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doi: 10.3934/jimo.2021189
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A new dynamic model to optimize the reliability of the series-parallel systems under warm standby components

1. 

Associate Professor, Department of Management, Faculty of Economics and Administrative sciences, Ferdowsi University of Mashhad, Iran

2. 

PhD Candidate in Operations Research, Department of Management, Faculty of Economics and Administrative sciences, Ferdowsi University of Mashhad, Iran

* Corresponding author: Amir Mohammad Fakoor Saghih

Received  January 2021 Revised  August 2021 Early access November 2021

Redundancy allocation problem (RAP) is a common technique for increasing the reliability of systems. In this paper, a new model for the RAP is introduced that takes into account the warm standby and mixed strategy, the model dynamics, and the type of the strategy in redundancy allocation problems. A recursive formula is first obtained for the reliability function in the dynamic warm standby and mixed redundancy strategies that leverages the success mode analysis and works for any arbitrary failure-time distribution. Failure rates for warm standby units change before and after their replacement with a damaged unit, and, therefore, the reliability function in warm standby varies with time (i.e., the model is dynamic). Although dynamic models are commonplace in practice, they are more challenging to assess than static models, which have been mainly considered in the literature. An optimization problem is then formulated to select the best redundancy strategy and redundancy levels. Genetic algorithm and particle swarm optimization are leveraged to solve the problem. Finally, the efficiency of the presented method is verified through a numerical example. The experimental results verify that the proposed model for RAP significantly improves the system reliability, which can be of vital importance for system designers.

Citation: Amir Mohammad Fakoor Saghih, Azam Modares. A new dynamic model to optimize the reliability of the series-parallel systems under warm standby components. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021189
References:
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M. A. Ardakan and A. Z. Hamadani, Reliability optimization of series-parallel systems with mixed redundancy strategy in subsystems, Reliability Engineering & System Safety, 130 (2014), 2-9.  doi: 10.1016/j.ress.2014.06.001.

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E. KilicS. S. AliG. W. Weber and R. Dubey, A value-adding approach to reliability under preventive maintenance costs and its applications,, Optimization: A Journal of Mathematical Programming and Operations Research, 63 (2014), 1805-1816.  doi: 10.1080/02331934.2014.917301.

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H. Kim and P. Kim, Reliability–redundancy allocation problem considering optimal redundancy strategy using parallel genetic algorithm, Reliability Engineering & System Safety, 159 (2017), 153-160.  doi: 10.1016/j.ress.2016.10.033.

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Z. LiH. Liao and D. W. Coit, A two stage approach for multi-objective decision making with applications to system reliability optimization, Reliability Engineering and System Safety, 94 (2009), 1585-1592.  doi: 10.1016/j.ress.2009.02.022.

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M. OuzinebM. Nourelfath and M. Gendreau, Tabu search for the redundancy allocation problem of homogenous series-parallel multi-state systems, Reliability Engineering and System Safety, 93 (2008), 1257-1272.  doi: 10.1016/j.ress.2007.06.004.

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M. SharifiM. ShahriariA. Khajehpoor and S. A. Mirtaheri, Reliability optimization of a $k$-out-of-$n$ series-parallel system with warm standby component, Scientia Iranica, 13 (2021), 171-188. 

[43]

R. Soltani, Reliability optimization of binary state non-repairable systems: A state of the art survey, International Journal of Industrial Engineering Computations, 5 (2014), 339-364.  doi: 10.5267/j.ijiec.2014.5.001.

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show all references

References:
[1]

S. V. Amari and G. Dill, Redundancy optimization problem with warm-standby redundancy, Proc. Annual. Reliability and Maintainability Symp, 39 (2010), 1-6.  doi: 10.1109/RAMS.2010.5448068.

[2]

M. A. Ardakan and A. Z. Hamadani, Reliability optimization of series-parallel systems with mixed redundancy strategy in subsystems, Reliability Engineering & System Safety, 130 (2014), 2-9.  doi: 10.1016/j.ress.2014.06.001.

[3]

M. A. ArdakanA. Z. Hamadani and A. Naghian, Optimizing bi-objective redundancy allocation problem with a mixed redundancy strategy, ISA Transactions, 55 (2014), 116-128.  doi: 10.1016/j.isatra.2014.10.002.

[4]

M. A. Ardakan and M. T. Rezvan, Multi-objective optimization of reliability-redundancy allocation problem with cold-standby strategy using NSGA-II, Reliability Engineering & System Safety, 172 (2018), 225-238.  doi: 10.1016/j.ress.2017.12.019.

