AN OPTIMIZATION APPROACH FOR DISASTER RELIEF NETWORK DESIGN UNDER UNCERTAINTY AND DISRUPTION WITH SUSTAINABILITY CONSIDERATIONS

. Human-made, natural, and unexpected disasters always cause human and financial losses to communities. Disaster management is a framework with proven performance to reduce the damage caused by disaster and supply chain disruptions. Transferring the injured people from affected areas to hospitals at the minimum possible time is a crucial goal in times of disaster. This paper develops a two-stage stochastic programming model to transport the injured people from affected areas to hospitals in the incidence of multiple disruptions at transportation links and facilities under uncertainties. Herein, economic, social, and environmental aspects of sustainability are considered, while simultaneous disruptions are managed to minimize the adverse impacts of the disasters. We aim to determine optimal locations to establish transfer points and flows between the relief network nodes with sustainability considerations. Ultimately, a case study in District 12 of Tehran, Iran is conducted to ensure the proposed model’s validity and performance. Various sensitivity analyses are also implemented to ensure the model’s effectiveness. The results indicate that disruptions in facilities and transportation links lead to increased relief time, hence has the most significant negative impact on relief operations.


Literature review
Due to the importance of relief chain and TPs in the disaster incidence, numerous researches are being published, and different models are proposed to locate relief facilities such as TPs to speed up the relief process. It is crucial to carry out rescue operations as soon as possible, as transporting survivors to shelters and the injured to hospitals at the minimum possible time would significantly reduce the severity of the impacts. Note that the wounded have to be transported as soon as possible in the post-disaster phase. However, it is sometimes difficult to land the helicopters due to geographical roughness in the AAs. Accordingly, TPs are located between the AAs and hospitals and are used to reduce relevant damage and facilitate relief operations in natural and human-made disasters. Vehicles used between different nodes need to transport the injured people at an increasing rate to save time as much as possible [3]. Therefore, optimally locating shelters and TPs is vital for disaster management.
For instance, Ghaderi and Momeni [17] proposed a multi-period maximal coverage model for locating ground and air emergency stations considering the inaccessibility of ambulances in light of the notion of backup coverage for providing services to the wounded and transporting them to treatment centers. Tzeng et al. [45] proposed a relief model to determine the absolute conditions of TPs in the distribution of goods. The objective functions include maximizing service satisfaction among the affected regions and minimizing the total cost and relief time. Two models were analyzed by Furuta and Tanaka [15] to minimax and minimize service times for locating TPs and helicopter stations in certain conditions, providing rapid and efficient relief services. Shahriari et al. [42] developed a bi-objective mathematical model that employed the compromise programming method to locate ground and air ambulance base and helipad locations under uncertainty. The main objectives of this model are the minimization of travel time and the maximization of level of service. Mansoori et al. [27] proposed a humanitarian relief network design problem with evacuation process under uncertainty and disruption caused by the earthquake. They aimed to minimize the unfulfilled demand for relief commodities, the total number of injured not transported to hospitals, remaining in the AA. Hassani and Mokhtari [19] designed a network for distributing relief commodities under operational risks. The objective functions include maximizing network population coverage and minimizing total costs and risks of the network. Akbarpour et al. [2] proposed a relief model for distribution of pharmaceutical items in AAs under uncertainty. They aimed to optimize coverage of AAs, pharmaceutical item's storage time, and mobility of the facilities.
The majority of information received in natural and human-made disasters has inherent uncertainties that can substantially influence the performance of relief operations; hence, the risk of rescue operations may increase sharply compared to deterministic environments [34]. Uncertainty is generally classified into random and epistemic categories. The random uncertainty arises from the random nature of the parameters. Stochastic programming approaches are usually used to tackle this type of uncertainty whenever statistical distribution is available for the random parameters [28,32]. Also, stochastic programming is employed in situations with a particular repetitive operation. Many modeling efforts on post-disaster relief operations use stochastic programming approaches to cope with uncertainty [16,44]. For instance, we can refer to models developed by Salmerón and Apte [41], Rawls and Turnquist [35], Bozorgi-Amiri et al. [6], Sheu and Pan [43], and Wang et al. [48].
Research on sustainable humanitarian supply chains is often qualitative [4,13,18,24]. Therefore, a limited number of studies proposed quantitative approaches and developed mathematical models [1,[6][7][8][9]20,21,25,31,33]. However, the most relevant studies do not sufficiently consider all aspects of sustainability under disruptions in relief operations. Accordingly, there is a research gap in incorporating all three aspects in humanitarian supply chain network design. For instance, Cao et al. [8] considered the total carbon emitted throughout the relief operations and the required time to distribute relief items as the environmental and social aspects of sustainability, respectively. Wei et al. [49] considered the pollutions emitted by vehicles that transport relief items as the environmental aspect, while from a social point of view, their model is similar to Cao et al. [8]. Finally, Boostani et al. [5] considered three factors as the aspects of sustainability in the humanitarian supply chain: minimizing total supply chain costs, maximizing the victims' satisfaction, maximizing the equity in the distribution of relief items, and minimizing the total environmental impacts associated with packaging and transporting relief items.
Research gaps are also identified by reviewing studies on robust and resilient relief network design. Researches reveal that most of the parameters associated with relief network design in natural and human-made disaster situations must be considered under uncertainty and disruption incidence. Multiple disruptions in facilities and transportation links between them are not adequately considered in relevant studies. The two-stage stochastic programming (TSSP) method is typically employed to consider the conditions resulting from the occurrence of each disruption mode under uncertainty; However, a robust scenario-based approach with the ability to exert the simultaneous impacts of perturbations and disruptions on the deterministic model is required while maintaining robustness at a desirable level. Also, due to the increasing importance of sustainable development, it is necessary to incorporate sustainability's social, economic, and environmental pillars in rescue operations planning.
Accordingly, this study applies a mixed-integer linear programming model for evacuating injured people from AAs to hospitals in times of disaster. The model aims to determine the location of TPs and the flow of injured people amongst network components under sustainability considerations and multiple disruptions in facilities (i.e. TPs and hospitals) as well as transportation links. Budget constraint, environmental impact limitation, and the objective function of the total relief time to the disaster points represent economic, environmental, and social sustainability, respectively. Also, AAs' demand is considered to be inherently uncertain, hence, the TSSP approach is employed to deal with it. Finally, a case study in District 12 of Tehran, Iran is applied to the proposed formulation for ensuring model applicability and validation. Sensitivity analyses are also implemented to verify the model.

