Basically we use a Data Mining strategy, in which every new individual has its route analyzed to extract patterns (sequence of customers) within a given range [minP atternSize, maxP atternSize]. Each pattern found is stored in a structure called patternsList along with the frequency it has appeared in the solutions already ana- lyzed. In addition to these information, we also keep record of the average cost of the route in which the pattern was found so as to improve the robustness of the eval- uation criteria that decides how good a pattern is. Therefore we have two types of data to evaluate a pattern: frequence and average cost. Good patterns have high frequency and low average cost. Since cost value is usually much higher than the frequency value, this data must be normalized. Lets call nF requency the normal- ized frequency value, nAvgCost the normalized average cost. Therefore we define qualityIndex = (1 − nAvgCost) + nF requency, as the value used to evaluate the pat- terns, since it considers both measures. The closer to 2 the better. This is not the first time an approach combining a heuristic and a data mining algorithm is proposed for a **vehicle** **routing** **problem**. In [28], Santos et al proposed 4 approaches for a single **vehicle** **routing** **problem**, including one that combines a Genetic Algorithm with the data mining algorithm Apriori. Our approach is not based on their approach and is fairly different from the algorithm they developed.

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Variações do 2E-CVRP também são encontradas na literatura. Crainic et al. (2009) apresentaram uma variação do 2E-CVRP, chamada de two-echelon, synchronized, scheduled, multi-depot, multiple-tour, heterogeneous vehi- cle **routing** **problem** with time windows (2SS-MDMT- VRPTW), ao tratar o gerenciamento da Logística Urbana. Esses autores desenvolveram um modelo e formulações ge- rais para a nova classe a partir de Programação Linear In- teira, mas não realizaram nenhum experimento computaci- onal para a mesma. Grangier et al. (2014) abordaram uma nova classe do 2E-CVRP, chamada two-echelon multiple- trip **vehicle** **routing** **problem** with sattelite synchronization (2E-MTVRP-SS) e utilizaram uma meta-heurística Adap- tive Large Neighborhood Search para resolução do pro- blema. Soysal et al. (2014) abordaram pela primeira vez a variação time-dependent em problemas 2E-CVRP, o Two- echelon Capacitated **Vehicle** **Routing** **Problem** with Time Dependent (2E-CVRPTD), assim como fatores que influen- ciam no consumo de combustível, como o tipo de veículo, a distância percorrida, a velocidade e a carga transportada pelo veículo. Esses autores desenvolveram um modelo ma- temético de PLIM baseada no modelo proposto por Jepsen et al. (2013) e testaram o modelo em um caso real, uma ca- deia de suprimentos localizada nos Países Baixos, com 1 depósito, 2 satélites e 16 clientes.

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A **vehicle** **routing** **problem** with time windows (VRPTW) is an important **problem** with many real applications in a transportation **problem**. The optimum set of routes with the minimum distance and vehicles used is determined to deliver goods from a central depot, using a **vehicle** with capacity constraint. In the real cases, there are other objective functions that should be considered. This paper considers not only the minimum distance and the number of vehicles used as the objective function, the customer’s satisfaction with the priority of customers is also considered. Additionally, it presents a new model for a bi-objective VRPTW solved by a revised multi-choice goal programming approach, in which the decision maker determines optimistic aspiration levels for each objective function. Two meta-heuristic methods, namely simulated annealing (SA) and genetic algorithm (GA), are proposed to solve large-sized problems. Moreover, the experimental design is used to tune the parameters of the proposed algorithms. The presented model is verified by a real-world case study and a number of test problems. The computational results verify the efficiency of the proposed SA and GA.

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Abstract: The paper deals with the design of a route elimination (RE) algorithm for the **vehicle** **routing** **problem** with time windows (VRPTW). The **problem** has two objectives, one of them is the minimal number of routes the other is the minimal cost. To cope with these objectives effectively two-phase solutions are often suggested in the relevant literature. In the first phase the main focus is the route elimination, in the second one it is the cost reduction. The algorithm described here is a part of a complete VRPWT study. The method was developed by studying the graph behaviour during the route elimination. For this purpose a model -called “Magic Bricks” was developed. The computation results on the Solomon **problem** set show that the developed algorithm is competitive with the best ones.

