Health Care Logistics Network Design and Analysis on Pandemic Outbreaks: Insights From COVID-19

Abstract

Health care systems throughout the world are under pressure as a result of COVID-19. It is over two years since the first case was announced in China and health care providers are continuing to struggle with this fatal infectious disease in intensive care units and inpatient wards. Meanwhile, the burden of postponed routine medical procedures has become greater as the pandemic has progressed. We believe that establishing separate health care institutions for infected and non-infected patients would provide safer and better quality health care services. The aim of this study is to find the appropriate number and location of dedicated health care institutions which would only treat individuals infected by a pandemic during an outbreak. For this purpose, a decision-making framework including two multi-objective mixed-integer programming models is developed. At the strategic level, the locations of designated pandemic hospitals are optimized. At the tactical level, we determine the locations and operation durations of temporary isolation centers which treat mildly and moderately symptomatic patients. The developed framework provides assessments of the distance that infected patients travel, the routine medical services expected to be disrupted, two-way distances between new facilities (designated pandemic hospitals and isolation centers), and the infection risk in the population. To demonstrate the applicability of the suggested models, we perform a case study for the European side of Istanbul. In the base case, seven designated pandemic hospitals and four isolation centers are established. In sensitivity analyses, 23 cases are analyzed and compared to provide support to decision makers.

Recently, a novel coronavirus emerged causing a pandemic. It came to be known as COVID-19 throughout the world ( 1 ). When Turkey reported its first case of COVID-19 on March 11, 2020 ( 2 ), there had already been more than 120,000 cases, causing 4,613 fatalities in 118 countries ( 3 ). About eight months after Turkey’s first case was reported, the total number of reported cases and deaths (by October 9, 2020) was 322,382 and 8,722, respectively ( 4 ). An epidemic outbreak has a huge impact on a society’s well-being, and its intensity is associated with the immune status of a person, person-to-person contact, the transmission rate, virulence, transportation modes, climate, and health services ( 5 ).

With regard to the deadly outbreaks over the last century, three influenza pandemics have been reported worldwide, and the recent outbreak (H1N1) has affected 13,591 people in Turkey alone ( 5 ). From the standpoint of the number of deaths, COVID-19 has already exceeded the number of recorded cases in the H1N1 pandemic outbreak by almost 24 times. Thus, effective protection methods, along with responsive health care systems, need to be vigorous if they are to control the rapid growth of COVID-19 and handle the case load in hospitals and other health care institutions. Because of the high case load in their health care institutions, several countries have struggled to maintain regular health care operations during the response stage of the COVID-19. For instance, the worldwide cancellation rate for surgical procedures between January and April 2020 is predicted to be 40% and 80% for cancer and benign diseases, respectively ( 6 ). Similarly, non-emergency operations have been cancelled or postponed in Turkey ( 7 ). These kinds of preventive action were taken because of the need for medical resources and to reduce the risk of spreading the pandemic in hospitals.

The demand for beds in both intensive care units (ICU) and non-intensive care units (non-ICU) and for access to ventilators is uncertain and unless fast and effective precautions are taken, a capacity deficiency in health care institutions is unavoidable. To mitigate this potential public health disaster, medical resource planning is essential. Even though cancelling routine health care services is considered a practical solution, its impact should not be underestimated. For instance, additional connected health problems and even deaths are likely to result from delays in time-sensitive cases such as cancer operations ( 6 ). These delays cannot be extended further as they could cause excessive demand on the capacity of health care institutions, just as they are regaining their pre-pandemic functionality. Health care institutions should, therefore, maintain their routine operations while responding to the demand increase caused by COVID-19. Even though this approach is accepted by some decision makers, determining designated hospitals ( 8 , 9 ) and directing COVID-19 patients to them is more effective in reducing the risk of contamination among health care professionals and other types of patient. Moreover, determining designated hospitals, which will treat individuals with moderate disease who only require non-ICU beds, while establishing isolation (Fangcang) hospitals for individuals with severe disease who need ICU beds or ventilators is another essential precaution that the authorities can take to keep the pandemic under control. Fangcang hospitals serve as field or shelter hospitals to isolate, treat, and monitor pandemic-related patients with minor symptoms. This is one of the most efficient approaches to control the spread of the disease and decrease deaths ( 10 ). Fangcang hospitals are established by transforming large public places into shelter hospitals, facilitating a rapid increase in the capacity of health care systems throughout an epidemic ( 10 ). Fang et al. ( 10 ) stated that the USA, the UK, and some other countries have been establishing hospitals similar to those in China.

This study primarily focuses on the key public health issues that are expected to increase in the response stage to the COVID-19 pandemic as authorities seek to offer health care services both to people who are infected with COVID-19 and to those who are not. One of the main objectives of this study is to improve the effectiveness of the response processes for the European side of Istanbul by establishing a methodology which provides location and resource optimization for health care institutions during pandemic outbreaks. In particular, the aim is to provide health care services to all infected patients—whether they have severe or moderate symptoms—within the limited capacity. Additionally, supplying isolation and treatment services to patients with mild symptoms is also a goal in the work to establish temporary isolation hospitals. Two multi-objective, multi-period mixed-integer mathematical programming models are developed, and sensitivity analyses are performed to imitate the nature of epidemic outbreaks.

The study is designed as follows: a literature review related to the logistical characteristics of a pandemic is provided. A problem statement, mathematical model formulations, and the lexicographic weighted Tchebycheff models are then proposed. We then outline the data used, and the case study conducted for the European side of Istanbul is discussed. Finally, we draw conclusions and discuss ideas for future research.

Literature Review

All around the world, the outbreak of the new infectious disease, COVID-19, badly affected logistics systems ( 11 ). New challenges in logistics and transportation operations brought by COVID-19 have motivated several recent studies. For example, Choi ( 12 ) cited disruptions to operations during COVID-19 and discussed business models for mobile service operations. Govindan et al. ( 13 ) proposed a decision support system for managing demand in health care. Further, Remko ( 14 ) and Choi ( 15 ) conducted studies on risk analysis in logistics associated with the COVID-19 pandemic. Ivanov ( 16 ) considered epidemic outbreaks as a special case of supply chain risk. The author examined the impact of current infectious diseases on supply chains by performing a simulation study. Queiroz et al. ( 17 ) analyzed the effects of COVID-19 on supply chains. These studies are examples of the literature that is already available dealing with COVID-19.

In addition, the significance of humanitarian supply chains has been increasing with the accelerating number of disasters around the world. As a natural disaster, an epidemic poses a severe threat to human life. Two comprehensive reviews have been conducted: Dasaklis et al. ( 18 ) on logistics in pandemics; and Adivar and Selen ( 19 ) on influenza.

In this section, Operations Research (OR) based applications for humanitarian and health care logistics are reviewed. We then analyze studies conducted on isolation hospitals. Mathematical modeling-based developments, such as linear programming (LP) ( 20 – 22 ), non-linear programming (NLP) ( 23 ), stochastic programming (SP) ( 24 – 28 ), and multi-objective programming (MOP) ( 29 – 33 ) to optimize solutions to logistic problems such as location, allocation, transportation, and routing are analyzed.

Managing resources is one of the main issues that authorities face during pandemic outbreaks. Within this context, a multi-objective model was proposed by Koyuncu and Erol ( 31 ) with the aim of allocating supplies for a possible influenza eruption. Similarly, a study was conducted on the transportation of medical supplies by Liu et al. ( 32 ). Besides supplies, optimizing the allocation and transportation of patients was examined by Sun et al. ( 34 ); they proposed a multi-period mathematical model to minimize the total patient travel time and the maximum distance to hospitals when ICU beds and ventilators are in limited supply. Specifically, for COVID-19, Sy et al. ( 22 ) developed an LP model to minimize the total number of deaths when COVID-19 medicines and other resources are limited in health care institutions. They analyzed the effects of different patient types in the context of how contagious they were.

Looking at location and allocation models in the context of epidemic responses, Jia et al. ( 35 ), Murali et al. ( 36 ), and Ramirez-Nafarrate et al. ( 37 ) developed tools to determine the location of dispensing facilities, in which mass medication strategies are considered. Jia et al. ( 35 ) applied traditional models to reopen mothballed emergency service sites built for bomb attacks, anthrax contamination, and smallpox. For the problem studied by Jia et al. ( 35 ), Lu and Hou ( 38 ) developed a heuristic algorithm based on an ant colony. The operational version of the model was studied by Murali et al. ( 36 ). To determine the number, capacities, and locations of temporary sites, Büyüktahtakın et al. ( 21 ) developed a multi-period mixed-integer model and performed a case study on the Ebola disease in the West Africa. The time taken to open and close these facilities is a key factor as shown by Liu et al. ( 23 ) who considered the H1N1 epidemic. Anparasan and Lejeune ( 39 ) developed an integer based LP model to determine the location of facilities, and the transportation of medical staff and patients for a cholera outbreak. As an optimization-based model for the COVID-19 outbreak, Yu et al. ( 33 ) proposed a location selection model for medical waste treatment centers, in which a multi-objective model was developed in Wuhan, China. The study achieved its objectives of minimzing the risks at facilities and the total costs.

Besides locating facilities, other issues that managers face are hospital management and planning during the epidemic outbreak. Some studies have looked at this area: Currie et al. ( 40 ) applied modeling and simulation based techniques to mitigate the unwanted impact of COVID-19. Weissman et al. ( 41 ) analyzed the impact in Philadelphia, USA of COVID-19 based demand on hospital resources such as ventilators and ICUs. They used a Monte Carlo simulation as their analyzing tool. Similarly, Moghadas et al. ( 42 ) used simulation to estimate hospital resources and emphasized the impact of self-isolation on the capacity of hospitals. Likewise, Keskinocak et al. ( 43 ) proposed an agent-based simulation methodology to analyze the epidemic-related surge in hospital admissions and the impact of non-pharmaceutical involvement policies in the case of COVID-19. Relating to the research proposed in this study, Swann et al. ( 44 ) analyzed the impact of dedicated health care services throughout an influenza outbreak. They also applied an agent-based simulation framework to support resource allocation.

Regular or field hospital location detection and resource planning are two important issues extensively studied in the literature. In addition to these essential problems, the location determination of isolation (or Fangcang) hospitals and health care services is another critical matter in pandemic outbreaks if the outbreak is to be brought under control and eventually ended. The importance of isolation hospitals has become better understood after the COVID-19 outbreak and the impact of these health care services has been analyzed extensively. As mentioned in Fang et al. ( 10 ), putting isolation hospitals into operation is one of the most effective approaches that can bring the transmission of disease under control and decrease the death rate. Isolation hospitals also support health care systems in the fight against pandemics by providing extra resources. In their study, Fang et al. ( 10 ) explain how they selected the location of Fangcang hospitals and their impact on the fight against COVID-19. Chen et al. ( 45 ) specified the advantages of Fangcang hospitals: they are easy to build; they fit into existing available spaces; they do not need heavy construction; and they isolate people from the public and their families and so limit transmission. They provide a place for medical treatment of infected people without imposing an excessive burden on other health care systems. Dickens et al. ( 46 ) claimed that institution-based isolation reduces transmission by approximately 57% compared with home-based isolation, which is high at about 20%. They also suggest that authorities should use dormitories or hotels if large spaces, such as stadiums, are not available. The model they proposed is validated by applying it to the city state of Singapore. Pan et al. ( 47 ) developed a differential equation-based model to define the spread of COVID-19 and to assess the impact of isolation hospitals on this progress. They concluded that the death rate provides evidence that isolation can significantly decrease the spread of the epidemic. Similar to Pan et al. ( 47 ), Cai et al. ( 48 ) analyzed the effects of isolation hospitals on the number of deaths. In their study, isolation hospitals are called “makeshift hospitals.” They also found that isolation hospitals have a great impact on the number of deaths. Besides mortality, they analyzed the effect of air temperature, relative humidity, and the air quality index on the spread of COVID-19 via a student t-test, a Mann-Whitney U test, Pearson’s analysis, and Spearman’s analysis. Zeng et al. ( 49 ) developed a revised version of a susceptible-exposed-infectious-removed (SEIR) model to analyze the impact of isolation hospitals on the mortality rate, as done in Pan et al. ( 47 ) and Cai et al. ( 48 ). They also suggested setting up isolation hospitals all over the world, since they significantly reduce the number of deaths compared with home-based isolation, as concluded in Dickens et al. ( 46 ). Shen et al. ( 50 ) aimed to explain the process of setting up an isolation hospital in the city of Hubei in China, which the author describes as the epicenter of the disease in China. They summarized the construction of drugstores, the allocation of chemists, the management of drugstores, and other medical services.

