Location and capacity allocation for emergency contact points in large-scale power outages

Abstract

Nowadays, industry and individuals alike are highly dependent on a reliable power supply. A large-scale power outage, commonly known as a “blackout” is caused by natural disasters, cyber attacks, technical failure, or human errors, and can lead to a variety of severe consequences. The far-reaching dynamics of blackouts can even result in the collapse of critical public service infrastructure reliant on electricity (e.g., communication, water supply, medical services, public safety). Particularly, the loss of information and communication infrastructure essential to reporting medical emergencies, and the collapse of the drinking water supply are two critical stressors for the population to cope with. One attempt to tackle this situation is to install temporary emergency contact points (ECPs) into existing infrastructure. These can be approached by the population to communicate with medical personnel and to receive drinking water. Different types of professional personnel, which is a limited resource, are required to run such ECPs. Our study introduces this tactical decision problem. We formulate it as an integer linear program for the optimal spatial allocation of ECPs, such that multiple types of human resources that are required for operating such locations can be efficiently assigned. A comprehensive numerical study, based on data of the City of Vienna, demonstrates how to reduce the walking distance of inhabitants while increasing the efficiency of resource allocation. Matrix pruning based on an enforced limit of the walking distances together with a decomposition approach is utilized to solve the considered instances.

1 Introduction

As today’s society is highly dependent on technology, the fear of a large-scale (or even partial) power outage, commonly known as blackout, continuously grows (Carreras et al. 2004). Phasing out of nuclear and coal power to fight climate change combined with a potential lack of personnel that operates power infrastructure (as, for instance, caused in different countries during the COVID-19 pandemic or during the armed conflicts in Ukraine (Tollefson 2022)) make such scenarios more likely to occur now than ever before (Carreras et al. 2016). Past events, including the massive electrical power failure across northeastern United States in 2003, the blackout in Texas, USA, during the winter of 2021, the series of blackouts in France in 2019, or blackouts in India in July 2017, demonstrate the far-reaching effects of extensive power outage on society, along with substantial long-term economic losses (Bo et al. 2015; Busby et al. 2021; Parihar and Bhaskar 2018; Mahdavian et al. 2020). In case of such incidents, natural calamities (storms, earthquakes, etc.), transmission and generation failures, cyber issues, or human errors trigger the disruption of power lines or the shutdown of entire power grids (Sharma et al. 2021). This, in turn, has cascading dynamics that finally result in the collapse of critical public service infrastructure which is reliant upon electricity; this includes communication, drinking water treatment, medical services, and public security (Beatty et al. 2006; Dobson et al. 2007). Card payments at supermarkets become disabled, frozen and perishable goods spoil quickly, gas stations cannot provide petrol to customers, homes are without heating, and public transport (including air and rail traffic, metro, and bus systems) is out service (Petermann et al. 2011).

For individual citizens, this means that basic daily needs can no longer be satisfied in the common way and services that are usually provided by public authorities are temporarily inaccessible for them. Encouraging inhabitants’ self-preparedness (storing food, maintaining battery-powered lighting, and using alternative ways of heating, etc.) is a basic preventive measure (Heidenstrøm and Kvarnlöf 2018). However, management of medical emergencies poses severe operational challenges for responsible authorities and involved disaster agencies (Freese et al. 2006). Compared to non-blackout situations, making an emergency call during such a crisis will only work as long as mobile phones are charged and cellular base stations are supplied by emergency power generators (Wang et al. 2019).

Only a limited number of stations are equipped with such generators, making a comprehensive communication network service almost impossible even during shorter blackouts that last only a few hours. A total failure of the information and communication infrastructure is inevitable once the emergency power generators run out of gas  (Petermann et al. 2011). Prominent examples in this regard are the communication network disruptions during hurricane Harvey affecting parts of Texas, USA, hurricane Irma causing 80% of lost cell sites in southern Florida, USA, or hurricane Maria, where up to 95% of the cell sites were out of order during a two-week-long blackout in Puerto Rico (Kwasinski 2018). This loss of information infrastructure and telecommunication is not only a considerable stressor among the population (Rubin and Rogers 2019), but might also results in a high number of untreated emergencies. Given the criticality of these situations, elaborating targeted preparedness and response activities to enable a blackout-proof communication system between inhabitants and authorities is urgently needed (Rudolph-Cleff et al. 2022).

One attempt to do so is the installation of stationary points of contact using existing infrastructure (in schools and other public buildings) that can be approached by inhabitants at shortest possible distance. These stationary points can be used to report medical emergencies and other critical incidents for further processing by the emergency medical service (EMS), the fire brigade, or the police. We refer to such points as emergency contact points (ECPs) in the rest of the article. Practitioners advise against locating ECPs within existing EMS stations as this could pose significant risks. Chaotic scenes resulting from the influx of individuals seeking care at EMS stations may disrupt internal processes and operations within the EMS stations.

The utilization of existing infrastructure to create temporary surge capacities during catastrophic events causing mass casualties is a well-explored topic in the literature, as evidenced by Hick et al. 2004 and Farahani et al. 2020. However, it is essential to note that the dynamics of mass casualties events significantly differ in the case of a blackout. In the scenario of a power outage, a fundamental assumption is that the number of incidents or casualties does not necessarily increase; rather, the major challenge lies in the complete absence of communication.

Primarily, ECPs should guarantee certain levels of medical care. Furthermore, they can represent supply locations to satisfy inhabitants’ urgent demand for essential goods. As water purification systems may not be fully functional, and pumps, used to transfer drinking water through the municipal water systems, may not work, drinking water quickly becomes the most demanded good during a blackout (Busby et al. 2021).

Enabling these essential services at ECPs requires thorough selection of the locations in order to ensure acceptable (preferably short) walking distances for the inhabitants. Also, the efficient use and allocation of human resources (physicians, paramedics, technicians, etc.) among the ECPs is critical because public authorities and other organizations cannot rely on sufficient backup resources during blackout response. They simply would be idle during non-disaster periods (Doan and Shaw 2019). Overcoming resource constraints by accumulating needed personnel through mutual support from other organizations in non-affected cities or regions is discussed to be rather difficult (Sarma et al. 2019), as large-scale blackouts usually have nationwide impacts on available resource infrastructure.

In light of those challenges and the recently issued warnings by several governments, e.g., in Switzerland, concerning the growing risk of large-scale blackouts to occur in the near future (Shields 2021), this study contributes as follows.

• We introduce and mathematically formulate the tactical decision problem that determines the optimal allocation of ECPs based on demand estimations, such that different types of human resources, which are required to run those locations, are assigned efficiently, while the reachability for the affected population is guaranteed.

• Citizens are involved in the response to a blackout such that they are asked to approach the ECPs on foot. In contrast to most existing models in the literature, we assume that there is no central authority controlling the flow of citizens that approach the ECPs during a blackout situation. Instead, our model is based on the rational behavior of the individual citizens. We assume that they will always walk to the nearest active ECP. Under this consideration, we aim to minimize the required resources to run the ECPs while always assigning the inhabitants of each residential building to the nearest active ECP.

• A methodology to estimate the demand for emergency care as well as for drinking water during blackout situations, which is based on census data and information from geographic information systems, is developed.

• An enforced limitation of the walking distances allows a highly effective matrix pruning approach, which removes at least 90.47% of the arcs. On top of that, a decomposition approach is used to solve the instances (after pruning). A comparative study shows that the quality of the results obtained through the decomposition approach is appropriate for the intended use case.

• We conduct an extensive computational study based on real-world data from the City of Vienna, Austria. Several methodological as well as managerial insights are derived. We demonstrate, for instance, that a carefully planned ECPs system can guarantee accessibility by foot, while efficient use of (human) resources is enabled.

In conclusion, our study forms a framework that can support authorities in their decision making such that basic services for citizens in case of a large-scale power outages can be maintained.

