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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.

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hospital placements allocation

Hospital placements allocation

Stephen Cresswell

and

Lee McCluskey

introduction
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
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
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
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
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
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.
model
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
Model: Capacity
  • Number of students allocated to a particular hospital, speciality and phase is within available capacity.
model reachability of hospitals
Model: Reachability of hospitals
  • Student can only be allocated to reachable hospitals
model non repetition
Model: Non-repetition
  • Don’t repeat same speciality – i.e. set of student’s allocated specialiaties has unique element for each time slot.
model alternation
Model: Alternation
  • Phase for student matches alloctation for student in all timeslots:
prolog solution
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
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 programming1
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 summary
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.
results
Results

Table shows #students in largest solved prob.

results 2
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.
goals1
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.
conclusions
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.
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