The cash in transit (CIT) problem is a version of the vehicle routing problem (VRP), which deals with the planning of money distribution from the depot(s) to the automated teller machines (ATMs) safely and quickly. This study investigates a novel CIT problem, which is a variant of time-dependent VRP with time windows. To establish a more realistic approach to the time-dependent CIT problem, vehicle speed varying according to traffic density is considered. The problem is formulated as a mixed-integer mathematical model. Artificial neural networks (ANNs) are used to forecast the money demand for each ATM. For this purpose, key factors are defined, and a formulation is proposed to determine the money deposited to and withdrawn into ATMs. The mathematical model is run for different scenarios, and optimum routes are obtained.