Photo Print Color Accuracy

Summary

This experiment investigates photo print color accuracy. Box-Behnken design to maximize color accuracy and minimize ink cost by tuning color profile, ink density, and paper type brightness.

The design varies 3 factors: profile gamma (gamma), ranging from 1.8 to 2.4, ink density pct (%), ranging from 80 to 120, and paper brightness (ISO), ranging from 90 to 100. The goal is to optimize 2 responses: delta e (dE) (minimize) and ink ml (mL/m2) (minimize). Fixed conditions held constant across all runs include printer = inkjet_8color, resolution dpi = 1440.

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 delta e, the most influential factors were paper brightness (43.2%), profile gamma (35.5%), ink density pct (21.3%). The best observed value was 3.8 (at profile gamma = 2.1, ink density pct = 100, paper brightness = 95).

For ink ml, the most influential factors were profile gamma (45.9%), paper brightness (38.3%), ink density pct (15.8%). The best observed value was 5.6 (at profile gamma = 2.1, ink density pct = 100, paper brightness = 95).

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
`profile_gamma`	1.8	2.4	gamma
`ink_density_pct`	80	120	%
`paper_brightness`	90	100	ISO

Fixed: printer = inkjet_8color, resolution_dpi = 1440

Responses

Response	Direction	Unit
`delta_e`	↓ minimize	dE
`ink_ml`	↓ minimize	mL/m2

Configuration

use_cases/155_photo_print_color/config.json

{
  "metadata": {
    "name": "Photo Print Color Accuracy",
    "description": "Box-Behnken design to maximize color accuracy and minimize ink cost by tuning color profile, ink density, and paper type brightness"
  },
  "factors": [
    {
      "name": "profile_gamma",
      "levels": [
        "1.8",
        "2.4"
      ],
      "type": "continuous",
      "unit": "gamma"
    },
    {
      "name": "ink_density_pct",
      "levels": [
        "80",
        "120"
      ],
      "type": "continuous",
      "unit": "%"
    },
    {
      "name": "paper_brightness",
      "levels": [
        "90",
        "100"
      ],
      "type": "continuous",
      "unit": "ISO"
    }
  ],
  "fixed_factors": {
    "printer": "inkjet_8color",
    "resolution_dpi": "1440"
  },
  "responses": [
    {
      "name": "delta_e",
      "optimize": "minimize",
      "unit": "dE"
    },
    {
      "name": "ink_ml",
      "optimize": "minimize",
      "unit": "mL/m2"
    }
  ],
  "settings": {
    "operation": "box_behnken",
    "test_script": "use_cases/155_photo_print_color/sim.sh"
  }
}

Experimental Matrix

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

Run	`profile_gamma`	`ink_density_pct`	`paper_brightness`
1	2.1	80	90
2	2.1	100	95
3	2.4	100	100
4	2.4	100	90
5	2.1	100	95
6	2.1	100	95
7	1.8	100	100
8	2.4	80	95
9	2.1	80	100
10	2.4	120	95
11	1.8	100	90
12	2.1	120	100
13	1.8	80	95
14	1.8	120	95
15	2.1	120	90

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/155_photo_print_color/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/155_photo_print_color/config.json \
    --output use_cases/155_photo_print_color/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/155_photo_print_color/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/155_photo_print_color/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/155_photo_print_color/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/155_photo_print_color/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/155_photo_print_color/config.json \
    --output use_cases/155_photo_print_color/results/report.html

Features Exercised

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

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: delta_e

Top factors: paper_brightness (43.2%), profile_gamma (35.5%), ink_density_pct (21.3%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
profile_gamma	2	5.7869	2.8934	3.791	0.0695
ink_density_pct	2	1.4379	0.7190	0.942	0.4292
paper_brightness	2	5.1979	2.5990	3.405	0.0852
Lack	of	Fit	6	7.3746	1.2291
Pure	Error	2	1.5267
Error	8	8.9013	0.7633
Total	14	21.3240	1.5231

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Half-Normal Plot of Effects

Model Diagnostics

Response: ink_ml

Top factors: profile_gamma (45.9%), paper_brightness (38.3%), ink_density_pct (15.8%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
profile_gamma	2	11.4079	5.7040	3.649	0.0748
ink_density_pct	2	1.2347	0.6174	0.395	0.6862
paper_brightness	2	10.4979	5.2490	3.358	0.0874
Lack	of	Fit	6	27.7368	4.6228
Pure	Error	2	3.1267
Error	8	30.8634	1.5633
Total	14	54.0040	3.8574

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.

delta e ink density pct vs paper brightness

delta e profile gamma vs ink density pct

delta e profile gamma vs paper brightness

ink ml ink density pct vs paper brightness

ink ml profile gamma vs ink density pct

ink ml profile gamma vs paper brightness

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 = 0.8690

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`delta_e`	1.5	0.8437	4.30 0.8437 4.30 dE	↓
`ink_ml`	1.0	0.9083	5.90 0.9083 5.90 mL/m2	↓

