Ceramic Glaze Firing Optimization

Summary

This experiment investigates ceramic glaze firing optimization. Taguchi orthogonal array with mixed-level factors for ceramic glaze quality.

The design varies 5 factors: clay type, ranging from porcelain to earthenware, glaze thickness, ranging from thin to thick, peak temperature (°C), ranging from 1100 to 1250, hold time (min), ranging from 30 to 90, and cooling rate, ranging from slow to fast. The goal is to optimize 2 responses: surface smoothness (score) (maximize) and color accuracy (deltaE) (minimize). Fixed conditions held constant across all runs include kiln type = electric.

The Taguchi Orthogonal Array produces 16 experimental runs.

Key Findings

For surface smoothness, the most influential factors were glaze thickness (33.0%), hold time (27.7%), peak temperature (20.7%). The best observed value was 86.91 (at clay type = porcelain, glaze thickness = thin, peak temperature = 1100).

For color accuracy, the most influential factors were hold time (32.9%), glaze thickness (21.9%), cooling rate (21.1%). The best observed value was 4.46 (at clay type = porcelain, glaze thickness = thin, peak temperature = 1250).

Recommended Next Steps

Consider whether any fixed factors should be varied in a future study.
The screening results can guide factor reduction — drop factors contributing less than 5% and re-run with a smaller, more focused design.

Experimental Setup

Factors

Factor	Low	High	Unit
`clay_type`	porcelain	earthenware
`glaze_thickness`	thin	thick
`peak_temperature`	1100	1250	°C
`hold_time`	30	90	min
`cooling_rate`	slow	fast

Fixed: kiln_type = electric

Responses

Response	Direction	Unit
`surface_smoothness`	↑ maximize	score
`color_accuracy`	↓ minimize	deltaE

Configuration

use_cases/302_ceramic_glaze_firing/config.json

{
  "metadata": {
    "name": "Ceramic Glaze Firing Optimization",
    "description": "Taguchi orthogonal array with mixed-level factors for ceramic glaze quality"
  },
  "factors": [
    {
      "name": "clay_type",
      "levels": [
        "porcelain",
        "stoneware",
        "earthenware"
      ],
      "type": "categorical",
      "unit": ""
    },
    {
      "name": "glaze_thickness",
      "levels": [
        "thin",
        "medium",
        "thick"
      ],
      "type": "ordinal",
      "unit": ""
    },
    {
      "name": "peak_temperature",
      "levels": [
        "1100",
        "1250"
      ],
      "type": "continuous",
      "unit": "\u00b0C"
    },
    {
      "name": "hold_time",
      "levels": [
        "30",
        "60",
        "90"
      ],
      "type": "continuous",
      "unit": "min"
    },
    {
      "name": "cooling_rate",
      "levels": [
        "slow",
        "fast"
      ],
      "type": "categorical",
      "unit": ""
    }
  ],
  "fixed_factors": {
    "kiln_type": "electric"
  },
  "responses": [
    {
      "name": "surface_smoothness",
      "optimize": "maximize",
      "unit": "score"
    },
    {
      "name": "color_accuracy",
      "optimize": "minimize",
      "unit": "deltaE"
    }
  ],
  "settings": {
    "operation": "taguchi",
    "test_script": "use_cases/302_ceramic_glaze_firing/sim.sh"
  }
}

Experimental Matrix

The Taguchi Orthogonal Array produces 16 runs. Each row is one experiment with specific factor settings.

