MBA management

Statistic topics:

Variation


Variation is the basis of Six Sigma. It’s defined as the fluctuation in the output of a process. Sometimes it’s called noise but it can be very expensive noise. It is a truism of Six Sigma that every repeatable process exhibits variation.

Any improvement of any process should reduce variation, so that the process can more consistently meet expectations of either internal customers (such as the employees responsible for subsequent processes) or external customers. In Six Sigma, teams must approach all projects from the perspective of understanding the variation in the inputs of a process, controlling them, and reducing or eliminating the defects.

But in order to reduce variation, we must be able to measure it.

So, how do we measure variation? There are several ways, each with advantages and disadvantages and disadvantages. Let’s take a very simple example to show how these methods work.

Your company produces widgets. There are two lines that assemble the components, A and B. You want to reduce the variation in assembly times so that the workers who package the components can work most efficiently not waiting for finished widgets, not falling behind, and not being forced to work so quickly that they make mistakes.

The first step is to track assembly times. You gather the following data;

Process A: 3.7, 6.5, 3.2, 3.2, 5.7, 7.4, 5.7, 7.7, 4.2, 2.9

Process B: 4.7, 5.3, 4.7, 5.4, 4.7, 4.4, 4.7, 5.8, 4.2, 5.7

Now, what do those figures mean? We can compare the two processes in several ways, using common statistical concepts. (In reality, You would be collecting much more data hundreds or even thousands of measurements.)

Mean, Median, Mode and Range


If we use the mean (the average, also known as the arithmetic mean or the simple average), we find that line A average 5.02 minutes and line B average 4.96 minutes. By that measure, the times measured for the two processes would be very close. But we don’t know which process varies more.

We can calculate the median value (the midpoint in our range of data).For A it’s 4.95 and for B it’s 4.7. By that measure, again, the two processes would be close, although not quite as close as close as when we use the mean.

We can also calculate the mode (the value that occurs most often). For A, it would be either 3.2 (two times) or 5.7 (two times) and for B it’s 4.7 (four times) .So, what does the mode tell us? Not much.

Based on these three measurements, what do we know about the variations in our two widget assembly lines? How do they compare? Which statistical concept best represents the variation in each line?

We quickly come to the conclusion that we don’t know much about our variation at this point. Fortunately, there are two more concepts that we can use: range and standard deviation.

Range is easy to calculate: it’s simply the spread of values, the difference between the highest and the lowest. The range for A is 4.8 (7.7-2.9) and the range for B is 1.6 (5.8-4.2). Now, that measure shows a considerable discrepancy between A and B. The variation in process A is much greater than in process B—at least if we use the range as our measure.

But range is a rough measure because it uses only maximum and minimum values. It seems to work OK in this case. But what if we had a third process, for which we measured the following values?

Process C : 3.2, 6.5, 3.4, 6.4, 6.5, 3.3, 3.7, 6.4, 6.5, 3.5

The range for this set of values is 3.3 (6.5-3.2), which suggests that there’s less variation in process C than in process A (range=4.8) and more variation than in process B (range =1.6).But common sense tells us that the values for C vary more than the values for A, even if less widely, and that vary only a little more than the values for B. But that’s just common sense: we need to quantify the variation for each process.

We need another concept, something more accurate than range for calculating and representing process variation. That concept is standard deviation.

Standard Deviation


The most accurate measure for quantifying variation is standard deviation, which is an indicator of the degree of variation in a set of measurements or a process calculated by measuring the average spread of the data around the mean. Calculating the standard deviation is more complicated than calculating the mean, the median, the mode, or the range, but it provides a far more accurate quantification of variation.

As with many formulas, the formula for the standard deviation seems more difficult because it uses symbols. But you already know one of those symbols, sigma (the Greek letter) is used to indicate the sum of some values. The third symbol is X and use it to represent the arithmetic mean, which, as we mentioned above, most of us know more commonly as the average. The final symbol in the equation for standard deviation is n, which stands for the number of times we measure something.

DISTRIBUTIONS AND CURVES


If we plot the values for a process and we have a large number of values we’ll likely find that the distribution of values forms some variant of a bell-shaped curve high in the middle around the mean and tapering off on both sides more or less symmetrically.

