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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.
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