THE TRAVELING SALESMAN PROBLEM: MATHEMATICAL MODELING AND OPTIMAL SOLUTIONS

Аннотация

Salesman issues trade in the field of products distribution , sales and transportation costs to calculate These issues trade representative planned routes and work efficiency increase for important is considered . Within the scope of the issue salesman work time , every one the store product sell opportunities , and various to cities to go time determination through total sold products or done affairs This type is issues not only trade representatives efficiency increase ,maybethe company general business strategies to optimizehelp gives . Key words :distances matrix , route , closed route , route length , whole valuable variables , graph , Hamilton outline ( Hamilton cycle ), networks and borders method , cited matrix , backpack ( bomberaircraft (issue about ),resources ,cargo, loads transportation ( flight ). Introduction . Processes research and optimal management – decision acceptance to doandsystems tooptimizescientific fields oriented .​ 1-Process research resources effective distribution for mathematician models , linear programming ,games​ theory andnetworks optimizationsuch as from methods uses . 2-Optimal management systems the most good management strategies determination with He is engaged in his work . main methods Pontryagin's Maximum principle and Bellman's dynamic programming . Literature analysis Salesman issue and his/her solution according to many scientific research , articles and books They exist . most issue solution for various mathematician and algorithmic approaches offer does. Dantzig , FG, Fulkerson, RW, & Johnson, S. (1954). "Solution of a large-scale traveling-salesman problem." This The work is about the TSP problem . historical beginning Danzig and​ his/her colleagues this issue mathematician optimization problem as seeing​ came out and him/her solution for special algorithms create necessity They emphasized . method through the issue of multiple cities for solved . Keller, DM, & Matheson, JE (1972). "The Traveling Salesman Problem: A Survey." This article TSP in solution used various kind approaches , that including heuristic​ and algorithmic methods about in detail information gives . TSP solution for used solutions efficiency analysis did​ Applegate, D., Bixby, R., Chvatal , V., & Cook, W. (2006). "The Traveling Salesman page1204 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue04,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai Problem: A Computational Study." In this book TSP solution for modern computer algorithmsapplication​ indetail statement done. Research methodology Salesman The Traveling Salesman Problem is one or one how many the city visit ordered , every one the city only one times pilgrimage so , the most short the way find The research is methodology this issue solution for various mathematician methods and algorithms using The problem is solved . mathematician Expression : Problem graph to the theory is based on. Each the city as a knot , their between​ distances and to look at as edges This problem is linear . programming and other mathematician methods is modeled using the heuristic Methods : Heuristic methods issue fast to solve help These methods​ every always the optimal solution even if it doesn't , it's fast and effective results For example , the Greedy algorithm or Simulation with to add methods used . Computer Simulation : In research computer programs and algorithms using issue solution practice Python or​ ​ other programming TSP algorithms in languages used . Experimental Analysis : At this stage various cities and roads​ for issues is created and theirThe effectiveness ofthealgorithms is tested . in practice how results to give from the test Analysis​ ​ ​ and Compare : In the study various algorithms and methods This is done using which method the most effective that determined.To practice Application : Research in the end , found solutions real world to the conditions applies and working release or logistics in the processes how benefit to take possibilityis displayed . Analysesandresults 1. Salesman There are cities . n Each​ city the rest transportation route with with connected . Cities between distances matrix C =(c ), i, j =1,n given ( i j c = , i =1,nwe assume ). Salesman somehow from the city out every one in the city ii only one once become initial to the city return need . To the cities to go route so choice must routeLet the length beminimal .This problem mathematician model to composefor 1, agar i -shahardan j -shaharga borilsa, x = (1) ij 0, aks holda. such as we define this on the ground The issue i, j =1,2,...,n, i j is as follows . is written : (1)byformula determined x of variables so values find must , ij n x =1, j =1,n, (2) ij i=1 n x =1, i =1,n, (3) ij j=1 u -u +nx n-1, i, j =1,n, i j, (4) i j i j conditionsbe doneandin this n n ( ) f x = C x (5) ij ij i=1 j=1 function to aminimumachieve . page1205 INTERNATIONAL JOURNAL OF ARTIFI