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Assembly Line Balancing: Conventional Methods and Extensions
7th IFAC Conference on Manufacturing Modelling, Management, and Control International Federation of Automatic Control June 19-21, 2013. Saint Petersburg, Russia
Assembly Line Balancing: Conventional Methods and Extensions Alexandre Dolgui* and Jean-Marie Proth** * Ecole Nationale Supérieure des Mines, UMR CNRS 6158 LIMOS 158 cours Fauriel, 42023 Saint-Etienne cedex 2, France (e-mail: [email protected]) **Institut National de Recherche en Informatique et Automatique (R), France (e-mail: [email protected]) Abstract: In this presentation the standard models from the OR literature are mentioned, as well as some simple available algorithms that provide a solution close to optimum. Then some extensions of previous algorithms are introduced, for example, the case where the operating times are stochastic. Bucketbrigades are briefly discussed. Finally, a list of problems that currently remain without solution are spotlighted, especially the problems of sensitivity analysis and robust optimization approaches under uncertainties. These areas could be the focus for promising future research. Keywords: Manufacturing systems, Assembly lines, Line design, Process planning, Line balancing. time average and maximum operation time. The same request may be repeated with several employees and the responses are averaged. Consider that the probability density is linear between two successive points, which represents the triangular density of the operation times. Techniques to estimate the probability that a station processing time exceed a given cycle are proposed and coupled with COMSOAL for optimization. Other approaches are also considered: robust optimization and sensitivity analysis.
1. INTRODUCTION Line balancing consists in assigning a set of operations (to obtain a finished product item) to assembly stations arranged in a line, in accordance with a partial order on the set of operations. The objective is to increase productivity while reducing the cost of production. In Section 2, two basic problems noted SALB-1 and SALB-2 (SALB is the acronym of Simple Assembly Line Balancing) are presented. In the first, the goal is to minimize the number of stations to achieve a given cycle time (also called takt time). In the second, the number of stations is given, and the objective is to minimize the takt time.
In Section 4, an interesting line configuration is presented where the work can be shared among different workers and stations - Bucket Brigades. They have the peculiarity of incorporating the treatment of randomness in their operating mode.
It will be shown how to adapt these algorithms to satisfy some additional constraints such as the usual need to perform an operation at a given station in order to use a specific tool, or the need to perform a given set of operations in the same station or the necessity to execute some operations at different stations because their incompatibility.
Section 5 presents the conclusion. 2. SIMPLE ASSEMBLY LINE BALANCING 2.1 General assumptions
Section 2 concludes with the presentation of a well-known algorithm: COMSOAL for “Computer Method of Sequencing Operations for Assembly Lines”. This algorithm can help to understand better the problems considered. In addition, often this algorithm is sufficient for practical use, in addition it can be easily extended for problems with supplementary constraints and new objective functions.
Problem SALB is characterized as follows: - Each operation is performed at a single station, and its operating time is deterministic, - An operation can be performed at any station, and the operating time does not depend on the station used, - There is a partial order on all operations. This determines the order in which they are processed,
In Section 3, the case where operating times are stochastic is examined. In the approach proposed, the point of view of management practitioners is adopted. Here it is no longer considered that it is possible to associate a referenced probability density (Gaussian distribution, for example) to each operation, but it refers to observations of practitioners. Employees are asked to give three pieces of information, namely: estimate of the minimum operation time, operation
978-3-902823-35-9/2013 © IFAC
- The stations are visited in a given order, - There is only a single type of product to be assembled on the line.
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2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
The goal is to assign all operations to stations minimizing one of the two criteria that are introduced below while satisfying the constraints on the partial order of operations.
- The three operations framed in blue are assigned to the third station. This station load is equal to 33 time units,
2.2 Problem SALB-1
- The three framed black operations are assigned to the fourth station. This station load is equal to 32 time units (8 units of idle time),
In this case, the cycle time C is given and the objective is to minimize the number of stations m used. To this end, the total idle time on all stations is minimized, that is to say:
- Finally, the operation framed in yellow is the only operation of the fifth station. This station load is equal to only nine units of time.
m
TW = m × C −
∑ Τi , where
Ti is the sum of the operation
Table 1. Data for the SALB-1 example
i =1
times related to the operations performed at station i . The following simple algorithm can be applied to search for an approximate solution. At each iteration, the algorithm selects from operations without predecessor or whose predecessors have already been assigned, the operation with the greatest operation time and which can be assigned to the station under consideration (i.e. whose operation time is less than or equal to the time still available for the station) . If no operation is found, a new station is opened. Of course, it is assumed that all the operation times are lower or equal to the cycle time; otherwise the problem would have no solution. This algorithm uses common sense. It is based on the fact that operations with significant operation times are more difficult to assign. So the algorithm tries to assign operations in the decreasing order of their operating times.
Operations
A
B
C
D
E
Operation times
10
12
7
8
20
Predecessors
/
/
/
A
B, C
Operations
F
G
H
I
J
Operation times
4
11
6
9
12
Predecessors
/
D
D, E, F
/
I, G
Operations
K
L
M
N
O
Operation times
15
13
9
8
9
Predecessors
G, H
J
J, K
M
L, N
I
It is possible to significantly improve this algorithm by doing the following: A
- Do not select operations with the greatest operating times to assign to the station being processed, but select at random an operation at each iteration from those whose times are less than or equal to the time left available for the current station,
D
G
J
L O
- Restart the algorithm K times ( K is at the discretion of the user) and keep the best solution among the solutions obtained.
B
E
C
F
H
K
M
N
This improved algorithm is called COMSOAL (Arcus, 1966). Fig. 1. Graph of the partial order on the set of operations and a feasible solution
The random selection introduced in this new version results in significant flexibility of the algorithm which when restarted many times achieves a near optimal solution.
