Hard Cider Fermentation

Summary

This experiment investigates hard cider fermentation. Central composite design to maximize flavor clarity and target ABV by tuning yeast pitch rate, fermentation temperature, and sugar addition.

The design varies 3 factors: pitch rate g L (g/L), ranging from 0.5 to 2.0, ferm temp c (C), ranging from 12 to 22, and sugar add g L (g/L), ranging from 0 to 50. The goal is to optimize 2 responses: flavor clarity (pts) (maximize) and abv pct (%) (maximize). Fixed conditions held constant across all runs include apple variety = mixed_cider, yeast = champagne.

A Central Composite Design (CCD) was selected to fit a full quadratic response surface model, including curvature and interaction effects. With 3 factors this produces 22 runs including center points and axial (star) points that extend beyond the factorial range.

Quadratic response surface models were fitted to capture potential curvature and factor interactions. The RSM contour plots below visualize how pairs of factors jointly affect each response.

Key Findings

For flavor clarity, the most influential factors were pitch rate g L (46.2%), sugar add g L (31.6%), ferm temp c (22.2%). The best observed value was 8.9 (at pitch rate g L = 2.61931, ferm temp c = 17, sugar add g L = 25).

For abv pct, the most influential factors were ferm temp c (53.2%), sugar add g L (24.9%), pitch rate g L (22.0%). The best observed value was 8.8 (at pitch rate g L = 1.25, ferm temp c = 17, sugar add g L = 25).

Recommended Next Steps

Run confirmation experiments at the predicted optimal settings to validate the model.
Consider whether any fixed factors should be varied in a future study.

Experimental Setup

Factors

Factor	Low	High	Unit
`pitch_rate_g_L`	0.5	2.0	g/L
`ferm_temp_c`	12	22	C
`sugar_add_g_L`	0	50	g/L

Fixed: apple_variety = mixed_cider, yeast = champagne

Responses

Response	Direction	Unit
`flavor_clarity`	↑ maximize	pts
`abv_pct`	↑ maximize	%

Configuration

use_cases/240_cider_making/config.json

{
  "metadata": {
    "name": "Hard Cider Fermentation",
    "description": "Central composite design to maximize flavor clarity and target ABV by tuning yeast pitch rate, fermentation temperature, and sugar addition"
  },
  "factors": [
    {
      "name": "pitch_rate_g_L",
      "levels": [
        "0.5",
        "2.0"
      ],
      "type": "continuous",
      "unit": "g/L"
    },
    {
      "name": "ferm_temp_c",
      "levels": [
        "12",
        "22"
      ],
      "type": "continuous",
      "unit": "C"
    },
    {
      "name": "sugar_add_g_L",
      "levels": [
        "0",
        "50"
      ],
      "type": "continuous",
      "unit": "g/L"
    }
  ],
  "fixed_factors": {
    "apple_variety": "mixed_cider",
    "yeast": "champagne"
  },
  "responses": [
    {
      "name": "flavor_clarity",
      "optimize": "maximize",
      "unit": "pts"
    },
    {
      "name": "abv_pct",
      "optimize": "maximize",
      "unit": "%"
    }
  ],
  "settings": {
    "operation": "central_composite",
    "test_script": "use_cases/240_cider_making/sim.sh"
  }
}

Experimental Matrix

The Central Composite Design produces 22 runs. Each row is one experiment with specific factor settings.

Run	`pitch_rate_g_L`	`ferm_temp_c`	`sugar_add_g_L`
1	1.25	17	25
2	2	12	50
3	0.5	22	0
4	1.25	26.1287	25
5	1.25	17	25
6	-0.119306	17	25
7	1.25	17	-20.6435
8	1.25	17	25
9	2	22	0
10	2.61931	17	25
11	1.25	17	25
12	1.25	7.87129	25
13	1.25	17	25
14	0.5	12	50
15	1.25	17	25
16	2	12	0
17	1.25	17	70.6435
18	2	22	50
19	1.25	17	25
20	0.5	12	0
21	0.5	22	50
22	1.25	17	25

