*Article* **Supported Evacuation for Disaster Relief through Lexicographic Goal Programming**

**Inmaculada Flores 1,2,\*, M. Teresa Ortuño 1,2 and Gregorio Tirado 2,3 and Begoña Vitoriano 1,2**


Received: 31 March 2020; Accepted: 19 April 2020; Published: 22 April 2020

**Abstract:** Disasters have been striking human-beings from the beginning of history and their managemen<sup>t</sup> is a global concern of the international community. Minimizing the impact and consequences of these disasters, both natural and human-made, involves many decision and logistic processes that should be optimized. A crucial logistic problem is the evacuation of the affected population, and the focus of this paper is the planning of supported evacuation of vulnerable people to safe places when necessary. A lexicographic goal programming model for supported evacuation is proposed, whose main novelties are the classification of potential evacuees according to their health condition, so that they can be treated accordingly; the introduction of dynamism regarding the arrival of potential evacuees to the pickup points, according to their own susceptibility about the disaster and the joint consideration of objectives such us number of evacuated people, operation time and cost, among which no trade-off is possible. The performance of the proposed model is evaluated through a realistic case study regarding the earthquake and tsunami that hit Palu (Indonesia) in September 2018.

**Keywords:** humanitarian logistics; evacuation; multi-criteria decision making; goal programming; disaster relief

#### **1. Introduction and Literature Review**

Disaster management, understood as the planning, organization and managemen<sup>t</sup> of all that is needed to deal with the humanitarian aspects of emergencies, disasters or catastrophes, in order to lessen their impact on population and infrastructures, has always been in the focus of the international community. In recent decades, this global concern has given birth to a growing literature on disaster managemen<sup>t</sup> and in humanitarian logistics, defined in the Humanitarian Logistics Conference, 2004, as the process of planning, implementing and controlling the efficient, cost-effective flow and storage of goods and materials as well as related information, from the point of origin to the point of consumption for the purpose of meeting the end requirements of beneficiaries and alleviate the suffering of vulnerable people. See the survey of Özdamar and Ertem [1], and the books of Tomasini and Van Wassenhove [2] and Vitoriano et al. [3], among others, for optimization problems addressed within the Operational Research community regarding disaster managemen<sup>t</sup> and humanitarian logistics.

The disaster managemen<sup>t</sup> cycle comprises four successive phases, see [2]: mitigation, preparedness, response and recovery. Each one of them comprises important logistic operations that must be planned in the most effective and efficient way. Research on these phases may focus on specific types of disasters, such as hurricanes, typhoons and cyclones [4], earthquakes [5], floods [6] or wildfires [7]; or it can address specific problems along the cycle, such as location [6], emergency mitigation [5,8], prepositioning of aid distribution centers [9], transportation and last mile distribution [10–12] or evacuation problems [13,14].

Evacuation is an activity long time considered in humanitarian logistics. According to UNDRR [15], an evacuation consists in moving people and assets temporarily to safer places before, during or after the occurrence of a hazardous event, in order to protect them. Evacuation may be required in buildings, regions or transportation means, such as trains, ships or airplanes. Due to the increasing height of buildings and the expansion of transport travel in the last century, initially, a large collection of works related with buildings, trains, airplanes and ships evacuation arose. Some examples can be found in [13,16–18]. In 2001, a preliminary state of the art review appeared, mainly based on building evacuation, [19]. Some years later, on the other hand, a systematic collection of network flow models applied to regional emergency evacuation and their applications came up, see [14].

