Operacioni Istrazuvanja - Zbirka Reseni Zadaci

 ³.  ´   ± 

     

 

, 2004

  

  



1.      x1  x2    F(x)=3x 1+x2    ,     : 2

(1)

x2 u 1

( 2)

x1 u

x1  x 2 u

3

(3)

x 2  x1 e

2

( 4)

4 x1



3x2

e

24

(5)

     ,       .           . (3)

x1

0

3

x2

3

0

( 4)

x1

0

2

x2

2

4

(5)

x1

0

6

x2

8

0

       ,            ,    . (:         .) 3 x1

 x2 !

0  ( Z 0 )

x2

x1

0

1

x2

0

3

10

8

(1)

(5)

(4) Z0

6 Z 4

2 (2)

 (2,1)

0 -2

0

   

!

3 x1

 x2 !

(3) 4

6

8

10 x1

-2 min

2

3 2 1 ! 7

  : x 1=2, x2=1,       min Z=7.

1

  

 

2.    1  2   1  2.  1     12  ,  2  6  .     1  3 ,   2  2 ,    1,     2,  1   3 ,   2 1 .      3  1  3   2.   1  3 ,   2  2 .         .              : max Z  ! 3 x1



2 x2

x1 e

3

(1)

x2 e

3

( 2)

3 x1



3x 2

e 12

2 x1

 x2 e

(3)

6

( 4) x82

6

(1)

Z

(4)

4 Z0

(2)  (2,2)

2

(3) 0 -2

0

2

4

6

x81

-2

max

¡   

!

3 x1



2x2

!

3 2  2 2 ! 10 .

  : x 1=2, x2=2,       max Z=10.

2

  

 

3.          : 18    , 16    B  24    C.       H1  H2,     :

A B C 

H1 6 2 2 8

H2 2 4 12 12

.        .              : min

¢   

!

8 x1

 12 x 2

6 x1



2x2

u 18

(1)

2 x1



4 x2

u 16

(2 )

2 x1

 12 x 2 u

24

(3)

12 x 2 (1) 10 8 6 (2) 4 (3) Z0

min

£   

!

8 x1

 12 x 2 !

8 2  12 3 ! 52 .

2 0

-2

0

2

4

6

Z 8

10

12 12

x1

-2 min Z  ! 8 x1

 12 x 2 !

8 2  12 3 ! 52 .

  : x 1=2, x2=3,       Z=52 .

3

  

 

4.       :

min Z  ! 15 x1



20 x2

x1  x 2 u 10

2 x1

 3x2 e

6

(2)

6

x1  x 2 u x1 , x 2 u

(1) (3)

0 x2

 (2,4) ¤  

x1

min Z  ! 15 x1



20 x 2

! 15

2  20 4 ! 110

  : x1=2, x2=4,          Z=110.

4

  

 

5.      : max Z  ! 10 x1  x1 

2 x2



20 x2

e 15

(1)

x1  x 2 e 12

5 x1



3x 2

x1 , x 2 u

e

(2) 45

(3)

0

16 x2 14 12

(1)

10 8 Z 6 4 Z0

2

(2)

(3) 0

-2

max

¥   

-2

! 10 x1 

0

20 x 2

2

! 15

4

6

8

10

12

1x41

3  20 9 ! 210

  : x 1=3, x2=9,          Z=210.

5

  

 

6.      :  : max Z  ! 4 x1 x1  x 2 e

2 x1

2 x2

4

 x2 e

x1 , x 2 u



(1) 6

( 2)

0

x82

6

Z (2)

Z0

4



2

(1) 0 -2

0

2

B

4

6

x81

-2         (2)  ,      .            AB .                  . max Z  ! 4 x1



2x2

!

4 2  2 2 ! 12

   

AB ,

      Z=12.

6

  

 

7.      :  : max Z  !

x1 

x1  x 2 u

4

2 x1



2 x2

x1 , x 2 u

e

2 x2 (1)

3

( 2)

0

x62

(1) 4

(2) 2 Z0 0 -2

0

2

4

x61

-2         ,   . :        ,       ,    ,     .)

