Optimized Staffing with variable demand

Management Science for Healthcare

INTRODUCTION INTO STAFFING MODELING WITH RANDOM DEMAND:

EXAMPLES AND PRINCIPLES

Alexander Kolker, PhD Hartford, WI

Alexander Kolker. All rights reserved

Outline

• Main Concept and Some Definitions. • The “newsvendor” framework approach. Staffing a nursing unit with variable census (demand) • Linear optimization framework approach. Minimizing staffing cost subject to variable constraints • Discrete event simulation framework approach. Staffing a unit with cross-trained staff • Key Points and Conclusions


• Using Data Analytics to Work Smarter • Workforce management involves data-driven decision-making. From

productivity measurements to patient safety outcomes to staff satisfaction metrics, healthcare executives must be focused on tracking and managing complex variables.

• Hospitals that use business analytics to make data-driven staffing decisions can optimize their most valuable resource, labor.

• Organizations that invest in a robust talent optimization solution, and then encourage system usage and compliance will be able to utilize real-time data to make better decisions regarding their valuable workforce.

• With real-time labor and staffing information available, healthcare providers can focus on managing productivity. That requires understanding the relationship between three key variables: (1) patient needs and acuity, (2) actual staffing, and (3) budgeted staffing Alexander Kolker

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• “Section 5.1 of the Baldrige Criteria for Performance Excellence asks "How do you assess your workforce capacity… including staffing levels?"

• The best answers to that question have all described approaches that were static and based on average arrivals, average demand for service and average length of stays.

• However, given the dynamic nature of healthcare systems, failure to understand patterns, anticipate variation and prepare for the uncertainty creates two types of problems: • one, excess staffing, which hurts margins; • and two, being understaffed, which requires overtime and/or

premium pay that also hurts margins and causes less than optimum quality of care.

• The latter problem adversely affects patients and staff satisfaction.”

From an ASQ Baldrige Application Examiner, 2012. Alexander

Kolker All rights reserved

Some definitions

• Staffing- determining the appropriate number of FTEs to be hired and retained in each skill set (RN, LPN, aides, MHA, MBA, etc..) in the most cost efficient way to provide high level of clinical outcomes (quality)

• Scheduling- allocation of care providers assigned on and off duty

by weeks, days and shifts; operational procedures • Reallocation-fine tunes of the previous decisions; daily and/or

shift by shift

Alexander Kolker

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45

Mid

nig

ht

Ce

nsu

s

Typical NICU Daily Census for the period 7/31/2010 - 9/30/2011 (Children’s

Hospital of Wisconsin)

Staff at this level ?

or Staff at these levels ?

The main root cause of staffing issues is VARIABILITY

Alexander Kolker

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.

Staff at this level ?

Typical PACU daily average census (on an annual basis)

or Staff on these levels ?

Alexander Kolker

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Key Points: • Nursing managers typically adjust nursing staffing needs

manually based on the past historical average number of patients (census)

Because of high variability of the patient census, the resulting staffing usually • (i) either is not enough to deliver proper quality of care

and it is not cost-effective due to excessive overtime, or call from the extra staff pool at a premium rate

• (ii) or excessive and results in idle time and/or pay under contractual obligation.

Alexander Kolker

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Optimized annual budgeted staffing level: “Newsvendor” framework (example 1)

Input Information: (i) census data (collected several times a day, usually at midnight, at

noon, or in the afternoon) (ii) patient: nurse ratio (PNR) broken down by patient acuity: Example: acuity 1: 28% of patients. Required PNR=1:1 acuity 2: 67% of patients. Required PNR=2:1 acuity 3: 5% of patients. Required PNR=3:1 The average PNR= 1.77

Problem Statement: Given the variable patient census, determine for the given long-time period the optimal (budgeted) staffing level that minimizes the cost of daily/shift fluctuations of too many nurses (calls-off) and not enough nurses (calls-on)

Alexander Kolker

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Solution: This type of problem – choosing an optimal minimal cost staffing level that faces random demand- is best addressed using a “newsvendor” type framework:

If the random demand follows the cumulative probability distribution function in a single time period, F(s), then the optimal staffing level, s*, that balances the cost of “too many” (overage cost, Co) and the cost of “too little” (underage cost, Cu) is calculated as:

F(s*)= Cu/(Cu+Co) (for derivation see Appendix 1)

Note: A similar equation that is used in retail, supply chain management, finance, etc. is: F(s*)= (p-w)/(p-v), where p is the retail price, w- is the wholesale price, and v is the salvage price (if available).

