Mortise & Tenon Fit — DOE Use Case

Summary

This experiment investigates mortise & tenon fit. Box-Behnken design to maximize joint strength and minimize assembly difficulty by tuning tenon thickness tolerance, shoulder depth, and glue type viscosity.

The design varies 3 factors: tolerance mm (mm), ranging from 0.05 to 0.5, shoulder mm (mm), ranging from 3 to 10, and glue viscosity (cP), ranging from 1000 to 8000. The goal is to optimize 2 responses: pull strength kn (kN) (maximize) and assembly score (pts) (maximize). Fixed conditions held constant across all runs include joint type = through_tenon, wood = white_oak.

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 pull strength kn, the most influential factors were tolerance mm (47.0%), glue viscosity (34.5%), shoulder mm (18.5%). The best observed value was 5.15 (at tolerance mm = 0.275, shoulder mm = 3, glue viscosity = 1000).

For assembly score, the most influential factors were tolerance mm (44.8%), glue viscosity (37.3%), shoulder mm (17.9%). The best observed value was 7.4 (at tolerance mm = 0.275, shoulder mm = 6.5, glue viscosity = 4500).

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
`tolerance_mm`	0.05	0.5	mm
`shoulder_mm`	3	10	mm
`glue_viscosity`	1000	8000	cP

Fixed: joint_type = through_tenon, wood = white_oak

Responses

Response	Direction	Unit
`pull_strength_kn`	↑ maximize	kN
`assembly_score`	↑ maximize	pts

Configuration

use_cases/202_mortise_tenon_fit/config.json

{
  "metadata": {
    "name": "Mortise & Tenon Fit",
    "description": "Box-Behnken design to maximize joint strength and minimize assembly difficulty by tuning tenon thickness tolerance, shoulder depth, and glue type viscosity"
  },
  "factors": [
    {
      "name": "tolerance_mm",
      "levels": [
        "0.05",
        "0.5"
      ],
      "type": "continuous",
      "unit": "mm"
    },
    {
      "name": "shoulder_mm",
      "levels": [
        "3",
        "10"
      ],
      "type": "continuous",
      "unit": "mm"
    },
    {
      "name": "glue_viscosity",
      "levels": [
        "1000",
        "8000"
      ],
      "type": "continuous",
      "unit": "cP"
    }
  ],
  "fixed_factors": {
    "joint_type": "through_tenon",
    "wood": "white_oak"
  },
  "responses": [
    {
      "name": "pull_strength_kn",
      "optimize": "maximize",
      "unit": "kN"
    },
    {
      "name": "assembly_score",
      "optimize": "maximize",
      "unit": "pts"
    }
  ],
  "settings": {
    "operation": "box_behnken",
    "test_script": "use_cases/202_mortise_tenon_fit/sim.sh"
  }
}

Experimental Matrix

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

Run	`tolerance_mm`	`shoulder_mm`	`glue_viscosity`
1	0.275	3	1000
2	0.275	6.5	4500
3	0.5	6.5	8000
4	0.5	6.5	1000
5	0.275	6.5	4500
6	0.275	6.5	4500
7	0.05	6.5	8000
8	0.5	3	4500
9	0.275	3	8000
10	0.5	10	4500
11	0.05	6.5	1000
12	0.275	10	8000
13	0.05	3	4500
14	0.05	10	4500
15	0.275	10	1000

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/202_mortise_tenon_fit/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/202_mortise_tenon_fit/config.json \
    --output use_cases/202_mortise_tenon_fit/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/202_mortise_tenon_fit/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/202_mortise_tenon_fit/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/202_mortise_tenon_fit/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/202_mortise_tenon_fit/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/202_mortise_tenon_fit/config.json \
    --output use_cases/202_mortise_tenon_fit/results/report.html

Features Exercised

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

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: pull_strength_kn

Top factors: tolerance_mm (47.0%), glue_viscosity (34.5%), shoulder_mm (18.5%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
tolerance_mm	2	3.9402	1.9701	3.738	0.0714
shoulder_mm	2	0.7717	0.3859	0.732	0.5105
glue_viscosity	2	2.3549	1.1775	2.234	0.1695
Lack	of	Fit	6	2.7683	0.4614
Pure	Error	2	1.0541
Error	8	3.8224	0.5270
Total	14	10.8892	0.7778

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for pull_strength_kn

Half-Normal Plot of Effects

Model Diagnostics

Response: assembly_score

Top factors: tolerance_mm (44.8%), glue_viscosity (37.3%), shoulder_mm (17.9%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
tolerance_mm	2	5.2764	2.6382	21.985	0.0006
shoulder_mm	2	1.1825	0.5913	4.927	0.0403
glue_viscosity	2	3.6982	1.8491	15.409	0.0018
Lack	of	Fit	6	3.7629	0.6271
Pure	Error	2	0.2400
Error	8	4.0029	0.1200
Total	14	14.1600	1.0114

