A multiobjective mathematical model for a humanitarian logistics multimodal transportation problem

1. Introduction

The need to respond efficiently to an earthquake with the only aim of minimizing the loss of human lives has led to the development of technologies and strategies capable of helping public and private institutions focused on this task. Researchers around the world have developed a wide number of strategies and mathematical models aimed at improving planning for an effective response. However, not all of them are useful because each region has its own economic, environmental and geographic limitations that prevent their use. In addition to these limitations are added the destructive consequences generated by each natural event. Determining the best response strategy to a catastrophic event in Latin America is a problem that has not yet been studied enough.

Public health emergencies require preparedness and response capacities from the government, public health systems and academic researchers (Lurie et al., 2013). From 2001 to 2013, 4 out of 32 major public global health emergencies were caused by earthquakes, e.g. the Bam earthquake in Iran (2013), the Sichuan earthquake occurred in 2008, the 2010 earthquake in Haiti and the Japan earthquake occurred in 2011. Earthquake destruction can cause further catastrophes; for example, in March 2011, a powerful earthquake occurred off the northeast coast of Honshu, a main Japan’s island, causing significant damage on land and, consequently, a series of large tsunamis devastating several coastal areas of the country producing a major nuclear accident at a power plant along the coast.

Due to earthquakes, road networks are frequently damaged, limiting terrestrial access, emergency shelters are often destroyed and water supplies and urgent medical attention are always required. Therefore, the adequate planning of logistical activities is essential to deliver services and commodities to the affected population Najafi et al. (2013). De la Torre et al. (2012) define disaster relief logistics, commonly known as humanitarian logistics, as the distribution of life-saving commodities to beneficiaries. Thomas and Mizushima (2005) define humanitarian logistics 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 point of origin to point of consumption for the purpose of meeting the end beneficiary’s requirements.” Researchers state that disasters present multiple logistics challenges, including damaged transportation infrastructure and limited communication/coordination of multiple agents. The main differences between humanitarian logistics and business logistics are the existence of unpredictable demand in terms of its location and the large number of products required of a wide variety of supplies, mainly medical and food and the delivery time (Balcik and Beamon, 2008; Kovács and Spens, 2009).

Four stages have been defined in emergency management (EM); these are mitigation, preparedness, response and recovery (Altay and Green, 2006; George D. Haddow and Coppola, 2007; McLoughlin, 1985; Thomas and Mizushima, 2005). Mitigation consists of reducing the vulnerability of risk areas. The preparedness stage educates people to face the disaster. The response stage consists of responding efficiently to a situation to minimize the risk of losing human lives. In contrast, the recovery stage works to repair the damage caused by a disaster. The International Federation of Red Cross and Red Crescent Societies defines disaster as “a sudden, calamitous event that seriously disrupts the functioning of a community or society and causes human, material, and economic or environmental losses that exceed the community’s or society’s ability to cope using its own resources.” Sometimes, in the response stage, relief supplies become excessive because many international donors send them to affected countries; this may seem very good. However, in most cases, the high quantity of supplies limits the proper organization, delaying its timely distribution and increasing its cost.

The delivery time of supplies in a response stage is one of the most important variables in humanitarian logistics because the number of injured people to assist or the loss of human lives after a disaster depends on it. The delivery time depends mainly on the speed of response of the government authorities, the modes of transportation used, the environmental conditions and accessibility to the points of consumption. Two examples are the earthquakes that occurred in Mexico (2017) and Haiti (2010), where road accessibility changed constantly and unpredictably due to the movement of debris and several roadblocks. To counteract the mentioned problems, in recent years, the use of unmanned aerial vehicles (UAVs), commonly called “drones”, have been considered as an alternative to traditional road vehicles, motivating the development of many single and multiobjective optimization (MOO) models to deliver medical and food supplies in humanitarian logistics problems (Arenzana et al., 2020). Some of these objectives include the minimization of traveling times, transportation costs, the number and types of vehicles used, the number of storage and transshipment facilities required and the maximization of materials and products transported from the points of origin (depots, collections centers, distribution centers, etc.) to the points of consumption. Escribano et al. (2020) show that UAVs can significantly improve response operations during disasters due to infrastructure damage, a key factor in relief distribution.

