Summary
This experiment investigates handmade soap formulation. Fractional factorial screening of coconut oil ratio, olive oil ratio, lye concentration, essential oil, and cure time for lather quality and hardness.
The design varies 5 factors: coconut pct (%), ranging from 15 to 40, olive pct (%), ranging from 30 to 70, lye concentration (%), ranging from 28 to 38, essential oil pct (%), ranging from 1 to 4, and cure weeks (weeks), ranging from 4 to 8. The goal is to optimize 2 responses: lather score (pts) (maximize) and hardness score (pts) (maximize). Fixed conditions held constant across all runs include superfat pct = 5, method = cold_process.
A fractional factorial design reduces the number of runs from 32 to 8 by deliberately confounding higher-order interactions. This is ideal for screening — identifying which of the 5 factors matter most before investing in a full study.
Key Findings
For lather score, the most influential factors were essential oil pct (35.7%), lye concentration (21.4%), olive pct (18.3%). The best observed value was 7.9 (at coconut pct = 40, olive pct = 30, lye concentration = 38).
For hardness score, the most influential factors were essential oil pct (31.6%), lye concentration (30.7%), olive pct (18.4%). The best observed value was 8.4 (at coconut pct = 40, olive pct = 30, lye concentration = 38).
Recommended Next Steps
- Follow up with a response surface design (CCD or Box-Behnken) on the top 3–4 factors to model curvature and find the true optimum.
- Consider whether any fixed factors should be varied in a future study.
- The screening results can guide factor reduction — drop factors contributing less than 5% and re-run with a smaller, more focused design.
Experimental Setup
Factors
| Factor | Low | High | Unit |
|---|---|---|---|
coconut_pct | 15 | 40 | % |
olive_pct | 30 | 70 | % |
lye_concentration | 28 | 38 | % |
essential_oil_pct | 1 | 4 | % |
cure_weeks | 4 | 8 | weeks |
Fixed: superfat_pct = 5, method = cold_process
Responses
| Response | Direction | Unit |
|---|---|---|
lather_score | ↑ maximize | pts |
hardness_score | ↑ maximize | pts |
Configuration
use_cases/144_soap_making/config.json
{
"metadata": {
"name": "Handmade Soap Formulation",
"description": "Fractional factorial screening of coconut oil ratio, olive oil ratio, lye concentration, essential oil, and cure time for lather quality and hardness"
},
"factors": [
{
"name": "coconut_pct",
"levels": [
"15",
"40"
],
"type": "continuous",
"unit": "%"
},
{
"name": "olive_pct",
"levels": [
"30",
"70"
],
"type": "continuous",
"unit": "%"
},
{
"name": "lye_concentration",
"levels": [
"28",
"38"
],
"type": "continuous",
"unit": "%"
},
{
"name": "essential_oil_pct",
"levels": [
"1",
"4"
],
"type": "continuous",
"unit": "%"
},
{
"name": "cure_weeks",
"levels": [
"4",
"8"
],
"type": "continuous",
"unit": "weeks"
}
],
"fixed_factors": {
"superfat_pct": "5",
"method": "cold_process"
},
"responses": [
{
"name": "lather_score",
"optimize": "maximize",
"unit": "pts"
},
{
"name": "hardness_score",
"optimize": "maximize",
"unit": "pts"
}
],
"settings": {
"operation": "fractional_factorial",
"test_script": "use_cases/144_soap_making/sim.sh"
}
}
Experimental Matrix
The Fractional Factorial Design produces 8 runs. Each row is one experiment with specific factor settings.
