Aplicacion del metodo simplex pdf
Transcript of Aplicacion del metodo simplex pdf
INVESTIGACION DE OPERACIONES
APICACION DEL METODO SIMPLEX
INGENIERIA INDUSTRIAL 4°
ISRAEL RIVAS ALVAREZ
ASESESOR:
ING. IND MIGUEL ANGEL RUBIO DAMAZO
Aplicacion del metodo simplex .......................................................................................... 3
Ejercicio #1 ............................................................................................................................... 3
Ejercicio #2 ............................................................................................................................... 4
3.- ZMAX = 4X1+3X2+6X3 4X1+3X2+6X3+0S1+0S2 ................................................... 5
EJERCICIO #4.- ....................................................................................................................... 7
EJERCICIO #5.- ....................................................................................................................... 8
EJERCICIO #6 .......................................................................................................................... 9
EJERCICO #7 ......................................................................................................................... 10
EJERCICIO #8 ........................................................................................................................ 11
EJERCICIO #9 ........................................................................................................................ 12
EJERCICIO #10 ..................................................................................................................... 13
EJERCICIO #11 ..................................................................................................................... 14
EJERCICIO #12 ..................................................................................................................... 15
EJERCICIO #13 ..................................................................................................................... 16
Aplicación del método simplex
Ejercicio #1
Zmax
= 2x1+3x
2………….. Z-2x
1-3x
2=0
-3x1+x2
2≤1……………… -3x
1+x
2+S
1=1
4x1+2x
2≤20……………..4x
1+2x
2+S
2=20
4x1-4x
2≤10………………4x
1-4x
2+S
3=10
-x1+2x
2≤5………………..-x
1+2x
2+S
4=5
x1,x
2≥0
z X1 X
2 S
1 S
2 S
3 S
4 R
S1 0 -3 1 1 0 0 0 1
S2 0 4 2 0 1 0 0 20
S3 0 4 -1 0 0 1 0 10
S4 0 -1 2 0 0 0 1 5
X2
1 -2 -3 0 0 0 0 0
0 -3 1 1 0 0 0 1
S2 10 0 -2 1 0 0 18
S3 0 1 0 0 0 1 0 11
S4 0 5 0 -2 0 0 1 3
z 1 -11 0 3 0 0 0 3
X2
0 0 1 -6/5 0 0 3/5 14/5
S2
0 0 0 2 1 0 -2 12
S3
0 0 0 2/5 0 1 -1/5 52/5
X1
0 1 0 -2/5 0 0 1/5 3/5
z 1 0 0 -7/5 0 0 11/5 48/5
Z=48/5 X1=3/5 X
2=14/5
Por lo tanto Zmax
= 2x1+3x
2…………. Zmax= 2(3/5)+3(14/5)
Zmax= 6/5+42/5
Zmax=48/5 o.k
Ejercicio #2
Zmin= 5x1+7x
2………5x
1+7x
2-0S
1-0S
2-0S
3+MA
1+MA
2+MA
3
2x1+3x
2 ≥ 147………..2x
1+3x
2-S
1+A
1=147
3x1+3x
2≥210………….3x
1+3x
2-S
2+A
2=210
X1+x
2≥63……………….x
1+x
2-S
3+A
3=63
X2,x
1≥0
5 7 0 0 0 M M M
Cj X1 X
2 S
1 S
2 S
3 A
1 A
2 A
3 b
MA1 2 3 -1 0 0 1 0 0 147
MA2 3 3 0 -1 0 0 1 0 210
MA3 1 1 0 0 -1 0 0 1 63
Zj 6M 7M -M -M M M M M 420
Cj-Zj 5-6M 7-7M -M M -M 0 0 0
3.- ZMAX = 4X1+3X2+6X3 4X1+3X2+6X3+0S1+0S2 3X1+X2+3X3≤30 3X1+X2+3X3+S1=30
2X1+2X2+3X3≤40 2X1+2X2+3X3+S1=40
CJ 4 3 6 0 0
X1 X2 X3 S1 S2 B
0S1 3 1 3 1 0 30
0S2 2 2 3 0 1 40
ZJ 0 0 0 0 0 0
CJ-ZJ 4 3 6 0 0 ---
CJ 4 3 6 0 0
X1 X2 X3 S1 S2 B
6X1 1 1/3 1 1/3 0 10
-3R1+R2 0S2 -1 1 0 -1 0 10
ZJ 6 2 6 0 0 60
CJ-ZJ -2 1 0 0 0 ---
CJ 4 3 6 0 0
X1 X2 X3 S1 S2 B
-1/3 R2+R1 6X3 4/3 0 6 1/3 0 20/3
3X2 -1 1 0 -1 0 10
ZJ 5 3 6 -1 0 70
CJ-ZJ -1 0 0 1 0 ---
CJ
4 3 6 0 0
X1 X2 X3 S1 S2 B
R2/1/3 0S1 4 0 18 1 0 20
R1+R2 3X2 3 1 0 0 0 30
ZJ 9 3 0 0 0 90
CJ-ZJ -5 0 6 0 0 ---
CJ
4 3 6 0 0
X1 X2 X3 S1 S2 B
6X3 4/18 0 1 1/18 0 20/18
3X2 3 1 0 0 0 30
ZJ 10 1/3 3 6 1/3 0 96 2/3
CJ-Z -6 1/3 6 0 -1/3 0 ---
X3=10
X2=10
S1=20
Zj=96 2/3
EJERCICIO #4.-
