TSP, knots are cities that have to be visited and edges
dimension the distance between two cities. The objec-
tive is to find the shortest round trip by visiting each
city exactly once. Both problems are very similar but
instead of our dynamic graph representation, the TSP
graph is static. Thus, the entire graph is traversable
every time and paths are fixed. Best performance and
efficiency for solving such represented minimization
problems are achieved by ant colony optimization. In
1991, the Italian mathematician Marco Dorigo pub-
lished the first ant algorithm for solving this problem
(Dorigo et al., 1991) (Dorigo and Stuetzle, 2004).
3 ANT COLONY OPTIMIZATION
Because the analogy of both problems, traveling
salesman and generation of process variants, we uti-
lize ant colony optimization. ACO is suitable for solv-
ing difficult discrete optimization problems that could
be described by a graph. Therefore, we use simple
agents, in this case artificial ants who communicate
with each other mediated by the environment.
Our previously discussed problems are classified
as static and dynamic combinatorial problems. TSP
is a static combinatorial problem, because the initially
given information cannot change. Against this, the
generation of process variants is a dynamic combi-
natorial problem. An example for changing informa-
tion are the intermediate products. Depending on the
preceding selected features, the resulting intermediate
product is quite different. In fact, shape aspects of one
feature could not be manufactured with a single oper-
ation because a particular machine is unavailable or
against this, a compound-feature that consists of two
or more atomic manufacturing features could be man-
ufactured with a single operation on a special milling
center. This forces our algorithm, to adapt the chang-
ing problem definition (Dorigo and Stuetzle, 2004).
Ant colony optimization was already used for solving
similar kinds of problems, like optimizing production
plans (Liu et al., 2010) or ad-hoc-networks (Kamali
and Opatrny, 2009). ACO summarizes a set of dis-
tinct algorithms that are based on the same approach
but optimized for special problems. For our purposes,
we apply the Ant Colony System (ACS).
3.1 Ant Colony System
Ant Colony Optimization is a nature analogue ap-
proach that imitates the behavior of Argentine ants.
Ants have limited opportunities to communicate with
each other. They use a chemical substance for com-
munication called pheromones and deposit these on
their way between anthill and source of food. Con-
sequently, succeeding ants can orient themselves on
the given trace. If there are two different ways with
different lengths between anthill and source of food
source, the first ant takes a random selection. A lit-
tle later, the pheromone concentration on the shorter
trace is higher than on the longer because of the length
this route could be more often passed. Consequently,
the probability of selecting the shorter route increases
with the pheromone concentration. But there always
remains a probability for selecting alternative routes
that characterizes a heuristic method. In fact, there is
never guaranteed that those algorithms find the opti-
mal solution but they always have an optimizing na-
ture. Hence, the natural approach was scientifically
adapted for solving combinatorial problems. Ant al-
gorithms consist of three phases, starting with solu-
tion construction, followed by pheromone update and
optional daemon-activities.
For creating a solution, an ant starts from the cur-
rent initial knot with a specific probability to a neigh-
bored knot. Afterwards, this task is repeated until
a termination criterion is met. During pheromone
update, the ant deposits pheromone by leaving the
edge. Daemon-activities are further optional activi-
ties e.g. deposition of pheromones on the entire fi-
nal ant path. Certain aspects of Ant Colony System
(ACS) differ from ant algorithm. To give an exam-
ple, ACS uses a global and a local pheromone update.
A global pheromone update addresses the deposition
and evaporation on the entire path of the current best
solution. Consequently, only the best evaluated ant is
authorized for a global pheromone update. Further-
more, there is a local pheromone update where every
ant reduces the pheromone concentration after leav-
ing an edge. This approach supports a wide search
that avoids a concentration on local optima as well as
premature convergence (Fischer, 2008).
Figure 4: Pseudo code for generating process variants.
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