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Int. J. Production Economics 96 (2005) 129–140

Assembly line balancing: Two resource constrained cases

K. ur- sad A& gpaka, Hadi G. ok@enb,*

aDepartment of Industrial Engineering, Faculty of Engineering, Gaziantep University, Gaziantep, Turkey

bDepartment of Industrial Engineering, Faculty of Engineering and Architecture, Gazi University, Maltepe, 06750 Ankara, Turkey

Received 15 April 2003; accepted 25 March 2004

Abstract

In this paper, a new approach on traditional assembly line balancing problem is presented. The goal of proposed

approach is to establish balance of the assembly line with minimum number of station and resources. For this purpose,

0–1 integer-programming models are developed. These models are solved using GAMS-CPLEX mathematical

programming software for a numerical example.

r 2004 Elsevier B.V. All rights reserved.

Keywords: Assembly line balancing; Resource constraint; 0–1 integer programming

1. Introduction

The idea of line balancing was first introduced by Bryton (1954) in his graduate thesis. The first published

scientific study belonged to Salveson (1955). For more than 45 years, many studies were made on this

subject. During this period various new balancing problem concepts such as U-type, two-sided, parallel,

flexible assembly line, etc., and solution algorithms for those problems have been produced. The common

thing for all these problems is using both the operator and the machine in the most efficient way, at the

same time providing flexibility in production.

The purpose of the cases presented in this paper is to provide flexibility in production while increasing the

productivity.

Nowadays assembly lines move towards cellular manufacturing in terms of variety of production. As a

result of this, usage of special equipment and/or professional workers, which are able to perform more than

one process, is increasing. In order to benefit from continuous productions’ advantages, these equipment

and workers must be added to the line in a way by which high efficiency measures (maximum usage,

minimum number of stations) can be achieved.

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*Corresponding author. Tel.: +90-312-231-74-00-2839; fax: +90-312-230-84-34.

E-mail address: hgokcen@gazi.edu.tr (H. G. ok@en).

0925-5273/$-see front matter r 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ijpe.2004.03.008

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In this paper, assembly line balancing problem, for which lots of variations has been examined till now, is

studied with a different perspective.

Efficient usage of resources that carry out assembly line operations has been targeted by balancing the

line with the minimum number of stations. In fact, the issue of line balancing with the minimum number of

resources has always been a serious problem in industry.

The paper is organized as follows: the second part will address the traditional assembly line balancing

problem and an integer programming model developed for the resource constrained cases. In the third part,

the solutions of traditional assembly line balancing and the resource constrained case on a numerical

example are given, and discussed. In the last part, conclusion and suggestions for future studies are

addressed.

2. Assembly line balancing problem

2.1. Traditional assembly line balancing problem and literature review

Assembly lines consist of successive workstations at which products are processed. Workstations are

defined as places where some tasks (operations) on products are performed. Products stay at each

workstation for the cycle time (C), which corresponds to the time interval between successively completed

units.

While designing the line, the list of tasks to be done, task times required to perform each task and the

precedence relations between them are analyzed. While the tasks are being grouped into stations based on

this analysis, the following goals are regarded:

1. Minimization of the number of workstations for a given cycle time.

2. Minimization of cycle time for a given number of work stations (Baybars, 1986).

A grouping which satisfies a determined goal, is called a balance. This problem is called the line balancing

problem. Up to now, many optimal or heuristic techniques have been developed for the solution of this

problem (Baybars, 1986; Ghosh and Gagnon, 1989; Erel and Sarin, 1998).

The problem analyzed in this paper is similar to the studies which is called Assembly System Design

Problems (ASDP) in the literature. ASDP has sought to optimize an economic criterion such as total cost

with machine selection (Ghosh and Gagnon, 1989; Nicosia et al., 2002; Pinnoi and Wilhelm, 1998; Graves

and Lamar, 1983; Yamada and Matsui, 2003). But in this study, workforce/machine assignment and line

balancing problem is analyzed with a different approach.

