## 2 Literature Review
In the first phase, all paths are generated. In the second phase, the problem is solved by column generation. [PERSON] et al. [39] proposed an approach using a pseudo-polynomial model. [PERSON] and [PERSON] [40] presented a branch-price-cut for a multi-trip vehicle routing problem; [PERSON] et al. [41] has two integer programs for the open vehicle routing problem and uses column generation to solve them; [PERSON] et al. [4] gave column generation embedded in branch and price algorithm to solve multi-trip vehicle routing problem with time windows using dynamic programming to generate all non-dominated paths by label correcting algorithm which are used in the subproblem; [PERSON] and [PERSON] [42] presented a branch and price algorithm to solve the multi-trip vehicle routing problem with time windows and driver work hours. [PERSON] et al. [43] used a branch-and-cut-and-price algorithm to solve the multi-trip separate pickup and delivery problem with time windows. A branch-cut-and-price algorithm is developed in [44] for the single and multi-trip two-echelon vehicle routing problem with time windows. A new variant of the multi-trip vehicle routing problem for the case of being in a queue while the unloading capacity is full is presented in [45] which is solved using a branch-and-price-and-cut algorithm. We are unaware of any work on exact methods for the multi-trip vehicle routing problem with time windows and the flexibility of having different wagons attached to service customers, as in this paper. In contrast to the previous works like [3,4,38] which give branch and price algorithms for \(MTVRP\), we formulate a new variant of \(MTVRP\) in which the capacity can be different. This requires a modification of the branch and price algorithms. We have a different master problem compared to the previous work. There are three sub-problems with three different objective functions, and each subproblem is solved by constructing a new route graph based on the capacity of the vehicle in which we look for all the non-dominated tours. Mathematical Model for Multi-Trip Vehicle Routing Problem with a Variable Number of Wagons and Time Windows  
An instance of this problem is defined by a set of customers \(C=\{1,2,\cdots,n\}\), and the depot is represented by the vertices \(0\) and \(n+1\). Depot \(0\) is the start depot, and \(n+1\) is the return depot. The set \(\{0,1,\ldots,n+1\}\) is denoted \(N\) in a complete directed graph \(G=(N,A)\), where \(A\) is a set of arcs \(\{(i,j):i\
eq j,i,j\in N\}\). A traveling time of \(t_{ij}\) is associated with each arc \((i,j)\in A\), which we consider as the distance of two vertices \(i\) and \(j\), where \(i,j\in N\). A fleet of wagons \(W\), with identical capacities \(q\) which can be connected as one wagon, two wagons or three wagons to organize a set of vehicles \(V\). Vehicles in \(V\) are used to serve the customers. \(\left\lvert\mspace{-1.0 mu }\right\rvert\) and \(\left\lvert\mspace{-1.0 mu }\right\rvert\) are the number of vehicles and wagons, respectively. The set of arcs \(A\) represents all the connections between customers and the depot. There are no arcs ending at vertex \(0\) or originating from vertex \(n+1\). A traveling time of \(t_{ij}\) is associated with arc \((i,j)\), where \(i\
eq j\). Any customer \(i\in C\) has a demand \(d_{i}\), a service time \(s_{i}\), and a time window \([a_{i},b_{i}]\), which means that a vehicle must arrive at the customer before \(b_{i}\). If it arrives before the time window opens, it has to wait until \(a_{i}\) to service the customer. The time windows for both depots are assumed to be \([a_{0},b_{0}]\), representing the scheduling horizon. The vehicles may not leave the depot before \(a_{0}\) and must return at the latest time \(b_{0}\). A route of a vehicle is a closed path that starts and ends at the depot. The vehicle starts at the depot, visits several customers in a specific order, and returns to the depot again. At every point on the route, the time windows and capacity constraints are satisfied.