ResearchPDF Available

Vehicle Routing to Minimize Time-Dependent Emissions in Urban Areas

Authors:

Abstract and Figures

This paper focuses on the problem of minimizing CO$_2$ emissions in the routing of vehicles in urban areas. While many authors have realized the importance of speed in minimizing emissions, most of the existing literature assumes that vehicles can travel at the emissions-minimizing speed on each arc in the road network. In urban areas, vehicles must travel at the speed of traffic, which is variable and time-dependent. The best routes also depend on the vehicle load. To solve the problem, we take advantage of previous work that transforms the stochastic shortest path subproblems into deterministic problems. While in general, these paths must be computed for each combination of start time and load, we introduce a result that identifies when the emissions-minimizing path between customers is the same for all loads. When this occurs, we can precompute the paths and store them in a lookup table which saves on runtime. To solve the routing problem, we adapt an existing tabu search algorithm. We test our approach on instances from a real road network dataset and 230 million speed observations. Experiments with different numbers of vehicles, vehicle weights, and pickup quantities demonstrate the value of our approach. We show that large savings in emissions can occur particularly in the suburbs, with heavier vehicles, and with heterogeneous pickup quantities as compared with routes created with more traditional objectives. We show that the savings in emissions are proportionally larger than the associated increases in duration, indicating improved emissions are achievable at a fairly low cost.
Content may be subject to copyright.
Vehicle Routing to Minimize Time-Dependent Emissions in Urban
Areas
Jan Fabian Ehmke1, Ann Melissa Campbell2, and Barrett W. Thomas2
1Business Information Systems, Freie Universit¨at Berlin, Garystr. 21, D-14195 Berlin
1Tel. +49-30-83858731, Fax +49-30-838458731, Email janfabian.ehmke@fu-berlin.de
2Department of Management Sciences, The University of Iowa, 52240 Iowa City, USA
October 14, 2015
Abstract
This paper focuses on the problem of minimizing CO2emissions in the routing of vehicles in urban
areas. While many authors have realized the importance of speed in minimizing emissions, most of
the existing literature assumes that vehicles can travel at the emissions-minimizing speed on each arc
in the road network. In urban areas, vehicles must travel at the speed of traﬃc, which is variable
and time-dependent. The best routes also depend on the vehicle load. To solve the problem, we take
advantage of previous work that transforms the stochastic shortest path subproblems into deterministic
problems. While in general, these paths must be computed for each combination of start time and load,
we introduce a result that identiﬁes when the emissions-minimizing path between customers is the same
for all loads. When this occurs, we can precompute the paths and store them in a lookup table which
saves on runtime. To solve the routing problem, we adapt an existing tabu search algorithm. We test our
approach on instances from a real road network dataset and 230 million speed observations. Experiments
with diﬀerent numbers of vehicles, vehicle weights, and pickup quantities demonstrate the value of our
approach. We show that large savings in emissions can occur particularly in the suburbs, with heavier
vehicles, and with heterogeneous pickup quantities as compared with routes created with more traditional
objectives. We show that the savings in emissions are proportionally larger than the associated increases
in duration, indicating improved emissions are achievable at a fairly low cost.
Key words: emissions, vehicle routing, green logistics, load dependency.
1 Introduction
The reduction of emissions from heavy-duty trucks has become an important part of worldwide eﬀorts to
reduce CO2emissions. As a result, as a way to reduce fuel usage and thus emissions, many delivery companies
have reported on their eﬀorts to lower the total number of miles traveled by their vehicles (Express, 2012,
UPS, 2013). However, emissions are nonlinear in speed, yielding higher emissions at both low and high
speeds (Demir et al., 2011). Thus, focusing on distance alone does not necessarily minimize emissions.
While many authors have realized the importance of speed in minimizing emissions, most of the existing
literature assumes that vehicles can travel at the emissions-minimizing speed on each arc in the road network.
Yet, in urban areas, vehicles must travel at the speed of traﬃc, which is both variable and time-dependent.
1
Further, because emissions are also a function of the load being carried by the truck, the order that customers
are visited should also be considered.
This paper focuses on the problem of minimizing expected CO2emissions in the routing of a ﬂeet of
capacitated vehicles in an urban area. We assume that each vehicle will visit multiple customers over the
course of a day, and the driver will not wait at any customer location. Each customer requires a pickup
of a particular weight, and as a result, the load of the vehicle changes as pickups are made. Traveling
between customers, the vehicle produces emissions. The expected emissions-minimized path between any
two customers can vary due to the impact of time-dependent speeds as well as the load on the vehicle. As
a result, the best path between each pair of customers cannot as a rule be precomputed as in most vehicle
routing problems (VRP).
To solve the problem, we adapt an existing local search procedure. The procedure is a tabu search heuris-
tic that was originally developed for time-dependent vehicle routing problems. In our case, the procedure is
adapted to include the computation of time-dependent, expected emissions-minimized paths between each
pair of customers on the route. We introduce a result that allows us to precompute and store a large number
of these paths. In the cases where we cannot precompute the path, we draw on previous work that demon-
strates that such paths can be eﬀectively determined by computing a time-dependent average emissions cost
for each arc and solving the resulting time-dependent, deterministic minimum emissions cost path problem.
This paper makes several important contributions to the literature. It is the ﬁrst paper to consider the
impact of travelling at the speed of traﬃc and the load on the vehicles in the minimization of emissions. We
propose a clever technique to reduce the computational burden and investigate the question of when it is
really important to incorporate the load of the vehicle into the optimization. We use instances derived from
a real road network dataset and 230 million speed observations. We show that large savings in emissions
can occur particularly in the suburbs, with heavier vehicles, and with heterogeneous pickup quantities as
compared with routes created with more traditional objectives. We also show that the savings in emissions
are proportionately larger than the associated increases in duration, indicating improved emissions may be
achievable at a fairly low cost.
The remainder of this paper is outlined as follows. In the next section, we review the existing literature.
2
In Section 3, we present a model of the problem. Section 4 presents our solution approach, and Section 5
introduces our experimental design. We present computational results in Section 6 and provide conclusions
in Section 7.
2 Literature Review
In the following, we provide an overview of the related literature. We begin by discussing existing models
for determining vehicle emissions. We then present literature on vehicle routing with emissions minimization
objectives. Unlike in traditional vehicle routing applications, we must also compute the shortest paths
between customers on the route. Consequently, we also oﬀer a brief review of the literature on determining
emissions-minimized paths.
2.1 Modeling of emissions
Demir et al. (2011) and Demir et al. (2014) provide an overview of existing models of emissions as well as
applications. A key feature of all of the models of emissions is that they are nonlinear in speed. These
nonlinearities can be particularly impactful in urban areas, where vehicles must travel at the speed of traﬃc
and are often slowed by congestion (van Woensel et al., 2001).
We focus on the Comprehensive Emissions Model (CEM) introduced in Barth and Boriboonsomsin (2008).
The CEM determines fuel consumption and thus emissions on a given arc as a function of speed, vehicle
weight, and numerous vehicle and arc-speciﬁc constants. The advantage of the CEM versus other emissions
models in the literature is that it accounts for the impact of a vehicle’s load on emissions. In our implemen-
tation, we focus on the time-dependent version of the the CEM ﬁrst presented in Franceschetti et al. (2013).
We also use the parameters presented in Franceschetti et al. (2013).
2.2 Emissions in Vehicle Routing
In recent years, there have been a number of papers that explore the minimization of emissions in the routing
of vehicles across multiple customers. Reviews of the literature considering emissions and vehicle routing
can be found in Demir et al. (2014) and Lin et al. (2014). Most of the existing literature assumes that the
vehicle can travel at the emissions-minimizing speed on each arc in the road network. We will ﬁrst present
the related literature that is not time-dependent, and then we will discuss the literature that includes time
3
dependency.
Similar to this paper, Bektas and Laporte (2011) use the CEM in addressing the VRP with an emissions-
minimizing objective. In contrast to the work in this paper, Bektas and Laporte (2011) determine the speed
for a given load that minimizes CO2emissions and assume the vehicle will travel at that speed. They only
consider speeds of 40 km/h or higher, and the authors note that environments where speeds are less than 40
km/h will require a diﬀerent approach. Their results show that the minimization of fuel and driver costs
do not necessarily minimize emissions. Extending the approach of Bektas and Laporte (2011), Demir et al.
(2012) solve larger instances using an Adaptive Large Neighborhood Search. They relax the restriction on
speeds below 40 km/h. For each route, they minimize the costs of fuel consumption and driver wages by
controlling the route length and the optimal speed on each arc.
Using an alternative emissions model that does not consider load, Figliozzi (2010) investigates the impact
of congestion on emissions in a network with time-dependent travel times and speeds. Travel speeds are
controlled by waiting at customer locations. The results demonstrate that congestion adversely aﬀects
emissions and travel times in vehicle routing. Xiao and Konak (2015) extend Figliozzi (2010) to include
time constraints on customer deliveries. Jabali et al. (2012) minimize travel time, fuel, and CO2emissions
costs (also using an emissions model that does not consider load) when travel times are time-dependent.
Jabali et al. (2012) assume that speeds are ﬁxed due to congestion during a rush hour period. Outside of
the congested period, speeds are set to emissions-minimizing levels. Their time-dependent routing problem
is solved by a tabu search procedure. Franceschetti et al. (2013) consider time-dependent speeds while
minimizing fuel and driver costs. Like Jabali et al. (2012), they consider just two time periods: a low-speed
congested period with relatively high emissions and a higher-speed uncongested period with relatively lower
emissions. They also use the CEM which is capable of accounting for the impact of load on emissions.
Franceschetti et al. (2013) allow waiting at the depot and post-service waiting to help the vehicles avoid
driving during the congested period. We consider more than two periods, do not allow vehicles to wait,
account for the variability in the travel times, and always assume that vehicles must travel at the speed of
traﬃc, as is common in urban areas.
Also related are the recent papers by Wen et al. (2014) and Qian and Eglese (2014). In Wen et al. (2014),
4
the authors solve a VRP where the total cost consists of fuel costs, driver costs, and a congestion charge.
Fuel charges are inﬂuenced by time-dependent speeds, and congestion charges are applied whenever a vehicle
enters a congestion charge zone during the day. Here, fuel charges do not estimate the impact of load, and
congestion charges are imposed once per day for a vehicle. By modeling the costs in this way, few vehicles
enter the congestion zone and incur the daily charge. In Qian and Eglese (2014), the authors create a single
route that minimizes emissions. They use an emissions model that does not account for load. They choose
the order that customers will be visited, as well as the vehicle speed and the amount of waiting time at the
customers. Vehicles may travel below the speed limit, and the speed limit varies with time. The authors
oﬀer two methods to solve the problems, one of which is based on dynamic programming. They compare
their solutions with the fastest routes serving the same customers and show emissions can be optimized with
9-10% more trip time. Unlike Qian and Eglese (2014), we consider multiple vehicles and account for the
impact of load, as well as account for the variability in travel times and do not allow vehicles to wait.
Another related problem is the energy minimization vehicle routing problem (EMVRP). In the EMVRP,
the cost of traveling an arc is the product of the vehicle load on that arc and the length of the arc. The
cost function is motivated by the fact that both fuel usage and emissions are correlated with vehicle weight
and distance traveled. The emissions function used in this work captures both of these elements as well as
the speed and variation in speeds on arcs. The EMVRP is introduced in Kara et al. (2007), Kramer and
Subramanian (2015) and Zachariadis et al. (2015) present metaheuristic approaches, and Fukasawa et al.
(2015) introduce a branch-cut-and-price approach.
2.3 Emissions-minimizing path computation
The existing literature on vehicle routing with emissions objectives that uses time-dependent speeds treats
these speeds as deterministic. In this paper, we assume that the vehicle must move at the speed of traﬃc,
which can vary. Further, because the loaded weight of the vehicle changes as the vehicle moves along
the route, the paths between customers must also account for the impact of vehicle weight in minimizing
emissions. We are aware of no work that addresses both of these issues in the computation of expected
emissions-minimized paths between customers.
The only work of which we are aware that determines expected emissions-minimized paths in urban areas
5
is the work by Hwang and Ouyang (2015) and Ehmke et al. (2015). Ignoring load, Hwang and Ouyang (2015)
seek to ﬁnd expected emissions-minimized shortest paths between origin and destination pairs. The authors
do not include time-dependent speeds, but do include a time limit on the length of the path. Hwang and
Ouyang (2015) propose two methods. To control label proliferation, one method discretizes the arrival time
distributions in a forward dynamic programming approach. Because of the computation time, the authors
also introduce a second approach that transforms the cost of each arc to an expected emissions value and
thus reduces the problem to its deterministic equivalent. The heuristics are tested on a relatively small road
network with randomly generated travel speed distributions.
In Ehmke et al. (2015), the authors incorporate real-world speed data with shortest path approaches to
determine a priori expected emission-minimized paths. As in this paper, Ehmke et al. (2015) draw their data
from the road network for Stuttgart, Germany, and 230 million speed observations from the years 2003-2005.
