As we know, the application of exact methods in the VRP **problem** solving is quite limited because of the combinatorial “explosion”. During the decades different successful metaheuristics have been developed, for instance Simulated Annealing, Evolutionary Algorithms, and Tabu Search (TS) etc. If we analyse the TS we must admit that despite its indisputable success it has problems in special cases when the **route** **elimination** goes together **with** considerable cost increment. Normally the object function of the TS is designed for finding cheaper solutions. Depending on the length of the tabu list the algorithm is able to reveal new regions. We can in the meantime change the object function and the length of the tabu-list but despite of these techniques it is difficult for the pure TS algorithm to get out from “deep valleys”, so the chance for eliminating a **route** is quite limited and the search is basically guided by the second priority objective. This topic is detailed in [2]. To leave such kind of deep valleys we have to find effective oscillation – sometimes it is called diversification – methods. In the **route** **elimination** respect – although it is the primary objective – the pure TS loses to other – lately developed – metaheuristics first of all hybrid metaheuristics [3]. The purpose of this part of the research and this article is to develop an effective **route** **elimination** phase.

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Objective function (1) minimizes a mixed cost, where the first term denotes the construction cost of charging stations, the second term the cost of electricity recharged at depot and stations, and the third term the diver wage associated **with** working **time**. Constraints (2) guarantee that each customer is visited exactly once. Constraints (3) ensure that the number of incoming arcs is equal to that of outgoing arcs for each vertex, except for the instances of the depot. Constraints (4) and (5) ensure that all the employed vehicles start and end routes at the depot. Constraints (6) and (7) enforce the fulfillment of demands at customer nodes, and Constraints (8) restrict the initial cargo load level of a **vehicle** to its capacity. Constraints (9) link the battery levels of the **vehicle** at the vertices i and j of a traveled arc ( , ) i j . Both Constraints (6) and (9) adopt the idea from the big-M method. Constraints (10) and (11) ensure that each **vehicle** leaves the depot or a located station **with** a fully charged battery. Constraints (12) confine the type of infrastructure located at a candidate site to be one. Constraints (13) define the simplified relationship between charging **time** and charging amount at located charging stations. Constraints (14) prevent a **vehicle** from charging at a vertex from a customer set. Constraints (15) guarantee that each **vehicle** has sufficient power to reach a located station or depot. Constraints (16) and (17) establish the relation between arrival times at vertices i and j if the arc ( , ) i j is traveled. Constraints (17) particularly cover the condition, where the arc ( , ) i j starts **with** a charging station. Constraints (18) ensure that all customer vertices are visited within their **time** **windows**. Constraints (19) to (21) define the natural features of the variables.

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Wireless sensors have been used for monitoring and tracking in several areas. **With** regard to their use in smart cities, these can be used to improve traffic control in large cities [13,14], collect information of passenger volume [15], provide optimal routes in real **time** [16], to reduce greenhouse gas (GHG) emissions [17], in rescue scenarios after a disaster [18] and even to aid the blind [19]. In [15], the system used for data gathering is discussed, but no further planning using such information is done. Our work on **vehicle** **routing** **with** backup provisioning goes further and uses the gathered data for **vehicle** **route** planning. The **problem** addressed in [16] is different from ours and can be considered more like a dynamic **vehicle** **routing** **problem** (DVRP), which has a wide range of real-world applications, as stated in [20]. In DVRP, real-**time** communication between vehicles and planners is required, and adjustments of the optimized routes can be performed during the execution process. This kind of **problem**, however, is not adequate when stops must be previously defined, which is the case that we are studying. The **problem** addressed in [17] falls into the category of green **vehicle** **routing** problems (GVRP), and the objective is to find routes while minimizing GHG emissions. In rescue scenarios, considered in [18], the demand-related information is quite limited in the initial rescue period and is intuitively unpredictable using historical data, and the emergency resources may be insufficient. The problems addressed in [17,18] are, therefore, different from the one considered in this article.

