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We introduce a new two-stage model for integrated quantile functions. This framework leads to a bilinear optimization problem (P), a special case of which reveals a new model for the VaR minimization problem. We discuss various techniques to under- and over-estimate the optimal value of (P), in turn developing new lower and upper-estimators for VaR. We provide computational comparisons over supply chain network problems. This approach helps constructing supply chain networks which are more resilient to probabilistic disruptions occurring in the network.