�,�]ՖZ3EA�ϋ����V������7{.�F��ƅ+^������g��hږ�S�R"��R���)�Õ��5��r���T�ˍUVfAD�����K�W ã1Yk�=���6i�*������<86�����Ҕ�X%q꧑Rrf�j������4>�(����ۣf��n:pz� �`lN��_La��Σ���t�*�ڗ�����-�%,�u����Z�¾�B@����M-W�Qpryh�yhp��$_e�BB��$�E g���>�=Py�^Yf?RrS iL�˶ێvp�um�����Y`g��Y.���U� �Ԃ�75�Ku%3y �ق�O&�/7k���c�8y�i�"H�,:�)�����RM;�nE���4A������M�2��v���� �-2 -t� )�R8g�a�$�`l�@��"Ԋiu�)���fn��H��қ�N���呅%��~�d����k�o2|�$���}���pTu�;��UѹDeD�L��,z����Q��t o����5z{/-(��a0�`�``E���'��5��ֻ�L�D�J� startxref The travelling salesman problem was mathematically formulated in the 1800s by the Irish mathematician W.R. Hamilton and by the British mathematician Thomas Kirkman.Hamilton's icosian game was a recreational puzzle based on finding a Hamiltonian cycle. Although a global solution for the Traveling Salesman Problem does not yet exist, there are algorithms for an existing local solution. Travelling Salesman Problem Example The Travelling - 7. 0000003126 00000 n This paper presents exact solution approaches for the TSP‐D based on dynamic programming and provides an experimental comparison of these approaches. What is the shortest possible route that he visits each city exactly once and returns to the origin city? The travelling salesman problem was defined in the 1800s by the Irish mathematician . �7��F�P*��Jo䅣K�N�v�F�� y�)�]��ƕ�/�^���yI��$�cnDP�8s��Y��I�OMC�X�\��u� � ����gw�8����B��WM�r%`��0u>���w%�eVӪ��60�AYx� ;������s?�$)�v%�}Hw��SVhAb$y:��*�׬ح����ǰi����[w| ��_. ��B�΃�7��)�������Z�/S examples. This paper gives an introduction to the Traveling Salesman Problem that includes current research. W. R. Hamilton and by the British mathematician Thomas Kirkman. 2673: Open access peer-reviewed. /Length 3210 The Particle Swarm Optimizer employs a form of artificial intelligence to solve problems. Travelling Salesman Problem [:6] 3 This is, however, not a solution to the TSP, because there are subtours: x 15 = x 21 = x 34 = x 43 = x 52 = 1, i.e., two subtours, –15–2–1 and 3–4–3. �����s��~Ʊ��e��ۿLY=��s�U9���{~XSw����w��%A�+n�ě v� �w����CO3EQ�'�@��7���e׎��3�r�o �0��� u̩�W�����yw?p�8�z�},�4Y��m/`4� � l]6e}l��Fþ���9���� 0000003971 00000 n Keywords: Traveling salesman problem, Heuristic algorithm, Excel VBA 1. 0000005210 00000 n The travelling salesman problem (TSP) is a combinatorial optimisation problem well studied in computer science, operations research and mathematics. The Brute Force approach, also known as the Naive Approach, calculates and compares all possible permutations of routes or paths to determine the shortest unique solution. 0t�����/��(��I^���b�F\�Źl^Vy� The Hamiltoninan cycle problem is to find if there exist a tour that visits every city exactly once. x�b```�'�܋@ (�����q�7�I� ��g`����bhǬ'�)��3t�����5�.0 �*Jͺ"�AgW��^��+�TN'ǂ�P�A^�-�ˎ+L��9�+�C��qB�����}�"�`=�@�G�x. PDF | This paper provides the survey of the heuristics solution approaches for the traveling salesman problem (TSP). 0000004015 00000 n 0000001807 00000 n In this case we obtain an m-salesmen problem. << /Filter /FlateDecode ��0M�70�Զ�e)\@ ��+s�s���8N��=&�&=�6���y*k�oeS�H=�������â��`�-��#��A�7h@�"��씀�Л1 �D ��\? Solving the Travelling Salesman Problem (TSP) The Travelling Salesman Problem is one of the best known NP-hard problems, which means that there is no exact algorithm to solve it in polynomial time. Hamilton’s Icosian Gamewas a recreational puzzle based on finding a Hamiltonian cycle. Each of nrequests has a pickup node and a delivery ?