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.