[5]

M. A. ArdakanM. SimaA. Zeinal Hamadani and D. W. Coit, A novel strategy for redundant components in reliability-redundancy allocation problems, IIE Transactions, 48 (2016), 1043-1057.  doi: 10.1080/0740817X.2016.1189631.

[6]

M. BaghelS. Agrawal and S. Silakari, Survey of meta-heuristic algorithms for combinatorial optimization, International Journal of Computer Applications, 58 (2012), 21-31.  doi: 10.5120/9391-3813.

[7]

N. BejiB. JarbouiM. Eddaly and H. Chanbchoub, A hybrid particle swarm optimization algorithm for the redundancy allocation problem, Journal of Computational Science, 1 (2017), 159-166.  doi: 10.1016/j.jocs.2010.06.001.

[8]

N. BejiaB. JarbouibP. Siarryc and H. Chabchoubb, A differential evolutional algorithm to solve redundancy allocation problems, International Transactions in Operational Research, 19 (2017), 809-824.  doi: 10.1111/j.1475-3995.2012.00856.x.

[9]

A. Birolini, On the Use of Stochastic Processes in Modeling Reliability Problems, Economics and Mathematical Systems, 2017. doi: 10.1007/978-3-642-46553-6.

[10]

P. Boddu and L. Xing, Reliability evaluation and optimization of series-parallel systems with k-out-of-n: G subsystems and mixed redundancy types, Proc. IMechE, Part O, Journal Risk Reliability, 227 (2013), 187-198.  doi: 10.1177/1748006X12473569.

[11]

P. G. BusaccaM. Marseguerra and E. Zio, Multi-objective optimization by genetic algorithms: Application to safety systems, Reliability Engineering and System Safety, 72 (2001), 59-74.  doi: 10.1016/S0951-8320(00)00109-5.

[12]

D. W. Coit, Maximization of system reliability with a choice of redundancy strategies, IIE Transportation, 35 (2003), 535-544.  doi: 10.1080/07408170304420.

[13]

D. W. Coit and J. C. Liu, System reliability optimization with k-out-of-n subsystems, International Journal of Reliability, Quality safety Engineering, 7 (2000), 129-142.  doi: 10.1142/S0218539300000110.

[14]

D. W. Coit and A. E. Smith, Reliability optimization of the series-parallel system using a genetic algorithm, IEEE Transactions on Reliability, 45 (1996), 254-260.  doi: 10.1109/24.510811.

[15]

V. EbrahimipourS. M. Asadzadeh and A. Azadeh, An emotional learning-based fuzzy inference system for improvement of system reliability evaluation in redundancy allocation problem, International Journal Advanced Manufacturing Technology, 66 (2013), 1657-1672.  doi: 10.1007/s00170-012-4448-x.

[16]

D. E. FyffeW. W. Hines and N. K. Lee, System reliability allocation and a computation algorithm, IEEE Transportation Reliability, 33 (1968), 64-69.  doi: 10.1109/TR.1968.5217517.

[17]

H. GargM. RaniS. P. Sharma and Y. Vishwakarma, Bi-objective optimization of the reliability-redundancy allocation problem for the series-parallel system, Journal of Manufacturing Systems, 33 (2014), 335-347.  doi: 10.1016/j.jmsy.2014.02.008.

[18]

H. Garg and S. P. Sharma, Multi-objective reliability-redundancy allocation problem using particle swarm optimization, Computers & Industrial Engineering, 64 (2013), 247-255.  doi: 10.1016/j.cie.2012.09.015.

[19]

H. Gholinejad and A. Z. Hamadani, A new model for the redundancy allocation problem with component mixing and mixed redundancy strategy, Reliability Engineering and System Safety, 164 (2017), 66-73.  doi: 10.1016/j.ress.2017.03.009.

[20]

M. GongH. Liu and R. Peng, Redundancy allocation of mixed warm and cold standby components in repairable $K$-out-of $N$ systems, Proceedings of the Institution of Mechanical Engineers, Journal of Risk and Reliability, 232 (2020), 38-51. 

[21]

L. GregoryL. Xing and Y. Dai, Optimization of component allocation/distribution and sequencing in warm standby series-parallel systems, IEEE Transactions on Reliability, 66 (2017), 980-988.  doi: 10.1109/TR.2016.2570573.

[22]

C. Ha and W. Kuo, Reliability redundancy allocation: An improved realization for non-convex nonlinear programming problems, European Journal of Operational Research, 171 (2006), 24-38.  doi: 10.1016/j.ejor.2004.06.006.

