A Hybrid Constraint Programming Approach to Nurse Rostering Problem

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1 A Hybrid Constraint Programming Approach to Nurse Rostering Problem Fang He*, Dr. Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) research group School of Computer Science University of Nottingham AI-2008 Twenty-eighth SGAI International Conference on Artificial Intelligence

2 Outline 1 Introduction The Nurse Rostering Problems Constraint Programming 2 The Formulation Variables Constraints Models 3 The Hybrid Approach Construction Stage Improvement Stage 4 Computational Experiment Results 5 Conclusions

3 Outline 1 Introduction The Nurse Rostering Problems Constraint Programming 2 The Formulation Variables Constraints Models 3 The Hybrid Approach Construction Stage Improvement Stage 4 Computational Experiment Results 5 Conclusions

4 The Nurse Rostering Problems Nurse Rostering Problems (Typical) In hospitals, nurses work in shift system Each nurse works at most one shift per day The demand on each day for each shift type is varying, but known Constraints are defined by regulations, working practice and preferences of nurses A roster(solution): an assignment of each nurse on each day to a shift which satisfies all constraints In reality,the constraints usually put into two groups: hard constraints and soft constraints

5 The Nurse Rostering Problems List of constraints All demanded shifts should be covered In the scheduling period, the nurse is not allowed to exceed his/her maximum work load The maximum number of night shift is 3 per period of 5 consecutive weeks An early shift after a day shift should be avioded Avoid sequence of shifts with length of 1...

6 The Nurse Rostering Problems Typical Solution

7 The Nurse Rostering Problems Objectives Find a feasible solution(roster) which satisfies all the hard constraints As well as: Balance work load of nurses Satisfy as many preferences as possible... Most real world NRPs are NP-hard(Karp,1972) Good solutions would provide significant benefits for hospital

8 Constraint Programming Constraint Programming Paradigm Emerged from a number of research areas: Artificial Intelligence Programming Language Symbolic Computing and Computing Logic Constraint Satisfaction arose from the research in Artificial Intelligence,applied to: Computer Graphics (Sutherland 1963, Borning 1981) Scene labelling (Waltz 1972) Combinatorial problem (Lassez, Jaffar 1987,Aggoun, Beldiceanu 1993)

9 Constraint Programming Constraint Programming Paradigm 1 Declarative description of problem with Variables which range within their domains Constraints over subsets of variables which restrict possible value combinations A solution is a value assignment which satisfies all constraints 2 Constraint filtering/propagation filtering: removing inconsistent values from variables domains Constraint can propagate due to one domain reduction can lead to new domain reduction 3 Search use strategies to assign values to variables each step triggers constraint propagation

10 Outline 1 Introduction The Nurse Rostering Problems Constraint Programming 2 The Formulation Variables Constraints Models 3 The Hybrid Approach Construction Stage Improvement Stage 4 Computational Experiment Results 5 Conclusions

11 Variables The Formulation Variables Constraints 2 models(csp Model,WCSP Model)

12 Variables Variables s ij, the shift type assigned to nurse i on day j domain: 0 Off 1 Early 2 Day 3 Late 4 Night

13 Constraints How to express the constraints? Simplistic constraints e.g. Nobody is allowed to work 3 consecutive night shifts if s ij == Night && s ij+1 == Night,then s ij+2 Night Problems Very limited propagation, no global view Massive number of simplistic constraints

14 Constraints Global Constraint Work on sets of variables Global point of view Building blocks Very strong propagation As general as possible Usable with other constraints Dictionary of constraints(hooker,2007) Global constraints(in NRPs) Cardinality(or Distribute) Stretch(or Sequence )

15 Constraints Cardinality(ILOG Solver) Cardinality(x/v,l,u) Bound how many times a variable can take a value(in its domain) variable x can take value v at least l and at most u times So, the maximum number of night shift constraint can be expressed as Cardinality(s ij,night,0,3)... Also can be used in production sequencing timetabling...

16 Constraints Stretch(ILOG Solver) Stretch(x/v,l,u,P) a stretch is consecutive variables that take the same value. it restricts the length of stretch within the range of [l,u] P is set of patterns, which restrict a stretch of value v j can immediately precedes a stretch of value v j Thus the stretch constraint puts bounds on how many consecutive days the nurse can work each shift, and which shifts can immediately follow another. So the constraint following a series of at least 2 night shifts, 2 days off is required can be defined as follows: Stretch(s ij, Night, 2, 3, P), P = (Night, Off) Also can be used in personnel assignment timetabling...

17 Constraints Problems Which global constraint to choose? Balance between expressiveness of the structure of the problem propagation power implementation effort How to implement them? Use solver package CHIP, ILOG Solver Roll your own see (Bourdais,2003) for algorithms

18 Models Model H1: Cardinality(s ij, K, D jk, D jk ), i I, j J H5: Cardinality(s ij, Night, 0, n 1 ), i I, j J H7: Stretch(s ij, Night, 2, 3, P), P = (Night, Off), i I, j J H8: Stretch(s ij, Night, 0, n 1, P), P = (Night, Off), i I, j J S9: this constraint can be expressed using a boolean implication constraint, which reflects a boolean logical relation between two variables as follows: s ij =Day s ij+1 Early, i I, j J s ij =Late s ij+1 Early, i I, j J s ij =Late s ij+1 Day, i I, j J...

