1. 3.4 Artificial starting solution Page103
Dr. Ayham Jaaron

2. Artificial starting solution
Simplex method offers a convenient starting basic feasible solution when constraints are all ≤ (i.e. Slacks).
What about (≥) or (=).
In such case (called ill-behaved LP) we need to add artificial slack variables that play the role of slacks at the first iteration!!
Remember: standard LP must contain equalities with slack variables.

3. M-Method
The M-method states that if constraint equation does not have a slack variable, an artificial variable, R, is added to form a starting solution similar to the all-slack basic solution.
Even if surplus is provided: you still need to add a slack variable to make it a standard LP form (remember: simplex method needs standard LP).
To understand this procedure, let us consider the following problem.

4. But first..Rule to remember..

5. Problem Minimize z = 4x1 + x2 Subject to: 3x1 + x2 = 3 4x1 + 3x2 ≥ 6

6. Problem: Q5 (a) Page 107

7. Two-Phase Method
When using computer for software solvers, M is a very large value that is difficult to program.
Also, it is hard to decide how big is M.
Therefore, to eliminate this difficulty of deciding about the value of M, the Two=Phase Method has been introduced.

8. Two-Phase method..continued
It solves the LP model in two phases:
Phase I:
Put the problem in a standard LP form.
Find a basic solution of the resulting equations that minimize the sum of the artificial variables (i.e. Minimize r= R1+R2)
Check feasibility. If r = positive value: no feasible solution. Otherwise go to phase II.
Phase II is used to solve the original problem by deleting the artificial variables first.

9. Example...Page 109
Consider the same problem we used in the M-method to explain the phases of the Two-Phase Method:
Minimize z = 4x1 + x2
Subject to: x1 + x2 = 3
4x1 + 3x2 ≥ 6
x1 + 2x2 ≤ 4
x1,x2 ≥ 0

10. Example…continued
Using r-row has helped us to solve Phase I of the problem. The optimal Tableau of this stage is as follows:

11. Problem 4: page 111
Solve the following LP model using the Two-Phase Method

12. Problem 4:Optimal Table after simplex iterations…

13. Special case: when Artificial variables are basic at end of Phase I

14. Problem: page 112