPS in [53]. Roth et al. (2016) [45] listed eight possible future research directions in OM after summarizing thedevelopment process of OM over the past 25 years, and pointed out that seru was one of the new research fields worthyof our attention. In this paper, we will study the seru scheduling problem considering both the sequence-dependent
setup time and DeJong’s learning effect for the first time, and hope that this research could improve the theoretical
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research in SPS and provide professional guidance to seru production managers.
According to the evolution of SPS, there are three types of seru in production practice, including divisional seru,
rotating seru and yatai (Akino, 1997 [5]). At the beginning, if the worker in an assembly line is partially crosstrained, which means that he/she can take charge of more than one task, then this assembly line can be converted into
a number of short lines. For example, divisional serus are constructed (example of divisional seru is seru 1 in Fig.
1). Subsequently, with more worker training and skills improving, some workers in divisional serus are completely
cross-trained, and they can handle all tasks of a job. The equipments are shared in the serus, and these completely
cross-trained workers move one after the other until a job is finished. The worker will return to the first workstationand start a new round in this seru when the job is finished. Hence, the rotating seru formation is accomplished(example of rotating seru is seru 2 in Fig. 1). Finally, as the technical and managerial skill of completely cross-trained
workers improving further, some rotating serus could evolve into yatais, which contain only one completely crosstrained worker who takes charge of all tasks from start to finish. Yatai is a small but highly autonomous single-person
production unit, and it is the highest evolution form of seru production mode implementation (example of yatai is
seru 3 in Fig. 1). The detailed description about these three types of seru can be found in Stecke et al. (2012) [50],
Liu et al. (2014) [32], Yu and Tang (2019) [69]. In this paper, the seru type is yatai since it is sensitive to the learning
effect. The divisional seru and rotating seru are left for future research.
In production practice, a new planning system, named just-in-time organization system (JIT-OS), is used to manage and control SPS, and practical industrial cases of JIT-OS are shown in Yin et al. (2008) [61] and Stecke et al.
(2012) [50]. As an extension of the Toyota’s traditional JIT material system (JIT-MS), the implementation mechanismof JIT-OS is similar: the correct serus, in the right place, at the appropriate time, in the exact amount (Stecke et al.,2012 [50]). The difference between JIT-OS and JIT-MS is that JIT-OS focuses on organizations (i.e., serus) while JITMS on materials (Liu et al., 2014 [32]). There are three decisions in JIT-OS, including seru formation, seru loading
and seru scheduling. First, a SPS with an appropriate number of serus is configured by seru formation. Kaku et al.
(2009) [25] studied the seru formation problem by computational experiments, and figured out that the appropriate
number of serus when an assembly line was converted into SPS and the appropriate number of workers allocated toeach seru in different cases. Liu et al. (2013) [31] constructed a bi-objective mathematical model for seru formation,and investigated the training and assignment problem of workers when an assembly line is reconfigured into SPS.
Yu et al. (2012, 2013, 2014) [65, 66, 67] constructed a series of mathematical models to evaluate the performanceof converting an assembly line to SPS, and the mathematical characteristics were also analyzed. Shao et al. (2016)[48] developed a multi-objective combinational optimization model based on queuing theory for seru formation problems with stochastic customer’s orders. Yu et al. (2017) [68] studied seru formation problem from different aspects,
including mathematical models, complexities, properties, solutions and insights. Ren and Wang (2019) [44] studied
the effect of seru formation problem on the waiting time from the customer perspective, and investigated the average
waiting queue length changed by the line-seru conversion. They pointed out that this conversion can reduce the average waiting queue length in multi-variety and small-batch production. Zhang et al. (2020) [70] designed a PSO-basedalgorithm for the seru formation problem in an unbalanced SPS by considering the lot splitting and setup time. Afterthe SPS is configured, the customer-ordered products are allocated to serus properly by seru loading. For seru loading
problem, Lian et al. (2012) [29] dealt with seru loading problem to minimize the variable production cost of all serus
in SPS, and designed a heuristic algorithm according to the earliest due date (EDD) principle. Luo et al. (2016) [35]proposed a combinatorial optimization model for seru loading problems in divisional seru in a single period, and took
the worker-operation assignment into account. To minimize the makespan and the total tardiness penalty cost, Luo etal. [36] studied the seru loading problem under uncertainty, where the proportional coefficient, setup time, tardinesspenalty coefficient were fuzzy random variables. Then, Luo et al. (2019) [37] constructed a bi-level programming
model to address the seru loading problem with worker assignment, and designed a simulated annealing and genetic
algorithm-based method as the solution method. With the objective of minimizing the makespan, Sun et al. (2019)[51] developed a cooperative co-evolution algorithm which combined genetic algorithm and ant colony optimization
algorithm for solving both seru formulation and seru loading problems at the same time. At last, the production plan
will be obtained within the due date by seru scheduling. Unfortunately, due to the complexity, the studies on seru
scheduling problem in SPS are still very rare.
In this situation, the methodology of parallel machine scheduling (PMS) problem inspires us to solve seru scheduling problems because SPS is a typical parallel production system. Generally speaking, there are two main categories
of problems in PMS, including identical parallel machine scheduling and unrelated parallel machine scheduling prob3

