ɎȿȾȿɊȺɅɖɇɈȿ ȺȽȿɇɌɋɌȼɈ ɉɈ ɈȻɊȺɁɈȼȺɇɂɘ ȽɈɋɍȾȺɊɋɌȼȿɇɇɈȿ ɈȻɊȺɁɈȼȺɌȿɅɖɇɈȿ ɍɑɊȿɀȾȿɇɂȿ ȼɕɋɒȿȽɈ ɉɊɈɎȿɋɋɂɈɇȺɅɖɇɈȽɈ ɈȻɊȺɁɈȼȺɇɂə «ȼɈɊɈɇȿɀɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɍɇɂȼȿɊɋɂɌȿɌ»
ȾɂɋɄɊȿɌɇȺə ɈɉɌɂɆɂɁȺɐɂə Ɇɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɤ ɤɭɪɫɚɦ «Ɇɨɞɟɥɢ ɢ ɦɟɬɨɞɵ ɞɢɫɤɪɟɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ», «ɂɫɫɥɟɞɨɜɚɧɢɟ ɨɩɟɪɚɰɢɣ» 3-ɟ ɢɡɞɚɧɢɟ, ɩɟɪɟɪɚɛɨɬɚɧɧɨɟ ɢ ɞɨɩɨɥɧɟɧɧɨɟ
ɋɨɫɬɚɜɢɬɟɥɢ: Ƚ.Ⱦ. ɑɟɪɧɵɲɨɜɚ, ɂ.ɇ. Ȼɭɥɝɚɤɨɜɚ
ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ 2007
ɍɬɜɟɪɠɞɟɧɨ ɧɚɭɱɧɨ-ɦɟɬɨɞɢɱɟɫɤɢɦ ɫɨɜɟɬɨɦ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ ȼȽɍ 25 ɫɟɧɬɹɛɪɹ 2007 ɝ., ɩɪɨɬɨɤɨɥ ʋ 1
Ɋɟɰɟɧɡɟɧɬ ɤɚɧɞ. ɮɢɡ.-ɦɚɬ. ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮ. ɧɟɥɢɧɟɣɧɵɯ ɤɨɥɟɛɚɧɢɣ ȼȽɍ Ɍ.ɂ. ɋɦɚɝɢɧɚ
ȼ ɩɨɫɨɛɢɢ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɨɫɧɨɜɧɨɣ ɤɪɭɝ ɛɚɡɨɜɵɯ ɡɚɞɚɱ ɞɢɫɤɪɟɬɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ. ɂɡɥɨɠɟɧɵ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɤɨɦɦɢɜɨɹɠɟɪɚ, ɨ ɧɚɡɧɚɱɟɧɢɹɯ, ɨɛɳɟɣ ɡɚɞɚɱɢ ɰɟɥɨɱɢɫɥɟɧɧɨɝɨ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ. ɉɪɢɦɟɧɟɧɢɟ ɤɚɠɞɨɝɨ ɦɟɬɨɞɚ ɢɥɥɸɫɬɪɢɪɭɟɬɫɹ ɪɟɲɟɧɢɹɦɢ ɬɢɩɨɜɵɯ ɩɪɢɦɟɪɨɜ. ɉɪɢɜɟɞɟɧɵ ɡɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ. ɉɨɫɨɛɢɟ ɩɨɞɝɨɬɨɜɥɟɧɨ ɧɚ ɤɚɮɟɞɪɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ ɢɫɫɥɟɞɨɜɚɧɢɹ ɨɩɟɪɚɰɢɣ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. Ɋɟɤɨɦɟɧɞɭɟɬɫɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 4 ɤɭɪɫɚ ɞ/ɨ ɢ 5 ɤɭɪɫɚ ɜ/ɨ.
Ⱦɥɹ ɫɩɟɰɢɚɥɶɧɨɫɬɢ 010501 – ɉɪɢɤɥɚɞɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɢ ɢɧɮɨɪɦɚɬɢɤɚ 2
1. ɆȺɌȿɆȺɌɂɑȿɋɄɂȿ ɆɈȾȿɅɂ ȾɂɋɄɊȿɌɇɈȽɈ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə I. ɐɟɥɨɱɢɫɥɟɧɧɵɟ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ 1) ɉɪɨɢɡɜɨɥɶɧɚɹ ɡɚɞɚɱɚ (ɐɁɅɉ). Ax b, x t 0, x j ɰɟɥɵɟ ɱɢɫɥɚ, n
¦c
j
j 1,2,..., n,
x j o max min .
(1.1) (1.2) (1.3)
(1.4)
j 1
Ɍɪɟɛɨɜɚɧɢɟ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ ɩɟɪɟɦɟɧɧɵɯ ɩɨɹɜɥɹɟɬɫɹ, ɧɚɩɪɢɦɟɪ, ɜ ɡɚɞɚɱɚɯ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɥɚɧɢɪɨɜɚɧɢɹ ɧɚ ɩɪɨɢɡɜɨɞɫɬɜɚɯ, ɫɩɟɰɢɚɥɢɡɢɪɭɸɳɢɯɫɹ ɩɨ ɜɵɩɭɫɤɭ ɲɬɭɱɧɵɯ ɢɡɞɟɥɢɣ. 2) Ɂɚɞɚɱɚ ɨ ɪɚɧɰɟ.
xj
^0,1`,
(1.5)
n
¦ajxj d A
,
(1.6)
j 1 n
¦ c j x j o max .
(1.7)
j 1
Ɍɚɤɚɹ ɡɚɞɚɱɚ ɜɨɡɧɢɤɚɟɬ ɩɪɢ ɜɵɛɨɪɟ ɧɚɛɨɪɚ ɩɪɟɞɦɟɬɨɜ ɦɚɤɫɢɦɚɥɶɧɨɣ ɫɬɨɢɦɨɫɬɢ ɞɥɹ ɩɨɝɪɭɡɤɢ ɜ ɧɟɤɨɬɨɪɭɸ ɬɚɪɭ (ɪɚɧɟɰ) ɩɪɢ ɨɝɪɚɧɢɱɟɧɢɹɯ ɧɚ ɨɛɴɟɦ ɢɥɢ ɧɚ «ɝɪɭɡɨɩɨɞɴɟɦɧɨɫɬɶ». Ɉɛɨɛɳɟɧɢɟɦ ɷɬɨɣ ɡɚɞɚɱɢ ɹɜɥɹɟɬɫɹ ɦɧɨɝɨɦɟɪɧɚɹ ɡɚɞɚɱɚ ɨ ɪɚɧɰɟ, ɜ ɤɨɬɨɪɨɣ ɩɪɢɫɭɬɫɬɜɭɟɬ ɧɟɫɤɨɥɶɤɨ ɨɝɪɚɧɢɱɟɧɢɣ. ȿɫɥɢ, ɤɪɨɦɟ ɬɨɝɨ, ɤɚɠɞɵɣ ɩɪɟɞɦɟɬ ɢɦɟɟɬɫɹ ɜ ɤɨɥɢɱɟɫɬɜɟ ɧɟɫɤɨɥɶɤɢɯ ɟɞɢɧɢɰ, ɬɨ ɬɪɟɛɨɜɚɧɢɟ (1.5) ɡɚɦɟɧɹɟɬɫɹ ɧɚ ɬɪɟɛɨɜɚɧɢɟ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ. ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ ɡɚɞɚɱɚ ɨ ɪɚɧɰɟ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɰɟɥɨɱɢɫɥɟɧɧɭɸ ɡɚɞɚɱɭ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɫ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨɣ ɦɚɬɪɢɰɟɣ ɨɝɪɚɧɢɱɟɧɢɣ. II. Ɂɚɞɚɱɢ ɤɨɦɛɢɧɚɬɨɪɧɨɝɨ ɬɢɩɚ 1) Ɂɚɞɚɱɚ ɨ ɧɚɡɧɚɱɟɧɢɹɯ (ɡɚɞɚɱɚ ɜɵɛɨɪɚ). xij ^0,1` ,
(1.8)
n
¦ xij
1, i 1,2,..., n ,
i 1
3
(1.9)
n
¦ xij
1,
j 1,2,..., n ,
(1.10)
j 1 n
n
¦¦ cij xij o min .
(1.11)
i 1j 1
Ɂɚɞɚɱɭ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɬɪɚɞɢɰɢɨɧɧɨ ɫɜɹɡɵɜɚɸɬ ɫ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɩɪɟɬɟɧɞɟɧɬɨɜ ɩɨ ɪɚɛɨɱɢɦ ɦɟɫɬɚɦ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɤɨɥɢɱɟɫɬɜɨ ɪɚɛɨɱɢɯ ɦɟɫɬ ɪɚɜɧɨ ɤɨɥɢɱɟɫɬɜɭ ɩɪɟɬɟɧɞɟɧɬɨɜ. Ɋɚɡɧɨɜɢɞɧɨɫɬɶɸ ɹɜɥɹɟɬɫɹ ɨɬɤɪɵɬɚɹ ɡɚɞɚɱɚ, ɜ ɤɨɬɨɪɨɣ ɱɢɫɥɨ ɪɚɛɨɱɢɯ ɦɟɫɬ ɧɟ ɫɨɜɩɚɞɚɟɬ ɫ ɱɢɫɥɨɦ ɩɪɟɬɟɧɞɟɧɬɨɜ. Ɂɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɨɫɨɛɵɦ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɬɪɚɧɫɩɨɪɬɧɨɣ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ. Ɉɫɨɛɟɧɧɨɫɬɶɸ ɹɜɥɹɟɬɫɹ ɫɢɥɶɧɚɹ ɜɵɪɨɠɞɟɧɧɨɫɬɶ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ, ɱɬɨ ɞɟɥɚɟɬ ɡɚɬɪɭɞɧɢɬɟɥɶɧɵɦ ɩɪɢɦɟɧɟɧɢɟ ɬɪɚɞɢɰɢɨɧɧɨɝɨ ɦɟɬɨɞɚ ɩɨɬɟɧɰɢɚɥɨɜ. 2) Ɂɚɞɚɱɚ ɤɨɦɦɢɜɨɹɠɟɪɚ. xij ^0,1` ,
(1.12)
n
¦ xij
1, i 1,2,..., n ,
(1.13)
¦ xij
1,
j 1,2,..., n .
(1.14)
i 1 n
j 1
Ⱦɨɩɨɥɧɢɬɟɥɶɧɨɟ ɬɪɟɛɨɜɚɧɢɟ ɨɬɫɭɬɫɬɜɢɹ ɩɨɞɰɢɤɥɨɜ n
(1.15)
n
¦¦ cij xij o min .
(1.16)
i 1j 1
Ɂɚɞɚɱɚ ɤɨɦɦɢɜɨɹɠɟɪɚ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɭɫɥɨɜɢɟɦ (1.15), ɤɨɬɨɪɨɟ ɜɨɡɧɢɤɚɟɬ ɜ ɫɜɹɡɢ ɫ ɬɟɦ, ɱɬɨ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɢɳɟɬɫɹ ɦɚɪɲɪɭɬ. ɂɦɟɟɬɫɹ ɬɪɟɛɨɜɚɧɢɟ ɩɪɨɯɨɠɞɟɧɢɹ ɱɟɪɟɡ ɤɚɠɞɵɣ ɢɡ n ɝɨɪɨɞɨɜ ɪɨɜɧɨ ɩɨ ɨɞɧɨɦɭ ɪɚɡɭ. ɉɪɢ ɷɬɨɦ ɰɟɥɟɜɚɹ ɮɭɧɤɰɢɹ (1.16) ɦɨɠɟɬ, ɧɚɩɪɢɦɟɪ, ɨɛɨɡɧɚɱɚɬɶ ɫɭɦɦɚɪɧɨɟ ɪɚɫɫɬɨɹɧɢɟ ɦɚɪɲɪɭɬɚ. ɍɫɥɨɜɢɟ (1.15) ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɚɧɚɥɢɬɢɱɟɫɤɢ, ɧɚɩɪɢɦɟɪ, ɜ ɜɢɞɟ ui u j nxi j d n 1, i, j 1, n, (1.15) ɝɞɟ ui 0,1, 2,..., n 2 — ɰɟɥɵɟ, i z j , i 1, n 1, j 1, n . ɉɪɢ ɩɨɫɬɪɨɟɧɢɢ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɱɚɫɬɨ ɭɫɥɨɜɢɟ (1.15) ɭɱɢɬɵɜɚɟɬɫɹ ɚɥɝɨɪɢɬɦɢɱɟɫɤɢ. Ɂɚɞɚɱɚ ɤɨɦɦɢɜɨɹɠɟɪɚ ɢɡɜɟɫɬɧɚ ɜ ɩɪɢɥɨɠɟɧɢɹɯ ɤɚɤ ɡɚɞɚɱɚ ɨ ɩɟɪɟɧɚɥɚɞɤɚɯ, ɜɨɡɧɢɤɚɸɳɚɹ ɩɪɢ ɨɩɬɢɦɢɡɚɰɢɢ ɡɚɝɪɭɡɤɢ ɨɛɨɪɭɞɨɜɚɧɢɹ. 3) Ɂɚɞɚɱɚ ɨ ɦɢɧɢɦɚɥɶɧɨɦ ɩɨɤɪɵɬɢɢ. aij ^0,1`,
xj
^0,1`, 4
(1.17) (1.18)
n
¦ aij x j t 1 ,
(1.19)
j 1 n
¦ x j o min .
(1.20)
j 1
Ɇɚɬɪɢɰɚ A, ɫɨɫɬɨɹɳɚɹ ɢɡ ɧɭɥɟɣ ɢ ɟɞɢɧɢɰ, ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɦɚɬɪɢɰɵ ɩɨɤɪɵɬɢɣ. Ɏɨɪɦɚɥɶɧɨ ɡɚɞɚɱɚ ɫɨɫɬɨɢɬ ɜ ɜɵɛɨɪɟ ɦɢɧɢɦɚɥɶɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɫɬɨɥɛɰɨɜ, ɨɛɴɟɞɢɧɟɧɢɟ ɤɨɬɨɪɵɯ ɩɨɤɪɵɜɚɟɬ ɜɫɟ ɫɬɪɨɤɢ ɦɚɬɪɢɰɵ (ɜ ɤɚɠɞɨɣ ɫɬɪɨɤɟ ɢɦɟɟɬɫɹ ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɨɞɧɚ ɟɞɢɧɢɰɚ). Ɍɚɤɚɹ ɡɚɞɚɱɚ ɜɨɡɧɢɤɚɟɬ, ɧɚɩɪɢɦɟɪ, ɩɪɢ ɬɟɫɬɢɪɨɜɚɧɢɢ, ɞɢɚɝɧɨɫɬɢɤɟ. III. Ɂɚɞɚɱɢ ɫ ɪɚɡɪɵɜɧɨɣ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɟɣ (ɬɪɚɧɫɩɨɪɬɧɚɹ ɡɚɞɚɱɚ ɫ ɮɢɤɫɢɪɨɜɚɧɧɵɦɢ ɞɨɩɥɚɬɚɦɢ). xij t 0 , (1.21) n
¦ xij
ai , i 1,2,..., m ,
(1.22)
bj ,
(1.23)
j 1
m
¦ xij
j 1,2,..., n ,
i 1 m n
¦¦ cij xij o min ,
(1.24)
°0, xij 0, cij xij ® °¯cij xij d ij , xij ! 0.
(1.25)
i 1j 1
Ⱦɚɧɧɚɹ ɡɚɞɚɱɚ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɫɬɚɧɞɚɪɬɧɨɣ ɬɪɚɧɫɩɨɪɬɧɨɣ ɩɪɟɞɩɨɥɨɠɟɧɢɟɦ ɨ ɬɨɦ, ɱɬɨ ɢɡɞɟɪɠɤɢ ɧɚ ɩɟɪɟɜɨɡɤɭ ɫɜɹɡɚɧɵ ɧɟ ɬɨɥɶɤɨ ɫ ɤɨɥɢɱɟɫɬɜɨɦ ɩɟɪɟɜɨɡɢɦɨɝɨ ɬɨɜɚɪɚ, ɧɨ ɬɚɤɠɟ ɫ ɞɨɩɨɥɧɢɬɟɥɶɧɵɦɢ ɞɨɩɥɚɬɚɦɢ, ɜɨɡɧɢɤɚɸɳɢɦɢ ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ɩɟɪɟɜɨɡɤɚ ɢɦɟɟɬ ɦɟɫɬɨ ɩɨ ɩɥɚɧɭ. ɂɡɧɚɱɚɥɶɧɨ ɜ ɡɚɞɚɱɟ ɨɬɫɭɬɫɬɜɭɟɬ ɭɫɥɨɜɢɟ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ ɩɟɪɟɦɟɧɧɵɯ. Ɉɞɧɚɤɨ ɩɭɬɟɦ ɜɜɟɞɟɧɢɹ ɧɨɜɵɯ ɩɟɪɟɦɟɧɧɵɯ (ɛɭɥɟɜɵɯ) ɨɧɚ ɦɨɠɟɬ ɛɵɬɶ ɩɟɪɟɩɢɫɚɧɚ ɤɚɤ ɡɚɞɚɱɚ ɞɢɫɤɪɟɬɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ. mɶ
n
¦¦ c
xij d ij yij o min,
(1.26)
ai , i 1,2,..., m ,
(1.27)
bj ,
(1.28)
ij
i 1 j 1
n
¦ xij j 1 m
¦x
ij
j 1,2,..., n,
i 1
xij d M ij yij , ɝɞɟ M ij
(1.29)
min^ai ,b j `, 5
xij t 0,
yij
^0,1`
i, j .
(1.30)
ȼɫɟ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɡɚɞɚɱɢ (ɤɪɨɦɟ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ) ɨɬɧɨɫɹɬɫɹ ɤ ɤɥɚɫɫɭ ɬɪɭɞɧɨɪɟɲɚɟɦɵɯ. Ⱦɥɹ ɢɯ ɪɟɲɟɧɢɹ ɧɟ ɢɡɜɟɫɬɧɵ ɚɥɝɨɪɢɬɦɵ ɩɨɥɢɧɨɦɢɚɥɶɧɨɣ ɫɥɨɠɧɨɫɬɢ. ȼɫɟ ɬɨɱɧɵɟ ɚɥɝɨɪɢɬɦɵ (ɢɳɭɳɢɟ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ) ɨɛɥɚɞɚɸɬ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɣ ɫɥɨɠɧɨɫɬɶɸ. Ⱦɥɹ ɨɬɵɫɤɚɧɢɹ ɬɨɱɧɨɝɨ ɪɟɲɟɧɢɹ ɢɫɩɨɥɶɡɭɸɬ, ɤɚɤ ɩɪɚɜɢɥɨ, ɦɟɬɨɞɵ ɬɢɩɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ, ɞɢɧɚɦɢɱɟɫɤɨɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ ɢ ɦɟɬɨɞɵ ɨɬɫɟɱɟɧɢɣ (ɫɟɤɭɳɢɯ ɩɥɨɫɤɨɫɬɟɣ). ɍɉɊȺɀɇȿɇɂə
1. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ ɥɸɛɚɹ ɰɟɥɨɱɢɫɥɟɧɧɚɹ ɡɚɞɚɱɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɷɤɜɢɜɚɥɟɧɬɧɨ ɩɟɪɟɩɢɫɚɧɚ ɤɚɤ ɡɚɞɚɱɚ ɫ ɛɭɥɟɜɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ. 2. Ɂɚɩɢɫɚɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɭɸ ɦɨɞɟɥɶ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɩɟɪɟɦɟɧɧɵɯ ° xij ® °¯ xij
1, ɟɫɥɢ i-ɣ ɝɨɪɨɞ ɫɬɨɢɬ ɜ ɩɟɪɟɫɬɚɧɨɜɤɟ ɧɚ j-ɦ ɦɟɫɬɟ, 0, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ.
3. ɋɮɨɪɦɭɥɢɪɨɜɚɬɶ ɭɫɥɨɜɢɟ ɫɨɜɦɟɫɬɢɦɨɫɬɢ ɜ ɡɚɞɚɱɚɯ ɨ ɪɚɧɰɟ, ɨ ɧɚɡɧɚɱɟɧɢɹɯ, ɨ ɦɢɧɢɦɚɥɶɧɨɦ ɩɨɤɪɵɬɢɢ.
2. ɁȺȾȺɑȺ Ɉ ɇȺɁɇȺɑȿɇɂəɏ. ȼȿɇȽȿɊɋɄɂɃ ɆȿɌɈȾ Ɋȿɒȿɇɂə 2.1. ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ. ɇɟɤɨɬɨɪɵɟ ɫɜɨɣɫɬɜɚ
ɉɭɫɬɶ ɢɦɟɸɬɫɹ n ɩɪɟɬɟɧɞɟɧɬɨɜ (ɤɚɠɞɨɦɭ ɢɡ ɧɢɯ ɨɬɜɟɱɚɟɬ ɢɧɞɟɤɫ i, i 1, n ) ɧɚ n ɦɟɫɬ ɪɚɛɨɬɵ (ɤɚɠɞɨɦɭ ɢɡ ɧɢɯ ɨɬɜɟɱɚɟɬ ɢɧɞɟɤɫ j , j 1, n ). ɉɪɢ ɷɬɨɦ ɢɡɜɟɫɬɧɚ ɫɬɨɢɦɨɫɬɶ cij ɡɚɬɪɚɬ, ɫɜɹɡɚɧɧɵɯ ɫ ɧɚɡɧɚɱɟɧɢɟɦ i -ɝɨ ɩɪɟɬɟɧɞɟɧɬɚ ɧɚ j-ɟ ɦɟɫɬɨ. Ɍɪɟɛɭɟɬɫɹ ɪɚɫɩɪɟɞɟɥɢɬɶ ɩɪɟɬɟɧɞɟɧɬɨɜ ɩɨ ɪɚɛɨɱɢɦ ɦɟɫɬɚɦ ɬɚɤ, ɱɬɨɛɵ ɤɚɠɞɵɣ ɩɪɟɬɟɧɞɟɧɬ ɡɚɧɹɥ ɨɞɧɨ ɦɟɫɬɨ, ɤɚɠɞɨɟ ɦɟɫɬɨ ɛɵɥɨ ɡɚɧɹɬɨ ɨɞɧɢɦ ɩɪɟɬɟɧɞɟɧɬɨɦ, ɢ ɬɚɤ, ɱɬɨɛɵ ɫɜɹɡɚɧɧɵɟ ɫ ɷɬɢɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɫɭɦɦɚɪɧɵɟ ɡɚɬɪɚɬɵ ɛɵɥɢ ɦɢɧɢɦɚɥɶɧɵɦɢ. Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɡɚɩɢɫɢ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɦɨɠɧɨ ɜɜɟɫɬɢ n 2 ɩɟɪɟɦɟɧɧɵɯ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: 1, ɟɫɥɢ i ɩɪɟɬɟɧɞɟɧɬ ɧɚɡɧɚɱɟɧ ɧɚ j-ɟ ɦɟɫɬɨ, xij ® ¯0, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ. Ɍɟɩɟɪɶ ɡɚɞɚɱɚ ɩɪɢɧɢɦɚɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ: 6
n
L( X )
n
¦ ¦ cij xij o min , i 1j 1
n
°¦ xij 1, j 1, n, °i 1 °n : ® ¦ xij 1, i 1, n, °j 1 ° x {0,1}, i, j 1, n. ° ij ¯ Ɂɚɦɟɱɚɧɢɟ. ȿɫɥɢ ɩɨɫɥɟɞɧɢɟ ɨɝɪɚɧɢɱɟɧɢɹ ɡɚɦɟɧɢɬɶ ɭɫɥɨɜɢɹɦɢ ɜɢɞɚ 0 d xij d 1, i, j , ɬɨ ɩɨɥɭɱɟɧɧɚɹ ɡɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɬɪɚɧɫɩɨɪɬɧɨɣ ɡɚɞɚɱɢ, ɭ ɤɨɬɨɪɨɣ, ɤɚɤ ɢɡɜɟɫɬɧɨ, ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɜɫɟɝɞɚ ɫɭɳɟɫɬɜɭɟɬ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɞɚɱɭ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɦɨɠɧɨ ɪɟɲɚɬɶ ɦɟɬɨɞɨɦ ɩɨɬɟɧɰɢɚɥɨɜ, ɩɪɢɱɟɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɩɟɰɢɮɢɤɨɣ ɷɬɨɝɨ ɦɟɬɨɞɚ ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɪɟɲɟɧɢɟɦ ɹɜɥɹɟɬɫɹ n 2 -ɦɟɪɧɵɣ ɜɟɤɬɨɪ, ɢɥɢ ɦɚɬɪɢɰɚ ɩɨɪɹɞɤɚ n×n, ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɣ ɪɚɜɧɵ 0 ɢɥɢ 1. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɨɥɭɱɟɧɧɵɣ ɨɬɜɟɬ ɛɭɞɟɬ ɬɚɤɠɟ ɹɜɥɹɬɶɫɹ ɨɬɜɟɬɨɦ ɜ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɟ ɨ ɧɚɡɧɚɱɟɧɢɹɯ. Ɉɞɧɚɤɨ ɧɚɱɚɥɶɧɚɹ ɛɚɡɢɫɧɚɹ ɬɨɱɤɚ, ɩɨɥɭɱɟɧɧɚɹ, ɧɚɩɪɢɦɟɪ, ɩɨ ɦɟɬɨɞɭ ɫɟɜɟɪɨɡɚɩɚɞɧɨɝɨ ɭɝɥɚ, ɫɨɞɟɪɠɢɬ ɧɟ m+n–1=2n–1, ɚ ɜɫɟɝɨ ɥɢɲɶ n ɧɟɧɭɥɟɜɵɯ ɤɨɦɩɨɧɟɧɬ ɪɚɜɧɵɯ 1, cɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ n t 2 ɷɬɨɬ ɩɥɚɧ ɫɬɚɧɨɜɢɬɫɹ ɜɵɪɨɠɞɟɧɧɵɦ. Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɷɬɨ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ ɫɭɳɟɫɬɜɟɧɧɨ ɭɫɥɨɠɧɹɟɬ ɜɵɱɢɫɥɢɬɟɥɶɧɭɸ ɩɪɨɰɟɞɭɪɭ ɪɟɲɟɧɢɹ ɬɪɚɧɫɩɨɪɬɧɨɣ ɡɚɞɚɱɢ. ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɫɭɳɟɫɬɜɭɸɬ ɫɩɟɰɢɚɥɶɧɵɟ ɦɟɬɨɞɵ. Ɋɚɫɫɦɨɬɪɢɦ ɨɞɢɧ ɢɡ ɧɢɯ, ɤɨɬɨɪɵɣ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɜɟɧɝɟɪɫɤɢɣ ɦɟɬɨɞ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɫɥɟɞɭɸɳɟɟ ɨɩɪɟɞɟɥɟɧɢɟ. Ɉɩɪɟɞɟɥɟɧɢɟ. Ʌɸɛɵɟ k ɷɥɟɦɟɧɬɨɜ ( k 2, n ) ɦɚɬɪɢɰɵ C= (cij ) ɩɨɪɹɞɤɚ n×n ɧɚɡɵɜɚɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ, ɟɫɥɢ ɜɫɹɤɢɟ ɞɜɚ ɢɡ ɧɢɯ ɪɚɫɩɨɥɚɝɚɸɬɫɹ ɜ ɪɚɡɧɵɯ ɫɬɪɨɤɚɯ ɢ ɜ ɪɚɡɧɵɯ ɫɬɨɥɛɰɚɯ. Ɍɟɩɟɪɶ ɦɨɠɧɨ ɩɟɪɟɮɨɪɦɭɥɢɪɨɜɚɬɶ ɡɚɞɚɱɭ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɫɪɟɞɢ n 2 ɷɥɟɦɟɧɬɨɜ ɞɚɧɧɨɣ ɦɚɬɪɢɰɵ C ɧɚɣɬɢ n ɧɟɡɚɜɢɫɢɦɵɯ
ɷɥɟɦɟɧɬɨɜ ci s j s , s 1, n , ɬɚɤɢɯ, ɱɬɨ ɫɭɦɦɚ
n
¦ ci
s 1
s js
ɦɢɧɢɦɚɥɶɧɚ.
