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Preference scheduling for nurses using column generation

https://doi.org/10.1016/j.ejor.2003.06.046Get rights and content

Abstract

The purpose of this paper is to present a new methodology for scheduling nurses in which several conflicting factors guide the decision process. Unlike manufacturing facilities where standard shifts and days off are the rule, hospitals operate 24 hours a day, 7 days a week and face widely fluctuating demand. A more flexible arrangement for working hours and days off is needed, especially in light of the growing nursing shortage. To improve retention, management must now take into account individual preferences and requests for days off in a way that is perceived as fair, while ensuring sufficient coverage at all times. This multi-objective problem is solved with a column generation approach that combines integer programming and heuristics. The integer program is formulated as a set covering-type problem whose columns correspond to alternative schedules that a nurse can work over the planning horizon. A double swapping heuristic is used to generate the columns. The objective coefficients are determined by the degree to which the individual preferences of a nurse are violated. As part of the computational scheme, feasible solutions are refined to minimize the use of outside nurses, but when gaps in coverage exist, the outside nurses are distributed as evenly as possible over the shifts. The methodology was tested on a series of problems with up to 100 nurses using data provided by a large hospital in the US. The results indicate that high-quality solutions can be obtained within a few minutes in the majority of cases.

Introduction

The current crisis in health care is forcing hospital executives to run their organizations in a more business-like manner. The constant challenge is to provide high-quality service at ever reduced cost. This problem is exacerbated by an acute shortage of nurses, said to be 120,000 today and expected to grow to 808,000 by 2020 in the United States (US) alone (USDHHS, 2002). Because the nursing service is one of the largest cost components in a hospital's budget, it is essential for every manager to develop an efficient operational plan that makes the best use of available resources. The decision is complicated by a welter of factors including hospital policies, labor laws, different nursing skill categories, the mix of part-timers and full-timers, random fluctuations in demand, and the desire to accommodate individual preferences.

Nursing skills are typically divided into at least four different categories: registered nurses (RN), licensed practical nurses (LPN), nurse aides, and technical nurses. Registered nurses are the most versatile and generally preferred because they can provide the widest range of care. LPNs are less flexible, while nurse aids have only limited training and skills. A technical nurse is usually needed to operate certain medical instruments specific to a unit, such as cardiology. For purposes of matching skills with requirements, some percentage of the average daily demand is assigned to each nurse category. The specification of this percentage is a management decision that represents a tradeoff between cost, availability, and quality of service.

While there are several levels of staff planning in the hospital environment, our focus is on the mid-term problem associated with weekly assignments. Decisions at this level are made on a regular basis to account for planned leave and expected departures from average demand. To accomplish this task in an equitable manner, information about demand, nurse availability, and the rules concerning priorities and individual requests must be gathered. The decision process must resolve the conflicting viewpoints of the hospital and the nursing personnel. Hospitals are required to provide some minimum level of care (in terms of staff by skill category) during each shift, while nurses want individualized schedules that take into account requests for days off, the exclusion of undesirable work patterns, and other personal considerations. The primary goal is to provide the nursing staff with high-quality schedules subject to demand requirements and cost considerations. This is known as preference scheduling.

The purpose of this paper is to present a robust methodology for solving the preference scheduling problem that can accommodate both the quantitative and qualitative detail that nurse managers must address. Rather than trying to minimize cost alone, our objective function is designed to balance contractual agreements and management prerogatives with the use of outside nurses (primarily floaters and agency nurses). The output is a set of rosters for the nursing staff that trades off individual preferences with personnel costs without violating any of the hard constraints in the system. Critical to the success of the procedure is the idea of fairness and the transparency of the resultant schedules. The short-term goal is adequate coverage at minimum cost, the long-term goal is staff retention.

In the next section, the scheduling environment is described, followed by a review of the nurse scheduling literature. We then present the optimization model and the solution approach. Because the full model was too big to solve with a commercial code, an iterative scheme based on column generation was developed, and then tested using data provided by a US hospital. The results indicate that problem instances with up to 100 nurses can be solved within 10 minutes for a 4-week planning horizon and within an hour for a 6-week planning horizon.

Section snippets

Background and literature review

There are three different approaches to solving the mid-term nurse scheduling problem. The first is called rotational or cyclical scheduling (Howell, 1998). In this approach, several sets of schedules are generated that collectively satisfy the demand requirements. Nurses are then rotated from one set of schedules to another in consecutive planning horizons. An exact solution of the cyclical scheduling problem can be found by using a simplified mathematical model. In this model, cyclic work

Problem statement and model formulation

Hospitals typically employ nurses to work either 8- or 12-hour shifts, giving rise to five standard shift types: three 8-hour shifts called Day (7 a.m.–3 p.m.), Evening (3 p.m.–11 p.m.), Night (11:00 p.m.– 7 a.m.) and two 12-hour shifts called AM (7 a.m.–7 p.m.) and PM (7 p.m.–7 a.m.). An AM shift starts at the same time as a Day shift and ends mid-way into the Evening shift. A similar interpretation exists for a PM shift. The problem will be first stated for a single nurse category using these

Model input

To a large extent, input requirements for model , , , , are a function of the algorithmic approach. Specific parameters, such as the maximum number of columns to be included in the formulation and the minimum number of requests permitted per nurse, are discussed in the next section where the column generation algorithm and its various components are presented. At the general level, we must specify the length of the planning horizon, the number and types of shifts to be assigned, the demand per

Solution methodology

The major component of the solution methodology is the procedure for generating candidate schedules. To initialize the problem, we start with either the sign-up schedule provided by each nurse or with individualized templates based on their contract. This gives us |N| columns for the structural constraints in (5b). If the full roster satisfies the demand this might be all that is needed; however, many of the nurses may not have been able to sign up for their desired shifts, implying that one or

Model enhancements

Several additional features have been incorporated in the algorithmic structure to ensure that the results are realistic and that the methodology is robust. The features considered were strongly influenced by the need to balance the computational effort with the quality of schedules produced.

Computational results

The column generation approach described in Section 5 was implemented in C++ and linked to the CPLEX 7.1 libraries to solve the IPs that arise at each major iteration. Run times were obtained through the clock function in C++. All computations were performed on a 1.1 GHz PC.

To test the effectiveness and usefulness of the approach, three sets of experiments were conducted on problems that included between 20 and 100 nurses. The first set was designed to see how well the algorithm performed on

Summary and conclusions

In this paper, a robust column generation procedure was presented for solving the nurse preference scheduling problem. At the center of the algorithm is a heuristic that identifies attractive schedules by adaptively swapping single and double periods. Although the computations were initiated with a sign-up schedule, other possibilities include the use of rotational schedules or an initialization procedure based on, say, the previous month's rosters. Solutions for problems with up to 100 nurses

Acknowledgements

This work was supported in part by the National Science Foundation under Grant No. DMI-0218701.

References (22)

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    This work was supported in part by the National Science Foundation under Grant No. DMI-0218701.

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