introduction to multicriteria optimisation

1
introduction to multicriteria optimisation

• previous slides have considered scalar objective
  functions, i.e. those that return a single
  (nominally real-valued) indicator of merit
• mutlicriteria optimisation (also referred to as
  multi-objective and vector evaluated) considers
  problems having more than one evaluation criteria
• typical of many engineering design problems
• for example, consider optimising an engine design
  based on criteria of acceleration and fuel
  consumption
• searching for a solution that maximises
  acceleration AND minimises fuel consumption
• there may be one single best solution, i.e. one
  that offers both the most acceleration and the
  least fuel consumption
• the fitness function can combine the criteria,
  e.g. by summation or multiplication
• solution can be readily found using a standard
  genetic algorithm
• it is more likely, however, that some solutions
  will offer better acceleration but worse fuel
  consumption and vice versa
• the optimal trade-off is generally to be selected
  by an expert user
• the objective of the search algorithm is to find
  a set of good solutions for the user to choose
  from

2
comparing fitness

• genetic algorithms are driven by bias in
  selection and/or replacement
• this requires the ability to compare absolute or
  relative fitness
• e.g. roulette wheel or tournament selection
• how does this work when there are more than one
  criteria?
• some solutions may be poor on all criteria
• others may be better on one criterion and worse
  on another

3
the optimal set

• individual A is said to dominate individual B
  when
• for all evaluation criteria A is at least as good
  as B
• for at least one criterion A is better than B
• the objective is to find all (or other
  satisfactory or feasible quantity) of
  non-dominated solutions
• often referred to as the Pareto-optimal set
• in continuous spaces this set may be infinite
• the number of solutions found will be limited by
  the population size
• a simple solution is to convert the multiple
  criteria to a single value
• for example, weights, wi, could be applied to
  each criterion to determine their relative
  importance to the application
• a standard genetic algorithm can then tackle the
  problem
• difficult to identify suitable values for the
  weights (implies considerable domain knowledge)
  and unexpected good solutions can be missed
• multiple runs could be made with different
  weights to find a set of solutions

4
simultaneous search

• because genetic algorithms are population based,
  they offer the possibility to simultaneously
  search for multiple elements of the optimal set
• no assumptions need to be made about the relative
  importance of criteria or about the properties of
  the Pareto-optimal front
• population size is a critical parameter to ensure
  satisfactory coverage
• algorithms adopt a range of measures including
• modifications to selection, typically based on
  dominance
• techniques to preserve diversity - the usual
  convergence of the population to a single
  solution is clearly a bad thing when a set of
  solutions is required
• preservation of dominant solutions discovered
• VEGA (vector evaluated genetic algorithm) is an
  early example
• divides the population into n sets
• selection within each set is based on a different
  criterion
• reproduction is carried out across the whole
  population
• tendency towards extreme solutions, i.e. those
  with best performance on individual criteria
  rather than good performance across all criteria
• no measures taken to preserve diversity or
  dominant solutions discovered
• the majority of algorithms now adopt some form of
  Pareto-ranking based on the number of solutions
  an individual dominates or is dominated by

5

• MOGA is an early example
• all non-dominated solutions are assigned a rank
  of 1
• all dominated solutions are assigned a rank based
  on the number of solutions to which they are
  inferior
• the population is sorted according to rank and
  fitness values assigned by interpolation, with
  scores averaged for individuals of the same rank
• most approaches also incorporate some form of
  niching technique to maintain diversity across
  the Pareto front
• the solution set is flat in these problems so
  there is no need to maintain stable
  sub-populations
• the population size is finite and generally small
  compared with the Pareto front so schemes that
  spread the population out across this front are
  useful
• different approaches are possible, but fitness
  sharing schemes and nearest neighbour density
  estimates are popular
• good solutions are often retained by elitist
  population structures
• alternatives include the use of an archive of
  know solutions
• the archive might simply provide a record of past
  solutions
• alternatively solutions from the archive could
  form part of the selection pool
• the archive typically contains the current
  approximation of the Pareto-optimal set, i.e. all
  non-dominated solutions
• some approaches exploit properties of
  coarse/fine-grained parallel models
• for example, local selection in fine grained
  models means that the Pareto-rank does not have
  to computed for the entire population, just the
  local neighbourhood