Research ArticleApplication of DEO Method to Solving FuzzyMultiobjective Optimal Control Problem

Latafat A Gardashova

Azerbaijan State Oil Academy Azadlyg avenue 20 Baku Az1010 Azerbaijan

Correspondence should be addressed to Latafat A Gardashova latshamyandexru

Received 29 November 2013 Revised 27 January 2014 Accepted 27 January 2014 Published 27 February 2014

Academic Editor Francesco Carlo Morabito

Copyright copy 2014 Latafat A Gardashova This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In the present paper a problem of optimal control for a single-product dynamical macroeconomic model is considered In thismodel gross domestic product is divided into productive consumption gross investment and nonproductive consumption Themodel is described by a fuzzy differential equation (FDE) to take into account imprecision inherent in the dynamics that may benaturally conditioned by influence of various external factors unforeseen contingencies of future and so forth The consideredproblems are characterized by four criteria and by several important aspects On one hand the problem is complicated by thepresence of fuzzy uncertainty as a result of a natural imprecision inherent in information about dynamics of real-world systemsOn the other hand the number of the criteria is not small and most of them are integral criteria Due to the above mentionedaspects solving the considered problem by using convolution of criteria into one criterion would lead to loss of information andalso would be counterintuitive and complex We applied DEO (differential evolution optimization) method to solve the consideredproblem

1 Introduction

The studies devoted to solving optimal control problemsfor dynamic economic models have a long history Themodels of economic growth attracted a large interest in thearea of mathematical economics [1ndash3] Recently intensiverevisiting of the growth models took place [4 5] as a resultof the improvements in existing models and progress indevelopment of the associated analytical techniques [6ndash8]

The intensive research in this direction was also con-ditioned by the support of the developed countries as thelatter were interested in construction of accurate economicgrowth models to improve their economic development [9ndash11] A lot of various mathematical methods of measuring theeffectiveness of economic growth were suggested In [12] fordescribing the change of production and accumulated RampDinvestment in a firm a nonlinear control model is consideredand analyzed They provide a solution of an optimal controlproblem with RampD investment rate as a control parameterThey also analyze an optimal dynamics of economic growthof a firm versus the current cost of innovation The resultsof 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 asingular arc

In [13] the authors provide results of comparison of solu-tion methods for dynamic equilibrium economies in termsof computing time implementation complexity and accu-racy The considered methods are the stochastic neoclassi-cal growth model the finite elements method Chebyshevpolynomials and value function iteration Authors conductanalysis of the obtained results

Modeling and stability analysis in fuzzy economics isdiscussed in [14] In this paper the author considers differentproblems of fuzzy economics mainly time path and stabilityof fuzzy dynamics of economic systems linguistic modellingof economic agents and neurofuzzy time-series forecastingof macroeconomic processes One of the main problemsconsidered in [14] is stability of various macroeconomicdynamical systems described by fuzzy differential equationswhich are used to take into account uncertainty inherent ineconomic dynamics

Hindawi Publishing CorporationApplied Computational Intelligence and So ComputingVolume 2014 Article ID 971894 7 pageshttpdxdoiorg1011552014971894

2 Applied Computational Intelligence and Soft Computing

In [15] a fuzzy multiobjective linear programming fordetermination of the optimal land use of regions by takinginto account economic social and environmental objectivesand implications

Reference [16] is devoted to the low-carbonized adjust-ment of energy structure in the area of world natural andcultural heritage The authors suggest a fuzzy multiobjectiveoptimization to investigate the relationship between energyand social economics and environment The use of multiob-jective approach is justified by the fact that sustainable devel-opment is obtained as a tradeoff between economic growthand environmental preservation and therefore is multi-dimensional concept that encompasses economic socialecological technological and ethical factors As a resultachievement of sustainable development requires dealingwith highly conflicting criteria

In [17] some important economic and environmentalcriteria for the problems of multiobjective programming andtheir conflicts are defined

In [18] some of the key insights of the field of interna-tional trade and economic growth are considered within aunified and tractable framework The authors study a seriesof theoretical models which have a common description oftechnology and preferences but are of different assumptionsconcerning trade frictions It is shown that on the baseof comparison of the predictions of these models one canidentify a variety of channels through which trade influ-ences the dynamics and geographical distribution of worldincome

The combination of fuzzy logic tools with multiobjectiveoptimization and multicriteria decision making has a greatrelevance in the literature [19ndash23] for example fuzzy sets areconsidered for interactive multiobjective optimization in [2223]

A single-product dynamical macroeconomic model wasfirst suggested by Kantorovich and Zhiyanov in [24] Theyalso suggested a principle of differential optimization for thedeveloped model Leontyev [25] suggested a single-productdynamic macroeconomic model in which gross domesticproduct is divided into productive consumption gross invest-ment and nonproductive consumption

In the existing works a qualitative analysis of economicsystems and development of analytic algorithms of optimiza-tion are based on sufficient conditions of optimality [26]or the maximum principle [27] Nowadays an analysis ofoptimal control problems on the base of the perturbationtheory attracted a high interest Especially as it is shown in[28 29] they consider application of asymptotic estimates tomodels to cope with nonlinearity computational instabilityhigh orders and so forth In [26] they study an optimalcontrol problem by using a small parametermethod and con-duct comparative analysis with the other existing approachesThe existing approaches to solving optimal control problemssuffer from a series of disadvantages From one side theexisting approaches are not developed to deal with uncer-tainty inherent in economic dynamics but are based classicalmathematical formalism Mainly there exist two types ofuncertainty related to economic dynamics probabilistic andpossibilistic (fuzzy) Modelling of probabilistic uncertainty

requires the use of very restrictive assumptions includingavailability of good statistical data Modelling of fuzzy uncer-tainty is a more realistic task and allows us to describeimprecise evaluation of variables of interest coming fromhuman expertise [30]

In the existing works economic uncertainty is mainlycaptured by using stochastic techniques However the use ofstochastic techniques is based on crucial assumptions whichnotably constrain its use for adequate modelling of real-world economic uncertainty These methods can be succes-sively applied when sufficient knowledge is available on theprobability density functions of all the variables involved inthe computations which is a rather rare case In contrastvery often prediction or estimation of variables of interest issubjective for example expert-opinion based The mainsources of possibilistic uncertainties include [31]

In order to take into account real-world uncertainty informalmodels they usually use sensitivity analysesHoweverthis is not sufficient to investigate interaction of uncertaintiesand their dynamics In order to deal with possibilistic Zadehsuggested the fuzzy set theory and fuzzy logic [32]

These theories have passed a long way and provided a lotof successful applications In particular control systems basedon fuzzy logic were successfully applied for control of variouscomplex and uncertain objects and provided better resultsthan their classical counterparts Fundamental approach tomodelling behaviour of a dynamical system under possi-bilistic uncertainty is fuzzy differential equations (FDEs)[33]

In the present paper we suggest an approach to decisionmaking in a single-product dynamic economic model Theconsidered problem is represented as a multiobjective opti-mal control problem of dynamics of capital under uncertainrelevant information The optimal control is based on fourobjective functions and two control variables The appli-cation of statistical approaches to handle uncertainty forthis problem requires imposing too restrictive assumptionswhich are unlikely to match real environment especially inthe light of the recent unsustainable and hardly predictablebehaviour of the present economic reality In order to modelpossibilistic uncertainty in dynamics of capital a linear FDE isused

The use of FDE allows us to take into account imprecisioninherent in the dynamics that may be naturally conditionedby influence of various external factors unforeseen contin-gencies of future and so forth

2 Statement of the Problem

Let us consider a single-product dynamic macroeconomicmodel which reflects interaction between factors of pro-duction when a gross domestic product (GDP) is dividedinto productive consumption gross investment and non-productive consumption as the performance of productionactivity In its turn productive consumption is assumed to becompletely consumed on capital formation and depreciationThese processes are complicated by a presence of possibilisticthat is fuzzy uncertainty which is conditioned by imprecise

Applied Computational Intelligence and Soft Computing 3

evaluation of future trends unforeseen contingencies andother vagueness and impreciseness inherent in economicalprocesses Under the above mentioned assumptions theconsidered dynamicmacroeconomicmodel can be describedby the following FDE

119889119909

119889119905=1

119902((1 minus 119886)

1minus 120583119909 minus

2) (1)

Here 119909 is a fuzzy variable describing imprecise informa-tion on capital that is fuzzy value of capital

1is a fuzzy value

of GDP (the first control variable) 2(the second control

variable) is a fuzzy value of a nonproductive consumptionand 119886 120583 119902 gt 0 are coefficients related to the produc-tive consumption net capital formation and depreciationrespectively

Let us consider a multiobjective optimal control problemof (1) within the period of planning [119905

0 119879]with four objective

functions (criteria) 1198691mdashprofit 119869

2mdashreduction of production

expenditures of GDP 1198693mdasha value of capital at the end of

period [1199050 119879] and 119869

4mdasha discount sum of a direct consump-

tion over [1199050 119879]The considered fuzzymultiobjective optimal

control problem is formulated below [34]

supuisinU

(1198691(u) =int

119879

1199050

119901 (119905) 2(119905) 119889119905 119869

2(u) = minus119888int

119879

1199050

10038161003816100381610038161(119905)1003816100381610038161003816 119889119905

1198693(u) = (119879) 119869

4(u) = int

119879

1199050

120579 (119905) 2(119905) 119889119905)

(2)

subject to

119889119909

119889119905=1

119902((1 minus 119886)

1minus 120583119909 minus

2)

119909 (119905) isin E1 119905 isin [119905

0 119879] 119909 (119905

0) = 1199090

119909 (119879) isinK (119879)

K (119879) = 119909 isin E1 119909 le 119909 (119879) le 119909

u = (1 2)119879

isin U = U1timesU2sub E2

U1= 1isin E1 1le 1(119905) le

U2= 2isin E1 2lowast le 2(119905) le

lowast

2

(3)

Here 119901(119905) is a price of production unit produced at thetime 119905 120579(119905) is the discount function [119905

0 119879] is the optimization

period and 119888 = const gt 0In this work we have decided to solve a problem of opti-

mal control for a single-product dynamical macroeconomicmodel with the help of a method of differential evolution

Recently many evolutionary algorithms have been pro-posed for global optimization of nonlinear nonconvex andnondifferential functions These methods are more flexiblethan classical as they do not require differentiability continu-ity or other properties to hold for optimizing functions Some

of such methods are genetic algorithm (GA) particle swarmoptimization (PSO) [35 36] and differential Evolution (DE)optimization [37ndash39]

Also all of them have their modified versions such asGroup Principle GA with multipoint crossover and muta-tion DE + Clustering [40] and Multifitness function GAAmong many other evolutionary algorithms DE is one of thehighest performance DE is a faster and improved versionof GA As opposite to GA which uses binary coding DEuses real coding of float point numbers The advantage ofDE over standard GA is higher performance efficienciesfor numerical problems not using time-consuming ldquocod-ingdecodingrdquo transformations chromosomegen structuresand other manipulations with binary structures

In [38] a heuristic approach for minimizing possiblynonlinear and nondifferentiable continuous space functionsis presented DEO method is a simple and efficient heuristicfor global optimization over continuous spaces By means ofan extensive test bed it is demonstrated that DEO methodconverges faster and with more certainty than many otheracclaimed global optimization methods The DEO methodrequires few control variables is robust easy to use and lendsitself verywell to parallel computation It should be noted thatdynamic programming is very computationally intensive andeven more for fuzzy problem Therefore here we use DEOmethod

3 Method of Solution

As a stochastic method DE algorithm uses initial populationrandomly generated by uniform distribution differentialmutation probability crossover and selection operators Thepopulation with ps individuals is maintained with eachgeneration A new vector is generated by mutation whichin this case is randomly selecting from the population 3individuals vector indexes 119903

1= 1199032= 1199033and adding a weighted

difference vector between two individuals to a third individ-ual (population member)

The mutated vector then undergoes crossover operationwith another vector generating new offspring vector

The selection process is done as follows If the resultingvector yields a lower objective function value than a predeter-mined population member the newly generated vector willreplace the vector with which it was compared with in thefollowing generation

