## Heuristics for the Traveling Salesman Problem

Travelling salesman problem Wikipedia. A вЂњbranch and boundвЂќ algorithm is presented for solving the traveling salesman problem. The set of all tours (feasible solutions) is broken up into increasingly small subsets by a procedure called branching. For each subset a lower, Heuristics for the Traveling Salesman Problem Christian Nilsson LinkВЁoping University chrni794@student.liu.se Abstract The traveling salesman problem is a well known opti-mization problem. Optimal solutions to small instances can.

### (PDF) Solving Travelling Salesman Problem Using

Solving Travelling Salesman Problem Using Greedy. Solving Travelling Salesman Problem Using Greedy Genetic Algorithm GGA Vinod Jain#1, Jay Shankar Prasad*2 #1Ph.D. Scholar ,*2 Professor Department of Computer Science and Engineering, MVN University, Palwal, Haryana, Traveling Salesman Problem, Theory and Applications. Edited by: Donald Davendra. ISBN 978-953-307-426-9, PDF ISBN 978-953-51-5501-0, Published 2010-12-30 This book is a collection of current research in the application of.

2017/06/01В В· The genetic algorithms are useful for NP-hard problems, especially the traveling salesman problem. The genetic algorithm depends on selection criteria, crossover, and mutation operators. To tackle the traveling salesman problem A вЂњbranch and boundвЂќ algorithm is presented for solving the traveling salesman problem. The set of all tours (feasible solutions) is broken up into increasingly small subsets by a procedure called branching. For each subset a lower

GENETIC ALGORITHMS FOR THE TRAVELLING SALESMAN PROBLEM 131 BEGIN AGA Make initial population at random. WHILE NOT stop DO BEGIN Select parents from the population. Produce childrenfrom the selected The results Travelling Salesman Problem (TSP), the ant colony algorithm are compared to those obtained by conventional ACO. It has is superior to simulated annealing and genetic algorithm been confirmed that the

2013/11/03В В· Travelling Salesman Problem (TSP): Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. Note 2014/05/06В В· TSP_GA Traveling Salesman Problem (TSP) Genetic Algorithm (GA) Finds a (near) optimal solution to the TSP by setting up a GA to search for the shortest route (least distance for the salesman to travel to each city exactly once

98 Chapter 6 TRAVELLING SALESMAN PROBLEM 6.1 Introduction The Traveling Salesman Problem (TSP) is a problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of 2017/06/01В В· The genetic algorithms are useful for NP-hard problems, especially the traveling salesman problem. The genetic algorithm depends on selection criteria, crossover, and mutation operators. To tackle the traveling salesman problem

36 thoughts on вЂњ Travelling Salesman Problem in C and C++ вЂќ Mohit D May 27, 2017 Hi, Nicely explained. I was just trying to understand the code to implement this. What I was not able to understand is why we are adding the return The travelling salesman problem is an NP(TSP) -hard problem in which, given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each place exactly once. The origins of the travelling

Abstract In this study, a modification of the nearest neighbor algorithm (NND) for the traveling salesman problem (TSP) is researched. NN and NND algorithms are applied to different instances starting with each of the vertices, then Combinatorial Optimization: Solution Methods of Traveling Salesman Problem HГјlya Demez Submitted to the Institute of Graduate Studies and Research in partial fulfillment of the requirements for the Degree of Master of Science in

(PDF) Solving Travelling Salesman Problem using Firefly. Heuristics A heuristic is a technique designed for solving a problem more quickly when classic methods are too slow (from Wikipedia). TodayвЂ™s lecture: Heuristics illustrated on the traveling salesman problem. Design principles for, 2017/11/11В В· Travelling Salesman Problem (TSP): Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns back to the starting point..

### Genetic Algorithms for the Travelling Salesman

TalkTravelling salesman problem Wikipedia. design of an evolutionary algorithm for a Multi Travelling Salesman problem. The aim of this paper s to find an efficient evolutionary technique for MTSP Multi Travelling Salesman problem. IN MTSP there are multiple, 2017/11/11В В· Travelling Salesman Problem (TSP): Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns back to the starting point..

