Capability Shortfall and the Order of Investment on a Filling Line
A beverage filler is specified at 500 plus or minus 6 grams. The process runs at a mean of 501 g with a standard deviation of 2.0 g. A customer now requires Cpk of at least 1.33 for continued supply, and an audit found two of the last 30 hourly subgroup averages outside the 3-sigma limits on an X-bar chart, both during a changeover to a new bottle supplier. Two options are quoted: recentering the filler nozzles for $8,000 would move the mean to 500 g; a servo retrofit for $45,000 would reduce the standard deviation to 1.5 g. Volume is 2.4 million units per year, an out-of-spec unit costs $0.85 in scrap or rework, and the contract is worth $1.9M of annual revenue at a 22% contribution margin. Net-content law also penalizes underfills.
1. Compute Cp, Cpk, and the defect rate as it stands, then evaluate each option separately and together against the customer requirement.
Read the worked answer
Current state
Cp = (506 - 494) / (6 x 2.0) = 12 / 12 = 1.00 Cpk = min(506 - 501, 501 - 494) / (3 x 2.0) = min(5, 7) / 6 = 0.83
Defects: upper z = (506 - 501) / 2.0 = 2.5, area 0.00621. Lower z = (494 - 501) / 2.0 = -3.5, area 0.00023. Total 0.00644, about 6,440 ppm, or roughly 15,500 units a year at $0.85 = $13,150 per year.
Option 1: recenter only ($8,000)
Mean 500, sigma 2.0: Cp = Cpk = 6 / 6 = 1.00. Still short of 1.33. Defects fall to about 2,700 ppm (6,480 units, $5,510 a year). Payback on scrap alone is roughly a year, but the customer requirement is not met.
Option 2: servo only ($45,000)
Mean stays at 501, sigma 1.5: Cpk = min(5, 7) / 4.5 = 1.11. Still short. Defects drop sharply to about 430 ppm because the near limit is now 3.3 sigma away.
Both ($53,000)
Mean 500, sigma 1.5: Cp = Cpk = 6 / 4.5 = 1.33. Requirement met. Defects fall to roughly 63 ppm.
Centering alone buys a perfect Cp but no margin; the variation is the binding constraint. Only the combination reaches 1.33. The $53,000 is trivially justified against $418,000 of annual contribution margin from the contract - the decision is not whether to spend, but in what order and on what evidence.
2. The process shows two out-of-control points. Should the company commit to the $45,000 retrofit now? Justify the sequence of action.
Read the worked answer
No. Fix control before capability. A capability index computed from an unstable process describes neither the process that will run tomorrow nor the one the customer will audit, and the audit finding is not incidental: both excursions coincided with a bottle-supplier changeover, which is a textbook special cause (neck-finish or fill-height variation from a new container).
Why this changes the economics of the retrofit.
- The 2.0 g estimate includes the excursions. Once the special cause is removed, common-cause sigma may already be below 2.0 g, and the $45,000 servo may be buying variation that has already gone away.
- Part of the observed variation may be measurement variation from the checkweigher. If a gage R&R shows measurement sigma of 1.0 g, the true process sigma is the square root of (4.0 - 1.0) = 1.73 g, and every number above changes.
- A servo retrofit installed on an unstable process simply automates an unstable process.
Sequence
- Investigate the two excursions and run a changeover study with the new bottle supplier; put a capability requirement on incoming containers.
- Complete an MSA on the checkweigher to separate measurement variation from process variation.
- Re-estimate mean and sigma from in-control data only, then re-run the Cp/Cpk arithmetic and re-choose the investment.
- Authorize the $8,000 recentering now. It is cheap, it pays back in about a year on scrap alone, and it addresses the underfill tail that net-content law punishes - the 3.5-sigma lower tail is small in count but legally and reputationally the most expensive defect on this line.
- Hold the servo retrofit until the post-control capability gap is known, then buy variation reduction against a real number rather than against a contaminated one.