Dog Kibble Formulation — DOE Use Case

Summary

This experiment investigates dog kibble formulation. Box-Behnken design to maximize palatability and coat condition by tuning protein content, fat percentage, and fiber content.

The design varies 3 factors: protein pct (%), ranging from 20 to 35, fat pct (%), ranging from 8 to 18, and fiber pct (%), ranging from 2 to 6. The goal is to optimize 2 responses: palatability (pts) (maximize) and coat score (pts) (maximize). Fixed conditions held constant across all runs include breed size = medium, life stage = adult.

A Box-Behnken design was chosen because it efficiently fits quadratic models with 3 continuous factors while avoiding extreme corner combinations — requiring only 15 runs instead of the 8 needed for a full factorial at two levels.

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 palatability, the most influential factors were fiber pct (49.2%), protein pct (35.8%), fat pct (15.1%). The best observed value was 7.7 (at protein pct = 35, fat pct = 8, fiber pct = 4).

For coat score, the most influential factors were fiber pct (47.8%), protein pct (38.9%), fat pct (13.3%). The best observed value was 6.9 (at protein pct = 35, fat pct = 18, fiber pct = 4).

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
`protein_pct`	20	35	%
`fat_pct`	8	18	%
`fiber_pct`	2	6	%

Fixed: breed_size = medium, life_stage = adult

Responses

Response	Direction	Unit
`palatability`	↑ maximize	pts
`coat_score`	↑ maximize	pts

Configuration

use_cases/167_dog_kibble_formulation/config.json

{
  "metadata": {
    "name": "Dog Kibble Formulation",
    "description": "Box-Behnken design to maximize palatability and coat condition by tuning protein content, fat percentage, and fiber content"
  },
  "factors": [
    {
      "name": "protein_pct",
      "levels": [
        "20",
        "35"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "fat_pct",
      "levels": [
        "8",
        "18"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "fiber_pct",
      "levels": [
        "2",
        "6"
      ],
      "type": "continuous",
      "unit": "%"
    }
  ],
  "fixed_factors": {
    "breed_size": "medium",
    "life_stage": "adult"
  },
  "responses": [
    {
      "name": "palatability",
      "optimize": "maximize",
      "unit": "pts"
    },
    {
      "name": "coat_score",
      "optimize": "maximize",
      "unit": "pts"
    }
  ],
  "settings": {
    "operation": "box_behnken",
    "test_script": "use_cases/167_dog_kibble_formulation/sim.sh"
  }
}

Experimental Matrix

The Box-Behnken Design produces 15 runs. Each row is one experiment with specific factor settings.

Run	`protein_pct`	`fat_pct`	`fiber_pct`
1	27.5	8	2
2	27.5	13	4
3	35	13	6
4	35	13	2
5	27.5	13	4
6	27.5	13	4
7	20	13	6
8	35	8	4
9	27.5	8	6
10	35	18	4
11	20	13	2
12	27.5	18	6
13	20	8	4
14	20	18	4
15	27.5	18	2

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/167_dog_kibble_formulation/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/167_dog_kibble_formulation/config.json \
    --output use_cases/167_dog_kibble_formulation/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/167_dog_kibble_formulation/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/167_dog_kibble_formulation/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/167_dog_kibble_formulation/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/167_dog_kibble_formulation/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/167_dog_kibble_formulation/config.json \
    --output use_cases/167_dog_kibble_formulation/results/report.html

Features Exercised

Feature	Value
Design type	`box_behnken`
Factor types	`continuous` (all 3)
Arg style	`double-dash`
Responses	2 (palatability ↑, coat_score ↑)
Total runs	15

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: palatability

Top factors: fiber_pct (49.2%), protein_pct (35.8%), fat_pct (15.1%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
protein_pct	2	2.1097	1.0549	0.619	0.5623
fat_pct	2	0.5122	0.2561	0.150	0.8628
fiber_pct	2	3.8379	1.9190	1.127	0.3706
Lack	of	Fit	6	6.4575	1.0762
Pure	Error	2	3.4067
Error	8	9.8641	1.7033
Total	14	16.3240	1.1660

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for palatability

Half-Normal Plot of Effects

Model Diagnostics

Response: coat_score

Top factors: fiber_pct (47.8%), protein_pct (38.9%), fat_pct (13.3%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
protein_pct	2	1.5847	0.7923	0.381	0.6950
fat_pct	2	0.1858	0.0929	0.045	0.9566
fiber_pct	2	2.6675	1.3338	0.641	0.5517
Lack	of	Fit	6	1.2313	0.2052
Pure	Error	2	4.1600
Error	8	5.3913	2.0800
Total	14	9.8293	0.7021

