XVIII Summer School "Francesco Turco" - Industrial Mechanical Plants
Assembly line balancing with equipment requirement and parallel workers:
an heuristic algorithm
Daria Battini, Maurizio Faccio, Mauro Gamberi,
Alessandro Persona, Francesco Pilati
Department of Management and Engineering, University of Padua, Stradella San Nicola 3, 36100 Vicenza – Italy
([email protected], [email protected], [email protected], [email protected],
[email protected])
Abstract: Production plant design is a key-aspect to compete in the manufacturing industry. Traditional approach
aim is to define a certain system configuration that maximizes a specific performance parameter. Furthermore the
literature traditionally focuses on the modeling of the aforementioned plant, rather than proposing techniques easy to
be exploited in real industry applications. Aim of this paper is to propose an assembly line balancing algorithm that
considers simultaneously several performance indices and able to design a complex industry system.
Purpose
Balancing assembly line means to assign assembly tasks to several stations, respecting precedence constraints so as to
complete the final product with the minimum expense of resources. Aim of this paper is to suggest an heuristic
algorithm to solve the balancing problem considering three important real aspects. First, each task, or elementary
assembly operation, requires specific equipment that differ from task to task. Second, large products (e.g. cars,
automatic machines or white goods), can be handled by more than one worker contemporarily processing different
tasks in the same station (parallel workers). And third, parallel workers operating in the same station, can generate
space or handling conflicts, considering task mounting position.
Design/methodology/approach
An heuristic algorithm is presented to design assembly lines considering a set of algorithm configuration parameters:
task ordering priority rule, maximum number of parallel workers per station, maximum worker idle time allowed and
maximum number of equipment per worker: varying the value of each parameter is possible to determine different
assembly line configurations. A Visual Basic application is developed to evaluate such configurations with respect to
four significant key performance indices: equipment requirements, employed workers, workers saturation, number of
assembly stations. The best solution can be chosen considering plant priorities.
Originality/value
The literature proposes several models that deal with the assembly line balancing problem, nevertheless few of them
investigate real world systems. This paper proposes a balancing heuristic algorithm considering the distinctive
features of real assembly lines, e.g. equipment requirement. This significant characteristic is usually ignored by the
literature, neglecting the equipment duplication. Practitioners and plant managers can benefit from this algorithm
exploiting its distinctive feature: the generation of several assembly line configurations. The different solutions can
be compared using the key performance indices assessed by the algorithm. The analyst can select the final
configuration considering the most relevant assembly line features with respect to the specific plant conditions.
Keywords: Assembly Line Balancing, Parallel workers, Equipment requirement, Assignment restriction,
Heuristic algorithm.
1. Introduction and literature review
with respect to the cycle time, that is the time available for
each station to complete all tasks assigned (Rekiek et al.,
2002, Shtub and Dar-el, 1990).
The order to perform the tasks along the stations must be
in accordance with the precedence diagram. It represents
the assembly precedence constraints, due to technological
Assembly lines are the traditional flow-oriented
production systems. Aim of the Assembly Line Balancing
Problem (ALBP) is to assign a set assembly operations
(tasks) to the work stations (Baybars, 1986). Traditional
purpose of the ALBP is to minimize the stations number,
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XVIII Summer School "Francesco Turco" - Industrial Mechanical Plants
or organizational restrictions and defines the sets of direct
predecessors and successors for each task (Scholl, 1999).
Most of the literature over the years has modelled and
solved the Simple Assembly Line Balancing Problem. This
problem is characterized by several assumptions: paced
line with fixed cycle time, deterministic task times, no
assignment restriction besides the precedence constraints,
serial line layout, all stations are equally equipped of
resources (Scholl and Becker, 2006). Nevertheless real
world ALBPs present particular characteristics that need a
further effort in the modeling phase (Boysen et al., 2007).
station. They execute different tasks simultaneously
respecting the mounting positions, avoiding any
interference (Boysen et al., 2008):
 The first study on parallel worker is carried out
by Akagi et al. (1983). He suggests a two-phase
heuristic method for small scale problem. In the
former phase the tasks are assigned to the
stations, enabled to have one or more workers.
In the latter one, for each station, the allotted
tasks are divided among the workers.
 The problem dimension is partially overcame by
Bukchin and Masin (2004). They present a
backtracking branch and bound algorithm and a
related heuristic version for large scale problems.
Nevertheless tasks partitioning among workers
of the same station is not considered.
