Jun 04 2013

# The Power Law

What do the recent tragic events in Boston and Sichuan in China have in common with the Cost of Quality (CoQ) at a manufacturing company? They all follow a statistical rule called the Power Law. Simply put, plotting the logarithm of the magnitude of an event against the logarithm of the probability that event will occur results in a straight line with a negative slope relationship. In the case of a terrorist event, the magnitude is often sadly measured by the number of casualties. Considerable research has shown that historical attacks have obeyed this law. In case of an earthquake, the relationship between magnitude and the probability of occurrence at a given time and region is described by the Gutenberg-Richter law, as the same type of Power Law distribution.

How are these related to CoQ? Figure 1 is an analysis of the warranty claim data of an Automotive Tier 1 supplier within the period of one year.

This data set indicates that larger claims (above \$10,000) follow the Power Law very well. The circled area shows smaller claims, likely indicating smaller sized defects that skipped the system, have a lower occurrence. Assuming these data are representative patterns, they are showing that the power constant is approximately equal to -1. This means that the occurrence of above \$100K claim is about 100 cases in a year, and those above \$1M claim are about 10 cases a year, those above \$10M claim are about once a year.

Studies on terror events all over the world have found that very similar relationships exist between casualty and the probability of occurrence. In fact, the power constant for terrorism is found to be about -2.5. In other words, the occurrence of a 200 casualty event such as the Boston bombing is approximately 10^2.5 = 316 times more likely than a casualty 2,000 and above event, such as the Sept 11 attack on the World Trade Center in New York.

# Why Do Quality Events Exhibit Power Law Behavior?

There are two major reasons why quality events follow the behavior suggested by the Power Law, both are tied to the networked nature of the supply chain.

1. Interdependency – Supply chain elements are highly interdependent. For example, I have experienced an instance where a small crack was discovered in a critical glass furnace at a remote factory in Japan. This turned out to be a devastating event. As a sole producer of glass substrate for storage media in magnetic drives, the crack disrupted the entire server and PC supply chain for days, costing manufacturers millions of dollars.
2. Positive feedback – Public awareness can distort the actual cost or danger of a defect. A now “famous” example is Toyota‘s “unintended acceleration” case, which ended up being settled by Toyota for \$1.1 billion dollars (story here). When Toyota identified a potential root cause to be floor mats made by certain suppliers, the number of reports increased exponentially and the publicity of the case became a media frenzy.

# Six Sigma and the Power Law

The Power Law behavior of CoQ offers important insights into how quality executives should deal with important quality events. This concept, however, might be counter-intuitive to those practicing Six Sigma, which has a foundation based on a normal distribution or the Bell curve. CoQ, however, observes the Power Law distribution, not the normal distribution. Here are some of the major differences:

• It is meaningless to talk about the average size of a warranty claim – the shape of the Power distribution has no mean or standard deviation (sigma)
• The most important data points are outliers – in our data set, the top 10 claims among all 412 contributed to over 50% of total warranty cost, but these large claims were outliners typically ignored by Six Sigma methodology
• Black Swan events occur – The theory was developed by Nassim Nicholas Taleb to describe highly unlikely events that determine the course of human history. According to extrapolation of the above data set, warranty claims that cost more than a billion dollars could occur about every century. Such an event, though rare, could easily lead a company to bankruptcy.

# The Power Law Strategy

Just like security gates alone cannot eliminate a terrorist event, implementing a traditional quality management system might not be enough to prevent quality defects when the Power Law is in effect.  A different strategy is needed that involves three major steps. The first is to enable detailed product traceability across the interdependency of the supply chain. Once this has been established, the second step would be to build an early warning system, perhaps based on “big data” enterprise manufacturing intelligence, which establishes granular traceability across the value chain. The third step is to tie such warning signals to a series of actions that involves the PDCA (Plan Do Check Act) cycle as well as a containment strategy.

These preventative measures could significantly lower the probability of an isolated event escalating into becoming catastrophic by self-reinforcing cycles of positive feedback. It is worth noting that traditional ROI analysis on annual return can rarely be used to justify investment to implement such strategies and solutions. When dealing with the potential catastrophic effect of the Power Law, executive leadership is required to set organizational direction. Seeking average annual return of such investment just does not make sense in a world with Black Swan events.