Choosing pivot column on simplex method when there are same values

brainyprb

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In case of minimization:

  1. Which column should be chosen as key column if there is multiple occurrence of same value? (i.e. same highest positive value occurs in more than one columns)
  2. When there are two values to be changed on same row to obtain an identity matrix, the first iteration changes one particular value and the other remains unchanged, how to change the other value also?
Examples (for both queries 1 and 2):
C x y z S1 S2 S3 A1 A2 Constant Ratio
R0 1 -5 -4 -3 0 0 0 -10 -10 0 -
R1 0 1 0 0 1 0 0 0 0 200 -
R2 0 0 1 0 0 0 -1 1 0 300 300
R3 0 0 0 1 0 1 0 0 0 400 -
R4 0 1 1 1 0 0 0 0 1 1000 1000
Table details:
Objective function:
Minimize C = 5x + 4y + 3z subjected to the following constraints

x <= 200
y >= 300
z <= 400
x + y + z = 1000
x, y, z >= 0
And,
x, y, z are no. of products.
S1, S2 are Slack variables and S3 is surplus.
A1 and A2 are artificial variables.
For query 1 I have to select a column with highest positive value. So 1 is the highest positive value but it occurs in multiple columns. Which column should I choose as key column?
For query 2, variables A1 and A2 in R0 have value of -10 which should be changed to zero to obtain an identity matrix. After selecting a key column how to change both the values to zero for identity matrix ?
 
In case of minimization:

  1. Which column should be chosen as key column if there is multiple occurrence of same value? (i.e. same highest positive value occurs in more than one columns)
  2. When there are two values to be changed on same row to obtain an identity matrix, the first iteration changes one particular value and the other remains unchanged, how to change the other value also?
Examples (for both queries 1 and 2):
C x y z S1 S2 S3 A1 A2 Constant Ratio
R0 1 -5 -4 -3 0 0 0 -10 -10 0 -
R1 0 1 0 0 1 0 0 0 0 200 -
R2 0 0 1 0 0 0 -1 1 0 300 300
R3 0 0 0 1 0 1 0 0 0 400 -
R4 0 1 1 1 0 0 0 0 1 1000 1000
Table details:
Objective function:
Minimize C = 5x + 4y + 3z subjected to the following constraints

x <= 200
y >= 300
z <= 400
x + y + z = 1000
x, y, z >= 0
And,
x, y, z are no. of products.
S1, S2 are Slack variables and S3 is surplus.
A1 and A2 are artificial variables.
For query 1 I have to select a column with highest positive value. So 1 is the highest positive value but it occurs in multiple columns. Which column should I choose as key column?
For query 2, variables A1 and A2 in R0 have value of -10 which should be changed to zero to obtain an identity matrix. After selecting a key column how to change both the values to zero for identity matrix ?

Personally, if there were two identical pivot possibilities, I would do it both ways and see if it makes a difference in this case. Then, you can theorize the complete answer.

Alternatively, I would check the setup for redundancy or inconsistency.
 
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