Геометричне моделювання території будівництва за показниками питомих вартостей при оптимізації зовнішніх мереж водопостачання

DOI: https://doi.org/10.32347/2077-3455.2020.57.206-215

ɍȾɄ        Ɉɪɟɥ ɘɥɿɹ Ɇɢɤɨɥɚʀɜɧɚ

ɚɫɩɿɪɚɧɬ ɤɚɮɟɞɪɢ ɚɪɯɿɬɟɤɬɭɪɧɢɯ ɤɨɧɫɬɪɭɤɰɿɣ Ʉɢʀɜɫɶɤɨɝɨ ɧɚɰɿɨɧɚɥɶɧɨɝɨ ɭɧɿɜɟɪɫɢɬɟɬɭ ɛɭɞɿɜɧɢɰɬɜɚ ɿ ɚɪɯɿɬɟɤɬɭɪɢ  orel.yulia.m@gmail.com http://orcid.org / 0000-0002-4754-1555

ȽȿɈɆȿɌɊɂɑɇȿ ɆɈȾȿɅɘȼȺɇɇə ɌȿɊɂɌɈɊȱȲ ȻɍȾȱȼɇɂɐɌȼȺ ɁȺ  ɉɈɄȺɁɇɂɄȺɆɂ ɉɂɌɈɆɂɏ ȼȺɊɌɈɋɌȿɃ ɉɊɂ ɈɉɌɂɆȱɁȺɐȱȲ 

ɁɈȼɇȱɒɇȱɏ ɆȿɊȿɀ ȼɈȾɈɉɈɋɌȺɑȺɇɇə 

Ⱥɧɨɬɚɰɿɹ: ɭ  ɫɬɚɬɬɿ  ɜɢɫɜɿɬɥɸɽɬɶɫɹ ɩɢɬɚɧɧɹ  ɜɢɪɿɲɟɧɧɹ  ɡɚɞɚɱɿ  ɨɩɬɢɦɿɡɚɰɿʀ  ɬɪɚɽɤɬɨɪɿɣ ɩɪɨɤɥɚɞɚɧɧɹ ɡɨɜɧɿɲɧɿɯ ɦɟɪɟɠ ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ ɦɟɬɨɞɚɦɢ ɞɢɫɤɪɟɬɧɨʀ  ɝɟɨɦɟɬɪɿʀ  ɧɚ  ɨɫɧɨɜɿ  ɿɬɟɪɚɰɿɣɧɨɝɨ  ɤɨɪɟɝɭɜɚɧɧɹ  ɤɨɟɮɿɰɿɽɧɬɿɜ   ɳɨ  ɜɿɞɨɛɪɚɠɚɸɬɶ  ɬɟɯɧɿɤɨ-ɟɤɨɧɨɦɿɱɧɭ  ɞɨɰɿɥɶɧɿɫɬɶ  ɩɪɨɤɥɚɞɚɧɧɹ  ɨɤɪɟɦɢɯ  ɥɚɧɨɤ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɧɚ  ɪɿɡɧɢɯ  ɬɟɪɢɬɨɪɿɹɯ  ɜɢɪɚɠɟɧɭ  ɱɟɪɟɡ  ɪɿɜɧɿ  ɩɢɬɨɦɨʀ  ɜɚɪɬɨɫɬɿ   Ⱦɥɹ  ɰɶɨɝɨ  ɫɤɚɥɹɪɧɟ  ɩɨɥɟ ɜɟɥɢɱɢɧ ɩɢɬɨɦɨʀ ɜɚɪɬɨɫɬɿ ɛɭɞɿɜɟɥɶɧɢɯ ɬɚ ɟɤɫɩɥɭɚɬɚɰɿɣɧɢɯ ɪɨɛɿɬ ɭ ɤɨɠɧɿɣ  ɬɨɱɰɿ  ɞɨɫɥɿɞɠɭɜɚɧɨʀ  ɬɟɪɢɬɨɪɿʀ  ɩɪɨɩɨɧɭɽɬɶɫɹ  ɡɚɞɚɜɚɬɢ  ɹɤ  ɿɧɬɟɪɩɨɥɹɰɿɣɧɭ  ɮɭɧɤɰɿɸ   ȼ  ɫɜɨɸ  ɱɟɪɝɭ   ɩɪɨɩɨɧɨɜɚɧɚ  ɿɧɬɟɪɩɨɥɹɰɿɣɧɚ  ɮɭɧɤɰɿɹ  ɛɭɞɭɽɬɶɫɹ  ɧɚ  ɨɫɧɨɜɿ  ɩɿɞɿɛɪɚɧɢɯ  ɩɿɞ  ɤɪɢɬɟɪɿʀ  ɡɚɞɚɱɿ   ɳɨ  ɞɨɫɥɿɞɠɭɽɬɶɫɹ   ɪɚɞɿɚɥɶɧɨ-ɛɚɡɢɫɧɢɯ  ɮɭɧɤɰɿɹɯ   Ɋɨɡɩɨɞɿɥ  ɩɢɬɨɦɢɯ  ɜɚɪɬɨɫɬɟɣ  ɡɟɦɟɥɶɧɢɯ  ɞɿɥɹɧɨɤ  ɛɭɞɟ  ɦɚɬɢ  ɛɟɡɩɨɫɟɪɟɞɧɿɣ  ɜɩɥɢɜ  ɧɚ  ɪɟɡɭɥɶɬɚɬɢ  ɤɨɪɟɝɭɜɚɧɧɹ  ɬɪɚɽɤɬɨɪɿʀ ɜɥɚɲɬɭɜɚɧɧɹ  ɥɚɧɨɤ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɬɚ  ɦɿɫɰɶ  ɪɨɡɦɿɳɟɧɧɹ  ʀɯ  ɫɬɢɤɭɜɚɧɧɹ  ɪɨɡɝɚɥɭɠɟɧɧɹ  ɦɟɪɟɠɿ  ɬɪɭɛɨɩɪɨɜɨɞɭ   Ɉɤɪɿɦ ɰɶɨɝɨ ɪɟɡɭɥɶɬɚɬɢ ɦɨɞɟɥɸɜɚɧɧɹ ɬɟɪɢɬɨɪɿʀ ɛɭɞɿɜɧɢɰɬɜɚ ɡɚ  ɩɨɤɚɡɧɢɤɚɦɢ  ɩɢɬɨɦɢɯ  ɜɚɪɬɨɫɬɟɣ  ʀʀ  ɞɿɥɹɧɨɤ  ɞɨɡɜɨɥɹɸɬɶ  ɜɢɤɨɧɚɬɢ  ɩɨɩɟɪɟɞɧɿɣ  ɩɟɪɟɞɩɪɨɟɤɬɧɢɣ ɚɧɚɥɿɡ ɜɢɯɿɞɧɢɯ ɞɚɧɢɯ  ɧɚ ɨɫɧɨɜɿ ɝɪɚɮɿɱɧɨɝɨ ɜɿɞɨɛɪɚɠɟɧɧɹ ɰɢɯ  ɩɨɤɚɡɧɢɤɿɜ   ȼ  ɤɿɧɰɟɜɨɦɭ  ɪɟɡɭɥɶɬɚɬɿ  ɩɪɨɜɟɞɟɧɧɹ  ɨɩɬɢɦɿɡɚɰɿʀ  ɜɢɡɧɚɱɟɧɧɹ  ɬɪɚɽɤɬɨɪɿʀ  ɬɪɚɫɭɜɚɧɧɹ  ɦɟɪɟɠɿ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɞɨɡɜɨɥɹɬɶ  ɹɤ  ɡɦɟɧɲɢɬɢ  ɞɨɜɠɢɧɢ  ɥɚɧɨɤ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ   ɬɚɤ  ɿ  ɡɦɟɧɲɢɬɢ  ɜɚɪɬɿɫɬɶ  ɛɭɞɿɜɟɥɶɧɨ-ɦɨɧɬɚɠɧɢɯ  ɪɨɛɿɬ   ɬɪɭɞɨɜɢɯ ɪɟɫɭɪɫɿɜ ɬɚ ɩɨɞɚɥɶɲɢɯ ɟɤɫɩɥɭɚɬɚɰɿɣɧɢɯ ɜɢɬɪɚɬ  ɐɟ ɩɨɹɫɧɸɽɬɶɫɹ ɬɢɦ   ɳɨ  ɫɤɨɪɟɝɨɜɚɧɚ  ɨɩɬɢɦɿɡɨɜɚɧɚ   ɫɯɟɦɚ  ɦɟɪɟɠɿ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɛɭɞɟ  ɜɨɥɨɞɿɬɢ  ɪɹɞɨɦ  ɩɟɪɟɜɚɝ  ɩɟɪɟɞ  ɤɥɚɫɢɱɧɢɦɢ  ɩɿɞɯɨɞɚɦɢ  ɩɪɨɟɤɬɭɜɚɧɧɹ   ɬɚɤɢɦɢ  ɹɤ  ɞɨɰɿɥɶɧɿɫɬɶ  ɜɢɤɨɪɢɫɬɚɧɧɹ  ɞɿɥɹɧɨɤ ɦɿɠɛɭɞɢɧɤɨɜɢɯ  ɬɟɪɢɬɨɪɿɣ   ɡɦɟɧɲɟɧɧɹ  ɧɚɜɚɧɬɚɠɟɧɶ  ɧɚ  ɨɤɪɟɦɿ  ɥɚɧɤɢ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ   ɦɨɠɥɢɜɿɫɬɶ  ɭɬɨɱɧɸɸɱɢɯ ɤɨɪɢɝɭɜɚɧɶ ɡɚ ɪɚɯɭɧɨɤ ɜɜɟɞɟɧɧɹ ɞɨɞɚɬɤɨɜɢɯ ɤɪɢɬɟɪɿʀɜ ɬɨɳɨ 

