Hot Air Balloon Flight Planning

Summary

This experiment investigates hot air balloon flight planning. Box-Behnken design to maximize flight duration and altitude ceiling by tuning burner output, envelope volume, and passenger count.

The design varies 3 factors: burner btu (BTU/hr), ranging from 6000000 to 12000000, envelope m3 (m3), ranging from 2000 to 4000, and passengers (count), ranging from 2 to 8. The goal is to optimize 2 responses: flight hrs (hrs) (maximize) and ceiling m (m) (maximize). Fixed conditions held constant across all runs include fuel kg = 100, ambient temp = 15C.

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 flight hrs, the most influential factors were passengers (42.0%), burner btu (32.9%), envelope m3 (25.1%). The best observed value was 2.8 (at burner btu = 6e+06, envelope m3 = 3000, passengers = 2).

For ceiling m, the most influential factors were passengers (43.1%), burner btu (36.7%), envelope m3 (20.2%). The best observed value was 729.0 (at burner btu = 6e+06, envelope m3 = 3000, passengers = 2).

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
`burner_btu`	6000000	12000000	BTU/hr
`envelope_m3`	2000	4000	m3
`passengers`	2	8	count

Fixed: fuel_kg = 100, ambient_temp = 15C

Responses

Response	Direction	Unit
`flight_hrs`	↑ maximize	hrs
`ceiling_m`	↑ maximize	m

Configuration

use_cases/270_hot_air_balloon/config.json

{
  "metadata": {
    "name": "Hot Air Balloon Flight Planning",
    "description": "Box-Behnken design to maximize flight duration and altitude ceiling by tuning burner output, envelope volume, and passenger count"
  },
  "factors": [
    {
      "name": "burner_btu",
      "levels": [
        "6000000",
        "12000000"
      ],
      "type": "continuous",
      "unit": "BTU/hr"
    },
    {
      "name": "envelope_m3",
      "levels": [
        "2000",
        "4000"
      ],
      "type": "continuous",
      "unit": "m3"
    },
    {
      "name": "passengers",
      "levels": [
        "2",
        "8"
      ],
      "type": "continuous",
      "unit": "count"
    }
  ],
  "fixed_factors": {
    "fuel_kg": "100",
    "ambient_temp": "15C"
  },
  "responses": [
    {
      "name": "flight_hrs",
      "optimize": "maximize",
      "unit": "hrs"
    },
    {
      "name": "ceiling_m",
      "optimize": "maximize",
      "unit": "m"
    }
  ],
  "settings": {
    "operation": "box_behnken",
    "test_script": "use_cases/270_hot_air_balloon/sim.sh"
  }
}

Experimental Matrix

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

Run	`burner_btu`	`envelope_m3`	`passengers`
1	9e+06	2000	2
2	9e+06	3000	5
3	1.2e+07	3000	8
4	1.2e+07	3000	2
5	9e+06	3000	5
6	9e+06	3000	5
7	6e+06	3000	8
8	1.2e+07	2000	5
9	9e+06	2000	8
10	1.2e+07	4000	5
11	6e+06	3000	2
12	9e+06	4000	8
13	6e+06	2000	5
14	6e+06	4000	5
15	9e+06	4000	2

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/270_hot_air_balloon/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/270_hot_air_balloon/config.json \
    --output use_cases/270_hot_air_balloon/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/270_hot_air_balloon/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/270_hot_air_balloon/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/270_hot_air_balloon/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/270_hot_air_balloon/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/270_hot_air_balloon/config.json \
    --output use_cases/270_hot_air_balloon/results/report.html

Features Exercised

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

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: flight_hrs

Top factors: passengers (42.0%), burner_btu (32.9%), envelope_m3 (25.1%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
burner_btu	2	0.6298	0.3149	0.482	0.6344
envelope_m3	2	0.2973	0.1486	0.227	0.8015
passengers	2	0.7373	0.3686	0.564	0.5899
Lack	of	Fit	6	4.8024	0.8004
Pure	Error	2	1.3067
Error	8	6.1090	0.6533
Total	14	7.7733	0.5552

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Half-Normal Plot of Effects

Model Diagnostics

Response: ceiling_m

Top factors: passengers (43.1%), burner_btu (36.7%), envelope_m3 (20.2%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
burner_btu	2	38454.0429	19227.0214	1.108	0.3760
envelope_m3	2	11595.9714	5797.9857	0.334	0.7255
passengers	2	38988.9714	19494.4857	1.124	0.3715
Lack	of	Fit	6	254570.7476	42428.4579
Pure	Error	2	34698.6667
Error	8	289269.4143	17349.3333
Total	14	378308.4000	27022.0286

