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TRANSPORTATION SCIENCE
Vol. 38, No. 2, May 2004, pp. 135–148
issn 0041-1655 eissn 1526-5447 04 3802 0135
inf
orms
®
doi 10.1287/trsc.1030.0068
© 2004 INFORMS
Real-Time Multivehicle Truckload Pickup and
Delivery Problems
Jian Yang
Department of Industrial and Manufacturing Engineering, New Jersey Institute of Technology,
Newark, New Jersey 07102, yang@adm.njit.edu
Patrick Jaillet
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, jaillet@mit.edu
Hani Mahmassani
Department of Civil and Environmental Engineering, The University of Maryland,
College Park, Maryland 20742, masmah@umd.edu
I
n this paper we formally introduce a generic real-time multivehicle truckload pickup and delivery problem.
The problem includes the consideration of various costs associated with trucks’ empty travel distances, jobs’
delayed completion times, and job rejections. Although very simple, the problem captures most features of the
operational problem of a real-world trucking fleet that dynamically moves truckloads between different sites
according to customer requests that arrive continuously.
We propose a mixed-integer programming formulation for the offline version of the problem. We then con-
sider and compare five rolling horizon strategies for the real-time version. Two of the policies are based on a
repeated reoptimization of various instances of the offline problem, while the others use simpler local (heuristic)
rules. One of the reoptimization strategies is new, while the other strategies have recently been tested for similar
real-time fleet management problems.
The comparison of the policies is done under a general simulation framework. The analysis is systematic and
considers varying traffic intensities, varying degrees of advance information, and varying degrees of flexibility
for job-rejection decisions. The new reoptimization policy is shown to systematically outperform the others
under all these conditions.
Key words: truckload trucking; vehicle routing; real-time fleet management; intelligent transportation systems
History: Received: December 2000; revisions received: November 2001, January 2002, June 2002,
November 2002; accepted: December 2002.
Introduction
Continuing developments in telecommunication and
information technologies provide unprecedented
opportunities for using real-time information to
enhance the productivity, optimize the performance,
and improve the energy efficiency of the logistics
and transportation sectors. Interest in the develop-
ment of dynamic models of fleet operations and fleet
management systems that are responsive to changes
in demand, traffic network, and other conditions is
emerging in many industries and for a wide variety
of applications. Managing and making use of the vast
quantities of real-time information made available
by navigation technologies, satellite positioning
systems, automatic vehicle identification systems,
and spatial geographic information systems (GIS)
databases require the development of new models
and algorithms.
The area of vehicle routing and scheduling, includ-
ing dynamic vehicle allocation and load assignment
models, has evolved rapidly in the past few years,
both in terms of underlying mathematical models
and actual commercial software tools. While some of
the approaches may well be adaptable to operations
under real-time information availability, underlying
existing formulations do not recognize the possible
additional decisions that become available under real-
time information.
In this paper we formally introduce a generic
real-time multivehicle truckload pickup and delivery
problem, called herafter TPDP. The problem includes
the consideration of various costs associated with
trucks’ empty travel distances, jobs’ delayed com-
pletion times, and job rejections. The TPDP captures
most features of the operational problem of a real-
world trucking fleet that moves truckloads between
different sites according to customer requests that
arrive continuously. On the other hand, the problem
is still a simplification of real-world problems in that
the latter also needs to address issues such as working
135
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
136 Transportation Science 38(2), pp. 135–148, © 2004 INFORMS
hour regulations, getting drivers home, and suitability
of the driver and the equipment for a load. Never-
theless, good solutions for this artificial TPDP should
provide good insights and building blocks for more
realistic real-time pickup and delivery problems.
We propose a mixed-integer programming formu-
lation for the offline version of the problem. We then
consider and compare five rolling horizon strategies
for the real-time version. Two of the policies are based
on a repeated reoptimization of various instances of
the offline problem, while the others use simpler local
(heuristic) rules. One of the reoptimization strategies
is new, while the others have recently been tested
for similar real-time fleet management problems. The
comparison of the policies is done under a general
simulation framework. The analysis is systematic and
considers varying traffic intensities, varying degrees
of advance information, and varying degrees of flexi-
bility for job-rejection decisions.
Before going into more details about the organiza-
tion and content of this paper, let us first provide an
overview of the related existing literature.
Vehicle routing problems (VRPs) are usually con-
cerned with efficiently assigning vehicles to jobs (such
as picking up and/or delivering given loads) in an
appropriate order so that these jobs are completed in
time and vehicles’ capacities are not exceeded. Deter-
ministic and static versions, with all the characteris-
tics of the jobs being known in advance and every
parameter assumed certain, have been widely studied
in the literature. Bodin et al. (1983), Christofides
(1985), Fisher (1995), Golden and Assad (1988), and
Solomon (1987) provide extensive surveys of the
various VRPs and solution techniques. Bienstock et al.
(1993), Bramel and Simchi-Levi (1996, 1997), and
Bramel et al. (1992, 1994), present probabilistic anal-
yses of many heuristics for deterministic and static
VRPs.
Stochastic and static versions of the vehicle routing
problem (SVRP) have also been widely studied.
Several authors have addressed the case in which
loads are random. Golden and Stewart (1978) tackle
problems with Poisson-distributed loads. Golden and
Yee (1979) consider other load distributions and
give theoretical explanations for the relations found
empirically by Golden and Stewart. Stewart (1981)
and Stewart and Golden (1983) formulate SVRP as
a stochastic programming problem with recourse.
Bastian and Rinnooy Kan (1992) show that with
one vehicle and independent, identically distributed
loads, SVRP could be reduced to the time-dependent
traveling salesman problem (Garfinkel 1985). Work in
this direction was also done by Tillman (1969), Dror
and Trudeau (1986), Yee and Golden (1980), Bertsimas
(1992), and Dror et al. (1989). Researchers have further
considered the case in which travel times between
jobs are random. Cook and Russell (1978) examine
a large SVRP with random travel times and random
loads. Berman and Simchi-Levi (1989) examine the
problem of finding the optimal depot in a network
with random travel times.
Some authors have considered cases in which the
number of and possibly the locations of the jobs are
not known in advance but are described instead by
probability distributions. The goal is to find optimal
a priori routes through all jobs and update these
routes at the time the specific subset of jobs to be
served is known. Such a problem, the probabilistic
traveling salesman problem, was first introduced by
Jaillet (1985, 1988), who develops an extensive analy-
sis of the case where all potential jobs have the same
probability to materialize. Jezequel (1985), Rossi and
Gavioli (1987), Bertsimas (1988), and Bertsimas and
Howell (1993) investigate additional theoretical prop-
erties and heuristics for the problem. Berman and
Simchi-Levi (1988) discuss the problem of finding an
optimal depot under a general job-appearing distribu-
tion. Laporte et al. (1994) formulate the problem as an
integer program and solve it using a branch-and-cut
approach.