Ʉɥɸɱɨɜɿ ɫɥɨɜɚ: ɞɢɫɤɪɟɬɧɟ ɦɨɞɟɥɸɜɚɧɧɹ  ɡɨɜɧɿɲɧɿ ɦɟɪɟɠɿ ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ   ɪɚɞɿɚɥɶɧɨ-ɛɚɡɢɫɧɿ ɮɭɧɤɰɿʀ 

ɉɨɫɬɚɧɨɜɤɚ  ɩɪɨɛɥɟɦɢ   əɤ  ɜɿɞɨɦɨ   ɩɪɨɰɟɫɢ  ɛɭɞɿɜɧɢɰɬɜɚ   ɪɟɦɨɧɬɭ  ɬɚ  ɩɨɞɚɥɶɲɨʀ  ɟɤɫɩɥɭɚɬɚɰɿʀ  ɡɨɜɧɿɲɧɿɯ  ɿɧɠɟɧɟɪɧɢɯ  ɦɟɪɟɠ  ɫɬɢɤɚɸɬɶɫɹ  ɡ  ɛɿɥɶɲɨɸ  ɤɿɥɶɤɿɫɬɸ  ɩɟɪɟɲɤɨɞ   ɚɧɿɠ  ɚɧɚɥɨɝɿɱɧɿ  ɩɪɨɰɟɫɢ   ɳɨ  ɩɨɜ¶ɹɡɚɧɿ  ɿɡ  ɜɧɭɬɪɿɲɧɿɦɢ  ɿɧɠɟɧɟɪɧɢɦɢ  ɫɢɫɬɟɦɚɦɢ  ɛɭɞɿɜɟɥɶ   Ⱦɥɹ  ɫɢɫɬɟɦɢ  ɡɨɜɧɿɲɧɶɨɝɨ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɞɚɧɚ ɫɢɬɭɚɰɿɹ ɧɟ ɽ ɜɢɤɥɸɱɟɧɧɹɦ  Ɂɨɜɧɿɲɧɿ  ɦɟɪɟɠɿ ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɧɚ ɜɿɞɦɿɧɭ  ɜɿɞ  ɜɧɭɬɪɿɲɧɿɯ  ɿɧɠɟɧɟɪɧɢɯ  ɤɨɦɭɧɿɤɚɰɿɣ   ɩɿɞ  ɱɚɫ  ɭɫɭɧɟɧɧɹ  ɚɜɚɪɿɣ  ɚɛɨ  ɩɪɨɜɟɞɟɧɧɹ  ɪɟɦɨɧɬɧɢɯ  ɪɨɛɿɬ  ɩɨɬɪɟɛɭɸɬɶ  ɛɿɥɶɲɟ  ɱɚɫɭ  ɬɚ  ɪɟɫɭɪɫɿɜ  ɞɥɹ  ɿɞɟɧɬɢɮɿɤɚɰɿʀ ɦɿɫɰɶ ɪɨɡɪɢɜɿɜ ɚɛɨ ɡɧɨɲɟɧɧɹ ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɜ ɛɿɥɶɲɨɫɬɿ ɜɢɩɚɞɤɿɜ   ɜɢɤɨɧɚɧɧɹ ɡɟɦɟɥɶɧɢɯ ɪɨɛɿɬ ɩɨ ɜɢʀɦɰɿ ʉɪɭɧɬɭ   ɬɚ ɭɫɭɧɟɧɧɹ ɞɟɮɟɤɬɿɜ  ɋɤɥɚɞɧɿɫɬɶ  ɪɨɛɿɬ  ɩɿɞɜɢɳɭɽɬɶɫɹ  ɬɚɤɨɠ  ɜɧɚɫɥɿɞɨɤ  ɬɚɤɢɯ  ɮɚɤɬɨɪɿɜ   ɹɤ  ɩɪɨɤɥɚɞɚɧɧɹ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɧɚ  ɞɿɥɹɧɤɚɯ  ɡɚɝɚɥɶɧɨɝɨ  ɤɨɪɢɫɬɭɜɚɧɧɹ   ɩɟɪɟɬɢɧ  ɜ  ɩɥɚɧɿ  ɬɪɚɧɫɩɨɪɬɧɢɯ  ɫɩɨɥɭɱɟɧɶ  ɬɚ  ɿɧɲɢɯ  ɿɧɠɟɧɟɪɧɢɯ  ɫɢɫɬɟɦ   ɚ  ɬɚɤɨɠ  ɮɚɤɬɨɪɭ  ɞɟɮɨɪɦɚɰɿʀ  ʉɪɭɧɬɿɜ   ɧɚ  ɹɤɢɯ  ɪɨɡɦɿɳɭɸɬɶɫɹ  ɬɪɭɛɨɩɪɨɜɨɞɢ   ɮɚɤɬɨɪɭ  ɥɿɧɿɣɧɢɯ  ɫɟɡɨɧɧɢɯ  ɞɟɮɨɪɦɚɰɿɣ  ɦɚɬɟɪɿɚɥɿɜ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɬɚ  ɿɧ    ȼɫɿ  ɰɿ  ɚɫɩɟɤɬɢ  ɚɤɬɭɚɥɿɡɭɸɬɶ  ɩɢɬɚɧɧɹ  ɨɩɬɢɦɿɡɚɰɿʀ  ɫɯɟɦ  ɪɨɡɦɿɳɟɧɧɹ  ɡɨɜɧɿɲɧɿɯ  ɦɟɪɟɠ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ ɳɟ ɧɚ ɟɬɚɩɿ ɩɪɨɟɤɬɭɜɚɧɧɹ 

Ⱥɧɚɥɿɡ  ɨɫɬɚɧɧɿɯ  ɞɨɫɥɿɞɠɟɧɶ  ɬɚ  ɩɭɛɥɿɤɚɰɿɣ  ɩɨɤɚɡɚɜ   ɳɨ  ɨɫɧɨɜɧɚ  ɭɜɚɝɚ  ɧɚɭɤɨɜɨ-ɩɪɚɤɬɢɱɧɢɯ  ɞɨɫɥɿɞɠɟɧɶ  ɡ    ɨɩɬɢɦɿɡɚɰɿʀ  ɫɢɫɬɟɦ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ >   ± 3], ɩɪɢɞɿɥɹɽɬɶɫɹ ɜɥɚɫɧɟ ɩɢɬɚɧɧɹɦ ɭɞɨɫɤɨɧɚɥɟɧɧɹ ɬɚ ɪɟɧɨɜɚɰɿʀ ɜɠɟ ɿɫɧɭɸɱɢɯ ɦɟɪɟɠ  ɬɚ  ɭɫɬɚɬɤɭɜɚɧɧɹ   Ɍɚɤɚ  ɨɩɬɢɦɿɡɚɰɿɹ  ɞɨɫɹɝɚɽɬɶɫɹ  ɲɥɹɯɨɦ  ɧɚɥɚɝɨɞɠɟɧɧɹ  ɪɟɠɢɦɿɜ  ɩɨɫɬɚɱɚɧɧɹ   ɜɫɬɚɧɨɜɥɟɧɧɹ  ɛɿɥɶɲ  ɟɮɟɤɬɢɜɧɨɝɨ  ɪɟɝɭɥɸɸɱɨɝɨ  ɨɛɥɚɞɧɚɧɧɹ  ɬɚ  ɚɜɬɨɦɚɬɢɡɚɰɿʀ   ɡɚɩɪɨɜɚɞɠɟɧɧɹ  ɡɚɯɨɞɿɜ  ɡ  ɦɨɧɿɬɨɪɢɧɝɭ  ɬɚ  ɤɨɧɬɪɨɥɸ  ɫɩɨɠɢɜɚɧɧɹ   ɪɟɫɭɪɫɿɜ ɬɚ ɡɦɟɧɲɟɧɧɹ ɡɚɛɪɭɞɧɟɧɧɹ ɧɚɜɤɨɥɢɲɧɶɨɝɨ ɫɟɪɟɞɨɜɢɳɚ  ɉɪɨɬɟ ɩɢɬɚɧɧɹ  ɪɨɡɪɨɛɤɢ ɩɪɨɟɤɬɧɢɯ ɪɿɲɟɧɶ ɜɥɚɲɬɭɜɚɧɧɹ ɿɧɠɟɧɟɪɧɢɯ ɫɢɫɬɟɦ ɬɚɤɢɦ ɱɢɧɨɦ  ɳɨɛ  ɞɨɫɹɝɬɢ  ɦɿɧɿɦɚɥɶɧɢɯ  ɜɢɬɪɚɬ  ɧɚ  ɡɜɟɞɟɧɧɹ ɟɤɫɩɥɭɚɬɚɰɿɸ  ɬɚ ɚɛɨ  ɦɚɤɫɢɦɚɥɶɧɢɯ  ɬɟɯɧɿɤɨ-ɟɤɨɧɨɦɿɱɧɢɯ  ɩɨɤɚɡɧɢɤɿɜ  ɫɢɫɬɟɦɢ  ɡɚ  ɪɚɯɭɧɨɤ  ɧɚɭɤɨɜɨ-ɨɛʉɪɭɧɬɨɜɚɧɨɝɨ  ɩɿɞɯɨɞɭ  ɞɨ  ɜɢɡɧɚɱɟɧɧɹ  ɨɩɬɢɦɚɥɶɧɢɯ  ɩɚɪɚɦɟɬɪɿɜ  ɪɨɡɝɚɥɭɠɟɧɨʀ  ɦɟɪɟɠɿ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɜ  ɞɨɫɬɚɬɧɿɣ  ɦɿɪɿ  ɧɟ  ɜɢɫɜɿɬɥɟɧɿ   ȼ  ɨɤɪɟɦɢɯ  ɜɢɩɚɞɤɚɯ   ɡɚɞɥɹ  ɡɚɛɟɡɩɟɱɟɧɧɹ  ɜɢɳɨʀ  ɩɪɨɞɭɤɬɢɜɧɨɫɬɿ  ɬɚ  ɟɮɟɤɬɢɜɧɨɫɬɿ  ɪɨɛɨɬɢ  ɿɧɠɟɧɟɪɧɢɯ  ɫɢɫɬɟɦ  ɩɪɨɩɨɧɭɽɬɶɫɹ  ɡɚɫɬɨɫɨɜɭɜɚɬɢ  ɦɚɬɟɦɚɬɢɱɧɟ  ɜɢɪɿɲɟɧɧɹ  ɥɨɤɚɥɶɧɢɯ  ɨɩɬɢɦɿɡɚɰɿɣɧɢɯ  ɡɚɞɚɱ  ɪɨɡɩɨɞɿɥɭ  ɚɛɨ  ɩɟɪɟɪɨɡɩɨɞɿɥɭ  ɪɟɫɭɪɫɿɜ   ɳɨ  ɩɨɫɬɚɱɚɸɬɶɫɹ    ɉɪɢ  ɰɶɨɦɭ  ɡɚɫɬɨɫɨɜɭɸɬɶɫɹ  ɩɟɪɟɜɚɠɧɨ  ɤɥɚɫɢɱɧɿ  ɦɟɬɨɞɢ  ɥɿɧɿɣɧɨɝɨ  ɩɪɨɝɪɚɦɭɜɚɧɧɹ  > @   ȼ  ɬɨɣ  ɠɟ  ɱɚɫ  ɩɪɢɧɰɢɩɢ  ɩɪɨɟɤɬɭɜɚɧɧɹ  ɦɟɪɟɠ  ɡɨɜɧɿɲɧɶɨɝɨ ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɪɟɤɨɦɟɧɞɨɜɚɧɿ ɞɥɹ ɿɧɠɟɧɟɪɿɜ-ɩɪɨɟɤɬɭɜɚɥɶɧɢɤɿɜ ɽ  ɫɬɚɧɞɚɪɬɧɢɦɢ > @ ɿ ɡɚɥɟɠɚɬɶ  ɜ ɛɿɥɶɲɿɣ ɦɿɪɿ  ɜɿɞ ɩɪɚɤɬɢɱɧɨɝɨ ɞɨɫɜɿɞɭ ɮɚɯɿɜɰɹ 

