33 10 Assembly Line Balancing Using Bacterial Foraging Optimization

8/12/2019 33 10 Assembly Line Balancing Using Bacterial Foraging Optimization

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ASSEMBLY LINE BALANCING USING BACTERIAL FORAGING OPTIMIZATION

ALGORITHM

Yakup Atasagun1*, Yakup Kara2

Department of Industrial Engineering, Selcuk University, Konya, Turkey1Phone: +903322232110 e-mail:[email protected]

2Phone: +903322232014 e-mail:[email protected]

Abstract: Assembly line balancing is the problem of assigning tasks to workstations by optimizing aperformance measure while satisfying precedence relations between tasks and cycle time restrictions. Manyexact, heuristic and metaheuristic approaches have been proposed for solving simple straight and U-shaped

assembly line balancing problems. In this study, a quite new optimization algorithm, Bacterial ForagingOptimization Algorithm (BFOA), based heuristic approach is proposed for solving simple straight and U-shaped assembly line balancing problems. The performance of the proposed algorithm is evaluated using a

well-known data set taken from the literature in which the number of tasks varies between 7 and 111. Theproposed algorithm yielded optimal solutions for 123 of 128 test problems in seconds.

Keywords: Assembly Line Balancing, U-Shaped Assembly Lines, Bacterial Foraging Optimization

Algorithm, Metaheuristics

1. Introduction

Assembly lines are the most important components of mass production systems. The improved laborproductivity is their essential significance for producers who have to produce high volume products in a fast

and cost effective manner. An assembly line consists of several successive workstations in which a group ofassembly operations (tasks) are performed in a limited duration (cycle time). The productivity level of an

assembly line generally depends on balancing performance. Assembly line balancing (ALB) is the problemof assigning tasks to successive workstations by satisfying some constraints and optimizing a performance

measure. This performance measure is usually the minimization of the number of workstations utilized over

the assembly line. The assignments should guarantee that each task is assigned to at least and at most oneworkstation (assignment constraints), all precedence relations among these tasks are satisfied (precedence

constraints) and the work content of a workstation does not exceed the predetermined cycle time (cycle timeconstraints). The cycle time is determined by means of demand rate of the product(s) in a planning horizon.Assembly lines can be categorized into several groups with regard to their shapes and number of differentproducts produced on the line. By means of the line shape, they are classified as traditional straight and U-

shaped assembly lines. Traditional straight assembly lines have been used in many mass productionindustries for approximately one century and provided significant productivity improvements. Moreover, in

many industries, producers have started to utilize U-shaped assembly lines to obtain the main benefits of theJIT (just-in-time) philosophy (Kara et al. 2010).

The ALB problem has been studied for all types of assembly lines. The simplest version of ALB is thesingle-model and straight assembly line balancing (SALB) which was first studied by Salveson (1955) and

has been studied by many researchers to date. The literature on SALB is too large to present here.Nevertheless, the review and assessment of SALB procedures can be found in Baybars (1986), Ghosh and

Gagnon (1989), Erel and Sarin (1998), Becker and Scholl (2006). Single-model and U-shaped assembly linebalancing (ULB) was first studied by Miltenburg and Wijngaard (1994). A small but growing literature ofULB includes Urban (1998), Scholl and Klein (1999), Erel et al. (2001), Aase et al. (2004), Gken andApak (2006).

*Corresponding author

Proceedings of the 41st International Conference on Computers & Industrial Engineering

