The objective of this work is to present a methodology for solving the Kolmogorov’s differential equations in fuzzy environment using Runga-Kutta and Biogeography-based optimization (RKBBO) algorithm. As most of the data are usually uncertain and hence the fuzzy set theory has been used for handling the uncertainties and therefore the corresponding differential equations becomes fuzzy Kolmogorov’s differential equations. The obtained differential equations has been solved by Runga-Kutta method and improvement of their solution has been found by formulating a nonlinear optimization problem and solved by BBO for finding the global or near to global solutions. Performance index has been used for finding the most critical component of the system for increasing the performance of system by considering reliability, availability and maintainability index. Final results are compared with the existing results. To show the application of the proposed method, a case of the thermal power plant, a repairable industrial system, has been taken for evaluating the fuzzy reliability of the system.
AbbasbandyS. and AllahviranlooT., Numerical solutions of fuzzy differential equations by taylor method, Journal of Computational Methods in Applied Mathematics2 (2002), 113–124.
2.
AggarwalA.K., KumarS., SinghV. and GargT.K., Markov modeling and reliability analysis of urea synthesis system of a fertilizer plant, Journal of Industrial Engineering International11(1) (2015), 1–14.
3.
AhmadianA., SalahshourS. and ChanC.S., A runga-kutta method with reduced number of function evaluations to solve hybrid fuzzy differential equations, Soft Computing19 (2015), 1051–1062.
4.
AllahviranlooT., AbbasbandyS. and BehzadiS., Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method, Soft Computing18(11) (2014), 2191–2200.
5.
AllahviranlooT., KianiN.A. and MotamediN., Solving fuzzy differential equatios by differential transformation method, Information Sciences179(7) (2009), 956–966.
6.
AllahviranlooT. and SalahshourS.A new approach for solving first order fuzzy differential equation, in:
HullermeierE., KruseR., HoffmannF. (eds.), Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications. vol. 81 of Communications in Computer and Information Science, Springer, Berlin Heidelberg, 2010, pp. 522–531.
7.
ArqubO.A., AL-SmadiM., MomaniS. and HayatT., Numerical solutions of fuzzy differential equations using reproducing kernel hilbert space method, Soft Computing (2015) 1–20. http://dx.doi.org/10.1007/s00500-015-1707-4
8.
BuckleyJ.J. and FeuringT., Introduction to fuzzy partial differential equations, Fuzzy sets and Systems105 (1999), 241–248.
9.
BuckleyJ.J. and FeuringT., Fuzzy differential equations, Fuzzy sets and Systems110 (2000), 43–54.
10.
CaoH., KangL. and ChenY., Evolutionary modeling of systems of ordinary differential equations with genetic programming, Genetic Programming and Evolvable Machines1 (2000), 309–337.
11.
ChakravertyS. and TapaswiniS., Standard orthogonal polynomialsbased solution of fuzzy differential equations, International Journal of Fuzzy Computation and Modelling1(1) (2014), 51–71.
12.
DhayabaranD.P. and KingstonJ.C., A method for solving fuzzy differential equations using runge-kutta method with harmonic mean of three quantities, International Journal of Engineering Science and Innovative Technology4 (2015), 90–96.
13.
DingZ., ZhouY. and ZhouM., A polynomial algorithm to performance analysis of concurrent systems via petri nets and ordinary differential equations, IEEE Transactions on Automation Science and Engineering12(1) (2015), 295–308.
14.
DongS.L., VuH. and NguyenP., The formulas of the solution for linear-order random fuzzy differential equations, Journal of Intelligent and Fuzzy Systems28(2) (2015), 795–807.
15.
GargH., An approach for analyzing fuzzy system reliability using particle swarm optimization and intuitionistic fuzzy set theory, Journal of Multiple-Valued Logic and Soft Computing21 (2013), 335–354.
16.
GargH., Performance analysis of reparable industrial systems using PSO and fuzzy confidence interval based methodology, ISA Transactions52(2) (2013), 171–183.
17.
GargH., An approach for analyzing the reliability of industrial system using fuzzy kolmogorov’s differential equations, The Arabian Journal for Science and Engineering40(3) (2015), 975–987.
18.
GargH., An efficient biogeography based optimization algorithm for solving reliability optimization problems, Swarm and Evolutionary Computation24 (2015), 1–10.
19.
GargH. and SharmaS.P., Stochastic behavior analysis of industrial systems utilizing uncertain data, ISA Transactions51(6) (2012), 752–762.
20.
GargH. and SharmaS.P., Multi-objective reliability-redundancy allocation problem using particle swarm optimization, Computers & Industrial Engineering64(1) (2013), 247–255.
21.
GongW., CaiZ., LingC. and LiH., A real-coded biogeographybased optimization with mutation, Applied Mathematics and Computation216(9) (2010), 2749–2758.
22.
KalevaO., Fuzzy differential equations, Fuzzy Sets and Systems24(3) (1987), 301–317.
23.
KumarA. and LataS., Reliability evaluation of condensate system using fuzzy markov model, Annals of Fuzzy Mathematics and Informatics4(2) (2012), 281–291.
24.
KumarP. and AggarwalK.K., Petri net modeling and reliability evaluation of distributed processing systems, Reliability Engineering and System Safety41(2) (1993), 167–176.
25.
MaH. and SimonD., Blended biogeography-based optimization for constrained optimization, Engineering Applications of Artificial Intelligence24(3) (2011), 517–525.
26.
MaM., FriedmanM. and KandelA., Numerical solutions of fuzzy differential equations, Fuzzy sets and Systems105 (1999), 133–138.
27.
MalinowskiM.T., Fuzzy stochastic differential equations of decreasing fuzziness: Approximate solutions, Journal of Intelligent and Fuzzy Systems29(3) (2015), 1087–1107.
28.
MastorakisN.E., Unstable ordinary differential equations: Solution via genetic algorithms and the method of nelder-mead, Stevens Point, in: Proceedings of the 6th WSEAS International Conference on Systems Theory & Scientific Computation, World Scientific and Engineering Academy and Society (WSEAS), Wisconsin, USA, 2007.
29.
ReichC., Simulation of imprecise ordinary differential equations using evolutionary algorithms, in: Proceedings of the 2000 ACM Symposium on Applied Computing1, 2000.
30.
SeikalaS., On the fuzzy intial value problem, Fuzzy Sets and Systems24 (1987), 319–330.
31.
SimonD., Biogeography - based optimization, IEEE Transactions on Evolutionary Computation12(6) (2008), 702–713.
32.
SimonD., ErgezerM., DuD. and RarickR., Markov models for biogeography-based optimization, IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics41(1) (2011), 299–306.
33.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
