Ideas for the Transportation Problem

1
Ideas for the Transportation Problem

• Although the textbooks initialization procedure
  has some flaws (which?), it can be reused to
  define sophisticated mutation operators and
  better initialization procedures (by taking a
  random number in 0,min instead of the minimum).
  Moreover, it can be used to develop randomized
  hill climbing style systems.
• If M1 and M2 are legal solutions, aM1 bM2
  (with a,bgt0 ab1) are also legal solutions. This
  provides as with a quite natural crossover
  operator. This operator is called arithmetical
  crossover in the EC numerical optimization
  literature.
• Boundary Mutation (that sets the value of one
  (possibly more) elements of the matrix to its
  minimum (0) or maximum possible value
  (min(source(i), dest(j))), might also have some
  merit.

2
Some Initial Thoughts on the Course Project

• Have a general theme
• Compare at least 2 approaches (could be similar)
• Run algorithms at least 3 times (you might just
  be unlucky)
• Report the results of running the benchmark
  transparently and completely
• Interpret your results (even if there is no clear
  evidence pointing into one direct) explain your
  results (could be speculative)
• Report the history of the project
• Be prepared to demo your program shortly after
  the due date.
• What was expected, what was unexpected?

3
Conducting Experiments in General and in the
Context of the Transportation Problem

• Things to observe when running an EC-system
• Average fitness
• Best solution found so far
• Diversity in the current population (expensive)
• Degree of change from generation to generation
• Visualizing the current best solutions could be
  helpful
• Size of searched solutions building blocks in
  the searched solutions
• Complexity runtime, storage, number of genetic
  operators applied,
• What parts of the search space are searched (hard
  to analyze)
• Things to report when summarizing experiments
• Experimental Environment Operators used and
  probabilities of their application, selection
  method, population size, best found solution,
  best average fitness.
• Observed Results Best solution found, best
  fitness/average fitness over time, diversity over
  time.

4
Requirements for TSP-Crossover Operators

• Edges that occur in both parents should not be
  lost.
• Introducing new edges that do not occur in any
  parent should be avoided.
• Producing offspring that are very similar to one
  of the parents but do not have any similarities
  with the other parent should be avoided.
• It is desirable that the crossover operator is
  complete in the sense that all possible
  combinations of the features occuring in the two
  parents can be obtained by a single or a sequence
  of crossover operations.
• The computational complexity of the crossover
  operator should be low.

ER
DR, TD
5
Donor-Receiver-Crossover (DR)

• 1) Take a path of significant length (e.g.
  between 1/4 and 1/2 of the chromosome length)
  from one parent called the donor this path will
  be expanded by mostly receiving edges from the
  other parent, called the receiver.
• 2) Complete the selected donor path giving
  preference to higher priority completions
• P1 add edges from the receiver at the end of the
  current path.
• P2 add edges from the receiver at the beginning
  of the current path.
• P3 add edges from the donor at the end of the
  current path.
• P4 add edges from the donor at the start of the
  current path.
• P5 add an edge including an unassigned city at
  the end of the path.
• The basic idea for this class of operator has
  been introduced by Muehlenbein.

6
Top-Down Edge Preserving Crossovers (TD)

• 1) Take all edges that occur in both parents.
• 2) Take legal edges from one parent alternating
  between parents, as long as
• possible.
• 3) Add edges with cities that are still missing.
• Michalewicz matrix crossover and many other
  crossover operators employ this scheme.

7
Typical TSP Mutation Operators

• Inversion (like standard inversion)
• Insertion (selects a city and inserts it a a
  random place)
• Displacement (selects a subtour and inserts it at
  a random place)
• Reciprocal Exchange (swaps two cities)
• Examples
• inversion transforms 123456789 into 127654389
• insertion transform 1gt23456789 into 134567289
• displacement transforms 1gt23456789 into
  156782349
• reciprocal exchange transforms 1gt23456gt789 into
  173456289

8
An Evolution Strategy Approach to TSP

• advocated by Baeck and Schwefel.
• idea solutions of a particular TSP-problem are
  represented by a real-valued vectors, from which
  a path is computed by ordering the numbers in the
  vector obtaining a sequence of positions.
• Example v???????????????????????????????respesen
  ts the sequence
• ???????????
• Traditional ES-operators are employed to conduct
  the seach for the best solution.

9
Non-GA Approaches for the TSP

• Greedy Algorithms
• Start with one city completing the path by adding
  the cheapest edge at he beginning or at the end..
• Start with ngt1 cities completing one path by
  adding the cheapest edge until all cities are
  included merge the obtained sub-routes.
• Local Optimizations
• Apply 2/3/4/5/... edge optimizations to a
  complete solution as long as they are
  beneficiary.
• Apply 1/2/3/4/.. step replacements to a complete
  solution as long as a better solution is
  obtained.
• ... (many other possibilities)
• Most approaches employ a hill-climbing style
  search strategy with mutation-style operators.

