fundamental-engineering-optimization-methods.pdf

# 2 if the reduced cost is negative but the pivot step

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2. If the reduced cost is negative but the pivot step cannot be completed due to all coefficients in the LBV column being negative, it reveals a situation where the cost function is unbounded below. 3. If, at some point during Simplex iterations, a basic variable attains a zero value (i.e., the rhs has a zero), it is called degenerate variable and the corresponding BFS is termed as degenerate solution, as the degenerate row hence forth becomes the pivot row, with no improvement in the objective function. While it is theoretically possible for the algorithm to fail by cycling between two degenerate BFSs, this is not known to happen in practice. An example for tableau implementation of the simplex method is presented below. .

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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 77 Linear Programming Methods Example 5.2: the Tableau method As an example of the tableau method, we resolve example 5.1 using tableaus, where the optimization problem is stated as: ǡ௫ ݖ ൌ ͵ݔ ൅ ʹݔ 6XEMHFW WR± ʹݔ ൅ ݔ ൑ ͳʹǡ ʹݔ ൅ ͵ݔ ൑ ͳ͸Ǣ ݔ ൒ Ͳǡ ݔ ൒ Ͳ The problem is first converted to the standard LP form. The constraints and cost function coefficients were entered in an initial simplex tableau, where the EBV, LBV, and the pivot element are identified underneath the tableau: %DVLF 5KV ¹ ¸ ¸ º ¸¹ ¹ » º ¸ ¸¼ ¶» ¶¹ º º Ͳ (%9± ݔ ³ /%9± ³ SLYRW± ´¸³¸µ The subsequent simplex iterations result in the series of tableaus appearing below: %DVLF UKV ¸ º²½ º²½ º ¼ º ¹ ¶¸ ¸ ¾ െࢠ º ¶º²½ ¸²½ º ¸¿ (%9± ݔ ³ /%9± ³ SLYRW± ´¹³¹µ %DVLF UKV ¸ º º²À½ ¶º²¹½ ½ º ¸ ¶º²½ º²½ ¹ െࢠ º º ¸²¹½ º²¹½ ¸Á At this point, since all reduced costs are positive, an optimum has been reached with: ݔ כ ൌ ͷǡ ݔ כ ൌ ʹǡ ݖ ௢௣௧ ൌ െͳͻ ² 5.3.1 Final Tableau Properties The final tableau from the simplex algorithm has certain fundamental properties that relate to the initial tableau. To reveal those properties, we consider the following optimization problem: ݖ ൌ ࢉ 6XEMHFW WR± ࡭࢞ ൑ ࢈ǡ ࢞ ൒ ૙ (5.5)
Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 78 Linear Programming Methods Adding surplus variables to the constraints results in the following standard LP problem: ݖ ൌ െࢉ 6XEMHFW WR± ࡭࢞ ൅ ࡵ࢙ ൌ ࢈ǡ ࢞ ൒ ૙ (5.6) An initial tableau for this problem is given as: %DVLF 5KV െࢠ െࢉ Ͳ Assuming that the same order of the variables is maintained, then at the termination of the Simplex algorithm the final tableau is given as: %DVLF UKV െࢠ ݖ כ

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