 Cevikcan et al. (2009) propose an integer
programming model that assigns the tasks to
each worker that belongs to a station. Due to the
complexity of the model an heuristic algorithm is
developed for large scale problems.
In the last years various researchers deal with parallel
workers assembly line balancing problem:
 Dimitriadis (2006) establishes the term multimanned station, to indicate a station with parallel
workers that simultaneously handle different
tasks.
 Roshani et al. (2013) and Fattahi et al. (2011), as
the previous author, validate respectively their
simulated annealing and ant colony heuristic
algorithms exploiting the problems collected by
Talbot and Patterson (1984). This data set
presents small and medium size problems, since
only one precedence diagram has more than 100
tasks (Arcus, 1963).
 Big size problems are tackled by Becker and
Scholl (2009) and Kellegöz and Toklu (2012).
The authors suggest two different branch and
bound procedures able to define the optimal
solution for small and medium size ALBP. For
large scale one an heuristic version of the
procedures is developed.
Nevertheless all these approaches are limited: the main
lack is the problem dimension of the analyzed case studies
that is always limited in term of task number. In fact, the
aforementioned approaches are tested only for two big
size problems: Bartholdi (1993) and Lee et al. (2001). They
propose two scenarios with more than 100 tasks, the
largest available in the literature. However these are too
simple if compared to real world problems.
In this context, aim of this paper is to model and define
an effective solution procedure for the balancing problem
well applicable in the real world considering assignment
restriction, equipment requirement, parallel workers and
numerous tasks. A procedure able to solve ALBP
handling all these aspect simultaneously is missing in the
literature and it is strongly encouraged by industries
(Falkenauer, 2005).
The remainder of this paper is organized as follows:
Section 2 proposes a Mixed Integer Programming model
for the Assembly Line Balancing Problem with Parallel
Assignment restriction. In addition to the precedence
diagram, several other restrictions can force or forbid
certain tasks to be assigned to a specific station or
particular tasks combinations (Scholl et al., 2010):
 Station restriction forces or forbid particular
tasks to be assigned or not to certain stations.
This constraint usually depends on the
impossibility to install the needed equipment in
different stations. Some resources have to be
necessary installed in particular stations, on the
contrary other resources can not be duplicated
(Johnson, 1983).
 Incompatible tasks are elementary operations
that can not be contemporary assigned to the
same station, e.g. a drilling operation is
incompatible with dimensional control (Agneti et
al., 1995, Bautista et al., 2000).
 Large products (e.g. cars, automatic machines,
white goods) require different mounting
positions to be handled by workers. As
suggested by Lapierre and Ruiz (2004) each
position enables certain tasks to be assigned to
the station and disables some others.
Equipment requirement. Assembly operations could require
specific tools, machines or worker technical capabilities to
be performed (Boysen et al., 2007):
 Rekiek et al. (2002) and Bukchin and Tzur (2000)
propose algorithms to assign a set of resources
to each station and allow it to handle a certain
group of tasks. Similarly, Boysen and Fliedner
(2008) and Nicosia et al. (2002) treat this
problem considering which device has to be
assigned to each station. Each device type is able
to perform different tasks and consequently it
defines a multipurpose flexible station.
 A complementary approach suggested by Amen
(2000) considers the worker specialization as a
constraint for the tasks to be assigned. Each
worker can handle only the tasks that require a
lower or equal specialization level.
 The branch and bound algorithm developed by
Bukchin and Rubinovitz (2003) overcomes the
distinction between human and equipment
resources. They define several equipment
alternatives that has to be chosen to define a
station. A station is classified accordingly to its
intensity of labor and equipment.
Parallel workers. Assembly lines that have to deal with large
products can benefit from assigning multiple workers per
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XVIII Summer School "Francesco Turco" - Industrial Mechanical Plants
Worker and Equipment Requirement (PAWERALBP).
Section 3 analyzes an heuristic algorithm as an effective
solution procedure for the balancing of these assembly
lines. Section 4 adopts the algorithm presented to solve a
well-known problem in the literature, proposing the key
inputs and the results as well as the assembly line
configuration. Finally Section 5 proposes the conclusions
and suggests further research opportunities.
2. Mixed Integer
PAWERALBP
Programming
model
∑
∑
∑
for
A Mixed Integer Programming (MIP) model is developed
for the balancing of assembly lines with assignment
restriction, equipment requirement and parallel workers.
Indices
j, h
w, v
k
m, l
e
Objective function (1) forces to minimize the workers
number of the entire assembly line. Constraint (2) ensures
that each worker belong exclusively to one station.
