A hybrid fix-and-optimize and simulated annealing approaches for nurse rostering problem

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Highlights

  • The Nurse Rostering Problem is solved via a hybrid algorithm.

  • The hybrid algorithm combines MIP-based heuristics and Simulated Annealing.

  • Initial solutions are found by the Fix-and-Relax heuristic.

  • Solutions are improved by the Fix-and-Optimize heuristic and Simulated Annealing.

  • Seven new best-known results are reported outperforming the state-of-the-art.

Abstract

The Nurse Rostering Problem (NRP) is a complex scheduling problem in which nurses must be assigned to shifts according to a set of constraints. The NRP is difficult to solve to optimality due to its combinatorial structure. In this paper, we propose a hybrid solution algorithm that combines Mixed Integer Programming (MIP) based heuristics—namely Fix-and-Relax (F&R) and Fix-and-Optimize (F&O)—and Simulated Annealing (SA) to solve the NRP. In MIP-based heuristics, a problem is decomposed into a set of smaller problems and each problem is iteratively solved to optimality. In the hybrid approach, high-quality initial solutions are obtained via the F&R algorithm and fed to the SA part of the algorithm. When solutions can no longer be improved, the F&O algorithm is inserted into the SA algorithm. This enables the hybrid algorithm to diversify the search space and provide better solutions. To assess the quality and efficiency of the hybrid approach, we use 24 publicly available test instances recently introduced in literature in the field. Computational results show that the hybrid method outperforms other advanced solution techniques in most of the test data and seven new best-known results are reported.

Introduction

Personnel scheduling problems have been studied by the Operations Research (OR) community since the 1950s. Scheduling today is notably different than it was that time, as many new features have been introduced to the process (Vanden Bergh et al., 2013). For example, employee satisfaction has become an important part of the scheduling effort. Employees are offered part-time and full-time opportunities as well as flexible working hours. Their preferences (i.e. desired shifts, days-off, planned vacations) are taken into consideration when a work schedule is developed.

According to Baker (1976), personnel scheduling problems are generally classified in three groups: time-of-day scheduling, day-of-week scheduling, and a combination of both. In time-of-day scheduling, shift start and end times are scheduled in a daily planning horizon. Weekly planning horizon is used in day-of-week scheduling in which a facility’s operating week may not match employees’ working week. Typical scenario is when employees work five days a week while a facility operates seven days a week. In the last group, scheduling must be done on both a daily and weekly basis. Hospitals are an example of this combined group. They must operate at all times, and employees have a variety of shifts in a day.

In hospitals, nurses are one of many scarce resources. Hospitals produce work schedules that ensure nurses are on staff 24 h a day, seven days a week. The quality of nurse schedules affects the quality of health care (Oldenkamp, 1996), but scheduling is generally done manually. Head nurses or hospital scheduling departments spend significant time constructing schedules that satisfy many constraints. Factors such as preset shifts, nurse requests, and last minute changes make the task tedious and time consuming (Cheang, Li, Lim, & Rodrigues, 2003). Therefore, nurse scheduling has attracted the attention of the OR community, which has developed the Nurse Rostering Problem (NRP)—also known as the Nurse Scheduling Problem (NSP)—to address these concerns.

The NRP generally has two types of constraints. Hard constraints (HC) are those that must be satisfied in order to generate feasible rosters. Soft constraints (SC) are not necessary conditions for feasible schedules, but violating them causes a penalty. For example, one HC is that a nurse cannot be assigned to a morning shift directly following a late shift because there must be rest time after a shift. An example of an SC would be a nurse asking to be assigned to a certain shift on a certain day. If this request is not fulfilled, a feasible roster can still be created, although it would result in a penalty because of the nurses’s dissatisfaction.

In the literature, solution techniques for the NRP can be divided into three categories: exact methods, heuristics, and hybrid solutions. The exact methods are integer programming (IP) and constraint programming (CP). These methods guarantee optimality, but computational times increase drastically as problem sizes increase. The heuristics category can provide high-quality solutions in faster processing times, but the solutions may not be optimal. Heuristics include many solution approaches such as variable neighborhood search (VNS), tabu search (TS), simulated annealing (SA), genetic algorithm (GA), ant colony optimization (ACO), electromagnetic algorithm (EM), scatter search (SS), memetic algorithm (MA), tailor-made heuristics, estimation of distribution algorithms (EDA), and case-based reasoning (CBR). The final category of solutions techniques, hybrid solutions, is relatively a new area of study. It combines different solution techniques to achieve greater strength and flexibility. For applications of several metaheuristic approaches, interested readers may refer to recent papers by Hamza, Abderazek, Lakhdar, Ferhat, and Yıldız (2018), and Yildiz, Abderazek, and Mirjalili (2019).