2.3 Problem SALB-2
Example: The following is taken from Dolgui and Proth (2010). The cycle time is equal to 40 units of time. In Table 1, the operating times and precedence constraints (if any) are given.
This problem is to minimize the cycle time knowing the number of stations. From a practical standpoint, this means that the investment is known and the goal is to use it optimally. (In the case of problem SALB-1, the investment is optimized while the desired productivity is known).
Figure 1 shows the result of optimization. In this figure: - The four operations framed in red are assigned to the first station. This station load is equal to 39 time units (1 unit of idle time),
Again, the following iterative heuristic algorithm can be applied: Whenever a new station is inserted, divide the sum of the operating times of the operations that are to be affected by the
- The four operations framed in green are assigned to the second station. This station load is equal to 40 time units (maximum possible occupancy), 44
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
number of stations that have not yet been used. Thus, a lower bound on the cycle time for stations not yet used is obtained.
- Let e be the average value of the operating time. These values are evaluated by users for each operation. The following probability density is then obtained:
If the current station is first, the algorithm assigns it the operations in a decreasing order of operating times for operations that are not yet assigned and still have no predecessor (or whose predecessors have all been assigned). If an assignment is not possible, a new station is opened. If stations have all been used, then it goes to step b) below.
0 if x ≤ a 2( x − a ) ( b − a) ( e − a ) if a ≤ x ≤ e f ( x) = 2(b− x) if e ≤ x ≤ b ( b − a ) ( b − e ) 0 if x ≥ b Figure 2 represents this density.
If the station under consideration is not the first one, then two cases may arise: a) if the new calculated cycle time is less than or equal to the preceding, then the procedure is as for the first station,
f(x)
b) if not, then one starts again the whole process after increasing the initial value of the cycle time of a "low" amount to give more flexibility to the system. The "low" amount in question here is at the discretion of the user.
2/(b–a)
2.4 Usual required adjustments a
Changes in the problem are sometimes necessary to allow the use of previous algorithms. The most common cases when using SALB-1 are as follows:
e
b
x
Fig. 2. Triangular density of probability
In the following sub-section, first, the case of an assembly line (for a single type of product) is considered. Then, the approach is generalized for a mix of different types of products.
- If it is needed to assign multiple operations to the same station, then just combine these operations into a single macro-operation. The operating time is equal to the sum of the operating times of the operations that compose it. The predecessors (resp. successors) are the set of predecessors (resp. successors) of the operations that compose this macrooperation. This type of situation can occur often, for example when there are specific tools for such operations at a station, or is necessary to protect the products from some pollutants (clean room), etc.
3.2 Probability of overflow
The basic problem is to decide (for each iteration) whether a set of operations may be assigned to the same station. A triangular probability density can be defined for each operation. It is necessary to determine whether the amount of time required to perform a set of operations is compatible with the cycle time C. Thus, it is necessary to verify that the total execution time for these operations can exceed the cycle time only with a probability less than a small value ε chosen by the user. For this, simulation can be used. The details of this approach are given in Dolgui and Proth (2010), section 8.3.2.
- Force the assignment of an operation at a given station. In this case, no modification of the algorithm is necessary. When all assignments are completed, simply identify the station that contains the operation as the given station. This occurs for example when an operation must be near a loading or unloading system, or a particular source of energy. - Prohibit two operations to be at the same station (vibrations, tolerances, etc.). In this case, it suffices to perform a modification of the algorithm when determining the candidates to assign.
If operating times are stochastic and if, in addition, the proportion of each type of product in the mix is also random, then to define the distribution of operations in the stations, it is also possible proceed by simulation.
3. STOCHASTIC OPERATING TIMES
For example, consider the case of a mix of three types of products. For each product, Table 2 provides the product structure and the triangular probability density associated to each operation.
3.1 Triangular density of probability
Assembly systems typically include a high proportion of operations performed manually, which results in variable operating times for the same operation. To take into account variations in processing times, one should proceed as follows:
Note that if some operations do not appear in all manufacturing processes, then, simply "empty" operations can be placed in the order common to all products and without predecessor.
- Let a be the smallest possible value of the operating time,
In Table 3, the probabilities of presence of each type of product in the mix are given.
- Let b be the largest possible value of the operating time, 45
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
Table 4. Number of stations versus ε
Table 2. Product structures and time probabilities
ε
Product of type 1 a e b Pred.
A 1 2 3 /
B 2 3 5 /
C 3 4 6 /
D 0 1 3 A
E 2 3 4 B, C
F 0 2 3 /
G 1 2 4 D
H 2 3 5 E, F
J 2 3 4 G
K 1 3 4 H
L 0 3 4 J
M 0 4 6 J, K
N 1 4 5 M
Number of stations
O 1 3 4 L, N
a e b Pred.
A 3 4 5 /
B 2 4 5 /
A 1 3 4 /
C 1 4 5 /
B 2 4 5 /
D 1 3 5 A
D 0 3 4 A
E 0 3 4 B, C
G 1 3 4 D
H 0 3 4 E
I 2 3 4 /
J K 3 2 5 5 6 6 G, I G, H
Product of type 3 G H I J 1 3 2 0 3 4 4 3 4 5 5 5 D D, E / G, I
E 2 4 5 B
K 1 3 4 G, H
L 1 3 4 J
L 3 4 5 J
M 0 2 3 J, K
N 0 3 4 M
O 1 3 4 L, N
M 1 3 5 J, K
N 2 4 5 M
O 1 3 4 L, N
1
2
Probabilities
0.2
0.7
0.1
0
1
2
3
Probabilities
0.1
0.5
0.3
0.1
0
1
2
Probabilities
0.3
0.5
0.2
Policy 1: All stations simultaneously begin processing the set of operations which they are responsible. This implies that all stations have finished processing all their operations when the next cycle is launched. Therefore, the average cycle time increases if ε decreases. Policy 2: When a station is not able to complete the assigned operations during the cycle time, these operations are performed by a relief assembly line. Of course, the next stations are delayed waiting for the end of operations processed by the relief assembly line, but the delay concerns certain operations only. When using this policy, the tendency is to oversize stations that are at the beginning of the manufacturing process. The cost of this solution is significant (investment for online backup and maintenance). Moreover, it is only effective if buffer stocks are maintained. Policy 3: It consists in storing one (or more) semi-finished products after each station, and using these stocks whenever the operations supported by the station are not completed at the end of a cycle time.
