I am currently reading a paper about vehicle routing problem with time windows and when formulating the main goals and restrictions the following is stated:
Objective: Minimize the number of vehicles and the total travel distance.
Assume N is a number of customers (1, 2, ..., n) that need to be serviced.
$c_{ij}$ - the transportation cost from the customer i to j.
$x_{ijk}$ - a variable taking value of 1 if the vehicle k is coming from the customer i to the customer j, and 0 if otherwise.
Objective function:
$$Z = \sum_{k\in V}\sum_{i\in N}\sum_{j\in N} c_{ij}x_{ijk} \to min$$
One of the restrictions is the following:
$$\sum_{j \in N}x_{0jk}=1,\forall k \in V$$
Doesn't the previous restriction imply that every vehicle will move from $0$ (depot) to a customer $j$?
If that was the case the number of vehicles wouldn't be minimized because it would force them all to be in use.
Yes, those constraints force every vehicle to be used.
One approach to minimize the number of vehicles is to start with $|V|=1$ and try to solve this problem for fixed $|V|$. Increment $|V|$ until the problem is feasible.
Another approach is to introduce a binary variable $y_k$, replace the RHS of constraints (4) and (5) with $y_k$, and minimize $\sum_k y_k$. You can also tighten constraint (3) by replacing the RHS with $q\cdot y_k$.