An efficient matheuristic for offline patient-to-bed assignment problems

Highlights

• We present integer programming models for the patient-bed assignment problem.

• We develop an efficient matheuristic to solve the NP-hard patient-bed assignment problem.

• We discuss on penalty values for soft models and improves all the best-known values in the literature.

Abstract

The bed assignment problem here addressed consists in assigning elective patients to beds by considering several requirements, such as patient clinical conditions and personal preferences, medical needs, bed availability in departments, length of stay and competing requests for beds. The rather complex combinatorial structure of the problem compels finding a solution for effective and efficient decision-making tools to support bed managers in making fast and accurate decisions. In this paper, we design and develop combinatorial optimization models for supporting the bed assignment decision-making process. Since the problem is $\mathcal{NP}$-hard, in order to solve the models efficiently, we propose and motivate a matheuristic solution framework based on a re-optimization approach. The matheuristic is implemented and tested on literature-based benchmark instances. It shows impressive computational performance for all the benchmark instances and the results improve all the best-known bounds of the state-of-the-art.

Introduction

In recent years, there has been an increasing interest of operational researchers in healthcare applications. Optimization and simulation models as well as innovative solution approaches have been devised and proposed with the aim to improve efficiency and quality of care in several complex decision-making processes (Brailsford, Vissers, 2011, Hulshof, Kortbeek, Boucherie, Hans, Bakker, 2012). Operating room planning and management problems, for instance, have attracted a lot of attention (Cardoen, Demeulemeester, Beliën, 2010, Guerriero, Guido, 2011). Usually, complex decision-making processes, like those concerning health care domains, are split into temporal and managerial levels, i.e. strategic, tactical and operational decision levels (Hulshof et al., 2012). The operational level refers to short-term decisions, and it is subdivided into offline and online decision-making. The offline situation reflects the in-advance decision making whereas the online one the real-time reactive decision-making in response to events that cannot be planned. For example, if each patient is classified as elective, urgent or in emergency, only admissions and discharges of elective patients can be planned and scheduled in advance.

Looking at the most difficult and challenging problems, the hospital care services delivery is of prominent significance. Within this context, the patient admission scheduling problem (PASP) and the patient bed assignment problem (PBAP) are decisional problems of notable interest at the operational decision level. The interest in these topics over the last decade is analyzed in Teixeira and De Oliveira (2015). The PASP is concerned with deciding which patient to admit and at what time (Barz & Rajaram, 2015). The PBAP, which is a sub-task of the PASP, aims to assign patients to suitable beds by taking into account of medical constraints, patient needs, and obviously bed availability. This specific problem has received minimal attention in the literature, as reported in Bachouch, Guinet, and Hajri-Gabouj (2012), Thomas et al. (2013), and usually it has been addressed only as a bed capacity problem (Hulshof et al., 2012). In the following, we first briefly review some papers that consider both the PASP and the PBAP; then we present an accurate literature review on more complex PBAPs.

Bekker and Koeleman (2011) present quantitative methods to determine admission quota for scheduled admissions and analyze how variability in scheduled admissions influences on required bed capacity. Adan and Vissers (2002) develop an integer linear programming (ILP) model and test it in a pilot setting of specialty orthopaedics: the authors define an admission profile by taking into account throughput and utilization of resources while satisfying some given restrictions. Conforti, Guerriero, Guido, Cerinic, and Conforti (2011) formulate an ILP model integrating patients admission and resource scheduling to improve clinical efficiency in a hospital rheumatology department: the main goal is to select elective patients, schedule their required medical equipment/devices and define their shorter length of stay (LOS) because of scarcity of beds. Lei, Na, Xin, and Fan (2014) propose a mixed ILP model with constraints on surgery capacity for tackling deterministic and stochastic LOS in the PASP and the PBAP. The scheduling result includes admitted or rejected patients, admission day, last hospitalization day and assigned bed number (all beds are identical). In these reviewed papers, beds are a limited hospital resource and the PBAP is reduced to bed capacity constraints as well as in other papers. Only a few papers consider medical requirements, patients’ room preference, bed equipment, hospital policies constraints in the formulated PBAP. Schmidt, Geisler, and Spreckelsen (2013) define a mathematical program based on up-to-date patients’ LOS and take into account of bed capacity and patients’ room preference (e.g., single, double); a dummy ward with high capacity is introduced to avoid infeasible solutions but its usage is penalized and interpreted as a dismissal. More complex is the decision support tool based on an ILP model of Bachouch et al. (2012), which aims to find available beds for elective and acute cases by considering patients’ expected LOS and several constraints (i.e., incompatibility between pathologies, no mixed-sex rooms, continuity of care, and contagious patients): if there is no available bed for an elective patient, the physician can change his/her admission day; if a there is no available bed for acute patient, he/she is transferred to another hospital or kept in the emergency ward while waiting for a bed. Mazier, Xie, and Sarazin (2010) tackle the emergency patient waiting time reduction problem in a real-time planning: the aim is to assign patients to hospital rooms without disruption in inpatient stay. Patients already assigned could be moved into other rooms or units but transfers are minimized. Thomas et al. (2013) propose a decision support system to manage the PASP and the PBAP in real-time and flexible way at the same time: a user can specify whether bed attributes, room-type requirements, gender mismatch requirements should be hard or soft constraints; hard constraints are on isolation requirements and capacity constraints, i.e., at most one patient is assigned to one bed and vice versa.

In the following, we review more complex PBAPs. In particular, we focus on the PBAP formalized in Demeester, Souffriau, De Causmaecker, and Vanden Berghe (2010).

