Revised Simplex Method Notes | EduRev

: Revised Simplex Method Notes | EduRev

 Page 1


Revised Simplex Method 09/23/04 page 1 of  22  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (includes both Phases I  &  II) 
 
 
 
 
© Dennis Bricker 
Dept of Mechanical & Industrial Engineering 
The  University of Iowa
Page 2


Revised Simplex Method 09/23/04 page 1 of  22  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (includes both Phases I  &  II) 
 
 
 
 
© Dennis Bricker 
Dept of Mechanical & Industrial Engineering 
The  University of Iowa
Revised Simplex Method 09/23/04 page 2 of  22  
 
 
1 2 34 56
Minimize z=3x 5 4 7 5 4 xx x x x ++ + + + 
subject to 
 
12 4 6
13 4 5 6
23 4 5
2310
33212
42 3 15
xx x x
xx x x x
xx x x
-+ + =
?
?
+- + + =
?
?
++ + =
?
 
 and  0   1, 6
j
x j =?= … 
 
Because of the lack of a slack variable in each constraint, we must use 
Phase I to find an initial feasible basis. 
 
Add variables X
9
, X
10
, X
11
  (artificial variables), and  
a Phase I objective of minimizing the sum of these three variables. 
 
Page 3


Revised Simplex Method 09/23/04 page 1 of  22  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (includes both Phases I  &  II) 
 
 
 
 
© Dennis Bricker 
Dept of Mechanical & Industrial Engineering 
The  University of Iowa
Revised Simplex Method 09/23/04 page 2 of  22  
 
 
1 2 34 56
Minimize z=3x 5 4 7 5 4 xx x x x ++ + + + 
subject to 
 
12 4 6
13 4 5 6
23 4 5
2310
33212
42 3 15
xx x x
xx x x x
xx x x
-+ + =
?
?
+- + + =
?
?
++ + =
?
 
 and  0   1, 6
j
x j =?= … 
 
Because of the lack of a slack variable in each constraint, we must use 
Phase I to find an initial feasible basis. 
 
Add variables X
9
, X
10
, X
11
  (artificial variables), and  
a Phase I objective of minimizing the sum of these three variables. 
 
Revised Simplex Method 09/23/04 page 3 of  22  
                                  Phase One 
   
 1  2 3  4 5 6  7  8 9 0 1  b  
 0  0 0  0 0 0  0  0 1 1 1  0    phase one objective  
 3  5 4  7 5 4  0  0 0 0 0  0    phase two objective 
 2 -1 0  1 0 3  0  0 1 0 0 10  
 1  0 3 -1 3 2 -1  0 0 1 0 12  
 0  4 2  3 1 0  0 -1 0 0 1 15  
 
Values of basic (artificial) variables are:  
 
   i   Xi    
   9   10   
  10   12   
  11   15   
 
Page 4


Revised Simplex Method 09/23/04 page 1 of  22  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (includes both Phases I  &  II) 
 
 
 
 
© Dennis Bricker 
Dept of Mechanical & Industrial Engineering 
The  University of Iowa
Revised Simplex Method 09/23/04 page 2 of  22  
 
 
1 2 34 56
Minimize z=3x 5 4 7 5 4 xx x x x ++ + + + 
subject to 
 
12 4 6
13 4 5 6
23 4 5
2310
33212
42 3 15
xx x x
xx x x x
xx x x
-+ + =
?
?
+- + + =
?
?
++ + =
?
 
 and  0   1, 6
j
x j =?= … 
 
Because of the lack of a slack variable in each constraint, we must use 
Phase I to find an initial feasible basis. 
 
Add variables X
9
, X
10
, X
11
  (artificial variables), and  
a Phase I objective of minimizing the sum of these three variables. 
 
Revised Simplex Method 09/23/04 page 3 of  22  
                                  Phase One 
   
 1  2 3  4 5 6  7  8 9 0 1  b  
 0  0 0  0 0 0  0  0 1 1 1  0    phase one objective  
 3  5 4  7 5 4  0  0 0 0 0  0    phase two objective 
 2 -1 0  1 0 3  0  0 1 0 0 10  
 1  0 3 -1 3 2 -1  0 0 1 0 12  
 0  4 2  3 1 0  0 -1 0 0 1 15  
 
Values of basic (artificial) variables are:  
 
   i   Xi    
   9   10   
  10   12   
  11   15   
 
Revised Simplex Method 09/23/04 page 4 of  22  
Page 5


Revised Simplex Method 09/23/04 page 1 of  22  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (includes both Phases I  &  II) 
 
 
 
 
© Dennis Bricker 
Dept of Mechanical & Industrial Engineering 
The  University of Iowa
Revised Simplex Method 09/23/04 page 2 of  22  
 
 
1 2 34 56
Minimize z=3x 5 4 7 5 4 xx x x x ++ + + + 
subject to 
 
12 4 6
13 4 5 6
23 4 5
2310
33212
42 3 15
xx x x
xx x x x
xx x x
-+ + =
?
?
+- + + =
?
?
++ + =
?
 
 and  0   1, 6
j
x j =?= … 
 
Because of the lack of a slack variable in each constraint, we must use 
Phase I to find an initial feasible basis. 
 
Add variables X
9
, X
10
, X
11
  (artificial variables), and  
a Phase I objective of minimizing the sum of these three variables. 
 
Revised Simplex Method 09/23/04 page 3 of  22  
                                  Phase One 
   
 1  2 3  4 5 6  7  8 9 0 1  b  
 0  0 0  0 0 0  0  0 1 1 1  0    phase one objective  
 3  5 4  7 5 4  0  0 0 0 0  0    phase two objective 
 2 -1 0  1 0 3  0  0 1 0 0 10  
 1  0 3 -1 3 2 -1  0 0 1 0 12  
 0  4 2  3 1 0  0 -1 0 0 1 15  
 
Values of basic (artificial) variables are:  
 
   i   Xi    
   9   10   
  10   12   
  11   15   
 
Revised Simplex Method 09/23/04 page 4 of  22  Revised Simplex Method 09/23/04 page 5 of  22  
  Iteration 1 
  
Current partition: (B = basis, N = non-basis) 
  
     B= {9 10 11}, N= {1 2 3 4 5 6 7 8}   
                   
Basis inverse is   
 
           1 0 0  
           0 1 0  
           0 0 1  
  
Simplex multipliers (dual solution): 
  
  i    p                
  1    1   
  2    1   
  3    1   
 
() []
1
10 0
1,1,1 0 1 0 [ 1 ,1 ,1]
00 1
B
B
cA
-
??
??
?p= = =
??
??
? ?
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