Saturday, June 10, 2023

Multi-criteria assignment problems for optimising the emergency medical services (EMS), considering non-homogeneous speciality of the emergency departments and EMS crews – Scientific Reports

Dispatching of ambulances to patients and then, patients to appropriate emergency departments, is a multi-stage decision process. On one hand, in an emergency situation, the help must come as soon as possible. On the other hand, however, the service must be well-suited to patient’s conditions. Currently, various EMS dispatching strategies may be used: dispatching of the closest idle unit, maximisation of the overall coverage, or maximisation of the preparedness of the EMS system1,2,3. Most importantly, the strategy of dispatching the closest ambulance has been proven to be sub-optimal already in 1972 and further confirmed by other research works4,5,6.

What is more, it should be noted that in many EMS systems ambulances differ as per the levels of speciality they can offer to patients. One example is the Polish national emergency medical system7, where ambulances are differentiated basing on the speciality they provide to the patients. Namely the following types of EMS units exist:

  • Basic—ambulance with at least 2 members of staff being paramedics or nurses,

  • Specialist—ambulance with at least 3 members of staff, one of them being a system doctor,

  • HEMS—helicopter emergency medical service, with at least 3 members of staff, one of them being a system doctor,

  • Collaborating units—organisations which normally do not provide public EMS services, yet might be dispatched if required (e.g. Order of Malta Ambulance Corps Poland).

In many European countries the process of handling a medical emergency call is as follows: first a caller dials an emergency number. All over the European Union they can dial 112—European general emergency number. If 112 is reached, the call would usually be taken by a non-medical dispatcher, who serves as the first triager. When the non-medical dispatcher decides that the call is medically valid, they would transfer the call to a dedicated professional medical dispatcher. The medical dispatcher would then investigate the call further, triage it appropriately and take care of assigning an appropriate EMS unit, if deemed necessary. This medical dispatcher would then also help the ambulance crew to find an appropriate destination hospital. For example such a model is present in Austria and Germany. In this approach other services (e.g. fire brigade) have their own dispatchers, who would be handling the call requiring their support. However, the decision problems faced by those dispatchers are out of scope of this paper.

In some countries it is also possible to bypass the 112 number and contact the professional EMS medical dispatcher directly, via a dedicated number. Example of such countries are: Poland, Romania and France. There is also possible another, much less common operational model, where the call is completely handled by the non-medical 112 dispatcher. Such a model is present in Finland8.

When present in the process, the medical dispatcher must face a decision-making problem through making a trade-off between the time requirement for the ambulance to arrive, and the speciality the crew can offer to the patients. Often, this process can be facilitated by the use of dedicated Medical Priority Dispatch Software, which is discussed further in the paper. The software however, helps in triaging and categorising the calls but does not optimise directly for which exact unit (in terms of its callsign) is best to respond. Decisions made may impact further treatment possibilities. For instance, dispatching an ambulance with no possibilities to teletransmit the ECG to regional specialist centre for consultations may result in misdiagnosis of serious cardiac pathologies, including ST-elevation myocardial infarction (STEMI)9. Thus, in optimal decisions of the ambulance-to-patient dispatching it is required to take into account both time-to-arrival and speciality of units.

Once the EMS unit is at the site, the team deepens the diagnosis of the patient condition. Then, based on the results, further decision must be made to select the appropriate Emergency Department (ED) by taking into consideration both its speciality required for the patient and the estimated time-to-arrival. In Poland, emergency departments are part of the national medical emergency system7. Just like the ambulances, EDs also offer different levels of specialities—local EDs, regional specialist centres, trauma centres. In this work we will be referring to the two last types as referential EDs and the local one as non-referential ED. Similarly to assigning of ambulances to patients, the problem of identifying the correct ED for a given patient is a nontrivial decision-making process that requires establishing a trade-off between the proximity to the ED and the speciality needed in the patient’s condition. According to the Polish regulations, establishing of the ED, to which the patient is to be taken, results from joint collaboration of the dispatcher with the chief of the emergency medical team caring for the patient.

Some acute conditions require highly specialised quick treatment in a referential unit within given time from symptom onset. Some examples of those are: aortic dissection (to be treated as soon as possible), STEMI (most effective treatment within 90 min. of first medical contact) or massive pulmonary embolism (most effective treatment within 48 h of onset)10,11,12. For treatment to be effective, the patient must be transferred to the referential hospital—either directly from the scene or via re-transferring from a non-referential unit. Yet, re-transferring may add some important delays on the time-to-treatment, making further treatment difficult to be effective. Therefore, it is necessary to find an optimal patient-to-hospital assignment strategy taking into both speciality and time-to-treatment.

