Математические методы исследования экономики
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ɉɪɢɧɰɢɩ ɨɩɬɢɦɚɥɶɧɨɫɬɢ Ʉɚɤɨɜɨ ɛɵ ɧɢ ɛɵɥɨ ɞɨɩɭɫɬɢɦɨɟ ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ xi 1 Xi 1 ɩɟɪɟɞ ɨɱɟɪɟɞɧɵɦ i-ɦ ɲɚɝɨɦ ɧɚɞɨ
ɜɵɛɪɚɬɶ ɞɨɩɭɫɬɢɦɨɟ ɍȼ ui Ui ɧɚ ɷɬɨɦ ɲɚɝɟ ɬɚɤ ɱɬɨɛɵ ɜɵɢɝɪɵɲ Wi
ɧɚ i-ɦ ɲɚɝɟ ɩɥɸɫ ɨɩɬɢɦɚɥɶɧɵɣ ɜɵɢɝɪɵɲ ɧɚ ɜɫɟɯ ɩɨɫɥɟɞɭɸɳɢɯ ɲɚɝɚɯ ɛɵɥ ɦɚɤɫɢɦɚɥɶɧɵɦ
ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɩɨɫɬɚɧɨɜɤɢ ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɩɪɨɞɨɥɠɢɦ ɪɚɫɫɦɨɬɪɟɧɢɟ ɡɚɞɚɱɢ ɭɩɪɚɜɥɟɧɢɹ ɮɢɧɚɧɫɢɪɨɜɚɧɢɟɦ ɝɪɭɩɩɵ ɩɪɟɞɩɪɢɹɬɢɣ ɉɭɫɬɶ ɜ ɧɚɱɚɥɟ i-ɝɨ ɝɨɞɚ
ɝɪɭɩɩɟ ɩɪɟɞɩɪɢɹɬɢɣ 1, 2 ,..., m ɜɵɞɟɥɹɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
ɫɪɟɞɫɬɜɚ |
u1i ,u2i ,...,umi .ɫɨɜɨɤɭɩɧɨɫɬɶ ɷɬɢɯ ɡɧɚɱɟɧɢɣ ɦɨɠɧɨ |
ɫɱɢɬɚɬɶ ɭɩɪɚɜɥɟɧɢɟɦ ɧɚ i-ɦ ɲɚɝɟ ɬɨ ɟɫɬɶ ui (u1i ,u2i ,...,umi ). |
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ɍɩɪɚɜɥɟɧɢɟ u |
ɩɪɨɰɟɫɫɨɦ ɜ ɰɟɥɨɦ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɨɜɨɤɭɩɧɨɫɬɶ |
ɜɫɟɯ ɲɚɝɨɜɵɯ ɭɩɪɚɜɥɟɧɢɣ ɬɨ ɟɫɬɶ u (u1,u2,...,uN ).
ɍɩɪɚɜɥɟɧɢɟ ɦɨɠɟɬ ɛɵɬɶ ɯɨɪɨɲɢɦ ɢɥɢ ɩɥɨɯɢɦ ɷɮɮɟɤɬɢɜɧɵɦ ɢɥɢ ɧɟɷɮɮɟɤɬɢɜɧɵɦ ɗɮɮɟɤɬɢɜɧɨɫɬɶ ɭɩɪɚɜɥɟɧɢɹ u ɨɰɟɧɢɜɚɟɬɫɹ ɩɨɤɚɡɚɬɟɥɟɦ S ȼɨɡɧɢɤɚɟɬ ɜɨɩɪɨɫ ɤɚɤ ɜɵɛɪɚɬɶ ɲɚɝɨɜɵɟ ɭɩɪɚɜɥɟɧɢɹ u1,u2,...,uN ɱɬɨɛɵ ɜɟɥɢɱɢɧɚ S ɨɛɪɚɬɢɥɚɫɶ ɜ ɦɚɤɫɢɦɭɦ "
ɉɨɫɬɚɜɥɟɧɧɚɹ ɡɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɡɚɞɚɱɟɣ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɚ ɭɩɪɚɜɥɟɧɢɟ ɩɪɢ ɤɨɬɨɪɨɦ ɩɨɤɚɡɚɬɟɥɶ S ɞɨɫɬɢɝɚɟɬ
ɦɚɤɫɢɦɭɦɚ ɧɚɡɵɜɚɟɬɫɹ ɨɩɬɢɦɚɥɶɧɵɦ Ɉɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ u * ɦɧɨɝɨɲɚɝɨɜɵɦ ɩɪɨɰɟɫɫɨɦ ɫɨɫɬɨɢɬ ɢɡ ɫɨɜɨɤɭɩɧɨɫɬɢ ɨɩɬɢɦɚɥɶɧɵɯ
ɲɚɝɨɜɵɯ ɭɩɪɚɜɥɟɧɢɣ
u * (u1*,u2*,...,uN* )
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɟɪɟɞ ɧɚɦɢ ɫɬɨɢɬ ɡɚɞɚɱɚ ɨɩɪɟɞɟɥɢɬɶ ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ui* (i 1 ɢ ɡɧɚɱɢɬ
ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ ɜɫɟɦ ɩɪɨɰɟɫɫɨɦ u * .
6 Ⱥɥɝɨɪɢɬɦ ɪɟɚɥɢɡɚɰɢɢ ɦɟɬɨɞɚ
Ɇɵ ɨɬɦɟɬɢɥɢ ɱɬɨ ɩɥɚɧɢɪɭɹ ɦɧɨɝɨɲɚɝɨɜɵɣ ɩɪɨɰɟɫɫ ɧɟɨɛɯɨɞɢɦɨ ɜɵɛɢɪɚɬɶ ɍȼ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ɫ ɭɱɟɬɨɦ ɟɝɨ ɛɭɞɭɳɢɯ ɩɨɫɥɟɞɫɬɜɢɣ ɧɚ ɟɳɟ ɩɪɟɞɫɬɨɹɳɢɯ ɲɚɝɚɯ Ɉɞɧɚɤɨ ɢɡ ɷɬɨɝɨ ɩɪɚɜɢɥɚ ɟɫɬɶ ɢɫɤɥɸɱɟɧɢɟ ɋɪɟɞɢ ɜɫɟɯ ɲɚɝɨɜ ɫɭɳɟɫɬɜɭɟɬ ɨɞɢɧ ɤɨɬɨɪɵɣ ɦɨɠɟɬ
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ɩɥɚɧɢɪɨɜɚɬɶɫɹ ɛɟɡ ɡɚɝɥɹɞɵɜɚɧɢɹ ɜ ɛɭɞɭɳɟɟ Ʉɚɤɨɣ ɷɬɨ ɲɚɝ" Ɉɱɟɜɢɞɧɨ ɩɨɫɥɟɞɧɢɣ - ɩɨɫɥɟ ɧɟɝɨ ɞɪɭɝɢɯ ɲɚɝɨɜ ɧɟɬ ɗɬɨɬ ɲɚɝ ɟɞɢɧɫɬɜɟɧɧɵɣ ɢɡ ɜɫɟɯ ɦɨɠɧɨ ɩɥɚɧɢɪɨɜɚɬɶ ɬɚɤ ɱɬɨɛɵ ɨɧ ɤɚɤ ɬɚɤɨɜɨɣ ɩɪɢɧɟɫ ɧɚɢɛɨɥɶɲɭɸ ɜɵɝɨɞɭ ɋɩɥɚɧɢɪɨɜɚɜ ɨɩɬɢɦɚɥɶɧɨ ɷɬɨɬ ɩɨɫɥɟɞɧɢɣ ɲɚɝ ɦɨɠɧɨ ɤ ɧɟɦɭ ɩɪɢɫɬɪɚɢɜɚɬɶ ɩɪɟɞɩɨɫɥɟɞɧɢɣ ɤ ɩɪɟɞɩɨɫɥɟɞɧɟɦɭ - ɩɪɟɞɩɪɟɞɩɨɫɥɟɞɧɢɣ ɢ ɬ ɞ
ɉɨɷɬɨɦɭ ɩɪɨɰɟɫɫ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɧɚ -ɦ ɷɬɚɩɟ ɪɚɡɜɨɪɚɱɢɜɚɟɬɫɹ ɨɬ ɤɨɧɰɚ ɤ ɧɚɱɚɥɭ ɬɨ ɟɫɬɶ ɪɚɧɶɲɟ ɜɫɟɯ ɩɥɚɧɢɪɭɟɬɫɹ ɩɨɫɥɟɞɧɢɣ