The remainder of this paper is organized as follows. In Section 2, we review related literature. Section 3 provides the problem description and introduces notations along with our model assumptions. Then, in Sect. 4, we propose the formal model which we apply within a numerical study for the City of Vienna. The latter is presented in Sect. 5. The applied solution approach to solve the problem is presented in Sect. 6. We discuss the results of the numerical study in 7. Finally, a discussion on the limitations of the study and an outlook to future research conclude the paper in Sect. 8.

2 Related work

The scientific literature on optimization models for disaster response is divided into four main areas, as outlined by Kamyabniya et al. (2024) in their review of related work: (i) distribution of relief items, (ii) location of relief facilities and temporary shelters, (iii) integrated relief items distribution and shelter location, (iv) transportation of the affected population. In our literature review, we initially focus on facility and resource allocation problems for different applications in disaster and emergency management. Afterwards, we discuss the scientific work in the field of blackout preparedness and response. This conclusively leads to the apparent gap in scientific literature and justifies the relevance of the study at hand.

2.1 Facility allocation in emergency management

Facility allocation models for coping with emergencies have been developed for single- or multi-objective settings by taking different facility types, such as supply distribution centers, shelters, or temporary EMS centers into account. Models can be further classified due to location capacity (capacitated, uncapacitated), solution space (continuous, discrete), or the number of new facilities (single, multiple), see Laporte et al. (2015). In Cavdur et al. (2016) a study on the optimal allocation of temporary disaster response facilities to minimize the walking distance for disaster victims, the amount of unmet demand and the number of opened facilities is presented. The authors consider a different set of commodities distributed at facilities for which they assume a demand probability based on guidelines presented by the World Health Organization. Locating distribution centers in the immediate aftermath of a disaster is subject of an analysis by Baharmand et al. (2019). Their models allow minimizing total logistics costs and total response times that consists of the required time for setting up distribution centers and the busy time during an operation.

Aside from optimally allocating supply points, literature also treats shelter allocation in the context of evacuation planning. In this respect, Bayram et al. (2015) present an allocation model for shelters with the objective of minimizing the total time that evacuees spend in the network, thereby ensuring that inhabitants are evacuated in a timely manner. The model is tested in a computational study with randomly created potential shelter sites based on the Istanbul road network. Randomly created demand for each origin node is considered. Recently, several contributions on shelter allocation for earthquake evacuation have been published. While the model by Xu et al. (2018) minimizes the total evacuation distance, Geng et al. 2021 follow multi-objective optimization to minimize the evacuation time and the number of opened sites. Both studies give examples of model application in real-world settings. Minimizing the shelter area and evacuation distance is considered by Ma et al. (2019), who also apply their model in a case study for the City of Beijing. Another recent study by Praneetpholkrang et al. (2021) presents a shelter location-allocation model with the target to minimize total costs, evacuation time for all victims, and number of shelters required to provide thorough service to victims.

The problem of locating disaster response and relief facilities in the city of Istanbul in preparation for a major earthquake is dissected by Görmez et al. (2011). A two-tier supply system shall utilize existing public facilities locally while newly established facilities act as regional supply points. A mathematical model to decide the locations of the regional “new” facilities with the objective of minimizing the average-weighted distance between the demand locations and the nearest facility is proposed. Additionally, distance limits, capacity limitations, and backup considerations are taken into consideration. Within Istanbul, 40 potential local facilities and 964 neighborhoods (representing the population) are considered. Analysis shows that five regional facilities can serve the local facilities with an average distance of 7 km.

The provision of medical care for disaster-affected populations is addressed by several publications that propose the optimal allocation of temporary EMS stations. In this regard, a study by Gao et al. (2017) suggests allocating EMS under consideration of minimizing the total travel time and the total mortality risk value of patients in the whole disaster area. The authors apply genetic algorithms to solve the model and show examples of model application. A recent study by Liu et al. (2019) also addresses the optimal allocation of medical service facilities considering the trade-off between the effectiveness of humanitarian medical service and its operational costs. In a bi-objective model, they maximize the number of expected survivals and minimize the total operational costs of ambulance and helicopters that transport casualties. Wang et al. (2020) concentrate on the optimal allocation of ambulance stations with the objective to minimize mean waiting time of victims and the response time between stations and affected locations. The presented model is tested in a computational study considering simulated demand locations and potential ambulance station sites over the area of Chaoyang, a district of Beijing. The optimal allocation of temporary disaster debris management sites to perform efficient sorting of debris generated after natural disasters is researched by Habib and Sarkar (2017).The proposed model is used to minimize the total transportation cost of debris to debris management sites. The performance of the model is demonstrated in a numerical study based on the City of Karachi, Pakistan. Dogan et al. (2016) present a model for locating preventive health care facilities based on 15-year forecasts of the population (age group and sex) on city district level. The travel distance to reach these locations is used as a measure of their accessibility.

The use of existing infrastructure (e.g. schools, parks, or stadiums) as alternative care facilities (ACFs) is a well-established approach to rapidly augment patient care capacity during catastrophic healthcare events, e.g., Hick et al. 2004 or Caunhye and Nie 2018. Major earthquakes or bomb attacks in metropolitan areas can result in a vast number of casualties, triggering a catastrophic healthcare event. The purpose of ACFs in such cases is to establish additional surge capacities when the still-functional hospitals are no longer capable of managing the large number of casualties. Efficient location-allocation of the ACFs is required due the uncertainties associated with the movements of self-evacuees (individuals seeking medical attention independently) and the decisions of emergency responders regarding casualty allocation for triage. In the context of earthquakes, a primary source of uncertainty stems from the sequence of aftershocks and the resulting damage. Disaster situations unfold dynamically, revealing more detailed information over time, and necessitating adaptive and robust planning. Earthquake casualties result from: (i) initial ground motion, (ii) damage to steel-frame buildings, (iii) fires following the earthquake, and (iv) transportation-related incidents. To address the complexities of ACF location-allocation in such dynamic and uncertain scenarios, Caunhye and Nie 2018 propose a three-stage stochastic programming model, aiming to optimally position ACFs and allocate casualties.

In their extensive literature review, Farahani et al. (2020) provide insights into mass casualty management in severe disaster scenes. They emphasize that the primary sources of uncertainty in such situations stem from: (i) the uncertainty regarding the location of the disaster, (ii) disruptions in transportation networks, (iii) resource scarcity, and (iv) the potential for casualties among the rescue and medical teams. Addressing these uncertainties necessitates the use of complex and dynamic models to effectively plan and respond to such challenging scenarios. However, in the case of a large-scale power outage, these factors are seldom applicable, and there are no indications that mass casualties may occur.

2.2 Resource allocation in emergency management

The problem of identifying resource needs and resource availabilities (e.g., food, water, shelter, medicine) in real time and optimally allocating them among a set of demand locations is tackled in Basu et al. (2022). Therefore, the authors propose a utility-driven resource allocation model based on demand information from micro-blogging websites, such as Twitter. The model has the objectives of minimizing overall resource deficits and total resource deployment times, which is the time to transport relief goods between warehouses and demand locations. A similar setting is considered by Liu et al. (2020), who present a multi-objective optimization model to allocate emergency material among demand points to minimize the losses and the economic cost arising from rescue operations. The validity of the model is proven in an experimental setup where material demand, storage capacities, transportation costs and the loss due to material shortage is randomly generated, and a Pareto optimal solution is determined. In Lu and Sun (2020), a model for the allocation of equipment resources to cope with different emergencies in metro systems is presented. In their work, the authors consider site management resources, life rescue resources, and engineering resources to be allocated among demand points under consideration of minimized penalty and allocation costs.

