Journal of the Operational Research Society

, Volume 61, Issue 5, pp 768–779

Modelling the size and skill-mix of hospital nursing teams


    • Cardiff University
  • N H Powell
    • University of Southampton
  • J E Williams
    • Cardiff University
Case-oriented Paper

DOI: 10.1057/jors.2009.43

Cite this article as:
Harper, P., Powell, N. & Williams, J. J Oper Res Soc (2010) 61: 768. doi:10.1057/jors.2009.43


Previously published work has described the development of a hospital capacity simulation tool, PROMPT. PROMPT has now been adopted by a number of hospitals in the UK and is used for both strategic and operational planning and management of key hospital resources. The work, as presented here, extends the PROMPT functionality to consider in more detail workforce issues. In particular, working with some of the current hospital users, the research has focussed on detailed planning for calculating the size and skill-mix of inpatient nursing teams. The chosen methodology utilizes both simulation and optimization. Outputs from the PROMPT three-phase discrete event simulation are fed into a stochastic programme which suggests the optimal number of nurses to employ (whole time equivalents) by skill-mix and the corresponding numbers by shift. A novel feature of the tool is the ability to predict and compare nursing needs based on different methods of capturing patient-to-nurse ratios as currently adopted across the UK National Health Service. Illustrative results from one hospital demonstrate that although the overall sizes of nursing teams on different wards are of an acceptable level and comparable to the outputs from the simulation phase of the work, often the number of nurses employed at different grades is not well matched to patient needs and the skill-mix should be reconsidered. Results from the optimization phase of the work suggest that it is cost beneficial to increase the number of permanently employed nurses to account for fluctuations in demand and corresponding high costs of temporary (agency) nurses. The scenario functionality of the tool permits for the study of changing size and skill-mix as a consequence of changes in patient volumes, patient case-mix, numbers of beds and length of stay.


simulationstochastic programminghealthcare modellingnurse skill-mix

1. Introduction

Published work by a number of authors in the healthcare modelling field, including those by Dodds (2005), Brailsford et al (2004), Harper (2002), Groothuis et al (2001), as well as more recent reviews such as Proudlove et al (2007) and Eldabi et al (2007), have highlighted the potential of simulation modelling for improving hospital flows and for capturing resulting capacity needs. Furthermore, Harper and Shahani (2002) demonstrated the pitfalls of using deterministic models for estimating hospital resource needs in the presence of variability, such is often the case when dealing with individual patients and systems with variable demand and service times. Harper (2002) presented the PROMPT model, a three-phase discrete event simulation model for the planning and management of hospital resources. At that time, PROMPT allowed the hospital users to predict bed and operating theatre needs, with some limited workforce functionality. Since this work was published, PROMPT has attracted the attention of a number of hospitals and is currently used by 10 National Health Service (NHS) Trusts across the UK to help both strategically and operationally plan their key resources. The success of PROMPT is due in part to generic nature of the tool allowing it to be fined-tuned to reflect local conditions. The benefits, and potential pitfalls to avoid, of generic simulation structures have been the subject of recent debate, for example Fletcher et al (2007) describe the use of a generic A&E model for use by the UK Department of Health.

The subject of hospital nurse staffing cutbacks and use of agency staff has attracted much attention over recent years both in the public media and in published academic reports. Reports by many authors including those by Shuldham (2004) and discussion papers by the Royal College of Nursing have highlighted serious concerns with current nurse staffing and skill-mix within many UK hospital wards. Furthermore, a recent study by Rafferty et al (2007) has shown that patient outcomes are much worse in UK hospitals where patients are nursed with inappropriate and non-optimal staffing levels and skill-mix. Despite such concerns, very little information and tools are available that allow hospital managers to more accurately match nurse staffing levels and appropriate skill-mix to patient case-mix and volumes. Most of the work-to-date that has been commissioned by the Department of Health has reviewed current staffing levels and possible methods for capturing size and skill-mix of nursing teams (Hurst, 2002, 2005).

The research presented in this paper extends the previously published work by Harper (2002) to address the above concerns on nurse staffing. A number of current NHS users of the PROMPT simulation tool have guided the development of an enhanced workforce module. The chosen methodology utilizes the discrete event simulation functionality of PROMPT and treats outputs from the simulation as an input into a stochastic programme that subsequently suggests the optimal number of nurses to employ (whole time equivalents) by grade (skill-mix) as well as the corresponding numbers to deploy each shift, at the ward, speciality and overall hospital levels. The simulation aspect is used to dynamically model individual patients through the hospital, and the output from this is the nursing need, for each type of nurse, at each time step (typically shift) of the simulation. The stochastic programming seeks to minimize the cost of employing both permanent and temporary nurses while satisfying the demand for each type of nurse in the output from the simulation. The scenario functionality of the tool permits the study of changing size and skill-mix as a consequence of changes in patient volumes, patient case-mix, number of beds and length of stay. Illustrative results from one UK hospital are presented and discussed.