APPLYING STATISTICS


Now that we’ve covered the basic statistics, let’s apply those concepts to our situation of assembling widgets. Your goal is to reduce the variation in your widget assembly processes. So you first need to determine how much variation is acceptable to your customers, the employees in the widget packaging group. Then, you use those values to set a lower specification limit (LSL) and an upper specification limit (USL). These are the lower and upper boundaries within which your assembly processes must operate, the values beyond which the performance of the processes is unacceptable. (The spec limits are sometimes also called the upper tolerance limit UTL and the lower tolerance limit LTL.)

If an aspect of a process, a product, or a service that customers consider critical to quality exceeds either specification limit, it‘s considered a defect. (A defect is measurable characteristic of the process or its output that is not within the acceptable and expected customer limits, i.e., not conforming to specifications.)

Causes of Variation


There are two categories of causes of variation common and special. A common cause is any source of unacceptable variation that is part of the random variation inherent in the process, when unknown factors result in a steady but random distribution of output around the average of the data. For this reason, it is also known as a random cause; common-cause variation is also called random variation, inherent variation, noise, non-controllable variation, and within-group variation, and within –group variation. Common cause variation is the result of many factors that are usually part of the process, acting at random, each independently many X’s, each with a small impact. Common cause variation is the standard deviation of the distribution. It is a measure of the process potential, how well the process can perform without any special cause variation.

A special cause is any source of unacceptable variation that lies outside the process. A special cause is sometimes also called an assignable cause. Special-cause variation is the result of a nonrandom event, intermittent and unpredictable, and the changes in the process output are occasional and unexpected a nonrandom distribution of output, a shift of a sample mean from the target. Also called assignable variation or exceptional variation, special-cause variation is a shift in output caused by a specific factor, such as environmental conditions or process input parameters few X’s with a big impact.

Here’s another way of expressing the difference between common causes and special causes. Variation from common causes is natural and variation from special causes is unnatural: on control charts, natural variation is variation between the UCL and the LCL and unnatural variation is variation above the UCL or below the LCL.

Why is it important to distinguish between the two? Because each category requires a different approach, different strategies. Common causes require a long-term strategy of process management to identify, understand, and reduce them. Special causes of variation require immediate action.

It is estimated that approximately 94 percent of problems are caused by common-cause variation. The less well-defined a process is, the more vulnerable it is random variation and the more defects result. However if there is any special cause of variation, the Six Sigma team must first eliminate it before working to stabilize the process, bring it into statistical control, and then improve it. If there are only common cause of variation, the output of a process forms a distribution that is stable over time.

If there are special cause of variation, a process cannot be stable over time. The team must first identify any special causes and eliminate them to bring the process into statistical control.

Process Capability


In addition to the lower and upper specification limits, there’s another pair of limits that should be plotted for any process—the lower control limit (LML) and the upper control limit (UCL). These values mark the minimum and maximum inherent limits of the process, based on data collected from the process .If the control limits are within the specification limits or align with them, then the process is considered to be capable of meeting the specification. If either or both of the control limits are outside the specification limits, then the process is considered incapable of meeting the specification.

Process capability is a statistical measure of inherent variation for a given characteristic in a stable process. In other words, it‘s the measure of the ability of a process to produce outputs that meet the specifications. It expresses the range of the natural variation as determined by common causes.

If you place the control limits on a process capability curve and the LCL is three sigma to the left of center and the UCL is there sigma to the right of center, the process capability is three sigma. The area of the curve between the two control limits represents the percentage of products or services that met the specifications- 99.73 percent. The area outside the control limits is the percentage that is out of spec-0.27 percentage. That percentage equates to 2,700 DPMO.

Shift


That might seem simple enough, but then statisticians add the complication of a shift. They have found that processes tend to shift from center over time. A process that has been improved is generally at its best immediately after the improvement. Data for the project is collected over a period of months; discrepancies are likely when it’s collected over years. Also, because the project tam should exclude special cause variation in order to focus on common cause variation, the short term data will usually indicate a higher process capability than the long term data, which is likely to be affected by special cause variation. Consequently, a process that’s really six sigma immediately after the project team has improved it would likely suffer some losses over time.

Controlling a process so that it remains on target in the longer term can be difficult. That’s why, in discussing process capability, it is common to distinguish between short term capability and long term capability. This distinction can seem complicated, but it’s simple idea.

