A reliable emergency logistics network for COVID-19 considering the uncertain time-varying demands

Abstract

The evolving COVID-19 epidemic pose significant threats and challenges to emergency response operations. This paper focuses on designing an emergency logistic network, including the deployment of emergency facilities and the allocation of supplies to satisfy the time-varying demands. A Demand prediction-Network optimization-Decision adjustment framework is proposed for the emergency logistic network design. We first present an improved short-term epidemic model to predict the evolutionary trajectory of the epidemic. Then, considering the uncertainty of the estimated demands, we construct a capacitated multi-period, multi-echelon facility deployment and resource allocation robust optimization model to improve the reliability of the decisions. To address the conservativeness of robust solutions during the evolution of the epidemic, an uncertainty budget adjustment strategy is proposed and integrated into the rolling horizon optimization approach. The results of the case study show that (i) the short-term prediction method has higher accuracy and the accuracy increases with the amount of observed data; (ii) considering the demand uncertainty, the proposed robust optimization model combined with uncertainty budget adjustment strategy can improve the performance of the emergency logistic network; (iii) the proposed solution method is more efficient than its benchmark, especially for large-scale cases. Moreover, some managerial insights related to the emergency logistics network design problem are presented.

1. Introduction

At the end of 2019, a novel coronavirus COVID-19 was diagnosed and subsequently spread globally. Since initial identification, despite active actions, now it becomes a global pandemic, which is a big threat and challenge to world health and economy. Many problems were exposed in the early stage of the epidemic due to the surging demand, such as limited supply, unreasonable and inefficiency allocation of prevention resources (such as masks, respirators, sickbeds, etc.)1 ^, 2 especially at the beginning of the outbreak (Bown, 2022). The emergency logistics network (ELN) bridges the supply side and the demand side. A disordered and inefficient ELN may exacerbate the shortage of emergency resources and delays in delivery. Therefore, it is necessary to conduct an in-depth study on the optimization of ELN to improve the efficiency of emergency relief and reduce the loss of people’s lives and properties. However, in the process of joint prevention and control of the epidemic, we are confronted with many difficulties in designing the ELN, such as estimating the time-varying demands and deploy the emergency facilities and resources. Motivated by that, this paper considers a joint problem of uncertain demand estimation and emergency resources deployment.

The first problem we need to concentrate on is the time-varying uncertain emergency demands. In terms of the theoretical foundation of epidemic transmission, existing literature proposes some methods to model the projections of the epidemic’s trajectory such as the medical prediction method, and differential equation models (Ekici et al., 2014). Among them, the susceptible–exposed–infectious–recovered–dead (SEIRD) model is widely used for characterizing epidemics. Nevertheless, the general static SEIRD model cannot fit well for real-time predictions due to the temporal variability of the epidemic. Considering the time-varying characteristics of the demands, mathematical modeling based on dynamical equations such as difference equations can provide more detailed insights into the prospect of epidemics. To settle this problem, we propose a dynamic updated SEIRD model (uSEIRD) based on the observation and make a more accurate short-term prediction. Thus, we divide the entire planning horizon into multiple periods and make short-term forecasts via the uSEIRD model in which the parameters are periodically fitted based on the updated data.

Another problem is the limited emergency facility location and resource allocation considering time-varying demands. Most of the studies construct multi-period optimization models for emergency facility location and resource allocation (Dönmez et al., 2021, Liu et al., 2023). But few of them consider the impacts of disaster evolution on the emergency facility deployment and resource allocation decisions, which usually leads to higher operational costs and transportation costs in reality. Therefore, dynamic configuration of emergency facilities and resources is needed to reduce relief cost and avoid runs on the medical system. We consider a capacitated multi-period, multi-echelon resource location–allocation problem, which is dynamic and responsive to the time-varying demands (Garrido et al., 2015, Garrido and Aguirre, 2020).

Although the demands can be estimated according to the predicted epidemic trend (Wu et al., 2020), uncertainty still exists. To ensure that the ELN remains operational as the pandemic evolves, the existing literature employs some methods to tackle the uncertainties. Stochastic optimization (SO) and robust optimization (RO) are two common methods to improve the feasibility of decisions. The SO method requires accurate probability distributions of uncertain parameters. However, due to the limited historical data, the distributions cannot be exactly known, so the SO method may not be applicable to the ELN design problem (Conejo et al., 2021). By contrast, RO relaxes the requirement of exact probability distribution and assumes the uncertain demands belong to an uncertainty set, which fits the context of this problem (Long et al., 2021, Cui et al., 2023). Furthermore, considering the time-varying demands during the epidemic, we use the multi-period RO approach to ensure the reliability of the plans by considering the worst-case (Li et al., 2020). However, the decision errors still cannot be eliminated in the decision process (such as the difference between the quantity of the allocated resources and actual demands). Therefore, we propose a dynamic adjustment strategy of uncertainty set after observing the realized uncertainty. Unlike affine and finite adaptability robust methods, which are mainly mathematically driven, our approach uses the difference between the allocated and actual demands in the current period to adjust the uncertainty set in the next decision cycle.

Usually, these problems incorporating time structure are very large and cannot be solved to global optimality by general methods such as Lagrangian decomposition and Benders decomposition within a reasonable time. Considering the multi-period nature of the optimization procedure, the rolling horizon (RH) approach is often adopted. This approach aims to solve the problem periodically, including additional information from the consequent periods. The reasons for choosing this method are as follows. First, under an emergency circumstance, managers may need to obtain an acceptable solution for the current decision point, which can be achieved by the RH method. Second, the RH method emphasizes flexibility of decisions using up-to-date information on uncertain parameters. As the iteration proceeds, decisions are obtained based on the fixed previous executed decisions, resulting in a reduction of the computation cost (Chand, 1983).

In summary, this paper focuses on designing an ELN in response to the COVID-19 epidemic considering the uncertain time-varying demands. The research objective of this study is to integrate short-term demand estimation, dynamic emergency facility deployment, resource allocation, decision feedback and adjustment into a framework to improve the operational efficiency of the ELN and the utilization of limited emergency supplies. This research has some practical implications: (i) based on the updated data, the more accurate short-term prediction can inform the managers in the judgment of the epidemic situation and guide their decision for emergency response; (ii) the multi-period emergency facility deployment and resource allocation decisions can save unnecessary operational costs and timely release medical resources for other types of patients; (iii) the proposed adjustable multi-period robust optimization model can reduce the shortage or idleness of emergency resources and thus improve resource utilization. The main contributions of this paper are summarized as follows and a graphical representation of the conceptual framework is shown in Fig. 1.

Fig. 1.

Conceptual framework of the main contributions.

• •We propose a Demand prediction-Network optimization-Decision adjustment (DND) framework to design the ELN by integrating the demand estimation, dynamic emergency facility deployment, resource allocation, decision feedback and adjustment. By combining epidemiological dynamics with operations research, our method sheds light on designing a responsive, multi-echelon emergency logistic network.
• •Different from the traditional method to predict epidemic trends, we first set the prediction interval based on the time required for decision implementation. Then we make near-term epidemic trend predictions through an improved SEIRD epidemic model in which the parameters are periodically fitted based on the updated data.
• •Considering the uncertainty of the estimated demands, we propose a robust ELN design model to deploy the emergency facilities and allocate the emergency resources periodically. Differently, we propose an uncertainty budget adjustment strategy based on the feedback from the deviation between the implemented decisions and actual demands, which is embedded in a customized rolling horizon solution approach to solve the model repeatedly.
• •The DND framework is applied to a real-world COVID-19 response case in Wuhan city. The results not only reveal that the multi-period robust model outperforms the deterministic static model, but also validate the computational power of the rolling horizon solution approach for large-scale problems. The DND method for ELN design can provide a practical decision-making framework for epidemic response operations.

The remainder of this paper is organized as follows. In the next section, we briefly review the most relevant literature. After stating the research questions, the demand estimation method and demand-based logistics model are illustrated in Section 3, and the solution method is presented in Section 4. Then the computational experiments are performed in Section 5, the result analyses are illustrated, and the managerial insights are derived. Finally, Section 6 concludes this paper and presents some potential future research directions.

2. Literature review

Our work focuses on demand prediction and resource location–allocation for epidemics. For the related works published in recent years, Queiroz et al. (2020) and Chowdhury et al. (2021) provide a comprehensive review of the research done on the resource supply operations considering the impacts of epidemic outbreaks. In this part, we focus on the relevant literature in the following aspects (i) uncertain demand predicting, (ii) resource location–allocation under epidemics, (iii) and the integration of them in the ELN design.

