The main goal of this paper is to define criteria for quality of a schedule for an elevator and study how some on-line algorithms perform. We have studied five on-line algorithms - Random (RAND), First Request First (FRF), Maximum Request First (MRF), Circular (CIRC), and Shortest Distance First (SDF). Our findings show that the CIRC algorithm gives the best performance in comparison to the other on-line algorithms. When the number of floors is low, the MRF algorithm also works well. We have made a serious attempt to implement an optimal off-line algorithm for the problem and compare the on-line algorithms against it. However, we have found that the task of computing an optimal off-line solution is non-trivial in terms of time complexity. Therefore, it has not been possible to make a reasonable comparison.