ɇɚɩɪɢɤɥɚɞ   ©ɬɪɚɧɫɩɨɪɬɧɚ  ɡɚɞɚɱɚª   ɳɨ  ɽ  ɨɞɧɿɽɸ  ɡ  ɧɚɣɛɿɥɶɲ  ɜɿɞɨɦɢɯ  ɤɥɚɫɢɱɧɢɯ  ɜɢɪɿɲɟɧɶ  ɨɩɬɢɦɿɡɚɰɿɣɧɨʀ  ɡɚɞɚɱɿ ɪɨɡɩɨɞɿɥɭ  ɛɭɞɶ-ɹɤɢɯ  ɪɟɫɭɪɫɿɜ  ɩɨɥɹɝɚɽ ɭ ɩɨɲɭɤɭ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɨɥɨɠɟɧɧɹ ɽɞɢɧɨʀ ɬɨɱɤɢ ɪɨɡɩɨɞɿɥɭ P₀(x₀, y₀)  ɞɨ 

ɹɤɨʀ  ɧɚɞɯɨɞɢɬɶ  ɜɟɫɶ  ɨɛɫɹɝ  ɪɟɫɭɪɫɿɜ  ɜɿɞ  ɩɨɫɬɚɱɚɥɶɧɢɤɚ   ɚ  ɞɚɥɿ  ɰɟɣ  ɪɟɫɭɪɫ  ɪɨɡɩɨɞɿɥɹɽɬɶɫɹ ɜɿɞ ɰɿɽʀ ɬɨɱɤɢ ɪɨɡɩɨɞɿɥɭ ɞɨ ɭɫɿɯ ɫɩɨɠɢɜɚɱɿɜ > @ 

Ɉɞɧɚɤ  ɧɚ  ɩɪɚɤɬɢɰɿ   ɫɢɫɬɟɦɚ  ɡɨɜɧɿɲɧɶɨɝɨ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɩɪɟɞɫɬɚɜɥɹɽ  ɫɨɛɨɸ ɫɤɥɚɞɧɭ ɬɚ ɞɨɫɢɬɶ ɪɨɡɝɚɥɭɠɟɧɭ ɦɟɪɟɠɭ  ɳɨ ɨɛɭɦɨɜɥɸɽɬɶɫɹ ɜɪɚɯɭɜɚɧɧɹɦ  ɪɹɞɭ ɦɿɫɬɨɛɭɞɿɜɧɢɯ ɭɦɨɜ ɬɚ ɨɛɦɟɠɟɧɶ  ɬɟɯɧɨɥɨɝɿɱɧɢɯ  ɿɧɠɟɧɟɪɧɨ-ɝɟɨɥɨɝɿɱɧɢɯ ɣ  ɿɧɲɢɯ  ɮɚɤɬɨɪɿɜ   ȼɿɞɩɨɜɿɞɧɨ   ɩɥɚɧɭɜɚɧɧɹ  ɩɨɥɨɠɟɧɧɹ ɜɭɡɥɿɜ  ɫɢɫɬɟɦɚ  ɡɨɜɧɿɲɧɶɨɝɨ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɡɚ  ɰɢɦ  ɩɿɞɯɨɞɨɦ  ɧɚ  ɩɪɚɤɬɢɰɿ  ɞɭɠɟ  ɫɤɥɚɞɧɨ  ɪɟɚɥɿɡɭɽɬɶɫɹ   ɚ  ɡɚ  ɭɦɨɜɢ  ɜɫɬɚɧɨɜɥɟɧɧɹ  ɮɭɧɤɰɿɨɧɚɥɶɧɨʀ  ɡɚɥɟɠɧɨɫɬɿ  ɩɨɥɨɠɟɧɧɹ  ɬɨɱɤɢ ɪɨɡɩɨɞɿɥɭ ɡ ɩɨɥɟɦ ɡɧɚɱɟɧɶ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɰɿɧɧɨɫɬɿ ɞɿɥɹɧɤɢ  ɧɚ ɹɤɿɣ ɛɭɞɟ  ɪɨɡɬɚɲɨɜɭɜɚɬɢɫɶ ɬɨɱɤɚ  ɫɬɚɽ ɳɟ ɫɤɥɚɞɧɿɲɨɸ   

ɉɨɩɟɪɟɞɧɿ  ɞɨɫɥɿɞɠɟɧɧɹ  ɩɨɤɚɡɚɥɢ   ɳɨ  ɧɚɣɛɿɥɶɲ  ɬɨɱɧɢɣ  ɬɚ  ɧɚɭɤɨɜɨ- ɨɛʉɪɭɧɬɨɜɚɧɢɣ  ɩɿɞɯɿɞ  ɞɨ  ɨɩɬɢɦɿɡɚɰɿʀ  ɫɤɥɚɞɧɢɯ  ɡɨɜɧɿɲɧɿɯ  ɦɟɪɟɠ  ɬɟɩɥɨɩɨɫɬɚɱɚɧɧɹ ɛɭɥɨ ɡɪɨɛɥɟɧɨ ɜ ɪɨɛɨɬɿ > @  ɞɟ ɛɭɥɚ ɨɞɟɪɠɚɧɚ ɫɢɫɬɟɦɚ ɪɿɜɧɹɧɶ   ɳɨ ɜɢɡɧɚɱɚɽ ɩɨɥɨɠɟɧɧɹ ɛɭɞɶ-ɹɤɨʀ ɤɿɥɶɤɨɫɬɿ ɜɭɡɥɿɜ ʀɯ ɪɨɡɝɚɥɭɠɟɧɧɹ 

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(1)

ɞɟ  k_i,j ± ɤɨɟɮɿɰɿɽɧɬ   ɳɨ  ɞɨɪɿɜɧɸɽ  ɜɟɥɢɱɢɧɿ  ɩɚɪɚɦɟɬɪɿɜ  ɩɢɬɨɦɢɯ  ɬɟɩɥɨɜɬɪɚɬ   ɜɿɞɧɟɫɟɧɢɯ ɞɨ ɞɨɜɠɢɧ ɨɤɪɟɦɢɯ ɞɿɥɹɧɨɤ  ɣ ɜɢɡɧɚɱɚɽɬɶɫɹ ɡɚ ɮɨɪɦɭɥɨɸ  

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i q K x x y y

k , (2)

ɞɟ  q_{l i,j} ɬɚ  K_SUPi,j ± ɥɿɧɿɣɧɚ  ɳɿɥɶɧɿɫɬɶ  ɬɟɩɥɨɜɨɝɨ  ɩɨɬɨɤɭ  ɣ  ɤɨɟɮɿɰɿɽɧɬ   ɹɤɢɣ 

ɜɪɚɯɨɜɭɽ ɡɛɿɥɶɲɟɧɧɹ ɬɟɩɥɨɜɬɪɚɬ ɱɟɪɟɡ ɨɩɨɪɢ ɿ ɩɿɞɜɿɫɤɢ  ɧɚ ɞɿɥɹɧɰɿ ɦɿɠ ɿ-ɦ ɬɚ j-ɦ  ɜɭɡɥɚɦɢ  

