Dynamic decision model for cyclical employee Scheduling
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Transcript of 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
19.12.14Dynamic decision model-employee Scheduling
Programming in LINGO (Optimization tool)
Parameter Inputs
Execution & Results
Conclusion
Achievements
The way forward
Applications
References
3
Introduction to Nurse Scheduling
19.12.14Dynamic decision model-employee Scheduling
Motivation for applying Operations Research for Employee Scheduling
Employee SchedulingConstraints
Management requirement
Employeesβ preferences
Conventional Register
Question on:
β’Tediousβ’Time β’Accuracyβ’Fairnessβ’Subjectivity
Mathematical Modeling
Advantages on:
β’Tediousβ’Time β’Accuracyβ’Fairnessβ’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
19.12.14Dynamic decision model-employee Scheduling
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:β’Phase 1 is to assign the working days and days off for each nurse whileβ’Phase 2 is to assign the shift types of their working days
β’Small set of constraints β’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 β 221)
Used a 0-1 goal programming thatapplied to one unit of a hospital with the considerations of the hospital policies and nursesβ preferences
β’2 week planning period β’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 hospitalβs objectives and the nursesβ preferences.
β’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 hospitalβs objectives as hard constraints and the nursesβ preferences as soft constraints to develop the schedules
β’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
β’ 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
β’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
β’Unable to incorporate uncertainty
Literature Review (cont..)
Research gap or opportunity for improvement
β’ Non-cyclical scheduling raising unfairness to the employeesβ’ Small set of constraints β frail solution for practical implementationβ’ Lack of incorporation of uncertainty raised by management and employees
Solution
Optimum schedulingβ’ Cyclicalβ’ Robust β’ Dynamic
H E A L T H C E N T R E , I I S c
19.12.14Dynamic decision model-employee Scheduling
NURSE SCHEDULING
Photo courtesy: Ms. Divya Choudhary
7
Observations at Health Centre, IISc
19.12.14Dynamic decision model-employee Scheduling
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
19.12.14Dynamic decision model-employee Scheduling
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? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?
This Spreadsheet is embedded with LINGO to feed the inputs for βData Setsβ & βAttributesβ and get output for all βDecision Variablesβ
9 19.12.14Dynamic decision model-employee Scheduling
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:β’ Fairness among nursesβ’ Considers nurses preferencesβ’ Lead to maximizing satisfactionβ’ Help Nurses to provide Quality of services
βThe right employees at the right time
and at the right cost while achieving a
high level of employee job satisfactionβ
19.12.14Dynamic decision model-employee Scheduling
Problem Formulation
β’ Notationsβ’ Constraintsβ’ Objective Function
11 19.12.14Dynamic decision model-employee Scheduling
Constraints
β’ Hard Constraints (Management)β’ 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 = 1β¦n
k = index for employees, k = 1β¦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 = 1β¦n
Ei = staff requirement for evening shift of day i, i = 1β¦n
Ni = staff requirement for night shift of day i, i = 1β¦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:
π΄πππ,π =1, if employee k is available for Morning shift for day i
0, therwise
π΄ππΈπ,π =1, if employee k is available for Evening shift for day i
0, therwise
π΄πππ,π =1, if employee k is available for Night shift for day i
0, therwise
19.12.14Dynamic decision model-employee Scheduling
Problem Formulation- Notation
13
Problem Formulation- Decision Variables
19.12.14Dynamic decision model-employee Scheduling
Decision Variables
ππ,π =1, if employee k is assigned a Morning shift for day i
0, therwise
ππ,π =1, if employee k is assigned a Evening shift for day i
0, therwise
ππ,π =1, if employee k is assigned a Night shift for day i
0, therwise
πΆπ,π =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
π=1
π
ππ,πβ π΄πππ,π β₯ ππ , π = 1,2,3, . . π
For Evening shift
π=1
π
ππ,πβ π΄ππΈπ,π β₯ πΈπ , π = 1,2,3, . . π
For Night shift
π=1
π
ππ,πβ π΄πππ,π β₯ ππ , π = 1,2,3, . . π
Set 2: Each nurse works only one shift a day:
ππ,π β π΄πππ,π + ππ,π β π΄ππΈπ,π + ππ,π β π΄πππ,π + πΆπ,π = 1, π = 1,2,3β¦ . π πππ π = 1,2,3β¦ . .π
19.12.14Dynamic decision model-employee Scheduling
Problem Formulation-Constraints
β¦.βnβ equations
β¦.βnβ equations
β¦.βnβ equations
β¦.βn*mβ equations
15
Hard Constraints:
Set 3: Each nurse works not more than 6 consecutive days:
Each Nurse has to have at least 1 βOffβ in 7 consecutive days
19.12.14Dynamic decision model-employee Scheduling
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 βKβ has to take position of βK+1β 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-
πΆπ,π + πΆπ+1,π + πΆπ+2,π + πΆπ+3,π + πΆπ+4,π + πΆπ+5,π + πΆπ+6,π β₯ 1, π = 1,2β¦π πππ π = 1,2, . . π
For all next 6 days and (m-1) employees
π=πβπ£
π
πΆπ,π +
π=1
6βπ£
πΆπ,π+1 β₯ 1, π£ = 0,1, . . 5 πππ π = 1,2, β¦ (π β 1)
For all next 6 days and mth employee
π=πβπ£
π
πΆπ,π +
π=1
6βπ£
πΆπ,1 β₯ 1, π£ = 0,1, . . 5
19.12.14Dynamic decision model-employee Scheduling 13.04.14Nurse Scheduling-IGP
β¦.β(n-6)*mβ equations
Problem Formulation-Constraints (continued..)
