#### DMCA

## Application of DEO Method to Solving Fuzzy Multiobjective Optimal Control Problem

### Citations

436 |
Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces
- Storn, Price
- 1997
(Show Context)
Citation Context ...them have their modified versions, such as Group Principle GA with multipoint crossover and mutation, DE + Clustering [40], and Multifitness function GA. Among many other evolutionary algorithms DE is one of the highest performance. DE is a faster and improved version of GA. As opposite to GA, which uses binary coding, DE uses real coding of float point numbers. The advantage of DE over standard GA is higher performance, efficiencies for numerical problems, not using time-consuming “coding/decoding” transformations, chromosome/gen structures, and other manipulations with binary structures. In [38] a heuristic approach for minimizing possibly nonlinear and nondifferentiable continuous space functions is presented. DEO method is a simple and efficient heuristic for global optimization over continuous spaces. By means of an extensive test bed it is demonstrated that DEO method converges faster and with more certainty than many other acclaimed global optimization methods. The DEO method requires few control variables, is robust, easy to use, and lends itself verywell to parallel computation. It should be noted that dynamic programming is very computationally intensive and even more for fuz... |

243 | A model of economic growth,” - Kaldor - 1957 |

203 | Innovation and Growth in the Global Economy - Grossman, Helpman - 1991 |

108 | Comparing solution methods for dynamic equilibrium economies
- Aruoba, Fernández-Villaverde, et al.
- 2006
(Show Context)
Citation Context ...ribing 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 equilibrium economies in terms of computing time, implementation complexity, and accuracy. The considered methods are the stochastic neoclassical growth model, the finite elements method, Chebyshev polynomials, and value function iteration. Authors conduct analysis of the obtained results. Modeling and stability analysis in fuzzy economics is discussed in [14]. In this paper, the author considers different problems of fuzzy economics, mainly time path and stability of fuzzy dynamics of economic systems, linguistic model... |

101 |
Fuzzy Sets and Interactive Multiobjective Optimization,
- Sakawa
- 1993
(Show Context)
Citation Context ... series of theoretical models which have a common description of technology and preferences but are of different assumptions concerning trade frictions. It is shown that on the base of comparison of the predictions of these models one can identify a variety of channels through which trade influences the dynamics and geographical distribution of world income. The combination of fuzzy logic tools with multiobjective optimization and multicriteria decision making has a great relevance in the literature [19–23]; for example, fuzzy sets are considered for interactive multiobjective optimization in [22, 23]. A single-product dynamical macroeconomic model was first suggested by Kantorovich and Zhiyanov in [24]. They also suggested a principle of differential optimization for the developed model. Leontyev [25] suggested a single-product dynamic macroeconomic model in which gross domestic product is divided into productive consumption, gross investment, and nonproductive consumption. In the existing works, a qualitative analysis of economic systems and development of analytic algorithms of optimization are based on sufficient conditions of optimality [26] or the maximum principle [27]. Nowadays, an... |

36 | Fuzzy multiple criteria decision making: recent developments,” - Carlsson, Fuller - 1996 |

31 | Application of control theory to economic growth,” - Arrow - 1968 |

28 |
Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems
- Bede, Gal
(Show Context)
Citation Context ...tainty is mainly captured by using stochastic techniques. However, the use of stochastic techniques is based on crucial assumptions which notably constrain its use for adequate modelling of realworld economic uncertainty. These methods can be successively applied when sufficient knowledge is available on the probability density functions of all the variables involved in the computations, which is a rather rare case. In contrast, very often prediction or estimation of variables of interest is subjective, for example, expert-opinion based. The main sources of possibilistic uncertainties include [31]. In order to take into account real-world uncertainty in formalmodels, they usually use sensitivity analyses.However, this is not sufficient to investigate interaction of uncertainties and their dynamics. In order to deal with possibilistic Zadeh suggested the fuzzy set theory and fuzzy logic [32]. These theories have passed a long way and provided a lot of successful applications. In particular, control systems based on fuzzy logic were successfully applied for control of various complex and uncertain objects and provided better results than their classical counterparts. Fundamental approach... |

26 | Differential Evolution—A Practical Approach To Global Optimization, Natural Computing Series, - Price, Storn, et al. - 2005 |

