Advancing QE Skills (12) | Cmk and Equipment Capability Acceptance: Quantitative Thresholds Before New Equipment Goes Online

1. Introduction: A "Qualified" Machine Fails After Three Months of Mass Production

A manufacturing company recently purchased a machining center. On the day of acceptance, the supplier's technician adjusted the machine on-site and tested 20 pieces, all of which met the dimensional specifications. The report was impressive, and the equipment was smoothly signed off. However, three months later, during the ramp-up of mass production, the same machine and the same process yielded a CPK of only 0.9, with frequent dimensional deviations and high rework rates. Upon reviewing the acceptance records, two issues were discovered: first, the samples were selected by the operator after the machine was adjusted, and only the "good-looking" pieces were sent for inspection; second, the acceptance only calculated the average and range, without any capability index. The supplier claimed, "The equipment is fine; it's your operators who are not using it correctly," while the company insisted, "The equipment is不合格." The argument lacked evidence because no quantifiable, reproducible evidence of the equipment's capability was left during the acceptance. Cmk is the tool that fills this gap: it answers not whether "this batch is good," but whether "the machine itself can produce conforming pieces."

2. Principle: What Distinguishes Cmk from Cpk

Cmk (Machine Capability Index, 设备能力指数) measures the variation caused solely by the equipment (including tooling and cutting tools) under controlled short-term conditions relative to the tolerance:

Cmk = min(USL − X̄, X̄ − LSL) / (3σ_short)

Three key qualifiers differentiate it from Cpk:

• Controlled Short-Term Conditions: The same operator, the same batch of materials, the same set of parameters, a single setup, continuous processing, and no mid-process adjustments. The goal is to minimize the variation from "people, materials, methods, and environment," leaving the remaining variation primarily from the equipment.
• Estimation of σ_short: The short-term standard deviation from continuous pieces. This can be the sample standard deviation s from 50 consecutive pieces or the σ from an I-MR chart, calculated as M̄R/1.128. It is neither the overall standard deviation used in Pp/Ppk nor the within-group standard deviation σ_within = R̄/d₂ commonly used in Cpk.
• Comparison with Tolerance Only: Cmk does not consider the long-term drift of the process center; it only looks at the position and dispersion in this snapshot.

A clearer comparison is as follows:

Therefore, for the same equipment, Cmk is usually higher than the Cpk during mass production (due to fewer sources of variation), but Cmk cannot be used to promise mass production capability—mass production introduces additional variations from shifts, batches, personnel, and environment.

3. Practical Steps: How to Design a Numerical Equipment Capability Acceptance

Step 1: Verify the Measurement System. If the measurement system is not up to standard, calculating Cmk is like dividing two unknowns. The gage GR&R should meet %GRR ≤ 10% (it is recommended not to relax this to 30% for new equipment acceptance), and the gage resolution should be ≤ 1/10 of the tolerance (judgment rule: for a tolerance of 0.05 mm, the reading resolution should be no less than 0.005 mm). Randomize the measurement sequence to avoid concentrating repeat positioning errors in the same period.

Step 2: Define the Sampling Plan. Continuous processing, at least 50 pieces (100 pieces are recommended for critical/safety characteristics), should be completed under the same operator, the same batch of materials, the same set of parameters, and a single setup. Mid-process adjustments, tool changes, and piece selection are prohibited. Initially, test-cut 5-10 pieces to reach thermal stability; these pieces are only for record-keeping and not included in the calculation. The 50 pieces should cover at least one hour of continuous operation to capture thermal drift; for fast cycle times (a few dozen seconds per piece), samples can be taken in three consecutive segments, but still within the same setup.

Step 3: Assess Stability Before Calculating the Index. Plot the 50 pieces in the order of processing on an I-MR chart or in groups of 5 on an X̄-R chart to assess stability: no points should exceed the control limits (I chart ±3σ, MR chart MR̄×3.267 upper limit), no 7 consecutive points should be rising or falling, and no 7 consecutive points should be on the same side of the center line. If any out-of-control signal appears, the Cmk for this run is not credible—an index calculated from unstable data is meaningless. Check for tool wear, thermal deformation, and clamping looseness first.

