Introduction:
Most important finding in the development of Linear Programming Problems is the existence of duality in linear programming problems.
Duality:
Example: The doctor advises a patient visited him that the patient is weak in his health due to shortage of two vitamins, i.e., vitamin X and vitamin Y. He advises him to take at least 40 units of vitamin X and 50 units of Vitamin Y every day. He also advises that these vitamins are available in two tonics A and B. Each unit of tonic A consists of 2 units of vitamin X and 3 units of vitamin Y. Each unit of tonic B consists of 4 units of vitamin X and 2 units of vitamin Y. Tonic A and B are available in the medical shop at a cost of Rs. 3 per unit of A and Rs. 2,50 per unit of B. The patient has to fulfill the need of vitamin by consuming A and B at a minimum cost.
Primal problem:
Minimize Z = 3a 2.5b. s.t.
2a 4b ≥ 40
3a 2b ≥ 50
Both a and b are ≥ 0.
Dual Problem: Maximize Z = 40x 50y s.t. 2x 3y ≤ 3 4x 2y ≤ 2.50 both x and y are ≥ 0.
Solution to Primal: (Minimization problem i.e., patient’s problem)