Nurse Scheduling-IGP 1

I N T E R G E R G O A L P R O G R A M M I N G

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NURSE SCHEDULING

Prepared by-Sowmiyan Morri

Swapnil SoniDoMS, IISc

Course-Applied Operations Research

Instructor-Prof M Mathirajan

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Index

Introduction to Nurse Scheduling Scheduling problem Motivation to adopt OR technique

Research and Literature work Literature Review The Paper

The Paper Parameters Problem Statement Problem Formulation

Notations & Decision Variables Constraints Objective Function

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Programming in LINGO (Optimization

tool)

Result Conclusion

Achievements The way forward Applications

Pilot Study at Health Centre, IISc Parameters Constraints Result

References

3

Introduction to Nurse Scheduling

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Motivation for applying Operations Research for Nurse Scheduling

Cyclical Nurse Schedule

Constraints

Hospitals requirement

Nurses’ preferences

Conventional RegisterQuestion on:

• Tedious•Time •Accuracy•Fairness

Mathematical ModelingAdvantages on:

• Tedious•Time •Accuracy•Fairness

Prescriptive Model

Cause Response

Variables of 1st order Linear

Variables with Binary values Integer

Constraints with priorities Goal

Liner Integer Goal Programming

Operations Research

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Literature Review

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Authors Reference Literature Limitations

Arthur & Ravindran

Arthur, J. L., & Ravindran, A., A Multiple Objective Nurse SchedulingModel, IIE Transactions, 13(1), pp. 55-60, 1981

Research on modelling Nurse Scheduling using goal programming has been studied which focused on two phases:•Phase 1 is to assign the working days and days off for each nurse while•Phase 2 is to assign the shift types of their working days

•Small set of constraints •Limited problem dimensions with the size of nurses is 4

Musa & Saxena

Musa, A. A., & Saxena, U., Scheduling Nurses Using Goal-ProgrammingTechniques, IIE Transactions, 16(3), pp. 216 – 221, 1984

Used a 0-1 goal programming thatapplied to one unit of a hospital with the considerations of the hospital policies and nurses’ preferences

•2 week planning period •1 one single shift

Ozkarahan & Bailey

Ozkarahan, I. & Bailey, J.E., Goal Programming Model Subsystem of AFlexible Nurse Scheduling Support System, IIE Transactions, 20(3), pp.306-316, 1988.

Nurse scheduling modelling showed theflexibility of goal programming in handling various goals which fulfilled the hospital’s objectives and the nurses’ preferences.

•Small set of constraints

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Authors Reference Literature Limitations

Azaiez & Al Sharif

Berrada, I., Ferland, J. A., & Michelon, P., A Multi-objective Approach toNurse Scheduling with Both Hard and Soft Constraints, Socio-EconomicPlanning Sciences, 30(3), pp. 183-193, 1996

Used the 0-1 goal programming approach with the considerations of hospital’s objectives as hard constraints and the nurses’ preferences as soft constraints to develop the schedules

•No cyclic scheduling

Harvey and Kiragu

Harvey, H.M., & Kiragu, M., Cyclic and Non-cyclic Scheduling of 12 hShift Nurses by Network Programming, European Journal of OperationalResearch, 104, pp. 582-592, 1998

Presented a mathematical model for cyclic and non-cyclic scheduling of 12hours shift nurses. The model is quite flexible and can accommodate a varietyof constraints

• With small requirements which are not appropriate to embed in real situations

Chan and Weil

Chan, P. & Weil, G., Cyclical Staff Scheduling Using Constraint LogicProgramming, Lecture Notes on Computer Sciences 2079, pp. 159-175,2001

Use of work cycles with various constraints to producetimetables of up to 150 people

•Small set of constraints

Literature Review

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The Paper

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Author From

Ruzzakiah JenalSchool of Information Technology, Faculty of Science and Information Technology,Universiti Kebangsaan Malaysia, Selangor, Malaysia

Wan Rosmanira Ismail School of Mathematical Sciences, Faculty of Science and Technology,Universiti Kebangsaan Malaysia, Selangor, Malaysia

Liong Choong Yeun

Ahmed Oughalime

Published By LPPM ITB, ISSN: 1978-3043

Accepted for Publication April 13th, 2011

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The Paper -Parameters

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Number of Nurses 18

Number of Days 21

Number of Shifts: 3 (Morning, Evening & Night)

Number of Decision Variables 18 X 21 X 4 (3 shifts+1 Off) = 1512

Type of Decision Variables Binary (0-1)

Parameters:

One Ward 18 nurses 3 Shifts

Morning ShiftAt least 4 nurses

Evening ShiftAt least 4

nurses

Night ShiftExactly 3 nurses

7:00 am-2:00pm

2:00pm-9:00pm

9:00pm-7:00am

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Problem StatementObjective:

Cyclic Nurse Scheduling: To allot shifts to each Nurse for each day thereby generating a schedule of working days and days off for each nurse in a ward of a hospital.

