Search results for: multi-vehicle routing problem

Authors: Cagri Memis, Muzaffer Kapanoglu

Abstract:

The school bus routing problem (SBRP) is a variant of the Vehicle Routing Problem (VRP) classified as a location-allocation-routing problem. In this study, the SBRP is decomposed into two sub-problems: (1) bus route generation and (2) bus stop selection to solve large instances of the SBRP in reasonable computational times. To solve the first sub-problem, we propose a genetic algorithm to generate bus routes. Once the routes have been fixed, a sub-problem remains of allocating students to stops considering the capacity of the buses and the walkability constraints of the students. While the exact method solves small-scale problems, treating large-scale problems with the exact method becomes complex due to computational problems, a deficiency that the genetic algorithm can overcome. Results obtained from the proposed approach on 150 instances up to 250 stops show that the matheuristic algorithm provides better solutions in reasonable computational times with respect to benchmark algorithms.

Keywords: genetic algorithm, matheuristic, school bus routing problem, vehicle routing problem

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Authors: Taehan Lee

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We consider a network design problem which has shortest routing restriction based on the values determined by the installed facilities on each arc. In conventional multicommodity network design problem, a commodity can be routed through any possible path when the capacity is available. But, we consider a problem in which the commodity between two nodes must be routed on a path which has shortest metric value and the link metric value is determined by the installed facilities on the link. By this routing restriction, the problem has a distinct characteristic. We present an integer programming formulation containing the primal-dual optimality conditions to the shortest path routing. We give some computational results for the model.

Keywords: integer programming, multicommodity network design, routing, shortest path

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Authors: Akshay Suhas Phalke, Manohar S. Chaudhari

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In this paper, we have discussed the multipath routing with its variants. Our purpose is to discuss the different types of the multipath routing mechanism. Here we also put the taxonomy of the multipath routing. Multipath routing is used for the alternate path routing, reliable transmission of data and for better utilization of network resources. We also discussed the multipath routing for topology hiding such as TOHIP. In multipath routing, different parameters such as energy efficiency, packet delivery ratio, shortest path routing, fault tolerance play an important role. We have discussed a number of multipath routing protocol based on different parameters lastly.

Keywords: multi-path routing, WSN, topology, fault detection, trust

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Authors: G. Sirbiladze, B. Ghvaberidze, B. Matsaberidze

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A fuzzy vehicle routing problem is considered in the possibilistic environment. A new criterion, maximization of expectation of reliability for movement on closed routes is constructed. The objective of the research is to implement a two-stage scheme for solution of this problem. Based on the algorithm of preferences on the first stage, the sample of so-called “promising” routes will be selected. On the second stage, for the selected promising routes new bi-criteria problem will be solved - minimization of total traveled distance and maximization of reliability of routes. The problem will be stated as a fuzzy-partitioning problem. Two possible solutions of this scheme are considered.

Keywords: vehicle routing problem, fuzzy partitioning problem, multiple-criteria optimization, possibility theory

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Authors: P. Parthiban, Sonu Rajak, R. Dhanalakshmi

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This paper presents an Ant Colony Optimization (ACO) approach, combined with a Speed Optimization Algorithm (SOA) to solve the Vehicle Routing Problem (VRP) with environmental considerations, which is well known as Pollution-Routing Problem (PRP). It consists of routing a number of vehicles to serve a set of customers, and determining fuel consumption, driver wages and their speed on each route segment, while respecting the capacity constraints and time windows. Since VRP is NP-hard problem, so PRP also a NP-hard problem, which requires metaheuristics to solve this type of problems. The proposed solution method consists of two stages. Stage one is to solve a Vehicle Routing Problem with Time Window (VRPTW) using ACO and in the second stage, a SOA is run on the resulting VRPTW solution. Given a vehicle route, the SOA consists of finding the optimal speed on each arc of the route to minimize an objective function comprising fuel consumption costs and driver wages. The proposed algorithm tested on benchmark problem, the preliminary results show that the proposed algorithm can provide good solutions within reasonable computational time.