One of the main contributions of this study is to empower health care services to cope with large-scale outbreaks by determining how to designate hospitals and health care facilities for infected and non-infected patients. During COVID-19, hospital-based resources have been largely allocated to infected individuals, and regular services have been limited in the face of the transmission risk. For the benefit of the sustainability of routine medical services at the lowest risk, we suggest establishing COVID-19 hospitals, treating only infected people, and COVID-19-free hospitals, serving as normal hospitals. In this scenario, we would propose, in the first stage, a multi-period, three-objective mixed-integer model to find the number and location of designated pandemic hospitals. In addition, one of the research questions investigated by the proposed study is where to allocate the pandemic-related patients under resource restrictions. To the best of authors’ knowledge, similar to this study, only Sun et al. ( 34 ) have considered the multi-objective, multi-period resource and patient allocation problem among multiple hospitals during a pandemic. Unlike Sun et al. ( 34 ), we develop an integrated three-objective, multi-period facility location and allocation model to select designated pandemic hospitals and allocate infected patients while considering the distance infected patients have to travel, the routine medical services expected to be disrupted, and the density of the potentially affected population in the area where hospitals are established. Determining separate hospitals and allocating COVID-19 patients to these facilities is an effective way to control the current health care crisis. However, the suggested hospital network design could be strengthened if the number of infected patients is under control. For this reason, powerful intervention measures are required to slow down the spread of disease. In the second stage, we aim to optimize the number and locations of temporary isolation facilities to separate those infected patients with mild symptoms from other people to reduce the infection rate, and thus bring the pandemic under control. The positive effects of institution-based isolation have been extensively analyzed in the literature ( 46 – 50 ). Even though several studies have been published on setting up isolation hospitals and the regional selection criteria, none of the studies have proposed a methodology to determine the location of these types of health care facility. The closest study was conducted by Hashemkhani Zolfani et al. ( 51 ), in which the location of a temporary hospital is selected. They applied importance criteria through inter-criteria correlation and combined compromise solution methods to the selection process. Thus, we propose a multi-period, bi-objective, mixed-integer mathematical model to determine the optimal location and operation of isolation hospitals, while minimizing two-way distance between new facilities and potentially affected population in the areas where facilities are established. By applying the lexicographic weighted Tchebycheff method, the results of this research exemplify how to design effective location optimization with respect to a pandemic disease logistics. The proposed model is generic and can be adjusted to other regions. We perform the mathematical model on a real case of the ongoing COVID-19 pandemic in the European side of Istanbul.

In the next section, the problem statement and two developed mathematical models are described in detail.

Problem Statement and Methodologies

The coronavirus pandemic has dramatically affected health care systems around the world. As a first response, hospital-based resources have been repurposed to cope with the surge of COVID-19 patients. In addition, the risk of disease transmission in health care facilities has limited non-infected patient admissions. For this reason, routine health care services have been disrupted during the COVID-19 outbreak. While the actions described have increased the responsiveness of health care system for infected patients, another public health crisis is now expected as a result of the delays in the delivery of routine services. The need for separate health care institutions has thus emerged as the pandemic has progressed. We take designated pandemic hospitals to be full-fledged health care institutions which cancel their regular inpatient and outpatient services and only admit COVID-19 patients. At the same time, non-pandemic (COVID-free) hospitals maintain only their routine health care services. Within this concept, we suggest establishing designated pandemic hospitals to cope with infected patients and facilitate the reorganization of routine medical services.

As well as the responsiveness of the health care system, non-pharmacological efforts such as social distancing are critical. A term of isolation is described by the Ministry of Health of Turkey as “keeping infected individuals separated from susceptible people to prevent the spread of a disease” ( 52 ). To cope with the risk of transmission of the virus in public, home isolation is important. On the other hand, the risk of household transmission cannot be underestimated, more so for the elderly than the young as they are more susceptible to the current disease ( 53 ). Home isolation might, therefore, be ill advised for infected individuals who live with older age groups and people with chronic diseases. As an alternative to home isolation, isolation institutions could be set up in public buildings. The pioneer in isolation centers was the creation of the Fangcang shelter hospitals in Wuhan, the capital city of Hubei province in China, where the new coronavirus disease was first reported. In Wuhan, stadiums and exhibition centers were converted into the Fangcang shelter hospitals to control the rapid transmission of COVID-19 ( 10 ). Unlike other emergency field hospitals, Fangcang hospitals were used for various purposes such as patient isolation, triage, medical treatment, monitoring, and social engagement ( 45 ). In this study, we also address the location problem of temporary health care facilities for isolation, treatment, and disease monitoring of individuals with both mild and moderate symptoms.

Considering the logistic network schema represented in Figure 1, we propose a framework to determine the number and location of designated pandemic hospitals and temporary isolation hospitals in a case of an infectious disease outbreak. Three echelons are modeled in the suggested logistic network configuration: demand points, designated pandemic hospitals, and temporary isolation hospitals. From now on, designated pandemic hospitals and temporary isolation hospitals are referred to as DPHs and TIHs. The locations and capacities of candidate health care facilities are known parameters. At the strategic level, we search for where to establish the health care facilities. At the tactical level, we determine when to open the isolation centers and the duration that they will treat patients during the considered time interval. The maximum capacity of a DPH is assigned in the first time period and remains the same for the rest of the periods. By contrast, a TIH can start serving at any time period within the planning time horizon. All patients are sent directly to DPHs from demand points, and then infected individuals who need institution-based isolation are transferred to TIHs. Please note that the full capacity of hospitals is allocated for infected patients. With regard to the illness severity of the infected patients, we consider three types of individual. Severely, moderately, and mildly ill patients are denoted as patients-I, patients-II, and patients-III, respectively. Infected individuals vary in the resources that they occupy in two different types of facility.

Figure 1.

General schema of the suggested logistic network configuration.

The following assumptions have been made in developing the mathematical model:

• A patient-I, who is infected by or has succumbed to the infectious disease, requires intensive care unit resources (ICU beds and ventilators). Some of the patients-I will die, receiving only ICU treatment before discharge. Recovered patients-I transfer to non-ICUs and complete their treatment. Note that patients-I are not transferred to isolation centers since they are considered to have recovered from infection after their long hospital stays.

• A patient-II only occupies a non-ICU bed at the DPH. To achieve efficient use of limited resources in DPHs, patients with moderate symptoms are discharged before they completely recover from the disease. Rather than completing the isolation period at home, they should be transferred to TIHs.

• A patient-III is sent to the DPH first. They do not occupy resources in ICU beds, ventilators, or non-ICU beds. They should be transferred to TIHs for institution-based isolation. To imitate the real-world environment, we assume that a certain amount of patients-III are transferred back to DPH because of their deteriorating health conditions. Please note that a patient-III who is sent back to DPH is called a transferred patient-III. The attribute of a transferred patient-III is the same as that of a patient-II.

Mathematical Model Formulations

In this subsection two different multi-objective, multi-period mixed-integer programming (MIP) models are proposed. The flow chart of the suggested methodology is illustrated in Figure 2.

Figure 2.

The flow chart of the suggested methodology.

Note: DPH = designated pandemic hospital; TIH = temporary isolation hospital.

Model-I: The Designated Pandemic Hospital (DPH) Location Model

The goal of the DPH location model is to find the optimal location decisions for DPHs and allocation decisions for infected patients from demand points to DPHs. The notifications and formulation of the three-objective mathematical model are as follows:

Indices and sets
$d,D$ :	Index and set of demand points, $d\subset D$
$h,H$ :	Index and set of candidate designated pandemic hospital (DPH) locations, $h\subset H$
$t,m,T$ :	Time period, $t,m=0,..,T$
$d^{\prime }$ :	Dummy temporary isolation hospital
Parameters
${\mathrm{DHA}}_d^{t,m}$ :	Demand of patients-I (need intensive medical care) in district $d$ who arrive at time period $t$ and are transferred to non-ICUs at time period $m$
${\mathrm{DDA}}_d^{t,m}$ :	Demand of patients-I (need intensive medical care) in district $d$ who arrive at time period $t$ and are departed (died) at time period $m$
${\mathrm{DA}}_d^t$ :	Total demand of patients-I in district $d$ and arrive at time period $t$ ,
	where $\sum _{m=t}^T{\mathrm{DHA}}_d^{\mathrm{tm}}+{\mathrm{DDA}}_d^{\mathrm{tm}}={\mathrm{DA}}_d^t,,\forall d\in D,\forall t\in T$
${\mathrm{DB}}_d^{\mathrm{tm}}$ :	Demand of patients-II (need non-intensive medical care) in district $d$ who arrive at time period $t$ and are discharged at time period $m$
${\mathrm{DC}}_d^t$ :	Demand of patients-III in district $d$ who arrive at time period $t$
${\mathrm{DTC}}_{d^{\prime }}^{t,m}$ :	Demand of transferred patients-III (sent back from TIHs) who arrive at time period $t$ and are discharged at time period $m$
$\mathrm{di}{s}_{d,h}$ :	Distance between demand point $d$ and DPH $h$
$\phi$ :	Average distance between TIHs and DPHs
${\alpha }_h$ :	Population density in the district where DPH $h$ is located
$\mathrm{ws}{r}_h$ :	Weighted service rate of DPH $h,(0<\mathrm{ws}{r}_h<1,\forall h\in D)$
$IC{U}_h^{}$ :	Total number of ICU beds at DPH $h$
$nonIC{U}_h^{}$ :	Total number of non-ICU beds at DPH $h$
$Ve{n}_h^{}$ :	Total number of ventilators at DPH $h$
$I{R}^{}$ :	Intubation rate
$\mathrm{ls}$ :	Hospital length of the stay of patients-I (including ICU and non-ICU stay)
$\ln$ :	A large number
Decision variables
${\mathrm{PHA}}_{d,h}^{t,m}$ :	Number of patients-I who are allocated to DPH $h$ from demand point $d$ at time period $t$ and transferred to non-ICUs at time period $m$
${\mathrm{PDA}}_{d,h}^{t,m}$ :	Number of patients-I who are allocated to DPH $h$ from demand point $d$ at time period $t$ and departed (died) at time period $m$
${\mathrm{PB}}_{d,h}^{t,m}$ :	Number of patients-II who are allocated to DPH $h$ from demand point $d$ at time period $t$ and discharged at time period $m$
${\mathrm{PC}}_{d,h}^t$ :	Number of patients-III who are allocated to DPH $h$ from demand point $d$ at time period $t$
${\mathrm{TPC}}_{d^{\prime },h}^{t,m}$ :	Number of transferred patients-III (sent back from TIHs) who are allocated to DPH $h$ at time period $t$ and discharged at time period $m$
${\mathrm{ArrivedA}}_h^t$ :	Number of patients-I who arrive in DPH $h$ at time period $t$
${\mathrm{DischargedA}}_h^m$ :	Number of patients-I who are discharged or depart from ICU in DPH $h$ at time period $m$
${\mathrm{ArrivedTA}}_h^t$ :	Number of transferred patients-I (carried in ICUs first and then transferred to non-ICUs) who arrive in non-ICU in DPH $h$ at time period $t$
${\mathrm{DischargedTA}}_h^m$ :	Number of transferred patients-I (carried in ICUs first and then transferred to non-ICUs) who are discharged from non-ICU in DPH $h$ at time period $m$
${\mathrm{ArrivedB}}_h^t$ :	Number of patients-II who arrive in DPH $h$ at time period $t$
${\mathrm{DischargedB}}_h^m$ :	Number of patients-II discharge from DPH $h$ at time period $m$
${\mathrm{ArrivedTC}}_h^t$ :	Number of transferred patients-III (sent back from TIHs) who arrive in DPH $h$ at time period $t$
${\mathrm{DischargedTC}}_h^m$ :	Number of transferred patients-III (sent back from TIHs) who are discharged from DPH $h$ at time period $m$
${\mathrm{AvICU}}_h^t$ :	Number of available ICUs at DPH $h$ at time period $t$
${\mathrm{AvVen}}_h^t$ :	Number of available ventilators at DPH $h$ at time period $t$
${\mathrm{AvnonICU}}_h^t$ :	Number of available non-ICU beds at DPH $h$ at time period $t$
$x_h$ :	$\{\begin{array}{c}1,\mathrm{if}\mathrm{candidate}\mathrm{hospital}hi\mathrm{s}\mathrm{chosen}\mathrm{as}\mathrm{a}\mathrm{DPH}\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$

Three-Objective DPH Location Mathematical Model

$$\mathbf{Min}{\mathit{f}}_1\left(\mathit{x}\right)=\sum _{d=1}^D\sum _{h=1}^H\mathrm{di}{s}_{d,h}(\sum _{t=1}^T\sum _{m=t}^T({\mathrm{PDA}}_{d,h}^{t,m}+{\mathrm{PHA}}_{d,h}^{t,m})+\sum _{t=1}^T\sum _{m=t}^T{\mathrm{PB}}_{d,h}^{t,m}+\sum _{t=1}^T{\mathrm{PC}}_{d,h}^t)+\phi \left(\sum _{h=1}^H\sum _{t=1}^T\sum _{m=t}^T{\mathrm{TPC}}_{d^{\prime },h}^{t,m}\right)$$ (obj1)

$$\mathbf{Min}{\mathit{f}}_2\left(\mathit{x}\right)=\sum _{h=1}^H{\alpha }_h{x}_h$$ (obj2)

$$\mathit{Min}{\mathit{f}}_3\left(\mathit{x}\right)=\sum _{h=1}^H\mathrm{ws}{r}_h{x}_h$$ (obj3)

