What are the Difference between primal and dual LPP?
Difference between Primal and dual LPP is, they are two related optimization problems that are used to solve a wide range of problems in operations research, economics, and engineering.
DIFFERENCE BETWEEN PRIMAL AND DUAL LPP TABLE:-
| Aspect | Primal LPP | Dual LPP |
| Objective Function | Minimize or Maximize | Minimize or Maximize |
| Variables | Decision variables in the primal problem | Dual variables in the dual problem |
| Constraints | Constraints in the primal problem | Variables in the dual problem |
| Objective Coefficients | Coefficients of the objective function in the primal problem | Coefficients of the constraint equations in the dual problem |
| Feasible Region | Set of feasible solutions for the primal problem | Set of feasible solutions for the dual problem |
| Optimality Condition | Satisfying the complementary slackness condition | Satisfying the complementary slackness condition |
| Weak Duality | The optimal value of the dual problem is always a lower bound on the optimal value of the primal problem | The optimal value of the primal problem is always an upper bound on the optimal value of the dual problem |
| Strong Duality | If the primal problem has an optimal solution, the dual problem has an optimal solution as well, and the optimal values are equal | If the dual problem has an optimal solution, the primal problem has an optimal solution as well, and the optimal values are equal |
Difference between primal and dual LPP
What exactly primal of a primal is
The “primal of a primal” refers to the second-level dual of a linear programming problem. To explain this more clearly:
- Primal Problem: In linear programming, the primal problem is the original optimization problem you’re trying to solve, either a maximization or minimization problem.
- Dual Problem: The dual of a primal problem is another linear programming problem derived from the primal, which gives the same optimal value under certain conditions. The dual problem provides a way to understand the primal problem through its constraints and objective function.
- Primal of a Primal (Second Dual): If you take the dual of the dual problem, you get what is called the “primal of a primal” (or sometimes the “second dual”). In theory, this should bring you back to the original primal problem, assuming the assumptions of strong duality hold (e.g., when the problem satisfies certain conditions like feasibility and boundedness).
Thus, the primal of a primal is effectively the same as the original primal problem, under the context of linear programming.
The main difference between primal and dual in Liner programming problems is in the objective function and the constraints.
Objective function:
The objective function of a primal LPP is to maximize or minimize a linear function of the decision variables subject to linear constraints.
The objective function of a dual LPP is to minimize or maximize a linear function of the dual variables subject to linear constraints.
Constraints:
The constraints in a primal LPP represent the limits or requirements of the problem, and they are expressed as linear inequalities or equations.
The constraints in a dual LPP are also expressed as linear inequalities or equations, but they represent the relationship between the decision variables and the objective function of the primal problem.
The number of variables and constraints:
The number of decision variables in a primal LPP is equal to the number of constraints in the dual LPP, and vice versa.
The number of constraints in a primal LPP is equal to the number of decision variables in the dual LPP, and vice versa.
Feasible solutions and optimal solutions:
A feasible solution to a primal LPP satisfies the constraints of the problem, while a feasible solution to a dual LPP satisfies the conditions of the dual problem.
An optimal solution to a primal LPP provides the maximum or minimum value of the objective function, while an optimal solution to a dual LPP provides the minimum or maximum value of the objective function of the dual problem.
In summary, the primal and dual linear programming problems are two sides of the same coin, and they are both important tools for solving optimization problems.
The primal problem is used to find the optimal solution to a given problem, while the dual problem provides additional insights into the structure of the problem and can be used to check the feasibility of the primal solution.
