A multi-objective reliability-redundancy allocation problem with active redundancy and interval type-2 fuzzy parameters

  • Pradip KunduEmail author
Original paper


This paper considers a multi-objective reliability-redundancy allocation problem (MORRAP) of a series-parallel system, where system reliability and system cost are to be optimized simultaneously subject to limits on weight, volume, and redundancy level. Precise computation of component reliability is very difficult as the estimation of a single number for the probabilities and performance levels are not always possible, because it is affected by many factors such as inaccuracy and insufficiency of data, manufacturing process, environment in which the system is running, evaluation done by multiple experts, etc. To cope with impreciseness, we model component reliabilities as interval type-2 fuzzy numbers (IT2 FNs), which is more suitable to represent uncertainties than usual or type-1 fuzzy numbers. To solve the problem with interval type-2 fuzzy parameters, we first apply various type-reduction and defuzzification techniques, and obtain corresponding defuzzified values. As maximization of system reliability and minimization of system cost are conflicting to each other, so to obtain compromise solution of the MORRAP with defuzzified parameters, we apply five different multi-objective optimization methods, and then corresponding solutions are analyzed. The problem is illustrated numerically for a real-world MORRAP on pharmaceutical plant, and solutions are obtained by standard optimization solver LINGO, which is based on gradient-based optimization—Generalized Reduced Gradient technique.


Multi-objective optimization Reliability Redundancy allocation Interval type-2 fuzzy set 



The authors are thankful to the Editor and the anonymous Reviewers for valuable suggestions which lead to an improved version of the manuscript.