The standard deviation and sigma level of a process are considered short-term values because the data is collected by the project team over a period of months and it shows the results of common cause variation only. In contrast, data collected over years will show the results of common cause variation and special-cause variation.

The Six Sigma pioneers at Motorola recognized this difficulty and adjusted for it. Thy calculated that, under the worst conditions, the performance of a process might suffer a degradation of as much as 1.5 sigma. So, thy decided to allow in their calculations for this worst case scenario and, to compensate, thy corrected in advance for a possible shift of 1.5 sigma. If you consult a standard normal distribution table, you’ll find that 6 sigma actually equates to about 2 defects per billion opportunities (0.002 DPMO). The figure usually used for 6 sigma, as in this book Figure 4, defects peer million opportunities really equates to 4.5 sigma. In other words, although statistical tables show 4, DPMO when the distribution between the man and the closest specification limit is 4.5 sigma, the target is raised to 6,0 sigma to compensate for process shifts over time and still achieve a maximum of only 3.4 DPMO.

Shifts in process averages will not always be as great as 1.5 sigma, but that figure allows enough leeway to ensure that the process will met the goal of 3.4 DPMO over the long term.

There’s nothing sacred about 1.5 sigma. Some practitioners advocate a lower figure, it’s possible to use a different figure when setting your sigma target for a process, if you have data experiences with similar processes. However, it’s simpler and safer to use the figures that have been adjusted to accommodate shifts of as much as 1.5 sigma. So, when a process that’s 3 sigma shifts 1.5 sigma from center, the capability declines from 99.73 percent to only 93.32 percent, as 6.68 percent of the area is now outside the control limits. This equates to 66,807 DPMO.

The goal of Six Sigma is to reduce the standard deviation of your process variation to the point that 6 sigma (six standard deviations) can fit within your specification limits. At that level of process capability, a shift of 1.5 sigma from center results in a defect rate of only 3.4 DPMO.

ROLLED THROUGHPUT YIELD


Six sigma is a performance target that applies to a single critical to quality characteristic (CTQ), not to an entire process, product, or service. A process capability of 6 sigma means that, on average, there will be only .4 defects in a critical-to-quality characteristic per million opportunities for a defect in this characteristic. The more complex a process, the more likely a defect will occur at some point.

As complicated as the concepts and formulas might b for sigma and process capability, they are simple calculations, actually, because the issue of variation is simple when we talk about a process as being simple. The sigma levels and DPMOs that we’ve been discussing apply to only on step and on specification. That is first-time yield (FYY) the number of good units coming out of a process or a step divided by the number of total units going into the process or step. (Yield is the percentage of units coming out of a process free of defects.)

However, virtually all processes consist of more than one step and involve more than on specification. Because variation is additive, we must consider and calculate cumulative effective effects of variation in each part of a process. In other words, we must think in terms of what could be called the worst-case scenario of variation. In Six Sigma, this is called rolled throughput yield (RTY).

RTY, also known as the rolling effect, is the probability that a single unit can pass through all the steps in a process free of defects. It’s the net result of the effect of all of the steps in a process.

In a five-step process in which each step has a first-time yield of 93.32 percent (three sigma), the process as a whole is not at 93.32 percent good. Instead, we must calculate the RTY as follows:

RTY=Step 1 Yield X Step2 Yield X Step 3 Yield X Step 4 Yield X Step 5 Yield

or

RTY + 0.9332 X 0.9332 X 0.9332 X 0.9332 X 0.9332 = 0.707739852

The RTY of 70.8 percent means that you have a 70.8 percent chance of getting through this five-step process without a defect. That’s just a little over 2 sigma at 69.1 percent, definitely not the 3 sigma (93.32 percent) of each step. In other words, the process would have a 29. 2 percent improvement gap, a 29.2 percent probability of creating a defect far from the 6.68 percent for each of the five steps in the process.

Now, imagine a process that consists of 50 steps, or of 100 steps, or of even more. What sigma levels would you need for each step in order to achieve a RTY that’s even acceptable, let alone impressive? Figure 5 shows some interesting calculations.

• What do the calculations in Figure 5 mean? Here are two examples. If you’re processing loan applications and the process consists of 30 steps, if each step of the process is at a 4 sigma level, only 82.95 percent of your applications will meet specifications.