N-ɣ ɲɚɝ Ⱥ ɤɚɤ ɟɝɨ ɫɩɥɚɧɢɪɨɜɚɬɶ ɟɫɥɢ ɦɵ ɧɟ ɡɧɚɟɦ ɱɟɦ ɤɨɧɱɢɥɫɹ ɩɪɟɞɩɨɫɥɟɞɧɢɣ" Ɉɱɟɜɢɞɧɨ ɧɭɠɧɨ ɫɞɟɥɚɬɶ ɜɫɟ ɜɨɡɦɨɠɧɵɟ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɨ ɬɨɦ ɱɟɦ ɤɨɧɱɢɥɫɹ ɩɪɟɞɩɨɫɥɟɞɧɢɣ (N - 1)-ɣ ɲɚɝ ɢ ɞɥɹ ɤɚɠɞɨɝɨ ɢɡ ɧɢɯ ɧɚɣɬɢ ɬɚɤɨɟ ɭɩɪɚɜɥɟɧɢɟ ɩɪɢ ɤɨɬɨɪɨɦ ɜɵɢɝɪɵɲɞɨɯɨɞ ɧɚ ɩɨɫɥɟɞɧɟɦ ɲɚɝɟ ɛɵɥ ɛɵ ɦɚɤɫɢɦɚɥɟɧ Ɋɟɲɢɜ ɷɬɭ ɡɚɞɚɱɭ ɦɵ ɧɚɣɞɟɦ ɭɫɥɨɜɧɨ ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ ɍɈɍ ɧɚ N-ɦ ɲɚɝɟ ɬ ɟ ɭɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɨɟ ɧɚɞɨ ɩɪɢɦɟɧɢɬɶ ɟɫɥɢ N - 1)-ɣ ɲɚɝ ɡɚɤɨɧɱɢɥɫɹ ɨɩɪɟɞɟɥɟɧɧɵɦ ɨɛɪɚɡɨɦ
ɉɪɟɞɩɨɥɨɠɢɦ ɱɬɨ ɷɬɚ ɩɪɨɰɟɞɭɪɚ ɜɵɩɨɥɧɟɧɚ ɬɨ ɟɫɬɶ ɞɥɹ ɤɚɠɞɨɝɨ ɢɫɯɨɞɚ
(N - 1)-ɝɨ ɲɚɝɚ ɦɵ ɡɧɚɟɦ ɍɈɍ ɧɚ N-ɦ ɲɚɝɟ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɟɦɭ ɭɫɥɨɜɧɨ ɨɩɬɢɦɚɥɶɧɵɣ ɜɵɢɝɪɵɲ ɍɈȼ Ɍɟɩɟɪɶ ɦɵ ɦɨɠɟɦ ɨɩɬɢɦɢɡɢɪɨɜɚɬɶ ɭɩɪɚɜɥɟɧɢɟ ɧɚ ɩɪɟɞɩɨɫɥɟɞɧɟɦ N - 1)-ɦ ɲɚɝɟ ɋɞɟɥɚɟɦ ɜɫɟ ɜɨɡɦɨɠɧɵɟ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɨ ɬɨɦ ɱɟɦ ɤɨɧɱɢɥɫɹ ɩɪɟɞɩɪɟɞɩɨɫɥɟɞɩɢɣ ɬɨ ɟɫɬɶ (N - 2)-ɣ ɲɚɝ ɢ ɞɥɹ ɤɚɠɞɨɝɨ ɢɡ ɷɬɢɯ ɩɪɟɞɩɨɥɨɠɟɧɢɣ ɧɚɣɞɟɦ ɬɚɤɨɟ ɭɩɪɚɜɥɟɧɢɟ ɧɚ (N - 1)-ɦ ɲɚɝɟ ɱɬɨɛɵ ɜɵɢɝɪɵɲ ɡɚ ɩɨɫɥɟɞɧɢɟ ɞɜɚ ɲɚɝɚ ɢɡ ɤɨɬɨɪɵɯ ɩɨɫɥɟɞɧɢɣ ɭɠɟ ɨɩɬɢɦɢɡɢɪɨɜɚɧ ɛɵɥ ɦɚɤɫɢɦɚɥɟɧ Ⱦɚɥɟɟ ɨɩɬɢɦɢɡɢɪɭɟɬɫɹ ɭɩɪɚɜɥɟɧɢɟ ɧɚ (N - 2)-ɦ ɲɚɝɟ ɢ ɬ ɞ
Ɉɞɧɢɦ ɫɥɨɜɨɦ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ɢɳɟɬɫɹ ɬɚɤɨɟ ɭɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɨɟ ɨɛɟɫɩɟɱɢɜɚɟɬ ɨɩɬɢɦɚɥɶɧɨɟ ɩɪɨɞɨɥɠɟɧɢɟ ɩɪɨɰɟɫɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɨɫɬɢɝɧɭɬɨɝɨ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɫɨɫɬɨɹɧɢɹ ɗɬɨɬ ɩɪɢɧɰɢɩ ɜɵɛɨɪɚ ɭɩɪɚɜɥɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɩɪɢɧɰɢɩɨɦ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɋɚɦɨ ɭɩɪɚɜɥɟɧɢɟ ɨɛɟɫɩɟɱɢɜɚɸɳɟɟ ɨɩɬɢɦɚɥɶɧɨɟ ɩɪɨɞɨɥɠɟɧɢɟ ɩɪɨɰɟɫɫɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɚɞɚɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɍɈɍ ɧɚ ɞɚɧɧɨɦ ɲɚɝɟ Ɍɟɩɟɪɶ ɩɪɟɞɩɨɥɨɠɢɦ ɱɬɨ ɍɈɍ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ɧɚɦ ɢɡɜɟɫɬɧɨ
ɦɵ ɡɧɚɟɦ ɱɬɨ ɞɟɥɚɬɶ ɞɚɥɶɲɟ ɜ ɤɚɤɨɦ ɛɵ ɫɨɫɬɨɹɧɢɢ ɧɢ ɛɵɥ ɩɪɨɰɟɫɫ ɤ ɧɚɱɚɥɭ ɤɚɠɞɨɝɨ ɲɚɝɚ Ɍɨɝɞɚ ɦɵ ɦɨɠɟɦ ɧɚɣɬɢ ɭɠɟ ɧɟ ɭɫɥɨɜɧɨɟ ɚ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɩɭɫɬɶ ɧɚɦ ɢɡɜɟɫɬɧɨ ɧɚɱɚɥɶɧɨɟ ɫɨɫɬɨɹɧɢɟ ɩɪɨɰɟɫɫɚ Ɍɟɩɟɪɶ ɦɵ ɭɠɟ ɡɧɚɟɦ ɱɬɨ ɞɟɥɚɬɶ ɧɚ ɩɟɪɜɨɦ ɲɚɝɟ ɧɚɞɨ ɩɪɢɦɟɧɢɬɶ ɍɈɍ ɧɚɣɞɟɧɧɨɟ ɞɥɹ ɩɟɪɜɨɝɨ ɲɚɝɚ ɢ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ȼ ɪɟɡɭɥɶɬɚɬɟ ɷɬɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɩɨɫɥɟ ɩɟɪɜɨɝɨ ɲɚɝɚ ɫɢɫɬɟɦɚ ɩɟɪɟɣɞɟɬ ɜ
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ɞɪɭɝɨɟ ɫɨɫɬɨɹɧɢɟ ɧɨ ɞɥɹ ɷɬɨɝɨ ɫɨɫɬɨɹɧɢɹ ɦɵ ɡɧɚɟɦ ɍɈɍ ɢ ɝ ɞ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɦɵ ɧɚɣɞɟɦ ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ ɩɪɨɰɟɫɫɨɦ ɩɪɢɜɨɞɹɳɟɟ ɤ ɦɚɤɫɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɨɦɭ ɜɵɢɝɪɵɲɭ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜ ɩɪɨɰɟɫɫɟ ɨɩɬɢɦɢɡɚɰɢɢ ɭɩɪɚɜɥɟɧɢɹ ɦɟɬɨɞɨɦ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɦɧɨɝɨɲɚɝɨɜɵɣ ɩɪɨɰɟɫɫɩɪɨɯɨɞɢɬɫɹ ɞɜɚɠɞɵ
ɉɟɪɜɵɣ ɪɚɡ - ɨɬ ɤɨɧɰɚ ɤ ɧɚɱɚɥɭ ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɧɚɯɨɞɹɬɫɹ ɍɈɍ_ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ɢ ɨɩɬɢɦɚɥɶɧɵɣ ɜɵɢɝɪɵɲ ɬɨɠɟ ɭɫɥɨɜɧɵɣ ɧɚ ɜɫɟɯ ɲɚɝɚɯ ɧɚɱɢɧɚɹ ɫ ɞɚɧɧɨɝɨ ɢ ɞɨ ɤɨɧɰɚ ɩɪɨɰɟɫɫɚ