In this paper we propose a novel multi-criteria optimisation problems towards both ambulance-to-patient and patient-to-hospital assignment problems, that take into consideration objectives such as time and speciality of the offered emergency service. Time and speciality requirements are not uniform across acute-state patients and depend greatly on their medical condition. We take this fact into consideration in our optimisation problems by optimising for both time-to-support and for speciality received by each patient. This is done for each patient individually (on a per-patient basis). In that sense, we aim to design an ambulance-to-patient and patient-to-hospital optimal assignment tool, that would pinpoint the best currently possible dispatch decisions taking into consideration clinical conditions of the patients. The tool is intended to facilitate dispatcher’s decisions by providing them with recommendations.

Dispatch in the paper is understood as establishing the best possible assignment of precise ambulances to precise patients, and further precise EDs to these patients. It is done taking into consideration the current operational state of the EMS system (e.g. number of ambulances available, number of hospital beds available, time-to-arrival of a given ambulance to the patient or time to arrive at a destination hospital). This is in contrast to understanding dispatch as triage and categorisation of emergency calls, which is sometimes found in literature. The optimisation problems proposed in this work aim to improve the decision support system in helping the medical dispatcher in assigning ambulances to acute-condition patients and then the patients to emergency departments, that can efficiently treat patients’ conditions. The proposed problem allows also for re-referral of patients between non-referential and referential hospitals. What is more, in this paper we also propose an embedding framework of the problems proposed into the current dispatching decision-making process.

The goal of this paper is to show the importance of considering not only a single criterion (mostly time) in the optimisation of ambulance-to-patient and patient-to-hospital assignments, but also other criteria related to speciality a given unit offers in treating a given urgent medical condition. We also aim to show the importance of considering patients’ medical requirements on those criteria expressed in terms of aspirations/reservations in the optimisation process. The paper outlines that it is both technically possible, and medically desirable, to incorporate the speciality criteria in the optimisation of assignment of resources. The outcomes of our work can be used in combination with currently used call categorisation software (e.g. Medical Priority Dispatch System—MPDS) and with currently existing patient transport protocols. These can be used as input to the optimisation problems proposed, enhancing the ability of assigning the appropriate unit—both in terms of time and speciality criteria.

To achieve this goal we propose two multi-criteria mixed integer linear programming (MILP) optimisation problems for optimising EMS assignment decisions. The first problem proposed yields a Pareto-optimal ambulance-to-patient dispatch, basing on patients’ requirements on ambulances’ time-to-arrival and speciality offered. These requirements are established on a per-patient basis with respect to their clinical condition. The second problem proposed yields a Pareto-optimal patient-to-hospital assignment, which also takes into consideration all patients’ requirements on time-to-arrival and ED’s speciality, estimated based on their clinical condition.

Clinical situation

To reduce the morbidity and mortality that can result from the acute phase of an illness or injury, it is essential that the ambulance response procedure is quickly ensured and that the patient is transported to the correct hospital, depending on the patient’s needs and the current capacity of the emergency medical services. To do this, the patient’s health condition and the maximum possible waiting time required to provide qualified medical first aid must be estimated13. The world’s leading causes of death include cardiovascular diseases. Research indicates that more than 4 million Europeans die each year for that reason. According to the research conducted in 2016–2017 in Katowice, Poland, the most common causes for Emergency Medical Service interventions were non-traumatic internal emergencies, which most often included: hypertension, atrial fibrillation, myocardial infarction, pulmonary edema, atrioventricular blocks, strokes, chronic obstructive pulmonary disease (COPD) and a diagnosis of bronchial asthma14. In addition, the most common medical emergencies include sudden cardiac arrest, which can be caused by hypoxia, cardiac tamponade, poisoning, ionic disturbances and shock. Symptoms such as abdominal pain, arm pain radiating to the jaw, unusual headache, severe bleeding, and confusion remain worrisome15.