2. Nurse staffing

All patients on a hospital ward require some form of nursing care with nursing needs typically differing between patients depending on individuals’ conditions. The ability of the nursing staff to meet the needs of the patients on the ward at any one moment in time will depend on which nursing staff have been rostered to work at that time. A roster is made up of a list of staff, by their grade (ie what they are qualified and able to do for the patient). Although there is a large academic literature on staff scheduling/rostering, and availability of (mostly) commercial rostering software tools for use within the NHS, such methods and tools need as prerequisite information on the numbers and grades of staff to roster. Unfortunately it is true that most hospital planners are not well informed about the best combination of staff (skill-mix) to roster in the first place (Aiken et al, 2002; Shuldham, 2004). Hence, although available tools and methods might be computationally efficient, they ultimately produce rosters that are based on incorrect assumptions of nurse-to-patient needs and thus are not well aligned to meet the needs of the patients. There may also be times, given the possibility of staff availability, changes in patient demand and patient case-mix, that hospital managers will need to make use of temporary (agency) staff, which usually come at a high cost to the hospital. In summary, trying to achieve the correct balance of appropriate numbers and grades of nurses to employ, while making use of temporary nurses as and when required, is a non-trivial task.

Currently there are nine basic grades of nurses in the UK (from Grade A—Auxiliary & Assistants to Grade I—Nurse Specialists) and within that, many different branches of nursing. Different nurses will have different skills, although most will share a common core. Hurst (2002) discusses five main methods developed over time and used (to varying degrees) within hospitals, for estimating the numbers and skill-mix of nurses needed. These are
  • professional judgement (or ‘Telford’ method);

  • nurses per occupied bed;

  • dependency-activity-quality (acuity-quality);

  • timed task/activity;

  • regression-based activity.

Strengths and weaknesses for each of the methods mean that none is regarded as wholly inferior or superior to the others, and Hurst notes that each should be interpreted before being applied to a particular situation. A brief description of the methods and the main strengths and weaknesses are shown in Table 1.
Table 1

Approaches to calculating the size and skill-mix of nurses (adapted from Hurst, 2002)





Professional judgement (Telford)

Expert healthcare professionals agree an appropriate size and mix for a ward.

Quick, simple and inexpensive to use; easy to update; little adjustment for other care groups; can deal with new and immeasurable variables easily.

The relationship between staffing and nursing quality is hard to explain using this method; insensitive to changing patient numbers and dependency mixes; viewed as too subjective by some managers.

Nurses per occupied bed

The number of nurses needed per occupied bed is aggregated from data collected over time for similar wards.

Simple; formulas use routinely collected data; quality is assured by deriving formulae from quality assured wards.

Formulae are insensitive to dependency changes; formulae are costly to update; routinely collected data can be error prone; poor practice could be replicated if quality assured wards are badly chosen; some hidden structures and processes (eg nurses doing non-ward based tasks) can be masked by the formulae.

Dependency-activity-quality (acuity-quality)

The overall time given to each patient in each category is recorded and converted to ratios. The workload is found from these ratios and the average patient mix.

Local values can be used, or database averages; numbers can easily be calculated for different time intervals; changing patient numbers and mix is easy; performance indicators are easily derived.

Complex; non-local data may not be acceptable; updating data can add to nurses’ workload; grade-mix proportions derived may not suit the individual ward; the relationship between nurse activity and nursing quality is complicated.

Timed task/activity

The average time needed for each care activity, for each type of patient is recorded. The total time is then calculated, based on the average patient mix.

As for acuity-quality; easily computerized; base information is easily updated.

As for acuity-quality; more complex than acuity-quality; commercial systems are expensive.


Predict the required number of nurses for a given level of activity made up of a number of independent variables.

Useful where predictions are possible; ease of use across specialities; data are easier to collect than some methods.

Many variables needing statistical analysis initially; difficulty of qualitative variables; lack of understanding by nurses; extrapolating to predict outside the model's observed range may be unsafe.

In a more recent paper, Hurst (2005) discusses further his data set on which his assessment of nursing workforce planning is made, but this time looking at the aspect of quality of care. The dependency-activity-quality method is the most popular of the five methods listed previously (Waters and Andalo, 2003). Hurst's analysis showed that lower-quality wards tended to have more fluctuating workloads and used more temporary (agency) nurses.

Building on this foundation, the Healthcare Commission (for England) has studied the impact of nurse staffing on outcome data such as pressure ulcer incidence, complaints about nursing care and drug errors. Nurse-sensitive indicators have been published, such as failure to rescue, the prevalence of pressure ulcers and falls, and other indicators more focussed on intensive care. These draw on the work of Aiken et al (2002, 2003), and Needleman et al (2002) in the USA who have shown that higher levels of nurse staffing have a positive impact on patient outcomes. A recent study by Rafferty et al (2007) has confirmed these findings in the UK, showing that deaths are lower where patients are nursed with optimal staffing levels and those wards that have less need for temporary nurses. This demonstrates the importance now placed on more accurately calculating staffing levels and appropriate skill-mix that is well matched to patient needs and which minimizes the need for temporary nurses and yet is cost-efficient for hospitals.

3. Previous workforce modelling approaches

As alluded to earlier, there has been much research interest in nurse rostering, and so only a brief overview is presented here. The fact that there is still much interest provides evidence to the fact that finding an efficient use of resources for an inherently variable demand is an intrinsically difficult problem. Siferd and Benton (1992) review nurse staffing and scheduling, with a direct examination of the role of operational research (OR) modelling. They predict that there is much potential for OR models to have a big impact and that it will interest researchers for many years to come.

A linear program to determine the optimum mix of different staff categories for a hospital medical unit is presented by Bordoloi and Weatherby (1999). Cost is minimized subject to constraints of patient demand and minimum staffing policies. Bordoloi and Weatherby further consider the managerial implications of the results and look at how the sensitivity analyses can be used effectively.