7

  

 

 

1.      : max Z  ! 3x1 x1 e

4

2 x2

e 12

3 x1



x1 u

2x2

0,

 5 x2

e 18

x2 u

0

              : : Z   3 x1  5 x2 !

4

x1  x 3 !

2 x2

 x 4 ! 12

3 x1



x1 u

2x2

0,

0

 x 5 ! 18

x2 u

0

..

..

Z

x1

x2

x3

x4

x5

..

Z

0

1

-3

-5

0

0

0

0

x3

1

0

1

0

1

0

0

4

x4

2

0

0

2

0

1

0

12

x5

3

0

3

2

0

0

1

18

Z

0'

1

-3

0

0

5/2

0

30

x3

1'

0

1

0

1

0

0

4

x2

2'

0

0

1

0

1/2

0

6

x5

3'

0

3

0

0

-1

1

6

Z

0"

1

0

0

0

3/2

1

36

x3

1"

0

0

0

1

1/3

-1/3

2

x2

2"

0

0

1

0

0.5

0

6

x1

3"

0

1

0

0

-1/3

1/3

2

    : x 1=2, x2=6,           Z=36.

8

  

 

2.     : max 4 x1

! x1  12 x 2  10 x 3

¦   

e

48

x1  1.5 x 2  3 x 3 e

24



6 x2



3x 3

3 x1  x 2 e 24               : : Z   x1  12 x 2  10 x 3 !

0

4 x1

 x4 !

48

x1  1,5 x 2  3 x 3  x 5 !

24

3 x1



6x2



3 x3

 x2  x6 !

x1 , x 2 , x 3 u

..

24

0

Z

.. 0

Z 1

x1 -1

x2 -12

x3 -10

x4 0

x5 0

x6 0

.. 0

x4

1

0

4

6

3

1

0

0

48

x5

2

0

1

3/2

3

0

1

0

24

x6

3

0

3

1

0

0

0

1

24

Z

0'

1

7

0

-4

2

0

0

96

x2

1'

0

2/3

1

1/2

1/6

0

0

8

x5

2'

0

0

0

9/4

-1/4

1

0

12

x6

3'

0

7/3

0

-1/2

-1/6

0

1

16

Z

0"

1

7

0

0

14/9

16/9

0

352/3

x2

1"

0

2/3 2/ 3

1

0

2/9

-2/9

0

16/3

x3

2"

0

0

0

1

-1/9

4/9

0

16/3

x6

3"

0

7/3 7/ 3

0

0

-2/9

2/9

1

56/3

    : x 1=0, x2=16/3, x3=16/3,       Z=352/3=117,33. Z=352/3=117,33.

9

  

 

3.     : max x1 

§   

3x2

e

x1  x 2 e x1 u

0,

2 x2

! x1 

8

4

x2 u

0

              : : ¨   

 x1  2 x2 !

0

x1  3 x2  x3 !

8

x1  x2  x4 !

..

4

Z

.. 0

Z 1

x1 -1

x2 -2

x3 0

x4 0

.. 0

x3

1

0

1

3

1

0

8

x4

2

0

1

1

0

1

4

Z

0'

1

-1/3

0

2/3

0

16/3

x2

1'

0

1/3

1

1/3

0

8/3

x4

2'

0

2/3

0

-1/3

1

4/3

Z

0"

1

0

0

1/2

1/2

6

x2

1"

0

0

1

1/2

-1/2

2

x1

2"

0

1

0

-1/2

3/2

2

    : x 1=2, x2=2,        Z=6.

10

  

 

4.     : max

©   

!

x1 e

4

x1 

3x2

2 x1

3 x1



2 x2

e 15

 x 2 e 10

x1 u

0,

x2 u

0

              : : Z   3 x1  2 x2 ! x1  x3 !

0

4

x1  3 x2  x4 ! 15

2 x1  x2  x5

! 10

x1 u

0

0,

x2 u

..

..

Z

x1

x2

x3

x4

x5

..