• If you’re building a widget and the process consists of 100 steps, a process capability of 4 sigma for each step means that only 5 percent of your widgets will meet specifications.

If you didn’t appreciate the huge differences between one sigma level and the next, you might appreciate it now, as that difference is magnified when the process in question consists of many steps. And even if these figures confuse you, it’s easy to understand that the fewer steps in a process, the less the RTY will decline.

For example, if you reduce the loan application process from 30 steps to 25 steps, you raise the success rate of your four-sigma process from 82.95 percent to 85.58 percent. That’s a good increase, considering that you didn’t even do anything to improve any of the steps. That’s money for nothing.

Percent Per Step   RTY, 10 Steps   RTY,20 Steps   RTY, 30 Steps   RTY,100 Steps
69.15%(2 sigma)   2.50 %   0.06 %   0.00 %   0.00 %
93.32% (3 sigma)   50.09 %   25.09 %   0.00 %   0.00 %
99.379% (4 sigma)   93.9607 %                    88.286 %   82.9542 %   53.6367 %
99.9767% (5 sigma)   99.7672 %   99.535 %   99.3034 %   97.69667 %
99.99966%(6 sigma)   99.9966 %   99.9932 %   99.9898 %   99.9660 %

The Cost Of Poor Quality: A Key Metric


Many believe that the cost of improving processes makes reaching six-sigma quality (3.4 defects per million opportunities for a defect) impractical. However , those companies that are striving for six sigma have realized that the net “cost” to reduce defects actually lowers as they approach six sigma, because as they dramatically reduce defects they can also dramatically redirect the resources they currently put into finding and fixing defects. In fact, the highest-quality producer of goods or services is actually the lowest-cost producer. On major reason is the metric called the cost of poor quality (COPQ).

This is one of the key business metric concepts of Six Sigma the cost of doing things wrong. The COPQ represents the visible and less visible costs of all the defects that exist in our processes. Every time we have any result that is not what the customer of a process needs, we consume time and resources to find, fix, and try to prevent these defects such as scrap, rework, inspection, warranty claims, and lost customer loyalty.

The COPQ represents opportunities for Six Sigma. The Six Sigma approach to managing is all about helping you identify ways to reduce the errors and rework that cost you time, money, opportunities, and customers. Six Sigma translates that knowledge into opportunities for business growth. As you improve the capability of your processes and boost your RTY, you not only decrease variation and defects, but also reduce the cost of running your process, sometimes dramatically.

Figure shows the cost in sales for five sigma levels. Note: This chart includes a 1/5 sigma shift, i.e., 6 sigma is 4.5 sigma.

Sigma Level (Process Capability)   Defects Per Million Opportunities   Cost of poor Quality (% of Sales)
2   308,537   30%-40%
3   66,807   20%-30%
4   6,210   15%-20%
5   233   10%-15%
6   34   < 10%


Consider an operation at a three-sigma level the baggage-handling process of a good airline. For every million pieces of baggage that airline handles, there’s a problem with more than 66,000 pieces. For each piece reported missing, the airline workers have to process a report, locate the piece, retrieve it, and deliver it which means time and money wasted just to right the wrong, to correct the defect. And the airline might not be able to recover the confidence of the passenger. When you translate the 6 percent probability gap of missing baggage into monetary terms, the hard cost of this defect can be much higher than 6 percent of the overall cost of handling baggage.

And the COPQ might not be completely obvious. Consider a typical manufacturer. What is the COPQ? There are warranty claim costs reported every month and may be maintenance costs incurred to fix failures in the field. There’s also the cost of scrap and rejects a waste of material, labor, and machine time, utilities, and wear. Add to that the cost of reworking defective parts. Then there are costs that are less obvious.

For example, when a process produces a lot of defective components, the time required to get completed components through the system increases. This increase in cycle time has a cost in terms of additional labor hours to get the work done. There is also the cost of all inspection and testing to try to catch the defects. Then, because some defects somehow escape detection, there’s the cost of lost customers and reduced customer loyalty important but hard to quantify.

To discover the COPQ, as in this example, might take a structured approach, such as the following:

• Internal failure costs resulting from defects found before the customer receives the product or service ( examples: scrap, rework, re inspection, retesting, downgrading, downtime, reduced productivity, failure analysis).

