Sequencing and scheduling appointments with weighted completion time minimization and waiting time tolerance

Abstract

This paper investigates a sequencing and scheduling problem incorporating waiting time tolerance. The objective is to determine an arrival time for each appointment to minimize the sum of weighted expected completion times of all appointments, while ensuring compliance with waiting time limits for each appointment. Firstly, we derive the optimal schedule given any predetermined sequence. Secondly, we establish the optimal sequence solution for a special case. Thirdly, we introduce a sequencing solution derived from an alternative problem. We prove that this solution results in at most twice the optimal objective value under a mild condition. The experimental results indicate that the proposed method exhibits strong performance across a variety of settings. In addition, our numerical study reveals that the optimal schedule follows a distinctive ”dome” shape structure when service durations are independent and identically distributed (i.i.d.). We observe that a larger threshold for the waiting time target corresponds to smaller optimal job allowances, reduced gaps between optimal job allowances and mean service durations, and increased gaps between completion time and cumulative optimal job allowances. Furthermore, our results indicate that patient no-shows contribute to greater variability in waiting and idle times, along with reduced job allowances. These findings offer valuable theoretical and practical insights into appointment scheduling problems with waiting time tolerance.