ZMAX =3X1+4X2+5X3 3X1+4X2+5X3+0S1+0S2+0S3
3X1+X2+5X3≤150 3X1+X2+5X3+S1=150
X1+4X2+X3≤120 X1+4X2+X3+S2=120
2X1+2X3≤105 2X1+2X3+S3=105
3 4 5 0 0 0 B
CJ X1 X2 X3 S1 S2 S3
0S1 3 1 5 1 0 0 150
0S2 1 4 1 0 1 0 120
0S3 2 0 2 0 0 1 105
ZJ 0 0 0 0 0 0 0
CJ-ZJ 3 4 5 0 0 0 -
3 4 5 0 0 0 B
CJ X1 X2 X3 S1 S2 S3
-5R2+R1
0S1
-2 -19 0 1 -5 0 -450
5X3 1 4 1 0 1 0 120
-2R2+R3
0S3
0 -8 0 0 -2 1 -135
ZJ 5 20 5 0 5 0 600
CJ-ZJ -2 -16 0 0 -5 0 -
X1=0
X2=0
X3=120
ZJ=600
EJERCICIO #5.-
Z MAX= 5X1+3X2+4X3 5X1+3X2+4X3+0S1+0S2
2X1+X2+X3≤120 2X1+X2+X3+S1=120
3X1+X2+2X3≤30 3X1+X2+2X3+S2=30
5 3 4 0 0
CJ X1 X2 X3 S1 S2 B
0S1 2 1 1 1 0 120
0S2 3 1 2 0 1 30
ZJ 0 0 0 0 0 0
CJ-ZJ 5 3 4 0 0 -
5 3 4 0 0
CJ X1 X2 X3 S1 S2 B
-2R2+R1
0S1
0 1/3 -1/3 1 -2/3 100
5X1 0 1/3 2/3 0 1/3 10
ZJ 5 5/3 10/3 0 5/3 50
CJ-ZJ 0 4/3 2/3 0 -5/3 -
5 3 4 0 0
CJ X1 X2 X3 S1 S2 B
-1/3R2+R1
3X2
-1 0 -1 1 -1 90
SX1 3 1 2 0 1 30
ZJ 12 5 7 3 2 420
CJ-ZJ -7 -2 -3 -3 -2 -
X1=10
X2=30
ZJ=420
EJERCICIO #6
ZMAX=2X1+4X2 +3X3…………2X1+4X2 +3X3+0S1+0S2+0S3
X1 +3X2+2X3 ≤ 30………… X1 +3X2+2X3+S1= 30
X1+X2+X3 ≤ 24………… X1+X2+X3+0s 2 =24
3X1+5X2+3X3 ≤ 60………… 3X1+5X2+3X3 +0S3 = 60
X1, X2, X3 ≤ 0
X1 X2 X3 S1 S2 S3 b Cj 2 4 3 0 0 0
0S1 1 3 2 1 0 0 30 30/10=10 0S2 1 1 1 0 1 0 24 24/1=24 0S3 1 5 3 0 0 1 60 60/5=12 Zj 0 0 0 0 0 0 0
Cj-Zj 2 4 3 0 0 0
X1 X2 X3 S1 S2 S3 b Cj 2 4 3 0 0 0
R1(1/3) 4X1 1/3 1 2/3 1/3 0 0 10 -R1+R2 S2 2/3 0 1/3 -1/3 1 0 14
-5R1+R3 S3 -2/3 0 -1/3 -5/3 0 1 10 Zj 4/3 4 8/3 4/3 0 0 40 Cj-Zj 2/3 0 1/3 -4/3 0 0
X1=0 X2=10 X3=0 Zj=40
EJERCICO #7
ZMIN=5000X1+7000X2 …………5000X1+7000X2 -0S1-0S2+MA1+MA2
-2X1 +X2≥1………… -2X1 +X2 -S1+A1= 1
X1 -2X2≥1………… X1 -2X2 –S2+A2= 1
X1, X2, X3 ≥0
X1 X2 X3 S1 S2 S3 b Cj 5000 7000 0 0 M M
MA1 -2 1 -1 0 1 0 1 1/-2=-1/2 MA2 1 -2 0 -1 0 1 1 1/1=1
Zj -M -M -M -M M M M Cj-Zj 5OOO-M 7000-M M M 0 0
X1 X2 X3 S1 S2 S3 b Cj 5000 7000 0 0 M M
2R2+R1 MA1 0 -3 -1 0 1 0 1 5000X2 1 -2 0 -1 0 1 1 Zj 5000 -3M-10000 -M -2M-5000 M 2M+5000 3M+5000 Cj-Zj 0 3M+17000 M 2M+5000 0 -M-5000
