statistical process control

Statistical Process Control

Nicola Mezzetti, Ph.D.

Department of Information Engineering and Computer ScienceUniversity of Trento

[email protected]

A.A. 2014/2015

Nicola Mezzetti, Ph.D. Statistical Process Control

”How much variation should we leave tochance?”

W. A. Shewhart


What is Statistical Process Control?

Statistical Process Control (SPC) is an industry standardmethodology for measuring and controlling quality during themanufacturing process.

Attribute data (measurements) is collected from products as theyare being produced.

By establishing upper and lower limits, variations in theprocesses are monitored before they result in a defectiveproduct,

reducing the amount of material scrap along with direct andindirect labor waste

eliminating the need for final inspection

increasing profitability


History of Statistical Process Control

In 1924 Walter Shewhart developed a simple graphical methodfor plotting collected data with predetermined control limits. Thiswas the first of a growing range of SPC charts, commissioned byBell Laboratories to improve the quality of telephonesmanufactured.

Understanding the causes of variation within an industrial processproved indispensable to identify actions to improve process andoutput. In the 1950’s, with the effective use of SPC, Demingconverted post war Japan into the world leader of manufacturingexcellence.

This approach is increasingly being applied in service industry bythinking of systems as processes. As well as providing a basis forquality improvement, SPC Charts also offer alternative methods ofdisplaying data.


When to use Statistical Process Control?

Are your quality costs really known?

Can current data be used to improve your processes, or is itjust data for the sake of data?

Are the right kinds of data being collected in the right areas?

Are decisions being made based on true data?

Can you easily determine the cause of quality issues?

Do you know when to perform preventative maintenance onmachines?

Can you accurately predict yields and output results?


About Statistical Process Control

Dr. W. Edwards Deming claimed that the majority of variation in aprocess is due to operator over adjustment.

SPC gives operators a tool to determine when a statisticallysignificant change has taken place in the process or when anseemingly significant change is just due to chance causes.


Why do Companies use SPC?

SPC itself will not make improvements.

SPC will give operating personnel a tool to identify when aspecial cause of variation has entered the process so that thespecial cause can be

eliminated (if the special cause has a negative impact on theprocess), or

built into the process (if the special cause has a positiveimpact on the process)

Moreover, SPC allows to

eliminate constant tweaking of the process

identify opportunities for improvement that can lead toreduced variation and processes that are better aimed at theirtarget


Control Charts

Control chart is a tool used to study how a process changesover time.

Measurements are plotted in time order. A control chart alwayshas

a central line for the average

an upper line for the upper control limit1

a lower line for the lower control limit

By comparing current data to these lines, you can drawconclusions about whether the process variation is in control oraffected by special causes of variation.

Control charts for variable data are used in pairs:

The top chart monitors the average (x̄ chart)

The bottom chart monitors the range (R chart)1Control limits are determined by the capability of the process, whereas

specification limits are determined by the customer’s needsNicola Mezzetti, Ph.D. Statistical Process Control

Process Mean Chart

Center Line

x̄ =

∑mi=1

∑nj=1 xij

mn

Control Limits

x̄ ± 3σ

where 99.73% of all datapoints should fall.

Plotted Statistics

x̄i =

∑nj=1 xij

n


Process Variation Chart

Center Line

R̄ =

∑mi=1 max(xij) −min(xij)

m

Upper Control Limit

D4R̄

Lower Control Limit

D3R̄

Plotted Statistics

Ri = max(xij) −min(xij)


How to use Control Charts?

Data is collected from the process, typically in subgroups of 3to 5, and the subgroup mean and range is plotted on the charts.Once a point is plotted the chart is interpreted to determine ifthe process is staying in-control or if the process is out-of-control.

Data that falls within the control limits indicates that everything isoperating as expected.

Any variation within the control limits is likely due to acommon cause, the natural variation that is expected as partof the process.

