dynamic decision model for cyclical employee scheduling

O P T I M I Z A T I O N O F E M P L O Y E E S C H E D U L I N G U S I N G O P E R A T I O N S R E S E A R C H T E C H N I Q U E - L I N E A R I N T E R G E R G O A L P R O G R A M M I N G

19.12.14Dynamic decision model-employee Scheduling

Dynamic decision model for cyclical employee Scheduling

Authored by-

Swapnil SoniSowmiyan Morri

V.KamalaDepartment of Management Studies,

Indian Institute of Science, Bangalore

Instructor-Dr M Mathirajan(Chief Research Scientist) Department of Management Studies,Indian Institute of Science, Bangalore

2

Index

Introduction to Employee Scheduling

Scheduling problem

Motivation to adopt OR technique

Research and Literature work

Literature Review

The latest reviewed Paper

Nurse Scheduling at Health Centre

Parameters

Problem Statement

Problem Formulation

Notations & Decision Variables

Constraints

Objective Function


Programming in LINGO (Optimization tool)

Parameter Inputs

Execution & Results

Conclusion

Achievements

The way forward

Applications

References

3

Introduction to Nurse Scheduling


Motivation for applying Operations Research for Employee Scheduling

Employee SchedulingConstraints

Management requirement

Employees’ preferences

Conventional Register

Question on:

•Tedious•Time •Accuracy•Fairness•Subjectivity

Mathematical Modeling

Advantages on:

•Tedious•Time •Accuracy•Fairness•Objectivity

Prescriptive ModelCause Response

Variables of 1st order Linear

Variables with Binaryvalues

Integer

Constraints with priorities Goal

Linear Integer Goal Programming

Operations Research

4

Literature Review


Authors Year Reference Literature Limitations

Arthur & Ravindran

1981

A Multiple Objective Nurse Scheduling Model

(IIE Transactions, 13(1), pp. 55-60)

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

1984

Scheduling Nurses Using Goal-Programming Techniques

(IIE Transactions, 16(3), pp. 216 – 221)

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 single shift

Ozkarahan& Bailey

1988

Goal Programming Model Subsystem of A Flexible Nurse Scheduling Support System

(IIE Transactions, 20(3), pp.306-316)

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

•Small set of constraints

5 19.12.14Dynamic decision model-employee Scheduling

Authors Year Reference Literature Limitations

Ferland& Michelon

1996

A Multi-objective Approach to Nurse Scheduling with Both Hard and Soft Constraints,(Socio-Economic Planning Sciences, 30 (3), pp. 183-193)

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 & Kiragu

1998

Cyclic and Non-cyclic Scheduling of 12 h Shift Nurses by Network Programming(European Journal of Operational Research, 104, pp. 582-592)

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

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

Chan & Weil

2001

Cyclical Staff scheduling Using Constraint Logic Programming(Lecture Notes on Computer Sciences 2079, pp. 159-175)

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

•Small set of constraints

Ruzzakiah Jenalet al.

2011A Cyclical Nurse Schedule Using Goal Programming(LPPM ITB, ISSN: 1978-3043)

Use of ILGP for cyclical Nurse scheduling

•Unable to incorporate uncertainty

Literature Review (cont..)

Research gap or opportunity for improvement

• Non-cyclical scheduling raising unfairness to the employees• Small set of constraints – frail solution for practical implementation• Lack of incorporation of uncertainty raised by management and employees

Solution

Optimum scheduling• Cyclical• Robust • Dynamic

H E A L T H C E N T R E , I I S c


NURSE SCHEDULING

Photo courtesy: Ms. Divya Choudhary

7

Observations at Health Centre, IISc


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 Shift

Evening Shift

Night Shift

6:00 am-2:00pm

2:00pm-10:00pm

10:00pm-6:00am

8

?=0,1 Nurse

DAYS SHIFT 1 2 3 4 5 6 7 8 9 10 11 DEMAND

1

M 5

E 3

N 1

2

M 5

E 3

N 1

3

M 5

E 3

N 1

4

M 5

E 3

N 1

5

M 5

E 3

N 1

6

M 5

E 3

N 1

14

M 5

E 3

N 1

Problem Statement


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? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?