Subject to

$$\sum _{h=1}^H{\mathrm{PHA}}_{d,h}^{t,m}={\mathrm{DHA}}_d^{t,m},\forall d\in D,t,m\in T$$ (1)

$$\sum _{h=1}^H{\mathrm{PDA}}_{d,h}^{t,m}={\mathrm{DDA}}_d^{t,m},\forall d\in D,t,m\in T$$ (2)

$$\sum _{h=1}^H{\mathrm{PB}}_{d,h}^{t,m}={\mathrm{DB}}_d^{t,m},\forall d\in D,t,m\in T$$ (3)

$$\sum _{h=1}^H{\mathrm{PC}}_{d,h}^t={\mathrm{DC}}_d^t,\forall d\in D,t\in T$$ (4)

$$\sum _{h=1}^H{\mathrm{TPC}}_{d^{\prime },h}^{t,m}={\mathrm{DTC}}_{d^{\prime }}^t,\forall t,m\in T$$ (5)

$$\sum _{d=1}^D\sum _{m=t}^T{\mathrm{PHA}}_{d,h}^{t,m}+{\mathrm{PDA}}_{d,h}^{t,m}={\mathrm{ArrivedA}}_h^t,\forall h\in H,t\in T$$ (6)

$$\sum _{d=1}^D\sum _{m=t}^T{\mathrm{PB}}_{d,h}^{t,m}={\mathrm{ArrivedB}}_h^t,\forall h\in H,t\in T$$ (7)

$$\sum _{m=t}^T{\mathrm{TPC}}_h^{t,m}={\mathrm{ArrivedTC}}_h^t,\forall h\in H,t\in T$$ (8)

$$\sum _{d=1}^D\sum _{t=1}^m{\mathrm{PHA}}_{d,h}^{t,m}+{\mathrm{PDA}}_{d,h}^{t,m}={\mathrm{DischargedA}}_h^m,\forall h\in H,m\in T$$ (9)

$$\sum _{d=1}^D\sum _{t=1}^m{\mathrm{PB}}_{d,h}^{t,m}={\mathrm{DischargedB}}_h^m,\forall h\in H,m\in T$$ (10)

$$\sum _{t=1}^m{\mathrm{TPC}}_h^{t,m}={\mathrm{DischargedTC}}_h^m,\forall h\in H,m\in T$$ (11)

$$\sum _{d=1}^D\sum _{t=1}^m{\mathrm{PHA}}_{d,h}^{t,m}={\mathrm{ArrivedTA}}_h^t,\forall h\in H,m\in T$$ (12)

$$\sum _{d=1}^D\sum _{m=t}^T{\mathrm{PHA}}_{d,h}^{t,m}={\mathrm{DischargedTA}}_h^{t+\mathrm{ls}},\forall h\in H,t\in T|t\le T-\mathrm{ls}$$ (13)

$${\mathrm{DischargedTA}}_h^t=0,\forall h\in H,t\in T|t\le \mathrm{ls}$$ (14)

$${\mathrm{AvICU}}_h^0={\mathrm{ICU}}_h^{\}{x}_h,\forall h\in H,}$$ (15)

$${\mathrm{AvVen}}_h^0={\mathrm{Ven}}_h^{\}{x}_h,\forall h\in H,}$$ (16)

$${\mathrm{AvnonICU}}_h^0={\mathrm{nonICU}}_h^{\}{x}_h,\forall h\in H}$$ (17)

$${\mathrm{AvICU}}_h^{t+1}={\mathrm{AvICU}}_h^t+{\mathrm{DischargedA}}_h^{t+1}-{\mathrm{ArrivedA}}_h^{t+1},\forall h\in H,t\in T|t\le T-1$$ (18)

$${\mathrm{AvVen}}_h^{t+1}={\mathrm{AvVen}}_h^t+\mathrm{IR}({\mathrm{DischargedA}}_h^{t+1}-{\mathrm{ArrivedA}}_h^{t+1}),\forall h\in H,t\in T|t\le T-1$$ (19)

$$\begin{array}{c}{\mathrm{AvnonICU}}_h^{t+1}={\mathrm{AvnonICU}}_h^t-\hfill \\ \multicolumn{1}{c}{({\mathrm{ArrivedB}}_h^{t+1}+{\mathrm{ArrivedTA}}_h^{t+1}+{\mathrm{ArrivedTC}}_h^{t+1})}\\ +({\mathrm{DischargedB}}_h^{t+1}+{\mathrm{DischargedTA}}_h^{t+1}+{\mathrm{DischargedTC}}_h^{t+1}),\hfill \\ \multicolumn{1}{c}{\forall h\in H,t\in T|t\le T-1}\end{array}$$ (20)

$${\mathrm{AvICU}}_h^t\le {\mathrm{AvICU}}_h^0,\forall h\in H,t\in T$$ (21)

$${\mathrm{AvVen}}_h^t\le {\mathrm{AvVen}}_h^0,\forall h\in H,t\in T$$ (22)

$${\mathrm{AvnonICU}}_h^t\le {\mathrm{AvnonICU}}_h^0,\forall h\in H,t\in T$$ (23)

$${\mathrm{ArrivedA}}_h^t\le {\mathrm{AvICU}}_h^0,\forall h\in H,t\in T$$ (24)

$$\mathrm{IR}({\mathrm{ArrivedA}}_h^t)\le {\mathrm{AvVen}}_h^0,\forall h\in H,t\in T$$ (25)

$${\mathrm{ArrivedB}}_h^t+{\mathrm{ArrivedTA}}_h^t+{\mathrm{ArrivedTC}}_h^t\le {\mathrm{AvnonICU}}_h^0,\forall h\in H,t\in T$$ (26)

$$\sum _{d=1}^D{\mathrm{PC}}_{d,h}^t\le \ln x_h,\forall h\in H,t\in T$$ (27)

$${\mathrm{PHA}}_{d,h}^{t,m}\ge 0,\forall d\in D,h\in H,t,m\in T$$ (28)

$${\mathrm{PDA}}_{d,h}^{t,m}\ge 0,\forall d\in D,h\in H,t,m\in T$$ (29)

$${\mathrm{PB}}_{d,h}^{t,m}\ge 0,\forall d\in D,h\in H,t,m\in T$$ (30)

$${\mathrm{TPC}}_{d^{\prime },h}^{t,m}\ge 0,\forall h\in H,t,m\in T$$ (31)

$${\mathrm{PC}}_{d,h}^t\ge 0,\forall d\in D,h\in H,t\in T$$ (32)

$${\mathrm{AvICU}}_h^t\ge 0,\forall h\in H,t\in T$$ (33)

$${\mathrm{AvVen}}_h^t\ge 0,\forall h\in H,t\in T$$ (34)

$${\mathrm{AvnonICU}}_h^t\ge 0,\forall h\in H,t\in T$$ (35)

$${\mathrm{ArrivedA}}_h^t\ge 0,\forall h\in H,t\in T$$ (36)

$${\mathrm{DischargedA}}_h^t\ge 0,\forall h\in H,t\in T$$ (37)

$${\mathrm{ArrivedTA}}_h^t\ge 0,\forall h\in H,t\in T$$ (38)

$${\mathrm{DischargedTA}}_h^t\ge 0,\forall h\in H,t\in T$$ (39)

$${\mathrm{ArrivedB}}_h^t\ge 0,\forall h\in H,t\in T$$ (40)

$${\mathrm{DischargedB}}_h^t\ge 0,\forall h\in H,t\in T$$ (41)

$${\mathrm{ArrivedTPC}}_h^t\ge 0,\forall h\in H,t\in T$$ (42)

$${\mathrm{DischargedTPC}}_h^m\ge 0,\forall h\in H,m\in T$$ (43)

$$x_h^{\}\in \{0,1\},\forall h\in H}$$ (44)

The first objective function (obj1) minimizes the summation of expected distance traveled by infected individuals. The aim of the second objective function (obj2) is to activate the DPHs in less dense areas. The third objective function (obj3) minimizes the normalized weighted medical service (wsr) rate. Here, the goal is to establish DPHs in the health care facilities where patient admissions and surgical operations are relatively low.

With the help of the constraints in (1) and (2), the total number of infected patients with severe (patient-I) symptoms, who are supposed to be discharged or have died, at each demand point are allocated to the DPHs in each time period. Constraint (3) assigns all infected patients with moderate (patient-II) symptoms to DPHs. Constraint (4) sends all infected patients with mild (patient-III) symptoms to DPHs. Here, P $C_{\mathrm{dh}}^t$ denotes the number of patients-III presenting in hospital $h$ from demand point $d$ at time period $t$ , and it is assumed that patients-III do not take up resources at DPHs but are transported directly to the TIHs in the same time period in which they arrived. Here, unlike ${\mathrm{PC}}_{\mathrm{dh}}^t$ , we deal with patients who initially show mild symptoms but then need to be sent to a full-fledged hospital as their symptoms deteriorate. Constraint (5) indicates that the demand of infected patients who are sent back from TIHs are satisfied in each time period. Here, ${\mathrm{DTC}}_{d^{\prime }}^t$ represents the demand of patients-III who are transferred from TIHs because their symptoms deteriorate from mild to severe. Further, we let ${\mathrm{TPC}}_{d^{\prime }h}^{\mathrm{tm}}$ denote the number of patients transferred from TIHs to each DPH. We take the average distance between all candidate DPHs and possible TIHs to minimize the distance traveled by patients-III transferred from TIHs to DPHs. Constraints (6), (7), and (8) determine the total number of hospitalized patients-I, patients-II, and transferred patients-III in each hospital $h$ at time period $t$ , respectively. Constraints (9), (10), and (11) calculate the total number of discharged patients-I, patients-II, and transferred patients-III in each hospital $h$ at time period $t$ . Please note that (9) implies discharges from ICUs. Constraint (12) calculates the number of patients-I discharged from ICU and transferred to non-ICU in each DPH at time period $t$ . Constraint (13) determines the number of patients-I discharged from each hospital. Constraint (14) prevents any transferred patients-I being assigned before the first discharged time period. For instance, if patients-I are expected to be discharged from non-ICUs at the fourth time period after they are admitted to ICUs, (14) prevents transferred patients-I being assigned at previous time periods. Constraints (15–17) ensure that the initial capacity of ICU beds, ventilators, and non-ICU beds are not assigned unless the candidate hospital is established as a DPH. Constraints (18–20) update the capacity of hospital-based resources: ICU beds, ventilators, and non-ICU beds, at each hospital in each time period. With the help of (21–23), the available capacity of hospital-based resources cannot exceed the initial capacity. Constraints (24–26) prevent the allocation of patients exceeding the initial capacity at each hospital $h$ . Constraint (27) ensures that a patient-III is not allocated to a hospital which is not established as a DHP. Constraints (28–43) and (44) represent non-negativity and integrity constraints, respectively.

