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Generalized maximum flow over time with intermediate storage

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Abstract

We consider two classical network flow problems. First, it is possible to store excess flow in the intermediate nodes to improve the total amount of flow that can be transported through the network. Second, we consider lossy network, where each arc has certain gain factor along with capacity and some portion of the flow is lost while traversing through such arcs. While both extensions have been considered separately in the literature, this will be the first time that they are considered simultaneously, which requires new mathematical model, analysis and solution methods. We consider that intermediate nodes have sufficient holding capacity and the excess flow is stored in the priority order within their capacity. We introduce the generalized maximum static and dynamic flow problems with intermediate storage in lossy network and present efficient algorithms and FPTAS for the solutions. Furthermore, notion of the generalized maximum dynamic flow with excess flow storage is extended to contraflow technique with symmetric and asymmetric arc travel times.

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References

  • Ahuja, R. K., Orlin, J. B., & Magnanti, T. L. (1993). Network flows: Theory, algorithms, and applications. Prentice-Hall.

    Google Scholar 

  • Arulselvan, A. (2009). Network model for disaster management. PhD thesis, University of Florida, USA.

  • Dantzig, G. B. (1963). Linear programming and extensions. Princeton University Press.

    Book  Google Scholar 

  • Dhamala, T. N., Pyakurel, U., & Dempe, S. (2018). A critical survey on the network optimization algorithms for evacuation planning problems. International Journal of Operations Research, 15(3), 101–133.

    Google Scholar 

  • Dhamala, T. N., Wagle, S., & Pyakurel, U. (2023). Flowloc problem with maximum excess flow. https://doi.org/10.3934/jimo.2023064

  • Dijkstra, E. (1959). A note on two problems in connection with graph. Numerische Mathematik, 1, 269–271.

    Article  Google Scholar 

  • Fleischer, L., & Tardos, E. (1998). Efficient continuous-time dynamic network flow algorithms. Operations Research Letters, 23, 71–80.

    Article  Google Scholar 

  • Fleischer, L., & Wayne, K. D. (2002). Fast and simple approximation schemes for generalized flow. Mathematical Programming, 91, 215–238.

    Article  Google Scholar 

  • Fleischer, L., & Skutella, M. (2007). Quickest flows over time. SIAM Journal of Computing, 36, 1600–1630.

    Article  Google Scholar 

  • Ford, L. R., & Fulkerson, D. R. (1962). Flows in networks. Princeton University Press.

    Google Scholar 

  • Gerasimenko, E., Kureichik, V., & Kuliev, E. (2022). Maximum dynamic flow model for hesitant fuzzy evacuation with intermediate storage at nodes. In C. Kahraman, S. Cebi, Onar S. Cevik, B. Oztaysi, A. C. Tolga, & I. U. Sari (Eds.), Intelligent and Fuzzy techniques for emerging conditions and digital transformation. INFUS 2021. Lecture notes in networks and systems. (Vol. 307). Springer.

    Google Scholar 

  • Glover, F., & Klingman, D. (1973). On the equivalence of some generalized network flow problems to pure network problems. Mathematical Programming, 4, 267–278.

    Article  Google Scholar 

  • Goldberg, A. V., Plotkin, S. A., & Tardos, É. (1991). Combinatorial algorithms for the generalized circulation problem. Mathematics of operations research, 16(2), 351–381.

    Article  Google Scholar 

  • Goldfarb, D., & Jin, Z. (1996). A faster combinatorial algorithm for the generalized circulation problem. Mathematics of Operations Research, 21, 529–539.

    Article  Google Scholar 

  • Goldfarb, D., Jin, Z., & Orlil, J. B. (1997). Polynomial time highest gain augmenting path algorithms for the generalized circulation problem. Mathematics of Operations Research, 22, 793–802.

    Article  Google Scholar 

  • Gondran, M., & Minoux, M. (1984). Graphs and algorithms. Wiley.