Ⱦɥɹ ɨɛɨɫɧɨɜɚɧɢɹ ɜɟɧɝɟɪɫɤɨɝɨ ɦɟɬɨɞɚ ɩɨɬɪɟɛɭɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɩɨɧɹɬɢɹ ɢ ɭɬɜɟɪɠɞɟɧɢɹ. Ɇɚɬɪɢɰɟɣ ɧɚɡɧɚɱɟɧɢɣ ɩɨɪɹɞɤɚ n×n ɧɚɡɵɜɚɟɬɫɹ ɦɚɬɪɢɰɚ, ɜ ɤɨɬɨɪɨɣ ɢɦɟɸɬɫɹ n ɧɟɡɚɜɢɫɢɦɵɯ ɟɞɢɧɢɰ ɢ n 2 n n( n 1) ɧɭɥɟɣ. ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɷɬɨ ɦɚɬɪɢɰɚ, ɭ ɤɨɬɨɪɨɣ ɜ ɤɚɠɞɨɣ ɫɬɪɨɤɟ ɢ ɜ ɤɚɠɞɨɦ ɫɬɨɥɛɰɟ ɢɦɟɟɬɫɹ ɪɨɜɧɨ ɨɞɧɚ ɟɞɢɧɢɰɚ, ɚ ɨɫɬɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɹɜɥɹɸɬɫɹ ɧɭɥɹɦɢ. 7
Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ : ɞɨɩɭɫɬɢɦɨɟ ɦɧɨɠɟɫɬɜɨ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɦɚɬɪɢɰ ɧɚɡɧɚɱɟɧɢɣ ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɬɚɤɢɯ ɦɚɬɪɢɰ ɫɨɫɬɚɜɥɹɟɬ ɞɨɩɭɫɬɢɦɨɟ ɦɧɨɠɟɫɬɜɨ : . Ɂɚɦɟɱɚɧɢɟ. ȼɫɟ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɪɚɡɦɟɪɚ n×n ɢɦɟɸɬ ɨɞɧɨ ɢ ɬɨ ɠɟ ɞɨɩɭɫɬɢɦɨɟ ɦɧɨɠɟɫɬɜɨ ɢ ɨɬɥɢɱɚɸɬɫɹ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɬɨɥɶɤɨ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ, ɬ. ɟ. ɦɚɬɪɢɰɟɣ C= (cij ) . Ɍɟɨɪɟɦɚ 1. ȿɫɥɢ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰ C ɢ D ɩɨɪɹɞɤɚ n×n ɫɜɹɡɚɧɵ ɪɚɜɟɧɫɬɜɚɦɢ d ij cij D i E j , ɬɨ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɫ ɞɚɧɧɵɦɢ ɦɚɬɪɢɰɚɦɢ C ɢ D ɷɤɜɢɜɚɥɟɧɬɧɵ, ɬ. ɟ. ɦɧɨɠɟɫɬɜɚ ɢɯ ɪɟɲɟɧɢɣ (ɨɩɬɢɦɚɥɶɧɵɯ ɬɨɱɟɤ) ɫɨɜɩɚɞɚɸɬ. Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ȼɨ-ɩɟɪɜɵɯ, ɤɚɤ ɨɬɦɟɱɚɥɨɫɶ ɜɵɲɟ, ɞɨɩɭɫɬɢɦɵɟ ɦɧɨɠɟɫɬɜɚ ɨɛɟɢɯ ɡɚɞɚɱ ɫɨɜɩɚɞɚɸɬ. ȼɨ-ɜɬɨɪɵɯ, ɫɪɚɜɧɢɦ ɡɧɚɱɟɧɢɹ ɰɟɥɟɜɵɯ ɮɭɧɤɰɢɣ ɨɛɟɢɯ ɡɚɞɚɱ, ɢɫɩɨɥɶɡɭɹ ɨɝɪɚɧɢɱɟɧɢɹ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ ɰɟɩɨɱɤɭ ɪɚɜɟɧɫɬɜ. n
n
n
n
n
n
n
n
¦ ¦ d ij xij ¦ ¦ cij xij ¦ ¦ D i xij ¦ ¦ E j xij i 1j 1
n
n
i 1j 1
n
n
i 1j 1
n
n
i 1j 1
n
n
n
n
n
n
¦ ¦ cij xij ¦ D i ¦ xij ¦ E j ¦ xij ¦ ¦ cij xij ¦ D i ¦ E j ¦ ¦ cij xij F , i 1j 1
i 1
j 1
j 1
i 1
i 1j 1
i 1 j 1
i 1j 1
const
ɢɡ ɤɨɬɨɪɨɣ ɫɥɟɞɭɟɬ, ɱɬɨ ɡɧɚɱɟɧɢɹ ɞɜɭɯ ɰɟɥɟɜɵɯ ɮɭɧɤɰɢɣ ɫ ɦɚɬɪɢɰɚɦɢ C ɢ D ɨɬɥɢɱɚɸɬɫɹ ɧɚ ɩɨɫɬɨɹɧɧɭɸ F. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɦɢɧɢɦɭɦɵ ɷɬɢɯ ɮɭɧɤɰɢɣ ɞɨɫɬɢɝɚɸɬɫɹ ɜ ɨɞɧɢɯ ɢ ɬɟɯ ɠɟ ɬɨɱɤɚɯ (ɧɚ ɨɞɧɢɯ ɢ ɬɟɯ ɠɟ ɦɚɬɪɢɰɚɯ ɧɚɡɧɚɱɟɧɢɣ). Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɢɞɚ d ij cij D i E j (ɞɨɛɚɜɥɟɧɢɟ ɤɨ ɜɫɟɦ ɷɥɟɦɟɧɬɚɦ ɥɸɛɨɣ ɫɬɪɨɤɢ ɢɥɢ ɥɸɛɨɝɨ ɫɬɨɥɛɰɚ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɱɢɫɥɚ) ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɷɤɜɢɜɚɥɟɧɬɧɵɦɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ. ɋɥɟɞɫɬɜɢɟ. ȼɫɟɝɞɚ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɜɫɟ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ C ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ, ɬ. ɟ. cij t 0, i, j. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɷɬɨɝɨ ɦɨɠɧɨ ɞɨɛɢɬɶɫɹ ɩɪɢɦɟɧɟɧɢɟɦ ɷɤɜɢɜɚɥɟɧɬɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ. Ɍɟɨɪɟɦɚ 2. ɉɭɫɬɶ ɜɫɟ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ C ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ, ɬ. ɟ. cij t 0, i, j. ȿɫɥɢ ɜ ɧɟɣ ɢɦɟɸɬɫɹ n ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɜɵɯ ɷɥɟɦɟɧɬɨɜ cij
0, ɬɨ ɢɯ ɫɭɦɦɚ ɹɜɥɹɟɬɫɹ ɦɢɧɢɦɚɥɶɧɨɣ. Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ʉɚɤɨɜɚ ɛɵ ɧɢ ɛɵɥɚ ɞɨɩɭɫɬɢɦɚɹ ɬɨɱɤɚ X : , n
L( X )
n
¦ ¦ cij xij t 0 .
ȼɜɟɞɟɦ ɦɚɬɪɢɰɭ ɧɚɡɧɚɱɟɧɢɣ X 0 ɫ ɟɞɢɧɢɰɚɦɢ
i 1j 1
ɢɦɟɧɧɨ ɧɚ ɬɟɯ ɦɟɫɬɚɯ, ɝɞɟ ɪɚɫɩɨɥɨɠɟɧɵ ɜɵɛɪɚɧɧɵɟ ɧɟɡɚɜɢɫɢɦɵɟ ɷɥɟɦɟɧɬɵ 8
cij 0 . Ɍɨɝɞɚ L( X 0 ) 0 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, X 0 ɨɩɬɢɦɚɥɶɧɚɹ ɬɨɱɤɚ. Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ. 2.2. ȼɟɧɝɟɪɫɤɢɣ ɦɟɬɨɞ
Ⱥɥɝɨɪɢɬɦ ɜɟɧɝɟɪɫɤɨɝɨ ɦɟɬɨɞɚ ɜɤɥɸɱɚɟɬ ɜ ɫɟɛɹ 4 ɷɬɚɩɚ. 1. ɉɪɢɜɟɞɟɧɢɟ ɦɚɬɪɢɰɵ. 2. ȼɵɱɢɫɥɟɧɢɟ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɱɢɫɥɚ k ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɣ ɜ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɬɪɢɰɟ. 3. ɉɨɥɭɱɟɧɢɟ ɧɨɜɵɯ ɧɭɥɟɣ ɩɪɢ k n . 4. Ɉɬɵɫɤɚɧɢɟ n ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɣ ɩɪɢ k n . Ɋɚɫɫɦɨɬɪɢɦ ɤɚɠɞɵɣ ɷɬɚɩ ɩɨɞɪɨɛɧɟɟ. ɗɬɚɩ 1 Ɇɚɬɪɢɰɚ C ɩɨɪɹɞɤɚ n u n ɧɚɡɵɜɚɟɬɫɹ ɩɪɢɜɟɞɟɧɧɨɣ, ɟɫɥɢ ɜɫɟ ɟɟ ɷɥɟɦɟɧɬɵ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ ɢ, ɤɪɨɦɟ ɬɨɝɨ, ɜ ɤɚɠɞɨɣ ɫɬɪɨɤɟ ɢ ɜ ɤɚɠɞɨɦ ɫɬɨɥɛɰɟ ɢɦɟɸɬɫɹ ɧɭɥɟɜɵɟ ɷɥɟɦɟɧɬɵ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢɜɟɞɟɧɧɚɹ ɦɚɬɪɢɰɚ C ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɞɜɭɦ ɭɫɥɨɜɢɹɦ: 1. cij t 0, i, j ;
2. cij
0, i ɢ cij
0, j .
Ⱦɥɹ ɩɪɢɜɟɞɟɧɢɹ ɦɚɬɪɢɰɵ C ɫ ɷɥɟɦɟɧɬɚɦɢ cij t 0 ɧɭɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɷɤɜɢɜɚɥɟɧɬɧɵɦɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ (4). ɉɪɢ ɷɬɨɦ ɜɧɚɱɚɥɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɪɢɜɟɞɟɧɢɟ ɦɚɬɪɢɰɵ ɩɨ ɫɬɪɨɤɚɦ (ɩɨ ɫɬɨɥɛɰɚɦ), ɬ. ɟ. ɢɳɟɬɫɹ ɧɚɢɦɟɧɶɲɢɣ ɷɥɟɦɟɧɬ ai ɜ ɤɚɠɞɨɣ ɫɬɪɨɤɟ ɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɟɪɟɯɨɞ ɤ ɦɚɬɪɢɰɟ C 1 ɫ ɷɥɟɦɟɧɬɚɦɢ c cij1
cij D1 . ȿɫɥɢ ɦɚɬɪɢɰɚ C 1 ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɩɪɢɜɟɞɟɧɧɨɣ ɩɨ ɫɬɨɥɛɰɚɦ (ɩɨ ɫɬɪɨɤɚɦ), ɬɨ ɢɳɟɬɫɹ ɧɚɢɦɟɧɶɲɢɣ ɜɚɪɢɚɧɬ B j ɜ ɤɚɠɞɨɦ ɫɬɨɥɛɰɟ ɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɟɪɟɯɨɞ ɤ ɦɚɬɪɢɰɟ C 2 , ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɣ ɜɵɱɢɫɥɹɸɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: cij2 cij1 E j . Ɇɚɬɪɢɰɚ C 2 ɩɨ ɩɨɫɬɪɨɟɧɢɸ ɹɜɥɹɟɬɫɹ ɩɪɢɜɟɞɟɧɧɨɣ.
9
ɉɪɢɦɟɪ 1. ɉɭɫɬɶ
C
§2 ¨ ¨1 ¨2 ¨ ¨1 ¨1 ©
1 2 1 3 5 7 7 8 6
0· ¸ 4¸ 9 ¸. ¸ 5 4 7 8¸ 3 4 9 10 ¸¹
C1
§2 ¨ ¨0 ¨0 ¨ ¨0 ¨0 ©
1 2 1 0· ¸ 2 4 6 3¸ 5 6 4 7¸ , ¸ 4 3 6 7¸ 2 3 9 9 ¸¹
C2
§2 ¨ ¨0 ¨0 ¨ ¨0 ¨0 ©
0 0 0 0· ¸ 1 2 5 3¸ 4 4 3 7¸ , ¸ 3 1 5 7¸ 1 1 7 9 ¸¹
Ʌɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ
ɩɪɢɱɟɦ ɦɚɬɪɢɰɚ C 2 ɹɜɥɹɟɬɫɹ ɩɪɢɜɟɞɟɧɧɨɣ. ɗɬɚɩ 2 ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɱɢɫɥɚ k ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɣ ɜ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɬɪɢɰɟ ɨɛɵɱɧɨ ɨɪɝɚɧɢɡɭɸɬ ɩɨɥɧɵɣ ɩɟɪɟɛɨɪ ɜɫɟɜɨɡɦɨɠɧɵɯ ɜɚɪɢɚɧɬɨɜ. ɉɪɢ ɷɬɨɦ ɧɟɨɛɯɨɞɢɦɚ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɫɥɟɞɭɸɳɢɦ ɭɬɜɟɪɠɞɟɧɢɟɦ. Ɍɟɨɪɟɦɚ 3. Ɇɚɤɫɢɦɚɥɶɧɨɟ ɱɢɫɥɨ ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɣ ɪɚɜɧɨ ɦɢɧɢɦɚɥɶɧɨɦɭ ɫɭɦɦɚɪɧɨɦɭ ɱɢɫɥɭ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ ɢ ɜɟɪɬɢɤɚɥɶɧɵɯ ɥɢɧɢɣ (ɫɬɪɨɤ ɢ ɫɬɨɥɛɰɨɜ), ɤɨɬɨɪɵɦɢ ɦɨɠɧɨ ɡɚɱɟɪɤɧɭɬɶ ɜɫɟ ɧɭɥɟɜɵɟ ɷɥɟɦɟɧɬɵ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɬɪɢɰɵ. ɉɪɢɦɟɪ 2. ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɩɪɢɜɟɞɟɧɧɚɹ ɦɚɬɪɢɰɚ § 2 0 0 0 0· ¸ ¨ ¨ 0 1 2 5 3¸ C 2 ¨0 4 4 3 7¸ ¸ ¨ ¨0 3 1 5 7¸ ¨0 1 1 7 9¸ ¹ © ɢɡ ɩɪɢɦɟɪɚ 1. ȼɫɟ ɟɟ ɧɭɥɢ ɫɨɞɟɪɠɚɬ 1-ɹ ɝɨɪɢɡɨɧɬɚɥɶ ɢ 1-ɹ ɜɟɪɬɢɤɚɥɶ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɡɞɟɫɶ k 2 5 . 10
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɛɭɞɟɦ ɩɪɨɜɨɞɢɬɶ ɭɤɚɡɚɧɧɵɟ ɥɢɧɢɢ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɧɟɤɨɬɨɪɵɟ ɷɥɟɦɟɧɬɵ ɨɤɚɠɭɬɫɹ ɡɚɱɟɪɤɧɭɬɵɦɢ, ɞɪɭɝɢɟ – ɡɚɱɟɪɤɧɭɬɵ ɞɜɚɠɞɵ ɢ ɨɫɬɚɥɶɧɵɟ – ɧɟɡɚɱɟɪɤɧɭɬɵɦɢ. ɗɬɚɩ 3 Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ ɫijr ɷɥɟɦɟɧɬ ɩɪɢɜɟɞɟɧɧɨɣ ɦɚɬɪɢɰɵ C 2 , ɡɚɱɟɪɤɧɭɬɵɣ r
ɪɚɡ (r 0,1,2) ɧɚ 2-ɦ ɷɬɚɩɟ, ɢ ɩɨɥɨɠɢɦ D min cij0 , ɝɞɟ ɦɢɧɢɦɭɦ ɛɟɪɟɬɫɹ ɩɨ ɜɫɟɦ i, j , ɬ. ɟ. ɢɳɟɬɫɹ ɧɚɢɦɟɧɶɲɢɣ ɢɡ ɧɟɡɚɱɟɪɤɧɭɬɵɯ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ C 2 . ȿɫɥɢ ɬɟɩɟɪɶ ɩɪɨɜɟɫɬɢ ɩɟɪɟɫɱɟɬ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ ɋ 2 ɩɨ ɮɨɪɦɭɥɚɦ cij0 D , °° cijɧɨɜ ®cij1 !, ° 2 °¯cij D , ɬɨ ɬɚɤɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɹɜɥɹɸɬɫɹ ɫɥɟɞɫɬɜɢɟɦ ɩɪɢɦɟɧɟɧɢɹ Ɍɟɨɪɟɦɵ 1. ɉɟɪɟɫɱɢɬɚɟɦ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ C 2 ɢɡ ɩɪɢɦɟɪɚ 2 ɩɪɢ D 1 , ɜ ɢɬɨɝɟ ɩɨɥɭɱɢɦ ɩɪɢɜɟɞɟɧɧɭɸ ɦɚɬɪɢɰɭ § 3 0 0 0 0· ¸ ¨ ¨ 0 0 1 4 2¸ ɋ3 ¨0 3 3 2 6¸ . ¸ ¨ ¨0 2 0 4 6¸ ¨0 0 0 6 8¸ ¹ © ȼɫɟ ɟɟ ɧɭɥɢ ɫɨɞɟɪɠɚɬ, ɧɚɩɪɢɦɟɪ, 1-ɹ ɝɨɪɢɡɨɧɬɚɥɶ ɢ ɬɪɢ ɜɟɪɬɢɤɚɥɢ: 1-ɚɹ, 2-ɹ, 3-ɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɡɞɟɫɶ k 4 5 ɢ D 2 . ȿɳɟ ɨɞɢɧ ɩɟɪɟɫɱɟɬ ɞɚɟɬ ɩɪɢɜɟɞɟɧɧɭɸ ɦɚɬɪɢɰɭ §5 2 2 0 0· ¸ ¨ ¨0 0 1 2 0¸ C1 ¨ 0 3 3 0 4 ¸ . ¸ ¨ ¨ 0 2 0 2 4¸ ¨0 0 0 4 6¸ ¹ © Ɂɞɟɫɶ ɭɠɟ k n 5.
ɗɬɚɩ 4 Ɉɬɵɫɤɚɧɢɟ n ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɣ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɟɪɟɛɨɪɨɦ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɜɚɪɢɚɧɬɨɜ. ɍɞɨɛɧɨ ɩɟɪɟɛɨɪ ɧɚɱɢɧɚɬɶ ɫ ɨɬɵɫɤɚɧɢɹ ɫɬɪɨɤ ɢɥɢ ɫɬɨɥɛɰɨɜ, ɫɨɞɟɪɠɚɳɢɯ ɟɞɢɧɫɬɜɟɧɧɵɣ ɧɭɥɟɜɨɣ ɷɥɟɦɟɧɬ, ɩɨɫɤɨɥɶɤɭ ɬɚɤɨɣ ɷɥɟɦɟɧɬ ɨɛɹɡɚɬɟɥɶɧɨ ɜɨɣɞɟɬ ɜ ɝɪɭɩɩɭ ɧɟɡɚɜɢɫɢɦɵɯ. ȼɵɛɪɚɜ ɷɥɟɦɟɧɬ cls , ɢɫɤɥɸɱɚɸɬ ɢɡ ɪɚɫɫɦɨɬɪɟɧɢɹ l -ɸ ɫɬɪɨɤɭ ɢ s -ɣ ɫɬɨɥɛɟɰ, ɩɨɫɥɟ ɱɟɝɨ ɩɟɪɟɯɨ11
ɞɹɬ ɤ ɨɬɵɫɤɚɧɢɸ ɫɥɟɞɭɸɳɟɝɨ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɧɭɥɟɜɨɝɨ ɷɥɟɦɟɧɬɚ ɫɬɪɨɤɢ ɢɥɢ ɫɬɨɥɛɰɚ ɨɫɬɚɜɲɟɣɫɹ ɱɚɫɬɶ ɦɚɬɪɢɰɵ. Ⱦɟɣɫɬɜɭɹ ɩɨɞɨɛɧɵɦ ɨɛɪɚɡɨɦ, ɦɨɠɧɨ ɫɬɨɥɤɧɭɬɶɫɹ ɫɨ ɫɥɟɞɭɸɳɢɦɢ ɞɜɭɦɹ ɫɢɬɭɚɰɢɹɦɢ. 1. ȼ ɪɟɡɭɥɶɬɚɬɟ ɭɞɚɟɬɫɹ ɜɵɛɪɚɬɶ n ɧɟɡɚɜɢɫɢɦɵɯ ɩɭɬɟɣ. ɈɋɌȺɇɈȼ. ȼɵɩɢɫɚɬɶ ɨɬɜɟɬ. 2. ɇɚ ɧɟɤɨɬɨɪɨɦ ɲɚɝɟ ɜɫɟ ɨɫɬɚɜɲɢɟɫɹ ɫɬɪɨɤɢ ɢ ɫɬɨɥɛɰɵ ɫɨɞɟɪɠɚɬ ɛɨɥɟɟ ɨɞɧɨɝɨ ɧɭɥɟɜɨɝɨ ɷɥɟɦɟɧɬɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɵɛɢɪɚɟɬɫɹ ɥɸɛɨɣ ɧɭɥɟɜɨɣ ɷɥɟɦɟɧɬ, «ɩɨɦɟɱɚɟɬɫɹ» ɧɨɦɟɪ ɫɬɪɨɤɢ, ɢɡ ɤɨɬɨɪɨɣ ɜɵɛɪɚɧ 0, ɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɟɪɟɯɨɞ ɤ ɜɵɛɨɪɭ ɫɥɟɞɭɸɳɟɝɨ ɧɭɥɟɜɨɝɨ ɷɥɟɦɟɧɬɚ. ȿɫɥɢ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɭɞɚɟɬɫɹ ɜɵɛɪɚɬɶ ɜɫɟ n ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɣ, ɬɨ ɈɋɌȺɇɈȼ. ȼɵɩɢɫɚɬɶ ɨɬɜɟɬ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɫɥɟɞɭɟɬ ɜɨɡɜɪɚɬɢɬɶɫɹ ɤ ɩɨɦɟɱɟɧɧɨɣ ɫɬɪɨɤɟ ɢ ɜɵɛɪɚɬɶ ɢɡ ɧɟɟ ɞɪɭɝɨɣ ɧɭɥɟɜɨɣ ɷɥɟɦɟɧɬ. ɉɨ ɷɬɨɣ ɫɯɟɦɟ ɧɚɞɨ ɞɟɣɫɬɜɨɜɚɬɶ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɧɟ ɩɨɥɭɱɢɦ n ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɣ. ɉɪɢɦɟɪ 4. ȼɡɹɜ ɦɚɬɪɢɰɭ C 4 ɢɡ ɩɪɢɦɟɪɚ 3, ɡɚɦɟɱɚɟɦ, ɱɬɨ ɧɟɡɚɜɢɫɢɦɵɦɢ ɡɞɟɫɶ ɛɭɞɭɬ ɧɭɥɢ ɫ ɢɧɞɟɤɫɚɦɢ (1,5), (2,2), (3,4), (4,3), (5,1). ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɨɬɜɟɬɨɦ ɜ ɡɚɞɚɱɟ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɫ ɦɚɬɪɢɰɟɣ C ɢɡ ɩɪɢɦɟɪɚ 1 ɹɜɥɹɟɬɫɹ ɦɚɬɪɢɰɚ ɧɚɡɧɚɱɟɧɢɹ ɜɢɞɚ §0 0 0 0 1· ¸ ¨ ¨ 0 1 0 0 0¸ 1 ¨ 0 0 0 1 0¸ , X ɨɩɬɢɦ ¸ ¨ ¨0 0 1 1 0¸ ¨1 0 0 0 0¸ ¹ ©
ɞɥɹ ɤɨɬɨɪɨɣ Lmin X 1 14 . ȼɚɠɧɨ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɧɚɪɹɞɭ ɫ ɭɤɚɡɚɧɧɵɦɢ ɡɞɟɫɶ ɛɭɞɭɬ ɬɚɤɠɟ ɧɟɡɚɜɢɫɢɦɵɦɢ ɧɭɥɢ ɫ ɢɧɞɟɤɫɚɦɢ (1,4), (2,5), (3,1), (4,3), (5,2). ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɬɚ ɡɚɞɚɱɚ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɢɦɟɟɬ ɟɳɟ ɨɞɢɧ ɨɬɜɟɬ — ɦɚɬɪɢɰɭ ɧɚɡɧɚɱɟɧɢɣ ɜɢɞɚ §0 0 0 1 0· ¸ ¨ ¨0 0 0 0 1¸ 2 ¨1 0 0 0 0¸ , X ɨɩɬɢɦ ¸ ¨ ¨0 0 1 0 0¸ ¨0 1 0 0 0¸ ¹ ©
ɩɪɢɱɟɦ Lmin X 2
14 .
