Essential Oil Therapeutic Blend

Summary

This experiment investigates essential oil therapeutic blend. Mixture simplex centroid design for optimizing a 4-component essential oil blend.

The design varies 4 factors: lavender (%), ranging from 0 to 100, eucalyptus (%), ranging from 0 to 100, peppermint (%), ranging from 0 to 100, and tea tree (%), ranging from 0 to 100. The goal is to optimize 2 responses: relaxation score (pts) (maximize) and antimicrobial zone (mm) (maximize). Fixed conditions held constant across all runs include carrier oil = jojoba, dilution = 5%.

The Simplex-Centroid Mixture Design produces 15 experimental runs.

Key Findings

For relaxation score, the most influential factors were eucalyptus (30.1%), lavender (25.6%), peppermint (22.2%). The best observed value was 86.7 (at lavender = 0, eucalyptus = 33.3333, peppermint = 33.3333).

For antimicrobial zone, the most influential factors were eucalyptus (31.5%), lavender (26.0%), tea tree (21.4%). The best observed value was 29.0 (at lavender = 50, eucalyptus = 50, peppermint = 0).

Recommended Next Steps

Consider whether any fixed factors should be varied in a future study.

Experimental Setup

Factors

Factor	High	Unit
`lavender`	100	%
`eucalyptus`	100	%
`peppermint`	100	%
`tea_tree`	100	%

Fixed: carrier_oil = jojoba, dilution = 5%

Responses

Response	Direction	Unit
`relaxation_score`	↑ maximize	pts
`antimicrobial_zone`	↑ maximize	mm

Configuration

use_cases/305_essential_oil_blend/config.json

{
  "metadata": {
    "name": "Essential Oil Therapeutic Blend",
    "description": "Mixture simplex centroid design for optimizing a 4-component essential oil blend"
  },
  "factors": [
    {
      "name": "lavender",
      "levels": [
        "0",
        "100"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "eucalyptus",
      "levels": [
        "0",
        "100"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "peppermint",
      "levels": [
        "0",
        "100"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "tea_tree",
      "levels": [
        "0",
        "100"
      ],
      "type": "continuous",
      "unit": "%"
    }
  ],
  "fixed_factors": {
    "carrier_oil": "jojoba",
    "dilution": "5%"
  },
  "responses": [
    {
      "name": "relaxation_score",
      "optimize": "maximize",
      "unit": "pts"
    },
    {
      "name": "antimicrobial_zone",
      "optimize": "maximize",
      "unit": "mm"
    }
  ],
  "settings": {
    "operation": "mixture_simplex_centroid",
    "test_script": "use_cases/305_essential_oil_blend/sim.sh"
  }
}

Experimental Matrix

The Simplex-Centroid Mixture Design produces 15 runs. Each row is one experiment with specific factor settings.

Run	`lavender`	`eucalyptus`	`peppermint`	`tea_tree`
1	0	50	0	50
2	0	33.3333	33.3333	33.3333
3	0	50	50	0
4	50	0	0	50
5	25	25	25	25
6	33.3333	0	33.3333	33.3333
7	50	0	50	0
8	0	0	100	0
9	0	0	50	50
10	0	0	0	100
11	50	50	0	0
12	33.3333	33.3333	0	33.3333
13	100	0	0	0
14	0	100	0	0
15	33.3333	33.3333	33.3333	0

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/305_essential_oil_blend/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/305_essential_oil_blend/config.json \
    --output use_cases/305_essential_oil_blend/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/305_essential_oil_blend/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/305_essential_oil_blend/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/305_essential_oil_blend/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/305_essential_oil_blend/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/305_essential_oil_blend/config.json \
    --output use_cases/305_essential_oil_blend/results/report.html

Features Exercised

Feature	Value
Design type	`mixture_simplex_centroid`
Factor types	`continuous` (all 4)
Arg style	`double-dash`
Responses	2 (relaxation_score ↑, antimicrobial_zone ↑)
Total runs	15

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: relaxation_score

Top factors: eucalyptus (30.1%), lavender (25.6%), peppermint (22.2%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
lavender	4	1317.2221	329.3055
eucalyptus	4	2116.1897	529.0474
peppermint	4	1273.8021	318.4505
tea_tree	4	941.4164	235.3541
Error	(Lenth	PSE)	0	0.0000	0.0000
Total	14	4487.6773	320.5484

