# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020020

## Maximizing reliability of the capacity vector for multi-source multi-sink stochastic-flow networks subject to an assignment budget

 Computer Science Branch, Mathematics Department, Faculty of Science, Aswan University, Aswan, Egypt

* Corresponding author: M. R. Hassan

Received  May 2019 Revised  August 2019 Published  January 2020

Many real-world networks such as freight, power and long distance transportation networks are represented as multi-source multi-sink stochastic flow network. The objective is to transmit flow successfully between the source and the sink nodes. The reliability of the capacity vector of the assigned components is used an indicator to find the best flow strategy on the network. The Components Assignment Problem (CAP) deals with searching the optimal components to a given network subject to one or more constraints. The CAP in multi-source multi-sink stochastic flow networks with multiple commodities has not yet been discussed, so our paper investigates this scenario to maximize the reliability of the capacity vector subject to an assignment budget. The mathematical formulation of the problem is defined, and a proposed solution based on genetic algorithms is developed consisting of two steps. The first searches the set of components with the minimum cost and the second searches the flow vector of this set of components with maximum reliability. We apply the solution approach to three commonly used examples from the literature with two sets of available components to demonstrate its strong performance.

Citation: M. R. Hassan. Maximizing reliability of the capacity vector for multi-source multi-sink stochastic-flow networks subject to an assignment budget. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020020
##### References:
 [1] A. Aissou, A. Daamouche and M. R. Hassan, Optimal components assignment problem for stochastic-flow networks, Journal of Computer Science, 15 (2019), 108-117.   Google Scholar [2] S. G. Chen, An optimal capacity assignment for the robust design problem in capacitated flow networks, Applied Mathematical Modelling, 36 (2012), 5272-5282.  doi: 10.1016/j.apm.2011.12.034.  Google Scholar [3] S. G. Chen, Optimal double-resource assignment for the robust design problem in multistate computer networks, Applied Mathematical Modelling, 38 (2014), 263-277.  doi: 10.1016/j.apm.2013.06.020.  Google Scholar [4] D. W. Coit and A. E. Smith, Penalty guided genetic search for reliability design optimization, Computers and Industrial Engineering, 30 (1996), 895-904.   Google Scholar [5] B. Dengiz, F. Altiparmak and A. E. Smith, Local search genetic algorithm for optimal design of reliable networks, IEEE Transactions on Evolutionary Computation, 10 (1997), 179-188.   Google Scholar [6] M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, 1$^{st}$ edition, Wiley Series in Engineering, Design, and Automation, 2000. Google Scholar [7] M. R. Hassan and H. Abdou, Multi-objective components assignment problem subject to lead-time constraints, Indian Journal of Science and Technology, 11 (2018), 1-9.   Google Scholar [8] M. R. Hassan, Solving a component assignment problem for a stochastic-flow network under lead-time constraint, Indian Journal of Science and Technology, 8 (2015), 1-5.   Google Scholar [9] M. R. Hassan, Solving flow allocation problems and optimizing system reliability of multisource multisink stochastic flow network, The International Arab Journal of Information Technology (IAJIT), 13 (2016), 477-483.   Google Scholar [10] C. C. Hsieh and Y. T. Chen, Reliable and economic resource allocation in an unreliable flow network, Computers and Operations Research, 32 (2005), 613-628.   Google Scholar [11] C. C. Hsieh and Y. T. Chen, Simple algorithms for updating multi-resource allocations in an unreliable flow network, Computers and Industrial Engineering, 50 (2006), 120-129.   Google Scholar [12] C. C. Hsieh and M. H. Lin, Reliability-oriented multi-resource allocation in a stochastic-flow network, Reliability Engineering and System Safety, 81 (2003), 155-161.   Google Scholar [13] Y.