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Surveys of existing methods for multi-objective problems were presented in Jozefowiez et al. (2008) and Zhou et al. (2011). In Jozefowiez et al. (2008), the authors examined multiobjective versions of several variants of the **Vehicle** **Routing** **Problem** (VRP) in terms of their objectives, their characteristics and the types of proposed algorithms to solve them. A survey of the state of the art of the multi-objective evolutionary algorithms was proposed by Zhou et al. (2011). This papers covers algorithms frameworks for multiobjective combinatorial problems during the last eight years. However, in the literature reviewed, there are few works considering the multi-objective version of the MDVRPB. Multiobjective metaheuristic approaches for combinatorial problems were presented in Doerner et al. (2004), Liu et al. (2006) and Lau et al. (2009). A multiobjective methodology by Pareto Ant Colony Optimization for solving a portfolio **problem** was introduced by Doerner et al. (2004). A multi-objective mixed zero-one integer-programming model for the **vehicle** **routing** **problem** **with** balanced workload and delivery **time** was introduced by Liu et al. (2006). In this work, a **heuristic**-based solution method was developed. A fuzzy multi-objective evolutionary algorithm for the **problem** of optimization of **vehicle** **routing** problems **with** multiple depots, multiple customers, and multiple products was proposed by Lau et al. (2009). In this work, two objectives were considered: minimization of the traveling distance and also the traveling **time**.

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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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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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Além do CVRP, outras versões do VRP são muito estudadas. O Problema de Roteamento de Veículos com Janelas de Tempo (VRPTW, do inglês **Vehicle** **Routing** **Problem** **with** **Time** **Windows**) é uma extensão do CVRP onde cada cliente deve ter seu atendimento iniciado em uma janela de tempo e o veículo associado deve atendê-lo durante um tempo previamente estipulado. Por sua vez, o Problema de Roteamento de Veículos com Backhauls (VRPB, do inglês **Vehicle** **Routing** **Problem** **with** Backhauls) consiste em um CVRP onde o conjunto de clientes é particionado em dois subconjuntos: linehaul e backhaul. O primeiro subconjunto consiste nos clientes que necessitam de itens a serem entregues, enquanto o segundo representa os clientes que dispõem de itens a serem coletados. No VRPB, todos os clientes linehaul devem ser visitados antes dos clientes backhaul. Uma outra variação do VRP é o Problema de Roteamento de Veículos com Coleta e Entrega (VRPPD, do inglês **Vehicle** **Routing** **Problem** **with** Pick-ups and Deliveries), onde uma requisição de transporte é associada a dois clientes, de tal forma que a demanda é coletada em um deles e entregue no outro. Nesse problema, uma solução viável requer que a coleta de uma requisição seja feita antes de sua entrega, e que ambas operações ocorram na mesma rota. Informações sobre os trabalhos propostos e os detalhes do VRPTW, VRPB e VRPPD, podem ser encontrados em Alvarenga et al. [2007], Toth & Vigo [2001c] e Desaulniers et al. [2001], respectivamente.

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Su¨ral and Bookbinder [30] are the first to directly address this **problem**. They present the **problem** using the notation α/β/γ, where α denotes the number of ve- hicles (1 for single and M for multiple), β the pickup service options (must or f ree if the pickup is respectively mandatory or optional) and γ the precedence order for visiting the customers (prec if all deliveries must precede the pickups, or any if they can be visited in any order). While the SVRPDSP is 1/free/any, according to this notation, the MVRPDSP can be described as M/free/any. They cited papers dealing **with** the 1/free/prec and 1/must/any problems, and claimed to be the first to address the 1/free/any. For the multiple **vehicle** versions, they list papers for the M/must/prec and M/free/prec, however no mention is made about the M/free/any, which is one of our objects of study. They propose a mixed integer linear programming formulation for the SVRPDSP along **with** some improvements on the constraints to strenghten the formulation, such as constraint disaggregation, coefficient improvement, cover and logi- cal inequalities, and lifted subtour **elimination** constraints. They modified 24 instances from the literature, **with** sizes of 10, 20 and 30 customers, to test their formulation. These instances were adapted for the SVRPDSP by setting some of the delivery de- mands as pickups in 3 different ways (20% of the customers reset as pickups, then 30% and 40%), generating a total of 72 instances, which were tested **with** some combina- tions of the formulation and the improvements resulting in about 75% of the instances optimally solved in a reasonable computational **time** for the best combination.

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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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The cement industry is not an exception. Cement is the second most consumed substance in the world and **with** the great number of trucks arriving at cement facilities, every day, the supply chain management of this industry must encompass this management as well. **With** the lack of assistance and guidance clients have inside the cement facilities, both companies incur in additional costs and clients experience reduced levels of service quality. To overcome these issues, three algorithms were developed and implemented. Each algorithm has different specifications and different goals. However, all the developed algorithms improve the service quality, guiding the truck drivers – the clients – inside the plants and giving the routes in shorter periods of **time**. One algorithm guides the trucks through the minimum distance **route** and will serve as a comparison term for the other two. The other two algorithms, named equilibrium approaches, are the main contribution of this dissertation. These dynamic algorithms consider not only the traveled distance, but also the workload both in the servers and in the roads. The entrance management in the facilities is also a crucial aspect cement companies must be aware of. Several thought policies are presented and an algorithm for the entrance management is developed and implemented. **With** a simulation software, the developed algorithms were tested and simulated. The simulation results are reported and discussed.

