Analysis Travel Times CO2 Emission VRP

Analysis of Travel Times and CO2 Emissions inTime-Dependent Vehicle Routing

O. JabaliHEC Montreal and CIRRELT, 3000 chemin de la Cote-Sainte-Catherine, H3T 2A7, Montreal, Canada, [email protected]

T. Van Woensel, A.G. de KokDepartment of Industrial Engineering and Innovation Sciences, Operations, Planning, Accounting and Control, Technische Universiteit

Eindhoven, P. O. Box 513, 5600 MB, Eindhoven, The Netherlands, [email protected], [email protected]

D ue to the growing concern over environmental issues, regardless of whether companies are going to voluntarilyincorporate 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 clientsis a common distribution problem. Accounting for time-dependent travel times between customers, we present a modelthat considers travel time, fuel, and CO2 emissions costs. Specifically, we propose a framework for modeling CO2 emis-sions in a time-dependent vehicle routing context. The model is solved via a tabu search procedure. As the amount ofCO2 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 run-ning into congestion periods where they incur high emissions. Clearly, considering this trade-off in the vehicle routingproblem has great practical potential. In the same line, we construct bounds on the total amount of emissions to be savedby making use of the standard VRP solutions. As fuel consumption is correlated with CO2 emissions, we show that reduc-ing emissions leads to reducing costs. For a number of experimental settings, we show that limiting vehicle speeds isdesired from a total cost perspective. This namely stems from the trade-off between fuel and travel time costs.

Key words: vehicle routing problems; time-dependent travel times; CO2 emissions; green logisticsHistory: Received: July 2009; Accepted: October 2010 by Luk van Wassenhove, after 2 revisions.

1. Introduction

The ever-growing concern over greenhouse gases(GHG) has led many countries to take policy actionsaiming at emissions reductions. Most notable is theKyoto Protocol, which requires countries to reduce abasket of the six major GHG by the year 2012 by 5.2%on average (compared to their 1990 emission levels).Next to this, a number of other initiatives haveemerged particularly to control CO2 emissions. Forexample, more than 12,000 industrial plants in the EUare subjected to CO2 caps, enabling the trade of emis-sions rights between parties. For a comprehensivesurvey of GHG trade market models, the reader isreferred to Springer (2004).The importance of environmental issues is continu-

ously translated into regulations, which potentiallyhave a tangible impact on supply chain management.As a consequence, there has been an increasingamount of research on the intersection between logis-tics and environmental factors. Sbihi and Eglese(2007) identified potential combinatorial optimization

problems where Green Logistics is relevant. Corbettand Kleindorfer (2001a, 2001b) discussed the integra-tion of environmental management in operationsmanagement. Kleindorfer et al. (2001) did so in thecontext of sustainable operations management. Leeand Klassen (2008) mapped factors that initiated andimproved environmental capabilities in small- andmedium-sized enterprises over time.European road freight transport uses considerable

amounts of energy (Baumgartner et al. 2008). Vanekand Campbell (1999) note that predictions are that theUnited Kingdom will meet the Kyoto targets. How-ever, they highlight that within the period from 1985until 1995, energy use across all sectors grew only by7% while transport energy use grew by 31%. Similarfindings were observed by Leonardi and Baumgart-ner (2004). They state that in the period 1991–2001road freight traffic in Germany increased by 40%.Moreover, in 2001 traffic was responsible for about6% of total CO2 emissions in Germany. More substan-tial CO2 increases are observed in India by Singhaet al. (2004), where the CO2 emissions from road

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Vol. 0, No. 0, xxxx–xxxx 2012, pp. 1–15 DOI 10.1111/j.1937-5956.2012.01338.xISSN 1059-1478|EISSN 1937-5956|12|0|0001 © 2012 Production and Operations Management Society

transport in the year 2000 have increased by almost400% compared to 1985. In the context of examiningscenarios for GHG, Yang et al. (2008) mention that inCalifornia the transportation sector accounted forover 40% of GHG for 2006, making it by far the largestcontributor. Baumgartner et al. (2008) acknowledgethat new vehicle designs are more efficient in terms ofemissions. However, this is outweighed by the trans-port growth rate in the EU. Ericsson et al. (2006) men-tion that CO2, which is directly related to theconsumption of carbon-based fuel, is regarded as oneof the most serious threats to the environmentthrough the greenhouse effect. Globally, transporta-tion accounts for about 21% of CO2 emissions. In thesame spirit, CO2 is identified as the most importantgreenhouse gas in the Netherlands, as it accountedfor 80% of total emissions in 1995 (see e.g., Kramer etal. 1999). Thus, there is a clear necessity to controlCO2 emissions.Road transport accounts for a large portion of CO2

emissions, of which goods transport constitutes asizeable portion. Thus, there is a need to address envi-ronmental concerns in the context of goods transport.Carrier companies may voluntarily adopt green poli-cies if this is aligned with profitability. This could bein the form of GHG trading mechanisms, or whenCO2 becomes a taxable commodity. Another reasonfor adopting green policies is the marketing potentialof a greener company image, for example, controllingthe carbon footprint. Furthermore, new regulationsmight force companies to change practices. It is worthmentioning that the department for environment,food and rural affairs in the United Kingdom valuesthe social cost of carbon between £ 35 and £ 140 pertonne. In essence, pricing carbon emissions leads toan assessment of its economic impact and regulationsmight be formed accordingly. In conclusion, either forexogenous or endogenous reasons, change is antici-pated in transportation management because of envi-ronmental factors. We argue that logistics serviceproviders should contemplate how to deal with theseissues.The focus of this article is on incorporating CO2-