ȼɬɨɪɨɣ ɪɚɡ - ɨɬ ɧɚɱɚɥɚ ɤ ɤɨɧɰɭ ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɧɚɯɨɞɹɬɫɹ ɨɩɬɢɦɚɥɶɧɵɟ ɭɩɪɚɜɥɟɧɢɹ ɧɚ ɜɫɟɯ ɲɚɝɚɯ ɩɪɨɰɟɫɫɚ
Ɇɨɠɧɨ ɫɤɚɡɚɬɶ ɱɬɨ ɩɪɨɰɟɞɭɪɚ ɩɨɫɬɪɨɟɧɢɹ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɦɟɬɨɞɨɦ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɪɚɫɩɚɞɚɟɬɫɹ ɧɚ ɞɜɟ ɫɬɚɞɢɢ ɩɪɟɞɜɚɪɢɬɟɥɶɧɭɸ ɢ ɨɤɨɧɱɚɬɟɥɶɧɭɸ ɇɚ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɣ ɫɬɚɞɢɢ ɞɥɹ ɤɚɠɞɨɝɨ ɲɚɝɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɍɈɍ ɡɚɜɢɫɹɳɟɟ ɨɬ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ ɞɨɫɬɢɝɧɭɬɨɝɨ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɪɟɞɵɞɭɳɢɯ ɲɚɝɨɜ ɢ ɭɫɥɨɜɧɨ ɨɩɬɢɦɚɥɶɧɵɣ ɜɵɢɝɪɵɲ ɧɚ ɜɫɟɯ ɨɫɬɚɜɲɢɯɫɹ ɲɚɝɚɯ ɧɚɱɢɧɚɹ ɫ ɞɚɧɧɨɝɨ ɬɚɤɠɟ ɡɚɜɢɫɹɳɢɣ ɨɬ ɫɨɫɬɨɹɧɢɹ ɇɚ ɨɤɨɧɱɚɬɟɥɶɧɨɣ ɫɬɚɞɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹɛɟɡɭɫɥɨɜɧɨɟ ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ ɞɥɹ ɤɚɠɞɨɝɨ ɲɚɝɚ ɉɪɟɞɜɚɪɢɬɟɥɶɧɚɹ ɭɫɥɨɜɧɚɹ ɨɩɬɢɦɢɡɚɰɢɹ ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɨ ɲɚɝɚɦ ɜ ɨɛɪɚɬɧɨɦ ɩɨɪɹɞɤɟ ɨɬ ɩɨɫɥɟɞɧɟɝɨ ɲɚɝɚ ɤ ɩɟɪɜɨɦɭ ɨɤɨɧɱɚɬɟɥɶɧɚɹɛɟɡɭɫɥɨɜɧɚɹ ɨɩɬɢɦɢɡɚɰɢɹ - ɬɚɤɠɟ ɩɨ ɲɚɝɚɦ ɧɨ ɜ ɟɫɬɟɫɬɜɟɧɧɨɦ ɩɨɪɹɞɤɟ ɨɬ ɩɟɪɜɨɝɨ ɲɚɝɚ ɤ ɩɨɫɥɟɞɧɟɦɭ ɂɡ ɞɜɭɯ ɫɬɚɞɢɣ ɨɩɬɢɦɢɡɚɰɢɢ ɧɟɫɪɚɜɧɟɧɧɨ ɛɨɥɟɟ ɜɚɠɧɨɣ ɢ ɬɪɭɞɨɟɦɤɨɣ ɹɜɥɹɟɬɫɹ ɩɟɪɜɚɹ ɉɨɫɥɟ ɨɤɨɧɱɚɧɢɹ ɩɟɪɜɨɣ ɫɬɚɞɢɢ ɜɵɩɨɥɧɟɧɢɟ ɜɬɨɪɨɣ ɬɪɭɞɧɨɫɬɢ ɧɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɨɫɬɚɟɬɫɹ ɬɨɥɶɤɨ ɩɪɨɱɟɫɬɶ ɪɟɤɨɦɟɧɞɚɰɢɢ ɭɠɟ ɡɚɝɨɬɨɜɥɟɧɧɵɟ ɧɚ ɩɟɪɜɨɣ ɫɬɚɞɢɢ
6.3. Ⱦɢɧɚɦɢɱɟɫɤɚɹ ɡɚɞɚɱɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɧɜɟɫɬɢɰɢɣ
Ɂɚɞɚɱɚ ɉɪɨɢɡɜɨɞɫɬɜɟɧɧɨɟ ɨɛɴɟɞɢɧɟɧɢɟ ɫɨɫɬɨɢɬ ɢɡ ɩɪɟɞɩɪɢɹɬɢɣ n Ɉɛɳɚɹ ɫɭɦɦɚ ɤɚɩɢɬɚɥɶɧɵɯ ɜɥɨɠɟɧɢɣ ɪɚɜɧɚ ɦɥɧ ɪɭɛ b ɜɵɞɟɥɹɟɦɵɟ ɩɪɟɞɩɪɢɹɬɢɟɦ ɫɭɦɦɵ ɤɪɚɬɧɵ ɦɥɧ ɪɭɛ ȿɫɥɢ j-ɟ ɩɪɟɞɩɪɢɹɬɢɟ ɩɨɥɭɱɚɟɬ ɢɧɜɟɫɬɢɰɢɢ ɜ ɨɛɴɟɦɟ x ɦɥɧ ɪɭɛ ɬɨ ɩɪɢɪɨɫɬ ɝɨɞɨɜɨɣ ɩɪɢɛɵɥɢ ɧɚ ɷɬɨɦ ɩɪɟɞɩɪɢɹɬɢɢ ɫɨɫɬɚɜɢɬ fj(x ɦɥɧ ɪɭɛ ɜ ɝɨɞ Ɂɧɚɱɟɧɢɹ ɮɭɧɤɰɢɣ fj(x ɩɪɢɜɟɞɟɧɵ ɜ ɬɚɛɥɢɰɟ
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100 |
200 |
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Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɚɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɢɧɜɟɫɬɢɰɢɣ ɦɟɠɞɭ ɩɪɟɞɩɪɢɹɬɢɹɦɢ ɤɨɬɨɪɨɟ ɦɚɤɫɢɦɢɡɢɪɭɟɬ ɫɭɦɦɚɪɧɵɣ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɜɫɟɯ ɩɪɟɞɩɪɢɹɬɢɹɯ ɜɦɟɫɬɟ
Ɋɟɲɟɧɢɟ ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɦɟɬɨɞɨɦ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɢ ɪɚɡɨɛɶɟɦ ɡɚɞɚɱɭ ɧɚ ɲɚɝɢ ȼɧɚɱɚɥɟ ɢɫɫɥɟɞɭɟɦ ɡɚɞɚɱɭ ɞɥɹ ɩɟɪɜɵɯ -ɯ ɩɪɟɞɩɪɢɹɬɢɣ ɢ ɪɚɫɫɦɨɬɪɢɦ ©ɨɛɴɟɞɢɧɟɧɧɨɟ ɩɟɪɜɨɟ ɫɨ ɜɬɨɪɵɦª ɩɪɟɞɩɪɢɹɬɢɟ ɧɚ ɫɥɟɞɭɸɳɟɦ ɷɬɚɩɟ ɢɫɫɥɟɞɭɟɦ ɡɚɞɚɱɭ ɞɥɹ -ɯ ɩɪɟɞɩɪɢɹɬɢɣ ɨɛɴɟɞɢɧɟɧɧɨɝɨ ɩɟɪɜɨɝɨ ɫɨ ɜɬɨɪɵɦ ɢ ɬɪɟɬɶɟɝɨ ɢ ɪɚɫɫɦɨɬɪɢɦ ©ɨɛɴɟɞɢɧɟɧɧɨɟ ɩɟɪɜɨɟ ɫɨ ɜɬɨɪɵɦ ɢ ɫ ɬɪɟɬɶɢɦª ɩɪɟɞɩɪɢɹɬɢɟ ɢ ɬ ɞ
Ɉɛɨɡɧɚɱɢɦ Fi(x ɦɚɤɫɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɵɣ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɩɟɪɜɵɯ i ɩɪɟɞɩɪɢɹɬɢɹɯ ɜɦɟɫɬɟ ɟɫɥɢ ɧɚ ɧɢɯ ɜ ɫɭɦɦɟ ɜɵɞɟɥɟɧɨ ɯ ɦɥɧ ɪɭɛ ɉɪɢ ɷɬɨɦ ɯi*(x ɛɭɞɟɬ ɨɡɧɚɱɚɬɶ ɫɭɦɦɭ ɞɚɸɳɭɸɫɹ i-ɦɭ ɩɪɟɞɩɪɢɹɬɢɸ Ɉɱɟɜɢɞɧɨ F1(x) = f1(x Ɉɩɪɟɞɟɥɢɦ F2(x).
