Application of DEO Method to Solving Fuzzy Multiobjective Optimal Control Problem

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.


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][2][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][7][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][10][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 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.
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 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 multicriteria decision making has a great relevance in the literature [19][20][21][22][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 to cope with nonlinearity, computational instability, high orders, and so forth.In [26] they study an optimal control problem by using a small parameter method 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 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 of possibilistic uncertainties include [31].
In order to take into account real-world uncertainty in formal models, 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 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 present economic reality.In order to model possibilistic uncertainty in dynamics of capital a linear FDE is used.
The use of FDE allows us 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.

Statement of the Problem
Let us consider a single-product dynamic macroeconomic model which reflects interaction between factors of production when a gross domestic product (GDP) is divided into productive consumption, gross investment, and nonproductive consumption as the performance of production activity.In its turn productive consumption is assumed to be completely consumed on capital formation and depreciation.These processes are complicated by a presence of possibilistic, that is, fuzzy uncertainty which is conditioned by imprecise evaluation of future trends, unforeseen contingencies and other vagueness and impreciseness inherent in economical processes.Under the above mentioned assumptions the considered dynamic macroeconomic model can be described by the following FDE: Here x is a fuzzy variable describing imprecise information on capital, that is, fuzzy value of capital, ũ1 is a fuzzy value of GDP (the first control variable), ũ2 (the second control variable) is a fuzzy value of a nonproductive consumption, and , ,  > 0 are coefficients related to the productive consumption, net capital formation, and depreciation, respectively.
Let us consider a multiobjective optimal control problem of (1) within the period of planning [ 0 , ] with four objective functions (criteria): J1 -profit, J2 -reduction of production expenditures of GDP, J3 -a value of capital at the end of period [ 0 , ], and J4 -a discount sum of a direct consumption over [ 0 , ].The considered fuzzy multiobjective optimal control problem is formulated below [34]: subject to Here () is a price of production unit produced at the time , () is the discount function, [ 0 , ] is the optimization period, and  = const > 0.
In this work we have decided to solve a problem of optimal control for a single-product dynamical macroeconomic model with the help of a method 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][38][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.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 very well to parallel computation.It should be noted that dynamic programming is very computationally intensive and even more for fuzzy problem.Therefore, here we use DEO method.

Method of Solution
As a stochastic method, DE algorithm uses initial population randomly generated by uniform distribution, differential mutation, probability crossover, and selection operators.The population with ps individuals is maintained with each generation.A new vector is generated by mutation which in this case is randomly selecting from the population 3 individuals: vector indexes  1 ̸ =  2 ̸ =  3 and adding a weighted difference vector between two individuals to a third individual (population member).
The mutated vector then undergoes crossover operation with another vector generating new offspring vector.
The selection process is done as follows.If the resulting vector yields a lower objective function value than a predetermined population member, the newly generated vector will replace the vector with which it was compared with in the following generation.
Extracting distance and direction information from the population to generate random deviations results in an adaptive scheme with excellent convergence properties.DE has been successfully applied to solve a wide range of problems such as image classification, clustering, optimization, and so forth.
Figure 1 shows the process of generation new trial solution  new vector from randomly selected population members  1 ,  2 ,  3 (Vector  4 is then the candidate for replacement by the new vector, if the former is better or lower in terms of the DE cost function).Here we assume that the solution vectors are of dimension 1 (i.e., 2 optimization parameters).

Simulation Results
Let us consider solving of the problem ( 1)-( 3) on the base of the method suggested in Section 4 under the following data: In addition, the following parameters are used in DE: the population size (NP), the crossover constant (CR), and the weighting factor () [38].
The program written on C# was used for computer simulation of the problem.The DE's parameters used were NP = 100, CR = 0.9, and  = 1.0.
The time used for solving the problem was 15-23 minutes.The problem was run for the crisp and fuzzy cases.
The results of crisp and fuzzy solutions are shown in Table 1 and Figure 2. Solution of the problem by using the DEO method ended up with the results in Table 1 and Figure 2.

Conclusions
We applied DEO method to solving fuzzy multiobjective optimal control problem for a single-product dynamical macroeconomic model.For such problems, application of the well-known existing approaches would be complex both form intuitive and computational points of view.Application

Figure 1 :
Figure 1: The process of generation new trial solution.