Taking into account these triangular probability densities and proportions of each type of product in the mix, simulation can be used to estimate the probabilities of overflow. Thus, COMSOAL can be applied where, each time when an assignment should be made, the simulation is repeated to check whether a set of operations can be executed in the same station with a risk of failure less than a given percentage (for example 1%).
3.4 Other approaches
Many theoretical studies on stochastic assembly lines can be found in the literature. There are various approaches, both through the models considered and the tools used. Some of them are mentioned below.
One of the feasible solutions for this example obtained with this method is represented in Fig. 3.
A survey of publications on design and balancing for stochastic assembly lines is presented in (Erel and Sarin, 1998). Kottas et al. (1981) consider design of paced production lines with stochastic task times to minimize the sum of incompletion cost and labour cost. Suresh et al. (1994) propose to use a simulated annealing approach for solving a stochastic assembly line balancing problem. Erel et al. (2005) consider a stochastic U-line problem to minimize the total expected cost of labour and expected incompletion costs.
I
A
D
G
J
L
O
B
E
H
K
M
N
C
F
0.3 4
Three policies are possible (Dolgui and Proth, 2010).
Product of type 3 Number of products
0.2 4
3.3 How to react when the cycle time is exceeded?
Product of type 2 Number of products
0.1 5
Whatever the solution, it is necessary to consider the possibility of exceeding the cycle time. How to react in this case?
Product of type 1 0
0.05 5
A tradeoff between the number of stations and probability of exceeding the cycle time can be sought. In fact, this means to find a compromise between investment (number of stations) and productivity (cycle time).
Table 3. Mix data Number of products
0.015 6
Of course, the number of stations decreases when ε increases, see Table 4 for the example considered.
Product of type 2 a e b Pred.
0.01 7
Fig. 3. A feasible solution
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2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
Dolgui and Kovalev (2012) studied a line balancing problem where operation times are uncertain in the sense that they belong to a given set of scenarios. The objective is to minimize the line cycle time for the worst scenario. Gurevsky et al. (2012) and (Hazir and Dolgui, 2013) studied the cases where interval uncertainty for operation times is assumed. The employed robust optimization approach does not require any distribution data. The worst-case performance of the system is optimized and decisions that perform well under the worst-case scenarios are sought.
5. DISCUSSION AND CONCLUSION The recent state of the art of research literature on assembly line balancing problems is presented in (Boysen et al., 2007) and (Battaïa and Dolgui, 2013). New extensions of line balancing problems appeared which take into account the possibility of parallel execution of operations at stations (Dolgui et al., 2008) as well as the necessity to select pieces of equipment for each station (Dolgui et al., 2012).
Sotskov et al. (2006), Gurevsky et al. (2012) and Gurevsky et al. (2013) develop sensitivity analysis approaches based on the calculation of stability radius for optimal solutions with respect to possible variations of the processing times. They search for the “maximal possible value of simultaneous and independent variations of the processing times” which keeps the optimality of the line balance for different line configurations and assumptions.
Some promising perspectives are addressed in concerning how to take into account uncertainties. Two new ways are explored: sensitivity analysis (Sotskov et al., 2006; Gurevsky et al., 2012; 2013) and robust optimisation (Dolgui and Kovalev, 2012; Hazir and Dolgui, 2013). The following two publications are also to be considered attentively: Falkenauer (2005) and Becker and Scholl (2009). Despite the extensions proposed in the second paper, it is undeniable that the work developed so far only partially meets industrial needs. In Falkenauer (2005), written by an experienced practitioner, significant observations are made. They are summarized below.
4. BUCKET-BRIGADE ASSEMBLY LINE A "bucket-brigade assembly line" has the advantage of integrating its adaptation to the system load and has the peculiarity of incorporating the treatment of randomness in its operating mode.
The managers of assembly lines certainly need to balance them initially, but how should they proceed when the conditions are modified (change of components, change in the mix of products assembled, emergence of new technologies, necessity to adapt workforce on demand)? It is not possible to ignore the existing system and look for the optimal balance of a new assembly line. The assembly line should be rebalanced for the lowest cost in order to better adapt to new conditions. It is interesting to note that a similar problem arises in scheduling.
The main advantage of this approach is summarized in (Barholdi et al., 1996), (Barholdi et al., 1999) and (Barholdi et al., 2001): “bucket brigades are a way of organizing workers on an assembly line so that the line balances itself”. A survey of publications on bucket brigades is presented in (Bratcu and Dolgui, 2005) and a simulation study is given in (Bratcu and Dolgui, 2009). Let us take an example of a bucket brigade with three employees. They move while working on an assembly. When employee number 3 reaches the end of the line, he/she deposits the finished product, and then returns to meet employee number 2. When they meet, employee number 3 takes the bucket of employee number 2 and continues its assembly. Then, employee number 2 goes to meet employee number 1 to take his/her charge. Employee number 1 then goes to the beginning of the line and starts a new assembly.
Most of the work reported in the literature belongs to one of two types: either they report theoretical developments related to balancing problems, and in this case they stray from applications, or they have been developed to meet specific industrial needs. In this case, they offer little theoretical reusable implications. Usually, theoretical developments generally ignore the specificity of stations. In this context, a station is neutral. It is supposed to be able to change assignment without a major problem that is to say at negligible cost. Practical developments are defined in relation to their environment (available space, ceiling height, disposal of the sources of energy, etc.) and these constraints are decisive in the choice of the solution to the point that they hided the general aspects of the method.
