HAL Id: hal-01018489
https://hal-enac.archives-ouvertes.fr/hal-01018489
Submitted on 4 Jul 2014
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Genetic algorithms for partitioning air space
Daniel Delahaye, Jean-Marc Alliot, Marc Schoenauer, Jean-Loup Farges
To cite this version:
Daniel Delahaye, Jean-Marc Alliot, Marc Schoenauer, Jean-Loup Farges. Genetic algorithms for partitioning air space. AI 1994, 10th Conference on Artificiel Intelligence for Applications, Mar 1994, San Antonia, United States. pp 291-297, �10.1109/CAIA.1994.323662�. �hal-01018489�
✏ ✑✒ ✓✔✕ ✏ ✔✕✑✖ ✑✗✔ ✘✔✑✒ ✙✚ ✑✛✜ ✢ ✕✕✓✣✤ ✚ ✑✛✜ ✥ ✜✖ ✣✔✒ ✑✦ ✔✛ ✘✔✑✒ ✙✧✣✦ ★ ✩✑✛✪✔✫
✬ ✭ ✮ ✢ ✯ ✭ ✮ ✢ ✬ ✰ ✬ ✚ ✢ ✱ ✲ ✳ ✬ ✭ ✴ ✵ ✶
✷✸ ✘✦ ✕✗ ✷✹✹✺
✻ ✼ ✽✾✿❀❁✾
❂❃ ❄❅❆❇ ❈❉❈❊❋● ❍ ❊ ❇❅■❍ ❅■❍ ❏❊❃❊❄❆❑ ❉▲❏■❋❆❄❅▼ ❇ ❑❉❃
◆
❊ ❖❇❊P ❄■ ❑■▼ ❈❖❄❊ ❉❖❄■▼ ❉❄❆❑❉▲▲◗ ❉
◆
❉▲❉❃ ❑❊P ❇❊❑❄■❋❆❃❏
■❘ ❉❆❋❇❈❉❑❊ ❄■ ❆❃ ❑❋❊❉❇❊ ❙ ❆❋ ❚❋❉❯ ❑ ❱■❃❄❋■▲ ❑❉❈ ❉❑❆❄◗ ❆❃
❅❆❏❅ P❊❃❇❆❄◗ ❉❋❊❉❇❲❳
❨ ❩ ❬❭❪ ❫❴❵ ❛ ❜❝❜❞❡❢ ❣ ❤✐❥❦❡❞❧♠ ♥ ♦❜❞♣ ❥❦q rs❦❞❡❞❡❥❝t
❡❝✐ ✉
✈ ❭✇ ①❫② ❨ ③④① ❵ ❣ ❡❦ ⑤❦s⑥ ❢ ⑦ ❥❝❞❦❥❤
