Determining the optimum location of facilities is critical in many fields, particularly in healthcare. This study proposes the application of a suitable location model for field hospitals during the novel coronavirus 2019 (COVID-19) pandemic. The used model is the most appropriate among the three most common location models utilized to solve healthcare problems (the set covering model, the maximal covering model, and the P-median model). The proposed nonlinear binary constrained model is a slight modification of the maximal covering model with a set of nonlinear constraints. The model is used to determine the optimum location of field hospitals for COVID-19 risk reduction. The designed mathematical model and the solution method are used to deploy field hospitals in eight governorates in Upper Egypt. In this case study, a discrete binary gaining–sharing knowledge-based optimization (DBGSK) algorithm is proposed. The DBGSK algorithm is based on how humans acquire and share knowledge throughout their life. The DBGSK algorithm mainly depends on two junior and senior binary stages. These two stages enable DBGSK to explore and exploit the search space efficiently and effectively, and thus it can solve problems in binary space.

Countries all over the world are working tirelessly to find new and better methods to face and defeat the novel coronavirus 2019 (COVID-19). Reference [

It is of great importance to place field hospitals geographically close to the places most vulnerable to infection by COVID-19, and to serve the largest number of patients who need healthcare under the constraints of limited available resources. The location of facilities is particularly critical where the implications of poor location decisions extend well beyond cost and patient service considerations. If too few facilities are utilized and/or they are not well located, mortality and morbidity rates can increase. Thus, facility location modeling takes on even greater importance when applied to the setting of healthcare facilities. Reference [

Reference [

Three classic facility location models are the basis for almost all facility location models that are used in healthcare applications. These are the set covering model, the maximal covering model, and the P-median model.

It is very important to note that location models are application-specific; that is, their structural form (the objectives, constraints, and variables) is determined by the particular location problem under study. Consequently, no general location model appropriate for all current or potential applications exists.

In this paper, a modified version of the maximal covering model is designed to be suitable for the optimum location of healthcare field hospitals for COVID-19. This designed model is a nonlinear binary mathematical programming model.

The structure of this paper is as follows. The first section presents an introduction and contains a concise point content for each subsequent section in this article. The second section presents a brief review of the three basic facility location models that are the most suitable for application in the healthcare field: The set covering model, the maximal covering model, and the P-median model. Section 3 introduces the location of field hospitals for COVID-19 to ease the patient burden on regular hospitals. Experience with field hospitals around the world proves that this method is an effective one when dealing with the crisis caused by the unbelievably quick spread of COVID-19. This section also clarifies that the formulation model is a modified version of the general location models that are appropriate for determining the location of healthcare facilities. Section 4 presents the model’s formulations, and clarifies that the model is a nonlinear binary constrained model. A real application is presented in section 5: The model is used to locate field hospitals in 8 governorates in Upper Egypt. In section 6, a novel discrete binary version of a recently developed gaining–sharing knowledge-based technique (GSK) is introduced to solve the mathematical model. GSK is augmented to become a discrete binary-GSK optimization algorithm (DBGSK), with two new discrete binary junior and senior stages. These stages allow DBGSK to inspect the problem search space efficiently. Section 7 represents experimental results, and Section 8 provides conclusions and suggestions for future research.

The set covering model, maximal covering model, and P-median model are discrete facility location models as opposed to continuous location models. Discrete location models assume that there is a finite set of candidate locations or nodes at which facilities can be sited. Conversely, continuous location models assume that facilities can generally be located anywhere in the region. Throughout this paper, discrete location models are strictly considered since they have been used more extensively in healthcare location problems.

The notion of coverage for the set covering and maximal covering models means that demands at a node are generally said to be covered by a facility located at some other node if the distance between the two nodes is less than or equal to some specified coverage distance.

The set covering location problem (SCLP) attempts to minimize the cost of the facilities that are selected so that all demand nodes are covered. Inputs to the model are the set of demand nodes, set of candidate facility sites, and fixed cost of locating a facility at candidate sites.