A detailed discussion of the dataset can be found in Lorkowski et al. (2004).
To make use of this data for constructing time-dependent expected emissions-minimized paths, Ehmke
et al. (2015) present two methods for determining such paths. The ﬁrst method is an adaptation of the A?
algorithm that samples the data to evaluate the emissions cost of traversing an arc. The A?algorithm is itself
a variant of Dijkstra’s algorithm that uses a heuristic estimate of future costs to determine node expansion.
Because of the computation time required for the sampling-based method, Ehmke et al. (2015) introduce
a second solution approach, referred to as the “arc-averaging” approach, that reduces the problem to a
time-dependent, deterministic shortest path problem. Because most emissions functions are convex in the
speed term and as a result of Jensen’s Inequality, using an average speed value for an arc does not translate
to an appropriate estimate of the average emissions. Thus, it is important to capture the variability in the
computation of the emissions. Ehmke et al. (2015) show that using sampled speeds from a particular time
bucket for a particular arc to generate emissions and then averaging these emissions values is an eﬀective
way to predict the emissions associated with using an arc at a speciﬁc time of day. The downside of the
deterministic method is that it does not allow for estimation of the arrival time distribution at each node in
the path. Yet, computational results in Ehmke et al. (2015) show the impact to be minimal, in part because
the paths are relatively short in an urban setting. Thus, in this paper in which we will need to solve many
6
emissions-minimized paths, we focus on the time-dependent, but deterministic method. We refer the reader
to Ehmke et al. (2015) for a discussion of algorithms for time-dependent, stochastic shortest path algorithms.
Related work includes Wen et al. (2014) and Yildirim and Catay (2014). In Wen et al. (2014), given a
starting time, the authors try to identify the least cost paths between nodes in a time-dependent network.
As in Wen and Eglese (2015), costs include fuel costs, driver costs, and congestion charges, imposed once per
day. Again, no variability in travel speeds is considered. The authors develop heuristics to identify paths
that either may enter the congestion charge zone or try to avoid the congestion charge zone. If the one that
enters the congestion zone is of lower cost, it is the only path recorded. These heuristics are used to identify
the paths between customers in Wen and Eglese (2015). Yildirim and Catay (2014) extend the shortest path
algorithm of Wen et al. (2014) to make use of upper bounds on the path cost. The resulting algorithm oﬀers
comparable solution quality to Wen et al. (2014) but with less computation time.
3 Model
In this section, we present our formal model of the problem being solved in this paper and the emissions
cost model we will use to evaluate the expected emissions cost of a particular path. We consider a set Cof
Ccustomers such that each customer c= 1, . . . , C is associated with a load of weight wc. The customers
are situated on a graph G= (N,A), where Nis a set of nodes, representing intersections and customers,
with C N , and Aa set of directed arcs connecting the nodes in N. We seek to serve the customers with
a ﬂeet of capacitated vehicles Mconsisting of Mvehicles..
To travel between two customers iand jin C, the vehicle travels a series of arcs, called a path. Traversing
an arc aincurs a time- and load-dependent expected emissions cost Fa(t, l), where tis the time at which
the vehicle begins traveling on arc aand lis the load of the vehicle when traversing a. In this sense, our
work contrasts with most of the routing literature in which the cost and required time to travel between
two customers is given by a parameter. We represent the cost of a path pij = (ap
1, . . . , ap
n) starting at time
τbetween customers iand jas φp(τ, l) = Pn
k=1 Fak(t(p, τ ), l), where t(p, τ) makes explicit that the time-
dependent cost of kth arc akdepends on the path phaving been begun at time τ. In this work, we assume
that there is no waiting at either customer or intersection nodes.
7
A route ris a sequence of customers in C. We assume that riis the ith customer on route r. The expected
cost of route rdepends on the start time of the route and the paths chosen for travel from customer to
customer on the route. We assume that the loads at each customer are being picked up, and thus, the load
on the vehicle increases over the route. Then, the expected cost of a route rstarting at time Tis
Φr(T) =
C1
X
k=1
φrk,rk+1 (τ(r, T ), l(r)),(1)
where τ(r, T ) makes explicit the dependence of the cost of the path from rkto rk+1 on the start time of
route rand l(r) the load on the vehicle after serving customer rkon r. The expected cost of a set of routes
Ris the sum of the expected costs of the routes rin R. We assume that there are Mroutes in R, one for
each vehicle in the ﬂeet M.
For a given start time T, the objective is ﬁnd a set of routes R?such that PrR?Φr(T)Pr0RΦr0(T)
for any other set of routes R.
Next, we discuss how we compute the expected emissions cost Fa(t, l) of traversing arc astarting at time
twith load l. Following Bektas and Laporte (2011) and Franceschetti et al. (2013), we use the CEM to
model emissions. We use the same equations and parameters as in Franceschetti et al. (2013) modiﬁed to
account for time-dependent speeds. The CEM takes as input the vehicle load l(in kg), distance da(in m)
associated with arc a, the speed on arc a(in m/s) at time t, which for our purposes is a random variable,
denoted va,t. The resulting time- and load-dependent expected emissions are
Fa(t, l) = EλkNeVda
va,t
+γβdav2
a,t +γα(µ+l)da,(2)
where Ne,V,µ,λ,k,γ,β, and αare parameters related to the vehicle and its engine. Following Franceschetti
et al. (2013), we set the engine speed Ne= 33, the engine displacement V= 5, the engine friction factor
k= 0.2, and the curb-weight µ= 6350 for a standard vehicle and µ= 12700 for a heavy vehicle, respectively.
Note that α,β,γand λrepresent the product of multiple factors presented in Franceschetti et al. (2013).
The corresponding values are α= 0.0981, β= 1.6487, γ= 0.0028, and λ= 1/32428. For a given truck, the
parameter values are assumed to be identical for each arc. As in Bektas and Laporte (2011) and Franceschetti
8
et al. (2013), we assume that the gradient of the road is zero.
4 Solution Methodology
In this section, we discuss our approach to ﬁnding expected emissions-minimized routes. The determination
of routes requires that we know the time- and load-dependent emissions-minimized path between any two
customers in the route for any given load. Because of our focus on both stochastic, time-dependent travel
times and emissions, our approach to ﬁnding these paths is tied to how we model the travel times and
emissions as well as their variations over the course of the day. As such, we ﬁrst discuss how we transform a
database of speed data into time-dependent, expected emissions and travel time values. We then discuss our
computation of expected load and time-dependent emissions-minimizing paths between customers as well as
the precomputation of such paths. Finally, we discuss the tabu search heuristic that we use to search the
space of expected emissions-minimizing routes.
4.1 Data Preparation and the Computation of Time-Dependent Expected Emissions-
Minimized Paths
As discussed in Section 2.3, Ehmke et al. (2015) show that high quality time-dependent expected emissions-
minimized paths can be eﬃciently determined by taking advantage of a transformation of the data that results
in a time-dependent, deterministic shortest path problem. To take advantage of the result, we ﬁrst need
expected time-dependent travel times and expected time-dependent emissions for each arc in the network.
In this paper, we derive these values using real speed data from the metropolitan area of Stuttgart, a major
city in southern Germany. The metropolitan area of Stuttgart is well-known for its congestion, especially
at peak times (Kr¨oger, 2013). We use a database of approximately 230 million speed observations from
the years 2003-2005. These speed observations were collected by the German Aerospace Center with FCD
technology using a ﬂeet of 700 taxis (Lorkowski et al., 2004). For details regarding ﬁltering of outliers and
incorrect measurements, see Ehmke et al. (2009). In addition, due to the speed limitations often imposed on
freight vehicles, we set a speed limit of 100 km/h, i.e., whenever we observe a speed larger than 100 km/h,
we set it to 100 km/h.
9
To account for time dependency, we follow standard practice in time-dependent routing and divide the
day into one-hour time buckets (06:00–07:00, 07:00–08:00, 08:00–09:00, etc.) (Ehmke et al., 2012). It is then
assumed that the travel time and emissions for traversing an arc starting at time tis given by the expected
travel times and expected emissions of the bucket that contains t. For example, if t= 06 : 30, then we would
use values associated with the bucket 06:00–07:00. If the neighboring bucket is entered while traversing an
arc (e.g., when the start time is close to the end of a bucket), the diﬀerent speed levels are linearized in the
transition area as described by Fleischmann et al. (2004).
Having divided the data by arc and time of day as prescribed by the one-hour time buckets, computing
the expected time-dependent travel times is straightforward. For a given arc aand time bucket b, for example
06:00–07:00, the expected time-dependent travel time is given by PKb
a
k=1 sa,bk
Kb
a, where sa,bkis the kth speed
observation in time bucket bfor arc aand Kb
ais the number of observations in bucket bfor arc a. We note
that these values can be precomputed for each arc and time bucket.
Precomputation of the expected time-dependent emissions is more challenging. Notably, we have the
additional challenge of the load of the vehicle as it crosses an arc. However, given the load land departure
time tfor a given arc a, we can rearrange Equation 2 such that we have
EλkNeVda
va,t
+γβdav2
a,t +γα(µ+l)da =λγα(µ+l)da+EλkNeVda
va,t
+γβdav2
a,t.(3)
Now, the expectation in Equation 3 no longer depends on the vehicle’s load, but only the arc and time at
which the vehicle enters the arc.
For convenience, we denote the terms λkNeVda
va,t +γβdav2
a,tin Equation 3 as g(va,t ). Because it
does not depend on the vehicle load, we can precompute the expectation in Equation 3. Then, we estimate
E[g(va,t)] as PKb
a
k=1 g(sa,bk)
Kb
a.
The just computed expected time-dependent travel times and emissions are the foundation of the “arc-
averaging” shortest path method introduced in Ehmke et al. (2015) and summarized in Section 2.3. Essen-
tially, with the expected time-dependent travel times and emissions, the problem is deterministic. In our
approach, we use a Dijkstra-like label-setting algorithm that eﬃciently ﬁnds shortest paths. One downside
of the approach is that the arrival time distributions at each node in the path are reduced to an expected
10
arrival time. However, the results in Ehmke et al. (2015) show that the approach is eﬀective given the short
path lengths often found in urban areas.
Yet, even with the eﬃciency of our shortest path approach, the need to account for load in Equation 2
suggests that the paths between customers must be computed at runtime. However, in Proposition 1, we
present a result that identiﬁes the condition under which a time-dependent path between two customers is
load invariant. The result allows us to reduce the computational challenge of ﬁnding the time- and load-
dependent paths between customers by making it possible to precompute expected time-dependent emissions
minimizing paths between some customers. For convenience of exposition, in the proof, we refer to the three
terms of Equation 2 as the engine module, the speed module, and the weight module.
Proposition 1. If a path p1between a pair of nodes is optimal in terms of emissions for an empty truck as
well as a full truck, p1is optimal in terms of emissions for all load sizes in between.
Proof. We deﬁne em as the weight of an empty truck and em +Las the weight of a full truck. Consider a
proof by contradiction where we assume that it is possible that there is a path p2that is optimal for some
load level between empty and full (em +gwhere 0 < g < L). For a particular path, if we sum Equation 2
across all arcs in the path, we get a sum of the engine modules, a sum of the speed modules, and a sum
of the weight modules. For any path, the sums of the engine and speed modules are not impacted by load,
so we need only to compute that sum once for all load levels. For conciseness, we represent the sum of the
engine and speed modules for a path piby bi. As shown in Equation 2, the impact of load on emissions is
determined by a scalar multiplied by the load. Thus, we can also sum the scalars across the arcs of a path
piand determine emissions by multiplying by the load value. Call this sum of the scalars ci. Then, the
optimality of p1for empty loads implies
b1+c1(em)b2+c2(em),(4)
and the optimality of p1for a full load implies
b1+c1(em +L)b2+c2(em +L).(5)
11
The optimality of p2for load em +gimplies
b2+c2(em +g)< b1+c1(em +g).(6)
The combination of Equation 4 and 6 implies that c1> c2. The combination of Equation 5 and 6 implies
that c1< c2. Because both cannot be true, we have a contradiction.
To implement Proposition 1, we compute the emissions-minimized paths for each time bucket and pair of
customer nodes with an empty vehicle and with a fully loaded vehicle. We assume that the departure time
of each path is the mid-point of a time bucket (06:30, 07:30, 08:30, etc.). Note that this is an approximation
and paths with a departure time near the beginning or end of a time bucket might diﬀer from those with
a departure time in the middle of the time bucket. In general, any eﬀects of this approximation can be
reduced by working with narrower time buckets, though that would increase the number of paths to be
computed and require more data to ensure there are adequate observations in each time bucket. We make
use of Proposition 1 and check to see if the paths computed for an empty vehicle and fully loaded vehicle
match. If they do, we can store the load-independent emissions costs of the path in a lookup table that will
be used by the routing algorithm. If they are not a match, no values are placed in the precomputed lookup
table, and we compute the cost of each arc in a path at runtime.
