This document presents a new multiple traveling salesman problem (MTSP*) that involves both ordinary cities that can be visited by any salesman and exclusive cities that can only be visited by a designated salesman. The document proposes using a genetic algorithm to solve MTSP*, with chromosomes encoding the cities and salesmen. Three types of crossover and mutation operators are designed for the genetic algorithm - simple city, simple salesman, and mixed city-salesman - to evolve solutions while ensuring the proper relationships between cities and salesmen. A simulation compares the performance of the genetic algorithm using the different operator modes, finding that using only city crossover and mutation achieves the best convergence.
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.
A new hybrid approach for solving travelling salesman problem using ordered c...eSAT Journals
Abstract Travelling Salesman Problem is a well known NP problem. It is an optimization problem. Genetic Algorithms are the evolution techniques to solve optimization problems. In this paper a new hybrid technique using ordered cross over 1 (OX1) and greedy approach has been proposed. Experiment results shows that the proposed hybrid cross over is better than the existing cross over operator as the new operator provide a better path when executed for the same number of iterations. Keywords:- Travelling Salesman Problem, ordered cross over 1 (OX1)
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.
The Optimizing Multiple Travelling Salesman Problem Using Genetic Algorithmijsrd.com
The traveling salesman problem (TSP) supports the idea of a single salesperson traveling in a continuous trip visiting all n cities exactly once and returning to the starting point. The multiple traveling salesman problems (mTSP) is complex combinatorial optimization problem, which is a generalization of the well-known Travelling Salesman Problem (TSP), where one or more salesman can be used in the path. In this paper mTSP has also been studied and solved with GA in the form of the vehicle scheduling problem. The existing model is new models are compared to both mathematically and experimentally. This work presents a chromosome methodology setting up the MTSP to be solved using a GA.
Particle Swarm Optimization to Solve Multiple Traveling Salesman ProblemIRJET Journal
This document proposes a new genetic ant colony optimization algorithm for solving the multiple traveling salesman problem (mTSP). The algorithm combines properties of genetic algorithms and ant colony optimization. Each salesman's route is determined using ant colony optimization, while the routes of different salesmen are combined into a complete solution controlled by the genetic algorithm. The algorithm is tested on benchmark problem instances and shown to perform efficiently compared to other existing algorithms for mTSP. Key aspects of the algorithm include the representation of solutions, crossover operators that always generate feasible solutions, and the integration of ant colony optimization and genetic algorithms.
The document discusses solving a traveling salesman problem (TSP) using two methods: a Hungarian heuristic approach and AMPL programming. The Hungarian method finds an optimal route of 770 miles for a salesman visiting 5 cities. Using AMPL, the optimal route is found to be 725 miles, showing AMPL finds a better solution that is 45 miles shorter. The conclusion is that AMPL programming provides a more optimal approach for solving TSP problems compared to the Hungarian heuristic method.
Quantum inspired evolutionary algorithm for solving multiple travelling sales...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
A new hybrid approach for solving travelling salesman problem using ordered c...eSAT Journals
Abstract Travelling Salesman Problem is a well known NP problem. It is an optimization problem. Genetic Algorithms are the evolution techniques to solve optimization problems. In this paper a new hybrid technique using ordered cross over 1 (OX1) and greedy approach has been proposed. Experiment results shows that the proposed hybrid cross over is better than the existing cross over operator as the new operator provide a better path when executed for the same number of iterations. Keywords:- Travelling Salesman Problem, ordered cross over 1 (OX1)
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.
The Optimizing Multiple Travelling Salesman Problem Using Genetic Algorithmijsrd.com
The traveling salesman problem (TSP) supports the idea of a single salesperson traveling in a continuous trip visiting all n cities exactly once and returning to the starting point. The multiple traveling salesman problems (mTSP) is complex combinatorial optimization problem, which is a generalization of the well-known Travelling Salesman Problem (TSP), where one or more salesman can be used in the path. In this paper mTSP has also been studied and solved with GA in the form of the vehicle scheduling problem. The existing model is new models are compared to both mathematically and experimentally. This work presents a chromosome methodology setting up the MTSP to be solved using a GA.
Particle Swarm Optimization to Solve Multiple Traveling Salesman ProblemIRJET Journal
This document proposes a new genetic ant colony optimization algorithm for solving the multiple traveling salesman problem (mTSP). The algorithm combines properties of genetic algorithms and ant colony optimization. Each salesman's route is determined using ant colony optimization, while the routes of different salesmen are combined into a complete solution controlled by the genetic algorithm. The algorithm is tested on benchmark problem instances and shown to perform efficiently compared to other existing algorithms for mTSP. Key aspects of the algorithm include the representation of solutions, crossover operators that always generate feasible solutions, and the integration of ant colony optimization and genetic algorithms.