ȼ ɫɥɟɞɭɸɳɟɣ ɬɚɛɥɢɰɟ ɤɚɠɞɚɹ ɫɟɜɟɪɨ-ɜɨɫɬɨɱɧɚɹ ɞɢɚɝɨɧɚɥɶ ɨɬɜɟɱɚɟɬ ɨɩɪɟɞɟɥɟɧɧɵɦ ɫɭɦɦɚɪɧɵɦ ɢɧɜɟɫɬɢɰɢɹɦ ɜ ɩɟɪɜɨɟ ɢ ɜɬɨɪɨɟ ɩɪɟɞɩɪɢɹɬɢɹ ɇɚɩɪɢɦɟɪ ɟɫɥɢ ɧɚ ɩɟɪɜɵɟ ɞɜɚ ɩɪɟɞɩɪɢɹɬɢɹ ɜɵɞɟɥɟɧɚ ɫɭɦɦɚ ɦɥɧ ɪɭɛ ɬɨ ɟɫɬɶ ɟɞɢɧɫɬɜɟɧɧɵɣ ɫɩɨɫɨɛ ɪɚɡɞɟɥɢɬɶ ɷɬɭ ɫɭɦɦɭ ɦɟɠɞɭ ɩɪɟɞɩɪɢɹɬɢɹɦɢ ɩɟɪɜɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɜɵɞɟɥɢɬɶ ɦɥɧ ɪɭɛ ɢ ɬɨɝɞɚ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɩɟɪɜɨɦ ɩɪɟɞɩɪɢɹɬɢɢ ɛɭɞɟɬ ɪɚɜɟɧ ɦɥɧ ɪɭɛ ɢ ɜɬɨɪɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɜɵɞɟɥɢɬɶ ɦɥɧ ɪɭɛ ɢ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɜɬɨɪɨɦ ɩɪɟɞɩɪɢɹɬɢɢ ɬɚɤɠɟ ɛɭɞɟɬ ɪɚɜɟɧ ɦɥɧ ɪɭɛ Ɂɧɚɱɢɬ F2(0)= 0, x2 = 0.
Ⱦɚɥɟɟ ɟɫɥɢ ɧɚ ɩɟɪɜɵɟ ɞɜɚ ɩɪɟɞɩɪɢɹɬɢɹ ɜɵɞɟɥɟɧɚ ɫɭɦɦɚ ɦɥɧ ɪɭɛ ɬɨ ɟɫɬɶ ɞɜɚ ɫɩɨɫɨɛɚ ɪɚɡɞɟɥɢɬɶ ɷɬɭ ɫɭɦɦɭ ɦɟɠɞɭ ɩɪɟɞɩɪɢɹɬɢɹɦɢ
-ɩɟɪɜɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɜɵɞɟɥɢɬɶ ɦɥɧ ɪɭɛ ɢ ɬɨɝɞɚ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɩɟɪɜɨɦ ɩɪɟɞɩɪɢɹɬɢɢ ɛɭɞɟɬ ɪɚɜɟɧ ɦɥɧ ɪɭɛ ɚ ɜɬɨɪɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɜɵɞɟɥɢɬɶ ɦɥɧ ɪɭɛ ɢ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɜɬɨɪɨɦ ɩɪɟɞɩɪɢɹɬɢɢ ɬɚɤɠɟ ɛɭɞɟɬ ɪɚɜɟɧ ɦɥɧ ɪɭɛ Ɂɧɚɱɢɬ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɭɦɦɚɪɧɵɣ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɞɜɭɯ ɩɪɟɞɩɪɢɹɬɢɹɯ ɛɭɞɟɬ ɪɚɜɟɧɦɥɧ ɪɭɛ
-ɩɟɪɜɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɜɵɞɟɥɢɬɶ ɦɥɧ ɪɭɛ ɢ ɬɨɝɞɚ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɩɟɪɜɨɦ ɩɪɟɞɩɪɢɹɬɢɢ ɛɭɞɟɬ ɪɚɜɟɧ ɦɥɧ ɪɭɛ ɚ ɜɬɨɪɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɜɵɞɟɥɢɬɶ ɦɥɧ ɪɭɛ ɢ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɜɬɨɪɨɦ ɩɪɟɞɩɪɢɹɬɢɢ ɛɭɞɟɬ ɪɚɜɟɧ ɦɥɧ ɪɭɛ Ɂɧɚɱɢɬ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɭɦɦɚɪɧɵɣ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɞɜɭɯ ɩɪɟɞɩɪɢɹɬɢɹɯ ɛɭɞɟɬ ɦɥɧ ɪɭɛ
54
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɦɨɠɧɨ ɨɛɟɫɩɟɱɢɬɶ ɦɚɤɫɢɦɚɥɶɧɵɣ ɫɭɦɦɚɪɧɵɣ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɧɚ ɞɜɭɯ ɩɪɟɞɩɪɢɹɬɢɹɯ F2 ɞɥɹ ɷɬɨɝɨ ɜɬɨɪɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɧɟɨɛɯɨɞɢɦɨ ɜɵɞɟɥɢɬɶ x2* (100) = 0.
