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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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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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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This work addresses the **Capacitated** **Vehicle** **Routing** **Problem** with two-dimensional loading constraints. Given a central depot and a set of clients, where each demands a speciﬁc amount of items, the **problem** aims to deﬁne minimum cost routes for a ﬂeet of homogeneous vehicles that performs customer service. The items have rectangular shapes, they must be transported in a way that there is no overlap between them, and in some cases, sequential loading restrictions, related to the order of visiting the customers, are required. To solve the **problem**, two hybrid approaches combining heuristics and Column Generation are proposed. Furthermore, the literature’s Branch-and-Cut was used to solve a reformulation of the original model of the **problem**. The methods developed were evaluated by means of the instances used in the literature and the results were compared with those previously published. The hybrid methods achieve satisfactory results, sometimes equal to the optima known, and the Branch-and-Cut could attest to optimality for several instances.

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In this work, we cross three combinatorial optimization problems – the Traveling Salesperson **Problem** (TSP), the **Capacitated** **Vehicle** **Routing** **Problem** (CVRP) and the Quadratic Assignment **Problem** (QAP) – against four different metaheuristics: the Greedy Randomized Adaptive Search Procedure (GRASP), the Iterated Local Search (ILS), the Tabu Search (TS) and the Variable Neighborhood Search (VNS). To do that, we took instance collections of each **problem** from public libraries and made ten independent runs with each one, using an execution time limit of 600 seconds. We looked for solutions with Optimal or Better-Known Values (OBKV), according to the more recent information available on the Internet.

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We consider in this chapter the **capacitated** **vehicle** **routing** **problem** (CVRP), in which a fixed fleet of delivery vehicles of the same capacity must service known customer demands for a single commodity from a common de- pot at minimum transit costs. The CVRP has been studied in a large number of separate works in the literature, but (to our knowledge) no work addresses all the available benchmarks together, since it means solving 160 different instances. We use such a large set of instances to test the behavior of our algorithm in many different scenarios in order to give a deep analysis of it and a general view of this **problem** not biased by any ad hoc selection of indi- vidual instances. The included instances are characterized by many different features: instances from real world, theoretically motivated ones, clustered, non-clustered, with homogeneous or heterogeneous demands on customers, with the existence of drop times or not, etc.

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As Restrições (4) e (5) estabelecem que k ônibus par- tem do nó 0 (depósito) e chegam no nó n 1 (nó virtual). As Restrições (6) e (7) definem que cada nó intermediário pode ser visitado uma única vez. A Restrição (8) foi introduzida em função do limite inferior e superior propostos, kmin e kmax, visando limitar o intervalo possível para a variável k e com isso reduzir o espaço de busca do modelo e, conse- quentemente, tornar menor o tempo computacional para en- contrar a solução ótima. As Restrições (9), (10) e (11) são restrições de capacidade baseadas no trabalho de Miller et al. (1960) para o **Capacitated** **Vehicle** **Routing** **Problem** (CVRP). Elas impõem que nenhum ônibus pode carregar mais que sua capacidade. As Restrições (9) evitam a forma- ção de subrotas nos nós intermediários. As Restrições (12), (13) e (14) são restrições de tempo, as quais limitam o

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delivery cash to and from each branch everyday in a given different time window. The most suitable methodology for this **problem** would be multiple depot **vehicle** **routing** problems with time window. However, the answer of this research is new routes which can handle changes in daily operations, i.e., change in demands, change in operation time, etc., therefore, N&C needs a methodology which can give results in a short processing time. The researcher decides to use easy and quick algorithm to solve this **problem**. There are two main methodologies used in this research. First, assignment **problem** is employed and **capacitated** **vehicle** **routing** **problem** with time window (VRPTW) is used later. The assignment **problem** clusters 377 branches into 3 groups each group belongs to each DC and VRPTW produces routes for each DC daily.

