Advances in Mathematical Models and Production Systems in Engineering
Quality Estimation of Assembly Line Balance
WALDEMAR GRZECHCA, MICHAŁ BŁACHUCIŃSKI
Department of Automation Control
Silesian University of Technology
44-100 Gliwice, ul Akademicka 16
POLAND
[email protected] [email protected] http://www.polsl.pl
Abstract: - In the paper authors discuss the quality estimation of assembly line balancing problem. Task are
assigned to workstations and the calculated measures show the quality of the final results. Because even
optimal solutions contain idle times, the estimation of quality of end results is very important. In the literature
we can find different measures but the most useful are: line efficiency, smoothness index and time of the line.
Authors present in the paper the difference between cycle time and maximum station time. Misunderstanding of
these values leads to wrong estimation of quality of assembly line balancing problem. To understand better the
problem a numerical example is discussed.
Key-Words: - assembly line balancing problem, heuristics, line efficiency, smoothness index, time of the line
to be dictated by implied rules set forth by the
production sequence [5]. For the manufacturing of
any product, there are some sequences of tasks that
must be followed. Since the process time of the
different tasks is usually not the same, an imbalance
occurs which generates losses. Therefore one tries to
balance the processing times. The assembly line
balancing problem (ALBP) originated with the
invention of the assembly line. Helgeson and Birnie
[6] were the first to propose the ALBP, and
Salveson [7] was the first to publish the problem in
its mathematical form. An ALBP generally consists
of finding a feasible line balance, i.e., an assignment
of each task to a station such that the cycle time
constraints, the precedence constraints and possible
further restrictions are fulfilled. The most popular
ALBP is called Simple Assembly Line Balancing
Problem (SALBP). It simplifies the more general
ALBP by introducing the following assumptions [810]:
− mass-production of one homogeneous product,
− all tasks are processed in a predetermined mode
(no processing alternatives exist),
− paced line with a fixed common cycle time
according to a desired output quantity,
− the line is considered to be serial with no feeder
lines or parallel elements,
− the processing sequence of tasks is subject to
precedence restrictions,
− deterministic task times,
− no assignment restrictions of tasks besides
precedence constraints,
1 Introduction
The manufacturing assembly line was first
introduced by Henry Ford in the early 1900’s. It was
designed to be an efficient, highly productive way
of manufacturing a particular product. The basic
assembly line consists of a set of workstations
arranged in a linear fashion, with each station
connected by a material handling device. The basic
movement of material through an assembly line
begins with a part being fed into the first station at a
predetermined feed rate. A station is considered any
point at the assembly line in which a task is
performed on the product usually joining one or
more new parts. These tasks can be performed by
machinery, robots, and/or human operators. Once
the product enters a station, the task is then
performed, and the product is moved to the next
station. The time it takes to complete a task at each
operation is known as the process time [1]. The
cycle time of an assembly line is predetermined by a
desired production rate. This production rate is to be
set so that the desired amount of the end product is
produced within a certain time period [2]. In order
for the assembly line to maintain a certain
production rate, the sum of the processing times at
each station (including the transfer time from station
to station) must not exceed the stations’ cycle time
[3]. If the sum of the processing times within a
station is less than the cycle time, idle time is said to
be present at that station [4]. One of the main issues
concerning the development of an assembly line is
how to arrange the tasks to be performed. This
arrangement may be somewhat subjective, but has
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Advances in Mathematical Models and Production Systems in Engineering
−
2.2 U – Shaped Lines
a task cannot be split among two or more
stations,
− all stations are equally equipped with respect to
machines and workers.
Two goals can be considered in addition to the
precedence relations between the tasks: the
minimization of the number of workstations for a
given cycle time (SALBP-I) and the minimization
of the cycle time for a given number of workstations
(SALBP-II). However, during the first forty years of
the assembly line’s existence, only trial-and-error
methods were used to balance the lines [4]. Since
then, there have been numerous methods developed
to solve the different forms of the ALBP. Salveson
[7] provided the first mathematical attempt by
solving the problem as a linear program. Gutjahr
and Nemhauser [10] have shown that the ALBP
problem falls into the class of NP-hard
combinatorial optimization problems. This means
that an optimal solution is not guaranteed for
problems of significant size. Therefore, heuristic
methods have become the most popular techniques
for solving the problem.
In order to deal with the problems of a serial line it
was redesigned to a form of U-shape (Fig. 2). In
such a line operators can work at more than one
station simultaneously. For example first operator
may both load and unload product units. As they are
included in more tasks during production process
they are gaining very important experience and
enlarge horizons. It is very helpful in case of just-intime production systems as it improves flexibility
which is crucial in dynamically changing demand
rates. What more, stations are closer together what
results in better communication between operators
and in case of emergency they are able to help each
other effectively.
1
2
….
M -1
Flow line direction
M
Flow line direction
K
K -1
….
