Chance-constrained stochastic assembly line balancing with branch, bound and remember algorithm

Abstract

Assembly lines are widely used mass production techniques applied in various industries from electronics to automotive and aerospace. A branch, bound, and remember (BBR) algorithm is presented in this research to tackle the chance-constrained stochastic assembly line balancing problem (ALBP). In this problem variation, the processing times are stochastic, while the cycle time must be respected for a given probability. The proposed BBR method stores all the searched partial solutions in memory and utilizes the cyclic best-first search strategy to quickly achieve high-quality complete solutions. Meanwhile, this study also develops several new lower bounds and dominance rules by taking the stochastic task times into account. To evaluate the performance of the developed method, a large set of 1614 instances is generated and solved. The performance of the BBR algorithm is compared with two mixed-integer programming models and twenty re-implemented heuristics and metaheuristics, including the well-known genetic algorithm, ant colony optimization algorithm and simulated annealing algorithm. The comparative study demonstrates that the mathematical models cannot achieve high-quality solutions when solving large-size instances, for which the BBR algorithm shows clear superiority over the mathematical models. The developed BBR outperforms all the compared heuristic and metaheuristic methods and is the new state-of-the-art methodology for the stochastic ALBP.

Full text

Annals of Operations Research (2024) 335:491–516
https://doi.org/10.1007/s10479-023-05809-1
ORIGINAL-COMPARATIVE COMPUTATIONAL STUDY
Chance-constrained stochastic assembly line balancing
with branch, bound and remember algorithm
Zixiang Li1,2 ·Celso Gustavo Stall Sikora3·Ibrahim Kucukkoc4
Received: 27 November 2022 / Accepted: 19 December 2023 / Published online: 1 February 2024
© The Author(s) 2024
Abstract
Assembly lines are widely used mass production techniques applied in various industries from
electronics to automotive and aerospace. A branch, bound, and remember (BBR) algorithm is
presented in this research to tackle the chance-constrained stochastic assembly line balancing
problem (ALBP). In this problem variation, the processing times are stochastic, while the
cycle time must be respected for a given probability. The proposed BBR method stores all
the searched partial solutions in memory and utilizes the cyclic best-first search strategy to
quickly achieve high-quality complete solutions. Meanwhile, this study also develops several
new lower bounds and dominance rules by taking the stochastic task times into account. To
evaluate the performance of the developed method, a large set of 1614 instances is generated
and solved. The performance of the BBR algorithm is compared with two mixed-integer
programming models and twenty re-implemented heuristics and metaheuristics, including
the well-known genetic algorithm, ant colony optimization algorithm and simulated annealing
algorithm. The comparative study demonstrates that the mathematical models cannot achieve
high-quality solutions when solving large-size instances, for which the BBR algorithm shows
clear superiority over the mathematical models. The developed BBR outperforms all the
compared heuristic and metaheuristic methods and is the new state-of-the-art methodology
for the stochastic ALBP.
Keywords Assembly line balancing ·Stochastic assembly line ·Branch and bound ·
Heuristic algorithms ·Meta-heuristics ·Chance-constraint
BIbrahim Kucukkoc
ikucukkoc@balikesir.edu.tr
Zixiang Li
zixiangliwust@gmail.com
Celso Gustavo Stall Sikora
celso.sikora@uni-hamburg.de
1Key Laboratory of Metallurgical Equipment and Control Technology of Ministry of Education,
Wuhan University of Science and Technology, Wuhan, Hubei, China
2Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan
University of Science and Technology, Wuhan, Hubei, China
3Institute for Operations Research, University of Hamburg, Hamburg, Germany
4Industrial Engineering Department, Balikesir University, Cagis Campus, Balikesir 10145, Turkey
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492 Annals of Operations Research (2024) 335:491–516
1 Introduction
Assembly lines are the production system of choice for several industries that deal with
the mass production of complex homogenous product families. A set of workstations (or
stations, shortly) is connected via a transportation system such as a conveyor or a moving
belt. The operations required to assemble a product are divided among the workstations, which
are specialized for a small subset of the tasks required in the assembly. This flow-oriented
process can be very efficient if the workload is well distributed among the workstations since
the production cycle time is limited by the system bottleneck.
The optimization problem of assigning tasks to workstations is a classic problem known
as the assembly line balancing problem (ALBP) and several variants and extensions of the
problem exist (Battaïa & Dolgui, 2013). Recent surveys by Boysen et al. (2022) and Battaïa
and Dolgui (2022) provide a comprehensive review and classification of the assembly line
balancing literature and identify potential future research directions. In its most basic version,
the simple ALBP can be described as distributing a set of tasks to a set of stations to minimize
the number of stations under capacity and precedence relationship constraints. The capacity
constraint ensures that the workload of each workstation is less than or equal to the cycle
time (CT) (Pitakaso et al., 2021). The precedence relationship constraint requires that a task
can only be performed after all its predecessors have been completed (Kucukkoc et al., 2018;
Sivasankaran & Shahabudeen, 2014).
In the traditional ALBP, the task times are deterministic and fixed in advance (Li,
Kucukkoc, & Tang, 2021). Nevertheless, in many real-world applications, the task times
are stochastic due to fatigue, breakdowns, erroneous entries, and workforce with insufficient
qualifications (A˘gpak & Gökçen, 2007; Sarin et al., 1999). Furthermore, real assembly lines
only seldom assemble a single product and must be adaptable to support the production of
a family of similar products. As the processing time of tasks belonging to different prod-
ucts may vary, assuming constant and deterministic times is not realistic. More advanced
approaches either consider the sequence of products as part of the decision process (Lopes
et al., 2020) or utilize uncertain processing times. This paper focuses on the latter formula-
tion, which was first introduced by (Vrat & Virani, 1976), who reduce the multiple-product
problem to a single-product problem with stochastic processing times.
Uncertainties in assembly lines can be found in the cycle time (Lopes, Michels, Sikora,
Brauner, & Magatão, 2021), and the product demand mix (Yuchen Li et al., 2023;Sikora,
2021b), although the great majority of the literature models uncertainty on the processing
time of tasks (Boysen et al., 2022). The uncertain task times might be defined as probability
distributions, where the normal distribution is usually assumed in most studies (A˘gpak &
Gökçen, 2007; Diefenbach & Stolletz, 2022; Urban & Chiang, 2006). The complication
of considering an uncertain processing time is noticed in the cycle time condition. Several
approaches were proposed to achieve a feasible or acceptable assignment when the workload
of stations is not deterministic. Such formulations may either be defined by general conditions
(such as a high completion probability) or application-oriented considerations or remedial
actions (such as the stoppage of the assembly line if a workstation requires longer than the
cycle time).
A recent survey on ALBP under uncertainty (Sikora, 2021a) classifies the literature on
stochastic single-product assembly lines in formulations that minimize product or sums of
non-completion probabilities and those that consider the cost of remedial actions. Examples
of remedial actions are the use of utility workers (Sikora, 2021b), stopping the line (Silverman
&Carter,1986), or considering incompletion costs or rework at the end of the line (Kottas &
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Annals of Operations Research (2024) 335:491–516 493
Lau, 1976). The methods that do not explicitly model a remedial action often use a chance
constraint approach for the cycle time. In the most common variation of the problem (Battaïa
& Dolgui, 2013), tasks assigned to a workstation are required to be completed within the
given CT with a probability equal to or larger than the predetermined limit. One advantage of
such a chance constraint-based stochastic programming approach is that a method developed
for the deterministic ALBP can often be extended to deal with the stochastic version. Other
approaches that have been used in literature to deal with uncertain task times are robust opti-
mization,stability analysis and uncertain programming (Lietal.,2020a). Brief information
on each of these methods will be provided below.
