Linear Complexity Algorithms for Visually Appealing Routes in the Vehicle Routing Problem

Abstract

The vehicle routing problem consists of finding cost-effective routes for fleets of trucks to serve customers. Logistics managers often prefer routes to also be “visually appealing” because of the better flexibility they provide in coping with small alterations, required due to last-minute or unforeseen events. Compactness of the routes is a key desirable feature, and it can be accomplished by minimizing the area enclosed by the routes. A common approach in the literature relies on imposing a penalty on the area of the convex hull. We propose to use new features which are well correlated with the convex hull area but are significantly easier to implement, having O(n) computational complexity instead of \(O(n \hbox {log} n)\). By accepting only a minimal loss of quality with respect to a primary objective function, like the routes’ total length, we show that area-type penalties can be effective in providing good guidance: construction methods which are based on insertion are naturally steered towards routes displaying more attractive shapes. Used in conjunction with an adaptive large neighbourhood search, our new proposed features lead to routes that exhibit similar compactness compared to using a convex hull area penalty. We also achieve good separation between routes.