8.9 Specific Systems with Dynamic Work Sharing
8.9.1 Bucket-brigade Assembly Lines
According to (Bartholdi and Eisenstein, 1996), (Bartholdi et al., 1999) and (Bar- tholdi et al., 2001): “Bucket brigades are a way of organizing workers on an as- sembly line so that the line balances itself.”
In this section, we consider this approach from a practical point of view and re- strict ourselves to the basis of the method with deterministic operation times.
8.9.1.1 Description of a Bucket-brigade Assembly Line
Consider a case where 3 workers w1,w2 and w3 are involved in the system. Each of them is working on a semi-finished product that is placed on a conveyor that moves at a constant speed V . Thus, the workers are moving at the same speed while working on the product.
As soon as the last worker w3 completes the product, he/she walks back up- stream to take over the work of the predecessor w2, who then goes upstream to free up the first worker w1, who then moves to the beginning of the assembly line and starts assembly of a new product.
Indeed, the model can be generalized to any number n of workers. In the fol- lowing, L denotes the length of the conveyor. This 3-worker bucket brigade is il- lustrated in Figure 8.12.
8.9.1.2 Particular Case
Let n denote the number of workers involved in the brigade. The goal is to define the speed V of the conveyor in order to complete the whole assembly process at the end of the line.
Assume that:
• The efficiency of each worker is steady. In other words that time ti needed by worker wi to assemble a whole product is constant.
• The speed v of workers wi when walking upstream is infinite. From a practi- cal point of view, it means that the time required for reaching a predecessor or the beginning of the line is negligible.
L
1 2 3
L
1 2 3
L
1 2 3
L
1 2 3
L
1 2 3
Figure 8.12 A 3-worker bucket brigade
Under these conditions, the time θi spent by worker wi on a unit of product does not depend on time either. Taking into account the fact that the backward speed v is infinite, we have θi=C, where C is the cycle time (takt time) that is to say the period between the completions of two consecutive products. The cycle time is also the working time of any worker on a unit of product.
The ratio
i i
i
t C t =
θ is the proportion of work done by wi on each unit of prod- uct.
Thus: 1
1
∑ =
= n
i i
i
t
θ (8.10)
and Equation 8.10 becomes:
1 1
1
∑ =
= n
i ti
C (8.11)
314 8 Advanced Line-balancing Approaches and Generalizations The following relation holds:
L C V
n =
This relation expresses the fact that the assembly is completed at the end of the conveyor.
Taking into account (8.11), this equality can be rewritten as:
n L t V
n
i i
∑=
= 1
1
(8.12) This relation provides the speed of the conveyor in steady state according to the
number of workers and their individual efficiencies if we want to complete assem- blies exactly at the end of the conveyor. Relation 8.12 also gives the maximum speed of the conveyor to complete the assembly processes before the end of the conveyor.
Furthermore, ∑
= n i1 ti
1 is the number of products assembled during one unit of time.
8.9.1.3 General Steady-state Model
This model is the same as the previous one, except that the walking speeds of the employees are no longer negligible. Since the speeds and working efficiencies do not depend on time, therefore in steady state, an employee meets his/her predeces- sor always at the same distance from the beginning of the conveyor.
Let us denote by xi,i=1,L,n−1 the meeting point of wi+1 and wi. To sim- plify the notations, we also set xn=L and x0=0.
We consider worker wi, i∈{1,L,n−1}. During each cycle time C, this worker performs 2 moves:
• he/she moves from xi−1 to xi at speed V while working,
• he/she walks upstream from xi to xi−1 at speed vi. Therefore, the following equations hold:
n v i
x V v x
x V
i i
i )(1 )for 2, ,
( ) 1
( 1
1
1 + = − − + = L (8.13)
Each member of Equation 8.13 is equal to the cycle time (takt time) multiplied by V .
Finally, an assembly is completed at time xn. Thus:
1
2
1 1
1 =
× + −
× ∑
= n −
i i
i i
t V
x x t
V x
or:
t V x x t
x n
i i
i
i− =
+∑
=
− 2
1 1
1 (8.14)
We also know that xn=L. So, the following algorithm leads to the solution.
Algorithm 8.10. (Bucket) 1. Read L,(ti,vi)i=1,L,n.
2. Initialize the speed of the conveyor using Equation 8.12.
3. Initialize x1=L/n.
4. Compute x2,x3,L,xn−1,xn =LC using Equations 8.13.
5. Compute V using Equation 8.14.
6. If (LC=L) then do:
6.1. Compute the cycle time V
v x V
L C
n
n ) (1 )/
( − 1 × +
= − .
6.2. Print VandC. 6.3. Stop.
7. If (LC≠L) then do:
7.1. x1=x1*L/LC. 7.2. Go to 4.
8.9.1.4 Numerical Example
Consider the problem defined as follows:
• L=50.
• 4 workers.
• v1=12,v2=14,v3=11,v4=15.
• t1=50,t2 =48,t3=46,t4=44.
We obtain V=1.06643 and C=12.6963. Furthermore, the meeting points are 359
. 37 , 016 . 25 , 4346 .
12 2 3
1= x = x =
x . If the times required to walk back up-
316 8 Advanced Line-balancing Approaches and Generalizations stream are negligible, then we obtain: V =1.06625 and C=11.72546. The meet- ing points are x1=12.5, x2=25, and x3=37.5.
8.9.1.5 Benefits Got from Bucket Brigades and Remarks The most important benefits can be summarized as follows:
• The system reaches an organization that naturally redistributes the work among workers, according to their efficiency.
• The flow of product is self-balancing. Thus, such a system needs little man- agement effort.
• Bucket brigades are agile and flexible. They adapt themselves quickly to unex- pected changes in demands.
• Handling activities are minimized because WIP is minimal. A beneficial con- sequence is quality improvement.
The restrictions resulting from the fact that workers move while they work, or transport products, remain an open field of study.
It must be noticed that the ambience in the workplace may deteriorate, due to the different effectiveness of the workers or their motivation. Some of them may be reluctant to work more than others.
Indeed, this section only skims over on the subject. For instance, conveyors are not used in many bucket brigades: employees move to take the product at the up- stream station. Note that the previous model also holds in this case.
For more information on bucket brigades, please see (Bratcu and Dolgui, 2005) and the web site of Bartholdi and Eisenstein: www.bucketbrigades.com