Solving PERT Problems Involving Type-2 Fuzzy Uncertainty: An Approach - Technologies and Applications of Artificial Intelligence - page 137

Several problems have been identified in the application of fuzzy PERT. Chen &
Wang [12] proposed an algorithm for finding multiple possible critical paths using fuzzy
sets. Fargier, Galvagnon & Dubois [8] found that while it is easy to compute fuzzy earli-
est starting dates of tasks in the critical path method, the problem of determining latest
starting dates and slack times is much more tricky, and he treated this problem with
serial-parallel graphs framed on the theory of possibility. Chanas & Zielinski [13] pro-
posed the application of the Zadeh's extension principle to the concept of criticality of
a route, and a method for calculating the degree of criticality through linear program-
ming. Chen [6] proposed an approach based on the fuzzy extension principle and a pair
of linear programs parameterized by possibility levels ʱ to analyze a critical path.
This paper proposes an extension to the PERT problem involving the opinion of
multiple experts who estimate activity times. The ambiguity of the experts and the un-
certainty associated with the discrepancy among them is represented by IT2FSs. Based
on the results of Chen [6], an optimization model was designed to address the PERT
problem with multiple experts, using a simulated scenario where multiple experts give
their opinions through optimist and pessimist activity times of a project.
This paper is structured as follows: in Section 2, we provide some basic concepts
about Type-2 fuzzy sets. In Section 3, we introduce the Fuzzy PERT problem and its
mathematical model. In Section 4, we propose a method and a model to solve fuzzy
PERT problem involving Type-2 fuzzy activity times alongside a numerical example.
Finally, concluding remarks are presented in Section 5.

2 Basic Concepts

A fuzzy set, namely A , is a generalization of a Crisp or Boolean set. It is defined over
a universe of discourse X by a Membership Function namely μ A ( x ) that takes values
in the interval [0,1]. A fuzzy set A may be represented as a set of ordered pairs of a
generic element x and its degree of membership, μ A ( x ),i.e.,

A = { ( x, μ A ( x )) | x ∈ X } (1)

2.1 α -Cuts

One of the most important concepts of fuzzy sets is the concept of ʱ - cut and the strong
ʱ - cut . Given a fuzzy set A defined over X and ʱ [0 , 1] the ʱ - cut , ʱ A ,andthe strong

∈

ʱ - cut , ʱ + A ,isdefinedasfollows

ʱ A =

(2)
That is, the ʱ - cut of a fuzzy set A is the crisp set ʱ A that contains all the elements
of the power set X whose memberships in A are greater or equal than ʱ (see Klir [15]).

{ x | A ( x ) ≥ ʱ }