-K. Lin and C. T. Yeh, A two-stage approach for a multi-objective component assignment problem for a stochastic-flow network, Engineering Optimization, 45 (2013), 265-285.  doi: 10.1080/0305215X.2012.669381.  Google Scholar [14] Y. K. Lin and C. T. Yeh, System reliability maximization for a computer network by finding the optimal two-class allocation subject to budget, Applied Soft Computing, 36 (2015), 578-588.   Google Scholar [15] Y. K. Lin and C. T. Yeh, Determining the optimal double-component assignment for a stochastic computer network, Omega, 40 (2012), 120-130.   Google Scholar [16] Y. K. Lin and C. T. Yeh, Evaluation of optimal network reliability under components-assignments subject to transmission budget, IEEE Transactions on Reliability, 59 (2010), 539-550.   Google Scholar [17] Y. K. Lin and C. T. Yeh, Maximal network reliability with optimal transmission line assignment for stochastic electric power networks via genetic algorithms, Applied Soft Computing, 11 (2011), 2714-2724.   Google Scholar [18] Y. K. Lin and C. T. Yeh, Multi-objective optimization for stochastic computer networks using NSGA-Ⅱ and TOPSIS, European Journal of Operational Research, 218 (2012), 735-746.  doi: 10.1016/j.ejor.2011.11.028.  Google Scholar [19] Y. K. Lin and C. T. Yeh, Multistate components assignment problem with optimal network reliability subject to assignment budget, Applied Mathematics and Computation, 217 (2011), 10074-10086.  doi: 10.1016/j.amc.2011.05.001.  Google Scholar [20] Y. K. Lin and C. T. Yeh, Optimal resource assignment to maximize multistate network reliability for a computer network, Computers and Operations Research, 37 (2010), 2229-2238.  doi: 10.1016/j.cor.2010.03.013.  Google Scholar [21] Y. K. Lin and C. T. Yeh, Computer network reliability optimization under double-source assignments subject to transmission budget, Information Sciences, 181 (2011), 582-599.   Google Scholar [22] Y. K. Lin, C. T. Yeh and P. S. Huang, A hybrid ant-tabu algorithm for solving a multistate flow network reliability maximization problem, Applied Soft Computing, 13 (2013), 3529-3543.   Google Scholar [23] Q. Liu, H. Z. Xiaoxian and Q. Zhao, Genetic algorithm-based study on flow allocation in a multicommodity stochastic-flow network with unreliable nodes, Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, 8 (2007), 576-581.   Google Scholar [24] Q. Liu, Q. Z. Zhao and W. K. Zang, Study on multi-objective optimization of flow allocation in a multi-commodity stochastic-flow network with unreliable nodes, Journal of Applied Mathematics Computing (JAMC), 28 (2008), 185-198.  doi: 10.1007/s12190-008-0093-9.  Google Scholar [25] M. J. Zuo, Z. Tian and H. Z. Huang, An efficient method for reliability evaluation of multistate networks given all minimal path vectors, IIE Transactions, 39 (2007), 811-817.   Google Scholar

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##### References:
 [1] A. Aissou, A. Daamouche and M. R. Hassan, Optimal components assignment problem for stochastic-flow networks, Journal of Computer Science, 15 (2019), 108-117.   Google Scholar [2] S. G. Chen, An optimal capacity assignment for the robust design problem in capacitated flow networks, Applied Mathematical Modelling, 36 (2012), 5272-5282.  doi: 10.1016/j.apm.2011.12.034.  Google Scholar [3] S. G. Chen, Optimal double-resource assignment for the robust design problem in multistate computer networks, Applied Mathematical Modelling, 38 (2014), 263-277.  doi: 10.1016/j.apm.2013.06.020.  Google Scholar [4] D. W. Coit and A. E. Smith, Penalty guided genetic search for reliability design optimization, Computers and Industrial Engineering, 30 (1996), 895-904.   Google Scholar [5] B. Dengiz, F. Altiparmak and A. E. Smith, Local search genetic algorithm for optimal design of reliable networks, IEEE Transactions on Evolutionary Computation, 10 (1997), 179-188.   Google Scholar [6] M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, 1$^{st}$ edition, Wiley Series in Engineering, Design, and Automation, 2000. Google Scholar [7] M. R. Hassan and H. Abdou, Multi-objective components assignment problem subject to lead-time constraints, Indian Journal of Science and Technology, 11 (2018), 1-9.   Google Scholar [8] M. R. Hassan, Solving a component assignment problem for a stochastic-flow network under lead-time constraint, Indian Journal of Science and Technology, 8 (2015), 1-5.   