Wood Stain & Finish — DOE Use Case

Summary

This experiment investigates wood stain & finish. Full factorial of stain dilution, number of coats, drying time between coats, and topcoat type to maximize color depth and durability.

The design varies 4 factors: stain dilution (%), ranging from 0 to 50, num coats (coats), ranging from 1 to 3, dry hrs (hrs), ranging from 2 to 24, and topcoat coats (coats), ranging from 1 to 3. The goal is to optimize 2 responses: color depth (pts) (maximize) and durability (pts) (maximize). Fixed conditions held constant across all runs include wood type = oak, stain color = walnut.

A full factorial design was used to explore all 16 possible combinations of the 4 factors at two levels. This guarantees that every main effect and interaction can be estimated independently, at the cost of a larger experiment (16 runs).

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 color depth, the most influential factors were stain dilution (38.2%), topcoat coats (35.3%), num coats (14.7%). The best observed value was 9.7 (at stain dilution = 50, num coats = 1, dry hrs = 2).

For durability, the most influential factors were num coats (33.1%), stain dilution (32.3%), topcoat coats (25.8%). The best observed value was 7.9 (at stain dilution = 0, num coats = 3, dry hrs = 2).

Recommended Next Steps

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

Experimental Setup

Factors

Factor	Low	High	Unit
`stain_dilution`	0	50	%
`num_coats`	1	3	coats
`dry_hrs`	2	24	hrs
`topcoat_coats`	1	3	coats

Fixed: wood_type = oak, stain_color = walnut

Responses

Response	Direction	Unit
`color_depth`	↑ maximize	pts
`durability`	↑ maximize	pts

Configuration

use_cases/141_wood_stain_finish/config.json

{
  "metadata": {
    "name": "Wood Stain & Finish",
    "description": "Full factorial of stain dilution, number of coats, drying time between coats, and topcoat type to maximize color depth and durability"
  },
  "factors": [
    {
      "name": "stain_dilution",
      "levels": [
        "0",
        "50"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "num_coats",
      "levels": [
        "1",
        "3"
      ],
      "type": "continuous",
      "unit": "coats"
    },
    {
      "name": "dry_hrs",
      "levels": [
        "2",
        "24"
      ],
      "type": "continuous",
      "unit": "hrs"
    },
    {
      "name": "topcoat_coats",
      "levels": [
        "1",
        "3"
      ],
      "type": "continuous",
      "unit": "coats"
    }
  ],
  "fixed_factors": {
    "wood_type": "oak",
    "stain_color": "walnut"
  },
  "responses": [
    {
      "name": "color_depth",
      "optimize": "maximize",
      "unit": "pts"
    },
    {
      "name": "durability",
      "optimize": "maximize",
      "unit": "pts"
    }
  ],
  "settings": {
    "operation": "full_factorial",
    "test_script": "use_cases/141_wood_stain_finish/sim.sh"
  }
}

Experimental Matrix

The Full Factorial Design produces 16 runs. Each row is one experiment with specific factor settings.

Run	`stain_dilution`	`num_coats`	`dry_hrs`	`topcoat_coats`
1	0	3	24	3
2	50	1	2	3
3	0	3	2	3
4	0	3	24	1
5	50	3	24	1
6	50	1	24	1
7	50	3	2	1
8	50	1	2	1
9	0	1	2	3
10	0	1	24	1
11	50	3	2	3
12	50	3	24	3
13	0	3	2	1
14	50	1	24	3
15	0	1	2	1
16	0	1	24	3

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/141_wood_stain_finish/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/141_wood_stain_finish/config.json \
    --output use_cases/141_wood_stain_finish/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/141_wood_stain_finish/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/141_wood_stain_finish/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/141_wood_stain_finish/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/141_wood_stain_finish/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/141_wood_stain_finish/config.json \
    --output use_cases/141_wood_stain_finish/results/report.html

