The provision of medical equipment during pandemics is one of the most crucial issues to be dealt with by health managers. This issue has revealed itself in the context of the COVID-19 outbreak in many hospitals and medical centers. Excessive demand for ventilators has led to a shortage of this equipment in several medical centers. Therefore, planning to manage critical hospital equipment and transfer the equipment between different hospitals in the event of a pandemic can be used as a quick fix. In this paper, a multi-objective optimization model is proposed to deal with the problem of hub network design to manage the distribution of hospital equipment in the face of epidemic diseases such as Covid-19. The objective functions of the model include minimizing transfer costs, minimizing the destructive environmental effects of transportation, and minimizing the delivery time of equipment between hospitals. Since it is difficult to estimate the demand, especially in the conditions of disease outbreaks, this parameter is considered a scenario-based one under uncertain conditions. To evaluate the performance of the proposed model, a case study in the eastern region of Iran is investigated and sensitivity analysis is performed on the model outputs. The sensitivity of the model to changing the cost parameters related to building infrastructure between hubs and also vehicle capacity is analyzed too. The results revealed that the proposed model can produce justified and optimal global solutions and, therefore, can solve real-world problems.
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1 Introduction
The global outbreak of Coronavirus since the beginning of 2020 caused the registration of more than 477 million cases and deaths of more than 6.11 million people by March 25, 2022 (Xie et al. 2022). One of the main consequences of the outbreak is the severe shortage of medical equipment such as ventilators and related drugs in various countries, especially in developing countries. According to the FDA FACT SHEET1 published on March 17, 2022, the lack of ventilators is one of the major problems in Covid-19 care centers. If the existing equipment utilization is not managed and new equipment is not provided, it can cause a severe disruption in the Covid-19 treatment procedure. Therefore, it is necessary to use management tools to deal with the shortage of equipment caused by a pandemic outbreak (Wang et al. 2022). In general, three approaches including 1) capacity expansion, 2) demand distribution and 3) equipment management to deal with disruptions caused by the lack of medical equipment in the context of the spread of infectious diseases are available. Applying each of the approaches depends on the creation of a reliable logistics infrastructure. The capacity expansion approach directly emphasizes the creation of new infrastructure as well as the provision of new equipment, which of course requires a lot of time and money (Alban et al. 2020). In this approach, it is believed that all the necessary equipment and facilities should be provided to meet the demand in each region and budget constraints should not be considered (Oleghe 2020). However, in the face of the Coronavirus outbreak, there was not enough time to build such infrastructure. In addition, many countries around the world faced new financial problems that limited the possibility of building facilities or purchasing new equipment. Therefore, this approach cannot be considered a rapid response tool in the face of epidemics. The focus of the demand distribution approach is on the transfer of applicants for medical services to different centers so that the empty capacity of all existing centers in an area can be used (Naudé 2020). It is clear that in the face of epidemics such as the Coronavirus outbreak, patient transfer faces many challenges, such as quarantine constraints and traffic bans, which make this approach realistically impossible to use. In other words, using such approaches in the context of outbreaks of diseases such as Covid-19 may lead to a higher prevalence of disease due to the increased traffic between different cities and neighborhoods (Feng et al. 2021). Therefore, only the equipment management approach can be considered an effective tool in addressing the shortage of medical equipment, which focuses on using the empty capacity of all medical equipment in an area (Ahmad et al. 2021). Notably, using this approach requires efficient logistics network management, which usually exists on land, air and sea in all parts of the world (Chen et al. 2021).
For this purpose, this paper investigates the problem of designing a logistics hub network for transferring medical equipment in the context of the COVID-19 outbreak. The main purpose of this paper is to answer the question of how the existing logistics capacity can be used to optimize the transfer of medical equipment in the face of pandemics to meet the demand in hospitals and medical centers. This study tries to find optimal solutions for hub location, allocation of hospitals to hubs along with equipment flow between the network nodes. To answer this question, a multi-objective optimization model is designed under uncertainty in which the objective functions include minimizing transportation costs, minimizing the destructive environmental effects of transportation, and minimizing equipment delivery time between hospitals. In addition, as demand forecasting is difficult, especially in terms of disease outbreaks, this parameter is considered an uncertain parameter in the proposed model. The mathematical model of the study can help healthcare policymakers, practitioners and planners to optimize the infrastructure and equipment productivity level. The model minimizes the delivery time of equipment to the hospital while minimizing the total negative environmental impact. Therefore, that could be useful for practitioners who are seeking more sustainable services in the healthcare industry. The rest of the paper is organized as follows:
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Section 2 provides a literature review in the area of research. The problem description and mathematical model are provided in Section 3 followed by case study explanations in Section 4. Section 5 is about discussions and managerial implications. Finally, conclusions are provided in Section 6.
2 Literature review
This section reviews the research conducted in the field of medical equipment management as well as the design of health logistics hub networks. Due to the high importance of medical equipment management during the Coronavirus outbreak, various tools have been used to optimize decisions, among which the use of mathematical optimization tools has a special place. Moreover, in some studies, other tools have been used to help operations research models to increase the efficiency of final decisions. For instance, Hubs are vital elements of communication and transportation networks and play an important role in interchanging the flows of information/passenger/goods. For this purpose, designing a highly reliable hub network is very critical, because the inefficiency of even a single hub across the network tends to reduce the efficiency of the whole network in transferring the flow appropriately. Eydi and Nasiri (2019) presented a bi-objective mathematical model designed to study the situations before and after hub failure. Considering reliability, the first objective was to maximize the flow through the network, and the second objective was to prevent wasting the flow due to a possible hub failure.
Aborujilah et al. (2021) used the Internet of Things to investigate the management of medical equipment during the Coronavirus outbreak, which greatly improves the transfer of information between medical centers. In this way, more complete information can be provided decision-makers to select the equipment distribution channel. In addition, Aborujilah et al. (2021) used blockchain technology to improve data transmission infrastructure to have more reliable inputs for the optimal design of available transmission networks. In this way, it can be ensured that the resulting outputs are also the most reliable to use in the real world. From a deeper perspective, equipment management problems and the transfer between medical centers can be considered equivalent to the problem of medical supply chain management (Sodhi and Tang 2021). It leads to more appropriate formulations to design the problem. For example, Khan et al. (2021) addressed the challenges of medical equipment management during the coronavirus outbreak from a supply chain development perspective. Based on the findings of this study, Vagner (2021) also examined the economic aspects of this issue, the results of which can be effective in optimizing managerial decisions on the efficient use of medical equipment.
