Assignment Problem With Multiple Assignments Meaning

The assignment problem is defined as follows:

There are a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment. It is required to perform all tasks by assigning exactly one task to each agent in such a way that the total cost of the assignment is minimized.

The number of tasks is larger than the number of agents.

My problem statement though imposes an additional constraint on the above.

Each task belongs to exactly one 'category'. Each 'category' has an associated maximum number of tasks that can be assigned. Enforce this constraint on the earlier definition.

For a layman's example, consider this -

A restaurant serves n customers (agents), and has on it's menu m possible dishes (tasks), with m > n. Each customer gives his preference for each of the m dishes, which is the cost for this particular assignment problem. Find a solution which minimizes cost i.e. which gives each customer a dish that is as high on their preference as possible.

Additionally, each dish belong to a certain cuisine (category). The restaurant can only make a certain number of dishes per cuisine. Enforce this constraint on the problem above.

I understand that this is a very specific problem, but any help would be appreciated.

I am currently solving the first part of the problem using the Hungarian Algorithm for assignment.


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