The document discusses solving a traveling salesman problem (TSP) using two methods: a Hungarian heuristic approach and AMPL programming. The Hungarian method finds an optimal route of 770 miles for a salesman visiting 5 cities. Using AMPL, the optimal route is found to be 725 miles, showing AMPL finds a better solution that is 45 miles shorter. The conclusion is that AMPL programming provides a more optimal approach for solving TSP problems compared to the Hungarian heuristic method.
Quantum inspired evolutionary algorithm for solving multiple travelling sales...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
The railway capacity optimization problem deals with the maximization of the number of trains running on
a given network per unit time. In this study, we frame this problem as a typical asymmetrical Travelling
Salesman Problem (ATSP), with the ATSP nodes representing the train arrival /departure events and the
ATSP total cost representing the total time-interval of the schedule. The application problem is then
optimized using the standard Ant Colony Optimization (ACO) algorithm. The simulation experiments
validate the formulation of the railway capacity problem as an ATSP and the ACO algorithm produces
optimal solutions superior to those produced by the domain experts.
A STUDY AND IMPLEMENTATION OF THE TRANSIT ROUTE NETWORK DESIGN PROBLEM FOR A ...cscpconf
The design of public transportation networks presupposes solving optimization problems,
involving various parameters such as the proper mathematical description of networks, the
algorithmic approach to apply, and also the consideration of real-world, practical
characteristics such as the types of vehicles in the network, the frequencies of routes, demand,
possible limitations of route capacities, travel decisions made by passengers, the environmental
footprint of the system, the available bus technologies, besides others. The current paper
presents the progress of the work that aims to study the design of a municipal public
transportation system that employs middleware technologies and geographic information
services in order to produce practical, realistic results. The system employs novel optimization
approaches such as the particle swarm algorithms and also considers various environmental
parameters such as the use of electric vehicles and the emissions of conventional ones.
A study and implementation of the transit route network design problem for a ...csandit
The design of public transportation networks presup
poses solving optimization problems,
involving various parameters such as the proper mat
hematical description of networks, the
algorithmic approach to apply, and also the conside
ration of real-world, practical
characteristics such as the types of vehicles in th
e network, the frequencies of routes, demand,
possible limitations of route capacities, travel de
cisions made by passengers, the environmental
footprint of the system, the available bus technolo
gies, besides others. The current paper
presents the progress of the work that aims to stud
y the design of a municipal public
transportation system that employs middleware techn
ologies and geographic information
services in order to produce practical, realistic r
esults. The system employs novel optimization
approaches such as the particle swarm algorithms an
d also considers various environmental
parameters such as the use of electric vehicles and
the emissions of conventional ones.
A New Approach To Solving The Multiple Traveling Salesperson Problem Using Ge...April Smith
This document proposes a new genetic algorithm chromosome and operators for solving the Multiple Traveling Salesperson Problem (MTSP). The MTSP involves scheduling multiple salespersons to visit locations while minimizing total distance traveled. Previous studies used standard TSP chromosomes which allowed redundant solutions. The proposed new chromosome and operators dramatically reduce redundancy, improving search efficiency. Computational testing shows the new approach often finds better solutions than previous techniques due to searching a smaller solution space.
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.
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 project aims to find optimal vacation routes that visit predetermined cities using graph theory and linear programming. Two problems are solved: the classic traveling salesperson problem to minimize distance and a "best trip" problem to maximize happiness factors. Cities are rated based on factors like cost of living, GDP, and attractions. The results discuss solutions in terms of cost and travel distance between 13 global cities.
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.
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.
Hybrid iterated local search algorithm for optimization route of airplane tr...IJECEIAES
The traveling salesman problem (TSP) is a very popular combinatorics problem. This problem has been widely applied to various real problems. The TSP problem has been classified as a Non-deterministic Polynomial Hard (NP-Hard), so a non-deterministic algorithm is needed to solve this problem. However, a non-deterministic algorithm can only produce a fairly good solution but does not guarantee an optimal solution. Therefore, there are still opportunities to develop new algorithms with better optimization results. This research develops a new algorithm by hybridizing three local search algorithms, namely, iterated local search (ILS) with simulated annealing (SA) and hill climbing (HC), to get a better optimization result. This algorithm aimed to solve TSP problems in the transportation sector, using a case study from the Traveling Salesman Challenge 2.0 (TSC 2.0). The test results show that the developed algorithm can optimize better by 15.7% on average and 11.4% based on the best results compared to previous studies using the TabuSA algorithm.
The document discusses the traveling salesman problem (TSP) which aims to find the shortest route for a salesman to visit each city in a list only once and return to the starting point. It provides an abstract, problem statement, and history of the TSP. It also includes a timeline of milestones in TSP research with increasing city sizes. Finally, it explains various solution methods to the TSP like brute force, branch and bound, greedy approach, and nearest neighbor algorithms.
