**1. Introduction**

After a disaster strikes a territory it is imperative to deliver relief items to the affected population to reduce human casualties and suffering. This is especially true in underdeveloped and developing countries that suffer from a lack of resources and reliable infrastructures. Additionally, poor planning and structural damage to the transportation network could hinder the relief effort or, in some cases, completely frustrate them. Damaged roads, resulting from the effect of the disaster, could completely disconnect affected populations from the rest of the transportation network, thus, making impossible to undertake any relief operation. This was the case in Haiti, after the 2010 earthquake, where, despite the large volume of emergency goods available, the victims could not receive support for a long time due to the damage suffered by the road network [1–4]. The 2011 Japan earthquake and tsunami interdicted about three-fourths of the highways in the region, hampering the emergency response operations [5]. More recently in Puerto Rico, after Hurricane Maria hit the island in 2017, shortages in vehicle drivers and disruptions to the road network made it impossible to deliver emergency supplies, including food, water, and medicine, resulting in at least 10,000 containers idly sitting at the port [6]. Locating emergency inventories and fortifying vulnerable elements of the road infrastructure prior to

an emergency helps to mitigate the impact of a disaster, facilitates the relief operations, and increases the effectiveness of the disaster response.

In this paper, a pre-disaster model that tackles these issues is proposed. The model jointly optimizes the fortification of elements of the distribution network, the location of emergency inventories, and the definition of their capacity. The goal is to support the distribution operations of relief goods in the best possible way. Due to the uncertainty that characterizes disasters, the model is stochastic in nature. Multiple objectives are considered. However, the objective of minimizing the expected unsatisfied demand has the highest priority, as not providing relief to populations affected by a disaster might result in human casualties. This objective has multiple optima, therefore, the optimal solutions are further evaluated according to two time measures: the expected maximum arrival time at a demand node and the expected total service time. The computational experiments show little trade-off between these time criteria. For this reason, the solutions are evaluated using a pay-off matrix, rather than by applying specific multicriteria methods. More generally, the methodology has been designed to be applicable in situations with limited data on the availability of resources and the status of the road infrastructure, which is especially relevant to underdeveloped and developing countries. In fact, the model has been tailored to a real case study on the 2018 tropical depression that hit the Nampula Province (Mozambique) and considers only the data that could be obtained from official sources. For this reason, and given its focus on pre-disaster operations, the model relies on a number of simplifying assumptions concerning the distribution operations. For instance, these are represented by an underlying flow model. As a consequence, most details of a real supply chain are disregarded, such as, multiple commodities, multiple transportation modes, vehicles availability and capacity, and distribution queues. However, this is consistent with the application context and the scope of the model.

#### *1.1. Literature Review*

According to Anysia and Kopczak [7], Humanitarian Logistics is "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 alleviating the suffering of vulnerable people." The seminal work by Aghani and Oh [8] presents a large-scale multi-commodity multi-modal network flow model with time windows in the context of disaster relief operations, which the authors solve using heuristics. In recent years, due to the relevance of the subject, the investigation in Humanitarian Logistics models has experienced a golden age and a wealth of research contributions have been published. Due to the large number of works in the field, the review presented in this paper focuses on relief distribution models that combine location operations with structural operations on the transportation network. However, pointers to exhaustive literature reviews are provided in the following. The interested reader is referred to the excellent literature review on operations managemen<sup>t</sup> in the context of Operations Research (OR) and Management Sciences (MS) by Altay and Green [9], which Ortuño et al. [10] and Galindo and Batta [11] expand and update. Liberatore et al. [12] provide an annotated bibliography specialized on uncertainty in humanitarian logistics. Çelik [4] reviews the literature on network restoration and recovery in humanitarian operations. Aslan and Çelik [13] give in their research paper an overview of recent studies on emergency inventory propositioning involving uncertainty. Finally, the review paper by Sabbaghtorkan, Batta, and He is specialized on prepositioning of assets and supplies in disaster operations managemen<sup>t</sup> [14].

