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

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

N. Ravi Shankar (GITAM University, India), B. Pardha Saradhi (Dr. L.B. College, India) and S. Suresh Babu (GITAM University, India)
Copyright: © 2017 |Pages: 18
DOI: 10.4018/978-1-5225-1908-9.ch069

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 978-1-5225-1908-9.ch069.m01 of U is defined by a membership function 978-1-5225-1908-9.ch069.m02 where 978-1-5225-1908-9.ch069.m03 is the grade of x in978-1-5225-1908-9.ch069.m04, 978-1-5225-1908-9.ch069.m05;

  • Definition 2: A fuzzy set 978-1-5225-1908-9.ch069.m06of universe set U is a fuzzy number if (i)978-1-5225-1908-9.ch069.m07is normal i.e., 978-1-5225-1908-9.ch069.m08 and (ii) 978-1-5225-1908-9.ch069.m09is convex i.e.;

    978-1-5225-1908-9.ch069.m10

and:
978-1-5225-1908-9.ch069.m11
;
  • Definition 3: The membership function of the real fuzzy number 978-1-5225-1908-9.ch069.m12 is given by:

    978-1-5225-1908-9.ch069.m13

where 978-1-5225-1908-9.ch069.m14 is a constant, a, b, c, d are real numbers and 978-1-5225-1908-9.ch069.m15978-1-5225-1908-9.ch069.m16 are two strictly monotonic and continuous functions. It is standard to write a fuzzy number as 978-1-5225-1908-9.ch069.m17. If 978-1-5225-1908-9.ch069.m18, then 978-1-5225-1908-9.ch069.m19 is a normalized fuzzy number, otherwise 978-1-5225-1908-9.ch069.m20 is said to be a generalized or non-normal fuzzy number if 978-1-5225-1908-9.ch069.m21;

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