References [

Reference [

In practice, at least two major problems occur with the set covering model. First, if minimizing the cost of the facilities that are selected is used as the objective function, the cost of covering all demands is often prohibitive. If minimizing the number of facilities that are located is used as the objective function, the number of facilities required to cover all demands is often too large. Second, the model fails to distinguish between demand nodes that generate a lot of demand per unit time and those that generate relatively little demand. Clearly, if all demands cannot be covered because the cost of doing so is prohibitive, it will be preferred to cover those demand nodes that generate a lot of demand rather than those that generate relatively little demand. These two concerns motivated the formulation of the maximal covering problem.

The maximal covering location problem (MCLP) is a classic model in the location science literature. The demand at each node and the number of facilities to locate are needed as inputs, and additional decision variables are added. The MCLP was formulated to address planning situations that have an upper limit on the number of facilities to be sited.

The objective function in this case is to maximize the number of covered demands. The constraints state that exactly

A variety of heuristic and exact algorithms have been proposed for this model. Lagrangian relaxation provides the most effective means of solving the problem when the following constraint is relaxed: That demand at a node cannot be counted as covered unless at least one facility that is able to cover that demand node is located. The problem can be divided into two separate problems: One for the coverage variables, and one for the location variables. The sub problem for the coverage variables can be solved by inspection, and the location variable sub problem requires only sorting. The Lagrangian relaxation approach can typically solve instances of the problem with hundreds of demand nodes and candidate sites in few seconds or minutes on today’s computers, even though the problem is technically NP-hard. Reference [

Reference [

In many cases, the average distance (or time) that a patient must travel to receive service or the average distance that a physician must travel to reach his/her patients is of interest. The

To formulate this problem, additional input and new decision variables are needed. The objective function aims at minimizing the demand-weighted total distance (or time). The constraints state that each demand node must be assigned to exactly one facility site, a demand node can only be assigned to open facility sites, and exactly

As in the case of the maximal covering problem, a variety of heuristic algorithms have been proposed for the

For moderate-sized problems, Lagrangian relaxation works quite well for the un-capacitated

The use of field hospitals is a relatively new experience for Egypt and the Middle East, and it is found to be greatly contributing to managing the crisis created by COVID-19. It is suggested that governments should focus on building field hospitals in parks and public places that are currently unutilized. In addition, establishing field hospitals in rural areas that have hospitals with limited capacities would be the most beneficial because of the great pressure those hospitals are facing. This great pressure eventually forces them to refuse taking any more patients who are in dire need of medical attention. Moreover, the continuing increase in the admission of coronavirus cases to regular hospitals is causing patients of other diseases to face medical negligence. Field hospitals would create space in other hospitals need to treat and focus on other critical diseases.

Field hospitals are prepared similarly to regular hospitals; they are equipped with all medical devices and beds required for the complete and efficient treatment of patients. For coronavirus cases, the patient needs a bed and medical attention if the condition is intermediate, and an artificial ventilator and intensive care if the condition is severe. The most important aspect of field hospitals is that their attention is directed to COVID-19 patients only; therefore, all necessary precautions will be taken to deal with those patients only (unlike other hospitals, which have other diseases to deal with as well).

International experiences with field hospitals have proved their effectiveness in dealing with the crisis caused by the unbelievably quick spread of COVID-19. In January 2020, the Chinese authorities in Wuhan constructed one of the largest field hospitals in only 10 days to welcome and treat the large numbers of COVID-19 patients. It was built on an area of about 60 thousand square meters and equipped with approximately a thousand beds and intensive care units. This hospital served a crucial role in providing medical care to coronavirus patients, in addition to the other 11 field hospitals that were built in cities that faced trouble in treating coronavirus patients. In Europe, France solved the problem of bed shortage in hospitals by announcing the building of a few field hospitals. Italy collaborated with Russia to prepare a field hospital in Pergamon that offered 145 beds to take care of coronavirus patients. Meanwhile, the Spanish authorities resorted to transforming exhibition halls in Madrid to field hospitals that were able to receive COVID-19 patients and provide 5,500 beds and 500 intensive care units. A field hospital was also opened in London, and it took only 9 days to construct within a huge conference center; initially, it offered only 500 beds, each equipped with an artificial ventilator and oxygen, but ended up offering 4,000 beds. In the United States, authorities in New York transformed Central Park into a field hospital with 68 medical beds. In North America, Brazil opened 9 field hospitals in Rio de Janeiro; the largest was built on an area of 13,000 square meters and contained 500 beds, including 100 beds in intensive care units. In the Middle East, Dubai’s government established a field hospital that provided about 3000 beds and 800 intensive care units, and in Saudi Arabia, a field hospital that provided treatment for 100 patients at a time was established in Mecca.