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The company currently defines its routes in a two step method. First, an instance **with** all the internal customers is created and is manually separated in geographical areas and in period of operation. The requests from the external customers, which are not period specific, are not considered during this phase because it is currently not possible to assign a **route** to each one of them on an operational level, even though they represent almost half of the total deliveries in one of the depots. Then, the nodes are used as input in a commercial **route** planner to define the robust routes for that period of operation. These routes are then used operationally until a new redefinition of the routes is performed, which does not have a defined frequency and is done very sporadically. All the customers that are not considered during the planning stage and, as such, do not have a specified **route** to be assigned to, are assigned daily on an operational level. A first rough assignment is made by having a manually created table that assigns every national postal code prefix to a **route**. However, some postal code prefixes correspond to a large area and some even have multiple routes going through them. In these cases, the drivers are given the task of reassigning some of the packages when they deem them to be in the wrong **route**. This procedure is **time** consuming and leads to mistakes and to deliveries that could be made in a better **route**. Moreover, the interchangeability of some deliveries, which may be delivered in any period of the day, is completely lost and every load is dispatched in the morning period unless it does not fit in the truck. This places a heavy burden on the morning routes, which often operate on tight schedules, and any delay in these early routes tends to spread to the remaining periods of the day as the drivers may be late for their next shift.

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Su¨ral and Bookbinder [30] are the first to directly address this **problem**. They present the **problem** using the notation α/β/γ, where α denotes the number of ve- hicles (1 for single and M for multiple), β the pickup service options (must or f ree if the pickup is respectively mandatory or optional) and γ the precedence order for visiting the customers (prec if all deliveries must precede the pickups, or any if they can be visited in any order). While the SVRPDSP is 1/free/any, according to this notation, the MVRPDSP can be described as M/free/any. They cited papers dealing **with** the 1/free/prec and 1/must/any problems, and claimed to be the first to address the 1/free/any. For the multiple **vehicle** versions, they list papers for the M/must/prec and M/free/prec, however no mention is made about the M/free/any, which is one of our objects of study. They propose a mixed integer linear programming formulation for the SVRPDSP along **with** some improvements on the constraints to strenghten the formulation, such as constraint disaggregation, coefficient improvement, cover and logi- cal inequalities, and lifted subtour **elimination** constraints. They modified 24 instances from the literature, **with** sizes of 10, 20 and 30 customers, to test their formulation. These instances were adapted for the SVRPDSP by setting some of the delivery de- mands as pickups in 3 different ways (20% of the customers reset as pickups, then 30% and 40%), generating a total of 72 instances, which were tested **with** some combina- tions of the formulation and the improvements resulting in about 75% of the instances optimally solved in a reasonable computational **time** for the best combination.

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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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resolution method used was a bi-objective tabu search algorithm. The first objective is to minimize the total number of vehicles used, and the second, to minimize the total cost, which is the weighted sum of the total distance traveled and the corresponding total **time**. Ref. [13], based on the work of [12], established a multi-criteria optimization model of long-haul VRP and scheduling integrating working hours rules. The solution method used was a bi- objective tabu search algorithm determining a set of **heuristic** non-dominated solutions. The mechanism consists of a single thread in which the weights assigned to the two objectives, namely, operating costs and driver inconvenience, are dynamically modified, and in which dominated solutions are eliminated throughout the search. Ref. [14] proposed a multi-depot VRP **with** a simultaneous delivery and pick-up model. The resolution method used was the iterated local search embedded adaptive neighborhood selection approach. Ref. [15] tested local search move operators on the VRP **with** split deliveries and **time** **windows**. To that end, they used eight local search opera-tors, in combinations of up to three of them, paired **with** a max-min ant system. Ref. [16] developed a dynamic model for solving the mixed integer programming of forest plant location and design, as well as production levels and flows between origins and destinations. Ref. [17] proposed a multi-depot forest transportation model solving the tactical **problem** of the flow between origins and destinations without solving the operational **problem** of VRP. The solution method used was column generation. Ref. [6] proposed a model for forest transportation, solving the **problem** of flow between origins and destinations, and involving a sedimentation constraint

