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Файл:Алгоритмизация в инженерных задачах. Учебное пособие
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2
0
=
3
0
y
=
x(fhk
;39,0
4
0
1
1
00
6
;392,0
Ʉɚɤɚɹ ɡɚɞɚɱɚ ɜ ɨɛɥɚɫɬɢ ɈȾɍ ɧɚɡɵɜɚɟɬɫɹ ɡɚɞɚɱɟɣ Ʉɨɲɢ?
1.
ɉɨɹɫɧɢɬɟ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɢɧɬɟɪɩɪɟɬɚɰɢɸ ɦɟɬɨɞɚ ɗɣɥɟɪɚ ɞɥɹ ɪɟɲɟɧɢɹ
2.
ɡɚɞɚɱɢ Ʉɨɲɢ.
ɉɨɹɫɧɢɬɟ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɢɧɬɟɪɩɪɟɬɚɰɢɸ ɦɟɬɨɞɚ ɗɣɥɟɪɚ-Ʉɨɲɢ ɞɥɹ
3.
ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ.
Ʉɚɤɨɣ ɦɟɬɨɞ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ ɹɜɥɹɟɬɫɹ ɛɨɥɟɟ ɬɨɱɧɵɦ ɢ ɩɨɱɟɦɭ?
4.
1
h
y,
2
1
000
2
5,1(25,0)k
h
2
1
y,
2
000
2
3
000
2
0
)kk2k2k(
0
0
6
1
4
3
001
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
375,0
2
39,0
5,1(25,0)k
=+=Δ+=
2
892,1392,05,1yyy
0
ɢ ɬ. ɞ.
25,0
05,1(
25,0
2
))
=−−−+⋅=++⋅=
2
;393,0)
=−−+⋅=++⋅=
;41,0)25,00393,05,1(25,0)ky,hx(fhk
=−−+⋅=++⋅=
)41,0393,0239,02375,0(
=+⋅+⋅+=+++=Δ
101
ȽɅȺȼȺ 8
ɋɈɋɌȺȼɅȿɇɂȿ ɄɈɆɉɊɈɆɂɋɋɇɕɏ
ɐȿɅȿȼɕɏ ɎɍɇɄɐɂɃ
8.1. Ɇɟɬɨɞɵ ɫɨɫɬɚɜɥɟɧɢɹ ɤɨɦɩɥɟɤɫɧɵɯ ɤɪɢɬɟɪɢɟɜ
ɨɩɬɢɦɢɡɚɰɢɢ
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɨɩɬɢɦɢɡɚɰɢɢ ɱɚɫɬɨ ɜɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɭɱɢɬɵɜɚɬɶ ɨɞɧɨɜɪɟɦɟɧɧɨ ɧɟɫɤɨɥɶɤɨ ɤɪɢɬɟɪɢɟɜ, ɩɪɨɬɢɜɨɪɟɱɚɳɢɯ ɞɪɭɝ ɞɪɭɝɭ (ɧɚɩɪɢɦɟɪ, ɦɚɤɫɢɦɢɡɢɪɨɜɚɬɶ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɩɪɢ ɦɢɧɢɦɢɡɚɰɢɢ ɫɟɛɟɫɬɨɢɦɨɫɬɢ).
ɋ ɰɟɥɶɸ ɨɩɬɢɦɢɡɚɰɢɢ ɬɚɤɢɯ ɩɪɨɰɟɫɫɨɜ ɩɪɢɦɟɧɹɸɬ ɨɛɨɛɳɟɧɧɵɟ (ɤɨɦɩɥɟɤɫɧɵɟ)
ɤɪɢɬɟɪɢɢ, ɤɨɬɨɪɵɟ ɫɨɞɟɪɠɚɬ ɧɟɫɤɨɥɶɤɨ ɱɚɫɬɧɵɯ ɤɪɢɬɟɪɢɟɜ ɜ ɨɞɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ.
Ɍɚɤ ɤɚɤ ɤɚɠɞɵɣ ɤɪɢɬɟɪɢɣ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɭɩɪɚɜɥɹɟɦɵɯ ɩɟɪɟɦɟɧɧɵɯ,
ɬɨ ɨɛɨɛɳɟɧɧɵɣ ɤɪɢɬɟɪɢɣ
ɗɬɭ ɮɭɧɤɰɢɸ ɧɚɡɵɜɚɸɬ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɟɣ.
Ʉɪɨɦɟ ɬɨɝɨ ɬɪɟɛɭɟɬɫɹ ɨɛɴɟɞɢɧɢɬɶ ɱɚɫɬɧɵɟ ɤɪɢɬɟɪɢɢ ɜ ɨɞɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ
(ɩɪɨɢɡɜɟɫɬɢ ɫɜɟɪɬɤɭ).
Ɇɟɬɨɞɵ ɫɜɟɪɬɵɜɚɧɢɹ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ
ɋɭɳɟɫɬɜɭɸɬ ɫɥɟɞɭɸɳɢɟ ɨɫɧɨɜɧɵɟ ɦɟɬɨɞɵ ɫɜɟɪɬɵɜɚɧɢɹ [22]:
Ⱥɞɞɢɬɢɜɧɵɣ. ȼ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɨɝɨ ɤɪɢɬɟɪɢɹ ɛɟɪɭɬ «ɜɡɜɟɲɟɧɧɭɸ»
1.
