Critical Path in Fuzzy Network Analysis
by Milan Mares
The project ‘Fuzzy Set Theoretical Models of Cooperative Behaviour
of Economic Subjects’, was solved in 1996-1998 at the Institute
of Information Theory and Automation of Sciences of the Czech
Republic. The project team has also included specialists from
the Faculty of Civil Engineering of the Slovak Technical University.
The widely concepted orientation of the grant has included, beside
others, the construction and investigation of the co-ordination
of activities in complex production and building processes which
is usually described by means of network analysis and critical
path method (CPM).
The deterministic version of the CPM algorithms is well known
for many years already and an effective software is commercially
available. Anyhow, the determinism of the model can be questioned
in many practical situations in which non-standard procedures
are used in the modelled sequence of activities. The main goal
of this part of research was to generalize the well known model
and include uncertain and vague phenomena. Namely, it was supposed
that the assumed durations of particular activities are not exactly
known in advance, and that the exclusive character of some of
them and objectively existing uncertainty of the technological
and economic environment influencing some others lead to only
uncertain idea about the time needed for their realization. Moreover,
this uncertainty is not based on statistical dispersion of possible
values and, consequently, application of probabilistic methods
is not effective. Such situation may appear, eg, if some activities
are realized by new technologies going beyond the stabilized experience,
if a non-standard object (atypical building, satelite or tanker)
is produced, or if the complete production process modelled by
the referred method is to last so long that some exactly unpredictable
changes of technologies cannot be excluded.
In such case a fuzzy set theoretical model of durations of activities
was used, fuzzy durations of paths and fuzzy float values were
derived, and it was also shown that the concept of critical path
itself is fuzzy. The described mathematical model of network analysis
with vague components is based on the paradigm due to which the
uncertainty of input data (durations of activities) generates
also uncertainty of some properties (criticism and sub-criticism
of paths, ordering of paths with respect to their durations) and
output data (floats of paths and activities). The derivation of
the characteristic of vagueness of the output data from the vagueness
of the input ones is based on the processing of fuzzy numbers.
From the point of interpretation the most interesting output is
the concept of fuzzy floats which indicates possible riscs of
delays with numerically structured possibilities. The fact that
being critical becomes a fuzzy property of paths means that the
critical paths form a fuzzy subset of the set of all paths to
which any path belongs with some possibility. Also the vague durations
of paths mean that for any pair of paths each of them can last
longer than the other one with some possibility and it is meaningful
to compute the floats regarding all possibly critical paths. The
possibility with which these floats reach negative values with
respect to all other paths indicates the possibility with which
a delay in realization can jeopardize the punctual fulfilment
of given time-limits. These possibilities of negative floats and
their structure offers an interesting information about the certainties
and riscs being related to the modelled production process and,
in this sense, the fuzzy set theoretical analysis of paths and
their durations offers a more relief and more finely structured
information than the deterministic model. The model is based on
some former works of the researchers and it probably opens the
possibility of development regarding further elements of the network
analysis.
The fuzzy set theoretical analysis of the critical path has illustrated
one methodological discrepancy of the operations with fuzzy quantities
based on the so called extension principle, namely the enormous
increasing of uncertainty extent if the algebraic operation of
summation is used. The paths in the network analysis which usually
consist of numerous activities show this problem in an illustrative
way. Such rapid increasing of uncertainty, moreover, does not
fully correspond with the everyday practical experience. It appears
to be useful to look for some alternative approach to the arithmetical
processing of fuzzy numbers. Such approach could be based on the
separation of the quantitative and fuzzy semantic component of
a vague number and on their separate processing by arithmetical
an fuzzy logical methods, respectively. One of the affiliated
outcomes of the referred research is a suggestion of an alternative
model of fuzzy quantities and their processing, respecting the
heuristic principle formulated above.
Further information (reprints of publications) is available on
request by e-mail.
Please contact:
Milan Mares - CRCIM
Tel: +420 2 688 4669
E-mail: mares@utia.cas.cz