2.1. Uncertain demand estimation based on epidemic models

The existing studies have developed various types of models for epidemics. There are some common ways to model the spread of an epidemic: sub-epidemic wave model, differential equations, simulation (agent-based) modeling, random graphs, and difference equations (Larson, 2007). Some studies forecast the epidemic’s trajectory by compartmental susceptible–infectious–recovered (SIR) model. By considering the scale of other compartments such as exposed, treated, or asymptomatic populations, researchers extend the basic epidemic model. Zhang and Enns (2022) develop a mathematical model reflecting COVID-19 transmission dynamics by considering the age categories and population exchange, and the impact and optimal timing of different mitigation approaches are evaluated. Wangping et al. (2020) propose a dynamic extended susceptible infected removed model, which covers the effects of different intervention measures in dissimilar periods and estimates the basic reproductive number using the Markov Chain Monte Carlo method. Yang et al. (2020) derive the epidemic curve by integrating the SEIR model and an artificial intelligence approach using the updated COVID-19 epidemiological data. They predict the COVID-19 epidemic peaks and propose some control measures in reducing the eventual COVID-19 epidemic size. Roosa et al. (2020) propose phenomenological models using the updated data to make forecasts of the epidemic trajectory in real time. They verify the real-time short-term forecasts generated from such models can be useful to guide resource allocation operations which is critical to bring the epidemic under control.

2.2. Emergency logistics network design

The second stream of relevant research is on emergency resource allocation and stockpiling in response to an emergency. Several surveys regarding OR/MS contribution to epidemics and disaster control can be found in Altay and Green III (2006). Studies mostly concentrate on the principles for the distribution of scarce resources or evacuation of evacuees in an emergency. Timely provision of medical resources plays a crucial role in emergency response. Sun et al. (2021) propose a bi-objective robust optimization model for strategic and operational response to decide the emergency resource location–allocation and casualty transportation plans. Specific to the prevention and control of infectious diseases, Ren et al. (2013) plan an effective response to infectious disease outbreaks and distribute limited vaccines to minimize the scale of fatalities. They consider the modification of subsequent transmission rates by past vaccination strategies. Enayati and Özaltın (2020) develop an epidemic model with multiple interacting subgroups and propose a mathematical model to make optimal vaccine allocation decisions considering vaccine coverage equity. They develop an exact solution approach based on multiparametric disaggregation. Liu et al. (2020) study the epidemic resource allocation problem in response to the H1N1 pandemic. They first construct a compartment model for analyzing epidemic dynamics and then propose a model to simultaneously determine when to open and close the new wards. Considering the time-varying nature of demands, Yarmand et al. (2014) use a simulation model to capture the epidemic dynamics in each region and then propose a two-stage stochastic linear program to determine the vaccine allocation plans. Assuming the demand at each period is stochastic and follows a normal distribution, Yazdekhasti et al. (2021) propose a multi-period multi-modal stochastic location–transportation model. Fadaki et al. (2022) propose a multi-period vaccine allocation model incorporating transshipment between facilities. They show that combining vaccine allocation and transportation into a comprehensive model can effectively mitigate the risk involved in such pandemics. Shang et al. (2022) focus on designing a network for the distribution of humanitarian aid. They study a multi-period hub location problem with serial demands under the assumption that the demand is realized.

As we can see from this part, related works that focus on the ELN design problem mostly ignore the impact of the time-varying property of the epidemics, which may make the solutions inefficient or even infeasible. Therefore, this paper optimizes the ELN dynamically based on the predicted demands to improve the response speed and resource utilization rate.

2.3. Integration of demand estimation and resource allocation

Several related studies investigate the integration of epidemic projection and logistics planning. Rachaniotis et al. (2012) propose a deterministic resource planning model in H1N1 control. They consider the limited available resource allocation problem in the infected areas and use a simple deterministic compartmental SIR model to describe the spread of epidemics. Wanying et al. (2016) study the logistics problem response to an anthrax attack. The proposed model can predict the daily demand according to the progression of anthrax and then distribute the medical resources to patients taking into account the different medical interventions. Büyüktahtakın et al. (2018) focus on the treatment capacity, migration, and spatial transmission rates and propose a new integrated epidemics-logistics model to optimize logistics for an epidemic. They assume that once a treatment facility is established, it stays open until the end of the planning horizon. Long et al. (2018) develop a two-stage model for optimizing when and where to assign treatment resources across areas during epidemic outbreaks. The scale of new cases is forecasted by a novel dynamic transmission model by considering connectivity among regions. Then different policies to allocate the treatment units are compared based on the forecasted demand. As a closely related work to our research, Ekici et al. (2014) study the food allocation plan during the influenza pandemic. They develop a disease spread model for demand prediction and then combine it with a facility location and resource distribution model, which inspires us in pandemic preparation such as how to allocate limited resources and respond dynamically. This paper differs from theirs in three aspects: (i) instead of using the agent-based method, we make a short-term prediction via the uSEIRD model, and (ii) without assuming the amount of demand is known and deterministic, we regard the estimated demands as uncertain parameters and propose the RO method to improve the reliability of the ELN, (iii) based on the observed uncertainties, the RH solution method is used to solve the model so that the decision is implementable in practice. In Appendix B, the related studies in each category are exhibited in Tables B.7, Table B.8, Table B.9 which summarizes the main methods, characteristics, and features.

As can be seen in the above-mentioned description, many researchers have endeavored to study the ELN design problem. Our literature review indicates the following aspects that deserve further study: (i) most of the studies consider the demand estimation and emergency resource deployment separately, which cannot guide the design of ELN effectively, (ii) most literature predicts the trajectory of epidemics statically based on the initially estimated parameters of the prediction models, and thus cannot accurately estimate the dynamic emergency demands, (iii) few studies consider the multi-period joint optimization of emergency facility location, resource allocation, and infected patient admission during epidemics considering the uncertain demands. The adjustment of robust optimization model parameters based on feedback from actual situations is often overlooked.

In this study, we propose a DND framework to design an emergency logistics network to respond to epidemics. We first apply an improved infectious disease model to estimate near-term demands. Then considering the uncertainties of the demands, we construct a multi-period robust ELN design model to deal with uncertainties. Based on the feedback from the actual demands, we incorporate the uncertainty budget adjustment strategy in the proposed RH algorithm to obtain better-performing solutions more quickly.

3. Problem description and model formulation

3.1. The research questions

The operations of ELN design are defined as a cyclical DND process. The first step is to predict future demands according to the reported epidemic-related data. Then the location of DCs (Distribution Centers, such as storage centers at airports and train stations allocate resources to downstream), hospitals (receive confirmed patients and medical resources), and LDPs (Local Distribution Points, such as schools, community, or subdistricts, to allocate general resources to the susceptibles), and the resource allocation plans are determined based on the estimated demands. The uncertainty is revealed over time, so the gap between the real demand and the determined decisions should be minimized to maintain the stability of the ELN in the next decision cycle. Then, the demand prediction is implemented again according to the new data, and the process continues until the end of the decision horizon. In the decision process, we are trying to handle two key research questions:

• (1)How to estimate the time-varying demands?

As an important input parameter for ELN design, the demand estimation problem should be addressed. After the COVID-19 outbreaks, the trend of infectious diseases is difficult to predict once and for all. So the demand estimating process should be implemented dynamically as the updated information in each decision cycle. We use an uSEIRD model to predict the epidemic spread trend, which can be described by a set of ordinary differential equations (ODE). The epidemics divide people into four classes: susceptible ($S$), incubation ($E$), infected ($I$), recovered ($R$) and dead ($D$). Therefore, we propose a short-term projection method that periodically fits its parameters based on updated data and then predicts the number of affected people and the demands. The projected time-varying demands can be used as input parameters to guide the design of the ELN.

• (2)How to design the ELN?

Based on the predicted demands, we introduce an integrated, time-dependent, and multilevel logistics network, which includes suppliers, DCs (Distribution Centers, for collecting and distributing resources), hospitals (for treating confirmed cases with corresponding medical resources), and LDPs (Local Distribution Points, for distributing general epidemic prevention resources). As shown in Fig. 2, a spatio-temporal network is presented to depict the structure of the ELN. In each decision cycle, the supplied emergency resources are sorted and stored in the opened DCs. When demands are generated, the resources are allocated to the operated hospitals and LDPs. Therefore, the following time-dependent decisions should be made: (i) the time-dependent opening decisions of DCs, hospitals, and LDPs, (ii) the resource allocation plans from suppliers to DCs, DCs to hospitals and LDPs, (iii) the number of unsatisfied people serviced by hospitals and LDPs. Focus on the responsiveness and the cost of rescue operations of the ELN, we minimize the facility opening and operating costs, confirmed cases and emergency resource transportation costs, supply shortage and unmet services penalty costs.