It should be noted that many articles have been published in the field of supply chain management, production and transmission of medical equipment affected by the outbreak of coronavirus, which has examined different dimensions of disorders in different countries since 2020 (Chowdhury et al. 2021). However, designing strategies for using the existing equipment and managing the transfer between medical centers to maximize coverage of patient demand has received less attention. Due to traffic constraints between cities during the outbreak, using classical transport infrastructure may not be efficient; therefore, it is necessary to use developed transport facilities such as hub networks (Armani et al. 2020). To this end, Marmolejo-Saucedo and Rojas-Arce (2020) proposed the problem of designing a logistics hub network for medical care during an epidemic, which determines the radius of medical services coverage based on population concentration. Macias et al. (2020) presented a two-step approach including a route optimization algorithm and a logistics hub selection algorithm. This is based on the use of flying equipment such as drones to perform operations quickly without the presence of people. In this way, the outbreak of the disease can be prevented to some extent. Alumur et al. (2021) examined the types of logistic hub network design structures to determine the advantages and disadvantages of each of them to determine the roadmap for the development of hub network design models in the new global context. Given the importance of proper formulation of logistics hub design problems, they examined different models in the same environment to ultimately determine the best mathematical formulation. The problem of designing a logistics hub network is not limited to the transfer of medical equipment and it also applies to other management issues, including the transfer of food. Achmad et al. (2021) designed a logistics hub network to transport food during coronavirus outbreaks. Since determining the level of demand faces many complexities, their model has been developed based on robust programming. In another application, Xu et al. (2021) designed a hub network to distribute the Covid-19 vaccine using mathematical optimization models. Li et al. (2023) considered a multimodal bi-objective hub network considering emergency relief, which aims to minimize the transportation time consumption and transportation costs, because they believed that when the corona virus spread, the epidemic areas without emergency relief They are primary and must be moved from one area to another.
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Multitask emergency logistics planning is a complex optimization problem in practice. When a disaster occurs, relief materials or rescue teams should be dispatched to destinations as soon as possible. In a nutshell, the problem can be described as an optimization of multi-point-to-multi-point transportation delivery problem in a given multimodal traffic network. Liu et al. (2022) present a multimodal traffic network is considered for emergency logistics transportation planning, and a mixed-integer programming (MIP) formulation is proposed to model the problem.
One of the contributions of this study is using the existing equipment without reinvestment, which can increase the efficiency of this model for many undeveloped or developing countries. In the present study, we try to use the existing capacities in the design of hub-based logistics networks in order to minimize the cost and time of operations. In order to better clarify the innovations of this research, Table 1 provides some of related conducted studies.
Table 1
Summary of research related to the field of research
According to the reviewed literature, it can be perceived that there is a lack of studies that develop a mathematical optimization model for designing a hub network while considering the location and equipment flow problem under uncertainty for medical equipment. In the hub network design for equipment management, facility location has been always the focus, which is a major part of strategic decisions. In the meantime, other key issues such as using logistics hub networks to enable direct transfer and sending through the hub network have not been studied enough. Depending on the specific circumstances of the outbreaks, the use of hub networks can reduce costs and time of operation. In addition, considering the uncertainty conditions in the demand parameter brings the issue closer to the real-world situation and provides more reliable results for decision-makers. Therefore, this study tries to cover some of the neglected points in the research literature. In this research, we are looking at the distribution of ventilators to the required areas during the outbreak. In comparison with the other studies in the literature, this research proposes a multi-objective mathematical model under uncertainty is developed that minimizes time and cost and environmental effects simultaneously which is considered the main contribution of the research. The main purpose of this paper is to answer the question of how the existing logistics capacity can be used to optimize the transfer of medical equipment in the face of pandemics to meet the demand in hospitals and medical centers. This study tries to find optimal solutions for hub location, allocation of hospitals to hubs, and equipment flow between the network nodes.
3 Problem description
In this paper, the distribution of medical equipment during the spread of epidemic diseases, such as Covid-19, is managed using a multi-objective mathematical optimization model under uncertainty to minimize the time of receiving hospital equipment demand. According to the available statistics, the lack of proper logistics infrastructure has been one of the main reasons for the delay in receiving hospital equipment demand such as ventilators. Therefore, some equipment must be sent specifically from dedicated routes. Nonetheless, creating dedicated routes for transporting hospital equipment has various problems, including the high cost of building the necessary infrastructure. Instead of creating dedicated transportation routes, it seems desirable to create structured transmission networks based on logistics hubs and also the existing transportation infrastructure. For this purpose, in this study, some hospitals that have more appropriate communication infrastructure are selected as hubs and the necessary equipment to be transferred to other hospitals is sent through communication networks between hubs. This will reduce both the cost and the supply time. It should be noted that the model designed in this study is based on the study (Alumur et al. 2012). To develop the basic model, some parameters and decision variables are added. In addition, due to changes in the structure of the mathematical model to use in hospital transportation, some constraints are removed from the basic model and others are replaced. A scenario-based approach is also developed to deal with uncertainty conditions in determining model parameters.
Considering uncertainty in input parameters is one of the important elements in the area of operations research and it causes several challenges for hospital managers in the face of epidemics (Darmian et al. 2021). Given that this study deals with the design of a hospital hub network, the determination of cost parameters is done accurately and with the least amount of error. The cost of building the hub and also the cost of transportation are fixed and the cost of building the infrastructure is clear. Therefore, the parameters related to operating costs do not have an uncertainty structure. The capacity of the vehicles is also known, and it is not uncertain. Based on the logistics structures in hub networks, the time along the path between two points is also known and certain. The only parameter that can be claimed to have an uncertainty structure and is reflected in real-world conditions, is the number of requests or the same number of equipment needed to be transported between two hospitals. Therefore, in this study, this parameter is considered under uncertainty. The following assumptions are made for the problem:
1.
The cost of transportation between the source node and the hub is known
2.
The fixed cost of the infrastructure is known
3.
The cost of unloading and loading is known.
4.
A node can be assigned to a hub when the desired hub is built.