related considerations in road freight distribution,specifically in the framework of the Vehicle RoutingProblem (VRP). CO2 emissions and fuel consumptionboth depend on the vehicle speed, which changesthroughout the day due to congestion. Thus, it is rele-vant to study the problem in conjunction with time-dependent travel times, i.e., where the travel timedepends on the time of day at which a distance istraversed. Time-dependency is modeled by partition-ing the planing horizon into free flow speed periodsand periods with congestion, that is, lower speeds.We introduce a new variant of the VRP that accountsfor travel time, fuel, and CO2 emissions costs. This

results in the Emissions-based Time-Dependent Vehi-cle Routing Problem, denoted by E-TDVRP. TheE-TDVRP builds on the possibility that carrier compa-nies limit the speed of their vehicles. Thus, the vehiclespeed limit is explicitly part of the optimization. Thetraditional Time-Dependent Vehicle Routing Problem(TDVRP) optimizes exclusively on travel times anddoes not consider limiting vehicle speed. However,we show that when accounting for fuel, travel time,and emissions, controlling vehicle speed is desirablefrom a total cost point of view.The E-TDVRP is clearly a complex problem, as in

addition to the complexity of the TDVRP, it alsodetermines the free flow speed. This implies that oneneeds to allocate customers to vehicles, determine theexact order in which customers are visited, and setthe free flow speed. The free flow speed impacts theresulting travel time function of each arc, and inreturn, affects the moment vehicles go into conges-tion. We assume that the congestion speed remainsconstant, as it is imposed by traffic conditions. Thisleads to a situation where speeds can be controlled inparticular time periods of the day, that is exclusivelyin free flow periods. The E-TDVRP can be reduced totwo sub-problems: one where only the CO2 emissionsare taken into account in the optimization (that is,a pure environmental model) and one where onlytravel times are considered (i.e., a pure logistical costmodel). As such, we study the trade-off between min-imizing CO2 emissions as opposed to minimizingtotal travel times. In addition, we develop bounds forthe potential reduction in CO2 emissions. Thesebounds are based on solutions of the standard time-independent VRP. Since most optimization tools usedin industry consider time-independent travel times,such bounds can concretely aid decision makers inevaluating the maximum reduction in emissions.The remainder of the article is organized as follows,

Section 2 reviews the relevant literature. Section 3describes the E-TDVRP model. It also introducesbounds for the potential savings in CO2 emissions.The solution method is discussed in section 4. Theexperimental settings and results are presented in sec-tion 5. Finally, section 6 highlights the main findingsand indicates directions for future research.

2. Literature Review

Van Woensel et al. (2001) highlighted the value oftraffic flow information in relation to emissions. Theirresults showed that calculating emissions under con-stant speed assumptions can be misleading, withdifferences of up to 20% in CO2 emissions on an aver-age day for gasoline vehicles and 11% for diesel vehi-cles. During congested periods of the day thesedifferences rose 40%. Similar results were shown by

Jabali, Van Woensel, and de Kok: Travel Times and CO2 Emissions in TDVRP2 Production and Operations Management 0(0), pp. 1–15, © 2012 Production and Operations Management Society

Palmer (2008), for a number of roads. Such results aremotivated mainly by the fact that CO2 emissions areproportional to fuel consumption (Kirby et al. 2000),and thus are speed dependent. The relation betweenemissions and vehicle speed leads to the study of theVRP problem in a time-dependent framework. Inwhat follows, we first discuss the literature concern-ing the TDVRP, and then review the literature dealingwith routing and emissions.Assigning and scheduling vehicles with a limited

capacity to service a set of clients with knowndemand, is a problem faced by numerous carrier com-panies. It has been extensively researched in the liter-ature as the well-known VRP. For a comprehensivereview, the reader is referred to Laporte (2007). In itsmost standard version, the problem deals with mini-mizing costs subject to demand satisfaction and vehi-cle capacity constraints. Each customer is visited by asingle vehicle, each vehicle starts and ends its route atthe depot. The relevance of the VRP to real-life prac-tice coupled with the hard nature of the problem hasattracted much research. To model reality more accu-rately, numerous features have been added to theproblem. One of these is including speed changesover the day, in an attempt to account for traffic con-gestion experienced in certain periods of the day.Lately, a number of researchers incorporated time-

dependent travel times between customers into theVRP, which gave rise to the TDVRP. The objective ofthe TDVRP, in most cases, is similar to that of theVRP, that is, minimizing costs (Hill and Benton 1992).However, in the TDVRP the travel time cost dependson the time of day a distance is traversed. Modelingtime-dependency is mostly done by associating differ-ent speeds to a number of time zones within the plan-ning horizon. Malandraki and Daskin (1992) motivatevariability in travel time by random events, such asaccidents and cyclic temporal variations in trafficflow. They propose a mixed integer programmingformulation for the TDVRP. Malandraki and Dial(1996) propose a dynamic programming formulationto solve the time-dependent Traveling SalesmanProblem (TSP). However, both these articles modeledtravel times by discrete step functions where traveltimes are associated with different time zones.Although this modeling approach does capture vari-ability in travel times, it enables the undesired effectof surpassing. This effect implies that a vehicledeparting at a certain time might surpass anothervehicle that started traveling earlier. This limitationwas discussed by Fleischmann et al. (2004), Ichouaet al. (2003), Nannicini et al. (2010), and Van Woenselet al. (2008). All these articles model time-dependen-cies complying with the First-In First-Out (FIFO)assumption, which does not allow for surpassing.This is done by using appropriate piecewise linear

