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October  2015, 11(4): 1375-1391. doi: 10.3934/jimo.2015.11.1375

## Optimal double-resource assignment for a distributed multistate network

 1 Department of Industrial Management, Tungnan University, New Taipei City, 222, Taiwan

Received  March 2014 Revised  October 2014 Published  March 2015

A distributed multistate network is a multistate network with the flows entering from multiple source nodes and exiting by multiple sink nodes. A multistate network is a network with its nodes and edges having multiple states (capacities) or failures. Such networks are different from the ones solved by the traditional methods in two aspects: the number of source/sink nodes is more than one, and the source nodes are also sink nodes. The optimal double-resource assignment problem for a distributed multistate network (ODRADMN) is to solve the optimal capacity assignment for nodes and edges in the network such that the total capacity requirement of the network is minimized while keeping the network still alive. Traditionally, multi-objective optimization methods are employed to solve such kind of problems. This paper proposes an elegant single-objective optimization method to solve the double-resource optimization problem in terms of network reliability. Several numerical examples are demonstrated to explain the proposed method.
Citation: Shin-Guang Chen. Optimal double-resource assignment for a distributed multistate network. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1375-1391. doi: 10.3934/jimo.2015.11.1375
##### References:
 [1] R. P. Agdeppa, N. Yamashita and M. Fukushima, An implicit programming approach for the road pricing problem with nonadditive route costs, Journal of Industrial and Management Optimization, 4 (2008), 183-197. doi: 10.3934/jimo.2008.4.183. [2] T. Aven, Reliability evaluation of multistate systems with multistate components, IEEE Transactions on Reliability, R-34 (1985), 473-479. doi: 10.1109/TR.1985.5222235. [3] M. O. Ball, Computational complexity of network reliability analysis: An overview, IEEE Transactions on Reliability, 35 (1986), 230-238. doi: 10.1109/TR.1986.4335422. [4] H. M. Bidhandi and R. M. Yusuff, Integrated supply chain planning under uncertainty using an improved stochastic approach, Applied Mathematical Modelling, 35 (2011), 2618-2630. doi: 10.1016/j.apm.2010.11.042. [5] S. G. Chen, Search for all minimal paths in a general directed flow network with unreliable nodes, International Journal of Reliability and Quality Performance, 2 (2011), 63-70. [6] 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. [7] 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. [8] S.-G. Chen and Y.-K. Lin, Search for all minimal paths in a general large flow network, IEEE Transactions on Reliability, 61 (2012), 949-956. [9] D. Coit and A. Smith, Reliability optimization of series-parallel systems using genetic algorithm, IEEE Transactions on Reliability, 45 (1996), 254-260, 266. doi: 10.1109/24.510811. [10] C. J. Colbourn, The Combinatorics of Network Reliability, Oxford University Press, UK, 1987. [11] I. Correia, S. Nickel and F. S. da Gama, Hub and spoke network design with single-assignment, capacity decisions and balancing requirements, Applied Mathematical Modelling, 35 (2011), 4841-4851. doi: 10.1016/j.apm.2011.03.046. [12] L. R. Ford and D. R. Fulkerson, Flows in Networks, NJ: Princeton University Press, 1962. [13] B. Gavish and I. Neuman, A system for routing and capacity assignment in computer communication networks, IEEE Transactions on Communications, 37 (1989), 360-366. doi: 10.1109/26.20116. [14] J. D. Glover, M. Sarma and T. Overbye, Power System Analysis & Design, 5th edition, Cengage