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

Figure 5: Three-level hierarchical fuzzy expert.

Figure 5 further illustrates the relation of these inputs to the proposed fuzzy logic
system. Level one of the hierarchical fuzzy expert system contains two fuzzy subsystems.
The first fuzzy subsystem aims to assess the ED’s demand status by evaluating the
ratio of patients in an ED waiting area to that emergency room’s capacity, and the
average patient complexity. Figure 6 illustrates the components of fuzzy subsystem I. The
first input to the fuzzy subsystem I is the ratio of waiting patients to ED capacity which is
characterized by four fuzzy membership functions; “Low”, “Medium”, “High”, and
“Very High”. To assess this input variable, trapezoidal functions are utilized to evaluate
the membership degree on an interval [0, 2]. The patient complexity, the second input to
the fuzzy subsystem I, is represented by three membership functions; “Low”, “Medium”,
and “High”. Similarly, a trapezoidal function is used for this input, evaluating the
membership degree on the interval [1, 5], which is adapted from the five levels of the
emergency severity index (Gilboy, Tanabe, Travers, Rosenau, & Eitel, 2005). Given
these fuzzy classes, the total number of fuzzy rules from this subsystem will be 12 fuzzy
rules (4×3). The output of fuzzy subsystem I is ED’s demand status, which is represented
by five membership functions; “Very Low”, “Low”, “Medium”, “High”, and “Very
High”. This output is evaluated with a triangular function for the interval [0, 100]. The
demand status is an intermediate variable rather than a final indicator, which feeds the
fourth and final fuzzy subsystem with a crisp value, to contribute to the final assessment
of the ED’s crowding level.
The second fuzzy logic subsystem, with two inputs and one output, is designed to
determine the level of ED staffing. Figure 7 presents the components of fuzzy subsystem
II. ED staffing status is subjective in nature and the membership functions that represent
this aspect of crowding reflect this subjectivity based on the knowledge from the health
care experts. The two inputs of this fuzzy subsystem are the level of ED physician
staffing and ED nurse staffing. Both inputs are represented by three membership