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lems. Due to the differences of both worker skill levels in SPS and production efficiency of each seru, the unrelated
PMS problem is more likely to be similar to a seru scheduling problem. However, the seru scheduling problem ismuch more complicated than the unrelated PMS problem. For example, even in the simplest seru type of yatai, where
only one completely cross-trained worker takes charge of all tasks from start to finish, it is still an unrelated PMS
problem with worker element consideration, including worker heterogeneity with different skill levels, learning effects, and so on. In divisional seru, apart from these worker elements, matching between workers and tasks, as well as
worker assignment are needed to be considered. While in rotating seru, apart from allocating workers to the appropriate seru, the production efficiency of each seru will be determined by the slowest worker since the workers move one
after the other in the seru. Hence, the unrelated PMS problem is a special case of seru scheduling problems, and its
related effective theory and methods could be tested and applied in the new-type SPS. In fact, for unrelated PMS problems, many effective models and algorithms have been proposed. Fanjul-Peyro and Ruiz (2010) [16] proposed a setof simple iterated greedy local search-based metaheuristics for unrelated PMS problems, and the generated solutions
presented a very good quality in a very short amount of time. Vallad and Ruiz (2011) [55] provided a genetic algorithm and included a fast local search and a local search enhanced crossover operator for the unrelated parallel machinescheduling problem, in which machine and job sequence-dependent setup times were considered. Lin et al. (2013)[30] proposed two heuristics and a genetic algorithm (GA) to obtain non-dominated solutions to multiple-objective
unrelated PMS problems, and the computational results showed that the proposed heuristics were computationallyefficient and provided solutions of reasonable quality. Fanjul-Peyro et al. (2019) [17] proposed new mixed integer
linear programs and a decomposition algorithm for unrelated PMS problems, and obtained optimal solutions for extremely large instances of up to 1000 jobs and 8 machines. Liu and Lei (2020) [33] designed an artificial bee colonyalgorithm for distributed unrelated PMS problems with preventive maintenance to minimize makespan, and the wholeswarm was divided into one employed bee colony and three onlooker bee colonies. Ewees et al. (2021) [15] modified
a salp swarm algorithm based on the firefly algorithm to enhance the solution quality of unrelated PMS problems, andcarried out an extensive comparison to several existing metaheuristic methods. Cheng and Sin (1990) [13], Mokotoff(2001) [38], Edis et al. (2013) [14] proposed a survey of PMS problem research. In this paper, we first employ themethodology of unrelated PMS problems for seru scheduling problems, which is a new idea in SPS. Further, it will
provide a broader application area for PMS theory and methods.
Moreover, in seru production scheduling problems, the consideration of setup time between jobs is an essential
issue because a minimum time must elapse between consecutive jobs executed in the same seru. Setup time is thetime for preparing necessary resources, such as workers or tools, to perform a task, i.e., operation or job (Salvendy,
2001 [47]), and it has been proved to be important in some industrial applications. For example, the reactors must
be cleaned in chemical plants when changing from processing one mixture to another; also, in printed circuit board
assembly, it was reported that from 20% to 50% loss of available capacity may arise from setup activities (Trovinger
and Bohn, 2005 [54]). Here are two types of setup time in scheduling problems, including sequence-independentand sequence-dependent, respectively. If the setup time depends only on the job to be processed, then it is sequenceindependent; otherwise, if the setup time depends on both the job to be processed and its immediately preceding job,
it is sequence-dependent (Wilbrecht and Prescott, 1969 [58]; Lee et al., 1997 [26]). In this paper, we will consider
sequence-dependent setup time because it is sensitive and significant in SPS. Actually, the necessity of considering
sequence-dependent setup time in production scheduling problems has been recognized widely in several studies. Ruiz
and Maroto (2006) [46] designed a genetic algorithm which incorporated new characteristics and four new crossover
operators for a complex generalized flowshop scheduling problem with sequence-dependent setup time. Pearn et al.
(2008) [42] addressed the multi-stage wafer probing scheduling problem with reentry and sequence-dependent setup
time, and proposed two strategies to solve this problem for minimizing the total workload. Alfieri (2009) [1] studied apractical multi-objective flowshop scheduling problem with sequence-dependent setup time in a cardboard company,
and presented a simulation-based environment where the production sequence can be found by a tabu-search basedheuristic algorithm interactively. Nishi and Hiranaka (2013) [39] applied the lagrangian relaxation and cut generation
technique to solve sequence-dependent setup time flowshop scheduling problems, and the proposed problem withadditional setup time constraints was solved by a novel dynamic programming effectively. Pan et al. (2017) [40]proposed a total of nine algorithms for the hybrid flowshop scheduling problem with sequence-dependent setup times,
and conducted a set of computational experiments to demonstrate the effectiveness of algorithms. Li et al. (2020) [28]designed a machine position-based mathematical model and proposed an improved artificial bee colony algorithm forthe distributed heterogeneous hybrid flowshop scheduling problem with sequence-dependent setup time. Allahverdi
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et al. (1999, 2008, 2015) reviewed the scheduling problems with setup time in [2, 3, 4], including the sequencedependent setup time.
Also, for a given job, it takes less processing time when scheduled later than an earlier time of the whole productlife cycle. In other words, the learning effect occurs (Biskup, 2008 [9]). The consideration of learning effects in
SPS is also necessary because the average product life cycle, such as for producing electronics products, is morethan six months (Yokoi, 2014) [64]. Under these circumstances, the learning effect of both partially and completely
cross-trained workers in serus is inevitable over a large time span. For example, given the assumption of assigning
10 yatais to assemble product A in seru production system (SPS), both the processing time and production efficiencyof SPS at the last day will differ from the first day due to worker learning effects during the product life cycle (i.e.,
180 days). Moreover, the production efficiency of each yatai varies due to the different improvements from workers’
learning effects, and the processing time required will be 60% or 80% of the original processing time. Hence, theseworker element considerations, including the worker learning effects, are essential differences of seru schedulingproblems compared to unrelated PMS problems, and considering learning effects in SPS is of great significancein practical productions. In fact, by following the first research on the learning effect from Wright (1936) [59] inaircraft industry manufacturing, many scholars considered the learning effect in production scheduling and proposeda large variety of position-based learning effect models. At the beginning, the learning effect in Wright’s model isdepicted as a log-linear cost model: C x = C1 xb , where C1 is the cost for producing the first unit product, C x is the
cumulative average cost for producing x units, and b ≤ 0 is the learning index. In this learning effect model, C x will
decrease when x increases evidently. In Biskup (1999) [8], the learning effect in a production scheduling problem
is: p jr = p̄ j ra , where p̄ j is the original processing time of job j, p jr is the actual processing time of job j in the rth
repetition (i.e., the position r of a schedule), and a ≤ 0 is the learning index. Similarly, Low and Lin (2011) [34]PP
used p jr = p̄ j ( nj=r p[ j] / nj=1 p[ j] )a br−1 to describe the position-weighted learning effect in a production schedulingPa bproblem, and p jr = p̄ j (1 + r−1
k=1 βk ln p̄[k] ) r in Cheng et al. (2013) [12]. Many other extensions of the position-basedlearning effect model have been proposed, such as Wang and Wang (2013) [56], Wu et al. (2016) [60], Cheng et al.
(2019) [11]. Unfortunately, all of these learning effect models mentioned above expose a common drawback: if there
are a large number of jobs, then p jr is close to zero if this job is processed in a later sequence. Obviously, that is
not going to happen in production practice. In this case, described by DeJong’s learning curve, a new learning effectmodel was constructed as
T s = T 1 (M + (1 − M)/sm )(1)
to cope with this defect (Badiru, 1992 [6]). In Eq (1), T 1 is the processing time for the first cycle of a batch, T s is theprocessing time for the sth cycle, 0 ≤ M ≤ 1 is the incompressibility factor, and 0 < m < 1 represents the reduction
exponent. When M = 0, Eq. (1) is transformed into Wright’s log-linear learning effect model to imply a completely
manual operation, and M = 1 represents a completely machine-dominated operation p jr = p j , respectively. Obviously,in Eq. (1), the processing time of the sth cycle will fall according to the increasing s, but it will be convergent to acertain limit T 1 M. Therefore, the drawback of other learning effect models are overcome by DeJong’s learning curve.
In this paper, DeJong’ model will be used to depict the learning effect in seru scheduling problem in SPS.
The remainder of this paper is organized as follows: the mixed-integer programming (MIP) model is formulated
in section 2, including a detailed description for the scheduling problem in seru production systems by considering
sequence-dependent setup time and DeJong’s learning effect. Then, the seru scheduling MIP model is decomposedby the logic-based Benders decomposition (LBBD) method in section 3, and the solution methodology is proposed
to include Benders cuts and find sub-optimal solutions in section 4. In section 5, computational experiments are
conducted and the results are reported and analyzed. The conclusions and further research are made in section 6.
2. Model formulation
The mixed-integer programming (MIP) model of the seru scheduling problem with the sequence-dependent setup
time and DeJong’s learning effect will be formulated in this section.
2.1. Problem description
In the seru scheduling problem of this paper, a set of jobs j ∈ J ≡ {1, 2, · · · , n J } will be scheduled on a set ofparallel serus i ∈ I ≡ {1, 2, · · · , nI } to minimize the makespan of SPS. Each seru starts from time zero onward and
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handles no more than one job at a time. Besides, preemption of jobs is not allowed in SPS. The jobs are processed
contiguously from time zero onward, and no seru is idle before all jobs are started. Each job has a processing timerequired to be processed, and the processing time of job j in the rth repetition on seru i considering DeJong’s learning
effect is pijr = pij (M + (1 − M)rb ), where pijr represents the processing time of job j in the rth repetition on seru i, pij
is the single processing time of job j on seru i, M is the incompressible factor 0 ≤ M ≤ 1, and b is the learning index−1 ≤ b ≤ 0. The setup time sij j0 considered in this paper are both sequence and seru dependent, i.e., the setup time on00

