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Nurse Scheduling Problem: An Integer Programming Model with a Practical Application

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Systems Analysis Tools for Better Health Care Delivery

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 74))

Abstract

We use a binary integer programming model to formulate and solve a nurse scheduling problem (NSP) which maximally satisfies nurse preferences. In a practical application of a VA hospital, besides considering the scheduling of two types of nurses (registered nurses and licensed practical nurses), two other types of employees (nursing assistants and health care techs), one nurse manager, and a clinical nurse leader are also included in our model. Most of these employees are working full-time. In addition, we distinguish the schedule of weekdays and weekends with different requirements and different preferences for employees. Besides the requirements for each shift, we consider requirements for specific employees in some shifts in practical situations. The seven shifts each day do not necessarily have the same length in our model. Vacation time of employees is also considered in our model. Thus, the requirements for nurse scheduling are complicated and the objective is to maximize the satisfaction of preferred schedules of all these employees, including both nurses and other staffs. The presented model is complex, but efficiently solvable, satisfying the set of requirements in a particular application in a VA hospital.

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Acknowledgment

This material is based upon work supported by the Office of Systems Redesign, Department of Veteran Affairs. This research has been partially supported by CAO funds.

Author information

Authors and Affiliations

  1. Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ, USA

    Neng Fan

  2. Department of Industrial and Systems Engineering, Center for Applied Optimization, University of Florida, Gainesville, FL, USA

    Syed Mujahid, Pando Georgiev, Petraq Papajorgji & Panos M. Pardalos

  3. Functional Neuroscience and Neurosurgery Laboratory, Department of Neurology/Neurosurgery, School of Medicine, The Johns Hopkins University, Baltimore, MD, USA

    Jicong Zhang

  4. Department of Mathematical Information Technology, University of Jyvaskyla, Jyvaskyla, Finland

    Ingrida Steponavice

  5. James A. Haley VAMC, 13000 Bruce B. Downs Blvd, Tampa, FL, 33612, USA

    Britta Neugaard

  6. College of Public Health, University of South Florida, Tampa, FL, USA

    Britta Neugaard

Authors
  1. Neng Fan
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  2. Syed Mujahid
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  3. Jicong Zhang
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  4. Pando Georgiev
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  5. Petraq Papajorgji
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  6. Ingrida Steponavice
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  7. Britta Neugaard
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  8. Panos M. Pardalos
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Corresponding author

Correspondence to Pando Georgiev .

Editor information

Editors and Affiliations

  1. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  2. Center for Applied Optimization, Dept. Industrial & Systems, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Pando G. Georgiev

  3. , Ministry of Education and Sciences, National Agency for Exams, Rruga Naim Frasheri, Nr. 37, Tirana, 32611, Albania

    Petraq Papajorgji

  4. James A. Haley VA Medical Center, Bruce B. Downs Blvd. 13000, Tampa, 33612, Florida, USA

    Britta Neugaard

Appendices

Appendix 1: Results of Nurse Scheduling with One Preference Shift in a Week

 

Preferences

Assignments

Satisfication

Weeks

1

2

3

4

1

2

3

4

1

2

3

4

Type 1 (RN)

