# Importance of Duality Concept in OR and Economic Importance of the Duality Concept

This is MBA assignment question for MB0032 of SMU. The question is – “Why Duality concept is important in OR? Describe the Economic importance of the Duality concept.”

The importance of duality concept is due to two main reasons:

a) If the primal contains a large number of constraints and a smaller number of variables, the labour of computation can be considerably reduced by converting it into the dual problem and then solving it.

b) The interpretation of the dual variable from the loss or Economic point of view proves extremely useful in making future decisions in the activities being programmed.

Economic interpretation of duality:

The linear programming problem can be thought of as a resource allocation model in which the objective is to maximize revenue or profit subject to limited resources. Looking at the problem from this point of view, the associated dual problem offers interesting Economic interpretations of the L.P resource allocation model.

We consider here a representation of the general primal and dual problems in which the primal takes the role of a resource allocation model.

From the above resource allocation model, the primal problem has n Economic activities and m resources. The coefficient cj in the primal represents the profit per unit of activity j. Resource i, whose maximum availability is bj, is consumed at the rate aij units per unit of activity j.

Economic interpretation of dual variables:

For any pair of feasible primal and dual solutions, (Objective value in the maximization problem) ≤ (Objective value in the minimization problem)

At the optimum, the relationship holds as a strict important.

This relationship implies that as long as the total return from all the activities is less than the worth of the resources, the corresponding primal and dual solutions are not optimal. Optimally is reached only when the resources have been exploited completely, which can happen only when the input equals the output (profit).

Economically the system is said to remain unstable (non optimal) when the input (worth of the resources) exceeds the output (return). Stability occurs only when the two quantities are equal. This is the most important aspect of Economic importance of dual variables for MBA assignments.

## Importance of Duality Concept in OR and Economic Importance of the Duality Concept

Duality is a fundamental concept in optimization theory that has important applications in Operations Research (OR) and Economics. It refers to the relationship between two related mathematical programs, known as the primal and dual programs. The primal program is the original optimization problem that we want to solve, while the dual program is a reformulation of the primal program that can help us gain additional insights and solve the problem more efficiently.

The duality concept is important in OR because it allows us to analyze and solve optimization problems from different perspectives. For example, if we are trying to minimize a cost function subject to certain constraints, we can formulate the corresponding dual problem that maximizes a related function subject to different constraints. By analyzing the dual problem, we can obtain additional information about the original problem, such as the upper bound on the optimal value or the sensitivity of the optimal solution to changes in the constraints.

The duality concept is also important in Economics because it provides a powerful tool for analyzing market equilibria and welfare implications of policies. In microeconomic theory, the duality between consumer and producer theory is a fundamental result that allows us to derive the demand and supply functions from the preferences and technologies of consumers and producers. This duality result enables us to study the efficiency and distributional effects of taxes, subsidies, and other policy interventions on market outcomes.

Moreover, the duality concept has important economic implications for the theory of the firm and industrial organization. The duality between cost minimization and profit maximization provides a framework for understanding the behavior of firms in different market structures, such as perfect competition, monopolistic competition, and oligopoly. The duality between production and cost functions also allows us to study the relationship between scale economies and market power, and to derive the optimal pricing and output decisions of firms in different market environments.

In summary, the duality concept is a powerful and versatile tool in optimization theory, OR, and Economics. It provides a complementary perspective on optimization problems and enables us to gain additional insights into market equilibria and welfare implications of policies. The importance of the duality concept is evidenced by its wide-ranging applications in many fields of study, including engineering, management science, finance, and public policy.

## For More Article

Liberty Magazine – Economic Website

———————————————————————————