Note that buckets are placed on a conveyor and the number of employees involved is not fixed: The greater the number of employees (within the limits of feasibility), the greater the productivity (that is to say the lower the average cycle time) and the lower the working time of an employee on a given assembly.
Taking into account the influence of internal transportation systems is essential, particularly when it comes to industries that require the handling of heavy and / or bulky objects, for example, in the automotive industry, aircraft manufacturing, agricultural equipment and, more generally, any assembly system that requires transporting objects of more than a few hundred grams. Indeed, it is difficult to restructure an assembly system that involves cranes and wire-guided carts, etc. when they are intended for specific products, among others. Also the rigidity of the assembly line - when the system for loading and unloading components is particular -
This system, however, can pose two types of problems: - Problems caused by the fact that employees are forced to move along the conveyor belt while working; - Possible discord among employees because of low efficiency for some employees results in overloads for others.
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2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
should be mentioned. In summary, internal transportation systems are factors that can make the adaptation of an assembly line to changes sometimes impossible and always very constraining. For the same reasons it is difficult, and even impossible, to eliminate stations.
production lines), International Journal of Production Research, 47(2), 369–388. Dolgui, A., Guschinsky, N., Levin, G., and Proth, J.-M. (2008). Optimisation of multi-position machines and transfer lines. European Journal of Operational Research, 185(3), 1375–1389. Dolgui, A., Guschinsky, N., and Levin, G. (2012). Enhanced mixed integer programming model for a transfer line design problem. Computers & Industrial Engineering, 62 (2), 570–578. Dolgui, A., and Kovalev, S. (2012). Scenario based robust line balancing: Computational complexity, Discrete Applied Mathematics, 160 (13–14), 1955–1963. Dolgui, A., and Proth, J.-M. (2010). Supply Chain Engineering: Useful Methods and Techniques. Chapters 7 and 8. Springer Verlag, London. Erel, E., and Sarin, S. (1998). A survey of the assembly line balancing procedures. Production Planning and Control, 9(5), 414–434. Erel, E., Sabuncuoglu, I., and Sekerci, H. (2005). Stochastic assembly line balancing using beam search. International Journal of Production Research, 43(7), 1411–1426. Falkenauer, E. (2005). Line balancing in the real world. Proceedings of the conference PLM-SP1-2005, pp. 360– 370. Gurevsky, E., Battaïa, O., and Dolgui, A. (2012). Balancing of simple assembly lines under variations of task processing times. Annals of Operations Research, 201, 265–286. Gurevsky, E., Battaïa, O., and Dolgui, A. (2013). Measure of stability for a generalized assembly line balancing problem, Discrete Applied Mathematics, 161(3), 377– 394. Gurevsky, E., Hazir, O., Battaïa, O., and Dolgui, A. (2012). Robust balancing of straight assembly lines with interval task times, Journal of Operational Research Society, doi: 10.1057/jors.2012.139 Hazir, O., and Dolgui, A. (2013). Assembly line balancing under uncertainty: Robust optimization models and an exact solution method. Computers & Industrial Engineering, doi: 10.1016/j.cie.2013. 03.004. Kottas, J.F., and Lau, H.-S. (1981). A stochastic line balancing procedure. International Journal of Production Research, 19(2), 177–193. Sotskov, Y., Dolgui, A., and Portmann, M.-C. (2006). Stability analysis of optimal balance for assembly line with fixed cycle time. European Journal of Operational Research, 168(3), 783–797. Suresh, G., and Sahu, S. (1994). Stochastic assembly line balancing using simulated annealing. International Journal of Production Research, 32(8), 1801–1810.
In practice, the choice of optimisation criteria is critical. It is no longer a question of minimizing the load of the busiest station, or to minimize the number of stations, but to ensure that the same workloads, or similar workloads, are assigned to all stations. Indeed, assembly lines are systems in which operations are generally performed manually, and it is important that employees consider their workloads are comparable. This remark is fundamentally changing the goal of balancing assembly lines. The influence of ergonomics in the design of a station becomes important, and this importance is justified because the influence of ergonomics on productivity and employee health is now recognized. This new factor tends to slow the modification or removal of a station. These remarks, although incomplete, show clearly than conventional models are insufficient to balance several assembly lines that the industry uses. On the other hand, in practice the problems of real life are heavily influenced by environmental constraints and the requirements of assembly activities. It remains to highlight the constraints that apply to the design of assembly lines, and to define realistic optima within these constraints. REFERENCES Arcus, A.L. (1966). A Computer Method of Sequencing Operations for Assembly Lines. International Journal of Production Research, 4, 259–277. Bartholdi, J.J., Eisenstein, D.D. (1996). A production line that balances itself. Operations Research, 44(1), 21–34. Bartholdi, J.J., Bunimovich, L.A., Eisenstein, D.D. (1999). Dynamics of 2- and 3- worker bucket brigade production lines. Operations Research, 47(3), 488–491. Bartholdi, J.J., Eisenstein, D.D., and Foley, R.D. (2001). Performance of bucket brigades when work is stochastic. Operations Research, 49(5), 710–719. Battaïa, O., and Dolgui, A. (2013). A taxonomy of line balancing problems and their solution approaches. International Journal of Production Economics, 142(2), 259–277. Becker, C., and Scholl, A. (2009). Balancing assembly lines with variable parallel workplaces: Problem definition and effective solution procedure. European Journal of Operational Research, 199(2), 359–374. Boysen, N., Fliedner, M., and Scholl, A. (2007). A classification of assembly line balancing problems. European Journal of Operational Research, 183(1), 674– 693. Bratcu, A., and Dolgui, A. (2005). A survey of the selfbalancing production lines (“Bucket Brigades”). Journal of Intelligent Manufacturing, 16(2), 139–158. Bratcu, A., and Dolgui, A. (2009). Some new results on the analysis and simulation of bucket brigades (self-balancing
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Assembly Line Balancing: Conventional Methods and Extensions Alexandre Dolgui* and Jean-Marie Proth** * Ecole Nationale Supérieure des Mines, UMR CNRS 6158 LIMOS 158 cours Fauriel, 42023 Saint-Etienne cedex 2, France (e-mail: [email protected]) **Institut National de Recherche en Informatique et Automatique (R), France (e-mail: [email protected]) Abstract: In this presentation the standard models from the OR literature are mentioned, as well as some simple available algorithms that provide a solution close to optimum. Then some extensions of previous algorithms are introduced, for example, the case where the operating times are stochastic. Bucketbrigades are briefly discussed. Finally, a list of problems that currently remain without solution are spotlighted, especially the problems of sensitivity analysis and robust optimization approaches under uncertainties. These areas could be the focus for promising future research. Keywords: Manufacturing systems, Assembly lines, Line design, Process planning, Line balancing. time average and maximum operation time. The same request may be repeated with several employees and the responses are averaged. Consider that the probability density is linear between two successive points, which represents the triangular density of the operation times. Techniques to estimate the probability that a station processing time exceed a given cycle are proposed and coupled with COMSOAL for optimization. Other approaches are also considered: robust optimization and sensitivity analysis.
