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Mathematics programmers - Nakonechny S.I.

6.5. Combinatory methods. The method is

In the fundamentals of combinators methods, пере perebir mozlivih varіantіv rozv'yakіv posavlenoї problemy. Kozhen them characterize pevnoyu poslіdovistyu pererobu varіantіv that rules viklyuchennya, scho dajut zmogu shche v proizvodstv rozv'yazuvannya problemy vijaviti neoptimalnі varianti without alternating conversions. Відносна эфективність різних метоів зажить від того, наскільки кожен з них уможливлює speed of the unbearable process of sorting out the variance of the results of the impregnated rules of exclusion.

Розглянемо one із комбінаторних метоів. For rozv'yazvannya problems tsilochislovogo programvannya efektivnim for the method Gomorye і method гілок і меж . Spotchatku, yak і v razi method Gomory, the simplex method of rozv'yazuyetsya indulged (without the minds of the problem) task. Potim to enter the rule of sorting.

Do not forget to know xj - the number of words, the meaning of xx = In the optimal plan for the relaxed tasks - the shotgun. Obviously, in the active part of a given point, such a value does not belong to the number of values, that one can exclude from the multiplicity of permissible plans of tasks in the populace. So promižkom є інтервал між найближчими до Number of values. Можна стверджувати, що на інтервалі Цілих значень немає.

Overlap, yakshtoo = 2,7 дістаємо інтервал , De, obviously , a lot of xj , yak nabuvaє tsilogo znachennya і optimalsnyi rozv'yazok budeyshihsya abo v intervalvali , Abo . Виключення проміжку From the multiplicity of permissible plans, one gets introduced to the system of interchange of tasks and dodatkovyh nerves. Tobto is permissible for the value of xj moe zdovolnyati one of the strains of the mind:

Abo .

Having written down the skin of the minds of minds up to tasks with indulgent obmezhennymi, distanama dvі, not affected by myself, tasks. Tobto, kopatkovu problem tsilochislovogo programvannya (6.1) - (6.4) podilimo dvі problemy y urahuvannyaam minds of the number of evil, meaning some in the optimal plan of relaxed tasks - shotguns. Tse means, by simplex method, rozv'yazuvatimo dvі soi tasks:

The first task: (6.14)

For the mind:

; (6.15)

; (6.16)

- the number of numbers, ; (6.17)

, (6.18)

Another task

(6.19)

For the mind:

, ; (6.20)

; (6.21)

- the number of numbers ; (6.22)

, (6.23)

De - the shot component of the problem of problems (6.1) - (6.4).

(6.14) - (6.18) and (6.19) - (6.23) the spool is sent positively, then it is rooted in the interchanges (6.17) and (6.22). If you know the optimal plan, you should satisfy your expectations, then you must meet the challenges of (6.1) - (6.4). Іnakse poshuk rozv'yaku zadachi trivaє. For a distant rozgaluzhennya vybierymo rozv'yazok tasks in the larger values ​​of tsil'ovoy funktsii, yakschod ydetsya pro maksimizatsyu, і navpaki - z mmenshim znachennyam tsil'ovoy funktsii in razi її мінімізації. Away rozgaluzhennya vikonuetsya doti, docks will not be put into the unhinderedness of the flooring of roses. Zdobuti ostnnіy plan - the optimal.

Rozv'yazuvannya tsilokshilovyh tasks by the method of gnilk and intercourse can be assigned to the enemy. Obviously, it is the task of the duck, the yaku otrimuyut in the process of rozv'yazuvannya vidriznyayatsya vid alternating between one obmezhennyam. Tom for poslovidnogo rozv'yazuvannya tasks mnoye sensu rozv'yazuvati їx simplex method of spoiling. Do you want to be able to use the new method of access (6.18) and (6.23) to the rest of the simplex tables for the different tasks and for the viluchiti (in times of need) nepotibnі "starі" obmezhennia.

(6.18) and (6.23) into the system of interchange of the pochtkovo problems, meaning the conduct of the hyperplanes (straight lines), and the deflection of the bagatogrone (bagatokutnik) of the permissible plans for the tasks of the linear programming in such a way that it can be included in the plan of finding the point of the quartz Bagatoknuta (Figure 6.4). It is permissible, but A is the point to the maximum, then for the method, that between the bagatoknik of the permissible tasks and problems ABCOD is subdivided into two parts directly That +1, scho viklyucha z rozglyadu point A, the coordinate of the Є do not count the number.

The algorithm for the method of heaps between is described:

• The simplex method is to solve the problem (6.1) - (6.3) (without permutation of the number of zymnites).

Yakshchoe sredi Elementov in the intellectually optimal plan for a lot of shotgun numbers, then Tse rozv'yazok - the optimal task plan for the numerical programming (6.1) - (6.4).

The problem (6.1) - (6.3) is not empty (the quotient function is not intermediate, but the system is not nonsingular), then the problem (6.1) - (6.4) is also not rosy.

• If in the intellectually optimal plan, the value of the fractions, then vibira- yut one of the non-numerically numbered ones І визначають її цілу частину .
• Write two obmezhennia, scho vidtinyayut netsilokloslovі rozvichy:

,

.