Ɉɬɜɟɬ: Lmin X 14 ɩɪɢ X
X 1 ɢɥɢ X
12
X 2.
ɍɉɊȺɀɇȿɇɂə
1. ɍɤɚɠɢɬɟ ɤɨɥɢɱɟɫɬɜɨ ɧɟɡɚɜɢɫɢɦɵɯ ɟɞɢɧɢɰ ɜ ɦɚɬɪɢɰɟ § 0 1 0 0· ¨ ¸ ¨ 0 0 1 0¸ ¨1 0 0 0¸ . ¨¨ ¸¸ ©0 0 0 1¹ 2. əɜɥɹɟɬɫɹ ɥɢ ɦɚɬɪɢɰɚ ɩɪɢɜɟɞɟɧɧɨɣ? ɍɤɚɠɢɬɟ ɤɨɥɢɱɟɫɬɜɨ ɧɟɡɚɜɢɫɢɦɵɯ ɧɭɥɟɣ ɜ ɧɟɣ 3 2 4· §0 1 § 2 0 0 0 0· ¨ ¸ ¨ ¸ 0 3¸ ¨1 1 6 ¨ 0 1 2 5 3¸ ¨0 4 4 3 7¸, b) ¨ 3 0 4 3 8 ¸ . a) ¨ ¸ ¨ ¸ 0 5 7¸ ¨2 5 ¨0 3 1 5 7¸ ¨4 3 ¨0 1 1 7 9¸ 8 7 0 ¸¹ © © ¹ 3. Ɋɟɲɢɬɟ ɡɚɞɚɱɭ ɨ ɧɚɡɧɚɱɟɧɢɹɯ ɪɚɡɦɟɪɚ 6 u 6 ɫ ɦɚɬɪɢɰɟɣ C ɜɢɞɚ §2 2 1 2 1 0· § 3 4 2 1 3 0· ¸ ¨ ¸ ¨ ¨1 3 5 6 9 4¸ ¨ 4 3 6 7 3 4¸ ¨2 7 8 5 3 6¸ ¨ 7 6 2 6 3 5¸ ¸, ¸, a) C ¨ b) C ¨ ¨1 5 4 7 8 3¸ ¨ 1 4 7 8 3 2¸ ¨1 3 4 9 5 6¸ ¨ 3 2 4 8 4 6¸ ¸ ¨ ¸ ¨ ¨2 4 6 5 3 1¸ ¨1 6 5 4 9 2¸ ¹ © ¹ ©
c) C
§2 ¨ ¨3 ¨6 ¨ ¨3 ¨8 ¨ ¨1 ©
3 4 2 4 0· ¸ 5 6 7 4 4¸ 7 2 6 2 5¸ ¸, 5 7 4 3 2¸ 3 5 2 1 6¸ ¸ 5 6 4 9 2 ¸¹
d) C
§4 ¨ ¨1 ¨4 ¨ ¨5 ¨3 ¨ ¨2 ©
3 2 1 2 3· ¸ 4 5 8 9 6¸ 5 8 9 6 7¸ ¸. 4 8 6 7 9¸ 4 9 8 5 7¸ ¸ 4 4 4 9 8 ¸¹
3. ɆȿɌɈȾ ȼȿɌȼȿɃ ɂ ȽɊȺɇɂɐ 3.1. Ɉɛɳɚɹ ɫɯɟɦɚ ɦɟɬɨɞɚ
ɋɪɟɞɢ ɤɨɦɛɢɧɚɬɨɪɧɵɯ ɦɟɬɨɞɨɜ ɨɫɨɛɨɣ ɩɨɩɭɥɹɪɧɨɫɬɶɸ ɩɨɥɶɡɭɟɬɫɹ ɦɟɬɨɞ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ. ɗɬɨ ɫɜɹɡɚɧɨ, ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ, ɫɨ ɫɥɟɞɭɸɳɢɦɢ ɨɫɨɛɟɧɧɨɫɬɹɦɢ ɷɬɨɣ ɝɪɭɩɩɵ ɦɟɬɨɞɨɜ. 13
1. ɋɬɪɭɤɬɭɪɚ ɦɟɬɨɞɚ ɞɨɫɬɚɬɨɱɧɨ ɭɧɢɜɟɪɫɚɥɶɧɚ ɢ ɦɨɠɟɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɚ ɞɥɹ ɪɟɲɟɧɢɹ ɲɢɪɨɤɨɝɨ ɤɥɚɫɫɚ ɡɚɞɚɱ. 2. Ɇɟɬɨɞ ɨɛɥɚɞɚɟɬ ɩɨɬɟɧɰɢɚɥɶɧɵɦɢ ɜɨɡɦɨɠɧɨɫɬɹɦɢ ɞɥɹ ɫɭɳɟɫɬɜɟɧɧɨɝɨ ɫɨɤɪɚɳɟɧɢɹ ɩɟɪɟɛɨɪɚ. 3. ȼɧɟɲɧɟɟ ɭɩɪɚɜɥɟɧɢɟ ɩɚɪɚɦɟɬɪɚɦɢ ɦɟɬɨɞɚ ɩɨɡɜɨɥɹɟɬ ɮɨɪɦɢɪɨɜɚɬɶ ɪɚɡɥɢɱɧɵɟ ɩɪɢɛɥɢɠɟɧɧɵɟ ɚɥɝɨɪɢɬɦɵ. Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɞɢɫɤɪɟɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɜɢɞɚ M ( x) o min, x:
(2.1)
ɝɞɟ : – ɧɟɤɨɬɨɪɨɟ ɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɢɡ R n . Ⱦɥɹ ɪɟɲɟɧɢɹ ɞɚɧɧɨɝɨ ɤɥɚɫɫɚ ɡɚɞɚɱ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɦɟɬɨɞ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ, ɜ ɨɫɧɨɜɟ ɤɨɬɨɪɨɝɨ ɥɟɠɚɬ ɫɥɟɞɭɸɳɢɟ ɨɫɧɨɜɧɵɟ ɦɨɞɭɥɢ: 1) ɩɨɫɬɪɨɟɧɢɟ ɞɟɪɟɜɚ ɞɨɩɭɫɬɢɦɵɯ ɜɚɪɢɚɧɬɨɜ; 2) ɫɨɫɬɚɜɥɟɧɢɟ ɨɰɟɧɨɱɧɵɯ ɡɚɞɚɱ; 3) ɨɩɪɟɞɟɥɟɧɢɟ ɩɪɚɜɢɥɚ ɨɛɯɨɞɚ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ; 4) ɨɬɛɪɚɫɵɜɚɧɢɟ «ɧɟɩɟɪɫɩɟɤɬɢɜɧɵɯ» ɦɧɨɠɟɫɬɜ ɜɚɪɢɚɧɬɨɜ; 5) ɩɪɨɜɟɪɤɚ ɧɚ ɨɫɬɚɧɨɜ. Ɋɚɫɫɦɨɬɪɢɦ ɩɨɞɪɨɛɧɨ ɤɚɠɞɵɣ ɢɡ ɭɤɚɡɚɧɧɵɯ ɦɨɞɭɥɟɣ. 1. ɉɨɫɬɪɨɟɧɢɟ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɪɚɡɛɢɟɧɢɹ ɦɧɨɠɟɫɬɜ ɧɚ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɩɨɞɦɧɨɠɟɫɬɜ. Ɏɚɤɬ ɪɚɡɛɢɟɧɢɹ ɦɧɨɠɟɫɬɜɚ ɧɚɡɵɜɚɟɬɫɹ ɜɟɬɜɥɟɧɢɟɦ ɢ ɩɪɨɢɡɜɨɞɢɬɫɹ ɧɚ ɨɱɟɪɟɞɧɨɦ ɲɚɝɟ ɦɟɬɨɞɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɦ ɡɚɪɚɧɟɟ ɩɪɚɜɢɥɨɦ. ɗɬɨ ɩɪɚɜɢɥɨ ɫɜɹɡɚɧɨ ɫɨ ɫɩɟɰɢɮɢɤɨɣ ɤɨɧɤɪɟɬɧɨɝɨ ɞɨɩɭɫɬɢɦɨɝɨ ɦɧɨɠɟɫɬɜɚ ɢɫɫɥɟɞɭɟɦɨɣ ɡɚɞɚɱɢ. Ɉɞɧɚɤɨ ɟɫɥɢ ɢɫɯɨɞɧɚɹ ɡɚɞɚɱɚ ɫɪɟɞɢ ɩɪɨɱɢɯ ɢɦɟɟɬ ɨɝɪɚɧɢɱɟɧɢɹ ɜɢɞɚ x j {1,0}, j 1, n (ɬ. ɟ. ɹɜɥɹɟɬɫɹ ɡɚɞɚɱɟɣ ɫ ɛɭɥɟɜɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ), ɬɨ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɭɧɢɜɟɪɫɚɥɶɧɵɦ ɩɪɚɜɢɥɨɦ ɜɟɬɜɥɟɧɢɹ: : : 1 : 2 , ɝɞɟ : 1 {x : : xi 1}; : 2 {x : : xi 0}; i – ɡɚɮɢɤɫɢɪɨɜɚɧɧɵɣ ɧɨɦɟɪ ɤɨɨɪɞɢɧɚɬɵ 1 d i d n . ɉɨɪɹɞɨɤ ɮɢɤɫɢɪɨɜɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɨɝɨɜɚɪɢɜɚɟɬɫɹ ɜ ɩɪɚɜɢɥɟ ɜɟɬɜɥɟɧɢɹ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɚɜɢɥɨ ɪɚɡɛɢɟɧɢɹ ɦɧɨɠɟɫɬɜɚ ɧɚ ɩɨɞɦɧɨɠɟɫɬɜɚ ɜ ɤɨɧɤɪɟɬɧɨɣ ɡɚɞɚɱɟ ɦɨɠɟɬ ɛɵɬɶ ɧɟ ɟɞɢɧɫɬɜɟɧɧɵɦ. Ɉɰɟɧɤɨɣ ɮɭɧɤɰɢɢ M (x ) ɡɚɞɚɱɢ (1) ɧɚ ɦɧɨɠɟɫɬɜɟ : ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɟ 2. ɱɢɫɥɨ [ [ (:) , ɱɬɨ M ( x) t [ , x : . Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɨɰɟɧɨɤ ɫɨɫɬɚɜɥɹɟɬɫɹ ɨɰɟɧɨɱɧɚɹ ɡɚɞɚɱɚ, ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɣ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɨɰɟɧɤɢ. Ʉ ɨɰɟɧɨɱɧɵɦ ɡɚɞɚɱɚɦ ɩɪɟɞɴɹɜɥɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɬɪɟɛɨɜɚɧɢɹ: ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɢɯ ɪɟɲɟɧɢɟ ɧɟ ɞɨɥɠɧɨ ɡɚɧɢɦɚɬɶ ɦɧɨɝɨ ɜɪɟɦɟɧɢ. ɇɨ ɜɦɟɫɬɟ ɫ ɬɟɦ ɨɰɟɧɤɢ, ɩɨɥɭɱɟɧɧɵɟ ɫ ɢɯ ɩɨɦɨɳɶɸ, ɞɨɥɠɧɵ ɛɵɬɶ ɤɚɤ ɦɨɠɧɨ ɬɨɱɧɟɟ (ɬ. ɟ. ɪɚɡɧɨɫɬɶ ( [ (:) min M ( x ) ) ɧɟ ɞɨɥɠɧɚ ɛɵɬɶ ɫɥɢɲɤɨɦ :
ɛɨɥɶɲɨɣ). Ɍɚɤɢɟ ɬɪɟɛɨɜɚɧɢɹ ɨɛɴɹɫɧɹɸɬɫɹ ɬɟɦ, ɱɬɨ ɢɦɟɧɧɨ ɨɰɟɧɤɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɪɢ ɨɬɛɪɚɫɵɜɚɧɢɢ ɧɟɩɟɪɫɩɟɤɬɢɜɧɵɯ ɦɧɨɠɟɫɬɜ (ɩɪɢ ɫɨɤɪɚɳɟɧɢɢ ɩɟɪɟɛɨɪɚ). ɉɪɢ ɫɨɫɬɚɜɥɟɧɢɢ ɨɰɟɧɨɱɧɵɯ ɡɚɞɚɱ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ 14
ɭɧɢɜɟɪɫɚɥɶɧɵɦ ɩɪɚɜɢɥɨɦ: ɨɬɛɪɨɫɢɬɶ ɜ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɟ ɧɚɢɛɨɥɟɟ «ɬɹɠɟɥɵɟ» (ɬɪɭɞɧɨɜɵɩɨɥɧɢɦɵɟ) ɨɝɪɚɧɢɱɟɧɢɹ (ɧɚɩɪɢɦɟɪ, ɬɪɟɛɨɜɚɧɢɟ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ). ɉɨɥɭɱɚɟɦɵɟ ɨɰɟɧɤɢ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɬɪɟɛɨɜɚɧɢɸ ɦɨɧɨɬɨɧɧɨɫɬɢ ɜ ɬɨɦ ɫɦɵɫɥɟ, ɱɬɨ ɨɰɟɧɤɢ ɩɨɞɦɧɨɠɟɫɬɜ ɧɟ ɞɨɥɠɧɵ ɛɵɬɶ ɦɟɧɶɲɟ ɨɰɟɧɤɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɪɚɡɜɟɬɜɥɟɧɧɨɝɨ ɦɧɨɠɟɫɬɜɚ (ɬɨ ɟɫɬɶ ɟɫɥɢ : : k , ɬɨ ɞɨɥɠɧɨ ɛɵɬɶ ɜɵɩɨɥɧɟɧɨ ɭɫɥɨɜɢɟ [ (:) d [ (: k )k ). k
3. ɉɪɚɜɢɥɨ ɨɛɯɨɞɚ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ ɜ ɩɪɨɰɟɫɫɟ ɪɚɛɨɬɵ ɚɥɝɨɪɢɬɦɚ ɧɚɡɵɜɚɸɬ ɫɬɪɚɬɟɝɢɟɣ ɨɛɯɨɞɚ ɞɟɪɟɜɚ. ɋɭɳɟɫɬɜɭɟɬ ɦɧɨɠɟɫɬɜɨ ɪɚɡɥɢɱɧɵɯ ɫɬɪɚɬɟɝɢɣ. ɇɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɦɢ ɹɜɥɹɸɬɫɹ ɬɪɢ ɢɡ ɧɢɯ. a. «ɉɨ ɦɢɧɢɦɭɦɭ ɨɰɟɧɤɢ». ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɥɹ ɩɨɫɥɟɞɭɸɳɟɝɨ ɪɚɡɛɢɟɧɢɹ ɜɵɛɢɪɚɟɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɨ, ɢɦɟɸɳɟɟ ɤ ɞɚɧɧɨɦɭ ɲɚɝɭ ɚɥɝɨɪɢɬɦɚ ɦɢɧɢɦɚɥɶɧɭɸ ɨɰɟɧɤɭ. b. Ɉɞɧɨɫɬɨɪɨɧɧɢɣ ɨɛɯɨɞ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ. Ⱦɥɹ ɩɨɫɥɟɞɭɸɳɟɝɨ ɪɚɡɛɢɟɧɢɹ ɜɵɛɢɪɚɟɬɫɹ ɨɞɧɨ ɢɡ ɩɨɞɦɧɨɠɟɫɬɜ, ɩɨɥɭɱɟɧɧɵɯ ɧɚ ɩɪɟɞɵɞɭɳɟɦ ɲɚɝɟ (ɧɚɩɪɢɦɟɪ, ɩɨɞɦɧɨɠɟɫɬɜɨ, ɜ ɤɨɬɨɪɨɦ xl 1 , 1 d l d n ). ȿɫɥɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɜɟɬɜɶ ɞɟɪɟɜɚ ɨɤɚɡɵɜɚɟɬɫɹ ɩɪɨɣɞɟɧɧɨɣ ɞɨ ɤɨɧɰɚ (ɢɥɢ ɨɬɛɪɨɲɟɧɚ ɤɚɤ ɧɟɩɟɪɫɩɟɤɬɢɜɧɚɹ), ɬɨ ɩɪɨɢɫɯɨɞɢɬ ɜɨɡɜɪɚɳɟɧɢɟ ɤ ɛɥɢɠɚɣɲɟɦɭ ɢɡ ɩɪɟɞɵɞɭɳɢɯ ɲɚɝɨɜ, ɝɞɟ ɫɨɯɪɚɧɢɥɚɫɶ ɚɥɶɬɟɪɧɚɬɢɜɚ. c. ɋɦɟɲɚɧɧɚɹ ɫɬɪɚɬɟɝɢɹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɥɹ ɩɪɨɞɜɢɠɟɧɢɹ «ɜɧɢɡ ɩɨ ɞɟɪɟɜɭ» ɢɫɩɨɥɶɡɭɟɬɫɹ ɨɞɧɨɫɬɨɪɨɧɧɢɣ ɨɛɯɨɞ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ, ɚ ɩɪɢ «ɩɨɞɴɟɦɟ ɜɜɟɪɯ» ɢɳɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɫ ɦɢɧɢɦɚɥɶɧɨɣ ɨɰɟɧɤɨɣ. 4. Ȼɭɞɟɦ ɧɚɡɵɜɚɬɶ ɦɧɨɠɟɫɬɜɨ ɧɟɩɟɪɫɩɟɤɬɢɜɧɵɦ, ɟɫɥɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɝɨ ɩɪɢɧɢɦɚɟɬɫɹ ɪɟɲɟɧɢɟ ɨ ɜɵɛɪɚɫɵɜɚɧɢɢ ɟɝɨ ɢɡ ɞɚɥɶɧɟɣɲɟɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ. Ɉɬɛɪɚɫɵɜɚɧɢɟ ɦɧɨɠɟɫɬɜ ɜ ɯɨɞɟ ɪɟɲɟɧɢɹ ɨɛɟɫɩɟɱɢɜɚɟɬ ɫɨɤɪɚɳɟɧɢɟ ɩɟɪɟɛɨɪɚ. ɂɫɤɥɸɱɟɧɢɟ ɦɧɨɠɟɫɬɜ ɜɚɪɢɚɧɬɨɜ ɢɡ ɞɚɥɶɧɟɣɲɟɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ ɩɪɨɢɡɜɨɞɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɨɰɟɧɨɤ ɷɬɢɯ ɦɧɨɠɟɫɬɜ ɢ ɪɟɤɨɪɞɚ. Ɋɟɤɨɪɞɨɦ (ɢɥɢ ɪɟɤɨɪɞɧɵɦ ɡɧɚɱɟɧɢɟɦ) ɧɚɡɵɜɚɸɬ ɡɧɚɱɟɧɢɟ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɜ «ɥɭɱɲɟɣ» (ɞɨɫɬɚɜɥɹɸɳɟɣ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ) ɢɡ ɩɨɥɭɱɟɧɧɵɯ ɞɨɩɭɫɬɢɦɵɯ ɬɨɱɟɤ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɧɚɱɚɥɶɧɨɝɨ ɪɟɤɨɪɞɚ ɪɟɤɨɦɟɧɞɭɟɬɫɹ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɤɚɤɢɦ-ɥɢɛɨ ɩɪɢɛɥɢɠɟɧɧɵɦ ɚɥɝɨɪɢɬɦɨɦ ɢɥɢ ɞɪɭɝɨɣ ɚɩɪɢɨɪɧɨɣ ɢɧɮɨɪɦɚɰɢɟɣ, ɟɫɥɢ ɨɧɚ ɢɦɟɟɬɫɹ. ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɪɟɤɨɪɞ ɨɛɧɨɜɥɹɟɬɫɹ. ɋɩɪɚɜɟɞɥɢɜɨ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ. ȿɫɥɢ ɨɰɟɧɤɚ ɧɟɤɨɬɨɪɨɝɨ ɩɨɞɦɧɨɠɟɫɬɜɚ ɛɨɥɶɲɟ ɢɦɟɸɳɟɝɨɫɹ ɪɟɤɨɪɞɚ, ɬɨ ɫɪɟɞɢ ɬɨɱɟɤ ɞɚɧɧɨɝɨ ɩɨɞɦɧɨɠɟɫɬɜɚ ɧɟɬ ɨɩɬɢɦɚɥɶɧɵɯ ɬɨɱɟɤ (ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ). ɗɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɩɨɡɜɨɥɹɟɬ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɨɫɧɨɜɧɨɟ ɩɪɚɜɢɥɨ ɫɨɤɪɚɳɟɧɢɹ ɩɟɪɟɛɨɪɚ: ɟɫɥɢ ɨɰɟɧɤɚ ɦɧɨɠɟɫɬɜɚ ɛɨɥɶɲɟ ɪɟɤɨɪɞɧɨɝɨ ɡɧɚɱɟɧɢɹ, ɬɨ ɬɚɤɨɟ ɦɧɨɠɟɫɬɜɨ ɜɚɪɢɚɧɬɨɜ ɜɵɛɪɚɫɵɜɚɟɬɫɹ ɢɡ ɞɚɥɶɧɟɣɲɟɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɦɨɠɟɬ ɧɟ ɩɨɞɜɟɪɝɚɬɶɫɹ ɩɨɫɥɟɞɭɸɳɟɦɭ ɜɟɬɜɥɟɧɢɸ ɢ ɩɨ ɞɪɭɝɢɦ ɩɪɢɱɢɧɚɦ: – ɟɫɥɢ ɩɪɢ ɪɟɲɟɧɢɢ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɢ ɧɚ ɞɚɧɧɨɦ ɦɧɨɠɟɫɬɜɟ ɛɵɥɚ ɩɨɥɭɱɟɧɚ ɬɨɱɤɚ, ɹɜɥɹɸɳɚɹɫɹ ɞɨɩɭɫɬɢɦɨɣ ɜ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɟ; 15
– ɟɫɥɢ ɞɨɩɭɫɬɢɦɨɟ ɦɧɨɠɟɫɬɜɨ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɢ ɩɭɫɬɨ. 5. Ɋɚɛɨɬɚ ɦɟɬɨɞɚ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɪɢɬɟɪɢɟɦ ɨɩɬɢɦɚɥɶɧɨɫɬɢ: ɨɰɟɧɤɢ ɜɫɟɯ ɩɨɞɦɧɨɠɟɫɬɜ ɧɟ ɦɟɧɶɲɟ ɢɦɟɸɳɟɝɨɫɹ ɪɟɤɨɪɞɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢɡɧɚɤɨɦ ɨɫɬɚɧɨɜɚ ɹɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɚɹ ɫɢɬɭɚɰɢɹ: ɧɟ ɨɫɬɚɥɨɫɶ ɧɢ ɨɞɧɨɝɨ ɩɟɪɫɩɟɤɬɢɜɧɨɝɨ ɦɧɨɠɟɫɬɜɚ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɛɵɬɶ ɩɨɞɜɟɪɝɧɭɬɨ ɩɨɫɥɟɞɭɸɳɟɦɭ ɜɟɬɜɥɟɧɢɸ. Ɋɟɲɟɧɢɟɦ ɩɪɢ ɷɬɨɦ ɹɜɥɹɟɬɫɹ ɬɨɱɤɚ, ɜ ɤɨɬɨɪɨɣ ɩɨɥɭɱɟɧɨ ɩɨɫɥɟɞɧɟɟ ɪɟɤɨɪɞɧɨɟ ɡɧɚɱɟɧɢɟ. ȼ ɩɪɟɞɫɬɚɜɥɟɧɧɵɯ ɜɵɲɟ ɩɹɬɢ ɩɭɧɤɬɚɯ ɨɩɢɫɚɧɵ ɨɫɧɨɜɧɵɟ ɦɨɞɭɥɢ, ɫ ɩɨɦɨɳɶɸ ɤɨɬɨɪɵɯ ɦɨɠɟɬ ɛɵɬɶ ɫɨɫɬɚɜɥɟɧɚ ɫɯɟɦɚ ɪɚɛɨɬɵ ɥɸɛɨɝɨ ɜɚɪɢɚɧɬɚ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ. Ʉɚɠɞɵɣ ɤɨɧɤɪɟɬɧɵɣ ɚɥɝɨɪɢɬɦɢɱɟɫɤɢɣ ɜɚɪɢɚɧɬ ɬɪɟɛɭɟɬ ɢɧɞɢɜɢɞɭɚɥɶɧɨɝɨ «ɧɚɩɨɥɧɟɧɢɹ» ɜɫɟɯ ɦɨɞɭɥɟɣ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɪɟɲɚɟɦɨɣ ɡɚɞɚɱɟ. Ⱥɥɝɨɪɢɬɦɢɱɟɫɤɚɹ ɫɯɟɦɚ ɦɟɬɨɞɚ