Fig. 2.

Time-space network of emergency resource allocation.

The assumptions of this problem are presented as follows:

• 1.The transmission among different conditions for the population does not change in a short time.
• 2.We do not consider the natural birth rate and death rate of the population, and the cross-regional transmission of individuals.
• 3.The hospitals referred to in this paper are mainly isolation wards of emergency designated hospitals or makeshift hospitals, which can provide related isolation and treatment services.
• 4.We divide the demands into two categories: medical kits such as protective clothing, surgical masks and related medicine for hospitalized confirmed patients; general epidemic prevention kits such as masks and surgical spirit for susceptible cases.
• 5.The status of each people changes dynamically during each decision cycle, so the type of demands they need will change. Therefore, this paper focuses on the demands in each decision cycle but does not consider the accumulated resource shortage.
• 6.The set of candidate locations for the facilities is known in advance. In reality, the location of confirmed cases is difficult to identify, so we assume the population of each city is concentrated in its centroid.

Note that some assumptions can be relaxed or altered, but in this paper, we use the assumptions above.

3.2. Short-term prediction of the COVID-19 trends

It is unpersuasive to use existing data to predict epidemic trends for a long time in the future. We make short-term forecasts of epidemic trends in the next few days using the uSEIRD model, where the parameters are periodically fitted based on updated data. Note that the prediction interval represents a periodic prediction step size, which predicts the trend of the epidemic in a future step, for example, 1 day. Then the emergency demands can be estimated based on the number of affected people of different types.

3.2.1. Epidemic model with time-varying transmission rates

Generally, the transmission rates between different statuses in the base SEIRD model are constant without intervention from outside. But in actual situations, the speed of transmission between different states may be changed. The daily updated epidemic data is the most useful information in predicting the trend of the epidemic. To estimate the number of people in different states more exactly, the parameters in the SEIRD model should be updated as the available data increases. Therefore, we propose an uSEIRD model with time-dependent transmission rates. First, we present the related notations in Table 1.

$$\frac{\mathrm{d}S\left(t\right)}{\mathrm{d}t}=-r\beta \left(t\right)I\left(t\right)S\left(t\right)/P_N-r_1{\beta }_1\left(t\right)E\left(t\right)S\left(t\right)/P_N$$$$\frac{\mathrm{d}E\left(t\right)}{\mathrm{d}t}=r\beta \left(t\right)I\left(t\right)S\left(t\right)/P_N+r_1{\beta }_1\left(t\right)E\left(t\right)S\left(t\right)/P_N-\alpha \left(t\right)E\left(t\right)$$$$\frac{\text{d}I\left(t\right)}{\text{d}t}=\alpha \left(t\right)E\left(t\right)-\gamma \left(t\right)I\left(t\right)-\eta \left(t\right)I\left(t\right)$$$$\frac{\text{d}R\left(t\right)}{\text{d}t}=\gamma \left(t\right)I\left(t\right)$$$$\frac{\text{d}D\left(t\right)}{\text{d}t}=\eta \left(t\right)I\left(t\right)$$ (1)

Table 1.

Notations of the SEIRD model.

Notations	Description
$P_N$	The population size of a region
$S\left(t\right)$	The number of susceptibles in $t\in \left\{1,\dots ,T\right\}$
$E\left(t\right)$	The number of exposed individuals in $t\in \left\{1,\dots ,T\right\}$
$I\left(t\right)$	The number of infected individuals in $t\in \left\{1,\dots ,T\right\}$
$R\left(t\right)$	The number of recovered individuals in $t\in \left\{1,\dots ,T\right\}$
$D\left(t\right)$	The number of dead individuals in $t\in \left\{1,\dots ,T\right\}$
$r\left(r_1\right)$	Daily number of contacts of confirmed (exposed) cases
$\alpha \left(t\right)$	Probability of transmission from exposed individuals in $t\in \left\{1,\dots ,T\right\}$
$\beta \left(t\right)\left({\beta }_1\left(t\right)\right)$	Probability of infection of $S$ individuals by $I$ ($E$) individuals in $t\in \left\{1,\dots ,T\right\}$
$\gamma \left(t\right)$	Recovery rate in $t\in \left\{1,\dots ,T\right\}$
$\eta \left(t\right)$	Death rate in $t\in \left\{1,\dots ,T\right\}$

The uSEIRD model (1) yields the rate of change in the number of $S,E,I,R,D$ individuals according to the entries and exits. As claimed by an academician of the Chinese Academy of Engineering in January 2020, patients in the latent status still have the possibility of transmitting the virus, which was later confirmed by the government. In this paper, we take into account the reality that the latent status is capable of transmitting viruses to people in a susceptible status. We introduce the daily number of contacts of exposed cases ($r_1$) and the probability of transmission (${\beta }_1\left(t\right)$) to represent it may turn more individuals from susceptible status into exposed status (Wang et al., 2021).

Under the assumption that the transmission rate does not significantly change in a short time, we introduce a concept of short-term prediction. As shown in Fig. 3, we first fit the parameters by training the uSEIRD model based on the updated data and then predict a short future outcome. For example, the parameters of the uSEIRD model can be fitted based on the data until decision point $t=a$, then we can predict the trend of the epidemics in a short term (such as $[a,a+b]$). Then the ELN design decisions can be determined and implemented in $[a,a+b]$. Analogously, at decision point $a+b$, we calibrate the parameters using the cumulative data until $t=a+b$ and predict the outcoming of the next interval $[a+b,a+2b]$. This loop continues until the end of the decision horizon. The parameter dynamic fitting method is introduced in Appendix A.

Fig. 3.

Parameter fitting with updated data.

Based on the predicted trend of the epidemic, in the next section, we formulate a mathematical model for the ELN design problem.

3.3. The robust resource allocation model

We first introduce the following notations in Table 2.

Table 2.

The notations.

Notations	Description
Index sets
$N,N_1,N_2,N_3$	Set of cities ($n\in N$) in Hubei Province, DCs ($i\in N_1$), hospitals ($j\in N_2$), and LDPs ($m\in N_3$), respectively
$T$	Set of decision cycles, indexed by $t\in \left\{1,\dots ,T\right\}$
$L$	The capacity level of opened DCs, hospitals and LDPs, indexed by $l\in L$ (the higher the level, the greater the capacity)
$K$	Types of resources indexed by $k\in K$
Parameters
$F_{il},F_{jl},F_{ml}$	Opening cost for facility $i\in N_1,j\in N_2,m\in N_3$ at capacity level $l\in L$
$f_{il},f_{jl},f_{ml}$	Operation cost of facility $i\in N_1,j\in N_2,m\in N_3$ at capacity level $l\in L$
${\varphi }_k$	Per head demand for medical resources $k\in K$
$\rho ,{\rho }^{\prime },{\rho }^{{}^{\prime \prime }}$	Unit penalty cost for un-hospitalized confirmed cases, un-satisfied hospitalizations served by hospitals, and unmet affected people served by LDPs, respectively
$c_{nj},c_{ij},c_{im}$	Unit transportation cost for confirmed cases from city $n\in N$ to hospital $j\in N_2$, for medical resources from DC $i\in N_1$ to hospital $j\in N_2$, and $c_{im}$, from DC $i\in N_1$ to LDP $m\in N_3$, respectively
$s_{kt}$	The supply capacity for resource type $k\in K$ in each decision cycle $t$
$V_{il}^{DC}$	The capacity of DC $i\in N_1$ with level $l\in L$
$V_{jl}^H$	The capacity of hospital $j\in N_2$ with level $l\in L$
$V_{ml}^{LDP}$	The capacity of LDP $m\in N_3$ with level $l\in L$
$I_t$	The forecasted number of confirmed cases in decision cycle $t$
Variables
$z_{ilt},z_{jlt},z_{mlt}$	1, if a new facility $i\in N_1$, $j\in N_2$, $m\in N_3$ is opened at the beginning of decision cycle $t$ in capacity level $l\in L$, 0, otherwise
$y_{ilt},y_{jlt},y_{mlt}$	1, if facility $i\in N_1,j\in N_2,m\in N_3$ is operating at level $l\in L$ in decision cycle $t$, 0, otherwise
$x_{ikt}^{\left(1\right)}$	Quantity of medical resources supplied to DC $i\in N_1$ indecision cycle $t$
$x_{njt}^{\left(2\right)}$	Number of confirmed cases transported from city $n\in N$ to hospital $j\in N_2$ in decision cycle $t$
$x_{ijkt}^{\left(3\right)},x_{imkt}^{\left(3\right)}$	Quantity of medical resources allocated from DC $i\in N_1$ to hospital $j\in N_2$ or LDP $m\in N_3$ in decision cycle $t$
$u_t$, $u_{jt}^H$, $u_t^{LDP}$	In decision cycle $t$, the number of un-hospitalized confirmed cases, the number of patients with unmet demands in hospital $j\in N_2$, the number of affected people with unmet demands served by the LDPs in decision cycle $t$, respectively

