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ISSUE 180, article 7

DOI:https://doi.org/10.15407/kvt180.02.084

Kibern. vyčisl. teh., 2015, Issue 179, pp 83-92.

Chernyshenko Sergei V., Dr (Biology), PhD (Phys. and Math.), Prof., Head of the Department of Applied Mathematics and Social Informatics of Khmelnytsky National University, st. Institutskaya, 11, Khmelnitsky, 29016, Ukraine, e-mail: svc@a-teleport.com

Ruzich Roman V., Assistant of the Department of Applied Mathematics and Social Informatics of Khmelnitsky National University, st. Institutskaya, 11, Khmelnitsky, 29016, Ukraine, e-mail: ninasus@gmail.com

DISCRETE EFFECTS IN CONTINUOUS MODELS OF SUCESSIONS

Introduction. A long-term ecological successions are considered as step-by-step process. The continuous model (model of open Eigen’s hypercycle) is used to describe this process.

The purpose of the paper is to investigate non-linear properties of the system, which define discrete processes that occur in the one.

Results. The multi-dimension case of the model of open Eigen’s hypercycle has been analyzed. It is shown that in many cases the consideration of dynamics of the -dimensional system can be simplified by partial reduction to -dimensional cases.

It is mathematically shown that evolution of system, which is described by the -dimensional model of open Eigen’s hypercycle has, as maximum, stages. Presence and duration of each stage are determined by the size of the ecological niche, as a characteristics of the environment. As an example: if the niche is very small (), there is only one association in the stable state of the ecosystem.

Conclusion. It is shown that the continuous model can describe discrete processes of sucessions. The quasi-discrete dynamics of the system is explained by its bifurcation properties, produced step-by-step changing of the system structure.

Keywords: succession, discrete process, continuous model, Eigen’s hypercycle, bifurcation.

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References

1 Clements F.E. Plant Succession: Analysis of the Development of Vegetation. Washington D.C.: Publ. Carnegi Inst. 1916. 512p. https://doi.org/10.5962/bhl.title.56234

2 Aaviksoo K. Simulating Vegetation Dynamics and Land Use in a Mire Landscape Using a Markov Model. Landscape and Urban Planning, 1995, vol. 31, pp. 129–142. https://doi.org/10.1016/0169-2046(94)01045-A

3 Logofet D.O., Golubyatnikov L.L., Denisenko E.A. Nonhomogeneous Markov Model of Vegetation Succession: A New Perspective of the Old Paradigm. Izvestiya RAS. Biology Series, 1997, no. 5, pp. 613–622 (in Russian).

4 Isaev A.S., Suhovolsky V.G., Buzykin A.I., Ovchinikov T.M. Successional Processes in Forest Communities: Models of Phase Transitions. Khvojnyje borealnoy zony, 2008,vol. XXV, no. 1–2, pp. 9–16 (in Russian).

5 Connell J.H., Slatyer R.O. Mechanisms of Succession in Natural Communities and Their Role in Community Stability and Organization. The American Naturalist, 1977, vol. 111, pp. 1119–1144. https://doi.org/10.1086/283241

6 Chakrabarti C.G., Ghosh S., Bhadra S. Non-equilibrium Thermodynamics of Lotka-Volterra Ecosystems: Stability and Evolution. Journal of Biological Physics, 1995, vol. 21, pp. 273–284. https://doi.org/10.1007/BF00700629

7 McNauhghton S.J., Wolf L.L. Dominance and the Niche in Ecological Systems. Science, 1970, vol. 167, pp. 131–139. https://doi.org/10.1126/science.167.3915.131

8 Rabotnov A.T. Phytocenology. Moscow: MGU Press. 1992. 352 p. (in Russian).

9 Chernyshenko S.V. Nonlinear Analysis of Forest Ecosystems Dynamics. Dnepropetrovsk: Dnepropetrovsk University Press. 2005. 500 p. (in Russian).

10 Chernyshenko S. V., Ruzich R. V. Bifurcation Model of Successions in Ecosystems. Proceedings of 27th European Conference on Modelling and Simulation ECMS 2013. May 27–30 2013. Alesund, Norway, pp. 767–774. https://doi.org/10.7148/2013-0769

11 Berezich I. S., Zitkov N. P. Methods of Calculations. Vol. 1–2. Moscow: Phys.-math. Liter. Main Press, 1959. 620 p. (in Russian).