ɫɭɦɦɭ ɱɚɫɬɧɵɯ ɤɪɢɬɟɪɢɟɜ Fi(X).
ɐɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɜ ɨɛɳɟɦ ɜɢɞɟ ɢɦɟɟɬ ɜɢɞ
ɝɞɟ n − ɤɨɥɢɱɟɫɬɜɨ ɨɛɴɟɞɢɧɹɟɦɵɯ ɱɚɫɬɧɵɯ ɤɪɢɬɟɪɢɟɜ;
ɟɧɬ i-ɝɨ ɱɚɫɬɧɨɝɨ ɤɪɢɬɟɪɢɹ;
o
)X(F
− i-ɣ ɧɨɪɦɢɪɭɸɳɢɣ ɞɟɥɢɬɟɥɶ; )X(f
i
ɱɚɫɬɧɨɝɨ ɤɪɢɬɟɪɢɹ.
ȼɟɫɨɜɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
ɪɢɹ ɧɚ ɪɟɡɭɥɶɬɚɬ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ (
ɟɧɬɵ ɦɨɠɧɨ ɨɩɪɟɞɟɥɹɬɶ ɦɟɬɨɞɨɦ ɷɤɫɩɟɪɬɧɵɯ ɨɰɟɧɨɤ ɥɢɛɨ ɩɪɢ ɚɧɚɥɢɡɟ ɪɟɡɭɥɶɬɚɬɨɜ, ɩɨɥɭɱɟɧɧɵɯ ɢɡ ɪɚɧɟɟ ɪɟɲɟɧɧɵɯ ɩɨɞɨɛɧɵɯ ɡɚɞɚɱ [22].
Ɇɭɥɶɬɢɩɥɢɤɚɬɢɜɧɵɣ. ȼ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɨɝɨ ɤɪɢɬɟɪɢɹ ɛɟɪɭɬ «ɜɡɜɟ-
2.
ɲɟɧɧɨɟ» ɩɪɨɢɡɜɟɞɟɧɢɟ ɱɚɫɬɧɵɯ ɤɪɢɬɟɪɢɟɜ Fi(X).
ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɭɩɪɚɜɥɹɟɦɵɯ ɩɟɪɟɦɟɧɧɵɯ.
n
i
C)X(F
¦¦
i
F
==
1i
i
n
)X(F
¦
ii
1i
i
n
=
i
=
1i
o
i
)X(F
− ɱɢɫɥɨɜɨɟ ɡɧɚɱɟɧɢɟ i-ɝɨ ɱɚɫɬɧɨɝɨ ɤɪɢɬɟɪɢɹ;
C ɨɩɪɟɞɟɥɹɸɬ ɫɬɟɩɟɧɶ ɜɥɢɹɧɢɹ ɱɚɫɬɧɨɝɨ ɤɪɢɬɟ-
i
max(min))X(fC
→==
− ɧɨɪɦɢɪɨɜɚɧɧɨɟ ɡɧɚɱɟɧɢɟ i-ɝɨ
1C
ɢɥɢ 100 %). ȼɟɫɨɜɵɟ ɤɨɷɮɮɢɰɢ-
[22],
C − ɜɟɫɨɜɨɣ ɤɨɷɮɮɢɰɢ-
i
102
m
ɐɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɡɚɩɢɫɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
∏
n
ii
=
1i
max(min))X(fC)X(F
→=
[22],
ɝɞɟ
C − ɜɟɫɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ i-ɝɨ ɱɚɫɬɧɨɝɨ ɤɪɢɬɟɪɢɹ; )X(f
i
− ɧɨɪɦɢɪɨɜɚɧɧɨɟ
i
ɡɧɚɱɟɧɢɟ i-ɝɨ ɱɚɫɬɧɨɝɨ ɤɪɢɬɟɪɢɹ;
Ɇɚɤɫɢɦɢɧɧɵɣ (ɦɢɧɢɦɚɤɫɧɵɣ). Ɉɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɢɞɟɟ ɪɚɜɧɨɦɟɪɧɨɫɬɢ:
3.
ɫɬɚɪɚɸɬɫɹ ɧɚɣɬɢ ɬɚɤɢɟ ɡɧɚɱɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ
{}
21
, ɩɪɢ ɤɨɬɨɪɵɯ
x,...,x,xX =
ɧɨɪɦɢɪɨɜɚɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜɫɟɯ ɱɚɫɬɧɵɯ ɤɪɢɬɟɪɢɟɜ ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ [22]:
ɝɞɟ
C − ɜɟɫɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ i-ɝɨ ɱɚɫɬɧɨɝɨ ɤɪɢɬɟɪɢɹ; )X(f
i
ii
− ɧɨɪɦɢɪɨɜɚɧɧɨɟ
i
,
K)x(fC
=
ɡɧɚɱɟɧɢɟ i-ɝɨ ɱɚɫɬɧɨɝɨ ɤɪɢɬɟɪɢɹ; Ʉ− ɤɨɧɫɬɚɧɬɚ.
