ModelRisks Six Sigma functions

The Industrial version of ModelRisk incorporates a set of functions that will return standard Six Sigma performance measurements for the random variable in a spreadsheet cell. The results are provided only after a simulation run has been completed. If no simulation has been completed yet, the functions return the message “No simulation results”.

Form of the Six Sigma functions

Each function takes the form:

VoseSixSigmaFunction(OutputCell, Parameter1, Parameter2, …, SimulationID)

where:

OutputCell is a cell reference (like ‘A1’ or a cell name) for which the Six Sigma metric is to be calculated;

Parameter1, Parameter2, … are parameters specific to the metric; and

SimulationID is an optional parameter used when running multiple simulations.

All of the Six Sigma functions use a sub-set of the following parameters:

• Lower Limit

• Upper Limit

• Target

• Long Term Shift

• Number of Standard Deviations

The functions make frequent use of the following:

m - the mean of the values generated in OutputCell;

s  - the standard deviation of the values generated in OutputCell;

Φ(∙) - the standard normal cumulative distribution function; and

Φ(∙) - the standard normal inverse cumulative distribution function.

Function list

The functions, in alphabetical order, are:

VoseSixSigmaCp(OutputCell, LowerLimit, UpperLimit, SimulationID)

This function calculates the ‘Process Capability’ Cp defined as:

VoseSixSigmaCpk(OutputCell, LowerLimit, UpperLimit, SimulationID)

This function calculates the ‘Process Capability Index’ Cpk defined as:

VoseSixSigmaCpkLower(OutputCell, LowerLimit, SimulationID)

This function calculates the ‘One-Sided Capability Index’ based on the lower specification limit and is defined as:

VoseSixSigmaCpkUpper(OutputCell, UpperLimit, SimulationID)

This function calculates the ‘One-Sided Capability Index’ based on the upper specification limit and is defined as:

VoseSixSigmaCpm(OutputCell, LowerLimit, UpperLimit, Target, SimulationID)

This function calculates the ‘Taguchi Capability Index’ defined as:

VoseSixSigmaDefectPPM(OutputCell, LowerLimit, UpperLimit, SimulationID)

This function calculates the ‘Defective Parts Per Million’ defined as:

VoseSixSigmaDefectShiftPPM(OutputCell, LowerLimit, UpperLimit, LongTermShift, SimulationID)

This function calculates the ‘Defective Parts Per Million’ with a shift and is defined as:

VoseSixSigmaDefectShiftPPMLower(OutputCell, LowerLimit, LongTermShift, SimulationID)

This function calculates the ‘Defective Parts Per Million’ below the lower specification limit with a shift and is defined as:

VoseSixSigmaDefectShiftPPMUpper(OutputCell, UpperLimit, LongTermShift, SimulationID)

This function calculates the ‘Defective Parts Per Million’ above the upper specification limit with a shift and is defined as:

VoseSixSigmaK(OutputCell, LowerLimit, UpperLimit, SimulationID)

This function calculates the Six Sigma ‘Measure of Process Center’ defined as:

VoseSixSigmaLowerBound(OutputCell, NumberOfStandardDeviations, SimulationID)

This function calculates the ‘Lower Bound’ as a specific number of standard deviations below the mean and is defined as:

VoseSixSigmaProbDefectShift(OutputCell, LowerLimit, UpperLimit, LongTermShift, SimulationID)

This function calculates the ‘Probability of Defect’ outside LowerLimit and UpperLimit with a shift and is defined as:

VoseSixSigmaProbDefectShiftLower(OutputCell, LowerLimit, LongTermShift, SimulationID)

This function calculates the ‘Probability of Defect’ below the LowerLimit with a shift and is defined as:

VoseSixSigmaProbDefectShiftUpper(OutputCell, UpperLimit, LongTermShift, SimulationID)

This function calculates the ‘Probability of Defect’ above the UpperLimit with a shift and is defined as:

VoseSixSigmaSigmaLevel(OutputCell, LowerLimit, UpperLimit, LongTermShift, SimulationID)

This function calculates the ‘Process Sigma Level’ with a shift and is defined as:

VoseSixSigmaUpperBound(OutputCell, NumberOfStandardDeviations, SimulationID)

This function calculates the ‘Upper Bound’ as a specific number of standard deviations above the mean and is defined as:

VoseSixSigmaYield(OutputCell, LowerLimit, UpperLimit, LongTermShift, SimulationID)

This function calculates the Six Sigma ‘Yield’ with a shift, i.e. the fraction of the process that is free of defects, and is defined as:

VoseSixSigmaZlower(OutputCell, LowerLimit, SimulationID)

This function calculates the number of standard deviations of the process that LowerLimit is below the mean of the process and is defined as:

VoseSixSigmaZmin(OutputCell, LowerLimit, UpperLimit, SimulationID)

This function calculates the minimum of Zlower and Zupper and is defined as:

VoseSixSigmaZupper(OutputCell, UpperLimit, SimulationID)

This function calculates the number of standard deviations of the process that UpperLimit is above the mean of the process and is defined as:

Assumptions

The Six Sigma functions are based on the assumption that samples generated by the OutputCell are approximately normally distributed. You can check this visually by running a simulation with OutputCell named as a simulation output and viewing the result in histogram form. Alternatively, you can check this numerically within the model by writing two formulae in the spreadsheet:

=VoseSimSkewness(OutputCell, SimulationID)

=VoseSimKurtosis(OutputCell, SimulationID)

where it is only necessary to specify SimulationID if it is also being used in the VoseSixSigma function. These functions will return values close to 0 and 3 respectively at the end of a simulation run if samples generated by the OutputCell are approximately normal and provided one has run a sufficiently large number of samples (1000 or so should be enough).

The most important vulnerability of the normality assumption is that it implies that distances about the mean of the distribution measured in standard deviations have a consistent probabilistic interpretation. For example, people with some limited knowledge of statistics can often quote that a range +/- two standard deviations about the mean contains 95% of the distribution but forget that this rule of thumb only applies for the normal distribution. Tchebysheff’s Rule expresses the more general, and far weaker, reality.

An additional assumption that is made in using the LongTermShift parameter is that the process mean will drift by this number of standard deviations over time, but that the standard deviation itself will remain unchanged.