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Optimal flow control in acyclic networks with uncontrollable routings and precedence constraints

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Abstract

This paper introduces a novel optimal flow control problem that seeks to convey a specified amount of fluid to each of the nodes of an acyclic digraph with a single source node, while minimizing the total amount of fluid inducted into the network. Two factors complicating the aforementioned task are (i) the presence of nodes with uncontrollable routing of the traversing flow and (ii) a set of precedence constraints regarding the satisfaction of the nodal fluid requirements. It is shown that the considered problem can be naturally formulated as a continuous-time optimal control problem that can be reduced to a hybrid optimal control problem with controlled switching. This property subsequently enables the solution of the considered problem through a Mixed Integer Programming formulation. Additional results in the paper establish the NP-hardness of the considered problem, highlight its affinity to some well known scheduling problems, and offer guidelines that can alleviate the increased problem complexity.

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Notes

  1. These distributions are determined by the consumer behavior with respect to the considered product, which is an uncontrollable and unobservable part of the entire process.

  2. Or a sampling schedule.

  3. The assumption | ∙  V| = 1 can also be satisfied for problem instances with | ∙  V| > 1 through the addition of a new dummy node; the details are left to the reader.

  4. We also notice that when the network of Fig. 1 is perceived in the context of the stochastic DP problem that motivated this work, the controllable (white) nodes correspond to decision nodes, and the arcs emanating from them are the available decisions. On the other hand, the uncontrollable (black) nodes model the impact of the randomness that might determine the outcome of some of the applied decisions. Obviously, in the context of such an interpretation of the considered problem and its flow dynamics, uncontrollable nodes should have a zero fluid requirement (since the sampling process takes place only at the decision nodes).

  5. An explanation of this effect can be found in the discussion provided at the end of the following section.

  6. This time is known as the schedule makespan in the relevant theory.

  7. Furthermore, it is easy to see that when defined on such a flat graph structure, the optimal control problem of Eqs. 1–6 has a very simple optimal solution, and therefore, there is no need to resort to the solution of the MIP formulation of Section 3 and the deployment of the mode graph \({\cal G}\).

  8. The insignificance of the nodes with zero fluid requirements for the determination of the optimal solution of the MIP formulation developed in Section 3, is manifested in the numerical example provided in that section, by the fact that modes ν 13, ν 24 and ν 33 correspond to zero total flows. These three modes correspond respectively to the completion of the (zero) fluid requirements for nodes 9, 5 and 3 in the graph G of Fig. 1.

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Authors and Affiliations

  1. School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Spyros Reveliotis

  2. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, USA

    Theologos Bountourelis

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  1. Spyros Reveliotis
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  2. Theologos Bountourelis
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Corresponding author

Correspondence to Spyros Reveliotis.

Additional information

This work was partially supported by the NSF grant CMMI-0619978.

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Reveliotis, S., Bountourelis, T. Optimal flow control in acyclic networks with uncontrollable routings and precedence constraints. Discrete Event Dyn Syst 21, 499–518 (2011). https://doi.org/10.1007/s10626-011-0112-0

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