ɉɪɢ ɧɚɥɢɱɢɢ ɧɟɫɤɨɥɶɤɢɯ ɤɪɢɬɟɪɢɟɜ ɜɵɛɢɪɚɸɬ [22]:
ɚ) ɚɞɞɢɬɢɜɧɵɣ ɤɪɢɬɟɪɢɣ, ɟɫɥɢ ɫɭɳɟɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɸɬ ɚɛɫɨɥɸɬɧɵɟ
ɡɧɚɱɟɧɢɹ ɤɪɢɬɟɪɢɟɜ ɩɪɢ ɜɵɛɪɚɧɧɨɦ ɜɟɤɬɨɪɟ ɩɚɪɚɦɟɬɪɨɜ X;
ɛ) ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɵɣ ɤɪɢɬɟɪɢɣ, ɟɫɥɢ ɛɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɟɬ ɢɡɦɟɧɟɧɢɟ ɚɛɫɨɥɸɬɧɵɯ ɡɧɚɱɟɧɢɣ ɱɚɫɬɧɵɯ ɤɪɢɬɟɪɢɟɜ ɩɪɢ ɜɚɪɢɚɰɢɢ ɜɟɤɬɨɪɚ X;
ɜ) ɦɚɤɫɢɦɢɧɧɵɣ (ɦɢɧɢɦɚɤɫɧɵɣ) ɤɪɢɬɟɪɢɣ, ɟɫɥɢ ɫɬɨɢɬ ɡɚɞɚɱɚ
ɞɨɫɬɢɠɟɧɢɹ
ɪɚɜɟɧɫɬɜɚ ɧɨɪɦɢɪɨɜɚɧɧɵɯ ɡɧɚɱɟɧɢɣ ɩɪɨɬɢɜɨɪɟɱɢɜɵɯ (ɤɨɧɮɥɢɤɬɧɵɯ) ɱɚɫɬɧɵɯ
ɤɪɢɬɟɪɢɟɜ.
ɇɨɪɦɢɪɨɜɚɧɢɟ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ
Ɍɚɤ ɤɚɤ ɪɚɡɥɢɱɧɵɟ ɤɪɢɬɟɪɢɢ ɨɩɬɢɦɢɡɚɰɢɢ ɢɦɟɸɬ ɪɚɡɥɢɱɧɭɸ ɪɚɡɦɟɪɧɨɫɬɶ, ɱɬɨɛɵ ɩɪɢɦɟɧɹɬɶ ɢɯ ɜ ɨɞɧɨɣ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ, ɧɟɨɛɯɨɞɢɦɨ ɩɪɢɜɟɫɬɢ ɢɯ
ɤ ɛɟɡɪɚɡɦɟɪɧɨɦɭ ɜɢɞɭ. Ⱦɥɹ ɷɬɨɝɨ ɩɪɨɢɡɜɨɞɹɬ ɧɨɪɦɢɪɨɜɚɧɢɟ ɤɪɢɬɟɪɢɟɜ.
Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɧɨɪɦɢɪɨɜɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ i-ɝɨ ɱɚɫɬɧɨɝɨ ɤɪɢɬɟɪɢɹ
)X(F
ɞɟɥɹɬ ɤɚɠɞɵɣ ɤɪɢɬɟɪɢɣ
ɲɟɧɧɵɣ ɧɨɪɦɢɪɭɸɳɢɣ ɞɟɥɢɬɟɥɶ
ɧɚ ɧɟɤɨɬɨɪɵɣ ɭɫɥɨɜɧɨ ɩɪɢɧɹɬɵɣ ɫɪɟɞɧɟɜɡɜɟ-
i
o
)X(F
[22]:
i
)X(F
i
=
)X(f
i
.
o
)X(F
i
)X(f
i
ȼ ɤɚɱɟɫɬɜɟ ɧɨɪɦɢɪɭɸɳɢɯ ɞɟɥɢɬɟɥɟɣ ɦɨɝɭɬ ɩɪɢɦɟɧɹɬɶɫɹ ɡɧɚɱɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɩɪɢ ɧɟɤɨɬɨɪɵɯ ɮɢɤɫɢɪɨɜɚɧɧɵɯ ɡɧɚɱɟɧɢɹɯ ɨɩɬɢɦɢɡɢɪɭɟɦɵɯ ɩɚɪɚɦɟɬɪɨɜ. ɉɪɢ ɞɢɫɤɪɟɬɧɨɦ ɦɧɨɠɟɫɬɜɟ ɡɧɚɱɟɧɢɣ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɜ ɤɚɱɟɫɬɜɟ ɧɨɪɦɢɪɭɸɳɢɯ ɞɟɥɢɬɟɥɟɣ ɛɟɪɟɬɫɹ ɫɪɟɞɧɟɟ
ɚɪɢɮɦɟɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɟɜ (ɢɡ ɬɚɛɥɢɱɧɵɯ ɞɚɧɧɵɯ ɥɢɛɨ ɩɨɥɭɱɟɧɧɵɟ
ɷɦɩɢɪɢɱɟɫɤɢɦ ɩɭɬɟɦ).