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


Problem Statement

Objective:

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 ofscheduling horizon. At the end of the scheduling horizon each worker would have completed each patternexactly 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”


Problem Formulation

• Notations• Constraints• Objective Function


Constraints

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

Hard Constraints

Soft Constraints

Problem Formulation-Constraints Description

Hard Constraints-Must be satisfied

Soft Constraint-May be violated

For 3 shifts for 24 hours a day and 7 days a week.1. Minimum staff level requirement or demand must be satisfied.2. Each nurse works at most one shift a day.3. No nurse works for more than 6 consecutive days.4. No nurse can have more than given allowable holidays in a fortnight.5. Avoid working in Night shift followed by Morning shift or Evening shift of

the next day.6. Morning shift constitutes at least the given %age of the total workload.7. Evening shift constitutes at least the given %age of the total workload.8. Night shift should not exceed the given %age of the total workload.

1. Each nurse has at least one day off in one weekend. 2. All nurses have the same amount of total workload.

Goal Programming

12

Notations

n = number of days in the schedule (n = 14)

m = number of employees available (m = 11)

i = index for days, i = 1…n

k = index for employees, k = 1…m

Dynamic inputs to the Model

The values of the following variables will be provided by user of the model; so will incorporate the uncertainty and maintain flexibility in the model:

Mi = staff requirement for morning shift of day i, i = 1…n

Ei = staff requirement for evening shift of day i, i = 1…n

Ni = staff requirement for night shift of day i, i = 1…n

MAX_HOLIDAYS = Maximum allowable holidays in a fortnight

IDEAL_WD = Ideal Working Days (n- MAX_HOLIDAYS)

MIN_WL_M = Minimum allowable workload for Morning shift

MIN_WL_E = Minimum allowable workload for Evening shift

MAX_WL_M = Maximum allowable workload for Night shift

The following notations will be used to specify the availability of employees for a given shift of the day:

𝐴𝑉𝑀𝑖,𝑘 =1, if employee k is available for Morning shift for day i

0, therwise

𝐴𝑉𝐸𝑖,𝑘 =1, if employee k is available for Evening shift for day i

0, therwise

𝐴𝑉𝑁𝑖,𝑘 =1, if employee k is available for Night shift for day i

0, therwise


Problem Formulation- Notation

13

Problem Formulation- Decision Variables


Decision Variables

𝑋𝑖,𝑘 =1, if employee k is assigned a Morning shift for day i

0, therwise

𝑌𝑖,𝑘 =1, if employee k is assigned a Evening shift for day i

0, therwise

𝑍𝑖,𝑘 =1, if employee k is assigned a Night shift for day i

0, therwise

𝐶𝑖,𝑘 =1, if employee k is assigned a day off for day i

0, therwise

14

Hard Constraints:

Set 1: Minimum staff level requirement must be satisfied:

For Morning shift

𝑘=1

𝑚

𝑋𝑖,𝑘∗ 𝐴𝑉𝑀𝑖,𝑘 ≥ 𝑀𝑖 , 𝑖 = 1,2,3, . . 𝑛

For Evening shift

𝑘=1

𝑚

𝑌𝑖,𝑘∗ 𝐴𝑉𝐸𝑖,𝑘 ≥ 𝐸𝑖 , 𝑖 = 1,2,3, . . 𝑛

For Night shift

𝑘=1

𝑚

𝑍𝑖,𝑘∗ 𝐴𝑉𝑁𝑖,𝑘 ≥ 𝑁𝑖 , 𝑖 = 1,2,3, . . 𝑛

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

𝑋𝑖,𝑘 ∗ 𝐴𝑉𝑀𝑖,𝑘 + 𝑌𝑖,𝑘 ∗ 𝐴𝑉𝐸𝑖,𝑘 + 𝑍𝑖,𝑘 ∗ 𝐴𝑉𝑁𝑖,𝑘 + 𝐶𝑖,𝑘 = 1, 𝑖 = 1,2,3… . 𝑛 𝑎𝑛𝑑 𝑘 = 1,2,3… . .𝑚


