Solving PERT Problems Involving Type-2 Fuzzy Uncertainty: An Approach - Technologies and Applications of Artificial Intelligence - page 142

Ta b l e 2 . Results of the optimization process

D L−
α

D R−
α

D L +
α

D R +
α

ʱ CP CP CP CP

1 20 1-3-4-7-8-9 25 1-3-4-7-8-9 20 1-3-4-7-8-9 25 1-3-4-7-8-9

0.9 18.6052 1-3-4-7-8-9 26.27914 1-3-4-7-8-9 18.2354 1-3-4-7-8-9 26.8061 1-3-6-8-9

0.8 18.0368 1-3-4-7-8-9 27.04905 1-3-6-8-9 17.513 1-3-4-7-8-9 28.2183 1-3-6-8-9

0.7 17.5645 1-3-4-7-8-9 27.96728 1-3-6-8-9 16.9019 1-3-4-7-8-9 29.3713 1-3-6-8-9

0.6 17.1398 1-3-4-7-8-9 28.79961 1-3-6-8-9 16.342 1-3-4-7-8-9 30.4124 1-3-6-8-9

0.5 16.7449 1-3-4-7-8-9 29.59367 1-3-6-8-9 15.8092 1-3-4-7-8-9 31.4027 1-3-6-8-9

0.4 16.3707 1-3-4-7-8-9 30.38111 1-3-6-8-9 15.2888 1-3-4-7-8-9 32.3826 1-3-6-8-9

0.3 16.012 1-3-4-7-8-9 31.19447 1-3-6-8-9 14.7676 1-3-4-7-8-9 33.3926 1-3-6-8-9

0.2 15.6654 1-3-4-7-8-9 32.08625 1-3-6-8-9 14.2254 1-3-4-7-8-9 34.4981 1-3-6-8-9

0.1 15.3286 1-3-4-7-8-9 33.65486 1-3-6-8-9 13.6085 1-3-4-7-8-9 36.7508 1-3-4-7-8-9

0.01 15.0325 1-3-4-7-8-9 38.02307 1-3-4-7-8-9 12.558 1-3-4-7-8-9 42.9836 1-3-4-7-8-9

0.001 15.0033 1-2-4-7-8-9 41.79054 1-3-4-7-8-9 11.9045 1-3-4-7-8-9 48.4745 1-3-4-7-8-9

0.00001 15 1-2-4-7-8-9 48.6921 1-3-4-7-8-9 11.9 1-3-4-7-8-9 58.675 1-3-4-7-8-9

The uncertainty associated to multiple experts allows to know other critical path (1-
3-6-8-9) which is important in order to ensure a suitable development of the project.
The activity 3-6 has the biggest influence in the project when the perception of experts
is pessimistic with a membership degree from 0.1 to 0.9.
Note that every point (optimal solution) allowable into the support of D has a set of
memberships u J d (see Figure 3). This means that every solution is satisfactory to all

experts in different degrees.

∈

5 Concluding Remarks

IT2FSs are useful to represent the uncertainty associated to the opinion of multiple
experts, or the ambiguity of a single expert regarding their activities times. A PERT
problem can be solved when linguistic uncertainty appears through linear optimization
tools that lead to a set of possible choices which conforms the solution of the problem.
We have solved a PERT problem with no statistical information, using linguistic in-
formation coming from people considered as experts of the system. This kind of infor-
mation has been handled through IT2FSs and optimization tools to find a set of possible
choices of CPs. This means that the analyst can see the project in different scenarios
and predict the behavior of the system through its CPs.
The proposal obtains an FOU that characterizes the optimal solution, which means
that the decision maker can manage the project in a more suitable way based on infor-
mation coming from experts.
Different fuzzy measures can be used to solve the uncertain PERT problem. Other
representations can be performed using the fuzzy measures of an IT2FS (see Wu &
Mendel [17]), and centroid based optimization can be performed using the results of
Figueroa-Garcıa [19,20]. Different LP models can be formulated for this problem, so
our proposal is just one approach to the problem.