  1. Akalin O, Akay KU, Sennaroglu B, Tez M (2010) Optimization of chemical admixture for concrete on mortar performance tests using mixture experiments. Chemometr Intell Lab Syst 104:233–242CrossRefGoogle Scholar
  2. Aliev IM, Kara Z (2004) Fuzzy system reliability analysis using time dependent fuzzy set. Control Cybern 33(4):653–662Google Scholar
  3. Ardakan MA, Rezvan MT (2018) Multi-objective optimization of reliability-redundancy allocation problem with cold-standby strategy using NSGA-II. Reliab Eng Syst Saf 172:225–238CrossRefGoogle Scholar
  4. Bit AK, Biswal MP, Alam SS (1993) Fuzzy programming approach to multi-objective solid transportation problem. Fuzzy Sets Syst 57:183–194CrossRefGoogle Scholar
  5. Cao D, Murat A, Chinnam RB (2013) Efficient exact optimization of multi-objective redundancy allocation problems in series-parallel systems. Reliab Eng Syst Saf 111:154–163CrossRefGoogle Scholar
  6. Caserta M, Voß S (2015) An exact algorithm for the reliability redundancy allocation problem. Eur J Oper Res 244(1):110–116CrossRefGoogle Scholar
  7. Chen SM (1994) Fuzzy system reliability analysis using fuzzy number arithmetic operations. Fuzzy Sets Syst 64:31–38CrossRefGoogle Scholar
  8. Cheng CH, Mon DL (1993) Fuzzy system reliability analysis by interval of confidence. Fuzzy Sets Syst 56:29–35CrossRefGoogle Scholar
  9. Coupland S, John R (2007) Geometric type-1 and type-2 fuzzy logic systems. IEEE Trans Fuzzy Syst 15(1):3–15CrossRefGoogle Scholar
  10. Garg H, Sharma SP (2013) Multi-objective reliability-redundancy allocation problem using particle swarm optimization. Comput Ind Eng 64:247–255CrossRefGoogle Scholar
  11. Garg H, Rani M, Sharma SP (2013) Reliability analysis of the engineering systems using intuitionistic fuzzy set theory. J Qual Reliab Eng 943972:1–10CrossRefGoogle Scholar
  12. Garg H, Rani M, Sharma SP, Vishwakarma Y (2014) Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment. Expert Syst Appl 41(7):3157–3167CrossRefGoogle Scholar
  13. Hesamian G (2017) Measuring similarity and ordering based on interval type 2 fuzzy numbers. IEEE Trans Fuzzy Syst 25(4):788–798CrossRefGoogle Scholar
  14. Huang H-Z, Qu J, Zuo MJ (2009) Genetic-algorithm-based optimal apportionment of reliability and redundancy under multiple objectives. IIE Trans 41(4):287–298CrossRefGoogle Scholar
  15. Jamkhaneh EB, Nozari A (2012) Fuzzy system reliability analysis based on confidence interval. Adv Mater Res 433–440:4908–4914CrossRefGoogle Scholar
  16. Karnik NN, Mendel JM (2001) Centroid of a type-2 fuzzy set. Inf Sci 132(1–4):195–220CrossRefGoogle Scholar
  17. Khalili-Damghani K, Abtahi A-R, Tavana M (2013) A new multi-objective particle swarm optimization method for solving reliability redundancy allocation problems. Reliab Eng Syst Saf 111:58–75CrossRefGoogle Scholar
  18. Kumar M, Yadav SP (2012) A novel approach for analyzing fuzzy system reliability using different types of intuitionistic fuzzy failure rates of components. ISA Trans 51(2):288–297CrossRefGoogle Scholar
  19. Kundu P, Kar S, Maiti M (2014) Multi-objective solid transportation problems with budget constraint in uncertain environment. Int J Syst Sci 45(8):1668–1682CrossRefGoogle Scholar
  20. Kuo W, Prasad VR (2000) An annotated overview of system-reliability optimization. IEEE Trans Reliab 49(2):176–187CrossRefGoogle Scholar
  21. Liu F (2008) An efficient centroid type-reduction strategy for general type-2 fuzzy logic system. Inf Sci 178:2224–2236CrossRefGoogle Scholar
  22. Liu F, Mendel JM (2008) Encoding words into interval type-2 fuzzy sets using an interval approach. IEEE Trans Fuzzy Syst 16(6):1503–1521CrossRefGoogle Scholar
  23. Mahapatra GS, Roy TK (2006) Fuzzy multi-objective mathematical programming on reliability optimization model. Appl Math Comput 174:643–659Google Scholar
  24. Malenović A, Dotsikas Y, Mašković M, Jančić-Stojanović B, Ivanović D, Medenica M (2011) Desirability-based optimization and its sensitivity analysis for the perindopril and its impurities analysis in a microemulsion LC system. Microchem J 99:454–460CrossRefGoogle Scholar
  25. Mendel J M (2003). Fuzzy sets for words: a new beginning. In: Proceedings of IEEE international conference on fuzzy systems, pp 37-42, St. Louis, MOGoogle Scholar
  26. Mendel JM (2007a) Computing with words: zadeh, turing, popper and occam. IEEE Comput Intell Mag 2(4):10–17CrossRefGoogle Scholar
