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Минимизация навигационных рисков в эргатической системе «Интегрированная система мостика - судоводитель».

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ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 5, ʋ2, 2002 ɝ.
ɫɬɪ.183-186
Ɇɢɧɢɦɢɡɚɰɢɹ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɪɢɫɤɨɜ ɜ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ
"ɢɧɬɟɝɪɢɪɨɜɚɧɧɚɹ ɫɢɫɬɟɦɚ ɦɨɫɬɢɤɚ – ɫɭɞɨɜɨɞɢɬɟɥɶ"
ȼ.ɂ. Ɇɟɧɶɲɢɤɨɜ, ȼ.Ⱥ. ɑɤɨɧɢɹ
ɋɭɞɨɜɨɞɢɬɟɥɶɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɫɭɞɨɜɨɠɞɟɧɢɹ
Ⱥɧɧɨɬɚɰɢɹ. ɋ ɬɨɱɤɢ ɡɪɟɧɢɹ ɬɟɨɪɢɢ ɷɪɝɚɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɩɪɨɛɥɟɦɚ ɢɫɩɨɥɶɡɨɜɚɧɢɹ
ɬɟɯɧɢɱɟɫɤɢɯ ɫɪɟɞɫɬɜ ɫɭɞɨɜɨɠɞɟɧɢɹ ɩɪɢ ɪɟɲɟɧɢɢ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɡɚɞɚɱɢ ɫ ɦɢɧɢɦɢɡɚɰɢɟɣ ɧɚɜɢɝɚɰɢɨɧɧɵɯ
ɪɢɫɤɨɜ. ȼ ɫɜɟɬɟ ɩɪɢɧɹɬɨɣ Ɇɟɠɞɭɧɚɪɨɞɧɨɣ Ɇɨɪɫɤɨɣ Ɉɪɝɚɧɢɡɚɰɢɟɣ Ɋɟɡɨɥɸɰɢɢ MSC.64(67) ɞɨɤɚɡɵɜɚɟɬɫɹ
ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɪɚɡɪɚɛɨɬɤɢ ɢɧɬɟɥɥɟɤɬɭɚɥɶɧɵɯ ɦɨɞɟɥɟɣ ɷɮɮɟɤɬɢɜɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɢɧɬɟɝɪɢɪɨɜɚɧɧɨɣ
ɫɢɫɬɟɦɨɣ ɦɨɫɬɢɤɚ (ɂɋɆ). ɂɫɫɥɟɞɭɸɬɫɹ ɩɪɢɱɢɧɵ, ɨɝɪɚɧɢɱɢɜɚɸɳɢɟ ɷɮɮɟɤɬɢɜɧɨɫɬɶ ɭɩɪɚɜɥɟɧɢɹ ɂɋɆ, ɢ
ɞɚɸɬɫɹ ɪɟɤɨɦɟɧɞɚɰɢɢ ɩɨ ɞɚɥɶɧɟɣɲɟɦɭ ɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɸ ɷɬɨɣ ɫɢɫɬɟɦɵ.
Abstract. The problem of navigation means usage when solving the navigational task with minimization of
navigational hazards has been considered from the point of view of the ergonomics systems' theory. Considering
the International Marine Organization Resolution MSC.64(67) the authors have demonstrated the necessity of
intellectual models of the integrated bridge system effective control. The reasons confining the efficiency of the
system management have been researched and some recommendations on its further perfecting have been given.
1. ȼɜɟɞɟɧɢɟ
Ɉɫɧɨɜɧɵɦ ɧɚɡɧɚɱɟɧɢɟɦ ɢɧɬɟɝɪɢɪɨɜɚɧɧɨɣ ɫɢɫɬɟɦɵ ɦɨɫɬɢɤɚ ɹɜɥɹɟɬɫɹ ɫɛɨɪ ɢ ɨɛɪɚɛɨɬɤɚ
ɢɧɮɨɪɦɚɰɢɢ, ɤɨɬɨɪɚɹ ɩɨɫɬɭɩɚɟɬ ɧɚ ɟɟ ɜɯɨɞ ɨɬ ɬɟɯɧɢɱɟɫɤɢɯ ɫɪɟɞɫɬɜ ɫɭɞɨɜɨɠɞɟɧɢɹ, ɚ ɬɚɤɠɟ ɫɨɡɞɚɧɢɟ
ɦɨɞɟɥɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɜ ɪɟɚɥɶɧɨɦ ɦɚɫɲɬɚɛɟ ɜɪɟɦɟɧɢ (Ɋɟɡɨɥɸɰɢɹ ɂɆɈ MSC.64(67), 1997). ȼ
ɨɛɳɟɦ ɫɥɭɱɚɟ ɮɨɪɦɢɪɨɜɚɧɢɟ ɦɨɞɟɥɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɢ ɟɟ ɨɬɨɛɪɚɠɟɧɢɟ ɧɚ ɷɥɟɤɬɪɨɧɧɨɦ
ɧɨɫɢɬɟɥɟ ɜ ɂɋɆ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɜ ɪɚɦɤɚɯ ɫɥɟɞɭɸɳɟɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɞɟɣɫɬɜɢɣ:
ɫɛɨɪ ɢɧɮɨɪɦɚɰɢɢ ɨɬ ɩɟɪɢɮɟɪɢɣɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɢ ɩɨɫɬɚɜɤɚ ɟɟ ɜ ɢɧɬɟɝɪɢɪɨɜɚɧɧɭɸ ɫɢɫɬɟɦɭ,
ɫɨɡɞɚɧɢɟ ɦɨɞɟɥɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɧɚ ɨɫɧɨɜɟ ɢɧɮɨɪɦɚɰɢɢ, ɫɨɞɟɪɠɚɳɟɣɫɹ ɜ ɩɨɫɬɭɩɚɸɳɢɯ ɫɨɨɛɳɟɧɢɹɯ,
ɨɬɨɛɪɚɠɟɧɢɟ ɫɨɡɞɚɜɚɟɦɨɣ ɦɨɞɟɥɢ ɜ ɭɞɨɛɧɨɣ ɞɥɹ ɜɨɫɩɪɢɹɬɢɹ ɮɨɪɦɟ ɢ ɜɵɞɟɥɟɧɢɟ ɧɚɢɛɨɥɟɟ ɡɧɚɱɢɦɵɯ
ɫɨɛɵɬɢɣ, ɫɜɹɡɚɧɧɵɯ ɫ ɩɨɹɜɥɟɧɢɟɦ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɪɢɫɤɨɜ.
