as$em bljr: balancing by linear programming...

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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


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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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,

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(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)



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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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.

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Work Element

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Activity Time

6 5 8 9 5 4 5 6 10 5 6 2 5 4 12 10 5 15 10 5 6



Immediate Work Station Predecessor 1 2 3 4

- a(1 ,1) a(1,2) a(l,3) a(1,4) - a(2,1) a(2,2) a(2,3l a(2,4) - a(3,1) a(3,2) a(3,3) a(3,4) 1 a(4,1) a(4,2) a(4,3) a(4,4)

1, 2 a(5,1) a(5,2) a(5,3) a(5,4) 2 a(6,1) a(6,2) a(6,3) a(6,4) 3 a(7,1) a(7,2) a(7,3) a(7,4) 3 a(8,1) a(8,2) a(8,3) a(8,4) 4 a(9,1) a(9,2) a(9,3) a(9,4)

5, 6 a(lO,l) ~(10,2) a(l0,3) a(10,4) 8 a(l1,1) a(l1,2) a(l1,3) a(l1,4)

10, 7 a(l2,1) a(l2,2) a(l2,3) a(l2,4) 12 a(13,1) a(13,2) a(13,3) a(13,4) 13 a(14,1) a(14,2) a(l4,3) a(14,4)

9, 11,14 a(15,1) a(15,2) a(15,3) a(15,4) 15 a(l6,1) a(16,2) a(l6,3) a(l6,4) 16 a(17,1) a(17,2) a(l7,3) a(l7,4) 16 a(18,1) a(18,2) a(18,3) a(18,4) 16 a(l9,1) a(19,2) a(l9,3) a(l9,4) 17 a(20,1) a(20,2) a(20,3) a(20,4)

18,19,20 a(21,1) a_(21,2) a(21,3) a(21,4) Table 4.1: Precedence relation and task times of work elements.

Let us consider for our study a cycle time of 35(=C) time units. This

results in Nmin = 5 and Cmin = 29 and 29 :::::; C1 :::::; 35 with S1 = 35 - C1• The

5 a(1,5) a(2,5) a(3,5) a(4,5) a(5,5) a(6,5) a(7,5) a(8,5) a(9,5)

a(10,5)_ a(l1,5) a(l2,5) a(13,5) a(14,5)_ a(l5,5) a(16,5) a(l7,5) a(18,5)_ a(l9,5) a(20,5) a(21,5)



optimum configuration is presented in Table 5.2 where M value is 1.5 and

balancing loss is 18.3%.

c Work Station Work Station 2

Work Work Station Work Station M 1 Station 3 4 5

31 1,2,3,5,8 6,7,10,11,12,13 4,9,15 16,17,19 18,20,21 1.5 ,14

Table 4.2: Final Optimum Configuration

4.6 Conclusion

We have presented a mathematical programming approach for

balancing an assembly line with twin objectives of minimization of

balancing loss and system loss. From the solution set generated by the

optimization technique, final choice is made based on optimum number

of workstations and minimum value of a measure of system loss,

proposed herein.

This study is giving an ideal set of configurations because we

can ensure some amount of slackness (S1) in each workstation when the

trail cycle time is less than the targeted cycle time.

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as$em bljr: balancing by linear programming...

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