J. Eng. Sci., Univ. Riyadh, 6 (1), pp. 63-69 (1980)
Assembly Line Balancing·
A Modified Algorithm with Comparative Applications
G.M. Nawara and B.A. Elezaby
Associate Professor and Lecturer, College of Engineering, Riyad U niversit y, Saudi Arabia.
A computerized procedure for assembly line balancing is proposed. Among the well- known procedures the 10-SP p'rocedure produces results that are competitive with other methods. It requires less computing time, although it has been mentioned in research work that it is desirable to further develop this method.
The proposed algorithm extends the 100SP approach to 20-SP. This algorithm simply selects the best of twenty solutions, each obtained by using a difTerent ranking system for the selection of work elements. Ranking systems are obtained from the following four basic ranking methods:
a) ranked positional weight, b) total number of job following, c) largest candidate rule,
d) total number of immediate followers.
The proposed algorithm has been programmed and compared against difTerent methods using some case studies. The new algorithm gives favourable results. It yields unique minimum balance delay in most cases and contributes in the solution of other cases.
In 1954 came the appearance of the first analytical study related to the problem of assembly line balancing.
Salvason [lJ produced the first published paper on assembly line balancing. Since that time a substantial amount of res~arch has been devoted to the balancing of assembly lines. The majority of contributions to assembly line balancing has the following four directions: (1) linear programming; (2) dynamic programming; (3) network methods; (4) branching methods. The branching methods are subgrouped into branch-bound-techniques and heuristic techniques.
Most of the recent work is devoted to the heuristic techniques.
Different heuristic solutions have been developed, but despite this the balancing problem persists. The Illinois Institute of Technology carried out a study of industrial firms, reported by Lehman [2J, which showed that less than 30% of the firms surveyed used computerized algorithmic balancing techniques. More than 40% of the firms used a trial and error approach to
balancing. Davis and Simmons [3J and others mentioned several reasons. for the limited usage of balancing techniques. The dominant reason could be that the theoretical assumptions used in some techniques are not realistic in terms of practicability.
The development 'of heuristic solutions goes forward and several papers have been published on computer-oriented methods for solving single model assembly line balancing problems. Of the many computerized heuristic methods presented since 1960 it was noted that four procedures yielded good results.
These procedures are as follows:
1. Ranked positional weight (RPW) technique proposed by Helgeson and Birnie [4]. This procedure was manually developed and computerized thereafter by others.
2. COMOSOAL: A Computer Method of Sequencing Operations for Assembly Line whlcn was
Assembly Line Balancing a Modified Algorithm with Compa~ative Applications
developed by Arcus [5].
3. MALB, the backtracking method of Mansoor [6].
4. 10-SP, which selects ten single-pass solutions, mentioned by Mansoor [7].
The 10-SP approach produces favourable results compared to the other methods with less execution time on the computer. Mansoor [7] stated, "Computation time can be expected to be proportional to the number of ranking systems used". For example, the Arcus times were generally about ten times that for the 1 O-SP. This means that a 1oo-SP technique is still likely to have computation time compared to Arcus. But it is also possible that the "best" solutions from a 1OO-SP algorithm would give results superi()r to those from Arcus or possibly even MALB. Therefore it is desirable to develop this method.
The Proposed Algorithm
The proposed method extends the 10-SP approach to 20-SP. The algorithm simply selects the best of twenty solutions, each obtained by using a different ranking system for the selection of work elements:
Ranking systems were obtained from the following four basic ranking methods ..
a) Ranked Positional Weight (RPJ.t'): the positional weight of a work element is its own processing time added to those of all following work elements.
b) Total Number of Job Following (IF): This is the sum of all work elements that follow the work element under consideration.
c) lnrgest Candidate Rule (LC): the selection of the work element having the largest processing time that can be assigned.
d) TInal Number of Immediate Followers (IF): this selects that candidate work element having the largest number of immediate followers:
The twenty ranking systems are as follows:
RM l=RPW RM 6=RPW+TF+LC
RM 2=TF RM 7=RPW+TF+LC+IF
RM 3=RPW+TF RM 8=TF+LC RM 4=RPW+LC RM 9=TF+IF RM 5=RPW+IF RM lO=TF+LC+IF
RMll =RPW+LC+IF RM16=2RPW+TFtLCHF
RM l2=LC RM17=RPW+2TF+LCtIF
RM13=LC+IF RM18=RPW+2TFt2LC+2IF
RM14=2RPW+LC+IF RM 19=2TF+LC+IF RM15=2RPW+TF+2LC RM 20=2TF+2LC+IF Selection of Ranking Systems
In the application of 10-SP each single ranking method contributed in the solution different times than the other ranking systems but the basic ranking methods are combined in equal weights. In 20-SP, the first ten ranking systems are identical to those of 10-SP.
Moreover, another new ten ranking methods ha ve been used. The concept of weighted average ranking method has been introduced, each basic ranking method has a different weight.
Problem Definition
Given the operating cycle time C(N), it is required to minimize the number of stations. The algorithm is applied at C(N) minimum (the theoretical minimum cycle time for N stations), then at C(N)min + 1, C(N)min
+ 2, ... until minimum balance delay is achieved.