ɉɟɪɟɱɢɫɥɟɧɧɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɞɟɣɫɬɜɢɣ ɩɨɡɜɨɥɹɟɬ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɢɫɬɟɦɚ ɫɨɡɞɚɟɬ ɧɟ ɫɬɨɥɶɤɨ ɛɚɡɭ
ɞɚɧɧɵɯ, ɧɟɨɛɯɨɞɢɦɭɸ ɞɥɹ ɤɨɧɬɪɨɥɹ ɫɨɫɬɨɹɧɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɫɤɨɥɶɤɨ ɧɟɤɨɬɨɪɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ
ɡɧɚɧɢɣ, ɤɨɬɨɪɨɟ ɹɜɥɹɟɬɫɹ ɢɫɯɨɞɧɵɦ ɞɥɹ ɭɩɪɚɜɥɟɧɱɟɫɤɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ ɫɭɞɨɜɨɞɢɬɟɥɹ. Ɍɨɝɞɚ ɨɛɴɟɞɢɧɟɧɢɟ
"ɂɋɆ – ɫɭɞɨɜɨɞɢɬɟɥɶ" ɫɥɟɞɭɟɬ ɨɬɧɟɫɬɢ ɤ ɤɥɚɫɫɭ ɷɪɝɚɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦ. ɂɦɟɧɧɨ ɜ ɬɚɤɢɯ ɫɢɫɬɟɦɚɯ ɞɨɩɭɫɬɢɦɨ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɪɚɤɬɢɱɟɫɤɢ ɜɫɟ ɮɨɪɦɵ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɨɛɴɟɤɬɨɜ ɞɪɭɝ ɫ ɞɪɭɝɨɦ. ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɞɚɧɧɨɦɭ
ɫɥɭɱɚɸ ɧɟɨɛɯɨɞɢɦɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɧɚ ɢɧɮɨɪɦɚɰɢɨɧɧɨɦ ɢ ɫɢɥɨɜɨɦ ɭɪɨɜɧɹɯ.
Ɇɨɞɟɥɶ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ, ɫɨɫɬɚɜɥɟɧɧɨɣ ɧɚ ɨɛɴɟɞɢɧɟɧɢɢ "ɂɋɆ – ɫɭɞɨɜɨɞɢɬɟɥɶ"
Ɇɨɞɟɥɶ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ, ɫɨɫɬɚɜɥɟɧɧɨɣ ɧɚ ɨɛɴɟɞɢɧɟɧɢɢ "ɂɋɆ ɫɭɞɨɜɨɞɢɬɟɥɶ",
ɨɛɟɫɩɟɱɢɜɚɸɳɟɣ ɷɮɮɟɤɬɢɜɧɨɟ ɭɩɪɚɜɥɟɧɢɟ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɚɩɩɚɪɚɬɭɪɨɣ ɫ ɨɛɹɡɚɬɟɥɶɧɵɦ ɭɫɥɨɜɢɟɦ
ɦɢɧɢɦɢɡɚɰɢɢ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɪɢɫɤɨɜ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɤɚɤ ɝɪɚɮ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ (ɪɢɫ. 1).
Ƚɪɚɮɨɦ ɮɢɤɫɢɪɨɜɚɧɵ ɫɥɟɞɭɸɳɢɟ ɫɨɫɬɨɹɧɢɹ ɦɨɞɟɥɢ: 1 ɫɛɨɪ ɩɨɫɬɭɩɚɸɳɟɣ ɢɧɮɨɪɦɚɰɢɢ ɢ
ɨɩɪɟɞɟɥɟɧɢɟ ɦɟɪɵ ɟɟ ɜɚɠɧɨɫɬɢ, ɨɫɭɳɟɫɬɜɥɹɟɦɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɧɨɣ ɫɢɫɬɟɦɨɣ ɦɨɫɬɢɤɚ, 2 – ɜɨɫɩɪɢɹɬɢɟ
ɫɭɞɨɜɨɞɢɬɟɥɟɦ ɢɧɮɨɪɦɚɰɢɨɧɧɨɣ ɦɨɞɟɥɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɢɡ ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɧɚɧɢɣ SK, 3 ɩɪɢɧɹɬɢɟ
ɪɟɲɟɧɢɣ ɢ ɪɟɚɥɢɡɚɰɢɹ ɞɟɣɫɬɜɢɣ, ɨɩɪɟɞɟɥɟɧɧɵɯ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɪɟɲɟɧɢɣ SR ɢ ɧɚɩɪɚɜɥɟɧɧɵɯ ɧɚ ɷɮɮɟɤɬɢɜɧɨɟ
ɭɩɪɚɜɥɟɧɢɟ ɂɋɆ, 4 – ɜɵɩɨɥɧɟɧɢɟ ɤɨɧɬɪɨɥɶɧɵɯ ɦɟɪɨɩɪɢɹɬɢɣ ɢ ɤɨɪɪɟɤɬɢɪɭɸɳɢɯ ɞɟɣɫɬɜɢɣ, ɩɨɜɵɲɚɸɳɢɯ
ɷɮɮɟɤɬɢɜɧɨɫɬɶ ɷɤɫɩɥɭɚɬɚɰɢɢ ɚɩɩɚɪɚɬɭɪɵ ɢ ɪɟɚɥɢɡɭɟɦɵɯ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɤɨɧɬɪɨɥɹ SC.
Ɏɚɤɬ ɩɪɟɛɵɜɚɧɢɹ ɫɢɫɬɟɦɵ ɜ ɫɨɫɬɨɹɧɢɢ 4 ɯɚɪɚɤɬɟɪɢɡɭɟɬ
ɫɨɝɥɚɫɨɜɚɧɧɨɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ SR ɫ ɩɪɨɫɬɪɚɧɫɬɜɨɦ SK ɜɨ ɜɪɟɦɟɧɢ. Ɍɚɤɨɟ
ɫɨɝɥɚɫɨɜɚɧɢɟ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ, ɟɫɥɢ ɪɚɡɪɚɛɨɬɚɬɶ ɢ ɜɧɟɞɪɢɬɶ
ɢɧɬɟɥɥɟɤɬɭɚɥɶɧɵɟ ɦɨɞɟɥɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɫɭɞɨɜɨɞɢɬɟɥɹ ɫ ɚɩɩɚɪɚɬɭɪɨɣ,
ɨɛɴɟɞɢɧɟɧɧɵɟ ɜ ɟɞɢɧɵɣ ɨɪɝɚɧɢɡɚɰɢɨɧɧɵɣ ɩɪɨɟɤɬ. ɉɨɩɵɬɚɟɦɫɹ ɞɨɤɚɡɚɬɶ, ɱɬɨ
Ɋɢɫ. 1. Ƚɪɚɮ
ɬɚɤɨɣ ɨɪɝɚɧɢɡɚɰɢɨɧɧɵɣ ɩɪɨɟɤɬ ɞɨɥɠɟɧ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɨɞɧɨɦɭ ɞɨɫɬɚɬɨɱɧɨ
ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ
"ɠɟɫɬɤɨɦɭ" ɭɫɥɨɜɢɸ.
