Inventory Routing - Subtour Elimination
$S$ contains every subtour except the tour with the depot. And for each subtour a constraint has to be added.

The meaning of $k=argmax_j\{z_{jt}\}$ has been answered in https://math.stackexchange.com/questions/1290142/explanation-of-textrmargmax-jz-jt-and-how-to-implement-it/1290178#1290178

I solved this with a callback as soon as a subtour is found.