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Bacterial foraging optimization algorithm is a quite young but effective bionic algorithm which mimics the

foraging behavior of E. coli bacteria. It was presented by Passino (2002) and applied to some engineeringproblems such as harmonic estimation (Mishra 2005) and scheduling (Wu et al. 2007). Many exact, heuristic

and metaheuristic approaches have been proposed for solving SALB and ULB problems. However, to thebest knowledge of the authors, Bacterial Foraging Optimization Algorithm (BFOA) has not been applied toALB problems yet. So there is no information about its solution performance for ALB problems in the

literature. The main purpose of this study is to present a BFOA-based approach for SALB and ULBproblems and to evaluate its performance. In accordance with this purpose, a BFOA based heuristic approach

is proposed and the performance of the proposed algorithm is evaluated using a well-known data set takenfrom the literature which is known as Talbot et al. (1986) data set and in which the number of tasks variesbetween 7 and 111.The remainder of the paper is structured as follows. Section 2 provides a formaldescription and biological basis of BFOA. Proposed BFOA based heuristic approach is explained in Section

3. Computational results are presented in Section 4. Finally, Section 5 contains the conclusion of the study.

2. Biological Basis of the Bacterial Foraging Optimization Algorithm

To mimic the biological principles which exist in the foraging behavior of E. coli bacteria, Passino (2002)presented BFOA for distributed optimization and control. An optimization period of BFOA consists of threeevents: chemotactic event, reproduction event and elimination-dispersal event (Wu et al. 2007).

Chemotactic event:This process simulates the movement of an E. coli cell through swimming and tumblingvia flagella. Biologically, an E. coli bacterium can move in two different ways. It can swim for a period of

time in the same direction, or it may tumble, and alternate between these two modes of operation for theentire lifetime. Then in computational chemotaxis the movement of the bacterium may be represented by

Equation 1 (Passino 2002).

( + 1, , )= (, , ) + ()() (1)Where (, , )stands for the current position of the ith individual; j, k and l indicate for the numbers ofchemotactic events, duplicate iterations and elimination-dispersal events, respectively;() stands for the

new advancing direction decided by flagella swaying and ()for the step length (Dasgupta et al. 2009).

Reproduction event:After a period of food search, the foraging strategies of some bacteria appear inferiorevidently. These bacteria with inferior foraging strategies suffer a high probability to be removed out of thepopulation due to their low ability to find enough food. To keep the population size constant, a portion ofbacteria with superior foraging strategies are duplicated to take the places of removed ones (Wu et al. 2007).

Elimination-dispersal event: Gradual or sudden changes in the local environment where a bacterium

population lives may occur due to various reasons: e.g., a significant local rise of temperature may kill agroup of bacteria that are currently in a region with a high concentration of nutrient gradients. Events can

take place in such a fashion that all the bacteria in a region are killed or a group is dispersed into a newlocation. To simulate this phenomenon in BFOA, some bacteria are liquidated at random with a very small

probability while the new replacements are randomly initialized over the search space (Dasgupta et al. 2009).

There are three loops of optimization in the algorithm. The outer loop is elimination- dispersal event, the

middle loop is duplicate event and the inner loop is chemotactic event. The inner loop, chemotactic event, isthe core of the three loops. It corresponds to the direction selection scheme which is the central stepemployed by a living creature to search food and in charge of the decisions that whether or not enter into anew region, how long does the individual stay in the current region, which direction should be selected in the

next move. These decisions mean that the chemotactic event has important influence to the algorithmconvergence (Wu et al. 2007).


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3. Proposed Algorithm

In the proposed BFOA, each bacterium indicates a solution which consists of tasks of the problem andassignment information of those tasks. For solutions of both straight and U-shaped ALB problems, a task

sequence that contains a number of entries equals to the task number of the related problem is generated.Each position (entry) on the task sequence is denoted by a task number and assignment information of eachtask is saved on a memory.

When a task is placed to the next available position of the task sequence, it is assigned to a station at the

same time. While determining the task which will be placed to any position of the task sequence, atemporary set of assignable tasks is generated according to the tasks on the previous positions of thesequence and precedence relations among tasks. Then, assignable tasks to the current station within thetemporary set according to their task times are listed and one of them is selected and placed to the next

available position of the task sequence. If any task in the temporary set is available for current stationaccording to its task time, a new station is opened. At the end of this procedure a complete task sequence thatconsists of all tasks of the problem and station assignment information of those tasks are obtained.