Problem definition
Transportation of the injured people from AAs at a minimum possible time is the main priority during disasters such as earthquakes, floods, and fires. Note that the number of the injured in each region may be variable depending on the fabric of the affected region, which needs to be recognized as well. Besides, the severity of disaster determines the extent of the damage they cause; Therefore, a disaster with high severity causes disruptions such as the destruction of facilities and transportation links between them.
According to Figure 1, The AAs are considered as demand points needing relief operations for transporting the injured either directly by ambulance or indirectly by helicopter to speed up the rescue process. In other words, the injured are taken using an ambulance when hospitals are not very far from the AA, hence the injury is not further aggravated. Otherwise, if the distance is too far or the nearest hospitals have reached their maximum level of service, the injured are transported using helicopters via TPs. Transferring the injured by TP involves the following steps; At the request of the rescue team, the helicopter first flies from the central station to the nearest TP, then the injured are transported to the hospital. Note that the number of injured transferred directly depends on the predefined capacity of the helicopters and hospitals.
This study develops an optimization model to locate TPs with sustainability considerations under multiple disruptions in transportation links and facilities. The proposed model considers economic and environmental sustainability aspects in the form of budget, and environmental emissions constraints. Moreover, the objective function of minimizing the total transportation time of the injured and the constraint of maximum relief time to each AA implies the social dimension of sustainability. Also, some parameters such as the number of the injured in AAs, disrupted links between network nodes and lost capacity of the disturbed facilities are considered under uncertainty that is tackled employing TSSP method. Note that AAs and helicopter stations, hospitals, and TPs are capacitated and respectively considered as departure, destination, and intermediate points in the proposed model.