4.2 Routing Algorithm
To solve the routing problem, we use a tabu search algorithm. We use tabu search because it is considered
one of the most successful solution approaches for diﬀerent variants of the VRP (Vidal et al., 2013). In
particular, we implement the LANTIME tabu search algorithm, which was designed for time-dependent
variants of the VRP. Details of the algorithm are provided by Maden et al. (2010).
LANTIME requires a feasible initial solution as an input. We construct this initial feasible solution using
Solomon’s I1 heuristic (Solomon, 1987). In addition, as in the literature, our LANTIME implementation uses
the neighborhoods Adapted Cross Exchange,Insertion/Removal,One Exchange, and Swap. A neighboring
solution is considered superior if the number of vehicles is smaller (main objective for VRP optimization)
or the number of vehicles is the same and the total cost of all routes is smaller (secondary objective for
12
Algorithm 1 Interweaving shortest path computation and emissions routing
1: Input: A solution scontaining a set of routes and a start time τ
2: for all routes rin sdo
3: for all iin rdo
4: if lookup(i,i+ 1, b,l) = null then
5: Compute time and load-dependent emissions-minimized path
6: Update table with expected emissions of path for the given i,i+ 1, b,l
7: end if
8: end for
9: end for
VRP optimization, only objective for traveling salesman problem, TSP). For cost, we will minimize the total
expected emissions cost as deﬁned by the CEM.
Because the precomputation discussed above does not allow for every pair of customer nodes at every time
of day to use precomputed paths, we must sometimes compute the time- and load-dependent path within
the routing algorithm. More formally, if a search routine changes the solution from sto s0, we evaluate s0
in conjunction with our shortest path algorithm broadly as shown in Algorithm 1. Whenever the lookup
of shortest path costs between customer nodes iand i+ 1 fails for a given time bucket band load l, we
compute the costs of the emissions-minimized path online and store it for potential retrieval. Algorithm 1
is embedded both in Solomon’s I1 heuristic to ﬁnd the initial solution and LANTIME to ﬁnd the improved
solution. As with our precomputation, any paths that need to be determined during the execution of the
routing algorithm are found using the “arc-averaging” algorithm. As computational experiments will show,
the proposed combination of precomputation and shortest path computation allows for high-quality solutions
in a suﬃcient amount of runtime.
5 Experimental Design
As discussed in the previous section, we demonstrate and evaluate the construction of expected emissions-
minimized routes based on real speed data from the metropolitan area of Stuttgart, a major city in southern
Germany. For our experiments, we focus on Tuesday data, as Tuesday is considered a “typical” day by traﬃc
engineers (Ehmke, 2012). We pair the database with a digital roadmap and derive a network that consists
of 5385 nodes and 8629 arcs. In the following, we explain how we design our test sets and computational
experiments.
13
5.1 Test Sets
We develop several test sets that consider spatio-temporal dimensions relevant for freight transportation in
urban areas. In particular, we consider that inner city customers are closer to one another than suburban
customers are to one another, and that congestion patterns on inner city roads diﬀer from congestion patterns
connecting the inner city and the suburbs and from patterns connecting the suburbs. To this end, based on
their geographical location, we divide the set of 5385 nodes into an inner city and a suburban node set with
1160 and 4225 nodes, respectively. Using this node division, three categories of test sets are generated:
Inner city sets (I) each consisting of 30 nodes that are randomly drawn from the nodes located in the
inner city,
Suburban sets (S) each consisting of 30 nodes that are randomly drawn from the nodes located outside
the inner city, and
Mixed sets (M) each consisting of half of the nodes being contained in the inner city and suburban
sets.
For each category, we produce ﬁve instances, resulting in instances denoted by I1.. .I5, S1. . .S5, and
M1. .. M5. For an example visualization of instances I1 and S1 with 30 inner city and 30 suburban nodes,
respectively, see Figure 1. We also investigate smaller variants of these sets with 10 nodes, where the nodes
are the ﬁrst 10 from the 30-nodes instances, respectively. The depot is the same for all instances; it is located
in the suburbs in the North Western part of the city, but close to the inner city area.
To investigate the impact of minimizing emissions on route structures, we analyze the following parame-
ters. For the optimization of a single route (time-dependent TSP), we consider the above node sets I, S and
M with 10 and 30 customers. For each instance and number of customers, we consider four starting times:
two rush hour times (06:30, 15:30) and two non-rush hour times (12:30, 19:30). For each combination of
instance, number of customers, and start time, we consider both a standard vehicle (gross weight of 6350
kg) and a heavy vehicle (gross weight of 12700 kg). We assume that the capacity of the vehicle is equal to
the gross weight of the vehicle, so that, when fully loaded, the vehicle weighs twice as much as when empty.
14
Figure 1: Customers and depot for Instances I1 and S1. The inner city area is denoted by the black box,
the depot location by a circle, inner city nodes by ”I” and suburban nodes by ”S”. Aerial view provided by
15
Thus, each customer has a pickup with a weight of
gross weight of vehicle
total number of customers,
implying that a vehicle is “fully loaded” on the way from the last customer to the depot. In total, we have
We also consider instances with heterogeneous loads to test the impact of heavily varying load quantities
on route structures. To this end, based on the I, S, and M instances with 10 customers, we generate four load
distributions for each of the gross truck weights (standard and heavy). For three of the load distributions,
we assign the loads for three selected customers such that their load totals 90% of the vehicle’s capacity,
and the remaining 10% is evenly divided among the other seven customers. The three customers with heavy
load quantities are chosen as follows:
Random three heavy: random selection of three customers with heavy load quantities,
Farthest three heavy: identiﬁcation of the three customers farthest from the depot in terms of distance,
Closest three heavy: identiﬁcation of the three customers closest to the depot in terms of distance.
We assign higher weights to only three loads to help build insights on where heavy pickups can have the
most impact on solutions. For the fourth load distribution, we assign loads for all customers as follows:
Random decreasing load: all customers are given a load such that the second heaviest pickup is half the
heaviest pickup, the third heaviest pickup is half the second heaviest pickup, etc., and we scale the
sum of the pickup weights such that it equals the total capacity of the vehicle. These decreasing loads
are randomly assigned to the customers. We use this type of instance to represent a more typical case
where customers’ pickup quantities show a large diversity in sizes.
As with the homogeneous load instances, we run each of these heterogeneous load instances for two rush
hour times (06:30, 15:30) and two non-rush hour times (12:30, 19:30) and for two diﬀerent gross weights, i.e.
the standard and heavy vehicle. The result is 480 “heterogeneous load” TSP instances.
To see how emissions impact the solutions of a multiple vehicle problem, we also generate VRP instances.
The VRP instances are based on the 30-customer instances introduced above. We consider ﬂeets of three
16
vehicles. We consider both homogeneous ﬂeets (three standard or three heavy vehicles) and a mixed ﬂeet of
a heavy vehicle and two standard weight vehicles. We vary the load at each customer so that three vehicles
are required to process all pickups. Speciﬁcally, we set the load as follows:
Homogeneous loads: We deﬁne the mean weight as
Pvehicle type gross weight of vehicle ×number of trucks of each vehicle type
total number of customers .
Thus, when considering only standard (heavy) vehicles, the numerator will be three times the weight
of a standard (heavy) vehicle. For the mixed ﬂeet instances, the numerator will be the sum of two
standard weight vehicles and one heavy vehicle. For the Homogeneous loads experiments, all customers
are assigned a load equal to the mean weight.
For the mixed ﬂeet, we also experiment with heterogeneous pickup quantities. These experiments should
help us gain insight into the characteristics of pickups that are assigned to diﬀerent vehicle types when there
is a choice. We consider three diﬀerent load assignments for the mixed ﬂeet experiments:
Random loads: We randomly choose 15 customers to have a load of 1.5 times the mean weight and 15
customers to have a load of 0.5 times the mean weight.
Closest 15 heavy: The 15 customers closest to the depot have a load of 1.5 times the mean weight, and
the 15 customers farthest from the depot have a load of 0.5 times the mean weight.
Farthest 15 heavy: The 15 customers closest to the depot have a load of 0.5 times the mean weight, and
the 15 customers farthest from the depot have a load of 1.5 times the mean weight.
As with the TSP instances, for each I, S, and M VRP instance, we consider two rush hour times (06:30,
15:30) and two non-rush hour times (12:30, 19:30). This creates a total of 360 VRP experiments.
An overview of all test sets is provided in Table 1. Note that each test set is investigated for four diﬀerent
departure times (6:30, 12:30, 15:30, 19:30) and ﬁve inner city, suburban and mixed instances. We run the
same experiment ﬁve times.
17
type demand vehicle # cust detailed results type demand vehicle # cust detailed results
TSP homogeneous standard 10 Table 3 TSP closest three heavy heavy 10 Appendix, p. 10
TSP homogeneous heavy 10 Appendix, p. 2 TSP random decreasing load standard 10 Appendix, p. 11
TSP homogeneous standard 30 Appendix, p. 3 TSP random decreasing load heavy 10 Appendix, p. 12
TSP homogeneous heavy 30 Appendix, p. 4 VRP homogeneous 3 standard 30 Appendix, p. 13
TSP random three heavy standard 10 Appendix, p. 5 VRP homogeneous 3 heavy 30 Appendix, p. 14
TSP random three heavy heavy 10 Appendix, p. 6 VRP homogeneous 1 heavy, 2 standard 30 Appendix, p. 15
TSP farthest three heavy standard 10 Appendix, p. 7 VRP random 15 heavy 1 heavy, 2 standard 30 Appendix, p. 16
TSP farthest three heavy heavy 10 Appendix, p. 8 VRP closest 15 heavy 1 heavy, 2 standard 30 Appendix, p. 17
TSP closest three heavy standard 10 Appendix, p. 9 VRP farthest 15 heavy 1 heavy, 2 standard 30 Appendix, p. 18
Table 1: Overview of test sets
5.2 Experiments
For each of the TSP and VRP instances, we generate solutions for each of the following objectives using the
routing algorithm from Section 4.2 adapted for each objective:
Objective 1 minimizing distances, which is often taken as a proxy for minimizing emissions,
Objective 2 minimizing time-dependent travel times, focusing on the shortest working time, which is a
well-known objective in routing,
Objective 3 minimizing time-dependent emissions given the gross weight of an empty truck, which does
not consider variation of load in the course of a route and is hence computationally eﬃcient, and
Objective 4 minimizing time-dependent emissions given detailed information about the load at every arc
in the course of a route, which is computationally challenging, but the most realistic model with respect
to emissions computation. Objective 4 is the time- and load-dependent objective given in Equation 1.
To compare the emissions found using each of the objectives, we evaluate the solutions from Objectives 1,
2 and 3 using Objective 4. This provides the emissions from a route in terms of kilograms of CO2. We denote
the Objective 1 solutions as the distance solution and abbreviate it as DIST. We refer to the Objective 2
solutions as time-dependent travel time solution and abbreviate it as TT. We refer to the Objective 3
solutions as the load-independent solution and abbreviate it as EM-LI. The Objective 4 solutions are called
the load-dependent solutions and are abbreviated as EM-LD.
In addition to the emissions, we compare the four solutions for each instance in terms of the distance
traveled by the solution in kilometers, the total duration of the solution in minutes, the runtime needed to
ﬁnd the solution in seconds, and the percentage of online path computations that were needed to ﬁnd the
solution (EM-LD only). For each of the four objectives, the measures reported are for the best solution
18
found in ﬁve runs. The runtime is the total runtime needed to complete the ﬁve runs. To help facilitate
comparisons, the numbers reported for emissions, distance traveled, and total duration for TT, EM-LI, and
EM-LD are the percentage change relative to the values found with DIST. A positive percentage represents
an increase in the value relative to DIST, where a negative value represents a decrease.
The creation of lookup tables follows the structure outlined in Section 4.1. For Objectives 1, 2 and 3,
varying load does not need to be considered. We can thus precompute and store the costs of DIST, TT and
EM-LI paths for each time bucket and pair of customer nodes. As discussed previously, this is not the case
in general for EM-LD paths. In addition to using Proposition 1 to reduce the computation, however, we
further reduce the computational cost for Objective 4 by storing any EM-LD path found for a particular
bucket and particular load size in each of the ﬁve runs of each instance. Thus, for example, Run 5 beneﬁts
from all of the online path computations conducted in the previous four runs. The percentage of online path
computations reported is the number of online path computations required over all ﬁve runs relative to the
sum of shortest path requests over the ﬁve runs.
Our experiments are performed on a Windows 7 64-bit operating system with an Intel Core i5-3470
processor and 8 GB of RAM. Algorithms are coded and executed in Java 64-bit with a memory allocation
of 4 GB. We let the tabu search metaheuristic run for 1000 iterations per experiment.
6 Computational Results
In this section, we present our computational results. We ﬁrst examine the value that Proposition 1 provides
in terms of reducing online computation time. We then discuss results that demonstrate the value in explicitly
modeling emissions. We start with looking at the results with a single vehicle. We present a summary of the
results with homogeneous loads across diﬀerent numbers of customers and then look at more detailed results
for 10 customer instances. With 10 customers, we next examine results with heterogeneous load quantities.