Green (2004) writes that aside from beds, personnel, in particular nurses, form an important measure of hospital capacity. In most hospitals, ratios of patients to nurses are used to determine the number of nurses assigned to a unit. Green proceeds to discuss that although there have been many papers demonstrating the use of optimization models to determine nurse staffing, the basic data needed to use such models is often lacking from hospitals, thus impeding their practical use.

Dowsland (1998) tackles the problem of allocating nurses to shifts, such that they appear to be allocated evenly, using a tabu search algorithm. This work is then further developed in Dowsland and Thompson (2000) to enable larger (50 nurses) and more complex problems to be solved, by applying knapsack and network models before and after the tabu search to restrict the solution space, and to ensure feasibility. The model for this work has been called computer aided rostering environment and it has been developed into a commercial software package.

A comprehensive annotated bibliography of personnel scheduling and rostering is provided by Ernst et al (2004), including a large section on nurses which would make an excellent starting point for the interested reader.

Turning our attentions away from rostering and to that of evaluating the balance between temporary and permanent staff, Kao and Queyranne (1985) present eight linear programming models for budgeting hospital nursing workforce need. The most complex, and hence difficult to solve, model is a two-stage model with recourse where the first stage decides the permanent workforce and the second stage decides the use of overtime and nursing agencies over a 12-period time horizon. The other models are simplifications, in various ways. The smallest time period considered is a month, and the demand for total nursing hours in each period is approximated using a normal distribution whose parameters are derived from an autoregressive integrated moving average. In comparing the models, they find that ignoring the time-varying nature of demand is acceptable so long as demand uncertainty is taken into account.

Brusco et al (1993) present analysis of a linear programming model, with the objective to minimize labour costs, designed to explicitly determine both full time equivalent and supplemental nursing staff resources required, subject to nurse availability. Sadly, the details of the linear program are confined to an unpublished manuscript by Brusco and Showalter. They present a number of cost savings where the linear program is used, but stress that it is designed to supplement, not replace the current staffing process.

Jeang (1994) develops a linear program, whereby the objective is to minimize cost, while determining numbers of full-time and part-time staff and any overtime. The initial program was found to be difficult to solve, a problem which was overcome by making the number of full-time staff a fixed number, rather than a variable. The linear program was then solved for various numbers of full-time staff, and the results compared.

More recently, Griffiths et al (2005) used discrete event simulation to model the need for rostered and supplementary nurses in an intensive care unit of a hospital. They assume a one-to-one ratio of nurses to patients and no variation in nursing ability. Supplementary nurses were modelled as costing four times as much to employ as a rostered nurse. Using the simulation model, a number of nurses to roster per shift was suggested in order to minimize cost. The model was also used to look at a number of scenarios related to patients’ length of stay, as well as demand in future years.

This paper now proceeds to discuss the development of the workforce module within PROMPT and the link to a stochastic programme to suggest the optimal and skill-mix of nursing teams. Unlike previous work, some of which is described above, more attention is given to formalizing the link between patient need and nursing need. Furthermore, the approach of combining discrete event simulation and stochastic programming for this problem is seemingly novel. The rationalization of this approach is to include the stochastic conditions in modelling patient flows and in ascertaining corresponding workforce needs over time, which are then fed into the stochastic programme which, while accommodating for fluctuations in these needs, suggest an optimal permanent number of nurses to employ (by grade) to meet the majority of demand and allowing for temporary nurses to be sought as and when required at busy times. The formulation is based on ranges of demand predictions for nurses by grade from the simulation and then minimizing cost to the hospital based on associated salary costs of permanent and temporary staff. Unlike many of the previously published papers on this topic, we can genuinely claim that this work has developed as a result of collaboration between academic and healthcare professionals, and has found successful implementation as evidenced by use of the tool in different NHS Trusts.

4. Linking patient-to-nursing needs

PROMPT is used to model individual patients as they pass through the hospital over time; see Harper (2002) for a comprehensive description of the model. PROMPT inputs include user-defined patient groups with associated demand profiles (by shift, day, week and month), length of stay distributions and bed manager priority lists. The user creates ‘care units’ where patients stay, for example an assessment unit, day-case unit or ward, with associated available capacities. PROMPT produces a range of outputs including predictions on bed occupancies and refusal rates (or outlier rates to inappropriate care units). The model demonstrates the importance of taking into account variability in healthcare processes, and guidance to hospital managers on the nature of the non-linear relationship between numbers of beds, occupancy and refusal rates (Harper and Shahani, 2002).

Guided by a number of current users and hospital personnel, including human resource (HR) directors, directors of nursing, speciality managers and senior nurses, PROMPT has been adapted to capture shift-by-shift nursing needs across all user-defined care units (a care unit is any place where patients can stay, such as an inpatient ward, day-case unit, assessment unit or speciality bed unit). The underlying concept of PROMPT is to model the patient flows and to predict the numbers of patients in each care unit at each time slice, which is usually set to one shift (typically eight hours), depending on the level of detail that the hospital wishes to explore. Given that PROMPT already has a handle on bed occupancies as based on patient demand (allowed to vary by shift, weekday and month to reflect observed or predicted trends) and service times (sampled from fitted length of stay distributions), resulting workforce needs can be evaluated if it is possible to link patient need to nursing need. Reviewing the five methods as highlighted earlier (Hurst, 2002 and Table 1), it was decided to focus only on the two methods which are predominantly used (or perceived to be useful) across the NHS, namely the nurses per occupied bed (hereafter refereed to as occupied bed method) and dependency-activity-quality (hereafter referred to as acuity method). The other three rejected methods are expert judgement and time-task activity, which require manual input and so are not suited to this type of modelling (but nevertheless are useful for validating the tool) and regression-based methods which are viewed as rather black-box with corresponding distrust and are rarely, if ever, used in practice.