Received 17.03.2015

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ISSUE 180, article 6

DOI:https://doi.org/10.15407/kvt180.02.066

Kibern. vyčisl. teh., 2015, Issue 179, pp 66-82.

Kotsur Maksim P., Assistant of the Department of Mathematical Control Problems and Cybernetics of Yuriy Fedkovych Chernivtsi National University, Ul. Kotsyubynsky, 2, Chernivtsi, 58012, Ukraine, e-mail: piramidam@rambler.ru

Nakonechniy Аleksandr G., Dr (Phys. and Math.), Prof., Academician-Secretary of the Department of Cybernetics and Systems Analysis, Head of the Department of System Analysis and Decision Theory of the Taras Shevchenko Kiev National University, Pr. Glushkov, 4 D, Kyiv, 03187, Ukraine, e-mail: a.nakonechniy@gmail.com

OPTIMAL CONTROL BY TRANSIENT MODE OF STAGE THERMOELECTRIC COOLER

Introduction. A large variety of optimization problems is related to distributed parameter systems (DPSs), whose behavior is described by partial differential equations. The optimal control theory is widely employed for optimization of lumped parameter systems (LPSs), which are modeled by ordinary differential equations. Development of the theory and methods for obtaining optimal control functions for DPSs is much more difficult as compared with solving such problem for LPSs.

The purpose of the present paper is to obtain optimality conditions and to develop numerical methods for solving the optimization problem of an unsteady one-dimensional process with distributed parameters, as well as their application to optimization of transient thermoelectric cooling.

Methods. A method is proposed for solving of optimal control problem for DPS described by nonlinear partial differential equations of parabolic type with nonstandard boundary conditions. A method consists in coordinate discretization of distributed object and transition to the problem for LPS. Optimal control theory based on the Pontryagin maximum principle can be used for such system optimization.

Results. This method is applied for optimization of transient thermoelectric cooling process. Optimal dependences of current on time have been calculated for stage thermoelectric cooler power supply with the purpose of minimizing the cooling temperature within a preset time interval. Results of computer experiment for one- and two-stage coolers are presented.

Conclusion. The obtained results show that the implementation of optimal functions for cooler power supply can provide the cooling temperature in transient mode by 10 K lower than minimum temperature reached at direct current.

Keywords: distributed parameter system, optimal control, Pontryagin maximum principle, optimization, transient thermoelectric cooling.

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References

1 Ahmed N.U. Distributed Parameter Systems. Encyclopedia of Physical Science and Technology (Third Edition), Academic PRESS, 2003, pp. 561–587. https://doi.org/10.1016/B0-12-227410-5/00183-6

2 Alessandri S.A., Gaggero M., Zoppoli R. Feedback Optimal Control of Distributed Parameter Systems by Using Finite-Dimensional Approximation Schemes. IEEE Transactions on Neural Networks and Learning Systems, 2012, vol. 23, no. 6, pp. 984–995. https://doi.org/10.1109/TNNLS.2012.2192748

3 Guangcao Ji, Clyde Martin. Optimal Boundary Control of the Heat Equation with Target Function at Terminal Time. Applied Mathematics and Computation, 2002, vol. 127, pp. 335–345. https://doi.org/10.1016/S0096-3003(01)00011-X

4 Bokhari M.A., Sadek I.S. Optimal Control of Parabolic Systems with Infinite Time Horizons. Applied Mathematics and Computation, 2008, vol. 206, pp. 678–684. https://doi.org/10.1016/j.amc.2008.05.048

5 El-Farra N.H., Armaou A., Christofides P.D. Analysis and Control of Parabolic PDE Systems with Input Constraints. Automatica, 2003, vol. 39, pp. 715–725. https://doi.org/10.1016/S0005-1098(02)00304-7

6 Kucuka I., Sadeka I., Yilmazc Y. Optimal Control of a Distributed Parameter System with Applications to Beam Vibrations Using Piezoelectric Actuators. Journal of the Franklin Institute, 2014, vol. 351, issue 2, February, pp. 656–666. https://doi.org/10.1016/j.jfranklin.2012.10.008