2.2. Resource constrained assembly line balancing (RCALB) problem

This problem was met in a factory where some machinery is manufactured and assembled. In this

factory, there are limited number of specific machines and limited number of workers that can use these

machines. For example, there is a special cutting tool that can cut metals in a specific width and shape. In

this situation, the problem is assigning these tools/machines and workers to the stations. In assembly lines,

where specific operation robots are used, the importance of simultaneously balancing of the resources and

the assembly line can be understood better.

At this point, RCALB problem should be defined for the line balancing literature. This new problem

deals with ‘‘maximization of resource usage/minimization of number of resources used’’ for a given C and

maximum number of stations. The resources expressed in description may be workforce or machines.

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In practice, existence of limited amount of workforce/machine type, which can perform only some

specific tasks in a system, is common (for example there may be only a specific number of workforce/

machine, which/who are only able to weld or only able to drill).

The new model proposed for the assembly line balancing problems foresees that while tasks are being

assigned to the workstations, the tasks that can be performed by the same resource should be assigned to

the same workstation. This model enables resource saving. During this classification, number of resources is

minimized as much as possible. This resource constrained case also helps smoothing the production flow.

In part 3, traditional balance and the effect of proposed model are explained on a numerical example

more clearly.

In RCALB problem, we may come across with two cases depending on resource type:

Case 1: There is no task that can be assigned to different resources. In this case, the intersection of the

sets of tasks that can be performed by resources is an empty set. For example let us assume an assembly line

with 9 tasks; resource A can do tasks 1, 3, 5, 7, 9 and resource B can do tasks 2, 4, 6, 8. So there is no

common task which can be performed by resource A and B. This case will be called as RCALB Type 1

problem.

Case 2: There are some tasks that can be assigned to different resources, which means some tasks can be

performed by alternative resources. For example, let us assume an assembly line with 9 tasks, resource A

can perform tasks 1, 2, 5, 7, 9; resource B can perform tasks 2, 4, 5, 6, 8. As seen, tasks 2 and 5 can be

performed by both resource A and B. This type will be called as RCALB Type 2.

Traditional assembly line balancing problem without resource constrained is given as follows (Patterson

and Albracht, 1975; G. ok@en and Erel, 1998).

Objective function:

Minimum

X

mmax

j¼1

zj:

ð1Þ

Constraints:

X

X

La

Li

j¼Ei

ðxijÞ ¼ 1;

i ¼ 1;y;n;

ð2Þ

iAwk

tiðxijÞpC;

j ¼ 1;y;mmax;

ð3Þ

X

X

xij; zjAf0;1g8i; j:

j¼Ea

jðxajÞ ?

X

Lb

j¼Eb

jðxbjÞp0; for 8ða;bÞAP;

ð4Þ

iAWj

ðxijÞ ? jjWjjjzjp0;

j ¼ 1;y;mmax;

ð5Þ

Notations:

C: cycle time.

mmax: maximum number of stations which can be estimated from a heuristic procedure.

Wj: subset of all tasks that can be assigned to station j:

jjWjjj : number of tasks in set Wj:

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Ei: earliest station task i can be assigned to, given the precedence relations.

Li: latest station task i can be assigned to, given the precedence relations.

P: set of tasks that precedes from a task.

1;

if task i in the precedence diagram is assigned to workstation j;

0;

otherwise;

1;

if there is any task assigned to workstation j;

0;

otherwise:

xij¼

?

?

zj¼

Constraint (2) assures that all tasks are assigned to at most one station. Constraint (3) ensures that the

sum of task times assigned to each station does not exceed the cycle time. Constraint (4) ensures the

precedence relationships between the tasks are not violated. Lastly, the objective of the formulation is to

minimize the number of work stations.

