Staff planning for operating rooms with different surgical services lines

Abstract

We present a two-phase model for a staff planning problem in a surgical department. We consider the setting where staff, in particular nurse circulators and surgical scrub technicians, are assigned to one of different service lines, and while they can be ‘pooled’ and temporally assigned to other service line if needed, these re-assignments should belimited. In Phase I, we decide on the number of staff hours to budget for each service line, considering policies limiting staff pooling and overtime, and different demand scenarios. In Phase II, we determine how these budgeted staff hours should be allocated across potential work days and shifts, given estimated staff requirements and shift-related scheduling restrictions. We propose a heuristic to speed the model’s Phase II solution time. We implement the model using a hospital’s surgical data and compare the model’s results with the hospital’s current practices. Using a simulation model for the surgical operations, we find that our two-phase model reduces the delays caused by staff unavailability as well as staff pooling, without increasing the workforce size. Finally, we briefly describe a decision-support tool we developed with the objective of fine-tuning staff planning decisions.

Appendices

Appendix A: Phase II formulation when a staff member can be assigned to different shift lengths

In the Phase II original formulation (Section 3.2), we assume that a full-time staff member should be assigned to only one type of shift length l, each week. If we only assume that full-time staff should work an average number of hours per week, say H ^r, we can drop the index l in Z _s,l and introduce the following parameters and variables:

• Average number of regular time hours per week per staff member

• Number of full-time staff assigned to service line s

We replace constraints (17), (18), and (19) with constraints (35), (36), (37).

$$ Z_{s}\geq \frac{1}{H^{r}}\sum\limits_{j\in J,d \in D} H^{l}_{j}Y^{ft}_{s,j,d}\quad s\in S $$

(35)

$$ Z_{s}\geq \sum\limits_{j \in J} Y^{ft}_{s,j,d}\quad s\in S, d \in D $$

(36)

$$ \sum\limits_{j \in J,d \in D} H^{l}_{j}Y^{pt}_{s,j,d} + H^{r}Z_{s} \leq H^{std} B_{s} \quad \forall{s\in S} $$

(37)

Constraints (35) are necessary conditions for full-time staff not be scheduled more than H ^r hours per week on average, during regular time. This means no more than $\frac {H^{r}}{8}$ shifts per week, per full-time staff member, since full-time shifts are at least 8 hours long.

Appendix B: Other model parameters for the two-phase ORs staffing model

We define the remaining of the parameters for the two-phase ORs staffing model based on the hospital’s current processes and practices. These parameters include:

Phase I

• The standard number of hours per FTE per week is H ^std = 40, and the effective number of hours is H ^e = 37.5.

• Staff hours in excess of 40 hours per FTE per week are paid at an overtime rate. Overtime is paid at a 1.5 rate with respect of regular time, i.e., $C^{o}_{s}= 1.5C^{r}_{s}$.

• The costs to hire and to fire are $C^{h}_{s}=0$ and $C^{f}_{s}=0$. The initial workforce size is assumed to be zero $(X^{0}_{s}=0)$.

Since we do not have a value for the hospital’s maximum fractions of overtime and pooling, we define these parameters so that the resulting permanent FTEs (X _s ) would be similar to those implemented by the hospital for the same planning horizon, with the goal of making results more comparable. The baseline for the number of permanent FTEs for both circulators and scrub techs is obtained from the hospital’s staffing in December 2012. The staffing budget and structure do not change much throughout a year, so we believe that this is a reasonable approximation of the original staff planning decisions for 2012. Even if this was not the case, Phase I makes decisions by considering about the same or less information (i.e., historical data available prior to July 2011) compared to that available to the hospital’s OR manager when the initial staffing decisions are made. In December 2012, the number of FTEs for scrub techs and circulators were 23.8 and 20.7, respectively. We test different maximum pooling and overtime settings to match these numbers of FTEs. Using CPLEX, we run Phase I for circulators with 30 demand scenarios for a 48-weeks planning horizon in 2012 assuming that no pooling or overtime is allowed. We arrive to approximately the same number of permanent FTEs as the hospital, so we keep the zero pooling and no overtime assumption. Similarly, we run Phase I for scrub techs and look for pooling and overtime settings that would give similar results to those implemented by the hospital. In Fig. 12 we see that 23.8 FTEs (the hospital’s budget) result from a maximum average overtime $(\bar {\alpha }^{o}_{s} \cdot 100~\%)$ of 0.7 % and a maximum average pooling $(\bar {\alpha }^{p}_{s} \cdot 100~\%)$ of 0.0 %, or 0.2 % and 10 % respectively. We select the latter setting, when overtime is lower.

Fig. 12

Number of FTEs obtained from Phase I, under different settings for the maximum average percentage of pooling and overtime

• Shifts can start every 30 minutes from 6:00 AM to 2:30 PM, every day of the week. There are five shift lengths: {5, 8, 9, 10, 12}.

• There is a pre-fixed night shift from 7:00 PM to 6:30 AM with a minimum staffing requirement of one circulator and two scrub techs. Including this night shift, there are 91 potential shifts (N _J = 91).

• The less-than-full-time hours fraction $(\alpha ^{lf}_{s})$ is limited to 0.04 of effective permanent FTEs hours for scrub techs and zero for circulators.

• The part-time hours fraction $(\alpha ^{pt}_{s})$ is limited to 0.22 of effective permanent FTEs hours for circulators and scrub techs.

• We use the staffing levels from week 28 in 2010 to week 27 in 2011 as an estimate for the staffing levels during the first 48 weeks of 2012 (the planning horizon).

