Layered graph approaches for combinatorial optimization problems

Extending the concept of time-space networks, layered graphs associate information about one or multiple resource state values with nodes and arcs. While integer programming formulations based on them allow to model complex problems comparably easy, their large size makes them hard to solve for non-trivial instances. We detail and classify layered graph modeling techniques that have been used in the (recent) scientific literature and review methods to successfully solve the resulting large-scale, extended formulations. Modeling guidelines and important observations concerning the solution of layered graph formulations by decomposition methods are given together with several future research directions

On solving constrained tree problems and an adaptive layers framework

Zsfassung in dt. SpracheIn this thesis we consider selected combinatorial optimization problems arising in the field of network design. In many of these problems there is a central server sending out information to a set of recipients. A common objective is then to choose connections in the network minimizing the total costs. Besides this, current applications, e.g. in multimedia, usually ask for additional quality-of-service constraints, e.g. limiting the communication delay between the central server and the clients. In general, these problems can be modeled on a graph and in many cases an optimal solution corresponds to a rooted tree with minimum costs satisfying all the given constraints. The most relevant of these optimization problems are NP-hard and thus - provided that P is not equal to NP - it is usually not possible to obtain proven optimal solutions for medium- to large-sized problem instances. Therefore, heuristic approaches yielding high quality but in general sub-optimal solutions are of high practical interest. We present sophisticated preprocessing methods to reduce the problem size and new heuristic as well as exact state-of-the-art solution approaches for several of these optimization problems. The proposed heuristics include several construction and improvement methods embedded in various metaheuristics. Small- to medium-sized problem instances are tackled with exact algorithms, mostly concentrating on mathematical programming methods. We compare different modeling approaches, among them a transformation to a layered graph. Finally, we introduce a generally-applicable iterative adaptive layers framework which tries to partly overcome major computational issues of layered graph approaches.17

Exact methods for the traveling salesman problem with drone

Efficiently handling last-mile deliveries becomes more and more important nowadays. Using drones to support classical vehicles allows improving delivery schedules as long as efficient solution methods to plan last-mile deliveries with drones are available. We study exact solution approaches for some variants of the traveling salesman problem with drone (TSP-D) in which a truck and a drone are teamed up to serve a set of customers. This combination of truck and drone can exploit the benefits of both vehicle types: The truck has a large capacity but usually low travel speed in urban areas; the drone is faster and not restricted to street networks, but its range and carrying capacity are limited. We propose a compact mixed-integer linear program (MILP) for several TSP-D variants that is based on timely synchronizing truck and drone flows; such an MILP is easy to implement but nevertheless leads to competitive results compared with the state-of-the-art MILPs. Furthermore, we introduce dynamic programming recursions to model several TSP-D variants. We show how these dynamic programming recursions can be exploited in an exact branch-and-price approach based on a set partitioning formulation using ng-route relaxation and a three-level hierarchical branching. The proposed branch-and-price can solve instances with up to 39 customers to optimality outperforming the state-of-the-art by more than doubling the manageable instance size. Finally, we analyze different scenarios and show that even a single drone can significantly reduce a route's completion time when the drone is sufficiently fast

Overview of Optimization Problems in Electric Car-Sharing System Design and Management

International audience; Car-sharing systems are increasingly employing environmentally-friendly electric vehicles. The design and management of Ecar-sharing systems poses several additional challenges with respect to those based on traditional combustion vehicles, mainly related with the limited autonomy allowed by current battery technology. We review the main optimization problems arising in Ecar-sharing systems at strategic, tactical and operational levels, and discuss the existing approaches often developed for similar problems, for example in car-sharing systems with traditional vehicles. We also outline open problems and fruitful research directions

Extended formulations and branch-and-cut algorithms for the black-and-white traveling salesman problem

In this paper we study Integer Linear Programming models and develop branch-and-cut algorithms to solve the Black-and-White Traveling Salesman Problem (BWTSP) (Bourgeois, Laporte, & Semet, 2003) which is a variant of the well known Traveling Salesman Problem (TSP). Two strategies to model the BWTSP have been used in the literature. The problem is either modeled on the original graph as TSP using a single set of binary edge variables and with additional non-trivial hop and distance constraints between every pair of black nodes (see Ghiani, Laporte, & Semat, 2006) or as a sequence of constrained paths composed of white nodes connecting pairs of black nodes (see Muter, 2015). In this paper, we study and develop an intermediate approach based on the observation that it is sufficient to guarantee the required distance (and hop) limit of the path from a given black node to the next black node without explicitly stating which one it is. Thus, instead of stating the two constraints (after adding appropriately defined variables) for each pair of black nodes, they are stated for each black node only (that represents the source of each path). Based on this idea we develop several variants of position- and distance-dependent reformulations together with corresponding layered graph representations. Branch-and-cut algorithms are developed for all proposed formulations and empirically compared by an extensive computational study. The obtained results allow us to provide insights into individual advantages and disadvantages of the different models

Exact Approaches for Network Design Problems with Relays

In this article we consider the network design problem with relays (NDPR), which gives answers to some important strategic design questions in telecommunication network design. Given a family of origin-destination pairs and a set of existing links these questions are as follows: (1) What are the optimal locations for signal regeneration devices (relays) and how many of them are needed? (2) Could the available infrastructure be enhanced by installing additional links in order to reduce the travel distance and therefore reduce the number of necessary relays? In contrast to previous work on the NDPR, which mainly focused on heuristic approaches, we discuss exact methods based on different mixed-integer linear programming formulations for the problem. We develop branch-and-price and branch-price-and-cut algorithms that build upon models with an exponential number of variables (and constraints). In an extensive computational study, we analyze the performance of these approaches for instances that reflect different real-world settings. Finally, we also point out the relevance of the NDPR in the context of electric mobility