In the middle of the 19th century, the invention of a safety device that prevented elevators from falling enabled the construction of tall buildings and skyscrapers. In the middle of the 20th century, control systems started to serve the given calls automatically by relay technology, and later by electro-mechanical systems. In the 1970s-80 s, software-based control systems invaded elevator technology. Passenger service levels improved with the application of mathematical methods such as artificial intelligence. When the old relay boards of the skyscrapers in New York were modernized by software-based group controls, passenger waiting times dropped to less than half. In this millennium, the need to reduce elevator core space has further increased, since a significant number of buildings already exceed 300 m. The challenge in constructing tall buildings is that elevator groups can occupy the rentable area of a building. At the elevator planning stage, elevator core space can be decreased by zoning the building. The latest trends include systems with several elevator cars running in the same shaft. With modern control systems, passenger journey times can be decreased and handling capacity increased. This article deals with mathematical methods used in elevator dispatching problems. Building traffic simulation is utilized to search for an elevator arrangement that saves the most space in an example building. The design criteria of the ISO 8100–32 standard are used in selecting the elevator arrangements.