{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T21:37:01Z","timestamp":1770241021955,"version":"3.49.0"},"reference-count":83,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T00:00:00Z","timestamp":1770163200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T00:00:00Z","timestamp":1770163200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Supercomput"],"DOI":"10.1007\/s11227-026-08272-4","type":"journal-article","created":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T07:37:01Z","timestamp":1770190621000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimizing the activation sequence in a heterogeneous standby reliability\u2013redundancy allocation problem via markov modeling and a hybrid genetic algorithm"],"prefix":"10.1007","volume":"82","author":[{"given":"Ehsan Ramezani","family":"Dobani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mostafa Abouei","family":"Ardakan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Negar","family":"Hosseinzavieh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammad N.","family":"Juybari","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,4]]},"reference":[{"issue":"6","key":"8272_CR1","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1080\/07408170108936846","volume":"33","author":"DW Coit","year":"2001","unstructured":"Coit DW (2001) Cold-standby redundancy optimization for nonrepairable systems. IIE Trans 33(6):471\u2013478. https:\/\/doi.org\/10.1080\/07408170108936846","journal-title":"IIE Trans"},{"issue":"4","key":"8272_CR2","doi-asserted-by":"publisher","first-page":"2021","DOI":"10.1007\/s40997-023-00609-1","volume":"47","author":"A Kasiri","year":"2023","unstructured":"Kasiri A, Fani Saberi F (2023) Optimal configuration of four reaction wheels in momentum management performance and reliability point of view. Iran J Sci Technol Trans Mech Eng 47(4):2021\u20132043","journal-title":"Iran J Sci Technol Trans Mech Eng"},{"key":"8272_CR3","unstructured":"E. A. Elsayed, Reliability engineering, vol. 88. John Wiley & Sons, 2012."},{"issue":"2","key":"8272_CR4","doi-asserted-by":"publisher","first-page":"176","DOI":"10.1109\/24.877336","volume":"49","author":"W Kuo","year":"2000","unstructured":"Kuo W, Prasad VR (2000) An annotated overview of system-reliability optimization. IEEE Trans Reliab 49(2):176\u2013187","journal-title":"IEEE Trans Reliab"},{"issue":"5","key":"8272_CR5","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/0167-6377(92)90008-Q","volume":"11","author":"M-S Chern","year":"1992","unstructured":"Chern M-S (1992) On the computational complexity of reliability redundancy allocation in a series system. Oper Res Lett 11(5):309\u2013315","journal-title":"Oper Res Lett"},{"key":"8272_CR6","unstructured":"C. E. Ebeling, An introduction to reliability and maintainability engineering. Tata McGraw-Hill Education, 2004."},{"key":"8272_CR7","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.simpat.2013.12.013","volume":"42","author":"MA Ardakan","year":"2014","unstructured":"Ardakan MA, Hamadani AZ (2014) Reliability\u2013redundancy allocation problem with cold-standby redundancy strategy. Simul Model Pract Theory 42:107\u2013118","journal-title":"Simul Model Pract Theory"},{"issue":"1","key":"8272_CR8","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1002\/qre.3243","volume":"40","author":"AS Bhandari","year":"2024","unstructured":"Bhandari AS, Kumar A, Ram M (2024) Hybrid PSO-GWO algorithm for reliability redundancy allocation problem with cold standby strategy. Qual Reliab Eng Int 40(1):115\u2013130. https:\/\/doi.org\/10.1002\/qre.3243","journal-title":"Qual Reliab Eng Int"},{"issue":"3","key":"8272_CR9","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1109\/TR.2004.832816","volume":"53","author":"Y-C Liang","year":"2004","unstructured":"Liang Y-C, Smith AE (2004) An ant colony optimization algorithm for the redundancy allocation problem (RAP). IEEE Trans Reliab 53(3):417\u2013423","journal-title":"IEEE Trans Reliab"},{"issue":"8","key":"8272_CR10","doi-asserted-by":"publisher","first-page":"1257","DOI":"10.1016\/j.ress.2007.06.004","volume":"93","author":"M Ouzineb","year":"2008","unstructured":"Ouzineb M, Nourelfath M, Gendreau M (2008) Tabu search for the