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A new dynamic model to optimize the reliability of the series-parallel systems under warm standby components

  • * Corresponding author: Amir Mohammad Fakoor Saghih

    * Corresponding author: Amir Mohammad Fakoor Saghih 
Abstract / Introduction Full Text(HTML) Figure(9) / Table(10) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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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 $.
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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