Ɂɚ ɜɪɟɦɹ ɩɪɟɛɵɜɚɧɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ SK = {1,2} ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫɨɛɫɬɜɟɧɧɨ ɜɨɫɩɪɢɹɬɢɟ ɦɨɞɟɥɢ
ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɢ ɨɩɪɟɞɟɥɟɧɢɟ ɦɟɪɵ ɜɚɠɧɨɫɬɢ ɢɧɮɨɪɦɚɰɢɢ, ɡɚ ɜɪɟɦɹ ɩɪɟɛɵɜɚɧɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ
SR = {1, 2, 3} ɷɬɢ ɩɪɨɰɟɫɫɵ ɡɚɜɟɪɲɚɸɬɫɹ ɩɪɢɧɹɬɢɟɦ ɪɟɲɟɧɢɹ ɧɚ ɭɩɪɚɜɥɟɧɢɟ ɂɋɆ ɢ ɪɟɚɥɢɡɚɰɢɟɣ ɷɬɨɝɨ
ɪɟɲɟɧɢɹ ɫ ɩɨɦɨɳɶɸ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɞɟɣɫɬɜɢɹ. ɋɥɟɞɭɟɬ ɩɨɞɱɟɪɤɧɭɬɶ, ɱɬɨ ɫɨɫɬɨɹɧɢɟ 1 ɜ ɝɪɚɮɟ
2.
183
Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ. ɢ ɑɤɨɧɢɹ ȼ.Ⱥ. Ɇɢɧɢɦɢɡɚɰɢɹ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɪɢɫɤɨɜ ɜ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ...
ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɹɜɥɹɟɬɫɹ ɞɥɹ ɨɛɨɢɯ ɩɪɨɫɬɪɚɧɫɬɜ ɧɚɱɚɥɶɧɵɦ. ɉɪɢ ɧɚɥɢɱɢɢ ɷɮɮɟɤɬɢɜɧɨɝɨ
ɨɪɝɚɧɢɡɚɰɢɨɧɧɨɝɨ ɩɪɨɟɤɬɚ ɩɪɨɫɬɪɚɧɫɬɜɚ SK ɢ SR ɞɨɥɠɧɵ ɛɵɬɶ ɫɨɝɥɚɫɨɜɚɧɧɵɦɢ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɛɪɚɬɧɵɟ
ɩɟɪɟɯɨɞɵ ɜ ɦɨɞɟɥɢ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜɨɡɦɨɠɧɵ ɬɨɥɶɤɨ ɢɡ ɫɨɫɬɨɹɧɢɹ 3.
ȿɫɥɢ ɱɟɪɟɡ T ɢ Ji (i = 1,2,3) ɨɛɨɡɧɚɱɢɬɶ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜɪɟɦɹ ɦɟɠɞɭ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦɢ
ɦɨɦɟɧɬɚɦɢ ɩɨɫɬɭɩɥɟɧɢɹ ɨɱɟɪɟɞɧɵɯ ɩɨɪɰɢɣ ɜɯɨɞɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɢ ɜɪɟɦɹ ɩɪɟɛɵɜɚɧɢɹ ɫɢɫɬɟɦɵ ɜ
ɫɨɫɬɨɹɧɢɢ i, ɬɨ ɩɟɪɟɯɨɞ 3–4 (ɪɟɲɟɧɢɟ ɩɪɢɧɹɬɨ) ɩɪɨɢɡɨɣɞɟɬ ɥɢɲɶ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɭɫɥɨɜɢɹ:
J3 d T – (J1+ J2),
ɚ ɩɟɪɟɯɨɞ 3-1 (ɞɚɧɧɵɯ ɧɟɞɨɫɬɚɬɨɱɧɨ ɞɥɹ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɹ – ɡɚɩɪɨɫ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɢɧɮɨɪɦɚɰɢɢ) – ɟɫɥɢ
J3 > T – (J1 + J2).
Ɍɨɝɞɚ ɫɬɟɩɟɧɶ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɫɨɫɬɚɜɥɟɧɧɵɯ ɢɧɬɟɥɥɟɤɬɭɚɥɶɧɵɯ ɦɨɞɟɥɟɣ, ɨɛɴɟɞɢɧɟɧɧɵɯ ɜ ɨɪɝɚɧɢɡɚɰɢɨɧɧɵɣ
ɩɪɨɟɤɬ, ɦɨɠɟɬ ɛɵɬɶ ɨɰɟɧɟɧɚ ɩɨ ɪɚɡɧɨɫɬɢ
'W = W1(SR) W1(SK),
ɝɞɟ W1(SR) ɢ W1(SK) – ɜɪɟɦɹ ɩɪɟɛɵɜɚɧɢɹ ɦɨɞɟɥɢ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɢ ɪɟɲɟɧɢɣ
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɩɪɢ ɨɛɳɟɦ ɞɥɹ ɧɢɯ ɧɚɱɚɥɶɧɨɦ ɫɨɫɬɨɹɧɢɢ 1.
ɋ ɭɱɟɬɨɦ ɧɚɩɪɚɜɥɟɧɢɣ ɩɟɪɟɯɨɞɨɜ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɢɦ ɜɟɫɨɜ (ɜɟɪɨɹɬɧɨɫɬɟɣ) (ɪɢɫ. 1), ɫɪɟɞɧɹɹ
ɜɟɥɢɱɢɧɚ ɪɚɡɧɨɫɬɢ < 'W* > ɜ ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɜɟɥɢɱɢɧɚɯ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
< 'W* > = p31 / p34 [1 + T3* / (J1*+J2*)] + J3*/ (J1* + J2*),
ɩɪɢɱɟɦ
f f
T3* = T* (J1*+J2*);
p31 = 1 – p34 = ³ ^³FT (x + y)dFJ2(y)} dFJ3(x),
0
0
ɝɞɟ FT (t) ɢ FJi (t) – ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ T ɢ Ji ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
ɂɡ ɜɵɪɚɠɟɧɢɹ ɞɥɹ < 'W* > ɫɥɟɞɭɟɬ, ɱɬɨ ɷɬɨ ɡɧɚɱɟɧɢɟ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɞɜɭɯ ɤɨɦɩɨɧɟɧɬ, ɩɪɢɱɟɦ
ɩɟɪɜɚɹ ɤɨɦɩɨɧɟɧɬɚ ɨɩɪɟɞɟɥɹɟɬ ɜɪɟɦɹ ɧɚɯɨɠɞɟɧɢɹ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ ɫɨɫɬɨɹɧɢɢ 4 (ɪɢɫ. 1). ɗɬɨ
ɜɪɟɦɹ ɡɚɜɢɫɢɬ ɨɬ ɧɟɫɨɝɥɚɫɨɜɚɧɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜ SK ɢ SR ɢ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɬɚɤ
< 'W* > SC = p31 / p34 [T*/(J1* + J2*)].
Ɉɱɟɜɢɞɧɨ, ɢɧɬɟɥɥɟɤɬɭɚɥɶɧɵɟ ɦɨɞɟɥɢ, ɨɛɴɟɞɢɧɟɧɧɵɟ ɜ ɨɪɝɚɧɢɡɚɰɢɨɧɧɵɣ ɩɪɨɟɤɬ ɢ
ɨɛɟɫɩɟɱɢɜɚɸɳɢɟ ɫɨɝɥɚɫɨɜɚɧɧɨɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɧɚɧɢɣ ɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɪɟɲɟɧɢɣ, ɦɚɤɫɢɦɚɥɶɧɨ
ɪɟɚɥɢɡɭɸɬ ɫɜɨɢ ɜɨɡɦɨɠɧɨɫɬɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ < 'W* >SC = 0, ɬ.ɟ. ɤɨɝɞɚ
f
0
f
³ 0^ ³ FT (x + y)dFJ2(y)}dFJ3(x) = 0.