Physical Constraints:

(A) Hard ConstraintMeeting management objectives

(B)Soft constraintsSatisfaction of employees(Nurses), work/life balance

Logical Constraints:

(C) Cyclic SchedulingA cyclic schedule consists of a set of work patterns which is rotated among a group of workers over a set of scheduling horizon. At the end of the scheduling horizon each worker would have completed each pattern exactly once.

Advantages:• Fairness among nurses• Considers nurses preferences • Lead to maximizing satisfaction• Help Nurses to provide Quality of services

“The right employees at the right time and at the right cost while achieving a high level of employee job satisfaction”

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Problem Statement

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Morning Shift                              

?=0,1Nurse

Demand1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Days

1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 42 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 43 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 44 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 45 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 46 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 47 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 48 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 49 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 4

10 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 411 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 412 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 413 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 414 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 415 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 416 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 417 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 418 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 419 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 420 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 421 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 4

Total Shift ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Similarly for:Evening, Day & Off Shift

This Excel sheet is linked with LINGO to feed the inputs for ‘Data Sets’ & ‘Attributes’ and get output for all ‘Decision Variables’

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Constraints•Hard Constraints (Management)•Soft Constraints (Nurse Specific)

Hard Constraints

•Each unit is covered by 3 shifts for 24 hours a day and 7 days a week.•Minimum staff level requirement must be satisfied.•Each nurse works at most one shift a day.•Avoid any isolated days patterns of “off-on-off”.•Each nurse must have three days off after having three consecutive night

shifts.•Each nurse works between 12 to 14 days per schedule.•Each nurse works not more than 6 consecutive days.•Evening shift constitutes at least 25% of total workload.•Morning shift constitutes at least 30% of total workload.

Soft Constraints

•Avoid working in an evening shift followed by a morning shift or a nightshift the next day.

•Avoid working in a morning shift followed by an evening shift or a night shift the next day.

•Each nurse has at least one day off in one weekend.•All nurses have the same amount of total workload.

Problem Formulation-Constraints Description

Hard Constraints-Must be satisfied

Soft Constraint-May be violated

Goal Programmin

g

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NotationsThe following notations are used to specify the model: n = number of days in the schedule (n = 21) m = number of nurses available for the unit of interest (m = 18) i = index for days, i = 1…n k = index for nurses, k = 1…m Pi = staff requirement for morning shift of day i, i = 1…n Ti = staff requirement for evening shift of day i, i = 1…n Mi = staff requirement for night shift of day i, i = 1…n

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Problem Formulation- Notation & Decision Variables

Decision Variables

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Hard Constraints:

Set 1: Minimum staff level requirement must be satisfied: For Morning shift (Where Pi=4)

For Evening shift (Where Ti=4)

For Night shift (Where Mi=3)

Set 2: Each nurse works only one shift a day:13.04.14

Problem Formulation-Constraints

….“n” equations

….“n” equations

….“n” equations

….“n*m” equations

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Hard Constraints:

Set 3: Avoid any isolated days patterns of “off-on-off” :

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Problem Formulation-Constraints (continued..)

….“(n-2)*m” equations

Day1 Day2 Day3Off On OffC1 X2/Y2/Z2 C3 Sum

Unacceptable1 1 1 3

Acceptable0 0 0 00 0 1 10 1 0 10 1 1 21 0 0 11 0 1 21 1 0 2

  Yes    No

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Nurse Scheduling-IGP

Hard Constraints:

Set 4: Each nurse works 3 consecutive days of night shift and followed by 3 days off. Each nurse will be assigned to their night shifts and off days as follow:

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Problem Formulation-Constraints (continued..)

….“m” equations

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Hard Constraints:

Set 5: Each nurse works between 12 to 14 days per schedule:

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Problem Formulation-Constraints (continued..)

….“2*m” equations

For each Nurse total Sum of all working shift should lie between 12 & 14

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Hard Constraints: Set 6: Each nurse works not more than 6 consecutive days:

Each Nurse has to have at least 1 “Off” in 7 consecutive days

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Problem Formulation-Constraints (continued..)

