1. Off the peg or made to measure
Off the peg or made to measure! Timetabling and scheduling with SA and TS.
Kath Dowsland
Gower Optimal Algorithms Ltd
(talk available at
Bespoke tailoring
High street stores ?

2. Main focus. ? Generic implementation. Change in specification
Problem
Specific
Generic
Problem
Specific ?
Generic implementation.
Easy to develop
Solution quality OK
Problem specific implementation.
Development effort
Change in specification
Generic approach may need minor modification
Problem specific approach may have to start afresh.

3. Secondary focus.
Reading widely
Talks / seminars
Solving my problem

4. Overview. Introduction Laboratory scheduling Examination timetabling
Nurse rostering
Conclusions

5. In an ideal world! Solution Models Problem Lots of published results?
Problem
In an ideal world!
Lots of published results
Solution
Algorithm

6. X In the real world! OR Models Either Problem Problem
Lots of published results
Largest problem solved
Models ?
Problem
Either X ?
Problem OR
In the real world!

7. ??? Answer: Use a metaheuristic
Lots of published results
Answer:
Use a metaheuristic
Local search based e.g. simulated annealing, tabu search, GRASP
Population based e.g. genetic algorithm, ant colony optimisation, scatter search

8. Local Search(minimisation).
Problem specified in terms of a solution space with an evaluation function, f, and neighbourhood structure, N, defined on it.
From any starting solution S find a neighbour Snew that is better in terms of f. Replace S by Snew.
Continue until no improving neighbours exist.
Well suited to scheduling type problems as they have a natural neighbourhood defined by changing the time (or other attribute) of an event or swapping 2 events.
But: converges to local, rather than, global optimum.
Simulated annealing (SA) and tabu search (TS) designed to overcome this problem by accepting ‘uphill’ moves.
SA accepts such moves with a probability depending on the change in the evaluation function and the search time.
At a local optimum TS accepts the ‘best’ neighbour. Some moves are classified as tabu to avoid cycling.
For both techniques smooth, gently sloping solution landscapes are best.

9. A Computer Laboratory Scheduling Problem
The development of AILSA
An Intelligent Laboratory Scheduling Algorithm

10. The problem. Given: A laboratory of known capacity, Q
A set of m timeslots when the laboratory is available
A set of n students together with the times when each is free to attend
Find:
An allocation of each student to a suitable timeslot
Such that:
The number of sessions utilised is minimised
The capacity is not exceeded
Spare capacity is spread evenly

11. Related models. If we ignore capacity we have a set covering problem.
Let Tj be the set of students who can attend at time j. Minimise the number of sets, Tj, needed to cover all the students.
Large NP-hard problem.
If we know which timeslots to use we have a transportation problem.
Let each timeslot define a supply node with available supply = Q, each student define a demand node with a demand of 1. Find a feasible allocation of demand nodes to supply nodes.
Easy to solve using a variety of techniques (e.g. network flow, LP)
Other location type models also provide partial solutions

12. A local search approach.
Solution space: allocations of students to a time-slot they are free to attend
Neighbourhood move: move a student to another session
Cost:
Assumption: p =
Define the ideal occupancy as C = n/p
Cost then based on sine curve.
|Sin(x)| C
 0
Occupancy

13. Problem 1. Search gets stuck in valleys separated by high hills.?
Search gets stuck in valleys separated by high hills.
Question 1: Can we escape intelligently?
Question 2: Even if we can escape, how do we know there is a feasible solution?

14. Causes
Converged towards a set of p slots that do not cover everyone.
No allocation of students to selected p slots satisfying capacity constraints.
Solutions
Identify when search is converging towards p slots and:
Include a penalty term relating to relevant constraint from set covering model into cost.
Solve transportation problem.
if feasible, problem solved
if not feasible add penalty term to drive the search away from the current set of slots.
Can also determine infeasibility by solving small set covering problems defined by the constraints identified in (1) and restarting with new p if infeasible.

15. Problem 2. Search slow to get to point of convergence.?
Search slow to get to point of convergence.
Cause: biased sampling.
Solution: sample student first / slot first in alternate iterations.

16. Is it robust? Changes in problem size/data. Preferences on time-slots.
More than one solution.
Minimising slots for a subset of students.
Combining new objective into cost not successful.
Solved in 2 phases.
Solve for subset first and then add additional slots around this solution.
Problem specific information enables search to seek out several local optima quickly and efficiently.

17. The Examination Timetabling Problem
The development of TISSUE
The Intelligent Scheduling System for University Exams

18. The problem. Given A set of k time slots A set of n exams
Other resources (rooms, invigilators etc)
A list of candidates for each exam
A list of requirements (including time windows) for each exam
Find
An allocation of exams to time-slots
Such that
No student is timetabled to sit 2 exams at the same time
The number of back-to-back exams (known as second order conflict) is minimised
Other secondary objectives / constraints are taken into account.

19. Related model.
If the only objectives are to avoid clashes and meet the time-window constraints then the problem is a graph colouring problem.
Vertices = set of exams  set of timeslots
Edges connect:
All pairs of timeslot vertices
Pairs of exam vertices with a student in common
Exam vertices to timeslot vertices at infeasible times
Objective: find a feasible colouring of G(V,E) in k or less colours.

20. A simulated annealing approach.
Local search framework
Solution space: allocation of exams to time-slots subject to capacity constraints
Neighbourhood move: move one exam
Cost: weighted cost of student clashes, second order-conflict + other objectives.
Problem:
Failure to converge to good feasible solutions.

