Abstract: Several transportation problems like traffic jam, crowded public transportation, parking shortage and pollution is caused by the actual scenario of urban mobility. The transport of passengers by charter is an alternative to improve the quality of urban mobility avoiding traffic jam and reducing pollution. Several companies offer as a benefit to their employees this type of transport to carry them to the company from their home and vice versa. Thus, it is proposed in this paper an adaptation of a mathematical model based on **Open** **Vehicle** **Routing** **Problem** (OVRP) for planning the transport of employees by a chartered bus fleet in order to reduce the total cost spent by the company. The model was applied to a company located in Vitória-ES and the results obtained by the model indicated a reduction in the cost of transportation when compared to the currently paid by the company.

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More recently, a variant of the classical VRP, called the **open** **vehicle** **routing** **problem** (OVRP), attracted the attention of practitioners and researchers. In this case, vehicles are not required to return to the depot after serving the last customer on a route [9]. This usually arises in real-world problems, like the planning of train services or bus routes (see [10]), or when industries do not own a **vehicle** fleet or their private fleet is inadequate to fully satisfy customer demand, and distribution services (or part of them) are either entrusted to external contractors or assigned to a hired **vehicle** fleet. In these cases, vehicles are not required to return to the central depot after their deliveries have been satisfied. The main difference between VRP and OVRP is that in VRP, the routes are Hamiltonian cycles, and in the OVRP, the routes are Hamiltonian paths originated at the depot and ending at one of the customers, so the shortest Hamiltonian path **problem** with a fixed source node has to be solved for each **vehicle** in the OVRP. The traveling salesperson **problem**, known to be NP-hard, consists of finding the Hamiltonian cycle with the lowest cost. This, together with the fact that the Hamiltonian cycle **problem** (HCP) is NP-hard and can be reduced to the Hamiltonian path **problem** (HPP) [11], allows us to conclude that the shortest HPP is NP-hard. Consequently, the OVRP is also an NP-hard **problem**, justifying the development of heuristics and meta-heuristics (see [12], where a new swarm intelligence approach is proposed). The **vehicle** **routing** with backup provisioning, under discussion here, can be seen as a variant of the OVRP applied to the transportation of persons, considering multiple depots and having the possibility of backup provision to certain critical stops. Therefore, the **vehicle** **routing** with backup provisioning is NP-hard.

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Distribution is an indispensable component of logistics and supply chain management. Location-**Routing** **Problem** (LRP) is an NP-hard **problem** that simultaneously takes into consideration location, allocation, and **vehicle** **routing** decisions to design an optimal distribution network. Multi-layer and multi-product LRP is even more complex as it deals with the decisions at multiple layers of a distribution network where multiple products are transported within and between layers of the network. This paper focuses on modeling a complicated four-layer and multi-product LRP which has not been tackled yet. The distribution network consists of plants, central depots, regional depots, and customers. In this study, the structure, assumptions, and limitations of the distribution network are defined and the mathematical optimization programming model that can be used to obtain the optimal solution is developed. Presented by a mixed-integer programming model, the LRP considers the location **problem** at two layers, the allocation **problem** at three layers, the **vehicle** **routing** **problem** at three layers, and a transshipment **problem**. The mathematical model locates central and regional depots, allocates customers to plants, central depots, and regional depots, constructs tours from each plant or **open** depot to customers, and constructs transshipment paths from plants to depots and from depots to other depots. Considering realistic assumptions and limitations such as producing multiple products, limited production capacity, limited depot and **vehicle** capacity, and limited traveling distances enables the user to capture the real world situations.

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Several approaches were made to solve the VRPTW problems. The use of meta-heuristics is a common solution [Mou08, TM08, BG02, LYL11, GTA99, GGLM03]. Other solutions in- clude heuristics like the one for the distribution of fresh vegetables presented in [OS08] in which the perishability represents a critical factor. The **problem** was formulated as a VRPTW with time-dependent travel-times, where the travel-times between two locations depend on both the distance and the time of the day. The **problem** was solved using a heuristic approach based on the Tabu Search and performance was veriﬁed using modiﬁed Solomon’s problems. A somewhat similar work was proposed in [TK02], which deals with distribution **problem** formulated as an **open** multi-depot **vehicle** **routing** **problem** encountered by a fresh meat distributor. To solve the **problem**, a stochastic search meta-heuristic algorithm, termed as the list-based threshold accepting algorithm, was proposed. In [AS07] a generalization of the asymmetric capacitated **vehicle** **routing** **problem** with split delivery was considered. The solution determines the dis- tribution plan of two types of products, namely: fresh/dry and frozen food. The **problem** was solved using a mixed-integer programming model, followed by a two-step heuristic procedure.

