algo-mfg-doe
設計並分析因子實驗，找出關鍵製程因子
中文翻譯
---
name: "algo-mfg-doe"
description: "設計並分析因子實驗，找出顯著的製程因子並最佳化設定。當使用者需要有系統地測試因子效應、最佳化製造製程，或判斷哪些變數最重要時使用，即使他們說的是『which factors affect quality』『optimize process settings』或『design an experiment』。"
metadata:
category: "WP-48 製造演算法"
tags: ["manufacturing", "doe", "factorial-design", "optimization"]
---
# 實驗設計（DOE）
## 概述
DOE 有系統地改變製程因子，找出它們對反應變數的影響。完全因子設計測試所有組合；部分因子設計測試經過策略挑選的子集。能找出主效應和交互作用。比一次只改一個因子（OFAT）更有效率，因為 OFAT 會漏掉交互作用。用 ANOVA 分析。
## 何時使用
**觸發條件：**
- 找出哪些製程因子對品質 / 良率有顯著影響
- 為目標表現最佳化製程設定
- 從很多因子中篩選出關鍵的少數
**不該使用的情況：**
- 製程不穩定時（先用 SPC 穩定下來）
- 只有觀察資料、無法操控因子時
## 演算法
```
IRON LAW: One-Factor-At-A-Time (OFAT) MISSES Interactions
Changing one factor while holding others fixed cannot detect
interactions (where the effect of A depends on the level of B).
Full factorial or fractional factorial designs test ALL main effects
AND interactions in fewer runs than OFAT. A 2³ factorial (8 runs)
gives more information than 6 OFAT runs at lower cost.
```
### 階段 1：輸入驗證
定義：反應變數、因子（實務上 2-7 個）、每個因子的水準數（篩選通常用 2，最佳化用 3）、限制條件、雜訊因子。
**關卡：** 因子與水準已定義，且實際上能跑完所有實驗條件。
### 階段 2：核心演算法
**篩選（很多因子）：** 2^(k-p) 部分因子設計。選解析度 III 以上（主效應之間不會混淆）。
**最佳化（少數因子）：** 2^k 完全因子設計，或用中心合成設計（CCD）做反應曲面。
1. 產生設計矩陣（執行順序、因子水準指派）
2. 隨機化執行順序（對有效性至關重要）
3. 執行實驗，記錄反應值
4. 分析：用 ANOVA 看因子顯著性、效應圖、交互作用圖
5. 如果要最佳化：擬合反應曲面模型，找出最佳設定
### 階段 3：驗證
檢查：模型的 R² 足夠、殘差呈常態且隨機。在預測的最佳設定下做確認實驗，結果要符合預測。
**關卡：** 模型顯著、殘差正常、確認實驗通過。
### 階段 4：輸出
回傳顯著因子、效應和最佳設定。
## 輸出格式
```json
{
"significant_factors": [{"factor": "temperature", "effect": 12.5, "p_value": 0.001}, {"factor": "pressure", "effect": -8.2, "p_value": 0.008}],
"interactions": [{"factors": "temperature×time", "effect": 5.1, "p_value": 0.03}],
"optimal": {"temperature": 180, "pressure": 50, "time": 30, "predicted_response": 95.2},
"metadata": {"design": "2^3_full_factorial", "runs": 8, "replicates": 2, "r_squared": 0.94}
}
```
## 範例
### 輸入輸出範例
**輸入：** 3 個因子（溫度、壓力、時間），各 2 個水準，反應變數 = 良率
**預期：** 2³ = 8 次執行 + 重複。ANOVA 顯示溫度和溫度×壓力的交互作用顯著。
### 邊界情況
| 輸入 | 預期 | 原因 |
|-------|----------|-----|
| 7 個以上因子 | 部分因子設計 | 完全因子太貴（2⁷=128 次執行） |
| 因子有限制條件 | 受限設計 | 有些因子組合在物理上可能不可行 |
| 非線性反應 | CCD 或 Box-Behnken | 2 水準設計只能擬合線性模型 |
## 注意事項
- **隨機化很關鍵**：沒有隨機化的話，會隨時間變化的因子（操作員疲勞、環境溫度）會混淆結果。一定要隨機化執行順序。
- **Replication vs repetition**：Replication（重新設定並重跑）才能估計誤差。Repetition（同一次執行多次量測）不行。要納入真正的 replicate。
- **別名結構**：部分因子設計會混淆某些效應。解讀結果之前，要先知道哪些效應互為別名（被混淆）。
- **中心點**：在 2 水準設計中加入中心點，用很低的成本就能偵測曲率（非線性）。一定要放 3-5 個中心點。
- **實務顯著 vs 統計顯著**：一個因子可能統計上顯著（p<0.05），但實務上不重要（效應很小）。要看效應的「大小」，不能只看 p 值。
## 參考資料
- 部分因子設計表，見 `references/fractional-tables.md`
- 反應曲面法（RSM），見 `references/rsm.md`英文原文
---
name: "algo-mfg-doe"
description: "Design and analyze factorial experiments to identify significant process factors and optimize settings. Use this skill when the user needs to systematically test factor effects, optimize a manufacturing process, or determine which variables matter most — even if they say 'which factors affect quality', 'optimize process settings', or 'design an experiment'."
