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Mathematics programmers - Nakonechny S.I.

3.3.2. The other theorem of binary

Між розв'язками спряжених задач крім рівності значен цільових функцій існує тісніший взаємозв'язок. For yogo, the dosage is rozglyanemo dvі sometricheski tasks lennyi programvannya.

Direct task:

(3.20)

.

The two-fold problem:

(3.21)

For rozvjazuvannya tasks simplex method neobhіdno zvedi ikh to the canonical form, for which in the system obmezhen problems (3.20) and (3.21) neobhimdno enter in the form m m n nvіd'єmnih zmіnnih. It is posible to the skinheads of skin problems in the prevention of discomfort of the two tasks.

Analogous:

Otkrim taku vіdpііnnіst між змінними спряжених задач:

There is a theorem in Literature, as a rule, I call the theorems about extra dignity.

Theorem ( another theorem of duality for symmetric problems ) . In order for the plan X * ta Y * for the conjugate tasks of the boules to be optimal, neobhidno i dostatno, shkob vikonuvavisya umov dopovnymyuchoi nezhorstkosti:

(3.22)

. (3.23)

Beforehand . Unpaidness . Nekhai X * ta Y * is the optimal plan for the direct problem of (3.20) i (3.21). The first theorems of two-sidedness,

,

And such components of the vectors X * and Y * are satisfied by the system of the alternation of problems (3.20) and (3.21), then:

, (3.24)

. (3.25)

Multiply (3.24) by , And (3.25) by І підсумуємо праві та ліві частини. Отримаємо:

;

The rights of the detachment of the rest of the nerves are not affected, alle oskilki їh lіvі parts are the same, then the meaning is, at the same time, stink of vich for personal reasons, tobto:

;

Виконаємо переверрення для кожного рівняння:

; (3.26)

. (3.27)

Oskilki , Then in the skin component (3.26) , a , That vikonannya рівняння (3.26) mozhlove lishe at that that, if kozhny dodanka the mind . An analogous minimization is carried out for (3.27), since one can visnovuvati, . Otzhe, neobhіdnіst minds додаткової нежорсткості brought.

Dostatnist . For the sake of vikonuyutsya рівняння

,

, .

It is necessary to extend X * Y * Y * an optimal plan for the straightforward (3.20) of the two (3.21) problems.

At kozhnomu rіvnyannі rozrikєmo arches ta pіdsumuєmo perche rіvnyannya on , And a friend - on . Отримаємо:

;

.

Ліві частини цих рівнянь одкові, отже, . Todi for the first theorem of two-sidedness, the recognition of functions of numerical functions of problems is impossible, one can imagine that X * and Y * are optimal conjugations of symmetric problems. The theorem is completed.

The obvious vzajmozv'azok mizh optimal plans for direct ya dvuhstoy tasks in the setting of a different theory of binary.

Naslidok . Yaksho as a result of the installation of an optimal one-to-one task plan (directly to the system) to the system of exchanging tasks and tasks, then alternating with a strict severity, then the component of the optimal plan for the conjugate tasks is deregulated.

Yaksho і -a component of the optimal plan for one's tasks is dodatno, then it's і- å-å-å å ñ ñ å å å å å å å ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ê ê.

Economical zmіst different theorems bjudestostі stasovno optimal plan X * direct tasks . Yakshcho for vygotovlennya vseiєy produktsii in obyazi, scho viznachaetsya optimal plan X * , vitrati one i- th resource strictly menshi, nizh yogo zagalny otsyag , Then a resource for such a resource (The component of the optimal plan for the two tasks) will be zeroed, that such a resource for giving minds to virbitnitsva is not "centrally".

Якщо ж витрати ресурсу дорівнють його наявному обсягові , Tobto yogo vikoristano povnistyu, then vin і "tsіnnym" for vibrobnitsva, i yogo otsinka Bude strictly blishoju vid zero.

Economically tlmachennja drugoi theorems of the duality of the optimal plan Y * of the two tasks : in the case, in the act of j-that the vicariousness is viked by yak nerivnist, toto all the vitrati on the virginess of the j j- th type of product to be transferred to the product of j , І in the optimal plan of direct tasks and the scope of such products Dorivnyuyu zero.

Якщо витрати на виробництво j- го виде продукіії дорівнють ціні одниці продукції , Then її neobhіdіno vigotovlati v obyazі, yakі viznachaє the optimal plan of direct problems .