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for assembly_score

Half-Normal Plot of Effects

Model Diagnostics

Response Surface Plots

3D surfaces fitted with quadratic RSM. Red dots are observed data points.

assembly score shoulder mm vs glue viscosity

assembly score tolerance mm vs glue viscosity

assembly score tolerance mm vs shoulder mm

pull strength kn shoulder mm vs glue viscosity

pull strength kn tolerance mm vs glue viscosity

pull strength kn tolerance mm vs shoulder mm

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 = 0.6268

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`pull_strength_kn`	1.5	0.4653	3.60 0.4653 3.60 kN	↑
`assembly_score`	1.5	0.8444	7.00 0.8444 7.00 pts	↑

Recommended Settings

Factor	Value
`tolerance_mm`	0.5 mm
`shoulder_mm`	10 mm
`glue_viscosity`	4500 cP

Source: from observed run #10

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`assembly_score`	7.00	7.40	+0.40

Top 3 Runs by Desirability

Run	D	Factor Settings
#15	0.5790	tolerance_mm=0.275, shoulder_mm=6.5, glue_viscosity=4500
#3	0.5765	tolerance_mm=0.05, shoulder_mm=6.5, glue_viscosity=8000

Model Quality

Response	R²	Type
`assembly_score`	0.7208	quadratic

Full Multi-Objective Output

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

Overall desirability: D = 0.6268

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
pull_strength_kn             1.5       0.4653        3.60 kN   ↑
assembly_score               1.5       0.8444        7.00 pts   ↑

Recommended settings:
  tolerance_mm = 0.5 mm
  shoulder_mm = 10 mm
  glue_viscosity = 4500 cP
  (from observed run #10)

Trade-off summary:
  pull_strength_kn: 3.60 (best observed: 5.15, sacrifice: +1.55)
  assembly_score: 7.00 (best observed: 7.40, sacrifice: +0.40)

Model quality:
  pull_strength_kn: R² = 0.7334 (quadratic)
  assembly_score: R² = 0.7208 (quadratic)

Top 3 observed runs by overall desirability:
  1. Run #10 (D=0.6268): tolerance_mm=0.5, shoulder_mm=10, glue_viscosity=4500
  2. Run #15 (D=0.5790): tolerance_mm=0.275, shoulder_mm=6.5, glue_viscosity=4500
  3. Run #3 (D=0.5765): tolerance_mm=0.05, shoulder_mm=6.5, glue_viscosity=8000

Full Analysis Output

doe analyze
=== Main Effects: pull_strength_kn ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
tolerance_mm             1.2875       0.2277            47.0%
glue_viscosity           0.9464       0.2277            34.5%
shoulder_mm              0.5079       0.2277            18.5%

=== ANOVA Table: pull_strength_kn ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
tolerance_mm                 2       3.9402       1.9701      3.738     0.0714
shoulder_mm                  2       0.7717       0.3859      0.732     0.5105
glue_viscosity               2       2.3549       1.1775      2.234     0.1695
Lack of Fit                  6       2.7683       0.4614      0.875     0.6201
Pure Error                   2       1.0541       0.5270
Error                        8       3.8224       0.5270
Total                       14      10.8892       0.7778

=== Summary Statistics: pull_strength_kn ===

tolerance_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0.05                4     4.4900     0.7877     3.4600     5.1500
  0.275               7     3.4371     0.7882     2.3900     4.8200
  0.5                 4     3.2025     0.6733     2.2700     3.8100

shoulder_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  10                  4     3.3850     0.1636     3.1800     3.5600
  3                   4     3.5100     1.2362     2.2700     4.8200
  6.5                 7     3.8929     0.9533     2.3900     5.1500

glue_viscosity:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1000                4     3.7725     0.9903     2.6600     5.0600
  4500                7     3.2686     0.7239     2.2700     4.2900
  8000                4     4.2150     0.9033     3.3400     5.1500

=== Main Effects: assembly_score ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
tolerance_mm             1.4393       0.2597            44.8%
glue_viscosity           1.1964       0.2597            37.3%
shoulder_mm              0.5750       0.2597            17.9%

=== ANOVA Table: assembly_score ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
tolerance_mm                 2       5.2764       2.6382     21.985     0.0006
shoulder_mm                  2       1.1825       0.5913      4.927     0.0403
glue_viscosity               2       3.6982       1.8491     15.409     0.0018
Lack of Fit                  6       3.7629       0.6271      5.226     0.1693
Pure Error                   2       0.2400       0.1200
Error                        8       4.0029       0.1200
Total                       14      14.1600       1.0114