[23]

H. HadipourM. Amiri and M. Sharifi, Redundancy allocation in series-parallel systems under warm standby and active components in repairable subsystems, Reliability Engineering System Safety, 192 (2019), 106048.  doi: 10.1016/j.ress.2018.01.007.

[24]

Q. HeX. HuH. Ren and H. Zhang, A novel artificial fish swarm algorithm for solving large-scale reliability–redundancy application problem, ISA Transactions, 59 (2015), 105-113.  doi: 10.1016/j.isatra.2015.09.015.

[25]

J. H. Holland, Adaptation in natural and artificial systems, Michigan: An overview of the current state-of-the-art, European Journal Operation Research, 137 (2002), 1-9. 

[26]

Y.-C. Hsieh and P.-S. You, An effective immune-based two-phase approach for the optimal reliability– redundancy allocation problem, Applied Mathematics and Computation, 218 (2011), 1297-1307.  doi: 10.1016/j.amc.2011.06.012.

[27]

C.-L. Huang, A particle-based simplified swarm optimization algorithm for reliability redundancy allocation problems, Reliability Engineering & System Safety, 142 (2015), 221-230.  doi: 10.1016/j.ress.2015.06.002.

[28]

J. Kennedy and R. Eberhart, "Particle swarm optimization, " Proceedings of ICNN'95, International Conference on Neural Networks, 4 (1995), 1942-1948. 

[29]

E. KilicS. S. AliG. W. Weber and R. Dubey, A value-adding approach to reliability under preventive maintenance costs and its applications,, Optimization: A Journal of Mathematical Programming and Operations Research, 63 (2014), 1805-1816.  doi: 10.1080/02331934.2014.917301.

[30]

H. Kim and P. Kim, Reliability–redundancy allocation problem considering optimal redundancy strategy using parallel genetic algorithm, Reliability Engineering & System Safety, 159 (2017), 153-160.  doi: 10.1016/j.ress.2016.10.033.

[31]

S. Kulturel-KonakA. E. Smith and D. W. Coit, Efficiently solving the redundancy allocation problem using tabu search, IIE Transaction, 35 (2003), 515-526.  doi: 10.1080/07408170304422.

[32]

A. Kumar and M. Agarwal, A review of standby redundant systems, IEEE Transportation Reliability, 29 (1980), 290-294.  doi: 10.1109/TR.1980.5220842.

[33]

G. LevitinL. Xing and Y. Dai, Cold vs. hot standby mission operation cost minimization for 1-out-of-N systems,, European Journal Operation. Research, 234 (2014), 155-162.  doi: 10.1016/j.ejor.2013.10.051.

[34]

Z. LiH. Liao and D. W. Coit, A two stage approach for multi-objective decision making with applications to system reliability optimization, Reliability Engineering and System Safety, 94 (2009), 1585-1592.  doi: 10.1016/j.ress.2009.02.022.

[35]

Y.-C. Liang and Y.-C. Chen, Redundancy allocation of series-parallel systems using variable neighborhood search algorithms, Reliability Engineering and System Safety, 92 (2007), 323-331.  doi: 10.1016/j.ress.2006.04.013.

[36]

N. NahasM. Nourelfath and D. Ait-Kadi, Coupling ant colony and the degraded ceiling algorithm for the redundancy allocation problem of series-parallel systems, Reliability Engineering and System Safety, 92 (2007), 211-222.  doi: 10.1016/j.ress.2005.12.002.

[37]

M. OuzinebM. Nourelfath and M. Gendreau, Tabu search for the redundancy allocation problem of homogenous series-parallel multi-state systems, Reliability Engineering and System Safety, 93 (2008), 1257-1272.  doi: 10.1016/j.ress.2007.06.004.

[38]

D. PandeyM. Jacob and J. Yadav, Reliability analysis of a power loom plant with cold-standby for its strategic unit, Microelectron. Reliability, 36 (1996), 115-119.  doi: 10.1016/0026-2714(95)00013-R.

[39]

S. PantA. Kumar and S. B. Singh, A modified particle swarm optimization algorithm for nonlinear optimization, Nonlinear Studies, 24 (2017), 127-138. 

[40]

N. Ramachandran, Taguchi Method as a Tool for Reducing Costs, Management System Engineering, Virginia Polytechnic Institute, and State University, 2000.

[41]

P. RoyB. S. MahapatraG. S. Mahapatra and P. K. Roy, Entropy-based region reducing genetic algorithm for reliability redundancy allocation in interval environment, Expert Systems with Applications, 41 (2014), 6147-6160.  doi: 10.1016/j.eswa.2014.04.016.