1

6

6

3

1

6

6

3

1

Y

Y

Y

Y

2

1

1

6

6

1

1

6

6

Y

Y

Y

Y

3

6

3

3

3

6

3

3

1

Y

Y

Y

N

4

1

3

6

3

1

3

6

3

Y

Y

Y

Y

5

1

6

3

1

1

6

1

1

Y

Y

N

Y

6

6

1

1

6

6

1

1

6

Y

Y

Y

Y

7

6

6

1

1

6

6

1

1

Y

Y

Y

Y

8

3

1

6

6

3

1

6

6

Y

Y

Y

Y

9

1

6

1

6

1

6

1

6

Y

Y

Y

Y

10

6

1

6

3

6

1

6

3

Y

Y

Y

Y

11

3

3

6

1

3

3

6

1

Y

Y

Y

Y

12

3

3

6

3

3

3

6

3

Y

Y

Y

Y

13

6

6

6

3

6

6

6

3

Y

Y

Y

Y

14

3

3

1

6

3

3

1

6

Y

Y

Y

Y

15

6

6

1

1

6

6

1

1

Y

Y

Y

Y

16

3

1

1

6

3

1

1

6

Y

Y

Y

Y

17

3

1

1

1

3

1

1

1

Y

Y

Y

Y

18

3

3

1

1

3

3

1

1

Y

Y

Y

Y

19

1

1

6

6

1

1

6

6

Y

Y

Y

Y

20

1

6

1

6

1

6

1

6

Y

Y

Y

Y

21

6

3

6

6

6

3

6

6

Y

Y

Y

Y

22

3

3

3

1

3

3

3

1

Y

Y

Y

Y

23

6

1

6

3

6

1

6

3

Y

Y

Y

Y

24

6

3

3

1

6

3

1

1

Y

Y

N

Y

25

3

1

3

3

3

1

6

3

Y

Y

N

Y

26

3

1

3

1

3

1

1

1

Y

Y

N

Y

27

1

6

3

3

1

6

3

3

Y

Y

Y

Y

28

3

1

1

6

3

3

3

6

Y

N

N

Y

29

1

1

3

3

1

1

6

3

Y

Y

N

Y

Type 2 (LPN)

1

4

4

2

3

1

4

2

3

N

Y

Y

Y

2

5

5

2

5

5

5

4

5

Y

Y

N

Y

3

4

6

2

2

4

6

1

6

Y

Y

N

N

4

1

5

2

7

1

5

2

7

Y

Y

Y

Y

5

6

3

3

5

6

3

3

5

Y

Y

Y

Y

6

5

4

2

4

5

6

1

4

Y

N

N

Y

7

6

7

1

1

6

7

4

5

Y

Y

N

N

8

4

2

4

5

4

1

4

3

Y

N

Y

N

9

2

2

5

2

6

1

5

4

N

N

Y

N

10

5

5

6

7

5

5

6

7

Y

Y

Y

Y

11

2

7

4

2

2

6

4

6

Y

N

Y

N

Type 3 (NA)

1

7

5

2

2

7

5

1

7

Y

Y

N

N

2

7

5

5

5

1

5

5

6

N

Y

Y

N

3

2

7

2

5

2

7

4

6

Y

Y

N

N

4

2

5

7

5

2

5

7

7

Y

Y

Y

N

Type 4 (HCT)

1

7

2

5

2

3

3

5

7

N

N

Y

N

2

7

2

2

2

7

2

6

1

Y

Y

N

N

3

5

4

3

5

5

4

3

6

Y

Y

Y

N

Appendix 2: Results of Assignments with Respect to Each Day with One Preference Shift in a Week (Employees Who Do Not Appear in this Table Mean They Do Not Need to Work in that Corresponding Week)

Week 1

(ID.Shift)

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Type 1 RN

1.6

1

1

    

1

2.1

1

1

1

 

1

1

1

3.6

      

1

4.1

1

1

1

1

 

1

1

5.1

1

 

1

1

1

1

1

6.6

1

1

1

 

1

  

7.6

  

1

1

1

 

1

8.3

 

1

1

1

1

  

9.1

1

1

1

1

1

 

1

10.6

1

1

  

1

1

 

11.3

1

  

1

1

1

 

12.3

1

 

1

 

1

 

1

13.6

1

 

1

1

 

1

1

14.3

 

1

1

1

  

1

15.6

 

1

1

1

  

1

16.3

1

1

 

1

1

1

 

17.3

1

1

 

1

1

  

18.3

1

1

1

 

1

1

1

19.1

1

1

1

1

1

1

 

20.1

1

1

 

1

1

1

1

21.6

1

1

1

  

1

 

22.3

1

1

   

1

1

23.6

1

1

 

1

1

1

 

24.6

 

1

1

1

1

1

 

25.3

  

1

1

1

1

1

26.3

1

 

1

 

1

1

 

27.1

1

1

1

1

1

1

 

28.3

1

1

1

1

 

1

1

29.1

 

