How do you deal with a person who approaches you and tells you many complaints he has about your department ?

Only one thing, listen to him very carefully and give him an impression that you are here just listen him.
After that note down the issues then go for them if they are controllable and give him a commitment.

always remember that a customer is always right!!!! one thing you have to do is listen and understand his / her complaint.
try to probe more where is the customer coming from.
the reason this people are complaining is because they are frustrated at something and they wants someone who will listen.
then after them bursting there frustrations at you.
that's the time you resolve his her issue.
believe me after listening a customer will be pacified and would understand whatever you explain them.

Understanding these complaints is highly important, if he mentioned problems, i should consider them as problems to solve even from his point of view because his opinion will be similar to others whose think the same.
so these problems must be covered.

If some body come to me for complains about my department ! frist i will listen him care fully i will try to note them if possible but i will keep them in my mind and observe them weather all the complains are really concern to my department of not. it is the time to stay calm and handle the custmer carefully, no need to say who is wrong and who is right while listening or handleng the situation, but try to get solution. try to stay the customer calm the best way to calm the customer is offering some cofee, tea or cold drink then work on complains if these really concern to my department.

1- Listening carefully until the customer finish the complains.

2- Apologize to your customer.

3- Take the complains one by one and start working on it.

4- let the customer feels that you're really care.

5- Thank your customer that he's dealing with your company.

6- After solving the complains, Make sure that the customer is satisfied.