Information Technology Reference
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where
y
j
is the occurrence time of the node
j
, node
i
precedes node
j
,
y
n
is the starting
time of the last activity, and
y
1
is the starting time of the first activity,
S
is the set of
all active nodes of the project. The activity times are defined by a fuzzy set
T
:
A
ₒ
(
R
+
) is the set of non-negative fuzzy numbers, thus
T
ij
is the fuzzy duration
time of activity (
i,j
)
F
(
R
+
)
,
F
S
, and its membership function is
μ
T
ij
(
t
ij
).
This way, we propose a model and an algorithm based on the representation theorem
and the decomposition of fuzzy set to solve a PERT problem where activities times are
defined by the knowledge of multiple experts. Next sections show the obtained results.
∈
4
The Proposed Approach
Usually, activity networks are performed by multiple people who have different ideas
and perceptions about every activity. In addition, not all projects have available and/or
reliable statistical information about its activities, so the analyst has to use linguistic
information coming from experts to formulate a strategy for planning the project.
In the case of a single expert, the model proposed by Chen [6] can be applied to
obtain the CPs of the project. In the case of multiple experts, we have to model the
PERT problem and design an optimization method to obtain the CPs of the network,
based on the results of Figueroa-Garcıa [20]. We remark that the idea is not to solve
a PERT problem per expert but solve only one model that comprises the information
provided by all experts at the same time.
In order to represent the uncertainty associated with the opinion of multiples experts,
we assume an expected time of each activity of the project, and based on this value
we select a discrete group of experts to ask them about two sentences: optimist and
pessimist time of each activity. Once we get their opinion, we can build a Triangular
Type-1 Fuzzy Set (T1FS) for each expert. Now, having m-fuzzy sets for each activity
we can use the representation theorem to describe the set
T
(see Liu & Mendel [18]):
m
T
=
T
e
(10)
e
=1
where
m
is the total amount of T1FSs and
T
e
is a T1FS provided by the expert
e
(a.k.a.
embedded fuzzy set). Now, we represent every set
T
ij
using five parameters as follows:
T
ij
=(
t
L
+
,t
L−
ij
, t
ij
,t
R−
,t
R
+
ij
)
(11)
ij
ij
where
t
L
+
ij
is the minimum optimistic time,
t
L−
ij
is the maximum optimistic time,
t
ij
is the expected time,
t
R−
ij
is the minimum pessimistic time and
t
R
+
ij
is the maximum
pessimistic time provided by the experts.
Figure 1 shows an example of a triangular IT2FS bounded by a UMF
μ
T
(
t
ij
) and
LMF
μ
T
(
t
ij
). To build the membership function of the total duration time of the project
μ
D
(
d
), it is necessary to derive its
ʱ
-cuts. The
ʱ
-cut of
T
ij
is defined as follows:
ʱ
T
ij
=
(
T
ij
)
L
+
,
(
T
ij
)
L
ʱ
,
(
T
ij
)
R−
]
,
(
T
ij
)
R−
(12)
ʱ
ʱ
ʱ
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