Optimal Path Problem with Possibilistic Weights - Geoinformatics for Intelligent Transportation - page 46

4 Case Study

The case study shows rather simple example of a graph with weights represented as
fuzzy numbers or possibility distributions. This case study may describe real word
example of travelling in the city where weights show time necessary to reach the
node. Obviously time cannot be expressed exactly because it may depend on
external conditions that are unknown at the time, when the model was created. Such
conditions could be weather, time when the the travel should be made, amount of
traf

c etc. Because none of those conditions can be know in advance, it is rea-
sonable to model them as possibility distributions.
The example used in case study is obvious. A simple graph containing 4 nodes
and 5 directed arcs (Fig. 2 ). Node a is a starting point of path and destination
node is d . From the visualization of the graph it is visible that 3 paths can be
identi ed: a → b → d, a → c → d and finally direct path from a → d. Since all the

weights of the graph are de they represents

triangular fuzzy numbers. Known property of fuzzy numbers is that if they are
aggregated the result is again triangular number [ 10 ]. So the Eq. ( 1 ) is applied only
for

ned as triplets in form ½A 0 ; A 1 ¼ A 1 ; A 0

values of 0 and 1. Obtained results are summarized in Table 1 .
Results can be best asses when they are visualized (Fig. 3 ). Now the ranking
needs to established. Since the interest is in

a

finding values that are smaller or equal
to the given value the Eqs. ( 2 ) and ( 3 ) will be used to calculate possibility and
necessity.
The comparison of the obtained results is summarized in Table 2 . From that can
be reasoned that solution a

d is the worst as it has possibility and necessity of at
least equality to both other solution equal to 1. Also a

→

d has both
possibility and necessity equal to 0 which excludes this solution from set of pos-
sible shortest paths. From comparison of solutions a

→ c → d ≥ a →

ditis
clear that there is no strict ordering of these solutions, because there is quite a big

→ b → d and a → c →

Fig. 2 Simple graph with fuzzy weights

Table 1 Resulting path and
their lengths of the case study

Path Triangular number

a → b → d [3.6, 6, 7.3]

a → c → d [4.5, 6.4, 8.2]

a → d [7.5, 8, 8.4]