All linear programming problems have another problem associated with them, which is known as its dual. In other words, every minimization problem is associated with a maximization problem and vice-versa. The original linear programming problem is known as primal problem, and the derived problem is known as its dual problem. The optimal solutions for the primal and dual problems are equivalent.
Conversion of primal to dual is done because of many reasons. The dual form of the problem, in many cases, is simple and can be solved with ease. Moreover, the variables of the dual problem contain information useful to management for analysis.
Procedure is as follows:
Step 1: Convert the objective function if maximization in the primal into minimization in the dual and vice versa. Write the equation considering the transpose of RHS of the constraints.
Step 2: The number of variables in the primal will be the number of constraints in the dual and vice versa.
Step 3: The co-efficient in the objective function of the primal will be the RHS constraints in the dual and vice versa.
Step 4: In forming the constraints for the dual, consider the transpose of the body matrix of the primal problems.
Sensitivity analysis involves ‘what if?’ questions. In the real world, the situation is constantly changing like change in raw material prices, decrease in machinery availability, increase in profit on one product, and so on. It is important to decision makers for find out how these changes affect the optimal solution. Sensitivity analysis can be used to provide information and to determine solution with these changes.
Sensitivity analysis deals with making individual changes in the co-efficient of the objective function and the right hand sides of the constraints. It is the study of how changes in the co-efficient of a linear programming problem affect the optimal solution.
We can answer questions such as:
1. How will a change in an objective function co-efficient affect the optimal solution?
2. How will a change in a right-hand side value for a constraint affect the optimal solution?