Last, we present a summary of results with multiple vehicles. Detailed results for all experiments can be
found in the electronic appendix.
19
6.1 A Comparison of Runtime with and without Precomputation of Time- and
In this section, we present a comparison of the runtime when we consider our advanced precomputation to
precompute time- and load-dependent paths. Table 2 provides an analysis of runtimes. The table gives the
number of customers in an instance (cust), the gross weight of the vehicle (wt), and the category of instances
(inst). In the next four columns, the table shows the number of online computations (# of online comp)
and the CPU times for the computation of the ﬁrst run of each instance in seconds when we (1) ignore
our analytical result, i.e., we only precompute emissions-minimized paths for an empty vehicle (standard
precomputation) and compute everything else online, versus (2) the same values considering our advanced
precomputation. To avoid prohibitive runtimes, we keep the emissions-minimized costs readily available in
memory for potential further lookups once we have computed them.
On average, we are able to reduce the number of online computations by 87% and 83% for standard
and heavy vehicles, respectively, on the 10 customer instances, and 88% and 83% for standard and heavy
vehicles, respectively, on the 30 customer instances. This reduction leads to an average decrease in runtimes
of 34-72 % for small instances and 10-64% for large instances. Note that the diﬀerences in runtimes are very
large for suburban and mixed instances, but relatively small for inner city instances where the additional
online computations can be done quite quickly because of the short distance of the paths.
Comparison of EM-LD run times with and without precomputation of shortest paths
cust wt inst # of online comp CPU # of online comp CPU
I 1195 4.85 76 3.01
S 1746 27.56 281 7.59
M 1118 15.26 162 4.86
I 1199 4.88 111 3.22
S 1791 27.98 391 9.60
M 1121 14.91 205 5.11
I 35472 485.79 2667 401.02
S 73599 1733.47 8784 628.38
M 39965 987.86 6402 498.98
I 35266 475.06 3749 425.50
S 69236 1681.34 12020 724.28
M 37605 947.58 7928 555.54
30
6350
12700
10
6350
12700
Table 2: Analysis of runtimes
Overall, as a consequence of Proposition 1, we are able to precompute an average of 85% of all paths
for standard vehicles and 81% for heavy vehicles. For both standard vehicle and heavy vehicles, Figure 2
shows the impact of time of day on our ability to precompute paths. Interestingly, the ﬁgure shows that the
ability to precompute the emissions-minimized paths varies by time of day. There is more path variability
20
65%
70%
75%
80%
85%
90%
95%
100%
Proportion of precomputed paths
Time of day
6350 kg 12700 kg
Figure 2: Proportion of paths that match in the course of the day compared to all paths contained in the
time-dependent lookup table (for two diﬀerent vehicle types)
in the morning (07:00-10:00) and afternoon buckets (16:00-18:00). Further, sometimes the proportion of
precomputed paths for the heavy vehicles can be higher than for the standard vehicles. This occurs because
the impact of speed variation becomes smaller with increasing weight in the CEM, so a larger weight does
not necessarily lead to a larger variation of path structures.
6.2 Summary of Results for Single Vehicle with Homogeneous Load Quantities
cust wt inst dur dist em CPU dur dist em CPU dur dist em CPU dur dist em CPU paths
I 156.45 19.40 5.24 13.34 -2.37% 4.13% -2.42% 11.05 -2.10% 1.53% -3.08% 13.48 -2.04% 1.37% -3.10% 14.35 0.002%
S 248.81 70.14 16.48 10.50 -8.56% 9.83% 2.58% 11.96 -6.12% 2.34% -2.64% 12.83 -5.42% 2.16% -3.22% 18.93 0.004%
M 222.94 54.30 13.14 10.29 -5.86% 10.13% 1.55% 11.73 -3.82% 1.79% -2.26% 12.11 -2.98% 2.49% -3.79% 16.31 0.003%
I 156.45 19.40 6.79 14.22 -2.37% 4.13% -1.00% 11.77 -1.93% 1.01% -2.10% 14.55 -1.88% 1.08% -2.15% 15.29 0.003%
S 248.81 70.14 21.98 10.90 -8.56% 9.83% 5.32% 12.75 -5.98% 1.88% -0.87% 12.63 -4.91% 1.71% -2.15% 21.68 0.004%
M 222.94 54.30 17.39 10.13 -5.86% 10.13% 3.58% 11.63 -3.49% 1.32% -1.42% 11.24 -2.64% 2.40% -3.40% 16.16 0.003%
I 388.28 30.35 8.24 1908.98 -1.44% 4.47% -2.05% 1793.74 -1.29% 1.76% -2.63% 1985.78 -1.29% 2.40% -2.68% 2033.49 0.001%
S 518.18 121.91 27.40 1948.89 -3.26% 6.44% 2.55% 1827.95 -1.89% 1.00% -1.29% 2149.47 -1.75% 1.10% -1.44% 2442.34 0.004%
M 501.44 95.57 22.44 1541.62 -3.94% 8.46% 2.39% 1880.94 -2.56% 1.64% -1.81% 1933.47 -2.00% 1.97% -3.45% 2087.18 0.002%
I 388.39 30.35 10.67 1895.06 -1.47% 4.33% -0.71% 1833.01 -1.22% 1.29% -1.62% 1954.49 -1.14% 1.59% -1.79% 2057.66 0.001%
S 518.77 122.04 37.13 2052.85 -3.36% 6.34% 3.52% 1962.38 -1.78% 0.35% -0.85% 2158.80 -1.70% 0.82% -1.33% 2788.51 0.004%
M 501.57 95.57 29.93 1665.28 -3.96% 8.39% 4.20% 1904.07 -2.35% 1.05% -0.84% 2008.51 -1.62% 2.03% -3.70% 2188.15 0.002%
time-dependent emissions (EM-LI) time-dependent emissions (EM-LD)time-dependent travel times (TT)
10
30
distances (DIST)
6350
12700
6350
12700
Table 3: Summary TSP results
Table 3 presents a summary of the results from our TSP experiments with homogeneous load quantities.
As with Table 2, the table gives the number of customers in an instance (cust), the gross weight of the vehicle
(wt), and the category of instances (inst). We present the average results from the ﬁve customer instances
per category (inner city/suburban/mixed) over the four departure times. We label the metrics associated
21
with each objective as follows: the average total duration of the route (dur ), the average total distance of
the route (dist), the average total emissions (em), and the average runtime of the routing algorithm (CPU ).
As indicated in Section 5.2, for TT, EM-LI, and EM-LD, we report the results for duration, distance, and
emissions in terms of the percentage change in the values relative to the values found when computing DIST
routes. For the EM-LD routes, we also report the percentage of online path computations that were needed
to ﬁnd the solution (paths).
First, we see that, as expected, the actual emissions increase with the number of customers for each type
of instance, but there does not seem to be an increasing percentage savings from optimizing emissions. The
number of customers clearly has a direct impact, though, on the CPU time. For all four objectives, the
runtimes get longer with more customers because the routing problem becomes harder to solve. For the
EM-LD routes, the online path computations cause some increase in the time to produce EM-LD routes over
EM-LI routes (e.g. 2442 seconds versus 2149 seconds for 30 customer suburban instances with a standard
weight vehicle).
Across the diﬀerent types of datasets, we consistently see that the most emissions occur with the suburban
and mixed instances. This is expected since vehicles are traveling further than with inner city instances.
Interestingly, we see that the savings found from optimizing emissions is consistently highest for the mixed
instances, even though, on average, suburban instances yield higher total emissions. For example, with
10 customers and a heavy weight vehicle, the savings in emissions with EM-LD are 2.15% for inner city
instances, 2.15% for suburban instances, and 3.40% for mixed instances. These results imply that it is
particularly important to consider emissions objectives for mixed instances.
In comparing the EM-LI and EM-LD routes, the incorporation of load translates to lower average emis-
sions (and larger savings) in all of these tests. In general, the savings found from EM-LD versus EM-LI tends
to be higher for the mixed and suburban instances and is the highest for the instances with heavy weight
vehicles. For example, for 10 customers and the heavy weight vehicle, the savings from EM-LD versus EM-LI
are 2.15% versus 2.10% for inner city instances, 2.15% versus 0.87% for suburban instances, and 3.40% versus
1.42% for mixed instances. These results indicate the consideration of load is less important for inner city
instances than for suburban and mixed instances and is particularly important with heavy weight vehicles.
22
The average runtime for the EM-LD experiments is always larger than for the EM-LI experiments, with
the largest percentage gains in runtime occurring with the suburban instances. Thus, the more accurate
modeling of EM-LD leads to reduced emissions but at the cost of runtime.
Table 3 also indicates an interesting tradeoﬀ between distance, duration, and emissions. Since DIST
routes minimize distance, it is not surprising that the distance increases when emissions are optimized. It is
interesting, though, that the percentage changes in distance are usually less than the percentage changes in
emissions. For example, with 10 customers/standard weight vehicles, the distance increases, on average, by
1.37% to create 3.10% savings in emissions for inner city instances with EM-LD, the distance increases by
2.16% to create 3.22% savings in emissions for suburban instances with EM-LD, and the distance increases
by 2.49% to create 3.79% savings in emissions for mixed instances with EM-LD. The EM-LD solutions are
using paths that are slightly longer but are faster to travel. For example, for the same 10 customer/standard
weight vehicle instances, these distance increases are accompanied by decreases in duration of 2.04% for
inner city instances, 5.42% for suburban instances, and 2.98% for mixed instances. To optimize emissions,
these results indicate that a company would need to pay workers to drive less time than with optimizing for
distance, potentially yielding an additional cost savings depending on the speciﬁc costs involved.
In comparing EM-LD with TT routes, we see many opposite changes as compared with DIST routes. The
emissions-optimized routes are shorter in distance and longer in duration, as expected, than TT routes. TT
routes do yield improvements in emissions as compared with DIST routes for inner city instances, but result
in worse emissions as compared with DIST routes for suburban and mixed instances. This indicates that
the TT objective, which is a common objective used for urban route planning, is a bad proxy for emissions
for suburban and mixed instances, but can yield a reasonable improvement in emissions versus DIST routes
for inner city instances. EM-LD solutions, though, still yield better savings in emissions even for the inner
city instances.
23
6.3 Detailed Results for 10 Customers with Single Vehicle and Homogeneous
Next, we look in more detail at how the optimization of emissions aﬀects the construction of single routes
with homogeneous load quantities across diﬀerent instances and departure times. Table 4 shows the results
for the ﬁve inner city, suburban and mixed instances (inst) with 10 customers and a standard vehicle. The
results in Table 4 follow the structure of Table 3, but the results are speciﬁc to a particular instance/departure
time.
In Table 4, we see that the savings in emissions can vary quite a bit among instances. Of note are
instances such as M3 with a departure time of 06:30 that oﬀers an improvement of 13.04% in emissions
with an EM-LD route. We also see that the savings in emissions can vary quite a bit among the diﬀerent
departure times for the same instance. For example, instance S5 has savings in emissions of 1.81% at 6:30,
4.29% at 12:30, 3.95% at 15:30, and 1.30% at 19:30 when using an EM-LD route. For the DIST routes,
the highest emissions tend to occur when start times lead to driving during rush hour periods, which is not
surprising because DIST routes tend to take paths that are shorter, but more likely to include emissions
causing congestion. The emissions objectives create consistent improvements in emissions, but there does
not seem to be a pattern with regard to what time of day the most emissions savings can be obtained.
In looking at these detailed results, we see that the EM-LI routes can sometimes yield much higher
emissions than the DIST routes. For example, the EM-LI route for instance M4 at 19:30 has 6.51% higher
emissions than the DIST route. This compounds the importance of considering load for suburban and mixed
instances. Table 3 shows that EM-LI and EM-LD yield average diﬀerences in emissions of around 1-2%.
This average is based on instances where both methods choose the same routes (and thus no diﬀerence in
savings) and instances with much higher diﬀerences in emissions, such as a diﬀerence of 8.39% for the M4
instance at 19:30. However, these improvements can come at a cost in terms of CPU time. For example,
instance S2 at 12:30 has a runtime 10 seconds longer when load is considered (25.97 vs. 15.72 seconds).
