deletions | additions       diff --git a/section_A_two_stage_stochastic__.tex b/section_A_two_stage_stochastic__.tex  index 699853c..5d8af2a 100644  --- a/section_A_two_stage_stochastic__.tex  +++ b/section_A_two_stage_stochastic__.tex 
...
\section{A two-stage stochastic programming model for transportation network protection}  Liu et al. \cite{LiuEtal2009} focus on improving the resilience and robustness of transportation system in mitigation of a disaster in uncertain disaster occurrence environment. In particular, for identifying bridges in seismic retrofitting, a two stage stochastic formulation is presented in order to minimize total structural and travel delay loss. In the stochastic model, in the first stage, retrofit decisions (bridges for retrofitting) are made, while flow of each link is determined in second stage.  The authors also prioritize network retrofit strategies based on the importance of facilities and available budgets. The problem is solved via   An efficient algorithm is developed, via extending the well-known L-shaped method using generalized benders decomposition, to efficiently handle the binary integer variables in the first stage and the nonlinear recourse in the second stage of the model formulation. for  the first stage of solution methodology,  the retrofit problem L-shaped technique  is to make retrofit decisions before the earthquake happens, while the second stage utilized. The model  is to evaluate the total loss due to evaluated on  a realized earthquake including repair cost real data from Sioux Fall City network with 24 nodes  and increased travel delay in the network. 76 links as well as Alameda County network with 510 nodes and 1424 links.