Model-II: The Temporary Isolation Hospital (TIH) Location Model

The aim of the TIH location model is to optimize the TIH location and infected patient allocation from DPHs to TIHs. Additional notations and the formulations of the bi-objective mathematical model are as follows:

Additional indices and sets
$t^{\prime },T$ :	Time period, $t^{\prime }=0,..,T$
$f,F$ :	Index and set of candidate temporary isolation hospital (TIH) locations, $f\subset F$
Additional parameters
${\mathrm{DIB}}_h^{t,m}$ :	Demand of patients-II in hospital $h$ who need institution-based isolation at time period $t$ and discharge at time period $m$
${\mathrm{DIC}}_h^{t,m}$ :	Demand of patients-III in hospital $h$ who need institution-based isolation at time period $t$ and discharge at time period $m$
$Ibe{d}_f^{}$ :	Total number of beds at TIH $f$
$p_{\mathrm{hf}}$ :	Distance between DPH $h$ and TIH $f$
${\theta }_{\mathrm{fh}}$ :	Distance between TIH $f$ and DPH $h$
${\beta }_f$ :	Population density in the district where TIH $f$ is located
$\epsilon$ :	Threshold value for minimum operating time period of open TIH
$\omega$ :	Threshold value for minimum usage of operating TIH $(0<\omega <1$ )
Additional decision variables
${\mathrm{IB}}_{h,f}^{t,m}$ :	Number of patients-II who are allocated to TIH $f$ from DPH $h$ at time period $t$ and discharged at time period $m$
${\mathrm{IC}}_{h,f}^{t,m}$ :	Number of patients-III who are allocated to TIH $f$ from DPH $h$ at time period $t$ and discharged at time period $m$
${\mathrm{ArrivedIB}}_f^t$ :	Number of patients-II who arrive in TIH $f$ at time period $t$
${\mathrm{DischargedIB}}_f^m$ :	Number of patients-II who are discharged from TIH $f$ at time period $m$
${\mathrm{ArrivedIC}}_f^t$ :	Number of patients-III who arrive in TIH $f$ at time period $t$
${\mathrm{DischargedIC}}_f^m$ :	Number of patients-III who discharge from TIH $f$ at time period $m$
${\mathrm{Avbed}}_f^t$ :	Number of available bed at TIH $f$ at time period $t$
$o_f^t$ :	$\{\begin{array}{c}1,\mathrm{if}\mathrm{candidate}\mathrm{facility}f\mathrm{starts}\mathrm{serving}\mathrm{as}\mathrm{a}\mathrm{TIH}\mathrm{at}\mathrm{time}\mathrm{period}t\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$
$s_f^t$ :	$\{\begin{array}{c}1,\mathrm{if}\mathrm{candidate}\mathrm{facility}f\mathrm{stays}\mathrm{open}\mathrm{at}\mathrm{time}\mathrm{period}t\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$

Bi-Objective TIH Location Mathematical Model

$$\mathbf{Min}{\mathit{f}}_4\left(\mathit{x}\right)=\sum _{h=1}^H\sum _{f=1}^F\sum _{t=1}^T(p_{h,f}+{\theta }_{f,h})o_f^t$$ (obj4)

$$\mathbf{Min}{\mathit{f}}_5\left(\mathit{x}\right)=\sum _{f=1}^F\sum _{t=1}^T{\beta }_f{o}_f^t$$ (obj5)

Subject to

$$\sum _{f=1}^F{\mathrm{IB}}_{h,f}^{t,m}={\mathrm{DIB}}_h^{t,m},\forall h\in H,t,m\in T$$ (45)

$$\sum _{f=1}^F{\mathrm{IC}}_{h,f}^{t,m}={\mathrm{DIC}}_h^{t,m},\forall h\in H,t,m\in T$$ (46)

$$\sum _{h=1}^H\sum _{m=t}^T{\mathrm{IB}}_{h,f}^{t,m}={\mathrm{ArrivedIB}}_f^t,\forall f\in F,t\in T$$ (47)

$$\sum _{h=1}^H\sum _{m=t}^T{\mathrm{IC}}_{h,f}^{t,m}={\mathrm{ArrivedIC}}_f^t,\forall f\in F,t\in T$$ (48)

$$\sum _{h=1}^H\sum _{t=1}^m{\mathrm{IB}}_{h,f}^{t,m}={\mathrm{DischargedIB}}_f^m,\forall f\in F,m\in T$$ (49)

$$\sum _{h=1}^H\sum _{t=1}^m{\mathrm{IC}}_{h,f}^{t,m}={\mathrm{DischargedIC}}_f^m,\forall f\in F,m\in T$$ (50)

$$\sum _{t=0}^T{o}_f^t\le 1,\forall f\in F$$ (51)

$$\sum _{m=0}^t{o}_f^m\ge s_f^{t+1},\forall f\in F,t\in T|t\le T-1$$ (52)

$$s_f^t-s_f^{t+1}\le 1-s_f^{t+m},\forall f\in F,t=1,..,T-m,m=2,..,T-1$$ (53)

$$o_f^t+s_f^t\le 1,\forall f\in F,t\in T$$ (54)

$$s_f^0=0,\forall f=1,..,F$$ (55)

$${\mathrm{Avbed}}_f^t\ge {\mathrm{Ibed}}_f^t,\forall f\in F,t\in T$$ (56)

$${\mathrm{Avbed}}_f^t\le {\mathrm{Ibed}}_f^{\}(o_f^t+s_f^t),\forall f\in F,t\in T}$$ (57)

$${\mathrm{ArrivedIB}}_f^t+{\mathrm{ArrivedIC}}_f^t\le {\mathrm{Ibed}}_f^t,\forall f\in F,t\in T$$ (58)

$$\begin{array}{l}Ibe{d}_f^{}\left(1-\left(s_f^{t+1}+o_f^{t+1}\right)\right)+Avbe{d}_f^{t+1}\ge Avbe{d}_f^t-\left(ArrivedI{B}_f^{t+1}+ArrivedI{C}_f^{t+1}\right)\\ +\left(DischargedI{B}_f^{t+1}+DischargedI{C}_f^{t+1}\right),\forall f\in F,t\in T|t\le T-1\end{array}$$ (59)

$$\begin{array}{c}{\mathrm{Avbed}}_f^{t+1}\le {\mathrm{Avbed}}_f^t-({\mathrm{ArrivedIB}}_f^{t+1}+{\mathrm{ArrivedIC}}_f^{t+1})\hfill \\ \multicolumn{1}{c}{+({\mathrm{DischargedIB}}_f^{t+1}+{\mathrm{DischargedIC}}_f^{t+1})}\\ +{\mathrm{Ibed}}_f^{\}(1-\sum _{m=0}^t{o}_f^m),\forall f\in F,t\in T|t\le T-1}\hfill \end{array}$$ (60)

$$1-(\frac{{\mathrm{Avbed}}_f^t}{{\mathrm{Ibed}}_f^{\}}})\ge \omega s_f^t,\forall f\in F,t\in T$$ (61)

$$\sum _{t=1}^T{s}_f^t\ge \epsilon \left(\sum _{t=0}^T{o}_f^t\right),\forall f\in F,t\in T$$ (62)

$${\mathrm{ArrivedIC}}_f^t\ge 0,\forall f\in F,t\in T$$ (63)

$${\mathrm{DischargedIC}}_f^m\ge 0,\forall f\in F,m\in T$$ (64)

$${\mathrm{ArrivedIB}}_f^t\ge 0,\forall f\in F,t\in T$$ (65)

$${\mathrm{DischargedIB}}_f^m\ge 0,\forall f\in F,m\in T$$ (66)

$${\mathrm{IC}}_{h,f}^{t,m}\ge 0,\forall h\in H,\forall f\in F,t,m\in T$$ (67)

$${\mathrm{IB}}_{h,f}^{t,m}\ge 0,\forall h\in H,\forall f\in F,t,m\in T$$ (68)

$${\mathrm{Avbed}}_f^t\ge 0,\forall f\in F,t\in T$$ (69)

$$o_f^t\in \{0,1\},\forall f\in F,t\in T$$ (70)

$$s_f^t\in \{0,1\},\forall f\in F,t\in T$$ (71)

The two objectives of the proposed model are demonstrated in equations (obj4) and (obj5). For the sake of the rapid transportation of infected patients to full-fledged hospitals and sharing of resources such as medical personnel, the accessibility to TIHs from DPHs is considered. The first objective function minimizes two-way distances between DPHs and TIHs. The second objective function is related to the density of the districts in which candidate TIHs are located. It is preferable to avoid placing isolation centers in densely populated areas. Here, the second objective aims to establish isolation hospitals in less densely populated areas. Constraints (45) and (46) ensure that the demand of patients-II and patients-III respectively are fulfilled. The constraints in (47) and (48) calculate the total number of arrivals (patients-II and patients-III) in each time period. With the help of constraints (49) and (50), the total number of discharged patients at each time period is determined. Constraints (51–58) assign the location of TIHs and the time interval in which that new facilities function. Constraint (51) prevents TIHs from starting to serve more than once during the outbreak. Constraint (52) implies that each TIH operating at the time period $t$ should have been open previously. Constraint (53) states that if a facility operates at time period t and stops accepting patients at time period t+1, it cannot treat new patients in the subsequent time periods. Constraint (54) prevents TIHs from starting to serve and operate at the same time. At the beginning of the pandemic, rather than being available, TIHs may initiate the services. Here, (55) controls each dorm not operating in time period 0 (t0). Constraints (56) and (57) assign the initial capacity of each TIH at the time period that facilities start treating patients. The given constraints also ensure that the total demand assigned to each dorm in time period $t$ cannot exceed its capacity. Constraint (58) prevents the model from assigning any entity to non-operating facilities. The constraints given also ensure that the number of patients admitted does not exceed the initial capacity. With the help of (59) and (60), the available capacity is updated in each facility. Constraint (61) ensures that the use of operating dorms does not exceed the identified threshold value. Otherwise, it would not function. Constraint (62) control the minimum total time period that an open TIH operates. Constraint (62) indicates that if a candidate facility is open, it should operate for a duration of at least $\epsilon$ time. Constraints (63–69) are non-negativity, and (70, 71) are integrity constraints.

The Lexicographic Weighted Tchebycheff Method

Since three and two objective functions are considered respectively for DPH and TIH location models and to determine efficient results from the Pareto frontier, a well-known method is used as applied in Samanlioglu ( 54 ). In multi-objective problems, objectives usually, conflict and only concession solutions are achievable ( 54 ). Therefore, to reach concession solutions the decision makers’ preferences and viewpoints need to be brought to a point of compromise. The lexicographic weighted Tchebycheff method is, therefore, used and the beneficial descriptions linked to the multi-objective problems are presented here as in Samanlioglu ( 54 ). Consider a multi-objective problem with $\omega ,\omega >1$ conflicting objectives, $minf\left(x\right)=\{f_1\left(x\right),f_2\left(x\right),\dots ,f_{\omega }(x)\}\mathrm{subject}\mathrm{to}x\in X$ that need to be optimized (minimized) concurrently. A Pareto optimal solution is defined as “supported” if there are non-negative multipliers ${\rho }_1,{\rho }_2,\dots ,{\rho }_{\omega }$ in a way that the solution gained is optimal, subject to a linear combination of objective functions: $min\sum _{\sigma =1}^{\omega }{\rho }_{\sigma }{f}_{\sigma }\left(x\right),\mathrm{subject}\mathrm{to}x\in X$ with the importance weights of ${\rho }_1,{\rho }_2,\dots ,{\rho }_{\omega }$ , else it is defined as “non-supported.” The lexicographic weighted Tchebycheff equations for three-objective DPH and bi-objective TIH location models can be developed as:

$$\mathrm{lex}\min \{\delta ,{\varsigma }^T(f\left(x\right)-f^{*}\left(x\right))\}$$ (72a)

$$\mathrm{subject}\mathrm{to}\delta \ge {\rho }_1(f_1\left(x\right)-{f_1}^{*}\left(x\right))$$ (72b)

$$\delta \ge {\rho }_2(f_2\left(x\right)-{f_2}^{*}\left(x\right))$$ (72c)

$$\delta \ge {\rho }_3(f_3\left(x\right)-{f_3}^{*}\left(x\right))$$ (72d)

$$\mathrm{and}(\mathrm{Obj}1,\mathrm{Obj}2,\mathrm{Obj}3\mathrm{and}1-44)$$ (72e)

where ${\rho }_{\omega }>0$ is the importance weights of objectives and $\sum _{\sigma =1}^3{\rho }_{\sigma }=1$ , and ${f_{\sigma }}^{*}\left(x\right),\sigma =1,2,3$ , are the “utopia” values, which are determined as ${f_{\sigma }}^{*}\left(x\right)=\underset{x\in X}{min}{f_{\sigma }}^{*}\left(x\right)-{\epsilon }_{\sigma },\sigma =1,2,3,({\epsilon }_{\sigma }>0)$ and ${\varsigma }^T=(111)$ .

$$\mathrm{lex}\min \{\delta ,{\varsigma }^T(f\left(x\right)-f^{*}\left(x\right))\}$$ (72f)

$$\mathrm{subject}\mathrm{to}\delta \ge {\rho }_4(f_4\left(x\right)-{f_4}^{*}\left(x\right))$$ (72g)

$$\delta \ge {\rho }_5(f_5\left(x\right)-{f_5}^{*}\left(x\right))$$ (72h)

$$\mathrm{and}(\mathrm{Obj}4,\mathrm{Obj}5\mathrm{and}45-71)$$ (72i)

where ${\rho }_{\omega }>0$ is the importance weights of objectives and $\sum _{\sigma =4}^5{\rho }_{\sigma }=1$ , and ${f_{\sigma }}^{*}\left(x\right),\sigma =4,5$ are the “utopia” values, which are determined as ${f_{\sigma }}^{*}\left(x\right)=\underset{x\in X}{min}{f_{\sigma }}^{*}\left(x\right)-{\epsilon }_{\sigma },\sigma =4,5({\epsilon }_{\sigma }>0)$ and ${\varsigma }^T=(11)$ .

Application: Data, Results, Sensitivity Analysis, and Discussion

Data Acquisition

For computational studies, the COVID-19 outbreak data from the European side of Istanbul in Turkey are used in the case study. We collected the data set recorded between June 29, 2020 and October 4, 2020. The specified time interval is divided into 14 periods. The data of infected patients were obtained from the official website of Republic of Turkey Ministry of Health (MoH) COVID-19 information page ( 55 ), and the official Turkey Radio and Television Corporation (TRT) ( 56 ). The demographic statistics of the part of the city being considered were taken from Istanbul Statistic Office ( 57 ), which were published by the Istanbul Metropolitan Municipality (IMM). Further, the information on public hospitals (PBH) and training and research hospitals (TRH) were gathered from the Statistic Report of Public Hospitals ( 58 ) published by the MoH of Turkey in 2017. For the data on the hospitals established for the COVID-19 outbreak, we used various sources such as the website of MoH of Turkey ( 59 , 60 ) and gazette news ( 61 , 62 ). Lastly, National Education Statistics ( 63 ), published by the Turkey’s Ministry of National Education (MoNE) in 2020, were used to collect the data on recommended temporary isolation hospitals (TIHs).