Extracting distance and direction information from thepopulation to generate random deviations results in an adap-tive scheme with excellent convergence properties DE hasbeen successfully applied to solve a wide range of problemssuch as image classification clustering optimization and soforth

Figure 1 shows the process of generation new trial solu-tion 119883new vector from randomly selected population mem-bers 119883

1199031 1198831199032 1198831199033

(Vector 1198834is then the candidate for

replacement by the new vector if the former is better orlower in terms of the DE cost function) Here we assume thatthe solution vectors are of dimension 1 (ie 2 optimizationparameters)

4 Applied Computational Intelligence and Soft Computing

X1

X2

Xr2

Xr3

Xr4

Xr1

Xr1 minus Xr2

Xnew

Figure 1 The process of generation new trial solution

4 Simulation Results

Let us consider solving of the problem (1)ndash(3) on the base ofthe method suggested in Section 4 under the following data

1198691= Δ

119873

sum

119899=0

119901(1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

2) 997888rarr max

1198692= minus119888Δsum

100381610038161003816100381610038161003816100381610038161003816

1198861(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

1

100381610038161003816100381610038161003816100381610038161003816

997888rarr max

1198693=

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119873sdotΔ 997888rarr max

1198694= Δ

119873minus1

sum

119899=0

119890119903119899Δ(1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ)+ 119887

2)997888rarr max

Table 1 Crisp solution

1198861

1198862

1198871

1198872

119869119894 119894 = 1 4

0108092 137119864 minus 05 5785196 5499741 1198951= 181500564

235119864 minus 10 248119864 minus 07 600 5499995 1198952= minus6599999

0048176 248119864 minus 19 5036475 5500001 1198953= 590870970

0100707 163119864 minus 09 5789494 550 1198954= 115110534

600 le 1198861(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

1le 800

450 le 1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

2le 550

119899 = 0 9 119888 = 05 Δ = 02 119873 = 10

119886 = 005 119901 = 1500 120583 = 008 119902 = 095

119903 = 005 0= (1900 2000 2100)

600 = (580 600 620) 800 = (780 800 820)

450 = (430 450 470) 550 = (530 550 570)

1198861ge 0 119886

2ge 0

(4)

In addition the following parameters are used in DE thepopulation size (NP) the crossover constant (CR) and theweighting factor (119865) [38]

The program written on C was used for computersimulation of the problem The DErsquos parameters used wereNP = 100 CR = 09 and 119865 = 10

The time used for solving the problemwas 15ndash23minutesThe problem was run for the crisp and fuzzy cases

The results of crisp and fuzzy solutions are shown inTable 1 and Figure 2 Solution of the problem by using theDEO method ended up with the results in Table 1 andFigure 2

5 Conclusions

We applied DEO method to solving fuzzy multiobjectiveoptimal control problem for a single-product dynamicalmacroeconomic model For such problems application ofthe well-known existing approaches would be complex bothform intuitive and computational points of view Application

Applied Computational Intelligence and Soft Computing 5

Fuzzy a2

0010203040506070809

1

minus00003 minus00001 00001 00003

Fuzzy b1

0010203040506070809

1

5499740 5499741 5499742 5499743

Fuzzy a1

0010203040506070809

1

01079 01080 01081 01082

Fuzzy b2

5784000 5785000 57860000

010203040506070809

1Fuzzy J1

18120 18140 18160 181800

010203040506070809

1

00000 10000

Fuzzy a1

0010203040506070809

1

00000 10000

Fuzzy a2

0010203040506070809

1

Fuzzy b1

580 588 596 604 612 6200

010203040506070809

1Fuzzy b2

53000 54000 55000 56000 570000

010203040506070809

1

Fuzzy a1

00000 00200 00400 00600 008000

010203040506070809

1Fuzzy a2

minus00060 minus00020 00020 000600

010203040506070809

1

Fuzzy b1

503557 503607 503657 503707 5037570

010203040506070809

1Fuzzy b2

549700 549900 550100 5503000

010203040506070809

1Fuzzy J3

2000000 12000000 220000000

010203040506070809

1

Fuzzy a1

00980 01000 01020 010400

010203040506070809

1Fuzzy a2

00000 02000 04000 06000 08000 100000

010203040506070809

1

Fuzzy J2

minus1320 minus660 0 660 13200

010203040506070809

1

(a)

Figure 2 Continued

6 Applied Computational Intelligence and Soft Computing

Fuzzy b1

57887 57892 57897 579020

010203040506070809

1Fuzzy b2

549970 549990 550010 5500300

010203040506070809

1Fuzzy J4

1150950 1151050 1151150 11512500

010203040506070809

1

(b)

Figure 2 Fuzzy solution

of DEO method allowed obtaining intuitively meaningfulsolution for the considered problem complicated by fourconflicting criteria and nonstochastic uncertainty intrinsic toreal-world economic problemsThe fuzzy approach allows usto consider many internal and external effects The solutionobtained in fuzzymodel is more sustainable and the objectivefunctions in this case are less sensitive for changes Besides theexperience has shown that in DE the processing time is rela-tively smallThe results obtained in the paper show validity ofthe applied approach The applied approach is characterizedby a low computational complexity as compared with theexisting methods for solving multiobjective optimal controlproblems

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgment

The author would like to thank Professor R A Aliyev forproviding useful advice and a great support in implementingthis research work

References

[1] K J Arrow ldquoApplication of control theory to economic growthrdquoin Mathematics of the Decision Sciences vol 12 of Lecture onApplied Mathematics part 2 pp 83ndash119 American Mathemati-cal Society Providence RI USA 1968

[2] M Intrilligator Mathematical Optimization and EconomicsTheory Prentice Hall Englenwood Clirs NJ USA 1981

[3] GM Grossman and E Helpman Innovation and Growth in theGlobal Economy MIT Press Cambridge UK 1991

[4] C Watanabe ldquoFactors goverining a firmrsquos RampD investment Acomparison between individual and aggregate datardquo in IIASATIT OECD Technical Meeting Paris France 1997

[5] C Watanabe ldquoSystems factors goverining afirmrsquos RampD invest-ment A systems perspective of inter-sectorial technologyspilloverrdquo in IIASA TIT OECD Technical Meeeting LaxenburgAustria 1998

[6] A Kryazhimskii C Watanabe and Yu Tou ldquoThe reachabilityof techno-labor homeostasis via regulation of investments inlabor RampD a model-based analysisrdquo Interium Report IR-02-026 IIASA Laxenburg Austria 2002

[7] A M Tarasyev and C Watanabe ldquoOptimal dynamics of inno-vation in models of economic growthrdquo Journal of OptimizationTheory and Applications vol 108 no 1 pp 175ndash203 2001

[8] S A Reshmin A M Tarasyev and C Watanabe ldquoOptimaltrajectories of the innovation process and their matchingwith econometric datardquo Journal of Optimization Theory andApplications vol 112 no 3 pp 639ndash685 2002

[9] B Kwintiana C Watanabe and A M Tarasyev ldquoDynamicoptimization of RampD intensity under the effeect of technologyassimilation econometric identification for Japanrsquos Automotiveindustryrdquo Interium Report IR-04-058 IIASA Laxenburg Aus-tria 2004

[10] N Kaldor ldquoA model of economic growthrdquo The Economic Jour-nal vol 67 no 268 pp 591ndash624 1957

[11] A Kryazhimskii and C Watanabe Optimization of Technologi-cal Growth GENDAITOSHO Kanagawa Japan 2004

[12] E V Grigorieva and E N Khailov ldquoOptimal control of anonlinear model of Economic growthrdquo Journal of Discrete andContinuous Dynamical Systems Supplement pp 456ndash466 2007

[13] S B Aruoba J Fernandez-Villaverde and J F Rubio RamırezldquoComparing solution methods for dynamic equilibrium econ-omiesrdquo Journal of Economic Dynamics and Control vol 30 no12 pp 2477ndash2508 2006

[14] R A Aliev ldquoModelling and stability analysis in fuzzy eco-nomicsrdquo Applied and Computational Mathematics vol 7 no 1pp 31ndash53 2008

[15] S A Mohaddes M Ghazali K A Rahim M Nasir and AV Kamaid ldquoFuzzy environmental-economic model for landuse planningrdquo American-Eurasian Journal of Agricultural ampEnvironmental Sciences vol 3 no 6 pp 850ndash854 2008

[16] Y Deng and J Xu ldquoA fuzzy multi-objective optimization modelfor energy structure and its application to the energy structureadjustmentmdashinworld natural and cultural heritage areardquoWorldJournal of Modelling and Simulation vol 7 no 2 pp 101ndash1122011

[17] M Pesic S Radukic and J Stankovic ldquoEconomic and envi-ronmental criteria in Multi-objective programming problemsrdquoFacta Universitatis vol 8 no 4 pp 389ndash400 2011

[18] J Ventura ldquoA global view of economic growthrdquo inHandbook ofEconomic Growth P Aghion and S N Durlauf Eds vol 1 pp1420ndash1492 Elsevier 2005

[19] J Kacprzyk and S A Orlovski EdsOptimizationModels UsingFuzzy Sets and Possibility Theory Reidel Publishing CompanyDordrecht The Netherlands 1987

[20] C Carlsson and R Fuller ldquoMultiobjective optimization withlinguistic variablesrdquo in Proceedings of the 6th European Congresson Intelligent Techniques and Soft Computing (EUFIT rsquo98) vol2 pp 1038ndash1042 Veralg Mainz Aachen Germany 1998

Applied Computational Intelligence and Soft Computing 7

[21] C Carlsson and R Fuller ldquoFuzzy multiple criteria decisionmaking recent developmentsrdquo Fuzzy Sets and Systems vol 78no 2 pp 139ndash153 1996

[22] R Fuller C Carlsson and S Giove ldquoOptimization under fuzzylinguistic rule constraintsrdquo in Proceedings of Eurofuse-SIC rsquo99pp 184ndash187 Budapest Hungary 1999

[23] M Sakawa Fuzzy Sets and Interactive Multiobjective Optimiza-tion Plenum Press London UK 1993

[24] L V Kantorovich and V I Zhiyanov ldquoThe one-productdynamic model of economy considering structure of fundsin the presence of technical progressrdquo Reports of Academy ofSciences of the USSR vol 211 no 6 pp 1280ndash1283 1973

[25] V Leontyev Input-Output Economies Oxford University PressNew York NY USA 1966

[26] V F Krotov Ed Bases of the Theory of Optimum Control TheHigher School 1990

[27] L S Pontryagin V G Boltyansky R V Gamkrelidze and EF Mishchenko Mathematical Theory of Optimum ProcessesHayka 1969

[28] A B Vasilyeva and V F Butuzov Asymptotic Decomposition ofDecisions Singulyarno Indignant Equations Science 1973

[29] A B Vasilyeva and M G Dmitriev ldquoSingulyarnye of indigna-tion in problems of optimum controlscience and equipmentResults It is grayrdquo Matem analysis T 20 VINITI pp 3ndash771982

[30] R Ostermark ldquoA fuzzy control model (FCM) for dynamicportfolio managementrdquo Fuzzy Sets and Systems vol 78 no 3pp 243ndash254 1996

[31] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[32] L A Zadeh ldquoToward extended fuzzy logic-a first steprdquo FuzzySets and Systems vol 160 no 21 pp 3175ndash3181 2009

[33] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand applications World Scientific Singapoure 1994

[34] R S Gurbanov L A Gardashova O H Huseynov and K RAliyeva ldquoDecision making problem of a one-product dynamiceconomic modelrdquo in Proceedings of the of the 10th InternationalConference on Application of Fuzzy Systems and Soft Computing(ICAFS rsquo12) pp 161ndash174 Lisbon Portugal 2012

[35] W Pedrycz and K Hirota ldquoFuzzy vector quantization withthe particle swarm optimization a study in fuzzy granulation-degranulation information processingrdquo Signal Processing vol87 no 9 pp 2061ndash2074 2007

[36] Y-J Wang J-S Zhang and G-Y Zhang ldquoA dynamic clusteringbased differential evolution algorithm for global optimizationrdquoEuropean Journal of Operational Research vol 183 no 1 pp 56ndash73 2007

