Two-phase simplex method

Definition

The two-phase simplex method is a technique used to solve linear programming problems that may be infeasible in the initial setup. It breaks down the optimization process into two phases: the first phase focuses on finding a feasible solution, while the second phase optimizes the objective function. This method is particularly important when dealing with special cases like infeasibility, degeneracy, and unboundedness, as it provides a systematic approach to navigate these challenges.

5 Must Know Facts For Your Next Test

1. The first phase of the two-phase simplex method aims to minimize the sum of artificial variables to find a feasible region for the problem.
2. If the minimum value of artificial variables is zero after Phase 1, it indicates that a feasible solution has been found for the original problem.
3. In Phase 2, the original objective function is optimized using the feasible solution obtained from Phase 1.
4. The two-phase method can handle cases where initial solutions are not obvious or when artificial variables are necessary due to infeasibility.
5. This method helps prevent cycling in degeneracy cases by ensuring a systematic approach through the phases.

Review Questions

• How does the two-phase simplex method address infeasibility in linear programming problems?
  • The two-phase simplex method addresses infeasibility by first introducing artificial variables and focusing on finding a basic feasible solution in Phase 1. By minimizing the sum of these artificial variables, the method checks if a feasible region exists. If the minimum value reaches zero, it confirms that an actual feasible solution exists for the original constraints, allowing for transition into Phase 2 where optimization occurs.
• Discuss how the two-phase simplex method differentiates from the traditional simplex method when dealing with degenerate solutions.
  • The two-phase simplex method provides a structured approach for dealing with degenerate solutions by separating the process into two distinct phases. In Phase 1, it minimizes artificial variables, which helps avoid complications caused by cycling. By establishing a feasible solution before entering Phase 2 for optimization, it reduces the likelihood of encountering multiple optimal solutions that can complicate traditional simplex processes.
• Evaluate the effectiveness of the two-phase simplex method in solving linear programming problems with unboundedness issues.
  • The effectiveness of the two-phase simplex method in tackling unboundedness lies in its ability to identify whether a feasible region extends infinitely. By first establishing feasibility in Phase 1 and then optimizing in Phase 2, it can clearly determine if no upper bound exists for the objective function. This structured process allows for precise diagnosis of unboundedness without losing sight of potential feasible solutions that may exist at extreme points.

Related terms

Artificial Variables: Extra variables added to a linear programming problem to facilitate finding an initial basic feasible solution when none is readily available.

Basic Feasible Solution:

A solution to a linear programming problem that satisfies all constraints and has a corresponding basis in the tableau.

Degeneracy:

A situation in linear programming where multiple optimal solutions exist, often resulting in cycling during the simplex method.