Small Assignment App, Web Based.

Would like to design an app (web based if possible so accessible from multiple computers) that assigned an incoming patient to a physician (multiple physicians working simultaneously). The assignment algorithm is based on the current time (of patient arrival), and previous assignment history. There are simple rules governing the algorithm. If possible, the app should be modifiable such that if physician shift times change (or are different based on day of week), the assignment rules will still be the same but the app will be able to handle these changes (for example: if a physician shift is added or shift hours lengthened, the algorithm will re-distribute patients based on the same rules accordingly). The current algorithm is based in a google spreadsheet (and very clunky)