algo-mfg-cpk
計算 Cpk 製程能力指數，判斷製程能否達到規格
中文翻譯
---
name: "algo-mfg-cpk"
description: "計算 Cpk 製程能力指數，評估製程是否符合規格要求。當使用者需要評估製程能力、比較不同製程，或判斷品質目標能不能達成時使用，即使他們說的是『can our process meet spec』『process capability』或『Cpk calculation』。"
metadata:
category: "WP-48 製造演算法"
tags: ["manufacturing", "cpk", "process-capability", "quality"]
---
# Cpk 製程能力指數
## 概述
Cpk 衡量製程落在規格界限內的程度，同時考慮變異（散布）和置中程度。Cpk = min((USL - μ) / 3σ, (μ - LSL) / 3σ)。通常要求 Cpk ≥ 1.33，關鍵特性要求 Cpk ≥ 1.67。和 Cp 不同，Cpk 會懲罰偏離中心的製程。
## 何時使用
**觸發條件：**
- 評估製造製程能不能符合客戶規格
- 比較不同製程、機台或時間區間的能力
- 為量產就緒做製程驗證
**不該使用的情況：**
- 製程沒有處於統計管制狀態時（先用 SPC 穩定下來）
- 非常態分布又沒有做轉換時
## 演算法
```
IRON LAW: Cpk Is Only Valid for a STABLE, IN-CONTROL Process
Computing Cpk on an unstable process gives a meaningless number.
The process MUST be in statistical control (per SPC charts) before
capability analysis. An unstable process with Cpk=2.0 today may
produce defects tomorrow when it shifts.
```
### 階段 1：輸入驗證
收集：來自穩定製程的 100 筆以上量測值。確定：USL、LSL（客戶規格）。確認製程在管制狀態（SPC 管制圖顯示穩定）。
**關卡：** 製程在管制中、規格已定義、有 100 筆以上資料。
### 階段 2：核心演算法
1. 計算製程平均：μ = Σxᵢ / n
2. 計算製程標準差：σ 用 R-bar/d₂ 或 S-bar/c₄（組內）估計，不是整體標準差
3. Cp = (USL - LSL) / 6σ（潛在能力，不考慮置中）
4. Cpk = min((USL - μ) / 3σ, (μ - LSL) / 3σ)（實際能力）
5. 從 Cpk 估計不良 PPM（例如 Cpk=1.33 → 約 63 PPM）
### 階段 3：驗證
檢查：Cp 和 Cpk 的差距代表置中問題（Cp >> Cpk = 偏離中心）。分布大致呈常態（直方圖、常態性檢定）。
**關卡：** 已算出能力指數、已評估置中程度、已驗證常態性。
### 階段 4：輸出
回傳能力指數，附上不良率估計。
## 輸出格式
```json
{
"capability": {"cp": 1.8, "cpk": 1.45, "ppm_defective": 27},
"centering": {"mean": 50.2, "target": 50.0, "offset_pct": 0.4},
"specs": {"usl": 55, "lsl": 45, "target": 50},
"metadata": {"samples": 200, "sigma_method": "rbar_d2", "normality_p": 0.35}
}
```
## 範例
### 輸入輸出範例
**輸入：** USL=55、LSL=45、μ=50.2、σ=1.5
**預期：** Cp = (55-45)/(6×1.5) = 1.11。Cpk = min((55-50.2)/4.5, (50.2-45)/4.5) = min(1.07, 1.16) = 1.07。低於 1.33 的目標。
### 邊界情況
| 輸入 | 預期 | 原因 |
|-------|----------|-----|
| μ 剛好在目標值 | Cp = Cpk | 完美置中 |
| μ 超出規格 | Cpk < 0 | 製程平均已超過規格界限 |
| 只有單邊規格 | 只用那一側算 Cpk | 例如表面粗糙度只有 USL |
## 注意事項
- **σ 的估計方法**：要用組內 σ（R̄/d₂），不是整體 σ。整體 σ 包含組間變異，會把 σ 墊大、低估 Cpk。
- **非常態資料**：Cpk 假設常態。偏態資料（表面粗糙度、同心度）要用 Box-Cox 轉換或非參數的能力指數。
- **短期 vs 長期**：Cp/Cpk 是短期的（組內變異）。Pp/Ppk 用整體變異（長期）。客戶常常要的是 Ppk。
- **樣本數的信心**：用 30 筆樣本算出的 Cpk，信賴區間很寬。報告點估計的同時要附上信賴區間。
- **Cpk ≠ 零缺陷**：就算 Cpk=2.0 理論上也有不良率（約 0.002 PPM）。超高關鍵的應用需要更高的 Cpk 或製程驗證。
## Scripts
| Script | 說明 | 用法 |
|--------|-------------|-------|
| `scripts/cpk.py` | 從製程資料計算 Cp、Cpk、Cpm 和不良 PPM | `python scripts/cpk.py --help` |
執行 `python scripts/cpk.py --verify` 來跑內建的健全性測試。
## 參考資料
- Cp/Cpk/Pp/Ppk 的比較，見 `references/capability-indices.md`
- 非常態的能力分析，見 `references/non-normal-capability.md`英文原文
---
name: "algo-mfg-cpk"
description: "Calculate Cpk process capability index to assess whether a process meets specification requirements. Use this skill when the user needs to evaluate process capability, compare processes, or determine if quality targets are achievable — even if they say 'can our process meet spec', 'process capability', or 'Cpk calculation'."