103
t
t
8.2. Ɉɩɪɟɞɟɥɟɧɢɟ ɨɝɪɚɧɢɱɟɧɢɣ
Ʉɚɱɟɫɬɜɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɩɪɨɰɟɫɫɚ ɨɩɬɢɦɢɡɚɰɢɢ ɢ ɟɟ ɞɨɫɬɨɜɟɪɧɨɫɬɶ ɡɚɜɢɫɹɬ ɨɬ ɜɵɛɨɪɚ ɬɟɯɧɢɱɟɫɤɢɯ ɨɝɪɚɧɢɱɟɧɢɣ. ɉɪɢ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɬɟɯɧɨɥɨɝɢɢ ɢɡɝɨɬɨɜɥɟɧɢɹ ɞɟɬɚɥɟɣ ɜɵɛɢɪɚɸɬ ɫɥɟɞɭɸɳɢɟ ɫɬɚɧɞɚɪɬɧɵɟ ɨɝɪɚɧɢɱɟɧɢɹ [9, 22, 23]:
ɫɬɨɣɤɨɫɬɶ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ
1)
ɦɨɳɧɨɫɬɶ ɗȾ ɩɪɢɜɨɞɚ ɝɥɚɜɧɨɝɨ ɞɜɢɠɟɧɢɹ
2)
ɦɢɧɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɭɸ ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ
3)
ɦɚɤɫɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɭɸ ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ
4)
ɦɢɧɢɦɚɥɶɧɭɸ ɩɨɞɚɱɭ
5)
ɦɚɤɫɢɦɚɥɶɧɭɸ ɩɨɞɚɱɭ
6)
7)
ɩɪɨɱɧɨɫɬɶ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ
ɠɟɫɬɤɨɫɬɶ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ
8)
ɠɟɫɬɤɨɫɬɶ ɡɚɝɨɬɨɜɤɢ
9)
ɬɪɟɛɭɟɦɭɸ ɲɟɪɨɯɨɜɚɬɨɫɬɶ ɩɨɜɟɪɯɧɨɫɬɢ.
10)
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɫɬɪɨɟɧɢɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɨɝɪɚɧɢɱɟɧɢɣ ɞɥɹ ɩɪɨɞɨɥɶɧɨɝɨ
ɬɨɱɟɧɢɹ.
ɋɬɨɣɤɨɫɬɶ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ. ɗɬɨ ɨɝɪɚɧɢɱɟɧɢɟ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɜɡɚ-
1.
ɢɦɨɫɜɹɡɶ ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ɪɟɡɚɧɢɹ, ɨɩɪɟɞɟɥɹɟɦɨɣ ɩɪɢ ɭɫɬɚɧɨɜɥɟɧɧɨɣ ɫɬɨɣɤɨɫɬɢ ɢɧɫɬɪɭɦɟɧɬɚ, ɪɟɠɢɦɟ ɨɛɪɚɛɨɬɤɢ.
ɋɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ, ɦ/ɦɢɧ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɮɨɪɦɭɥɵ [9, 12, 22]:
V =
v
,
k
v
yxm
StT
C
ɝɞɟ C
, kv, m, x, y − ɷɦɩɢɪɢɱɟɫɤɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ [12], Ɍ − ɩɟɪɢɨɞ ɫɬɨɣɤɨɫɬɢ
v
ɪɟɡɰɚ, ɦɢɧ, t − ɝɥɭɛɢɧɚ ɪɟɡɚɧɢɹ, ɦɦ, S − ɩɨɞɚɱɚ, ɦɦ/ɨɛ.
ɋɨɝɥɚɫɧɨ ɤɢɧɟɦɚɬɢɤɟ ɫɬɚɧɤɚ ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ, ɦ/ɦɢɧ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ
ɮɨɪɦɭɥɟ
dnVπ
=
,
1000
ɝɞɟ d − ɞɢɚɦɟɬɪ ɬɨɱɟɧɢɹ, ɦɦ, n − ɱɚɫɬɨɬɚ ɜɪɚɳɟɧɢɹ ɲɩɢɧɞɟɥɹ ɩɪɢ ɬɨɱɟɧɢɢ,
ɨɛ/ɦɢɧ.
ɉɪɢɪɚɜɧɹɜ ɜɵɪɚɠɟɧɢɹ, ɩɨɥɭɱɢɦ
ɨɬɤɭɞɚ
C
v
yxm
T
S
y
=
nS
π
dn
,
=
k
v
1000
kC1000
vv
.
xm
dT
π
104
2.
Ɇɨɳɧɨɫɬɶ ɷɥɟɤɬɪɨɞɜɢɝɚɬɟɥɹ ɩɪɢɜɨɞɚ ɝɥɚɜɧɨɝɨ ɞɜɢɠɟɧɢɹ. ɗɬɢɦ ɨɝɪɚ-
ɧɢɱɟɧɢɟɦ ɭɫɬɚɧɚɜɥɢɜɚɸɬ ɜɡɚɢɦɨɫɜɹɡɶ ɦɟɠɞɭ ɷɮɮɟɤɬɢɜɧɨɣ ɦɨɳɧɨɫɬɶɸ ɪɟɡɚɧɢɹ
(ɡɚɬɪɚɱɢɜɚɟɦɨɣ ɧɚ ɩɪɨɰɟɫɫ ɨɛɪɚɛɨɬɤɢ) ɢ ɦɨɳɧɨɫɬɶɸ ɷɥɟɤɬɪɨɞɜɢɝɚɬɟɥɹ ɩɪɢɜɨɞɚ
ɝɥɚɜɧɨɝɨ ɞɜɢɠɟɧɢɹ.
