Something old, something new: a hybrid approach to a nurse scheduling problem.

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Something old, something new a hybrid approach
to a nurse scheduling problem.

• Kath Dowsland
• Gower Optimal Algorithms Ltd.

Modern Heuristic
Robust
Classical IP
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Outline

• The problem
• Why heuristic
• Ensuring feasibility
• Other problems with the landscape and their
  solution
• Performance

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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

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• 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.
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Types A full-time nurse works 5 days or 4
nights. Part-timers work other combinations
e.g. (4,3), (3,3), (3,2). Days off and study
days yield yet more combinations.
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Grades There are 3 grade bands. Covering
requirements are given cumulatively for each
band.
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Patterns
Each nurse type defines a set of feasible shift
patterns. A full time nurse defines 2125
feasible day patterns and 35 night 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

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Where. xij 1 if nurse i works pattern j, 0
otherwise. pij is the penalty associated with
nurse i working pattern j F(i) is the set of
patterns feasible for nurse i. ajk 1 if pattern
j covers shift k Gr 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.
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Why heuristic?

• Large number of variables
• Would need advanced options
• Solution to be implemented on each ward
• Possible changes in problem specification
• Requirement for a set of different solutions

BUT
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Problems not always feasible. Additional nurses
imply a cost. If search finds it difficult to
locate a feasible solution how long should we
keep on trying?
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• Ensuring feasibility.

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Let E and D be the number of night and day shifts
required, ei and di the number of nights and
days worked by a nurse of type i, and Ni the
number of nurses of type i available. Then the
problem is feasible if there is a set of integer
variables yi satisfying
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Or if
is at least E. This is a standard bounded
knapsack problem and can easily be solved using a
straightforward branch and bound algorithm.
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• Constraints for higher grades can be included by
• taking a feasible solution for all 3 grades.
• formulating knapsack problem for subset of grades
  required
• adjusting upper/lower bound according to the
  numbers of nurses available and those used in the
  solution.

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A local search approach.
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Local search framework

• Solution space allocations of an allowable shift
  pattern to each nurse.
• Neighbourhood move change the shift pattern of a
  single nurse
• Cost combination of cost for undercovered shifts
  and pij penalty values.
• Possible approach find feasible solution and
  then search feasible solutions for improvement.

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Problem

• Solution space is large
• Bias towards day shifts.
• Many paths of slow descent
• Few paths of steeper descent

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Solution

• Ignore early and late allocations
• Allocate early and late for good solutions only

• Find allocation using network flow.

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Days
Nurses
LB min. earlies UB max. earlies Cost 0
LB 0 UB 1 Cost penalty for early, -
penalty for late
LB req. earlies UB tot-req. lates. Cost 0
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Relaxing balance
LB 0 UB max. earlies Cost penalty
LB 0 UB di max earlies Cost penalty
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Adding grades
LB, UB based on grade 1
From grade 1 nurses
LB, UB based on grade 2
From grade 2 nurses
From grade 3 nurses
LB, UB based on grade overall reqs
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Result smaller solution space for local search
Landscape investigated with aggressive
descentAccept a move if covering cost
decreases and penalty does not increase.
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Problem
Infeasible local optima separated from perfect
cover by high mountains.
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Solution.

• Such local optima are caused by an imbalance of
  day/night cover.
• Restrict moves to those moving towards the right
  balance.
• Select least cost such move.

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Problem
Flat bottomed valleys relating to moves between
patterns of similar attractiveness.
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Solution

• Use specially designed chains of moves to shift
  any extra cover to under covered shifts.
• Shift-chain moves change a subset of nurses
  patterns by one day to form a chain of days.
• Nurse chain moves move each nurse onto the
  pattern worked by the previous nurse in the chain.

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• Combining these moves into an aggressive descent
  and then allowing an uphill move subject to a
  tabu list of length 1, based on the list of
  nurses, consistently finds feasible solutions.
• These must then be improved in terms of penalty
  cost.

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Problem


















Space of feasible solutions disconnected.
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Solution

• Use chain moves to define cycles of moves
  maintaining feasibility.

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Problem
Large basins separated by high-hills.
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Solution

• A tabu list of day-night / night-day moves.
• Frequency based diversification.
• Bounds on cost of day/night mix to force early
  escape from high valleys.

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Performance
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Optimal v. mean penalty cost over 10 random starts
120
100
10 optimal (41)
9 optimal(4)
80
Mean
7 optimal(5)
60
6 optimal(1)
40
20
0
0
50
100
150
Optimal
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Changes to specification

• Two most highly qualified nurses to work at most
  one w/e shift
• Spread surplus evenly to meet preferred cover
• Addition of some mixed shift contracts
• Team working

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Conclusion
Researching the solution landscape by studying
problem structure to allow the exploitation
of Has led to an aggressive downhill search
that can seek out many good local optima
resulting in a a fast and flexible solution
approach.

• classical models
• newer ingredients

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OR While an off-the-peg solution may look OK,
?