Features Exercised

Feature	Value
Design type	`full_factorial`
Factor types	`continuous` (all 4)
Arg style	`double-dash`
Responses	2 (color_depth ↑, durability ↑)
Total runs	16

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: color_depth

Top factors: stain_dilution (38.2%), topcoat_coats (35.3%), num_coats (14.7%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
stain_dilution	1	3.8025	3.8025	0.551	0.4914
num_coats	1	0.5625	0.5625	0.081	0.7868
dry_hrs	1	0.3600	0.3600	0.052	0.8284
topcoat_coats	1	3.2400	3.2400	0.469	0.5238
stain_dilution*num_coats	1	0.7225	0.7225	0.105	0.7594
stain_dilution*dry_hrs	1	9.0000	9.0000	1.303	0.3053
stain_dilution*topcoat_coats	1	0.3600	0.3600	0.052	0.8284
num_coats*dry_hrs	1	3.6100	3.6100	0.523	0.5021
num_coats*topcoat_coats	1	1.2100	1.2100	0.175	0.6929
dry_hrs*topcoat_coats	1	0.0625	0.0625	0.009	0.9279
Error	5	34.5275	6.9055
Total	15	57.4575	3.8305

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Half-Normal Plot of Effects

Model Diagnostics

Response: durability

Top factors: num_coats (33.1%), stain_dilution (32.3%), topcoat_coats (25.8%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
stain_dilution	1	17.6400	17.6400	5.869	0.0599
num_coats	1	18.4900	18.4900	6.152	0.0558
dry_hrs	1	1.3225	1.3225	0.440	0.5365
topcoat_coats	1	11.2225	11.2225	3.734	0.1112
stain_dilution*num_coats	1	0.4225	0.4225	0.141	0.7231
stain_dilution*dry_hrs	1	1.2100	1.2100	0.403	0.5536
stain_dilution*topcoat_coats	1	2.5600	2.5600	0.852	0.3984
num_coats*dry_hrs	1	3.6100	3.6100	1.201	0.3231
num_coats*topcoat_coats	1	1.2100	1.2100	0.403	0.5536
dry_hrs*topcoat_coats	1	0.0025	0.0025	0.001	0.9781
Error	5	15.0275	3.0055
Total	15	72.7175	4.8478

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.

color depth dry hrs vs topcoat coats

color depth num coats vs dry hrs

color depth num coats vs topcoat coats

color depth stain dilution vs dry hrs

color depth stain dilution vs num coats

color depth stain dilution vs topcoat coats

durability dry hrs vs topcoat coats

durability num coats vs dry hrs

durability num coats vs topcoat coats

durability stain dilution vs dry hrs

durability stain dilution vs num coats

durability stain dilution vs topcoat coats

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 = 0.9421

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`color_depth`	1.5	0.9545	9.70 0.9545 9.70 pts	↑
`durability`	1.0	0.9237	7.70 0.9237 7.70 pts	↑

Recommended Settings

Factor	Value
`stain_dilution`	50 %
`num_coats`	3 coats
`dry_hrs`	2 hrs
`topcoat_coats`	3 coats

Source: from observed run #1

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`durability`	7.70	7.90	+0.20

Top 3 Runs by Desirability

Run	D	Factor Settings
#3	0.7380	stain_dilution=0, num_coats=1, dry_hrs=24, topcoat_coats=1
#12	0.6259	stain_dilution=50, num_coats=3, dry_hrs=24, topcoat_coats=1

Model Quality

Response	R²	Type
`durability`	0.3904	linear

Full Multi-Objective Output

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

Overall desirability: D = 0.9421

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
color_depth                  1.5       0.9545        9.70 pts   ↑
durability                   1.0       0.9237        7.70 pts   ↑

Recommended settings:
  stain_dilution = 50 %
  num_coats = 3 coats
  dry_hrs = 2 hrs
  topcoat_coats = 3 coats
  (from observed run #1)

Trade-off summary:
  color_depth: 9.70 (best observed: 9.70, sacrifice: +0.00)
  durability: 7.70 (best observed: 7.90, sacrifice: +0.20)