1

1

1

1

1

1

Type 2 (LPN)

2.5

   

1

1

1

1

3.4

1

1

 

1

1

1

1

4.1

      

1

5.6

1

 

1

1

   

6.5

1

1

1

1

1

1

1

7.6

    

1

1

1

8.4

  

1

    

9.2

1

    

1

1

10.5

1

1

1

1

1

 

1

11.2

1

1

1

1

1

1

1

Type 3 (NA)

1.7

  

1

1

1

1

 

3.2

 

1

1

1

1

1

 

4.2

1

1

1

1

1

 

1

Type 4 (HCT)

1.7

1

1

     

2.7

      

1

3.5

1

1

1

  

1

 

Week 2

 
 

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Type 1 (RN)

1.6

1

1

1

1

1

1

 

3.3

     

1

1

4.3

 

1

1

1

 

1

 

5.6

1

1

 

1

1

  

6.1

1

 

1

1

1

1

1

7.6

1

1

1

1

 

1

1

8.1

1

1

1

1

1

1

 

9.6

  

1

1

1

1

 

10.1

1

1

1

 

1

1

1

11.3

1

1

 

1

1

1

1

12.3

1

1

 

1

1

1

1

13.6

1

1

1

 

1

 

1

14.3

1

1

1

1

1

1

 

15.6

1

1

1

1

1

1

 

16.1

1

 

1

1

1

 

1

17.1

1

1

1

 

1

1

1

18.3

1

 

1

 

1

1

 

19.1

1

 

1

1

1

  

20.6

 

1

1

1

  

1

21.3

1

1

1

1

1

 

1

22.3

1

1

1

1

1

 

1

23.1

1

1

 

1

 

1

1

24.3

 

1

1

1

1

 

1

25.1

 

1

 

1

1

1

1

26.1

1

1

1

1

 

1

1

27.6

1

   

1

1

1

29.1

 

1

 

1

 

1

1

Type 2 (LPN)

1.4

  

1

 

1

1

1

2.5

1

  

1

1

  

3.6

      

1

4.5

1

1

1

1

1

1

 

5.3

1

1

1

1

1

1

1

7.7

  

1

1

1

1

1

8.2

1

1

 

1

1

1

1

9.2

1

 

1

1

1

  

10.5

      

1

Type 3 (NA)

1.5

 

1

1

   

1

2.5

1

1

1

1

1

1

1

3.7

1

1

     

4.5

     

1

 

Type 4 (HCT)

1.2

 

1

1

 

1

1

1

2.2

1

1

1

1

 

1

1

3.4

1

1

 

1

   

Week 3

 
 

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Type 1 (RN)

1.3

1

1

1

1

1

1

 

2.6

 

1

1

 

1

1

1

3.3

 

1

   

1

1

4.6

1

 

1

 

1

1

 

5.3

1

 

1

1

 

1

 

6.1

1

1

 

1

1

1

1

7.1

 

1

1

1

 

1

1

8.6

1

  

1

 

1

1

9.1

1

 

1

1

1

1

1

10.6

 

1

 

1

1

1

 

11.6

 

1

 

1

1

 

1

12.6

1

 

1

 

1

1

 

13.6

1

1

1

1

   

14.1

1

1

1

 

1

1

1

15.1

1

1

 

1

1

  

16.1

1

1

1

1

1

1

 

17.1

1

 

1

1

1

1

1

18.1

1

1

1

   

1

19.6

1

1

1

 

1

1

1

20.1

 

1

1

1

1

1

1

21.6

1

 

1

1

  

1

22.3

1

1

1

1

1

1

 

23.6

1

1

 

1

1

  

24.3

1

 

1

1

1

 

1

25.3

1

1

  

1

 

1

26.3

1

1

1

1

1

 

1

27.3

 

1

1

1

1

1

 

29.3

1

 

1

 

1

 

1

Type 2 (LPN)

1.2

 

1

1

  

1

1

3.2

   

1

   

4.2

1

1

1

1

1

1

1

8.4

1

 