Alexander Kolker

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Calculation of the understaffing cost , Cu, (in excess to the regular nursing pay rate, R, $/hr) If too few nurses are scheduled, then an additional nurse can be called from: (i) the internal float pool at no extra cost (if a trained nurse is

available-30% of time in this example), or (ii) an off-duty nurse pool / staffing agency at a premium 60% above

normal pay (in this example). Thus, understaffing cost per shift per nurse, Cu=(1-0.3)*0.6*R = 0.7 *0.6*R. Note: This cost can be somewhat underestimated because float nurses are usually less efficient than the crew staff nurses.

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Calculation of the overstaffing cost , Co, (in excess to the regular nursing pay rate, R, $/hr) If too many nurses are scheduled, then the extra nurses can be: (i) Floated out to another unit if the need arise (about 40% of time in

this example), or (ii) Offered to take paid/unpaid vacation day, or (iii) Put on-call for the contractual pay of 25% of the base rate, R Thus, overstaffing cost per shift per nurse, Co=(1-0.4)*0.25*R . Note: Being sent home after showing up for work is not popular and impacts nurse satisfaction; therefore the true cost of overstaffing can be somewhat underestimated.

Thus, the right-hand side of the optimal staffing equation is: Cu/(Cu+Co)=0.7*0.6*R/(0.7*0.6*R + 0.6*0.25*R)=0.74

Alexander Kolker

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Frequency Cumulative %

1 0.22%

0 0.22%

3 0.88%

4 1.75%

7 3.28%

11 5.69%

13 8.53%

19 12.69%

23 17.72%

20 22.10%

36 29.98%

29 36.32%

35 43.98%

36 51.86%

40 60.61%

40 69.37%

48 79.87%

41 88.84%

31 95.62%

12 98.25%

2 98.69%

6 100.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

55.00%

60.00%

65.00%

70.00%

75.00%

80.00%

85.00%

90.00%

95.00%

100.00%

0

10

20

30

40

50

60

Fre

qu

en

cy

staffing, FTE

Staffing Distribution

Calculation staffing cumulative distribution function

Optimal core staffing: 24 FTE (Cu>Co)

Average staffing: 22 FTE (Cu=Co)

Optimal core staffing: 19 FTE (Cu<Co)

Alexander Kolker

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NOTE: If, for example, • the internal float nurse is available 80% of time (instead of

30%), and • the regular rate must be paid for nurses that were

scheduled and showed up for work (instead of 25% of the regular pay for those sent home), then the ratio

Cu/(Cu+Co)= 0.2*0.6/(0.2*0.6+0.6*1)=0.166, and the optimal core staffing level will be 19 FTE vs. the average level of 22 FTE

Alexander Kolker All rights reserved


15

St. John Hospital, MBU unit: Optimized budgeted staffing on a monthly basis (Example 2)


16

Annual budgeted staffing level

Key Points • Choosing the optimal long-term core staffing level that

face random demand is best addressed using a “newsvendor” type framework.

• Depending on the ratio of the overage and underage costs, the optimal core staffing level can be higher or lower than the average.

• Only if the overage and underage costs are same, then the optimal staffing will be close to the average.

• The optimal staffing provides a trade-off between “ too few nurses” and “too many nurses” (this improves the quality of care and staff utilization, thus reducing the overall cost of doing business).

Alexander Kolker

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Linear Optimization framework

Optimization and Scheduling of a Clinical Unit Staff for 24/7

Three-Shift Operations: Is Staffing Cost Minimized?

•A typical full time unit nurse (such as an ICU nurse) usually works five days a week with two consecutive rotating days off and rotating shifts. •Usually three 8-hours shifts per day should be covered. • A typical clinical unit has some minimal staffing requirement based on the average shift patient census and the assumed nurse : patient ratio. This ratio is based on assessed patient acuity level or external regulations. •Suppose that the pay rate is $50/hour (base wages and overhead) with 50% pay rate increase for Saturday and Sunday shifts

Alexander Kolker

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Problem statement: Develop a staffing schedule to meet the minimum coverage for each day and shift with five work days and two consecutive days off for each staff member in such a way that the total weekly staffing cost is minimized .

