Here, we consider the generalized travelling salesman problem (GTSP), which is a generalization of the travelling salesman problem (TSP). This problem has several real-life applications. Since the problem is complex and NP-hard, solving this problem by exact methods is very difficult. Therefore, researchers have applied several heuristic algorithms to solve this problem. We propose the application of genetic algorithms (GAs) to obtain a solution. In the GA, three operators—selection, crossover, and mutation—are successively applied to a group of chromosomes to obtain a solution to an optimization problem. The crossover operator is applied to create better offspring and thus to converge the population, and the mutation operator is applied to explore the areas that cannot be explored by the crossover operator and thus to diversify the search space. All the crossover and mutation operators developed for the TSP can be used for the GTSP with some modifications. A better combination of these two operators can create a very good GA to obtain optimal solutions to the GTSP instances. Therefore, four crossover and three mutation operators are used here to develop GAs for solving the GTSP. Then, GAs is compared on several benchmark GTSPLIB instances. Our experiment shows the effectiveness of the sequential constructive crossover operator combined with the insertion mutation operator for this problem.

Vehicle Routing Problem (VRP) is important in the transportation and logistics industries. Vehicle Routing Problem with Time Window (VRPTW) is a kind of VRP with the additional time windows constraint in the model and is classified as an NP-hard problem. In this study, we proposed Stas crossover in Genetic Algorithm (GA) to solve VRPTW by developing the problem with K-mean clustering. The experiments use the standard Solomon’s benchmark problem instances for VRPTW. The results with K-mean clustering are shown to perform better for minimum distance and average distance than without K-mean clustering. In the case of location and dispersion characteristics of the customer, the paths with K-mean clustering are arranged into groups and are orderly, but the paths without K-mean clustering are disordered. After that, this paper shows the comparison of the crossover operator performance on instances of Solomon benchmark, and appropriate crossover operators are recommended for each type of problem. The results of the proposed algorithm are better than the best-known solutions from the previous studies for some instances. Moreover, our proposed research will serve as a guideline for a real-world case study.

The genetic algorithm (GA) is a natural selection-inspired optimization algorithm. It is a population-based search algorithm that utilizes the concept of survival of the fittest. This study creates a new crossover operator called “Stas Crossover” that is a combination of four crossover operators, including Single point crossover, Two points crossover, Arithmetic crossover, and Scattered crossover, and then presents the performance of this crossover operator. The area size and probability of Stas crossover can be adjusted.GA is used to find the optimal solution for this multi-product and multi-period aggregate production planning (APP) problem, which was used to test the algorithm, which provides optimal levels of inventory, backorders, overtime and regular production rates, and other controllable variables. According to the findings of this study, the benefit of stable crossover is that it allows for more variety in the way offspring are created and increases the opportunity for offspring to obtain good genetic information directly.