We formulate a logistics model for resource allocation in response to epidemics based on the predicted demands. Aiming at total operational cost minimization, the optimization model is presented as follows:

$$min\sum _{g\in N_1\cup N_2\cup N_3}\sum _{l\in L}\sum _{t=1}^T\left(F_{gl}{y}_{glt}+f_{gl}{z}_{glt}\right)+\sum _{n\in N}\sum _{j\in N_2}\sum _{t=1}^T{c}_{nj}{x}_{njt}^{\left(2\right)}+\sum _{i\in N_1}\sum _{g\in N_2\cup N_3}\sum _{k\in K}\sum _{t=1}^T{c}_{ig}{x}_{igkt}^{\left(3\right)}+\sum _{t=1}^T\left(\rho u_t+\sum _{j\in N_2}{\rho }^{\prime }{u}_{jt}^H+{\rho }^{″}{u}_t^{LDP}\right)$$ (2)

The objective function (2) minimizes the total cost: the transportation cost of confirmed cases from cities to hospitals; the transportation cost of resources from DC to hospitals and LDPs; the facility (DCs, hospitals and LDPs) open and operation cost; the penalty costs for un-hospitalized confirmed cases, un-satisfied hospitalizations in hospitals and individuals not served by LDPs. The model is subject to the following constraints:

$$\sum _{n\in N}\sum _{j\in N_2}{x}_{njt}^{\left(2\right)}⩾I_t-u_t,\forall t\in \left\{1,\dots ,T\right\},$$ (3)

Constraints (3) denote the flow balance of hospital $j\in N_2$ in decision cycle $t$.

$$\sum _{n\in N}\sum _{j\in N_2}{x}_{njt}^{\left(2\right)}⩽\sum _{l\in L}{y}_{jlt}\left(V_{jl}^H-P_{jt}^H\right),\forall j\in N_2,t\in \left\{1,\dots ,T\right\},$$ (4)

Constraints (4) ensure the capacity limitation of each hospital for receiving confirmed patients, where $P_{jt}^H={\sum }_{n\in N}{\sum }_{\tau =0}^{t-1}{x}_{nj\tau }^{\left(2\right)}\left(1-{\gamma }_t-{\eta }_t\right),\forall j\in N_2,t\in \{1,\dots ,T\}$ denotes the cumulated number of remaining patients in hospital $j\in N_2$ in decision cycle $t$.

$$\sum _{i\in N_1}{x}_{ijkt}^{\left(3\right)}⩾\left(P_{jt}^H-u_{jt}^H\right){\varphi }_k,\forall j\in N_2,k=1,t\in \left\{1,\dots ,T\right\},$$ (5)

In practice, the hospitalized confirmed patients may still suffer from shortages of medical resources. So we use constraints (5) to present the flow balance on the amount of supplies transported to the hospital for hospitalized patients.

$$\sum _{m\in N_3}\sum _{i\in N_1}{x}_{imkt}^{\left(3\right)}⩾{\varphi }_k\left(S_t-u_t^{LDP}\right),\forall ,k=2,t\in \left\{1,\dots ,T\right\},$$ (6)

Constraints (6) present the flow balance on the amount of supplies transported to LDP $m\in N_3$ for susceptibles.

$$\sum _{i\in N_1}{x}_{imkt}^{\left(3\right)}⩽\sum _{l\in L}{V}_{ml}^{LDP}{y}_{mlt},\forall m\in N_3,k=2,t\in \left\{1,\dots ,T\right\},$$ (7)

Constraints (7) denote the capacity limitation of LDP $m\in N_3$.

$$\sum _{i\in N_1}{x}_{ikt}^{\left(1\right)}⩽s_{kt},\forall k\in K,t\in \left\{1,\dots ,T\right\},$$ (8)

Constraints (8) represent the supply capacity in decision cycle $t$.

$$\sum _{k\in K}{x}_{ikt}^{\left(1\right)}+\sum _{k\in K}{A}_{ikt}^{DC}⩽\sum _{l\in L}{V}_{il}^{DC}{y}_{ilt},\forall i\in N_1,t\in \left\{1,\dots ,T\right\},$$ (9)

Constraints (9) present the capacity limitation in each decision cycle $t$, where $A_{ikt}^{DC}={\sum }_{\tau =0}^{t-1}{x}_{ik\tau }^{\left(1\right)}-{\sum }_{j\in N_2}{\sum }_{\tau =0}^{t-1}{x}_{ijk\tau }^{\left(3\right)}-{\sum }_{m\in N_3}{\sum }_{\tau =0}^{t-1}{x}_{imk\tau }^{\left(3\right)},\forall i\in N_1,k\in K,t\in \{1,\dots ,T\}$denotes the available capacity of DC $i\in N_1$ in decision cycle $t$.

$$\sum _{j\in N_2}{x}_{ijkt}^{\left(3\right)}+\sum _{m\in N_3}{x}_{imkt}^{\left(3\right)}⩽\sum _{k\in K}{x}_{ikt}^{\left(1\right)}+A_{ikt}^{DC},\forall i\in N_1,k\in K,t\in \left\{1,\dots ,T\right\}$$ (10)

Constraints (10) ensure the flow balance on the amount of supplies transported to hospitals and LDPs.

$$M\sum _{l\in L}{y}_{jlt}⩾P_{jt}^H,\forall j\in N_2,t\in \left\{1,\dots ,T\right\},$$ (11)

Constraints (11) ensure the operation of hospital $j\in N_2$ when there are remaining patients.

$$z_{glt}⩾y_{glt}-y_{gl\left(t-1\right)},\forall g\in N_1,N_2,N_3,t\in \left\{1,\dots ,T\right\}$$ (12)

Constraints (12) restrict service to open facilities.

$$\sum _{l\in L}{y}_{gl0}=0,\forall g\in N_1,N_2,N_3,$$ (13)

Constraints (13) set the initial status of DCs, hospitals, and LDPs, respectively.

$$\sum _{l\in L}{y}_{glt}⩽\text{1,}\forall g\in N_1,N_2,N_3,t\in \left\{1,\dots ,T\right\},$$ (14)

Constraints (14) ensure that each DC can only operate at one capacity level.

$$u_t,u_{jt}^H,u_t^{LDP}⩾\text{0,}\forall j\in N_2,t\in \left\{1,\dots ,T\right\},$$ (15)

$$x_{ikt}^{\left(1\right)},x_{njt}^{\left(2\right)},x_{igkt}^{\left(3\right)}⩾0,\forall n\in N,i\in N_1,j\in N_2,g\in N_2,N_3,k\in K,t\in \left\{1,\dots ,T\right\},$$ (16)

$$y_{glt},z_{glt}\in \{0,1\},\forall g\in N_1,N_2,N_3,l\in L,t\in \left\{1,\dots ,T\right\},$$ (17)

$$x_{ik0}^{\left(1\right)}=0,x_{nj0}^{\left(2\right)}=0,x_{igk0}^{\left(3\right)}=0,\forall n\in N,i\in N_1,j\in N_2,g\in N_2,N_3.$$ (18)

Constraints (15)–(18) state the variable range requirements.

3.3.1. Uncertainty on the predicted number of confirmed cases

By predicting the size of different types of affected population, their corresponding demands can be estimated. But as mentioned in Lamas and Garrido (2011), the forecasted demand is still uncertain and the demand changes with the evolution of the disaster. Therefore, the actual situation cannot be described exactly by these deterministic demands, that is, uncertainty still exists. The admission of confirmed cases is the most important work in ELN, and in reality the predicted number of confirmed cases in each decision cycle can be easily verified by the reported data. Therefore, this paper focuses on the hospital capacity requirements of uncertain number of confirmed case (${\tilde{I}}_t$). In the absence of related data, we apply a distribution-free RO approach to deal with the uncertain right-hand side of constraints (3).

