# Routing problem

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### A mathematical modeling proposal for a Multiple Tasks Periodic Capacitated Arc Routing Problem

The countless accidents and incidents occurred at dams at the last years, propelled the development of politics related with dams safety. One of the strategies is related to the plan for instrumentation and monitoring of dams. The monitoring demands from the technical team the reading of the auscultation data, in order to periodically monitor the dam. The monitoring plan of the dam can be modeled as a problem of mathematical program of the periodical capacitated arcs routing program (PCARP). The PCARP is considered as a generalization of the classic problem of routing in capacitated arcs (CARP) due to two characteristics: 1) Planning period larger than a time unity, as that vehicle make several travels and; 2) frequency of associated visits to the arcs to be serviced over the planning horizon. For the dam's monitoring problem studied in this work, the frequent visits, along the time horizon, it is not associated to the arc, but to the instrument with which is intended to collect the data. Shows a new problem of Multiple tasks Periodic Capacitated Arc Routing Problem and its elaboration as an exact mathematical model. The new main characteristics presented are: multiple tasks to be performed on each edge or edges; different frequencies to accomplish each of the tasks; heterogeneous fleet; and flexibility for more than one vehicle passing through the same edge at the same day. The mathematical model was implemented and examples were generated randomly for the proposed model's validation.
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### A multi-objective Pareto ant colony algorithm for the Multi-Depot Vehicle Routing problem with Backhauls

The Multi-Depot Vehicle Routing Problem with Backhauls (MDVRPB) is an operational problem of the supply chain management. The MDVRPB considers a supply chain involving two echelons: depots and customers. The MDVRPB is an NP-hard problem, since it is a generalization of the two well-known NP- hard problems: the Multi-Depot Vehicle Routing Problem (MDVRP) (for further details see Escobar et al. 2014a) and the Vehicle Routing Problem with Backhauls (VRPB). The MDVRPB has many realistic applications in Transportation and Logistics. The features of the customers, depots and vehicles, as well as different operating constraints on the performed routes, leads to different variants of MDVRPB: (i) simultaneous collecting and dispatching of products; (ii) collecting first, following the delivery of products; and (iii) collecting after of the delivering process. We have considered a specific version for which the collection of products must be performed after the backhaul customers have been served.
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### Optimized Crossover Genetic Algorithm for Vehicle Routing Problem with Time Windows

The Vehicle Routing Problem with Time Windows (VRPTW) which is an extension of Vehicle Routing Problems (VRPs) arises in a wide array of practical decision making problems. Instances of the VRPTW occur in rail distribution, school bus routing, mail and newspaper delivery, airline and railway fleet routing and etc. In general the VRPTW is defined as follows: given a set of identical vehicles V = {1, 2, … , K} , a central depot node, a set of customer nodes

### A mixed load rural school bus routing problem with heterogeneous fleet: a study for the Brazilian problem

The implemented RRT algorithm (Li et al., 2007), illustrated in Algorithm 3, resembles a deterministic simulated annealing methodology which iterates between two phases: an Uphill (lines 5-8), which can be considered as a perturbation stage; and a Downhill (lines 11-20), which can be seen as an intensification, refinement step. In both phases, the neighborhood structures are used as a local search method to improve a given solution. Though instead of functioning as in a VND algorithm, each neighborhood structure is applied only once in the order prescribed by the permutation P given rise to the function OnePass (lines 6 and 12). Further, each phase adopts a different acceptance criterion for updating the current best solution. While during the Uphill phase, solutions worst than the best current solution, but within a given threshold (δ), are accepted as candidate solutions to be further improved; on the Downhill phase only solutions better than the overall best are accepted. The method iterates for a maximum of MaxIter iterations without improvements; while the Uphill phase lasts for MaxPert iterations. The simplicity behind the method and its fast running times for generating good solutions for instances of the classical vehicle routing problem with hundreds of nodes are the main motivations for the RRT procedure to be here adapted for the mixed load rural school bus routing problem with heterogeneous fleet.
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### The mixed capacitated arc routing problem with non-overlapping routes

Real world applications for vehicle collection or delivery along streets usually lead to arc routing problems, with additional and complicating constraints. In this paper we focus on arc routing with an additional constraint to identify vehicle service routes with a limited number of shared nodes, i.e. vehicle service routes with a limited number of intersections. This constraint leads to solutions that are better shaped for real application purposes. We propose a new problem, the bounded overlapping MCARP (BCARP), which is deﬁned as the mixed capacitated arc routing problem (MCARP) with an additional constraint imposing an upper bound on the number of nodes that are common to diﬀerent routes. The best feasible upper bound is obtained from a modiﬁed MCARP in which the minimization criteria is given by the overlapping of the routes. We show how to compute this bound by solving a simpler problem. To obtain feasible solutions for the bigger instances of the BCARP heuristics are also proposed. Computational results taken from two well known instance sets show that, with only a small increase in total time traveled, the model BCARP produces solutions that are more attractive to implement in practice than those produced by the MCARP model.
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### Route Elimination Heuristic for Vehicle Routing Problem with Time Windows

