Keywords

1 Introduction

Emergencies such as mass casualty traffic incidents, natural disasters or pandemics are major challenges for the healthcare system, local and state authorities, etc. In emergency situations, a large number of problems arise which must be solved simultaneously within a relatively short time. The most important problems include the appropriate triage of patients (affected/infected), ensuring quick and secure transportation to a specific hospital, providing appropriate hospital treatment and care, etc. Hospitals in a given area are very often unprepared for a large influx of a specific type of patients within a short period of time. For this reason, the so-called field hospitals are often set up, or selected hospitals are adapted for treating only a specific group of patients (from a disaster, pandemic, etc.).

Concerning patient triage in emergency situations, various methods are applied, such as Pandemic Influenza Triage Algorithm (PITA) [1], etc. The secure transportation of patients to the nearest appropriate hospital (i.e. one with available beds, medical specialists and equipment, such as ventilators, etc.) is usually the responsibility of the emergency ambulance service, which may be sometimes aided by military medical transportations services. Reorganization applies also to emergency departments which must be prepared for processing a larger influx of patients. Patient traffic within the hospital must also be modified accordingly.

Implementing the measures mentioned above is hindered by constrained resources, i.e. the number of doctors and nurses, the number of hospital beds of specific type, the number of ambulances and, last but not least, time. It should also be remembered that the hospitals and emergency ambulance service in a given area also provide treatment and transportation to other patients who are not casualties of a disaster or pandemic.

Consequently, the question arises: How to optimally allocate (designate and transport) patients to hospitals in view of the abovementioned constraints?

In order to answer this and similar questions, the paper proposes a mathematical model of optimizing patient allocation to the existing hospital network using medical transportation services, taking into account the existing constrained medical and logistic resources. To implement the model, GUROBI [2] mathematical programming and constrained programming environment and AMPL [3] modeling language were used.

2 Literature Review

Modeling the issues of resources allocation in healthcare has been broadly discussed in the literature of the subject, both in the context of pandemics of e.g. influenza-like diseases, and for situations not related to pandemics. Planning and responding during pandemics of diseases such as seasonal influenza, N1H1 or, most recently, the SARS-CoV-2 coronavirus, involves a number of aspects, including the disease spread forecast, organizing mass vaccinations (if a vaccine is available), planning and resupplying hospital resources, reorganizing the hospital network, or organizing medical transport. Mathematical models of the spread of influenza-like diseases have been presented in a number of studies [4, 5]. Hospital resources planning is another important area addressed in the literature of the subject. Using planning software, such as FluAid and FluSurge, to forecast the impact of a pandemic on the operation and resources of hospitals and other healthcare facilities was discussed in [6]. A model for optimizing nurse allocation in a hospital during an outbreak of influenza pandemic using FluSurge 2.0 software is presented in [7]. The literature concerning the issue of allocation in a hospital network (i.e. patient allocation, allocation/re-allocation of hospital resources, etc.) is quite broad and covers the issues of both long-term planning (e.g. planning for the hospital network) and short-term planning (e.g. ambulance allocation, emergency response to disasters, etc.). Concerning the former category, the literature of the subject provides a variety of location-allocation models (optimal healthcare facility locations and patient-to-facility allocation), e.g. in the studies in [8, 9]. In terms of short-term planning, there is a number of studies on modeling issues such as immediate emergency response following an earthquake, flood, hurricane or terrorist attack; ambulance allocation; allocation of the available medical resources to specific areas of operation, etc. The most interesting studies in this category include those in [10, 11]. The models related to the issues discussed herein are mostly focused on patient-to-hospital allocation/distribution, taking into account the availability of hospital beds and the patient transport time as the optimization criterion.

There are no models which would include not only the availability of hospital beds, but also of the healthcare staff, the type of available hospital beds, e.g. ventilator beds, patient classification, e.g. based on various types of TRIAGE, the medical transport capacity, etc.

The main contribution of the presented work is to propose a formal model of patient-to-hospital allocation during pandemic (Sect. 3) and an iterative method of solving it, which enables decision support in the field of patient transport and allocation (Sect. 4). The proposed model takes into account not only the limited availability of hospital beds, but also takes into account the type of available beds, the number of available healthcare staff with specific qualifications, the availability of a specific type of medical transport, etc. Moreover, the proposed model can be used both in reactive (decisions made immediately by the operator) and proactive (decisions related to the possibility of serving a specific group of patients at a specific time, moving equipment and personnel, etc.) mode. Given the high parameterization of the model, it is possible to easily improve its functionality by altering the values and meaning, and the structure of individual parameters.

3 State of the Problem and the Mathematical Model

The following assumptions were adopted in building the mathematical model of optimizing patient-to-hospital allocation:

  1. 1.

    A hospital network h ∈ H is given in the specific area/metropolitan area or smaller towns and countryside – distances between healthcare facilities from several do several dozen kilometers.

  2. 2.

    The hospital has various types of beds b ∈ B, e.g. for general treatment, for pandemic patients, with ventilators, etc. The type of hospital bed is also the designation of the patient’s category. Each hospital at the given time has a specific number of beds of the given type available wh,b.

