[Paper Review] New artificial-free phase 1 simplex method

Nasiruddin Khan, Syed Inayatullah+2 more|arXiv (Cornell University)|Apr 8, 2013

Advanced Optimization Algorithms Research11 references3 citations

TL;DR

This paper introduces an artificial-free phase 1 simplex method that eliminates the need for artificial variables and constraints by allowing negative variables in the basis, maintaining the same sequence of corner points as the traditional method. The approach avoids stalling and reduces degenerate pivots, improving efficiency in solving linear programming problems without introducing auxiliary variables.

ABSTRACT

This paper presents new and easy to use versions of primal and dual phase 1 processes which obviate the role of artificial variables and constraints by allowing negative variables into the basis. During the process new method visits the same sequence of corner points as the traditional phase 1 does. The new method is artificial free so, it also avoids stalling and saves degenerate pivots in many cases of linear programming problems.

Motivation & Objective

• To eliminate the use of artificial variables and constraints in phase 1 of the simplex method.
• To develop a primal and dual phase 1 process that is simpler and more intuitive to apply.
• To avoid stalling and reduce degenerate pivots in linear programming problems.
• To maintain the same sequence of corner points as the traditional phase 1 method while improving computational behavior.
• To provide a more efficient and practical alternative to standard phase 1 in the simplex algorithm.

Proposed method

• The method allows negative variables to enter the basis during phase 1, bypassing the need for artificial variables.
• It uses a modified pricing rule that enables entering variables even when they are negative, maintaining feasibility.
• The algorithm follows the same path of corner points as the traditional phase 1, ensuring correctness.
• The dual phase 1 process is adapted to handle infeasibility detection without artificial constraints.
• The method leverages standard simplex pivot rules but modifies the basis entry condition to allow negative variables.
• It maintains dual or primal feasibility throughout the process by carefully selecting entering and leaving variables.

Experimental results

Research questions

• RQ1 Can phase 1 of the simplex method be performed without artificial variables or constraints?
• RQ2 How can negative variables be incorporated into the basis to eliminate artificial variables?
• RQ3 Does the new method preserve the same sequence of corner points as the traditional phase 1?
• RQ4 Can the method reduce stalling and degenerate pivots in practice?
• RQ5 Is the proposed method simpler and more efficient than existing phase 1 approaches?

Key findings

• The proposed method successfully eliminates artificial variables and constraints, achieving an artificial-free phase 1 process.
• The algorithm visits the same sequence of corner points as the traditional phase 1, ensuring correctness and consistency.
• By allowing negative variables in the basis, the method avoids stalling and reduces the number of degenerate pivots in many cases.
• The approach maintains primal or dual feasibility throughout the phase 1 process without auxiliary constraints.
• The method is simpler to implement and apply compared to standard phase 1 with artificial variables.
• The computational behavior is improved, particularly in problems prone to degeneracy or cycling.