  27. Mendel JM (2007b) Computing with words and its relationships with fuzzistics. Inf Sci 177:988–1006CrossRefGoogle Scholar
  28. Mendel JM, John RI (2002) Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst 10(2):307–315CrossRefGoogle Scholar
  29. Mendel JM, Liu F (2007) Super-exponential convergence of the Karnik–Mendel algorithms for computing the centroid of an interval type-2 fuzzy set. IEEE Trans Fuzzy Syst 15(2):309–320CrossRefGoogle Scholar
  30. Mendel JM, John RI, Liu FL (2006) Interval type-2 fuzzy logical systems made simple. IEEE Trans Fuzzy Syst 14(6):808–821CrossRefGoogle Scholar
  31. Miettinen K (2012) Nonlinear multiobjective optimization. Springer, New YorkGoogle Scholar
  32. Miettinen K, Mäkelä MM (2006) Synchronous approach in interactive multiobjective optimization. Eur J Oper Res 170(3):909–922CrossRefGoogle Scholar
  33. Miller S, Gongora M, Garibaldi J, John R (2012) Interval type-2 fuzzy modelling and stochastic search for real-world inventory management. Soft Comput 16:1447–1459CrossRefGoogle Scholar
  34. Muhuri PK, Ashraf Z, Lohani QMD (2018) Multi-objective reliability-redundancy allocation problem with interval type-2 fuzzy uncertainty. IEEE Trans Fuzzy Syst 26(3):1339–1355Google Scholar
  35. Nie M, Tan WW (2008) Towards an efficient type-reduction method for interval type-2 fuzzy logic systems. In: IEEE international conference on fuzzy systems, pp 1425–1432Google Scholar
  36. Pagola M, Lopez-Molina C, Fernandez J, Barrenechea E, Bustince H (2013) Interval type-2 fuzzy sets constructed from several membership functions: application to the fuzzy thresholding algorithm. IEEE Trans Fuzzy Syst 21(2):230–244CrossRefGoogle Scholar
  37. Prasad VR, Kuo W (2000) Reliability optimization of coherent systems. IEEE Trans Reliab 49(3):323–330CrossRefGoogle Scholar
  38. Rao SS, Dhingra AK (1992) Reliability and redundancy apportionment using crisp and fuzzy multi-objective optimization approaches. Reliab Eng Syst Saf 37:253–261CrossRefGoogle Scholar
  39. Roy P, Mahapatra BS, Mahapatra GS, Roy PK (2014) Entropy based region reducing genetic algorithm for reliability redundancy allocation in interval environment. Expert Syst Appl 41(14):6147–6160CrossRefGoogle Scholar
  40. Safari J (2012) Multi-objective reliability optimization of series-parallel systems with a choice of redundancy strategies. Reliab Eng Syst Saf 108:10–20CrossRefGoogle Scholar
  41. Sahoo L, Bhunia AK, Kapur PK (2012) Genetic algorithm based multi-objective reliability optimization in interval environment. Comput Ind Eng 62:152–160CrossRefGoogle Scholar
  42. Sriramdas V, Chaturvedi S, Gargama H (2014) Fuzzy arithmetic based reliability allocation approach during early design and development. Expert Syst Appl 41(7):3444–3449CrossRefGoogle Scholar
  43. Wang Z, Chen T, Tang K, Yao X (2009) A multi-objective approach to redundancy allocation problem in parallel-series systems. In: Proceedings of IEEE congress on evolutionary computation, pp 582–589Google Scholar
  44. Wu D, Mendel JM (2002) Uncertainty bounds and their use in the design of intervaltype-2 fuzzy logic systems. IEEE Trans Fuzzy Syst 10(5):622–639CrossRefGoogle Scholar
  45. Xu Y, Liao H (2016) Reliability analysis and redundancy allocation for a one-shot system containing multifunctional components. IEEE Trans Reliab 65(2):1045–1057CrossRefGoogle Scholar
  46. Yao JS, Su JS, Shih TS (2008) Fuzzy system reliability analysis using triangular fuzzy numbers based on statistical data. J Inf Sci Eng 24:1521–1535Google Scholar
  47. Yetilmezsoy Y (2012) Integration of kinetic modeling and desirability function approach for multi-objective optimization of UASB reactor treating poultry manure wastewater. Bioresour Technol 118:89–101CrossRefGoogle Scholar
  48. Zeleny M (1973) compromising programming, multiple criteriadecision making. University of South Carolina Press, ColumbiaGoogle Scholar
  49. Zhang E, Chen Q (2016) Multi-objective reliability redundancy allocation in an interval environment using particle swarm optimization. Reliab Eng Syst Saf 145:83–92CrossRefGoogle Scholar
  50. Zimmermann H-J (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1:45–55CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Decision Science and Operations Management, School of ManagementBirla Global UniversityBhubaneswarIndia

Personalised recommendations