• External failure costs resulting from defects found after the customer receives the product or service ( examples: warranty charges, complaint adjustments, returned material, allowances, replacements, compensation, damage to reputation).

• Appraisal costs of determining the degree of conformance to quality requirements ( examples: inspection, testing, process control, quality audits).

• Prevention costs of minimizing failure and appraisal costs (: example quality planning, policies and procedures, new design reviews, in-process inspections and testing, supplier evaluations, education and training, preventive maintenance)

• Non-value added activities costs of any steps or processes that don’t add value from the customers’ perspective.

This approach can be deployed using a step-by-step technique, or it might us a concurrent method to expedite the process.

The cost of poor quality has a personal side as well. People who work in an organization that has problems with quality might be affected in various ways: poor morale, conflicts, decreased productivity, increased absenteeism, health problems related to stress, burnout, and higher turnover. These human consequences add to the cost of poor quality.

Costs That Are Hidden---and Even Accepted and Allowed

Sometimes it seems that the only people who recognize or suspect that there are “hidden” losses are CEOs and some employees closest to the processes. R they just worrying too much, being too suspicious? After all, surely any loss will be exposed in some kind of reconciliation exercise. Well, may be. But consider the following.

The traditional way to measure process performance is end yield (not rolled throughput yield). Mature processes have predictable end yields and known process step yields. When you know what the individual process step yields are, you also know what their losses are. The losses become predictable.

In traditional unit cost calculations, losses are accounted for by applying a loss factor, such as material losses and reject levels. When these predicable losses (scaled for production forecasts) become factors, they become invisible. From there, they are built into budgets. For example, in a manufacturing environment, a typical loss factor might be that 10 percent of the units produced would be considered as a loss: in a fast food restaurant, it might be an over capacity of labor cost of 20 percent caused by an inability to predict customer demand levels.

Middle managers are responsible for maintaining their budgets. So, not only are the losses invisible, the managers actually allow for them. There is no incentive for middle managers to reduce these losses. They are both hidden and even accepted and allowed. It doesn’t matter what kind of business you are in. Any hidden waste streams in any of your processes ultimately siphon off dollars that should b going to your bottom line.

As you implement the core Six Sigma methodology, you will be armed with the tools that enable you to identify, correct, and control the critical-to –quality (CTO) elements so important to your customers and reduce the cost of poor quality (COPQ). Once you start implementing the method full-time with your black belts and project teams, your projects will start revealing costs that are hidden and returning that money to the company.

Money is generally the most important reason for using Six Sigma processes that are inefficient waste time and other resources and organizations pay a lot for poor quality: the COPQ for traditionally managed organizations has been estimated at between 20 percent and 40 percent of budget. This means that a company that has annual revenues of $1 million has waste and defects in its systems that are costing between $ 200,000 and $ 400,000 in potential benefit. That would seem to be enough of a “burning platform” to convince the people at the top to start Six Sigma immediately.

We conclude this discussion of COPQ with a warning: don’t focus solely on COPQ. Consider the experience of one company that implemented Six Sigma. During the first year of implementation, the teams focused on COPQ alone. Then they realized that it was smart to broaden their focus, because they were finding that focusing on COPQ alone drove them to work on internally focused projects.

That just makes sense, because COPQ is the voice of the business, about saving money, and not the voice of the customers. The company discovered that internally focused projects were not resulting in the breakthrough impact that it was expecting.

Are You Committed to Quality?

Business leaders often say, “We are committed to quality.” That is a standard claim. But what does it mean exactly? How can you verify that? How do you quantify that?

You measure the extent to which goods and services are meeting customer expectations. After all, that’s basic criterion for quality. You measure very aspect of the goods, services, and processes that affect quality. By doing this, you remove opinions and emotions from the equation and replace them with facts and figures that verify or refute that claim of commitment to quality.

Traditional management often operates by the “seat of the pants”-by tradition, impression, reaction to event, gut instincts. The essence of Six Sigma management is to use objective data to make decisions.

Enough Already

A friend who was studying statistics had a young daughter who referred to the subject as “sadistic.”

From the mouths of babes….

At this point, you might not be able to perform all of these calculations and others used in Six Sigma. That’s why there’s statistical software. The industry standard that is used pervasively throughout the Six Sigma world is MINITABTM. You might not understand all the ins and outs of these concepts. That’s why training is essential to any Six Sigma initiative.

