Summary
This experiment investigates fermented hot sauce formulation. Fractional factorial screening of pepper type, salt concentration, garlic ratio, fermentation days, and vinegar addition for heat balance and umami depth.
The design varies 5 factors: pepper shu (SHU), ranging from 5000 to 100000, salt pct (%), ranging from 2 to 6, garlic pct (%), ranging from 2 to 10, ferm days (days), ranging from 7 to 90, and vinegar pct (%), ranging from 5 to 25. The goal is to optimize 2 responses: heat balance (pts) (maximize) and umami depth (pts) (maximize). Fixed conditions held constant across all runs include ferm temp = 22, jar size = 1L.
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 heat balance, the most influential factors were vinegar pct (50.8%), garlic pct (35.1%), pepper shu (13.1%). The best observed value was 6.5 (at pepper shu = 5000, salt pct = 6, garlic pct = 10).
For umami depth, the most influential factors were vinegar pct (46.2%), ferm days (23.8%), garlic pct (11.9%). The best observed value was 8.8 (at pepper shu = 5000, salt pct = 2, garlic pct = 2).
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 |
|---|---|---|---|
pepper_shu | 5000 | 100000 | SHU |
salt_pct | 2 | 6 | % |
garlic_pct | 2 | 10 | % |
ferm_days | 7 | 90 | days |
vinegar_pct | 5 | 25 | % |
Fixed: ferm_temp = 22, jar_size = 1L
Responses
| Response | Direction | Unit |
|---|---|---|
heat_balance | ↑ maximize | pts |
umami_depth | ↑ maximize | pts |
Configuration
use_cases/96_fermented_hot_sauce/config.json
{
"metadata": {
"name": "Fermented Hot Sauce Formulation",
"description": "Fractional factorial screening of pepper type, salt concentration, garlic ratio, fermentation days, and vinegar addition for heat balance and umami depth"
},
"factors": [
{
"name": "pepper_shu",
"levels": [
"5000",
"100000"
],
"type": "continuous",
"unit": "SHU"
},
{
"name": "salt_pct",
"levels": [
"2",
"6"
],
"type": "continuous",
"unit": "%"
},
{
"name": "garlic_pct",
"levels": [
"2",
"10"
],
"type": "continuous",
"unit": "%"
},
{
"name": "ferm_days",
"levels": [
"7",
"90"
],
"type": "continuous",
"unit": "days"
},
{
"name": "vinegar_pct",
"levels": [
"5",
"25"
],
"type": "continuous",
"unit": "%"
}
],
"fixed_factors": {
"ferm_temp": "22",
"jar_size": "1L"
},
"responses": [
{
"name": "heat_balance",
"optimize": "maximize",
"unit": "pts"
},
{
"name": "umami_depth",
"optimize": "maximize",
"unit": "pts"
}
],
"settings": {
"operation": "fractional_factorial",
"test_script": "use_cases/96_fermented_hot_sauce/sim.sh"
}
}
Experimental Matrix
The Fractional Factorial Design produces 8 runs. Each row is one experiment with specific factor settings.
| Run | pepper_shu | salt_pct | garlic_pct | ferm_days | vinegar_pct |
|---|---|---|---|---|---|
| 1 | 5000 | 6 | 10 | 7 | 5 |
| 2 | 100000 | 2 | 2 | 7 | 5 |
| 3 | 100000 | 6 | 2 | 90 | 5 |
| 4 | 100000 | 6 | 10 | 90 | 25 |
| 5 | 5000 | 6 | 2 | 7 | 25 |
| 6 | 100000 | 2 | 10 | 7 | 25 |
| 7 | 5000 | 2 | 2 | 90 | 25 |
| 8 | 5000 | 2 | 10 | 90 | 5 |
Step-by-Step Workflow
1
Preview the design
Terminal
$ doe info --config use_cases/96_fermented_hot_sauce/config.json
2
Generate the runner script
Terminal
$ doe generate --config use_cases/96_fermented_hot_sauce/config.json \
--output use_cases/96_fermented_hot_sauce/results/run.sh --seed 42
3
Execute the experiments
Terminal
$ bash use_cases/96_fermented_hot_sauce/results/run.sh
4
Analyze results
Terminal
$ doe analyze --config use_cases/96_fermented_hot_sauce/config.json
5
Get optimization recommendations
Terminal
$ doe optimize --config use_cases/96_fermented_hot_sauce/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/96_fermented_hot_sauce/config.json --multi
7
Generate the HTML report
Terminal
$ doe report --config use_cases/96_fermented_hot_sauce/config.json \
--output use_cases/96_fermented_hot_sauce/results/report.html
Features Exercised
| Feature | Value |
|---|---|
| Design type | fractional_factorial |
| Factor types | continuous (all 5) |
| Arg style | double-dash |
| Responses | 2 (heat_balance ↑, umami_depth ↑) |
| Total runs | 8 |
Analysis Results
Generated from actual experiment runs using the DOE Helper Tool.
