Understanding Duality in Linear Programming Problems

Introduction

Linear Programming is a powerful mathematical tool used to optimize complex systems by maximizing or minimizing a linear objective function subject to linear equality and inequality constraints. One of the key concepts in Linear Programming is duality, which plays a crucial role in providing insights into the problem and aiding in finding optimal solutions. In this article, we will explore the meaning of duality in Linear Programming Problems.

What is Duality in Linear Programming?

Duality in Linear Programming refers to the relationship between two related optimization problems known as the Primal and the Dual. The Primal problem is the original linear programming problem that we seek to solve, while the Dual problem is derived from the Primal and provides additional information about the problem.

Primal and Dual Problems

The Primal problem can be formulated as:

• Maximize or Minimize: Z = c₁x₁ + c₂x₂ + … + c_nx_n
• Subject to: a₁₁x₁ + a₁₂x₂ + … + a_1nx_n ≤ b₁,
• a₂₁x₁ + a₂₂x₂ + … + a_2nx_n ≤ b₂,
• …
• a_m1x₁ + a_m2x₂ + … + a_mnx_n ≤ b_m,
• x₁, x₂, …, x_n ≥ 0

The Dual problem can be formulated as:

• Maximize or Minimize: W = b₁y₁ + b₂y₂ + … + b_my_m
• Subject to: a₁₁y₁ + a₂₁y₂ + … + a_m1y_m ≤ c₁,
• a₁₂y₁ + a₂₂y₂ + … + a_m2y_m ≤ c₂,
• …
• a_1ny₁ + a_2ny₂ + … + a_mny_m ≤ c_n,
• y₁, y₂, …, y_m ≥ 0

Duality Theorem

The Fundamental Duality Theorem states that for any Linear Programming problem, the optimal values of the Primal and Dual problems are equal. That is, if the Primal problem has an optimal solution, then the Dual problem also has an optimal solution with the same objective function value.

Importance of Duality

The concept of duality in Linear Programming is essential for several reasons:

• Provides alternative perspectives on the problem
• Helps in verifying the optimality of solutions
• Facilitates sensitivity analysis
• Aids in understanding the structure of the problem

Example

Consider a manufacturing company that produces two types of products, A and B. The company can produce a maximum of 100 units of product A and 150 units of product B. The profit per unit of product A is $5, and the profit per unit of product B is $7. The company wants to maximize its profit. The constraints on the production capacity and profit can be modeled using Linear Programming.

Conclusion

Duality in Linear Programming provides a deeper insight into the optimization problem and helps in exploring alternative solutions. Understanding the relationship between the Primal and Dual problems can lead to more efficient and effective decision-making processes.