1. 1 Nurse Rostering with CARE. Jonathan Thompson 1, Kath Dowsland 2 & Bill Dowsland 2. 1. Cardiff University 2.Gower Optimal Algorithms Ltd. These slides & Adobe PDF copies of published academic papers: http://www.goweralg.co.uk/talks/ http://www.goweralg.co.uk/talks/ Details of the CARE software: http://www.goweralg.co.uk/CARE/http://www.goweralg.co.uk/CARE/

2. 2 Outline The problem A quantitative model Solution approach Performance quality and flexibility.

3. 3 The problem. To produce weekly schedules of work for all nurses on the ward so that: minimum covering requirements are met nurses’ preferences and requests are considered schedules are deemed to be fair

4. 4 The day: A day is made up of 3 shifts: earlies (7.5 hour day) lates (7.5 hour day) nights (9.5 or 9 hour night) Nurses work either a whole week of days (usually a mixture of earlies and lates) or a whole week of nights.

5. 5 Types: A full-time nurse works 5 days or 4 nights. Part-timers work other combinations e.g. (4,3), (3,3), (3,2). Sickness, leave and study days yield yet more combinations.

6. 6 Grades: There are 3 grade bands. Covering requirements are given cumulatively for each band. For example Grade G+2 Grade E/F4 Grade D8

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9. 9 Each nurse type defines a set of feasible shift patterns. A full time nurse defines 21*2 5 feasible day patterns and 35 night patterns. (Amalgamating earlies and lates gives 21 patterns) Each nurse-pattern pair has a penalty cost given by: general quality of pattern meeting of requests for days off history of patterns worked recently Patterns:

10. 10 Pattern cost. A cost of 0,1,2 or 3 depending on the desirability of the pattern. Good features include consecutive days off, nights without a gap. Bad features include on/off/on/off etc. Default pattern costs can be overwritten for an individual nurse. Some preference for early or late but less important.

11. 11 Request cost. Request are graded into 5 classes Essential.  Top priority18 High priority12 Medium priority 8 Low priority 3 Where requests for the following week are available their consequences are included in this weeks costs

12. 12 History cost. Based on the previous 3 weeks and ensures compliance with working practices and fairness. Patterns exceeding successive day limit banned. Patterns resulting in on/off/on/off penalised Decreasing penalty for recent weekend/night working Increased penalty this week if undesirable shift last week. Can be overridden with respect to specific contracts.

13. 13 Where. x ij = 1 if nurse i works pattern j, = 0 otherwise. p ij is the penalty associated with nurse i working pattern j F(i) is the set of patterns feasible for nurse i. a jk = 1 if pattern j covers shift k G r is the set of nurses of grade-band r or above R(k,r) is the minimum acceptable number of nurses of grade r or above for shift k. Formulation.

14. 14 Why heuristic? Large number of variables I.P. approach would need ‘advanced’ options Solution to be implemented on each ward Possible changes in problem specification Requirement for a set of different solutions BUT

15. 15 Potential problems. Problem may not be feasible (call in bank nurses). Large solution space (early/late allocation just fine-tuning). Solution. Preprocessing with a knapsack model to determine over/under covering and invite response Postprocessing with a network flow model to allocate earlies/lates to those working days. Result. Heuristic can focus on difficult part of problem.

16. 16 Let E and D be the number of night and day shifts required, e i and d i the number of nights and days worked by a nurse of type i, and N i the number of nurses of type i available. Then the problem is feasible if there is a set of integer variables y i such that: The knapsack model. is at least E. This is a standard bounded knapsack problem and can easily be solved using a straightforward branch and bound algorithm. The model / solution process can be modified to include grades.

17. 17 Nurses Days LB = min. earlies UB = max. earlies Cost = 0 LB = req. earlies UB = tot-req. lates. Cost = 0 LB = 0 UB = 1 Cost = penalty for early, - penalty for late The network model. This model can easily be adapted to deal with grades and to balance the earlies/lates for each nurse.

18. 18 LB = 0 UB = max. earlies Cost = penalty Relaxing balance LB = 0 UB = d i – max earlies Cost = penalty

19. 19 From grade 1 nurses From grade 2 nurses From grade 3 nurses LB, UB based on grade 1 LB, UB based on grade 2 LB, UB based on grade overall reqs Adding grades

20. 20 Local search framework Solution space: feasible solutions Neighbourhood move: change the shift pattern of a single nurse Cost: sum of p ij penalty values.