Occupied bed: The nurse-to-patient ratio is defined for each grade of nurse. This ratio indicates the nursing need for every occupied bed. Note that this is regardless of the status of the patient in each bed and is simply an average need. Based on surveys from 83 hospitals, Hurst (2002) has reported helpful occupied bed ratios for this purpose. These are specified by type of ward (medical, surgical etc) and for all nursing grades. When the simulation is run, the number of occupied beds is recorded at each time step (shift). The direct nursing need is a multiple of the occupied bed numbers using the defined ratio. The ratios are usually defined in terms of direct care; that is the amount of nursing needed for ‘hands-on’ care. Indirect care, such as administrative duties, coffee breaks etc needs to be factored in when translating these figures to numbers of nurses to staff on each shift. Furthermore, annual leave, training and sickness levels need to be considered when translating to whole time equivalents (WTEs) to employ.

Acuity: The patient's length of stay is divided into various user-defined dependency levels to reflect that a patient will typically need differing amounts of care, and given by different grades of nurse, at different times during their stay in hospital. For example, consider a patient that has just moved onto a ward from a high-dependency unit. Early in their stay, this patient will require a high ratio of care from more senior nurses, and as they gradually recover over the pursuing days, their nursing needs will decrease and can typically be met by less senior staff. In essence, each patient passes through different (user-defined) dependency levels (with a dwelling time in each level expressed as a percentage of their overall stay) and a nurse-to-patient ratio for each nurse grade is defined for each level. During the model runtime, the simulation will keep track for every occupied bed the dependency level of the patient in that bed at any given time. Hence it is possible to calculate the associated staffing needs based on corresponding ratios. In addressing some of the perceived weaknesses of this method (Table 1), integration of the acuity method into PROMPT was performed in such a way as to provide greater transparency in the dependency levels and associated ratios, easier maintainability (dependency grids can be saved, any minor changes made and the model re-run), the availability of benchmark data grids (see below) and the ability to view results generated using the acuity and occupied bed methods alongside each other for easy comparison.

The PROMPT workforce module permits the user to define all of the necessary workforce parameters and to select either or both of the occupied bed and acuity methods to evaluate workforce need. The principles of a generic structure of PROMPT have been maintained where possible in the workforce module. For the acuity method, any number of user-defined dependency grids (eg adult medicine, paediatrics) may be created and within each grid the number of dependency levels stated (for example 3 levels where 1 is defined to be low dependency, 2 medium and 3 high). The majority of Trusts that collaborated on this work already had in place well-defined dependency levels that were used in the model. For each user-defined patient group, the percentage of length of stay that is spent in each dependency level is captured. For example, a fractured neck of femur (hip) patient may spend 40% of their time in level 3, 40% in level 2 and 20% in level 1, whereas for some other minor orthopaedic procedures, patients may spend 100% of their time in level 1. Finally, the user defines corresponding nurse-to-patient ratios for each nursing grade and level. In practice, defining patient groups, levels, ratios and patient time spent in each level was carried out by a steering group at each Trust comprising of nurses, senior managers and HR personnel. This was informed by published work and benchmark ratios by Hurst (2002, 2005). Engaging all levels of staff in the decision-making process, including lower-grade nurses, was seen as critical in achieving model credibility, and was based on the author's previous experiences and suggested approach (Harper and Pitt, 2004).

For the occupied bed method, more straightforward parameters are required. These are simply the ratio of nurses-to-patient by nurse grade, and may differ across specialities. Again these were agreed by the steering group and may be derived by suitable consolidation of acuity ratios. Once more, the work by Hurst (2002, 2005) was a useful reference point.

Other workforce parameters, independent of whichever method is chosen, are required to translate results to numbers to staff by grade on a shift and WTEs to cover all shifts. These parameters are for indirect nursing time that is captured as a percentage of a shift spent on other necessary non-direct care (admin etc) as well as coffee breaks and other time-outs. For example, Hurst's (2002) survey of 83 hospitals found that on average nurses spend only 42% of their time on direct (hands-on patient care) activities. To translate shift needs to WTEs to employ, PROMPT requires parameter values for sickness levels, professional development (training) days and holiday entitlement by nurse grade (all expressed as an average number of days per year) as well as contracted working hours by grade. All workforce model parameters are captured in PROMPT through user-friendly designed forms.

5. Model formulation and structure

In order to determine the optimal number of nurses and associated skill-mix to employ, one approach might be to run the simulation for a number of scenarios, with a different level of resources (nurses) for each scenario. The respective results could then be analysed. Although this approach would take into account the availability of the different nurses, the respective costs associated with each type of nurse would not be considered, although the costs could be calculated at the end. We adopt a stochastic programming approach, which takes the demand for the different types of nurses over the year and evaluates the optimal number to be employed on a permanent basis, and how many temporary (or agency) nurses should be used and when they might be needed. As the nursing need is variable, it was felt that a stochastic program was appropriate.