Infected Patients and Demand Points

The MoH started to publish the COVID-19 daily situation reports on June 29, 2020 ( 50 , 55 ). These official reports include data on new cases, hospitalizations, intubations, and discharges recorded in Turkey. Likewise, the number of infected cases, hospital admissions, and discharges reported in Istanbul are provided in the same official documents. Even though the number of daily hospitalizations has been published, intubation, ICU, and death rates have not been provided for the city of Istanbul. We, therefore, use the general data of Turkey to estimate the statistics for infected patients in Istanbul. Figure 3a gives the COVID-19 data for Turkey, while Figure 3b gives the data for Istanbul, both for the time period being studied. According to the data obtained from the official situation reports, 55,796 people were hospitalized, and 6,349 people were intubated in Turkey within the considered time period. The total number of COVID-19-related deaths reported between Jun 29, 2020 and October 4, 2020 is given as 3,344 on the official website of TRT ( 51 ).

Figure 3.

COVID-19 data of: (a) Turkey; and (b) Istanbul.

Together with the situation reports, the statistics published by TRT ( 56 ) are also taken into account. We assessed in particular the current number of intubated patients, the “intubation/death” ratio, and the daily number of recorded deaths. We conclude that all of the deaths recorded are from infected patients. The reason for this is that the product of the current number of intubated patients and the daily “intubation/death” ratio is equal to the daily number of deaths. By considering the data we gathered, we make the following assumptions in this study:

• 11% of daily hospitalized infected patients were intubated.

• 51% of intubated patients lost their lives to the disease.

• 50% of infected patients who needed intensive care were intubated (assumed from TRT Haber [56]).

In Table 1, we summarize the statistics that are used to generate the number of hospitalized patients. The European side of the city has a population of more than 10 million, approximately 65% of the whole city. First, we gather the daily COVID-19 data for Istanbul from the related reports. To project the number of infected individuals in the European side of Istanbul, all calculated values are multiplied by 0.65. To prevent loss in the number of infected patients, the values are not rounded up to the nearest integer.

Table 1.

Assumptions Based on Hospitalized Infected Patient Groups

Patient group	Percentage in daily hospitalization (%)	Death (%)	Recovery (%)
Patients-I (including intubated patients)	22	26	74
Patients-I (intubated)	11	51	49
Patients-II	78	0	100

As mentioned previously, recorded infections are classified into three different groups based on the severity of disease symptoms. Each patient group has different attributes in the considered system. Given the inadequacy of the local statistics, we take the analysis done by Weissman et al. ( 41 ) into account. Similar to Weissman et al. ( 41 ), we assume that the ICU and hospital length of stay of patients-I and patients-II follow a gamma distribution. Readers can find detailed information on the model parameters in the supplementary material of Weissman et al. ( 41 ). As in Weissman et al. ( 41 ), we agree that patients-I stay in ICUs for, on average, eight days, longer than one period; those who are transferred to non-ICUs complete their treatment within 21 days (three time periods) after they are admitted to ICUs. The mean time of hospitalization of patients-II is taken to be 12 days. We determine the length of the stay of patients-II at TIHs by considering the home isolation standards of the MoH. It is stated that if there is no indication for hospitalization, home isolation is terminated on the 14th day (two time periods) after discharge from hospital (64). Similarly, it is assumed that the length of stay of patients-III at TIHs is 14 days (two time periods). Transferred patients-III, whose symptoms deteriorate from mild to moderate, are supposed to arrive from isolation facilities. They take up the same resources for the same length of time as a patient-II uses at a DPH. Please note that we assume they are sent back to DPHs within the same time period that they are admitted in TIHs. We summarize the data used for the length of hospitalization in Table 2.

Table 2.

Length of Stay of Patient Groups at DPHs and TIHs

Patient group	Designated pandemic hospital (DPH)		Temporary isolation hospital (TIH)
	ICU stay (days)	Hospital stay (days)	Isolation (days)
Patients-I	gamma (32.47,0.27)	21	NA
Patients-II	NA	gamma (136.21,0.09)	14
Patients-III	NA	NA	14
Transferred patient-III	NA	gamma (136.21,0.09)	14

Note: DPH = designated pandemic hospital; TIH = temporary isolation hospital; ICU = intensive care unit; NA = not applicable.

We apply a simulation tool and generate stochastic parameters for the length of the stay of patients-I, patients-II, and transferred patients-III. Here, random numbers are obtained by using MATLAB R2018a (9.4.0.813654) 64-bit (maci64).

The European side of the city has 25 districts or municipalities and these are chosen as demand points. We distribute the estimated number of infected cases to districts by assigning percentages. Here, the number of residents and the population density are taken as base parameters. We calculate a rate for each demand point by following the notations and equations given below.

Parameters:

Formulations:

$\mathrm{re}{s}_d$ : Total number of the resident of district $d$ $\mathrm{su}{r}_d$ : Surface of district $d$ $\mathrm{de}{n}_d$ : Density of district $d$ $\mathrm{pe}{r}_d$ : Population percentage in district $d$ $n_d$ : Normalized density of district $d$ $\mathrm{cas}{e}_d$ : Percentage of infected cases emerged in district $d$ $In{f}_{}^{t,m}$ : Total number of infected people who arrive at time period $t$ and recover or die in time period m in the European side of Istanbul. ${\alpha }_d^{t,k}$ : $\mathrm{Number}\mathrm{of}$ infected people in district $d$ who arrive at time period $t$ and recover or die in time period k.

$\mathrm{de}{n}_d=\frac{\mathrm{re}{s}_d}{\mathrm{su}{r}_d},\forall d\in D$ (73a) $\mathrm{pe}{r}_d=\frac{\mathrm{re}{s}_d}{\sum _{d=1}^D\mathrm{re}{s}_d},\forall d\in D$ (73b) $n_d=\frac{\mathrm{Dde}{n}_d}{\sum _{d=1}^D\mathrm{de}{n}_d},\forall d\in D$ (73c) $\mathrm{cas}{e}_d=\frac{\mathrm{pe}{r}_d{n}_d}{\sum _{d=1}^D{n}_d},\forall d\in$ (73d) ${\alpha }_d^{t,m}=cas{e}_{d}In{f}_{}^{t,m},\forall d\in D,t,m\in T$ (73e)

The percentages obtained from (73d) are then multiplied by the total number of infections expected for the European side of Istanbul (73e). The demographic information of the considered part of the city is given in the Appendix. Figure 4 shows the distribution rate of infected people across the districts.

Figure 4.

Distribution of infected patients among the demand points.

Health Care Institutions

Two different types of health care facility, DPH and TIH, are considered to treat infected individuals during the COVID-19 pandemic. Figure 5 shows the location of candidate DPHs, which are colored red with an “H” sign, and candidate TIHs, which are colored blue with bed sign.

Figure 5.

Locations of candidate designated pandemic hospitals (DPHs) and temporary isolation hospitals (TIHs).

Candidate Designated Pandemic Hospitals

Twenty-three proper government-owned hospitals were assessed as candidate DPHs. Looking at resource availability and hospital specialties, 11 public hospitals (PBH) (h1–h11) and eight training and research hospitals (TRH) (h12–h19) were taken into the consideration. Furthermore, three large hospitals (h20, h21, h23) and one medium-capacity (h22) hospital were opened in the European side of Istanbul during the COVID-19 outbreak. By April 3, 2020 Istanbul Prof. Dr. Cemil Taşcıoğlu City Hospital (CH) (h20), which was undergoing reconstruction, had been reopened. The new numbers of hospital-based non-ICU and ICU bed resources were obtained from A-news ( 61 ). The annual patient admission and operation numbers were taken to be the same as the statistics given in the Public Hospitals Statistics Report ( 58 ). The capacity of Yeşilköy Prof. Dr. Murat Dilmener Emergency Hospital (h21) and Hadımköy Dr. İsmail Niyazi Kurtulmuş Hospital (h23) were taken from the website of the MoH and gazette news ( 59 , 62 ). Last, we obtain the data of Istanbul Başakşehir Çam and Sakura City Hospital (CH) from the website of the MoH ( 60 ). To estimate annual patient admissions and surgical operations of new established hospitals, we take the approximate number of beds in the hospital into consideration. Nevertheless, given the large capacity of h23, we multiply the annual medical services of h18 by three times to estimate the related statistics of h23. In DPHs, non-intensive care units (non-ICU beds), intensive care unit beds (ICU beds), and ventilators are taken into account as scarce resources. Please note that the number of ventilators is taken to be the same as the number of ICU beds and the number given for ICUs is the total of adult and pediatric ICUs. Here, we assume that all hospitals are equally qualified. Moreover, we consider that it is not preferable to establish pandemic hospitals in densely populated areas because of the risk of disease transmission. Therefore, the location and related demographic information on candidate hospitals’ location are also taken into account. Furthermore, to demonstrate the disruption to routine health care services, the weighted medical service rate (wsr), which illustrates the intensity of annual medical services at each hospital is calculated by following notations and equations (74a–74c):

Parameters:
$\mathrm{pa}{d}_h$ :	Annual patient admissions of DPH $h$
$o{p}_h$ :	Annual surgical operations of DPH $h$
$\mathrm{norpa}{d}_h$ :	Normalized annual patient admissions of candidate DPH $h$
$\mathrm{noro}{p}_h$ :	Normalized annual surgical operations of DPH $h$
$\mathrm{ws}{r}_h$ :	Annual medical service rate of DPH $h$
Formulation:
	$\mathrm{norpa}{d}_h=\frac{\mathrm{pa}{d}_h}{\sum _{h=1}^H\mathrm{pa}{d}_h},\forall h\in H$ (74a)
	$\mathrm{noro}{p}_h=\frac{o{p}_h}{\sum _{h=1}^{H}o{p}_h},\forall h\in H$ (74b)
	$\mathrm{ws}{r}_h=(0.5)\mathrm{norpa}{d}_h+(0.5)\mathrm{noro}{p}_h,\forall h\in H$ (74c)

Table 3 shows the service capacity, annual medical services with calculated weighted medical service rate (wsr), and location and demographic data of the candidate DPHs’ location.

Table 3.

Candidate Designated Pandemic Hospitals (DPHs)

ID	Candidate hospital	Capacity			Annual medical services			Location and population density
		Non-ICU bed	ICU bed	Total	Patient admission	Surgical operation	wsr	District	Population density (residents/ $\mathrm{k}{\mathrm{m}}^2$ )
h1	Arnavutköy PBH	201	16	217	1,077,196	3,543	0.018	Arnavutköy	624
h2	Avcılar Murat Kölük PBH	100	9	109	679,589	4,494	0.015	Avcılar	8,978
h3	Bahçelievler PBH	205	13	218	1,114,930	11,919	0.031	Bahçelievler	35,945
h4	Başakşehir PBH	100	8	108	888,825	6,229	0.020	Başakşehir	4,301
h5	Bayrampaşa PBH	100	4	104	1,124,800	4,747	0.020	Bayrampaşa	30,526
h6	Çatalca İlyas Çokay PBH	100	13	113	408,165	1,719	0.007	Çatalca	65
h7	Esenyurt Necmi Kadioğlu PBH	199	18	217	1,695,209	6,059	0.028	Esenyurt	22,200
h8	Eyüpsultan PBH	140	14	154	1,023,044	4,632	0.019	Eyüpsultan	1,757
h9	İstinye PBH	128	9	137	628,840	3,803	0.013	Sarıyer	1,962
h10	Kağıthane PBH	51	4	55	687,754	1,402	0.010	Kağıthane	29,868
h11	Silivri PBH	223	9	232	890,048	4,684	0.017	Silivri	6,679
h12	Şişli Hamidiye Etfal TRH	756	54	810	2,093,703	22,265	0.059	Şişli	7,563
h13	Bağcılar TRH	498	48	546	2,512,420	22,002	0.063	Bağcılar	32,397
h14	Bakırköy Dr. Sadi Konuk TRH	612	79	691	2,837,298	20,091	0.063	Bakırköy	7,905
h15	Gaziosmanpaşa Taksim TRH	600	63	663	1,397,639	9,013	0.030	Gaziosmanpaşa	40,997
h16	Haseki TRH	554	55	609	2,112,182	17,296	0.051	Fatih	29,539
h17	İstanbul TRH	507	49	556	1,864,927	16,922	0.048	Fatih	29,539
h18	Kanuni Sultan Süleyman TRH	1,010	33	1,043	3,539,366	25,252	0.079	Küçükçekmece	18,019
h19	Yedikule Chest Diseases and Thoracic Surgery TRH	350	22	372	459,131	3,877	0.011	Zeytinburnu	24,465
h20	Prof. Dr. Cemil Taşcıoğlu CH (Okmeydanı TRH)	709	81	790	2,558,136	20,557	0.061	Şişli	7,563
h21	Yeşilköy Prof. Dr. Murat Dilmener EH	576	432	1,008	3,539,366	25,252	0.079	Bakırköy	7,905
h22	Hadımköy Dr. İsmail Niyazi Kurtulmuş Hospital	42	59	101	1,124,800	4,747	0.02	Arnavutköy	624
h23	Başakşehir Çam and Sakura CH	2,682	490	3,172	10,618,098	75,756	0.238	Başakşehir	4,301

Note: PBH = public hospital; TRH = training and research hospital; CH = city hospital; EH = emergency hospital; ICU = intensive care unit; non-ICU = Non-intensive care unit; wsr = weighted service rate.

