The vehicle routing problem with arrival time diversification on a multigraph

Abstract Efficiency and security are the two major concerns in cash-in-transit transportation planning. Whereas efficiency is generally achieved by finding short routes, security can be improved by generating dissimilar visit patterns. To achieve a good balance between these two objectives, the vehicle routing problem with arrival time diversification (VRPATD) aims to find optimized routing plans, over multiple periods, subject to a minimum difference between visit times at each customer. Since the customer visits are constrained by time windows and no waiting time is allowed en route, the number of feasible solutions is generally limited. To explore a larger set of feasible route options, we propose to consider alternative paths with different distances between visit locations. The resulting multigraph VRPATD better captures the characteristics of urban networks. Moreover, the extra flexibility achieved with the alternative paths helps finding better routing plans while meeting time constraints. To solve this complex problem, we introduce an adaptive large neighborhood search, which exploits piecewise-linear penalty functions for insertion evaluations, efficient local searches, and an adaptive destruction rate. This method produces remarkable results on classical instances for the simple-graph VRPATD. Moreover, our theoretical results and our experiments on real-life instances obtained from an application case in Vienna show that the multigraph problem extension leads to very significant distance savings subject to the same arrival-time diversification constraints.