ɗɮɮɟɤɬɢɜɧɚɹ ɦɨɳɧɨɫɬɶ, ɤȼɬ, ɧɟɨɛɯɨɞɢɦɚɹ ɞɥɹ ɨɛɪɚɛɨɬɤɢ, ɨɩɪɟɞɟɥɹɟɬɫɹ
ɩɨ ɮɨɪɦɭɥɟ [9, 12, 22]
VP
z
N
=
ɷ
,
60102
⋅
− ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɢɥɵ ɪɟɡɚɧɢɹ, ɇ,
ɝɞɟ P
z
1nyx
,
kVStCPz =
pz
p
ɝɞɟ C
, x, y, n1, k
pz
− ɷɦɩɢɪɢɱɟɫɤɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ [9, 12, 22].
p
ɗɮɮɟɤɬɢɜɧɚɹ ɦɨɳɧɨɫɬɶ ɞɨɥɠɧɚ ɨɛɟɫɩɟɱɢɜɚɬɶɫɹ ɡɚɹɜɥɟɧɧɨɣ ɩɨ ɩɚɫɩɨɪɬɭ ɦɨɳɧɨɫɬɶɸ ɩɪɢɜɨɞɚ ɝɥɚɜɧɨɝɨ ɞɜɢɠɟɧɢɹ
,
η≤ NNɷ
85,0=η
ɝɞɟ Ș − ɄɉȾ ɩɪɢɜɨɞɚ ɝɥɚɜɧɨɝɨ ɞɜɢɠɟɧɢɹ (
).
ɉɪɢɪɚɜɧɢɜɚɹ ɩɪɚɜɵɟ ɱɚɫɬɢ ɧɟɪɚɜɟɧɫɬɜɚ, ɩɨɥɭɱɢɦ
1nyx
VkVStC
pz
p
60102
,
η≤⋅N
+
yx
pz
)11n(
VkStC
p
⋅
60102
,
η≤
N
π
§
¨
1000
©
dn
11n
+
·
¸
¹
yx
kStC
p
pz
⋅
60102
,
η≤
N
+
11ny
nS
≤
η⋅
x
π
p
pz
.
11n
+
)d(ktC
+
11n
)1000(N60102
3. Ɇɢɧɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɚɹ ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ
,nn
min_ɫɬ
n
ɝɞɟ
ɨɛ/ɦɢɧ.
≥
− ɦɢɧɢɦɚɥɶɧɚɹ ɱɚɫɬɨɬɚ ɜɪɚɳɟɧɢɹ ɲɩɢɧɞɟɥɹ ɩɨ ɩɚɫɩɨɪɬɭ ɫɬɚɧɤɚ,
min_ɫɬ
105
4.
Ɇɚɤɫɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɚɹ ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ
,nn
max_ɫɬ
n
ɝɞɟ
ɨɛ/ɦɢɧ.
≤
− ɦɚɤɫɢɦɚɥɶɧɚɹ ɱɚɫɬɨɬɚ ɜɪɚɳɟɧɢɹ ɲɩɢɧɞɟɥɹ ɩɨ ɩɚɫɩɨɪɬɭ ɫɬɚɧɤɚ,
max_ɫɬ
5. Ɇɢɧɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɚɹ ɩɨɞɚɱɚ
,SS
min_ɫɬ
ɝɞɟ
≥
S
− ɦɢɧɢɦɚɥɶɧɚɹ ɩɨɞɚɱɚ ɩɨ ɩɚɫɩɨɪɬɭ ɫɬɚɧɤɚ, ɦɦ/ɨɛ.
min_ɫɬ
6. Ɇɚɤɫɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɚɹ ɩɨɞɚɱɚ
,SS
max_ɫɬ
ɝɞɟ
≤
S
− ɦɚɤɫɢɦɚɥɶɧɚɹ ɩɨɞɚɱɚ ɩɨ ɩɚɫɩɨɪɬɭ ɫɬɚɧɤɚ, ɦɦ/ɨɛ.
max_ɫɬ
7. ɉɪɨɱɧɨɫɬɶ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ. ɗɬɨ ɨɝɪɚɧɢɱɟɧɢɟ ɭɫɬɚɧɚɜɥɢɜɚɟɬ
ɜɡɚɢɦɨɫɜɹɡɶ ɪɚɫɱɟɬɧɨɣ ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ, ɩɨɞɚɱɢ ɫ ɞɨɩɭɫɬɢɦɨɣ ɩɪɨɱɧɨɫɬɶɸ ɢɧɫɬɪɭɦɟɧɬɚ. ɉɪɟɞɟɥ ɩɪɨɱɧɨɫɬɢ ɦɚɬɟɪɢɚɥɚ ɞɟɪɠɚɜɤɢ ɪɟɡɰɚ ɩɪɢ ɢɡɝɢɛɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨɦ
8.