3.1 The framework of robust optimization model
Robust optimization obtains a set of solutions that are robust against parameter fluctuations (input data). Due to the difficulty of estimating the exact value of some input parameters, there is always a definite/indefinite level of uncertainty that causes changes in the output values. In order to deal with the uncertainty, various approaches have been used by researchers, among which robust programming has a special place (Franco et al. 2018). Different structures of robust programming have been defined, one of the most basic of which is the robust optimization approach proposed by Mulvey, which is able to take the decision of incompatibility risk or service level function (Mulvey et al. 1995). This approach also provides a set of solutions that are less sensitive to data acquisition in the scenario set (model feasibility). In this approach, two types of robustness are introduced, each of which is described below. 1) solution robustness (near-optimal solution in all scenarios) and 2) model robustness (as feasible solutions as possible in all scenarios) (Farughi et al. 2019). Solution robustness means that if the input parameters of the model change, the value of the decision variables will still be close to the global optimal value. In addition, it is called a robust solution if the final solutions are as feasible as possible for small changes in the input data. This robust optimization approach includes two distinct constraints: 1) structural constraint 2) control constraint (Farughi and Mostafayi 2016). Structural constraints are a concept of linear programming and input data are fixed and far from any disturbance, while control constraints are formulated as auxiliary constraints that are affected by uncertain data. Suppose \(x\epsilon {R}^{{n}_{1}}\) is a vector of design variables and \(y\epsilon {R}^{{n}_{2}}\) is a vector of control variables. The form of the robust optimization model is as follows:
$$Min {c}^{T}x+{d}^{T}y$$
(1)
$$Ax=b$$
(2)
$$Bx+Cy=e$$
(3)
$$x , y\ge 0$$
(4)
Constraint (1) is a structural constraint and the coefficients are fixed. Constraint (2) is a control constraint and the coefficients are affected by the scenario and are uncertain. Constraint (3) also ensures that the variables are non-negative. The formulation of the robust optimization problem involves a set of scenarios \(\uptau =\{\mathrm{1,2},\dots \mathrm{S}\}\). Under each scenario τ ϵ S, the coefficients for the control constraints with a constant probability of \({P}_{s}\) are \(\left\{{d}_{s},{B}_{s},{C}_{s},{e}_{s}\right\}\) where \({P}_{s}\) represents the probability of occurrence of each scenario so that: \(\sum_{s}{P}_{s}=1\). It is clear that the optimal solution will be a robust solution if it remains close to the global optimal solution for each scenario of τ set, and thus the robustness of the model is achieved. It should be noted that in some cases, there are situations in which the resulting solutions may not be both feasible and optimal for all S ϵ τ scenarios at the same time. In these cases, the relationship between solution robustness and model robustness is determined using a control parameter between zero and one.
To formulate this structure, we first consider the control variable \({Y}_{s}\) for each scenario \(S\upepsilon \tau\) and the error vector \({\delta }_{s}\), which measures the allowable level of infeasibility in the control constraints under scenario s. Due to the presence of uncertain parameters in the model, the final solution may be infeasible for some scenarios. Thus \({\delta }_{s}\) indicates that the model under scenario s is infeasible. It is clear that if the model is feasible, \({\delta }_{s}\) will be equal to zero, otherwise \({\delta }_{s}\) will take a positive value according to constraint (7). In fact, it measures the robustness of the unmet demand model. The robust optimization model based on the mathematical programming problem (5) to (8) is formulated as follows:
Since this approach considers multiple scenarios, the first expression of the objective function of the previous model is formulated as the objective function of Eq. (7). In stochastic linear programming formulation, the mean value \(\sigma \left(0\right)=\sum_{s}{\beth }_{s} {P}_{s}\) is used, which shows the solution's robustness. The expression \(\rho \left({\delta }_{1},{\delta }_{2},\dots ,{\delta }_{s}\right)\) is also a function of the penalty for the conflict of value feasible for the constraints of the problem. Violation of control constraints means that infeasible solutions are obtained under a set of scenarios. Using the weight ω, the relationship between the solution robustness, which is measured from the first expression σ (0), and the model robustness, which is measured from the penalty function ρ (0), is obtained. For example, if ω = 0, the goal would be to minimize the expression σ (0), and the resulting solution might be infeasible. On the other hand, if ω is considered large enough, the expression ρ (0) becomes more important and leads to create more cost in the objective function. The expression \(\upsigma \left(x,{y}_{1},\dots ,{y}_{s}\right)\) is given by Mulvey as Eq. (9):
To show the solution robustness, the variance of Eq. (7) indicates that the decision has a high risk. In other words, a small variable in uncertain parameters can cause large changes in the value of the measurement function. Also, λ is the weight assigned to the solution variance. As can be seen, there is a quadratic expression in Eq. (8). To reduce computational operations, an absolute value expression is used instead of a quadratic expression, as shown below.
Moreover, the amount of conflict from the feasibility of the constraints is as follows.
$${\delta }_{is}={E}_{s}^{+}-{E}_{s}^{-}$$
(14)
where the index i represents the constraint of the mathematical model. In this way, the developed structure of the mathematical model of the problem can be explained using a robust programming approach.
