Candle Making Optimization

Summary

This experiment investigates candle making optimization. Box-Behnken design to maximize burn time and scent throw by tuning wick size, fragrance load, and pour temperature.

The design varies 3 factors: wick size (mm), ranging from 4 to 10, fragrance pct (%), ranging from 4 to 12, and pour temp c (C), ranging from 55 to 80. The goal is to optimize 2 responses: burn hrs (hrs) (maximize) and scent throw (pts) (maximize). Fixed conditions held constant across all runs include wax type = soy, container = 8oz_jar.

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 burn hrs, the most influential factors were wick size (45.9%), pour temp c (29.5%), fragrance pct (24.6%). The best observed value was 51.0 (at wick size = 7, fragrance pct = 12, pour temp c = 80).

For scent throw, the most influential factors were fragrance pct (46.1%), wick size (44.1%), pour temp c (9.8%). The best observed value was 7.7 (at wick size = 7, fragrance pct = 12, pour temp c = 55).

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
`wick_size`	4	10	mm
`fragrance_pct`	4	12	%
`pour_temp_c`	55	80	C

Fixed: wax_type = soy, container = 8oz_jar

Responses

Response	Direction	Unit
`burn_hrs`	↑ maximize	hrs
`scent_throw`	↑ maximize	pts

Configuration

use_cases/140_candle_making/config.json

{
  "metadata": {
    "name": "Candle Making Optimization",
    "description": "Box-Behnken design to maximize burn time and scent throw by tuning wick size, fragrance load, and pour temperature"
  },
  "factors": [
    {
      "name": "wick_size",
      "levels": [
        "4",
        "10"
      ],
      "type": "continuous",
      "unit": "mm"
    },
    {
      "name": "fragrance_pct",
      "levels": [
        "4",
        "12"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "pour_temp_c",
      "levels": [
        "55",
        "80"
      ],
      "type": "continuous",
      "unit": "C"
    }
  ],
  "fixed_factors": {
    "wax_type": "soy",
    "container": "8oz_jar"
  },
  "responses": [
    {
      "name": "burn_hrs",
      "optimize": "maximize",
      "unit": "hrs"
    },
    {
      "name": "scent_throw",
      "optimize": "maximize",
      "unit": "pts"
    }
  ],
  "settings": {
    "operation": "box_behnken",
    "test_script": "use_cases/140_candle_making/sim.sh"
  }
}

Experimental Matrix

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

Run	`wick_size`	`fragrance_pct`	`pour_temp_c`
1	7	4	55
2	7	8	67.5
3	10	8	80
4	10	8	55
5	7	8	67.5
6	7	8	67.5
7	4	8	80
8	10	4	67.5
9	7	4	80
10	10	12	67.5
11	4	8	55
12	7	12	80
13	4	4	67.5
14	4	12	67.5
15	7	12	55

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/140_candle_making/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/140_candle_making/config.json \
    --output use_cases/140_candle_making/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/140_candle_making/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/140_candle_making/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/140_candle_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/140_candle_making/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/140_candle_making/config.json \
    --output use_cases/140_candle_making/results/report.html

Features Exercised

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

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: burn_hrs

Top factors: wick_size (45.9%), pour_temp_c (29.5%), fragrance_pct (24.6%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
wick_size	2	146.1833	73.0917	10.442	0.0059
fragrance_pct	2	30.3262	15.1631	2.166	0.1771
pour_temp_c	2	51.9333	25.9667	3.710	0.0725
Lack	of	Fit	6	256.4905	42.7484
Pure	Error	2	14.0000
Error	8	270.4905	7.0000
Total	14	498.9333	35.6381

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Half-Normal Plot of Effects

Model Diagnostics

Response: scent_throw

Top factors: fragrance_pct (46.1%), wick_size (44.1%), pour_temp_c (9.8%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
wick_size	2	4.8198	2.4099	0.668	0.5394
fragrance_pct	2	7.3827	3.6913	1.023	0.4023
pour_temp_c	2	0.3138	0.1569	0.043	0.9577
Lack	of	Fit	6	21.1010	3.5168
Pure	Error	2	7.2200
Error	8	28.3210	3.6100
Total	14	40.8373	2.9170

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.