Response: antimicrobial_zone

Top factors: eucalyptus (31.5%), lavender (26.0%), tea_tree (21.4%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
lavender	4	109.0248	27.2562
eucalyptus	4	187.9924	46.9981
peppermint	4	81.1867	20.2967
tea_tree	4	58.0714	14.5179
Error	(Lenth	PSE)	0	20.5448	0.0000
Total	14	456.8200	32.6300

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 = 0.5422

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`relaxation_score`	1.5	0.5803	61.30 0.5803 61.30 pts	↑
`antimicrobial_zone`	1.0	0.4896	17.80 0.4896 17.80 mm	↑

Recommended Settings

Factor	Value
`lavender`	0 %
`eucalyptus`	0 %
`peppermint`	100 %
`tea_tree`	0 %

Source: from observed run #6

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`antimicrobial_zone`	17.80	29.00	+11.20

Top 3 Runs by Desirability

Run	D	Factor Settings
#4	0.5400	lavender=0, eucalyptus=33.3333, peppermint=33.3333, tea_tree=33.3333
#7	0.5258	lavender=33.3333, eucalyptus=33.3333, peppermint=33.3333, tea_tree=0

Model Quality

Response	R²	Type
`antimicrobial_zone`	0.0714	linear

Full Multi-Objective Output

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

Overall desirability: D = 0.5422

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
relaxation_score             1.5       0.5803       61.30 pts   ↑
antimicrobial_zone           1.0       0.4896       17.80 mm   ↑

Recommended settings:
  lavender = 0 %
  eucalyptus = 0 %
  peppermint = 100 %
  tea_tree = 0 %
  (from observed run #6)

Trade-off summary:
  relaxation_score: 61.30 (best observed: 86.70, sacrifice: +25.40)
  antimicrobial_zone: 17.80 (best observed: 29.00, sacrifice: +11.20)

Model quality:
  relaxation_score: R² = 0.1948 (linear)
  antimicrobial_zone: R² = 0.0714 (linear)

Top 3 observed runs by overall desirability:
  1. Run #6 (D=0.5422): lavender=0, eucalyptus=0, peppermint=100, tea_tree=0
  2. Run #4 (D=0.5400): lavender=0, eucalyptus=33.3333, peppermint=33.3333, tea_tree=33.3333
  3. Run #7 (D=0.5258): lavender=33.3333, eucalyptus=33.3333, peppermint=33.3333, tea_tree=0

Full Analysis Output

doe analyze
=== Main Effects: relaxation_score ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
eucalyptus              47.0333       4.6228            30.1%
lavender                39.9000       4.6228            25.6%
peppermint              34.6857       4.6228            22.2%
tea_tree                34.4000       4.6228            22.0%

=== ANOVA Table: relaxation_score ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
lavender                     4    1317.2221     329.3055                      
eucalyptus                   4    2116.1897     529.0474                      
peppermint                   4    1273.8021     318.4505                      
tea_tree                     4     941.4164     235.3541                      
Error (Lenth PSE)            0       0.0000       0.0000
Total                       14    4487.6773     320.5484
  Note: Error estimated using Lenth's pseudo-standard-error (unreplicated design)

=== Summary Statistics: relaxation_score ===

lavender:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   7    51.4857    11.6781    29.4000    63.1000
  100                 1    60.5000     0.0000    60.5000    60.5000
  25                  1    78.2000     0.0000    78.2000    78.2000
  33.3333             3    50.2667    32.1848    25.7000    86.7000
  50                  3    38.3000    11.8419    25.0000    47.7000

eucalyptus:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   7    55.8286    18.5120    25.0000    86.7000
  100                 1    55.5000     0.0000    55.5000    55.5000
  25                  1    78.2000     0.0000    78.2000    78.2000
  33.3333             3    31.1667     6.5317    25.7000    38.4000
  50                  3    48.9333    10.7240    42.2000    61.3000

peppermint:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   7    43.5143    13.9823    25.0000    60.5000
  100                 1    55.4000     0.0000    55.4000    55.4000
  25                  1    78.2000     0.0000    78.2000    78.2000
  33.3333             3    51.5000    30.8144    29.4000    86.7000
  50                  3    57.3667     8.4198    47.7000    63.1000

tea_tree:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   7    51.5714     8.9513    38.4000    61.3000
  100                 1    52.4000     0.0000    52.4000    52.4000
  25                  1    78.2000     0.0000    78.2000    78.2000
  33.3333             3    47.2667    34.2003    25.7000    86.7000
  50                  3    43.8000    19.0549    25.0000    63.1000

