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

for the low class was between 1 and 2.5. Similarly, for “medium” average complexity,
patients rated between 2.5 and 3.5 make up the core structure, and with the supports
assigned values between 2 and 2.5, and between 3.5 and 4, the entire class boundary lies
between 2 and 4. For “high” average complexity, the expert assigned values between 4
and 5 for the core area, with values between 3.5 and 4 for the support, making the
boundary for the high class between 3.5 and 5. The core areas of each class are consistent
in size, due to the data being taken from one expert instead of ten.

Figure 11: Membership function of patient Figure 12: Membership function of patient
demand. complexity.

The membership function for ED demand in Figure 13 represents the output for
subsystem one, which is considered the standard membership function for outputs. The
function is triangular, with membership degree values peaking at 1, and the boundaries
for different classes overlap the peaks of adjacent classes perfectly, demonstrating that
the membership function always obtains membership from two classes. This also means
that at any given point, the membership degree from two overlapping classes always
equals 1, but there are only five points where classes obtain membership completely.
These points occur at 0, 25, 50, 75, and 100 for “very-low”, “low”, “medium”, “high”,
and “very-high”, respectively.
In subsystem II, the membership functions for the physician staffing and nurse
staffing inputs are constructed with trapezoids for three classes. The output, ED staffing,
is also represented with a trapezoidal membership function, which features equally
spaced boundaries across three classes. Table 6 details the linguistic classes and fuzzy
numbers for subsystem II and its membership functions.
Physician staffing is represented in the membership functions in Figure 14. The three
classes overlap as seen in subsystem I, representing the regions where linguistic terms did
not reach full degrees of membership. For instance, the inadequate class core boundary
begins and ends at 0.06, representing full membership for the linguistic term
“inadequate”. The upper bound for the inadequate class is 0.12, where the linguistic term
“inadequate” achieves partial membership, and the lower bound for the partially adequate
class is 0.06, where its term also achieves partial membership. The boundaries for the
three classes are between 0 and 0.12 for the inadequate class, between .06 and 0.24 for
the partially adequate class, and between 0.16 and 0.32 for the adequate class. The