Google Scholar [9] M. R. Hassan, Solving flow allocation problems and optimizing system reliability of multisource multisink stochastic flow network, The International Arab Journal of Information Technology (IAJIT), 13 (2016), 477-483.   Google Scholar [10] C. C. Hsieh and Y. T. Chen, Reliable and economic resource allocation in an unreliable flow network, Computers and Operations Research, 32 (2005), 613-628.   Google Scholar [11] C. C. Hsieh and Y. T. Chen, Simple algorithms for updating multi-resource allocations in an unreliable flow network, Computers and Industrial Engineering, 50 (2006), 120-129.   Google Scholar [12] C. C. Hsieh and M. H. Lin, Reliability-oriented multi-resource allocation in a stochastic-flow network, Reliability Engineering and System Safety, 81 (2003), 155-161.   Google Scholar [13] Y.-K. Lin and C. T. Yeh, A two-stage approach for a multi-objective component assignment problem for a stochastic-flow network, Engineering Optimization, 45 (2013), 265-285.  doi: 10.1080/0305215X.2012.669381.  Google Scholar [14] Y. K. Lin and C. T. Yeh, System reliability maximization for a computer network by finding the optimal two-class allocation subject to budget, Applied Soft Computing, 36 (2015), 578-588.   Google Scholar [15] Y. K. Lin and C. T. Yeh, Determining the optimal double-component assignment for a stochastic computer network, Omega, 40 (2012), 120-130.   Google Scholar [16] Y. K. Lin and C. T. Yeh, Evaluation of optimal network reliability under components-assignments subject to transmission budget, IEEE Transactions on Reliability, 59 (2010), 539-550.   Google Scholar [17] Y. K. Lin and C. T. Yeh, Maximal network reliability with optimal transmission line assignment for stochastic electric power networks via genetic algorithms, Applied Soft Computing, 11 (2011), 2714-2724.   Google Scholar [18] Y. K. Lin and C. T. Yeh, Multi-objective optimization for stochastic computer networks using NSGA-Ⅱ and TOPSIS, European Journal of Operational Research, 218 (2012), 735-746.  doi: 10.1016/j.ejor.2011.11.028.  Google Scholar [19] Y. K. Lin and C. T. Yeh, Multistate components assignment problem with optimal network reliability subject to assignment budget, Applied Mathematics and Computation, 217 (2011), 10074-10086.  doi: 10.1016/j.amc.2011.05.001.  Google Scholar [20] Y. K. Lin and C. T. Yeh, Optimal resource assignment to maximize multistate network reliability for a computer network, Computers and Operations Research, 37 (2010), 2229-2238.  doi: 10.1016/j.cor.2010.03.013.  Google Scholar [21] Y. K. Lin and C. T. Yeh, Computer network reliability optimization under double-source assignments subject to transmission budget, Information Sciences, 181 (2011), 582-599.   Google Scholar [22] Y. K. Lin, C. T. Yeh and P. S. Huang, A hybrid ant-tabu algorithm for solving a multistate flow network reliability maximization problem, Applied Soft Computing, 13 (2013), 3529-3543.   Google Scholar [23] Q. Liu, H. Z. Xiaoxian and Q. Zhao, Genetic algorithm-based study on flow allocation in a multicommodity stochastic-flow network with unreliable nodes, Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, 8 (2007), 576-581.   Google Scholar [24] Q. Liu, Q. Z. Zhao and W. K. Zang, Study on multi-objective optimization of flow allocation in a multi-commodity stochastic-flow network with unreliable nodes, Journal of Applied Mathematics Computing (JAMC), 28 (2008), 185-198.  doi: 10.1007/s12190-008-0093-9.  Google Scholar [25] M. J. Zuo, Z. Tian and H. Z. Huang, An efficient method for reliability evaluation of multistate networks given all minimal path vectors, IIE Transactions, 39 (2007), 811-817.   Google Scholar
Modified uniform crossover
Mutation operation
Crossover operation
Mutation operation
Network with three source and two sink nodes
The minimum cost found at each generation
Two-source two-sink computer network
The minimum cost found at each generation
Network with two source and three sink nodes
The minimum cost found at each generation
Component information for the network in Figure 5