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/240_cider_making/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/240_cider_making/config.json \
    --output use_cases/240_cider_making/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/240_cider_making/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/240_cider_making/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/240_cider_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/240_cider_making/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/240_cider_making/config.json \
    --output use_cases/240_cider_making/results/report.html

Features Exercised

Feature	Value
Design type	`central_composite`
Factor types	`continuous` (all 3)
Arg style	`double-dash`
Responses	2 (flavor_clarity ↑, abv_pct ↑)
Total runs	22

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: flavor_clarity

Top factors: pitch_rate_g_L (46.2%), sugar_add_g_L (31.6%), ferm_temp_c (22.2%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
pitch_rate_g_L	4	4.1379	1.0345	1.724	0.2282
ferm_temp_c	4	2.6320	0.6580	1.097	0.4145
sugar_add_g_L	4	6.0854	1.5213	2.536	0.1134
Lack	of	Fit	2	0.0000	0.0000
Pure	Error	7	4.2000
Error	9	1.8542	0.6000
Total	21	14.7095	0.7005

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for flavor_clarity

Half-Normal Plot of Effects

Model Diagnostics

Response: abv_pct

Top factors: ferm_temp_c (53.2%), sugar_add_g_L (24.9%), pitch_rate_g_L (22.0%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
pitch_rate_g_L	4	3.6282	0.9070	0.548	0.7057
ferm_temp_c	4	12.6740	3.1685	1.913	0.1925
sugar_add_g_L	4	4.0140	1.0035	0.606	0.6685
Lack	of	Fit	2	4.7020	2.3510
Pure	Error	7	11.5950
Error	9	16.2970	1.6564
Total	21	36.6132	1.7435

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.

abv pct ferm temp c vs sugar add g L

abv pct pitch rate g L vs ferm temp c

abv pct pitch rate g L vs sugar add g L

flavor clarity ferm temp c vs sugar add g L

flavor clarity pitch rate g L vs ferm temp c

flavor clarity pitch rate g L vs sugar add g L

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 = 0.6758

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`flavor_clarity`	1.5	0.6364	7.50 0.6364 7.50 pts	↑
`abv_pct`	1.0	0.7397	7.50 0.7397 7.50 %	↑

Recommended Settings

Factor	Value
`pitch_rate_g_L`	1.25 g/L
`ferm_temp_c`	17 C
`sugar_add_g_L`	25 g/L

Source: from observed run #2

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`abv_pct`	7.50	8.80	+1.30

Top 3 Runs by Desirability

Run	D	Factor Settings
#12	0.6214	pitch_rate_g_L=1.25, ferm_temp_c=17, sugar_add_g_L=-20.6435
#14	0.5504	pitch_rate_g_L=1.25, ferm_temp_c=17, sugar_add_g_L=25

Model Quality

Response	R²	Type
`abv_pct`	0.6389	quadratic

Full Multi-Objective Output

doe optimize --multi
============================================================
MULTI-OBJECTIVE OPTIMIZATION
Method: Derringer-Suich Desirability Function
============================================================

Overall desirability: D = 0.6758

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
flavor_clarity               1.5       0.6364        7.50 pts   ↑
abv_pct                      1.0       0.7397        7.50 %   ↑

Recommended settings:
  pitch_rate_g_L = 1.25 g/L
  ferm_temp_c = 17 C
  sugar_add_g_L = 25 g/L
  (from observed run #2)

Trade-off summary:
  flavor_clarity: 7.50 (best observed: 8.90, sacrifice: +1.40)
  abv_pct: 7.50 (best observed: 8.80, sacrifice: +1.30)

Model quality:
  flavor_clarity: R² = 0.5962 (quadratic)
  abv_pct: R² = 0.6389 (quadratic)

Top 3 observed runs by overall desirability:
  1. Run #2 (D=0.6758): pitch_rate_g_L=1.25, ferm_temp_c=17, sugar_add_g_L=25
  2. Run #12 (D=0.6214): pitch_rate_g_L=1.25, ferm_temp_c=17, sugar_add_g_L=-20.6435
  3. Run #14 (D=0.5504): pitch_rate_g_L=1.25, ferm_temp_c=17, sugar_add_g_L=25