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Golden et al. (1984) implementaram heurísticas baseadas no método das economias de Clarke e Wright (1964) e no roteiro gigante (roteiriza depois agrupa) para o problema de dimensionamento e roteirização de uma frota heterogênea de veículos (Fleet Size and Mix **Vehicle** **Routing** **Problem** – FSMVRP). O objetivo é minimizar a soma dos custos fixos e variáveis dos veículos. Como os custos de viagem independem do tipo de veículo utilizado, os custos variáveis de roteirização são proporcionais à distância total percorrida. Foram implementadas também heurísticas de melhorias baseadas na troca de arcos do tipo 2-opt e 3- opt. As heurísticas de Golden et al. buscam superar a deficiência da heurística de economias para problemas com frota heterogênea, substituindo as distâncias por custos variáveis unitários multiplicados pela distância e adicionando o custo fixo do menor veículo capaz de atender a demanda solicitada. Os autores geraram um conjunto de instâncias de problemas para o FSMVRP.

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There are about 10 company’s proper terminals along the Brazilian coast, with around 20 berths in total, distributed heterogeneously by these terminals. The berths are speciic locations inside maritime terminals, where ships dock in order to perform the loading and unloading of cargo. Each one of the berths presents physical restrictions for draft and LOA that must be met so that the ships are allowed to berth (each ship occupies a single berth). However, in practice, the draft restriction may be relaxed in some speciic cases, and this is done by limiting the load on board to a value lower than the maximum capacity of the ship. Both platforms and terminals are technically called and referred to as “operating sites”. The pairs of pickup and delivery are pre-established by tactical planning, but **routing** and scheduling of ships is conigured as a decision to be supported by the model. This modeling approach is called “origin-destination”, since each one of the origins (platforms) is pre-matched with its respective destination (terminal). Importantly, this pickup and delivery **problem** differs from most cases in relation to the maritime transport of oil, which typically involve large distances. In most cases of oil exploration around the world, the transport occurs in several producing companies for several reineries with different rules and responsibility governing the freight. In Brazil, the same company produces, reines and plans the transportation, which considerably increases the possibilities of logistical gains. It is possible to ind similar characteristics in the operations performed in the North Sea and the Gulf of Mexico.

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The method proposed in section V has been applied to an actual case utilizing historical data concerning a motorcycle manufacturing enterprise in Paraguay [35]. The studied motorcycle factory did not apply a formal method in the planning of its **vehicle** routes when distributing their products, thus, this job was naturally tedious for the employee in charge of logistics, who was satisfied enough with being able to automatize the procedure as much as possible. The logistic area of the factory worked in the following way: the department in charge of logistics within the business collected weekly orders from their internal clients (branches) and continuously made empirical decisions without a mathematical model that would allow them to neither quantify their true costs nor take decisions that would allow the enterprise to optimize their distribution. In consequence, this work mathematically models the logistical **problem** with the distribution of motorcycles and proposes the utilization of the Generalized MOACS algorithm presented in the previous section to solve the already stated mathematical model.