Problem Formulation-Constraints

….“n” equations



….“n*m” equations

15

Hard Constraints:

Set 3: Each nurse works not more than 6 consecutive days:

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


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

Day

s

1 K K+1 K+1 K+1

2 K K K+1 K+1

3 K K K K+1

4 K K K K

5 K K K K K

6 K K K K K K

7 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

13 K K K K K

14 K K K K

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

16

Set 3: Each nurse works not more than 6 consecutive days

For all first (n-6) days and all m employees-

𝐶𝑖,𝑘 + 𝐶𝑖+1,𝑘 + 𝐶𝑖+2,𝑘 + 𝐶𝑖+3,𝑘 + 𝐶𝑖+4,𝑘 + 𝐶𝑖+5,𝑘 + 𝐶𝑖+6,𝑘 ≥ 1, 𝑖 = 1,2…𝑛 𝑎𝑛𝑑 𝑘 = 1,2, . . 𝑚

For all next 6 days and (m-1) employees

𝑖=𝑛−𝑣

𝑛

𝐶𝑖,𝑘 +

𝑖=1

6−𝑣

𝐶𝑖,𝑘+1 ≥ 1, 𝑣 = 0,1, . . 5 𝑎𝑛𝑑 𝑘 = 1,2, … (𝑚 − 1)

For all next 6 days and mth employee

𝑖=𝑛−𝑣

𝑛

𝐶𝑖,𝑚 +

𝑖=1

6−𝑣

𝐶𝑖,1 ≥ 1, 𝑣 = 0,1, . . 5

19.12.14Dynamic decision model-employee Scheduling 13.04.14Nurse Scheduling-IGP

….“(n-6)*m” equations


….“6*(m-1)” equations

….6 equations

17

Hard Constraints:

Set 4: No employee can have more than given allowable number of holidays in a fortnight:

𝑖=1

𝑛

𝐶𝑖,𝑘 ≤ 𝑀𝐴𝑋_𝐻𝑂𝐿𝐼𝐷𝐴𝑌𝑆, 𝑘 = 1,2. . . . 𝑚

Set 5: Avoid working in Night shift followed by Morning shift or Evening shift of the next day:

For all first (n-1) days and all m employees-

𝑍𝑖,𝑘 ∗ 𝐴𝑉𝑁𝑖,𝑘 + 𝑋𝑖+1,𝑘 ∗ 𝐴𝑉𝑀𝑖+1,𝑘 + 𝑌𝑖+1,𝑘 ∗ 𝐴𝑉𝐸𝑖+1,𝑘 ≤ 1, i = 1,2, … (n − 1) and k = 1,2,…m

For all first nth day and (m-1) employees-

𝑍𝑛,𝑘 ∗ 𝐴𝑉𝑁𝑛,𝑘 + 𝑋1,𝑘 ∗ 𝐴𝑉𝑀1,𝑘 + 𝑌1,𝑘 ∗ 𝐴𝑉𝐸1,𝑘 ≤ 1, i = 1,2, … (n − 1) and k = 1,2, … (m − 1)

For all first nth day and mth employee-𝑍𝑛,𝑚 ∗ 𝐴𝑉𝑁𝑛,𝑚 + 𝑋1,1 ∗ 𝐴𝑉𝑀1,1 + 𝑌1,1 ∗ 𝐴𝑉𝐸1,1 ≤ 1



….“m” equations

….“(n-1)*m” equations

….“1” equation

18

Set 6: Morning shift constitutes at least the given percentage of the total workload:

𝑖=1

𝑛

𝑋𝑖,𝑘 ∗ 𝐴𝑉𝑀𝑖,𝑘 ≥ 𝑀𝐼𝑁_𝑊𝐿_𝑀 ∗

𝑖=1

𝑛

𝑋𝑖,𝑘 ∗ 𝐴𝑉𝑀𝑖,𝑘 +

𝑖=1

𝑛

𝑌𝑖,𝑘 ∗ 𝐴𝑉𝐸𝑖,𝑘 +

𝑖=1

𝑛

𝑍𝑖,𝑘 ∗ 𝐴𝑉𝑁𝑖,𝑘 ,

𝑘 = 1,2… .𝑚

Set 7: Evening shift constitutes at least the given percentage of the total workload:

𝑖=1

𝑛

𝑌𝑖,𝑘 ∗ 𝐴𝑉𝐸𝑖,𝑘 ≥ 𝑀𝐼𝑁_𝑊𝐿_𝐸 ∗

𝑖=1

𝑛


𝑖=1

𝑛


𝑖=1

𝑛


𝑘 = 1,2… .𝑚

Set 8: Night shift should not exceed the given percentage of the total workload:

𝑖=1

𝑛

𝑌𝑖,𝑘 ≤ 𝑀𝐴𝑋_𝑊𝐿_𝑁 ∗

𝑖=1

𝑛


𝑖=1

𝑛


𝑖=1

𝑛


𝑘 = 1,2… .𝑚

19.12.14Dynamic decision model-employee Scheduling 13.04.14Nurse Scheduling-IGP





19

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:

d+ : Positive deviation

d- : Negative deviation

Set 1:Each employee has at least one day off in one weekend:𝐶7,𝑘 + 𝐶14,𝑘 ≥ 1, 𝑘 = 1,2… .𝑚

𝐶7,𝑘 + 𝐶14,𝑘 + 𝑑1− − 𝑑1+ = 1, 𝑘 = 1,2… .𝑚

Set 2: All employees have the same amount of total workload:

𝑖=1

𝑛


𝑖=1

𝑛


𝑖=1

𝑛

𝑍𝑖,𝑘 ∗ 𝐴𝑉𝑁𝑖,𝑘 = 𝐼𝐷𝐸𝐴𝐿_𝑊𝐷, 𝑘 = 1,2… .𝑚

𝑖=1

𝑛


𝑖=1

𝑛


𝑖=1

𝑛

𝑍𝑖,𝑘 ∗ 𝐴𝑉𝑁𝑖,𝑘 + 𝑑2− − 𝑑2+ = 𝐼𝐷𝐸𝐴𝐿_𝑊𝐷,

𝑘 = 1,2… .𝑚



….”m” equation

….”m” equation

Minimize d1 -

Minimize d2 + & d2 -


Problem Formulation-Objective Function:

Multi-objective Goal Programming model:

Subject to:

• Hard constraints(as mentioned)

• Soft constraints(as mentioned)

• Binary constraints:𝑋𝑖,𝑘 , 𝑌𝑖,𝑘 , 𝑍𝑖,𝑘 , 𝐶𝑖,𝑘 = 0 𝑜𝑟 1

• Non-negativity constraints:𝑑1+, 𝑑1−, 𝑑2+, 𝑑2− ≥ 0

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒

𝑘=1

𝑚

𝑑1− +

𝑘=1

𝑚

𝑑2+ +

𝑘=1

𝑚

𝑑2−

Objective Function:


Execution & Results

• Program• Input parameters• Output• Conclusion

22

Programming in LINGO


Defining Sets

Import & Export of Data with Excel

23

Parameter Inputs (dynamic)


(A) Availability of Nurses (𝐴𝑉𝑀𝑖,𝑘 , 𝐴𝑉𝐸𝑖,𝑘, , 𝐴𝑉𝑁𝑖,𝑘)

AV AvailableNA Not Available

Nurse

DAYS SHIFT 1 2 3 4 5 6 7 8 9 10 11

1

M AV AV AV AV AV AV AV AV AV AV AV

E AV AV AV AV AV AV AV AV AV AV AV

N AV AV AV AV AV AV AV NA NA AV AV

2

M NA AV AV AV AV AV AV AV AV AV AV

E NA AV NA AV AV AV AV AV AV AV AV

N NA AV AV AV AV AV AV AV AV AV AV

3

M AV AV AV AV AV NA AV AV AV AV AV

E AV AV AV AV AV NA AV AV AV AV AV

N AV AV AV AV AV NA AV AV NA AV AV

4

M AV AV AV AV AV AV AV AV AV AV AV

E AV AV AV NA NA AV AV AV AV AV AV

N AV AV AV AV AV AV AV AV AV AV AV

5

M AV AV AV AV AV AV AV NA NA AV NA

E AV AV AV AV AV AV AV AV AV AV NA

N AV NA AV AV AV AV AV AV AV AV NA

14M AV AV AV AV AV AV AV AV AV AV AV

E AV AV AV AV AV AV AV AV AV AV AV

N AV AV AV AV AV AV AV AV AV AV AV

As per dynamic requirement of Management and Employee inputs can be incorporated to generate the optimum schedule