(1)
ɉɨɥɭɱɟɧɧɨɟ ɭɫɥɨɜɢɟ ɹɜɥɹɟɬɫɹ ɨɫɧɨɜɧɵɦ ɩɪɢ ɫɨɫɬɚɜɥɟɧɢɢ ɢɧɬɟɥɥɟɤɬɭɚɥɶɧɵɯ ɦɨɞɟɥɟɣ ɢ ɩɨɞɱɟɪɤɢɜɚɟɬ
ɜɚɠɧɨɫɬɶ ɫɨɝɥɚɫɨɜɚɧɢɹ ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɧɚɧɢɣ ɫ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɪɟɲɟɧɢɣ. ɂɧɬɟɥɥɟɤɬɭɚɥɶɧɵɟ ɦɨɞɟɥɢ
ɨɩɢɫɵɜɚɸɳɢɟ "ɩɪɚɜɢɥɶɧɵɟ" ɢɧɮɨɪɦɚɰɢɨɧɧɵɟ ɢ ɫɢɥɨɜɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫɭɞɨɜɨɞɢɬɟɥɹ ɫ ɫɨɜɪɟɦɟɧɧɵɦɢ
ɬɟɯɧɢɱɟɫɤɢɦɢ ɫɪɟɞɫɬɜɚɦɢ ɫɭɞɨɜɨɠɞɟɧɢɹ (Ɍɋɋ), ɜɤɥɸɱɚɸɳɢɦɢ ɫɢɫɬɟɦɵ ɨɬɨɛɪɚɠɟɧɢɹ, ɜɵɱɢɫɥɢɬɟɥɶɧɭɸ
ɬɟɯɧɢɤɭ, ɚɩɩɚɪɚɬɧɵɟ ɢ ɩɪɨɝɪɚɦɦɧɵɟ ɫɪɟɞɫɬɜɚ, ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɭɫɥɨɜɢɸ (1). Ɉɞɧɚɤɨ ɞɚɠɟ "ɩɪɚɜɢɥɶɧɨ"
ɫɨɫɬɚɜɥɟɧɧɵɟ ɢɧɬɟɥɥɟɤɬɭɚɥɶɧɵɟ ɦɨɞɟɥɢ ɧɟ ɫɩɨɫɨɛɧɵ ɝɚɪɚɧɬɢɪɨɜɚɬɶ ɩɪɢɧɹɬɵɣ ɭɪɨɜɟɧɶ ɛɟɡɨɩɚɫɧɨɫɬɢ ɧɚɜɢɝɚɰɢɢ,
ɟɫɥɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɛɭɞɭɬ ɢɦɟɬɶ ɦɟɫɬɨ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹ ɦɟɠɞɭ ɪɟɚɥɶɧɵɦ
ɧɚɜɢɝɚɰɢɨɧɧɵɦ ɩɪɨɰɟɫɫɨɦ ɢ ɟɝɨ ɨɬɨɛɪɚɠɚɟɦɨɣ ɦɨɞɟɥɶɸ. Ɍɚɤɢɟ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹ ɦɨɝɭɬ ɝɟɧɟɪɢɪɨɜɚɬɶɫɹ ɡɚ ɫɱɟɬ
ɫɛɨɟɜ ɞɚɬɱɢɤɨɜ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ, ɫɨɩɪɹɝɚɟɦɵɦɢ ɫ ɂɋɆ.
3. ȼɥɢɹɧɢɟ ɫɛɨɟɜ ɞɚɬɱɢɤɨɜ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɧɚ ɩɪɨɫɬɪɚɧɫɬɜɨ ɡɧɚɧɢɣ
ɇɟɫɜɨɟɜɪɟɦɟɧɧɨɟ ɨɛɧɚɪɭɠɟɧɢɟ ɬɟɯɧɢɱɟɫɤɨɝɨ ɢɥɢ ɢɧɮɨɪɦɚɰɢɨɧɧɨɝɨ ɫɛɨɹ ɜ ɞɚɬɱɢɤɚɯ
ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ, ɤɚɤ ɩɪɚɜɢɥɨ, ɜɥɟɱɟɬ ɡɚ ɫɨɛɨɣ ɩɨɹɜɥɟɧɢɟ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɪɢɫɤɚ, ɫɩɨɫɨɛɧɨɝɨ
ɩɪɢɜɟɫɬɢ ɤ ɬɹɠɟɥɵɦ ɩɨɫɥɟɞɫɬɜɢɹɦ. ɉɨɷɬɨɦɭ ɩɥɚɧɢɪɨɜɚɧɢɟ ɢ ɩɨɞɞɟɪɠɚɧɢɟ ɛɟɡɨɩɚɫɧɨɫɬɢ ɧɚɜɢɝɚɰɢɢ ɧɚ
ɧɟɨɛɯɨɞɢɦɨɦ ɭɪɨɜɧɟ ɞɨɥɠɧɨ ɛɵɬɶ ɫɜɹɡɚɧɨ ɧɟ ɬɨɥɶɤɨ ɫ ɦɢɧɢɦɢɡɚɰɢɟɣ ɜɧɟɲɧɢɯ ɪɢɫɤɨɜ, ɧɨ ɢ ɫ
ɩɪɟɞɫɤɚɡɭɟɦɨɫɬɶɸ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ Ɍɋɋ, ɧɚ ɤɨɬɨɪɵɯ ɪɟɚɥɢɡɭɟɬɫɹ ɧɚɜɢɝɚɰɢɨɧɧɵɣ ɩɪɨɰɟɫɫ, ɬ.ɟ.
s(•)
¦ : Q o Q |P,
ɝɞɟ Ɋ  P0 – ɷɥɟɦɟɧɬɵ ɦɧɨɠɟɫɬɜɚ Ɍɋɋ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɟ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɛɟɡɨɩɚɫɧɨɫɬɶ ɧɚɜɢɝɚɰɢɨɧɧɨɦɭ
ɩɪɨɰɟɫɫɭ Q ɜ ɪɚɦɤɚɯ ɩɪɚɜɢɥ S(•).