The stochastic programming approach is more favorable when past data is available on
task processing times. Conversely, robust optimization may be preferred when implementing
a new product or utilizing a new set of tasks with no historical data. Robust optimization may
be employed to calculate the most adverse scenario when the execution times of tasks are
known within specific intervals (Hazır & Dolgui, 2013). Among others addressing the robust
optimization approaches for ALBP, Pereira and Miranda (2018) developed a formulation
and a branch and bound (referred to as BB hereafter) algorithm for the simple ALBP to
minimize the number of workstations. Moreira et al. (2015) considered the assignment of
heterogeneous workers as well as ALBP with worker-dependent and uncertain task execution
times, respecting a robustness measure. Hazir and Dolgui (2015) defined the robust U-shaped
ALBP and proposed an iterative approximate solution algorithm.
Stability analysis determines and evaluates the effect of uncertain task times on the opti-
mality of line balance (Gurevsky et al., 2012). Gurevsky et al. (2012) addressed the type-E
ALBP (aiming to maximize the line efficiency) and proposed two heuristics seeking a com-
promise between the efficiency and the desired stability measure. Gurevsky et al. (2013)
addressed the generalized formulation for an ALBP with several workplaces in a worksta-
tion and proposed a stability measure for feasible and optimal solutions when task processing
times may vary. Lai et al. (2019) also addressed the type-E ALBP and performed the sta-
bility analysis of an optimal line balance determining sufficient and necessary conditions
that the line balance to stay stable. More recently, Gurevsky et al. (2022) proposed a mixed-
integer linear program to maximize the stability factor in the general case and investigated
the relationship between the stability factor and stability radius.
Uncertain programming has emerged due to the mathematical intractability of stability
analysis. It relies on the belief degrees of experts, based on the uncertainty theory invented
by Liu (2007). Following the first uncertain programming model by Liu (2009), it has been
applied to various optimization problems, including machine scheduling (Li & Liu, 2017),
capacitated facility location-allocation (Wen et al., 2014), vehicle routing (Ning & Su, 2017)
and production control (Liu & Yao, 2015). Its applications on the ALBPs are still limited. Li
et al., (2019a) applied the uncertainty theory to model task time uncertainties and introduced
the belief reliability measure to the assembly line production to minimize cycle time. Li et al.,
(2020a) considered two types of uncertainties caused by task times through a new reliability
metric and proposed a mathematical model as well as neighborhood search methods to
maximize the reliability and efficiency of the line.
As mentioned above, the chance constraint-based stochastic programming approach is the
most studied method in the literature. Several exact, heuristic and metaheuristic methods have
been developed. The exact methods include the dynamic programming approach (Carraway,
1989), mixed-integer programming models (A˘gpak & Gökçen, 2007; Fathi et al., 2019;
Urban & Chiang, 2006), and BB (Diefenbach & Stolletz, 2022). Mixed-integer programming
models can solve small-size problems optimally, whereas they might not achieve high-quality
or feasible solutions within acceptable computation time when solving large-size instances.
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494 Annals of Operations Research (2024) 335:491–516
Hence, heuristic and metaheuristic methods have been developed to tackle this problem in
a short time. Several heuristics and metaheuristics were developed, including the multiple
single-pass heuristic algorithm (Gamberini et al., 2009), beam search (Erel et al., 2005),
simulated annealing algorithm (Suresh & Sahu, 1994), the single-run optimization algorithm
(JrJung, 1997), genetic algorithm (Baykaso˘glu & Özbakır, 2007), imperialist competitive
algorithm (Bagher et al., 2011), ant colony optimization algorithm (Celik et al., 2014), hybrid
evolutionary algorithm (Zhang et al., 2017), modified evolutionary algorithm (Zhang et al.,
2018), and particle swarm optimization (Aydo˘gan et al., 2019). Meanwhile, the modeling of
chance constraints for variations, and extensions of the stochastic ALBP have also attracted
the attention of many research papers. Specifically, the literature on other variants of the
stochastic ALBP includes the type-II stochastic ALBP (Liu et al., 2005;Pınarba¸sı & Alaka¸s,
2020), cost-oriented stochastic ALBP (Foroughi & Gökçen, 2019), multi-objective stochastic
ALBP (Cakir et al., 2011), stochastic U-shaped ALBP (Aydo˘gan et al., 2019;Bagheretal.,
2011; Baykaso˘glu & Özbakır, 2007; Celik et al., 2014; Chiang & Urban, 2006; Serin et al.,
2019; Urban & Chiang, 2006; Zhang et al., 2018), stochastic two-sided ALBP (Chiang et al.,
2015; Delice et al., 2016; Tang, Li, Zhang, & Zhang, 2017; Özcan, 2010), stochastic parallel
ALBP (Özbakır & Seçme, 2020; Özcan, 2018), mixed-model stochastic ALBP (McMullen
& Frazier, 1997;Zhaoetal.,2007) and others. Efficient methods for the stochastic ALBP
with one product can be extended to deal with more realistic variants.
Although the literature on stochastic ALBP is extensive, not many approaches can effi-
ciently solve large instances exactly. After the publication of the dynamic programming
approach (Carraway, 1989), there is a large hiatus in the literature. Recently, a new BB algo-
rithm (Diefenbach & Stolletz, 2022) was proposed for the exact solution of the problem. The
BB algorithm is based on a sampling method that draws random realizations for the processing
time of the tasks. In the numerical experiments, the authors used 10,000 realizations to model
the stochastic processing time as a set of possible scenarios. As it is easy to check whether
the cycle time restriction is obeyed for a specific scenario, the chance constraint reduces to
checking whether the solution is feasible for a given percentage of scenarios. The approach
is flexible enough to deal with any probability distribution (with or without correlation) but
showed satisfying results for only small and medium-size instances. The results show that
even some instances with 20 tasks cannot be solved within 10,000 s and only one instance
with 50 tasks has been solved. Efficient and fast solution procedures for larger instances are,
therefore, still required. In this manuscript, we propose a solution method for a specific yet
highly studied ALBP: the stochastic ALBP with normally distributed task processing times.
The branch, bound, and remember (BBR) algorithm is an exact method, which combines
the dynamic programming method with the BB algorithm (Li, Kucukkoc, & Tang, 2020;Li
et al., 2020b; Morrison et al., 2014; Sewell & Jacobson, 2012). BB algorithm utilizes bounds
to eliminate partial solutions which could not possibly achieve a better solution. Dynamic
programming method utilizes memory to eliminate redundant partial solutions. The main dif-
ference between BBR and BB algorithms is that BBR keeps every sub-problem created in the
BBR framework in memory. Before branching on a sub-problem, BBR checks all the partial
solutions in memory and a sub-problem is not explored when it is dominated by a previously
encountered subproblem (i.e. a stored partial solution). This powerful method is capable of
achieving all the optimal solutions of Scholl’s 269 instances within an average computation
time of one second, and it might be considered to be the state-of-the-art methodology for the
simple ALBP (Battaïa & Dolgui, 2013; Li, Kucukkoc, & Tang, 2020). Apart from the simple
ALBP, BBR algorithm also produces a competing performance in robust ALBP (Pereira &
Álvarez-Miranda, 2018), U-shaped ALBP (Li et al., 2018), two-sided ALBP (Li, Kucukkoc,
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Annals of Operations Research (2024) 335:491–516 495
& Zhang, 2020), integrated worker assignment and line balancing problem (Pereira, 2018;
Vilà & Pereira, 2014) and robotic ALBP (Borba et al., 2018).
