The article presents different approaches to finding the optimal solution for a problem, which extends the classical traveling salesman problem. Takes into consideration the possibility of choosing a toll highway and the standard road between two cities. Describes the experimentation system. Provides mathematical model, results of the investigation, and a conclusion.
This document summarizes a presentation on a hybrid approach to journey planning that minimizes environmental impact. The approach uses Dijkstra's algorithm to find the closest public transport nodes to the start and end points, and then builds a mathematical model to compute the optimal journey between the nodes. The model is a mixed integer linear program that minimizes a weighted combination of travel time and environmental cost. The approach was developed for the GreenYourMove project, which aims to create a multi-modal transport planning app that provides the most environmentally friendly routes.
Solving real world delivery problem using improved max-min ant system with lo...ijaia
This paper presents a solution to real-world delive
ry problems (RWDPs) for home delivery services wher
e
a large number of roads exist in cities and the tra
ffic on the roads rapidly changes with time. The
methodology for finding the shortest-travel-time to
ur includes a hybrid meta-heuristic that combines a
nt
colony optimization (ACO) with Dijkstra’s algorithm
, a search technique that uses both real-time traff
ic
and predicted traffic, and a way to use a real-worl
d road map and measured traffic in Japan. We
previously proposed a hybrid ACO for RWDPs that use
d a MAX-MIN Ant System (MMAS) and proposed a
method to improve the search rate of MMAS. Since tr
affic on roads changes with time, the search rate i
s
important in RWDPs. In the current work, we combine
the hybrid ACO method with the improved MMAS.
Experimental results using a map of central Tokyo a
nd historical traffic data indicate that the propos
ed
method can find a better solution than conventional
methods.
Presentation of GreenYourMove's hybrid approach in the 3rd Conference on Sust...GreenYourMove
The document summarizes a hybrid approach to solving the environmental multi-modal journey planning problem. It uses Dijkstra's algorithm to find the closest public transportation nodes to the starting and ending points, and then builds a mixed integer linear program (MILP) to compute the optimal journey between those nodes that minimizes both travel time and environmental costs. The proposed method provides a novel way to address the multi-criteria optimization challenge of journey planning across multiple transportation modes.
New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
In this paper, a proposed method, namely, zero average method is used for solving fuzzy transportation problems by assuming that a decision-maker is uncertain about the precise values of the transportation costs, demand, and supply of the product. In the proposed method, transportation costs, demand, and supply are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method, a numerical example is solved. The proposed method is easy to understand and apply to real-life transportation problems for the decision-makers.
Vehicle routing problem is a NP-hard problem, with the expansion of problem solving more difficult.
This paper proposes a hybrid behavior based on ant colony algorithm to solve the problem, ant to different
objectives in the first place as the path selection according to the analysis of the impact on the algorithm, then
define the ant behavior and design four concrete ant behavior by selecting different ways of ant behavior to
form different improved algorithm. Finally, experimental results show that the improved algorithm can solve
vehicle routing problems quickly and effectively.
A COMPARISON BETWEEN SWARM INTELLIGENCE ALGORITHMS FOR ROUTING PROBLEMSecij
Travelling salesman problem (TSP) is a most popular combinatorial routing problem, belongs to the class of NP-hard problems. Many approacheshave been proposed for TSP.Among them, swarm intelligence (SI) algorithms can effectively achieve optimal tours with the minimum lengths and attempt to avoid trapping in local minima points. The transcendence of each SI is depended on the nature of the problem. In our studies, there has been yet no any article, which had compared the performance of SI algorithms for TSP perfectly. In this paper,four common SI algorithms are used to solve TSP, in order to compare the performance of SI algorithms for the TSP problem. These algorithms include genetic algorithm, particle swarm optimization, ant colony optimization, and artificial bee colony. For each SI, the various parameters and operators were tested, and the best values were selected for it. Experiments oversome benchmarks fromTSPLIBshow that
artificial bee colony algorithm is the best one among the fourSI-basedmethods to solverouting problems like TSP.
FOLLOWING CAR ALGORITHM WITH MULTI AGENT RANDOMIZED SYSTEMijcsit
We present a new Following Car Algorithm in Microscopic Urban Traffic Models which integrates some real-life factors that need to be considered, such as the effect of random distributions in the car speed,acceleration, entry of lane… Our architecture is based on Multi-Agent Randomized Systems (MARS) developed in earlier publications
This document summarizes a presentation on a hybrid approach to journey planning that minimizes environmental impact. The approach uses Dijkstra's algorithm to find the closest public transport nodes to the start and end points, and then builds a mathematical model to compute the optimal journey between the nodes. The model is a mixed integer linear program that minimizes a weighted combination of travel time and environmental cost. The approach was developed for the GreenYourMove project, which aims to create a multi-modal transport planning app that provides the most environmentally friendly routes.
Solving real world delivery problem using improved max-min ant system with lo...ijaia
This paper presents a solution to real-world delive
ry problems (RWDPs) for home delivery services wher
e
a large number of roads exist in cities and the tra
ffic on the roads rapidly changes with time. The
methodology for finding the shortest-travel-time to
ur includes a hybrid meta-heuristic that combines a
nt
colony optimization (ACO) with Dijkstra’s algorithm
, a search technique that uses both real-time traff
ic
and predicted traffic, and a way to use a real-worl
d road map and measured traffic in Japan. We
previously proposed a hybrid ACO for RWDPs that use
d a MAX-MIN Ant System (MMAS) and proposed a
method to improve the search rate of MMAS. Since tr
affic on roads changes with time, the search rate i
s
important in RWDPs. In the current work, we combine
the hybrid ACO method with the improved MMAS.
Experimental results using a map of central Tokyo a
nd historical traffic data indicate that the propos
ed
method can find a better solution than conventional
methods.
Presentation of GreenYourMove's hybrid approach in the 3rd Conference on Sust...GreenYourMove
The document summarizes a hybrid approach to solving the environmental multi-modal journey planning problem. It uses Dijkstra's algorithm to find the closest public transportation nodes to the starting and ending points, and then builds a mixed integer linear program (MILP) to compute the optimal journey between those nodes that minimizes both travel time and environmental costs. The proposed method provides a novel way to address the multi-criteria optimization challenge of journey planning across multiple transportation modes.
