A colored traveling salesman problem (CTSP) is a generalization of the well-known multiple traveling salesman problem, which introduces colors to distinguish the accessibility of its cities to salesmen. This work proposes a city/customer-centric model called cumulative capacitated CTSP ( C2 -CTSP) to tackle some practical problems with fast response requirements. Its hypergraph and mathematical programming formulations are developed for the first time. A general variable neighborhood search (GVNS) metaheuristic is designed to solve it. Specifically, greedy backtracking is proposed to initialize a solution taking into account the cumulative cost and two constraints including colors and capacities. Next, 2-swap, reinsertion, and double-bridge operations are randomly selected and carried out to execute the perturbation. Moreover, neighborhood-list-2-opt, relocation move, and generalized partition crossover are organized as variable neighborhood descent to constitute the local search for better solutions. Extensive experiments are conducted to compare the proposed GVNS with four genetic algorithms, two hybrid ant colony systems, two variable neighborhood search methods, and a perturb-based local search in 20 regular and random cases. The statistical results demonstrate that GVNS is superior to all competitors tuned by irace package in terms of both search ability and convergence rate. In addition, the study of six GVNS variants lacking different operators validates the significant role of each corresponding operator in GVNS's outstanding performance.