Constraint (3) forces each task to be assigned uniquely to
one worker. Constraint (4) fastens the tasks to be assigned
only to the workers that belong to any station. Constraint
(5) enables a worker to handle a task only if equipped
properly.
Constraints (6) and (7) limit the task handling period: the
former ensures that any assembly operation has to be
finished within the cycle time, the latter guarantees the
precedence relations among the tasks.
Constraints (8), (9) and (10) represent the restrictions of
the environment where the line has to be installed, as well
as the designer preferences. The maximum number of
worker per station could depend on space or interference
constraint, the maximum equipment size per worker
preempts the resource duplication and the maximum
worker idle time ensures a certain workers saturation.
Constraints (11)-(15) ensure the feasibility of the
balancing, considering that several tasks are handled
simultaneously by different workers in the same station.
Constraint (11) guarantees that two tasks can not precede
each other simultaneously. Constraint (12) limits the
handling period of the tasks that has an additional
precedence relation represented by the decision variable
ujh. Constraint (13) ensures that each worker can handle
one task at a time. Constraint (14) guarantees each
mounting position to receive one task at a time. For the
duration of any task that requires a mounting position that
disables some others, constraint (15) disables the
incompatible mounting positions.
indices for task
indices for worker
index for station
indices for mounting position
index for equipment
Decision variables
{
{
{
{
Parameters
c
tj
cycle time
operation time of task j
{
{
{
{
mw
me
mi
maximum number of workers per station
maximum equipment size per worker
maximum worker idle time
Objective function
∑∑
3. Heuristic algorithm for PAWERALBP
Constraints
The MIP model previously described for PAWERALBP
is NP-hard, consequently an heuristic algorithm for
balancing of this category of assembly line is developed.
The algorithm opens one station at time, determines the
number of workers allocated to it, the equipment and the
tasks assigned to each worker. Figure 1 presents how the
generic k-th station is designed.
∑
∑
∑
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XVIII Summer School "Francesco Turco" - Industrial Mechanical Plants
Start
mw = mw - 1
Initialization
r = 1st w k
iw = 1, w k
im = 1, m
yew = 0, e, w k
j AT, h such that
sj = 1,
j
Feasible station
configuration
idle timew < mi,
OR
mw = 1
OR
n(AT) = 0
FALSE
phj = 1
xhv ∙ zvk’ = 1, with k’<k
TRUE
w
End
FALSE
AT
TRUE
AT ordering
Positional weight
OR
Resource index:
r < mw
AND
n(AT) > 0
n=0
n = n+1
j = nth task
AT
r = w, such that
TRUE
ir = ir + 1
Assignment evaluation
n > n(AT)
OR
sj + tj < c
sj < ir
djm ∙ sj < im
aml ∙ djm ∙ sj < il ,
ir > c
FALSE
FALSE
FALSE
n < n(AT)
TRUE
TRUE
Task assignment
IF [(bje = 1) AND (yer = 0)]
THEN yer = 1
ir = ir + tj
im = im + tj
AT update
sj = ir , j insert in AT
r = w, such that
,
e
Figure 1: PAWERALBP heuristic algorithm.
For each k-th station the algorithm selects one worker at
time (r) and evaluates the tasks assignment considering
each cycle time unit of time (i) consecutively, both for
workers (iw, ir) and for mounting positions (im, il).
The tasks available to be handled at a certain i-th instant
(AT) are decreasing ordered with respect to their
positional weight or resource index (ri), whose value is as
high as the task require equipment already assigned to the
worker considered and depends on parameters na and nb.
Considering the ordered AT, one task at time is evaluated
to be assigned at instant i, to the required mounting
position m and to worker r. A task is assigned if it can be
completed within the cycle time, if the worker, the
mounting and the incompatible positions are available for
the entire task duration and if the worker equipment size,
considering old and new resources, is within the bound.
The task assignment involves the updating of the worker
equipment (yer), available task set (AT), worker, mounting
position and tasks available instants (ir, im, sj), as well as
new worker r to consider for handling operation. He is the
one with the highest idle time.
If any available task is assignable to worker r at instant i r,
worker time index ir will be increased by one unit of time
and the assignment evaluation repeated. At time that i r is
greater than cycle time the worker considered is replaced
by the one with the highest idle time and index ir is set to
0.
The algorithm ends after that all workers and available
tasks have been evaluated. However if the maximum
worker idle time constraint does not hold even for one
worker, the maximum number of worker per station is
decreased by one and the algorithm computed again.