To understand how the diﬀerent objectives impact the underlying structure of the routes, we examine
the diﬀerent routes created for a sample instance and departure time. For instance S3 at 6:30, the routes
are shown in Figures 3(a), 3(b), 3(c) and 3(d). The DIST route goes counterclockwise around the city
24
TSP 10 Customer Instances, Homogeneous Load, Standard Vehicle
inst dep dur dist em CPU dur dist em CPU dur dist em CPU dur dist em CPU paths
06:30 155.14 18.27 5.02 12.34 -2.92% 5.04% -3.59% 10.22 -2.71% 0.99% -4.38% 13.32 -2.71% 0.99% -4.38% 15.71 0.002%
12:30 148.74 17.59 4.58 14.34 -0.49% 0.63% -0.66% 12.86 -0.45% 0.00% -0.87% 15.69 -0.45% 0.00% -0.87% 17.31 0.002%
15:30 155.18 17.59 4.96 14.75 -1.01% 0.34% -1.81% 14.46 -1.01% 0.34% -1.81% 16.32 -1.01% 0.34% -1.81% 17.97 0.003%
19:30 146.31 17.59 4.47 14.81 -1.72% 1.59% -2.68% 12.74 -1.71% 0.45% -2.91% 15.13 -1.71% 0.45% -2.91% 16.79 0.002%
06:30 158.29 19.39 5.34 15.82 -1.00% 7.27% 1.12% 12.19 -0.64% 0.41% -0.75% 15.26 -0.64% 0.41% -0.75% 15.12 0.002%
12:30 156.62 19.37 5.23 15.61 -0.89% 3.56% -0.19% 12.26 -0.73% 2.22% -0.38% 15.55 -0.73% 2.22% -0.38% 16.49 0.001%
15:30 162.59 19.04 5.54 16.48 -2.42% 8.98% -1.44% 12.75 -1.57% 2.36% -1.99% 15.34 -1.57% 2.36% -1.99% 16.05 0.001%
19:30 153.63 19.37 5.10 16.45 -2.13% 1.19% -3.14% 14.08 -2.08% 0.52% -3.33% 17.59 -2.08% 0.52% -3.33% 17.58 0.003%
06:30 153.25 21.26 5.31 10.64 -1.56% 3.34% -1.32% 13.09 -0.98% 1.22% -1.51% 15.13 -0.98% 1.22% -1.51% 14.61 0.003%
12:30 149.81 21.26 5.11 11.28 -0.20% 1.93% 0.78% 12.17 0.00% 0.00% 0.00% 13.28 0.00% 0.00% 0.00% 13.89 0.002%
15:30 158.20 21.18 5.57 11.35 -0.95% 1.70% -1.26% 11.85 -0.88% 1.09% -1.44% 13.43 -0.88% 1.09% -1.44% 14.19 0.002%
19:30 147.88 21.21 5.07 11.23 -0.24% 0.94% -0.20% 9.17 -0.24% 0.94% -0.20% 12.81 -0.24% 0.94% -0.20% 13.29 0.001%
06:30 156.75 18.39 5.11 13.70 -4.08% 4.73% -3.91% 9.28 -3.74% 1.14% -6.07% 11.66 -3.74% 1.14% -6.07% 14.33 0.004%
12:30 158.02 18.42 5.20 13.68 -3.78% 2.50% -5.77% 8.04 -3.76% 2.01% -5.77% 12.49 -3.50% 2.44% -5.77% 13.16 0.002%
15:30 162.22 18.36 5.42 14.70 -3.99% 8.44% -2.95% 9.66 -3.60% 2.18% -5.17% 11.63 -3.60% 2.18% -5.17% 12.92 0.001%
19:30 150.32 18.36 4.76 14.98 -2.15% 2.78% -2.94% 9.36 -1.92% 1.25% -2.73% 13.17 -2.15% 2.78% -2.94% 14.18 0.002%
06:30 164.28 20.36 5.80 10.77 -4.50% 6.29% -4.83% 8.57 -3.38% 1.52% -5.17% 9.72 -3.38% 1.52% -5.17% 9.22 0.005%
12:30 163.68 20.37 5.74 10.90 -4.80% 6.33% -5.40% 9.21 -4.69% 5.65% -5.57% 10.78 -3.52% 0.49% -5.75% 12.30 0.001%
15:30 170.42 20.24 6.12 11.75 -4.55% 7.36% -4.90% 10.84 -4.51% 4.30% -6.21% 10.85 -4.51% 4.30% -6.21% 10.83 0.002%
19:30 157.74 20.41 5.44 11.28 -4.09% 7.64% -3.31% 8.35 -3.42% 2.06% -5.33% 10.54 -3.42% 2.06% -5.33% 11.06 0.004%
Avg 156.45 19.40 5.24 13.34 -2.37% 4.13% -2.42% 11.05 -2.10% 1.53% -3.08% 13.48 -2.04% 1.37% -3.10% 14.35 0.002%
06:30 247.02 61.71 14.98 7.91 -7.41% 23.24% 9.28% 8.86 -6.77% 3.35% -3.67% 9.47 -6.77% 3.35% -3.67% 18.87 0.005%
12:30 255.61 61.64 15.40 8.24 -14.54% 2.81% -6.75% 8.35 -14.18% 1.78% -6.95% 10.69 -13.64% 3.11% -10.19% 15.96 0.004%
15:30 253.89 61.53 15.30 8.14 -9.59% 3.02% -7.12% 9.54 -9.29% 2.45% -7.19% 12.03 -9.29% 2.45% -7.19% 21.42 0.004%
19:30 221.59 61.38 13.58 9.90 -7.36% 4.45% -2.36% 9.67 -6.59% 2.44% -3.31% 8.96 -6.59% 2.44% -3.31% 17.40 0.003%
06:30 273.22 90.51 21.21 11.09 -8.84% 2.97% -2.55% 14.29 -7.35% 1.46% -3.06% 15.94 -7.18% 1.60% -3.06% 19.46 0.005%
12:30 276.23 92.98 21.30 13.39 -10.04% 7.63% -0.61% 13.23 -7.47% 3.37% -3.94% 15.72 -7.47% 3.37% -3.94% 25.97 0.004%
15:30 268.58 92.08 20.96 18.06 -6.03% 6.69% 1.77% 16.98 -3.76% 1.37% -1.15% 21.35 -3.76% 1.37% -1.15% 25.93 0.004%
19:30 258.34 99.54 21.34 14.89 -7.63% 5.07% 3.00% 12.78 -5.56% 1.93% 0.84% 16.03 -2.90% 0.65% -0.84% 25.23 0.003%
06:30 253.77 64.35 15.58 5.83 -8.15% 6.68% 3.53% 9.04 -6.86% 1.90% 0.13% 8.28 -5.52% 2.49% -2.57% 14.98 0.007%
12:30 256.89 64.43 15.87 5.24 -12.27% 10.29% 2.33% 13.64 -8.65% 3.90% -2.58% 12.49 -2.95% 0.54% -2.65% 10.83 0.005%
15:30 264.28 64.20 16.10 5.62 -9.72% 20.55% 13.79% 9.01 -7.68% 2.43% -0.99% 7.21 -4.83% 2.13% -3.79% 14.22 0.005%
19:30 221.39 64.41 13.94 5.13 -6.72% 9.56% 9.25% 9.84 -2.18% 0.50% -1.94% 8.37 -2.18% 0.50% -1.94% 15.03 0.002%
06:30 234.80 62.23 15.03 15.76 -5.74% 11.63% 2.93% 14.98 -5.59% 5.06% -1.26% 16.51 -3.74% 2.84% -1.80% 22.23 0.004%
12:30 231.87 62.86 14.93 15.95 -8.26% 2.99% -3.42% 15.51 -7.83% 1.94% -4.09% 16.67 -7.83% 1.94% -4.09% 20.46 0.003%
15:30 230.79 61.83 14.73 15.17 -3.81% 4.50% 0.68% 16.52 -1.95% 3.02% -0.20% 19.11 -3.35% 4.53% -0.75% 25.77 0.003%
19:30 210.01 63.20 13.92 16.03 -5.34% 1.71% -1.15% 15.67 -2.51% 0.30% -2.08% 18.06 -2.51% 0.30% -2.08% 20.85 0.002%
06:30 257.49 68.70 16.53 7.64 -8.52% 17.19% 7.44% 11.06 -3.43% 2.34% -1.81% 9.51 -3.43% 2.34% -1.81% 15.38 0.006%
12:30 264.38 68.17 16.80 9.02 -13.33% 23.24% 7.92% 10.56 -7.80% 4.75% -4.23% 10.16 -7.47% 4.72% -4.29% 16.83 0.005%
15:30 262.79 68.12 16.71 8.44 -9.71% 13.26% 2.27% 11.83 -4.87% 1.04% -3.95% 11.36 -4.87% 1.04% -3.95% 17.56 0.005%
19:30 233.35 68.94 15.35 8.55 -8.22% 19.10% 11.34% 7.90 -2.13% 1.48% -1.30% 8.73 -2.13% 1.48% -1.30% 14.25 0.006%
Avg 248.81 70.14 16.48 10.50 -8.56% 9.83% 2.58% 11.96 -6.12% 2.34% -2.64% 12.83 -5.42% 2.16% -3.22% 18.93 0.004%
06:30 236.89 56.00 14.08 7.79 -5.32% 11.77% 0.43% 10.00 -4.17% 3.20% -4.47% 11.85 -4.17% 3.20% -4.47% 13.98 0.003%
12:30 222.19 55.77 13.23 10.95 -4.37% 8.52% 2.04% 11.84 0.22% 0.48% -1.28% 12.90 0.22% 0.48% -1.28% 15.07 0.003%
15:30 234.52 55.95 13.90 9.29 -3.54% 8.36% 2.09% 11.25 -1.66% 0.59% -1.37% 13.07 -1.66% 0.59% -1.37% 15.83 0.005%
19:30 207.10 56.09 12.56 8.45 -2.77% 8.88% 4.46% 10.45 1.51% 0.46% -0.32% 11.11 1.51% 0.46% -0.32% 11.02 0.003%
06:30 233.42 62.18 15.65 10.09 -5.26% 7.88% -0.06% 9.88 -3.20% 1.11% -2.36% 10.91 -2.95% 5.73% -4.47% 11.96 0.006%
12:30 227.66 65.36 15.88 7.73 -4.57% 4.42% 0.31% 8.92 -2.58% 0.89% -1.51% 10.03 -0.82% 0.54% -6.93% 16.33 0.005%
15:30 247.48 66.21 15.67 14.80 -8.90% 8.13% 1.72% 14.52 -6.87% 1.86% -1.98% 15.95 -6.87% 1.86% -1.98% 18.62 0.002%
19:30 220.76 71.02 15.15 12.83 -1.94% 1.87% -0.53% 11.98 -1.52% 0.34% -1.12% 15.05 -1.52% 0.34% -1.12% 20.11 0.002%
06:30 218.55 38.43 11.20 13.61 -11.12% 10.82% -12.14% 11.45 -9.16% 4.63% -10.71% 12.81 -8.86% 5.39% -13.04% 16.36 0.002%
12:30 200.91 38.09 9.50 11.81 -5.10% 4.33% 0.00% 9.81 -4.49% 3.49% 0.32% 14.84 -3.61% 1.94% -3.47% 19.51 0.001%
15:30 213.40 38.35 10.23 11.73 -7.51% 9.62% -2.44% 14.63 -4.52% 3.75% 0.00% 11.23 -4.16% 2.82% -4.59% 17.92 0.002%
19:30 186.75 38.26 8.77 11.10 -4.07% 3.24% -2.85% 12.68 -3.73% 1.10% -3.65% 16.79 -3.73% 1.10% -3.65% 17.89 0.005%
06:30 231.44 58.45 13.76 6.44 -5.48% 2.58% -4.22% 9.60 -5.40% 1.71% -4.51% 6.79 -5.40% 1.71% -4.51% 16.62 0.003%
12:30 225.55 59.46 13.56 5.34 -6.86% 14.08% 6.34% 12.00 -4.21% 1.18% -3.24% 5.85 -4.21% 1.18% -3.24% 13.08 0.005%
15:30 232.74 58.27 13.85 6.95 -6.17% 11.88% 4.40% 12.19 -6.11% 1.75% -5.20% 9.68 -6.11% 1.75% -5.20% 17.24 0.003%
19:30 209.49 59.56 12.74 6.07 -6.31% 4.87% 7.54% 11.12 -4.70% 2.40% 6.51% 6.96 -2.11% 0.69% -1.88% 13.44 0.002%
06:30 225.65 52.26 13.22 10.34 -3.58% 18.03% 5.37% 13.93 -1.19% 0.67% -0.91% 13.71 0.27% 6.24% -1.97% 20.76 0.002%
12:30 227.52 52.33 13.27 12.45 -6.76% 17.62% 3.47% 11.28 -3.71% 2.12% -1.51% 12.94 -0.32% 3.48% -3.32% 17.06 0.003%
15:30 251.80 51.43 14.34 16.28 -12.34% 22.96% 0.21% 13.14 -8.77% 3.32% -6.42% 15.41 -8.02% 7.97% -7.25% 18.07 0.002%
19:30 205.04 52.61 12.30 11.78 -5.28% 22.66% 14.80% 13.97 -2.08% 0.72% -1.54% 14.33 2.90% 2.36% -1.71% 15.27 0.002%
Avg 222.94 54.30 13.14 10.29 -5.86% 10.13% 1.55% 11.73 -3.82% 1.79% -2.26% 12.11 -2.98% 2.49% -3.79% 16.31 0.003%
M5
S4
S5
M1
M2
M3
M4
S3
distances (DIST) time-dependent travel times (TT) time-dependent emissions (EM-LI) time-dependent emissions (EM-LD)
I1
I2
I3
I4
I5
S1
S2
Table 4: Results for TSP Instances with 10 customers (standard vehicle)
25
(Figure 3(a) with a total length of 64.35 km and total emissions of 15.58 kg CO2). This route uses shortcuts
wherever possible. The TT and EM-LI routes switch to clockwise customer visits (Figure 3(b), 3(c)) and
include some highways, which extends the total distance (68.65/65.57 km) and increases emissions compared
to the DIST solution (16.13/15.60 kg CO2). Finally, the EM-LD route (Figure 3(d)) switches back to
counterclockwise again, following diﬀerent paths than the DIST solution and assembling more customers,
and thus more load, at the end of the route (65.95 km, 15.18 kg CO2).
