An exact branch-and-price approach for the medical student scheduling problem
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.
Section snippets
Related literature
The medical student scheduling problem has been studied in different forms for which different dedicated solution methodologies have been proposed in the literature.
Problem definition and formulation
The Medical Student Scheduling problem assigns graduate students to a specific set of medical disciplines over the course of the academic year to fulfil his/her clinical training. A training program is designed by a medical school or university and embodies a list of disciplines related to relevant medical specialties. In order to fulfil their individual training program, students are assigned to disciplines across different wards and hospitals, where they work during one or multiple
Solution methodology
In this study, we present a dedicated branch-and-price solution methodology that exploits different problem-specific mechanisms to reduce the size of the branching tree and to speed up the computational performance, enabling the procedure to solve larger-sized problem instances. The procedure relies upon the Dantzig-Wolfe decomposition of the compact problem formulation to tighten the linear programming relaxation. The decomposition is related to the scheduling of individual students, which is
Computational experiments
In this section, we provide computational insights into the proposed procedure. In Section 5.1, we describe the relevant parameter settings and the methodology based on which a synthetic dataset of test instances is constructed. In the next sections, we validate the computational impact of the different aspects of the proposed methodology. In Section 5.2, we analyse the applied branching rule and search strategy. Section 5.3 discusses the performance of the lower bound calculations. In Section
Conclusions
In this paper, we proposed a branch-and-price algorithm to solve the medical student scheduling problem that assigns students to medical disciplines and hospitals over the academic year, providing equitably-efficient solutions regarding to the student preferences. The proposed procedure is able to find optimal solutions within an acceptable time span outperforming standard optimisation approaches both in terms of computational effort and solution quality. The contribution of this paper is
CRediT authorship contribution statement
Babak Akbarzadeh: Conceptualization, Methodology, Software, Formal analysis, Data curation, Writing - original draft. Broos Maenhout: Conceptualization, Methodology, Investigation, Visualization, Writing - original draft, Writing - review & editing, Supervision, Project administration, Funding acquisition.
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