seru i between job j and j is different from that on the same seru i between job j and j. Moreover, the setup time000
between job j and j on seru i is different from that between job j and j on seru i . Generally, the setup time in SPS
also comply with the triangle inequality sij j0 ≤ sij j00 + sij00 j0 .
Now, define a partial schedule on a seru to be a schedule which is formed by a subset of J jobs on this seru.
Thus, a schedule for a seru scheduling problem consists of nI partial schedules, i.e., one for each seru, where nI is
the quantities of serus in SPS. Further, for a given seru scheduling problem, there is a predetermined job ordering
restriction in SPS: for each job j, a set of jobs in J must be scheduled before or after job j. Hence, we can define afeasible partial schedule on a seru as a partial schedule on this seru and this partial schedule satisfies the given job
ordering restriction. Let
0

0

0

0

Aij = { j ∈ J| job j can succeed job j in a feasible partial schedule on seru i}
Bij = { j ∈ J| job j can precede job j in a feasible partial schedule on seru i}and(

1, if job j is processed first on seru i;0, otherwise.

(

1, if job j is processed last on seru i;0, otherwise.

yi0 j =
yij,nJ +1 =

(2)

(3)

where n J is the quantity of jobs needing to be scheduled in SPS. Therefore, the seru scheduling optimization problem
in this paper concerns two parts: (1) determine how to assign the jobs to serus, (2) determine the job sequence
processed on each seru, where the sequence-dependent setup time and DeJong’s learning effect are considered tominimize the makespan.
2.2. Notation(1) Indicesi seru index, i ∈ I ≡ {1, 2, · · · , nI }jjob index, j ∈ J ≡ {1, 2, · · · , n J }rposition index, r ∈ {1, 2, · · · , n J }(2) Parameterspij
normal processing time of job j in seru i
pijr

actual processing time of job j at the rth position in seru i considering the learning effectMincompressible factor, 0 ≤ M ≤ 1blearning index, −1 ≤ b ≤ 0

cjcompletion time of job j

LCT ilatest completion time of jobs in seru i

sij j0setup time from job j to j on seru ist jsetup time factor of job j, st j ≥ 0

Cmax

maximum completion time of the whole SPS (makespan)VJ0

a large positive number
set of jobs to be scheduled with an additional dummy node which is indexed by 0

0

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(3) Decision variables(
1, if job j is assigned to seru i;ixj =0, otherwise.
(0
1, if job j is processed immediately after job j in seru i;
yij j0 =0, otherwise.
(
1, if job j is assigned in position r in seru i;zijr =
0, otherwise.
2.3. Modeling
The objective of the seru scheduling problem considered in this paper is to minimize the makespan, which isusually used in the parallel production system because the schedules with low makespan tend to balance the workloadin the whole system (Pinedo, 1995 [43]). Since the makespan is denoted as the maximal completion time of jobs inall serus, i.e., Cmax = maxi∈I {LCT i }, hence:
min Cmax(4)
where
LCT i =