redundancy allocation problem of homogenous series\u2013parallel multi-state systems. Reliab Eng Syst Saf 93(8):1257\u20131272","journal-title":"Reliab Eng Syst Saf"},{"issue":"4","key":"8272_CR11","first-page":"1297","volume":"218","author":"Y-C Hsieh","year":"2011","unstructured":"Hsieh Y-C, You P-S (2011) An effective immune based two-phase approach for the optimal reliability\u2013redundancy allocation problem. Appl Math Comput 218(4):1297\u20131307","journal-title":"Appl Math Comput"},{"issue":"11","key":"8272_CR12","doi-asserted-by":"publisher","first-page":"2688","DOI":"10.1016\/j.cor.2012.02.002","volume":"39","author":"T-J Hsieh","year":"2012","unstructured":"Hsieh T-J, Yeh W-C (2012) Penalty guided bees search for redundancy allocation problems with a mix of components in series\u2013parallel systems. Comput Oper Res 39(11):2688\u20132704","journal-title":"Comput Oper Res"},{"key":"8272_CR13","doi-asserted-by":"crossref","unstructured":"T.-J. Hsieh, \u201cReliability-redundancy allocation in k-out-of-n (G) systems with mixed redundancy and component mixing strategies by using a tree-profiled artificial bee colony algorithm,\u201d Ann. Oper. Res., pp. 1\u201339, 2025.","DOI":"10.1007\/s10479-025-06819-x"},{"issue":"1","key":"8272_CR14","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1016\/j.simpat.2010.07.010","volume":"19","author":"ID Lins","year":"2011","unstructured":"Lins ID, Droguett EL (2011) Redundancy allocation problems considering systems with imperfect repairs using multi-objective genetic algorithms and discrete event simulation. Simul Model Pract Theory 19(1):362\u2013381","journal-title":"Simul Model Pract Theory"},{"issue":"4","key":"8272_CR15","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/s41872-021-00170-3","volume":"10","author":"PP Guilani","year":"2021","unstructured":"Guilani PP, Ardakan MA, Guilani PP, Lehdarboni SA (2021) An exact Markov chain model for reliability\u2013redundancy allocation problem with a choice of redundancy strategy. Life Cycle Reliab Saf Eng 10(4):373\u2013385","journal-title":"Life Cycle Reliab Saf Eng"},{"issue":"4","key":"8272_CR16","first-page":"853","volume":"238","author":"F Kayedpour","year":"2024","unstructured":"Kayedpour F, Amiri M, Rafizadeh M, Nia AS, Sharifi M (2024) A Markov chain-based genetic algorithm for solving a redundancy allocation problem for a system with repairable warm standby components. Proc Inst Mech Eng part o J risk Reliab 238(4):853\u2013872","journal-title":"Proc Inst Mech Eng part o J risk Reliab"},{"key":"8272_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2022.108394","volume":"222","author":"A Zaretalab","year":"2022","unstructured":"Zaretalab A, Sharifi M, Guilani PP, Taghipour S, Niaki STA (2022) A multi-objective model for optimizing the redundancy allocation, component supplier selection, and reliable activities for multi-state systems. Reliab Eng Syst Saf 222:108394. https:\/\/doi.org\/10.1016\/j.ress.2022.108394","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2022.108843","volume":"229","author":"J Zhang","year":"2023","unstructured":"Zhang J, Lv H, Hou J (2023) A novel general model for RAP and RRAP optimization of k-out-of-n:G systems with mixed redundancy strategy. Reliab Eng Syst Saf 229:108843. https:\/\/doi.org\/10.1016\/j.ress.2022.108843","journal-title":"Reliab Eng Syst Saf"},{"issue":"4","key":"8272_CR19","first-page":"786","volume":"239","author":"H Ge","year":"2025","unstructured":"Ge H, Gao H, Li X (2025) Reliability optimization of reliability-redundancy allocation problems based on K-mixed strategy. Proc Inst Mech Eng Part O J Risk Reliab 239(4):786\u2013801","journal-title":"Proc Inst Mech Eng Part O J Risk Reliab"},{"issue":"7","key":"8272_CR20","doi-asserted-by":"publisher","first-page":"7595","DOI":"10.1007\/s12652-020-02481-2","volume":"12","author":"TK Sharma","year":"2021","unstructured":"Sharma TK (2021) Enhanced butterfly optimization algorithm for reliability optimization problems. J Ambient Intell Humaniz Comput 12(7):7595\u20137619","journal-title":"J Ambient Intell Humaniz Comput"},{"key":"8272_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109807","volume":"244","author":"MAM Chowdury","year":"2024","unstructured":"Chowdury