Z

0

1

-3

-2

0

0

0

0

x3

1

0

1

0

1

0

0

4

x4

2

0

1

3

0

1

0

15

x5

3

0

2

1

0

0

1

10

Z

0'

1

0

-2

3

0

0

12

x1

1'

0

1

0

1

0

0

4

x4

2'

0

0

3

-1

1

0

11

x5

3'

0

0

1

-2

0

1

2

Z

0"

1

0

0

-1

0

2

16

x1

1"

0

1

0

1

0

0

4

x4

2"

0

0

0

5

1

-3

5

x2

3"

0

0

1

-2

0

1

2

Z

0'''

1

0

0

0

0.2

1.4

17

x1

1'''

0

1

0

0

-0.2

0.6

3

x3

2'''

0

0

0

1

0.2

-0.6

1

x2

3'''

0

0

1

0

0.4

-0.2

4

    : x 1=3, x2=4,           Z=17.

11

  

 

5.     : max

2 x1

   

!

 x2

x 2 e 10

2 x1



5x 2

e

60

x1  x 2 e 18

3 x1

 x2 e

x1 u

0,

44

x2 u

0

              : : Z  

2 x1

 x2 !

0

x 2  x 3 ! 10

2 x1



5x 2

 x4 !

60

x1  x 2  x 5 ! 18

3 x1

 x2  x6 !

44

..

..

Z

x1

x2

x3

x4

x5

x6

..

Z

0

1

-2

-1

0

0

0

0

0

x3

1

0

0

1

1

0

0

0

10

x4

2

0

2

5

0

1

0

0

60

x5

3

0

1

1

0

0

1

0

18

x6

4

0

3

1

0

0

0

1

44

Z

0'

1

0

-1/3

0

0

0

2/3

88/3

x3

1'

0

0

1

1

0

0

0

10

x4

2'

0

0

13/3

0

1

0

-2/3

92/3

x5

3'

0

0

2/3

0

0

1

-1/3

10/3

x1

4'

0

1

1/3

0

0

0

1/3

44/3

Z

0"

1

0

0

0

0

1/2

1/2

31

x3

1"

0

0

0

1

0

-3/2

1/2

5

x4

2"

0

0

0

0

1

-13/2

3/2

9

x2

3"

0

0

1

0

0

3/2

-1/2

5

x1

4"

0

1

0

0

0

-1/2

1/2

13

    : x 1=13, x2=5,       Z=31.

12

  

 

6.     : max 3 x1

2 x1

   

!



5x2

 x 2 e 12

x2 e

3,5

x1 

3x2

e 12

x1  x 2 e x1 u

0,

5

x2 u

0

              : :  2 x1  5 x 2 !

   

3 x1  x2  x3 x2  x4 !

0

! 12

3,5

x1  3 x2  x5 ! 12 x1  x 2  x6 !

5

..

..

Z

x1

x2

x3

x4

x5

x6

..

Z

0

1

-2

-5

0

0

0

0

0

x3

1

0

3

1

1

0

0

0

12

x4

2

0

0

1

0

1

0

0

3.5

x5

3

0

1

3

0

0

1

0

12

x6

4

0

1

1

0

0

0

1

5

Z

0'

1

-2

0

0

5

0

0

17.5

x3

1'

0

3

0

1

-1

0

0

8.5

x2

2'

0

0

1

0

1

0

0

3.5

x5

3'

0

1

0

0

-3

1

0

1.5

x6

4'

0

1

0

0

-1

0

1

1.5

Z

0"

1

0

0

0

-1

2

0

21

x3

1"

0

0

0

1

8

-3 -3

0

4

x2

2"

0

0

1

0

1

0

0

3.5

x1

3"

0

1

0

0

-3

1

0

1.5

x6

4"

0

0

0

0

2

-1 -1

1

0

    : x 1=1,5; x2=3,5;       Z=21. (:                 x5  x6          x5    x6.

13

  

 

7.     : max

! 10 x1 

   

 x1 

2 x2

20 x 2

e 15

x1  x 2 e 12

5 x1



x1 u

3x2

0,

e

45

x2 u

0

              : :    

 10 x1  20 x2 !

0

 x1  2 x2  x3 ! 15

x1  x2  x4 ! 12

5 x1  3 x2

 x5 !

