BÀI TOÁN QUY HOẠCH TUYẾN TÍNH

§

0.06kg
0.08kg

0.04kg
0

x1; x2 ; x3

xi  0, i  1,3 .
0.06 x1  0.04 x2  0.07 x3
0.08x1  0.x2  0.04 x3
2 x1  1.7 x2  1.8x3

1

f  x   2 x1  1.7 x2  1.8 x3  max

2

0.06 x1  0.04 x2  0.07 x3  500

 0.08 x1  0.x2  0.04 x3  300

3

x j  0, j  1.3

0.06 0.04 0.07
A

0
0.04
0.08
-1-

0.07kg
0.04kg


500
B

300
x  x1; x2 ; x3 
x  x1; x2 ; x3 
x  x1; x2 ; x3 

p2
p3

XN

1

3.5m
2.8m

2

20h 4m

10h 2.6m

3

16h 3.8m
12h 2.5m

1)

x1; x2 ; x3

x j  0, j  1,3
2)
3)

35x1  40 x2  43x3 ,
45x1  42 x2  30 x3 ,

45x1  42 x2  30 x3  35x1  40 x2  43x3  10 x1  2 x2  13x3  0
35x1  40 x2  43x3 ,
4)
-2-

18h
15h


5)

3.5  35x1  4  40x2  3.8  43x3


2.8  45x1  2.6  42x2  2.5  30x3
248.5 x1

 269.2 x2  238.4 x3

20  35x1  16  40x2  18  43x3

10  45x1  12  42x2  15  30x3
1150x1

 1144x2  1224x3

x1  x2  x3

1 f x   x1  x2  x3  min
10 x1  2 x2  13x3  0


35 x1  40 x2  43x3  1500
2
248.5 x1  269.2 x2  238.4 x3  10000
 1150 x1  1144 x2  1224 x3  52000

3x j  0, j  1,3

2
 13 
 10

 0 
 35
 1500 
40
43 



A
, B
248.5 269.2 238.4
10000 




 1150 1144 1224 
52000

1

2

C2, C3

-3-

1,

CT
km
Kho

C1

C2

C3

15T

20T

25T

7 km

5km

K1
20T

4km

x11

2km
x12

3km

6km

x13

K2
40T

x22

x21

-

x23

T  km
xij i  1,2; j  1,2,3

Ki  C j

xij  0

x11  x12  x13
3
x21  x22  x23
x11  x21
x12  x22

1

2

1

2

3

x13  x23
T  km :
7.
5x11  7 x12  2 x13  4 x21  3x22  6 x23

-4-


1 f x   5 x11  7 x12  2 x13  4 x21  3x22  6 x23  min
 x11  x12  x13  20
 x  x  x  40
22
23
 21
2 x11  x21  15
 x  x  20
12
22

 x13  x23  25

3xij  0 i  1,2; j  1,2,3

-5-


§
t:
n

1 f x    c j x j  min max 
j 1

 n
  aij x j  bi
 j 1
 n
2  aij x j  bi
 j 1
 n
  aij x j  bi
 j 1
3x j  0  j  J1 ; x j  0  j  J 2 ; x j tu  y

- Vector x  x1; x2 ;; xn 
-

y   j  J 3 ; J1  J 2  J 3  1;2;; n

-

1 f x   3x1  x2  2 x3  x4  5 x5  max
2 x1  x2  x3  2 x4  x5  17

4 x1  2 x2  x3  20
2
 x1  x2  2 x3  x5  18
 x1  x2  2 x3  x4  100
3x1; x4  0; x2 ; x5  0; x3 tu  y y
n

1 f x    c j x j  min max 
j 1

n

2  aij x j  bi
j 1

i  1, m

3x j  0  j  1, n 

-6-


1 f x   3x1  x2  x3  3x4  x5  min
 2 x1  x2  x3  3x4  0
2 x2  x3  x4  x5  18
 x  2 x  x  17

