Donetsk National Technical University Chesnokov Artem

Solution of problems of multicriterial optimization of dinamic systems in the area of mathematic modelation Scilab

Problems with a lot of criteria meet in different fiels of science and technology but more often in mechanical engineering.

Peculiarities of the method wich is used. Distinction of the method wich is used it is a systematical inspection of manydimensioned fields: as trail dots in a space of porameters are used dots of the sequences which are proportinaly distributed. For this aim was used so called ЛПт - sequences, wich have the best characteristics of proportion among all knowen nowdays sequences wich are proportionaly diatributed.

Othe pacularity of the method consists of next: the designer should not "combine" it means he should not guess what advantage of one criteria can give discounts of othe criteria, it is found out during the dialogue with the programme.

When was given the mathematical model of reseached or designed system, this model is dependent on n pfrfmeters A1 . . . , An, wich can be physical quantities. N – dimensioned space, wich consists of dot A with decart co-ordinate (A1 ..., An) is called the space of parameters. Disigners can point out reasonable limits of change of each parameter, which we shall call parametrical limitation. Except of parametrical limitations the factor of the problem includes functional limitations.

The criterion of quality is called the characteristic of the system which is connected with its quality of monotone dependence. In other words, with other equil factors the system is better when bigger (smaller) criterion.

The reseaching is hold on in three stages.

The first stage: making the table of testings. This stage is hold on without the inteference of the disigner. You choose in consecutive way N of trail dots A 1, wich are proportionaly situated in G. In each dot A 1 is calculated the system and meaning of all criteria F 1 (A1) ..., F k (A1). On each criteria is made the table of testings, in which meaning situate in the order of increasing and point out numbers i1, i2..., in of corresponding testing dots.

The second stage: the choice of the criterial limitation. During this stage the inteference of the disigner takes place. Looking through each table of testing it has to point out a limitation Fv for each criteria.

The third stage: a checking nonemptiness D. Thes stage is made automaticaly. Some criteria are recorded, for example F 1 (A) and corresponding table of testings is looked through.

With the help of strumming of meanings of all criteria in dots A i1,..., Ais you can check is situated here at least one dot wich satisfy all criterial limitations. If such dot A is situated when multitude D is not empty and the problem is soluted.

In other way you should turn tj the second stage and insist on discount during the nomination F v**. If such discounts are impossible when you should turn to the first stage and increase the quantaty N of testing dots, to repeat the second and the third stages with tables of testing of great volume.

At last, when repeated increas of N takes place, dots Aij, wich belong to D are not found we can think that criterial limitation F v**, that we have chosen, are not compatible. Лтг – sequences are the most proportional distributed sequences.

It is often thought that the most proportional inspection of n – dimensioned of a cube are supplied by the cubic trellis, which consists of N = Mn dots. But it is a mistake. Such trellis is used just in onedimensioned way when n = 1. But when n = 2 it is not good and with increas of n "proportion" become worth.

For development of the programme, which realize a manydimensioned way of problems of optimization was chosen Scilab of 4.0 version.

Scilab is a language of high leveled mathematic calculation. Scilab includes a set of instruments and interactive documents. For the first time Scilab seems to be an ancommersial analogue of Matlab. About main distinctions between them you can read on the site of the disigner of Scilab.

Scilab is free disseminated with primary codes. Use, copying, change, dissemination are free. Scilab is a way of mathematical modelation which is powerfull enough. Paying attention to a nessesity of later dissemination of developed programme modul, which take to nessesity to solute systems of differential equation. It becomes clear that a powerfull way of solution of mathematic calculation is nessesery.

Now the modul which make construction of Лпг - sequences is realized. It makes possible a construction of a space of parameters and reseaching of dots which are proportionaly distributed in this space. This modul may be used durind the divelopment of other programme products in the area of Scilab.

Later moduls which realize dialogue with user will be developed. A valuable programme product wich solute a problem of multicriterial optimization of difficult systems will be developed.