[42]

M. SharifiM. ShahriariA. Khajehpoor and S. A. Mirtaheri, Reliability optimization of a $k$-out-of-$n$ series-parallel system with warm standby component, Scientia Iranica, 13 (2021), 171-188. 

[43]

R. Soltani, Reliability optimization of binary state non-repairable systems: A state of the art survey, International Journal of Industrial Engineering Computations, 5 (2014), 339-364.  doi: 10.5267/j.ijiec.2014.5.001.

[44]

S. K. Srinivasan and R. Subramanian, Reliability analysis of a three-unit warm standby redundant system with repair, Annals of Operations Research, 143 (2006), 227-235.  doi: 10.1007/s10479-006-7384-z.

[45]

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Figure 1.  Series-parallel system
Figure 2.  Mean $ S\big/ N $ ratios of the parameters of GA
Figure 3.  Mean $ S\big/ N $ ratios of the parameters of PSO
Figure 4.  Chromosome representation (solution encoding)
Figure 5.  Representation of a solution
Figure 6.  Max-min crossover operator
Figure 7.  Max-min mutation operator
Figure 8.  Convergence diagram of the best implementation of the GA
Figure 9.  Convergence diagram of the best implementation of the PSO
Table 1.  Notations in proposed model
Sets
$ A $ The set of subsystems that use active strategy.
$ S $ The set of subsystems that use standby strategy.
$ N $ The set of subsystems that do not use any redundancy.
$ M $ The set of subsystems that use mixed redundancy.
$ Z $ The set of chosen component.
Decision variables
$ z_{i} $ Index of chosen component for the subsystem $ i $.
$ n_{i} $ The total number of components that are used in subsystem $ i $.
$ n_{S, i} $ The number of warm standby components of the subsystem $ i $.
$ n_{A, i} $ Number of active redundant components in the subsystem $ i $.
Parameters
$ n_{\max, i} $ Upper bound for $ n_i $.
$ m_i $ The number of components to be chosen from for subsystem $ i $.
$ C_{iz_{i} }, w_{iz_{i} } $ Cost and weight for subsystem $ i $ for the $ z_{i}^{th} $ available component.
$ R_{iz_{i} } (t) $ Reliability at time $ t $ for subsystem $ i $ for the $ z_{i}^{th} $ available component.
$ \lambda _{iz_{i} } $ The failure rate for subsystems $ i $ for the$ z_{i}^{th} $ component.
$ \lambda _{iz_{i} }^{-} $ The reduced failure rate for subsystems $ i $ for the $ j^{th} $ component.
$ W $ Upper bound for weight.
$ C $ Upper bound for the cost.
$ t $ Mission time.
$ T_{iz_{i} }^{'nS} $ The lifetime of the $ z_{i}^{th} $ component of $ n^{th} $ standby for the subsystem $ i $ in standby mode.
$ T_{iz_{i} }^{nS} $ The lifetime of the $ z_{i}^{th} $ standby component of nth standby for the subsystem$ i $ in operation mode.
$ T_{iz_{i} }^{{\rm active}} $ The lifetime of the $ z_{i}^{th} $ all active component of for subsystem$ i $.
$ T_{\max, iz_{i} }^{{\rm active}} $ The lifetime of the $ z_{i}^{th} $ all active component of for subsystem $ i $.
$ f_{iz_{i} }^{nS} (t) $ Pdf for the $ n^{th} $ warm standby failure arrival of the $ z_{i}^{th} $ component for the subsystem$ i $.