As mentioned, cardiac arrhythmias and cardiovascular diseases are the most common reasons for the Emergency Medical Service interventions. Direct threats to life include acute coronary syndromes, pulmonary embolism or abdominal aortic aneurysm, which, if untreated, can lead to death in a short period of time. Cardiovascular diseases continue to be world’s leading causes of death, of which 50% are caused by ischaemic heart disease16. According to the Institute for Health Metrics and Evaluations data from 2017, 1.6 million people in Poland developed ischaemic heart disease. On the other hand, the data made available by the National Health Fund show that more than 85,000 acute coronary syndromes were recorded in Poland in 2021. Cases of acute coronary syndromes have also been reported, with nearly 67,000 myocardial infarctions17.

Acute coronary syndromes (ACS) are mainly caused by an imbalance between the myocardial oxygen demand and its supply. The cause of the oxygen limitation is most often the presence of atherosclerotic plaque in the coronary arteries, but there may also be the presence of cardiac arrhythmias, and complications after hemorrhagic shock. ACS include ST-elevation myocardial infarction (STEMI), non-ST-elevation myocardial infarction (NSTEMI) and unstable angina18. The main symptom with which patients visit the ED is sudden pain or chest tightness, usually localized retrosternally with radiation to the shoulders, angle of the jaw and elbows19. The diagnosis is based on the record of the received electrocardiogram (ECG), which should be performed within 10 minutes of the first contact with the health care system and on the basis of clinical symptoms. Currently, ambulances are equipped with an ECG recording machine, which allows for a quick diagnosis. If there is an ST-segment elevation, we diagnose STEMI; if there is a non-ST segment elevation, we should measure the level of troponins, elevated levels of which may indicate myocardial infarction. Once ST-segment elevation is recognized, the patient requires rapid reperfusion therapy according to the latest European guidelines or percutaneous coronary intervention (PCI)20. Patients diagnosed with myocardial infarction should be transported by the Emergency Medical Service to a PCI-capable facility as soon as possible. For a patient not capable of primary PCI, fibrinolytic therapy should be started within less than 10 min. Current recommendations say that the patient should be transported to the nearest hemodynamics centre on 24-h duty, and not to the nearest hospital. When a patient with ST-segment elevation MI (STEMI) arrives at a non-ICU hospital, he or she should be immediately transported to an invasive cardiology unit21. A patient presenting to a hospital where PCI can be performed should receive treatment within no more than 60–90 min if fibrinolytic treatment fails, however, the maximum delay from STEMI diagnosis to reperfusion during PCI according to the Polish cardiac society is 120 min if a primary PCI strategy is chosen instead of fibrinolytic treatment. When immediate PCI is not possible, pharmacotherapy with invasive treatment should be considered, where coronary angiography is performed within 24 h22.

Apart from the above, nearly 5% of patients arriving in the ED are those with neurological symptoms. Sang-Beom et al. in their research distinguished a significant predominance of patients with stroke symptoms, epileptic seizures and status epilepticus among neurological emergencies. Among strokes, 80–90% of cases are patients with ischaemic stroke due to embolism or extra-cerebral vascular pathology, and misdiagnosis has been associated with increased mortality rates23. Ischaemic stroke is the second most common cause of death and long-term disability of adults worldwide, and the incidence of that disease increases with age. Fibrinolytic therapy is an effective treatment for stroke patients and the therapeutic window for intravenous tissue-type plasminogen activator therapy is 3–4.5 h from the onset of the first symptoms; however, only about 25% of patients on the ward receive thrombolytic treatment within the indicated time window24,25. Diagnosis and implementation of treatment of patients with symptoms of acute central nervous system injury determines the effectiveness of planned therapy, but often patients arrive at an intermediate hospital that does not have a stroke unit or lacks diagnostic and therapeutic capabilities, which delays the timing of thrombolysis. In order not to delay the therapeutic window, the emergency team should notify the stroke unit staff to reduce the occurrence of in-hospital delays, while inexperienced and unequipped centres for neuroimaging in the treatment of stroke patients have an indication to use remote consultation with reference centres26. As a result, only appropriate hospitals can provide treatment for stroke patients.

The introduction of intravenous thrombolysis with recombinant tissue-type plasminogen activator (rtPA, alteplase) to treat acute ischaemic stroke required a revolution in the organisation of stroke care. Recognition that “time is brain” drove effective public and prehospital awareness campaigns, such as the “Face, Arm, Speech, Time” (FAST) test27 and rapid prehospital triage to designated centres.