Learning, Stamford, CT, USA, 2008. [15] W. S. Griffith, Multistate reliability models, Journal of Applied Probability, 17 (1980), 735-744. doi: 10.2307/3212967. [16] 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. doi: 10.1016/j.cor.2003.08.008. [17] J. C. Hudson and K. C. Kapur, Reliability analysis for multistate systems with multistate components, IIE Transactions, 15 (1983), 127-135. doi: 10.1080/05695558308974623. [18] J. C. Hudson and K. C. Kapur, Reliability bounds for multistate systems with multistate components, Operations Research, 33 (1985), 153-160. doi: 10.1287/opre.33.1.153. [19] C. C. Jane and Y. W. Laih, A practical algorithm for computing multi-state two-terminal reliability, IEEE Transactions on Reliability, 57 (2008), 295-302. [20] G. Levitin and A. Lisnianski, A new approach to solving problems of multi-state system reliability optimization, Quality Reliability Engineering International, 17 (2001), 93-104. doi: 10.1002/qre.388. [21] Y.-K. Lin, System capacity for a two-commodity multistate flow network with unreliable nodes and capacity weight, Computers and Operations Research, 34 (2007), 3043-3054. doi: 10.1016/j.cor.2005.11.013. [22] Y.-K. Lin and C.-T. Yeh, Optimal resource assignment to maximize multistate network reliability for a computer network, Computers & Operations Research, 37 (2010), 2229-2238. doi: 10.1016/j.cor.2010.03.013. [23] Y.-K. Lin and C.-T. Yeh, Computer network reliability optimization under double-resource assignments subject to a transmission budget, Information Sciences, 181 (2011), 582-599. doi: 10.1016/j.ins.2010.09.036. [24] Y.-K. Lin and C.-T. Yeh, Determine the optimal double-component assignment for a stochastic computer network, Omega, 40 (2012), 120-130. doi: 10.1016/j.omega.2011.04.002. [25] K. Murakami and H. S. Kim, Joint optimization of capacity and flow assignment for self-healing ATM networks, in IEEE International Conference on Communications, 1 (1995), 216-220. doi: 10.1109/ICC.1995.525168. [26] D. W. Pentico, Assignment problems, a golden anniversary survey, European Journal of Operational Research, 176 (2007), 774-793. doi: 10.1016/j.ejor.2005.09.014. [27] Y. Shen, A new simple algorithm for enumerating all minimal paths and cuts of a graph, Microelectronics and Reliability, 35 (1995), 973-976. doi: 10.1016/0026-2714(94)00121-4. [28] E. D. Sykas, On the capacity assignment problem in packet-switching computer networks, Applied Mathematical Modelling, 10 (1986), 346-356. doi: 10.1016/0307-904X(86)90094-6. [29] W. L. Winston, Introduction to Mathematical Programming: Application and Algorithms, Duxbury Press, California, 1995. [30] J. Xue, On multistate system analysis, IEEE Transactions on Reliability, 34 (1985), 329-337. [31] Q. Yang, S. Song and C. Wu, Inventory policies for a partially observed supply capacity model, Journal of Industrial and Management Optimization, 9 (2013), 13-30. doi: 10.3934/jimo.2013.9.13. [32] W. C. Yeh, Search for minimal paths in modified networks, Reliability Engineering & System Safety, 75 (2002), 389-395. doi: 10.1016/S0951-8320(01)00128-4. [33] W. C. Yeh, A new approach to evaluating reliability of multistate networks under the cost constraint, Omega, 33 (2005), 203-209. doi: 10.1016/j.omega.2004.04.005. [34] A. F. Zantuti, Algorithms for capacities and flow assignment problem in computer networks, in 19th International Conference on Systems Engineering, (2008), 315-318. doi: 10.1109/ICSEng.2008.24. [35] X. Zhang, D. Wang, H. Sun and K. S. Trivedi, A BDD-based algorithm for analysis of multistate systems with multistate components, IEEE Transactions on Computers, 52 (2003), 1608-1618. [36] 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. doi: 10.1080/07408170601013653.

