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

this interval of [0,100], this output is evaluated in the same interval, and its membership value
is assessed with a triangular function. The ED workload is an intermediate variable that feeds
the fourth fuzzy subsystem, and represents a major determinate of crowding by containing
four of the seven inputs alone. Combined with the output of subsystem I and the final input,
the output of subsystem III will contribute to subsystem IV’s assessment of emergency
department crowding.
In review, the first level of the hierarchical fuzzy expert system was composed of two
fuzzy logic subsystems, with the second level containing one subsystem, which is also
detailed in Figure 5. Level three of the hierarchical fuzzy expert system contains the fourth
and final fuzzy logic subsystem, which receives inputs in some manner from every previous
subsystem.
This fourth fuzzy logic subsystem is the main component of this hierarchical fuzzy expert
system which aims to assess the ED crowding level. The three inputs of this fuzzy subsystem
include the two previously mentioned indicators ED demand status and ED workload, and the
third, new input, which is the seventh independent input of the entire hierarchical system, is
ED boarding status. The components of fuzzy subsystem IV are illustrated in Figure 9. The
first input to this subsystem, the ED demand status, as previously described, is represented by
five triangular membership functions; “Very Low”, “Low”, “Medium”, “High”, and “Very
High”, with an interval of [0, 100]. The second input, the ED workload is represented by four
triangular membership functions; “Low”, “Medium”, “High”, and “Very High”. Its interval of
the crisp value is [0,100]. The third input, ED boarding status, is an independent variable,
which is derived from the ratio of boarded patients to the capacity of the emergency room.
This input has four fuzzy classes as the second input, but is evaluated with a trapezoidal
membership function on an interval of [0, 0.4]. With the three sets of membership indicators
in this subsystem, the number of fuzzy rules is 80 (4 ×5). The output of the fourth fuzzy logic
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subsystem is the ED crowding level, and is the final output for the entire hierarchical system.
It is represented by five membership functions; “Insignificant”, “Low”, “Medium”, “High”,
and “Extreme”, which are used to indicate the degree of crowding in emergency departments.
Like other outputs, the interval of the crisp value for the final output is [0,100], and is
evaluated with a triangular function.
Utilizing the hierarchical fuzzy system appears to be the most appropriate approach for
this study, rather than the standard fuzzy system. This approach creates different indicators,
such as demand status, workload, and staffing indicators, while reducing the total number of
fuzzy rules from 5184 (under the standard fuzzy system) to just 137 rules. This difference
represents a great reduction in calculation and simplifies the process of acquiring knowledge
from experts, and potentially reduces the threshold for academic access to meaningful results.