Resource constraint which express encounter of first case (Type I RCALB problem), for proposed model

is given below:

xij? jjKjrjjMjrp0;

r ¼ 1;y;R;

X

iAKjr

ð6Þ

where Kjris the set of tasks that can be performed in workstation j with resource r: Mjrdefines the resource

r in workstation j: If there is resource r in workstation j; Mjrvalue is equal to 1, in otherwise, this value is 0.

jjKjrjj is number of elements in set Kjr: Constraint (6) ensures that if at least one task is done in workstation j

with resource r; then resource r is used in workstation j; and Mjrvalue gets 1.

Now, the objective of the proposed model can be defined as minimization of the number of resources that

is assigned to workstations:

Minimum

X

R

r¼1

X

mmax

j¼1

Mjr:

ð7Þ

Complete 0–1 integer programming formulation of proposed model for Type I RCALB is given below:

Minimum

X

R

r¼1

X

mmax

j¼1

Mjr:

ð8Þ

Constraints:

Eqs. (2)–(5),

X

iAKjr

xij? jjKjrjjMjrp0;

r ¼ 1;y;R;

ð9Þ

X

xij; zjAf0;1g8i;j:

mmax

j¼1

zjpmmax;

ð10Þ

If the problem is Type II RCALB, common tasks that can be performed by different resources, are

shown as a separate set. So, the resource constraint in Type I RCALB model is modified to constraint (11):

xij? jjNjrjjMjrp0;

r ¼ 1;y;R;

X

iANjr

ð11Þ

where Njris the set of tasks that can be done in workstation j with resource r excluding all the common

tasks that can be performed by different resources. Assigning the common tasks to stations is satisfied by

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new constraint (12):

xij?

X

rAVi

Mjrp0;

r ¼1;y;R; 8 iAfKjr-Njrg;

j ¼ 1;y;mmax;

ð12Þ

where Vi is the set of resources that can do task i: Constraint (12) ensures that if task i is assigned to

workstation j; then it is sufficient that at least one of the resources that can perform task i has been or will

be assigned to workstation j:

Proposed model for Type II RCALB including constraints (11) and (12) is given below:

Minimum

X

R

r¼1

X

mmax

j¼1

Mjr:

ð13Þ

Constraints:

Eqs. (2)–(5),

X

xij?

iANjr

xij? jjNjrjjMjrr0;

r ¼ 1;y;R;

ð14Þ

X

rAVi

Mjrp0;

r ¼ 1;y;R; 8 iAfKjr-Njrg;

j ¼ 1;y;mmax;

ð15Þ

X

xij; zjAf0;1g 8 i; j:

mmax

j¼1

zjpmmax;

ð16Þ

3. Numerical example

A precedence diagram with 11 tasks is given in Fig. 1. The performance times of the tasks and the

resources are presented in Table 1. The problem is solved for both the traditional ALB model and the

proposed model, and the results are compared. All models in this study have been solved using GAMS-

CPLEX mathematical programming software package.

While solving the problem, cycle time was assumed as 9 and the maximum number of stations was

estimated as 7 (note that the theoretical minimum number of stations for numerical example is calculated as

6). Table 2 shows the results obtained when the model given in Appendix A is solved.

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1

2

5

4

7

3

9

8

6

10

11

Fig. 1. Precedence diagram with 11 tasks.

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When the same problem is solved using the traditional assembly line balancing model, we get the results

given in Table 3.

As seen from the Tables 2 and 3, a total of 8 resources (4 units of resource A and 4 units of resource B)

are being used in the traditional line balancing model, while this number could be reduced to 6 (3 units of

resource A and 3 units of resource B) with the proposed model. Thus, 2 units of resources (one unit of

resource A and one unit of resource B) are saved. Besides, the required number of stations is calculated as 6

in the line balanced with the proposed model. This number is same with the theoretical minimum number of

stations, which means that the solution is also optimal in terms of station assignment. When the number of

station is optimal, the ever best situation for the number of resources and stations is satisfied if the number

of resources is equal to the number of stations, and this situation is achieved in our solution. Consequently,

the solution is the best also in terms of the number of resources, and while reaching this best, the best

number of stations is also achieved.

