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Operations

## How to Determine the Worst Case for a Process

### Have confidence in the confidence interval

Published: Wednesday, June 29, 2016 - 15:56

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 x1 = 20, s1 = 3; x2 = 30, s2 = 5; and x3 = 60, s3 = 9, respectively. Since variation (variance) is additive, the variance of the entire process is therefore:
S2Process = 32 + 52 + 92 = 9 +25 + 81 = 115, and the process standard deviation is:
SProcess = 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 0.00152 = 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.

### Ken Levine

Ken Levine is a Lean Six Sigma management consultant. He retired as director and lead instructor in the Lean Six Sigma certification program at Georgia State University in Atlanta in 2019. He holds a doctorate in business administration. Levine retired from The Coca-Cola Company in 2000 where he held the position of director of continuous improvement in the Coca-Cola USA division for three years. Levine is a Six Sigma Master Black Belt and Certified Purchasing Manager. He has previously published “Root Cause Analysis Takes Too Long,” “How to Determine the Worst Case for a Process,” “Recycling Your Meeting Waste”, “What Really is a Stretch Objective”, and “Ensuring LSS Success with a Robust Define Phase” in Quality Digest.