Line Efficiency
The efficiency of assembly line balance is evaluated· by the total idle time in the line and the measure used for solution efficiency is balance delay (BO).
Assembly line capacity -
BO% = Total work content in line x 100 Assembly line capacity BO% N.C(N)-T x100
N.C(N) where
N Number of work stations
C(N) Operating cycle time for N stations T Total work content in the assembly Journal of Eng. Sci.-Vol. 6-No. 1-1980-Col/ege of Eng., Univ. of Riyad.
C.M. Nawara and H.A. Elezaby
Algorithm Description
The proposed algorithm consists of two phases. In the first phase, the basic data are prepared for the next phase where the solution of the assembly line balancing problem is found by computer.
Phase 1 consists of the following steps:
1. Construct the precedence diagram In manner described by Jackson [8].
the 2. The corresponding time of each element in the precedence diagram is determined.
3. The half precedence matrix is then prepared to describe precedence relationships. This half matrix is a matrix of zeros and one, where Pij= 1 if element number i must precede element number j, and Pij=O if element number i and element number j are not ordered by any precedence relation.
4. Assign the total number of immediate followers and the immediate predecessors for each element.
5. Calculate the theoretical minimum operating cycle time. The theoretical minimum cycle time (C(N)min)=T/N. C(N)min is the initial cycle time at which the balance delay is equal to zero.
Phase 2: During this phase, the prepared input data are fed to computer, then the optimum solutions for each varying number of feasible work stations are found.
A FORTRAN program has been used to calculate the minimum balance delay for specific number of stations.
The program is run on a HP 3000 computer.
Program Steps
The main program whose flow diagram is given in Fig. 1 consists of the following steps:
1. Read input data (cycle time, number of elements. maximum number of predecessors, ... etc.).
2. Call subroutine PW. This subroutine calculates the rank of each work element according to the ranked positional weight technique (RPW).
3. Call sub-routine TF. The rank of each work element is estimated according to total number of jobs following (TF) rule.
4. Call subroutine RM 34. Here, the weight of each work element is computed by using two basic ranking systems. These are the largest candidate (LC) rule, and total number of immediate followers (IF).
5. Calculate the rank of each work using the twenty different ranking systems.
Read no. of elements. max. no.
of predecessors, cycle time.
Calculate the rank of each element according to 20 SP method.
Fig. 1: Block diagram of the main program.
6. Call subroutine MS. This subroutine evaluates the balance delay for specific cycle time 20 times, using each time different ranking system.
This subroutine whose flow diagram is given in Fig. 2 consists of the following steps.
1. Sort work elements in decreasing order of their weights in a certain ranking system.
2. Select the work eleme)1t with the highest rank and check the following conditions.
2.1 The work element has no immediate predecessor.
2.2 The work element time is less than or equal to cumulative station time.
Assembly Line Balancing a Modified Algorithm with Comparative Applications
MS
IC=O
ort work elements in decreasin order, according to their weight in ranking system.
Increase stations
NO
Assign zeroes to immediate pre- decessor matrix for that work element, assign zero to rank of work el ement.
Fig. 2: Block diagram of subroutine MS.
If the above two conditions are satisfied proceed to step 3, if not go to step 6.
3. Assign the work element to the work station.
4. Increase cummulative station time by work element time.
5. Assign zeros to the immediate predecessor of that work element in immediate predecessor matrix.
6. Select the work element with the next highest rank and attempt to assign it to the work station repeating steps 2 through 5. If the work element violates rule (2.1) and lor (2.2) proceed to step 7.
7. Continue to select, check and assign, if possible, until one of the two dollowing conditions has been met:
a) All work elements have been assigned.
b) No unassigned work element remains, that can satisfy (2.1) and/or (2.2).
8. Assign the unassigned work element with the highest rank to the next station and repeat steps 2 through 7.
9. Continue assigning work elements until all work elements have been assigned.
10. Calculate the balance delay.
11. Repeat the whole procedure for all ranking systems.
12. If minimum balance delay has been obtained from any ranking system, stop. If not, increase C(N)min by one and repeat steps 1 through 11 until minimum balance delay has been achieved.
Case Studies
To examine the applicability of the proposed method, two case problems are solved by this method and satisfying balancing solutions are obtained.
First, the 20-SP has been applied to the example introduced by Kilbridge [9J which was initially suggested by Salveson [1]. This example has the advantage that it yields zero balance delay for six cycle times. Therefore, the comparison between Kilbridge and the 20SP would indicate explicity the importance of the new method.
Secondly, the proposed method has been applied to Mukherjee's [10J problem with minor changes in input data. The tested assembly line involved light electronic consumer goods. The line consisted of 95 work elements. The balance delay has been evaluated at number of stations N
=
2,3,4,5,6,7,8,10,12. The choice of this example is based on two facts; (1) the application of Journal of Eng. Sci.- Vol. 6-No. 1-1980-Col/ege of Eng., Univ. of Riyad.G.M. Nawara and H.A. Elezaby
the new algorithm in industrial areas would prove the reliability of this method in real life situations; and (2) the large size problem would extend the applicability of the proposed method to dilTerent assembly line sizes.