[37] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo Tech Rep TR-95-012 ICSI 1995

[38] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[39] K Price R Storn and J Lampinen Differential EvolutionmdashAPractical Approach To Global Optimization Natural ComputingSeries Springer Berlin Germany 2005

[40] L A Gardashova ldquoFuzzy neural network and DEO based fore-castingrdquo Journal of Automatics Telemechanics and Communica-tion no 29 pp 36ndash44 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Applied Computational Intelligence and Soft Computing

In [15] a fuzzy multiobjective linear programming fordetermination of the optimal land use of regions by takinginto account economic social and environmental objectivesand implications

Reference [16] is devoted to the low-carbonized adjust-ment of energy structure in the area of world natural andcultural heritage The authors suggest a fuzzy multiobjectiveoptimization to investigate the relationship between energyand social economics and environment The use of multiob-jective approach is justified by the fact that sustainable devel-opment is obtained as a tradeoff between economic growthand environmental preservation and therefore is multi-dimensional concept that encompasses economic socialecological technological and ethical factors As a resultachievement of sustainable development requires dealingwith highly conflicting criteria

In [17] some important economic and environmentalcriteria for the problems of multiobjective programming andtheir conflicts are defined

In [18] some of the key insights of the field of interna-tional trade and economic growth are considered within aunified and tractable framework The authors study a seriesof theoretical models which have a common description oftechnology and preferences but are of different assumptionsconcerning trade frictions It is shown that on the baseof comparison of the predictions of these models one canidentify a variety of channels through which trade influ-ences the dynamics and geographical distribution of worldincome

The combination of fuzzy logic tools with multiobjectiveoptimization and multicriteria decision making has a greatrelevance in the literature [19ndash23] for example fuzzy sets areconsidered for interactive multiobjective optimization in [2223]

A single-product dynamical macroeconomic model wasfirst suggested by Kantorovich and Zhiyanov in [24] Theyalso suggested a principle of differential optimization for thedeveloped model Leontyev [25] suggested a single-productdynamic macroeconomic model in which gross domesticproduct is divided into productive consumption gross invest-ment and nonproductive consumption

In the existing works a qualitative analysis of economicsystems and development of analytic algorithms of optimiza-tion are based on sufficient conditions of optimality [26]or the maximum principle [27] Nowadays an analysis ofoptimal control problems on the base of the perturbationtheory attracted a high interest Especially as it is shown in[28 29] they consider application of asymptotic estimates tomodels to cope with nonlinearity computational instabilityhigh orders and so forth In [26] they study an optimalcontrol problem by using a small parametermethod and con-duct comparative analysis with the other existing approachesThe existing approaches to solving optimal control problemssuffer from a series of disadvantages From one side theexisting approaches are not developed to deal with uncer-tainty inherent in economic dynamics but are based classicalmathematical formalism Mainly there exist two types ofuncertainty related to economic dynamics probabilistic andpossibilistic (fuzzy) Modelling of probabilistic uncertainty

requires the use of very restrictive assumptions includingavailability of good statistical data Modelling of fuzzy uncer-tainty is a more realistic task and allows us to describeimprecise evaluation of variables of interest coming fromhuman expertise [30]

In the existing works economic uncertainty is mainlycaptured by using stochastic techniques However the use ofstochastic techniques is based on crucial assumptions whichnotably constrain its use for adequate modelling of real-world economic uncertainty These methods can be succes-sively applied when sufficient knowledge is available on theprobability density functions of all the variables involved inthe computations which is a rather rare case In contrastvery often prediction or estimation of variables of interest issubjective for example expert-opinion based The mainsources of possibilistic uncertainties include [31]

In order to take into account real-world uncertainty informalmodels they usually use sensitivity analysesHoweverthis is not sufficient to investigate interaction of uncertaintiesand their dynamics In order to deal with possibilistic Zadehsuggested the fuzzy set theory and fuzzy logic [32]

These theories have passed a long way and provided a lotof successful applications In particular control systems basedon fuzzy logic were successfully applied for control of variouscomplex and uncertain objects and provided better resultsthan their classical counterparts Fundamental approach tomodelling behaviour of a dynamical system under possi-bilistic uncertainty is fuzzy differential equations (FDEs)[33]

In the present paper we suggest an approach to decisionmaking in a single-product dynamic economic model Theconsidered problem is represented as a multiobjective opti-mal control problem of dynamics of capital under uncertainrelevant information The optimal control is based on fourobjective functions and two control variables The appli-cation of statistical approaches to handle uncertainty forthis problem requires imposing too restrictive assumptionswhich are unlikely to match real environment especially inthe light of the recent unsustainable and hardly predictablebehaviour of the present economic reality In order to modelpossibilistic uncertainty in dynamics of capital a linear FDE isused

The use of FDE allows us to take into account imprecisioninherent in the dynamics that may be naturally conditionedby influence of various external factors unforeseen contin-gencies of future and so forth

2 Statement of the Problem

Let us consider a single-product dynamic macroeconomicmodel which reflects interaction between factors of pro-duction when a gross domestic product (GDP) is dividedinto productive consumption gross investment and non-productive consumption as the performance of productionactivity In its turn productive consumption is assumed to becompletely consumed on capital formation and depreciationThese processes are complicated by a presence of possibilisticthat is fuzzy uncertainty which is conditioned by imprecise

Applied Computational Intelligence and Soft Computing 3

evaluation of future trends unforeseen contingencies andother vagueness and impreciseness inherent in economicalprocesses Under the above mentioned assumptions theconsidered dynamicmacroeconomicmodel can be describedby the following FDE

119889119909

119889119905=1

119902((1 minus 119886)

1minus 120583119909 minus

2) (1)

Here 119909 is a fuzzy variable describing imprecise informa-tion on capital that is fuzzy value of capital

1is a fuzzy value

of GDP (the first control variable) 2(the second control

variable) is a fuzzy value of a nonproductive consumptionand 119886 120583 119902 gt 0 are coefficients related to the produc-tive consumption net capital formation and depreciationrespectively

Let us consider a multiobjective optimal control problemof (1) within the period of planning [119905

0 119879]with four objective

functions (criteria) 1198691mdashprofit 119869

2mdashreduction of production

expenditures of GDP 1198693mdasha value of capital at the end of

period [1199050 119879] and 119869

4mdasha discount sum of a direct consump-

tion over [1199050 119879]The considered fuzzymultiobjective optimal

control problem is formulated below [34]

supuisinU

(1198691(u) =int

119879

1199050

119901 (119905) 2(119905) 119889119905 119869

2(u) = minus119888int

119879

1199050

10038161003816100381610038161(119905)1003816100381610038161003816 119889119905

1198693(u) = (119879) 119869

4(u) = int

119879

1199050

120579 (119905) 2(119905) 119889119905)

(2)

subject to

119889119909

119889119905=1

119902((1 minus 119886)

1minus 120583119909 minus

2)

119909 (119905) isin E1 119905 isin [119905

0 119879] 119909 (119905

0) = 1199090

119909 (119879) isinK (119879)

K (119879) = 119909 isin E1 119909 le 119909 (119879) le 119909

u = (1 2)119879

isin U = U1timesU2sub E2

U1= 1isin E1 1le 1(119905) le

U2= 2isin E1 2lowast le 2(119905) le

lowast

2

(3)

Here 119901(119905) is a price of production unit produced at thetime 119905 120579(119905) is the discount function [119905

0 119879] is the optimization

period and 119888 = const gt 0In this work we have decided to solve a problem of opti-

mal control for a single-product dynamical macroeconomicmodel with the help of a method of differential evolution

Recently many evolutionary algorithms have been pro-posed for global optimization of nonlinear nonconvex andnondifferential functions These methods are more flexiblethan classical as they do not require differentiability continu-ity or other properties to hold for optimizing functions Some

of such methods are genetic algorithm (GA) particle swarmoptimization (PSO) [35 36] and differential Evolution (DE)optimization [37ndash39]

Also all of them have their modified versions such asGroup Principle GA with multipoint crossover and muta-tion DE + Clustering [40] and Multifitness function GAAmong many other evolutionary algorithms DE is one of thehighest performance DE is a faster and improved versionof GA As opposite to GA which uses binary coding DEuses real coding of float point numbers The advantage ofDE over standard GA is higher performance efficienciesfor numerical problems not using time-consuming ldquocod-ingdecodingrdquo transformations chromosomegen structuresand other manipulations with binary structures

In [38] a heuristic approach for minimizing possiblynonlinear and nondifferentiable continuous space functionsis presented DEO method is a simple and efficient heuristicfor global optimization over continuous spaces By means ofan extensive test bed it is demonstrated that DEO methodconverges faster and with more certainty than many otheracclaimed global optimization methods The DEO methodrequires few control variables is robust easy to use and lendsitself verywell to parallel computation It should be noted thatdynamic programming is very computationally intensive andeven more for fuzzy problem Therefore here we use DEOmethod

3 Method of Solution

As a stochastic method DE algorithm uses initial populationrandomly generated by uniform distribution differentialmutation probability crossover and selection operators Thepopulation with ps individuals is maintained with eachgeneration A new vector is generated by mutation whichin this case is randomly selecting from the population 3individuals vector indexes 119903

1= 1199032= 1199033and adding a weighted

difference vector between two individuals to a third individ-ual (population member)

The mutated vector then undergoes crossover operationwith another vector generating new offspring vector

The selection process is done as follows If the resultingvector yields a lower objective function value than a predeter-mined population member the newly generated vector willreplace the vector with which it was compared with in thefollowing generation

Extracting distance and direction information from thepopulation to generate random deviations results in an adap-tive scheme with excellent convergence properties DE hasbeen successfully applied to solve a wide range of problemssuch as image classification clustering optimization and soforth

Figure 1 shows the process of generation new trial solu-tion 119883new vector from randomly selected population mem-bers 119883

1199031 1198831199032 1198831199033

(Vector 1198834is then the candidate for

replacement by the new vector if the former is better orlower in terms of the DE cost function) Here we assume thatthe solution vectors are of dimension 1 (ie 2 optimizationparameters)

4 Applied Computational Intelligence and Soft Computing

X1

X2

Xr2

Xr3

Xr4

Xr1

Xr1 minus Xr2

Xnew

Figure 1 The process of generation new trial solution

4 Simulation Results

Let us consider solving of the problem (1)ndash(3) on the base ofthe method suggested in Section 4 under the following data

1198691= Δ

119873

sum

119899=0

119901(1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

2) 997888rarr max

1198692= minus119888Δsum

100381610038161003816100381610038161003816100381610038161003816

1198861(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

1

100381610038161003816100381610038161003816100381610038161003816

997888rarr max

1198693=

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119873sdotΔ 997888rarr max

1198694= Δ

119873minus1

sum

119899=0

119890119903119899Δ(1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ)+ 119887

2)997888rarr max

Table 1 Crisp solution

1198861

1198862

1198871

1198872

119869119894 119894 = 1 4

0108092 137119864 minus 05 5785196 5499741 1198951= 181500564

235119864 minus 10 248119864 minus 07 600 5499995 1198952= minus6599999

0048176 248119864 minus 19 5036475 5500001 1198953= 590870970

0100707 163119864 minus 09 5789494 550 1198954= 115110534

600 le 1198861(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

1le 800

450 le 1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

2le 550

119899 = 0 9 119888 = 05 Δ = 02 119873 = 10

119886 = 005 119901 = 1500 120583 = 008 119902 = 095

119903 = 005 0= (1900 2000 2100)

600 = (580 600 620) 800 = (780 800 820)

450 = (430 450 470) 550 = (530 550 570)

1198861ge 0 119886

2ge 0

(4)

In addition the following parameters are used in DE thepopulation size (NP) the crossover constant (CR) and theweighting factor (119865) [38]

The program written on C was used for computersimulation of the problem The DErsquos parameters used wereNP = 100 CR = 09 and 119865 = 10

The time used for solving the problemwas 15ndash23minutesThe problem was run for the crisp and fuzzy cases

