On the heterogeneous vehicle routing problem under demand uncertainty

In this paper we study the heterogeneous vehicle routing problem under demand uncertainty, on which there has been little research to our knowledge. The focus of the paper is to provide a strong formulation that also easily allows tractable robust and chance-constrained counterparts. To this end, we propose a basic Miller-Tucker-Zemlin (MTZ) formulation with the main advantage that uncertainty is restricted to the right-hand side of the constraints. This leads to compact and tractable counterparts of demand uncertainty. On the other hand, since the MTZ formulation is well known to provide a rather weak linear programming relaxation, we propose to strengthen the initial formulation with valid inequalities and lifting techniques and, furthermore, to dynamically add cutting planes that successively reduce the polyhedral region using a branch-and-cut algorithm. We complete our study with extensive computational analysis with different performance measures on different classes of instances taken from the literature. In addition, using simulation, we conduct a scenario-based risk level analysis for both cases where either unmet demand is allowed or not

Thirty years of heterogeneous vehicle routing

It has been around thirty years since the heterogeneous vehicle routing problem was introduced, and significant progress has since been made on this problem and its variants. The aim of this survey paper is to classify and review the literature on heterogeneous vehicle routing problems. The paper also presents a comparative analysis of the metaheuristic algorithms that have been proposed for these problems

Gemischt-autonome Flotten in der urbanen Logistik

We consider a city logistics application in which a service provider seeks a repeatable plan to transport commodities from distribution centers to satellites. The service provider uses a mixed autonomous fleet that is composed of autonomous vehicles and manually operated vehicles. The autonomous vehicles are only able to travel independently on feasible streets of the heterogeneous infrastructure but elsewhere need to be pulled by manually operated vehicles in platoons. We introduce the service network design problem with mixed autonomous fleets to determine a tactical plan that minimizes the total costs over a medium-term time horizon. The tactical plan determines the size and mix of the fleet, schedules transportation services, and decides on the routing or outsourcing of commodities. We model this problem as an integer program on a time-expanded network and study the impact of different problem characteristics on the solutions. To precisely depict the synchronization requirements of the problem, the time-expanded networks need to consider narrow time intervals. Thus, we develop an exact solution approach based on the dynamic discretization discovery scheme that refines partially time-expanded networks containing only a fraction of the nodes and arcs of the fully time-expanded network. Further methodological contributions of this work include the introduction of valid inequalities, two enhancements that exploit linear relaxations, and a heuristic search space restriction. Computational experiments show that all evaluated variants of the solution approach outperform a commercial solver. For transferring a tactical plan to an operational solution that minimizes the transshipment effort on a given day, we present a post-processing technique that specifically assigns commodities to vehicles and vehicles to platoons. Finally, we solve a case study on a real-world based network resembling the city of Braunschweig, Germany. Analyzing the tactical and operational solutions, we assess the value of using a mixed autonomous fleet and derive practical implications.Wir betrachten eine Anwendung der urbanen Logistik, bei der ein Dienstleister einen wiederholbaren Plan für den Gütertransport von Distributionszentren zu Satelliten anstrebt. Dafür setzt der Dienstleister eine gemischt-autonome Flotte ein, die sich aus autonomen Fahrzeugen und manuell gesteuerten Fahrzeugen zusammensetzt. Die autonomen Fahrzeuge können nur auf bestimmten Straßen der heterogenen Infrastruktur selbstständig fahren, außerhalb dieser müssen sie von manuell gesteuerten Fahrzeugen mittels Platooning gezogen werden. Wir führen das „service network design problem with mixed autonomous fleets“ ein, um einen taktischen Plan zu ermitteln, der die Gesamtkosten über einen mittelfristigen Zeithorizont minimiert. Der taktische Plan bestimmt die Größe und Zusammensetzung der Flotte, legt die Transportdienste fest und entscheidet über das Routing oder das Outsourcing von Gütern. Wir modellieren dieses Problem als ganzzahliges Programm auf einem zeiterweiterten Netzwerk und untersuchen die Auswirkungen verschiedener Problemeigenschaften auf die Lösungen. Um die Synchronisationsanforderungen des Problems präzise darzustellen, müssen die zeiterweiterten Netzwerke kleine Zeitintervalle berücksichtigen. Daher entwickeln wir einen exakten Lösungsansatz, der auf dem Schema des „dynamic discretization discovery“ basiert und partiell zeiterweiterte Netzwerke entwickelt, die nur einen Teil der Knoten und Kanten des vollständig zeiterweiterten Netzwerks enthalten. Weitere methodische Beiträge dieser Dissertation umfassen die Einführung von Valid Inequalities, zweier Erweiterungen, die lineare Relaxationen verwenden, und einer heuristischen Suchraumbegrenzung. Experimente zeigen, dass alle evaluierten Varianten des Lösungsansatzes einen kommerziellen Solver übertreffen. Um einen taktischen Plan in eine operative Lösung zu überführen, die die Umladevorgänge an einem bestimmten Tag minimiert, stellen wir eine Post-Processing-Methode vor, mit der Güter zu Fahrzeugen und Fahrzeuge zu Platoons eindeutig zugeordnet werden. Schließlich lösen wir eine Fallstudie auf einem realitätsnahen Netzwerk, das der Stadt Braunschweig nachempfunden ist. Anhand der taktischen und operativen Lösungen bewerten wir den Nutzen einer gemischt-autonomen Flotte und leiten Implikationen für die Praxis ab

A solution approach for multi-trip vehicle routing problems with time windows, fleet sizing, and depot location