New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
In this paper, a proposed method, namely, zero average method is used for solving fuzzy transportation problems by assuming that a decision-maker is uncertain about the precise values of the transportation costs, demand, and supply of the product. In the proposed method, transportation costs, demand, and supply are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method, a numerical example is solved. The proposed method is easy to understand and apply to real-life transportation problems for the decision-makers.
Vehicle routing problem is a NP-hard problem, with the expansion of problem solving more difficult.
This paper proposes a hybrid behavior based on ant colony algorithm to solve the problem, ant to different
objectives in the first place as the path selection according to the analysis of the impact on the algorithm, then
define the ant behavior and design four concrete ant behavior by selecting different ways of ant behavior to
form different improved algorithm. Finally, experimental results show that the improved algorithm can solve
vehicle routing problems quickly and effectively.
A COMPARISON BETWEEN SWARM INTELLIGENCE ALGORITHMS FOR ROUTING PROBLEMSecij
Travelling salesman problem (TSP) is a most popular combinatorial routing problem, belongs to the class of NP-hard problems. Many approacheshave been proposed for TSP.Among them, swarm intelligence (SI) algorithms can effectively achieve optimal tours with the minimum lengths and attempt to avoid trapping in local minima points. The transcendence of each SI is depended on the nature of the problem. In our studies, there has been yet no any article, which had compared the performance of SI algorithms for TSP perfectly. In this paper,four common SI algorithms are used to solve TSP, in order to compare the performance of SI algorithms for the TSP problem. These algorithms include genetic algorithm, particle swarm optimization, ant colony optimization, and artificial bee colony. For each SI, the various parameters and operators were tested, and the best values were selected for it. Experiments oversome benchmarks fromTSPLIBshow that
artificial bee colony algorithm is the best one among the fourSI-basedmethods to solverouting problems like TSP.
FOLLOWING CAR ALGORITHM WITH MULTI AGENT RANDOMIZED SYSTEMijcsit
We present a new Following Car Algorithm in Microscopic Urban Traffic Models which integrates some real-life factors that need to be considered, such as the effect of random distributions in the car speed,acceleration, entry of lane… Our architecture is based on Multi-Agent Randomized Systems (MARS) developed in earlier publications
Robot Three Dimensional Space Path-planning Applying the Improved Ant Colony ...Nooria Sukmaningtyas
This document describes an improved ant colony optimization (ACO) algorithm for 3D robot path planning to avoid obstacles. It proposes modifying the pheromone updating and transition probability functions of the ACO algorithm. Specifically, it introduces a distance factor to the transition probability function to encourage paths closer to a direct line between start and end points. It also uses fuzzy control to vary the pheromone amount based on iteration count and path length, rather than a fixed value. Simulation results show the improved ACO finds better paths with fewer iterations than the conventional ACO algorithm.
Travelling salesman problem using genetic algorithms Shivank Shah
This document describes using a genetic algorithm to solve the traveling salesman problem. It defines the traveling salesman problem as finding the shortest route for a salesman to visit each city once and return to their starting city. The method uses a genetic algorithm with operations like generating a random initial population, calculating fitness, selection for crossover using probabilities, crossover using techniques like PMX, and mutation techniques like swapping or flipping parts of routes. The goal is to evolve routes with shorter distances over multiple generations to minimize the total travel distance.
This document provides an overview of discrete choice analysis and nested logit models. It begins with a review of binary and multinomial logit models and the independence from irrelevant alternatives property. It then introduces nested logit models as a way to address IIA violations when alternatives are correlated or choices are multidimensional. The document provides an example of a nested logit specification and calculation of choice probabilities. It concludes with extensions like mixed logit models and an appendix on additional model specifications.
This document provides a review of fuzzy microscopic traffic flow models. It discusses how fuzzy logic can be used to model traffic flow and driver behavior by introducing uncertainty into variables like speed and headway. It describes fuzzy cellular automata models that represent traffic as vehicles characterized by fuzzy numbers for position and velocity. It also covers fuzzy logic car-following models that use linguistic terms and rules to model car-following behavior, and fuzzy route choice models that calculate possibility indexes to determine the most likely route. The goal of these fuzzy models is to more realistically simulate traffic flow and account for the imprecise nature of traffic data.
Adjusting the flow in crucial areas can maximize the overall throughput of traffic along a stretch of road. This is of particular interest in regions of high traffic density, which may be caused by high volume peak time traffic, accidents or closure of one or more lanes of the road.
ENTROPY BASED ASSESSMENT OF HYDROMETRIC NETWORK USING NORMAL AND LOG-NORMAL D...mathsjournal
This document describes a methodology for assessing and optimizing a hydrometric network using entropy theory. It involves computing marginal entropy, conditional entropy, and transinformation index values using normal and log-normal distributions. The technique is applied to the upper Bhima basin network in India. The results identify the station with highest marginal entropy as the highest priority station. Transinformation matrices are developed to identify redundant information between stations and derive an optimal network configuration. The first priority stations were identified as Pargaon using normal distribution and Dattawadi using log-normal distribution.
We all make choices between alternatives every day in many contexts - not just transport. There is theory to help planners forecast those decisions, but it is generally poorly understood. The aim of this presentation is to be of particular relevance to all PhD students and early career researchers - who should know something about DCM even if not planning to work in that area. No prior knowledge necessary.
Tony Fowkes first joined ITS in September 1976, coming from the University's School of Economic Studies, where he had been lecturing. Initially he worked on Car Ownership Forecasting, before working in a wide variety of areas of Transport Planning. In 1982 he joined the first UK Value of Time study, as well as a parallel project on Business Travel which led to pioneering work on Business Value of Time. On both those projects he helped to develop the new technique of Stated Preference estimation. In 1984 he began 4 years here as British Railways Senior Rail Research Fellow. He then moved to a mix of teaching and research, jointly with LUBS. He has published widely and contributed to many influential reports for government bodies. He retired in October 2016 as Reader in Transport Econometrics, and is now a Visiting Reader at ITS.