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Table 1

Task times and resources

Task no.Task time Resource

1

2

3

4

5

6

7

8

9

6

2

5

7

1

2

3

6

5

5

4

A

B

A

B

A

B

A

B

A

B

A

10

11

Table 2

Balance with proposed model (C ¼ 9; mmax¼ 7)

No. of stationTaskResource

1

2

3

4

5

6

1

2, 4

3, 5, 7

6, 8

10

9, 11

A

B

A

B

B

A

Table 3

Balance with traditional model (C ¼ 9; mmax¼ 7)

No. of stationTask Resource

1

2

3

4

5

6

1, 2, 5

6, 8

3

4

7, 9

10, 11

A, B

B

A

B

A

A, B

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In the example, if we assume that both resources can perform the tasks of 2 and 5, the problem becomes

RCALB Type 2. The model given in Appendix B is solved with same cycle time and maximum number of

station value given before (the theoretical minimum number of stations is 6), and the obtained results are

given in Table 4.

As seen from Table 4, the balance is achieved with 6 stations and 6 resources. Task 5, which was

previously performed by resource A, along with the task 4 are performed by resource B. Similarly, it is

considerable that the task 2, which was previously performed by B, along with the task 1 are performed by

resource A. When the problem is solved without any resource constrained case, the number of stations is

again found as 6, while the number of resources needed being calculated as 8 (Table 3). So, it can be stated

that the optimal number of resources has been found under the optimal number of stations.

4. Conclusion and future work

Although, resource constrained cases are widely experienced in practice, there has not been sufficient

interest in the literature. This paper presents a new approach on traditional assembly line balancing

problems. The goal of proposed approach is to establish balance of assembly line with minimum number of

stations and resources. For this purpose, 0–1 integer-programming models are developed. With this

proposed model, an important need with regard to line balancing while maintaining the flexibility of

production is met. Because assembly line balancing problems are NP-hard nature (Ghosh and Gagnon,

1989), large-scale problems are also quite hard to solve using this proposed model.

In the future studies, the models explained here can also be modified as a goal programming model by

adding the deviational variables. Especially, when there is limited number of resources, the objective

function (Eqs. (7) and (13)) can be defined as a goal constraint:PR

as a goal by adding deviational variables:

Pmmax

Also, several problems such as minimization of number of the stations for a given number of resources,

and minimization of cycle time for a given number of the stations and resources can be examined, and some

heuristic algorithms can be developed for the large-scale problems.

r¼1

Pmmax

j¼1Mjr? dþ

Mþ d?

MpMmax; where

Mmaxis the maximum number of resources. The station constraints (Eqs. (10) and (16)) can be expressed

j¼1zj? dþ

deviational variables for number of stations and resources.

STþ d?

STpmmax; where dþ

ST; d?

ST; dþ

M; d?

Mare

Acknowledgements

This research was supported in part by the State Planning Organization (DPT) of Turkish Prime

Ministry under Grant no. 2002K120250.

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Table 4

Balance with proposed model (C ¼ 9; mmax¼ 7)

No. of station TaskResource

1

2

3

4

5

6

1, 2

4, 5

6, 8

10

3, 7

9, 11

A

B

B

B

A

A

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Appendix A

Eiand Livalues:

Ei¼

tiþ

X

jAPi

tj

!

=C

"#þ

;

Li¼mmaxþ 1 ?

tiþ

X

jASi

tj

!

