If you are expecting a review of which bathtubs relieve the stress of an Asset Manager at the end of the day, you will be disappointed in this blog.
Instead, we are here today to pay tribute to the Normal Probability Density Function (PDF):
Oh geez, more math. Alright, let's skip the math and talk about curves.
In my opinion, the bathtub curve is a cornerstone of asset management. Traditional reliability analysis generally assumed that asset failures occurred at a constant rate or a straight line. Classic Weibull Analysis proved otherwise and used a normal statistical distribution as noted above.
You may be familiar with the common statistical term known as the “bell curve”. In asset management, curves better define asset degradation than a straight line. My cheesy graphic below illustrates the aptly deserved name.
Startup or Break-In Period
The moment an asset is put into service, it has an incredibly high probability of failure due to many things; manufacturer defect, improper installation, excessive pressure or current, etc. Therefore, the Failure Rate is potentially at its highest point in the asset life cycle as shown in Figure 1.
Figure 1. The Break-In Period
After the first second, minute, an hour, or whatever is the recommended break-in procedure, the failure rate plummets as the asset settles in for its normal operating service.
Normal Operating or Useful Life Period
After the Break-In period, an asset serves its Normal Operating period, or designed Useful Life. This is the classic Operations and Maintenance (O&M) budget period. Normally this is the longest period of the asset life cycle. The duration is a function of the asset class characterized by material properties, in-situ conditions, capacity management needs, etc. Failures can randomly occur during the normal course of operating the asset, however, for years the Failure Rate remains flat or relatively constant as shown in Figure 2.
Figure 2. Normal Operating Period
Whether a failure data point is logged depends on what has been defined as a “failure” for the specific asset class. A classic example is a water main leak vs a break. The primary difference is a loss in service. The AWWA Asset Management Definitions Guidebook Version 1.0 defines a water main break as: “The structural failure of the primary water main conduit or associated joint, tap, fitting or lateral resulting in a loss of service”. A water main leak conversely does NOT result in a loss of service, i.e. water flow or pressure. Many water main leaks go unnoticed for long periods of time (this is why you should have an active Non-Revenue Water Loss Program). So is a leak a failure? Depends on the organization.
I realize I am mixing more than the semantics of the individual probability of failure at any given time vs the actual failure rate over time, but what designates the end of the normal operating period is the inflection point entering the Wear-Out period where the probability of failure increases, so does the rate, and invariable the intensity. I have taken artistic liberty in denoting this as the “First Failure” point in Figure 2. Only after the fact can data analytics demonstrate whether that is indeed the inflection point of an asset entering the Wear-Out period.
An asset enters its Wear-Out period when the failure rate begins to increase as shown in Figure 3. The actual number of observed failures is variable to the asset class, and quite honestly to the organization’s tolerance to failure. This tolerance is guided by service levels, customer satisfaction, available O&M budget for repairs, etc. When an organization has had enough repairs, the “Last Failure” is a subjective endpoint when an organization is finally ready to renew, rehabilitate, or replace the asset and allocate those expensive Capital dollars.
Figure 3. Wear-Out Period
Remaining Useful Life
Calculating the Remaining Useful Life (RUL) of an asset is the million-dollar value benefit of asset management. Figure 4 cleans up the graph and shows RUL as that variable timeframe to the end of the Wear-Out period. In reality, RUL is actually pertinent the second after an asset is placed in service. You can check out the AWWA definition of RUL in the guidebook referenced above.
Figure 4: Remaining Useful Life
Also in Figure 4 are example formulas in calculating the RUL. Many asset management software systems use the simplistic linear regression, or slope of the line in the Wear-Out period to calculate Business Risk Exposure (BRE). Accountants use this for asset depreciation in calculating book value. What is found in engineering practice is more of a polynomial curve. Determining the shape of this curve is a bit of science and art. The general shape of the curve can be assumed for a given asset class, yet the actual curve of any given asset can be skewed by the same properties mentioned above; material, in-situ conditions, etc. Calculating material degradation characteristics is an industry in its own right. Active condition assessment programs calibrated by real-time sensor diagnostics should be applied to override the many assumptions in these calculations.
While engineers love straight lines, curves are where it’s at for calculating the RUL of an asset. Investigating what affects the shape of the curve is almost as important as the formula of the curve itself. Understanding the inflection points of the Bathtub Curve provides real-world insight into the operational aspects of your system.
A word of caution is that any formula or statistic has a number of assumptions. Real-time sensors and new innovative technologies providing in-situ diagnostics reflect more accurate approximations of the condition and therefore RUL, yet no organization to date has been able to afford the sensoring-up or inspection of 100% of their system. However, seasoned statistical methods such as the Bathtub Curve give an organization valuable insight into managing their assets.