Recent advances in system reliability optimization driven by importance measures

doi:10.1007/s42524-020-0112-6

Front. Eng

2020, Vol. 7

Issue (3) : 335-358 https://doi.org/10.1007/s42524-020-0112-6

REVIEW ARTICLE

Recent advances in system reliability optimization driven by importance measures

Shubin SI¹(

), Jiangbin ZHAO¹, Zhiqiang CAI¹, Hongyan DUI²

¹. School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China; Ministry of Industry and Information Technology Key Laboratory of Industrial Engineering and Intelligent Manufacturing, Northwestern Polytechnical University, Xi’an 710072, China
². School of Management Engineering, Zhengzhou University, Zhengzhou 450001, China

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Abstract

System reliability optimization problems have been widely discussed to maximize system reliability with resource constraints. Birnbaum importance is a well-known method for evaluating the effect of component reliability on system reliability. Many importance measures (IMs) are extended for binary, multistate, and continuous systems from different aspects based on the Birnbaum importance. Recently, these IMs have been applied in allocating limited resources to the component to maximize system performance. Therefore, the significance of Birnbaum importance is illustrated from the perspective of probability principle and gradient geometrical sense. Furthermore, the equations of various extended IMs are provided subsequently. The rules for simple optimization problems are summarized to enhance system reliability by using ranking or heuristic methods based on IMs. The importance-based optimization algorithms for complex or large-scale systems are generalized to obtain remarkable solutions by using IM-based local search or simplification methods. Furthermore, a general framework driven by IM is developed to solve optimization problems. Finally, some challenges in system reliability optimization that need to be solved in the future are presented.

Keywords importance measure system performance reliability optimization optimization rules optimization algorithms

Corresponding Author(s): Shubin SI

Just Accepted Date: 19 April 2020 Online First Date: 27 May 2020 Issue Date: 06 August 2020

Cite this article:

Shubin SI,Jiangbin ZHAO,Zhiqiang CAI, et al. Recent advances in system reliability optimization driven by importance measures[J]. Front. Eng, 2020, 7(3): 335-358.

URL:

https://academic.hep.com.cn/fem/EN/10.1007/s42524-020-0112-6
https://academic.hep.com.cn/fem/EN/Y2020/V7/I3/335

Fig.1 Classifications and extensions of Birnbaum importance for system reliability optimization.

References	Problems	Systems	IMs	Rules
^{Barabady and Kumar (2007)}	ROP	Binary	$I λ A$ or $I μ A$	Ranking
^{Zio et al. (2007)}	ROP	Multistate	$I α PAW$	Ranking
^{Gupta et al. (2013)}	ROP	Binary	$I CEIM$	Ranking
^{Wu et al. (2016)}	ROP	Binary	$I j \| i M$	Ranking
^{Roychowdhury and Bhattacharya (2019)}	ROP	Multistate	$I j MCP$	Ranking
^{Boland et al. (1988)}	RAP	Binary	$I PR$	Ranking
^{Shen and Xie (1990)}	RAP	Binary	$I PR$	Ranking
da Costa Bueno (2005)	RAP	Binary	$I T BM$	Ranking
Ramirez-Marquez and Coit (2007)	RAP	Multistate	$I MAD$	Ranking
^{Bhattacharya and Roychowdhury (2014)}	RAP	Binary	$I S BM$ or $I BM$	Ranking
^{Bhattacharya and Roychowdhury (2016)}	RAP	Binary	$I Bay$	Ranking
^{Zuo and Kuo (1990)}	CAP	Binary	$I CAP BM$	Heuristic
^{Lin and Kuo (2002)}	CAP	Binary	$I CAP BM$	Heuristic
^{Yao et al. (2011)}	CAP	Binary	$I CAP BM$	Heuristic
^{Zhu et al. (2017)}	CAP	Binary	$I CAP BM$	Heuristic
^{Qiu et al. (2018)}	CAP	Binary	$I [?] BM$	Heuristic

Tab.1 Reference analysis of optimization rules based on IMs

Fig.2 Procedures of ranking-based optimization rules for ROP.

Fig.3 Procedures of ranking-based optimization rules for RAP.

Steps	Description
I	Evaluate the $I MAD$ of each component by simulation with the max-flow min-cut algorithm
II	(a) Determine the number of redundant components based on the cost per unit increase in the value of $I MAD$ (b) Update each binary minimal cut vector
III	Judge the stopping rules, if the rules are not satisfied, the process goes to Step I; otherwise, stop this heuristic

Tab.2 Process of the MAD-based heuristic

Steps	Description
I	Generate an initial arrangement randomly, $π = (π 1, π 2, ..., π i, ..., π n)$
II	Calculate $I CAP BM (i)$ for all positions from position 1 to position n by Eq. (7)
III	For k = 1 to n- 1, do the loop (a) Find positions m and r such that $π m = k$ and $π r = k + 1$ (b) If $I CAP BM (m) > I CAP BM (r)$ and $R (P, π) > R (P, π (m, r))$ , exchange the assignments of components $π m$ and $π r$
IV	If there is no exchange in Step III, output the final assignment; otherwise, go to Step II

