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Maximum flows in generalized processing networks

Author

Listed:
  • Michael Holzhauser

    (University of Kaiserslautern)

  • Sven O. Krumke

    (University of Kaiserslautern)

  • Clemens Thielen

    (University of Kaiserslautern)

Abstract

Processing networks (cf. Koene in Minimal cost flow in processing networks: a primal approach, 1982) and manufacturing networks (cf. Fang and Qi in Optim Methods Softw 18:143–165, 2003) are well-studied extensions of traditional network flow problems that allow to model the decomposition or distillation of products in a manufacturing process. In these models, so called flow ratios $$\alpha _e \in [0,1]$$ α e ∈ [ 0 , 1 ] are assigned to all outgoing edges of special processing nodes. For each such special node, these flow ratios, which are required to sum up to one, determine the fraction of the total outgoing flow that flows through the respective edges. In this paper, we generalize processing networks to the case that these flow ratios only impose an upper bound on the respective fractions and, in particular, may sum up to more than one at each node. We show that a flow decomposition similar to the one for traditional network flows is possible and can be computed in strongly polynomial time. Moreover, we show that there exists a fully polynomial-time approximation scheme (FPTAS) for the maximum flow problem in these generalized processing networks if the underlying graph is acyclic and we provide two exact algorithms with strongly polynomial running-time for the problem on series–parallel graphs. Finally, we study the case of integral flows and show that the problem becomes $${\mathcal {NP}}$$ NP -hard to solve and approximate in this case.

Suggested Citation

  • Michael Holzhauser & Sven O. Krumke & Clemens Thielen, 2017. "Maximum flows in generalized processing networks," Journal of Combinatorial Optimization, Springer, vol. 33(4), pages 1226-1256, May.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:4:d:10.1007_s10878-016-0031-y
    DOI: 10.1007/s10878-016-0031-y
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    References listed on IDEAS

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    1. Prahalad Venkateshan & Kamlesh Mathur & Ronald H. Ballou, 2008. "An efficient generalized network-simplex-based algorithm for manufacturing network flows," Journal of Combinatorial Optimization, Springer, vol. 15(4), pages 315-341, May.
    2. Haiyan Lu & Enyu Yao & Liqun Qi, 2006. "Some further results on minimum distribution cost flow problems," Journal of Combinatorial Optimization, Springer, vol. 11(4), pages 351-371, June.
    3. Chou-Hong J. Chen & Michael Engquist, 1986. "A Primal Simplex Approach to Pure Processing Networks," Management Science, INFORMS, vol. 32(12), pages 1582-1598, December.
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    Cited by:

    1. Jan Boeckmann & Clemens Thielen, 2023. "New Ways in Municipal Flood Mitigation: a Mixed-Integer Programming Approach and its Practical Application," SN Operations Research Forum, Springer, vol. 4(4), pages 1-68, December.

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