The results of crisp and fuzzy solutions are shown inTable 1 and Figure 2 Solution of the problem by using theDEO method ended up with the results in Table 1 andFigure 2

5 Conclusions

We applied DEO method to solving fuzzy multiobjectiveoptimal control problem for a single-product dynamicalmacroeconomic model For such problems application ofthe well-known existing approaches would be complex bothform intuitive and computational points of view Application

Applied Computational Intelligence and Soft Computing 5

Fuzzy a2

0010203040506070809

1

minus00003 minus00001 00001 00003

Fuzzy b1

0010203040506070809

1

5499740 5499741 5499742 5499743

Fuzzy a1

0010203040506070809

1

01079 01080 01081 01082

Fuzzy b2

5784000 5785000 57860000

010203040506070809

1Fuzzy J1

18120 18140 18160 181800

010203040506070809

1

00000 10000

Fuzzy a1

0010203040506070809

1

00000 10000

Fuzzy a2

0010203040506070809

1

Fuzzy b1

580 588 596 604 612 6200

010203040506070809

1Fuzzy b2

53000 54000 55000 56000 570000

010203040506070809

1

Fuzzy a1

00000 00200 00400 00600 008000

010203040506070809

1Fuzzy a2

minus00060 minus00020 00020 000600

010203040506070809

1

Fuzzy b1

503557 503607 503657 503707 5037570

010203040506070809

1Fuzzy b2

549700 549900 550100 5503000

010203040506070809

1Fuzzy J3

2000000 12000000 220000000

010203040506070809

1

Fuzzy a1

00980 01000 01020 010400

010203040506070809

1Fuzzy a2

00000 02000 04000 06000 08000 100000

010203040506070809

1

Fuzzy J2

minus1320 minus660 0 660 13200

010203040506070809

1

(a)

Figure 2 Continued

6 Applied Computational Intelligence and Soft Computing

Fuzzy b1

57887 57892 57897 579020

010203040506070809

1Fuzzy b2

549970 549990 550010 5500300

010203040506070809

1Fuzzy J4

1150950 1151050 1151150 11512500

010203040506070809

1

(b)

Figure 2 Fuzzy solution

of DEO method allowed obtaining intuitively meaningfulsolution for the considered problem complicated by fourconflicting criteria and nonstochastic uncertainty intrinsic toreal-world economic problemsThe fuzzy approach allows usto consider many internal and external effects The solutionobtained in fuzzymodel is more sustainable and the objectivefunctions in this case are less sensitive for changes Besides theexperience has shown that in DE the processing time is rela-tively smallThe results obtained in the paper show validity ofthe applied approach The applied approach is characterizedby a low computational complexity as compared with theexisting methods for solving multiobjective optimal controlproblems

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgment

The author would like to thank Professor R A Aliyev forproviding useful advice and a great support in implementingthis research work

References

[1] K J Arrow ldquoApplication of control theory to economic growthrdquoin Mathematics of the Decision Sciences vol 12 of Lecture onApplied Mathematics part 2 pp 83ndash119 American Mathemati-cal Society Providence RI USA 1968

[2] M Intrilligator Mathematical Optimization and EconomicsTheory Prentice Hall Englenwood Clirs NJ USA 1981

[3] GM Grossman and E Helpman Innovation and Growth in theGlobal Economy MIT Press Cambridge UK 1991

[4] C Watanabe ldquoFactors goverining a firmrsquos RampD investment Acomparison between individual and aggregate datardquo in IIASATIT OECD Technical Meeting Paris France 1997

[5] C Watanabe ldquoSystems factors goverining afirmrsquos RampD invest-ment A systems perspective of inter-sectorial technologyspilloverrdquo in IIASA TIT OECD Technical Meeeting LaxenburgAustria 1998

[6] A Kryazhimskii C Watanabe and Yu Tou ldquoThe reachabilityof techno-labor homeostasis via regulation of investments inlabor RampD a model-based analysisrdquo Interium Report IR-02-026 IIASA Laxenburg Austria 2002

[7] A M Tarasyev and C Watanabe ldquoOptimal dynamics of inno-vation in models of economic growthrdquo Journal of OptimizationTheory and Applications vol 108 no 1 pp 175ndash203 2001

[8] S A Reshmin A M Tarasyev and C Watanabe ldquoOptimaltrajectories of the innovation process and their matchingwith econometric datardquo Journal of Optimization Theory andApplications vol 112 no 3 pp 639ndash685 2002

[9] B Kwintiana C Watanabe and A M Tarasyev ldquoDynamicoptimization of RampD intensity under the effeect of technologyassimilation econometric identification for Japanrsquos Automotiveindustryrdquo Interium Report IR-04-058 IIASA Laxenburg Aus-tria 2004

[10] N Kaldor ldquoA model of economic growthrdquo The Economic Jour-nal vol 67 no 268 pp 591ndash624 1957

[11] A Kryazhimskii and C Watanabe Optimization of Technologi-cal Growth GENDAITOSHO Kanagawa Japan 2004

[12] E V Grigorieva and E N Khailov ldquoOptimal control of anonlinear model of Economic growthrdquo Journal of Discrete andContinuous Dynamical Systems Supplement pp 456ndash466 2007

[13] S B Aruoba J Fernandez-Villaverde and J F Rubio RamırezldquoComparing solution methods for dynamic equilibrium econ-omiesrdquo Journal of Economic Dynamics and Control vol 30 no12 pp 2477ndash2508 2006

[14] R A Aliev ldquoModelling and stability analysis in fuzzy eco-nomicsrdquo Applied and Computational Mathematics vol 7 no 1pp 31ndash53 2008

[15] S A Mohaddes M Ghazali K A Rahim M Nasir and AV Kamaid ldquoFuzzy environmental-economic model for landuse planningrdquo American-Eurasian Journal of Agricultural ampEnvironmental Sciences vol 3 no 6 pp 850ndash854 2008

[16] Y Deng and J Xu ldquoA fuzzy multi-objective optimization modelfor energy structure and its application to the energy structureadjustmentmdashinworld natural and cultural heritage areardquoWorldJournal of Modelling and Simulation vol 7 no 2 pp 101ndash1122011

[17] M Pesic S Radukic and J Stankovic ldquoEconomic and envi-ronmental criteria in Multi-objective programming problemsrdquoFacta Universitatis vol 8 no 4 pp 389ndash400 2011

[18] J Ventura ldquoA global view of economic growthrdquo inHandbook ofEconomic Growth P Aghion and S N Durlauf Eds vol 1 pp1420ndash1492 Elsevier 2005

[19] J Kacprzyk and S A Orlovski EdsOptimizationModels UsingFuzzy Sets and Possibility Theory Reidel Publishing CompanyDordrecht The Netherlands 1987

[20] C Carlsson and R Fuller ldquoMultiobjective optimization withlinguistic variablesrdquo in Proceedings of the 6th European Congresson Intelligent Techniques and Soft Computing (EUFIT rsquo98) vol2 pp 1038ndash1042 Veralg Mainz Aachen Germany 1998

Applied Computational Intelligence and Soft Computing 7

[21] C Carlsson and R Fuller ldquoFuzzy multiple criteria decisionmaking recent developmentsrdquo Fuzzy Sets and Systems vol 78no 2 pp 139ndash153 1996

[22] R Fuller C Carlsson and S Giove ldquoOptimization under fuzzylinguistic rule constraintsrdquo in Proceedings of Eurofuse-SIC rsquo99pp 184ndash187 Budapest Hungary 1999

[23] M Sakawa Fuzzy Sets and Interactive Multiobjective Optimiza-tion Plenum Press London UK 1993

[24] L V Kantorovich and V I Zhiyanov ldquoThe one-productdynamic model of economy considering structure of fundsin the presence of technical progressrdquo Reports of Academy ofSciences of the USSR vol 211 no 6 pp 1280ndash1283 1973

[25] V Leontyev Input-Output Economies Oxford University PressNew York NY USA 1966

[26] V F Krotov Ed Bases of the Theory of Optimum Control TheHigher School 1990

[27] L S Pontryagin V G Boltyansky R V Gamkrelidze and EF Mishchenko Mathematical Theory of Optimum ProcessesHayka 1969

[28] A B Vasilyeva and V F Butuzov Asymptotic Decomposition ofDecisions Singulyarno Indignant Equations Science 1973

[29] A B Vasilyeva and M G Dmitriev ldquoSingulyarnye of indigna-tion in problems of optimum controlscience and equipmentResults It is grayrdquo Matem analysis T 20 VINITI pp 3ndash771982

[30] R Ostermark ldquoA fuzzy control model (FCM) for dynamicportfolio managementrdquo Fuzzy Sets and Systems vol 78 no 3pp 243ndash254 1996

[31] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[32] L A Zadeh ldquoToward extended fuzzy logic-a first steprdquo FuzzySets and Systems vol 160 no 21 pp 3175ndash3181 2009

[33] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand applications World Scientific Singapoure 1994

[34] R S Gurbanov L A Gardashova O H Huseynov and K RAliyeva ldquoDecision making problem of a one-product dynamiceconomic modelrdquo in Proceedings of the of the 10th InternationalConference on Application of Fuzzy Systems and Soft Computing(ICAFS rsquo12) pp 161ndash174 Lisbon Portugal 2012

[35] W Pedrycz and K Hirota ldquoFuzzy vector quantization withthe particle swarm optimization a study in fuzzy granulation-degranulation information processingrdquo Signal Processing vol87 no 9 pp 2061ndash2074 2007

[36] Y-J Wang J-S Zhang and G-Y Zhang ldquoA dynamic clusteringbased differential evolution algorithm for global optimizationrdquoEuropean Journal of Operational Research vol 183 no 1 pp 56ndash73 2007

[37] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo Tech Rep TR-95-012 ICSI 1995

[38] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[39] K Price R Storn and J Lampinen Differential EvolutionmdashAPractical Approach To Global Optimization Natural ComputingSeries Springer Berlin Germany 2005

[40] L A Gardashova ldquoFuzzy neural network and DEO based fore-castingrdquo Journal of Automatics Telemechanics and Communica-tion no 29 pp 36ndash44 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 3

evaluation of future trends unforeseen contingencies andother vagueness and impreciseness inherent in economicalprocesses Under the above mentioned assumptions theconsidered dynamicmacroeconomicmodel can be describedby the following FDE

119889119909

119889119905=1

119902((1 minus 119886)

1minus 120583119909 minus

2) (1)

Here 119909 is a fuzzy variable describing imprecise informa-tion on capital that is fuzzy value of capital

1is a fuzzy value

of GDP (the first control variable) 2(the second control

variable) is a fuzzy value of a nonproductive consumptionand 119886 120583 119902 gt 0 are coefficients related to the produc-tive consumption net capital formation and depreciationrespectively

Let us consider a multiobjective optimal control problemof (1) within the period of planning [119905

0 119879]with four objective

functions (criteria) 1198691mdashprofit 119869

2mdashreduction of production

expenditures of GDP 1198693mdasha value of capital at the end of

period [1199050 119879] and 119869

4mdasha discount sum of a direct consump-

tion over [1199050 119879]The considered fuzzymultiobjective optimal

control problem is formulated below [34]

supuisinU

(1198691(u) =int

119879

1199050

119901 (119905) 2(119905) 119889119905 119869

2(u) = minus119888int

119879

1199050

10038161003816100381610038161(119905)1003816100381610038161003816 119889119905

1198693(u) = (119879) 119869

4(u) = int

119879

1199050

120579 (119905) 2(119905) 119889119905)

(2)

subject to

119889119909

119889119905=1

119902((1 minus 119886)

1minus 120583119909 minus

2)

119909 (119905) isin E1 119905 isin [119905

0 119879] 119909 (119905

0) = 1199090

119909 (119879) isinK (119879)

K (119879) = 119909 isin E1 119909 le 119909 (119879) le 119909

u = (1 2)119879

isin U = U1timesU2sub E2

U1= 1isin E1 1le 1(119905) le

U2= 2isin E1 2lowast le 2(119905) le

lowast

2

(3)

Here 119901(119905) is a price of production unit produced at thetime 119905 120579(119905) is the discount function [119905

