Hospital placements allocation

Stephen Cresswell

and

Lee McCluskey

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.

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.

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.

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.

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.

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.

Example scheduleStudent t1 t2 t3 t4 t5 t6

1 BUPA BUPA Dewsbury Dewsbury LGI LGI

Ortho Ortho General General Gynae Gynae

Surgery Anaesth Surgery Anaesth Surgery Anaesth

2 Bradford Bradford Calderdale Calderdale Bradford Bradford

Ortho Ortho Urology Urology General General

Anaesth Surgery Anaesth Surgery Anaesth Surgery

3 …

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 <h,sp,ph>

Model: Capacity

• Number of students allocated to a particular hospital, speciality and phase is within available capacity.

),(,,),(:.... sphcapphsphtstallocsttphsph

Model: Reachability of hospitals

• Student can only be allocated to reachable hospitals

),(

,,),(..

...

hstreachable

phsphtstallocphsp

htst

Model: Non-repetition

• Don’t repeat same speciality – i.e. set of student’s allocated specialiaties has unique element for each time slot.

3,,),(...:. phsphtstallocphhtspst

Model: Alternation

• Phase for student matches alloctation for student in all timeslots:

phsphtstallocsphtphst ,,),(.....

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 <h,sp,ph,t> tracked.– Reachability– Non-repetition– Alternation: checked via phase variable for each student.

Constraint Programming

• A finite domain variable for each alloc(st,t).• Each tuple <h,sp,ph> 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.

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).

ILP model

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/continuousCollapse timeIgnore phase (A-S or S-A)

• Appropriate for optimising an objective function rather than finding any feasible solution.

Results

Table shows #students in largest solved prob.

Pure

Prolog

CLP ILP for

Relaxed prob.

Prob1 66 69 73

Prob2 71 72 75

Optimal Optimal

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 <hosp,sp,ph> tuples.

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.

Further work

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.

Scrapyard

phsphtstalloctphsp

streachableh

hst

,,),(...

)(

..