Vehicle route scheduling and transportation cost minimization in a latex indu...IJRES Journal
The vehicle route scheduling problem is concerned with the determination of routes and schedules for a fleet of vehicles to satisfy the demands of a set of customers. The goal of vehicle routing is to schedule multiple suppliers from various places. Vehicle routing has existed since the advent of the Industrial age, when large-scale production became possible. As the complexity and scale of the manufacturing world increased, the task of optimizing vehicle routing grew. The vehicle routing problem is a combinatorial optimization and integer programming problem seeking to service a number of customers with a fleet of vehicles. Often the context is that of delivering goods located at a central depot to customers who have placed orders for such goods or vice-versa. Implicit is the goal of minimizing the cost of distributing the goods. Many methods have been developed for searching for good solutions to the problem, however even for the smallest problems, finding global minimum for the cost function is computationally complex. The paper presents an optimization algorithm using Particle Swarm Optimization (PSO) for the vehicle routing that would enable the logistic manager of a latex industry to minimize the transportation cost and maximize the collection using minimum number of vehicles.
In this project, the travelling salesman problem, its complexity, variations and its applications in various domains was studied. Here, we proposed GACO to solve the complex problem and compare the result with the nearest Neighbour method, metaheuristics such as Simulated Annealing, Tabu Search and Evolutionary Algorithms like Genetic Algorithm and Ant Colony Optimization. The experimental results demonstrated that the HYBRID GACO approach of finding the solution gives the best result in terms of the optimal route travelled by the salesman as compared to other heuristics used in this project. The minimum distance travelled by the salesman is the least for GACO.
Uav route planning for maximum target coveragecseij
Utilization of Unmanned Aerial Vehicles (UAVs) in military and civil operations is getting popular. One of
the challenges in effectively tasking these expensive vehicles is planning the flight routes to monitor the
targets. In this work, we aim to develop an algorithm which produces routing plans for a limited number of
UAVs to cover maximum number of targets considering their flight range.
The proposed solution for this practical optimization problem is designed by modifying the Max-Min Ant
System (MMAS) algorithm. To evaluate the success of the proposed method, an alternative approach,
based on the Nearest Neighbour (NN) heuristic, has been developed as well. The results showed the success
of the proposed MMAS method by increasing the number of covered targets compared to the solution based
on the NN heuristic.
Mixed Integer Linear Programming Formulation for the Taxi Sharing Problemjfrchicanog
The document presents a mixed integer linear programming (MILP) formulation for solving the taxi sharing problem. The taxi sharing problem aims to optimize taxi routes by allowing passengers with similar pick-up and drop-off locations to share taxis. The formulation models the problem as sequences of passenger locations that represent taxi rides. Experiments on real-world taxi trip data show the MILP formulation finds lower cost solutions than a parallel evolutionary algorithm, especially on medium and large problem instances, demonstrating the benefits of the exact MILP approach.
The document summarizes a proposed hybrid approach to solve the environmental multi-modal journey planning problem. It combines Dijkstra's algorithm to find the closest public transportation nodes to the start and end points, with a mixed integer linear program (MILP) to compute the optimal journey between those nodes that minimizes both travel time and environmental cost. The MILP uses constraints and an objective function involving costs, times, and decision variables to model potential journeys as an optimization problem and select the best option. Future work aims to improve the model and algorithm.
The document summarizes research on using ant colony optimization (ACO) to solve the travelling salesman problem (TSP). It provides background on TSP, describes how ACO was applied to find optimal routes between cities. The researchers tested their ACO approach on sample problems with 5 cities/ants and 8 cities/ants, finding the optimal route in 31 steps. They conclude ACO provides relatively good results in a short time, making it useful for practical applications where exact methods require too much computation time.
Comparison Study of Multiple Traveling Salesmen Problem using Genetic AlgorithmIOSR Journals
This document compares solving the multiple traveling salesman problem (MTSP) using a genetic algorithm. MTSP is an extension of the traveling salesman problem where multiple salesmen must visit cities and return to a depot. The genetic algorithm represents solutions as sequences of cities visited and uses crossover and mutation operators to evolve better solutions. Experimental results on different datasets show the genetic algorithm can find good quality MTSP solutions in reasonable time, especially for large problems.
This document summarizes a research paper that proposes using a genetic algorithm to solve the travelling salesman problem (TSP). It begins by defining the TSP and explaining that it is NP-hard. The document then reviews various existing approaches that have used genetic algorithms and other metaheuristics to solve TSP. It proposes a genetic algorithm with tournament selection, two-point crossover, and interchange mutation operators. The algorithm is tested on sample problems with 15 cities and is shown to find optimal or near-optimal solutions. In conclusion, the document argues that genetic algorithms can efficiently find good solutions to TSP, especially when combined with knowledge from heuristic methods.
Presentation of GreenYourMove's hybrid approach in 3rd International Conferen...GreenYourMove
Presentation of the Journey planning problem and GreenYourMove's hybrid approach.
Dr. Georgios Saharidis, Fragogios Antonis, Rizopoulos Dimitris, Chrysostomos Chatzigeorgiou
The document summarizes a presentation on a proposed hybrid approach to solve the multi-modal journey planning problem. The approach combines mathematical programming and heuristic methods like Dijkstra's algorithm. It develops a mixed integer linear program model to minimize travel time and environmental cost. Future work aims to improve the algorithm by reducing the model's dimensionality and constraints to enhance computational speed for online applications.
This document summarizes recent research on trajectory planning algorithms for autonomous vehicles. It discusses graph search algorithms like A* that plan optimal paths but have limitations in dynamic environments. Improvements like D* and Focused D* allow recomputing only changed portions of the path. Kinematic A* adds vehicle constraints to generate smoother, safer paths. Overall, the document analyzes how these algorithms aim to enable reliable trajectory planning in unknown, changing environments.
The document discusses using the Floyd-Warshall algorithm to find the shortest paths between stoppage points in a public transportation system on a real road network. It implements the algorithm on a sample map of Pune, India, finding the shortest distances between all stoppage point pairs and then allocating vehicles to routes based on passenger needs and vehicle capacities. The results show the total distances and times required to cover the shortest paths for each allocated vehicle.
Algorithms And Optimization Techniques For Solving TSPCarrie Romero
The document discusses three algorithms - simulated annealing, ant colony optimization, and genetic algorithm - for solving the traveling salesman problem (TSP). It analyzes each algorithm's approach, parameters used, and results of experiments on 15 and 50 randomly generated cities. Simulated annealing had average distances of 4.1341 and 20.1316 units for 15 and 50 cities respectively. Ant colony optimization yielded average distances of 3.9102 units for 15 cities, running faster than simulated annealing. Genetic algorithm was tested on 15 cities in Brazil.