=== Main Effects: antimicrobial_zone ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
eucalyptus              12.8333       1.4749            31.5%
lavender                10.6000       1.4749            26.0%
tea_tree                 8.7000       1.4749            21.4%
peppermint               8.6000       1.4749            21.1%

=== ANOVA Table: antimicrobial_zone ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
lavender                     4     109.0248      27.2562                      
eucalyptus                   4     187.9924      46.9981                      
peppermint                   4      81.1867      20.2967                      
tea_tree                     4      58.0714      14.5179                      
Error (Lenth PSE)            0      20.5448       0.0000
Total                       14     456.8200      32.6300
  Note: Error estimated using Lenth's pseudo-standard-error (unreplicated design)

=== Summary Statistics: antimicrobial_zone ===

lavender:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   7    18.8143     3.1002    14.5000    23.0000
  100                 1    15.1000     0.0000    15.1000    15.1000
  25                  1    12.4000     0.0000    12.4000    12.4000
  33.3333             3    19.9333    11.4256     7.1000    29.0000
  50                  3    23.0000     3.8105    20.8000    27.4000

eucalyptus:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   7    18.0286     6.3466     7.1000    27.4000
  100                 1    14.5000     0.0000    14.5000    14.5000
  25                  1    12.4000     0.0000    12.4000    12.4000
  33.3333             3    25.2333     3.2808    23.0000    29.0000
  50                  3    19.7333     1.6773    17.8000    20.8000

peppermint:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   7    21.0000     5.5405    14.5000    29.0000
  100                 1    20.9000     0.0000    20.9000    20.9000
  25                  1    12.4000     0.0000    12.4000    12.4000
  33.3333             3    17.9333     9.3885     7.1000    23.7000
  50                  3    17.9667     2.7538    15.3000    20.8000

tea_tree:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   7    19.0857     3.3919    14.5000    23.7000
  100                 1    19.6000     0.0000    19.6000    19.6000
  25                  1    12.4000     0.0000    12.4000    12.4000
  33.3333             3    19.7000    11.3168     7.1000    29.0000
  50                  3    21.1000     6.0655    15.3000    27.4000

Optimization Recommendations

doe optimize
=== Optimization: relaxation_score ===
Direction: maximize

Best observed run: #13
  lavender = 0
  eucalyptus = 33.3333
  peppermint = 33.3333
  tea_tree = 33.3333
  Value: 86.7

RSM Model (linear, R² = 0.3839, Adj R² = 0.1375):
  Coefficients:
    intercept                      -13961316.3728
    lavender                       -6980690.5053
    eucalyptus                     -6980672.9243
    peppermint                     -6980685.3898
    tea_tree                       -6980678.4536

  Predicted optimum (from linear model, at observed points):
    lavender = 0
    eucalyptus = 33.3333
    peppermint = 33.3333
    tea_tree = 33.3333
    Predicted value: 67.0164

  Surface optimum (via L-BFGS-B, linear model):
    lavender = 0
    eucalyptus = 0
    peppermint = 0
    tea_tree = 0
    Predicted value: 13961410.9001

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

Factor importance:
  1. lavender  (effect: 28.8, contribution: 29.3%)
  2. peppermint  (effect: 24.7, contribution: 25.1%)
  3. tea_tree  (effect: 23.4, contribution: 23.8%)
  4. eucalyptus  (effect: 21.5, contribution: 21.8%)

=== Optimization: antimicrobial_zone ===
Direction: maximize

Best observed run: #1
  lavender = 50
  eucalyptus = 50
  peppermint = 0
  tea_tree = 0
  Value: 29.0

RSM Model (linear, R² = 0.4107, Adj R² = 0.1750):
  Coefficients:
    intercept                      +6579566.4093
    lavender                       +3289774.4731
    eucalyptus                     +3289771.2093
    peppermint                     +3289774.0334
    tea_tree                       +3289771.1938

  Predicted optimum (from linear model, at observed points):
    lavender = 100
    eucalyptus = 0
    peppermint = 0
    tea_tree = 0
    Predicted value: 24.4459

  Surface optimum (via L-BFGS-B, linear model):
    lavender = 100
    eucalyptus = 100
    peppermint = 100
    tea_tree = 100
    Predicted value: 19738657.3188

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

Factor importance:
  1. tea_tree  (effect: 9.7, contribution: 27.5%)
  2. peppermint  (effect: 9.6, contribution: 27.1%)
  3. lavender  (effect: 9.0, contribution: 25.4%)
  4. eucalyptus  (effect: 7.0, contribution: 19.9%)