Ɉɞɧɚɤ  ɤɨɥɢ ɦɨɜɚ ɣɞɟ  ɩɪɨ ɫɢɫɬɟɦɢ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ ɣ ɨɫɧɨɜɧɨɸ ɰɿɥɶɨɜɨɸ  ɮɭɧɤɰɿɽɸ  ɽ  ɬɟɯɧɿɤɨ-ɟɤɨɧɨɦɿɱɧɿ  ɩɨɤɚɡɧɢɤɢ  ɛɭɞɿɜɧɢɰɬɜɚ  ɣ  ɟɤɫɩɥɭɚɬɚɰɿʀ   ɚ  ɧɟ  ɟɧɟɪɝɨɜɢɬɪɚɬɢ   ɬɨ  ɞɨɰɿɥɶɧɿɲɟ  ɜ  ɹɤɨɫɬɿ  ɤɨɟɮɿɰɿɽɧɬɚ k_i,j ɩɪɢɣɧɹɬɢ  ɜɟɥɢɱɢɧɭ   ɹɤɚ  ɜɿɞɨɛɪɚɠɚɬɢɦɟ  ɪɿɜɟɧɶ  ɩɢɬɨɦɨʀ  ɜɚɪɬɨɫɬɿ  ɩɪɨɤɥɚɞɚɧɧɹ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɧɚ  ɪɿɡɧɢɯ  ɞɿɥɹɧɤɚɯ ɞɨɫɥɿɞɠɭɜɚɧɨʀ ɬɟɪɢɬɨɪɿʀ 

Ɇɟɬɚ  ɫɬɚɬɬɿ. ȼ  ɪɚɦɤɚɯ  ɡɚɩɪɨɩɨɧɨɜɚɧɨɝɨ ɩɿɞɯɨɞɭ  ɞɨ  ɨɩɬɢɦɿɡɚɰɿʀ  ɫɯɟɦ  ɩɪɨɤɥɚɞɚɧɧɹ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɦɟɪɟɠ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɡ  ɭɪɚɯɭɜɚɧɧɹɦ  ɯɚɪɚɤɬɟɪɢɫɬɢɤ  ɬɟɯɧɿɤɨ-ɟɤɨɧɨɦɿɱɧɨʀ  ɞɨɰɿɥɶɧɨɫɬɿ   ɜɢɤɨɧɚɬɢ  ɝɟɨɦɟɬɪɢɱɧɟ  ɦɨɞɟɥɸɜɚɧɧɹ ɬɟɪɢɬɨɪɿʀ ɛɭɞɿɜɧɢɰɬɜɚ ɡɚ ɩɨɤɚɡɧɢɤɚɦɢ ɩɢɬɨɦɨʀ ɜɚɪɬɨɫɬɿ  ɰɿɧɧɨɫɬɿ   ʀʀ  ɨɤɪɟɦɢɯ  ɞɿɥɹɧɨɤ   Ɍɨɛɬɨ  ɧɟɨɛɯɿɞɧɨ  ɡɚɩɪɨɩɨɧɭɜɚɬɢ  ɬɚɤɭ  ɝɟɨɦɟɬɪɢɱɧɭ  ɦɨɞɟɥɶ  ɪɨɡɩɨɞɿɥɭ  ɩɢɬɨɦɢɯ  ɜɚɪɬɨɫɬɟɣ  ɞɿɥɹɧɨɤ   ɹɤɚ  ɛ  ɦɚɤɫɢɦɚɥɶɧɨ  ɜɿɞɩɨɜɿɞɚɥɚ  ɨɛ¶ɽɤɬɭ  ɞɨɫɥɿɞɠɟɧɧɹ 

ȼɢɤɥɚɞ  ɨɫɧɨɜɧɨɝɨ  ɦɚɬɟɪɿɚɥɭ  Ɋɨɡɝɥɹɧɟɦɨ  ɫɢɫɬɟɦɭ  ɪɿɜɧɹɧɶ       ɡ  ɬɨɱɤɢ  ɡɨɪɭ  ɩɪɢɤɥɚɞɧɨʀ  ɝɟɨɦɟɬɪɿʀ   Ɍɚɤɢɦ  ɱɢɧɨɦ   ɪɨɡɜ¶ɹɡɚɧɧɹ  ɫɢɫɬɟɦɢ      ɞɨɡɜɨɥɹɽ 

ɜɢɡɧɚɱɢɬɢ  ɤɨɨɪɞɢɧɚɬɢ  ɦɿɧɿɦɚɥɶɧɨʀ  ɩɨ  ɡɚɝɚɥɶɧɿɣ  ɞɨɜɠɢɧɿ  ɥɚɧɨɤ   ɫɿɬɤɢ  ɿɡ  ɭɪɚɯɭɜɚɧɧɹɦ  ɤɨɟɮɿɰɿɽɧɬɿɜ  ɳɿɥɶɧɨɫɬɿ  ɜɡɚɽɦɨɞɿʀ  ɦɿɠ  ɨɤɪɟɦɢɦɢ  ɜɭɡɥɚɦɢ  k_i,j).

əɤɳɨ ɠ ɞɨ     ɡɚɫɬɨɫɭɜɚɬɢ ɚɩɪɨɤɫɢɦɚɰɿɸ ɡɪɿɜɧɨɜɚɠɟɧɨɸ ɫɿɬɤɨɸ  ɫɮɨɪɦɨɜɚɧɨɸ  ɜɿɞɩɨɜɿɞɧɨ  ɞɨ  ɫɬɚɬɢɤɨ-ɝɟɨɦɟɬɪɢɱɧɨɝɨ  ɦɟɬɨɞɭ  ɞɢɫɤɪɟɬɧɨʀ  ɝɟɨɦɟɬɪɿʀ  > @   ɬɨ  ɜ  ɪɟɡɭɥɶɬɚɬɿ  ɨɬɪɢɦɚɽɦɨ  ɤɨɨɪɞɢɧɚɬɢ  ɫɿɬɤɢ   ɹɤɚ  ɛɭɞɟ  ɦɚɬɢ  ɞɨɫɬɚɬɧɶɨ  ɜɢɫɨɤɭ  ɬɨɱɧɿɫɬɶ  ɬɚ  ɧɚɛɥɢɠɚɬɢɫɶ  ɞɨ  ɦɿɧɿɦɚɥɶɧɨʀ   ȼɿɞɩɨɜɿɞɧɨ  ɞɨ  ɰɶɨɝɨ  ɦɟɬɨɞɭ   ɤɨɟɮɿɰɿɽɧɬɢ     ɭ ɛɿɥɶɲɨɫɬɿ ɜɢɩɚɞɤɚɯ ɩɪɢɣɦɚɸɬɶɫɹ ɫɬɚɥɢɦɢ ɜɟɥɢɱɢɧɚɦɢ  ɬɨɛɬɨ   k_i,j = FRQVW   ȼ  ɞɚɧɨɦɭ  ɜɢɩɚɞɤɭ  k_i,j ɽ  ɧɚɩɟɪɟɞ  ɡɚɞɚɧɢɦɢ  ɤɨɟɮɿɰɿɽɧɬɚɦɢ  ɩɪɨɩɨɪɰɿɣɧɨɫɬɿ  ɳɨ ɯɚɪɚɤɬɟɪɢɡɭɸɬɶ ɫɢɥɭ ɜɡɚɽɦɨɞɿʀ ɦɿɠ ɨɤɪɟɦɢɦɢ ɜɭɡɥɚɦɢ  ɹɤɿ  ʀɯ ɫɩɨɥɭɱɚɸɬɶ  

Ⱦɥɹ ɫɢɫɬɟɦɢ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɨɫɧɨɜɧɨɸ  ɰɿɥɶɨɜɨɸ  ɮɭɧɤɰɿɽɸ  ɽ  ɬɟɯɧɿɤɨ- ɟɤɨɧɨɦɿɱɧɿ  ɩɨɤɚɡɧɢɤɢ  ɛɭɞɿɜɧɢɰɬɜɚ  ɣ  ɟɤɫɩɥɭɚɬɚɰɿʀ   ɚ  ɧɟ  ɟɧɟɪɝɨɜɢɬɪɚɬɢ   ɬɨɦɭ  ɧɚɣɤɪɚɳɢɦ  ɽ  ɩɪɢɣɧɹɬɢ  ɜ  ɹɤɨɫɬɿ  ɤɨɟɮɿɰɿɽɧɬɚ k_i,j ɜɟɥɢɱɢɧɭ   ɹɤɚ  ɜɿɞɨɛɪɚɠɚɬɢɦɟ  ɪɿɜɟɧɶ  ɩɢɬɨɦɨʀ  ɜɚɪɬɨɫɬɿ  ɩɪɨɤɥɚɞɚɧɧɹ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɧɚ  ɪɿɡɧɢɯ  ɞɿɥɹɧɤɚɯ  ɞɨɫɥɿɞɠɭɜɚɧɨʀ  ɬɟɪɢɬɨɪɿʀ  ɩɿɞ  ɛɭɞɿɜɧɢɰɬɜɨ   Ɉɬɠɟ   ɤɨɟɮɿɰɿɽɧɬ  ɩɪɨɩɨɪɰɿɣɧɨɫɬɿ k_i,j ɛɭɞɟ ɡɚɥɟɠɚɬɢ ɜɿɞ ɤɨɨɪɞɢɧɚɬ ɩɨɱɚɬɤɭ ɿ ɤɿɧɰɹ ɤɨɠɧɨʀ ɥɚɧɤɢ  ɬɨɛɬɨ ɞɟɹɤɨɝɨ ɿ-ɝɨ ɬɚ  j-ɝɨ ɜɭɡɥɿɜ ɿ ɦɨɠɟ ɛɭɬɢ ɩɨɞɚɧɢɣ ɮɭɧɤɰɿɨɧɚɥɶɧɨɸ ɡɚɥɟɠɧɿɫɬɸ     

).