When information on jobs is gradually known in
the course of the system’s operation, real-time tech-
niques become increasingly important. In his review
of dynamic VRPs, Psaraftis (1988) points out that
very little had been published on real-time VRPs
as opposed to classical VRPs. Powell et al. (1995)
present a survey of dynamic network and routing
models and identify general issues associated with
modeling dynamic problems. (For more recent sur-
veys on dynamic VRPs and related routing problems,
see Psaraftis 1995, Bertsimas and Simchi-Levi 1996,
and Gendreau and Potvin 1998.)
Bertsimas and van Ryzin (1991, 1993a, 1993b) ana-
lyze a dynamic routing problem in the Euclidean
plane with random onsite service times. They use
queuing models to compare the impact of various
dispatching rules on the average time spent by the
customer in the system. They derive the asymptotic
behavior of the optimal system time under heavy
traffic and find several policies that result in asymp-
totic system times that are within constant factors of
that of the optimal one in heavy traffic.
For the more general dynamic VRP with time win-
dows, Gendreau et al. (1999) have proposed a general
heuristic strategy (a continuously running tabu search
attempting to improve on the current best solution,
interrupted by a local search heuristic for insert-
ing newly arrived demand). Their objective takes
into account job rejections, operational cost due to
vehicle travel distances, and cost due to customer
waiting. Ichoua et al. (2000) further consider methods
that allow vehicle diversions for these VRPs with
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
Transportation Science 38(2), pp. 135–148, © 2004 INFORMS 137
time windows. Empirical tests show a reduction in
the number of unserved customers if diversion is
allowed. Due to computational limitations and the
notorious difficulty of the offline VRP with time win-
dows, it is unclear that a reoptimization-based strat-
egy similar to the one proposed in this paper could
also be effective for this problem and improve on this
tabu search procedure.
Closer to the class of problems considered here,
Bookbinder and Sethi (1980), Powell et al. (1984),
Powell (1986, 1987, 1988, 1996), Dejax and Crainic
(1987), and Frantzekakis and Powell (1990) all address
the dynamic vehicle allocation problem (DVA) for
which a fleet of vehicles is assigned to a set of
locations with dynamically occuring demands. In all
these models, both locations and decision epochs are
discrete. Dimensionality causes the models to have
limited time horizons, and they cannot effectively
address the issue of job delays. Most effective DVA
models are of a multistage stochastic programming
type. Frantzekakis and Powell (1990) use linear func-
tions to approximate separable convex recourse objec-
tive functions and solve the problem at each decision
epoch using backward recursion. Powell (1996) shows
that it is advantageous to take forecasted demands
into consideration when deciding on the vehicle loca-
tion assignment, compared to a model that reacts after
new demands have arrived. This, however, presumes
that one can accurately predict future demands.
More recently, Powell et al. (2000a) consider a
dynamic assignment of drivers to known tasks.
Their formulation includes many practical issues and
driver-related constraints and generalizes the offline
version of the problem we consider in this paper.
Two primal-dual iterative methods are developed
to solve the offline problem. Powell et al. (2000b)
implement the previous primal-dual approaches into
a dynamic driver assignment problem where there
are three sources of uncertainty—customer demands,
travel time, and user noncompliance—and compare
these with simpler, nonoptimal local rules. They find
that the increase in future uncertainty may reduce
the benefit of fully reoptimizing the offline problem
each time a new request comes. This contrasts with
some of our findings, which indicate that fully reop-
timizing each time leads to an overall better per-
formance, under our various testing situations. We
should, however, be very cautious in comparing these
results because the problems and the comparison set-
tings are quite different.
Regan et al. (1995, 1996a, 1996b, 1998) evaluate
vehicle diversion as a real-time operational strategy
for similar truckload pickup and delivery problems
and investigate various local rules for the dynamic
assignment of vehicles to loads under real-time infor-
mation. That approach features relatively simple,
easy-to-implement, and fast-to-execute local rules that
might not always take full advantage of the existing
past and present information. The empirical analy-
sis of these local rules is conducted using a lim-
ited exploratory simulation framework, typically with
small fleet sizes and under the objective of mini-
mizing total empty distance. Reoptimization real-time
policies for truckload pickup and delivery problems
are further introduced and tested under a more gen-
eral objective function in Yang et al. (1998).
In this paper, we build on this previous work and
use computer simulation to experimentally identify
and test good strategies under varying situations. The
main contributions of the paper are the introduction
of a new optimization-based policy (OPTUN) for the
TPDP and its comparison with the simple local rules
of Regan et al. (1998) and other strategies introduced
in Yang et al. (1998). The comparison is done under
a general framework in which the objective function
relaxes the hard constraints associated with the deliv-
ery of a job and introduces a penalty function for
delay beyond the due time. The analysis is systematic
and considers the performance of the policies under
varying traffic intensities, varying degrees of advance
information, and varying degrees of flexibility for job-
rejection decisions. The OPTUN policy turns out to be
the best performing policy under all these different
conditions and clearly outperforms the simple local
rules and other strategies.
The paper is organized as follows. In §1, we present
a detailed definition of the problem. In §2, we discuss
formulations for the offline problem corresponding to
our real-time problem. In §3, we discuss the various
policies and present the detailed mechanism of how
a trucking company would operate under these poli-
cies. In §4, we present the details of our simulation
studies. In §5, we present results and conclusions
from the simulation studies. Section 6 concludes the
paper.
1. Problem Statement
Overview
In this stylized problem, we consider a trucking com-
pany with a fleet of K trucks. The company faces a
sequence of future and unknown requests for truck-
load moves, hereafter called jobs, within a predefined
region. Each truck can carry only one job at a time and
cannot serve another job until the current job is deliv-
ered to its final destination. When the request arrives,
the company is given the pickup location, the delivery
location, the earliest pickup time, and the latest deliv-
ery time of the job. The company can either accept or
reject a job request within a small prescribed amount
of time. The revenue generated from a given accepted
job is proportional to the length of the job, defined
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
138 Transportation Science 38(2), pp. 135–148, © 2004 INFORMS
as the distance between its pickup and delivery loca-
tions. Completion beyond the latest delivery time is
allowed but penalized, and the penalty is propor-
tional to both the job’s length and the amount of delay
occured. In case a job request is not accepted, the cost
of rejection is the gross revenue the company would
have otherwise obtained had it accepted the job. Over
the course of serving the sequence of requests, the
company incurs additional operating costs propor-
tional to the empty distance traveled by trucks to
serve the accepted jobs. Finally, we assume that the
trucks all move at the same constant unit speed.