9 |
A dynamic clustering based differential evolution algorithm for global optimization,”
- Wang, Zhang, et al.
- 2007
(Show Context)
Citation Context ...0 , ... |

6 | Multiobjective optimization with linguistic variables,” - Carlsson, Fuller - 1998 |

5 |
A fuzzy control model (FCM) for dynamic portfolio management,”
- Ostermark
- 1996
(Show Context)
Citation Context ...rom a series of disadvantages. From one side, the existing approaches are not developed to deal with uncertainty inherent in economic dynamics but are based classical mathematical formalism. Mainly, there exist two types of uncertainty related to economic dynamics: probabilistic and possibilistic (fuzzy). Modelling of probabilistic uncertainty requires the use of very restrictive assumptions including availability of good statistical data. Modelling of fuzzy uncertainty is a more realistic task and allows us to describe imprecise evaluation of variables of interest coming from human expertise [30]. In the existing works, economic uncertainty is mainly captured by using stochastic techniques. However, the use of stochastic techniques is based on crucial assumptions which notably constrain its use for adequate modelling of realworld economic uncertainty. These methods can be successively applied when sufficient knowledge is available on the probability density functions of all the variables involved in the computations, which is a rather rare case. In contrast, very often prediction or estimation of variables of interest is subjective, for example, expert-opinion based. The main sources ... |

4 |
Toward extended fuzzy logic-a first step,” Fuzzy Sets and Systems,
- Zadeh
- 2009
(Show Context)
Citation Context ...ble on the probability density functions of all the variables involved in the computations, which is a rather rare case. In contrast, very often prediction or estimation of variables of interest is subjective, for example, expert-opinion based. The main sources of possibilistic uncertainties include [31]. In order to take into account real-world uncertainty in formalmodels, they usually use sensitivity analyses.However, this is not sufficient to investigate interaction of uncertainties and their dynamics. In order to deal with possibilistic Zadeh suggested the fuzzy set theory and fuzzy logic [32]. These theories have passed a long way and provided a lot of successful applications. In particular, control systems based on fuzzy logic were successfully applied for control of various complex and uncertain objects and provided better results than their classical counterparts. Fundamental approach to modelling behaviour of a dynamical system under possibilistic uncertainty is fuzzy differential equations (FDEs) [33]. In the present paper we suggest an approach to decision making in a single-product dynamic economic model. The considered problem is represented as a multiobjective optimal con... |

3 | Optimal dynamics of innovation in models of economic growth,” - Tarasyev, Watanabe - 2001 |

3 | Optimal trajectories of the innovation process and their matching with econometric data,” - Reshmin, Tarasyev, et al. - 2002 |

3 | Fuzzy environmental-economic model for land use planning,”
- Mohaddes, Ghazali, et al.
- 2008
(Show Context)
Citation Context ... stability of fuzzy dynamics of economic systems, linguistic modelling of economic agents, and neurofuzzy time-series forecasting of macroeconomic processes. One of the main problems considered in [14] is stability of various macroeconomic dynamical systems described by fuzzy differential equations which are used to take into account uncertainty inherent in economic dynamics. Hindawi Publishing Corporation Applied Computational Intelligence and So Computing Volume 2014, Article ID 971894, 7 pages http://dx.doi.org/10.1155/2014/971894 2 Applied Computational Intelligence and Soft Computing In [15] a fuzzy multiobjective linear programming for determination of the optimal land use of regions by taking into account economic, social, and environmental objectives and implications. Reference [16] is devoted to the low-carbonized adjustment of energy structure in the area of world natural and cultural heritage. The authors suggest a fuzzy multiobjective optimization to investigate the relationship between energy and social economics and environment. The use of multiobjective approach is justified by the fact that sustainable development is obtained as a tradeoff between economic growth and e... |