, , ,

,j ( i j i j

i F x x y y

k (3)

Ⱦɥɹ ɧɚɣɩɪɨɫɬɿɲɨɝɨ ɜɢɩɚɞɤɭ ɦɨɠɟɦɨ ɡɚɩɢɫɚɬɢ  ).

, ( ) , ( ) , , ,

(x_i x_j y_i y_j f x_i y_i f x_j y_j

F (4)

ȼɿɞɩɨɜɿɞɧɨ ɞɨ     ɮɭɧɤɰɿɹ f(x,y  ɜɿɞ ɤɨɨɪɞɢɧɚɬ ɿ-ɝɨ ɬɚ j-ɝɨ ɜɭɡɥɿɜ ɜɢɡɧɚɱɚɽ  ɩɨɥɟ  ɩɢɬɨɦɢɯ  ɩɨɤɚɡɧɢɤɿɜ  ɜɚɪɬɨɫɬɿ  ɛɭɞɿɜɧɢɰɬɜɚ   ɪɟɦɨɧɬɭ  ɣ  ɟɤɫɩɥɭɚɬɚɰɿʀ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ ɧɚ ɩɥɨɳɢɧɿ  

ɇɚ  ɩɪɚɤɬɢɰɿ  ɩɨɥɟ  ɩɢɬɨɦɢɯ  ɩɨɤɚɡɧɢɤɿɜ  ɜɚɪɬɨɫɬɿ  ɛɭɞɿɜɧɢɰɬɜɚ   ɪɟɦɨɧɬɭ  ɣ  ɟɤɫɩɥɭɚɬɚɰɿʀ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɦɚɬɢɦɟ  ɧɟɥɿɧɿɣɧɢɣ  ɯɚɪɚɤɬɟɪ   Ɍɨɦɭ  ɮɭɧɤɰɿɹ f(x,y) ɦɨɠɟ  ɛɭɬɢ  ɩɪɟɞɫɬɚɜɥɟɧɚ  ɞɟɹɤɨɸ  ɿɧɬɟɪɩɨɥɹɰɿɣɧɨɸ  ɚɛɨ  ɚɩɪɨɤɫɢɦɚɰɿɣɧɨɸ  ɩɨɜɟɪɯɧɟɸ  

Ⱦɨɫɜɿɞ  ɝɟɨɦɟɬɪɢɱɧɨɝɨ  ɦɨɞɟɥɸɜɚɧɧɹ  ɫɤɥɚɞɧɢɯ  ɫɬɪɭɤɬɭɪɨɜɚɧɢɯ  ɝɟɨɦɟɬɪɢɱɧɢɯ ɩɨɜɟɪɯɨɧɶ >  @ ɫɜɿɞɱɢɬɶ ɩɪɨ ɞɨɰɿɥɶɧɿɫɬɶ  ɜɢɤɨɪɢɫɬɚɧɧɹ ɭ ɹɤɨɫɬɿ  ɚɩɪɨɤɫɢɦɚɰɿɣɧɢɯ  ɮɭɧɤɰɿɣ  ɞɥɹ  ɞɚɧɨɝɨ  ɬɢɩɭ  ɡɚɞɚɱ  ɫɚɦɟ  ɪɚɞɿɚɥɶɧɨ-ɛɚɡɢɫɧɢɯ  ɮɭɧɤɰɿɣ  > @   ɉɪɢɤɥɚɞɨɦ  ɧɚɣɛɿɥɶɲ  ɜɠɢɜɚɧɢɯ  ɪɚɞɿɚɥɶɧɨ-ɛɚɡɢɫɧɢɯ  ɮɭɧɤɰɿɣ  ɽ  ɡɜɨɪɨɬɧɿ ɤɜɚɞɪɚɬɢɱɧɿ     ɚɛɨ ɦɭɥɶɬɢɤɜɚɞɪɚɬɢɱɧɿ     ɮɭɧɤɰɿʀ 

, )]

) ( ) ((

1 [ )

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1

2 2

M 2 i

i i

i x x y y

z y

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f (5)

. ) ) ( ) ((

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1

2 2

M 2 i

i i

i x x y y

z y

x

f (6)

ȼ ɮɨɪɦɭɥɚɯ     ɬɚ     ɜɢɤɨɪɢɫɬɚɧɿ ɧɚɫɬɭɩɧɿ ɩɨɡɧɚɱɟɧɧɹ  z_i ± ɡɧɚɱɟɧɧɹ ɩɢɬɨɦɢɯ  ɩɨɤɚɡɧɢɤɿɜ  ɜɚɪɬɨɫɬɿ  ɭ  ɨɩɨɪɧɢɯ  ɬɨɱɤɚɯ  ɡ  ɤɨɨɪɞɢɧɚɬɚɦɢ x_i ɬɚ y_i; ± ɤɨɟɮɿɰɿɽɧɬ  ɝɥɚɞɤɨɫɬɿ  ɚɩɪɨɤɫɢɦɚɰɿʀ  ɦɿɠ  ɨɩɨɪɧɢɦɢ  ɬɨɱɤɚɦɢ  ɮɭɧɤɰɿʀ  f(x,y); Ɇ ± ɤɿɥɶɤɿɫɬɶ 

ɨɩɨɪɧɢɯ ɬɨɱɨɤ 

ȼ ɹɤɨɫɬɿ ɨɫɧɨɜɧɨʀ ɪɚɞɿɚɥɶɧɨ-ɛɚɡɢɫɧɨʀ ɮɭɧɤɰɿʀ ɩɪɢɣɦɟɦɨ ɮɭɧɤɰɿɸ      Ⱦɚɧɚ  ɮɭɧɤɰɿɹ ɦɨɠɟ ɛɭɬɢ ɜ ɞɟɹɤɿɣ ɦɿɪɿ ɦɨɞɢɮɿɤɨɜɚɧɚ  ȼɢɤɨɧɚɜɲɢ ɪɹɞ ɩɟɪɟɬɜɨɪɟɧɶ ɬɚ  ɚɧɚɥɿɡɭɸɱɢ  ɪɟɡɭɥɶɬɚɬɢ  ɝɪɚɮɿɱɧɢɯ  ɩɨɛɭɞɨɜ  ɛɭɥɨ  ɜɢɤɨɧɚɧɨ  ɩɨɲɭɤ  ɧɚɣɟɮɟɤɬɢɜɧɿɲɨʀ  ɜɚɪɿɚɰɿʀ  ɡ  ɬɨɱɤɭ  ɡɨɪɭ    ɜɿɞɩɨɜɿɞɧɨɫɬɿ  ɮɚɤɬɢɱɧɨɦɭ  ɪɨɡɩɨɞɿɥɭ  ɬɟɯɧɿɤɨ-ɟɤɨɧɨɦɿɱɧɢɯ  ɯɚɪɚɤɬɟɪɢɫɬɢɤ  ɞɿɥɹɧɨɤ  ɛɭɞɿɜɟɥɶɧɨʀ  ɬɟɪɢɬɨɪɿʀ   Ɍɨɛɬɨ   ɦɨɞɢɮɿɤɨɜɚɧɚ  ɦɭɥɶɬɢɤɜɚɞɪɚɬɢɱɧɚ  ɪɚɞɿɚɥɶɧɨ-ɛɚɡɢɫɧɚ  ɮɭɧɤɰɿɹ ɦɨɠɟ  ɛɭɬɢ  ɡɚɩɢɫɚɧɚ ɭ ɜɢɝɥɹɞɿ ɮɭɧɤɰɿʀ     

M

i

r i

i

i x x y y

z y

x f

1

2

2 ( )

) ( )

,

( . (7)

ɞɟ  r ± ɩɨɤɚɡɧɢɤ  ɫɬɭɩɟɧɹ  ɦɭɥɶɬɢɤɜɚɞɪɚɬɢɱɧɨʀ  ɛɚɡɢɫɧɨʀ  ɮɭɧɤɰɿʀ   ɧɚɣɨɩɬɢɦɚɥɶɧɿɲɟ  ɡɧɚɱɟɧɧɹ  ɞɥɹ  ɜɢɪɿɲɟɧɧɹ  ɩɨɫɬɚɜɥɟɧɨʀ  ɡɚɞɚɱɿ  ɛɭɥɨ  ɨɬɪɢɦɚɧɨ  ɩɪɢ k=    ɪɟɲɬɚ ɩɨɡɧɚɱɟɧɶ ɡɚɥɢɲɢɥɢɫɶ ɛɟɡ ɡɦɿɧ 

Ɋɨɡɝɥɹɧɟɦɨ  ɜɢɩɚɞɨɤ  ɪɨɡɩɨɞɿɥɭ  ɩɢɬɨɦɢɯ  ɜɚɪɬɨɫɬɟɣ   -ɢ  ɡɟɦɟɥɶɧɢɯ  ɞɿɥɹɧɨɤ  ɛɭɞɿɜɟɥɶɧɨʀ ɬɟɪɢɬɨɪɿʀ  ɇɟɯɚɣ ɜɿɞɨɦɿ ɪɟɡɭɥɶɬɚɬɚɦɢ ɚɧɚɥɿɡɭ ɬɟɪɢɬɨɪɿʀ ɛɭɞɿɜɧɢɰɬɜɚ  ɦɟɪɟɠɿ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ ɡɚ  ɩɨɤɚɡɧɢɤɚɦɢ  ɜɚɪɬɨɫɬɿ  ɰɿɧɧɨɫɬɿ   ɡɟɦɟɥɶɧɢɯ ɞɿɥɹɧɨɤ   Ɂɧɚɱɟɧɧɹ ɩɢɬɨɦɢɯ ɜɚɪɬɨɫɬɟɣ ɡɟɦɟɥɶɧɢɯ ɞɿɥɹɧɨɤ ɡɜɟɞɟɧɨ ɜ ɬɚɛɥɢɰɿ   