Cases for 7 Consecutive days for Kth Nurse

Case-1

Case-2

Case-3

Case-4

Case-5

Case-6

Case-7

Case-8

Case-9

Case-10

Case-11

Case-12

Case-13

Case-14

Case-15

Case-16

Case-17

Case-18

Case-19

Case-20

Case-21

Days

1 K K+1 K+1 K+1 K+1 K+1 K+12 K K K+1 K+1 K+1 K+1 K+13 K K K K+1 K+1 K+1 K+14 K K K K K+1 K+1 K+15 K K K K K K+1 K+16 K K K K K K K+17 K K K K K K K 8 K K K K K K K 9 K K K K K K K

10 K K K K K K K 11 K K K K K K K 12 K K K K K K K 13 K K K K K K K 14 K K K K K K K 15 K K K K K K K 16 K K K K K K K 17 K K K K K K K 18 K K K K K K K 19 K K K K K K K 20 K K K K K K K 21 K K K K K K K

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

Due to Cyclic constraint, Nurse “K” has to take position of “K+1” in each next cycle

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Set 6: Each nurse works not more than 6 consecutive days For 1st 15 Days, 18 Nurses (in following eq “i” can take maximum of 15 value)

For next 6 days, 17 Nurses (in following eq “k” can take maximum of 17 value)

For next 6 days, 18th Nurses 13.04.1413.04.141

7Nurse Scheduling-IGP

6 ….“(n-6)*m” equations

Problem Formulation-Constraints (continued..)

….“6*(m-1)” equations

….6 equations

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Set 7: Evening shift constitutes at least 25% of total workload:

Sum of all Evening shifts for a nurse >=25% of Total worked shifts

Set 8: Morning shift constitutes at least 30% of total workload:

o Sum of all Morning shifts for a nurse >=30% of Total worked shifts

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Problem Formulation-Constraints (continued..)

0.25* ….“m” equations

0.30* ….“m” equations

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Soft Constraints:

Soft constraints are arising out of Nurses’ preferences so these can be treated as Goals for our Integer Liner Programming.

The deviation for each goal are christened: ρ : Positive Deviation η : Negative Deviation

Set 1: Avoid working in an evening shift followed by a morning shift or a night shift the next day:

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Day1 Day2Evening Morning/Night

Y1 X2/Z2 SumUnacceptable

1 1 2Acceptable

0 0 00 1 11 0 1

Yes No

Problem Formulation-Constraints (continued..)

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Set 1: Avoid working in an evening shift followed by a morning shift or a night shift the next day:

For 1st 20 Days, 18 Nurses (in following eq “i” can take maximum of 20 value)

For 21st & 1st days, 17 Nurses (in following eq “k” can take maximum of 17 value)

For 21st & 1st days, 18th & 1st Nurses

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….“(n-1)*m” equations

Problem Formulation-Constraints (continued..)

….“(m-1)” equations

….1 equation

Goal-1: Minimize

=

=

=

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Set 2: Avoid working in an Morning shift followed by a Evening shift or a night shift the next day:

For 1st 20 Days, 18 Nurses (in following eq “i” can take maximum of 20 value)

For 21st & 1st days, 17 Nurses (in following eq “k” can take maximum of 17 value)

For 21st & 1st days, 18th & 1st Nurses

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….“(n-1)*m” equations

Problem Formulation-Constraints (continued..)

….“(m-1)” equations

….1 equation

Goal-2: Minimize

=

=

=

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Set 3: Each nurse has at least one weekend off:

Sum of above heighted weekends >=1 (for each Nurse)

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Problem Formulation-Constraints (continued..)

Nurse1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Days

1                                    2                                    3                                    4                                    5                                    6                                    7                                    8                                    9                                    

10                                    11                                    12                                    13                                    14                                    15                                    16                                    17                                    18                                    19                                    20                                    21                                    

Each Nurse has to have at least one Off here out of highlighted 3 weekends

….“m” equations

Goal-3: Minimize

=

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Set 4: All nurses have the same amount of total workload: In Hard Constraint Set-5, it has been seen that Management preference for

total work load should be between 12 & 14. But Nurses prefer to have equal work load. Thus trade off is to have work load of 13 for each nurse.

Sum of all shifts for each Nurse = 13

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Problem Formulation-Constraints (continued..)

….“m” equations

Goal-4: Minimize

Binary Constraints: For each nurse and for each shift (Morning, Evening, Night, Off), value can be either 1 or 0.

