Don’t lose it – use it!
(using problem specific information in the solution of
real-life combinatorial optimisation problems).
Kath Dowsland
Gower Optimal Algorithms Ltd
& ASAP Research Group, University of Nottingham.
[email protected]
http://www.goweralg.co.uk/talks/
Models
Problem
?
Solution
In an ideal
world!
Algorithm
Lots of published results
Models
Either
Problem
?
OR
In the real
world!
Problem
Lots of published results
?
Largest
problem
solved
Result.
Lots of published results
BUT:
We can still use information from the model in conjunction with that from the
physical problem.
Outline
• Ways in which information from the model/problem can help
– For any solution approach
– For branch and bound
– For heuristic search
• Real examples
– When the model fits but the problem is large
• The pallet loading problem
• A shipper rationalisation problem
– When the model does not fit
• Examination scheduling
• A laboratory scheduling problem
– Exploiting data characteristics
• Nurse rostering.
Any solution approach
Reduce solution space
Speed up calculations by preprocessing.
Identify special cases.
Split into sub-problems
Day 4:
Days 2-3:
Day 1:
With branch and bound
Order the branches
1
5
?
4
2
3
6
Tighten the bounds

With heuristic search
Smooth the landscape
Diversify intelligently

Improve neighbourhoods.

Search large neighbourhoods efficiently
Highest point in
Cumbria
When the model fits but typical problems are
large.
Pallet loading problem.
Given a pallet and box of known size maximise the number of boxes that
can be packed orthogonally onto the pallet.
Collate
Reduces to solving many
problems of the form:
Given a small a x b rectangle
and a larger X x Y rectangle
optimise the number of copies
of the a x b rectangle that can be
fitted orthogonally in the X x Y
rectangle.
Stable?
Clampable?
Which design?
We will denote a problem with
dimensions X,Y,a and b by
P(X,Y,a,b)
Graph theoretic model
(clique or independent set).
V = set of potential box positions (x,y,r) where r = orientation.
(vi,vj)  E if boxes placed in positions i and j don’t overlap.
Optimal solutions  maximum cliques.
BUT
|V| = (X+1-a)(Y+1-b) + (X+1-b)(Y+1-a) > 1,000,000
Best algorithms at the time: enhancements to Bron and Kerbosch
branch and bound approach for 200-300 vertices max.
A useful property
Let E(S,a,b) = {(n,m): n,m  ℕ, na + mb  S < na + (m+1)b}
We call E(S,a,b) the set of efficient partitions of S in a and b.
Property.
The efficient partitions define a set of equivalence classes on
the set of pallet loading problems in that all problems with the
same set of efficient partitions of both X and Y in a and b have
the same set of feasible, and hence the same set of optimal,
solutions.
Reductions.
1.
Let Q{X,Y,a,b} be the set of problems in the same equivalence class
as P(X,Y,a,b)
Select the problem from Q{X,Y,a,b} that minimises |V|
RESULT: <18,000 vertices
2.
Restrict set of feasible positions to those at distances
d = na + mb,where n,mℕ, from the nearest pallet edge.
RESULT: < 700 vertices.
REMARK: this is equivalent to the following max clique reduction
rule:
If  (v,w) E such that (v)  (w) discard v. Computing the
reduction directly is obviously far more efficient.
Performance on 1000 randomly generated problems.
Problem size:
mean
max.
min
|V|
186
580
10
Density
86%
98.64
54%
Number out of 1000 randomly
generated problems solving in
reasonable time.
0
300
Performance of clique algorithm at
different densities
|V|
10 seconds
100 seconds
200 seconds
60%
200
70%
80%
150
1
Re-ordering the branches
100
1
0
700
0
500
1000 1500 2000 2500 3000
Solution time (secs)
A global upper bound.
1. Area based bound:
 XY 
 ab 
2. Number theoretic bound due to Barnes
Quality: Approx 20% match with good heuristic
Improvement:
•
Find dimensions that minimise area based bound over Q{X,Y,a,b}
(reduces to minimising a/b within a known interval)
•
Apply Barnes’ bound to this problem
Quality: > 90% match with good heuristic
Result: bound good enough to use as target.
Local upper bound.
The natural ordering allows us to calculate a local lower bound bound based on
the waste in the current partial layout.
1
2
3
4
5
6
7
8
11
12
9
10
13
14
15
16
21
17
18
19
20
22
As well as actual waste we can define ‘future’ waste
Additional bounds avoid reflections/rotations of previous solutions.
Result.
•
•
•
•
98% of problems solve to proven optimality within time-limit
> 90% solve within 10 seconds (on mid 1980’s pc)
Some fail because bound is slack
Others are computationally expensive with right bound.
