Selection of entering variable for each of the

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Selection of entering variable For each of the nonbasic variables, calculate the coefficient ( WP - c ) , where, P is the corresponding column vector associated with the nonbasic variable at hand, c is the cost coefficient associated with that nonbasic variable and W = C S S -1 . For maximization (minimization) problem, nonbasic variable, having the lowest negative (highest positive) coefficient, as calculated above, is the entering variable.
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D Nagesh Kumar, IISc Optimization Methods: M3L5 9 Revised Simplex method: Iterative steps 2. Selection of departing variable a) A new column vector U is calculated as U = S -1 B b) Corresponding to the entering variable, another vector V is calculated as V = S -1 P , where P is the column vector corresponding to entering variable. c) It may be noted that length of both U and V is same (= m). For i = 1,…, m , the ratios, U ( i )/ V ( i ), are calculated provided V ( i ) > 0. i = r , for which the ratio is least, is noted. The r th basic variable of the current basis is the departing variable. If it is found that V ( i ) < 0 for all i , then further calculation is stopped concluding that bounded solution does not exist for the LP problem at hand.
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D Nagesh Kumar, IISc Optimization Methods: M3L5 10 Revised Simplex method: Iterative steps 3. Update to new Basis Old basis S , is updated to new basis S new , as S new = [ E S -1 ] -1 where ( 29 ( 29 ( 29 = = r i r V r i r V i V i for 1 for η and r th column 1 2 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 r m m η η η η η - = E L L L L M M M L M M M M L L M M M M L M M M L L L L
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D Nagesh Kumar, IISc Optimization Methods: M3L5 11 Revised Simplex method: Iterative steps S is replaced by S new and steps1 through 3 are repeated. If all the coefficients calculated in step 1, i.e., is positive (negative) in case of maximization (minimization) problem, then optimum solution is reached The optimal solution is X S =S -1 B and z = CX S
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D Nagesh Kumar, IISc Optimization Methods: M3L5 12 Duality of LP problems Each LP problem (called as Primal in this context) is associated with its counterpart known as Dual LP problem. Instead of primal, solving the dual LP problem is sometimes easier in following cases a) The dual has fewer constraints than primal Time required for solving LP problems is directly affected by the number of constraints, i.e., number of iterations necessary to converge to an optimum solution, which in Simplex method usually ranges from 1.5 to 3 times the number of structural constraints in the problem b) The dual involves maximization of an objective function It may be possible to avoid artificial variables that otherwise would be used in a primal minimization problem.
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D Nagesh Kumar, IISc Optimization Methods: M3L5 13 Finding Dual of a LP problem Inequality sign of i th Constraint: if dual is maximization if dual is minimization x i > 0 j th variable j th constraint i th constraint i th variable Maximization Minimization Minimization Maximization Dual Primal …contd. to next slide
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