Recommended Settings

Factor	Value
`profile_gamma`	2.1 gamma
`ink_density_pct`	80 %
`paper_brightness`	90 ISO

Source: from observed run #13

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`ink_ml`	5.90	5.60	+0.30

Top 3 Runs by Desirability

Run	D	Factor Settings
#9	0.8449	profile_gamma=2.1, ink_density_pct=120, paper_brightness=90
#7	0.7928	profile_gamma=2.4, ink_density_pct=80, paper_brightness=95

Model Quality

Response	R²	Type
`ink_ml`	0.6998	quadratic

Full Multi-Objective Output

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

Overall desirability: D = 0.8690

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
delta_e                      1.5       0.8437        4.30 dE   ↓
ink_ml                       1.0       0.9083        5.90 mL/m2   ↓

Recommended settings:
  profile_gamma = 2.1 gamma
  ink_density_pct = 80 %
  paper_brightness = 90 ISO
  (from observed run #13)

Trade-off summary:
  delta_e: 4.30 (best observed: 3.80, sacrifice: +0.50)
  ink_ml: 5.90 (best observed: 5.60, sacrifice: +0.30)

Model quality:
  delta_e: R² = 0.0763 (linear)
  ink_ml: R² = 0.6998 (quadratic)

Top 3 observed runs by overall desirability:
  1. Run #13 (D=0.8690): profile_gamma=2.1, ink_density_pct=80, paper_brightness=90
  2. Run #9 (D=0.8449): profile_gamma=2.1, ink_density_pct=120, paper_brightness=90
  3. Run #7 (D=0.7928): profile_gamma=2.4, ink_density_pct=80, paper_brightness=95

Full Analysis Output

doe analyze
=== Main Effects: delta_e ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
paper_brightness         1.5750       0.3187            43.2%
profile_gamma            1.2929       0.3187            35.5%
ink_density_pct          0.7750       0.3187            21.3%

=== ANOVA Table: delta_e ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
profile_gamma                2       5.7869       2.8934      3.791     0.0695
ink_density_pct              2       1.4379       0.7190      0.942     0.4292
paper_brightness             2       5.1979       2.5990      3.405     0.0852
Lack of Fit                  6       7.3746       1.2291      1.610     0.4313
Pure Error                   2       1.5267       0.7633
Error                        8       8.9013       0.7633
Total                       14      21.3240       1.5231

=== Summary Statistics: delta_e ===

profile_gamma:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.8                 4     6.1500     1.1619     5.0000     7.7000
  2.1                 7     4.9571     0.9727     3.8000     6.7000
  2.4                 4     6.2500     1.3916     5.0000     7.9000

ink_density_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  100                 7     5.4857     1.3825     3.8000     7.7000
  120                 4     6.1250     1.5370     4.3000     7.9000
  80                  4     5.3500     0.6658     4.8000     6.3000

paper_brightness:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  100                 4     4.9500     0.4509     4.3000     5.3000
  90                  4     6.5250     1.2285     4.8000     7.7000
  95                  7     5.4857     1.3533     3.8000     7.9000

=== Main Effects: ink_ml ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
profile_gamma            2.0250       0.5071            45.9%
paper_brightness         1.6893       0.5071            38.3%
ink_density_pct          0.6964       0.5071            15.8%

=== ANOVA Table: ink_ml ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
profile_gamma                2      11.4079       5.7040      3.649     0.0748
ink_density_pct              2       1.2347       0.6174      0.395     0.6862
paper_brightness             2      10.4979       5.2490      3.358     0.0874
Lack of Fit                  6      27.7368       4.6228      2.957     0.2742
Pure Error                   2       3.1267       1.5633
Error                        8      30.8634       1.5633
Total                       14      54.0040       3.8574

=== Summary Statistics: ink_ml ===

profile_gamma:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.8                 4     9.8250     1.8319     8.0000    11.5000
  2.1                 7     7.8857     2.2327     5.6000    11.0000
  2.4                 4     7.8000     0.9345     6.4000     8.3000

ink_density_pct:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  100                 7     8.1286     2.4445     5.6000    11.5000
  120                 4     8.3750     1.9466     6.3000    11.0000
  80                  4     8.8250     1.3598     7.7000    10.8000

paper_brightness:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  100                 4     9.1750     2.3229     6.3000    11.3000
  90                  4     9.1500     2.4906     6.4000    11.5000
  95                  7     7.4857     1.2048     5.6000     8.5000

Optimization Recommendations

doe optimize
=== Optimization: delta_e ===
Direction: minimize

Best observed run: #7
  profile_gamma = 2.1
  ink_density_pct = 100
  paper_brightness = 95
  Value: 3.8

RSM Model (linear, R² = 0.3965, Adj R² = 0.2319):
  Coefficients:
    intercept                      +5.6200
    profile_gamma                  +0.6250
    ink_density_pct                -0.3750
    paper_brightness               +0.7250