The organisation of stroke care depends upon local geography, but the implementation of dedicated acute stroke pathways varies widely. Comprehensive stroke centres provide all aspects of acute stroke care. Triage of patients eligible for endovascular thrombectomy directly to a comprehensive stroke centre (the “mothership” model) may improve the likelihood of good outcome, even if other hospitals are closer. Primary stroke centres are usually smaller centres that initiate intravenous thrombolysis and transfer patients eligible for endovascular thrombectomy to a comprehensive stroke centre, the so-called “drip-and-ship” model28. The key aspect of any stroke service model is that patients can access specialist expertise, neuroimaging and stroke unit care without delay29.

Worldwide, there exist accepted guidelines and dedicated protocols for the treatment of patients in life-threatening conditions and their transfer to dedicated centres. The European Resuscitation Council’s 2021 guidelines, indicate that a patient suffering from a cardiac arrest should be transported to a dedicated centre for treatment of reversible causes of cardiac arrest, basing on local guidelines30. Local guidelines are then developed for many locations. For example, in the US state-wide local transport protocols have been developed. These are present for instance in: Alabama31 and Delaware32. They are briefly discussed in this section.

In Delaware guidelines for patients diagnosed with ST-segment elevation myocardial infarction are based on the same indications to transport the patient as soon as possible to a facility capable of performing percutaneous coronary intervention PCI with concomitant pharmacological treatment. For paediatric patients, the guidelines point to the notion of effective chest compressions followed by transporting the paediatric patient from the scene to an ECMO-equipped facility as quickly as possible. Similarly, the state of Alabama has also adopted a protocol for bypassing primary care hospitals for patients with acute coronary syndromes and myocardial infarction with STEMI to hospitals with an accessible catheterization (PCI) laboratory.

Let us now consider guidelines for stroke patients. Delaware recommends to immediately transfer a stroke patient to the nearest specialised stroke center certified by the state of Delaware. To this end, criteria were adopted for VAN (Vision, aphasia, neglect) negative and LKW (last known well) patients with a time when they were last seen without stroke symptoms of less than 4.5 h, admission to the nearest specialized stroke center should be considered. For VAN positive and LKW patients more than 4.5 h, transport of the patient directly to a certified thrombectomy center should be considered. Similarly, the same procedures are adopted for stroke patients in Alabama.

Apart from cardiac and stroke cases, guidelines on bypassing the local facility also exist for trauma and burn patients. Patients assessed with the Glasgow Coma Score < 13 and low systolic pressure and respiratory count < 13 should be transported first to a highly specialised centre. It is also advised in Delaware that in case of an obvious injury, the patient is transported to the highest-level trauma centre. Detailed list of obvious injuries can be found in Ref.32. Similar guidelines on trauma handling are also found in the Alabama protocol. However, the protocol requires that the patient is diverted to the closest ED in case of: loss of airway, haemodynamic instability with no vascular access and external uncontrolled bleeding.

When it comes to burns, patients are required to be transported to a burn centre bypassing the nearest centre  basing on the percentage of burn area and on whether respiratory burns occurred. Assessment of whether a given patient is to be transported to a burn centre can be made using the rule of nines, also given in the protocols.

There are many emergency conditions that can lead to death. Hence, it is crucial to take action in the pre-hospital setting when transporting the patient to the hospital. Many of the acute conditions have a therapeutic window, i.e. a maximum time to implement therapy from the time of the first worrying symptoms. Delaying appropriate medical care in a specialised unit, in a serious condition practically does not guarantee survival. If a patient is transported to a hospital that has no specialised equipment and personnel, we delay the time to provide treatment at the cost of transporting the patient to a specialised centre.

Literature review

Organising, operating and forecasting of Emergency Medical Services is a topic of extensive research. Computer-based systems might help in making well-suited, timely decisions to support operations of the whole EMS system, e.g. in assigning of ambulances to calls, assigning ambulances to EDs, ambulance routing, medical documentation handling or patient drop-off procedures and in notifications of staff required to handle a given emergency33,34.

Within that field, substantial number of research works focusing on the use of operational research (OR) methods for this purpose have been published. Authors of Ref.35 identified that researchers focus on applying OR in the following problems of EMS organisation: location of ambulances with their further relocation, dispatching and routing of ambulances, interplay of EMS with general health system as well as forecasting of calls and availability and crew scheduling. They also note that an important research area is development of simulation/validation tools. These observations have been backed up by the authors of another review paper2, who underlined also the necessity of staff hiring and fleet operations optimisation and of Ref.1 who reviewed the problems in EMS logistics.