If data falls outside of the control limits, this indicates that anassignable cause is likely the source of the product variation

something within the process should be changed to fix theissue before defects occur.


Interpreting Control Charts

The most common patterns to watch out for are:

One point outside of the control limits

Eight points in a row on either side of the center line

Eight points in a row trending in the same direction

Cycles or recurring trends


Combining Variability and Mean Charts

The R chart is examined before the x̄ chart:

if the R chart indicates the sample variability is in statisticalcontrol, then the x̄ chart is examined to determine if thesample mean is also in statistical control

if the sample variability is not in statistical control, then theentire process is judged to be not in statistical control


Control Points

Before initiating any SPC program it is necessary to identify whatto count, that is control points. Control points can be related to

Process

Product

Financials


When to Use a Control Chart

Putting spec limits on control charts

Using control charts only to satisfy customer needs

Plotting data after the process has already been run

Using the wrong type of control chart for the process

Not reviewing control charts and how they are used on aregular basis

Not first conducting a process capability study

Not taking random samples from the process, or not using asampling frequency or sample size that captures the variationin the process


When to Use a Control Chart

When controlling ongoing processes by finding and correctingproblems as they occur

When predicting the expected range of outcomes from aprocess

When determining whether a process is stable (in statisticalcontrol)

When analyzing patterns of process variation from specialcauses (non-routine events) or common causes (built into theprocess)

When determining whether your quality improvement projectshould aim to prevent specific problems or to makefundamental changes to the process


The ”Seven Step Process”

The use of a Seven Step Process improves statistical processcontrol. Proper application of SPC will improve process, productand financial results.

Investigation and benchmarking of current process

Identification of appropriate measurable variables

Estimation of available resources and project cost

Estimation of project time line

Application of appropriate statistical techniques

Implementation of corrective action

Statistical monitoring of identified variables


Process Capability Indices

We sometimes talk about Process Capability and define it as the”six sigma” spread.

The term ”Process Capability” means the ability of the processspread to fit within the tolerance spread.

For the comparison of two or more processes, we’ll need some kindof index number to help us compare ”apples to apples.”


Basic Capability Indices

Pp = Process Performance, a simple and straightforwardindicator of process performance.

Ppk = Process Performance Index, adjustment of Pp for theeffect of non-centered distribution.

Cp = Process Capability, a simple and straightforward indicatorof process capability.

Cpk = Process Capability Index, adjustment of Cp for the effectof non-centered distribution.


Process Performance (Pp)

For the Pp index we take a sampling from the process, measure thecharacteristic in question, and calculate the average and standarddeviation using the standard formulas.

The average ±3σ will account for 99.73% of the entire population.So 6σ will essentially represent all of the product.

For the Pp index, we want to see how well this 6σ spread could fitinto the tolerance spread.

Let’s suppose our tolerance is ±5 units and our σ is 1. Thetolerance spread is 10 and the process spread is 6; if we dividethe tolerance by the process we get 1.67 Pp.

Since the process spread is the denominator in this equation, anynumber greater than 1 is good and any number less than 1 is poor.


Considerations on Pp

But is the process centered in the tolerance zone?

Since there is more of the population closer to the average inthe normal distribution, it is important to have the average inthe middle of the tolerance.

The formula for the Pp index does not consider this in anyway; in theory, you could have a good Pp and run 100% scrap.

Think of this index as the potential capability of the process.

If we can center the process in the tolerance zone perfectly, itwill achieve the quality represented by the Pp index.


Process Performance Index Ppk

To keep our values consistent with the Pp values, we’ll only use thesmallest (or minimum) of these two indices for the Ppk value.And we have now arrived at our textbook formula for Ppk

Ppk =min(USL− ¯̄X , ¯̄X − LSL)

3σ


Process Capability: Cp and Cpk

Process Capability is an indicator of the process’ stability.

if a process was stable2 then we can trust it to maintain thePp (or Ppk) value for a longer period of time.

