Stronger multi-commodity flow formulations of the capacitated vehicle routing problem

The Capacitated Vehicle Routing Problem is a much-studied (and strongly NP-hard) combinatorial optimization problem, for which many integer programming formulations have been proposed. We present two new multi-commodity flow (MCF) formulations, and show that they dominate all of the existing ones, in the sense that their continuous relaxations yield stronger lower bounds. Moreover, we show that the relaxations can be strengthened, in pseudo-polynomial time, in such a way that all of the so-called knapsack large multistar (KLM) inequalities are satisfied. The only other relaxation known to satisfy the KLM inequalities, based on set partitioning, is strongly NP-hard to solve. Computational results demonstrate that the new MCF relaxations are significantly stronger than the previously known ones

A Solution Proposal to Vehicle Routing Problem with Integer Linear Programming: A Distributor Company Sample

It was aimed to minimize the total distance of the routes under the capacity constraint of the routes that a distributor company has drawn in the direction of the demands. To this end, a route to Gebze-based steel production and distribution was drawn up to meet all the demands of a fabrication plant. In order to determine the minimum total distance routes, the solution recommendation by adapting the Capacity Constrained Vehicle Routing Problem (CVRP) which is one of the basic route problems using Branch and Cut algorithm of 0-1 Integer Linear Programming (ILP) was introduced. Distances between the nodes that make up the route are measured via Google Maps. Optimal solutions were obtained by using LINDO computer software to solve the problem.

Multi-depot rural postman problems

The final publication is available at Springer via http://dx.doi.org/10.1007/s11750-016-0434-zThis paper studies multi-depot rural postman problems on an undirected graph. These problems extend the well-known undirected rural postman problem to the case where there are several depots instead of just one. Linear integer programming formulations that only use binary variables are proposed for the problem that minimizes the overall routing costs and for the model that minimizes the length of the longest route. An exact branch-and-cut algorithm is presented for each considered model, where violated constraints of both types are separated in polynomial time. Despite the difficulty of the problems, the numerical results from a series of computational experiments with various types of instances illustrate a quite good behavior of the algorithms. When the overall routing costs are minimized, over 43 % of the instances were optimally solved at the root node, and 95 % were solved at termination, most of them with a small additional computational effort. When the length of the longest route is minimized, over 25 % of the instances were optimally solved at the root node, and 99 % were solved at termination.Peer ReviewedPostprint (author's final draft

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

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

Problemas de rotas com custos cumulativos em distribuição logística

Mestrado em Controlo de Gestão e dos NegóciosNesta dissertação aborda-se um importante problema de distribuição logística, o problema da determinação de rotas ótimas designadas na literatura da especialidade por Vehicle Routing Problem (VRP). Estamos perante um mercado cada vez mais competitivo, em que a satisfação das necessidades dos clientes aparece em primeiro lugar, assim, o tempo de entregas toma especial importância, mas do mesmo modo a otimização de recursos e redução de custos é cada vez mais importante para a sobrevivência das organizações. Esta dissertação apresenta, num contexto de logística, o problema de otimização de rotas com custos cumulativos, designado na literatura da especialidade por Cumulative Vehicle Routing Problem (CumVRP). Este problema, para além de permitir uma melhor otimização de rotas atendendo aos seus custos num todo, permite também, face a um VRP normal, obter o tempo mínimo de chegada aos clientes. E o problema mais indicado em termos de Logística Humanitária, em que é preciso gerir um conjunto de processos e sistemas envolvidos na mobilização de pessoas, recursos, habilidades e conhecimentos para socorrer vitimas de desastres, com o objetivo de estabelecer rotas que permitam no menor tempo possível chegar às áreas afetadas afim de poder prestar o socorro necessário.This dissertation addresses an important problem of logistic distribution, the problem of determination of optimal routes designated in the specialty literature for Vehicle Routing Problem (VRP). We are facing an increasingly competitive market, in which the satisfaction of customers’ needs is vital, thus the delivery time takes special importance, but likewise the resource optimization and cost reduction is increasingly important for the survival of organizations. This dissertation presents, in a context of logistics, the route optimization problem with cumulative costs, designated in the specialty literature by Cumulative Vehicle Routing Problem (CumVRP). This problem, in addition to allow a best route optimization in view of its costs in all, allows, in face of a normal VRP, the minimization of the arrival time to clients. This dissertation presents, in a context of logistics, the route optimization problem with cumulative costs, designated in the specialty literature by Cumulative Vehicle Routing Problem (CumVRP). This problem, in addition to allow a best route optimization in view of its costs in all, allows, in the face of a normal VRP, the minimization of arrival time to clients. Is the most suitable in terms of humanitarian logistics, where is necessary to manage a set of processes and systems involved in mobilizing people, resources, skills and knowledge to assist victims of disasters, with the aim of establishing routes that allow in the shortest time possible to reach the affected areas in order to be able to provide the help necessary.info:eu-repo/semantics/publishedVersio

Location of charging stations in electric car sharing systems

Electric vehicles are prime candidates for use within urban car sharing systems, both from economic and environmental perspectives. However, their relatively short range necessitates frequent and rather time-consuming recharging throughout the day. Thus, charging stations must be built throughout the system's operational area where cars can be charged between uses. In this work, we introduce and study an optimization problem that models the task of finding optimal locations and sizes for charging stations, using the number of expected trips that can be accepted (or their resulting revenue) as a gauge of quality. Integer linear programming formulations and construction heuristics are introduced, and the resulting algorithms are tested on grid-graph-based instances, as well as on real-world instances from Vienna. The results of our computational study show that the best-performing exact algorithm solves most of the benchmark instances to optimality and usually provides small optimality gaps for the remaining ones, whereas our heuristics provide high-quality solutions very quickly. Our algorithms also provide better solutions than a sequential approach that considers strategic and operational decisions separately. A cross-validation study analyzes the algorithms' performance in cases where demand is uncertain and shows the advantage of combining individual solutions into a single consensus solution, and a simulation study investigates their behavior in car sharing systems that provide their customers with more flexibility regarding vehicle selection

A p-step formulation for the capacitated vehicle routing problem

We introduce a _p_-step formulation for the capacitated vehicle routing problem (CVRP). The parameter _p_ indicates the length of partial paths corresponding to the used variables. This provides a family of formulations including both the traditional arc-based and path-based formulations. Hence, it is a generalization which unifies arc-based and path-based formulations, while also providing new formulations. We show that the LP bound of the _p_-step formulation is increasing in _p_, although not monotonically. Furthermore, we prove that computing the set partitioning bound is NP-hard. This is a meaningful result in itself, but combined with the _p_-step formulation this also allows us to show that there does not exist a strongest compact formulation for the CVRP, if _P ≠ NP_. While ending the search for a strongest compact formulation, we propose th