100 Chapter 4 As$em blJr: Line Balancing by Linear Programming Appro. ach . 4.1 Introduction The need for high volume and low cost production has resulted in re pl acement of traditional production metho ds by Assembly Lines. The general line balancing problem is a difficult optimi zation problem where
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100
Chapter 4
As$em blJr: Line Balancing by Linear
Programming Appro.ach ..
4.1 Introduction
The need for high vo lume and low cost production has resulted in
replacement of traditional production methods by Assembly Lines. The
general line balancing problem is a di ffic ult optimization problem where
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In Search of Optimum Assembly Line Balancing
Chapter 4: ALB by LP approach
constraints are not restricted to precedence constraints only. There may be
problems of divisibility of a work element. Also, zonal constraints may
be operative. To solve a line balancing problem, the approach followed in
OR is to simplify the same by bringing down the level of complexity to a
solvable state.
The classic OR definition of the line balancing problem, given by
Becker and Scholl (2006), is as follows: Given a set of tasks of various
durations, a set of precedence constraints among the tasks, and a set of
workstations, assign each task to exactly one workstation in such a way
that no precedence constraint is violated and the assignment is optimal.
This simplified problem of optimization can be classified mainly
into two categories.
( 1) The number of workstation is to be minimized for a given cycle
time. This cycle time can not be exceeded by the total task time of work
elements assigned to any ofthe workstations.
(2) The cycle time is to be minimized given the number of
workstation. This cycle time is equal to the largest total task time of the
work elements assigned to the workstations.
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It is by far these two variants of line balancing that have been
widely researched. Attempts to solve these optimization problems started
during 1950s. That time, the focus of attention was on the core problem
of configuration, which is the assignment of tasks to stations. Bowman
(1960) was the first to consider the linear programming approach to
arrive at an optimum solution to the line balancing problem. During the
last few decades, several researchers handled this problem of line
balancing by using different optimization techniques. Hoffman (1963),
Mansoor and Yadin (1971) and Geoffrion (1976) used mathematical
programming to present a clear formulation of the problem and solve the
same. Later, Van Assche and Herroelen (1979) have proposed an optimal
procedure for the single-model deterministic assembly line balancing
problem. Integer programming procedur~ was used by Graves and
Lamer (1983) for designing an assembly system. Infact, in the mid 80's
some researchers gave emphasis on application part like application of
operational research models and techniques in flexible manufacturing
systems (Kusiak, 1986) and application of a hierarchical approach for
solving machine grouping and loading problems of flexible
manufacturing systems (see Stecke, 1986). Berger et al (1992) adopted
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Branch-and-bound algorithms for the multi-product assembly line
balancing problem. In 1998, Pinnoi and Wilhelm dealt with the problem
of system design using the branch and cut approach. Nicosia et al (2002)
introduced the concept of cost and studied the problem of assigning
operations to an ordered sequence of non-identical workstations, which
also took into consideration the precedence relationships and cycle time
restrictions. The purpose was to minimize the cost of the assignment by
using a dynamic programming algorithm. They also introduced several
rules to reduce the number of states in the dynamic programming
formulation. In 2006, Bukchin and Rabinowitch proposed a branch-and
bound based solution for the mixed-model assembly line-balancing
problem for minimizing stations and task duplication costs.
Problem description
The balancing problems studied in all the above mentioned methods are
oriented towards minimization of either balancing loss or cost of
assignment. These methods can be best used in transfer lines where work
elements are preferably performed by machines. Assembly lines
involving human elements have another pressing problem. "The losses
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resulting from workers' variable operation times" is known as System
loss (see Ray Wild, 2004) which is more important than balancing loss.
Our objective in this current work is to design an assembly line
where dual objectives of minimization of balancing loss and system loss
can be met. For this purpose, we first propose a measure for system loss
(MSL) and then install an optimization method through Linear
Programming approach.
4.2 Notation
K number of jobs
N number of workstations
ti task time or assembly time of i1h job
wj lh workstation
a(i,j) binary measure for assignment oftask ito workstationj
Lj idle time oft work station
Nmin minimum number of workstation for a given cycle time
C cycle time
C1 trial cycle time
Cmin minimum cycle time for a given K
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S1 slackness for trial cycle time Cto i.e., S1 = C - C1
B balancing loss, i.e., {(NC- I Ti) /NC}*lOO%
R range of idle times L1, L2, ••••• , LN
M measure of system loss (MSL) = R I minimum idle time
4.3 Methodology
Balancing loss occurs due to the uneven allocation of work to station.
