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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),

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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.

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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) =

C−1

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 Pr∈R?Φr(T)≤Pr0∈RΦr0(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

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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)

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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

Google Earth.

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

240 “homogeneous load” TSP instances.

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

Load-Dependent Paths

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

with advanced precomputationwith standard precomputation

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

SummaryTSP10and30Customerswithhomogeneousloads

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

Load Quantities

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

loads for comparison.

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

Google Earth.

27

SummaryTSP10Customerswithhomogeneousandheterogeneousloads

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

decreasing load

6350

12700

farthest three

heavy

closest three

heavy

6350

time-dependent emissions (EM-LI) time-dependent emissions (EM-LD)time-dependent travel times (TT)

homogeneous load

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)

equal load

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

loads.

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 Sch¨afer, 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