of 15
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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#$
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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%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= + +
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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$
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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his is the otimal tableau$
timal solution' x1 12, x2 6
timal value ' z 102
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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onsi#er the %%'
Minimize 321 352 xxxz ++=Subject to the constraints
0,,
5042202
321
321
321
=++
+
xxx
xxx
xxx
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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%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= +
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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$
7/24/2019 L07_Artificial Variable techniques - Two Phase Method(Continued).ppt
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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