The above discussed articles primarily focus on tangible resources and ignore the allocation of human resources. This is the subject of analysis in Su et al. (2016), who use an integer linear programming model to allocate disaster response coalitions consisting of different resource types (members of the police force or the fire brigade) to concurrent emergencies. The model follows the objective of minimizing the weighted sum of the total travel time of disaster response coalitions and the total costs of the allocated emergency resources. In an example of application, the authors compare the performance of the proposed solution algorithm and demonstrate its practical feasibility in emergency response. Another study by Tirkolaee et al. (2020) also addresses the allocation and scheduling of disaster rescue teams of different types to enable efficient planning for rescue units in the immediate response to disasters. The authors present a mixed-integer linear programming model to minimize the total weighted time and complete all the rescue operations as well as to minimize the total weighted delay in all the rescue operations when sending rescue units from a center to demand locations. A numerical example based on the province of Mazandaran, Iran, is used to show the applicability of the proposed model. Equipment as well as personnel resources for coping with simultaneous disasters are considered by Doan and Shaw (2019), for which the authors present two optimization models. The models allow allocating existing resources to achieve best performance and to determine the optimal resource capacity to manage parallel disasters. The authors also take budgetary limitations into consideration when allocating additional resources.

Rest and Hirsch (2022) introduce a decision support that shall help securing the provision of home health care during times of disaster (epidemics, blackout, heatwaves, etc.) where there is increased stress on these types of services. However, the authors acknowledge that using a software-based decision support system makes it prone to blackout situations.

2.3 Facility and resource allocation in emergency management

Research that combines both—the optimal allocation of facilities and resources—is quite limited. A study by Paul and MacDonald (2016) proposes a model for determining the location and capacities of distribution centers for emergency stockpiles in preparedness of earthquake disasters having the objective of minimizing the total expected costs (comprising costs for facilities, supplies, and fatalities). The demand, respectively, the number of casualties that needs to be satisfied is modeled as a function of earthquake severity and its interaction with factors such as the resistance of building infrastructure and geological characteristics of a given region. In Wang et al. (2020), the authors investigate the location allocation problem to determine the selection of shelters, medical centers, and supply distribution centers in combination with the allocation of commodities and medical supplies at demand nodes. Demand for respective resources is modeled under consideration of different priority evacuees with injuries that are roughly estimated. The developed programming model follows the objective to minimize the total evacuation distance, total cost, and total unmet resources. An example of application on Beijing, China is used to validate the proposed method. A similar approach is presented in Zhang and Cui (2021), who illustrate a facility location and resource allocation model for disaster preparedness and response. In detail, the model determines the optimal location of service locations and quantities of stocked emergency supplies (e.g., food, medical kits). The victims’ demand for supplies is estimated following the concept of deprivation cost, indicating that demand patterns follow an exponential increase before supply arrival and a linear decrease after supply arrival. The model which is used to minimize the total system costs is tested in a real-world setting using data of the US Gulf Coast area. In Escudero et al. (2018), a model for the optimal warehouse location and capacity assignment of emergency commodities (such as blankets, water, and clothing) is presented. The considered objective function minimizes the total expected costs. An illustrative example shows the potential added value of the suggested methodology. El Tonbari et al. (2024) analyze a two-stage natural disaster management problem using a stochastic programming model. The first stage involves a facility location decision, where facilities are opened and resources like medical and food kits are pre-allocated. The second stage addresses a fixed-charge transportation problem, focusing on routing resources to disaster-affected areas after the event has occurred. Acknowledging the lack of historical data, a classical stochastic programming approach seems ill-suited for this application. As a result, the authors propose a two-stage distributionally robust formulation using a Wasserstein ambiguity set. This formulation accounts for distributions consistent with historical data and includes a tunable parameter to manage risk aversion. The effectiveness of their approach is demonstrated through a case study of hurricane threats in the Gulf of Mexico, utilizing a column-and-constraint generation approach. Aringhieri et al. (2022) focus on fairness in providing ambulance services in the post-disaster stage. The paper is concerned with the problem of finding the best ambulance tours to transport the patients in relief operations while considering fairness and equity to deliver services to patients. In the publication by Hongzhong Jia and Dessouky (2007) models to locate emergency medical facilities that maximize efficiency are proposed. The model seeks to find the minimum number of facilities that cover the demand focusing primarily on the challenges posed by terrorist attacks involving dirty bombs or anthrax. Further, both p-median and p-center models are explored. The case study based on the Los Angeles area utilized a relatively strong level of aggregation to handle the complexity of urban settings and potential disaster scenarios. The city is divided into squares, and each square’s center represents an aggregated demand point, covering an area of $5\times 5$ miles.

2.4 Research on blackout preparedness and response and power restoration

So far, little has been reported on specific preparedness and response activities to cope with blackout scenarios. Prezant et al. (2005) analyze the impact on emergency medical services and hospitals during the New York City blackout in August 2003. The study reveals a significant increase in EMS calls and hospital activity, with respiratory issues surging by 189% due to failures in respiratory devices among community-based patients. The findings highlight how blackouts can easily overwhelm emergency services. It suggests that disaster preparedness could be improved by providing backup power systems to community-based patients.

Misumi et al. (2021) take up the problem of disrupted communication infrastructure during power outages and addresses the need to distribute information across the population at minimum remaining electricity consumption. The authors suggest using optimally allocated information boxes (i.e., devices having a function to receive and store disaster relevant information) to spread information while reducing the energy consumption of mobile terminals belonging to disaster victims.

After reviewing the vast body of existing literature on facility and resource allocation for emergency and disaster response, it is apparent that problems have not yet been treated in the context of blackout scenarios. To the best of our knowledge, this is the first study that introduces ECPs to enable the provision of medical treatment and drinking water supply to the public during blackout situations. Consequently, we can derive recommendations that have not been communicated before, thereby contributing to the infant stream of research on blackout preparedness and response.

Scheduling repair, reconfiguration tasks, and dispatching crews to restore power networks after a system outage is an emerging research area. This domain encompasses various interconnected planning tasks, as highlighted in recent studies, e.g., (Arif et al. 2018; Chen et al. 2019; Morshedlou et al. 2018; Tan et al. 2019; Simon et al. 2012). Further, Makarov et al. (2005) argue that in emergency situations of a cascading nature, such as blackouts, automatic emergency control systems should play a major role. The paper outlines the Russian principles and systems of dispatch and emergency control, describing a multilevel defense system designed to protect the Russian power grid from developing cascade failures.

3 ECPs for blackout response

ECPs are stationary points of contact located at suitable existing infrastructure (schools or other public buildings) that are activated and operated in case of a blackout. Short activation times (1–2 h) are essential to effectively provide basic services to the population during such an event. Further, this kind of locations must be operational for periods of up to one week.

ECPs are supplied with electricity from (portable) emergency power generators and, therefore, their operability is independent from the public power grid. ECPs can be approached by citizens to report medical emergencies and critical incidents (crimes, fires, etc.) for further processing by the responsible authorities. Relevant emergency/incident information is forwarded via radio and backup communication means to command and control centers that coordinate help with other organizations. ECPs should not serve as EMS, police or fire base stations from where resources, such as ambulances and medical personnel, are dispatched to demand locations (emergency sites). In the event of a blackout, EMS providers will gather resources at their base stations as these locations are typically equipped with emergency power generators. However, quick intervention at an ECP—in case patients in urgent medical need arrive—should be guaranteed by allocating medical staff (a physician and a paramedic) on site.

ECPs must be reachable by foot as public transportation will most probably be out of service and the road network will likely suffer from severe congestion due to non-functional traffic lights (Alvarez and Blas 2020). Walking distances to reach the closest ECP exceeding 1750 m (one-way) are deemed unacceptable. This distance equates to a walking time of approximately 20–25 min for an average fit individual. Prolonged walking distances and times are considered unreasonable, particularly for individuals with disabilities or injuries. ECPs should further offer a basic supply of drinking water as, depending on the specifics of the local water supply network, water purification systems and electric pumps may become dysfunctional due to lack of power which may lead to a collapse of the water supply (Clark-Ginsberg et al. 2021).