1. INTRODUCTION Line balancing consists in assigning a set of operations (to obtain a finished product item) to assembly stations arranged in a line, in accordance with a partial order on the set of operations. The objective is to increase productivity while reducing the cost of production. In Section 2, two basic problems noted SALB-1 and SALB-2 (SALB is the acronym of Simple Assembly Line Balancing) are presented. In the first, the goal is to minimize the number of stations to achieve a given cycle time (also called takt time). In the second, the number of stations is given, and the objective is to minimize the takt time.
In Section 4, an interesting line configuration is presented where the work can be shared among different workers and stations - Bucket Brigades. They have the peculiarity of incorporating the treatment of randomness in their operating mode.
It will be shown how to adapt these algorithms to satisfy some additional constraints such as the usual need to perform an operation at a given station in order to use a specific tool, or the need to perform a given set of operations in the same station or the necessity to execute some operations at different stations because their incompatibility.
Section 5 presents the conclusion. 2. SIMPLE ASSEMBLY LINE BALANCING 2.1 General assumptions
Section 2 concludes with the presentation of a well-known algorithm: COMSOAL for “Computer Method of Sequencing Operations for Assembly Lines”. This algorithm can help to understand better the problems considered. In addition, often this algorithm is sufficient for practical use, in addition it can be easily extended for problems with supplementary constraints and new objective functions.
Problem SALB is characterized as follows: - Each operation is performed at a single station, and its operating time is deterministic, - An operation can be performed at any station, and the operating time does not depend on the station used, - There is a partial order on all operations. This determines the order in which they are processed,
In Section 3, the case where operating times are stochastic is examined. In the approach proposed, the point of view of management practitioners is adopted. Here it is no longer considered that it is possible to associate a referenced probability density (Gaussian distribution, for example) to each operation, but it refers to observations of practitioners. Employees are asked to give three pieces of information, namely: estimate of the minimum operation time, operation
978-3-902823-35-9/2013 © IFAC
- The stations are visited in a given order, - There is only a single type of product to be assembled on the line.
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The goal is to assign all operations to stations minimizing one of the two criteria that are introduced below while satisfying the constraints on the partial order of operations.
- The three operations framed in blue are assigned to the third station. This station load is equal to 33 time units,
2.2 Problem SALB-1
- The three framed black operations are assigned to the fourth station. This station load is equal to 32 time units (8 units of idle time),
In this case, the cycle time C is given and the objective is to minimize the number of stations m used. To this end, the total idle time on all stations is minimized, that is to say:
- Finally, the operation framed in yellow is the only operation of the fifth station. This station load is equal to only nine units of time.
m
TW = m × C −
∑ Τi , where
Ti is the sum of the operation
Table 1. Data for the SALB-1 example
i =1
times related to the operations performed at station i . The following simple algorithm can be applied to search for an approximate solution. At each iteration, the algorithm selects from operations without predecessor or whose predecessors have already been assigned, the operation with the greatest operation time and which can be assigned to the station under consideration (i.e. whose operation time is less than or equal to the time still available for the station) . If no operation is found, a new station is opened. Of course, it is assumed that all the operation times are lower or equal to the cycle time; otherwise the problem would have no solution. This algorithm uses common sense. It is based on the fact that operations with significant operation times are more difficult to assign. So the algorithm tries to assign operations in the decreasing order of their operating times.
Operations
A
B
C
D
E
Operation times
10
12
7
8
20
Predecessors
/
/
/
A
B, C
Operations
F
G
H
I
J
Operation times
4
11
6
9
12
Predecessors
/
D
D, E, F
/
I, G
Operations
K
L
M
N
O
Operation times
15
13
9
8
9
Predecessors
G, H
J
J, K
M
L, N
I
It is possible to significantly improve this algorithm by doing the following: A
- Do not select operations with the greatest operating times to assign to the station being processed, but select at random an operation at each iteration from those whose times are less than or equal to the time left available for the current station,
D
G
J
L O
- Restart the algorithm K times ( K is at the discretion of the user) and keep the best solution among the solutions obtained.
B
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C
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H
K
M
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This improved algorithm is called COMSOAL (Arcus, 1966). Fig. 1. Graph of the partial order on the set of operations and a feasible solution
The random selection introduced in this new version results in significant flexibility of the algorithm which when restarted many times achieves a near optimal solution.
2.3 Problem SALB-2
Example: The following is taken from Dolgui and Proth (2010). The cycle time is equal to 40 units of time. In Table 1, the operating times and precedence constraints (if any) are given.
