A Representation for Permutation
Optimization with a
Combinatorial Genetic Algorithm
Robert E. Smith 
Dan Holtkamp 
Engineering Science and Mechanics 
HewlettPackard 
The 


Email: dgh@hpfitst2.fc.hp.com 


Email: rob@comec4.mh.ua.edu 

Abstract
This paper presents a
way of representing problems of sequence in a traditional, combinatorial
genetic algorithm (GA). This representation differs from socalled orderbased
(or permutational) GA representations. A combinatorial representation is
advantageous, given that orderbased GAs are complex, and have limited
analytical and empirical success when compared to combinatorial GAs. The
representation involves letting the genetic algorithm specify a rank for each
element of the problem, and then sequencing the elements according to the
rankings. Ties in rank are resolved either at random or by a problem specific
(knowledge augmented) operator. Promising results are presented using this sort
of representation for traveling salesperson problems. Extension of this sort of
representation to other problems is also discussed.
Introduction
In 1985, Goldberg and
Lingle were examining the problem of adjusting linkages between bits in genetic
algorithm (GA) representations. Usually, GAs are combinatorial optimizers that are based on the mechanics of natural
genetics. In their exploration, Goldberg and Lingle developed a method for
rearranging bits in a GA through a crossoverlike operator. Their efforts to
test this method led to a new area of application for GAs: problems of sequence
or permutation problems. The resulting GAs are often called orderbased GAs.
This development, and
the popularity of certain NPcomplete permutation problems (chiefly the traveling salesperson problem or TSP, which Goldberg and Lingle used as a
test problem), led to a flurry of developments in orderbased GAs. Many direct
representations of permutations, with special crossover operators (like
Goldberg and Lingle’s PMX (1985)) that preserved valid permutations (Oliver,
Smith & Holland, 1987). Some used combinatorial representations, with
specialized and often complex decoding and constraint satisfaction operators to
insure valid permutations (Starkweather, McDaniel, Mathias, Whitley &
Whitley, 1991).
Orderbased GAs have met
with mixed success when compared to GAs applied to combinatorial problems.
Although theory on their operation exists, it is somewhat less firm than the
theory for combinatorial GAs. This paper presents a new, simple representation
that allows a combinatorial GA to operate on a permutation problem in a natural
way.
Combinatorial
Versus Permutational GAs
In a GA, one must
combine different possible solutions as a part of the search process. The
typical GA recombination operator is crossover, where half of one solution is
spliced to half of another. For instance, given two bits strings that represent
solutions to a problem:
1 0 1 0 1 0 1
0 1 1 0 1 1 0
One could split the
strings at a randomly selected crossover point (say, between the second and
third bits), and create the strings
0 1 1 0 1 0 1
1 0 1 0 1 1 0
This example problem is
combinatorial: GA solutions are represented by combinations of bits. Theory and
empirical evidence suggest that this sort of GA is robust. That is, it is
effective across a broad class of problems (Goldberg, 1989, Holland, 1975).
To see how things differ
in a problem of sequence, consider the TSP. This problem is specified by a list
of city locations. The goal of the search is to find a route that visits each
city once and only once, and has minimum length.
A natural representation
for this problem is a list of city names, or, more generally, city numbers. For
instance, the string
1 2 4 3
could represent a tour
(or subtour) that visits city 1, followed by city 2, followed by city 4, and
finally visiting city 3.
Now consider two such
solutions:
1 2 4 3
2 3 4 1
If one attempts to cross
these strings at their midpoint, the following two strings result:
1 2 4 1
2 3 4 3
Neither of these tours
is valid, since they violate the restriction of visiting each city once and
only once.
To prevent this
quandary, new forms of GAs were developed. They used special crossover
operators and repair schemes to insure that no invalid tours were generated.
The results from these efforts have been mixed (Fox & McMahon, 1991;
Goldberg & Lingle, 1985; Oliver, Smith & Holland, 1987; Starkweather,
McDaniel, Mathias, Whitley & Whitley, 1991). Given their variety and
complexity, these techniques are not discussed in detail here.
A
New, Combinatorial Representation
Using permutational
representations in GAs leads to the need for complex operators to insure that
the GA generates valid solutions. Despite their complexity, these techniques
have met with mixed results. This section introduces a combinatorial
representation scheme for problems of sequence in GAs.
In the suggested
combinatorial representation, a binary string is used to describe a path of all
cities needing to be visited. This string is divided into n fields of length log_{2}(n)
each. Each field represents an integer rating for the associated city. The
GA constructs a tour by visiting the cities in an order from lowest rating to
highest rating. Ties are resolved at random.
As an example of this representation,