functions for travel times. In this article, we adopt tra-vel time functions similar to the ones used by Ichouaet al. (2003), and thus adhere to the FIFO assumption.Ichoua et al. (2003) modeled the TDVRP by parti-

tioning the day into three speed zones, where thespeed differences due to congestion were determinedby different factors of the free flow speeds. The traveltime profiles were constructed by piecewise linearfunctions. Fleischmann et al. (2004) discuss a generalframework for integrating time-dependent traveltimes in a number of VRP algorithms. Furthermore,they provide an overview of traffic information sys-tems from which data can be collected. Modeling oftime-dependent travel times would benefit from thesedata. Based on empirical traffic data, queueing mod-els were developed by Van Woensel (2003) to modelcongestion, where the model parameters were set toincorporate different traffic flows and weather condi-tions. The analysis of the data resulted in averagespeeds for different time zones (see also Van Woenseland Vandaele 2006). These speeds were later used inVan Woensel et al. (2008) to solve the TDVRP withtabu search. Jung and Haghani (2001) proposed amodified genetic algorithm to solve the TDVRP.Not much research has been conducted on the VRP

under minimizing emissions. Cairns (1999) studiedthe environmental impact of grocery home delivery,but this was done by converting distance into emis-sions, irrespective of speed changes. The effect ofspeed changes is incorporated in the work of Palmer(2008). He studied emissions in the context of groceryhome delivery vehicles, where real traffic data wereused to derive fuel consumption and emissions. Asimilar detailed methodology was used by Ericssonet al. (2006). Both Palmer (2008) and Ericsson et al.(2006) considered CO2 minimization as an optimiza-tion criteria. Bektas and Laporte (2011) introduced thepollution routing problem where they accounted forthe amount of greenhouse emissions, fuel, traveltimes, and their costs. The authors present a compre-hensive formulation of the problem and solve moder-ately sized instances. Scott et al. (2010) investigatedthe influence of the minimization of CO2 emissions onthe solutions to shortest path and traveling salesmanproblems in freight delivery applications. Real lifeexamples are studied using the road network ofScotland. In a more aggregate view, Sugawara andNiemeier (2002) presented an emissions-based tripassignment optimization model. They also exploredpotential emission reduction under the assumptionthat drivers choose emission minimizing routes.Assessing vehicle emissions can be very complex, asemissions depend on factors such as the age of thevehicle, engine state, engine size, speed, type of fuel,and weight (Taniguchi et al. 2001). For our study, weuse speed emission functions from the MEET model

Jabali, Van Woensel, and de Kok: Travel Times and CO2 Emissions in TDVRPProduction and Operations Management 0(0), pp. 1–15, © 2012 Production and Operations Management Society 3

(European Commission 1999). In this article, we focuson CO2 and leave other pollutants for additionalresearch.

3. Description of E-TDVRP

This section starts with a complete description of theE-TDVRP model and details the most relevant partsof this model. Section 3.1 elaborates on the computa-tion of the travel times. Section 3.2 explains the com-putation of the CO2 emissions and fuel consumptions.Finally, in section 3.3 the bounds on CO2 emissionsare presented.In general, the VRP is described by a complete

directed graph G = (V,A) where V = {0,1,…,n} is a setof vertices and A = {(i,j):i<>j ∈ V} is the set of direc-ted arcs. The vertex 0 denotes the depot; the rest ofthe vertices represent customers. For each customer,a non-negative demand qi is given (q0 ¼ 0). A non-negative cost cij is associated with each arc (i,j). Theobjective is to find, for a given number of vehicles N,the minimum costs where the following conditionshold: every customer is visited exactly once by onevehicle, all vehicle routes start and end at the depot,and every route has a total demand not exceeding thevehicle capacity Q. This definition is valid for theE-TDVRP model presented in this study.The E-TDVRP differs from the VRP as it considers

time-dependent travel times. Furthermore, it consid-ers fuel and emissions costs. The objective function ofthe E-TDVRP is the sum of all these costs. Limitingthe free flow speed of the vehicles is examined in thisproblem. Therefore, in addition to the routing solu-tion the E-TDVRP has a decision variable vf corre-sponding to the upper limit of the vehicle speed. Thespeed limit vf is considered a tactical choice of the car-rier company. Thus, vf is imposed on all vehicles, thatis, all routes have the same limit.We define a solution as a set S with s routes

fR1;R2; . . .;Rsg where s � N, Rr ¼ ð0; . . .; i; . . .; 0Þ,that is, each route begins and ends at the depot. Wewrite i 2 Rr, if the vertex i � 1 is part of the route Rr

(each vertex belongs to exactly one route). We writeði; jÞ 2 Rr, if i and j are two consecutive vertices in Rr.An E-TDVRP solution is defined by the solution set Scoupled with a vehicle speed limit vf .A solution (S; vf ) results in travel times TTðS; vfÞ

and emissions EðS; vfÞ. The computation of TTðS; vfÞ isexplained in section 3.1. The computation of EðS; vfÞis discussed in section 3.2.We define three cost factors in the E-TDVRP: the

hourly cost of a driver (with a cost of a €/hour), thecost of fuel (costing b €/liter), and the cost of CO2

emissions (c €/kg). Since fuel consumption is directlyrelated to CO2 emission, we use the factor of h (equalto 1

2:7 liter/kg) for converting CO2 emissions into fuel

consumption, similar to Palmer (2008). The objectivefunction for E-TDVRP is given by Equation (1).