45

..

..

Z

x1

x2

x3

x4

x5

..

Z

0

1

-10

-20

0

0

0

0

x3

1

0

-1

2

1

0

0

15

x4

2

0

1

1

0

1

0

12

x5

3

0

5

3

0

0

1

45

Z

0'

1

-20

0

10

0

0

150

x2

1'

0

-1/2

1

1/2

0

0

15/2

x4

2'

0

3/2

0

-1/2

1

0

9/2

x5

3'

0

13/2

0

-3/2

0

1

45/2

Z

0"

1

0

0

3

13

0

210

x2

1"

0

0

1

1/3

1/3

0

9

x1

2"

0

1

0

-1/3

2/3

0

3

x5

3"

0

0

0

2/3

-13/3

1

3

    : x 1=3, x2=9,           Z=210.

14

  

 

 





1.      : min Z  ! 3 x1 2 x1

 x2 

2 x1



3x 2



2 x2

3x3



!

5x3

x1 , x 2 , x 3 u



4 x3

60

u 120

0

                     : 

   



3 x1



2 x1

 x2 

2 x1



3x2

2x2

3x3

4 x3



 x4 !

 M 

x 4  M  x 6 !

0

60

(0) (1)

 5 x 3  x 5  x 6 ! 120

( 2)

                    , (0)-·(1)-·(2),        :    





(3  5M ) x1



(2  4 M ) x 2



(4  8M  ) x3

 M 

x5 ! 180 M 

           .

Z

.. 0

Z -1

x1 3-5M

x2 2-4M

x3 4-8M

x4 0

x5 M

x6 0

.. -180M

x4

1

0

2

1

3

1

0

0

60

x6

2

0

3

3

5

0

-1

1

120

Z

0'

-1

0

-4/3+8M/3

M

0

-80-20M

x3

1'

0

2/3

1/3

1

1/3

0

0

20

x6

2'

0

-1/3

4/3 4/ 3

0

-5/3

-1

1

20

Z

0"

-1

1/2

0

0

-1/2+M -1/2+M

1/2

-1/2+M

-90

x3

1"

0

3/4

0

1

3/4

1/4

-1/4

15

x2

2"

0

-1/4

1

0

-5/4

-3/4

3/4

15

..

1/3+M/3 2/3-4M/3

    : x 1=0, x2=15, x3=15,       Z=90.

15

  

 

2.      : min

!

   

x1 

3 x1

2 x2



2 x2

e 12

2 x1



3x 2

2 x1

 x2 u

x1 , x 2 u

! 12

8

0

            : 

 3 x1 

   

x1 

2x2

2x2

 M 

 x 3 ! 12

3x2

0 (0) (1)

2 x1



2 x1

 x 2  x5  x 6 !

x1 , x 2 u

x 4  M  x 6 !

 x 4 ! 12

(2 ) 8

(3)

0

    (2)  (3),              , (0)-·(2)-·(3),  : 

   

 (3 

4M ) x1



(2  4M ) x 2

 M  x5 ! 20M 

0

x5 M

x6 0

.. -20M

1

0

0

0

12

3

0

1

0

0

12

2

1

0

0

-1

1

8

-1

5/3-4M/3

0

0

-2/3+4M/3

M

0

-8-4M

1'

0

-1/3

0

1

-2/3

0

0

4

x2

2'

0

2/3

1

0

1/3

0

0

4

x6

3'

0

4/3

0

0

-1/3

-1

1

4

Z

0"

-1

0

0

0

-1/4+M

5/4

-5/4+M

-13

x3

1"

0

0

0

1

-3/4

-1/4

1/4

5

x2

2"

0

0

1

0

1/2

1/2

-1/2

2

x1

3"

0

1

0

0

-1/4

-3/4

3/4

3

..

Z -1

x1 3-4M

x2 2-4M

x3 0

x4

Z

.. 0

x3

1

0

1

2

x4

2

0

2

x6

3

0

Z

0'

x3

    : x 1=3, x2=2         Z=13.