$ f_{iz_{i} }^{{\rm active}} (t) $ Pdf for the active failure arrival of the $ z_{i}^{th} $ component for the subsystem$ i $.
$ f_{iz_{i} }^{\max, nA_{i} } (t) $ Pdf for the maximum failure times of $ nA_i $ the number of the $ z_{i}^{th} $ component for the subsystem $ i $.
$ R_{iz_{i} }^{nS} $ Reliability for $ n^{th} $ warm standby component of the $ z_{i}^{th} $ component for the subsystem $ i $.
$ R_{iz_{i} }^{'nS} $ Reliability for $ n^{th} $ warm standby component of the $ z_{i}^{th} $ component for the subsystem $ i $.
$ R_{iz_{i} }^{{\rm Switch}} $ Switching reliability of the $ z_{i}^{th} $ component for subsystem $ i $ at a time $ t $.
$ R(t;z, n_{A}, n_{S}) $ Pdf for the maximum failure times of $ nA_i $ number of the $ z_{i}^{th} $ component for the subsystem $ i $.
Sets
$ A $ The set of subsystems that use active strategy.
$ S $ The set of subsystems that use standby strategy.
$ N $ The set of subsystems that do not use any redundancy.
$ M $ The set of subsystems that use mixed redundancy.
$ Z $ The set of chosen component.
Decision variables
$ z_{i} $ Index of chosen component for the subsystem $ i $.
$ n_{i} $ The total number of components that are used in subsystem $ i $.
$ n_{S, i} $ The number of warm standby components of the subsystem $ i $.
$ n_{A, i} $ Number of active redundant components in the subsystem $ i $.
Parameters
$ n_{\max, i} $ Upper bound for $ n_i $.
$ m_i $ The number of components to be chosen from for subsystem $ i $.
$ C_{iz_{i} }, w_{iz_{i} } $ Cost and weight for subsystem $ i $ for the $ z_{i}^{th} $ available component.
$ R_{iz_{i} } (t) $ Reliability at time $ t $ for subsystem $ i $ for the $ z_{i}^{th} $ available component.
$ \lambda _{iz_{i} } $ The failure rate for subsystems $ i $ for the$ z_{i}^{th} $ component.
$ \lambda _{iz_{i} }^{-} $ The reduced failure rate for subsystems $ i $ for the $ j^{th} $ component.
$ W $ Upper bound for weight.
$ C $ Upper bound for the cost.
$ t $ Mission time.
$ T_{iz_{i} }^{'nS} $ The lifetime of the $ z_{i}^{th} $ component of $ n^{th} $ standby for the subsystem $ i $ in standby mode.
$ T_{iz_{i} }^{nS} $ The lifetime of the $ z_{i}^{th} $ standby component of nth standby for the subsystem$ i $ in operation mode.
$ T_{iz_{i} }^{{\rm active}} $ The lifetime of the $ z_{i}^{th} $ all active component of for subsystem$ i $.
$ T_{\max, iz_{i} }^{{\rm active}} $ The lifetime of the $ z_{i}^{th} $ all active component of for subsystem $ i $.
$ f_{iz_{i} }^{nS} (t) $ Pdf for the $ n^{th} $ warm standby failure arrival of the $ z_{i}^{th} $ component for the subsystem$ i $.
$ f_{iz_{i} }^{{\rm active}} (t) $ Pdf for the active failure arrival of the $ z_{i}^{th} $ component for the subsystem$ i $.
$ f_{iz_{i} }^{\max, nA_{i} } (t) $ Pdf for the maximum failure times of $ nA_i $ the number of the $ z_{i}^{th} $ component for the subsystem $ i $.