Different types of evacuation are stated by the London Resilience Partnership in [20], comprising self-, assisted and supported evacuation. Self-evacuation is understood as individuals moving from an unsafe place to a safer one by their own means, without any kind of assistance. In assisted evacuation the individuals are capable of traveling by themselves but require certain support (for example, information) from agencies. Finally, supported evacuation is needed if individuals require greater and specific support (for example, an ambulance) to be transported from the unsafe areas to safer ones. Self-evacuation and assisted evacuation (both often referred as car-based evacuation) have been vastly studied in the literature, facing problems like traffic congestion, clogging, bottlenecks, density waves, oscillations, patterns and panic, usually solved by nonlinear techniques and queuing theory [18,21–25]. However, and according to Houston et al. [26], sometimes self evacuation is not an option. An example can be found at Hurricane Katrina (New Orleans, 2005), where over 40% of victims did not evacuate, either due to physical disability or because they were caring for a person with one. Support can be needed for different reasons, from people with no access to an adequate vehicle to those that require special transport means (mainly ambulances). In addition, each type of disaster often requires a different evacuation process. For example, in Hamacher and Tjandra [19] two evacuation scenarios are considered, precautionary and life-saving operations. Furthermore, Amideo et al. [27] stablished that hurricanes and wildfires require preventive evacuation while earthquakes and floods demand immediate post-disaster evacuation. Pyakurel et al. [28] established that an evacuation optimizer looks for a plan on an evacuation network for an efficient transfer of the maximum quantity of evacuees from the dangerous points (sources) to safer ones (sinks) as quickly as possible. Other analyses related with evacuation planning in order to save human lives and support humanitarian relief can be found in [29,30].

In terms of regional evacuation, and according to Amideo et al. [27], the related literature prior to 2011 used to be focused on shelter location-allocation and self-evacuation. The first paper on regional supported (or bus-based) evacuation is [31], where this type of evacuation is presented as a variant of the vehicle routing problem. After this paper, few works can be found until 2017, when Shahparvari and Abbasi [32] proposed a stochastic model for a supported evacuation to determine the required vehicles, scheduling and routing under uncertainty, with time windows and bushfire propagation and considering road availability and disruptions. Besides that, Shahparvari et al. [33] developed a capacitated vehicle routing solution to evacuate under short-notice. Finally, the identification of the required number of vehicles and the safest routes and schedules for late evacuees it approached in [34].

In this work, a new model for supported evacuation of vulnerable people after a disaster is presented. The model seeks to evacuate people from pick-up points to shelters with limited capacity, and it is multimodal, considering different types of vehicles; dynamic, because people reach the pick up points at different periods along the time horizon; and multi-objective. It is based on the coordination of network flows of vehicles and people, as stated in [35], where there is a collection of evacuation models based in networks flows with discretized time. We also consider shelters with limited capacity, as in the location-allocation model presented by Sherali et al. [6], that selects a set of candidate shelters according to the available resources and minimizes the total congestion in a car based evacuation. Capacity constraints in arcs and nodes can also be found in [36]. Regarding the optimization criteria, a wide variety of them have been employed in the humanitarian logistics literature, see [37]. It is important to note that the use of multi-criteria optimization methods has experienced a significant growth in the last years, mainly because they are able to help the decision maker select the best option between several alternatives, evaluating criteria that may generate conflict. An interesting example that takes into consideration multiple criteria such as cost, time, equity, reliability, security or priority can be found in [38]. In particular, for evacuation models, different criteria have been considered in the literature. Shahparvari et al. [7] highlighted the fact that most existing studies have focused on minimizing just the total evacuation time, and they consider two additional objectives: maximize the amount of evacuated people and minimize the usage of resources. Mejia-Argueta et al. [39] proposed to minimize the maximum evacuation and distribution time and the total cost of relief operations, while Alçada-Almeida et al. [40] minimized the total distance required for all the population to reach a shelter, the fire risk faced while traveling, the risks associated with staying in the shelter and the total time from the shelters to the hospital. Our model considers the maximization of the number of evacuees of different types regarding health condition and the minimization of the evacuation time and operation costs.

The main contributions of our research are the following:


The structure of the remainder of this paper is as follows. Section 2 presents the detailed description of the supported evacuation problem considered, whose mathematical formulation is proposed in Section 3 through a lexicographic goal programming model. Section 4 introduces the case study used to evaluate the performance of the model and analyzes the computational results provided by its resolution. Finally, Section 5 draws some conclusions from this work.

#### **2. Problem Description**

A detailed definition of the problem approached in this paper, focused on regional supported evacuation, will be given in what follows. As stated in the previous section, to mitigate the possible effects of some disasters, agencies may determine certain areas that are necessary to evacuate for security reasons and some other areas that are safe and available to host the evacuated population. It happens frequently that a certain amount of people, for a variety of reasons, cannot evacuate from the compromised areas to safe ones by their own means, requiring additional support from public agencies to be able to evacuate. The way these vulnerable people is evacuated is the main focus of our model.