3
4
5


3x j  0; j  1,5
n

1 f x    c j x j  min max 
j 1

 x1

2



 a1m 1 xm 1   a1n xn  b1
 a2m 1 xm 1   a2n xn  b2
........................................

x2

xm

 amm 1 xm 1   amn xn  bm

3x j  0  j  1, n ; bi  0i  1, m 
x1

x2

xm

xm 1

xn

 1 0 ...... 0 a1m 1 ... a1n 


0
1
......
0
a
...
a


2
m

1
2
n

A  
   


 


0
0
......
1
a
...
a

mm 1
mn 








bi  0 i  1, m .

x1; x2 ;; xm

-

x1; x2 ;; xm ; xm1;; xn   b1; b2 ;; bm ;0;;0
-7-

bi  0, i  1, m


1 f x   3x1  x2  x3  3x4  x5  min
 2 x1  2 x4  x5  20
2 3x1  4 x2  4 x4  x6  0
 x  2 x  x  3 x  28
2
3
4
 1

3x j  0; j  1,6
x1

x2

x3

x4

x5

x6

 2 0 0 2 1 0


A   3 4 0  4 0 1
 1 2 1 3 0 0



5

x1, x2 , x3, x4 , x5 , x6,   0,0,28,0,20,0
6

3

-8-


§3
n

 aij x j  bi

j 1

xi 1  0
n

 aij x j  xn 1  bi

j 1

n

 aij x j  bi

j 1

xi 1  0

-

n

 aij x j  xn 1  bi

j 1

x j  0 ta thay x j  t j , t j  0
x j tu  y y ta thay x j  xj  xj

, xj , xj  0

:

1 f x   2 x1  x2  2 x3  x4  2 x5  min

 x1  2 x2  x3  2 x4  x5  7
 x1  2 x2  x3  2 x4  x5  7a 


x2  2 x3  x4  1
 x2  2 x3  x4  1b 


2


2 x3  x4  3x5  10
2 x3  x4  3x5  10c 


 x1  x2  2 x3  x4  20
 x1  x2  2 x3  x4  20d 

3x1; x5  0; x4  0; x2 ; x3 tu  y

y


x6  0 .
x7  0 .


-1
x8  0
 Thay x4  t4 ; t4  0
 Thay x2  x2  x2 ; x2  0 x2  0
 Thay x3  x3  x3 ; x3  0 x3  0
-9-


1 f x   2 x1  x2  x2   2x3  x3   t 4  2 x5  0.x6  0 x7  0 x8  min
 x1  2 x2  x2    x3  x3   2t 4  x5  x6  7a 

  x2  x2   2 x3  x3   t 4  x7  1b 
2

2 x3  x3   t 4  3x5  x8  10c 


x1   x2  x2   2 x3  x3   t 4  20d 
3x1; x5  0; t 4  0; x2 ; x2 ; x3 ; x3 ; x6 ; x7 ; x8  0

x10 , x20 , x20 , x30 , x30 ,t40 , x50 , x60 , x70 , x80 
x10 , x20 , x30 , x40 , x50 with x20  x20  x20 , x30  x30  x30 , x40  t40
bi  0, i  1, m )
n

1 f x    c j x j  min max 
j 1

a11 x1    a1n x n  b1
a x    a x  b
11 1
2
2 11 1

 
a11 x1    a11 x1  bm

3x j  0  j  1, n 

xn i  0




f x   min



f x   max

- 10 -

–M


n

m

j 1

i 1

1 f x    c j x j  M  xn  i  min max 
 b1
 a11 x1    a1n x n  xn 1
a x    a x 
xn  2
 b2
2n n
2 21 1


 

am1 x1    amn xn 
xn  m  bm

3x j  0  j  1, n  m 
 a11 x1
a x
A   21 1
 

am1 x1

0
0

   

 amn xn 0 0  1 
 a1n xn
 a2 n xn



1
0

0
1






xn  i i  1, m

1 f x   2 x1  x2  x3  x4  max
 x1  5 x2  5 x4  25
2 4 x2  x3  6 x4  18
 3 x  8 x  28
2
4