Tab.3 Process of the ZKA heuristic

Steps	Description
I	Assign component 1 to all positions that are set $P i 1 = P 11$ and $π i = 1$ for $i = 1, 2, ..., n$
II	Let $S = {1, 2..., n}$ , which is the set of available positions that could receive other components
III	For k = n to 2, do the loop (a) Calculate $I CAP BM (i)$ for all $i ∈ S$ by Eq. (7) (b) Find the position $m ∈ S$ , which meets that $I CAP BM (m) = m a x i ∈ S I CAP BM (i)$ (c) Let $S = S / {m}$ , assign component k to position m
IV	If there are no components in S, output the final assignment; otherwise, go to Step II

Tab.4 Process of LKA heuristic

Heuristics	Step III	Step III(a)	Step III(b)
ZKA	1 to n - 1	$π r = k + 1$	$I CAP BM (m) > I CAP BM (r)$
ZKB	1 to n - 1	$I C A P B M (r) = min ? i : P i 1 > P m 1 I C A P B M (i)$	$I CAP BM (m) > I CAP BM (r)$
ZKC	n to 2	$π r = k − 1$	$I CAP BM (m) < I CAP BM (r)$
ZKD	n to 2	$I CAP BM (r) = m a x i : P i 1 < P m 1 I CAP BM (i)$	$I CAP BM (m) < I CAP BM (r)$

Tab.5 Differences between ZK-type heuristics

Heuristics	Step I	Step III	Step III(b)	Step III(d)
LKA	Component 1 to all positions	n to 2	$I C A P B M (m) = max ? i e s I C A P B M (i)$	Component k to position m
LKB	Component n to all positions	1 to n - 1	$I C A P B M (m) = min ? i e s I C A P B M (i)$	Component k to position m
LKC	Component 1 to all positions	2 to n	$I C A P B M (m) = min ? i e s I C A P B M (i)$	Component k to positions in S
LKD	Component n to all positions	1 to n - 1	$I C A P B M (m) = max ? i e s I C A P B M (i)$	Component k to positions in S

Tab.6 Differences between LK-type heuristics

Tab.7 Process of the BIT heuristic

References	Problems	Systems	IMs	Algorithms
^{Wang et al. (2018)}	ROP	Binary	$I BM$ and $I Δ$	Local search
^{Si et al. (2019)}	ROP	Binary	$I GB$	Local search
^{Zio and Podofillini (2007)}	ROP	Any states	$I B$	Simplification
^{Cai et al. (2018)}	ROP	Continuous	$I U$	Local search
^{Xiong et al. (2017)}	RAP	Binary	$I PRW$	Simplification
^{Shojaei and Mahani (2019)}	RAP	Binary	$I BM$	Simplification
^{Zhao et al. (2019c)}	RAP	Binary	$I BM$	Local search
^{Yao et al. (2014)}	CAP	Binary	$I BM$	Local search
^{Cai et al. (2016)}	CAP	Binary	$I BM$	Local search
^{Zhang et al. (2019)}	CAP	Binary	$I BM$	Local search
^{Dui et al. (2018)}	CAP	Binary	$I opt$	Simplification
^{Zhao et al. (2019b)}	CAP	Binary	$I BM$	Simplification
^{Nguyen et al. (2017)}	CP	Binary	$I S BM$	Simplification
^{Du et al. (2019)}	CP	Binary	$I c-IM$ and $I p-IM$	Simplification
^{Xing and Dugan (2002)}	CP	Multistate	$I BS$	Simplification
^{Li et al. (2015)}	CP	Binary	$I k PM$	Local search
^{Wu and Wu (2017)}	CP	Binary	$I C PR$	Local search
^{Wu et al. (2018)}	CP	Binary	$I GC$	Simplification

Tab.8 Reference analysis of optimization algorithms based on IMs

Procedures	Description
I	Select the component modules with the highest $I BM (i)$
II	Perform the reliability adjustment strategy 1 or 2 randomly (1) Strategy 1: Increasing component reliability after decreasing the component redundancy (2) Strategy 2: Decreasing component reliability after increasing the component redundancy
III	Identify the solution after the adjustment

Tab.9 Procedures of the IM-based local search method

Tab.10 Reference analysis of the optimization algorithms by IM-based local search methods

Fig.4 General procedures of optimization algorithms by IM-based local search method.

Tab.11 Reference analysis of optimization algorithms by IM-based simplification methods

Fig.5 General procedures of optimization algorithms by IM-based simplification methods.

Fig.6 General optimization framework driven by IMs.