Response: heat_balance
Top factors: vinegar_pct (50.8%), garlic_pct (35.1%), pepper_shu (13.1%).
ANOVA
| Source | DF | SS | MS | F | p-value |
|---|---|---|---|---|---|
| Source | DF | SS | MS | F | p-value |
| pepper_shu | 1 | 0.7812 | 0.7812 | 32.051 | 0.0024 |
| salt_pct | 1 | 0.0013 | 0.0013 | 0.051 | 0.8298 |
| garlic_pct | 1 | 5.6113 | 5.6113 | 230.205 | 0.0000 |
| ferm_days | 1 | 0.0013 | 0.0013 | 0.051 | 0.8298 |
| vinegar_pct | 1 | 11.7613 | 11.7613 | 482.513 | 0.0000 |
| pepper_shu*salt_pct | 1 | 0.0012 | 0.0012 | 0.051 | 0.8298 |
| pepper_shu*garlic_pct | 1 | 11.7612 | 11.7612 | 482.513 | 0.0000 |
| pepper_shu*ferm_days | 1 | 0.0012 | 0.0012 | 0.051 | 0.8298 |
| pepper_shu*vinegar_pct | 1 | 5.6112 | 5.6112 | 230.205 | 0.0000 |
| salt_pct*garlic_pct | 1 | 0.0312 | 0.0312 | 1.282 | 0.3089 |
| salt_pct*ferm_days | 1 | 0.7812 | 0.7812 | 32.051 | 0.0024 |
| salt_pct*vinegar_pct | 1 | 0.1512 | 0.1512 | 6.205 | 0.0551 |
| garlic_pct*ferm_days | 1 | 0.1512 | 0.1512 | 6.205 | 0.0551 |
| garlic_pct*vinegar_pct | 1 | 0.7812 | 0.7812 | 32.051 | 0.0024 |
| ferm_days*vinegar_pct | 1 | 0.0312 | 0.0312 | 1.282 | 0.3089 |
| Error | (Lenth | PSE) | 5 | 0.1219 | 0.0244 |
| Total | 7 | 18.3387 | 2.6198 |
Pareto Chart
Main Effects Plot
Normal Probability Plot of Effects
Half-Normal Plot of Effects
Model Diagnostics
Response: umami_depth
Top factors: vinegar_pct (46.2%), ferm_days (23.8%), garlic_pct (11.9%).
ANOVA
| Source | DF | SS | MS | F | p-value |
|---|---|---|---|---|---|
| Source | DF | SS | MS | F | p-value |
| pepper_shu | 1 | 1.1250 | 1.1250 | 0.667 | 0.4513 |
| salt_pct | 1 | 0.6050 | 0.6050 | 0.359 | 0.5754 |
| garlic_pct | 1 | 1.4450 | 1.4450 | 0.856 | 0.3972 |
| ferm_days | 1 | 5.7800 | 5.7800 | 3.425 | 0.1234 |
| vinegar_pct | 1 | 21.7800 | 21.7800 | 12.907 | 0.0157 |
| pepper_shu*salt_pct | 1 | 5.7800 | 5.7800 | 3.425 | 0.1234 |
| pepper_shu*garlic_pct | 1 | 21.7800 | 21.7800 | 12.907 | 0.0157 |
| pepper_shu*ferm_days | 1 | 0.6050 | 0.6050 | 0.359 | 0.5754 |
| pepper_shu*vinegar_pct | 1 | 1.4450 | 1.4450 | 0.856 | 0.3972 |
| salt_pct*garlic_pct | 1 | 2.0000 | 2.0000 | 1.185 | 0.3260 |
| salt_pct*ferm_days | 1 | 1.1250 | 1.1250 | 0.667 | 0.4513 |
| salt_pct*vinegar_pct | 1 | 0.6050 | 0.6050 | 0.359 | 0.5754 |
| garlic_pct*ferm_days | 1 | 0.6050 | 0.6050 | 0.359 | 0.5754 |
| garlic_pct*vinegar_pct | 1 | 1.1250 | 1.1250 | 0.667 | 0.4513 |
| ferm_days*vinegar_pct | 1 | 2.0000 | 2.0000 | 1.185 | 0.3260 |
| Error | (Lenth | PSE) | 5 | 8.4375 | 1.6875 |
| Total | 7 | 33.3400 | 4.7629 |
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.