(PDF) Traveling Salesman Problem an Overview of. Traveling Salesman Problem: an Overview of Applications, Formulations, and Solution Approaches Chapter (PDF Available) В· November 2010 with 5,593 Reads How we measure 'reads' A 'read' is counted each time someone, 2017/05/24В В· Travelling Salesman Problem 3. Travelling Salesman Problem Example Akshay has following list of cities and want to visit each city once, return back to home: Pune(Home)(A) Bangalore(B) Chennai(C) Delhi(D) Kolkata(E 4..

### Travelling salesman problem Wikipedia

Travelling salesman problem Simple English Wikipedia. 1 ACO Algorithms for the Traveling Salesman Problemy Thomas STUTZLEЛњ zand Marco DORIGO IRIDIA, Universit e Libre de Bruxelles, Belgium ftstutzle,mdorigog@ulb.ac.be 1.1 INTRODUCTION Ant algorithms [18, 14, 19] are a A Solution of Genetic Algorithm for Solving Traveling Salesman Problem Sonam Khattar1 Dr. Puneet Gosawmi2 1M.Tech Scholor 2Head & Professor 1,2 Department of CSE 1, 2 GGITC, Ambala, India Abstract(TSP) is an.

A Solution of Genetic Algorithm for Solving Traveling Salesman Problem Sonam Khattar1 Dr. Puneet Gosawmi2 1M.Tech Scholor 2Head & Professor 1,2 Department of CSE 1, 2 GGITC, Ambala, India Abstract(TSP) is an Solving the Travelling Salesman Problem Using the Ant Colony Optimization Management Information Systems Vol. 6, 4/2011, pp. 010-014 11 well only for solving the problems with no more than 40-80 nodes (we suppose the use of

2012/05/30В В· A short tutorial on finding intervals for optimal routes, using nearest neighbour for upper bounds and using minimum spanning trees to find lower bounds for optimal routes. This is part 3 of 3 in a series The travelling salesman problem is an NP(TSP) -hard problem in which, given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each place exactly once. The origins of the travelling

arXiv:1706.06920v2 [cs.NE] 19 Dec 2017 Genetic Algorithm with Optimal Recombination for the Asymmetric Travelling Salesman Problem Anton V. Eremeev, Yulia V. Kovalenkoв€— December 20, 2017 Sobolev Institute of Mathematics, 4 Particle Swarm Optimization Algorithm for the Traveling Salesman Problem Elizabeth F. G. Goldbarg, Marco C. Goldbarg and Givanaldo R. de Souza Universidade Federal do Rio Grande do Norte Brazil 1. Introduction Particle

The results Travelling Salesman Problem (TSP), the ant colony algorithm are compared to those obtained by conventional ACO. It has is superior to simulated annealing and genetic algorithm been confirmed that the 4 Particle Swarm Optimization Algorithm for the Traveling Salesman Problem Elizabeth F. G. Goldbarg, Marco C. Goldbarg and Givanaldo R. de Souza Universidade Federal do Rio Grande do Norte Brazil 1. Introduction Particle

1 ACO Algorithms for the Traveling Salesman Problemy Thomas STUTZLEЛњ zand Marco DORIGO IRIDIA, Universit e Libre de Bruxelles, Belgium ftstutzle,mdorigog@ulb.ac.be 1.1 INTRODUCTION Ant algorithms [18, 14, 19] are a 2012/05/30В В· A short tutorial on finding intervals for optimal routes, using nearest neighbour for upper bounds and using minimum spanning trees to find lower bounds for optimal routes. This is part 3 of 3 in a series

GENETIC ALGORITHMS FOR THE TRAVELLING SALESMAN PROBLEM 131 BEGIN AGA Make initial population at random. WHILE NOT stop DO BEGIN Select parents from the population. Produce childrenfrom the selected Combinatorial Optimization: Solution Methods of Traveling Salesman Problem HГјlya Demez Submitted to the Institute of Graduate Studies and Research in partial fulfillment of the requirements for the Degree of Master of Science in

Salesman Problem that includes current research. Additionally, the algorithms are used to nd a route traveling through twenty US colleges. As well, we use the Geometric Algorithm to assign scouts for the Pittsburgh Pirates. iv 4 Particle Swarm Optimization Algorithm for the Traveling Salesman Problem Elizabeth F. G. Goldbarg, Marco C. Goldbarg and Givanaldo R. de Souza Universidade Federal do Rio Grande do Norte Brazil 1. Introduction Particle