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.

coat score fat pct vs fiber pct

coat score protein pct vs fat pct

coat score protein pct vs fiber pct

palatability fat pct vs fiber pct

palatability protein pct vs fat pct

palatability protein pct vs fiber 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 = 0.9382

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`palatability`	1.5	0.9545	7.70 0.9545 7.70 pts	↑
`coat_score`	1.5	0.9221	6.80 0.9221 6.80 pts	↑

Recommended Settings

Factor	Value
`protein_pct`	27.5 %
`fat_pct`	13 %
`fiber_pct`	4 %

Source: from observed run #10

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`coat_score`	6.80	6.90	+0.10

Top 3 Runs by Desirability

Run	D	Factor Settings
#15	0.8102	protein_pct=20, fat_pct=8, fiber_pct=4
#12	0.7983	protein_pct=27.5, fat_pct=8, fiber_pct=2

Model Quality

Response	R²	Type
`coat_score`	0.1063	linear

Full Multi-Objective Output

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

Overall desirability: D = 0.9382

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
palatability                 1.5       0.9545        7.70 pts   ↑
coat_score                   1.5       0.9221        6.80 pts   ↑

Recommended settings:
  protein_pct = 27.5 %
  fat_pct = 13 %
  fiber_pct = 4 %
  (from observed run #10)

Trade-off summary:
  palatability: 7.70 (best observed: 7.70, sacrifice: +0.00)
  coat_score: 6.80 (best observed: 6.90, sacrifice: +0.10)

Model quality:
  palatability: R² = 0.6683 (quadratic)
  coat_score: R² = 0.1063 (linear)

Top 3 observed runs by overall desirability:
  1. Run #10 (D=0.9382): protein_pct=27.5, fat_pct=13, fiber_pct=4
  2. Run #15 (D=0.8102): protein_pct=20, fat_pct=8, fiber_pct=4
  3. Run #12 (D=0.7983): protein_pct=27.5, fat_pct=8, fiber_pct=2

Full Analysis Output

doe analyze
=== Main Effects: palatability ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
fiber_pct                1.3750       0.2788            49.2%
protein_pct              1.0000       0.2788            35.8%
fat_pct                  0.4214       0.2788            15.1%

=== ANOVA Table: palatability ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
protein_pct                  2       2.1097       1.0549      0.619     0.5623
fat_pct                      2       0.5122       0.2561      0.150     0.8628
fiber_pct                    2       3.8379       1.9190      1.127     0.3706
Lack of Fit                  6       6.4575       1.0762      0.632     0.7194
Pure Error                   2       3.4067       1.7033
Error                        8       9.8641       1.7033
Total                       14      16.3240       1.1660

=== Summary Statistics: palatability ===

protein_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  20                  4     6.7000     1.2302     5.0000     7.7000
  27.5                7     6.0286     0.9844     4.4000     7.0000
  35                  4     5.7000     1.1343     4.2000     6.8000

fat_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  13                  7     5.9286     0.9725     4.4000     6.9000
  18                  4     6.3500     1.0344     5.0000     7.5000
  8                   4     6.2250     1.5196     4.2000     7.7000

fiber_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4     5.3750     0.4787     5.0000     6.0000
  4                   7     6.1857     1.3957     4.2000     7.7000
  6                   4     6.7500     0.1915     6.6000     7.0000

=== Main Effects: coat_score ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
fiber_pct                1.0750       0.2163            47.8%
protein_pct              0.8750       0.2163            38.9%
fat_pct                  0.3000       0.2163            13.3%

=== ANOVA Table: coat_score ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
protein_pct                  2       1.5847       0.7923      0.381     0.6950
fat_pct                      2       0.1858       0.0929      0.045     0.9566
fiber_pct                    2       2.6675       1.3338      0.641     0.5517
Lack of Fit                  6       1.2313       0.2052      0.099     0.9881
Pure Error                   2       4.1600       2.0800
Error                        8       5.3913       2.0800
Total                       14       9.8293       0.7021

