An exact approach for the vehicle routing problem with two-dimensional loading constraints

We consider a special case of the symmetric capacitated vehicle routing problem, in which a fleet of K identical vehicles must serve n customers, each with a given demand consisting in a set of rectangular two-dimensional weighted items. The vehicles have a two-dimensional loading surface and a maximum weight capacity. The aim is to find a partition of the customers into routes of minimum total cost such that, for each vehicle, the weight capacity is taken into account and a feasible two-Dimensional allocation of the items into the loading surface exists. The problem has several practical applications in freight transportation, and it is -hard in the strong sense. We propose an exact approach, based on a branch-and-cut algorithm, for the minimization of the routing cost that iteratively calls a branch-and-bound algorithm for checking the feasibility of the loadings. Heuristics are also used to improve the overall performance of the algorithm. The effectiveness of the approach is shown by means of computational results

Vehicle routing with multi-dimensional loading constraints

Zwei der wichtigsten Problemstellungen in der Transportlogistik behandeln einerseits das Verladen von Produkten auf LKWs und andererseits die ressourceneffiziente Belieferung der Kunden auf dem gegebenen Straßennetz. Bis dato wurden diese zwei Probleme mit Hilfe von kombinatorischer Optimierung getrennt von einander behandelt und es existieren zahlreiche Publikationen zu beiden Themen in den einschlägigen Fachzeitschriften. Erst in den letzten drei Jahren wurde einem integrierten Ansatz, der beide Problemstellungen zu einem Optimierungsproblem vereint betrachtet. Somit werden die Bestellungen einzelner Kunden nicht bloß über ihre Gewichte, sondern auch über ihre Abmessungen definiert. Der klare Vorteil dieses Ansatzes liegt darin, dass die einzelnen LKW Routen auch tatsächlich so gefahren werden können, da die tatsächliche Beladung auch berücksichtigt wurde. Andererseits steigt die kombinatorische Komplexität drastisch, weil das kapazitierte Vehicle Routing Problem (CVRP) mit Bin Packing Problemen (BPP) kombiniert wird und beide Probleme für sich alleine NP schwer sind. Diese Dissertation präsentiert drei verschiedene Probleme, die sich neben der Frage welches Fahrzeug beliefert welchen Kunden auch der Frage widmet, wie die bestellten Produkte auf den LKW geladen werden können. - Das Multi-Pile Vehicle Routing Problem (MP-VRP) bindet in das klassische CVRP eine Beladekomponente ein, die zwischen eindimensionalem und zweidimensionalem Bin Packing Problem angesiedelt ist. Die Problemstellungen wurden durch einen österreichischen Holzzulieferer motiviert. - Beim kapazitierten Vehicle Routing Problem mit zweidimensionalen Beladenebenbedingungen (2L-CVRP) bestellt jeder Kunden rechteckige Objekte, welche auf der rechteckigen Beladefläche des LKWs untergebracht werden müssen. - Das allgemeinste Beladeproblem stellt das dreidimensionale Bin Packing Problem dar. Hier bestellt jeder Kunde dreidimensionale Objekte, welche auf dem dreidimensionalen Laderaum des LKWs untergebracht werden müssen. Das klassische dreidimensionale Bin Packing Problem wird durch zusätzliche Beladenebenbedingungen erweitert. Momentan gibt es zu diesen kombinierten Problemen nur wenige Publikationen. Exakte Ansätze gibt es momentan nur zwei, einen für das MP-VRP (hier können Probleme bis zu 50 Kunden gelöst werden) und für das 2L-CVRP (hier können Probleme bis zu 30 Kunden exakt gelöst werden). Für Realweltanwendungen müssen jedoch Heuristiken gefunden werden, welche größere Probleminstanzen lösen können. In dieser Arbeit wird für alle drei Problemstellungen ein Ameisenalgorithmus verwendet und mit bestehenden Lösungsansätzen aus dem Bereich der Tabu-Suche (TS) verglichen. Es wird gezeigt, dass der präsentierte Ameisenansatz für die zur Verfügung stehenden Benchmarkinstanzen die besten Ergebnisse liefert. Darüber hinaus wird der Einfluss verschiedener Beladenebenbedingungen auf die Lösungsgüte untersucht, was eine wichtige Entscheidungsgrundlage für Unternehmen darstellt.Two of the most important problems in distribution logistics concern the loading of the freight into the vehicles, and the successive routing of the vehicles along the road network, with the aim of satisfying the demands of the clients. In the combinatorial optimization field, these two loading and routing problems have been studied intensively but separately yielding a large number of publications either for routing or packing problems. Only in recent years some attention has been brought to their combined optimization. The obvious advantage is that, by considering the information on the freight to be loaded, one can construct more appropriate routes for the vehicles. The counterpart is that the combinatorial difficulty of the problem increases consistently. One must not forget that both the vehicle routing problem and the bin packing problem are NP hard problems! This thesis presents three different problems concerning the combination of routing and loading (packing) problems. - The Multi-Pile Vehicle Routing Problem (MP-VRP) incorporates an interesting loading problem situated between one dimensional and two dimensional bin packing. This problem has been inspired by a real world application of an Austrian wood distributing company. - The Capacitated Vehicle Routing Problem with Two-Dimensional Loading Constraints (2L-CVRP) augments the classical Capacitated Vehicle Routing Problem by requiring the satisfaction of general two dimensional loading constraints. This means that customers order items represented by rectangles that have to be feasibly placed on the rectangular shaped loading surface of the used vehicles. - The most general packing problem to be integrated is the Three Dimensional Bin Packing Problem (3DBPP) resulting in the Capacitated Vehicle Routing Problem with Three-Dimensional Loading Constraints (3L-CVRP). Here the order of each customer consists of cuboid shaped items that have to be feasibly placed on the loading space of the vehicle. A feasible placement is influenced by additional constraints that extend the classical 3DBPP. Concerning the literature solving these problems with exact methods it becomes clear that this is only possible to some very limited extent (e.g.: the MP-VRP can be solved up to 50, the 2L-CVRP can be solved exact up to 30 customers, for the 3L-CVRP no exact approach exists). Nevertheless for real world applications the problem instances are much larger which justifies the use of (meta-)heuristics. The rank-based Ant System was modified and extended to solve the combined problem by integrating different packing routines. The designed methods outperform the existing techniques for the three different problem classes. The influence of different loading constraints on the objective value is investigated/is intensively studied to support the decision makers of companies facing similar problems