metadata:
category: "WP-48 製造演算法"
tags: ["manufacturing", "doe", "factorial-design", "optimization"]
---
# Design of Experiments (DOE)
## Overview
DOE systematically varies process factors to identify their effects on responses. Full factorial tests all combinations; fractional factorial tests a strategic subset. Identifies main effects and interactions. More efficient than one-factor-at-a-time (OFAT) which misses interactions. Uses ANOVA for analysis.
## When to Use
**Trigger conditions:**
- Identifying which process factors significantly affect quality/yield
- Optimizing process settings for target performance
- Screening many factors to find the vital few
**When NOT to use:**
- When the process is not stable (stabilize with SPC first)
- For observational data with no ability to manipulate factors
## Algorithm
```
IRON LAW: One-Factor-At-A-Time (OFAT) MISSES Interactions
Changing one factor while holding others fixed cannot detect
interactions (where the effect of A depends on the level of B).
Full factorial or fractional factorial designs test ALL main effects
AND interactions in fewer runs than OFAT. A 2³ factorial (8 runs)
gives more information than 6 OFAT runs at lower cost.
```
### Phase 1: Input Validation
Define: response variable(s), factors (2-7 practical), levels per factor (usually 2 for screening, 3 for optimization), constraints, noise factors.
**Gate:** Factors and levels defined, practical to run all experimental conditions.
### Phase 2: Core Algorithm
**Screening (many factors):** 2^(k-p) fractional factorial. Choose resolution III+ (main effects not confounded with each other).
**Optimization (few factors):** 2^k full factorial or central composite design (CCD) for response surface.
1. Generate design matrix (run order, factor level assignments)
2. Randomize run order (critical for validity)
3. Execute experiments, record responses
4. Analyze: ANOVA for factor significance, effect plots, interaction plots
5. If optimizing: fit response surface model, find optimal settings
### Phase 3: Verification
Check: R² of model is adequate, residuals are normally distributed and random. Confirmation runs at predicted optimal settings match prediction.
**Gate:** Model is significant, residuals OK, confirmation runs pass.
### Phase 4: Output
Return significant factors, effects, and optimal settings.
## Output Format
```json
{
"significant_factors": [{"factor": "temperature", "effect": 12.5, "p_value": 0.001}, {"factor": "pressure", "effect": -8.2, "p_value": 0.008}],
"interactions": [{"factors": "temperature×time", "effect": 5.1, "p_value": 0.03}],
"optimal": {"temperature": 180, "pressure": 50, "time": 30, "predicted_response": 95.2},
"metadata": {"design": "2^3_full_factorial", "runs": 8, "replicates": 2, "r_squared": 0.94}
}
```
## Examples
### Sample I/O
**Input:** 3 factors (temperature, pressure, time), each at 2 levels, response = yield
**Expected:** 2³ = 8 runs + replicates. ANOVA reveals temperature and temp×pressure interaction are significant.
### Edge Cases
| Input | Expected | Why |
|-------|----------|-----|
| 7+ factors | Fractional factorial | Full factorial too expensive (2⁷=128 runs) |
| Factors with constraints | Constrained design | Some factor combinations may be physically impossible |
| Non-linear response | CCD or Box-Behnken | 2-level designs only fit linear models |
## Gotchas
- **Randomization is critical**: Without randomization, time-varying factors (operator fatigue, ambient temperature) confound results. ALWAYS randomize run order.
- **Replication vs repetition**: Replication (re-setup and re-run) estimates error. Repetition (multiple measurements from one run) does not. Include true replicates.
- **Alias structure**: Fractional factorials confound some effects. Know which effects are aliased (confounded) before interpreting results.
- **Center points**: Adding center points to a 2-level design detects curvature (non-linearity) at minimal cost. Always include 3-5 center points.
- **Practical significance vs statistical significance**: A factor can be statistically significant (p<0.05) but practically unimportant (tiny effect). Focus on effect SIZE, not just p-values.
## References
- For fractional factorial design tables, see `references/fractional-tables.md`
- For response surface methodology (RSM), see `references/rsm.md`