We model a novel Profit-Maximizing Pickup and Delivery Selection Problem (PPDSP). PPDSP is solved optimally by our graph search, which is faster than GUROBI. We identify factors that influence computation time and optimal profit of PPDSP. We develop two efficient heuristics: a randomized search and a greedy heuristic. Homogeneous fleets are modeled as Maximum Set Packing for faster optimization.