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Mathematics programmers - Nakonechny S.I.

3.6. The simplex method

Як відомо з поперепоніх параграів даного розділу, кожній задачі лінійного програмування you can postpone the performance of the binary task. By the theorems of the binary, the zv'azok mizh rozv'yazkami straightforward dyachistoy tasks. For the know-how of rozv'yaku odnієї zi conjugating tasks, you can go to the two-fold, vikorostoyuchi її optimal plan, vysnachiti optimal plan pochatkoy.

The transition to the two tasks is not obovazykovy. Easily pomititi, scho zvichayna simplex table in stovpchikah mistit kopatkovu task, and in rows - dvoistu. Оцінками to the plan of direct tasks є row ( ), And the plans for the two-way plan are the "Plan" with components of the vector of the old members of the system. Otzhe, rozv'yazyuchi direct task, the simplex method of letting you know one-hour know-how and rozv'yaz dvoystoy tasks. However, the task can be the same for the table, it is written straightforward, and the optimal plan of the two tasks is addressed, at the same time, the distribution of tasks is assigned. Such a process of solving problems of linear programming is called a duplex simplex method . Straight that way, the simplex method of dealing with itself.

Nekha neobhіdno rozv'yati problem lіnіynogo programvannya, I will submit in the canonical form:

, (3.60)

, (3.61)

. (3.62)

Todi is the same as before:

(3.63)

. (3.64)

Behind the algorithm of a binary simplex method, the first basic plan is to select the permissive permissive rozv'azok dvuhstoy tasks (in idiots in literary yoga nazivayut "pseudo-plan"), and zegergeyatsya yogo admissibility for bodystoy tasks and simplification of all crocs.

It is permissible, however, that the basic basis is stored in m m vectors , And one of the components of the vector Vіd'ємна. Nehai , Odnak sravazhuyutsya criteriya optim'nostі plan, toto all otsynki vectorіv ( ). On the basis of the first theorems of the two-pronged plan of the two tasks, one should observe the following: . Tsei plan is not optimal for direct tasks, it does not satisfy the expectations of the unknown (3.62) and it is not optimal for the two tasks, but all the vectors in the optimal plan for the two tasks do not need to be maddened.

Otzhe, vector, Що відповідає компоненті , Potrybno viklyuchi z basis pochatkoy problems, and vector dvoystoy tasks, sho vidpoviday vіd'єmnіy otsінці, include up to the basis of binary.

In the direct simplex method, the spearlet vyyavlyayut zmіnnu, yaku slіd enter at the basis, and in the dual simplex method navpaki - spoletka viznachayut zmіnnu, yaku viklyuchayut z basisu, and potim zmіnnu, yaku enter at the basis.

In the literature [19, 22, 28, 31] rasterize variance of a two-sided simplex method, which does not maim the principles of visions. Розглянемо such algorithm of a two -dashed simplex method :

1. It is necessary to start all tasks with tasks to the " ", To introduce additions of non-existent zmіnnі, vysnachiti povchatkovyi basis that first support plan .

2. Якщо всі оцінки vectorsів І components of a vector-стоппчика "Plan" For all , Then the problem is unknown. I also need to vibrate after the module component І Відповідну змінну Vice versa in the basis.

3. Yaksho in l- mu row, scho vіdpovyde zmіnnіy , Do not run cold , Then the central function of the two tasks is not assigned to the Bagatograns rozv'yakiv, but the task of roving is not masculine. Інакше існують деякі І тоді for відповідних стовпчиків viznachayut analogously to the simple simplex method оцінки :

( ),

So let's make a vibrator vector, which is included in the basis.

4. After the conviction of the Croix method of the old Wiccluscans of the Jordan-Gauss, pass to the onset of the simplex table (Pass to point 2).

Signifi- cantly, for tasks, the value of the maximum value of the function is imposed on the algorithm by changing to the function , Abo desho zmіniti algorithm itself.

The detailed analysis of all the crooked algorithms for a simplex simplex method is outlined in [10].

Know the minimum value of functions

For the mind:

.

Розв'язання : Multiply to each other nervivnost on (- 1) і vvedlimo додаткові змінні.

.

The cobwebby basis is the vectors A4 and A5. Pseudoplan .

Foldable to the ears is a simplex table.

Oskilki , Then in the basis it is necessary to introduce the vector A 5 і іпіпповідну змінну .

For the vector of the vector, u, is introduced into the basis, . In the other row there are two visions of oneself , Yaki propagate to the vectors A1 and A3 . Viznachimo, a kind of vectors need to enter into the basis.

.

The oil, , It means that the vector A1 is introduced in the basis of the field. The rozv'zazuvalnim element will be a21 (-1).

In the result of realizing the method of the Giordana-Gauss wiklucen, for two crooks, an optimal plan is produced.

Zgіdno z ostannogo simplex table maemo such an optimal plan pochatkoy tasks:

,

The optimal plan for the two tasks:

, .

For rozraunku optimal plan for the two tasks, necessary neat Home-shops on (- 1), osklіki tі spryazhenі problemy є sometricchnymi.

Zauvazhimo, scho zdebilshogo bjastii simplex method for kilkistyu ітерацій not кращий, ніж звичайний. However, in the overloaded tasks, let's discover the rozrawns. In the induced hang of the application for the best-known design by the simplex method, neobhidno boulo b introduce a piece of zmіnnu. Zavshyaki dvuhistomu simplex method, rozv'yazuvannya zadachi i prostіshe, zmenshena kilkіst ітерацій.

In addition, the simplex method of simplex is a wiggling for rozvjazuvannya tasks, which is to accumulate already rozv'yanykh, napriklad, yakshcho introduced to a few newcomers to clarify the problem of abdominal prištosovuyot її to zmіnenih real minds.