The total number of confirmed cases $I_t$ is composed of the cases from each city ($I_{nt},n\in N_t$, where $N_t$ is the set of cities). Therefore, the uncertain right-hand side ${\tilde{I}}_t$ can be seen as the summation of uncertain confirmed cases in all cities (${\tilde{I}}_t={\sum }_{n\in N_t}{\tilde{I}}_{nt}$). Here, it represents the set of uncertain number of confirmed case whose values can be changed at time $t$. We set the predicted number of confirmed cases to be the nominal value ${\overline{I}}_t$. In fact, ${\overline{I}}_t={\sum }_{n\in N_t}{\overline{I}}_{nt}$, which indicates that the predicted number of confirmed cases of Hubei Province is the summation of the predicted number of confirmed cases in all cities. Let ${\hat{I}}_{nt}$ denote the maximum deviation from their corresponding nominal values. The budget of uncertainty, denoted by ${\Gamma }_t$, takes values within the interval $[0,\left|N_t\right|]$, where $\left|N_t\right|$ is the number of cities in Hubei Province. ${\Gamma }_t$ controls the tradeoff between the protection level of the constraints and the degree of conservatism of the solutions at time $t$. According to Bertsimas and Sim (2004), the budget of uncertainty approach guarantees that there are at most $\lfloor {\Gamma }_t\rfloor$ cities in which the number of confirmed cases is allowed to change by ${\hat{I}}_{nt}$, and there is one city on which the number of confirmed cases changes by $\left({\Gamma }_t-\lfloor {\Gamma }_t\rfloor \right){\hat{I}}_{nt}$, then the solution of this problem will remain feasible with high probability. So constraints (3) can be rewritten as follows:

$$\sum _{n\in N}\sum _{j\in N_2}{x}_{njt}^{\left(2\right)}+u_t⩾{\tilde{I}}_t=\sum _{n\in N_t}{\tilde{I}}_{nt},\forall t.$$ (19)

We reformulate the constraints and get the robust counterpart via the following proposition.

Proposition 1

The robust counterparts of constraint (3) can be rewritten as follows:

$$\sum _{n\in N}\sum _{j\in N_2}{x}_{njt}^{\left(2\right)}+u_{nt}⩾{\overline{I}}_t+\sum _{n\in N_t}{\alpha }_{nt}+{\gamma }_t{\Gamma }_t,\forall t\in 1,\dots ,T$$ (20a)

$${\alpha }_{nt}+{\gamma }_t⩾{\hat{I}}_{nt},\forall n\in N_t,t\in 1,\dots ,T,$$ (20b)

$${\alpha }_{nt}⩾0,\forall n\in N_t,t\in 1,\dots ,T,$$ (20c)

$${\gamma }_t⩾0,\forall t\in 1,\dots ,T.$$ (20d)

Proof

We define a protective function $\beta \left(N,{\Gamma }_t\right)$ to achieve feasible solution when up to $\lfloor {\Gamma }_t\rfloor$ right-hand sides change and one uncertain parameter ${\hat{I}}_{n_{t}t}$ changes by $\left({\Gamma }_t-\lfloor {\Gamma }_t\rfloor \right){\hat{I}}_{n_{t}t}$. Then the right-hand side can be written as follows:

$${\tilde{I}}_t=\sum _{n\in N_t}{\tilde{I}}_{nt}=\sum _{n\in N_t}{\overline{I}}_{nt}+\beta \left(N_t,{\Gamma }_t\right).$$ (21)

To achieve feasible solutions in the worst case of allowed changes, $\beta \left(N_t,{\Gamma }_t\right)$ is defined as follows:

$$\beta \left(N_t,{\Gamma }_t\right)=\underset{\left\{S_t\cup n_t\left|S_t\in N_t\right),\left|S_t\right|=\lfloor {\Gamma }_t\rfloor ,n_t\in N_t\setminus S_t\right\}}{max}\left\{\sum _{n\in N_t}{\hat{I}}_{nt}+\left({\Gamma }_t-\lfloor {\Gamma }_t\rfloor \right){\hat{I}}_{n_{t}t}\right\}.$$ (22)

According to constraints (21), constraints (19) can be rewritten as follows:

$$\sum _{j\in N_2}{x}_{njt}^{\left(2\right)}+u_t⩾\sum _{n\in N_t}{\overline{I}}_{nt}+\beta \left(N_t,{\Gamma }_t\right)\Rightarrow \sum _{j\in N_2}{x}_{njt}^{\left(2\right)}+u_t⩾{\overline{I}}_t+\beta \left(N_t,{\Gamma }_t\right).$$ (23)

To convert the above constraints to a linear one, we define a linear equivalent of $\beta \left(N_t,{\Gamma }_t\right)$, which equals to the following linear problem:

$$\beta \left(N_t,{\Gamma }_t\right)=max\sum _{n\in N_t}{\hat{I}}_{nt}{\xi }_{nt}$$ (24a)

$$\sum _{n\in N_t}{\xi }_{nt}⩽{\Gamma }_t,$$ (24b)

$$0⩽{\xi }_{nt}⩽1,\forall n\in N_t.$$ (24c)

We can get the dual of the above model:

$$min\sum _{n\in N_t}{\alpha }_{nt}+{\gamma }_t{\Gamma }_t$$ (25a)

$${\alpha }_{nt}+{\gamma }_t⩾{\hat{I}}_{nt},\forall t,n\in N_t,$$ (25b)

$${\alpha }_{nt}⩾0,\forall t,n\in N_t,$$ (25c)

$${\gamma }_t⩾0,\forall t,$$ (25d)

where ${\alpha }_{nt},{\gamma }_t$ are dual variables of constraints (24b), (24c).

According to the strong duality theorem, model (24) is feasible and bounded for all ${\Gamma }_t$, so the dual model (25) is also feasible, and bounded and their optimal objective function values are equal. Therefore, the constraints (3) finally reformulated as shown in Proposition 1. □

Corollary 1

Finally, we get the reformulation:

( $RELN$ ): Objective function: (2) ,

subject to (4) – (18) , (20) .

The model may be difficult to solve when the scale of the problem becomes large. In the next section, we introduce a solution method to handle this problem.

4. Rolling horizon approach with uncertainty adjustment strategy

4.1. The rolling horizon approach

Since the model $RELN$ is NP-hard, commercial solvers cannot solve large-scale problems in a reasonable time. Considering the periodicity of this problem, the entire plan horizon can be decomposed into several decision cycles and solved one after another. This is consistent with the concept of the RH approach, which subdivides the entire planning horizon into several discrete decision cycles.

The framework of RH is shown in Fig. 4. The entire planning horizon $T$ from the beginning to the end of the resource allocation process is subdivided into several decision cycles. In general, the reliability of prediction decreases with the length of the prediction interval, so the short-term prediction can fit well with this method. The RH approach considers a prediction horizon (estimate the uncertain demands) and a control horizon (implement the optimized decisions based on the predicted demands). Note that in this paper, the length of the prediction horizon and control horizon is set to be the same because we ignore the non-immediate resource allocation decisions determined from the long-term not-as-reliable predictions. In the elapsed horizons, the decisions are fixed as constants for the following decision cycles. The RH method does not require re-solving the model to obtain the decisions of the preceding decision cycle, thus it can reduce the computational complexity.

Fig. 4.

Concepts associated with the rolling horizon approach.

For this paper, the solution is implemented only for the control horizon and then rolled forward by a time interval to the next iteration. Based on the updated data, the demands are predicted and used to design the ELN for the next control horizon. This procedure is repeated until the end of the planning horizon. Considering that the emergency transportation of patients and supplies in Hubei Province can be completed within one day, so we set the prediction horizon and control horizon to be one day.3

4.2. Adjustment policy of uncertainty budget

In fact, when solving this periodic problem, a fixed value of uncertainty budget may lead to a more conservative solution. This problem can be mitigated by dynamically adjusting the uncertainty budget based on the deviation of gradually revealed information and the already implemented decisions. The policy is incorporate into the rolling horizon progress to reduce the conservatism of the model. The adjustment of ${\Gamma }_t$ (within the interval $[0,\left|N_t\right|]$) is based on the gap between the reported actual data of each city in Hubei Province and the implemented decisions in each decision cycle. The steps are as follows:

• 1.At decision cycle $t$, the implemented patient transportation decision $x_{njt}^{\left(2\right)}$ and the number of reported confirmed cases in each city $I_{nt}$ are revealed.
• 2.Let $u$ and $o$ represent the number of cities that the decisions underestimated (${\sum }_{j\in N_2}{x}_{njt}^{\left(2\right)}-I_{nt}<0,\forall n\in N$) or overestimate (${\sum }_{j\in N_2}{x}_{njt}^{\left(2\right)}-I_{nt}>0,\forall n\in N$) the actual number of confirmed cases, respectively.
• 3.Considering the predicted evolution trajectory does not significantly change in a short time, so in decision cycle $t+1$, we adjust the value of ${\Gamma }_{t+1}=max\left\{{\Gamma }_t+(u-o),0\right\}$ to reduce the number of cities that is permitted to change.
• 4.Updating the uncertainty budget value ${\Gamma }_{t+1}$ and the predicted number of confirmed cases ${\tilde{I}}_{n(t+1)}$. Then solving the model at $t+1$, and the process continues until the end of the entire decision horizon.