From the literature review, BBR has not been utilized for the stochastic ALBP, although
it has produced a competing performance in several variations of ALBP. Meanwhile, to the
best of the authors’ knowledge, there is no exact method which is capable of solving the
large-size stochastic ALBP effectively in an acceptable computation time. Therefore, a BBR
algorithm, which is an exact method, is developed to tackle the stochastic ALBP, where the
task times satisfy the normal distribution, to minimize the number of workstations. In short,
this study presents two contributions as follows. 1) This study develops new dominance rules
and lower bounds for the stochastic ALBP, which differentiate the proposed BBR algorithm
from those in the published studies. Regarding the dominance rules, the maximal load rule
and the extended Jackson rule (originally developed for the deterministic ALBP) are mod-
ified by taking the probability constraint into account as they cannot be directly applied to
the stochastic ALBP. Meanwhile, this study also proves the correctness of the new extended
Jackson rule. As for the lower bounds, LB1, LB2 and LB3 in the deterministic situation are
transferred into LBS
1,LBS
2and LBS
3by taking the stochastic operation time into account.
Notice that, the developed LBS
1,LBS
2and LBS
3are different from that in Diefenbach and
Stolletz (2022). The LBS
1,LBS
2and LBS
3are very fast; the corresponding lower bounds in
Diefenbach and Stolletz (2022) are sampling-based lower bounds and they consume much
more computation time. Hence, the proposed BBR is not a simple extension of the determin-
istic BBR and all the segments of the BBR algorithm are modified to solve this problem. 2)
This study re-implements two mixed-integer programming models and twenty heuristics and
metaheuristics, including the well-known tabu search algorithm, simulated annealing algo-
rithm, genetic algorithm, particle swarm optimization algorithm and ant colony optimization
algorithm, and conducts a comprehensive comparative study to evaluate the performance of
the proposed method. The computational results demonstrate that the mathematical models
cannot achieve high-quality solutions when solving the large-size instances and the BBR
algorithm shows clear superiority over the mathematical models for the large-size instances.
The BBR algorithm outperforms all the implemented heuristic and metaheuristic methods
and is the state-of-the-art methodology for the stochastic ALBP.
The remainder of this paper is organized as follows. Section 2presents the problem
description and the mathematical formulation. Section 3illustrates the main procedure of
the BBR algorithm along with the main segments. The computational study is provided in
Sect. 4to evaluate the developed BBR algorithm. Conclusions are given in Sect. 5together
with some future research directions.
2 Problem formulation
The stochastic ALBP and its mathematical formulation are provided in this section.
2.1 Problem description
This study addresses the stochastic ALBP to minimize the number of workstations. The
chance-constrained formulation is given first to describe the problem. Task times are normally
distributed with known means and variances (ti∼Nμi,σ2
i) as in most of the studies in
the literature, e.g. see Urban and Chiang (2006)andA˘gpak and Gökçen (2007). For the
stochastic ALBP, a set of tasks is partitioned into a set of connected stations under the
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496 Annals of Operations Research (2024) 335:491–516
Table 1 The task times and
precedence relationships of the
illustrated example
Tasks Successors μiσ2
i
1 2,4 1 0.0598
2 3,5 5 0.0817
3 – 4 0.1566
4 7 3 0.3090
5 6 5 0.6931
6 – 6 0.1526
7 – 5 0.3806
probability constraint (cycle time constraint) and the precedence constraint. The precedence
constraint requires that the predecessors of one task must be allocated to the former station or
be operated before the successors when they are allocated to the same station. The probability
constraint demands that the tasks on stations can be completed within the given CT with a
probability equal to or larger than the predetermined limit α(greater than 0.5).
Assuming the station-load Y(Y=ti) has a completion probability larger than or
equal to α, the probability constraint can be expressed with Pti≤CT≥α.Asti∼
Nμi,σ2
i, the operation time of station-load Ycan be defined as Y∼Nμi,σ2
iand
hence the probability constraint can be rewritten using the normal distribution as presented
in Eq. (1). For the normal distribution, the critical value for Zis available numerically for
a value of αgiven. For instance, Z0.95 =1.6449, Z0.975 =1.9600 and Z0.99 =2.3263.
Hence, the probability constraint for a station-load is then described utilizing Eq. (2).
P⎛
⎝Z≤CT −μi
σ2
i
⎞
⎠≥α(1)
μi+zα·σ2
i≤CT (2)
To exhibit the features of the stochastic ALBP, one example with 7 tasks is given here,
where the precedence relation and task times are illustrated in Table 1. Figure 1depicts the
optimal solutions of this illustrated example on the traditional (deterministic) assembly line
and stochastic assembly line. Specifically, Fig. 1a illustrates the optimal solution when the
stochastic operation time is not considered (ti=μi) within a cycle time of 7 time-units.
Fig. 1 Optimal assignment of tasks on the traditional and stochastic assembly lines
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Annals of Operations Research (2024) 335:491–516 497
Figure 1b illustrates the optimal solution when the stochastic operation time is considered
(ti∼Nμi,σ2
i) with αequals to 0.9.
As seen in Fig. 1, five stations are utilized when ti=μion the traditional assembly
line and six stations are utilized when ti∼Nμi,σ2
ion the stochastic assembly line. The
reason lies behind this is that task 4 and task 3 are allocated to station 3 on the traditional
assembly line, whereas task 3 cannot be allocated to station 3 on the stochastic assembly
line as stochastic operation time is involved. Namely, the cycle time constraint is satisfied on
the traditional assembly line when μi≤CT; the probability constraint is satisfied on the
stochastic assembly line when the tasks on each station could be completed within the cycle
time with a probability equal to or larger than 0.9 (μi+zα·σ2
i≤CT).
2.2 Mathematical model
The notations used in the mathematical model are first explained as follows.
i,h Index of tasks, i∈{1,··· ,nt},where nt is the number of tasks
I Set of tasks, I={1,··· ,nt }
j,k Index of stations, j∈{1,··· ,mmax },wheremmax is the maximum number of
stations that can be utilized
J Set of stations, J={1,··· ,mmax }
tiOperation time of task i, where ti∼Nμi,σ2
i
PSet of precedence relations of tasks, where (i,h)∈Pwhen task i is the immediate
predecessor of task h
CT Cycle time
xij 1, if task i is assigned to station j; 0, otherwise
sj1, if there is at least one task assigned to station j; 0, otherwise
The model by A˘gpak and Gökçen (2007) is formulated utilizing Eqs. (3)–(7)hereusing
the notation given above. Equation (3) minimizes the number of workstations. Equation (4)
ensures that each task is allocated to exactly one workstation. Equation (5) is the precedence
constraint, which indicates that the predecessors of one task should be allocated to the former
or the same station in which the task has been assigned. Equation (6) is the probability
constraint, indicating that the sum of the processing times of tasks on each workstation does
not exceed the cycle time with a probability of α. Equation (7) defines the domain of each
variable.
Min 
j∈J
sj(3)

j∈J
xij =1fori∈I(4)

j∈J
j·xij ≤
k∈J
k·xhk for (i,h)∈P(5)
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498 Annals of Operations Research (2024) 335:491–516

i∈I
μi·xij +Zα·
i∈Iσ2
i·xij≤CT ·sjfor j∈J(6)
xij,sj∈{0,1}for all i,j(7)
As the above model is a non-linear integer programming model, this study proposes two
integer programming models. The first model is the direct linear approach (LA) by A˘gpak
and Gökçen (2007) to provide an approximate solution. It utilizes Eq. (8) to replace Eq. (6).