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Problem Formulation-Objective Function:

Preemptive Goal Programming for this model:

Subject to:

• Hard constraints• Soft constraints• Binary Constraints• Non-negativity constraints

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Programming in LINGO

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Defining Sets

Import & Export of Data with Excel

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Program Execution

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Time Line Analysis

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1 2 3 4 5 6 7 80

500

1000

1500

2000

2500

3000

3500

4000

4500

No. of Variables Vs Time to solve

Tim

e to

sol

ve (

in M

inut

es)

(for 21 Days)

Exponential increase in time to solve the problem w.r.t. No. of Nurses

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Result-Optimal Solution

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OVERALL SCHEDULE       

Nurse Total Nurses in Morning

Shift

Total Nurses in Evening

Shift

Total Nurses in Night

Shift

Total Nurses in all Shifts1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Days

1 N OFF OFF OFF E E N OFF M E E M N OFF M M OFF OFF 4 4 3 112 N OFF M OFF E E N OFF E E OFF OFF N OFF M M M OFF 4 4 3 113 N OFF M OFF E OFF N OFF OFF E M OFF N OFF E E M M 4 4 3 114 OFF E E E OFF N OFF M M OFF E N OFF M OFF OFF M N 4 4 3 115 OFF E E E M N OFF E M M OFF N OFF M OFF OFF OFF N 4 4 3 116 OFF E E OFF M N OFF E OFF M OFF N OFF M E OFF M N 4 4 3 117 E E OFF M N OFF OFF E M OFF N OFF M M E OFF N OFF 4 4 3 118 E E M M N OFF OFF OFF M OFF N OFF E E E M N OFF 4 5 3 129 OFF E E M N OFF M M OFF OFF N OFF E E OFF M N OFF 4 4 3 11

10 E OFF E N OFF OFF M M E N OFF M OFF OFF M N OFF E 4 4 3 1111 E M OFF N OFF OFF E M E N OFF M OFF E M N OFF E 4 5 3 1212 OFF M OFF N OFF M E M OFF N OFF M E E OFF N OFF E 4 4 3 1113 OFF M N OFF M M OFF E N OFF M E E OFF N OFF OFF E 4 4 3 1114 OFF M N OFF M E E E N OFF M OFF OFF M N OFF OFF E 4 4 3 1115 E OFF N OFF OFF E E OFF N OFF M E M M N OFF M OFF 4 4 3 1116 E N OFF E M OFF OFF N OFF M E E M N OFF M E OFF 4 5 3 1217 OFF N OFF E M M OFF N OFF E OFF E OFF N OFF M E M 4 4 3 1118 M N OFF OFF E M M N OFF E M OFF OFF N OFF E OFF E 4 4 3 1119 N OFF OFF M OFF M N OFF M E E OFF N OFF M E E OFF 4 4 3 1120 N OFF E M OFF OFF N OFF E OFF E M N OFF M OFF E M 4 4 3 1121 N OFF E E OFF M N OFF OFF M OFF M N OFF OFF E E M 4 4 3 11

Total Morning Shifts 1 4 3 5 6 6 3 5 6 4 5 6 3 6 6 6 5 4Total Evening Shifts 6 6 7 5 4 4 4 5 4 6 5 4 4 4 4 4 5 6

Total Night Shifts 6 3 3 3 3 3 6 3 3 3 3 3 6 3 3 3 3 3Total Off's 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

Total Working Days 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13

Hard Constraints1) Demand is met

2) Each nurse works at most one shift a day

3) Avoid any isolated days patterns of “off-on-off”.

4) Each nurse must have three days off after having three consecutive night

5) Each nurse works between 12 to 14 days per schedule.

6) Each nurse works not more than 6 consecutive days

7) Evening shift constitutes at least 25% of total workload

Soft Constraints1) Avoid working in an evening shift followed by a morning shift or a nightshift the next day

3) Each nurse has at least one day off in one weekend.

4) All nurses have the same amount of total workload

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Conclusion

Achievements

The developed model with various constraints and goals using the 0-1 goal programming technique gives the optimum solution that showed both the hard constraints and soft constraints are satisfied.

The pattern will be rotated among the nurses and each nurse will be working according to each schedule’s pattern. After completing 18 schedules, then each nurse will revisit the starting schedule.

Cyclical nurse scheduling rotates equally through the desirable and undesirable work stretches among the nurses and requires relatively less scheduling effort of the head nurse.

The schedule satisfies the factors of completeness and continuity. While the fairness factor is dealt with since the schedule’s pattern is going to rotate among the nurses.

All nurses will have the opportunity to work with the satisfactory and unsatisfactory schedule’s patterns.

With this cyclical scheduling, it gives nurses more control over their work life because they know the type of shift schedule in the future which should have a positive effect on their job satisfaction.

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The way forward

New schedule will only need to be produced when changes occur in its average daily staff requirements.

For further research, one of possible work is to embed the model into user friendly software that would be easy to use and reliable.