Each task has an assignment weight which is used as a priority rule for assigning tasks to the stations. A taskis selected from the assignable tasks list according to those assignment weights. Initially, the assignmentweights of each task for each station equal to cycle time value of the problem. Therefore, while initial

solutions are being generated, a task which is going to be placed to the next available position of the tasksequence is selected randomly. The number of available stations equals to maximum number of stations.

Maximum number of stations is determined by multiplying the theoretical minimum number of stations by1.3 and rounding the result to the smallest integer greater than or equal to it.

The formula which is used to calculate the cost of a solution for a problem is given below:

=( 1) + (2)whereJdenotes the cost of the solution (objective function value), ndenotes the number of utilized stations,

C denotes the cycle time of the problem and tlast denotes the workload of the last station on the line. Byminimizing Equation 2 as an objective function which was used by Miltenburg (1998), all stations except the

last station are loaded as much as possible and workload of the last station can become zero within theprocess so total number of utilized stations is decreased.

Chemotaxis of proposed BFOA:At the beginning of a chemotaxis, a number of random vectors equal tomaximum number of stations are generated in the range of [-1,1] for each bacterium. A random vector of abacterium for each of the stations contains a number of entries equal to the number of tasks. By using thoserandom vectors a number of direction vectors equal to maximum number of stations are obtained for each of

the stations by Equation 3.

() = ()()()

(3)

where (i)denotes the direction vector of ith bacterium and (i)denotes the randomly generated vector for

ith bacterium. Then, assignment weights of the bacterium for each of the stations are updated by multiplyingthe related direction vector by the step length.

While a task is being placed to the next available position of the task sequence the task which has the

greatest assignment weight for the current station within the assignable tasks list is selected. After all tasksare placed and a complete solution is obtained, the objective function value of that solution is compared withthe previous objective function value of the related bacterium. If the new solution is better, it is accepted asthe current solution of the bacterium and the same direction vectors are applied to the assignment weights

again. Thus, the bacterium moves one more step to the same direction. This procedure is repeated until thenew solution of the bacterium does not get better or step count limit is reached.


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Reproduction event of proposed BFOA:All of the bacteria are ranked regarding to the sum of the objective

function values that are achieved during the lifetimes of the bacteria, and then the last half individuals areremoved out and each of the first half are duplicated one copy for each other to keep a constant population

size.

Elimination Dispersal event of proposed BFOA:For the elimination dispersal event of the proposed

BFOA, a random number in the range of (0,1) is generated for each bacterium. If the generated number issmaller than a predetermined elimination-dispersal probability value, all assignment weights of that

bacterium are set equal to the cycle time as in the initial solution and a new random solution is generated.

3.1. Illustrative ExampleIn this section one chemotaxis process of a bacterium are briefly explained on a problem with 7 tasks for the

cycle time value of 10 (C=10). Suppose that task times of the problem are 1, 5, 4, 3, 5, 6 and 5 respectively.

Sum of the task times is 29. Theoretical minimum number of the stations for the problem is [29/10] = 3.And maximum number of stations is determined as 4. Therefore, initial assignments weights of the tasks for

each of these 4 stations are set to cycle time (10). In the chemotactic event, four random vectors of size 7 inthe range of [-1,1] are generated for each of the stations. Then, by using those random vectors, four directionvectors are obtained by Equation 4. These direction vectors for each station are given in Table 1.