PnJ P
r=1

i∈I

P

j∈J



yij j0 sij j0 + zijr pijr , ∀i ∈ I(5)Cmax ≥ LCT i , ∀i ∈ I(6)pijr = pij (M + (1 − M)rb ), i ∈ I, j ∈ J, r ∈ {1, 2, ..., n J }(7)

sij j0 = st j

r−1
X

0

pijυ , i ∈ I, j ∈ J, j ∈ J\ j, r ∈ {2, 3, · · · , n J }

(8)

υ=1

In Eq. (5), for each seru i, the latest completion time is equal to the sum of processing time and setup time for all jobs
in seru i. For Eq. (6), the makespan Cmax is the maximal completion time of all serus, hence, Cmax is greater than orequal to LCT i .
Further, to ensure that each job is assigned to only one seru, the set of constraints are employed.
IX

xij = 1, ∀ j ∈ J

(9)

i=1

where x0i = 1 places the dummy job in a seru, i.e., signifying the start and end of a sequence of jobs. Also, for seru iin SPS, each position r can be only occupied by one job j and each job j can only appear in one position r, so
Pn J ir=1 z jr = 1, ∀i ∈ I, j ∈ J
Pn J i(10)
j=1 z jr = 1, ∀i ∈ I, r ∈ {1, 2, · · · , n J }Moreover, since each job j has only a single predecessor and successor in seru i, thusXX0yij j0 +yij j0 = 1, i ∈ I, j ∈ J, j ∈ J\ jj0 ∈Aij ∪{0}where ifPij0 ∈Aij ∪{0} y j j0

(11)

j0 ∈Bij ∪{0}

= 1, then job j is scheduled early; and ifP

j0 ∈Bij ∪{0}yi 0jj= 1, then job j is scheduled tardily. In

addition, the condition that only one job can be scheduled first in each seru can be guaranteed byX
yi0 j ≤ 1, ∀i ∈ I, j ∈ J
j∈J

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(12)

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A job j can only have a predecessor in a seru i if this job also has a successor in the same seru,XX0yij j0 =yij0 j , ∀i ∈ I, j ∈ J, j ∈ J\ jj0 ∈Bij ∪{0}(13)

j0 ∈Aij ∪{n J +1}00And, if job j precedes job j in seru i, then the earliest completion time of job j must be greater than or equal to the00sum of the completion time of job j, the setup time from job j to job j and the processing time of job j , hence,0

c j0 + V(1 − yij j0 ) ≥ c j + sij j0 + zij0 r × pij0 r , i ∈ I, j ∈ J, j ∈ J\ j, r ∈ {1, 2, · · · , n J }(14)c0 = 0(15)

0From Eq. (14), we know that if job j is processed immediately after job j in seru i, then yij j0 = 1, 1 − yij j0 = 0, and0
this constraint is simplified as c j0 ≥ c j + sij j0 + zij0 r × pij0 r . On the contrary, if job j is not processed immediately after
job j in seru i, then yij j0 = 0, and the large positive number V makes this constraint non-binding. Therefore, Eq. (14)ensures that a valid job sequence will be scheduled in each seru, and the processing time overlap can be avoided.
Based on the discussions above, the mixed-integer programming (MIP) model for the seru scheduling problem
considering sequence-dependent setup time and DeJong’s learning effect can be constructed as:
min Cmax








Pn J P P  i i

i i




LCT
=
y
0 s 0 + z jr p jr , ∀i ∈ I

i

i∈I
j∈J
r=1


jj jj










C
≥
LCT
,
∀i
∈
I


max
i













pijr = pij (M + (1 − M)rb ), i ∈ I, j ∈ J, r ∈ {1, 2, ..., n J }









P

0



i


sij j0 = st j r−1υ=1 p jυ , i ∈ I, j ∈ J, j ∈ J\ j, r ∈ {2, 3, · · · , n J }







PI i








i=1 x j = 1, ∀ j ∈ J






Pn J i










r=1 z jr = 1, ∀i ∈ I, j ∈ J



P


nJ

i


s.t. 

j=1 z jr = 1, ∀i ∈ I, r ∈ {1, 2, · · · , n J }






P
P

0


i
i





j0 ∈Aij ∪{0} y j j0 +j0 ∈Bij ∪{0} y j j0 = 1, i ∈ I, j ∈ J, j ∈ J\ j







P



i





j∈J y0 j ≤ 1, ∀i ∈ I, j ∈ J







P
P
0



i
i





j0 ∈Bij ∪{0} y j j0 =j0 ∈Aij ∪{n J +1} y j0 j , ∀i ∈ I, j ∈ J, j ∈ J\ j








0





c j0 + V(1 − yij j0 ) ≥ c j + sij j0 + zij0 r × pij0 r , i ∈ I, j ∈ J, j ∈ J\ j, r ∈ {1, 2, · · · , n J }















c0 = 0










0




 xij ∈ {0, 1}, yij j0 ∈ {0, 1}, zijr ∈ {0, 1}, i ∈ I, j ∈ J, j ∈ J\ j, r ∈ {1, 2, · · · , n J }(16)

The proposed MIP model (16) has four decision variables, and they represent the decision making associated witha job: xij , yij j0 , zijr and c j , and define the seru that a job is processed on, the sequence of processing, the position of job
in the seru, and the completion time, respectively.
3. Logic-based Benders decomposition (LBBD) method
As a generalization of Benders decomposition (Benders, 1962 [7]), LBBD was introduced by Hooker (2000)[21] and refined by Hooker and Ottosson (2003) [22] for solving highly combinatorial problems, such as planingand scheduling (Hooker, 2007 [23]). In fact, LBBD has been applied successfully to a wide range of combinatorial
optimization problems, including bin-packing (Pisinger and Sigurd, 2007 [41]), location-allocation (Fazel-Zarandi
and Beck, 2012 [18]), inventory-location (Wheatley et al., 2015 [57]), scheduling problem (Hooker, 2007 [23]; Sunet al., 2019 [52]), home health care delivery (Heching et al., 2019 [19]), etc. In this section, the LBBD method is
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also used to decompose the seru scheduling MIP model into a master problem and a set of independent single seru
scheduling subproblems.
Generally, to decompose a problem by the LBBD, the first step is to partition the decision variables into two
vectors x and y, and then the problem can be viewed as
min f (x, y)(17)s.t. (x, y) ∈ C(18)

x ∈ D x , y ∈ Dy

(19)

where f is a real-valued objective function, C is the feasible set defined by the collection of constraints containing
variables x, y, D x and Dy are the domains of x and y, respectively. Fix x to be a given value xh ∈ D x , and the following
subproblem is obtained:
min f (xh , y)(20)s.t. (xh , y) ∈ C(21)y ∈ Dy(22)

