In a seaport, vessels need the assistance of tugboats when mooring and unmooring. Tugboats assist a vessel by pushing or towing the vessel's tug points, and the vessel can moor (or unmoor) successfully only if each of the tug points is operated with sufficient horsepower. For a busy port where vessels frequently require the service of tugboats, effectively scheduling tugboats for serving incoming and outgoing vessels is a key to successful execution of the vessels' berth plans. In this paper, we study a tugboat scheduling problem in a busy port, where incoming and outgoing vessels frequently require the assistance of tugboats but the number of available tugboats is limited. We make use of a network representation of the problem and develop an integer programming formulation, which takes into account the berth plans of vessels, the tug points of vessels for different move types, and the horsepower requirements of the tug points, so as to minimize the weighted sum of the berthing and departure tardiness of vessels, the operating cost of tugboats, and the number of vessels that cannot be served successfully. We analyze the computational complexity of the problem, and develop a novel iterative solution method, which combines Lagrangian relaxation and Benders decomposition, for generating near-optimal solutions. Computational performance of the proposed solution method is evaluated on problem instances generated from the operational data of a container port in Shanghai.