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Abstract: **Problem** statement: In this study, we considered the application of a genetic algorithm to **vehicle** **routing** **problem** with time windows where a set of vehicles with limits on capacity and travel time are available to service a set of customers with demands and earliest and latest time for serving. The objective is to find routes for the vehicles to service all the customers at a minimal cost without violating the capacity and travel time constraints of the vehicles and the time window constraints set by the customers. Approach: We proposed a genetic algorithm using an optimized crossover operator designed by a complete undirected bipartite graph that finds an optimal set of delivery routes satisfying the requirements and giving minimal total cost. Various techniques have also been introduced into the proposed algorithm to further enhance the solutions quality. Results: We tested our algorithm with benchmark instances and compared it with some other heuristics in the literature. The results showed that the proposed algorithm is competitive in terms of the quality of the solutions found. Conclusion/Recommendations: This study presented a genetic algorithm for solving **vehicle** **routing** **problem** with time windows using an optimized crossover operator. From the results, it can be concluded that the proposed algorithm is competitive when compared with other heuristics in the literature.

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transporte de passageiros por fretamento surge como alternativa para a solução dos problemas de mobilidade podendo cola- borar para diminuir os congestionamentos e a emissão de poluentes. Algumas empresas oferecem como benefício a seus empregados o transporte de ida e volta do trabalho por ônibus fretados. Assim, é proposto nesse artigo uma adaptação de um modelo matemático baseado no Open **Vehicle** **Routing** **Problem** (OVRP) para o planejamento do transporte de empregados por meio de uma frota de ônibus fretada visando à redução do custo total gasto pela empresa. O modelo foi aplicado a uma empresa localizada em Vitória-ES e os resultados obtidos pelo modelo indicaram uma redução no custo de transporte quando comparado ao atualmente pago pela empresa.

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One of the most important extensions of the CVRP is the **Vehicle** **Routing** **Problem** with Time Window (VRPTW) which is each customer must be served within a specific time window. The objective is to minimize the **vehicle** fleet with the sum of travel time and waiting time needed to supply all customers in their required hour [10], [12]. A variety of exact algorithms and efficient heuristics have already been proposed for VRPTW by many researchers as shown in Table 1. In addition, Table 2 represents the various methods applying in exact algorithm, classical heuristic algorithms and metaheuristic algorithms for various type of VRP.

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Concerning oil exploitation, there is a class of onshore wells called artiﬁcial lift wells where the use of auxiliary methods for the elevation of ﬂuids (oil and water) is necessary. In this case, a ﬁxed system of beam pump is used when the well has a high productivity. Because oil is not a renewable product, the production of such wells will diminish until the utilization of equipment permanently allocated to them will become economically unfeasible. The exploitation of low productivity wells can be done by mobile equipment coupled to a truck. This **vehicle** has to perform daily tours visiting wells, starting and ﬁnishing at the oil treatment station (OTS), where separation of oil from water occurs. Usually the mobile collector is not able to visit all wells in a single day. In this context, arises the **problem** called oil collecting **vehicle** **routing** **problem** (OCVRP). In this **problem**, the objective is to collect the maximum amount of oil in a single day, starting and

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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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Approaches for multi-objective versions of the VRPB have been proposed by Anbuudayasankar et al. (2012), García-Nájera et al. (2015) and Yalcın & Erginel (2015). Three heuristics approaches for solving a bi-objective **vehicle** **routing** **problem** with forced backhauls were introduced by Anbuudayasankar et al. (2012). In particular, two heuristics are based on the well-known savings algorithm and the third heuristic is based on a Genetic Algorithm (GA). Finally, an evolutionary approach and a fuzzy programming for the multi-objective **vehicle** **routing** problems with backhauls were presented by García- Nájera et al. (2015) and Yalcın and Erginel (2015), respectively. Other multiobjective algorithms proposed for solving related logistic combinatorial problems could be consulted in Nezhad et al. (2013), Mortezaei and JabalAmeli (2011), Mohammadi et al. (2011), Rao and Patel (2014), Yazdian and Shahanaghi (2011), Escobar et al. (2013), Escobar et al. (2014b), Escobar et al. (2015) and Bolaños et al. (2015).