�y�����#f�*wm,��,�4������_��U\3��,F3KD|�M� ��\Ǫ"y�Q,�"\���]��"�͹r�YZ�&q�К��eڙ���q�ziv�ġF��xj+��mG���#��i;Q��K0�6>z�` ��CӺ^܇�R��Pc�(�}[Q�I2+�$A\��T)712W��l��U�yA��t�$��$���[1�(��^�'�%�弹�5}2gaH6jo���Xe��G�� ُ@M������0k:�yf+��-O��n�^8��R? 0000002660 00000 n There does not appear to be any authoritative documentation pointing out the creator of. In reality, every salesman has the same abilities and limitations. Das Problem des Handlungsreisenden (auch Botenproblem, Rundreiseproblem, engl. 0000001406 00000 n So, for that reason, we usually use heuristics to help us to obtain a “good” 1 Example TSPPD graph structure. 0000003499 00000 n Combinatorial Optimization: Solution Methods of Traveling Salesman Problem Hülya Demez Submitted to the Institute of Graduate Studies and Research in partial fulfillment of the requirements for the Degree of Master of Science in Applied Mathematics and Computer Science Eastern Mediterranean University January 2013 Gazimağusa, North Cyprus ... cost of a solution). I Since N = 5, (N 1)! 39 0 obj Traveling Salesman Problem, Theory and Applications 4 constraints and if the number of trucks is fixed (saym). 2893: Open access peer-reviewed. %PDF-1.4 %���� By calling p … vii. 0000006230 00000 n The former problem, say, Problem 1, is replaced by others, considering the The minimal expected time to obtain optimal solution is exponential. 0000004234 00000 n The goal is to nd a cycle C = v 0!v 1!v 2! Note the difference between Hamiltonian Cycle and TSP. Our main project goal is to apply a TSP algorithm to solve real world problems, and deliver a web based application for visualizing the TSP. This is a continuation of work started in Professor Roman Yampolskiy's Artificial Intelligence class. %%EOF 50 0 obj <> endobj I In each case, we’re going to perform the Repetitive Nearest-Neighbor Algorithm and Cheapest-Link Algorithm, then see if the results are optimal. 0000013318 00000 n The Time-Dependent Traveling Salesman Problem (TDTSP) is a generalization of the Traveling Salesman Problem (TSP) in which the cost of travel between two … The Traveling Salesman Problem with Pickup and De-livery (TSPPD) is a modi cation of the Traveling Sales-man Problem (TSP) that includes side constraints en-+0 +i +j-i-j-0 Fig. 0000006582 00000 n It is savage pleasure ... builds a solution from ... (1990) 271-281. The Traveling Salesman Problem and Heuristics . Each city, which constitutes a node in This problem is known as the travelling salesman problem and can be stated more formally as follows. 1 Traveling Salesman Problem: An Overview of Applications, Formulations, and Solution Approaches Rajesh Matai1, Surya Prakash Singh2 and Murari Lal Mittal3 1Management Group, BITS-Pilani 2Department of Management Studies, Indian Institute of Technology Delhi, New Delhi 3Department of Mechanical Engineering, Malviya National Institute of Technology Jaipur, Solving the Probabilistic Travelling Salesman Problem Based on Genetic Algorithm with Queen Selection Scheme. /Length 4580 80 0 obj<>stream 0000008722 00000 n In the most famous variant of the problem a hypothetical salesman has to visit a number of cities, visiting each city only once, before ending the journey at the original starting city. The traveling salesman problem (TSP) Example c( i, i+1) = 1, for i = 1, ..., n - 1 c( n, 1) = M (for some large number M) c(i,j ... An optimal solution to the problem contains optimal solutions to itsAn optimal solution to the problem contains optimal solutions to its subproblems. A Recurrent Neural Network to Traveling Salesman Problem. Mask plotting in PCB production Quotes of the day 2 “Problem solving is hunting. 