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This research mainly focuses on a less studied VRP extension which is the conVRP. This optimization **problem** demands the definition of **vehicle** routes for several periods, maintaining a certain level of consistency on pre-selected metrics. For instance, when distribution companies make an agreement for the deliveries to be made always by same driver, they are adding consis- tency constraints in order to take into account customer satisfaction. Therefore, the objective is to achieve minimum cost **routing** plans satisfying the classical **routing** constraints as well as con- sistency requirements taking into account customer satisfaction. Generally, this type of customer- oriented **routing** considers two types of consistency for customer satisfaction: driver consistency, and time consistency (Kovacs et al., 2014a). Driver consistency is measured by the number of different drivers that visit a customer whereas time consistency is related to the maximum dif- ference between the earliest visit and the latest arrival at each customer. The conVRP arises in many industries where customer satisfaction is considered as a distinctive factor of competitive- ness. Particularly in industries transporting small packages, providing a standard service with a single driver and approximately at the same time of the day enables the customers to prepare them- selves for a delivery, strengthening supplier/customer relationships (Kovacs et al., 2014b). Since the conVRP considers several periods, it can be seen as a tactical extension of the classical VRP with customer-focused routes.

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In the Single **Vehicle** **Routing** **Problem** with Deliveries and Selective Pickups (SVR- PDSP) there are a set of customers to be served and a depot from where a **vehicle** departs to serve the customers. Each customer has a certain demand of goods either to be delivered or to be picked up, which generates a revenue if collected. It is possible for a customer to have both demands. In such case, if both are going to be served, they can be performed simultaneously or in two different visits, each completely fulfilling one of the demands. The **vehicle** that departs from the depot shall perform a route that visits a subset of customers performing deliveries and pickups, then return to the depot. All delivery demands must be fulfilled exactly once. The pickup demands, however, are not mandatory, therefore they are only performed if there is enough space in the **vehicle** and if they are profitable. Serving a pickup demand is profitable if the revenue generated by collecting it is greater than the additional **routing** cost. One can notice that some pickups might not be served at all and it is possible to argue that they would need to be performed at some point. To address this issue these pickup demands could either be delayed to be served in the following day, or a third party service can be used to collect these pickups, which could be a less costly option than forcing all pickups to be fulfilled or sending another **vehicle** only to perform a few pickups. The objective is to find a route that minimizes the total **routing** cost, which is the travel cost to visit the customers minus the revenue generated by the collected pickups. Fig- ure 1.1a shows a small example with 8 customers and a **vehicle** with capacity equals to 35. In the figure, d is the delivery demand of a customer, while p stands for the pickup demand and r is the revenue generated by performing the respective pickup demand of the customer. The transportation cost of the solution presented is equal to 5 + 4 + 4 + 1 + 10 + 8 + 5 + 4 = 41 and the total revenue generated by the three pickups collected is 5 + 20 + 8 = 33. Therefore the total cost of this solutions is 41 − 33 = 8. In

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The aim of this work is to present some alternatives to improve the performance of an evolutionary algorithm applied to the **problem** known as the oil collecting **vehicle** **routing** **problem**. Some proposals based on the insertion of local search and data mining (DM) modules in a genetic algorithm (GA) are presented. Four algorithms were developed: a GA, a GA with a local search procedure, a GA including a DM module and a GA including local search and DM. Experimental results demonstrate that the incorporation of DM and local search modules in GA can improve the solution quality produced by this method.

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tasks in the sequence {15,16, … , 25} should be transferred to another **vehicle** agent. Then, the **vehicle** agent starts to examine which one of the 11 remaining tasks should be transferred. By examining task 16, it notes that there are prospective gains due to two types of reduction: (a) reduced mileage, and (b) reduced down time when visiting client 16, which should be transferred from the route of the regular **vehicle** agent. The gain is the difference between the cost generated by the remaining basic **routing** sequence, and the new **routing** that excludes the selected task. Therefore, by analyzing every single task on sequence {15,16, … , 25} the system will select the tasks to be transferred that will mostly reduce cost. The third form of selecting the task to be trans- ferred is an extension of the previous form. After some at- tempts to transfer a task from the **routing**, such as task 16, the VRP algorithm is once again applied to the other activ- ities, resetting the **routing** sequence but considering all the other remaining tasks except number 16. Finally, the task that most reduces mileage in the regular VRP sequence is chosen.

