# TSP subtour elimination by assigning distance traveled

Given a set $$S$$ which we need to travel following TSP rules.

I was wondering if this sub tour elimination method is good enough or not?

Let $$b_{i,j}$$ denote edge from $$i$$ to $$j$$ is taken or not and $$d_{i,j} > 0$$ denotes distance from $$i$$ to $$j$$.

\begin{align}\min&\quad\sum_{i,j \in S} d_{i,j} \cdot b_{i,j}\\\text{s.t.}&\quad\sum_{j \in S} b_{j,i} - \sum_{k \in S} b_{i,k} = 0\\&\quad\sum_{j \in S} b_{j,i} = 1\end{align}

Let $$s_0$$ be the start node. Now use a continuous variable $$DS_i$$ to store the distance at node $$i$$, with $$DS_{s_0} = 0$$.

$$\forall j \in S \setminus \{s_0\} \quad DS_{j} = \sum_{i} b_{i,j} \cdot (DS_{i} + d_{i,j})$$

The last constraint eliminates the sub-tour in the path.

My question is how efficient this sub tour elimination constraint is and how to calculate it.