The objective is to find a good strategy for han-
dling this sequence of future unknown requests to
maximize the overall net revenue. The strategy needs
to address job acceptance/rejection decisions in “real
time” as well as job-truck assignment decisions for
currently accepted jobs, not knowing the timing and
characteristics of possible future requests.
Formal Notation and Model Statement
The time evolution of the system is indexed by a con-
tinuous variable t ∈ 0 . Initially, at time t = 0, all K
trucks are empty and idle at a common depot. Job
pickup and delivery locations, as well as truck posi-
tions at any given time t ≥ 0, are assumed to be points
in a bounded region of a metric space. For sim-
plicity, we assume that this space is the Euclidean
plane and that the distance between any two points in
that plane is the Euclidean distance hereafter denoted
D··.
The exogeneous stimulus of the system is provided
by a sequence of job requests. Formally it is rep-
resented by a sequence of increasing real numbers
ARV
i
i≥1
, where
ARV
i
denotes the arrival time of job
i (we assume that jobs are labeled according to their
order of arrival). At the arrival time of job i, its char-
acteristics are then revealed through a 5-tuple I
i
≡
request time earliest pickup
time
earliest delivery
time
latest acceptance
latest delivery
time
RES
i
T
ADV
i
T
SLK
i
T
i
W
ARV
i
τ
AVL
i
τ
DLN
i
τ
t
Figure 1 Illustration of Time Elements in I
i
o
i
d
i
T
ADVAVG
i
T
SLK
i
T
RES
i
with the following defini-
tions, (some also being illustrated in Figure 1):
• o
i
and d
i
are the pickup and delivery locations,
respectively. The corresponding distance between
these two locations, that is, the length of job i,is
denoted W
i
.
• T
ADV
i
measures the time between the arrival
epoch of job i and its earliest pickup time. In other
words, if
AVL
i
is the earliest pickup time, then
AVL
i
=
ARV
i
+ T
ADV
i
.
• T
SLK
i
is the slack time between the earliest pos-
sible and latest allowed delivery time and captures
the tightness of job i’s completion deadline. In other
words, if
DLN
i
is the time of latest delivery, then
DLN
i
=
AVL
i
+ W
i
+ T
SLK
i
.
• Finally, T
RES
i
is the time within which the com-
pany needs to respond to a job request with a
final acceptance or rejection decision. In other words,
the latest time for the trucking company to decide
whether to accept or reject job i is
ARV
i
+ T
RES
i
.
The joint sequence
ARV
i
, I
i
i≥1
completely charac-
terizes the job requests. Let
t
t≥0
be the (informa-
tion) filtration generated by the sequence
ARV
i
, I
i
i≥1
.
Informally,
t
represents the known information up
to time t, as contained in all the past requests j such
that
ARV
j
≤ t.
Facing this sequence of job requests, the company
responds with a series of decisions including job
acceptance/rejection decisions and job-truck assign-
ment decisions on accepted jobs. We call such a series
of decisions a policy or a strategy. We restrict the
policies to be
t
-adapted; that is, any decisions at
time t must depend only on the information up to
time t (decisions are
t
measurable). Because exo-
geneous information is updated only at job-arrival
epochs, a policy can be described in a rolling-horizon
fashion: At any time that is not a job-arrival epoch,
there is a previously agreed assignment plan being
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
Transportation Science 38(2), pp. 135–148, © 2004 INFORMS 139
carried out. At every job-arrival epoch, the previ-
ous plan is interrupted and a new plan is decided
for the time to come. For this reason, we also call a
job-arrival epoch a decision epoch. At every decision
epoch, a myopic policy is to optimize a new assign-
ment plan without recognizing that it may not be fully
carried out because of future unknown requests. We
also require that, at any decision epoch , the new
plan decided by a policy does not change the pre-
vious acceptance/rejection decisions associated with
any job i such that
ARV
i
+ T
RES
i
≤ . Indeed, the
acceptance/rejection decision status of job i becomes
permanent by time
ARV
i
+ T
RES
i
and cannot then be
changed.
Under such a policy , each truck k,1≤ k ≤ K,
is at any time t, either idle, moving empty, or mov-
ing loaded. We formally represent this status with
an integer variable s
k
t with three possible values:
s
k
t = 0 if idle; −1 if moving empty; and +1 if mov-
ing loaded. Let l
k
t be truck k’s current location at
time t (a two-dimensional real-valued vector in case
of cartesian coordinates). Let Q
k
t be the current
(time t) ordered list of noncompleted jobs assigned
to truck k under the last updated assignment plan
associated with the policy . Q
k
t = if and only
if s
k
t = 0. For Q
k
t =, let q
k
t be the first ele-
ment of the ordered list and LQ
k
t be the remain-
ing other elements of the list (i.e., when nonempty,
Q
k
t = q
k
t ∪ LQ
k
t). Finally, let L
TEMP
t be the
current (time t) set of jobs temporarily rejected under
the last updated assignment plan associated with the
policy .
Together with the fact that vehicles move at a
constant unit speed, it should be clear that s
k
t,
l
k
t Q
k
t,1≤ k ≤ K, and L
TEMP
t allow a full
description of the dynamics of the system under pol-
icy . The details are as follows. Assume a sequence
of requests
ARV
i
, I
i
i≥1
and a given policy to serve
these requests. Consider that we are at time t
−
in a
state described by s
k
t
−
l
k
t
−
Q
k
t
−
,1≤ k ≤ K,
and L
TEMP
t
−
.
(1) Assume first that t is a decision epoch, that is,
t =
ARV
j
for a given job j. A new assignment plan
is then made according to the policy . The param-
eters s
k
tl
k
t Q
k
t,1≤ k ≤ K, and L
TEMP
t are
then fully updated according to the specifics of the
policy , which depends only on the past up to
time t.
(2) Assume now that t is not a decision epoch. Let
ARV
>tbe the next job arrival. The dynamics are then
as follows:
• Update on the set of temporarily rejected jobs:
For all j ∈ L
TEMP
t
−
such that t ≤
ARV
j
+ T
RES
j
= t
≤
ARV
, L
TEMP
t
= L
TEMP
t
−
\j.
• Update on the idle vehicles: For any k such
that s
k
t
−
= 0, we have, for t ≤ t
≤
ARV
, s
k
t
= 0,
l
k
t
= l
k
t, and Q
k
t
=.
