An exact branch-and-price approach for the medical student scheduling problem

Highlights

• Scheduling of student internships over the academic year.

• An efficient exact branch-and-price with dedicated acceleration mechanisms.

• Dedicated two-stage pricing procedure to reduce the tailing-off effect.

• New composite alternating branching rule.

• Novel strategy to search branching tree.

Abstract

In this paper, we consider the medical student scheduling problem, which is a tactical scheduling problem assigning medical students to specific disciplines and hospitals in order to ensure an appropriate training over a one-year scheduling horizon. These internship positions are offered by local hospitals that specify minimum and maximum staffing requirements. To some extent, students can customise their own training program as they need to select a subset of the disciplines from an elective list and they are able to express their preferences related to these disciplines and the hospitals in which they would like to carry out their training program. The objective is to find a feasible student roster satisfying all educational and hospital requirements while optimising the student preferences and the equity between students. We propose an efficient branch-and-price algorithm to solve the problem under study. To that purpose, we include different dedicated mechanisms to accelerate the algorithm performance to find the exact solution in an acceptable time span. In this way, we propose a two-stage pricing procedure finding a diverse set of suitable columns to reduce the tailing-off effect in column generation. The pricing algorithm relies on dynamic programming and different pruning and symmetry breaking rules. In addition, the branch-and-price tree is constructed based on a mixed-shifting branching rule that executes different individual branching rules in a specific order. The created nodes in the branching tree are visited in the order determined by a novel right-first search strategy. We conducted different computational experiments to show the performance of the proposed branch-and-price and the implemented speed-up techniques and optimisation principles. We have benchmarked the branching and pricing methodology with other approaches and show the contribution of the different components in the algorithm.

Introduction

Upon graduation from medical school or university, students need to complete a clinical training to fulfil the specialty board certification requirements to work as an independent physician. This training implies that during different academic years students follow a one-year residency training program, organised by the medical school. Such a one-year training program, which is linked to the seniority level of the student, embodies a list of disciplines (e.g. critical care, gynaecology, general surgery) that should be carried out at local hospitals under the supervision of an attending physician. The available number of student positions offered by the hospitals are constrained and may vary over time. To fulfil the training program, students must attend all or a subset of these disciplines and practice these during a certain period. The disciplines are typically organised in a group of mandatory disciplines and a group of elective disciplines out of which the student can choose different disciplines. Moreover, students can indicate their availability and their preferences to carry out a specific discipline in a particular hospital. In this way, the students have the opportunity to focus on one or more (sub-) specialties according to their interests and the training program is customised to individual students. Incorporating these student preferences is highly important since the quality of the schedule has a significant impact on the students’ life and education and on the delivered care to patients (Cohn et al., 2009, Rose et al., 2015). In addition, according to Colbert et al., 2017, not only the student preferences but also the fairness between the training programs of individual students need to be considered in order to safeguard the care quality delivery and effective learning.

In this study, we address the medical student scheduling problem, which is a tactical scheduling problem and assigns medical students to a (predefined) set of medical disciplines over a given time horizon in order to ensure the students receive an appropriate medical training. The individual training schedules are constructed in line with the student preferences and the (hard) educational requirements stipulated by the medical school. These regulations define the duration of the training program and of the relevant individual disciplines, the required (number of) disciplines and the precedence relationships between different disciplines. The individual training schedules for all students build the student roster, which should satisfy the (hard) minimum and maximum student staffing requirements stipulated by the hospitals. The objective of the problem is to find equitably-efficient schedules, optimising on the one hand the total student desire and on the other hand the fairness across students as the quality of the worst student schedule is maximised. In this way, the problem considers the elementary conditions for defining a student scheduling problem, identified by Guo et al. (2016), enriched by other relevant constraints and objectives that are recognised as indispensable in academic literature and real life (Akbarzadeh and Maenhout, 2020).

In contrast to the related research of Akbarzadeh and Maenhout, 2020, who developed a heuristic procedure to solve large-scale real-life problems, the focus of this paper is to propose an efficient exact solution method that is able to solve problems of reasonable size in an exact manner and an acceptable time span. To that purpose, we decompose the original problem formulation using the Dantzig-Wolfe decomposition in a master problem relying on column variables, which represent feasible student schedules, and a subproblem employing the original decision variables to define feasible student schedules. The decomposed problem is solved using a dedicated branch-and-price procedure, which applies a column generation algorithm in every node of the branching tree to find the optimal LP solution and a branching method to drive the LP solution, when fractional, to integrality. In order to enhance the branch-and-price algorithm, we introduce several dedicated mechanisms, which improve the computational performance dramatically. First, to solve the pricing problem, the most time consuming component of the algorithm, we propose an efficient two-stage algorithm based on dynamic programming. The first stage aims to generate a diverse set of promising columns in a heuristic manner to stabilise the column generation algorithm and to reduce the tailing-off effect. The second stage aims to find the best column in an exact manner. Second, we implement different pruning rules and a symmetry breaking rule in order to speed up the performance of the dynamic programming algorithm. Third,in each node of the branching tree, we determine high-quality solutions quickly based on the optimal LP solution using a greedy heuristic and a Hungarian-based heuristic. Fourth, we customise the search for an integer solution based on the optimal LP relaxation and the imposed branching constraints. The exploration of the branching tree in search for the optimal integer solution relies on a mixed-shifting branching rule that alternates between different individual branching rules based on the allocated capacity and the original decision variables. The nodes in the branching tree are explored in the order defined by a novel right-first search strategy. In order to validate these optimisation principles in a computational manner, we conduct experiments on a synthetic dataset that is generated in a controlled manner, guided by the problem features encountered in real life and academic literature. We analyse the design choices of the proposed algorithm component by component and show how each component contributes to the performance of the algorithm, reducing the required computational effort and/or the optimality gap. In this analysis, the proposed algorithm is benchmarked with different other solution methodologies suggested by the literature.

The remainder of the paper is organised as follows. In Section 2, we discuss the relevant literature concerning the medical student scheduling problem and solution approaches. In Section 3, we describe the problem under study. Section 4 provides the proposed solution methodology. In Section 5, we discuss the computational performance of the algorithm. Section 6 provides concluding remarks and the contribution of the proposed research.