The PROMPT model is run for a user-defined number of iterations, which we denote by N, each lasting a year in period. PROMPT captures the numbers of required nursing staff by grade for each day i(i=1,…,365). The N iterations have identical input parameters and thus may be considered equally likely. The resulting N sets of outputs (nursing needs by day) are then fed into the stochastic program (the optimization software used was Xpress-MP and data handling from PROMPT using VBA). An alternative and feasible approach would be to run N iterations of the simulation, with different input parameters for each iteration. The stochastic programme formulation would then require associated probabilities of the likelihood of each of the different input parameters to be defined.

After some experimentation and sensitivity analysis, 30 iterations seemed a sensible choice as the average results of the simulation showed no improved accuracy when more runs were performed and that this number kept the stochastic program and run-times to an acceptable and manageable size.

We assume that we employ nurses of grade j where j=1,…,J. In practice J may be of the region 5–10 as evidenced by the collaborating NHS Trusts. Some Trusts prefer to use the standard set of nursing grades (A, B, C etc) whereas others have evolved their own local definitions. We assume also that the higher the value of j, the more senior the nurse is with value J being the most competent nurse available.

When permanent staff are not able to meet the demand, temporary nurses may be used. After discussions with collaborating NHS Trusts, we consider two sources of temporary nurse; NHS Professionals (an internal agency) and externally run agencies. Within each of these two sources, nurses are further considered at two levels of competency: fully trained (denoted by the subscript T in the formulation below) and healthcare assistant (denoted by the subscript H). The two separate sources have been identified and formulated as availability of staff and daily staff costs from each source will typically vary. Naturally, the formulation below could easily be extended to a more generalizable case with more than two sources and categories of temporary nurse available.

We define the variables as follows:
  • Let xj be the number of WTE permanent staff of type j to employ, where j=1,…,J.

  • Let niks be the number of staff from NHS Professionals required on day i for staff competency level k, and iteration s, where i=1,…,365, k=T,H, and s=1,…,N.

  • Let aiks be the number of staff from external agencies required on day i for staff competency level k, and iteration s, where i=1,…,365 and s=1,…,N.

We define the constants as follows:
  • Let dijs be the demand on day i for nurse type j for scenario s, where i=1,…,365, and s=1,…,N.

  • Let cj be the annual salary per WTE nurse of type j=1,…,J

  • Let gk be the daily wage for nurses from NHS Professionals, k=T,H.

  • Let hk be the daily wage for nurses from external agencies, k=T,H.

We also introduce the notion of efficiency and denote e∈[0,1] to be the efficiency ratio for the temporary nursing staff such that when e=1 temporary nursing staff are as efficient as permanent staff. Although there is currently no comprehensive quantitative evidence to support that permanent staff are in some ways more efficient than agency staff, research such as that by Rafferty et al (2007) has shown that patient outcomes are likely to be worse in hospitals that make greater use of agency nurses. It therefore does not seem unreasonable to suppose that patient contact with permanent staff, whom they see regularly, will be more efficient, such as by reducing any unnecessary length of stay, than for patients being treated by many different agency nurses during their stay, each one of whom will need time to appreciate the patients’ medical history and previous care provided. For this reason, we experiment with different values of e as no numeric value has been suggested in the literature.

The rate payable to NHS Professionals is in fact always less than that of the external agencies. However, staff from NHS Professionals are not always available and sometimes the external agencies have to be used. As we can assume no difference in standard of care given from each source, clearly an optimal solution will always chose staff with the lesser cost. We therefore make an assumption about the percentage of time that staff from NHS Professionals are available. After consultation with NHS staff, an initial figure of 70% was agreed upon, that is, NHS Professionals staff are available 70% of the time that a request is made. We can therefore remove one of the variables by setting aiks=3niks/7.

Finally, we formulate the stochastic program as a combination of an objective and constraints. In this formulation the need for the temporary staff, niks, are the re-course variables. Each scenario occurs with probability ps. As there are N iterations (scenarios) and we assume that each scenario is equally likely because there is no difference in input parameters, ps=1/N. However, as stated earlier, it would be possible to run N iterations of the simulation with different input parameters for each, which would then require each iteration to be weighted by the corresponding likelihood ps (s=1,…,N) such that ∑s=1Nps=1.

The objective is to minimize costs and the constraints ensure that the demand for each level of nursing care is met, allowing for a nurse with a higher competency level to be used as a substitute for a nurse with a lower competency level but not vice versa. The formulation is as follows:
with the inequality repeated for y=1,…,(J−1)
Note that in the constraint, the fraction 10/7 results from consolidating the two types of temporary nurse to one variable, and that NHS Professionals staff are available 70% of the time that a request is made, that is, we update the number of NHS Professionals to the total number of temporary staff, which is equivalent to the number of NHS Professionals divided by 0.7. Likewise, in the objective, the total number of external agency staff required is 0.3 multiplied by the number NHS Professionals divided by 0.7, which is 3/7 multiplied by NHS Professionals.

For converting between number of nurses to roster on each shift and WTEs to employ, the relationship is set out below and PROMPT will perform these calculations depending on parameter values as chosen by the user. These parameters may be easily changed and results are interactively updated to allow for experimentation and scenario analysis by the user at the end of the simulation run. Figures in brackets are suggested initial parameter values chosen by the majority of participating NHS Trusts.