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.

ceiling m burner btu vs envelope m3

ceiling m burner btu vs passengers

ceiling m envelope m3 vs passengers

flight hrs burner btu vs envelope m3

flight hrs burner btu vs passengers

flight hrs envelope m3 vs passengers

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 = 0.9545

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`flight_hrs`	1.5	0.9545	2.80 0.9545 2.80 hrs	↑
`ceiling_m`	1.0	0.9545	729.00 0.9545 729.00 m	↑

Recommended Settings

Factor	Value
`burner_btu`	1.2e+07 BTU/hr
`envelope_m3`	3000 m3
`passengers`	8 count

Source: from observed run #15

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`ceiling_m`	729.00	729.00	+0.00

Top 3 Runs by Desirability

Run	D	Factor Settings
#4	0.8373	burner_btu=9e+06, envelope_m3=4000, passengers=2
#10	0.7735	burner_btu=6e+06, envelope_m3=2000, passengers=5

Model Quality

Response	R²	Type
`ceiling_m`	0.0917	linear

Full Multi-Objective Output

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

Overall desirability: D = 0.9545

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
flight_hrs                   1.5       0.9545        2.80 hrs   ↑
ceiling_m                    1.0       0.9545      729.00 m   ↑

Recommended settings:
  burner_btu = 1.2e+07 BTU/hr
  envelope_m3 = 3000 m3
  passengers = 8 count
  (from observed run #15)

Trade-off summary:
  flight_hrs: 2.80 (best observed: 2.80, sacrifice: +0.00)
  ceiling_m: 729.00 (best observed: 729.00, sacrifice: +0.00)

Model quality:
  flight_hrs: R² = 0.1952 (linear)
  ceiling_m: R² = 0.0917 (linear)

Top 3 observed runs by overall desirability:
  1. Run #15 (D=0.9545): burner_btu=1.2e+07, envelope_m3=3000, passengers=8
  2. Run #4 (D=0.8373): burner_btu=9e+06, envelope_m3=4000, passengers=2
  3. Run #10 (D=0.7735): burner_btu=6e+06, envelope_m3=2000, passengers=5

Full Analysis Output

doe analyze
=== Main Effects: flight_hrs ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
passengers               0.5250       0.1924            42.0%
burner_btu               0.4107       0.1924            32.9%
envelope_m3              0.3143       0.1924            25.1%

=== ANOVA Table: flight_hrs ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
burner_btu                   2       0.6298       0.3149      0.482     0.6344
envelope_m3                  2       0.2973       0.1486      0.227     0.8015
passengers                   2       0.7373       0.3686      0.564     0.5899
Lack of Fit                  6       4.8024       0.8004      1.225     0.5142
Pure Error                   2       1.3067       0.6533
Error                        8       6.1090       0.6533
Total                       14       7.7733       0.5552

=== Summary Statistics: flight_hrs ===

burner_btu:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.2e+07             4     1.1750     0.9708     0.3000     2.4000
  6e+06               4     1.1750     0.4787     0.8000     1.8000
  9e+06               7     1.5857     0.7777     0.3000     2.8000

envelope_m3:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2000                4     1.2750     0.6652     0.3000     1.8000
  3000                7     1.5143     0.8802     0.3000     2.8000
  4000                4     1.2000     0.7071     0.5000     2.1000

passengers:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4     1.0000     0.5598     0.3000     1.5000
  5                   7     1.4857     0.7581     0.5000     2.8000
  8                   4     1.5250     0.9394     0.3000     2.4000

=== Main Effects: ceiling_m ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
passengers             137.0000      42.4437            43.1%
burner_btu             116.8929      42.4437            36.7%
envelope_m3             64.2143      42.4437            20.2%

=== ANOVA Table: ceiling_m ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
burner_btu                   2   38454.0429   19227.0214      1.108     0.3760
envelope_m3                  2   11595.9714    5797.9857      0.334     0.7255
passengers                   2   38988.9714   19494.4857      1.124     0.3715
Lack of Fit                  6  254570.7476   42428.4579      2.446     0.3184
Pure Error                   2   34698.6667   17349.3333
Error                        8  289269.4143   17349.3333
Total                       14  378308.4000   27022.0286