Some interesting usages of OR models for emergency medical system planning are given in Refs.36,37,38,39, some also investigating fairness measures40. A number of significant papers have also been published in the field of forecasting41 and in management of patients once in ED or hospital42,43,44. Those however are not directly linked to the scope of this paper and thus are given only as a reference for an interested reader.

From statement perspective this paper builds on ambulance dispatch/allocation/routing problems. These problems have been of significant research interest. Jangtenberg with co-authors studied the dispatching of ambulances as applied to the Dutch practice6,45. Not only did they propose a new dispatch strategy outperforming the closest idle, but further proposed a benchmark model for offline optimal dispatching of ambulances. EMS dispatching taking into consideration equity call prioritisation was studied in Refs.46,47, where Enayati et al. focused also on simultaneous optimal location of ambulances. The notion of simultaneous optimisation of dispatch and location of ambulances was also applied in Ref.48. Authors depicted on the example of EMS data from Portugal that using OR tools with more advanced dispatch strategies can give better results than doing this by hand under closest idle criterion. Relocation optimisation and dispatch policies was also studied by Siong Lim et al.49, who reviewed dynamic ambulance relocation models from the perspective of dispatch policies. Their paper presents also a comparison of different EMS dispatch policies. Boutilier et al.50 however proposed to combine optimisation of location and routing of ambulances in the city of Dhaka, Bangladesh.

Interesting notion in dispatch optimisation is integration of considering different types (specialities) of ambulances51,52,53, i.e. (ALS)—Advance Life Support and (BLS)—Basic Life Support which are assigned to emergency calls basing on case severity. Knight et al.54 assess the severity with means of survival probability functions and operate the EMS system in order to maximise their expected value. As shown by Stout et al.55 , the fact of operating an all-ALS EMS system it is possible to reduce the complexity of triaging of calls and of defining what sort of unit should respond. What is more, in such systems there is no possible need of secondary triage on-scene (e.g. calling a different ambulance type for support). This however comes at the cost of possible prolongation of time-to-arrival and of dilution of certain paramedic skills. The latter is specifically important, since according to Stout et al., in only 10% of calls ALS skills are required.

When describing current state of the art in ambulance dispatching, we should mention the general Emergency Medical Dispatch software, and specifically the Medical Priority Dispatch System (MPDS). It is a software system, which aims to categorise emergency medical calls into numerical complaint-based categories and to assign them a given handling priority. The system provides the dispatcher with detailed questions, which are then asked to the caller. Basing on their answers the system categorises the call, assigns the handling priority. Then, the calls can have a sub-group and a modifier assigned to help responders in knowing the details of the case they are to deal with. The category, priority, sub-group and modifier together form the so-called MPDS determinant56. The MPDS is widely used across the world and in Europe itself for triage and categorisation of the calls57. It has been proven that the use of MPDS system has high sensitivity but moderate to low specificity in sending appropriate units to patients requiring ALS58,59. Despite this problem, Dong et al. showed that the use of an optimised version of MPDS in China led to an increased diagnosis consistency of the Acute Coronary Syndrome and reduced the call-to-patient arrival time60.

The classical version of the tool however, stops at categorising the calls, and not naming (in terms of exact callsign) the best unit to respond61. Since optimisation methods look at identifying the best possible decisions, combining them with MPDS may be a good idea. One could first categorise the call using MPDS and then find the best exact ambulance which should respond to the call via mathematical optimisation. Similar approach was proposed in Ref.47 , where authors perform multicriteria ambulance assignment (dispatch) optimisation considering different levels of priority of the emergency calls received. Although they do not state that the priorities are assigned using MPDS, one can easily deduct that MPDS could be a good candidate to perform this task.

After the EMS crew finishes stabilising the condition of the patient, the correct emergency department is to be identified. These problems have been studied in literature as well, mostly as ambulance routing or allocation problems. Talarico et al.62 investigated routing of ambulances transporting patients with different levels of acuity, yet they have not distinguished EDs basing on speciality they can offer to patients. This has been included as an additional criterion through weighted sum scalarisation in Ref.63. ED competence in ambulance allocation optimisation considering possible ED overcrowding was also included by Acuna et al.64. The authors have considered the speciality through constraints in the optimisation problem. An important contribution in the field of emergent cases assignment to EDs was given by Leo et al.65, where the authors included both speciality of units (as additional criterion, with weighted sum scalarisation) combined with the ED workload management.