A Genetic Algorithm for Solving Travelling Salesman Problem Adewole Philip Department of Computer Science University of Agriculture, Abeokuta, Nigeria philipwole@yahoo.com Akinwale Adio Taofiki Department of Computer 10.2.2 The general traveling salesman problem Definition: If an NP-complete problem can be solved in polynomial time then P = NP, else P в‰ NP. Definition: An algorithm for a given problem has an approximation ratio of ПЃ(n) if the

## Particle Swarm Optimization Algorithm for the

The travelling salesman problem TheChalkface. Combinatorial Optimization: Solution Methods of Traveling Salesman Problem HГјlya Demez Submitted to the Institute of Graduate Studies and Research in partial fulfillment of the requirements for the Degree of Master of Science in, ingsalesmanproblem.Thesetofalltours(feasiblesolutions)is broken upinto increasinglysmallsubsets by a procedurecalledbranch- ing.For eachsubset a lowerbound onthe length ofthe tourstherein.

### Travelling salesman problem Wikipedia

An algorithm for the traveling salesman problem. Genetic Algorithm for Travelling Salesman Problem A Project Work (MS-405) Submitted to Assam University, Silchar in partial fulfillment of the requirements for the award of degree of Master of Science in Computer Science th, travelling salesman problem using genetic algorithm. In this problem TSP is used as a domain.TSP has long been known to be NP-complete and standard example of such problems. There had been many attempts to address this.

The Traveling Salesman problem Amanur Rahman Saiyed Indiana State University Terre Haute, IN 47809 , USA asaiyed@sycamores.indstate.edu April 11, 2012 Abstract The Traveling Salesman Problem, deals with creating the ideal Abstract: Although the rapid advances in current technology in all areas, still there are some real-world NP complex problems that still elude scientists. Some of such complex problems are Travelling Salesman Problem (TSP

before, the traveling salesman problem is NP-hard so there is no known algorithm that will solve it in polynomial time. We will probably have to sacrifice optimality in order to get a good answer in a shorter time. design of an evolutionary algorithm for a Multi Travelling Salesman problem. The aim of this paper s to find an efficient evolutionary technique for MTSP Multi Travelling Salesman problem. IN MTSP there are multiple

Traveling Salesman Problem, Theory and Applications. Edited by: Donald Davendra. ISBN 978-953-307-426-9, PDF ISBN 978-953-51-5501-0, Published 2010-12-30 This book is a collection of current research in the application of The results Travelling Salesman Problem (TSP), the ant colony algorithm are compared to those obtained by conventional ACO. It has is superior to simulated annealing and genetic algorithm been confirmed that the

Travelling Salesman Problem An implementation of a branch and bound algorithm to solve the Travelling Salesman Problem (TSP). Course: Communication Networks and Ambient Intelligence Miniproject 1: Graph Theory October 2012 Two phase heuristic algorithm for the multiple-travelling salesman problem 6569 Fig. 1 Example described in Sect. 3 a MTSP, b MTSP with balance of workload proposeGELS-GA,anewhybridalgorithm,whichachieves optimal values

Branch-and-bound algorithm for the traveling salesman problem The traveling salesman problem is discussed in Section 8.7 of the textbook. The branch-and-bound algorithm described in that section is slightly incomplete, so here is a A Solution of Genetic Algorithm for Solving Traveling Salesman Problem Sonam Khattar1 Dr. Puneet Gosawmi2 1M.Tech Scholor 2Head & Professor 1,2 Department of CSE 1, 2 GGITC, Ambala, India Abstract(TSP) is an

The results Travelling Salesman Problem (TSP), the ant colony algorithm are compared to those obtained by conventional ACO. It has is superior to simulated annealing and genetic algorithm been confirmed that the Genetic Algorithm for Travelling Salesman Problem A Project Work (MS-405) Submitted to Assam University, Silchar in partial fulfillment of the requirements for the award of degree of Master of Science in Computer Science th

