Adaptive Large Neighborhood Search for the Periodic Capacitated Arc Routing Problem with Inventory Constraints Juan-Pablo Riquelme-Rodríguez and André Langevin Department of Mathematics and Industrial Engineering, École Polytechnique de Montréal, Canada Department of Mathematical and Industrial Engineering, École Polytechnique de Montréal, Interuniversity Research Center on Enterprise Networks, Logistics and Transportation (CIRRELT), Montréal, Canada Michel Gamache Department of Mathematics and Industrial Engineering, École Polytechnique de Montréal, Canada Department of Mathematics and Industrial Engineering, École Polytechnique de Montréal, Research Group in Decision Analysis (GERAD), Montréal, Canada This article describes the problem in which the edges of a network represent customers, and a quantity of material is delivered to them so that each one achieves a desired inventory level while finding the lowest-cost route of delivery. Routing and inventory decisions are made at the same time. An example of an application of this prob- lem is dust suppression in open-pit mines. A fleet of trucks spray water along the roads of a mine. Humid- ity increases the effectiveness of dust-particle retention. Because the level of humidity decreases, replenishment is done periodically. Other examples of applications include dust suppression in forest roads and plants watering in street medians and sidewalks. We develop a mathematical model that combines two objectives: An inventory objective that minimizes the penalty for the lack of humidity and a routing objective that minimizes watering and traversing costs. Due to the complexity of the mathematical model, we developed an adaptive large neighborhood search algorithm that combines several destroy and repair operators dynamically. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 64(2), 125–139 2014 Keywords: periodic capacitated arc routing problem; vehicle routing; adaptive large neighborhood search; open-pit mine Received January 2013; accepted January 2014 Correspondence to: J.-Pablo Riquelme-Rodríguez; e-mail: juan-pablo. riquelme@polymtl.ca Contract grant sponsor: National Council for Science and Technology (CONACYT) of the Government of Mexico DOI 10.1002/net.21562 Published online 1 September 2014 in Wiley Online Library (wileyonlinelibrary.com). © 2014 Wiley Periodicals, Inc. 1. INTRODUCTION An arc routing problem is a routing problem in which the service activity takes place on the arcs of a network [1]. A special case in the category of arc routing problems, is the periodic capacitated arc routing problem (PCARP). This problem was first described in [11] for a garbage collection problem. The vehicles in charge of providing the service to the arcs, in this case, garbage trucks, have a limited capacity. Because garbage accumulates at different rates on different streets, the frequency of service also varies. Thus, some of the streets visited on the first day, may not be visited on subse- quent days. This means that the solution found for one period of time does not apply to the rest of them in a given time hori- zon. Therefore, a suitable solution for the whole time horizon is needed. A periodic problem is characterized by a different frequency of service depending on the customer’s needs and the service spacing between visits. PCARP was shown to be NP-hard [11] because it includes the capacitated arc routing problem (CARP) as a particular case when the number of periods is one. CARP was shown to be NP-hard in [9]. Other applications of PCARP include road monitor- ing [14] and road watering [13]. Mathematical models for PCARP applications were proposed in [3] and [15] for the garbage collection problem and in [17] for the PCARP with irregular services. Because of the complexity of the prob- lem, heuristic algorithms were also proposed to solve large instances. Heuristic methods for the PCARP include the algo- rithm for the periodic rural postman problem [7], a memetic NETWORKS—2014—DOI 10.1002/net