Alexander Kolker

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Solution. Step 1. Identify decision variables For a seven-days week and three shifts per day there are total 21 different schedules possible These schedules are presented in Table along with the average shift census and the minimal staff demands for each shift assuming nurse to patient ratio 1:2 for all shifts (only Monday to Thursday are shown in this Table. Friday, Saturday and Sunday are structured similarly but not shown here to save space). Decision variables for this problem are the number of nurses, Xs (s=1,.., 21) assigned to each of s=21 schedules

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Binary index variable for s=21 schedules and days and shifts: 1- on shift, 0-off shift. Nurse to patient ratio is assumed to be 1:2 for all shifts

Schedule

sDays Off

Mon_

shift_1

Mon_

shift_2

Mon_

shift_3

Tue_

shift_1

Tue_

shift_2

Tue_

shift_3

Wed_

shift_1

Wed_

shift_2

Wed_

shift_3

Thu_

shift_1

Thu_

shift_2

Thu_

shift_3

1 Sat, Sun 1 0 0 1 0 0 1 0 0 1 0 0

2 Sat, Sun 0 1 0 0 1 0 0 1 0 0 1 0

3 Sat, Sun 0 0 1 0 0 1 0 0 1 0 0 1

4 Sun, Mon 0 0 0 1 0 0 1 0 0 1 0 0

5 Sun, Mon 0 0 0 0 1 0 0 1 0 0 1 0

6 Sun, Mon 0 0 0 0 0 1 0 0 1 0 0 1

7 Mon, Tue 0 0 0 0 0 0 1 0 0 1 0 0

8 Mon, Tue 0 0 0 0 0 0 0 1 0 0 1 0

9 Mon, Tue 0 0 0 0 0 0 0 0 1 0 0 1

10 Tue, Wed 1 0 0 0 0 0 0 0 0 1 0 0

11 Tue, Wed 0 1 0 0 0 0 0 0 0 0 1 0

12 Tue, Wed 0 0 1 0 0 0 0 0 0 0 0 1

13 Wed, Thu 1 0 0 1 0 0 0 0 0 0 0 0

14 Wed, Thu 0 1 0 0 1 0 0 0 0 0 0 0

15 Wed, Thu 0 0 1 0 0 1 0 0 0 0 0 0

16 Thu, Fri 1 0 0 1 0 0 1 0 0 0 0 0

17 Thu, Fri 0 1 0 0 1 0 0 1 0 0 0 0

18 Thu, Fri 0 0 1 0 0 1 0 0 1 0 0 0

19 Fri, Sat 1 0 0 1 0 0 1 0 0 1 0 0

20 Fri, Sat 0 1 0 0 1 0 0 1 0 0 1 0

21 Fri, Sat 0 0 1 0 0 1 0 0 1 0 0 1

Average Census

per Shift10 24 20 21 18 15 8 15 24 18 20 21

Minimal Staff

Demand per Day

and Shift,

N ds, MinDemand

5 12 10 10 9 7 4 7 12 9 10 10Alexander


Step 2. Identify the Objective function Objective function is the total weekly staffing cost for all

shifts that should be minimized by placing the right staff in the right shift:

Cost= Mon_staffing * pay rate ($/hour) * shift length (hour) +

Tue_staffing * pay rate ($/hour) * shift length (hour) + etc.

(for each day of the week) -> MIN

Alexander Kolker

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Step 3. Identify Constraints •Constraints for decision variables Xs (s=1,.., 21) are

the minimal total staff demand for each day and shift,

Nds, Min Demand

•These values are calculated using the average census

per shift (indicated in Table) and the nurse to patient

ratio (1:2 in this case).

