## 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

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