Notice that the achieved solution by the LA might not be the true optimal solution, but the
achieved solution is a valid upper bound.

i∈I
(μi+Zα·σi)·xij ≤CT ·sjfor j∈J(8)
The second model is the model with pure linear transformation (MLTT) by A˘gpak and
Gökçen (2007). The MLTT consists of Eqs. (3)–(5) and the following, i.e., Eqs. (9)–(13). The
MLTT has a limited capacity to solve large-size instances optimally based on our preliminary
experiments.
CT2−2·CT ·
i∈I
μi·xij +
i∈I
μ2
i·xij +2·
i∈I
h|h∈Ih>i
μi·μh·uihj
−Z2
α·
i∈I
σ2
i·xij≥0for j∈J(9)

i∈I
μi·xij ≤CT ·sjfor j∈J(10)
xij +xhj −uihj ≤1for j∈J,i= h(11)
xij +xhj −2·uihj ≥0for j∈J,i= h(12)
xij,sj,uihj ∈{0,1}for all i,h,j(13)
3 Proposed method (BBR)
BBR is an exact algorithm which integrates the dynamic programming method with the BB
algorithm (Li, Kucukkoc, & Tang, 2020;Lietal.2020b;Morrisonetal.,2014;Sewell&
Jacobson, 2012). This exact method shows a competing performance in solving the simple
ALBP (Battaïa & Dolgui, 2013; Li, Kucukkoc, & Tang, 2020) and many others (Li et al.,
2018; Li, Kucukkoc, & Zhang, 2020). Nevertheless, there is no application of this powerful
exact method in solving the stochastic ALBP. Meanwhile, there is no exact method which can
tackle the large-size stochastic ALBP effectively within an acceptable amount of computation
time. Therefore, this section develops the BBR algorithm to deal with the large-size stochastic
ALBP optimally for the first time. The procedure and the main components of the proposed
BBR method are explained in this section.
3.1 The main procedure applied by the BBR algorithm
Algorithm 1 illustrates the procedure of the developed BBR. The proposed BBR algorithm
consists of three phases: Phase I, Phase II and Phase III. Firstly, Phase I achieves a high-quality
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Annals of Operations Research (2024) 335:491–516 499
upper bound (UB) utilizing the modified Hoffman heuristic (called MHH hereafter). Here, the
lower bound at the root LBroot is calculated with LBroot =max{LBS
1,LBS
2,LBS
3,BPLB},
where LBS
1,LBS
2and LBS
3are three lower bounds and BPLB is the bin packing lower bound
(see Sect. 3.3 for further explanation). If the achieved solution is not optimal or is not verified
to be optimal, Phase II conducts the modified cyclic best-first search (MCBFS) with the
developed new lower bounds and dominance rules (see Sect. 3.3 for further explanation).
Phase II aims at achieving the solution with the smaller station number and proving the
optimality of the achieved solution. Finally, Phase III conducts the breadth-first search (BrFS)
when Phase II is unable to prove the optimality of the solution achieved. Phase III proves
the optimality by enumerating all the possible station-loads and it is relatively slow for the
large-size instances. However, on the basis of the strict upper bound by Phase II, Phase III
is utilized mainly for proving the optimality of the achieved solutions only for some “hard”
instances.
Algorithm 1 The main procedure of the BBR method
The differences between the proposed BBR method and the reliability-based BB algorithm
in Diefenbach and Stolletz (2022) can be clarified as follows:
(1) The proposed BBR method utilizes the MCBFS whereas the BB algorithm utilizes the
depth-first search (DFS) strategy.
(2) The proposed BBR method utilizes the LBS
1,LBS
2and LBS
3, which are very fast. The
BB algorithm utilizes the sampling-based lower bounds which consume much more
computation time.
(3) The proposed BBR method stores all the searched partial solutions in memory, and it
utilizes the memory-based dominance rule, whereas the BB algorithm does not store all
the searched partial solutions in memory.
(4) The proposed BBR method also utilizes the extended Jackson rule and the no-successor
rule, which are not utilized in the BB algorithm.
This study also tests various BBR methods with different search strategies, lower bounds,
and dominance rules in Sect. 4.2 to prove the advantages of the differences used in the
developed BBR method.
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500 Annals of Operations Research (2024) 335:491–516
3.2 Branching method and upper bound
There are two branching methods in the published BB algorithms. They are station-oriented
branching and task-oriented branching (Li, Kucukkoc, & Tang, 2020). Task-oriented branch-
ing obtains partial solutions by assigning one task to the current station when all the constraints
are met or to a newly opened station when the probability constraint is conflicted. Station-
oriented branching creates partial solutions by assigning a station-load consisting of several
tasks to one newly opened station. Suppose that a partial solution utilizes mstations. This
partial solution can be expressed with ℘=(A,U,S1,S2,··· ,Sm),whereAand Uare the
set of assigned tasks and unassigned tasks, respectively. Smholds the set of tasks assigned to
workstation m. As station-oriented branching assigns a station-load together, the new partial
solution at the deeper depth can be described with ℘=(A,U,S1,S2,··· ,Sm,Sm+1).This
study selects the station-oriented branching due to its superior performance demonstrated in
the published study (Li, Kucukkoc, & Tang, 2020). To avoid tremendous time to achieve all
the station-loads, the maximum number of generated station-loads is set to 10,000 in Phase
II as in Sewell and Jacobson (2012).
On the basis of the station-oriented branching, this study utilizes the MHH (Sewell &
Jacobson, 2012) to obtain a high-quality initial solution as UB. MHH generates a feasible
solution from station to station and the main procedure of the proposed MHH is described
as follows. Firstly, MHH generates a number of station-loads for the first station and selects
the most promising station-load utilizing the selection criterion as the selected station-load.
This procedure is applied to the latter station subsequently and terminates when the complete
solution is achieved. Specifically, for a selected partial solution ℘=(A,U,S1,S2,··· ,Sm),
MHH generates a set of new partial solutions ℘=A,U,S1,S2,··· ,Sm,Sm+1for
station m+1. Subsequently, the most promising station-load with the maximum value of
i∈Sm+1(μi+α·wi+β·|Fi|−γ)is selected as the station-load for station m+1, where
Sm+1is the task set assigned to station m+1, Fiis the set of immediate successors of task
i,|Fi|is the number of tasks in set Fi,F∗
iis the set of all successors of task iand wiis the
positional weight of task icalculated with wi=μi+h∈F∗
iμh. The terms α,βand γare
three parameters, where the values of them are set to α∈{0,0.005,0.010,0.015,0.020},
β∈{0,0.005,0.010,0.015,0.020}and γ∈{0,0.01,0.02,0.03}. To obtain a high-quality
UB, this study tests all the combinations of these three parameters following Sewell and
Jacobson (2012) and the minimal station number among the ones by all the combinations is
selected as the output of MHH.