21. Lessons from previous experience.
Solve in phases.
Find feasible solution first.
Then search space of feasible solutions.
Reasonable results.
Sampling.
Natural sampling: select exam, then slot.
Other possibilities:
select ‘from’ slot, then exam, then ‘to’ slot
Select ‘from’ slot, then ‘to’ slot, then exam.
140
120
Cost
100 80 60 40 20 1 2 3
Sampling policy

22. Problem. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Space of feasible solution disconnected (or very sparsely connected)

23. Solution.
Paper on local search for graph colouring found improved results with Kempe chain neighbourhoods.
Given a graph and a feasible colouring (i-j)-Kempe chains are the connected components in the sub-graphs induced by the vertices coloured i and j.
Swapping the colours in such a chain will preserve the feasibility of the colouring.
Comparison of natural and Kempe chain neighbourhoods
(the numbers on the bars indicate the percentage savings)
36.17
23.76
35.19
42.80
30.89
46.77
36.50
20.59
Data sets
second order costs
Natural
Kempe
500
1000
1500
2000
2500

24. Is it robust? Used to schedule exams at Swansea since 1992
Produces good solutions on a range of bench-mark data
Works well with different objectives and constraints.

25. The development of CARE Computer Aided Rostering Environment
The Nurse Rostering Problem
The development of CARE
Computer Aided Rostering Environment

26. The problem.
Objective: to produce weekly schedules of work for all nurses on each ward so that:
minimum covering requirements are met
nurses’ preferences and requests are considered
schedules are deemed to be fair
Detail:
The day is made up of 3 shifts: hour day shifts (earlies and lates) and a longer night shift.
Nurses work either days or nights in a given week with more days than nights comprising a week’s work.
Different contracts give different numbers of days/nights worked e.g. (5,4), (4,3), (3,3).
Covering requirements are given cumulatively for 3 grade-bands.
Nurses can submit requests for days/nights on/off.
Bank nurses can be used if cover cannot be met without.
After several consultations with the hospital we were able to put numeric values on the different non-binding constraints / secondary objectives.

27. Solve in 3 phases. Phase 1: determine the number of bank nurses.
Solve as knapsack problem:
Maximise the number of day shifts available subject to the constraint that the number of night-shifts sacrificed is sufficiently low.
Phase 2: allocate nurses to to day/night/day-off for each day.
See later.
Phase 3: allocate nurses on days to ‘earlies’ and ‘lates’.
Solve as network flow problem:
Nodes represent the nurses and days/grades. Flow variables represent working ‘earlies’. Appropriate costs and bounds ensure preference costs are optimised subject to nurses’ availability and cover constraints.

28. A local search approach to phase 2.
Work with shift patterns and solve in phases.
Phase 2.1: Find a feasible solution
Phase 2.2: Search space of feasible solutions
Solution space: allocation of nurses to feasible shift patterns
Neighbourhood move: change pattern of a single nurse
Cost: total shortfall in cover or sum of preference costs depending on phase.
BUT

29. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Both phases: valleys that are difficult to escape ? ?
Both phases: large plateau-like areas
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Phase 2.2: disconnected solution space

30. Solutions.
In phase 2.1:
Some deep valleys due to infeasible day-night mix – restrict moves to those that correct this.
Negotiate plateaux and cross mountains using chains of moves – good chains correspond to shortest paths in small graphs.
In phase 2.2:
Mountains, plateaux and some connection problems solved by extending a. and b. above to finding negative cost circuits so that cover is unchanged and preference cost improves.
Remaining connection problems avoided by crossing back into infeasible region and repeatedly oscillating through phases 2.1 and 2.2.
In order to ensure component likely to give good solutions, cost in phase 2.1 includes a preference element.
Search carried out as an aggressive TS with tabu lists and diversification strategies based on day/night mix

31. Is it robust? Changes to specification.
Addition of nurses working both days and nights in the same week.
Spreading of over-cover
Constraints on number of highly paid nurses working weekends
Introduction of team working.
All incorporated with minor changes due to ability to seek out many good local optima.

32. analysing the physical problem
Conclusion.
‘A general algorithm is like a size 48 coat. It can cover everybody but it doesn’t fit most people very well.’
And
analysing the physical problem
Researching the underlying model(s)
A made-to-measure solution.
Potential benefits
Better quality solutions
Obtained more quickly
and more consistently
Can also prove robust.

33. Social life Travel Reading widely Talks / seminars
(talk available at

34. References (talk available at www.goweralg.co.uk/talks/)
The talk is based on:
K.A Dowsland,’Off-the-peg or made to measure: timetabling and scheduling with SA and TS' in Practice and Theory of Automated Timetabling II (Burke, E. and Carter, M. eds) LNCS 1408, (1997 )
The laboratory scheduling problem.
K.A.Dowsland, ‘Using simulated annealing for efficient allocation of students to practical classes’ in Applied Simulated Annealing (Vidal R.V.V. ed.) LNEMS 396, (1993) Springer.
The examination scheduling problem.
J.M. Thompson & K.A. Dowsland ‘Variants of simulated annealing for the examination scheduling problem’ Annals of Operations Research 63 (1996)
J.M. Thompson & K.A. Dowsland ‘General cooling schedules for a simulated annealing based timetabling system’ in Practice and Theory of Automated Timetabling, (Burke E. and Ross, P. eds.), LNCS 1153 (1996) , Springer.
J.M. Thompson and K.A. Dowsland ‘A robust simulated annealing based examination timetabling system’ Comput. & Opns. Res. 25 (1998)
The nurse scheduling problem.
K.A. Dowsland ‘Nurse scheduling with tabu search and strategic oscillation’ EJOR 106 (1998)
K.A. Dowsland & JM Thompson ‘Solving a nurse scheduling problem with knapsacks, networks and tabu search’ J. Opl. Res. Soc. 51 (2000)
(talk available at