heat balance ferm days vs vinegar pct
heat balance garlic pct vs ferm days
heat balance garlic pct vs vinegar pct
heat balance pepper shu vs ferm days
heat balance pepper shu vs garlic pct
heat balance pepper shu vs salt pct
heat balance pepper shu vs vinegar pct
heat balance salt pct vs ferm days
heat balance salt pct vs garlic pct
heat balance salt pct vs vinegar pct
umami depth ferm days vs vinegar pct
umami depth garlic pct vs ferm days
umami depth garlic pct vs vinegar pct
umami depth pepper shu vs ferm days
umami depth pepper shu vs garlic pct
umami depth pepper shu vs salt pct
umami depth pepper shu vs vinegar pct
umami depth salt pct vs ferm days
umami depth salt pct vs garlic pct
umami depth salt pct vs vinegar pct
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 = 0.8245
Per-Response Desirability
| Response | Weight | Desirability | Predicted | Dir |
|---|---|---|---|---|
heat_balance |
1.5 |
0.7121
|
5.30 0.7121 5.30 pts | ↑ |
umami_depth |
1.5 |
0.9545
|
8.80 0.9545 8.80 pts | ↑ |
Recommended Settings
| Factor | Value |
|---|---|
pepper_shu | 5000 SHU |
salt_pct | 2 % |
garlic_pct | 10 % |
ferm_days | 90 days |
vinegar_pct | 5 % |
Source: from observed run #4
Trade-off Summary
Sacrifice = how much worse than single-objective best.
| Response | Predicted | Best Observed | Sacrifice |
|---|---|---|---|
umami_depth | 8.80 | 8.80 | +0.00 |
Top 3 Runs by Desirability
| Run | D | Factor Settings |
|---|---|---|
| #8 | 0.7971 | pepper_shu=100000, salt_pct=6, garlic_pct=10, ferm_days=90, vinegar_pct=25 |
| #3 | 0.6286 | pepper_shu=100000, salt_pct=2, garlic_pct=2, ferm_days=7, vinegar_pct=5 |
Model Quality
| Response | R² | Type |
|---|---|---|
umami_depth | 0.5187 | linear |
Full Multi-Objective Output
doe optimize --multi
============================================================
MULTI-OBJECTIVE OPTIMIZATION
Method: Derringer-Suich Desirability Function
============================================================
Overall desirability: D = 0.8245
Response Weight Desirability Predicted Direction
---------------------------------------------------------------------
heat_balance 1.5 0.7121 5.30 pts ↑
umami_depth 1.5 0.9545 8.80 pts ↑
Recommended settings:
pepper_shu = 5000 SHU
salt_pct = 2 %
garlic_pct = 10 %
ferm_days = 90 days
vinegar_pct = 5 %
(from observed run #4)
Trade-off summary:
heat_balance: 5.30 (best observed: 6.50, sacrifice: +1.20)
umami_depth: 8.80 (best observed: 8.80, sacrifice: +0.00)
Model quality:
heat_balance: R² = 0.5593 (linear)
umami_depth: R² = 0.5187 (linear)
Top 3 observed runs by overall desirability:
1. Run #4 (D=0.8245): pepper_shu=5000, salt_pct=2, garlic_pct=10, ferm_days=90, vinegar_pct=5
2. Run #8 (D=0.7971): pepper_shu=100000, salt_pct=6, garlic_pct=10, ferm_days=90, vinegar_pct=25
3. Run #3 (D=0.6286): pepper_shu=100000, salt_pct=2, garlic_pct=2, ferm_days=7, vinegar_pct=5
Full Analysis Output
doe analyze
=== Main Effects: heat_balance ===
Factor Effect Std Error % Contribution
--------------------------------------------------------------
vinegar_pct -2.4250 0.5723 50.8%
garlic_pct 1.6750 0.5723 35.1%
pepper_shu -0.6250 0.5723 13.1%
salt_pct -0.0250 0.5723 0.5%
ferm_days -0.0250 0.5723 0.5%
=== ANOVA Table: heat_balance ===
Source DF SS MS F p-value
-----------------------------------------------------------------------------
pepper_shu 1 0.7812 0.7812 32.051 0.0024
salt_pct 1 0.0013 0.0013 0.051 0.8298
