Presentation on theme: "3.4 Artificial starting solution Page103"— Presentation transcript:
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-MethodThe 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.
7 Two-Phase MethodWhen 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 109Consider the same problem we used in the M-method to explain the phases of the Two-Phase Method:Minimize z = 4x1 + x2Subject to: x1 + x2 = 34x1 + 3x2 ≥ 6x1 + 2x2 ≤ 4x1,x2 ≥ 0
10 Example…continuedUsing 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 111Solve 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