Demeester et al. (2010) present a PBAP similar to that addressed in Thomas et al. (2013) but formalized as an offline challenging problem, which hospitals with a central planning unit have to manage. This combinatorial optimization problem aims to find optimal bed assignments for elective patients by considering hospital departments, specialties, bed equipment, quality of care and patient characteristics such as pathology, mandatory bed equipment, preferred bed equipment, gender, age and room preferences. Vancroonenburg, Della Croce, Goossens, and Spieksma (2014) demonstrate that the PBAP defined in Demeester et al. (2010) is $\mathcal{NP}$-hard. The best patient bed assignment, which consists in matching patients’ characteristics and room characteristics among all possible alternatives, becomes difficult to determine and hard to solve manually. Hence, the interest in developing efficient quantitative approaches to support hospital admission administrators. The ILP model formulated in Demeester et al. (2010) assigns patients to beds and is a soft version of the original problem because violations on mandatory bed equipment, age policy and gender policy, which are hard constraints in the PBAP definition, are penalized. Transfers in and out of rooms are allowed but highly penalized. The authors constructed six PBAP instances, related to Belgian hospitals, and developed a hybrid tabu-search method for solving them because of the high computational time with common solvers (in some cases more than 24 hours for finding a feasible solution). Bilgin, Demeester, Misir, Vancroonenburg, and Vanden Berghe (2012) in their follow up paper design several heuristic approaches for improving the results on the above six benchmark instances and report that no optimal solution was found after a week of computations with integer programming approaches. They provide also seven new and more complex benchmark instances. The contribution of Ceschia and Schaerf (2011) to this problem is notable because they reduce the number of decision variables by formulating the PBAP as a patient-to-room assignment problem, compute different kinds of lower bounds by removing some constraints, and improve considerably the best-known upper bound values by a simulated annealing approach. These new best values are found by solving the same soft version of the original PBAP formulated in Demeester et al. (2010) and allowing at most only one transfer. Misir, Verbeeck, Causmaecker, and Berghe (2013) analyze several hyper-heuristics with varying characteristics on the first six instances and their best results are similar or worse than those in Ceschia and Schaerf (2011). Range, Lusby, and Larsen (2014) tackle the problem with a Dantzig–Wolfe decomposition in a set-partitioning problem as master problem, and a set of room scheduling problems as pricing problem. They are the first authors who present computational results on the original PBAP (i.e., hard constraints on mandatory bed equipment, age and gender policies) and improve some of the known upper bounds but only of the small sized benchmark instances because of the increased complexity and size of the model with the larger instances. As reported in the literature, some of the benchmark instances require more than 24 hours for finding a feasible solution. Turhan and Bilgen (2017) decompose the problem in subproblems by employing time and patient decomposition approaches, and the feasible solutions, found in low computational times, have a gap from the best-known values in the range 15–21%.

Successive developments on PBAP are due to Ceschia and Schaerf (2012) and Vancroonenburg, De Causmaecker, and Vanden Berghe (2016) who extend the original PBAP formulation to an online PASP to manage some dynamic aspects such as arrival of urgent patients and uncertainty of LOS. Patients admission and discharge dates are not fixed a priori but they have to be defined in order to improve bed utilization and minimize penalty costs. Both papers propose two-index patient/room formulations. Vancroonenburg et al. (2016) test their models on four benchmark instances of the PBAP by adding a random registration date and an expected departure date; Ceschia and Schaerf (2012) propose a simplified optimization model formulation for the PASP under uncertainty, that they named PASU, introduced a binary matrix for feasible/infeasible assignments, and provided benchmark instances for PASU. Lusby, Schwierz, Range, and Larsen (2016) solve these instances by an efficient and flexible approach based on a simulated annealing framework and an adaptive large neighborhood search procedure. They report improvements in the range of 3–14% for the large instances; improvements in schedule quality for the short family instances are found by setting suitable penalty values in Guido, Solina, and Conforti (2017). Finally, a more elaborate model for a PASP and PBAP with a flexible planning horizon and operating room resources and a solution approach, which explores the search space using a composite neighborhood search algorithm, are proposed in Ceschia and Schaerf (2016).

In Table 1, we summarize the main characteristics of the several PBAPs addressed in the reviewed papers, i.e., online PBAP, offline PBAP and whether constraints on gender policies, mandatory equipment, preferred equipment, age policies, required unit are only hard, only soft, or both hard and soft. We also detail if there is a solved PASP, static LOS, dynamic LOS, and a dummy room. The check and hyphen symbols denote the presence and absence, respectively, of a given characteristic.

The last six papers address the problem defined in Demeester et al. (2010) even though they refer to it as PASP, and present results on the benchmark instances.

The contribution of this study is fourfold: (1) we address operational bed management challenges of the PBAP defined in Demeester et al. (2010) by developing specific ILP models, (2) owing to the multiobjective nature of the problem, we give some tips about how to set suitable penalty values, (3) we design a matheuristic solution approach based on solving a sequence of hierarchical optimization subproblems, (4) we test our approach on the PBAP benchmarks and improve all the best-known bounds. Some optimal value are also found.

The paper is organized as follows: Section 2 presents the PBAP definition, its main features, and our ILP models; Section 3 discusses the main differences between our optimization models and those reported in the literature and provides useful tips to set penalty values; Section 4 presents a short review on heuristic and hybrid approaches relevant to our solution approach; then we introduce our matheuristic approach, which is a combination of fix and optimize heuristic, neighborhood searches, and ILP solvers. We evaluate it on the benchmark datasets of PBAP by carrying out computational experiments based on the default penalty values and on suitable setting values. Results are reported and compared with those of the literature in Section 5; conclusions are drawn and future directions outlined in Section 6.