Sensors are being extensively adopted for use in smart cities in order to monitor various parameters, so that any anomalous behaviours manifesting in the deployment area, can be easily detected. Sensors in a deployment area have two functions, sensing/coverage and communication, with this paper focusing on the former. Over the years, several coverage models have been proposed which utilizes the Set Cover based problem formulation. This formulation unfortunately has a drawback, in the sense that it lacks unique identification capability for the location where anomalous behavior is sensed. This limitation can be overcome through utilization of Identifying Code. The optimal solution of the Identifying Code problem provides the minimum number of sensors that will be needed to uniquely identify the location where anomalous behavior is sensed. In this paper, we introduce a novel budget constrained version of the problem, whose goal is to find the largest number of locations that can be uniquely identified with the sensors that can be deployed within the specified budget. We provide an Integer Linear Programming formulation and a Maximum Set-Group Cover (MSGC) formulation for the problem and prove that the MSGC problem cannot have a polynomial time approximation algorithm with a 1/k factor performance guarantee unless P = NP.