Solving Linear Programming Problems Using the Dual Simplex Method When Primal Feasibility Is Unattainable

Introduction

When working with linear programming (LP) problems, it is sometimes necessary to employ the dual simplex method, especially in cases where the initial feasible basis for the primal method is not directly available or feasible. This article will explore how to apply the dual simplex method effectively and what steps are involved when the dual feasibility condition is not initially satisfied.

Practical Applications of the Dual Simplex Method

Most advanced LP solvers, such as CPLEX, XPRESS, Gurobi, and MOSEK, automatically handle dual feasibility conditions. These solvers allow you to input your model and solve the problem directly without the need for a manual phase I or phase II procedure. This is particularly useful in large-scale and complex optimization problems where manual intervention is impractical.

Nevertheless, understanding the underlying concepts is crucial for effective problem-solving. The dual simplex method operates similarly to phase I of the primal simplex method but is applied to the dual problem implicitly. This method helps in resolving situations where the initial starting basis is not feasible but still allows for efficient and accurate solutions.

Dual Simplex Method: The Conceptual Framework

The dual simplex method is a powerful technique in the realm of linear programming. It is designed to efficiently handle cases where the initial feasible solution to the dual problem is not readily available. The method involves constructing an auxiliary problem (an artificial problem) designed to reduce the infeasibility of the dual solution as much as possible. This auxiliary problem has a known feasible starting solution and aims to minimize the sum of the absolute values of the violations of the original problem.

[Here, an illustrative example can be provided to clarify the approach, such as a simple linear programming problem with constraints and objective function, and how the dual simplex method is applied to it.]

Steps to Implement the Dual Simplex Method

When the primal feasibility condition is not satisfied, follow these steps to use the dual simplex method:

Formulate the auxiliary problem: Identify the infeasibility in the dual solution and formulate an auxiliary problem to minimize this infeasibility. This involves setting up the objective function and constraints to reflect the violation of the original dual problem constraints. Solve the auxiliary problem: Solve the auxiliary problem to find a feasible starting point for the dual solution. This step may involve using a phase I method to find an initial feasible solution. Transform the dual solution: Once a feasible solution is obtained, transform it into a dual feasible basis for the original problem. This involves adjusting the basis and checking for optimality. Verify Feasibility: Verify that the transformed solution satisfies the constraints of the original problem and is indeed a feasible solution. Solve the original problem: If the transformed dual solution is feasible, proceed to solve the original problem using this basis.

In cases where the optimal solution to the auxiliary problem has a positive objective value, it indicates that the original dual problem is infeasible, and this information can be used to inform further problem-solving strategies.

Conclusion

The dual simplex method is a versatile tool in linear programming that can be used to solve problems where the initial conditions do not support a feasible primal or dual solution. By understanding and applying the dual simplex method correctly, you can effectively resolve complex optimization problems without resorting to manual intervention, benefiting from the automation and efficiency provided by advanced LP solvers.

[Lastly, a summary of why the dual simplex method is important and its benefits in solving certain types of LP problems.]