This strategy is integrated into the RH method and illustrated in Algorithm 1.

5. Numerical study and result analysis

5.1. Parameter setting and data sources

In this section, we demonstrate the validity of the proposed model using the example of the COVID-19 epidemic. All computational processes are conducted on a PC with a 2.5 GHz Intel i7 CPU and 8G memory. In the uSEIRD model fitting part, the real-time epidemiological data of COVID-19 are mainly obtained from Surging News Network and WHO.4 The details of the daily number of confirmed cases, and deaths contained therein can be used directly. And the initial values of transmission rates in the uSEIRD model are set as He et al. (2020). In the logistics modeling part, the inputs of parameters are estimated to closely reflect the real situation based on the data collected from other literature. We consider two types of emergency medical kits that should be allocated to the hospitals for confirmed cases and LDPs for other affected cases (5 and 3 units per person per day, respectively, 50 units/m${}^3$). In this paper, 16 cities in Hubei Province, 32 DCs, 87 hospitals, and 92 LDPs are considered in response to the epidemics. The capacity level (cubic meters) of a DC and LDP is proportional to the total population, such as $\varphi P_N/50\left|N_1\right|$ and $\varphi P_N/50\left|N_3\right|$, respectively. The opening and operating cost of DC and LDP is proportional to their capacities. We set the basic capacity of hospitals as the reported available beds according to Wuhan Municipal Health Commission.5 The capacity levels of hospitals are set to be proportional to the number of beds, such as 10%, 30%, and 80% for level 1, 2 and 3, respectively. The penalty cost incurred by confirmed cases who are not hospitalized and the resource shortage in hospitals or LDPs are based on the opening cost of facilities. Transportation costs are proportional to the Euclidean distance between the facilities. This paper focuses on Hubei Province, the area where the epidemic is most severe in early 2020, so the numerical analysis also focuses on the prediction of the epidemic trend in Hubei, as well as decisions on facility location and resource allocation. If there is no special explanation hereinafter, the results of numerical analysis are for the city of Hubei as an example. Considering the critical period for epidemic prevention and control, we set the planning horizon from January 24th, 2020 to April 27th, 2020. The initial value settings of the uSEIRD model parameters are shown in Table B.10 in the Appendix. In this paper, the used reported data about COVID-19 are provided on Github.6

5.2. Analysis of the performance of epidemic projections

5.2.1. The prediction accuracy and generality of uSEIRD model

To demonstrate the advantages of the proposed method in predicting the epidemic spread over time, we compare the uSEIRD method with a data-driven logistic model (LM) (Reddy et al., 2021). We take the outbreak in Wuhan from 2020 January 24 to April 27 as an example. Fig. 5 shows the results, the prediction error of uSEIRD method for the number of active cases (active=confirmed–recovered–death), recovered cases, and dead cases is 5.91%, 6.72%, and 3.82%, respectively. While the corresponding prediction error of the LM method with default setting is 31.72%, 41.92%, and 3.84%, respectively. This indicates that dynamic fitting of uSEIRD model parameters performs better than the LM approach.

Fig. 5.

The uSEIRD method compared with the LM method.

To test the generality of the proposed uSEIRD model, we analyze the difference between the number of reported and predicted confirmed cases in different regions (the other two cities in Hubei Province). As can be seen from Fig. 6, when the prediction interval is 1 day, the prediction error can be controlled within an acceptable range (less than 6% on average), verifying the validity of the uSEIRD model in different regions.

Fig. 6.

The prediction results of the uSEIRD model for different cities.

5.2.2. The impacts of data volume on epidemic prediction

The volume of reported data has a significant impact on the accuracy of predictions. We estimate the parameters of the uSEIRD model using different scales of training data (accumulated data for 10, 20, or 30 days, respectively) and then forecast new active cases until 2020 April 27. As shown in Fig. 7, the volume of data has a significant impact on the performance of the prediction. For example, only 10 days of reported data can hardly predict the number of new cases after February 10, which makes the results have little indicative value. By contrast, having 20 days of reported data can significantly reduce projection error (see 7(a)). While the amount of data is increased to 30 days, the prediction error can further be reduced by 97.8%. This indicates that the accuracy of the uSEIRD model gradually improves with the cumulative updates of the data.

Fig. 7.

Active cases forecasting based on 10, 20, 30 days of data.

5.2.3. The advantages of short-term prediction

Demand estimation is a prerequisite for the ELN design. In this part, we analyze the effect of different prediction intervals on the accuracy of predicting the number of active cases. We select different time points to check the prediction performance on the different check points and the results are shown in Table 3. We apply root-mean-square error (RMSE) to measure the differences between the observed active cases and the predicted number of active cases. The value of RMSE is calculated from $\sqrt{\frac{1}{Q}{\sum }_{q=1}^Q{\left(O_q-P_q\right)}^2}$, where $Q$ is the number of check points, and $O_q$ and $P_q$ are the number of observed active cases and the number of predicted active cases at check point $q$, respectively. We can see that when the prediction interval is 1 day, the value of RMSE is 16.99% of that when the prediction interval is 3 days, and 10.74% of that when the prediction interval is 15 days. The RMSE value grows with the length of the prediction interval, that is, the shorter the prediction interval, the better the prediction accuracy.

Table 3.

Comparison between reported and projected active cases under different prediction intervals (days).

Check points	Observed	Prediction interval
		1 day	3 day	15 day
The 10th day	3710	3581	4084	6085
The 20th day	17358	17385	20642	23787
The 30th day	36594	36902	33598	33598
The 40th day	24106	24549	26134	26134
The 50th day	12358	12891	12391	12391
The 60th day	4700	4990	3914	3514
The 70th day	1128	1309	935	635
The 80th day	302	354	255	85
The 90th day	97	85	80	10
RSME	–	281.32	1655.70	2619.73

Furthermore, we compare the active, recovered, and dead cases by varying the prediction interval. As shown in Fig. 8(b), when the prediction interval is three days, the prediction error is obvious at the end of each prediction interval, and it can be amended timely by the updated data. When the prediction interval is shortened to 1 day, we can get higher prediction accuracy as shown in Fig. 8(a). This demonstrates that the prediction error increases with the prediction interval, and the decision-makers can choose a shorter prediction interval as far as possible according to the time required to implement the emergency response decisions.

Fig. 8.

Prediction results under different prediction interval (days).

5.3. Joint impacts of facility capacity and supply

In this section, we analyze the impacts of the crucial factors in the ELN design: capacity of emergency facilities and material supply ability. In this paper, the basic quantity of daily supply is set to 25% of the capacity of the sum of all DCs. We set $Ms$ as the multiplying factor of the basic quantity of daily supply. Analogously, the basic value of the capacity of the hospital and LDP is set to 25% of their capacities. The corresponding multiplying factors are denoted by $Mh$ and $Ml$, respectively. Then we analyze the effect of different values of supply and capacity on the results. All the multiplying factors used in this part and their associated basic value settings are listed in Table 4.

Table 4.

The description of multiplying factors.

Multiplying factor	Related to	Description
$Ms$	basic daily supply	$25\text{\%}{\sum }_i{V}_{il}^{DC},l=3$
$Mh$	basic capacity of hospital	$25\text{\%}{V}_{jl}^H,\forall j,l=3$
$Ml$	basic capacity of LDP	$25\text{\%}{V}_{ml}^{LDP},\forall m,l=3$

The number of confirmed cases received by hospitals is mainly affected by the capacity, while the satisfaction rate of demands is mainly affected by the supply. We analyze the impact of different capacities of hospitals on the received number of patients. As shown in Fig. 9, all confirmed cases are hospitalized under higher capacity ($Mh=4$). By contrast, when we limit their capacity ($Mh=0.5$), the number of hospitalized cases reduced significantly on February 4th, and since then only a few beds are vacated by discharged patients for new patients.

Fig. 9.

The number of hospitalizations varying with hospital capacity.

Then we analyze the joint impact of resource supply and capacity of LDPs. The results shown in Fig. 10 show that when one of the two factors is at a low level, increasing the value of the other factor cannot significantly improve the resource shortage. Such as when the capacity of LDPs is at a low level (e.g., $Me=0.5$), increasing the daily supply cannot significantly reduce the resource shortage due to the capacity limitation of the LDP. On the other hand, when one of the two factors is large enough, increasing the value of another factor can significantly improve the shortage of relief resources. Therefore, to improve the effectiveness of ELN, we should consider the combined effects of these factors.

Fig. 10.

Impacts of supply and capacity of LDP on resource shortage in LDP.

