Cp / Cpk Capability

1 Introduksi

1.1 Latar Belakang

Process capability index $C_p$ dan $C_{pk}$ mengkuantifikasi kemampuan proses memenuhi customer specifications (USL dan LSL). $C_p$ measures spread vs spec width tanpa memperhitungkan centering; $C_{pk}$ measures worst case dari kedua side. Diperkenalkan di Juran's Quality Handbook (Juran, 1951) dan diadopsi sebagai Six Sigma metric standar (Pyzdek & Keller, 2018; Montgomery, 2020).

Interpretasi praktis (rule of thumb industri):

$C_p < 1.0$: process incapable (variabilitas > spec width).
$C_p = 1.0$: process exactly fits spec ($3\sigma$ each side).
$C_p = 1.33$: minimum acceptable industri Western ($4\sigma$).
$C_p = 1.67$: aerospace / pharmaceutical ($5\sigma$).
$C_p = 2.0$: Six Sigma quality ($6\sigma$, 3.4 DPMO).

1.2 Tujuan Modul

Modul Cp / Cpk Capability di SQalytics ditujukan untuk:

Menerima measurement data + USL/LSL spec.
Menghitung $C_p$, $C_{pk}$, $C_{pu}$, $C_{pl}$, $P_p$, $P_{pk}$ (short-term vs long-term).
Estimasi DPMO (Defects Per Million Opportunities) dari fitted normal distribution.
Menampilkan histogram + spec lines + capability stripe.
Audiens: praktisi quality engineering, supplier audit, Six Sigma practitioners.

1.3 Posisi di Antara Alternatif

Pilih Cp / Cpk Capability untuk process capability vs spec. Untuk time-series monitoring stability, pakai SPC Xbar / R Charts. Untuk lot release decision, pakai Acceptance Sampling. Untuk statistical comparison antar batch, pakai Compare Many Groups.

2 Metode

2.1 Dasar Teoretis

Process capability indices (Juran, 1974; Pyzdek & Keller, 2018):

$C_p$ (potential capability, ignore centering):

$$C_p = \frac{\text{USL} - \text{LSL}}{6 \hat{\sigma}}$$

$C_{pk}$ (actual capability, account centering — worst of two sides):

$$C_{pk} = \min(C_{pu}, C_{pl}) = \min\!\left(\frac{\text{USL} - \mu}{3 \hat{\sigma}}, \frac{\mu - \text{LSL}}{3 \hat{\sigma}}\right)$$

$P_p$ dan $P_{pk}$ — long-term performance index pakai overall $s$ (vs $\hat{\sigma}$ short-term dari SPC $\bar{R}/d_2$).

Short-term $\hat{\sigma}$ (dari Xbar-R baseline): $\hat{\sigma}_{\text{ST}} = \bar{R} / d_2$.

Long-term $s$ (dari semua data): $s = \sqrt{\sum (x_i - \bar{x})^2 / (n - 1)}$.

Bila proses stable, $\hat{\sigma}_{\text{ST}} \approx s$ → $C_p \approx P_p$. Bila tidak (drift, shift), $s > \hat{\sigma}_{\text{ST}}$ → $C_p > P_p$ (proses unstable).

DPMO estimation (assume normal distribution):

$$\text{DPMO} = 10^6 \cdot [P(X > \text{USL}) + P(X < \text{LSL})] = 10^6 \cdot [(1 - \Phi(z_u)) + \Phi(z_l)]$$

dengan $z_u = (\text{USL} - \mu)/\sigma$ dan $z_l = (\text{LSL} - \mu)/\sigma$.

Six Sigma reference:

$C_p$	Sigma level	DPMO (centered)
0.50	1.5σ	668,000
1.00	3σ	66,800
1.33	4σ	6,210
1.67	5σ	233
2.00	6σ	3.4 (Six Sigma target)

$C_{pm}$ (Taguchi loss function) — penalize off-target:

$$C_{pm} = \frac{\text{USL} - \text{LSL}}{6 \sqrt{\sigma^2 + (\mu - T)^2}}$$

dengan $T$ target nominal.

2.2 Persamaan Inti

$C_p$: $(\text{USL} - \text{LSL}) / (6\sigma)$
$C_{pk}$: $\min[(\text{USL} - \mu), (\mu - \text{LSL})] / (3\sigma)$
Short-term $\sigma$: $\bar{R} / d_2$
DPMO: $10^6 \cdot [(1 - \Phi(z_u)) + \Phi(z_l)]$
$C_{pm}$ (Taguchi): include $(\mu - T)^2$ term.

