Diana and dessouky 2004 have applied a parallel

Diana and Dessouky ( 2004 ) have applied a parallel regret based insertion procedure to solve
a version of the DARP in which the objective is a weighted sum of distance, excess ride
time over direct travel time, and vehicle idle time. They have reported results on instances
of sizes 500 and 1000.
In 2006, four heuristics were published on the multi-vehicle DARP. In the paper by
Rekiek et al. ( 2006 ), the main objective is the minimization of the number of vehicles used.

38 Ann Oper Res (2007) 153: 29–46
The authors propose a genetic algorithm for the clustering phase and an insertion mecha-
nism for the routing phase. They report good results on data provided by the City of Brussels
( 100 ≤ n ≤ 164 ) . Xiang et al. ( 2006 ) solve an elaborate version of the DARP in which the
objective is the minimization of a combination of vehicle fixed costs, vehicle variable costs,
driver costs, waiting time, and service time under several operating constraints. Insertions
and inter-route exchanges are used to construct the routes. The authors introduce an element
of diversification in their search mechanism by using a secondary objective function focused
on idle times. Instances containing between 50 and 2000 requests were solved. The objec-
tive considered by Wong and Bell ( 2006 ) is the minimization of a linear combination of
total operating time, passenger ride time and taxi cost for unassigned requests. The authors
work with several vehicle types and maximum route durations. Some vehicles are equipped
for wheelchair access but the capacities reserved for wheelchair users and non-wheelchair
users are not substitutable. The authors propose a parallel insertion procedure. Users are first
ranked according to an index that measures the difficulty and inconvenience caused to other
requests when they are inserted into a route. The insertions are then made by considering
the most difficult requests first. This is followed by a post-optimization phase consisting of
trip reinsertions in other routes and trip exchanges. The algorithm was tested on artificial
instances involving 150 requests. Wolfler Calvo and Colorni ( 2006 ) have devised a heuristic
for a version of the DARP in which the number of available vehicles is fixed and windows
are imposed on pickup and delivery times. A hierarchical objective function is used: the
algorithm first attempts to service as many users as possible and then minimizes user in-
convenience expressed as the sum of waiting time and excess ride time. The heuristic first
constructs a set of m routes and a number of subtours by solving an assignment problem.
A routing phase is then performed to insert the subtours in the m routes and to resequence
the vertices within the routes. Tests were carried out on instances involving between 10 and
180 users.
Two exact branch-and-cut algorithms were recently proposed for the version of the
DARP considered by Cordeau and Laporte ( 2003a ). Cordeau ( 2006 ) and Ropke et al. ( 2007 )
both formulate the problem as an integer linear program (with some continuous variables in
the first case). These two formulations are provided in Sects. 3.1 and 3.2 , respectively. Sev-
eral families of valid inequalities are also proposed for each model. Initially, the models are
solved by relaxing some of the constraints. During the branching process, separation algo-
rithms are applied to identify violated constraints among those that were initially relaxed
or among the valid inequalities. These constraints are then introduced and the process ends