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3.ΓΡΑΜΜΙΚΟΣ ΠΡΟΓΡΑΜΜΑΤΙΣΜΟΣ
Citation preview
-11-
3.
3.1
. , , .
,
.
:
max=xv+...+xv+xv=V nn2211 (3.1)
bx+...+x+x
.......................................
bx+...+x+x
bx+...+x+x
mnmn22m11m
2nn2222121
1nn1212111
(3.2)
)n,...,2,1=i(0 x i (3.3)
. (3.1) , (3.2)
, (3.3)
.
,
.
(3.2)
. (3.3)
x1, x2,..., xn
. ,
,
.
3.2,
b1, b2, ..., bn
.
-12-
3.2
.
, (
)
.
.
0,4 /m3 0,2 /m3 .
0,6 /m3 0,4 /m3
.
4 .
.
15
.m3.
,
.
.
,
. . .
:
max=x4+x6=V 21 (3.4)
40x2+x4 21 (3.5)
15x+x 21 (3.6)
0 x
0 x
2
1 (3.7)
x1 x2
.
-13-
.
x1 x2, (. 3.1),
(3.5), (3.6) (3.7).
,
(3.4).
.
, ,
V
. V
. V,
.
0 5
5
10
10
15
15
20
A
B
x1
x2
3.1
x1+x2=15
4x1 +
2x2 =
40
6x1+4x2=V
.
, . ,
(4),
( )
V.
x1=5, x2=10
-14-
. 5 .
m3 10 . m3 ,
, ( 7 . )
40 . .,
15 . m3.
3.3 1
1 , , .
,
, 1
, .
1
( . 3.2),
.
, , 4
8)
0 5
5
10
10
15
15
20
A
B
x1
x2
x1 +x
2 =15
4x1 +
2x2 =
40
1x
1+4x2=V (
1
-15-
1 > 8, ( 3.2)
(x1 = 10, x2 = 0), .
), 1 < 4,
( 3.2) (x1 = 0, x2 =
15, . ).
, 1, .
,
( 1 > 8,
4 < 1 < 8 1 < 4).
.
1 = 4 1 = 8
. (..
1 = 4) ,
.
.
.
. 1 .
, .
.
3.4 2
,
(. 3.4, 3.5, 3.6 3.7) :
-16-
. .
max=x4+x6=V 21 (3.8)
40=x+x2+x4 321 (3.9)
15=x+x+x 421 (3.10)
0 x
0 x
0 x
0 x
4
3
2
1
(3.11)
x3 x4,
(3.5) (3.6) (3.9) (3.10).
x3 x4
, .
,
, ,
, .
.
2 .
(. 3.8, 3.9, 3.10, 3.11) 4
x1, x2, x3 x4, 2 x1, x2
.
. 4
-17-
. ,
, n
( ) m
, n-m
.
3.5
.
H
,
. ,
,
.
.
.
.
.
.
2 , .
. ,
,
, .
,
,
.
-18-
.
.
-18-
.
.
3.6
. .
+xv+...+xv+xv=V kk2211
max=x.0+...+x.0+x.0+ m+k2+k1+k (3.12)
b=x+...+0+0+x+...+x+x
..............................................................
b=0+...+x+0+x+...+x+x
b=0+...+x+x+...+x+x
mm+kkmk22m11m
22+kkk2222121
11+kkk1212111
(3.13)
(3.13) xk+1, xk+2, ..., xk+m,
.
,
n-m=k+m-m=k . , ,
:
mm+k22+k11+kk21 b=x,...,b=x,b=x,0=x=...=x=x
. .
(3.13)
:
-19-
m
2
1
mk
2k
1k
k
2
1
mk2m1m
k22221
k11211
b
.
.
.
b
b
x
.
x
x
x
.
x
x
1...00...
............
....10...
0...01...
(3.14)
(3.14). m
, m
,
.
. (3.14)
m21mk2k1kk21 b,...,b,b,0,...,0,0x,...,x,x,x,...,x,x
.
,
,
. ..
x1 xk+1,
(3.14) 11,
i1/11. ( i1 21, 31,...,
m1). (3.14) :
11
1m1m
11
2112
11
1
mk
2k
1k
k
2
1
11
1m
11
1mk1mk
11
1m122m
11
21
11
21k1k2
11
211222
1111
k1
11
12
bb
.
.
