The orienteering problem with time windows (OPTW) deals with the problem about selecting a set of points of interest and then determining the route to visit them under the time window constraints. In the classical OPTW each candidate point of interest is associated with a profit value, and the objective is to maximize the total profit. In this study, we extend the problem and allow each point to have multiple profit values, which could reflect different aspects of consideration. We propose an ant colony optimization (ACO) algorithm to solve the multiobjective OPTW (MOOPTW) with the goal of finding the set of Pareto optimal solutions. To our best knowledge, this is the first study to address the MOOPTW with comprehensive numerical experiments. Our algorithm is a decomposition-based one, which decomposes the multiobjective optimization problem into single-objective sub-problems. Pheromone matrices are associated with sub-problems. We also incorporate path-relinking and propose several strategies. We apply our algorithm to solve 76 public benchmark instances and offer the list of non-dominated solutions to facilitate performance comparison in future researches.