3x j  0; j  1,4

0 5
1 5
A  0  4  1 6 


0 3
0 8

1 f x   2 x1  x2  x3  x4  Mx5  Mx6  max
 x1  5 x2  5 x4  25
2 4 x2  x3  6 x4  x5  18
 3 x  8 x  x  28
2
4
6



3x j  0; j  1,6

0 5 0 0
1 5
A  0  4  1 6 1 0 


0 3
0 8 0 1
- 11 -




a)
b)

x  x1, x2 ,, xn 
x  x1, x2 ,...,xn ,0,...,0




x 0  x10 , x20 ,, xn0 ,0...,0
x 0  x10 , x20 ,, xn0



x 0  x10 , x20 ,, xn0 ,0...,0
x 0  x10 , x20 ,, xn0

c)







d)

1)

1

1

1
2
3
4
5

2
0
8
1
9

0
7
0
6
2

2
4
3
2
0

1

c
2)

I: 1.2m

1
2
3

0.55m
1
2
0

0.8m

0
0
0
- 12 -

0.45m
1
0
2

0.2
0.1
0.3


II: 1.5m

1
2
3
4
1
2
3
4

III: 1.8m

1
1

0
0
1
0
0
0

1
0
1
0
1
1
2
0

n.

1)
1 f x   min
 2 x1  x2  3x3  4 x5  17
4 x1  2 x3  x4  5 x5  20

2

3x j  0 j  1,5
2)
1 f x   max

 x1  x2  2 x3  x4  x5  32

2
x2  x3  2 x4  16
 x  x  2 x  x  32
2
3
4
 1

3x1, x3  0; x2  0; x4 , x5 tu  y

y

3)
1 f x   min
 2 x1  x2  3 x3  4 x5  17
24 x1  2 x3  x4  5 x5  20
  2x  4x  x  6
1
5
6


3x j  0

 j  1,6

4)

- 13 -

0
2
1
3
1
2
0
4

0.15
0.05
0.25
0.15
0
0.1
0.2
0


1 f x   max
 x1  x2  2 x3  5
2 x1  2 x2  x3  x4  15
 2 x  x  3x  8
1
2
3


3x j  0

 j  1,4

5)
1 f x   min
 x1  2 x2  2 x3  7
2 x2  2 x3  x4  15
 2 x  3x  x  8
3
4
 1

3x j  0

 j  1,4

a)
b)
c)

- 14 -


I:
§1

f x   min

P

c1

c2

cr

cm

x1

x2

xr

xm

cm 1  cv  cn
xm 1  xv  xn

i

c1
c2

x1
x2

b1
b2

1

0

0  0  0
1  0  0

a1m 1
a2m 1

 a1v
 a 2v

 a1n
 a2n

1
2

cr

xr

br

0

0  1  0

a3m 1





r

cm

xm

bm

0

f0

1

0  0  1 amm 1  amv  amn
2
r
 m  m 1   v   n

f x 

m

m

i 1

i 1

f 0   ci bi &  j   ci aij  c j
j 0

a)
b)

j t

 j  0

f x   f 0
ma  aij  0 i  1, m





 v  max  j thi xv
j

b
i  i with aiv  0
aiv

If

r  min i then xr out
i

-Jordan)
- 15 -

arv

arn


1. Thay xr

xv

2.
3.
4.

 aiv 
 v 
f x   max
g x    f x   min

xr )/ arv .

