# 3 selection of exiting variable the basic variable

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3. Selection of exiting variable: The basic variable with the highest negative value is the exiting variable. If there are two candidates for exiting variable, any one is selected. The row of the selected exiting variable is marked as pivotal row. 4. Selection of entering variable: Cost coefficients, corresponding to all the negative elements of the pivotal row, are identified. Their ratios are calculated after changing the sign of the elements of pivotal row, i.e., The column corresponding to minimum ratio is identified as the pivotal column and associated decision variable is the entering variable. × - = row pivotal of Elements ts Coefficien Cost ratio 1

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D Nagesh Kumar, IISc Optimization Methods: M3L5 21 Dual Simplex Method: Iterative steps… contd. 5. Pivotal operation: Pivotal operation is exactly same as in the case of simplex method, considering the pivotal element as the element at the intersection of pivotal row and pivotal column. 6. Check for optimality: If all the basic variables have nonnegative values then the optimum solution is reached. Otherwise, Steps 3 to 5 are repeated until the optimum is reached.
D Nagesh Kumar, IISc Optimization Methods: M3L5 22 Dual Simplex Method: An Example Consider the following problem: 1 2 12 3 4 24 4 3 2 to subject 2 Minimize 2 1 2 1 2 1 1 2 1 + - + + + = x x x x x x x x x Z

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D Nagesh Kumar, IISc Optimization Methods: M3L5 23 Dual Simplex Method: An Example…contd. After introducing the surplus variables the problem is reformulated with equality constraints as follows: 1 2 12 3 4 24 4 3 2 to subject 2 Minimize 6 2 1 5 2 1 4 2 1 3 1 2 1 - = + - - = + - - = + + - = + - + = x x x x x x x x x x x x x Z
D Nagesh Kumar, IISc Optimization Methods: M3L5 24 Dual Simplex Method: An Example…contd. Expressing the problem in the tableau form:

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D Nagesh Kumar, IISc Optimization Methods: M3L5 25 Dual Simplex Method: An Example…contd. Successive iterations:
D Nagesh Kumar, IISc Optimization Methods: M3L5 26 Dual Simplex Method: An Example…contd. Successive iterations:

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D Nagesh Kumar, IISc Optimization Methods: M3L5 27 Dual Simplex Method: An Example…contd. Successive iterations: As all the b r are positive, optimum solution is reached. Thus, the optimal solution is Z = 5.5 with x 1 = 2 and x 2 = 1.5
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

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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 ,..
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