7 Zuyev A. Partial Asymptotic Stabilization of Nonlinear Distributed Parameter Systems. Automatica, 2005, vol. 41, pp. 1–10. https://doi.org/10.1016/S0005-1098(04)00240-7 https://doi.org/10.1016/j.automatica.2004.08.009

8 Zuazua E. Propagation, Observation and Control of Waves Approximated by Finite Difference Methods. SIAM Rev., 2005, vol. 47, no. 2, pp. 197–243. https://doi.org/10.1137/S0036144503432862

9 Krstic M., Guo B.-Z., Smyshlyaev A. Boundary Controllers and Observers for the Linearized Schrodinger Equation. SIAM J. Control Opt., 2011, vol. 49, no. 4, pp. 1479–1497. https://doi.org/10.1137/070704290

10 Subas M. An Otimal Control Problem Governed by the Potential of a Linear Schrodinger Equation. Applied Mathematics and Computation, 2002, vol. 131, pp. 95–106. https://doi.org/10.1016/S0096-3003(01)00161-8

11 Alvarez-Vazquez L.J., Fernandez F.J. Optimal Control of a Bioreactor. Applied Mathe-matics and Computation, 2010, vol. 216, pp. 559–575. https://doi.org/10.1016/j.amc.2010.03.097

12 Ryu Sang-Uk, Yagi Atsushi. Optimal Control for an Adsorbate-Induced Phase Transition Model. Applied Mathematics and Computation, 2005, vol. 171, pp. 420–432. https://doi.org/10.1016/j.amc.2005.01.044

13 Hoffman K.H., Jiang L. Optimal Control of a Phase Field Model for Solidification. Numer. Funct. Anal. and Optimiz., 1992, vol. 13, no. 1&2, pp. 11–27. https://doi.org/10.1080/01630569208816458

14 Muresan V., Abrudean M., Unguresan M., Colosi T. Cascade Control of a Residual Water Blunting System. Advances in Electrical and Computer Engineering, 2014, vol. 14, no. 2, pp. 135–144. https://doi.org/10.4316/AECE.2014.02022

15 Egorov A.I. Optimal control by thermal and diffuse processes. Moscow: Science, 1978, 463 p. (in Russian).

16 Geering Hans P. Optimal Control with Engineering Applications. NewYork, Berlin, Heidelberg: Springer-Verlag, 2007. 134 p.

17 Sethi Suresh P. Optimal Control Theory. Applications to Management Science and Economics. Springer Science+Business Media, Inc., 2000. 505 p.

18 Anatychuk L.I. Optimal Control by Properties of Thermoelectric Materials and Devices. Chernovtsy: Prut, 1992. 263 p. (in Russian).

19 Pontryagin L.S., Boltyanski V.G., Gamkrelidze R.S., Mischenko E.F. The Mathematical Theory of Optimal Processes. Moscow: Nauka, 1976. 392 p. (in Russian).

20 Butkovskii A.G. Theory of Optimal Control by Distributed Parameters Systems. Moscow: Nauka, 1965. 474 p. (in Russian).

21 Sirazetdinov T.K. Optimization of Distributed Parameters Systems. Moscow: Nauka, 1977. 479 p. (in Russian).

22 Fursikov A.V. Optimal Control by Distributed Systems. Theory and Application. Novosibirsk: Nauchnaya Kniga, 1999. 352 p. (in Russian). https://doi.org/10.1090/mmono/187

23 Shevyakov A.A., Yakovleva R.V. Control by Thermal Objects with Distributed Parameters. Moscow: Energoatomizdat, 1986. 208 p. (in Russian).

24 Anatychuk L.I. Thermoelectricity. Vol. II. Thermoelectric Energy Convectors. Kiev, Chernovtsy: Institute of Thermoelectricity, 2003. 376 p.

25 Yang Ronggui, Chen Gang, Kumar A. Ravi, Snyder G. Jeffrey, Fleurial Jean-Pierre. Transient Cooling of Thermoelectric Coolers and its Applications for Microdevices. Energy Conversion and Management, 2005, vol. 46, pp.1407–1421. https://doi.org/10.1016/j.enconman.2004.07.004

26 Kotsur M.P. Approximate Method of Optimal Control in Problems of Transient Thermoelectric Cooling. Journal of Computational and Applied Mathematics, 2013, vol. 114, no. 4, pp. 37-47. (in Ukrainian)

27 One-stage thermoelectric modules. Available at: http://www.kryothermtec.com/ru/standsard-single-stage-thermoelectric-coolers.html. Two-stage thermoelectric modules. Available at: http://www.kryothermtec.com/ru/2-stage-thermoelectric-coolers.html.