β¦.β6*(m-1)β equations
β¦.6 equations
17
Hard Constraints:
Set 4: No employee can have more than given allowable number of holidays in a fortnight:
π=1
π
πΆπ,π β€ ππ΄π_π»ππΏπΌπ·π΄ππ, π = 1,2. . . . π
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-
ππ,π β π΄πππ,π + ππ+1,π β π΄πππ+1,π + ππ+1,π β π΄ππΈπ+1,π β€ 1, i = 1,2, β¦ (n β 1) and k = 1,2,β¦m
For all first nth day and (m-1) employees-
ππ,π β π΄πππ,π + π1,π β π΄ππ1,π + π1,π β π΄ππΈ1,π β€ 1, i = 1,2, β¦ (n β 1) and k = 1,2, β¦ (m β 1)
For all first nth day and mth employee-ππ,π β π΄πππ,π + π1,1 β π΄ππ1,1 + π1,1 β π΄ππΈ1,1 β€ 1
19.12.14Dynamic decision model-employee Scheduling
Problem Formulation-Constraints (continued..)
β¦.βmβ equations
β¦.β(n-1)*mβ equations
β¦.β1β equation
18
Set 6: Morning shift constitutes at least the given percentage of the total workload:
π=1
π
ππ,π β π΄πππ,π β₯ ππΌπ_ππΏ_π β
π=1
π
ππ,π β π΄πππ,π +
π=1
π
ππ,π β π΄ππΈπ,π +
π=1
π
ππ,π β π΄πππ,π ,
π = 1,2β¦ .π
Set 7: Evening shift constitutes at least the given percentage of the total workload:
π=1
π
ππ,π β π΄ππΈπ,π β₯ ππΌπ_ππΏ_πΈ β
π=1
π
ππ,π β π΄πππ,π +
π=1
π
ππ,π β π΄ππΈπ,π +
π=1
π
ππ,π β π΄πππ,π ,
π = 1,2β¦ .π
Set 8: Night shift should not exceed the given percentage of the total workload:
π=1
π
ππ,π β€ ππ΄π_ππΏ_π β
π=1
π
ππ,π β π΄πππ,π +
π=1
π
ππ,π β π΄ππΈπ,π +
π=1
π
ππ,π β π΄πππ,π ,
π = 1,2β¦ .π
19.12.14Dynamic decision model-employee Scheduling 13.04.14Nurse Scheduling-IGP
β¦.βmβ equations
Problem Formulation-Constraints (continued..)
β¦.βmβ equations
β¦.βmβ equations
19
Soft Constraints:
Soft constraints are arising out of Nursesβ 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:πΆ7,π + πΆ14,π β₯ 1, π = 1,2β¦ .π
πΆ7,π + πΆ14,π + π1β β π1+ = 1, π = 1,2β¦ .π
Set 2: All employees have the same amount of total workload:
π=1
π
ππ,π β π΄πππ,π +
π=1
π
ππ,π β π΄ππΈπ,π +
π=1
π
ππ,π β π΄πππ,π = πΌπ·πΈπ΄πΏ_ππ·, π = 1,2β¦ .π
π=1
π
ππ,π β π΄πππ,π +
π=1
π
ππ,π β π΄ππΈπ,π +
π=1
π
ππ,π β π΄πππ,π + π2β β π2+ = πΌπ·πΈπ΄πΏ_ππ·,
π = 1,2β¦ .π
19.12.14Dynamic decision model-employee Scheduling
Problem Formulation-Constraints (continued..)
β¦.βmβ equation
β¦.βmβ equation
Minimize d1 -
Minimize d2 + & d2 -
20 19.12.14Dynamic decision model-employee Scheduling
Problem Formulation-Objective Function:
Multi-objective Goal Programming model:
Subject to:
β’ Hard constraints(as mentioned)
β’ Soft constraints(as mentioned)
β’ Binary constraints:ππ,π , ππ,π , ππ,π , πΆπ,π = 0 ππ 1
β’ Non-negativity constraints:π1+, π1β, π2+, π2β β₯ 0
πππππππ§π
π=1
π
π1β +
π=1
π
π2+ +
π=1
π
π2β
Objective Function:
19.12.14Dynamic decision model-employee Scheduling
Execution & Results
β’ Programβ’ Input parametersβ’ Outputβ’ Conclusion
22
Programming in LINGO
19.12.14Dynamic decision model-employee Scheduling
Defining Sets
Import & Export of Data with Excel
23
Parameter Inputs (dynamic)
19.12.14Dynamic decision model-employee Scheduling
(A) Availability of Nurses (π΄πππ,π , π΄ππΈπ,π, , π΄πππ,π)
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
24 19.12.14Dynamic decision model-employee Scheduling
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
19.12.14Dynamic decision model-employee Scheduling
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
19.12.14Dynamic decision model-employee Scheduling
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 βgoalsβ 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 rotaryscheduleβs 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.
19.12.14Dynamic decision model-employee Scheduling
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
19.12.14Dynamic decision model-employee Scheduling
Conclusion (continued..)
29 19.12.14Dynamic decision model-employee Scheduling
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