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The MDVRPB can be defined as the following graph theory **problem**. Let � = (�, �) be a complete undirected graph, where � = {1. . . . . �} is the set of vertices, and � is the set of edges. The set � is partitioned into two subsets: the set of customers � = {1. . . . . �} and the set of potential depots � = {1. . . . . �}. Additionally, the set � is divided into a subset of Linehaul nodes (Linehaul customers - �), and the Backhaul nodes (Backhaul customers – �). Therefore, � = � ∪ �. The Linehaul customers ask for delivering products while Backhaul customers require the collection of products. Each customer has a nonnegative amount � � ( � ∈ �) of product to be delivered (� ∈ �) or to be picked up (� ∈ �). Each depot has a fictitious demand. i.e. � � = 0, with � ∈ �. A set of � identical vehicles with a given capacity � � is initially placed at each depot. It must be clarified that all vehicles are not necessarily used. For each

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No processo de distribuição é necessário fazer a entrega de bens e serviços para clientes dispersos geograficamente, nesse processo encontra-se o **Vehicle** **Routing** **Problem** (VRP). O VRP, ou Problema de Roteirização de Veículos, é o nome de uma classe de problemas para definir a sequência de visita a clientes dispersos geograficamente com um conjunto finito de veículos a partir de um depósito comum. Para resolver este problema e analisar os resultados obtidos, foi desenvolvido um algoritmo utilizando a meta-heurística Variable Neighborhood Descent (VND), ou Descida em Vizinhança Variável, o qual foi aplicado em instâncias conhecidas na literatura e realizado um benchmarking com outros algoritmos. Esse problema aplica-se na prática em coleta de peças automobilísticas, coleta de lixo industrial, coleta de lixo residencial, limpeza de ruas, e entre outras situações. Os VRPs receberam muita atenção nos últimos anos devido a sua aplicabilidade e sua importância econômica na determinação de estratégias eficientes, com o objetivo de reduzir os custos operacionais. Os resultados obtidos com o algoritmo proposto foram próximos dos algoritmos estudados no benchmarking realizado. Contudo, superando alguns destes algoritmos em apenas uma das instâncias das oito instâncias utilizadas.

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Rochat and Taillard22 have developed an adaptive memory mechanism for the capacity and route duration constrained VRP and for the VRP with time windows, based on the earlier [r]

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extension is done by adding a parameter for setting a minimum value of the tabu list size tls called Threshold. The variation of this parameter improves the exploration of the search space by varying the compromise between intensification and diversification. It allows us to get a dynamic compromise between intensification and diversification. In summary, the more the same solutions found are repeated, the more the tabu list size increases, and vice versa; conversely, the more the solutions are different, the more the tabu list size decreases. This mechanism whereby the number of tabu solutions is increased when reaching local optima allows us to avoid the local optima trap by exploring other solutions in this case because all neighbors have become tabu. The optimization technique for the Reactive tabu with a variable threshold aimed at improving the initial solution (improvement) is developed (Fig. 3) in order to find the best compromise (optimal) solution of the **problem**. It can quickly check the feasibility of the movement suggested, and then react to the repetition to guide the search. This algorithm is performed via a tabu list size (tls) update mechanism elaborated in five steps, as shown in Fig. 3. The counters and parameters used in Reactive tabu with a variable threshold are defined as follows, and initialized to the following values.

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Infrastructure based **routing**: Wu et al. (2013) proposed a moving direction and destination location based **routing** (MEDAL) algorithm, which takes the moving directions of vehicles and the destination location to select a neighbor **vehicle** as the next hop for forwarding data. Nzouonta et al. (2009) proposed a set of Road-Based Vehicular Traffic **routing** (RBVT) protocols, areactive protocol RBVT-R and a proactive protocol RBVT-P that leverage real-time vehicular traffic information to create paths consisting of successions of road intersections. Punithavathi and Duraiswamy (2010) proposed a Client-Server based mobile agent for fast reponse and information reteival. However their protocol requires more server units to store and backup the data. Though most of these algorithm ssupports both V2V and V2I communications, they requires all vehicles to store the periodic hello beacons of other vehicles and also depends on the support of intersections.

