Information Technology Reference
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These intervals indicate where the values of t ij lie at possibility ʱ . Based on the
Zadeh's extension principle, the fuzzy set μ D ( d ) can be defined as:
μ T ij |
y 1
μ D ( d )=
sup
d∈R + , ( ij ) ∈S
min
( ij ) ∈S
d = y n
(13)
To find the membership function μ D ( d ), we need to compute the left external bound
D L +
ʱ
, the left internal bound D L−
ʱ
, the right internal bound D R−
ʱ
and the right external
bound D R ʱ of the ʱ -cut of μ D ( d ), using the following models:
D L +
ʱ
y i +( T i,j ) L +
=min
{
y n
y 1 |
y j
; y i ,y j R
( i,j )
S
}
(14)
ʱ
D L−
ʱ
y i +( T i,j ) L−
=min
{
y n
y 1 |
y j
; y i ,y j R
( i,j )
S
}
(15)
ʱ
D R−
ʱ
y i +( T i,j ) R−
=min
{
y n
y 1 |
y j
; y i ,y j R
( i,j )
S
}
(16)
ʱ
D R +
ʱ
y i +( T i,j ) R +
=min
{
y n
y 1 |
y j
; y i ,y j R
( i,j )
S
}
(17)
ʱ
where y j represents the occurrence time of the node j , the node i precedes the node j ,
and the value of y n
y 1 represents the duration of the project, since y n is the starting
time of the last activity, and y 1 is the starting time of the first activity. The objective of
the dual problem is to find the required minimal duration time such that all precedence
relationships among the different activities are satisfied, and each constraint specifies
the precedence relationships among activities.
4.1
An Example
Consider a project whose network is shown in Figure 2 (proposed by Chen [6]). The
activity times are IT2FSs coming from the opinion of eight experts who provided their
optimistic and pessimistic perceptions which composes the FOU of each T ij ,andthe
fuzzy activity times, shapes and parameters are presented in Table 1.
Here, L ij ( x ) is the left shape, and R ij ( x ) is the right shape of the set T ij .After
solving the models proposed in (14), (15), (16), and (17), we have obtained the results
shown in Table 2.
Now, we can derive μ D ( d ) through the extension principle, D L−
,D R−
ʱ
,D L +
ʱ
,and
ʱ
D R +
ʱ
. The obtained results are shown in Figure 3.
Fig. 2. The network structure of the example (taken from Chen [6])
 
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