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Operations Research

Operations Research. Chapter 4 Linear Programming Duality and Sensitivity analysis. Linear Programming Duality and Sensitivity analysis. Dual Problem of an LPP Definition of the dual problem Examples (1,2 and 3) Sensitivity Analysis Examples (1 and 2). Dual Problem of an LPP.

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Operations Research

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  1. Operations Research Chapter 4 Linear Programming Duality and Sensitivity analysis

  2. Linear Programming Duality and Sensitivity analysis • Dual Problem of an LPP • Definition of the dual problem • Examples (1,2 and 3) • Sensitivity Analysis • Examples (1 and 2)

  3. Dual Problem of an LPP Given a LPP (called the primal problem), we shall associate another LPP called the dual problem of the original (primal) problem. We shall see that the Optimal values of the primal and dual are the same provided both have finite feasible solutions. This topic is further used to develop another method of solving LPPs and is also used in the sensitivity (or post-optimal) analysis.

  4. Definition of the dual problem Given the primal problem (in standard form) Maximize subject to

  5. the dual problem is the LPP Minimize subject to

  6. If the primal problem (in standard form) is Minimize subject to

  7. Then the dual problem is the LPP Maximize subject to

  8. We thus note the following: 1. In the dual, there are as many (decision) variables as there are constraints in the primal. We usually say yi is the dual variable associated with the ith constraint of the primal. 2. There are as many constraints in the dual as there are variables in the primal.

  9. 3. If the primal is maximization then the dual is minimization and all constraints are  If the primal is minimization then the dual is maximization and all constraints are  • In the primal, all variables are  0 while in the dualall the variables are unrestricted in sign.

  10. 5. The objective function coefficients cjof the primal are the RHS constants of the dual constraints. 6. The RHS constants bi of the primal constraints are the objective function coefficients of the dual. 7. The coefficient matrix of the constraints of the dual is the transpose of the coefficient matrix of the constraints of the primal.

  11. Example1 Write the dual of the LPP Maximize subject to

  12. Thus the primal in the standard form is: Maximize subject to

  13. Hence the dual is: Minimize subject to

  14. Example2 Write the dual of the LPP Minimize subject to

  15. Thus the primal in the standard form is: Minimize subject to

  16. Hence the dual is: Maximize subject to

  17. Example3 Write the dual of the LPP Maximize subject to

  18. Thus the primal in the standard form is: Maximize subject to

  19. Hence the dual is: Minimize subject to

  20. Theorem: The dual of the dual is the primal (original problem). Proof. Consider the primal problem (in standard form) Maximize subject to

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