burn hrs fragrance pct vs pour temp c

burn hrs wick size vs fragrance pct

burn hrs wick size vs pour temp c

scent throw fragrance pct vs pour temp c

scent throw wick size vs fragrance pct

scent throw wick size vs pour temp c

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 = 0.8204

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`burn_hrs`	1.0	0.6166	44.31 0.6166 44.31 hrs	↑
`scent_throw`	1.5	0.9924	7.94 0.9924 7.94 pts	↑

Recommended Settings

Factor	Value
`wick_size`	10 mm
`fragrance_pct`	11.2 %
`pour_temp_c`	55 C

Source: from RSM model prediction

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`scent_throw`	7.94	7.70	-0.24

Top 3 Runs by Desirability

Run	D	Factor Settings
#14	0.6357	wick_size=10, fragrance_pct=8, pour_temp_c=55
#7	0.5846	wick_size=7, fragrance_pct=8, pour_temp_c=67.5

Model Quality

Response	R²	Type
`scent_throw`	0.9330	quadratic

Full Multi-Objective Output

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

Overall desirability: D = 0.8204

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
burn_hrs                     1.0       0.6166       44.31 hrs   ↑
scent_throw                  1.5       0.9924        7.94 pts   ↑

Recommended settings:
  wick_size = 10 mm
  fragrance_pct = 11.2 %
  pour_temp_c = 55 C
  (from RSM model prediction)

Trade-off summary:
  burn_hrs: 44.31 (best observed: 51.00, sacrifice: +6.69)
  scent_throw: 7.94 (best observed: 7.70, sacrifice: -0.24)

Model quality:
  burn_hrs: R² = 0.2611 (linear)
  scent_throw: R² = 0.9330 (quadratic)

Top 3 observed runs by overall desirability:
  1. Run #12 (D=0.6600): wick_size=4, fragrance_pct=12, pour_temp_c=67.5
  2. Run #14 (D=0.6357): wick_size=10, fragrance_pct=8, pour_temp_c=55
  3. Run #7 (D=0.5846): wick_size=7, fragrance_pct=8, pour_temp_c=67.5

Full Analysis Output

doe analyze
=== Main Effects: burn_hrs ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
wick_size                7.0000       1.5414            45.9%
pour_temp_c              4.5000       1.5414            29.5%
fragrance_pct            3.7500       1.5414            24.6%

=== ANOVA Table: burn_hrs ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
wick_size                    2     146.1833      73.0917     10.442     0.0059
fragrance_pct                2      30.3262      15.1631      2.166     0.1771
pour_temp_c                  2      51.9333      25.9667      3.710     0.0725
Lack of Fit                  6     256.4905      42.7484      6.107     0.1474
Pure Error                   2      14.0000       7.0000
Error                        8     270.4905       7.0000
Total                       14     498.9333      35.6381

=== Summary Statistics: burn_hrs ===

wick_size:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  10                  4    37.0000     3.5590    34.0000    41.0000
  4                   4    38.7500     6.5000    33.0000    48.0000
  7                   7    44.0000     5.5976    36.0000    51.0000

fragrance_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  12                  4    42.2500     5.9652    38.0000    51.0000
  4                   4    38.5000     8.4261    33.0000    51.0000
  8                   7    41.1429     4.9809    34.0000    48.0000

pour_temp_c:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  55                  4    43.5000     7.5939    34.0000    51.0000
  67.5                7    39.0000     4.5461    33.0000    45.0000
  80                  4    41.0000     7.0711    36.0000    51.0000

=== Main Effects: scent_throw ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
fragrance_pct            1.5929       0.4410            46.1%
wick_size                1.5250       0.4410            44.1%
pour_temp_c              0.3393       0.4410             9.8%

=== ANOVA Table: scent_throw ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
wick_size                    2       4.8198       2.4099      0.668     0.5394
fragrance_pct                2       7.3827       3.6913      1.023     0.4023
pour_temp_c                  2       0.3138       0.1569      0.043     0.9577
Lack of Fit                  6      21.1010       3.5168      0.974     0.5864
Pure Error                   2       7.2200       3.6100
Error                        8      28.3210       3.6100
Total                       14      40.8373       2.9170

