Appointment scheduling for multi-stage sequential service systems with limited distributional information

Highlights

• Study multi-stage appointment scheduling problems with limited distributional information.

• Establish linear relationships among waiting times, idle times and overtimes in different stages.

• Formulate the problem as a min–max distributionally robust (DR) model.

• Develop an efficient cutting-plane approach to solve the DR model.

• Reformulate the inner maximization problem as a MILP.

Abstract

We study a multi-stage appointment scheduling problem with limited distributional information of stochastic service times, and formulate the distributionally robust (DR) model. In our studied problem, only the means and supports of service times are known to the planner, and the planner has to determine a job allowance for each appointment in the first stage. The objective is to minimize the maximum total expected weighted costs of customers’ waiting times and service providers’ idle times and overtimes over multiple stages under the worst-case distribution of service times. For the DR multi-stage appointment scheduling model, we first establish linear relationships among performance indicators (i.e., waiting times, idle times and overtimes), and then develop a cutting-plane approach to solve it. To overcome the difficulty of implementing the cutting-plane approach, we reformulate the inner maximization problem in the DR model as a mixed-integer linear program. Finally, computational experiments are conducted to evaluate the computational and simulational performance of the DR model, investigate the structure of the optimal schedule and examine the efficiency of some potential sequencing heuristics.

Introduction

In recent decades, appointment scheduling is widely applied in service industries, such as outpatient care, tax consulting and so on. The key problem in appointment scheduling (see, Denton and Gupta, 2003, Hassin and Mendel, 2008) is to optimize appointment schedule in advance. Through it, service providers could make full use of their working time, and the average waiting time of customers can be reduced (Robinson and Chen, 2003, Denton et al., 2007). Up to now, most works on appointment scheduling problems are offline scheduling studies, and are carried out in single-stage service systems, which involve only one service process with a single service provider (see, Erdogan and Denton, 2013, Begen et al., 2012, Ge et al., 2013, Mancilla and Storer, 2013).

However, besides the single-stage service systems, multi-stage service systems also exist extensively in our daily life. In recent years, there are many emerging studies on appointment scheduling in multi-stage service systems, where customers need to go through multiple service processes (Chien et al., 2008, Pérez et al., 2013, Saremi et al., 2013). In multi-stage service systems, there is a common situation that customers have to go through all stages in the same sequence, but they just need to make appointments in the first stage and go through the remaining stages in a First In First Served (FIFS) order, and only a single service provider offers service in each stage. We call this kind of service systems in above situation multi-stage sequential service systems. For example, when having childhood vaccinations in community hospitals, children have to go through the registration and pre-examination (like body temperature measurement), and the vaccinations. And there is only one server (physician or nurse) provides service in each stage. In fact, the setting in multi-stage sequential service system is the same as the setting in flow shop scheduling (Pinedo, 2016). For appointment scheduling problems in multi-stage sequential service systems, the most studied objective is to determine job allowances for all appointments in the first stage, so as to minimize the total expected costs associated with patients’ waiting times and service providers’ idle times over multiple stages (Kuiper and Mandjes, 2015, Klassen and Yoogalingam, 2018, Zhou and Yue, 2019).

Uncertainties cannot be overlooked in appointment systems and other healthcare operations, such as in emergency medical system (EMS) (Boujemaa et al., 2020) and blood supply chain management (Abbasi et al., 2020). Thus, uncertainties are usually taken into consideration in single-stage and multi-stage appointment scheduling problems, such as the random service times (Kaandorp and Koole, 2007, Berg et al., 2014, Begen and Queyranne, 2011), the random no-show behaviors (Robinson and Chen, 2010, Zacharias and Pinedo, 2014, Zacharias and Pinedo, 2017), and so on. Since the planner in reality could be access to different levels of distributional information about uncertainties, various assumptions and solution approaches are developed to incorporate them in existing works on appointment scheduling problems. In some cases, the planner can be fully access to the distributional information of those uncertainties. Thus, some studies assume either deterministic service times (Robinson and Chen, 2010, Zacharias and Pinedo, 2014, Zacharias and Pinedo, 2017), or precise distribution pattern of service times or no-shows (Jiang et al., 2019, Bendavid et al., 2018, Zhou and Yue, 2019). To solve the corresponding appointment scheduling problems, Stochastic Programming (SP) optimization approach is often used first to incorporate uncertainties. On the basis, some studies derive tractable formulations (Mak et al., 2014, Shehadeh et al., 2019), some works focus on developing efficient algorithms to achieve (near) optimal solutions, like Benders decomposition method (Jiang et al., 2019), simulation-based sequential algorithms (Bendavid et al., 2018) and L-shaped algorithm (Denton and Gupta, 2003, Zhou and Yue, 2019). Furthermore, some works examine the structural properties of the optimal schedule, such as the “no hole” structure (Robinson and Chen, 2010) and the “dome” shape (Wang, 1993, Wang, 1997, Hassin and Mendel, 2008). Here, the “no hole” structure means that for the case where a day is divided into many time slots, if a time slot is occupied by scheduled patients, then all the previous time slots should be occupied by scheduled patients in an optimal schedule. The “dome” shape means that the job allowances increase for the first few appointments, then remain stable, and finally descend for the last few appointments.