FðS; vf ; a; b; cÞ ¼ aTTðS; vfÞ þ ðbhþ cÞEðS; vfÞ ð1Þ

The first part of the objective function considers thetravel time costs. The second part considers the com-posite costs combining fuel and emissions. We explic-itly consider both, as the cost parameters for fuel (b)and emissions (c) are different.The E-TDVRP can be reduced into two special cases.

Setting a to zero the model minimizes solely the costsof fuel and CO2, which is equivalent to minimizingEðS; vfÞ, with an objective function value FðS;vf ;0;b;cÞ.Similarly, setting both b and c to zero results in amodel that minimizes TTðS;vfÞ, with an objective func-tion value FðS;vf ;a;0;0Þ. These special cases facilitate atrade-off analysis showing the additional travel timerequired to obtainminimal CO2 emissions.For completeness, we summarize the speed-related

notation we will use in the remainder of this section:

� vf : The speed limit set in E-TDVRP.� vc: The congestion speed imposed by traffic.� v�: The speed that yields the optimal CO2 emis-

sion per km value.� v�f : The optimal speed for EðS; vfÞ.� vh: The upper bound value of v�f .� v0: The speed used for computing the upper

bound on EðS; vfÞ.

3.1. Determining TTðS; vfÞThe time-dependency of travel times throughout theplanning horizon, is in essence, driven by changingspeeds in different time zones. We subject all links togiven speed profiles. This stems from the notion thatmost motorways, on average, follow the same patternof having morning and evening congestion periods(see Ichoua et al. [2003] for a similar reasoning). More-over, collecting data for each link and each time zoneis infeasible from an operational standpoint.Figure 1 provides an illustration of how speeds are

translated into travel time profiles. The left sidedepicts a speed profile that starts with a congestionspeed vc until time a. After time a the vehicle cantravel at free flow speed vf . Subjecting a givendistance d to the speed profile on the left side of thefigure results in the travel time profile on the rightside of Figure 1. Note that while the x-axis for the speedprofile is the time of day, the x-axis for the travel timefigure is starting time. For a given distance d and forevery starting time, the figure on the right providesthe travel time. The main intuition for modeling theprofile is that during the first period (up to a � TTc) ittakes TTc time units to traverse d since, throughoutthat period the vehicle will be driving with speed vc


along the entire link. However, starting from timea � TTc up to point a, the vehicle will be in the tran-sient zone, where it will start traversing the link withspeed vc and the remainder with speed vf . Starting atpoint a the speed will remain constant at vf with thetravel time equaling TTf . Thus, the travel time is acontinuous piecewise linear function over the startingtimes.The linearity in the transient zone stems from the

stepwise speed change which imposes different

speeds over time. The slope can be defined asTTf �TTc

TTc.

By substituting TTc with dvcand TTf with d

vf, we obtain

that the slope is equal tovc � vfvf

, which is independent of

d. However, the intersection with the travel time axis is

a function of the distance and it is equal toðvf �vcÞ

vfa þ d

vf.

Therefore, the travel time function between custom-ers i and j depends on the distance dij between these

customers. We define g(t) as the travel time functionassociated with any starting time t for Figure 1.

gðtÞ ¼TTc if t�a�TTcvc�vfvf

tþðvf�vcÞvf

aþTTf if a�TTc\t\a

TTf if t�a

8<:

For starting times within ð0; a � TTcÞ the FIFO issatisfied since the speed associated with these startingtimes is constant. Given two starting times within thisinterval t and t + D, both will arrive TTc time unitsafter their departure. Similarly, FIFO holds for startingtimes after a. The line in the transient zone ða � TTc; aÞcan be viewed as a consequence of the FIFO assump-tion as well. Given two starting times t and t + D, bothin the transient zone, the arrival times are t + g(t) andt + D + g(t + D), respectively, the difference betweenthe arrival times of t + D and t is D þ vc � vf

vfD [ 0.

Thus, the FIFO assumption holds in the transient zone.Note that in a speed decreasing situation the FIFOassumption holds even for step travel time changes.Nonetheless, for consistency, speeddrops are constructedexactly in the samemanner as speed increases.We furthernote that the construction of travel times is equivalentto integrating the distance over the different speeds.

The proposed model is convenient since it requiresspeed values and the points in time at which thespeed changes. TTijt represents the travel timebetween node i and node j when starting at time t.The starting time t belongs to a time zone from the setft1; t2; . . .; tkg. The travel time is computed dependingon the specific zone that t belongs to. Equation (2)considers the total travel time associated with a solu-tion S and vf .

TTðS; vfÞ ¼Xr

Xt

Xi;jð Þ2Rr

TTijt ð2Þ

3.2. Determining E(S,vf)In this article, we use emission functions provided inthe MEET report (European Commission 1999). Thefunction h(v) provides the emissions in grams per kilo-meter for speed v. Equation (3) depicts the amount ofemissions per km, given that a vehicle is at speed v.

hðvÞ ¼ K þ avþ bv2 þ cv3 þ d1

vþ e

1

v2þ f

1

v3ð3Þ

The coefficients (K,a,…,f) differ per vehicle type andsize. Here, we focus on heavy duty trucks weighing32–42 tons. The coefficients for the CO2 emissionsfor this specific vehicle category are (K,a,b,c,d,e,f) =(1576, �17.6, 0, 0.00117, 0, 36067, 0). Figure 2 depictsthis CO2 (kg/km) emissions function. This func-tion has a unique minimum and we define v� as theinteger speed which achieves the lowest CO2 emis-sions (v� ¼ 71 km/hour).

