1. A Hospital Placements Allocation Problem Stephen Cresswell and Lee McCluskey (paper to be presented at PlanSIG 2005)

2. Introduction Problem belongs to University of Huddersfield’s course on Operating Department Practice (ODP). 1st year students are sent on 6 placements to hospitals in Yorkshire. Previously, each student has been allocated to a single hospital for the year – so hospitals have taken the responsibility for organising a suitable programme.

3. Introduction (2) ODP wanted to organise placements centrally. –Don’t want the hospitals to ‘own’ the students. –Work around bottlenecks to increase the capacity of the course. A side-effect of this change is that allocating students to placements has become a problem too difficult to manage without computer support. In the rest of the talk describe the problem and our approaches to solving it.

4. The placements problem We tackle the problem of allocating 1st year of ODP course. –60 students (approx.) –11 hospitals –6 timeslots –4 specialities, from which each student takes any 3. For each speciality selected, the student has 1 anaesthetic placement and 1 surgery placement.

5. Constraints Capacity: Each hospital has a limited capacity (usually 0-2) for the number of placement students that can be accepted in each speciality. Reachability of hospitals: Hospital must be within reasonable commuting distance from home location of student. Non-repetition: Each of 6 placements is in either anaesthetics or surgery in one of 4 specialities. Alternation: A student should not have –2 consecutive placements of anaesthetic, or –2 consecutive placements of surgery.

6. Example allocation Studentt1t2t3t4t5t6 1BUPA Dewsbury LGI Ortho General Gynae SurgeryAnaesthSurgeryAnaesthSurgeryAnaesth 2Bradford Calderdale Bradford Ortho Urology General AnaesthSurgeryAnaesthSurgeryAnaesthSurgery 3…

7. Simplifying assumptions Pair timeslots so that students take –Surgery then Anaesthetic, or –Anaesthetic then Surgery in the same speciality. Student has same phase for all placements. We then have 3 timeslots, and we must allocate 3 from 4 specialities.

8. Goals Can we produce an allocation of students to placements which meets all the constraints? How many more students can be accommodated under the central placements system? –The availability of placements is the main factor limiting the expansion of the course.

9. Model Symbols: h - hospital,st - student, sp - speciality,t - timeslot, ph - phase cap(h,sp) –Integer capacity of hospital h in speciality sp reachable(st) –Set of hospitals reachable by student st alloc(st,t) –Allocation of student st at time t, –Allocation is tuple

10. Model: Capacity Number of students allocated to a particular hospital, speciality and phase is within available capacity.

11. Model: Reachability of hospitals Student can only be allocated to reachable hospitals

12. Model: Non-repetition Don’t repeat same speciality –i.e. set of student’s allocated specialities has unique element for each time slot.

13. Model: Alternation Phase for student matches allocation for student in all timeslots:

14. Related Problems Scheduling Timetabling Hospitals residents problem

15. 3 approaches Programmed heuristic solution –Depth first search with a priori ordering of variables. Constraint programming –Generalisation of programmed heuristic solution –Propagate constraints to maintain domains of variables Integer linear programming –Modelled relaxed version of problem. –Mainly useful for giving bound on optimal solution.

16. Programmed heuristic solution For each alloc(st,t) we have a Prolog term t(H,Sp), where H and Sp are initially uninstantiated variables. Use Prolog built-in depth-first search with heuristic ordering determining solution order for students. Constraints checked as allocations made: –Capacity: total for of each tracked. –Alternation: checked via phase variable for each student.

17. Constraint Programming A finite domain variable for each alloc(st,t). Each tuple represented by an integer value. Constraint types: –Capacity - ‘atmost’ constraint –Reachability - a priori pruning of domain –Non-repetition - ‘alldifferent’ constraint –Alternation - element constraint linking a phase variable for student with indexes of compatible tuples.

18. Constraint Programming Post constraints first, then impose search strategy. Finds schedule with (almost) no backtracking. Default search strategy was “fail first” heuristic. –Select variable with smallest domain. –Not so different from heuristic used in Prolog prog. There are some symmetries - e.g. between timeslots and between some sets of students. We didn’t try breaking those symmetries. (Implemented in Oz).

19. ILP model Integer linear programming can be used where all constraints can be expressed as linear (in)equalities Tools: –eclipse + xpress-mp –Oz + lp_solve Model is a bit different, using alloc(st,h,sp,t), which is solved for 0 or 1 Various relaxations of the problem tried.

20. ILP model An allocation for each student in each timeslot Capacity

21. ILP model Non-repetition Reachability

22. ILP summary Phase constraints are not naturally encoded as linear inequations, so not all constraints handled. Solving a relaxed version of the problem is good for detecting infeasibility. Relaxations: Integer/continuous Ignore phase (A-S or S-A) Appropriate for optimising an objective function rather than finding any feasible solution.

23. Initial results Table shows #students in largest solved prob. Prog. heuristic CLPILP for Relaxed prob. Prob1666973 Prob2717275

24. Initial results (2) Programmed heuristic solution is faster. CLP approach found solutions for more students. Prolog and CLP programs, used very similar heuristics –Prolog a priori ordering of students according to number reachable hospitals –CLP program used ‘fail first’ heuristic – dynamically ordering variables to select var with smallest domain – i.e. the smallest choice of tuples.

25. Extra Constraints After discussing initial results with ODP course organisers, additional constraints emerged: –Student preferences: Students select 3 hospitals in order of preference. –Fluctuating capacity (capacity varies over timeslots). –Flexible phase (capacity to be distributed over phases). –Extra specialities (12 instead of 4). –Combined specialities (e.g. gynae & urology). –Even distribution of allocation. –Forced moves.

26. Delivered system Constraint programming: –Variable ordering heuristic: fail-first –Value ordering heuristic: Student preferences, then random. –Search algorithm: biased towards revising choices near root of search tree, c.f. Depth-Bounded Discrepency Search. User-friendly interface. Implemented in Mozart/Oz.

27. Summary & Discussion We have tackled an allocation problem arising in a real application. A relatively simple constraint model has been adequate for finding feasible solutions. This approach has also been flexible enough to adapt to changing user requirememts.

28. Further Work Compare other solution techniques - e.g. evolutionary algorithms. Treat as optimisation problem with objective based on student preferences. Consider the problem of detecting infeasible sub-problems. Improve user interface to make mixed manual / automatic allocation easier.

29. Leftover slides

30. Model Summary Capacity Reachability Non-repetition Alternation

31. Extra Constraints Fluctuating Capacity: The capacity is dependent on time and phase. Capacity can be split between phases.

32. Extra Constraints Forced moves

33. Goals Can we produce an allocation of students to placements which meets all the constraints? –Yes! How many more students can be accommodated under the central placements system? –Current capacity of the course is 56 students. –We can produce schedules for up to 69 students, assuming additional students can travel anywhere. –There could be solutions up to 73 students.

34. (Old) Conclusions Problem is easy to solve for the number of students currently involved. Maximising number of students is more challenging. Software can be used for Huddersfield ODP problem, and hopefully also elsewhere.