10
Hillis Sorting Networks Coevolution

• The presented material is taken from Melanie
  Mitchells textbook pages 19-25.
• Sorting Networks
• they employ the following basic operation
• OP(i,j) compare i-th and j-th
  element and swap if out of order.
• have been designed for a particular integer n
  (e.g. n16)
• our discussion rely on a particular sorting
  scheme Batcher-SortKnuth 1973
• theoretical problem find a network with the
  minimum number of comparisons for sorting a set
  of integers of cardinality n.
• significant efforts were spend on finding the
  optimal network for n16
• In 1962, Bose/Nelson employed general methodology
  to achieve 65 comparisons.
• Knuth/Floyd 63 comparisons in 1964.
• Shapiro reduced it to 62 in 1969.
• Green reduced it further to 60 in the early 70s
  --- no proof of optimality was given
• Hillis took up the challenge of finding better
  networks in 1990, relying on an EP approach.

11
Hillis EP-Appoach to the n16 Sorting Problem

• Chromosomal representation sorting networks were
  represented as sequences of integers --- then
  determine parallelism in the specified sequence
  of operations to derive the complete sorting
  network.
• The sequence length of solutions ranged between
  60 and 120 (cannot find better solutions than 60
  comparisons).
• Hillis employed diploid chromomes and his
  crossover operator employed techniques that
  resemble natural reproduction in biological
  systems.
• Initial population consisted of a randomly
  generated set of strings.
• Fitness was defined as the percentage of cases
  the network sorted correctly based on random
  samples of testcases.
• Solutions were placed in a 2-dimensional lattice
  restricting breeding to individuals that are not
  too far from each other. The less fitter half of
  a population was replaced by individuals obtained
  by breeding the top half of the popolation.
  Mutation was applied to an individual with
  probability 0.001.

12
Hillis EP-Approach (continued)

• Hillis appraoch only found moderately good
  solutions with 65 and more comparisons.
• Hillis employed coevolution to obtain better
  results.
• not only the algorithms but also the testing
  examples were evolved.
• fitness of test cases was measured by number of
  failures the testcase caused in the population of
  networks.
• classical genetic operators were employed to
  evolve the test-cases
• The author claims that the new appoach resulted
  in the discovery of a network that needs 61
  comparisons for n16 a significant improvement,
  but still frustrating considering Greens
  solution that requires only 60 comparisons.

13
Other Examples of Coevolution

• In games with diverse roles (e.g. hunters and
  escapees) were strategies of each role is defined
  by its performance against strategies of the
  other group.
• Evolving the architecture as well as solutions
  under this architecture (eultural algorithms).
• Evolving main programs as well as subprogramms,
  as it is the case in the ADF approach.
• Evolving local decision makers as well as global
  decision makers that combine the evidence of
  local decision makers (similar to the
  meta-learning approach).
• Simulating preditor/prey systems in biology.
• Evolving complex objects that are decomposed of
  objects of different types for example the best
  software team that is decomposed of programmers,
  managers, secretaries,... Fitness might be
  defined how well these different objects
  cooperate.
• Simulation of sexual preferences and mating
  behavior.
• Evolving rule-sets as well as rules inside a
  rule-set.

14
Remarks Project Distance Preserving Mappings

• If you have empirical results, explain clearly
  how do you mearsure performance. Integrate tables
  with the text. It was almost impossible to
  compare different results from different
  students.
• Quality of reports varied significantly size of
  the projects varied significantly.
• Topics that were explored included
• impact of different coding schemes on the
  performance.
• run the system for lower order dimensions and use
  learnt solutions for initial population and other
  purposes for higher order solutions.
• analysis on how to speed up the EP approach.
• a genetic programming with somewhat restricted
  tree structures (solutions do not seem to be much
  worse than those obtained with EP, but slowness
  of the GP appraoch seems to be serious weakness
  of the appraoch).
• experimentation with various mutation rates
• influence of scaling (error almost doubles from
  15 to 135 (e.g. 0.041 to 0.078))
• equilibrium search versus GA search (reducing the
  number of variables in a GA-system)

15
Reverse Initialization Algorithm

• Let row(I) the sum of elements in the I-th row
• Let col(j) the sum of the elements in the j-th
  column
• Let sour(I) the sum of the supplies of the I-th
  row
• Let dest(j) the demand for the j-th colums
• Let dest(j)ltcol(j) and source(I)ltrow(I) for
  I,j
• Visit the matrix elements (possibly excluding
  some elements) in some randomly selected order
  and do the following with the visited element vij
  with its current value v
• maxred min(col(j)-dist(j), row(i)-sour(i))
• r min(v, maxred)
• vijvij-r row(i)row(i)-r col(j)col(j)

16
Boundary Mutation

• 2 2 0 0 0 2
  6
• 1 1 2 4 0 2 0
  6
• 1 0 0 3 2 0
  6
• 0 1 2 0 1 0 2
  6
• Boundary Mutation
• Selection an element of the matrix
• Set it to its maximum possible value (4 in the
  example)
• Rerun a reverse initialization algorithm (the
  normal initialization algorithm) that reduces the
  elements of a matrix until the source and
  destination amounts are correct.