A simheuristic for routing electric vehicles with limited driving ranges and stochastic travel times

Green transportation is becoming relevant in the context of smart cities, where the use of electric vehicles represents a promising strategy to support sustainability policies. However the use of electric vehicles shows some drawbacks as well, such as their limited driving-range capacity. This paper analyses a realistic vehicle routing problem in which both driving-range constraints and stochastic travel times are considered. Thus, the main goal is to minimize the expected time-based cost required to complete the freight distribution plan. In order to design reliable Routing plans, a simheuristic algorithm is proposed. It combines Monte Carlo simulation with a multi-start metaheuristic, which also employs biased-randomization techniques. By including simulation, simheuristics extend the capabilities of metaheuristics to deal with stochastic problems. A series of computational experiments are performed to test our solving approach as well as to analyse the effect of uncertainty on the routing plans.Peer Reviewe

On the use of biased-randomized algorithms for solving non-smooth optimization problems

Soft constraints are quite common in real-life applications. For example, in freight transportation, the fleet size can be enlarged by outsourcing part of the distribution service and some deliveries to customers can be postponed as well; in inventory management, it is possible to consider stock-outs generated by unexpected demands; and in manufacturing processes and project management, it is frequent that some deadlines cannot be met due to delays in critical steps of the supply chain. However, capacity-, size-, and time-related limitations are included in many optimization problems as hard constraints, while it would be usually more realistic to consider them as soft ones, i.e., they can be violated to some extent by incurring a penalty cost. Most of the times, this penalty cost will be nonlinear and even noncontinuous, which might transform the objective function into a non-smooth one. Despite its many practical applications, non-smooth optimization problems are quite challenging, especially when the underlying optimization problem is NP-hard in nature. In this paper, we propose the use of biased-randomized algorithms as an effective methodology to cope with NP-hard and non-smooth optimization problems in many practical applications. Biased-randomized algorithms extend constructive heuristics by introducing a nonuniform randomization pattern into them. Hence, they can be used to explore promising areas of the solution space without the limitations of gradient-based approaches, which assume the existence of smooth objective functions. Moreover, biased-randomized algorithms can be easily parallelized, thus employing short computing times while exploring a large number of promising regions. This paper discusses these concepts in detail, reviews existing work in different application areas, and highlights current trends and open research lines