PROBLEMAS DE MÉTODO SIMPLEX REUSELTOS

1. INVESTIGACION DE OPERACIONES
APICACION DEL METODO SIMPLEX
INGENIERIA INDUSTRIAL 4°
ISRAEL RIVAS ALVAREZ
ASESESOR:
ING. IND MIGUEL ANGEL RUBIO DAMAZO

2. 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

3. Aplicación del método simplex
Ejercicio #1
Zmax= 2x1+3x2………….. Z-2x1-3x2=0
-3x1+x22≤1……………… -3x1+x2+S1=1
4x1+2x2≤20……………..4x1+2x2+S2=20
4x1-4x2≤10………………4x1-4x2+S3=10
-x1+2x2≤5………………..-x1+2x2+S4=5
x1,x2≥0
S2
S3
S4
z
X1
-3
4
4
-1
-2
X2
1
2
-1
2
-3
S1
1
0
0
0
0
S2
0
1
0
0
0
S3
0
0
1
0
0
S4
0
0
0
1
0
R
1
20
10
5
0
0
S1
S2
S3
S4
X2
z
0
0
0
0
1
-3
10
1
5
-11
1
0
0
0
0
1
-2
0
-2
3
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
1
18
11
3
3
0
1
-6/5
0
0
3/5
14/5
0
0
2
1
0
-2
12
0
0
2/5
0
1
-1/5
52/5
1
0
-2/5
0
0
1/5
3/5
0
0
-7/5
0
0
11/5
48/5
0
0
1
0
X2
0
S2
0
S3
0
X1
1
z
Z=48/5
X1=3/5
X2=14/5
Por lo tanto Zmax= 2x1+3x2…………. Zmax= 2(3/5)+3(14/5)
Zmax= 6/5+42/5
Zmax=48/5 o.k

4. Ejercicio #2
Zmin= 5x1+7x2………5x1+7x2-0S1-0S2-0S3+MA1+MA2+MA3
2x1+3x2 ≥ 147………..2x1+3x2-S1+A1=147
3x1+3x2≥210………….3x1+3x2-S2+A2=210
X1+x2≥63……………….x1+x2-S3+A3=63
X2,x1≥0
Cj
MA1
MA2
MA3
Zj
Cj-Zj
5
X1
2
3
1
6M
5-6M
7
X2
3
3
1
7M
7-7M
0
S1
-1
0
0
-M
-M
0
S2
0
-1
0
-M
M
0
S3
0
0
-1
M
-M
M
A1
1
0
0
M
0
M
A2
0
1
0
M
0
M
A3
0
0
1
M
0
b
147
210
63
420

5. 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
0S1
4
X1
3
3
X2
1
6
X3
3
0
S1
1
0
S2
0
B
30
0S2
2
2
3
0
1
40
ZJ
CJ-ZJ
0
4
0
3
0
6
0
0
0
0
0
---
CJ
6X1
-3R1+R2 0S2
4
X1
1
-1
3
X2
1/3
1
6
X3
1
0
0
S1
1/3
-1
0
S2
0
0
B
10
10
ZJ
CJ-ZJ
6
-2
2
1
6
0
0
0
0
0
60
---
CJ
4
X1
4/3
-1
3
X2
0
1
6
X3
6
0
0
S1
1/3
-1
0
S2
0
0
B
20/3
10
ZJ
5
3
6
-1
0
70
CJ-ZJ
-1
0
0
1
0
---
4
X1
3
X2
6
X3
0
S1
0
S2
B
CJ
-1/3 R2+R1 6X3
3X2
CJ
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
---

6. 4
X1
CJ
3
X2
6
X3
0
S1
0
S2
B
20/18
6X3
4/18
0
1
1/18
0
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
---

7. 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
CJ
0S1
0S2
0S3
ZJ
CJ-ZJ
CJ
-5R2+R1
0S1
5X3
-2R2+R3
0S3
ZJ
CJ-ZJ
X1=0
X2=0
X3=120
ZJ=600
3
X1
3
1
2
0
3
4
X2
1
4
0
0
4
5
X3
5
1
2
0
5
0
S1
1
0
0
0
0
0
S2
0
1
0
0
0
0
S3
0
0
1
0
0
B
3
X1
-2
4
X2
-19
5
X3
0
0
S1
1
0
S2
-5
0
S3
0
-450
1
0
4
-8
1
0
0
0
1
-2
0
1
120
-135
5
-2
20
-16
5
0
0
0
5
-5
0
0
600
-
150
120
105
0
-
B