M+1
Fig. 2. U - Shaped assembly line structure
2 Assembly Line Structures
2.3 Parallel Lines
There exists also a classification regarding plant
layout which is used to describe the arrangement of
physical facilities in a production plant [9]. Five
types of layout can be distinguished:
• serial lines,
• U-shaped lines,
• parallel lines,
• parallel stations,
• two-sided lines.
In order to deal with problems described in case of a
serial line it might be a good idea to create several
lines doing the same or similar tasks (Fig. 3).
The advantages of such a solution [9]:
• increased flexibility for mixed-model
systems,
• flexibility due to changing demand rates,
• lowered risk of machine breakdown
stopping the whole production,
• cycle time can be more flexibly chosen
which leads to more feasible solutions.
The optimal number of lines is however a subject of
discussion for every single case separately.
2.1 Serial (Single) Lines
This is a very basic layout of a flow line production
system (Fig. 1). It is determined by the flow of
materials. It is mostly used for small size products.
These lines have several disadvantages:
• monotone work,
• sensibility due to failures,
• inflexibility due to changing demand rates.
Flow line direction
1
2
….
K-1
K
K-1
K
K-1
K
K
1
2
….
Flow line direction
K-1
K
1
2
….
Fig. 3. Parallel assembly lines structures
Fig. 1. Serial assembly line structure
ISBN: 978-960-474-387-2
….
Flow line direction
Flow line direction
1
2
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Advances in Mathematical Models and Production Systems in Engineering
2.3 Parallel Stations
Smoothness index (SI) describes relative smoothness
for a given assembly line balance. Perfect balance is
indicated by smoothness index 0. This index is
calculated in the following manner:
As an extension of serial lines bottlenecks are
replaced with parallel stations (Fig.4). Tasks
performed on parallel stations are the same and
throughput is this way increased.
SI =
2
Flow line direction
1
This kind of flow lines is mainly used in case of
heavy workpieces when it is more convenient to
operate on both sides of a workpiece rather than
rotating it. Instead of single working-place, there are
pairs of two directly facing stations such as 1 and 2
(Fig. 5) Such a solution makes the line much more
flexible as the workpiece can be accessed either
from left or right. In comparison to serial lines:
• it can shorten the line length,
• reduce unnecessary work reaching to the
other side of the workpiece.
K-3
3
K-2
4
K-1
LT = c ⋅ (Km − 1) + Max{t(S K ), t(S K −1 )
K
(4)
As far as smoothness index and line efficiency are
concerned, its estimation, on contrary to LT, is
performed without any change to original version.
These criterions simply refer to each individual
station, despite of parallel character of the method.
But for more detailed information about the balance
of right or left side of the assembly line additional
measures will be proposed:
K
(1)
Smoothness index of the left side
⋅ 100%
c⋅K
where:
K - total number of workstations,
c - cycle time.
ISBN: 978-960-474-387-2
}
where:
Km – number of mated-stations,
K – number of assigned single stations,
t(SK) – processing time of the last single station.
Some measures of solution quality have appeared in
line balancing problem. Below are presented three
of them [9]:
Line efficiency (LE) shows the percentage utilization
of the line. It is expressed as ratio of total station
time to the cycle time multiplied by the number of
workstations:
i
(3)
In two – sided assembly line balancing method
within mated-stations, tasks are intended to perform
its operations at the same time to the both sides. In
consequence, modification has to be introduced to
line time parameter which is the consequence of
parallelism.. We must treat those stations as two
double ones (mated-stations), rather than individual
ones Sk. Accepting this line of reasoning, new
formula is presented below:
3 Measures
LE =
(2)
where:
c - cycle time,
K -total number of workstations.
Fig. 5. Two – sided assembly line
i =1
2
LT = c ⋅ (K − 1) + TK
Flow line direction
∑ ST
− ST i )
Time of the line (LT) describes the period of time
which is need for the product to be completed on an
assembly line:
2.4 Two –sided Lines
2
max
where:
STmax = maximum station time (in most cases cycle
time),
STi = station time of station i.
K
Fig. 4. Parallel stations
1
∑ (ST
i =1
K-1
2
K
SI L =
K
∑ (ST
maxL
i =1
45
− ST iL )
2
(5)
Advances in Mathematical Models and Production Systems in Engineering
where:
SIL- smoothness index of the left side of two-sided
line,
STmaxL- maximum of duration time of left allocated
stations,
STiL- duration time of i-th left allocated station.
1
4
6
2
5
7
Smoothness index of the right side
3
SI R =
K
∑ (ST
maxR
− ST iR )
2
8
9
12
10
13
11
14
15
Fig. 6. Precedence graph of a numerical example
(6)
i =1
Table 1. Processing times of a numerical example
where:
SIR- smoothness index of the right side of two-sided
line,
STmaxR- maximum of duration time of right allocated
stations,
STiR- duration time of i-th right allocated station.
