Chapter 8: Measuring and Controlling Quality pt2

Measuring and Controlling Quality

Process Capability Indexes

• Definitions:
  • The process capability index measures the relationship between specifications and natural variation.
  • Formula:
  • ( C_p = \frac{USL - LSL}{6 \sigma} )
    • Where USL = Upper Specification Limit, LSL = Lower Specification Limit, and ( \sigma ) = standard deviation.
• Interpretation of C_p:
  • ( C_p > 1 ): Process can meet specifications (process variation < specification range).
  • ( C_p < 1 ): Process is incapable of producing conforming output.

One-Sided Indexes

• Used for process centering:
  • Upper one-sided index: ( C_{pu} = \frac{USL - \mu}{3 \sigma} )
  • Lower one-sided index: ( C_{pl} = \frac{\mu - LSL}{3 \sigma} )
  • Combined: ( C{pk} = \min(C{pl}, C{pu}) = Cp(1 - k) )
  • Where ( k = \frac{2 \times |\mu - T|}{USL - LSL} )
• Confidence Interval for C_pk:
  • ( C{pk} \pm z{\alpha/2}\sqrt{\frac{1}{9n} + \frac{C_{pk}^2}{2(n-2)}} )

Example: Mach4 Tool Co.

• Initial Specification:
  • Dimensions: 0.575 ± 0.007 inch
• Statistics from First Sample:
  • Mean ( \bar{x} = 0.5740 ), Standard deviation ( \sigma = 0.0067 )
  • Calculating Capability:
  1. ( C_p = \frac{0.582 - 0.568}{6 \times 0.0067} = 0.3483 ) (Not capable)
  2. Upper index: ( C_{pu} = \frac{0.582 - 0.574}{3 \times 0.0067} = 0.398 ) (Not capable)
  3. Lower index: ( C_{pl} = \frac{0.574 - 0.568}{3 \times 0.0067} = 0.299 ) (Not capable)
  4. Combined: ( C_{pk} = 0.299 ) (Not capable)
• Post-Adjustment Statistics:
  • Mean ( \bar{x} = 0.5755 ), Standard deviation ( \sigma = 0.0017 )
  • New Capability Calculations:
  1. ( C_p = \frac{0.582 - 0.568}{6 \times 0.0017} = 1.373 ) (Capable)
  2. Upper index: ( C_{pu} = \frac{0.582 - 0.5755}{3 \times 0.0017} = 1.28 ) (Capable)
  3. Lower index: ( C_{pl} = \frac{0.5755 - 0.568}{3 \times 0.0017} = 1.47 ) (Capable)
  4. Combined: ( C_{pk} = 1.28 ) (Capable)

Statistical Process Control (SPC)

• Definition:
  • Methodology for monitoring processes to identify variations and implement corrective actions.
• Control Charts:
  • Visual tools with horizontal lines (UCL, LCL) indicating control limits.

Process Control States

• Process is in control: Only common causes of variation present.
• Process is out of control: Special causes present.

Developing Control Charts

• Steps:
  1. Collect data when the process is in control.
  2. Determine control limits.
  3. Analyze the chart.
  4. Use for ongoing control.
• Improvements often follow the introduction of control charts.

Control Chart Construction

• Selection depends on sample size, characteristics of the data, and intended monitoring (e.g., variable vs. attribute).
  • Types of charts include X-bar, R-chart, p-chart, c-chart.

Example: River Bottom Fire Department

• Purpose: Evaluate response times using control charts.
• Data Collection: 30 samples, calculate means, ranges, control limits.
• Results:
  • Control limits for x-chart:
  • UCL: ( 4.731 ), LCL: ( 4.357 )
  • Control limits for R-chart:
  • UCLR: 0.776, LCLR: 0

p-Charts

• Definition: Monitor the fraction of nonconforming units.
• Calculations:
  • Daily samples of nonconformances, estimate the standard deviation.
  • Control limits: UCL, LCL based on proportion ( p ) and standard deviation.

Example: Altodiez Package Co.

• Data: 200 orders inspected daily, errors counted for 25 days.
• Center line and control limits:
  • Center line: ( p = 0.0256 )
  • UCL: 0.0591, LCL: 0 (correcting for negative values).

Patterns in Control Charts

• Key Indicators of Control:
  1. No points outside control limits.
  2. Random distribution of points around the center line.
  3. Most points near the center line, few near limits.
• Unusual Patterns:
  • Out-of-control points, shifts, cycles, trends.

Sampling Considerations

• Homogeneity in Samples:
  • Rational subgroups minimize within-sample variability.
• Sample Size Implications:
  • Small samples reduce cost but are less informative. Larger samples better for detecting smaller shifts.

Control Limits and Sampling Frequency

• Adjustment of Control Limits:
  • Effects cost of false conclusions.
• Frequency of Sampling:
  • Balance between cost and timely detection of changes.

Cost Considerations for Control Limits and Sample Size

1. High investigation costs favor wider limits.
2. Significant defect costs favor narrower limits.
3. When both costs are high, wider limits and larger sample sizes recommended.

Conclusion

• Understanding process capabilities and control is crucial for quality management.
• Properly constructed control charts and careful consideration of sampling size and frequency improve quality monitoring.