1 f x   2 x1  5 x2  4 x3  x4  x5  min
 x1  6 x2  2 x4  9 x5  32

22 x2  x3  1 x4  3 x5  30
2
2


3
x

x

36
2
5


3x j  0; j  1,5

 s tan dard form
1 f x   2 x1  5 x2  4 x3  x4  x5  0.x6  min
 x1  6 x2  2 x4  9 x5  32

22 x2  x3  1 x4  3 x5  30
2
2

3 x2  x5  x6  36


3x j  0; j  1,6

1  6 0  2  9 0
1
3



A  0 2 1
0
2
2
0 3 0 0
1
1


- 16 -


2
4
0

x1
x3
x6
f x 
 j  0, j 

2

5

4

1

-5

0

x1

x2

x3

x4

x5

x6

32
30
36

1
0
0

-6
2
3

0

1
0

-2
½
0

-9
3/2
1

0
0
1

184

0

-9
0
-3
-7
x1, x2 , x3 , x4 , x5   32,0,30,0,0

0

1 f x   6 x1  x2  x3  3x4  x5  7 x6  7  min
  x1  x2  x4  x6  15
2 2 x1  x3  2 x6  9

4 x  2 x  x  3 x  2
4
5
6
 1

3x j  0; j  1,6

 s tan dard form
1 f x   6 x1  x2  x3  3x4  x5  7 x6  7  min
  x1  x2  x4  x6  15
2  2 x1  x3  2 x6  9
4 x  2 x  x  3 x  2
4
5
6
 1

3x j  0; j  1,6

1 1 0 1 0 1 
A    2 0 1 0 0  2


 4 0 0 2 1  3

- 17 -


1

1
1

-7
1
1

x2
x3
x5
f x 
x6
x3
x5
f x 

6

1

1

3

1

-7

x1

x2

x3

x4

x5

x6

15
9
2

-1
-2
4

1
0
0

0
1
0

-1
0
2

0
0
1

(1)
-2
-3

26+7
15
39
47

1
-5
-1
-4
1

2
0
1
2
3

3
0
0
1
0

4
-2
-1
-2
-1

5
0
0
0
1

6
(3)
1
0
0

-19+7

-2

-3

0

(1)

0

0

max  j   6  3  x6 in 
arv  1  x2 out 
max  j   4  1but ai 4  0,  1,2,1 

- 18 -


1 f x   2 x1  6 x2  4 x3  2 x4  3x5  max
 x1  2 x2  x3  52
24 x2  2 x3  x4  60
 3 x  x  36
2
5


3x j  0; j  1,5

 s tan dard form
1g ( x)   f x   2 x1  6 x2  4 x3  2 x4  3x5  min
 x1  2 x2  4 x3  52
24 x2  2 x3  x4  60
 3 x  x  36
2
5


3x j  0; j  1,5

1 2 4 0 0
A  0 4 2 1 0 


0 3 0 0 1

2
2
-3

x1
x4
x5
g x 

x3
x4
x5
g x 
x3
x2
x5
g x 

2

-6

-4

2

-3

x1

x2

x3

x4

x5

52
60
36

1
0
0

2
4
3

(4)
2
0

0
1
0

0
0
1

116
13
34
36

1
0
1/4
-1/2
0

2
9
1/2
(3)
3

3
(16)
1
0

0

4
0
0
1
0

5
0
0
0
1

-92
22/3
34/3
2

-4
1/3
-1/6
1/2

(1)
0
1
0

0

1
0
0

0
-1/6
1/3
-1

0
0
0
1

-310/3

-23/6

0

0

-1/3

0

- 19 -


max  j   3  16  x3 in 

52
 13
4
a rv  4  x1 out 
min i  1 

max  j   2  1  x2 in 
34
min i  2 
3
arv  3  x4 out 
 j  0, j 

x1, x2 , x3 , x4 , x5    0, 34 , 22 ,0,2 


3

3



f0   g0 

310
3

- 20 -

§

1.
2.
3.