ɉɪɨɛɥɟɦɚ ɩɨɫɬɪɨɟɧɢɹ ɫɢɫɬɟɦɵ ɩɥɚɧɢɪɨɜɚɧɢɹ ɢ ɩɨɞɞɟɪɠɚɧɢɹ ɩɪɢɧɹɬɨɝɨ ɭɪɨɜɧɹ ɛɟɡɨɩɚɫɧɨɫɬɢ
ɧɚɜɢɝɚɰɢɢ, ɭɱɢɬɵɜɚɸɳɟɣ ɩɨɹɜɥɟɧɢɟ ɬɟɯɧɢɱɟɫɤɢɯ ɢɥɢ ɢɧɮɨɪɦɚɰɢɨɧɧɵɯ ɫɛɨɟɜ ɜ ɨɞɧɨɦ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ
ɷɥɟɦɟɧɬɚɯ, ɚ ɬɚɤɠɟ ɢɯ ɪɟɡɟɪɜɢɪɨɜɚɧɢɟ, ɹɜɥɹɟɬɫɹ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɧɚɢɦɟɧɟɟ ɢɫɫɥɟɞɨɜɚɧɧɨɣ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɭɸ ɫɯɟɦɭ ɢɡ Ɋ ɷɥɟɦɟɧɬɨɜ. ȿɟ ɦɨɠɧɨ ɨɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦɢ
ɨɛɴɟɤɬɚɦɢ: ɦɧɨɠɟɫɬɜɨ ɜɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ X Ž X0; ɦɧɨɠɟɫɬɜɨ ɜɵɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ Z Ž Z0; ɦɧɨɠɟɫɬɜɨ
184
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 5, ʋ2, 2002 ɝ.
ɫɬɪ.183-186
ɜɧɭɬɪɟɧɧɢɯ ɫɨɫɬɨɹɧɢɣ Y Ž Y0, ɝɞɟ X0, Y0, Z0 – ɩɨɥɧɵɟ ɦɧɨɠɟɫɬɜɚ. Ɍɨɝɞɚ ɞɥɹ ɮɭɧɤɰɢɨɧɢɪɭɸɳɟɣ ɛɟɡ ɫɛɨɟɜ
ɫɯɟɦɵ ɢɡ Ɋ ɷɥɟɦɟɧɬɨɜ ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɮɭɧɤɰɢɸ ɩɟɪɟɯɨɞɨɜ ; ɢ ɮɭɧɤɰɢɸ ɜɵɯɨɞɨɜ ::
; : X u Y o Y,
: : X u Y o Z.
ɇɚɥɢɱɢɟ ɜ ɫɯɟɦɟ ɬɟɯɧɢɱɟɫɤɢɯ ɢɥɢ ɢɧɮɨɪɦɚɰɢɨɧɧɵɯ ɫɛɨɟɜ ɩɪɢɜɨɞɢɬ ɤ ɢɡɦɟɧɟɧɢɹɦ ɤɚɤ
ɜɧɭɬɪɟɧɧɟɝɨ ɫɨɫɬɨɹɧɢɹ ɫɚɦɨɣ ɫɯɟɦɵ, ɬɚɤ ɢ ɟɟ ɜɵɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ, ɤɨɬɨɪɵɟ ɛɭɞɭɬ ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɬɟɯ,
ɤɨɬɨɪɵɟ ɛɵɥɢ ɡɚɩɥɚɧɢɪɨɜɚɧɵ ɢɥɢ ɩɨɞɞɟɪɠɢɜɚɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɣ ;, : . Ɉɞɧɚɤɨ ɢɡɦɟɧɟɧɧɵɟ
ɜɧɭɬɪɟɧɧɟɟ ɫɨɫɬɨɹɧɢɟ ɢ ɜɵɯɨɞɧɵɟ ɩɚɪɚɦɟɬɪɵ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ, ɟɫɥɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɜɜɟɫɬɢ:
; ' : X u Y o YF
ɮɭɧɤɰɢɸ ɩɟɪɟɯɨɞɨɜ ɜ ɫɯɟɦɟ ɫɨ ɫɛɨɟɦ f
ɮɭɧɤɰɢɸ ɜɵɯɨɞɨɜ ɫɯɟɦɵ ɫɨ ɫɛɨɟɦ f
: ' : X u Y o Z F,
ɩɪɢɱɟɦ f  F, ɝɞɟ F – ɡɚɞɚɧɧɵɣ ɤɥɚɫɫ ɢɧɮɨɪɦɚɰɢɨɧɧɵɯ ɢ ɬɟɯɧɢɱɟɫɤɢɯ ɧɟɫɨɨɬɜɟɬɫɬɜɢɣ, ɢɞɭɳɢɯ ɜ ɂɋɆ
ɨɬ ɫɨɩɪɹɠɟɧɧɵɯ ɫ ɧɟɸ ɞɚɬɱɢɤɨɜ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ.
Ɏɭɧɤɰɢɨɧɢɪɨɜɚɧɢɟ ɫɯɟɦɵ ɢɡ Ɋ ɷɥɟɦɟɧɬɨɜ, ɨɛɟɫɩɟɱɢɜɚɸɳɟɣ ɛɟɡɨɩɚɫɧɭɸ ɪɟɚɥɢɡɚɰɢɸ
ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɫɥɟɞɭɟɬ ɩɪɢɡɧɚɬɶ ɩɪɨɝɧɨɡɢɪɭɟɦɵɦ ɩɨ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹɦ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ
ɞɥɹ ɥɸɛɨɝɨ ɫɛɨɹ ɢɡ ɤɥɚɫɫɚ F ɢ ɩɪɢ ɥɸɛɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɜɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ X, ɟɫɥɢ ɜɵɩɨɥɧɹɸɬɫɹ
ɨɩɟɪɚɰɢɢ ɜɢɞɚ:
(2)
Y ŀ ɏF = ij; Y U YF = Y0,
Z ŀ ZF = ij; Z U ZF = Z0.
(3)
ɗɬɨ ɹɜɥɹɟɬɫɹ ɥɢɲɶ ɧɟɨɛɯɨɞɢɦɵɦ ɭɫɥɨɜɢɟɦ ɞɥɹ ɩɪɨɝɧɨɡɚ ɫɨɫɬɨɹɧɢɹ ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɧɚɧɢɣ. Ⱦɥɹ
ɬɨɝɨ, ɱɬɨɛɵ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɛɵɥɢ ɩɪɨɝɧɨɡɢɪɭɟɦɵ ɧɚ ɭɪɨɜɧɟ ɞɨɫɬɚɬɨɱɧɨɫɬɢ ɩɪɢ
ɥɸɛɨɦ ɫɛɨɟ ɢɡ ɤɥɚɫɫɚ F ɢ ɩɪɢ ɥɸɛɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɜɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ X, ɧɟɨɛɯɨɞɢɦɨ ɨɩɟɪɚɰɢɢ
(2) ɢ (3) ɞɨɩɨɥɧɢɬɶ ɟɳɟ ɪɹɞɨɦ ɭɫɥɨɜɢɣ. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɩɪɨɝɧɨɡɢɪɨɜɚɧɢɹ ɧɚ ɭɪɨɜɧɟ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɢ
ɞɨɫɬɚɬɨɱɧɨɫɬɢ ɧɟɫɨɨɬɜɟɬɫɬɜɢɣ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɡɚ ɫɱɟɬ ɫɛɨɟɜ, ɦɟɧɹɸɳɢɯ ɜɧɭɬɪɟɧɧɟɟ ɫɨɫɬɨɹɧɢɟ
ɫɯɟɦɵ ɢɡ Ɋ ɷɥɟɦɟɧɬɨɜ, ɤ ɨɩɟɪɚɰɢɢ (2) ɧɭɠɧɨ ɞɨɛɚɜɢɬɶ ɫɥɟɞɭɸɳɢɣ ɩɟɪɟɱɟɧɶ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɭɫɥɨɜɢɣ:
; ' (x,y) = ;(x,y) ɢɥɢ ; ' (x,y) = yf , yf  YF;
1. f  F, y  Y, x  X
2. f  F, yf  YF, x  X
; ' (x,yf) = yf', yf' YF.
