Lower bounds for the Periodic Capacitated Arc Routing Problem Feng CHU, Nacima LABADI (corresponding author), Christian PRINS LOSI, University of Technology of Troyes, BP 2060, F-10010 Troyes Cedex, France fax.: +(33) 3 25 71 56 49, e-mail: {Feng.Chu, Nacima.Labadi, Christian.Prins}@utt.fr Abstract The Periodic Capacitated Arc Routing Problem (PCARP) generalizes the well-known CARP by extending the planning horizon to P days. PCARP involves the assignment of tasks to days and routing of vehicles, under frequency constraints, so that the total cost over the whole horizon is minimal. This new problem is encountered in various applications such as waste collection, mail delivery, etc. In this paper, this very hard problem is studied. Two lower bounds based on a graph transformation are proposed. The results are compared to the best existing upper bounds. 1. Introduction The CARP considers an undirected graph in which a fleet of identical vehicles with limited capacity is based at a depot node. Each edge has a non-negative traversal cost and can be traversed any number of times. A subset of edges, the tasks, must be traversed to be serviced by a vehicle. Each task has a non- negative weight or demand. The CARP consists of determining a set of feasible vehicle trips that minimizes the total cost of traversed edges. Each trip starts and ends at the depot, each edge is serviced by one single vehicle and the total demand serviced by any trip must not exceed vehicle capacity. The CARP is NP-hard and the existing exact methods [2] are limited to 20 edges. So, larger instances must be tackled in practice using constructive heuristics like Path-Scanning [3] and Augment-Merge [4], or by recent metaheuristics like the tabu search method of Hertz et al. [5] and the hybrid genetic algorithm proposed by Lacomme et al [6], which is currently the most efficient solution method. The Periodic CARP generalizes the classical CARP by extending the planning to P days. Over these P days, each task must be served a given number of times. The goal is to assign tasks to days and to compute a set of feasible trips minimizing the total cost over the whole horizon. This new problem was introduced recently by Lacomme et al [7]. Our paper is organised as follows. Section 2 describes the PCARP. Lower bounds are presented in section 3. The fourth and last section compares our lower bounds with the best-known upper bounds and mentions some concluding remarks. 2. Problem description The PCARP is defined on a multi-period horizon H of P periods (“days”) and an undirected connected network G = (X,E) where X is a set of n nodes (crossroads) and E a set of m edges (streets). Node 1 is a depot that contains a fleet of V identical vehicles with limited capacity W. The number of vehicles V is a decision variable. Each edge e has a traversal cost C e and a daily production q ep for each day p. A subset R of t edges, the tasks, must be serviced. Each task e has a frequency f e (number of services required in H) and a set of allowed day combinations comb(e). A combination is a set of f e possible service days. The demands are known for each task e, each combination of comb(e) and each day of this combination. The goal is to determine one day combination for each task and one set of vehicle trips for each day to minimize the total cost of the trips, subject to the following constraints: a) each task e is serviced f e times over the horizon, but at most once in each day, b) each trip starts and ends at the depot, c) a task assigned to a day is serviced by one single trip in that day, and d) vehicle capacity is respected. The PCARP is obviously NP-Hard since it boils down to the CARP if P = 1. In a first study [1], we designed the first mathematical model and the first heuristics to the problem. The heuristics were evaluated on our benchmark obtained by converting the well-known DeArmon instances. The results show that the best heuristic is that we called LBH (Lower Bound Heuristic). It is a two-phase heuristic in which we assign tasks to day combinations then compute the CARP solutions for the P days thanks to CARP GA of Lacomme et al. The first phase starts with P empty clusters, one for each non-idle day in the horizon. It uses a good lower bound for the CARP based on matching techniques and developed by Pearn, see [8] for details. At each iteration, all the tasks not yet served and their allowed day combinations are examined. The task which increases the less possible the sum of the CARP lower bounds is then added to the