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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The second strand of the relevant literature consists of the **routing** problems that consider the limited driving range of vehicles and the possibility of refueling en route. Conrad and Figliozzi introduced the recharging VRP, in which vehicles **with** a limited range are allowed to recharge at certain customer stations within a fixed **time** [23]. Erdogan and Miller-Hookers presented the green VRP (G-VRP) for **routing** AFVs and solved the **problem** **with** two algorithms. In G-VRP, refueling stations are assumed to be independent of customer sites, and an AFV may refuel at these stations within a fixed **time** [24]. Later, Schneider et al. incorporated the **time** window constraint into the G-VRP and proposed the EV-**routing** **problem** **with** **time** **windows** and recharging station (E-VRPTW) [7]. The charging **time** in E-VRPTW is not fixed but instead is related to the battery charge of an EV upon arrival at the station. To address the **problem**, they developed a hybrid heuristic that combines the VNS **with** the TS **algorithm** (VNS/TS). Schneider et al. then introduced the VRP **with** intermediate stops (VRPIS), which generalized the G-VRP, and solved the **problem** by AVNS [29]. Five route selection methods and three vertex sequence selection methods were utilized in the adaptive shaking phase of AVNS. Felipe et al. proposed several heuristics to address the G-VRP **with** multiple technologies and partial recharges. The **problem** extends the G-VRP by incorporating different technologies for battery recharge and the possibility of partial recharges [25]. Goeke and Schneider combined the E- VRPTW **with** a mixed fleet of EVs and ICVs, and utilized realistic energy consumption functions in their **problem** [26]. The resulting **problem** was solved by an ALNS **with** a local search for intensification. Yang and Sun adopted the simultaneous optimization idea from the LRP to the context of EV and proposed the BSS location-**routing** **problem** of EVs [6]. The **problem** is intended to minimize infrastructure and shipping costs by determining the station location and **vehicle**-**routing** plan jointly under a driving range limitation. For the solution method, they employed the concept of solving separate sub-problems iteratively from the LRP and proposed two hybrid heuristics [22]. In detail, one **algorithm** called TS-modified Clarke–Wright saving (MCWS) combines the TS **algorithm** for location strategy and the MCWS method for the **routing** decision. The other approach named SIGALNS includes four main phases: initialization, location sub-**problem**, **routing** sub-**problem**, and improvement. Iterative greedy (IG) is utilized in the location phase, and an ALNS in the **routing** phase.

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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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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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In computer networks, the task of finding a path from source node to destination node is known as **routing**. For a given network, it consists of more than one path. Based on the shortest path, we have to find **routing** to a given network. Examples of such algorithms are Dijkstra’s & Bellman Ford algorithms. The alternative methods for shortest path **routing** algorithms have been find out by researchers. One such alternative method is the use of **Genetic** **Algorithm**, which is a multi-purpose search & optimization **algorithm**. Here it encodes the **problem** into a chromosome which has several genes and a group of chromosomes referred as a population is represented as a solution to this **problem**. For every iteration, the chromosomes in population will undergo one or more **genetic** operations such as **crossover** and mutation. The result of the **genetic** operations are the next generations of the solution. The process continues until a solution is found or a termination condition exists. The idea behind **genetic** **algorithm** is to have the chromosomes in the population to slowly converge to an optimal solution. At the same **time**, the **algorithm** is supposed to maintain enough diversity so that it can search a large search space. Based on these two characterstics, **Genetic** **Algorithm** is called as a good search and optimization **algorithm**. The earliest GeneticAlgorithm-based **algorithm** was proposed by Munetomo, to generate alternative paths in case of link failures. In the proposed **algorithm**, the **algorithm** chromosome is encoded as a list of node id’s from source to destination path. The chromosomes will be of variable length because different paths can have different number of nodes. This **algorithm** employs **crossover**, mutation and migration **genetic** operators in generating the next generation of solutions. Chang also proposed a GA-based **routing** **algorithm** which is similar to Munetomo’s **algorithm** but differs in its implementation. The **algorithm** proposed by Chang had several advantages, the first is that GA is insensitive to variations in network topologies **with** respect to route optimality and convergence speed. The second is that GA-based **routing** **algorithm** is scalable which means that the real computation size does not increase very much as the network size gets larger. However **genetic** **algorithm** is not fast enough for real **time** computation. In order to achieve a really fast computation **time** in **genetic** **algorithm**, it requires a hardware implementation. In this paper, we propose a parallel **genetic** **algorithm** for the shortest path **routing** **problem**. The notion behind this is, parallel implementation of **genetic** **algorithm** should improve its computation **time** and it can be implemented on a MPI(message passing interface) cluster.