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A fim de considerar o engarrafamento no planeja- mento da distribuição física, dentre os diversos conceitos de roteamento de veículos já estudados, o que mais se adere a esta realidade é o Time Dependent **Vehicle** **Routing** **Problem** (TDVRP). No TDVRP tem-se uma frota de veículos com capacidade limitada que deve coletar ou entregar cargas a clientes a partir de um depósito central. Os clientes devem ser designados aos veículos que realizam rotas, de forma que o tempo total gasto seja minimizado. O tempo de via- gem entre dois clientes ou entre um cliente e o depósito de- pende de suas distâncias e também do momento do dia que o transporte é feito; por exemplo, nos horários de pico o tempo para deslocamento é maior devido ao congestiona- mento. As janelas de tempo para servir os clientes, ou seja, o período que os clientes podem ser atendidos, devem ser consideradas assim como a máxima duração permitida para cada rota (horário de trabalho do motorista) (Malandraki e Daskin, 1992). O TDVRP é, então, uma extensão do Pro- blema de Roteamento de Veículos (VRP) que pode levar em

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13.1 The **Vehicle** **Routing** **Problem** 177 The contribution of this work is then to define a powerful yet simple cMA capable of competing with the best known approaches for solving CVRP in terms of accuracy (final cost) and computational effort (the number of evalua- tions made). For that purpose, we test our algorithm over the mentioned large selection of instances (160), which will allow us to guarantee deep and mean- ingful conclusions. Besides, we compare our results against the best existing ones in the literature, some of which we even improve. In [11] the reader can find a seminal work with a comparison between our algorithm and some other known heuristics for a reduced set of 8 instances. In that work, we showed the advantages of embedding local search techniques into a cGA for solving CVRP, since our hybrid cGA was the best algorithm out of all those compared in terms of accuracy and time. Cellular GAs represent a paradigm much simpler to comprehend and customize than others such as tabu search (TS) [97, 249] and similar (very specialized or very abstract) algorithms [37, 207]. This is an important point too, since the greatest emphasis on simplicity and flexibility is nowadays a must in research to achieve widely useful contributions [52].

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The Multi-Depot **Vehicle** **Routing** **Problem** (MDVRP) is an important variant of the classical **Vehicle** **Routing** **Problem** (VRP), where the customers can be served from a number of depots. This paper introduces a coop- erative coevolutionary algorithm to minimize the total route cost of the MDVRP. Coevolutionary algorithms are inspired by the simultaneous evolution process involving two or more species. In this approach, the prob- lem is decomposed into smaller subproblems and individuals from different populations are combined to create a complete solution to the original **problem**. This paper presents a **problem** decomposition approach for the MDVRP in which each subproblem becomes a single depot VRP and evolves independently in its do- main space. Customers are distributed among the depots based on their distance from the depots and their distance from their closest neighbor. A population is associated with each depot where the individuals rep- resent partial solutions to the **problem**, that is, sets of routes over customers assigned to the corresponding depot. The ﬁtness of a partial solution depends on its ability to cooperate with partial solutions from other populations to form a complete solution to the MDVRP. As the **problem** is decomposed and each part evolves separately, this approach is strongly suitable to parallel environments. Therefore, a parallel evolution strategy environment with a variable length genotype coupled with local search operators is proposed. A large num- ber of experiments have been conducted to assess the performance of this approach. The results suggest that the proposed coevolutionary algorithm in a parallel environment is able to produce high-quality solutions to the MDVRP in low computational time.

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Abstract: **Problem** statement: The Capacitated **Vehicle** **Routing** **Problem** (CVRP) is a well-known combinatorial optimization **problem** which is concerned with the distribution of goods between the depot and customers. It is of economic importance to businesses as approximately 10-20% of the final cost of the goods is contributed by the transportation process. Approach: This **problem** was tackled using an Ant Colony Optimization (ACO) combined with heuristic approaches that act as the route improvement strategies. The proposed ACO utilized a pheromone evaporation procedure of standard ant algorithm in order to introduce an evaporation rate that depends on the solutions found by the artificial ants. Results: Computational experiments were conducted on benchmark data set and the results obtained from the proposed algorithms shown that the application of combination of two different heuristics in the ACO had the capability to improve the ants’ solutions better than ACO embedded with only one heuristic. Conclusion: ACO with swap and 3-opt heuristic has the capability to tackle the CVRP with satisfactory solution quality and run time. It is a viable alternative for solving the CVRP.