Define W to be the number of hours in the working week for each WTE nurse (37.5), which includes all meal breaks. We assume each day is divided approximately into three shifts (which we call early, late and night), each of eight hours duration. In the PROMPT model, the user may, if they wish, define all associated occupied bed and acuity ratios for each shift. However this adds yet an extra level of data need. A more straightforward approach is to assume (which in practice is often the case) that the early shift is the busiest shift of the day. Hence we define ratios only for the benchmarked early shift and subsequently define PL and PN to be the percentage reduction in nursing needs, compared to the early shift, for late and night shifts, respectively (initial suggested values of 10% and 20%, respectively). Define the direct care needed on an early shift to be D (an output of the simulation model as an associated number of nurses required to provide all hands-on direct care to all patients), and the proportion of indirect time as a user-defined parameter to be I (58%), then the total care need per day, taking into account the indirect care time component is given by T, where
Let R be the total proportion of the year that a nurse is either on leave, off sick or on study leave (20%), then the total number of WTEs to employ, E, in order to sufficiently cover all the shifts in a week (assuming here 7 days per week; simple adjustments can be made if the ward is closed for part of the week) while accounting for typical estimates of the proportion of time permanent nurses might be away from work, is given by
It should be noted that if the variables W, I or R differ for different grades of nurse, it is necessary to convert the demand for each type of nurse into WTEs either before performing the stochastic program, or alternatively as multiplying the right-hand side of each of the constraints by a positive constant is equivalent to multiplying the objective function by the same constant, it is possible to perform this conversion after the stochastic program.

6. Illustrative case study

To illustrate the use of the workforce module together with the stochastic programme, we present here one case study from one of the participating NHS Trusts. Due to confidentiality requirement and sensitivity over results, we are unable to explicitly name the Trust, however it has around 1000 in-patient beds and is typical of a medium-to-large size NHS hospital serving a population in excess of half a million.

The PROMPT model was used by the Trust to model nursing needs for different specialities. We present here results from the orthopaedic department. A common modelling approach across all specialities, including that for orthopaedics, was adopted. This involved small steering group panels comprising of the speciality manager, senior nurses and representatives from HR. Prior analysis of historical patient data was presented at the meetings, which showed volumes of care (patient episodes) by procedure for the last year. The panel considered all procedures separately for emergency and elective admissions, and subsequently grouped all procedures into clinically meaningful groups, that is, those procedures which have a similar profile for nursing needs. In total, nine emergency and eight elective groups were defined, as listed in Table 2.
Table 2

Orthopaedic groupings and acuity data



Dependency level (% of LoS)

Number of patients

Average LoS








Emergency A

Fractured neck of femur and major joint replacements








Emergency B

Spinal procedures








Emergency C

Small closed reductions of fractures







Emergency D

Primary open reductions of fractures








Emergency E

Intravenous therapy







Emergency F

Major closed reductions of fractures








Emergency G

Minor procedures







Emergency X

Other procedures







Emergency N

Non-surgical treatment







Elective A

Hip and knee replacements








Elective B

Other major joints








Elective C

Minor joints







Elective D

Decompressions and spinal fusions








Elective E









Elective F

Hand surgery







Elective X

Other procedures







Elective N

Non-surgical treatment







All emergencies









All electives









For each group, demand profiles (by day and month) together with fitted length of stay distributions were used to populate the PROMPT model. For the acuity method, dependency levels for all adults (irrespective of speciality) were already in use by the Trust and thus adopted for our study. There were five dependency levels, ranging from level 1, lowest dependency, to level 5, critical (usually reserved for those in critical care). Furthermore, for each group the panel agreed the percentage of the length of stay that patients spend in each dependency level. Given that length of stay for each patient is itself sampled from an appropriate distribution, use of fixed percentages indicates the relative time a patient will spend of their total length of stay in each dependency level. The steering group felt that the actual time a patient spends in each level is dependent on their overall length of stay, thus the use of fixed percentages by level was deemed to make more clinical sense than the use of fixed time estimates alone. For example, it is more sensible to state that a patient spends 10% of their stay in dependency level 4 than to specify that all patients spend exactly one day in level 4. An alternative way to consider this percentage is to estimate, or observe over a sufficiently long period, the average percentage of patients in orthopaedic beds in each level. For example, if across 50 orthopaedic beds on average 20 patients were level 1 and the remaining 30 level 2, then equivalently this would represent that patients spent on average 40% of their length of stay in level 1 and 60% in level 2. National benchmarking values by Hurst (2002, 2005) were also helpful for this purpose and the hospital used activity and survey data to agree the percentages shown in Table 2 for each orthopaedic group. Naturally, these values could be easily changed in the PROMPT model by other hospitals to reflect the use of different dependency levels or local conditions and case-mix.