Table 1. Direction vectors of a bacteriumTasks

1 2 3 4 5 6 7

Direction vector for station 1: [ 0.12 0.26 -0.14 0.44 -0.19 -0.52 0.02]

Direction vector for station 2: [ 0.04 -0.30 -0.41 -0.22 0.82 0.11 0.63]Direction vector for station 3: [ 0.32 0.13 0.54 -0.42 0.11 -0.56 -0.21]

Direction vector for station 4: [ 0.17 -0.46 -0.81 0.19 0.32 0.05 -0.66]

Each entry of those direction vectors are multiplied by the step length and the obtained result is added to the

assignment weight of the related task to the related station. In the proposed algorithm, for all of the problems,step length is set to the cycle time of the problem. So, step length is 10 for the example problem. Assignment

weights of each task for each station which are updated by this way are given in Table 2.Table 2. Updated assignment weights of tasks for stations

Task

1 2 3 4 5 6 7

Station

1 11,2 12,6 8,6 14,4 8,1 4,8 10,22 10,4 7 5,9 7,8 18,2 11,1 16,3

3 13,2 11,3 15,4 5,8 11,1 4,4 7,94 11,7 5,4 1,9 11,9 13,2 10,5 3 ,4

While a task is being placed to the next available position of the task sequence the task which has the

greatest weight for current station is selected from the assignable tasks list.

4. Computational ResultsIn this section, performance of the proposed BFOA is evaluated using a well-known data set and the results

are presented. The data set which consists of 64 ALB problems is taken from www.assembly-line-balancing.de which is known as Talbot et al. (1986) data set in the literature and in which the number oftasks varies between 7 and 111. These 64 problems are solved using the proposed BFOA for both straight

and U-shaped line configurations. Optimal station numbers, BFOA solutions, CPU times and relatedparameter levels of the BFOA for straight and U-shaped line configurations are given in Table 3.

Brackets at the Optimal Station Numbercolumn denotes that optimal station number of the problem is not

known and the values in the brackets are lower and upper bounds for the number of stations. Punctuationmark * at theBFOA Resultcolumn denotes that optimal solution is not achieved by the proposed BFOA.

The results show that, proposed BFOA yielded optimal solutions for 63 of 64 test problems for straight lineconfiguration and for 60 of 64 test problems for U-shaped line configuration.


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Table 3. Straight and U-shaped ALB solutions of the proposed BFOAProblem Name

(Number of

tasks)

Cycle

time

Straight Line Configuration U-Shaped Line ConfigurationOptimal

Stations

BFOA

Result

Parameters CPU

(sec)

Optimal

Stations

BFOA

Result

Parameters CPU

(sec)S Nc Nr Ned S Nc Nr Ned

Merten (7)

6 6 6 2 2 0 0 0,03 6 6 2 2 0 0 0,01

7 5 5 2 2 0 0 0,01 5 5 2 2 0 0 0,028 5 5 2 2 0 0 0,01 5 5 2 2 0 0 0,01

10 3 3 2 2 0 0 0,01 3 3 2 2 0 0 0,01

15 2 2 2 2 0 0 0,01 2 2 2 2 0 0 0,0118 2 2 2 2 0 0 0,01 2 2 2 2 0 0 0,01

Bowman (8) 20 5 5 2 2 0 0 0,01 4 4 2 2 0 0 0,02

Jaeschke (9)

6 8 8 2 2 0 0 0,02 8 8 2 2 0 0 0,027 7 7 2 2 0 0 0,01 7 7 2 2 0 0 0,028 6 6 2 2 0 0 0,01 6 6 2 2 0 0 0,02

10 4 4 2 2 0 0 0,01 4 4 2 2 0 0 0,02

18 3 3 2 2 0 0 0,01 3 3 2 2 0 0 0,02

Mansoor (11)

48 4 4 2 2 0 0 0,01 4 4 2 2 0 0 0,03

62 3 3 2 2 0 0 0,01 3 3 2 4 0 0 0,0594 2 2 2 2 0 0 0,01 2 2 2 2 0 0 0,03

Jackson (11)

7 8 8 2 2 0 0 0,02 7 7 2 4 0 0 0,04

9 6 6 2 2 0 0 0,01 6 6 2 2 0 0 0,0310 5 5 2 2 0 0 0,01 5 5 2 2 0 0 0,04

13 4 4 2 2 0 0 0,02 4 4 2 2 0 0 0,0514 4 4 2 2 0 0 0,01 4 4 2 2 0 0 0,0321 3 3 2 2 0 0 0,02 3 3 2 2 0 0 0,04