This document proposes an improved hybrid behavior ant colony algorithm to solve vehicle routing problems. It defines four types of ant behaviors - random, greedy, pheromone-based, and a hybrid behavior considering factors like distance, saving value, and vehicle load. The algorithm allows ants to select behaviors and routes probabilistically based on these factors. Simulation experiments on a 31-city dataset show the hybrid behavior outperforms basic ant colony and other variants, finding better solutions on average. The results demonstrate this improved algorithm can effectively solve vehicle routing problems.
Robot Three Dimensional Space Path-planning Applying the Improved Ant Colony ...Nooria Sukmaningtyas
This document describes an improved ant colony optimization (ACO) algorithm for 3D robot path planning to avoid obstacles. It proposes modifying the pheromone updating and transition probability functions of the ACO algorithm. Specifically, it introduces a distance factor to the transition probability function to encourage paths closer to a direct line between start and end points. It also uses fuzzy control to vary the pheromone amount based on iteration count and path length, rather than a fixed value. Simulation results show the improved ACO finds better paths with fewer iterations than the conventional ACO algorithm.
Travelling salesman problem using genetic algorithms Shivank Shah
This document describes using a genetic algorithm to solve the traveling salesman problem. It defines the traveling salesman problem as finding the shortest route for a salesman to visit each city once and return to their starting city. The method uses a genetic algorithm with operations like generating a random initial population, calculating fitness, selection for crossover using probabilities, crossover using techniques like PMX, and mutation techniques like swapping or flipping parts of routes. The goal is to evolve routes with shorter distances over multiple generations to minimize the total travel distance.
This document provides an overview of discrete choice analysis and nested logit models. It begins with a review of binary and multinomial logit models and the independence from irrelevant alternatives property. It then introduces nested logit models as a way to address IIA violations when alternatives are correlated or choices are multidimensional. The document provides an example of a nested logit specification and calculation of choice probabilities. It concludes with extensions like mixed logit models and an appendix on additional model specifications.
This document provides a review of fuzzy microscopic traffic flow models. It discusses how fuzzy logic can be used to model traffic flow and driver behavior by introducing uncertainty into variables like speed and headway. It describes fuzzy cellular automata models that represent traffic as vehicles characterized by fuzzy numbers for position and velocity. It also covers fuzzy logic car-following models that use linguistic terms and rules to model car-following behavior, and fuzzy route choice models that calculate possibility indexes to determine the most likely route. The goal of these fuzzy models is to more realistically simulate traffic flow and account for the imprecise nature of traffic data.
Adjusting the flow in crucial areas can maximize the overall throughput of traffic along a stretch of road. This is of particular interest in regions of high traffic density, which may be caused by high volume peak time traffic, accidents or closure of one or more lanes of the road.
ENTROPY BASED ASSESSMENT OF HYDROMETRIC NETWORK USING NORMAL AND LOG-NORMAL D...mathsjournal
This document describes a methodology for assessing and optimizing a hydrometric network using entropy theory. It involves computing marginal entropy, conditional entropy, and transinformation index values using normal and log-normal distributions. The technique is applied to the upper Bhima basin network in India. The results identify the station with highest marginal entropy as the highest priority station. Transinformation matrices are developed to identify redundant information between stations and derive an optimal network configuration. The first priority stations were identified as Pargaon using normal distribution and Dattawadi using log-normal distribution.
We all make choices between alternatives every day in many contexts - not just transport. There is theory to help planners forecast those decisions, but it is generally poorly understood. The aim of this presentation is to be of particular relevance to all PhD students and early career researchers - who should know something about DCM even if not planning to work in that area. No prior knowledge necessary.
Tony Fowkes first joined ITS in September 1976, coming from the University's School of Economic Studies, where he had been lecturing. Initially he worked on Car Ownership Forecasting, before working in a wide variety of areas of Transport Planning. In 1982 he joined the first UK Value of Time study, as well as a parallel project on Business Travel which led to pioneering work on Business Value of Time. On both those projects he helped to develop the new technique of Stated Preference estimation. In 1984 he began 4 years here as British Railways Senior Rail Research Fellow. He then moved to a mix of teaching and research, jointly with LUBS. He has published widely and contributed to many influential reports for government bodies. He retired in October 2016 as Reader in Transport Econometrics, and is now a Visiting Reader at ITS.
Vehicle route scheduling and transportation cost minimization in a latex indu...IJRES Journal
The vehicle route scheduling problem is concerned with the determination of routes and schedules for a fleet of vehicles to satisfy the demands of a set of customers. The goal of vehicle routing is to schedule multiple suppliers from various places. Vehicle routing has existed since the advent of the Industrial age, when large-scale production became possible. As the complexity and scale of the manufacturing world increased, the task of optimizing vehicle routing grew. The vehicle routing problem is a combinatorial optimization and integer programming problem seeking to service a number of customers with a fleet of vehicles. Often the context is that of delivering goods located at a central depot to customers who have placed orders for such goods or vice-versa. Implicit is the goal of minimizing the cost of distributing the goods. Many methods have been developed for searching for good solutions to the problem, however even for the smallest problems, finding global minimum for the cost function is computationally complex. The paper presents an optimization algorithm using Particle Swarm Optimization (PSO) for the vehicle routing that would enable the logistic manager of a latex industry to minimize the transportation cost and maximize the collection using minimum number of vehicles.
In this project, the travelling salesman problem, its complexity, variations and its applications in various domains was studied. Here, we proposed GACO to solve the complex problem and compare the result with the nearest Neighbour method, metaheuristics such as Simulated Annealing, Tabu Search and Evolutionary Algorithms like Genetic Algorithm and Ant Colony Optimization. The experimental results demonstrated that the HYBRID GACO approach of finding the solution gives the best result in terms of the optimal route travelled by the salesman as compared to other heuristics used in this project. The minimum distance travelled by the salesman is the least for GACO.
Uav route planning for maximum target coveragecseij
Utilization of Unmanned Aerial Vehicles (UAVs) in military and civil operations is getting popular. One of
the challenges in effectively tasking these expensive vehicles is planning the flight routes to monitor the
targets. In this work, we aim to develop an algorithm which produces routing plans for a limited number of
UAVs to cover maximum number of targets considering their flight range.
The proposed solution for this practical optimization problem is designed by modifying the Max-Min Ant
System (MMAS) algorithm. To evaluate the success of the proposed method, an alternative approach,
based on the Nearest Neighbour (NN) heuristic, has been developed as well. The results showed the success
of the proposed MMAS method by increasing the number of covered targets compared to the solution based
on the NN heuristic.
