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In this research we introduce Local Area (LA) routes for improving theefficiency and tightness of column generation (CG) methods for solving vehiclerouting problems (VRP). LA-routes rely on pre-computing the lowest costelementary sub-route (called an LA-arc) for each tuple consisting of thefollowing: (1) a (first) customer where the LA-arc begins, (2) a distantcustomer (from the first) where the LA-arc ends, and (3) a set of intermediatecustomers near the first customer. LA-routes are constructed by concatenatingLA-arcs where the final customer in a given LA-arc is the first customer in thesubsequent LA-arc. A Decremental State Space Relaxation method is applied overLA-routes to construct the lowest reduced cost elementary route during thepricing step of CG. LA-route based solvers can be used to efficiently tightenthe standard set cover VRP using a variant of subset row inequalities, which donot alter the structure of pricing. We incorporate LA-arcs into a novel CGstabilization scheme. Specifically each column generated during pricing ismapped to an ordered list of customers consistent with that column. An LA-arcis consistent with an ordering if the first/last customer in the arc comebefore/after all other customers in the LA-arc in the associated orderingrespectively. Each such ordering is then mapped to a multi-graph where nodescorrespond to (customer/demand) and edges correspond to LA-arcs consistent withthat ordering. Hence any path from source to sink on the multi-graph is afeasible elementary route. The ordering for a column places customers spatiallynearby in nearby positions on the ordering so that routes can be generated soas to permit spatially nearby customers to be visited without traveling faraway first. We solve the restricted master problem over these graphs, which hasspecial structure allowing for fast solution.

Local Area Routes and Valid Inequalities for Efficient Vehicle Routing