We can think of the Pp (or Ppk) as a snapshot of the processcapability at a given moment (short term capability indices).

If we want to know the capability of a process over the long term,we’d like to know how stable that process is.

The classic test for stability is the control chart.

2A process is stable if it will stay at the same average and standarddeviation for a reasonable period.


Computing Cp and Cpk

For Pp (or Ppk) we estimated the standard deviation using themathematical formula:

σs =

√∑ni=1

∑mj=1 (Xij − ¯̄X )2

nm − 1

The only difference mathematically between the Cp/Cpk and thePp/Ppk is how you estimate the standard deviation.

you use the average Range to estimate the standard deviationby dividing it by the d2 constant factor3

σR̄/d2=

R̄

d2

Cpk would be the long term performance capability index.

3d2 = 2.059Nicola Mezzetti, Ph.D. Statistical Process Control

Potential and Performance Capability Indices

Many people consider the Pp and Cp indices the potentialcapability of the process and the Ppk and Cpk the ”performance”capability indices, so

Pp would be the short term potential capability index

Cpk would be the long term performance capability index

But... why does anyone use the Ppk index anymore?

On a new production part, during the initial phases ofproduction, you have yet to get the control chart establishedenough to enforce stability: the only choice you have is thePpk index based upon the small sample you have at this time!


The ppm equivalent Capability Index (Cpppm)

There are numerous other indices.

One that is actually quite useful is the Cpppm4 which has the

advantage of being comparable in application to the Cpk (orPpk) index.

4the ppm superscript stands for parts per million.Nicola Mezzetti, Ph.D. Statistical Process Control

The ppm equivalent Capability Index: Examples

Let’s make an example! What does having Cpk = 1.00 represent interms of parts per million rejected?

We know that X̄ ± 3σ corresponds to 99.73% of thepopulation

In a million parts, this would equal 997.300 parts, leaving2.700 parts rejected

An equivalent Cpppm of 1.00 should have 2.700 ppm rejected

Having a Cpk = 1.33 is the same as a process spread of ±4σ

We know that X̄ ± 4σ corresponds to 99.9937% of thepopulation

In a million parts, this would equal 999.937 parts, leaving 63parts rejected

An equivalent Cpppm of 1.33 should have 63 ppm rejected


More on the ppm equivalent Capability Index

Suppose your customer wants a capability index Ppk (or Cpk) equalto 1.33 but you only have attribute data. You’ve sorted 55.000parts and found 3 parts defective.

Can you tell your customer you are supplying parts at Cpk of 1.33?

We know that a Cpk of 1.33 is equivalent to 64ppm rejected. Thismeans that in 55.000 parts at a Cpk of 1.33 we should find 3, 52defects.

We found 3, so it sounds like we are doing better than anequivalent Cpk of 1.33 and there is good basis for declaring we aresatisfying the requirement, BUT...

... this is true only if your data is normally distributed5

5All of the capability indices are based upon the normal distribution. If yourdata isn’t normal, then your capability is in doubt.


Benefits of Statistical Process Control

With real-time SPC you can:

Dramatically reduce variability and scrap

Scientifically improve productivity

Reduce costs

Uncover hidden process personalities

Instant reaction to process changes

Make real-time decisions on the shop floor


Measuring the ROI on Statistical Process Control

To measure the ROI on your SPC investment, start by identifyingthe main areas of waste and inefficiency at your facility. Commonareas of waste include:

scrap

rework

over inspection

inefficient data collection

incapable machines and/or processes

paper based quality systems

inefficient production lines


References

Montgomery, D. C., ”Introduction to Statistical ProcessControl, 6th ed.”, Wiley and sons, 2009.

Steiner, S., Abraham, B., MacKay, J., ”Understanding ProcessCapability Indices”, Institute for Improvement of Quality andProductivity, Department of Statistics and Actuarial Science,Universty of Waterloo.


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