OPTIMIZATION APPROACHES AND APPLICATION OF MATHEMATICAL MODEL OF THE TRAVELING SALESMAN PROBLEM IN TOURISM COMPANIES

Аннотация

В данной статье рассматривается задача коммивояжёра (Travelling Salesman Problem, TSP) и подходы к её оптимизации, которые играют важную роль в эффективной деятельности туристических компаний. Особенно актуальной эта проблема становится при планировании туристических маршрутов, снижении транспортных расходов и экономии времени.В статье приводится классическая формулировка задачи TSP, а также рассматриваются математическая модель и методы комбинаторной оптимизации. В частности, анализируются такие подходы, как динамическое программирование, генетические алгоритмы и аппроксимационные алгоритмы, а также их применение в деятельности туристических фирм на основе практических примеров.Кроме того, в работе представлены результаты программной симуляции, разработанной на основе реальных данных туристической компании, что позволяет продемонстрировать эффективность современных алгоритмов при выборе оптимальных маршрутов.Результаты исследования могут быть полезны для повышения конкурентоспособности в туристической отрасли, совершенствованиялогистических системи улучшениякачества предоставляемых услуг. Annotatsiya: Mazkur maqolada turizm firmalarining samarali faoliyat yuritishida muhim ahamiyatga ega bo‘lgan kommivoyajyor masalasi (travelling salesman problem, TSP) va uning optimallashtirish yondashuvlari tahlil qilinadi. Ayniqsa, sayyohlik yo‘nalishlarini rejalashtirish, transport xarajatlarini kamaytirish va vaqtni tejash kabi omillar bu masalaning dolzarbligini oshiradi. Maqolada TSP masalasining klassik formulasi keltirilib, uning matematik modeli va kombinator https://ijmri.de/index.php/jmsi volume 4,issue3,2025 731 optimallashtirish usullari asosida yechim usullari ko‘rib chiqiladi. Shu jumladan, dinamik dasturlash, genetika algoritmi, yaqinlashtirish algoritmlari kabi yondashuvlar tahlil qilinib, ularning turizm firmalari faoliyatiga qo‘llanish imkoniyatlari amaliy misollar orqali yoritiladi.Shuningdek, maqolada real hayotdagi turizm firmasi ma'lumotlari asosida ishlab chiqilgan matematik modelning dasturiy tajribasi (simulyatsiyasi) natijalari taqdim etiladi va u orqali optimal yo‘nalishlar aniqlashda zamonaviy algoritmlarning samaradorligi asoslanadi.Maqola natijalari turizm sohasida raqobatbardoshlikni oshirish, logistika tizimlarini takomillashtirish vaxizmat ko‘rsatish sifatiniyuksaltirishdafoydali bo‘lishimumkin. Keywords: Travelling Salesman Problem (TSP), Optimization, Combinatorics, Optimal routes, Tourismcompanies, Mathematical model, Travel routes, Timesaving, Algorithm efficiency. Ключевые слова: Задача коммивояжёра (TSP), Оптимизация, Комбинаторика, Оптимальные маршруты, Туристические компании, Математическая модель, Туристическиемаршруты,Экономия времени,Эффективностьалгоритмов. Kalit so‘zlar: Kommivoyajyor masalasi (TSP), optimallashtirish, kombinatorika, optimal yo‘nalishlar, turizm firmalari, matematik model, sayyohlik yo‘nalishlari, vaqtni tejash, algoritmsamaradorligi. Introduction. The Traveling Salesman Problem (TSP) is a popular and research-intensive problem in combinatorial optimization. Its main goal is to find the shortest route that will take you from one city to another and return to your starting point. This problem is used in many areas, including transportation logistics, manufacturing process optimization, electronic circuit design, and computer network planning. The TSP is, in fact, of great importance in finding solutions to real-world problems and creating efficient systems. Optimization approaches to this problem also play a significant role in the organization of tourism companies and travel services, as the need to plan tourist routes, reduce costs and save time is increasing day by day. There are various methods for solving it, such as: Bruteforce (Full Review), dynamic programming (Held- Karp algorithm), Greedy algorithm (Greedy Algorithm), genetic algorithm, simulated annealing (SimulatedAnnealing), linear programming and chess (Branch and Bound) The goal of the traveling salesman problem is to find the shortest and most efficient route for a salesman to visit a given number of cities. Each city must be visited only once, and at the end, the salesman must return to the starting city. The main goal of the problem is to minimize the distance traveled to the cities, thereby reducing time and cost. Solving the TSP, by optimizing the objective, increases efficiency in real-world applications such as transportation, logistics, and resource management. The algorithms and methods developed for solving the TSP can also be applied to manyotheroptimization problems. Find a cycle that can traverse the given nine cities in the shortest time (distance, cost) by visiting each of them only once. In this case, the number of cycles is at most ta. This problem is related to the problem of finding a Hamilton cycle of minimum length. The "Networks and Boundaries" method can be used to solve the traveling salesman problem. This method is carried out using a graph with nocycles andapath, and by