• Update on the vehicles moving empty: For
any k such that s
k
t
−
=−1, let i be q
k
t
−
. Moving
at unit constant speed, vehicle k would reach the ori-
gin of job i at time t
i
= t + Dl
k
to
i
. So for t ≤
t
≤ min
ARV
t
i
we have s
k
t
=−1, l
k
t
= l
k
t +
o
i
− l
k
tt
− t/t
i
− t, and Q
k
t
= Q
k
t
−
. Then if
t
i
≤
ARV
, s
k
t
i
=+1, l
k
t
i
= o
i
, and Q
k
t
i
= Q
k
t
−
.
Otherwise, at t
=
ARV
we are back in case 1 above.
• Update on the vehicles moving loaded: For
any k such that s
k
t
−
=+1, let i be q
k
t
−
. Moving at
unit constant speed, vehicle k would reach the desti-
nation of job i at time t
i
= t + Dl
k
td
i
. So for t ≤
t
≤ min
ARV
t
i
we have s
k
t
=+1, l
k
t
= l
k
t +
d
i
− l
k
tt
− t/t
i
− t, and Q
k
t
= Q
k
t
−
. Then if
t
i
≤
ARV
, s
k
t
i
= 0ifLQ
k
t
−
=, s
k
t
i
=−1; other-
wise, l
k
t
i
= d
i
, and Q
k
t
i
= Q
k
t
−
\i. Otherwise,
at t
=
ARV
we are back in case 1 above.
We have made no assumption on how to model the
uncertainty associated with the sequence of requests
because we want to devise real-time strategies assum-
ing little, if any, knowledge (deterministic or prob-
abilistic) of the future requests. Of course, present
actions influence the company’s performance in the
future. The difficulty is making decisions based only
on the past and current requests. The approach usu-
ally taken is to assume some probabilistic model of
the future and act on this basis. This is the start-
ing point of the theory of Markov decision pro-
cesses (see, e.g., Heyman and Sobel 1984). Another
approach is to devise and evaluate strategies under
the worst possible scenario using the concept of “com-
petitive analysis,” now well known in the analysis
of online problems and algorithms (see, e.g., Borodin
and El-Yaniv 1998).
In this paper we are taking a middle ground. We
assume that the strategies have to be developed with
no knowledge of the future (with the exception of
one of the proposed strategies, OPTUN, which uses
some minimal probabilistic information on the loca-
tion of job requests, as explained in §3). The analysis
and comparison of the proposed strategies, however,
are performed under some very specific probabilistic
assumptions. Specifically, we consider a probability
space under which is defined a Poisson pro-
cess N
t
t≥0
with intensity . The sequence of job-
arrival epochs
ARV
i
i≥1
corresponds to the Poisson
process arrival times. We also define in this probabil-
ity space a stochastic process I
i
i≥1
with values in
5
describing the sequence of job characteristics.
Under such probabilistic assumptions one can go
further in properly defining an objective function for
the evaluation of strategies. First we define a time-
dependent set of random variables that record the
system’s performance when a certain stationary pol-
icy is adopted. For each vehicle k we let
k
t =
1s
k
t < 0 be a 0–1 random variable indicating
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
140 Transportation Science 38(2), pp. 135–148, © 2004 INFORMS
whether or not truck k is moving empty at time t.
Now let Nt = i
ARV
≤ t be the set of jobs that
have been requested by time t. Let Mt= i
ARV
i
+
T
RES
i
≤ t be the subset of jobs in Nt for which a
final acceptance or rejection decision is mandatory by
time t. Finally, let Rt be the subset of jobs in Nt
that have been fully served by time t. Note that for all
s ≤ s
, Ns⊂ Ns
, Ms⊂ Ms
, and Rs ⊂ Rs
. For
each job i ∈ Mt, we let y
i
t be a 0–1 random vari-
able indicating whether job i has been permanently
rejected. Note that the policy imposes consisten-
cies in the sense that for each job i ∈ Mt we have
y
i
t
= y
i
t for all t
≥ t. For each job i ∈ Rt,welet
COM
i
t ≤ t be the time of completion of job i.
We will assume that as a function of t,
k
t
t
,
y
i
t
t
, and
COM
i
t
t
are well-defined stochastic
processes with right-continuous left-limit sample
paths.
Let us now specify applicable cost parameters.
Let $ be the operational cost per unit distance of
truck-empty movement, and let % be the penalty cost
per unit of time delay and per unit of job length
(5 units of time delay for a job i of length W
i
= 10
costs 50%; i.e., the longer the job the more costly pro-
portionally it is to delay its final delivery).
Because of unit speed, the total empty distance cov-
ered by truck k up to time t is the random variable
t
0
k
s ds. Under policy and up to time t, the
cumulative cost C
t is a random variable defined as
C
t ≡ $
K
k=1
t
0
k
sds+%
i∈Rt
W
i
COM
i
t−
DLN
i
+
+
i∈Mt
W
i
y
i
t' (1)
C
t captures the fleet’s operational cost of empty
movement, the loss of customer goodwill due to
delay, and the loss of revenue from job rejections.
For the second cost term, accounting at time t is
done for completed jobs; for the third term, account-
ing at time t is done for those jobs whose accep-
tance/rejection decisions have been finalized.
We assume that all the policies under considera-
tion in this paper are stable ergodic policies, by which
we mean that there exists a constant c
such that
lim
t→
C
t
t
= c
(a.s.) and lim
t→
E
C
t
t
= c
' (2)
Mathematically the overall optimization problem is
to find a
opt
among the set of all stable ergodic poli-
cies * such that
c
opt
= inf
∈*
c
'
This is how an optimal policy is defined in this paper.
One can define an equivalent optimization prob-
lem. For any integer n, let
∗
n
= inft for all 1 ≤
i ≤ n, y
i
t = 1ori ∈ Rt.
∗
n
is the smallest time t
by which all n first jobs have either been served or
rejected.) For stable ergodic policies as defined above,
we then have
lim
n→
C
∗
n
n
= c
a
(a.s.) and
lim
n→
E
C
∗
n
n
= c
a
' (3)
Note that for any ∈ * the two constants c
and c
a
defined in (2) and (3) respectively are such that c
a
=
c
/. The two problems inf
∈*
c
and inf
∈*
c
a
are
thus equivalent. The constant c
a
can be interpreted as
the long-run average cost per requested job.
For each stable ergodic policy introduced in this
paper, we numerically estimate the constant c
a
by
considering a finite approximation. More precisely, for
a large enough n, we will assume that the following
measure
E
C
∗
n
n
(4)
is a good approximation of the constant c
a
to mini-
mize. It is this approximate objective function (4) that
we numerically estimate via our simulation experi-
ments in §5 and that we call AvgCost. Section 4 pre-
cisely describes how AvgCost relates to (4).
