## 4 Methodology
### A Tracing Example of the Algorithm  
In the following, a tracing example is given for a certain patient that has comorbidities according to Table 2. The patient is diabetic with a mild increase in body temperature (37.8) and his \(O_{2}\) saturation is 96. The patient is not known to have any other diseases. This example has been tested over nine hospitals that have medical preparations regarding kidney machines, ventilators, and Intensive Care Units (ICUs) according to Table 3. Also, Table 3 presents the time to reach and time to admit to the patient under the test. In our experiments, the time to reach and time to admit are stored in the patient record as these times differ for each patient. Table 3 has the number of available beds that can accept the patients, which is used to determine whether the hospital is full or not. Table 4 presents a tracing example of the execution of the algorithm over the patient identified in Table 2 to the hospitals presented in Table 3. Table 4 gives in each row an execution step of the algorithm. It shows the iteration number, reach time of the patient to the hospital, admission time of the patient to the hospital, maximum of reach and admission times, match value of patient comorbidities and hospital preparations, and the best hospitals to admit the patient. ```
Initialization. Let the number of patients \(P_{no}\), number of hospitals \(H_{no}\), patient's comorbidities \(P_{com}\), \(BestSol\) to be the best solution, \(h_{r}\) time for the patient to reach the hospital, \(t_{ad}\) the time needed for the patient to be admitted in the hospital, and empty the non-dominated set of solutions. Main Loop. for\(p=1,\ldots,P_{no}\)do for\(h=1,\ldots,H_{no}\)do  Set \(Max_{t}=Max(h_{t_{r}},h_{t_{ad}})\)  Set \(m=Match(P_{com},H(h)_{md})\)  Set \(Sol\) to be the set of (\(h\), \(m\),\(Max_{t}\)). Add \(Sol\) to non-dominated solutions according to dominance criteria of \(PO\). endfor if\(P(p_{status})\) is  Severe then  Set \(BestSol\) to be the hospital with the smallest admission time \(Max_{t}\) regardless of the patient's comorbidities and the hospital's medical devices. else  Set \(BestSol\) to be the hospital with the highest matching score and least admission time. endif endfor Return \(BestSol\)
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**Algorithm 1** Pseudocode of Pareto optimization algorithm applied for the hospital admission problem  
In iteration 1, the maximum time of reach and admission is 10 and the match value is 1 between patient and hospital. The hospital number is considered to be the best hospital and is added to the non-dominated solutions because there are no other solutions. At iteration 2, hospital number 2 has the same maximum time and match value so, the non-dominated list adds hospital number 2. At iteration number 3, hospital 3 has a maximum time of 8 and zero matching criterion. As hospital 3 has less time, it will be added to the non-dominated set without considering the zero-match value as this hospital could be valid for severe cases. The rules of accepting the previous hospital or not can be changed according to the real-life decision-makers. Iteration 4 has the same case as in iterations (1,2). At iteration number 5, the hospital is rejected as it has a maximum time greater than those in the non-dominated solutions. Hospitals number (6,7) are rejected as their maximum time is greater than the maximum time of hospital number 3. Hospital number 8 has been added to the non-dominated solutions and eliminated hospital number 3 as it has zero matches and has less maximum time than hospital number 3. At iteration number 9, the hospital dominates the hospitals (1,2,4) as it has the same match value and has less maximum time.