The bathtub curve is one of the most famous graphical representations in reliability. It shows the typical trend of failure rate over time (Figure 1). This curve can be divided into three parts corresponding to decreasing, constant and increasing failure rate.
High but decreasing failure rate in the first part of the curve (1) indicates products defective or prone to fail due to manufacturing or design issues. The middle section (2) is characterized by low and constant failure rate. Failures in this phase are caused by unpredictable external events like flatting tire or power surge causing electronic failures. The right-hand side of the bathtub curve (3) shows increasing failure rate. In this stage, failures occur due to wear-out damage mechanisms such as fatigue, creep, ageing, wear and corrosion.
Figure 1. Schematic representation of the bathtub curve. The three different zones refer to 1) decreasing failure rate; 2) constant failure rate; 3) increasing failure rate.
Often, the shape of the bathtub curve is said to reproduce mortality in three main phases of human life from birth to senility: 1) infant mortality; 2) mortality related to unintentional injury like accidents; 3) mortality due to ageing-associated diseases. Failure rate of products and human mortality seem to be similar. Moreover, probably for sketching purposes, the bathtub curve is represented symmetric. After all, if you have a bathtub at home, this is most likely symmetric.
The schematic example in Figure 1 also follows this assumption. It seems that it is somehow in the nature of things that mortality of infants is similar to that of elderly individuals. As father of three children and reliability engineer, I was concerned about that. Especially at the childbirth. According to the bathtub curve, infant mortality is highest at childbirth. Childbirth is such a critical moment in life.
I decided to analyze the data on mortality in Italy from the Human Mortality Database (HMD) [1]. Mortality is calculated as the number of deaths each year at age “x” divided by the population at that age over the same period. Figure 2 shows mortality in the range 0-95 years in Italy in the year 2020. The color and the size of the circular markers indicate the number of deaths and the number of alive individuals for a certain year, respectively. I was relieved to note that mortality of infants is not the same as for a 80 year old person. The curve does not seem like a bathtub curve, we introduced in Figure 1. It is pretty flat until the age 70 years. After this point, the gradient of the curve increases gradually, and accelerates more rapidly from the age of 80 years.
Figure 2. Mortality over age (Country: Italy; Year: 2020; Age interval 0 – 95 years) Color: number of deaths, Size: number of alive individuals. Source: [1].
Zooming into the initial part of Figure 2, one can see that infant mortality is higher at childbirth than later (Figure 3). For infants, the risk of death is highest just after birth, but mortality is orders of magnitude lower than that of elderly individuals.
Figure 3. Mortality over age (Country: Italy; Year: 2020; Age interval 0 – 10 years). Color: number of deaths. Size: number of alive individuals. Source: [1].
In the past the situation was worse. Figure 4 shows that in 1872, the year when the records began, mortality at childbirth was 27.8% with 227.888 fatalities. In 2020, mortality decreases to 0.2% with 1016 fatalities (a reduction of 99.1%!). According to HMD, the quality of data for 1872-1905 is lower than in later years. However, for the purpose of this article, I want to highlight the significant improvement in terms of reduction of infant mortality over the last 150 years. Even without going so far back in time, comparing mortality at the time when I was born (1987) with present, the result is impressive. In this period, the reduction of mortality at childbirth is 78%.
Figure 4. Mortality over year (Country: Italy; Age: 0 year). Source: [1]
So, can we finally say that the bathtub curve does not reproduce human life? Only if countries invest in healthcare regardless of people’s income, race and national origin. Otherwise, terrible trends like those shown by Italy before 1900 could be observed (Figure 5). For example, in the year 1872, mortality of children was dramatically high and not just at the childbirth. It took more than 5 years to reach the plateau.
Figure 5. Mortality over age (Country: Italy; Year: 1872; Age interval 0 – 95 years); Color: number of deaths. Size: number of alive individuals. Source: [1].
What about using the concept of the bathtub curve for modelling the failure rate of products? Is it still valid? For new products or new production processes we might see an increase of the failure rate at the beginning of the product life due to lack of design and process optimization. Defective products are delivered to the customer who immediately calls service for replacement. Companies track early failures with specific KPIs such as the Service Call Rate (SCR). SCR indicates the number of products for which an intervention by the service was needed in a specific time range.
Comparing the mortality curves in 1872 (Fig. 5) and 2020 (Fig. 2), we can derive the main tasks for reliability engineers:
- Apply risk mitigation approaches such as FMEA and DRBFM in order to minimize failure rate at beginning of the product life.
- Minimize failure rate in the central part of the curve implementing countermeasures aimed to limit the impact of unpredictable external events on reliability.
- Design the product so that the change from constant to increasing failure rate fulfills customer’s expectations.
Concluding, infant mortality can and must be minimized. Ageing can just be delayed. With these two principles in mind, we can design products lasting the time expected by the consumers without creating a lot of waste due to defective products.
[1] HMD. Human Mortality Database. Max Planck Institute for Demographic Research (Germany), University of California, Berkeley (USA), and French Institute for Demographic Studies (France). Available at www.mortality.org (data downloaded on 12/11/2023).
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