Document Type : Original Article

**Authors**

Department of Mathematics, Sidho-Kanho-Birsha University, Purulia

**Abstract**

In some reliability optimization problem the constraints relations have probabilistic nature. These constraints are called the chance constraints and are difficult to handle up to some extent. The aim of this paper is to solve the reliability-redundancy allocation problem involving chance constraints in precise and imprecise environments. The component reliabilities of the system are imprecise numbers and further the constraints are stochastic type i.e., chance constraints. The genetic algorithm incorporated with stochastic simulation approach is implemented to optimize the system reliability. We introduced the fuzzy and intuitionistic fuzzy numbers to consider the impreciseness. In particular, component reliabilities are assumed to be triangular fuzzy numbers and triangular intuitionistic fuzzy numbers in two different environments. The simulation technique known as Monte Carlo Simulation is used to find the deterministic constraints from the stochastic ones. To transform the constrained optimization problem into unconstrained one we make use of the effective Big-M penalty approach. The problems are coded with real coded genetic algorithm. We have taken up some numerical examples to show the performance of the proposed method and the sensitivities of the GA parameters are also presented graphically.

**Keywords**

Angelov P.(1995). Crispification: defuzzification over intuitionistic fuzzy sets. Bull Stud Exch Fuzz Appl.; 64: 51-55.

Bhunia AK, Sahoo L, Roy D. ( 2010). Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm. Appl Math Comput; 216: 929-39.

Bhunia AK, Sahoo L, Roy D. (2010). Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm. Appl Math Comput; 216: 929-39.

Charnes A, Cooper WW, Symonds GH. (1958) Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil. Manage Sci.; 4: 235-63.

Chern MS.( 1992). On the computational complexity of reliability redundancy allocation in a series system. Oper Res Lett.; 11(5): 309-15.

Chiang J, Yao JS, and Lee HM. Fuzzy inventory with backorder Defuzzification by signed distance method. J Inf Sci Eng. 2005; 21: 673-94.

Goldberg DE.(1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Boston. Addison-Wesley Longman Publishing Co.;.

Gupta RK, Bhunia AK, Roy D. (2009). A GA based penalty function technique for solving constrained redundancy allocation problem of series system with interval valued reliabilities of components. J Comput Appl Math.; 232(2): 275-84.

Hikita M Y, Nakagawa K, Nakashima, Narihisa H. (1992). Reliability optimization of systems by a surrogate-constraints algorithm. IEEE T Reliab.; 41(3): 473-80.

Holland JH.(1975). Adaptation in Natural and Artificial Systems. Ann Arbor. University of Michigan Press;.

Iwamura K, Liu B. (1996). A genetic algorithm for chance constraint programming. J Inf Opt Sci.; 17(2): 409-22.

Jana RK, Biswal MP.(2004). Stochastic simulation based genetic algorithm for chance constraint programming problems with continuous random variables. Int J Comput Math. b; 81(12): 1455-63.

Jana RK, Biswal MP.(2004). Stochastic simulation based genetic algorithm for chance constraint programming problems with continuous random variables. Int J Comput Math. a; 81(9): 1069-76.

Kall P, Wallace SW. (1994). Stochastic programming. Chichester. John Wiley & Sons;.

Kuo W, Lin H, Xu Z, Zhang W.(1987). Reliability optimization with the Lagrange-multiplier and Branch-and-Bound technique. IEEE T Reliab. 36(5): 624-30.

Kuo W, Prasad VR. (2000). An annotated overview of system reliability optimization. IEEE T Reliab.; 49(2): 487-93.

Kuo, W, Prasad VR, Tillman FA, Hwang CL.( 2001). Optimal Reliability Design: Fundamentals and Applications. London. Cambridge University Press;.

Mahapatra GS, Roy TK. Reliability Evaluation using Triangular Intuitionistic Fuzzy Numbers Arithmetic Operations. Int J Comput Elec Auto Cont Inf Eng. 2009; 3(2): 574-81.

Mahato SK, Bhattacharyee N, Paramanik R. (2020). Fuzzy Reliability Redundancy Optimization with Signed Distance Method for Defuzzification Using Genetic Algorithm*. *Int J Oper Res.; 37(3): 307-23.

Mahato SK, Bhunia AK.(2016). Reliability Optimization in Fuzzy and Interval Environments. Saarbrucken. LAP LAMBERT Academic Publishing..

Mahato SK, Sahoo L, Bhunia AK. (2012). Reliability-redundancy optimization problem with interval valued reliabilities of components via genetic algorithm. J Infor Comput Sci.; 7(4): 284-95.

Miettinen K, Mäkelä MM, Toivanen J. (2003). Numerical comparison of some penalty-based constraint handling techniques in genetic algorithms. J Glob Opt.; 27: 427-46.

Miller LB, Wagner H.(1965). Chance-constrained programming with joint constraints. Oper Res.; 13: 930-45.

Misra B, Sharma U. (1991). An efficient algorithm to solve integer-programming problems arising in system-reliability design. IEEE T Reliab.; 40(1): 81-91.

Nakagawa Y, Miyazaki S.(1981) Surrogate constraints algorithm for reliability optimization problems with two constraints. IEEE T Reliab.; 30(2): 181-84.

Nayagam VLG, Silverman SJG.( 2016). Complete Ranking of Intuitionistic Fuzzy numbers. Fuzzy Inf Eng. 8: 237-54.

Ravi V, Murty BSN, Reddy PJ. (1997). Non equilibrium simulated-annealing algorithm applied to reliability optimization of complex systems. IEEE T Reliab.; 46(2): 233-39.

Rubinstein RY. (1981).Simulation and Monte Carlo method. New York. John Wiley and Sons;.

Sahoo L, Bhunia AK, Kapur PK. (2012). Genetic algorithm based multi-objective reliability optimization in interval environment. Comput Ind Eng. a; 62(1): 152-60.

Sahoo L, Bhunia AK, Roy D. (2010). A genetic algorithm based reliability redundancy optimization for interval valued reliabilities of components. J Appl Quan Meth.; 5(2): 270-87.

Sahoo L, Bhunia AK, Roy D. (2012). An application of genetic algorithm in solving reliability optimization problem under interval component Weibull parameters. Mex J Oper Resb; 1(1): 2-19.

Sahoo L, Bhunia AK, Roy D.( 2013). Reliability optimization in Stochastic Domain via Genetic Algorithm. Int J Qual Reliab Managb. http://dx.doi.org/10.1108/IJQRM-06-2011-0090.

Sahoo L, Mahato SK, Bhunia AK.( 2013). Optimization of System Reliability for Series System with Fuzzy Component Reliabilities by Genetic Algorithm. J Uncertain Syst. a; 8(2): 136-48.

Sheikhalishahi MV, Ebrahimipour H, Zaman, Jeihoonian M. (2013). A hybrid GA-PSO approach for reliability optimization in redundancy allocation problem. Int J Adv Manuf Tech.; 68: 317-38.

Sun X, Li D. (2002). Optimal condition and Branch and Bound algorithm for constrained redundancy optimization in series system. Optim Eng; 3: 53-65.

Sung CS, Cho YK. (1999). Branch and Bound redundancy optimization for a series system with multiple-choice constraints. IEEE T Reliab.; 48(2): 108-17.

Autumn 2020

Pages 25-47