•Nds, Min Demand values are indicated in the last row of

Table

•Thus, the optimal solution Xs (s=1,.., 21) must satisfy these conditions Alexander


Step 4. Identify parameters Model parameters are: •The nurse to patient ratio (1:2) •The average census per shift • Pay rates ($50/hr weekdays, $75/hr weekend) • Shift length (8 hrs)

Alexander Kolker

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Alexander Kolker

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Solution Schedule ID Days off

1 Sat, Sun

2 Sat, Sun 3 Sat, Sun

4 Sun, Mon

5 Sun, Mon

6 Sun, Mon

7 Mon, Tue

8 Mon, Tue

9 Mon, Tue

10 Tue, Wed

11 Tue, Wed

12 Tue, Wed

13 Wed, Thu

14 Wed, Thu

15 Wed, Thu 16 Thu, Fri 17 Thu, Fri 18 Thu, Fri

19 Fri, Sat 20 Fri, Sat 21 Fri, Sat

Mon_ shift_1

Mon_ shift_2

Mon_ shift_3

Tue_ shift_1

Tue_ shift_2

Tue_ shift_3

Wed_ shift_1

Wed_ shift_2

Wed_ shift_3

2 2 2 7 7 7 8 8 8 5 5 1 1 1 1 1 1 4 1 1 1 1 1 2 2 2 2 2 2 5 12 10 10 9 11 10 9 12 <-Sub-total

Objective function: Goal to minimize total cost per week $ 77,800


26

Key Points

Linear Optimization framework for staffing is preferred if: • the primary goal is the minimal staffing pool needed for

continuous time (shift) coverage, such as the number of nurses, residents (or attending physicians) that should always be available for the specific time length or shift due to safety or legal regulations regardless of their actual workload.

Alexander Kolker

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Discrete Event Simulation framework Staffing of the Unit with Cross-Trained Staff

Problem description: • A hospital case management department performs three types of

transactions: reservation, urgent admissions and pre-registration. • About 14% of all transactions are reservation, 32% is urgent admissions and 52% is pre-registration. • Case management specialists for urgent admissions and pre-

registration are cross-trained and could substitute each other if needed. Reservation specialists work independently and they are not involved in performing other type of transactions.

• The department works Monday to Friday from 8 am to 4:30 pm, with

a 30 min lunch time and two 15 min breaks during a typical day. Alexander Kolker

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Actual 2009

Projected 2010

Projected 2011

Projected 2012

Transaction volume

49,559 50,763 52,286 53,855

Transaction volumes

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Analysis and Estimation FTEs

• Transaction time data have been collected for: • Reservation IP • Urgent_Admission IP • Pre_Registration IP

• Urgent_Adm and Pre_Reg are mutually interchangeable

• Case related activity time, as well as non-case related activity time have been collected.

•Percentage of case related activity time has been estimated as the availability metrics

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Transaction type Time, min

Mean Median

Reservation 4.9 3.0

Urgent Admission 8.7 6.0

Pre-Registration 4.5 3.0

Transaction time data were collected over some representative

period and summarized in Table

Summary of Mean and Median Time per Transaction

Problem Statement: Develop minimal staffing (FTE) requirement for each transaction type that allows performing the annual transaction volume without overtime. Alexander


Traditional FTE estimation based on the average transaction time

• Reservation: FTE(2009)=49559*0.14*4.9 min/(255*8*60)/0.87 = 0.32

• Urg_Adm: FTE(2009)=49559*0.32*8.7 min/(255*8*60)/0.85= 1.33 • Pre_Reg: FTE(2009)=49559*0.54*4.5 min/(255*8*60)/0.78= 1.26

Total average-based calculated(2009): 2.9 Actual (2009): 3.5

NOTE: Cross-trained staff cannot be taken into account by the simple formulas based on average transaction time 32

Actual Time study: Reservation time distribution

Mean=4.9 min Median=3 min

33

Actual Time study: Urgent_Adm time distribution


34

Actual Time study: Pre_Reg_IP


35

Simulation Process Model layout

Interchangeable (cross-trained) staff

36

FTEs Required to Meet Annual IP_Verification Patient Volume Target

2009- Baseline: Model Validation

Conclusion0.6 FTE

Reservation

(Time study

availability

87%)

1 FTE

Urg_Adm

(Time study

availability

85%)

2 FTE

Pre_Reg

(Time study

availability

68%)

Total 3.6

FTEsAnnual

Transaction

volume range

(Target 49,559)

99% CI:

6892-6974

99% CI:

15763-15902

99% CI:

26716-26856

99% CI:

49543-

49559

2010

Conclusion

0.5 FTE

Reservation

1 FTE

Urg_Adm

2.5 FTE

Pre_Reg

Total 4

FTEsAnnual

Transaction

volume range

(Target 50,763)