Sets
\(i,j\in \left\{N\right\}\) Set of all hospitals
\(k,l\in \left\{H\right\}\) Set of potential hospitals for the construction of distribution hubs
\(s\in \left\{S\right\}\) Set of all scenarios
Parameters
\({f}_{ijs}\) The number of vehicles required to transport between points i and j under scenario s
\({C}_{ij}\) The cost of transportation between two hospitals in the hub network
\({oc}_{k}\) Operating costs in the hospital k
\({FH}_{k}\) Fixed cost of building the necessary infrastructure in a potential hospital k as a hub
\({HL}_{kl}\) Fixed cost of establishing a direct connection between hubs k and l
\({t}_{ij}\) transport time between two hospitals in case of direct transport
\({ot}_{k}\) Operating time in hospital k
\({\beta }\) Maximum time available to perform the entire operation
\({{\alpha }_{c}}\) cost discount coefficient between hubs
\({{\alpha }_{t}}\) time discount coefficient between hubs
\(p\) Maximum hubs to be built
\(M\) A large enough number
\({\widehat{C}}_{ij}\) The cost of transportation between two hospitals in case of direct transportation
\({cap}\) Capacity of vehicles to carry equipment
\({CV}\) Fixed cost of using vehicles
\({d}_{kl}\) unloading and loading costs between hubs h and k
\({EV}_{ij}\) The amount of bio-pollutants created by transportation between two hospitals
\({ES}_{ij}\) The amount of noise pollutants caused by transportation between two hospitals
\({\alpha }\) time discount coefficient between hubs
\({SB}_{ij}\) Time range for sending equipment between two hospitals
\({P}_{s}\) Probability of occurrence of any scenario
\(\lambda\) solution robustness coefficient
\(\omega\) Model robustness coefficient
\({ST}_{ijs}\) Shipment time of cargo between two hospitals
\({ICG}_{kls}\) The cost of moving goods between hubs k and l
Variables
\({X}_{iks}\) If hospital i is assigned to hub k under scenario s, one; Otherwise, zero
\({H}_{ks}\) If a hospital at potential location k is selected as a hub under scenario s, one; Otherwise, zero
\({Y}_{ijkls}\) If the transport between two hospitals is via a hub link k and l under scenario s, one; Otherwise, zero
\({Z}_{kls}\) If a hub link is created between two hubs k and l under scenario s, one; Otherwise, zero.
\({\widehat{Y}}_{ijs}\) If the transport between the two hospitals is done directly, one; Otherwise, zero
\({num}_{kls}\) Number of locomotives required between k and l
\({TFM}_{kls}\) Number of shipments between k and l
\({Q}_{s}^{+},{Q}_{s}^{-}\) Linearization variable to calculate the violation of solution robustness under each scenario
\({E}_{s}^{+},{E}_{s}^{-}\) Linearization variable to calculate the violation of model robustness under each scenario
$$Min \sum_{i,j:i\ne i , k,l:k\ne l,s\in S }\left({EV}_{ik}+{EV}_{kj}\right){Y}_{ijkls}{f}_{ijs}+\sum_{i,j:i\ne i , k,l:k\ne l,s\in S }\left({ES}_{ik}+{ES}_{kj}\right){Y}_{ijkls}{f}_{ijs}$$
(16)
$$Min \sum_{i}\sum_{j}\sum_{s\in S}{ST}_{ijs}$$
(17)
$$\begin{array}{l}Subject\;to:\\{TC}_{s}=\sum\limits_{k}{FH}_{k}{H}_{ks}+\sum\limits_{k,l:k\ne l}{HL}_{kl}{Z}_{kls}+\sum\limits_{i,j}{f}_{ijs}{\widehat{C}}_{ij}{\widehat{Y}}_{ijs}\\\quad\quad\quad+\sum\limits_{i,j:i\ne i , k,l:k\ne l }\left({C}_{ik}+{C}_{kj}\right){Y}_{ijkls}{f}_{ijs}\\\quad\quad\quad+\sum\limits_{k,l:k\ne l\ne }{ICG}_{kls}+\sum\limits_{k,l:k\ne l}\left(\left(\sum\limits_{i,j :i\ne j}{f}_{ijs}{Y}_{ijkls}\right)\left({C}_{kl}+{d}_{kl}\right)\right)\end{array}\;\forall s\in S$$
$$\sum\limits_{l:l\ne k ,}{Z}_{kls}\ge 1+M\left({X}_{kks}-1\right)\quad\forall k\in H,s\in S$$
(22)
$$\sum\limits_{k,l:k\ne l }{Y}_{ijkls}=1-{\widehat{Y}}_{ijs}\quad\forall i,j\in N:i\ne j,s\in S$$
(23)
$${num}_{kls}\ge \frac{{TFM}_{kls}}{cap}\quad\forall k,l\in H :k\ne l,s\in S$$
(24)
$${ICG}_{kls}={num}_{kls}{CV}\quad\forall k,l\in H :k\ne l,s\in S$$
(25)
$${TFM}_{kls}-\sum_{i,j :i\ne j}{f}_{ijs}{Y}_{ijkls}={E}_{s}^{+}-{E}_{s}^{-}\quad\forall k,l\in H :k\ne l,s\in S$$
(26)
$$\begin{array}{l}{ST}_{ijs}=\sum\limits_{k :k\ne i}{tt}_{ik}{X}_{iks}+\sum\limits_{k.l :k\ne l , t}\left({ot}_{k}+{\alpha }{tt}_{ij}+{ot}_{l}\right){Y}_{ijkls}\\\quad\quad\quad+\sum\limits_{k :k\ne j}{tt}_{kj}{X}_{kjs}+{tt}_{ij}{\widehat{Y}}_{ijs}\end{array}\quad\forall i,j\in N :i\ne j,s\in S$$
(27)
$${ST}_{ijs}\le {SB}_{ijs}\quad\forall i,j\in N :i\ne j,s\in S$$
(28)
$${\widehat{Y}}_{ijs}\in \left\{\mathrm{0,1}\right\}\quad\forall i,j\in N :i\ne j,s\in S$$
(29)
$${TFG}_{kls}\ge 0\quad\forall k,l\in H :k\ne l,s\in S$$
(30)
$${num}_{kls}\ge 0\quad\forall k,l\in H :k\ne l,s\in S$$
(31)
$${ICG}_{kls}\ge 0\quad\forall k,l\in H :k\ne l,s\in S$$
(32)
$${TFM}_{kls}\ge 0\quad\forall k,l\in H :k\ne l,s\in S$$
(33)
$${ICM}_{kls}\ge 0\quad\forall k,l\in H :k\ne l,s\in S$$
(34)
$${ST}_{ijs}\ge 0\quad\forall i,j\in N :i\ne j,s\in S$$
(35)
The first objective function minimizes the costs of building a hub network and transferring equipment between hospitals. The second objective function minimizes the environmental pollutants produced in inter-hospital transportation. For this purpose, the first sentence minimizes biological pollutants and the second sentence minimizes noise pollutants. The third objective function also minimizes the total delivery time between hospitals. It is clear that as the delivery time decreases; it increases the utility of the applicant hospitals; therefore, this objective function is known as the social dimension of sustainable development. Constraint (18) calculates the costs of creating a hub and transporting equipment under different scenarios. The left side of this constraint has six sentences. In the first part, the fixed cost of constructing a hub is calculated.
