What would be a good system to use for vacation and personal days for employees, preferably one that would not be abused by said employees?

I work at a 24hr Emergency Veterinary clinic and when someone calls in (for whatever reason) Someone else has to come in to cover that shift and more than likely they are placed in overtime. My senior dosent want employees abusing any personal days they may be given, he does not want to have to pay O/T because someone is not ethical and has a hangover. We do have Vacation Time but he won’t allow that to be used for illnesses. Or he will when pressed about it but really hates to do it. Help me Please!