8. 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
0S1
0S2
ZJ
CJ-ZJ
X1
2
3
0
5
X2
1
1
0
3
X3
1
2
0
4
S1
1
0
0
0
S2
0
1
0
0
B
120
30
0
-
CJ
5
X1
0
3
X2
1/3
4
X3
-1/3
0
S1
1
0
S2
-2/3
B
100
0
0
1/3
5/3
4/3
2/3
10/3
2/3
0
0
0
1/3
5/3
-5/3
10
50
-
5
X1
-1
3
X2
0
4
X3
-1
0
S1
1
0
S2
-1
B
90
3
12
-7
1
5
-2
2
7
-3
0
3
-3
1
2
-2
30
420
-
-2R2+R1
0S1
5X1
ZJ
CJ-ZJ
CJ
-1/3R2+R1
3X2
SX1
ZJ
CJ-ZJ
X1=10
X2=30
ZJ=420
5

9. 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
Cj
0S1
0S2
0S3
Zj
Cj-Zj
R1(1/3)
-R1+R2
-5R1+R3
X1
2
1
1
1
0
2
X2
4
3
1
5
0
4
X3
3
2
1
3
0
3
S1
0
1
0
0
0
0
S2
0
0
1
0
0
0
S3
0
0
0
1
0
0
Cj
4X1
S2
S3
Zj
Cj-Zj
X1
2
1/3
2/3
-2/3
4/3
2/3
X2
4
1
0
0
4
0
X3
3
2/3
1/3
-1/3
8/3
1/3
S1
0
1/3
-1/3
-5/3
4/3
-4/3
S2
0
0
1
0
0
0
X1=0
X2=10
X3=0
b
30
24
60
0
S3
0
0
0
1
0
0
Zj=40
30/10=10
24/1=24
60/5=12
b
10
14
10
40

10. 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
Cj
MA1
MA2
Zj
Cj-Zj
2R2+R1
X1
5000
-2
1
-M
X2
7000
1
-2
-M
5OOO-M
Cj
MA1
5000X2
Zj
Cj-Zj
7000-M
X3
0
-1
0
-M
M
S1
0
0
-1
-M
M
S2
M
1
0
M
0
S3
M
0
1
M
0
X1
5000
0
1
X2
7000
-3
-2
X3
0
-1
0
S1
0
0
-1
S2
M
1
0
S3
M
0
1
b
5000
-3M-10000
-2M-5000
3M+5000
3M+17000
M
0
2M+5000
0
-M
M
X1=1
X2=0
2M+5000
Zj=3M+5000
b
1
1
M
-M-5000
1/-2=-1/2
1/1=1
1
1

11. 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
Cj
0S1
0S2
0S3
Zj
Cj-Zj
-3R2+R1
1/4R2
-R1+R3
2R1
-1/2R1 +R2
-3/2R1 +R3
X1
5
1
3
2
0
5
X2
9
3
4
1
0
0
X3
7
2
2
2
0
7
S1
0
1
0
0
0
0
S2
0
0
1
0
0
0
S3
0
0
0
1
0
0
Cj
0S1
9X2
0S3
Zj
Cj-Zj
X1
5
-5/4
3/4
5/4
27/4
-7/4
X2
9
0
1
0
9
0
X3
7
1/2
1/2
3/2
9/2
5/2
S1
0
1
0
0
0
0
S2
0
-3/4
1/4
-1/4
9/4
-9/4
S3
0
0
0
1
0
0
Cj
7X3
9X2
0S3
Zj
Cj-Zj
X1
5
-5/2
2
5
1/2
9/2
X2
9
0
1
0
9
9
X3
7
1
0
0
7
0
S1
0
2
-1
-3
5
-5
S2
0
-6/4
1
2
-3/2
-3/2
S3
0
0
0
1
0
0
X1=0
X2=2
X3=2
b
10
12
8
0
Zj=32
10/3=3.3
12/4=3
8/1=8
b
1
3
5
27
b
2
2
2
32
1/1/2= 2
3/1/2 = 6
5/ ½ = 10

12. EJERCICIO #9
Z MAX= 3x1+5x2+6x3
3x1+5x2+6x3
2x1+x2+x3 ≤ 4
ZJ=16
2x1+x2+x3+s1 = 4
X3 =2
X1+2x2+x3 + s2= 4
X1+2x2+x3 ≤ 4
X2 =2
X1+x2+2x3 ≤ 4
X1+x2+2x3+ s3 = 4
X1+x2+x3 ≤ 3
X1+x2+x3+s4 = 3
X1=0
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
CJ-ZJ
3
5
6
0
0
0
0
3
X1
3/2
½
½
½
3
0
1
0
0
1
5
-2
5
X2
1/2
½
½
½
3
2
0
0
0
1
5
0
6
X3
0
0
1
0
6
0
0
0
1
0
6
0
0
S1
1
0
0
0
0
0
1
0
0
0
0
0
0
S2
0
1
0
0
0
0
0
1
0
0
0
0
0
S3
-1/2
-1/2
½
-1/2
3
0
0
0
1
-1
1
-1
0
S4
0
0
0
1
0
0
-2
-2
-2
2
-2
2
S4
CJ
ZJ
CJ-ZJ
ZJ
CJ-ZJ
-
b/R
2
0
2
1
12
1
1
1
1
2
16
-

