How to interpret sub tour elimination constraint in travelling salesman problem in cplex?

Question

I have written the following code:

How to interpret auxiliary constraints and sub tour elimination constraints in the following formulation?

Answer 1

The interpretation of the MTZ subtour elimination approach is actually quite simple. Assign numbers u(i) to each node in the order in which they are visited. The constraint

u(i) - u(j) + n x(i,j) <= n - 1

can be interpreted as

u(j) >= u(i) + 1 - M*(1-x(i,j))

or

x(i,j) = 1 ==>   u(j) >= u(i) + 1

If you have a subtour say 2-3-2 this cannot hold:

2-3 means u(3) >= u(2)+1
3-2 means u(2) >= u(3)+1 

As the whole tour should be allowed, we just drop the first node from this check.