3.1. Multi-Objective Problems (MOPs)
A
MOP can be described as a set of
m objectives as:
where
m represents the number of objective functions of the
MOP. Each objective can be described mathematically as a set of equality or inequality equations as [
23]:
where
H represents the equality functions and the
G represents the inequality functions. These functions assess the quality of solutions. Each parameter of an objective has a search space that can be described as:
where
n is the number of variables,
l and
u represent the lower and upper bounds of each parameter
i respectively. There are a limited number of algorithms that can deal with
MOPs, which include
PO [
6] and the lexicographic method [
7]. The lexicographic approach starts with ordering the objectives of the problem according to their importance. Then, it tries to find the optimal solution for the problem according to the order of the objectives.
3.2. Pareto Optimization
PO is a common method that has been widely used to solve
MOPs [
6,
8,
12,
23]. It tries to satisfy the whole objectives simultaneously.
PO can compare among multiple solutions of a
MOP based on different objectives and identify the non-dominated solutions or the Pareto Front (
PF) solutions. The
PF or non-dominated solutions are the solutions that cannot be further optimized. Optimizing one objective in the non-dominated solutions results in deteriorating the degree of optimization of other objectives.
PO applies the dominance concept between two solutions
v and
z by varying among the objectives in two solutions, in case all objectives in
v are as good as in
z and at least one objective in
v is better than the same objective in
z, then solution
v is considered to dominate solution
z. This comparison can be described mathematically as [
5]:
where
v,
z are two different solutions, and
m represents the number of objectives. These two rules must hold to ensure that solution
v dominates solution
z.
3.3. 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
and the admission time of the patient to get the medical care
. The second objective focuses on finding the hospital, equipped with the best medical devices
compared with the patient comorbidities
. De Nardo 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
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 , reach times, and has the most suitable medical devices for the patient’s comorbidities . The problem has a set of constraints, which can be summarized into the existence of a limited number of hospitals , a limited number of medical devices in each hospital , 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:
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
and admission time
. This function takes the maximum time of the two times (
,
) 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.
3.4. Problem’s Constraints
From our perspective, the main constraints of the admission problem can be concluded into: (1) the number of available beds in a hospital at time
t and (2) the admission time should be minimum. This analysis has been deducted based on multiple recent studies [
17,
25,
26]. As we can see in each method, different constraints are considered. This returns to the problem itself and the perspective of the authors to the problem. The first constraint can be mathematically defined as:
This function ensures that the total number of patients’ requests do not exceed the total number of beds in all the hospitals . In case the number of requests exceeds the total number of beds, the patients’ requests are added to a queue to be handled later.
Another important constraint is the response time of the system to identify the most suitable hospital for the patient. This can be defined mathematically as:
where
p represents the patient,
h is the identified hospital, and
is the demand of patient
p at time
t. This constraint ensures that patients should not wait for more than the threshold
to know to which hospital they would be admitted.