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**Published: **10/04/2010

In my August column, “How to Turn Capability Indexes Into Dollars,” and my September column, “The Gaps Between Performance and Potential,” I showed how to convert capability indexes into the *effective cost of production and use* (ECP&U), and how to use these costs to quantify the payback for various improvement options. In this column, I will show how the ECP&U defines what is required to operate in the zone of economic production.

Figure 1 shows how the *ECP&U* varies with the capability ratio, *C** _{p}*, for the case where all nonconforming product is scrapped. The first curve is labeled with a zero and represents the case where the process is perfectly centered within the specifications. The region to the left of this curve is the impossible zone and the region to the right is for situations where the process is not centered within the specifications.

The second curve in figure 1 is labeled 0.9. This curve represents the relationship between the ECP&U and the capability ratio when the process is 0.9-sigma off center. The third curve, labeled 1.5, represents the relationship between the ECP&U and the capability ratio when the process is 1.5-sigma off center. In a similar manner, the remaining curves show what happens when the process is 3.0-sigma off center, 4.5-sigma off center, etc. Each of these curves is a direct result of the rigorous mathematical argument presented in my August column. As we move to the right along any one of these curves, we see that they all flatten out as they approach the limiting value of 100 percent for the ECP&U.

This nonlinear behavior identifies a diminishing return. As the ECP&U gets closer to 100 percent, the increase in the capability ratio will need to be greater in order to see further reductions in the ECP&U value. Thus, it is reasonable to acknowledge this diminishing return by defining some region close to 100 percent as the zone where further improvements are no longer required. This will be the zone of economic production. I have chosen to draw this line at 110 percent in figure 2. Some might prefer to draw this line higher or lower, but to me this value seems to be consistent with how we often act in practice.

From figure 2 we see that a process that is perfectly centered within the specifications will enter the zone of economic production when the capability ratio is about 1.05. A process that is operated 0.9-sigma off center will enter the zone of economic production when the capability ratio is about 1.4. A process that is operated 1.5-sigma off center will enter the zone of economic production when the capability ratio is about 1.9.

All of the results above can be computed with mathematical precision. Unfortunately, our processes are seldom so exact and precise. Processes change, and we have to face the consequences of these changes. So how do the mathematical results given above translate into practice? To answer this, we need to consider how our ability to detect a change will depend upon the size of that change. Published tables of the power functions of various statistical procedures computed by this author and others have all shown that smaller shifts are always harder to detect than larger shifts. For a simple process behavior chart we can summarize this result as shown in figure 3.

The vertical scale in figure 3 shows the probability of detecting a shift in the process location within 10 observations following that shift. The horizontal scale shows the size of the process shift. Detection Rule One is a point falling outside the three-sigma limits of the X chart.

When the shift size is zero, there is no shift and we want a small probability of a false alarm, as shown in figure 3. In the region marked A, the shift size is less than 0.9 sigma. There will be a small probability of detecting a process shift in this region. Shifts of this size will be hard to detect in a timely manner.

In the region B, the size of the shift is between 0.9 sigma and 1.5 sigma. Here, the probability of detecting a shift triples over what it was in region A. We will be slow to detect our process changes in this region.

In the region marked C, the size of the shift is between 1.5 and 3.0 sigma. Here, the probability of detecting our process shift rapidly increases up to a virtual certainty. Shifts of this size and larger will be rapidly and reliably detected. Figure 4 shows how the regions of figure 3 relate to the curves of figures 1 and 2.

No real process is ever exactly the same over time. Things change, and these changes will result in shifts in the product stream. Even when a process is operated predictably, the many common causes of routine variation will buffet the process and shift the process stream around slightly. When we make allowance for this buffeting and our inability to detect small shifts in a timely manner, we have to conclude that when a process is operated predictably and on target, it will generally inhabit region A, with occasional excursions into region B. Therefore, when a process is operated predictably and on target, it is safe to say that the process average will be within 1.5 sigma(X) of the target most of the time. When larger excursions occur, they will generally be detected in a timely manner and the process can be returned to operating on target.