16

  

 

3.      : min

2 x2

x1 

2 x1

! 15 x1 

   



u 10

3x2

x1  x 2 u x1 u

0,

20 x 2

e

6

6 x2 u

0

            : 

 15 x1 

!   

2 x2

x1 

2 x1

20 x 2

 M 

6

x1  x 2  x 6  x 7 ! x1 u

0,

x2 u

0

10

 x3  x 4 !

 3 x 2  x5 !

x 4  M  x 7 !

6

0

                  ,     : "   





(15  2M ) x1



(20  3M  ) x2

 Mx3  Mx6 ! 16 M 

..

..

Z

x1

x2

x3

x4

x5

x6

x7

..

Z

0

-1

15-2M

20-3M

M

0

0

M

0

-16M

x4

1

0

1

2

-1

1

0

0

0

10

x5

2

0

2

-3

0

0

1

0

0

6

x7

3

0

1

1

0

0

0

-1

1

6

Z

0'

-1

5-M/2

0

0

M

0

-100-M

x2

1'

0

½

1

-1/2

1/2

0

0

0

5

x5

2'

0

7/2

0

-3/2

3/2

1

0

0

21

x7

3'

0

½

0

1/2

-1/2

0

-1

1

1

Z

0"

-1

0

0

5

-5+M

0

10

-10+M

-110

x2

1"

0

0

1

-1

1

0

1

-1

4

x5

2"

0

0

0

-5

5

1

7

-7

14

x1

3"

0

1

0

1

-1

0

-2

2

2

10-M/2 -10+3M/2

  : x 1=2, x2=4         Z=110.

17

  

 

4.      : min Z  ! 4 x1

 5 x2

3 x1

 x2 e

5 x1



5x 2

!

60

6 x1



4x2

u

60

x1 u

0,

27

x2 u

0

            :  Z  

4 x1

 5x2 

x4 

#    

3 x1

 x2  x3 !

5 x1



5x2

 x4 !

6 x1



4x2

 x5  x 6 !

x1 u

0,

x2 u

x6 !

#    

0

27 60 60

0

                ,     :  Z  

( 4  11M  ) x1 Z



(5  9 M  ) x 2

..

..

x1

Z

0

-1 4-11M

x3

1

0

x4

2

x6

 Mx 5 ! 120 M 

x2

x3

x4

x5

x6

..

5-9M

0

0

M

0

-120M - 120M

3

1

1

0

0

0

27

0

5

5

0

1

0

0

60

3

0

6

4

0

0

-1

1

60

Z

0'

-1

0

11/3-16M/3

-4/3+11M/3

0

M

0

-36-21M

x1

1'

0

1

1/3

1/3

0

0

0

9

x4

2'

0

0

10/3

-5/3

1

0

0

15

x6

3'

0

0

2

-2

0

-1

1

6

Z

0"

-1

0

0

7/3-5M/3

0

11/6-5M/3

-11/6+8M/3

-47-5M

x1

1"

0

1

0

2/3

0

1/6

-1/6

8

x4

2"

0

0

0

5/3

1

5/3

-5/3

5

x2

3"

0

0

1

-1

0

-1/2

1/2

3

Z

0'''

-1

0

0

1

-11/10+M

0

M

-105/2

x1

1'''

0

1

0

1/2

-1/10

0

0

15/2

x5

2'''

0

0

0

1

3/5

1

-1

3

x2

3'''

0

0

1

-1/2

3/10

0

0

9/2

  : x1=4,5; x2=7,5;       Z=52,5.

18

  

 

    1.      : min Z  ! 3 x1



2x2

2 x1

 x2  3x3 !

2 x1



3x2



4 x3

60

 5 x 3 u 120

x1 , x 2 , x 3 u

0

      ,        ,       : : 2 x1

 x2 

2 x1



3x2

3x 3



 x4 !

5 x3

60

 x 5  x 6 ! 120

 I:

               .                 0.        : 

$   

 x4  x6 !

0

            ,      : 

%   



4 x1



4 x2

 8 x 3  x 5 ! 180

..

..

Z

x1

x2

x3

x4

x5

x6

..