Keywords: ant colony optimization, CO2 emissions, speed optimization, vehicle routing

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Authors: Eki Ruskartina, Vincent F. Yu, Budi Santosa, A. A. N. Perwira Redi

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This paper introduces symbiotic organism search (SOS) for solving capacitated vehicle routing problem (CVRP). SOS is a new approach in metaheuristics fields and never been used to solve discrete problems. A sophisticated decoding method to deal with a discrete problem setting in CVRP is applied using the basic symbiotic organism search (SOS) framework. The performance of the algorithm was evaluated on a set of benchmark instances and compared results with best known solution. The computational results show that the proposed algorithm can produce good solution as a preliminary testing. These results indicated that the proposed SOS can be applied as an alternative to solve the capacitated vehicle routing problem.

Keywords: symbiotic organism search, capacitated vehicle routing problem, metaheuristic

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Authors: Ahmed Tarek, Ahmed Alveed

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Vehicle routing problem is a well-known re-search avenue in computing. Modern vehicle routing is more focused with the GPS-based coordinate system, as the state-of-the-art vehicle, and trucking systems are equipped with digital navigation. In this paper, a new two dimensional coordinate-based algorithm for addressing the vehicle routing problem for a supply chain network is proposed and explored, and the algorithm is compared with other available, and recently devised heuristics. For the algorithms discussed, which includes the pro-posed coordinate-based search heuristic as well, the advantages and the disadvantages associated with the heuristics are explored. The proposed algorithm is studied from the stand point of a small supermarket chain delivery network that supplies to its stores in four different states around the East Coast area, and is trying to optimize its trucking delivery cost. Minimizing the delivery cost for the supply network of a supermarket chain is important to ensure its business success.

Keywords: coordinate-based optimal routing, Hamiltonian Circuit, heuristic algorithm, traveling salesman problem, vehicle routing problem

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Authors: P. Parthiban, Sonu Rajak, N. Kannan, R. Dhanalakshmi

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This paper deals with the Vehicle Routing Problem (VRP) with environmental considerations which is called Pollution Routing Problem (PRP). The objective is to minimize the operational and environmental costs. It consists of routing a number of vehicles to serve a set of customers, and determining fuel consumption, driver wages and their speed on each route segment, while respecting the capacity constraints and time windows. In this context, we presented an Ant Colony Optimization (ACO) approach, combined with a Speed Optimization Algorithm (SOA) to solve the PRP. The proposed solution method consists of two stages. Stage one is to solve a Vehicle Routing Problem with Time Window (VRPTW) using ACO and in the second stage a SOA is run on the resulting VRPTW solutions. Given a vehicle route, the SOA consists of finding the optimal speed on each arc of the route in order to minimize an objective function comprising fuel consumption costs and driver wages. The proposed algorithm tested on benchmark problem, the preliminary results show that the proposed algorithm is able to provide good solutions.

Keywords: ant colony optimization, CO2 emissions, combinatorial optimization, speed optimization, vehicle routing

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Authors: Abdolsalam Ghaderi

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In this paper, the single-depot capacitated location-routing problem under uncertainty is presented. The problem aims to find the optimal location of a single depot and the routing of vehicles to serve the customers when the parameters may change under different circumstances. This problem has many applications, especially in the area of supply chain management and distribution systems. To get closer to real-world situations, travel time of vehicles, the fixed cost of vehicles usage and customers’ demand are considered as a source of uncertainty. A combined approach including robust optimization and stochastic programming was presented to deal with the uncertainty in the problem at hand. For this purpose, a mixed integer programming model is developed and a heuristic algorithm based on Variable Neighborhood Search(VNS) is presented to solve the model. Finally, the computational results are presented and future research directions are discussed.

Keywords: location-routing problem, robust optimization, stochastic programming, variable neighborhood search

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Authors: Lydia Wahid, Mona F. Ahmed, Nevin Darwish

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The vehicle routing problem (VRP) consists of a group of customers that needs to be served. Each customer has a certain demand of goods. A central depot having a fleet of vehicles is responsible for supplying the customers with their demands. The problem is composed of two subproblems: The first subproblem is an assignment problem where the number of vehicles that will be used as well as the customers assigned to each vehicle are determined. The second subproblem is the routing problem in which for each vehicle having a number of customers assigned to it, the order of visits of the customers is determined. Optimal number of vehicles, as well as optimal total distance, should be achieved. In this paper, an approach for solving the first subproblem (the assignment problem) is presented. In the approach, a clustering algorithm is proposed for finding the optimal number of vehicles by grouping the customers into clusters where each cluster is visited by one vehicle. Finding the optimal number of clusters is NP-hard. This work presents a polynomial time clustering algorithm for finding the optimal number of clusters and solving the assignment problem.

