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# Thus the optimal solution is z 55 with x 1 2 and x 2

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Thus, the optimal solution is Z = 5.5 with x 1 = 2 and x 2 = 1.5

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D Nagesh Kumar, IISc Optimization Methods: M3L5 28 Solution of Dual from Primal Simplex 0 , , 2 2 2 2 1 2 4 4 5 2 to subject 4 0 6 ' Minimize 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 - + - - - + + + + = y y y y y y y y y y y y y y y Z Primal Dual 0 , , 4 2 2 5 0 2 4 6 2 2 to subject 2 4 Maximize 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 - - + - + + + - = x x x x x x x x x x x x x x x Z y 1 y 2 y 3 Z
D Nagesh Kumar, IISc Optimization Methods: M3L5 29 Sensitivity or post optimality analysis Changes that can affect only Optimality Change in coefficients of the objective function, C 1 , C 2 ,.. Re-solve the problem to obtain the solution Changes that can affect only Feasibility Change in right hand side values, b 1 , b 2 ,.. Apply dual simplex method or study the dual variable values Changes that can affect both Optimality and Feasibility Simultaneous change in C 1 , C 2 ,.. and b 1 , b 2 ,.. Use both primal simplex and dual simplex or re-solve

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D Nagesh Kumar, IISc Optimization Methods: M3L5 30 Sensitivity or post optimality analysis A dual variable, associated with a constraint, indicates a change in Z value (optimum) for a small change in RHS of that constraint. j i Z y b ∆ = where y j is the dual variable associated with the i th constraint, b i is the small change in the RHS of i th constraint, Z is the change in objective function owing to b i .
D Nagesh Kumar, IISc Optimization Methods: M3L5 31 Sensitivity or post optimality analysis: An Example Let, for a LP problem, ith constraint be and the optimum value of the objective function be 250. RHS of the i th constraint changes to 55, i.e., i th constraint changes to Let, dual variable associated with the i th constraint is y j , optimum value of which is 2.5 (say). Thus, b i = 55 – 50 = 5 and y j = 2.5 So, Z = y j b i = 2.5x5 = 12.5 and revised optimum value of the objective function is 250 + 12.5 = 262.5. 1 2 2 50 x x + 1 2 2 55 x x +

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D Nagesh Kumar, IISc Optimization Methods: M3L5 32 Thank You
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Thus the optimal solution is Z 55 with x 1 2 and x 2 15 D...

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