As noted in Section 1, location models are application-dependent, which means that their mathematical model is formulated related to the specific location problem under consideration. The formulation of the location of field hospitals for COVID-19 is like the maximal covering location problem. Thus, this is the problem adopted in the present study.

The proposed mathematical model for the location of field hospitals is formulated in this section.

Define the following decision variables:

In addition, define the following inputs:

i) Number of Hospitals Constraint:

The number of field hospitals is limited by the available resources (budget, doctors, other medical staff, equipment, and consumed materials); see

ii) Covered Demand Constraints:

Demand at governorate _{c}_{c}

^{(2.513459 * 10(−8))}, as shown in

Then, the following covered demand constraints can be obtained:

When _{j}_{c}

iii) Binary Constraints: All decision variables are binary (see

Constraint

iv) The Objective Function:

In this location problem, the objective is to maximize the number of patients covered by the established field hospitals (see

A real application of the model is presented here. The model is applied to the 8 governorates in Upper Egypt, namely, Al-Fayoum, Beni Suef, El-Minya, Assiut, Sohag, Qena, Luxor, and Aswan. In view of the increase in cases of COVID-19 infections, it is necessary to increase the capacity of healthcare facilities by establishing additional field hospitals for medical quarantine and treating. This is to be done under limited finances, medical staff, and necessary equipment. The 8 governorates are briefly described below.

—Al-Fayoum:

Al-Fayoum governorate is located in Upper Egypt in a low location of the Western Desert, southwest of Cairo; it has an area of about 1,827 km^{2} and a population of 3.9 million people [

—Beni Suef:

Beni Suef governorate is located on the western bank of the River Nile, 110 km south of Cairo, and has a population of about 3.4 million people [

—El-Minya:

El-Minya governorate is located in the center of Egypt, on the River Nile. Its population is about 5.9 million [

—Assiut:

Assiut governorate is located in the center of Egypt on the River Nile. This governorate’s population is about 4.8 million people [

—Sohag:

Sohag governorate is located in Upper Egypt, south of the Assiut governorate and north of the Qena governorate. Its area is nearly 1547 km^{2}, and it has a population of about 5.4 million people [

—Qena:

Qena governorate is located in Upper Egypt, 5–6 km from the River Nile and between the Arab and Libyan deserts. The area of the Qena governorate is about 1.851 km^{2}, and it is home to about 3.5 million people [

—Luxor:

The city of Luxor was built on the ruins of the ancient capital city of Taibeh, Luxor governorate is located in Upper Egypt, its area is about 55 km^{2}, and its population is about 1.4 million people [

—Aswan:

Aswan governorate has an area of about 679 km^{2} and is located directly on the eastern bank of the Nile River under the first waterfall; its population is about 1.6 million people [

According to reference [

The complete mathematical model for the application case study is formulated according to mathematical expressions (

Metaheuristic approaches have been developed for complex optimization problems with continuous variables [

There are many constraint handling techniques in the literature [

An optimization problem with constraints is worked out as follows:

Here,

The GSK algorithm contains two stages: Junior and senior gaining and sharing stages. All people acquire knowledge and share their views with others. People in the early stages of life (junior stage) gain knowledge from their networks, such as family members, relatives, and neighbors, and want to share this knowledge with others who might not be in their networks (this stems from a curiosity to explore others). These may not have the experience to categorize the people. In the same way, people in middle or later stages of life (senior stage) enhance their knowledge by interacting with various circles, such as friends, colleagues, and social media friends, and share their knowledge with the most suitable person (this stems from a desire to teach others). These people have the experience to judge other people and can categorize them as good or bad. The process described above can be mathematically formulated through the following steps.