Applications of quantitative models in NRP date back to the 1970s and have been addressed widely in the literature. An interested reader may also refer to Ernst et al., 2004, Vanden Bergh et al., 2013, and De Bruecker, Van den Bergh, Beliën, and Demeulemeester (2015) for a detailed bibliography of personnel rostering publications; Cheang et al., 2003, Burke et al., 2004, De Causmaecker and Berghe, 2011 for the detailed nurse rostering literature. Besides nurses, the scheduling of other types of medical personnel like physicians have been subject to many research studies. In a recent survey paper, Erhard, Schoenfelder, Fügener, and Brunner (2018) provide a review on physician scheduling problem. Very recently, Kletzander and Musliu (2020) propose a new framework for general employee scheduling that allows independent handling of various constraints in a unified way.

Table 1 represents a comprehensive summary of studies in the NRP field in the last several decades.

Two of these studies are particularly notable because they use similar NRP models and experiment on the same data sets as in our work. Curtois and Qu (2014) use ejection chain (EC) and branch and price (B&P) algorithms, and the Gurobi optimizer to solve the test data. Their results indicate that the B&P method is effective on smaller instances but it is inadequate on larger test data, and it runs out of memory at times. Conversely, the EC metaheuristic finds good solutions on the larger data sets but is outperformed by the B&P method on smaller sets. The second study experimenting on the same instances as our work is by Rahimian, Akartunali, and Levine (2017b), who hybridize the VNS algorithm using IP to solve the NRP. In the study, initial solutions are generated using a greedy heuristic and improved using the hybrid method. Five different neighborhoods are applied to improve schedules during the VNS phase and an IP based ruin-and-recreate framework is embedded into the process to further improve solutions. The study reports new best-known results and compares findings with studies in the literature. It also generates new test data and makes the instances publicly available for other researchers for benchmarking.

Most of the studies in the literature utilize stand-alone heuristic approaches such as (Burke et al., 1999, Aickelin and Dowsland, 2000, Burke et al., 2008, Awadallah et al., 2015) or exact solution techniques such as (Azaiez and Al Sharif, 2005, Burke and Curtois, 2014, Dowsland and Thompson, 2000, Maenhout and Vanhoucke, 2010). There are still only a handful of publications (Burke et al., 2010a, Qu and He, 2008, Rahimian et al., 2017a, Rahimian et al., 2017b) that investigate the combination of the both approaches. Moreover, studies applying hybrid approaches generally use VNS for the heuristic part. Our study proposes a hybrid solution methodology that combines MIP based heuristics with SA.

In our study, we propose a hybrid approach to solve the NRP. Our approach integrates Mixed Integer Programming (MIP)-based Fix-and-Relax (F&R) and Fix-and-Optimize (F&O) heuristics with Simulated Annealing (SA). In MIP-based heuristics, a problem is decomposed into a set of sub-problems, and then each sub-problem is optimized. This process continues iteratively until all the sub-problems are solved. In our proposed NRP implementation framework, we use F&R heuristic as a starting point to find high-quality initial solutions. The NRP is decomposed based on the number of nurses and weeks. The initial solution obtained using the F&R heuristic is then used in the SA part of the framework. Many neighborhood structures are applied during the subsequent iterations to improve the initial solution. When solutions can no longer be improved, the F&O heuristic is injected into the process as a vehicle to diversify the search space. Low cost days are fixed to their existing values and other days are optimized. This interaction between the SA and F&O algorithms often results in better solutions and leads to intensification. Even when the F&O heuristic produces worse solutions, the diversification of the search space improves the performance of the SA algorithm and the neighborhoods. Results are reported when a termination criterion is met in the SA algorithm. Finally, we use data instances previously reported in the literature to assess the performance of the proposed solution and compare our results to other solution techniques.

The main contributions of the paper to the literature are two-fold: our study is the first implementation of the both MIP-based heuristics to the NRP problem and computational results outperform some of the state-of-the-art solution techniques studied in the literature.

The remainder of this paper is organized as follows. In Section 2, the description, notation, and the mathematical programming formulation of the NRP are defined. Section 3 describes the proposed algorithm and illustrates the implementation with an example. In Section 4, the computational experiments are presented and compared to results previous reported in the literature. Finally, in Section 5, conclusions are drawn and possible future research opportunities are discussed.

Section snippets

Problem description

Before describing the problem in detail, some of the useful terms and essential definitions of the NRP literature are provided in the following paragraphs.

Nurse: Person who has completed a program of nursing education and is authorized to care for patients.

Shift: Time period during which nurses perform their duties. There can be multiple shifts in a day (i.e. early shift, day shift, night shift). Shifts have pre-defined start and end times.

Cover: Minimum number of nurses per shift needed to

Solution methodology

This paper proposes a hybrid solution methodology to solve the NRP. It is a combination of the MIP based F&R and F&O heuristics and SA. The F&R heuristic, which is first introduced by Dillenberger, Escudero, Wollensak, and Zhang (1994), finds quick, high-quality initial solutions. During this algorithm, the problem is decomposed into a set of smaller sub-problems. The method of decomposition is based on the size of the problem. For smaller problem sizes, we choose to decompose by weeks (WD).

