Efficiently representing and generating combinations can allow the seamless visualization, sampling, and evaluation of combinatorial architectures. In this paper, being relevant to tackle resource allocation problems ubiquitously, we address the subset sum problem by (1) using gradient-free optimization with a number-based representation of the combinatorial search space and by (2) generating combinations with minimal change order through parallel reductions in the GPU. Our computational experiments consisting of a relevant set of problem instances and gradient-free optimization algorithms show that (1) it is possible to generate combinations in the GPU efficiently, with quasi-linear complexity, (2) it is possible to tackle instances of the subset sum problem within a reasonable number of function evaluations, and (3) Particle Swarm Optimization with Fitness Euclidean Ratio converges faster. Since the search space of number-based representations is one-dimensional and amenable to parallelization schemes (e.g., GPU), we believe our work opens the door to tackle further combinatorial problems.