Keywords: vehicle routing problems, clustering algorithms, Clarke and Wright Saving Method, agglomerative hierarchical clustering

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Authors: M. K. Benhaoua, A. E. H. Benyamina, T. Djeradi, P. Boulet

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In this paper, we propose adaptive routing that considers the routing of communications in order to optimize the overall performance. The routing technique uses a newly proposed Algorithm to route communications between the tasks. The routing we propose of the communications leads to a better optimization of several performance metrics (time and energy consumption). Experimental results show that the proposed routing approach provides significant performance improvements when compared to those using static routing.

Keywords: multi-processor systems-on-chip (mpsocs), network-on-chip (noc), heterogeneous architectures, adaptive routin

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Authors: K. Prasanth, S. Hafeezullah Khan, B. Haribalakrishnan, D. Arun, S. Jayapriya, S. Dhivya, N. Vijayarangan

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The main goal of our approach is to find the optimum positions for the sensor nodes, reinforcing the communications in points where certain lack of connectivity is found. Routing is the major problem in sensor network’s data transfer between nodes. We are going to provide an efficient routing technique to make data signal transfer to reach the base station soon without any interruption. Clustering and routing are the two important key factors to be considered in case of WSN. To carry out the communication from the nodes to their cluster head, we propose a parameterizable protocol so that the developer can indicate if the routing has to be sensitive to either the link quality of the nodes or the their battery levels.

Keywords: clusters, routing, wireless sensor networks, three phases, sensor networks

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Authors: Katherine Carl Payne, Moshe Dror

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The problem of minimum cost routing has been extensively explored in a variety of contexts. While there is a prevalence of routing applications based on least distance, time, and related attributes, exertion-based routing has remained relatively unexplored. In particular, the network structures traditionally used to construct minimum cost paths are not suited to representing exertion or finding paths of least exertion based on road gradient. In this paper, we introduce a topographical network or “topograph” that enables minimum cost routing based on the exertion metric on each arc in a given road network as it is related to changes in road gradient. We describe an algorithm for topograph construction and present the implementation of the topograph on a road network of the state of California with ~22 million nodes.

Keywords: topograph, RPE, routing, GIS

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Authors: Jawid Ahmad Baktash, Mursal Dawodi, Tomokazu Nagata

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With the rapid growth of Cloud Computing, Formal Methods became a good choice for the refinement of message specification and verification for Dynamic Routing in Cloud Computing. Cloud-based Dynamic Routing is becoming increasingly popular. We propose feedback in Formal Methods for Dynamic Routing and Cloud Computing; the model and topologies show how to send messages from index zero to all others formally. The responsibility of proper verification becomes crucial with Dynamic Routing in the cloud. Formal Methods can play an essential role in the routing and development of Networks, and the testing of distributed systems. Event-B is a formal technique that consists of describing the problem rigorously and introduces solutions or details in the refinement steps. Event-B is a variant of B, designed for developing distributed systems and message passing of the dynamic routing. In Event-B and formal methods, the events consist of guarded actions occurring spontaneously rather than being invoked.

Keywords: cloud, dynamic routing, formal method, Pro-B, event-B

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Authors: Tarek Aboueldahab, Hanan Farag

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Cost minimization of Multiple Vehicle Routing Problem becomes a critical issue in the field of transportation because it is NP-hard optimization problem and the search space is complex. Many researches use the hybridization of artificial intelligence (AI) models to solve this problem; however, it can not guarantee to reach the best solution due to the difficulty of searching the whole search space. To overcome this problem, we introduce the hybrid model of Discrete Particle Swarm Optimization (DPSO) with a passive congregation which enable searching the whole search space to compromise between both local and global search. The practical experiment shows that our model obviously outperforms other hybrid models in cost minimization.

Keywords: cost minimization, multi-vehicle routing problem, passive congregation, discrete swarm, passive congregation

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Authors: Samira Kalantary, Zohre Saatzade

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Wireless Mesh Networks (WMNs) are low-cost access networks built on cooperative routing over a backbone composed of stationary wireless routers. WMNs must deal with the highly unstable wireless medium. Thus, routing metrics and protocols are evolving by designing algorithms that consider link quality to choose the best routes. In this work, we analyse the state of the art in WMN metrics and propose taxonomy for WMN routing protocols. Performance measurements of a wireless mesh network deployed using various routing metrics are presented and corroborate our analysis.