A Genetic Algorithm for Solving Travelling Salesman Problem Adewole Philip Department of Computer Science University of Agriculture, Abeokuta, Nigeria philipwole@yahoo.com Akinwale Adio Taofiki Department of Computer Salesman Problem that includes current research. Additionally, the algorithms are used to nd a route traveling through twenty US colleges. As well, we use the Geometric Algorithm to assign scouts for the Pittsburgh Pirates. iv

algorithm Travelling Salesman Remarks The Travelling Salesman Problem is the problem of finding the minimum cost of travelling through N vertices exactly once per vertex. There is a cost cost[i][j] to travel from vertex i to vertex j. A вЂњbranch and boundвЂќ algorithm is presented for solving the traveling salesman problem. The set of all tours (feasible solutions) is broken up into increasingly small subsets by a procedure called branching. For each subset a lower

GENETIC ALGORITHMS FOR THE TRAVELLING SALESMAN PROBLEM 131 BEGIN AGA Make initial population at random. WHILE NOT stop DO BEGIN Select parents from the population. Produce childrenfrom the selected algorithm Travelling Salesman Remarks The Travelling Salesman Problem is the problem of finding the minimum cost of travelling through N vertices exactly once per vertex. There is a cost cost[i][j] to travel from vertex i to vertex j.

2013/11/03В В· Travelling Salesman Problem (TSP): Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. Note A Survey on Travelling Salesman Problem Sanchit Goyal Department of Computer Science University of North Dakota Grand Forks, North Dakota 58203 sanchitgoyal01@gmail.com Abstract The Travelling Salesman problem is one of

A Solution of Genetic Algorithm for Solving Traveling Salesman Problem Sonam Khattar1 Dr. Puneet Gosawmi2 1M.Tech Scholor 2Head & Professor 1,2 Department of CSE 1, 2 GGITC, Ambala, India Abstract(TSP) is an 36 thoughts on вЂњ Travelling Salesman Problem in C and C++ вЂќ Mohit D May 27, 2017 Hi, Nicely explained. I was just trying to understand the code to implement this. What I was not able to understand is why we are adding the return

98 Chapter 6 TRAVELLING SALESMAN PROBLEM 6.1 Introduction The Traveling Salesman Problem (TSP) is a problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of VI Contents A Multi-World Intelligent Genetic Algorithm to Optimize Delivery Problem with Interactive-Time 137 Yoshitaka Sakurai and Setsuo Tsuruta An Efп¬Ѓ cient Solving the Travelling Salesman Problem: Global Optimization of

2017/11/11В В· Travelling Salesman Problem (TSP): Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns back to the starting point. In this paper, we have solved Travelling Salesman Problem using Genetic algorithm approach. System starts from a matrix of the calculated Euclidean distances between the cities to be visited by the travelling salesman and

before, the traveling salesman problem is NP-hard so there is no known algorithm that will solve it in polynomial time. We will probably have to sacrifice optimality in order to get a good answer in a shorter time. ingsalesmanproblem.Thesetofalltours(feasiblesolutions)is broken upinto increasinglysmallsubsets by a procedurecalledbranch- ing.For eachsubset a lowerbound onthe length ofthe tourstherein

Genetic Algorithm for Travelling Salesman Problem A Project Work (MS-405) Submitted to Assam University, Silchar in partial fulfillment of the requirements for the award of degree of Master of Science in Computer Science th GENETIC ALGORITHMS FOR THE TRAVELLING SALESMAN PROBLEM 131 BEGIN AGA Make initial population at random. WHILE NOT stop DO BEGIN Select parents from the population. Produce childrenfrom the selected

### Chapter 6 TRAVELLING SALESMAN PROBLEM

Heuristics for the Traveling Salesman Problem. travelling salesman problem using genetic algorithm. In this problem TSP is used as a domain.TSP has long been known to be NP-complete and standard example of such problems. There had been many attempts to address this, In this paper, we have solved Travelling Salesman Problem using Genetic algorithm approach. System starts from a matrix of the calculated Euclidean distances between the cities to be visited by the travelling salesman and.

Two phase heuristic algorithm for the multiple. Heuristics for the Traveling Salesman Problem Christian Nilsson LinkВЁoping University chrni794@student.liu.se Abstract The traveling salesman problem is a well known opti-mization problem. Optimal solutions to small instances can, Branch-and-bound algorithm for the traveling salesman problem The traveling salesman problem is discussed in Section 8.7 of the textbook. The branch-and-bound algorithm described in that section is slightly incomplete, so here is a.