1	M Abouei Ardakan, A Zeinal Hamadani (2014). Reliability-redundancy allocation problem with cold-standby redundancy strategy. Simulation Modelling Practice and Theory, 42: 107–118 https://doi.org/10.1016/j.simpat.2013.12.013
2	M Abouei Ardakan, M T Rezvan (2018). Multi-objective optimization of reliability-redundancy allocation problem with cold-standby strategy using NSGA-II. Reliability Engineering & System Safety, 172: 225–238 https://doi.org/10.1016/j.ress.2017.12.019
3	H Aliee, S Vitzethum, M Glaß, J Teich, E Borgonovo (2016). Guiding Genetic Algorithms using importance measures for reliable design of embedded systems. In: Proceedings of International Symposium on Defect and Fault Tolerance in VLSI and Nanotechnology Systems. Storrs, CT: IEEE, 53–56
4	S Arora, S Singh, K Yetilmezsoy (2018). A modified butterfly optimization algorithm for mechanical design optimization problems. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(1): 21 https://doi.org/10.1007/s40430-017-0927-1
5	T Aven, U Jensen (2000). Stochastic Models in Reliability. 2nd ed. New York: Springer
6	J Barabady, U Kumar (2007). Availability allocation through importance measures. International Journal of Quality & Reliability Management, 24(6): 643–657 https://doi.org/10.1108/02656710710757826
7	R E Barlow, F Proschan (1975). Importance of system components and fault tree events. Stochastic Processes and Their Applications, 3(2): 153–173 https://doi.org/10.1016/0304-4149(75)90013-7
8	H Baroud, K Barker (2018). A Bayesian kernel approach to modeling resilience-based network component importance. Reliability Engineering & System Safety, 170: 10–19 https://doi.org/10.1016/j.ress.2017.09.022
9	L A Baxter (1984). Continuum structures I. Journal of Applied Probability, 21(4): 802–815 https://doi.org/10.2307/3213697
10	L A Baxter (1986). Continuum structures II. Mathematical Proceedings of the Cambridge Philosophical Society, 99(2): 331–338 https://doi.org/10.1017/S0305004100064240
11	D Bhattacharya, S Roychowdhury (2014). Redundancy allocation using component-importance measures for maximizing system reliability. American Journal of Mathematical and Management Sciences, 33(1): 36–54 https://doi.org/10.1080/01966324.2013.877363
12	D Bhattacharya, S Roychowdhury (2016). Bayesian importance measure-based approach for optimal redundancy assignment. American Journal of Mathematical and Management Sciences, 35(4): 335–344 https://doi.org/10.1080/01966324.2016.1203850
13	Z W Birnbaum (1969). On the importance of different components in a multicomponent system. Multivariate Analysis, II: 581–592
14	P J Boland, E El-Neweihi, F Proschan (1988). Active redundancy allocation in coherent systems. Probability in the Engineering and Informational Sciences, 2(3): 343–353 https://doi.org/10.1017/S0269964800000899
15	E Borgonovo, H Aliee, M Glaß, J Teich (2016). A new time-independent reliability importance measure. European Journal of Operational Research, 254(2): 427–442 https://doi.org/10.1016/j.ejor.2016.03.054
16	A S Bretas, R J Cabral, R C Leborgne, G D Ferreira, J A Morales (2018). Multi-objective MILP model for distribution systems reliability optimization: A lightning protection system design approach. International Journal of Electrical Power & Energy Systems, 98: 256–268 https://doi.org/10.1016/j.ijepes.2017.12.006
17	Z Q Cai, S B Si, Y Liu, J B Zhao (2018). Maintenance optimization of continuous state systems based on performance improvement. IEEE Transactions on Reliability, 67(2): 651–665 https://doi.org/10.1109/TR.2017.2743225
18	Z Q Cai, S B Si, S D Sun, C T Li (2016). Optimization of linear consecutive-k-out-of-n system with a Birnbaum importance-based genetic algorithm. Reliability Engineering & System Safety, 152: 248–258 https://doi.org/10.1016/j.ress.2016.03.016
19	M Čepin (2019). Evaluation of the power system reliability if a nuclear power plant is replaced with wind power plants. Reliability Engineering & System Safety, 185: 455–464 https://doi.org/10.1016/j.ress.2019.01.010
20	A Chakri, X S Yang, R Khelif, M Benouaret (2018). Reliability-based design optimization using the directional bat algorithm. Neural Computing & Applications, 30(8): 2381–2402 https://doi.org/10.1007/s00521-016-2797-3
21	M S Chern (1992). On the computational complexity of reliability redundancy allocation in a series system. Operations Research Letters, 11(5): 309–315 https://doi.org/10.1016/0167-6377(92)90008-Q
22	D W Coit, A E Smith (1996). Reliability optimization of series-parallel systems using a genetic algorithm. IEEE Transactions on Reliability, 45(2): 254–260 https://doi.org/10.1109/24.510811