99% CI:

7100-7200

99% CI:

16138-16255

99% CI:

27332-27497

99% CI:

50760-

50763

2011

Conclusion0.5 FTE

Reservation

1.5 FTE

Urg_Adm

2.5 FTE

Pre_Reg

Total 4.5

FTEsAnnual

Transaction

volume range

(Target 52,286)

99% CI:

7266-7361

99% CI:

16623-16768

99% CI:

28211-28341

99% CI:

52284-

52286

2012

Conclusion0.6 FTE

Reservation

1.5 FTE

Urg_Adm

2.6 FTE

Pre_Reg

Total 4.7

FTEsAnnual

Transaction

volume range

(Target 53,855)

99% CI:

7516-7636

99% CI:

17121-17301

99% CI:

28995-29139

99% CI:

53854-

53856

4 FTEs

required;

4.5 FTEs

required;

4.7 FTEs

required;

3.6 FTEs

required;

current level

3.5 FTEs.

The Model is

Valid

37

Key Points • The benefits of cross-training exhibits the law of

diminishing return: just a little bit of cross-training (staff flexibility) goes a long way.

• However, there are non-technical (staff comfort zone) issues with cross-training.

3.5

3.7

3.9

4.1

4.3

4.5

4.7

4.9

49000 50000 51000 52000 53000 54000

FTE

Annual transaction volume

FTE with cross-training

Alexander Kolker

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Overall Key Points of the Staffing Modeling and Simulation

• Proper staffing and scheduling can mean the difference between profitability and business failure • There are three main methodology frameworks for modeling staffing with variable demand:

‘newsvendor’ framework- best for determining the optimal staffing level in the specified time period

linear optimization (including integer, mix-integer and stochastic)-best for determining the optimal staffing if the objective function and constraints can be presented as linear functions of the decision variables

discrete event simulation- best for highly stochastic systems with

multiple staff types, shared and cross-trained staff Alexander Kolker

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APPENDIX 1


41

The Newsvendor equation derivation If the distribution density function of demand, D, is f(D), then the average loss due to overage, i.e. when staffing S > D (loss from the overstaffing) is:

𝐶0 𝑆 − 𝐷 𝑓 𝐷 𝑑𝐷𝑆

0 , where Co is the unit cost of overage;

The average loss due to underage, i.e. when staffing S < D (loss from understaffing-unmet demand) is:

𝐶𝑢 𝐷 − 𝑆 𝑓 𝐷 𝑑𝐷𝐷𝑚𝑎𝑥

𝑆, where Cu is the unit cost of underage;

Total loss function is:

L(S)=𝐶0 𝑆 − 𝐷 𝑓 𝐷 𝑑𝐷𝑆

0+𝐶𝑢 𝐷 − 𝑆 𝑓 𝐷 𝑑𝐷

𝐷𝑚𝑎𝑥

𝑆

and it should be minimized with respect to staffing, S, by taking the derivative and making it equal to 0.

42

𝑑𝐿

𝑑𝑆= 𝐶0 𝑓 𝐷 𝑑𝐷 − 𝐶𝑢 𝑓 𝐷 𝑑𝐷

𝐷𝑚𝑎𝑥

𝑆

𝑆

0=0;

Here, the derivative of L(S) with respect to S is taken using the general rule:

If 𝑓 𝑝 = 𝑓 𝑥, 𝑝 𝑑𝑥, 𝑡ℎ𝑒𝑛𝑏(𝑝)

𝑎(𝑝)

𝑑𝑓

𝑑𝑝=

𝜕𝑓(𝑥,𝑝)

𝜕𝑝

𝑏

𝑎dx+𝑓(𝑏, 𝑝)

𝑑𝑏

𝑑𝑝 - f(a,p)

𝑑𝑎

𝑑𝑝

The 2-nd derivative is 𝑑2𝐿

𝑑𝑆2= 𝐶0f(S)+𝐶𝑢 f(S)>0, hence this is minimum.

Thus,

𝐶0F(S)-𝐶𝑢(1-F(S))=0; and F(S opt)=Cu/(Cu+Co)

APPENDIX 2


Key Points

Management Engineering/Science is indispensable in addressing the following typical hospital issues:

- Capacity: How many beds are required for a department or unit? How many procedure rooms, operating rooms or pieces of equipment are needed for different services?