garlic_pct 1 5.6113 5.6113 230.205 0.0000
ferm_days 1 0.0013 0.0013 0.051 0.8298
vinegar_pct 1 11.7613 11.7613 482.513 0.0000
pepper_shu*salt_pct 1 0.0012 0.0012 0.051 0.8298
pepper_shu*garlic_pct 1 11.7612 11.7612 482.513 0.0000
pepper_shu*ferm_days 1 0.0012 0.0012 0.051 0.8298
pepper_shu*vinegar_pct 1 5.6112 5.6112 230.205 0.0000
salt_pct*garlic_pct 1 0.0312 0.0312 1.282 0.3089
salt_pct*ferm_days 1 0.7812 0.7812 32.051 0.0024
salt_pct*vinegar_pct 1 0.1512 0.1512 6.205 0.0551
garlic_pct*ferm_days 1 0.1512 0.1512 6.205 0.0551
garlic_pct*vinegar_pct 1 0.7812 0.7812 32.051 0.0024
ferm_days*vinegar_pct 1 0.0312 0.0312 1.282 0.3089
Error (Lenth PSE) 5 0.1219 0.0244
Total 7 18.3387 2.6198
Note: Error estimated using Lenth's pseudo-standard-error (unreplicated design)
=== Interaction Effects: heat_balance ===
Factor A Factor B Interaction % Contribution
------------------------------------------------------------------------
pepper_shu garlic_pct 2.4250 39.1%
pepper_shu vinegar_pct -1.6750 27.0%
salt_pct ferm_days 0.6250 10.1%
garlic_pct vinegar_pct 0.6250 10.1%
salt_pct vinegar_pct -0.2750 4.4%
garlic_pct ferm_days -0.2750 4.4%
salt_pct garlic_pct 0.1250 2.0%
ferm_days vinegar_pct 0.1250 2.0%
pepper_shu salt_pct 0.0250 0.4%
pepper_shu ferm_days 0.0250 0.4%
=== Summary Statistics: heat_balance ===
pepper_shu:
Level N Mean Std Min Max
------------------------------------------------------------
100000 4 4.8750 0.4425 4.4000 5.3000
5000 4 4.2500 2.3784 2.0000 6.5000
salt_pct:
Level N Mean Std Min Max
------------------------------------------------------------
2 4 4.5750 1.5756 2.4000 6.1000
6 4 4.5500 1.9053 2.0000 6.5000
garlic_pct:
Level N Mean Std Min Max
------------------------------------------------------------
10 4 3.7250 1.7689 2.0000 5.3000
2 4 5.4000 1.0551 4.4000 6.5000
ferm_days:
Level N Mean Std Min Max
------------------------------------------------------------
7 4 4.5750 1.8910 2.0000 6.5000
90 4 4.5500 1.5927 2.4000 6.1000
vinegar_pct:
Level N Mean Std Min Max
------------------------------------------------------------
25 4 5.7750 0.6292 5.2000 6.5000
5 4 3.3500 1.3404 2.0000 4.6000
=== Main Effects: umami_depth ===
Factor Effect Std Error % Contribution
--------------------------------------------------------------
vinegar_pct -3.3000 0.7716 46.2%
ferm_days 1.7000 0.7716 23.8%
garlic_pct -0.8500 0.7716 11.9%
pepper_shu -0.7500 0.7716 10.5%
salt_pct 0.5500 0.7716 7.7%
=== ANOVA Table: umami_depth ===
Source DF SS MS F p-value
-----------------------------------------------------------------------------
pepper_shu 1 1.1250 1.1250 0.667 0.4513
salt_pct 1 0.6050 0.6050 0.359 0.5754
garlic_pct 1 1.4450 1.4450 0.856 0.3972
ferm_days 1 5.7800 5.7800 3.425 0.1234
vinegar_pct 1 21.7800 21.7800 12.907 0.0157
pepper_shu*salt_pct 1 5.7800 5.7800 3.425 0.1234
pepper_shu*garlic_pct 1 21.7800 21.7800 12.907 0.0157
pepper_shu*ferm_days 1 0.6050 0.6050 0.359 0.5754
pepper_shu*vinegar_pct 1 1.4450 1.4450 0.856 0.3972
salt_pct*garlic_pct 1 2.0000 2.0000 1.185 0.3260
salt_pct*ferm_days 1 1.1250 1.1250 0.667 0.4513
salt_pct*vinegar_pct 1 0.6050 0.6050 0.359 0.5754
garlic_pct*ferm_days 1 0.6050 0.6050 0.359 0.5754
garlic_pct*vinegar_pct 1 1.1250 1.1250 0.667 0.4513
ferm_days*vinegar_pct 1 2.0000 2.0000 1.185 0.3260
Error (Lenth PSE) 5 8.4375 1.6875
Total 7 33.3400 4.7629
Note: Error estimated using Lenth's pseudo-standard-error (unreplicated design)