From medical point of view, many patient transport protocols have been developed. These documents give guidelines to the responding teams on where to transport a given patient. Some examples of those are given for Alabama31 and for Delaware32. They give information on where and how to transport a given patient, basing on certain clinical criteria. For example, in Alabama it is recommended that the ambulance crew considers transporting a patient with STEMI to a hospital with catheterisation lab available. Yet, if the ambulance crew is unsure of the appropriate destination hospital, Online Medical Directors (OLMD) should be contacted for support. Similarly, in Delaware such a patient should be transported when practical to a PCI-capable facility, bypassing the closest hospital. A little bit more strict are the Polish Emergency Medical System plans, established for each of the 16 Polish voivodeships. As an example—in the Swietokrzyskie voivodeship exact addresses of hospitals capable of performing a given emergency medical procedures are named. The plan leaves choosing the most appropriate unit for a given patient X to the joint discretion of the medical dispatcher and of the chief of the medical team66.

Unfortunately not for all conditions such protocols exist, and not everywhere they were established. The authors of Ref.67 outlined that 78% of US states had implemented EMS triage and destination plans for trauma, around 33% for burns, stroke, and STEMI, while only 10% for cardiac arrest. This is in line with further findings of Authors of Ref.68 , who identified only 16 states with specific transport protocols for patients with stroke caused by large vessel occlusion (LVO). What is more, even if protocols are well adopted with dedicated nation-wide patient-care networks established, misdirection of patients can also happen. This is reported for European countries when referring to STEMI patients, for whom quick intervention in a PCI-capable hospital is crucial to reduce the mortality69,70. What is more, the protocols themselves provide guidelines on when to bypass the closest ED and transport the patient directly to a referential unit. In that sense, they do not assign a given ED (in terms of its exact address) to a given, precised patient X. Neither do they take into consideration the current operational state of the EMS system, e.g. in terms of current availability of hospital beds. That is why these protocols should be considered as input to optimisation procedures, which take care of assigning a very precise hospital to a given patient in urgency.

Our literature review outlined that, there exist some currently used interesting dispatch systems (MPDS) and EMS transport protocols. Dispatch systems however, focus mostly on performing the triage of calls and assigning a given priority to them. They do not perform the dispatch as understood by the OR community, i.e. do not give exact information on which unit (identified through its callsign) is best to respond to a given emergency. When it comes to EMS transport protocols, they give guidelines on with what sort of emergency should the ambulance crew consider taking the patient to a specialised ED. The protocols do not tell exactly that a given patient X is to be taken to the hospital Y, considering current operational state of the complete EMS system. These systems and protocols can integrate well with optimisation techniques. They can act as an input guidelines—by either estimating the priority of the call or by setting standards on what speciality should the destination hospital offer to a patient suffering from a specific medical condition. Then, taking this medical input, operational research (OR) techniques can be applied to determine and assign the currently best unit to respond to an emergency (either ambulance or ED). Our paper intends to fill this gap, by combining OR methods which allow for assigning exact units to exact patients in a Pareto-optimal way. This is done considering clinical condition of the patients and the current operational state of the EMS system.

Despite the fact that OR in EMS organisation is a topic of extensive research, the majority of papers mostly consider the time criterion in the ambulance-to-patient and patient-to-hospital dispatch. There exist, however some notable research works that include also the speciality levels of ambulances or EDs. From what we have found, it is mostly included in the optimisation problems as constraints or criterion with weighted sum scalarisation. We believe that inclusion of speciality in the form of constraints might greatly restrict the feasible set of the problem, and in some situations even make the dispatch infeasible. When it comes to weighted sum scalarisation however, we believe that assignment of appropriate weights to criteria might be a nontrivial task, especially for a medical dispatcher, who is not an expert in OR. Thus, this scalarisation might not be the easiest to be applied. What is more, to the best of our knowledge, we have not identified any paper that considered possible re-referrals of patients between a unit with lower speciality and the one with its higher level. In that sense our paper intends to fill the gap identified, as well as applies the Reference Point Method scalarisation, which we believe is well-suited for applications in services of general interest.

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