Received 18.02.2015

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ISSUE 180, article 5

DOI:https://doi.org/10.15407/kvt180.02.045

Kibern. vyčisl. teh., 2015, Issue 179, pp 45-65.

Pavlov Vadim V., Dr (Engineering), Prof., Head of the Department of Intellectual Control of International Research and Training Center for Information Technologies and Systems of National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, av. Acad. Glushkova, 40, Kiev, 03187, Ukraine, e-mail: dep185@irtc.org.ua

Volkov Aleksandr E., PG (Postgraduate) of the Department of Intellectual Control of International Research and Training Center for Information Technologies and Systems of National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, av. Acad. Glushkova, 40, Kiev, 03187, Ukraine, e-mail: alexvolk@ukr.net

Voloshenyuk Dmitrii A., PG (Postgraduate) of the Department of Intellectual Control of International Research and Training Center for Information Technologies and Systems of National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, av. Acad. Glushkova, 40, Kiev, 03187, Ukraine, e-mail: P-h-o-e-n-i-x@ukr.net

INVARIANT NET-CENTRIC CONTROL SYSTEM FOR CONFLICT AVOIDANCE OF AIRCRAFTS IN THE LANDING PHASE

Introduction. The question of the need to create a control system of conflict situations between the aircrafts in the landing phase is discusses.

The purpose of this research is to create a method and system of conflict resolution between the aircrafts on the route of flight, takeoff and landing phases with the condition to provide a high and guaranteed level of flight safety. The approach considered in the article will be based on the principles of using the network-centric technologies and the theory of invariance.

Results. The expected result of this work is the creation of a new universal control system of conflict situations between the aircrafts based on network-centric technologies and principles of the theory of invariance, which will meet all the requirements of modern air traffic management (ATM) to provide a guaranteed level of safety.

Conclusion. It is shown that a new approach to the problem of creating a control system of conflict situations between the aircrafts based on research in the field of differential games and the theory of invariance is effective.

Keywords: net-centric system, flight safety, invariance, conflict situations, differential games, free flight.

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References

  1. Eurocontrol. Airspace Strategy for the ECAC States. ІСАО: 2001, 91 p. (in Russian).
  2. Harchenko V.P., Argunov G.F, Zakora S.A. et al. The risks of collision and the flight level of aircrafts. Кiev: NAU, 2011, 326 p. (in Russian).
  3. Zakora S.A. Classification of сonflict resolution modeling methods for free flight. Bulletin of the National Aviation University, 2005, no. 1, pp. 42–74 (in Russian).
  4. The ICAO Global Air Navigation Plan for 2013–2028 years. ІСАО: Canada, 2013.
  5. Krasovskiy N.N., Subbotin A.I. The positional differential games. Moscow: Science, 1974, 458 p. (in Russian).
  6. Krasovskiy N.N. Game Problems of counter movements. Moscow: Science, 1970, 424 p. (in Russian).
  7. Chikriy A.A. The guaranteed result in game problems of traffic control. Proceedings of the RAS Institute of Mathematics, 2010, pp. 223–232. (in Russian).
  8. Pshenichnyiy B.N., Chikriy A.A. The problem of collision avoidance in differential games. Bulletin of Computational Mathematics and Physics, 1974, no. 6, pp. 1416—1426 (in Russian).
  9. Bodner V.A. Aircraft Control System. Moscow: Mashinostroenie, 1973, 501 p. (in Russian).
  10. Ayzeks R. The differential games. Moscow: Mir, 1967, 480 p. (in Russian).
  11. Pavlov V.V. The conflicts in technical systems. Кiev: Vyscha shkola, 1982, 183 p. (in Russian).
  12. Kuntsevich V.M. Optimal control of convergence of conflicting moving objects under uncertainty. Cybernetics and systems analysis, 2002, no. 2, pp. 95–104 (in Russian).
  13. Zolotuhin V.V. Some actual problems of air traffic control. Proceedings of MFTI, 2009, no. 3, pp. 94–114 (in Russian).
  14. Harchenko V.P. Aircraft conflicts resolution by course maneuvering. Bulletin of the National Aviation University, 2011, no. 2, pp. 15–20 (in Russian).
  15. Mhitaryan A.M. Aircraft flight dynamics. Moscow: Mashinostroenie, 1978. 424 p. (in Russian).
  16. Bochkarev V.V., Kravtsov V.F., Kryizhanovskiy G.A. The concept and systems of CNS/ATM in civil aviation. Moscow: Akademkniga, 2003, 415 p. (in Russian).
  17. Pavlov V.V. The invariance and autonomy of nonlinear control systems. Кiev: Naukova Dumka, 1971, 272 p. (in Russian).