6.4 Summary of Results for Single Vehicle with Heterogeneous Load Quantities
As described in Section 5, we also experiment with heterogeneous load sizes to understand if this impacts
the savings from minimizing emissions. We again focus on 10 customer instances. Table 5 describes the type
of load size considered (load), the gross weight of the vehicle (wt), and the category of instances (inst ). The
solution values are presented as in Table 3. In the table, we also present our prior results from homogeneous
For homogeneous loads, the gain from EM-LD versus EM-LI routes is less than 2% for suburban instances
with heavy vehicles. In Table 5, for random three heavy, the additional savings are almost 3%. For farthest
three heavy, the additional savings are over 4%. For closest three heavy, the savings are more than 4.5%. For
random decreasing load, the additional savings are over 7% . Overall, these tests show that larger savings
in emissions can occur with heterogeneous loads, and it is particularly important to optimize for loads when
they are heterogeneous and vehicles are heavy.
Because the EM-LI routes do not consider varying loads on the arcs, the average distance and duration of
these routes do not change with heterogeneous versus homogeneous loads. EM-LD routes with heterogeneous
loads are on average longer and take slightly longer to travel than those with homogeneous loads. Essentially,
the routes are slightly longer so that the heavy loads can be handled more eﬃciently in terms of emissions,
which sometimes means driving longer distances with less weight.
To understand the changes in the underlying structure that occur with heterogeneous loads, we examine
Figures 4(a), 4(b), 5(a), and 5(b). In Figures 4(a) and 4(b), we examine S1 at 12:30 with the three closest
customers to the depot having heavy loads, and the vehicle is heavy weight. The EM-LI route serves the
heavy customers in the early part of the route (position 1, 4 and 5). In the EM-LD route, the heavy customers
26
(a) DIST route for S3, departure 06:30 (b) TT route for S3, departure 06:30
(c) EM-LI route for S3, departure 06:30 (d) EM-LD route for S3, departure 06:30
Figure 3: Routes for S3 at 06:30 with Homogeneous Loads and Standard Vehicle. Aerial view provided by
27
load wt inst dur dist em CPU dur dist em CPU dur dist em CPU dur dist em CPU paths
I 156.45 19.40 5.24 13.34 -2.37% 4.13% -2.42% 11.05 -2.10% 1.53% -3.08% 13.48 -2.04% 1.37% -3.10% 14.35 0.002%
S 248.81 70.14 16.48 10.50 -8.56% 9.83% 2.58% 11.96 -6.12% 2.34% -2.64% 12.83 -5.42% 2.16% -3.22% 18.93 0.004%
M 222.94 54.30 13.14 10.29 -5.86% 10.13% 1.55% 11.73 -3.82% 1.79% -2.26% 12.11 -2.98% 2.49% -3.79% 16.31 0.003%
I 156.45 19.40 6.79 14.22 -2.37% 4.13% -1.00% 11.77 -1.93% 1.01% -2.10% 14.55 -1.88% 1.08% -2.15% 15.29 0.003%
S 248.81 70.14 21.98 10.90 -8.56% 9.83% 5.32% 12.75 -5.98% 1.88% -0.87% 12.63 -4.91% 1.71% -2.15% 21.68 0.004%
M 222.94 54.30 17.39 10.13 -5.86% 10.13% 3.58% 11.63 -3.49% 1.32% -1.42% 11.24 -2.64% 2.40% -3.40% 16.16 0.003%
I 156.45 19.40 5.26 13.46 -2.37% 4.13% -2.37% 11.06 -2.09% 1.49% -2.93% 13.50 -1.85% 2.23% -3.48% 14.38 0.007%
S 248.81 70.14 16.46 10.59 -8.56% 9.83% 0.77% 12.34 -6.11% 2.32% -3.41% 12.56 -5.42% 2.94% -5.31% 31.92 0.009%
M 222.94 54.30 12.91 10.30 -5.86% 10.13% 1.63% 11.91 -3.82% 1.75% -1.33% 11.95 -2.26% 2.33% -5.97% 25.64 0.010%
I 156.45 19.40 6.82 14.31 -2.37% 4.13% -0.89% 11.67 -1.93% 1.03% -1.76% 14.68 -1.54% 2.20% -3.23% 15.98 0.009%
S 248.81 70.14 21.94 10.72 -8.56% 9.83% 2.63% 12.46 -5.99% 1.90% -2.69% 12.33 -4.74% 2.53% -5.63% 38.73 0.009%
M 222.94 54.30 16.92 9.59 -5.86% 10.13% 3.84% 11.19 -3.49% 1.33% -0.70% 10.81 -1.54% 3.10% -7.11% 24.38 0.007%
I 156.45 19.40 5.27 13.15 -2.37% 4.13% -2.48% 10.96 -2.09% 1.49% -3.04% 13.07 -1.92% 1.86% -3.46% 13.88 0.007%
S 248.81 70.14 16.51 10.33 -8.56% 9.83% 3.11% 12.05 -6.07% 2.29% -2.37% 12.42 -5.34% 2.93% -4.55% 32.82 0.011%
M 222.94 54.30 13.22 10.92 -5.86% 10.13% 1.88% 11.54 -3.81% 1.75% -2.49% 11.70 -3.70% 2.11% -3.72% 21.99 0.011%
I 156.45 19.40 6.84 14.33 -2.37% 4.13% -1.11% 11.92 -1.94% 1.06% -2.07% 14.46 -1.50% 1.62% -3.23% 15.83 0.009%
S 248.81 70.14 22.04 10.93 -8.56% 9.83% 6.10% 12.69 -6.00% 1.91% -0.21% 12.57 -3.01% 3.32% -4.58% 41.47 0.011%
M 222.94 54.30 17.54 10.24 -5.86% 10.13% 4.12% 11.36 -3.46% 1.28% -0.98% 11.16 -3.16% 2.26% -3.49% 23.89 0.009%
I 156.45 19.40 5.22 13.15 -2.37% 4.13% -2.56% 10.95 -2.08% 1.42% -3.29% 12.93 -1.94% 3.14% -3.64% 14.67 0.006%
S 248.81 70.14 16.52 10.37 -8.56% 9.83% 1.13% 11.95 -6.12% 2.28% -3.40% 12.11 -5.45% 2.87% -5.75% 34.25 0.011%
M 222.94 54.30 13.05 10.68 -5.86% 10.13% 1.64% 11.44 -3.80% 1.78% -2.10% 11.55 -2.44% 2.93% -5.60% 23.80 0.009%
I 156.45 19.40 6.74 14.15 -2.37% 4.13% -1.18% 11.79 -1.94% 1.04% -2.29% 14.34 -1.81% 3.28% -3.39% 15.90 0.010%
S 248.81 70.14 22.07 10.84 -8.56% 9.83% 3.12% 12.69 -5.97% 1.88% -2.15% 12.50 -3.59% 4.92% -6.68% 42.96 0.012%
M 222.94 54.30 17.21 10.14 -5.86% 10.13% 3.81% 11.40 -3.49% 1.33% -2.09% 10.94 -1.56% 4.61% -6.63% 24.70 0.008%
I 156.45 19.40 5.17 13.55 -2.37% 4.13% -1.66% 11.20 -2.06% 1.38% -2.98% 13.35 -1.92% 1.96% -3.57% 19.36 0.034%
S 248.81 70.14 16.36 10.49 -8.56% 9.83% 5.18% 12.16 -6.16% 2.34% -1.16% 12.43 -4.71% 4.02% -4.75% 90.68 0.042%
M 222.94 54.30 13.43 10.50 -5.86% 10.13% -0.05% 11.59 -3.86% 1.77% -3.09% 11.86 -3.07% 3.40% -5.62% 64.93 0.060%
I 156.45 19.40 6.64 14.25 -2.37% 4.13% 0.26% 11.78 -1.95% 1.06% -2.36% 14.90 -1.63% 1.80% -3.19% 21.75 0.036%
S 248.81 70.14 21.73 10.91 -8.56% 9.83% 9.40% 12.76 -5.98% 1.88% 1.85% 12.67 -3.76% 4.22% -5.47% 120.80 0.044%
M 222.94 54.30 17.96 10.13 -5.86% 10.13% 1.15% 11.79 -3.42% 1.27% -1.60% 11.55 -1.73% 2.90% -6.44% 72.03 0.057%
12700
6350
12700
random
6350
12700
farthest three
heavy
closest three
heavy
6350
time-dependent emissions (EM-LI) time-dependent emissions (EM-LD)time-dependent travel times (TT)
random three
heavy
distances (DIST)
6350
12700
6350
12700
Table 5: TSP Results for 10 Customers with Heterogeneous Loads
are shifted to the end of the route (position 6, 9 and 10), making a diﬀerence of 3.5 kg CO2between these
routes. In Figures 5(a) and 5(b), we examine S5 at 19:30 with the three farthest customers having heavy
loads, and the vehicle is of standard weight. The EM-LI route goes around the city clockwise, picking up
the heavy ones early (position 2, 4 and 8). The EM-LD partially changes the direction shifting the heavy
customers later in the route (position 3, 7, and 9) and saving an additional 0.7 kg of CO2.
6.5 Summary of Results for Multiple Vehicles
To understand the impact of emissions in the case of multiple vehicles, Table 6 presents a summary of results
of VRP optimization with diﬀerent objectives. Except for the ﬁrst column, the metrics are comparable
to those reported for the TSP. As described in Section 5, VRP optimization is based on the 30-customer
instances. However, the pickup quantity at each customer is varied in a way such that three vehicles are
required. We present the results where the ﬂeet includes three standard vehicles, three heavy vehicles, and
a mixed ﬂeet comprised of one heavy and two standard vehicles.
For the ﬂeets of three standard and three heavy vehicles, we can compare the results in Table 6 with
the 30 customer, one vehicle results in Table 3. The average total duration, distance, and emissions increase
when the number of customers per vehicle decreases and the number of vehicles increases. The CPU time,
28
(a) EM-LI route with Three Closest Customers with
Heavy Loads (position 1, 4 and 5)
(b) EM-LD route with Three Closest Customers with
Heavy Loads (position 6, 9 and 10)
Figure 4: Emissions-Minimized Routes for S1 at 12:30 with Heterogeneous Loads and Heavy Vehicle
(a) EM-LI route with Three Farthest Customers with
Heavy Loads (position 2, 4 and 8)
(b) EM-LD route with Three Farthest Customers with
Heavy Loads (position 3, 7 and 9)
Figure 5: Emissions-Minimized Routes for S5 at 19:30 with Heterogeneous Loads and Standard Vehicle
29
SummaryVRP
wt inst dur dist em CPU dur dist em CPU dur dist em CPU dur dist em CPU paths
I 424.15 43.74 11.63 208.22 -1.68% 3.79% -1.86% 203.61 -1.56% 2.15% -2.31% 232.78 -1.48% 1.85% -2.54% 241.79 0.007%
S 579.16 141.62 33.06 171.74 -5.10% 6.97% 0.98% 185.78 -3.40% 1.65% -2.38% 189.03 -3.03% 1.88% -2.69% 296.55 0.012%
M 544.66 109.67 26.44 216.36 -4.13% 7.52% 0.64% 236.50 -2.87% 1.84% -2.32% 253.31 -2.56% 2.24% -3.28% 292.58 0.006%
I 424.15 43.74 15.09 217.61 -1.68% 3.68% -0.54% 212.50 -1.42% 1.40% -1.41% 236.19 -1.32% 1.20% -1.76% 247.37 0.007%
S 580.42 141.60 44.51 176.18 -5.32% 7.59% 2.39% 194.16 -3.22% 1.06% -1.72% 198.55 -2.86% 2.03% -2.55% 336.57 0.012%
M 544.49 109.64 35.17 217.04 -4.11% 7.59% 2.09% 231.33 -2.63% 1.24% -1.43% 243.84 -1.99% 2.53% -3.10% 296.02 0.007%
I 420.41 42.61 12.77 253.62 -1.54% 3.79% -1.64% 218.86 -1.30% 1.86% -3.17% 239.68 -1.18% 2.35% -3.39% 240.48 0.014%
S 570.13 134.85 36.76 199.94 -5.07% 7.76% 2.63% 189.13 -2.65% 1.99% -1.95% 198.32 -2.55% 2.58% -3.08% 477.97 0.023%
M 539.46 106.77 30.29 236.50 -4.51% 8.02% 1.33% 241.05 -2.08% 3.26% -9.24% 196.43 -1.88% 4.01% -9.52% 342.54 0.016%
time-dependent emissions (EM-LD)time-dependent emissions (EM-LI)time-dependent travel times (TT)distances (DIST)
1 heavy, 2
standard
3 standard
3 heavy
Table 6: Summary VRP Results
SummaryVRP
veh wt inst dur dist em CPU dur dist em CPU dur dist em CPU dur dist em CPU paths
I 420.41 42.61 12.77 253.62 -1.54% 3.79% -1.64% 218.86 -1.30% 1.86% -3.17% 239.68 -1.18% 2.35% -3.39% 240.48 0.014%
S 570.13 134.85 36.76 199.94 -5.07% 7.76% 2.63% 189.13 -2.65% 1.99% -1.95% 198.32 -2.55% 2.58% -3.08% 477.97 0.023%
M 539.46 106.77 30.29 236.50 -4.51% 8.02% 1.33% 241.05 -2.08% 3.26% -9.24% 196.43 -1.88% 4.01% -9.52% 342.54 0.016%
I 423.06 43.44 13.06 209.33 -1.85% 2.54% -2.17% 189.69 -1.56% 1.58% -3.24% 208.60 -1.34% 2.20% -3.63% 259.25 0.035%
S 576.24 137.58 37.70 156.14 -4.94% 9.20% 3.76% 150.08 -2.50% 1.94% -2.62% 159.76 -2.21% 3.29% -3.46% 712.78 0.052%
M 543.34 108.35 30.21 216.18 -4.53% 9.49% 5.07% 209.33 -1.48% 3.48% -8.23% 173.62 -1.10% 4.40% -8.90% 484.34 0.038%
I 422.01 43.16 13.21 262.96 -1.80% 3.91% -1.17% 206.00 -1.29% 2.92% -4.32% 210.28 -0.63% 4.82% -5.15% 262.65 0.046%
S 574.78 137.74 38.75 195.69 -5.02% 7.97% 4.12% 177.19 -1.41% 5.36% -6.65% 181.85 -1.05% 5.87% -7.82% 991.90 0.064%
M 542.45 107.77 30.00 238.78 -4.64% 9.27% 6.33% 229.28 -1.89% 3.16% -9.19% 200.31 -1.57% 3.85% -10.25% 534.71 0.035%
I 425.42 43.85 13.22 217.53 -2.06% 4.48% -2.55% 196.34 -1.80% 2.56% -3.88% 213.77 -1.82% 2.59% -4.04% 257.08 0.033%
S 587.33 142.29 38.07 153.13 -5.78% 9.05% 2.63% 145.24 -3.53% 1.92% -2.57% 159.86 -3.30% 1.89% -2.97% 660.98 0.052%
M 545.92 109.83 30.53 161.64 -4.41% 7.86% 3.46% 189.96 -2.74% 1.78% -2.16% 188.02 -1.71% 4.49% -3.21% 454.98 0.029%
Equal load: every customer has the same pickup weight.