Full Analysis Output

doe analyze
=== Main Effects: flavor_clarity ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
pitch_rate_g_L           1.9250       0.1784            46.2%
sugar_add_g_L            1.3167       0.1784            31.6%
ferm_temp_c              0.9250       0.1784            22.2%

=== ANOVA Table: flavor_clarity ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
pitch_rate_g_L               4       4.1379       1.0345      1.724     0.2282
ferm_temp_c                  4       2.6320       0.6580      1.097     0.4145
sugar_add_g_L                4       6.0854       1.5213      2.536     0.1134
Lack of Fit                  2       0.0000       0.0000      0.000     1.0000
Pure Error                   7       4.2000       0.6000
Error                        9       1.8542       0.6000
Total                       21      14.7095       0.7005

=== Summary Statistics: flavor_clarity ===

pitch_rate_g_L:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  -0.119306           1     6.0000     0.0000     6.0000     6.0000
  0.5                 4     7.0250     0.2217     6.8000     7.3000
  1.25               12     6.5833     0.7234     4.9000     7.5000
  2                   4     7.2250     1.2473     5.9000     8.9000
  2.61931             1     5.3000     0.0000     5.3000     5.3000

ferm_temp_c:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  12                  4     7.2250     1.2580     5.9000     8.9000
  17                 12     6.4250     0.8125     4.9000     7.5000
  22                  4     7.0250     0.1500     6.9000     7.2000
  26.1287             1     6.3000     0.0000     6.3000     6.3000
  7.87129             1     6.9000     0.0000     6.9000     6.9000

sugar_add_g_L:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  -20.6435            1     7.5000     0.0000     7.5000     7.5000
  0                   4     6.6250     0.4856     5.9000     6.9000
  25                 12     6.3083     0.7217     4.9000     7.1000
  50                  4     7.6250     0.8539     7.1000     8.9000
  70.6435             1     7.1000     0.0000     7.1000     7.1000

=== Main Effects: abv_pct ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
ferm_temp_c              4.6000       0.2815            53.2%
sugar_add_g_L            2.1500       0.2815            24.9%
pitch_rate_g_L           1.9000       0.2815            22.0%

=== ANOVA Table: abv_pct ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
pitch_rate_g_L               4       3.6282       0.9070      0.548     0.7057
ferm_temp_c                  4      12.6740       3.1685      1.913     0.1925
sugar_add_g_L                4       4.0140       1.0035      0.606     0.6685
Lack of Fit                  2       4.7020       2.3510      1.419     0.3038
Pure Error                   7      11.5950       1.6564
Error                        9      16.2970       1.6564
Total                       21      36.6132       1.7435

=== Summary Statistics: abv_pct ===

pitch_rate_g_L:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  -0.119306           1     5.9000     0.0000     5.9000     5.9000
  0.5                 4     5.1750     1.1325     3.5000     6.0000
  1.25               12     5.7500     1.5826     3.3000     8.8000
  2                   4     5.6750     0.7274     5.0000     6.7000
  2.61931             1     4.0000     0.0000     4.0000     4.0000

ferm_temp_c:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  12                  4     5.3000     1.3880     3.5000     6.7000
  17                 12     5.4917     1.2838     3.3000     7.6000
  22                  4     5.5500     0.1000     5.4000     5.6000
  26.1287             1     8.8000     0.0000     8.8000     8.8000
  7.87129             1     4.2000     0.0000     4.2000     4.2000

sugar_add_g_L:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  -20.6435            1     7.5000     0.0000     7.5000     7.5000
  0                   4     5.3500     1.3379     3.5000     6.7000
  25                 12     5.4917     1.5582     3.3000     8.8000
  50                  4     5.5000     0.4163     5.0000     6.0000
  70.6435             1     5.5000     0.0000     5.5000     5.5000

Optimization Recommendations

doe optimize
=== Optimization: flavor_clarity ===
Direction: maximize

Best observed run: #12
  pitch_rate_g_L = 2.61931
  ferm_temp_c = 17
  sugar_add_g_L = 25
  Value: 8.9

RSM Model (linear, R² = 0.0244, Adj R² = -0.1382):
  Coefficients:
    intercept                      +6.6955
    pitch_rate_g_L                 -0.0884
    ferm_temp_c                    +0.0397
    sugar_add_g_L                  +0.1227