Model quality:
  color_depth: R² = 0.1116 (linear)
  durability: R² = 0.3904 (linear)

Top 3 observed runs by overall desirability:
  1. Run #1 (D=0.9421): stain_dilution=50, num_coats=3, dry_hrs=2, topcoat_coats=3
  2. Run #3 (D=0.7380): stain_dilution=0, num_coats=1, dry_hrs=24, topcoat_coats=1
  3. Run #12 (D=0.6259): stain_dilution=50, num_coats=3, dry_hrs=24, topcoat_coats=1

Full Analysis Output

doe analyze
=== Main Effects: color_depth ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
stain_dilution          -0.9750       0.4893            38.2%
topcoat_coats           -0.9000       0.4893            35.3%
num_coats               -0.3750       0.4893            14.7%
dry_hrs                 -0.3000       0.4893            11.8%

=== ANOVA Table: color_depth ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
stain_dilution               1       3.8025       3.8025      0.551     0.4914
num_coats                    1       0.5625       0.5625      0.081     0.7868
dry_hrs                      1       0.3600       0.3600      0.052     0.8284
topcoat_coats                1       3.2400       3.2400      0.469     0.5238
stain_dilution*num_coats     1       0.7225       0.7225      0.105     0.7594
stain_dilution*dry_hrs       1       9.0000       9.0000      1.303     0.3053
stain_dilution*topcoat_coats    1       0.3600       0.3600      0.052     0.8284
num_coats*dry_hrs            1       3.6100       3.6100      0.523     0.5021
num_coats*topcoat_coats      1       1.2100       1.2100      0.175     0.6929
dry_hrs*topcoat_coats        1       0.0625       0.0625      0.009     0.9279
Error                        5      34.5275       6.9055
Total                       15      57.4575       3.8305

=== Interaction Effects: color_depth ===
Factor A             Factor B              Interaction   % Contribution
------------------------------------------------------------------------
stain_dilution       dry_hrs                    1.5000            39.0%
num_coats            dry_hrs                    0.9500            24.7%
num_coats            topcoat_coats             -0.5500            14.3%
stain_dilution       num_coats                 -0.4250            11.0%
stain_dilution       topcoat_coats              0.3000             7.8%
dry_hrs              topcoat_coats             -0.1250             3.2%

=== Summary Statistics: color_depth ===

stain_dilution:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   8     6.5500     1.7452     3.9000     9.7000
  50                  8     5.5750     2.1493     3.1000     8.7000

num_coats:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1                   8     6.2500     2.4605     3.4000     9.7000
  3                   8     5.8750     1.4400     3.1000     8.1000

dry_hrs:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   8     6.2125     2.3443     3.1000     9.7000
  24                  8     5.9125     1.6313     3.6000     8.7000

topcoat_coats:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1                   8     6.5125     2.2662     3.4000     9.7000
  3                   8     5.6125     1.6155     3.1000     8.2000

=== Main Effects: durability ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
num_coats               -2.1500       0.5504            33.1%
stain_dilution          -2.1000       0.5504            32.3%
topcoat_coats            1.6750       0.5504            25.8%
dry_hrs                  0.5750       0.5504             8.8%

=== ANOVA Table: durability ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
stain_dilution               1      17.6400      17.6400      5.869     0.0599
num_coats                    1      18.4900      18.4900      6.152     0.0558
dry_hrs                      1       1.3225       1.3225      0.440     0.5365
topcoat_coats                1      11.2225      11.2225      3.734     0.1112
stain_dilution*num_coats     1       0.4225       0.4225      0.141     0.7231
stain_dilution*dry_hrs       1       1.2100       1.2100      0.403     0.5536
stain_dilution*topcoat_coats    1       2.5600       2.5600      0.852     0.3984
num_coats*dry_hrs            1       3.6100       3.6100      1.201     0.3231
num_coats*topcoat_coats      1       1.2100       1.2100      0.403     0.5536
dry_hrs*topcoat_coats        1       0.0025       0.0025      0.001     0.9781
Error                        5      15.0275       3.0055
Total                       15      72.7175       4.8478

