Robust models for after disaster delivery schedules

Abstract

Emergency management has a four stage life cycle: mitigation, preparation, response and recovery (Beamon, et al., 2004, Wassenhove, et al., 2006). The disaster response stage aims to provide critical resources and services to the affected people in a very short time window. Thus this stage requires accessing the areas impacted by disasters to enable proper disaster responses.

A subproblem is then set up to solve explicitly the value of the parameters given a specific instance of variables. From there, a closed form solution can be found with the final optimization problem to be a convex mixed integer second order cone problem. The center of the parameters of that constraint are determined through linear regression model. We constructed separable piecewise linear functions to approximate the convex constraint. We also require separate linear pieces to be connected at the specific breaking points. Several formulations of the optimization problem are proposed to explore and accommodate detailed problem features.

We apply our approach to response stage scheduling problem. Part of the objective function is the deprivation cost. We also apply the method to a Seattle highway network problem to compute a system optimum with uncertain parameters. Keywords: Robust optimization, humanitarian logistics, piecewise linear regression

One important progress in disaster research is to include concepts of human suffering from welfare economics. Holģuin-Veras , et al. (2013) demonstrate the importance to use deprivation cost functions to appropriately accommodate inherent insufficiencies of the commercial logistic functions traditionally used in post-disaster humanitarian-logistics (PD-HL). The impact from the deprivation time is modeled by deprivation cost (DC) functions. In a deprivation cost function, the period of time in which the population affected by disasters live without critical resources and service is denominated as deprivation time (DT). Also, the deprivation cost functions are convex nonlinear nondecreasing functions. Therefore, they are very suitable functions for optimization models.

The deprivation cost function proposed by Holģuin-Veras, et al. (2013) contains some parameters with specific fixed values. We robustize the parameters of the deprivation cost function by setting up a nonlinear convex constraint. The degree of robustness is managed through a unifying $\epsilon$ value. We set up a model to minimize a convex combination of logistics cost and deprivation cost. Our model also takes into consideration of logistics cost by minimizing a convex combination of logistics cost and deprivation cost.