Abstract
This paper proposes a credibility based chance constrained programming approach for project scheduling problems with fuzzy activity durations where the objective is to minimize the fuzzy project completion time. This paper expresses the fuzzy events such as a project activity’s duration or project completion time with fuzzy chance constraints and the chance of a fuzzy event is illustrated with fuzzy credibility distribution. Due to uncertainty in durations of a project, fuzzy sets and fuzzy numbers can be used in order to illustrate the uncertainty and find a solution space for the problem. Therefore, fuzzy credibility based chance constraint technique is investigated for project scheduling problems with fuzzy activity durations considering the uncertainty or chance of a fuzzy event within a closed interval. In this paper, a fuzzy mathematical model and its crisp equivalent by using credibility measure and chance-constrained programming are given for project scheduling problems with fuzzy activity durations.
Highlights
Basic definitionsWe present some of basic definitions and notations of fuzzy numbers and measures of a fuzzy event such as credibility, possibility, and necessity for the readers
In order to express activity’s durations in an interval or to encode the uncertainty in activity durations, fuzzy sets, and fuzzy numbers can be used. Fuzzy events such as a project activity’s duration or project completion time is less than a certain real number are expressed with fuzzy chance constraints and the chance of a fuzzy event is illustrated with fuzzy credibility distribution
This paper investigates a project scheduling problem with fuzzy activity durations where the objective is to minimize fuzzy project completion time
Summary
We present some of basic definitions and notations of fuzzy numbers and measures of a fuzzy event such as credibility, possibility, and necessity for the readers. For a TFN A with A (x) , Possibility and Necessity measures are calculated as follows: 0, r AL r − AL ( A r ) = Sup μ(x) =. Credibility measure [47] is a self-dual fuzzy measure and it is average of Possibility and Necessity measures as follows: Cr ( A r ) = 1 ( ( A r ) + N ( A r )) (4). The credibility distribution function (x) is a strictly increasing function on the real axis and it has an inverse function −1 ( ) that is unique for any α confidence level. For a TFN Ã, the credibility distribution function and inverse credibility distribution function (see Figure 1) are calculated as follows: 0,
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