 p Capacity Cost 0 1 2 3 4 1 0.02 0.04 0.14 0.80 0.00 1 2 0.04 0.06 0.10 0.15 0.65 3 3 0.02 0.03 0.05 0.90 0.00 4 4 0.05 0.08 0.87 0.00 0.00 2 5 0.01 0.04 0.10 0.85 0.00 3 6 0.02 0.05 0.15 0.78 0.00 2 7 0.05 0.10 0.85 0.00 0.00 1 8 0.04 0.06 0.15 0.75 0.00 4 9 0.03 0.05 0.12 0.80 0.00 1 10 0.01 0.04 0.05 0.15 0.75 3 11 0.03 0.05 0.07 0.85 0.00 1 12 0.01 0.02 0.07 0.90 0.00 1
 p Capacity Cost 0 1 2 3 4 1 0.02 0.04 0.14 0.80 0.00 1 2 0.04 0.06 0.10 0.15 0.65 3 3 0.02 0.03 0.05 0.90 0.00 4 4 0.05 0.08 0.87 0.00 0.00 2 5 0.01 0.04 0.10 0.85 0.00 3 6 0.02 0.05 0.15 0.78 0.00 2 7 0.05 0.10 0.85 0.00 0.00 1 8 0.04 0.06 0.15 0.75 0.00 4 9 0.03 0.05 0.12 0.80 0.00 1 10 0.01 0.04 0.05 0.15 0.75 3 11 0.03 0.05 0.07 0.85 0.00 1 12 0.01 0.02 0.07 0.90 0.00 1
The initial population for the network example in Figure 5
 No. The components$\boldsymbol{(}\mathcal{B}$) The Flow vector (F) ${\boldsymbol{Z}}_{\boldsymbol{obj}}\left(\mathcal{B}\right)$ $\boldsymbol{R}\boldsymbol{(}{\boldsymbol{X}}_{\boldsymbol{F}}\boldsymbol{(}\mathcal{B}\boldsymbol{)}$ 1 2 10 3 12 7 1 8 11 6 9 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0067 0.259087 2 1 7 4 11 6 8 9 10 2 3 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0005 0.288896 3 6 12 8 11 1 10 9 3 2 4 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0164 0.236032 4 12 6 9 7 11 1 3 8 10 2 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.293038 5 8 5 4 9 7 1 12 6 10 2 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0036 0.269935 6 10 7 12 9 1 5 6 3 2 4 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.292652 7 1 9 6 10 5 3 11 4 2 12 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0000 0.312330 8 2 1 9 8 12 4 3 6 10 11 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0012 0.282898 9 9 3 12 5 4 1 6 10 2 7 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.293636 10 2 8 4 1 7 5 11 9 10 3 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 23.0000 0.314006
 No. The components$\boldsymbol{(}\mathcal{B}$) The Flow vector (F) ${\boldsymbol{Z}}_{\boldsymbol{obj}}\left(\mathcal{B}\right)$ $\boldsymbol{R}\boldsymbol{(}{\boldsymbol{X}}_{\boldsymbol{F}}\boldsymbol{(}\mathcal{B}\boldsymbol{)}$ 1 2 10 3 12 7 1 8 11 6 9 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0067 0.259087 2 1 7 4 11 6 8 9 10 2 3 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0005 0.288896 3 6 12 8 11 1 10 9 3 2 4 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0164 0.236032 4 12 6 9 7 11 1 3 8 10 2 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.293038 5 8 5 4 9 7 1 12 6 10 2 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0036 0.269935 6 10 7 12 9 1 5 6 3 2 4 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.292652 7 1 9 6 10 5 3 11 4 2 12 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0000 0.312330 8 2 1 9 8 12 4 3 6 10 11 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0012 0.282898 9 9 3 12 5 4 1 6 10 2 7 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.293636 10 2 8 4 1 7 5 11 9 10 3 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 23.0000 0.314006
The component information for the network of Figure 7.
 p Capacity 0 1 2 3 4 5 6 7 8 9 1 0.001 0.001 0.003 0.004 0.005 0.005 0.006 0.007 0.010 0.015 2 0.001 0.003 0.003 0.004 0.005 0.007 0.007 0.008 0.009 0.010 3 0.002 0.002 0.003 0.006 0.007 0.007 0.010 0.012 0.015 0.017 4 0.001 0.001 0.002 0.003 0.005 0.008 0.010 0.011 0.012 0.015 5 0.001 0.002 0.009 0.012 0.020 0.040 0.050 0.060 0.806 0.000 6 0.001 0.002 0.002 0.005 0.010 0.012 0.015 0.017 0.020 0.025 7 0.001 0.001 0.002 0.005 0.008 0.010 0.012 0.015 0.015 0.017 8 0.001 0.002 0.005 0.005 0.007 0.008 0.010 0.012 0.015 0.015 9 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.008 0.009 0.010 10 0.002 0.003 0.005 0.006 0.007 0.009 0.012 0.015 0.941 0.000 11 0.002 0.002 0.003 0.005 0.007 0.008 0.010 0.011 0.020 0.030 12 0.001 0.002 0.003 0.005 0.008 0.009 0.010 0.012 0.015 0.040 13 0.001 0.001 0.003 0.005 0.005 0.010 0.011 0.017 0.018 0.020 14 0.001 0.001 0.002 0.002 0.003 0.005 0.007 0.009 0.016 0.021 15 0.001 0.001 0.002 0.003 0.004 0.005 0.007 0.008 0.009 0.011 16 0.001 0.002 0.002 0.004 0.005 0.006 0.007 0.009 0.014 0.017 17 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.007 0.009 0.011 18 0.001 0.001 0.002 0.002 0.002 0.003 0.003 0.004 0.005 0.007 19 0.001 0.001 0.002 0.003 0.005 0.008 