Discrete Optimization
Integrating location and network restoration decisions in relief networks under uncertainty

https://doi.org/10.1016/j.ejor.2019.06.012Get rights and content

Highlights

  • We integrate location and restoration models for effective disaster response.

  • We use two-stage stochastic programming to incorporate disaster scenarios.

  • We generate scenarios considering the spatial correlation between nodes.

  • We solve our model by the sample average approximation with concentration sets.

  • We identify concentration sets using information obtained from disaster scenarios.

Abstract

Prepositioning emergency relief items in emergency response facilities before an anticipated disaster is a common strategy to increase the effectiveness of relief distribution. In this paper, we assume that relief distribution activities are hampered due to damaged roads, which can be restored by repair teams using restoration equipment. We propose a two-stage stochastic programming model integrating facility location and network restoration decisions. Our integrated model decides on the location of restoration equipment prior to the disaster in addition to the facility location decisions. Moreover, decisions related to relief item distribution and network restoration are made jointly after the disaster for each disaster scenario. We capture uncertainty in the network availability by incorporating the repair times required to restore the damaged roads. To solve our integrated model efficiently, we develop a sample average approximation method with concentration sets motivated by Rosing and ReVelle’s (1997) Heuristic Concentration. These concentration sets are comprised of promising locations identified by information obtained from disaster scenarios. We limit our solution space in the first stage to concentration sets to reduce the problem size without sacrificing the solution quality significantly. Our computational results show significant improvement in unmet demand and cost measures by integrating location and network restoration models.

Introduction

When a disaster strikes, emergency relief items should be delivered to the disaster victims as soon as possible. These relief items can be procured from local suppliers or global suppliers after the disaster; however, the high degree of uncertainty following the disaster makes these two sourcing options more challenging. For example, supplies from local sources may be insufficient to meet the total demand. Moreover, it may be expensive and time consuming to obtain supplies from global sources. Therefore, locating emergency response facilities and prepositioning emergency relief items in these facilities prior to an anticipated disaster is a common practice to increase the effectiveness of the disaster response. In this study, we consider the problem of locating emergency response facilities, which serve as warehouses to store emergency relief items before the disaster. These relief items are transported to disaster victims from the emergency response facilities using the road network after the disaster. Some of these victims cannot be reached from any of the emergency response facilities due to the damage in the road network. To reestablish connectivity between the facilities and demand points, some of the roads are restored by repair teams. Therefore, relief item distribution and network restoration operations proceed together. We assume that the repair teams utilize restoration equipment to restore damaged roads. This equipment cannot pass over damaged roads until they are repaired; consequently, it may not have access to some of the damaged roads from its initial position in the network. Our aim in this study is to increase the amount of demand satisfied for emergency relief items after the disaster with lower costs by integrating facility location decisions and network restoration decisions.

Damaged roads can pose a great challenge for last-mile distribution. For example, despite the abundance of supplies, the 2010 Haiti earthquake victims could not receive relief items for a long period since the damage in the road network hampered transportation activities severely (Çelik, 2016). After the 2011 Japan earthquake and tsunami, around three-fourths of the highways in the region were not operational which impeded emergency response activities (Asaly & Salman, 2014). Mountains of emergency supplies waited idle at the main port of Puerto Rico due to the shortage of truckers and the damaged infrastructure after Hurricane Maria in 2017 (Gillespie, Romo, & Santana, 2017). These examples show that some damaged roads should be repaired during the response phase. The success of last-mile distribution depends both on the location of emergency response facilities and the allocation of restoration equipment to the damaged network. Therefore, considering these two problems separately may lead to suboptimal solutions. By integrating location and network restoration decisions, more beneficiaries are reached during the early stages of the response and the total cost of performing pre-disaster and post-disaster activities decreases.