19 Outline 1 Introduction The Nurse Rostering Problems Constraint Programming 2 The Formulation Variables Constraints Models 3 The Hybrid Approach Construction Stage Improvement Stage 4 Computational Experiment Results 5 Conclusions

20 The Hybrid Approach Example Basic CP search method s limitation: exponential search space We need more intelligence search procedure The idea of the approach is based on the observation that high quality nurse rosters consist of high quality shift sequences(brucker et al,2008). Mon Tue Wed Thu Fri Sat Sun Cost O O D D N N N 0 N N O O O E E 0 E E E O O L L 0 D D L L L O O 0...

21 The Hybrid Approach Example Basic CP search method s limitation: exponential search space We need more intelligence search procedure The idea of the approach is based on the observation that high quality nurse rosters consist of high quality shift sequences(brucker et al,2008). Mon Tue Wed Thu Fri Sat Sun Cost O O D D N N N 0 N N O O O E E 0 E E E O O L L 0 D D L L L O O 0...

22 Outline Construction stage: Shift sequence generation and iterative forward search Improvement stage: Variable Neighbourhood Search to make further improvement

23 Construction Stage Shift Sequence Generation By decomposing the problem into solvable sub-problem for CP, then using iterative forward search to extend the partial solution to complete solution. Use Constraint Satisfaction Problem(CSP) model to generate these high quality shift sequences Variable: s ij, the shift type assigned to nurse i on day j Constraint: shift sequence related constraints building block for next step

24 Construction Stage Iterative Forward Search Based on the high quality shift sequences Weighted CSP model Variable:s iw, one week length shift sequence assigned to nurse i in week w Constraints(soft constraints) have associated costs, which depends on the assignment of the variables. The goal is to minimize the sum of costs: Minimize w i P(x i ) Use it to find optimal(near optimal) solution which satisfy all the hard constraints, at the same time, satisfy as many soft constraints as possible

25 Construction Stage Iterative Forward Search

26 Construction Stage Iterative Forward Search Variable& Value selection rules Randomly selected Selected by heuristic Variable: first-fail principle, heavier work-load nurse is selected first Value: night shift sequence first

27 Improvement Stage Variable Neighbourhood Search Two neighborhoods defined: re-assign a shift to a different nurse working on the same day swap two shifts assigned to two nurses on the same day

28 Outline 1 Introduction The Nurse Rostering Problems Constraint Programming 2 The Formulation Variables Constraints Models 3 The Hybrid Approach Construction Stage Improvement Stage 4 Computational Experiment Results 5 Conclusions

29 Pure CP without decomposition VS hybrid Table: Pure CP VS hybrid Data No. of Variables No. of Constraints Pure CP Hybrid A B GPOST ORTEC ORTEC ORTEC ORTEC indicates no feasible solutions can be obtained within 24 hours running

30 Variable & Value Selection Rule Table: Variable Value Selection Rule Problem Random Selection Heurictic Selection ORTEC ORTEC ORTEC ORTEC

31 Table: Hybrid compared to other approaches Problem instances Hybrid GA Hybrid VNS Hybrid CP Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec within 1/2 hour running,vns restrict in 10 minutes average of 370 seconds to generate shift sequences

32 Outline 1 Introduction The Nurse Rostering Problems Constraint Programming 2 The Formulation Variables Constraints Models 3 The Hybrid Approach Construction Stage Improvement Stage 4 Computational Experiment Results 5 Conclusions

33 Conclusions CP is a general technique, can encapsulate a lot of work CP allows the use of symbolic representation The performance of search depends on the search strategies Hybrid CP is superior to pure CP for large problems

34 Appendix Some of References P. Brucker, R. Qu, E. K. Burke and G. Post. A decomposition, construction and post-processing approach for a specific nurse rostering problem, MISTA 05, New York, USA, Jul 2005 L. Hellsten, G. Pesant, and P. van Beek. A domain consistency algorithm for the stretch constraint. Principles and Practice of Constraint Programming - CP 2004, Lecture Notes in Computer Science, Vol. 3258, , 2004, Springer, Berlin J. Hooke., Integrated Methods for Optimization, Springer, , 2006 J. C. Regin. Generalized arc consistency for global cardinality constraint. National Conference on Artificial Intelligence (AAAI 1996), AAAI Press, , 1996

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A Constraint-directed Local Search Approach to Nurse Rostering Problems

A Constraint-directed Local Search Approach to Nurse Rostering Problems Fang He & Rong Qu Automated Scheduling, Optimization and Planning (ASAP) Group School of Computer Science University of Nottingham,