(4)
Ⱥɧɚɥɨɝɢɱɧɵɣ ɩɟɪɟɱɟɧɶ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɭɫɥɨɜɢɣ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɢ ɞɥɹ ɨɩɟɪɚɰɢɢ (3), ɩɨɞɬɜɟɪɠɞɚɹ
ɭɪɨɜɟɧɶ ɞɨɫɬɚɬɨɱɧɨɫɬɢ ɩɪɨɝɧɨɡɚ ɧɟɫɨɨɬɜɟɬɫɬɜɢɣ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɡɚ ɫɱɟɬ ɫɛɨɟɜ ɩɚɪɚɦɟɬɪɨɜ ɜɵɯɨɞɚ zf  ZF
ɜ ɫɯɟɦɟ ɢɡ Ɋ ɷɥɟɦɟɧɬɨɜ ɫ ɥɸɛɵɦ ɫɛɨɟɦ f  F ɢ ɥɸɛɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɸ ɢɡ X. Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɞɥɹ
ɩɪɨɝɧɨɡɢɪɨɜɚɧɢɹ ɢ ɩɨɞɞɟɪɠɚɧɢɹ ɧɚ ɡɚɞɚɧɧɨɦ ɭɪɨɜɧɟ ɛɟɡɨɩɚɫɧɨɫɬɢ ɧɚɜɢɝɚɰɢɢ ɜ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɫ ɭɱɟɬɨɦ
ɫɛɨɟɜ ɜ ɫɯɟɦɟ Ɍɋɋ, ɫɨɞɟɪɠɚɳɟɣ Ɋ  50 ɷɥɟɦɟɧɬɨɜ, ɧɟɨɛɯɨɞɢɦɨ ɢ ɞɨɫɬɚɬɨɱɧɨ, ɱɬɨɛɵ ɩɟɪɜɨɟ ɧɟɫɨɨɬɜɟɬɫɬɜɢɟ ɜ
ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ, ɚ ɬɚɤɠɟ ɜɫɟ ɩɨɫɥɟɞɭɸɳɢɟ ɛɵɥɢ ɫɝɟɧɟɪɢɪɨɜɚɧɵ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ ɢɫɬɨɱɧɢɤɨɦ,
ɨɩɪɟɞɟɥɟɧɧɵɦ ɤɚɤ ɨɛɴɟɞɢɧɟɧɢɟ ɜɢɞɚ W = YF U ZF, ɢ ɡɚ ɫɱɟɬ ɷɥɟɦɟɧɬɨɜ ɢɡ ɤɥɚɫɫɚ F. Ɉɞɧɚɤɨ ɜɨɡɦɨɠɧɨɫɬɢ
ɩɪɨɝɧɨɡɢɪɨɜɚɧɢɹ ɧɟɫɨɨɬɜɟɬɫɬɜɢɣ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɟɳɟ ɧɟ ɢɫɱɟɪɩɵɜɚɸɬ ɜɫɟɣ
ɫɥɨɠɧɨɫɬɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɦɟɠɞɭ ɂɋɆ ɢ ɫɭɞɨɜɨɞɢɬɟɥɟɦ. ɋ ɩɪɚɤɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɜɚɠɧɨ ɡɧɚɬɶ ɜɪɟɦɹ, ɜ
ɬɟɱɟɧɢɟ ɤɨɬɨɪɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɨ ɡɧɚɧɢɣ ɛɭɞɟɬ ɞɚɜɚɬɶ ɫɭɞɨɜɨɞɢɬɟɥɸ ɨɲɢɛɨɱɧɵɟ ɞɚɧɧɵɟ ɢ ɧɟɜɟɪɧɵɟ
ɪɟɤɨɦɟɧɞɚɰɢɢ, ɨɫɨɛɟɧɧɨ ɜ ɱɚɫɬɢ ɫɨɛɵɬɢɣ, ɫɜɹɡɚɧɧɵɯ ɫ ɩɨɹɜɥɟɧɢɟɦ ɫɭɳɟɫɬɜɟɧɧɵɯ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɪɢɫɤɨɜ.
4. ȼɪɟɦɹ ɧɟɧɚɞɟɠɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɞɚɧɧɵɯ ɜ ɩɪɨɫɬɪɚɧɫɬɜɨ ɡɧɚɧɢɣ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ
Ɋɚɫɫɦɨɬɪɢɦ ɫɬɚɰɢɨɧɚɪɧɵɣ ɪɟɠɢɦ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɧɚɧɢɣ, ɤɨɬɨɪɨɟ ɮɨɪɦɢɪɭɟɬ
ɫɯɟɦɚ Ɍɋɋ Ɋ ɫ ɩɨɦɨɳɶɸ ɂɋɆ, ɩɪɢ ɧɚɥɢɱɢɢ ɜ ɷɬɨɣ ɫɯɟɦɟ ɫɛɨɟɜ ɢɡ ɤɥɚɫɫɚ F. ɉɭɫɬɶ ɫɯɟɦɚ, ɧɚɛɪɚɧɧɚɹ ɢɡ
ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ Ɋ  Ɋ0, ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɨɞɧɨɥɢɧɟɣɧɭɸ ɫɢɫɬɟɦɭ, ɜ ɤɨɬɨɪɨɣ ɬɟɯɧɢɱɟɫɤɢɟ ɢ
ɢɧɮɨɪɦɚɰɢɨɧɧɵɟ ɫɛɨɢ f  F ɫɥɟɞɭɸɬ ɫ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ, ɪɚɜɧɨɣ O. Ɉɛɧɚɪɭɠɟɧɢɟ ɫɛɨɟɜ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ
ɡɧɚɧɢɣ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫɭɞɨɜɨɞɢɬɟɥɟɦ, ɚ ɢɯ ɭɫɬɪɚɧɟɧɢɟ ɩɪɨɢɡɜɨɞɢɬɫɹ ɜ ɫɥɭɱɚɣɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ,
ɩɨɞɱɢɧɹɸɳɢɟɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɸ ɜɢɞɚ ȼ(W). ɉɨɫɤɨɥɶɤɭ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɡɚɜɢɫɹɬ ɨɬ
ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɢ ɢɫɬɨɱɧɢɤɚ ɫɛɨɟɜ, ɢɦɟɸɬ ɩɪɨɢɡɜɨɥɶɧɵɣ ɩɟɪɢɨɞ ɫɥɟɞɨɜɚɧɢɹ, ɚ ɫɚɦɢ ɫɛɨɢ ɩɨɞɱɢɧɹɸɬɫɹ
ɭɫɥɨɜɢɹɦ (4), ɬɨ ɦɨɠɧɨ ɨɬɤɚɡɚɬɶɫɹ ɨɬ ɤɚɤɢɯ-ɥɢɛɨ ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ ɨɱɟɪɟɞɧɨɫɬɶ ɨɛɧɚɪɭɠɟɧɢɹ ɢ ɭɫɬɪɚɧɟɧɢɟ
ɧɟɫɨɨɬɜɟɬɫɬɜɢɣ ɫɭɞɨɜɨɞɢɬɟɥɟɦ.