Run	`clay_type`	`glaze_thickness`	`peak_temperature`	`hold_time`	`cooling_rate`
1	stoneware	thin	1100	60	slow
2	earthenware	medium	1250	90	slow
3	stoneware	medium	1100	30	slow
4	stoneware	thick	1250	30	fast
5	porcelain	thick	1250	30	slow
6	earthenware	thick	1100	60	fast
7	porcelain	thin	1250	60	slow
8	earthenware	thin	1100	30	fast
9	porcelain	medium	1250	60	fast
10	porcelain	thick	1100	90	slow
11	porcelain	medium	1100	30	fast
12	porcelain	thin	1100	90	fast
13	stoneware	thin	1250	90	fast
14	earthenware	thin	1250	30	slow
15	porcelain	thin	1100	30	slow
16	porcelain	thin	1250	30	fast

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/302_ceramic_glaze_firing/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/302_ceramic_glaze_firing/config.json \
    --output use_cases/302_ceramic_glaze_firing/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/302_ceramic_glaze_firing/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/302_ceramic_glaze_firing/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/302_ceramic_glaze_firing/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/302_ceramic_glaze_firing/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/302_ceramic_glaze_firing/config.json \
    --output use_cases/302_ceramic_glaze_firing/results/report.html

Features Exercised

Feature	Value
Design type	`taguchi`
Factor types	`continuous` (2), `categorical` (2)
Arg style	`double-dash`
Responses	2 (surface_smoothness ↑, color_accuracy ↓)
Total runs	16

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: surface_smoothness

Top factors: glaze_thickness (33.0%), hold_time (27.7%), peak_temperature (20.7%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
clay_type	2	189.4248	94.7124	3.001	0.1249
glaze_thickness	2	725.7416	362.8708	11.499	0.0089
peak_temperature	1	395.1150	395.1150	12.521	0.0122
hold_time	2	376.2926	188.1463	5.962	0.0375
cooling_rate	1	2.8477	2.8477	0.090	0.7740
peak_temperature*cooling_rate	1	121.2752	121.2752	3.843	0.0976
Error	6	189.3369	31.5561
Total	15	2000.0337	133.3356

Response: color_accuracy

Top factors: hold_time (32.9%), glaze_thickness (21.9%), cooling_rate (21.1%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
clay_type	2	0.6933	0.3467	0.055	0.9470
glaze_thickness	2	4.5282	2.2641	0.359	0.7126
peak_temperature	1	3.4410	3.4410	0.545	0.4881
hold_time	2	13.0643	6.5322	1.035	0.4110
cooling_rate	1	6.9696	6.9696	1.104	0.3338
peak_temperature*cooling_rate	1	1.4520	1.4520	0.230	0.6484
Error	6	37.8646	6.3108
Total	15	68.0131	4.5342

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 = 0.7417

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`surface_smoothness`	1.0	0.5081	66.47 0.5081 66.47 score	↑
`color_accuracy`	1.5	0.9545	4.46 0.9545 4.46 deltaE	↓

Recommended Settings

Factor	Value
`clay_type`	earthenware
`glaze_thickness`	thin
`peak_temperature`	1100 °C
`hold_time`	30 min
`cooling_rate`	fast

Source: from observed run #10

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`color_accuracy`	4.46	4.46	+0.00

Top 3 Runs by Desirability

Run	D	Factor Settings
#7	0.7143	clay_type=porcelain, glaze_thickness=thin, peak_temperature=1250, hold_time=60, cooling_rate=slow
#5	0.6669	clay_type=stoneware, glaze_thickness=medium, peak_temperature=1100, hold_time=30, cooling_rate=slow

Model Quality

Response	R²	Type
`color_accuracy`	0.3525	linear

Full Multi-Objective Output

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

Overall desirability: D = 0.7417

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
surface_smoothness           1.0       0.5081       66.47 score   ↑
color_accuracy               1.5       0.9545        4.46 deltaE   ↓

Recommended settings:
  clay_type = earthenware
  glaze_thickness = thin
  peak_temperature = 1100 °C
  hold_time = 30 min
  cooling_rate = fast
  (from observed run #10)

Trade-off summary:
  surface_smoothness: 66.47 (best observed: 86.91, sacrifice: +20.44)
  color_accuracy: 4.46 (best observed: 4.46, sacrifice: +0.00)

