Filling-Line Capability and Tampering
On a bottling line filling 500 mL bottles, the fill volume specification is 495-505 mL. Operators collect 25 subgroups of five bottles every 30 minutes. The Xbar-bar is 500.2 mL and the average range R-bar is 3.0 mL. For subgroup size 5, A2 = 0.577, D4 = 2.114, D3 = 0, and d2 = 2.326. The last five subgroups show Xbar values of 500.8, 501.1, 501.4, 501.8, and 502.1 mL; the last range is 6.8 mL. The customer requires a minimum Cpk of 1.33. The line has a history of filler-nozzle wear, and a new fruit pulp supplier changed viscosity last week. The production supervisor wants to adjust the filler pressure after every high bottle and has already changed it three times during the shift.
1. Calculate the Xbar and R control limits and determine whether the process is in statistical control. What does the trend and last range tell you?
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Xbar limits: UCL = Xbar-bar + A2 x R-bar = 500.2 + 0.577 x 3.0 = 501.93 mL; LCL = 500.2 - 1.731 = 498.47 mL. R limits: UCL = D4 x R-bar = 2.114 x 3.0 = 6.34 mL; LCL = D3 x R-bar = 0. The last Xbar value, 502.1 mL, is above the UCL, and the last range, 6.8 mL, is above the R UCL. In addition, the five-point increasing run (500.8 to 502.1) is a trend signal. The process is not in statistical control. The trend plus the high range suggests an assignable cause such as progressive filler-nozzle wear, a viscosity change from the new pulp supplier, or a gauge/calibration problem. Adjusting pressure after every high bottle is tampering and will increase variation.
2. Calculate Cp and Cpk using the within-subgroup sigma estimate. Is the process capable? What improvement is needed to meet Cpk at least 1.33?
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Sigma estimate = R-bar/d2 = 3.0/2.326 = 1.29 mL. Cp = (USL - LSL)/(6 sigma) = (505 - 495)/(7.74) = 1.29. Cpk = min[(505 - 500.2)/(3 x 1.29), (500.2 - 495)/(3 x 1.29)] = min(4.8/3.87, 5.2/3.87) = min(1.24, 1.34) = 1.24. The process is not capable against the customer requirement of 1.33. Centering the mean at 500.0 would give Cpk = 5.0/3.87 = 1.29, still short. To reach 1.33 after centering, sigma must be no more than 5.0/(3 x 1.33) = 1.25 mL, which corresponds to R-bar <= 1.25 x 2.326 = 2.91 mL. So both centering and variation reduction are needed, with variation reduction being the larger gap.
3. Recommend a corrective-action sequence and explain why the supervisor's three pressure adjustments are risky. What records should be updated?
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Sequence: (1) stop routine adjustment and put the filler on hold or 100% inspection; (2) quarantine the affected product and check the last good control record; (3) investigate the assignable cause: inspect and replace worn nozzles, verify pulp viscosity and temperature, check the pressure gauge and weigh cell calibration; (4) run a controlled trial or DOE to confirm the cause; (5) implement the fix and verify with a capability study; (6) recalculate control limits only after the process is stable and the changes are validated; (7) update the control plan, SOP and operator training; (8) record the CAPA, root cause and verification in the quality system. The supervisor's three pressure changes are risky because they react to common-cause noise and can create new special causes; they also make it impossible to know which change produced any improvement.