23	D W Coit, E Zio (2019). The evolution of system reliability optimization. Reliability Engineering & System Safety, 192: 106259 https://doi.org/10.1016/j.ress.2018.09.008
24	M Compare, L Bellani, E Zio (2019). Optimal allocation of prognostics and health management capabilities to improve the reliability of a power transmission network. Reliability Engineering & System Safety, 184: 164–180 https://doi.org/10.1016/j.ress.2018.04.025
25	M Compare, M Bellora, E Zio (2017). Aggregation of importance measures for decision making in reliability engineering. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 231(3): 242–254
26	V da Costa Bueno (2005). Minimal standby redundancy allocation in a k-out-of-n: F system of dependent components. European Journal of Operational Research, 165(3): 786–793 https://doi.org/10.1016/j.ejor.2003.01.004
27	C Derman, G J Lieberman, S M Ross (1974). Optimal allocations in the construction of k-out-of-n reliability systems. Management Science, 21(3): 241–250 https://doi.org/10.1287/mnsc.21.3.241
28	Y J Du, S B Si, T D Jin (2019). Reliability importance measures for network based on failure counting process. IEEE Transactions on Reliability, 68(1): 267–279 https://doi.org/10.1109/TR.2018.2864563
29	H Y Dui, L W Chen, S M Wu (2017a). Generalized integrated importance measure for system performance evaluation: Application to a propeller plane system. Maintenance and Reliability, 19(2): 279–286 https://doi.org/10.17531/ein.2017.2.16
30	H Y Dui, S M Li, L D Xing, H L Liu (2019). System performance-based joint importance analysis guided maintenance for repairable systems. Reliability Engineering & System Safety, 186: 162–175 https://doi.org/10.1016/j.ress.2019.02.021
31	H Y Dui, S B Si, S D Sun, Z Q Cai (2013). Gradient computations and geometrical meaning of importance measures. Quality Technology & Quantitative Management, 10(3): 305–318 https://doi.org/10.1080/16843703.2013.11673416
32	H Y Dui, S B Si, S M Wu, R C M Yam (2017b). An importance measure for multistate systems with external factors. Reliability Engineering & System Safety, 167: 49–57 https://doi.org/10.1016/j.ress.2017.05.016
33	H Y Dui, S B Si, R C M Yam (2017c). A cost-based integrated importance measure of system components for preventive maintenance. Reliability Engineering & System Safety, 168: 98–104 https://doi.org/10.1016/j.ress.2017.05.025
34	H Y Dui, S B Si, R C M Yam (2018). Importance measures for optimal structure in linear consecutive-k-out-of-n systems. Reliability Engineering & System Safety, 169: 339–350 https://doi.org/10.1016/j.ress.2017.09.015
35	J F Espiritu, D W Coit, U Prakash (2007). Component criticality importance measures for the power industry. Electric Power Systems Research, 77(5–6): 407–420 https://doi.org/10.1016/j.epsr.2006.04.003
36	C Fang, F Marle, M Xie (2017). Applying importance measures to risk analysis in engineering project using a risk network model. IEEE Systems Journal, 11(3): 1548–1556 https://doi.org/10.1109/JSYST.2016.2536701
37	Y Fang, N Pedroni, E Zio (2016). Resilience-based component importance measures for critical infrastructure network systems. IEEE Transactions on Reliability, 65(2): 502–512 https://doi.org/10.1109/TR.2016.2521761
38	Y Fu, T Yuan, X Zhu (2019a). Importance-measure based methods for component reassignment problem of degrading components. Reliability Engineering & System Safety, 190: 106501 https://doi.org/10.1016/j.ress.2019.106501
39	Y Fu, T Yuan, X Zhu (2019b). Optimum periodic component reallocation and system replacement maintenance. IEEE Transactions on Reliability, 68(2): 753–763 https://doi.org/10.1109/TR.2018.2874187
40	H Garg, S P Sharma (2013). Multi-objective reliability-redundancy allocation problem using particle swarm optimization. Computers & Industrial Engineering, 64(1): 247–255 https://doi.org/10.1016/j.cie.2012.09.015
41	S Ghambari, A Rahati (2018). An improved artificial bee colony algorithm and its application to reliability optimization problems. Applied Soft Computing, 62: 736–767 https://doi.org/10.1016/j.asoc.2017.10.040
42	K Gopal, K K Aggarwal, J S Gupta (1980). A new method for solving reliability optimization problem. IEEE Transactions on Reliability, R-29(1): 36–37 https://doi.org/10.1109/TR.1980.5220700
43	W S Griffith (1980). Multistate reliability models. Journal of Applied Probability, 17(3): 735–744 https://doi.org/10.2307/3212967
44	S Gupta, J Bhattacharya, J Barabady, U Kumar (2013). Cost-effective importance measure: A new approach for resource prioritization in a production plant. International Journal of Quality & Reliability Management, 30(4): 379–386 https://doi.org/10.1108/02656711311308376
45	X He, Y Yuan (2019). A framework of identifying critical water distribution pipelines from recovery resilience. Water Resources Management, 33(11): 3691–3706 https://doi.org/10.1007/s11269-019-02328-2
