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With the assumption that each appointment patient up in clinic arrives on time, this paper investigates the appointment scheduling problem with no-show. An integer programming model is established with the numbers of appointment patient and probability distribution of remaining patients at each slot as optimization variables. The objective function includes the benefits of serving patients, the patient waiting time costs, and system overtime expenses. By relaxing the coupled constraints about probability distribution of remaining patients at each slot, this paper proposes a Lagrangian relaxation method to solve the model with a dynamic programming algorithm solving the relaxation problem and a sub-gradient method solving the dual problem. Numerical experiments show that the algorithm can find the optimal solution for small scale problems, and that the best solution from the proposed algorithm is better than the one in the related literature for large scale problems. [ABSTRACT FROM AUTHOR]