This document discusses using a genetic algorithm to solve the travelling salesman problem (TSP). It begins with an abstract that outlines representing TSP solutions as chromosomes, using crossover and mutation genetic operators, and selecting chromosomes with minimum costs for the next generation. It then provides more details on the genetic algorithm steps, including initializing a population, selecting parents via roulette wheel selection, applying one-point crossover and mutation, and iterating until finding an optimal solution. Experimental results on an 8 city TSP problem are presented showing minimum, maximum and average costs decreasing over generations as the genetic algorithm converges on an optimal tour.
The Traveling salesman problem (TSP) is proved to be NP-complete in most cases. The genetic algorithm
(GA) is one of the most useful algorithms for solving this problem. In this paper a conventional GA is
compared with an improved hybrid GA in solving TSP. The improved or hybrid GA consist of
conventional GA and two local optimization strategies. The first strategy is extracting all sequential
groups including four cities of samples and changing the two central cities with each other. The second
local optimization strategy is similar to an extra mutation process. In this step with a low probability a
sample is selected. In this sample two random cities are defined and the path between these cities is
reversed. The computation results show that the proposed method also finds better paths than the
conventional GA within an acceptable computation time.
The well-known Vehicle Routing Problem (VRP) consist of assigning routeswith a set
ofcustomersto different vehicles, in order tominimize the cost of transport, usually starting from a central
warehouse and using a fleet of fixed vehicles. There are numerousapproaches for the resolution of this kind of
problems, being the metaheuristic techniques the most used, including the Genetic Algorithms (AG). The
number of approachesto the different parameters of an AG (selection, crossing, mutation...) in the literature is
such that it is not easy to take a resolution of a VRP problem directly. This paper aims to simplify this task by
analyzing the best known approaches with standard VRP data sets, and showing the parameter configurations
that offer the best results.
AN IMPROVED GENETIC ALGORITHM WITH A LOCAL OPTIMIZATION STRATEGY AND AN EXTRA...ijcseit
The Traveling salesman problem (TSP) is proved to be NP-complete in most cases. The genetic algorithm
(GA) is one of the most useful algorithms for solving this problem. In this paper a conventional GA is
compared with an improved hybrid GA in solving TSP. The improved or hybrid GA consist of
conventional GA and two local optimization strategies. The first strategy is extracting all sequential
groups including four cities of samples and changing the two central cities with each other. The second
local optimization strategy is similar to an extra mutation process. In this step with a low probability a
sample is selected. In this sample two random cities are defined and the path between these cities is
reversed. The computation results show that the proposed method also finds better paths than the
conventional GA within an acceptable computation time.
This paper analyzes the swap rates issued by the China Inter-bank Offered Rate(CHIBOR) and
selects the one-year FR007 daily data from January 1st, 2019 to June 30th, 2019 as a sample. To fit the data,
we conduct Monte Carlo simulation with several typical continuous short-term swap rate models such as the
Merton model, the Vasicek model, the CIR model, etc. These models contain both linear forms and nonlinear
forms and each has both drift terms and diffusion terms. After empirical analysis, we obtain the parameter
values in Euler-Maruyama scheme and relevant statistical characteristics of each model. The results show that
most of the short-term swap rate models can fit the swap rates and reflect the change of trend, while the CKLSO
model performs best.
With the widespread of smart mobile devices and the
availability of many applications that provide maps, many programs
have spread to find the closest and fastest routes between
two points on the map. While the exactness and effectiveness of
best path depend on the traffic circumstances, the system needs to
add more parameters such as real traffic density and velocity in
road. In addition, because of the restricted resources of phone devices,
it is not reasonable to be used to calculate the exact optimal
solutions by some familiar deterministic algorithms, which are
usually used to find the shortest path with a map of reasonable
node number. To resolve this issue, this paper put forward to use
the genetic algorithm to reduce the computational time. The proposed
system use the genetic algorithm to find the shortest path
time with miscellaneous situations of real traffic conditions. The
genetic algorithm is clearly demonstrate excellent result when applied
on many types of map, especially when the number of nodes
increased.
TRUSTS Mobile App Demo Poster (AAMAS 2013)Samantha Luber
The document summarizes the TRUSTS system and mobile app for randomized patrol strategies to deter fare evasion on transit systems. It describes how the system uses a transition graph and Markov decision process to model patrol officer actions and generate randomized patrol plans. The mobile app allows officers to view their assigned patrol plan and record data from their shifts. Real-world testing on Los Angeles metro lines showed the MDP approach generates more robust plans than the original system by accounting for execution uncertainty. Collected shift data may also be used to improve future plan quality.
With the expanding of database of the watch list of anti-money laundering, improving the speed in
matching between the watch list and the database of account holders and clients’ transaction is especially
important. This paper proposes an improved AC-BM Algorithm, a matching algorithm of subsection, to
improve the speed of matching. Experiment results show the time performance of the improved algorithm
is better than traditional BM algorithm, AC algorithm and the AC-BM algorithm. It can improve the
efficiency of on-line monitoring of anti-money laundering.