Random 15 heavy: Based on the average pickup weight from ”equal load”, 15 randomly selected customers have 50% more and the others 50% less.
Closest 15 heavy: Based on the average pickup weight from ”equal load”, the 15 closest customers from the depot have 50% more and the others 50% less.
Farthest 15 heavy: Based on the average pickup weight from ”equal load”, the 15 farthest customers from the depot have 50% more and the others 50% less.
time-dependent emissions (EM-LD)
farthest 15
heavy
mixed fleet (1 heavy, 2 standard)
random 15
heavy
time-dependent emissions (EM-LI)time-dependent travel times (TT)distances (DIST)
closest 15
heavy
Table 7: Summary VRP Results with Heterogeneous Loads
though, decreases as the routing problems become easier when the 30 customers are spread over three vehicles
instead of just one. With more vehicles, there does not seem to be a signiﬁcant pattern with regard to the
savings in emissions for the ﬂeet of three standard and three heavy vehicles as compared with the one vehicle
case. What stands out in Table 6 is the increased savings in emissions that result with the use of a mixed
ﬂeet. For example, with the mixed ﬂeet, the savings from EM-LD for the mixed instances is 9.52%. This
indicates that with a mixed ﬂeet it is particularly important to consider emissions in the route planning.
With more vehicles, the performance of EM-LI versus EM-LD routes do not seem to exhibit any notable
changes from the one vehicle case. The EM-LD routes always yield better emissions, on average, but the EM-
LI results are usually within 1-2%. It is somewhat surprising that the gap between the emissions found from
EM-LI versus EM-LD is not larger for the mixed ﬂeet. The reason why EM-LI seems to perform relatively
well is that it does consider the empty weight of the vehicles when assigning customers to the diﬀerent
vehicles. This consideration is enough to capture a signiﬁcant portion of the diﬀerences in emissions when
using mixed ﬂeet.
For the mixed ﬂeet, we also experiment with heterogeneous loads to help build insights into what types
of loads are assigned to the diﬀerent vehicle types. These results are presented in Table 7. Table 7 reveals
some of the highest average savings from optimizing emissions in all of the experiments. For example, the
results for closest 15 heavy demonstrate a savings in emissions of 10.25% for EM-LD versus DIST routes for
30
mixed instances. As with the single vehicle experiments, the most savings in emissions typically come from
the mixed instances with one major exception. For the farthest 15 heavy instances, the savings in emissions
are higher for inner city instances than with suburban and mixed instances. This indicates that the DIST
routes are not a very good proxy for emissions even in the inner city with mixed ﬂeets and heterogeneous
Across the diﬀerent types of load assignments, we see the biggest savings in emissions from EM-LD come
from these closest 15 heavy instances. To understand what is happening in such instances, we examine
Figures 6(a), 6(b), and 6(c). In these pictures, we examine the route that is assigned to the heavy vehicle for
EM-LI and EM-LD routes and compare it to the DIST route as a benchmark. Figures 6(a) shows that, in the
DIST solution, the heavy vehicle makes 15 stops and travels a long route. From a distance perspective, this
makes sense as a bigger vehicle can make more deliveries, creating a lower total distance but high emissions.
Note that we do not show the TT route here because it also assigns many customers to the ﬁrst vehicle
because of its larger capacity. As shown in Figure 6(b), the heavy vehicle remains close to the depot in the
EM-LI solution to pick up the heavy customers and travels much less distance, creating less emissions. The
diﬀerence between the EM-LI and DIST solutions comes from the fact that the EM-LI solution recognizes
the weight of the heavy vehicle, even if it does not account for the impact of customers’ loads. In Figure 6(c),
the consideration of load by EM-LD route leads to an even tighter/shorter route than the one chosen by
EM-LI, which translates to even less emissions for EM-LD. As with single vehicle results, these results imply
that optimizing for emissions is particularly important for heterogeneous loads. The main diﬀerence is that
with the mixed ﬂeet the resulting percentage change in emissions can be much higher than in the single
vehicle case.
7 Conclusions
In this paper, we explore the minimization of expected emissions from vehicle routing in urban areas, where
minimization of emissions will be one of the major challenges in the short- and mid-term future. A key
challenge in the work is the development of a computationally tractable way of considering detailed load
information. Unlike in the case of traditional vehicle routing objectives, the presence of the load requires
31
(a) DIST route – heavy customers at position 1, 2, 3, 4,
10, 11, 12, 13
(b) EM-LI route – all shown customers are heavy
(c) EM-LD route – all shown customers are heavy
Figure 6: Heterogeneous ﬂeet – M2, departure 19:30, closest 15 heavy customers, routes for the heavy vehicle
32
customer-to-customer expected time-dependent emissions-minimizing paths to be computed online. We
introduce an analytical result that allows us to precompute the majority of such paths.
Through our experiments, we identify that signiﬁcant savings in emissions can be achieved when opti-
mizing emissions for mixed customer instances, while inner city tours show signiﬁcantly less improvements,
mainly because routes are relatively short. We show that it is also more important to minimize emissions
when trucks are heavier and can potentially handle more load. The most signiﬁcant savings in emissions
with single vehicles were noted for routes with heterogeneous loads. For heterogeneous loads, our solution
framework shifts heavy pickups to the end of a route, ensuring that we carry most of the load over a short
distance. In general, we ﬁnd that savings in emissions can be found with relatively small increases in tour
durations, indicating that big savings in emissions can be possible without large costs to companies. We also
extend our results to multiple vehicles. We ﬁnd large savings with mixed ﬂeets since vehicle weights are a
big driver in emissions. For the mixed ﬂeet, we see particular savings with heterogeneous loads, especially
when heavy customers are close to the depot. Our emissions-optimized solutions ﬁnd that the heavy vehicle
stays close to the depot using a short route to pick up the heavy customers.
There exist a number of interesting avenues of future work. For one, this work does not include the
possibility that vehicles might wait at the depot or at customers for favorable road conditions. Adding
waiting would change the objective, and, depending on the costs involved, may change the routes that
are selected. Further, it would be interesting to see how diﬀerent emissions models impact the structure
of resulting routes. In addition, an interesting path for future research would be to extend local search
procedures such that deliveries can be handled as eﬃciently as pickups. So far, local-search based evaluation
of a route for deliveries would lead to re-evaluation of costs for the the whole route, which is computationally
ineﬃcient. Finally, the functions describing emissions are similar to those that describe fuel consumption. As
a result, the methods discussed in this paper could be applied to routing problems that seek to incorporate
the costs of fuel in urban areas with uncertain speeds.
33
References
Barth, M. and Boriboonsomsin, K. (2008). Real-World Carbon Dioxide Impacts of Traﬃc Congestion.
Transportation Research Record: Journal of the Transportation Research Board, (2058):163–171.
Bektas, T. and Laporte, G. (2011). The Pollution-Routing Problem. Transportation Research Part B:
Methodological, 45(8):1232–1250.
Demir, E., Bektas, T., and Laporte, G. (2011). A comparative analysis of several vehicle emission models for
road freight transportation. Transportation Research Part D: Transport and Environment, 16(5):347–357.
Demir, E., Bektas, T., and Laporte, G. (2012). An Adaptive Large Neighborhood Search Heuristic for the
Pollution-Routing Problem. European Journal of Operational Research, 223(2):346–359.
Demir, E., Bekta¸s, T., and Laporte, G. (2014). A review of recent research on green road freight transporta-
tion. European Journal of Operational Research, 237(3):775–793.
Ehmke, J. F. (2012). Integration of Information and Optimization Models for Routing in City Logistics,
volume 177 of International Series in Operations Research & Management Science. Springer, New York.
Ehmke, J. F., Campbell, A. M., and Thomas, B. W. (2015). Data-driven approaches for emissions-minimized
paths in urban areas. Accepted for publication in Computers and Operations Research.
Ehmke, J. F., Meisel, S., and Mattfeld, D. C. (2009). Data Chain Management for Planning in City Logistics.
International Journal of Data Mining, Modelling and Management, 1(4):335–356.
Ehmke, J. F., Meisel, S., and Mattfeld, D. C. (2012). Floating car based travel times for city logistics.
Transportation Research Part C: Emerging Technologies, 21(1):338–352.
Express, T. (2012). TNT Express Annual Report 2012. Technical report, TNT Express.
Figliozzi, M. (2010). Vehicle Routing Problem for Emissions Minimization. Transportation Research Record:
Journal of the Transportation Research Board, (2197).
Fleischmann, B., Gietz, M., and Gnutzmann, S. (2004). Time-varying travel times in vehicle routing.
Transportation Science, 38(2):160–172.
34
Franceschetti, A., Honhon, D., Van Woensel, T., Bekta¸s, T., and Laporte, G. (2013). The time-dependent
pollution-routing problem. Transportation Research Part B: Methodological, 56:265–293.
Fukasawa, R., He, Q., and Song, Y. (2015). A branch-cut-and-price algorithm for the energy minimization
vehicle routing problem. To appear in Transportation Science.
Hwang, T. and Ouyang, Y. (2015). Urban freight truck routing under stochastic congestion and emission
considerations. Sustainability, 7(6):6610–6625.
Jabali, O., Van Woensel, T., and de Kok, A. (2012). Analysis of Travel Times and CO2 Emissions in
Time-Dependent Vehicle Routing. Production and Operations Management, 21(6):1060–1074.
Kara, ˙
I., Kara, B. Y., and Yetis, M. K. (2007). Energy minimizing vehicle routing problem. In Dress, A.,
Xu, Y., and Zhu, B., editors, Combinatorial Optimization and Applications, volume 4616 of Lecture Notes
in Computer Science, pages 62–71. Springer Berlin Heidelberg.
Kramer, R. and Subramanian, A. (2015). A matheuristic approach for the Pollution-Routing Problem.
European Journal of Operational Research, 243(2):523–539.
Kr¨oger, M. (2013). Stau-Analyse: Stuttgart qu¨alt seine Pendler am meisten (Analysis of traﬃc jams:
Stuttgart tortures its commuters the most). Spiegel Online.
Lin, C., Choy, K., Ho, G., Chung, S., and Lam, H. (2014). Survey of Green Vehicle Routing Problem: Past
and future trends. Expert Systems with Applications, 41(4):1118–1138.