3.3 New lower bounds and dominance rules
Lower bounds are utilized to reduce the number of nodes explored in the enumeration. All the
lower bounds of the simple ALBP with the μias the operation time are the lower bounds of
the stochastic ALBP. Nevertheless, these lower bounds might be weak, and this study extends
LB1, LB2 and LB3 (the three well-known lower bounds), to the stochastic ALBP with normal
distributed processing times. The developed LBS
1,LBS
2and LBS
3in the stochastic situation
are calculated in Eqs. (14)–(17). If σiis equal to 0.0 for all tasks, the LBS
1,LBS
2and LBS
3
are equal to the LB1, LB2 and LB3 in the deterministic situation. Optimally, this study also
utilizes the bin packing lower bound (BPLB) as in Sewell and Jacobson (2012) with the μias
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Annals of Operations Research (2024) 335:491–516 501
the operation time. This is because BPLB is more powerful than LB1, LB2 and LB3. BPLB
helps further prone the sub-problem when the variances of the task times are not large and
hence it is utilized here.
LBS
1=⎡
⎢
⎢
⎢i∈Iti+zα·i∈Iσ2
i
CT ⎤
⎥
⎥
⎥
+
(14)
LBS
2=i∈I|ti>CT
2or(ti=CT
2and σi>0)
+i∈i|ti=CT
2and σi=0
2+
(15)
LBS
3=i∈Ivi+(16)
vi=⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
1if ti>2·CT
3or (ti=2·CT
3and σi>0)
2/3if ti=2·CT
3and σi=0
1/2ifCT/3<ti<2·CT/3or (ti=CT
3and σi>0)
1/3ifti=CT
3and σi=0
0other wise
(17)
Dominance rules determine whether a generated new partial solution is dominated, and
whether the dominated partial solution should be pruned. This study utilizes and modifies
four dominance rules, which are maximal load, extended Jackson, no-successor and memory-
based dominance rules.
3.3.1 Maximal load rule
One partial solution is pruned when 1) this partial solution contains a station-load Sjand an
unallocated task i;2)taskican be allocated to station junder the precedence constraint and
probability constraint. Suppose that the load on station jbe expressed as Y∼Nμj,σ2
j,
then maximal load rule is applied only when μj+μi+Zα·σ2
j+σ2
i≤CT.
3.3.2 Extended Jackson rule
One partial solution is pruned when 1) there is a task iassigned to the last station j;2)
there is an unallocated task hwhich potentially dominates task i;3)taskhcan replace
task iwithout violation of the precedence constraint and probability constraint. Here, task
hdominates task ionly when task iand task hhas no precedence relation, μi≤μh,
μi+Zα·σ2
i≤μh+Zα·σ2
hand F∗
i⊆F∗
h.
3.3.3 No-successor rule
One partial solution is pruned when 1) the assigned tasks on the last station have no successors;
2) there exists an unassigned task which can be allocated to the last station and has at least
one successor.
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502 Annals of Operations Research (2024) 335:491–516
3.3.4 Memory-based dominance rule
One partial solution is pruned when 1) the assigned task set of this partial solution is equivalent
to the task set of a previously identified partial solution; 2) the current partial solution requires
no smaller station number than that of a previously identified partial solution.
When the maximal load rule is applied to two sub-problems with the same station number
(X =S1#S2#···#Sj···,Y=S1#S2#···#S
j··· and #Sj∈#S
j), only the
sub-problem Y with more tasks is preserved. Hence, the maximal load rule never prunes an
optimal solution. When the memory-based dominance rule is applied to two sub-problems
with the same number of tasks assigned, only the sub-problem with an equivalent or better
solution is preserved. Hence, the memory-based dominance rule never prunes an optimal
solution (Morrison et al., 2014). Hence, this study only proves the correctness of the extended
Jackson rule and the no-successor rule here.
Lemma 1: For a given partial solution, a sub-problem containing this partial solution can
be pruned if (1) there is a task i assigned to the last station j;(2)there is an unallocated
task h which potentially dominates task i ;(3)task h can replace task i without violation of
the precedence constraint and probability constraint.
Proof 1: Let X=S1#S2#···#Sj#Sj+1#··· be a feasible solution, where i∈Sj
and h∈#Sj+1#···. One new solution is achieved by exchanging the positions of task i
and task h(Y=S1#S2#···#Sj#Sj+1#···,h∈Sjand i∈#Sj+1#···)when1)
task iand task hhave no precedence relation, μi≤μh,μi+Zα·σ2
i≤μh+Zα·σ2
h
and F∗
i⊆F∗
h;2)taskhcan replace task iwithout violation of the probability constraint
and precedence constraint. Suppose that task his allocated to station kin solution Xand
the remaining station-load after removing task his expressed as Y∼Nμk,σ2
k,itis
satisfied that μk+μh+Zα·σ2
k+σ2
h≤CT. After exchanging the positions of task
iand task h, solution Yis a feasible solution when μk+μi+Zα·σ2
k+σ2
i≤CT.
When μi≤μhand σ2
i≤σ2
h, clearly it is satisfied that μk+μi+Zα·σ2
k+σ2
i≤
μk+μh+Zα·σ2
k+σ2
h≤CT. When μi≤μhand σ2
i>σ
2
h,μi+Zα·σ2
i≤μh+Zα·σ2
h
can be transferred into 0 ≤Zα·σ2
i−Zα·σ2
h≤μh−μi. Subsequently, it can be obtained
that 0 ≤Zα·σ2
k+σ2
i−Zα·σ2
k+σ2
h≤Zα·σ2
i−Zα·σ2
h≤μh−μi. After transferring,
it is clear that μi+Zα·σ2
k+σ2
i≤Zα·σ2
k+σ2
h+μhand μk+μi+Zα·σ2
k+σ2
i≤
μk+μh+Zα·σ2
k+σ2
h≤CT. Namely, solution Yis a feasible solution after exchanging
the positions of task iand task hunder the given condition. The two solutions Xand Y
have the same station number and hence deleting solution Xby pruning the corresponding
partial solution with the extended Jackson rule will never prevent the discovery of the optimal
solution.
Lemma 2:
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Annals of Operations Research (2024) 335:491–516 503
For a given partial solution, a sub-problem containing this partial solution can be pruned
if 1) the assigned tasks on the last station have no successors; 2) there exists an unassigned
task which can be allocated to the last station and has at least one successor.
Proof 2: Let X =S1#S2#···#Sj···#Sk#··· be a feasible solution, where Sjand Sk
are task sets on station jand station k, respectively. There must be one solution by exchanging
the positions of task sets Sjand Sk:Y=S1#S2#···#Sk···#Sj#··· if 1) the tasks
in Sjhave no successors; 2) there exists an unassigned task in Skwith at least one successor.
The two solutions Xand Yhave the same station number and hence deleting solution Xby
pruning the corresponding partial solution with the no-successor rule will never prevent the
discovery of the optimal solution.
3.4 Search strategy and selection criterion
A search strategy is utilized to decide the order of the explored partial solutions and it has
a big effect on the consumed time of the BBR algorithm. Among the studies, there are four
main search strategies: depth-first search (DFS), best-first search (BFS), breadth-first search
(BrFS) and cyclic best-first search (CBFS) (Li, Kucukkoc, & Tang, 2020). DFS starts with
selecting one partial solution at depth 1 and later selects one partial solution at a deeper depth
in sequence until reaching the deepest depth. After exhausting the solutions at the deepest
depth, DFS returns to the second deepest depth and finally returns to the top of the search tree
after exhausting all the partial solutions at deeper depths. BFS selects the most promising
partial solution utilizing one selection criterion, and the effectiveness of the BFS mainly
depends on the selection criterion. BrFS generates all the partial solutions at depth 1, depth
2,···, and the deepest depth and it terminates only when the complete optimal solution is
obtained. As BrFS tests all the possible nodes, BrFS is very slow when solving the large-size
instances and it is utilized to prove the optimality of some instances in Phase III with the
strict upper bound provided.