The feasible set C is relaxed as C, which is the constraint that results from fixing x = x in D x . The inference dualof the subproblem is the problem of inferring the tightest possible lower bound on f (xh , y) from C. Different fromthe classical Benders decomposition, there are no structural restrictions in LBBD, such as linearity, on the different
components of the decomposition. Formally, in iteration h, the master problem can be redefined as (Hooker, 2007[23]):
min z
(23)
h

s.t. x ∈ C
z ≥ Bxh (x),(24)

h = 1, 2, · · · H − 1(25)z ∈ R, x ∈ D x(26)

where z is a real-valued decision variable, Bxh (x) is a Benders cut on the objective function f in iteration h, x , x , · · · xH−1
are solutions of the previous H − 1 master problems, and constraints (25) are derived from solving the subproblem.
The process of solving a LBBD model is as follows: in iteration h, the solution xh is produced by solving themaster problem to optimality. Then, xh is used to formulate the subproblems, and each subproblem is solved byproducing bounding functions, i.e., Benders cuts. Let yh be the subproblem solution, and if the h-th master problem
solution satisfies all the Benders cuts gained from iteration 1 to h, then the process will converge to a globally optimal
solution (xh , yh ). Otherwise, solve the master problem again and h := h + 1. Repeat the process iteratively until themaster problem and the subproblems are convergent.
1

2

3.1. Master problem
In the master problem, all jobs are assigned to serus in SPS by the decision variable xij . Different from the seru
scheduling MIP (16), the master problem is a relaxation of (16), which means that when assigning all jobs to serus,
multiple disjoint sequences are allowed instead of a single determined sequence of jobs in each seru. Essentially, thedecision variable c j , and constraints (14) and (15) are removed from the proposed seru scheduling MIP model in themaster problem. Moreover, the decision variables yij j0 and zijr are relaxed to be any real valued number between 0 and
1. Thus, the master problem is:
min Cmax(27)
s.t. constraints (5) − (13)Benders cut(28)

xij ∈ {0, 1}(29)

0 ≤ yij j0 ≤ 1(30)

0 ≤ zijr ≤ 1(31)

Constraint (28) is the Benders cut, which will be defined in subsection 4.1. When solving the problem, if the makespanof any subproblem is larger than that of its master problem, the Benders cut will be added.
93.2. Sequencing subproblems
According to the assignment from the master problem, the subproblems will find the optimal schedules in eachih∗ ih∗h∗ih∗seru. Let xih∗j , y j j0 , z jr , and C max be the solution obtained from the master problem in iteration h, where x j provides anassignment of each job to one of the nI serus. Hence, nI separate subproblems will be created, and each subproblem
represents one seru and contains only these assigned jobs, for example, for seru i, only jobs j where xih∗
= 1 isjcontained. Further, given a fixed assignment xih∗,thesequence
of
jobs
in
a
seru
does
not
affect
any
other
seru
in
SPS,
j
therefore, nI subproblems could be solved independently.
Since a subproblem of the seru scheduling problem needs the sequence of all the assigned jobs to minimize themakespan, it is similar to find a Hamiltonian cycle, i.e., a tour that passes through all the nodes, with the minimal
sum of edge weights. Let the complete graph G = (V, E, W) denote the subproblem, where V is the set of job nodes,
0
E = ( j, j ) is the set of edges, and W is the edge weight which is equal to pijr + sij j0 . If j is the first job to be processedin seru i, then the edge weight from the dummy node 0 to job j is equal to the setup time of job j, i.e, si0 j ; if j is thelast job to be processed in seru i, then the edge weight from job j to the dummy node 0 is equal to the processing
time of job j, i.e, pijr . Due to the different W between any two jobs, the subproblem can be finally modelled as an
asymmetric travelling salesman problem (ATSP) similarly. The ATSP representation example for a subproblem of theseru scheduling problem containing four jobs is shown in Fig. 2.
p1ir  s12ii
p2i r  s21
i
s02

p2i r

i
s01

i
p2i r  s24i
p3i r  s31

p1ir

job 0

p3i r
i
s03
job 3

job 2

i
p4i r  s42

job 1

p1ir  s13i

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i
s04

p
i
p3i r  s34ii
p4 r  s43

i
4r

job 4

Figure 2: ATSP representation for a subproblem

From Fig. 2, it can be seen clearly that if the order of jobs in seru i to be processed is the sequence 1 → 3 → 4 → 2,then the distance travelled would besi01 + pi11 + si13 + pi32 + si34 + pi43 + si42
In other words, the tour distance is equal to the makespan of processing four jobs in this order. It is also observed thata solution to the subproblem corresponds to a minimal length sequence of jobs.
4. Solution methodology
In this section, the solution methodology will be provided for the LBBD model, including developing the Benders
cut and finding sub-optimal solutions.
4.1. Benders cut
When solving nI subproblems, if the makespan of a seru i is less than or equal to the makespan Cmax of the masterproblem, then the solution is feasible regarding the master problem and no cut needs to be added. Otherwise, theBenders cut is created and the master problem is updated.
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Given a set of jobs that are assigned to the same seru i from iteration h of the master problem, and it is denotedhas J ih = { j : xih∗
j = 1}. In order to define the cut, the maximal setup time (i.e., max S T j ) for job j is introduced first,
where job j directly succeeds another job that is assigned in the master problem to the same seru i in iteration h , i,e, 
max S T hj = 0 max0 sij0 j(32)j ∈J hi ; j , j