Figure 1 The Conversion of Speed into Travel Times

Figure 2 CO2 Emissions vs. Speed


We denote the amount of CO2 emissions in gramsproduced by traversing arc (i,j) at time t with a freeflow speed vf by EijtðvfÞ. The average speeds areused to calculate the emissions per km by Equation(3). Multiplication of emissions per km by distancestraversed yields the total amount of CO2 emissions.We define X½dij; t; vf � as the average speed for tra-versing arc (i,j) when leaving i at time zone t, withspeed limit vf . Consequently, EijtðvfÞ is given byEquation (4).

EijtðvfÞ ¼ h X½dij; t; vf ��

dij ð4Þ

The objective function for term EðS; vfÞ is defined inEquation (5) summing the total CO2 emissions pro-duced by the different routes r in solution S withspeed limit vf .

EðS; vfÞ ¼Xr

Xt

Xi;jð Þ2Rr

EijtðvfÞ ð5Þ

Both from a practical and an optimization point, wechoose to work with integer speeds. Optimizing onEðS; vfÞ implies then finding an optimal integer freeflow speed applied to all routes r in S. The depen-dence of Eijt on the speeds associated with dij makesthe free flow speed vf a decision variable. We empha-size that limiting the speed of a vehicle means that infree flow the vehicle’s speed is limited. However, incongestion the vehicle is restricted by the congestionspeed. In essence, while congestion zones can be seenas constraints, in free flow zones the maximum speedis decided upon.Modifying the speed profile affects the travel time

profiles. As explained in section 1, the change ofspeed at point a starts affecting the travel timesalready at point a � TTc. If there is a speed drop atpoint a from a certain free flow speed vf to a conges-tion speed vc, travel times for a given distance d willstart to increase if it is traversed after time a � d

vf. If

the free-flow speed is decreased, the travel time pro-file is affected, since the vehicle will start experiencingcongestion at earlier starting times. We demonstratethe effect of free flow speed on the total emissions

produced with the example depicted in Figure 3. Letvc ¼ 40 km/hour and consider two options for freeflow speed: v1 ¼ 71 km/hour, and v2 ¼ 72 km/hour,with starting time 200 and a = 500 (in minutes). Set-ting the free flow speed to v1 produces 2.8 kg moreCO2 emissions than setting it to v2. Moreover, settingthe free flow speed to v1 results in an increase of15 minutes in travel times, with respect to v2.Considering optimizing on EðS; vfÞ, that is, only

on the amount of emissions, let the optimal speedbe v�f . Proposition 1 shows that only a sub-set ofpotential speeds need to be taken into account for theEðS; vfÞ optimization. This proposition makes it possi-ble to considerably reduce the search space for thisproblem.

PROPOSITION 1. Given vc \ v�, there exists a vh [ v�

such that hðvcÞ ¼ hðvhÞ and v� � v�f � vh.

For the case of h(v) being convex having a uniqueminimum as is in the case of the CO2 emission func-tion considered, Proposition 1 is straightforward toshow. Since vc is smaller than v� there exists anotherspeed vh which is higher than v� such that hðvcÞ ¼hðvhÞ. We show that vc \ v�f \ vh for the setting inquestion.

PROOF. Arguing by contradiction, assume thatv�f [ vh. The shape of the emission per km functionimplies that hðv�f Þ [ hðvhÞ ¼ hðvcÞ. In such a case,being in free flow zones, that is, where the speed isset to v�f , will produce higher emissions than in con-gestion. Any ~v 2 ½vc; vh� will produce less emissionssince hð~vÞ\ hðv�f Þ and thus v�f is not optimal.Again by contradiction, assume that vc \ v�f \ v�.

For any ~v 2 ½vc; v�� there exists a v [ v� such thathðvÞ ¼ hð~vÞ. Since for v the total time spent in con-gestion will be lower than that for ~v, we concludethat v� \ v�f \ vh. h

As previously mentioned, we only consider integervalues for speeds. Thus, vh is rounded up to the near-est integer value.

Figure 3 Example for Two Free Flow Speeds


3.3. Bounds for EðS; vfÞThe EðS; vfÞ on its own is a difficult problem tohandle: one needs to find a solution in a time-depen-dent setting in combination with setting the free flowspeed. In this section we present bounds using thesolutions of the standard VRP (i.e., time-indepen-dent). Throughout the years various solution tech-niques have been developed to tackle this problem.Moreover, from a practical standpoint, most carriercompanies developed software solutions for the stan-dard VRP. Hence, bounds based on these basic VRPsolutions may be easily computed in practice as well.The lower bound is realized by traversing the total

distance in the VRP solution with a speed v�, sincethis implies the minimum distance traversed usingthe optimal emissions speed. In essence, the upperbound can be constructed by subjecting the VRP solu-tions to a speed profile with a single speed drop, simi-lar to Profile I in Figure 4. However, our TDVRPsetting is similar to Profile II in Figure 4. For a givendistance to be traversed starting at time zero, optimiz-ing CO2 emissions under Profile II would generateresults superior or equivalent to the same optimiza-tion under profile I. For minimizing emissions, theoptimal performance under profile I is an upperbound on the performance of Profile II. Thus, weconstruct an upper bound on the total CO2 emissionsby assuming a single speed decrease that is outper-formed by a profile consisting of a speed dropfollowed by a speed increase.We analyze the case of subjecting the VRP solutions

to Profile I. We denote the distance of an arbitraryroute by d. We distinguish two cases:

(i) If the vehicle is not fast enough to avoid con-gestion, that is, v < d/a, then the total emis-sions are given by the emissions in free flowh(v)av plus the emissions in congestion whichare hðvcÞðd � avÞ.