Computational experiments

In this section, we present an experimental evaluation for the hybrid algorithm. The purpose of section is to answer the following questions:

  • i.

    Under what circumstances does the algorithm provide high quality initial and final solutions?

  • ii.

    What parameters do lead to better results?

  • iii.

    How does each part of the hybrid algorithm contribute to the final solution?

  • iv.

    How do results of the algorithm compare to the results of the state-of-the-art methods?

The software of the proposed hybrid algorithm is written in

Conclusion

The efficient usage of scarce resources, especially nurses, in health care is an extremely important task. Yet many scheduling departments and head nurses still create schedules manually. Automated scheduling solutions will improve the utilization of and fairness among nurses. The NRP problem addresses this critical need by automating the assignment of nurses to shifts and days according to hospital needs and nurses’ preferences.

This study proposes a hybrid solution for the NRP by integrating

CRediT authorship contribution statement

Aykut Melih Turhan: Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft. Bilge Bilgen: Conceptualization, Methodology, Writing - review & editing, Supervision.

Acknowledgements

Data gathering is a time consuming task. When it comes to health care, it is especially difficult, as personal health data is very sensitive. For this reason, we are thankful to Curtois and Qu (2014) for providing the datasets, explanations, and related links to proper sites in a single website.

References (71)

  • X. Cai et al.

    A genetic algorithm for scheduling staff of mixed skills under multi-criteria

    European Journal of Operational Research

    (2000)
  • B. Cheang et al.

    Nurse rostering problems – A bibliographic survey

    European Journal of Operational Research

    (2003)
  • P. De Bruecker et al.

    Workforce planning incorporating skills: State of the art

    European Journal of Operational Research

    (2015)
  • C. Dillenberger et al.

    On practical resource allocation for production planning and scheduling with period overlapping setups

    European Journal of Operational Research

    (1994)
  • K.A. Dowsland

    Nurse scheduling with tabu search and strategic oscillation

    European Journal of Operational Research

    (1998)
  • F.F. Easton et al.

    A distributed genetic algorithm for deterministic and stochastic labor scheduling problems

    European Journal of Operational Research

    (1999)
  • M. Erhard et al.

    State of the Art in physician scheduling

    European Journal of Operational Research

    (2018)
  • A. Ernst et al.

    Staff scheduling and rostering: A review of applications, methods, and models

    European Journal of Operations Research

    (2004)
  • C.A. Glass et al.

    The nurse rostering problem: A critical appraisal of the problem structure

    European Journal of Operational Research

    (2010)
  • W.J. Gutjahr et al.

    An aco algorithm for a dynamic regional nurse-scheduling problem in Austria

    Computers & Operations Research

    (2007)
  • S. Helber et al.

    A fix-and-optimize approach for the multi-level capacitated lot sizing problem

    International Journal of Production Economics

    (2010)
  • Z. Liu et al.

    Simulated annealing for a multi-level nurse rostering problem in hemodialysis service

    Applied Soft Computing

    (2018)
  • Z. Lu et al.

    Adaptive neighborhood search for nurse rostering

    European Journal of Operational Research

    (2012)
  • R. M’Hallah et al.

    Scheduling of nurses: A case study of a Kuwaiti health care unit

    Operations Research for Health Care

    (2013)
  • E. Rahimian et al.

    A hybrid integer programming and variable neighborhood search algorithm to solve nurse rostering problems

    European Journal of Operational Research

    (2017)
  • E. Rahimian et al.

    A hybrid integer and constraint programming approach to solve nurse rostering problems

    Computers & Operations Research

    (2017)
  • F. Sahling et al.

    Solving a multi-level capacitated lot sizing problem with multi-period setup carry-over via a fix-and-optimize heuristic

    Computers & Operations Research

    (2009)
  • R. Soto et al.

    Modeling NRPs with soft and reified constraints

    AASRI Procedia

    (2013)
  • I.X. Tassopoulos et al.

    A two-phase adaptive variable neighborhood approach for nurse rostering

    Computers & Operations Research

    (2015)
  • C.-C. Tsai et al.

    A two-stage modeling with genetic algorithms for the nurse scheduling problem

    Expert Systems with Applications

    (2009)
  • C. Valouxis et al.

    A systematic two phase approach for the nurse rostering problem

    European Journal of Operational Research

    (2012)
  • J.V.D. Vanden Bergh et al.

    Personnel scheduling: A literature review

    European Journal of Operational Research

    (2013)
  • H. Vermuyten et al.

    Integrated staff scheduling at a medical emergency service: An optimisation approach

    Expert Systems with Applications

    (2018)
  • U. Aickelin et al.

    Exploiting problem structure in a genetic algorithm approach to a nurse rostering problem

    Journal of Scheduling

    (2000)
  • U. Aickelin et al.

    An estimation of distribution algorithm for nurse scheduling

    Annals of Operations Research

    (2007)
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