CBFS is a relatively new search strategy by combing the DFS and the BFS. CBFS starts
with selecting one most promising station-load at depth 1, and later selects the most promising
station-load at depth 2, depth 3 and the deepest depth. Once the deepest depth is reached,
CBFS comes back to depth 1 and this procedure is repeatedly carried out until the termination
criterion is met. To avoid generating too many partial solutions, Li et al. (2018) proposed
a modified CBFS (namely, MCBFS), where a sub-problem at depth lis not selected when
there are many unsearched partial solutions at depth l+1. Due to the superiority of the
MCBFS (Li, Kucukkoc, & Tang, 2020), this study utilizes the MCBFS in Phase II of the
BBR method as presented in Algorithm 2. Here, the MCBFS utilizes the selection criterion
of b(℘)=LB(U)+IT/m−λ|U|,where℘=(A,U,S1,S2,··· ,Sm), LB is the minimal
value of the LBS
1,LBS
2,LBS
3andBPLBoftheremainingtasksetU,IT is the idle time on
the former mstations, |U|is the number of unassigned tasks and λis an input parameter (λ
is set to 0.02 as in Li, Kucukkoc, & Tang (2020)).
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504 Annals of Operations Research (2024) 335:491–516
Algorithm 2 Modified cyclic best-first search (MCBFS)
Another important issue is the utilization sequence of the lower bounds and domi-
nance rules when conducting the MCBFS and BFS. The procedure of conducting the
search strategy is provided in Algorithm 3. In this procedure, for any partial solution
YA,U,S1,S2,··· ,Sm,Sm+1;LBS
1,LBS
2and LBS
3are applied first. After that, the max-
imal load rule, extended Jackson rule, no-successor rule and memory-based dominance rule
are applied in sequence. Here, for the memory-based dominance rule, the maximally loaded
partial solutions (where the sub-problems not maximally loaded are pruned by the maximal
load rule), are compared and the sub-problem with equivalent or better solutions is preserved.
In short, for any partial solution YA,U,S1,S2,··· ,Sm,Sm+1, it is stored when it is not
dominated by the lower bounds LBS
1,LBS
2and LBS
3and dominance rules. Recall that, in
Phase II the maximum number of generated station-loads is set to 10,000 whereas in Phase III
there is no limit on the maximum number of generated station-loads. Namely, the complete
enumeration is utilized in Phase III and the proposed algorithm is an exact method.
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Annals of Operations Research (2024) 335:491–516 505
Algorithm 3 The procedure of conducting the search strategy
4 Computational results
This section aims at testing the performance of the developed algorithm. Section 4.1 presents
the utilized instances, compared algorithms and the running environments. Section 4.2 con-
ducts the structural parameter evaluation to evaluate the proposed structural parameter values.
Section 4.3 compares the performance of the BBR method with the mathematical formula-
tions. Section 4.4 provides a comparative study between BBR method and the re-implemented
heuristic and metaheuristic methods. Finally, Sect. 4.5 conducts the comparison on multiple
instance characteristics to clarify differences according to multiple instance characteristics.
4.1 Experimental design
To evaluate the implemented method, this study generates a set of instances on the basis of the
Scholl’s 269 instances utilizing the method in Carraway (1989). To observe the performance
of the proposed method on instances with different variances, this study tests two types
of task variances (low task variance and high task variance) and three types of αlevels
(α∈{0.90,0.95,0.975}), where the corresponding values of zαare 1.28, 1.645 and 1.96,
respectively. All the two variances and three different αlevels are tested, leading to a total
number of 269 ×2×3=1614 instances. Specifically, the means (μi) of task times are set to
the original task times and the variances are randomly generated within [0, (ti/4)2]forthelow
task variance and [0, (ti/2)2] for the high task variance. The detailed precedence relations and
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506 Annals of Operations Research (2024) 335:491–516
the original operations times of tasks are available at the website (http://www.assembly-line-
balancing.de). Nevertheless, the feasible instances cannot be achieved for many instances
in the preliminary experiments and this study sets that μi+zα·$σi2≤CT for any task
iwhen generating the instances, where αis set to 0.975. This modification ensures that
there are feasible solutions for all the generated instances. The instances are divided into two
sets: small-size instances with 70 tasks at maximum (135 ×6=690 instances in total) and
large-size instances with more than 70 tasks.
To observe the performance of the proposed methodology on the different instances, this
study also generates a set of instances based on the instance from Otto et al. (2013) utilizing
the above method. The selected cases are Otto-20 with 20 tasks, Otto-50 with 50 tasks, Otto-
100 with 100 tasks and Otto-1,000 with 1,000 tasks. The number of test instances in each
set Otto-20, Otto-50, Otto-100 and Otto-1,000 is 525. For each instance, this study generates
six new instances for the stochastic ALBP with two variances and three different αlevels.
In total, a total number of 525 ×4×2×3=12,600 instances are generated based on the
instance in Otto et al. (2013). This large set of instances could help to clarify differences
according to multiple instance characteristics. All the generated instances are available upon
request.
To evaluate the performance of the proposed BBR method, BBR is compared with the
two mathematical models given in Sect. 2: direct linear approach (LA) and the model with
pure linear transformation (MLTT). In addition, the proposed BBR method is also compared
with twenty heuristic and metaheuristic methods. These methods include two heuristics in
ALBP, including the random search (RS) and random task priority search (RTPS). Here, RS
creates solutions by selecting tasks randomly (Pape, 2015) and RTPS utilizes the roulette
wheel selection to select a task based on the task priorities (Pape, 2015). The metaheuristic
methods contain four recent and effective evolutionary algorithms: teaching–learning-based
optimization (TLBO) (Rao et al., 2011), migrating birds optimization (MBO) (Duman et al.,
2012), grey wolf optimizer (GWO) (Mirjalili et al., 2014), and whale optimization algorithm
(WOA) (Mirjalili & Lewis, 2016). Meanwhile, this study also includes some algorithms
in solving variants of ALBP, including late acceptance hill-climbing algorithm (LAHC)
(Yuan et al., 2015), simulated annealing algorithm (SA) (Suresh & Sahu, 1994), tabu search
algorithm (TS) (Özcan & Toklu, 2009), genetic algorithm (GA) (Baykaso˘glu & Özbakır,
2007), particle swarm optimization algorithm (PSO) (Hamta et al., 2013), discrete particle
swarm optimization algorithm (DPSO) (Li et al., 2016), artificial bee colony algorithm (ABC)
(Lietal.,2019b), improved artificial bee colony algorithm-1 (IABC1) (Li et al., 2019b),
improved artificial bee colony algorithm-2 (IABC2) (Li et al., 2019b), bees algorithm (BA)
(Li et al., 2019b), cuckoo search algorithm (CS) (Li et al., 2019b), discrete cuckoo search
algorithm (DCS) (Li et al., 2019b), improved migrating birds optimization algorithm (IMBO)
(Li et al., 2019b) and ant colony optimization algorithm (ACO) (Celik et al., 2014). All the
algorithms (including BBRs) have been coded in C ++programming language on Microsoft
Visual Studio 2015 and run utilizing the same configuration. The main procedures of all these
methods are not presented due to page limits but they are available upon request.