Neal Elbaum Shares Top 5 Trends Shaping the Logistics Industry in 2024Neal Elbaum
In the ever-evolving world of logistics, staying ahead of the curve is crucial. Industry expert Neal Elbaum highlights the top five trends shaping the logistics industry in 2024, offering valuable insights into the future of supply chain management.
The railway capacity optimization problem deals with the maximization of the number of trains running on
a given network per unit time. In this study, we frame this problem as a typical asymmetrical Travelling
Salesman Problem (ATSP), with the ATSP nodes representing the train arrival /departure events and the
ATSP total cost representing the total time-interval of the schedule. The application problem is then
optimized using the standard Ant Colony Optimization (ACO) algorithm. The simulation experiments
validate the formulation of the railway capacity problem as an ATSP and the ACO algorithm produces
optimal solutions superior to those produced by the domain experts.
A STUDY AND IMPLEMENTATION OF THE TRANSIT ROUTE NETWORK DESIGN PROBLEM FOR A ...cscpconf
The design of public transportation networks presupposes solving optimization problems,
involving various parameters such as the proper mathematical description of networks, the
algorithmic approach to apply, and also the consideration of real-world, practical
characteristics such as the types of vehicles in the network, the frequencies of routes, demand,
possible limitations of route capacities, travel decisions made by passengers, the environmental
footprint of the system, the available bus technologies, besides others. The current paper
presents the progress of the work that aims to study the design of a municipal public
transportation system that employs middleware technologies and geographic information
services in order to produce practical, realistic results. The system employs novel optimization
approaches such as the particle swarm algorithms and also considers various environmental
parameters such as the use of electric vehicles and the emissions of conventional ones.
A study and implementation of the transit route network design problem for a ...csandit
The design of public transportation networks presup
poses solving optimization problems,
involving various parameters such as the proper mat
hematical description of networks, the
algorithmic approach to apply, and also the conside
ration of real-world, practical
characteristics such as the types of vehicles in th
e network, the frequencies of routes, demand,
possible limitations of route capacities, travel de
cisions made by passengers, the environmental
footprint of the system, the available bus technolo
gies, besides others. The current paper
presents the progress of the work that aims to stud
y the design of a municipal public
transportation system that employs middleware techn
ologies and geographic information
services in order to produce practical, realistic r
esults. The system employs novel optimization
approaches such as the particle swarm algorithms an
d also considers various environmental
parameters such as the use of electric vehicles and
the emissions of conventional ones.
A New Approach To Solving The Multiple Traveling Salesperson Problem Using Ge...April Smith
This document proposes a new genetic algorithm chromosome and operators for solving the Multiple Traveling Salesperson Problem (MTSP). The MTSP involves scheduling multiple salespersons to visit locations while minimizing total distance traveled. Previous studies used standard TSP chromosomes which allowed redundant solutions. The proposed new chromosome and operators dramatically reduce redundancy, improving search efficiency. Computational testing shows the new approach often finds better solutions than previous techniques due to searching a smaller solution space.
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.
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 project aims to find optimal vacation routes that visit predetermined cities using graph theory and linear programming. Two problems are solved: the classic traveling salesperson problem to minimize distance and a "best trip" problem to maximize happiness factors. Cities are rated based on factors like cost of living, GDP, and attractions. The results discuss solutions in terms of cost and travel distance between 13 global cities.
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.
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.
Hybrid iterated local search algorithm for optimization route of airplane tr...IJECEIAES
The traveling salesman problem (TSP) is a very popular combinatorics problem. This problem has been widely applied to various real problems. The TSP problem has been classified as a Non-deterministic Polynomial Hard (NP-Hard), so a non-deterministic algorithm is needed to solve this problem. However, a non-deterministic algorithm can only produce a fairly good solution but does not guarantee an optimal solution. Therefore, there are still opportunities to develop new algorithms with better optimization results. This research develops a new algorithm by hybridizing three local search algorithms, namely, iterated local search (ILS) with simulated annealing (SA) and hill climbing (HC), to get a better optimization result. This algorithm aimed to solve TSP problems in the transportation sector, using a case study from the Traveling Salesman Challenge 2.0 (TSC 2.0). The test results show that the developed algorithm can optimize better by 15.7% on average and 11.4% based on the best results compared to previous studies using the TabuSA algorithm.
The document discusses the traveling salesman problem (TSP) which aims to find the shortest route for a salesman to visit each city in a list only once and return to the starting point. It provides an abstract, problem statement, and history of the TSP. It also includes a timeline of milestones in TSP research with increasing city sizes. Finally, it explains various solution methods to the TSP like brute force, branch and bound, greedy approach, and nearest neighbor algorithms.
