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Generalized Hose uncertainty in single-commodity robust network design

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Abstract

Single-commodity network design considers an edge-weighted, undirected graph with a supply/demand value at each node. It asks for minimum weight capacities such that each node can exactly send (or receive) its supply (or demand). In the robust variant, the supply or demand values may assume any realization in a given uncertainty set. One popular set is the well-known Hose polytope, which specifies an interval for the supply/demand at each node, while ensuring that the total supply and demand are balanced across the whole network. While previous work has established the Hose uncertainty set as a tractable choice, it can yield unnecessarily expensive solutions because it admits many unlikely supply and demand scenarios. In this paper, we propose a generalization of the Hose polytope that more realistically captures existing interdependencies among nodes in real life networks, and we show how to extend the state-of-the-art cutting plane algorithm for solving the single-commodity robust network design problem in view of this new uncertainty set. Our computational studies across multiple robust network design instances illustrate that the new set can provide significant cost savings without sacrificing numerical tractability.

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Notes

  1. For ease of exposition, we assume that each node is contained in some partition. This is w.l.o.g., since whenever we do not want to impose cumulative bounds on certain nodes, we can define a partition \(X \subseteq V\) that contains all these nodes and simply postulate trivial bounds \(B^L := b^L(X)\) and \(B^U := b^U(X)\).

  2. If that were not the case, i.e.,if \(b^*(S) < 0\), we could have equivalently considered the complement set, \(\overline{S}\), which exhibits the same absolute value and satisfies the requirement \(b^*(\overline{S}) > 0\) (recall that, by construction, \(b^*(S) + b^*(\overline{S}) = b^*(V) = 0\)).

  3. Even though there should always be at least one scenario in any practical setting, the non-emptiness of (3) can be readily checked with any LP solver. Furthermore, for instances in which (3) is indeed empty, it can be easily shown that the trivial solution that assigns zero capacity to all edges is optimum.

  4. More precisely, we chose \(\beta \in \{3,5,8\}\) for \(|V|=50\), \(\beta \in \{5,10,16\}\) for \(|V|=100\), \(\beta \in \{8,16,24\}\) for \(|V|=150\), and \(\beta \in \{10,21,32\}\) for \(|V|=200\), which approximately resulted in the desired graph densities.

  5. We chose \(\overline{b}_i \in \{0,\ldots ,5\}\) uniformly at random and chose \(\hat{b}_i\) from a geometric distribution with mean 2.

  6. We independently chose \(\overline{b}_i, \hat{b}_i \in \{0,\ldots ,5\}\) uniformly at random.

  7. We remark that, since computing \(WCD(\xi )\) exactly is cumbersome, we approximated it by sampling 15,000 points from within the uncertainty set \(\mathcal{H}^+\) and by solving the resulting deterministic instances. All points corresponded to random vertices of \(\mathcal{H}^+\), which we located by optimizing randomly chosen linear objective functions over \(\mathcal{H}^+\). These objective functions featured coefficients that were chosen uniformly at random from the interval \([-1,+1]\).

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Acknowledgements

Daniel R. Schmidt was supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD).

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Authors and Affiliations

  1. Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213, USA

    Chrysanthos E. Gounaris

  2. Institut für Informatik, Universität zu Köln, Albertus-Magnus-Platz, 50923, Cologne, Germany

    Daniel R. Schmidt

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  1. Chrysanthos E. Gounaris
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Gounaris, C.E., Schmidt, D.R. Generalized Hose uncertainty in single-commodity robust network design. Optim Lett 14, 925–944 (2020). https://doi.org/10.1007/s11590-019-01427-8

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