LOSI, Université de Technologie de Troyes, BP 2060, 10010 Troyes Cedex, France
Phone: +33 3 25 71 56 26 – Fax: +33 3 25 71 56 49 – Web: http://www-losi.utt.fr
Research Report
LOSI-2001-01
Competitive Memetic Algorithms
for Arc Routing Problems
Philippe LACOMME
Christian PRINS
Wahiba RAMDANE-CHERIF
First version: 07/11/2001 – Revised: 21/04/2002
Submitted for publication to Annals of Operations Research
Competitive memetic algorithms for arc routing problems – P. Lacomme et al.
RR LOSI-2001-01 – Page 1
Competitive Memetic Algorithms
for Arc Routing Problems
Philippe LACOMME
LIMOS, Université Blaise Pascal
Campus Universitaire des Cézeaux
63177 Aubière Cedex, France
e-mail: lacomme@sp.isima.fr
Christian PRINS (corresponding author)
LOSI, Université de Technologie de Troyes
12 Rue Marie Curie, BP 2060
10010 Troyes Cedex, France
e-mail: prins@utt.fr, phone (33) 3 25 71 56 41, fax (33) 3 25 71 56 49
Wahiba RAMDANE-CHERIF
LOSI, Université de Technologie de Troyes
12 Rue Marie Curie, BP 2060
10010 Troyes Cedex, France
ramdane@utt.fr
Competitive memetic algorithms for arc routing problems – P. Lacomme et al.
RR LOSI-2001-01 – Page 2
Abstract
The Capacitated Arc Routing Problem or CARP arises in applications like waste collection or
winter gritting. Metaheuristics are tools of choice for solving large instances of this NP-hard
problem. The paper presents basic components that can be combined into powerful memetic
algorithms (MAs) for solving an extended version of the CARP (ECARP). The best resulting
MA outperforms all known heuristics on three sets of benchmark files containing in total 81
instances with up to 140 nodes and 190 edges. In particular, one open instance is broken by
reaching a tight lower bound designed by Belenguer and Benavent, 26 best-known solutions
are improved, and all other best-known solutions are retrieved.
Keywords
: Capacitated Arc Routing Problem, CARP, metaheuristic, memetic algorithm.
Competitive memetic algorithms for arc routing problems – P. Lacomme et al.
RR LOSI-2001-01 – Page 3
1. INTRODUCTION
Contrary to the well-known Vehicle Routing Problem (VRP), in which goods must be
delivered to client nodes in a network, the Capacitated Arc Routing Problem (CARP) consists
of visiting a subset of edges. CARP applications include for instance urban waste collection,
winter gritting and inspection of power lines. From now on, to make the paper more concrete
without loss of generality, examples are inspired by municipal refuse collection.
The basic CARP of literature tackles undirected networks. Each edge models a two-way street
whose both sides are treated in parallel and in any direction (bilateral collection), a common
practice in residential areas with narrow streets. A fleet of identical vehicles of limited
capacity is based at a depot node. Each edge can be traversed any number of times, with a
known traversal cost. Some edges are required, i.e., they have a non-zero demand (amount of
waste) to be collected by a vehicle. The CARP consists of determining a set of vehicle trips of
minimum total cost, such that each trip starts and ends at the depot, each required edge is
serviced by one single trip, and the total demand processed by a trip fits vehicle capacity.
The CARP is NP-hard, even in the single-vehicle case called Rural Postman Problem (RPP).
Since exact methods are still limited to 20-30 edges (Hirabayashi et al., 1992), heuristics are
required for solving large instances, e.g. Augment-Merge (Golden and Wong, 1981), Path-
Scanning (Golden et al., 1983), Construct-and-strike (Pearn's improved version, 1989),
Augment-Insert (Pearn, 1991) and Ulusoy's tour splitting algorithm (1985).
The first metaheuristic for the CARP, a simulated annealing procedure, was designed by
Eglese in 1994 for solving a winter gritting problem. Several tabu search (TS) algorithms are
also available, both for particular cases like the undirected RPP (Hertz et al., 1999) or the
mixed RPP (Corberan et al., 2000) and for the CARP itself (Eglese, 1996; Hertz et al., 2000).
All these metaheuristics and classical heuristics may be evaluated thanks to lower bounds,
generally based on linear programming formulations, see Benavent et al. (1992), Belenguer
and Benavent (1998), Amberg and Vo
β
(2001). On most instances, the best-known lower
bound is obtained by a cutting-plane algorithm (Belenguer and Benavent, to appear).