0 119879] is the optimization

period and 119888 = const gt 0In this work we have decided to solve a problem of opti-

mal control for a single-product dynamical macroeconomicmodel with the help of a method of differential evolution

Recently many evolutionary algorithms have been pro-posed for global optimization of nonlinear nonconvex andnondifferential functions These methods are more flexiblethan classical as they do not require differentiability continu-ity or other properties to hold for optimizing functions Some

of such methods are genetic algorithm (GA) particle swarmoptimization (PSO) [35 36] and differential Evolution (DE)optimization [37ndash39]

Also all of them have their modified versions such asGroup Principle GA with multipoint crossover and muta-tion DE + Clustering [40] and Multifitness function GAAmong many other evolutionary algorithms DE is one of thehighest performance DE is a faster and improved versionof GA As opposite to GA which uses binary coding DEuses real coding of float point numbers The advantage ofDE over standard GA is higher performance efficienciesfor numerical problems not using time-consuming ldquocod-ingdecodingrdquo transformations chromosomegen structuresand other manipulations with binary structures

In [38] a heuristic approach for minimizing possiblynonlinear and nondifferentiable continuous space functionsis presented DEO method is a simple and efficient heuristicfor global optimization over continuous spaces By means ofan extensive test bed it is demonstrated that DEO methodconverges faster and with more certainty than many otheracclaimed global optimization methods The DEO methodrequires few control variables is robust easy to use and lendsitself verywell to parallel computation It should be noted thatdynamic programming is very computationally intensive andeven more for fuzzy problem Therefore here we use DEOmethod

3 Method of Solution

As a stochastic method DE algorithm uses initial populationrandomly generated by uniform distribution differentialmutation probability crossover and selection operators Thepopulation with ps individuals is maintained with eachgeneration A new vector is generated by mutation whichin this case is randomly selecting from the population 3individuals vector indexes 119903

1= 1199032= 1199033and adding a weighted

difference vector between two individuals to a third individ-ual (population member)

The mutated vector then undergoes crossover operationwith another vector generating new offspring vector

The selection process is done as follows If the resultingvector yields a lower objective function value than a predeter-mined population member the newly generated vector willreplace the vector with which it was compared with in thefollowing generation

Extracting distance and direction information from thepopulation to generate random deviations results in an adap-tive scheme with excellent convergence properties DE hasbeen successfully applied to solve a wide range of problemssuch as image classification clustering optimization and soforth

Figure 1 shows the process of generation new trial solu-tion 119883new vector from randomly selected population mem-bers 119883

1199031 1198831199032 1198831199033

(Vector 1198834is then the candidate for

replacement by the new vector if the former is better orlower in terms of the DE cost function) Here we assume thatthe solution vectors are of dimension 1 (ie 2 optimizationparameters)

4 Applied Computational Intelligence and Soft Computing

X1

X2

Xr2

Xr3

Xr4

Xr1

Xr1 minus Xr2

Xnew

Figure 1 The process of generation new trial solution

4 Simulation Results

Let us consider solving of the problem (1)ndash(3) on the base ofthe method suggested in Section 4 under the following data

1198691= Δ

119873

sum

119899=0

119901(1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

2) 997888rarr max

1198692= minus119888Δsum

100381610038161003816100381610038161003816100381610038161003816

1198861(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

1

100381610038161003816100381610038161003816100381610038161003816

997888rarr max

1198693=

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119873sdotΔ 997888rarr max

1198694= Δ

119873minus1

sum

119899=0

119890119903119899Δ(1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ)+ 119887

2)997888rarr max

Table 1 Crisp solution

1198861

1198862

1198871

1198872

119869119894 119894 = 1 4

0108092 137119864 minus 05 5785196 5499741 1198951= 181500564

235119864 minus 10 248119864 minus 07 600 5499995 1198952= minus6599999

0048176 248119864 minus 19 5036475 5500001 1198953= 590870970

0100707 163119864 minus 09 5789494 550 1198954= 115110534

600 le 1198861(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

1le 800

450 le 1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

2le 550

119899 = 0 9 119888 = 05 Δ = 02 119873 = 10

119886 = 005 119901 = 1500 120583 = 008 119902 = 095

119903 = 005 0= (1900 2000 2100)

600 = (580 600 620) 800 = (780 800 820)

450 = (430 450 470) 550 = (530 550 570)

1198861ge 0 119886

2ge 0

(4)

In addition the following parameters are used in DE thepopulation size (NP) the crossover constant (CR) and theweighting factor (119865) [38]

The program written on C was used for computersimulation of the problem The DErsquos parameters used wereNP = 100 CR = 09 and 119865 = 10

The time used for solving the problemwas 15ndash23minutesThe problem was run for the crisp and fuzzy cases

The results of crisp and fuzzy solutions are shown inTable 1 and Figure 2 Solution of the problem by using theDEO method ended up with the results in Table 1 andFigure 2

5 Conclusions

We applied DEO method to solving fuzzy multiobjectiveoptimal control problem for a single-product dynamicalmacroeconomic model For such problems application ofthe well-known existing approaches would be complex bothform intuitive and computational points of view Application

Applied Computational Intelligence and Soft Computing 5

Fuzzy a2

0010203040506070809

1

minus00003 minus00001 00001 00003

Fuzzy b1

0010203040506070809

1

5499740 5499741 5499742 5499743

Fuzzy a1

0010203040506070809

1

01079 01080 01081 01082

Fuzzy b2

5784000 5785000 57860000

010203040506070809

1Fuzzy J1

18120 18140 18160 181800

010203040506070809

1

00000 10000

Fuzzy a1

0010203040506070809

1

00000 10000

Fuzzy a2

0010203040506070809

1

Fuzzy b1

580 588 596 604 612 6200

010203040506070809

1Fuzzy b2

53000 54000 55000 56000 570000

010203040506070809

1

Fuzzy a1

00000 00200 00400 00600 008000

010203040506070809

1Fuzzy a2

minus00060 minus00020 00020 000600

010203040506070809

1

Fuzzy b1

503557 503607 503657 503707 5037570

010203040506070809

1Fuzzy b2

549700 549900 550100 5503000

010203040506070809

1Fuzzy J3

2000000 12000000 220000000

010203040506070809

1

Fuzzy a1

00980 01000 01020 010400

010203040506070809

1Fuzzy a2

00000 02000 04000 06000 08000 100000

010203040506070809

1

Fuzzy J2

minus1320 minus660 0 660 13200

010203040506070809

1

(a)

Figure 2 Continued

6 Applied Computational Intelligence and Soft Computing

Fuzzy b1

57887 57892 57897 579020

010203040506070809

1Fuzzy b2

549970 549990 550010 5500300

010203040506070809

1Fuzzy J4

1150950 1151050 1151150 11512500

010203040506070809

1

(b)

Figure 2 Fuzzy solution

of DEO method allowed obtaining intuitively meaningfulsolution for the considered problem complicated by fourconflicting criteria and nonstochastic uncertainty intrinsic toreal-world economic problemsThe fuzzy approach allows usto consider many internal and external effects The solutionobtained in fuzzymodel is more sustainable and the objectivefunctions in this case are less sensitive for changes Besides theexperience has shown that in DE the processing time is rela-tively smallThe results obtained in the paper show validity ofthe applied approach The applied approach is characterizedby a low computational complexity as compared with theexisting methods for solving multiobjective optimal controlproblems

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgment

The author would like to thank Professor R A Aliyev forproviding useful advice and a great support in implementingthis research work

References

[1] K J Arrow ldquoApplication of control theory to economic growthrdquoin Mathematics of the Decision Sciences vol 12 of Lecture onApplied Mathematics part 2 pp 83ndash119 American Mathemati-cal Society Providence RI USA 1968

[2] M Intrilligator Mathematical Optimization and EconomicsTheory Prentice Hall Englenwood Clirs NJ USA 1981

[3] GM Grossman and E Helpman Innovation and Growth in theGlobal Economy MIT Press Cambridge UK 1991

[4] C Watanabe ldquoFactors goverining a firmrsquos RampD investment Acomparison between individual and aggregate datardquo in IIASATIT OECD Technical Meeting Paris France 1997

[5] C Watanabe ldquoSystems factors goverining afirmrsquos RampD invest-ment A systems perspective of inter-sectorial technologyspilloverrdquo in IIASA TIT OECD Technical Meeeting LaxenburgAustria 1998

[6] A Kryazhimskii C Watanabe and Yu Tou ldquoThe reachabilityof techno-labor homeostasis via regulation of investments inlabor RampD a model-based analysisrdquo Interium Report IR-02-026 IIASA Laxenburg Austria 2002

[7] A M Tarasyev and C Watanabe ldquoOptimal dynamics of inno-vation in models of economic growthrdquo Journal of OptimizationTheory and Applications vol 108 no 1 pp 175ndash203 2001

[8] S A Reshmin A M Tarasyev and C Watanabe ldquoOptimaltrajectories of the innovation process and their matchingwith econometric datardquo Journal of Optimization Theory andApplications vol 112 no 3 pp 639ndash685 2002

[9] B Kwintiana C Watanabe and A M Tarasyev ldquoDynamicoptimization of RampD intensity under the effeect of technologyassimilation econometric identification for Japanrsquos Automotiveindustryrdquo Interium Report IR-04-058 IIASA Laxenburg Aus-tria 2004

[10] N Kaldor ldquoA model of economic growthrdquo The Economic Jour-nal vol 67 no 268 pp 591ndash624 1957

[11] A Kryazhimskii and C Watanabe Optimization of Technologi-cal Growth GENDAITOSHO Kanagawa Japan 2004

[12] E V Grigorieva and E N Khailov ldquoOptimal control of anonlinear model of Economic growthrdquo Journal of Discrete andContinuous Dynamical Systems Supplement pp 456ndash466 2007

[13] S B Aruoba J Fernandez-Villaverde and J F Rubio RamırezldquoComparing solution methods for dynamic equilibrium econ-omiesrdquo Journal of Economic Dynamics and Control vol 30 no12 pp 2477ndash2508 2006

[14] R A Aliev ldquoModelling and stability analysis in fuzzy eco-nomicsrdquo Applied and Computational Mathematics vol 7 no 1pp 31ndash53 2008

[15] S A Mohaddes M Ghazali K A Rahim M Nasir and AV Kamaid ldquoFuzzy environmental-economic model for landuse planningrdquo American-Eurasian Journal of Agricultural ampEnvironmental Sciences vol 3 no 6 pp 850ndash854 2008

[16] Y Deng and J Xu ldquoA fuzzy multi-objective optimization modelfor energy structure and its application to the energy structureadjustmentmdashinworld natural and cultural heritage areardquoWorldJournal of Modelling and Simulation vol 7 no 2 pp 101ndash1122011

[17] M Pesic S Radukic and J Stankovic ldquoEconomic and envi-ronmental criteria in Multi-objective programming problemsrdquoFacta Universitatis vol 8 no 4 pp 389ndash400 2011

[18] J Ventura ldquoA global view of economic growthrdquo inHandbook ofEconomic Growth P Aghion and S N Durlauf Eds vol 1 pp1420ndash1492 Elsevier 2005

[19] J Kacprzyk and S A Orlovski EdsOptimizationModels UsingFuzzy Sets and Possibility Theory Reidel Publishing CompanyDordrecht The Netherlands 1987

[20] C Carlsson and R Fuller ldquoMultiobjective optimization withlinguistic variablesrdquo in Proceedings of the 6th European Congresson Intelligent Techniques and Soft Computing (EUFIT rsquo98) vol2 pp 1038ndash1042 Veralg Mainz Aachen Germany 1998

Applied Computational Intelligence and Soft Computing 7

[21] C Carlsson and R Fuller ldquoFuzzy multiple criteria decisionmaking recent developmentsrdquo Fuzzy Sets and Systems vol 78no 2 pp 139ndash153 1996

[22] R Fuller C Carlsson and S Giove ldquoOptimization under fuzzylinguistic rule constraintsrdquo in Proceedings of Eurofuse-SIC rsquo99pp 184ndash187 Budapest Hungary 1999