$ R_{iz_{i} }^{nS} $ Reliability for $ n^{th} $ warm standby component of the $ z_{i}^{th} $ component for the subsystem $ i $.
$ R_{iz_{i} }^{'nS} $ Reliability for $ n^{th} $ warm standby component of the $ z_{i}^{th} $ component for the subsystem $ i $.
$ R_{iz_{i} }^{{\rm Switch}} $ Switching reliability of the $ z_{i}^{th} $ component for subsystem $ i $ at a time $ t $.
$ R(t;z, n_{A}, n_{S}) $ Pdf for the maximum failure times of $ nA_i $ number of the $ z_{i}^{th} $ component for the subsystem $ i $.
Table 2.  Redundancy strategies
Scenarios $ nA_{i} $ $ nS_{i} $ Redundancy strategy
S1 $ =1 $ $ =0 $ No redundancy
S2 $>1 $ $ =0 $ Active redundancy strategy
S3 $ =1 $ $ \ge 1 $ warm-standby redundancy strategy
S4 $>1 $ $ \ge 1 $ Mixed redundancy strategy
Scenarios $ nA_{i} $ $ nS_{i} $ Redundancy strategy
S1 $ =1 $ $ =0 $ No redundancy
S2 $>1 $ $ =0 $ Active redundancy strategy
S3 $ =1 $ $ \ge 1 $ warm-standby redundancy strategy
S4 $>1 $ $ \ge 1 $ Mixed redundancy strategy
Table 3.  Controllable factors and their levels
Parameters Notations Levels Optimal levels
Level 3 Level 2 Level 1
GA Popsize A 150 100 50 100
$ p_{c} $ B 0.8 0.5 0.4 0.8
$ p_{m}^{1} $ C 0.3 0.2 0.1 0.3
$ p_{m}^{2} $ D 0.3 0.2 0.1 0.2
PSO $ C_{1} $ A 2 1.5 1 1.5
$ C_{2} $ B 2 1.5 1 1.5
$ W_{\max } $ C 0.9 0.8 0.7 0.9
$ W_{\min } $ D 0.4 0.3 0.2 0.3
Parameters Notations Levels Optimal levels
Level 3 Level 2 Level 1
GA Popsize A 150 100 50 100
$ p_{c} $ B 0.8 0.5 0.4 0.8
$ p_{m}^{1} $ C 0.3 0.2 0.1 0.3
$ p_{m}^{2} $ D 0.3 0.2 0.1 0.2
PSO $ C_{1} $ A 2 1.5 1 1.5
$ C_{2} $ B 2 1.5 1 1.5
$ W_{\max } $ C 0.9 0.8 0.7 0.9
$ W_{\min } $ D 0.4 0.3 0.2 0.3
Table 4.  Taguchi experimental results on test problem for the GA
Exp NO. $ {\rm Popsize} $ $ p_{c} $ $ p_{m}^{1} $ $ p_{m}^{2} $ $ S\big/ N $
1 1 1 1 1 $ -0.47 $
2 1 2 2 1 $ -0.43 $
3 1 3 3 1 $ -0.32 $
4 2 1 2 2 $ -0.24 $
5 2 2 3 2 $ -0.23 $
6 2 3 1 2 $ -0.44 $
7 3 1 3 3 $ -0.41 $
8 3 2 1 3 $ -0.46 $
9 3 3 2 3 $ -0.36 $
Exp NO. $ {\rm Popsize} $ $ p_{c} $ $ p_{m}^{1} $ $ p_{m}^{2} $ $ S\big/ N $
1 1 1 1 1 $ -0.47 $
2 1 2 2 1 $ -0.43 $
3 1 3 3 1 $ -0.32 $
4 2 1 2 2 $ -0.24 $
5 2 2 3 2 $ -0.23 $
6 2 3 1 2 $ -0.44 $
7 3 1 3 3 $ -0.41 $
8 3 2 1 3 $ -0.46 $
9 3 3 2 3 $ -0.36 $
Table 5.  Taguchi experimental results on test problem for the PSO
Exp NO. $ c_{1} $ $ c_{2} $ $ w_{\max } $ $ w_{\min } $ $ S\big/ N $
1 1 1 1 1 $ -0.47 $
2 1 2 2 1 $ -0.24 $
3 1 3 3 1 $ -0.41 $
4 2 1 2 2 $ -0.34 $
5 2 2 3 2 $ -0.23 $
6 2 3 1 2 $ -0.24 $
7 3 1 3 3 $ -0.27 $
8 3 2 1 3 $ -0.41 $
9 3 3 2 3 $ -0.23 $
Exp NO. $ c_{1} $ $ c_{2} $ $ w_{\max } $ $ w_{\min } $ $ S\big/ N $
1 1 1 1 1 $ -0.47 $
2 1 2 2 1 $ -0.24 $
3 1 3 3 1 $ -0.41 $
4 2 1 2 2 $ -0.34 $
5 2 2 3 2 $ -0.23 $
6 2 3 1 2 $ -0.24 $
7 3 1 3 3 $ -0.27 $
8 3 2 1 3 $ -0.41 $
9 3 3 2 3 $ -0.23 $
Table 6.  Data for the illustrative example
Choice 1 ($ j=1 $) Choice 2 ($ j=2 $)
$ i $ $ \lambda _{ij} $ $ \lambda _{ij}^{-} $ $ c_{ij} $ $ w_{ij} $ $ \lambda _{ij} $ $ \lambda _{ij}^{-} $ $ c_{ij} $ $ w_{ij} $
