PYTHON · NUMPY

Yield, scrap & process capability

Compute a line's yield and process-capability index (Cp/Cpk) from a batch, and raise yield above a target by centering and/or narrowing the process.

01Challenge

Run a 500-part batch on a 3-station line whose bore station ships off-center (μ=12.07, σ=0.020) against spec Ø12.00±0.05. Read first-pass yield, scrap, and Cpk, then change only the bore station's settings (shift μ, reduce σ) until first-pass yield ≥ 98%. Most novices reach for 'make it more precise' (expensive) before noticing the mean is just in the wrong place; that discovery is the point.

Predict firstA process is already perfectly centered in its tolerance band. To halve the scrap rate you should…

02Model

You ran a perfect plan on a perfect machine and still got bad parts, because every process is a spread and your spec is a band. Yield is just the green overlap: the fraction of parts inside the band; scrap is the tails. A line is a chain: line yield is the product of every station's pass rate, so one weak station drags the whole line down, exactly like the bottleneck dragged throughput, but for quality.

Capability adds margin: Cp = (USL−LSL)/(6σ) asks if your spread even fits; Cpk = min((USL−μ),(μ−LSL))/(3σ) adds 'are you centered?' The gap between Cp and Cpk is pure off-centering, usually the cheapest thing to fix. Two moves: center the process (often a free offset), or narrow it (costs time/tooling). Reach for centering first.

▶ Live · scrub & hover — the process spread and center against the spec limits set the yield

03Guided practice

Step 1 (worked): run the metrics cell on the shipped state → μ=12.07, yield 15.87%, Cp=0.83, Cpk=−0.33 (negative Cpk = mean outside a spec limit's 3σ reach).
Step 2 (guided): apply bore_offset = −0.070 to re-center → yield ≈98.8% at zero cost; the expensive narrow move was never needed.
Step 3 (independent + twist): a seed already centered at 12.000 but σ=0.030; centering does nothing; lower feed_per_rev to narrow σ and note the cost (cycle time rises 4.0→5.3 s).

04Feedback

PASS WHEN Base yield ≥0.98 reached by centering (no paid narrowing), reported Cpk within ±0.03 of computed, and on the twist seed yield ≥0.98 with σ ≤0.019 plus a noted cycle-time cost, on seeds 5101/5102.

If your run fails, check:

μ = 12.070, center = 12.000 → mean is +0.070 above the band. Yield is capped at 16% by off-centering, not spread. Apply bore_offset ≈ −0.070 before touching feed.
Cpk you reported (0.41) ≠ computed (0.83): Cpk uses min of BOTH (USL−μ) and (μ−LSL) over 3σ; did you divide by 3σ, not 6σ?
Yield 98% reached, but you got there by lowering feed (cycle 4.0→5.3 s). The base process was only off-center; re-centering hits target at zero cost.
Twist line: μ is centered, Cp=0.56, so centering can't help; you must narrow σ. Lower feed_per_rev until σ ≤ 0.019 and Cpk ≥ 1.0.

05Retrieve & space

From M3.2: a bimodal batch histogram. Name the most likely systematic cause and one fix. → tool wear mid-batch / fixture swap; split or re-zero.
From M4.2: line yield = 0.998×0.995×0.942. What was the throughput analogue of that weakest 0.942 station? → the bottleneck; both are governed by the worst station.
From M3.1: tightening the bore spec from ±0.05 to ±0.03 raises Cpk for free, true or false? → false; a narrower spec makes capability worse.

06Mastery & project

In sim, on an unseen 3-station line, correctly compute yield/scrap/Cp/Cpk, raise first-pass yield from sub-90% to ≥98% by the correct lever (center when off-center, narrow when centered), and state the cost of any narrowing. Unlocks 5.2.

Yield and Cpk become two of the three capstone scoreboard targets; 'center before you narrow' and 'line yield = product of station yields' are direct capstone levers.

← Balancing the Line Optimizing Under Competing Targets →

PYTHON · NUMPY

Yield, scrap & process capability

The Yield & Capability panel streams the batch's measured bore into numpy; you compute yield, Cp, Cpk yourself, then center or narrow Station C. Edit the offset/feed and run.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you