[23] M Sakawa Fuzzy Sets and Interactive Multiobjective Optimiza-tion Plenum Press London UK 1993

[24] L V Kantorovich and V I Zhiyanov ldquoThe one-productdynamic model of economy considering structure of fundsin the presence of technical progressrdquo Reports of Academy ofSciences of the USSR vol 211 no 6 pp 1280ndash1283 1973

[25] V Leontyev Input-Output Economies Oxford University PressNew York NY USA 1966

[26] V F Krotov Ed Bases of the Theory of Optimum Control TheHigher School 1990

[27] L S Pontryagin V G Boltyansky R V Gamkrelidze and EF Mishchenko Mathematical Theory of Optimum ProcessesHayka 1969

[28] A B Vasilyeva and V F Butuzov Asymptotic Decomposition ofDecisions Singulyarno Indignant Equations Science 1973

[29] A B Vasilyeva and M G Dmitriev ldquoSingulyarnye of indigna-tion in problems of optimum controlscience and equipmentResults It is grayrdquo Matem analysis T 20 VINITI pp 3ndash771982

[30] R Ostermark ldquoA fuzzy control model (FCM) for dynamicportfolio managementrdquo Fuzzy Sets and Systems vol 78 no 3pp 243ndash254 1996

[31] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[32] L A Zadeh ldquoToward extended fuzzy logic-a first steprdquo FuzzySets and Systems vol 160 no 21 pp 3175ndash3181 2009

[33] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand applications World Scientific Singapoure 1994

[34] R S Gurbanov L A Gardashova O H Huseynov and K RAliyeva ldquoDecision making problem of a one-product dynamiceconomic modelrdquo in Proceedings of the of the 10th InternationalConference on Application of Fuzzy Systems and Soft Computing(ICAFS rsquo12) pp 161ndash174 Lisbon Portugal 2012

[35] W Pedrycz and K Hirota ldquoFuzzy vector quantization withthe particle swarm optimization a study in fuzzy granulation-degranulation information processingrdquo Signal Processing vol87 no 9 pp 2061ndash2074 2007

[36] Y-J Wang J-S Zhang and G-Y Zhang ldquoA dynamic clusteringbased differential evolution algorithm for global optimizationrdquoEuropean Journal of Operational Research vol 183 no 1 pp 56ndash73 2007

[37] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo Tech Rep TR-95-012 ICSI 1995

[38] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[39] K Price R Storn and J Lampinen Differential EvolutionmdashAPractical Approach To Global Optimization Natural ComputingSeries Springer Berlin Germany 2005

[40] L A Gardashova ldquoFuzzy neural network and DEO based fore-castingrdquo Journal of Automatics Telemechanics and Communica-tion no 29 pp 36ndash44 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Applied Computational Intelligence and Soft Computing

X1

X2

Xr2

Xr3

Xr4

Xr1

Xr1 minus Xr2

Xnew

Figure 1 The process of generation new trial solution

4 Simulation Results

Let us consider solving of the problem (1)ndash(3) on the base ofthe method suggested in Section 4 under the following data

1198691= Δ

119873

sum

119899=0

119901(1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

2) 997888rarr max

1198692= minus119888Δsum

100381610038161003816100381610038161003816100381610038161003816

1198861(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

1

100381610038161003816100381610038161003816100381610038161003816

997888rarr max

1198693=

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119873sdotΔ 997888rarr max

1198694= Δ

119873minus1

sum

119899=0

119890119903119899Δ(1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ)+ 119887

2)997888rarr max

Table 1 Crisp solution

1198861

1198862

1198871

1198872

119869119894 119894 = 1 4

0108092 137119864 minus 05 5785196 5499741 1198951= 181500564

235119864 minus 10 248119864 minus 07 600 5499995 1198952= minus6599999

0048176 248119864 minus 19 5036475 5500001 1198953= 590870970

0100707 163119864 minus 09 5789494 550 1198954= 115110534

600 le 1198861(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

1le 800

450 le 1198862(

1198871(1 minus 119886) minus 119887

2

(1 minus 119886) 1198861minus (120583 + 119886

2)

+ (0+

1198872minus 1198871(1 minus 119886)

(1 minus 119886) 1198861minus (120583 + 119886

2))

times 119890((1minus119886)1198861minus(120583+1198862))sdot119899sdotΔ) + 119887

2le 550

119899 = 0 9 119888 = 05 Δ = 02 119873 = 10

119886 = 005 119901 = 1500 120583 = 008 119902 = 095

119903 = 005 0= (1900 2000 2100)

600 = (580 600 620) 800 = (780 800 820)

450 = (430 450 470) 550 = (530 550 570)

1198861ge 0 119886

2ge 0

(4)

In addition the following parameters are used in DE thepopulation size (NP) the crossover constant (CR) and theweighting factor (119865) [38]

The program written on C was used for computersimulation of the problem The DErsquos parameters used wereNP = 100 CR = 09 and 119865 = 10

The time used for solving the problemwas 15ndash23minutesThe problem was run for the crisp and fuzzy cases

The results of crisp and fuzzy solutions are shown inTable 1 and Figure 2 Solution of the problem by using theDEO method ended up with the results in Table 1 andFigure 2

5 Conclusions

We applied DEO method to solving fuzzy multiobjectiveoptimal control problem for a single-product dynamicalmacroeconomic model For such problems application ofthe well-known existing approaches would be complex bothform intuitive and computational points of view Application

Applied Computational Intelligence and Soft Computing 5

Fuzzy a2

0010203040506070809

1

minus00003 minus00001 00001 00003

Fuzzy b1

0010203040506070809

1

5499740 5499741 5499742 5499743

Fuzzy a1

0010203040506070809

1

01079 01080 01081 01082

Fuzzy b2

5784000 5785000 57860000

010203040506070809

1Fuzzy J1

18120 18140 18160 181800

010203040506070809

1

00000 10000

Fuzzy a1

0010203040506070809

1

00000 10000

Fuzzy a2

0010203040506070809

1

Fuzzy b1

580 588 596 604 612 6200

010203040506070809

1Fuzzy b2

53000 54000 55000 56000 570000

010203040506070809

1

Fuzzy a1

00000 00200 00400 00600 008000

010203040506070809

1Fuzzy a2

minus00060 minus00020 00020 000600

010203040506070809

1

Fuzzy b1

503557 503607 503657 503707 5037570

010203040506070809

1Fuzzy b2

549700 549900 550100 5503000

010203040506070809

1Fuzzy J3

2000000 12000000 220000000

010203040506070809

1

Fuzzy a1

00980 01000 01020 010400

010203040506070809

1Fuzzy a2

00000 02000 04000 06000 08000 100000

010203040506070809

1

Fuzzy J2

minus1320 minus660 0 660 13200

010203040506070809

1

(a)

Figure 2 Continued

6 Applied Computational Intelligence and Soft Computing

Fuzzy b1

57887 57892 57897 579020

010203040506070809

1Fuzzy b2

549970 549990 550010 5500300

010203040506070809

1Fuzzy J4

1150950 1151050 1151150 11512500

010203040506070809

1

(b)

Figure 2 Fuzzy solution

of DEO method allowed obtaining intuitively meaningfulsolution for the considered problem complicated by fourconflicting criteria and nonstochastic uncertainty intrinsic toreal-world economic problemsThe fuzzy approach allows usto consider many internal and external effects The solutionobtained in fuzzymodel is more sustainable and the objectivefunctions in this case are less sensitive for changes Besides theexperience has shown that in DE the processing time is rela-tively smallThe results obtained in the paper show validity ofthe applied approach The applied approach is characterizedby a low computational complexity as compared with theexisting methods for solving multiobjective optimal controlproblems

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgment

The author would like to thank Professor R A Aliyev forproviding useful advice and a great support in implementingthis research work

References

[1] K J Arrow ldquoApplication of control theory to economic growthrdquoin Mathematics of the Decision Sciences vol 12 of Lecture onApplied Mathematics part 2 pp 83ndash119 American Mathemati-cal Society Providence RI USA 1968

[2] M Intrilligator Mathematical Optimization and EconomicsTheory Prentice Hall Englenwood Clirs NJ USA 1981

[3] GM Grossman and E Helpman Innovation and Growth in theGlobal Economy MIT Press Cambridge UK 1991

[4] C Watanabe ldquoFactors goverining a firmrsquos RampD investment Acomparison between individual and aggregate datardquo in IIASATIT OECD Technical Meeting Paris France 1997

[5] C Watanabe ldquoSystems factors goverining afirmrsquos RampD invest-ment A systems perspective of inter-sectorial technologyspilloverrdquo in IIASA TIT OECD Technical Meeeting LaxenburgAustria 1998

[6] A Kryazhimskii C Watanabe and Yu Tou ldquoThe reachabilityof techno-labor homeostasis via regulation of investments inlabor RampD a model-based analysisrdquo Interium Report IR-02-026 IIASA Laxenburg Austria 2002

[7] A M Tarasyev and C Watanabe ldquoOptimal dynamics of inno-vation in models of economic growthrdquo Journal of OptimizationTheory and Applications vol 108 no 1 pp 175ndash203 2001

[8] S A Reshmin A M Tarasyev and C Watanabe ldquoOptimaltrajectories of the innovation process and their matchingwith econometric datardquo Journal of Optimization Theory andApplications vol 112 no 3 pp 639ndash685 2002

[9] B Kwintiana C Watanabe and A M Tarasyev ldquoDynamicoptimization of RampD intensity under the effeect of technologyassimilation econometric identification for Japanrsquos Automotiveindustryrdquo Interium Report IR-04-058 IIASA Laxenburg Aus-tria 2004

[10] N Kaldor ldquoA model of economic growthrdquo The Economic Jour-nal vol 67 no 268 pp 591ndash624 1957

[11] A Kryazhimskii and C Watanabe Optimization of Technologi-cal Growth GENDAITOSHO Kanagawa Japan 2004

[12] E V Grigorieva and E N Khailov ldquoOptimal control of anonlinear model of Economic growthrdquo Journal of Discrete andContinuous Dynamical Systems Supplement pp 456ndash466 2007

[13] S B Aruoba J Fernandez-Villaverde and J F Rubio RamırezldquoComparing solution methods for dynamic equilibrium econ-omiesrdquo Journal of Economic Dynamics and Control vol 30 no12 pp 2477ndash2508 2006

[14] R A Aliev ldquoModelling and stability analysis in fuzzy eco-nomicsrdquo Applied and Computational Mathematics vol 7 no 1pp 31ndash53 2008

[15] S A Mohaddes M Ghazali K A Rahim M Nasir and AV Kamaid ldquoFuzzy environmental-economic model for landuse planningrdquo American-Eurasian Journal of Agricultural ampEnvironmental Sciences vol 3 no 6 pp 850ndash854 2008

[16] Y Deng and J Xu ldquoA fuzzy multi-objective optimization modelfor energy structure and its application to the energy structureadjustmentmdashinworld natural and cultural heritage areardquoWorldJournal of Modelling and Simulation vol 7 no 2 pp 101ndash1122011

[17] M Pesic S Radukic and J Stankovic ldquoEconomic and envi-ronmental criteria in Multi-objective programming problemsrdquoFacta Universitatis vol 8 no 4 pp 389ndash400 2011

[18] J Ventura ldquoA global view of economic growthrdquo inHandbook ofEconomic Growth P Aghion and S N Durlauf Eds vol 1 pp1420ndash1492 Elsevier 2005

[19] J Kacprzyk and S A Orlovski EdsOptimizationModels UsingFuzzy Sets and Possibility Theory Reidel Publishing CompanyDordrecht The Netherlands 1987

[20] C Carlsson and R Fuller ldquoMultiobjective optimization withlinguistic variablesrdquo in Proceedings of the 6th European Congresson Intelligent Techniques and Soft Computing (EUFIT rsquo98) vol2 pp 1038ndash1042 Veralg Mainz Aachen Germany 1998

Applied Computational Intelligence and Soft Computing 7

[21] C Carlsson and R Fuller ldquoFuzzy multiple criteria decisionmaking recent developmentsrdquo Fuzzy Sets and Systems vol 78no 2 pp 139ndash153 1996

