This paper studies the order acceptance and scheduling (OAS) problem in an identical parallel machine environment where orders are characterised by their revenues, processing times and weights. A mixed integer linear programming (MILP) model is presented for the problem with the objective of maximising the revenue minus the scheduling cost. The problem is NP- hard and a branch and bound (B&B) algorithm is developed to solve the problem. An extension of the B&B algorithm is proposed to solve very large problem instances to obtain ∈-optimal solutions. The B&B algorithm and the extended B&B are evaluated for their performances against the solutions obtained by solving the MILP problem formulation using CPLEX solver through computational experiments. Copyright © 2018 Inderscience Enterprises Ltd.