2.3 Asumsi & Batas Validitas

Asumsi	Konsekuensi jika dilanggar	Cara cek di SQalytics
Process in control (stable)	Capability misleading	Cek SPC first
Normal distribution	DPMO bias non-normal	Shapiro-Wilk; pakai non-normal $C_p$
$\geq 30$ data points	Estimate noisy	Modul flag
USL/LSL realistic (engineering)	Numbers misleading	Domain validation
Random sampling	Bias systematic	SOP sampling

3 Cara Kerja

3.1 Step-by-Step di SQalytics

Buka Cp / Cpk Capability dari domain Mutu dan Analisis Lanjutan.
Muat tabel: Sample, Measurement.
Input USL, LSL, Target (Target opsional, untuk $C_{pm}$).
(Opsional) Tampilkan $C_p$ vs $P_p$ (short-term vs long-term).
Aktifkan non-normal flag bila data clearly skew.
Klik Calculate Capability.
Tinjau Tab Indices Table — $C_p$, $C_{pk}$, $C_{pu}$, $C_{pl}$, $C_{pm}$, $P_p$, $P_{pk}$, DPMO.
Cek Tab Histogram + Spec Plot, Tab Normality Check (Shapiro-Wilk + QQ-plot), Tab Recommendation.

3.2 Template Tabel Input + Contoh Data Sintetis

Kolom	Tipe	Wajib	Catatan
`Sample`	category	✓	ID unit
`Measurement`	numeric	✓	Quality variable

Contoh data sintetis (fill weight, USL = 255 g, LSL = 245 g, Target = 250 g):

Sample	Weight_g
S001	251.2
S002	249.8
S003	252.5
S004	248.5
S005	250.9
… ($n = 100$, mean = 250.4 g, SD = 1.2 g)

SYNTHETIC Fill weight data: $n = 100$, mean = 250.4 g, SD = 1.2 g. USL = 255 g, LSL = 245 g, Target = 250 g. Mean sedikit off-center (+0.4 g dari target) — menyebabkan $C_{pk} < C_p$. CSV setara: docs/assets/example-data/id/quality-advanced/template_quality_cpk.csv.

3.3 Contoh Luaran

Capability indices:

Index	Value	Interpretation
$C_p$	$10 / (6 \cdot 1.2) = 1.39$	Spread acceptable (> 1.33)
$C_{pu}$	$(255 - 250.4) / (3 \cdot 1.2) = 1.28$	—
$C_{pl}$	$(250.4 - 245) / (3 \cdot 1.2) = 1.50$	—
$C_{pk}$	$\min(1.28, 1.50) = 1.28$	Slightly below 1.33 — improvement needed
$C_{pm}$	$10 / (6 \sqrt{1.44 + 0.16}) = 1.32$	Off-target penalty kecil
$P_p$	1.35	Long-term spread
$P_{pk}$	1.25	Long-term centered
Sigma level	~4.1σ	Acceptable industrial
DPMO (estimated)	250 (per million)	Acceptable

Cp / Cpk Capability — figure 01 — Gambar 1. *Panel* (a) Histogram fill weight dengan USL/LSL spec lines dan normal fit; *panel* (b) capability indices ($C_p$, $C_{pk}$, $C_{pm}$) *bar chart* dengan 1.33 target line.

Kesimpulan ringkas: $C_p = 1.39$ acceptable spread, tetapi $C_{pk} = 1.28 < 1.33$ karena mean (250.4) sedikit off-center (target 250). Recommendation: shift mean ke 250.0 via re-calibration filler — akan naik $C_{pk}$ ke 1.39 sama dengan $C_p$. Lanjut ke SPC Xbar / R Charts untuk verify stability post-recalibration. Bila target Six Sigma (DPMO < 4), perlu reduce $\sigma$ ke ~ 0.8 g.

4 Kesimpulan

4.1 Relevansi Real-World

Supplier qualification — $C_{pk} \geq 1.33$ acceptance criteria.
GMP pharmaceutical compliance — Content uniformity dengan $C_{pk}$ assessment.
Aerospace/medical device — $C_{pk} \geq 1.67$ requirement.
Six Sigma Black Belt projects — DPMO reduction target.
Customer Quality Agreement — capability metric reporting.

4.2 Where to Go from Here

SPC Xbar / R Charts — ensure process stable sebelum capability.
Acceptance Sampling — lot decision dengan capability data.
Compare Many Groups — uji capability antar shift / operator.
Pyzdek & Keller (2018) untuk Six Sigma comprehensive.

⚙ Troubleshooting Cepat

$C_p$ tinggi tapi $C_{pk}$ rendah. Process off-center; re-calibrate centering.

Both $C_p$ rendah. Reduce variability (DOE root cause).

Non-normal data. Pakai Box-Cox transform atau non-normal $C_p$.

i Riwayat Revisi

Tanggal	Revisi	Penulis
2026-05-12	Draft v2 publikasi (KaTeX $C_p$/$C_{pk}$/$C_{pm}$ + DPMO + sigma level + APA Juran/Pyzdek/Montgomery)	Claude
2026-05-12	Konversi MD → HTML (W5 quality-advanced batch)	Claude

4 Referensi

Juran, J. M. (Ed.). (1974). Quality control handbook (3rd ed.). McGraw-Hill.
Pyzdek, T., & Keller, P. (2018). The Six Sigma handbook (5th ed.). McGraw-Hill.
Montgomery, D. C. (2020). Introduction to statistical quality control (8th ed.). John Wiley & Sons.
Kotz, S., & Lovelace, C. R. (1998). Process capability indices in theory and practice. Arnold.
Kane, V. E. (1986). Process capability indices. Journal of Quality Technology, 18(1), 41–52. https://doi.org/10.1080/00224065.1986.11978984