Ɋɟɲɚɟɬɫɹ ɡɚɞɚɱɚ ɜɢɞɚ: M ( x) o min . x:0
ɒɚɝ 1. ɂɧɢɰɢɚɥɢɡɚɰɢɹ. Ɂɚɞɚɬɶ ɧɚɱɚɥɶɧɨɟ ɪɟɤɨɪɞɧɨɟ ɡɧɚɱɟɧɢɟ R. ȿɫɥɢ ɨɬɵɫɤɚɧɢɟ ɧɚɱɚɥɶɧɨɝɨ ɪɟɤɨɪɞɚ ɡɚɬɪɭɞɧɢɬɟɥɶɧɨ, ɩɨɥɨɠɢɬɶ R f ɉɨɥɨɠɢɬɶ I – ɦɧɨɠɟɫɬɜɨ ɧɨɦɟɪɨɜ ɩɨɞɦɧɨɠɟɫɬɜ, ɩɨɞɥɟɠɚɳɢɯ ɜɟɬɜɥɟɧɢɸ, J {0} — ɦɧɨɠɟɫɬɜɨ ɧɨɦɟɪɨɜ ɩɨɞɦɧɨɠɟɫɬɜ, ɞɥɹ ɤɨɬɨɪɵɯ ɛɭɞɭɬ ɪɟɲɚɬɶɫɹ ɨɰɟɧɨɱɧɵɟ ɡɚɞɚɱɢ. ɒɚɝ 2. ȼɵɱɢɫɥɟɧɢɟ ɨɰɟɧɨɤ. Ɋɟɲɢɬɶ ɨɰɟɧɨɱɧɵɟ ɡɚɞɚɱɢ ɞɥɹ ɦɧɨɠɟɫɬɜ : j , ɝɞɟ j J . ȼɵɱɢɫɥɢɬɶ
[ (: j ) [ J , j J . ɒɚɝ 3. Ɉɛɧɨɜɥɟɧɢɟ ɪɟɤɨɪɞɚ. ȿɫɥɢ ɧɚ ɲɚɝɟ 2 ɩɨɥɭɱɟɧɵ ɞɨɩɭɫɬɢɦɵɟ ɬɨɱɤɢ x p , p 1, P , ɬɨ ɩɨɥɨɠɢɬɶ
R= min{R, min M ( x p )} . p
ɒɚɝ 4. ɋɨɤɪɚɳɟɧɢɟ ɩɟɪɟɛɨɪɚ. Ɉɫɭɳɟɫɬɜɢɬɶ ɡɚɤɪɵɬɢɟ ɧɟɩɟɪɫɩɟɤɬɢɜɧɵɯ ɦɧɨɠɟɫɬɜ (ɜɤɥɸɱɚɹ ɬɟ ɧɨɦɟɪɚ i , ɞɥɹ ɤɨɬɨɪɵɯ [ i R ). ɍɞɚɥɢɬɶ ɢɯ ɧɨɦɟɪɚ ɢɡ ɦɧɨɠɟɫɬɜ I ɢ J. ɉɨɥɨɠɢɬɶ I I J , J . ȿɫɥɢ I , ɬɨ ɩɟɪɟɣɬɢ ɤ ɲɚɝɭ 7. ɒɚɝ 5. Ɋɟɚɥɢɡɚɰɢɹ ɫɬɪɚɬɟɝɢɢ. ȼɵɛɪɚɬɶ ɢɡ ɦɧɨɠɟɫɬɜɚ I ɧɨɦɟɪ k – ɢɧɞɟɤɫ ɩɨɞɦɧɨɠɟɫɬɜɚ : k , ɩɨɞɥɟɠɚɳɟɝɨ ɜɟɬɜɥɟɧɢɸ ɧɚ ɞɚɧɧɨɦ ɷɬɚɩɟ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɮɢɤɫɢɪɨɜɚɧɧɵɦ ɩɪɚɜɢɥɨɦ. ɒɚɝ 6. ȼɟɬɜɥɟɧɢɟ. ɧɚ ɩɨɞɦɧɨɠɟɫɬɜɚ Ɉɫɭɳɟɫɬɜɢɬɶ ɪɚɡɛɢɟɧɢɟ ɦɧɨɠɟɫɬɜɚ :k : k : ks . ɉɨɥɨɠɢɬɶ J J {k s , s 1, S} . ɉɟɪɟɣɬɢ ɤ ɲɚɝɭ 2. s 1,S
16
ɒɚɝ 7. Ɉɫɬɚɧɨɜ, M min R . Ɂɚɦɟɬɢɦ, ɱɬɨ ɬɚɤɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɲɚɝɨɜ ɦɨɠɟɬ ɛɵɬɶ ɧɟ ɪɚɰɢɨɧɚɥɶɧɨɣ ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ɩɪɢ ɜɵɱɢɫɥɟɧɢɢ ɨɰɟɧɨɤ ɡɚɜɟɞɨɦɨ ɧɟ ɦɨɝɭɬ ɩɨɹɜɢɬɶɫɹ ɞɨɩɭɫɬɢɦɵɟ ɬɨɱɤɢ (ɧɚɩɪɢɦɟɪ, ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ). ɍɉɊȺɀɇȿɇɂə
1. Ⱦɨɤɚɠɢɬɟ ɫɜɨɣɫɬɜɨ ɦɨɧɨɬɨɧɧɨɫɬɢ ɨɰɟɧɨɤ ɜ ɭɫɥɨɜɢɹɯ, ɩɪɢ ɤɨɬɨɪɵɯ ɜɟɬɜɥɟɧɢɟ ɢ ɫɨɫɬɚɜɥɟɧɢɟ ɨɰɟɧɨɱɧɵɯ ɡɚɞɚɱ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɨ ɩɪɚɜɢɥɚɦ, ɭɤɚɡɚɧɧɵɦ ɜ ɩɩ. 1 ɢ 2 ɨɩɢɫɚɧɢɹ ɨɫɧɨɜɧɵɯ ɦɨɞɭɥɟɣ. 2. ɉɪɟɞɥɨɠɢɬɟ ɞɪɭɝɢɟ ɫɬɪɚɬɟɝɢɢ ɨɛɯɨɞɚ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ. 3. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɨɫɧɨɜɧɨɝɨ ɩɪɚɜɢɥɚ ɨɬɛɪɚɫɵɜɚɧɢɹ ɧɟɩɟɪɫɩɟɤɬɢɜɧɵɯ ɦɧɨɠɟɫɬɜ (ɩ. 4) ɧɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨɬɟɪɢ ɪɟɲɟɧɢɹ. 4. ȼ ɩɪɟɞɥɨɠɟɧɧɨɣ ɚɥɝɨɪɢɬɦɢɱɟɫɤɨɣ ɫɯɟɦɟ ɨɬɵɫɤɢɜɚɟɬɫɹ ɨɞɧɨ ɪɟɲɟɧɢɟ, ɞɚɠɟ ɟɫɥɢ ɨɧɨ ɜ ɡɚɞɚɱɟ ɧɟ ɟɞɢɧɫɬɜɟɧɧɨ. Ƚɞɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨɬɟɪɹ ɞɪɭɝɢɯ ɪɟɲɟɧɢɣ? ɂɫɩɪɚɜɶɬɟ ɚɥɝɨɪɢɬɦ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɩɨɹɜɢɥɚɫɶ ɜɨɡɦɨɠɧɨɫɬɶ ɨɬɵɫɤɚɧɢɹ ɜɫɟɯ ɪɟɲɟɧɢɣ ɡɚɞɚɱɢ. 3.2. Ɇɟɬɨɞ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ
ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ ɫɨɫɬɨɢɬ ɜ ɫɥɟɞɭɸɳɟɦ. ɂɦɟɟɬɫɹ n ɝɨɪɨɞɨɜ. Ɂɚɞɚɧɚ ɦɚɬɪɢɰɚ ɪɚɫɫɬɨɹɧɢɣ ɦɟɠɞɭ ɧɢɦɢ: C (cij ), i, j 1, n . Cɱɢɬɚɟɦ, ɱɬɨ cij t 0, i, j . ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɜɨɡɦɨɠɧɨ, ɱɬɨ cij z c ji . Ʉɪɨɦɟ ɬɨɝɨ, ɛɭɞɟɦ ɩɨɥɚɝɚɬɶ, ɱɬɨ cii f, i . ɂɳɟɬɫɹ ɤɪɚɬɱɚɣɲɢɣ ɡɚɦɤɧɭɬɵɣ ɦɚɪɲɪɭɬ (ɰɢɤɥ), ɩɪɨɯɨɞɹɳɢɣ ɱɟɪɟɡ ɤɚɠɞɵɣ ɝɨɪɨɞ ɪɨɜɧɨ ɨɞɢɧ ɪɚɡ ɢ ɦɢɧɢɦɢɡɢɪɭɸɳɢɣ ɫɭɦɦɚɪɧɨɟ ɩɪɨɣɞɟɧɧɨɟ ɪɚɫɫɬɨɹɧɢɟ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɚ, ɧɚɩɪɢɦɟɪ, ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. n
n
¦¦ c
L
ij
xij o min ,
i 1 j 1
n
¦ xij
1,
j 1, n,
i 1 n
¦ xij
1, i 1, n,
j 1
xij {0,1}, i, j 1, n. ȼ ɷɬɨɣ ɩɨɫɬɚɧɨɜɤɟ ɧɟ ɭɱɢɬɵɜɚɟɬɫɹ ɟɫɬɟɫɬɜɟɧɧɨɟ ɬɪɟɛɨɜɚɧɢɟ ɫɜɹɡɧɨɫɬɢ ɦɚɪɲɪɭɬɚ (ɨɬɫɭɬɫɬɜɢɹ ɩɨɞɰɢɤɥɨɜ), ɧɨ ɜ ɞɚɥɶɧɟɣɲɟɦ ɨɧɨ ɛɭɞɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɚɥɝɨɪɢɬɦɢɱɟɫɤɢ. 17
Ɉɩɪɟɞɟɥɟɧɢɟ. Ɇɚɬɪɢɰɚ C ɧɚɡɵɜɚɟɬɫɹ ɩɪɢɜɟɞɟɧɧɨɣ, ɟɫɥɢ ɜɫɟ ɟɟ ɷɥɟɦɟɧɬɵ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ, ɚ ɤɚɠɞɚɹ ɫɬɪɨɤɚ ɢ ɤɚɠɞɵɣ ɫɬɨɥɛɟɰ ɫɨɞɟɪɠɚɬ ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɩɨ ɨɞɧɨɦɭ ɧɭɥɟɜɨɦɭ ɷɥɟɦɟɧɬɭ. ɉɪɢɜɟɞɟɧɢɟ ɦɚɬɪɢɰɵ ɦɨɠɟɬ ɛɵɬɶ ɨɫɭɳɟɫɬɜɥɟɧɨ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. C (cij ), cij t 0, i, j 1, n . ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɦɚɬɪɢɰɚ ɇɚɣɞɟɦ
min cij j
D i , cij ' cij D i , i, j. ɉɨɥɭɱɢɦ ɦɚɬɪɢɰɭ C ' (cij ' ) , ɤɨɬɨɪɚɹ ɜ ɤɚɠ-
ɞɨɣ ɫɬɪɨɤɟ ɫɨɞɟɪɠɢɬ ɧɭɥɟɜɵɟ ɷɥɟɦɟɧɬɵ. ɇɚɣɞɟɦ ɞɚɥɟɟ min cij E j , cij ' ' cij ' E j , i, j. ɉɨɥɭɱɟɧɧɚɹ ɦɚɬɪɢɰɚ C ' ' ɹɜɥɹɟɬɫɹ ɩɪɢɜɟɞɟɧi
¦D i ¦ E j
ɧɨɣ, ɚ ɫɭɦɦɚ S
i
ɧɚɡɵɜɚɟɬɫɹ ɫɭɦɦɨɣ ɩɪɢɜɨɞɹɳɢɯ ɤɨɧɫɬɚɧɬ.
j
Ɇɚɬɪɢɰɚ C ' ɨɩɪɟɞɟɥɹɟɬ ɧɨɜɭɸ ɡɚɞɚɱɭ ɤɨɦɦɢɜɨɹɠɟɪɚ, ɤɨɬɨɪɚɹ ɜ ɤɚɱɟɫɬɜɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɬɭ ɠɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɝɨɪɨɞɨɜ. Ɇɟɠɞɭ ɜɟɥɢɱɢɧɚɦɢ L ɢ L' ' (ɞɥɢɧɚɦɢ ɨɩɬɢɦɚɥɶɧɵɯ ɦɚɪɲɪɭɬɨɜ) ɛɭɞɟɬ ɫɭɳɟɫɬɜɨɜɚɬɶ ɫɥɟɞɭɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ: L L' ' S . Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɨɱɟɜɢɞɧɨɟ ɧɟɪɚɜɟɧɫɬɜɨ: L t S , ɬɨ ɟɫɬɶ ɫɭɦɦɚ ɩɪɢɜɨɞɹɳɢɯ ɤɨɧɫɬɚɧɬ ɹɜɥɹɟɬɫɹ ɧɢɠɧɟɣ ɨɰɟɧɤɨɣ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ. Ʉɨɧɤɪɟɬɢɡɢɪɭɟɦ ɬɟɩɟɪɶ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɞɚɧɧɨɣ ɡɚɞɚɱɟ. ɉɭɫɬɶ : 0 — ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɦɚɪɲɪɭɬɨɜ. ȼɟɬɜɥɟɧɢɟ. ɉɪɢ ɜɟɬɜɥɟɧɢɢ ɨɱɟɪɟɞɧɨɟ ɦɧɨɠɟɫɬɜɨ : k ɪɚɡɛɢɜɚɟɬɫɹ ɧɚ 1. ɞɜɚ ɩɨɞɦɧɨɠɟɫɬɜɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. ȼ ɦɚɬɪɢɰɟ C k , ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ
ɪɚɡɜɟɬɜɥɹɟɦɨɦɭ ɦɧɨɠɟɫɬɜɭ, ɞɥɹ ɤɚɠɞɨɝɨ ɧɭɥɟɜɨɝɨ ɷɥɟɦɟɧɬɚ cijk ɜɵɱɢɫɥɹɟɬɫɹ ɱɢɫɥɨ S ijk ɤɚɹ ɱɬɨ
S qrk
ɭɫɥɨɜɢɹ xqr
min cilk min cljk . Ɂɚɬɟɦ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɚɪɚ ɢɧɞɟɤɫɨɜ ( q, r ) , ɬɚl
max S ijk i, j
l
. ɉɟɪɜɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ : k1 ɮɨɪɦɢɪɭɟɬɫɹ ɞɨɛɚɜɥɟɧɢɟɦ
1 (ɢɡ q -ɝɨ ɝɨɪɨɞɚ ɢɞɬɢ ɜ r -ɣ), ɜɬɨɪɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ : k2 ɫɨ-
ɞɟɪɠɢɬ ɭɫɥɨɜɢɟ xqr
0.
2. ȼɵɱɢɫɥɟɧɢɟ ɨɰɟɧɨɤ. ɉɭɫɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɟɞɵɞɭɳɢɦ ɩɭɧɤɬɨɦ ɩɪɨɢɡɜɟɞɟɧɨ ɪɚɡɛɢɟɧɢɟ : k : k1 : k2 . Ɋɚɫɫɦɨɬɪɢɦ ɩɪɚɜɢɥɨ ɩɟɪɟɯɨɞɚ ɨɬ ɦɚɬɪɢɰɵ C k ɤ ɦɚɬɪɢɰɚɦ C k 1 ɢ Ck2 . Ɇɚɬɪɢɰɚ Ck2 ɫɨɞɟɪɠɢɬ ɬɟ ɠɟ ɫɬɪɨɤɢ ɢ
°cijk , (i, j ) z (q, r ), . ɉɪɢɦɟɧɹɹ ɤ ɩɨɥɭɱɟɧ® °¯ f, (i, j ) (q, r ) ɧɨɣ ɦɚɬɪɢɰɟ Ck ɩɪɨɰɟɞɭɪɭ ɩɪɢɜɟɞɟɧɢɹ, ɩɨɥɭɱɢɦ ɦɚɬɪɢɰɭ Ck2 . ɉɪɢ ɷɬɨɦ ɫɬɨɥɛɰɵ, ɱɬɨ ɢ C k . ɉɨɥɨɠɢɦ cijk
ɫɭɦɦɚ ɩɪɢɜɨɞɹɳɢɯ ɤɨɧɫɬɚɧɬ ɛɭɞɟɬ ɪɚɜɧɚ S qrk . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɰɟɧɤɨɣ 18
ɦɧɨɠɟɫɬɜɚ : k2 ɛɭɞɟɬ [ (: k2 ) [ (: k ) S qrk . Ɉɩɪɟɞɟɥɢɦ ɬɟɩɟɪɶ ɩɪɚɜɢɥɨ ɩɨɫɬɪɨɟɧɢɹ ɦɚɬɪɢɰɵ C k . ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɦɧɨɠɟɫɬɜɨ : k1 ɡɚɜɟɞɨɦɨ ɫɨɞɟɪɠɢɬ ɩɟɪɟɯɨɞ ɢɡ q -ɝɨ ɝɨɪɨɞɚ ɜ r -ɣ. ɉɨɷɬɨɦɭ ɜ ɦɚɬɪɢɰɟ C k 1 ɫɥɟɞɭɟɬ ɜɵɱɟɪɤɧɭɬɶ q -ɸ ɫɬɪɨɤɭ ɢ r -ɣ ɫɬɨɥɛɟɰ. Ⱦɚɥɟɟ ɫɥɟɞɭɟɬ ɡɚɩɪɟɬɢɬɶ ɜɨɡɦɨɠɧɨɫɬɶ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɩɨɞɰɢɤɥɨɜ (ɡɚɦɵɤɚɧɢɹ ɮɪɚɝɦɟɧɬɨɜ ɦɚɪɲɪɭɬɚ). ɋ ɷɬɨɣ ɰɟɥɶɸ ɩɨɥɚɝɚɟɦ ɪɚɜɧɵɦɢ f ɜɫɟ ɷɥɟɦɟɧɬɵ, ɜɜɟɞɟɧɢɟ ɤɨɬɨɪɵɯ ɜ ɦɚɪɲɪɭɬ ɞɚɫɬ ɧɚɥɢɱɢɟ ɩɨɞɰɢɤɥɚ (ɧɚɩɪɢɦɟɪ, crq f ). Ʉ ɩɨɥɭɱɟɧɧɨɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɦɚɬɪɢɰɟ ɫɥɟɞɭɟɬ ɩɪɢɦɟɧɢɬɶ ɩɪɨɰɟɫɫ ɩɪɢɜɟɞɟɧɢɹ ɢ, ɧɚɣɞɹ ɫɭɦɦɭ ɩɪɢɜɨɞɹɳɢɯ ɤɨɧɫɬɚɧɬ S , ɩɨɫɱɢɬɚɬɶ ɨɰɟɧɤɭ [ (: k1 ) [ (: k ) S . ɉɪɚɜɢɥɨ ɨɛɯɨɞɚ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ, ɜɵɛɨɪ ɩɟɪɫɩɟɤɬɢɜɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɩɪɢ ɜɟɬɜɥɟɧɢɢ ɢ ɩɪɨɜɟɪɤɚ ɤɪɢɬɟɪɢɹ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɨɫɭɳɟɫɬɜɥɹɸɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɛɳɟɣ ɫɯɟɦɨɣ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ. 1
ɋɯɟɦɚ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɞɥɹ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ ɒɚɝ 1. Ɉɩɪɟɞɟɥɟɧɢɟ ɧɚɱɚɥɶɧɨɝɨ ɪɟɤɨɪɞɚ. (ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɦɨɠɧɨ ɜɡɹɬɶ ɞɥɢɧɭ ɥɸɛɨɝɨ ɦɚɪɲɪɭɬɚ). ɉɪɢɜɟɞɟɧɢɟ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ. Ɂɚɞɚɬɶ k = 0. ɒɚɝ 2. ȼɵɛɨɪ ɩɚɪɵ ( q, r ) . ɒɚɝ 3. ȼɟɬɜɥɟɧɢɟ : k : k1 : k2 . ɒɚɝ 4. ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɦɚɬɪɢɰɵ C k . ȼɵɱɢɫɥɟɧɢɟ ɦɚɬɪɢɰ C k 1 ɢ Ck2 . ȿɫɥɢ ɤɚɤɚɹ-ɬɨ ɢɡ ɷɬɢɯ ɦɚɬɪɢɰ ɢɦɟɟɬ ɪɚɡɦɟɪ 2 u 2, ɬɨ ɩɟɪɟɯɨɞ ɤ ɲɚɝɭ 7. ɒɚɝ 5. ȼɵɱɢɫɥɟɧɢɟ ɨɰɟɧɨɤ [ (: k2 ) ɢ [ (: k1 ) .
ɒɚɝ 6. ȼɵɛɨɪ ɩɟɪɫɩɟɤɬɢɜɧɨɝɨ ɦɧɨɠɟɫɬɜɚ : s ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɬɪɚɬɟɝɢɟɣ. ɉɨɥɨɠɢɬɶ k s. ɉɟɪɟɯɨɞ ɤ ɲɚɝɭ 2. ɒɚɝ 7. ɉɨɥɭɱɟɧɢɟ ɞɨɩɭɫɬɢɦɨɝɨ ɦɚɪɲɪɭɬɚ, ɜɨɡɦɨɠɧɚɹ ɫɦɟɧɚ ɪɟɤɨɪɞɚ ɢ ɫɨɤɪɚɳɟɧɢɟ ɩɟɪɟɛɨɪɚ. ɒɚɝ 8. ɉɪɨɜɟɪɤɚ ɤɪɢɬɟɪɢɹ ɨɩɬɢɦɚɥɶɧɨɫɬɢ. ȿɫɥɢ ɨɧ ɜɵɩɨɥɧɟɧ, ɬɨ ɨɫɬɚɧɨɜ. ɂɧɚɱɟ ɩɟɪɟɯɨɞ ɤ ɲɚɝɭ 6. Ɂɚɦɟɱɚɧɢɟ. ȼ ɦɨɦɟɧɬ ɩɨɥɭɱɟɧɢɹ ɦɚɬɪɢɰɵ 2 u 2 ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɚɦɵɤɚɸɳɚɹ ɩɚɪɚ ɝɨɪɨɞɨɜ ɞɥɹ ɨɛɪɚɡɨɜɚɧɢɹ ɞɨɩɭɫɬɢɦɨɝɨ ɦɚɪɲɪɭɬɚ. ɉɪɢɦɟɪ. Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɤɨɦɦɢɜɨɹɠɟɪɚ ɫ n 8 ɢ ɦɚɬɪɢɰɟɣ ɪɚɫɫɬɨɹɧɢɣ
19
C
3 2 4 5 2 7 · § f 5 ¸ ¨ 7 1 6 7 8 9 6 ¸ f ¨ ¨ 3 2 f 3 2 4 5 7 ¸ ¸ ¨ 6 2 f 1 3 6 2 ¸ ¨ 4 . ¨ 4 5 1 2 f 4 2 3 ¸ ¸ ¨ ¨ 3 8 9 6 4 f 1 5 ¸ ¸ ¨ 6 6 4 2 1 f 0 ¸ ¨ 1 ¨ 7 8 7 2 8 3 0 f ¸¹ ©
ȼɵɛɟɪɟɦ ɫɬɪɚɬɟɝɢɸ «ɩɨ ɦɢɧɢɦɭɦɭ ɨɰɟɧɤɢ». ɉɨɥɨɠɢɦ R f . Ɉɫɭɳɟɫɬɜɢɜ ɨɩɟɪɚɰɢɸ ɩɪɢɜɟɞɟɧɢɹ, ɩɨɥɭɱɚɟɦ ɦɚɬɪɢɰɭ C 0 .
ɋ0=
ʋ 1 2 3 4 5 6 7 8 1 2 2 5 0 0 1 +f 3 5 +f 5 6 6 8 5 0 2 1 1 3 5 0 0 0 +f 3 2 5 1 +f 0 1 5 1 4 2 4 1 +f 2 1 2 0 5 1 7 8 5 3 +f 4 0 6 6 6 4 2 0 0 +f 0 7 6 8 7 2 8 2 0 +f 8 E
1
0
0
0
0
1
0
0
D 2 1 2 1 1 1 0 0 10
.