Before we can describe the five proposed online
policies for TPDP, it is important to first understand
the following corresponding offline problem: Given a
set of trucks and known jobs, find an optimal plan to
serve these jobs, assuming no future requests. Even
though we introduce the offline problem as a prob-
lem being repeatedly solved at decision epochs by a
myopic policy for the real-time problem, it models a
specific and interesting problem in its own right. This
is the subject of the next section.
2. The Offline Problem
In this offline problem we consider a problem with K
trucks. We assume that truck k is first available at
time
0
k
and at location l
k
. We assume that there
are N known jobs, each being characterized by an
arrival epoch and a 5-tuple I as described above. For
notational simplicity, we let D
k
0i
be the distance from
truck k’s location l
k
to job i’s pickup location and D
ij
be the distance from job i’s delivery location to job j’s
pickup location. Out of the N jobs, we assume that
a subset A of these has to be served. The other jobs
could be rejected, if it is economically optimal to do
so. For an arbitrary choice of A the given offline prob-
lem could be infeasible.
Note that when the offline problem is the problem
solved at a decision epoch in an online strategy as
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
Transportation Science 38(2), pp. 135–148, © 2004 INFORMS 141
described in the previous section,
0
k
and l
k
are either
the current time and location of truck k if it is idle
or moving empty, or the time and location at which
truck k will finish its current job if it is moving loaded.
Also, the N jobs would be those in the real-time prob-
lem that are already known at t and have neither
been picked up nor been permanently rejected yet.
In this setting some jobs may have been permanently
accepted and form the elements of the set A. Since the
offline problem is always called at the arrival epoch of
a new job and we can always reject the new job and
keep the previous plan, introducing this set A does
not make the offline problem infeasible.
We have looked at two equivalent formulations for
the problem. The first formulation is of a multicom-
modity network-flow type and has been inspired by
the work of Desrochers et al. (1988). All the nodes
except for one dummy node, node 0, represent jobs.
All the arcs except for those linking job nodes to the
dummy node represent possible connections in real
services. A truck’s route is represented by a flow unit
from the dummy node, through some job nodes, and
then back to the dummy node. Empirically, this first
formulation is not as competitive as the second one,
so we omit going into details here.
In the chosen formulation, we model the problem
as an assignment problem with timing constraints.
The assignment problem, in turn, consists of finding
a least-cost set of cycles going through all the nodes
of 1'''KK+ 1'''K+ N, where node k for k =
1'''K corresponds to truck k and node K + i for
i = 1'''N corresponds to job i. In the formulation,
we use binary variable x
uv
for u v = 1'''K+ N to
indicate whether arc uv is selected in one of the
cycles. In the truck-job terminology, x
kK+i
indicates
whether truck k first serves job i, x
K+i K+j
indicates
whether there is a truck that serves jobs i and j con-
secutively, x
kk
= 1 means that truck k serves no job,
and x
K+i K+i
= 1 means that job i is rejected. We also
use continuous variables
PICK
i
and 1
i
to represent the
pickup time and amount of delay of job i, respectively.
The timing constraints presented below prevent
cycles from being formed with job nodes only. As
a result, there is a clear interpretation of a feasible
cycle using our truck-job terminology. For instance, if
a cycle goes as 1, K + 1, K + 2, 2, K + 3, K + 4, K + 5, 1,
the interpretation is that truck 1 serves jobs 1 and 2
and truck 2 serves jobs 3, 4, and 5. The mixed-integer
programming formulation is presented below:
min $
K
k=1
N
i=1
D
k
0i
x
kK+i
+
N
i=1
N
j=1j=i
D
ij
x
K+i K+j
+ %
N
i=1
W
i
1
i
+
N
i=1
W
i
x
K+i K+i
subject to
K+N
v=1
x
uv
= 1 ∀u = 1'''K+ N (5)
K+N
v=1
x
vu
= 1 ∀u = 1'''K+ N (6)
x
uv
= 0 1 ∀u v = 1'''K+ N (7)
−
K
k=1
D
k
0i
+
0
k
x
kK+i
+
PICK
i
≥ 0 ∀i = 1'''N (8)
−Tx
K+i K+j
−
PICK
i
+
PICK
j
≥−T + W
i
+ D
ij
∀ij = 1'''N i= j (9)
PICK
i
≥
AVL
i
∀i = 1'''N (10)
1
i
−
PICK
i
≥ W
i
−
DLN
i
∀i = 1'''N (11)
1
i
≥ 0 ∀i = 1'''N (12)
x
K+i K+i
= 0 ∀i ∈ A' (13)
Constraints (5), (6), and (7) are classical assignment
constraints. Constraints (8), (9), and (10) are the tim-
ing constraints, with T a large number. Constraints (8)
ensure that truck k arrives at the pickup location of
job i after D
k
0i
+
0
k
if i is the first job being served
by k. Constraints (9) ensures that the truck arrives at
the pickup location of job j at least W
i
+ D
ij
amount
of time after reaching job i’s pickup location if j
is to be served after i. Because T is large enough,
when x
K+i K+j
= 0, the constraints are nonrestrictive.
We note that constraints (8) and (9) are those that pre-
vent cycles without a truck. Constraints (10) simply
enforce that a job’s pickup time is no earlier than its
earliest pickup time. Constraints (11) and (12) specify
ranges of the amount of delay. Constraints (13) pre-
vent rejection of jobs in the specified subset A.
3. Real-Time Policies
In all the policies considered in this paper, a truck
remains idle at the destination of its last job when not
assigned to a new job. Under any given plan, a truck k
is assigned a queue of jobs that has been (permanently
or tentatively) accepted. If truck k is currently idle, the
queue is empty. If truck k is currently moving empty,
the queue has at least one job waiting and truck k is
moving toward the pickup location of the first waiting
job. Finally, if truck k is moving loaded, the queue
contains at least the job being currently served. For
all policies, queues are nonpreemptive: Once a job is
picked up, it is delivered without disruption.
3.1. A Simple Benchmark Policy
The first policy considered in this paper, BENCH,
reflects what a company might do without the aid
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
142 Transportation Science 38(2), pp. 135–148, © 2004 INFORMS
of sophisticated decision-support systems. At a job-
arrival epoch, BENCH decides whether this new job is
accepted and, if accepted, assigns it to the queue of a
specific vehicle k. These decisions are permanent and
are based on a sequential evaluation. For each truck k,
BENCH calculates the marginal cost of serving this
new job if inserted at the end of its queue. When all
marginal costs are higher than the cost of rejection,
the job is rejected. Otherwise, it is assigned at the end
of the queue of the truck k with the lowest marginal
cost.