21. 21 Potential problems with solution landscape. Disconnected or sparsely connected regions (as many moves violate covering restrictions) Plateau-like areas (as many patterns have similar costs) Valleys separated by high ‘mountains’ (due to day/night balance) Solutions. Relaxation of covering constraints when necessary using strategic oscillation. Intelligent use of variable neighbourhoods. Intelligent diversification using tabu lists and frequency based criteria to change day/night allocation. Result. Aggressive search process that is able to seek out good local optima within very fast computing time.

22. 22 Shift-Chain Move Nurse 11111100 Nurse 20110101 Nurse 31011101 Days 3 and 5 have too many staff, day 6 has 2 too few staff Changing individual shifts increase penalty costs. Nurse 10111110 Nurse 21010101 Nurse 31101101 Instead, we change chains of shifts.

23. 23 Nurse-Chain Move Nurse 1 1111100 0000000 Nurse 2 0000000 0111100 Nurse 3 1101101 0000000 Days 3 and 5 have too many staff, day 6 has 2 too few staff Changing individual shifts increase penalty costs. Nurse 1 1101101 0000000 Nurse 2 1101110 0000000 Nurse 3 0000000 0111100 Instead, we change chains of nurses

24. 24 Phase 2. Reduce preference cost while remaining feasible. Random descent over 3 neighbourhoods. neighbour- hood candidate list evaluation function selection 1 standard no CC increase PC first decrease 2 shift-chain all PC first decrease 3 nurse-chain all PC first decrease When a local optima is reacheda single phase 3 move reduces the penalty cost at the expense of covering cost.

25. 25 0 20 40 60 80 100 120 050100150 Optimal Mean 10 optimal (41) 9 optimal (4) 7 optimal (5) 6 optimal ( 1) Optimal v. mean penalty cost over 10 random starts

26. 26 Changes to specification. Change: Two most highly qualified nurses to work at most one w/e shift. Solution: Treat as a covering constraint. In general this would restrict the flexibility of the larger neighbourhoods but is OK because only a small number of nurses involved. Change: Introduction of contracts to work both days and nights in a single week. Solution: Add to pattern base and adjust knapsack model to account for these nurses.

27. 27 Changes to specification (cont.). Change: ‘Earlies’ covering requirement often for preferred cover. Real requirement to cover as many days (sometimes with preference for certain days) with this cover and the remainder with one less. Solution: Use the knapsack pre-processor to calculate the number of days where preferred cover is possible. Introduce a ‘dummy’ nurse to cover the remainder and allocate patterns to meet any preferences between days.

28. 28 Changes to specification (cont.). Change: Nurses allocated to one of 2 teams. At least one nurse from each team required to be on duty at all times. Note: Including this constraint as a covering constraint would severely restrict the power of the larger neighbourhoods. Solution: Include in the moves using the basic neighbourhood only and ignore violations elsewhere. Because the combination of aggressive downhill pressure and diversification ingredients ensure that many good local optima are visited this is sufficient to nudge the search towards optimal solutions that also satisfy the team constraint.

29. 29 Summary A combination of a knapsack pre-processor, network post- processor and a tabu search algorithm with intelligent use of strategic oscillation, variable neighbourhoods and frequency based diversification has resulted in a powerful solution tool. Many additional objectives and constraints can be incorporated with minimal additional development. Where this is not possible the ability of the search to seek out many good local optima means that solutions satisfying additional objectives and constraints are likely to be found quickly and easily.

30. 30 CARE in the ‘community’ Developed by academics and successfully implemented in several wards of a major UK hospital. Proven over a period of many months to be adaptable to the changing requirements encountered in the practical environment. Able to be operated by appropriate nursing / administrative staff and to produce good schedules from both operational and nursing perspectives. Has been subjected to extensive further testing on practical and artificially generated data sets – proving the power and adaptability of its algorithmic core. Details published in articles in leading scientific journals. HOWEVER!

31. 31 CARE in the ‘hospital’ The original version of CARE had a strong algorithmic basis that was flexible enough to solve the REAL problem. But lacked: A full modern Windows interface. Customer support by a proven software company. Platform independent operation. Marketing support. In recent months the software interface has been re-written to incorporate enhancements to the algorithms, interface and operating environment.

32. 32 CARE in the future. Gower Optimal Algorithms Ltd: for 20 years involved in the commercial implementation of OR techniques in logistics & scheduling. We intend, in partnership with the original developers, to further develop the ideas & techniques described for this together with other scheduling / rostering problems. www.goweralg.co.uk/CARE/

33. 33 These slides & Adobe PDF copies of published academic papers: http://www.goweralg.co.uk/talks/ http://www.goweralg.co.uk/talks/ Details of the CARE software: http://www.goweralg.co.uk/CARE/ http://www.goweralg.co.uk/CARE/