.
bb
b
x
.
x
x
x
.
x
x
1...00...0
............
....10...0
0...01
...1
(3.15)
(3.15)
-20-
mk1kk21 x,...,x,x,...,x,x
11
1m1m
11
2112
11
1
bb,....,
bb,0,0,...,0,
b
. x1 xk+1.
.
,
.
3.7
.
V
,
,
V .
.
,
.
(3.13)
,
, V
. xi
(3.13) yi. (3.13) :
-21-
mmkmkkkmk222m111m
22k2kkkk222221121
11k1kkkk122121111
byx...00yx...yxyx
.....................................................................................................
b0...yx0yxa...yx)yx(
b0...0yxyx...yxyx
(3.13)
:
0y...00y...yy
............................................................
00...y0y...yy
00...0yy...yy
mkkmk22m11m
2kkk2222121
1kkk1212111
(3.16)
. (3.16) yi
(3.13).
yi (3.16)
xi+yi (3.13).
xi+yi V1
kkk2221111 yxv...yxvyxvV
mkmk1k1k yx0...yx.0
. xi+yi
mk1kkk2211 y.0...y.0yv...yvyvV (3.17)
xi (i = 1
k) , . xi
yi V .
x1 y1 (.
y1 = 1), y2 = y3 = ... = yk = 0, (3.16)
-22-
1mmk
212k
111k
y
................
y
y
(3.18)
. (3.17)
:
1m11k211 .0....00.v...0.vvV (3.19)
,
xj yj (. yj = 1),
, . (3.16)
mjmk
j22k
j11k
y
...............
y
y
(3.20)
. (3.17)
s
ssjjj vvV (3.21)
s
, . .
vj,
, .
xj vj
,
, ( vj)
.
-23-
.
, , ,
.
xj. ..
. (3.14)
x1, v1 > v2, v3,..., vk.
xk+1,xk+2,...,xk+m. xk+1,
x1 xk+1, . (3.14)
. (3.15). . (3.15) x1=(b1/11).
xk+2,
. (3.14) . (3.15),
x1=(b2/21). xk+1
x1=(bi/1).
[x1 = (b1/i1)]
.
, . (bi/i1),
. ,
(bi/i1) .
x1,
.
.
,
. .
. (3.14)
, ,
.
,
-24-
. ,
,
.
-24-
. ,
,
.
3.8 Simplex
Simplex,
:
1
( )
.
,
. .
xk+1, xk+2,..., xk+m ,
.
.
.
2 . (3.21) =1, Vj
. (3.21)
. .
vj .
, s
. ,
vk+1=vk+2=...=vk+m=0, V1=v1, V2=v2, ....., Vk=vk.
v1, v2,..., vk,
-25-
vk+1,...., vk+m .
3 Vj.
,
Vj
. v1>v2,v3,....,vk
x1 .
x1 x2 ...... xk xk+1 xk+2 ...... xk+m
1 xk+1 1
1
12 ...... 1k 1 0 ...... 0 b1
11
1b
2 xk+2 2
1
22 ...... 2k 0 1 ...... 0 b2
21
2b
3 xk+3 3
1
32 ...... 3k 0 0 ...... 0 b3
31
3b
... ... ... ... ...... ... ... ... ...... ... ... ...
m xk+m m
1
m
2
...... m
k
0 0 ...... 1 bm
1m
mb
v1 v2 ...... vk 0 0 ...... 0 V
v1 v2 ...... vk - - ...... - Vj
4 b1, b2,...., bm
. (
, b1, b2,..., bm 11, 21,..., m1).
b1/11, b2/21,..., bm/m1
.
5 ,
-26-
.
b2/21
xk+2.
x1 x2 ...... xk xk+1 xk+2 ...... xk+m
1 xk+1 0 21
11
2212
-
...... 21
11
k2k1
-
1 21
11
-
...... 0 21
11
2-1
bb
21
11
2212
21
11
2-1
-
bb
2 x1 1 21
22
...... 21
k2
0 21
1
...... 0 21
2
b
21
22
21
2
b
3 xk+3 0 21
31
2232
-
...... 21
31
k2k3
-
0 21
31
-
...... 0 21
31
23
b-b
21
31
2232
21
31
23
-
b-b
... ... ... ... ...... ... ... ... ...... ... ... ...
m xk+m 0 211m
222m
-
...... 21
1m
k2mk
-
0 21
1m
-
...... 1 21
1m
2-m
bb
21
1m
222m
21
1m
2m
-
b-b
v1 v2 ...... vk 0 0 ...... 0 v
-
21
22
122
v-v=V
...... 21
k2
1-kk
vv=V
- - ...... - vj
II
6 ( )
.