show all references

##### References:
 [1] R. P. Agdeppa, N. Yamashita and M. Fukushima, An implicit programming approach for the road pricing problem with nonadditive route costs, Journal of Industrial and Management Optimization, 4 (2008), 183-197. doi: 10.3934/jimo.2008.4.183. [2] T. Aven, Reliability evaluation of multistate systems with multistate components, IEEE Transactions on Reliability, R-34 (1985), 473-479. doi: 10.1109/TR.1985.5222235. [3] M. O. Ball, Computational complexity of network reliability analysis: An overview, IEEE Transactions on Reliability, 35 (1986), 230-238. doi: 10.1109/TR.1986.4335422. [4] H. M. Bidhandi and R. M. Yusuff, Integrated supply chain planning under uncertainty using an improved stochastic approach, Applied Mathematical Modelling, 35 (2011), 2618-2630. doi: 10.1016/j.apm.2010.11.042. [5] S. G. Chen, Search for all minimal paths in a general directed flow network with unreliable nodes, International Journal of Reliability and Quality Performance, 2 (2011), 63-70. [6] 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. [7] 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. [8] S.-G. Chen and Y.-K. Lin, Search for all minimal paths in a general large flow network, IEEE Transactions on Reliability, 61 (2012), 949-956. [9] D. Coit and A. Smith, Reliability optimization of series-parallel systems using genetic algorithm, IEEE Transactions on Reliability, 45 (1996), 254-260, 266. doi: 10.1109/24.510811. [10] C. J. Colbourn, The Combinatorics of Network Reliability, Oxford University Press, UK, 1987. [11] I. Correia, S. Nickel and F. S. da Gama, Hub and spoke network design with single-assignment, capacity decisions and balancing requirements, Applied Mathematical Modelling, 35 (2011), 4841-4851. doi: 10.1016/j.apm.2011.03.046. [12] L. R. Ford and D. R. Fulkerson, Flows in Networks, NJ: Princeton University Press, 1962. [13] B. Gavish and I. Neuman, A system for routing and capacity assignment in computer communication networks, IEEE Transactions on Communications, 37 (1989), 360-366. doi: 10.1109/26.20116. [14] J. D. Glover, M. Sarma and T. Overbye, Power System Analysis & Design, 5th edition, Cengage Learning, Stamford, CT, USA, 2008. [15] W. S. Griffith, Multistate reliability models, Journal of Applied Probability, 17 (1980), 735-744. doi: 10.2307/3212967. [16] 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. doi: 10.1016/j.cor.2003.08.008. [17] J. C. Hudson and K. C. Kapur, Reliability analysis for multistate systems with multistate components, IIE Transactions, 15 (1983), 127-135. doi: 10.1080/05695558308974623. [18] J. C. Hudson and K. C. Kapur, Reliability bounds for multistate systems with multistate components, Operations Research, 33 (1985), 153-160. doi: 10.1287/opre.33.1.153. [19] C. C. Jane and Y. W. Laih, A practical algorithm for computing multi-state two-terminal reliability, IEEE Transactions on Reliability, 57 (2008), 295-302. [20] G. Levitin and A. Lisnianski, A new approach to solving problems of multi-state system reliability optimization, Quality Reliability Engineering International, 17 (2001), 93-104. doi: 10.1002/qre.388. [21] Y.-K. Lin, System capacity for a two-commodity multistate flow network with unreliable nodes and capacity weight, Computers and Operations Research, 34 (2007), 3043-3054. doi: 10.1016/j.cor.2005.11.013. [22] Y.-K. Lin and C.-T. Yeh, Optimal resource assignment to maximize multistate network reliability for a computer network, Computers & Operations Research, 37 (2010), 2229-2238. doi: 10.1016/j.cor.2010.03.013. [23] Y.-K. Lin and C.-T. Yeh, Computer network reliability optimization under double-resource assignments subject to a transmission budget, Information Sciences, 181 (2011), 582-599. doi: 10.1016/j.ins.2010.09.036. [24] Y.-K. Lin and C.-T. Yeh, Determine the optimal double-component assignment for a stochastic computer network, Omega, 40 (2012), 120-130. doi: 10.1016/j.omega.2011.04.002. [25] K. Murakami and H. S. Kim, Joint optimization of capacity and flow assignment for self-healing ATM networks, in IEEE International Conference on Communications, 1 (1995), 216-220. doi: 10.1109/ICC.1995.525168. [26] D. W. Pentico, Assignment problems, a golden anniversary survey, European Journal of Operational Research, 176 (2007), 774-793. doi: 10.1016/j.ejor.2005.09.014. [27] Y. Shen, A new simple algorithm for enumerating all minimal paths and cuts of a graph, Microelectronics and Reliability, 35 (1995), 973-976. doi: 10.1016/0026-2714(94)00121-4. [28] E. D. Sykas, On the capacity assignment problem in packet-switching computer networks, Applied Mathematical Modelling, 10 (1986), 346-356. doi: 10.1016/0307-904X(86)90094-6. [29] W. L. Winston, Introduction to Mathematical Programming: Application and Algorithms, Duxbury Press, California, 1995. [30] J. Xue, On multistate system analysis, IEEE Transactions on Reliability, 34 (1985), 329-337. [31] Q. Yang, S. Song and C. Wu, Inventory policies for a partially observed supply capacity model, Journal of Industrial and Management Optimization, 9 (2013), 13-30. doi: 10.3934/jimo.2013.9.13. [32] W. C. Yeh, Search for minimal paths in modified networks, Reliability Engineering & System Safety, 75 (2002), 389-395. doi: 10.1016/S0951-8320(01)00128-4. [33] W. C. Yeh, A new approach to evaluating reliability of multistate networks under the cost constraint, Omega, 33 (2005), 203-209. doi: 10.1016/j.omega.2004.04.005. [34] A. F. Zantuti, Algorithms for capacities and flow assignment problem in computer networks, in 19th International Conference on Systems Engineering, (2008), 315-318. doi: 10.1109/ICSEng.2008.24. [35] X. Zhang, D. Wang, H. Sun and K. S. Trivedi, A BDD-based algorithm for analysis of multistate systems with multistate components, IEEE Transactions on Computers, 52 (2003), 1608-1618. [36] 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. doi: 10.1080/07408170601013653.
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