Candidate Temporary Isolation Hospitals

We follow the specified characteristics of Fangcang hospitals ( 10 ), which are classified as as having:

• large indoor and outdoor space;

• proper infrastructure to host large numbers of people who need medical care;

• location not in high-density areas; and

• accessibility appropriate for a full-fledged hospital.

Because of the benefit of established infrastructure and large indoor spaces, the candidate TIHs are selected from dormitories dependent on the Higher Education Loans and Dormitories Institution. To select the locations suitable for rapid vehicular entrance, the possible locations were examined subjectively by searching on Google Maps ( 65 ) and the IBB city map ( 66 ). One of the critical features of isolation centers is that it is not appropriate to establish them in densely populated areas. Capacity and related demographical data on TIHs are given in Table 4.

Table 4.

Candidate Temporary Isolation Hospitals (TIHs)

ID	Candidate facility	Capacity	Location and population density
			District	Population density (residents/ $\mathrm{k}{\mathrm{m}}^2$ )
f1	Atatürk Öğrenci Yurdu	3,555	Zeytinburnu	24,465
f2	Fatih Sultan Mehmet Öğrenci Yurdu	3,240	Gungoren	41,349
f3	Florya-Beşyol Öğrenci Yurdu	1,205	Küçükçekmece	18,019
f4	Kadırga Erkek Öğrenci Yurdu	723	Fatih	29,539
f5	Şişli Erkek Öğrenci Yurdu	198	Şişli	7,563
f6	Mahir İz Öğrenci Yurdu	500	Esenyurt	22,200
f7	Bahçeköy Yurdu	506	Sarıyer	1,962
f8	Kanuni Sultan Süleyman Yurdu	2,492	Başakşehir	4,301
f9	Sadabad Öğrenci Yurdu	752	Kağıthane	29,868

Others

The MoH has regularized the treatment protocols for COVID-19 ( 67 , 68 ). Although different approaches might be implemented by health care providers dealing with unusual symptoms or exceptional circumstances in a patient’s history, similar treatment protocols were adopted at all government-owned hospitals. We assigned the service for patients-I, both those who recover and those who do not, patients-II, and transferred patients-III, $|\mathrm{qh}{a}_h,|q{b}_h,|\mathrm{qd}{a}_h,|\mathrm{qt}{c}_h$ , as zero. However, the proposed model is generic, and decision makers can easily reflect the impact of different service rates of hospitals as long as proper parameters are available. Further, the locations of candidate hospitals (DPHs and TIHs) and the distances between demand points, DPHs, and TIHs were gathered from Google Maps ( 65 ). Please note that the shortest distances between the nodes are taken into the consideration.

Results

In this subsection, we provide the results for both three-objective and bi-objective, multi-period mathematical models, which were developed to locate DPHs and TIHs. Both mixed-integer optimization problems are solved in GAMS (33.1.0) by using the CPLEX optimization program. All generated cases are solved to optimality. First, we analyze the base cases. Then, Pareto optimal solutions constructed on the combinations of different weights, based on ( 54 , 69 ), are represented. Finally, we conduct a sensitivity analysis to investigate the effects of the non-pharmacological intervention, isolation, on facility location decisions.

Base Case Study

The base case is generated to research the location decisions of the health care facilities expected to be designated as DPHs in the European side of Istanbul. We first optimize the three-objective DPH location problem. Next, we solve the bi-objective TIH location model by using the results of the DPH location model. Different solutions based on various weights are also investigated, and the detailed evaluations are given in the following section. Depending on the precedence of decision makers, the most ideal case can be implemented. In this section, we investigate a sample of an efficient solution, Case 1 of DPH (DPH-C1), where weight vectors assigned as (1/3, 1/3, 1/3) and Case 1 of TIH (TIH-C1), where weight vectors assigned as (1/2, 1/2) are explained.

Designated Pandemic Hospitals

For normalization purposes, three objective functions are optimized individually, and we determine the utopia and nadir points of objectives as given in Table 5. The proposed three-objective problem is then altered to a single-objective mathematical model.

Table 5.

Solutions of Individually Optimized Objective Functions of DPH Location Model

	Min. ${\mathit{f}}_1(\mathit{x})$ : Total traveled distance	Min. ${\mathit{f}}_2(\mathit{x})$ : Density	Min. ${\mathit{f}}_3(\mathit{x})$ : Weighted service rate	${\mathit{f}}_{\mathit{l}}^{\mathit{u}}$	${\mathit{f}}_{\mathit{l}}^{\mathit{n}}$
${\mathit{f}}_1\left(\mathit{x}\right)$ (km)	48,260.683	262,950.834	198,124.065	48,260.683	2,62,950.834
${\mathit{f}}_2\left(\mathit{x}\right)$ (population/ $\mathit{k}{\mathit{m}}^2$ )	281,420	4,301	66,086	4,301	281,420
${\mathit{f}}_3(\mathit{x})$ (service rate)	0.598	0.238	0.061	0.061	0.598

Note: DPH = designated pandemic hospital; Min. = minimum.

In the base case (DPH-C1), as shown in Table 9, the total expected distance traveled by infected individuals is determined as 89,976.127 km while the total population density around the new established DPHs is recorded as 56,894 population/km². With this combination of priority weights, wsr, which gives the expected disruption of the medical services at DPHs, is measured as 0.164. In the considered case, Arnavutköy PBH (h1), Avcılar Murat Kölük PBH (h2), Bahçelievler PBH (h3), Çatalca İlyas Çokay PBH (h6), Eyüpsultan PBH (h8), İstinye PBH (h9), and Prof. Dr. Cemil Taşcıoğlu PBH (h20) are selected as DPHs. Also, 16 hospitals are selected as non-COVID hospitals and dedicated to non-infected individuals. The allocation plan for infected patients from hospitals to isolation centers is shown in Table 6.

Table 9.

Cases and Solutions of DPH Location Model

Case ID	Weight vectors			Objective function values				CPU (second)	DPH
	$w_1$	$w_2$	$w_3$	$z_1$	$f_1(x)$ (km)	$f_2(x)$ (population/ $k{m}^2$ )	$f_3\left(x\right)$ (service rate)
DPH-C1	1/3	1/3	1/3	0.065	89,976.127	56,894	0.164	174.180	h1, h2, h3, h6, h8, h9, h20
DPH-C2	0.50	0.25	0.25	0.069	77,772.163	80,487	0.208	85.726	h6, h7, h8, h14, h15, h20
DPH-C3	0.25	0.50	0.25	0.056	96,276.558	34,513	0.148	62.190	h1, h6, h7, h8, h9, h14
DPH-C4	0.25	0.25	0.50	0.056	96,236.136	56,327	0.121	93.115	h7, h8, h14, h19
DPH-C5	0.60	0.30	0.10	0.082	77,772.163	80,487	0.208	59.622	h6, h7, h8, h14, h15, h20
DPH-C6	0.10	0.60	0.30	0.032	116,193.472	18,640	0.113	141.871	h2, h8, h21
DPH-C7	0.30	0.10	0.60	0.049	83,385.798	140,318	0.104	77.268	h2, h3, h6, h10, h15, h19
DPH-C8	0.80	0.10	0.10	0.051	61,958.432	145,304	0.255	41.184	h1, h3, h7, h8, h11, h15, h16, h20
DPH-C9	0.10	0.80	0.10	0.033	119,786.727	15,276	0.147	68.368	h1, h4, h6, h8, h14, h22
DPH-C10	0.10	0.10	0.80	0.027	106,955.296	75,064	0.076	69.828	h2, h15, h19, h22
DPH-C11	0.40	0.40	0.20	0.073	87,422.088	52,991	0.239	119.250	h1, h4, h7, h8, h9, h11, h14, h20
DPH-C12	0.20	0.40	0.40	0.048	99,311.652	31,927	0.117	81.138	h6, h7, h8, h14
DPH-C13	0.40	0.20	0.40	0.061	80,824.180	88,366	0.120	141.976	h1, h2, h3, h6, h8, h15
DPH-C14	0.70	0.20	0.10	0.071	70,064.898	102,807	0.307	47.991	h4, h6, h7, h8, h14, h15, h18, h20
DPH-C15	0.20	0.10	0.70	0.046	97,549.292	86,544	0.096	124.341	h2, h3, h15, h22
DPH-C16	0.10	0.70	0.20	0.033	119,165.504	16,614	0.142	115.834	h4, h6, h8, h9, h14, h22
Average								93.9926

Note: CPU = Computational Processing Unit; DPH = designated pandemic hospital. $z_1$ = Weighted objective function value of DPH location model.

Table 6.

Suggested DPHs and Health Care Services

Established DPH		Health care services
ID	Name	Allocated demand points	Admission
			Patients-I	Patients-II	Transferred patients-III	Patients-III
h1	Arnavutköy PBH	d1, d6, d20, d23	47	31	60	88
h2	Avcılar Murat Kölük PBH	d2,d9, d14, d20	112	429	84	1,162
h3	Bahçelievler PBH	d3, d4, d5, d13, d18, d20, d25	180	1,218	155	3,269
h6	Çatalca İlyas Çokay PBH	d9, d11, d14, d22	18	24	0	68
h8	Eyüpsultan PBH	d7, d13, d15, d16, d17, d18, d23, d25	193	1,038	24	3,399
h9	İstinye PBH	d21	3	10	38	28
h20	Prof. Dr. Cemil Taşcıoğlu CH	d2, d3, d4, d5, d7, d8, d10, d13, d15, d16, d17, d18, d19, d24, d25	369	446	91	958
Total			922	3,196	452	8,972

Note: DPH = designated pandemic hospital; PBH = public hospital; TRH = training and research hospital; CH = city hospital.

The results shown in the table are rounded to the nearest integer.

Given the central location of Bahçelievler (h3), Eyüp PBH (h8), and Prof. Dr. Cemil Taşcıoğlu CH (h20), they treat patients sent from various demand points. Compared with h3, h8, and h20, fewer districts are served by Arnavutköy (h1), Çatalca İlyas Çokay PBH (h6), and İstinye PBH (h9). On the other hand, h1, h6, and h9 have the advantage of a lower weighted service rate (wsr). This means that the impact of postponed medical operations is relatively low in the considered hospitals. The location decisions for h1, h6, and h9 do not take advantage of service accessibility. However, a trade-off is made for the benefit of routine health care services. Moreover, we conclude that h3, h8, and h20 have the benefit of ease of access while the population density is relatively low at the locations where h1, h6, h8, and h9 are established. We calculate two measures of service level: the number of patients admitted and use of hospital resources. Figure 6 shows the percentage allocation of infected patients according to their illness severity. Our analysis demonstrates that almost 40% of the infected patients who require intensive care support (patients-I) are sent to h20, which has the largest capacity among other DPHs. Respectively, h3 and h8 accept 20% and 21% of patients who are severely symptomatic. When we consider the number of moderately ill patients (patients-II and transferred patients-III) who only occupy non-ICU beds at DPHs, we see that h3 and h8 gain importance, and these hospitals meet most of the demand from moderately symptomatic patients. Please note that we also include transferred patients-III, who are expected to be sent back from TIHs, in our analysis. We, therefore, conclude that the burden of moderately symptomatic patient admissions is mainly shared between h3 and h8. On the other hand, it is also important to note the number of patients transferred to non-ICUs from ICUs. Therefore, considering the percentage of patients-I admitted to h20, this hospital also has an important role in non-ICU services. Since mildly symptomatic patients (patients-III) do not take up resources at DPHs, they are assigned to the closest one. When we assess the percentage of admitted patients-III, the location advantage of h3 and h8 produces the largest number of patient-III admissions. Last, we conclude that almost 7% and 5% of patients who need ICU and non-ICU, respectively, are sent to h1, h6, and h9. Accordingly, because of their location advantages, h2, h3, h8, and h20 treat higher number of infected patients than h1, h6, and h9.

Figure 6.

Allocation of infected patients.

Note: PBH = public hospital; CH = city hospital.