,
. xk+2
x1.
7
,
.
21, 22,...,2k, 0, 1,..., 0, b2 21
x1.
-27-
8
,
. ,
.
9 . .
2 ,
Vj
.
.
vj ,
( ).
, vj , ,
,
. .
.
3.9
. .
:
max=xv+...+xv+xv=V kk2211 (3.22)
1kk1212111 b x+...+x+x (3.23)
2kk2222121 b x+...+x+x (3.24)
b x+...+x+x
.......................................
b x+...+x+x
mkmk22m11m
3kk3232131
(3.25)
-28-
0 x
........
0 x
0 x
k
2
1
-28-
0x
........
0x
0x
k
2
1
(3.26) (2
. (3.1), (3.2) (3.3),
(3.23) (3.24) ,
(3.25) .
.
(3.23), (3.24), (3.25) (3.26)
:
1m+k3+k2+k1+kkk1212111 b=x.0+...+x.0+x.0+xx+...+x+x (3.27)
2m+k3+k2+k1+kkk2222121 b=x.0+...+x.0+xx.0+x+...+x+x (3.28)
b=x+...+x.0+x.0+x.0+x+...+x+x
......................................................................................
b=x.0+...+x+x.0+x.0+x+...+x+x
mm+k3+k2+k1+kkmk22m11m
3m+k3+k2+k1+kkk3232131
(3.29)
0x
.......
0x
0x
.......
0x
0x
m+k
1+k
k
2
1
(3.30)
. (27) (28) . (13),
xk+1 xk+2,
. , (23) (24)
.
x1=0, x2=0,...,
xk=0, xk+1=-b1, xk+2=-b2, xk+3=b3,..., xk+m=bm. b1,
-29-
Simplex. .
b2,..., bm ,
xk+1 xk+2 .
(3.30)
,
, . ,
.
() .
(3.22) (3.27),
(3.28), (3.29) (3.30) :
+...+x.0+x.0+xv+...+xv+xv=V 2+k1+kkk2211
max=x.M-x.M-x.0+ 2+m+km+km+k (3.31)
+...+x.0+x.0+x-x+...+x+x 3+k2+k1+kkk1212111
12+m+k1+m+km+k b=x.0+x+x.0+ (3.32)
+...+x.0+x-x.0+x+...+x+x 3+k2+k1+kkk2222121
22+m+k1+m+km+k b=x+x.0+x.0+ (3.33)
b=x.0+x.0+x+...+x.0+
+x.0+x.0+x+...+x+x
...............................................................
b=x.0+x.0+x.0+...+x+
+x.0+x.0+x+...+x+x
m2+m+k1+m+km+k3+k
2+k1+kkmk22m11m
32+m+k1+m+km+k3+k
2+k1+kkk3232131
(3.34)
0x,x,x,...,x,x,x,...,x,x 2+m+k1+m+km+k2+k1+kk21 (3.35)
-30-
(3.32) (3.33) xk+m+1 xk+m+2
. (3.31)
.
.
. ,
.
,
, -xk+m+1, -
Mxk+m+2 -.
, .
,
.
.
Simplex,
. , .
(3.32), (3.33) (3.34),
x1=0, x2=0, ..., xk=0, xk+1=0, xk+2=0, xk+3=b3, ..., xk+m=bm, xk+m+1=b1,
xk+m+2=b2, (3.35),
.
Simplex.
Simplex
. .
,
, (3.35)
-31-
Simplex.
3.10
.
.
3.106
2,2.106 m
3,
1,5.106 2,8.106 m3 .
,
2:1.
(
) : 0,32 /m3,
0,24 /m3, ,
0,26 /m3, 0,2 /m3,
.
, .
.
x1 x2 ( m3
100.000)
, x3 x4
,
:
min=x2,0+x26,0+x24,0+x32,0=V 4321
min=x5+x5,6+x6+x8=V 4321 (/ 25) (3.36)
-32-
15 x+x 31 ( ) (3.37)
28x+x 42 ( ) (3.38)
30x+x 21 ( )
22x+x 43 ( ) (3.39)
0x2+x- 31 ( )
0 x i (i = 1, 2, 3, 4) (3.40)
. .