ȼ ɩɨɫɥɟɞɧɟɦ ɫɬɨɥɛɰɟ ɢ ɧɢɠɧɟɣ ɫɬɪɨɤɟ ɡɚɩɢɫɚɧɵ ɩɪɢɜɨɞɹɳɢɟ ɤɨɧɫɬɚɧɬɵ. ɂɯ ɫɭɦɦɚ S=10, ɬɨ ɟɫɬɶ [ (: 0 ) 10 . Ⱦɥɹ ɤɚɠɞɨɝɨ ɧɭɥɟɜɨɝɨ ɷɥɟɦɟɧɬɚ ɦɚɬɪɢɰɵ C 0 ɫɱɢɬɚɟɦ S ij0 : S170
0 S 31
0 S 35
0 S 71
0, S140
0 S45
0 S53
S670
0 0 0 0 S760 S780 1, S 87 . ɋɥɟ2, S 32 3 , S 23 5 . ɂɦɟɟɦ max{0,1,2,3,5} 5 S 23 ɞɨɜɚɬɟɥɶɧɨ, (q, r ) (2,3) . Ɏɨɪɦɢɪɭɟɦ ɦɧɨɠɟɫɬɜɚ :1 ɢ : 2 , ɞɨɛɚɜɥɹɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɭɫɥɨɜɢɹ x23 1 ɢ x23 0 . ȼɵɱɢɫɥɹɟɦ ɦɚɬɪɢɰɵ C1 ɢ C 2 .
ɋ1=
ʋ 1 2 4 5 6 7 8 2 6 5 0 0 1 +f 0 1 1 3 5 0 +f 0 3 2 2 +f 1 5 1 0 4 1 1 1 0 0 +f 0 5 1 4 5 3 +f 4 0 6 3 4 2 0 0 +f 0 7 6 5 2 8 2 0 +f 8 E
0
3
0
0 20
0
0
0
D 0 0 0 1 0 0 0 4
,
ʋ 1 2 3 4 5 6 7 8 1 2 2 5 0 0 1 +f 3 1 1 3 0 +f +f 0 0 2 1 1 3 5 0 0 0 +f 3 2 5 1 +f 1 5 1 0 4 2 4 1 +f 2 1 2 0 5 4 1 7 8 5 3 +f 0 6 6 6 4 2 0 0 +f 0 7 6 8 7 2 8 2 0 +f 8
ɋ2=
E
0
0
0
0
0
0
0
D 0 5 0 0 0 0 0 0 5
0
.
ɋɱɢɬɚɟɦ ɨɰɟɧɤɢ: [ (:1 ) 10 4 14 , [ (: 2 ) 10 5 15 . ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ ɞɚɥɶɧɟɣɲɟɦɭ ɜɟɬɜɥɟɧɢɸ ɩɨɞɥɟɠɢɬ ɦɧɨɠɟɫɬɜɨ :1 . Ⱦɥɹ ɤɚɠɞɨɝɨ ɧɭɥɟɜɨɝɨ ɷɥɟɦɟɧɬɚ ɦɚɬɪɢɰɵ C1 ɫɱɢɬɚɟɦ S ij1 : 1 S12
1 S14
1 S17
1 S 21
1 S 25
1 S 45
1 S 67
1 S 76
1 S 78
1, S 87
1 S 52
1 S 54
1 S 57
1 S 71
1 S 78
0,
2 . ɂɦɟɟɦ max{0,1,2} 2
1 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, S87
(q, r ) (8,7) . Ɏɨɪɦɢɪɭɟɦ ɦɧɨɠɟɫɬɜɚ : 3 ɢ : 4 , ɞɨɛɚɜɥɹɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɭɫɥɨɜɢɹ x87 1 ɢ x87 0 . ȼɵɱɢɫɥɹɟɦ ɦɚɬɪɢɰɵ C 3 ɢ C 4 .
ɋ3=
ɋ4=
ʋ 1 2 4 5 6 8 1 +f 0 2 2 4 0 3 1 4 0 +f 1 0 4 2 2 +f 0 1 0 5 1 0 0 +f 1 0 6 3 4 2 +f 2 0 7 3 4 2 0 0 +f E 0 0 0 0 0 1
ʋ 1 2 4 5 6 7 8 1 +f 0 2 2 5 0 0 3 1 3 5 0 +f 1 0 4 2 2 +f 0 1 5 1 5 1 1 0 0 +f 1 0 6 1 4 5 3 +f 0 4 7 3 4 2 0 0 +f 0 8 4 3 6 0 0 +f +f E 0 0 0 0 0 0 0
D
0 0 0 0 1 0 2
,
D
0 0 0 0 0 0 2 2
.
ȼɵɱɢɫɥɹɟɦ ɨɰɟɧɤɢ: [ (: 3 ) 14 2 16 , [ (: 4 ) 14 2 16 . 21
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɬɪɚɬɟɝɢɟɣ ɞɚɥɶɧɟɣɲɟɦɭ ɜɟɬɜɥɟɧɢɸ ɩɨɞɥɟɠɢɬ ɦɧɨɠɟɫɬɜɨ : 2 , ɬɚɤ ɤɚɤ ɨɧɨ ɢɦɟɟɬ ɧɚɢɦɟɧɶɲɭɸ ɨɰɟɧɤɭ [ (: 2 ) 15 . Ⱥɧɚɥɨɝɢɱɧɨ ɞɥɹ ɜɫɟɯ ɧɭɥɟɜɵɯ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ C 2 ɫɱɢɬɚɟɦ S ij2 ɢ ɨɩɪɟɞɟɥɹɟɦ, ɱɬɨ (q, r ) (3,2) . Ɏɨɪɦɢɪɭɟɦ ɦɧɨɠɟɫɬɜɚ : 5 ɢ : 6 , ɞɨɛɚɜɥɹɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɭɫɥɨɜɢɹ x32 1 ɢ x32 0 . Ɇɚɬɪɢɰɵ C 5 ɢ C 6 ɢɦɟɸɬ ɜɢɞ: ʋ 1 3 4 5 6 7 8 D 1 +f 1 2 2 5 0 0 0 2 0 +f 0 1 3 1 0 0 4 2 1 +f 0 1 5 1 0 ɋ5= 5 2 1 +f 2 1 2 0 0 6 1 8 5 3 +f 0 4 0 7 0 6 4 2 0 +f 0 0 8 6 7 2 8 2 0 +f 0 E 0 0 0 0 0 0 0 0
ɋ6=
ʋ 1 2 3 4 5 6 7 8
1 +f 0 0 2 2 1 0 6
2 0 +f +f 2 1 4 3 5
3 1 +f +f 1 0 8 6 7
4 0 0 1 +f 1 5 4 2
5 2 1 0 0 +f 3 2 8
6 2 1 1 1 2 +f 0 2
7 0 3 3 5 1 0 +f 0
8 5 0 5 1 2 4 0 +f
,
D
0 5 0 0 0 0 0 0
.
ȼɵɱɢɫɥɹɟɦ ɨɰɟɧɤɢ: [ (: 5 ) 15 0 15 , [ (: 6 ) 15 3 18 . Ⱦɚɥɶɧɟɣɲɟɦɭ ɜɟɬɜɥɟɧɢɸ ɩɨɞɥɟɠɢɬ ɦɧɨɠɟɫɬɜɨ : 5 . Ⱦɟɥɢɦ : 5 ɧɚ ɩɨɞɦɧɨɠɟɫɬɜɚ : 7 ɢ : 8 ɩɨ ɩɚɪɟ (q, r ) (8,7) . Ɋɚɫɫɱɢɬɚɜ ɦɚɬɪɢɰɵ C 7 ɢ C8 , ɨɩɪɟɞɟɥɹɟɦ [ (: 7 ) 15 1 16 , [ (: 8 ) 15 2 17 . Ɇɢɧɢɦɚɥɶɧɭɸ ɨɰɟɧɤɭ 16 ɢɦɟɟɬ ɬɪɢ ɩɨɞɦɧɨɠɟɫɬɜɚ: : 3 , : 4 ɢ :7 . ȼɵɛɟɪɟɦ ɞɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɜɟɬɜɥɟɧɢɹ : 3 . Ⱦɟɥɢɦ ɟɝɨ ɧɚ ɩɨɞɦɧɨɠɟɫɬɜɚ : 9 ɢ :10 ɩɨ ɩɚɪɟ (q, r ) (6,1) . ɉɨɥɭɱɢɦ [ (: 9 ) 16 , [ (:10 ) 18 . Ⱦɚɥɟɟ ɞɟɥɢɦ : 9 ɧɚ ɩɨɞɦɧɨɠɟɫɬɜɚ :11 ɢ :12 ɩɨ ɩɚɪɟ (q, r ) (7,6) . ȼɵɱɢɫɥɹɟɦ [ (:11 ) 16 , [ (:12 ) 19 . Ⱦɟɥɢɦ :11 ɩɨ ɩɚɪɟ (q, r ) (3,5) ɧɚ :13 ɢ :14 . ɉɪɢ ɷɬɨɦ ɩɨɥɭɱɚɟɦ ɦɚɬɪɢɰɭ C13 ɪɚɡɦɟɪɚ 2 u 2. ȼ ɪɟɡɭɥɶɬɚɬɟ ɜɵɩɢɫɵɜɚɟɦ ɞɨɩɭɫɬɢɦɭɸ ɬɨɱɤɭ (ɦɚɬɪɢɰɭ X)
22
x ij
§0 ¨ ¨0 ¨0 ¨ ¨0 ¨0 ¨ ¨1 ¨ ¨0 ¨0 ©
1 0 0 0 0 0 0· ¸ 0 1 0 0 0 0 0¸ 0 0 0 1 0 0 0¸ ¸ 0 0 0 0 0 0 1¸ 0 0 1 0 0 0 0¸ ¸ 0 0 0 0 0 0 0¸ ¸ 0 0 0 0 1 0 0¸ 0 0 0 0 0 1 0 ¸¹
ɫɨ ɡɧɚɱɟɧɢɟɦ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ L 16 . ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɦɚɪɲɪɭɬ: 1-2-3-5-4-8-7-5-1. Ɇɟɧɹɟɦ ɪɟɤɨɪɞ: R 16 , ɢ ɩɨɞɦɧɨɠɟɫɬɜɚ : 4 , : 6 , : 7 , : 8 , :10 , :12 , :14 ɜɵɛɪɚɫɵɜɚɸɬɫɹ ɢɡ ɪɚɫɫɦɨɬɪɟɧɢɹ ɤɚɤ ɧɟɩɟɪɫɩɟɤɬɢɜɧɵɟ (ɢɯ ɨɰɟɧɤɢ ɩɪɟɜɵɲɚɸɬ ɢɥɢ ɪɚɜɧɵ ɪɟɤɨɪɞɧɨɦɭ ɡɧɚɱɟɧɢɸ). Ɍɚɤ ɤɚɤ ɛɨɥɶɲɟ ɩɨɞɦɧɨɠɟɫɬɜ ɞɥɹ ɜɟɬɜɥɟɧɢɹ ɧɟ ɨɫɬɚɥɨɫɶ, ɬɨ ɧɚɣɞɟɧɧɚɹ ɬɨɱɤɚ ɹɜɥɹɟɬɫɹ ɨɩɬɢɦɚɥɶɧɵɦ ɪɟɲɟɧɢɟɦ. Ɉɞɧɚɤɨ ɨɧɨ ɦɨɠɟɬ ɛɵɬɶ ɧɟ ɟɞɢɧɫɬɜɟɧɧɵɦ, ɬɚɤ ɤɚɤ ɩɨɞɦɧɨɠɟɫɬɜɚ : 4 , : 7 ɬɚɤ ɠɟ ɢɦɟɸɬ ɨɰɟɧɤɭ 16. ɑɬɨɛɵ ɜɵɹɫɧɢɬɶ, ɫɭɳɟɫɬɜɭɸɬ ɥɢ ɞɪɭɝɢɟ ɪɟɲɟɧɢɹ, ɧɭɠɧɨ ɨɫɭɳɟɫɬɜɥɹɬɶ ɞɚɥɶɧɟɣɲɟɟ ɜɟɬɜɥɟɧɢɟ ɷɬɢɯ ɦɧɨɠɟɫɬɜ. Ⱦɟɪɟɜɨ ɜɚɪɢɚɧɬɨɜ ɜ ɪɚɫɫɦɨɬɪɟɧɧɨɦ ɩɪɢɦɟɪɟ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ. x23=1 x23=0 10 14
x87=1
x87=0
15
x32=1 16
16
x76=1
x76=0
16
18
x87=1 16
19
16
x35=1
16
x61=0
x61=1
x35=0 17
23
15
x32=0 x87=0 17
18
ɍɉɊȺɀɇȿɇɂə
1. Ɋɟɲɢɬɶ ɪɚɫɫɦɨɬɪɟɧɧɵɣ ɜ ɩ. 3.2 ɩɪɢɦɟɪ, ɢɫɩɨɥɶɡɭɹ ɫɬɪɚɬɟɝɢɸ «ɥɟɜɨɫɬɨɪɨɧɧɟɝɨ ɨɛɯɨɞɚ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ». 2. ȼ ɩɪɨɝɪɚɦɦɧɨɣ ɪɟɚɥɢɡɚɰɢɢ ɚɥɝɨɪɢɬɦɚ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ ɢɫɩɨɥɶɡɭɟɬɫɹ, ɤɚɤ ɩɪɚɜɢɥɨ, ɥɟɜɨɫɬɨɪɨɧɧɢɣ ɨɛɯɨɞ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ. Ɉɬɦɟɬɶɬɟ ɩɪɟɢɦɭɳɟɫɬɜɨ ɷɬɨɣ ɫɬɪɚɬɟɝɢɢ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ. 3. Ɋɟɲɢɬɟ ɡɚɞɚɱɭ ɤɨɦɦɢɜɨɹɠɟɪɚ ɫ ɦɚɬɪɢɰɟɣ: a)
b)
ɋ
93 13 33 9 57 · § f 3 ¨ ¨ 4 f 77 42 21 16 34 ¨ 45 17 f 36 16 28 25 ¨ ɋ 7 91 ¨ 39 90 80 f 56 ¨ 28 46 88 33 f 25 57 ¨ ¨ 3 88 18 46 92 f 7 ¨ © 44 26 33 27 84 39 f ¹
Ɉɬɜɟɬ: 1-4-6-7-3-5-2-1, L = 126.
Ɉɬɜɟɬ: 4-3-5-6-2-1-4, L = 63.
ɫ)
ɋ
§ f 27 43 16 30 26 · ¸ ¨ 1 30 25 ¸ ¨ 7 f 16 ¨ 20 13 f 35 5 0 ¸ ¸ ¨ ¨ 21 16 25 f 18 18 ¸ ¨ 12 46 27 48 f 5 ¸ ¸ ¨ ¨ 23 5 5 9 5 f ¸¹ ©
d) 9 8 10 6 · § f 14 ¸ ¨ 4 11 5 ¸ ¨ 10 f 8 ¨ 11 10 f 10 20 11 ¸ ¸ ¨ 4 4 f 8 4 ¸ ¨ 12 ¨ 13 10 9 7 f 10 ¸ ¸ ¨ ¨ 11 9 8 3 4 f ¸¹ ©
ɋ
Ɉɬɜɟɬ: 3-1-6-5-4-2-3, L = 39.
10 13 4 8 · § f 4 ¸ ¨ 7 6 7 ¸ ¨ 2 f 9 ¨ 8 5 f 5 5 9 ¸ ¸ ¨ 8 5 f 7 10 ¸ ¨ 5 ¨ 6 4 4 9 f 4 ¸ ¸ ¨ ¨ 5 1 4 8 3 f ¸¹ ©
Ɉɬɜɟɬ: 2-1-5-3-4-6-2, L = 26.
24
3.3. Ɇɟɬɨɞ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɞɥɹ ɥɢɧɟɣɧɵɯ ɡɚɞɚɱ ɰɟɥɨɱɢɫɥɟɧɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ
Ɋɚɫɫɦɨɬɪɢɦ ɰɟɥɨɱɢɫɥɟɧɧɭɸ ɡɚɞɚɱɭ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ (ɐɁɅɉ): n
M( x )
¦c
j
x j o min ,
j 1
n °¦ aij x j db i , i 1, m, :0 ® j 1 °0 d x d d , x =, j 1, n. j j j ¯
ɉɪɨɰɟɫɫ ɩɨɥɭɱɟɧɢɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɧɚɱɢɧɚɟɬɫɹ ɫ ɪɟɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɟɣ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɫ ɨɬɛɪɨɲɟɧɧɵɦ ɭɫɥɨɜɢɟɦ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ. ȿɫɥɢ ɩɨɥɭɱɟɧɧɚɹ ɩɪɢ ɷɬɨɦ ɨɩɬɢɦɚɥɶɧɚɹ ɬɨɱɤɚ x 0 ɢɦɟɟɬ ɬɨɥɶɤɨ ɰɟɥɵɟ ɤɨɨɪɞɢɧɚɬɵ, ɬɨ ɨɧɚ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɡɧɚɱɟɧɢɟ M ( x 0 ) ɞɚɟɬ ɧɢɠɧɸɸ ɨɰɟɧɤɭ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ. Ʉɨɧɤɪɟɬɢɡɢɪɭɟɦ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɞɚɧɧɨɣ ɡɚɞɚɱɟ. 1. ȼɟɬɜɥɟɧɢɟ. ɉɭɫɬɶ x k ( x1k ,..., xnk ) – ɨɩɬɢɦɚɥɶɧɚɹ ɬɨɱɤɚ, ɩɨɥɭɱɟɧɧɚɹ ɩɪɢ ɪɟɲɟɧɢɢ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɢ ɧɚ ɦɧɨɠɟɫɬɜɟ : k . ȿɫɥɢ x k ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɬɪɟɛɨɜɚɧɢɹɦ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ, ɬɨ ɦɧɨɠɟɫɬɜɨ : k ɢɫɤɥɸɱɚɟɬɫɹ ɢɡ ɞɚɥɶɧɟɣɲɟɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ (ɧɚɣɞɟɧɨ ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɧɚ ɧɟɦ). Ɂɧɚɱɟɧɢɟ M ( x k ) ɫɪɚɜɧɢɜɚɟɬɫɹ ɫ ɪɟɤɨɪɞɧɵɦ (ɩɪɢ ɷɬɨɦ ɜɨɡɦɨɠɧɨ ɫɦɟɧɚ ɪɟɤɨɪɞɚ). ȿɫɥɢ ɠɟ ɟɫɬɶ ɧɟ ɰɟɥɚɹ ɤɨɦɩɨɧɟɧɬɚ x kj , ɬɨ ɦɧɨɠɟɫɬɜɨ : k ɪɚɡɛɢɜɚɟɬɫɹ ɧɚ ɞɜɚ : k : k1 : k2 , ɩɨɞɦɧɨɠɟɫɬɜɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: : k1 {x : k , x j d [ x kj ]} , : k2 {x : k , x j t [ x kj ] 1} , ɝɞɟ ɫɢɦɜɨɥɨɦ [] ɨɛɨɡɧɚɱɟɧɚ ɰɟɥɚɹ ɱɚɫɬɶ ɱɢɫɥɚ. 2. ȼɵɱɢɫɥɟɧɢɟ ɨɰɟɧɨɤ. Ɉɰɟɧɨɱɧɨɣ ɡɚɞɚɱɟɣ ɧɚ ɦɧɨɠɟɫɬɜɟ : k ɛɭɞɟɬ ɹɜɥɹɬɶɫɹ ɡɚɞɚɱɚ, ɜ ɤɨɬɨɪɨɣ ɨɬɛɪɨɲɟɧɨ ɬɪɟɛɨɜɚɧɢɟ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ ɩɟɪɟɦɟɧɧɵɯ. ɉɪɢ ɪɟɲɟɧɢɢ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɢ ɩɨɥɭɱɚɟɦ ɨɩɬɢɦɚɥɶɧɭɸ ɬɨɱɤɭ x k . Ɂɧɚɱɟɧɢɟ M ( x k ) ɞɚɟɬ ɧɢɠɧɸɸ ɨɰɟɧɤɭ ɦɧɨɠɟɫɬɜɚ : k (ɦɨɠɧɨ ɜ ɤɚɱɟɫɬɜɟ ɨɰɟɧɤɢ ɛɪɚɬɶ ɰɟɥɭɸ ɱɚɫɬɶ M ( x k ) ). ɉɪɚɜɢɥɨ ɨɛɯɨɞɚ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ, ɜɵɛɨɪ ɩɟɪɫɩɟɤɬɢɜɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɩɪɢ ɜɟɬɜɥɟɧɢɢ ɢ ɩɪɨɜɟɪɤɚ ɤɪɢɬɟɪɢɹ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɨɫɭɳɟɫɬɜɥɹɸɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɬɚɧɞɚɪɬɧɨɣ ɫɯɟɦɨɣ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ.
25
ɋɯɟɦɚ ɪɚɛɨɬɵ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɞɥɹ ɐɁɅɉ ɒɚɝ 1. Ɉɩɪɟɞɟɥɟɧɢɟ ɧɚɱɚɥɶɧɨɝɨ ɪɟɤɨɪɞɚ. (ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɢɧɮɨɪɦɚɰɢɢ R = f ). Ɂɚɞɚɬɶ k = 0. ɒɚɝ 2. Ɋɟɲɟɧɢɟ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɢ ɧɚ ɦɧɨɠɟɫɬɜɟ : 0 . ȿɫɥɢ ɩɨɥɭɱɟɧɧɚɹ
ɬɨɱɤɚ x 0 ɰɟɥɨɱɢɫɥɟɧɧɚɹ, ɬɨ ɪɟɲɟɧɢɟ ɧɚɣɞɟɧɨ. Ɉɫɬɚɧɨɜ. ɒɚɝ 3. ȼɵɛɨɪ ɢɧɞɟɤɫɚ j ɬɚɤɨɝɨ, ɱɬɨ ɤɨɨɪɞɢɧɚɬɚ xkj — ɧɟ ɰɟɥɚɹ. ɒɚɝ 4. ȼɟɬɜɥɟɧɢɟ : k : k1 : k2 . ɒɚɝ 5. Ɋɟɲɟɧɢɟ ɨɰɟɧɨɱɧɵɯ ɡɚɞɚɱ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɢ ɜɵɱɢɫɥɟɧɢɟ ɨɰɟɧɨɤ [ (: k2 ) ɢ [ (: k1 ) . ȿɫɥɢ ɤɚɤɚɹ-ɬɨ ɢɡ ɩɨɥɭɱɟɧɧɵɯ ɨɩɬɢɦɚɥɶɧɵɯ ɬɨɱɟɤ ɰɟɥɨɱɢɫɥɟɧɧɚɹ, ɬɨ ɩɟɪɟɯɨɞ ɤ ɲɚɝɭ 7. ɒɚɝ 6. ȼɵɛɨɪ ɩɟɪɫɩɟɤɬɢɜɧɨɝɨ ɦɧɨɠɟɫɬɜɚ : s ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɬɪɚɬɟɝɢɟɣ. ɉɨɥɨɠɢɬɶ k s. ɉɟɪɟɯɨɞ ɤ ɲɚɝɭ 2. ɒɚɝ 7. ȼɨɡɦɨɠɧɚɹ ɫɦɟɧɚ ɪɟɤɨɪɞɚ ɢ ɫɨɤɪɚɳɟɧɢɟ ɩɟɪɟɛɨɪɚ ɡɚ ɫɱɟɬ ɨɬɛɪɚɫɵɜɚɧɢɹ ɧɟɩɟɪɫɩɟɤɬɢɜɧɵɯ ɦɧɨɠɟɫɬɜ. ɒɚɝ 8. ɉɪɨɜɟɪɤɚ ɤɪɢɬɟɪɢɹ ɨɩɬɢɦɚɥɶɧɨɫɬɢ. ȿɫɥɢ ɨɧ ɜɵɩɨɥɧɟɧ, ɨɫɬɚɧɨɜ. ɂɧɚɱɟ ɩɟɪɟɯɨɞ ɤ ɲɚɝɭ 6. x1 ɉɪɢɦɟɪ. 7 (2) x1 x2 o min,
2 x1 11x2 d 38, 1 (3) °x x d 7, 2 ° 1 2 :0 ® 3 3 °4 x1 5 x2 d 5, x0 °¯ x1 , x2 t 0, x1 , x2 =. ɉɨɥɚɝɚɟɦ R = f . Ɋɟɲɚɟɦ ɢɫɯɨɞɧɭɸ ɡɚɞɚɱɭ ɛɟɡ ɬɪɟɛɨɜɚɧɢɹ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ. Ƚɪɚ5 ɮɢɱɟɫɤɚɹ ɢɥɥɸɫɬɪɚɰɢɹ ɪɟɲɟɧɢɹ ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫɭɧɤɟ. Ɉɩɬɢɦɚɥɶɧɨɣ ɬɨɱɤɨɣ ɹɜɥɹɟɬɫɹ
(1) x2 7
x0
(4 94 ,2 95 ) ,
0
[ ( : 0 ) M ( x ) 7 . Ɍɚɤ ɤɚɤ ɬɨɱɤɚ ɧɟ ɰɟɥɨɱɢɫɥɟɧɧɚɹ, ɨɫɭɳɟɫɬɜɥɹɟɦ ɜɟɬɜɥɟɧɢɟ (ɧɚɩɪɢɦɟɪ, ɩɨ ɩɟɪɜɨɣ ɤɨɨɪɞɢɧɚɬɟ): : 0 :1 : 2 , :1 {x : 0 , x1 d 4} , : 2 {x : 0 , x1 t 5} . Ɋɟɲɚɟɦ ɞɜɟ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ, ɡɚɤɥɸɱɚɸɳɢɟɫɹ ɜ ɦɢɧɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɢ M (x ) ɧɚ ɦɧɨɠɟɫɬɜɚɯ :1 ɢ : 2 ɛɟɡ ɬɪɟɛɨɜɚɧɢɹ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ. ɉɨɥɭɱɚɟɦ [ (: 2 ) f .