As most water networks are gravity-based, it is expected that (semi-)detached houses can be supplied with water even during extended blackout situations. Citizens living in multi-storey buildings are thus more likely to be immediately affected by water shortage, making residents dependent on drinking water supplied at ECPs. Offering other supplies at ECPs, especially fresh food, is highly impractical from a logistics viewpoint as perishable goods require cooling systems which are typically not available at ECPs. Moreover, pre-positioning of supply inventories at ECPs is not possible due to capacity constraints of buildings. Clearly, shipments from supermarkets to ECPs are not feasible due to limited transport capacities and paralyzed traffic networks. Anyway, we assume that it is more practical to distribute groceries in a controlled manner directly at the supermarkets where they are stored rather than relocating them. This strategy is now also pursued by Austrian food retailers (ORF 2022).

Clearly, the number of established ECPs is limited by the available (human) resources required to operate them. This is a phenomenon that is often observed in disaster management (Doan and Shaw 2019). Along with medical personnel and water and sanitation (WatSan) experts, there are also administrative, security, and technical resources required to maintain operations of an ECP. Certain coordination tasks, security-related issues (chaos prevention), and maintenance activities fall into the scope of responsibilities of ECP personnel. When selecting potential ECP locations, officials must assess the infrastructure’s suitability based on its size and layout.

4 Formal model

In this section, we first focus on the demand types that are generated at residential buildings as well as on the resource types required to operate an ECP. We then provide a formal model of the problem, which has been introduced in Sect. 3. Primarily, the model is concerned with the limited availability of resources in case of a blackout as well as the distance—walking paths—between residential buildings (as demand locations) and potential ECP locations. As costs for establishing the ECPs and providing resources cannot be determined, they are neglected in the model. Costs may be implicitly contained in the model in form of limited resources.

4.1 Demand at residential buildings

In case of a blackout, residential buildings create demand that must be satisfied by the established active ECPs. In detail, we consider the set of all residential buildings $\mathcal {J}$, where each building $j \in \mathcal {J}$ creates three types of demand with the following enumeration $\mathcal {K}=\{1\equiv \text {base}, ~ 2\equiv \text {WatSan}, ~ 3\equiv \text {medical} \}$.

Note that we assume that medical and WatSan demand occurs only at residential buildings and not at any other places. It is assumed that, during a blackout, most inhabitants are likely to remain at their place of residence and refrain from moving elsewhere (such as workplaces). The used notation is summarized in Table 1. As the demand values are often based on estimations, as elaborated in Sect. 5.2, defining them as integers would be too restrictive.

Table 1 Overview of input parameters and decision variables

Base Demand A residential building must be within reasonable walking distance of an active ECP such that any critical incident (for instance a medical emergency) can be reported in a timely manner. However, we do not explicitly quantify this type of demand in our model. Accordingly, we set $d_{j1}=1$ and $s_1=\infty$.

WatSan Demand Value $d_{j2} \in \mathbb {R}^{+}$ defines the number of people (living in the residential building) that must be supplied with drinking water through an ECP.

Medical Demand Value $d_{j3} \in \mathbb {R}^{+}$ defines the average number of medical incidents per day that occur at the residential building. This number depends on the number of residents and their age.

4.2 Resources and capacities at ECPs

The set of candidate locations for ECPs $\mathcal {I}$ must be determined via a manual process of finding appropriate options. The walking distance between a residential building j and an potential ECP i is $w_{ij} \in \mathbb {R}^{+}$.

The demand induced by the residential buildings must be covered by installing sufficient capacities (same as demand types). Those three resource types are as follows.

Base Team Some basic personnel is required to run an ECP. Such a team consists of the following roles: (i) a general manager, (ii) a staff manager, (iii) a police officer, (iv) a security employee, (v) a radio operator (taking care of the communication with the headquarters), and (vi) a technician (to run and maintain the infrastructure). Each base team can handle $s_1$ inhabitants that are assigned to the respective ECP, and each ECP candidate location can host exactly one such team, i.e., $T_{i1}=1$.

WatSan Team WatSan experts (as maintained by several emergency response units world wide, see IFRC 2022), guarantee certain quality standards of water distributed to inhabitants. Such a team consists of (i) a WatSan expert, that is backed by (ii) a security employee. An ECP i can host at most $T_{i2} \in \mathbb {N}$ WatSan teams. When determining the maximum number of teams that can be allocated to candidate locations, it is important to consider both the size and layout of these locations. Each team can provide water for $s_2$ people per day. This number is strongly dependent on the given equipment. There are many different configurations of water purification and distribution systems available on the market, where the largest can provide drinking water for up to 45,000 people per day. Usually, two team members run the water purifier while the actual distribution of water is organized via a rack of tap stands.

Medical Team A team consisting of (i) a physician supported by (ii) a paramedic can provide immediate care for $s_3$ medical incidents per day. An ECP i can host at most $T_{i3} \in \mathbb {N}$ medical teams.

4.3 Integer linear model

Primarily, one must decide which ECPs are selected—being active—among the candidate locations and, secondly, the allocation of the teams to active ECPs.

Hence, we are facing a location and capacity allocation problem (LCAP). The capacity of an ECP is determined by the number of assigned WatSan and medical teams. Sufficient medical and WatSan capacities imply availability of the corresponding supplies. The objective is to efficiently utilize the capacities, i.e., to minimize the excess capacities at each active ECP, while keeping the walking distances for the inhabitants low. Hence, the objective function implicitly minimizes the number of used teams. In general, we seek to cover all demand using a minimal number of teams. This is based on the fact that coordination, and therefore activation times (such as the setup of equipment), increases with the number of people. Also, space at the ECPs is often limited. When running the ECPs for longer periods, personnel must work in shifts, and it is therefore preferable to have excess capacities available. We assume that the local authorities have total control of the installation and resource assignments of ECPs.

In contrast to most existing models in the literature, we assume that there is no central authority controlling the flow of citizens approaching the ECPs during a blackout situation. Similar to Kongsomsaksakul et al. (2005), we assume that the authorities do not have control of the route choices of the inhabitants when approaching ECPs. Instead, our model is based on the rational behavior of the individual citizens. We assume that the citizens will always walk to the nearest active ECP. Hence, inhabitants will most probably not adhere to any predefined ECP assignments if it is not the closest active ECP. Under this assumption, we aim to minimize the required resources to run the ECPs while ensuring that inhabitants (of each residential building) are assigned to the nearest active ECP.

In that sense, our model is strongly related to the obnoxious p-median problem (OpMP) (Labbé et al. 2001), which is concerned with allocating a set of obnoxious facilities, e.g., nuclear power plant or waste dumpsites. As one typically aims to avoid having these kind of facilities near the population, respective distances are to be minimized (Lin and Chiang 2021). The OpMP requires that in the mathematical model the nearest active location to a population (clients) is taken into account. From a mathematical point of view, our formulation uses a set of constraints to enforce this specific property as suggested in the OpMP model by Lei and Church (2015).

$$\begin{aligned}{} & {} \min \quad \sum _{i\in \mathcal {I}} \sum _{k \in \mathcal {K}} z_{ik} \end{aligned}$$

(1)

$$\begin{aligned}{} & {} w_{ij}x_{ij} \le W \quad \forall (i,j) \in \mathcal {A} \end{aligned}$$

(2)

$$\begin{aligned}{} & {} \sum _{(i,j) \in \mathcal {A}, ~j\in \mathcal {J}} d_{jk}x_{ij} \le s_k z_{ik} \quad \forall i \in \mathcal {I}, \ k \in \mathcal {K} \end{aligned}$$

(3)

$$\begin{aligned}{} & {} z_{ik} \le T_{ik}y_{i} \quad \forall i \in \mathcal {I}, \ k \in \mathcal {K} \end{aligned}$$

