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doi: 10.3934/jimo.2021101
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## Stochastic comparisons of series-parallel and parallel-series systems with dependence between components and also of subsystems

 1 Department of Statistics, University of Zabol, Sistan and Baluchestan, Iran 2 Department of Mathematics, University of Zabol, Sistan and Baluchestan, Iran 3 Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

Received  October 2020 Revised  January 2021 Early access May 2021

Fund Project: The first author is supported by University of Zabol, grant number: GR-UOZ:3389

In this paper, we consider series-parallel and parallel-series systems comprising dependent components that are drawn from a heterogeneous population consisting of $m$ different subpopulations. The components within each subpopulation are assumed to be dependent, and the subsystems themselves are also dependent, with their joint distribution being modeled by an Archimedean copula. We consider a very general setting in which we assume that the subpopulations have different Archimedean copulas for their dependence. Under such a general setup, we discuss the usual stochastic, hazard rate and reversed hazard rate orders between these systems and present a number of numerical examples to illustrate all the results established here. Finally, some concluding remarks are made. The results established here extend the recent results of Fang et al. (2020) in which they have assumed all the subsystems to be independent.

Citation: Ghobad Barmalzan, Ali Akbar Hosseinzadeh, Narayanaswamy Balakrishnan. Stochastic comparisons of series-parallel and parallel-series systems with dependence between components and also of subsystems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021101
##### References:
 [1] A. Billionnet, Redundancy allocation for series-parallel systems using integer linear programming, IEEE Transactions on Reliability, 57 (2008), 507-516.  doi: 10.1109/TR.2008.927807.  Google Scholar [2] D. W. Coit and A. E. Smith, Reliability optimization of series-parallel systems using a genetic algorithm, IEEE Transactions on Reliability, 45 (1996), 254-260.  doi: 10.1109/24.510811.  Google Scholar [3] A. Di Crescenzo and F. Pellerey, Stochastic comparisons of series and parallel systems with randomized independent components, Operations Research Letters, 39 (2011), 380-384.  doi: 10.1016/j.orl.2011.07.004.  Google Scholar [4] L. Fang, N. Balakrishnan and Q. Jin, Optimal grouping of heterogeneous components in series-parallel and parallel-series systems under Archimedean copula dependence, Journal of Computational and Applied Mathematics, 377 (2020), 112916. doi: 10.1016/j.cam.2020.112916.  Google Scholar [5] N. K. Hazra, M. Finkelstein and J. H. Cha, On optimal grouping and stochastic comparisons for heterogeneous items, Journal of Multivariate Analysis, 160 (2017), 146-156.  doi: 10.1016/j.jmva.2017.06.006.  Google Scholar [6] S. Kotz, N. Balakrishnan and N. L. Johnson, Continuous Multivariate Distributions-Vol. 1, 2$^{nd}$ edition, John Wiley & Sons, New York, 2000.  Google Scholar [7] G. Levitin and S.V. Amari, Optimal load distribution in series-parallel systems, Reliability Engineering & System Safety, 94 (2009), 254-260.  doi: 10.1016/j.ress.2008.03.001.  Google Scholar [8] X. Li and R. Fang, Ordering properties of order statistics from random variables of Archimedean copulas with applications, Journal of Multivariate Analysis, 133 (2015), 304-320.  doi: 10.1016/j.jmva.2014.09.016.  Google Scholar [9] X. Ling, Y. Wei and P. Li, On optimal heterogeneous components grouping in series-parallel and parallel-series systems, Probability in the Engineering and Informational Sciences, 33 (2019), 564-578.  doi: 10.1017/S0269964818000499.  Google Scholar [10] A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and its Applications, 2$^{nd}$ edition, Springer, New York, 2011., doi: 10.1007/978-0-387-68276-1.  Google Scholar [11] A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Hoboken, John Wiley & Sons, New Jersey, 2002.  Google Scholar [12] R. B. Nelsen, An Introduction to Copulas, Springer, New York, 2006. doi: 10.1007/s11229-005-3715-x.  Google Scholar [13] J. E. Ramirez-Marquez, D. W. Coit and A. Konak, Redundancy allocation for series-parallel systems using a max-min approach, IIE Transactions, 36 (2004), 891-898.  doi: 10.1080/07408170490473097.  Google Scholar [14] A. M. Sarhan, A. S. Al-Ruzaiza, I. A. Alwasel and A. I. El-Gohary, Reliability equivalence of a series-parallel system, Applied Mathematics and Computation, 154 (2004), 257-277.  doi: 10.1016/S0096-3003(03)00709-4.  Google Scholar [15] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5.  Google Scholar [16] M. X. Sun, Y. F. Li and E. Zio, On the optimal redundancy allocation for multi-state series-parallel systems under epistemic uncertainty, Reliability Engineering & System Safety, 192 (2019), 106019. doi: 10.1016/j.ress.2017.11.025.  Google Scholar [17] P. Zhao, P. S. Chan and H. K. T. Ng, Optimal allocation of redundancies in series systems, European Journal of Operational Research, 220 (2012), 673-683.  doi: 10.1016/j.ejor.2012.02.024.  Google Scholar [18] P. Zhao, Y. Zhang and L. Li, Redundancy allocation at component level versus system level, European Journal of Operational Research, 241 (2015), 402-411.  doi: 10.1016/j.ejor.2014.08.040.  Google Scholar