x1
(4,2 118 ) , [ (:1 ) 6 , : 2 26
Ø. ɉɨɥɚɝɚɟɦ
Ɍɚɤ ɤɚɤ ɰɟɥɨɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɨɩɹɬɶ ɧɟ ɩɨɥɭɱɟɧɨ, ɨɫɭɳɟɫɬɜɥɹɟɦ ɜɟɬɜɥɟɧɢɹ ɦɧɨɠɟɫɬɜɚ :1 : : 3 {x :1 , x2 d 2} , : 4 {x :1 , x2 t 3} . Ɋɟɲɚɟɦ ɨɰɟɧɨɱɧɵɟ ɡɚɞɚɱɢ ɧɚ ɦɧɨɠɟɫɬɜɚɯ : 3 ɢ : 4 . ɉɨɥɭɱɚɟɦ
x3
(3 34 ,2) , [ (: 3 ) 5 , x 4 (2 12 ,3) , [ (: 4 ) 5 . ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɫɬɪɚɬɟɝɢɟɣ ɨɞɧɨɫɬɨɪɨɧɧɟɝɨ ɨɛɯɨɞɚ ɢ ɜɵɛɟɪɟɦ ɞɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɜɟɬɜɥɟɧɢɹ ɦɧɨɠɟɫɬɜɨ : 3 . ɉɨɥɭɱɢɦ: : 3 : 5 : 6 , : 5 {x : 3 , x1 d 3} , : 6 {x : 3 , x1 t 4} . Ɋɟɲɚɟɦ ɨɰɟɧɨɱɧɵɟ ɡɚɞɚɱɢ ɧɚ ɦɧɨɠɟɫɬɜɚɯ : 5 ɢ : 6 . ɉɨɥɭɱɚɟɦ x5
(3,2) , [ (: 5 ) M ( x 5 ) 5 , : 6 Ø. ɉɨɥɭɱɟɧɨ ɰɟɥɨɱɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ. ɉɪɨɢɫɯɨɞɢɬ ɫɦɟɧɚ ɪɟɤɨɪɞɚ R = –5. Ɇɧɨɠɟɫɬɜɚ : 6 , : 4 ɢ : 2 ɜɵɛɪɚɫɵɜɚɟɦ ɢɡ ɪɚɫɫɦɨɬɪɟɧɢɹ ɤɚɤ ɧɟɩɟɪɫɩɟɤɬɢɜɧɵɟ (ɢɯ ɨɰɟɧɤɢ ɩɪɟɜɵɲɚɸɬ ɢɥɢ ɪɚɜɧɵ ɪɟɤɨɪɞɧɨɦɭ ɡɧɚɱɟɧɢɸ). Ɍɚɤ ɤɚɤ ɩɟɪɫɩɟɤɬɢɜɧɵɯ ɦɧɨɠɟɫɬɜ ɞɥɹ ɜɟɬɜɥɟɧɢɹ ɧɟ ɨɫɬɚɥɨɫɶ, ɬɨ ɨɫɬɚɧɨɜ, ɩɨɥɭɱɟɧɨ ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ: x min (3,2) , M min 5 . Ⱦɟɪɟɜɨ ɜɚɪɢɚɧɬɨɜ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
x1 d 4 -6
-5
-5
x1 t 5 +f
x2 t 3
x2 d 2
x1 d 3
-7
-5
x1 t 4 +f
ɍɉɊȺɀɇȿɇɂə
1. ɉɨɤɚɠɢɬɟ, ɱɬɨ ɥɸɛɚɹ ɡɚɞɚɱɚ ɰɟɥɨɱɢɫɥɟɧɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɷɤɜɢɜɚɥɟɧɬɧɨ ɩɟɪɟɩɢɫɚɧɚ ɤɚɤ ɡɚɞɚɱɚ ɫ ɛɭɥɟɜɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ. 2. ɉɨɥɶɡɭɹɫɶ ɨɩɪɟɞɟɥɟɧɢɟɦ ɨɰɟɧɤɢ, ɞɨɤɚɠɢɬɟ, ɱɬɨ ɡɚɞɚɱɚ ɫ ɨɬɛɪɨɲɟɧɧɵɦ ɭɫɥɨɜɢɟɦ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɨɰɟɧɨɱɧɨɣ ɤ ɢɫɯɨɞɧɨɣ. 3. Ɋɟɲɢɬɶ ɐɁɅɉ:
27
b ) 3x1 4 x2 o min,
c ) x1 4 x2 o min,
x1 3x2 d 11,
x1 5 x2 d 15,
x1 2 x2 d 10 ,
2 x1 2 x2 d 13,
x1 x2 d 11 ,
x1 x2 d 7 12 ,
2 x1 x2 d 11,
2 x1 x2 d 10,
2 x1 x2 d 10 ,
x1 , x2 t 0, x1 , x2 =.
x1 , x2 t 0, x1 , x2 =.
x1 , x2 t 0 , x1 , x2 =.
a ) 2 x1 3 x2 o min,
Ɉɬɜɟɬ: x
min
=(2,4).
1 3
Ɉɬɜɟɬ: x
min
=(7,4).
Ɉɬɜɟɬ: x min =(2,5).
3.4. ɉɚɪɚɦɟɬɪɢɡɚɰɢɹ ɚɥɝɨɪɢɬɦɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ
Ɇɨɞɭɥɶɧɚɹ ɫɯɟɦɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɜɵɞɟɥɢɬɶ ɩɚɪɚɦɟɬɪɵ, ɜɧɟɲɧɟɟ ɭɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɵɦɢ ɩɨɡɜɨɥɹɟɬ ɮɢɤɫɢɪɨɜɚɬɶ ɬɭ ɢɥɢ ɢɧɭɸ ɜɟɪɫɢɸ ɚɥɝɨɪɢɬɦɚ. Ʉɚɤ ɜɢɞɧɨ ɢɡ ɩɚɪɚɝɪɚɮɚ 1, ɬɚɤɢɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɦɨɝɭɬ ɛɵɬɶ ɫɩɨɫɨɛɵ ɜɟɬɜɥɟɧɢɹ, ɩɪɚɜɢɥɚ ɜɵɱɢɫɥɟɧɢɹ ɨɰɟɧɨɤ, ɫɬɪɚɬɟɝɢɹ ɨɛɯɨɞɚ ɞɟɪɟɜɚ ɜɚɪɢɚɧɬɨɜ ɢ ɞɪ. ɉɪɚɜɢɥɨ ɜɵɛɨɪɚ ɩɚɪɚɦɟɬɪɨɜ ɦɨɠɟɬ ɨɫɭɳɟɫɬɜɥɹɬɶɫɹ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ɜɟɬɜɥɟɧɢɹ ɫ ɭɱɟɬɨɦ ɪɟɫɭɪɫɨɜ ɗȼɆ. Ʉɪɨɦɟ ɬɨɝɨ, ɫɯɟɦɚ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɩɨɡɜɨɥɹɟɬ ɫɬɪɨɢɬɶ ɩɪɢɛɥɢɠɟɧɧɵɟ ɚɥɝɨɪɢɬɦɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɞɢɫɤɪɟɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢɢ. Ⱦɥɹ ɷɬɨɝɨ ɞɨɫɬɚɬɨɱɧɨ, ɧɚɩɪɢɦɟɪ, ɜɧɟɫɬɢ ɢɡɦɟɧɟɧɢɹ ɜ ɫɩɨɫɨɛɵ ɨɬɫɟɜɚ ɩɟɪɫɩɟɤɬɢɜɧɵɯ ɦɧɨɠɟɫɬɜ, ɜ ɤɪɢɬɟɪɢɢ ɨɫɬɚɧɨɜɚ. ɇɚɩɪɢɦɟɪ, ɩɪɚɜɢɥɨ ɨɬɛɪɚɫɵɜɚɧɢɹ ɦɨɠɧɨ ɫɤɨɪɪɟɤɬɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ (ɩɪɢ ɨɬɵɫɤɚɧɢɢ ɦɚɤɫɢɦɭɦɚ). [k R d e , ɝɞɟ e – ɧɚɩɟɪɟɞ ɡɚɞɚɧɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ. Ⱦɥɹ ɩɪɨɝɪɚɦɦɧɨɣ ɩɚɪɚɦɟɬɪɢɡɚɰɢɢ ɦɟɬɨɞɚ ɜɜɟɞɟɦ ɫɥɟɞɭɸɳɢɟ ɨɩɪɟɞɟɥɟɧɢɹ. ɉɚɪɚɦɟɬɪ N ɧɚɡɨɜɟɦ ɝɥɭɛɢɧɨɣ ɩɚɦɹɬɢ ɚɥɝɨɪɢɬɦɚ, ɟɫɥɢ ɜ ɬɟɱɟɧɢɟ N ɲɚɝɨɜ ɚɥɝɨɪɢɬɦ ɪɚɛɨɬɚɟɬ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɬɚɧɞɚɪɬɧɨɣ ɫɯɟɦɨɣ (ɛɟɡ ɢɡɦɟɧɟɧɢɣ, ɫ ɡɚɮɢɤɫɢɪɨɜɚɧɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ). ɇɚɡɨɜɟɦ ɱɢɫɥɨ A ɢɧɞɢɤɚɬɨɪɨɦ ɩɟɪɫɩɟɤɬɢɜɧɨɫɬɢ ɚɥɝɨɪɢɬɦɚ, ɟɫɥɢ ɨɬɛɪɚɫɵɜɚɧɢɟ ɩɨɞɦɧɨɠɟɫɬɜ ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɨ ɩɪɚɜɢɥɭ: : k ɜɵɛɪɚɫɵɜɚɟɬɫɹ ɢɡ ɞɚɥɶɧɟɣɲɟɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ, ɟɫɥɢ [ : k d A . Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɚɧɢɩɭɥɹɰɢɹ ɪɚɡɥɢɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɷɬɢɯ ɩɚɪɚɦɟɬɪɨɜ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɚɬɶ ɫɨɜɨɤɭɩɧɨɫɬɶ ɩɪɢɛɥɢɠɟɧɧɵɯ ɚɥɝɨɪɢɬɦɨɜ ɪɚɡɧɨɣ ɬɨɱɧɨɫɬɢ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, «ɛɨɥɶɲɢɟ ɡɧɚɱɟɧɢɹ» N ɢ A R , ɝɞɟ R – ɜɟɥɢɱɢɧɚ ɪɟɤɨɪɞɚ, ɩɪɢɜɨɞɹɬ ɤ ɬɨɱɧɨɦɭ ɚɥɝɨɪɢɬɦɭ ɪɟɲɟɧɢɹ, ɡɧɚɱɟɧɢɹ N 1 ɢ A max [ k e , ɝɞɟ e ! 0 – k
ɦɚɥɨɟ ɱɢɫɥɨ, ɩɨɪɨɠɞɚɸɬ ɩɪɢɛɥɢɠɟɧɧɵɣ ɚɥɝɨɪɢɬɦ ɩɪɨɯɨɞɚ ɩɨ ɞɟɪɟɜɭ ɜɚɪɢɚɧɬɨɜ, ɩɨ ɦɧɨɠɟɫɬɜɚɦ ɫ ɦɚɤɫɢɦɚɥɶɧɨɣ ɨɰɟɧɤɨɣ ɞɨ ɩɟɪɜɨɝɨ ɩɨɥɭɱɟɧɢɹ ɞɨɩɭɫɬɢɦɨɣ ɬɨɱɤɢ. ȿɫɥɢ ɚɥɝɨɪɢɬɦ ɩɪɟɞɭɫɦɚɬɪɢɜɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɩɚɪɚɦɟɬɪɨɜ ɜ ɩɪɨɰɟɫɫɟ ɪɟɲɟɧɢɹ, ɬɨ ɜɨɡɦɨɠɧɨ ɜɧɟɲɧɟɟ ɭɩɪɚɜɥɟɧɢɟ ɬɨɱɧɨɫɬɶɸ ɚɥɝɨɪɢɬɦɚ. ȼɨɡɦɨɠɧɨɫɬɶ ɭɩɪɚɜɥɟɧɢɹ ɫɬɪɚɬɟɝɢɟɣ ɢ ɬɨɱɧɨɫɬɶɸ ɚɥɝɨɪɢɬɦɚ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɚɬɶ ɦɧɨɠɟɫɬɜɨ ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɩɪɢɛɥɢɠɟɧɧɵɯ ɚɥɝɨ28
ɪɢɬɦɨɜ, ɜ ɨɫɧɨɜɟ ɤɨɬɨɪɵɯ ɥɟɠɢɬ ɫɯɟɦɚ ɦɟɬɨɞɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ. ȼ ɦɨɞɟɥɶɧɭɸ ɫɯɟɦɭ ɚɥɝɨɪɢɬɦɚ ɩɪɢ ɷɬɨɦ ɜɧɨɫɹɬɫɹ ɫɥɟɞɭɸɳɢɟ ɢɡɦɟɧɟɧɢɹ. Ɇɨɞɟɥɶɧɚɹ ɫɯɟɦɚ ɩɚɪɚɦɟɬɪɢɡɨɜɚɧɧɨɝɨ ɚɥɝɨɪɢɬɦɚ (ɞɥɹ ɩɨɢɫɤɚ ɦɚɤɫɢɦɭɦɚ) ɒɚɝ 1. Ɂɚɞɚɬɶ: R — ɪɟɤɨɪɞɧɨɟ ɡɧɚɱɟɧɢɟ; A — ɢɧɞɢɤɚɬɨɪ ɩɟɪɫɩɟɤɬɢɜɧɨɫɬɢ; N — ɝɥɭɛɢɧɚ ɩɚɦɹɬɢ. Ɉɪɝɚɧɢɡɨɜɚɬɶ ɫɩɢɫɤɢ ɢɧɞɟɤɫɨɜ I ɢ J ; i I — ɧɨɦɟɪɚ ɧɟɡɚɤɪɵɬɵɯ ɦɧɨɠɟɫɬɜ ɫ ɢɡɜɟɫɬɧɵɦɢ ɭɠɟ ɨɰɟɧɤɚɦɢ, j J — ɧɨɦɟɪɚ ɦɧɨɠɟɫɬɜ, ɨɰɟɧɤɢ ɤɨɬɨɪɵɯ ɟɳɟ ɧɟ ɜɵɱɢɫɥɟɧɵ. ɉɨɥɨɠɢɬɶ I 0 , J ^0`, k 1 , k — ɫɱɟɬɱɢɤ ɱɢɫɥɚ ɲɚɝɨɜ ɚɥɝɨɪɢɬɦɚ (ɱɢɫɥɨ ɜɟɬɜɥɟɧɢɣ). ɒɚɝ 2. ȼɵɱɢɫɥɢɬɶ ɨɰɟɧɤɢ ɦɧɨɠɟɫɬɜ : j , j J . ɒɚɝ 3. ȼɨɡɦɨɠɧɨɟ ɨɛɧɨɜɥɟɧɢɟ ɪɟɤɨɪɞɚ ½ R : max ® R; max M x S ¾, sS ¯ ¿ s ɝɞɟ x , s S — ɢɡɜɟɫɬɧɵɟ ɤ ɷɬɨɦɭ ɦɨɦɟɧɬɭ ɞɨɩɭɫɬɢɦɵɟ ɬɨɫɤɢ ɢɡ : 0 . ɒɚɝ 4. ȿɫɥɢ k t N 1 , ɬɨ ɩɟɪɟɣɬɢ ɤ ɲɚɝɭ 6. ɒɚɝ 5. ɉɨɥɨɠɢɬɶ R1 ɪɚɜɧɵɦ R . ɉɟɪɟɣɬɢ ɤ ɲɚɝɭ 7. ɒɚɝ 6. ɉɨɥɨɠɢɬɶ R1 ɪɚɜɧɵɦ A , k 0 . ɒɚɝ 7. ɍɞɚɥɢɬɶ ɢɡ ɫɩɢɫɤɚ I ɧɨɦɟɪɚ i , ɞɥɹ ɤɨɬɨɪɵɯ M i d R1 . ɍɞɚɥɢɬɶ ɢɡ ɫɩɢɫɤɚ J ɧɨɦɟɪɚ ɡɚɤɪɵɬɵɯ ɦɧɨɠɟɫɬɜ (ɩɨɥɭɱɟɧɚ ɞɨɩɭɫɬɢɦɚɹ ɬɨɱɤɚ ɩɪɢ ɪɟɲɟɧɢɢ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɢ, ɜɟɬɜɶ ɩɪɨɣɞɟɧɚ ɩɨ ɜɫɟɣ ɝɥɭɛɢɧɟ ɞɟɪɟɜɚ, ɞɨɩɭɫɬɢɦɨɟ ɦɧɨɠɟɫɬɜɨ ɜ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɟ ɨɤɚɡɚɥɨɫɶ ɩɭɫɬɵɦ, ɨɰɟɧɤɢ H j ɭɞɨɜ-
ɥɟɬɜɨɪɹɸɬ ɧɟɪɚɜɟɧɫɬɜɚɦ H j d R1). Ɉɫɬɚɥɶɧɵɟ ɧɨɦɟɪɚ ɩɟɪɟɧɟɫɬɢ ɢɡ ɫɩɢɫɤɚ J ɜ ɫɩɢɫɨɤ I . ɒɚɝ 8. ȿɫɥɢ I 0 , ɬɨ ɩɟɪɟɣɬɢ ɤ ɲɚɝɭ 12. ɒɚɝ 9. ɉɨɥɨɠɢɬɶ k ɪɚɜɧɵɦ k 1 . ɒɚɝ 10. Ɉɩɪɟɞɟɥɢɬɶ ɩɨɞɦɧɨɠɟɫɬɜɨ, ɩɨɞɥɟɠɚɳɟɟ ɞɚɥɶɧɟɣɲɟɦɭ ɜɟɬɜɥɟɧɢɸ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɫɬɪɚɬɟɝɢɟɣ ɜɵɛɪɚɬɶ ɬɪɟɛɭɟɦɵɣ ɧɨɦɟɪ ɢɡ ɦɧɨɠɟɫɬɜɚ I . ɒɚɝ 11. Ɉɫɭɳɟɫɬɜɢɬɶ ɪɚɡɛɢɟɧɢɟ ɦɧɨɠɟɫɬɜɚ, ɜɵɛɪɚɧɧɨɝɨ ɧɚ ɩɪɟɞɵɞɭɳɟɦ ɲɚɝɟ, ɧɚ ɩɨɞɦɧɨɠɟɫɬɜɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɚɜɢɥɨɦ ɜɟɬɜɥɟɧɢɹ : p : pl . lL
Ɂɚɧɟɫɬɢ ɧɨɦɟɪɚ pl , l L ɜ ɫɩɢɫɨɤ J . ɒɚɝ 12. Ɉɫɬɚɧɨɜ. ȼɵɞɚɱɚ ɪɟɡɭɥɶɬɚɬɨɜ ɩɨ ɢɧɮɨɪɦɚɰɢɢ, ɫɜɹɡɚɧɧɨɣ ɫ ɪɟɤɨɪɞɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɢ ɬɨɱɤɚɦɢ, ɜ ɤɨɬɨɪɵɯ ɷɬɢ ɡɧɚɱɟɧɢɹ ɩɨɥɭɱɟɧɵ.
29
ɍɉɊȺɀɇȿɇɂə
1. ɋɨɫɬɚɜɶɬɟ ɤɨɧɤɪɟɬɧɭɸ ɜɵɱɢɫɥɢɬɟɥɶɧɭɸ ɫɯɟɦɭ ɚɥɝɨɪɢɬɦɚ, ɨɛɟɫɩɟɱɢɜɚɸɳɟɝɨ ɜ ɤɚɱɟɫɬɜɟ ɨɬɜɟɬɚ ɩɟɪɜɭɸ ɩɨɩɚɜɲɭɸɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɜɵɛɪɚɧɧɨɣ ɫɬɪɚɬɟɝɢɢ ɞɨɩɭɫɬɢɦɭɸ ɬɨɱɤɭ: ɚ) ɢɫɩɨɥɶɡɭɣɬɟ ɫɬɪɚɬɟɝɢɸ «ɩɨ ɦɚɤɫɢɦɭɦɭ ɨɰɟɧɤɢ», ɛ) ɢɫɩɨɥɶɡɭɣɬɟ ɫɬɪɚɬɟɝɢɸ ɨɞɧɨɫɬɨɪɨɧɧɟɝɨ ɨɛɯɨɞɚ ɞɟɪɟɜɚ. 2. ɉɪɟɞɥɨɠɢɬɟ ɪɟɤɨɦɟɧɞɚɰɢɢ ɩɨ ɭɩɪɚɜɥɟɧɢɸ ɩɚɪɚɦɟɬɪɚɦɢ N ɢ A ɜ ɩɪɨɰɟɫɫɟ ɪɚɛɨɬɵ ɚɥɝɨɪɢɬɦɚ (ɤɚɤ ɪɟɚɤɰɢɸ ɧɚ ɩɨɫɬɭɩɚɸɳɭɸ ɢɧɮɨɪɦɚɰɢɸ ɜ ɩɪɨɰɟɫɫɟ ɪɚɛɨɬɵ ɚɥɝɨɪɢɬɦɚ). 3. ɋɨɫɬɚɜɶɬɟ ɚɥɝɨɪɢɬɦ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɫ [ max [ min ɰɟɥɨɱɢɫɥɟɧɧɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ ɩɪɢ N 2 , A . 2 4. ɋɨɫɬɚɜɶɬɟ ɚɥɝɨɪɢɬɦɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ ɩɪɢ
[ max [ min
ɚ) N 1 , A R ( A ɛ) N
2
);
4 , A [ max H , H ! 0 .
5. ɋɨɫɬɚɜɶɬɟ ɚɥɝɨɪɢɬɦ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɞɥɹ ɤɨɧɜɟɣɟɪɧɨɣ ɫɢɫɬɟɦɵ ɩɪɢ ɚ) N
5 , A [0 H ,H ! 0 ;
ɛ) N
5 , A [ min H , H ! 0 ;
ɜ) N
5, A
ɝ) N
5, A
[ max [ min 2
;
[ max max [ 0 , [ min 2
.
6. ȼ ɤɚɤɢɯ ɫɥɭɱɚɹɯ ɦɨɠɟɬ ɛɵɬɶ ɧɟ ɩɨɥɭɱɟɧɨ ɧɢ ɨɞɧɨɣ ɞɨɩɭɫɬɢɦɨɣ ɬɨɱɤɢ? 4. ɆȿɌɈȾɕ ɈɌɋȿɑȿɇɂɃ
Ɉɫɧɨɜɧɚɹ ɢɞɟɹ ɦɟɬɨɞɨɜ ɨɬɫɟɱɟɧɢɣ (ɦɟɬɨɞɨɜ ɫɟɤɭɳɢɯ ɩɥɨɫɤɨɫɬɟɣ) ɫɨɫɬɨɢɬ ɜ ɫɥɟɞɭɸɳɟɦ. ɂɫɯɨɞɧɚɹ ɡɚɞɚɱɚ «ɩɨɝɪɭɠɚɟɬɫɹ» ɜ ɡɚɞɚɱɭ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ, ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɚ, ɧɚɩɪɢɦɟɪ, ɫɬɚɧɞɚɪɬɧɵɦ ɫɢɦɩɥɟɤɫɧɵɦ ɦɟɬɨɞɨɦ. ɉɨɥɭɱɟɧɧɨɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɟ ɩɪɨɜɟɪɹɟɬɫɹ ɧɚ ɞɨɩɭɫɬɢɦɨɫɬɶ ɜ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɟ. ȿɫɥɢ ɩɨɥɭɱɟɧɧɚɹ ɬɨɱɤɚ ɞɨɩɭɫɬɢɦɚ, ɬɨ ɨɧɚ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ. 30
ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɫɬɪɨɹɬɫɹ ɨɬɫɟɤɚɸɳɢɟ ɩɥɨɫɤɨɫɬɢ (ɜɜɨɞɹɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɥɢɧɟɣɧɵɟ ɨɝɪɚɧɢɱɟɧɢɹ), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɫɥɟɞɭɸɳɢɦ ɬɪɟɛɨɜɚɧɢɹɦ: – ɜɫɟ ɞɨɩɭɫɬɢɦɵɟ ɬɨɱɤɢ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɧɨɜɵɦ ɨɝɪɚɧɢɱɟɧɢɹɦ; – ɩɨɥɭɱɟɧɧɨɟ ɬɨɥɶɤɨ ɱɬɨ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɷɬɢɦ ɨɝɪɚɧɢɱɟɧɢɹɦ (ɨɬɫɟɤɚɟɬɫɹ). ɋɭɳɟɫɬɜɭɸɬ ɪɚɡɥɢɱɧɵɟ ɜɟɪɫɢɢ ɬɚɤɨɝɨ ɪɨɞɚ ɚɥɝɨɪɢɬɦɨɜ, ɜ ɤɨɬɨɪɵɯ ɩɪɟɞɥɚɝɚɸɬɫɹ ɪɚɡɧɵɟ ɩɨɫɬɪɨɟɧɢɹ ɨɬɫɟɤɚɸɳɢɯ ɩɥɨɫɤɨɫɬɟɣ. 4.1. ɉɟɪɜɵɣ ɚɥɝɨɪɢɬɦ Ƚɨɦɨɪɢ
Ɋɚɫɫɦɨɬɪɢɦ ɰɟɥɨɱɢɫɥɟɧɧɭɸ ɡɚɞɚɱɭ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɜ ɫɥɟɞɭɸɳɟɣ ɮɨɪɦɟ: n
¦c
M ( x)
j
x j o min,
(4.1)
j 1
n
¦a
ij
x j db i , i 1, m,
(4.2)
j 1
x j t 0,
(4.3)
x j =, j 1, n.