UAVs are used for many different purposes, including searching for missing people, aerial photography/cartography, fire prevention/control, security/military applications, agriculture, business logistics and EM. For EM, UAVs can distribute different types of essential supplies such as food, water and health care products such as medical toolboxes, medicine, defibrillators, blood samples, oxygen masks, vaccines and insulin injections (Scott and Scott, 2008; Kim et al., 2017), some examples are the use of UAVs in sanitation duties, temperature testing and vaccine delivery during the COVID-19 pandemic in rural medical centers around the world.

The use of UAVs in humanitarian logistics has its origin in 2006 after the devastation caused by Hurricane Katrina, when the Federal Aviation Administration in the USA authorized their use over civil airspace for rescue and disaster relief operations. The main benefits of using UAVs are that they allow for reducing labor costs by minimizing the number of staff needed (Dorling et al., 2017; Thiels et al., 2015); they require much less space to take-off and landing (Haidari et al., 2016); they reduce the carbon emission (Figliozzi, 2017; Goodchild and Toy, 2018; Arenzana et al., 2020); and they avoid road networks, which are vulnerable to disasters and traffic congestion (Dorling et al., 2017; Arenzana et al., 2020; Skorup and Haaland, 2020; Rabta et al., 2018), besides, they are cheaper to maintain than trucks. UAVs can also provide up-to-the-minute updates, which makes them useful for emergency response in fires, oil spills and earthquakes. Nevertheless, UAVs have significant weaknesses; the main one is their transportation capabilities, which are considerably smaller than ground vehicles (Dorling et al., 2017), followed by their limited operations range as well as limited flight times due to energy constraints (Rabta et al., 2018). In addition, their efficiency depends on climatic and environmental conditions, which makes them unusable in some disasters such as hurricanes and volcanic eruptions.

Humanitarian logistics is one of the most complex logistics operations because it takes place in an environment of severe disaster. A wrong decision can cause an increase in deaths. Even though there exists research on transshipment problems in this domain as well as on routing problems (milk runs problems) to deliver medical and food supplies that propose an interaction between UAVs and ground vehicles, MOO models that combine these two configurations of transportation have not been widely studied.

This work addresses the study of a three-echelon humanitarian logistics problem related to the response stage after an earthquake. A bi-objective optimization model for a transportation system that combines a full-truck-load transshipment problem [based on the transshipment model proposed by Olivares-Benitez et al. (2013)] with a heterogeneous fleet capacitated vehicle routing problem (an adapted version of Arenas Vasco (2018) is proposed . To evaluate the performance of the proposed model, this is applied to the case study of the earthquake that occurred in Mexico on September 19, 2017, in which approximately 370 people died and more than 3,000 were injured. The first objective of the model minimizes the total transportation costs among echelons and the cost associated with opening facilities; the second objective minimizes the delivery time of supplies. These two objectives are used because many countries, including Mexico, have budget constraints to respond to an emergency. To solve the model, the ϵ-constraint method is used. This method allows to build a set of nondominated solutions commonly called Pareto-Front. A Pareto-Front is useful for decision-makers because it shows the strategic balance between the objectives of the model in different scenarios (Yv et al., 1971; Caramia and Dell’Olmo, 2008).

The contribution of this paper is twofold. First, a mathematical model for a multiobjective three-echelon transportation problem, which considers the use of a heterogeneous fleet composed of drones and land vehicles, is proposed. Second, an extensive computational experience, considering a case study on humanitarian logistics, is provided to show the performance of the proposed method.

The paper is structured as follows. Section 2 discusses previous studies related to the problem. In Section 3, the methodology of the study is developed. Section 4 describes the case study used to evaluate the proposed model. In Section 5, the results of the computational experience are presented and discussed. Finally, in Section 6, some conclusions and suggestions for further research are outlined.