(4)

$$\begin{aligned}{} & {} x_{ij} \le y_i \quad \forall (i,j) \in \mathcal {A} \end{aligned}$$

(5)

$$\begin{aligned}{} & {} \sum _{(i,j) \in \mathcal {A}, ~i \in \mathcal {I}} x_{ij} = 1 \quad \forall j \in \mathcal {J} \end{aligned}$$

(6)

$$\begin{aligned}{} & {} w_{ij}x_{ij} \le w_{i'j} + |w_{ij} - w_{i^{'}j}|\left( 2-y_i-y_{i'}\right) \quad \forall (i,j) \in \mathcal {A},~ (i^{'},j) \in \mathcal {A} \end{aligned}$$

(7)

$$\begin{aligned}{} & {} x_{ij} \in \{0,1\} \quad \forall i \in \mathcal {I}, ~j \in \mathcal {J} \end{aligned}$$

(8)

$$\begin{aligned}{} & {} y_{i} \in \{0,1\} \quad \forall i \in \mathcal {I} \end{aligned}$$

(9)

$$\begin{aligned}{} & {} z_{ik} \in \{0,1,2,\ldots \} \quad \forall i \in \mathcal {I}, k \in \mathcal {K} \end{aligned}$$

(10)

The objective function (1) minimizes the number of installed teams (of all three types). Equivalently, this can be written as

$$\begin{aligned} \min \quad \sum _{k \in \mathcal {K}} \frac{1}{s_k} \left( \sum _{i\in \mathcal {I}} s_k z_{ik} - \sum _{j\in \mathcal {J}} d_{jk} \right) . \end{aligned}$$

(11)

which minimizes the weighted excessive capacities; where $\sum _{j\in \mathcal {J}} d_{jk}$ is a constant.

Inequalities (2) ensure that the walking distance from residential building j to the allocated ECP i must be lower than a certain distance W. Inequalities (3) ensure that the demand (of each resource type k) imposed to each ECP i can be satisfied by the allocated teams. Inequalities (4) ensure that the number of teams allocated to ECP i does not exceed the maximum number of teams that can be hosted. Further, it guarantees that teams can only be allocated to active ECPs i. Inequalities (5) ensure that a residential building j can only be allocated to an active ECP i, while Eq. (6) assure that each residential building j is allocated to exactly one ECP i. Finally, inequalities (7) allocate the nearest active ECP i to building j. The variable domains are defined in (8)–(10).

One may want to weaken the demand satisfaction constraints (3). In that case, we add a term that penalizes the unsatisfied demand (with weight $\beta \in \mathbb {R}^{+}$) to the objective function (1).

5 Numerical study

In this section, we apply the proposed location allocation model (see Sect. 4.3) within a numerical study that aims at establishing ECPs in the City of Vienna, Austria, which covers an area of $414.87\,\hbox {km}^{2}$. The total population counts 1,920,949 people (according to official statistics from 2021 (Magistrat der Stadt Wien 2021)).

Moreover, we describe how to derive the demand estimation on a granular level (individual buildings) from the publicly available data found in geographic information systems (GIS) and census data. We first explain how the data is organized and processed. Here, we put explicit focus on the identification of residential buildings and the estimation of the number of inhabitants. Then, we describe the process of estimating medical and WatSan demand that arises in a residential building based on the number of inhabitants (and their age). Additionally, we outline the selection process for ECP candidate locations, how the walking distances are derived, and give an overview on the resulting model size. We provide the key numbers of the LCAP instance of Vienna in Table 6 which can be found in the Appendix A. A detailed analysis of the results is presented in Sect. 7.

5.1 Data preparation

In this section, we give details on the used census data as well as GIS data.

5.1.1 Census data

As described above, we estimate the number of inhabitants living in a residential building as well as the medical and WatSan demand based on census data provided by the City of Vienna. Historical data on both—the number of inhabitants per residential building and medical/WatSan demand—is either not accessible due to data protection guidelines or simply non existent. Most authorities publish census data on an aggregated level, i.e., the number of inhabitants of each age group is reported for relatively small defined geographic areas. In the UK, for instance, the spatial partitions are defined by the Office for National Statistics  (2016). There, the smallest of the areas with published data have between 100 and 625 inhabitants (i.e., 40–250 households). Only little data is available on this level. The United States census divides the country into census blocks, which is the smallest census bureau geographic entity (United States Census Bureau 2021). Generally, census blocks are bounded by streets, streams, and the boundaries of legal and statistical entities. Typically, around 4000 inhabitants live within a census block.

In Austria, where our study is conducted, the census is based on rather small geographic areas. Vienna’s 23 municipal districts are subdivided into $|\mathcal {C}|={1368}$ such (non-overlapping) census areas, so-called Zählgebiete (Stadt Wien 2021a). The population numbers of each census area are provided at a granularity of 5 years (i.e., age groups 0–4, 5–9, ..., 79–84, and $>85$).

5.1.2 Identification of residential buildings

Prior to estimating medical and WatSan demand, one must identify the residential buildings as demand generating locations. The city of Vienna provides precise building and land use data¹ under Creative Commons License (CC BY 4.0). From this, we use the boundaries (Stadt Wien 2020a) of the 23 districts to divide our data set. A digital city map of Vienna (Baukörpermodell LOD0.4) (Stadt Wien 2021b) provides 3D-models of all buildings, bridges, streets, etc. Among all entities, buildings can be identified having the tag F_KLASSE=11. Also, the height of each building (gutter height) and its elevation above the sea level is provided in the data. Hence, a building can be represented by several polygons and/or multipolygons that have different building heights. In the first step, we simplify each building by putting everything into a single multipolygon. Areas that belong to the same building can be uniquely identified via a common building id (BW_GEB_ID). For this “flattened” building representation, we take the largest building height value of all areas. Finally, we simplify the representation of the buildings to a single point as this is sufficient for our study.

To distinguish residential buildings from other building categories, we rely on the actual land use map (Realnutzungskartierung), that is based on interpretation of aerial pictures (Stadt Wien 2005). From the 42 available categories, we are interested in categories having the identifiers KG (Kleingärten, allotment), WM (Wohnen mit Garten, residential areas with garden), and WO (Wohnmischgebiete inklusive Pensionistenheimen usw., mixed residential areas including retirement homes). We assume that these three categories represent residential areas. Clearly, the zones are represented by polygons.

Finally, we join the point representations of the buildings with the land use map and filter for the residential areas. In that way, we retrieve all residential buildings, their ground area $f_{j}$, their height h(j), and their elevation e(j) (above sea level). Similarly, we assign each building to its respective census area.

5.2 Demand estimations

We first provide information on how to estimate the number of people living in residential buildings and, furthermore, explain estimations of medical- and WatSan demand.

5.2.1 Estimation of people living in a residential building

The medical and WatSan demand estimations are based on the residents (and their age) of a building. To estimate the number of residents, we define a set of age groups $\mathcal {G}$, where $|\mathcal {G}|=3$ and $g\in \mathcal {G}$ (in accordance with Jánošíková et al. 2021). These are $0-14$ ($g=1$), $15-64$ ($g=2$), and $>65$ ($g=3$). In each residential building j, there are $p_{gj}\in \mathbb {R}_{0}^{+}$ inhabitants of age group g.

Considering the distribution of the age groups within a census area, we estimate the number of people living in a residential building based on the ground area of the considered building in proportion to the total ground area of all identified residential buildings in the census area. From the census data, we obtain the number of inhabitants $p_{gc}\in \mathbb {N}_{0}$ in age group g living in census area $c\in \mathcal {C}$, where C is the set of census areas. As our study is based on estimations, both $p_{gj}$ and $p_{gc}$ can take fractional values.