1 0.00532 0.00420 1 3 0.000726 0.000516 1 4
2 0.00818 0.00630 2 8 0.000619 0.000415 1 1
3 0.0133 0.0025 2 7 0.011 0.001 3 5
4 0.00741 0.00430 3 5 0.0124 0.0013 4 6
5 0.00619 0.00319 2 4 0.00431 0.00240 2 3
6 0.00436 0.00215 3 5 0.00567 0.00350 3 4
7 0.0105 0.0003 4 7 0.00466 0.00432 4 8
8 0.015 0.003 3 4 0.00105 0.0006 5 7
9 0.00268 0.00165 2 8 0.000101 0.00002 3 9
10 0.0141 0.0025 4 6 0.00683 0.00467 4 5
11 0.00394 0.00186 3 5 0.00355 0.00255 4 6
12 0.00236 0.00167 2 4 0.00769 0.00543 3 5
13 0.00215 0.00135 2 5 0.00436 0.00258 3 5
14 0.011 0.001 4 6 0.00834 0.00542 4 7
Choice 3 ($ j=3 $) Choice 4 ($ j=4 $)
$ i $ $ \lambda _{ij} $ $ \lambda _{ij}^{-} $ $ c_{ij} $ $ w_{ij} $ $ \lambda _{ij} $ $ \lambda _{ij}^{-} $ $ c_{ij} $ $ w_{ij} $
1 0.00499 0.00134 2 2 0.00818 0.0065 2 5
2 0.00431 0.00363 1 9 - - - -
3 0.0124 0.0013 1 6 0.00466 0.0046 4 4
4 0.00683 0.00458 5 4 - - - -
5 0.00818 0.00542 3 5 - - - -
6 0.00268 0.00156 2 5 0.000408 0.000304 2 4
7 0.00394 0.0014 5 9 - - - -
8 0.0105 0.0004 6 6 - - - -
9 0.000408 0.000265 4 7 0.000943 0.000870 3 8
10 0.00105 0.0005 5 6 - - - -
11 0.00314 0.00021 5 6 - - - -
12 0.0133 0.0032 4 6 0.011 0.001 5 7
13 0.00665 0.00546 4 6 - - - -
14 0.00355 0.00143 2 6 0.00436 0.00154 6 9
Choice 1 ($ j=1 $) Choice 2 ($ j=2 $)
$ i $ $ \lambda _{ij} $ $ \lambda _{ij}^{-} $ $ c_{ij} $ $ w_{ij} $ $ \lambda _{ij} $ $ \lambda _{ij}^{-} $ $ c_{ij} $ $ w_{ij} $
1 0.00532 0.00420 1 3 0.000726 0.000516 1 4
2 0.00818 0.00630 2 8 0.000619 0.000415 1 1
3 0.0133 0.0025 2 7 0.011 0.001 3 5
4 0.00741 0.00430 3 5 0.0124 0.0013 4 6
5 0.00619 0.00319 2 4 0.00431 0.00240 2 3
6 0.00436 0.00215 3 5 0.00567 0.00350 3 4
7 0.0105 0.0003 4 7 0.00466 0.00432 4 8
8 0.015 0.003 3 4 0.00105 0.0006 5 7
9 0.00268 0.00165 2 8 0.000101 0.00002 3 9
10 0.0141 0.0025 4 6 0.00683 0.00467 4 5
11 0.00394 0.00186 3 5 0.00355 0.00255 4 6
12 0.00236 0.00167 2 4 0.00769 0.00543 3 5
13 0.00215 0.00135 2 5 0.00436 0.00258 3 5
14 0.011 0.001 4 6 0.00834 0.00542 4 7
Choice 3 ($ j=3 $) Choice 4 ($ j=4 $)
$ i $ $ \lambda _{ij} $ $ \lambda _{ij}^{-} $ $ c_{ij} $ $ w_{ij} $ $ \lambda _{ij} $ $ \lambda _{ij}^{-} $ $ c_{ij} $ $ w_{ij} $
1 0.00499 0.00134 2 2 0.00818 0.0065 2 5
2 0.00431 0.00363 1 9 - - - -
3 0.0124 0.0013 1 6 0.00466 0.0046 4 4
4 0.00683 0.00458 5 4 - - - -
5 0.00818 0.00542 3 5 - - - -
6 0.00268 0.00156 2 5 0.000408 0.000304 2 4
7 0.00394 0.0014 5 9 - - - -
8 0.0105 0.0004 6 6 - - - -
9 0.000408 0.000265 4 7 0.000943 0.000870 3 8
10 0.00105 0.0005 5 6 - - - -
11 0.00314 0.00021 5 6 - - - -
12 0.0133 0.0032 4 6 0.011 0.001 5 7
13 0.00665 0.00546 4 6 - - - -
14 0.00355 0.00143 2 6 0.00436 0.00154 6 9
Table 7.  Numerical results of model by GA
Redundancy allocation problem
$ i $ $ z_{i} $ $ {n_{A} } $ $ n_{s} $ Redundancy
1 1 3 0 Active
2 2 2 1 Standby
3 4 3 0 Active
4 3 1 1 Standby
5 2 2 4 Mixed
6 2 2 0 Active
7 2 1 1 Standby
8 2 2 2 Mixed
9 3 2 2 Mixed
