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**Solving Linear Programming Problems**

The Primal-Dual Relationship Shubhagata Roy

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Concept of Duality One part of a Linear Programming Problem (LPP) is called the Primal and the other part is called the Dual. In other words, each maximization problem in LP has its corresponding problem, called the dual, which is a minimization problem. Similarly, each minimization problem has its corresponding dual, a maximization problem. For example, if the primal is concerned with maximizing the contribution from the three products A, B, and C and from the three departments X, Y, and Z, then the dual will be concerned with minimizing the costs associated with the time used in the three departments to produce those three products. An optimal solution from the primal and the dual problem would be same as they both originate from the same set of data.

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**Rules for Constructing the Dual from Primal**

1. A dual variable is defined for each constraint in the primal problem,i.e, the no. of variables in the dual problem is equal to no. of constraints in the primal problem and vice-versa. If there are m constraints and n variables in the primal problem then there would be m variables and n constraints in the dual problem. 2. The RHS of primal,i.e, b1,b2,……,bm, become the coefficients of dual variables(Y1,Y2,……,Ym) in the dual objective function(ZY). Also the coefficients of primal variables(X1,X2,……,Xn),i.e, c1,c2,……,cn, become RHS of the dual constraints. 3. For a maximization primal problem(with all < or = constraints), there exists a minimization dual problem(with all > or = constraints) and vice-versa.

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4.The matrix of coefficients of variables in dual problem is the transpose of matrix of coefficients in the primal problem and vice-versa. 5. If any of the primal constraint (say ith) is an equality then the corresponding dual variable is unrestricted in sign and vice-versa The Primal-Dual Relationship

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**Example: Maximize ZX = 6000X1+4000X2 s. t**

Example: Maximize ZX = 6000X1+4000X2 s.t. 4X X2 <or= X X2 <or= X1 + 3X2 <or= X1 +40X2 <or= X1,X2 >or= 0

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Hence the Dual Problem looks like, Minimize ZY = 12Y1+20Y2+18Y3+40Y4 subject to the constraints, 4Y1+9Y2+7Y3+10Y4 >or= Y1+ Y2+3Y3+40Y4 >or= Y1,Y2 >or= 0

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Summary of complementary slackness: 1. If x i ≠ 0, i= 1, 2, …, n the ith dual equation is tight. 2. If equation i of the primal is not tight, y i =0. 1.

Summary of complementary slackness: 1. If x i ≠ 0, i= 1, 2, …, n the ith dual equation is tight. 2. If equation i of the primal is not tight, y i =0. 1.

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