In the present paper a problem of optimal control for a single-product dynamical macroeconomic model is considered. In this model gross domestic product is divided into productive consumption, gross investment, and nonproductive consumption. The model is described by a fuzzy differential equation (FDE) to take into account imprecision inherent in the dynamics that may be naturally conditioned by influence of various external factors, unforeseen contingencies of future, and so forth. The considered problems are characterized by four criteria and by several important aspects. On one hand, the problem is complicated by the presence of fuzzy uncertainty as a result of a natural imprecision inherent in information about dynamics of real-world systems. On the other hand, the number of the criteria is not small and most of them are integral criteria. Due to the above mentioned aspects, solving the considered problem by using convolution of criteria into one criterion would lead to loss of information and also would be counterintuitive and complex. We applied DEO (differential evolution optimization) method to solve the considered problem. 1. Introduction The studies devoted to solving optimal control problems for dynamic economic models have a long history. The models of economic growth attracted a large interest in the area of mathematical economics [1–3]. Recently, intensive revisiting of the growth models took place [4, 5] as a result of the improvements in existing models and progress in development of the associated analytical techniques [6–8]. The intensive research in this direction was also conditioned by the support of the developed countries as the latter were interested in construction of accurate economic growth models to improve their economic development [9–11]. A lot of various mathematical methods of measuring the effectiveness of economic growth were suggested. In [12] for describing the change of production and accumulated R&D investment in a firm, a nonlinear control model is considered and analyzed. They provide a solution of an optimal control problem with R&D investment rate as a control parameter. They also analyze an optimal dynamics of economic growth of a firm versus the current cost of innovation. The results of analytical investigation showed that the optimal control can be of the two types depending on the model parameters: (a) piecewise constant with at most two switchings and (b) piecewise constant with two switchings and containing a singular arc. In [13] the authors provide results of comparison of solution methods for dynamic
References
[1]
K. J. Arrow, “Application of control theory to economic growth,” in Mathematics of the Decision Sciences, vol. 12 of Lecture on Applied Mathematics, part 2, pp. 83–119, American Mathematical Society, Providence, RI, USA, 1968.
[2]
M. Intrilligator, Mathematical Optimization and Economics Theory, Prentice Hall, Englenwood Clirs, NJ, USA, 1981.
[3]
G. M. Grossman and E. Helpman, Innovation and Growth in the Global Economy, MIT Press, Cambridge, UK, 1991.
[4]
C. Watanabe, “Factors goverining a firm's R&D investment. A comparison between individual and aggregate data,” in IIASA, TIT, OECD Technical Meeting, Paris, France, 1997.
[5]
C. Watanabe, “Systems factors goverining afirm's R&D investment. A systems perspective of inter-sectorial technology spillover,” in IIASA, TIT, OECD Technical Meeeting, Laxenburg, Austria, 1998.
[6]
A. Kryazhimskii, C. Watanabe, and Yu. Tou, “The reachability of techno-labor homeostasis via regulation of investments in labor R&D: a model-based analysis,” Interium Report IR-02-026, IIASA, Laxenburg, Austria, 2002.
[7]
A. M. Tarasyev and C. Watanabe, “Optimal dynamics of innovation in models of economic growth,” Journal of Optimization Theory and Applications, vol. 108, no. 1, pp. 175–203, 2001.
[8]
S. A. Reshmin, A. M. Tarasyev, and C. Watanabe, “Optimal trajectories of the innovation process and their matching with econometric data,” Journal of Optimization Theory and Applications, vol. 112, no. 3, pp. 639–685, 2002.
[9]
B. Kwintiana, C. Watanabe, and A. M. Tarasyev, “Dynamic optimization of R&D intensity under the effeect of technology assimilation : econometric identification for Japan's Automotive industry,” Interium Report IR-04-058, IIASA, Laxenburg, Austria, 2004.
[10]
N. Kaldor, “A model of economic growth,” The Economic Journal, vol. 67, no. 268, pp. 591–624, 1957.
[11]
A. Kryazhimskii and C. Watanabe, Optimization of Technological Growth, GENDAITOSHO, Kanagawa, Japan, 2004.
[12]
E. V. Grigorieva and E. N. Khailov, “Optimal control of a nonlinear model of Economic growth,” Journal of Discrete and Continuous Dynamical Systems Supplement, pp. 456–466, 2007.
[13]
S. B. Aruoba, J. Fernández-Villaverde, and J. F. Rubio-Ramírez, “Comparing solution methods for dynamic equilibrium economies,” Journal of Economic Dynamics and Control, vol. 30, no. 12, pp. 2477–2508, 2006.
[14]
R. A. Aliev, “Modelling and stability analysis in fuzzy economics,” Applied and Computational Mathematics, vol. 7, no. 1, pp. 31–53, 2008.