Let T AT ih
j be the total assembly time of job j in seru i, andhi
T AT ihj = max S T j + p jr

(33)

Then, the cut used in this paper is
ih∗Cmax ≥ Cmax−X
1 − xij T AT ihj(34)

j∈J ihih∗where Cmax
is the makespan found in iteration h for seru i.
Thus, depending on the jobs that are assigned, this cut places a lower bound (LB) on the makespanin nextP
iterations. It means that if the same assignment is given to the subproblem, then xij = 1, and j∈J ih 1 − xij T AT ih
j =0ih∗
in Eq. (34). In this case, the makespan of subproblem Cmax is a new LB on the makespan of master problem Cmax .
Otherwise, a different assignment is made to the subproblems, and at least one of the xij = 0. Thus, the makespan in thesubsequent iteration is bounded by the makespan found in the subproblem minus the corresponding T AT ih
j value(s),


P
ih
ih∗
i
i.e., Cmax − j∈J ih 1 − x j T AT j . The cut employed in this paper presents the two following properties shown in
Theorem 4.1 and 4.2.

Theorem 4.1. The cut proposed in Eq. (34) must remove the current solution from the master problem space.
Proof. Suppose that in the subsequent iteration, there is an exactly same set of jobs being assigned to seru i. Then,the master problem must increase the value of the makespan. Otherwise, for seru i, a change must be made to thejob assignment. Therefore, the current solution in iteration h is removed from the master problem space in either
case.
Theorem 4.2. The cut proposed in Eq. (34) does not remove any globally optimal solution of the seru schedulingproblem.
Proof. In order to prove Theorem 4.2, an assumption that there is a globally optimal schedule violating the cut in Eq.
(34) is made first, and then the contradiction holds.
Let J i be a set of jobs assigned to seru i in the current iteration, and C J i be the optimal makespan of seru i. Assume
that there is a globally optimal schedule which violates the cut generated from Eq. (34) for seru i in iteration h. Now,
let J i∗ be the set of jobs assigned to seru i corresponding to this globally optimal schedule, and C J i∗ be the makespan.
Since there are J¯i := J i − J i∗ jobs not being assigned from J i to seru i, so for j ∈ J¯i , xij = 0. Hence we have thefollowing property due to the violation:X
C J i∗ < C J i −T AT ih(35)
jj∈ J¯i

Given the schedule corresponding to C J i∗ , define a reduced schedule that contains Jˆi := J i∗ ∩ J i jobs, which areassigned in the same order as the globally optimal schedule. Let C Jˆi be the makespan of the reduced schedule in seru
i, because the setup time satisfies sij j0 ≤ sij j00 + sij00 j0 , so C Jˆi ≤ C J i∗ . Hence, the reduced schedule also violates the cut
Eq. (34), i.e.,
X
C Jˆi < C J i −T AT ih(36)
jj∈ J¯iAt the end of the reduced schedule, each job is placed one by one in J¯i , and the reduced schedule will be extended to0
a schedule containing all J i jobs now. Because the max S T hj is the maximal setup time of job j, so the makespan C J i
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satisfies:

C J i < C Jˆi +X

= C Jˆi +X

0pijr + max S T hjj∈ J¯i(37)

T AT ihjj∈ J¯iSo we have0

C Jˆi ≥ C J i −X

T AT ihj(38)j∈ J¯i0

Further, C J i is the optimal, i.e., minimal, makespan of seru i, hence, C J i ≤ C J i . Then,X
C Jˆi ≥ C J i −T AT ihjj∈ J¯i(39)

Therefore, the contradiction occurs between Eq. (36) and Eq. (39), and we can conclude that the cut from Eq. (34)will not remove any globally optimal solution.
Based on Theorems 4.1 and 4.2, we know that the cut from Eq. (34) used in this paper is a valid cut.
4.2. Sub-optimal solutions
In this subsection, a method is provided for finding sub-optimal solutions of LBBD to obtain the globally feasible
solutions. This method is about maintaining the best solution found up to now. When solving the master problem, afeasible solution will assign job j to seru i and nI subproblems are constructed. Ignore the Cmax value in the masterproblem, and the maximal makespan over all nI subproblems is globally feasible. Thus, the globally feasible solution
for each feasible master solution can be found, and the best schedule found so far can be tracked. Following thisway, a feasible schedule for the global seru scheduling problem exists as long as the first feasible master solution is
obtained and nI subproblems are solved.
Also, this method offers another stopping criterion of LBBD except for the terminate condition mentioned in
section 3, i.e., when an optimal master solution is found and its makespan is equal to the best global solution found so
far, LBBD can also be stopped.
5. Computational experiments
To test the performance of the LBBD method compared to solving the MIP model in Eq. (16) directly for theseru scheduling problem, computational experiments are made and the performances are analyzed. Both the LBBD
method and MIP model are coded in MATLAB R2019a by combining with Concorde TSP, and they are tested on 10th
GEN Intel Core i7-10510U CPU (in 32-bit mode), 16 GB main memory, 1TB SSD, running on Windows 10.
5.1. Experiment settings
The parameters of test problems for LBBD are generated in Table 1.