Received 23.02.2015

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ISSUE 180, article 4

DOI:https://doi.org/10.15407/kvt180.02.034

Kibern. vyčisl. teh., 2015, Issue 179, pp 34-44.

Mashchenko Sergei O., Dr (Phys. & Math.), Professor of the Deparment of System Analysis and Decision Theory of Taras Shevchenko National University of Kyiv, Acad.Glushkova Ave., 4, Kiev, 03187, Ukraine, e-mail: msomail@yandex.ua

Shusharin Yurii V., PhD (Phys. & Math.), Associate Professor of the Department of Higher Mathematics of the Vadim Hetman Kiev National Economic University, Victory Ave., 54/1, Kyiv, 03680, Ukraine, e-mail: shusharin@meta.ua

MINIMAX REGRET CRITERION IN DECISION MAKING PROBLEMS WITH THE FUZZY SET OF STATES OF THE ENVIRONMENT

Introduction. In this work, the problem of alternatives rational choice in the conditions of uncertainties with the fuzzy set of states of the environment is considered. A need for the solving of such problems arises, when a decision making person cannot expressly indicate, what states of nature will affect the consequences of the alternatives choice in a problem which was appeared at the moment of the decision making. In this case, it can only set the belonging function of a fuzzy set of relevant nature states.

The purpose of the article is the application of the known minimax regret principle for construction of a fuzzy criterion of minimax regret in the conditions of uncertainties with the fuzzy set of nature states.

Methods. The methods of the fuzzy set theory, fuzzy mathematical programming, multicriterion optimization are used in this work.

Results. It is suggested to estimate every alternative of the decision making problem by a fuzzy set of the guaranteed regrets, which can be obtained as a set of maximal values function of results utility on the fuzzy set of nature states. The solving of the problem with a fuzzy set of nature states by the fuzzy criterion of minimax regret is determined as a fuzzy set of non-dominated alternatives according to a specially built fuzzy relation of preference. The method of choice of the best-by-regret alternative with the degree of membership to the fuzzy set of decisions not less than the set number is offered. For this purpose, it is suggested to solve the system of two problems of mathematical programming of the special kind.

Conclusion. The offered method of problem solving in the conditions of uncertainties with the fuzzy set of nature states can be easily generalized in the case of a fuzzy set of alternatives and fuzzy estimations of results utility. For this purpose, it is enough to use the known technique of defuzzification.

Keywords: decision making, uncertainties, fuzzy set, the L. Savage criterion, maximizing decision.

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References

  1. Truhaev R.I. Models of decision making in the conditions of uncertainties. Moscow: Science, 1981, 258 p. (in Russian).
  2. Mashchenko S.O. Generalization Germeyer’s criterion in the decision making problem in conditions of uncertainty with the fuzzy set of nature states. Journal of Automation and Information Sciences, 2012, no. 5, pp. 102–110 (in Russian). https://doi.org/10.1615/JAutomatInfScien.v44.i10.20
  3. Orlovsky S.A. Problems of decision making at fuzzy initial information. Moscow: Science, 1981, 208 p. (in Russian).

Received 17.02.2015

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ISSUE 180, article 3

DOI:https://doi.org/10.15407/kvt180.02.025

Kibern. vyčisl. teh., 2015, Issue 179, pp 25-33.