1

1

1

1

1

9.5

1

1

1

1

1

1

1

11.4

 

1

     

Type 3 (NA)

1.2

  

1

    

2.5

1

1

1

1

1

1

1

3.2

1

   

1

1

1

4.7

1

1

1

1

1

1

1

Type 4 (HCT)

1.5

1

1

1

1

1

1

1

2.2

1

1

 

1

1

  

3.3

 

1

1

1

 

1

1

Week 4

 
 

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Type 1 (RN)

1.1

1

    

1

1

2.6

    

1

  

4.3

1

1

1

 

1

1

1

5.1

1

1

1

 

1

1

1

6.6

1

 

1

  

1

1

7.1

1

 

1

1

1

 

1

8.6

1

1

1

1

1

 

1

9.6

1

1

 

1

1

  

10.3

1

1

1

1

1

1

 

11.1

 

1

1

1

1

1

1

12.3

1

1

 

1

1

1

1

13.3

1

1

1

1

1

 

1

14.6

1

1

 

1

 

1

 

15.1

1

1

 

1

1

1

1

16.6

1

 

1

1

  

1

17.1

1

1

 

1

 

1

 

18.1

1

1

1

1

1

1

 

19.6

 

1

1

1

 

1

 

20.6

1

 

1

 

1

1

 

21.6

1

1

1

1

1

 

1

22.1

1

 

1

 

1

 

1

23.3

 

1

1

1

1

1

1

24.1

 

1

1

1

1

 

1

25.3

1

1

1

1

1

1

 

26.1

 

1

1

1

 

1

 

27.3

1

 

1

1

 

1

1

28.6

 

1

  

1

1

1

29.3

1

1

1

1

1

 

1

Type 2 (LPN)

2.5

1

1

1

1

1

1

1

3.2

1

1

1

1

1

1

 

5.5

1

1

1

1

1

1

1

6.4

1

1

1

1

1

1

1

8.5

1

      

10.7

1

1

1

1

1

1

1

11.2

1

1

1

1

 

1

1

TYPE 3 (NA)

1.2

1

1

 

1

1

1

1

3.5

 

1

1

  

1

 

Type 4 (HCT)

2.2

  

1

 

1

 

1

3.5

   

1

1

 

1

Appendix 3: Results of Nurse Scheduling with Two Preferences Shift (One for Weekdays and One for Weekends)

 

Preferences

Assignments

Satisfication

Weeks

1

2

3

4

1

2

3

4

1

2

3

4

Type 1 (RN)

1

(6,6)

(6,1)

(3,6)

(1,3)

(6,6)

(6,1)

(3,6)

(1,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

2

(1,6)

(1,3)

(6,1)

(6,6)

(1,1)

(1,3)

(6,1)

(6,6)

(Y,N)

(Y,Y)

(Y,Y)

(Y,Y)

3

(6,6)

(3,1)

(3,6)

(1,6)

(3,6)

(3,1)

(3,6)

(1,6)

(N,Y)

(Y,Y)

(Y,Y)

(Y,Y)

4

(1,3)

(3,3)

(6,6)

(3,1)

(1,3)

(3,3)

(6,6)

(3,1)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

5

(1,6)

(6,6)

(1,6)

(1,3)

(1,6)

(6,6)

(1,6)

(1,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

6

(6,1)

(1,3)

(1,1)

(6,3)

(6,1)

(1,3)

(1,1)

(6,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

7

(6,6)

(6,3)

(1,1)

(1,3)

(6,6)

(6,3)

(1,1)

(1,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

8

(3,6)

(1,3)

(6,6)

(6,1)

(3,6)

(1,3)

(6,6)

(6,1)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

9

(1,6)

(6,1)

(1,6)

(6,6)

(1,6)

(6,1)

(1,6)

(6,6)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

10

(6,1)

(1,3)

(6,3)

(3,3)

(6,1)

(1,3)

(6,3)

(3,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

11

(3,3)

(3,1)

(6,6)

(1,1)

(3,3)

(3,1)

(6,6)

(1,1)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

12

(3,3)

(3,1)

(6,3)

(3,6)