(ii) If the vehicle travels fast enough to avoid con-gestion, that is, v � d/a, then the total emis-sions would simply be given by h(v) times thecovered distance d.

According to the above distinction, define E1ðd; vÞand E2ðd; vÞ as follows:

E1ðd; vÞ ¼ hðvÞavþ hðvcÞðd� avÞE2ðd; vÞ ¼ hðvÞd:

Then, total emissions as a function of distance d canbe written as a function of speed:

EdðvÞ :¼ E1ðd; vÞ if v 2 ð0; da�E2ðd; vÞ if v 2 ½da ;þ1Þ

�

In our case E1ðd; vÞ, E2ðd; vÞ, and h(v) are convex andhave a unique minimum. The problem of finding anoptimal speed involves finding the minimum of Ed:

min minv2ð0;da�

E1ðd; vÞ; minðv2½da;þ1Þ

E2ðd; vÞ( )

: ð6Þ

LEMMA 1. There exists a universal speed v0 [ v� suchthat for all a,d > 0 there exists a unique solution v�f toproblem (6) given by:

v�f ¼v� if d� av�

d=a if av�\d\av0

v0 if d� av0.

8<:

PROOF. E1ðd; vÞ has a unique optimal v0. This can beeasily seen by differentiation

@E1ðd; vÞ@v

¼ aðh0ðvÞvþ hðvÞ � hðvcÞÞ:

As earlier observed E2ðd; vÞ has a unique optimalspeed v�. Furthermore E1ðd; d=aÞ ¼ E2ðd; d=aÞ, sothat Ed is continuous. Now, since by convexity weknow that

� E1 is decreasing for v\ v0 and increasing forv [ v0,

Figure 4 Speed Profiles for the Bounds on E-TDVRP


� E2 is decreasing for v\ v� and increasing forv [ v�,

it is sufficient to put together E1 and E2, distinguish-ing between the following three cases.

Case ðiÞ : d � av�.

EdðvÞ ¼E1ðvÞ; decreasing if v� d=aE2ðvÞ; decreasing if d=a\v\v�

E2ðvÞ; increasing if v� � v

8<:

so the minimum value is reached when v ¼ v�.Case ðiiÞ : av� \ d\ av0.

EdðvÞ ¼ E1ðvÞ; decreasing if v� d=aE2ðvÞ; increasing if d=a\v

�

so the minimum value is reached when v = d/a.Case ðiiiÞ : av0 � d.

EdðvÞ ¼E1ðvÞ; decreasing if v� v0

E1ðvÞ; increasing if v0\v\d=aE2ðvÞ; increasing if d=a� v

8<:

so the minimum value is reached when v ¼ v0. h

The importance of Lemma 1 is that for large enoughdistances, d � av0 there exists a single speed v0 whichis optimal under Profile I. We use v0 to construct theupper bound on EðS; vfÞ.Let S0 be the optimal solution for the standard VRP.

Let d0i be the distance of route Ri in S0. We arrange thedifferent routes in descending order, with respect totheir distance, S0 ¼ fd0½1�; . . .; d0½s�g. Equation (7) repre-sents a lower bound on the amount of CO2 emissionsproduced. The lower bound assumes that all dis-tances can be traversed in v�, so that absolute mini-mum emissions can be achieved.

Xsi¼1

d0½i�hðv�Þ�EðS; vfÞ ð7Þ

The upper bound is constructed by imposing Pro-file I with vf ¼ v0 onto S0. We define u as the largestindex such that d½u� [ av0. Thus, routes fd0½1�; . . .; d0½u�gwill run into congestion. However, the routesfd0½uþ1�; . . .; d

0½s�g will not suffer congestion. Equation (8)

depicts this upper bound.

EðS; vfÞ� hðv0Þuav0 þ hðvcÞXui¼1

ðd0i � av0Þ þ hðv0ÞXsi¼uþ1

d0i

� ½hðv0Þ � hðvcÞ�uav0 þ hðvcÞXui¼1

d0i þ hðv0ÞXsi¼uþ1

d0i

ð8ÞGiven the standard VRP solution, these bounds are

rather straightforward to calculate, and enable deci-

sion makers to easily assess the maximal reduction inCO2 emissions. Furthermore, these bounds are usedto validate the proposed solution procedure.

4. Solution Method

We propose a tabu search procedure for theE-TDVRP. Tabu search was first introduced by Glover(1989a, 1989b). It makes use of adaptive memory toescape local optima. The method has been extensivelyused for solving the VRP; see Gendreau et al. (1994,1996), and Hertz et al. (2000) for examples. Tabusearch is also used to deal with the time-dependentversion of the VRP (see Ichoua et al. 2003, Jabali et al.2009, Van Woensel et al. 2008). The E-TDVRP modeldiffers from the TDVRP as it includes the free flowspeed vf as a decision variable. We chose to adapt atabu search procedure to fit the E-TDVRP. Essentially,the procedure works on local search principles whileupdating the vf . The algorithm searches a neighbor-hood with vf . After a number of iterations it checksthe performance of the best solution so far for differ-ent integer values of vf . If a value which yields betterresults is found, the speed is updated and the searchcontinues. Next, we describe the main components ofthe algorithm. The overall procedure is described inpseudo-code in Algorithm 1.