3.2. Advanced Policies
In these policies, initial acceptance/rejection decisions
are not necessarily permanent, and a job being
accepted or rejected at one decision epoch could be
reconsidered before a permanent decision has to be
made (based on the extra time T
RES
). As introduced
in §1, we use a list L
TEMP
t to represent at any time t
the tentatively rejected jobs whose acceptance deci-
sion deadlines have not expired yet.
At a job-arrival (decision) epoch , we first per-
manently remove jobs in L
TEMP
whose response
deadlines have expired (they become permanently
rejected). For all remaining jobs in L
TEMP
, as well as
the current new job (which triggered the current deci-
sion epoch), we need to decide whether to tentatively
accept or reject them. For convenience, we refer to
these jobs as the pending jobs. At the same time, some
waiting jobs in some vehicles’ queue will have passed
their acceptance decision deadlines and hence become
permanently accepted. The other waiting jobs (tenta-
tively accepted) could be potentially rejected as well
at this new decision epoch. Out of the four advanced
policies presented below, the last three will consider
this as an option.
The first two policies are local in the sense that, like
BENCH, they evaluate the insertion of each pending
job, one at a time, into each truck’s queue, also one at
a time. Pending jobs are evaluated in the decreasing
order of their arrival epochs. Each pending job is
either inserted into a particular queue, if the cor-
responding marginal cost is the smallest among all
queues and is smaller than the cost of rejection, or is
tentatively rejected otherwise. In the latter case, the
pending job is added (back) to L
TEMP
.
The two local policies differ in how they consider
insertion of a pending job in a truck’s queue. NS does
not modify the relative ordering of the jobs already in
the queue and only considers all possible insertions
in between these jobs. It also does not consider reject-
ing a tentatively accepted job in a queue while trying
to insert the pending job. On the contrary, SE evalu-
ates all possible orderings of the original waiting jobs
together with the current pending job, and does so
by solving a one-truck instance of the offline problem.
The optimal solution determines whether the current
pending job and previously tentatively accepted jobs
of the queue are accepted and, if accepted, in what
order they should be served. If the pending job or
a previously tentatively accepted job becomes tenta-
tively rejected, it is added to L
TEMP
.
Strategies BENCH, NS, and SE are similar to strate-
gies evaluated previously by Regan et al. (1998), albeit
with very different implementations (in particular,
here we allow rejection of a job based on cost con-
siderations, and acceptance/rejection decisions may
not be immediately permanent). These strategies have
also been considered in a previous article (Yang et al.
1998).
We also propose two reoptimization policies that
consider, in one optimization run, all trucks, all accep-
tance/rejection and allocation decisions of pending
and tentatively accepted waiting jobs, and all realloca-
tion decisions of permanently accepted waiting jobs.
MYOPT optimizes the acceptance and (re-)allocation
decisions as if no future new job would ever be
requested. It corresponds to solving a full instance of
the offline problem. Conceivably, this policy should
perform better than any of the local policies. How-
ever, this remains an empirical question and is inves-
tigated using a systematic simulation framework
introduced in §4.
OPTUN operates in almost the same way as
MYOPT. The only difference is that OPTUN intro-
duces opportunity costs of serving jobs, somewhat
accounting for future job requests. It assumes some
knowledge about the probability law of future job
pickup (and delivery) locations. More precisely, let
Da be the expected distance from a random point to
point a and
D be the expected distance between two
independent random points, where random points
are distributed according to the probability law of job
pickup and delivery locations. Instead of using D
k
0i
s,
D
ij
s, and W
i
s in the formulation of the offline prob-
lem, OPTUN uses C
k
0i
s, C
ij
s, and 2W
i
s, respectively.
The new parameters are:
C
k
0i
≡ D
k
0i
+ K
O
1
Do
i
−
Dl
k
+ K
O
2
Dd
i
−
D
C
ij
≡ D
ij
+ K
O
1
Do
j
−
Dd
i
+ K
O
2
Dd
j
−
D
and
2 = 1 + K
O
3
K
k=1
N
i=1
C
k
0i
+
N
i=1
N
j=1j=i
C
ij
K
k=1
N
i=1
D
k
0i
+
N
i=1
N
j=1j=i
D
ij
where K
O
1
, K
O
2
, and K
O
3
are exogeneous parameters.
In the expressions of C
k
0i
and C
ij
, the term associ-
ated with K
O
1
is to influence the vehicle-job assign-
ment decision, and the term associated with K
O
2
is to
influence the job acceptance/rejection decision. The
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
Transportation Science 38(2), pp. 135–148, © 2004 INFORMS 143
rationale behind these corrective terms is based on the
following crude heuristic arguments, illustrated here
for C
k
0i
(the arguments being similar for C
ij
):
The multiplicative term of K
O
1
, that is,
Do
i
−
Dl
k
,
represents a measure of the change in the opportu-
nity cost for an empty truck moving from l
k
to o
i
.
One can think of
Da as the average distance between
a and a potential future job’s pickup location, and is
thus a measure of how isolated the point a is. If the
truck moves from l
k
to o
i
, and a new request comes
when the truck is a fraction 0 ≤ 3 ≤ 1 away from its
departure, its new position has an “isolation” measure
of 1 − 3
Dl
k
+ 3
Do
i
. The difference between this
and the isolation of the initial starting point is then
1 − 3
Dl
k
−
Do
i
. The parameter K
O
1
is exoge-
neously chosen and partially reflects what 1 − 3
would be on average.
The multiplicative term of K
O
2
, that is,
Dd
j
−
D,
represents a measure of the resulting action of accept-
ing job i and, after serving it up, of ending up in a
location d
i
with an isolation measure,
Dd
j
, signifi-
cantly different from the one of a random point,
D.
This corrective term results in penalizing remote loca-
tions and favoring central locations.
Finally, the term 2 and the associated parameter K
O
3
are simply used to compensate for the inflation in the
other parameters in the formulation.
4. Simulation Setup
The goal of the simulation experiments is to compare
the proposed policies under both typical probabilis-
tic settings and various parameters. Because of the
heavy computational requirements of individual sim-
ulation runs, a full factorial experimental design is
neither practical nor necessary. Therefore, the policies
are tested under several typical scenarios rather than
the full range of possible occurences.
Throughout the simulation study, we assume that
(1) job-arrival rate is 1/T
INT
; (2) pickup and destina-
tion locations of the jobs are independent, identically
distributed uniform random variables in a unit square;
and (3) T
ADV
i
s, T
SLK
i
s, and T
RES
i
s are all drawn indepen-
dently from uniform distributions with mean T
ADV
,
T
SLK
, and T
RES
, and ranges 0 2T
ADV
, 0 2T
SLK
, and
0 2T
RES
, respectively.
