Computational Techniques of the Simplex Method - download pdf or read online

By István Maros

Computational strategies of the Simplex Method is a scientific remedy all for the computational problems with the simplex technique. It presents a entire assurance of crucial and winning algorithmic and implementation concepts of the simplex technique. it's a specific resource of crucial, by no means mentioned information of algorithmic components and their implementation. at the foundation of the e-book the reader could be in a position to create a hugely complex implementation of the simplex approach which, in flip, can be utilized at once or as a construction block in different answer algorithms.

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This approach (not the only imaginable one) is called the two-phase simplex method. In phase-1 it looks for a feasible solution to the problem. It does it by creating and solving an auxiliary problem, the so called phase-l problem. If the search is unsuccessful we can conclude that the problem is infeasible. If this is the case, the optimal solution of the phase-1 problem points out which constraints could not be satisfied. It does not necessarily mean that these constraints are the real cause of infeasibility.

This can be achieved by solving the following LP problem: m minimize Za = L Xn+i i=l n subject to L a;xj + Xn+i = bi, i = 1, ... 49) j=l Xj 2 0, j = 1, ... ,n + m. 47). 49) the number of variables is n + m. An important difference is the objective function. Now it is the sum of the artificial variables. The original objective function is ignored at this stage. 49) by setting B = {n + 1, ... , n + m} and B = I. The initialization step of PSM-l will 'determine' B- 1 = I and Xs = b. The iterative steps of the algorithm can be carried out unchanged.

Of course, while y is always dual feasible, XB is feasible only when an optimal solution is reached. The same question arises here as in the primal. How do we get a dual feasible basis? In some cases it is very simple. 60) are all nonnegative then the unit matrix of logical variables is a dual feasible basis. Some methods have been worked out for the case when such an obvious choice is not available. The most typical of them is the dual big-M method. In this case an extra '$' type constraint 'Ej=1 Xj + X n +1 = M, with X n +1 ~ 0, is added to the original constraints.

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