Simplex : ) :1- : . 2- .3- .4- 1947 Dantzig 2005 .5-
) : ( ) , :
:
( ) ( )
Zj
Cj-Zj
:1- .2- ( .3- .4- .# : ./ / .1/ :Max Z= 40 X1 + 50 X2 St: X1 + 2
X2 =0 :
X1 X2 S1 S240 50 0 0
401 2 1 0S1 0
1204 3 0 1S2 0
(0x40)+(0x120)=0(0x1)+(0x4)=0 0 0 0Zj
(40 0)=40 50 0 0
Cj-Zj
: / / () ./ / Max Cj-Zj . Min Cj-Zj ./ / Max : Cj-Zj Min :
Cj-Zj/ / .
*** :
: X1=24 , X2=8 : 1360X1 X2 S1 S240 50 0 0
401 2 1 0S1 0
1204 3 0 1S2 0
(0x40)+(0x120)=0(0x1)+(0x4)=0 0 0 0Zj
(40 0)=40 50 0 0
Cj-Zj
(40/2)=20(1/2)= (2/2)= (1/2)= (0/2)= 0.5 1 0.5 0 X2 50
120-(3x20)=604-(3x 0.5)= 3-(3x1)= 0-(3x0.5)= 1-(3x0)= 2.5 0 -1.5
1 S2 0
1000 25 50 25 0Zj
15 0 -25 0Cj-Zj
8 0 1 0.74 -0.2X2 50
X1 40
24 1 0 -0.48 0.4
1360 40 50 17.8 6Zj
0 0 -17.8 -6Cj-Zj
2/ :Max Z= 8 X1 + 2 X2 St: 2 X1 + X2 =0 : X1=5 , X2=0 , S1=0 ,
S2=10 , Z=40 ( )( Big M): ( ) . Max . Min . .
3/ :Max Z= 8 X1 + 4 X2 St: 4 X1 + 2 X2 >=9 X1 + X2 >=3X1 ,
X2 >=0/ :Max Z=8 X1 + 4 X2 - MA1 - MA2 St: 4 X1 + 2 X2 - S1 +A1
=9 X1 + X2 - S2 +A2 =3X1 , X2 ,S1,S2 >=0
: . :-
X1 X2 S1 S2 A1 A2 8 4 0 0 -m -m
9 4 2 -1 0 1 0A1 -m
3 1 1 0 -1 0 1A2 -m
-12m-5m -3m m m -m -mZj
8+5m 4-3m -m -m 0 0Cj-Zj
2.25 1 0.5 -0.25 0 0X1 8
0.75 0 0.5 0.25 -1 1A2 -m
18-0.75m 8 4-0.5m -2-0.25m m -mZj
0 0.5m 2+0.25m -m 0Cj-Zj
1.5 1 0 -0.5 1X1 8
1.5 0 1 0.5 -2 X2 4
18 8 4 -2 0Zj
0 0 2 0Cj-Zj
:-1- Cj-Zj Max Min . : X1 X2 S1 S2 40 50 0 0
8 0 1 0.74 -0.2X2 50
X1 40
24 1 0 -0.48 0.4
1360 40 50 17.8 6Zj
0 0 -17.8 -6Cj-Zj
2- : Cj-Zj . 32 : X1 X2 S1 S2 1 -1 0 0
2 1 0 -1 2X1 1
X2 -1
1 0 1 -1 1
1 1 -1 0 1 Zj
0 0 0 -1Cj-Zj
: . 32 : X1 X2 S1 S2 1 1 0 0
2 1 0 -1 2X1 1
X2 1
1 0 1 -1 1
3 1 1 -2 3 Zj
0 0 2 -3Cj-Zj
- : ( ) . Max Cj-Zj .
_:
:Max Z= 5X1 + 3X2St: 4X1 + 2X2 =4 X2 >=6X1 , X2 >=0
/ :Max Z= 5X1 + 3X2 mA1 mA2 St: 4X1 + 2X2 +S1 =8 X1 - S2 + A1 =4
X2 - S3 + A2 =6 X1 , X2 , S1 , S2 , S3 >=0
.X1 X2 S1 S2 S3 A1 A25 3 0 0 0 -m -m
84 2 1 0 0 0 0S1 0
41 0 0 -1 0 1 0A1 -m
60 1 0 0 -1 0 1A2 -m
-10m-m -m 0 m m -m -mZj
5+m 3+m 0 -m -m 0 0Cj-Zj
21 0.5 0.25 0 0 0 0X1 5
20 -0.5 -0.25 -1 0 1 0 A1 -m
60 1 0 0 -1 0 1A2 -m
10-8m5 5/2-m/2 5/4+m/4 -m -m 0 0Zj
0 1/2+m/2 5/4-m/4 -m -m 0 0Cj-Zj
42 1 0.5 0 0 0 0X2 3
41 0 0 -1 0 1 0A1 -m
2-2 0 - 0.5 0 -1 0 1A2 -m
12-6m6+m 3 3/2+m/2 m m -m -mZj
-1-m 0 -3/2-m/2 -m -m 0 0Cj-Zj