(3,3)

(3,1)

(6,3)

(3,6)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

13

(6,3)

(6,1)

(6,3)

(3,3)

(6,3)

(6,1)

(6,3)

(3,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

14

(3,1)

(3,3)

(1,1)

(6,1)

(3,1)

(3,3)

(1,1)

(6,1)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

15

(6,6)

(6,6)

(1,6)

(1,6)

(6,6)

(6,6)

(1,6)

(1,6)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

16

(3,1)

(1,1)

(1,3)

(6,1)

(3,1)

(1,1)

(1,3)

(6,1)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

17

(3,1)

(1,3)

(1,6)

(1,6)

(3,1)

(1,3)

(1,6)

(1,6)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

18

(3,3)

(3,1)

(1,1)

(1,6)

(3,3)

(3,1)

(1,1)

(1,6)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

19

(1,3)

(1,1)

(6,3)

(6,1)

(1,3)

(1,1)

(6,3)

(6,1)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

20

(1,1)

(6,3)

(1,1)

(6,3)

(1,1)

(6,3)

(1,1)

(6,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

21

(6,1)

(3,1)

(6,6)

(6,6)

(6,1)

(3,1)

(6,6)

(6,6)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

22

(3,3)

(3,6)

(3,1)

(1,3)

(3,3)

(3,6)

(3,1)

(1,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

23

(6,1)

(1,3)

(6,6)

(3,3)

(6,1)

(1,3)

(6,6)

(3,3)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

24

(6,1)

(3,3)

(1,6)

(1,1)

(6,1)

(3,1)

(3,6)

(1,1)

(Y,Y)

(Y,N)

(N,Y)

(Y,Y)

25

(3,1)

(1,6)

(6,3)

(3,1)

(3,1)

(1,6)

(3,3)

(3,1)

(Y,Y)

(Y,Y)

(N,Y)

(Y,Y)

26

(3,6)

(1,3)

(1,1)

(1,6)

(3,6)

(1,3)

(3,1)

(1,6)

(Y,Y)

(Y,Y)

(N,Y)

(Y,Y)

27

(1,3)

(6,1)

(3,3)

(3,1)

(1,3)

(6,1)

(3,3)

(3,1)

(Y,Y)

(Y,Y)

(Y,Y)

(Y,Y)

28

(3,6)

(3,6)

(3,1)

(6,6)

(3,6)

(1,1)

(1,1)

(6,6)

(Y,Y)

(N,N)

(N,Y)

(Y,Y)

29

(1,3)

(1,1)

(6,1)

(3,3)

(1,3)

(1,1)

(3,1)

(3,3)

(Y,Y)

(Y,Y)

(N,Y)

(Y,Y)

Type 2 (LPN)

1

(1,6)

(4,6)

(2,4)

(3,4)

(2,6)

(7,6)

(2,4)

(2,2)

(N,Y)

(N,Y)

(Y,Y)

(N,N)

2

(5,3)

(5,6)

(4,4)

(5,5)

(5,4)

(2,6)

(7,1)

(5,5)

(Y,N)

(N,Y)

(N,N)

(Y,Y)

3

(4,6)

(6,5)

(1,5)

(6,7)

(4,1)

(4,5)

(5,5)

(5,5)

(Y,N)

(N,Y)

(N,Y)

(N,N)

4

(1,7)

(5,3)

(2,1)

(7,7)

(2,7)

(5,2)

(2,5)

(2,7)

(N,Y)

(Y,N)

(Y,N)

(N,Y)

5

(6,3)

(3,4)

(3,5)

(5,6)

(5,3)

(3,4)

(3,1)

(5,1)

(N,Y)

(Y,Y)

(Y,N)

(Y,N)

6

(5,6)

(6,3)

(1,5)

(4,4)

(5,1)

(6,3)

(5,5)

(4,4)

(Y,N)

(Y,Y)

(N,Y)

(Y,Y)

7

(6,5)

(7,7)

(4,4)

(5,6)

(4,5)

(7,7)

(1,1)

(1,1)

(N,Y)

(Y,Y)