Ɍɚɛɥɢɰɹ    ȼɢɯɿɞɧɿ ɡɧɚɱɟɧɧɹ ɩɢɬɨɦɢɯ ɜɚɪɬɨɫɬɟɣ ɡɟɦɟɥɶɧɢɯ ɞɿɥɹɧɨɤ

ʿ̸̨̦̖̦̦̌́̚ ̸̨̛̦̖̦̟̏̌̚ 

̡̛̞̣̦̔́ ̌̚ ̨̡̡̛̛̪̦̥̌̌̚ 

̨̨̛̪̯̥̟ ̨̬̯̭̯̞̏̌ 

ʿ̨̡̡̛̛̦̌̚ ̨̨̛̪̯̥̟ 

̨̬̯̭̯̞̏̌ ;̶̨̞̦̦̭̯̞Ϳ 

̵̛̖̥̖̣̦̽̚ ̨̡̞̣̦̔́ 

ʶ̨̨̛̛̬̦̯̥̔̌̌ ̶̖̦̯̬̱ ̏̌̐ ̡̛̞̣̦̔́ 

džŝ  LJŝ 

̡̞̣̦̔́̌    z_Ⱥ ϰ͕Ϯ  x_Ⱥ Ϯϰ  y_Ⱥ ϲϯ 

̡̞̣̦̔́̌    z_B ϯ͕ϭ  x_B ϱϬ  y_B ϳϱ 

̡̞̣̦̔́̌    z_C ϱ͕ϰ  xC ϭϭϬ  yC ϯϭ 

̡̞̣̦̔́̌    zD Ϯ͕Ϭ  xD ϴϬ  yD Ϯϲ 

̡̞̣̦̔́̌    z_E ϭ͕ϯ  x_E ϲϳ  y_E ϱϮ 

Ⱦɥɹ ɜɿɡɭɚɥɿɡɚɰɿʀ ɝɟɨɦɟɬɪɢɱɧɨʀ ɦɨɞɟɥɿ  ɪɢɫ     ɪɨɡɩɨɞɿɥɭ ɩɢɬɨɦɢɯ ɜɚɪɬɨɫɬɟɣ  ɞɿɥɹɧɨɤ ɛɭɞɿɜɟɥɶɧɨʀ ɬɟɪɢɬɨɪɿʀ ɜɢɤɨɪɢɫɬɚɽɦɨ ɡɜɚɠɟɧɿ ɮɭɧɤɰɿʀ ɡɝɿɞɧɨ ɡ     

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ɉɪɢ  ɰɶɨɦɭ  ɜɜɟɞɟɦɨ  ɧɚɫɬɭɩɧɿ  ɤɪɚɣɨɜɿ  ɭɦɨɜɢ   x 0;150 ɬɚ  y 0;150 . Ʉɨɟɮɿɰɿɽɧɬ ɝɥɚɞɤɨɫɬɿ ɚɩɪɨɤɫɢɦɚɰɿʀ  = 0,00001.

Ɋɢɫ    ɉɨɜɟɪɯɧɹ z(x,y) ɡɧɚɱɟɧɶ ɩɢɬɨɦɢɯ ɜɚɪɬɨɫɬɟɣ ɡɟɦɟɥɶɧɢɯ  ɦɿɫɬɨɛɭɞɿɜɧɢɯ  ɞɿɥɹɧɨɤ  ɚ  ɜ ɩɥɚɧɿ  ɛ  ɜ ɚɤɫɨɧɨɦɟɬɪɿʀ 

̌Ϳ 

̍Ϳ 

ȼɢɫɧɨɜɤɢ   Ɋɨɡɩɨɞɿɥ  ɩɢɬɨɦɢɯ  ɜɚɪɬɨɫɬɟɣ  ɡɟɦɟɥɶɧɢɯ  ɞɿɥɹɧɨɤ  ɛɭɞɟ  ɦɚɬɢ  ɛɟɡɩɨɫɟɪɟɞɧɿɣ  ɜɩɥɢɜ  ɧɚ  ɪɟɡɭɥɶɬɚɬɢ  ɤɨɪɟɝɭɜɚɧɧɹ  ɬɪɚɽɤɬɨɪɿʀ  ɜɥɚɲɬɭɜɚɧɧɹ ɥɚɧɨɤ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɬɚ  ɦɿɫɰɶ  ɪɨɡɦɿɳɟɧɧɹ  ʀɯ  ɫɬɢɤɭɜɚɧɧɹ  ɪɨɡɝɚɥɭɠɟɧɧɹ  ɦɟɪɟɠɿ  ɬɪɭɛɨɩɪɨɜɨɞɭ   Ɉɤɪɿɦ ɰɶɨɝɨ ɪɟɡɭɥɶɬɚɬɢ ɦɨɞɟɥɸɜɚɧɧɹ ɬɟɪɢɬɨɪɿʀ ɛɭɞɿɜɧɢɰɬɜɚ ɡɚ  ɩɨɤɚɡɧɢɤɚɦɢ  ɩɢɬɨɦɢɯ  ɜɚɪɬɨɫɬɟɣ  ʀʀ  ɞɿɥɹɧɨɤ  ɞɨɡɜɨɥɹɸɬɶ  ɜɢɤɨɧɚɬɢ ɩɨɩɟɪɟɞɧɿɣ  ɩɟɪɟɞɩɪɨɟɤɬɧɢɣ ɚɧɚɥɿɡ ɜɢɯɿɞɧɢɯ ɞɚɧɢɯ  ɧɚ ɨɫɧɨɜɿ ɝɪɚɮɿɱɧɨɝɨ ɜɿɞɨɛɪɚɠɟɧɧɹ ɰɢɯ  ɩɨɤɚɡɧɢɤɿɜ   ȼ  ɤɿɧɰɟɜɨɦɭ  ɪɟɡɭɥɶɬɚɬɿ  ɩɪɨɜɟɞɟɧɧɹ  ɨɩɬɢɦɿɡɚɰɿʀ  ɜɢɡɧɚɱɟɧɧɹ  ɬɪɚɽɤɬɨɪɿʀ  ɬɪɚɫɭɜɚɧɧɹ  ɦɟɪɟɠɿ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɞɨɡɜɨɥɹɬɶ  ɹɤ  ɡɦɟɧɲɢɬɢ  ɞɨɜɠɢɧɢ  ɥɚɧɨɤ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ   ɬɚɤ  ɿ ɡɦɟɧɲɢɬɢ  ɜɚɪɬɿɫɬɶ  ɛɭɞɿɜɟɥɶɧɨ-ɦɨɧɬɚɠɧɢɯ  ɪɨɛɿɬ   ɬɪɭɞɨɜɢɯ ɪɟɫɭɪɫɿɜ ɬɚ ɩɨɞɚɥɶɲɢɯ ɟɤɫɩɥɭɚɬɚɰɿɣɧɢɯ ɜɢɬɪɚɬ  ɐɟ ɩɨɹɫɧɸɽɬɶɫɹ ɬɢɦ   ɳɨ  ɫɤɨɪɟɝɨɜɚɧɚ  ɨɩɬɢɦɿɡɨɜɚɧɚ   ɫɯɟɦɚ  ɦɟɪɟɠɿ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ  ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ  ɛɭɞɟ  ɜɨɥɨɞɿɬɢ  ɪɹɞɨɦ  ɩɟɪɟɜɚɝ  ɩɟɪɟɞ  ɤɥɚɫɢɱɧɢɦɢ  ɩɿɞɯɨɞɚɦɢ  ɩɪɨɟɤɬɭɜɚɧɧɹ   ɬɚɤɢɦɢ  ɹɤ  ɞɨɰɿɥɶɧɿɫɬɶ  ɜɢɤɨɪɢɫɬɚɧɧɹ  ɞɿɥɹɧɨɤ  ɦɿɠɛɭɞɢɧɤɨɜɢɯ  ɬɟɪɢɬɨɪɿɣ   ɡɦɟɧɲɟɧɧɹ  ɧɚɜɚɧɬɚɠɟɧɶ  ɧɚ  ɨɤɪɟɦɿ  ɥɚɧɤɢ  ɬɪɭɛɨɩɪɨɜɨɞɿɜ   ɦɨɠɥɢɜɿɫɬɶ  ɭɬɨɱɧɸɸɱɢɯ ɤɨɪɢɝɭɜɚɧɶ ɡɚ ɪɚɯɭɧɨɤ ɜɜɟɞɟɧɧɹ ɞɨɞɚɬɤɨɜɢɯ ɤɪɢɬɟɪɿʀɜ ɬɨɳɨ 

Ʌɿɬɟɪɚɬɭɪɚ 

1. Rak J.R. Selected problems of water supply safety / J.R. Rak // Environment Protection Engineering, 35, 2, 2009. ± Ɋ    -28.

2. Nahman J.M. Dependability of engineering systems ± modeling and evaluation / J. M. Nahman // Springer-Verlag Berlin Heidelberg, 2002, 192 pp..

DOI: 10.1007/978-3-662-04892-4.

3. Ostfeld A. Reliability simulation of water distribution systems ± single and multiquality / Avi Ostfeld, Dimitri Kogan, Uri Shamir // Urban Water  ʋ , 2002 ± Ɋ   53±61. DOI: 10.1016/S1462-0758(01)00055-3.

4. Iske A. Radial basis functions: basics, advanced topics and meshfree methods for transport problems / A. Iske // Rendiconti del Seminario Matematico, Univ. Pol.

7RULQR        ʋ         ± P. 247±284.