- Staffing: How many nurses, physicians and other providers are needed for a particular shift in a unit (department) in order to best achieve operational and service performance objectives?

- Scheduling: What are the optimized staff schedules that help not only delivering a safe and efficient care for patients but also take into account staff preferences and convenience?

- Patient flow: What patient wait time at the service stations is acceptable (if any at all) in order to achieve the system throughput goals? Alexander


Key Points (cont.)

- Resource allocation: Is it more efficient to use specialized resources or pooled (interchangeable) resources (operating/procedure rooms, beds, equipment, and staff)?

- Forecasting: How to forecast the future patient volumes (demand) or transaction volumes for the short- and long-term budget and other planning purposes?

And the ultimate goal (and the holy grail of Management Engineering):

Given the variable patient volume and patient mix, design and manage hospital operations efficiently, i.e. increase profitability (reduce operating expenses, increase revenue) while keeping high quality, safety and outcomes standards for patients.

Alexander Kolker

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One of the Most Powerful Methodologies of Operations Management is Simulation Modeling

A Simulation Model is the computer model that mimics the behavior of a

real complex system as it evolves over the time in order to visualize and

quantitatively analyze its performance in terms of:

• Cycle times

• Wait times

• Value added time

• Throughput capacity

• Resources utilization

• Activities utilization

• Any other custom collected process information

•The Simulation Model is a tool to perform ‘what-if’ analysis and play

different scenarios of the model behavior as conditions and process

parameters change

•This allows one to build various experiments with the computer model

and test the effectiveness of various solutions (changes) before

implementing the change Alexander Kolker

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Main Steps for Application of ME methodology

• Describe your process • Depict the logical flow and the boundaries of the process in

the graphical form by developing a flow map

• Define Performance Objectives

• Examples: target utilization; target wait time; the number of

served patients or transactions; target net revenue; etc.

• Develop a simulation model layout and action logic

using model building software package

Alexander Kolker

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Cont.

• Collect the data required to feed the model: •Examples of required data: •Entities, their quantities and arrival times: periodic, random, scheduled, daily pattern, etc •Time the entities spend in the activities: This is usually not a fixed time but a statistical distribution. The wider the time distribution the higher the variability of the system behavior • Capacity of each activity: The max number of entities that can be processed concurrently in the activity •The size of input and output queues for the activities (if needed) •The routing type or the logical conditions for a specific routing •Resource Assignments: the number of resources, their availability, and/or resources shift schedule • Validate the model

• Compare the model’s outcome/prediction with the actual baseline process performance

Alexander Kolker

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Healthcare Institutions that widely use ME methodology:

• Mayo Clinic has defined the Science of Healthcare Delivery as one of its four strategic directions. The Center for the Science of Healthcare Delivery was created, that will focus on developing new approaches to how healthcare is delivered www.mayoclinic.org/news2011-rst/6151.html • Texas Children’s Hospital. Clinic Flow Optimization & Patient Scheduling using Process Simulation. www.Createasoft.com • Texas Medical Center, Houston, TX. A new center of Systems Engineering in

Healthcare has recently opened. The program called ‘Systems and Workflow Improvements Flexible Team’ will be working on Medical Centers throughout the region. An integral part of this program will be Industrial (Systems) Engineers

• Cincinnati Children’s Medical Center. CEO and CFO provided the Webcast: http://www.ihi.org/knowledge/Pages/AudioandVideo/WIHIAllHospitalsinFavorofSavingMoneySayPatientFlow.aspx • Southern California Region. Cooperatively work on the use of ME for Operations Improvement in 20+ Regional Hospitals (Led by University SC) • York Hospital, PA. ED & Health Services Design •University of Iowa, Carver College of Medicine, Department of Anesthesia-Operations Research for Surgical Services. Alexander


http://www.mayoclinic.org/news2011-rst/6151.html



http://www.createasoft.com/

http://www.ihi.org/knowledge/Pages/AudioandVideo/WIHIAllHospitalsinFavorofSavingMoneySayPatientFlow.aspx

http://www.ihi.org/knowledge/Pages/AudioandVideo/WIHIAllHospitalsinFavorofSavingMoneySayPatientFlow.aspx

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Optimized Staffing with variable demand