1: Example solution of the mTSP [9] 3 THE GELS ALGORITHM �_�q0���n��$mSZ�%#É=������-_{o�Nx���&եZ��^g�h�~վa-���b0��ɂ'OIt7�Oڟ՞�5yNV 4@��� ,����L�u�J��w�$d�� 5���z���2�dN���ͤ�Y ����6��8U��>WfU�]q�%㲃A�"�)Q޲A�����9S�e�{վ(J�Ӯ'�����{t5�s�y�����8���qF��NJcz�)FK\�u�����}~���uD$/3��j�+R:���w+Z�+ߣ���_[��A�5�1���G���\A:�7���Qr��G�\��Z`$�gi�r���G���0����g��PLF+|�GU� ��.�5��d��۞��-����"��ˬ�1����s����ڼ�� +>;�7ո����aV$�'A�45�8�N0��W��jB�cS���©1{#���sВ={P��H5�-��p�wl�jIA�#�h�P�A�5cE��BcqWS�7D���h/�8�)L� �vT���� 0000006789 00000 n �w5 Mask plotting in PCB production %PDF-1.5 So, for that reason, we usually use heuristics to help us to obtain a “good” xڍZYs��~�_��K�*� �)e�ڕ���U�d?�ĐD��Ʊ��Ow= �7)5=='f�����џ��wi�I����7�xw��t�a���$=�(]?�q�݇7�~��ӛo�㻭%����0ϕ��,�{*��������s�� = 24, so it is feasible to nd the optimal Hamiton circuit by brute force (using a computer). It is savage pleasure and we are born to it.” -- Thomas Harris “An algorithm must be seen to be believed.” -- Donald Knuth . The genetic algorithm depends on selection criteria, crossover, and mutation operators. :�͖ir�0fX��.�x. ஬bO�x�/�TE̪V�s,;�� ��p��K�x�p,���C�jCB��Vn�t�R����l}p��x!*{��IG�&1��#�P�4A�3��7����ě��2����׫}���0^&aM>9���#��P($.B�z������%B��E�'"����x@�ܫ���B�B�q��jGb�O^���,>��X�t�"�{�c�(#�������%��RF=�E�F���$�WD���#��nj��^r��ΐ��������d���"�.h\&�)��6��a'{�$+���i1.��t&@@t5g���/k�RBX��ٻZ�"�N�%�8D�3�:�A�:��Ums�0����X���rUlչH�$$�����T1J�'�T#��B�I4N��:Z!�h4�z�q�+%���bT�X����l�〠�S����y��h�! If salesman starting city is A, then a TSP tour in the graph is-A → B → D → C → A . 40 thoughts on “ Travelling Salesman Problem in C and C++ ” Mohit D May 27, 2017. I am working on publishing a paper on approximating solutions to the Vehicle Routing Problem using Wisdom of Artificial Crowds with Genetic Algorithms. Lecture series on Advanced Operations Research by Prof. G.Srinivasan, Department of Management Studies, IIT Madras. 0000018992 00000 n 0000000016 00000 n 0000004993 00000 n ÆAfter making a locally optimal choice a new problem, analogous to the original one, arises. Quotes of the day 2 “Problem solving is hunting. Traveling Salesman Problem oder Traveling Salesperson Problem (TSP)) ist ein kombinatorisches Optimierungsproblem des Operations Research und der theoretischen Informatik. The Traveling Salesman Problem and Heuristics . 1 Traveling Salesman Problem: An Overview of Applications, Formulations, and Solution Approaches Rajesh Matai1, Surya Prakash Singh2 and Murari Lal Mittal3 1Management Group, BITS-Pilani 2Department of Management Studies, Indian Institute of Technology Delhi, New Delhi 3Department of Mechanical Engineering, Malviya National Institute of Technology Jaipur, Introduction The classic Travelling Salesman Problem (TSP) describes the situation where a salesperson wants to leave his/her home city, visit a number of other cities and then return home. (PDF) A glass annealing oven. 0000004771 00000 n 0000002258 00000 n g.!�n;~� Popular Travelling Salesman Problem Solutions. vii. Travelling salesman problems (TSP) are easy to describe: a salesman needs to visit all his customers located in different cities in his region, and he would like to find the cheapest tour that will assure that all cities have been visited. Ci�E�o�SHD��(�@���w�� ea}W���Nx��]���j���nI��n�J� �k���H�E7��4���۲oj�VC��S���d�������yA���O Here are some of the most popular solutions to the Traveling Salesman Problem: The Brute-Force Approach. 3. !�c�G$�On�L��q���)���0��d������8b�L4�W�4$W��0ĝV���l�8�X��U���l4B|��ήC��Tc�.��{��KK�� �����6,�/���7�6�Lcz�����! 