5. Ɍɭɝɚɣ Ⱥ Ɇ  Ɋɨɡɪɚɯɭɧɨɤ ɿ  ɩɪɨɟɤɬɭɜɚɧɧɹ ɫɩɨɪɭɞ ɫɢɫɬɟɦ ɜɨɞɨɩɨɫɬɚɱɚɧɧɹ    Ɍɭɝɚɣ Ⱥ Ɇ   Ɍɟɪɧɨɜɰɟɜ ȼ Ɉ   Ɍɭɝɚɣ ə Ⱥ   / ɇɚɜɱ   ɩɨɫɿɛɧɢɤ  Ʉ   ɄɇɍȻȺ        ±     ɫ 

6. Ƚɸɥɶɲɬɟɣɧ  ȿ Ƚ   Ɂɚɞɚɱɢ  ɥɢɧɟɣɧɨɝɨ  ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ  ɬɪɚɧɫɩɨɪɬɧɨɝɨ  ɬɢɩɚ   ȿ Ƚ  Ƚɸɥɶɲɬɟɣɧ  Ⱦ Ȼ  ɘɞɢɧ   ɍɱɟɛɧɢɤ  Ɇ.  ɇɚɭɤɚ        ±     ɫ 

7. Ȼɪɨɧɲɬɟɣɧ ɂ ɇ   ɋɟɦɟɧɞɹɟɜ Ʉ  Ⱥ  ɋɩɪɚɜɨɱɧɢɤ ɩɨ ɦɚɬɟɦɚɬɢɤɟ ɞɥɹ ɢɧɠ  ɢ  ɭɱɚɳɢɯɫɹ ɜɭɡɨɜ  M., ɇɚɭɤɚ, 1980. ± 976 c.

8. ɋɤɨɱɤɨ  ȼ ȱ   ɋɤɨɪɨɱɟɧɧɹ  ɬɟɩɥɨɜɬɪɚɬ  ɫɢɫɬɟɦ  ɬɟɩɥɨɩɨɫɬɚɱɚɧɧɹ  ɲɥɹɯɨɦ  ɨɩɬɢɦɿɡɚɰɿʀ  ʀɯ  ɝɟɨɦɟɬɪɢɱɧɢɯ  ɦɨɞɟɥɟɣ  ɩɪɢ  ɩɪɨɟɤɬɭɜɚɧɧɿ     ȼ ȱ  ɋɤɨɱɤɨ   ȼ Ɉ  ɉɥɨɫɤɢɣ  Ⱥ Ⱦ  Ƚɟɝɟɪ  Ʌ Ɉ  ɋɤɨɱɤɨ    ȿɧɟɪɝɨɟɮɟɤɬɢɜɧɿɫɬɶ ɜ ɛɭɞɿɜɧɢɰɬɜɿ ɬɚ 

ɚɪɯɿɬɟɤɬɭɪɿ   Ʉ    ɄɇɍȻȺ          ȼɢɩ        ± ɋ     -28. DOI: 10.32347/2310- 0516.2018.10.15-28

9. Ʉɨɜɚɥɶɨɜ  ɋ Ɇ   ɉɪɢɤɥɚɞɧɚ  ɝɟɨɦɟɬɪɿɹ  ɬɚ  ɿɧɠɟɧɟɪɧɚ  ɝɪɚɮɿɤɚ   ɋɩɟɰɿɚɥɶɧɿ  ɪɨɡɞɿɥɢ   ȼɢɩɭɫɤ        ɋ Ɇ  Ʉɨɜɚɥɶɨɜ   Ɇ ɋ  ȱɝɭɦɟɧ   ɋ ɂ  ɉɭɫɬɸɥɶɝɚ   ȼ ȯ  Ɇɢɯɚɣɥɟɧɤɨ  ɬɚ  ɿɧ        ɇɚɜɱ   ɩɨɫɿɛ    ɡɚ  ɪɟɞ   ȼ   ȯ   Ɇɢɯɚɣɥɟɧɤɚ   Ʌɭɰɶɤ   Ɋɟɞɚɤɰɿɣɧɨ-ɜɢɞɚɜɧɢɱɢɣ ɜɿɞɞɿɥ ɅȾɌɍ        ±     ɫ 

10. ɉɥɨɫɤɢɣ  ȼ Ɉ   Ⱦɨɫɥɿɞɠɟɧɧɹ  ɦɟɬɨɞɿɜ  ɝɟɨɦɟɬɪɢɱɧɨɝɨ  ɦɨɞɟɥɸɜɚɧɧɹ  ɬɨɩɨɝɪɚɮɿɱɧɢɯ  ɩɨɜɟɪɯɨɧɶ     ȼ Ɉ   ɉɥɨɫɤɢɣ   ɋ Ⱥ   Ʉɨɠɟɞɭɛ     Ƚɟɨɦɟɬɪɢɱɧɟ  ɬɚ  ɤɨɦɩ¶ɸɬɟɪɧɟ ɦɨɞɟɥɸɜɚɧɧɹ  ɏɚɪɤɿɜ  ɏȾɍɏɌ        ȼɢɩ     ± ɋ   -17.

11. Dyn N. Interpolation and approximation by radial and related functions / N. Dyn // Approximation Theory, 1989, ± P. 211-234.

12. Rudin W. Functional Analysis / W. Rudin // McGraw-Hill, New York, 1973. ± P. 424.

References

1. Rak J.R. (2009). Selected problems of water supply safety. Environment Protection Engineering, 35 (2), 23-28. (in English).

2. Nahman J.M. (2002). Dependability of engineering systems ± modeling and evaluation, 192 pp. DOI: 10.1007/978-3-662-04892-4. (in English).

3. Ostfeld A. Reliability simulation of water distribution systems ± single and multiquality / Avi Ostfeld, Dimitri Kogan, Uri Shamir // Urban Water  ʋ , 2002 ± Ɋ  53±61. DOI: 10.1016/S1462-0758(01)00055-3. (in English).

4. Iske A. (2003). Radial basis functions: basics, advanced topics and meshfree methods for transport problems. Rendiconti del Seminario Matematico, ʋ        , 247-284. (in English).

5. Tuhai Ⱥ Ɇ   Ternovtsev V.O., Tuhai Ya.A. (2001). Calculation and design of water supply systems [Rozrakhunok i proektuvannya sporud system vodopostachannya], 256 pp. (in Ukrainian).

6. Gyul'shteyn Ye.G., Yudin D.B. (1969). Transport type linear programming problems [Zadachi lineynogo programmirovaniya transportnogo tipa], 384 pp. (in Russian).

7. Bronshteyn I.N., Semendyayev K. A. (1980). A handbook of mathematics for engineers and students [Spravochnik po matematike dlya inzhenerov i uchashchikhsya vuzov], 976 pp. (in Russian).

8. Skochko V.I. Ploskyi V.O., Heher A.D., Skochko L.O. (2018). [Skorochennya teplovtrat system teplopostachannya shlyakhom optymizatsiyi yikh heometrychnykh modeley pry proektuvanni] Energy-Efficiency in Civil Engineering and Architecture.

Vol. 10, 15-28. DOI: 10.32347/2310-0516.2018.10.15-28. (in Ukrainian).

9. Kovalov S.M., Ihumen M.S., Pustyulha S.Y., Mykhaylenko V.Ye., ets. (2006).

Applied geometry and engineering graphics. Special sections. Issue 1 [Prykladna heometriya ta inzhenerna hrafika. Spetsialni rozdily. Vypusk 1], 256 pp. (in Ukrainian).

10. Ploskyi V.O., Kozhedub S.A. (2009). Research of methods of geometric modeling of topographic surfaces [Doslidzhennya metodiv heometrychnoho modelyuvannya topohrafichnykh poverkhon] Geometric and computer modeling, Vol. 22, 11-17. (in Ukrainian).

11. Dyn N. (1989) Interpolation and approximation by radial and related functions.

Approximation Theory, 211-234. (in English).

12. Rudin W. (1973). Functional Analysis, 424 pp. (in English).

Ⱥɧɧɨɬɚɰɢɹ

Ɉɪɟɥ  ɘɥɿɹ  Ɇɢɤɨɥɚʀɜɧɚ   ɚɫɩɢɪɚɧɬɤɚ  ɤɚɮɟɞɪɵ  ɚɪɯɢɬɟɤɬɭɪɧɵɯ  ɤɨɧɫɬɪɭɤɰɢɣ   Ʉɢɟɜɫɤɢɣ  ɧɚɰɢɨɧɚɥɶɧɵɣ  ɭɧɢɜɟɪɫɢɬɟɬ  ɫɬɪɨɢɬɟɥɶɫɬɜɚ  ɢ  ɚɪɯɢɬɟɤɬɭɪɵ 

Ƚɟɨɦɟɬɪɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɬɟɪɪɢɬɨɪɢɢ ɫɬɪɨɢɬɟɥɶɫɬɜɚ ɩɨ  ɩɨɤɚɡɚɬɟɥɹɦ ɭɞɟɥɶɧɵɯ ɫɬɨɢɦɨɫɬɟɣ ɩɪɢ ɨɩɬɢɦɢɡɚɰɢɢ ɜɧɟɲɧɢɯ ɫɟɬɟɣ 

ɜɨɞɨɫɧɚɛɠɟɧɢɹ.