3 | Optimization under fuzzy linguistic rule constraints,”
- Fuller, Carlsson, et al.
- 1999
(Show Context)
Citation Context ... series of theoretical models which have a common description of technology and preferences but are of different assumptions concerning trade frictions. It is shown that on the base of comparison of the predictions of these models one can identify a variety of channels through which trade influences the dynamics and geographical distribution of world income. The combination of fuzzy logic tools with multiobjective optimization and multicriteria decision making has a great relevance in the literature [19–23]; for example, fuzzy sets are considered for interactive multiobjective optimization in [22, 23]. A single-product dynamical macroeconomic model was first suggested by Kantorovich and Zhiyanov in [24]. They also suggested a principle of differential optimization for the developed model. Leontyev [25] suggested a single-product dynamic macroeconomic model in which gross domestic product is divided into productive consumption, gross investment, and nonproductive consumption. In the existing works, a qualitative analysis of economic systems and development of analytic algorithms of optimization are based on sufficient conditions of optimality [26] or the maximum principle [27]. Nowadays, an... |

3 |
Fuzzy vector quantization with the particle swarm optimization: a study in fuzzy granulationdegranulation information processing,”
- Pedrycz, Hirota
- 2007
(Show Context)
Citation Context ...0 , ... |

2 |
Factors goverining a firm’s R&D investment. A comparison between individual and aggregate data,”
- Watanabe
- 1997
(Show Context)
Citation Context ... 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 ana... |

2 | The reachability of techno-labor homeostasis via regulation of investments in labor R&D: a model-based analysis,” Interium Report IR-02-026, - Tou - 2002 |

2 | Optimization of Technological Growth, - Kryazhimskii, Watanabe - 2004 |

2 |
Modelling and stability analysis in fuzzy economics,”
- Aliev
- 2008
(Show Context)
Citation Context ...nding 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 equilibrium economies in terms of computing time, implementation complexity, and accuracy. The considered methods are the stochastic neoclassical growth model, the finite elements method, Chebyshev polynomials, and value function iteration. Authors conduct analysis of the obtained results. Modeling and stability analysis in fuzzy economics is discussed in [14]. In this paper, the author considers different problems of fuzzy economics, mainly time path and stability of fuzzy dynamics of economic systems, linguistic modelling of economic agents, and neurofuzzy time-series forecasting of macroeconomic processes. One of the main problems considered in [14] is stability of various macroeconomic dynamical systems described by fuzzy differential equations which are used to take into account uncertainty inherent in economic dynamics. Hindawi Publishing Corporation Applied Computational Intelligence and So Computing Volume 2014, Article ID 971894, 7 pages ... |

2 |
A global view of economic growth,” inHandbook of Economic Growth,
- Ventura
- 2005
(Show Context)
Citation Context ...ship between energy and social economics and environment. The use of multiobjective approach is justified by the fact that sustainable development is obtained as a tradeoff between economic growth and environmental preservation and, therefore, is multidimensional concept that encompasses economic, social, ecological, technological, and ethical factors. As a result, achievement of sustainable development requires dealing with highly conflicting criteria. In [17] some important economic and environmental criteria for the problems of multiobjective programming and their conflicts are defined. In [18] some of the key insights of the field of international trade and economic growth are considered within a unified and tractable framework. The authors study a series of theoretical models which have a common description of technology and preferences but are of different assumptions concerning trade frictions. It is shown that on the base of comparison of the predictions of these models one can identify a variety of channels through which trade influences the dynamics and geographical distribution of world income. The combination of fuzzy logic tools with multiobjective optimization and multicr... |

2 | Eds.,OptimizationModels Using Fuzzy Sets and Possibility Theory, - Kacprzyk, Orlovski - 1987 |

2 |
Kloeden,Metric Spaces of Fuzzy Sets, Theory and applications, World Scientific, Singapoure,
- Diamond, P
- 1994
(Show Context)
Citation Context ..., this is not sufficient to investigate interaction of uncertainties and their dynamics. In order to deal with possibilistic Zadeh suggested the fuzzy set theory and fuzzy logic [32]. These theories have passed a long way and provided a lot of successful applications. In particular, control systems based on fuzzy logic were successfully applied for control of various complex and uncertain objects and provided better results than their classical counterparts. Fundamental approach to modelling behaviour of a dynamical system under possibilistic uncertainty is fuzzy differential equations (FDEs) [33]. In the present paper we suggest an approach to decision making in a single-product dynamic economic model. The considered problem is represented as a multiobjective optimal control problem of dynamics of capital under uncertain relevant information. The optimal control is based on four objective functions and two control variables. The application of statistical approaches to handle uncertainty for this problem requires imposing too restrictive assumptions which are unlikely to match real environment especially in the light of the recent unsustainable and hardly predictable behaviour of the ... |

