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Ta b l e 1 . Parameters and shapes of the example
T ij UMF LMF L ij ( x ) R ij ( x )
T 12 = (1,1.5,1,1) (1, 1.5, 0.8, 0.8) L 12 ( x )=max(1 − x 2 , 0) R 12 ( x )=max(0 , 1 − x )
T 13 = (2,3,0.3,2.4)
L 13 ( x )=exp −x
(2,3,0,2)
R 13 ( x )=max(0 , 1 − x )
T 24 = (0,0,0,0)
(0,0,0,0)
L 24 ( x )=max(0 , 1 − x ) R 24 ( x )=max(0 , 1 − x )
T 25 = (2,3,1.4,2.3)
L 25 ( x )=max(0 , 1 − x 4 )
R 25 ( x )=exp −x
(2,3,1,2)
T 34 = (0,0,0,0)
L 34 ( x )=max(0 , 1 − x ) R 34 ( x )=max(1 − x 2 , 0)
(0,0,0,0)
T 36 = (6,7,0.5,2.5)
L 36 ( x )=exp −x 2
R 36 ( x )=max(1 − x 2 , 0)
(6,7,0,2)
T 46 = (5,5,1.1,1.2)
L 46 ( x )=max(0 , 1 − x ) R 46 ( x )=max(0 , 1 − x 4 )
(5,5,1,1)
T 47 = (9,9,1.4,1.6)
L 47 ( x )=max(0 , 1 − x 4 )
R 47 ( x )=exp −x
(9,9,1,1)
T 59 = (8,9,2.3,4.7)
L 59 ( x )=max(0 , 1 − x 4 ) R 59 ( x )=max(1 − x 2 , 0)
(8,9,2,4)
T 68 = (4,4,2.6,2.8)
L 68 ( x )=max(1 − x 2 , 0) R 68 ( x )=max(0 , 1 − x 4 )
(4,4,2,2)
T 78 = (3,4,2.5,0.3)
L 78 ( x )=max(0 , 1 − x ) R 78 ( x )=max(0 , 1 − x 4 )
(3,4,2,0)
T 89 = (6,9,2.2,3.7)
R 89 ( x )=exp −x 2
L 89 ( x )=max(1 − x 2 , 0)
(6,9,2,3)
1
0.8
FOU of D
J d
0.6
0.4
0.2
0
10
15
20
25
30
35
40
45
50
55
60
d
D
Fuzzy set of optimal solutions
Fig. 3. Fuzzy set D composed from optimal project times
4.2
Analysis of Results
The shape of the left function for the optimal solution shows the same critical path (1-
3-4-7-8-9) for several ʱ -cuts, the minimal optimistic time for the external left function
and the internal left function is approximately 12 and 15 respectively. Table 2 shows
that the project has three possible CPs (1-2-4-7-8-9), (1-3-4-7-8-9), (1-3-6-8-9) which
can occur anytime in the project, so the analyst should pay attention to CPs and their
activities when performing the project to identify when a CP will switch to another one.
The internal right bound presents a different CP (1-3-6-8-9) from 0.1 to 0.8 ʱ -cut
and its maximal pessimistic time is approximately 49, the external right bound presents
a similar behavior changing the CP from 0.2 to 0.9 ʱ -cut and its maximal pessimistic
time is approximately 59.
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