ȼ ɫɬɚɬɶɟ  ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ  ɜɨɩɪɨɫɵ  ɪɟɲɟɧɢɹ  ɡɚɞɚɱɢ  ɨɩɬɢɦɢɡɚɰɢɢ  ɬɪɚɟɤɬɨɪɢɣ  ɩɪɨɥɨɠɟɧɢɹ  ɜɧɟɲɧɢɯ  ɫɟɬɟɣ  ɜɨɞɨɫɧɚɛɠɟɧɢɹ  ɦɟɬɨɞɚɦɢ  ɞɢɫɤɪɟɬɧɨɣ  ɝɟɨɦɟɬɪɢɢ ɧɚ ɨɫɧɨɜɟ ɢɬɟɪɚɰɢɨɧɧɨɝɨ ɤɨɪɪɟɤɬɢɪɨɜɚɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɤɨɬɨɪɵɟ  ɨɬɨɛɪɚɠɚɸɬ ɬɟɯɧɢɤɨ-ɷɤɨɧɨɦɢɱɟɫɤɭɸ ɰɟɥɟɫɨɨɛɪɚɡɧɨɫɬɶ ɩɪɨɥɨɠɟɧɢɹ ɨɬɞɟɥɶɧɵɯ  ɡɜɟɧɶɟɜ  ɬɪɭɛɨɩɪɨɜɨɞɨɜ  ɧɚ  ɪɚɡɥɢɱɧɵɯ  ɬɟɪɪɢɬɨɪɢɹɯ   ɜɵɪɚɠɟɧɧɭɸ  ɱɟɪɟɡ  ɭɪɨɜɧɢ  ɭɞɟɥɶɧɨɣ  ɫɬɨɢɦɨɫɬɢ   Ⱦɥɹ  ɷɬɨɝɨ  ɫɤɚɥɹɪɧɨɟ  ɩɨɥɟ  ɜɟɥɢɱɢɧ  ɭɞɟɥɶɧɨɣ  ɫɬɨɢɦɨɫɬɢ  ɫɬɪɨɢɬɟɥɶɧɵɯ  ɢ  ɷɤɫɩɥɭɚɬɚɰɢɨɧɧɵɯ  ɪɚɛɨɬ  ɜ  ɤɚɠɞɨɣ  ɬɨɱɤɟ  ɢɫɫɥɟɞɭɟɦɨɣ  ɬɟɪɪɢɬɨɪɢɢ  ɩɪɟɞɥɚɝɚɟɬɫɹ  ɡɚɞɚɜɚɬɶ  ɤɚɤ  ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɭɸ  ɮɭɧɤɰɢɸ   ȼ  ɫɜɨɸ  ɨɱɟɪɟɞɶ   ɩɪɟɞɥɚɝɚɟɦɚɹ  ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɚɹ  ɮɭɧɤɰɢɹ  ɫɬɪɨɢɬɫɹ  ɧɚ  ɨɫɧɨɜɟ  ɩɨɞɨɛɪɚɧɧɵɯ  ɩɨɞ  ɤɪɢɬɟɪɢɢ  ɡɚɞɚɱɢ   ɪɚɞɢɚɥɶɧɨ-ɛɚɡɢɫɧɵɯ  ɮɭɧɤɰɢɣ.

Ɋɚɫɩɪɟɞɟɥɟɧɢɟ  ɭɞɟɥɶɧɵɯ  ɫɬɨɢɦɨɫɬɟɣ  ɡɟɦɟɥɶɧɵɯ  ɭɱɚɫɬɤɨɜ  ɛɭɞɟɬ  ɢɦɟɬɶ  ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɜɥɢɹɧɢɟ ɧɚ ɪɟɡɭɥɶɬɚɬɵ ɤɨɪɪɟɤɬɢɪɨɜɤɢ ɬɪɚɟɤɬɨɪɢɢ ɭɫɬɪɨɣɫɬɜɚ  ɡɜɟɧɶɟɜ  ɬɪɭɛɨɩɪɨɜɨɞɨɜ  ɢ  ɦɟɫɬ  ɪɚɡɦɟɳɟɧɢɹ  ɢɯ  ɫɬɵɤɨɜɤɢ  ɪɚɡɜɟɬɜɥɟɧɢɹ  ɫɟɬɢ  ɬɪɭɛɨɩɪɨɜɨɞɚ    Ʉɪɨɦɟ  ɷɬɨɝɨ  ɪɟɡɭɥɶɬɚɬɵ  ɦɨɞɟɥɢɪɨɜɚɧɢɹ  ɬɟɪɪɢɬɨɪɢɢ  ɫɬɪɨɢɬɟɥɶɫɬɜɚ  ɩɨ  ɩɨɤɚɡɚɬɟɥɹɦ  ɭɞɟɥɶɧɵɯ  ɫɬɨɢɦɨɫɬɟɣ  ɟɟ  ɭɱɚɫɬɤɨɜ  ɩɨɡɜɨɥɹɸɬ  ɜɵɩɨɥɧɢɬɶ  ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɣ  ɩɪɟɞɩɪɨɟɤɬɧɵɣ  ɚɧɚɥɢɡ  ɢɫɯɨɞɧɵɯ  ɞɚɧɧɵɯ   ɧɚ  ɨɫɧɨɜɟ  ɝɪɚɮɢɱɟɫɤɨɝɨ  ɨɬɨɛɪɚɠɟɧɢɹ  ɷɬɢɯ  ɩɨɤɚɡɚɬɟɥɟɣ   ȼ  ɤɨɧɟɱɧɨɦ  ɪɟɡɭɥɶɬɚɬɟ  ɩɪɨɜɟɞɟɧɢɟɟ ɨɩɬɢɦɢɡɚɰɢɢ  ɬɪɚɟɤɬɨɪɢɢ  ɬɪɚɫɫɢɪɨɜɤɢ  ɫɟɬɢ  ɬɪɭɛɨɩɪɨɜɨɞɨɜ  ɩɨɡɜɨɥɢɬ  ɤɚɤ  ɭɦɟɧɶɲɢɬɶ  ɞɥɢɧɵ  ɡɜɟɧɶɟɜ  ɬɪɭɛɨɩɪɨɜɨɞɨɜ   ɬɚɤ  ɢ  ɭɦɟɧɶɲɢɬɶ  ɫɬɨɢɦɨɫɬɶ  ɫɬɪɨɢɬɟɥɶɧɨ-ɦɨɧɬɚɠɧɵɯ  ɪɚɛɨɬ   ɬɪɭɞɨɜɵɯ  ɪɟɫɭɪɫɨɜ  ɢ  ɞɚɥɶɧɟɣɲɢɯ 

ɷɤɫɩɥɭɚɬɚɰɢɨɧɧɵɯ  ɪɚɫɯɨɞɨɜ   ɗɬɨ  ɨɛɴɹɫɧɹɟɬɫɹ  ɬɟɦ   ɱɬɨ  ɫɤɨɪɪɟɤɬɢɪɨɜɚɧɧɚɹ  ɨɩɬɢɦɢɡɢɪɨɜɚɧɧɚɹ   ɫɯɟɦɚ  ɫɟɬɢ  ɬɪɭɛɨɩɪɨɜɨɞɨɜ ɜɨɞɨɫɧɚɛɠɟɧɢɹ  ɛɭɞɟɬ ɨɛɥɚɞɚɬɶ  ɪɹɞɨɦ  ɩɪɟɢɦɭɳɟɫɬɜ  ɩɟɪɟɞ  ɤɥɚɫɫɢɱɟɫɤɢɦɢ  ɩɨɞɯɨɞɚɦɢ  ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ   ɬɚɤɢɦɢ  ɤɚɤ  ɰɟɥɟɫɨɨɛɪɚɡɧɨɫɬɶ  ɢɫɩɨɥɶɡɨɜɚɧɢɹ  ɭɱɚɫɬɤɨɜ  ɨɛɳɟɞɨɫɬɭɩɧɵɯ ɬɟɪɪɢɬɨɪɢɣ   ɭɦɟɧɶɲɟɧɢɟ  ɧɚɝɪɭɡɨɤ  ɧɚ  ɨɬɞɟɥɶɧɵɟ  ɡɜɟɧɶɹ  ɬɪɭɛɨɩɪɨɜɨɞɨɜ   ɜɨɡɦɨɠɧɨɫɬɶ  ɭɬɨɱɧɹɸɳɢɯ ɤɨɪɪɟɤɬɢɪɨɜɨɤ ɡɚ ɫɱɟɬ ɜɜɟɞɟɧɢɹ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɤɪɢɬɟɪɢɟɜ ɢ ɬ.ɩ..

Ʉɥɸɱɟɜɵɟ  ɫɥɨɜɚ  ɞɢɫɤɪɟɬɧɨɟ  ɦɨɞɟɥɢɪɨɜɚɧɢɟ   ɜɧɟɲɧɢɟ  ɫɟɬɢ  ɜɨɞɨɫɧɚɛɠɟɧɢɹ  ɪɚɞɢɚɥɶɧɨ-ɛɚɡɢɫɧɵɟ ɮɭɧɤɰɢɢ 

Annotation

Yulia Orel, graduate student, Kiyv National University of Construction and Architecture.

Geometric modeling of the construction site in terms of unit costs for optimizing external water supply networks.

The article describes the issues of solving the problem of optimizing the trajectories of laying external water supply networks. This problem has solving by using discrete geometry methods based on iterative correction of the coefficients that reflect the technical and economic feasibility of pipeline links on some territories, expressed through the unit cost levels. For this, the scalar field of values of the unit cost of construction and maintenance work at each point of the study area is proposed to be set as an interpolation function. In turn, the proposed interpolation function is constructed on the basis of radial-basis functions selected for the criteria of the problem. The distribution of the specific values of land plots will have a direct impact on the results of adjusting the trajectory of pipeline links and the locations of their docking (branching of the pipeline network). In addition, the results of modeling the construction site in terms of the specific values of its plots make it possible to perform a preliminary pre-design analysis of the initial data. These analysis based on the graphical display of indicators of the unit cost. As a result, the optimization of the trajectory of the routing of pipeline network will allow both to reduce the length of the pipeline links and to reduce the cost of construction and installation work, labor resources and further operating costs. This is due to the fact that the adjusted (optimized) scheme of the water supply pipeline network will have a number of advantages over classical design approaches, such as the expediency of using sections of public areas, reducing the load on individual pipeline links, the possibility of making adjustments by introducing additional criteria, etc.

Keywords: discrete modeling, external water supply networks, radial-basis functions.