In the second part, the cost of constructing infrastructure between hubs is determined. In the third part, the cost of transportation in case of direct transportation of equipment is calculated. In the fourth, fifth and sixth parts, the cost of transportation in case of using a hub network is determined. Constraint (19) also calculates the amount of distance from the optimal solution in different scenarios. Constraint (20) establishes a relationship between two types of facilitator construction variables. Constraint (21) indicates that a node can be assigned to a hub when the hub in question is constructed. Constraint (22) states that if one point is selected as a hub, it will be connected to another hub with an inter-hub connection. In constraint (23), we choose between direct and hub-based transportation methods. Constraint (24) determines the number of vehicles required and constraint (25) calculates the cost of transportation between hubs. In Constraint (26), the total number of shipments transported between hubs is calculated. Constraint (27) calculates the total service time between two hospitals. Constraint (28) specifies the corresponding boundary. Constraints (29) to (35) indicate the domain of the decision variables.
3.2 Solution method
In solving multi-objective optimization problems, instead of one optimal solution, there is a set of solutions called the optimal Pareto front. Various methods have been proposed to discover the optimal Pareto front, among which the enhanced epsilon constraint method is more efficient. Suppose a multi-objective optimization model has a general structure as a model (36).
where x is the vector of the decision variables, \({f}_{1}\left(x\right),{f}_{2}\left(x\right),\dots ,{f}_{p}\left(x\right)\) are the objective functions, and S is the feasible solution space. According to the structure proposed by (Mavrotas and Florios 2013), in order to discover the optimal Pareto solutions, model (36) is converted to model (37).
where \({e}_{2},{e}_{3},\dots ,{e}_{p}\) are the values to the right, \({r}_{2},{r}_{3},\dots ,{r}_{p}\) are the domain of objective functions, s \({s}_{2},{s}_{3},\dots ,{s}_{p}\) are the auxiliary variables of the constraint and eps can be in the range \(\left[{10}^{-6},{10}^{-3}\right]\). Figure 1 shows the implementation steps of the enhanced epsilon constraint method based on (Mavrotas and Florios 2013).
Fig. 1
Flowchart the of enhanced epsilon constraint method
×
\({lb}_{k}\):
Where:
\({e}_{k}={lb}_{k}+ {i}_{k}\times {step}_{k}\)
\(lb_k\): lower bound for objective function k
\({step}_{k}=\frac{{r}_{k}}{{g}_{k}}\): step for the objective function k
\({g}_{k}\): number of intervals for objective function k
\({n}_{p}\): number of Pareto optimal solutions
4 Case study
In this section, medical equipment transfer management in 10 eastern cities of Iran is discussed as a case study. According to the districting plan of the Iranian health system, these cities are located in one health district (Farughi et al. 2020; Darmian et al. 2021); therefore, the transfer of equipment in order to cover the maximum demand is part of their executive services. It is noteworthy that the proposed model is solved in GAMS 27 commercial software environment with the help of CPLEX 12.2.1 solver in Windows 10 environment. In this case study, the management of medical sciences hospitals in Birjand (1), Nehbandan (2), Fariman (3), Boshruyeh (4), Khosf (5), Kashmar (6), Mashhad (7), Torbat Heydariyeh (8), Gonabad (9) and Bajestan (10) intend to share their hospital equipment, which mainly includes ventilators, by creating a hub network to provide the maximum possible coverage.
Hospitals 1, 4, 6, 8 and 10 are considered as distribution hubs due to their existing communication capacities. It should be noted that the selection of these hospitals is based on the geographical location and also the possibility of exchanging equipment by means of transportation and land grabbing. According to Fig. 2, Mashhad has the best communication infrastructure compared to other cities, but its selection as a hub cannot have a proper management aspect because it is located at the top of the geographical structure. The relevant data was gathered based on the available historical data gathered from the health care authority. The relevant input data are available in Appendix (Tables 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 and 29). It is noteworthy that the time discount coefficient α is 0.206. The capacity of vehicles to carry equipment (cap) is 20 and the cost of using vehicles (CV) is 50. Also, the probability of occurrence of each scenario is equal to 0.5, 0.2 and 0.3, respectively. The value of the solution robustness coefficient is 50 and the model robustness coefficient is 200. These parameters can be included in the structure of sensitivity analysis to provide different values. After solving the model under uncertainty, numerical results are obtained as follows. Table 2 shows the objective functions values for different scenarios.
Fig. 2
Geographical structure of eastern cities of Iran
Table 2
Objective functions values
Total cost
The first expression
The second expression
The third expression
The fourth expression
The fifth expression
The sixth expression
1st scenario
141396
15358
10734
53778
12389
2655
46482
2nd scenario
149685
15358
10734
59156
10136
2655
51646
3th scenario
154310
12565
10734
59156
12389
2655
56811
×
The values of the objective function expressions are completely different for each scenario. The reason for this can be considered in the different flow parameters of ventilators between cities. But the value of the final objective function of the scenario-based model is reported to be 621,488. It should be noted that each numerical scenario has a different output structure and therefore only one scenario needs to be used as the final solution to implement in real-world conditions. For this purpose, in this study, according to the experts, the solutions related to Scenario 1 are explained. Due to the volume of demand for sending ventilators from different cities, it is necessary to create several hub centers. Therefore, in scenario 1, cities 4, 6, 8 and 10 are considered as hubs. But in some cities, ventilators are shipped directly without the use of a hub network, as shown in Table 3.
Table 3
Direct delivery of ventilators between hospitals
City 1
City 2
City 3
City 4
City 5
City 6
City 7
City 8
City 9
City 10
City 1
✔
City 2
✔
City 3
✔
City 4
✔
City 5
✔
City 6
✔
City 7
✔
City 8
✔
✔
City 9
✔
✔
City 10
✔
The equipment required for other cities has been provided through the networks created, which are presented in Fig. 3.
Fig. 3
Hub network created to transfer ventilators
×
The optimal allocation of each city or hospital to the hubs is also presented in Table 4.
Table 4
Allocation of hospitals to constructed hubs
City 1
City 2
City 3
City 4
City 5
City 6
City 7
City 8
City 9
City 10
City (hub) 4
1
1
1
City (hub) 6
1
1
City (hub) 8
1
1
1
City (hub) 10
1
1
According to constraint (2) in the mathematical model, each city is assigned to only one hub. Moreover, based on constraint (5), if a hub is created in a city, that city is assigned to the hub created in the same city. The structure of the inter-hub network is also in accordance with Table 5.