Compared to the VRP, the CARP has been relatively neglected for a long time but it attracts
more and more researchers: successful applications are reported (Mourão and Almeida, 2000)
and extensions are now investigated, for instance the directed RPP with turn penalties
(Benavent and Soler, 1999), the multi-depot CARP (Amberg et al., 2000) and the CARP with
intermediate facilities (Ghianni et al., 2001).
This paper presents powerful memetic algorithms (MAs) for an extended CARP. Compared
to an earlier GA for the mixed CARP with forbidden turns (Lacomme et al., 2001), they
handle other objectives, like the makespan or the number of vehicles used, and extensions like
parallel arcs, turn penalties, a maximum trip length and a limited fleet. Several possible bricks
for each MA step are designed with a low complexity and tested, e.g. a generational approach
and a partial replacement procedure. The best resulting MA is twice faster, it improves 26
best-known solutions and tackles large instances with 140 nodes and 190 edges.
The extended problem (ECARP) is presented in section 2. Three classical constructive
heuristics are extended to the ECARP in section 3 to provide good initial solutions. Section 4
describes possible components for each step of memetic algorithms. Section 5 is devoted to
computational evaluations: the best MA structure is defined after a preliminary testing and
results are reported for three sets of benchmark instances.
Competitive memetic algorithms for arc routing problems – P. Lacomme et al.
RR LOSI-2001-01 – Page 4
2. EXTENDED CARP MODEL (ECARP)
2.1 Extensions considered and street modelling
For the sake of clarity, this subsection presents without mathematical symbols our extended
problem and the modelling technique for the streets of a real network. Subsections 2.2 to 2.4
are respectively devoted to the required notation, to some complications raised by forbidden
turns, and to the representation of solutions. The ECARP tackles the following extensions:
a)
mixed multigraph with two kinds of links (edges and arcs) and parallel links,
b)
two distinct costs per link (deadheading and collecting),
c)
prohibited turns (e.g., U-turns) and turn penalties (e.g., to penalize left turns)
d)
maximum trip length (an upper limit on the cost of any trip).
Like in the basic CARP, the depot is unique, the fleet is homogeneous, and no split collection
is allowed. The number of vehicles is a decision variable. To ensure the existence of feasible
solutions, the maximum trip length allows a vehicle to reach any required link, collect it, and
return to the depot. The cost of a trip comprises collecting costs (for each link collected) and
deadheading costs (for each link traversed without collection), see 2.4 for a formula. The goal
is to find a set of trips of minimum total cost, covering all required links.
A mixed graph allows to model non-required streets and three kinds of required streets. A
non-required street is modelled either as one arc (one-way street) or two opposite arcs (two-
way streets). The three types of required streets are: i) two-way streets collectable in any
direction (giving one edge), ii) two-way streets with sides collected separately (giving two
opposite arcs) and iii) one-way streets (modelled as one arc). We use a mixed multigraph to
handle more complicated cases: for instance, two parallel arcs can model a one-way street too
wide for bilateral collection and requiring two traversals, one for each side.
To ease algorithmic design, the mixed multigraph is coded as a fully directed graph in which
each edge is replaced by two arcs with opposite directions. Only one of these arcs must be
collected in any feasible solution. To ensure this, both arcs are linked by a pointer variable:
when an algorithm selects one direction, both arcs can be marked "collected".
2.2 Reference list of mathematical symbols
Table 1 provides a quick reference for the remainder of the paper. The mixed multigraph is
coded as a fully directed graph G = (N, A) with m arcs indexed from 1 to m (pairs of nodes are
ambiguous for parallel arcs). The required arcs are the ones with a non-zero demand q(u)
(amount of waste). They have a service cost w(u), generally greater than the deadheading cost
c(u) in waste management applications. By convention, w(u) = 0 if u is not required. All costs
and demands are non-negative integers.
As explained in 2.1, a pointer inv is used to link two arcs u and v coding an edge. In that case,
inv(u) = v, inv(v) = u and edge data are copied on each arc: q(u) = q(v), c(u) = c(v) and
w(u) = w(v). If an arc u is required but does not code an edge, or if it is not required, then
inv(u) = 0. We call tasks the
τ
required links in the mixed multigraph. They comprise
ε
edge-
tasks and
α
arc-tasks. Since each edge-task is coded as two arcs in A, the number of required
arcs in A is
ρ
= 2
ε
+
α
.
τ
and
ρ
have an impact on the complexity of our algorithms.
