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Mathematics programmers - Nakonechny S.I.

5.5.2. The method of potentiating the roving of the transport tasks

A duality theorem for problems (5.1) - (5.4) and (5.22) - (5.23) is formulated to a friend.

To ensure that the planning of the conjugate tasks of the boule is optimal, it is necessary and sufficient, the shoe is vikonuvalisya umov dopovnymyuchoi nezhorstkost:

1) ; (5.24)

2) (5.25)

Zauvazhimo, scho friend of the group of minds for transport tasks is automatic, all tasks are assigned to є рівняннями.

Persha is able to vikonuetsya in two vipadks:

A) yakshto . Other spiivmetshnik , For example (5.23) ( );

B) Yakshtso , Then for the purpose of transport tasks , Тоді ( ).

Necessity and availability of the vikonannya of such minds for the optimality of the plan of straightforward tasks of the boolean is brought to the forefront. 3. Otzhe, yak naslidok druzhye theorems dvuhistosti for transport problems, otrimali neobhіdnі ta takotnyi umvi optimnostnosti planu.

The theorem (the essence of optimality of the basic plan of transport problems ). For an active support program X * = ( xij *), find the numbers ui and vj , for some of them,

1) ui + vj = cij , xij > 0,

2) ui + vj ≤ cij , xij = 0

For all That , Then він є the optimal plan for transport tasks.

Використовойчи наведені исвиіінунуняня розв'язку транспортної задачі, methods of motivating the basic planes and the optimal way of supporting the transport tasks, formulating the algorithm to the method of potentials, which is repeated on the basis of the algorithm of the simplex method.

Algorithm to the method of potentiation is to be stored in these stages:

• Viznachennya type of transport tasks (in response to the shutdown). For neobihodnosti slid zvezit problem before zakrito type.
• Pobudova of the first basic plan of transport tasks in one of the methods.
• Perevirka of the basic task plan for virodzhenst. For neobhіdnosty enter zero postovanya.
• Perevirka the plan of transport tasks for optimal.
• 4.1. Viznachennya potentsialyv for the skin row і стовпчика таблиці транспортної задачі. Potentsiali of the basic plan viznachayut із sistemi рівнянь ui + vj = cij , yakі zapusayutu v vsyh zapovnenyh klіotinok transportnoї tables, kіlkіst yakih dorіvnyuє , And кількість невідомих - . Кількість рівнянь на одне менша, ніж невідомих, to that system is not identified, but to potential alone to give zero value. Після цього всі інші potentііали rozrachovuyut unequivocally.

4.2. Перевірка виконання умови оптимальності для пустих клітин. For the add-on of the potentials, I will overhaul the optimal ui + vj ≤ cij for the nezavovnenii clay tables. Yaksho hocha b for odnієї klіtinіc tsy umowa not vikonuєtsya, tobto ui + vj > cij , then the final plan is not optimal, but it is necessary to go to the new basic plan.

4.3. Вибір змінної for entering into the basis on the offensive kroci. The rule for the transition from one basic plan to the other is in that, in the baseline, vivodjat the song of the world (vector), and on the ії місце enter іншу змінну (vector), яка має покращити значення цільової функції. Analogical operation is performed in the algorithm of the method of potentialities.

Перехід від one basic plan to the next виконють заповненням клітинки, for the anchored impulse of optimality. Yakshto such klitinok kilka, then for zapovnennya vibirajut taku, scho moe naybishe pograshnnia, tobto .

4.4. Pobudova cycle і перехід to the offensive support plan. Vibrana empty klіtina at once in the least prone to become , Otzhe, zich klitin obovyazykov set cycle (the theorem of nalidok § 5.2). In the interiors of the cycle, the construction of a pererahuvanya, yaki, leads to a postoperative postachan production. The leather upper vertex and the cycle sign the first sign, prichomu vіlnіy klіtintsi - the sign "+", and all the others - for chervgostyu signs "-" that "+". In the case of clocks with the "-" sign, the value q І to transfer yogo from empty to clay. One-hour number to add up to the number of digits, you can hide in clichés with the "+" sign, that you can see numbers that are known by the "-" sign. Yakshto znachennuu q vіdpovyde kіlka novoevyh perevezen, then at vidnimanni zalishemo u vidpovidnih klіtinkah zeroovyi magnitudes, transporting in such a way, so that you can save the invisibility of the basic plan.

Vnasledok vovedenogo rule viboru q distaemo noviy proezhny plan, yaky not mestit vіd'єmnih transport and і zadovolnya іmvi transport tasks. Оскільки кількість всіх клітин таблиці, що go in for the cycle, є парною і to half of them, those are the number q to be given, and for half of the meeting, then zagalna suma is carried across all the columns and rows to be filled with nezmіchnyu.

Acyclicity of the new plan is brought. The vector of the minds, which is imprinted on the basis of the cli- nite, the linear combination of the vectors in the basis, which I call at once the cycle, the vectors enter the logical combination in the form of the coefficients with the help of theorems in §5.2. Turn on the cycle of one of the basic vectors to lead to the news system Linearly non-squeezed vectors, the introduction of new basic vectors of a vector mavi two separate distributions through vectors of a transverse basis, but inconclusively. A system of linearly independent vectors in order to acyclical sukinnit klitin tables of transport problems, but po tribno bulo bring.

Otzhe, klіtinka, scho bola vlynoyu, sta zapovnenoyu, and vidpodіnna klіtinka z mіnimalnoyu size xij vvazhaetsya vozhnyoyu. At the result of such a re-routing, the transport of products of a new basic plan of transport tasks.

5. Perevirka nachiv optimichnosti inaccurate support plan. Yaksho ostav optimality vikonuetsya - maєmo optimal plan of transport problems, and then it is necessary to go to the offensive support plan (tobto turn to point 3 of this algorithm).

Zauvazhimo, scho analogous to rozvazyvanynyam zagalnoy problemy linyinogo programuvannya simplex method, yakshcho for perevirki optimal plan for transport tasks for active clitin Vikonuetsya rіvnist , Then tse means, but the problem is alternatives to optimal planning. Otrimati їх mozhna, yakschoo poduduvati cyclis pererozpodilu obyagіv perevezen for vidpovіdnih klіtin.