=C

"#þ

:

Pi: The set of tasks which must proceed task i:

Si: The set of tasks which must succeed task i:

Task:

Ei:

Li:

1

1

2

2

1

5

3

2

6

4

2

5

5

1

6

6

2

6

7

3

6

8

2

6

9

3

7

10

3

7

11

6

7

Objective:

Minimum Z ¼MAKð1;1Þ þ MAKð1;2Þ þ MAKð1;3Þ þ MAKð1;4Þ þ MAKð1;5Þ

þ MAKð1;6Þ þ MAKð1;7Þ þ MAKð2;1Þ þ MAKð2;2Þ þ MAKð2;3Þ

þ MAKð2;4Þ þ MAKð2;5Þ þ MAKð2;6Þ þ MAKð2;7Þ

Constraints

Assignment constraints:

Xð1;1Þ þ Xð1;2Þ ¼ 1;

Xð2;1Þ þ Xð2;2Þ þ Xð2;3Þ þ Xð2;4Þ þ Xð2;5Þ ¼ 1;

Xð3;2Þ þ Xð3;3Þ þ Xð3;4Þ þ Xð3;5Þ þ Xð3;6Þ ¼ 1;

Xð4;2Þ þ Xð4;3Þ þ Xð4;4Þ þ Xð4;5Þ ¼ 1;

Xð5;1Þ þ Xð5;2Þ þ Xð5;3Þ þ Xð5;4Þ þ Xð5;5Þ þ Xð5;6Þ ¼ 1;

Xð6;2Þ þ Xð6;3Þ þ Xð6;4Þ þ Xð6;5Þ þ Xð6;6Þ ¼ 1;

Xð7;3Þ þ Xð7;4Þ þ Xð7;5Þ þ Xð7;6Þ ¼ 1;

Xð8;2Þ þ Xð8;3Þ þ Xð8;4Þ þ Xð8;5Þ þ Xð8;6Þ ¼ 1;

Xð9;3Þ þ Xð9;4Þ þ Xð9;5Þ þ Xð9;6Þ þ Xð9;7Þ ¼ 1;

Xð10;3Þ þ Xð10;4Þ þ Xð10;5Þ þ Xð10;6Þ þ Xð10;7Þ ¼ 1;

Xð11;6Þ þ Xð11;7Þ ¼ 1:

Cycle time constraints:

6?Xð1;1Þ þ 2?Xð2;1Þ þ Xð5;1Þp9;

6?Xð1;2Þ þ 2?Xð2;2Þ þ 5?Xð3;2Þ þ 7?Xð4;2Þ þ Xð5;2Þ þ 2?Xð6;2Þ þ 6?Xð8;2Þp9;

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2?Xð2;3Þ þ 5?Xð3;3Þ þ 7?Xð4;3Þ þ Xð5;3Þ þ 2?Xð6;3Þ þ 3?Xð7;3Þ þ 6?Xð8;3Þ þ 5?Xð9;3Þ

þ 5?Xð10;3Þp9;

2?Xð2;4Þ þ 5?Xð3;4Þ þ 7?Xð4;4Þ þ Xð5;4Þ þ 2?Xð6;4Þ þ 3?Xð7;4Þ þ 6?Xð8;4Þ þ 5?Xð9;4Þ

þ 5?Xð10;4Þp9;

2?Xð2;5Þ þ 5?Xð3;5Þ þ 7?Xð4;5Þ þ Xð5;5Þ þ 2?Xð6;5Þ þ 3?Xð7;5Þ þ 6?Xð8;5Þ þ 5?Xð9;5Þ

þ 5?Xð10;5Þp9;

5?Xð3;6Þ þ Xð5;6Þ þ 2?Xð6;6Þ þ 3?Xð7;6Þ þ 6?Xð8;6Þ þ 5?Xð9;6Þ

þ 5?Xð10;6Þ þ 4?Xð11;6Þp9;

5?Xð9;7Þ þ 5?Xð10;7Þ þ 4?Xð11;7Þp9:

Resource constraints:

Xð1;1Þ þ Xð5;1Þ ? 2?MAKð1;1Þp0;

Xð2;1Þ ? MAKð2;1Þp0;

Xð1;2Þ þ Xð3;2Þ þ Xð5;2Þ ? 3?MAKð1;2Þp0;

Xð2;2Þ þ Xð4;2Þ þ Xð6;2Þ þ Xð8;2Þ ? 4?MAKð2;2Þp0;