⑧❬① ❬⑨ ⑩❵ ❶ r ❜❦s❞❡❥❝ s❤ ♠ ❥ ❢qt❷ r ✉
❸ ❹ ❭③❬❵ ❺ ♠ s❝❻❼❜s❦
❩✇ ❪① ❴❬❵ ❽❝❢❦❜s❾❜ ❢srs❢❡❞❼ ❥❿ ❣ ❡❦ ⑤❦s⑥ ❢ ⑦❥❝❞❦❥❤
❾❼❾❞❜♠ ❾ ❡❝ ❧❡✐❧ ➀❜❝❾❡❞❼ s❦❜s❾ ➁❼ ➁ s❤s❝❢❡❝✐ ❤❥s➀ ❥❿ ❾❜❢t
❞❥❦❾✉
➂ ➃➄➅➆ ➇ ➈ ➉ ➊➅ ➋➇ ➄
⑤ ❧❜ ⑦ ➌ ♦ ❣ ❡❾ ❞❧ ❜ ❥❦✐s❝❡❾♠ ❡❝ ❢❧s❦✐❜ ❿❥❦ ❾❞❷➀❡❜❾
s❝ ➀ ❦❜❾❜s❦❢❧ ❿❥❦ ❜❝❧s❝❢❡❝✐ ❞❧❜ ➍❦❜❝ ❢❧ ❣⑤ ⑦ ❾❼❾❞❜♠ ❾✉
➎
❞❷➀❡❜❾ ❥❝ ❞❧❜ ❷ ❾❜ ❥❿ ✐❜❝❜❞❡❢ s❤✐❥❦❡❞❧♠ ❾ ❿❥❦ ❢❥❝➏❡❢❞
❦❜❾❥❤❷❞❡❥❝ ❧s➐❜ ✐❡➐❜❝ ❜❝ ❢❥❷ ❦s✐❡❝✐ ❦❜❾❷ ❤❞❾ ➑➒➓♥ s❝ ➀ s
❝❜♣ ❾❞❷➀❼ ❧s❾ ➁ ❜❜❝ ❿❷❝ ➀❜➀ ❞❥ ❾❥❤➐❜ ❞❧ ❜ ❤❥s➀ ➁s❤s❝ ❢❡❝✐
r❦❥➁ ❤❜♠ ❿❥❦ ❣ ❡❦ ⑤❦s⑥ ❢ ⑦❥❝❞❦❥❤✉ ➔ ❧ ❜❝ →❥❡❝❡❝✐ ❞♣ ❥
s❡❦r ❥❦❞❾♥ s❝ s❡❦❢❦s❿❞ ♠ ❷❾❞ ❿❥❤❤❥♣ ❦❥❷❞❜❾ s❝➀ ➁❜s❢❥❝ ❾ ➣
❞❧❜❾❜ ➁ ❜s❢❥❝❾ s❦❜ ❝❜❢❜❾❾s❦❼ ❿❥❦ r❡❤❥❞❾ ❞❥ q❝❥♣ ❞❧❜❡❦
r❥❾❡❞❡❥❝ ➀❷ ❦❡❝✐ ❝ s➐❡✐s❞❡❥❝ s❝ ➀ ➁ ❜❢s❷ ❾❜ ❥❿ ❞❧❜ ❾♠ s❤❤
❝❷♠ ➁❜❦ ❥❿ ➁❜s❢❥❝ ❾ ❥❝ ❞❧❜ ✐❦❥❷❝ ➀ ❞❧❜❼ ❥❿❞❜❝ ❦❜r❦❜❾❜❝❞
❢❦❥❾❾❡❝ ✐ r ❥❡❝❞❾ ❥❿ ➀❡↔❜❦❜❝❞ s❡❦♣ s❼❾✉
⑦ ❦❥❾❾❡❝✐ r❥❡❝❞❾ ♠ s❼ ✐❜❝❜❦s❞❜ ❢❥❝➏ ❡❢❞❾ ➁❜❞♣ ❜❜❝ s❡❦t
❢❦s❿❞ ♣❧ ❜❝ ❞❧❜❡❦ ❞❦s→❜❢❞❥❦❡❜❾ ❢❥❝➐❜❦✐❜ ❥❝ ❡❞ s❞ ❞❧❜ ❾s♠ ❜
❞❡♠ ❜ s❝➀ ❡❝ ➀❷❢❜ s ❦❡❾q ❥❿ ❢❥❤❤❡❾❡❥❝ ✉
❣ ❞ ❞❧ ❜ ➀ s♣❝ ❥❿ ❢❡➐❡❤ s➐❡s❞❡❥❝ ♥ r❡❤❥❞❾ ❦❜❾❥❤➐❜➀ ❢❥❝t