(4.4)
ɂɞɟɹ ɦɟɬɨɞɨɜ ɨɬɫɟɱɟɧɢɣ ɫɨɫɬɨɢɬ ɜ ɫɥɟɞɭɸɳɟɦ. ɉɪɨɰɟɫɫ ɩɨɥɭɱɟɧɢɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɧɚɱɢɧɚɟɬɫɹ ɫ ɪɟɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɟɣ ɨɰɟɧɨɱɧɨɣ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ (4.1) — (4.3) ɫ ɨɬɛɪɨɲɟɧɧɵɦ ɭɫɥɨɜɢɟɦ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ. ȿɫɥɢ ɩɨɥɭɱɟɧɧɚɹ ɩɪɢ ɷɬɨɦ ɨɩɬɢɦɚɥɶɧɚɹ ɬɨɱɤɚ x 0 ɢɦɟɟɬ ɬɨɥɶɤɨ ɰɟɥɵɟ ɤɨɨɪɞɢɧɚɬɵ, ɬɨ ɨɧɚ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (4.1) — (4.4). ȿɫɥɢ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ ɧɟɰɟɥɨɱɢɫɥɟɧɧɨɟ, ɬɨ ɜɜɨɞɢɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɨɝɪɚɧɢɱɟɧɢɟ, ɤɨɬɨɪɨɟ ɨɬɫɟɤɚɟɬ ɱɚɫɬɶ ɨɛɥɚɫɬɢ ɞɨɩɭɫɬɢɦɵɯ ɪɟɲɟɧɢɣ ɡɚɞɚɱɢ (4.1) — (4.3) ɜɦɟɫɬɟ ɫ ɧɚɣɞɟɧɧɵɦ ɧɟɰɟɥɨɱɢɫɥɟɧɧɵɦ ɪɟɲɟɧɢɟɦ, ɫɨɯɪɚɧɹɹ ɩɪɢ ɷɬɨɦ ɜɫɟ ɰɟɥɨɱɢɫɥɟɧɧɵɟ ɬɨɱɤɢ. Ɂɚɬɟɦ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ (4.1) — (4.3) ɫ ɭɱɟɬɨɦ ɧɨɜɨɝɨ ɨɝɪɚɧɢɱɟɧɢɹ. ȿɫɥɢ ɨɧɨ ɧɟɰɟɥɨɱɢɫɥɟɧɧɨɟ, ɬɨ ɜɜɨɞɢɬɫɹ ɧɨɜɨɟ ɨɝɪɚɧɢɱɟɧɢɟ, ɢ ɬɚɤ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɧɟ ɛɭɞɟɬ ɧɚɣɞɟɧɨ ɰɟɥɨɱɢɫɥɟɧɧɨɟ ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ. ɉɪɚɜɢɥɚ ɮɨɪɦɢɪɨɜɚɧɢɹ ɨɬɫɟɤɚɸɳɢɯ ɨɝɪɚɧɢɱɟɧɢɣ ɛɵɥɢ ɪɚɡɪɚɛɨɬɚɧɵ ɚɦɟɪɢɤɚɧɫɤɢɦ ɭɱɟɧɵɦ Ɋ. Ƚɨɦɨɪɢ. ɂɞɟɹ ɦɟɬɨɞɚ, ɧɚɡɜɚɧɧɨɝɨ ɩɟɪɜɵɦ ɚɥɝɨɪɢɬɦɨɦ Ƚɨɦɨɪɢ, ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ. ɉɭɫɬɶ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (4.1) — (4.3) ɫɢɦɩɥɟɤɫ-ɦɟɬɨɞɨɦ ɩɨɥɭɱɟɧɚ ɧɟɰɟɥɨɱɢɫɥɟɧɧɚɹ ɨɩɬɢɦɚɥɶɧɚɹ ɬɨɱɤɚ. Ɂɚɤɥɸɱɢɬɟɥɶɧɚɹ ɫɢɦɩɥɟɤɫɬɚɛɥɢɰɚ ɫɨɞɟɪɠɢɬ ɭɪɚɜɧɟɧɢɹ ɜɢɞɚ: (4.5) bi xi ¦ aij x j , j J
31
i I , ɝɞɟ I — ɦɧɨɠɟɫɬɜɨ ɛɚɡɢɫɧɵɯ ɩɟɪɟɦɟɧɧɵɯ, J — ɦɧɨɠɟɫɬɜɨ ɧɟɛɚɡɢɫɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɡɚɞɚɱɢ, bi ɢ aij — ɩɟɪɟɫɱɢɬɚɧɧɵɟ ɤ ɞɚɧɧɨɦɭ ɲɚɝɭ ɡɧɚɱɟɧɢɹ bi ɢ aij . ȼɵɛɟɪɟɦ ɢɧɞɟɤɫ i I ɬɚɤɨɣ, ɱɬɨ ɛɚɡɢɫɧɚɹ ɤɨɨɪɞɢɧɚɬɚ xi bi — ɞɪɨɛɧɚɹ. ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɰɟɥɨɣ ɱɚɫɬɢ ɱɢɫɥɚ ɢɦɟɟɦ: y t [ y ] . Ɍɚɤ ɤɚɤ x j t 0, j , ɬɨ
¦ aij x j t ¦ [aij ]x j .
j J
bi t xi
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫ ɭɱɟɬɨɦ ɪɚɜɟɧɫɬɜɚ (4.5) ɢɦɟɟɦ:
j J
¦ [aij ]x j
. ȿɫɥɢ ɩɟɪɟɦɟɧɧɵɟ x j =, j 1, n , ɬɨ ɩɨɫɥɟɞɧɟɟ ɧɟɪɚɜɟɧ-
j J
ɫɬɜɨ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɬɚɤ: [bi ] t xi ¦ [aij ]x j .
(4.6)
jJ
ȼɵɱɢɬɚɹ ɢɡ ɪɚɜɟɧɫɬɜɚ (3.5) ɧɟɪɚɜɟɧɫɬɜɨ (3.6), ɩɨɥɭɱɚɟɦ ɧɟɪɚɜɟɧɫɬɜɨ (4.7) {bi } d ¦ {aij }x j , jJ
ɝɞɟ ɫɢɦɜɨɥɨɦ {y} ɨɛɨɡɧɚɱɟɧɚ ɞɪɨɛɧɚɹ ɱɚɫɬɶ ɱɢɫɥɚ y. ɇɟɪɚɜɟɧɫɬɜɨ (4.7) ɜɟɪɧɨ ɞɥɹ ɜɫɟɯ ɞɨɩɭɫɬɢɦɵɯ ɰɟɥɵɯ ɬɨɱɟɤ ɜ ɡɚɞɚɱɟ (4.1) — (4.3). Ɉɞɧɚɤɨ ɤɨɨɪɞɢɧɚɬɵ ɩɨɥɭɱɟɧɧɨɝɨ ɧɟɰɟɥɨɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɷɬɨɦɭ ɨɝɪɚɧɢɱɟɧɢɸ, ɬɚɤ ɤɚɤ ɞɥɹ ɧɟɝɨ ɜɫɟ ɧɟɛɚɡɢɫɧɵɟ ɩɟɪɟɦɟɧɧɵɟ x j 0, j J , ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, {bi } d 0 , ɱɬɨ ɧɟɜɨɡɦɨɠɧɨ ɞɥɹ ɧɟɰɟɥɨɝɨ ɱɢɫɥɚ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɞɪɨɛɧɨɣ ɱɚɫɬɢ. ɇɟɪɚɜɟɧɫɬɜɨ (4.7) ɦɨɠɟɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɨ ɞɥɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɝɨ ɨɬɫɟɤɚɸɳɟɝɨ ɨɝɪɚɧɢɱɟɧɢɹ. ȼ ɩɪɢɜɟɞɟɧɧɭɸ ɤ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɟ ɡɚɞɚɱɭ ɟɝɨ ɜɜɨɞɹɬ ɜ ɜɢɞɟ (4.8) ¦ {aij }x j xn m 1 {bi }, j J
ɝɞɟ xn m 1 – ɞɨɩɨɥɧɢɬɟɥɶɧɚɹ ɩɟɪɟɦɟɧɧɚɹ (ɩɨɫɥɟ ɩɪɢɜɟɞɟɧɢɹ ɡɚɞɚɱɢ (4.1) – (4.3) ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ ɩɟɪɟɦɟɧɧɵɯ ɜ ɧɟɣ ɛɭɞɟɬ n m ). 4.2. Ⱥɥɝɨɪɢɬɦɢɱɟɫɤɚɹ ɫɯɟɦɚ ɦɟɬɨɞɚ ɒɚɝ 1. ɇɚɣɬɢ ɬɨɱɤɭ x1 — ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ (4.1) – (4.3) (ɛɟɡ ɭɫɥɨɜɢɹ ɰɟɥɨɱɢɫɥɟɧɧɨɫɬɢ). ɉɨɥɨɠɢɬɶ ɤ = 1. ɒɚɝ 2. ȿɫɥɢ x k — ɰɟɥɨɱɢɫɥɟɧɧɚɹ, ɬɨ ɪɟɲɟɧɢɟ ɧɚɣɞɟɧɨ, ɨɫɬɚɧɨɜ. ɂɧɚɱɟ ɡɚɮɢɤɫɢɪɨɜɚɬɶ ɦɧɨɠɟɫɬɜɚ I ɢ J — ɢɧɞɟɤɫɨɜ ɛɚɡɢɫɧɵɯ ɢ ɧɟɛɚɡɢɫɧɵɯ ɩɟɪɟɦɟɧɧɵɯ. ɒɚɝ 3. ȼɵɛɪɚɬɶ ɧɨɦɟɪ i I ɬɚɤɨɣ, ɱɬɨ ɤɨɨɪɞɢɧɚɬɚ xik — ɞɪɨɛɧɚɹ. ɒɚɝ 4. Ⱦɨɛɚɜɢɬɶ ɜ ɫɢɦɩɥɟɤɫ-ɬɚɛɥɢɰɭ ɧɨɜɨɟ ɨɝɪɚɧɢɱɟɧɢɟ: ¦ {aij }x j xn m k {bi } . j J
32
ɒɚɝ 5. ɋ ɭɱɟɬɨɦ ɞɨɛɚɜɥɟɧɧɨɝɨ ɨɝɪɚɧɢɱɟɧɢɹ ɨɬɵɫɤɚɬɶ ɧɨɜɨɟ ɪɟɲɟɧɢɟ x k 1 . ɉɨɥɨɠɢɬɶ k = k+1. ɉɟɪɟɣɬɢ ɧɚ ɲɚɝ 2. Ɂɚɦɟɱɚɧɢɟ. ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɨɛɵɱɧɨɝɨ ɫɢɦɩɥɟɤɫ-ɦɟɬɨɞɚ ɩɪɢ ɪɟɲɟɧɢɢ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɧɟɭɞɨɛɧɨ, ɬɚɤ ɤɚɤ ɞɨɛɚɜɥɟɧɢɟ ɧɨɜɨɝɨ ɨɝɪɚɧɢɱɟɧɢɹ ɤɚɠɞɵɣ ɪɚɡ ɛɭɞɟɬ ɩɪɢɜɨɞɢɬɶ ɤ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɜɵɡɨɜɚ ɦɟɬɨɞɚ ɢɫɤɭɫɫɬɜɟɧɧɨɝɨ ɛɚɡɢɫɚ. Ȼɨɥɟɟ ɩɪɟɞɩɨɱɬɢɬɟɥɶɧɵɦ ɜ ɞɚɧɧɨɣ ɫɢɬɭɚɰɢɢ ɹɜɥɹɟɬɫɹ ɞɜɨɣɫɬɜɟɧɧɵɣ ɫɢɦɩɥɟɤɫ-ɚɥɝɨɪɢɬɦ. ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ ɧɨɜɨɟ ɨɝɪɚɧɢɱɟɧɢɟ ɜɜɨɞɢɬɫɹ ɜ ɫɢɫɬɟɦɭ ɜ ɜɢɞɟ: ¦ { aij }x j xn m k { bi } , jJ
ɢ ɩɟɪɟɦɟɧɧɚɹ xn m k ɫɬɚɧɨɜɢɬɫɹ ɛɚɡɢɫɧɨɣ. ɋɨɝɥɚɫɧɨ ɞɜɨɣɫɬɜɟɧɧɨɦɭ ɫɢɦɩɥɟɤɫ-ɦɟɬɨɞɭ ɢɫɤɥɸɱɟɧɢɸ ɢɡ ɛɚɡɢɫɚ ɩɨɞɥɟɠɢɬ ɬɚ ɩɟɪɟɦɟɧɧɚɹ xl , ɡɧɚɱɟɧɢɟ ɤɨɬɨɪɨɣ ɨɬɪɢɰɚɬɟɥɶɧɨ. (ȿɫɥɢ ɜɫɟ ɛɚɡɢɫɧɵɟ ɩɟɪɟɦɟɧɧɵɟ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ, ɬɨ ɢɦɟɸɳɟɟɫɹ ɪɟɲɟɧɢɟ ɨɩɬɢɦɚɥɶɧɨ). Ⱦɥɹ ɜɜɨɞɚ ɜ ɛɚɡɢɫ ɜɵɛɢɪɚɟɬɫɹ ɬɚ ɢɡ ɧɟ'k ɛɚɡɢɫɧɵɯ ɩɟɪɟɦɟɧɧɵɯ xk , ɞɥɹ ɤɨɬɨɪɨɣ alk 0 ɢ ɨɬɧɨɲɟɧɢɟ ɦɢɧɢ| alk | ɦɚɥɶɧɨ. ɉɪɢɦɟɪ. Ɋɟɲɢɬɶ ɡɚɞɚɱɭ ɰɟɥɨɱɢɫɥɟɧɧɨɝɨ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ. M ( x ) x1 4 x2 o max, x1 2 x2 d 10 ,
x1 x2 d 15 / 2 , 2 x1 x2 d 10 , x1 , x2 t 0 , x1 , x2 =. Ɋɟɲɟɧɢɟ. ɉɪɢɜɟɞɟɦ ɡɚɞɚɱɭ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ (ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɭɦɧɨɠɢɜ ɜɬɨɪɨɟ ɨɝɪɚɧɢɱɟɧɢɟ ɧɚ 2). x1 4 x2 o max, x1 2 x2 x3 10 , 2 x1 2 x2 x4 15 , 2 x1 x2 x5 10 , x j t 0 , x j = , j 1,5.
33
Ɉɮɨɪɦɢɦ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ ɫɢɦɩɥɟɤɫ-ɬɚɛɥɢɰɵ. 1 4 0 0 0 0 0 CB b x1 x2 x3 x4 x5 x6 x7 x3 0 10 -1 2 1 0 0 B
x4
0
15
2
2
0
1
0
x5 'j x2 x4
0
10
4 0
5 5
-1 -1 -1/2 3
-1 -4 1 0
0 0 1/2 -1
0 0 0 1
1 0 0 0
x5 'j x2 x1 x5 'j x6 x2 x1 x5 x3 'j x7 x2 x1 x5 x3 x6 'j x8 x2 x1 x5 x3 x6 x4 'j
0
15
4 1 0
35/6 5/3 25/2
0 4 1 0 0
-2/3 11/2 2 23/2 1
0 4 1 0 0 0
-1/2 5 5/2 10 5/2 1
0 4 1 0 0 0 0
-1/2 5 2 11 2 1 1
3/2 -3 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0
0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
1/2 0 2 0 1/3 1/6 -1/3 1/3 1 -1/2 1 1 -2/3 -1/3 0 0 0 1/2 0 -1 1 1/2 0 1/2 0 0 0 0 0 1/2 0 -1 1 1/2 0 0 0 1/2 0 -1/2 0 0 0 0 0 0 1 0 0 0 0 1 0 0 34
1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0
0 x8
4
Ʉɨɦɦɟɧɬɚɪɢɣ
10 /2 15 /2 -
5/ 3 10 0 0 0 1 1/2 -1/2 3/2 -3/2 3/2 -1/2 0 0 0 0 1 0 0 0 0 0 0 1 0 0
ɇɚɣɞɟɧɚ ɬɨɱɤɚ x1 ! 0 0 0 0 1 1 -1 -6 6 -2 3 0 1 -1 -6 6 -2 0 3
ɇɚɣɞɟɧɚ ɬɨɱɤɚ x2 ! 0 0 0 0 0 1 0 1 -2 1 0 -2 1
ɇɚɣɞɟɧɚ ɬɨɱɤɚ x3 !
ɇɚɣɞɟɧɨ ɰɟɥɨɱɢɫɥ. ɪɟɲɟɧɢɟ!
ɇɚ 3-ɣ ɢɬɟɪɚɰɢɢ ɫɢɦɩɥɟɤɫ-ɦɟɬɨɞɚ ɧɚɣɞɟɧɨ ɧɟɰɟɥɨɱɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ 5 35 25 5 x1 ( , ,0,0, ) . ȼɵɛɢɪɚɟɦ, ɧɚɩɪɢɦɟɪ, x11 – ɞɪɨɛɧɭɸ ɛɚɡɢɫɧɭɸ ɤɨ3 6 2 3 ɨɪɞɢɧɚɬɭ – ɢ ɩɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɫɬɪɨɤɟ ɬɚɛɥɢɰɵ ɮɨɪɦɢɪɭɟɦ ɧɨɜɨɟ ɨɝɪɚ1
5
1
ɧɢɱɟɧɢɟ: {3}x3 {3}x4 x6 {3}. ɂɥɢ: 3 x3 3 x 4 x6 3 . Ⱦɨɛɚɜɥɹɟɦ ɷɬɨ ɨɝɪɚɧɢɱɟɧɢɟ ɜ ɬɚɛɥɢɰɭ ɢ ɨɫɭɳɟɫɬɜɥɹɟɦ ɨɞɧɭ ɢɬɟɪɚɰɢɸ ɞɜɨɣɫɬɜɟɧɧɨɝɨ ɫɢɦɩɥɟɤɫ-ɦɟɬɨɞɚ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ ɬɨɱɤɭ 11 23 11 2 2 x (2, ,1,0, ,0) . ȼɵɛɢɪɚɟɦ ɞɪɨɛɧɭɸ ɤɨɨɪɞɢɧɚɬɭ x2 ɢ ɞɨɛɚɜɥɹɟɦ 2 2 2 ɨɝɪɚɧɢɱɟɧɢɟ
12 x6 x7 12
1
1
2
ɢ ɬ. ɞ.
ɇɚ ɩɨɫɥɟɞɧɟɣ ɢɬɟɪɚɰɢɢ ɩɨɥɭɱɟɧɚ ɬɨɱɤɚ x 4 ɹɜɥɹɟɬɫɹ ɰɟɥɨɱɢɫɥɟɧɧɨɣ. Ɉɫɬɚɧɨɜ. Ɉɬɜɟɬ: x* (2, 5) , M ( x* ) 22.
(2, 5, 2,1,11,1,0, 0) , ɤɨɬɨɪɚɹ
ɍɉɊȺɀɇȿɇɂə
1. Ⱦɨ ɩɪɢɜɟɞɟɧɢɹ ɡɚɞɚɱɢ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ ɜɫɟ ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɞɨɥɠɧɵ ɛɵɬɶ ɩɪɢɜɟɞɟɧɵ ɤ ɰɟɥɵɦ. ɉɨɱɟɦɭ? 2. Ɋɟɲɢɬɶ ɐɁɅɉ ɦɟɬɨɞɨɦ ɨɬɫɟɱɟɧɢɣ: a ) 3 x1 4 x2 o max,
b ) x1 2 x2 o max,
c ) x1 4 x2 o max,
x1 x2 d 7 / 2 ,
2 x1 x2 d 4 ,
x1 2 x2 d 10 ,
2 x1 d 11,
x1 5 x2 d 40 ,
x1 x2 d 7 21 ,
7 x1 11x2 d 0 ,
6 x1 x2 d 48 ,
2 x1 x2 d 10 ,
x1 , x2 t 0 , x1 , x2 =.
x1 , x2 t 0 , x1 , x2 =.
x1 , x2 t 0 , x1 , x2 =.
Ɉɬɜɟɬ: x
max
=(5,3).
Ɉɬɜɟɬ: x
max
=(6,9).
Ɉɬɜɟɬ: x max =(2,5).