=== Summary Statistics: ceiling_m ===

burner_btu:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.2e+07             4   407.2500   238.4832   194.0000   686.0000
  6e+06               4   446.2500    64.2048   355.0000   505.0000
  9e+06               7   524.1429   161.6915   239.0000   729.0000

envelope_m3:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2000                4   436.5000   135.5126   239.0000   525.0000
  3000                7   500.7143   175.5588   224.0000   729.0000
  4000                4   458.0000   205.5043   194.0000   696.0000

passengers:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4   394.5000   134.1405   224.0000   525.0000
  5                   7   482.7143   157.1355   194.0000   729.0000
  8                   4   531.5000   213.8450   239.0000   696.0000

Optimization Recommendations

doe optimize
=== Optimization: flight_hrs ===
Direction: maximize

Best observed run: #15
  burner_btu = 6e+06
  envelope_m3 = 3000
  passengers = 2
  Value: 2.8

RSM Model (linear, R² = 0.4384, Adj R² = 0.2852):
  Coefficients:
    intercept                      +1.3667
    burner_btu                     -0.1250
    envelope_m3                    -0.1875
    passengers                     -0.6125

RSM Model (quadratic, R² = 0.6839, Adj R² = 0.1148):
  Coefficients:
    intercept                      +1.4000
    burner_btu                     -0.1250
    envelope_m3                    -0.1875
    passengers                     -0.6125
    burner_btu*envelope_m3         -0.2000
    burner_btu*passengers          +0.3500
    envelope_m3*passengers         -0.0750
    burner_btu^2                   +0.1125
    envelope_m3^2                  -0.4625
    passengers^2                   +0.2875

  Curvature analysis:
    envelope_m3                    coef=-0.4625  concave (has a maximum)
    passengers                     coef=+0.2875  convex (has a minimum)
    burner_btu                     coef=+0.1125  convex (has a minimum)

  Notable interactions:
    burner_btu*passengers          coef=+0.3500  (synergistic)

  Predicted optimum (from linear model, at observed points):
    burner_btu = 9e+06
    envelope_m3 = 2000
    passengers = 2
    Predicted value: 2.1667

  Surface optimum (via L-BFGS-B, linear model):
    burner_btu = 6e+06
    envelope_m3 = 2000
    passengers = 2
    Predicted value: 2.2917

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

Factor importance:
  1. passengers  (effect: 1.2, contribution: 56.9%)
  2. envelope_m3  (effect: 0.7, contribution: 31.5%)
  3. burner_btu  (effect: 0.2, contribution: 11.6%)

=== Optimization: ceiling_m ===
Direction: maximize

Best observed run: #15
  burner_btu = 6e+06
  envelope_m3 = 3000
  passengers = 2
  Value: 729.0

RSM Model (linear, R² = 0.3065, Adj R² = 0.1174):
  Coefficients:
    intercept                      +472.2000
    burner_btu                     -16.1250
    envelope_m3                    -60.0000
    passengers                     -103.1250

RSM Model (quadratic, R² = 0.6521, Adj R² = 0.0259):
  Coefficients:
    intercept                      +483.3333
    burner_btu                     -16.1250
    envelope_m3                    -60.0000
    passengers                     -103.1250
    burner_btu*envelope_m3         -82.0000
    burner_btu*passengers          +50.7500
    envelope_m3*passengers         -75.5000
    burner_btu^2                   +61.4583
    envelope_m3^2                  -113.7917
    passengers^2                   +31.4583

  Curvature analysis:
    envelope_m3                    coef=-113.7917  concave (has a maximum)
    burner_btu                     coef=+61.4583  convex (has a minimum)
    passengers                     coef=+31.4583  convex (has a minimum)

  Notable interactions:
    burner_btu*envelope_m3         coef=-82.0000  (antagonistic)
    envelope_m3*passengers         coef=-75.5000  (antagonistic)
    burner_btu*passengers          coef=+50.7500  (synergistic)

  Predicted optimum (from linear model, at observed points):
    burner_btu = 9e+06
    envelope_m3 = 2000
    passengers = 2
    Predicted value: 635.3250

  Surface optimum (via L-BFGS-B, linear model):
    burner_btu = 6e+06
    envelope_m3 = 2000
    passengers = 2
    Predicted value: 651.4500

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

Factor importance:
  1. passengers  (effect: 206.2, contribution: 43.9%)
  2. envelope_m3  (effect: 180.4, contribution: 38.4%)
  3. burner_btu  (effect: 83.5, contribution: 17.8%)