➏ ❡❢❞❾ ❞❧ ❜♠ ❾❜❤➐❜❾ ➁❜❢s❷❾❜ ❞❧❜❼ s❤♣ s❼❾ ➏❜♣ ❡❝ ✐❥❥➀
♣❜s❞❧❜❦ ❢❥❝➀❡❞❡❥❝❾ ↕✐❥❥➀ ➐❡❾❡➁❡❤❡❞❼ ➙ ♣ ❡❞❧ ❤❥♣ ❾r❜❜➀
s❡❦❢❦s❿❞✉ ❶ ❝ ❞❧❜ ❥❞❧ ❜❦ ❧ s❝ ➀ ♥ ♠ ❥➀❜❦❝ →❜❞ s❡❦❢❦s❿❞ ➀❥
❝❥❞ ❜❝s➁ ❤❜ r❡❤❥❞❾ ❞❥ ❦❜❾❥❤➐❜ ❢❥❝➏ ❡❢❞❾ ➁ ❜❢s❷ ❾❜ ❥❿ ❞❧❜❡❦
❧❡✐❧ ❾r ❜❜➀ s❝➀ ❞❧❜❡❦ s➁ ❡❤❡❞❼ ❞❥ ➏ ❼ ♣ ❡❞❧ ➁ s➀ ➐❡❾❡➁ ❡❤❡t
❞❼✉ ⑤ ❧❜❦❜❿❥❦❜♥ r❡❤❥❞❾ ♠ ❷❾❞ ➁❜ ❧❜❤r❜➀ ➁❼ s❝ s❡❦ ❞❦s⑥ ❢
❢❥❝❞❦❥❤❤❜❦ ❥❝ ❞❧❜ ✐❦❥❷❝ ➀ ♣ ❧❥ ❧s❾ s ✐❤❥➁ s❤ ➐❡❜♣ ❥❿ ❞❧ ❜
❢❷ ❦❦❜❝❞ ❞❦s⑥ ❢ ➀❡❾❞❦❡➁❷❞❡❥❝ ❡❝ ❞❧❜ s❡❦❾rs❢❜ s❝ ➀ ❢s❝ ✐❡➐❜
❥❦➀❜❦❾ ❞❥ ❞❧❜ r❡❤❥❞❾ ❞❥ s➐❥❡➀ ❢❥❤❤❡❾❡❥❝❾✉
➛➜➝➞➟➠➝ ➡➢➤➟➥➡➝➦ ➡➝ ➧➨ ➩➨➫➭➯➨➟➭➲➞ ➳➝➵➠➭➝➞➞➝
➸
➤➺➲➧➝ ➩➨➟➭➲➞➨➧➝ ➡➝ ➧➢➳ ➫➭➨➟➭➲➞ ➜➭➫➭➧➝
➻
➜➝➞➟➠➝ ➡➝ ➼ ➨➟➽➝➵➾ ➨➟➭➚➥➝➦➨➪➪➧➭➚➥➝➵➝➦ ➡➝ ➧➢➶
➹
➜➝➞➟➠➝ ➡➢➤➟➥➡➝➦ ➝➟ ➡➝ ➘➝➺➽➝➠➺➽➝ ➡➝ ➴➲➥➧➲➥➦➝
➷
❣ ❾ ❞❧❜❦❜ s❦❜ ❤❥❞ ❥❿ r ❤s❝ ❜❾ ❾❡♠ ❷ ❤❞s❝ ❜❥❷❾❤❼ r ❦❜❾❜❝❞ ❡❝
❞❧❜ ❾q❼♥ s ❾❡❝ ✐❤❜ ❢❥❝❞❦❥❤❤❜❦ ❡❾ ❝❥❞ s➁ ❤❜ ❞❥ ♠ s❝ s✐❜ s❤❤ ❥❿
❞❧❜♠ ✉ ❽❝ ➍❦s❝❢❜♥ s❡❦❾rs❢❜ ❡❾ rs❦❞❡❞❡❥❝❜➀ ❡❝❞❥ ➀❡↔❜❦❜❝❞
❾❜❢❞❥❦❾♥ ❜s❢❧ ❥❿ ❞❧❜♠ ➁ ❜❡❝✐ s❾❾❡✐❝ ❜➀ ❞❥ s ❢❥❝❞❦❥❤❤❜❦✉
➎❜❢❞❥❦❡❝✐ ❡❾ ❢❷❦❦❜❝❞❤❼ ➀❥❝❜ ❡❝ s❝ ❜♠ r ❡❦❡❢s❤ ♣ s❼
➁❼ ❾❥♠ ❜ s❡❦❾rs❢❜ ❜Ï r❜❦❞❾ ♣ ❧❥ srr ❤❼ ❦❷❤❜❾ ❞❧ ❜❼ ❧s➐❜