=== Summary Statistics: scent_throw ===

wick_size:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  10                  4     4.4500     1.5022     2.7000     5.8000
  4                   4     5.9750     1.6860     3.8000     7.7000
  7                   7     5.0000     1.8583     1.9000     7.5000

fragrance_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  12                  4     3.9500     1.9330     1.9000     5.6000
  4                   4     5.5250     1.8554     3.7000     7.7000
  8                   7     5.5429     1.4093     3.7000     7.5000

pour_temp_c:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  55                  4     4.8750     0.9777     3.8000     5.8000
  67.5                7     5.2143     1.9412     2.7000     7.7000
  80                  4     5.1750     2.2396     1.9000     6.8000

Optimization Recommendations

doe optimize
=== Optimization: burn_hrs ===
Direction: maximize

Best observed run: #7
  wick_size = 7
  fragrance_pct = 12
  pour_temp_c = 80
  Value: 51.0

RSM Model (linear, R² = 0.5216, Adj R² = 0.3911):
  Coefficients:
    intercept                      +40.7333
    wick_size                      +0.6250
    fragrance_pct                  +1.3750
    pour_temp_c                    +5.5000

RSM Model (quadratic, R² = 0.7266, Adj R² = 0.2344):
  Coefficients:
    intercept                      +37.6667
    wick_size                      +0.6250
    fragrance_pct                  +1.3750
    pour_temp_c                    +5.5000
    wick_size*fragrance_pct        +1.0000
    wick_size*pour_temp_c          -1.2500
    fragrance_pct*pour_temp_c      +0.7500
    wick_size^2                    +0.6667
    fragrance_pct^2                +0.1667
    pour_temp_c^2                  +4.9167

  Curvature analysis:
    pour_temp_c                    coef=+4.9167  convex (has a minimum)
    wick_size                      coef=+0.6667  convex (has a minimum)
    fragrance_pct                  coef=+0.1667  convex (has a minimum)

  Notable interactions:
    wick_size*pour_temp_c          coef=-1.2500  (antagonistic)
    wick_size*fragrance_pct        coef=+1.0000  (synergistic)
    fragrance_pct*pour_temp_c      coef=+0.7500  (synergistic)

  Predicted optimum (from linear model, at observed points):
    wick_size = 7
    fragrance_pct = 12
    pour_temp_c = 80
    Predicted value: 47.6083

  Surface optimum (via L-BFGS-B, linear model):
    wick_size = 10
    fragrance_pct = 12
    pour_temp_c = 80
    Predicted value: 48.2333

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

Factor importance:
  1. pour_temp_c  (effect: 11.0, contribution: 73.3%)
  2. fragrance_pct  (effect: 2.8, contribution: 18.3%)
  3. wick_size  (effect: 1.2, contribution: 8.3%)

=== Optimization: scent_throw ===
Direction: maximize

Best observed run: #10
  wick_size = 7
  fragrance_pct = 12
  pour_temp_c = 55
  Value: 7.7

RSM Model (linear, R² = 0.6137, Adj R² = 0.5084):
  Coefficients:
    intercept                      +5.1133
    wick_size                      +0.5000
    fragrance_pct                  +0.1875
    pour_temp_c                    -1.6875

RSM Model (quadratic, R² = 0.6807, Adj R² = 0.1060):
  Coefficients:
    intercept                      +5.6667
    wick_size                      +0.5000
    fragrance_pct                  +0.1875
    pour_temp_c                    -1.6875
    wick_size*fragrance_pct        -0.0500
    wick_size*pour_temp_c          +0.4500
    fragrance_pct*pour_temp_c      +0.2750
    wick_size^2                    -0.4458
    fragrance_pct^2                -0.5208
    pour_temp_c^2                  -0.0708

  Curvature analysis:
    fragrance_pct                  coef=-0.5208  concave (has a maximum)
    wick_size                      coef=-0.4458  concave (has a maximum)
    pour_temp_c                    coef=-0.0708  negligible curvature

  Notable interactions:
    wick_size*pour_temp_c          coef=+0.4500  (synergistic)

  Predicted optimum (from linear model, at observed points):
    wick_size = 10
    fragrance_pct = 8
    pour_temp_c = 55
    Predicted value: 7.3008

  Surface optimum (via L-BFGS-B, linear model):
    wick_size = 10
    fragrance_pct = 12
    pour_temp_c = 55
    Predicted value: 7.4883

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

Factor importance:
  1. pour_temp_c  (effect: 3.4, contribution: 66.9%)
  2. wick_size  (effect: 1.0, contribution: 19.8%)
  3. fragrance_pct  (effect: 0.7, contribution: 13.3%)