| Run | coconut_pct | olive_pct | lye_concentration | essential_oil_pct | cure_weeks |
|---|---|---|---|---|---|
| 1 | 15 | 70 | 38 | 1 | 4 |
| 2 | 40 | 30 | 28 | 1 | 4 |
| 3 | 40 | 70 | 28 | 4 | 4 |
| 4 | 40 | 70 | 38 | 4 | 8 |
| 5 | 15 | 70 | 28 | 1 | 8 |
| 6 | 40 | 30 | 38 | 1 | 8 |
| 7 | 15 | 30 | 28 | 4 | 8 |
| 8 | 15 | 30 | 38 | 4 | 4 |
Step-by-Step Workflow
1
Preview the design
Terminal
$ doe info --config use_cases/144_soap_making/config.json
2
Generate the runner script
Terminal
$ doe generate --config use_cases/144_soap_making/config.json \
--output use_cases/144_soap_making/results/run.sh --seed 42
3
Execute the experiments
Terminal
$ bash use_cases/144_soap_making/results/run.sh
4
Analyze results
Terminal
$ doe analyze --config use_cases/144_soap_making/config.json
5
Get optimization recommendations
Terminal
$ doe optimize --config use_cases/144_soap_making/config.json
6
Multi-objective optimization
With 2 competing responses, use --multi to find the best compromise via Derringer–Suich desirability.
Terminal
$ doe optimize --config use_cases/144_soap_making/config.json --multi
7
Generate the HTML report
Terminal
$ doe report --config use_cases/144_soap_making/config.json \
--output use_cases/144_soap_making/results/report.html
Features Exercised
| Feature | Value |
|---|---|
| Design type | fractional_factorial |
| Factor types | continuous (all 5) |
| Arg style | double-dash |
| Responses | 2 (lather_score ↑, hardness_score ↑) |
| Total runs | 8 |
Analysis Results
Generated from actual experiment runs using the DOE Helper Tool.
Response: lather_score
Top factors: essential_oil_pct (35.7%), lye_concentration (21.4%), olive_pct (18.3%).
ANOVA
| Source | DF | SS | MS | F | p-value |
|---|---|---|---|---|---|
| Source | DF | SS | MS | F | p-value |
| coconut_pct | 1 | 0.9800 | 0.9800 | 0.403 | 0.5533 |
| olive_pct | 1 | 2.6450 | 2.6450 | 1.088 | 0.3446 |
| lye_concentration | 1 | 3.6450 | 3.6450 | 1.500 | 0.2752 |
| essential_oil_pct | 1 | 10.1250 | 10.1250 | 4.167 | 0.0967 |
| cure_weeks | 1 | 1.4450 | 1.4450 | 0.595 | 0.4755 |
| coconut_pct*olive_pct | 1 | 10.1250 | 10.1250 | 4.167 | 0.0967 |
| coconut_pct*lye_concentration | 1 | 1.4450 | 1.4450 | 0.595 | 0.4755 |
| coconut_pct*essential_oil_pct | 1 | 2.6450 | 2.6450 | 1.088 | 0.3446 |
| coconut_pct*cure_weeks | 1 | 3.6450 | 3.6450 | 1.500 | 0.2752 |
| olive_pct*lye_concentration | 1 | 1.6200 | 1.6200 | 0.667 | 0.4513 |
| olive_pct*essential_oil_pct | 1 | 0.9800 | 0.9800 | 0.403 | 0.5533 |
| olive_pct*cure_weeks | 1 | 2.0000 | 2.0000 | 0.823 | 0.4059 |
| lye_concentration*essential_oil_pct | 1 | 2.0000 | 2.0000 | 0.823 | 0.4059 |
| lye_concentration*cure_weeks | 1 | 0.9800 | 0.9800 | 0.403 | 0.5533 |
| essential_oil_pct*cure_weeks | 1 | 1.6200 | 1.6200 | 0.667 | 0.4513 |
| Error | (Lenth | PSE) | 5 | 12.1500 | 2.4300 |
| Total | 7 | 22.4600 | 3.2086 |
Pareto Chart
Main Effects Plot
Normal Probability Plot of Effects
Half-Normal Plot of Effects
Model Diagnostics
Response: hardness_score
Top factors: essential_oil_pct (31.6%), lye_concentration (30.7%), olive_pct (18.4%).