Table 1: Parameters used in the experiments

Parameters
nI
nJ
pijr
learning index
setup time factor
V

Value{2,5,8}[10, 50], by the increment of 10 jobs
uniform distribution U[5, 100]b = −0.7, M = 0.5
st j = 0.5
10612In addition, to obtain the sequence-dependent setup time by satisfying the triangular inequality, the Manhattan distance is used in this paper. Supposed that for each seru i in SPS, each job j in seru-job pair (i, j) is given two different
sets of coordinates (xij1 , yij1 ), (xij2 , yij2 ) on the Cartesian plane according to the triangular inequality assumption, where
the coordinates along the x and y axis are generated by the uniform distribution U[0, 50]. The asymmetric setup time00of job j to j are the Manhattan distance from the coordinates of j to j . Let l and u be the lower and upper boundof the setup time, respectively, then the Manhattan distance is used to provide the setup time by linearly scaling a
distance of 0 to l of the setup time distribution and 100 to u. In this paper, set l = 25 and u = 50, respectively.
5.2. Results
The scatter plots of the pairwise comparisons between the LBBD and MIP model are shown in Fig. 3, and thetime limit is set to be 4 hours (14400 seconds) for each instance. In Fig. 3, each point corresponds to an instance,
and the time is recorded at which an optimal solution are found. The format of the graphs means that points below
the y = x line demonstrate a superior performance. The points at the right side (x = 14400) in Fig. 3 represent the
instances that the MIP model can not find optimal solutions within 4 hours. Obviously, the LBBD method presents
a greater improvement than the MIP model. Except for two instances which are solved in less than 10 seconds, the
LBBD method outperforms MIP model.
Performance comparision
100000

10000

1000

LBBD

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100

10

1

1

10

100

1000

10000

100000

MIP model

Figure 3: Pairwise runtime comparison between LBBD and MIP model

Table 2 shows more detailed results. It can be seen that the CPU runtime of both the LBBD and MIP modelincreases dramatically along with the growing quantity of jobs n J . The LBBD method provides a vast improvement
over the MIP model, which can only solve instances with up to n J = 20. Even with only 20 jobs, and 5 or more serus,
the MIP model can not gain the optimal solutions for all instances. On the contrary, the LBBD method can solve all
instances of the seru scheduling problem within an acceptable time. Hence, it can be concluded that the LBBD has afaster calculation speed than the MIP for exact solutions.
5.3. Test on large instances
In order to show the superiority of LBBD, large instances with nI = {10, 15, 20} and n J = {200, 600, 100} aretested, and other parameters are set as the same in Table 1. Meanwhile, a metaheuristic algorithm, i.e. adaptive
genetic algorithm (A-GA), is designed for a comparative analysis. Because the crossover probability pc and mutation
probability pm will affect the convergence directly, self-learning pc and pm are adopted to calibrate A-GA parameters
so as to prevent GA from falling into local optimal solutions (Ho et al., 2007 [20]). The adaptive adjustments of pc
and pm are as follows:(pcmax −pcmin )(Fita −Fit s ), Fita > Fit s pcmax −Fita −Fitmin
pc = 
(40) p, Fit ≤ Fit
cmax

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Table 2: CPU runtime and unsolved instances comparisons between LBBD and MIP model

nJ

nI

10

2
5
8
2
5
8
2
5
8
2
5
8
2
5
8

20

30

40

50

LBBD
average runtime
number of(seconds)
unsolved instances
0.57
0
2.36
0
5.13
0
3.91
0
88.67
0
302.5007.93
0553.2201562.19014.96
01026.1003685.46
059.37
02089.6105211.980MIPaverage runtime(seconds)
1.22
14.74
30.85
1010.427984.3710246.81
6927.35
14400.00
14400.00
14400.00
14400.00
14400.00
14400.00
14400.00
14400.00

(pmmax −pmmin )(Fita −Fitcan ) pmmax −Fita −Fitmin
pm = 
 p
mmax

number ofunsolved instances
0
0
0
0
6
13
8
22
22
22
22
22
22
22
22

,

Fita > Fit s,

Fita ≤ Fit s(41)

where pcmax , pmmax and pcmin , pmmin are the upper and lower bounds of pc and pm , and equal to 0.9, 0.5, 0.6, 0.1,
respectively (Chen et al., 2020 [10]). Fita is the average fitness of the population, Fit s is the smaller fitness valueof any two crossover chromosomes, Fitmin is the best fitness of the current population, and Fitcan in Eq. (41) is thefitness value of the candidate mutation individual. Further, the deviation (RD) indicator of makespan obtained by the
LBBD and A-GA is employed, andC A−GA − C LBBD
RD = max LBBDmax × 100%
Cmax
After 600 runs of the A-GA with pop size = 300, GEN = 500, the average runtime of both the LBBD and A-GA,
as well as RD are reported in Table 3.
Table 3: Results of large instances for LBBD and A-GA

nJ

nI

200

10
15
20
10
15
20
10
15
20

600

1000

Average runtime (seconds)
LBBD
A-GA
9124
4678
11026
5071
13579
5935
9684
7939
12697
8447
13970
9241
12166
9626
14002
10177
14400
11595

RD (%)
8.65
9.33
18.61
3.39
16.02
10.68
20.45
4.39
–

The results from Table 3 show that LBBD is still effective and able to handle larger instances for seru schedulingproblems (except for the largest case with n J = 1000 and nI = 20, and it is also solvable but the runtime time exceeds
14400 seconds). It is interesting to observe that the runtime required by the LBBD grows significantly as the numberof nI increases, but it is more stable in A-GA. Fig. 4 presents the scatter diagram of RD. It can be seen that the solution
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returned by A-GA is not always satisfying compared to the optimal solution obtained by the LBBD method, and theunstable RD indicates that it does not have any specific rule to follow. Overall speaking, the average runtime of the
A-GA is less than that of the LBBD method, but the quality of solution returned by the A-GA is worse. Thus, in real
production practice, the LBBD method is indeed a very good choice for managers because apart from the scalability
for large instances, it can obtain exact solutions of seru scheduling problems with an acceptable time compared toother exact methods (such as MIP model), and can get higher-quality solutions compared to metaheuristic algorithms(such as A-GA).