Mixed Integer Linear Programming Formulation for the Taxi Sharing Problemjfrchicanog
The document presents a mixed integer linear programming (MILP) formulation for solving the taxi sharing problem. The taxi sharing problem aims to optimize taxi routes by allowing passengers with similar pick-up and drop-off locations to share taxis. The formulation models the problem as sequences of passenger locations that represent taxi rides. Experiments on real-world taxi trip data show the MILP formulation finds lower cost solutions than a parallel evolutionary algorithm, especially on medium and large problem instances, demonstrating the benefits of the exact MILP approach.
The document summarizes a proposed hybrid approach to solve the environmental multi-modal journey planning problem. It combines Dijkstra's algorithm to find the closest public transportation nodes to the start and end points, with a mixed integer linear program (MILP) to compute the optimal journey between those nodes that minimizes both travel time and environmental cost. The MILP uses constraints and an objective function involving costs, times, and decision variables to model potential journeys as an optimization problem and select the best option. Future work aims to improve the model and algorithm.
The document summarizes research on using ant colony optimization (ACO) to solve the travelling salesman problem (TSP). It provides background on TSP, describes how ACO was applied to find optimal routes between cities. The researchers tested their ACO approach on sample problems with 5 cities/ants and 8 cities/ants, finding the optimal route in 31 steps. They conclude ACO provides relatively good results in a short time, making it useful for practical applications where exact methods require too much computation time.
Comparison Study of Multiple Traveling Salesmen Problem using Genetic AlgorithmIOSR Journals
This document compares solving the multiple traveling salesman problem (MTSP) using a genetic algorithm. MTSP is an extension of the traveling salesman problem where multiple salesmen must visit cities and return to a depot. The genetic algorithm represents solutions as sequences of cities visited and uses crossover and mutation operators to evolve better solutions. Experimental results on different datasets show the genetic algorithm can find good quality MTSP solutions in reasonable time, especially for large problems.
This document summarizes a research paper that proposes using a genetic algorithm to solve the travelling salesman problem (TSP). It begins by defining the TSP and explaining that it is NP-hard. The document then reviews various existing approaches that have used genetic algorithms and other metaheuristics to solve TSP. It proposes a genetic algorithm with tournament selection, two-point crossover, and interchange mutation operators. The algorithm is tested on sample problems with 15 cities and is shown to find optimal or near-optimal solutions. In conclusion, the document argues that genetic algorithms can efficiently find good solutions to TSP, especially when combined with knowledge from heuristic methods.
Presentation of GreenYourMove's hybrid approach in 3rd International Conferen...GreenYourMove
Presentation of the Journey planning problem and GreenYourMove's hybrid approach.
Dr. Georgios Saharidis, Fragogios Antonis, Rizopoulos Dimitris, Chrysostomos Chatzigeorgiou
The document summarizes a presentation on a proposed hybrid approach to solve the multi-modal journey planning problem. The approach combines mathematical programming and heuristic methods like Dijkstra's algorithm. It develops a mixed integer linear program model to minimize travel time and environmental cost. Future work aims to improve the algorithm by reducing the model's dimensionality and constraints to enhance computational speed for online applications.
This document summarizes recent research on trajectory planning algorithms for autonomous vehicles. It discusses graph search algorithms like A* that plan optimal paths but have limitations in dynamic environments. Improvements like D* and Focused D* allow recomputing only changed portions of the path. Kinematic A* adds vehicle constraints to generate smoother, safer paths. Overall, the document analyzes how these algorithms aim to enable reliable trajectory planning in unknown, changing environments.
The document discusses using the Floyd-Warshall algorithm to find the shortest paths between stoppage points in a public transportation system on a real road network. It implements the algorithm on a sample map of Pune, India, finding the shortest distances between all stoppage point pairs and then allocating vehicles to routes based on passenger needs and vehicle capacities. The results show the total distances and times required to cover the shortest paths for each allocated vehicle.
Algorithms And Optimization Techniques For Solving TSPCarrie Romero
The document discusses three algorithms - simulated annealing, ant colony optimization, and genetic algorithm - for solving the traveling salesman problem (TSP). It analyzes each algorithm's approach, parameters used, and results of experiments on 15 and 50 randomly generated cities. Simulated annealing had average distances of 4.1341 and 20.1316 units for 15 and 50 cities respectively. Ant colony optimization yielded average distances of 3.9102 units for 15 cities, running faster than simulated annealing. Genetic algorithm was tested on 15 cities in Brazil.
This document proposes an improved hybrid behavior ant colony algorithm to solve vehicle routing problems. It defines four types of ant behaviors - random, greedy, pheromone-based, and a hybrid behavior considering factors like distance, saving value, and vehicle load. The algorithm allows ants to select behaviors and routes probabilistically based on these factors. Simulation experiments on a 31-city dataset show the hybrid behavior outperforms basic ant colony and other variants, finding better solutions on average. The results demonstrate this improved algorithm can effectively solve vehicle routing problems.
A comparative study of initial basic feasible solution methodsAlexander Decker
This document compares three methods for obtaining an initial basic feasible solution for transportation problems: Vogel's Approximation Method (VAM), a Proposed Approximation Method (PAM), and a new Minimum Transportation Cost Method (MTCM). It applies all three methods to solve a sample transportation problem with 4 sources and 6 destinations. All three methods produce the same optimal solution and total transportation cost of 450. The document concludes VAM, PAM, and the new MTCM all provide viable options for obtaining the initial basic feasible solution for this transportation problem.
A comparative study of initial basic feasible solution methodsAlexander Decker
This document compares three methods for finding an initial basic feasible solution for transportation problems: Vogel's Approximation Method (VAM), a Proposed Approximation Method (PAM), and a new Minimum Transportation Cost Method (MTCM). It presents the algorithms for each method and applies them to a sample transportation problem. The MTCM provides not only the minimum transportation cost but also an optimal solution, unlike VAM and PAM which sometimes only find a close to optimal solution. The document aims to evaluate which initial basic feasible solution method works best.
This document discusses multi-modal journey planning and describes a proposed solution approach. It summarizes the multi-modal journey planning problem, characteristics, previous work, and proposes a hybrid approach using a mathematical programming model combined with heuristic methods like Dijkstra's algorithm. The approach involves using the programming model to solve the multi-modal journey planning problem after applying Dijkstra's algorithm and graph techniques to pre-process the data.