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In this paper, we propose an exact method, based on a model that simul- taneously designs sectors and builds routes, and a heuristic solution method which sequentially solves the two problems. In fact, by solving a MCARP with an upper bound on the number of overlapping areas, we are able to de- sign “nice” routes. These methods may then directly be applied to a sectors design **problem**, if each sector is assigned to one **vehicle**.

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In [9] a simple classification of the PCARP is made and a **problem** in mixed arcs is proposed, suggesting a Memetic Algorithm (MA) to solve the **problem**. A Memetic Algorithm, according to the author is a hybrid genetic algorithm with a local search. The name was proposed by [10], being adopted in the work because the hybrid genetic algorithms are very diffuse, since the algorithm has been hybridized with many other techniques, such as neural networks and simulated annealing. A linear programming mathematical model is not stated, but computational tests have been executed to evaluate the proposed heuristic comparing with the periodical **vehicle** **routing** **problem** (PVRP).

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A network is normally represented by a graph that is composed of a set of nodes and edges. The task of network clustering is to divide a network into different clusters based on certain principles. Each cluster is called a community. The LRP combines two classical planning tasks in logistics, that is, optimally locating depots and planning **vehicle** routes from these depots to geographically scattered customers [8]. These two interdependent problems have been addressed separately for a long time, which often leads to suboptimal planning results. The idea of LRP started in the 1960s, when the interdependence of the two problems was pointed out [9,10]. The variants of the LRP have been frequently studied in recent years. Such variants include the **capacitated** LRP (CLRP) with constraints on depots and vehicles [20,21], the LRP with multi-echelon of networks [11,12], the LRP with inventory management [13,14], and the LRP with service time windows [15–17]. For the variant **problem** with time windows, Semet and Taillard incorporated the time window constraint to the LRP for a special case of the road–train- **routing** **problem** [15]. Zarandi et al. studied the CLRP with fuzzy travel time and customer time windows, in which a fuzzy chance-constrained mathematical program was used to model the **problem** [16]. Later, they extended the **problem** by adding the fuzzy demands of customers and developed a cluster-first route-second heuristic to solve the **problem** [17]. A detailed review of the LRP variants can be found in two recent surveys [18,19].

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As far as we know, the coevolutionary paradigm has never been applied to the MDVRP. With regard to **vehicle** **routing** in general, a large scale **capacitated** arc **routing** **problem** is addressed in Mei, Li, and Yao (2014) using a coevolutionary algorithm. In this work, the routes are grouped into different subsets to be optimized and prob- lem instances with more than 300 edges are solved. A multi-objective **capacitated** arc **routing** **problem** is also studied in Shang et al. (2014). A coevolutionary algorithm is presented in Wang and Chen (2013b) for a pickup and delivery **problem** with time windows. To minimize the number of vehicles and the total traveling distance, the authors use two populations: one for diversiﬁcation purposes and the other for intensiﬁcation purposes. In the scheduling domain, a competi- tive coevolutionary quantum genetic algorithm for minimizing the makespan of a job shop scheduling **problem** is reported in Gu, Gu, Cao, and Gu (2010).

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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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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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Over the past 20 years, the **Vehicle** **Routing** **Problem** (VRP) was mainly solved through the use of meta- heuristics (see [7] and [8]). Ref. [9] carried out a taxonomic review of VRP characterizing this research field, and conducted a detailed classification of variants with many examples. Following the review of previous classifications and taxonomies, major journals having published articles on the subject issue are listed, and a taxonomy is proposed. They conducted a classification by type of study, scenario characteristics, physical characteristics of the **problem**, and by characteristics of information and data used. Ref. [10] in turn conducted a review of biologically-inspired algorithms used to handle the VRP. He highlighted the different variants of the **problem** and the different methodologies used to solve them. These include evolutionary algorithms, ant colonies, particle swarm optimization, neural networks, artificial immune systems and hybrid algorithms. Ref. [11] for their part conducted a review of the state of the art of large scale VRP, indicating the difficulty of solving the problems of more than 100 customers with exact methods. They criticized the major works on large scale VRP by highlighting the techniques used. The review compared the performance of different algorithms and conducted an analysis based on key attributes such as effectiveness, efficiency, simplicity, and flexibility.