The surface area $f_{j} \in \mathbb {N}$ of building j is estimated from its polygon representation. In general, a room height of at least 2.5 m is recommended in the Austrian construction guidelines (Österreichisches Institut für Bautechnik (OIB) 2019). Therefore, we assume that 3 m of building height are equivalent to one floor (accounting for the thickness of the ceiling and the flooring). We obtain an estimate of the total area $\hat{f}_{j} \in \mathbb {N}$ of a building having height h(j) as

$$\begin{aligned} \hat{f}_{j}= \max \left( \left\lfloor \frac{h(j)}{3} \right\rfloor , 1 \right) f_j. \end{aligned}$$

(12)

Finally, let $\mathcal {J}_{c} \subset \mathcal {J}$ denote the subset of buildings located in census area c. The number of inhabitants (of age group a) is then estimated as

$$\begin{aligned} p_{gj} = p_{gc} \frac{ \hat{f}_j }{ \sum \nolimits _{j \in \mathcal {J}_{c}} \hat{f}_j }. \end{aligned}$$

(13)

5.2.2 Estimation of WatSan demand

Water supply is independent from pumps in most parts as well as for most residential buildings in Vienna Stadt (Wien 2022a). The City of Vienna is supplied with drinking water by two main pipelines that originate at different springs in the nearby mountain regions. The two pipelines end in two water cisterns located at 251 m above sea level (cistern Rosenhügel) and 327 m above sea level (cistern Lainz), respectively. From there, the water is further distributed through a network of pipelines and smaller cisterns across the entire city infrastructure.

The historical parts of the city are supplied with water from the cisterns; hence, all historical buildings that are located in those areas are supplied without pumps and based on gravity. According to the building codes from the years 1859 and 1868, it was required that the gutter height of such a building does not exceed 233 m above sea level to ensure water supply (back then, only the first cistern existed).

Consequently, we can assume that residential buildings that are supplied by the first or the second cistern can be supplied with water based on gravity up to a gutter height (above sea level) of 231 m or 307 m (after subtracting a constant value of 20 m that accounts for the minimal height difference), respectively.

For those parts of the city where it is not completely clear from which cistern they are supplied from, we assume a threshold of 269 m (average value between both cistern heights).

We can therefore decide if WatSan demand will occur (indicated by $v_{j}$) at a residential building j based on its elevation e(j) and building height h(j) as follows ($\theta _j$ denotes the required threshold for building j for being served by one of the cisterns).

$$\begin{aligned} v_{j}= {\left\{ \begin{array}{ll} 1 \quad \quad \text {if}~ e(j) + h(j) < \theta _j \\ 0\quad \quad \text {else}. \end{array}\right. } \end{aligned}$$

(14)

Consequently, the WatSan demand imposed by building j is

$$\begin{aligned} d_{j2} = \sum _{\mathcal {G}} p_{gj}v_{j} . \end{aligned}$$

(15)

In Vienna, WatSan demand would only occur for some outskirts that are located at higher elevation and few high-rise buildings (such as the United Nations Office). However, for most other cities which do not benefit from this kind of topology and water network design (i.e., water supply based on gravity), WatSan demand may occur for the majority of the residential buildings. Typically, buildings of up to two storeys (8–12 m) can be supplied by gravity, but higher buildings may require electric pressure booster systems depending on the topology (World Health Organization 2006). Digital elevation models and OpenStreetMap data can be used to estimate indicator $v_j$.

5.2.3 Estimation of medical demand

The medical demand arising from a residential building j, i.e., the average number of emergencies per day ($d_3$), can be estimated from the number and age of its inhabitants $p_{gj}$. In Jánošíková et al. (2021), $\texttt {rate}_g$, the number of emergency cases per 1000 people in age group g, is reported.

We translate this into the probability $e_g$ of an individual person having a medical emergency on a given day as

$$\begin{aligned} e_g= \texttt {rate}_g\frac{1}{1000} \cdot \frac{1}{365} \end{aligned}$$

(16)

and summarize the data in Table 2. Consequently, we estimate the medical demand of a residential building j as

$$\begin{aligned} d_{j3} = \sum _{\mathcal {G}} p_{gj}e_g. \end{aligned}$$

(17)

Table 2 Values $\texttt {rate}_g$ give the number of emergencies per 1000 people per year in age group g. Values $e_g$ define the probability of a person of age group g having a medical emergency per day (Jánošíková et al. 2021)

5.3 Candidate locations for ECPs

We select public schools, sport facilities, and fire stations as candidate locations in our analysis. As the majority of public schools and fire stations are owned by the city or the federal government, they can be easily adapted to serve as ECPs. The same holds true for sport facilities, especially gyms, which are also well suited to host ECPs due to their size. Most gyms are attached to schools anyway. The City of Vienna provides the locations of all schools Stadt (Wien 2020b) and all sport facilities (Stadt Wien 2020c). From that data, we select all public schools and all gyms (sport facility type = 6) and compare them against the digital city map of Vienna (Baukörpermodell LOD0.4) (Stadt Wien 2021b). We observe that, often, several different types of schools and gyms are hosted in the same building. As possible ECP locations, we select all buildings that intersect with at least one school or gym. Additionally, we restrict our data to buildings having more than $300\,\hbox {m}^{2}$ ground area. The fire stations had to be identified in a manual process from a publicly available source, see Stadt Wien (2022b).

The resulting instance, shown in Fig. 1, consists of 490 ECP candidate locations and 141,494 residential buildings. Likewise, we give an overview of the capacity limitations at the ECP candidates in Table 3.

Finally, we retrieve exact values for the walking distances $w_{ij}$. We query these values from the Open Source Routing Machine (OSRM) (Luxen and Vetter 2011; Geofabrik GmbH 2022) using the routing profile for pedestrians (foot.lua).

Fig. 1

Overview of the LCAP instance generated for Vienna. ECP candidates are shown in red, residential buildings posing WatSan demand are indicated in blue, and all other residential buildings are shown in black

Table 3 Summary of the ECP capacity limitations per facility type

6 Solution method

One intuitive attempt to reduce the size of the considered LCAP instance may be to split the City of Vienna into its 23 districts. Preliminary analyses show that a remarkable number of the arcs (that are below the distance limit) cross district borders, i.e., candidate ECP $i \in \mathcal {I}$ and residential building $j \in \mathcal {J}$ are located in different (mostly neighboring) districts. We observer that 7.43%, 18.98%, and 37.66% of the arcs $\mathcal {A}$ cross district borders for $W=250\,\hbox {m}$, $W=1000\,\hbox {m}$, and $W=1750\,\hbox {m}$, respectively (see also Sect. 6.1).

Based on these findings, we strongly recommend to not split the LCAP instances among any administrative borders in order to ensure the significance of the derived solutions. Moreover, reducing the size of the instance by creating artificial demand points to represent building clusters is problematic. Forming meaningful clusters of demand points created would introduce a problem on its own. The location of these artificially created demand points does significantly affect the walking distances calculated by the routing engine, thereby undermining the validity of any subsequent analysis.

To deal with the occurring instance sizes, we propose a two-step approach in which we first prune our instances (Sect. 6.1), to reduce the number of constraints without reducing solution space. After the matrix pruning, we use the overlapping decomposition optimization method described in Sect. 6.2 to solve the problem.

6.1 Matrix pruning

The walking distance between an ECP $i \in \mathcal {I}$ and a residential building $j \in \mathcal {J}$ is a parameter which we can use to reduce the size of the LCAP instance. The full model contains $|\mathcal {A}|=|\mathcal {I}| \cdot |\mathcal {J}|$ arcs. All neglectable arcs, i.e. decision variables $x_{ij}$ which will be 0 in any feasible solution as they violate the walking distance constraint (2), can be safely removed without reducing the solution space. Hence, we remove all arcs which are longer than a given threshold W. In case a building j is further away from any ECP candidate location i than W, we connect it to the closest ECP candidate (removing all other arcs). Consequently, we impose constraints (2), (7), and any other constraints that contain variable $x_{ij}$ only if this variable is not excluded due to above reasoning. Set $\mathcal {A}^W \subseteq \mathcal {A}$ denotes pairs (i, j), which are considered in the pruned matrix. Accordingly, we replace $\mathcal {A}$ by $\mathcal {A}^{W}$ for the remainder of this study.