Step 1:

To get a starting point of the optimization problem, the initial population must be obtained. The initial population is created randomly within the boundary constraints (

Here, t is the number of populations; and

Step 2:

The dimensions of junior and senior stages should be computed through

Here,

Step 3:

Junior gaining–sharing knowledge stage: In this stage, the young people gain knowledge from their small networks and share their views with other people who may or may not belong to their network.

Thus, individuals are updated as follows.

According to the objective function values, individuals are arranged in ascending order. For every _{t −1}) and worst (_{t+1}) to gain knowledge, and also choose randomly (_{r}

_{f}

Step 4:

Senior gaining–sharing knowledge stage: This stage comprises the impact and effect of other people (good or bad) on the individual. The updated individual can be determined as follows.

The individuals are classified into three categories (best, middle, and worst) after sorting individuals into ascending order (based on the objective function values).

The

For every individual _{t}

To solve problems in discrete binary space, a novel discrete binary gaining–sharing knowledge-based optimization technique (DBGSK) is presented. In DBGSK, the new initialization and the working mechanism of both stages (junior and senior gaining–sharing stages) are introduced over discrete binary space, and the remaining algorithms remain the same. The working mechanism of DBGSK is described below.

Discrete Binary Initialization:

A first population is obtained in GSK using

The round operator is used to convert the decimal number into the nearest binary number.

Discrete Binary Junior Gaining and Sharing stage:

This stage is based on the original GSK with _{f}

Case 1. When

There are three different vectors (_{t −1}, _{t+1}, _{r}^{3} combinations are possible (see

Results | Modified results | ||||
---|---|---|---|---|---|

Subcase (a) | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 1 | 1 | 1 | |

1 | 1 | 0 | 0 | 0 | |

1 | 1 | 1 | 1 | 1 | |

Subcase (b) | 1 | 0 | 0 | 1 | 1 |

1 | 0 | 1 | 2 | 1 | |

0 | 1 | 0 | −1 | 0 | |

0 | 1 | 1 | 0 | 0 |

Subcase (a): If _{t −1} is equal to _{t+1}, the result is equal to _{r}

Subcase (b): When _{t −1} is not equal to _{t+1}, then the result is the same as _{t −1} by taking −1 as 0 and 2 as 1.

The mathematical formulation of Case 1 is as follows:

Case 2. When

There are four different vectors (_{t −1}, _{t}_{t+1}, _{r}^{4} possible combinations (see

Results | Modified results | |||||
---|---|---|---|---|---|---|

Subcase (c) | 1 | 1 | 0 | 0 | 3 | 1 |

1 | 0 | 0 | 0 | 1 | 1 | |

0 | 1 | 1 | 1 | 0 | 0 | |

0 | 0 | 1 | 1 | −2 | 0 | |

Subcase (d) | 0 | 0 | 0 | 0 | 0 | 0 |

0 | 1 | 0 | 0 | 2 | 1 | |

0 | 0 | 1 | 0 | −1 | 0 | |

0 | 0 | 0 | 1 | −1 | 0 | |

1 | 0 | 1 | 0 | 0 | 0 | |

1 | 0 | 0 | 1 | 0 | 0 | |

0 | 1 | 1 | 0 | 1 | 1 | |

0 | 1 | 0 | 1 | 1 | 1 | |

1 | 1 | 1 | 0 | 2 | 1 | |

1 | 0 | 1 | 1 | −1 | 0 | |

1 | 1 | 0 | 1 | 2 | 1 | |

1 | 1 | 1 | 1 | 1 | 1 |

These 16 combinations can be divided into two subcases [(c) and (d)]; subcases (c) and (d) have 4 and 12 combinations, respectively.

Subcase (c): If _{t −1} is not equal to _{t+1}, but _{t+1} is equal to _{r}_{t −1}.

Subcase (d): If any of the condition arises where _{t −1} = _{t+1} = _{r}_{t}

The mathematical formulation of Case 2 is

Discrete Binary Senior gaining and sharing stage:

The working mechanism of the discrete binary senior gaining and sharing stage is the same as the binary junior gaining and sharing stage with _{f}

Case 1.

