## 3 Background
### Problem Definition  
The admission problem of Covid-19 patients to hospitals can be formulated of two objectives: the first objective is concerned with the time that a patient needs to get admitted and the second objective is concerned with matching the patient status with the hospital medical preparations. The first objective has two main factors: the time that a patient needs to reach a hospital \(t_{r}\) and the admission time of the patient to get the medical care \(t_{ad}\). The second objective focuses on finding the hospital, equipped with the best medical devices \(h_{md}\) compared with the patient comorbidities \(p_{com}\). [PERSON] et al. [24] have identified 11 comorbidities to prioritize the patients infected with Covid-19 virus. The proposed model has used seven comorbidities of them, which include diabetes mellitus, heart failure, chronic pulmonary disease, chronic liver disease, chronic kidney disease, Temperature, and \(O_{2}\) saturation. These comorbidities are used to evaluate the patient's status to identify the devices that should be in a hospital to treat the patients.  
The best solution to the problem can be identified as the hospital \(h\) with the least admission \(h_{t_{ad}}\), reach \(h_{t_{r}}\) times, and has the most suitable medical devices \(h_{t_{md}}\) for the patient's comorbidities \(P_{com}\). The problem has a set of constraints, which can be summarized into the existence of a limited number of hospitals \(h=\left\{1,2,\ldots,H\right\}\), a limited number of medical devices in each hospital \(md=\left\{1,2,\ldots,MD\right\}\), and the response time of the system to admit patients should not exceed a specific threshold.  
To choose the best hospital for patient admission, the hospital should have the most suitable medical devices for the patient and does not require a long time to admit the patient. This clarifies the existence of two conflicting objectives that a solution needs to satisfy. The first objective is related to the admission time, which is a minimization function. The second objective is concerned with assessing how suitable the medical devices in a hospital compared with the patient status, which is a maximization function. According to this scenario, the problem can be defined as:  
\[\begin{array}{ccccc}&T_{p}(h)=Maximum(h_{t_{ad}},h_{t_{r}})&Minimization,\\ &F_{p}(h)=Match(h_{md},P_{com})&Maximization.\end{array} \tag{6}\]  
The first function is used to identify the nearest hospital to the patient location and has the least admission time based on two times: reach time \(h_{t_{ad}}\) and admission time \(h_{t_{r}}\). This function takes the maximum time of the two times (\(h_{t_{r}}\), \(h_{t_{ad}}\)) to determine the time needed for a patient to be admitted to hospital \(h\). The second function is applied to identify the hospital that has the maximum number of medical devices needed to treat the patient. This ensures that the patient will have the whole medical care needed in one hospital. The matching function makes a comparison of the patient's comorbidities with the medical devices of each hospital. \(PO\) can be applied to obtain the whole non-dominated solutions to the problem. The best solution will be determined from the non-dominated solutions based on each patient's situation. To describe the main parameters used in the problem, Table 1 presents the notations of these parameters and their description.