ANOVA
| Source | DF | SS | MS | F | p-value |
|---|---|---|---|---|---|
| Source | DF | SS | MS | F | p-value |
| coconut_pct | 1 | 0.7200 | 0.7200 | 0.218 | 0.6604 |
| olive_pct | 1 | 2.2050 | 2.2050 | 0.667 | 0.4513 |
| lye_concentration | 1 | 6.1250 | 6.1250 | 1.852 | 0.2317 |
| essential_oil_pct | 1 | 6.4800 | 6.4800 | 1.959 | 0.2205 |
| cure_weeks | 1 | 0.5000 | 0.5000 | 0.151 | 0.7134 |
| coconut_pct*olive_pct | 1 | 6.4800 | 6.4800 | 1.959 | 0.2205 |
| coconut_pct*lye_concentration | 1 | 0.5000 | 0.5000 | 0.151 | 0.7134 |
| coconut_pct*essential_oil_pct | 1 | 2.2050 | 2.2050 | 0.667 | 0.4513 |
| coconut_pct*cure_weeks | 1 | 6.1250 | 6.1250 | 1.852 | 0.2317 |
| olive_pct*lye_concentration | 1 | 1.1250 | 1.1250 | 0.340 | 0.5851 |
| olive_pct*essential_oil_pct | 1 | 0.7200 | 0.7200 | 0.218 | 0.6604 |
| olive_pct*cure_weeks | 1 | 4.5000 | 4.5000 | 1.361 | 0.2960 |
| lye_concentration*essential_oil_pct | 1 | 4.5000 | 4.5000 | 1.361 | 0.2960 |
| lye_concentration*cure_weeks | 1 | 0.7200 | 0.7200 | 0.218 | 0.6604 |
| essential_oil_pct*cure_weeks | 1 | 1.1250 | 1.1250 | 0.340 | 0.5851 |
| Error | (Lenth | PSE) | 5 | 16.5375 | 3.3075 |
| Total | 7 | 21.6550 | 3.0936 |
Pareto Chart
Main Effects Plot
Normal Probability Plot of Effects
Half-Normal Plot of Effects
Model Diagnostics
Response Surface Plots
3D surfaces fitted with quadratic RSM. Red dots are observed data points.
hardness score coconut pct vs cure weeks
hardness score coconut pct vs essential oil pct
hardness score coconut pct vs lye concentration
hardness score coconut pct vs olive pct
hardness score essential oil pct vs cure weeks
hardness score lye concentration vs cure weeks
hardness score lye concentration vs essential oil pct
hardness score olive pct vs cure weeks
hardness score olive pct vs essential oil pct
hardness score olive pct vs lye concentration
lather score coconut pct vs cure weeks
lather score coconut pct vs essential oil pct
lather score coconut pct vs lye concentration
lather score coconut pct vs olive pct
lather score essential oil pct vs cure weeks
lather score lye concentration vs cure weeks
lather score lye concentration vs essential oil pct
lather score olive pct vs cure weeks
lather score olive pct vs essential oil pct
lather score olive pct vs lye concentration
Multi-Objective Optimization
When responses compete, Derringer–Suich desirability finds the best compromise. Each response is scaled to a 0–1 desirability, then combined via a weighted geometric mean.
Overall Desirability
D = 0.9545
Per-Response Desirability
| Response | Weight | Desirability | Predicted | Dir |
|---|---|---|---|---|
lather_score |
1.5 |
0.9545
|
7.90 0.9545 7.90 pts | ↑ |
hardness_score |
1.5 |
0.9545
|
8.40 0.9545 8.40 pts | ↑ |
Recommended Settings
| Factor | Value |
|---|---|
coconut_pct | 40 % |
olive_pct | 70 % |
lye_concentration | 38 % |
essential_oil_pct | 4 % |
cure_weeks | 8 weeks |
Source: from observed run #6
Trade-off Summary
Sacrifice = how much worse than single-objective best.
