My supervisor told me to select an article so that I implement that as my graduate research undertaking. After a year, I am in a deadlock now. Reasons:
the article is abstractly written (apparently, the writer didn't write the article very clearly in order to stop it from being replicated).
After spending a year, you found an article which is not be replicated. Why didn't you choose an article which is written in simple English and lesser complexity, and easy to implement. Many won't agree with your statement on apparently, the writer didn't write the article very clearly in order to stop it from being replicated -- This is a strong statement. What if the other way "You lack serious fundamentals of the methods to understand the article.".
my supervisor is prepared to do nothing. His opinion is it's my job, and I am the only one who should come up with the solution.
Yes, he is right. It is your job to understand the article fully and then present in front of him. Then, ask for advice on further research or possible extension. Or, working on the demerit part of the article. Even, you better take advice on the article itself "whether should you change it now, as it is difficult to follow through?"
Is there any other alternative way to undertake a graduate research rather than implementing a published research article (this form of research seems to be very rigid in nature, as I need to replicate the exact result which has been talked about in the article)?
Different supervisors have different styles of working. From your comment, it seems that you are doing MSc. I feel this kind of research is pretty reasonable, as you would not be knowing a lot of things to continue independently on a modeling. If you have idea, then defend this situation with that idea and start working on the idea. But, be careful! You might fail and at that time your supervisor might step back.
What precautions could have I taken in order to avoid any more deadlock in future (as I am planning to change my graduate school due to livelihood-related issues)?
Just don't follow yourself all the time. Supervisors are rude, selfish and strong at times, but they are there to help the students. Analyzing your present situation, I see that your supervisor is a decent person. "What precautions ... issues)?" -- This is a separate question itself. Ask another question instead.