Model quality:
  surface_smoothness: R² = 0.4908 (linear)
  color_accuracy: R² = 0.3525 (linear)

Top 3 observed runs by overall desirability:
  1. Run #10 (D=0.7417): clay_type=earthenware, glaze_thickness=thin, peak_temperature=1100, hold_time=30, cooling_rate=fast
  2. Run #7 (D=0.7143): clay_type=porcelain, glaze_thickness=thin, peak_temperature=1250, hold_time=60, cooling_rate=slow
  3. Run #5 (D=0.6669): clay_type=stoneware, glaze_thickness=medium, peak_temperature=1100, hold_time=30, cooling_rate=slow

Full Analysis Output

doe analyze
=== Main Effects: surface_smoothness ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
glaze_thickness         15.8550       2.8868            33.0%
hold_time               13.3075       2.8868            27.7%
peak_temperature        -9.9388       2.8868            20.7%
clay_type                8.1450       2.8868            16.9%
cooling_rate             0.8438       2.8868             1.8%

=== ANOVA Table: surface_smoothness ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
clay_type                    2     189.4248      94.7124      3.001     0.1249
glaze_thickness              2     725.7416     362.8708     11.499     0.0089
peak_temperature             1     395.1150     395.1150     12.521     0.0122
hold_time                    2     376.2926     188.1463      5.962     0.0375
cooling_rate                 1       2.8477       2.8477      0.090     0.7740
peak_temperature*cooling_rate    1     121.2752     121.2752      3.843     0.0976
Error                        6     189.3369      31.5561
Total                       15    2000.0337     133.3356

=== Interaction Effects: surface_smoothness ===
Factor A             Factor B              Interaction   % Contribution
------------------------------------------------------------------------
peak_temperature     cooling_rate               5.5063           100.0%

=== Summary Statistics: surface_smoothness ===

clay_type:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  earthenware         4    73.6375    11.6386    62.7400    86.9100
  porcelain           8    65.4925    13.0390    45.2900    85.6800
  stoneware           4    66.1650     8.4486    59.8400    78.3700

glaze_thickness:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  medium              4    79.3575     9.3822    66.4700    86.9100
  thick               4    63.5025     3.6233    61.2000    68.8300
  thin                8    63.9637    11.7767    45.2900    79.8200

peak_temperature:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1100                8    72.6663     8.7369    59.8400    85.6800
  1250                8    62.7275    12.3669    45.2900    86.9100

hold_time:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  30                  8    68.8725    13.1456    45.2900    85.6800
  60                  4    59.8675     6.8581    50.4200    66.4700
  90                  4    73.1750     9.5394    65.2100    86.9100

cooling_rate:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  fast                8    67.2750    12.3137    45.2900    85.6800
  slow                8    68.1188    11.5623    50.4200    86.9100

=== Main Effects: color_accuracy ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
hold_time                2.0600       0.5323            32.9%
glaze_thickness          1.3700       0.5323            21.9%
cooling_rate             1.3200       0.5323            21.1%
peak_temperature        -0.9275       0.5323            14.8%
clay_type                0.5875       0.5323             9.4%

=== ANOVA Table: color_accuracy ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
clay_type                    2       0.6933       0.3467      0.055     0.9470
glaze_thickness              2       4.5282       2.2641      0.359     0.7126
peak_temperature             1       3.4410       3.4410      0.545     0.4881
hold_time                    2      13.0643       6.5322      1.035     0.4110
cooling_rate                 1       6.9696       6.9696      1.104     0.3338
peak_temperature*cooling_rate    1       1.4520       1.4520      0.230     0.6484
Error                        6      37.8646       6.3108
Total                       15      68.0131       4.5342

=== Interaction Effects: color_accuracy ===
Factor A             Factor B              Interaction   % Contribution
------------------------------------------------------------------------
peak_temperature     cooling_rate              -0.6025           100.0%