Xð3;3Þ þ Xð5;3Þ þ Xð7;3Þ þ Xð9;3Þ ? 4?MAKð1;3Þp0;

Xð2;3Þ þ Xð4;3Þ þ Xð6;3Þ þ Xð8;3Þ þ Xð10;3Þ ? 5?MAKð2;3Þp0;

Xð3;4Þ þ Xð5;4Þ þ Xð7;4Þ þ Xð9;4Þ ? 4?MAKð1;4Þp0;

Xð2;4Þ þ Xð4;4Þ þ Xð6;4Þ þ Xð8;4Þ þ Xð10;4Þ ? 5?MAKð2;4Þp0;

Xð3;5Þ þ Xð5;5Þ þ Xð7;5Þ þ Xð9;5Þ ? 4?MAKð1;5Þp0;

Xð2;5Þ þ Xð4;5Þ þ Xð6;5Þ þ Xð8;5Þ þ Xð10;5Þ ? 5?MAKð2;5Þp0;

Xð3;6Þ þ Xð5;6Þ þ Xð7;6Þ þ Xð9;6Þ þ Xð11;6Þ ? 5?MAKð1;6Þp0;

Xð6;6Þ þ Xð8;6Þ þ Xð10;6Þ ? 3?MAKð2;6Þp0;

Xð9;7Þ þ Xð11;7Þ ? 2?MAKð1;7Þp0;

Xð10;7Þ ? MAKð2;7Þp0:

Station constraints:

Xð1;1Þ þ Xð2;1Þ þ Xð5;1Þ ? 3?ISTð1Þp0;

Xð1;2Þ þ Xð2;2Þ þ Xð3;2Þ þ Xð4;2Þ þ Xð5;2Þ þ Xð6;2Þ þ Xð8;2Þ ? 7?ISTð2Þp0;

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Xð2;3Þ þ Xð3;3Þ þ Xð4;3Þ þ Xð5;3Þ þ Xð6;3Þ þ Xð7;3Þ þ Xð8;3Þ þ Xð9;3Þ þ Xð10;3Þ ? 9?ISTð3Þp0;

Xð2;4Þ þ Xð3;4Þ þ Xð4;4Þ þ Xð5;4Þ þ Xð6;4Þ þ Xð7;4Þ þ Xð8;4Þ þ Xð9;4Þ þ Xð10;4Þ ? 9?ISTð4Þp0;

Xð2;5Þ þ Xð3;5Þ þ Xð4;5Þ þ Xð5;5Þ þ Xð6;5Þ þ Xð7;5Þ þ Xð8;5Þ þ Xð9;5Þ þ Xð10;5Þ ? 9?ISTð5Þp0;

Xð3;6Þ þ Xð5;6Þ þ Xð6;6Þ þ Xð7;6Þ þ Xð8;6Þ þ Xð9;6Þ þ Xð10;6Þ þ Xð11;6Þ ? 8?ISTð6Þp0;

Xð9;7Þ þ Xð10;7Þ þ Xð11;7Þ ? 3?ISTð7Þp0;

ISTð1Þ þ ISTð2Þ þ ISTð3Þ þ ISTð4Þ þ ISTð5Þ þ ISTð6Þ þ ISTð7Þp7:

Precedence constraints:

Xð1;1Þ þ 2?Xð1;2Þ ? Xð2;1Þ ? 2?Xð2;2Þ ? 3?Xð2;3Þ ? 4?Xð2;4Þ ? 5?Xð2;5Þp0;

Xð1;1Þ þ 2?Xð1;2Þ ? 2?Xð3;2Þ ? 3?Xð3;3Þ ? 4?Xð3;4Þ ? 5?Xð3;5Þ ? 6?Xð3;6Þp0;

Xð1;1Þ þ 2?Xð1;2Þ ? 2?Xð4;2Þ ? 3?Xð4;3Þ ? 4?Xð4;4Þ ? 5?Xð4;5Þp0;