metadata:
category: "WP-48 製造演算法"
tags: ["manufacturing", "cpk", "process-capability", "quality"]
---
# Cpk Process Capability Index
## Overview
Cpk measures how well a process fits within specification limits, accounting for both variation (spread) and centering. Cpk = min((USL - μ) / 3σ, (μ - LSL) / 3σ). Cpk ≥ 1.33 is typically required; Cpk ≥ 1.67 for critical characteristics. Unlike Cp, Cpk penalizes off-center processes.
## When to Use
**Trigger conditions:**
- Assessing whether a manufacturing process can meet customer specifications
- Comparing capability across processes, machines, or time periods
- Qualifying a process for production readiness
**When NOT to use:**
- When the process is not in statistical control (stabilize first with SPC)
- For non-normal distributions without transformation
## Algorithm
```
IRON LAW: Cpk Is Only Valid for a STABLE, IN-CONTROL Process
Computing Cpk on an unstable process gives a meaningless number.
The process MUST be in statistical control (per SPC charts) before
capability analysis. An unstable process with Cpk=2.0 today may
produce defects tomorrow when it shifts.
```
### Phase 1: Input Validation
Collect: 100+ measurements from a stable process. Determine: USL, LSL (customer specifications). Verify process is in control (SPC charts show stability).
**Gate:** Process in control, specifications defined, 100+ data points.
### Phase 2: Core Algorithm
1. Compute process mean: μ = Σxᵢ / n
2. Compute process standard deviation: σ = estimated from R-bar/d₂ or S-bar/c₄ (within-subgroup) — NOT overall std dev
3. Cp = (USL - LSL) / 6σ (potential capability, ignoring centering)
4. Cpk = min((USL - μ) / 3σ, (μ - LSL) / 3σ) (actual capability)
5. Estimate PPM defective from Cpk (e.g., Cpk=1.33 → ~63 PPM)
### Phase 3: Verification
Check: Cp vs Cpk difference indicates centering issue (Cp >> Cpk = off-center). Distribution is approximately normal (histogram, normality test).
**Gate:** Capability computed, centering assessed, normality verified.
### Phase 4: Output
Return capability indices with defect rate estimates.
## Output Format
```json
{
"capability": {"cp": 1.8, "cpk": 1.45, "ppm_defective": 27},
"centering": {"mean": 50.2, "target": 50.0, "offset_pct": 0.4},
"specs": {"usl": 55, "lsl": 45, "target": 50},
"metadata": {"samples": 200, "sigma_method": "rbar_d2", "normality_p": 0.35}
}
```
## Examples
### Sample I/O
**Input:** USL=55, LSL=45, μ=50.2, σ=1.5
**Expected:** Cp = (55-45)/(6×1.5) = 1.11. Cpk = min((55-50.2)/4.5, (50.2-45)/4.5) = min(1.07, 1.16) = 1.07. Below 1.33 target.
### Edge Cases
| Input | Expected | Why |
|-------|----------|-----|
| μ exactly at target | Cp = Cpk | Perfectly centered |
| μ outside specs | Cpk < 0 | Process mean beyond specification limit |
| One-sided spec only | Use Cpk for that side only | e.g., surface finish has only USL |
## Gotchas
- **σ estimation method**: Use within-subgroup σ (R̄/d₂), NOT overall σ. Overall σ includes between-subgroup variation that inflates σ and understates Cpk.
- **Non-normal data**: Cpk assumes normality. For skewed data (surface finish, concentricity), use Box-Cox transformation or non-parametric capability indices.
- **Short-term vs long-term**: Cp/Cpk are short-term (within subgroup variation). Pp/Ppk use overall variation (long-term). Customers often want Ppk.
- **Sample size confidence**: Cpk from 30 samples has wide confidence intervals. Report confidence intervals alongside point estimates.
- **Cpk ≠ defect-free**: Even Cpk=2.0 has a theoretical defect rate (~0.002 PPM). For ultra-critical applications, higher Cpk or process validation is required.
## Scripts
| Script | Description | Usage |
|--------|-------------|-------|
| `scripts/cpk.py` | Compute Cp, Cpk, Cpm, and PPM defective from process data | `python scripts/cpk.py --help` |
Run `python scripts/cpk.py --verify` to execute built-in sanity tests.
## References
- For Cp/Cpk/Pp/Ppk comparison, see `references/capability-indices.md`
- For non-normal capability analysis, see `references/non-normal-capability.md`