0.009 0.011 0.013 0.014 20 0.002 0.002 0.003 0.006 0.007 0.007 0.010 0.013 0.015 0.020 10 11 12 13 14 15 16 17 18 19 20 0.060 0.150 0.733 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.943 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.919 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.015 0.016 0.020 0.856 0.025 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.891 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.020 0.022 0.025 0.030 0.817 0.000 0.000 0.000 0.000 0.000 0.000 0.016 0.020 0.884 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.011 0.015 0.016 0.017 0.019 0.020 0.857 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.902 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.895 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.025 0.031 0.853 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.024 0.025 0.030 0.035 0.040 0.060 0.719 0.000 0.000 0.000 0.000 0.015 0.017 0.020 0.027 0.870 0.000 0.000 0.000 0.000 0.000 0.000 0.020 0.022 0.025 0.030 0.035 0.040 0.761 0.000 0.000 0.000 0.000 0.015 0.017 0.018 0.019 0.020 0.022 0.844 0.017 0.017 0.000 0.000 0.008 0.009 0.011 0.013 0.014 0.014 0.015 0.000 0.000 0.019 0.020 0.015 0.017 0.020 0.030 0.851 0.000 0.000 0.000 0.000 0.000 0.000 0.915 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 21 22 23 24 25 26 27 28 29 30 31 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.019 0.020 0.023 0.025 0.026 0.740 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
 p Capacity 0 1 2 3 4 5 6 7 8 9 1 0.001 0.001 0.003 0.004 0.005 0.005 0.006 0.007 0.010 0.015 2 0.001 0.003 0.003 0.004 0.005 0.007 0.007 0.008 0.009 0.010 3 0.002 0.002 0.003 0.006 0.007 0.007 0.010 0.012 0.015 0.017 4 0.001 0.001 0.002 0.003 0.005 0.008 0.010 0.011 0.012 0.015 5 0.001 0.002 0.009 0.012 0.020 0.040 0.050 0.060 0.806 0.000 6 0.001 0.002 0.002 0.005 0.010 0.012 0.015 0.017 0.020 0.025 7 0.001 0.001 0.002 0.005 0.008 0.010 0.012 0.015 0.015 0.017 8 0.001 0.002 0.005 0.005 0.007 0.008 0.010 0.012 0.015 0.015 9 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.008 0.009 0.010 10 0.002 0.003 0.005 0.006 0.007 0.009 0.012 0.015 0.941 0.000 11 0.002 0.002 0.003 0.005 0.007 0.008 0.010 0.011 0.020 0.030 12 0.001 0.002 0.003 0.005 0.008 0.009 0.010 0.012 0.015 0.040 13 0.001 0.001 0.003 0.005 0.005 0.010 0.011 0.017 0.018 0.020 14 0.001 0.001 0.002 0.002 0.003 0.005 0.007 0.009 0.016 0.021 15 0.001 0.001 0.002 0.003 0.004 0.005 0.007 0.008 0.009 0.011 16 0.001 0.002 0.002 0.004 0.005 0.006 0.007 0.009 0.014 0.017 17 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.007 0.009 0.011 18 0.001 0.001 0.002 0.002 0.002 0.003 0.003 0.004 0.005 0.007 19 0.001 0.001 0.002 0.003 0.005 0.008 0.009 0.011 0.013 0.014 20 0.002 0.002 0.003 0.006 0.007 0.007 0.010 0.013 0.015 0.020 10 11 12 13 14 15 16 17 18 19 20 0.060 0.150 0.733 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.943 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.919 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.015 0.016 0.020 0.856 0.025 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.891 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.020 0.022 0.025 0.030 0.817 0.000 0.000 0.000 0.000 0.000 0.000 0.016 0.020 0.884 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.011 0.015 0.016 0.017 0.019 0.020 0.857 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.902 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.895 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.025 0.031 0.853 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.024 0.025 0.030 0.035 0.040 0.060 0.719 0.000 0.000 0.000 0.000 0.015 0.017 0.020 0.027 0.870 0.000 0.000 0.000 0.000 0.000 0.000 0.020 0.022 0.025 0.030 0.035 0.040 0.761 0.000 0.000 0.000 0.000 0.015 0.017 0.018 0.019 0.020 0.022 0.844 0.017 0.017 0.000 0.000 0.008 0.009 0.011 0.013 0.014 0.014 0.015 0.000 0.000 0.019 0.020 0.015 0.017 0.020 0.030 0.851 0.000 0.000 0.000 0.000 0.000 0.000 0.915 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 21 22 23 24 25 26 27 28 29 30 31 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.019 0.020 0.023 0.025 0.026 0.740 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
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