(N,N)

(N,N)

8

(4,6)

(1,4)

(4,2)

(3,6)

(7,4)

(2,2)

(4,2)

(4,2)

(N,N)

(N,N)

(Y,Y)

(N,N)

9

(6,2)

(1,7)

(5,2)

(4,6)

(2,2)

(2,2)

(5,2)

(5,5)

(N,Y)

(N,N)

(Y,Y)

(N,N)

10

(5,4)

(5,4)

(6,2)

(7,6)

(5,4)

(5,6)

(2,2)

(7,2)

(Y,Y)

(Y,N)

(N,Y)

(Y,N)

11

(2,7)

(6,2)

(4,7)

(6,3)

(2,1)

(4,2)

(2,7)

(2,2)

(Y,N)

(N,Y)

(N,Y)

(N,N)

Type 3 (NA)

1

(7,3)

(5,3)

(1,6)

(7,4)

(7,5)

(5,3)

(2,2)

(7,2)

(Y,N)

(Y,Y)

(N,N)

(Y,N)

2

(1,1)

(5,5)

(5,3)

(6,5)

(4,1)

(5,5)

(5,2)

(2,5)

(N,Y)

(Y,Y)

(Y,N)

(N,Y)

3

(2,4)

(7,5)

(4,2)

(6,3)

(2,2)

(7,5)

(4,2)

(2,5)

(Y,N)

(Y,Y)

(Y,Y)

(N,N)

4

(2,3)

(5,2)

(7,1)

(7,1)

(2,5)

(5,2)

(7,4)

(2,2)

(Y,N)

(Y,Y)

(Y,N)

(N,N)

Type 4 (HCT)

1

(3,5)

(3,4)

(5,1)

(7,4)

(2,5)

(2,2)

(5,5)

(7,5)

(N,Y)

(N,N)

(Y,N)

(Y,N)

2

(7,2)

(2,6)

(6,5)

(1,1)

(5,2)

(2,6)

(2,5)

(2,5)

(N,Y)

(Y,Y)

(N,Y)

(N,N)

3

(5,2)

(4,7)

(3,4)

(6,3)

(5,2)

(4,1)

(3,5)

(4,4)

(Y,Y)

(Y,N)

(Y,N)

(N,N)

Appendix 4: Results of Assignments with Respect to Each Day with Two Preferences for Weekdays and Weekends (Employees Who Do Not Appear in this Table Mean They Do Not Need to Work in that Corresponding Week)

Week 1

 

(ID.Shift)

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Type 1 (RN)

1.6

1

 

1

1

 

2.1

1

 

1

1

1

1

 

3.6

      

1

4.1

1

1

1

1

1

  

4.3

      

1

5.1

1

1

1

1

1

  

5.6

      

1

6.1

     

1

1

6.6

 

1

1

1

   

7.6

1

1

 

1

1

1

 

8.3

  

1

1

1

  

8.6

     

1

 

9.1

1

1

1

1

   

10.1

      

1

10.6

  

1

1

1

  

11.3

1

 

1

1

1

1

 

12.3

1

1

   

1

1

13.3

      

1

13.6

1

1

1

 

1

  

14.1

     

1

1

14.3

1

1

1

    

15.6

1

1

1

1

1

 

1

16.3

1

1

1

1

1

  

17.1

     

1

1

17.3

1

 

1

1

   

18.3

  

1

1

1

1

 

19.1

1

1

1

1

1

  

19.3

     

1

 

20.1

1

1

1

1

1

1

 

21.1

     

1

1

21.6

1

 

1

    

22.3

1

1

 

1

1

1

1

23.1

     

1

1

23.6

1

1

1

1

   

24.1

      

1

24.6

 

1

1

1

1

  

25.3

1

1

1

1

   

26.3

 

1

1

 

1

  

26.6

     

1

1

27.1

 

1

1

1

1

  

27.3

     

1

1

28.3

1

1

 

1

1

  

28.6

     

1

1

29.1

1

1

 

1

1

  

29.3

      

1

Type 2 (LPN)

1.2

1

1

     