• The penalty for unmet staffing level by service line is constant for all time buckets and all service lines (Π _s,t = 1).

• An additional 0.5 penalty $({\Pi }^{\prime }_{t}=0.5)$ is incurred for overall unmet staffing levels.

Appendix C: OR simulation

We build the simulation model using Arena. Figure 13 shows a snapshot of the Arena model, with the identification of the main parts:

• Read staff schedule: This module reads a file with the number of available staff at any time for each service line, based on the staffing structure under consideration.

• Read cases information: This module reads a file with the surgical cases information and times.

• Patient arrival and paperwork: This module generates patient arrivals according to the arrival time distributions. After a patient’s arrival, paperwork is completed at reception. The patient is prepared for surgery when the estimated procedure start time is closer (<2 hours).

• Blood drawn and assessment: Blood may be drawn, and the patient is assessed by a nurse.

• Waiting for scheduled time and OR: The patient waits for the scheduled procedure start time if it is the first case for the OR, the anesthesiologist, or the surgeon; otherwise, the patient waits for the OR to be ready.

• Seize anesthesia and OR staff: The case patient seizes the required anesthesia and OR staff. OR staff assigned to the case’s service line is chosen first if available, if not, OR staff form other service lines can be pooled and used.

• Procedure: The patient is taken to the OR and the surgical procedure starts. The patient is taken out of the room after the procedure is completed.

• Room turnover: The OR is cleaned and prepared for the next case by the patient care assistants and the OR staff.

• Write output: This module writes the surgical case time stamps in a file for analysis.

Fig. 13

Arena simulation snapshot

To obtain the time distribution for the patient’s arrival relative to the case scheduled start time, the percentage of cases with blood being drawn, and the patient assessment and preparation time distributions, we use a 16-days time study done in November 2010 that includes the time stamps of the more than 280 OR patients from their arrival to the end of their preparation for the surgical procedure. We link these time stamps with data provided by the surgical information system, including the type of patient and procedure, the scheduled start and end times for the case, and the patient’s wheels-in and -out times. The cases assigned OR, sequence, and actual duration are taken directly from the surgical data. All the time distribution fittings are done using Expert Fit. If there is not a distribution with a good fit, an empirical distribution based on the available data is used. We use empirical time distributions for outpatients and inpatients paperwork, blood work and nurse assessment, and for outpatients arrival time; and we fit a Normal distribution (AD p-value > 0.25) for inpatients arrival time. We use the surgical data from 2009 and 2010 to calculate the turnover time distributions by case type. We assume that the turnover for a case starts at the wheels-out of the previous patient in the room and ends with the wheels-in of the new patient. Only the turnover for cases with some delay are considered, so that the case waiting time for its scheduled start time is not included in the calculation. Table 12 shows the best fitted distributions for each case type. AD p-values > 0.25 and KS p-values > 0.15, for all case types (except for Colon-Rectal, with a KS p-value > 0.10).

Table 12 Distribution results for OR turnover time

1.1 C.1 OR Simulation Validation

To validate the simulation, we analyze the simulation results given the hospital’s CP staffing structure and compare them with the actual surgical data for the planning horizon, i.e, the first 48 weeks of 2012. In particular, we look at the percentage of delayed cases and an OR last wheels-out time of the day, since these statistics reflect the surgical schedule characteristics and the resource availability dynamics.

According to surgical data, the percentage of delayed cases (those cases for which the wheels-in occurs 10 minutes or later after the scheduled time) is 42.4 %, whereas according to the simulation this percentage is 41.6 %, a non-statistically significant difference of 0.8 % (Fisher’s exact test p-value > 0.31). The average OR last wheels-out time in a day (for those days with more than 5 ORs open) is 3:37PM according to surgical data and 3:30PM according to the simulation, a non-statistically significant difference of 7 minutes (two-sample t-test with p-vale > 0.65). The average OR last wheels-out time distributions for the actual surgical data and the simulation results are shown in Fig. 14. We perform a KS 2-sample test under the null hypothesis that these distributions are not different. According to this test there is not sufficient evidence to reject this hypothesis at a 95 % confidence.

Fig. 14

Simulated and actual average OR last wheels-out empirical distribution functions

Appendix D: A decision-support tool

We developed a decision-support tool with the objective of helping the OR manager to make adjustments to the staff budget and the staffing structure by: reassigning staff to another service line, changing the number of people assigned to a shift, adding a new shift, etc. The tool was implemented using Microsoft Excel, and it was automated for ease of use.

Tool Input

• Historical surgical data (from automatic reports from the surgical information system).

• Current staffing structure.

• Classification of surgical services to one of the three service lines.

• OR staff requirements per case type.

• User can select: time baseline, staff type, one or a group of ORs.

Tool Output

• Budget by service line and staff type.

• Weekly volume trends by service line, compared with the budget. See Fig. 15.

• Current staffing levels vs. observed demand patterns, overall and by service line. See Fig. 16.

Fig. 15

Weekly volume trends by service line in OR staff hours

Fig. 16

Staffing levels of circulators compared with the demand patterns for all service lines

Although this tool does not make automated staff planning decisions (it is not integrated with Phase I and Phase II yet), it gives useful statistics and graphs based on easily available surgical data, which are helpful to analyze and fine-tune previous decisions. For example, if the OR manager observes that the demand of a particular service line exhibits a decreasing trend, and the demand of another service line increases, he/she can reassign staff. Similarly, if the aggregated demand trend is increasing, the manager can request an increase in the total FTEs budget. Also, if the demand patterns suggest that a service line may be overstaffed on one day and understaffed on another, the OR manager can restructure shifts.