1 | Mathematical Optimization and Economics Theory, Prentice Hall, Englenwood Clirs, - Intrilligator - 1981 |

1 |
Systems factors goverining afirm’s R&D investment. A systems perspective of inter-sectorial technology spillover,”
- Watanabe
- 1998
(Show Context)
Citation Context ... 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 ana... |

1 | Dynamic optimization of R&D intensity under the effeect of technology assimilation : econometric identification for Japan’s Automotive industry,” - Kwintiana, Watanabe, et al. - 2004 |

1 | Optimal control of a nonlinear model of Economic growth,”
- Grigorieva, Khailov
- 2007
(Show Context)
Citation Context ...ed 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... |

1 | A fuzzy multi-objective optimization model for energy structure and its application to the energy structure adjustment—inworld natural and cultural heritage area,”World
- Deng, Xu
- 2011
(Show Context)
Citation Context ...14] is stability of various macroeconomic dynamical systems described by fuzzy differential equations which are used to take into account uncertainty inherent in economic dynamics. Hindawi Publishing Corporation Applied Computational Intelligence and So Computing Volume 2014, Article ID 971894, 7 pages http://dx.doi.org/10.1155/2014/971894 2 Applied Computational Intelligence and Soft Computing In [15] a fuzzy multiobjective linear programming for determination of the optimal land use of regions by taking into account economic, social, and environmental objectives and implications. Reference [16] is devoted to the low-carbonized adjustment of energy structure in the area of world natural and cultural heritage. The authors suggest a fuzzy multiobjective optimization to investigate the relationship between energy and social economics and environment. The use of multiobjective approach is justified by the fact that sustainable development is obtained as a tradeoff between economic growth and environmental preservation and, therefore, is multidimensional concept that encompasses economic, social, ecological, technological, and ethical factors. As a result, achievement of sustainable devel... |

1 |
Economic and environmental criteria in Multi-objective programming problems,”
- Pesic, Radukic, et al.
- 2011
(Show Context)
Citation Context ...ure in the area of world natural and cultural heritage. The authors suggest a fuzzy multiobjective optimization to investigate the relationship between energy and social economics and environment. The use of multiobjective approach is justified by the fact that sustainable development is obtained as a tradeoff between economic growth and environmental preservation and, therefore, is multidimensional concept that encompasses economic, social, ecological, technological, and ethical factors. As a result, achievement of sustainable development requires dealing with highly conflicting criteria. In [17] some important economic and environmental criteria for the problems of multiobjective programming and their conflicts are defined. In [18] some of the key insights of the field of international trade and economic growth are considered within a unified and tractable framework. The authors study a series of theoretical models which have a common description of technology and preferences but are of different assumptions concerning trade frictions. It is shown that on the base of comparison of the predictions of these models one can identify a variety of channels through which trade influences th... |

1 |
The one-product dynamic model of economy considering structure of funds in the presence of technical progress,”
- Kantorovich, Zhiyanov
- 1973
(Show Context)
Citation Context ...rent assumptions concerning trade frictions. It is shown that on the base of comparison of the predictions of these models one can identify a variety of channels through which trade influences the dynamics and geographical distribution of world income. The combination of fuzzy logic tools with multiobjective optimization and multicriteria decision making has a great relevance in the literature [19–23]; for example, fuzzy sets are considered for interactive multiobjective optimization in [22, 23]. A single-product dynamical macroeconomic model was first suggested by Kantorovich and Zhiyanov in [24]. They also suggested a principle of differential optimization for the developed model. Leontyev [25] suggested a single-product dynamic macroeconomic model in which gross domestic product is divided into productive consumption, gross investment, and nonproductive consumption. In the existing works, a qualitative analysis of economic systems and development of analytic algorithms of optimization are based on sufficient conditions of optimality [26] or the maximum principle [27]. Nowadays, an analysis of optimal control problems on the base of the perturbation theory attracted a high interest. ... |

