Column-generation-based heuristic approaches to stochastic surgery scheduling with downstream capacity constraints

Highlights

• A stochastic programming model for advance surgery scheduling is formulated.

• Capacity constraints of operating rooms and downstream resources are incorporated.

• Uncertainties in surgery duration and length-of-stay are considered.

• Several column-generation-based heuristic approaches are developed.

• The performances of the proposed approaches are validated in numerical experiments.

Abstract

This paper addresses an advance surgery scheduling problem in an operating theater composed of multiple operating rooms (ORs) and a downstream surgical intensive care unit (SICU). Uncertainties in surgery durations and postoperative length-of-stays are taken into consideration. The decisions are made on a weekly basis and consist of three parts: determining the surgical blocks to open, selecting the surgeries to be performed from a waiting list, and assigning the selected surgeries to available surgical blocks. The objective is to minimize the patient-related cost as well as the hospital-related cost while respecting the SICU capacity constraints. We propose a two-stage stochastic programming model with recourse to address the studied problem. Sample average approximation is employed to translate the stochastic programming model into a deterministic integer linear programming (DILP) model, which is then solved by column-generation-based heuristic (CGBH) approaches. The CGBH approaches developed in this paper reformulate the DILP model in a column-oriented way and adopt multiple column-generation strategies and heuristic rules to improve computational efficiency. The experimental results illustrate that the proposed CGBH approaches require significantly less computation time than the conventional algorithm, and that the gaps between the resulting near-optimal solutions and the exact ones are below 1%. Moreover, numerical experiments carried out with large test problems validate the capability of the CGBH approaches in solving realistically sized cases.

Introduction

Over the past decades, the aging population and the growing quality of life have been driving the demand for healthcare service to increase, requiring the hospital managers to improve the quality and efficiency of healthcare activities. In a hospital, the operating theater (OT), consisting of operating rooms (ORs) and surgical intensive care units (SICUs), is generally considered as the major revenue center as well as the most expensive department that consumes the largest part of the hospital’s budget (Min and Yih, 2010, Wang et al., 2015, Monteiro et al., 2015). Therefore, the management of OT and the scheduling of surgeries have drawn much attention from researchers and practitioners.

The research on OT managing and surgery scheduling can be classified into three hierarchical decision levels: strategic level, tactical level, and operational level (Guerriero and Guido, 2011, Zhu et al., 2018). The operational level can further be divided into two stages: advance scheduling and allocation scheduling. The former addresses the assignments of patients to specific surgical blocks (a surgical block is a combination of an OR and a workday), while the latter determines the specific starting time of each surgery (Zhu et al., 2018). In this paper, we address an advance scheduling problem with consideration of downstream facilities and uncertainties, whereas the intra-day sequencing of surgeries are not considered. We assume that the allocation of surgical resources among different specialties has already been determined at the strategic level, and that a master surgery schedule (MSS) specifying the pre-assignments of surgical blocks to specialties has been fixed under the block scheduling strategy (Guerriero and Guido, 2011) at the tactical level. The outputs of the two decision levels serve as fixed parameters in the studied problem of this paper.

The complexity of advance scheduling results from various factors. First, each surgery is associated with an uncertain surgery duration and an uncertain length-of-stay (LOS) in the downstream facility. These uncertainties are difficult to predict and strongly affect the utilization and availability of surgical resources (Batun et al., 2011, Molina-Pariente et al., 2018). Explicitly incorporating uncertainties into the mathematical modeling helps to improve the quality of the schedule, but significantly augments the computational complexity. Second, the interests of different stakeholders are usually contradictory. Specifically, the hospital administrators and the surgical staff want to achieve low costs and low overtime work, while the patients only desire short waiting times. A policy that schedules too many surgeries in a planning period leads to short waiting times and high satisfactions of patients, but may cause severe over-utilization of surgical resources and increase the hospital’s expenses drastically; on the other hand, if too few surgeries are scheduled in the present planning period, the hospital’s expenses and the surgical staff’s workload are reduced, but the patients have to suffer from long waiting times. Hence, an optimal surgical schedule should well balance the interests of both the hospital and the patients. Third, the existing research has revealed that planning the capacity of ORs independently does not yield high-quality schedules, since the unavailability of downstream facilities, such as surgical intensive care units (SICU) and post-anaesthesia care units (PACU), may block the postoperative patients in ORs and deteriorate the surgery schedule (Min and Yih, 2010, Jebali and Diabat, 2015). Therefore, the capacities of ORs and downstream facilities should be jointly planned, which brings downstream capacity constraints with complicated structures into the studied problem.

The aforementioned difficulties show the necessity of developing operations research methodologies to solve stochastic advance scheduling problems with downstream capacity constraints. Relevant research can be found in the literature (e.g., Min and Yih, 2010, Saadouli et al., 2015, Jebali and Diabat, 2015, Jebali and Diabat, 2017, Neyshabouri and Berg, 2017). In this paper, we consider the advance scheduling of elective surgeries in an OT with multiple ORs and multiple recovery beds in SICU. Emergency patients are not considered since they are assumed to be treated in dedicated facilities (refer to the dedicated policy described in Van Riet and Demeulemeester (2015)). Uncertainties in surgery durations and LOSs are taken into account. At the beginning of each week, we need to determine the surgical blocks to open, select the surgeries to be performed from a waiting list, and assign these selected surgeries to open surgical blocks. Our objective is to minimize the total cost incurred by performing and postponing surgeries as well as opening and overusing surgical blocks, meanwhile the downstream capacity constraints should be respected. The studied problem is formulated as a two-stage stochastic programming model with recourse, which is translated into a deterministic model through sample average approximation (SAA) and then solved by the column-generation-based heuristic (CGBH) approaches developed in this paper. Intensive numerical experiments with various problem sizes are carried out to evaluate the computational performance and the solution quality of the CGBH approaches.

The main contributions of this paper can be summarized as follows. First, for the advance scheduling of elective surgeries, we propose a comprehensive stochastic programming formulation which incorporates the capacity constraints of ORs and SICU, uncertain surgery durations and LOSs, opening decisions of surgical blocks, time-dependent surgery priorities, and multiple specialties with different surgery characteristics. To the best of our knowledge, this formulation is the first mathematical model that incorporates all these elements together. Second, we develop several CGBH approaches to solve the deterministic model which is translated from the stochastic programming model by SAA. In the literature, CGBH approaches are rarely adopted to solve stochastic programming models for surgery scheduling problems. This paper shows that the combination of CGBH and SAA can efficiently compute near-optimal solutions for large instances that cannot be tackled by conventional methodologies. Third, we perform numerical experiments with a number of instances to compare the computational performances of the CGBH approaches and a commercial optimization solver (GUROBI). The results validate that the CGBH approaches can solve realistically sized problems with reasonable computation time and provide high-quality near-optimal solutions.

The remainder of this paper is organized as follows. Section 2 provides a brief review of the relevant literature. In Section 3, we describe the advance surgery scheduling problem studied in this paper and formulate the stochastic programming model, then employ the SAA algorithm to convert the stochastic programming model to a solvable deterministic one. In Section 4, we elaborate the proposed CGBH approaches with different column-generation strategies and heuristic rules. Then, the experimental results are presented in Section 5 to validate the performances of the CGBH approaches. Finally, Section 6 gives the conclusions and possible future extensions.