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This research explores the effectiveness of heuristic techniques for solving combinatorial optimization problems, with a particular focus on the Traveling Salesman Problem (TSP). Combinatorial optimization is a critical area of study, especially in fields like computer science, engineering, and economics, where finding optimal solutions from a finite set of possibilities is crucial. However, the NP-hard nature of many combinatorial problems, such as the TSP, makes traditional exact methods like Branch-and-Bound and Dynamic Programming computationally expensive and inefficient for larger problem sizes. The primary objective of this research is to evaluate the performance of heuristic methods, including Simulated Annealing (SA), Genetic Algorithms (GA), and Iterative Computation techniques, such as Tabu Search (TS) and Particle Swarm Optimization (PSO). These methods are tested for their ability to provide approximate solutions efficiently. The findings reveal that while ACO provided the best solution quality, it had the longest runtime. TS was the fastest, though with slightly lower solution quality. SA and GA demonstrated a balance between solution quality and computational efficiency, but their performance heavily depended on parameter tuning. The hybridization of SA and GA showed potential for improving solution quality but introduced additional complexity. The research concludes that heuristic methods, especially when combined, offer viable solutions for large-scale combinatorial optimization problems, though the trade-off between solution quality and computational time must be considered when selecting an algorithm.

Optimizing distribution routes is an important step for MSMEs in increasing operational efficiency and customer satisfaction. This research was conducted on Bolen Crispy Mak Tin MSMEs which face distribution challenges with routes that are not yet optimal, causing increased transportation costs and the risk of decreasing product quality. This research uses the Branch and Bound and Nearest Neighbor algorithms to solve the Traveling Salesman Problem (TSP) problem in determining efficient distribution routes. The results of data processing are optimal routes that have the minimum distance with a total distance of 282.5 KM with route P-1-2-7-4-5-6-3-0 for the branch and Bound algorithm and 239 km with route P- 2-3-4-5-6-7-1-P for Nearest Neighbor This result is more optimal when compared to the previous route, namely P-1-2-3-4-5-6-7-P with a distance of 291 km analysis shows that Method Nearest Neighbor is able to provide an optimal solution by minimizing travel distance and distribution costs, while the Branch and Bound algorithm also provides an optimal solution but is less efficient. and distribution cost efficiency from Rp. 570,320.9 to 565,511.36 or 0.84% ​​more savings for the Branch and Bound Algorithm and 540,795.45 or 5.18% more savings for Nearest Neighbor

Combinatorial optimization is a fundamental area in operations research and computer science, focusing on identifying optimal solutions from a finite set of possibilities. This study explores the integration of branch and bound methods with heuristic algorithms to address optimization problems in graph theory and discrete mathematics. Python was employed for algorithm implementation due to its flexibility and comprehensive computational libraries, enabling efficient data analysis and visualization. Several benchmark problems were examined, including the Traveling Salesman Problem (TSP), Minimum Spanning Tree (MST), and Graph Coloring. Simulations were conducted using datasets of varying sizes (small, medium, and large) to evaluate performance across different scales. The results demonstrate that the hybrid approach achieves a balance between solution quality and computational efficiency, outperforming brute-force methods in terms of speed while maintaining near-optimal accuracy. Tabulated results and graphical comparisons highlight the reduction in computation time and improved scalability of the proposed method. The findings suggest that combining systematic search strategies with heuristics offers a practical and effective framework for solving complex combinatorial optimization problems. Recommendations for future research include testing scalability with larger datasets, incorporating advanced metaheuristics, and applying the approach to real-world domains such as logistics and network design.