Z

0

-1

-4

-4

-8

0

1

0

-180

x4

1

0

2

1

3

1

0

0

60

x6

2

0

2

3

5

0

-1

1

120

Z

0'

-1

4/3

-4/3

0

8/3

1

0

-20

x3

1'

0

2/3

1/3

1

1/3

0

0

20

x6

2'

0

-4/3

4/3

0

-5/3

-1

1

20

Z

0"

-1

0

0

0

1

0

1

0

x3

1"

0

1

0

1

3/4

1/4

-1/4

15

x2

2"

0

-1

1

0

-5/4

-3/4

3/4

15

19

  

 

 II:

            . &   



 3 x1 

2x2



4 x3

!

0

(*)

        x 2  x3,          0,    .             x 2  x3    , (*)  2 ( 2" )  4 (1" ) ,     : '   



 x1 

1 2

x5 ! 90

Z -1

x1 1

x2 0

x3 0

x5

..

Z

.. 0

1/2

-90

x3

1

0

1

0

1

1/4

15

x2

2

0

-1

1

0

-3/4

15

..

         ,    .   : x 1=0; x2=15; x3=15;        Z=90.

20

  

 

2.      : min Z  ! 3 x1 x1 

2 x2



2 x2

e 12

2 x1



3x 2

2 x1

 x2 u 8

x1 , x 2 u

! 12

0

 I:

        :

(   



 4 x1  4 x2  x5 ! 20

..

..

Z

x1

x2

x3

x4

x5

x6

..

Z

0

-1

-4

-4

0

0

1

0

-20

x3

1

0

1

2

1

0

0

0

12

x4

2

0

2

3

0

1

0

0

12

x6

3

0

2

1

0

0

-1

1

8

Z

0'

-1

0

-2

0

0

-1

2

-4

x3

1'

0

0

3/2

1

0

1/2

-1/2

8

x4

2'

0

0

2

0

1

1

-1

4

x1

3'

0

1

1/2

0

0

-1/2

1/2

4

Z

0"

-1

0

0

0

1

0

1

0

x3

1"

0

0

0

1

-3/4

-1/4

1/4

5

x2

2"

0

0

1

0

1/2

1/2

-1/2

2

x1

3"

0

1

0

0

-1/4

-3/4

3/4

3

 II:

 (0)       (*)     : (*)  3 (3" )  2 (2" )

..

Z x3 x2 x1

.. 0 1 2 3

Z -1 0 0 0

x1 0 0 0 1

x2 0 0 1 0

x3 0 1 0 0

x5 5/4 -1/4 1/2 -3/4

.. -13 5 2 3

  : x 1=3; x2=2;        Z=13.

21

  

 

3.      : min Z  ! 15 x1 x1 

2 x1

2 x2

20 x 2

u 10

 3x2 e

x1  x 2 u x1 u



0,

6

6 x2 u

0

 I:

      :

 Z  

2 x1

 3 x 2  x3  x 6 ! 16

..

..

Z

x1

x2

x3

x4

x5

x6

x7

..

Z

0

-1

-2

-3

1

0

0

1

0

-16

x4

1

0

1

2

-1

1

0

0

0

10

x5

2

0

2

-3

0

0

1

0

0

6

x7

3

0

1

1

0

0

0

-1

1

6

Z

0'

-1

-1/2

0

-1/2

3/2

0

1

0

-1

x2

1'

0

1/2

1

-1/2

1/2

0

0

0

5

x5

2'

0

7/2

0

-3/2

3/2

1

0

0

21

x7

3'

0

1/2

0

1/2

-1/2

0

-1

1

1

Z

0"

-1

0

0

0

1

0

0

1

0

x2

1"

0

0

1

-1

1

0

1

-1

4

x5

2"

0

0

0

-5

5

1

7

-7

14

x1

3"

0

1

0

1

-1

0

-2

2

2

 II:

 (0)       (*)     : (*)  15 (3" )  20 (1" ) ..

Z

.. 0

Z -1

x1 0

x2 0

x3 5

x5 0

x6 10

.. -110

x2

1

0

0

1

-1

0

1

4

x5

2

0

0

0

-5

1

7

14

x1

3

0

1

0

1

-2

2

0

  : x 1=2, x2=4         Z=110.

