The problem of sequential detection of anomalous processes among K independent processes is considered. At each time, only a subset of the processes can be observed, and the observations from each chosen process follow two different distributions, depending on whether the process is normal or abnormal. Each anomalous process incurs a cost per unit time until its anomaly is identified and fixed. Different anomalous processes may incur different costs depending on their criticality to the system. Switching between processes and state declarations are allowed at all times, while decisions are based on all past observations and actions. The objective is a sequential search strategy that minimizes the total expected cost, incurred by all the processes during the detection process, under reliability constraints. We develop a simple closed-loop policy (i.e., decisions depend on past observations and actions) for the anomaly detection problem. Asymptotic optimality of the proposed policy is shown when a single process is observed at a time and strong performance are demonstrated by simulation examples under multi-processes probing.