| Response | Predicted | Best Observed | Sacrifice |
|---|---|---|---|
hardness_score | 8.40 | 8.40 | +0.00 |
Top 3 Runs by Desirability
| Run | D | Factor Settings |
|---|---|---|
| #4 | 0.7523 | coconut_pct=15, olive_pct=30, lye_concentration=38, essential_oil_pct=4, cure_weeks=4 |
| #2 | 0.5046 | coconut_pct=40, olive_pct=30, lye_concentration=28, essential_oil_pct=1, cure_weeks=4 |
Model Quality
| Response | R² | Type |
|---|---|---|
hardness_score | 0.4343 | linear |
Full Multi-Objective Output
doe optimize --multi
============================================================
MULTI-OBJECTIVE OPTIMIZATION
Method: Derringer-Suich Desirability Function
============================================================
Overall desirability: D = 0.9545
Response Weight Desirability Predicted Direction
---------------------------------------------------------------------
lather_score 1.5 0.9545 7.90 pts ↑
hardness_score 1.5 0.9545 8.40 pts ↑
Recommended settings:
coconut_pct = 40 %
olive_pct = 70 %
lye_concentration = 38 %
essential_oil_pct = 4 %
cure_weeks = 8 weeks
(from observed run #6)
Trade-off summary:
lather_score: 7.90 (best observed: 7.90, sacrifice: +0.00)
hardness_score: 8.40 (best observed: 8.40, sacrifice: +0.00)
Model quality:
lather_score: R² = 0.5931 (linear)
hardness_score: R² = 0.4343 (linear)
Top 3 observed runs by overall desirability:
1. Run #6 (D=0.9545): coconut_pct=40, olive_pct=70, lye_concentration=38, essential_oil_pct=4, cure_weeks=8
2. Run #4 (D=0.7523): coconut_pct=15, olive_pct=30, lye_concentration=38, essential_oil_pct=4, cure_weeks=4
3. Run #2 (D=0.5046): coconut_pct=40, olive_pct=30, lye_concentration=28, essential_oil_pct=1, cure_weeks=4
Full Analysis Output
doe analyze
=== Main Effects: lather_score ===
Factor Effect Std Error % Contribution
--------------------------------------------------------------
essential_oil_pct 2.2500 0.6333 35.7%
lye_concentration 1.3500 0.6333 21.4%
olive_pct -1.1500 0.6333 18.3%
cure_weeks 0.8500 0.6333 13.5%
coconut_pct 0.7000 0.6333 11.1%
=== ANOVA Table: lather_score ===
Source DF SS MS F p-value
-----------------------------------------------------------------------------
coconut_pct 1 0.9800 0.9800 0.403 0.5533
olive_pct 1 2.6450 2.6450 1.088 0.3446
lye_concentration 1 3.6450 3.6450 1.500 0.2752
essential_oil_pct 1 10.1250 10.1250 4.167 0.0967
cure_weeks 1 1.4450 1.4450 0.595 0.4755
coconut_pct*olive_pct 1 10.1250 10.1250 4.167 0.0967
coconut_pct*lye_concentration 1 1.4450 1.4450 0.595 0.4755
coconut_pct*essential_oil_pct 1 2.6450 2.6450 1.088 0.3446
coconut_pct*cure_weeks 1 3.6450 3.6450 1.500 0.2752
olive_pct*lye_concentration 1 1.6200 1.6200 0.667 0.4513
olive_pct*essential_oil_pct 1 0.9800 0.9800 0.403 0.5533
olive_pct*cure_weeks 1 2.0000 2.0000 0.823 0.4059
lye_concentration*essential_oil_pct 1 2.0000 2.0000 0.823 0.4059
lye_concentration*cure_weeks 1 0.9800 0.9800 0.403 0.5533
essential_oil_pct*cure_weeks 1 1.6200 1.6200 0.667 0.4513
Error (Lenth PSE) 5 12.1500 2.4300
Total 7 22.4600 3.2086
Note: Error estimated using Lenth's pseudo-standard-error (unreplicated design)
=== Interaction Effects: lather_score ===
Factor A Factor B Interaction % Contribution