    Google Scholar 

  • Groß, M. (2014). Approximation algorithms for complex network flow over time problems. PhD Thesis, Technical University, Berlin.

  • Groß, M., & Skutella, M. (2012). Generalized maximum flows over time. In International workshop on approximation and online algorithms (pp. 247–260).

  • Gupta, S. P., Pyakurel, U., & Dhamala, T. N. (2022). Generalized dynamic contraflow with non-symmetric transit times. American Journal of Computational and Applied Mathematics, 11(1), 12–17.

    Google Scholar 

  • Gupta, S. P., Pyakurel, U., & Dhamala, T. N. (2023). Multicommodity-flow problem on lossy network with partial lane reversals. Annals of Operation Research. https://doi.org/10.1007/s10479-023-05210-y

    Article  Google Scholar 

  • Hoppe, H., & Tardos, É. (1998). The quickest transshipment problem. Mathematics, 25, 36–62.

    Google Scholar 

  • Jewell, W. S. (1962). New methods in mathematical programming—Optimal flow through networks with gains. Operations Research, 10(4), 476–499.

    Article  Google Scholar 

  • Khanal, D. P., Pyakurel, U., & Dhamala, T. N. (2021). Maximum multicommodity flow with intermediate storage. Mathematical Problems in Engineering. https://doi.org/10.1155/2021/5063207

    Article  Google Scholar 

  • Minieka, E. (1973). Maximal, lexicographic, and dynamic network flows. Operations Research, 21, 517–527.

    Article  Google Scholar 

  • Nath, H. N., Pyakurel, U., & Dhamala, T. N. (2021). Network reconfiguration with orientation dependent travel times. International Journal of Mathematics and Mathematical Sciences. https://doi.org/10.1155/2021/6613622

    Article  Google Scholar 

  • Nath, H. N., Pyakurel, U., Dhamala, T. N., & Dempe, S. (2021). Dynamic network flow location models and algorithms for quickest evacuation planning. Journal of Management and Optimization, 17, 2943–2970.

    Google Scholar 

  • Olver, N., & Végh, L.A. (2017). A simpler and faster strongly polynomial algorithm for generalized flow maximization problem. In Proceeding of the 49th Annual ACM SIGACT Symposium on Theory of Computing (pp. 1100–111). https://doi.org/10.1145/3055399.3055439

  • Onaga, K. (1966). Dynamic programming of optimum flows in lossy communication nets. IEEE Transactions on Circuit Theory, 13(3), 282–287.

    Article  Google Scholar 

  • Onaga, K. (1967). Optimum flows in general communication networks. Journal of the Franklin Institute, 283(4), 308–327.

    Article  Google Scholar 

  • Pyakurel, U., & Dempe, S. (2020). Network flow with intermediate storage: Models and algorithms. SN Operations Research Forum, 1(4), 1–23.

    Article  Google Scholar 

  • Pyakurel, U., & Dempe, S. (2021). Universal maximum flow with intermediate storage for evacuation planning. In I. S. Kotsireas, A. Nagurney, P. M. Pardalos, & A. Tsokas (Eds.), Dynamics of disasters. Springer optimization and its applications. (Vol. 169). Springer. https://doi.org/10.1007/978-3-030-64973-9_14

    Chapter  Google Scholar 

  • Pyakurel, U., & Dhamala, T. N. (2016). Continuous time dynamic contraflow models and algorithms. Advances of Operations Research. https://doi.org/10.1155/2016/7902460

    Article  Google Scholar 

  • Pyakurel, U., & Dhamala, T. N. (2017). Continuous dynamic contraflow approach for evacuation planning. Annals of Operations Research, 253(1), 573–598.

    Article  Google Scholar 

  • Pyakurel, U., Dhamala, T. N., & Dempe, S. (2017). Efficient continuous contraflow algorithms for evacuation planning problems. Annals of Operations Research, 254(1 & 2), 335–364.