Keywords: wireless mesh networks, routing protocols, routing metrics, bioinformatics

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Authors: Anna Sikharulidze, Gia Sirbiladze

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For the evaluation of unreliability levels of movement on the closed routes in the vehicle routing problem, the fuzzy operators family is constructed. The interactions between routing factors in extreme conditions on the roads are considered. A multi-criteria decision-making model (MCDM) is constructed. Constructed aggregations are based on the Choquet integral and the associated probability class of a fuzzy measure. Propositions on the correctness of the extension are proved. Connections between the operators and the compositions of dual triangular norms are described. The conjugate connections between the constructed operators are shown. Operators reflect interactions among all the combinations of the factors in the fuzzy MCDM process. Several variants of constructed operators are used in the decision-making problem regarding the assessment of unreliability and possibility levels of movement on closed routes.

Keywords: vehicle routing problem, associated probabilities of a fuzzy measure, choquet integral, fuzzy aggregation operator

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Authors: Mohanad Al-Behadili, Djamila Ouelhadj

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In this paper, a specific type of vehicle routing problem under stochastic demand (SVRP) is considered. This problem is of great importance because it models for many of the real world vehicle routing applications. This paper used a robust optimisation model to solve the problem along with the novel Simulation-Particle Swarm Optimisation (Sim-PSO) approach. The proposed Sim-PSO approach is based on the hybridization of the Monte Carlo simulation technique with the PSO algorithm. A comparative study between the proposed model and the Sim-PSO approach against other solution methods in the literature has been given in this paper. This comparison including the Analysis of Variance (ANOVA) to show the ability of the model and solution method in solving the complicated SVRP. The experimental results show that the proposed model and Sim-PSO approach has a significant impact on the obtained solution by providing better quality solutions comparing with well-known algorithms in the literature.

Keywords: stochastic vehicle routing problem, robust optimisation model, Monte Carlo simulation, particle swarm optimisation

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Authors: Akhil Dubey, Rajnesh Singh

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In modern days there occur many rapid modifications in field of ad hoc network. These modifications create many revolutionary changes in the routing. Predictive energy efficient routing is inspired on the bee’s behavior of swarm intelligence. Predictive routing improves the efficiency of routing in the energetic point of view. The main aim of this routing is the minimum energy consumption during communication and maximized intermediate node’s remaining battery power. This routing is based on food searching behavior of bees. There are two types of bees for the exploration phase the scout bees and for the evolution phase forager bees use by this routing. This routing algorithm computes the energy consumption, fitness ratio and goodness of the path. In this paper we review the literature related with predictive routing, presenting modified routing and simulation result of this algorithm comparison with artificial bee colony based routing schemes in MANET and see the results of path fitness and probability of fitness.

Keywords: mobile ad hoc network, artificial bee colony, PEEBR, modified predictive routing

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Authors: R. Poonkuzhali, M. Y. Sanavullah, M. R. Gurupriya

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Opportunistic routing is used, where the network has the features like dynamic topology changes and intermittent network connectivity. In Delay Tolerant network or Disruption tolerant network opportunistic forwarding technique is widely used. The key idea of opportunistic routing is selecting forwarding nodes to forward data and coordination among these nodes to avoid duplicate transmissions. This paper gives the analysis of pros and cons of various opportunistic routing techniques used in MANET.

Keywords: ETX, opportunistic routing, PSR, throughput

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Authors: Khaled Moh. Alhamad

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This paper describes a tabu search heuristic for a ship routing and scheduling problem (SRSP). The method was developed to address the problem of loading cargos for many customers using heterogeneous vessels. Constraints relate to delivery time windows imposed by customers, the time horizon by which all deliveries must be made and vessel capacities. The results of a computational investigation are presented. Solution quality and execution time are explored with respect to problem size and parameters controlling the tabu search such as tenure and neighbourhood size.

Keywords: heuristic, scheduling, tabu search, transportation

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Authors: Kaddi Mohammed, Benahmed Khelifa D. Benatiallah

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A sensor network consists of a large number of sensors deployed in areas to monitor and communicate with each other through a wireless medium. The collected routing data in the network consumes most of the energy of the sensor nodes. For this purpose, multiple routing approaches have been proposed to conserve energy resource at the sensors and to overcome the challenges of its limitation. In this work, we propose a new low energy consumption routing protocol for wireless sensor networks based on a meta-heuristic methods. Our protocol is to operate more fairly energy when routing captured data to the base station.