5.3.1. Operational decisions of facilities with the trend of epidemics

Despite increasing the demand satisfaction rate of the affected cases, the decision-makers also concentrate on the operation cost. One of the features of our model is making dynamic deployment of the emergency facilities during the planning horizon rather than a one-shot decision. Fig. 11 illustrates the opening, operating, and capacity level decisions of the emergency facilities based on the epidemic trends. We can see from Fig. 11, Fig. 11, in the early days of the outbreaks, the huge demand generated by the susceptible individuals leads to opening more high-capacity DCs and LDPs. As the epidemic subsides, the number of open facilities decreases significantly and are mostly converted to small capacity levels. As shown in Fig. 11(c), the number of initially confirmed cases is low, so only a few low-capacity hospitals are open. With the outbreak of the epidemic, more large-capacity hospitals are opened to receive the confirmed patients. While in the recession of the epidemic, the number and capacity levels of hospitals are gradually reduced as the contained epidemic and the cured patients. The hospitals will stop service when all the hospitalized patients are discharged. Moreover, the calculation results show that the dynamic emergency facility deployment method can reduce operating cost by about 11% compared to its single-period version that fixes facility location decisions throughout the whole decision horizon.

Fig. 11.

Operating state of DC, LDP and Hospital during the decision horizon.

5.3.2. Impacts of the penalty cost on emergency response efforts

We analyze the impacts of penalty cost on decreasing the number of un-served people by varying the multiplying factor ($Mp\in \left\{0.5,1,2,4\right\}$) of the value of penalty cost. As we can see from Table 5, lower penalty cost (e.g., $Mp=0.5$) leads to more resource shortages, but when the penalty cost is high (e.g., $Mp=4$), the shortages are significantly reduced. This is because the penalty cost is greater than the cost of opening a new facility, so the model opens new facilities to reduce the total cost. This result provides a reference for managers, who can choose the value of penalty cost according to their decision-making attitude.

Table 5.

Impacts of penalty costs.

$Mp$	${\sum }_t{u}_t$ (person)	${\sum }_{j,t}{u}_{jt}^H$ (person)	${\sum }_t{u}_t^{LDP}$ (person)
0.5	29723	5543	5.34E＋07
1	13624	2164	1.64E＋05
2	4642	897	1.02E＋03
4	0	0	0

5.4. Performance of the solution methods

In this section, we verify the validity of the RO method and the performance of the uncertain budget adjustment strategy (ROA). The computational performance of the RH approach is also compared with the direct solution method.

5.4.1. Performance of the RO method with uncertainty set adjustment

Similar to this paper, Ekici et al. (2014) study the food distribution planning problem during an influenza pandemic. The facility deployment decisions and food distribution decisions are determined periodically based on the food demand estimated by the disease spread model. However, they regard the demand as deterministic, and the model parameters are not adjusted in each decision cycle according to the feedback from the actual demands.

To validate the DND framework proposed in this paper, we apply the method used in Ekici et al. (2014) to the ELN design problem in this paper. The performance of ELN designed by our method and the deterministic method used in Ekici et al. (2014) (denoted as DM) is shown in Table 6. When using the RO method (e.g., $\Gamma =5$), the number of un-hospitalized confirmed cases, the number of hospitalizations with unmet demands, and the number of affected individuals with unmet demand served by the LDPs are lower than that of DM (58.76%, 60.38%, and 70.61%, respectively). This is because the RO method considers the uncertain demands and improves the robustness of the solutions.

More obviously, when applying the uncertainty budget adjustment strategy, the above three results are reduced by 2.21%, 5.97%, and 2.63% respectively compared to the RO method with a high value of uncertainty budget (e.g., $\Gamma =10$). This is because the dynamically adjusted value of the uncertainty budget according to the actual demand feedback can reduce the shortage or idleness of emergency resources and thus improve resource utilization.

Moreover, due to the reduction of penalty cost caused by resource shortage, the objective function value of the ROA method has a significant decrease (96.96% and 28.53% respectively) compared to that of the DM and RO ($\Gamma =5$) methods, which illustrates the advantage of the proposed ROA method. To further demonstrate the performance of the ROA method in the ELN design, additional experiments (Zhejiang and Hunan province) are conducted and shown in Table B.11 in the Appendix.

Table 6.

Performance of the DM and RO model.

	DM	RO		ROA
		$\Gamma =5$	$\Gamma =10$
${\sum }_t{u}_t$ (person)	35214	14521	5342	5224
${\sum }_{jt}{u}_{jt}^H$ (person)	9820	3891	2011	1891
${\sum }_t{u}_t^{LDP}$ (person)	8.54E＋07	2.51E＋07	1.52E＋06	1.48E＋06
Total Cost (CNY)	7.33E＋11	3.21E＋11	3.12E＋10	2.23E＋10

5.4.2. Performance of the uncertainty budget adjustment strategy

In each decision cycle $t$, there will be a decision error (positive or negative), which reflects the difference between the actual demand and the implemented resource allocation decision. We compare the dynamic performance of the static RO method with the uncertainty budget adjustment strategy throughout the decision horizon. As can be seen from Fig. 12, fixing the value of $\Gamma$ leads to a large decision error. This is because the decisions obtained under the static value of $\Gamma$ cannot cope with the dynamic error between it and the realized actual demands. In contrast, adjusting the value of $\Gamma$ according to the decision error results in a better performance of the ELN. For example, when there is a huge demand on February 25, a large decision error is detected. The ROA method can adjust the value of $\Gamma$ to reduce decision error in the next decision cycle.

Fig. 12.

Comparison of surplus resource with static versions.

5.4.3. Performance of the RH method

We generate test cases for different scales, including the different number of nodes (DCs, hospitals, and LDPs) and decision cycles, to verify the computational advantages of the RH approach over the direct solving method (DS). As shown in Fig. 13, the solution time of the RH approach increases linearly with the number of nodes and decision cycles, while the solution time of DS grows significantly as the problem size increases. This is because when only a partial decision horizon is considered in the RH method, a smaller scale of variables is needed to be optimized in each decision cycle. Combined with the uncertainty set adjustment strategy, the RH method can obtain better-performing solutions quickly.

Fig. 13.

Solution time of RH and DS method under different problem size.

5.5. Discussion and managerial implications

The results of this case study show that our proposed DND decision framework is effective and practical. The three parts of the DND framework are derived from reality and can be applied in practice to inform decision-makers.

First, the demands are estimated by the projected number of affected people in different categories, and the interval for short-term prediction is set to ensure the implementation of the decisions in reality. Second, the multi-period optimization model can reduce operating costs, release more medical resources and avoid runs on the medical system. This is also in line with reality, such as fever clinics and makeshift hospitals opening and closing with the trend of the epidemic. Third, the decisions such as confirmed patient transportation and emergency supplies allocation made in this paper are all designed based on reality, which is also of concern to decision-makers in practice.

There are some managerial implications for decision-makers when designing ELN. In the projection of the epidemic’s trajectory part, the results prompt the decision-makers to make a short-term prediction. The prediction interval should be selected according to the time required for decision implementation. Moreover, the amount of historical data has a significant impact on the accuracy of demand prediction, so it is important to collect as much effective data as possible in emergency response to lay the foundation for decision-making. In the ELN optimization part, the results show that the facility capacity and material supply jointly affect the operating efficiency of ELN. It is suggested that decision-makers should try to improve the two factors together to avoid the buckets effect of ELN. The facility deployment and resource allocation decisions are not one-size-fits-all, so the multi-period robust optimization approach is recommended to improve the robustness of the solution, save unnecessary operational costs, and release medical resources for other types of patients promptly. Moreover, the DND framework proposed in this paper can also be applied to other related problems with dynamically changing uncertainties, such as vaccine allocation problems or food distribution problems during a pandemic.

Inevitably, this paper has some limitations. For example, in the assumption, we did not consider the impact of population mobility on the design of ELN. We also did not consider changes in infection status and demands of individuals due to the failure to meet their demands in time. First, the earlier lockdown of Wuhan played a key role in controlling the spread of the epidemic. Therefore, the assumption that does not consider inter-regional population migration is mild in our case study. But this assumption is stronger for other areas that do not adopt this lockdown action, even though it is very useful. However, when considering the population migration and individual infection status tracking, this problem will become very complex and maybe a topic of future research. Second, at each decision cycle, considering the changed statuses of the people whose demands in the previous decision cycle are unmet, the optimization model is difficult to construct and calculate. Moreover, in the early stages of the epidemic, we cannot track and determine the current statuses of all individuals before making decisions. However, we believe that with the development of big data and Internet of Things technologies, this problem will be solved in the future.