This document discusses using a genetic algorithm to solve the travelling salesman problem (TSP). It begins with an abstract that outlines representing TSP solutions as chromosomes, using crossover and mutation genetic operators, and selecting chromosomes with minimum costs for the next generation. It then provides more details on the genetic algorithm steps, including initializing a population, selecting parents via roulette wheel selection, applying one-point crossover and mutation, and iterating until finding an optimal solution. Experimental results on an 8 city TSP problem are presented showing minimum, maximum and average costs decreasing over generations as the genetic algorithm converges on an optimal tour.
The Traveling salesman problem (TSP) is proved to be NP-complete in most cases. The genetic algorithm
(GA) is one of the most useful algorithms for solving this problem. In this paper a conventional GA is
compared with an improved hybrid GA in solving TSP. The improved or hybrid GA consist of
conventional GA and two local optimization strategies. The first strategy is extracting all sequential
groups including four cities of samples and changing the two central cities with each other. The second
local optimization strategy is similar to an extra mutation process. In this step with a low probability a
sample is selected. In this sample two random cities are defined and the path between these cities is
reversed. The computation results show that the proposed method also finds better paths than the
conventional GA within an acceptable computation time.
The well-known Vehicle Routing Problem (VRP) consist of assigning routeswith a set
ofcustomersto different vehicles, in order tominimize the cost of transport, usually starting from a central
warehouse and using a fleet of fixed vehicles. There are numerousapproaches for the resolution of this kind of
problems, being the metaheuristic techniques the most used, including the Genetic Algorithms (AG). The
number of approachesto the different parameters of an AG (selection, crossing, mutation...) in the literature is
such that it is not easy to take a resolution of a VRP problem directly. This paper aims to simplify this task by
analyzing the best known approaches with standard VRP data sets, and showing the parameter configurations
that offer the best results.
AN IMPROVED GENETIC ALGORITHM WITH A LOCAL OPTIMIZATION STRATEGY AND AN EXTRA...ijcseit
The Traveling salesman problem (TSP) is proved to be NP-complete in most cases. The genetic algorithm
(GA) is one of the most useful algorithms for solving this problem. In this paper a conventional GA is
compared with an improved hybrid GA in solving TSP. The improved or hybrid GA consist of
conventional GA and two local optimization strategies. The first strategy is extracting all sequential
groups including four cities of samples and changing the two central cities with each other. The second
local optimization strategy is similar to an extra mutation process. In this step with a low probability a
sample is selected. In this sample two random cities are defined and the path between these cities is
reversed. The computation results show that the proposed method also finds better paths than the
conventional GA within an acceptable computation time.
This paper analyzes the swap rates issued by the China Inter-bank Offered Rate(CHIBOR) and
selects the one-year FR007 daily data from January 1st, 2019 to June 30th, 2019 as a sample. To fit the data,
we conduct Monte Carlo simulation with several typical continuous short-term swap rate models such as the
Merton model, the Vasicek model, the CIR model, etc. These models contain both linear forms and nonlinear
forms and each has both drift terms and diffusion terms. After empirical analysis, we obtain the parameter
values in Euler-Maruyama scheme and relevant statistical characteristics of each model. The results show that
most of the short-term swap rate models can fit the swap rates and reflect the change of trend, while the CKLSO
model performs best.
With the widespread of smart mobile devices and the
availability of many applications that provide maps, many programs
have spread to find the closest and fastest routes between
two points on the map. While the exactness and effectiveness of
best path depend on the traffic circumstances, the system needs to
add more parameters such as real traffic density and velocity in
road. In addition, because of the restricted resources of phone devices,
it is not reasonable to be used to calculate the exact optimal
solutions by some familiar deterministic algorithms, which are
usually used to find the shortest path with a map of reasonable
node number. To resolve this issue, this paper put forward to use
the genetic algorithm to reduce the computational time. The proposed
system use the genetic algorithm to find the shortest path
time with miscellaneous situations of real traffic conditions. The
genetic algorithm is clearly demonstrate excellent result when applied
on many types of map, especially when the number of nodes
increased.
TRUSTS Mobile App Demo Poster (AAMAS 2013)Samantha Luber
The document summarizes the TRUSTS system and mobile app for randomized patrol strategies to deter fare evasion on transit systems. It describes how the system uses a transition graph and Markov decision process to model patrol officer actions and generate randomized patrol plans. The mobile app allows officers to view their assigned patrol plan and record data from their shifts. Real-world testing on Los Angeles metro lines showed the MDP approach generates more robust plans than the original system by accounting for execution uncertainty. Collected shift data may also be used to improve future plan quality.
With the expanding of database of the watch list of anti-money laundering, improving the speed in
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2. II. PROBLEM STATEMENT
MTSP* with m salesmen and n cities can be formally
defined over a complete graph ( , )
G V E , where the vertex
set 1 2
{ , , , }
n
V v v v
" corresponding to cities is partitioned into
m+1 sets 0 1
, , , m
V V V
" , and each edge ( , ) ,
i j
v v E i j
z , is
associated with a weight ij
c C
that represents a visit cost
(distance) from cities i to j . Its objective is to determine m
sequences of Hamiltonian cycles or circuits on G with the least
total cost such that any vertex of each exclusive set is visited
exactly once by a specified salesman and any vertex of the
public set is visited by any salesman exactly once.