In this study, we propose an integrated facility location and network restoration model. Our model is a two-stage stochastic programming model in which the first stage and second stage correspond to the preparedness phase and the response phase of a disaster, respectively. In the first stage, we determine where to locate emergency response facilities and restoration equipment. In addition to the location decisions, we also determine the capacity of emergency response facilities as well as the number of pieces of restoration equipment. In the second stage, we make decisions related to relief item distribution and network restoration jointly. In our stochastic programming model, we consider demand uncertainty, supply availability uncertainty and network availability uncertainty. One limitation is that the deterministic equivalent MIP of our stochastic programming model cannot be solved optimally for large problems. We use the sample average approximation (SAA) method to reduce the scenario set to a manageable size. Moreover, we use concentration sets to limit the number of first-stage decision variables, which allows us to reduce the problem size even further.

We make the following contributions with this paper:

  • 1.

    We propose an integrated facility location and network restoration model to locate both emergency response facilities and restoration resources prior to an anticipated disaster. By integrating location and network restoration decisions, our model satisfies more demand for emergency relief items promptly with lower cost. Our model also ensures that only operational roads are used when transporting both emergency relief items and restoration equipment.

  • 2.

    We use the SAA method with concentration sets to reduce the size of the problem. To identify the nodes which are likely to be in the optimal solution, we use multiple solutions obtained by solving the problem for smaller scenario groups. We select P nodes to include in the concentration set from among these promising nodes using the P-median problem formulation. Our SAA algorithm supported by the concentration sets allows us to solve our integrated model efficiently without degrading the solution quality substantially.

  • 3.

    We propose a scenario generation algorithm which takes into consideration the spatial correlation between neighbor nodes. We use our scenario generation algorithm to generate random samples of scenarios for an anticipated earthquake in Istanbul.

The organization of the rest of the paper is as follows: in Section 2, we review the literature on prepositioning and network restoration models. We present our integrated stochastic programming model in Section 3 and we explain our solution approach to solve this model in Section 4. We introduce a case study for an earthquake in Section 5 where we also discuss our computational results. In Section 6, we give our conclusions and we sketch our future study plan.

Section snippets

Literature review

Integrating facility location and network design models have been shown to be useful in a number of applications where changing the underlying network is less costly compared to adding a new facility (Melkote and Daskin, 2001a, Melkote and Daskin, 2001b). These integrated models are deterministic and they assume that both facility location and network design decisions can be made at the same time. However, these decisions should be made at different points in time for the problem environment

Model

Our literature review points out that two-stage stochastic programming is a common approach to model the prepositioning problem. The models proposed in the literature decide on the number and location of the emergency response facilities in the first stage and focus on the distribution of emergency relief items from these facilities to demand points in the second stage; however, these studies disregard network restoration efforts. In this study, we propose a two-stage stochastic programming

Solution approach

In our model, we consider three types of uncertainty. Even if we ignore demand and damage ratio uncertainties, we end up with (t+1)|A| different scenarios assuming that the repair time of a damaged arc can be any integer value from 1 to t time periods. Therefore, evaluating the exact value of the expected second-stage cost becomes quite challenging as the number of arcs increases. There exist several scenario generation and scenario reduction methods to overcome this difficulty. In this study,

Case study

Turkey is located in a seismically active region. It has been hit by several major earthquakes throughout its history. The 1999 Marmara Earthquake was one of these earthquakes. The magnitude of the earthquake was 7.4 on the Richter scale with an epicenter close to Istanbul, which is the biggest city of Turkey. The earthquake killed approximately 17,500 people and damaged 77,000 residences and workplaces severely (Özmen, 2000). Studies show that Istanbul is threatened by another strong

Conclusions

In this study, we propose a two-stage stochastic programming model to locate emergency response facilities and road restoration equipment prior to a disaster. The facilities are used to distribute emergency relief items to disaster victims after the disaster. We integrate facility location decisions with network restoration decisions so that paths between facilities and inaccessible demand nodes become available throughout the time horizon. We avoid allocating restoration resources to

Acknowledgments

This work was funded by a variety of internal University of Michigan funding sources.

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