The Trust uses five locally meaningful grades of nursing, which were adopted in the model, although each grade does map directly to the standard set of pay bands used throughout the UK. Nursing grades and current annual salaries (correct at the time of the study), as adopted by the Trust and subsequently used in the stochastic program, are shown in Table 3. Efficiency of agency staff was initially set to e=0.9.
Table 3

Adopted nursing grades and salary bands

Nurse grade

Local definition

NHS band equivalent

Annual salary


Trained nurse experienced

Grades F, G, H, I

£28 087


Trained nurse competent

Grades D, E

£19 372


Trained nurse inexperienced

Grade C

£15 514


Healthcare assistant experienced

Grade B

£13 549


Healthcare assistant inexperienced

Grade A

£11 528

Acuity and occupied bed ratios were defined. Table 4 presents the acuity ratios that were adopted across all adult specialties after meetings and agreement with senior staff. These ratios were carefully considered and validated against known benchmarks, such as those by Hurst, while reflecting local differences based on experience from previous time-tasked studies undertaken across a selection of wards. Different acuity and occupied bed ratios were defined for paediatrics and a few other specialities where it was deemed appropriate, such as maternity services.
Table 4

Nurse-to-patient direct care ratios

Nurse grade

Patient dependency level






























The orthopaedic speciality consists of 120 beds that are divided into 6 wards. Hourly costs for temporary nurses from NHS Professionals were set at £14.11 and £8.63 for trained and assistants, respectively. Equivalent costs from external agencies were £20.69 and £12.65. As discussed in Section 5, it is assumed that requests for temporary staff from NHS Professionals are met 70% of the time.

The PROMPT model was populated and run to simulate 1 year with 30 iterations. Daily nursing needs (by ward and across all orthopaedic beds) were calculated in PROMPT using the equations given in Section 5. The daily nursing needs, for each nursing grade and for each iteration, were fed into the stochastic program. Illustrative results are presented below and show numbers of suggested permanent WTE nurses to employ by grade under both occupied bed and acuity methods, and compared to those using national UK data from Hurst and current numbers employed by the Trust. Furthermore, we experiment with the efficiency factor e to show how total nursing needs may change. Model validation is made more difficult as we are predicting what we believe are true nursing needs based on a bottom-up patient needs’ analysis compared to those currently staffed based on historical reasons. Nevertheless, it was possible to compare outputs for predicted and current staffing levels, by nursing grade, and to discus findings with the steering group panel to ensure that results seemed reasonable. On a few occasions and for a handful of specialities, there was a need to run through a few iterations of the model by adjusting the nurse-to-patient ratios accordingly to obtain more acceptable ranges of nursing needs. For these specialities, it was agreed that the common hospital patient dependencies did not always reflect local differences in particular patient needs and fine-tuning was justified. Comparing these results against those derived using Hurst's benchmarked national data set provided for an additional validation test and increased confidence in the model outputs.

Table 5 presents predicated nursing results from the PROMPT model, using occupied bed ratios, acuity ratio and Hurst ratios, compared to the hospital's current establishment (WTEs). Note that the model predictions shown are average nursing needs across the entire year. Of course these needs will fluctuate over days and weeks and PROMPT is able to predict these needs dynamically and shows them in both graphical and tabular form. In the second phase of modelling, the stochastic program accounts for these fluctuations when suggesting an optimal number of permanent nurse to employ to meet the demand predictions from the simulation while trying to minimize costs. The optimal numbers of nurses are shown in Table 5 in the last column.
Table 5

Model predictions and current establishment staffing levels (WTEs)

Nurse grade

Occupied bed



Hospital establishment




































































Predictions using the occupied bed and acuity methods are broadly in agreement with a total average need of 129.2 and 132.8 WTEs, respectively (less than 3% difference). They also suggest similar needs by grade, although interestingly the occupied bed method tends to suggest less need for the most qualified staff (TNEs and TNCs). This pattern was observed for many of the specialties across the Trust and suggests that the more detailed acuity method (compared to the simplistic averages of the occupied bed method) is better able to pick up subtle differences between grades given the bottom-up planning approach.

When comparing these two methods to the predictions using national benchmarks by Hurst, again they are all in broad agreement, with Hurst ratios predicting an average need of 130.5 WTEs which falls in between the numbers suggested by the occupied bed and acuity methods. Hurst acuity figures tend to reinforce that the occupied bed methods are under-estimating needs for the most qualified staff.

The most interesting findings emerge when comparing the simulation outputs to actual current staffing levels. Although the total number of WTEs are similar (current actual total of 125.5 WTEs), the suggested skill-mix (division of staff between grades) is somewhat different. In particular, current levels of TNEs (most qualified) and HAI (least qualified) are significantly higher (shown using suitable chi-squared tests) than those predicted by all three methods. As a consequence, actual staffing levels for the middle bands (especially TNI and HAE) are significantly lower than those predicted by the model. This demonstrates that although the overall size of nursing teams on different wards are of an acceptable level, the number of nurses employed at different grades is not well matched to patient needs and the skill-mix should be reconsidered. This observation was made across a number of specialities. It highlighted to the steering groups that the hospital is tending to under-invest in nurses and assistants within the middle grades (HAE through to TNC) bands (which make up the majority of nursing team employees). It would therefore be more appropriate to examine ways in which these nurses, given the right training, could progress up the career ladder into higher bands. There is also the sensitive issue to address that the orthopaedic speciality currently employs more than twice the number of the highest qualified nurses (TNEs) than the predicted needs form the model under all methods. The panel felt that this could be addressed at appropriate times by not replacing retirements/leavers or through career progression from TNE to more senior Trust roles.