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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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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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The Mixed Fleet VRP (MFVRP), implies vehicles **with** different capabilities (or heterogeneous capabilities), **with** known fixed and variable costs related to each **vehicle** in a fleet that must serve a series of consumers **with** known demands. In [15], Golden, Assad, Levy and Gheysens describe a series of effective **heuristic** procedures for the **problem** of **routing** **with** a heterogeneous fleet, **with** the objective of determining the optimal truck fleet size and its capabilities, minimizing a cost function. The authors, Subramanian, Penna, Uchoa, and Ochi [16] studied the optimal composition of a fleet of vehicles through a hybrid algorithm, as well as determining the routes that would minimize travel expenses. Similarly, Salhi and Rand, in [17], and Taillard in [18], also attempt to find the ideal composition for a fleet of vehicles by solving the MFVRP. The authors of [19], Wassan and Osman, developed new Tabu Search (TS) variants in order to solve the heterogeneous fleet **problem**. At the same **time**, the article by Chen and Ching [20] suggests the alternative of employing an Ant Colony Optimization algorithm in order to solve the heterogeneous fleet **routing** **problem**, proving that ACO is a competitive algorithm for this VRP variant; another factor influencing its adoption for this work.

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This work studies the implementation of heuristics and scatter search (SS) metaheuristic in a real heterogeneous fleet **vehicle** **routing** **problem** **with** **time** **windows** and split deliveries (HFVRPTWSD) in Brazil. In the **vehicle** **routing** **problem** **with** **time** **windows** and split deliveries (VRPSD) each client can be supplied by more than one **vehicle**. The **problem** is based in a single depot, the demand of each client can be greater than the vehicle’s capacity and beyond the **time** **windows** constraints, and there are also **vehicle** capacity and accessibility constraints (some customers cannot be served by some vehicles). The models were applied in one of the biggest retail market in Brazil that has 519 stores distributed in 12 Brazilian states. Results showed improvements over current solutions in a real case, reducing up to 8% the total cost of the operation.

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Stage 3. Post optimization - The next stage is a cycle which is computed until all routes are closed by the ERP/administrator or all PFIH parameters are tested. The cycle starts by getting (and removing) the most promising solution from P F IHSet and setting an ejection rate value. A tabu list, T , is started which will contain all computed solutions before applying post-optimization, for each ejection rate. Then try at most M axT ries times to improve the solution by applying: (a) a 2-Opt operator (which iterates through all routes, one by one, and tries to rearrange the sequence by which the customers are visited in order to reduce the **route** distance, maintaining feasibility [CGF + 08]); (b) a cross **route** operator (similar to the One Point Crossover operator of the Genetic Algorithms [MaM13], receives two paths as input, and tries to ﬁnd a point where the routes can be crossed, thus improving the total distance and without losing feasibility); and (c) a band ejection operator which is a generalization of the radial ejection [SSSWD00](selects a **route** and, based on the proximity and similarity of the nodes, for each customer located in the **route** ejects it and a certain number of geographical neighbors which are then reinserted in other routes, without violating the problem’s constraints). Please refer to [CSME15] for a more detailed explanation. The ﬁrst two operators are capable of diminishing the total distance, i.e., doing **route** optimization. However, they are not capable of reducing the number of routes present in the original solution, which can be achieved using the third operator. The ﬁrst two stages are quite fast. Therefore it is during the last stage that new orders arriving from the i3FR-Hub to the i3FR-Opt are treated. On other words, the i3FR-Opt/Server thread is responsible for the continuous communications **with** the i3FR- Hub, and whenever new orders arrive they are placed in shared memory. After each cycle the i3FR-Opt/Optimizer checks the shared memory for new orders that will be treated as ejected customers, i.e., it tries to insert them in the existing routes or creates a new **route** if that is not possible. As mentioned, during the process, improved solutions are sent from the i3FR-Opt to the i3FR-Hub which in turn resends them to the ERP/administrator.

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The **Vehicle** **Routing** **Problem** (VRP) is a well known combinatorial optimization **problem** and many studies have been dedicated to it over the years since solving the VRP optimally or near-optimally for very large size problems has many practical applications (e.g. in various logistics systems). **Vehicle** **Routing** **Problem** **with** hard **Time** **Windows** (VRPTW) is probably the most studied variant of the VRP **problem** and the presence of **time** **windows** requires complex techniques to handle it. In fact, finding a feasible solution to the VRPTW when the number of vehicles is fixed is an NP-complete **problem**. However, VRPTW is well studied and many different approaches to solve it have been developed over the years.

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