Mitchell (21)

15 8 8 2 2 0 0 0,06 8 8 2 2 0 0 0,13

14 8 8 4 10 0 0 0,37 8 8 2 2 0 0 0,1221 5 5 2 2 0 0 0,06 5 5 2 2 0 0 0,18

26 5 5 2 2 0 0 0,06 5 5 2 2 0 0 0,0935 3 3 2 2 0 0 0,06 3 3 2 2 0 0 0,1439 3 3 2 2 0 0 0,05 3 3 2 2 0 0 0,19

Heskia (28)

138 8 8 2 2 0 0 0,14 8 8 2 2 0 0 0,35205 5 5 20 20 1 0 14,04 5 5 4 10 0 0 1,63216 5 5 2 2 0 0 0,12 5 5 2 2 0 0 0,28

256 4 4 10 20 1 0 7,06 4 4 2 4 0 0 0,37324 4 4 2 2 0 0 0,12 4 4 2 2 0 0 0,26

342 3 3 2 4 0 0 0,32 3 3 2 2 0 0 0,31

Sawyer (30)

25 14 14 2 2 0 0 0,15 14 14 2 2 0 0 0,2827 13 13 2 10 0 0 0,50 13 13 2 2 0 0 0,4130 12 12 2 2 0 0 0,14 11 11 10 12 1 0 10,39

36 10 10 2 2 0 0 0,16 9 10* 2 2 0 0 0,24

41 8 8 16 20 1 0 14,47 8 8 2 4 0 0 0,5954 7 7 2 2 0 0 0,17 6 6 10 14 1 0 12,14

75 5 5 2 2 0 0 0,12 5 5 2 2 0 0 0,35

K. & Wester

(45)

57 10 10 2 4 0 0 0,93 10 10 2 2 0 0 1,1579 7 7 20 20 1 0 56,06 7 7 8 10 0 0 12,20

92 6 6 10 10 1 0 15,67 6 6 8 10 0 0 12,51

110 6 6 2 2 0 0 0,59 6 6 2 2 0 0 1,14138 4 4 2 6 0 0 1,03 4 4 2 4 0 0 1,61

184 3 3 2 6 0 0 1,03 3 3 2 2 0 0 1,00

Tonge (70)

176 21 21 10 10 1 0 58,62 [20,21] 21 2 2 0 0 3,71364 10 10 2 2 0 0 1,49 10 10 2 2 0 0 3,56

410 9 9 2 2 0 0 1,47 9 9 2 2 0 0 3,66468 8 8 2 2 0 0 1,39 8 8 2 2 0 0 3,91527 7 7 2 2 0 0 1,88 7 7 2 2 0 0 3,66

Arcus (83)

5048 16 16 2 2 0 0 2,24 16 16 2 2 0 0 5,815853 14 14 2 2 0 0 2,32 13 14* 2 2 0 0 5,13

6842 12 12 2 2 0 0 2,65 12 12 2 2 0 0 5,167571 11 11 2 2 0 0 2,29 11 11 2 2 0 0 4,368412 10 10 2 2 0 0 2,31 10 10 2 2 0 0 5,20

8898 9 9 2 2 0 0 2,27 9 9 2 2 0 0 5,1810816 8 8 2 2 0 0 2,63 7 8* 2 2 0 0 5,83

Arcus (111)

5755 27 28* 2 2 0 0 5,83 27 28* 2 2 0 0 15,52

8847 18 18 2 2 0 0 6,32 [17,18] 18 2 2 0 0 14,4310027 16 16 2 2 0 0 5,46 [15,16] 16 2 2 0 0 13,0810743 15 15 2 2 0 0 5,87 [14,15] 15 2 2 0 0 17,04