    Article  Google Scholar 

  • Pyakurel, U., Hamacher, H. W., & Dhamala, T. N. (2014). Generalized maximum dynamic contraflow on lossy network. International Journal of Operations Research Nepal, 3(1), 27–44.

    Google Scholar 

  • Pyakurel, U., Khanal, D. P., & Dhamala, T. N. (2022). Abstract network flow with intermediate storage for evacuation planning. European Journal of Operational Research. https://doi.org/10.1016/j.ejor.2022.06.054

    Article  Google Scholar 

  • Pyakurel, U., Nath, H. N., & Dhamala, T. N. (2019). Partial contraflow with path reversals for evacuation planning. Annals of Operations Research, 283, 591–612.

    Article  Google Scholar 

  • Radzik, T. (1998). Faster algorithms for the generalized network flow problem. Mathematics of Operations Research, 23, 69–100.

    Article  Google Scholar 

  • Rebennack, S., Arulselvan, A., Elefteriadou, L., & Pardalos, P. M. (2010). Complexity analysis for maximum flow problems with arc reversals. Journal of Combinatorial Optimization, 19(2), 200–216.

    Article  Google Scholar 

  • Shigeno, M. (2004). A survey of combinatorial maximum flow algorithms on a network with gains. Journal of the Operations Research, Society of Japan, 47(4), 244–264.

    Article  Google Scholar 

  • Tardos, É., & Wayne, K. D. (1998). Simple generalized maximum flow algorithms. In 6th international integer programming and combinatorial optimization conferences (pp. 310–324).

  • Truemper, K. (1977). On max flows with gains and pure min-cost flows. SIAM Journal on Applied Mathematics, 32(2), 450–456.

    Article  Google Scholar 

  • Végh, L. A. (2016). A strongly polynomial algorithm for generalized flow maximization problem. Mathematics of Operations Research, 42(1), 179–211. https://doi.org/10.1287/moor.2016.0800

    Article  Google Scholar 

  • Wayne, K. D. (1999). Generalized maximum flow algorithms. PhD Thesis, Cornell University.

Download references

Acknowledgements

The first author (Tanka Nath Dhamala) thanks to the Alexander von Humboldt Foundation for the support of his research visits at TU Bergakademie Freiberg (at different stages and during June 1–August 31, 2023). The third author (Durga Prasad Khanal) thanks to the German Academic Exchange Service—DAAD for Research Grants—Bi-nationally Supervised Doctoral Degrees/Cotutelle at TU Bergakademie Freiberg (October 1, 2021–September 30, 2023) and University Grants Commission Nepal for PhD Research Fellowship (2020–2023). Similarly, the fourth author (Urmila Pyakurel) was supported by the Alexander von Humboldt Foundation under Remote Cooperation Abroad fellowship (March 1–August 31, 2022), during which main part of this paper was conceptualized.

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Authors and Affiliations

  1. Central Department of Mathematics, Tribhuvan University, Kathmandu, Nepal

    Tanka Nath Dhamala, Mohan Chandra Adhikari & Urmila Pyakurel

  2. Saraswati Multiple Campus, Tribhuvan University, Kathmandu, Nepal

    Durga Prasad Khanal

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  1. Tanka Nath Dhamala
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  2. Mohan Chandra Adhikari
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  3. Durga Prasad Khanal
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  4. Urmila Pyakurel
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Correspondence to Tanka Nath Dhamala.

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Tanka Nath Dhamala, Mohan Chandra Adhikari and Durga Prasad Khanal are deeply shocked to report the untimely demise of fourth author Prof. Dr. Urmila Pyakurel and dedicate this work to her, who passed away on April 12, 2023 at the age of 42. She was an energetic Nepalese woman role model with an outstanding research career in mathematics. Her passing leaves the scientific community with a severe loss.

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Dhamala, T.N., Adhikari, M.C., Khanal, D.P. et al. Generalized maximum flow over time with intermediate storage. Ann Oper Res 335, 111–134 (2024). https://doi.org/10.1007/s10479-023-05773-w

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