l07_artificial variable techniques - two phase method(continued).ppt

7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt

1/15

Minimize21

75 xxz +=

Subject tothe constraints

0,

18

6043

4232

21

21

21

21

+

+

+

xx

xx

xx

xx

We shall solve this roblem b! t"o hase

metho#$


2/15

%hase &'

Minimize

Subject to the constraints

1 2 1 1

1 2 2 2

1 2 3 3

1 2 1 2 3 1 2 3

2 3 42

3 4 60

18

, , , , , , , 0

x x s R

x x s R

x x s R

x x s s s R R R

+ + =

+ + =

+ + =

(ere s1,s2,s3are surlus variables) R1,R2,R3

are arti*icial variables$

1 2 3r R R R= + +


3/15

r 1 0 0 0 0 0 +1 +1 +1 0

asic r -1 -2 s1 s2 s3 .1 .2 .3 Sol

6 8 +1 +1 +1 0 0 0 120

.1 0 2 3 +1 0 0 1 0 0 42

.2 0 3 4 0 +1 0 0 1 0 60

.3 0 1 1 0 0 +1 0 0 1 18

-2 0 2/3 1 +1/3 0 0 1/3 0 0 14

r 1 2/3 0 5/3 +1 +1 +8/3 0 0 8

.2 0 1/3 0 4/3 +1 0 +4/3 1 0 4

.3 0 1/3 0 1/3 0 +1 +1/3 0 1 4


4/15

r 1 2/3 0 5/3 +1 +1 +8/3 0 0 8

asic r -1 -2 s1 s2 s3 .1 .2 .3 Sol

-2 0 2/3 1 +1/3 0 0 1/3 0 0 14

.2 0 1/3 0 4/3 +1 0 +4/3 1 0 4

.3 0 1/3 0 1/3 0 +1 +1/3 0 1 4

-2 0 3/4 1 0 +1/4 0 0 1/4 0 15

r 1 1/4 0 0 1/4 +1 +1 +5/4 0 3

s1 0 1/4 0 1 +3/4 0 +1 3/4 0 3

.3 0 1/4 0 0 1/4 +1 0 +1/4 1 3


5/15

r 1 1/4 0 0 1/4 +1 +1 +5/4 0 3

asic r -1 -2 s1 s2 s3 .1 .2 .3 Sol

-2 0 3/4 1 0 +1/4 0 0 1/4 0 15

s1 0 1/4 0 1 +3/4 0 +1 3/4 0 3

.3 0 1/4 0 0 1/4 +1 0 +1/4 1 3

-2 0 0 1 0 +1 3 0 1 +3 6

r 1 0 0 0 0 0 +1 +1 +1 0

s1 0 0 0 1 +1 1 +1 1 +1 0

-1 0 1 0 0 1 +4 0 +1 4 12


6/15

his is otimal tableau$ hus %hase & has

en#e# an# "e no" start %hase && "ith the

startin S as the otimal solution o*

%hase &$

%hase &&'

Minimize 21 75 xxz +=

subject to the same constraints as iven

in the oriinal roblem$


7/15

z 1 +5 +7 0 0 0 0

asic z -1 -2 s1 s2 s3 .1 .2 .3 Sol

0 0 0 +2 1 102

-2 0 0 1 0 +1 3 6s1 0 0 0 1 +1 1 0

-1 0 1 0 0 1 +4 12

-2 0 0 1 +3 2 0 6

z 1 0 0 +1 +1 0 102

s3 0 0 0 1 +1 1 0

-1 0 1 0 4 +3 0 12


8/15

his is the otimal tableau$

timal solution' x1 12, x2 6

timal value ' z 102


9/15

onsi#er the %%'

Minimize 321 352 xxxz ++=Subject to the constraints

0,,

5042202

321

321

321

=++

+

xxx

xxx

xxx


10/15

%hase &'

Minimize

Subject to the constraints

1 2 3 1 1

1 2 3 2

1 2 3 1 1 2

2 20

2 4 50

, , , , , 0

x x x s R

x x x R

x x x s R R

+ + =

+ + + =

(ere s1 is a surlus variable) R1,R2 are

arti*icial variables$

1 2r R R= +


11/15

r 1 0 0 0 0 +1 +1 0

asic r -1 -2 -3 s1 .1 .2 Sol

3 2 2 +1 0 0 70

.1 0 1 +2 1 +1 1 0 20

.2 0 2 4 1 0 0 1 50

-1 0 1 +2 1 +1 1 0 20

r 1 0 8 +1 2 +3 0 10

.2 0 0 8 +1 2 +2 1 10


12/15

r 1 0 8 +1 2 +3 0 10

asic r -1 -2 -3 s1 .1 .2 Sol

-1 0 1 +2 1 +1 1 0 20

.2 0 0 8 +1 2 +2 1 10

-1 0 1 0 3/4 +1/2 1/2 1/4 45/2

r 1 0 0 0 0 +1 +1 0

-2 0 0 1 +1/8 1/4 +1/4 1/8 5/4


13/15


14/15

z 1 +2 +5 +3 0 0

asic z -1 -2 -3 s1 .1 .2 Sol

-1 0 1 0 3/4 +1/2 45/2

-2 0 0 1 +1/8 1/4 5/4

-1 0 1 2 1/2 0 25

z 1 0 +1 +2 0 50

s1 0 0 4 +1/2 1 5

0 0 +17/8 1/4 205/4

hus the otimal solution is ' x1 25, x2 0, x3 0

n# the otimal z Min z 50$


15/15

Minimize321

32 xxxz ++=

Subject tothe constraints

0,,

623

824

321

21

321

+

++

xxx

xx

xxx

ns' timal Solution' x1 4/5, x2 /5, x3 0

timal value' z 7

l07_artificial variable techniques - two phase method(continued).ppt

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