Model formulation
TSSP is employed to formulate the problem under uncertainty so that the decision variables are considered pre-event (first stage or scenario independent) and post-event (second stage or scenario dependent). Pre-event decisions include establishing TPs, and post-event decisions involve determining the number of injured transported between AAs and TPs, AAs and hospitals, and TPs and hospitals in the proposed model. Parameters and variables associated with the proposed two-stage linear model are explained below.

Sets
Set of scenarios, indexed as ∈ Set of TPs, indexed as ∈ Set of hospitals, indexed as ℎ ∈ Set of helicopter stations, indexed as ∈ Set of AAs, indexed as ∈

Parameters
The population of AA ∈ must be transferred under scenario ∈ Transfer time from AA ∈ to TP ∈ ′ ℎ Transfer time from AA ∈ to hospital ℎ ∈

Mathematical model
The objective function and constraints are provided based on TSSP as follows.

Case study
The case study is conducted in Iran, particularly, district 12 of its capital city Tehran. District 12 is located in the historical center of Tehran. Tehran Grand Bazaar, numerous state institutions, ministries, and embassies are the district's most important attractions and buildings. According to Tehran's administrative divisions, the investigated district is divided into six zones, and the centers of these zones are considered as AAs in the relief network. Table 1 illustrates the estimation of the injured needing to be transported from each AA due to the context of each zone, the severity of the disaster that may take place, and other influential factors. Three independent scenarios, including optimistic, most likely, and pessimistic, are generated considering disaster severity, and the possibility of occurrence for each scenario is estimated, as shown in Table 2. Table 3 shows that there are 18 hospitals as existing facilities and nine candidate locations for establishing TPs are considered in the case study. Note that the number of established TPs must not exceed a maximum of seven. The capacity of a candidate TP for evacuating the injured is considered to be 2000; however, the capacities of hospitals are variable based on their difference in their equipment and facilities, that are outlined in Table 4. The proposed model addresses simultaneous disruptions in facilities and transportation links; therefore, due to the distinctive fabric of the AAs and severity of the disaster, some routes will not be exploitable for relief processes under particular disruption scenarios. The travel times between nodes are calculated based on their distance via the Google Map platform. Also, the speed coefficients of ambulances and helicopters relative to standard vehicles are 0.3 and 0.1, respectively.  An overall observation implies that the use of the facilities (i.e. TPs and hospitals) is sharply increased under pessimistic scenario; The rationale behind this behavior is that the occurrence of the discussed scenario leads to the rescue chain facing the greatest number of the injured, and given that the network intends to perform rescue operations in minimum possible time, therefore, the use of facilities sharply increases. For instance, nearly 100% of the Sevom Shaban hospital's capacity was used under the pessimistic scenario, whereas that facility was not employed under the most likely and optimistic scenarios. Also, about 50% of Haqani park TP's capacity, is used    to transport the injured from the AAs under the optimistic and most likely scenarios, while the used capacity of the aforementioned facility is close to 100% under the pessimistic scenario. The Mahdieh and Amir A'lam hospitals that have the maximum capacity among existing hospitals, provide services to the injured with total capacity under all scenarios, so the used capacity of these two hospitals is completely scenario-independent. Similarly, Figure 2 indicates that the Heidarnia Stadium TP serves the injured from the AAs at total capacity under all scenarios; the rationale behind this behavior is that the mentioned TP has the shortest distance to two high-demand AAs Mehdi Alley and Sadi Street.
Disruptions can significantly affect the time it takes to rescue and transport the injured, so relief processes may face numerous problems. This study addresses partial disruptions in facilities, including TPs, hospitals, and complete disruptions in transportation links between different nodes. Figure 5 shows how the previously mentioned disruptions affect the total transportation or rescue time, indicating exciting and significant results. Note that the expected total transportation time changes are separately calculated under each disruption to evaluate the severity of the disturbances on the relief time. Results indicate a complete disruption in transportation links, particularly, the disrupted links between AAs and TPs, dramatically increases the transportation time. Also, when all possible disruptions are simultaneously considered, the relief time increases by about 13% compared to the situation without disturbance. Accordingly, results imply that considering complete disruptions in transportation links leads to a more significant increase in the total rescue and transportation time compared to partial disturbances in facilities.   The sensitivity analysis is implemented on the capacity of TPs and hospitals to ensure the verification and application of the proposed model. Figures 6 and 7, respectively imply that an increase in the capacity of hospitals and TPs leads to a remarkable reduction in the total transportation and relief time.
The impact of changing the maximum allowable number of TPs on the total relief operations' time is evaluated below. Figure 8 illustrates that increasing the parameter significantly decreases the total transportation time. The rationale behind this behavior is that some TPs may not be available under multiple disruptions in facilities and links connecting them. So, the more TPs are established, the more the injured can be transferred from the AAs to the hospitals via TPs. Accordingly, reducing the number of transfers by helicopters and ambulances leads to a remarkable decrease in the total transfer time.
Furthermore, the total transportation time is optimized separately under each scenario. Note that the optimistic scenario, in contrast with the pessimistic one, considers a situation in which the severity of disruption is less in terms of negative impacts. Figure 9 illustrates that an increment in disaster's severity remarkably increases the total transportation time. The logic behind this behavior is that facilities and transportation links are simultaneously disrupted in the relief network, and the more severe the disruption, the greater its harmful effects, which further reduces the available capacity of facilities and closes more transportation links. This, consequently, slows down the transfer of the injured. Thus, the total transportation time sharply increases by incrementing the disaster severity.