Table 4, where we report the resulting model sizes for different values of W, shows that this pruning procedure is highly effective in removing neglectable arcs. In our instance, we consider 490 ECPs and 141,494 residential buildings, which lead to $|\mathcal {A}|=|\mathcal {I}| \cdot |\mathcal {J}| = 6.933\times 10^{7}$ arcs. Column $|\mathcal {A}^W|$ lists the number of arcs of the pruned matrix, while columns $\frac{|\mathcal {A}^W|}{ |\mathcal {A}|}$ list the relative number of considered pairs (i, j). It is remarkable that, especially in instances with small values of W, the number of variables is considerably reduced. However, also in larger instances, we see a significant reduction of arcs, as e.g., in instances $W=1750\,\hbox {m}$, where only 9.53% of arcs need to be considered.

Table 4 Number of arcs $|\mathcal {A}^W|$ considered in the LCAP Vienna instance for different values of W

6.2 Decomposition approach

Although pruning the matrix removes at least 90.47% of the arcs, solving the model as a whole is still rendered impossible. Hence, we apply a decomposition approach to solve the LCAP.

We decompose the problem instance by creating a subproblem for each ECP i. Each such subproblem includes the ECPs and the residential buildings that are located within a certain walking distance to i. This method creates an overlapping structure, which we then solve sequentially by applying a greedy approach. The full procedure, summarized in Algorithm 1, is given as follows.

First, we create an initial solution $L_0$ by allocating each residential building to its nearest ECP. If the capacity of an ECP is exceeded, a repair approach allocates a residential building to the next nearest ECP with residual capacity until we reach an feasible solution. Then, we use the result to apply a greedy approach wherein we sort the active ECPs ascending by their resource utilization rate of the initial solution $L_0$. We then solve for each ECP $i \in \mathcal {I}$ a decomposed LCAP (defined in Sect. 4.3), by fixing selected decision variables to the values obtained in the previous iteration.

To obtain solution $L_i$, we determine all ECPs $\mathcal {I}^{*}_{i}$ which are located within a walking distance of 2W to ECP i. Next, we identify all arcs $\mathcal {A}_{i}^W$, which are contained in the pruned matrix and are connected to one of the ECPs in $\mathcal {I}^{*}_{i}$. We solve the LCAP by fixing all decision variables $x_{ij}, ~ (i,j) \in \mathcal {A} {\setminus } \mathcal {A}_{i}^W, ~ y_i, ~ i \in \mathcal {I} {\setminus } \mathcal {I}^{*}_{i} \text { and}, ~ z_{ik},~ i \in \mathcal {I} {\setminus } \mathcal {I}^{*}_{i}, ~ k\in \mathcal {K}$ to the values obtained in the previous iteration. We repeat this approach for all ECPs  (following the sorting). The solution ($L_{i-1}$) obtained in the previous step serves as the starting solution for obtaining $L_i$.

This decomposition approach requires pair-wise distances between ECPs, while the base problem only requires the pair-wise distances between ECPs and demand locations. The limit of 2W concerns the pair-wise distance between ECPs. Accordingly, we need to allow the distance of 2W to avoid removing feasible solutions. Constraints (2), which enforce the maximum walking distance, are omitted in this approach, as any edges that would violate these constraint are removed by the matrix pruning.

Algorithm 1

SolveModel

7 Results and discussion

We use R language to gather and process all the data and to build the model (see Sect. 4) using sparse matrices. We solve the model using Gurobi Optimizer (version 10.0.0) on a Linux machine equipped with an Intel Xeon W-2195 @ 2.30 GHz (16 core) and 512 GB RAM.

At first we analyze the solution quality of the decompostion approach. Hence, we compare the results using the decomposition approach (Algorithm 1) and the exact approach where we solve the complete LCAP model. Note that the exact approach uses the solution obtained by the decomposition approach to start with. The results are summarized in Table 5, which also hold the key results of the study. Using Algorithm 1, the LCAP instance with $W=250\%$ was solved in only 1 h and 16 min, while the instance with $W=1750\,\hbox {m}$ took more than 324 h. Concerning the easy instances $W\le 750\,\hbox {m}$, the exact approach only requires at most 15% additional computation time, but only showing none or neglectable improvement of at most 0.32% of the objective. For instances with $W=1000\,\hbox {m}$ and $W=1250\,\hbox {m}$, we notice an improvement of 4.5% and 3%, respectively. However, this enhancement in performance comes at the cost of an additional 109% and 319% increase in computation times, respectively. Note, that for the instances $W\ge 1500\,\hbox {m}$, the exact approach could not obtain any solution within a time out of 14 days.

Therefore, in order to maintain consistency and comparability, we utilize the results derived from the decomposition approach (Algorithm 1) for all subsequent analyses and evaluations. We depict the number of required teams (resources) per category compared to the parameterization W in Fig. 2. Remarkably, the total number of required teams accounts to 1012 for $W=250\,\hbox {m}$, while there are 234 teams needed for $W=1750\,\hbox {m}$. In detail, the number of base teams is identical with the number of medical teams. This is due to the fact that one base and one medical team per ECP is sufficient to cover the demand posed in all instances. In general, we observe a strong link between the number of required teams and parameter W. With regard to the WatSan demand, we can observe that there is only a slight decrease in the required WatSan teams when increasing W. In absolute values, this means a drop from 48 to 34 WatSan teams from $W=250\,\hbox {m}$ to $W={1750}\,\hbox {m}$. One reason could be that, mostly, the western outskirt pose WatSan demand and that this area is rather sparsely covered with ECP candidates (see Fig. 1). Additionally, we notice several (high-rise) buildings scattered around the entire city that require WatSan. Clearly, each of those needs an ECP (within the acceptable walking distance) that is staffed with a WatSan team. Hence, policymakers should concentrate more on base- and medical resources than on WatSan in selecting the appropriate W. Base- and medical teams are more sensitive to different values of W as shown in Fig. 2.

Figure 3 illustrates a box plot of the distribution of the actual walking distances for each instance. For $W=250\,\hbox {m}$ the median is 579 m, while for $W=1750\,\hbox {m}$ it is 966 m. Interestingly, for instances with $250\,\hbox {m}\le W \le 1000\,\hbox {m}$, the 75% quartile is nearly identical, which indicates that the actual walking distance to the closest ECP is lower than 953 m for 75% of residential buildings. Furthermore, the medians show only little variation among the $W \le 500\,\hbox {m}$ instances. However, the actual walking distance increases for the remaining instances with a 75% quartile of 1245 m in ($W=1750\,\hbox {m}$). Even for large values of parameter W (1500 m and 1750 m, the median actual walking distances remains below $1000\,\hbox {m}$ while the 90% quantile of demand locations $\mathcal {J}$) is still slightly below ${1510}\,\hbox {m}$.

Figure 4 shows empirical densities and distributions of the actual walking distances between residential buildings and closest ECPs. Among the individual residential buildings, this implies very diverse actual walking distances, especially in $W=1750\,\hbox {m}$. Also, we can see that a certain number of residential buildings (approximately 5%) show relatively large actual walking distances in all instances. The reason is that there are certain residential buildings that are located in the outskirts, where there are only few ECP candidates. Accordingly, we observe in Fig. 4 that the density functions (for all settings of W) nearly overlap above the 95% line. This issue may be addressed by adding additional ECP candidate locations (other than schools, gyms, or fire stations) or finding an alternative form of blackout handling for the concerned residential buildings. Once identified, officials can tailor solutions to those under-supplied demand locations based on their local characteristics.