The inhabitants that cannot self-evacuate by their own means go directly to previously designated pickup points, if they are able to, according to their own feeling regarding the disaster, or are taken there by police or search and rescue teams. According to Özdamar and Ertem [1], many factors influence the feeling of the population regarding the catastrophic situation, including demographic characteristics such as age, health status or gender, and socioeconomic ones, such as homeownership, financial conditions, prior experiences and awareness. In addition, these factors may vary significantly along time. As a result, the arrival of potentially vulnerable evacuees to the designated pick up points is clearly dynamic, and this is a key aspect of the problem.

The analysis of real cases of disasters shows that classifying the population in need of supported evacuation is necessary to prioritize appropriately the attention from the involved agencies. Houston et al. [26] indicate that people with disabilities, medical conditions, homeless or pregnan<sup>t</sup> women should be considered as high priority population and be evacuated with a higher urgency, due to their special vulnerability. Other population groups are usually classified as normal priority evacuees. However, the particular classification considered for the design of the evacuation plans may vary significantly from one case to another, so that it represents the existing situation as faithfully as possible.

Safe areas contain hospitals, for population requiring medical care, and temporary shelters, for the rest of the population, that are associated to the nodes of the network. An additional distinction between different types of shelters, as suggested in [26], into general population shelters and shelters for people with special needs or medical needs shelters, may also be performed. All hospital and shelters have a certain capacity to accommodate evacuees that cannot be exceeded.

Transportation to safe areas during emergencies is critical in order to evacuate people who either have specific mobility issues or do not have access to transportation. For this purpose, different types of vehicles with diverse characteristics will be available. Vehicles may traverse the arcs individually or forming convoys and they can visit one or more locations several times, even though not all of them must be visited mandatorily. Each vehicle type has a certain capacity for the transportation of people that cannot be exceeded. Other characteristics, different for each vehicle type, are the fixed cost by distance unit, the variable cost depending on the amount of people transported, the possibility to traverse certain arcs depending on their conditions and the travel time required to traverse these arcs.

The operational area is a dynamic network composed by a region to be evacuated and a safe region. The streets or roads connecting the nodes are represented by arcs/edges and may be classified, depending on the requisites of the particular case, as paths/unpaved roads, local roads or within towns, highways, freeways, etc., according to their type; and as blocked or shattered, seriously damaged, partially damaged, usable, etc., according to their state and the environmental or traffic conditions. These elements determine whether a certain vehicle can traverse a certain arc or not and its maximum speed when travelling. There may also be a certain flow capacity associated to each arc that represents the maximum number of evacuees or vehicles that can traverse it per time unit.

In short, a solution of the problem must comprise a set of itineraries for the available vehicles and the corresponding flow of people moving from the affected area to the safe one, in such a way that they are compatible with each other and verify all capacity constraints given by the network. The resulting evacuation plans can be evaluated according to different criteria and, as a result, several objectives will be considered. The number of people evacuated successfully, that is to be maximized, seems the most important one. However, the evacuation time, to be minimized, is also crucial. Additionally, the operational cost, even though may not be the main focus of the decision maker, should also be minimized. All these criteria will be considered jointly within a lexicographical goal programming model, as detailed in the next section.

#### **3. The Proposed Evacuation Model**

The network of the problem is represented by a graph G(N , E), where N is the set of nodes, which may represent areas, cities or locations within them, and E is the set of arcs (if directed) or edges (if undirected), which correspond to roads or streets that communicate the nodes. Pick up nodes from where the compromised population will be transferred to the safe areas, N A, are located at known places at the unsafe area. Meanwhile, temporary shelters nodes, which usually correspond to schools, universities or governmen<sup>t</sup> buildings, N S; and hospitals or medical centers, N H, are located at the safe areas. Finally, some transit nodes, N T , may also exist in the network. As stated in the previous section, edges or arcs may be classified according to the requirements of the particular case study or disaster.

Evacuees are classified into different categories depending on the criticality of their health state and the space they require on vehicles or nodes at the secured area, since they may require different transport and assistance conditions. The population is to be transported by fleets of different types of vehicles with different characteristics. All evacuation operations must be carried out within a certain monetary budget and a given time span.

#### *3.1. Parameters of the Model and Decision Variables*

#### **Sets and indices**

