Appointment Scheduling with Delay Tolerance Heterogeneity

In this study, we investigate an appointment sequencing and scheduling problem with heterogeneous user delay tolerances under service time uncertainty. We aim to capture the delay tolerance effect with heterogeneity, in an operationally effective and computationally tractable fashion, for the appointment scheduling problem. To this end, we first propose a Tolerance-Aware Delay (TAD) index that incorporates explicitly the user tolerance information in delay evaluation. We show that the TAD index enjoys decision-theoretical rationale in terms of Tolerance sensitivity , monotonicity , and convexity and positive homogeneity , which enables it to incorporate the frequency and intensity of delays over the tolerance in a coherent manner. Specifically, the convexity of TAD index ensures a tractable modeling of the collective delay dissatisfaction in the appointment scheduling problem. Using the TAD index, we then develop an appointment model with known empirical service time distribution that minimizes the overall tolerance-aware delays of all users. We analyze the impact of delay tolerance on the sequence and schedule decisions and show that the resultant TAD appointment model can be reformulated as a mixed-integer linear program (MILP). Furthermore, we extend the TAD appointment model by considering service time ambiguity. In particular, we encode into the TAD index a moment ambiguity set and a Wasserstein ambiguity set, respectively. The former captures effectively the correlation among service times across positions and user types, whereas the latter captures directly the service time data information. We show that both of the resultant TAD models under ambiguity can be reformulated as polynomial-sized, mixed-integer conic programs (MICPs). Finally, we compare our TAD models with some existing counterpart approaches and the current practice using synthetic data and a case of real hospital data, respectively. Our results demonstrate the effectiveness of the TAD appointment models in capturing the user delay tolerance with heterogeneity and mitigating the worst-case delays.