This problem is to minimize the cycle time knowing the number of stations. From a practical standpoint, this means that the investment is known and the goal is to use it optimally. (In the case of problem SALB-1, the investment is optimized while the desired productivity is known).
Figure 1 shows the result of optimization. In this figure: - The four operations framed in red are assigned to the first station. This station load is equal to 39 time units (1 unit of idle time),
Again, the following iterative heuristic algorithm can be applied: Whenever a new station is inserted, divide the sum of the operating times of the operations that are to be affected by the
- The four operations framed in green are assigned to the second station. This station load is equal to 40 time units (maximum possible occupancy), 44
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
number of stations that have not yet been used. Thus, a lower bound on the cycle time for stations not yet used is obtained.
- Let e be the average value of the operating time. These values are evaluated by users for each operation. The following probability density is then obtained:
If the current station is first, the algorithm assigns it the operations in a decreasing order of operating times for operations that are not yet assigned and still have no predecessor (or whose predecessors have all been assigned). If an assignment is not possible, a new station is opened. If stations have all been used, then it goes to step b) below.
0 if x ≤ a 2( x − a ) ( b − a) ( e − a ) if a ≤ x ≤ e f ( x) = 2(b− x) if e ≤ x ≤ b ( b − a ) ( b − e ) 0 if x ≥ b Figure 2 represents this density.
If the station under consideration is not the first one, then two cases may arise: a) if the new calculated cycle time is less than or equal to the preceding, then the procedure is as for the first station,
f(x)
b) if not, then one starts again the whole process after increasing the initial value of the cycle time of a "low" amount to give more flexibility to the system. The "low" amount in question here is at the discretion of the user.
2/(b–a)
2.4 Usual required adjustments a
Changes in the problem are sometimes necessary to allow the use of previous algorithms. The most common cases when using SALB-1 are as follows:
e
b
x
Fig. 2. Triangular density of probability
In the following sub-section, first, the case of an assembly line (for a single type of product) is considered. Then, the approach is generalized for a mix of different types of products.
- If it is needed to assign multiple operations to the same station, then just combine these operations into a single macro-operation. The operating time is equal to the sum of the operating times of the operations that compose it. The predecessors (resp. successors) are the set of predecessors (resp. successors) of the operations that compose this macrooperation. This type of situation can occur often, for example when there are specific tools for such operations at a station, or is necessary to protect the products from some pollutants (clean room), etc.
3.2 Probability of overflow
The basic problem is to decide (for each iteration) whether a set of operations may be assigned to the same station. A triangular probability density can be defined for each operation. It is necessary to determine whether the amount of time required to perform a set of operations is compatible with the cycle time C. Thus, it is necessary to verify that the total execution time for these operations can exceed the cycle time only with a probability less than a small value ε chosen by the user. For this, simulation can be used. The details of this approach are given in Dolgui and Proth (2010), section 8.3.2.
- Force the assignment of an operation at a given station. In this case, no modification of the algorithm is necessary. When all assignments are completed, simply identify the station that contains the operation as the given station. This occurs for example when an operation must be near a loading or unloading system, or a particular source of energy. - Prohibit two operations to be at the same station (vibrations, tolerances, etc.). In this case, it suffices to perform a modification of the algorithm when determining the candidates to assign.
If operating times are stochastic and if, in addition, the proportion of each type of product in the mix is also random, then to define the distribution of operations in the stations, it is also possible proceed by simulation.
3. STOCHASTIC OPERATING TIMES
For example, consider the case of a mix of three types of products. For each product, Table 2 provides the product structure and the triangular probability density associated to each operation.
3.1 Triangular density of probability
Assembly systems typically include a high proportion of operations performed manually, which results in variable operating times for the same operation. To take into account variations in processing times, one should proceed as follows:
Note that if some operations do not appear in all manufacturing processes, then, simply "empty" operations can be placed in the order common to all products and without predecessor.
- Let a be the smallest possible value of the operating time,
In Table 3, the probabilities of presence of each type of product in the mix are given.
- Let b be the largest possible value of the operating time, 45
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
Table 4. Number of stations versus ε
Table 2. Product structures and time probabilities
ε
Product of type 1 a e b Pred.
A 1 2 3 /
B 2 3 5 /
C 3 4 6 /
D 0 1 3 A
E 2 3 4 B, C
F 0 2 3 /
G 1 2 4 D
H 2 3 5 E, F
J 2 3 4 G
K 1 3 4 H
L 0 3 4 J
M 0 4 6 J, K
N 1 4 5 M
Number of stations
O 1 3 4 L, N
a e b Pred.
A 3 4 5 /
B 2 4 5 /
A 1 3 4 /
C 1 4 5 /
B 2 4 5 /
D 1 3 5 A
D 0 3 4 A
E 0 3 4 B, C
G 1 3 4 D
H 0 3 4 E
I 2 3 4 /
J K 3 2 5 5 6 6 G, I G, H
Product of type 3 G H I J 1 3 2 0 3 4 4 3 4 5 5 5 D D, E / G, I
E 2 4 5 B
K 1 3 4 G, H
L 1 3 4 J
L 3 4 5 J
M 0 2 3 J, K
N 0 3 4 M
O 1 3 4 L, N
M 1 3 5 J, K
N 2 4 5 M
O 1 3 4 L, N
1
2
Probabilities
0.2
0.7
0.1
0
1
2
3
Probabilities
0.1
0.5
0.3
0.1
0
1
2
Probabilities
0.3
0.5
0.2
Policy 1: All stations simultaneously begin processing the set of operations which they are responsible. This implies that all stations have finished processing all their operations when the next cycle is launched. Therefore, the average cycle time increases if ε decreases. Policy 2: When a station is not able to complete the assigned operations during the cycle time, these operations are performed by a relief assembly line. Of course, the next stations are delayed waiting for the end of operations processed by the relief assembly line, but the delay concerns certain operations only. When using this policy, the tendency is to oversize stations that are at the beginning of the manufacturing process. The cost of this solution is significant (investment for online backup and maintenance). Moreover, it is only effective if buffer stocks are maintained. Policy 3: It consists in storing one (or more) semi-finished products after each station, and using these stocks whenever the operations supported by the station are not completed at the end of a cycle time.