[22] R Fuller C Carlsson and S Giove ldquoOptimization under fuzzylinguistic rule constraintsrdquo in Proceedings of Eurofuse-SIC rsquo99pp 184ndash187 Budapest Hungary 1999

[23] M Sakawa Fuzzy Sets and Interactive Multiobjective Optimiza-tion Plenum Press London UK 1993

[24] L V Kantorovich and V I Zhiyanov ldquoThe one-productdynamic model of economy considering structure of fundsin the presence of technical progressrdquo Reports of Academy ofSciences of the USSR vol 211 no 6 pp 1280ndash1283 1973

[25] V Leontyev Input-Output Economies Oxford University PressNew York NY USA 1966

[26] V F Krotov Ed Bases of the Theory of Optimum Control TheHigher School 1990

[27] L S Pontryagin V G Boltyansky R V Gamkrelidze and EF Mishchenko Mathematical Theory of Optimum ProcessesHayka 1969

[28] A B Vasilyeva and V F Butuzov Asymptotic Decomposition ofDecisions Singulyarno Indignant Equations Science 1973

[29] A B Vasilyeva and M G Dmitriev ldquoSingulyarnye of indigna-tion in problems of optimum controlscience and equipmentResults It is grayrdquo Matem analysis T 20 VINITI pp 3ndash771982

[30] R Ostermark ldquoA fuzzy control model (FCM) for dynamicportfolio managementrdquo Fuzzy Sets and Systems vol 78 no 3pp 243ndash254 1996

[31] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[32] L A Zadeh ldquoToward extended fuzzy logic-a first steprdquo FuzzySets and Systems vol 160 no 21 pp 3175ndash3181 2009

[33] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand applications World Scientific Singapoure 1994

[34] R S Gurbanov L A Gardashova O H Huseynov and K RAliyeva ldquoDecision making problem of a one-product dynamiceconomic modelrdquo in Proceedings of the of the 10th InternationalConference on Application of Fuzzy Systems and Soft Computing(ICAFS rsquo12) pp 161ndash174 Lisbon Portugal 2012

[35] W Pedrycz and K Hirota ldquoFuzzy vector quantization withthe particle swarm optimization a study in fuzzy granulation-degranulation information processingrdquo Signal Processing vol87 no 9 pp 2061ndash2074 2007

[36] Y-J Wang J-S Zhang and G-Y Zhang ldquoA dynamic clusteringbased differential evolution algorithm for global optimizationrdquoEuropean Journal of Operational Research vol 183 no 1 pp 56ndash73 2007

[37] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo Tech Rep TR-95-012 ICSI 1995

[38] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[39] K Price R Storn and J Lampinen Differential EvolutionmdashAPractical Approach To Global Optimization Natural ComputingSeries Springer Berlin Germany 2005

[40] L A Gardashova ldquoFuzzy neural network and DEO based fore-castingrdquo Journal of Automatics Telemechanics and Communica-tion no 29 pp 36ndash44 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 5

Fuzzy a2

0010203040506070809

1

minus00003 minus00001 00001 00003

Fuzzy b1

0010203040506070809

1

5499740 5499741 5499742 5499743

Fuzzy a1

0010203040506070809

1

01079 01080 01081 01082

Fuzzy b2

5784000 5785000 57860000

010203040506070809

1Fuzzy J1

18120 18140 18160 181800

010203040506070809

1

00000 10000

Fuzzy a1

0010203040506070809

1

00000 10000

Fuzzy a2

0010203040506070809

1

Fuzzy b1

580 588 596 604 612 6200

010203040506070809

1Fuzzy b2

53000 54000 55000 56000 570000

010203040506070809

1

Fuzzy a1

00000 00200 00400 00600 008000

010203040506070809

1Fuzzy a2

minus00060 minus00020 00020 000600

010203040506070809

1

Fuzzy b1

503557 503607 503657 503707 5037570

010203040506070809

1Fuzzy b2

549700 549900 550100 5503000

010203040506070809

1Fuzzy J3

2000000 12000000 220000000

010203040506070809

1

Fuzzy a1

00980 01000 01020 010400

010203040506070809

1Fuzzy a2

00000 02000 04000 06000 08000 100000

010203040506070809

1

Fuzzy J2

minus1320 minus660 0 660 13200

010203040506070809

1

(a)

Figure 2 Continued

6 Applied Computational Intelligence and Soft Computing

Fuzzy b1

57887 57892 57897 579020

010203040506070809

1Fuzzy b2

549970 549990 550010 5500300

010203040506070809

1Fuzzy J4

1150950 1151050 1151150 11512500

010203040506070809

1

(b)

Figure 2 Fuzzy solution

of DEO method allowed obtaining intuitively meaningfulsolution for the considered problem complicated by fourconflicting criteria and nonstochastic uncertainty intrinsic toreal-world economic problemsThe fuzzy approach allows usto consider many internal and external effects The solutionobtained in fuzzymodel is more sustainable and the objectivefunctions in this case are less sensitive for changes Besides theexperience has shown that in DE the processing time is rela-tively smallThe results obtained in the paper show validity ofthe applied approach The applied approach is characterizedby a low computational complexity as compared with theexisting methods for solving multiobjective optimal controlproblems

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgment

The author would like to thank Professor R A Aliyev forproviding useful advice and a great support in implementingthis research work

References

[1] K J Arrow ldquoApplication of control theory to economic growthrdquoin Mathematics of the Decision Sciences vol 12 of Lecture onApplied Mathematics part 2 pp 83ndash119 American Mathemati-cal Society Providence RI USA 1968

[2] M Intrilligator Mathematical Optimization and EconomicsTheory Prentice Hall Englenwood Clirs NJ USA 1981

[3] GM Grossman and E Helpman Innovation and Growth in theGlobal Economy MIT Press Cambridge UK 1991

[4] C Watanabe ldquoFactors goverining a firmrsquos RampD investment Acomparison between individual and aggregate datardquo in IIASATIT OECD Technical Meeting Paris France 1997

[5] C Watanabe ldquoSystems factors goverining afirmrsquos RampD invest-ment A systems perspective of inter-sectorial technologyspilloverrdquo in IIASA TIT OECD Technical Meeeting LaxenburgAustria 1998

[6] A Kryazhimskii C Watanabe and Yu Tou ldquoThe reachabilityof techno-labor homeostasis via regulation of investments inlabor RampD a model-based analysisrdquo Interium Report IR-02-026 IIASA Laxenburg Austria 2002

[7] A M Tarasyev and C Watanabe ldquoOptimal dynamics of inno-vation in models of economic growthrdquo Journal of OptimizationTheory and Applications vol 108 no 1 pp 175ndash203 2001

[8] S A Reshmin A M Tarasyev and C Watanabe ldquoOptimaltrajectories of the innovation process and their matchingwith econometric datardquo Journal of Optimization Theory andApplications vol 112 no 3 pp 639ndash685 2002

[9] B Kwintiana C Watanabe and A M Tarasyev ldquoDynamicoptimization of RampD intensity under the effeect of technologyassimilation econometric identification for Japanrsquos Automotiveindustryrdquo Interium Report IR-04-058 IIASA Laxenburg Aus-tria 2004

[10] N Kaldor ldquoA model of economic growthrdquo The Economic Jour-nal vol 67 no 268 pp 591ndash624 1957

[11] A Kryazhimskii and C Watanabe Optimization of Technologi-cal Growth GENDAITOSHO Kanagawa Japan 2004

[12] E V Grigorieva and E N Khailov ldquoOptimal control of anonlinear model of Economic growthrdquo Journal of Discrete andContinuous Dynamical Systems Supplement pp 456ndash466 2007

[13] S B Aruoba J Fernandez-Villaverde and J F Rubio RamırezldquoComparing solution methods for dynamic equilibrium econ-omiesrdquo Journal of Economic Dynamics and Control vol 30 no12 pp 2477ndash2508 2006

[14] R A Aliev ldquoModelling and stability analysis in fuzzy eco-nomicsrdquo Applied and Computational Mathematics vol 7 no 1pp 31ndash53 2008

[15] S A Mohaddes M Ghazali K A Rahim M Nasir and AV Kamaid ldquoFuzzy environmental-economic model for landuse planningrdquo American-Eurasian Journal of Agricultural ampEnvironmental Sciences vol 3 no 6 pp 850ndash854 2008

[16] Y Deng and J Xu ldquoA fuzzy multi-objective optimization modelfor energy structure and its application to the energy structureadjustmentmdashinworld natural and cultural heritage areardquoWorldJournal of Modelling and Simulation vol 7 no 2 pp 101ndash1122011

[17] M Pesic S Radukic and J Stankovic ldquoEconomic and envi-ronmental criteria in Multi-objective programming problemsrdquoFacta Universitatis vol 8 no 4 pp 389ndash400 2011

[18] J Ventura ldquoA global view of economic growthrdquo inHandbook ofEconomic Growth P Aghion and S N Durlauf Eds vol 1 pp1420ndash1492 Elsevier 2005

[19] J Kacprzyk and S A Orlovski EdsOptimizationModels UsingFuzzy Sets and Possibility Theory Reidel Publishing CompanyDordrecht The Netherlands 1987

[20] C Carlsson and R Fuller ldquoMultiobjective optimization withlinguistic variablesrdquo in Proceedings of the 6th European Congresson Intelligent Techniques and Soft Computing (EUFIT rsquo98) vol2 pp 1038ndash1042 Veralg Mainz Aachen Germany 1998

Applied Computational Intelligence and Soft Computing 7

[21] C Carlsson and R Fuller ldquoFuzzy multiple criteria decisionmaking recent developmentsrdquo Fuzzy Sets and Systems vol 78no 2 pp 139ndash153 1996

[22] R Fuller C Carlsson and S Giove ldquoOptimization under fuzzylinguistic rule constraintsrdquo in Proceedings of Eurofuse-SIC rsquo99pp 184ndash187 Budapest Hungary 1999

[23] M Sakawa Fuzzy Sets and Interactive Multiobjective Optimiza-tion Plenum Press London UK 1993

[24] L V Kantorovich and V I Zhiyanov ldquoThe one-productdynamic model of economy considering structure of fundsin the presence of technical progressrdquo Reports of Academy ofSciences of the USSR vol 211 no 6 pp 1280ndash1283 1973

[25] V Leontyev Input-Output Economies Oxford University PressNew York NY USA 1966

[26] V F Krotov Ed Bases of the Theory of Optimum Control TheHigher School 1990

[27] L S Pontryagin V G Boltyansky R V Gamkrelidze and EF Mishchenko Mathematical Theory of Optimum ProcessesHayka 1969

[28] A B Vasilyeva and V F Butuzov Asymptotic Decomposition ofDecisions Singulyarno Indignant Equations Science 1973

[29] A B Vasilyeva and M G Dmitriev ldquoSingulyarnye of indigna-tion in problems of optimum controlscience and equipmentResults It is grayrdquo Matem analysis T 20 VINITI pp 3ndash771982

[30] R Ostermark ldquoA fuzzy control model (FCM) for dynamicportfolio managementrdquo Fuzzy Sets and Systems vol 78 no 3pp 243ndash254 1996

[31] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[32] L A Zadeh ldquoToward extended fuzzy logic-a first steprdquo FuzzySets and Systems vol 160 no 21 pp 3175ndash3181 2009

[33] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand applications World Scientific Singapoure 1994

[34] R S Gurbanov L A Gardashova O H Huseynov and K RAliyeva ldquoDecision making problem of a one-product dynamiceconomic modelrdquo in Proceedings of the of the 10th InternationalConference on Application of Fuzzy Systems and Soft Computing(ICAFS rsquo12) pp 161ndash174 Lisbon Portugal 2012

[35] W Pedrycz and K Hirota ldquoFuzzy vector quantization withthe particle swarm optimization a study in fuzzy granulation-degranulation information processingrdquo Signal Processing vol87 no 9 pp 2061ndash2074 2007

[36] Y-J Wang J-S Zhang and G-Y Zhang ldquoA dynamic clusteringbased differential evolution algorithm for global optimizationrdquoEuropean Journal of Operational Research vol 183 no 1 pp 56ndash73 2007