• The skin of the possessed nerves is attached to the obtuse of the entangled tasks. As a result, the two new tasks are solved in linear programming.
• Have be-yakiy poslidovnost rozv'yazuyut obydvі tasks. At different times, if the number of rozvjazok hocha b odnієї із tasks, znachennа tsіlової functіії цієї tasksі зіщеннять з почаковим значенням. Якщо різниця не більша від a given number e , then the process of rozvozyavannya може бути закінчено. At different times, if the numbers of rozvyazoks are gained in oboj problems, then from the outset, the image of the one that is so much more important than the value of the function. Yakshtogo in both tasks is not enough to solve, then for the future, it is the task for which the problem is more important than the function of the past and the future to the crook.

The problem is solved by the method and between the problem and the butt 6.1.

Rozvjazannya . Vidkynuvshi umovu tsilochislovostі, dystanemo roz'yazok: x 1 = 1, x 2 = . Otzhe, admissible tsіle znachennya x 2 maє zadovolnyati one z hіrіvnostej Abo . Приеднуємо до початкової problems / tasks / tasks / /, /, /, /, /, /,

For tasks I (with menstruation) ) Optimal buddy roz'yazok , , And for tasks II (to the ) - rozv'azok , . Oskilki tsilocheslovogo plan is not known, the process neobhimno prodovzhiti, having taken for a distant rozgaluzhenna persha problem, the optimal plan for the future is more important than functional. It is rooted in the task of I, priemno prienechichi to neї obmezhenna: I . Отримуємо такі дві задачі:

Rozv'yazkom zadachi ІІІ є the plan , , And tasks IV plan , . Obidva rozv'yazki і tsilokshislovi, protesh krasche znachenny tsil'ovoy funkcii zabezpechuet roz'yazok problemi IV. The optimal plan for the target-oriented tasks , , Scho zbigayatsya z rozv'yazkom, otrimanim for the method of Gomory.

The scheme for the process of isolating tasks from the buttstock 6.1 (Figure 6.5) is to explain in absentia the name of the method, Pochatkova problem rozdilyaetsya (gіkluyatsya) on dvі prostіshi, i, yakshcho sredi them not існує tasks with the optimal number of rozvozykom, then the process of sale is sold. Otzhe, vsyo vygljagnuti dії mozhna zobraziti at viglyadі "tree":

Fig. 6.5.

The skin element of such a "tree" is the objective task, which is the optimum plan. Pislya ostderzhannya netsiloksislovogo rozv'yaku slylablenia (tobto without uchivichilochilosvosty) pochatkoy tasks reread Ії on dvі інші з додатковими умов. For them, we hide the problems of the tasks, I, one of the wolves of the bouves are not numbered, then they pushed the process out of the water. The task I introduced the add-ons to the interrogation was reworked in task III of that task IV. Optimal plans for both tasks and tasks, and the plan of tasks IV is more important than the functional, that is, the optimal plan for the tasks of the tasks, the tasks of IV.

Briefly rozglyanemo without applying a single method, a method that can be applied to the type of combinatorial (yogo detail in the literature [12]) - the method of post-analysis analizu variants. The legal scheme for this method has been decimated by the Ukrainian interpretation of VS Mikhalevich, a kind of pratsyuvav at the Kiev Institute of Cybernetics. Ідея кього метод полягає в послідовному повторенні such procedures:

1) rozbittya mnzhini varіantіv rozv'yakіv zadachi na kol'ka pidmnozhin, kozhna z yakih myo specifichnі authorities;

2) vichoristana vischezaznacheniyh vlastostivosti for poshuku logicchnyh superechnosti in the description of multiple copies;

3) viklyuchennya із distant rozglyadu tih pidmnozhin varіantіv rozv'yakіv, in the descriptions of some є logisticny superechnosty.

Otzhe, a technique poslіdnogo analizu varіantіv base on vіdesіvі unpromising varіantіv shche up їх дови. Oskilki u razі vіdsіvu nesperspektivnyh pochatkivnyh partyn varіantіv vіdsіvaєєsya і entire procedure продовжень їх розрахунків, it is possible to reach the economical hour through the speed of the calculating operations. Відсів неперспективних елементів відбувається yak for obmezhenni, so і for tsil'ovoyu funktsiєyu. The basis of the method of the post-mortem analogy variants in the field is viznachennyi rules, for some way zdіysnyuvatisya vіsіv unpromising meaning zmіnnih, the result of chоgo postіyno zvozhuvatimetsya mnozhina znachen, yakії відшукуть the optimum.

Zrozumilo, scho for the skin type of tasks of the quantitative programmvannya formulyuyut specifichny rules for vіdsіvu varіantіv.

Method poslidovogo analizu varіantіv uspіshno zastosovuva дляsya for rozv'yazuvannya rіznomantіnih tasks optimal іnuvannya ta proektovannya. Napriklad, for rozraunku transport hozhizh, rozmіschennya na mnezhi type of wood, the project of the rozpodilnyh elektricchnyh measurer, the optimal parameters of the magistralnyh gas pipelines in particular.