46	P Hilber, L Bertling (2004). Monetary importance of component reliability in electrical networks for maintenance optimization. In: Proceedings of International Conference on Probabilistic Methods Applied to Power Systems. Ames, IA: IEEE, 150–155
47	T Jiang, Y Liu, Y X Zheng (2019). Optimal loading strategy for multi-state systems: Cumulative performance perspective. Applied Mathematical Modelling, 74: 199–216 https://doi.org/10.1016/j.apm.2019.04.043
48	C Kim, L A Baxter (1987). Reliability importance for continuum structure functions. Journal of Applied Probability, 24(3): 779–785 https://doi.org/10.2307/3214108
49	C R Knight (1991). Four decades of reliability progress. In: Proceedings of Annual Reliability and Maintainability Symposium. Orlando, FL: IEEE, 156–160
50	S Kulturel-Konak, A E Smith, D W Coit (2003). Efficiently solving the redundancy allocation problem using tabu search. IIE Transactions, 35(6): 515–526 https://doi.org/10.1080/07408170304422
51	W Kuo, H H Lin, Z K Xu, W X Zhang (1987). Reliability optimization with the Lagrange-multiplier and branch-and-bound technique. IEEE Transactions on Reliability, R-36(5): 624–630 https://doi.org/10.1109/TR.1987.5222487
52	W Kuo, V R Prasad (2000). An annotated overview of system-reliability optimization. IEEE Transactions on Reliability, 49(2): 176–187 https://doi.org/10.1109/24.877336
53	W Kuo, X Zhu (2012). Importance Measures in Reliability, Risk and Optimization: Principles and Applications. 1st ed. Chichester: John Wiley & Sons
54	G Levitin, M Finkelstein, Y Dai (2017a). Redundancy optimization for series-parallel phased-mission systems exposed to random shocks. Reliability Engineering & System Safety, 167: 554–560 https://doi.org/10.1016/j.ress.2017.07.006
55	G Levitin, L Xing, S V Amari (2012). Recursive algorithm for reliability evaluation of non-repairable phased-mission systems with binary elements. IEEE Transactions on Reliability, 61(2): 533–542 https://doi.org/10.1109/TR.2012.2192060
56	G Levitin, L Xing, Y Dai (2017b). Optimization of component allocation/distribution and sequencing in warm standby series-parallel systems. IEEE Transactions on Reliability, 66(4): 980–988 https://doi.org/10.1109/TR.2016.2570573
57	G Levitin, L Xing, Y Dai (2017c). Reliability versus expected mission cost and uncompleted work in heterogeneous warm standby multiphase systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47(3): 462–473 https://doi.org/10.1109/TSMC.2015.2505643
58	R Y Li, J F Wang, H T Liao, N Huang (2015). A new method for reliability allocation of avionics connected via an airborne network. Journal of Network and Computer Applications, 48: 14–21 https://doi.org/10.1016/j.jnca.2014.10.005
59	X Y Li, H Z Huang, Y F Li (2018). Reliability analysis of phased-mission system with non-exponential and partially repairable components. Reliability Engineering & System Safety, 175: 119–127 https://doi.org/10.1016/j.ress.2018.03.008
60	Y C Liang, A E Smith (2004). An ant colony optimization algorithm for the redundancy allocation problem (RAP). IEEE Transactions on Reliability, 53(3): 417–423 https://doi.org/10.1109/TR.2004.832816
61	F H Lin, W Kuo (2002). Reliability importance and invariant optimal allocation. Journal of Heuristics, 8(2): 155–171 https://doi.org/10.1023/A:1017908523107
62	M S Lin, D J Chen (1997). The computational complexity of the reliability problem on distributed systems. Information Processing Letters, 64(3): 143–147 https://doi.org/10.1016/S0020-0190(97)00150-6
63	S Lin, X Fang, F Lin, Z Yang, X Wang (2018). Reliability of rail transit traction drive system: A review. Microelectronics and Reliability, 88–90: 1281–1285 https://doi.org/10.1016/j.microrel.2018.07.037
64	S Y Liu, R She, P Y Fan, K B Letaief (2018). Non-parametric message importance measure: Storage code design and transmission planning for big data. IEEE Transactions on Communications, 66(11): 5181–5196 https://doi.org/10.1109/TCOMM.2018.2847666
65	M Marseguerra, E Zio, L Podofillini, D W Coit (2005). Optimal design of reliable network systems in presence of uncertainty. IEEE Transactions on Reliability, 54(2): 243–253 https://doi.org/10.1109/TR.2005.847279
66	A Mettas (2000). Reliability allocation and optimization for complex systems. In: Proceedings of Annual Reliability and Maintainability Symposium. International Symposium on Product Quality and Integrity. Los Angeles, CA: IEEE, 216–221
67	J H Mi, Y F Li, W Peng, H Z Huang (2018). Reliability analysis of complex multi-state system with common cause failure based on evidential networks. Reliability Engineering & System Safety, 174: 71–81 https://doi.org/10.1016/j.ress.2018.02.021