RSM Model (quadratic, R² = 0.3220, Adj R² = -0.1865):
  Coefficients:
    intercept                      +6.8112
    pitch_rate_g_L                 -0.0884
    ferm_temp_c                    +0.0397
    sugar_add_g_L                  +0.1227
    pitch_rate_g_L*ferm_temp_c     -0.2250
    pitch_rate_g_L*sugar_add_g_L   +0.1500
    ferm_temp_c*sugar_add_g_L      -0.4500
    pitch_rate_g_L^2               +0.1871
    ferm_temp_c^2                  -0.1729
    sugar_add_g_L^2                -0.1879

  Curvature analysis:
    sugar_add_g_L                  coef=-0.1879  concave (has a maximum)
    pitch_rate_g_L                 coef=+0.1871  convex (has a minimum)
    ferm_temp_c                    coef=-0.1729  concave (has a maximum)

  Notable interactions:
    ferm_temp_c*sugar_add_g_L      coef=-0.4500  (antagonistic)

  Predicted optimum (from linear model, at observed points):
    pitch_rate_g_L = 0.5
    ferm_temp_c = 22
    sugar_add_g_L = 50
    Predicted value: 6.9463

  Surface optimum (via L-BFGS-B, linear model):
    pitch_rate_g_L = 0.5
    ferm_temp_c = 22
    sugar_add_g_L = 50
    Predicted value: 6.9463

  Model quality: Weak fit — consider adding center points or using a different design.

Factor importance:
  1. pitch_rate_g_L  (effect: 3.3, contribution: 60.9%)
  2. sugar_add_g_L  (effect: 1.2, contribution: 21.2%)
  3. ferm_temp_c  (effect: 1.0, contribution: 17.9%)

=== Optimization: abv_pct ===
Direction: maximize

Best observed run: #17
  pitch_rate_g_L = 1.25
  ferm_temp_c = 17
  sugar_add_g_L = 25
  Value: 8.8

RSM Model (linear, R² = 0.0804, Adj R² = -0.0728):
  Coefficients:
    intercept                      +5.5591
    pitch_rate_g_L                 -0.2816
    ferm_temp_c                    -0.3356
    sugar_add_g_L                  +0.0943

RSM Model (quadratic, R² = 0.5977, Adj R² = 0.2960):
  Coefficients:
    intercept                      +6.3749
    pitch_rate_g_L                 -0.2816
    ferm_temp_c                    -0.3356
    sugar_add_g_L                  +0.0943
    pitch_rate_g_L*ferm_temp_c     -0.4250
    pitch_rate_g_L*sugar_add_g_L   +0.8000
    ferm_temp_c*sugar_add_g_L      -0.3000
    pitch_rate_g_L^2               -0.3079
    ferm_temp_c^2                  -0.7129
    sugar_add_g_L^2                -0.2029

  Curvature analysis:
    ferm_temp_c                    coef=-0.7129  concave (has a maximum)
    pitch_rate_g_L                 coef=-0.3079  concave (has a maximum)
    sugar_add_g_L                  coef=-0.2029  concave (has a maximum)

  Notable interactions:
    pitch_rate_g_L*sugar_add_g_L   coef=+0.8000  (synergistic)
    pitch_rate_g_L*ferm_temp_c     coef=-0.4250  (antagonistic)

  Predicted optimum (from quadratic model, at observed points):
    pitch_rate_g_L = 2
    ferm_temp_c = 12
    sugar_add_g_L = 50
    Predicted value: 6.8244

  Surface optimum (via L-BFGS-B, quadratic model):
    pitch_rate_g_L = 0.5
    ferm_temp_c = 18.3656
    sugar_add_g_L = 0
    Predicted value: 6.9046

  Model quality: Moderate fit — use predictions directionally, not precisely.

Factor importance:
  1. ferm_temp_c  (effect: 2.5, contribution: 61.7%)
  2. pitch_rate_g_L  (effect: 1.0, contribution: 23.2%)
  3. sugar_add_g_L  (effect: 0.6, contribution: 15.1%)