The proposed BBR and mathematical models terminate when the optimal solution is
achieved and verified, or the computation time reaches 500 s (s). The implemented algorithms
terminate when the achieved station number is equal to the lower bound at the root or the
computation time reaches 30 s, 180 s, 300 s and 500 s. The utilization of four computation
times allows the observation of the algorithms’ performance under different computation
times. The tested models are solved utilizing the CPLEX solver of the IBM ILOG CPLEX
Optimization Studio 12.6.1. The experiments are conducted on a tower type of server with
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Annals of Operations Research (2024) 335:491–516 507
two Intel Xeon E5-2680 v2 processors and 64 GB RAM. The real experiments have been
run on a set of virtual computers and each of them has one processor with 2 GB of RAM.
4.2 Structural parameter evaluation
This section evaluates the structural parameters by solving all the instances generated based
on Scholl’s 269 dataset. Table 2provides the results by the proposed BBR and the variants
of the BBR method. In this table, #OPT is the number of optimal solutions achieved and
#ARPD is the average gap or the average relative percentage deviation (RPD) values for all
the tested instances in one run. The RPD for one instance is calculated utilizing RPD =
100·(UB
some−LB)/LB,whereUB
some is the achieved number of stations by a method and
LBis the theoretical lower bound on the number of stations. Time is the average computation
time for all the tested instances in seconds. In addition, for the dual feasible solution method,
this study utilizes the values of k =1, 2, …, 100 as proposed in Fekete and Schepers (2001).
Sampling LB1, sampling LB2, sampling LB3 and sampling LB6 are the new LB1, LB2,
LB3 and LB6 for the stochastic ALBP based on the sampling approach with a sample size
of 10,000 in Diefenbach and Stolletz (2022).
For the lower bounds, it is observed that the proposed BBR outperforms the variants
of the BBR method with other lower bounds. Although the sampling LB1, sampling LB2,
sampling LB3, sampling LB6 and sampling dual feasible solution methods could possibly
prune a greater number of partial solutions, their utilization leads to poor performance as
more computation time is needed to calculate the sampling lower bounds. For the dominance
rules, it is observed that the maximal load rule, extended Jackson rule, and no-successor rule
obtain minor improvements whereas the memory-based dominance rule could achieve great
improvement. For the search strategy, the proposed search strategy outperforms the original
CBFS and BFS strategies.
Table 2 Results by different BBR methods
Methods #OPT #ARPD Time(s)
Proposed BBR 911 8.7414 216.13
BBR with lower bounds of the simple ALBP (such as LB1, LB2 and
LB3)
911 8.7445 229.23
BBR with dual feasible solution method of the simple ALBP 911 8.7536 217.22
BBR with sampling LB1, sampling LB2 and sampling LB3 897 9.6029 346.48
BBR with sampling LB6 875 9.7023 350.24
BBR with sampling dual feasible solution method 863 9.7359 283.35
BBR without maximal load rule 911 8.8179 231.85
BBR without extended Jackson rule 911 8.7644 233.51
BBR without no-successor rule 911 8.7448 235.53
BBR without memory-based dominance rule 840 9.3346 258.11
BBR with the original CBFS strategy 911 8.8022 215.27
BBR with BFS strategy 907 9.3821 214.86
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508 Annals of Operations Research (2024) 335:491–516
Table 3 Results by BBR and two models
Method #OPT #Feasible #Worse-than-BBR #Equal-to-BBR #Better-than-BBR
BBR 683 690 – – –
LA 154 690 536 154 0
MLTT 343 463 347 343 0
In short, the comparative study shows that the proposed BBR outperforms all the variants
of the BBR method, indicating that the proposed lower bounds, dominance rules and search
strategy are effective and efficient for the stochastic ALBP.
4.3 Comparison with mathematical formulations
A comparative study between BBR and two mathematical models (LA and MLTT) is pre-
sented in this section. As LA and MLTT cannot obtain satisfying or feasible solutions for most
large-size problems within the given computation time, this section mainly tests the problems
generated based on Scholl’s 269 instances with 70 tasks at maximum (135×6=690 instances
in total). Table 3illustrates the results by the BBR and two models under 500 s, where #OPT
is the number of optimal solutions achieved, #Feasible is the number of feasible solutions
achieved within the given computation time, #Worse-than-BBR is the number of instances
where the BBR outperforms the other method, #Equal-to-BBR is the number of instances
where the other method shows the same performance with the BBR, and #Better-than-BBR
is the number of instances where the other method outperforms the BBR.
The results presented in this Table 3indicate that the proposed BBR outperforms LA
and MLTT in terms of both #OPT and #Feasible. Specifically, BBR obtains 683 optimal
solutions, whereas LA and MLTT achieve 154 and 343 optimal solutions, respectively. For
the #Feasible, BBR and LA achieve 690 solutions, whereas MLTT only obtains 463 solutions.
It is also observed that the BBR outperforms LA and MLTT for 536 and 347 instances,
respectively. LA and MLTT cannot show better performance than BBR for any of the instances
tested. Notice that, for the large-size instances, LA and MLTT cannot obtain satisfying or
feasible solutions within the given computation time and the proposed BBR is capable of
achieving high-quality feasible solutions and shows the same or superior performance than
LA and MLTT in all large-size instances. This comparative study shows the superiority
of BBR which can reach much more optimal solutions. Therefore, it outperforms the two
mathematical models by a significant margin.
4.4 Comparison with heuristics and metaheuristics
This section provides the experimental study to compare the performances of BBR and
other implemented heuristics and metaheuristics, where all the instances generated based on
Scholl’s 269 instances are solved here. The results by BBR and other implemented methods
under different computation time limits are presented in Table 4.Inthistable,#ARPD is
the average gap or the average relative percentage deviation (RPD) values for all the tested
instances in 10 runs. As can be seen in the table, BBR is the best performer under all
the termination criteria. Specifically, BBR is the best performer, ACO is the second-best
performer and RTPS is the third-best performer. Despite the superiority of the BBR in terms
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Annals of Operations Research (2024) 335:491–516 509
Table 4 Results by BBR and other heuristics and metaheuristics
Method #ARPD
30 s 180 s 300 s 500 s
BBR 9.133 8.866 8.821 8.769
RS 11.373 11.051 10.968 10.892
RTPS 9.581 9.446 9.415 9.386
TLBO 9.907 9.772 9.726 9.679
MBO 10.787 10.694 10.678 10.665
GWO 10.149 9.882 9.846 9.830
WOA 10.420 10.139 10.091 10.039
LAHC 10.543 10.251 10.194 10.134
SA 10.664 10.292 10.203 10.118
TS 10.624 10.325 10.251 10.176
GA 10.927 10.634 10.580 10.548
PSO 10.073 9.874 9.846 9.813
DPSO 10.697 10.456 10.406 10.324
ABC 10.742 10.400 10.331 10.248
IABC1 10.571 10.324 10.252 10.195
IABC2 10.544 10.288 10.216 10.139
BA 10.632 10.410 10.338 10.293
CS 10.493 10.289 10.217 10.158
DCS 10.571 10.295 10.211 10.145
IMBO 10.548 10.305 10.228 10.166
ACO 9.520 9.439 9.396 9.376
*Best in bold
of the #ARPD, BBR is an exact method, and it can achieve the optimal solution and verify
the optimality of the achieved solution.
To have a better assessment of the BBR and other methods, Table 4presents a comparison
between the BBR and other implemented heuristics and metaheuristics in one run with the
termination criterion of 500 s. In this table, the meanings of #OPT,#Feasible,#Worse-than-
BBR,#Equal-to-BBR and #Better-than-BBR are the same as that presented above in Table 3.