68	F Mohamad, J Teh (2018). Impacts of energy storage system on power system reliability: A systematic review. Energies, 11(7): 1749 https://doi.org/10.3390/en11071749
69	K A Nguyen, P Do, A Grall (2017). Joint predictive maintenance and inventory strategy for multi-component systems using Birnbaum’s structural importance. Reliability Engineering & System Safety, 168: 249–261 https://doi.org/10.1016/j.ress.2017.05.034
70	J Onishi, S Kimura, R J W James, Y Nakagawa (2007). Solving the redundancy allocation problem with a mix of components using the improved surrogate constraint method. IEEE Transactions on Reliability, 56(1): 94–101 https://doi.org/10.1109/TR.2006.884602
71	S Pant, D Anand, A Kishor, S B Singh (2015). A particle swarm algorithm for optimization of complex system reliability. International Journal of Performability Engineering, 11(1): 33–42
72	S Papastavridis (1987). The most important component in a consecutive-k-out-of-n: F system. IEEE Transactions on Reliability, R-36(2): 266–268 https://doi.org/10.1109/TR.1987.5222364
73	R Peng, Q Q Zhai, L D Xing, J Yang (2016). Reliability analysis and optimal structure of series-parallel phased-mission systems subject to fault-level coverage. IIE Transactions, 48(8): 736–746 https://doi.org/10.1080/0740817X.2016.1146424
74	V R Prasad, W Kuo (2000). Reliability optimization of coherent systems. IEEE Transactions on Reliability, 49(3): 323–330 https://doi.org/10.1109/24.914551
75	S Q Qiu, M Sallak, W Schön, H X G Ming (2018). Extended LK heuristics for the optimization of linear consecutive-k-out-of-n: F systems considering parametric uncertainty and model uncertainty. Reliability Engineering & System Safety, 175: 51–61 https://doi.org/10.1016/j.ress.2018.01.016
76	J E Ramirez-Marquez, D W Coit (2004). A heuristic for solving the redundancy allocation problem for multi-state series-parallel systems. Reliability Engineering & System Safety, 83(3): 341–349 https://doi.org/10.1016/j.ress.2003.10.010
77	J E Ramirez-Marquez, D W Coit (2007). Multi-state component criticality analysis for reliability improvement in multi-state systems. Reliability Engineering & System Safety, 92(12): 1608–1619 https://doi.org/10.1016/j.ress.2006.09.014
78	J E Ramirez-Marquez, C M Rocco, B A Gebre, D W Coit, M Tortorella (2006). New insights on multi-state component criticality and importance. Reliability Engineering & System Safety, 91(8): 894–904 https://doi.org/10.1016/j.ress.2005.08.009
79	M Rausand, A Høyland (2003). System Reliability Theory: Models, Statistical Methods and Applications. 2nd ed. Hoboken: John Wiley & Sons
80	S Roychowdhury, D Bhattacharya (2019). Performance improvement of a multi-state coherent system using component importance measure. American Journal of Mathematical and Management Sciences, 38(3): 312–324 https://doi.org/10.1080/01966324.2018.1551733
81	J Shen, L Cui (2015). Reliability and Birnbaum importance for sparsely connected circular consecutive-k systems. IEEE Transactions on Reliability, 64(4): 1140–1157 https://doi.org/10.1109/TR.2015.2413374
82	J Shen, L Cui, S Du (2015). Birnbaum importance for linear consecutive-k-out-of-n systems with sparse d. IEEE Transactions on Reliability, 64(1): 359–375 https://doi.org/10.1109/TR.2014.2337074
83	K Shen, M Xie (1990). On the increase of system reliability by parallel redundancy. IEEE Transactions on Reliability, 39(5): 607–611 https://doi.org/10.1109/24.61320
84	M Shojaei, A Mahani (2019). Efficient reliability-redundancy allocation with uniform importance measure in presence of correlated failure. International Journal of Computers and Applications, 41(5): 378–391 https://doi.org/10.1080/1206212X.2018.1442135
85	S B Si, H Y Dui, Z Q Cai, S D Sun (2012a). The integrated importance measure of multi-state coherent systems for maintenance processes. IEEE Transactions on Reliability, 61(2): 266–273 https://doi.org/10.1109/TR.2012.2192017
86	S B Si, H Y Dui, Z Q Cai, S D Sun, Y F Zhang (2012b). Joint integrated importance measure for multi-state transition systems. Communications in Statistics—Theory and Methods, 41(21): 3846–3862 https://doi.org/10.1080/03610926.2012.688158
87	S B Si, H Y Dui, X B Zhao, S G Zhang, S D Sun (2012c). Integrated importance measure of component states based on loss of system performance. IEEE Transactions on Reliability, 61(1): 192–202 https://doi.org/10.1109/TR.2011.2182394
88	S B Si, G Levitin, H Y Dui, S D Sun (2013). Component state-based integrated importance measure for multi-state systems. Reliability Engineering & System Safety, 116: 75–83 https://doi.org/10.1016/j.ress.2013.02.023
