CIMSA 2003 - International Symposium on
Computational Intelligence for Measurement Systems and Applications
Lugano, Switzerland, 29-31 July 2003
Integration of a Robust Shortest Path Algorithm with a Time Dependent Vehicle
Routing Model and Applications
Alberto V. Donati, Roberto Montemanni,
Luca M. Gambardella, Andrea E. Rizzoli
Istituto Dalle Molle di Studi sull'Intelligenza Artificiale (IDSIA)
Galleria 2, 6928 Manno, Switzerland
Phone: +41-(0)91-610-8568, Fax: +41-(0)91-610-8661, Email: alberto@idsia.ch

Abstract – This paper describes a way of combining two
techniques, in order to define a framework that can deal with
the following problem: find an optimized set of routes when the
customers set is a proper subset of an entire network, and
variable traffic conditions have to be taken into account.
This is accomplished on one hand by extending the vehicle
routing problem (VRP) to a time dependent case (TDVRP), on
the other hand by using an appropriate algorithm, the robust
shortest path (RPS) that can provide itineraries when moving to
a location to another, and guarantee good performance under
any possible road network situation.
Once a proper description of the TDVRP model is given, we
discuss the optimization technique, based on the ant colony
system (ACS), and the robust shortest path (RPS) algorithm.
Different ways of integrating these techniques are discussed.
The one presented here consists in using the RPS algorithm for
the calculation of the paths among each pair of customers, so
that this information can to be used by the TDVRP optimization
in a very efficient way.
In the case of a real road network, some tests have been made,
that show that the optimal solutions obtained for the classic
VRP case are sub optimal when considered in a time dependent
context, revealing that the approximation of constant speeds is
sometimes inadequate.

I. INTRODUCTION AND PROBLEM DEFINITION
The vehicle routing problem (VRP) has been largely
studied because of the interest in its applications in logistic
and supply-chains management. Different versions of this
problem have been formulated to take into account
different aspects/issues.
In the Time Dependent Vehicle Routing Problem, TDVRP,
as in the case of the classic Vehicle Routing Problem with
time windows, VRP/TW, a fleet of vehicles of fixed
capacity has to be scheduled to visit a given set of
customers, each by a time no later than the ending time of
the customer’s time window. If the arrival time at the
customer is earlier than the time window, the vehicle will
incur in a wait at the customer’s location. Each customer is
also characterized by a service time, the time to complete
the unload/work. Other assumptions of the problem are
that 1. the quantity requested by a customer must be
delivered in full, and at one time; 2. all tours must
originate and end at the depot, within the depot opening
time; 3. the total quantity delivered in each tour can not
exceed Q. The novelty of the model is that instead of
having fixed travel times, like in the VRP case, the travel
times are dependent on time.
The problem can be represented with an incomplete
directed graph G(V, A), where A is a set of oriented arcs
connecting pairs of nodes, and V is the set of nodes of
which one represent a depot, and the rest the customers.
Each node is characterized by a location, and each
customer
i
c , i=1,..,N has a requested quantity
i
q , a the
delivery time window ],[
iii
ebtw = , and a service
time
i
wt . For the depot the opening and closing
times ],[
co
TT and the fixed truck capacity Q, are given.
For each oriented arc, a speed distribution v(t) is given,
defined on ],[
co
TT , where t is the time when the travel
begins from the starting node.
A feasible solution is a set of routes not violating any of
the described constraints, and visiting all the customers
once. The optimization consists in finding the solution that
minimizes the number of tours and (then) the total
traveling time.
The TDVRP optimization is a combinatorial NP-hard
optimization problem, so an exact approach, if exists, is
often inconvenient if relative short times are available for
computations. Heuristic algorithms can find solutions of
very high quality, close to an exact algorithm, but in a
considerably shorter time.

II. ANT COLONY OPTIMIZATION
Ant Colony Optimization (Dorigo, Di Caro, and
Gambardella in [1]) is based on the idea that a large
number of simple artificial agents are able to build
solutions via low-level based communication, inspired by
the collaborative behavior of a colony of ants. A variety of
ACO algorithms has been proposed to solve NP
optimization problems, and have been successfully applied
to the traveling salesman problem (TSP), the quadratic
assignment problem, graph-coloring problem, job-shop,
flow-shop, sequential ordering, and vehicle routing
problems. As showed by Dorigo, Maniezzo, Colorini in
[2], on the TSP problem, the prototype of the NP-hard