ɍɫɥɨɜɧɵɟ ɫɬɚɰɢɨɧɚɪɧɵɟ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɨɫɬɨɹɧɢɣ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɸɬɫɹ ɬɚɤ
pk*(W) = pk(W) / B0(W),
(5)
ɝɞɟ B0(W) = 1 B(W), ɚ ɷɬɢ ɫɨɫɬɨɹɧɢɹ ɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ ɫɥɟɞɭɸɳɟɣ ɫɢɫɬɟɦɨɣ ɭɪɚɜɧɟɧɢɣ:
dp1*(W)/dW = O p1*(W);
dpk*(W)/dW = Opk*(W) + Opk-1*(W).
ɑɬɨɛɵ ɩɟɪɟɣɬɢ ɜ (6) ɨɬ ɭɫɥɨɜɧɵɯ ɩɥɨɬɧɨɫɬɟɣ ɤ ɛɟɡɭɫɥɨɜɧɵɦ, ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ (5):
dpk*(W)/dW = [p'k(W)B0(W) pk(W)B0'(W)] / B0 2(W) = [pk'(W) + P(W)pk(W)] / B0(W),
185
(6)
Ɇɟɧɶɲɢɤɨɜ ȼ.ɂ. ɢ ɑɤɨɧɢɹ ȼ.Ⱥ. Ɇɢɧɢɦɢɡɚɰɢɹ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɪɢɫɤɨɜ ɜ ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ...
ɝɞɟ P(W) = B'(W) / B0(W) – ɦɝɧɨɜɟɧɧɚɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɨɛɧɚɪɭɠɟɧɢɹ ɢ ɭɫɬɪɚɧɟɧɢɹ ɨɬɤɚɡɚ f  F ɬɟɯɧɢɱɟɫɤɢɯ
ɫɪɟɞɫɬɜ ɫɭɞɨɜɨɠɞɟɧɢɹ, ɢɫɤɚɠɚɸɳɟɝɨ ɩɪɨɫɬɪɚɧɫɬɜɨ ɡɧɚɧɢɣ. ɉɨɞɫɬɚɜɥɹɹ ɪɟɡɭɥɶɬɚɬ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɜ
(6), ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɛɟɡɭɫɥɨɜɧɵɯ ɩɥɨɬɧɨɫɬɟɣ:
dp1(W)/dW = [P(W) + O]p1(W);
dpk(W)/dW = [P(W) + O]pk(W) + Opk - 1(W).
(7)
ɋɢɫɬɟɦɚ (7) ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɚ, ɟɫɥɢ ɡɚɞɚɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ
f
f
pk*(0) = ³ pk+1*(W)dB(W);
p1*(0) = Op0 + ³ p2* (W)dB(W)
0
0
ɢɥɢ ɩɟɪɟɩɢɫɚɬɶ ɢɯ ɬɚɤ
pk(0) = ³ P(W)pk+1(W)dW;
p1(0) = Op0 + ³ P(W)p2(W)dW.
(7a)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɬɪɚɞɢɰɢɨɧɧɨɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ (7) ɞɥɹ ɬɚɤɢɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɨɛɵɱɧɨ ɨɬɵɫɤɢɜɚɟɬɫɹ ɩɪɢ ɞɜɭɯ ɢ
ɛɨɥɟɟ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹɯ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ, ɬ.ɟ. k = 2,3,..., ɚ ɬɚɤɠɟ ɞɨɩɭɳɟɧɢɢ, ɱɬɨ ɫɪɟɞɧɹɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶ
ɨɛɧɚɪɭɠɟɧɢɹ ɢ ɭɫɬɪɚɧɟɧɢɹ ɫɛɨɟɜ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ, ɧɚɯɨɞɹɳɟɦɫɹ ɜ k-ɨɦ ɫɨɫɬɨɹɧɢɢ, ɪɚɜɧɚ
P* = ³P(W)pk(W)dW.
Ɉɩɢɪɚɹɫɶ ɧɚ ɬɪɚɞɢɰɢɨɧɧɵɣ ɩɪɢɟɦ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (7), ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɷɬɭ ɫɢɫɬɟɦɭ
ɨɬ ɧɭɥɹ ɞɨ ɛɟɫɤɨɧɟɱɧɨɫɬɢ. ɉɪɢ ɧɚɥɢɱɢɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɞɜɭɯ ɢ ɛɨɥɟɟ ɧɟɫɨɨɬɜɟɬɫɬɜɢɣ ɢɦɟɟɦ
f
f
f
pk(f) pk(0) = ³P(W)pk(W)dW O³pk(W)dW + O³pk-1(W)dW.
0
0
0
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɞɥɹ ɜɫɟɯ k = 2, 3, … ɫɨɛɥɸɞɚɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɪɚɜɟɧɫɬɜɚ
pk = ³ p(W)dW.
pk (f) = 0;
Ɂɧɚɱɢɬ, ɩɪɢ ɥɸɛɨɦ k ! 2
pk(0) = ³P(W) pk(W) dW + Opk Opk-1.
ɋɪɚɜɧɢɜ ɫ (7ɚ), ɩɨɥɭɱɢɦ
f
f
³P(W)pk+1(W)dW Opk = ³P(W)pk(W)dW Opk-1.
0
(8)
0
ɋɨɜɪɟɦɟɧɧɵɟ Ɍɋɋ, ɩɨɫɬɚɜɥɹɸɳɢɟ ɢɧɮɨɪɦɚɰɢɸ ɞɥɹ ɂɋɆ, ɨɛɥɚɞɚɸɬ ɜɟɫɶɦɚ ɜɵɫɨɤɢɦɢ ɩɨɤɚɡɚɬɟɥɹɦɢ
ɧɚɞɟɠɧɨɫɬɢ. ɉɨɷɬɨɦɭ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨ ɧɚɥɢɱɢɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ ɞɜɭɯ ɢ ɛɨɥɟɟ ɧɟɫɨɨɬɜɟɬɫɬɜɢɣ,
ɫɝɟɧɟɪɢɪɨɜɚɧɧɵɯ ɫɛɨɹɦɢ ɜ ɫɯɟɦɟ Ɋ, ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɦɚɥɨɜɟɪɨɹɬɧɨɟ, ɯɨɬɹ ɢ ɜɨɡɦɨɠɧɨɟ ɫɨɛɵɬɢɟ.
