## 4 Column Generation for \(Mtvp\)-\(Vw\)-\(Tw\)
### Solving the Pricing Subproblems
We will extend the model for split delivery with time windows, multiple wagons, and multiple trips per vehicle. The model can be extended to a multi-objective problem as well. In addition to these two types of the \(VRP\)s, the model can be extended to other variants of the \(VRP\)s. We can develop heuristic, exact and approximate approaches to solve the model. Conceptualization, L.K. ; methodology, L.K. ; software, validation, formal analysis L.K. and C.N.F. ; writing--original draft preparation, L.K. ; writing--review and editing, L.K. and C.N.F. All authors have read and agreed to the published version of the manuscript. Funding Funding support from NSERC, Canada, in the form of research assistantships, is gratefully acknowledged. Support for experimentation on a compute cluster from Compute Canada is also acknowledged. Institutional Review Board Statement: Not Applicable. Informed Consent Statement: Not Applicable. Data Availability Statement: [PERSON]'s (100 customers) instances for Euclidean VRPTW from [[https://www.sintef.no/projectweb/top/vrptw/100-customers/](https://www.sintef.no/projectweb/top/vrptw/100-customers/)]([https://www.sintef.no/projectweb/top/vrptw/100-customers/](https://www.sintef.no/projectweb/top/vrptw/100-customers/)) (accessed on 20 August 2022) are modified to evaluate the model. Acknowledgments: The authors would like to thank [PERSON] and [PERSON] for numerous discussions and helpful comments on a draft version of this manuscript. Conflicts of Interest: There is no conflict of interest. ## Abbreviations

The following abbreviations are used in this manuscript: \begin{tabular}{l l} VRP & Vehicle Routing Problem \\ VRPTW & Vehicle Routing Problem with Time Windows \\ MTVRP & Multi-trip vehicle routing problem \\ MTVRPTW & Multi-trip Vehicle Routing Problem with Time Windows \\ MTVRP-VW-TW & Multi-trip VRP with a Variable number of Wagons and Time Windows \\ RMP & Restricted Master Problem \\ RLMP & LP-relaxation of RMP \\ \end{tabular} 

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