Daily predicted nursing needs from the acuity method, for all 30 iterations, were than fed into the stochastic program. The suggested number of permanent staff is shown in the final column of Table 5. The optimal number is predicted as 141.0 WTE, which is 6.2% and 12.4% more staff than the acuity method and current staffing levels, respectively. These results indicate that in order to minimize costs, a hospital should employ more nurses than the average need would indicate. This is due to the stochastic nature of occupancy and corresponding nursing needs and that those occasions when the hospital has insufficient staff on duty (ie when demand is above the average) are costly as expensive temporary staff are required. These results help to demonstrate the importance of modelling in the presence of variability and reinforce that average values tend to under-estimate resource needs (Harper and Shahani, 2002).

The stochastic program was run for differing levels of efficiency of the temporary staff to see the effect that this had on the permanent members of staff. The relationship between the optimal number of permanent staff and the efficiency of the temporary staff is illustrated in Figure 1. The results suggest that in practice there will always be a need to rely on temporary nursing staff to meet peaks in demand, as the number of permanent staff needed when the efficiency of the temporary staff is set to zero (so effectively they do not exist) is significantly more than for other values of e.
Figure 1

Relationship between efficiency of temporary staff and optimal number of permanent staff to employ.

We were able to obtain figures from the hospital showing how much was spent in the year on agency nurses. At around £350 000 (or 14% of the total orthopaedic nursing costs), this equates to an average of three agency nurses per shift. The optimized model predicts the total annual cost of agency staff at less than £20 000 (or a need of just one nurse on average every five shifts). Figure 2 shows predicted orthopaedic nursing costs for the optimized model compared to current staffing levels and for those if the hospital used acuity predictions alone. Figure 2 helps to demonstrate that despite an increased cost of permanent staff for some grades, an optimal nursing team configuration makes significant savings in agency staff and hence lower overall costs. One might consider these cost savings to be a lower bound as in reality the hospital will, on occasions, not be able to obtain agency nurses for last minute requests. Hence there will be times throughout the year when despite having insufficient numbers or inappropriate skill-mix of staff on a shift, those present will simply have to make do with available resources. In turn, it is possible that this may put patients at risk as well as cause further inefficiencies such as inappropriate extensions in lengths of stay. Therefore the true cost of agency staff, if all requests were met and all shifts had satisfactory and appropriate staffing levels, would indeed be higher. In our analysis we assume that all requests made for agency staff under the optimized model will be met.
Figure 2

Nursing costs (by permanent and agency categories) for different scenarios.

7. Conclusions

The subject of hospital nurse staffing cutbacks and use of agency staff has attracted much attention over recent years both in the public media and in published academic reports. A number of reports have highlighted serious concerns with current nurse staffing and skill-mix within many UK hospital wards, and some have shown that patient outcomes are worse where patients are nursed with inappropriate and non-optimal staffing levels and skill-mix. The majority of published OR papers on this topic propose staff rostering algorithms. These tools however require as a pre-requite information on numbers and skill-mix of staff to roster. Hence, there is a need to more accurately predict the size and skill-mix of nursing teams based on patient need, which is addressed in this paper.

This paper focuses on the development and use of a hospital nursing module within an existing capacity tool, PROMPT. The approach utilizes both the discrete event simulation functionality of PROMPT combined with a stochastic program add-on to produce optimal nursing needs by staff grade. The optimization accounts for costs incurred for both permanent staff and the use of temporary (agency) staff as necessary due to fluctuations over time (driven by variability) of patient demands and resulting nursing needs. The flexible and generic structure of PROMPT allows the approach to be generalizable and adopted at different NHS Trusts.

Use of the tool has been demonstrated for an orthopaedic speciality comprising 120 beds. A comparison has been made between current staffing levels at the hospital and those predicted using two patient-to-nurse methods, namely the occupied bed and acuity methods. Results demonstrate that the predicted size and skill-mix for both methods are similar, although the more detailed acuity method is probably better able to detect subtle differences between grades given the bottom-up planning approach. Comparison against a benchmarked national data set provided for an additional validation test and increased confidence in the model outputs. The simulation phase of the work demonstrated to the hospital that although the overall size of nursing teams on different wards are of an acceptable level, the number of nurses employed at different grades is not well matched to patient needs and the skill-mix should be reconsidered.

In the second phase of modelling, the stochastic program accounted for fluctuations in nursing needs when suggesting an optimal number of permanent nurses to employ to meet the demand predictions from the simulation while trying to minimize costs. The stochastic program suggests that the optimal size of the orthopaedic nursing team needs to be 12.4% larger than the current size, but where the skill-mix is very different. Although overall more permanent staff is suggested at an increased annual cost to the hospital, significant savings are made on the spending of agency staff, hence the total staffing cost is minimized. These results help to demonstrate the importance of modelling in the presence of variability and that average values tend to under-estimate resource needs and fail to account for fluctuations in demand, which are important for staffing issues when both size and skill-mix are important and whereby temporary agency cover is expensive.

This methodology could be used to evaluate other hospital HR needs, such as for doctors and allied health professionals (eg physiotherapists). However at present it is much harder, at least in the UK, to get agreement on ratios of care for these staff. If patient-to-staff ratios become more readily available, or a national data set is possible, then PROMPT could well be useful. Finally, this work has focussed on costs when considering optimality of nursing teams. However as mentioned in this paper, there is emerging evidence linking patient outcomes to inappropriate size and skill-mix of nursing teams, and so another important research theme should be to examine such relationships and to further demonstrate the importance of considering patient needs when employing the right nursing team configuration and not simply to cut back on nursing staff when healthcare budgets become stretched.

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© Operational Research Society 2010