ɂɬɚɤ ɜ ɷɬɨɣ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɬɚɛɥɢɰɟ ɦɵ ɡɧɚɱɟɧɢɹ f2(x) ɫɤɥɚɞɵɜɚɟɦ ɫɨ ɡɧɚɱɟɧɢɹɦɢ F1(x-x2)=f1(x-x2 ɢ ɧɚ ɤɚɠɞɨɣ ɫɟɜɟɪɨɜɨɫɬɨɱɧɨɣ ɞɢɚɝɨɧɚɥɢ ɧɚɯɨɞɢɦ ɧɚɢɛɨɥɶɲɟɟ ɱɢɫɥɨ ɤɨɬɨɪɨɟ ɜɵɞɟɥɹɟɦ ɠɢɪɧɵɦ ɢ ɭɤɚɡɵɜɚɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɡɧɚɱɟɧɢɹ x2* ɯ
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84 |
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300 |
45 |
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45 |
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65 |
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79 |
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91 |
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98 |
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400 |
62 |
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62 |
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82 |
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96 |
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108 |
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500 |
78 |
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78 |
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98 |
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112 |
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600 |
90 |
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90 |
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110 |
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700 |
98 |
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98 |
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ɉɨɫɥɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɩɨɥɧɹɟɦ ɨɫɧɨɜɧɭɸ ɬɚɛɥɢɰɭ ɞɥɹ F2(x) |
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ɢ x2* ɯ |
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Ɍɚɛɥ |
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ɏ |
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0 |
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100 |
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200 |
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300 |
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400 |
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500 |
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600 |
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700 |
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F2(x) |
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0 |
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20 |
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38 |
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52 |
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65 |
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82 |
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98 |
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112 |
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x2* ɯ |
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0 |
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0 |
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100 |
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100 |
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300 |
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400 |
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500 |
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500 |
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ɉɪɨɞɨɥɠɚɹ ɩɪɨɰɟɫɫ ɬɚɛɭɥɢɪɭɟɦ ɮɭɧɤɰɢɢ F3(x), x3* ɯ ɢ ɬ ɞ Ɍɚɛɥ |
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ɯ3 ɯ- |
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0 |
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100 |
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200 |
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400 |
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500 |
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700 |
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ɯ3 |
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f3(x3)/ |
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0 |
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20 |
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38 |
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52 |
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65 |
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82 |
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98 |
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112 |
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F2(x-x3) |
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0 |
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0 |
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0 |
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20 |
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38 |
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52 |
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65 |
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82 |
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98 |
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112 |
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100 |
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25 |
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25 |
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45 |
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63 |
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77 |
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90 |
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107 |
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123 |
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200 |
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41 |
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41 |
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61 |
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79 |
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93 |
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106 |
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123 |
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300 |
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52 |
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52 |
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72 |
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90 |
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104 |
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117 |
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400 |
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74 |
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74 |
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94 |
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112 |
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126 |
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500 |
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82 |
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82 |
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102 |
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120 |
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600 |
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88 |
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88 |
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108 |
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700 |
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90 |
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90 |
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55
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Ɍɚɛɥ |
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ɏ |
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0 |
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100 |
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200 |
300 |
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400 |
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500 |
600 |
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700 |
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F3(x) |
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0 |
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25 |
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45 |
63 |
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79 |
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94 |
112 |
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126 |
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x3* ɯ |
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0 |
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100 |
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100 |
100 |
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200 |
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400 |
400 |
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400 |
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Ɍɚɛɥ |
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x4 ɯ- |
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0 |
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100 |
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200 |
300 |
400 |
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500 |
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600 |
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700 |
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ɯ4 |
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f4(x4)/ |
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0 |
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25 |
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45 |
63 |
79 |
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94 |
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112 |
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126 |
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F3(x-x4) |
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0 |
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0 |
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126 |
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100 |
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30 |
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142 |
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200 |
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52 |
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146 |
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300 |
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76 |
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155 |
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400 |
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90 |
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153 |
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500 |
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104 |
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149 |
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600 |
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116 |
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141 |
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700 |
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125 |
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125 |
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ȼ |
ɩɨɫɥɟɞɧɟɣ ɬɚɛɥɢɰɟ ɡɚɩɨɥɧɹɟɦ ɬɨɥɶɤɨ ɨɞɧɭ ɞɢɚɝɨɧɚɥɶ ɞɥɹ |
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ɡɧɚɱɟɧɢɹ ɯ ɇɚɢɛɨɥɶɲɟɟ ɱɢɫɥɨ ɧɚ ɷɬɨɣ ɞɢɚɝɨɧɚɥɢ |
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|
z* |
ɦɥɧ ɪɭɛ |
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ɩɪɢɱɟɦ ɱɟɬɜɟɪɬɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɞɨɥɠɧɨ ɛɵɬɶ ɜɵɞɟɥɟɧɨ x4* = x4* ɦɥɧ ɪɭɛ
ɧɚ ɞɨɥɸ ɨɫɬɚɥɶɧɵɯ ɬɪɟɯ ɩɪɟɞɩɪɢɹɬɢɣ ɨɫɬɚɟɬɫɹ ɦɥɧ ɪɭɛ ɂɡ ɬɚɛɥ ɜɢɞɧɨ ɱɬɨ ɬɪɟɬɶɟɦɭ ɩɪɟɞɩɪɢɹɬɢɸ ɞɨɥɠɧɨ ɛɵɬɶ ɜɵɞɟɥɟɧɨ
x3* = x3*(700-ɯ4* ɯ3* ɦɥɧ ɪɭɛ
ɩɪɨɞɨɥɠɚɹ ɨɛɪɚɬɧɵɣ ɩɪɨɰɟɫɫ ɧɚɯɨɞɢɦ ɯ2* ɯ2*(700- ɯ4*- ɯ3* ɯ2* ɦɥɧ ɪɭɛ ɧɚ ɞɨɥɸ ɩɟɪɜɨɝɨ ɩɪɟɞɩɪɢɹɬɢɹ ɨɫɬɚɟɬɫɹ ɯ1* = 700- ɯ4*- ɯ3*- ɯ2* ɦɥɧ ɪɭɛ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɧɚɢɥɭɱɲɢɦ ɹɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɟɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ
ɤɚɩɢɬɚɥɶɧɵɯ ɜɥɨɠɟɧɢɣ ɩɨ ɩɪɟɞɩɪɢɹɬɢɹɦ ɯ1* ɯ2* ɯ3* ɯ4*=300.
Ɉɧɨ ɨɛɟɫɩɟɱɢɜɚɟɬ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɨɦɭ ɨɛɴɟɞɢɧɟɧɢɸ ɧɚɢɛɨɥɶɲɢɣ ɜɨɡɦɨɠɧɵɣ ɩɪɢɪɨɫɬ ɩɪɢɛɵɥɢ ɦɥɧ ɪɭɛ
Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ ɢ ɩɪɨɰɟɫɫɨɜ ɜ ɪɚɦɤɚɯ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɹɜɥɹɟɬɫɹ ɜɚɠɧɵɦ ɢɧɫɬɪɭɦɟɧɬɨɦ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ Ɉɧɨ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɩɨɥɭɱɢɬɶ ɱɟɬɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨɛ ɢɫɫɥɟɞɭɟɦɨɦ ɨɛɴɟɤɬɟ ɨɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɢ ɤɨɥɢɱɟɫɬɜɟɧɧɨ ɨɩɢɫɚɬɶ ɟɝɨ ɜɧɭɬɪɟɧɧɸɸ ɫɬɪɭɤɬɭɪɭ ɢ ɜɧɟɲɧɢɟ ɫɜɹɡɢ
56
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ ɐɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɩɨɬɪɟɛɥɟɧɢɹ ɉɨɫɬɪɨɟɧɢɟ Ɏɭɧɤɰɢɢ ɫɩɪɨɫɚ
ɐɟɥɶ ɢɫɩɨɥɶɡɭɹ ɦɟɬɨɞɵ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɫ ɩɨɦɨɳɶɸ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɩɨɬɪɟɛɥɟɧɢɹ ɧɚɭɱɢɬɶɫɹ ɧɚɯɨɞɢɬɶ ɨɩɬɢɦɚɥɶɧɵɣ ɧɚɛɨɪ ɛɥɚɝ ɩɨɬɪɟɛɢɬɟɥɹ ɮɭɧɤɰɢɢ ɫɩɪɨɫɚ ɧɚ ɛɥɚɝɚ ɩɨ ɰɟɧɟ ɮɭɧɤɰɢɢ ɫɩɪɨɫɚ ɩɨ ɞɨɯɨɞɭ ɫ ɩɨɦɨɳɶɸ ɗȼɆ
Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɨɝɨ ɩɨɬɪɟɛɢɬɟɥɹ ɤɨɬɨɪɵɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɜɨɟɝɨ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɩɨɬɪɟɛɥɹɟɬ ɧɟɤɨɬɨɪɵɟ ɛɥɚɝɚ ɍɪɨɜɟɧɶ ɭɞɨɜɥɟɬɜɨɪɟɧɢɹ ɩɨɬɪɟɛɧɨɫɬɟɣ ɩɨɬɪɟɛɢɬɟɥɹ ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ U. ɉɪɟɞɩɨɥɨɠɢɦ ɱɬɨ ɢɦɟɟɬɫɹ n ɜɢɞɨɜ ɛɥɚɝ Ȼ1 Ȼ2 Ȼn ɉɭɫɬɶ ɤɨɥɢɱɟɫɬɜɨ ɩɨɬɪɟɛɥɟɧɢɹ ɤɚɠɞɨɝɨ ɛɥɚɝɚ ɪɚɜɧɨ ɯ1, ɯ2 ,…, ɯn ɐɟɥɟɜɨɣ ɮɭɧɤɰɢɟɣ ɩɨɬɪɟɛɥɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɡɚɜɢɫɢɦɨɫɬɶ U U (x1, x2 ,..., xn ) Ʉɚɠɞɵɣ
ɩɨɬɪɟɛɢɬɟɥɶ ɫɬɪɟɦɢɬɫɹ ɦɚɤɫɢɦɢɡɢɪɨɜɚɬɶ ɭɪɨɜɟɧɶ ɭɞɨɜɥɟɬɜɨɪɟɧɢɹ ɩɨɬɪɟɛɧɨɫɬɟɣ ɬɨ ɟɫɬɶ U max Ɉɛɨɡɧɚɱɢɦ ɰɟɧɭ ɧɚ ɟɞɢɧɢɰɭ ɤɚɠɞɨɝɨ ɛɥɚɝɚ ɱɟɪɟɡ ɪ1, ɪ2 ,…, ɪn ɚ ɞɨɯɨɞ ɩɨɬɪɟɛɢɬɟɥɹ ɱɟɪɟɡ D Ɍɨɝɞɚ ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɛɸɞɠɟɬɧɨɟ ɨɝɪɚɧɢɱɟɧɢɟ p1x1 p2 x2 ... pn xn D ȼ
ɪɟɡɭɥɶɬɚɬɟ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɨɩɬɢɦɚɥɶɧɨɝɨ ɧɚɛɨɪɚ ɛɥɚɝ ɧɟɨɛɯɨɞɢɦɨ ɪɟɲɚɬɶ ɡɚɞɚɱɭ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ
U (x1, x2 ,..., xn ) max;
p1x1 p2 x2 ... pn xn D;
xi 0, (i 1,2,...,n).