❤❜s❦❝❜➀ ♣ ❡❞❧ ❜Ï r❜❦❡❜❝ ❢❜✉ ❣ ❢❞❷s❤❤❼♥❞❧ ❜ ❾❜❢❞❥❦❡❝✐ ♠ ❥➀t
❡Ð ❢s❞❡❥❝ ❾ s❦❜ ❷❾❷ s❤❤❼ ➀❷❜ ❞❥ ❞❦s⑥ ❢ ❜➐❥❤❷ ❞❡❥❝ ❥❝ ❤❥❝✐
r ❜❦❡❥➀ s❝➀ ♣ ❧❜❝ s ❾❜❢❞❥❦ ❡❾ ❦❜✐❷❤s❦❤❼ ❥➐❜❦❤❥s➀❜➀ ❡❞ ❧s❾
❞❥ ➁ ❜ ♠ ❥➀❡Ð ❜➀ ✉ ⑤❥ ❦❜s❢❧ ❞❧❡❾ s❡♠ ♥s❝ s➀ ❧❥❢ ❢❥♠ ♠ ❡❾t
❾❡❥❝ ♠ ❜❜❞❾ ❞❥ ❜❤s➁ ❥❦s❞❜ ❝ ❜♣ ❿❦❥❝❞❡❜❦❾ ❿❥❦ ❞❧ ❜ ❾❜❢❞❥❦❾ ❡❝
❥❦➀❜❦ ❞❥ ➁ s❤s❝ ❢❜ ❞❧ ❜ ♣❥❦q❤❥s➀ ✉ ❣❿❞❜❦♣ s❦➀ ❾❜❢❞❥❦❡❝✐ ❡❾
❷r ➀s❞❜➀ ↕❷❝❞❡❤ ❝❜♣ r ❦❥➁❤❜♠ ❾ s❦❡❾❜➙✉
⑤ ❧❡❾ ♣ s❼ ❥❿ ♠ sq❡❝✐ ❡❾ ❦❜❤❜➐s❝❞ ➁ ❜❢s❷ ❾❜ ❡❞ ❞sq❜❾ ❡❝❞❥
s❢❢❥❷❝❞ s ❤❥❞ ❥❿ r❦s❢❞❡❢s❤ s❾r❜❢❞❾ ➁❷❞ ❧s❾ s ❤❡♠ ❡❞❜➀ ❜❿t
❿❜❢❞ ❥❝ ❞❧❜ ❤❥❢s❤ Ñ❥❝❜ ❡❞ ❞❦❜s❞❾✉ ❶ ❝❜ ❢s❝ ❞❦❼ ❞❥ ❡♠ r❦❥➐❜
s❝➀ ❢❥♠ r❤❜❞❜ ❞❧ ❡❾ r ❦❥ ❢❜❾❾ ♣ ❡❞❧ s❝ s❷❞❥♠ s❞❡❢ sr r❦❥s❢❧
❡❝ ❥❦➀❜❦ ❞❥ ✐❡➐❜ s ❾❥❤❷❞❡❥❝ ❞❥ ❞❧❜ ❾❜❢❞❥❦❡❝✐ r❦❥➁❤❜♠ ❡❝
❞❧❜ ♣ ❧❥❤❜ s❡❦❾r s❢❜ s❝ ➀ ❞❧ s❞ ❢❥❷❤➀ ➁ ❜ ❦❜Ð ❝❜➀ ➁❼ ❜Ït
r ❜❦❞❾✉
⑤ ❧❡❾ Ð❦❾❞ s❷ ❞❥♠ s❞❡❢ srr❦❥s❢❧ ❡❾ ➐❜❦❼ ❝ ❜♣ s❝ ➀ ❾❞❡❤❤
♠ s❼ ➁ ❜ ❡♠ r ❦❥➐❜➀ ➁❷ ❞ ❡❞ ✐❡➐❜❾ ➐❜❦❼ ❜❝ ❢❥❷ ❦s✐❡❝✐ ❦❜❾❷ ❤❞❾✉
❽❝ ❞❧ ❡❾ r sr ❜❦ ♣ ❜ ❾❧❥♣ ❧❥♣ ♣❜❤❤ ❛ ❜❝❜❞❡❢ ❣ ❤✐❥❦❡❞❧♠ ❾
♠ s❝ s✐❜ ❞❧❡❾ r❦❥➁❤❜♠ ↕s❿❞❜❦ ❾❥♠ ❜ ❾❡♠ r❤❡Ð ❢s❞❡❥❝❾ s❝➀