This document discusses intelligent traffic light control using multi-agent reinforcement learning. It summarizes three research papers on the topic. The first paper proposes a distributed Q-learning approach that considers both motorized and non-motorized traffic to achieve near-global optimization. The second designs a two-stage negotiation system where traffic lights determine green times based on real-time traffic conditions. The third applies particle swarm optimization to find optimal light cycles for large vehicular networks under various scenarios.
This document summarizes a study that uses Ant Colony Optimization (ACO) to optimize the capacity of a railway terminal station. The capacity problem is formulated as a Travelling Salesman Problem (TSP) where arrival/departure events are nodes and the schedule length is the tour length. The ACO algorithm is applied to find an optimal schedule that maximizes the number of trains departing per hour while satisfying constraints. Simulation results show ACO produces superior solutions to domain experts and validate formulating the capacity problem as a TSP. The study contributes an application of soft computing techniques to solve a combinatorial optimization problem in transportation planning.
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1. 1
Experimentation system for supporting
solving Time and Cost Dependent TSP
Problem
Traveling Eco-Salesman
Adrian Aleksander Strugala
Faculty of Electronics
Wroclaw University of Science and Technology
Wrocław, Poland
209227@student.pwr.wroc.edu.pl
Abstract— The article presents different approaches to finding the optimal solution for a problem, w hich extendsthe classical
traveling salesman problem. Takes into consideration possibility of choosing toll highw ay and the standard road betw een two
cities. Describes experimentation system. Provides mathematical model, results of the investigation, and conclusion.
Index Terms— Traveling Salesman Problem; Time and Cost dependent; Ant Colony Optimization; Nearest Neighbour; God;
artificialintelligence; AI
—————————— ——————————
1 INTRODUCTION
Observing the past centuries we can conclude, that
the humanity tends to search for optimal roads.
Merchants, travelers, and explorers were creating more
and more detailed maps [1]. In the digital era, everyone
can use free and open real-time maps found on the
Internet. Some of them provide a feature for finding the
shortest road connecting a set of cities [2], but very few
can answer common question “Is it really worth to pay
for toll highway or should I just stay on the free road?” A
solution of Traveling Eco-Salesman Problem provides
an answer to this question.
The Traveling Salesman Problem describes a
theoretical problem of finding the least expensive road
(in the meaning of time, money or distance) connecting
given a collection of cities [3] [4].
Traveling Eco-Salesman Problem introduces a new
extension to the classical problem: each of the city can
be connected to each other both by regular – free road
or toll highway. This is modification inspired by [5].
Following assumptions are made: every city is
connected with each other by free road. Additionally,
cities may be connected by toll highway, but they do not
have to. Cost of traveling toll road is greater than 0 and
time of traveling toll road is shorter than the time of
traveling the corresponding free road. There is given
initial and ending city. Each city has to be visited once
and cannot be visited more than once. The goal is to find
the optimal (based on time and cost of travel) road
connecting all the cities [6].
Four different approaches to cost management are
being taken into consideration:
The first is; only free roads can be chosen. This is
classical TSP.
The second is; toll roads are chosen whenever
possible. The result of this approach is the fastest
possible solution but uses the highest amount of money.
The third is; there is given additional parameter – limit
of combined cost of all chosen toll roads. The limit must
not be exceeded.
The final is; the computed solution is optimal in the
meaning of minimizing Goal Function (7), where goal
depends on the cost of the chosen roads and time
difference between toll highway and free road for every
chosen toll road.
Three algorithms for solving TSP are implemented
and compared: Ant Colony Optimization, Nearest
Neighbour, and God.
Following sections, covers listed topics. 2 – statement
of the problem with the introduction of the mathematical
model. 3 – description of four different approaches to
finding the optimal solution for the problem and
description of three implemented TSP engines. 4 –
description of experimentation system with examples. 5
– results of the investigation. 6 - conclusion, and plans
for future work and 7 - references.
2 PROBLEM STATEMENT
The whole system is represented as a directed graph
G(V,E).
Vertices of this graph are a representation of cities.
𝑉 = ( 𝑣0, 𝑣1,… , 𝑣𝑛
) (1)
A number of vertices is equal to n.
2. 2
Each edge is represented as a value of time needed
for travel from origin todestination city. They are two sets
of arcs – one for free roads and one for the toll. They are
collected in matrices. For free roads, it is written as a
matrix 𝑇𝑅n×n. Analogical, distances for toll roads are 𝑇𝑇 n×n.
To answer a question when toll road is worth to
choose, a correlation between the cost of the road and
time of traveling this road had to be formulated.
This correlation is a statement of Goal Factor (GP)
(2) for a single path:
𝐺𝑃 = 𝐶 × 𝑇 × 𝐼 [€ × 𝑠] (2)
Where GP is the value of Goal Factor, C is the cost of
the travel, T is a time of the travel for a single path and I
is practical importance factor based on the value of
saved time.
Empirical meaning of the Goal Factor is to find a road
connecting two cities with the possible minimal time and
cost of the travel.
Following assumptions are made:
1) Time of traveling free road (TF) is always longer
than traveling on the toll road (TT):
∀( 𝑎 < 𝑛, 𝑏 < 𝑛) 𝑇𝑅 𝑎,𝑏
> 𝑇 𝑇 𝑎,𝑏
(3)
2) Cost of traveling free road (CF) is always lower
than traveling on the toll road (CT) :
∀( 𝑎 < 𝑛, 𝑏 < 𝑛) 𝐶 𝑅 𝑎,𝑏
< 𝐶 𝑇 𝑎,𝑏
(4)
3) Cost of traveling free road is a cost of gasoline
only (CG). Cost of gasoline is calculated as the
multiplication of distance, average combustion
and price of fuel:
𝐶 𝐹 = 𝐶 𝐺 = 𝑠 × 𝑐𝑜𝑚𝑏𝑢𝑠𝑡𝑖𝑜𝑛 × 𝑓𝑢𝑒𝑙 𝑝𝑟𝑖𝑐𝑒 [€] (5)
4) Cost of traveling toll road is a cost of highway
payment (CP) and gasoline (CG). Combustion on
the highway is 1.25 times higher than on the
national road, because of higher average
velocity:
𝐶 𝑇 = 𝐶 𝐺 × 1.25 + 𝐶 𝑃[€] (6)
The final Goal Function (7) is the minimal sum of
Goal Factors for single paths. This is the key function of
the future research:
𝐺 = min∑ 𝐺 𝑃(𝑎,𝑏)
[€ × 𝑠] (7)
3 SOLUTION TO THE PROBLEM
The basis for solving the problem is to find optimal
route for TSP. This is done by Ant Colony Optimization
algorithm, [6] Nearest Neighbour [9] or God [10]
algorithms. Pseudocode of Ant Colony Optimization is
presented in Figure 1., pseudocode of Nearest
Neighbour is presented in Figure 2. and pseudocode of
God is presented in Figure 3.
