In the aftermath of the clarifications provided by you, I endorse the suggestions given by the members. You may like to consider the following as well.
Considerations for Employee Sleeping on Duty
Dismissing an employee when caught sleeping on duty, right in the first instance, could be disastrous, as pointed out. You mentioned some tasks were assigned, but they are still of no avail. Do the employees go to sleep after completing the tasks or after ignoring/leaving them incomplete? This should be reprimanded first, after careful consideration.
If the jobs are completed and then the employee goes to sleep, then increase the tasks assigned. Is the sleeping problem confronted with all the employees or a particular employee? If it is all the employees, then it is Management's turn to think of improving the work environment. If it is one or a handful of employees, then they can be given counseling.
Lastly, apart from incomplete tasks, is there any customer complaint? If there is, it should be given priority.
Regards,
V. Raghunathan
Navi Mumbai