RSM Model (quadratic, R² = 0.7608, Adj R² = 0.3303):
  Coefficients:
    intercept                      +5.4000
    profile_gamma                  +0.6250
    ink_density_pct                -0.3750
    paper_brightness               +0.7250
    profile_gamma*ink_density_pct  -0.7250
    profile_gamma*paper_brightness +0.6750
    ink_density_pct*paper_brightness -0.4250
    profile_gamma^2                -0.4125
    ink_density_pct^2              +0.0375
    paper_brightness^2             +0.7875

  Curvature analysis:
    paper_brightness               coef=+0.7875  convex (has a minimum)
    profile_gamma                  coef=-0.4125  concave (has a maximum)
    ink_density_pct                coef=+0.0375  negligible curvature

  Notable interactions:
    profile_gamma*ink_density_pct  coef=-0.7250  (antagonistic)
    profile_gamma*paper_brightness coef=+0.6750  (synergistic)
    ink_density_pct*paper_brightness coef=-0.4250  (antagonistic)

  Predicted optimum (from quadratic model, at observed points):
    profile_gamma = 2.4
    ink_density_pct = 100
    paper_brightness = 100
    Predicted value: 7.8000

  Surface optimum (via L-BFGS-B, quadratic model):
    profile_gamma = 1.8
    ink_density_pct = 80
    paper_brightness = 93.4921
    Predicted value: 3.9784

  Model quality: Good fit — general trends are captured, some noise remains.

Factor importance:
  1. paper_brightness  (effect: 1.5, contribution: 43.5%)
  2. profile_gamma  (effect: 1.2, contribution: 35.3%)
  3. ink_density_pct  (effect: 0.8, contribution: 21.2%)

=== Optimization: ink_ml ===
Direction: minimize

Best observed run: #1
  profile_gamma = 2.1
  ink_density_pct = 100
  paper_brightness = 95
  Value: 5.6

RSM Model (linear, R² = 0.2173, Adj R² = 0.0039):
  Coefficients:
    intercept                      +8.3800
    profile_gamma                  -0.0250
    ink_density_pct                -1.1875
    paper_brightness               -0.2375

RSM Model (quadratic, R² = 0.8132, Adj R² = 0.4769):
  Coefficients:
    intercept                      +6.6333
    profile_gamma                  -0.0250
    ink_density_pct                -1.1875
    paper_brightness               -0.2375
    profile_gamma*ink_density_pct  -1.6250
    profile_gamma*paper_brightness +0.7250
    ink_density_pct*paper_brightness -0.0500
    profile_gamma^2                +0.2083
    ink_density_pct^2              +0.8833
    paper_brightness^2             +2.1833

  Curvature analysis:
    paper_brightness               coef=+2.1833  convex (has a minimum)
    ink_density_pct                coef=+0.8833  convex (has a minimum)
    profile_gamma                  coef=+0.2083  convex (has a minimum)

  Notable interactions:
    profile_gamma*ink_density_pct  coef=-1.6250  (antagonistic)
    profile_gamma*paper_brightness coef=+0.7250  (synergistic)

  Predicted optimum (from quadratic model, at observed points):
    profile_gamma = 2.1
    ink_density_pct = 80
    paper_brightness = 90
    Predicted value: 11.0750

  Surface optimum (via L-BFGS-B, quadratic model):
    profile_gamma = 2.4
    ink_density_pct = 120
    paper_brightness = 94.499
    Predicted value: 4.8656

  Model quality: Good fit — general trends are captured, some noise remains.

Factor importance:
  1. ink_density_pct  (effect: 2.4, contribution: 49.8%)
  2. paper_brightness  (effect: 2.3, contribution: 49.1%)
  3. profile_gamma  (effect: 0.1, contribution: 1.0%)

Run	`profile_gamma`	`ink_density_pct`	`paper_brightness`
1	2.1	80	90
2	2.1	100	95
3	2.4	100	100
4	2.4	100	90
5	2.1	100	95
6	2.1	100	95
7	1.8	100	100
8	2.4	80	95
9	2.1	80	100
10	2.4	120	95
11	1.8	100	90
12	2.1	120	100
13	1.8	80	95
14	1.8	120	95
15	2.1	120	90

Run	`profile_gamma`	`ink_density_pct`	`paper_brightness`
1	2.1	80	90
2	2.1	100	95
3	2.4	100	100
4	2.4	100	90
5	2.1	100	95
6	2.1	100	95
7	1.8	100	100
8	2.4	80	95
9	2.1	80	100
10	2.4	120	95
11	1.8	100	90
12	2.1	120	100
13	1.8	80	95
14	1.8	120	95
15	2.1	120	90

Run	`profile_gamma`	`ink_density_pct`	`paper_brightness`
1	2.1	80	90
2	2.1	100	95
3	2.4	100	100
4	2.4	100	90
5	2.1	100	95
6	2.1	100	95
7	1.8	100	100
8	2.4	80	95
9	2.1	80	100
10	2.4	120	95
11	1.8	100	90
12	2.1	120	100
13	1.8	80	95
14	1.8	120	95
15	2.1	120	90