Taking into account these triangular probability densities and proportions of each type of product in the mix, simulation can be used to estimate the probabilities of overflow. Thus, COMSOAL can be applied where, each time when an assignment should be made, the simulation is repeated to check whether a set of operations can be executed in the same station with a risk of failure less than a given percentage (for example 1%).
3.4 Other approaches
Many theoretical studies on stochastic assembly lines can be found in the literature. There are various approaches, both through the models considered and the tools used. Some of them are mentioned below.
One of the feasible solutions for this example obtained with this method is represented in Fig. 3.
A survey of publications on design and balancing for stochastic assembly lines is presented in (Erel and Sarin, 1998). Kottas et al. (1981) consider design of paced production lines with stochastic task times to minimize the sum of incompletion cost and labour cost. Suresh et al. (1994) propose to use a simulated annealing approach for solving a stochastic assembly line balancing problem. Erel et al. (2005) consider a stochastic U-line problem to minimize the total expected cost of labour and expected incompletion costs.
I
A
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O
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C
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0.3 4
Three policies are possible (Dolgui and Proth, 2010).
Product of type 3 Number of products
0.2 4
3.3 How to react when the cycle time is exceeded?
Product of type 2 Number of products
0.1 5
Whatever the solution, it is necessary to consider the possibility of exceeding the cycle time. How to react in this case?
Product of type 1 0
0.05 5
A tradeoff between the number of stations and probability of exceeding the cycle time can be sought. In fact, this means to find a compromise between investment (number of stations) and productivity (cycle time).
Table 3. Mix data Number of products
0.015 6
Of course, the number of stations decreases when ε increases, see Table 4 for the example considered.
Product of type 2 a e b Pred.
0.01 7
Fig. 3. A feasible solution
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2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
Dolgui and Kovalev (2012) studied a line balancing problem where operation times are uncertain in the sense that they belong to a given set of scenarios. The objective is to minimize the line cycle time for the worst scenario. Gurevsky et al. (2012) and (Hazir and Dolgui, 2013) studied the cases where interval uncertainty for operation times is assumed. The employed robust optimization approach does not require any distribution data. The worst-case performance of the system is optimized and decisions that perform well under the worst-case scenarios are sought.
5. DISCUSSION AND CONCLUSION The recent state of the art of research literature on assembly line balancing problems is presented in (Boysen et al., 2007) and (Battaïa and Dolgui, 2013). New extensions of line balancing problems appeared which take into account the possibility of parallel execution of operations at stations (Dolgui et al., 2008) as well as the necessity to select pieces of equipment for each station (Dolgui et al., 2012).
Sotskov et al. (2006), Gurevsky et al. (2012) and Gurevsky et al. (2013) develop sensitivity analysis approaches based on the calculation of stability radius for optimal solutions with respect to possible variations of the processing times. They search for the “maximal possible value of simultaneous and independent variations of the processing times” which keeps the optimality of the line balance for different line configurations and assumptions.
Some promising perspectives are addressed in concerning how to take into account uncertainties. Two new ways are explored: sensitivity analysis (Sotskov et al., 2006; Gurevsky et al., 2012; 2013) and robust optimisation (Dolgui and Kovalev, 2012; Hazir and Dolgui, 2013). The following two publications are also to be considered attentively: Falkenauer (2005) and Becker and Scholl (2009). Despite the extensions proposed in the second paper, it is undeniable that the work developed so far only partially meets industrial needs. In Falkenauer (2005), written by an experienced practitioner, significant observations are made. They are summarized below.
4. BUCKET-BRIGADE ASSEMBLY LINE A "bucket-brigade assembly line" has the advantage of integrating its adaptation to the system load and has the peculiarity of incorporating the treatment of randomness in its operating mode.
The managers of assembly lines certainly need to balance them initially, but how should they proceed when the conditions are modified (change of components, change in the mix of products assembled, emergence of new technologies, necessity to adapt workforce on demand)? It is not possible to ignore the existing system and look for the optimal balance of a new assembly line. The assembly line should be rebalanced for the lowest cost in order to better adapt to new conditions. It is interesting to note that a similar problem arises in scheduling.
The main advantage of this approach is summarized in (Barholdi et al., 1996), (Barholdi et al., 1999) and (Barholdi et al., 2001): “bucket brigades are a way of organizing workers on an assembly line so that the line balances itself”. A survey of publications on bucket brigades is presented in (Bratcu and Dolgui, 2005) and a simulation study is given in (Bratcu and Dolgui, 2009). Let us take an example of a bucket brigade with three employees. They move while working on an assembly. When employee number 3 reaches the end of the line, he/she deposits the finished product, and then returns to meet employee number 2. When they meet, employee number 3 takes the bucket of employee number 2 and continues its assembly. Then, employee number 2 goes to meet employee number 1 to take his/her charge. Employee number 1 then goes to the beginning of the line and starts a new assembly.
Most of the work reported in the literature belongs to one of two types: either they report theoretical developments related to balancing problems, and in this case they stray from applications, or they have been developed to meet specific industrial needs. In this case, they offer little theoretical reusable implications. Usually, theoretical developments generally ignore the specificity of stations. In this context, a station is neutral. It is supposed to be able to change assignment without a major problem that is to say at negligible cost. Practical developments are defined in relation to their environment (available space, ceiling height, disposal of the sources of energy, etc.) and these constraints are decisive in the choice of the solution to the point that they hided the general aspects of the method.