In a unit square, the average distance between two
points is approximately 0'522. So the maximum pos-
sible service rate per truck is 4 1/0'522 1'916 (this
maximum service rate corresponds to a very high
job-arrival rate for which the empty distance from a
job’s destination to a next job’s origin can be made
arbitrarily small in expectation). We define the traf-
fic intensity 5 to be 1/KT
INT
4. Without job rejection,
5 should be below 1 for the system to be stable. To
be realistic and allow trucks to have some operational
flexibility, we have chosen 5 = 0'5 as a default value.
For given values of K and 5, the interarrival time for
demands is chosen as T
INT
= 1/K54.
Finally, we assume that $ = 1'0 and % = 0'2. This
choice of $ implies that the cost per unit of empty dis-
tance has the same weight as the loss revenue per unit
of loaded distance. Also, % = 0'2 implies that 5 units of
delay would offset the revenue from any accepted job.
For every input parameter vector, and for every
policy under investigation, we simulate R = 10 inde-
pendent runs. Each policy experiences the same
10 independent runs. Each run starts with all trucks
located at the central depot and simulates the arrivals
of n = 100 × K jobs. Let C
n r
denote the value of
the function C
∗
n
/n (see (3)) that we record for
the rth run. C
n r
is computed in our simulation as
follows. For each truck, we have a double-precision
variable ET k, which records the truck’s total empty
travel distance at decision and job-completion epochs.
For each job i, we have a double-precision vari-
able TCOMi, which records the job’s completion
time, and a binary variable REJ i, which indicates
whether this job has been permanently rejected, at
decision and job-completion epochs. When
∗
n
has
been reached in this rth run, it is straightforward
for us to use (1) and the three arrays of variables to
calculate the corresponding C
n r
. The sample mean
AvgCost =
R
r=1
C
n r
/R serves as our approximation of
the policy measure (4) defined in §1. From extensive
initial tests, we find that this number of simulated
arrivals is sufficient to guarantee steady-state behav-
iors and remove the effects of initial conditions. Also,
due to the option of job rejection, the actual traffic
intensity of the system is much smaller than 1 and so,
for every policy and every batch of 10 independent
runs, the sample variation of various results across
runs stays well below 1% of their corresponding sam-
ple means.
The SE policy and the two reoptimization policies
need to call CPLEX to solve instances of the offline
problem. To guarantee robustness and timeliness of
the solutions, we limit the number of jobs involved in
each optimization to a fixed upper-bound N
B
(10 for
the SE policy and 20 for the reoptimization policies).
To do this, we both limit the size of L
TEMP
t to N
B
− 1
(if the output of an offline problem optimization leads
to TR > N
B
− 1 tentatively rejected jobs, then TR −
N
B
+ 1 of them are picked at random and perma-
nently rejected) and, if needed, consider only a few
waiting jobs at the end of each queue (for the reopti-
mization strategies, this is done as evenly as possible
across all trucks, by keeping on average only the last
N
B
− 1 − L
TEMP
t/K jobs per queue). We also limit
the total amount of time the SE and reoptimization
policies spend solving each optimization problem to
a fixed T
LIM
(20 seconds). When the SE policy is used
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
144 Transportation Science 38(2), pp. 135–148, © 2004 INFORMS
Table 1 Values of Parameters in Main Simulation Results
KT
SLK
T
ADV
T
RES
Default 10 2.0 1.0 0.2 0.5 0.0 0.0
Table 2 10 2.0 1.0 0.2 0.5 0.0 0.0
Table 3 10 2.0 0.2 1.0 0.5 0.0 0.0
Table 4 10 0.5 1.0 0.2 0.5 0.0 0.0
Figure 2 10 2.0 1.0 0.2 02 ∼ 08 0.0 0.0
Figure 3 10 2.0 1.0 0.2 0.5 00 ∼ 15 0.0
Figure 4 10 2.0 1.0 0.2 0.5 0.25 00 ∼ 0375
with N pending jobs, each optimization is allocated a
maximum time of T
LIM
/K ∗ N. When a reoptimiza-
tion policy is used, each optimization is allocated a
maximum time of T
LIM
.
Table 1 lists various values of the parameters used
in our main comparisons of the five proposed policies.
The default parameter values in Table 1 are used as
starting points to find good values for the remaining
parameters associated with the policies under inves-
tigation. We find that OPTUN works best when
K
O
1
= 0'12, K
O
2
= 0'10, and K
O
3
= 0'06. Finally, as men-
tioned before, for the SE and reoptimization policies,
we always let T
LIM
be 20.0 seconds and N
B
be 10 for
SE and 20 for the reoptimization policies.
Assuming that all these parameters are given,
each simulation is now parameterized by a vector
K T
SLK
$%5T
ADV
T
RES
. In our implementation,
the instances of the offline problem are solved by the
commercial CPLEX 6.5 solver. The simulation source
code is written in C language. All the runs have
been conducted on a Dell OptiPlex machine with a
Pentium II processor.
All the input parameters are:
R: number of independent runs; its default
value: 10.
K: number of trucks; its default value: 10.
n: number of jobs; its default value: 100 × K =
1000.
5: traffic intensity; its reasonable range: 0'2 ∼ 0'8;
its default value: 0.5.
$: relative weight of cost due to empty traveling
versus cost due to job rejection; its default value: 1.0.
%: relative weight of cost due to job waiting versus
cost due to job rejection; its default value: 0.2.
T
ADV
: the average T
ADV
i
; its reasonable range:
0 ∼ 1'5.
T
SLK
: the average T
SLK
i
; its default value: 2.0.
T
RES
: the average T
RES
i
; its default value: 0.0.
N
B
: maximumly allowed number of jobs to be
involved in each optimization; its default values:
10 for the SE policy and 20 for the reoptimization
policies.
T
LIM
: maximumly allowed amount of optimization
time to be spent during one decision epoch in the
SE and reoptimization policies; its default value:
20 seconds.
Table 2 Performance of Policies Under Typical Parameters
Policy RjcRate EmpDist DelayWt RjLDist AvgCost
BENCH 0.154 0.197 0.061 0.236 0.213
NS 0.097 0.177 0.091 0.221 0.198
SE 0.101 0.174 0.107 0.226 0.199
MYOPT 0.092 0.155 0.050 0.209 0.169
OPTUN 0.076 0.155 0.047 0.188 0.166
Note. RjcRate is the average rate of jobs being rejected; EmpDist is the aver-
age empty distance traveled by the trucks per accepted job; DelayWt is the
average weighted delay per accepted job; RjLDist is the average distance of
the rejected jobs; AvgCost is the average cost incurred per requested job.