Savchenko Evgenia A., PhD (Engineering), Senior Researcher of Department of Information Technologies of inductive modeling, International Research and Training Center for Information Technologies and Systems of National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, av. Acad. Glushkova, 40, Kiev, 03187, Ukraine, e-mail: savchenko_e@meta.ua

TECHNOLOGY FOR SOLVING THE PROBLEM OF MODELING AND FORECASTING BASED ON INDUCTIVE APPROACH

Introduction. The advantage of inductive algorithms is in their ability to automatically find dependencies hidden in a sample of experimental data. Combinatorial algorithms of GMDH (group method of data handling) are the main inductive modeling algorithms. These algorithms applied to real problems showed that it’s not always possible to unambiguously determine a model by one criterion. Method of a model after-determination based on the Combinatorial GMDH algorithm is developed for such case. A technology based on the combinatorial GMDH algorithm and the after-determination method was developed for the modeling and forecasting.

The purpose of this article is to develop the methodology and technology for modeling and forecasting on the experimental data sample based on the combinatorial algorithm GMDH method and the after-determination method. They will help to find the optimal model in real applications.

Results. A technology for solving the problem of modeling and forecasting on the basis of the inductive approach was developed and described. This approach is based on the combinatorial algorithm GMDH method and completions. This technology, based on a sample of experimental data, automatically finds the object model or process using two external selection criteria: accuracy and bias. The developed computer technology was tested in solving applied problems of modeling and prediction: in problems of diabetes in a home-based monitoring and problems of modeling the interaction of ions with the surface of the jet gas materials.

Conclusion. A computer technology that provides an effective solution for the problems of modeling and prediction of the experimental data was developed and described. Numerical examples demonstrate its efficiency. This technology provides increased noise immunity models due to the consistent application of external criteria GMDH: the criterion of regularity and bias. This technology was used in real applications for modeling and forecasting and its effectiveness has been confirmed.

Keywords: inductive approach, combinatorial algorithm of group method of data handling, modeling, forecasting, technology.

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References

  1. Ivakhnenko A.G. Noise–immunity of modeling. Kiev: Naukova Dumka, 1985, 215 p.(in Russian).
  2. Madala H.R. Inductive Learning Algorithms for Complex Systems Modeling. Boca Raton: CRC Press Inc., 1994, 384 p.
  3. Ivakhnenko A.G. Group method of data handling as competitor to the method of stochastic approximation. Automatics, 1968, no. 3, pp. 64–78 (In Russian).
  4. Stepashko V.S. Combinatorial GMDH algorithm with optimal scheme of models sorting. Automatics, 1981, no. 3, pp. 31–36 (In Russian).
  5. Ivakhnenko A.G. Investigation of Efficiency of Additional Determination Method of the Model Selection in the Modeling Problems by Application of the GMDH Algorithm. Journal of Automation and Information sciences. Begell House: Inc. Publishers, 2008, vol. 40, no. 3, pp. 47–58.
  6. Savchenko E.A. Analytical and Numerical Study of the Selective Properties of the Errors Unbiasedness Criterion in the Problems of Inductive Modeling. Journal of Automation and Information sciences. Begell House: Inc. Publishers, 2012, vol. 44, no. 4, pp. 1–12.
  7. Ivahnenko A.G. Conception of the successive algorithmic approaching (lowering) to the exact decision of interpolation tasks of artificial intelligence. Cybernetics and computing engineering, 1999, vol. 124, pp. 40–60 (In Russian).
  8. Savchenko E.A. Preprocessing of data sample in inductive modeling problem. Control Systems and Computers, 2015, N2, pp. 82–87.
  9. Ivakhnenko A.G. Application of Algorithms of the Method of Batch Assessment of Arguments for Recovering Missed Data and Prediction of the Glucose Level in Blood on at Home Diabetes Monitoring. Journal of Automation and Information sciences. Begell House: Inc. Publishers, 2002, vol. 34, no. 6, pp.123–133.
  10. Savchenko E.А. Analysis of the selective properties of the GMDH criteria when applying them consistently. Modeling and control as ecological and economic systems of the region. Kiev: IRTC ITS, 2008, no. 4, pp. 199–210 (In Russian).

Received 25.03.2015

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ISSUE 180, article 2

DOI:https://doi.org/10.15407/kvt180.02.015

Kibern. vyčisl. teh., 2015, Issue 179, pp 15-24.