The existence of exclusive and ordinary city groups,
makes MTSP* different from TSP and MTSP. A vertex from
an exclusive group in MTSP* can be next to another in the
same or ordinary group. Similarly, a vertex in an ordinary
group can be next to another in the same or an exclusive one.
MTSP* is not a simple composition of TSP and MTSP.
Figure 2. Example of MTSP*
An example MTSP* is shown Fig. 2. It has 4 salesmen
and 51 cities where city 0 is the depot. It is modified from the
classic example-eil51 by preserving its original city coordinate
information. 1
V represents the exclusive city s to be
exclusively visited by salesman 1, which contains cities 1-7;
2
V represents the exclusive city set 2 containing cities 8-14 to
be visited by salesman 2; 3
V represents the exclusive one by
salesman 3, which contains cities 15-22; 4
V the exclusive one
by salesman 4, which includes cities 23-30; and 0
V represents
the ordinary city set for all salesmen, which contains cities 31-
50. The objective of this problem is to determine a minimum
route that all salesmen start form the depot and finally return
to it. Meanwhile, each exclusive city must be visited exactly
once by a specified salesman and each ordinary city is visited
exactly once by any salesman.
Fig. 2 shows a feasible but may not be the best route.
Routes 1-4 for salesmen 1-4 are˖
1) 0ė35ė39ė4ė1ė2ė7ė5ė3ė49ė6ė45ė41ė
31ė0;
2) 040473810128141193413
48420;
3) 03236162017221521181943
33370;
4) 0ė24ė44ė30ė26ė25ė27ė23ė29ė28ė46ė
50ė0.
Ordinary city 44 can be next to exclusive city 24 in route 4
and exclusive city 6 next to ordinary city 49 in route 1.
III. GA DESIGN
A genetic algorithm (GA) is used to search solutions based
on the evolutionary principle. Since it was firstly introduced
by Holland [6], it has been successfully applied to a variety of
NP-hard combinatorial optimization problems, such as TSP
and MTSP. To apply GA to solve MTSP*, the focus is to
establish an effective coding and decoding scheme and design
suitable selection, crossover and mutation operators to ensure
better population evolution.
A. Chromosome coding for MTSP*
Chromosome representation is a crucial basic work when
applying a GA. It directly determines the performance of the
algorithm. The GAs designed to solve MTSP of m traveling
salesman n cities mainly exploit three encoding schemes. The
first is the one-chromosome scheme. It uses a single
chromosome of length n+m-1. In it, n cities are represented by
a permutation of integers from 1 to n. This permutation is
partitioned into m sub-tours by the insertion of m-1 negative
integers from -1 to -(m-1) that represent the transition from
one salesman to the next. The second is the two-part-
chromosome technique proposed by Arthur and Cliff [9]. The
first part of a chromosome is a permutation of n cities while
the second part is of length m and represents the number of
cities assigned to each of m salesmen. The values assigned to
the second part are constrained to be m positive integers that
must sum to the number of cities to be visited (n). The third
one is the two-chromosome technique. It requires two
chromosomes, each being of length n, to represent a solution.
The first one provides a permutation of n cities while the
second one assigns a salesman to each of the cities in the
corresponding position of the first one.
For MTSP*, it is d to use the first two schemes mentioned
above. The third one with two types of chromosomes can be
adapted to MTSP* via some modification. Particularly, the
city chromosome consists of a permutation of integers from 1
to n while a genetic value of salesman chromosome is the
number of a salesman that corresponds to an exclusive or
ordinary city in the same position of the city chromosome. The
exclusive city is assigned to the specified salesman, and the
ordinary city is randomly assigned to the salesman.
A coding example of MTSP* with three salesmen and ten
cities is shown Fig. 3, where six cities are exclusive ones and
three salesmen are assigned three exclusive ones. The rest
cities are ordinary. Namely, exclusive cities of salesmen 1-3
are 1-2, and 3-4, and 5-6. Respectively, ordinary cities are
cities 7-10. According to two chromosomes in Fig. 3, cities 2,
7 and 1 (in that order) are visited by salesman 1. Similarly,
cities 10, 4 and 3 (in that order) are visited by salesman 2, and
cities 9, 5 and 6 (in that order) are visited by salesman 3.
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628
3. Figure 3. Example of MTSP* coding
B. Selection operator
Roulette Wheel method and the elitist strategy [11] are
adopted as the selection operation in this work. The former
simply chooses a chromosome in a statistical fashion based
solely upon its fitness value. The elitist strategy copies the
individual with the best fitness at the present generation to the
next one. It can prevent the optimized individuals from being
eliminated after a selection, crossover or mutation operation. It
is critical to ensure the convergence of a GA. A GA containing
an elitist strategy is proven to be globally convergent.
