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Mathematics programmers - Nakonechny S.I.

1.4. Багатокритеріальна оптимізація

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Zauvazhimo, scho in the classic statement of problems of mathematical programming, one of the functions is vain, yaka kilkisno is vysnachena. In real economy systems for the role of criterion optimality (efektivnosti) claim kilka tjatkіv pokaznik. Napriklad, the maximum net income for the implementation of the viral production, the maximum of the profitability, the minimum amount of the viril prodigious products, the minimum of the vitrification of the deficit resources. In addition, bazhanim є zastosuvannya kіlokh kriterіїv one-hourly, prichomu vony mozut bouti vzagali nessimnimi. Napriklad, vimoga dosyagti vysokogo efektivnostі vibrobnitsva for mіnіmalnikh vitrat resursіv za gladu posobie matematicheskoi problemy є nekorektnoyu. Мінімальні витрати ресурсів - це нульові витрати, що мають місце за повної відсутності be-yogo process vibrobitvtva. Analogically, the maximum efektivnist can be meted out to the Lisha in different regions of the singers' landscapes (zvichayno not nullovyh) resources. To that korektnimi є statement of tasks of such type: досягти maximum efektivnosti at set vitatah chi досягти given ефекту for мінімальних витрат.

Oskilki not існує єдиного універсального критерію економічної ефективності, then often confuse yourself with the sight of the bagatrictriereal optimization . Wanting the problem of mathematical programming, one of the functional functions is parsed, the mathematical methods are decoupled, and the future planning of the compromise plan is given, and the object is the balancing algorithm of optimization.

Naichastishe way vikoristana bugatokh kriterіїv in the problems of mathematical programming to get up to a piece of objednannya kilkok vichnyh otkaznikiv in one. The number of such methods is [7].

Nehaj u voprosy vverno m criterіv optim'nost'i Fi . Overwhelming criterism may mothers vihlyad sumi okremih otraznikіv efektivnostі з відповідними коефіцієнтами:

, (1.5)

De - додатні чи від'ємні кефіцієнти. Додатні коефіцієнти відповідають тим критеріям, які потрівно максимізувати, and від'ємні - тим, які мінімізуються. Absolute value of medical devices Відповідають prіоритету (важливості) of that chin іншого показника.

On the other hand, it is a task to solve the task of the vibrochnic, then, in order to do so, it is necessary to enter into such quantities, yak otsyag pributku, otrimanoj vid realizatii goods, that ogg, z vid'emnimi - vitrati resursiv (hour, pratsi), sobivartіst oditytsi produkcii.

Crises of criticism can be submitted to the vigil for a shot, de bougie is supposed to have a dolphin showing, yaki nehohimno maximizuvati, pripemo , And in the denomination - dobutok quietly, yakі potrіbno мінімізувати :

(1.6)

Overwhelmingly shortcomings of criterions (1.5), (1.6) є te, scho існує можливість недотню ефективність one критерію компенсувати іншим. Napriklad, znizhennya znachennya vikonannya krestonnyh zamovlen (in (1.6) bude in the chapel) can mozhe kompensuvaetsya zmenshennyam vikoristannya resursiv (znamennik drobu (1.6)). Oskilki okremi magnitudes in the chapel to that banner exchanged proportionally, then the meaning of the shot does not resemble, the prototype of the warehouse on the basis of such rozraunkіv plan can be called to negative nalldikiv.

Takі criterii porіvnjut із unlocking Leo Tolstim jargom "criterium otsinki people" in vyglyadі drobu, de in the tea-pot zaznachayut sravzhnі dostigny people, and in the flag-sign - dumku about himself. Otzhe, yakzscho lyudina mayzhe mnoye dostoinstv (chiselnik drobu budem malim number) і vodnochas y nezovsim vіsutsyanya zarozumіlіst (in the znamenniku - mayzhe zero), then vona budi mati neskincchenno veliku tsinnnost (oskilki be-yake number, podilenee na neskinchenno malu value, let Нескінченність).

Otzhe, before vichoristana zagnyschih udivіv formivaniya tsil'ovikh funktsii neobhidno pіdhoditi zvazheno that is thought out.

There is one method that I asked І. Nikovski [24]. Optimal plan to know beyond the skin of vibrant criterias, giving a lot of meaning to the meaning of the rules of function . For the rest of the year, the problem with a single criterion is posed:

, (1.7)

De - the value of the i-th criterion of optimality in the optimal compromise plan. For such a task, the tasks of the tasks are to be determined for the criterion, which minimizes the minimum value of the modules, often the value of the skin functions, the compromise plan, the optimal values ​​for the optimal values, but all the criterias are important. For vrahuvannya perevag some critics over the oldest dotsilno zastosovuvati zazagalenny criterias this kind:

. (1.8)

Недоліками цих двох способів є, по-перше, жорстке співвідношення між значеннями відхилень критеріїв оптимильності, що значно звужує многожину допустимих планів; In a different way, one value of an active criterion can intersecute a multiplicity of others, a prichomu of such, for some optimal plan for a more economical view of the effec- tive; On-tretє, vіsutnja a technique obsektivnogo viznachennya koefіtsієntіv .

Zvedennya bagatokriterialnyh tasks to tasks in one criterion can be done through a view from a vibrant set of displays of one, a kind of invaluable nayvazhvlyshim - Fk і namagayutsya dopyagti yogo maximum znachennya (yakshcho neobhidno znayte mininim, then dos zmіniti znamenik sign). All інші pokazniki (критерії) є oddryadryadnymi, і on them покладаються обмеження вид: , De Є the lower boundary of the icon of the display, , Yakschoo neobhidno, shchob znachennya otpazhchika not perevishchualo . For vibrotic tasks it is possible to see yak nayvazhlivshy pokaznik efektivnosti pributok і, maximizing the yogo value, dodatkovo enter omegzhennya shodo-profitability vibrobitvtva not lower than the property not in the bottom of the pinnacle. So it's time to enter before the system of poach-like minds of tasks.

Ostannii rozgljanemo so ranks "a method poslidovnih an act ". All of the criteria are necessary for ranking: for declining, it is important: the spoiler of the head, the tell-tale F1 , the impulse of the important F2, and so on. It is important that you need to maximize the value for all criterias (you need to know the minimum, then change the sign of the display). Spokchatka rozv'yazuyetsya task with one parent criterion (znachoditsya znachennya ), Potim recognize the act is small for the absolute value of "action" , The yaku can zmіniti (zmenshiti) znachennya kriteria In addition, the level of maximum (bolshogo) value for the offensive criterion F2 . The magnitude of the "actions" to lie in the control of the accuracy of the rozrachnik in the dostovirnosti of the obchatkovyh danyh. Potim up to sistemi pochatkovichh obmezhen zadachi priedniyot obmezhennia, scho vstanovlyuet riven mozhlivogo vidhilennya show: , І rozv'yazyut new problem with criterion of optimality F2 і etc. The process of launching tasks in such a way of showing, the price of some "act" to reach the results.

Obviously, the bugokotraternal tasks of mathematical programming do not mimic the universal way of outgoing. Otzhe, vibir that korektne zastosuvannya be-yakogo n navedenih hozyativ zalishaetsya for sub'ekt prinyattya rishen. Zavdannya mathematicheskogo programuvannya polagayє in zabezpechennі potrobnuyu kilkistyu naukovoobrazovannoї iznformitsii, on pіdstavі yakії здісьнюється вибір управлінського рішення.