22

  

 

4.      : min ! 4 x1  5 x 2 0   

3 x1

 x2 e

5 x1



5x 2

!

60

6 x1



4x2

u

60

x1 u

0,

27

x2 u

0

 I:         :



 11x1  9 x 2  x 5 ! 120

)   

..

..

Z

x1

x2

x3

x4

x5

x6

..

Z

0

-1

-11

-9

0

0

1

0

-120

x3

1

0

3

1

1

0

0

0

27

x4

2

0

5

5

0

1

0

0

60

x6

3

0

6

4

0

0

-1

1

60

Z

0'

-1

0

-16/3

11/3

0

1

0

-21

x1

1'

0

1

1/3

1/3

0

0

0

9

x4

2'

0

0

10/3

-5/3

1

0

0

15

x6

3'

0

0

2

-2

0

-1

1

6

Z

0"

-1

0

0

-5/3

0

-5/3

8/3

-5

x1

1"

0

1

0

2/3

0

1/6

-1/6

8

x4

2"

0

0

0

5/3

1

5/3

-5/3

5

x2

3"

0

0

1

-1

0

-1/2

1/2

3

Z

0

-1

0

0

0

1

0

1

0

x1

1

0

1

0

0

-2/5

-1/2

1/2

6

x3

2

0

0

0

1

3/5

1

-1

3

x2

3

0

0

1

0

3/5

1/2

-1/2

6

 II:  (0)  : (*)  4 (1" )  5 (3" ) ..

..

Z

x1

x2

x3

x5

..

Z

0

-1

0

0

0

-1/2

-54

x1

1

0

1

0

0

-1/2

6

x3

2

0

0

0

1

1

3

x2

3

0

0

1

0

1/2

6

Z

0

-1

0

0

1/2

0

-105/2

x1

1

0

1

0

1/2

0

15/2

x3

2

0

0

0

1

1

3

x2

3

0

0

1

-1/2

0

9/2

  : x 1=4,5; x2=7,5;          Z=52,5.

23

  

 

      1.  4        3 .    ,            : 

1

        

1 2 3 

2

3

4

8

7

5

2

5

2

1

3

6

4

3

5

40

60

20

60

 60 80 40 180

        ( ),          .                   ,     . min( 40,60)

!

40,

x11 !

40

min( 20,60)

!

20,

x12 !

20

min( 40,80) ! 40,

x 22 !

40

min( 40,20)

!

20,

x 23 !

20

min( 20,60)

!

20,

x 24 !

20

min( 40,40)

!

40,

x34 !

40

         

1 1 2 3 

2

8

7

40 5

3 5

2

1

3

 60

20 2

40 6

4

4

20 3

20 5

40 40

60

20

60

80 40 180

      : Z  !

40 8  20 7  40 2  20 1  20 3  40 5 ! 820..

24

  

 

    1.                 . 

1

        

1 2 3 

2

3

4

8

7

5

2

5

2

1

3

6

4

3

5

40

60

20

60

 60 80 40 180

                            .                   . min c 23

!

40

min( 20,80) ! 20,

x 23 !

20

min c14

!

60

min(60,60) ! 60,

x14 !

60

min c 22

!

40

min(60,60) ! 60,

x 22 !

60

min c 31

!

40

min( 40, 40)

x31 !

40

!

20,

         

1 1 2 3 

2

8

3

7

4

5



2

60 5

2

1

60 6

3

20

4

3

5

60

20

60

40 40

60 80 40 180

      : Z  !

60 2  20 1  60 2  40 6 ! 500..

25

  

 

   1.         . 

1

        

1 2 3 

2

3

4

8

7

5

2

5

2

1

3

6

4

3

5

40

60

20

60

 60 80 40 180

          ,       .             .               .          

1 1 2 3

8

3

7

4

5

 

2

60 5

2

1

3

60 6

20

4

3

5

60

20

60

40

 40 

2

1

2

2

60

3

80

1

40

1

180 1

      : Z  !

60 2  20 1  60 2  40 6 ! 500..