Xð1;1Þ þ 2?Xð1;2Þ ? Xð5;1Þ ? 2?Xð5;2Þ ? 3?Xð5;3Þ ? 4?Xð5;4Þ ? 5?Xð5;5Þ ? 6?Xð5;6Þp0;

Xð2;1Þ þ 2?Xð2;2Þ þ 3?Xð2;3Þ þ 4?Xð2;4Þ þ 5?Xð2;5Þ ? 2?Xð6;2Þ ? 3?Xð6;3Þ ? 4?Xð6;4Þ

? 5?Xð6;5Þ ? 6?Xð6;6Þp0;

2?Xð3;2Þ þ 3?Xð3;3Þ þ 4?Xð3;4Þ þ 5?Xð3;5Þ þ 6?Xð3;6Þ ? 3?Xð7;3Þ ? 4?Xð7;4Þ ? 5?Xð7;5Þ

? 6?Xð7;6Þp0;

2?Xð4;2Þ þ 3?Xð4;3Þ þ 4?Xð4;4Þ þ 5?Xð4;5Þ ? 3?Xð7;3Þ ? 4?Xð7;4Þ ? 5?Xð7;5Þ ? 6?Xð7;6Þp0;

Xð5;1Þ þ 2?Xð5;2Þ þ 3?Xð5;3Þ þ 4?Xð5;4Þ þ 5?Xð5;5Þ þ 6?Xð5;6Þ ? 3?Xð7;3Þ ? 4?Xð7;4Þ

? 5?Xð7;5Þ ? 6?Xð7;6Þp0;

2?Xð6;2Þ þ 3?Xð6;3Þ þ 4?Xð6;4Þ þ 5?Xð6;5Þ þ 6?Xð6;6Þ ? 2?Xð8;2Þ ? 3?Xð8;3Þ ? 4?Xð8;4Þ

? 5?Xð8;5Þ ? 6?Xð8;6Þp0;

3?Xð7;3Þ þ 4?Xð7;4Þ þ 5?Xð7;5Þ þ 6?Xð7;6Þ ? 3?Xð9;3Þ ? 4?Xð9;4Þ ? 5?Xð9;5Þ

? 6?Xð9;6Þ ? 7?Xð9;7Þx0;

2?Xð8;2Þ þ 3?Xð8;3Þ þ 4?Xð8;4Þ þ 5?Xð8;5Þ þ 6?Xð8;6Þ ? 3?Xð10;3Þ ? 4?Xð10;4Þ ? 5?Xð10;5Þ

? 6?Xð10;6Þ ? 7?Xð10;7Þp0;

3?Xð9;3Þ þ 4?Xð9;4Þ þ 5?Xð9;5Þ þ 6?Xð9;6Þ þ 7?Xð9;7Þ ? 6?Xð11;6Þ ? 7?Xð11;7Þp0;

3?Xð10;3Þ þ 4?Xð10;4Þ þ 5?Xð10;5Þ þ 6?Xð10;6Þ þ 7?Xð10;7Þ ? 6?Xð11;6Þ ? 7?Xð11;7Þp0:

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Appendix B

Objective:

Minimum Z ¼MAKð1;1Þ þ MAKð1;2Þ þ MAKð1;3Þ þ MAKð1;4Þ þ MAKð1;5Þ

þMAKð1;6Þ þ MAKð1;7Þ þ MAKð2;1Þ þ MAKð2;2Þ þ MAKð2;3Þ

þ MAKð2;4Þ þ MAKð2;5Þ þ MAKð2;6Þ þ MAKð2;7Þ

Constraints

Assignment constraints, Precedence constraints, Station constraints and Cycle time constraints are the

same as those given in the model at Appendix A.