=== Interaction Effects: durability ===
Factor A             Factor B              Interaction   % Contribution
------------------------------------------------------------------------
num_coats            dry_hrs                    0.9500            29.7%
stain_dilution       topcoat_coats              0.8000            25.0%
stain_dilution       dry_hrs                   -0.5500            17.2%
num_coats            topcoat_coats             -0.5500            17.2%
stain_dilution       num_coats                  0.3250            10.2%
dry_hrs              topcoat_coats              0.0250             0.8%

=== Summary Statistics: durability ===

stain_dilution:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0                   8     5.8625     2.0619     2.0000     7.9000
  50                  8     3.7625     1.9018     2.1000     7.1000

num_coats:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1                   8     5.8875     2.1203     2.3000     7.9000
  3                   8     3.7375     1.8031     2.0000     7.1000

dry_hrs:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   8     4.5250     2.4259     2.0000     7.9000
  24                  8     5.1000     2.0771     2.1000     7.1000

topcoat_coats:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1                   8     3.9750     2.2334     2.0000     7.7000
  3                   8     5.6500     1.9486     3.0000     7.9000

Optimization Recommendations

doe optimize
=== Optimization: color_depth ===
Direction: maximize

Best observed run: #1
  stain_dilution = 50
  num_coats = 1
  dry_hrs = 2
  topcoat_coats = 1
  Value: 9.7

RSM Model (linear, R² = 0.3856, Adj R² = 0.1622):
  Coefficients:
    intercept                      +6.0625
    stain_dilution                 +0.3625
    num_coats                      -0.3875
    dry_hrs                        -0.0250
    topcoat_coats                  -1.0500

RSM Model (quadratic, R² = 0.4671, Adj R² = -6.9931):
  Coefficients:
    intercept                      +1.2125
    stain_dilution                 +0.3625
    num_coats                      -0.3875
    dry_hrs                        -0.0250
    topcoat_coats                  -1.0500
    stain_dilution*num_coats       -0.0625
    stain_dilution*dry_hrs         +0.2250
    stain_dilution*topcoat_coats   -0.2500
    num_coats*dry_hrs              -0.1000
    num_coats*topcoat_coats        -0.1250
    dry_hrs*topcoat_coats          +0.3875
    stain_dilution^2               +1.2125
    num_coats^2                    +1.2125
    dry_hrs^2                      +1.2125
    topcoat_coats^2                +1.2125

  Curvature analysis:
    stain_dilution                 coef=+1.2125  convex (has a minimum)
    dry_hrs                        coef=+1.2125  convex (has a minimum)
    topcoat_coats                  coef=+1.2125  convex (has a minimum)
    num_coats                      coef=+1.2125  convex (has a minimum)

  Notable interactions:
    dry_hrs*topcoat_coats          coef=+0.3875  (synergistic)

  Predicted optimum (from linear model, at observed points):
    stain_dilution = 50
    num_coats = 1
    dry_hrs = 2
    topcoat_coats = 1
    Predicted value: 7.8875

  Surface optimum (via L-BFGS-B, linear model):
    stain_dilution = 50
    num_coats = 1
    dry_hrs = 2
    topcoat_coats = 1
    Predicted value: 7.8875

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

Factor importance:
  1. topcoat_coats  (effect: -2.1, contribution: 57.5%)
  2. num_coats  (effect: -0.8, contribution: 21.2%)
  3. stain_dilution  (effect: 0.7, contribution: 19.9%)
  4. dry_hrs  (effect: -0.1, contribution: 1.4%)

=== Optimization: durability ===
Direction: maximize

Best observed run: #12
  stain_dilution = 0
  num_coats = 3
  dry_hrs = 2
  topcoat_coats = 1
  Value: 7.9