MAM, Nath R, Shukla AK, Rauniyar A, Muhuri PK (2024) Multi-task optimization in reliability redundancy allocation problem: a multifactorial evolutionary-based approach. Reliab Eng Syst Saf 244:109807. https:\/\/doi.org\/10.1016\/j.ress.2023.109807","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2022.117161","volume":"200","author":"M Sharifi","year":"2022","unstructured":"Sharifi M, Taghipour S (2022) Redundancy allocation problem of a multi-state system with binary-state continuous performance level components. Expert Syst Appl 200:117161","journal-title":"Expert Syst Appl"},{"issue":"13","key":"8272_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/s11227-025-07708-7","volume":"81","author":"M Akhtar","year":"2025","unstructured":"Akhtar M, Mandal G, Bhunia AK (2025) A hybrid algorithm based on differential evolution for reliability\u2013redundancy allocation problem of a series system under multiple brands. J Supercomput 81(13):1250","journal-title":"J Supercomput"},{"issue":"1","key":"8272_CR24","doi-asserted-by":"publisher","first-page":"990","DOI":"10.1016\/j.eswa.2011.07.099","volume":"39","author":"SJ Sadjadi","year":"2012","unstructured":"Sadjadi SJ, Soltani R (2012) Alternative design redundancy allocation using an efficient heuristic and a honey bee mating algorithm. Expert Syst Appl 39(1):990\u2013999. https:\/\/doi.org\/10.1016\/j.eswa.2011.07.099","journal-title":"Expert Syst Appl"},{"key":"8272_CR25","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1016\/j.cie.2017.04.009","volume":"108","author":"AE Jahromi","year":"2017","unstructured":"Jahromi AE, Feizabadi M (2017) Optimization of multi-objective redundancy allocation problem with non-homogeneous components. Comput Ind Eng 108:111\u2013123","journal-title":"Comput Ind Eng"},{"issue":"3","key":"8272_CR26","doi-asserted-by":"publisher","first-page":"1064","DOI":"10.1016\/j.apm.2013.07.035","volume":"38","author":"R Soltani","year":"2014","unstructured":"Soltani R, Sadjadi SJ, Tofigh AA (2014) A model to enhance the reliability of the serial parallel systems with component mixing. Appl Math Model 38(3):1064\u20131076. https:\/\/doi.org\/10.1016\/j.apm.2013.07.035","journal-title":"Appl Math Model"},{"key":"8272_CR27","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/j.amc.2015.08.069","volume":"271","author":"R Soltani","year":"2015","unstructured":"Soltani R, Safari J, Sadjadi SJ (2015) Robust counterpart optimization for the redundancy allocation problem in series-parallel systems with component mixing under uncertainty. Appl Math Comput 271:80\u201388. https:\/\/doi.org\/10.1016\/j.amc.2015.08.069","journal-title":"Appl Math Comput"},{"key":"8272_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.knosys.2014.03.011","volume":"64","author":"W-C Yeh","year":"2014","unstructured":"Yeh W-C (2014) Orthogonal simplified swarm optimization for the series\u2013parallel redundancy allocation problem with a mix of components. Knowl Based Syst 64:1\u201312. https:\/\/doi.org\/10.1016\/j.knosys.2014.03.011","journal-title":"Knowl Based Syst"},{"issue":"24","key":"8272_CR29","doi-asserted-by":"publisher","first-page":"5109","DOI":"10.1080\/03610926.2013.813046","volume":"44","author":"J Zhuang","year":"2015","unstructured":"Zhuang J, Li X (2015) Allocating redundancies to k-out-of-n systems with independent and heterogeneous components. Commun Stat Theory Methods 44(24):5109\u20135119. https:\/\/doi.org\/10.1080\/03610926.2013.813046","journal-title":"Commun Stat Theory Methods"},{"key":"8272_CR30","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.ress.2016.08.023","volume":"157","author":"M Feizabadi","year":"2017","unstructured":"Feizabadi M, Jahromi AE (2017) A new model for reliability optimization of series-parallel systems with non-homogeneous components. Reliab Eng Syst Saf 157:101\u2013112. https:\/\/doi.org\/10.1016\/j.ress.2016.08.023","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR31","doi-asserted-by":"crossref","unstructured":"J. Shen, R. Zheng, X. Zhu, and Y. Ma, \u201cReliability analysis for multi-component systems with self-healing states and unobservable downtime,\u201d Qual. Technol. Quant. Manag., pp. 1\u201321, 