RÉSUMÉ: We present a solution approach for a multi-trip vehicle routing problem with time windows in which the locations of a prescribed number of depots and the fleet sizes must also be optimized. Given the complexity of the task, we divide the problem into subproblems that are solved sequentially. First, we address strategic decisions, which are solved once and remain constant thereafter. Depots are allocated by solving a p-median problem and fleet sizes are determined by identifying the vehicle requirements of several worst-case demand instances. Then, we address the operational planning aspect: optimizing the vehicle routes on a daily basis to satisfy the fluctuating customer demand. We assign customers to depots based on distance and “routing effort,” and for the routing problem we combine a tailor-made branch-and-cut algorithm with a heuristic consisting of a route construction phase and packing of routes into vehicle trips. Our strategic decision models are robust in the sense that when applied to unseen data, all customers could be visited with the allocated fleet sizes and depot locations. Our operational routing methods are both time and cost-effective. The exact method yields acceptable optimality gaps in 20 min and the heuristic runs in less than 2min, finding optimal or near-optimal solutions for small instances. Finally, we explore the trade-off between depot and fleet costs, and routing costs to make recommendations on the optimal number of depots. Our solution approach was entered into the 12th AIMMS-MOPTA Optimization Modeling Competition and was awarded the first prize

The multi-depot VRP with vehicle interchanges

In real-world logistic operations there are a lot of situations that can be exploited to get better operational strategies. It is important to study these new alternatives, because they can represent significant cost reductions to the companies working with physical distribution. This thesis defines the Multi-Depot Vehicle Routing Problem with Vehicle Interchanges (MDVRPVI). In this problem, both vehicle capacities and duration limits on the routes of the drivers are imposed. To favor a better utilization of the available capacities and working times, it is allowed to combine pairs of routes at predefined interchange locations. The objective of this thesis is to analyze and solve the Multi-Depot Vehicle Routing Problem adding the possibility to interchange vehicles at predefined points. With this strategy, it is possible to reduce the total costs and the number of used routes with respect to the classical approach: The Multi-Depot Vehicle Routing Problem (MDVRP). It should be noted that the MDVRP is more challenging and sophisticated than the single-depot Vehicle Routing Problem (VRP). Besides, most exact algorithms for solving the classical VRP are difficult to adapt in order to solve the MDVRP (Montoya-Torres et al., 2015). From the complexity point of view, the MDVRPVI is NP-Hard, since it is an extension of the classical problem, which is already NP-Hard. We present a tight bound on the costs savings that can be attained allowing interchanges. Three integer programming formulations are proposed based on the classical vehicle-flow formulations of the MDVRP. One of these formulations was solved with a branch-and-bound algorithm, and the other two formulations, with branch-and-cut algorithms. Due to its great symmetry, the first formulation is only able to solve small instances. To increase the dimension of the instances used, we proposed two additional formulations that require one or more families of constraints of exponential size. In order to solve these formulations, we had to design and implement specific branch-and-cut algorithms. For these algorithms we implemented specific separation methods for constraints that had not previously been used in other routing problems. The computational experience performed evidences the routing savings compared with the solutions obtained with the classical approach and allows to compare the efficacy of the three solution methods proposed.En les operacions logístiques del món real es donen situacions que poden ser explotades per obtenir millors estratègies operacionals. És molt important estudiar aquestes noves alternatives, perquè poden representar una reducció significativa de costos per a les companyies que treballen en distribució de mercaderies. En aquesta tesi es defineix el Problema d'Enrutament de Vehicles amb Múltiples Dipòsits i Intercanvi de Vehicles (MDVRPVI). En aquest problema, es consideren tant la capacitat dels vehicles com els límits de duració de les rutes dels conductors. Per tal de millorar la utilització de les capacitats i temps de treball disponibles, es permet combinar parelles de rutes en punts d'intercanvi predefinits. L'objectiu d'aquesta tesi és analitzar i resoldre el problema d'Enrutament de Vehicles amb Múltiples Dipòsits, on es permet l'intercanvi de vehicles. Amb aquesta estratègia, és possible reduir els costos totals i el nombre de les rutes utilitzades respecte l'enfocament clàssic: el problema d'Enrutament de Vehicles amb Múltiples Dipòsits (MDVRP). Cal assenyalar que el MDRVP és més desafiant i sofisticat que el problema d'Enrutament de Vehicles d'un únic dipòsit (VRP). A més, molts algoritmes exactes per resoldre el VRP clàssic son complicats d'adaptar per resoldre el MDVRP (Montoya-Torres et al., 2015). Des del punt de vista de la complexitat, el MDRVPVI és NP-Dur, perquè és una extensió del problema clàssic, que també ho és. Presentem una cota ajustada de l'estalvi en els costos de distribució que es pot obtenir permetent els intercanvis. Es proposen tres formulacions de programació sencera basades en la formulació clàssica “vehicle-flow” del MDVRP. La primera formulació, degut a la seva grandària i la seva simetria, només permet resoldre instàncies molt petites. Per augmentar la dimensió de les instàncies abordables, es proposen dues formulacions addicionals que requereixen una o vàries famílies de restriccions de mida exponencial. Per això, per tal de resoldre el problema amb aquestes formulacions, ha calgut dissenyar i implementar sengles algorismes de tipus branch-and-cut. En aquests algorismes s'han implementat mètodes de separació específics per a les restriccions que no s'havien utilitzat prèviament en altres problemes de rutes. L’experiència computacional realitzada evidencia els estalvis obtinguts comparació amb les solucions corresponents l'enfocament clàssic. També es compara l’eficàcia dels tres mètodes propostes a l'hora de resoldre el problema.Postprint (published version