[15]
S. A. Mohaddes, M. Ghazali, K. A. Rahim, M. Nasir, and A. V. Kamaid, “Fuzzy environmental-economic model for land use planning,” American-Eurasian Journal of Agricultural & Environmental Sciences, vol. 3, no. 6, pp. 850–854, 2008.
[16]
Y. Deng and J. Xu, “A fuzzy multi-objective optimization model for energy structure and its application to the energy structure adjustment—in world natural and cultural heritage area,” World Journal of Modelling and Simulation, vol. 7, no. 2, pp. 101–112, 2011.
[17]
M. Pe？i？, S. Raduki？, and J. Stankovi？, “Economic and environmental criteria in Multi-objective programming problems,” Facta Universitatis, vol. 8, no. 4, pp. 389–400, 2011.
[18]
J. Ventura, “A global view of economic growth,” in Handbook of Economic Growth, P. Aghion and S. N. Durlauf, Eds., vol. 1, pp. 1420–1492, Elsevier, 2005.
[19]
J. Kacprzyk and S. A. Orlovski, Eds., Optimization Models Using Fuzzy Sets and Possibility Theory, Reidel Publishing Company, Dordrecht, The Netherlands, 1987.
[20]
C. Carlsson and R. Fuller, “Multiobjective optimization with linguistic variables,” in Proceedings of the 6th European Congress on Intelligent Techniques and Soft Computing (EUFIT '98), vol. 2, pp. 1038–1042, Veralg Mainz, Aachen, Germany, 1998.
[21]
C. Carlsson and R. Fullér, “Fuzzy multiple criteria decision making: recent developments,” Fuzzy Sets and Systems, vol. 78, no. 2, pp. 139–153, 1996.
[22]
R. Fuller, C. Carlsson, and S. Giove, “Optimization under fuzzy linguistic rule constraints,” in Proceedings of Eurofuse-SIC '99, pp. 184–187, Budapest, Hungary, 1999.
[23]
M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, London, UK, 1993.
[24]
L. V. Kantorovich and V. I. Zhiyanov, “The one-product dynamic model of economy considering structure of funds in the presence of technical progress,” Reports of Academy of Sciences of the USSR, vol. 211, no. 6, pp. 1280–1283, 1973.
[25]
V. Leontyev, Input-Output Economies, Oxford University Press, New York, NY, USA, 1966.
[26]
V. F. Krotov, Ed., Bases of the Theory of Optimum Control, The Higher School, 1990.
[27]
L. S. Pontryagin, V. G. Boltyansky, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimum Processes, Hayka, 1969.
[28]
A. B. Vasilyeva and V. F. Butuzov, Asymptotic Decomposition of Decisions Singulyarno Indignant Equations, Science, 1973.
[29]
A. B. Vasilyeva and M. G. Dmitriev, “Singulyarnye of indignation in problems of optimum control//science and equipment Results. It is gray,” Matem. analysis., T. 20. VINITI, pp. 3–77, 1982.
[30]
R. ？stermark, “A fuzzy control model (FCM) for dynamic portfolio management,” Fuzzy Sets and Systems, vol. 78, no. 3, pp. 243–254, 1996.
[31]
B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005.
[32]
L. A. Zadeh, “Toward extended fuzzy logic-a first step,” Fuzzy Sets and Systems, vol. 160, no. 21, pp. 3175–3181, 2009.
[33]
P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, Theory and applications, World Scientific, Singapoure, 1994.
[34]
R. S. Gurbanov, L. A. Gardashova, O. H. Huseynov, and K. R. Aliyeva, “Decision making problem of a one-product dynamic economic model,” in Proceedings of the of the 10th International Conference on Application of Fuzzy Systems and Soft Computing (ICAFS '12), pp. 161–174, Lisbon, Portugal, 2012.
[35]
W. Pedrycz and K. Hirota, “Fuzzy vector quantization with the particle swarm optimization: a study in fuzzy granulation-degranulation information processing,” Signal Processing, vol. 87, no. 9, pp. 2061–2074, 2007.
[36]
Y.-J. Wang, J.-S. Zhang, and G.-Y. Zhang, “A dynamic clustering based differential evolution algorithm for global optimization,” European Journal of Operational Research, vol. 183, no. 1, pp. 56–73, 2007.
[37]
R. Storn and K. Price, “Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces,” Tech. Rep. TR-95-012, ICSI, 1995.
[38]
R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” Journal of Global Optimization, vol. 11, no. 4, pp. 341–359, 1997.
[39]
K. Price, R. Storn, and J. Lampinen, Differential Evolution—A Practical Approach To Global Optimization, Natural Computing Series, Springer, Berlin, Germany, 2005.
[40]
L. A. Gardashova, “Fuzzy neural network and DEO based forecasting,” Journal of Automatics, Telemechanics and Communication, no. 29, pp. 36–44, 2012.
