DOI: https://doi.org/10.15588/1607-3274-2018-3-7

### TWO METHODS FOR CONSTRUCTION OF SUBOPTIMISTIC AND SUBPESSIMISTIC SOLUTIONS OF THE INTERVAL PROBLEM OF MIXED-BOOLEAN PROGRAMMING

#### Abstract

object of the study was a model of the integer programming.

Objective. Development of methods for constructing suboptimistic and subpessimistic solutions of the mixed Boolean

programming interval problem. Two methods for constructing suboptimistic and subpessimistic solutions of mixed Boolean programming problems with interval initial data are introduced. These methods are based on some economic interpretation of the model considered.

Method. Two methods for constructing suboptimistic and subpessimistic solutions of mixed Boolean programming problems

with interval initial data are introduced. These methods are based on some economic interpretation of the considered model. In the

first method a criterion of selecting unknowns for assigning values, which is based on the principle of profit maximum for each unit

of expenditure is introduced. Since the coefficients of the problem are intervals, two strategies are chosen: optimistic and pessimistic.

In the optimistic strategy, the idea of choosing unknowns is used, which corresponds to the maximum ratio of the corresponding

maximum profit to the minimum expenditure. And in the pessimistic strategy, the idea of maximum ratio of the minimum profit to

the maximum expenditure is used. In the second method, the concept of a non-linearly increasing penalty (price) for using a unit of

the remaining resources is introduced, that on the right side is bounded. Taking into account the principles of the above first and

second methods, using this concept of penalty (price), methods for constructing suboptimistic and subpessimistic solutions have been

developed.

Results. The algorithms for constructing suboptimistic and subpessimistic solutions to the interval problem of mixed Boolean

programming are developed.

Conclusions. A software package was developed for constructing suboptimistic and subpessimistic solutions to the interval problem

of mixed Boolean programming. A number of computational experiments have been carried out over random problems of various

dimensions.

#### Keywords

#### Full Text:

PDF (Українська)#### References

Gary M., Johnson D. M. Vichislitelniye mashini i trudnoreshayemiye zadaci. Mir, 1982, 416 p.

Aho А., Hopcroft J., Ullman J. Postroyeniye i analiz

vichislitelnix alqoritmov. Mir, 1979, 536 p.

Libura M. Integer programming problems with inexact

objective function, Control And Cybernetics, 1980,

Vol. 9, No. 4, pp. 189–202.

Roshin V. А., Semenova N. V., Sergiyenko I. V. Dekompozicionniy podxod k resheniyu nekotorix zadac celocislennoqo proqrammirovaniya s netocnimi dannimi,

Journal Vicislitelnoy Matematiki i Matematiceskoy Fiziki,

, Vol. 30, No. 5, pp. 786–791.

Devyaterikova М. V., Kolokolov А. А., Kolosov А. P.

Alqoritmi perebora L-klassov dla bulevoy zadaci o ryukzake s

intervalnimi dannimi. Materiali III Vserossiyskoy konferencii

“Problemi optimizacii I ekonomiceskoye prilojeniye”. Omsk,

Izd-vo Ом ГТУ, 2006, P. 87.

Emelichev V. A., Podkopaev D. P. Quantitative stability analysis for vector problems of 0–1 programming, Discrete Optimitation, 2010, No. 7, pp. 48–63.

Mamedov К. Sh., Маmedli N. О. Metodi postroyeniya suboptimisticeskoqo i subpessimisticeskoqo resheniy chasticno-Bulevoy zadaci o ranche s intervalnimi dannimi, Izv. NAN Azerbaijan, 2016, No. 6, pp. 6–13.

Mamedov К. Sh., Мamedova А. H. Ponyatiya suboptimisticeskoqo I subpessimisticeskoqo resheniy i postroyeniye ix v intervalnoy zadace Bulevoqo proqrammirovaniya, Radio Electronics,

Computer Science, Control, 2016, No. 3, pp. 99–107.

Martello S., Toth P. Knapsack problems, Algorithm and Computers implementations. Chichster, John Wiley & Sons, 1990, 296 p.

Kovalev M. М. Diskretnaya optimizaciya (celocislennoye proqrammirovaniye). Мoscow, U.RSS, 2003, 192 p.

Babayev J. А. Мamedov К. Sh., Mextiyev М. Q. Metodi postroyeniya suboptimalnix resheniy mnoqomernoy zadaci o

rance, Journal Vicislitelnoy Matematiki i Matematiceskoy

Fiziki, 1978, Vol.28, No. 6, pp. 1443–1453.

Аlefeld Q., Herzberger J. Vvedeniye v intervalniye vicisleniya. Translation from eng. Мoscow, Mir, 1987, 360 p.

Toyoda Y. A simplified algorithm for obtaining approximate solutions to zero-one programming problems, Management Science, 1975, No. 12, pp. 1417–1427.

#### GOST Style Citations

Copyright (c) 2018 К. Sh. Mamedov, N. O. Mammadli

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

**Address of the journal editorial office:**

Editorial office of the journal «Radio Electronics, Computer Science, Control»,

Zaporizhzhya National Technical University,

Zhukovskiy street, 64, Zaporizhzhya, 69063, Ukraine.

Telephone: +38-061-769-82-96 – the Editing and Publishing Department.

E-mail: rvv@zntu.edu.ua

**The reference to the journal is obligatory in the cases of complete or partial use of its materials.**