Ɇ5,1
ɢɡɝ
[]
≥σ
ɢ
W
,
ɝɞɟ
ɸɳɢɣ ɦɨɦɟɧɬ ɜ ɦɟɫɬɟ ɡɚɤɪɟɩɥɟɧɢɹ ɞɟɪɠɚɜɤɢ ɪɟɡɰɚ, ɇɦ;
− ɞɥɢɧɚ ɜɵɥɟɬɚ ɪɟɡɰɚ, ɦɦ (ɪɢɫ. 35); W − ɦɨɦɟɧɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɟɱɟɧɢɹ ɞɟɪ-
l
ɜɪ
ɠɚɜɤɢ ɪɟɡɰɚ, ɦɦ
− ɩɪɟɞɟɥ ɩɪɨɱɧɨɫɬɢ ɦɚɬɟɪɢɚɥɚ ɞɟɪɠɚɜɤɢ ɪɟɡɰɚ, Ɇɉɚ;
[]
σ
ɢɡɝ
3
;
W
2
ɇȼ
⋅
=
, ɦɦ3; ȼ, ɇ, ɦɦ − ɲɢɪɢɧɚ ɢ ɜɵɫɨɬɚ ɞɟɪɠɚɜɤɢ ɪɟɡɰɚ
6
Ɇ − ɢɡɝɢɛɚ-
ɢɡɝ
lPɆ ⋅=
ɜɪzɢɡɝ
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
1nyx
,
kVStC10P =
pzz
p
1nyx
⋅⋅
pz
2
BH
⋅
lkVStC1065,1
ɜɪp
,
[]
σ≤
ɢ
1nyx
pz
⋅⋅π⋅⋅
21n
⋅⋅
BH1000
lk)nd(StC1065,1
ɜɪp
,
[]
σ≤
ɢ
,
106
1n2
⋅⋅⋅σ
1000BH
π⋅⋅⋅
p
.
1n
l)d(ktC1065,1
ɜɪ
≤
[]
ɢ
x
pz
1ny
nS
Ɋɢɫ. 35. Ɋɚɫɱɟɬɧɚɹ ɫɯɟɦɚ ɨɩɪɟɞɟɥɟɧɢɹ ɠɟɫɬɤɨɫɬɢ
ɢ ɩɪɨɱɧɨɫɬɢ ɪɟɡɰɚ
ɀɟɫɬɤɨɫɬɶ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ. ɗɬɨ ɨɝɪɚɧɢɱɟɧɢɟ ɭɫɬɚɧɚɜɥɢɜɚɟɬ
9.
ɜɡɚɢɦɨɫɜɹɡɶ ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ ɢ ɩɨɞɚɱɢ ɫ ɞɨɩɭɫɬɢɦɨɣ ɠɟɫɬɤɨɫɬɶɸ ɢɧɫɬɪɭɦɟɧɬɚ.
Ɇɚɤɫɢɦɚɥɶɧɚɹ ɧɚɝɪɭɡɤɚ, ɞɨɩɭɫɤɚɟɦɚɹ ɠɟɫɬɤɨɫɬɶɸ ɪɟɡɰɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
3f E I
P= , ɇ,
[]
⋅⋅
3
l
ɜɪ
ɝɞɟ f − ɞɨɩɭɫɬɢɦɚɹ ɫɬɪɟɥɚ ɩɪɨɝɢɛɚ ɪɟɡɰɚ, ɦɦ (
f=0,05ɦɦ
ɪɟɡɰɚ (ȿ =
ɞɥɹ ɱɢɫɬɨɜɵɯ ɦɟɬɨɞɨɜ); ȿ − ɦɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ ɦɚɬɟɪɢɚɥɚ ɞɟɪɠɚɜɤɢ
5
102 ⋅ Ɇɉɚ); I − ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɞɟɪɠɚɜɤɢ ɪɟɡɰɚ, ɦɦ
ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ ɞɟɪɠɚɜɤɢ ɪɟɡɰɚ,
ɞɟɪɠɚɜɤɢ ɪɟɡɰɚ); l
− ɞɥɢɧɚ ɜɵɥɟɬɚ ɪɟɡɰɚ, ɦɦ.
ɜɪ
f=0,1ɦɦ
ɞɥɹ ɱɟɪɧɨɜɵɯ ɦɟɬɨɞɨɜ,
4
=
I
(
4
ɞɥɹ ɤɪɭɝɥɨɝɨ ɫɟɱɟɧɢɹ
d05,0I =
12
3
⋅
HB
Ɉɝɪɚɧɢɱɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨɦ
[]
.PPz ≤
107
⋅⋅
p
y
l
l
pz
ɜɪ
≤
,
IEf3
3
l
1000
π
,
ɜɪ
⋅⋅
IEf3
,
3
1n
,
1nx
k)d(tC
p
1nyx
≤
kVStC
π
(StC
1000
l
dn
ɜɪ
p
1nyx
k)
⋅⋅
IEf3
⋅
3
pz
pz
1ny
≤
nS
ɀɟɫɬɤɨɫɬɶ ɡɚɝɨɬɨɜɤɢ. ɗɬɨ ɨɝɪɚɧɢɱɟɧɢɟ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɜɡɚɢɦɨɫɜɹɡɶ
10.
ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ ɢ ɩɨɞɚɱɢ ɫ ɞɨɩɭɫɬɢɦɨɣ ɠɟɫɬɤɨɫɬɶɸ ɡɚɝɨɬɨɜɤɢ. Ɉɝɪɚɧɢɱɟɧɢɟ
ɤɥɚɫɫɢɱɟɫɤɢɦ ɜɵɪɚɠɟɧɢɟɦ ɞɥɹ ɪɚɫɱɟɬɚ ɠɟɫɬɤɨɫɬɢ ,ɇ/ɦɦ, (ɪɢɫ. 36) ɢɦɟɟɬ ɜɢɞ
P
j
=
Δ
lΔ
− ɫɬɪɟɥɚ ɩɪɨɝɢɛɚ, ɦɦ; Py − ɪɚɞɢɚɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɢɥɵ ɪɟɡɚɧɢɹ, ɇ.