Table 5
The structure created for the inter hub network
City (hub) 4
City (hub) 6
City (hub) 8
City (hub) 10
City (hub) 4
0
1
1
1
City (hub) 6
1
0
1
1
City (hub) 8
1
1
0
1
City (hub) 10
1
1
1
0
The number of vehicles required between two cities is also shown in Table 6.
Table 6
Number of vehicles required \({num}_{kl}\)
City 4
City 6
City 8
City 10
City 4
0
6
8
5
City 6
4
0
5
3
City 8
9
3
0
3
City 10
5
3
5
0
Other information related to the cost of moving between hubs, the total number of shipments between hubs, and finally the shipping time between hubs is shown in Table 7.
Table 7
Total number of shipments between hubs \({TFM}_{kl}\) consider uncertainty
City 4
City 6
City 8
City 10
City 4
0
104
146
84
City 6
68
0
92
51
City 8
179
46
0
55
City 10
82
45
93
0
As shown in Table 8, by considering the robust optimization method and considering the uncertainty conditions, a lower amount of equipment transfer with the same efficiency has been done, which shows the fact that the issue of ventilator transfer by considering the uncertainty can reduce the cost and transfer time.
Table 8
Total number of shipments between hubs \({TFM}_{kl}\) without of uncertainty
City 4
City 6
City 8
City 10
City 4
0
112
162
96
City 6
86
0
98
63
City 8
221
56
0
65
City 10
90
55
95
0
According to the solutions provided in Table 9, it can be said that the solutions provided are feasible; therefore, the performance of the mathematical model is correct.
Table 9
Time of equipment transportation between two cities \({ST}_{ij}\)
City 1
City 2
City 3
City 4
City 5
City 6
City 7
City 8
City 9
City 10
City 1
0
93.355
121.064
0
88.355
99.473
93.236
117.064
0
101.591
City 2
101.473
0
0
115.827
91.355
98.355
108.591
0
103.473
92.236
City 3
115.827
0
0
121.945
93.355
116.827
118.827
0
113.709
102.473
City 4
0
105.591
100.355
100.591
99.355
117.827
92.236
0
122.064
City 5
102.709
101.709
112.945
104.709
111.945
93.355
92.473
108.827
0
City 6
126.182
125.182
124.064
107.591
103.709
0
0
95.355
115.827
116.945
City 7
111.827
90.236
117.945
101.473
97.591
0
0
109.827
109.709
110.827
City 8
123.182
0
0
125.182
117.182
99.473
105.591
0
96.355
113.945
City 9
0
113.945
112.827
0
108.945
91.236
122.064
88.236
0
101.591
City 10
115.945
102.591
109.709
105.591
0
121.064
110.709
97.473
109.709
0
5 Discussion and managerial implications
In this section, a detailed discussion of the results, along with the sensitivity of the model and managerial implications, is provided.
5.1 Analysis of changes in vehicle capacity (Cap)
The purpose of this section is to investigate the effect of changes in vehicle capacity on objective function expressions. For this purpose, according to Table 10, the amount of capacity between 10 to 15 units is considered and the changes of the expressions of the first objective function are calculated.
Table 10
The effect of capacity changes on the values of objective function expressions
Capacity
The total value of the objective function of the scenario-based model
Cap = 50
410253
Cap = 45
427741
Cap = 40
438940
Cap = 35
505579
Cap = 30
590079
Cap = 25
619828
Cap = 20
621488
Cap = 15
712447
Cap = 10
813024
Based on the results, as the number of capacities increases, the values of the objective function expressions decrease. This is exactly true because if the capacity is increased, a lower number of vehicles will be needed; therefore, the costs will be reduced. Therefore, practitioners and healthcare managers should consider vehicle capacity for equipment transportation. Higher capacity of the vehicles reduces the number of transportation and consequently reduces the negative environmental impacts (Fig. 4).
Fig. 4
Sensitivity of the objective function values to capacity changes under uncertainty
×
Regarding the total number of vehicles used in case of capacity change, one can also refer to the information presented in Table 11.
Table 11
Changes in the number of vehicles \({(num}_{kl}\) to changes in capacity level
Capacity
Number of vehicles
Capacity
Number of vehicles
The first scenario
The second scenario
The third scenario
The first scenario
The second scenario
The third scenario
Cap = 50
25
27
26
Cap = 25
50
54
51
Cap = 45
29
32
33
Cap = 20
57
59
60
Cap = 40
30
31
33
Cap = 15
65
68
69
Cap = 35
34
35
37
Cap = 10
72
70
68
Cap = 30
43
46
40
It is clear that as capacity increases, the number of vehicles to respond to each scenario separately decreases. But a specific trend for the Table 12 shows the changes in the total number of vehicles to changes in capacity.
Table 12
Changes of objective function expressions based on parameter values \({HL}_{kl}\)
The total value of the first objective function (scenario-based)
Decrease
60%
225318
40%
232248
20%
333966
Increase
20%
342702
40%
510963
60%
542160
5.2 Analysis of changes in the cost of construction of inter-hub infrastructure \({HL}_{kl}\)
One of the most important costs in transportation planning in hub networks is the cost of creating the necessary infrastructure for the development of communications in this type of networks. This section examines the effects of increasing and decreasing this cost on the values of the first objective function as well as the number of hubs created. For this purpose, the cost of interstitial network connection in the range of 20%, 40% and 60% increase / decrease and the results are described.
As expected, as the cost of building infrastructure increases / decreases, the entire system cost increases / decreases. Interestingly, the value of other expressions of the objective function also increased / decreased, which indicates the great impact of this parameter on the model. It is expected that as the cost increases or decreases, the number of selected hub centers decreases or increases; therefore, the total flow of shipments in the model changes.
Regarding the number of hubs created, it can be said that increasing or decreasing the value of the parameter \({HL}_{kl}\) causes the number of hubs to increase or decrease. This trend is shown in Table 13 and Fig. 7.
