15.2 Process Capability Analysis

Process capability analysis helps businesses ensure their products meet quality standards. It involves calculating indices like Cp and Cpk, which measure how well a process meets specifications. These indices compare the spread of data to the allowable range.

Interpreting capability indices is crucial for quality control. Values above 1 indicate a capable process, while values below 1 suggest improvements are needed. The analysis assumes normal distribution, but methods exist for non-normal data too.

Process Capability Analysis

Calculation of capability indices

• Process capability indices quantify how well a process meets specifications
  • Cp: Process Capability Index measures the potential capability of a process to meet specifications
    • Formula: $Cp = \frac{USL - LSL}{6\sigma}$
      • USL: Upper Specification Limit (maximum allowable value)
      • LSL: Lower Specification Limit (minimum allowable value)
      • $\sigma$: Process standard deviation (measure of variability)
  • Cpk: Process Capability Index adjusted for process centering measures the actual capability of a process to meet specifications
    • Formula: $Cpk = \min(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma})$
      • $\mu$: Process mean (average value)
    • Cpk considers the process center relative to the specification limits (how well the process is centered between USL and LSL)

Interpretation of capability indices

• Cp and Cpk values indicate the process capability
  • Cp and Cpk > 1: Process is capable of meeting specifications (producing parts within the allowable range)
    • Higher values indicate greater capability (more room for error)
  • Cp and Cpk < 1: Process is not capable of meeting specifications (producing out-of-spec parts)
  • Cp > Cpk: Process is not centered within the specification limits (off-center, closer to one limit)
  • Cp = Cpk: Process is centered within the specification limits (equal room for error on both sides)
• Minimum acceptable values for Cp and Cpk depend on the industry and criticality of the product
  • Typical minimum values range from 1.33 (less critical) to 2.00 (highly critical, like aerospace)

Process capability for normal distributions

• Process capability analysis assumes that the data follows a normal distribution (bell-shaped curve)
• Steps to determine process capability for normally distributed data:
  1. Collect data on the process characteristic of interest (measurements, dimensions)
  2. Test the data for normality using graphical methods or statistical tests
     • Graphical methods: histogram (shape), normal probability plot (straight line)
     • Statistical tests: Anderson-Darling, Shapiro-Wilk (p-value > 0.05 indicates normality)
  3. If the data is normally distributed, calculate the process mean ($\mu$) and standard deviation ($\sigma$)
  4. Calculate Cp and Cpk using the formulas provided
  5. Interpret the results based on the calculated values and the specific requirements of the process

Process capability for non-normal data

• If the data is not normally distributed, process capability analysis can still be performed with some adjustments
• Methods for assessing process capability with non-normal data:
  • Transform the data to achieve normality
    • Box-Cox transformation (power transformation to stabilize variance and make data more normal)
    • After transformation, follow the steps for normally distributed data
  • Use non-parametric methods, such as percentile-based indices
    • Pp: Process Performance Index
      • Formula: $Pp = \frac{USL - LSL}{6s}$, where $s$ is the sample standard deviation
    • Ppk: Process Performance Index adjusted for process centering
      • Formula: $Ppk = \min(\frac{USL - \tilde{x}}{3s}, \frac{\tilde{x} - LSL}{3s})$, where $\tilde{x}$ is the sample median
  • Use distribution-specific capability indices
    • Cpm for Weibull distribution (used for reliability analysis)
• Interpret the results based on the chosen method and the specific requirements of the process