## 1 Introduction  
The Vehicle Routing Problem (\(VRP\)) initially emerged when [PERSON] and [PERSON] formulated and resolved the problem of supplying fuel to service stations around the end of the fifties of the last century [1].  
The \(VRP\) definition states that \(n\) customers with discrete quantities of goods must be served by \(m\) vehicles initially located at a depot. A \(VRP\) is to determine the optimal routes taken by a group of vehicles while serving a group of users. The objective is to minimize the overall transportation cost. The solution to the classical \(VRP\) problem is a set of routes visiting all the customers exactly once that all begin and end in the depot. The transportation cost is improved by reducing the total traveled distance [2].  
The Multi-Trip Vehicle Routing Problem with Time Windows (\(MTVRPTW\)) is a type of the classical Vehicle Routing Problem with Time Windows (\(VRPTW\)) with more than one trip for a vehicle during a workday. A trip is a timed route when more than one route can be allocated to a vehicle. The multi-trip feature is needed when the vehicle fleet size is limited. In this case, a benefit is a reduced number of drivers and vehicles. Besides, in practice, industries cannot provide an unlimited number of vehicles to serve all customers, and they tend to prefer a limited number of vehicles to do more than one trip. Despite its apparent practical relevance, this variant of the classical VRP has not been the subject of a large number of studies. Refs. [3, 4, 5] are a few papers worked on Multi trip vehicle routing problem (\(MTVRP\)).  
Multi-trip vehicle routing problem with a variable number of wagons and time windows defines a variant of the classical vehicle problem in which the capacity of vehicles can be determined given the total demand of the route when a vehicle is prepared to leave the depot. In this situation, one, two, or three wagons can be attached to make a vehicle ready to service the customers. The number of wagons and vehicles is limited, and the vehicle configuration will stay the same during all vehicle trips. This new methodology is suitable to decrease time and cost by reducing the number of vehicles, drivers, and fuel consumption, which is specifically more critical in distributing goods over large distances like two different cities or from large cities to rural areas.  
The main contribution of this work is to introduce a mathematical model for Multi-trip Vehicle Routing Problem with a Variable number of Wagons and Time Windows (MTVRP-VW-TW). We develop a Branch and Price method to find an optimal solution. Column generation is used for each iteration of the branch and pricing. Various subproblems are formed to choose the appropriate column based on the various capacities. We used modified [PERSON]'s instances [6] to test the algorithm. It is the first time the model has been presented and solved with exact methods.  
The rest of the paper is organized as follows. We review the relevant research in Section 2. We give the definition and mathematical model for MTVRP-VW-TW in Section 3. Column generation for MTVRP-VW-TW is presented in Section 4. MTVRP-VW-TW is solved using the branch and price algorithm in Section 5. The detailed experimental study is discussed in Section 6. Section 7 concludes this paper with a discussion of the limitations of this work and possible future extensions.