Appointment Scheduling Under a Service-Level Constraint

Abstract

Problem definition: This paper studies an appointment system where a finite number of customers are scheduled to arrive in such a way that (1) the expected waiting time of each individual customer cannot exceed a given threshold; and (2) the appointment times are set as early as possible (without breaking the waiting time constraint). Methodology/results: First, we show that, under the service-level constraint, a prospective schedule can be obtained from a sequential scheduling approach. In particular, we can schedule the appointment time of the next customer based on the scheduled appointment times of the previous customers. Then, we use a transient queueing-analysis approach and apply the theory of majorization to analytically characterize the structure of the optimal appointment schedule. We prove that, to keep the expected waiting time of each customer below a certain threshold, the minimum inter-appointment time required increases with the arrival sequence. We further identify additional properties of the optimal schedule. For example, a later arrival has a higher chance of finding an empty system and is more likely to wait less than the duration of his expected service time. We show the convergence of the service-level-constrained system to the D/M/1 queueing system as the number of arrivals approaches infinity and propose a simple, yet practical, heuristic schedule that is asymptotically optimal. We also develop algorithms that can help system managers determine the number of customers that can be scheduled in a fixed time window. We compare the service-level-constrained appointment system with other widely studied systems (including the equal-space and cost-minimization systems). We show that the service-level-constrained system leads to a lower upper bound on each customer’s waiting time; ensures a fair waiting experience among customers; and performs quite well in terms of system overtime. Finally, we investigate various extended settings of our analysis, including customer no-shows; mixed Erlang service times; multiple servers; and probability-based service-level constraints. Managerial implications: Our results provide guidelines on how to design appointment schedules with individual service-level constraints. Such a design ensures fairness and incorporates the threshold-type waiting perception of customers. It is also free from cost estimation and can be easily applied in practice. In addition, under the service-level-constrained appointment system, customers with later appointment times can have better waiting experiences, in contrast to the situation under other commonly studied systems. Funding: Z. Yan was partly supported by a Nanyang Technological University startup grant; the Ministry of Education Academic Research Fund Tier 1 [Grant RG17/21] and Tier 2 [Grant MOE2019-T2-1-045]; and Neptune Orient Lines [Fellowship Grant NOL21RP04]. Supplemental Material: The online supplement is available at https://doi.org/10.1287/msom.2022.1159 .

Please don't make me wait! Influence of customers' waiting preference and no‐show behavior on appointment systems

“…To our knowledge, for most appointment systems studied in the literature, the customer's waiting time follows a similar increasing pattern to the ES system; that is, later arrivals in general expect to wait longer than earlier arrivals (e.g., Cayirli & Veral, 2003). The only exception is a novel equal-waiting (EW) appointment system recently proposed by Benjaafar et al (2022). In the EW system, customers are scheduled in a way that, except for the first handful of appointments with customers scheduled to arrive together in the very beginning, the expected waiting time is equal for all other customers, irrespective of their appointment positions.…”

“…Given the total number of customers to be scheduled (N), when the waiting time threshold 𝜃 is set, the corresponding EW system is well defined. There is a one-to-one correspondence between the waiting time threshold 𝜃 and the total length of arrival period T. That is, lower 𝜃 leads to higher T (Lemma 1 of Benjaafar et al, 2022). This conclusion is sensible because, in order to achieve a shorter waiting time for all customers, appointments must be spread out over a longer arrival period.…”

“…The EW system Benjaafar et al (2022) propose the EW system that preserves the equality (or fairness) in waiting time for later arrivals. Under the EW system, customers are scheduled to arrive at the earliest possible times such that the expected waiting time of each individual customer in the system does not exceed a given threshold 𝜃 (assuming all customers show up).…”

“…The only exception is a novel equal‐waiting (EW) appointment system recently proposed by Benjaafar et al. (2022). In the EW system, customers are scheduled in a way that, except for the first handful of appointments with customers scheduled to arrive together in the very beginning, the expected waiting time is equal for all other customers, irrespective of their appointment positions.…”

“…In fact, Benjaafar et al. (2022) specifically calls for a further investigation into customers' appointment preferences under the EW system, a question this research aims to address.…”