5. ȾɂɇȺɆɂɑȿɋɄɈȿ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂȿ
ȼ ɞɚɧɧɨɦ ɪɚɡɞɟɥɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ, ɨɛɥɚɞɚɸɳɢɟ ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣɫɬɜɚɦɢ: ɩɪɨɰɟɫɫ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɛɢɬ ɧɚ ɨɬɞɟɥɶɧɵɟ ɷɬɚɩɵ (ɲɚɝɢ); ɪɟɲɟɧɢɟ, ɩɪɢɧɢɦɚɟɦɨɟ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ, ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɫɨɫɬɨɹɧɢɹ ɭɩɪɚɜɥɹɟɦɨɝɨ ɨɛɴɟɤɬɚ ɧɚ ɞɚɧɧɨɦ ɲɚɝɟ; ɫɨɫɬɨɹɧɢɟ ɨɛɴɟɤɬɚ ɜ ɤɨɧɰɟ ɤɚɠɞɨɝɨ ɲɚɝɚ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɫɨɫɬɨɹɧɢɹ ɜ ɧɚɱɚɥɟ ɲɚɝɚ ɢ ɨɬ ɪɟɲɟɧɢɹ, ɩɪɢɧɢɦɚɟɦɨɝɨ ɧɚ ɷɬɨɦ ɲɚɝɟ; 35
ɰɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɞɨɥɠɧɚ ɛɵɬɶ ɚɞɞɢɬɢɜɧɨɣ: G x
N
¦ g i xi . i 1
ɍɤɚɡɚɧɧɵɟ ɫɜɨɣɫɬɜɚ ɩɨɡɜɨɥɹɸɬ ɨɫɭɳɟɫɬɜɢɬɶ ɢɧɜɚɪɢɚɧɬɧɨɟ ɩɨɝɪɭɠɟɧɢɟ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ ɜ ɫɟɦɟɣɫɬɜɨ ɚɧɚɥɨɝɢɱɧɵɯ ɡɚɞɚɱ. ɋɪɟɞɢ ɡɚɞɚɱ ɷɬɨɝɨ ɫɟɦɟɣɫɬɜɚ ɧɚɯɨɞɹɬɫɹ, ɜ ɱɚɫɬɧɨɫɬɢ, ɡɚɞɚɱɢ ɫ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ. Ɍɚɤɢɟ ɡɚɞɚɱɢ, ɤɚɤ ɩɪɚɜɢɥɨ, ɥɟɝɤɨ ɪɟɲɚɸɬɫɹ. ɋ ɩɨɦɨɳɶɸ ɪɟɤɭɪɪɟɧɬɧɵɯ ɫɨɨɬɧɨɲɟɧɢɣ (ɭɪɚɜɧɟɧɢɣ Ȼɟɥɥɦɚɧɚ) ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɫɜɹɡɶ ɦɟɠɞɭ ɡɚɞɚɱɚɦɢ ɛɨɥɶɲɢɯ ɪɚɡɦɟɪɨɜ ɫ ɡɚɞɚɱɚɦɢ ɦɟɧɶɲɢɯ ɪɚɡɦɟɪɨɜ: ɡɚɞɚɱɢ ɫ ɞɜɭɦɹ ɩɟɪɟɦɟɧɧɵɦɢ ɪɟɲɚɸɬɫɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɨɞɧɨɦɟɪɧɵɯ, ɡɚɞɚɱɢ ɫ ɬɪɟɦɹ ɩɟɪɟɦɟɧɧɵɦɢ – ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɞɜɭɦɟɪɧɵɯ ɢ ɬ. ɞ. ɉɪɢ ɷɬɨɦ ɤɚɠɞɵɣ ɪɚɡ ɪɟɲɚɸɬɫɹ ɡɚɞɚɱɢ ɩɨɢɫɤɚ ɩɨ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ. 5.1. ɉɪɢɧɰɢɩ ɨɩɬɢɦɚɥɶɧɨɫɬɢ Ȼɟɥɥɦɚɧɚ
ɉɪɢɧɰɢɩ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ, ɤɚɤɨɜɵ ɛɵ ɧɢ ɛɵɥɢ ɫɨɫɬɨɹɧɢɟ ɨɛɴɟɤɬɚ ɜ ɧɚɱɚɥɟ ɥɸɛɨɝɨ ɲɚɝɚ ɢ ɪɟɲɟɧɢɟ, ɩɪɢɧɢɦɚɟɦɨɟ ɧɚ ɷɬɨɦ ɲɚɝɟ, ɩɨɫɥɟɞɭɸɳɟɟ ɪɟɲɟɧɢɟ ɞɨɥɠɧɨ ɛɵɬɶ ɨɩɬɢɦɚɥɶɧɵɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɜ ɤɨɬɨɪɨɟ ɩɟɪɟɯɨɞɢɬ ɨɛɴɟɤɬ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɪɢɧɹɬɨɝɨ ɪɟɲɟɧɢɹ. ɗɬɨ ɫɜɨɣɫɬɜɨ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɨɜɟɞɟɧɢɹ. ɇɚ ɟɝɨ ɨɫɧɨɜɟ ɡɚɩɢɫɵɜɚɟɬɫɹ ɪɟɤɭɪɪɟɧɬɧɨɟ ɭɪɚɜɧɟɧɢɟ Ȼɟɥɥɦɚɧɚ (ɞɥɹ ɡɚɞɚɱɢ ɦɚɤɫɢɦɢɡɚɰɢɢ). f k p k max^g k xk f k 1 T p k , xk `. xk
Ɂɞɟɫɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ: p k – ɫɨɫɬɨɹɧɢɟ ɨɛɴɟɤɬɚ ɜ ɧɚɱɚɥɟ k-ɝɨ ɲɚɝɚ; xk – ɪɟɲɟɧɢɟ, ɩɪɢɧɢɦɚɟɦɨɟ ɧɚ k-ɦ ɲɚɝɟ; f k pk – ɨɩɬɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɜ ɡɚɞɚɱɟ ɫ k ɩɟɪɟɦɟɧɧɵɦɢ; f k 1 T p k , xk — ɨɩɬɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɜ ɡɚɞɚɱɟ ɫ (k-1)-ɣ ɩɟɪɟɦɟɧɧɨɣ; g k ( xk ) — ɡɧɚɱɟɧɢɟ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ, ɩɨɥɭɱɟɧɧɨɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ, ɩɪɢɧɹɬɨɝɨ ɧɚ ɞɚɧɧɨɦ ɲɚɝɟ; pk 1 T pk , xk — ɭɪɚɜɧɟɧɢɟ ɩɪɨɰɟɫɫɚ, ɤɨɬɨɪɨɟ ɩɨɤɚɡɵɜɚɟɬ, ɤɚɤ ɦɟɧɹɟɬɫɹ ɫɨɫɬɨɹɧɢɟ ɨɛɴɟɤɬɚ pk 1 ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɪɢɧɹɬɨɝɨ ɪɟɲɟɧɢɹ xk . ɉɪɨɰɟɫɫ ɪɟɲɟɧɢɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɫɧɚɱɚɥɚ ɪɟɲɚɸɬɫɹ ɡɚɞɚɱɢ, ɨɛɨɡɧɚɱɟɧɧɵɟ ɱɟɪɟɡ f1 p1 , ɝɞɟ p1 — ɥɸɛɨɟ ɜɨɡɦɨɠɧɨɟ ɫɨɫɬɨɹɧɢɟ (ɞɨɩɭɫɬɢɦɨɟ ɦɧɨɠɟɫɬɜɨ ɫɨɫɬɨɹɧɢɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɧɞɢɜɢɞɭɚɥɶɧɨ ɜ ɤɚɠɞɨɣ ɡɚɞɚɱɟ). Ɂɚɬɟɦ, ɢɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ, ɫ ɩɨɦɨɳɶɸ ɪɟɤɭɪɪɟɧɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɪɟɲɚɸɬɫɹ ɡɚɞɚɱɢ ɫ ɞɜɭɦɹ ɩɟɪɟɦɟɧɧɵɦɢ 36
f 2 p2 max^g 2 x2 f1 T p2 , x2 ` ɞɥɹ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɫɨɫɬɨɹɧɢɣ ɨɛɴx2
ɟɤɬɚ p2 . ɂ ɬɚɤ ɞɚɥɟɟ ɞɨ ɩɨɥɭɱɟɧɢɹ ɪɟɲɟɧɢɹ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ f N p0 , ɝɞɟ N — ɤɨɥɢɱɟɫɬɜɨ ɲɚɝɨɜ ɜ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɟ, p0 — ɧɚɱɚɥɶɧɨɟ ɫɨɫɬɨɹɧɢɟ ɨɛɴɟɤɬɚ. 5.2. Ɂɚɞɚɱɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɪɟɞɫɬɜ ɦɟɠɞɭ ɩɪɟɞɩɪɢɹɬɢɹɦɢ
ɋɨɜɟɬ ɞɢɪɟɤɬɨɪɨɜ ɪɚɫɫɦɚɬɪɢɜɚɟɬ ɩɪɟɞɥɨɠɟɧɢɹ ɩɨ ɧɚɪɚɳɢɜɚɧɢɸ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɦɨɳɧɨɫɬɟɣ ɞɥɹ ɭɜɟɥɢɱɟɧɢɹ ɜɵɩɭɫɤɚ ɨɞɧɨɪɨɞɧɨɣ ɩɪɨɞɭɤɰɢɢ ɧɚ ɱɟɬɵɪɟɯ ɩɪɟɞɩɪɢɹɬɢɹɯ, ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɮɢɪɦɟ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɜɵɞɟɥɹɸɬɫɹ ɫɪɟɞɫɬɜɚ ɜ ɨɛɴɟɦɟ 80 ɦɥɧ ɪ. ȼɚɪɢɚɧɬɵ ɜɥɨɠɟɧɢɣ ɤɪɚɬɧɵ 20 ɦɥɧ ɪ. (0; 20; 40; 60; 80). ɉɪɢɪɨɫɬ ɜɵɩɭɫɤɚ ɩɪɨɞɭɤɰɢɢ ɡɚɜɢɫɢɬ ɨɬ ɜɵɞɟɥɟɧɧɨɣ ɫɭɦɦɵ. ɗɬɢ ɡɧɚɱɟɧɢɹ ɡɚɞɚɧɵ ɬɚɛɥɢɰɟɣ. ɇɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɪɟɞɫɬɜ ɦɟɠɞɭ ɩɪɟɞɩɪɢɹɬɢɹɦɢ, ɨɛɟɫɩɟɱɢɜɚɸɳɟɟ ɦɚɤɫɢɦɚɥɶɧɵɣ ɩɪɢɪɨɫɬ ɜɵɩɭɫɤɚ ɩɪɨɞɭɤɰɢɢ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɧɚ ɨɞɧɨ ɩɪɟɞɩɪɢɹɬɢɟ ɦɨɠɧɨ ɨɫɭɳɟɫɬɜɢɬɶ ɧɟ ɛɨɥɟɟ ɨɞɧɨɣ ɢɧɜɟɫɬɢɰɢɢ.
ȼɨɡɦɨɠɧɵɟ ɜɥɨɠɟɧɢɹ 20 40 60 80
ɉɪɢɪɨɫɬ ɜɵɩɭɫɤɚ ɩɪɨɞɭɤɰɢɢ gi ( xi ) , ɦɥɧ ɪ. ɩɪɟɞɩɪɢɹɬɢɟ ɩɪɟɞɩɪɢɹɬɢɟ ɩɪɟɞɩɪɢɹɬɢɟ ɩɪɟɞɩɪɢɹɬɢɟ 1 2 3 4 8 10 12 11 16 20 21 23 25 28 27 30 36 40 38 37
ɗɬɭ ɡɚɞɚɱɭ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɨɪɦɟ g1 x1 g 2 x2 g 3 x3 g 4 x4 o max .
ɉɪɢ ɨɝɪɚɧɢɱɟɧɢɹɯ
x1 x2 x3 x4
80,
xi 0; 20; 40; 60, i 1,2,3,4. ɑɟɪɟɡ xi ɨɛɨɡɧɚɱɟɧɵ ɤɨɥɢɱɟɫɬɜɚ ɫɪɟɞɫɬɜ, ɜɵɞɟɥɹɟɦɵɟ ɤɚɠɞɨɦɭ ɩɪɟɞɩɪɢɹɬɢɸ, S = 80 – ɨɛɳɚɹ ɫɭɦɦɚ ɩɨɞɥɟɠɚɳɚɹ ɪɚɫɩɪɟɞɟɥɟɧɢɸ, g i xi – ɨɠɢɞɚɟɦɵɣ ɩɪɢɪɨɫɬ ɩɪɨɞɭɤɰɢɢ ɨɬ ɜɵɞɟɥɟɧɧɵɯ ɫɪɟɞɫɬɜ (ɦɥɧ ɪ.)
37
Ɋɟɲɟɧɢɟ. Ɋɚɡɛɢɜɚɟɦ ɩɪɨɰɟɫɫ ɪɟɲɟɧɢɹ ɧɚ ɱɟɬɵɪɟ ɷɬɚɩɚ (ɩɨ ɤɨɥɢɱɟɫɬɜɭ ɩɪɟɞɩɪɢɹɬɢɣ). I ɷɬɚɩ. Ɉɛɫɭɠɞɚɸɬɫɹ ɢɧɜɟɫɬɢɰɢɢ ɧɚ ɩɟɪɜɨɟ ɩɪɟɞɩɪɢɹɬɢɟ. ȼɨɡɦɨɠɧɵɟ ɡɧɚɱɟɧɢɹ ɩɟɪɟɦɟɧɧɨɣ ɩɪɢ ɫɨɫɬɨɹɧɢɢ S = 80 ɦɨɝɭɬ ɛɵɬɶ ɪɚɜɧɵɦɢ 0, 20, 40, 60, 80. ɉɨɫɤɨɥɶɤɭ ɜ ɡɚɞɚɱɟ ɫ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ ɜɵɛɨɪ ɨɬɫɭɬɫɬɜɭɟɬ, ɬɨ ɜɤɥɚɞɵɜɚɟɦɚɹ ɫɭɦɦɚ x1 ɛɭɞɟɬ ɪɚɜɧɚ ɜɵɞɟɥɟɧɧɵɦ ɫɪɟɞɫɬɜɚɦ S ɢ ɨɩɬɢɦɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɛɭɞɭɬ ɪɚɜɧɵ f1 S g1 x1 . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, f1 20 8, f1 40 16, f1 60 25, f1 80 36.
II ɷɬɚɩ. ȼɵɞɟɥɟɧɧɭɸ ɫɭɦɦɭ S ɪɚɫɩɪɟɞɟɥɹɟɦ ɦɟɠɞɭ ɩɟɪɜɵɦ ɢ ɜɬɨɪɵɦ ɩɪɟɞɩɪɢɹɬɢɹɦɢ. Ɋɟɤɭɪɪɟɧɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢ ɷɬɨɦ ɢɦɟɟɬ ɜɢɞ
f 2 S max^g 2 x2 f1 S x2 `. x2
ɉɨɢɫɤ ɜɟɞɟɬɫɹ ɩɨ ɩɟɪɟɦɟɧɧɨɣ x2 , ɤɨɬɨɪɚɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɱɢɫɥɚ S ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ 0; 20;…S. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ: ɩɪɢ S = 20 f 2 20 max^8 0, 0 10` 10 ɩɪɢ x2 20; ɩɪɢ S = 40 f 2 40 max^16, 8 10, 20` 20 ɩɪɢ x2 40; ɩɪɢ S = 60 f 2 60 max^25, 16 10, 8 20, 28` 28 ɩɪɢ x2 40 , ɚ ɬɚɤɠɟ ɩɪɢ x2 60 ; ɩɪɢ S = 80 ɦɚɤɫɢɦɭɦ ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ x2 80 . III ɷɬɚɩ. Ɋɟɲɚɟɬɫɹ ɡɚɞɚɱɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɪɟɞɫɬɜ ɦɟɠɞɭ ɬɪɟɦɹ ɩɪɟɞɩɪɢɹɬɢɹɦɢ. Ɋɟɤɭɪɪɟɧɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ f 3 S max^g 3 x3 f 2 S x3 `. x3
ɉɨɢɫɤ ɜɟɞɟɬɫɹ ɩɨ ɩɟɪɟɦɟɧɧɨɣ x3 , ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ 0, 20,…S. ɩɪɢ S = 20 f 3 20 max^10, 12` 12 ɩɪɢ x3 20; ɩɪɢ S = 40 f 3 40 max^20, 10 12, 21` 22 ɩɪɢ x3 20; ɩɪɢ S = 60 f 3 60 max^28, 20 12, 10 21, 27` 28 ɩɪɢ x3 0; ɩɪɢ S = 80 f 3 80 max^40, 28 12, 20 21, 10 27, 38` 41 ɦɚɤɫɢɦɭɦ ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ x3 40. 38
IV ɷɬɚɩ. ɇɚ ɞɚɧɧɨɦ ɷɬɚɩɟ ɪɟɲɚɟɬɫɹ ɢɫɯɨɞɧɚɹ ɡɚɞɚɱɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɭɦɦɵ ɜ 80 ɦɥɧ ɪ. ɦɟɠɞɭ ɱɟɬɵɪɶɦɹ ɩɪɟɞɩɪɢɹɬɢɹɦɢ ɧɚ ɨɫɧɨɜɟ ɪɟɤɭɪɪɟɧɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɢɞɚ
f 4 80 max^g 4 x4 f 3 80 x4 `. x4
ɉɨɢɫɤ ɜɟɞɟɬɫɹ ɩɨ ɩɟɪɟɦɟɧɧɨɣ x4 , ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ 0, 20, 40, 60, 80. f 4 80 max^41, 11 28, 23 22, 30 12, 37` 45 ɩɪɢ x4 40. ɂɬɚɤ, ɩɨɥɭɱɟɧ ɨɠɢɞɚɟɦɵɣ ɦɚɤɫɢɦɚɥɶɧɵɣ ɩɪɢɪɨɫɬ ɜɵɩɭɫɤɚ ɩɪɨɞɭɤɰɢɢ, ɪɚɜɧɵɣ 45 ɦɥɧ ɪ. Ⱦɚɥɟɟ ɫɥɟɞɭɟɬ ɨɩɪɟɞɟɥɢɬɶ, ɩɪɢ ɤɚɤɢɯ ɜɚɪɢɚɧɬɚɯ ɜɥɨɠɟɧɢɣ ɩɨɥɭɱɟɧ ɷɬɨɬ ɪɟɡɭɥɶɬɚɬ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɣɬɢ ɨɛɪɚɬɧɵɦ ɩɭɬɟɦ ɨɬ ɱɟɬɜɟɪɬɨɝɨ ɷɬɚɩɚ ɤ ɩɟɪɜɨɦɭ ɢ ɩɪɨɫɥɟɞɢɬɶ, ɤɚɤ ɩɨɥɭɱɟɧɨ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ. ɇɚ ɱɟɬɜɟɪɬɨɦ ɷɬɚɩɟ ɩɨɥɭɱɟɧ ɦɚɤɫɢɦɚɥɶɧɵɣ ɜɚɪɢɚɧɬ ɩɪɢ x4 40 . Ɏɢɤɫɢɪɭɟɦ ɷɬɨ ɡɧɚɱɟɧɢɟ ɩɟɪɟɦɟɧɧɨɣ. Ɂɚɦɟɱɚɟɦ, ɱɬɨ 45 = 23 + 22, ɝɞɟ 22 f 3 40 . ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ ɩɨɥɭɱɟɧ ɧɚ ɬɪɟɬɶɟɦ ɷɬɚɩɟ ɩɪɢ x 3 20 . Ɏɢɤɫɢɪɭɟɦ ɷɬɨ ɡɧɚɱɟɧɢɟ ɩɟɪɟɦɟɧɧɨɣ. Ⱦɚɥɟɟ ɡɚɦɟɱɚɟɦ, ɱɬɨ f 3 40 12 10 , ɝɞɟ 10 f 2 20 . ɗɬɨ ɡɧɚɱɟɧɢɟ ɩɨɥɭɱɟɧɨ ɩɪɢ x2 20 . Ⱥɧɚɥɨɝɢɱɧɨ ɩɨɥɭɱɚɟɦ, ɱɬɨ x1 0 . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɧɜɟɫɬɢɰɢɢ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɜɵɞɟɥɢɬɶ ɜɬɨɪɨɦɭ, ɬɪɟɬɶɟɦɭ ɢ ɱɟɬɜɟɪɬɨɦɭ ɩɪɟɞɩɪɢɹɬɢɹɦ ɜ ɤɨɥɢɱɟɫɬɜɟ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ 20; 20; 40 ɦɥɧ ɪ. Ɉɩɬɢɦɚɥɶɧɵɣ ɩɪɢɪɨɫɬ ɫɨɫɬɚɜɢɬ 45 ɦɥɧ ɪ. ɍɉɊȺɀɇȿɇɂə
1. Ɇɟɬɨɞɨɦ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɪɟɲɢɬɶ ɡɚɞɚɱɭ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɪɟɫɭɪɫɨɜ ɦɟɠɞɭ ɩɪɟɞɩɪɢɹɬɢɹɦɢ. 40 ɦɥɧ ɪ. ɧɟɨɛɯɨɞɢɦɨ ɪɚɫɩɪɟɞɟɥɢɬɶ ɦɟɠɞɭ ɱɟɬɵɪɶɦɹ ɩɪɟɞɩɪɢɹɬɢɹɦɢ ɬɚɤ, ɱɬɨɛɵ ɩɨɥɭɱɢɬɶ ɦɚɤɫɢɦɚɥɶɧɵɣ ɩɪɢɪɨɫɬ ɜɵɩɭɫɤɚ ɩɪɨɞɭɤɰɢɢ. Ⱦɨɯɨɞɧɨɫɬɢ ɨɬ ɜɥɨɠɟɧɢɣ gi ( xi ) ɡɚɞɚɧɵ ɬɚɛɥɢɰɟɣ. ȼɥɨɠɟɧɢɹ ɤɪɚɬɧɵ 8 ɦɥɧ ɪ. g i xi xi
8 16 24 32 40
g1 x1
g 2 x2
g 3 x3
g 4 x4
A 57 120 150 180
28 B 122 146 175
35 67 C 144 180
27 73 125 D 178
39
Ɂɧɚɱɟɧɢɹ A, B, C, D ɞɚɧɵ ɜ ɬɚɛɥɢɰɟ. ȼɚɪɢɚɧɬ: 1 2 3 4 5 6 7 8 9 10 40 42 39 41 46 45 38 47 50 48 A 61 65 59 68 64 66 62 63 67 60 B 119 123 124 126 118 122 125 120 128 130 C 176 175 179 181 174 178 177 173 180 182 D
2. ɉɪɟɞɥɨɠɢɬɶ ɧɚ ɨɫɧɨɜɟ ɪɟɤɭɪɪɟɧɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ȼɟɥɥɦɚɧɚ ɚɥɝɨɪɢɬɦɢɱɟɫɤɭɸ ɫɯɟɦɭ ɪɟɲɟɧɢɹ ɜ ɡɚɞɚɱɟ ɨ ɪɚɧɰɟ. 3. Ɂɚɩɢɫɚɬɶ ɪɟɤɭɪɪɟɧɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɞɜɭɦɟɪɧɨɣ ɡɚɞɚɱɟ ɨ ɪɚɧɰɟ.
40
ɅɂɌȿɊȺɌɍɊȺ Ɉɫɧɨɜɧɚɹ: 1. Ⱥɬɬɟɬɤɨɜ Ⱥ.ȼ. Ɇɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ / Ⱥ.ȼ. Ⱥɬɬɟɬɤɨɜ, ɋ.ȼ. Ƚɚɥɤɢɧ, ȼ.ɋ. Ɂɚɪɭɛɢɧ. – Ɇ. : ɂɡɞ-ɜɨ ɆȽɌɍ ɢɦ. Ȼɚɭɦɚɧɚ, 2001. – 480 ɫ. 2. Ʌɟɬɨɜɚ Ɍ.Ⱥ. Ɇɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ / Ɍ.Ⱥ. Ʌɟɬɨɜɚ, Ⱥ.ȼ. ɉɚɧɬɟɥɟɟɜ. – M. : ȼɵɫɲɚɹ ɲɤɨɥɚ, 2002. – 544 ɫ. 3. Ʌɟɫɢɧ ȼ.ȼ. Ɉɫɧɨɜɵ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ: ɭɱɟɛ. ɩɨɫɨɛɢɟ / ȼ.ȼ. Ʌɟɫɢɧ, ɘ.ɉ. Ʌɢɫɨɜɟɰ. – Ɇ. : ɂɡɞ-ɜɨ ɆȺɂ, 1998. – 344 ɫ. 4. ɋɢɝɚɥ ɂ.ɏ. ȼɜɟɞɟɧɢɟ ɜ ɩɪɢɤɥɚɞɧɨɟ ɞɢɫɤɪɟɬɧɨɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ: ɦɨɞɟɥɢ ɢ ɜɵɱɢɫɥɢɬɟɥɶɧɵɟ ɚɥɝɨɪɢɬɦɵ / ɂ.ɏ. ɋɢɝɚɥ, Ⱥ.ɉ. ɂɜɚɧɨɜɚ. – Ɇ. : Ɏɢɡɦɚɬɥɢɬ, 2003. – 240 ɫ. Ⱦɨɩɨɥɧɢɬɟɥɶɧɚɹ: 1. Ʉɨɪɛɭɬ Ⱥ.Ⱥ. Ⱦɢɫɤɪɟɬɧɨɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ / Ⱥ.Ⱥ. Ʉɨɪɛɭɬ, ɘ.ɘ Ɏɢɧɤɟɥɶɲɬɟɣɧ. – Ɇ. : ɇɚɭɤɚ, 1978. – 368c. 2. ɉɚɩɚɞɢɦɢɬɪɢɭ ɏ. Ʉɨɦɛɢɧɚɬɨɪɧɚɹ ɨɩɬɢɦɢɡɚɰɢɹ (ɚɥɝɨɪɢɬɦɵ ɢ ɫɥɨɠɧɨɫɬɶ) / ɏ. ɉɚɩɚɞɢɦɢɬɪɢɭ, Ʉ. ɋɬɚɣɝɥɢɰ .– M. : Ɇɢɪ, 1985. – 352 ɫ. 3. Ȼɟɥɥɦɚɧ Ɋ. Ⱦɢɧɚɦɢɱɟɫɤɨɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ ɢ ɫɨɜɪɟɦɟɧɧɚɹ ɬɟɨɪɢɹ ɭɩɪɚɜɥɟɧɢɹ / Ɋ. Ȼɟɥɥɦɚɧ, Ɋ. Ʉɚɥɚɛɚ – Ɇ. : ɇɚɭɤɚ, 1969. – 118 ɫ.
41
ɋɈȾȿɊɀȺɇɂȿ
1. Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɞɢɫɤɪɟɬɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ 2. Ɂɚɞɚɱɚ ɨ ɧɚɡɧɚɱɟɧɢɹɯ. ȼɟɧɝɟɪɫɤɢɣ ɦɟɬɨɞ ɪɟɲɟɧɢɹ 2.1. ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ. ɇɟɤɨɬɨɪɵɟ ɫɜɨɣɫɬɜɚ 2.2. ȼɟɧɝɟɪɫɤɢɣ ɦɟɬɨɞ 3. Ɇɟɬɨɞ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ 3.1. Ɉɛɳɚɹ ɫɯɟɦɚ ɦɟɬɨɞɚ 3.2. Ɇɟɬɨɞ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɤɨɦɦɢɜɨɹɠɟɪɚ 3.3. Ɇɟɬɨɞ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ ɞɥɹ ɥɢɧɟɣɧɵɯ ɡɚɞɚɱ ɰɟɥɨɱɢɫɥɟɧɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ 3.4. ɉɚɪɚɦɟɬɪɢɡɚɰɢɹ ɚɥɝɨɪɢɬɦɚ ɜɟɬɜɟɣ ɢ ɝɪɚɧɢɰ 4. Ɇɟɬɨɞɵ ɨɬɫɟɱɟɧɢɣ 4.1. ɉɟɪɜɵɣ ɚɥɝɨɪɢɬɦ Ƚɨɦɨɪɢ 4.2. Ⱥɥɝɨɪɢɬɦɢɱɟɫɤɚɹ ɫɯɟɦɚ ɦɟɬɨɞɚ 5. Ⱦɢɧɚɦɢɱɟɫɤɨɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ 5.1. ɉɪɢɧɰɢɩ ɨɩɬɢɦɚɥɶɧɨɫɬɢ Ȼɟɥɥɦɚɧɚ 5.2. Ɂɚɞɚɱɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɪɟɞɫɬɜ ɦɟɠɞɭ ɩɪɟɞɩɪɢɹɬɢɹɦɢ Ʌɢɬɟɪɚɬɭɪɚ
42
3 6 6 9 13 13 17 25 28 30 31 32 35 36 37 41
ɍɱɟɛɧɨɟ ɢɡɞɚɧɢɟ
ȾɂɋɄɊȿɌɇȺə ɈɉɌɂɆɂɁȺɐɂə Ɇɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɤ ɤɭɪɫɚɦ «Ɇɨɞɟɥɢ ɢ ɦɟɬɨɞɵ ɞɢɫɤɪɟɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ», «ɂɫɫɥɟɞɨɜɚɧɢɟ ɨɩɟɪɚɰɢɣ»
ɋɨɫɬɚɜɢɬɟɥɢ: ɑɟɪɧɵɲɨɜɚ Ƚɚɥɢɧɚ Ⱦɦɢɬɪɢɟɜɧɚ, Ȼɭɥɝɚɤɨɜɚ ɂɪɢɧɚ ɇɢɤɨɥɚɟɜɧɚ
ɉɨɞɩɢɫɚɧɨ ɜ ɩɟɱɚɬɶ 17.09.07. Ɏɨɪɦɚɬ 60×84/16. ɍɫɥ. ɩɟɱ. ɥ. 2,5. Ɍɢɪɚɠ 100 ɷɤɡ. Ɂɚɤɚɡ 1890. ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. 394000, ɝ. ȼɨɪɨɧɟɠ, ɩɥ. ɢɦ. Ʌɟɧɢɧɚ, 10. Ɍɟɥ. 208-298, 598-026 (ɮɚɤɫ) http://www.ppc.vsu.ru; e-mail:
[email protected] Ɉɬɩɟɱɚɬɚɧɨ ɜ ɬɢɩɨɝɪɚɮɢɢ ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɨɝɨ ɰɟɧɬɪɚ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. 394000, ɝ. ȼɨɪɨɧɟɠ, ɭɥ. ɉɭɲɤɢɧɫɤɚɹ, 3. Ɍɟɥ. 204-133.
43
