1. Hospital placements allocation Stephen Cresswell and Lee McCluskey

2. Introduction Problem belongs to Human & Health Department, who run on course on Operating Department Practi{c|s}e (ODP). 2-year course, in which students are sent on 7 hospital placements per year. Up to now, 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 would like to organise placements centrally. –Work around bottlenecks to increase the capacity of the course –Give students experience of more than one hospital A side-effect of this change will be that allocating students to placements has become a combinatorial problem seemingly too difficult to do by hand. In the rest of the talk, we describe the problem and our approaches to solving it.

4. Constraints Reachability of hospitals: Placements must be within reasonable commuting distance from home location of student. Participating hospitals: –Leeds(3), Bradford, Huddersfield, Halifax, Dewsbury, Wakefield, Pontefract, Keighley, Harrogate. Non-repetition: Each of 6 placements is in either anaesthetics or surgery in one of 4 specialities: – General Surgery, Gynaecology, Urology, Orthopaedics.

5. More constraints Capacity: Each hospital has a limited capacity (usually 0-2) for the number of placement students that can be accepted in each speciality. Alternation: A student should not have –2 consecutive placements of anaesthetic, or –2 consecutive placements of surgery.

6. 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.

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

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 specialiaties has unique element for each time slot.

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

14. Prolog 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. –Reachability –Non-repetition –Alternation: checked via phase variable for each student.

15. 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.

16. 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 Lee’s heuristic There are some symmetries - e.g. between timeslots and between some sets of students. We didn’t try breaking those symmetries. (Implemented in Oz).

17. ILP model

18. ILP summary Some of the constraints are not naturally encoded as linear inequations, and this defeats the solver. Solving a relaxed version of the problem is good for detecting infeasibility. Relaxations: Integer/continuous Collapse time Ignore phase (A-S or S-A) Appropriate for optimising an objective function rather than finding any feasible solution.

19. Results Table shows #students in largest solved prob. Pure Prolog CLPILP for Relaxed prob. Prob1666973 Prob2717275

20. Results(2) Pure Prolog 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.

21. 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.

22. Further work

23. 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.

24. Scrapyard