# Fuzzy Critical Path Method Based on a New Approach of Ranking Fuzzy Numbers using Centroid of Centroids

N. Ravi Shankar (GITAM University, Visakhapatnam, AP, India), B. Pardha Saradhi (Dr. L.B. College, Visakhapatnam, AP, India) and S. Suresh Babu (GITAM University, Visakhapatnam, AP, India)
DOI: 10.4018/ijfsa.2013040102

## Abstract

The Critical Path Method (CPM) is useful for planning and control of complex projects. The CPM identifies the critical activities in the critical path of an activity network. The successful implementation of CPM requires the availability of clear determined time duration for each activity. However, in practical situations this requirement is usually hard to fulfil since many of activities will be executed for the first time. Hence, there is always uncertainty about the time durations of activities in the network planning. This has led to the development of fuzzy CPM. In this paper, a new approach of ranking fuzzy numbers using centroid of centroids of fuzzy numbers to its distance from original point is proposed. The proposed method can rank all types of fuzzy numbers including crisp numbers with different membership functions. The authors apply the proposed ranking method to develop a new fuzzy CPM. The proposed method is illustrated with an example.
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## Fuzzy Basic Definitions

In this section, some basic fuzzy basic definitions on fuzzy sets are presented (Kaufmann & Gupta, 1985):

• Definition 1: Let U be a Universe set. A fuzzy set of U is defined by a membership function where is the grade of x in, ;

• Definition 2: A fuzzy set of universe set U is a fuzzy number if (i)is normal i.e., and (ii) is convex i.e.;

and:;
• Definition 3: The membership function of the real fuzzy number is given by:

where is a constant, a, b, c, d are real numbers and are two strictly monotonic and continuous functions. It is standard to write a fuzzy number as . If , then is a normalized fuzzy number, otherwise is said to be a generalized or non-normal fuzzy number if ;

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