Keywords: WSN, routing, energy, heuristic

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Authors: I. Baddari, A. Riahla, M. Mezghich

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Today in the literature, we discover a lot of routing algorithms which some have been the subject of normalization. Two great classes Routing algorithms are defined, the first is the class reactive algorithms and the second that of algorithms proactive. The aim of this work is to make a comparative study between some routing algorithms. Two comparisons are considered. The first will focus on the protocols of the same class and second class on algorithms of different classes (one reactive and the other proactive). Since they are not based on analytical models, the exact evaluation of some aspects of these protocols is challenging. Simulations have to be done in order to study their performances. Our simulation is performed in NS2 (Network Simulator 2). It identified a classification of the different routing algorithms studied in a metrics such as loss of message, the time transmission, mobility, etc.

Keywords: ad-hoc network routing protocol, simulation, NS2, delay, packet loss, wideband, mobility

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Authors: Mukhtiar Singh, Sumeet Nagar

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Vehicle routing problem (VRP) is a combinatorial optimization and nonlinear programming problem aiming to optimize decisions regarding given set of routes for a fleet of vehicles in order to provide cost-effective and efficient delivery of both services and goods to the intended customers. This paper proposes the application of integrated particle swarm optimization (PSO) and genetic optimization algorithm (GA) to address the Vehicle routing problem in sugarcane industry in India. Suger industry is very prominent agro-based industry in India due to its impacts on rural livelihood and estimated to be employing around 5 lakhs workers directly in sugar mills. Due to various inadequacies, inefficiencies and inappropriateness associated with the current vehicle routing model it costs huge money loss to the industry which needs to be addressed in proper context. The proposed algorithm utilizes the crossover operation that originally appears in genetic algorithm (GA) to improve its flexibility and manipulation more readily and avoid being trapped in local optimum, and simultaneously for improving the convergence speed of the algorithm, level set theory is also added to it. We employ the hybrid approach to an example of VRP and compare its result with those generated by PSO, GA, and parallel PSO algorithms. The experimental comparison results indicate that the performance of hybrid algorithm is superior to others, and it will become an effective approach for solving discrete combinatory problems.

Keywords: fuzzy logic, genetic algorithm, particle swarm optimization, vehicle routing problem

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Authors: Aydin Sipahioglu, Gokhan Celik

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Helicopter routing problem (HRP) is finding good tours for helicopter so as to pick up and deliver personnel or material among specified nodes, mutually. It can be encountered in case of being lots of supply and demand points for different commodities and requiring delivering commodities with helicopter. For instance, to deliver personnel or material from shore to oil rig is a good example. In fact, HRP is a branch of vehicle routing problem with pickup and delivery (VRPPD). However, it has additional constraints such that fuel capacity, performance of helicopter in different altitude and temperature, and the number of maximum takeoff and landing allowed. This kind of pickup and delivery problems can be classified into 3 groups, basically. 1-1 (one to one), M-M (many to many) and 1-M-1 (one to many to one). 1-1 means each commodity has only one supply and one demand point. M-M means there can be more than one supply and demand points for each kind of commodity. 1-M-1 means commodities at depot are delivered to demand points and commodities at customers are delivered to depot. In this case helicopter takes off from its own base, complete its tour and return to its own base. In this study, we define 1-M-M-1 type HRP. That means helicopter takes off from its home base, deliver commodities among the nodes as well as between depot and customers and return to its home base. These problems have NP-hard nature. Therefore, obtaining a good solution in a reasonable time is not easy. In this study, a model is offered for 1-M-M-1 type HRP. It is shown on small scale test instances that the model can find the optimal solution.