Ɋɚɫɫɦɨɬɪɢɦ ɦɟɬɨɞɵ ɟɟ ɪɟɲɟɧɢɹ ɧɚ ɩɪɢɦɟɪɟ ɉɊɂɆȿɊ ɉɭɫɬɶ ɱɢɫɥɨ ɛɥɚɝ ɪɚɜɧɨ ɬɪɟɦ ɚ ɮɭɧɤɰɢɹ ɩɨɬɪɟɛɥɟɧɢɹ
ɪɚɜɧɚ U (x1, x2 , x3) x1 x2 x3 ɉɪɟɞɩɨɥɨɠɢɦ ɱɬɨ ɰɟɧɚ ɧɚ ɟɞɢɧɢɰɭ
ɩɟɪɜɨɝɨ ɛɥɚɝɚ ɪɚɜɧɚ ɜɬɨɪɨɝɨ ɢ ɬɪɟɬɶɟɝɨ ɚ ɞɨɯɨɞ ɩɨɬɪɟɛɢɬɟɥɹ ɫɨɫɬɚɜɥɹɟɬ Ɍɨɝɞɚ ɡɚɞɚɱɚ ɩɪɢɦɟɬ ɜɢɞ
x1 x2 x3 max;
15x1 10x2 15x3 500;.
x1,2,3 0.
ɉɨɞɝɨɬɨɜɢɦ ɞɚɧɧɵɟ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜ Excel ɫɨɝɥɚɫɧɨ ɪɢɫɭɧɤɭ
57
ȼɜɨɞɢɦ ɜ ɹɱɟɣɤɭ ȼ © ɄɈɊȿɇɖ % & ' ª ɤɚɜɵɱɤɢ ɧɟ ɜɜɨɞɢɬɶ ɚ ɜ ȼ © % % & & ' ' ª Ɂɚɩɭɫɤɚɟɦ ɋȿɊȼɂɋ ɉɈɂɋɄ Ɋȿɒȿɇɂə (Solver Add – in ȼ ɹɱɟɣɤɭ ©ɍɫɬɚɧɨɜɢɬɶ ɰɟɥɟɜɭɸª Set Target Cell ɭɫɬɚɧɚɜɥɢɜɚɟɦ ɫɫɵɥɤɭ ɧɚ ȼ ɩɪɨɜɟɪɢɬɶ ɱɬɨ ɮɥɚɠɨɤ ɧɢɠɟ ɩɨɥɹ ɫɬɨɢɬ ɧɚɩɪɨɬɢɜ ɧɚɞɩɢɫɢ ©Ɋɚɜɧɨɣ ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɸª Equal to … Max … Value of ɉɨɫɥɟ ɫɬɚɜɢɦ ɤɭɪɫɨɪ ɜ ɩɨɥɟ ©ɂɡɦɟɧɹɹ ɹɱɟɣɤɢª By Changing Cell ɢ ɨɛɜɨɞɢɦ ɹɱɟɣɤɢ ɫ ɩɟɪɟɦɟɧɧɵɦɢ ȼ ɋ ɢ D Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɜɜɟɫɬɢ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚɠɢɜɚɸɬ ɤɧɨɩɤɭ ©Ⱦɨɛɚɜɢɬɶª Add ɨɬɤɪɨɟɬɫɹ ɨɤɧɨ ©Ⱦɨɛɚɜɥɟɧɢɟ ɨɝɪɚɧɢɱɟɧɢɹª Add Constraints ȼ ɥɟɜɨɦ ɩɨɥɟ ©ɋɫɵɥɤɚ ɧɚ ɹɱɟɣɤɭª (Cell Reference ɜɜɨɞɹɬ ɫɫɵɥɤɭ ɧɚ ɥɟɜɭɸ ɱɚɫɬɶ ɩɟɪɜɨɝɨ ɨɝɪɚɧɢɱɟɧɢɹ – ɹɱɟɣɤɭ ȼ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɨɤɧɟ ɨɩɪɟɞɟɥɹɟɦ ɡɧɚɤ ɢ ɜ ɩɪɚɜɨɦ ©Ɉɝɪɚɧɢɱɟɧɢɹª Constraints ɞɟɥɚɟɦ ɫɫɵɥɤɭ ɧɚ ɞɨɯɨɞ D Ⱦɥɹ ɜɜɨɞɚ ɜɬɨɪɨɝɨ ɨɝɪɚɧɢɱɟɧɢɹ ɜɧɨɜɶ ɧɚɠɢɦɚɟɦ ©Ⱦɨɛɚɜɢɬɶª Add ɫɬɚɜɢɦ ɤɭɪɫɨɪ ɜ ɥɟɜɨɟ ɩɨɥɟ ɢ ɨɛɜɨɞɢɦ ɹɱɟɣɤɢ ȼ ɋ ɢ D ɜ ɫɪɟɞɧɟɦ ɨɤɧɟ ɫɬɚɜɢɦ © ª ɢ ɜ ɩɪɚɜɨɦ ɱɢɫɥɨ ɇɚɠɢɦɚɟɦ ©ȼɵɩɨɥɧɢɬɶª Solve), ɩɨɞɬɜɟɪɠɞɚɟɦ ɪɟɡɭɥɶɬɚɬɵ ɜɵɛɢɪɚɹ ©ɋɨɯɪɚɧɢɬɶ ɧɚɣɞɟɧɧɨɟ ɪɟɲɟɧɢɟª
(Keep |
Solver |
Solution |
ɢ ©ɈɄª ɩɨɥɭɱɚɟɦ ɪɟɡɭɥɶɬɚɬ |
x1 11,1; |
x2 16,7; |
x3 11,1; |
ɰɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɪɚɜɧɚ |
Ɋɟɲɢɦ ɬɟɩɟɪɶ ɡɚɞɚɱɭ ɧɚɯɨɠɞɟɧɢɹ ɮɭɧɤɰɢɢ ɫɩɪɨɫɚ ɩɨ ɰɟɧɟ ɇɚɣɞɟɦ ɧɚɩɪɢɦɟɪ ɫɩɪɨɫ ɧɚ ɜɬɨɪɨɟ ɛɥɚɝɨ ɞɥɹ ɪɚɡɧɵɯ ɰɟɧ ɧɚ ɟɞɢɧɢɰɭ ɷɬɨɝɨ ɛɥɚɝɚ Ȼɭɞɟɦ ɡɚɞɚɜɚɬɶ ɰɟɧɭ ɧɚ ɜɬɨɪɨɟ ɛɥɚɝɨ ɨɬ ɞɨ ɢ ɮɢɤɫɢɪɨɜɚɬɶ ɫɩɪɨɫ x2 ɩɪɢ ɷɬɢɯ ɰɟɧɚɯ ȼɜɟɞɟɦ ɜ ɫɬɨɥɛɟɰ F ɰɟɧɭ ɛɥɚɝɚ
ɚ ɜ ɫɬɨɥɛɟɰ G ɫɩɪɨɫ ɧɚ ɧɟɝɨ ɋɬɚɜɢɦ ɤɭɪɫɨɪ ɜ F1 ɢ ɜɜɨɞɢɦ ɩɨɞɩɢɫɶ ©ɐɟɧɚª ɚ ɜ ɹɱɟɣɤɭ G1 ɜɜɨɞɢɦ ɩɨɞɩɢɫɶ ©ɋɩɪɨɫª ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɰɟɧɚ ɜɬɨɪɨɝɨ ɛɥɚɝɚ ɫɨɫɬɚɜɥɹɟɬ ɞɟɧɟɠɧɵɯ ɟɞɢɧɢɰ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ ɫɩɪɨɫ ɧɚ ɷɬɨ ɛɥɚɝɨ ɫɨɫɬɚɜɥɹɟɬ x2 16,7 ȼɜɨɞɢɦ
