Before revision:  Here w\inR^{|X|} is a predefined non-increasing non-negative vector, and }c  . In the fault-tolerant problem, every client jhas a requirement r_j and needs to be assigned r_j distinct facilities.
After revision:  Here w\inR^{|X|} is a predefined non-increasing non-negative vector, and }c^{\downarrow . In the fault-tolerant problem, every client jhas a requirement r_j and needs to be assigned r_j distinct facilities.