Figure 1.
Modified uniform crossover
-
Previous Article
Time-consistent investment-reinsurance strategy with a defaultable security under ambiguous environment
- JIMO Home
- This Issue
-
Next Article
Impacts of horizontal mergers on dual-channel supply chain
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 |
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.
Keywords:
Multi-source multi-sink stochastic-flow networks,
components assignment problem,
reliability of the Capacity Vector.
Mathematics Subject Classification:
Primary: 90B25; Secondary: 90B15.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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 |
show all references
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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 |
Figure 2.
Mutation operation
Figure 3.
Crossover operation
Figure 4.
Mutation operation
Figure 5.
Network with three source and two sink nodes
Figure 6.
The minimum cost found at each generation
Figure 7.
Two-source two-sink computer network
Figure 8.
The minimum cost found at each generation
Figure 9.
Network with two source and three sink nodes
Figure 10.
The minimum cost found at each generation
| 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 |
| No. | The components |
The Flow vector (F) | ||
| 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 |
The Flow vector (F) | ||
| 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 |
| 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 |
| [1] |
Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2020034 |
| [2] |
Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021013 |
| [3] |
Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020171 |
| [4] |
Ömer Arslan, Selçuk Kürşat İşleyen. A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021002 |
| [5] |
Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130 |
| [6] |
Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021025 |
| [7] |
Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291 |
| [8] |
Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021004 |
| [9] |
Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 |
| [10] |
Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109 |
| [11] |
Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117 |
| [12] |
Nicholas Geneva, Nicholas Zabaras. Multi-fidelity generative deep learning turbulent flows. Foundations of Data Science, 2020, 2 (4) : 391-428. doi: 10.3934/fods.2020019 |
| [13] |
Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055 |
| [14] |
Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047 |
| [15] |
Hongyu Cheng, Shimin Wang. Response solutions to harmonic oscillators beyond multi–dimensional brjuno frequency. Communications on Pure & Applied Analysis, 2021, 20 (2) : 467-494. doi: 10.3934/cpaa.2020222 |
| [16] |
Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134 |
| [17] |
Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 |
| [18] |
Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021006 |
| [19] |
Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021005 |
| [20] |
A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020441 |
2019 Impact Factor: 1.366
Tools
Article outline
Figures and Tables
[Back to Top]