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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%.