Ants are released from the hive. Their goal is to
solve Traveling Salesman Problem. According to a real-
life example, at first, they are choosing the route
randomly and leaving pheromones. The strongest
pheromone trail is left on the most commonly used
route, which is considered as the optimal path.
Figure 1. Pseudocode of Ant Colony Optimization
Nearest Neighbour was one of the first solutions to
TSP problem [9]. The idea is very straightforward.
Starting from the initial city, the program chooses the
nearest city already considered. Then adds it to the path
and marks as visited. City, which is set as an ending
point, is simply added as the last city in the path.
Figure 2. Pseudocode of Attention Whore
God algorithm comes from human considerations
of the beginning of the word and life [10]. One of the
theories says that at beginning of everything God
created our universe. In this theory, God created an
infinite number of universes, with random laws of
physics, rules of chemistry and other rules governing
the world. Universes were collapsing one after another,
and only one of them survived – with the optimal rules.
This is our universe.
Adjusting God to the TSP is as follows. The program
chooses random route connecting all the cities,
numerous times. It uses all the strengths of parallel
programming – drawing random routes may occur in the
same. The only limit is a computational power of the
Input: distance matrix, no. of ants, no. of cities
Output: shortest route
1: current shortest route = initial solution;
2: while (ant can chose different path than current shortest)
3: while (ant < no. of ants)
4: while (no. of cities in current path != no. of cities)
5: select city i with probability based on distance and
pheromone matrices;
6: add city i to shortest path for this ant;
7: leave pheromone trial;
8: shortest path in this iteration = compare paths for each ant
and take the shortest;
9: if (shortest path in this iteration < current shortest path)
10: current shortest path = minimum path in this iteration;
11: evaporate pheromone trials;
12: shortest route = current shortest path;
Input: distance matrix, no. of cities
Output: shortest route
1: current shortest route = initial solution;
2: add starting city as the first;
2: for (no. of cities -2)
3: find nearest neighbour (exclude already visited);
4: add nearest neighbour to path;
5: mark nearest neighbour as visited;
6: add ending city as the last;
7: shortest route = found path;
3. 3
computer.
Figure 3. Pseudocode of God
The field of investigations consists of finding the most
optimal, in the meaning of minimizing Goal (7), solution.
This is done by approaching the problem of cost
management in four different ways.
Following scenarios were implemented based on Ant
Colony Optimization algorithm. Each of the scenarios
puts different constraints on the TSP algorithms and
changes its behavior significantly.
Only free roads. The first idea is to answer the
question if choosing toll roads is profitable in any case.
In this scenario the only cost of the travel is the cost of
fuel and importance parameter is equal to 1:
𝐼 = 1 (11)
Toll roads whenever possible. The second idea is
to investigate this scenario: Choosing toll road is always
profitable. This is opposite approach to the first one.
I parameter from equation (2) is calculated as follows:
𝐼 =
𝐶 𝐹
𝐶 𝑇
(12)
Cost limit. The third idea is to introduce additional
constraint of limiting overall cost of travel. This is the
most flexible approach and the most practical for future
users.
∑ 𝐶(𝑎,𝑏) < 𝐿 (13)
This scenario goal is to prioritize section of roads with
the highest gain over those the least profitable. The Goal
(7) is strongly dependent on financial capabilities of the
user. This is the most promising approach for future
commercial development.
Evaluation of the cost of time. The fourth and final
idea is to take into consideration the relationship
between cost and time parameters calculated for the
Goal Factors (2). Then, for every part of the road chose
separately if it is worth to use toll or free road.
During execution of the TSP engine, the algorithm
builds path based on this evaluation of the cost of time.
4 EXPERIMENTATION SYSTEM
All the investigations are made in dedicated
experimentation system, named Traveling Eco-
Salesman. Prefix “eco” comes from economical
approach underlined in solutions presented in this
article.
The program is written in C#, Microsoft object-
oriented programming language and uses REST
protocol to receive information from the Internet in JSON
format. It communicates with Google Maps API to get
real-time information about the time needed to travel
between chosen cities [7]. That information populates
matrices 𝑇𝑅 and 𝑇𝑇 .
Similarly, matrix CP is filled by actual data of cost
downloaded from Michelin Maps API, ViaMichelin [8].
The program takes a list of cities as an input, for
which user wants to calculate optimal route. The first city
is an origin, last is a final destination.
The initial user interface is represented in Figure 4.
The position of each city is represented by a dot on the
cartographic grid.
The user has an access to 6 different features in form
of solutions described in Section 3 – four approaches to
cost management and two additional TSP engines.
What is more, the user is given an ability to display the
optimal road on the screen.
Figure 4. Initial user interface
After selecting one of the options, the screen shows
calculated route. Optimal path, considering all factors is
shown in Figure 5. In this example, God TSP engine was
chosen.
Figure 5. Example of God functionality
Input: distance matrix, no. of cities, no. of universes
Output: shortest route
1: current shortest route = initial solution;
2: parallel for (no. no universes)
3: draw random route
4: shortest route = find shortest route in random routes;
4. 4
Section of the road is represented as a continuous
line for free road or dotted line for toll one.
Application displays route between cities, which
specification if a toll or free way is preferable on every
part of the route. The application gives information on
duration of the travel and cost of toll fees. For test
purposes, additionally, the value of Goal Function and
duration of execution of the TES algorithm are visible.
5 INVESTIGATION
The research was done on three various datasets
with unique features. The first was set of cities: Como,
Verona, Florence, Pisa, Turin, Milan, Genoa, and
Bergamo. All the cities are located in northern Italy. This
region is very well communicated and, because of
mountainous terrain, the time difference between
traveling toll and the free road is significant. This set is
named Italy for a shortcut.