=== Summary Statistics: assembly_score ===

tolerance_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0.05                4     5.0750     1.0340     4.1000     6.2000
  0.275               7     6.5143     0.8092     4.9000     7.4000
  0.5                 4     6.0250     0.7632     5.3000     7.1000

shoulder_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  10                  4     6.2500     0.2887     5.9000     6.6000
  3                   4     6.2750     1.1786     4.9000     7.4000
  6.5                 7     5.7000     1.1944     4.1000     7.0000

glue_viscosity:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1000                4     5.9000     1.3736     4.3000     7.4000
  4500                7     6.4714     0.5707     5.7000     7.1000
  8000                4     5.2750     0.9743     4.1000     6.3000

Optimization Recommendations

doe optimize
=== Optimization: pull_strength_kn ===
Direction: maximize

Best observed run: #14
  tolerance_mm = 0.275
  shoulder_mm = 3
  glue_viscosity = 1000
  Value: 5.15

RSM Model (linear, R² = 0.1865, Adj R² = -0.0354):
  Coefficients:
    intercept                      +3.6553
    tolerance_mm                   +0.2225
    shoulder_mm                    -0.4512
    glue_viscosity                 +0.0263

RSM Model (quadratic, R² = 0.5637, Adj R² = -0.2216):
  Coefficients:
    intercept                      +3.0733
    tolerance_mm                   +0.2225
    shoulder_mm                    -0.4512
    glue_viscosity                 +0.0262
    tolerance_mm*shoulder_mm       +0.0950
    tolerance_mm*glue_viscosity    +0.6300
    shoulder_mm*glue_viscosity     +0.1175
    tolerance_mm^2                 -0.0729
    shoulder_mm^2                  +0.5396
    glue_viscosity^2               +0.6246

  Curvature analysis:
    glue_viscosity                 coef=+0.6246  convex (has a minimum)
    shoulder_mm                    coef=+0.5396  convex (has a minimum)
    tolerance_mm                   coef=-0.0729  negligible curvature

  Notable interactions:
    tolerance_mm*glue_viscosity    coef=+0.6300  (synergistic)

  Predicted optimum (from linear model, at observed points):
    tolerance_mm = 0.5
    shoulder_mm = 3
    glue_viscosity = 4500
    Predicted value: 4.3291

  Surface optimum (via L-BFGS-B, linear model):
    tolerance_mm = 0.5
    shoulder_mm = 3
    glue_viscosity = 8000
    Predicted value: 4.3553

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

Factor importance:
  1. shoulder_mm  (effect: 1.0, contribution: 47.2%)
  2. glue_viscosity  (effect: 0.6, contribution: 30.7%)
  3. tolerance_mm  (effect: 0.4, contribution: 22.1%)

=== Optimization: assembly_score ===
Direction: maximize

Best observed run: #4
  tolerance_mm = 0.275
  shoulder_mm = 6.5
  glue_viscosity = 4500
  Value: 7.4

RSM Model (linear, R² = 0.1820, Adj R² = -0.0411):
  Coefficients:
    intercept                      +6.0000
    tolerance_mm                   -0.0125
    shoulder_mm                    +0.5625
    glue_viscosity                 -0.0750

RSM Model (quadratic, R² = 0.6378, Adj R² = -0.0142):
  Coefficients:
    intercept                      +6.7333
    tolerance_mm                   -0.0125
    shoulder_mm                    +0.5625
    glue_viscosity                 -0.0750
    tolerance_mm*shoulder_mm       -0.1500
    tolerance_mm*glue_viscosity    -0.6750
    shoulder_mm*glue_viscosity     +0.1250
    tolerance_mm^2                 +0.1333
    shoulder_mm^2                  -0.9667
    glue_viscosity^2               -0.5417

  Curvature analysis:
    shoulder_mm                    coef=-0.9667  concave (has a maximum)
    glue_viscosity                 coef=-0.5417  concave (has a maximum)
    tolerance_mm                   coef=+0.1333  convex (has a minimum)

  Notable interactions:
    tolerance_mm*glue_viscosity    coef=-0.6750  (antagonistic)

  Predicted optimum (from quadratic model, at observed points):
    tolerance_mm = 0.5
    shoulder_mm = 6.5
    glue_viscosity = 1000
    Predicted value: 7.0625

  Surface optimum (via L-BFGS-B, quadratic model):
    tolerance_mm = 0.05
    shoulder_mm = 7.92584
    glue_viscosity = 6602.98
    Predicted value: 7.2046

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

Factor importance:
  1. shoulder_mm  (effect: 1.5, contribution: 64.9%)
  2. glue_viscosity  (effect: 0.6, contribution: 24.1%)
  3. tolerance_mm  (effect: 0.3, contribution: 11.0%)