Resource constraints:

Xð1;1Þ ? MAKð1;1Þp0;

Xð1;2Þ þ Xð3;2Þ ? 2?MAKð1;2Þp0;

Xð4;2Þ þ Xð6;2Þ þ Xð8;2Þ ? 3?MAKð2;2Þp0;

Xð3;3Þ þ Xð7;3Þ þ Xð9;3Þ ? 3?MAKð1;3Þp0;

Xð4;3Þ þ Xð6;3Þ þ Xð8;3Þ þ Xð10;3Þ ? 4?MAKð2;3Þp0;

Xð3;4Þ þ Xð7;4Þ þ Xð9;4Þ ? 3?MAKð1;4Þp0;

Xð4;4Þ þ Xð6;4Þ þ Xð8;4Þ þ Xð10;4Þ ? 4?MAKð2;4Þp0;

Xð3;5Þ þ Xð7;5Þ þ Xð9;5Þ ? 3?MAKð1;5Þp0;

Xð4;5Þ þ Xð6;5Þ þ Xð8;5Þ þ Xð10;5Þ ? 4?MAKð2;5Þp0;

Xð3;6Þ þ Xð7;6Þ þ Xð9;6Þ þ Xð11;6Þ ? 4?MAKð1;6Þp0;

Xð6;6Þ þ Xð8;6Þ þ Xð10;6Þ ? 3?MAKð2;6Þp0;

Xð9;7Þ þ Xð11;7Þ ? 2?MAKð1;7Þp0;

Xð10;7Þ ? MAKð2;7Þp0;

Xð2;1Þ ? MAKð1;1Þ ? MAKð2;1Þp0;

Xð2;2Þ ? MAKð1;2Þ ? MAKð2;2Þp0;

Xð2;3Þ ? MAKð1;3Þ ? MAKð2;3Þp0;

Xð2;4Þ ? MAKð1;4Þ ? MAKð2;4Þp0;

Xð2;5Þ ? MAKð1;5Þ ? MAKð2;5Þp0;

Xð5;1Þ ? MAKð1;1Þ ? MAKð2;1Þp0;

Xð5;2Þ ? MAKð1;2Þ ? MAKð2;2Þp0;

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Xð5;3Þ ? MAKð1;3Þ ? MAKð2;3Þp0;

Xð5;4Þ ? MAKð1;4Þ ? MAKð2;4Þp0;

Xð5;5Þ ? MAKð1;5Þ ? MAKð2;5Þp0;

Xð5;6Þ ? MAKð1;6Þ ? MAKð2;6Þp0:

References

Baybars,’I., 1986. A survey of exact algorithms for the simple assembly line balancing problem. Management Science 32 (8), 909–932.

Bryton, B., 1954. Balancing of a continuous production line. M.S. Thesis, Northwestern University, Evanston, IL.

Erel, E., Sarin, S.C., 1998. A survey of the assembly line balancing procedures. Production Planning and Control 9 (5), 414–434.

Ghosh, S., Gagnon, J., 1989. A comprehensive literature review and analysis of the design, balancing and scheduling of assembly

systems. International Journal of Production Research 27 (4), 637–670.

G. ok@en, H., Erel, E., 1998. Binary integer formulation for mixed model assembly line balancing problem. Computers and Industrial

Engineering 34 (2), 451–461.

Graves, S.C., Lamar, B.W., 1983. An integer programming procedure for assembly system design problems. Operations Research 31

(3), 522–545.

Nicosia, G., Pacciarelli, D., Pacifici, A., 2002. Optimally balancing assembly lines with different workstations. Discrete Applied

Mathematics 118, 99–113.

Patterson, J.H., Albracht, J.J., 1975. Assembly line balancing: Zero–one programming with fibonacci search. Operations Research 23,

166–172.

Pinnoi, A., Wilhelm, W.E., 1998. Assembly system design: A branch and cut approach. Management Science 44 (1), 103–118.

Salveson, M.E., 1955. The assembly line balancing problem. Journal of Industrial Engineering 6 (3), 18–25.

Yamada, T., Matsui, M., 2003. A management design approach to assembly line systems. International Journal of Production

Economics 84, 193–204.

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