ȼɵɪɚɠɟɧɢɟ (8) ɨɩɢɫɵɜɚɟɬ ɜɟɪɨɹɬɧɨɫɬɧɭɸ ɜɡɚɢɦɨɫɜɹɡɶ ɦɟɠɞɭ ɧɟɫɤɨɥɶɤɢɦɢ ɧɟɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ
ɫɨɫɬɨɹɧɢɹɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɧɚɧɢɣ ɢ ɧɟ ɫɩɨɫɨɛɧɨ ɨɩɢɫɚɬɶ ɩɨɹɜɥɟɧɢɟ ɨɞɢɧɨɱɧɨɝɨ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹ. Ɉɞɧɚɤɨ,
ɜɡɹɜ ɟɝɨ ɡɚ ɨɫɧɨɜɭ ɢ ɜɵɩɨɥɧɢɜ ɚɧɚɥɨɝɢɱɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫ ɩɟɪɜɵɦ ɭɪɚɜɧɟɧɢɟɦ ɢɡ (7), ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ
f
ɪ1(0) = ³P(W)p2(W)dW + Op0.
(9)
0
ɂɡ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɨ ɧɚɣɬɢ ɜɪɟɦɹ W, ɜ ɬɟɱɟɧɢɟ ɤɨɬɨɪɨɝɨ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɧɚɧɢɣ
ɷɪɝɚɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɛɭɞɟɬ ɫɭɳɟɫɬɜɨɜɚɬɶ ɧɟɫɨɨɬɜɟɬɫɬɜɢɟ, ɩɨɪɨɠɞɚɸɳɟɟ ɧɚɜɢɝɚɰɢɨɧɧɵɣ ɪɢɫɤ ɢ
ɫɧɢɠɚɸɳɟɟ ɭɪɨɜɟɧɶ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɛɟɡɨɩɚɫɧɨɫɬɢ. Ɉɰɟɧɤɭ ɫɪɟɞɧɟɝɨ ɜɪɟɦɟɧɢ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹ
ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɧɚɧɢɣ ɪɟɚɥɶɧɨ ɫɤɥɚɞɵɜɚɸɳɟɣɫɹ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɫɢɬɭɚɰɢɢ, ɩɨɞɥɟɠɚɳɭɸ ɭɱɟɬɭ ɩɪɢ
ɩɥɚɧɢɪɨɜɚɧɢɢ ɢ ɩɨɞɞɟɪɠɚɧɢɢ ɡɚɞɚɧɧɨɝɨ ɭɪɨɜɧɹ ɛɟɡɨɩɚɫɧɨɫɬɢ ɧɚɜɢɝɚɰɢɢ, ɫɥɟɞɭɟɬ ɢɫɤɚɬɶ, ɨɱɟɜɢɞɧɨ, ɬɚɤ:
(10)
< W > = ³ W B(W)dW.
ɉɨɥɭɱɟɧɧɚɹ ɨɰɟɧɤɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɜɪɟɦɹ, ɜ ɬɟɱɟɧɢɟ ɤɨɬɨɪɨɝɨ ɂɋɆ ɧɟ ɫɩɨɫɨɛɧɚ ɨɬɨɛɪɚɡɢɬɶ ɪɟɚɥɶɧɭɸ
ɦɨɞɟɥɶ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɢ ɫɮɨɪɦɢɪɨɜɚɬɶ ɚɞɟɤɜɚɬɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɡɧɚɧɢɣ, ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟ
ɫɩɨɫɨɛɧɚ ɢɧɮɨɪɦɚɰɢɨɧɧɨ ɩɨɞɞɟɪɠɚɬɶ ɩɪɨɰɟɞɭɪɭ ɩɨ ɦɢɧɢɦɢɡɚɰɢɢ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɪɢɫɤɨɜ.
5. Ɂɚɤɥɸɱɟɧɢɟ
ɗɮɮɟɤɬɢɜɧɨɟ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɂɋɆ ɢ ɫɨɩɪɹɝɚɟɦɵɯ ɫ ɧɟɸ ɬɟɯɧɢɱɟɫɤɢɯ ɫɪɟɞɫɬɜ ɫɭɞɨɜɨɠɞɟɧɢɹ
ɩɪɟɞɭɫɦɚɬɪɢɜɚɟɬ ɫɨɝɥɚɫɨɜɚɧɢɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɧɚɧɢɣ ɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɪɟɲɟɧɢɣ, ɤɨɬɨɪɨɟ ɪɟɚɥɢɡɭɟɬɫɹ ɩɭɬɟɦ
ɪɚɡɪɚɛɨɬɤɢ ɢ ɜɧɟɞɪɟɧɢɹ ɜ ɩɪɚɤɬɢɤɭ ɫɭɞɨɜɨɠɞɟɧɢɹ ɢɧɬɟɥɥɟɤɬɭɚɥɶɧɵɯ ɦɨɞɟɥɟɣ, ɨɛɴɟɞɢɧɟɧɧɵɯ ɜ
ɨɪɝɚɧɢɡɚɰɢɨɧɧɵɣ ɩɪɨɟɤɬ. Ɉɞɧɚɤɨ ɬɚɤɨɣ ɩɪɨɟɤɬ ɧɟ ɫɩɨɫɨɛɟɧ ɩɨɥɧɨɫɬɶɸ ɢɫɤɥɸɱɢɬɶ ɧɚɜɢɝɚɰɢɨɧɧɵɟ ɪɢɫɤɢ,
ɨɫɨɛɟɧɧɨ ɬɟ, ɤɨɬɨɪɵɟ ɨɛɭɫɥɨɜɥɟɧɵ ɫɛɨɹɦɢ ɞɚɬɱɢɤɨɜ ɢɧɮɨɪɦɚɰɢɢ, ɫɨɩɪɹɝɚɟɦɵɯ ɢɥɢ ɜɯɨɞɹɳɢɯ ɜ ɤɨɦɩɥɟɤɬ ɂɋɆ.
Ɋɟɚɥɶɧɚɹ ɦɢɧɢɦɢɡɚɰɢɹ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɪɢɫɤɨɜ, ɫɜɹɡɚɧɧɚɹ ɫ ɭɦɟɧɶɲɟɧɢɟɦ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ (10), ɜɨɡɦɨɠɧɚ
ɥɢɲɶ ɟɫɥɢ ɜ ɧɨɜɵɟ ɷɤɫɩɥɭɚɬɚɰɢɨɧɧɵɟ ɬɪɟɛɨɜɚɧɢɹ ɛɭɞɭɬ ɜɤɥɸɱɟɧɵ ɩɭɧɤɬɵ, ɨɛɹɡɵɜɚɸɳɢɟ ɩɪɨɢɡɜɨɞɢɬɟɥɟɣ
ɩɪɨɟɤɬɢɪɨɜɚɬɶ ɢ ɜɵɩɭɫɤɚɬɶ ɂɋɆ ɫ ɩɪɢɡɧɚɤɚɦɢ ɩɨɥɧɨɣ ɫɚɦɨɩɪɨɜɟɪɹɟɦɨɫɬɢ.
Ʌɢɬɟɪɚɬɭɪɚ
Ɋɟɡɨɥɸɰɢɹ ɂɆɈ MSC.64(67) ɨɬ 5 ɞɟɤɚɛɪɹ 1996 ɝɨɞɚ. M., Ɍɪɚɧɫɩɨɪɬ, 237 c., 1997.
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