Process capability and control

Cp and CpkSPC formulassix sigma mathsis the process capableOEE

Cp, Cpk, control limits, DPMO, rolled throughput yield and OEE: whether a process holds tolerance and what the losses multiply to.

Process Capability (Cp)

C_p = \frac{\mathit{USL} - \mathit{LSL}}{6\sigma}

How the tolerance band compares with the spread of the process: the width of the specification divided by six standard deviations. Says nothing about whether the process is centred.

Process Capability Index (Cpk)

C_{pk} = \min\!\left(\frac{\mathit{USL} - \mu}{3\sigma},\ \frac{\mu - \mathit{LSL}}{3\sigma}\right)

Capability measured to the nearer specification limit, so an off-centre process is penalised: the smaller of the two one-sided indices. Cpk can never exceed Cp, and equals it only when the process is perfectly centred.

X-bar Chart Upper Control Limit

\mathit{UCL} = \bar{\bar{x}} + A_2 \bar{R}

Upper control limit of an X-bar chart, built from the average range rather than a computed sigma: grand average plus the tabulated A2 factor times the average range. The lower limit is the same product subtracted.

Defects Per Million Opportunities (DPMO)

\mathit{DPMO} = \frac{D}{U \times O} \times 10^{6}

Defect rate on a scale that makes very good processes comparable: defects found, divided by the units inspected times the chances to go wrong on each unit, scaled to a million.

Rolled Throughput Yield

\mathit{RTY} = y^{\,n}

Fraction of units that pass every step of an n-step process with no rework, when each step has the same first-pass yield: the yield raised to the number of steps.

Overall Equipment Effectiveness (OEE)

\mathit{OEE} = A \times P \times Q

The single number total productive maintenance is built on: availability times performance times quality. Three losses multiplied, never averaged.

Machine Utilisation

U = \frac{T_o}{T_a}

Share of the time a machine was actually running: operating time divided by the time it was available to run. The answer depends entirely on which hours you count as available.

Variance and Standard Deviation

\sigma^2 = \sigma \cdot \sigma

Converts between variance and standard deviation: the variance is the square of the standard deviation, and the deviation is its square root.

How they fit together

There is an order of operations here that quality systems get wrong constantly. Stability comes first. The X-bar chart control limits are computed from the process itself and have nothing to do with the customer's tolerance, and until the chart says the process is stable, a capability index is arithmetic performed on a moving target. Control limits describe what the process does. Specification limits describe what you wanted. Putting spec limits on a control chart is the single most common error in the field.

Once stable, measure capability. Cp compares the tolerance band with six standard deviations and assumes the process is centred, which it never quite is. Cpk measures to the nearer limit and so charges you for being off centre. The gap between them is pure centring error, and it is the cheapest thing to fix, because shifting a mean is usually an adjustment while shrinking a sigma is a project. Both depend entirely on standard deviation, which is why it sits in this set: every capability number is only as trustworthy as the sample it came from.

DPMO and rolled throughput yield are the same distribution translated into language a plant manager uses, and RTY is the one that changes minds. Twenty steps at 99% each look excellent on any individual report and deliver 82% first-pass yield overall. OEE does the same multiplication to equipment: availability times performance times quality, never averaged. Three respectable-looking 90% figures give 73%, and world class is 85%. Machine utilisation is the availability term on its own, and it is the easiest number in this set to inflate, because the answer depends entirely on which hours you agreed to count as available.