Algorithm 1 E-TDVRP algorithmic structure

Construct initial solution S0 with vf ¼ v0 and computeF ðS0; vf ; a;b; cÞ.Set the best solution Sb and the best feasible solution S� both to S0.Set S to S0si = r.for 1 � i � Imax do

Generate the neighborhood of SChoose the solution S 0 that minimizes F2ðS ; vf ; a;b; cÞ and is nottabu or satisfies its aspiration criteria.

Update the tabu list.if S 0 is feasible and it is better than the current best feasiblesolution S� thenUpdate the best feasible solution, i.e., set S� to S 0.Set si = i + φr.

end ifif S 0 is not feasible and is better than the current best solution Sb thenUpdate the best solution i.e., set Sb to S 0.

end ifUpdate w if necessary according to the feasibility status of S 0.

if i > si thenSet the current solution S to the best solution Sb .Check for the new best speed within ½vf � s; vf þ s� andupdate vf accordingly.Set si = i + r.

end ifif new best solution found and Imax � i \w thenset Imax ¼ i þ 2w

end ifUpdate S to S 0.

end forreturn the best feasible solution S�


The initial solution S0 is the output of a nearestneighbor heuristic with vf ¼ v0. A neighborhood isevaluated by considering all possible 1-interchangesas proposed by Osman (1993). The (0,1) interchangesof node vr 2 Rr to route Rp are considered only if Rp

contains one of g nearest neighbors of vr. The (1,1)interchanges between nodes vr 2 Rr and vp 2 Rp areconsidered if Rp contains one of g nearest neighbor ofvr and Rr contains one of g nearest neighbor of vp.After completing the neighborhood search, a 2-optintra-route move is executed for each altered route. Ifnode vr is removed from Rr, reinserting vr back intoRr is tabu for ‘ iterations. ‘ is randomly chosenbetween ½‘min; ‘max�. However, we use an aspirationcriterion similar to Cordeau et al. (2003). This criterionrevokes the tabu status of a move if it yields a solutionwith lower costs.Similar to Gendreau et al. (1994), we allow capac-

ity-infeasible solutions, i.e., routes with total demandexceeding the vehicle capacity. Such infeasible solu-tions are penalized, in proportion to the capacity vio-lation, by the following objective function, extendingFðS; vf ; a; b; cÞ:

F2ðS; vf ; a; b; cÞ ¼ FðS; vf ; a; b; cÞ

þ wXR2Z

Xi2R

qi

!�Q

" #þð9Þ

In Equation (9) each unit of excess demand is penal-ized by a factor w. This penalty w is decreased by mul-tiplication with a factor m after / consecutive feasiblemoves. Similarly, w is increased (multiplied by factorm�1) after / infeasible iterations.Every r iterations, we consider the best found solu-

tion ðS; vfÞ and evaluate the solution for speeds withinthe interval ½vf � s; vf þ s�. The speed that minimizesF2ðS; � ; a; b; cÞ is chosen as the new vf . If a new bestfeasible solution is found then vf is kept for anotherrφ iterations. The algorithm is run for Imax iteration.However, if a best solution is found and the remain-ing number of iterations is less than w, the remainingnumber of iterations is updated to 2w.

5. Experimental Settings

We conducted two types of experiments. Section 1describes our experiments with a single speed profile.The other experiments include two speed profiles andare presented in section 2. We experiment with setsfrom Augerat et al. (1998). The number of nodes inthese sets ranges between 32 and 80. To achieve morerealistic travel times, the coordinates of these setswere multiplied by 4.9. Note that a test set named“32 k5” means 32 customers including the depot withfive vehicles. Optimal VRP solutions for these sets areavailable from Ralphs (2011).

5.1. Single Speed ProfileWe set up the speed profile with two congested peri-ods, while throughout the rest of the day the speed isset to the free flow speed (Figure 5, left). Many Euro-pean roads face such a morning and afternoon con-gestion period (see Van Woensel and Vandaele 2006).The right side of Figure 5 depicts the travel time func-tion for each starting time for an arbitrary distance.We set v2 and v4 to congestion speed vc. Furthermore,v5 ¼ v3 ¼ v1 are considered as free flow speeds, thatis, they correspond to the decision variable vf inFðS; vf ; a; b; cÞ. The points a, b, c, d, e correspond to6:00, 9:00, 16:00, 19:00, and 0:00 hours. All links aresubjected to this profile. Based on empirical data, thecongestion speed is set to 50 km/hour (Lecluyse et al.2009).Observe that a starting time after 9:00 hours is supe-

rior to starting time before 9:00 hours, since the latterrisks going twice into congestion. Consequently, weassume that all vehicles leave the depot at 9:00 hours.To compute the upper bound, we set a in Equation (8)to c � b. Considering vc ¼ 50 together with this pro-file and based on Proposition 1 leads to the conclusionthat 71 � v�f � 91. Furthermore, by Lemma 1, thespeed for computing the upper bound on EðS; vfÞ isv0 ¼ 74 km/hour.The parameters chosen for the costs and the tabu

search algorithm in section 4 are summarized inTable 1. We note that c is based on the actual carbon

Figure 5 Speed and Travel Time Profile for the Experimental Setting


market cost of the first half of April 2009 (PointCarbon 2009). All runs are performed on a Intel CoreDuo with 2.4 GHz and 2 GB of RAM.