89	S B Si, G Levitin, H Y Dui, S D Sun (2014). Importance analysis for reconfigurable systems. Reliability Engineering & System Safety, 126: 72–80 https://doi.org/10.1016/j.ress.2014.01.012
90	S B Si, M L Liu, Z Y Jiang, T D Jin, Z Q Cai (2019). System reliability allocation and optimization based on generalized Birnbaum importance measure. IEEE Transactions on Reliability, 68(3): 831–843 https://doi.org/10.1109/TR.2019.2897026
91	N D Singpurwalla (2006). Reliability and Risk: A Bayesian Perspective. Hoboken: John Wiley & Sons
92	H Su, J J Zhang, E Zio, N Yang, X Y Li, Z J Zhang (2018). An integrated systemic method for supply reliability assessment of natural gas pipeline networks. Applied Energy, 209: 489–501 https://doi.org/10.1016/j.apenergy.2017.10.108
93	Z G Tian, M J Zuo, H Z Huang (2008). Reliability-redundancy allocation for multi-state series-parallel systems. IEEE Transactions on Reliability, 57(2): 303–310 https://doi.org/10.1109/TR.2008.920871
94	F A Tillman, C L Hwang, W Kuo (1977). Optimization techniques for system reliability with redundancy: A review. IEEE Transactions on Reliability, R-26(3): 148–155 https://doi.org/10.1109/TR.1977.5220100
95	H C Vu, P Do, A Barros (2016). A stationary grouping maintenance strategy using mean residual life and the Birnbaum importance measure for complex structures. IEEE Transactions on Reliability, 65(1): 217–234 https://doi.org/10.1109/TR.2015.2455498
96	L H Wang, D H Ma, X Han, W Wang (2019). Optimization for postearthquake resilient power system capacity restoration based on the degree of discreteness method. Mathematical Problems in Engineering, 5489067 https://doi.org/10.1155/2019/5489067
97	N Wang, J B Zhao, Z Y Jiang, S Zhang (2018). Reliability optimization of systems with component improvement cost based on importance measure. Advances in Mechanical Engineering, 10(11): 1–15 https://doi.org/10.1177/1687814018809781
98	S M Wu, L Y Chan (2003). Performance utility-analysis of multi-state systems. IEEE Transactions on Reliability, 52(1): 14–21 https://doi.org/10.1109/TR.2002.805783
99	S M Wu, Y Chen, Q T Wu, Z L Wang (2016). Linking component importance to optimization of preventive maintenance policy. Reliability Engineering & System Safety, 146: 26–32 https://doi.org/10.1016/j.ress.2015.10.008
100	S M Wu, F P A Coolen (2013). A cost-based importance measure for system components: An extension of the Birnbaum importance. European Journal of Operational Research, 225(1): 189–195 https://doi.org/10.1016/j.ejor.2012.09.034
101	X Y Wu, X Y Wu (2017). An importance based algorithm for reliability-redundancy allocation of phased-mission systems. In: Proceedings of International Conference on Software Quality, Reliability and Security Companion. Prague: IEEE, 152–159
102	X Y Wu, X Y Wu, N Balakrishnan (2018). Reliability allocation model and algorithm for phased-mission systems with uncertain component parameters based on importance measure. Reliability Engineering & System Safety, 180: 266–276 https://doi.org/10.1016/j.ress.2018.07.022
103	T F Xiahou, Y Liu, T Jiang (2018). Extended composite importance measures for multi-state systems with epistemic uncertainty of state assignment. Mechanical Systems and Signal Processing, 109: 305–329 https://doi.org/10.1016/j.ymssp.2018.02.021
104	S H Xiang, J Yang (2018). Performance reliability evaluation for mobile ad hoc networks. Reliability Engineering & System Safety, 169: 32–39 https://doi.org/10.1016/j.ress.2017.08.001
105	R S Xiao, Y M Xiang, L F Wang, K G Xie (2018). Power system reliability evaluation incorporating dynamic thermal rating and network topology optimization. IEEE Transactions on Power Systems, 33(6): 6000–6012 https://doi.org/10.1109/TPWRS.2018.2829079
106	M Xie, K Shen (1989). On ranking of system components with respect to different improvement actions. Microelectronics and Reliability, 29(2): 159–164 https://doi.org/10.1016/0026-2714(89)90564-7
107	L D Xing, J B Dugan (2002). Analysis of generalized phased-mission system reliability, performance, and sensitivity. IEEE Transactions on Reliability, 51(2): 199–211 https://doi.org/10.1109/TR.2002.1011526
108	G Z Xiong, C Zhang, F Zhou (2017). A robust reliability redundancy allocation problem under abnormal external failures guided by a new importance measure. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 231(2): 180–199
109	Z P Xu, J E Ramirez-Marquez, Y Liu, T F Xiahou (2020). A new resilience-based component importance measure for multi-state networks. Reliability Engineering & System Safety, 193: 106591 https://doi.org/10.1016/j.ress.2019.106591
110	Q Yao, X Zhu, W Kuo (2011). Heuristics for component assignment problems based on the Birnbaum importance. IIE Transactions, 43(9): 633–646 https://doi.org/10.1080/0740817X.2010.532856