1.6

      

1

2.4

      

1

2.5

    

1

  

3.4

1

1

1

 

1

  

4.7

     

1

1

5.3

     

1

1

5.5

1

 

1

1

   

6.1

      

1

6.5

1

1

 

1

1

  

7.4

   

1

   

7.5

     

1

1

8.4

     

1

 

8.7

1

1

     

9.2

    

1

1

1

10.5

 

1

1

 

1

  

11.2

1

1

 

1

1

  

Type 3 (NA)

1.5

      

1

1.7

  

1

1

1

  

3.2

  

1

  

1

 

4.2

1

1

1

1

1

  

4.5

     

1

 

Type 4 (HCT)

1.2

  

1

1

   

1.5

     

1

1

2.2

      

1

2.5

  

1

1

   

3.2

     

1

1

3.5

1

1

     

Week 2

 
 

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Type 1 (RN)

1.1

      

1

1.6

1

1

1

1

1

  

2.3

      

1

3.1

     

1

1

4.3

  

1

1

1

1

 

5.6

 

1

 

1

1

 

1

6.1

1

 

1

1

1

  

6.3

     

1

 

7.3

      

1

7.6

 

1

1

1

1

  

8.1

1

1

1

1

1

  

8.3

      

1

9.1

     

1

1

9.6

1

1

1

1

   

10.1

1

1

1

1

1

  

10.3

     

1

 

11.3

1

1

1

1

1

  

12.1

     

1

1

12.3

1

1

1

1

   

13.6

1

1

1

1

1

  

14.3

1

1

1

 

1

1

 

15.6

1

   

1

1

1

16.1

1

1

  

1

1

1

17.1

1

1

1

1

1

  

18.1

      

1

18.3

1

1

1

1

1

  

19.1

1

1

  

1

1

 

20.3

     

1

 

20.6

1

 

1

1

   

21.1

      

1

21.3

1

1

1

1

1

  

22.3

1

  

1

1

  

22.6

     

1

 

23.1

1

  

1

   

23.3

     

1

1

24.3

1

1

1

1

1

  

25.1

1

1

1

1

   

25.6

     

1

1

26.1

 

1

1

1

1

  

26.3

      

1

27.1

     

1

1

27.6

 

1

1

    

29.1

 

1

1

1

1

1

 

Type 2 (LPN)

1.6

     

1

1

2.2

 

1

1

1

1

  

2.6

     

1

 

3.4

    

1

  

3.5

     

1

1

4.5

1

1

1

1

1

  

5.3

 

1

     

5.4

     

1

1

6.3

      

1

6.6

1

   

1

  

7.7

   

1

1

1

1

8.2

1

1

 

1

1

  

9.2

1

 

1

 

1

 

1

10.5

1

1

  

1

  

10.6

     

1

 

11.2

     

1

1

11.4

   

1

   

Type 3 (NA)

1.3

     

1

 

1.5

  

1

1

   

2.5

1

1

1

1

1

1

1

3.5

     

1

1

3.7

1

1

1

    

4.2

     

1

 

Type 4 (HCT)

1.2

   

1

 

1

1

2.2

1

1

1

    

2.6

      

1

3.4

1

1

1

    

Week 3

 
 

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Type 1 (RN)

1.3

1

1

1

1

1

  

2.1

      

1

2.6

    

1

  

3.3

   

1

1

  

4.6

1

 

1

1

1

1

 

5.1

 

1

1

1

1

  

6.1

1

1

 

1

1

1

1

7.1

 

1

 

1

1

1

1

8.6

1

1

 

1

1

 

1

9.1

1

 

1

1

1

  

9.6

      

1

10.3

     

1

1

10.6

 

1

 

1

   

11.6

1

  

1

1

1

1

12.3

     

1

1

12.6

 

1

1

1

   

13.3

     

1

1

13.6

1

1

1

    

14.1

1

1

  

1

1

1

15.1

 

1

1

1

1

  

15.6

     

1

 

16.1

1

1

1

1

   

16.3

     

1

1

17.1

1

1

1

1

   