=== Summary Statistics: color_accuracy ===

clay_type:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  earthenware         4     9.4575     1.7940     7.9000    11.2000
  porcelain           8     9.7238     2.5640     4.4600    13.0600
  stoneware           4    10.0450     1.9702     9.0400    13.0000

glaze_thickness:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  medium              4     8.8325     3.5855     4.4600    13.0000
  thick               4    10.2025     1.0820     9.1000    11.4000
  thin                8     9.9575     1.7487     7.9000    13.0600

peak_temperature:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1100                8    10.2012     1.7312     7.9000    13.0000
  1250                8     9.2737     2.4955     4.4600    13.0600

hold_time:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  30                  8    10.6150     1.8406     7.9000    13.0600
  60                  4     8.5550     2.8240     4.4600    10.8100
  90                  4     9.1650     1.5609     7.9200    11.4000

cooling_rate:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  fast                8     9.0775     2.4767     4.4600    13.0600
  slow                8    10.3975     1.6083     7.9200    13.0000

Optimization Recommendations

doe optimize
=== Optimization: surface_smoothness ===
Direction: maximize

Best observed run: #7
  clay_type = porcelain
  glaze_thickness = thin
  peak_temperature = 1100
  hold_time = 30
  cooling_rate = slow
  Value: 86.91

RSM Model (linear, R² = 0.1937, Adj R² = -0.2095):
  Coefficients:
    intercept                      +68.1761
    clay_type                      -2.2088
    glaze_thickness                +1.9202
    peak_temperature               -0.7744
    hold_time                      +3.8370
    cooling_rate                   +2.9181

  Predicted optimum (from linear model, at observed points):
    clay_type = porcelain
    glaze_thickness = thick
    peak_temperature = 1100
    hold_time = 90
    cooling_rate = slow
    Predicted value: 75.7056

  Surface optimum (via L-BFGS-B, linear model):
    clay_type = porcelain
    glaze_thickness = 1.0000
    peak_temperature = 1100
    hold_time = 60
    cooling_rate = fast
    Predicted value: 79.8346

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

Factor importance:
  1. glaze_thickness  (effect: 11.9, contribution: 34.3%)
  2. hold_time  (effect: 11.0, contribution: 31.7%)
  3. cooling_rate  (effect: 5.8, contribution: 16.8%)
  4. clay_type  (effect: 4.4, contribution: 12.7%)
  5. peak_temperature  (effect: -1.5, contribution: 4.5%)

=== Optimization: color_accuracy ===
Direction: minimize

Best observed run: #10
  clay_type = porcelain
  glaze_thickness = thin
  peak_temperature = 1250
  hold_time = 60
  cooling_rate = slow
  Value: 4.46

RSM Model (linear, R² = 0.4404, Adj R² = 0.1605):
  Coefficients:
    intercept                      +9.9327
    clay_type                      +0.6212
    glaze_thickness                -1.1718
    peak_temperature               -0.2962
    hold_time                      -0.3909
    cooling_rate                   -0.7362

  Predicted optimum (from linear model, at observed points):
    clay_type = porcelain
    glaze_thickness = medium
    peak_temperature = 1100
    hold_time = 30
    cooling_rate = fast
    Predicted value: 12.5280

  Surface optimum (via L-BFGS-B, linear model):
    clay_type = porcelain
    glaze_thickness = 1.0000
    peak_temperature = 1250
    hold_time = 60
    cooling_rate = fast
    Predicted value: 6.7163

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

Factor importance:
  1. glaze_thickness  (effect: 2.2, contribution: 31.0%)
  2. clay_type  (effect: 1.6, contribution: 23.0%)
  3. cooling_rate  (effect: -1.5, contribution: 20.7%)
  4. hold_time  (effect: 1.2, contribution: 16.9%)
  5. peak_temperature  (effect: -0.6, contribution: 8.3%)