Note that AvgCost is the value of the objective minimized (expressed on a
per requested job basis), and is therefore the ultimate figure of merit in this
evaluation.
5. Simulation Results
The first set of simulation experiments is performed
with the default parameter values. The results are
shown in Table 2. Under the default parameters,
the reoptimization policies appear to outperform the
more limited policies by a significant margin. The
results confirm the value of seeking optimal solutions
at each decision epoch, even when the formulation is
limited to consideration of only those loads that have
already materialized.
The policies are compared under a different com-
bination of parameter values, in which delay time is
given greater weight by increasing the value of % from
0.2 to 1.0, and the empty distance is correspondingly
de-emphasized by reducing $ from 1.0 to 0.2. The
results, shown in Table 3, again indicate that the reop-
timization policies outperform the local policies.
In the next set of experiments, all parameters are
kept at their default levels, with the exception of
the average time until latest pickup, T
SLK
, which is
reduced from 2.0 to 0.5, reflecting tighter pickup win-
dows and greater job urgency than the default sce-
nario. The results are shown in Table 4, indicating that
reoptimization policies again outperform local poli-
cies, though by a smaller margin (about 10% in terms
of AvgCost) than in the less-constrained cases. The
simulation experiments shown here clearly indicate
that optimization over available job requests at each
decision epoch leads to better overall (over the entire
sequence of load requests) job acceptance/rejection
decisions, and shorter empty distance than the more
Table 3 Performance of Policies when Delay Penalty Is Relatively
More Important
Policy RjcRate EmpDist DelayWt RjLDist AvgCost
BENCH 0.013 0.226 0.014 0.179 0.061
NS 0.006 0.211 0.012 0.163 0.054
SE 0.006 0.210 0.012 0.087 0.054
MYOPT 0.004 0.181 0.008 0.065 0.045
OPTUN 0.004 0.180 0.007 0.064 0.043
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
Transportation Science 38(2), pp. 135–148, © 2004 INFORMS 145
Table 4 Performance of Policies when Jobs Are Very Urgent
Policy RjcRate EmpDist DelayWt RjLDist AvgCost
BENCH 0.168 0.194 0.162 0.251 0.231
NS 0.132 0.182 0.191 0.247 0.224
SE 0.133 0.180 0.196 0.252 0.224
MYOPT 0.122 0.166 0.170 0.228 0.203
OPTUN 0.102 0.167 0.160 0.217 0.201
local strategies considered here. Under all situations
considered, applying reoptimization policies appears
to produce significant savings in operating costs.
The next set of simulations examines how the poli-
cies fare under varying degrees of relative satura-
tion in the system, captured by the index 5. The
results are shown in Figure 2. The most striking phe-
nomenon here is the widening of the gap between
the respective performance of the local and reopti-
mization policies up to 5 = 0'7. When 5 increases,
the average number of jobs at each decision epoch
increases and action on one job affects more jobs. Also
reoptimization policies generate even better payoffs
under higher traffic intensities. Finally, the increase of
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.2 0.3
0.4 0.5 0.6 0.7 0.8
AVGCOST
RHO
BENCH
NS
SE
MYOPT
OPTUN
Figure 2 Performance of Policies when Is Varying
5 makes the knowledge about future jobs more impor-
tant. From the widening of the gap between OPTUN
and MYOPT in the first half of the experiment, we
see that OPTUN utilizes the distributional informa-
tion about jobs in a more efficient way. The jump in
the average cost under both reoptimization policies
at highest saturation rate (from 5 = 0'7to5 = 0'8)
is due to the computational limitations imposed on
the solution of individual problem instances at each
decision epoch. In fact, in these experiments (with a
20-second limit on any problem instance), only about
14% of the optimizations reached duality gaps within
one percent.
Next, we investigate the effect of advance informa-
tion. With all other parameters at their default lev-
els, we vary the average time that a job is requested
prior to its earliest pickup time, T
ADV
. The results are
shown in Figure 3. BENCH is not very sensitive to
the change of T
ADV
. Its performance even degrades
when T
ADV
becomes too big. This degradation can be
partially explained by the fact that this policy inserts
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0 0.2
0.4 0.6 0.8 1 1.2
1.4
AVGCOST
T
ADV
BENCH
NS
SE
MYOPT
OPTUN
Figure 3 Value of Advance Information: Comparative Performance of
Policies when T
ADV
Is Varying
Yang, Jaillet, and Mahmassani: Real-Time Pickup and Delivery Problems
146 Transportation Science 38(2), pp. 135–148, © 2004 INFORMS
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0 0.05 0.1 0.15
0.2 0.25 0.3
0.35
AVGCOST
T
RES
BENCH
NS
SE
MYOPT
OPTUN
Figure 4 Performance of Policies when T
RES
Is Varying
jobs at the end of the queues, even though they could
be available for pickup earlier than for any other jobs
currently in the queues.
The performance of SE and the two reoptimiza-
tion policies improves as T
ADV
increases. For SE, the
range where T
ADV
has a visible effect is from 0 to
0.7 and the maximal improvement is about 3%. For
the two reoptimization policies, the range where T
ADV
has visible effects is from 0 to 1.0 and the maximal
improvements are about 10%.
Finally, we conduct another simulation to study the
effect of T
RES
. In this simulation, all the parameters
stay typical, except that we let T
ADV
= 0'25 and T
RES
vary. The results are shown in Figure 4.
By definition, BENCH is not affected by T
RES
at
all, since its decisions are made permanently at job-
arrival epochs. For all other policies, changes brought
by the varying T
RES
are visible yet not remarkable.
6. Concluding Remarks
In this paper, we have introduced and studied a
generic real-time truckload pickup and delivery prob-
lem in a very general framework, taking into account
various costs due to job rejection, empty travel of
trucks, and delay time of job completions. The frame-
work also facilitates investigation of the value of
advanced information.
We have evaluated several rolling-horizon policies
based on various heuristics either previously intro-
duced in the literature or proposed here for the first
time. We found that the policies based on fully opti-
mizing the offline model of the problem perform very
competitively with other policies under typical cost
structures. The best policy we found is the one that
takes some future job distribution into consideration.
We also found that advanced information is very use-
ful for some of the policies.
We think future research should concentrate on the
search for better policies that utilize some information
about future jobs more efficiently. From the improve-
ment of OPTUN over MYOPT, we believe that there
is still much potential for progress left uncovered.
Acknowledgments
The authors thank two referees and Teo Crainic for com-
ments that greatly helped improve the quality and clarity of
the paper. The first author was supported by the National
Center of Transportation and Industrial Productivity Grant
992518. The second author was supported by National Sci-
ence Foundation Grant DMI-9713682.
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