X1=1 X2=0 Zj=3M+5000
EJERCICIO #8
ZMAX=5X1+9X2 +7X3…………5X1+9X2 +7X3+0S1+0S2+0S3
X1 +3X2+2X3 ≤ 10………… X1 +3X2+2X3+S1= 10
3X1+4X2+2X3 ≤ 12………… 3X1+4X2+2X3+0S2 =12
2X1+X2+2X3 ≤ 8………… 2X1+X2+2X3+0S3 = 8
X1, X2, X3 ≤ 0
X1 X2 X3 S1 S2 S3 b Cj 5 9 7 0 0 0
0S1 1 3 2 1 0 0 10 10/3=3.3 0S2 3 4 2 0 1 0 12 12/4=3 0S3 2 1 2 0 0 1 8 8/1=8 Zj 0 0 0 0 0 0 0
Cj-Zj 5 0 7 0 0 0
X1 X2 X3 S1 S2 S3 b Cj 5 9 7 0 0 0
-3R2+R1 0S1 -5/4 0 1/2 1 -3/4 0 1 1/1/2= 2 1/4R2 9X2 3/4 1 1/2 0 1/4 0 3 3/1/2 = 6 -R1+R3 0S3 5/4 0 3/2 0 -1/4 1 5 5/ ½ = 10
Zj 27/4 9 9/2 0 9/4 0 27 Cj-Zj -7/4 0 5/2 0 -9/4 0
X1 X2 X3 S1 S2 S3 b Cj 5 9 7 0 0 0
2R1 7X3 -5/2 0 1 2 -6/4 0 2 -1/2R1 +R2 9X2 2 1 0 -1 1 0 2 -3/2R1 +R3 0S3 5 0 0 -3 2 1 2
Zj 1/2 9 7 5 -3/2 0 32 Cj-Zj 9/2 9 0 -5 -3/2 0
X1=0 X2=2 X3=2 Zj=32
EJERCICIO #9
Z MAX= 3x1+5x2+6x3 3x1+5x2+6x3
2x1+x2+x3 ≤ 4 2x1+x2+x3+s1 = 4
X1+2x2+x3 ≤ 4 X1+2x2+x3 + s2= 4
X1+x2+2x3 ≤ 4 X1+x2+2x3+ s3 = 4
X1+x2+x3 ≤ 3 X1+x2+x3+s4 = 3
CJ-ZJ 3 5 6 0 0 0 0 -
CJ
3 X1
5 X2
6 X3
0 S1
0 S2
0 S3
0 S4
b/R
3/2 1/2 0 1 0 -1/2 0 2
½ ½ 0 0 1 -1/2 0 0
½ ½ 1 0 0 ½ 0 2
½ ½ 0 0 0 -1/2 1 1
ZJ 3 3 6 0 0 3 0 12
CJ-ZJ 0 2 0 0 0 0 0 1
1 0 0 1 0 0 -2 1
0 0 0 0 1 0 -2 1
0 0 1 0 0 1 -2 1
1 1 0 0 0 -1 2 2
ZJ 5 5 6 0 0 1 -2 16
CJ-ZJ -2 0 0 0 0 -1 2 -
CJ
3 X1
5 X2
6 X3
0 S1
0 S2
0 S3
0 S4
b/R
S1 2 1 1 1 0 0 0 4/1= 4
S2 1 2 1 0 1 0 0 4/1=4
S3 1 1 2 0 0 1 0 4/2=2
1 1 1 0 0 0 1 3/1=3
ZJ 0 0 0 0 0 0 0 0
ZJ=16
X3 =2
X2 =2
X1=0
S4
EJERCICIO #10
Z MIN= 2X1+3X2+X3 2X1+3X2+X3 -0S1 -0S 2 +0S3+MA1+MA2
X1+4X2+2X3 ≥ 8 X1+4X2+2X3-S1+A1 = 8
3X1+2X2 ≥ 6 3X1+2X2 - S2+A2 = 6
X1, X2, X3 ≥ 0
CJ
3 X1
5 X2
6 X3
0 S1
0 S2
0 A1
0 A2
b/R
MA1 1 4 2 -1 0 1 0 8
MA2 3 2 0 0 -1 0 1 6
ZJ 4M 6M 2M -M -M M M 14M
CJ-ZJ 2-4M
3-6M 1-2M M M -M -M -
-R2+R1
MA1
0 10/3 2 -1 1/3 1 -1/3 6
2X1 1 2/3 0 0 -1/3 1 -1/3 6
ZJ 2 4/3+10/3M 2M -M -2/3+1/3M 4 2/3+(-1/3M)
4+6M
CJ-ZJ 0 3-(2/3+10/3M)
1-2M M -(-2/3+1/3M) -4 -(2/3(-1/3M)
-
R1/ 10/3 3X2
0 1 -3/10 -3/10 1/10 3/10 -1/10 9/5