Lorkowski, S., Mieth, P., Thiessenhusen, K.-U., Chauhan, D., Passfeld, B., and Scafer, R.-P. (2004). To-
wards Area-Wide Monitoring-Applications derived from Probe Vehicle Data. In AATT 2004, pages 389–
394.
Maden, W., Eglese, R., and Black, D. (2010). Vehicle routing and scheduling with time-varying data: A
case study. Journal of the Operational Research Society, 61:515–522.
Qian, J. and Eglese, R. (2014). Finding least fuel emission paths in a network with time-varying speeds.
Networks, 63(1):96–106.
35
Solomon, M. M. (1987). Algorithms for the vehicle routing and scheduling problems with time window
constraints. Operations Research, 35:254–265.
UPS (2013). Saving Fuel: UPS Saves Fuel and Reduces Emissions the ”Right” Way by Avoiding Left Turns.
van Woensel, T., Creten, R., and Vandaele, N. (2001). Managing the Environmental Externalities of Traﬃc
Logistics: the Issue of Emissions. Production and Operations Management, 10(2):207–223.
Vidal, T., Crainic, T. G., Gendreau, M., and Prins, C. (2013). Heuristics for multi-attribute vehicle routing
problems: A survey and synthesis. European Journal of Operational Research, 231(1):1–21.
Wen, L., C¸ atay, B., and Eglese, R. (2014). Finding a minimum cost path between a pair of nodes in a time-
varying road network with a congestion charge. European Journal of Operational Research, 236(3):915–923.
Wen, L. and Eglese, R. (2015). Minimum cost VRP with time-dependent speed data and congestion charge.
Computers & Operations Research, 56:41–50.
Xiao, Y. and Konak, A. (2015). Green Vehicle Routing Problem with Time-Varying Traﬃc Congestion. In
Proceedings of the 14th INFORMS Computing Society Conference, pages 134–148, Richmond, Virginia.
Yildirim, M. U. and Catay, B. (2014). A fast algorithm for ﬁnding the greenest path on road networks. In
Proceedings of the 7th International Symposium on Transportation Technologies, pages 233–242, Istanbul,
Turky.
Zachariadis, E. E., Tarantilis, C. D., and Kiranoudis, C. T. (2015). The load-dependent vehicle routing
problem and its pick-up and delivery extension. Transportation Research Part B: Methodological, 71:158–
181.
36
Article
Full-text available
In this paper, we investigate the Pollution Routing Problem in dynamic environments (DPRP). It consists in determining the routing plan of a fleet of vehicles supplying a set of customers, while minimizing the traveled distance and CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CO_2$$\end{document} emissions. The dynamic character of the problem is manifested by the occurrence of new customer demands when the working plan is in progress. Consequently, the planned routes have to be adapted in real time to include the locations of the new customers. In order to efficiently manage the trade-off between the two considered objectives, a new vector evaluated evolutionary algorithm augmented with an exploitation phase and hyper-mutation is proposed. This combination aims to reinforce the refinement of compromised solutions, and to speed up adaptation after the occurrence of a change in the problem inputs. An experimental study is conducted to test the proposed approaches on mono-objective and bi-objective test problems, and against well known approaches from the literature. The obtained results show that our proposal performs well and is highly competitive compared with the competing meta-heuristics.
Article
Full-text available
This study designs a new variant of the capacitated vehicle routing problem (CVRP) under a fuzzy environment. In CVRP, several vehicles start their journey from a central depot to provide services to different cities and finally return to the depot. This paper introduces an additional time beyond the service time at each city to fulfill the pre-ordered demands. The need for this excess service time is to provide the services to new customers who are not enlisted at the start of the process. It is a market enhancement step. The proposed model’s main objective is to find the maximum time-dependent profit by using the optimum number of vehicles in an appropriate route and spending optimum excess service time in each city. The model considers travel time and travel cost as fuzzy numbers. An expected value model (EVM) is formulated using the credibility approach on fuzzy variables. A hybrid meta-heuristic method combining a genetic algorithm (GA) and bacteria foraging optimization algorithm (BFOA) is designed to solve the proposed model. The proposed model is explained with the help of some numerical examples. Sensitivity analyses based on different independent parameters of the algorithms are also conducted.
Article
Full-text available
The transportation science research community has contributed to numerous practical and intellectual innovations and improvements over the last decades. Technological advancements have broadened and amplified the potential impacts of our field. At the same time, the world and its communities are facing greater and more serious challenges than ever before. In this paper, we call upon the transportation science research community to work on a research agenda that addresses some of the most important of these challenges. This agenda is guided by the sustainable development goals outlined by the United Nations and organized into three areas: (1) well‐being, (2) infrastructure, and, (3) natural environment. For each area, we identify current and future challenges as well as research directions to address those challenges.
Conference Paper
Full-text available
We present a linear mixed integer programming model for the time-dependent heterogeneous green vehicle routing and scheduling problem (GVRSP) with the objective of minimizing total carbon dioxide emissions and weighted tardiness. Instead of discrete time intervals, the proposed model takes the traveled distances of arcs in different time periods as decision variables to determine the travel schedules of vehicles. We propose an exact dynamic programming method to calculate the optimal discrete departure/arriving time for the GVRSP. The dynamic programming method significantly reduces the computational complexity of the GVRSP when applying existing heuristic algorithms to solve large-sized problems. A genetic algorithm with dynamic programming (GA-DP) is developed to solve the formulated problem. Computational experiments are carried out to study the efficiency of the proposed hybrid solution approach with promising results.
Article
Full-text available
Freight trucks are known to be a major source of air pollutants as well as greenhouse gas emissions in U.S. metropolitan areas, and they have significant effects on air quality and global climate change. Emissions from freight trucks during their deliveries should be considered by the trucking service sector when they make routing decisions. This study proposes a model that incorporates total delivery time, various emissions including CO2, VOC, NOX, and PM from freight truck activities, and a penalty for late or early arrival into the total cost objective of a stochastic shortest path problem. We focus on urban transportation networks in which random congestion states on each link follows an independent probability distribution. Our model finds the best truck routing on a given network so as to minimize the expected total cost. This problem is formulated into a mathematical model, and two solution algorithms including a dynamic programming approach and a deterministic shortest path heuristic are proposed. Numerical examples show that the proposed approach performs very well even for the large-size U.S. urban networks.
Article
Concerns about air quality and global warming have led to numerous initiatives to reduce emissions. In general, emissions are proportional to the amount of fuel consumed, and the amount of fuel consumed is a function of speed, distance, acceleration, and weight of the vehicle. In urban areas, vehicles must often travel at the speed of traffic, and congestion can impact this speed particularly at certain times of day. Further, for any given time of day, the observations of speeds on an arc can exhibit significant variability. Because of the nonlinearity of emissions curves, optimizing emissions in an urban area requires explicit consideration of the variability in the speed of traffic on arcs in the network. We introduce a shortest path algorithm that incorporates sampling to both account for variability in travel speeds and to estimate arrival time distributions at nodes on a path. We also suggest a method for transforming speed data into time-dependent emissions values thus converting the problem into a time-dependent, but deterministic shortest path problem. Our results demonstrate the effectiveness of the proposed approaches in reducing emissions relative to the use of minimum distance and time-dependent paths. In this paper, we also identify some of the challenges associated with using large data sets.
Article
We study a variant of the capacitated vehicle routing problem where the cost over each arc is defined as the product of the arc length and the weight of the vehicle when it traverses that arc. We propose two new mixed-integer linear programming formulations for the problem: an arc-load formulation and a set partitioning formulation based on q-routes with additional constraints. A family of cycle elimination constraints are derived for the arc-load formulation. We then compare the linear programming (LP) relaxations of these formulations with the two-index one-commodity flow formulation proposed in the literature. In particular, we show that the arc-load formulation with the new cycle elimination constraints gives the same LP bound as the set partitioning formulation based on 2-cycle-free q-routes, which is stronger than the LP bound given by the two-index one-commodity flow formulation. We propose a branch-and-cut algorithm for the arc-load formulation, and a branch-cut-and-price algorithm for the set partitioning formulation strengthened by additional constraints. Computational results on instances from the literature demonstrate that a significant improvement can be achieved by the branch-cut-and-price algorithm over other methods.
Article
A heuristic algorithm, called LANCOST, is introduced for vehicle routing and scheduling problems to minimize the total travel cost, where the total travel cost includes fuel cost, driver cost and congestion charge. The fuel cost required is influenced by the speed. The speed for a vehicle to travel along any road in the network varies according to the time of travel. The variation in speed is caused by congestion which is greatest during morning and evening rush hours. If a vehicle enters the congestion charge zone at any time, a fixed charge is applied. A benchmark dataset is designed to test the algorithm. The algorithm is also used to schedule a fleet of delivery vehicles operating in the London area.
Article
City logistics routing requires time-dependent travel times for each network link. We rely on the concept of Floating Car Data (FCD) to develop and provide such travel times. Different levels of aggregation in the determination of time-dependent travel times from a database of historical FCD are presented and evaluated with regard to routing quality. Furthermore, a Data Mining approach is introduced, allowing for a substantial reduction of the volume of input data required for city logistics routing. The different approaches are investigated and evaluated by a huge amount of FCD collected for the urban area of Stuttgart, Germany. The results show that the Data Mining approach enables efficient provision of time-dependent travel times without a significant loss of routing quality for city logistics applications.
Article
The Time-Dependent Pollution-Routing Problem (TDPRP) consists of routing a fleet of vehicles in order to serve a set of customers and determining the speeds on each leg of the routes. The cost function includes emissions and driver costs, taking into account traffic congestion which, at peak periods, significantly restricts vehicle speeds and increases emissions. We describe an integer linear programming formulation of the TDPRP and provide illustrative examples to motivate the problem and give insights about the tradeoffs it involves. We also provide an analytical characterization of the optimal solutions for a single-arc version of the problem, identifying conditions under which it is optimal to wait idly at certain locations in order to avoid congestion and to reduce the cost of emissions. Building on these analytical results we describe a novel departure time and speed optimization algorithm for the cases when the route is fixed. Finally, using benchmark instances, we present results on the computational performance of the proposed formulation and on the speed optimization procedure.
Article
The present paper examines a Vehicle Routing Problem (VRP) of major practical importance which is referred to as the Load-Dependent VRP (LDVRP). LDVRP is applicable for transportation activities where the weight of the transported cargo accounts for a significant part of the vehicle gross weight. Contrary to the basic VRP which calls for the minimization of the distance travelled, the LDVRP objective is aimed at minimizing the total product of the distance travelled and the gross weight carried along this distance. Thus, it is capable of producing sensible routing plans which take into account the variation of the cargo weight along the vehicle trips. The LDVRP objective is closely related to the total energy requirements of the vehicle fleet, making it a credible alternative when the environmental aspects of transportation activities are examined and optimized. A novel LDVRP extension which considers simultaneous pick-up and delivery service is introduced, formulated and solved for the first time. To deal with large-scale instances of the examined problems, we propose a local-search algorithm. Towards an efficient implementation, the local-search algorithm employs a computational scheme which calculates the complex weighted-distance objective changes in constant time. Solution results are presented for both problems on a variety of well-known test cases demonstrating the effectiveness of the proposed solution approach. The structure of the obtained LDVRP and VRP solutions is compared in pursuit of interesting conclusions on the relative suitability of the two routing models, when the decision maker must deal with the weighted distance objective. In addition, results of a branch-and-cut procedure for small-scale instances of the LDVRP with simultaneous pick-ups and deliveries are reported. Finally, extensive computational experiments have been performed to explore the managerial implications of three key problem characteristics, namely the deviation of customer demands, the cargo to tare weight ratio, as well as the size of the available vehicle fleet.
Article
Due to the growing concern over environmental issues, regardless of whether companies are going to voluntarily incorporate green policies in practice, or will be forced to do so in the context of new legislation, change is foreseen in the future of transportation management. Assigning and scheduling vehicles to service a pre-determined set of clients is a common distribution problem. Accounting for time-dependent travel times between customers, we present a model that considers travel time, fuel, and CO2 emissions costs. Specifically, we propose a framework for modeling CO2 emissions in a time-dependent vehicle routing context. The model is solved via a tabu search procedure. As the amount of CO2 emissions is correlated with vehicle speed, our model considers limiting vehicle speed as part of the optimization. The emissions per kilometer as a function of speed are minimized at a unique speed. However, we show that in a time-dependent environment this speed is sub-optimal in terms of total emissions. This occurs if vehicles are able to avoid running into congestion periods where they incur high emissions. Clearly, considering this trade-off in the vehicle routing problem has great practical potential. In the same line, we construct bounds on the total amount of emissions to be saved by making use of the standard VRP solutions. As fuel consumption is correlated with CO2 emissions, we show that reducing emissions leads to reducing costs. For a number of experimental settings, we show that limiting vehicle speeds is desired from a total cost perspective. This namely stems from the trade-off between fuel and travel time costs.
Article
Road freight transportation is a major contributor to carbon dioxide equivalent emissions. Reducing these emissions in transportation route planning requires an understanding of vehicle emission models and their inclusion into the existing optimization methods. This paper provides a review of recent research on green road freight transportation.