Table 13
Changes in the number of hubs based on changes in \({HL}_{kl}\)
The incremental amount of\({{\varvec{H}}{\varvec{L}}}_{{\varvec{k}}{\varvec{l}}}\)
Number of hubs
The first scenario
The second scenario
The third scenario
Decrease
60%
7
7
5
40%
5
6
7
20%
6
7
5
Increase
20%
4
5
4
40%
4
3
3
60%
3
3
2
It can be seen that the number of hubs has always had an unprecedented increase or decrease trend, and this is the reason for the correct operation of the model as expected. According to the analysis, it can be said that the behavior of the model is consistent with reality and, therefore, its performance is correct.
5.3 Multi-objective model analysis
In this section, the proposed multi-objective model is solved. It should be noted that in the second objective function, it minimizes the environmental effects, including the negative environmental impact on the environment. Therefore, the numerical values related to these two parameters are presented in Tables 14 and 15. It should be noted that the values of these parameters are numerical coefficients between zero and one.
Table 14
The amount of bio-pollutants created by transportation between two cities \({EV}_{ij}\)
City 1
City 2
City 3
City 4
City 5
City 6
City 7
City 8
City 9
City 10
City 1
0
0.26
0.08
0.46
0.01
0.36
0.46
0.53
0.08
0.43
City 2
0.26
0
0.52
0.15
0.11
0.24
0.19
0.6
0.5
0.02
City 3
0.08
0.52
0
0.23
0.04
0.47
0.43
0.42
0.26
0.47
City 4
0.46
0.15
0.23
0
0.39
0.31
0.04
0.51
0.13
0.28
City 5
0.01
0.11
0.04
0.39
0
0.46
0.01
0.03
0.29
0.34
City 6
0.36
0.24
0.47
0.31
0.46
0
0.59
0.53
0.07
0.47
City 7
0.46
0.19
0.43
0.04
0.01
0.59
0
0.23
0.23
0.42
City 8
0.53
0.6
0.42
0.51
0.03
0.53
0.23
0
0.47
0.11
City 9
0.08
0.5
0.26
0.13
0.29
0.07
0.23
0.47
0
0.51
City 10
0.43
0.02
0.47
0.28
0.34
0.47
0.42
0.11
0.51
0
Table 15
The amount of noise pollutants caused by transportation between two cities \({ES}_{ij}\)
City 1
City 2
City 3
City 4
City 5
City 6
City 7
City 8
City 9
City 10
City 1
0
0.27
0.34
0.21
0.1
0.09
0.47
0.43
0.08
0.01
City 2
0.27
0
0.6
0.6
0.15
0.17
0
0.49
0.14
0.38
City 3
0.34
0.6
0
0.34
0.56
0.33
0.19
0.51
0
0.59
City 4
0.21
0.6
0.34
0
0.47
0.08
0.31
0.04
0.33
0.43
City 5
0.1
0.15
0.56
0.47
0
0.16
0.31
0.22
0.42
0.16
City 6
0.09
0.17
0.33
0.08
0.16
0
0.11
0.09
0.3
0.21
City 7
0.47
0
0.19
0.31
0.31
0.11
0
0.27
0.14
0.19
City 8
0.43
0.49
0.51
0.04
0.22
0.09
0.27
0
0.53
0.42
City 9
0.08
0.14
0
0.33
0.42
0.3
0.14
0.53
0
0.35
City 10
0.01
0.38
0.59
0.43
0.16
0.21
0.19
0.42
0.35
0
It is noteworthy that the other input parameters are the same values as for the prime numerical example. After solving the problem with the help of the Epsilon Constraint method, 597 Pareto members are obtained, the diagram of which is shown in Fig. 5.
Fig. 5
Pareto front diagram obtained by solving a numerical example
×
It is clear that the produced solutions are non-dominated. In fact, no solution dominates the others. However, it should be noted that the analysis of model behavior in multi-objective problems is difficult and the various dimensions cannot be examined like single-objective problems. Therefore, in this study, in order to evaluate the results of solving the mathematical model, various analyses are proposed using the MID index. The structure of this index is such that the Euclidean distance between the final non-dominated solutions generated by CPLEX is calculated according to the Eq. (38).
where \({f}_{i}^{j}\) represents the i-th solution and j-th objective function. Also, \({f}_{best}^{j}\) is the ideal point to the j-th objective function, and \({f}_{max}^{j}\) and \({f}_{min}^{j}\) are the maximum and minimum values of all Pareto solutions for the j-th objective function, respectively. Moreover, | Q | is the number of points in the optimal Pareto front and \({n}_{obj}\) is the number of objective functions. Figure 6 shows the conceptual view of this index.
Fig. 6
Conceptual structure of MID index
×
Table 16 describes the numerical results. It should be noted that in order to deepen the analysis of the results, different amounts of change in demand, indicated by the symbol (bal), are considered in different values of 0.2, 0.3 and 0.35%. In this way, sensitivity can be examined.
Table 16
Comparison of the results of solving the mathematical model
bal = 0.2
bal = 0.3
bal = 0.35
Capacity
MID
MID
MID
Cap = 50
1.1
11.66
12.31
Cap = 45
1.901
12.16
12.99
Cap = 40
6.18
12.19
14.24
Cap = 35
6.64
12.89
15.08
Cap = 30
9.29
14.68
17.15
Cap = 25
13.49
15.9
18.08
Cap = 20
14.42
16.16
19.22
Cap = 15
14.89
16.91
19.48
Cap = 10
158
17.5
19.51
When it comes to MID values, it can also be seen that in the large capacities of the vehicles, the Pareto front hardly changes. But at smaller capacities, the difference is shown and the structure of the front changes. The Fig. 7 shows the trend of MID changes for bal values.
Fig. 7
Changes in the value of MID in different values of bal
×
As shown in 20, as the capacity decreases, the value of MID increases in all values of bal. The important point is that at the lowest capacity level, the MID value is almost equal for all bal values; Because in this view, the generated solutions do not have significant differences in different points. Finally, in Table 17, in order to investigate the behavior of the model in solving the research case under conditions of uncertainty in the multi-objective mode, sensitivity analysis on capacity changes is performed.