2. Literature review

Over the years, many strategies and mathematical models have been developed with the aim of minimizing the loss of human lives in the response stage of a natural disaster. The literature review suggests that, in general terms, these models fall into two categories: facility locations and relief distribution (Caunhye et al., 2012; Habib et al., 2016; Nolz et al., 2011). Most of these models consider only one or two echelons. Other categories include evacuation and casualty transportation. Most of the facility location models in humanitarian logistics integrate the operations of location (building new facilities or choosing among existing ones), evacuation or relief distribution (Caunhye et al., 2012). These models are mainly based on mixed-integer programs using binary variables to identify the facilities (shelters, warehouses, collection centers, depots, distribution centers, etc.), routes and several transportation modes to be used. Their main objectives are to minimize delivery times and operating costs. In addition, they consider budget constraints, facility expansion, damage and injury to emergency equipment and personnel. Models focused on relief distribution consist of bringing relief (medical supplies, shelters, manpower, sanitation and other resources) to the affected zones (Caunhye et al., 2012). Most of the research available in the literature focuses exclusively on relief distribution; in addition, these models have been developed mainly to solve problems for a single objective because they are easier to solve. Both facility location and relief distribution models generally contemplate the use of land vehicles for their operations. Some of the most representative studies regarding the use of drones in the health-care system are described below.

Scott and Scott (2017) developed two models concerned with the design of a drone-based health-care-delivery network that facilitates time and cost-effective delivery. Escribano et al. (2020) share a stochastic vehicle routing problem that coordinates UAVs with land vehicles and demonstrates the benefits of their use in terms of costs. Arenzana et al. (2020) proposed an optimization model to design UAV-based hospital-delivery networks, which minimizes drone-traveling time, battery-consumption levels, CO₂-emission levels, vehicle investment and infrastructure costs. Arenzana et al. (2020) considered UAVs during humanitarian crises. They developed a mathematical model for a stochastic vehicle routing problem, which uses UAVs to reveal road damage conditions and trucks to subsequently deliver commodities during a relief distribution operation. Numerous benefits were obtained in the selection of truck routes compared with deterministic methods; in this work, UAVs limitations were highlighted. Skorup and Haaland (2020) identified the potential benefits of UAVs in reducing social interaction in daily tasks and slowing down the spread of COVID-19. Other authors (Chang et al., 2007; Duran et al., 2011; Iakovou et al., 1997; McCall, 2006; Psaraftis et al., 1986; Wilhelm and Srinivasa, 1996) have developed models that combine the location of facilities, the relief distribution for single-echelon and single-objective humanitarian logistics problems, which suggest the use of only land vehicles. However, other researchers (Mete and Zabinsky, 2010; Rawls and Turnquist, 2010) have developed two-echelon models with one and two objectives to meet the needs in the response stage of a disaster, and these models also were focused on the use of land vehicles. The model proposed in this document differs from previous works by offering a study of a three-echelon bi-objective problem, which considers UAVs as one mode of transport in the third echelon. Table 1 summarizes the literature found for emergency logistics models used in facility locations with relief distribution in the response stage of a disaster.

3. Methodology

4. Case study

The proposed model was implemented for a case a study occurred in Mexico on September 19, 2017, where a Mw 7.1 earthquake occurred with an epicenter at 55 km (34 mi) from the south of the city of Puebla that lasted for approximately 20 s (Working Group of the National Seismological Service and UNAM, 2017). The earthquake caused damage in Guerrero, Mexico City, Morelos, Oaxaca and Puebla. A total of 369 people died, and more than 7,000 people were injured (Criales and Mota, 2019).

In Mexico, immediately after a natural disaster, an extraordinary emergency declaration is established and resources of the Fund for Emergency Attention (FONDEN) are immediately distributed to address the primary needs of the affected population (food, shelter and health).

Fuerza México is a digital platform that contains data from the Ministry of Finance and Public Credit of different agencies and entities involved in the reconstruction tasks from the earthquakes occurred on September 7 and 19, 2017. This platform does not provide information, documentary support on donations or actions carried out by the private sector. According to the general rules of FONDEN, entities must follow four main steps after a natural disaster:

1. Request emergency support to meet the immediate needs of the population.

2. Request immediate partial supports (charged to FONDEN) to obtain resources for the execution of emerging actions, the restoration of communications, basic services and the removal of debris from the affected area to avoid further damage and protect the population.