  3. 3.

    The hospital also has the fleet of available ambulances a ∈ A. The parameter fa,b specifies whether ambulance a can transport type b patients (fa,b= 1 Yes; fa,b= 0 No).

  4. 4.

    The hospital has employees with specific qualifications to which the relevant types of positions g ∈ G are assigned. The applicable standards specify the maximum number of patients per the given g type employee; this is determined by the value of parameter dg.

  5. 5.

    Each hospital h has a specific number of g type employees. This is determined by the value of parameter uh,g. The parameter zh determines the current number of patients treated in hospital h.

  6. 6.

    There is a given set of patients (p ∈ P). The patients are transported to the hospital by ambulances after calling the central emergency phone number, or, if the patient’s condition allows it, they report directly to the emergency department. Each patient undergoes initial triage and is assigned a specific category b (type of hospital bed required).

  7. 7.

    In the case of patients reporting directly to the ED, if the hospital is full (no available beds of the required type b), the patients may be transported to another hospital.

  8. 8.

    The time of transporting each patient to each hospital ep,h is known, as is the required time in which the patient should be taken to the hospital jp.

  9. 9.

    In order to ensure model cohesion and reduce its size, the so-called virtual ambulance (transporting the patient within the given hospital from the ED to the relevant ward) and virtual hospital (for all patients transported to the ED by ambulance) are introduced.

  10. 10.

    The developed model applies to a specific situation in time used for the so-called reactive planning. Therefore, the model can be run iteratively following an update of the resources in specified intervals, or after receiving a specific number of reports from the patients. It can also be used for proactive planning, i.e. simulating a pandemic threat in a given area, and relocating or resupplying resources as appropriate.

Considering the abovementioned assumptions for the modelled problem, a fundamental question emerges: How to quickly and safely put the patients in the available hospitals? Thus, two parameters are crucial: quickness, i.e. the time of transportation to the hospital, and safety, i.e. using the appropriate means of transport, and transporting the patient to the appropriate hospital, i.e. one with the available required medical personnel, hospital beds, etc. An overall diagram of the modelled problem for a case of 4 hospitals, 6 patients and transportation provided by the emergency ambulance service, is provided on Fig. 1.

Fig. 1.
figure 1

Illustration of an example of data instance of the modelled problem (4 hospitals, 6 patients, centralized medical transportation

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On the basis of those premises, a model of optimizing patient-to-hospital allocation during a pandemic has been formulated in the form of a binary integer programming (BIP) model (1)…(10). Minimizing the patient transport time is adopted as the goal function (1). The goal function is complemented with penalties for failing to deliver the patient to the hospital (np), or failing to deliver the patient to the hospital on time (kp). The model has significant constraints (2)…(9), discussed in Table 2. The key constraints arise from limited availability of resources such as hospital beds, healthcare staff, ambulances, etc. Table 1 presents indices, parameters and decisive variables of the model.

Table 1. Indices, sets, parameters and decision variables
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Table 2. Description of the problem constraints
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In the development of the model, an innovation in modeling decision variables of the problem was implemented. Basically, modeling the decision variable Xa,p,h,t would suffice. With the introduction of two additional decision variables Yp and Up, the model will always have a solution. Obviously, if the additional variables are different than zero, the solution will not always be satisfactory, but with model of decision variables, we eliminate the risk of an “NSF” (no solution found) response of the solver and thus no information on the solution. Obtaining non-zero values of Yp and Up in the solution informs the decision-maker of specific missing resources, thus enabling them to make the decision on emergency resupplying and delivering the patient to the hospital.

$$ \hbox{min} \sum\limits_{a \in A} {\sum\limits_{p \in P} {\sum\limits_{h \in H \cup s} {\sum\limits_{t \in T} {e_{p,h} \cdot X_{a,p,h,t} } } } } + \sum\limits_{p \in P} {(k_{p} \cdot Y_{p} + n_{p} \cdot U_{p} )} $$
(1)
$$ \sum\limits_{a \in A \cup i} {\sum\limits_{p \in P} {\sum\limits_{h \in H \cup s} {X_{a,p,h,t} } } } + z_{t} \le o_{t,g} \cdot d_{g} \forall t \in H \cup s,g \in G $$
(2)
$$ \sum\limits_{p \in P} {\sum\limits_{h \in H \cup s} {\sum\limits_{t \in H} {X_{a,p,h,t} } } } \le 1\forall a \in A $$
(3)
$$ X_{a,p,h,t} = 0\forall a \in A,p \in P,h \in H \cup s,t \in H \wedge s_{p,a} = 0 $$
(4)
$$ \sum\limits_{a \in A \cup i} {\sum\limits_{h \in H \cup s} {\sum\limits_{t \in H} {r_{p,h} \cdot X_{a,p,h,t} } } } + U_{p} = 1\forall p \in P $$
(5)
$$ X_{i,p,h,t} = 0\forall p \in P,h \in H,t \in H \wedge h \ne t $$
(6)
$$ X_{a,p,h,t} \cdot e_{p,t} - m \cdot Y_{p} \le j_{p} \forall a \in A \cup i,p \in P,h \in H \cup s,t \in H \cup s $$
(7)
$$ \sum\limits_{a \in A} {\sum\limits_{p \in P} {\sum\limits_{h \in H \cup s} {c_{p,b} X_{a,p,h,t} } } } \le w_{t,b} \forall t \in H,b \in B $$
(8)
$$ s_{p,a} = \sum\limits_{b \in B} {f_{a,b} } \cdot c_{p,b} \forall p \in P,a \in A $$
(9)
$$ X_{a,p,h,t} \in \{ 0,1\} \forall a \in A \cup i,p \in P,h \in H \cup s,t \in H $$
(10)
$$ Y_{p} ,U_{p} \in \{ 0,1\} \forall p \in P $$