In Figs. 5 and 6 we show solutions for the selected areas, showing active ECPs and the residential buildings assigned to them, for parameters $W=250\,\hbox {m}$ and $W=1750\,\hbox {m}$, respectively. Additional map views showing the whole area of Vienna can be found in the appendix, see Figs. 7 and 8.

We observe that constraints (7) effectively generate areas having clear boundaries. In preliminary experiments, we noticed that optimizing for the minimal weighted distance (as in the classic p-median problem) leads to rather fuzzy borders between the areas assigned to different active ECPs. Such results may lead to confusion among the citizens (and the authorities) in case of a blackout.

What can be also seen is the higher number of active ECPs  (in total 482) in Fig. 5 compared to Fig. 6 (with $W=1750\,\hbox {m}$). With this the number of assigned residential buildings per ECP is lower and average walking distances are higher in comparison to $W=250\,\hbox {m}$. For the entire study region, we can observe that the number of active ECPs does not change considerably between $W=250\,\hbox {m}$ and $W=1750\,\hbox {m}$ in the outskirts, especially in the north-eastern part of the city. This is again based on the fewer number of ECP candidates in the suburban area. Similar situations might be observed for many other cities of similar size.

With this, policymakers are shown that particularly in the city under investigation, they have a higher planning flexibility in the center than in the outskirts where the number of ECP candidates is rather small. The results underline that instances $W < 1000\,\hbox {m}$ show similar distributions of the actual walking distances encountered by the population while a decrease in required resources can be observed (51% reduction). Consequently, it is not recommended to apply a threshold $W < 1000\,\hbox {m}$. This is valuable information for decision makers given the prevalent resource bottlenecks that they might face during a blackout or other disasters. Notably, they have to be careful when selecting parameter W as the actual walking distances increase above a certain threshold. Decreasing W, however, causes an increase of the number of ECPs that must be activated which, in consequence, requires more resources to operate each of them. The framework presented in this study provides a highly data-driven and quantitative method for finding a set of ECP locations that ensure reasonable accessibility for citizens.

Table 5 Summary of the obtained solutions for different values of W showing computational times and number of utilized teams using the exact and the decomposition (decomp) approach. Column run time gives the computational times. For the exact approach we report the additional solve time that was needed when the solution obtained via decomp was used as the warm start solution. Columns Base, WatSan and Medical list the overall demand of each resource category. Columns Total displays the sum of the number of required teams. TO indicates that the exact approach could not produce a solution within 336 h

Fig. 2

Line plot of the number of required teams (per type) for solutions with different settings of W

Fig. 3

Box plots of the actual walking distance for solutions obtained with different values of W. The x-axis is cropped to 2000 m

Fig. 4

Empirical densities and distributions of the actual walking distances obtained from the solutions with different settings of W. The x-axis is cropped to 2000 m

Fig. 5

Map view (zoomed in) of the solution for $W=250\,\hbox {m}$. The color indicates the assignment of residential buildings to active ECPs  (same color)

Fig. 6

Map view (zoomed in) of the solution for $W=1750\,\hbox {m}$. The color indicates the assignment of residential buildings to active ECPs  (same color)

8 Conclusion

From an academic point of view, this paper is valuable as it is the first that proposed the optimal allocation of ECPs under consideration of efficient resource distribution for blackout response. While the current body of knowledge is dominated by optimization models that either focus on facility or resource allocation, this study combined both aspects in a single model and introduced it in a relatively untouched research context.

In our LCAP model, we aimed to minimize the number of teams that are required to operate the active ECPs, while assigning each residential building to the nearest active ECP. In contrast, the most classic capacitated location allocation problems minimize the weighted sum of all distances between the demand locations and the allocated facilities in order to minimize overall transportation cost.

Hence, the LCAP formulation has strong parallels to the OpMP, which is concerned with allocating a set of obnoxious facilities. As the OpMP typically aims to avoid having these kind of facilities near the population, it aims to maximize distance to them. Exact approaches for solving the OpMP, available in the current scientific literature, allow tackling benchmark instances of up to 300 (candidate) facilities and 300 clients with a cardinality p of up to 50% (of the candidate facilities being selected) within reasonable time, see Lin and Chiang 2021. As there are no constraints concerning the maximal distance between facilities and clients in the OpMP, the complete graph must be considered. In our work, we solved the proposed LCAP model for instances of 490 ECP candidate locations and 141,494 residential buildings.

Due to the proposed matrix pruning approach (see Sect. 6.1), we were able to significantly reduce the problem size to 0.85% (for $W=250\,\hbox {m}$) and 9.55% (for $W=1750\,\hbox {m}$). This, together with the proposed decomposition approach (see Sect. 6.2), allowed us to tackle large-scale LCAP instances. We were able to solve these instances within a few hours (for $W \le 1250\,\hbox {m}$) and within a few days (for $W \le 1750\,\hbox {m}$), which is still acceptable for tactical planning problems.

There are several issues that limit this research and that are worth of further consideration. Firstly, the study focuses on two demand types, i.e., medical and WatSan demand. Other types of demand may occur during blackout situations as, for instance, residents could also run out of petrol for heating. Food supply may also become critical, leading to another type of demand to be satisfied at ECPs. In this regard, further analysis to estimate required food supply and corresponding resources is required. Secondly, in reality, the medical as well as the WatSan demand follows a distribution that describes how many cases and how much drinking water demand occurs over the entire day. However, as there is no such data available, we settled for daily average values. Taking dynamic stochastic demand into consideration is thus up to future research. Thirdly, in order to increase resource utilization, the walking distance constraint (2) could be set individually for each residential building based on the age of the inhabitants. Additional mobile units could be used to serve some areas—especially in the outskirts of a city—to increase resource utilization. Further, as the terrain of the city is relatively flat, we did not include changes in altitude for determining parameter W as well as for calculating the average walking distance for citizen that approach ECPs. Anyway, when applying the framework to cities or regions with hilly (or even mountainous) topologies, elevation differences should be taken into account. Therefore, digital elevation models can be used to properly estimate the elevation difference along the foot paths, see, e.g., Truden et al. 2022. In terms of data usage, the study underlines the value of census data and GIS application for emergency planning. Nevertheless, one must take into account that incorrect data may severely affect the outcome of the study (cf. Grubesic and Helderop 2022). With the study at hand, we, however, provide practitioners with a planning framework for blackout preparedness and response that enables them (i) to estimate medical as well as WatSan demand, (ii) to determine the optimal allocation of ECPs over a given geographical region under consideration of maximum walking distances for citizens, and (iii) to efficiently distribute human resources (i.e. base, WatSan, and medical teams) among activated ECPs. In practice, the presented approach can form the basis of future blackout planning efforts or complement already existing emergency plans. This supports practitioners in better controlling resource uncertainties and in lowering risks associated with ECP site selection. In this sense, the generated results may raise the practitioners’ awareness towards the criticality of profound resource planning in such extraordinary events. This knowledge gain can help adequately utilize scarce resources while aiming to realize acceptable service levels for citizens. Further, if responsible decision makers (e.g., NGOs) are aware of certain vulnerable and elderly populations, they might consider setting the parameter W at a higher granularity (e.g., the individual building level) by modifying Eq. (2). Above all, the study might contribute to more organized and structured help for citizens in need when it comes to blackout scenarios, which is the most important implication of all.

Appendix A Map views and census data

See Figs. 7, 8 and Table 6.

Fig. 7

Map view for $W=250\,\hbox {m}$. The color indicates the assignment of residential buildings to active ECPs  (same color)

Fig. 8

Map view for $W=1750\,\hbox {m}$. The color indicates the assignment of residential buildings to active ECPs  (same color)

Table 6 Counts of residential buildings and ECPs candidate locations aggregated by district and type. Additionally, the estimated medical demand (patients/day), WatSan demand (people/day) is given