Hello Logapriya,
There are various universities and institutes in Chennai that offer a course in Hospital Management/Health Service Management. Some of these Institutions are as follows:
Apollo Institute of Hospital Management and Allied Sciences
Amrita International Institute of Hotel Management
Saveetha Institute of Medical and Technical Sciences
St. Peter’s Institute of Higher Education and Research
Rajaran Academy  
Now answering the second part of your question; the University of Madras does not have an MBA specialization in Hospital Management/Health Service Management.
Hope this was helpful!

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