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customer to be looked for. Finally the window violation is resolved by either reordering customers before the window violation or moving customers from the preceding positions to a new position that follows the violated customer. The number of infeasible trials can be decided by the user. If none of the trials of the given customer were successful then compute all the possible swaps of the customer **with** the earlier mentioned cost limit and try the whole procedure **with** the replaced customer. The evolution of maybe circles must be blocked by storing all the already executed swaps on the suitable list. The depth of the search gives the number of consecutive swaps as it was described at the main features of the **algorithm**. If the insertion is unsuccessful and so far no other unsuccessful insertion has been made a repair **algorithm** is initiated.

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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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The simple multi-objective method is to form a composite objective function as the weighted sum of the objectives, where a weight for an objective is proportional to the preference factor assigned to that particular objective. This method of secularizing an objective vector into a single composite objective function converts the multi-objective optimization **problem** into a single-objective optimization **problem**. In an ideal multi-objective optimization procedure, multiple trade-off solutions are found. Higher level information is used to choose one of the trade-off solutions. It is realized that, single-objective optimization is a degenerate case of multi-objective optimization. Srinivas and Deb (1994) developed Non-dominated Sorting **Genetic** **Algorithm** (NSGA) in which a ranking selection method emphasizes current non-dominated solutions and a niching method maintains diversity in the population. Chitra and Subbaraj (2010) applied NSGA to shortest path **routing** **problem** and compared its validity **with** single-objective optimization. However, NSGA suffers from three weaknesses: computational complexity, non-elitist approach and the need to specify a sharing parameter.

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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 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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A typical cement plant receives hundreds of trucks every day. Each one of them has one or more locations to visit, in order to load or unload materials, depending on each truck. This process is, in this sense, unpredictable, due to the fact that it is not possible to know the locations each truck must visit before arriving at the plant. Besides this, the truck driver usually does not know the plants ’ map, due to their big dimensions. Even if the driver already knows the facility, the choice of the route will be made only by what he knows of it. Either way, the driver will much probably follow a disadvantageous route, forcing him to stay more **time** inside the plant, causing delays to him and to other truck drivers that already are inside, or who will still enter the plant. Additionally, the driver may load or unload the materials in wrong locations, causing delays, additional costs to the company, etc. One other big **problem** caused by the trucks is the congestion in the roads of the plant. Each truck driver chooses its own route, and this ‘irreflective’ choice will overload some roads in the plant.

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This paper dealt **with** the one-dimensional cutting stock **problem** considering two conflicting objectives. In practice, hundreds of different items may need to be cut, requiring thousands of different cutting patterns, which makes even the minimization of loss **problem** difficult to solve. When other practical aspects, such as reducing production **time** lost to set-ups are taken into account, the resulting nonconvex multi-objective function is far more complex. In order to deal **with** this difficulty, we have proposed a straightforward multi-objective **genetic** **algorithm**. The proposed method is very simple, flexible in terms of the objective function to be minimized and relatively efficient in terms of computational **time**. Considering the quality of the produced solution, the method is competitive **with** other methods from the literature and it has the advan- tage of producing a set of non-dominated solutions, from which the decision maker can choose according to the many factors which influence businesses.

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Those traditional algorithms such as Cupidity **Algorithm**, Dynamic Programming **Algorithm**, are all facing the same obstacle, which is when the **problem** scale N reaches to a certain degree, the so- called “Combination Explosion” will occur. A lot of algorithms have been proposed to solve TSP. Some of them (based on dynamic programming or branch and bound methods) provide the global optimum solution. Other algorithms are heuristic ones, which are much faster, but they do not guarantee the optimal solutions. The TSP was also approached by various modern heuristic methods, like simulated annealing, evolutionary algorithms and tabu search, even neural networks. In this paper, we proposed a new **algorithm** based on Inver-over operator, for traveling salesman problems. In the new **algorithm** we will use new strategies including selection operator, replace operator and some new control strategy, which have been proved to be very efficient to accelerate the converge speed.[5]