17.6

      

1

18.1

1

 

1

1

 

1

1

19.3

     

1

1

19.6

1

 

1

1

   

20.1

1

1

1

 

1

1

1

21.6

 

1

1

1

1

1

 

22.1

      

1

22.3

1

1

1

1

1

  

23.6

1

1

1

 

1

  

24.3

1

  

1

1

  

24.6

     

1

1

25.3

 

1

1

 

1

1

1

26.1

     

1

1

26.3

1

1

1

    

27.3

1

1

1

1

1

  

29.1

     

1

 

29.3

1

1

1

1

   

Type 2 (LPN)

1.2

   

1

   

2.7

    

1

  

3.5

     

1

1

4.2

1

1

1

1

   

4.5

      

1

5.3

 

1

 

1

1

  

6.5

 

1

1

1

1

1

1

8.2

     

1

1

8.4

1

 

1

1

   

9.5

1

1

1

1

1

  

10.2

1

1

  

1

1

 

11.2

1

1

     

11.7

     

1

1

Type 3 (NA)

1.2

  

1

1

1

1

 

2.2

      

1

2.5

1

 

1

    

3.2

      

1

3.4

 

1

  

1

  

4.4

     

1

1

4.7

1

1

1

1

   

Type 4 (HCT)

1.5

1

1

 

1

1

  

2.2

  

1

 

1

  

2.5

     

1

 

3.3

1

 

1

 

1

  

Week 4

 
 

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Type 1 (RN)

1.1

1

  

1

   

1.3

     

1

1

2.6

 

1

  

1

1

1

3.6

      

1

4.1

      

1

4.3

 

1

1

1

1

  

5.1

1

1

 

1

1

  

5.3

     

1

1

6.3

      

1

6.6

1

 

1

 

1

  

7.1

1

1

  

1

  

7.3

     

1

1

8.1

     

1

1

8.6

1

1

 

1

   

9.6

 

1

 

1

1

1

1

10.3

1

1

1

1

1

 

1

11.1

  

1

1

1

1

1

12.3

1

1

1

1

1

  

13.3

1

1

1

 

1

1

 

14.1

     

1

 

14.6

1

1

1

1

   

15.1

1

1

1

 

1

  

15.6

     

1

 

16.1

     

1

1

16.6

  

1

1

   

17.1

 

1

1

1

1

  

17.6

      

1

18.1

1

1

1

 

1

  

18.6

     

1

 

19.1

     

1

1

19.6

1

1

1

    

20.3

      

1

20.6

1

 

1

 

1

  

21.6

1

1

1

1

1

  

22.1

1

1

1

1

   

23.3

1

1

1

1

1

1

 

24.1

 

1

1

1

 

1

1

25.1

     

1

1

25.3

1

1

 

1

   

26.1

1

1

1

1

1

  

27.3

1

1

1

1

1

  

28.6

   

1

1

1

1

29.3

1

 

1

1

1

1

 

Type 2 (LPN)

1.2

1

  

1

 

1

 

2.5

1

1

1

1

 

1

1

3.5

1

1

1

1

1

  

4.7

     

1

1

5.5

1

1

1

1

1

  

6.4

   

1

   

8.2

     

1

 

8.4

 

1

     

9.5

    

1

 

1

10.2

     

1

1

10.7

    

1

  

11.2

1

   

1

 

1

Type 3 (NA)

1.2

      

1

1.7

 

1

1

    

2.2

1

1

1

    

2.5

     

1

 

3.2

 

1

1

1

1

  

4.2

    

1

  

Type 4 (HCT)

1.5

      

1

1.7

1

  

1

   

2.2

 

1

1

1

   

2.5

     

1

 

3.4

1

 

1

 

1

1

1

       

1

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Fan, N. et al. (2013). Nurse Scheduling Problem: An Integer Programming Model with a Practical Application. In: Pardalos, P., Georgiev, P., Papajorgji, P., Neugaard, B. (eds) Systems Analysis Tools for Better Health Care Delivery. Springer Optimization and Its Applications, vol 74. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5094-8_5

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