C. Crossover operator
A crossover operator exchanges parts of the genes from
two parent individuals to form two new individuals. It is one
of important features that distinguish a genetic algorithm from
other ones. In MTSP a crossover operator may be one of
partially matched crossover (PMX), ordered crossover (OX),
cycle crossover (CX), two-point crossover, etc.
We design three modes of crossover operators
corresponding to the adopted chromosome coding style, i.e.,
city crossover (CC), salesman crossover (SC), and city-
salesman crossover (CSC).
1) City crossover
In this paper, we ameliorate PMX as a city crossover
operator. It requires randomly selecting two crossing points to
determine a matching section. The corresponding matching
sections in two parents are swapped, thereby resulting in two
new descendants. Then two new individuals are checked if the
exclusive cities are assigned to the specified salesman. If not,
the particular genes in a salesman individual should be
corrected.
Figure 4. Example of CC
The crossover of two chromosomes is shown in Fig. 4. In
Step 1, given two parents, we randomly select a section of a
city individual, then swap its genes with those of another
individual and produce two individual of descendants as
shown in Step 2. The mapping relationship of the selected
sections in two city individuals is 8—3, 9—8, 5—2, 4—7, 7—
1, and 1—10. In Step 3, exchange the redundant genes
according to the selected section, then find that exclusive cities
5, 3, 7, 1, and 6 in the left chromosome and cities 2, 5, and 4 in
the right one are assigned to the wrong salesmen. Then, the
exclusive cities are reassigned to the correct salesmen and two
reasonable generations are produced as shown in Step 4.
2) Salesman crossover
To avoid a number of duplicate genes appearing in a
salesman chromosome, this work adopts traditional two-point
crossover. It also requires randomly determining a matching
section. Then the corresponding sections of two chromosomes
are swapped to generate two descendants. At end, the
matching relationship between the exclusive cities should be
checked and a wrong salesman should be corrected to the
specified one.
An SC process is shown in Fig. 5. In Step 1, there are two
parents. In particular, the randomly selected matching sections
are marked in gray color in the salesman chromosome. After
swapping of the two marked sections, two descendants are
produced in Step 2. However, it is obvious that exclusive cities
5 and 4 of the left chromosome and exclusive cities 3, 2, and 1
of the right chromosome are assigned to the wrong salesmen.
In Step 3, it is corrected by reassigning exclusive cities to the
specified salesmen and obtained two correct descendants.
Figure 5. Example of SC
3) City-salesman crossover
For CSC, a city chromosome applies PMX, while a
salesman chromosome adopts the two-point crossover scheme.
Figure 6. Example of CSC
An example of CSC is shown in Fig. 6. In Step 1, we
select for each kind of parents a couple of random crossing
sections and swap each couple of sections, respectively. It
results in two new city generations and two new salesman
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4. ones as shown in Step 2. In Step 3, exchange the redundant
genes of the city chromosome, and find that exclusive cities 5
and 3 of the left city chromosome and 2 of the right city one
are assigned to a wrong salesman. After the correction of
salesman genes, the result is shown in Step 4.
D. Mutation operator
A mutation operator plays an import role in improving
local search ability and maintaining variability of the
population. It also prevents the premature termination in GA.
This work adopts swapping mutation. It requires random
selection of two crossing points, and then swaps the selected
points. Finally, check whether the exclusive cities of a new
descendant match with their corresponding salesmen.
Corresponding to the above three crossover schemes, we
design three mutation operators, i.e., city mutation (CM),
salesman mutation (SM), and city-salesman mutation (CSM).
1) City mutation
In CM, only a city-chromosome applies swapping
mutation. For example, first, two selected swapping gene
points are cities 8 and 7, as shown in Fig. 7. After swapping
them, the exclusive cities are proven to be in accord with
salesmen and the mutation is over.
Figure 7. Example of CM
2) Salesman mutation
In the operation of SM swapping mutation only applied for
a salesman chromosome. An example is given in Fig. 8, where
the selected swapping points are genes 2 and 1 marked in gray
color in the salesman-chromosome. A correct descendant is
generated by swapping them.
Figure 8. Example of SM
3) City-salesman mutation
Swapping mutation takes place in both city and salesman
chromosomes in CSM. An example is shown in Fig. 9, where
the swapping genes marked in gray are swapped pairwise,
which results in new feasible generations.
Considering that crossover and mutation should not apply
into different types of chromosomes, this work selects three
compositions of reasonable crossover and mutation operators
from nine ones. By selecting CC as crossover and CM as
mutation operator, we have CC CM (called CCM), SC
SM (called SCM), and CSC CSM (called CSCM).
Figure 9. Example of CSM
E. Fitness function
A fitness function is used in a GA to judge the chance that
an individual (a route) can be selected into the next generation.
It is a limiting factor to the efficiency of evolution. For a GA,
many selection strategies based on the proportion of fitness
require a non-negative fitness and the larger fitness the better
individual. Hence, for a problem with a minimum solution as
its optimization objective, it needs to turn it to a maximum one.
