Optimizing Nurse Practitioner Staffing to Cover Nursing Home Patients
Angela Chupa
Spring 2014
Background
For the past few years, University of Wisconsin (UW) Health has been working on an initiative
to reduce patient readmissions to the hospital. Through extensive data analysis, it was assessed
that the elderly population has a higher chance of readmission, not surprising of course, since
they are typically the sickest people in the patient population and usually have multiple
complications. In addition, not all patients are discharged to home. In fact, many of the elderly
patients are discharged to nursing homes, known as skilled nursing facilities (SNF).
Once a patient is discharged from the hospital, it is urgent that they receive follow-up care as
soon as possible. Data shows that of all the elderly patient readmissions, the majority occur
within the first week of discharge. In attempt to reduce readmissions for this population, UW
Health has invested in a unique program to hire nurse practitioners (NP) to visit patients in the
SNFs and provide the appropriate care needed.
Under the transitional care program, there are nine SNFs in the Dane County area where the
NPs would go to care for patients. Each SNF is a privately-run facility and has a unique number
of beds. Appendix A shows the location of the various SNFs in addition to a matrix showing the
driving distances to and from each location.
Problem
Since it is critical to visit all patients that are discharged to the nine participating skilled nursing
facilities, UW Health must ensure that the optimal number of nurse practitioners is hired. More
simply, the number of nurse practitioners hired must be able to cover at most the total number
of beds in the Dane County area, which is 815 (see Appendix B). Since the median salary for a
nurse practitioner is $92,6701, hiring an unnecessary worker is a cost burden. Conversely, not
hiring enough nurse practitioners could result in patient readmissions to the hospital.
Under this program, a nurse practitioner will be “stationed” at a specific SNF. Therefore, this
optimization problem aims to determine the minimum number of nurse practitioners needed
to maximize the number of patients that are within a three mile distance of a SNF that has a
stationed nurse practitioner. Three miles is chosen as the distance because it is critical that
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nurse practitioners do not spend too much of their time driving to multiple facilities, as this is a
non-value added activity and takes away from the direct patient care that could be occurring.
In order to model this problem, several assumptions are made:
1. Each nurse practitioner works at the same pace and can visit the same number of
patients
2. Exact travel times are not considered, only distances.
Optimization Model
The following basic shell is used to solve this problem:
Sets:
SNF = 1,2,…,9 → each SNF is assigned an ID (see Appendix B)
Parameters:
Patients(SNF) → number of beds used (see Appendix B)
Distance(SNF i, SNF j) → driving distance matrix used (see Appendix A)
Variables:
Z [free variable] = number of patients reached
X(i) [binary variable] = 1 if NP is stationed at SNF i, otherwise 0
Objective:
maximize 𝑍 = ∑(𝑖,𝑗) 𝑝𝑎𝑡𝑖𝑒𝑛𝑡𝑠(𝑗) ∗ 𝑥(𝑖)
Subject to:
∑𝑖 𝑥(𝑖) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑁𝑃𝑠
∑𝑖 𝑥(𝑖) ≤ 1
Results
Nine models, one for each different number of NPs, were run using mixed integer
programming. The various objective values (Z) for each model are as follows:
Table 1: Objective Values
Number of NPs
1
2
3
4
5
6
7
8
9
Number of Patients Serviced
418
523
618
705
773
815
815
815
815
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It is evident that based on the z.l values, the maximum number of patients is reached when at
least six nurse practitioners are hired. Considering the costs associated with hiring at least six
nurses, the optimal value that maximizes the number of patients serviced while minimizing cost
is six nurse practitioners. When the program has six nurse practitioners, they will be stationed
at the following SNFs:






Capitol Lakes
Karmenta
Oakwood West
Oakwood East
Sunny Hill
St. Mary’s
In addition, the total number of beds at these six locations is 815, approximately 71.2% of the
total beds to be serviced. Furthermore, seeing that it would take six nurse practitioners to
service 815 patients, it is critical to assess the feasibility of this:
815 𝑝𝑎𝑡𝑖𝑒𝑛𝑡𝑠
𝑤𝑒𝑒𝑘
1 𝑤𝑒𝑒𝑘
𝑝𝑎𝑡𝑖𝑒𝑛𝑡𝑠
𝑝𝑎𝑡𝑖𝑒𝑛𝑡𝑠
∗ 6 𝑛𝑢𝑟𝑠𝑒 𝑝𝑟𝑎𝑐𝑡𝑖𝑡𝑖𝑜𝑛𝑒𝑟𝑠 = 135.8𝑛𝑢𝑟𝑠𝑒 𝑝𝑟𝑎𝑐𝑡𝑖𝑡𝑖𝑜𝑛𝑒𝑟∗𝑤𝑒𝑒𝑘 = 2.42𝑛𝑢𝑟𝑠𝑒 𝑝𝑟𝑎𝑐𝑡𝑖𝑡𝑖𝑜𝑛𝑒𝑟∗ℎ𝑜𝑢𝑟
An hourly rate of 2.42 patients is reasonable, considering these patients are still receiving fulltime care at the SNFs; the additional care received by nurse practitioners will enable the
patient’s medications, physical therapy orders, and lab orders to be changed if needed.
Future Extensions
Variations of this problem can be run to achieve results using additional information. For
example, various patient arrivals throughout the week could be modeled to determine exactly
how many patients are needed on each day. This would allow for part-time FTE, thus reducing
cost. However, this would take substantive past data to find trends on discharges from the
hospital. The model can also be run with a greater distance, that is, the distance between
patients and a SNF with a stationed NP. A trial was completed with a distance of 5 miles
(instead of 3). The results indicate that only three nurse practitioners are needed, but this level
is not feasible given working hours each nurse practitioner and the patient demand. Overall,
there is a great need for operations research in healthcare, as it can help to improve patient
outcomes and reduce costs simultaneously.
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Appendix A
Figure 1: Map of Skilled Nursing Facilities in Dane County
Table 2: Driving Distances between Skilled Nursing Facilities
Distances (mi)
Capitol Lakes
Karmenta
Oakwood West
Oakwood East
Sunny Hill
Attic Angels
St. Mary's
Oak Park
Belmont
Capitol
Lakes
5.2
6.4
8.8
5.8
10.9
8.9
7
5
Karmenta
5.2
15.8
5.5
11.9
17.2
15.2
2.6
0.5
Oakwood Oakwood Sunny Attic
St.
Oak
West
East
Hill
Angels Mary's Park Belmont
6.4
8.8
5.6
10.9
8.9
7
5
15.8
5.5
11.9
17.2
15.2
2.6
0.5
20.9
3.4
3.4
5.4 15.5
14.6
20.9
17.4
22.7
20.7
7
5.5
3.4
17.4
6
3.9
13
12.1
3.4
22.7
6
6.3 17.9
17
5.4
20.7
3.9
6.3
15.9
15
15.5
7
13
17.9
15.9
2.9
14.6
5.5
12.1
17
15
2.9
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Appendix B
Table 3: Number of Beds at each SNF2
ID
1
2
3
4
5
6
7
8
9
SNF
Capitol Lakes
Karmenta
Oakwood West
Oakwood East
Sunny Hill
Attic Angels
St. Mary's
Oak Park
Belmont
Total
2
Number of Beds
87
105
95
42
68
44
184
100
90
815
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