Table 17
Obtained results for different values (bal) with MID index
bal = 0.2
bal = 0.3
bal = 0.35
Scenario 1
Scenario 2
Scenario 3
Scenario 1
Scenario 2
Scenario 3
Scenario 1
Scenario 2
Scenario 3
Cap = 50
1573.6
1856.848
2209.649
1255.8
1494.402
1793.282
1816.3
2034.256
2237.682
Cap = 45
1101.7
1211.87
1369.413
2174.7
2392.17
2703.152
2973.1
3300.141
3828.164
Cap = 40
2664.5
3037.53
3432.409
2598.8
2936.644
3347.774
2838.1
3235.434
3850.166
Cap = 35
2366.7
2769.039
3212.085
1017.5
1159.95
1333.943
1594.3
1833.445
2071.793
Cap = 30
1006.5
1187.67
1306.437
1321.1
1506.054
1777.144
1056.2
1256.878
1432.841
Cap = 25
1586.3
1744.93
1919.423
2386
2672.32
3180.061
1459.1
1634.192
1961.03
Cap = 20
2340.3
2738.151
3094.111
1850.6
2202.214
2444.458
1998.3
2258.079
2551.629
Cap = 15
2961.5
3464.955
3811.451
1994.2
2313.272
2683.396
843.3
936.063
1076.472
Cap = 10
855.8
958.496
1073.516
1877.2
2083.692
2500.43
1352.2
1609.118
1786.121
As can be seen, no specific trend can be provided for MID changes for different values of bal in each scenario. The reason for this is that the Pareto structure in each scenario is completely different for any given value of bal, and in fact, it is not possible to decide on a superior scenario. Analytical diagrams of model behavior are provided for different values of bal for each scenario as Figs. 8, 9 and 10.
Fig. 8
Changes in MID values for different bal values for the first scenario
Fig. 9
Changes in MID values for different bal values for the second scenario
Fig. 10
Changes in MID values for different bal values for the third scenario
×
×
×
5.4 Managerial implications
Although several studies have been conducted on hub network design, less attention has been devoted to proposing a robust mathematical model for designing a hub network for Covid-19 Medical Equipment Management. Therefore, this study contributes toward knowledge by proposing a mathematical model for hub network design for COVID-19 medical equipment management under uncertain conditions. This study has significant managerial implications. The outcome of the model helps healthcare decision-makers to optimize the distribution of hospital equipment in the face of epidemic diseases such as Covid under conditions of uncertainty. The mathematical model aims to minimize the costs of building a hub network and transferring equipment between hospitals, environmental pollutants produced in inter-hospital transportation, and the total delivery time between hospitals by finding the best allocations from hub centres to hospitals. This would help the practitioners to manage the limited equipment efficiently during a pandemic, such as COVID-19.
In this study, equipment management is considered as one of the strategies for dealing with logistics disruptions when a pandemic like COVID-19 surges. This leads to using the existing equipment without reinvestment, which can increase the efficiency of this model for many undeveloped or developing countries. This would be useful for the managers in the eastern cities of Iran as the other response strategy like capacity expansion is not feasible due to financial problems. In the present study, we try to use the existing capacities in the design of hub-based logistics networks to minimize the cost and time of operations.
As there are inherent uncertainties with the parameters of the hub network design problem, a robust optimization technique was used to deal with the situation. Different scenarios were defined to make the problem as realistic as possible. Therefore, the results of the study would be useful for healthcare managers during a pandemic outbreak. The result of the model based on the case study data reveals that the capacity of the vehicles has a great impact on the productivity of the equipment management in the case. Increasing the vehicle’s capacity would improve the delivery time and environmental impacts. Therefore, practitioners and healthcare managers need to consider vehicles with a higher capacity to transfer the equipment.
The proposed mathematical models for the hub design help healthcare managers to improve resource allocation when a pandemic surges. This consists of the allocation of equipment, staff, and other resources. Because when the hospitals know about the arrival times and the number of available equipment like ventilators on different days, they can do better planning for staff and their patients. As a result, a well-designed hub for healthcare equipment management helps managers to ensure that resources are available at the right time in the right place. In addition, with the designed hub for equipment management, healthcare organizations can optimize the utilization of their equipment. This can lead to better patient care and productivity improvement. Therefore, the proposed model helps healthcare organizations, during a pandemic, can enhance service quality and dependability while reducing costs and negative environmental impact.
6 Conclusion, limitations, and future suggestions
Based on the previous studies provided by medical centers one of the most important problems in the equipment supply sector is the lack of ventilators for patients to use. The rate of using ventilators is not the same in all medical centers because the number of patients with Covid-19 in different cities is different. When one hospital in one city faces a shortage of ventilators, there may be unused ventilators in other cities. Therefore, if there is a fast and reliable transportation system, it is possible to meet the demand in hospitals and provide the maximum level of coverage for patient’s needs for this vital device. However, the lack of proper management planning has prevented the implementation of such projects in various countries, including Iran. This study has contributed to theory and practice by developing a robust model under uncertainty for equipment management under uncertainty in the face of pandemics like Covid-19 to bridge the gap in the literature. The model aims to find the optimal locations of hubs and flow between the network nodes while minimizing the total delivery time, total cost, and negative environmental impacts. A scenario-based robust optimization model to design a hub network for Covid-19 medical equipment management is developed to deal with the inherent uncertainty of the problem. In terms of contribution towards practice, a case health study of the eastern region of Iran was used to show the proficiency and applicability of the model. The model shows practitioners how to manage their scarce equipment such as ventilators in the face of a pandemic when capacity expansion is not an option.
Based on the results of the optimization model, a series of hospitals with higher capacity were identified as ventilator supply hubs, cities 1, 4, 6, 8, and 10. In addition, the flow of ventilators between hubs and hospitals in different cities is determined. The results of the scenario-based model show that the costs and delivery time are optimized while the total environmental impact of the hub network minimized. The proposed model as a technique for equipment management would help the healthcare managers to have the ventilators at the right time and at the right place which helps to reduce the mortality and the spread of the disease. As the results showed, it is possible to reduce the cost by increasing the capacity, which requires investment in the future to deal with such pandemics. By increasing the capacity, the need for a vehicle will also be reduced and it will save and justify the investment. We also had limitations in this article due to the complex conditions of Corona, the possibility of visiting the facilities in terms of their impact had limitations, and also the investigation of the problem in other cities due to the dispersion. It is suggested to use meta-heuristic solution methods to solve the problem. There would be another avenue of research to develop a mathematical model to consider both temporary capacity expansions and equipment management simultaneously.
Declarations
Competing interests
The authors have no competing interests to declare that are relevant to the content of this article.
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