Keywords: helicopter routing problem, vehicle routing with pickup and delivery, integer programming

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Authors: Jawid Ahmad Baktash, Tadashi Shiroma, Tomokazu Nagata, Yuji Taniguchi, Morikazu Nakamura

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The goal of this paper is to explore the use of formal methods for Dynamic Routing, The purpose of network communication with dynamic routing is sending a massage from one node to others by using pacific protocols. In dynamic routing connections are possible based on protocols of Distance vector (Routing Information Protocol, Border Gateway protocol), Link State (Open Shortest Path First, Intermediate system Intermediate System), Hybrid (Enhanced Interior Gateway Routing Protocol). The responsibility for proper verification becomes crucial with Dynamic Routing. Formal methods can play an essential role in the Routing, development of Networks and testing of distributed systems. Event-B is a formal technique consists of describing rigorously the problem; introduce solutions or details in the refinement steps to obtain more concrete specification, and verifying that proposed solutions are correct. The system is modeled in terms of an abstract state space using variables with set theoretic types and the events that modify state variables. Event-B is a variant of B, was designed for developing distributed systems. In Event-B, the events consist of guarded actions occurring spontaneously rather than being invoked. The invariant state properties must be satisfied by the variables and maintained by the activation of the events.

Keywords: dynamic rout RIP, formal method, event-B, pro-B

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Authors: Amir Mohammad Fathollahi Fard, Mostafa Hajiaghaei-Keshteli, Mohammad Mahdi Paydar

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With increasing life expectancy in developed countries, the role of home care services is highlighted by both academia and industrial contributors in Home Health Care Supply Chain (HHCSC) companies. The main decisions in such supply chain systems are the location of pharmacies, the allocation of patients to these pharmacies and also the routing and scheduling decisions of nurses to visit their patients. In this study, for the first time, an integrated model is proposed to consist of all preliminary and necessary decisions in these companies, namely, location-allocation-routing model. This model is a type of NP-hard one. Therefore, an Imperialist Competitive Algorithm (ICA) is utilized to solve the model, especially in large sizes. Results confirm the efficiency of the developed model for HHCSC companies as well as the performance of employed ICA.

Keywords: home health care supply chain, location-allocation-routing problem, imperialist competitive algorithm, optimization

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Authors: Erika Ruíz, Luis Amaya, Diego Carreño

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The main objective of this study is to design a mathematical model for the logistics of mining collection centers in the northern region of the department of Boyacá (Colombia), determining the structure that facilitates the flow of products along the supply chain. In order to achieve this, it is necessary to define a suitable design of the distribution network, taking into account the products, customer’s characteristics and the availability of information. Likewise, some other aspects must be defined, such as number and capacity of collection centers to establish, routes that must be taken to deliver products to the customers, among others. This research will use one of the operation research problems, which is used in the design of distribution networks known as Location Routing Problem (LRP).

Keywords: location routing problem, logistic, mining collection, model

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Authors: Khadija Raissi, Bechir Ben Gouissem

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In this paper, we perform the investigation of some routing protocols in Vehicular Ad-Hoc Network (VANET) context. Indeed, we study the efficiency of protocols like Dynamic Source Routing (DSR), Ad hoc On-demand Distance Vector Routing (AODV), Destination Sequenced Distance Vector (DSDV), Optimized Link State Routing convention (OLSR) and Vehicular Multi-hop algorithm for Stable Clustering (VMASC) in terms of packet delivery ratio (PDR) and throughput. The performance evaluation and comparison between the studied protocols shows that the VMASC is the best protocols regarding fast data transmission and link stability in VANETs. The validation of all results is done by the NS3 simulator.

Keywords: VANET, smart city, AODV, OLSR, DSR, OLSR, VMASC, routing protocols, NS3

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Authors: Fatemeh Torfi

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Estimation of stochastic demand in physical distribution in general and efficient transport routs management in particular is emerging as a crucial factor in urban planning domain. It is particularly important in some municipalities such as Tehran where a sound demand management calls for a realistic analysis of the routing system. The methodology involved critically investigating a fuzzy least-squares linear regression approach (FLLRs) to estimate the stochastic demands in the vehicle routing problem (VRP) bearing in mind the customer's preferences order. A FLLR method is proposed in solving the VRP with stochastic demands. Approximate-distance fuzzy least-squares (ADFL) estimator ADFL estimator is applied to original data taken from a case study. The SSR values of the ADFL estimator and real demand are obtained and then compared to SSR values of the nominal demand and real demand. Empirical results showed that the proposed methods can be viable in solving problems under circumstances of having vague and imprecise performance ratings. The results further proved that application of the ADFL was realistic and efficient estimator to face the stochastic demand challenges in vehicle routing system management and solve relevant problems.

Keywords: fuzzy least-squares, stochastic, location, routing problems

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