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The main problem of the Traveling Salesman Problem is that a salesman travels to several places to go with a known distance and then returns to his original place by using the shortest route from his journey, and all the places the salesman goes to are only allowed once. This research focuses on the problem of distributing goods at PT. The Medan Nugraha Ekakurir (JNE) route with the destination delivery address in the Medan area. The Cheapest Insertion Heuristic Algorithm is an algorithm used to form tours (travels) by gradually building the shortest path route with minimal weight, by adding new points one at a time. One. The Nearest Neighbor Algorithm is a simple and fast algorithm to build a feasible initial tour length from TSP where the technique takes the shortest distance from the initial position regardless of other distances. This study resulted in the conclusion that the application of the cheapest insertion heuristic and nearest neighbor algorithms in terms of finding the distance to the problem of shipping goods at PT. The Medan Nugraha Ekakurir (JNE) route starts with finding the distance between addresses with the help of google maps, then continues with the help of the WinQSB software. Based on the research results obtained using the cheapest insertion heuristic and nearest neighbor algorithms, it is obtained that the search for the shortest route distance for shipping goods at PT. The smaller Medan Nugraha Ekakurir (JNE) route is generated by the nearest neighbor algorithm. This shows that the nearest neighbor algorithm is more effective in terms of finding the traveling distance on the Traveling Salesman Problem problem of shipping goods at PT. Medan's Nugraha Ekakurir (JNE) Line.

Quantum-Inspired Algorithms (QIAs) combine principles of quantum computing with classical evolutionary strategies to address complex optimization problems. This research explores the potential of QIAs in improving optimization processes, particularly in combinatorial and multi-objective optimization scenarios. The study focuses on the application of Quantum-Inspired Genetic Algorithms (QIGAs) and Quantum-Inspired Evolutionary Algorithms (QIEAs), assessing their effectiveness in solving classical problems like the Traveling Salesman Problem (TSP) and Minimum Spanning Tree (MST). Through computational simulations, the research compares the time convergence and solution accuracy of QIAs against traditional classical algorithms. The findings demonstrate that QIAs achieve faster convergence rates and higher-quality solutions, with accuracy levels reaching 98-99% of the global optimal solutions, while significantly reducing computational time. These results underline the advantages of QIAs in solving large and complex optimization problems, making them a promising alternative to traditional algorithms. Additionally, QIAs excel in avoiding local minima, a common pitfall of classical methods, due to their ability to explore the solution space more efficiently through quantum principles like superposition and interference. The implications of this study suggest that QIAs can be a valuable tool for tackling real-world optimization challenges, with potential applications in fields such as finance, logistics, telecommunications, and energy management. The research also indicates the necessity for further improvements in quantum-inspired algorithms' scalability and hardware integration, particularly for larger, more intricate optimization problems, to fully realize their potential in practical industrial applications.

Travelling Salesman Problem (TSP) is a problem that is often encountered by a salesman who must travel exactly once to all consumers in a route and will return to the starting point of departure. Algorithm Genetic Algorithm is one way to find heuristic solutions based on the evolutionary ideas of natural selection and genetics. The aim is to find the optimal route for the distribution of bottled water products produced by PT. Mual Natio Maju Bersama. To find a solution, the chromosomes processed by the genetic algorithm are represented through the stages in the Genetic Algorithm individual initialization, fitness value, linear fitness ranking, roulette whell selection, crossover, and mutation. In order to achieve the optimum solution, namely The best path obtained is PT Mual Tio Maju Bersama –BUMDES Sait ni Huta - UD. Alvaro - UD. Lancelhot – UD. Alris – UD. Jamel – Toko Kelontong SRC Resi 2 – Toko Notra – UD. B Siringoringo – Toko Dahlia Siahaan – UD. Purba – UD. Cahaya – UD. Hutapea – UD. Gabe – UD. Setia II – UD. Larisma II – UD. Antoni – UD. Bona Siahaan – UD. Sederhana – Toko Manalu – UD. Setia I – Toko Ferdinan – UD. Alboy – Wisma Daun Mas – UD. Top Jaya – UD. Mega Silaban – BUMDES Silaitlait – UD. Rika – UD. Panamot – Piltik Coffee and Homestay Bandar Udara Silangit – UD. Rolas Boy – UD. Salamat Karya – UD. Simpang Jaya – UD. Lambok - Piltik Coffee and Homestay Siborongborong – UD. Bahagia – UD. Marlinca – UD. Heri Joel Pasaribu – UD. Ebenezer – UD. Mawar – UD. A Saudara – UD. SP Perdana – PDAM Mual Na Tio – UD. Rokkap - PT Mual Tio Maju Bersama. The best path length is 125.2700 cartesian units and the best fitness value is 0.008000.