2025.","DOI":"10.1080\/16843703.2025.2547376"},{"key":"8272_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109835","volume":"244","author":"R Nath","year":"2024","unstructured":"Nath R, Muhuri PK (2024) A novel evolutionary solution approach for many-objective reliability-redundancy allocation problem based on objective prioritization and constraint optimization. Reliab Eng Syst Saf 244:109835. https:\/\/doi.org\/10.1016\/j.ress.2023.109835","journal-title":"Reliab Eng Syst Saf"},{"issue":"13","key":"8272_CR33","doi-asserted-by":"publisher","first-page":"4514","DOI":"10.1080\/03610926.2021.1995433","volume":"52","author":"M Sharifi","year":"2023","unstructured":"Sharifi M, Pourkarim Guilani P, Zaretalab A, Abhari A (2023) Reliability evaluation of a system with active redundancy strategy and load-sharing time-dependent failure rate components using Markov process. Commun Stat Methods 52(13):4514\u20134533","journal-title":"Commun Stat Methods"},{"key":"8272_CR34","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109580","volume":"241","author":"L Gu","year":"2024","unstructured":"Gu L, Wang G, Zhou Y, Peng R (2024) Reliability optimization of multi-state systems with two performance sharing groups. Reliab Eng Syst Saf 241:109580","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109310","volume":"236","author":"N Wang","year":"2023","unstructured":"Wang N, Xiao Y, Tian T, Yang J (2023) The optimal 5G base station location of the wireless sensor network considering timely reliability. Reliab Eng Syst Saf 236:109310","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR36","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109769","volume":"242","author":"N Wang","year":"2024","unstructured":"Wang N, Tian T, He J, Zhang C, Yang J (2024) Transmission reliability evaluation of wireless sensor networks considering channel capacity randomness and energy consumption failure. Reliab Eng Syst Saf 242:109769","journal-title":"Reliab Eng Syst Saf"},{"issue":"1","key":"8272_CR37","doi-asserted-by":"publisher","first-page":"446","DOI":"10.1002\/qre.3429","volume":"40","author":"T Tian","year":"2024","unstructured":"Tian T, Yang J, Wang N (2024) Reliability assessment of two\u2010level balanced systems with common bus performance sharing subjected to epistemic uncertainty. Qual Reliab Eng Int 40(1):446\u2013471","journal-title":"Qual Reliab Eng Int"},{"issue":"1","key":"8272_CR38","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1109\/TR.2014.2337073","volume":"64","author":"Q Zhai","year":"2015","unstructured":"Zhai Q, Yang J, Peng R, Zhao Y (2015) A study of optimal component order in a general 1-out-of-$ n $ warm standby system. IEEE Trans Reliab 64(1):349\u2013358","journal-title":"IEEE Trans Reliab"},{"issue":"4","key":"8272_CR39","doi-asserted-by":"publisher","first-page":"570","DOI":"10.1016\/j.cie.2013.05.001","volume":"65","author":"G Levitin","year":"2013","unstructured":"Levitin G, Xing L, Dai Y (2013) Optimal sequencing of warm standby elements. Computers Ind Eng 65(4):570\u2013576. https:\/\/doi.org\/10.1016\/j.cie.2013.05.001","journal-title":"Computers Ind Eng"},{"issue":"2","key":"8272_CR40","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1002\/asmb.704","volume":"24","author":"JH Cha","year":"2008","unstructured":"Cha JH, Mi J, Yun WY (2008) Modelling a general standby system and evaluation of its performance. Appl Stoch Models Bus Ind 24(2):159\u2013169","journal-title":"Appl Stoch Models Bus Ind"},{"key":"8272_CR41","doi-asserted-by":"crossref","unstructured":"S. V Amari and G. Dill, \u201cA new method for reliability analysis of standby systems,\u201d in Reliability and Maintainability Symposium, 2009. RAMS 2009. Annual, IEEE, 2009, pp. 417\u2013422.","DOI":"10.1109\/RAMS.2009.4914713"},{"key":"8272_CR42","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.ress.2013.04.010","volume":"118","author":"G Levitin","year":"2013","unstructured":"Levitin G, Xing L, Dai Y (2013) Cold-standby sequencing optimization considering mission cost. Reliab Eng Syst Saf 118:28\u201334","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR43","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.ress.2016.10.019","volume":"159","author":"H Kim","year":"2017","unstructured":"Kim H, Kim P (2017) Reliability