The second is set of cities in Poland: Wroclaw,
Gdansk, Krakow, Poznan, Warsaw, and Lublin. This
region is not that well communicated and toll fee is
relatively high. This set is named Poland for a shortcut.
The last set of cities are locations all over Europe:
Moscow, Warsaw, Barcelona, London, Prague,
Budapest, Paris, Amsterdam, Madrid, Oslo, Berlin,
Vienna, Athens, Rome, Stockholm, Istanbul, and Lisbon
Those cities are located in different countries and tests
also connection between different regions in Europe.
This set is named Europe for a shortcut.
The first field of investigation was the influence of
limit (13) in the Limit approach on the value of Goal
Function (7). This investigation was done based on Italy
dataset.
In Table 1. results of the exaction of the TSP with the
limit approach are collected. Based on those results
influence of limit parameter (L) on the value of Goal
Function (7) is clearly visible in Figure 9.
Table 1. Influence of parameter L on the value of Goal Function
Limit [€] Cost [€] Goal [€ × s]
0 0 380.586
7 3.5 388.3
14 12.2 385.553
21 12.2 385.553
28 27.9 313.413
35 27.9 313.413
42 27.9 313.413
49 43.1 270.51
56 43.1 270.51
63 61.2 292.605
70 61.2 292.605
77 61.2 292.605
84 77.8 254.318
Figure 6. Chart showing the influence of parameter L to value of
Goal Function
The general trend is, that with increasing limit of
cost, the value of Goal Function (7) decreases. For
some intervals of the cost, the value of Goal Function
(7) stays on the same level.
The upper limit of research range was the value of
cost received in the approach of using toll roads
whenever possible.
The second field of investigation was an analysis of
the optimal approach to management of the cost.
Dataset used in this comparison is set of cities from
northern Italy. Results collected in Table 2 shows the
best-received results for 30 executions of the program.
Table 2. Comparison of results for different approaches to cost
management
Duration
[hh:mm:ss]
Cost
[€]
Goal [€
× s]
Time of
execution
[ms]
Free 20:04:25 0 380.586 816
Toll 12:05:01 77.8 254.318 933
Limit 12:05:01 77.8 254.318 610
Evaluation 13:10:16 55.2 223.604 692
Considering the value of Goal Function (7) as a
criterion for choosing the optimal solution to handling
the cost of travel; evaluation of the cost of time is the
most promising approach.
Further research, deciding the most promising TSP
engine were performed based on the very same
mechanism of automatically choosing if travel between
two cities should be done on toll or free road.
Mean values of 30 executions of the program for
0
50
100
150
200
250
300
350
400
450
0 20 40 60 80 100
Goal (L)
5. 5
different TSP engines and different datasets are shown
in Table 3, Table 4 and Table 5.
The key parameter, informing about the performance of
each engine is the value of Goal Function (7).
Table 3. Comparison of results for different TSP engines for Italy
dataset
Italy
Duration
[hh:mm:ss]
Cost
[€]
Goal [€ ×
s]
Time of
executio
n [ms]
ACO 14:13:46 93.21 339.804 640
NN 14:13:50 93.60 340.319 0.01
God 13:10:16 55.20 223.600 693
The result of research performed on Italy dataset
shows that outcome of Ant Colony Optimization and
Nearest Neighbour are very similar.
Because of simplicity of implementation, time of
execution of NN engine is significantly shorter. This is a
true statement for every performed experiment.
The best TSP engine used in this experiment was
God algorithm. Value of Goal Function (7) is 33% lower
than in rest of the engines. Hence, time and cost of
travel have also lower values.
Table 4. Comparison of resultsfor different TSP engines for Poland
dataset
Poland
Duration
[hh:mm:ss]
Cost
[€]
Goal [€
× s]
Time of
execution
[ms]
ACO 22:38:57 10.71 680.569 403
NN 23:15:58 10.26 712.960 0.002
God 18:58:04 0 396.122 747
For Poland dataset, once more, the optimal
algorithm was God. In this experiment value of Goal
Function (7) for God was 42% lower than for Ant Colony
Optimization and 44% lower than for Nearest
Neighbour.
Table 5. Comparison of resultsfor different TSP engines for Europe
dataset
Europ
e
Duration
[hh:mm:ss
]
Cost
[€]
Goal [€ ×
s]
Time of
executio
n [ms]
ACO 227:19:36
166.9
2
34132.98
8
2523
NN 233::21:45
176.6
9
37021.90
0
0.01
God 218:20:44 115.58
22331.69
3
548
The third experiment performed on the list of cities
placed in all over Europe confirmed, that God algorithm
provides the optimal results between implemented TSP
engines.
In this case, the result was also drastically better.
Mean value of Goal Function for God was 35% lower
than for ACO and 40% lower than for Nearest
Neighbour.
6 CONCLUSION
The optimal approach for cost management is an
evaluation of the cost of the time performed by the
program, based on values of Goal Factors (2).
The optimal TSP engine implemented in the
Traveling Eco-Salesman program is God algorithm.
This extension to TSP problem provides a number of
possible future investigation fields.
Firstly, the proposition of new formula for Goal
Function (7). Goal presented in this article allows for
finding profitable routes. Reformulation of Goal though
may allow finding even more optimal paths.
Secondly, the proposition of another scenario for
deciding when to how to manage the cost of the travel.
Analogical to presented in Section 3.
Thirdly, implementing another algorithm for solving
TSP problem. Another possible improvement in TSP
engines is calibration of used parameters.
And finally, commercialize Traveling Eco-Salesman
program. The most promising way is to propose it as an
extension of existing Internet maps. It may also succeed
as a standalone application.
7 REFERENCES
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prob. 1400 BC
[2] E. Balais, The traveling salesman problem and its variations, Springer,
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[5] S. Qin, S. Liu, Q. Zhang, The Traveling Salesman Problem with
Profits Considering Time-Dependent Cost and Multi-Trip Mode,
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[6] S. Oliveira, M. S. Hussin, A. Roli, M. Dorigo, T. Stützle, Analysis
of the population-based ant colonyoptimization algorithm for
the TSP and the QAP 2017 IEEE Congress of Evolutionary
Computation, pp. 1734-1741, 2017.
[7] https://developers.google.com/maps/documentation/, 2017
[8] http://dev.viamichelin.com/getting-started-rest.html, 2017
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greedy: domination analysis of greedy-type heuristics for the
TSP, Discrete Applied Mathematics 117, pp. 81-86, 2002
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