algo-mfg-spc
用統計製程管制圖監控製程穩定度
中文翻譯
---
name: "algo-mfg-spc"
description: "實作統計製程管制（SPC）管制圖，監控生產製程的穩定性。當使用者需要偵測製程偏移、設定管制界限，或區分共同原因與特殊原因變異時使用，即使他們說的是『process monitoring』『control chart』或『is our process in control』。"
metadata:
category: "WP-48 製造演算法"
tags: ["manufacturing", "spc", "control-chart", "quality"]
---
# 統計製程管制
## 概述
SPC 用管制圖監控製程隨時間的穩定性。管制上限與下限（UCL/LCL）設在離製程平均 ±3σ 的位置。點落在界限內 = 共同原因變異（穩定）。點落在界限外或呈現特定模式 = 特殊原因變異（要調查）。主要的圖：X-bar/R、X-bar/S、I-MR、p-chart、c-chart。
## 何時使用
**觸發條件：**
- 監控生產製程的穩定性並偵測偏移
- 為品質指標設定以統計為基礎的管制界限
- 區分正常變異和可歸因的原因
**不該使用的情況：**
- 評估製程能力時（用 Cpk）
- 針對已知問題做根因分析時（用魚骨圖 / 5-why）
## 演算法
```
IRON LAW: Control Limits Are NOT Specification Limits
Control limits (±3σ) describe what the process IS doing.
Specification limits describe what the process SHOULD do.
A process can be in statistical control (stable) but still produce
out-of-spec products (incapable). Conversely, a capable process may
be out of control (drifting). Monitor control FIRST, then assess capability.
```
### 階段 1：輸入驗證
收集：25 組以上的量測子群組（X-bar/R 通常每組 5 個）。確認：量測系統足夠（gauge R&R < 10%）、資料依時間順序收集。
**關卡：** 子群組數足夠、資料依時間排序、量測系統已驗證。
### 階段 2：核心演算法
**X-bar/R 管制圖（子群組資料）：**
1. 計算各子群組的平均（X̄）和全距（R）
2. 計算總平均（X̄̄）和平均全距（R̄）
3. UCL_X̄ = X̄̄ + A₂×R̄，LCL_X̄ = X̄̄ - A₂×R̄（A₂ 依子群組大小查統計表）
4. UCL_R = D₄×R̄，LCL_R = D₃×R̄
5. 畫出各點，套用 Western Electric 規則找失控訊號
### 階段 3：驗證
檢查：點超出界限、連串（連續 7 點以上在同一側）、趨勢（連續 7 點以上上升 / 下降）、3 點中有 2 點超過 2σ、5 點中有 4 點超過 1σ。
**關卡：** 管制圖已建立，失控訊號已辨識。
### 階段 4：輸出
回傳管制圖資料，附上訊號與穩定性評估。
## 輸出格式
```json
{
"chart": {"type": "xbar_r", "center_line": 50.2, "ucl": 52.1, "lcl": 48.3},
"signals": [{"subgroup": 18, "rule": "point_beyond_ucl", "value": 52.8}],
"stability": "out_of_control",
"metadata": {"subgroups": 30, "subgroup_size": 5}
}
```
## 範例
### 輸入輸出範例
**輸入：** 25 個子群組，每組 5 筆量測值，全部落在 ±3σ 內，沒有任何模式
**預期：** 製程在管制中。沒有觸發任何訊號。
### 邊界情況
| 輸入 | 預期 | 原因 |
|-------|----------|-----|
| 一個點剛好超出 UCL | 有訊號，但可能是誤報 | 即使在管制中，每個點仍有約 0.27% 的機率 |
| 緩慢的上升趨勢 | 觸發趨勢規則 | 製程在漂移，要調查 |
| 所有點都靠近中心 | 可疑，檢查資料 | 可能代表資料被動過手腳或量測有問題 |
## 注意事項
- **合理的子群組劃分**：子群組必須在相似條件下收集（同一個班別、機台、操作員）。子群組劃分不好會墊高組內變異，讓界限太寬。
- **重新計算界限**：不要每新增一次資料就重算界限。用穩定的基準期建立界限，並固定不動，直到已知的製程變更發生。
- **管制圖類型的選擇**：計量資料（量測值）→ X-bar/R 或 I-MR。計數資料（個數 / 比例）→ p-chart、np-chart、c-chart、u-chart。圖選錯 = 界限算錯。
- **常態假設**：X-bar 圖對非常態相當穩健（中央極限定理）。個別值管制圖（I-MR）需要大致常態，要用直方圖檢查。
- **過度調整**：對每個小變動都做反應（tampering）會「增加」變異。只調查特殊原因的訊號，不要去處理共同原因的變異。
## 參考資料
- 管制圖常數表，見 `references/chart-constants.md`
- Western Electric 規則與模式偵測，見 `references/we-rules.md`英文原文
---
name: "algo-mfg-spc"
description: "Implement Statistical Process Control charts to monitor production process stability. Use this skill when the user needs to detect process shifts, set control limits, or distinguish common cause from special cause variation — even if they say 'process monitoring', 'control chart', or 'is our process in control'."