=== Interaction Effects: umami_depth ===
Factor A Factor B Interaction % Contribution
------------------------------------------------------------------------
pepper_shu garlic_pct 3.3000 30.0%
pepper_shu salt_pct -1.7000 15.5%
salt_pct garlic_pct -1.0000 9.1%
ferm_days vinegar_pct -1.0000 9.1%
pepper_shu vinegar_pct 0.8500 7.7%
salt_pct ferm_days 0.7500 6.8%
garlic_pct vinegar_pct 0.7500 6.8%
pepper_shu ferm_days -0.5500 5.0%
salt_pct vinegar_pct 0.5500 5.0%
garlic_pct ferm_days 0.5500 5.0%
=== Summary Statistics: umami_depth ===
pepper_shu:
Level N Mean Std Min Max
------------------------------------------------------------
100000 4 5.3750 2.7379 2.4000 8.8000
5000 4 4.6250 1.8007 3.2000 7.2000
salt_pct:
Level N Mean Std Min Max
------------------------------------------------------------
2 4 4.7250 2.2911 2.4000 7.2000
6 4 5.2750 2.3796 3.6000 8.8000
garlic_pct:
Level N Mean Std Min Max
------------------------------------------------------------
10 4 5.4250 2.5902 3.2000 8.8000
2 4 4.5750 1.9805 2.4000 7.2000
ferm_days:
Level N Mean Std Min Max
------------------------------------------------------------
7 4 4.1500 1.5588 2.4000 6.1000
90 4 5.8500 2.5994 3.2000 8.8000
vinegar_pct:
Level N Mean Std Min Max
------------------------------------------------------------
25 4 6.6500 1.8120 4.5000 8.8000
5 4 3.3500 0.7550 2.4000 4.2000
Optimization Recommendations
doe optimize
=== Optimization: heat_balance ===
Direction: maximize
Best observed run: #1
pepper_shu = 5000
salt_pct = 6
garlic_pct = 10
ferm_days = 7
vinegar_pct = 5
Value: 6.5
RSM Model (linear, R² = 0.5445, Adj R² = -0.5941):
Coefficients:
intercept +4.5625
pepper_shu -0.1875
salt_pct -0.3125
garlic_pct +0.0625
ferm_days +0.1875
vinegar_pct -1.0375
Predicted optimum (from linear model, at observed points):
pepper_shu = 5000
salt_pct = 2
garlic_pct = 10
ferm_days = 90
vinegar_pct = 5
Predicted value: 6.3500
Surface optimum (via L-BFGS-B, linear model):
pepper_shu = 5000
salt_pct = 2
garlic_pct = 10
ferm_days = 90
vinegar_pct = 5
Predicted value: 6.3500
Model quality: Moderate fit — use predictions directionally, not precisely.
Factor importance:
1. vinegar_pct (effect: 2.1, contribution: 58.0%)
2. salt_pct (effect: -0.6, contribution: 17.5%)
3. pepper_shu (effect: 0.4, contribution: 10.5%)
4. ferm_days (effect: 0.4, contribution: 10.5%)
5. garlic_pct (effect: -0.1, contribution: 3.5%)
=== Optimization: umami_depth ===
Direction: maximize
Best observed run: #4
pepper_shu = 5000
salt_pct = 2
garlic_pct = 2
ferm_days = 90
vinegar_pct = 25
Value: 8.8
RSM Model (linear, R² = 0.6506, Adj R² = -0.2230):
Coefficients:
intercept +5.0000
pepper_shu -0.7500
salt_pct -0.3750
garlic_pct -0.5000
ferm_days +1.3250
vinegar_pct -0.0500
Predicted optimum (from linear model, at observed points):
pepper_shu = 5000
salt_pct = 2
garlic_pct = 2
ferm_days = 90
vinegar_pct = 25
Predicted value: 7.9000
Surface optimum (via L-BFGS-B, linear model):
pepper_shu = 5000
salt_pct = 2
garlic_pct = 2
ferm_days = 90
vinegar_pct = 5
Predicted value: 8.0000
Model quality: Moderate fit — use predictions directionally, not precisely.
Factor importance:
1. ferm_days (effect: 2.7, contribution: 44.2%)
2. pepper_shu (effect: 1.5, contribution: 25.0%)
3. garlic_pct (effect: 1.0, contribution: 16.7%)
4. salt_pct (effect: -0.8, contribution: 12.5%)
5. vinegar_pct (effect: 0.1, contribution: 1.7%)