ɝɞɟ
Ɋɢɫ. 36. ɋɯɟɦɚ ɞɟɮɨɪɦɚɰɢɢ ɡɚɝɨɬɨɜɤɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɪɚɞɢɚɥɶɧɨɣ
ɫɨɫɬɚɜɥɹɸɳɟɣ ɫɢɥɵ ɪɟɡɚɧɢɹ
ɉɪɨɝɢɛ ɡɚɝɨɬɨɜɤɢ ɞɨɥɠɧɚ ɛɵɬɶ ɦɟɧɶɲɟ ɩɨɥɨɜɢɧɵ ɜɟɥɢɱɢɧɵ ɞɨɩɭɫɤɚ ɧɚ
ɪɚɡɦɟɪ δ:
δ≤Δ 5,0l ,
3
)xl(P
−
pɡy
fl
==Δ
(ɩɪɢ ɭɫɬɚɧɨɜɤɟ ɡɚɝɨɬɨɜɤɢ ɜ ɰɟɧɬɪɚɯ),
IE48
⋅⋅
py
1nyx
⋅⋅
)1000(IE48
108
3
−π
)xl(k)dn(StC10
pɡp
1n
,
δ≤
5,0
4
,
d05,0I =
r
1n
)1000(IE485,0
1ny
nS
≤
x
py
⋅⋅⋅δ
pɡp
.
1n3
)d()xl(ktC10
π−
11. Ɍɪɟɛɭɟɦɚɹ ɲɟɪɨɯɨɜɚɬɨɫɬɶ ɩɨɜɟɪɯɧɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɜɵɪɚɠɟɧɢɸ ȼ.
Ʌ. ɑɟɛɵɲɟɜɚ
2
S
,
≥⋅
Rz001,0
8
ɝɞɟ Rz − ɬɪɟɛɭɟɦɚɹ ɩɨ ɱɟɪɬɟɠɭ ɜɵɫɨɬɚ ɩɪɨɮɢɥɹ ɲɟɪɨɯɨɜɚɬɨɫɬɢ ɩɨ ɞɟɫɹɬɢ ɬɨɱɤɚɦ, ɦɤɦ, r − ɪɚɞɢɭɫ ɩɪɢ ɜɟɪɲɢɧɟ ɪɟɡɰɚ, ɦɦ.
Ɉɬɤɭɞɚ
.rRz008,0S ⋅≤
8.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 6.1.
ȼ ɨɬɞɟɥɟ ɬɟɯɧɢɱɟɫɤɨɝɨ ɤɨɧɬɪɨɥɹ (ɈɌɄ) ɧɟɤɨɬɨɪɨɣ ɮɢɪɦɵ ɪɚɛɨɬɚɸɬ ɤɨɧɬɪɨɥɟɪɵ ɪɚɡɪɹɞɨɜ 1 ɢ 2. ɇɨɪɦɚ ɜɵɪɚɛɨɬɤɢ ɈɌɄ ɡɚ 8-ɱɚɫɨɜɨɣ ɪɚɛɨɱɢɣ
ɞɟɧɶ ɫɨɫɬɚɜɥɹɟɬ ɧɟ ɦɟɧɟɟ 1800 ɢɡɞɟɥɢɣ. Ʉɨɧɬɪɨɥɟɪ ɪɚɡɪɹɞɚ 1 ɩɪɨɜɟɪɹɟɬ 25 ɢɡɞɟɥɢɣ ɜ ɱɚɫ, ɩɪɢɱɟɦ ɧɟ ɨɲɢɛɚɟɬɫɹ ɜ 98 % ɫɥɭɱɚɟɜ. Ʉɨɧɬɪɨɥɟɪ ɪɚɡɪɹɞɚ 2 ɩɪɨɜɟɪɹɟɬ 15 ɢɡɞɟɥɢɣ ɜ ɱɚɫ; ɟɝɨ ɬɨɱɧɨɫɬɶ ɫɨɫɬɚɜɥɹɟɬ 95 %.
Ɂɚɪɚɛɨɬɧɚɹ ɩɥɚɬɚ ɤɨɧɬɪɨɥɟɪɚ ɪɚɡɪɹɞɚ 1 ɪɚɜɧɚ 4$ ɜ ɱɚɫ, ɤɨɧɬɪɨɥɟɪ ɪɚɡɪɹɞɚ
2 ɩɨɥɭɱɚɟɬ 3$ ɜ ɱɚɫ. ɉɪɢ ɤɚɠɞɨɣ ɨɲɢɛɤɟ ɤɨɧɬɪɨɥɟɪɚ ɮɢɪɦɚ ɧɟɫɟɬ ɭɛɵɬɨɤ ɜ ɪɚɡɦɟɪɟ 2$. Ɏɢɪɦɚ ɦɨɠɟɬ ɩɪɢɦɟɧɹɬɶ 8 ɤɨɧɬɪɨɥɟɪɨɜ ɪɚɡɪɹɞɚ 1 ɢ 10 ɤɨɧɬɪɨɥɟɪɨɜ
ɪɚɡɪɹɞɚ 2. Ɋɭɤɨɜɨɞɫɬɜɨ ɮɢɪɦɵ ɯɨɱɟɬ ɨɩɪɟɞɟɥɢɬɶ ɨɩɬɢɦɚɥɶɧɵɣ ɫɨɫɬɚɜ ɈɌɄ, ɩɪɢ
ɤɨɬɨɪɨɦ ɨɛɳɢɟ ɡɚɬɪɚɬɵ ɧɚ ɤɨɧɬɪɨɥɶ ɛɭɞɭɬ ɦɢɧɢɦɚɥɶɧɵɦɢ.
