Thursday, August 25, 2016

A Generalized Nash Equilibrium Model for Post-Disaster Humanitarian Relief - Case Study on Hurricane Katrina and Beyond

The devastating floods in Baton Rouge, Louisiana, 11 years after Hurricane Katrina,  which have resulted in the worst natural disaster in the US since Hurricane Sandy,  as well as the 6.2 magnitude earthquake that hit central Italy about 100 miles from Rome recently, demonstrate the impact that disasters have on societies. The response phase is essential to finding survivors and saving victims.

Every spring I teach my Humanitarian Logistics and Healthcare class at the Isenberg School of Management and it continues to be one of the most fascinating courses that I have ever taught. On this blog I have written posts about many of the truly special guest speakers that have come to share  their experiences in emergency management and disaster response with the students.

This past Spring, one of the students in the class, Emilio Alvarez Flores, who graduated with a degree in Operations and Information Management from UMass Amherst in May 2016, was also working on his honors dissertation, since he was a member of the Commonwealth Honors College. I had the pleasure of co-supervising his dissertation, along with Professor Ceren Soylu of the Economics Department at UMass Amherst. The title of his thesis was: Optimizing Non-Governmental Organizations’ Operations and Fundraising: A Game-Theoretical Supply Chain Approach. Emilio defended his thesis at the Undergraduate Research Conference. Emilio was honored for his thesis (one of about a dozen) with a Deans Honors Award from the Commonwealth Honors College.
Emilio had been hard at work for about one year modeling the integration of supply chain aspects as well as financial funds raised by nongovernmental organizations with a focus on integrating in a model both supply chain aspects and financial fund aspects. Together wtih Professor Soylu we spent hours discussing and working out various aspects and versions of the model. One of the unique (and challenging) aspects of disaster relief is that NGOs and governments are nonprofits and derive some utility from helping victims post-disasters. At the same time, the former compete with one another for financial funds and, depending upon their response to disasters and visibility, they may get more or less of the financial funds. Also, NGOs need to minimize their costs since waste is not something that donors and stakeholders look kindly on.  Victims, on the other hand, need water, food, and medical supplies, as well as protection from the elements as soon as possible and, hopefully, no later than 72 hours. 

There have been numerous instances of surpluses of one kind of relief item being delivered to victims post-disasters, whereas shortages arise of other supplies. Hurricane Katrina, which struck southern parts of the United States in August 2005, is a vivid case. It was the costliest natural disaster in US history. Making landfall in August of 2005, Katrina caused extensive damages to property and infrastructure, left 450,000 people homeless, and took 1,833 lives in Florida, Texas, Mississippi, Alabama, and Louisiana. 63% of all insurance claims were in Louisiana with overall damages assessed as being in the rang of  $105 -$150 billion. In Louisiana alone, over 1.3 million people were affected, with Katrina being responsible for 300,000 jobs lost, 200,000 people left homeless, and over 1,500 fatalities. The New York Times reported that the Red Cross mismatched supplies with the victim’s needs; thereby, leading to obsolete inventory. 

In developing a model that is computable and would be based on data it was clear that the model had to be a game theory one. However, most models in the disaster relief and humanitarian logistics arenas are optimization models. Moreover, we were interested in evaluating policies in order to minimize materiel convergence and to assist in the delivery of the needed amounts of relief supplies to the destinations.

 Although a Nash Equilibrium model could be developed and we have a lot of experience in formulating, analyzing, and solving Nash Equilibrium models in a spectrum of supply chain applications from the pharmaceutical industry to food supply chains, the behavior there would be that of profit-maximization, which is not appropriate in the case of NGOs in disaster relief. Moreover, we wanted to explore what the possible impacts might be if there was a coordinating body, such as a supra NGO or governmental authority that would provide data as to the relief item needs in terms of lower and upper bounds at different points of demand. With such complicating constraints, which would be shared by the NGOs, the model would have to be a Generalized Nash Equilibrium model, and as far as we are aware there are no such models in the humanitarian relief sphere.

But if you have passion for a problem to be solved and, frankly, you are obsessed with the formulation and solution, you will figure out a way and you will get it done.

The result is the paper, A Generalized Nash Equilibrium Network Model for Post-Disaster Humanitarian Relief, Anna Nagurney, Emilio Alvarez Flores, and Ceren Soylu,  which has now been accepted for publication in Transportation Research Part E: Logistics and Transportation Review. The paper contains a case study on Hurricane Katrina and also demonstrates how much better the solutions are under a Generalized Nash Equilibrium framework than under simply a Nash Equilibrium one in which each NGO just has to satisfy its own constraints.  

As our case study in the paper reveals: It is immediately clear that there is a large contrast between the relief item product flow patterns under the Generalized Nash and Nash Equilibria. For example, the Nash Equilibrium flow pattern results in about $500 million less in donations. While this has strong implications about how collaboration between NGOs can be beneficial for their fundraising efforts, the differences in the general flow pattern highlights a much stronger point. Under the Nash Equilibrium, NGOs successfully maximize their utility. Overall, the Nash Equilibrium solution leads to an increase of utility of roughly 21% when compared to the relief item flow patterns under the Generalized Nash Equilibrium. But they do so at the expense of those in need. In the Nash Equilibrium, each NGO chooses to supply relief items such that costs can be minimized. On the surface, this might be a good thing, but recall that, given the nature of disasters, it is usually more expensive to provide aid to demand points with the greatest needs. With this in mind, one can expect oversupply to the demand points with lower demand levels, and undersupply to the most affected under a purely competitive scheme. This behavior can be seen explicitly in our results. For example, St. Charles Parish in Louisiana receives roughly 795% of its upper demand, while Orleans Parish only receives about 30.5% of its minimum requirements. That means that much of the 21% in ‘increased’ utility is in the form of waste. In contrast, the supply chain product relief item flows under the Generalized Nash Equilibrium guarantee that minimum requirements will be met and that there will be no waste; that is to say, as long as there is a coordinating authority that can enforce the upper and lower bound constraints, the humanitarian relief flow patterns under this bounded competition will be significantly better than under untethered competition.

This paper, we believe, has big policy implications and we expect that it will also generate further research. It was quite the journey doing this research but with inspired collaborators with great passion it was also thrilling and we are ecstatic that the paper has been accepted for publication and in one of my favorite journals!

In late June, I gave an invited seminar at Lancaster University in England: Disaster Relief Supply Chains: Network Models, Algorithm, and Case Studies, in which I highlighted some of our research in this area and the last part of the presentation presents highlights from our Generalized Nash Equilibrium Model for Post-Humanitarian Relief paper.