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The present study is set upon solving a merchandise distribution **problem**, in which m establishments must be supplied from a central warehouse, utilizing a heterogeneous fleet consisting of k vehicles always leaving from the same central warehouse, where orders can be fulfilled in more than just one trip (split delivery). This particular **problem** is called SD/MF-VRP, and it has already been studied in the specialized literature, for instance, by Belfiore and Yoshizaki [21] and [22], in a mono-objective context; i.e., optimizing a single objective function at a **time**. However, this work proposes to solve for the first **time**, an SD/MF-VRP variant, taking into consideration the simultaneous optimization of several objective functions in a purely multi-objective context, where no objective function is necessarily more important than the others, facing a variant of the SD/MF-VRP **problem** **with** a concrete utility in a motorcycle factory in Paraguay, considering specific third-world restriction, for example, that not all vehicles may transit on any road due to the condition of the various routes, or because of limitations related to weight and/or height.

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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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This work is dedicated to a proposal of a multicriteria evaluation scheme for heuristic algorithms, which we called the Weight Evaluation Method (WOM). It involves an application of the Con- dorcet ranking technique, presented in Item 1.2. The initial discussion of WOM is the object of Item 1.3. Sections 2 and 3 present, respectively, quick explanations on the three problems and the four metaheuristics used in the tests. The use of the evaluation technique is detailed in Sec- tion 4 **with** the aid of an example. Section 5 presents the results of the comparison among the metaheuristics when used **with** the three problems. The conclusions are exposed in Section 6. The use of metaheuristics to find good quality solutions for discrete optimization problems has the double advantage of working **with** algorithms based on models already known and the effi- ciency of the methods themselves. This is very important when dealing **with** problems that have

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Part B illustrates a comparison of results between RPGA and two different versions of this **algorithm**, the outcomes from the introduction of two frequently used crossovers for the TSP: the PMX (partially mapped **crossover**) defined by Goldberg [9] and the OX (order **crossover**) proposed by Davis [5]. PMX builds children by choosing a subsequence of a tour from one parent and preserving the order of as many genes as possible from the other parent. OX builds children by choosing a subsequence of a tour from one parent and preserving the relative order of genes from the other parent. The difference between these crossovers is crucial as it explains why the OX performs much better than the PMX for the RPP.

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The first term of the equation corresponds to van der Waals interaction and electrostatic interaction between the protein and the ligand molecule, and the last two terms cor- respond to the ligand internal energy interaction, which also have one term for van der Waals interaction and one term for electrostatic interaction. The ligand-protein dock- ing **problem** involves millions of energy evaluations, and the computational cost of each energy evaluation increases **with** the number of the atoms of the complex ligand-protein which has thousands of atoms. To reduce the computational cost, we implemented a grid-based methodology where the protein active site is embedded in a 3D rectangular grid and on each point of the grid the electrostatic interaction energy and the van der Waals terms for each ligand atom type are pre-computed and stored, taking into account all the protein atoms. In this way the protein contribution at a given point is obtained by tri-linear interpolation in each grid cell. A random initial population of individuals is generated inside the grid. For translational genes, random values between the maximum and minimum grid sizes are generated. For flexible docking, we also generated the initial population using a Cauchy distribution. The individual translational genes are generated by adding a random perturbation (drawn from a Cauchy distribution) to the grid center coor- dinates. In this way individuals are generated **with** higher probability near the grid center, while still permitting that individuals be generated far from the center. The Cauchy distribution is given by:

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However, if the destination is quite farther from the source node, the cluster-head inserts its sequence number and rebroadcasts the RReq to all its gateway nodes. Once the RReq packet reaches the destination it will send anRRep back to the source node. In case, if the source cluster-head detects more than one route to the desired destination, it selects the most stable route rather than selecting a shortest one. In order to select a stable route, we adopt a metric called stability function proposed by Barghi et al. (2009), which wasori-ginally derived from Link Expiration **Time** (LET) tech-nique. LET is a mobility prediction metric that considers the current distance and relative velocity between two nodes. Let nodes i and j are within the transmission range defined by MAC protocol and (x i ,y i ), (νx i ,νy i ) be

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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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