Three different vectors (

Results | Modified results | ||||
---|---|---|---|---|---|

Subcase (a) | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 1 | 1 | 1 | |

1 | 1 | 0 | 0 | 0 | |

1 | 1 | 1 | 1 | 1 | |

Subcase (b) | 1 | 0 | 0 | 1 | 1 |

1 | 0 | 1 | 2 | 1 | |

0 | 1 | 0 | −1 | 0 | |

0 | 1 | 1 | 0 | 0 |

Subcase (a): If

Subcase (b): If

Case 1 can be mathematically formulated in the following way:

Case 2.

There are four different binary vectors (

Results | Modified results | |||||
---|---|---|---|---|---|---|

Subcase (c) | 1 | 1 | 0 | 0 | 3 | 1 |

1 | 0 | 0 | 0 | 1 | 1 | |

0 | 1 | 1 | 1 | 0 | 0 | |

0 | 0 | 1 | 1 | −2 | 0 | |

Subcase (d) | 0 | 0 | 0 | 0 | 0 | 0 |

0 | 1 | 0 | 0 | 2 | 1 | |

0 | 0 | 1 | 0 | −1 | 0 | |

0 | 0 | 0 | 1 | −1 | 0 | |

1 | 0 | 1 | 0 | 0 | 0 | |

1 | 0 | 0 | 1 | 0 | 0 | |

0 | 1 | 1 | 0 | 1 | 1 | |

0 | 1 | 0 | 1 | 1 | 1 | |

1 | 1 | 1 | 0 | 2 | 1 | |

1 | 0 | 1 | 1 | −1 | 0 | |

1 | 1 | 0 | 1 | 2 | 1 | |

1 | 1 | 1 | 1 | 1 | 1 |

Subcase (c): When

Subcase (d): If any case arises other than (c), then the obtained result is equal to _{t}

The mathematical formulation of Case 2 is

The pseudo-code of DBGSK is presented in

The proposed mathematical model employs the proposed novel DBGSK algorithm, the parameters of which are presented in

Parameters of DBGSK | Considered values |
---|---|

NP | 300 |

k | 10 |

_{r} |
0.9 |

p | 0.1 |

_{f} |
1 |

Max number of iterations | 200 |

DBGSK runs on an Intel

Algorithm | Best (maximum) | Median | Average | Worst (minimum) | Standard deviation |
---|---|---|---|---|---|

DBGSK | 552 | 552 | 552 | 552 | 0.00 |

It can be obviously seen in

The optimum solutions for the problem with different numbers of field hospitals to be established (1, 2, or 3) are presented in

The main contributions of this paper can be summarized as follows.

A variant of the maximal coverage location model to formulate establishing of field hospitals for COVID-19 problem is proposed. The model is designed to suit a special problem formulation for placing some field hospitals in candidate locations while maximizing the number of covered patients.

A nonlinear binary constrained model is formulated for the given problem. The binary decision variables are establishing field hospitals in the chosen candidate sites and covering patients in different governorates.

The designed mathematical model and method for obtaining the optimum solution are applied to 8 governorates in Upper Egypt with different numbers of hospitals to be established.

The problem is solved by a novel discrete binary gaining–sharing knowledge-based optimization algorithm (DBGSK), which involves two main stages: Discrete binary junior and senior gaining and sharing stages, with a knowledge factor _{f}

DBGSK can find the solutions of the problem, with good robustness and convergence.

Suggestions for future research are as follows.

To propose other mathematical models’ formulations for the same problem comprising designing the objective function(s), decision variables, and constraints, and then to compare various proposed mathematical models.

To continue applying the problem to other regions of the country, to the whole country, and to other countries.

To build an online decision support system that can handle repetitive situations with timely updated data and, in turn, update the locations of field hospitals for COVID-19 to reflect the updated data.

To check the performance of the DBGSK approach in solving different complex optimization problems, and with different kinds of constraint handling methods.

The authors are grateful to the Deanship of Scientific Research, King Saud University, KSA, for funding through the Vice Deanship of Scientific Research Chairs.