OPS 385 Week 4 Process Capability and Control Charts Example

A Cpk of 0.60 at the Voltage Limit: Process Capability and Control Charts for Rebuilt Alternators

[Student Name]

University of Phoenix

OPS/385: Lean Six Sigma and Process Management

Week 4 Assignment

[Instructor Name]

[Date]

Delta Rebuild Industries, its measurements, limits and indexes are composites written for a model paper.

In Week 3, Delta Rebuild Industries, the fictional Memphis alternator and starter remanufacturer in this course, found that high-voltage failures were concentrated in units built with regulators from a newer, cheaper supplier. Regulated output voltage is a critical-to-quality characteristic: customers need it to stay inside the 13.8 to 14.6 volt band so batteries charge without boiling. This paper asks how capable the voltage process is today and sets up control charts to monitor it as improvements are made.

Stability First

Capability describes what a stable process can deliver. Shewhart (1931), who developed the control chart, separated the background variation every process carries from variation with a specific, traceable source that can be found and removed. A process affected by assignable causes is not stable, and capability figures for it are unreliable. The team collected 30 days of data, five consecutive alternators each day, and plotted them on X-bar and R charts. No points fell outside the control limits and no unusual runs appeared, so the process, though mediocre, is stable. The data also looked roughly bell-shaped on a histogram, supporting the normal-based capability calculation.

Capability Indexes

The overall mean voltage was 14.31 volts and the standard deviation 0.16 volts. Kane (1986) described the process capability indexes and their interpretation, including the index that adjusts for the process mean not being centered.

Cp compares the specification width with six standard deviations: (14.6 minus 13.8) divided by (6 × 0.16) = 0.8 / 0.96 = 0.83.

Cpk uses the distance from the mean to the nearer limit, divided by three standard deviations. Upper side: (14.6 minus 14.31) / (3 × 0.16) = 0.29 / 0.48 = 0.60. Lower side: (14.31 minus 13.8) / 0.48 = 1.06. Cpk is the smaller, 0.60.

What the Numbers Mean

A Cp of 0.83 says the process spread is wider than the specification even if perfectly centered. A Cpk of 0.60 says the process is also shifted toward the upper limit. Under a normal distribution, the upper limit sits about 1.8 standard deviations above the mean, which implies roughly 3.6 percent of units above 14.6 volts, close to the share of high-voltage test failures observed. A common target for an established process is a Cpk of at least 1.33. Reaching that would require both centering the mean near 14.2 volts and cutting the standard deviation to about 0.10.

The customer sets the limits; the process decides how often it stays inside them.

X-bar and R Charts

For daily subgroups of five, the standard constants are A2 = 0.577, D3 = 0 and D4 = 2.114. The average of the daily means, X-double-bar, was 14.31 volts and the average daily range, R-bar, was 0.37 volts.

X-bar chart: center line 14.31; upper control limit 14.31 + 0.577 × 0.37 = 14.52; lower control limit 14.31 minus 0.214 = 14.10.

R chart: center line 0.37; upper control limit 2.114 × 0.37 = 0.78; lower control limit 0.

These limits describe the current process. The upper control limit on the X-bar chart, 14.52, sits below the 14.6-volt specification, but individual units vary more than daily averages, which is why individual units still fail. Specification limits are never drawn on the X-bar chart, since it plots averages.

A p-Chart for Overall Failures

For the overall test failure rate, a p-chart tracks the daily proportion of failed units among about 42 tested a day. With a baseline average proportion of 0.074, the standard error works out to √(0.074 × 0.926 ÷ 42), about 0.040, so the upper control limit is about 0.074 + 0.121 = 0.195 and the lower limit is zero. After improvements, the limits will be recalculated from the new average.

Signals and Responses

Woodall (2000) reviewed debates in statistical process control and stressed the difference between using charts to assess whether a process was stable in the past and using them to monitor it going forward, along with the trade-off between catching real shifts and raising false alarms. Delta uses a short rule set: any point outside a control limit; seven daily averages in a row above, or in a row below, the middle line; or a steady trend of six points. A signal on the X-bar or R chart triggers a check of the regulator lot and test bench calibration that day by the test technician; a p-chart signal triggers a review by the rebuild supervisor. Each signal and its cause are logged.

Why Not Just Inspect Every Unit?

Every alternator is already tested, so a manager might ask why charts are needed at all. Testing sorts good units from bad after the fact; it does not tell anyone that the process has started drifting. A control chart on voltage shows a shift in the average while most units still pass, giving the team a chance to act before failures rise. It also prevents overreaction: without limits, a technician who sees two high readings in a morning may adjust the process or blame a rebuilder when nothing has actually changed. The chart separates the noise every process has from a signal that something new has happened.

Capability After Improvement

The Week 5 changes aim to center the mean near 14.2 volts and reduce spread. The team will recalculate Cp and Cpk once the new process has run stably for at least 20 subgroups, and will recompute control limits from the new data rather than carrying the old ones forward.

Setting Up the Charts on the Floor

The charts live on a screen beside the test bench, updated automatically from bench readings, with a printed copy posted daily for rebuilders. The test technician was trained to read them, and the supervisor reviews them at the shift start meeting.

Conclusion

The voltage process at Delta is stable but not capable: a Cp of 0.83 and a Cpk of 0.60, explained mainly by a mean shifted toward the upper limit and excess spread, consistent with the regulator problem found in Week 3. X-bar and R charts and a p-chart, with clear response rules, will show whether the improvements planned in Week 5 move the process and whether the gains hold.

References

Kane, V. E. (1986). Process capability indices. Journal of Quality Technology, 18(1), 41-52. https://doi.org/10.1080/00224065.1986.11978984

Shewhart, W. A. (1931). Economic control of quality of manufactured product. Van Nostrand.

Woodall, W. H. (2000). Controversies and contradictions in statistical process control. Journal of Quality Technology, 32(4), 341-350. https://doi.org/10.1080/00224065.2000.11980013