Parameter Inputs (dynamic)

DAYS 1 2 3 4 5 6 7 8 9 10 14

SHIFT M E N M E N M E N M E N M E N M E N M E N M E N M E N M E N M E N

DEMAND 5 3 1 5 3 1 5 3 1 5 3 1 5 3 1 5 3 1 5 3 1 5 3 1 5 3 1 5 3 1 5 3 1

(C) Maximum allowed holidays to nurses in a fortnight (MAX_HOLIDAYS)

3 Days11 Ideal working days for each Nurse (IDEAL_WD)

(D) Minimum allowable workload for the Morning shift (MIN_WL_M)30% of the total workload

(E) Minimum allowable workload for the Evening shift (MIN_WL_E)25% of the total workload

(F) Maximum allowable workload for the Night shift (MIN_WL_N)30% of the total workload

(B) Minimum staff level requirement or demand the Health centre (Mi, Ei, Ni)

25

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

Day

s

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Total Morning Shifts 4 5 5 4 4 8 6 7 9 9 9

Total Evening Shifts 4 3 5 5 5 3 5 3 3 3 3

Total Night Shifts 3 3 2 3 2 0 0 1 0 0 0

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

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

Total Off's in weekends 1 0 0 0 0 0 1 1 1 0 0

Execution & Result


Hard Constraints1) Demand is met

2) Each nurse works at most one shift a day

3) No nurse works for more than 6 consecutive days

Soft Constraints1) Each employee has at least one day off in one weekend

4) No employee can have more than given allowable number of holidays in a fortnight

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

6,7,8) Shift wise Workloads have to be as per given distribution

2) All employees have the same amount of total workload

26

Time Line Analysis


1 2 3 4 5 6 7 8

No of Nurses 5 6 7 8 9 10 11 12

Time to Solve (min) 16 19 81 134 212 901 1498 3980

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Tim

e t

o s

olv

e (

in M

inu

tes

)

No. of Variables Vs Time to solve

NP Hard (Non-polynomial Non-deterministic) problem:

Exponential increase in time to solve the problem w.r.t. increase in number of Nurses

5 11 15 18 20 22 24 26

0 0.1 81 134 212 901 1498 3980

27

Conclusion

Achievements

Management Objectives

The developed model with various constraints and goals using the 0-1 goal programming techniquegives the optimum solution that showed that the hard constraints are strictly satisfied.

Employees Preferences

The developed model elucidates that employees preferences are incorporated as ‘goals’ andoptimally met. (Yet all the goals are not achieved)

Factors of completeness, continuity & fairness

The optimal cyclical schedule delivered factors of completeness, continuity & fairness asemployees will have the opportunity to work with the satisfactory and unsatisfactory rotaryschedule’s patterns.

Employee productivity

With this cyclical scheduling, it gives employees more control over their work life because theyknow the type of shift schedule in the future which should have a positive effect on their jobsatisfaction.

Dynamic capability

The program incorporates the uncertainty and can be tailored as per requirement up to a certainlimit.


28

The way forward

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

To avoid NP Hard issue, Heuristic approach can be explored to attain near optimal solution.

Applications

Transportation

Call centres

Health care

Emergency services

Civic services and utilities

Venue management

Financial services

Hospitality and tourism

Manufacturing


Conclusion (continued..)


Websites

www.lindo.com

www.sciencedirect.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

Book

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

http://www.worldbank.org/




Thank you!

Feedback & Discussion!

dynamic decision model for cyclical employee scheduling

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