A detailed analysis of the use of ICUs and non-ICUs at each DPH is illustrated in Figure 7. Our base case has 14 time periods, corresponding to 14 weeks between July 29 and October 4, 2020. We observe that hospital-based resources were used more efficiently at h2, h3, and h8. Here, h2, h3, and h8 saw above-average usage of their ICU and non-ICU resources compared with all DPHs in each time period. Considering the total capacities of the hospitals, Prof.Dr. Cemil Taşcıoğlu CH (h20) is the most capacious health care facility among other DPHs. Here, please note that h1, h2, h3, h6, h8, and h9 are around a quarter of the size of h20. Despite the high demand from patients-I satisfied at h20, the use of ICUs at the largest facility fluctuated mostly below the average of all DPHs. Similarly, the usage at h1, h6, and h9 varied below the average. On the other hand, usage at h3 and h8 was at full capacity during the considered time period. Here, the use of ventilators (Figure 7b) reached 50% at most, leading us to assume that only 50% of the patients in ICUs needed a ventilator.

Figure 7.

Usage of hospital-based resources: (a) ICU utilization; (b) Ventilator utilization; (c) Non-ICU utilization.

Note: PBH = public hospital; CH = city hospital.

Temporary Isolation Hospitals

The two-way distance minimization and the population density minimization are considered as objective functions for solving the TIH location problem. In seeking to achieve resource efficiency, frequent changes of location and low use of facilities are not desirable. For this reason, we establish two parameters in the TIH location model: the threshold value of facility usage, and operation duration of new facilities. By changing the mentioned parameters, decision makers can evaluate various scenarios through the model and decide on the most appropriate plan. Please note that we generate the base case where each functioning TIH operates at least three consecutive time periods. Also, we set the usage threshold as 60 % at operating facilities. The optimization of the bi-objective TIH location model is carried out in the same manner as we apply in the three-objective DPH location model. The results of single-objective function values, which are minimized individually, are given in Table 7. We evaluate the sample solution, TIH-C1, where the weight vectors are equal (1/2, 1/2). In Table 10, the total expected two-way distance between DPHs and TIHs is determined as 1,342 km while the total population density in the districts where TIHs established is recorded as 77,321 population/ km².

Table 7.

Solutions of Individually Optimized Objective Functions of TIH Location Model

	Min. ${\mathit{f}}_4(\mathit{x})$ : Two-way distance	Min. ${\mathit{f}}_5(\mathit{x})$ : Density	${\mathit{f}}_{\mathit{l}}^{\mathit{u}}$	${\mathit{f}}_{\mathit{l}}^{\mathit{n}}$
${\mathit{f}}_4(\mathit{x})$ (km)	1,263.100	1,378.200	1,263.100	1,378.200
${\mathit{f}}_5(\mathit{x})$ (population/ km2)	84,989	57,083	57,083	84,989

Note: Min. = minimum.

Table 10.

Cases and Solutions of TIH Location Model

TIH Case ID	Weight vectors		Objective function values			CPU (second)
	$w_4$	$w_5$	$z_2$	$f_4(x)$ (km)	$f_5(x)$ (population/km2)
TIH-C1	0.50	0.50	0.363	1,342	77,321	26.861
TIH-C2	0.40	0.60	0.315	1,353.7	57,412	15.878
TIH-C3	0.20	0.80	0.157	1,353.7	57,412	10.393
TIH-C4	0.60	0.40	0.295	1,317.5	77,650	22.961
TIH-C5	0.80	0.20	0.20	1,263.1	84,989	43.281
TIH-C6	0.70	0.30	0.30	1,263.1	84,989	56.458
TIH-C7	0.30	0.70	0.236	1,353.7	57,412	17.954
Average						27.6837

Note: TIH = temporary isolation hospital: CPU = Computational Processing Unit; $z_2$ = weighted objective function value of TIH location model.

Please note that the number of patients who are expected to be discharged from DPHs is kept the same as that given in Case 1 (DPH-C1) of the DPH location model. Figure 8 shows the expected demand for institution-based isolation. We assume that all moderately symptomatic patients (patients-II and transferred patients-III) who only take up non-ICU beds at DPHs, and mildly ill patients (patients-III) require institution-based isolation. All patients are directly transferred to TIHs. In addition, we assume that a certain number of patients-III are expected to be transferred back to full-fledged DPHs within the same time period they are admitted to TIHs. We, therefore, take the 5% of the mildly symptomatic patients (patients-III) out of the expected number of demands. Patients-II and patients-III who are transferred from DPHs act completely the same as each other at TIHs and stay at TIHs for two time periods, which correspond to 14 days.

Figure 8.

Number of infected patients who reqiure instution-based isolation in each DPH.

Note: DPH = designated pandemic hospital; PBH = public hospital; CH = city hospital.

In the considered case, Florya Beşyol Erkek Öğrenci Yurdu (f3), Kadırga Erkek Öğrenci Yurdu (f4), Şişli Erkek Öğrenci Yurdu (f5), and Mahir İz Öğrenci Yurdu (f6) are established as new TIHs. In Table 8, we observe that all the new established TIHs accept infected individuals from all DPHs excluding Şişli Erkek Öğrenci Yurdu (f5). Each facility treats infected patients at different time periods. F3, f4, f5, and f6 are opened at time period 0 (t0), 1 (t1), 0 (t0), and 10 (t10), respectively. As shown in Table 8, approximately 60% of infected patients are sent to f3. Here, f3 starts receiving infected patients at time period 1 (t1) and treats them the end (t14). In addition, f4 is opened at time period 1 (t1) and starts accepting patient at time period 2 (t2) and treats them for 13 time periods. On the other hand, f5 and f6 operate for only six and three time periods, and they are opened at different time periods. In Figure 8, we show that the number of patients who need institution-based isolation tends to decrease at time period 5 (t5) and 6 (t6), when f5 stops operating. Furthermore, a meaningful increase in demand at time period 11 (t11) causes f6 to be operating.

Table 8.

Suggested TIHs and Health Care Services

		Allocated demand points	Health care services
Established TIHs				Service interval
ID	Name		Number of admissions	Open	Operation interval	Service duration (period)
f3	Florya-Beşyol Erkek Öğrenci Yurdu	h1, h2, h3, h6, h8, h9, h20	7,076	t0	t1–t14	14
f4	Kadırga Erkek Öğrenci Yurdu	h1, h2, h3, h6, h8, h9, h20	3,443	t1	t2–t14	13
f5	Şişli Erkek Öğrenci Yurdu	h1, h2, h3, h8, h9	594	t0	t1–t6	6
f6	Mahir İz Öğrenci Yurdu	h1, h2, h3, h6, h8, h9, h20	678	t10	t11–t13	3
Total			11,791

Note: TIH = temporary isolation hospitals.

Numbers shown in the table are rounded to the nearest integer.

Figure 9 illustrates the operating TIHs and DPHs where infected patients are sent at each time period. TIHs are allowed to be open for one time period. Moreover, infected patients are accepted as long as the TIH status is shown as “operating.” No infected individuals are sent to the related facility once the status is turned to “not operating.” For instance, the opening period of Şişli Erkek Öğrenci yurdu (f5) is time period 0 (t0). The relevant facility starts admitting infected patients at time period 1 (t1). It functions for six time periods and stops admitting new patients at time period 7 (t7). Please note that the considered facility does not admit new patients while operating in time period 2 (t2), but its resources are occupied by patients admitted during previous time periods. It is assumed that the patients who are admitted but do not complete the isolation term before time period 6 (t6), and are discharged after they complete their isolation term. These are not represented in Figure 9.

Figure 9.

Time duration TIHs are operating and accepted hospitals.

Note: TIH = temporary isolation hospitals.

We conclude that opening only seven DPHs and four TIHs is sufficient to serve all infected patients. To ensure the efficient use of TIHs, they can be established whenever they are needed. In the considered case, f3 and f4 treat patients for 14 and 13 time periods, while f5 and f6 are admitting patients for six and three time periods in the planning time horizon. We select the DPH location at the strategic level, which is set at the beginning since establishing the infrastructure of full-fledged hospitals is more complicated. On the other hand, the location and operation duration decisions of TIHs is made at strategic and tactical levels respectively.

Sensitivity Analysis

First, we analyze DPH location model and the results of 16 cases generated by varying weight vectors. Seven more cases are then constructed to evaluate the results of TIH location model. Finally, we discuss the positive impact of isolation hospitals by decreasing the number of infected individuals.

In Table 9, the weight vectors and 16 different cases, along with their optimal objective function values and selected DPHs, are represented. It is assumed that the rest of the resources, not shown in the Table 9, are dedicated to non-infected patients.

To interpret the relationship between the first and second objective function values, we first evaluate the cases where the third objective function weights are the same. In Case 14 (DPH-C14), Case 8 (DPH-C8), Case 9 (DPH-C9), and Case 5 (DPH-C5) where the weight of the third objective function is 0.1, the number of opened DPHs decreases respectively. When the priority is given to distance minimization, the number of opened DPHs is expected to increase. However, a comparison of Case 5 (DPH-C5) with Case 9 (DPH-C9) demonstrates that the number of opened hospitals does not directly influence the distance traveled by infected patients. Here, the density of the areas where DPHs are established gains importance. For instance, instead of establishing h1, h4, and h22 where the density is relatively low, h15, h17, and h20 are found to be more appropriate to achieving the first objective function value. A similar situation is observed in Case 2 (DPH-C2) and Case 3 (DPH-C3). It is demonstrated that more hospitals are established at areas of lower population density. However, for the sake of distance minimization, the trade-off is not made. Further, we analyze the interchanged dispersed weight vectors between $f_1$ and $f_3$ while keeping $w_2$ as 0.1. In Case 7 (DPH-C7), Case 8 (DPH-C8), Case 10 (DPH-C10), and Case 15 (DPH-C15), where $w_2$ is taken as 0.1, the relationship between the number of opened DPHs and $f_3$ is relatively evident. As the third objective function values reduce, the number of established DPHs decreases as well. Here, the objective function value of $f_1$ increases in parallel. Case 2 (DPH-C2) and Case 4 (DPH-C4) demonstrate similar consequences. In Case 6 (DPH-C6), Case 9 (DPH-C9), Case 10 (DPH-C10), and Case 16 (DPH-C16), where $w_1$ is assigned as 0.1, the interchanged weight vectors between $f_2$ and $f_3$ affect the location of DPHs instead of the total number. We conclude that the decisions made for the DPH location model by changing weight vectors, do not have a straight impact on the total number, yet the location decisions vary in each case.

Table 10 shows the conducted cases for the TIH location model and their solutions. For the TIH location model, seven cases were conducted based on the represented weight vectors given in Table 10. Please note that here we use the parameters of the base case of the DPH location model.

In Figure 10, the green solid double and single lines represent nadir and utopia values of the first objective function. Furthermore, green round dots stand for the first objective function values obtained in each case. Similarly, nadir and utopia values of the second objective function are demonstrated by the yellow solid double and single lines. Round dots colored yellow represent second objective function values. Sharp alternations between two objective function values are not observed. For instance, in Case 2 (TIH-C2) and Case 3 (TIH-C3), where $w_4$ is assigned as 0.4 and 0.2, respectively, the same objective function values are produced. As shown in Table 11, f3, f5, f7, and f9 are opened and operate almost at the same time interval in the considered cases (TIH-C2 and TIH-C3). Furthermore, it is interesting to note that in Case 5 (TIH-C5) and Case 6 (TIH-C6), where the priority is given to distance minimization, the first objective function value reaches its utopia point, and the second objective function value reaches its nadir point.

Figure 10.

Objective function values of TIH location model.

Note: TIH = temporary isolation hospitals.

Table 11.

Suggested TIHs and Operation Periods in Cases

Case ID	Dispersed weight vector		TIHs and operation periods
	$w_4$	$w_5$	ID	Open	Operating time interval	Duration (time period)
TIH- C1	0.5	0.5	f3	t0	t1–t14	14
			f4	t1	t2–t14	13
			f5	t0	t1–t6	6
			f6	t10	t11–t13	3
TIH- C2	0.4	0.6	f3	t0	t1–t14	14
			f5	t3	t4–t6	3
			f7	t10	t11–t13	3
			f9	t1	t2–t14	13
TIH- C3	0.2	0.8	f3	t0	t1–t14	14
			f5	t1	t2–t6	3
			f7	t10	t11–t14	4
			f9	t1	t2–t14	13
TIH- C4	0.6	0.4	f3	t0	t1–t14	14
			f5	t3	t4–t6	3
			f6	t10	t11–t14	3
			f9	t1	t2–t13	12
TIH- C5	0.8	0.2	f3	t0	t1–t14	14
			f4	t1	t2–t13	12
			f5	t0	t1–t4	4
			f9	t11	t12–t14	3
TIH- C6	0.7	0.3	f3	t0	t1–t13	13
			f4	t11	t12–t14	3
			f5	t0	t1–t4	4
			f9	t1	t2–t14	13
TIH- C7	0.3	0.7	f3	t0	t1–t13	13
			f5	t2	t3–t5	3
			f7	t11	t12–t14	3
			f9	t1	t2–t14	13

Note: TIH = temporary isolation hospital.