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We assume a large-scale heterogeneous WSN on a sensing field A is composed of three types of devices, Sensor Nodes (SNs), Relay Nodes (RNs) and a Base Station (BS). A SN senses the environment, generates data, and periodically transmits the data to an active RN1 , which functions as a cluster head (CH), in a single hop. It has limited energy and a fixed transmission radius rSN. It has no relaying function or at least traffic relaying is not a routine function of a SN for the following reasons. First, relaying traffic demands high intelligence, such as security and **routing**, which leads to higher device cost. Second, extra communication leads to faster energy dissipation. A RN is also energy constrained and has fixed transmission range rRN, where typically rRN is a few times larger than rSN. A RN works as a CH when active, which groups the SNs in its proximity into a cluster. It also coordinates and schedules the MAC layer access within its cluster so that the energy overhead, e.g., retransmissions due to collisions, is minimized. After receiving the data from SNs, it aggregates the traffic.The aggregation diminishes the redundant information from multiple nodes and reduces the network traffic. In the end, it transmits the aggregated data to the next hop active RN according to the **routing** algorithm running on these active RNs. The aggregated traffic won’t be aggregated again while passing through other RNs. We assume the traffic is light compared with the available bandwidth, or the traffic is well scheduled so that there is no traffic congestion in

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O Problema do Caixeiro Viajante com Vizinhança (Traveling Salesman **Problem** with Neigh- borhood, TSPN), introduzido por Arkin & Hassin (1994), é uma variante do clássico Problema do Caixeiro Viajante. No TSPN, os n clientes são representados por figuras geométricas. Uma solução para o problema consiste em um ciclo hamiltoniano mínimo sobre pontos presentes em todas as n regiões. O custo de deslocamento é dado pela distância euclidiana entre todos os pontos do passeio, entretanto, há casos em que o problema adota funções assimétricas não negativas para valorar o comprimento das arestas. de Berg et al. (2005) apresentaram um algo- ritmo aproximativo para casos em que as regiões de visitação do TSPN são disjuntas e convexas, além de um algoritmo polinomial que encontra o valor ótimo para casos em que as regiões de visitação são sobrepostas e não convexas. O problema, neste caso, é classificado como APX-difícil. No trabalho de Gentilini et al. (2013) foram mostradas duas formulações com distância dada pela norma euclidiana, seus modelos foram testados no Solver não linear COIN-OR (2016) em instâncias com até 16 pontos. Para alguns problemas testes fez-se necessário mais de qua- tro horas de processamento em um computador Intel Xeon 3.33 GHz com 12GB de memória RAM.

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Em 2002, Toth & Vigo referiam que, “o interesse nos VRP é motivado pela sua relevância prática e a sua dificuldade considerável de implementação: as maiores instâncias de VRP que podem ser consistentemente resolvidas pelos algoritmos exatos mais eficientes existentes no momento contêm 50 clientes, sendo que problemas maiores podem ser resolvidos otimamente apenas em casos particulares” (p. xvii). Já em 2012, Lin, Chou, Lee, & Lee dizem que “o VRPTW [um tipo de VRP que se analisará posteriormente] é NP-completo e instâncias com 100 clientes ou mais são muito dificeis de resolver optimamente” (p. 11). Tendo em conta estes testemunhos e o enunciado relativamente simples dos VRP, o que os torna difíceis de solucionar? Ralphs, Hartman, & Galati (2001) afirmam que uma das razões é o facto de este tipo de problemas ser a interseção de dois problemas difíceis: Traveling Salesman **Problem** (Problema do Caixeiro Viajante - encaminhamento ou roteamento); e o Bin Packing **Problem** (empacotamento).

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Operational planning within public transit companies has been extensively tackled but still remains a challenging area for operations research models and techniques. This phase of the planning process comprises **vehicle** scheduling, crew scheduling and rostering problems. In this paper, a new integer mathematical formulation to describe the integrated **vehicle**-crew-rostering **problem** is presented. The method proposed to solve this multi-objective **problem** is a sequential algorithm considered within a preemptive goal programming framework that starts from the solution of an integrated **vehicle** and crew scheduling **problem** and ends with the solution of a driver rostering **problem**. Feasible solutions for the **vehicle** and crew scheduling **problem** are obtained by combining a column generation scheme with a branch-and-bound method. These solutions are the input of the rostering **problem**, which is tackled through a mixed binary linear programming approach. An application to real data of a Portuguese bus company is reported and shows the importance of integrating the three scheduling problems .

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