♠ ❥➀❜❤❡❝ ✐➙✉ ❽❝ ❞❧❜ Ð ❦❾❞ rs❦❞ ♣❜ ➀❜❾❢❦❡➁❜ ♠ ❥❦❜ r❦❜❢❡❾❜❤❼
❥❷❦ r ❦❥➁❤❜♠ s❝ ➀ ♠ sq❜ ❾❥♠ ❜ ❦❜❤❜➐s❝❞ ❾❡♠ r❤❡Ð ❢s❞❡❥❝ ❞❥
➀❜➐❜❤❥r s ♠ s❞❧❜♠ s❞❡❢s❤ ♠ ❥➀❜❤✉ ❽❝ ❞❧ ❜ ❾❜❢❥❝➀ rs❦❞
♣❜ r❦❜❾❜❝❞ ❢❥♠ r❤❜❞❜ ❜Ï s♠ r❤❜❾ ❥❿ ❦❜❾❥❤❷ ❞❡❥❝ ♣ ❡❞❧ ❞❧❜
➀❡↔❜❦❜❝❞ ❛ ❜❝ ❜❞❡❢ ❣ ❤✐❥❦❡❞❧♠ ❾✉
Ò Ó Ô➋Õ Ö ×➋Ø Ù ➈ Õ ➇ ➈ Ù×
ÚÛÜ ÝÞ
✾✿ß à á ❁✾âß
Þ
ã ❜❿❥❦❜ ➀❥❡❝✐ s ♠ s❞❧❜♠ s❞❡❢s❤ ➀❜❾❢❦❡r ❞❡❥❝ ❥❿ ❥❷❦
r❦❥➁❤❜♠ ♥ ❡❞ ❡❾ ❝❜❢❜❾❾s❦❼ ❞❥ ❾❞❦❜❾❾ ❥❷❞ ❥❷ ❦ ❿❦s♠ ❜♣ ❥❦q
❞❥ ❡❝❞❦❥➀❷ ❢❜ ❾❥♠ ❜ ❾❡♠ r❤❡Ð❢s❞❡❥❝❾ ❿❥❦ ❥❷❦ ♠ ❥➀❜❤✉ ➎❡❝❢❜
❡❞ ❡❾ ➐❜❦❼ ❤❥❝ ✐ ❞❥ ❞❦s❡❝ s❝ s❡❦ ❞❦s⑥ ❢ ❢❥❝❞❦❥❤❤❜❦ ❥❝ ❧❡❾
❾❜❢❞❥❦ ↕❿❦❥♠ ä ❞❥ å ♠ ❥❝❞❧ ❾➙♥♣ ❜ ♠ ❷❾❞ ❝❥❞ ❡❝➐❜❾❞❡✐s❞❜ s
❦❜s❤ ❞❡♠ ❜ ❾❜❢❞❥❦❡❝✐ ❥r❞❡♠ ❡Ñs❞❡❥❝ s❢❢❥❦➀❡❝✐ ❞❥ ❞❧❜ ➐s❦❡t
s❞❡❥❝❾ ❥❿ ❞❧❜ ❞❦s⑥ ❢ ❤❥s➀ ✉ ❽❝❾❞❜s➀ ♣❜ ❧s➐❜ ❞❥ ❢❥❝❾❡➀❜❦
s ❦❜✐❡❾❞❜❦❜➀ ♠ sÏ ❡♠ ❷♠ ❤❥s➀ ❞❦s⑥ ❢ r ❜❦❡❥ ➀ ❥❝ ❞❧ ❜ ♣ ❥❦qt
❡❝ ✐ ❝❜❞♣❥❦q ✉ ❶ ❷❦ r ❦❥➁❤❜♠ ❡❾ ❞❧ ❜❝ ❞❥ rs❦❞❡❞❡❥❝ ❞❧❜ s❡❦
❾rs❢❜ ❞❥ ✐❜❞ s ➁ s❤s❝ ❢❜➀ ❡❝➀ ❷❢❜➀ ❢❥❝❞❦❥❤ ♣ ❥❦q❤❥s➀ ✉
➔ ❧ ❜❝ ❜Ïs♠ ❡❝❡❝✐ ❞❧❜ r❧❼❾❡❢s❤ s❡❦ ❞❦s⑥ ❢ ❝ ❜❞♣❥❦q ♥♣ ❜
❝❥❞❡❢❜ ❞❧ s❞ s❡❦♣ s❼❾ s❦❜ ❾❷ r❜❦r❥❾❡❞❡❥❝ ❥❿ ❾❜➐❜❦s❤ ❦❥❷❞❜❾