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198 Abdulrahman Albar, Ahmad Elshennawy, Mohammed Basingab et al.

HIERARCHICAL FUZZY SYSTEM

Hierarchical fuzzy systems (HFSs) are implemented by researchers for two main
purposes. First, they help in minimizing the total number of fuzzy rules in the knowledge
base which feed into the fuzzy inference engine. Second, the HFSs are effective in
building the logical relationship among different crisp input variables in complex
systems, unlike Standard Fuzzy Systems (SFSs), which become exponentially
complicated as the number of variables and their fuzzy sets’ levels increase. Figure 2 and
Figure 3illustrate the difference between applying traditional standard fuzzy logic
approach versus applying hierarchical fuzzy logic approach to construct and determine
the relationship between a fuzzy subsystem’s crisp outputs and the main fuzzy system,
where On stands for the crisp output of fuzzy subsystem n, and Of stands for the crisp
output of the main fuzzy system [7]. In the case of SFSs, the total number of fuzzy rules
related to the number of crisp inputs is exponentially proportional, whereas it is linearly
proportional in HFSs. For instance, supposing that there are five crisp variables, and each
variable encompasses five fuzzy sets, then for utilizing a SFS, the total number of fuzzy
rules for the whole fuzzy system is (55 = 3125 rules), whereas in a four-level HFS with
four fuzzy subsystems, each encompassing two crisp inputs, the total number of fuzzy
rules for the complete fuzzy system is (52 = 100 rules). It is clear that utilizing HFSs
significantly reduces the total number of fuzzy rules necessary to construct the
knowledge bases for the whole fuzzy system. Thus, utilizing HFSs in this study makes it
possible to analyze the complicated nature of emergency health care systems, which if
studied through SFSs, could involve too many fuzzy rules and computations for an
effective analysis. It is also notable that using HFSs detailed in Figure 3, will help in
determining the relationship between outputs of the fuzzy subsystems and the main fuzzy
system, and in specifying the relationship among fuzzy subsystems as well.

Figure 2: Standard fuzzy logic system. Figure 3: Hierarchical fuzzy systems.