Results presented in Table 5indicate that BBR performs the best in terms of #OPT ,
and so it outperforms all the implemented twenty methods. Regarding #Worse than BBR,it
is observed that BBR outperforms the other methods in many instances. Specifically, BBR
outperforms RS, RTPS, TLBO, MBO, GWO, WOA, LAHC, SA, TS, GA, PSO, DPSO, ABC,
IABC1, IABC2, BA, CS, DCS, IMBO, and ACO for 702, 301, 408, 676, 460, 514, 529, 525,
544, 636, 464, 577, 560, 542, 530, 585, 551, 536, 535 and 297 instances, respectively. So,
BBR shows superior performance over all the methods tested here. For #Better-than-BBR
results, BBR is only outperformed by RTPS and ACO for 1 and 2 instances, respectively,
and the remaining methods cannot outperform BBR for any of the instances tested. In short,
this comparative study demonstrates that the proposed BBR outperforms all the implemented
heuristic and metaheuristic methods. Therefore, BBR is the new state-of-the-art methodology
for the stochastic ALBP.
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510 Annals of Operations Research (2024) 335:491–516
Table 5 Comparison between BBR and other implemented heuristics and metaheuristics
Method #OPT #Feasible #Worse-than- BBR #Equal-to- BBR #Better-than- BBR
BBR 911 1614 – – –
RS 656 1614 702 912 0
RTPS 823 1614 301 1312 1
TLBO 808 1614 408 1206 0
MBO 670 1614 676 938 0
GWO 777 1614 460 1154 0
WOA 751 1614 514 1100 0
LAHC 790 1614 529 1085 0
SA 790 1614 525 1089 0
TS 780 1614 544 1070 0
GA 705 1614 636 978 0
PSO 775 1614 464 1150 0
DPSO 743 1614 577 1037 0
ABC 754 1614 560 1054 0
IABC1 779 1614 542 1072 0
IABC2 785 1614 530 1084 0
BA 725 1614 585 1029 0
CS 747 1614 551 1063 0
DCS 784 1614 536 1078 0
IMBO 784 1614 535 1079 0
ACO 824 1614 297 1315 2
*Best in bold
4.5 Comparison based on multiple instance characteristics
This section utilizes the instances derived from Otto et al. (2013) to clarify differences accord-
ing to multiple instance characteristics. The proposed BBR solves all the 12,600 instances
derived from Otto et al. (2013) as described in Sect. 4.1 and terminates when the optimal
solution is achieved and verified, or the computation time reaches 500 s.
Tabl e 6provides the results for different instance sizes, where Avg-RPD,Min-RPD and
Max-RPD are the average value, minimum value, and maximum value of RPD values and Avg-
t,Min-t and Max-t are the average value, minimum value and maximum value of computation
Table 6 Results for different instance sizes
Instance size nt #OPT Avg-RPD Min-RPD Max-RPD Avg-t Min-t Max-t
Small 20 3150 0.00 0.00 0.00 0.39 0.10 0.69
Medium 50 2759 2.34 0.00 55.56 93.94 0.28 502.01
Large 100 979 18.92 0.00 79.25 377.60 0.66 502.91
Very large 1000 0 35.26 8.57 81.40 501.07 500.51 505.80
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Annals of Operations Research (2024) 335:491–516 511
Table 7 Results for different task variances and αlevels
Tas k
variance
αlevels #OPT Avg-RPD Min-RPD Max-RPD Avg-t Min-t Max-t
Low 0.9 1196 8.61 0.00 48.06 231.74 0.10 501.62
Low 0.95 1185 10.44 0.00 56.19 235.91 0.12 505.80
Low 0.975 1166 12.08 0.00 62.30 239.90 0.11 501.31
High 0.9 1132 14.86 0.00 70.14 247.00 0.11 502.91
High 0.95 1111 18.07 0.00 77.12 252.25 0.12 501.56
High 0.975 1098 20.73 0.00 81.40 252.70 0.12 501.76
times in seconds. It is observed that the proposed BBR is capable of solving all the small-size
instances with 20 tasks optimally. Whereas the BBR fails to find the optimal solution or
verify the optimality of the achieved solution for very large-size instances with 1,000 tasks.
In short, it is more difficult for the proposed BBR to solve the instances with larger number
of tasks.
Tabl e 7provides the results for different task variances and αlevels. It is observed that
the BBR method performs better for the instances with lower task variance. Meanwhile, it
is also prominent that BBR achieves higher numbers of optimal solutions with smaller α
levels. In short, it is more difficult for the proposed BBR to solve the instances with higher
task variance and larger αlevels.
Due to page limits, this section mainly investigates the differences in terms of instance
size, task variance and αlevel. All the results are available upon request and interested
researchers could also study the differences in terms of other instance characteristics based
on the detailed results, such as the order strength and the distribution of task times.
5 Conclusions and future research
This research develops the branch, bound, and remember (BBR) algorithm to solve the
stochastic ALBP. The proposed BBR method stores all the searched partial solutions in
memory and utilizes the modified cyclic best-first search strategy to achieve high-quality
complete solutions quickly. Meanwhile, this study also develops numerous new dominance
rules and lower bounds by taking the stochastic task times into account. To better assess
the performance of the BBR algorithm, this study generates a tremendous number of test
instances based on Scholl’s 269 benchmark problems as well as those derived from Otto
et al. (2013). The proposed BBR method is compared with two mathematical models and
twenty re-implemented heuristics and metaheuristics, including the well-known simulated
annealing, tabu search, particle swarm optimization, ant colony optimization and genetic
algorithms. The structural parameters of the proposed BBR have also been validated through
an experimental study including several variants of the BBR.
Computational results confirm that the BBR method shows superior performance over
the two models in terms of both the number of optimal solutions and the number of feasible
solutions achieved. The comparative studies between BBR and other implemented heuristics
and metaheuristics show that BBR outperforms all the other methods in terms of the number of
achieved optimal solutions and the average relative percentage deviation. Especially, the BBR
123
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512 Annals of Operations Research (2024) 335:491–516
algorithm obtains the same or better results than all these twenty heuristic and metaheuristic
methods for almost all the tested instances. As a result, the proposed BBR algorithm can be
considered to be the new state-of-the-art methodology for the stochastic ALBP.
One of the main limitations of the chance-constrained approach is the requirement for
past data to characterize the uncertain parameter. A sufficient amount of data is needed to
accurately represent the distribution of uncertain task processing times. Another limitation
could be the increase in the problem complexity when dealing with multiple stochastic
constraints. It is also needed to define an appropriate αlevel, which directly has an impact
on the feasibility of the solution obtained eventually.
The proposed methodology might help line managers to obtain high-quality solutions
where stochastic operation times are involved. Future studies might thoroughly study the
influence of different search parameters on the BBR algorithm to obtain a fine-tuned method
in solving the stochastic ALBP. The proposed BBR algorithm can also be applied to other
assembly and disassembly line balancing problems (e.g., stochastic U-shaped and multi-
manned assembly line balancing problems) with some adaptations.
Author contributions All authors contributed to the conceptual design and writing the first draft of the
manuscript. Software development and tests were performed by ZL with comments from all authors and
the results were also analyzed by CGSS and IK. All authors have contributed to the revision of the article and
approved the final manuscript.
Funding Open access funding provided by the Scientific and Technological Research Council of Türkiye
(TÜB˙
ITAK). This project is partially supported by the National Natural Science Foundation of China under
grant numbers 62173260 and 61803287.
Data availability All data generated or analyzed during this study are available upon request.
Declarations
Conflict of interests The authors have no relevant financial or non-financial interests to disclose.
Ethical approval This article does not contain any studies with human participants or animals performed by
any of the authors.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included in the
article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is
not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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