=== Summary Statistics: coat_score ===

protein_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  20                  4     6.1000     0.5292     5.6000     6.8000
  27.5                7     5.5429     1.0533     4.1000     6.9000
  35                  4     5.2250     0.4992     4.7000     5.9000

fat_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  13                  7     5.5857     0.9245     4.1000     6.9000
  18                  4     5.4750     0.6397     4.8000     6.2000
  8                   4     5.7750     1.0468     4.6000     6.8000

fiber_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4     4.9250     0.4573     4.6000     5.6000
  4                   7     5.7714     1.0161     4.1000     6.9000
  6                   4     6.0000     0.3367     5.8000     6.5000

Optimization Recommendations

doe optimize
=== Optimization: palatability ===
Direction: maximize

Best observed run: #10
  protein_pct = 35
  fat_pct = 8
  fiber_pct = 4
  Value: 7.7

RSM Model (linear, R² = 0.0466, Adj R² = -0.2135):
  Coefficients:
    intercept                      +6.1200
    protein_pct                    -0.1000
    fat_pct                        +0.1500
    fiber_pct                      -0.2500

RSM Model (quadratic, R² = 0.6997, Adj R² = 0.1592):
  Coefficients:
    intercept                      +6.3667
    protein_pct                    -0.1000
    fat_pct                        +0.1500
    fiber_pct                      -0.2500
    protein_pct*fat_pct            -0.5250
    protein_pct*fiber_pct          -0.8250
    fat_pct*fiber_pct              -0.7250
    protein_pct^2                  +0.5792
    fat_pct^2                      -0.1208
    fiber_pct^2                    -0.9208

  Curvature analysis:
    fiber_pct                      coef=-0.9208  concave (has a maximum)
    protein_pct                    coef=+0.5792  convex (has a minimum)
    fat_pct                        coef=-0.1208  concave (has a maximum)

  Notable interactions:
    protein_pct*fiber_pct          coef=-0.8250  (antagonistic)
    fat_pct*fiber_pct              coef=-0.7250  (antagonistic)
    protein_pct*fat_pct            coef=-0.5250  (antagonistic)

  Predicted optimum (from quadratic model, at observed points):
    protein_pct = 20
    fat_pct = 18
    fiber_pct = 4
    Predicted value: 7.6000

  Surface optimum (via L-BFGS-B, quadratic model):
    protein_pct = 20
    fat_pct = 18
    fiber_pct = 3.8371
    Predicted value: 7.6061

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

Factor importance:
  1. fiber_pct  (effect: 1.2, contribution: 53.3%)
  2. protein_pct  (effect: 0.8, contribution: 33.4%)
  3. fat_pct  (effect: 0.3, contribution: 13.3%)

=== Optimization: coat_score ===
Direction: maximize

Best observed run: #3
  protein_pct = 35
  fat_pct = 18
  fiber_pct = 4
  Value: 6.9

RSM Model (linear, R² = 0.1806, Adj R² = -0.0429):
  Coefficients:
    intercept                      +5.6067
    protein_pct                    +0.2250
    fat_pct                        +0.3750
    fiber_pct                      -0.1750

RSM Model (quadratic, R² = 0.8209, Adj R² = 0.4986):
  Coefficients:
    intercept                      +5.6000
    protein_pct                    +0.2250
    fat_pct                        +0.3750
    fiber_pct                      -0.1750
    protein_pct*fat_pct            -0.3250
    protein_pct*fiber_pct          -0.8250
    fat_pct*fiber_pct              +0.0750
    protein_pct^2                  +0.2875
    fat_pct^2                      +0.4375
    fiber_pct^2                    -0.7125

  Curvature analysis:
    fiber_pct                      coef=-0.7125  concave (has a maximum)
    fat_pct                        coef=+0.4375  convex (has a minimum)
    protein_pct                    coef=+0.2875  convex (has a minimum)

  Notable interactions:
    protein_pct*fiber_pct          coef=-0.8250  (antagonistic)
    protein_pct*fat_pct            coef=-0.3250  (antagonistic)

  Predicted optimum (from quadratic model, at observed points):
    protein_pct = 20
    fat_pct = 18
    fiber_pct = 4
    Predicted value: 6.8000

  Surface optimum (via L-BFGS-B, quadratic model):
    protein_pct = 20
    fat_pct = 18
    fiber_pct = 5.01754
    Predicted value: 6.9844

  Model quality: Good fit — general trends are captured, some noise remains.

Factor importance:
  1. fiber_pct  (effect: 0.9, contribution: 40.6%)
  2. fat_pct  (effect: 0.8, contribution: 36.4%)
  3. protein_pct  (effect: 0.5, contribution: 23.0%)