We study a new network design model combining ring and tree structures under capacity constraints. The solution topology of this capacitated ring tree problem (CRTP) is based on ring trees which are the union of trees and 1-trees. The objective is the minimization of edge costs but could also incorporate other types of measures. This overall problem generalizes prominent capacitated vehicle routing and Steiner tree problem variants. Two customer types have to be connected to a distributor ensuring single and double node connectivity, respectively, while installing optional Steiner nodes. The number of ring trees and the number of customers supplied by such a single structure are bounded. After embedding this combinatorial optimization model in existing network design concepts, we develop a mathematical formulation and introduce several valid inequalities for the CRTP that are separated in our exact algorithm. Additionally, we use local search techniques to tighten the obtained upper bounds. For a set of literature-derived instances we consider various reliability scenarios and present computational results.

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A ring tree is a tree graph with an optional additional edge that closes a unique cycle. Such a cycle is called a ring and the nodes on it are called ring nodes. The capacitated ring tree problem (CRTP) asks for a network of minimal overall edge cost that connects given customers to a depot by ring trees. Ring trees are required to intersect in the depot which has to be either a ring node in a ring tree or a node of degree one if the ring tree does not contain a ring. Customers are predefined as of type 1 or type 2. The type 2 customers have to be ring nodes, whereas type 1 customers can be either ring nodes or nodes in tree sub-structures. Additionally, optional Steiner nodes are given which can be used as intermediate network nodes if advantageous. Capacity constraints bound both the number of the ring trees as well as the number of customers allowed in each ring tree. In this paper we present first advanced neighborhood structures for the CRTP. Some of them generalize existing concepts for the TSP and the Steiner tree problem, others are CRTP-specific. We also describe models to explore these multi-node and multi-edge exchange neighborhoods in one or more ring trees efficiently. Moreover, we embed these techniques in a heuristic multi-start framework and show that it produces high quality results for small and medium size literature instances.

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