To the best of the authors' knowledge, the first model to consider the coordination of the activities of relief transportation, road recovery, and inventory prepositioning is by Wisetjindawat et al. [15]. The authors consider ongoing road repairs as constraints on the availability of roads, rather than as part of the decision making process, i.e., all links in the transportation network must be available after 24 h. Therefore, the problem is formulated as a location-routing problem that explicitly consider routes that satisfy the availability time constraint, at a given confidence level. The approach proposed

by the authors is an interesting combination of simulation, vehicle routing, and location analysis. However, it does not explicitly incorporate neither the stochastic elements of the problem nor the recovery operations in the formulation.

Later, Iloglu and Albert [16] study the relationship between network restoration and emergency response operations in the early stages of a disaster. A multi-period p-median model determines the location of emergency responders that attend emergency calls. The formulation includes scheduling constraints that plan the repair team operations of network recovery. The problem should be used right after a disaster strikes, therefore no stochasticity is considered. However, the formulation allows repair crews to access arcs that are completely disconnected from the rest of the network, which is unrealistic.

This initial model is later expanded by the authors in [17], where the underlying multi-period p-median model is substituted by a multi-period maximal multiple coverage model. As the emergency responders can be relocated at every period, this more recent formulation limits the choices for relocation to those within a certain radius from the previous one. Also, the multiple coverage model considers allows for backup emergency service during large volume of emergency service requests after disasters. Obviously, the new formulation improves the realism and applicability of the model. However, it still allows repair teams to access arcs disconnected from the rest of the network. Therefore, despite the update, the major flaw of the model is still present.

In another recent study, Aslan and Çelik [13] propose a two-stage stochastic programming model to design a multi-echelon humanitarian response network. In the first stage, the model defines the location of the inventories and their capacity for each commodity. In the second stage, relief transportation decisions and road repair operations are made jointly. A limitation of this contribution is that the model assumes that there are sufficient restoration resources and these resources can start repairing any damaged arc immediately after the disaster. The authors recognize the limited realism of this approach, however, they claim that the assumptions help to simplify the solution of the problem and understand its structure. More recently, Sanci and Daskin [18] introduce an additional level of complexity in the model. Their article expands on the previous contributions by integrating facility location and network restoration models to locate both emergency response facilities and restoration equipment before a disaster. Also, it allows to allocate more than one restoration resource to a damaged arc to reduce the recovery time. Finally, the model ensures that damaged arcs can be repaired only if they are accessible from the initial locations of the repair resources, therefore, addressing the major flaw in [16,17]. Interestingly, the model can find a balance between unmet demand and transportation costs by changing the value of a penalization coefficient. This approach clearly contradicts the rationale behind Humanitarian Logistics, that gives absolute priority to the minimization of human casualties and suffering. The authors sugges<sup>t</sup> setting the penalty for unmet demand to a value that ensures that all the demand is attended. However, this clearly shows that the authors recognize that the formulation is indeed flawed, and that a lexicographical approach should have adopted instead of using a penalization coefficient.

All the studies reviewed consider post-disaster network recovery operations. The model proposed in this article, however, optimizes pre-disaster network fortification operations, rather than recovery operations. Therefore, to the best of the authors' knowledge, this is the first fully pre-disaster model to combine relief transportation, road recovery, and inventory prepositioning. Also, the formulation presented in this study considers a simpler supply chain than those of the previous contributions: one commodity, three types of nodes, and unlimited capacities. This is a desired feature. Differently from the papers presented in the review, our model has been designed to being applicable in contexts with limited information availability, such as underdeveloped and developing countries.

## *1.2. Contributions*

The contribution of this research is twofold. In particular, this work extends the literature by introducing:


The rest of the paper is organized as follows. Section 2 presents the optimization model and the formulation. The dataset on the Nampula Province case study is detailed in Section 3. Section 4 is devoted to the computational experiments and provides a discussion of the results. Finally, the paper is concluded with some insights and guidelines for future research (Section 5).