The suggested TIHs and operation periods in each case are provided in Table 11. In all cases, it is suggested that f3 and f5 be opened. When the second objective function is prioritized (TIH-C2, TIH-C3, and TIH-C7), f7 and f9 are established. Instead of f7, f6 or f4 is preferred when distance minimization is considered first. Evaluating the time interval that facilities functioning, each facility varies. For instance, f3 and f5 operate for between 13 and 14 and between three and six time periods in all cases. F6 operates for three time periods in TIH-C1 while the same facility is available for three time periods in TIH-C4. F9 operates in 13 time periods in TIH-C2, TIH-C3, TIH-C6, and TIH-C7 and demonstrates different treating periods in TIH-C4 and TIH-C5. In TIH-C2, TIH-C3, and TIH-C7, f7 operates for three, four, and three time periods, respectively.

The average CPU times of the 16 cases of Model-I and the seven cases of Model-II are received as 93.99 s and 26.68 s, respectively. In Model-I, the shortest results take about 41.18 s (DPH-C8), and the longest case is received in 174.18 s (DPH-C1). Likewise, in Model-II, the shortest and longest cases are reported as 10.39 s and 43.28 s, respectively. Decisions at the strategic level apply a long-term planning horizon of more than three years ( 70 ). Both of multi-objective multi-period optimization problems, where long-lasting decisions are obtained, are solved in less than 3 min. Since we consider strategic decisions in this research paper, which will not be solved repeatedly, the computational times of mathematical models are acceptable ( 54 , 71 ). We are able to solve the proposed optimization problems in a reasonable time with CPLEX methodology, a heuristic can be developed to solve larger problems, which is a future research direction of this study.

Impact of Temporary Isolation Hospitals

One of the effective approaches in the control of disease transmission is isolating infected individuals from susceptible people in the community. The study done by Dickens et al. ( 46 ) investigated the effect of home-based and institution-based isolation on the total number of infections. They indicated that home-based isolation causes a 20% reduction while a 57% reduction is observed if institution-based isolation is adopted. In Turkey, the current policy strongly suggests home-based isolation intervention for confirmed cases with mild symptoms ( 56 ). Since home-based isolation has been adopted in Turkey, we make assumptions about the effects of institution-based isolation on the number of new infections. By following the estimate made by Keskinocak et al. ( 43 ), we accept that a 46% decline in the number of COVID-19 patients is expected during the considered time interval. Table 12 represents the solution of individually optimized objective function values of the DPH and TIH location models. Three-objective and bi-objective problems are altered to a single-objective mathematical model. Please note that the usage threshold of TIH is taken as 0.6 and TIH should operate at least three time periods as long as they are established.

Table 12.

Solutions of Individually Optimized Objective Functions of DPH and TIH Location Models

DPH location model
	Min. ${\mathit{f}}_1(\mathit{x})$ : Total traveled distance	Min. ${\mathit{f}}_2(\mathit{x})$ : Density	Min. ${\mathit{f}}_3(\mathit{x})$ : Weighted service rate	${\mathit{f}}_{\mathit{l}}^{\mathit{u}}$	${\mathit{f}}_{\mathit{l}}^{\mathit{n}}$
${\mathit{f}}_1\left(\mathit{x}\right)$ (km)	26,031.334	195,734.657	276,803.324	26,031.334	276,803.324
${\mathit{f}}_2\left(\mathit{x}\right)$ (population/ km2)	311,024	3,070	41,062	3,070	311,024
${\mathit{f}}_3(\mathit{x})$ (service rate)	0.653	0.064	0.037	0.037	0.653
TIH location model
	Min ${\mathit{f}}_4(\mathit{x})$ : Two-way distance		Min ${\mathit{f}}_5(\mathit{x})$ : Density	${\mathit{f}}_{\mathit{l}}^{\mathit{u}}$	${\mathit{f}}_{\mathit{l}}^{\mathit{n}}$
${\mathit{f}}_4(\mathit{x})$ (km)	800.600		1,764.337	800.600	1,764.337
${\mathit{f}}_5(\mathit{x})$ (population/ km2)	89,170		4,844.829	4,844.829	89,170

Note: DPH = designated pandemic hospital; TIH = temporary isolation hospital; Min. = minimum.

Table 13 illustrates the results of the DPH and TIH location models and the suggested facilities. Only four DPHs and three TIHs are recommended if the total amount of infected individuals decreases by 46%. Here, we obtain the fourth objective function value as 885.5 and the fifth objective function value as 61,264 when we optimize the TIH location model. In the considered case, f4 and f5 operate for 13 and 12 time periods, while f6 and f7 admit patients for three and four time periods. F4, f5, f6, and f7 are opened at t1, t3, t12, and t2, respectively.

Table 13.

Solutions of DPH and TIH Location Models

Weight vectors			Objective function values				CPU (second)	Facilities
${\mathit{w}}_1$	${\mathit{w}}_2$	${\mathit{w}}_3$	${\mathit{z}}_1$	${\mathit{f}}_1(\mathit{x})$ (km)	${\mathit{f}}_2\left(\mathit{x}\right)$ (population/ $\mathit{k}{\mathit{m}}^2$ )	${\mathit{f}}_3\left(\mathit{x}\right)$ (service rate)
DPH location model
1/3	1/3	1/3	0.039	55,288.382	18,705	0.104	155.977	h2, h6, h8, h14
${\mathit{w}}_4$		${\mathit{w}}_5$	${\mathit{z}}_2$	${\mathit{f}}_4(\mathit{x})$ (km)	${\mathit{f}}_5(\mathit{x})$ (population/ $\mathit{k}{\mathit{m}}^2$ )
TIH location model
0.50		0.50	0.335	885.5	61,264		17.094	f4,f5, f6, f7

Note: DPH = designated pandemic hospital; TIH = temporary isolation hospital; CPU = Computational Processing Unit.

Discussions

The COVID-19 outbreak has posed a severe threat to human life, not only as a result of the danger of coronavirus-2 itself but also the illnesses caused by disrupted medical services. At the beginning of the ongoing pandemic, health care providers all over the world took preparatory actions for the influx of infected patients. For instance, hospital-based resources such as operating theaters were repurposed for infected individuals needing intensive care support. In parallel, given the risk and fear of transmission, a sharp drop in hospital admissions of non-infected patients emerged. During the time period when resources have largely been allocated to infected individuals, the backlog of non-COVID patients has grown. It has been almost two years since the first case was announced. However, the battle with COVID-19, in which health care providers and professionals are engaged, is not over yet. Besides, the burden placed on health care systems has become heavier because of the delays to medical operations. Scholars have highlighted the impact of the COVID-19 pandemic on health care systems and routine medical services ( 6 , 72 – 75 ). Furthermore, studies ( 8 , 9 , 76 , 77 ) have emphasized the need to separate health care services and resources between COVID and non-COVID patients. The coronavirus disease has been a black box for health care professionals. It was not possible to establish separate hospitals for infected patients because of huge unknowns about the novel coronavirus and scarcity of existing resources. Today, health care managers and professionals have gained experience and knowledge about how to struggle with the disease. They now have a better understanding of prevention, diagnosis, and treatment methods. As well as separate pandemic hospitals, isolation centers can flatten the rate of disease transmission and help to control the ongoing outbreak. Therefore, separate health care institutions such as COVID-19, COVID-19-free hospitals, and isolation centers can be established as long as the safe measures are strictly followed.

As the coronavirus 2019 disease emerged, dedicated COVID-19 hospitals and health care facilitates took on an important role in controlling the outbreak. For example, China implemented institution-based isolation in Wuhan, where the first COVID-19 case was reported. Large public venues were converted into isolation hospitals, called Fangcang shelter hospitals, to isolate and monitor patients with mild infections ( 10 ). Together with isolation hospitals, designated hospitals where patients with moderate and critical conditions are treated have also been established. Similar measures were taken in countries such as South Korea and India ( 78 , 79 ). Furthermore, the impact of institution-based isolation has been widely examined by scholars ( 46 – 50 ). Real-life experience from these facilities demonstrates the applicability of the suggested methodology. As well as its application, the studies which investigated the effects of facility-based isolation demonstrate the non-negligible effect of dedicated health care facilities on the spread of the disease.

Our study, therefore, supports the experience gained in the ongoing pandemic. Even though this policy has not been adopted in Istanbul, these findings could support such an approach in future in the part of the city we studied.

Managerial Implications

The results of this study lead to beneficial suggestions for health care managers in the era of COVID-19. The increasing number of infected patients and the risk of the spread of the disease have caused serious disruption to routine health care services. Because of the backlog of non-infected patients, health care professionals except huge loss in the near future. One of the most beneficial managerial implications of this study is that health care managers are able to select the appropriate number and location of DPHs, treating only COVID-19-related patients. Decision makers can evaluate the outcomes to reduce the circulation of infected patients, the impact of COVID-19 on routine health care services, and address the risk of transmission of the disease in areas where COVID-19 hospitals are established. Nevertheless, as long as the number of infected patients grows, COVID-19-free hospitals cannot be maintained against the backdrop of the resource scarcity caused by excessive numbers of infected patients being admitted to hospital. The transmission of COVID-19 should, therefore, also be controlled by effective intervention measures such as facility-based isolation. In the second stage, the results present the strategy and implications for health care managers seeking to control the transmission of COVID-19, while deciding on the location of TIHs. Health care managers can also use the TIH location model to efficiently manage the operation, duration, and usage of new health care facilities. The findings of this study could also assist the authorities in controlling the transmission of the disease. These effective epidemic measures not only affect the logistics of health care systems but also suppliers, facilities, and markets in other industries as well. The weight vectors, 16 and 7, are generated for DPH and TIH location models, respectively. By doing so, and with regard to the significance of objective functions, managers will be able to prioritize each objective function. Although the two mathematical models are integrated, managers will be free to apply either one of them depending on their preference. However, we encourage the decision makers to consider both practices.

Conclusions

In this research paper, we provide a framework for health care logistic networks to determine the number and location of DPHs and temporary isolation hospitals (TIHs). Two integrated multi-objective location and allocation models are developed. First, a three-objective mathematical model for DPH location selection is shown. Specifically, the suggested model considers: (1) the distance traveled by infected patients; (2) the population density around the facilities; and (3) the state of usage routine service at hospitals. First, we aim to mitigate the transmission risk of disease in the community while infected individuals are traveling to hospitals. Second, the developed model enables DPHs to be established in areas that are not densely populated. Third, health care hospitals where regular health services, such as outpatient and inpatient activities are not in high demand are considered better options to become DPHs. Furthermore, the sharing of resources between patients who are severely and moderately symptomatic is taken into account. We reflect the uncertainty of the length of hospitalization of infected patients by conducting a simulation analysis. In the second stage of this study, a TIH location mathematical model selects the appropriate isolation facilities under capacity, usage, and demand constraints. There are two objectives that can be achieved simultaneously, namely the minimization of both the two-way distance traveled between TIHs and DPHs and the population density in the areas where the TIHs are established. Here we give due regard to resource sharing between two different facilities and the transportation of infected individuals whose health status improves from severe from mild. As long as the proper facilities are available, TIHs can be set up promptly whenever needed during an outbreak. The performance of the suggested mathematical model is analyzed with a case study based on the COVID-19 pandemic in the European side of Istanbul, Turkey. The considered part of the city has multiple districts, which are considered demand points. In a case an outbreak of an infectious disease, the expected demand in each district will differ according to its population size and density. We construct our solution model considering the impact of these demographic features of the demand points. The base case is assessed, and then different cases are generated based on various weight vectors to show the effectiveness of multi-objectivity. By doing so, we enable decision makers to prioritize various criteria. Lastly, a sensitivity analysis is conducted to predict the positive effect of institution-based isolation on facility location decisions.

We encountered some limitations in this study. Even though using simulation tools for the length of hospital stay demonstrates the stochastic changes in resource usage at DPHs, a similar approach cannot be performed for TIH location model because of insufficient information. To achieve efficient resource planning, we consider patients who are transferred to the different departments. However, patients can be admitted to various departments for several time periods stochastically. Even though we demonstrate the impact of institution-based isolation on location decisions by decreasing the certain amount of demand, the infection attract rate of pandemic in the population may vary depending on various parameters. To imitate prospective studies and achieve more realistic results for institution-based isolation, various modeling approaches, compartmental models, generic programming, and machine learning technics can be conducted. In addition, given the overwhelming number of infected patients, variables are set as continuous and the results obtained are rounded to the nearest integer. A heuristic methodology can be developed to overcome this limitation. For future studies, stochastic parameters such as patients retransferred to ICUs and to non-ICUs may be included to simulate the capacity uncertainty. Lastly, separating pandemic hospitals and stopping regular services can be challenging for health care managers since patients with chronic diseases tend to need treatment in the place where that are treated regularly. Furthermore, establishing isolation centers in various facilities at various time periods requires a strong planning strategy. Putting this into practice can be tough. However, the decrease in both transportation time and infection risk, along with the increase in service quality will be worthwhile in the long term.