models for a nonrepairable system with heterogeneous components having a phase-type time-to-failure distribution. Reliab Eng Syst Saf 159:37\u201346. https:\/\/doi.org\/10.1016\/j.ress.2016.10.019","journal-title":"Reliab Eng Syst Saf"},{"issue":"Supplement C","key":"8272_CR44","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1016\/j.ress.2017.10.020","volume":"170","author":"H Kim","year":"2018","unstructured":"Kim H (2018) Maximization of system reliability with the consideration of component sequencing. Reliab Eng Syst Saf 170(Supplement C):64\u201372. https:\/\/doi.org\/10.1016\/j.ress.2017.10.020","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR45","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2025.110990","volume":"260","author":"L Gu","year":"2025","unstructured":"Gu L, Wang G, Zhou Y (2025) Reliability analysis and optimization of multi-state tree-structured systems with performance sharing mechanism. Reliab Eng Syst Saf 260:110990","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR46","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1016\/j.ress.2017.03.009","volume":"164","author":"H Gholinezhad","year":"2017","unstructured":"Gholinezhad H, Zeinal Hamadani A (2017) A new model for the redundancy allocation problem with component mixing and mixed redundancy strategy. Reliab Eng Syst Saf 164:66\u201373. https:\/\/doi.org\/10.1016\/j.ress.2017.03.009","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR47","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2019.106660","volume":"193","author":"PP Guilani","year":"2020","unstructured":"Guilani PP, Juybari MN, Ardakan MA, Kim H (2020) Sequence optimization in reliability problems with a mixed strategy and heterogeneous backup scheme. Reliab Eng Syst Saf 193:106660. https:\/\/doi.org\/10.1016\/j.ress.2019.106660","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR48","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109632","volume":"241","author":"M Sharifi","year":"2024","unstructured":"Sharifi M, Taghipour S (2024) Redundancy allocation problem with a mix of components for a multi-state system and continuous performance level components. Reliab Eng Syst Saf 241:109632. https:\/\/doi.org\/10.1016\/j.ress.2023.109632","journal-title":"Reliab Eng Syst Saf"},{"issue":"9","key":"8272_CR49","doi-asserted-by":"publisher","first-page":"3794","DOI":"10.1016\/j.eswa.2012.12.093","volume":"40","author":"LD Afonso","year":"2013","unstructured":"Afonso LD, Mariani VC, dos Santos Coelho L (2013) Modified imperialist competitive algorithm based on attraction and repulsion concepts for reliability-redundancy optimization. Expert Syst Appl 40(9):3794\u20133802","journal-title":"Expert Syst Appl"},{"key":"8272_CR50","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1016\/j.ress.2015.06.002","volume":"142","author":"C-L Huang","year":"2015","unstructured":"Huang C-L (2015) A particle-based simplified swarm optimization algorithm for reliability redundancy allocation problems. Reliab Eng Syst Saf 142:221\u2013230","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR51","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/j.isatra.2015.09.015","volume":"59","author":"Q He","year":"2015","unstructured":"He Q, Hu X, Ren H, Zhang H (2015) A novel artificial fish swarm algorithm for solving large-scale reliability\u2013redundancy application problem. ISA Trans 59:105\u2013113","journal-title":"ISA Trans"},{"issue":"1","key":"8272_CR52","first-page":"91","volume":"13","author":"A Samanta","year":"2019","unstructured":"Samanta A, Basu K (2019) A novel particle swarm optimization with fuzzy adaptive inertia weight for reliability redundancy allocation problems. Intell Decis Technol 13(1):91\u201399","journal-title":"Intell Decis Technol"},{"key":"8272_CR53","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2019.106531","volume":"191","author":"PK Muhuri","year":"2019","unstructured":"Muhuri PK, Nath R (2019) A novel evolutionary algorithmic solution approach for bilevel reliability-redundancy allocation problem. Reliab Eng Syst Saf 191:106531","journal-title":"Reliab Eng Syst Saf"},{"issue":"11","key":"8272_CR54","doi-asserted-by":"publisher","first-page":"2728","DOI":"10.1080\/03610926.2019.1580741","volume":"49","author":"N