26

  

 

   STEPPING STONE  1.        ,           : 

1

        

1 2 3 

2

8

3

7

40

4

5

2

1

3

 60

20

5

2

40 6

4

20 3

80

20 5

40

40 40

60

20

60

180

(1,1)

(1,2)

(1,2)

(1,3)

(2,2)

(2,4)

-

+

-

+

-

+

+

-

+

-

+

-

(2,1)

(2,2)

(2,2)

(2,3)

(3,2)

(3,4)

d 21 ! C 21  C 11  C 12  C 22 d 21 !

58 7  2 ! 2

d 13 ! C 13  C 23  C 22  C 12 d 13 !

d 32 !

5  1  2  7 ! 1

(2,3)

(2,4)

(1,2)

(1,4)

-

+

-

+

+

-

+

-

(3,3)

(3,4)

(2,2)

(2,4)

d 33 ! C 33  C 23  C 24  C 34 d 33 !

3 11 3 ! 0

d 32 ! C 32  C 22  C 24  C 34

4  2 1 3 ! 0

d 14 ! C 14  C 24  C 22  C 12 d 14 !

23 27

! 6

               :

27

  

 

         

1 1 2 3 

2

8

3

7

4

5

2

40 5

20 2

1

3

60 6



4

80

20 3

5

40 40

60

20

60

60

40 180

           : Z  !

40 8  20 2  60 2  20 1  40 5 ! 700..

     : d 31 ! C 31  C 11  C 14  C 34 d 31 !

6  8  2  5 ! 5          

1 1 2 3 

2

8

3

7

4

5



2

60 5

2

1

60 6

3

20

4

3

5

60

20

60

40 40

60 80 40 180

               : Z  !

60 2  20 1  60 2  40 6 ! 500..

28

  

 

    2.      : 

1         

1 2 3 4

2

3

4

5

6

4

2

2



1

3

3

4

2

1

2

1

3

2



20

10

10

 10 20 20 10

20

60

           : 

1         

1 2 3 4

2

5

3

4

6

4

2



1

3

2

1

 10

10 2

10 3

20

10 4

10 2

1

3

10 2

10



20

10

10

20

20 10 60

      : Z  ! 10

5  10 2  10

 10

2  10 1  10 2 ! (120  10

)..

29

  

 

    2.                 . 

1         

1 2 3 4

2

3

4

5

6

4

2

2



1

3

3

4

2

1

2

1

3

2



20

10

10

 10 20 20 10

20

60

           : 

1         

1 2 3 4 

2

5

3

4

6

4

2



1

3

 10

10 2

10 3

20

10 4

2

1

20 2

1

3

2

10

10 20

10

10

20

20

60

      : Z  ! 10

5  10 2  10 1  10 1  20 1 ! 110 ..

30

  

 

   2.         . 

1         

1 2 3 4

2

3

4

5

6

4

2

2



1

3

3

4

2

1

2

1

3

2



20

10

10

 10 20 20 10

20

60

          : 

1         

1 2 3 4 

2

5

3

6

4

4



2

10 2

1



3

10 3

20

10 4

2

1

10 2

10 1

3

2

10

10

20 10

10 20

10

20

60

      : Z  ! 10

2  10 2  10 1  10 3  10 1  10 1 ! 100..

31

  

 

STEPPING STONE  2.        ,           : 

1         

1 2 3 4 

2

5

3

6

4

4



2

10

10 2

1



3

10 3

20

10 4

2

1

10 2

20

10 1

3

2

10

10 20

10

10

20

60

(2,1)

(3,2)

(2,1)

(2,4)

(1,1)

(1,4)

+

-

-

+

+

-

-

+

+

-

-

+

(3,1)

(3,3)

(3,1)

(3,4)

(3,1)

(3,4)

d 33 ! C 33  C 31  C 21  C 32 d 33 !

2  3  2 1 ! 0

d 24 ! C 24  C 34  C 31  C 21

d 11 ! C 11  C 14  C 34  C 31

d 24 !

d 32 !

3 1  3  2 ! 3

5  2 1 3 ! 1

               : Z  ! 100 ..

32