By Yamal 
The Dual Simplex Method is an optimization technique used in linear programming, particularly when dealing with changes in constraints rather than the objective function. It is an extension of the traditional Simplex Method and is specifically designed to handle situations where the primal problem is feasible, but the dual problem may not be.
Here’s a breakdown of the Dual Simplex Method and its significance in linear programming:
Basics of Linear Programming
Linear programming involves optimizing a linear objective function subject to a set of linear inequalities or equations, known as constraints. Each linear programming problem has two forms: the primal form and the dual form. The primal problem is often about maximizing or minimizing a linear function, while the dual provides an alternative perspective on resource allocation.
Primal and Dual Problems
– Primal Problem: The original linear programming problem framed in terms of maximizing or minimizing some objective function subject to constraints.
– Dual Problem: Formulated from the primal problem, the dual problem helps interpret the constraints and can provide insights into resource valuation. It involves its own objective function and constraints derived from the primal problem.
When to Use the Dual Simplex Method
The Dual Simplex Method is particularly useful in scenarios where:
– The primal solution is feasible, but the dual solution becomes infeasible due to changes in constraints.
– Situations arise in post-optimality analysis where the primal feasible region remains valid while dual feasibility is violated, such as during sensitivity analysis or when you add new constraints to the primal model.
The Procedure of the Dual Simplex Method
The Dual Simplex Method follows these general steps:
– Initialization: Begin with a basic feasible solution for the primal problem. Identify the dual infeasibility.
– Pivoting: Select a pivot element in the tableau that represents the variable to enter the basis (which can help restore dual feasibility) and the variable to leave the basis (which is causing the dual infeasibility).
– Iteration: Update the tableau using the pivot element. Continue this process until all dual constraints are satisfied and the optimal solution is reached.
Advantages of the Dual Simplex Method
– Efficient Handling of Changes: It is particularly efficient for problems where constraints are modified after a feasible solution has already been found.
– Stable Solutions: In many cases, it retains the primal feasibility while adjusting to restore dual feasibility, making it beneficial in dynamic situations.
Applications
The Dual Simplex Method can be applicable in various fields, such as:
– Operations Research: Solving transportation, assignment, and scheduling problems.
– Finance: Portfolio optimization and resource allocation.
– Manufacturing: Production scheduling and supply chain optimization.
Comparison with Regular Simplex Method
While the standard Simplex Method focuses on improving the primal objective and ensures primal feasibility, the Dual Simplex Method prioritizes dual feasibility while maintaining primal feasibility. This makes the Dual Simplex Method a powerful alternative for specific scenarios in linear programming.
Conclusion
The Dual Simplex Method offers a distinct and effective approach to solving linear programming problems, especially in complex and dynamic environments. Understanding its principles and applications helps practitioners in various fields optimize their processes and resource allocations efficiently. By leveraging this technique, organizations can ensure they adapt to constraints effectively while still achieving their objectives.