Once a process is under control the question arises, "to what extent does the long-term performance of the process comply with engineering requirements or managerial goals?" In more general terms, the question is, "how capable is our process (or supplier's process) in terms of producing items within the specification limits?" 

Process capability measurements allow us to summarize the process capability in terms of meaningful percentages and indices. While control charts demonstrate the reproducibility of the process, CpK demonstrates the effectiveness of the process.

CpK = Process Capability with Kurtosis (pointedness)

CpK measurements are almost always beneficial. Some reasons to determine supplier CpK measurements are as follows:

1. To set realistic cost effective part specifications based upon the customer's needs and the costs associated by the supplier at meeting those needs.

2. To understand and or predict hidden supplier costs. Suppliers may not know or hide their natural free capability limits in an effort to keep business. This could mean that unnecessary costs could occur such as a sorting or improvement efforts made to meet process factors that already in actuality meet customer needs.

3. To be pro-active. For example, a CpK made using injection molding barrel temperature measurements may help reveal a faulty heating element about to break before part measurements vary out of spec. The same CpK may help reveal special causes of variation such as doors left open next to the machine where cool incoming air contributes to process variation.  CpK is not a substitute for SPC but when one point in time is desired, CpK is as good tool.  

4. To save time and costs during problem solving. The CpK is a measurement of what a process is fully capable of producing. Problem solving situations when parts are measured out of specification may be easily understood with a quick comparison of the measurements against the process CpK.

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The software package Statistica lists the following edited information regarding Cpk:   Most of the procedures and indices described here were only recently introduced to the US by Ford Motor Company (Kane, 1986). They allow us to summarize the process capability in terms of meaningful percentages and indices.

To calculate a process CpK, follow the instructions for either formula below.

 

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Here is simple formula from net-citizen Keith Laubscher:
Cpk = the smaller value of Cpu or Cpl
Cpu = (Upper spec limit - x)/(3s)
Cpl = (x - lower spec limit)/3s
where x = average, s= sigma (standard deviation from calculator or spreadsheet)

The following page is of Kim's notes from a two day seminar put on by Allied Signal for it's vendors and taught by Keki R. Bhote (author of Supply management: How to Make U.S. Suppliers Competitive, World Class Quality, Beyond Customer Satisfaction to Customer Loyalty: The Key to Greater Profitability, and others).

To calculate the process CpK, one figures out formula #1, then #2, then #3. CpK values of 1.33 or greater are considered to be stable as an "industry standard". This means that the process is contained within four standard deviations of the process specifications. See other notes below the examples.

CpK's for all CRITICAL PRODUCT MEASUREMENTS deemed important by the customer should be produced at the beginning of the initial production to determine expected production costs and general ability to meet the customer's specifications. Then from time to time over the life of the production process, CpK's should be re-generated for comparison and to look for potential problems with tool wear, operators in need of training, etc.

CpK's for CRITICAL PROCESS FACTOR MEASUREMENTS are often as important as they may be critical in controlling important aspects of the product. 

Process width for these formulas are calculated with the assumptions that all data comes from a normal distribution and with an even distribution of error. Therefore, the best quick way to calculate the process width is to multiply the standard deviation of the process by 3 to 4.

When calculating the CpK keep in mind that it is only as accurate as the data used to calculate it. Care should be made as possible to acquire data over the range of all natural variation, considering the following possibilities:

1.      Part variation:  piece-to-piece, raw material lot-to-lot, etc. 

2.      Tooling variation: cavity-to-cavity, tool-to-tool, wear over time, etc.

3.      Human variation: operator-to-operator variation, supervisor-to-supervisor, number of other tasks performed at the same time, etc.

4.      Time variation:  shift to shift, day-to-day, week-to-week, month-to-month, season-to-season, year-to-year variation, etc.

5.      Location variation:  machine to machine, building-to-building, plant-to-plant, state-to-state, country to country, etc.

 

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Computer programs such as Stat Graphics 4.0 (plots shown below) allow for plotting and additional information to be obtained. For example, the following CpK could have come from a pre-production lot made by an anxious to please vendor that enabled the following information to be determined:

1. The lot was sorted at the 3.0 measurement based upon the high side non-normal overall left skewed data truncated at the 3.0 measurement area.

2. The lot received is of a bi-modal distribution typical of sorted lots where the majority of unsure parts got thrown into the good pile and many good parts get measured as bad.

3. The lot was sorted 100% based on the same patterns, some calculations, the lack of out liers, and the bi-nomial distribution.

4. Much can be determined about how the sorting took place (i.e. visual, no-go gage, etc. as opposed to an accurate measurement).

5. Through calculation that 0.0000% of the lot are likely to exceed the more important 3.75 measurement at the 95% confidence level.

6. Through calculation that 8.05% of the lot are likely to exceed the lower 2.4 specification at the 95% confidence level.

7. The lot was not sorted for the lower specification due to the data in that region being normally distributed (see normal distribution plot below).

8. Other details such as the probability of measurements at all points within the specification and what the ideal specification for the vendor would be can easily be calculated.

From this example, it would be beneficial for the customer and the vendor to sit down and work out either a process improvement program and or specification changes so that cost, quality, delivery, and profitability would improve for both parties.

This plot above shows all the data points and how the data at the low end of the specification is normally distributed as it follows the line well.

Summary Statistics for MR092299

Count = 83 Average = 2.67145 Median = 2.64 Mode = Geometric mean = 2.66441 Variance = 0.037654 Standard deviation = 0.194046 Standard error = 0.0212994 Minimum = 2.25 Maximum = 2.98

Range = 0.73 Lower quartile = 2.52 Upper quartile = 2.85 Interquartile range = 0.33 Skewness = -0.138101 Stnd. skewness = -0.513641 Kurtosis = -1.02187 Stnd. kurtosis = -1.90033 Coeff. of variation = 7.26372% Sum = 221.73

Confidence Intervals for MR092299
95.0% confidence interval for mean: 2.67145 +/- 0.0423713 [2.62907,2.71382]
95.0% confidence interval for standard deviation: [0.168354,0.229065]

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