MTSP* takes the minimum length of all salesmen as its
optimization objective. Hence this work takes the reciprocal of
the length as its fitness.
Taking f(x) as the length of the solution, the fitness
function is given as:
1
( )=
1+ ( )
F x
f x
IV. SIMULATION AND RESULT
This paper takes the revised eil51 as an example given in
Section 2 to verify the correctness and performance of our GA
with the three pairs of crossover and mutation operators.
A. Convergence of GA
Convergence is an important indicator to show the
performance of a GA. If a GA is convergent, it indicates its
stability and evolution towards a correct direction. The quality
of the solution is better and better as the evolution goes on. To
examine a GA’s convergence, we conduct the following
experiment, where the generation count is 2000, the crossover
probability 0.6, mutation probability 0.1, the size of population
is 50, and CCM is selected.
The result as shown in Fig. 10 indicates that the
convergence of our GA is good without considerable
fluctuation. The length of total routes is optimized from
1198.23 km of the initial population to 558.511 km of the
2000-th generation. It implies that the effect of optimization is
notable. The evolution process can be divided into three main
stages. The first one is from the initial generation to the 560-th
one, where the GA convergence is the fastest, and the total
length of routes descends from 1198.23 km to 666.21 km. The
second one from the 561-th to 1560-th generations shows a
630
630
5. slow change of the total length form 666.21 km to 558.51 km.
In the last one from the 1561-th to 2000-th generations, the
result tends to be stable. Hence, our GA has a good
convergence. For the other two pairs of crossover and
mutation operators, i.e., SCM and CSCM, we can obtain the
similar results.
Figure 10. Result of the example of MTSP*
B. Comparison
To compare the performance of our GA with different
crossover and mutation operators, three groups of experiments
are designed where three pairs of crossover and mutation
operators are adopted in turn. The other parameters are set, i.e.,
the size of population 30, the crossover probability 0.6, the
mutation probability 0.1, and the generation count 2000. Each
experiment is carried out for ten times by using a DELL
Inspiron620s computer with Windows 7 and Inter Core i3
CPU at 3.30GHZ. The data are from the revised eil51 example
in Section 2. The experimental result is shown in TABLE
and Fig. 11.
TABLE I. Result of the three group experiments Unit˖Km
Group CCM SCM CSCM
1 558.25 575.26 568.45
2 556.85 576.45 572.12
3 561.47 580.56 573.89
4 565.25 576.65 567.45
5 558.36 570.25 570.35
6 557.68 590.68 566.25
7 562.45 586.45 576.87
8 568.56 573.89 565.54
9 556.35 584.57 575.58
10 560.58 571.64 566.4
Average 560.58 578.64 570.29
Figure 11. Result of different operators
The best, the worst and the average solutions are observed
and shown in TABLE .
TABLE II. Data analysis of each operator
Crossover and
mutation
operator
best solution
(km)
worst
solution
(km)
Average
solution
(km)
CCM 556.35 568.56 560.58
SCM 570.25 590.68 578.64
CSCM 565.54 576.87 570.29
It shows that the performance of CCM is the best among
three pairs of crossover and mutation operators while that of
SCM is the worst. By observing the results we find that by
SCM, crossover and mutation merely take place in the
salesman chromosomes. It results in a small solution space and
weakens genic recombination due to large duplicate salesman
individuals. Using CCM only the city chromosomes are
crossed and mutated and there is no duplicate gene. Thus the
extent of genic recombination is stronger and the solution
space is larger than that of SCM. Namely, the population
diversity of CCM is better than that of SCM, as shown in the
experiment data. Both a city chromosome and a salesman
chromosome are crossed and mutated by CSCM. The
dynamics of genic recombination is stronger than SCM. Hence,
the performance of CSCM is better than that of SCM.
However, crossover and mutation of both city chromosomes
and salesman chromosomes may destroy the better solution
due to too strong dynamics of genic recombination. Thus,
CSCM is worse than CCM as observed. As for time
consumption, three groups have small differences from each
other. Therefore, we should select CCM in a GA for MTSP*
.
V. CONCLUSIONS
This paper presents a new multiple traveling salesman
problems with exclusive and ordinary cities. To solve it, we
design a genetic algorithm. It revises the two-chromosome
scheme and design three pairs of crossover and mutation
operators, i.e., simple city crossover and mutation, simple
salesman crossover and mutation, and mixed city-salesman
631
631
6. crossover and mutation. To show the performance of our
algorithm with different operators, we design three groups of
experiments. The results indicate that the proposed GA is
suitable to solve MTSP* and CCM deploying city crossover
and city mutation has the best performance among three
compositions of crossover and mutation operators.
VI. ACKNOWLEDGEMENT
This work was supported in part by the China National
Natural Science Foundation under Grants 61004035 and
61175113.
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