The model also should be extended to account for other important scheduling aspects such as requested day off in order to being acceptable to all parties.

Applications

Transportation Call centres Health care Emergency services Civic services and utilities Venue management Financial services Hospitality and tourism Manufacturing

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Conclusion (continued..)

Nurse Scheduling-IGP 31

HEALTH CENTRE, IISc

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PILOT STUDY- NURSE SCHEDULING

Photo courtesy: Ms. D. Choudhary

32Nurse Scheduling-IGP

Pilot Study at Health Centre IISc

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Number of Nurses 11

Number of Days 14 (2 Weeks)

Number of Shifts: 3 (Morning, Day & Night)

Number of Decision Variables 11 X 14 X 4 (3 shifts+1 Off) = 616

Type of Decision Variables Binary (0-1)

Health Centre 11 nurses 3 Shifts

Morning ShiftAt least 5 nurses

Evening ShiftAt least 2

nurses

Night ShiftExactly 1 nurses

6:00 am-1:00pm

1:00pm-8:00pm

8:00pm-6:00am

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Constraints•Hard Constraints (Management)•Soft Constraints (Nurse Specific)

Hard Constraints

•Each unit is covered by 3 shifts for 24 hours a day and 7 days a week.•Minimum staff level requirement must be satisfied.•Each nurse works at most one shift a day.•Each nurse works not more than 6 consecutive days.• Each nurse can’t have more than 3 holidays fortnightly.

Soft Constraints

•Avoid working in Night shift followed by Morning shift or Evening shift of the next day.

•Each nurse has at least one day off in one weekend. (could not be met)

Problem Formulation-Constraints Description

Hard Constraints-Must be satisfied

Soft Constraint-May be violated

Goal Programmin

g

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Execution & Result

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OVERALL SCHEDULE PROPOSED FOR HEALTH CENTRE, IISc        

Nurses Total Nurses in Morning

Shift

Total Nurses in Evening

Shift

Total Nurses in

Night Shift

Total Nurses in all Shifts1 2 3 4 5 6 7 8 9 10 11

Days

1 E E M M M N M E M E E 5 5 1 11

2 E M E E M N M E E M M 5 5 1 11

3 M N M E OFF OFF OFF M M M E 5 2 1 8

4 M N OFF E M E M OFF OFF M M 5 2 1 8

5 M OFF E E M E E N M M M 5 4 1 10

6 OFF E M OFF E M M N M M OFF 5 2 1 8

7 OFF M M E OFF M E N M OFF M 5 2 1 8

8 M M OFF N E OFF M OFF M M E 5 2 1 8

9 E M M OFF M E OFF N OFF M M 5 2 1 8

10 M N M E M E M OFF M E OFF 5 3 1 9

11 N OFF M M E M E M E OFF M 5 3 1 9

12 N E M M OFF M OFF E M M M 6 2 1 9

13 OFF N E OFF M E M M E M M 5 3 1 9

14 M OFF OFF M M E M M OFF E N 5 2 1 8

Total Morning Shifts 6 4 8 4 8 4 8 4 8 9 8

Total Evening Shifts 3 3 3 6 3 6 3 3 3 3 3

Total Night Shifts 2 4 0 1 0 2 0 4 0 0 1

Total Off's 3 3 3 3 3 2 3 3 3 2 2

Total Working Days 11 11 11 11 11 12 11 11 11 12 12

Hard Constraints1) Demand is met

2) Each nurse works at most one shift a day

3) Each nurse works not more than 6 consecutive days

4) Each nurse can’t have more than 3 holidays fortnightly

Soft Constraints1) Avoid working in Night shift followed by Morning shift or Evening shift of the next day

35Nurse Scheduling-IGP 13.04.14

Websites www.lindo.com www.journal.itb.ac.id

Research Papers A Cyclic Nurse Schedule using Goal Programming By Ruzzakiah Jenal et.al. A Multiple Objective Nurse Scheduling Model By Arthur & Ravidran Scheduling Nurses Using Goal-Programming Techniques By Musa & Saxena Goal Programming Model Subsystem of A Flexible Nurse Scheduling Support System

By Ozkarahan & Bailey

Books An Introduction to Management Science By Anderson Sweeney Williams

Tools used Microsoft Encarta (Encyclopedia for offline references) Microsoft Excel (Data embedding) Industrial LINGO (Linear Integer Programming)

References

Nurse Scheduling-IGP 13.04.1436

Thank you!

They said it….

“There’s a fundamental distinction between strategy and operational effectiveness” (Michael Porter)

Leanings….

• Practical application of Operations Research• Optimization Software- LINGO and its limitations• Literature Review of Research Paper