3. Once immediate partial supports have been received, a damage evaluation committee is installed to evaluate and quantify the damages in different sectors. Entities may request resources to cover expenses derived from the damage assessment work carried out from the natural event.

4. The necessary works and actions for the reconstruction are determined.

5. Computational experience

The mathematical model was programmed with Python using Anaconda Navigator – Spyder 4.0.1 on a MacBook Pro with a 2.7 GHz Intel Core i5 and an 8-GB 1867 MHzDDR3 with Gurobi solver version 9 implemented.

6. Conclusions and recommendations for future research

Historically, the use of UAVs in logistics activities has great benefits, and their flexibility makes them excellent modes of transportation. However, in practice, their economic, political and social limitations should always be considered. This makes regulating the use of UAVs a priority issue to avoid violating the privacy and rights of citizens when they are used.

In this research, a mathematical model for a humanitarian logistics supply-chain problem using UAVs as an additional transportation mode is developed. The model has been applied to the case study of the Earthquake on September 19, 2017, in Mexico.

The results of the study suggest that when the delivery time is restricted, the transportation costs increase, and in the last echelon, an increased number of routes are assigned to the UAVs (because there is less response time to deal with the emergency). When a larger delivery time is considered, transportation costs decrease, and routes are assigned to transportation modes with larger capacity, regardless of their speed. Although the proposed model is limited to large-scale applications, it provides a good representation of a transportation network that can be designed and optimized to increase the performance of the given response during a natural disaster, extending it to consider the potential use of UAVs to address this problem, which is one of the major contribution of this research.

Alternatively, UAVs can be used to decrease delivery time in the last mile to regions with limited access. Although some researchers highlight the low maintenance cost of UAVs, their implementation can result in high costs because it is a relatively new technology, and therefore, implementing them is more feasible with a large emergency budget. Furthermore, the model can be adapted to problems that have arisen from the COVID-19 pandemic. For example, they can be used to respond to the high demand for oxygen tanks and distribute them in areas that have difficult access, as well as for the distribution of vaccines through cold chains. For problems where demand is greater than UAVs’ capacity, modifications should be made to the proposed model so that final users can receive their products in more exhibitions, or it could be considered not to use drones at all as a transportation mode.

This research has different implications in real-life situations. Although the model can be adapted to any system anywhere in the world for problems beyond humanitarian logistics, its application may be limited by the economic, geographical and social situation (security and crime rates) where it is implemented, as well as regulations and policies for the use of UAVs.

The economic impact of this research can be visualized in both the short and long term. In the short term, investment in UAVs is large, whereas, in the long term, as explained in the literature review, their use can have a positive impact on associated costs as well as on CO₂ emissions. These economic implications are consistent with the results obtained in this research, as the more routes assigned to UAVs, the cost is higher.

The use of UAVs during natural disasters can have a positive and negative social impact. A positive impact can be achieved if UAVs are seen as reliable vehicles that can help during any emergency in the shortest possible time. However, a negative impact is associated with privacy violations and vehicle damage due to the crime rate in the area. Thus, regulations should be made to address these problems and UAVs insurance services. Regarding their economic impact, currently, UAVs are relatively high-cost vehicles. However, with its continued development, it is expected that the viability of its use will increase in a few years.

For future research, the model can be adapted to allow vehicles to make additional routes doing an adequacy of the vehicle routing problem with multiple routes in the last echelon. To expand the model, more parameters should be considered, such as vehicle–distance ratio and the number of times a demand node can be visited by a vehicle. Furthermore, because for larger scale instances, the proposed model, solved by using a general-purpose solver, was not able to obtain optimal (even feasible) solutions in a reasonable computing time, it is recommended to explore the design and the development of metaheuristics to obtain good quality solutions efficiently. Some of the most used meta-heuristics for multiobjective problems are GA-based solution methods, which can be a good starting point for future research lines.