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Three types of dynamic events can be highlighted: 1) the first are related to the business itself and are handled by decision makers personally because they require human in- tervention in order to adjust the planning process (e.g. new pick-up requests, activity priority changes , etc.); 2) random operational events that occur in the process and can be rep- resented by probability distributions (e.g. such as machine breakdown, **vehicle** breakdown, etc.); and 3) other stochas- tic situations which could be identified after some observa- tions are made (like traffic jams, weather conditions, etc.). The second and third types of event do not require human intervention, and can be represented by probability distribu- tions and identified by sensors. In this work, we made a pro- posal where vehicles could deal with traffic jams (an event of the third type).

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The first class is often referreded as **Vehicle** **Routing** **Problem** with Backhauls (VRPB). In such **problem** customers have either a delivery demand (linehaul) or a pickup demand(backhaul). However all delivery demands must be performed before the pickup demands, thus mixed load are not allowed at any point. This constraint is important when the vehicles are rear-loaded and the rearrangement of the loads at the customers is not deemed feasible, as stated by Goetschalckx and Jacobs-Blecha in [13]. The multiple **vehicle** variant of this **problem** is the most studied in the literature, approached in several papers. Brand˜ao in [3] proposed a Tabu Search (TS), and Ganesh and Narendran proposed a constructive heuristic and a Genetic Algorithm (GA) in [12]. In addition, in 2009, Gajpal and Abad [11] have presented a Multi-ant Colony System (MACS). Recently Zachariadis and Kiranoudis have developed an effective local search approach in [32]. Some authors have also approached the VRPB with time windows constraints. Among the approaches for this variant are an Insertion Based Ant System proposed by Reimann et al. in [26] and a heuristic based on Large Neighborhood Search, proposed by Ropke and Pisinger in [27].

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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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Evolutionary algorithms and genetic algorithms (GA), its most popular representative, are part of the research area of artiﬁcial intelligence inspired by the natural evolution theory and genetics, known as evolutionary computation. Those algorithms try to simulate some aspects of Darwin’s natural selection and have been used in several areas to solve problems considered intractable (NP-complete and NP-hard). Although these methods provide a general tool for solving optimization problems, their traditional versions [26,11,15] do not demonstrate much efﬁciency in the resolution of high complexity combinatorial optimization (CO) problems. This deﬁciency has led researchers to propose new hybrid evolutionary algorithms (HEA) [8,24,5], sometimes named ‘‘memetic algorithms’’ ([20,21], which usually combine better con- structive algorithms, local search and specialized crossover operators. The outcome of these hybrid versions is generally better than independent versions of these algo- rithms. In this paper we propose an HEA for a **routing** **problem** which incorporates all features cited before plus an additional module of data mining (DM), which tries to

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In this work, we propose and analyze a **routing** and scheduling **problem** of oil tankers from offshore platforms to supply terminals in the Brazilian coast. The **problem** is majorly motivated by operations of an oil company in Brazil. An optimization approach based on mixed integer programming is presented to properly represent the oil pickup at the platforms and delivery at the terminals in the context of this company. A relax-and-ix heuristic is also explored using the GAMS/CPLEX optimization package, with a view to capturing business requirements and speciic aspect of the company’s operations. Although there are other related works in the literature along this strand of research, authors are not aware of other studies exploring optimization models and solution methods addressing this pickup and delivery **problem** with all its particular operating features, except for our other work in Rodrigues (2016) that also proposed a time decomposition heuristic procedure for this **problem**. These studies were conducted in strong collaboration with a Brazilian oil company, so that the **problem** could be well deined both in the terms of market characteristics and validity of

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