Dotsenko Sergey I., PhD (Phys. & Math.), Senior Researcher, Associate Professor, Department of Operations Research, Faculty of Cybernetics of the Taras Shevchenko National University of Kyiv, Pr. Glushkov, 4 D, Kyiv, 03187, Ukraine, e-mail: sergei204@ukr.net

SOLUTION OF THE PROBLEM OF OPTIMAL CHOICES WITH A GROUP BROWSING BY A GAME-THEORETIC APPROACH

Introduction. The problem of optimal choice in the case when the objects are divided into groups and carried out the simultaneous viewing of candidates in each group was considered by Bruss T. If watching the group of candidates is similar to the classical problem and the group is presented by the best candidate among all previously viewed items (such an element is called a maximum) to make a decision — choose this candidate and finish viewing or reject it and continue — the returning to the previously rejected candidates prohibited.

In this case, the optimal rule for selecting the best candidate is based on the so-called «choice theorem» (or «Bruss theorem»).

For the particular case of two groups the search strategy is trivial — namely to ignore the smaller group and to view the bigger one. However, if this case is considered as two person game, the problem appeared to be intriguing.

The purpose of the article is to find Nash equilibrium for two persons game, associated to group search secretary problem at the following set of rules.

1)  Each player makes his choice at his own set of elements.

2)  At the beginning the set of searched elements are divided at random into two subsets according to uniform distribution.

3)  Each of two players searches the best element (i.e. solve the secretary problem for two groups). After that, the prize is paid to one or two players according to the following rules. If one of the players made his lucky choice and the other one not, then the first one got 1. If the both players made their lucky choice at the same group then they share the price and got 1/2 each. If one of the players got his lucky choice at the first group, and another one at the second group, then the first one got 1, and the second one got nothing.

Results. For the considered game the Bayes-Nash equilibrium is obtained for three different cases. Equilibrium points are shown at two-dimensional diagram. Depending on the problem statement, Nash equilibrium area may take different shapes — either single point (cases 1 and 2) or family of points inside the closed curve (case 3). In first two cases, the slight effect of anarchy is observed.

Conclusion. The general principle, that the game situation solutions are based on even trivial optimization problems, makes these solutions to be complicated. The equilibrium situations of the game were found based on the concept of a complex rational behavior for each of the cases.

Keywords: problem of optimal choice, threshold strategy, group search, Bayes-Nash equilibrium, the price of anarchy.

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References

  1. Thomas Bruss. Sum the odds to one and stop. The annals of probability, 2000, vol. 28, no. 3, pp. 1384–1391. https://doi.org/10.1214/aop/1019160340
  2. V. Mazalov. Mathematical theory of games and its applications. Saint Peterburg: Lan, 2010, 446 p. (in Russian).

Received 30.03.2015

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ISSUE 180, article 1

DOI:https://doi.org/10.15407/kvt180.02.004

Kibern. vyčisl. teh., 2015, Issue 179, pp 4-14.

Surovtsev Igor V., PhD (Engineering), Senior Researcher of System Modeling Department of International Research and Training Center for Information Technologies and Systems of National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, av. Acad. Glushkova, 40, Kiev, 03187, Ukraine, e-mail: igorsur52@gmail.com

TRANSFORMATION OF DATA STRUCTURE IN DETERMINING THE CONCENTRATION BY METHODS OF INVERSION CHRONOPOTENTIOMETRY

Introduction. The complexity of direct measurement of the inversion time for the original signal in determining the concentration of toxic elements by inversion chronopotentiometry in the sample solution was not possible to determine its less than 0.1 mkg/ml.

Purpose. Necessary to create information technology of measurement the concentration of toxic elements in liquid tests of objects surrounding by methods of inversion chronopotentiometry, which let possibility essentially to increase sensitiveness and reliability in determent of the concentration.

Results. Using work out of information technology determining the concentration of toxic elements in liquid tests of inversion chronopotentiometry in apparatus the analyzer allow to increase until 14 elements, to increase sensitiveness until 0,0001 mkg/ml and to improve repetition of measurements the concentration.

Conclusion. Information technology has a universal character and can be applied for the analysis of signals of different nature, in which the values are monotonically increasing or decreasing.

Keywords: transformation of the data structure, methods of inversion chronopotentiometry, modeling, information technology.

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References

  1. Karnaukhov A.I., Grynevych V.V., Skobets E.M. Differential variant of inversion chronopotentiometry with a given resistance in oxidative circuit. Ukrainian chemical journal, 1973, no. 39, pp. 710–714 (in Ukrainian).
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Received 03.03.2015