Solution.
•
Pre-process difficult equivalence classes and store indexed appropriately.
Shipper rationalisation problem.
Given
•
•
•
A set of product (boxes) of known dimensions
Annual demand for each product
A number, p, of shippers required
Find
•
A set of p shippers for storage/transportation such that the total annual volume
of shippers used is minimised.
A p-median model
Where:
n m
min   cij xij
xij = 1 if product i is allocated to shipper j
i 1 j 1
n
 xij  1
st
j 1
xij  y j
i
i, j
(1)
( 2)
m
 yj  p
j 1
xij , yi binary i, j
(3)
yj = 1 if a shipper of size i is used
cij is the annual shipper volume required if product i is
allocated to shipper j
V j * di
cij 
f ij
Vj is the volume of shipper j
di is the annual demand for product i
fij is the number of product i that can be fitted into shipper j.
BUT
•
•
Calculation of cij implies the solution of a pallet loading problem for each
shipper/product pair
Data to nearest mm  large m (>109)
Using information from the problem
Pre-processing cost part of cost calculation.
• For each product identify one shipper base from each equivalence class.
• Solve and store results indexed appropriately.
Result: mean time for 20,000 local search iterations for a 46 product problem
over 5 runs for p = 3, 6 and 9 reduced from 481.4 secs to 32.97 secs.
Reducing the size of the solution space.
• Reduce set of potential base dimensions to integer combinations of lengths and
widths of some product.
• Reduce potential heights to integer multiplesof product heights.
Result: m reduced from 1.25 x 1011 to 1.25 x 108
 solution space reduction by factor of approx 103p.
A fast post-processor.
• Reduce each shipper so that each of its dimensions is a linear combination of
one of the products allocated to it.
Local search and neighbourhood size.
Local search based on Teitz and Bart.
Natural neighbourhood:
• Swap a shipper for one of different dimensions.
Problem:
• Very large neighbourhoods
Solutions:
• Change one or 2 dimensions at a time
• Limit size of change
Result: based on p = 10
original neighbourhood size: 1.25 x 109
neighbourhood based on changing 1 dimensions: 1.5 x 104
Final implementation.
Reduced solution space / neighbourhoods small enough to extend local search.
• Not clear which is best
• Solution space reduction raises questions of sampling strategy
Use a simulated annealing heuristic with several neighbourhoods and select
current neighbourhood / sampling rule using hyper heuristic.
Performance of shipper rationalisation algorithm on 3 data sets.
100
98
96
% utilisation
94
92
original
90
noise
88
combined
86
84
82
80
3
4
5
6
7
8
no. shippers
9
10
11
12
What if the model does not fit?
Examination scheduling.
Given a set of exams and a set of constraints including the number of sessions
available, number of desks available, time-window restrictions etc find a feasible
timetable that maximises timetable quality (e.g minimises some measure of backto-back, same day or other frequencies of exams.)
Graph colouring model:
V= set of exams
(vi,vj) E if exams i and j cannot take place at the same time.
Feasible colourings  clash-free timetables.
BUT
Graph colouring: find feasible colouring in minimum no. colours
Exam scheduling: find feasible colouring in q colours such that
•
quality objectives are optimised
•
additional constraints are satisfied.
Using information from the model.
Graph colouring heuristics are frequently used for constructive approaches to time-tabling
problems.
An SA approach:
•
Find a clash-free timetable by solving colouring problem by appropriate means.
•
Search space of clash-free solutions wrt quality objective / additional constraints
Problem: space now sparsely connected.
Solution: a review of the literature revealed an SA approach to graph-colouring using
neighbourhoods based on Kempe-chains.
Comparison of natural and Kempe chain neighbourhoods
(the numbers on the bars indicate the percentage savings)
second order costs
2500
42.80
2000
46.77
Natural
36.50
1500
Kempe
30.89
1000
500
36.17
23.76 35.19
20.59
Data sets
Laboratory scheduling.
Given: a laboratory of known capacity, a set of timeslots when the laboratory is
available and a set of students together with the times when each is free to attend,
Find: an allocation of each student to a suitable timeslot in order to minimise the
number of sessions utilised.
Secondary objectives vary from year to year and include:
• including student preferences
• uniform utilisation of used sessions
• preference ordering of sessions
• minimising sessions used by a given subset of students
Related models.
•
•
•
If we ignore capacity we have a set covering problem.
If we can use any number of sessions (or know which sessions to use) we have
a transportation problem
If we know the number of sessions required we have a capacitated p-median
problem.
