The economic interpretation of the dual linear programming problem is valuable as it provides insights into the value of resources and the impact of changes in resource availability on the optimal solution of the primal problem. It helps in understanding the sensitivity of the primal problem to changes in constraints and assists decision-makers in resource allocation and management.
Economic interpretations of the dual linear programming problem:
Shadow Prices (Dual Variables):
The dual variables in the dual problem are often referred to as shadow prices or prices of resources. Each dual variable is associated with a constraint in the primal problem and represents the rate of change in the optimal objective function value of the primal problem with respect to a one-unit increase in the corresponding right-hand side (RHS) of the constraint.
Interpretation: The shadow price of a constraint indicates the marginal value of an additional unit of the corresponding resource in terms of the objective function. It represents how much the objective function value will increase or decrease if there is an additional unit of the resource available.
Optimality of Resource Allocation:
In the dual problem, the objective function represents the cost of obtaining additional resources to satisfy the constraints in the primal problem. The objective is to minimize the cost of these resources while maintaining feasibility.
Interpretation: The optimal dual solution (the values of dual variables that minimize the objective function) provides the most cost-efficient way to obtain the necessary resources to satisfy the primal problem’s constraints. It tells us how to allocate resources optimally to minimize the cost while achieving the primal’s objectives.
Interpretation of Dual Constraints:
The constraints in the dual problem are derived from the primal problem’s objective function coefficients and represent the available resources or capacities associated with each constraint in the primal problem.
Interpretation: The dual constraints provide bounds on the resource values and reflect the trade-offs between different constraints in the primal problem. They tell us the maximum amount by which the objective function coefficient in the primal problem can be increased while still maintaining feasibility.
Complementary Slackness:
The complementary slackness condition states that at the optimal solution of both the primal and dual problems, the product of each primal constraint’s slack variable and the corresponding dual variable is zero.
Interpretation: This condition indicates that when a constraint is binding (active) in the primal problem, the corresponding dual variable is positive, and the shadow price is meaningful. When a constraint is not binding (slack), the dual variable is zero, and the shadow price is irrelevant.