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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### Combining an evolutionary algorithm with data mining to solve a single-vehicle routing problem

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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### Contributions to the single and multiple vehicle routing problem with deliveries and selective pickup

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

A computer network is an interconnected group of computers with the ability to exchange data. Today, computer networks are the core of modern communication. Routing problem is one of the important research issues in communication networks (Jayakumar and Gopinath, 2008). An ideal routing algorithm should strive to find an optimal path for packet transmission within a specified time so as to satisfy the Quality of Service (QoS). The objective functions related to cost, time, reliability and risk are appropriated for selecting the most satisfactory route in many communication network optimization problems. Traditionally, the routing problem has been a single-objective problem of minimization of either cost or delay. However, it is necessary to take into account that many real world problems are multi-objective in nature and so is the shortest path routing problem in computer networks.
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### Multiple Charging Station Location-Routing Problem with Time Window of Electric Vehicle

This study introduces the EV multiple charging station location-routing problem with time window (EV-MCS- LRPTW). The problem incorporates the location decision and the type selection of charging infrastructure with the vehicle-routing plan，and considers customer time windows and the loading and battery capacities of vehicles. Each type of charging infrastructure is associated with a particular construction cost and charging rate. The problem is intended to provide an optimal solution with a minimal cost for the logistic enterprise that plans to adopt EVs for distribution and to construct its own charging stations. The problem minimizes the total cost, including the construction cost of the charging station, the charging cost of EVs, and the labor cost of drivers. Given that the problem extends the location- routing problem (LRP), exact solution methods are incapable of addressing realistically sized instances within reasonable time periods [8]. Thus, we develop a hybrid heuristic named adaptive variable neighborhood search (AVNS)/tabu search (TS) to solve the problem, which
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### Modelos matemáticos baseados no Time Dependent Vehicle Routing Problem para planejamento da logística urbana sob a ótica ambiental

Abstract: Urban logistics companies are seeking solutions to reduce their cost, but must of them are not paying attention to environmental issues. This is due to the belief that environmentally friendly solutions are more expensive. However, with the growing of environmental concerns, companies have been taking into account the environmental factors, seeking for their social responsibility. Thus, this paper presents two mathematical models, both based on the Time Dependent Vehicle Routing Problem (TDVRP), one to evaluate the reduction in the time of the routes and the other to evaluate the reduction of greenhouse gas emissions. In order to evaluate the model, a real case of a food distribution company in the metropolitan area of Vitória, ES was done. CPLEX 12.6 was used to run both models considering scenarios based on data from a real company. The results showed that environmentally friendly solution may be also financially advantageous for the company.
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### A solution for a real-time stochastic capacitated vehicle routing problem with time windows

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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### Modelo matemático Two-echelon Capacitated Vehicle Routing Problem para a logística de distribuição de encomendas

Abstract: Many cities are facing difficulties in urban mobility and therefore are imposing restrictions on the movement of larger trucks. Thus, logistics companies developed a two level logistics strategy based on Urban Distribution Centers (CDU) that receives larger trucks and split the cargo to put in small trucks to distribute to customers. To support this type of logistics planning, this paper presents an adaptation of a mathematical model based on the Two-echelon capacitated Vehicle Routing Problem (2E-CVRP) to plan the routes from the central depot to the satelites and from these to the clients. The model was applied to the logistics of Correios in the metropolitan area of the Espírito Santo, Brazil, and instances with up to 4 CDU and 25 clients were tested using CPLEX solver 12.6 obtaining routes for deliveries at both levels.
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### REPOSITORIO INSTITUCIONAL DA UFOP: A cooperative coevolutionary algorithm for the multi-depot vehicle routing problem.

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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### Ant Colony Optimization for Capacitated Vehicle Routing Problem

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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### Modeling a four-layer location-routing problem

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

In computer networks the routing is based on shortest path routing algorithms. Based on its advantages, an alternative method is used known as Genetic Algorithm based routing algorithm, which is highly scalable and insensitive to variations in network topology. Here we propose a coarse-grained parallel genetic algorithm to solve the shortest path routing problem with the primary goal of computation time reduction along with the use of migration scheme. This algorithm is developed and implemented on an MPI cluster. The effects of migration and its performance is studied in this paper.
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### Assignment Problem and Vehicle Routing Problem for an Improvement of Cash Distribution

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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### 13 Logistics: The Vehicle Routing Problem

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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