Mahdavi-Nasab","year":"2020","unstructured":"Mahdavi-Nasab N, Abouei Ardakan M, Mohammadi M (2020) Water cycle algorithm for solving the reliability-redundancy allocation problem with a choice of redundancy strategies. Commun Stat Methods 49(11):2728\u20132748","journal-title":"Commun Stat Methods"},{"key":"8272_CR55","doi-asserted-by":"crossref","unstructured":"S. Gupta, K. Deep, and A. Assad, \u201cReliability\u2013redundancy allocation using random walk gray wolf optimizer,\u201d in Soft Computing for Problem Solving: SocProS 2018, Volume 1, Springer, 2019, pp. 941\u2013959.","DOI":"10.1007\/978-981-15-0035-0_75"},{"key":"8272_CR56","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1016\/j.ress.2018.09.005","volume":"181","author":"Z Ouyang","year":"2019","unstructured":"Ouyang Z, Liu Y, Ruan S-J, Jiang T (2019) An improved particle swarm optimization algorithm for reliability-redundancy allocation problem with mixed redundancy strategy and heterogeneous components. Reliab Eng Syst Saf 181:62\u201374. https:\/\/doi.org\/10.1016\/j.ress.2018.09.005","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR57","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109815","volume":"242","author":"H Gholinezhad","year":"2024","unstructured":"Gholinezhad H (2024) A new model for reliability redundancy allocation problem with component mixing. Reliab Eng Syst Saf 242:109815. https:\/\/doi.org\/10.1016\/j.ress.2023.109815","journal-title":"Reliab Eng Syst Saf"},{"issue":"1","key":"8272_CR58","first-page":"152","volume":"237","author":"M Sharifi","year":"2023","unstructured":"Sharifi M, Sayyad A, Taghipour S, Abhari A (2023) Optimizing a joint reliability-redundancy allocation problem with common cause multi-state failures using immune algorithm. Proc Inst Mech Eng Part O J Risk Reliab 237(1):152\u2013165","journal-title":"Proc Inst Mech Eng Part O J Risk Reliab"},{"key":"8272_CR59","doi-asserted-by":"publisher","DOI":"10.1016\/j.cie.2025.111438","author":"A Najmi","year":"2025","unstructured":"Najmi A, Guilani PP (2025) Optimization of a bi-objective reliability redundancy allocation problem with heterogeneous components and strategy selection. Comput Ind Eng. https:\/\/doi.org\/10.1016\/j.cie.2025.111438","journal-title":"Comput Ind Eng"},{"key":"8272_CR60","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2019.106563","volume":"191","author":"ER Dobani","year":"2019","unstructured":"Dobani ER, Ardakan MA, Davari-Ardakani H, Juybari MN (2019) RRAP-CM: a new reliability-redundancy allocation problem with heterogeneous components. Reliab Eng Syst Saf 191:106563","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR61","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1016\/j.ress.2014.06.001","volume":"130","author":"M Abouei Ardakan","year":"2014","unstructured":"Abouei Ardakan M, Zeinal Hamadani A (2014) Reliability optimization of series\u2013parallel systems with mixed redundancy strategy in subsystems. Reliab Eng Syst Saf 130:132\u2013139. https:\/\/doi.org\/10.1016\/j.ress.2014.06.001","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR62","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109509","volume":"239","author":"S Wang","year":"2023","unstructured":"Wang S, Yao Y, Ge D, Lin Z, Wu J, Yu J (2023) Reliability evaluation of standby redundant systems based on the survival signatures methods. Reliab Eng Syst Saf 239:109509","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR63","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109548","volume":"240","author":"C Wang","year":"2023","unstructured":"Wang C, Wang X, Xing L, Guan Q, Yang C, Yu M (2023) Efficient reliability approximation for large k-out-of-n cold standby systems with position-dependent component lifetime distributions. Reliab Eng Syst Saf 240:109548. https:\/\/doi.org\/10.1016\/j.ress.2023.109548","journal-title":"Reliab Eng Syst Saf"},{"issue":"Supplement C","key":"8272_CR64","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/j.ress.2016.10.033","volume":"159","author":"H Kim","year":"2017","unstructured":"Kim H, Kim P (2017) Reliability\u2013redundancy allocation problem considering optimal redundancy strategy using parallel genetic algorithm. Reliab