A lower bound for the Node, Edge, and Arc Routing Problem

Abstract

The Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004. This problem generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It captures important aspects of real-life routing problems that were not adequately modeled in previous Vehicle Routing Problem (VRP) variants. The authors also proposed a memetic algorithm procedure and defined a set of test instances called the CBMix benchmark. The NEARP definition and investigation contribute to the development of rich VRPs. In this paper we present the first lower bound procedure for the NEARP. It is a further development of lower bounds for the CARP. We also define two novel sets of test instances to complement the CBMix benchmark. The first is based on well-known CARP instances; the second consists of real life cases of newspaper delivery routing. We provide numerical results in the form of lower and best known upper bounds for all instances of all three benchmarks. For three of the instances, the gap between the upper and lower bound is closed. The average gap is 25.1%. As the lower bound procedure is based on a high quality lower bound procedure for the CARP, and there has been limited work on approximate solution methods for the NEARP, we suspect that the main reason for the rather large gaps is the quality of the upper bound. This fact, and the high industrial relevance of the NEARP, should motivate more research on approximate and exact methods for this important problem. AbstractThe Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004. This problem generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It captures important aspects of real-life routing problems that were not adequately modeled in previous Vehicle Routing Problem (VRP) variants. The authors also proposed a memetic algorithm procedure and defined a set of test instances called the CBMix benchmark. The NEARP definition and investigation contribute to the development of rich VRPs. In this paper we present the first lower bound procedure for the NEARP. It is a further development of lower bounds for the CARP. We also define two novel sets of test instances to complement the CBMix benchmark. The first is based on well-known CARP in-1 stances; the second consists of real life cases of newspaper delivery routing. We provide numerical results in the form of lower and best known upper bounds for all instances of all three benchmarks. For three of the instances, the gap between the upper and lower bound is closed. The average gap is 25.1%. As the lower bound procedure is based on a high quality lower bound procedure for the CARP, and there has been limited work on approximate solution methods for the NEARP, we suspect that the main reason for the rather large gaps is the quality of the upper bound. This fact, and the high industrial relevance of...

An Adaptive Iterated Local Search for the Mixed Capacitated General Routing Problem

“…In each line of Table 4, we give the name of the instance addressed, the number of tasks τ , and the best known lower bound LB, obtained as the maximum value among those presented by Bach et al (2013) and Bosco et al (2013). For MA, we give the solution value z it obtains, and the number of CPU seconds required to run to completion, sec tot .…”

“…Five MCGRP benchmarks were used, namely, CBMix proposed by Prins and Bouchenoua (2005), BHW and DI-NEARP proposed by Bach et al (2013), and mggdb and mgval by Bosco et al (2013). The CBMix benchmark consists of 23 randomly generated instances on mixed graphs that imitate real street networks.…”

“…The resulting problem, denoted the Capacitated General Routing Problem (CGRP), was solved with a route-firstcluster-second heuristic. The algorithm was tested on random test instances inspired from curb-side waste collection in residential areas, and on random instances from the Capacitated Chinese Postman Problem The first lower bounding procedure for the MCGRP was proposed by Bach et al (2013). It was obtained by adapting a procedure originally developed for the CARP by Wøhlk (2006), based on the solution of a matching problem.…”

“…The MCGRP is also of interest in combinatorial optimization, because it generalizes both the CVRP, the CARP and many other routing problems, as described in Section 2. Its resulting combinatorial complexity is, however, very high, and solving it to optimality is a difficult task even for moderate-size instances, see Bach et al (2013) and Bosco et al (2013).…”

The bi-objective mixed capacitated general routing problem with different route balance criteria

“…The first integer programming formulation was defined in Bosco et al (2013) who use a branch-and-cut algorithm to find optimal solutions for small problem instances. Lower bounds for the MCGRP was identified in Bach et al (2013). The MCGRP with stochastic demands is introduced in Beraldi et al (2015).…”