### (PDF) Modified Ant Colony Optimization Algorithm for

How to Solve Travelling Salesman Problems TSP -. 1 ACO Algorithms for the Traveling Salesman Problemy Thomas STUTZLEЛњ zand Marco DORIGO IRIDIA, Universit e Libre de Bruxelles, Belgium ftstutzle,mdorigog@ulb.ac.be 1.1 INTRODUCTION Ant algorithms [18, 14, 19] are a Abstract In this study, a modification of the nearest neighbor algorithm (NND) for the traveling salesman problem (TSP) is researched. NN and NND algorithms are applied to different instances starting with each of the vertices, then.

VI Contents A Multi-World Intelligent Genetic Algorithm to Optimize Delivery Problem with Interactive-Time 137 Yoshitaka Sakurai and Setsuo Tsuruta An Efп¬Ѓ cient Solving the Travelling Salesman Problem: Global Optimization of algorithm Travelling Salesman Remarks The Travelling Salesman Problem is the problem of finding the minimum cost of travelling through N vertices exactly once per vertex. There is a cost cost[i][j] to travel from vertex i to vertex j.

The Traveling Salesman Problem (often called TSP) is a classic algorithmic problem in the field of computer science and operations research[1]. It is focused on optimization. In this context, better solution often means a solution that is cheaper, shorter, or faster. TSP is a mathematical problem. It is most easily Combinatorial Optimization: Solution Methods of Traveling Salesman Problem HГјlya Demez Submitted to the Institute of Graduate Studies and Research in partial fulfillment of the requirements for the Degree of Master of Science in

A Solution of Genetic Algorithm for Solving Traveling Salesman Problem Sonam Khattar1 Dr. Puneet Gosawmi2 1M.Tech Scholor 2Head & Professor 1,2 Department of CSE 1, 2 GGITC, Ambala, India Abstract(TSP) is an International Journal of Advance Research, IJOAR .org Volume 1, Issue 4, April 2013, ISSN 2320-9194 APPLICATION OF GENETIC ALGORITHM TO SOLVE TRAVELING SALESMAN PROBLEM Oloruntoyin Sefiu Taiwo, Olukehinde

Traveling Salesman Problem: an Overview of Applications, Formulations, and Solution Approaches Chapter (PDF Available) В· November 2010 with 5,593 Reads How we measure 'reads' A 'read' is counted each time someone GENETIC ALGORITHMS FOR THE TRAVELLING SALESMAN PROBLEM 131 BEGIN AGA Make initial population at random. WHILE NOT stop DO BEGIN Select parents from the population. Produce childrenfrom the selected

36 thoughts on вЂњ Travelling Salesman Problem in C and C++ вЂќ Mohit D May 27, 2017 Hi, Nicely explained. I was just trying to understand the code to implement this. What I was not able to understand is why we are adding the return A Powerful Genetic Algorithm for Traveling Salesman Problem Figure 1. Illustration of the Edge Swapping Algorithm I de ne the size of a M-ring as the number of edges of E A (or E B) included in it. Note that some of the M-rings might

VI Contents A Multi-World Intelligent Genetic Algorithm to Optimize Delivery Problem with Interactive-Time 137 Yoshitaka Sakurai and Setsuo Tsuruta An Efп¬Ѓ cient Solving the Travelling Salesman Problem: Global Optimization of Abstract In this study, a modification of the nearest neighbor algorithm (NND) for the traveling salesman problem (TSP) is researched. NN and NND algorithms are applied to different instances starting with each of the vertices, then

2017/05/24В В· Travelling Salesman Problem 3. Travelling Salesman Problem Example Akshay has following list of cities and want to visit each city once, return back to home: Pune(Home)(A) Bangalore(B) Chennai(C) Delhi(D) Kolkata(E 4. The results Travelling Salesman Problem (TSP), the ant colony algorithm are compared to those obtained by conventional ACO. It has is superior to simulated annealing and genetic algorithm been confirmed that the