[37] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo Tech Rep TR-95-012 ICSI 1995

[38] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[39] K Price R Storn and J Lampinen Differential EvolutionmdashAPractical Approach To Global Optimization Natural ComputingSeries Springer Berlin Germany 2005

[40] L A Gardashova ldquoFuzzy neural network and DEO based fore-castingrdquo Journal of Automatics Telemechanics and Communica-tion no 29 pp 36ndash44 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Applied Computational Intelligence and Soft Computing

Fuzzy b1

57887 57892 57897 579020

010203040506070809

1Fuzzy b2

549970 549990 550010 5500300

010203040506070809

1Fuzzy J4

1150950 1151050 1151150 11512500

010203040506070809

1

(b)

Figure 2 Fuzzy solution

of DEO method allowed obtaining intuitively meaningfulsolution for the considered problem complicated by fourconflicting criteria and nonstochastic uncertainty intrinsic toreal-world economic problemsThe fuzzy approach allows usto consider many internal and external effects The solutionobtained in fuzzymodel is more sustainable and the objectivefunctions in this case are less sensitive for changes Besides theexperience has shown that in DE the processing time is rela-tively smallThe results obtained in the paper show validity ofthe applied approach The applied approach is characterizedby a low computational complexity as compared with theexisting methods for solving multiobjective optimal controlproblems

Conflict of Interests

The author declares that there is no conflict of interests

Acknowledgment

The author would like to thank Professor R A Aliyev forproviding useful advice and a great support in implementingthis research work

References

[1] K J Arrow ldquoApplication of control theory to economic growthrdquoin Mathematics of the Decision Sciences vol 12 of Lecture onApplied Mathematics part 2 pp 83ndash119 American Mathemati-cal Society Providence RI USA 1968

[2] M Intrilligator Mathematical Optimization and EconomicsTheory Prentice Hall Englenwood Clirs NJ USA 1981

[3] GM Grossman and E Helpman Innovation and Growth in theGlobal Economy MIT Press Cambridge UK 1991

[4] C Watanabe ldquoFactors goverining a firmrsquos RampD investment Acomparison between individual and aggregate datardquo in IIASATIT OECD Technical Meeting Paris France 1997

[5] C Watanabe ldquoSystems factors goverining afirmrsquos RampD invest-ment A systems perspective of inter-sectorial technologyspilloverrdquo in IIASA TIT OECD Technical Meeeting LaxenburgAustria 1998

[6] A Kryazhimskii C Watanabe and Yu Tou ldquoThe reachabilityof techno-labor homeostasis via regulation of investments inlabor RampD a model-based analysisrdquo Interium Report IR-02-026 IIASA Laxenburg Austria 2002

[7] A M Tarasyev and C Watanabe ldquoOptimal dynamics of inno-vation in models of economic growthrdquo Journal of OptimizationTheory and Applications vol 108 no 1 pp 175ndash203 2001

[8] S A Reshmin A M Tarasyev and C Watanabe ldquoOptimaltrajectories of the innovation process and their matchingwith econometric datardquo Journal of Optimization Theory andApplications vol 112 no 3 pp 639ndash685 2002

[9] B Kwintiana C Watanabe and A M Tarasyev ldquoDynamicoptimization of RampD intensity under the effeect of technologyassimilation econometric identification for Japanrsquos Automotiveindustryrdquo Interium Report IR-04-058 IIASA Laxenburg Aus-tria 2004

[10] N Kaldor ldquoA model of economic growthrdquo The Economic Jour-nal vol 67 no 268 pp 591ndash624 1957

[11] A Kryazhimskii and C Watanabe Optimization of Technologi-cal Growth GENDAITOSHO Kanagawa Japan 2004

[12] E V Grigorieva and E N Khailov ldquoOptimal control of anonlinear model of Economic growthrdquo Journal of Discrete andContinuous Dynamical Systems Supplement pp 456ndash466 2007

[13] S B Aruoba J Fernandez-Villaverde and J F Rubio RamırezldquoComparing solution methods for dynamic equilibrium econ-omiesrdquo Journal of Economic Dynamics and Control vol 30 no12 pp 2477ndash2508 2006

[14] R A Aliev ldquoModelling and stability analysis in fuzzy eco-nomicsrdquo Applied and Computational Mathematics vol 7 no 1pp 31ndash53 2008

[15] S A Mohaddes M Ghazali K A Rahim M Nasir and AV Kamaid ldquoFuzzy environmental-economic model for landuse planningrdquo American-Eurasian Journal of Agricultural ampEnvironmental Sciences vol 3 no 6 pp 850ndash854 2008

[16] Y Deng and J Xu ldquoA fuzzy multi-objective optimization modelfor energy structure and its application to the energy structureadjustmentmdashinworld natural and cultural heritage areardquoWorldJournal of Modelling and Simulation vol 7 no 2 pp 101ndash1122011

[17] M Pesic S Radukic and J Stankovic ldquoEconomic and envi-ronmental criteria in Multi-objective programming problemsrdquoFacta Universitatis vol 8 no 4 pp 389ndash400 2011

[18] J Ventura ldquoA global view of economic growthrdquo inHandbook ofEconomic Growth P Aghion and S N Durlauf Eds vol 1 pp1420ndash1492 Elsevier 2005

[19] J Kacprzyk and S A Orlovski EdsOptimizationModels UsingFuzzy Sets and Possibility Theory Reidel Publishing CompanyDordrecht The Netherlands 1987

[20] C Carlsson and R Fuller ldquoMultiobjective optimization withlinguistic variablesrdquo in Proceedings of the 6th European Congresson Intelligent Techniques and Soft Computing (EUFIT rsquo98) vol2 pp 1038ndash1042 Veralg Mainz Aachen Germany 1998

Applied Computational Intelligence and Soft Computing 7

[21] C Carlsson and R Fuller ldquoFuzzy multiple criteria decisionmaking recent developmentsrdquo Fuzzy Sets and Systems vol 78no 2 pp 139ndash153 1996

[22] R Fuller C Carlsson and S Giove ldquoOptimization under fuzzylinguistic rule constraintsrdquo in Proceedings of Eurofuse-SIC rsquo99pp 184ndash187 Budapest Hungary 1999

[23] M Sakawa Fuzzy Sets and Interactive Multiobjective Optimiza-tion Plenum Press London UK 1993

[24] L V Kantorovich and V I Zhiyanov ldquoThe one-productdynamic model of economy considering structure of fundsin the presence of technical progressrdquo Reports of Academy ofSciences of the USSR vol 211 no 6 pp 1280ndash1283 1973

[25] V Leontyev Input-Output Economies Oxford University PressNew York NY USA 1966

[26] V F Krotov Ed Bases of the Theory of Optimum Control TheHigher School 1990

[27] L S Pontryagin V G Boltyansky R V Gamkrelidze and EF Mishchenko Mathematical Theory of Optimum ProcessesHayka 1969

[28] A B Vasilyeva and V F Butuzov Asymptotic Decomposition ofDecisions Singulyarno Indignant Equations Science 1973

[29] A B Vasilyeva and M G Dmitriev ldquoSingulyarnye of indigna-tion in problems of optimum controlscience and equipmentResults It is grayrdquo Matem analysis T 20 VINITI pp 3ndash771982

[30] R Ostermark ldquoA fuzzy control model (FCM) for dynamicportfolio managementrdquo Fuzzy Sets and Systems vol 78 no 3pp 243ndash254 1996

[31] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[32] L A Zadeh ldquoToward extended fuzzy logic-a first steprdquo FuzzySets and Systems vol 160 no 21 pp 3175ndash3181 2009

[33] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand applications World Scientific Singapoure 1994

[34] R S Gurbanov L A Gardashova O H Huseynov and K RAliyeva ldquoDecision making problem of a one-product dynamiceconomic modelrdquo in Proceedings of the of the 10th InternationalConference on Application of Fuzzy Systems and Soft Computing(ICAFS rsquo12) pp 161ndash174 Lisbon Portugal 2012

[35] W Pedrycz and K Hirota ldquoFuzzy vector quantization withthe particle swarm optimization a study in fuzzy granulation-degranulation information processingrdquo Signal Processing vol87 no 9 pp 2061ndash2074 2007

[36] Y-J Wang J-S Zhang and G-Y Zhang ldquoA dynamic clusteringbased differential evolution algorithm for global optimizationrdquoEuropean Journal of Operational Research vol 183 no 1 pp 56ndash73 2007

[37] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo Tech Rep TR-95-012 ICSI 1995

[38] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[39] K Price R Storn and J Lampinen Differential EvolutionmdashAPractical Approach To Global Optimization Natural ComputingSeries Springer Berlin Germany 2005

[40] L A Gardashova ldquoFuzzy neural network and DEO based fore-castingrdquo Journal of Automatics Telemechanics and Communica-tion no 29 pp 36ndash44 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 7

[21] C Carlsson and R Fuller ldquoFuzzy multiple criteria decisionmaking recent developmentsrdquo Fuzzy Sets and Systems vol 78no 2 pp 139ndash153 1996

[22] R Fuller C Carlsson and S Giove ldquoOptimization under fuzzylinguistic rule constraintsrdquo in Proceedings of Eurofuse-SIC rsquo99pp 184ndash187 Budapest Hungary 1999

[23] M Sakawa Fuzzy Sets and Interactive Multiobjective Optimiza-tion Plenum Press London UK 1993

[24] L V Kantorovich and V I Zhiyanov ldquoThe one-productdynamic model of economy considering structure of fundsin the presence of technical progressrdquo Reports of Academy ofSciences of the USSR vol 211 no 6 pp 1280ndash1283 1973

[25] V Leontyev Input-Output Economies Oxford University PressNew York NY USA 1966

[26] V F Krotov Ed Bases of the Theory of Optimum Control TheHigher School 1990

[27] L S Pontryagin V G Boltyansky R V Gamkrelidze and EF Mishchenko Mathematical Theory of Optimum ProcessesHayka 1969

[28] A B Vasilyeva and V F Butuzov Asymptotic Decomposition ofDecisions Singulyarno Indignant Equations Science 1973

[29] A B Vasilyeva and M G Dmitriev ldquoSingulyarnye of indigna-tion in problems of optimum controlscience and equipmentResults It is grayrdquo Matem analysis T 20 VINITI pp 3ndash771982

[30] R Ostermark ldquoA fuzzy control model (FCM) for dynamicportfolio managementrdquo Fuzzy Sets and Systems vol 78 no 3pp 243ndash254 1996

[31] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzydifferential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005

[32] L A Zadeh ldquoToward extended fuzzy logic-a first steprdquo FuzzySets and Systems vol 160 no 21 pp 3175ndash3181 2009

[33] P Diamond and P KloedenMetric Spaces of Fuzzy Sets Theoryand applications World Scientific Singapoure 1994

[34] R S Gurbanov L A Gardashova O H Huseynov and K RAliyeva ldquoDecision making problem of a one-product dynamiceconomic modelrdquo in Proceedings of the of the 10th InternationalConference on Application of Fuzzy Systems and Soft Computing(ICAFS rsquo12) pp 161ndash174 Lisbon Portugal 2012

[35] W Pedrycz and K Hirota ldquoFuzzy vector quantization withthe particle swarm optimization a study in fuzzy granulation-degranulation information processingrdquo Signal Processing vol87 no 9 pp 2061ndash2074 2007

[36] Y-J Wang J-S Zhang and G-Y Zhang ldquoA dynamic clusteringbased differential evolution algorithm for global optimizationrdquoEuropean Journal of Operational Research vol 183 no 1 pp 56ndash73 2007

[37] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient adaptive scheme for global optimization over continuousspacesrdquo Tech Rep TR-95-012 ICSI 1995

[38] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[39] K Price R Storn and J Lampinen Differential EvolutionmdashAPractical Approach To Global Optimization Natural ComputingSeries Springer Berlin Germany 2005

[40] L A Gardashova ldquoFuzzy neural network and DEO based fore-castingrdquo Journal of Automatics Telemechanics and Communica-tion no 29 pp 36ndash44 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014