Managerial insights
Herein, the most important practical and managerial insights from the aforementioned numerical results are listed. Due to the inherent uncertainty caused by supervenient disasters, important parameters related to the relief process, such as the number of injured in AAs, cannot be accurately estimated, thus, the relief network management needs to adopt a robust approach for resilient planning in such situations. Initial observations indicate that disruption in links has the most negative impacts on relief times and endangers the survival of the injured, so fortifying relief infrastructures and considering backup transportation links can be perfectly helpful in disaster incidence. Due to the lack of available resources and time for relief operations in disaster situations, it is necessary to make appropriate investments to optimally increase the capacity of relief facilities in order to serve more injured people in a shorter time, while lowering costs.

Conclusion
The most important priority of disaster management decision-makers is to carry out rescue operations in the minimum possible time because relief time directly affects the severity of injuries and the death rate. One of these operations is to evacuate the injured from the AAs to the hospitals. In this regard, this study addresses a mixed-integer linear programming model for locating TPs under disruption and uncertainties considering sustainability considerations. Environmental pollution and budget constraints were considered as environmental and economic pillars of sustainability in the model. Also, the objective function of minimizing the total time required to transport the injured and the constraint of maximum relief time for each AA represent the social dimension of sustainability. Due to the negative impact of the disaster on relief chain management, it is assumed that the facilities, including TPs and hospitals, are partially disrupted, and the transportation links connecting different network nodes may be completely disturbed. Due to the lack of precise data on the hypothetical number of individuals injured, the share of the population of each AA that need to be transferred to hospitals is considered to be uncertain, and a TSSP approach is employed to deal with uncertainties. Moreover, the capacity constraint for TPs and hospitals is incorporated to further adapt the model to real-world situations. Finally, a case study of Iran, particularly, district 12 of Tehran province is conducted to ensure the practical application as well as validation of the proposed model and the results and various sensitivity analyses are reported.
The results indicate that disrupted facilities and transportation links lead to increased relief time, so disruption in links has the most significant negative impact on relief operations. Accordingly, decision-makers aim to use facilities with sufficient capacity to optimize available time and resources to provide the most effective rescue operations to all AAs in the minimum possible time. Therefore, sensitivity analysis was implemented on the capacity of hospitals and TPs, and the maximum number of TPs allowed. The impact of different scenarios, including pessimistic, most likely, and optimistic scenarios, on the evacuation and transportation of the injured people among components of the relief network is also evaluated. Results illustrate that those facilities with the highest capacity and minimum distance to AAs are employed to achieve the rescue network's minimum relief time.
Despite the intended contributions, the present research is not without limitations. This study can be contributed in several directions, such as incorporating the distribution network of relief commodities to the proposed model, considering shelters for covering and protecting healthy people, and considering fairness in relief operations. Also, adopting both reactive and proactive resilience strategies to manage disruptions and applying appropriate robust optimization approaches to cope with operational risks arising from the inherent uncertainty of parameters can further reinforce the contributions of this study.