10 3 1 1 Standby
11 3 3 1 Mixed
12 1 1 2 Standby
13 2 1 1 Standby
14 3 2 0 Active
System Reliability 0.9823
System weight 170
System cost 116
Redundancy allocation problem
$ i $ $ z_{i} $ $ {n_{A} } $ $ n_{s} $ Redundancy
1 1 3 0 Active
2 2 2 1 Standby
3 4 3 0 Active
4 3 1 1 Standby
5 2 2 4 Mixed
6 2 2 0 Active
7 2 1 1 Standby
8 2 2 2 Mixed
9 3 2 2 Mixed
10 3 1 1 Standby
11 3 3 1 Mixed
12 1 1 2 Standby
13 2 1 1 Standby
14 3 2 0 Active
System Reliability 0.9823
System weight 170
System cost 116
Table 8.  Numerical results of model by PSO
Redundancy allocation problem
$ i $ $ z_{i} $ $ {n_{A} } $ $ n_{S} $ Redundancy
1 3 3 0 active
2 2 1 1 Standby
3 4 2 0 active
4 3 2 4 Mixed
5 2 4 0 Active
6 4 1 0 Active
7 2 2 0 Active
8 2 1 1 Standby
9 1 2 2 Mixed
10 3 1 1 Standby
11 3 2 0 Active
12 1 1 2 Mixed
13 1 1 2 Mixed
14 3 2 0 Active
System Reliability 0.9432
System weight 170
System cost 118
Redundancy allocation problem
$ i $ $ z_{i} $ $ {n_{A} } $ $ n_{S} $ Redundancy
1 3 3 0 active
2 2 1 1 Standby
3 4 2 0 active
4 3 2 4 Mixed
5 2 4 0 Active
6 4 1 0 Active
7 2 2 0 Active
8 2 1 1 Standby
9 1 2 2 Mixed
10 3 1 1 Standby
11 3 2 0 Active
12 1 1 2 Mixed
13 1 1 2 Mixed
14 3 2 0 Active
System Reliability 0.9432
System weight 170
System cost 118
Table 9.  Comparison between the computational of GA and PSO
PSO GA Algorithm
0.9432 0.9823 System reliability
170 170 Resource consumed cost
118 116 Resource consumed Weight
PSO GA Algorithm
0.9432 0.9823 System reliability
170 170 Resource consumed cost
118 116 Resource consumed Weight
Table 10.  Comparison results among proposed mixed strategy and other redundancy strategies
Strategy; $ i $ Warm standby [58] (GA) Warm standby [58] (HGA) Proposed mixed (GA) Proposed mixed (PSO)
$ Z_{i} $ $ n_{i} $ $ Z_{i} $ $ n_{i} $ $ Z_{i} $ $ n_{A, i} $ $ n_{s, i} $ $ Z_{i} $ $ n_{active} $ $ n_{s} $
1 3 2 3 2 1 3 0 3 3 0
2 1 2 1 2 2 2 1 2 1 1
3 4 2 4 1 4 3 0 4 2 0
4 3 3 3 3 3 1 1 3 2 4
5 1 1 2 1 2 2 4 2 4 0
6 2 2 2 2 2 2 0 4 1 0
7 3 1 2 1 2 1 1 2 2 0
8 1 3 1 3 2 2 2 2 1 1
9 3 3 3 3 3 2 2 1 2 2
10 2 4 2 4 3 1 1 3 1 1
11 1 4 1 4 3 3 1 3 2 0
12 1 2 1 2 1 1 2 1 1 2
13 2 2 2 2 2 1 1 1 1 2
14 3 3 3 4 3 2 0 3 2 0
System 0.4269 0.4403 0.9823 0.9432
reliability
System weight 118 118 116 118
System cost 170 170 170 170
Strategy; $ i $ Warm standby [58] (GA) Warm standby [58] (HGA) Proposed mixed (GA) Proposed mixed (PSO)
$ Z_{i} $ $ n_{i} $ $ Z_{i} $ $ n_{i} $ $ Z_{i} $ $ n_{A, i} $ $ n_{s, i} $ $ Z_{i} $ $ n_{active} $ $ n_{s} $
1 3 2 3 2 1 3 0 3 3 0
2 1 2 1 2 2 2 1 2 1 1
3 4 2 4 1 4 3 0 4 2 0
4 3 3 3 3 3 1 1 3 2 4
5 1 1 2 1 2 2 4 2 4 0
6 2 2 2 2 2 2 0 4 1 0
7 3 1 2 1 2 1 1 2 2 0
8 1 3 1 3 2 2 2 2 1 1
9 3 3 3 3 3 2 2 1 2 2
10 2 4 2 4 3 1 1 3 1 1
11 1 4 1 4 3 3 1 3 2 0
12 1 2 1 2 1 1 2 1 1 2
13 2 2 2 2 2 1 1 1 1 2
14 3 3 3 4 3 2 0 3 2 0
System 0.4269 0.4403 0.9823 0.9432
reliability
System weight 118 118 116 118
System cost 170 170 170 170
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