Fluctuations on Continuously Moving Assembly Lines 2—Waiting Times

In my previous post I posed the problem of decoupling fluctuations on continuously moving assembly lines. You cannot decouple using inventory; you can decouple only long-term fluctuations using capacity, while all other fluctuations are decoupled using time. In my last post I introduced the topic and urged you not to have workers between different stations to handle fluctuations. In this post I will look in more detail at the waiting time of the workers due to fluctuations in the work duration for the stations.

The System

To take a deeper look at how fluctuations affect a continuously moving assembly line, I will model such an assembly line. For this, I need a couple of assumptions:

• It is a continuously moving assembly line that stops if one station cannot make the cycle time (i.e., an Andon system). Hence, if one station is delayed, all stations are delayed.
• The line has 10 stations. I will do 10 000 iterations or cycles.
• The workers don’t speed up or slow down their work. In reality, if a worker notices that he slows down everybody else, he may speed up.
• We have a number of stations, where the random distribution of the work duration has a Weibull distribution. For simplicity, all stations have the same distribution, although of course in reality they differ. Here, I use a Weibull distribution with a shape parameter of 1.5 and a scale parameter of 1. The whole distribution is offset by 5 time units, and then multiplied by 12. Hence, the distribution starts at 60 time units, and has a mean of 70.82 time units. If the time unit its seconds, then this is within the range of automotive assembly lines, with a shape that is also commonly found in manufacturing processes. The standard deviation is 7.35 seconds. The distribution is plotted below.

I will introduce more assumptions as I show you different examples of how to look at this problem.

Assumption: Line Speed Infinite

First, we assume an infinite line speed. Since our random distribution has a minimum of 60 seconds, every worker will slow down the line every time. Through this I can see the actual line speed based purely on the speed of the slowest of the ten workers. Every time, the randomly slowest of the ten workers will define when the line will continue. Repeating this 10 000 times gives me a histogram of the actual line speed.

The diagram below shows again the Weibull distribution for a single worker, and the histogram of the time of the line to do one cycle. This is the slowest random time of the ten workers on the line. While a worker on its own could do the work in average within 70.82 seconds, the line itself needs 84 seconds, since they always have to wait for the slowest worker.

Since we cannot use inventory or capacity to decouple the workers fluctuations, we only can use time (i.e., all workers but one have to wait for the slowest one at every cycle). Since the individual worker in average needs 70.82 seconds, but the combined system with ten workers need 84.0 seconds, at every cycle the worker has an average waiting time of 13.2 seconds. This is the best possible situation with the given system…

…but in reality this is not achievable. Again, we have the assumption of an infinite speed. Hence, the line stops immediately at time zero, and waits until the last worker is done. Once the last worker is done, the entire line moves at an infinite speed to the next station. And that’s where the problem is: In reality, you cannot move the line at infinite speed. First, it is physically not possible, and even a very fast speed may be dangerous. Hence, for a continuously moving line, we need to set the speed. And, no matter what speed we set, it will be sometimes too slow for some and too fast for others. (Note: For slower lines this may be a useful option. If you move the part once every 8 hours, a moving time of 10 minutes is comparatively small. If you move the part every 60 seconds, a moving time of 10 seconds is comparatively large.)

Plus, there are other reasons to have a reasonable speed. If it is too slow, the worker has to work a lot at the beginning of his station. If it is too fast, he has to work at the end of his station. A fixed speed also helps to keep a steady takt, and psychologically gives a structure that sets an expectation by the management on the speed. When setting the line speed, you can set the takt and hence the target output of the line. It also makes line balancing easier.

Assumption: Line Speed 90 Seconds

Next, we set the line to a reasonable speed of 90 seconds per cycle. Why 90? Because at that speed, most cycles will be within the target line speed of 90 seconds for all workers. Only 17% of all cycles are delayed, and 83% of all cycles are completed within the cycle. While this means that 83% of the time, all workers will have to wait for the continuously moving assembly line, you avoid a constant starting and stopping of the line.

Important: For this example, I put the horse on backwards. I looked at the random distributions and then set the speed of the line. In reality, you should start with the customer takt and then assign the work to the stations (i.e., creating the random distributions of the work time) to fit the line speed. Here, I did this backwards to make the explanation easier.

Overall, on this example you can see how fluctuations impact your continuously moving assembly line and why it is so important to reduce fluctuations. In my next post I will show you a super secret Toyota miracle tool…

Okay… I lied… It is not so secret. It’s the team leaders for small groups that support four to five operators. But since many Western companies don’t understand fluctuations, they don’t see the value of the team leaders, but only the expense, and hence the team leaders have gotten long since axed. But more about this in my next post. Now, go out, set a good speed for your assembly line that matches your need, then fit the work (and its randomness) to your needs, and organize your industry!