The algorithm is implemented in a Visual Basic
application to facilitate the assembly line balancing.
Varying the value of the algorithm parameters (mw, me,
mi, na, nb and available task ordering priority rule) is
possible to determine different assembly line
configurations and evaluate them with respect to four
significant key performance indices: equipment
requirements, employed workers, workers saturation and
number of assembly stations.
3. Case study and results
To test the proposed algorithm the 148 tasks ALBP
presented by Bartholdi (1993) is used with some
modifications: tasks equipment requirement are shown in
Table 1, whereas Table 2 illustrates tasks mounting
positions.
The assembly line is balanced applying the proposed
algorithm with parameters mw=3, me=3, mi=0.3∙c,
na=-1, nb=0.25. Figure 2 shows the solution obtained.
The configured assembly line consists in 9 stations, 15
workers and 32 resources. 3 stations have one worker, 6
stations two workers.
As depicted in Figure 2, the algorithm achieves the
minimum number of worker (equal to the absolute lower
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XVIII Summer School "Francesco Turco" - Industrial Mechanical Plants
bound), indeed the sum of workers idle time is lower than
the cycle time. Furthermore the algorithm fastens the
workers utilization and prevents the equipment
duplication: 13 workers on 15 are distinguished by an idle
time lower or equal to 6% of cycle time and just 6 workers
on 15 are equipped with all the available resources (X, Y,
Z).
Table 1. Tasks equipment requirements.
Table 2. Tasks mounting positions.
Tasks
Equipment
required
Tasks
Mounting
position
x
2 4 6 8 10 12 14 15 17 18 22 23 27 28 29 32
34 42 43 44 45 46 47 49 50 51 52 53 54 56
57 58 60 62 65 67 73 75 77 79 81 83 85 87
89 91 93 95 97 99 101 103 105 107 109 111
112 113 115 117 119 121 123 125 127 128
129 130 131 133 135 137 139 141 143 144
147
Left
1 3 5 7 9 11 13 16 19 20 21 24 25 26 30 31
33 35 36 37 38 39 40 41 48 55 59 61 63 64
66 68 69 70 71 72 74 76 78 80 82 84 86 88
90 92 94 96 98 100 102 104 106 108 110
114 116 118 120 122 124 126 132 134 136
138 140 142 145 146 148
Right
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18 19 20 21 22 23 24 25 26 27 28 29 30 31
32 33 34 36 41 42 48 49 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69 70 71 72 73
74 75 82 83 84 85 86 87 88 89 90 91 92 93
94 95 96 97 98 101 102 103 104 105 110
111 114 115 120 121 122 123 124 125 126
127 128 129 130 135 136 137 138 139 140
143 146 147 148
35 37 38 39 40 43 44 45 46 47 112 113
116 117 118 119
76 77 78 79 80 81 99 100 106 107 108 109
141 142 144 145
x,y
x,z
50 51 52 53 131 132 133 134
x,y,z
0%
1%
3%
4%
2%
0%
0%
5%
Station 1
Station 2
Idle time
Station 3
Station 4
6%
Equipment assigned
of cycle
X% time
X
Y
Z
Station 5
2%
46%
4%
4%
2%
12%
Station 9
Station 8
Station 7
Station 6
Figure 2: Assembly line configuration.
4. Conclusions and further research
assembly line configurations and evaluate them with a set
of key performance indices.
The aforementioned algorithm is tested with a real
assembly line balancing problem proposed by the
literature. The suggested line configuration is
distinguished by the minimum number of worker, a
negligible workers idle time and limited equipment
duplication.
Further research has to validate the algorithm with
additional real world case studies, even with a greater
number of tasks, as well as to propose a cost objective
functions, to consider simultaneously worker salary,
equipment purchase and station cost.
This paper deals with the balancing of assembly line that
presents real world characteristics such as assignment
restriction, equipment requirement, parallel workers and
numerous tasks.
A mixed integer programming model is proposed:
objective function forces to minimize workers number
subject to equipment size, incompatible mounting
positions, workers idle time and number of workers per
station constraints.
Due to model intrinsic infeasibility (NP-hard) an heuristic
algorithm is developed to solve the Assembly Line
Balancing Problem with Parallel Worker and Equipment
Requirement (PAWERALBP). The algorithm is
implemented in a Visual Studio application: varying the
algorithm parameters is possible to obtain several
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XVIII Summer School "Francesco Turco" - Industrial Mechanical Plants
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