Operational Research: Making Things Better

1
Operational Research Making Things Better

• The Mathematics of Improvement
• Professor Colin Reeves
• Coventry University

2
Operational Research

• Began in WWII
• How best to use limited military resources
• Radar station location and operation
• Convoy protection
• ASW search patterns

3
Ideas of better

• Reduce cost
• Increase profit
• Reduce time
• Raise the probability of survival
• Save energy
• Reduce (average) error

4
Travelling Salesman Problem
5
TSP Good Solution
6
TSP-Bad Solution
7
TSP How Many Routes?

• 7 cities (including start point) 6!
  6x5x4x3x2x1720 alternative routes (if direction
  matters otherwise 360)
• In general (n-1)! routes, if direction matters
• E.g., 40!8.2x1047 (for comparison, no. of water
  molecules on Earth 3.3x1045)

8
Global Search

• List all possibilities and pick the best.
• OK for 7 cities (maybe up to 15 , about 1million
  million routes?)
• Clearly not OK for 40 cities
• Real TSPs thousands of cities
• Smart global search (e.g., branch-and-bound
  methods) not fundamentally any better

9
Local Search

• Compare neighbours of initial solution
• Replace with a better one
• Repeat until there is no better one
• TSP neighbours of C-F-A-E-G-B-D-C
• C-A-F-E-G-B-D-C
• C-F-E-A-G-B-D-C
• C-F-A-G-E-B-D-C
• C-F-A-E-B-G-D-C
• C-F-A-E-G-D-B-C

10
Local Search
11
Local Search Properties

• Neighbourhood size relatively small
• Number of steps normally also quite small
• Doesnt guarantee optimality just finds a
  local optimum
• Local optima are guaranteed to be better than
  average, thats all

12
Global Optimum
13
Local Optimum
14
Case Study

• Radiotherapy Treatment Planning
• Multiple beams send a stream of radioactive
  particles at a tumour
• Required to
• choose beam paths
• choose beam intensities
• In order to
• kill the tumour
• preserve any vital organs at risk
• reduce damage to healthy tissues

15
Radiotherapy Treatment Problem
16
Search Space

• 9 beams, e.g., 5 degrees in 1 degree steps
• 4 intensity levels for each beam
• 11 x 4 44 combinations for each beam
• 449 6.18 x 1014 combinations
• at 1 eval./sec, 20 million years of computing
  time

17
Optimization Approach

• Mathematical modelling of the intensity
  distribution
• Solution of the inverse problem by mixed
  continuous/discrete optimization methods (genetic
  algorithms)

18
Result
19
Genetic (or Evolutionary) Algorithms

• Start with a population of solution vectors
• Evaluate their fitness
• Select (some of) the best ones
• Breed new solutions
• Repeat until bored

20
Crossover

• Parents
• P1 a1 a2 a3 a4 a5 a6 a7
• P2 b1 b2 b3 b4 b5 b6 b7
• Children
• Mask 1 0 0 0 1 0 1
• C1 a1 b2 b3 b4 a5 b6 a7
• C2 b1 a2 a3 a4 b5 a6 b7

21
Mutation

• Usually carried out with small probability at
  each gene
• S a1 a2 a3 a4 a5 a6 a7
• Mask 0 0 1 0 0 0 1
• S' a1 a2 b3 a4 a5 a6 b7

22
GAs The Myth

• GAs are global optimizers
• because they avoid local optima
• because they use populations
• because they use crossover

23
Quotations

• The GA is easy to use and always finds the
  global optimum of a problem. (Oil Industry promo)
• The genetic algorithm has the ability to avoid
  being trapped in a local optimum, and is
  designed to locate the global optimum. (US Govt
  website)
• The GA is all but immune to some of the
  difficulties - false peaks, discontinuities, high
  dimensionality, etc. - that commonly attend
  complex problems. (Holland, 1986)
• Local search is a prescription for becoming
  trapped on a false peakGAs counter this by
  adopting a population approach. (Goldberg, 1989)

24
No Free Lunch

• On average, all methods have the same performance
• GAs are not exempt!
• Horses for courses? Are GAs better for some
  interesting class of problem instances?

25
GAs - The Maths

• GA is a dynamical system
• Population vector p
• p' G(p)
• G is an operator containing the effects of
  selection, crossover and mutation
• G transforms one population into a new one

26
Solution vector length 2Population size 2
27
Dynamical System Model
28
Dynamical System Iteration

• Suppose x is a number between 0 and 1
• A new x is generated by x' 2x(1-x)
• Try x0.3 x' 0.42
• Try x0.6 x' 0.48
• Try x0.5 x' ???
• x 0.5 is a fixed point
• Not all fixed points are stable (x0)

29
GA Fixed Points

• Vose-Wright Theorem gives general conditions for
  stable fixed points
• Vose-Wright Conjecture claims that stable fixed
  points for crossoveronly GAs are homogeneous
  populations
• Applies to large populations, N??

30
Implications of Vose/Wright Theorem
31
Further Implications

• Fixed points of a GA are equivalent to local
  optima
• Some local optima are not fixed points
• But remember large N real GAs have relatively
  small populations
• Bottom line real GAs cant avoid local optima,
  and may do even worse if populations arent large
  enough!

32
Are GAs Useless?

• GAs may do better than local search, but may do
  worse
• Need careful design and domain knowledge to work
  effectively
• Better for some applications than others
• Multiple objectives
• Computationally expensive or subjective
  objectives
• Few constraints

33
Reflections

• Why so many unguarded, unproven yet extravagant
  statements?
• Computation is easier than maths
• Rhetoric over rigour
• Funding pressures
• Story-telling for media consumption
• Narrative as science
• Lessons

34
Questions