Ant Colony Optimization for
Real-world Vehicle Routing Problems
Dario Coltorti, dario.coltorti@antoptima.ch, AntOptima SA, Lugano, Switzerland
Andrea E. Rizzoli, andrea.rizzoli@antoptima.ch, AntOptima SA, Lugano, Switzerland
Metaheuristics like ant colony optimization (ACO) can be used to solve
combinatorial optimization problems. In this paper we refer to its suc-
cessful application to the vehicle routing problem (VRP). At the beginning,
we introduce the VRP and some of its variants. The variants of VRP were
designed to reproduce the kind of situations faced in the real-world. Fur-
ther, we introduce the fundamentals of ant colony optimization, and we
present in few words its application to the solution of the VRP. At the end,
we discuss the applications of ACO to a number of real-world problems:
a VRP with time windows for a major supermarket chain in Switzerland; a
VRP with pickup and delivery for a leading distribution company in Italy
and an on-line VRP in the city of Lugano, Switzerland, where clients’ or-
ders arrive during the delivery process.
Introduction
Most logistics problems are particularly challenging as their search space
grows exponentially with the problem dimensions and no efficient algo-
rithms to explore such space are known. For these problems, which are
technically known as NP-hard, the time required to find an optimal solu-
tion might be simply too high for practical purposes.
Heuristics methods have been devised to explore parts of the search
space, concentrating in those parts that appear to be most promising,
thus reducing the time required to obtain a sub-optimal, but still good
enough, solution. A heuristic makes use of peculiar characteristics of a
problem and exploits them to find a solution. Therefore a heuristic has to
be especially devised for each new problem.
A metaheuristic is a set of concepts that can be used to define heuris-
tic methods that can be applied to a wide set of different problems
[16]. Well known examples of metaheuristics include simulated anneal-
ing (SA), tabu search (TS), iterated local search (ILS), evolutionary algo-
rithms (EC), and ant colony optimization (ACO), the subject of this paper.
Ant Colony Optimization (ACO) is based on the observation that ants can
find the optimal path between a food source and their nest exploiting a
mix of probabilistic behavior and pheromone depositing. In fact, in ACO
a set of artificial ants somehow simulate the behavior of real ants; the
artificial ants move on the graph representation of a combinatorial opti-
mization problem and build solutions probabilistically. The probabilities
are biased by artificial pheromones that ants deposit while building so-
lution (for a recent overview of ACO see [3]; for a detailed description
[8]). In this paper we discuss how ACO can be successfully applied to the
solution of real-world vehicle routing problems.
The Vehicle Routing Problem
The vehicle routing problem can be designed as a combinatorial opti-
mization problem: Finding optimal routes for a fleet of vehicles perform-
ing assigned tasks on a number of geographically sectored clients. An
answer to this problem is the best route serving all clients using a fleet of
vehicles, respecting all operational constraints, such as vehicle capacity
and the driver’s maximum working time, and minimizing the total trans-
portation cost.
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EDITORIAL
There are 3 main factors that define and constrain each model of the VRP:
the road network, specifying the relatedness among clients and depots,
the vehicles, transporting goods between clients and depots on the road
network; the clients, which place orders and receive goods.
Joining the various factors of the problem, we can define a whole set of
different VRPs (for a detailed overview of the various VRPs see [20]). All
these variants have been created in order to bring the VRP closer to the
kind of situations faced in the real-world. Table 1 shows some important
VRP starting from basic version, continuing by static case (VRP with time
windows, VRP with time windows and pick-up and delivery constraints)
and finishing by dynamic case (time dependent VRP like on-line VRP).
Optimization Framework Inspired By Ants
Ant colony optimization [5] is a metaheuristic inspired by the observa-
tion, made by ethologists, that ants are able to find the shortest path
to a food source by laying and following chemical trails. The chemi-
cal substance which ants use to communicate information regarding the
shortest path to food is called pheromone. Communicate means that a
moving ant lays some pheromone on the ground, thus marking a path
with a trail of this substance. In the majority of cases an isolated ant
moves randomly and when it discovers a previously laid pheromone trail
it can decide, with high probability, to follow it, thus reinforcing the trail
with its own pheromone. The group behavior that results is a form of
self-organisational process where the more ants follow a trail, the more
attractive for other ants it becomes. The process running by basic rules
is characterised by a positive feedback loop, where the probability with
which an ant chooses a path increases with the number of ants that
previously chose the same path. Other positive characteristics of the
above process are the flexibility (adaptability) and the robustness (sys-
tem doesn’t depend on one ant). This group behavior of ants with its pos-
itive attributes inspired the ACO metaheuristic. The main factors are arti-
ficial ants (called from now on ants), simple computational agents that in-
dividually and iteratively construct solutions on a graph, which has been
modeled depending on the specific problem. A problem solution is an or-
dered sequence of nodes connected by edges visited by exploring ants.
Ants compute a solution in parallel, deploying the search process over
several constructive computational threads. A dynamic memory struc-
ture, inspired by the pheromone laying process, guides the construction
process of each thread.
The memory structure incorporates information on the effectiveness of
previously obtained results. Intermediate partial problem solutions are
seen as states; at each iteration k of the algorithm each ant moves from
state x k (i) to x k+1 (j), enlarging the partial solution from node i adding
node j.
Based on these elements the first ACO algorithm to be proposed was Ant
System (AS) [7]. It is organized in two main stages: construction of a so-
lution, and update of the pheromone trail. Since its publication different
variants have been proposed to improve the solutions of combinatorial
optimization problems: elitist ant system [4], rank-based ant system [1],
and Max – Min ant system [19] are variants, where the algorithm dif-
fers from the original mainly in the pheromone update rule. On the other
hand, extensions of AS display more substantial changes in the algorithm
structure. Ant Colony System (ACS, [6]) is one of them. ACS differs from
AS for a revised rule used in the tour construction algorithm, and for the
use of both local and global updates of the pheromone trails.
ACS has been shown to be very efficient in solving problems of the ve-
hicle routing class, ranging from the static case (VRP with time windows,
and VRP with time windows and pick-up and delivery constraints) to the
dynamic case (on-line VRP). In the next section we describe how ACO has
been applied in a number of cases to solve real world logistic problems.
Major supermarket chains:
Distribution of goods from inventory stores to shops
In this business case one of the major supermarket chains in Switzerland
has the following challenge: Palletized goods must be distributed to more
than 600 stores, all over Switzerland. To replenish their local stocks each
store orders daily quantities of goods, which have to be delivered within
time windows. So each store can plan and allocate efficiently according
to the daily availability of its personnel and the time requested for in-
ventory management tasks. Further there are three types of vehicles:
trucks (capacity: 17 pallets), trucks with trailers (35 pallets), and trac-
tor units with semi-trailers (33 pallets). One practical restriction is the
access of vehicles to the store, which depends on the store location. In
some cases the truck with trailer can leave the trailer at a previous store
and then continue to other less accessible locations. Moreover the num-
ber of vehicles is assumed to be infinite, since transport services can be
purchased on the market according to the needs.
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