What’s important her is that you understand the basic concepts of Six Sigma measurements and better appreciate the importance of establishing merits to track variation so you improve processes. With that quick overview of the essentials, we can leave our imaginary example of widgets and return to the very real situation of your business.

SUMMARY

Because it uses statistical terminology, people tend to believe that Six Sigma is a statistics and measurement program. This is not true. Statistics are used only for interpreting and clarifying data, to turn it into information.

Variation is the basis of Six Sigma. It’s defined as the fluctuation in the output of a process. Every repeatable process exhibits variation. Any improvement of any process should reduce variation so that the process can more consistently meet the expectations of the customers. To reduce variation, it‘s first necessary to be able to measure it. There are several ways, each with advantages and disadvantages mean, median, mode, range, standard deviation, and variance.

If a large number of values for a process are plotted, the distribution of values will generally from some variant of a bell-shaped curve high in the middle around the mean and tapering off on both sides more or less symmetrically. This is considered a normal distribution.

A normal distribution can be described in terms of its mean and its standard deviation. A normal distribution is symmetrical around its man and the mean, the median, and the mode are equal. In a normal distribution, 68 percent of the values lie within one standard deviation (+_1 sigma) of the mean, 95 percent of the values li within two standard deviations (+_ 2 sigma) of the mean, and 99.73 percent of the values li within three standard deviation (+_ 3 sigma) of the mean.

A lower specification limit (LSL) and an upper specification limit (USL) are set as the boundaries within which a process must operate, thee minimum and maximum values beyond which the performance of the process is unacceptable to the customers a defect. The objective is to e reduce the variation in the processes so that 99.99966 percent of the outputs will fall between the LSL and the USL. In other words, the processes will be producing at most 3.4 DPMO.

There are two categories of causes of variation common and special. A common cause is any source of unacceptable variation that is part of the random variation inherent in the process. A special cause is any source of unacceptable variation that lies outside the process. Variation from common causes is natural and variation from special causes is unnatural: on control charts, natural variation is variation between the UCL and the LCL and unnatural variation is variation above the UCL or below the LCL. Each category requires a different approach: special causes of variation require immediate action and common cause require a long-term strategy of process management. If there are special causes of variation, a process cannot be stable over time. The team must first identify and eliminate any special causes to bring the process into statistical control.

The lower control limit (LCL) and the upper control limit (UCL) mark the minimum and maximum inherent limits of a process, based on data collected from the process. If the control limits are within the specification limits or align with them, then the process is considered capable of meeting the specification. If either or both of the control limits are outside the specification limits, then the process is considered incapable of a meeting the specification. Process capability is the ability of a process to produce outputs that meet the specifications.

Over time, processes tend to shift from center, and special causes are likely to occur. In order to compensate for this shift, the Six Sigma pioneers at Motorola calculated for a worst-cause scenario shift of 1.5 sigma. So, sigma capability charts generally allow for this 1.5 sigma difference.

Two indices are most commonly used to measure process capability Cp and Cpk. (Cpk is sometimes called process performance, to distinguish it from Cp, process capability.) Cp is a measure of the width of a distribution of outputs of the process. Cpk tells us the same thing, but also how close the average value is to the target value.

Six Sigma is a performance target that applies to a single critical-to-quality characteristic, not to an entire process, product, or service. It’s actually a measure of first-time yield the number of good units coming out of a process or a step divided by the number of total units going into the process or step. The more complex a process, the more likely a defect will occur at some point. That’s why a better metric is rolled throughput yield the probability that a single unit can pass through all the steps in a process free of defects. It’s the net result of the effect of all of the steps in a process.

The cost of poor quality is one of the key business metric concepts of Six Sigma the cost of doing things wrong, the total of all the costs of all the defects in the processes. COPQ represents opportunities for Six Sigma project teams.
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Review Questions
  • 1. What is variation?
  • 2. Define: Mean, Median, Mode and Range.
  • 3. What is the midpoint in a range of data?
  • 4. Define Standard deviation?
  • 5. Explain in detail about normal distribution?
  • 6. Explain briefly about special cause and common cause?
  • 7. What is first-time yield (FTY) and Rolled throughput yield (RTY)?
  • 8. What is the rolled throughout yield for a three-step process with a 90 percent yield for each step?
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