1 f x   x1  2 x2  x4 _ 5 x5  min



 3 x3  9 x4  0
0 0 3 9 0 


2 x2  7 x3  5 x4  2 x5  5 , A  0 1  7  5  2

1 2
4
1 

1
2
4
1
2
1

x

x


x

x

x


 1 3 2 3 3 3 4 5 5 3
3 3
3
5 


3x j  0; j  1,5
 s tan dard

form

1 f x   x1  2 x2  x4 _ 5 x5  Mx6  Mx7  min

 3 x3  9 x4  x6  0

2 x2  7 x3  5 x4  2 x5  x7  5

1
2
4
1
2

 x1  3 x2  3 x3  3 x4  3 x5  3

3x j  0; j  1,7


0 0  3  9 0
A  0 1  7  5  2

1 2
4
1
1  3 3
3
5


1 0
0 1

0 0


- 21 -


M
M
1

x6

x7
x1
f x 
x6
x2
x1
f x 

1

2

0

1

-5

x1

x2

x3

x4

x5

0
5

2/3

0
0
1

0
(1)
-1/3

-3
-7
2/3

-9
-5
4/3

0
-2
1/3

5M+2/3
0
5
7/3

1
0
0

0
1

2
(-7/3+M)
0
1
0

3
2/3-10M
-3
-7
-5/3

4
1/3-14M
-9
-5
-1/3

5
16/3-2M
0
-2
-1/3

37/3

0

0

-47/3-3M

-34/3-9M

2/3

7
max  j   2    M  x2 in 
3
5
min i  2 
1
arv  1  x7 out 

max  j   5 

2
 0 but
3

1

ai5  0,2,  0 
3


- 22 -

1 f x   16 x1  7 x2  9 x3  min
1
1  2
 2
 x1  x2  x3 

2 3
3
3 3
 5
  5 x1  5 x2  7


3x j  0; j  1,3
 s tan dard

1 
1
3 
5 0



form

1 f x   16 x1  7 x2  9 x3  Mx4  min
1
1

 2
 x1  x2  x3 
2 3
3
3
  5 x1  5 x2  x4  7

3x j  0; j  1,4
 2

A 3
 5


1
1
3
5 0



0

1


-16

7

9

x1

x2

x3

1/3
7

-2/3
-5

-1/3
(5)

1
0

f x 
x3
x2

7M+3
12/15
7/5

1
10-5M

-1
-1

2
(-10+5M)
0
1

3
0
1
0

f x 

17

0

0

0

b

9
M

x3
x4

max  j   2  10  5M  x2 in 
7
5
arv  5  x4 out 
min i  2 

 j  0, j 
- 23 -


x1, x2 , x3    0, 7 , 12 
 5 15 

f 0  17

1 f x   2 x1  4 x2  2 x3  min
 x1  2 x2  x3  27
22 x1  x2  2 x3  50
 x  x  x  18
2
3
 1

3x j  0; j  1,3
 s tan dard

form

1 f x   2 x1  4 x2  2 x3  0.x4  min

 x1  2 x2  x3  27
2 2 x1  x2  2 x3  50
 x  x  x  x  18
2
3
4
 1

3x j  0; j  1,4

1  2 1 0 
A  2 1
2 0


1  1  1 1
 s tan dard form

1 f x   2 x1  4 x2  2 x3  0.x4  Mx5  Mx6  min
 x1  2 x2  x3  x5  27
22 x1  x2  2 x3  x6  50
 x  x  x  x  18
2
3
4
 1

3x j  0; j  1,6

1  2 1 0

A  2 1
2 0

1  1  1 1

1 0
0 1

0 0

- 24 -


M
M
0

1

2

0

1

x1

x2

x3

x4

27
50
18

1
2
1

-2
1
-1

1
(2)
-1

0
0
1

77M
2
25
43

1
-2+3M

0
1
2

2
-4-M
-5/2
1/2
-1/2

3
(2+3M)
0
1
0

4
0
0
0
1

-50+2M

-4

-5-5M/2

0

0

x5
x6
x4
f x 
x5
x3
x4
f x 

max  j   3  2  3M  x3 in 
min i  2  25
arv  2  x6 out 
 j  0 but

x5  2  0 

.

- 25 -