Step 4: Calculate Cmk and Provide a Judgment. Use the continuous data from the stable segment to calculate X̄ and s (or M̄R/1.128), and substitute into the formula. The judgment criteria should be written into the acceptance terms and be enforceable:

• Cmk ≥ 1.67: Acceptance passed, ready for direct production.
• 1.33 ≤ Cmk < 1.67: Conditional acceptance, with a specified rectification period (e.g., 30 days) and retesting. During the rectification period, stricter inspection (increased sampling or 100% first article inspection with SPC monitoring) should be implemented.
• Cmk < 1.33: Rejection, the supplier must rectify and retest the capability. The costs of retesting and production downtime should be agreed upon in the contract.
• Safety/Regulatory Characteristics: The threshold is uniformly set to Cmk ≥ 2.00, with no conditional acceptance allowed.

Step 5: Document the "Three Essentials" of Acceptance. ① Measurement system report (GR&R plus resolution); ② Sampling and stability evidence (I-MR chart plus original 50 pieces data); ③ Cmk calculation and judgment conclusion (with signature). These three documents should be archived with the equipment records and retested after relocation, major overhaul, or changes in the spindle or tooling supplier.

4. Common Pitfalls

1. Using Mass Production Data to Calculate Cmk as Equipment Capability. Using a few days of mass production data (across shifts, batches, and mid-process tool changes) to calculate the index and label it as Cmk will attribute process variation to the equipment, leading to falsely low values and incorrect rejection of the equipment. Conversely, using a single run of Cmk in PPAP as a mass production capability commitment will result in falsely high values and subsequent embarrassment. These three elements—sampling conditions, σ estimation, and thresholds—cannot be interchanged.
2. Sampling Immediately After Adjustment or Piece Selection. Taking samples based on the results of machine adjustments or selecting "good-looking" pieces visually will artificially lower σ, leading to falsely high Cmk values. The correct approach is to number the pieces in the order of processing, blind-sample, and test all pieces according to their numbers.
3. Calculating Capability Without MSA. The variation from the measurement system itself will directly enter the denominator. When %GRR reaches 30%, the calculated Cmk can be reduced by more than 20%: a machine that should have passed is deemed不合格, or it becomes "mysteriously"合格 after GR&R improvement.
4. Focusing on a Single Characteristic or Only the Value, Not Stability. When a machine has multiple critical dimensions, Cmk should be calculated for each characteristic, and the worst characteristic should be used for judgment. If there is a wear trend (e.g., cutting tools, grinding wheels), it will manifest as a monotonic drift in the average value. The Cmk snapshot may appear合格, but after a few hours of operation, the center may shift out of tolerance. Stability charts and periodic retesting must be combined.
5. Using 100% Inspection as a Substitute for Equipment Rectification. "If Cmk is insufficient, just do 100% inspection" is the most expensive mistake: 100% inspection only blocks nonconforming products from leaving, but does not address the equipment's inherent capability defects. Long-term labor costs and the risk of missed inspections far exceed the cost of a single rectification, and this issue will certainly be pursued during customer process audits.
6. Accepting Equipment Without Specifying Judgment Criteria. The acceptance form states "operates normally, dimensions合格," but does not include the Cmk value, sample size, sampling conditions, and judgment criteria. This makes it impossible to trace back in the future, leading to disputes between the supplier and the company.

5. Self-Check List

• Has MSA been performed for each critical characteristic of this acceptance, with %GRR ≤ 10% and gage resolution ≤ 1/10 of the tolerance?
• Is the sampling continuous ≥50 pieces under the same operator, the same batch of materials, and a single setup, excluding the initial thermal stabilization pieces, selected pieces, and mid-process adjustments?
• Are there no out-of-control signals on the stability chart (I-MR or X̄-R), and have thermal drift and trends been confirmed or ruled out?
• Are the Cmk judgment criteria clearly stated (≥1.67 pass, 1.33～1.67 conditional, <1.33 reject; safety characteristics ≥2.00), with specified rectification and retesting periods?
• Have the measurement system report, original data, stability chart, and Cmk judgment conclusion been archived, and are the retesting conditions after relocation or major overhaul clearly stated?

Acceptance requires evidence of equipment capability, not just a good-looking dimensional report.

Knowledge code: 6.3.2

Version: v20260922

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools to quality management practitioners, helping companies continuously improve their quality capabilities.