1 |
Input-Output Economies,
- Leontyev
- 1966
(Show Context)
Citation Context ...tions of these models one can identify a variety of channels through which trade influences the dynamics and geographical distribution of world income. The combination of fuzzy logic tools with multiobjective optimization and multicriteria decision making has a great relevance in the literature [19–23]; for example, fuzzy sets are considered for interactive multiobjective optimization in [22, 23]. A single-product dynamical macroeconomic model was first suggested by Kantorovich and Zhiyanov in [24]. They also suggested a principle of differential optimization for the developed model. Leontyev [25] suggested a single-product dynamic macroeconomic model in which gross domestic product is divided into productive consumption, gross investment, and nonproductive consumption. In the existing works, a qualitative analysis of economic systems and development of analytic algorithms of optimization are based on sufficient conditions of optimality [26] or the maximum principle [27]. Nowadays, an analysis of optimal control problems on the base of the perturbation theory attracted a high interest. Especially, as it is shown in [28, 29], they consider application of asymptotic estimates to models t... |

1 |
Bases of the Theory of Optimum Control, The Higher School,
- Krotov, Ed
- 1990
(Show Context)
Citation Context ...eractive multiobjective optimization in [22, 23]. A single-product dynamical macroeconomic model was first suggested by Kantorovich and Zhiyanov in [24]. They also suggested a principle of differential optimization for the developed model. Leontyev [25] suggested a single-product dynamic macroeconomic model in which gross domestic product is divided into productive consumption, gross investment, and nonproductive consumption. In the existing works, a qualitative analysis of economic systems and development of analytic algorithms of optimization are based on sufficient conditions of optimality [26] or the maximum principle [27]. Nowadays, an analysis of optimal control problems on the base of the perturbation theory attracted a high interest. Especially, as it is shown in [28, 29], they consider application of asymptotic estimates to models to cope with nonlinearity, computational instability, high orders, and so forth. In [26] they study an optimal control problem by using a small parametermethod and conduct comparative analysis with the other existing approaches. The existing approaches to solving optimal control problems suffer from a series of disadvantages. From one side, the exist... |

1 |
Asymptotic Decomposition of Decisions Singulyarno Indignant Equations, Science,
- Vasilyeva, Butuzov
- 1973
(Show Context)
Citation Context ...ple of differential optimization for the developed model. Leontyev [25] suggested a single-product dynamic macroeconomic model in which gross domestic product is divided into productive consumption, gross investment, and nonproductive consumption. In the existing works, a qualitative analysis of economic systems and development of analytic algorithms of optimization are based on sufficient conditions of optimality [26] or the maximum principle [27]. Nowadays, an analysis of optimal control problems on the base of the perturbation theory attracted a high interest. Especially, as it is shown in [28, 29], they consider application of asymptotic estimates to models to cope with nonlinearity, computational instability, high orders, and so forth. In [26] they study an optimal control problem by using a small parametermethod and conduct comparative analysis with the other existing approaches. The existing approaches to solving optimal control problems suffer from a series of disadvantages. From one side, the existing approaches are not developed to deal with uncertainty inherent in economic dynamics but are based classical mathematical formalism. Mainly, there exist two types of uncertainty relat... |

1 |
Decision making problem of a one-product dynamic economic model,”
- Gurbanov, Gardashova, et al.
- 2012
(Show Context)
Citation Context ...control variable) is a fuzzy value of a nonproductive consumption, and ... |

1 |
Fuzzy neural network and DEO based forecasting,”
- Gardashova
- 2012
(Show Context)
Citation Context ...hod of differential evolution. Recently many evolutionary algorithms have been proposed for global optimization of nonlinear, nonconvex, and nondifferential functions. These methods are more flexible than classical as they do not require differentiability, continuity, or other properties to hold for optimizing functions. Some of such methods are genetic algorithm (GA), particle swarm optimization (PSO) [35, 36], and differential Evolution (DE) optimization [37–39]. Also all of them have their modified versions, such as Group Principle GA with multipoint crossover and mutation, DE + Clustering [40], and Multifitness function GA. Among many other evolutionary algorithms DE is one of the highest performance. DE is a faster and improved version of GA. As opposite to GA, which uses binary coding, DE uses real coding of float point numbers. The advantage of DE over standard GA is higher performance, efficiencies for numerical problems, not using time-consuming “coding/decoding” transformations, chromosome/gen structures, and other manipulations with binary structures. In [38] a heuristic approach for minimizing possibly nonlinear and nondifferentiable continuous space functions is presented.... |