,
(3.36).
,
-1.
(3.37), (3.38) (3.39)
5
, (3.37) (3.38)
2 ,
(). x5 x6
x7 x8 (3.37) (3.38).
(3.39) x9, x10 x11.
.
maxx.0x.0x.0Mx-Mx-x.0x.0x5-x5,6-x6-x8-V 1110987654321 (3.41)
0=x+x.0+x.0+x.0+x.0+x.0+x.0+x.0+x.2+x.0+x
22=x.0+x+x.0+x.0+x.0+x.0+x.0+x+x+x.0+x.0
30=x.0+x.0+x+x.0+x.0+x.0+x.0+x.0+x.0+x+x
28=x.0+x.0+x.0+x+x.0+x-x.0+x+x.0+x+x.0
15=x.0+x.0+x.0+x.0+x+x.0+x-x.0+x+x.0+x
1110987654321
1110987654321
1110987654321
1110987654321
1110987654321
(3.42)
-33-
0x i (i = 1, 2, ..., 11) (3.43)
x7 x8 .
( . (41), (42) (43))
Simplex
:
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
1 x7 1 0 1 0 -1 0 1 0 0 0 0 15
2
x8 0 1 0 1 0 -1 0 1 0 0 0 28 28
3 x9 1 1 0 0 0 0 0 0 1 0 0 30
4 x10 0 0 1 1 0 0 0 0 0 1 0 22 22
5 x11 -1 0 2 0 0 0 0 0 0 0 1 0 0/0
-8 -6 -6,5 -5 0 0 -M -M 0 0 0 -43M
-8+M -6+M -6,5+M -5+M -M -M - - - - -
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
1 x7 1 0 1 0 -1 0 1 0 0 0 0 15
2 x8 0 1 -1 0 0 -1 0 1 0 -1 0 6 6
3 x9 1 1 0 0 0 0 0 0 1 0 0 30 30
4 x4 0 0 1 1 0 0 0 0 0 1 0 22
5 x11 -1 0 2 0 0 0 0 0 0 0 1 0 0/0
-8 -6 -6,5 -5 0 0 -M -M 0 0 0 -21M-110
-8+M -6+M -1,5 - -M -M - - - - -
-34-
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
1 x7 1 0 1 0 -1 0 1 0 0 0 0 15 15
2 x2 0 1 -1 0 0 -1 0 1 0 -1 0 6
3 x9 1 0 1 0 0 1 0 -1 1 1 0 24 24
4 x4 0 0 1 1 0 0 0 0 0 1 0 22 22
5 x11 -1 0 2 0 0 0 0 0 0 0 1 0 0
-8 -6 -6,5 -5 0 0 -M -M 0 0 0 -15M-146
-8+M - -7,5+M - -M -6 - -M+6 - -1 -
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
1 x7 1,5 0 0 0 -1 0 1 0 0 0 -0,5 15 10
2 x2 -0,5 1 0 0 0 -1 0 1 0 -1 0,5 6
3 x9 1,5 0 0 0 0 1 0 -1 1 1 -0,5 24 16
4 x4 0,5 0 0 1 0 0 0 0 0 1 -0,5 22 44
5 x3 -0,5 0 1 0 0 0 0 0 0 0 0,5 0
-8 -6 -6,5 -5 0 0 -M -M 0 0 0 -15M-146
1,5M-11,75 - - - -M -6 - -M+6 - -1 -0.5M+3,75
-35-
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
1 x1 1 0 0 0 -0.67 0 0.67 0 0 0 -0.33 10
2 x2 0 1 0 0 -0.33 -1 0.33 1 0 -1 0.33 11
3 x9 0 0 0 0 1 1 -1 -1 1 1 0 9
4 x4 0 0 0 1 0.33 0 -0.33 0 0 1 -0.33 17
5 x3 0 0 1 0 -0.33 0 0.33 0 0 0 0.33 5
8 -6 -6,5 -5 0 0 -M -M 0 0 0
- - - - -7,835 -6 -M+7,835 -M+6 - -1 -0,165
A . (3.41), (3.42)
(3.43) : x1=10, x2=11, x3=5, x4=17, x9=9,
.
1.106 m3 0,5.106
m3 . 1,1.106 m3 1,7.106
m3 . (1,054.106 .)
.