But what happens when a process is not operated predictably? Part of the smoke and mirror computations of various Six Sigma programs has been a completely unsupported assumption that an unpredictable process will not shift around more than ±1.5 *s*igma(X*)*. Real data completely refute this assumption.

Even a well-behaved process that is operated predictably and on target will experience occasional excursions that are substantially greater than 1.5 sigma(X)*.* A good example of this is the Tokai Rika Cigar Lighter process used my August column. The complete chart is shown in chapter seven of *Understanding Statistical Process Control, Third Edition* (SPC Press, 2010). Even though this is one of the most predictable processes you will ever see, the complete chart shows several excursions in the 2 to 3 sigma(X) range, and one excursion of 6.7 sigma(X)*. *If a reasonably predictable process can suffer repeated excursions at magnitudes of two sigma, three sigma, and even six sigma, what hope would a truly unpredictable process have of always staying within 1.5 sigma(X) of a target value?

To answer this question, consider the batch weight data for one week’s production, which were given in my September column. The XmR chart for these data is shown in figure 5. There we see a process that shifts around from 8 sigma(X) below the average to 9 sigma(X) above the average within a 24-hour period. Moreover, on Friday afternoon, this process oscillates from more than 7 sigma(X) above the average to more than 5 sigma(X) below the average *for several successive batches*.

What part of the word unpredictable is not clear? There is simply *no way* to place a bound on the size of excursions that may be seen when a process is operated unpredictably*. *Hence, all computations and all arguments that are based on the assumption that an unpredictable process will not shift around more than ±1.5 sigma(X) are completely spurious.

When we combine the facts of life and the mathematical results given above, we end up with the picture shown in figure 6. We simply cannot *begin* to operate within the zone of economic production until we have a process with a capability ratio that exceeds 1.10 that is operated predictably and on target.

Making allowance for those hard-to-detect process excursions into region A, we cannot be *assured* of operation in the zone of economic production until we have a process with a capability ratio in excess of 1.50 that is operated predictably and on target.

Making allowance for occasional undetected excursions into region B, we cannot *guarantee* operation in the zone of economic production until we have a process with a capability ratio in excess of 1.90 that is operated predictably and on target.

Thus, with the rigorous approach to making sense of our capability indexes that I have outlined in this and my two previous columns, we find that the minimum requirements for assured operation within the zone of economic production are:

1. You will need to operate your process predictably.

2. You will need to operate your process on target.

3. You will need to have a capability ratio in excess of 1.50.

Operating a process predictably is a necessity simply because this is the only way to get your process to operate up to its full potential. To operate a process predictably is to operate that process with minimum variance. Unpredictable operation will inevitably increase the variation, which will lower the capability indexes and increase the *effective cost of production and use*.

Operating on target is a necessity simply because, regardless of how large your capability ratio might be, operating off target can take you out of the zone of economic production.

Therefore, any attempt to define economic operation based on capability indexes alone is not sufficient. No matter how we might dress our capability indexes up as “process sigma-levels” or talk about the “parts-per-million defective” or the “defects per million,” we still cannot define economic operation without reference to predictable and on-target operation. Figure 6 provides the first and only rigorous definition of what it takes to operate in the “six-sigma zone.” It is what has been missing from many Six Sigma programs.

As I have shown in this and my August and September columns, the *effective cost of production and use* allows us to convert capability indexes and performance indexes into the language of management. Moreover, when we follow the mathematics behind this conversion to its logical conclusion we end up with a definition of those conditions needed to operate economically.

• A good capability ratio is necessary to operate economically, but a good capability ratio is not sufficient to guarantee economical operation.

• A predictable process is necessary to operate economically, but a predictable process is not sufficient to guarantee economical operation.

• A way of operating a process on target is necessary to operate economically, but operating on target is not sufficient to guarantee economical operation.

All three of these necessary conditions must be present in order to guarantee economical operation. Do not focus on one condition while ignoring the other two.

This article and my two previous columns were excerpted from my new book *Reducing Production Costs*, and are presented here with the permission of SPC Press. In the book, you will find complete explanations as well as tables for converting capability indexes into *effective costs of production and use* or *excess costs of production*, or *excess costs of use*.

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