We consider a variant of the vehicle routing problems with time windows, where the objective includes the inconvenience cost modeled by a convex function on each node. We formulate this mixed integer convex program using a novel set partitioning formulation, by considering all combinations of routes and block structures over the routes. We apply a branch-and-price framework to solve this formulation. To solve the pricing problem, we leverage a recursive algorithm that solves the optimal service time scheduling problem given a fixed route, and develop a labeling algorithm with a dominance rule that is easy to check. In our computational study, we show the effectiveness of the labeling algorithm compared to state-of-the-art mixed integer convex programming solvers on an extensive set of vehicle routing problems with time windows and quadratic inconvenience costs.
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