13. EJERCICIO #10
Z MIN= 2X1+3X2+X3
X1=2
2X1+3X2+X3 -0S1 -0S 2 +0S3+MA1+MA2
X2=9/5
X1+4X2+2X3 ≥ 8
X1+4X2+2X3-S1+A1 = 8
3X1+2X2 ≥ 6
X3=3
3X1+2X2 - S2+A2 = 6
ZJ=7
X1, X2, X3 ≥ 0
3
X1
1
3
4M
24M
0
5
X2
4
2
6M
3-6M
6
X3
2
0
2M
1-2M
0
S1
-1
0
-M
M
0
S2
0
-1
-M
M
0
A1
1
0
M
-M
0
A2
0
1
M
-M
b/R
8
6
14M
-
10/3
2
-1
1/3
1
-1/3
6
1
2
2/3
4/3+10/3M
0
2M
0
-M
-1/3
-2/3+1/3M
1
4
CJ-ZJ
0
1-2M
M
-(-2/3+1/3M)
R1/ 10/3
3X2
-2/3R1+R2
2X2
ZJ
CJ-ZJ
R1/2/3 X3
2/5R1+R2
2X1
ZJ
CJ-ZJ
0
3(2/3+10/3M)
1
-3/10
-3/10
1
0
-2/5
2
0
0
1
3
0
5/3
2/3
2
0
3
0
CJ
MA1
MA2
ZJ
CJ-ZJ
-R2+R1
MA1
2X1
ZJ
6
4+6M
1/10
-1/3
2/3+(1/3M)
-4
-(2/3(1/3M)
3/10 -1/10
1/5
-2/5
-1/5
1/15
4/5
-1/2
-1/2
1
0
-1/2
1/2
-1/2
0
-1/2
1/2
1/6
-1/3
1/2
-1/2
½
0
-1/6
1/6
-1/6
-2/15
7
3
2
1
0
-1/2
1/2
-1/2
1/2
1/2
-1/2
-13/30
13/30
7
-
9/5

14. 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
2
1
3
0
0
M
CJ
X1
X2
X3
0S1
0S2
A1
R
0S1
5
2
7
1
0
0
420
420/7=60
MA1
3
2
5
0
-1
1
280
280/5=56
Zj
3M
2M
5M
0
-M
M
280M
CJ-Zj
2-3M
1-2M
3-5M
0
+M
0
--------
2
1
3
0
0
M
CJ
X1
X2
X3
0S1
0S2
A1
R
0S1
4/5
-4/5
0
1
7/5
-7/5
60
-7(R2)+R1
3x3
3/5
2/5
1
0
-1/5
1/5
56
R2(1/5)
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
--------
Zj=168
X1=60
X2=0
X3=56

15. 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
3
2
4
0
0
M
CJ
X1
X2
X3
0S1
0S2
A1
R
0S1
2
1
3
1
0
0
60
60/3=20
MA1
3
3
5
0
1
1
120
120/5=24
Zj
3M
3M
5M
0
M
M
120M
CJ-Zj
3-3M
2-3M
4-5M
0
-M
-M
--------
3
2
4
0
0
M
CJ
X1
X2
X3
0S1
0S2
A1
R
0S1
2/3
1/3
1
1/3
0
0
20
R1(1/3)
4x3
-1/3
1/3
0
-5/3
1
1
24
-5R1+R2
Zj
-4/3
16/3
0
-20/3
4
4
96
CJ-Zj
13
-10/3
4
20/3
4
4M
--------
Zj=96
X3=24
X2=0
X1=0

16. 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
X2
2
3
X3
1
1
-S1
0
-1
-S2
0
0
-S3
0
0
R
0S1
X1
-1
0
120
120/3=40
0S2
1
-1
-4
0
-1
0
80
180/-1=-180
0S2
-3
1
2
0
0
-1
100
100/-1=-100
Zj
0
0
0
0
0
0
Cj-Zj
-1
2
1
0
0
0
CJ
X2
2
1
X3
1
1/3
-S1
0
-1/3
-S2
0
0
-S3
0
0
R
2X2
X1
-1
0
40
R1(1/3)
0S2
1
0
-11/3
-1/3
-1
1
120
R1+R2
0S2
-3
0
5/3
1/3
0
-1
Zj
0
2
2/3
-2/3
0
0
Cj-Zj
-1
0
1/3
2/3
0
0
ZJ=80
X1=0
X2=40
X3=0
-1R1+R3
80