As we can notice there are a lot of wrong
calculations and mistakes in final results measures
because of Equation 1 and Equation 3. The formulae
depend on a number of workstations and cycle time.
But as we can notice in Equation 2 cycle time is
considered as STmax (maximum workstation time).
Therefore the formulae should be modified and
correct equations are Equation 7 and Equation 8. In
this way we can avoid mistakes and
misunderstanding results.
LE =
i =1
i
STmax ⋅ K
(7)
⋅ 100%
and
LT = STmax ⋅ (K − 1) + TK
(8)
The modified equation for line time of two-sided
assembly line balancing problem is:
LT = STmax ⋅ (Km − 1) + Max{t(S K ), t(S K −1 )
}
(9)
4 Numerical Example
A numerical example with 15 tasks will be
considered. Its precedence graph is represented in
Fig. 6 and processing times are given in Table 1.
ISBN: 978-960-474-387-2
Time
Task
Time
Task
Time
1
4
6
5
11
5
2
3
7
5
12
7
3
7
8
5
13
3
4
2
9
3
14
1
5
1
10
1
15
2
The goal of our calculations is to find a feasible
assignment of our tasks in assembly line structure.
To find a number of workstations of assembly line
we calculated the balance using well-known Ranked
Positional Weight method [6]. Below in Fig. 7 and
Fig. 8 station and idle times are presented. In first
experiment the balance for cycle time c=9 was
calculated. As a result we got a feasible solution
with 8 workstations and different station efficiency
for each of them. In the second step the cycle time
value was changed to 12. Now we got a solution
with 5 workstations. None of workstations had idle
time equal to zero what means none of workstations
is uploaded with 100% processing time. In this case
we should be very careful with estimation of final
results quality. Very often the cycle time value is
changed to maximum workstation time (smoothness
index Eq. 2) but other measures still include in their
formulae cycle time value what leads to mistakes.
Therefore instead of Eq. 1 and Eq. 3 in practice
Eq. 7 and Eq. 8 are used. Now both situations –
maximum workstation time equal to cycle time
value and maximum workstation time different than
cycle time value are considered without mistakes. It
is very important for production engineers and
managers to deal with detailed knowledge about
manufacturing processes. It always helps to take the
right decision.
K
∑ ST
Task
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Advances in Mathematical Models and Production Systems in Engineering
5 Conclusions
Assembly line balancing is still an important
problem in scheduling and sequencing theory. The
way how we assign tasks to the workstations
decides about quality of the final result. Because the
problem belongs to the NP-hard class of complexity,
very often heuristic methods are only the one which
solve the problem in reasonable time. Therefore
different measures ensure a good choice of feasible
end results. Authors of the paper proposed modified
formulae for line efficiency and time of the line
what can in the future avoid mistakes in detailed
information about balance process.
This publication was supported by the Human
Capital Operational Programme and was co-financed
by the European Union from the financial resources
of the European Social Fund, project no.
POKL.04.01.02-00-209/11.
Fig. 7. Station and idle times for cycle c=9
References:
[1] Sury, R.J., Aspects of assembly line balancing,
Int. Journal of Production Research, 9, 1971,
pp. 8-14.
[2] Baybars, I., 1986, A survey of exact algorithms
for simple assembly line balancing problem,
Management Science, 32,1986, pp. 11-17.
[3] Fonseca D.J., Guest C.L., Elam M., Karr C.L.,
A fuzzy logic approach to assembly line
balancing, Mathware & Soft Computing
12,2005, pp. 57-74.
[4] Erel E, , Sarin S.C., A survey of the assembly
line
balancing
procedures,
Production
Planning and Control, 9, 1998, pp. 34-42.
[5] Kao, E.P.C., A preference order dynamic
program for stochastic assembly line
balancing, Management Science, 22, 1976, pp.
19-24.
[6] Halgeson W. B., Birnie D. P., Assembly line
balancing using the ranked positional
weighting technique, Journal of Industrial
Engineering, 12, 1961, pp. 18-27.
[7] Salveson, M.E., The assembly line balancing
problem, Journal of Industrial Engineering,
1955, pp. 62-69.
[8] Baybars I., A Survey of Exact Algorithms for
the Simply Assembly Line Balancing Problem,
Management Science, 32, 1986, pp. 909-932.
[9] Scholl A., Balancing and sequencing of
assembly lines, Physica-Verlag
[10] Gutjahr, A.L., Neumhauser G.L., An
algorithm for the balancing problem,
Management Science, 11, 1964, pp. 23-35.
Fig. 8. Station and idle times for cycle c=12
In Table 2 final measures for c=9 and c=12 are
given.
Table 2. Final measures for different cycle times
Cycle c
LE %
SI
LT
9
75
9,7
64
12
90
2,83
58
11
98,18
1
54
Fig. 9. Station and idle times for cycle c=11
ISBN: 978-960-474-387-2
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