I am planning to conduct a simple random sampling for a population of $N$ patients with index $1,...N$. I have enough money to interview $n$ patients (sample size=$n$). In order to determine the sample patients, I first generate $n$ random integers(with replacement) and ask the patients with such index whether they would like to join the interview. However, due to some reasons, several patients (non-responder) were not able to join.
My question is: what shall I do if in this case I have patients who can not join, but I also want to maximize the use of $n$ samples?
I can think of:
If 3 patients can not join, re-generate 3 random integers and assign three new patients. Repeat this process until all patients are willing to join. However, I will still record information from the 3 non-responders, and the total sample size is $n+3$. The non-responder's information will be analyzed later to see if there is any bias.
Is this method reasonable?