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Ch.4 Duality and Post Optimal Analysis Dr. Ayham Jaaron Introduction • One of the most important discoveries in the early development of linear programming was the concept of duality and its many important ramifications. • This discovery revealed that every linear programming problem has associated with it another linear programming problem called the dual. The relationships between the dual problem and the original problem (called the primal) prove to be extremely useful in a variety of ways. • We shall describe many valuable applications of duality theory in this chapter. Definition of the Dual problem • The dual problem is an LP defined directly and systematically from the primal (original) LP model. • The two problems are so closely related that the primal solution of one problem automatically provides the optimal solution to the other. • The primal problem represents a resource allocation case where the dual problem represents a resource valuation problem. • Duality help simplification of the simplex problem. Rules for constructing the dual problem Primal Problem Dual Problem Objective Objective Constraints Type Variable sign Maximization Minimization unrestricted Minimization Maximization ≥ ≤ unrestricted Example (1) • Write the dual for the following primal problem: • Maximize Z= 5x1 + 12x2 + 4x3 Subject to: x1+2x2+x3 ≤ 10 2x1- x2 + 3x3 = 8 x1,x2,x3 ≥ 0 What if you considered artificial variables to change to standard form rather than equation form???.....Try Example (2) • Write the dual for the following primal problem Minimize Z= 15x1+ 12x2 Subject to x1 + 2x2 ≥ 3 2x1 - 4x2 ≤ 5 x1,x2 ≥ 0 Example (3) • Maximize Z = 5x1 + 6x2 • Subject to: x1 + 2x2 = 5 -x1 + 5x2 ≥ 3 4x1 + 7x2 ≤ 8 x1 unrestricted, x2 ≥ 0 Optimal Dual Solution • This section provides two methods for solving the optimal of the dual problems. • However, dual of the dual is itself the primal, which means that the dual solution can also be used to yield the optimal primal solution automatically. Optimal dual solution..... Problem • For the following primal problem, find the optimal dual solution: • Maximize Z= 5x1 + 12x2 + 4x3 Subject to: x1+2x2+x3 ≤ 10 2x1- x2 + 3x3 = 8 x1,x2,x3 ≥ 0 Problem 3- page 162 Write the dual and determine its optimal solution in two ways Problem 4- Page 163 Problem6- page 163 Problem 6- Page 163 Verification methods Examples The following table represents the optimal primal solution for the above LP model. Using the optimal inverse provided in the table, verify that the given table represents a correct solution for the original LP model? Example..continued.. Example...continued..solution.. Primal-dual objective values Primal-dual relationship Book problems- page 166 Continued.... Book Problems- Page 167 Problem 4- continued..... Continued...optimal simplex table is