Mostly, one uses the concept of balancing loss, B to examine the
efficiency of an assembly line. Our proposed work is a multi-objective
one where minimization of balancing loss is to be addressed along with
system loss. According to Ray Wild (2004) this System loss arises out of
workers' variable operation time. However, no standard measure has
been proposed so far on the system loss. We propose to consider the
difference between maximum idle time and minimum idle time as a
measure of the system loss with this expectation that a system will be
stable if the idle time of each workstation is more or less same. However,
in the extreme case when there is no idle time for any ofthe work stations
this difference will be zero. But that situation will lead to high system
In Search of Optimum Assembly Line Balancing
Chapter 4: ALB by LP approach . .
loss. Thus, a minimum idle time is needed in the system and if the
minimum idle time increases then the system loss decreases. Keeping
these two issues in mind, system loss can be measured through range with
lower value preferred over higher value and can also be measured
through minimum idle time with higher value preferred over lower value.
To make them unidirectional and suggest a unit free measure, we
consider a combined range based measure of system loss as M = range I
minimum idle time.
Given a choice of C, it may be noted that the theoretical minimum
number of workstations, Nmin, must satisfy the following constraints:
K K ,LT/C ::; Nmin::; ,LT/C +1, i=l i=l
from where we arrive at Cmim the minimum value of C with the same
balancing loss, as
Cmin=[ rTi/ Nmin+ 0 . I=l J
So, for a given a cycle time, C, one may conceptually start from a
trial cycle time, C1 which satisfies the condition Cmin :::; C1 ::; C. In that
case each workstation can be provided with at least a minimum slack
time S1• Then with C1 as the trial cycle time one can arrive at the set of
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In Search of Optimum Assembly Line Balancing
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optimum workstation configurations and maintain the targeted cycle time
C by uniformly adding to each workstation a slackness St to Ct. For
optimum configuration of each trial cycle time, we get a value of the
measure M. The configuration that gives minimum value ofM is the final
optimal solution. We next develop a mathematical formulation of the
problem for arriving at the minimum M value.
4.4 Mathematical Formulation
Since our aim is to minimize M, the ratio of range of idle times and
minimum idle time, we propose to minimize {p- q}/{C- Ct + q} where
p is the maximum idle time, q is the minimum idle time under trial cycle
time Ct. Thus the objective function is z = {p- q}/{C- Ct + q} and our
objective is to minimize z.
Let us consider the binary variable a(i, j) such that
a(i,j) ~ {: if i E wj i th task is assigned to Wj,
if i ~ Wj i th task is assigned to Wj,
andistruefor i= 1,2, ..... ,K, j = 1,2, ..... ,N.
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Chapter 4: ALB by LP approach ----·~~,--~~--····-·~·~
Then, under the condition that the ith task can be assigned to only one
workstation
N
L a(i., j) =1 must hold for all i = 1, 2, ..... , K. j=l
Also, according to precedence constraints if task i' is to be
assigned before assigning task i, that is i' < i , then
j
a(i,j) ::;La(i',r) r=l
. ' . v l < l
Under the condition that p is the maximum idle time for trial cycle time
Ct. we have,
p :<: [ C,- t. t,a(i, j)] j = 1, 2, ..... , N.
Similarly, for q as the minimum idle time for trial cycle time Ct.
j = 1, 2, ..... , N.
Thus, an integer programming formulation of the optimization problem
can be written as:
Minimize z = {p- q}/{C- C1 + q}
Subject to constraints,
109
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
In Search of Optimum Assembly Line Balancing
Chapter 4: ALB by LP approach
N
La(i,j) = 1. \/ i j=l
a(i, j) :S I a(i', r) \/ i' < i r=l
p2 [ C,-t tia(i,j)]
[ C,- ttia(i,j)] 2 q
p:SC
q20
a(i,j) = 0,1 \/ i, j
Ct= Cmim Cmin +1, ...... ,C
4.5 Worked Out Example
To explain how the proposed algorithm works, we consider in Figure
4.1 an assembly line balancing problem from Ray Wild (2004). A figure
within a circle represents task number and that close to a circle
represents corresponding task time. Precedence constraints are
represented by arrows.
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In Search of Optimum Assembly Line Balancing
Chapter 4: ALB by LP approach
Figure 4.1: Precedence diagram of workstations along with the task times.
This problem can be summarized in a tabular form in terms of the
binary variables a(i,j)s and is given in the Table 4.1.