To solve the vehicle routing problem with simultaneous pickup–delivery and time windows (VRPSDPTW), a sine cosine and firefly perturbed sparrow search algorithm (SFSSA) is presented. Based on the standard sparrow search algorithm, the initial population uses tent chaotic mapping to change the population diversity; then, the discoverer location is updated using the sine cosine fluctuation range of the random weight factor, and finally the global population location is updated using the firefly perturbation strategy. In this study, SFSSA was compared with a genetic algorithm (GA), parallel simulated annealing algorithm (p-SA), discrete cuckoo search algorithm (DCS), and novel mimetic algorithm with efficient local search and extended neighborhood (MATE) adopting improved Solomon’s benchmark test cases. The computational results showed that the proposed SFSSA was able to achieve the current optimal solutions for 100% of the nine small-to-medium instances. For large-scale instances, SFSSA obtained the current optimal solutions for 25 out of 56 instances. The experimental findings demonstrated that SFSSA was an effective method for solving the VRPSPDTW problem.