111	Q Yao, X Zhu, W Kuo (2014). A Birnbaum-importance based genetic local search algorithm for component assignment problems. Annals of Operations Research, 212(1): 185–200 https://doi.org/10.1007/s10479-012-1223-1
112	W C Yeh (2019). A novel boundary swarm optimization method for reliability redundancy allocation problems. Reliability Engineering & System Safety, 192: 106060
113	W C Yeh, T C Chu (2018). A novel multi-distribution multi-state flow network and its reliability optimization problem. Reliability Engineering & System Safety, 176: 209–217 https://doi.org/10.1016/j.ress.2018.04.006
114	W C Yeh, T J Hsieh (2011). Solving reliability redundancy allocation problems using an artificial bee colony algorithm. Computers & Operations Research, 38(11): 1465–1473 https://doi.org/10.1016/j.cor.2010.10.028
115	H Yu, J Yang, J Lin, Y Zhao (2017). Reliability evaluation of non-repairable phased-mission common bus systems with common cause failures. Computers & Industrial Engineering, 111: 445–457 https://doi.org/10.1016/j.cie.2017.08.002
116	J W Yu, S L Zheng, H Pham, T Chen (2018). Reliability modeling of multi-state degraded repairable systems and its applications to automotive systems. Quality and Reliability Engineering International, 34(3): 459–474 https://doi.org/10.1002/qre.2265
117	A Zaretalab, V Hajipour, M Tavana (2020). Redundancy allocation problem with multi-state component systems and reliable supplier selection. Reliability Engineering & System Safety, 193: 106629 https://doi.org/10.1016/j.ress.2019.106629
118	S Zhang, J B Zhao, H G Li, N Wang (2017). Reliability optimization and importance analysis of circular-consecutive k-out-of-n system. Mathematical Problems in Engineering, 1831537 https://doi.org/10.1155/2017/1831537
119	S Zhang, J B Zhao, W J Zhu, L Du (2019). Reliability optimization of linear consecutive k-out-of-n: F systems with Birnbaum importance-based quantum genetic algorithm. Advances in Mechanical Engineering, 11(4): 1–16 https://doi.org/10.1177/1687814019842996
120	J B Zhao, Z Q Cai, W T Si, S Zhang (2019a). Mission success evaluation of repairable phased-mission systems with spare parts. Computers & Industrial Engineering, 132: 248–259 https://doi.org/10.1016/j.cie.2019.04.038
121	J B Zhao, S B Si, Z Q Cai (2019b). A multi-objective reliability optimization for reconfigurable systems considering components degradation. Reliability Engineering & System Safety, 183: 104–115 https://doi.org/10.1016/j.ress.2018.11.001
122	J B Zhao, S B Si, Z Q Cai, M Su, W Wang (2019c). Multiobjective optimization of reliability-redundancy allocation problems for serial parallel-series systems based on importance measure. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 233(5): 881–897
123	X B Zhao, S B Si, H Y Dui, Z Q Cai, S D Sun (2013). Integrated importance measure for multi-state coherent systems of k level. Journal of Systems Engineering and Electronics, 24(6): 1029–1037 https://doi.org/10.1109/JSEE.2013.00120
124	X Y Zhu, Q Z Yao, W Kuo (2011). Birnbaum importance in solving component assignment problems. In: Proceedings of Annual Reliability and Maintainability Symposium. Lake Buena Vista, FL: IEEE,1–6
125	X Y Zhu, Y Q Fu, T Yuan, X Y Wu (2017). Birnbaum importance based heuristics for multi-type component assignment problems. Reliability Engineering & System Safety, 165: 209–221 https://doi.org/10.1016/j.ress.2017.04.018
126	E Zio, M Marella, L Podofillini (2007). Importance measures-based prioritization for improving the performance of multi-state systems: Application to the railway industry. Reliability Engineering & System Safety, 92(10): 1303–1314 https://doi.org/10.1016/j.ress.2006.07.010
127	E Zio, L Podofillini (2003a). Importance measures of multi-state components in multi-state systems. International Journal of Reliability Quality and Safety Engineering, 10(3): 289–310 https://doi.org/10.1142/S0218539303001159
128	E Zio, L Podofillini (2003b). Monte Carlo simulation analysis of the effects of different system performance levels on the importance of multi-state components. Reliability Engineering & System Safety, 82(1): 63–73 https://doi.org/10.1016/S0951-8320(03)00124-8
129	E Zio, L Podofillini (2007). Importance measures and genetic algorithms for designing a risk-informed optimally balanced system. Reliability Engineering & System Safety, 92(10): 1435–1447 https://doi.org/10.1016/j.ress.2006.09.011
130	M Zuo, W Kuo (1990). Design and performance analysis of consecutive-k-out-of-n structure. Naval Research Logistics, 37(2): 203–230 https://doi.org/10.1002/1520-6750(199004)37:2<203::AID-NAV3220370203>3.0.CO;2-X

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