4 Methods and Implementation

The proposed mathematical model (1) .. (10) of the problem (Sect. 3) has become a central element of the decision support system regarding the allocation of patients in hospitals. In order to take into account the dynamics of changes in the availability of beds, medical transport and personnel, medical equipment etc., an iterative method of solving the modeled problem in successive time intervals (τ) was proposed with the simultaneous update of the model parameters resulting from the obtained solution. For this reason, all model parameters were divided into two sets. The first parameter_1 specifies the parameters whose values change when the problem is solved for the period τ. The second set named parameters_2 defines parameters with the same value for all periods τ. The diagram of the iterative approach is shown in Fig. 2. The methods of mathematical programming (including Branch & Bound method) and constraint programming (in particular constraint propagation as presolving method) were used to solve the model. Due to the nature of the modeled problem (NP-hard), with its larger size, the proposed methods may turn out to be ineffective. In the future, the hybrid approaches will be proposed, using dedicated heuristics, metaheuristics and exact methods and algorithms like constraint logic programming [12], accelerated cuckoo optimization algorithm [13], etc.

Fig. 2.
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Iterative approach to solving modeled problem and generate decisions in period τ

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The model was implemented using AMPL modeling language, NoSQL graph database (Neo4j), into which the data instances of the modeled problem (parameters, obtained outcomes, etc.) were uploaded, and GUROBI mathematical and constraint programming solver. The method of implementation and optimization of the model is illustrated schematically on Fig. 3.

Fig. 3.
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Model implementation and optimization scheme

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5 Computational Experiments

In order to verify the proposed model, a number of computational experiments were conducted. In the first phase, optimization was performed for the selected problem data instance Example_1(characterized by the following parameters: 4 hospitals, 15 patients, of which 7 reported to emergency departments by themselves, 10 ambulances, etc. – details are provided in Table 3). The results are provided in Table 4 and as a data base graph on Fig. 4. Seven patients were transported to the hospitals, 2 were transported from the emergency departments to other hospitals, and the remaining 6 patients were admitted to the hospitals to which they reported. In the second phase of the experiments, the efficiency of the proposed model and implementation thereof was analyzed. The calculations were performed for 10 data instances which differed in terms of the number of hospitals, ambulances, patients, etc. The time of calculations turned out to be very short, i.e. 1–2 s. In some cases, the solution was not satisfactory, because the hospitals ran out of resources, as evident from the non-zero values of additional variables (instance 5, 7, 8 and 9). The results are provided in Table 5. Calculations were performed on a workstation with the following hardware configuration: Intel (R) core (TM) i5 4200 M CPU @ 2.50 GHZ processor, 8 GB RAM, Windows 10.

Table 3. Parameter values for Example_1
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Table 4. Results for Example_1 (i-virtual ambulance, s-virtual hospital)
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Fig. 4.
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Results for Example_1 in the form of a database graph

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Table 5. Results for second part of calculation experiments -number of non-zero additional variables in brackets (Yp-Up)
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6 Conclusion

The proposed resource-constrained model of optimizing patient-to-hospital allocation will support numerous decisions in the fields of logistics and resources directly related to saving the lives of patients during a pandemic. There are two main ways the model can be used. First, to reactively support planning decisions. This arises from the fact that the proposed implementation of the model makes it possible to integrate it with a database, which can be updated on a continuous basis by hospitals, emergency ambulance service, etc. Furthermore, as shown by the calculation experiments, the model is characterized by short time-to-solution (optimization), which allows the decision-maker/operator to make ongoing optimal decisions on patient allocation, or to identify resource deficits (non-zero values of Up, Yp), in order to make the appropriate allocation. The second way the model can be used is to proactively support planning decisions in the case of a hypothetical pandemic. It will be possible to determine which hospitals have sufficient resources, and identify and address potential deficits, whether the ambulance service has the necessary fleet of ambulances for a specific type of pandemic, i.e. a specific number of patients in a given category in subsequent periods of time.

In future studies, it is planned to introduce mechanisms to the model which will make it possible to take into account the potential absences of healthcare personnel (infections, work shifts, etc.). In subsequent versions of the implementation, it is also planned to use a proprietary hybrid approach [12, 14,15,16], fuzzy logic [17, 18], and neural networks [19] etc.