## The Duality Concept

Pairs of related maximization and minimization problems are known as primal and dual linear programming problems. The concept of duality demonstrates the symmetry between the value of a firm's products and the value of resources used in production. With the duality concept, it is possible to show that value maximization can be attained by focusing on either resource requirements and the revenue-generating capability of a firm's products or on the cost of resources and their productivity.

In addition to providing valuable insight into the economics of optimal resource employment, duality provides the key to solving difficult constrained optimization problems. Because primal solution

Input for short-run operating decisions dual solution

### Input for long-range planning shadow prices

Implicit values associated with linearprogramming-problem decision variables of the symmetry between primal and dual problem specifications, either one can be constructed from the other and the solution to either problem can be used to solve both. This is helpful because it is sometimes easier to obtain the solution to the dual problem than to the original or primal problem.

Finally, the duality concept also allows one to evaluate the solution to a constrained decision problem in terms of the activity required for optimization and in terms of the economic impact of constraint conditions. Analysis of the constraint conditions and slack variable solutions frequently provides important information for long-range planning. In fact, the primal solution is often described as a tool for short-run operating decisions, whereas the dual solution is often seen as a tool for long-range planning. The duality concept shows how operating decisions and long-range planning are related.

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