consider the following bit string for 4 cities in a 16 city TSP:
0101011110101000 ...
This gives city 1's
rating = 5, city 2's rating = 7, city 3's rating = 10, and city 4's rating = 8.
Given these ratings, the subsequence ordering city 1, city 2, city 4, and city3.
As
another example, consider the bit string
0111011110101000 …
which give city 1's
rating = 5, city 2's rating = 5, city 3's rating = 10, and city 4's rating = 8.
In this case, two subsequences are possible, depending on a random decision
between city 1 and city 2 for first place. In other words, the subsequence
city 1, city 2, city 4, and city 3, and the subsequence city 2, city 1, city
4, and city 3 are equally likely.
Note that this
representation does not give a onetoone mapping between encoded bit strings
and actual TSP tours. For most search techniques, this would hopelessly
complicate the search process. However, GAs search by selecting and recombining
building blocks based on emergent average fitness values. When the underlying
mapping between encodings and solutions is not onetoone, the GA will simply
average the effects on building blocks, and select accordingly. It is well
known that the combinatorial GA can handle a great deal of noise in its evaluation
function, and still perform well. The GA sees noise in the decoding process
(i.e., the random breaking of rating ties) simply as evaluation noise.
Therefore, if the GA can tie the space of ratings in this representation to
meaningful building blocks, it should be capable of constructing good
solutions.
One need not use a
purely random strategy for resolving ratings ties in this representation. For
instance, in the TSP, one resolve ties by simply picking the option with the
shortest local path. In problems other than the TSP, it may be necessary to use
more sophisticated methods to resolve ties and insure a valid permutation
solution.
The suggested
combinatorial representation can be easily extended to other problems of
sequence. For instance, consider a simple job shop scheduling problem. N jobs must be assigned to M machines. Any given job must spend a
certain amount of time of each machine, with possible constraints on the order
in which the machines a visited. One could construct an combinatorial
representation by assigning each job a ranking for each machine. Then, a
separate schedule builder could construct the complete job shop schedule by
ordering the jobs by rank, breaking ties either by random or by some
knowledgeaugmented method, and insuring that no job is assigned to a machine
before it is ready. In other words, each machine monitors for a set of jobs
that are ready for its processing, and selects from this set based on rank.
First
Experiments
In a preliminary
investigation, TSP experiments have been conducted with combinatorial
representation presented above. Both random and knowledge augmented
tiebreaking methods have been explored . Results from two separate problems
are presented here. The first problem uses the coordinates of the 48 capitals
of the continental US. This problem was
selected because a known optimum result exists. The x, y coordinates of the 48
capitals were provided by Gerhard Reinelt at the University of Augsburg and Bob
Bixby at Rice University. The second problem uses 32 cities with randomly
generated coordinates.
In the 48 capitals
problem, the knowledge augmented technique performs best. This is an intuitive
result, since one would think a “greedy” interpretation of the chromosome would
yield most accurate fitness values. Results from the 48 capitals run are shown
in Figure 1.
Figure 1:
Best capital city tour found by the GA.
The GA run that
generated these results used a population size of 1200, single point crossover
with probability of 0.8, and single point mutation with a probability of
0.001667. The results are within five percent of the known optima. This
solution is very similar to the known optima for this problem, which is given
in Figure 2.
Figure 1:
Optimal capital city tour.
Results on the 32 random
cities showed better performance when random tie breaking was used. This
counter intuitive result points out that being greedy may in fact bias the GA
search, rather than aiding it. Best results on the 32 random cities run are shown in Figure 3.
Figure 2:
Best tour through 32 random cities found by the GA.
The GA that generated
these results used a population size of 800, a crossover rate of 0.9, and a mutation rate of 0.00125.
Final
Comments
The results presented
here are certainly preliminary. They employed the simplest variety of
combinatorial GA, and the simplest form of fitness function (tour length).
However, they do emphasize an important point. The GA approach can handle
ambiguous representations, and generate nearoptimal results. Sometimes an
ambiguous (that is, not onetoone) representation will allow for a more
natural, and more easily encoded attempt at a GA solution. This allows
standard, wellresearched combinatorial GAs to be used in a broader class of
problems, including permutation problems like the TSP. Trials comparing
combinatorial and permutational GAs on more complex problems of sequence are an
important area for future investigation.
References
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