Steve Beeler

You have a goal…I have a way to get you there.

X-bar & R Chart

Lake Michigan Water Levels

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Lake Michigan shore line

Record high Lake Michigan water levels and dune erosion have been big concerns along the “north coast” in recent years.  As beaches disappeared under the rising water, homes near the shoreline were threatened.

Barge with crane on Lake Michigan

Property owners resorted to armoring the coastline with huge boulders to dissipate wave energy.  In some cases, cranes on barges were required to place the boulders.

I finally got around to doing a statistical analysis on Lake Michigan water levels using data from the United States Army Corps of Engineers.  For years, control charts have been in my operations engineering toolkit.  Why not apply statistical process control to a natural phenomenon?

An X-bar & R chart is commonly used to measure the magnitude of common cause (random) variation and to assess statistical stability (constant probabilities).  For a sample plan, I used one data point per year in subgroups of five.  That is, each data point on the X-bar chart is an average of five years and each data point on the range chart is difference between the highest and lowest value in that five-year period.  With 103 years of data (1918 to 1921), this sample plan provided twenty data points, the minimum required to calculate control limits.

Control chart of Lake Michigan water levels

The range chart (bottom) is in control.  All points are within the control limits and there are no trends.  This means the magnitude of year-to-year water level variation has been constant over the last 100+ years.

On the other hand, the X-bar chart (top) does not exhibit statistical control.  While it is mean centered with no trends, there are multiple points (4 above and 6 below) outside the control limits.  This suggests that the five-year sample plan has underestimated the range.  In other words, the within subgroup variation did not entirely capture the common cause variation in Lake Michigan water levels.

In retrospect, this is not surprising.  While 103 years seems like such a long time, it is not even an instant from a geologist’s perspective.  A sample plan with a longer duration appears to be called for but then there would not be enough subgroups to calculate control limits.

Given that the range chart is stable, I suspect that the X-bar chart (i.e., average water levels) would be too if we had data sampled over a sufficiently long period of time.

Others are interested in the Great Lakes.  Enough, apparently, to warrant a scientific publication, the Journal of Great Lakes Research.  In it, I found an abstract that supports my hypothesis of stable Lake Michigan water levels: “Historical Variation of Water Levels in Lakes Erie and Michigan-Huron” by Craig T. Bishop.

From this abstract I learned that the earliest “reliable” Great Lakes water level data were recorded in 1819.  From historical and archeological evidence, Bishop concludes that “… over at least the last 1,800 years, climate-related variations in maximum mean annual water levels have probably not exceeded those measured on Lakes Erie and Michigan-Huron since A.D. 1819.”

That sounds like Lake Michigan water levels have been stable for a long time.  In 2022, the water is down from record levels and beaches are reappearing.  The spectacular sunsets never left.

Lake Michigan sunset

If you would like to perform your own statistical analysis on Great Lakes water levels, click HERE for a link to the United States Army Corps of Engineers data set.

Simulation Output Reports

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Discrete Event Simulation is a portal to the future: find constraints, test strategies to break them, improve performance to the goal, maximize investment returns, and reduce risk. Shown below are three powerful simulation output reports.

Simulation Output Reports

Time in State. Time in state simulation output reports are especially useful in finding the constraint in the process. Before the constraint, machine elements are generally blocked (blue) more than they are starved. After the constraint, they are starved (yellow) more than they are blocked. The machine element with the most uptime (green) is likely the constraint.  Knowledge of the constraint’s location is a key factor in improving the process. Focus on the constraint for opportunities to increase throughput. Look to non-constraints for opportunities to reduce operating costs.

Simulation Output Reports

Volume Histogram. The volume histogram is the first to two methods used to validate a simulation model against its real-world process. How production counts vary over time is an important metric…the less uncertainty the better.  The volume histogram provides a qualitative comparison of the hour-to-hour variation in volume throughput.

Simulation Output Reports

X-bar & R Chart. The production count X-bar & R chart provides a more quantitative comparison between the simulation and the real-world process. Statistical process control charts not only quantify the magnitude of the common cause process variation but also identify special cause events. Characterizing the output variation as common cause vs special cause is an important factor in validating the simulation model. It is difficult if not impossible to simulate special cause events. That is because discrete event simulation “engines” utilize constant probability random number streams. If the real-world process variation is being driven by non-random events, then the root causes of the special cause events will have to be removed before the simulation what-if results will predict future performance.

These three simulation output reports are the foundation for a Plan-Do-Check-Act continuous improvement of the simulated manufacturing or business process. Simulate, validate, and experiment. A robust solution will follow.

Control Charts

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To provide a rational means to categorize (common cause vs special cause) and measure variation, Walter A. Shewhart invented control charts circa 1925 – 1931. W. Edwards Deming took these basic statistical concepts and tools with him to Japan after World War II and planted the seeds for the quality revolution. The rest is history.

Control charts are best maintained at the job station by the production team to provide those closest to the process guidance on when and when not to take action. Once a process is in statistical control (a very fine achievement in itself), a control chart can verify actions taken to reduce variability.

The primary signal of an out-of-control process is a point outside the control limits. Given constant probability (an requirement for process control), the chance of a point outside the control limits is very, very small. Other out-of-control signals include seven points in a row above or below the mean and runs of seven up or down. If these signals are present, then a special cause has visited the process. Look for it right away, before the trail grows cold!

Control Charts

Here is an example of an X-bar & R chart for variables data. An assembly plant body shop was requiring significant overline overtime to make production and the management team was attributing the volume shortfalls to “catastrophic” breakdowns at its constraint. A control chart for hour-to-hour production counts was in control (no special cause events) but with an unacceptably high level of common cause variation. The management team was “recalibrated” to focus their effort on the many small production loss causal factors occurring every hour: speed losses, absenteeism, operator pacing, untrained operators, material deliveries, weld faults, tip changes, etc.

Control Charts

Control charts also work with attribute data when variables data is not available. Here is an example of a U-chart of incomplete operations at an inspection and repair station. This chart has been out of control for quite awhile: the mean number of repairs have shifted up. In retrospect, the team was slow in finding the special cause in this case a defective bumper shield clip.

Control Charts

Out-of-control signals can be good. Let’s say an adjustment has been made to a process running above the specification target. A run of seven points below the averages chart mean is verification that the adjustment has moved the process mean. Or maybe a material spec change has been made. A run of seven points below the range chart mean is verification that the material change has reduced common cause variation. In both cases, new control limits must be calculated to reflect the process changes.

Terminology

Variables Data = quantitative value that can be measured (572 feet, 2403 lbs, 59.6 seconds, etc)
Attributes Data = qualitative value that can be counted (pass or fail, black or white, etc)