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O Problema de Roteamento de Veículo Suficientemente Próximo (Close-Enough **Vehicle** **Routing** **Problem**, CEVRP), apresentado por W. K. Mennell (2009), foi inspirado na necessidade de rotear VANTs para monitoramento de áreas em um plano euclidiano. Neste caso, o veículo precisa alcançar certa distância do centro do alvo para realizar serviços, tais como fotografar, gravar vídeo, etc. W. K. Mennell (2009) formulou o CEVRP como um modelo não linear e não convexo, e mostrou que para o caso em que as distâncias das demandas ao centro do alvo são reduzidas a zero, o CEVRP é transformado no VRP no espaço euclidiano e, portanto, um problema NP-difícil.

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Despite the advantages of adopting consistent routes, few papers have addressed the conVRP and most approaches resort to approximation methods. Groer et al. (2009) formulate the conVRP as a Mixed-Integer Program (MIP) and improve the algorithm used by Li et al. (2005) to solve very large VRPs. A real-world data set is used to generate instances with up to 700 customers which are solved by the algorithm. The obtained consistent routes are less than 10% longer on average, compared to inconsistent routes. Recently, Ridder (2014) shows that some optimal solu- tions provided by Groer et al. (2009) are not feasible because service times were not considered. The author develops an algorithm that improves solutions provided by the latter paper. Tarantilis et al. (2012) propose a Tabu Search (TS) algorithm to iteratively generate template routes and to improve the daily routes that are derived from the template routes. These routes are used as the basis to construct the **vehicle** routes and service schedules for both frequent and non-frequent customers over multiple days. The best reported cumulative and mean results over all conVRP- benchmark instances is improved. Kovacs et al. (2014b) construct template routes by means of an Adaptive Large Neighbourhood Search (ALNS), which uses several operators in order to destroy and repair a given solution. It is shown that solving daily VRPs may lead to inconsistent routes whereas consistent long-term solutions can be generated by using historic template routes. Kovacs et al. (2014a) state that assigning one driver to each customer and bound the variation in the arrival times over a given planning horizon may be too restrictive in some applications. They propose the generalized conVRP in which a customer is visited by a limited number of drivers and the vari- ation in the arrival times is penalized in the objective function. A Large Neighbourhood Search (LNS) metaheuristic generates solutions without using template routes. The computational results on different variants of the conVRP prove the efficiency of the algorithm, as it outperforms all published algorithms. Sungur et al. (2010) consider a real-world courier delivery **problem** where customers appear probabilistically. Although the authors do not call it a conVRP, their assump- tions are completely in line with this type of **problem**. The proposed approach generates master plans and daily schedules with the objective of maximizing both the coverage of customers and the similarity between the routes performed in each day. In order to deal with uncertain service times, it is assumed that the master plans serves frequent customers with the worst-case service times found in historical data. Once again, a mathematical formulation is proposed but the real-world **problem** is tackled by means of a two-phase heuristic based on insertion and TS.

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geograficamente dispersas, encontrar o melhor conjunto de rotas para atender todos os. clientes[r]

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Abstract: The transport activities usually involves several actors and vehicles spread out on a network of streets. This complex system intricate the techniques to deal with dynamic events usually present in transport operations. In this context, as could be noted in the literature review, the use of multi-agent systems (MAS) seems suitable to support the autonomous decision-making. This work presents an agent based system to deal with a dynamic **vehicle** **routing** **problem**, more precisely, in a pick-up **problem**, where some tasks assigned to vehicles at the beginning of the operation could be transferred to others vehicles. The task transfer happens when the **vehicle** agents perceive that the cycle time can exceed the daily limit of working hours, and is done through a negotiation protocol called Vickrey. The proposed system allows a collaborative decision- making among the agents, which makes possible adjustments during the course of the planned route.

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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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When trying to solve a Split Delivery/ Mixed Fleet – **Vehicle** **Routing** **Problem** or SD/MF-VRP for a motorcycle distribution company in Paraguay, we found ourselves with a need to consider various unconventional restrictions (such as vehicles that cannot transit on certain roads) and to simultaneously minimize 4 objectives: (1) the total cost, (2) total travel time, (3) total delivery time, and (4) unsatisfied demand, which entails solving a pretty complex practical **problem**, today known as many-objective optimization **problem** [34]. In consequence, after carefully analyzing the state of the art, a well-known MOACO was chosen to solve the **problem** at hand, the MOACS [26] proposed in 2003 to solve a bi-objective TSP **problem**. In consequence, said algorithm was modified to be able to treat a generic number u of objective functions (u ≥ 2). This generalized version of the MOACS explained in detailed in section V, was then successfully tested at a Paraguayan motorcycle factory, as explained in section VI, proving the viability of the proposal.

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