IEOPER Quiz 2 Procedures

Question 1

Simplex Method

Answer 1

1. Set all variables to the left side and the constants to the right (convert to standard form)
2. Add slack variables to constraints
3. Get n-m variables and set (n-m) variables to 0
4. Reflect the initial feasible solution in the simplex tableau
5. Indicate the EV by locating in the z-row the most negative coefficient if it is a maximization problem and the most positive coefficient if it is a minimization problem.
6. Indicate the LV row by getting the ratio of a solution divided by the pivot column.
7. Replace the LV with the EV in the tableau, then get the intersection as this becomes the pivot element.
8. Divide the pivot row by the pivot element
9. Make all other coefficients in the pivot column zero using the Gauss-Jordan technique.
10. Repeat the process until optimality, wherein all the coefficients of the nonbasic variables are non-negative for the maximization problem and non-positive for the minimization problem.

Question 2

Name all submatrices and their meaning

Answer 2

XB = Current basic variables
XN = Current Non-basic variables
CB = Objective function of the coefficient of current BVs
CN = Objective function of the coefficient of current NBVs
B = Constraint coefficient of current BVs
N = Constraint coefficient of current NBVs

Question 3

Name the original matrices and their respective sub matrices

Answer 3

X - XB, XN
C - CB, CN
A - B, N

Question 4

Formula for constraints for revised simplex

Answer 4

AX = b

Derived version:
I XB + B^-1 N XN = B^-1 b

Question 5

Formula for objective function for revised simplex

Answer 5

Z = CX

Derived version:
Z + (CB B^-1 - CN) XN = CB B^-1 b

Question 6

Formula for Basic variable values

Answer 6

XB = B^-1 b ; basically the constraints equation but let XN = 0

Question 7

Formula for Objective function value

Answer 7

Z = CB B^-1 b; basically the objective function equation but let XN = 0.

Question 8

Optimality Test for Revised Simplex

Answer 8

Max: CB B^-1N - CN >= 0
Min: CB B^-1 N - CN <= 0

Question 9

Ratio formula for determining LV in revised simplex

Answer 9

B^-1 b / B^-1 Nj; then get the lowest number or the least ratio

Question 10

The formula for determining EV in revised simplex

Answer 10

CB B^-1 N - CN

For max problem, get the most negative coefficient.

For min problem, get the most positive coefficient

Question 11

Steps for Big M:

Answer 11

1. Transform the constraints into standard form. Add artificial variables (R) to constraints with >+ or = signs.
2. Indicate n (including both Sn and Rn variables) and m and set (n-m) variables into NBV. Choose the negative Sn variables as NBV.
3. Add MRi to the untransposed objective function if its a minimization problem. Subtract MRi to the untransposed objective function if its a maximization problem. Transform objective function into standard form.
4. Isolate Rn from constraints and substitute into standard form of objective function.
5. Simplify the objective function
6. Perform simplex method

Question 12

Steps for Two-Phase Method:

Answer 12

1. Transform given into standard form
2. Do Phase 1 (Min R0 = R1+R2)
3. Do Simplex until you get optimal solution
4. Do Phase 2 (Use Original objective function)
5. Iterate until optimal solution