20.45%
18.61%
16.02%

10.68%

9.33%
8.65%

4.39%

3.39%

10

15

20

10

15

20

10

15

Figure 4: Scatter diagram of the relative deviation (RD)5.4. Sensitivity analysis
In order to scrutinise the management insights to production practice, sensitivity analysis for both the sequencedependent setup time and Dejong’s learning effect are conducted in this subsection.
5.4.1. Change of sequence-dependent setup time
To verify the effect of the sequence-dependent setup time on seru scheduling problems, a sensitive analysis ofsetup time factor st j is made. Without loss of generality, st j = {0.2, 0.4, 0.6, 0.8} are selected to compare with original
st j = 0.5. Four combinations of serus and jobs n J = 10, nI = 2, 5, 8; n J = 50, nI = 2, 5, 8; n J = 200, nI = 10, 15, 20;n J = 1000, nI = 10, 15, 20 are tested. The results are shown in Table 4, and the mean of absolute deviation is equal tost =0.5jCmax

st =0.2

j− Cmaxst =0.5

j+ Cmaxst =0.4

j− Cmaxst =0.5

j+ Cmaxst =0.6

j− Cmaxst =0.5

j+ Cmaxst =0.8

j− Cmaxst =0.5j
4 × Cmax

The results in Table 4 demonstrate that the sequence-dependent setup time has a significant impact on the systemrobustness performance of SPS. Along with the change of setup time factor st j from 0.2 to 0.8, the mean of absolutedeviation of the makespan Cmax is changed only from 0.0066 to 0.0497. Meanwhile, when the job number becomeslarger, the deviation is more stable. Overall speaking, although the proportion of the sequence-dependent setup timeto processing time will lead to the longer flow time, the robustness of SPS is guaranteed. Hence, it can be concluded
that the sequence-dependent setup time should be given as an explicit consideration in seru scheduling problems.
5.4.2. Change of Dejong’s learning effect
In order to test the Dejong’s learning effect on seru scheduling problems explicitly, a sensitive analysis of theincompressibility factor M ( 0 ≤ M ≤ 1) is made. When M = 0, the learning effect is the strongest, and Dejong’s
learning effect is transformed into Wright’s log-linear learning effect to imply a completely manual operation. On
the contrary, when M = 1 there is no learning effect, and it represents a completely machine-dominated operation,
respectively. Thus, we also select n J = 10, nI = 2, 5, 8; n J = 50, nI = 2, 5, 8; n J = 200, nI = 10, 15, 20; n J =1000, nI = 10, 15, 20 and M ∈ [0, 1] (with an increment of 0.1) to study the sensitivity analysis of M to the makespan
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Table 4: Results summary of four combinations

nJ

nI

10

2
5
8
2
5
8
10
15
20
10
15
20

50

200

1000

st j = 0.5
41.33
14.25
6.07
374.31211.97138.02319.93
192.96115.01
11743.217096.794947.33

Makespan Cmax
st j = 0.2 st j = 0.4 st j = 0.6
40.13
40.69
41.75
13.99
14.08
14.62
6.02
6.05
6.10
368.24372.18380.82206.38208.70216.77133.94135.79142.28308.62315.27325.67187.28190.35199.92110.32112.49120.9811364.24 11597.41 11987.22
6885.196927.57
7241.364762.874821.585036.81st j = 0.8
42.02
14.98
6.13
383.26220.84149.61331.78202.36124.7112057.39
7448.215247.33

Mean ofabsolute deviation
0.01780.02680.00660.01580.02660.04010.02620.03190.04970.02310.03090.0354

Cmax for seru scheduling problems. The detailed results for each case with different values of M are shown in Fig. 5
to 8.
In general, the learning effect has a significant influence on the makespan Cmax for seru scheduling problems.
Cmax usually obtains the maximum value, i.e., the worst one, when M = 1 (no learning effect at all) in each case.
Meanwhile, Cmax and the processing time do not decrease continuously, but stabilize to a fixed value eventually eventhough there are a large number of jobs such as n J = 1000 in Fig. 8. Hence, the advantages of DeJong’s learning
effect compared to other learning effects are also verified. Moreover, compared with the slope of learning curves
from n J = 10 to n J = 1000, it is obvious that with more n J jobs, the learning effect becomes more evident. In
addition, we also find that with more evenly number of jobs assigned to each seru, the makespan Cmax is smaller. For
example, Cmax (n J = 10, nI = 2) − Cmax (n J = 10, nI = 5) > Cmax (n J = 10, nI = 5) − Cmax (n J = 10, nI = 8) andCmax (n J = 1000, nI = 10) − Cmax (n J = 1000, nI = 15) > Cmax (n J = 1000, nI = 15) − Cmax (n J = 1000, nI = 20). Thisphenomenon complies with the ‘group balance principle’ that suppose there are n J jobs to be assigned to nI serus,
to achieve group balance of SPS, the number of jobs in each group is either d nnJI e or d nnJI e − 1 . Therefore, production
managers of SPS should make full considerations of ratio nnJI to achieve a SPS balance and gain high production
efficiency.
6. Conclusion
This paper focuses on the scheduling problem in seru production system considering the sequence-dependent
setup time and DeJong’s learning effect to minimize the makespan. A mixed-integer programming (MIP) model isdeveloped, then logic-based Benders decomposition (LBBD) method is applied to reformulated the proposed model.
Computational experiments are made, and the results indicate that the LBBD method has a good scalability andperformance to generate optimal solutions for seru scheduling problems. Compared with MIP model in small cases,
the LBBD method has a faster calculation speed for exact solutions. In addition, compared with A-GA in large cases,
LBBD method can get higher-quality solutions.
Future research should concentrate on improving the computational speed of the LBBD method, especially on
analyzing properties of the subproblems which can be transformed into an asymmetric travelling salesman problem.
Also, applying the proposed model and LBBD method to divisional seru and rotating seru should be concerned. Bothareas are important and should be studied in the future.
Acknowledgement
This research was sponsored by National Natural Science Foundation of China (Grant No. 71401075, 71801129,
71871175), the Natural Science Foundation of Jiangsu Province (Grant No. BK20180452), and the Fundamental
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Figure 5: Cmax with different values of M
(n J = 10, nI = 2, 5, 8)

Figure 6: Cmax with different values of M
(n J = 50, nI = 2, 5, 8)

Figure 7: Cmax with different values of M
(n J = 200, nI = 10, 15, 20)

Figure 8: Cmax with different values of M
(n J = 1000, nI = 10, 15, 20)

Research Funds for the Central Universities (Grant No. 30920010021). We would like to give our great appreciation
to all the reviewers and editors who contributed this research.
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