Results and Discussions
Tables 1 and 2 represent the summary of results obtained from the 45 and 95 work element examples respectively. For the 45 work element example zero balance delay was achieved by one or more cycle times of the new suggested algorithm. In the case of 6 aod 8 work stations a balance delay very close to zero is achieved. Although the solution introduced by Kilbridge [9J reached zero balance delay for all stations, the suggested 20-SP is still superior, since Kilbridge's heuristic method is a manual procedure and consumes considerable time especially for large scale problems.
Table 1. Summary of the 45 Work Element Example
No. of stations
8
6
4
3
2
Cycle Balance time Delay%
69 lUI 70
}
12.38 1.43*92 14.29 93
}
15.21 1.08*138
}
20.0 0.0"184
}
0.0"25.0 276 0.0"
33.33
*. zero balance delay
* minimum balance delay
Ranking Method
all 20 ranking methods 2,5,8,9,10,15,18,20
the rest of the 20 ranking methods all 20 ranking methods
2,6,7,8,9,10,11 through 20 1,3,4,5,11
11
all the other 19 ranking methods 12
all the other 19 ranking methods 11,12, and 13
all the other 17 ranking methods
The 95 element example solved using the 20-SP gave better results than the Mukerjee's method [10]. In the latter case minimum balance delay was only achieved at N = 10 stations with 0.75%. But the 20-SP reaches 2.59, 0.99, 1.5,0.52,0.28,0.09,0.09 and 0.5% at N
=
12, 10, 8, 6, 5, 4, 3, 2 stations respectively. Also Mukherjee uses only the ranked positional weight technique which is only one of the ranking methods used in the 20 SP. This proves that the suggested method is efficient.Table 2. Summary of the 95 Work Element Example
No. of Cycle Balance stations time Delay %
l2 3.51 7.78
3.52 8.04 3.53 8.3 3.54 8.56 3.55 8.82 3.56 9.08 3.57 9.33 3.58 9.58 3.59 9.84 3.60 2.59·
4.24 9.78 4.25 0.99*
10
8 5.26 11.11 5.27 11.28 5.28 11.45 5.29 11.62 5.30 l1.78 5.31 11.95 5.32 -12.11 5.33 12.28 5.34 1.5*
6 7.02 14.37 7.03 14.50 7.04 14.61 7.05 0.52*
5 8.42 16.71 8.43 16.8 8,44 0.28*
4 10.52 20 10.53 0.09·
3 14.03 25 14.04 0.09*
2 21.04 33.3 21.05 0.05*
Conclusions
Ranking Method
all the 20 ranking methods
13
all the 20 ranking methods
6,10,18
8,10,13
all the 20 ranking methods
1,2,3,5,7,8,10,11,12,14,16 through 20.
all the 20 ranking methods 2,8
all the 20 ranking methods 7,8,10,12,16 through 20.
all the 20 ranking methods 1,2,9,10,18,19,20
A 20-SP procedure for assembly-line balancing is . proposed and programmed in Fortran IV. The suggested procedure has been tested with well known case problems and yields favourable results. It is very likely to prove worthwhile to extend the 20-SP approach to 100-SP and analyze the correlation between the dilTerent ranking methods as applied to different size problems.
Assembly Line Balancing a Modified Algorithm with Comparative Applications
References
I. Salveson, M.E., The assembly line balancing problem, 1. Ind.
2.
3.
4.
5.
Engg.,3, 18-25 (1955).
Lehman, M., What's going on in production assembly, Ind.
Engg., 1,41 (1969).
Davis, K.R. and Simmons, L.F., Improving assembly line efficiency: a dynamic programming - heuristic approach.
Computer and Operations Research, 4, 75-87 (1973).
Helegeson, W.P. and Birnie, D.P., Assembly line balancing using ranked positional weight technique. 1. Ind. Engg., 12,6, 394-398 (1961).
Arcus, A.L., CO MOSOL - a computer method of sequencing operations for assembly lines. Inc. 1. Prod. Res., 4, 4,259-277 (1966).
6.
7.
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10.
Dar EI (Mansoor), E.M., MALB - a heuristic technique for balancing large single-model assembly lines. AI I E Transactions, 7, 3, 302-310 (1973).
Mansoor, E.M., Solving large single - model assembly line balancing problems - A comparative study, AIlE Transactions, Vol. 7, No.3, 302-310 (1975).
Jackson, J.R., Computing procedure for a line-balancing problem, Mgmc Science, 2, 3,261 (1956).
Kilbridge, M.D. and Wester, 1., A heuristic method of assembly line balancing, 1. Ind. Engg., 124, 292-298 (1961).
Mukherjee, S.K. and Basu, S.K., Application of heuristic method of assembly line balancing in an Indian industry.
Proc. I. Mecl!. Eng., 78, I, 11,277-292 (1964).
Journal of Eng. Sci.-Vol. 6-No. J-J980-Col/ege of Eng., Univ. of Riyad.
G.M. Nawara and H.A. Elezaby
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