6. Conclusion and future research

This paper studies the ELN design problem in response to epidemics. A robust ELN design model is presented to periodically deploy the emergency facilities and allocate the emergency resources while minimizing the transportation cost and penalty cost caused by unmet demand.

This research pointed out the value of the proposed Demand prediction-Network design-Decision adjustment emergency logistics network design framework. First, we improve the traditional SEIRD epidemic model by fitting its parameters periodically based on the updated data and predicting the near-term trend of the epidemic. The results show that the value of RSME for short-term prediction is significantly smaller than that of long-term prediction, which can inform the decision-makers in the judgment of the epidemic situation and guide them in making decisions. Second, considering the uncertainty of the estimated demands, the proposed multi-period robust ELN design model can make emergency facility location and resource allocation dynamically. The results show that compared with the static deterministic model, it can reduce the operating cost significantly. Third, a customized rolling horizon method is proposed to iteratively solve a series of linked cycle-based problems. We improve the optimization model by allowing the uncertainty budget to be adjusted based on feedback from the actual situation and incorporated into the iterative steps of the algorithm. The results show that this solution method can significantly shorten the solution time and reduce the cumulative decision error caused by the fixed uncertainty budget. Moreover, the results of the case study illustrate the benefit of increasing the data volume for predicting the trajectory of the epidemic. The capacity of emergency facilities and material supply capacity should be considered together to improve the effectiveness of the ELN.

In summary, the proposed ELN design framework can improve the operational efficiency of the ELN and the utilization of limited emergency supplies. The DND decision framework is derived from reality and can be applied in practice and provide references for decision-makers. It can also be extended to other problems that suffer from dynamic uncertainties, such as hurricane disasters, the operation of the power system, etc.

In terms of future research, we can pay more attention to cross-regional transmission with different transition rates when building the model. As for epidemic prevention and control, the influence of the disruption of human behavior on decision-making, and the impact of real-time tracking of population infection status for accurate countermeasures should also be taken into consideration.

CRediT authorship contribution statement

Jianghua Zhang: Conceptualization, Formal analysis, Funding acquisition, Investigation, Project administration, Resources, Supervision, Validation, Writing – original draft, Writing – review & editing. Daniel Zhuoyu Long: Conceptualization, Software, Visualization, Writing – review & editing. Yuchen Li: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Visualization, Writing – original draft, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Dynamic parameter fitting for the uSEIRD model

To solve the time-varying parameters based on the updated data, we first present the simplified resulting ODE of uSEIRD with time-dependent parameters in a matrix form:

$$\frac{dO\left(t\right)}{dt}=A\left(t\right)\cdot O\left(t\right)+F\left(t\right)=f\left(O,A,F\right),$$ (A.1)

where $O\left(t\right)$ represents the condition of the system, $O\left(t\right)={\left[S\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right),D\left(t\right)\right]}^T$, and $A$ means the transmission rate matrix:

$$A\left(t\right)=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\alpha \left(t\right)\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \alpha \left(t\right)\hfill & \hfill -\gamma \left(t\right)-\eta \left(t\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \gamma \left(t\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \eta \left(t\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],$$

and $F$ completes the whole uSEIRD model:

$$F\left(t\right)=S\left(t\right)\cdot {\left[\begin{array}{cccc}\hfill \frac{-r\beta \left(t\right)I\left(t\right)}{P_N}-\frac{r_1{\beta }_{1}E\left(t\right)}{P_N},\hfill & \hfill \frac{r\beta \left(t\right)I\left(t\right)}{P_N}+\frac{r_1{\beta }_1\left(t\right)E\left(t\right)}{P_N},\hfill & \hfill 0,\hfill & \hfill 0\hfill \end{array}\right]}^T.$$

In this paper, the equation $\frac{dO\left(t\right)}{dt}=A\left(t\right)\cdot O\left(t\right)+F\left(t\right)$ is solved via 4th order Runge–Kutta (RK4) method, which is a high-precision single-step algorithm for solving nonlinear ordinary differential equations. RK4 method divides the prediction interval into four subintervals and solves ODE by finding four approximations of slopes for each of the population functions at different statuses per time step and weighting them as the average slope. We use RK4 method to fit the parameters of the uSEIRD model as follows:

$$O_{t+1}=O_t+\frac{dt}{6}\left(e_1+2e_2+2e_3+e_4\right),$$ (A.2)

where $dt$ is the step-size, $e_1,e_2,e_3,e_4$ respectively indicates the slopes of the four subintervals of $O\left(t\right)$, which can be calculated by the following formulas:

$$\left\{\begin{array}{c}{e}_1=f\left(O,A,F\right)\hfill \\ e_2=f\left(O+\frac{dt}{2}{e}_1,A,F\right)\hfill \\ e_3=f\left(O+\frac{dt}{2}{e}_2,A,F\right)\hfill \\ e_4=f\left(O+\frac{dt}{2}{e}_3,A,F\right).\hfill \end{array}\right)$$

Now based on the differential equation (A.1), a set of given initial condition $O_0=O\left(t=a\right)$ and the updated data, we can obtain the time-varying parameters by the RK4 method mentioned above and then predict the number of individuals in different infection status.

Appendix B. Tables

See Table B.7, Table B.8, Table B.11, Table B.9, Table B.10.

Table B.7.

Prediction based on epidemic models.

	Epidemic model	Features	Events
Zhang and Enns (2022)	SEIR	Age categories & population exchange	COVID-19
Wangping et al. (2020)	SEIR	Markov Chain Monte Carlo method	COVID-19
Yang et al. (2020)	SEIR	Artificial intelligence approach	COVID-19
Roosa et al. (2020)	Phenomenological model	Real-time prediction	COVID-19
Enayati and Özaltın (2020)	SEIQR	Interacting among subgroups	H1N1
Liu et al. (2020)	SEIHR-A	Considering the hospitalized individuals	H1N1

S: susceptible. E: exposed. I: infected. R: recovered. Q: quarantined. T: Treatment. H: hospitalized.

Table B.8.

Emergency logistics network design.

	Methods	Features
Sun et al. (2021)	Bi-objective robust optimization	Resource location–allocation
Ren et al. (2013)	Epidemic based vaccine allocation model	Vaccine distribution
Yarmand et al. (2014)	Two-stage stochastic optimization	Vaccine allocation
Enayati and Özaltın (2020)	Epidemic based vaccine allocation model	Resource coverage equity
Liu et al. (2020)	Epidemic prediction based facility location model	Dynamic facility location

Table B.9.

Integration of demand estimation and resource allocation.

	Methods	Features
Dasaklis et al. (2012)	Epidemics-allocation model	Control actions in vaccination
Wanying et al. (2016)	Prediction & allocation	Multi-period
Büyüktahtakın et al. (2018)	Epidemics-logistics model	Geographically varying
Long et al. (2018)	Prediction-allocation model	Geographic Proximity & Human Behavior
Ekici et al. (2014)	Epidemics-facility location model	Food allocation
Liu et al. (2020)	Epidemic prediction based facility location model	Facility dynamic operational decisions
Sun et al. (2021)	Multi-objective optimization	Patients and resources allocation

Table B.10.

Initial value of the parameters.

Variable	Initial value	Source
$S\left(1\right)$	1.24E＋07	National Health Commission of the PRCa
$E\left(1\right)$	1523	Roda et al. (2020)
$I\left(1\right)$	495	National Health Commission of the PRC
$R\left(1\right)$	31	National Health Commission of the PRC
$D\left(1\right)$	23	National Health Commission of the PRC
$r\left(1\right)$	20	Zhang and Enns (2022)
$r_1\left(1\right)$	20	Zhang and Enns (2022)
$\alpha \left(1\right)$	0.631	Roda et al. (2020)
$\beta \left(1\right)$	0.018	Roda et al. (2020)
${\beta }_1\left(1\right)$	0.005	Zhang and Enns (2022)
$\gamma \left(1\right)$	0.001	Zhang and Enns (2022)
$\eta \left(1\right)$	0.0055	Zhang and Enns (2022)

^ahttp://en.nhc.gov.cn/.

Table B.11.

Results of DM and RO model in different regions.

	Zhejiang			Hunan
	DM	RO ($\Gamma =10$)	ROA	DM	RO ($\Gamma =10$)	ROA
${\sum }_t{u}_t$ (person)	1232	120	20	720	91	11
${\sum }_{jt}{u}_{jt}^H$ (person)	601	59	11	318	28	2
${\sum }_t{u}_t^{LDP}$ (person)	9.34E＋08	6.53E＋07	3.31E＋07	3.12E＋08	4.52E＋07	2.32E＋07
Total Cost (CNY)	7.99E＋11	7.98E＋09	4.12E＋09	3.89E＋11	2.07E＋09	1.86E＋09