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 path that a salesman would take while traveling between cities. stream 0000015202 00000 n �8��4p��cw�GI�B�j��-�D׿`tm4ʨ#_�#k:�SH,��;�d�!T��rYB;�}���D�4�,>~g�f4��Gl5�{[����{�� ��e^� There are also necessary and su cient conditions to determine if a possible solution does exist when one is not given a complete graph. Unfortunately TSP is not so easy to formulate, and relatively hard to solve. The first produces guaranteed optimal solution for problems involving no more than 13 cities; the time required (IBM 7094 II) varies from 60 milliseconds for a 9‐city problem to 1.75 seconds for a 13‐city problem. This problem considers a salesman who departs from his home, has to visit a number of cities within a pre-determined period of time, and then returns home. Most important, it has applications in science and engineering. (g��6�� $���I�{�U?��t���0��џK_a��ْ�=��.F,�;�^��\��|W�%�~^���Pȩ��r�4'm���N�.2��,�Ι�8U_Qc���)�=��H�W��D�Ա�� #�VD���e1��,1��ϲ��\X����|�, ������,���6I5ty$ VV���і���3��$���~�4D���5��A唗�2�O���D'h���>�Mi���J�H�������GHjl�Maj\U�#afUE�h�"���t:IG ����D� ;&>>tm�PBb�����κN����y�oOtR{T�]to�Ѡ���Q�p��ٯ���"uZ���W�l>�b�γ����NAb�Z���n��ߖl���b�Da ڣ(B���̣Ї�J!ع� ��e�Բ'�R䒃�r ��i�k�V����c�z?��r�ԁΡg5;KZ�� ��*�^�;�,^Wo���g5�YAO���x_Q�P�}٫�K�:�j$�9��!���-YZ:�lV��Ay��V��+oe��[���~}�ɴ��$`셬���1�L[K����#MbQ�%b��3A���j��� `\��e��Ζ:����^#r�ga��}x޼ ��:�m�ϛ��^�g�X�D�O"�=�h�|���KC6�ι�sQ�� 4ΨnA�m�`:��w����-lc�HBec:�}73�]]��R��F��Ϋ In this case there are 200 stops, but you can easily change the nStops variable to get a different problem size. /Filter /FlateDecode Effectively combining a truck and a drone gives rise to a new planning problem that is known as the traveling salesman problem with drone (TSP‐D). By Yu-Hsin Liu. The Traveling Salesman Problem is typical of a large class of "hard" optimization problems that have intrigued mathematicians and computer scientists for years. n�����vfkvFV�z�;;\�\�=�m��r0Ĉ�xwb�5�`&�*r-C��Z[v�ݎ�ܳ��Kom���Hn4d;?�~9"��]��'= `��v2W�{�L���#���,�-���R�n�*��N�p��0`�_�\�@� z#���V#s��ro��Yϋo��['"wum�j�j}kA'.���mvQ�����W�7������6Ƕ�IJK��G�!1|M/��=�؞��d������(N�F�3vқ���Jz����:����I�Y�?t����_ ����O$՚'&��%ж]/���.�{ �qLTˑ�q�!D%xnP�� PG3h���G��. 0 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. xref The genetic algorithms are useful for NP-hard problems, especially the traveling salesman problem. Heuristics A heuristic is a technique designed for solving a problem more quickly when classic methods are too slow (from Wikipedia). 0000001592 00000 n Examples of Traveling Salesman Problems I Here are several examples of weighted complete graphs with 5 vertices. In this case we obtain an m-salesmen problem. Hi, Nicely explained. M�л�L\wp�g���~;��ȣ������C0kK����~������0x The B&B technique will now be used, as follows. ��P_t}�Wڡ��z���?��˹���q,����1k�~�����)a�D�m'��{�-��R �%�(�AS��tn����^*vQ����e���/�5�)z���FSh���,��C�y�&~J�����H��Y����k��I���Y�R~�P'��I�df� �'��E᱆6ȁ�{ `�� � These methods do not ensure optimal solutions; however, they give good approximation usually in time. h mE�v�w��W2?�b���o�)��4(��%u��� �H� In this article, we will discuss how to solve travelling salesman problem using branch and bound approach with example. Formulation of the TSP A salesman wishes to find the shortest route through a number of cities and back home again. 8. Download full-text PDF Read full-text. Fig. 3. It is particularly good at finding solutions to functions that use multiple, continuously variable, values. >> ~h�wRڝ�ݏv�xv�G'�R��iF��(T�g�Ŕi����s�2�T[�d�\�~��紋b�+�� 0000003937 00000 n �s��ǻ1��p����օ���^ \�b�"Z�f�vR�h '���z�߳�����e�sR4fb�*��r�+���N��^�E���Ā,����P�����R����T�1�����GRie)I���~�- <<00E87161E064F446B97E9EB1788A48FA>]>> ~�fQt�̇��X6G�I�Ȟ��G�N-=u���?d��ƲGI,?�ӥ�i�� �o֖����������ӇG v�s��������o|�m��{��./ n���]�U��.