Time Constraints: The Cost of Sustainability - Sustainable Urban Logistics: Concepts, Methods and Information Systems - page 167

cost increase of 8 %. Also, the initial situation led to 4 % more CO 2 emissions,
with the softer access time window scenario only caused a 2.5 % increase with
respect to the no-restriction case.

2 The Vehicle Routing Problem with Access Time Window

2.1 Problem Formulation

Operations Research techniques have been increasingly applied over the last years
to deal with planning problems related to urban freight deliveries (Crainic et al.
2004 ). More specifically, the optimization of vehicle routing in cities, due to its
specific characteristics, has usually been associated with the time-dependent
vehicle routing problem (Malandraki and Daskin 1992 ; Donati et al. 2008 ), real-
time dynamic vehicle routing (Gendreau et al. 1999 ; Fleischmann 2004 )ora
combination of both (Chen et al. 2006 ). Also using a case-study analysis, Figliozzi
( 2010 ) assessed the influence of urban congestion on the cost of freight vehicle
tours. Finally, the aforementioned works by Groothedde et al. ( 2003 ), van Rooijen
et al. ( 2007 ), Quak ( 2008 ) and Deflorio et al. ( 2012 ) applied VRPTW techniques to
assess the influence of access time windows, often considering a multi-town
environment.
However, when considering a single town with many customers, a specific
development is required. We will show that the characteristics of the routing
problem that delivery companies have to face when operating in a city with access
time windows does not correspond purely to a classical VRPTW formulation, and
the cost estimations calculated through the application of VRPTW techniques are
approximations to the exact evaluation provided by the Vehicle Routing Problem
with Access Time Windows (VRPATW).
When formulating the VRPATW, we consider our vehicle routing problem
defined on a graph: [N, L], where N is the set of nodes and L is the set of links
communicating them. The set of nodes N contains one node d with a positive level
of supply (depot), a su bset C of nodes with a positive level of demand (customers),
and anoth er subset C of nodes with zero levels of supply and demand, so that
N ¼ C [ C [ d. A number V of vehicles (where V is a variable) will travel
through the graph visiting all the different customers, only one vehicle per cus-
tomer. We do not consider capacity restrictions on vehicles, which is a realistic
assumption in the case of less-than-truckload urban freight deliveries, where
vehicles are rarely full.
The problem is defined inside a predefined time horizon, corresponding to the
day's working hours, and the objective is to minimize the number of vehicles that
need to be used and the cost (in time units) of transporting goods from the dep ot
d to the nodes of C, crossing along the way the necessary nodes of the subset C.