Eng Syst Saf 159(Supplement C):153\u2013160. https:\/\/doi.org\/10.1016\/j.ress.2016.10.033","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR65","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/j.ress.2017.12.019","volume":"172","author":"M Abouei Ardakan","year":"2018","unstructured":"Abouei Ardakan M, Rezvan MT (2018) Multi-objective optimization of reliability\u2013redundancy allocation problem with cold-standby strategy using NSGA-II. Reliab Eng Syst Saf 172:225\u2013238. https:\/\/doi.org\/10.1016\/j.ress.2017.12.019","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR66","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109780","volume":"242","author":"M Oszczypa\u0142a","year":"2024","unstructured":"Oszczypa\u0142a M, Konwerski J, Zi\u00f3\u0142kowski J, Ma\u0142achowski J (2024) Reliability analysis and redundancy optimization of k-out-of-n systems with random variable k using continuous time Markov chain and Monte Carlo simulation. Reliab Eng Syst Saf 242:109780. https:\/\/doi.org\/10.1016\/j.ress.2023.109780","journal-title":"Reliab Eng Syst Saf"},{"key":"8272_CR67","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2024.112041","volume":"165","author":"M Oszczypa\u0142a","year":"2024","unstructured":"Oszczypa\u0142a M, Zi\u00f3\u0142kowski J, Ma\u0142achowski J (2024) Redundancy allocation problem in repairable k-out-of-n systems with cold, warm, and hot standby: a genetic algorithm for availability optimization. Appl Soft Comput 165:112041. https:\/\/doi.org\/10.1016\/j.asoc.2024.112041","journal-title":"Appl Soft Comput"},{"key":"8272_CR68","doi-asserted-by":"crossref","unstructured":"R. Zheng, \u201cOptimal replacement for a partially observable system subject to dependent failure modes,\u201d Nav. Res. Logist., 2025.","DOI":"10.1002\/nav.70024"},{"key":"8272_CR69","doi-asserted-by":"crossref","unstructured":"L. Gu, G. Wang, S. Zhang, F. Duan, and F. Wu, \u201cReliability modeling for two-phase degradation products based on the inverse Gaussian process,\u201d Proc. Inst. Mech. Eng. Part O J. Risk Reliab., p. 1748006X251348686, 2024.","DOI":"10.1177\/1748006X251348686"},{"issue":"11","key":"8272_CR70","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1080\/0740817X.2016.1189631","volume":"48","author":"M Abouei Ardakan","year":"2016","unstructured":"Abouei Ardakan M, Sima M, Zeinal Hamadani A, Coit DW (2016) A novel strategy for redundant components in reliability--redundancy allocation problems. IIE Trans 48(11):1043\u20131057. https:\/\/doi.org\/10.1080\/0740817X.2016.1189631","journal-title":"IIE Trans"},{"issue":"4","key":"8272_CR71","doi-asserted-by":"publisher","first-page":"576","DOI":"10.1109\/24.249589","volume":"41","author":"AK Dhingra","year":"1992","unstructured":"Dhingra AK (1992) Optimal apportionment of reliability and redundancy in series systems under multiple objectives. IEEE Trans Reliab 41(4):576\u2013582","journal-title":"IEEE Trans Reliab"},{"issue":"1","key":"8272_CR72","doi-asserted-by":"publisher","first-page":"515","DOI":"10.2478\/ntpe-2018-0064","volume":"1","author":"W Tarelko","year":"2018","unstructured":"Tarelko W (2018) Redundancy as a way increasing reliability of ship power plants. New Trends Prod Eng 1(1):515\u2013522","journal-title":"New Trends Prod Eng"},{"issue":"5","key":"8272_CR73","doi-asserted-by":"publisher","first-page":"1562","DOI":"10.1016\/j.cor.2008.02.017","volume":"36","author":"A Azaron","year":"2009","unstructured":"Azaron A, Perkgoz C, Katagiri H, Kato K, Sakawa M (2009) Multi-objective reliability optimization for dissimilar-unit cold-standby systems using a genetic algorithm. Comput Oper Res 36(5):1562\u20131571","journal-title":"Comput Oper Res"},{"key":"8272_CR74","doi-asserted-by":"crossref","unstructured":"G. Latouche and V. Ramaswami, Introduction to matrix analytic methods in stochastic modeling, vol. 5. Siam, 1999.","DOI":"10.1137\/1.9780898719734"},{"issue":"1","key":"8272_CR75","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1016\/j.bjbas.2015.02.003","volume":"4","author":"H Garg","year":"2015","unstructured":"Garg H (2015) An approach for solving constrained reliability-redundancy allocation problems using cuckoo search algorithm. Beni-Suef Univ J Basic Appl Sci 