Ɋɚɡɪɚɛɨɬɤɚ ɦɨɞɟɥɢ. ɉɭɫɬɶ ɯ1 ɢ ɯ2 ɨɛɨɡɧɚɱɚɸɬ ɤɨɥɢɱɟɫɬɜɨ ɤɨɧɬɪɨɥɟɪɨɜ
ɪɚɡɪɹɞɨɜ 1 ɢ 2 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɑɢɫɥɨ ɤɨɧɬɪɨɥɟɪɨɜ ɤɚɠɞɨɝɨ ɪɚɡɪɹɞɚ ɨɝɪɚɧɢɱɟɧɨ,
ɬ, ɟ. ɢɦɟɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɨɝɪɚɧɢɱɟɧɢɹ:
81ɯ ≤ (ɪɚɡɪɹɞ 1),
102ɯ ≤ (ɪɚɡɪɹɞ 2).
ȿɠɟɞɧɟɜɧɨ ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɜɟɪɹɬɶ ɧɟ ɦɟɧɟɟ 1800 ɢɡɞɟɥɢɣ, ɩɨɷɬɨɦɭ ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
18002ɯ1201ɯ2002ɯ1581ɯ258 ≥+=⋅+⋅ ɢɥɢ 452ɯ31ɯ5 ≥+ .
ɉɪɢ ɩɨɫɬɪɨɟɧɢɢ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɧɟɨɛɯɨɞɢɦɨ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɪɚɫɯɨɞɵ
ɮɢɪɦɵ, ɫɜɹɡɚɧɧɵɟ ɫ ɤɨɧɬɪɨɥɟɦ, ɫɨɞɟɪɠɚɬ ɞɜɟ ɫɨɫɬɚɜɥɹɸɳɢɟ:
1) ɡɚɪɩɥɚɬɭ ɤɨɧɬɪɨɥɟɪɨɜ
2) ɭɛɵɬɤɢ, ɜɵɡɜɚɧɧɵɟ ɨɲɢɛɤɚɦɢ ɤɨɧɬɪɨɥɟɪɨɜ.
109
Ɋɚɫɯɨɞɵ ɧɚ ɨɞɧɨɝɨ ɤɨɧɬɪɨɥɟɪɚ ɪɚɡɪɹɞɚ 1 ɫɨɫɬɚɜɥɹɸɬ
$/ɱɟɥ.
502,02524 =⋅⋅+
Ɋɚɫɯɨɞɵ ɧɚ ɨɞɧɨɝɨ ɤɨɧɬɪɨɥɟɪɚ ɪɚɡɪɹɞɚ 2 ɪɚɜɧɵ
$/ɱɟɥ.
5,405,01523 =⋅⋅+
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɢɧɢɦɢɡɢɪɭɟɦɚɹ ɰɟɥɟɜɚɹ ɮɭɧɤɰɢɹ, ɜɵɪɚɠɚɸɳɚɹ ɟɠɟɞɧɟɜɧɵɟ ɪɚɫɯɨɞɵ ɧɚ ɤɨɧɬɪɨɥɶ, ɢɦɟɟɬ ɜɢɞ
.2ɯ361ɯ40)2ɯ5,41ɯ5(8Z +=+=
ȼ ɢɬɨɝɟ ɩɨɥɭɱɟɧɚ ɡɚɞɚɱɚ ɦɢɧɢɦɢɡɚɰɢɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ:
min,2ɯ361ɯ40Z ⎯→⎯+=
;81x
≤
°
°
°
®
°
°
°
¯
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. Ⱦɥɹ ɱɟɝɨ ɩɪɢɦɟɧɹɟɬɫɹ ɫɜɟɪɬɤɚ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ ɜ ɡɚɞɚɱɚɯ ɨɩɬɢ-
ɦɢɡɚɰɢɢ?
ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɦɟɬɨɞɵ ɫɜɟɪɬɤɢ.
2.
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɧɨɪɦɢɪɨɜɚɧɢɹ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ?
3.
Ʉɚɤ ɜɵɛɢɪɚɸɬɫɹ ɧɨɪɦɢɪɭɸɳɢɟ ɞɟɥɢɬɟɥɢ ɞɥɹ ɨɛɴɟɞɢɧɟɧɢɹ ɱɚɫɬɧɵɯ
4.
ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ ɜ ɨɞɢɧ ɤɨɦɩɥɟɤɫɧɵɣ?
ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɜɢɞɵ ɨɝɪɚɧɢɱɟɧɢɣ ɩɪɢ ɨɩɬɢɦɢɡɚɰɢɢ ɩɚɪɚɦɟɬɪɨɜ
5.
ɪɟɠɢɦɚ ɪɟɡɚɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ ɞɟɬɚɥɟɣ ɦɚɲɢɧ.
;102x
≤
;452x31x5
≥+
;01x
≥
.02x
≥
110
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