metadata:
category: "WP-48 製造演算法"
tags: ["manufacturing", "spc", "control-chart", "quality"]
---
# Statistical Process Control
## Overview
SPC uses control charts to monitor process stability over time. Upper and Lower Control Limits (UCL/LCL) are set at ±3σ from the process mean. Points within limits = common cause variation (stable). Points outside or showing patterns = special cause variation (investigate). Primary charts: X-bar/R, X-bar/S, I-MR, p-chart, c-chart.
## When to Use
**Trigger conditions:**
- Monitoring production process for stability and detecting shifts
- Setting statistically-based control limits for quality metrics
- Distinguishing normal variation from assignable causes
**When NOT to use:**
- For process capability assessment (use Cpk)
- For root cause analysis of known problems (use fishbone/5-why)
## Algorithm
```
IRON LAW: Control Limits Are NOT Specification Limits
Control limits (±3σ) describe what the process IS doing.
Specification limits describe what the process SHOULD do.
A process can be in statistical control (stable) but still produce
out-of-spec products (incapable). Conversely, a capable process may
be out of control (drifting). Monitor control FIRST, then assess capability.
```
### Phase 1: Input Validation
Collect: 25+ subgroups of measurements (5 per subgroup typical for X-bar/R). Verify: measurement system is adequate (gauge R&R < 10%), data collected in time order.
**Gate:** Sufficient subgroups, time-ordered data, measurement system verified.
### Phase 2: Core Algorithm
**X-bar/R Chart (subgroup data):**
1. Compute subgroup means (X̄) and ranges (R)
2. Compute grand mean (X̄̄) and average range (R̄)
3. UCL_X̄ = X̄̄ + A₂×R̄, LCL_X̄ = X̄̄ - A₂×R̄ (A₂ from statistical tables by subgroup size)
4. UCL_R = D₄×R̄, LCL_R = D₃×R̄
5. Plot points, apply Western Electric rules for out-of-control signals
### Phase 3: Verification
Check for: points outside limits, runs (7+ consecutive on one side), trends (7+ consecutive increasing/decreasing), 2 of 3 beyond 2σ, 4 of 5 beyond 1σ.
**Gate:** Chart constructed, out-of-control signals identified.
### Phase 4: Output
Return control chart data with signals and stability assessment.
## Output Format
```json
{
"chart": {"type": "xbar_r", "center_line": 50.2, "ucl": 52.1, "lcl": 48.3},
"signals": [{"subgroup": 18, "rule": "point_beyond_ucl", "value": 52.8}],
"stability": "out_of_control",
"metadata": {"subgroups": 30, "subgroup_size": 5}
}
```
## Examples
### Sample I/O
**Input:** 25 subgroups of 5 measurements each, all within ±3σ, no patterns
**Expected:** Process in control. No signals triggered.
### Edge Cases
| Input | Expected | Why |
|-------|----------|-----|
| One point just outside UCL | Signal, but may be false alarm | ~0.27% chance per point even when in control |
| Gradual upward trend | Trend rule triggered | Process drifting, investigate |
| All points near center | Suspicious — check data | May indicate data manipulation or measurement issue |
## Gotchas
- **Rational subgrouping**: Subgroups must be collected under similar conditions (same shift, machine, operator). Poor subgrouping inflates within-group variation, making limits too wide.
- **Recalculating limits**: Don't recalculate limits every time you add data. Establish limits from a stable baseline period and keep them fixed until a known process change.
- **Chart type selection**: Variables data (measurements) → X-bar/R or I-MR. Attribute data (counts/proportions) → p-chart, np-chart, c-chart, u-chart. Wrong chart type = wrong limits.
- **Normality assumption**: X-bar chart is robust to non-normality (central limit theorem). Individual charts (I-MR) require approximate normality — check with histogram.
- **Over-adjustment**: Reacting to every small variation (tampering) INCREASES variability. Only investigate special cause signals, not common cause variation.
## References
- For control chart constants tables, see `references/chart-constants.md`
- For Western Electric rules and pattern detection, see `references/we-rules.md`