However, in some cases, it is difficult for the planner to measure the exact probability distribution of uncertainties. On the one hand, due to the lack of data, the precise probability distribution of uncertainties is hard to estimate. For example, Denton et al. (2007) report that only 21 data points available per surgery type on average at a health center of Vermont and upstate New York. On the other hand, the probability distribution of uncertainties may exhibit different patterns during the serving process. For example, Shehadeh et al. (2020) find that colonoscopy durations follow one of two different probability distributions with respect to quality of pre-procedure bowel preparation. To address appointment scheduling problems in the absence of exact distributional information about uncertainties, the alternative approach, Distributionally Robust (DR) optimization method, is applied instead of Stochastic Programming (SP), which requires full distributional information of uncertainties. Compared with the SP approach, the DR model requires less distributional information of uncertainties, such as the means and the supports (Jiang et al., 2017, Kong et al., 2013, Mak et al., 2015, Wiesemann et al., 2014). Furthermore, the DR model considers the risk aversion nature of the planner, because it aims to provide an optimal solution that performs well for all possible realizations of uncertainties and hedges against worst-case scenario by minimizing the maximum expected costs under the worst-case distribution of uncertainties, which is chosen from a family of distributions of uncertainties (defined as the ambiguity set). Whereas, the SP approach optimizes expected system performance across all potential scenarios.

Among literature on appointment scheduling problems, the DR model has been applied in single-stage service system environment (Jiang et al., 2017, Kong et al., 2013, Mak et al., 2015), while no works exploit the DR model in multi-stage service systems. However, the difficulty of obtaining exact probability distribution of uncertainties also exists in multi-stage service systems. This observation motivates us to exploit the DR model to study appointment scheduling problem in multi-stage service systems when the planner cannot be fully access to the distributional information of uncertainties.

In this work, we consider a multi-stage sequential appointment scheduling problem with limited distributional information of random service times, and take the risk aversion nature of the decision-maker into consideration by exploiting the DR model. Specifically, we study the DR multi-stage appointment scheduling model by using the means and supports of service times to construct the ambiguity set, i.e., the family of distributions of random service times. There are two reasons why we use the means and supports. First, the means and supports of service times can be easily estimated based on available data. Second, with the ambiguity set characterized by means and supports, we can develop an efficient algorithm to solve the studied problem. The objective of the problem we consider is to determine a job allowance for each customer in the first stage, so as to minimize the maximum total expected costs of patients’ waiting times, service providers’ idle times and overtimes under the worst-case distribution of random service times.

The main contributions of our study are twofold.

• First, in view of literature, we make contributions to the emerging studies on multi-stage appointment scheduling. Specifically, we apply the DR approach to model the multi-stage appointment scheduling problem with limited distributional information of random service times. To the best of our knowledge, we are the first to study the DR multi-stage appointment scheduling model.

• Second, from a methodological point of view, we establish linear relationships among performance indicators (i.e., waiting times, idle times and overtimes) for multi-stage appointment system. Based on this, we develop an efficient cutting-plane approach to solve the DR model.

The remainder of this paper is organized as follows. In Section 2, we briefly review the relevant literature. We formally formulate the distributionally robust model for our studied problem in Section 3. In Section 4, we establish linear relationships among performance indicators, and develop an efficient cutting-plane algorithm to solve the DR model. Numerical analyses are conducted in Section 5 to study the computational and simulation performance of the DR model, investigate the structure of the optimal schedule, and examine the efficiency of some potential sequencing heuristics. This paper concludes with future available research topics in Section 6.