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##### References:
 [1] A. Billionnet, Redundancy allocation for series-parallel systems using integer linear programming, IEEE Transactions on Reliability, 57 (2008), 507-516.  doi: 10.1109/TR.2008.927807.  Google Scholar [2] D. W. Coit and A. E. Smith, Reliability optimization of series-parallel systems using a genetic algorithm, IEEE Transactions on Reliability, 45 (1996), 254-260.  doi: 10.1109/24.510811.  Google Scholar [3] A. Di Crescenzo and F. Pellerey, Stochastic comparisons of series and parallel systems with randomized independent components, Operations Research Letters, 39 (2011), 380-384.  doi: 10.1016/j.orl.2011.07.004.  Google Scholar [4] L. Fang, N. Balakrishnan and Q. Jin, Optimal grouping of heterogeneous components in series-parallel and parallel-series systems under Archimedean copula dependence, Journal of Computational and Applied Mathematics, 377 (2020), 112916. doi: 10.1016/j.cam.2020.112916.  Google Scholar [5] N. K. Hazra, M. Finkelstein and J. H. Cha, On optimal grouping and stochastic comparisons for heterogeneous items, Journal of Multivariate Analysis, 160 (2017), 146-156.  doi: 10.1016/j.jmva.2017.06.006.  Google Scholar [6] S. Kotz, N. Balakrishnan and N. L. Johnson, Continuous Multivariate Distributions-Vol. 1, 2$^{nd}$ edition, John Wiley & Sons, New York, 2000.  Google Scholar [7] G. Levitin and S.V. Amari, Optimal load distribution in series-parallel systems, Reliability Engineering & System Safety, 94 (2009), 254-260.  doi: 10.1016/j.ress.2008.03.001.  Google Scholar [8] X. Li and R. Fang, Ordering properties of order statistics from random variables of Archimedean copulas with applications, Journal of Multivariate Analysis, 133 (2015), 304-320.  doi: 10.1016/j.jmva.2014.09.016.  Google Scholar [9] X. Ling, Y. Wei and P. Li, On optimal heterogeneous components grouping in series-parallel and parallel-series systems, Probability in the Engineering and Informational Sciences, 33 (2019), 564-578.  doi: 10.1017/S0269964818000499.  Google Scholar [10] A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and its Applications, 2$^{nd}$ edition, Springer, New York, 2011., doi: 10.1007/978-0-387-68276-1.  Google Scholar [11] A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Hoboken, John Wiley & Sons, New Jersey, 2002.  Google Scholar [12] R. B. Nelsen, An Introduction to Copulas, Springer, New York, 2006. doi: 10.1007/s11229-005-3715-x.  Google Scholar [13] J. E. Ramirez-Marquez, D. W. Coit and A. Konak, Redundancy allocation for series-parallel systems using a max-min approach, IIE Transactions, 36 (2004), 891-898.  doi: 10.1080/07408170490473097.  Google Scholar [14] A. M. Sarhan, A. S. Al-Ruzaiza, I. A. Alwasel and A. I. El-Gohary, Reliability equivalence of a series-parallel system, Applied Mathematics and Computation, 154 (2004), 257-277.  doi: 10.1016/S0096-3003(03)00709-4.  Google Scholar [15] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5.  Google Scholar [16] M. X. Sun, Y. F. Li and E. Zio, On the optimal redundancy allocation for multi-state series-parallel systems under epistemic uncertainty, Reliability Engineering & System Safety, 192 (2019), 106019. doi: 10.1016/j.ress.2017.11.025.  Google Scholar [17] P. Zhao, P. S. Chan and H. K. T. Ng, Optimal allocation of redundancies in series systems, European Journal of Operational Research, 220 (2012), 673-683.  doi: 10.1016/j.ejor.2012.02.024.  Google Scholar [18] P. Zhao, Y. Zhang and L. Li, Redundancy allocation at component level versus system level, European Journal of Operational Research, 241 (2015), 402-411.  doi: 10.1016/j.ejor.2014.08.040.  Google Scholar
Plots of the reliability functions under Clayton copula for the components and copulas considered in Parts (ⅰ)-(ⅲ) for the subsystems, from left to right
Plots of the reliability functions under Clayton copula for the components and copulas considered in Parts (ⅰ)-(ⅲ) for the subsystems, from left to right
Plots of the distribution functions of parallel-series systems under Ali-Mikhail-Haq copula for the components and copulas considered in Parts (ⅰ)- (ⅲ) for the subsystems, from left to right
Plots of the distribution functions of parallel-series systems under Ali-Mikhail-Haq copula for the components and copulas considered in Parts (ⅰ)-(ⅲ) for the subsystems, from left to right
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