RSM Model (linear, R² = 0.0634, Adj R² = -0.2771):
  Coefficients:
    intercept                      +4.8125
    stain_dilution                 -0.4250
    num_coats                      -0.2500
    dry_hrs                        -0.1875
    topcoat_coats                  -0.1000

RSM Model (quadratic, R² = 0.6018, Adj R² = -4.9733):
  Coefficients:
    intercept                      +0.9625
    stain_dilution                 -0.4250
    num_coats                      -0.2500
    dry_hrs                        -0.1875
    topcoat_coats                  -0.1000
    stain_dilution*num_coats       +0.0125
    stain_dilution*dry_hrs         +0.4750
    stain_dilution*topcoat_coats   -0.3375
    num_coats*dry_hrs              +0.4750
    num_coats*topcoat_coats        -0.4375
    dry_hrs*topcoat_coats          +1.3000
    stain_dilution^2               +0.9625
    num_coats^2                    +0.9625
    dry_hrs^2                      +0.9625
    topcoat_coats^2                +0.9625

  Curvature analysis:
    stain_dilution                 coef=+0.9625  convex (has a minimum)
    topcoat_coats                  coef=+0.9625  convex (has a minimum)
    num_coats                      coef=+0.9625  convex (has a minimum)
    dry_hrs                        coef=+0.9625  convex (has a minimum)

  Notable interactions:
    dry_hrs*topcoat_coats          coef=+1.3000  (synergistic)
    num_coats*dry_hrs              coef=+0.4750  (synergistic)
    stain_dilution*dry_hrs         coef=+0.4750  (synergistic)
    num_coats*topcoat_coats        coef=-0.4375  (antagonistic)
    stain_dilution*topcoat_coats   coef=-0.3375  (antagonistic)

  Predicted optimum (from linear model, at observed points):
    stain_dilution = 0
    num_coats = 1
    dry_hrs = 2
    topcoat_coats = 1
    Predicted value: 5.7750

  Surface optimum (via L-BFGS-B, linear model):
    stain_dilution = 0
    num_coats = 1
    dry_hrs = 2
    topcoat_coats = 1
    Predicted value: 5.7750

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

Factor importance:
  1. stain_dilution  (effect: -0.8, contribution: 44.2%)
  2. num_coats  (effect: -0.5, contribution: 26.0%)
  3. dry_hrs  (effect: -0.4, contribution: 19.5%)
  4. topcoat_coats  (effect: -0.2, contribution: 10.4%)

Run	`stain_dilution`	`num_coats`	`dry_hrs`	`topcoat_coats`
1	0	3	24	3
2	50	1	2	3
3	0	3	2	3
4	0	3	24	1
5	50	3	24	1
6	50	1	24	1
7	50	3	2	1
8	50	1	2	1
9	0	1	2	3
10	0	1	24	1
11	50	3	2	3
12	50	3	24	3
13	0	3	2	1
14	50	1	24	3
15	0	1	2	1
16	0	1	24	3

Run	`stain_dilution`	`num_coats`	`dry_hrs`	`topcoat_coats`
1	0	3	24	3
2	50	1	2	3
3	0	3	2	3
4	0	3	24	1
5	50	3	24	1
6	50	1	24	1
7	50	3	2	1
8	50	1	2	1
9	0	1	2	3
10	0	1	24	1
11	50	3	2	3
12	50	3	24	3
13	0	3	2	1
14	50	1	24	3
15	0	1	2	1
16	0	1	24	3

Run	`stain_dilution`	`num_coats`	`dry_hrs`	`topcoat_coats`
1	0	3	24	3
2	50	1	2	3
3	0	3	2	3
4	0	3	24	1
5	50	3	24	1
6	50	1	24	1
7	50	3	2	1
8	50	1	2	1
9	0	1	2	3
10	0	1	24	1
11	50	3	2	3
12	50	3	24	3
13	0	3	2	1
14	50	1	24	3
15	0	1	2	1
16	0	1	24	3