11378 14 14 2 2 0 0 6,56 14 14 2 2 0 0 11,27

17067 9 9 2 2 0 0 7,23 9 9 2 2 0 0 14,94


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5. Conclusion

Performance of assembly lines, that are generally the last stage of production processes, has an importanteffect on general performance of entire production systems. Thus, it is very important to obtain effective

solutions for ALB problems in a reasonable time. For this reason, the literature on ALB problems has to beimproved by applying recently presented metaheuristic approaches because of the NP-hard nature of theproblem. In this study, a quite new optimization algorithm, Bacterial Foraging Optimization Algorithm is

applied to simple straight and U-shaped ALB problems. The proposed algorithm reveals competitiveperformance on a well-known data set and it can be thought that the BFOA can be applied to other versions

of the ALB problems successfully. Better solutions can be obtained for more complex or larger scaledversions of the problem by making some improvements on the initial solutions or the chemotactic eventnamely the core of the algorithm. In addition, BFOA can be hybridized with well-known metaheuristicapproaches to obtain more effective solutions for ALB problems.

ReferencesAase, G.R., Olson, J.R., and Schniederjans, M.J., 2004. U-shaped assembly line layouts and their impact on

labor productivity: An experimental study.European Journal of Operational Research, 156 (3), 698711.Baybars, I., 1986. A survey of exact algorithms for the simple line balancing problem.Management Science,32 (8), 909932.Becker, C. and Scholl, A., 2006. A survey on problems and methods in generalized assembly line balancing.

European Journal of Operational Research, 168 (3), 694715.

Dasgupta, S., Das, S., Abraham, A., Biswas, A., 2009. Adaptive Computational Chemotaxis in

Bacterial Foraging Optimization: An Analysis. IEEE Transactions on Evolutionary Computation,

13(4), 919-941.Erel, E. and Sarin, S.C., 1998. A survey of the assembly line balancing procedures.Production Planning and

Control, 9 (5), 414434.Erel, E., Sabuncuolu, I., and Aksu, B.A., 2001. Balancing of U-type assembly systems using simulated

annealing.International Journal of Production Research, 39 (13), 30033015.

Ghosh, S. and Gagnon, J., 1989. A comprehensive literature review and analysis of the design, balancing andscheduling of assembly systems.International Journal of Production Research, 27 (4), 637670.Gken, H. and Apak, K., 2006. A goal programming approach to simple U-line balancing problem.

European Journal of Operational Research, 171 (2), 577585.Kara, Y., Gken, H. and Atasagun, Y., 2010. Balancing parallel assembly lines with precise and fuzzygoals.International Journal of Production Research, 48(6), pp.1685-1703.Miltenburg, J., 1998. Balancing U-Lines in a Multiple U-Line Facility, European Journal of Operational

Research109 (1)-23Miltenburg, J. and Wijngaard, J., 1994. The U-line balancing problem.Management Science, 40 (10), 13781388.

Mishra, S., 2005. A hybrid least square-fuzzy bacteria foraging strategy for harmonic estimation. IEEETrans. Evol. Comput.9 (1), 6173.

Passino, K.M., 2002. Biomimicry of bacterial foraging for distributed optimization and control, IEEEControl Systems Magazine, 5267.

Salveson, M.E., 1955. The assembly line balancing problem. Journal of Industrial Engineering, 6 (3), 1825.Scholl, A. and Klein, R., 1999. Ulino: Optimally balancing U-shaped JIT assembly lines. International

Journal of Production Research, 37 (4), 721736.Talbot, F.B., Patterson, J.H., Gerhlein, W.V., 1986. A comparative evaluation of heuristic line balancingtechniques.Management Science32 (4),430-454.

Urban, T.L., 1998. Optimal balancing of U-shaped assembly lines.Management Science, 44 (5), 738741.Wu, C., Zhang, N., Jiang, J., Yang, J., Liang, Y., 2007. Improved Bacterial Foraging Algorithms and Their

Applications to Job Shop Scheduling Problems.ICANNGA 2007, Part I, LNCS 4431, 562 569.


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33 10 Assembly Line Balancing Using Bacterial Foraging Optimization

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