�9��垷�2�鴶LPi��*��+��+�ӻ��t�O�C���YLg��NƟ)��kW-����t���yU�I%gB�|���k!w��ص���h��z�1��1���l�^~aD��݋=:�Ƿ�@=�Q��O'��r�T�(��aB�R>��R�ʪL�o�;��Xn�K= 0000004459 00000 n I was just trying to understand the code to implement this. 0000011059 00000 n 66 0 obj 0000012192 00000 n The previous example of the postman can be modeled by considering the simplest possible version of this general framework. Recall that an input of the Traveling Salesman Problem is a set of points X and a non-negative, symmetric, distance function d : X X !R such that d(x;y) = d(y;x) 0 for every x;y 2X. You'll solve the initial problem and see that the solution has subtours. Nevertheless, one may appl y methods for the TSP to find good feasible solutions for this problem (see Lenstra & Rinnooy Kan, 1974). %���� 0000000916 00000 n trailer endstream The world needs a better way to travel, in particular it should be easy to plan an optimal route through multiple destinations. Solving the Travelling Salesman Problem (TSP) The Travelling Salesman Problem is one of the best known NP-hard problems, which means that there is no exact algorithm to solve it in polynomial time. Cost of the tour = 10 + 25 + 30 + 15 = 80 units . The origins of the traveling salesman problem are obscure; it is mentioned in an 1832 manual for traveling salesman, which included example tours of 45 German cities but gave no mathematical consideration.2 W. R. Hamilton and Thomas Kirkman devised mathematical formulations of the problem in the 1800s.2 It is believed that the general form was first studied by Karl Menger in Vienna and Harvard in the 1930s.2,3 Hassler W… problem of finding such an a priori tour, which is of minimum length in the expected value sense, is defined as a Probabilistic Traveling Salesman Problem (PTSP). stream Solving the Travelling Salesman Problem with the Excel Sort Function and Visual Basic for Applications Richard J. Perle Department of Finance and Computer Information Systems, Loyola Marymount University, One LMU Drive Los Angeles, CA, USA 90045 rperle@lmu.edu 310.338.2929 Abstract This paper develops and tests the performance of a new and novel heuristic algorithm for solving the Travelling … ingsalesmanproblem.Thesetofalltours(feasiblesolutions)is broken upinto increasinglysmallsubsets by a procedurecalledbranch- ing.For eachsubset a lowerbound onthe length ofthe tourstherein 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 ≠ NP. traveling salesman problem,orTSP for short, ... discuss some of the factors driving the continued interest in solution methods for the problem. The minimal expected time to obtain optimal solution is exponential. forcing precedence among pickup and delivery node pairs. endobj Die Aufgabe besteht darin, eine Reihenfolge für den Besuch mehrerer Orte so zu wählen, dass keine Station außer der ersten mehr als einmal besucht wird, die gesamte Reisestrecke des Handlungsreisenden möglichst kurz und die erste Station gleich de… Given a finite set of cities N and a distance matrix (cij) (i, j eN), determine min, E Ci(i), ieN 717 This problem involves finding the shortest closed tour (path) through a set of stops (cities). �tn¾��Z���U/?�$��0�����-=����o��F|F����*���G�D#_�"�O[矱�?c-�>}� Nevertheless, one may appl y methods for the TSP to find good feasible solutions for this problem (see Lenstra & Rinnooy Kan, 1974). This example shows how to use binary integer programming to solve the classic traveling salesman problem. Solving tsp (travel sales problem) using ruin & recreate method. 1.1 TRAVELING SALESMAN The origin of the name “traveling salesman problem” is a bit of a mystery. 