4(1):14\u201325. https:\/\/doi.org\/10.1016\/j.bjbas.2015.02.003","journal-title":"Beni-Suef Univ J Basic Appl Sci"},{"issue":"11","key":"8272_CR76","doi-asserted-by":"publisher","first-page":"1273","DOI":"10.1080\/0305215X.2012.729055","volume":"45","author":"E Valian","year":"2013","unstructured":"Valian E, Valian E (2013) A cuckoo search algorithm by L\u00e9vy flights for solving reliability redundancy allocation problems. Eng Optim 45(11):1273\u20131286","journal-title":"Eng Optim"},{"issue":"1","key":"8272_CR77","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/j.trit.2017.01.002","volume":"2","author":"S Padhy","year":"2017","unstructured":"Padhy S, Panda S (2017) A hybrid stochastic fractal search and pattern search technique based cascade PI-PD controller for automatic generation control of multi-source power systems in presence of plug in electric vehicles. CAAI Trans Intell Technol 2(1):12\u201325. https:\/\/doi.org\/10.1016\/j.trit.2017.01.002","journal-title":"CAAI Trans Intell Technol"},{"issue":"Supplement C","key":"8272_CR78","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.ress.2016.03.019","volume":"152","author":"MA Mellal","year":"2016","unstructured":"Mellal MA, Zio E (2016) A penalty guided stochastic fractal search approach for system reliability optimization. Reliab Eng Syst Saf 152(Supplement C):213\u2013227. https:\/\/doi.org\/10.1016\/j.ress.2016.03.019","journal-title":"Reliab Eng Syst Saf"},{"issue":"6","key":"8272_CR79","first-page":"544","volume":"4","author":"M Sohal","year":"2015","unstructured":"Sohal M, Singh A, Virk RS (2015) A framework for optimizing distributed database queries based on stochastic fractal search. Int J Comput Sci Mob Comput 4(6):544\u2013551","journal-title":"Int J Comput Sci Mob Comput"},{"issue":"Supplement C","key":"8272_CR80","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.knosys.2014.07.025","volume":"75","author":"H Salimi","year":"2015","unstructured":"Salimi H (2015) Stochastic fractal search: a powerful metaheuristic algorithm. Knowl Based Syst 75(Supplement C):1\u201318. https:\/\/doi.org\/10.1016\/j.knosys.2014.07.025","journal-title":"Knowl Based Syst"},{"key":"8272_CR81","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1016\/j.advengsoft.2017.07.002","volume":"114","author":"S Mirjalili","year":"2017","unstructured":"Mirjalili S, Gandomi AH, Mirjalili SZ, Saremi S, Faris H, Mirjalili SM (2017) Salp swarm algorithm: a bio-inspired optimizer for engineering design problems. Adv Eng Softw 114:163\u2013191","journal-title":"Adv Eng Softw"},{"key":"8272_CR82","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/j.ijpe.2016.10.022","volume":"183","author":"H-B Kwon","year":"2017","unstructured":"Kwon H-B (2017) Exploring the predictive potential of artificial neural networks in conjunction with DEA in railroad performance modeling. Int J Prod Econ 183:159\u2013170. https:\/\/doi.org\/10.1016\/j.ijpe.2016.10.022","journal-title":"Int J Prod Econ"},{"issue":"4","key":"8272_CR83","doi-asserted-by":"publisher","first-page":"1711","DOI":"10.1007\/s12008-023-01256-1","volume":"17","author":"I Aqel","year":"2023","unstructured":"Aqel I, Mellal MA (2023) Optimal reliability allocation of heterogeneous components in pharmaceutical production plant. Inter J Interactive Design Manufacturing (IJIDeM) 17(4):1711\u20131720","journal-title":"Inter J Interactive Design Manufacturing (IJIDeM)"}],"container-title":["The Journal of Supercomputing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11227-026-08272-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11227-026-08272-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11227-026-08272-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T09:02:40Z","timestamp":1770195760000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11227-026-08272-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,4]]},"references-count":83,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["8272"],"URL":"https:\/\/doi.org\/10.1007\/s11227-026-08272-4","relation":{},"ISSN":["1573-0484"],"issn-type":[{"value":"1573-0484","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,4]]},"assertion":[{"value":"24 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"125"}}