------------------------------------------------------------------------
coconut_pct olive_pct 2.2500 20.8%
coconut_pct cure_weeks 1.3500 12.5%
coconut_pct essential_oil_pct -1.1500 10.6%
olive_pct cure_weeks 1.0000 9.3%
lye_concentration essential_oil_pct 1.0000 9.3%
olive_pct lye_concentration -0.9000 8.3%
essential_oil_pct cure_weeks -0.9000 8.3%
coconut_pct lye_concentration 0.8500 7.9%
olive_pct essential_oil_pct 0.7000 6.5%
lye_concentration cure_weeks 0.7000 6.5%
=== Summary Statistics: lather_score ===
coconut_pct:
Level N Mean Std Min Max
------------------------------------------------------------
15 4 5.0000 2.2672 2.6000 7.9000
40 4 5.7000 1.4213 4.1000 7.4000
olive_pct:
Level N Mean Std Min Max
------------------------------------------------------------
30 4 5.9250 1.5798 4.1000 7.9000
70 4 4.7750 2.0271 2.6000 7.4000
lye_concentration:
Level N Mean Std Min Max
------------------------------------------------------------
28 4 4.6750 0.7411 4.0000 5.5000
38 4 6.0250 2.3922 2.6000 7.9000
essential_oil_pct:
Level N Mean Std Min Max
------------------------------------------------------------
1 4 4.2250 1.4841 2.6000 6.2000
4 4 6.4750 1.3817 5.1000 7.9000
cure_weeks:
Level N Mean Std Min Max
------------------------------------------------------------
4 4 4.9250 2.2336 2.6000 7.9000
8 4 5.7750 1.4198 4.0000 7.4000
=== Main Effects: hardness_score ===
Factor Effect Std Error % Contribution
--------------------------------------------------------------
essential_oil_pct 1.8000 0.6218 31.6%
lye_concentration 1.7500 0.6218 30.7%
olive_pct -1.0500 0.6218 18.4%
coconut_pct 0.6000 0.6218 10.5%
cure_weeks -0.5000 0.6218 8.8%
=== ANOVA Table: hardness_score ===
Source DF SS MS F p-value
-----------------------------------------------------------------------------
coconut_pct 1 0.7200 0.7200 0.218 0.6604
olive_pct 1 2.2050 2.2050 0.667 0.4513
lye_concentration 1 6.1250 6.1250 1.852 0.2317
essential_oil_pct 1 6.4800 6.4800 1.959 0.2205
cure_weeks 1 0.5000 0.5000 0.151 0.7134
coconut_pct*olive_pct 1 6.4800 6.4800 1.959 0.2205
coconut_pct*lye_concentration 1 0.5000 0.5000 0.151 0.7134
coconut_pct*essential_oil_pct 1 2.2050 2.2050 0.667 0.4513
coconut_pct*cure_weeks 1 6.1250 6.1250 1.852 0.2317
olive_pct*lye_concentration 1 1.1250 1.1250 0.340 0.5851
olive_pct*essential_oil_pct 1 0.7200 0.7200 0.218 0.6604
olive_pct*cure_weeks 1 4.5000 4.5000 1.361 0.2960
lye_concentration*essential_oil_pct 1 4.5000 4.5000 1.361 0.2960
lye_concentration*cure_weeks 1 0.7200 0.7200 0.218 0.6604
essential_oil_pct*cure_weeks 1 1.1250 1.1250 0.340 0.5851
Error (Lenth PSE) 5 16.5375 3.3075
Total 7 21.6550 3.0936
Note: Error estimated using Lenth's pseudo-standard-error (unreplicated design)
=== Interaction Effects: hardness_score ===
Factor A Factor B Interaction % Contribution
------------------------------------------------------------------------
coconut_pct olive_pct 1.8000 16.7%
coconut_pct cure_weeks 1.7500 16.2%
olive_pct cure_weeks 1.5000 13.9%
lye_concentration essential_oil_pct 1.5000 13.9%
coconut_pct essential_oil_pct -1.0500 9.7%
olive_pct lye_concentration -0.7500 6.9%
essential_oil_pct cure_weeks -0.7500 6.9%