ɜ ɹɱɟɣɤɭ F7 ɡɧɚɱɟɧɢɟ ɰɟɧɵ ɚ ɜ ɫɨɫɟɞɧɸɸ G7 - ɫɩɪɨɫ Ɋɚɫɫɱɢɬɚɟɦ ɬɟɩɟɪɶ ɫɩɪɨɫ ɩɪɢ ɰɟɧɟ ɂɫɩɪɚɜɥɹɟɦ ɜ ɋ ɡɧɚɱɟɧɢɟ ɧɚ ɜɵɡɵɜɚɟɦ ɋȿɊȼɂɋ ɉɈɂɋɄ Ɋȿɒȿɇɂə (Solver Add – in) ɧɚɠɢɦɚɟɦ ©ȼɵɩɨɥɧɢɬɶª Solve ɩɨɞɬɜɟɪɠɞɚɟɦ ɪɟɡɭɥɶɬɚɬɵ ȼɢɞɢɦ ɜ ɹɱɟɣɤɟ ɋ ɧɨɜɨɟ ɡɧɚɱɟɧɢɟ ɫɩɪɨɫɚ - x2 15,2 ȼɜɨɞɢɦ ɜ F8 ɜɪɭɱɧɭɸ ɱɢɫɥɨ ɜ
G8 ɱɢɫɥɨ Ɍɨɱɧɨ ɬɚɤɠɟ ɨɛɹɡɚɬɟɥɶɧɨ ɩɪɨɞɟɥɚɬɶ ɧɚ ɗȼɆ
58
ɢɡɦɟɧɹɟɦ ɜ ɋ ɡɧɚɱɟɧɢɹ ɧɚ ɢ ɡɚɩɢɫɚɜ ɷɬɢ ɠɟ ɡɧɚɱɟɧɢɹ ɜ F9-F12 ɤɚɠɞɵɣ ɪɚɡ ɡɚɩɭɫɤɚɟɦ ɧɚɞɫɬɪɨɣɤɭ ©ɉɨɢɫɤ ɪɟɲɟɧɢɹª ɩɨɥɭɱɚɟɦ ɧɨɜɵɟ ɪɟɡɭɥɶɬɚɬɵ ɜ ɋ ɡɚɩɢɫɵɜɚɟɦ ɢɯ ɜɪɭɱɧɭɸ ɧɟ ɤɨɩɢɪɨɜɚɧɢɟɦ ɨɤɪɭɝɥɹɹ ɞɨ ɞɟɫɹɬɵɯ ɜ G9-G Ⱦɚɥɟɟ ɪɚɫɫɱɢɬɵɜɚɟɦ ɡɧɚɱɟɧɢɹ ɫɩɪɨɫɚ ɞɥɹ ɰɟɧɵ ɦɟɧɶɲɟɣ ɟɞɢɧɢɰ Ⱦɥɹ ɷɬɨɝɨ ɢɡɦɟɧɹɟɦ ɜ ɋ ɡɧɚɱɟɧɢɹ ɧɚɢ ɡɚɩɢɫɚɜ ɷɬɢ ɡɧɚɱɟɧɢɹ ɜ F2-F ɤɚɠɞɵɣ ɪɚɡ ɡɚɩɭɫɤɚɟɦ ɧɚɞɫɬɪɨɣɤɭ ©ɉɨɢɫɤ ɪɟɲɟɧɢɹª ɩɨɥɭɱɚɟɦ ɧɨɜɵɟ ɪɟɡɭɥɶɬɚɬɵ ɜ ɋ ɡɚɩɢɫɵɜɚɟɦ ɢɯ ɜ G2-G ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɪɢ ɩɪɚɜɢɥɶɧɨɦ ɜɵɩɨɥɧɟɧɢɢ ɜɫɟɯ ɞɟɣɫɬɜɢɣ ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɢɟ ɪɟɡɭɥɶɬɚɬɵ
əɱɟɣɤɚ |
F2 |
F3 |
F4 |
F5 |
F6 |
F7 |
F8 |
F9 |
F10 |
F11 |
F12 |
Ɂɧɚɱɟɧɢɟ |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
əɱɟɣɤɚ |
G2 |
G3 |
G4 |
G5 |
G6 |
G7 |
G8 |
G9 |
G10 |
G11 |
G12 |
Ɂɧɚɱɟɧɢɟ |
33,3 |
27,8 |
23,8 |
20,8 |
18,5 |
16,7 |
15,2 |
13,9 |
12,8 |
11,9 |
11,1 |
ɉɨɫɬɪɨɢɦ ɩɨ ɩɨɥɭɱɟɧɧɵɦ ɞɚɧɧɵɦ ɮɭɧɤɰɢɸ ɫɩɪɨɫɚ Ⱦɥɹ ɷɬɨɝɨ ɫɬɚɜɢɦ ɤɭɪɫɨɪ ɜ ɥɸɛɭɸ ɫɜɨɛɨɞɧɭɸ ɹɱɟɣɤɭ ɜɵɡɵɜɚɟɦ ɦɚɫɬɟɪ ɞɢɚɝɪɚɦɦ (ȼɋɌȺȼɄȺ ȾɂȺȽɊȺɆɆȺ ɜɵɛɢɪɚɟɦ ɬɢɩ ɞɢɚɝɪɚɦɦɵ ©Ƚɪɚɮɢɤª ɜɢɞ ©Ƚɪɚɮɢɤ ɫ ɦɚɪɤɟɪɚɦɢª ɥɟɜɵɣ ɜɬɨɪɨɣ ɫɜɟɪɯɭ ɧɚɠɢɦɚɟɦ ©Ⱦɚɥɟɟª ɋɬɚɜɢɦ ɤɭɪɫɨɪ ɜ ɩɨɥɟ ©Ⱦɢɚɩɚɡɨɧª ɢ ɨɛɜɨɞɢɦ ɹɱɟɣɤɢ G2-G12. ɉɟɪɟɯɨɞɢɦ ɧɚ ɡɚɤɥɚɞɤɭ ©Ɋɹɞª ɢ ɫɬɚɜɢɦ ɤɭɪɫɨɪ ɜ ɩɨɥɟ ©ɉɨɞɩɢɫɢ ɨɫɢ ɏª ɨɛɜɨɞɢɦ ɹɱɟɣɤɢ F2-F12 ɧɚɠɢɦɚɟɦ ©Ƚɨɬɨɜɨª ɉɨɥɭɱɚɟɦ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɫɩɪɨɫɚ ɩɨ ɰɟɧɟ Ɍɨɱɧɨ ɬɚɤɠɟ ɦɨɠɧɨ ɢɫɫɥɟɞɨɜɚɬɶ ɫɩɪɨɫ ɢ ɧɚ ɩɟɪɜɨɟ ɢ ɬɪɟɬɶɟ ɛɥɚɝɨ
ɇɚɣɞɟɦ ɬɟɩɟɪɶ ɮɭɧɤɰɢɸ ɫɩɪɨɫɚ ɩɨ ɞɨɯɨɞɭ ɧɚ ɜɬɨɪɨɟ ɛɥɚɝɨ Ⱦɥɹ ɷɬɨɝɨ ɛɭɞɟɦ ɦɟɧɹɬɶ ɞɨɯɨɞ ɜ ɞɢɚɩɚɡɨɧɟ - ɱɟɪɟɡ ɟɞɢɧɢɰ ɮɢɤɫɢɪɭɹ ɫɩɪɨɫ ɜ ɹɱɟɣɤɟ ɋ ȼɜɨɞɢɦ ɜ H1 ɩɨɞɩɢɫɶ ©Ⱦɨɯɨɞª ɚ ɜ I1 ɩɨɞɩɢɫɶ ©ɋɩɪɨɫª ɂɫɩɪɚɜɥɹɟɦ ɜ ɋ ɰɟɧɭ ɧɚ ɚ ɜ D4 ɫɬɚɜɢɦ ɞɨɯɨɞȼɵɡɵɜɚɟɦ ɢ ɡɚɩɭɫɤɚɟɦ ɧɚɞɫɬɪɨɣɤɭ ɉɈɂɋɄ Ɋȿɒȿɇɂə (Solver Add – in) ȼɢɞɢɦ ɱɬɨ ɫɩɪɨɫ ɧɚ ɜɬɨɪɨɟ ɛɥɚɝɨ ɪɚɜɟɧ ȼɜɨɞɢɦ ɜ H2 ɡɧɚɱɟɧɢɟ ɞɨɯɨɞɚ ɚ ɫɩɪɨɫ ɜɜɨɞɢɦ ɜ I Ⱦɚɥɟɟ ɩɨ ɚɧɚɥɨɝɢɢ ɢɡɦɟɧɹɟɦ ɜ D ɞɨɯɨɞ ɧɚ ɡɚɧɨɫɹ ɷɬɢ ɞɚɧɧɵɟ ɜ H3-H10 ɤɚɠɞɵɣ ɪɚɡ ɡɚɩɭɫɤɚɟɦ ɧɚɞɫɬɪɨɣɤɭ ɉɈɂɋɄ Ɋȿɒȿɇɂə (Solver Add – in) ɩɨɥɭɱɟɧɧɵɣ ɜ ɋ ɫɩɪɨɫ ɜɧɨɫɢɦ ɜ ɹɱɟɣɤɢ I3-I10 ɉɪɢ ɩɪɚɜɢɥɶɧɨɦ ɪɚɫɱɟɬɟ ɪɟɡɭɥɶɬɚɬɵ ɛɭɞɭɬɉɨ ɩɨɥɭɱɟɧɧɵɦ ɞɚɧɧɵɦ ɬɚɤɠɟ ɤɚɤ ɢ ɞɥɹ ɮɭɧɤɰɢɢ ɫɩɪɨɫɚ ɩɨ ɰɟɧɟ ɫɬɪɨɢɦ ɝɪɚɮɢɤ ȼɢɞɧɨ ɱɬɨ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɝɪɚɮɢɤ ɫɩɪɨɫɚ ɩɨ ɞɨɯɨɞɭ ɩɪɹɦɚɹ ɥɢɧɢɹ
ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ ɱɬɨ ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɮɭɧɤɰɢɸ ɩɟɪɟɤɪɟɫɬɧɨɝɨ ɫɩɪɨɫɚ ɧɚ ɨɞɧɨ ɛɥɚɝɨ ɩɨ ɰɟɧɟ ɧɚ ɞɪɭɝɨɟ
59
Ɂɚɞɚɧɢɟ ɑɟɬɵɪɟɯɮɚɤɬɨɪɧɭɸ ɰɟɥɟɜɭɸ ɮɭɧɤɰɢɸ ɩɨɬɪɟɛɥɟɧɢɹ U U (x1,x2,x3,x4) ɰɟɧɵ ɧɚ ɛɥɚɝɚ p1, p2, p3, p4 ɢ ɞɨɯɨɞ D ɜɡɹɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɚɪɢɚɧɬɨɦ ɢɡ ɬɚɛɥɢɰɵ
ɋɨɫɬɚɜɢɜ ɢ ɪɟɲɢɜ ɡɚɞɚɱɭ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɧɚɣɬɢ ɨɩɬɢɦɚɥɶɧɵɣ ɧɚɛɨɪ ɛɥɚɝ
ɋɨɫɬɚɜɢɬɶ ɮɭɧɤɰɢɸ ɫɩɪɨɫɚ ɧɚ ɜɬɨɪɨɟ ɛɥɚɝɨ ɨɬ ɟɝɨ ɰɟɧɵ ɜɡɹɜ ɰɟɥɵɯ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɰɟɧɵ ɞɨ ɢ ɩɨɫɥɟ ɬɨɣ ɤɚɤɚɹ ɭɤɚɡɚɧɚ ɜ ɬɚɛɥɢɰɟ
ɋɨɫɬɚɜɢɬɶ ɮɭɧɤɰɢɸ ɫɩɪɨɫɚ ɧɚ ɬɪɟɬɶɟ ɛɥɚɝɨ ɩɨ ɞɨɯɨɞɭ ɜɡɹɜ ɩɨ ɱɟɬɵɪɟ ɡɧɚɱɟɧɢɹ ɞɨɯɨɞɚ ɞɨ ɢ ɩɨɫɥɟ ɭɤɚɡɚɧɧɨɣ ɜ ɬɚɛɥɢɰɟ ɫ ɲɚɝɨɦ
ȼɚɪ |
U (x1,x2,x3,x4) |
p1 |
p2 |
p3 |
p4 |
D |
ȼɚɪ |
U (x1 |
,x2,x3,x4) |
p1 |
p2 |
p3 |
p4 |
D |
||||||||||||
1. |
ln(x1 x2 x3 x4) |
15 |
10 |
14 |
15 |
400 |
7. |
x1 |
|
|
x2 x3 x4 |
20 |
10 |
8 |
17 |
950 |
||||||||||
2. |
x (x |
2 |
7)x x |
4 |
21 |
7 |
15 |
13 |
400 |
8. |
x x |
2 |
x x |
4 |
14 |
12 |
8 |
9 |
500 |
|||||||
|
1 |
|
|
3 |
|
1 |
|
|
3 |
|
||||||||||||||||
3. |
3 x1 x2 |
x3 x4 |
|
|
17 |
19 |
16 |
21 |
350 |
9. |
ln(x1 x2 x3 x4) |
9 |
17 |
11 |
10 |
800 |
||||||||||
4. |
x1 |
x2 |
x3 x4 |
|
|
15 |
10 |
9 |
15 |
500 |
10. |
x1(x2 1)x3x4 |
15 |
10 |
9 |
8 |
600 |
|||||||||
5. |
x1x2x3(x4 3) |
|
16 |
18 |
11 |
21 |
500 |
11. |
x1x2 |
(x3 5)x4 |
11 |
19 |
13 |
11 |
650 |
|||||||||||
6. |
x |
x |
2 |
|
x x |
4 |
17 |
14 |
18 |
20 |
650 |
12. |
x x |
2 |
|
x |
x |
4 |
8 |
16 |
14 |
20 |
700 |
|||
|
1 |
|
|
|
3 |
|
|
1 |
|
|
3 |
|
||||||||||||||
ɉɊɂɆȿɑȺɇɂȿ ȼ ɹɱɟɣɤɟ ɫ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɟɣ ȼ ɞɨɥɠɧɚ ɫɨɞɟɪɠɚɬɶɫɹ ɮɭɧɤɰɢɹ ɜɢɞɚ
1. |
=LN(B3*C3*D3*E3) |
7. |
ɄɈɊȿɇɖ ɋ D ȼ ȿ |
2. |
ɄɈɊȿɇɖ B3*(C3+7)*D3*E3) |
8. |
ɄɈɊȿɇɖ B ɋ D ȿ |
3. |
ɋɌȿɉȿɇɖ B3*C3*D3*E3; 0,33) |
9. |
=LN(B3*C3*D3*E3) |
4. |
ɄɈɊȿɇɖ B3)*C3*D3*E3 |
10. |
=B3*(C3+1)*D3*E3 |
5. |
=B3*C3*D3*(E3+3) |
11. |
=B3*C3*(D3+5)*E3 |
6. |
ɄɈɊȿɇɖ B3*D ɋ ȿ |
12. |
ɄɈɊȿɇɖ D3)*B3*C3*E3 |
Ɉɬɱɟɬ ɞɨɥɠɟɧ ɫɨɞɟɪɠɚɬɶ ɨɩɬɢɦɚɥɶɧɵɣ ɧɚɛɨɪ ɛɥɚɝ x1, x2 , ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɫɩɪɨɫɚ ɧɚ ɜɬɨɪɨɟ ɛɥɚɝɨ ɨɬ ɟɝɨ ɰɟɧɵ x2(p2) ɢ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ ɫɩɪɨɫɚ ɧɚ ɬɪɟɬɶɟ ɛɥɚɝɨ ɩɨ ɞɨɯɨɞɭ x3(D) .
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