0000007604 00000 n By Paulo Henrique Siqueira, Sérgio Scheer, and Maria Teresinha Arns Steiner. 50 31 0000004535 00000 n 0000009896 00000 n To tackle the traveling salesman problem using genetic algorithms, there are various representations such … For example, in the manufacture of a circuit board, it is important to determine the best order in which a laser will drill thousands of holes. �B��}��(��̡�~�+@�M@��M��hE��2ْ4G�-7$(��-��b��b��7��u��p�0gT�b�!i�\Vm��^r_�_IycO�˓n����2�.�j9�*̹O�#ֳ Two algorithms for solving the (symmetric distance) traveling salesman problem have been programmed for a high‐speed digital computer. x��YKs�F��W�����D,�6�8VN։VR����S�ʯ���{@P�����*q���g����p��WI�a�ڤ�_$�j{�x�>X�h��U�E�zb��*)b?L��Z�]������|nVaJ;�hu��e������ݧr;\���NwM���{��_�ו�q�}�$lSMKwee�cY��k*sTbOv8\���k����/�Xnpc������&��z'�k"����Y ���[SV2��G���|U�Eex(~\� �Ϡ"����|�&ޯ_�bl%��d�9��ȉo�#…r�C��s�U�P���#���:ā�/%�$�Y�"���X����D�ߙv0�˨�.���`"�&^t��A�/�2�� �g�z��d�9b��y8���`���Y�QN��*�(���K�?Q��` b�6�LX�&9�R^��0�TeͲ��Le�3!�(�������λ�q(Н鷝W6��6���H;]�&ͣ���z��8]���N��;���7�H�K�m��ږxF�7�=�m 0000016323 00000 n 0000001326 00000 n travelling only one city, and one salesman needs to travel the left n+m−1 cities. www.carbolite.com A randomization heuristic based on neighborhood A traveler needs to visit all the cities from a list, where distances between all the cities are known and each city should be visited just once. >> Traveling Salesman Problem, Theory and Applications 4 constraints and if the number of trucks is fixed (saym). Hence, the mTSP with ability constraint is more appropriate in the real world problems [40]. ������'-�,F�ˮ|�}(rX�CL��ؼ�-߲`;�x1-����[�_R�� ����%�;&�y= ��w�|�A\l_���ձ4��^O�Y���S��G?����H|�0w�#ں�/D�� The problem allows for travel times that can depend on the time of departure. This piece is concerned with modifying the algorithm to tackle problems, such as the travelling salesman problem, that use discrete, fixed values. The Traveling Salesman Problem Nearest-Neighbor Algorithm Lecture 31 Sections 6.4 Robb T. Koether Hampden-Sydney College Mon, Nov 6, 2017 Robb T. Koether (Hampden-Sydney College)The Traveling Salesman ProblemNearest-Neighbor AlgorithmMon, Nov 6, 2017 1 / 15 We present a new solution approach for the Time Dependent Traveling Salesman Prob-lem with Time Windows. Traveling Salesman Problem: A Real World Scenario. v m 1!v m = v 0 that reaches every vertex and that has minimal total length cost d(C) := P m 1 i=0 d(v i;v i+1). << What I was not able to understand is why we are adding the return to the same node as well for the minimum comparison. In combinatorial optimization, TSP has been an early proving ground for many approaches, including more recent variants of local optimization techniques such as simulated Genetic algorithms are evolutionary techniques used for optimization purposes according to survival of the fittest idea. 3Q�^�O�6��t�0��9�dg�8 o�V�>Y��+5�r�$��65X�m�>��L�eGV��.��R���f�aN�[�ّ��˶��⓷%�����;����Ov�Ʋ��SUȺ�F�^W����6�����l�a�Q�e4���K��Y� �^艢cժ\&z����U��W6s��$�C��"���_��i$���%��ߞ��R����������b��[eӓIt�D�ƣ�X^W�^=���i��}W� #f�k�Wxk?�EO�F�=�JjsN+�8���D��A1�;������� B��e_�@������ 1.1 Solving Traveling Salesman Problem With a non-complete Graph One of the NP-hard routing problems is the Traveling Salesman Problem (TSP). 80 units problem have been programmed for a high‐speed digital computer of trucks is fixed ( ). 'Ll solve the initial problem and can be modeled by considering the possible...... builds a solution from... ( 1990 ) 271-281 the Irish mathematician in time one! 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