This article examines a potential solution to the well-known Travelling Salesman Problem (TSP), which is classified as an NP-hard problem. We also provide a theoretical synopsis of several approaches that have been employed to tackle this problem. A prominent example of a combinatorial problem is the traveling salesman problem (TSP). To address the fundamental PSO algorithm's premature convergence issue and stagnation behavior on TSP, a scout characteristic-based PSO algorithm is suggested.We address Particle Swarm Optimization (PSO), a member of the evolutionary methods class, and outline the methodology for applying PSO to the TSP. Among population-based metaheuristic optimization methods, Particle Swarm Optimization (PSO) is one of the most widely used. Scientific domains such as engineering, chemistry, medicine, advanced physics, and humanities have all effectively employed PSO. Numerous theoretical and empirical results on the convergence and parameterization of PSO versions have been produced as a result of the method's extensive investigation since its introduction in 1995. Hundreds of PSO versions have been developed. It is well recognized that population size has a significant impact on the effectiveness of metaheuristics; nevertheless, no comprehensive research has been done on the appropriate selection of PSO swarm size to date.Through the application of this approach, we examine the effects of various control settings. The ideal solution and the quality of the solution are contrasted.    

Splazz drinking water depot is a refill drinking water depot and drinking water supplier in the Yogyakarta area which has a distribution/delivery route for drinking water to consumers in several different locations. The distribution carried out does not take into account optimal routes so that the costs incurred in distribution are not optimal. By Therefore, research was carried out at the Splazz drinking water depot to optimize routes and costs for distributing drinking water to consumers using the saving matrix method and the branch and bound method. The initial route for the distribution/delivery of drinking water carried out at depots based on days from the data collected shows a total distance of 66.6 Km to 35 delivery points. Processing the data using two different methods resulted in a route saving of 16 - 18% of the delivery distance with a total distance savings of 54.9 Km for the saving matrix method and 55.8 Km for the branch and bound method, the distribution vehicle used uses a modified motorbike with a carrying capacity of 5 gallons. Then calculate the amount of cost savings after saving the distance on the distribution/delivery routes, resulting in fuel cost savings of 16 – 17%.

Splazz drinking water depot is a refill drinking water depot and drinking water supplier in the Yogyakarta area which has a distribution/delivery route for drinking water to consumers in several different locations. The distribution carried out does not take into account optimal routes so that the costs incurred in distribution are not optimal. By Therefore, research was carried out at the Splazz drinking water depot to optimize routes and costs for distributing drinking water to consumers using the saving matrix method and the branch and bound method. The initial route for the distribution/delivery of drinking water carried out at depots based on days from the data collected shows a total distance of 66.6 Km to 35 delivery points. Processing the data using two different methods resulted in a route saving of 16 - 18% of the delivery distance with a total distance savings of 54.9 Km for the saving matrix method and 55.8 Km for the branch and bound method, the distribution vehicle used uses a modified motorbike with a carrying capacity of 5 gallons. Then calculate the amount of cost savings after saving the distance on the distribution/delivery routes, resulting in fuel cost savings of 16 – 17%.

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.

Tujuan pemecahan masalah penjual atau wira niaga keliling (traveling salesman problem, TSP) adalah menentukan lintasan atau rute dengan total jarak atau biaya yang paling minimum. Penelitian ini berkenaan dengan proses optimalisasi pencarian alternatif lintasan atau rute pada masalah TSP. Penjual mencari rute atau lintasan untuk mengunjungi semua kota yang ada sebanyak satu kali kunjungan. Proses optimalisasi pencarian alternatif menggunakan algoritma backtracking. Terdapat lima buah kasus TSP yang akan digunakan yaitu TSP1 memiliki 4 buah kota dengan 6 lintasan (jalur), TSP2 memiliki 5 buah kota dengan 10 lintasan, TSP3 memiliki 6 buah kota dengan 15 lintasan, TSP4 memiliki 7 buah kota dengan 21 lintasan, dan TSP5 memiliki 8 buah kota dengan 28 lintasan. Dari hasil pengujian diperoleh algoritma runut-balik (backtracking) dapat digunakan untuk menghasilkan jumlah alternatif lintasan yang lebih sedikit jika dibandingkan dengan total kombinasi alternatif lintasan. Semakin besar jumlah node (kota) dan jumlah lintasan (jalur) maka prosentase pengurangan variasi (alternatif) akan lintasan semakin besar.