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Operations

Published: Wednesday, June 29, 2016 - 16:56

How do you determine the “worst case” scenario for a process? Is it by assuming the worst case for each process task or step? No. The reason is that the probability of every step having its worst case at the same time is practically zero. What we’re looking for is a value that will occur a very small percentage of the time, but still be a possibility.

In statistics, we do this with a confidence interval, typically plus or minus three standard deviations from the mean to achieve 99.7-percent confidence.

For example, let’s say that we have a three-step process, with means and standard deviations of x_{1} = 20, s_{1 }= 3; x_{2} = 30, s_{2} = 5; and x_{3} = 60, s_{3 }= 9, respectively. Since variation (variance) is additive, the variance of the entire process is therefore:

S^{2}_{Process} = 3^{2} + 5^{2} + 9^{2} = 9 +25 + 81 = 115, and the process standard deviation is:

S_{Process} = SQRT(115) = 10.7.

If the measure of x is time in minutes, or another measure where a high value is undesirable, the “worst case” should be the mean time plus three standard deviations for the process, or (20 + 30 + 60) + 3*10.7 = 110 + 32.1 = 142.1. (This assumes that there is no wait or lead time between the process steps—not a good assumption, but for now, this will simplify the main point attempting to be made here.)

By contrast, taking the upper limit of each individual process step and adding them together would result in an overly pessimistic worst case of:

[20 +3(3)] + [30 +3(5)] + [60 + 3(9)] = 29.0 + 45.0 + 87.0 = 161.0 minutes.

This difference can be significant, as overly pessimistic worst-case assumptions are expensive. They often result in excessive inventory and carrying costs, as well as other related costs.

In general, if we use 99.7-percent confidence, then the probability of the worst case for a single process step would be (1.00 - 0.997)/2, or 0.0015. If the process has two steps, the probability of each step experiencing its worst case at the same time would be 0.0015^{2} = 0.00000225 (or, expressed in scientific notation, 2.3E-6).

Note that the probability of a defect when a process is at Six Sigma quality is 3.4 defects per million, or 3.4E-6. So experiencing the worst case simultaneously in each step of a two-step process is roughly equivalent to the very small probability of getting a defect when a process is at Six Sigma quality.

If there are four steps in the process, this probability is 5.1E-12, or thirteen zeroes after the decimal point before a non-zero value occurs. And, of course, the number of steps in a process chosen for process improvement is often much more than four. Clearly, the probability of this occurrence cannot be called “zero,” but it is so small that it would be inappropriate to use it for worst case contingency planning in a decision-making situation.

Earlier in my career, I was the assistant controller at a major bank in Atlanta. Each month, the controller calculated the worst possible case for the following month by calculating the worst case for each income statement item. I tried to explain to him that the worst case would most likely never happen. He didn’t get it.

Don’t fall into this same trap. The worst case is often not as bad as it might seem.

## Comments

## Worst Case

There's lies, damn lies and statistics. As a junior engineer, I worked in a factory that started making reject. I was told to keep away while "experts" were brought in to fix it. They made reject for a month and filled a warehouse with product with "hold" tags. Suddenly the factory started making good product again. No one knew why. The experts were sent home.

Where's the probability figures for that?

Several months later, exactly the same problem occurred during the middle of the night when I was on as shift foreman, in control of the whole factory and two others. First step was to find the cause, which I did. The "experts" had forgotten this key action. My next step was to design a solution. By morning I'd made major changes to the process and was making good product again. When management arrived and saw the changes I'd made to the process, I was crucified.