Note that buckets are placed on a conveyor and the number of employees involved is not fixed: The greater the number of employees (within the limits of feasibility), the greater the productivity (that is to say the lower the average cycle time) and the lower the working time of an employee on a given assembly.
Taking into account the influence of internal transportation systems is essential, particularly when it comes to industries that require the handling of heavy and / or bulky objects, for example, in the automotive industry, aircraft manufacturing, agricultural equipment and, more generally, any assembly system that requires transporting objects of more than a few hundred grams. Indeed, it is difficult to restructure an assembly system that involves cranes and wire-guided carts, etc. when they are intended for specific products, among others. Also the rigidity of the assembly line - when the system for loading and unloading components is particular -
This system, however, can pose two types of problems: - Problems caused by the fact that employees are forced to move along the conveyor belt while working; - Possible discord among employees because of low efficiency for some employees results in overloads for others.
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2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
should be mentioned. In summary, internal transportation systems are factors that can make the adaptation of an assembly line to changes sometimes impossible and always very constraining. For the same reasons it is difficult, and even impossible, to eliminate stations.
production lines), International Journal of Production Research, 47(2), 369–388. Dolgui, A., Guschinsky, N., Levin, G., and Proth, J.-M. (2008). Optimisation of multi-position machines and transfer lines. European Journal of Operational Research, 185(3), 1375–1389. Dolgui, A., Guschinsky, N., and Levin, G. (2012). Enhanced mixed integer programming model for a transfer line design problem. Computers & Industrial Engineering, 62 (2), 570–578. Dolgui, A., and Kovalev, S. (2012). Scenario based robust line balancing: Computational complexity, Discrete Applied Mathematics, 160 (13–14), 1955–1963. Dolgui, A., and Proth, J.-M. (2010). Supply Chain Engineering: Useful Methods and Techniques. Chapters 7 and 8. Springer Verlag, London. Erel, E., and Sarin, S. (1998). A survey of the assembly line balancing procedures. Production Planning and Control, 9(5), 414–434. Erel, E., Sabuncuoglu, I., and Sekerci, H. (2005). Stochastic assembly line balancing using beam search. International Journal of Production Research, 43(7), 1411–1426. Falkenauer, E. (2005). Line balancing in the real world. Proceedings of the conference PLM-SP1-2005, pp. 360– 370. Gurevsky, E., Battaïa, O., and Dolgui, A. (2012). Balancing of simple assembly lines under variations of task processing times. Annals of Operations Research, 201, 265–286. Gurevsky, E., Battaïa, O., and Dolgui, A. (2013). Measure of stability for a generalized assembly line balancing problem, Discrete Applied Mathematics, 161(3), 377– 394. Gurevsky, E., Hazir, O., Battaïa, O., and Dolgui, A. (2012). Robust balancing of straight assembly lines with interval task times, Journal of Operational Research Society, doi: 10.1057/jors.2012.139 Hazir, O., and Dolgui, A. (2013). Assembly line balancing under uncertainty: Robust optimization models and an exact solution method. Computers & Industrial Engineering, doi: 10.1016/j.cie.2013. 03.004. Kottas, J.F., and Lau, H.-S. (1981). A stochastic line balancing procedure. International Journal of Production Research, 19(2), 177–193. Sotskov, Y., Dolgui, A., and Portmann, M.-C. (2006). Stability analysis of optimal balance for assembly line with fixed cycle time. European Journal of Operational Research, 168(3), 783–797. Suresh, G., and Sahu, S. (1994). Stochastic assembly line balancing using simulated annealing. International Journal of Production Research, 32(8), 1801–1810.
In practice, the choice of optimisation criteria is critical. It is no longer a question of minimizing the load of the busiest station, or to minimize the number of stations, but to ensure that the same workloads, or similar workloads, are assigned to all stations. Indeed, assembly lines are systems in which operations are generally performed manually, and it is important that employees consider their workloads are comparable. This remark is fundamentally changing the goal of balancing assembly lines. The influence of ergonomics in the design of a station becomes important, and this importance is justified because the influence of ergonomics on productivity and employee health is now recognized. This new factor tends to slow the modification or removal of a station. These remarks, although incomplete, show clearly than conventional models are insufficient to balance several assembly lines that the industry uses. On the other hand, in practice the problems of real life are heavily influenced by environmental constraints and the requirements of assembly activities. It remains to highlight the constraints that apply to the design of assembly lines, and to define realistic optima within these constraints. REFERENCES Arcus, A.L. (1966). A Computer Method of Sequencing Operations for Assembly Lines. International Journal of Production Research, 4, 259–277. Bartholdi, J.J., Eisenstein, D.D. (1996). A production line that balances itself. Operations Research, 44(1), 21–34. Bartholdi, J.J., Bunimovich, L.A., Eisenstein, D.D. (1999). Dynamics of 2- and 3- worker bucket brigade production lines. Operations Research, 47(3), 488–491. Bartholdi, J.J., Eisenstein, D.D., and Foley, R.D. (2001). Performance of bucket brigades when work is stochastic. Operations Research, 49(5), 710–719. Battaïa, O., and Dolgui, A. (2013). A taxonomy of line balancing problems and their solution approaches. International Journal of Production Economics, 142(2), 259–277. Becker, C., and Scholl, A. (2009). Balancing assembly lines with variable parallel workplaces: Problem definition and effective solution procedure. European Journal of Operational Research, 199(2), 359–374. Boysen, N., Fliedner, M., and Scholl, A. (2007). A classification of assembly line balancing problems. European Journal of Operational Research, 183(1), 674– 693. Bratcu, A., and Dolgui, A. (2005). A survey of the selfbalancing production lines (“Bucket Brigades”). Journal of Intelligent Manufacturing, 16(2), 139–158. Bratcu, A., and Dolgui, A. (2009). Some new results on the analysis and simulation of bucket brigades (self-balancing
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