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. The cost of client j is the sum of distances to its assigned facilities .
After 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 \geq 1 distinct facilities in S. The service cost of j is now the sum of distances to its assigned facilities .