-2/3R1+R2
2X2
1 0 -2/5 1/5 -2/5 -1/5 1/15 4/5
ZJ 2 3 -1/2 -1/2 -1/2 1/2 -1/6 7
CJ-ZJ 0 0 -1/2 1/2 1/2 -1/2 1/6 -
R1/2/3 X3 0 5/3 1 -1/2 1/6 ½ -1/6 3
2/5R1+R2
2X1
1 2/3 0 0 -1/3 0 -2/15 2
ZJ 2 3 1 -1/2 -1/2 1/2 -13/30 7
CJ-ZJ 0 0 0 1/2 1/2 -1/2 13/30 -
X1=2
X2=9/5
X3=3
ZJ=7
EJERCICIO #11
Zmin= 2X1+ X2+ 3X3------------------------------2X1+ X2+ 3X3+0S1-0S2+MA1
5X1+2X2+7X3=420----------5X1+2X2+7X3+0S1=420
3X1+2X2+5X3≥280---------3X1+2X2+5X3-0S2+A1=280
X1,X2,X3≥0
Zj=168
X1=60
X2=0
X3=56
CJ X1 X2 X3 0S1 0S2 A1 R
0S1 5 2 7 1 0 0 420
MA1 3 2 5 0 -1 1 280
Zj 3M 2M 5M 0 -M M 280M
CJ-Zj 2-3M 1-2M 3-5M 0 +M 0 --------
CJ X1 X2 X3 0S1 0S2 A1 R
0S1 4/5 -4/5 0 1 7/5 -7/5 60
3x3 3/5 2/5 1 0 -1/5 1/5 56
Zj 9/5 6/5 3 0 -3/5 3/5 168
CJ-Zj 1/5 -1/5 0 0 +3/5 m-3/5 --------
2 1 3 0 0 M
420/7=60
280/5=56
-7(R2)+R1
R2(1/5)
2 1 3 0 0 M
EJERCICIO #12
Zmin= 3X1+ 2X2+ 4X3------------------------------3X1+ 2X2+ 4X3+0S1-0S2+MA1
2X1+X2+3X3=60----------2X1+X2+3X3+S1=60
3X1+5X2+5X3=120---------3X1+3X2+5X3-S2+A1=120
X1,X2,X3≥0
Zj=96
X3=24
X2=0
X1=0
CJ X1 X2 X3 0S1 0S2 A1 R
0S1 2 1 3 1 0 0 60
MA1 3 3 5 0 1 1 120
Zj 3M 3M 5M 0 M M 120M
CJ-Zj 3-3M 2-3M 4-5M 0 -M -M --------
CJ X1 X2 X3 0S1 0S2 A1 R
0S1 2/3 1/3 1 1/3 0 0 20
4x3 -1/3 1/3 0 -5/3 1 1 24
Zj -4/3 16/3 0 -20/3 4 4 96
CJ-Zj 13 -10/3 4 20/3 4 4M --------
60/3=20
120/5=24
3 2 4 0 0 M
R1(1/3)
-5R1+R2
3 2 4 0 0 M
EJERCICIO #13
Zmax= -X1+ 2X2+ X3------------------------------3X1+ 2X2+ 4X3+0S1-0S2+MA1
+3X2+X3≤120----------3X2+X3-S1 =120
X1-X2-4X3≤120--------- X1-X2-4X3-S2=80
-3X1+X2+2X3≤100-------3X1+X2+2X3-S3=100
X1,X2,X3≥0
CJ X1 -1
X2 2
X3 1
-S1 0
-S2 0
-S3 0
R
0S1 0 3 1 -1 0 0 120
0S2 1 -1 -4 0 -1 0 80
0S2 -3 1 2 0 0 -1 100
Zj 0 0 0 0 0 0
Cj-Zj -1 2 1 0 0 0
CJ X1 -1
X2 2
X3 1
-S1 0
-S2 0
-S3 0
R
2X2 0 1 1/3 -1/3 0 0 40
0S2 1 0 -11/3 -1/3 -1 1 120
0S2 -3 0 5/3 1/3 0 -1
Zj 0 2 2/3 -2/3 0 0 80
Cj-Zj -1 0 1/3 2/3 0 0
ZJ=80
X1=0
X2=40
X3=0
180/-1=-180
120/3=40
100/-1=-100
R1+R2
R1(1/3)
-1R1+R3