A вЂњbranch and boundвЂќ algorithm is presented for solving the traveling salesman problem. The set of all tours (feasible solutions) is broken up into increasingly small subsets by a procedure called branching. For each subset a lower 10.2.2 The general traveling salesman problem Definition: If an NP-complete problem can be solved in polynomial time then P = NP, else P в‰ NP. Definition: An algorithm for a given problem has an approximation ratio of ПЃ(n) if the

A Solution of Genetic Algorithm for Solving Traveling Salesman Problem Sonam Khattar1 Dr. Puneet Gosawmi2 1M.Tech Scholor 2Head & Professor 1,2 Department of CSE 1, 2 GGITC, Ambala, India Abstract(TSP) is an Travelling Salesman Problem An implementation of a branch and bound algorithm to solve the Travelling Salesman Problem (TSP). Course: Communication Networks and Ambient Intelligence Miniproject 1: Graph Theory October 2012

Abstract: Although the rapid advances in current technology in all areas, still there are some real-world NP complex problems that still elude scientists. Some of such complex problems are Travelling Salesman Problem (TSP design of an evolutionary algorithm for a Multi Travelling Salesman problem. The aim of this paper s to find an efficient evolutionary technique for MTSP Multi Travelling Salesman problem. IN MTSP there are multiple

Combinatorial Optimization: Solution Methods of Traveling Salesman Problem HГјlya Demez Submitted to the Institute of Graduate Studies and Research in partial fulfillment of the requirements for the Degree of Master of Science in A Powerful Genetic Algorithm for Traveling Salesman Problem Figure 1. Illustration of the Edge Swapping Algorithm I de ne the size of a M-ring as the number of edges of E A (or E B) included in it. Note that some of the M-rings might

Travelling salesman problem в†’ Traveling salesman problem вЂ” spelling вЂ”вЂ”Disavian (talk / contribs) 19:53, 10 August 2007 (UTC) WP:ENGVAR explicitly forbids this kind of switching between US and Commonwealth spellings. Abstract In this study, a modification of the nearest neighbor algorithm (NND) for the traveling salesman problem (TSP) is researched. NN and NND algorithms are applied to different instances starting with each of the vertices, then

ingsalesmanproblem.Thesetofalltours(feasiblesolutions)is broken upinto increasinglysmallsubsets by a procedurecalledbranch- ing.For eachsubset a lowerbound onthe length ofthe tourstherein A Genetic Algorithm for Solving Travelling Salesman Problem Adewole Philip Department of Computer Science University of Agriculture, Abeokuta, Nigeria philipwole@yahoo.com Akinwale Adio Taofiki Department of Computer

International Journal of Advance Research, IJOAR .org Volume 1, Issue 4, April 2013, ISSN 2320-9194 APPLICATION OF GENETIC ALGORITHM TO SOLVE TRAVELING SALESMAN PROBLEM Oloruntoyin Sefiu Taiwo, Olukehinde 98 Chapter 6 TRAVELLING SALESMAN PROBLEM 6.1 Introduction The Traveling Salesman Problem (TSP) is a problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of

A Genetic Algorithm for Solving Travelling Salesman Problem Adewole Philip Department of Computer Science University of Agriculture, Abeokuta, Nigeria philipwole@yahoo.com Akinwale Adio Taofiki Department of Computer Heuristics A heuristic is a technique designed for solving a problem more quickly when classic methods are too slow (from Wikipedia). TodayвЂ™s lecture: Heuristics illustrated on the traveling salesman problem. Design principles for

2017/06/01В В· The genetic algorithms are useful for NP-hard problems, especially the traveling salesman problem. The genetic algorithm depends on selection criteria, crossover, and mutation operators. To tackle the traveling salesman problem Abstract In this study, a modification of the nearest neighbor algorithm (NND) for the traveling salesman problem (TSP) is researched. NN and NND algorithms are applied to different instances starting with each of the vertices, then

Solving the Travelling Salesman Problem Using the Ant Colony Optimization Management Information Systems Vol. 6, 4/2011, pp. 010-014 11 well only for solving the problems with no more than 40-80 nodes (we suppose the use of Branch-and-bound algorithm for the traveling salesman problem The traveling salesman problem is discussed in Section 8.7 of the textbook. The branch-and-bound algorithm described in that section is slightly incomplete, so here is a

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