A local search approach
Solution space: allocations of students to a session they are free to attend
Neighbourhood move: move a student to another session
Cost: main term to encourage full or empty time-slots + weighted terms for other
objectives. (This assumes p = lower bound)
Problems:
• convergence to infeasible sets of p slots
• don’t know if p is big enough
Solution from underlying models:
• Use local search to identify promising sets of p slots
• Identify covering constraints wrt uncovered students and use to:
• Penalise all solutions not satisfying constraint(s)
• Build (small) set covering problem to check for feasibilty
• If all covered optimise allocation using transportation model.
Further penalties ensure no return to previous solutions and p is increased when
necessary by decreasing capacity.
Result: good solutions in < 1 minute run time.
Exploiting data characteristics.
Nurse scheduling.
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
The day is made up of 3 shifts: earlies, lates and nights.
Nurses work either days or nights in a given week with more days than nights
comprising a weeks work. e.g. (5,4), (4,3), (3,3).
Nurses can submit requests for days/nights on/off.
For each nurse we can define a set of feasible weekly work patterns and allocate a
numeric score to each.
Model (multiple choice covering problem).
n
min z    pij xij (1)
i 1 jF ( i )
s.t.  xij  1 i (2)
j F ( i )
  a jk xij  R(k , r ) r , k (3)
iGr jF ( i )
xij  0 or 1 (4)
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.
Partitioning the problem.
Data characteristic: Contributions to penalty of different magnitudes of scale with
P(bank nurses) >> P(days/nights worked) >> P(early/late)
Partition problem into 3 sub-problems.
1) Determine the number of bank nurses required using a small knapsack
model.
2) Allocate nurses to days, nights, and days-off to minimise penalty while
maintaining cover (see later)
3) Allocate those on days to earlies or lates using a network flow model.
Result:
• Feasibility guaranteed.
• Size of problem reduced e.g. no. variables per full time nurse reduced from 707 to
56.
• Remaining problem is solved using tabu search with strategic oscillation using
neighbourhood moves that change the shift pattern of a single nurse. But the
solution landscape can exhibit several problems:
• Hills separating areas of solution space
• Flat-bottomed valleys
• Sparse connectedness.
Improving the tabu search.
Data characteristic: requests form a significant component of P(days/nights worked).
 moves based on moving just 1 shift may have significant effect on penalty cost.
Therefore:
Incorporate larger neighbourhoods based on chains of moves involving changing 1
shift for several nurses. Good moves can be found using shortest path routines.
Data characteristic: allocations to days/nights can be critical to feasibility.
Therefore:
• include a swap neighbourhood to improve day/night balance
• use a tabu list to ensure that the search stabilises in a new day/night allocation before
reversing it
• use a frequency based diversification to move to new day/night allocations after a
given number of iterations.
Result:
Search consistently finds optimal solutions (as confirmed by IP solution) for
52 real data sets in a few seconds computing time.
Conclusion
‘A general algorithm is like a size 48 coat. It can cover everybody but it
doesn’t fit most people very well.’
And the physical
problem
Research the
underlying model(s)
A made-to-measure solution.
Potential benefits
•
Better quality solutions
•
Obtained more quickly
•
and more consistently
References
The pallet loading problem.
K.A. Dowsland ‘A graph-theoretic approach to a pallet loading problem’ NZOR 13 77-86 (1985)
K.A. Dowsland ‘Determining an upper bound for a class of rectangle packing problems’ Comput & Opns Res
12 (1985) 201-205
K.A. Dowsland ‘A combined data-base and algorithmic approach to the pallet loading problem’ A J. Opl.
Res. Soc 38 (1987) 441-345
K.A. Dowsland. ‘An exact algorithm for the pallet loading problem’ EJOR 31 (1987) 78-84
The shipper rationalisation problem.
K.A. Dowsland, E. Soubeigaa & E.K. Burke ‘Solving a shipper rationalisation problem with a simulated
annealing based hyperheuristic.’ Submitted to EJOR (2004)
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) 105-128
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)
345-363, Springer.
J.M. Thompson and K.A. Dowsland ‘A robust simulated annealing based examination timetabling system’
Comput. & Opns. Res. 25 (1998) 637-648
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) 125-150 Springer.
The nurse scheduling problem.
K.A. Dowsland ‘Nurse scheduling with tabu search and strategic oscillation’ EJOR 106 (1998) 393-407
K.A. Dowsland & JM Thompson ‘Solving a nurse scheduling problem with knapsacks, networks and tabu
search’ J. Opl. Res. Soc. 51 (2000) 825-833
http://www.goweralg.co.uk/talks/