Figure 5 plots the fNPV in three dimensions when the discount rate varies from 0% to 15%.
Figure 6 provides a closer insight as to how the fIRR can be formed by the intersection of [P.sub.xz] (at y = 0) with the fuzzy sets of fNPV. It shows 25 NPV membership function graphs that correspond to discount rates from 4% to 8.8% at 0.2 increments.
It is also evident how the fIRR is formed from the intersection of various fNPV membership functions with [P.sub.xz] (NPV = 0).
The ability to plot and study fNPV in three dimensions is very promising.
As such, the fuzzy project performance indicators (fNPV and fIRR) of alternative projects can be compared with Equations (3)-(5).
Overall, the fuzzy methodology presents the following advantages: (a) computational efficiency: the results of fNPV and fIRR are derived through a single analytical calculation, contrary to Monte Carlo analysis that requires thousands of iterations; (b) repeatability: in Monte Carlo analysis, the results of every simulation are moderately different from each other due to the randomness of numbers that are generated by computers.
With three-dimensional fNPV and fIRR graphs, project evaluation and selection can be seen for the first time from a different perspective.
Finally, future research can be directed in formulating a holistic methodology that incorporates fIRR and fNPV with fuzzy project scheduling, fuzzy cash flow analysis, and benefits realization in an advanced management system.