Dvlf 5kv ¹ ¹ º º ¹ ¹ ¼ º º ͳ 9 ݕ ଵ ³ 9

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%DVLF 5KV ¶¹ ¶¹ ¸ º ¶» ¶¸ ¶» º ¸ ¶¹ െ࢝ ¸¹ ¸¼ º º Ͳ (%9± ݕ ³ /%9± ³ SLYRW± ´¸³¸µ %DVLF 5KV ¸ ¸ ¶¸Â¹ º »Â¹ º ¶¹ ¶¸Â¹ ¸ ¶¸Â¹ െ࢝ º ¾ ¼ º ¶¸¿ /%9± ³ (%9± ݕ ³ SLYRW± ´¹³¹µ %DVLF ܛ ܛ 5KV ¸ º ¶»Â¾ ¸Â¹ ½Â¾ º ¸ ¸Â¾ ¶¸Â¹ ó െ࢝ º º ½ ¹ ¶¸Á At this point the dual LP problem is solved and the optimal solution is: ݕ ൌ ͳǤʹͷǡ ݕ ൌ ͲǤʹͷǡ ݓ ௢௣௧ ൌ ͳͻ ² We note that the first feasible solution obtained above is also the optimal solution. We further note that: a) The optimal value of the objective function for (D) is the same as the optimal value for (P). b) The optimal values for the basic variables for (P) appear as reduced costs associated with non-basic variables in (D). As an added advantage, the dual simplex method obviates the need for the two-phase simplex method to obtain a solution when an initial BFS is not readily available. This is illustrated by re-solving Example 5.3 using the dual simplex algorithm. Example 5.8: Dual Simplex algorithm We consider the dual problem of Example 5.3. The original LP problem is stated as: ǡ௫ ݖ ൌ ͵ݔ ൅ ʹݔ 6XEMHFW WR± ͵ݔ ൅ ʹݔ ൒ ͳʹǡ ʹݔ ൅ ͵ݔ ൑ ͳ͸ǡ ݔ ൒ Ͳǡ ݔ ൒ Ͳ
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Download free eBooks at Fundamental Engineering Optimization Methods 96 Linear Programming Methods The GE constraint in the problem is first multiplied by –1; the problem is then converted to dual problem using the symmetric form of duality. The dual optimization problem is given as: ǡ௬ ݖ ൌ െͳʹݕ ൅ ͳ͸ݕ 6XEMHFW WR± െ͵ݕ ൅ ʹݕ ൒ ͵ǡ െʹݕ ൅ ͵ݕ ൒ ͳǢ ݕ ൒ Ͳǡ ݕ ൒ Ͳ The series of tableaus leading to the optimal solution via the dual simplex method is given below: %DVLF 5KV » ¶¹ ¸ º ¶» ¹ ¶» º ¸ ¶¹ െ࢝ ¶¸¹ ¸¼ º º Ͳ /%9± ³ (%9± ݕ ³ SLYRW± ´¸³¹µ %DVLF 5KV ¶»Â¹ ¸ ¶¸Â¹ º »Â¹ ¶½Â¹ º ¶»Â¹ ¸ ½Â¹ െ࢝ ¸¹ º ¿ º ¶¹¾ At this point the dual LP problem is solved with the optimal solution: ݕ כ ൌ Ͳǡ ݕ כ ൌ ͳǤͷǡ ݓ כ ൌ ʹͶ ² We note that this is the same solution obtained for Example 5.3. We further note that the reduced costs for nonbasic variables match with the optimal values of the primal basic variables. The final dual Simplex example involves a problem with equality constraints.
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