olive_pct essential_oil_pct 0.6000 5.6%
lye_concentration cure_weeks 0.6000 5.6%
coconut_pct lye_concentration -0.5000 4.6%
=== Summary Statistics: hardness_score ===
coconut_pct:
Level N Mean Std Min Max
------------------------------------------------------------
15 4 4.7250 2.4663 3.3000 8.4000
40 4 5.3250 0.9465 4.7000 6.7000
olive_pct:
Level N Mean Std Min Max
------------------------------------------------------------
30 4 5.5500 1.9740 3.9000 8.4000
70 4 4.5000 1.6083 3.3000 6.7000
lye_concentration:
Level N Mean Std Min Max
------------------------------------------------------------
28 4 4.1500 0.6807 3.3000 4.7000
38 4 5.9000 2.1710 3.3000 8.4000
essential_oil_pct:
Level N Mean Std Min Max
------------------------------------------------------------
1 4 4.1250 0.9743 3.3000 5.2000
4 4 5.9250 2.0271 3.9000 8.4000
cure_weeks:
Level N Mean Std Min Max
------------------------------------------------------------
4 4 5.2750 2.1854 3.3000 8.4000
8 4 4.7750 1.5086 3.3000 6.7000
Optimization Recommendations
doe optimize
=== Optimization: lather_score ===
Direction: maximize
Best observed run: #6
coconut_pct = 40
olive_pct = 30
lye_concentration = 38
essential_oil_pct = 1
cure_weeks = 8
Value: 7.9
RSM Model (linear, R² = 0.7941, Adj R² = 0.2793):
Coefficients:
intercept +5.3500
coconut_pct -0.4250
olive_pct -0.8250
lye_concentration +0.8750
essential_oil_pct -0.7500
cure_weeks +0.2000
Predicted optimum (from linear model, at observed points):
coconut_pct = 40
olive_pct = 30
lye_concentration = 38
essential_oil_pct = 1
cure_weeks = 8
Predicted value: 7.5750
Surface optimum (via L-BFGS-B, linear model):
coconut_pct = 15
olive_pct = 30
lye_concentration = 38
essential_oil_pct = 1
cure_weeks = 8
Predicted value: 8.4250
Model quality: Good fit — general trends are captured, some noise remains.
Factor importance:
1. lye_concentration (effect: 1.8, contribution: 28.5%)
2. olive_pct (effect: -1.6, contribution: 26.8%)
3. essential_oil_pct (effect: -1.5, contribution: 24.4%)
4. coconut_pct (effect: -0.9, contribution: 13.8%)
5. cure_weeks (effect: 0.4, contribution: 6.5%)
=== Optimization: hardness_score ===
Direction: maximize
Best observed run: #6
coconut_pct = 40
olive_pct = 30
lye_concentration = 38
essential_oil_pct = 1
cure_weeks = 8
Value: 8.4
RSM Model (linear, R² = 0.6839, Adj R² = -0.1063):
Coefficients:
intercept +5.0250
coconut_pct +0.2500
olive_pct -0.5250
lye_concentration +0.9000
essential_oil_pct -0.7500
cure_weeks +0.3750
Predicted optimum (from linear model, at observed points):
coconut_pct = 40
olive_pct = 30
lye_concentration = 38
essential_oil_pct = 1
cure_weeks = 8
Predicted value: 7.8250
Surface optimum (via L-BFGS-B, linear model):
coconut_pct = 40
olive_pct = 30
lye_concentration = 38
essential_oil_pct = 1
cure_weeks = 8
Predicted value: 7.8250
Model quality: Moderate fit — use predictions directionally, not precisely.
Factor importance:
1. lye_concentration (effect: 1.8, contribution: 32.1%)
2. essential_oil_pct (effect: -1.5, contribution: 26.8%)
3. olive_pct (effect: -1.0, contribution: 18.7%)
4. cure_weeks (effect: 0.8, contribution: 13.4%)
5. coconut_pct (effect: 0.5, contribution: 8.9%)