5.1.1. Validating the Proposed Model. Table 2shows the results for the reduced model where wefocus on emissions, that is, with the objective func-tion FðS; vf ; 0; 0; 1Þ ¼ EðS; vfÞ. Table 2 gives the CO2

emissions (kg), the speed limit vf , and the resultingtravel time, denoted by TTðFðS; vf ; 0; 0; 1ÞÞ; the lastcolumn gives the run times. There is a relationbetween the amount of CO2 emissions and the traveltime. For example, set 33 k5 achieves the lowestemissions and the lowest travel time. The speedsachieving lowest emissions range between 72 and77 km/hour.We tested the performance of our algorithm on the

standard time-independent VRP from Augerat et al.(1998). The algorithm reached an average optimalitygap of 4% over the different sets. Furthermore, weevaluated the performance when setting the free flowspeed to 71 km/hour (F(S,71;0,0,1)), which is thespeed that minimized the CO2 emissions function asobserved in Figure 2. The results showed that adopt-ing this speed leads to an average increase of 2.9%and 6.0% in emissions and travel times, respectively,when compared to the results in Table 2 (see OnlineAppendix S1 for further details).

To validate the model, we compare the results ofFðS; vf ; 0; 0; 1Þ with the upper bound (UB) and thelower bound (LB) presented in section 3. We gener-ate the bounds using the optimal VRP solutionsfrom Ralphs (2011) for the Augerat sets. Table 3compares the amount of CO2 emissions producedusing the FðS; vf ; 0; 0; 1Þ ¼ EðS; vfÞ objective functionwith the bounds. FðS; vf ; 0; 0; 1Þ/LB compares theemissions produced by the E-TDVRP with the lowerbound. On average the E-TDRVP is within 4.3%from the lower bound. We note that the LB is com-puted under the assumption of no congestion. Thelast column in Table 3 quantifies the gap betweenUB and LB. The quality of the bounds is good asthe average gap is only 4.7%. Based on this analysis,we conclude that the solution method is efficientand that the bounds are useful in real-life decision-making.

5.1.2. Numerical Results for FðS; vf ; a; b; cÞ. Table4 gives an overview of the cost-based solutions for allsets (a = 20 €/hour, b = 1.2 €/l and c = 11 €/ton).The last three columns give the allocation of costsbetween its three major components.The vast majority of costs are attributed to driving

cost and fuel cost, which together on average accountfor 98.6% of the costs. The cost of CO2 is rather

Table 1 Cost and Tabu Search Parameter Values

Cost Parameter Description Value

a Travel time cost (€ /hour) 20b Fuel cost € /l (diesel) 1.2c Carbon cost € / ton 11

TabuSearchParameter Description Value

g Number of closest customersconsidered for the neighborhoodsearch, where n is the number ofnodes

[0.4n]

‘ The tabu tenure length Randomly chosen inthe interval[15, 25]

w The penalty for infeasible solutions Initial objectivefunction valuedivided by 30

/ Number of iterations considered inupdating w

5

m Updating factor for w 0.75Imax The maximum number of iterations 640w Threshold for updating Imax 160r Number of iterations keeping the

same speed120

φ Updating factor for r 1.5τ Half-width of the interval of speed

search1

Table 2 Results for the F (S ; vf ; 0; 0; 1) Model (One Speed Profile)

Instance F ðS ; vf ; 0; 0; 1Þ vf TT ðF ðS ; vf ; 0; 0; 1ÞÞRun time(minutes)

32 k5 2976 72 3420 4.833 k5 2489 74 2836 3.333 k6 2777 73 3168 4.234 k5 2929 73 3377 3.836 k5 3088 75 3449 2.337 k5 2516 72 2911 8.537 k6 3613 74 4122 4.938 k5 2819 73 3209 5.039 k5 3175 72 3852 6.639 k6 3151 73 3603 7.744 k6 3575 72 4122 8.345 k6 3644 74 4162 3.045 k7 4466 74 4918 2.746 k7 3458 74 3912 4.648 k7 4156 73 4727 7.453 k7 3875 72 4500 13.654 k7 4563 77 5023 4.055 k9 4174 74 4713 4.060 k9 5362 72 6230 16.061 k9 4060 74 4536 8.462 k8 5104 73 5871 13.263 k9 6209 73 7130 13.363 k10 5115 73 5869 12.264 k9 5379 73 6168 13.665 k9 4666 75 5228 6.969 k9 4508 75 4996 31.180 k10 7062 73 8097 30.9Average 4034 73 4598 9.0


insignificant when compared to others: 1.4% on aver-age. Specifically, the current market cost of CO2 hasno tangible economic impact on the results. However,due to the direct relation between fuel consumptionand CO2, speeds associated with this model arebetween 80 and 87 km/hour. In fact, in all instancesthe resulting speed is strictly less than 90 km/hour.This implies that even with the current cost structurespeeds are likely to be <90 km/hour.

5.1.3. The Reduced Models: FðS; vf ; 0; 0; 1Þ andFðS; vf; 1; 0; 0Þ. We analyze the results for the tworeduced models. The first one optimizes only on theemissions (FðS; vf ; 0; 0; 1Þ), the second one focuses ontravel times (FðS; vf ; 1; 0; 0Þ). This facilitates a trade-offanalysis between travel times and emissions. We ranthe settings F(S,80;1,0,0) and F(S,90;1,0,0), that is, wefixed the speed limits to 80 and 90 respectively. Thetruck speed limit in most European countries is 80 or90 km/hour, so these two specific settings depict aninteresting European benchmark.Let TTðFðS; vf ; 0; 0; 1ÞÞ be the travel times resulting

from the (reduced) model with objective functionFðS; vf ; 0; 0; 1Þ. Specifically, the objective function onlydepends on emissions. The resulting solution (S; vf ) isthen evaluated in terms of the travel time resulting inTTðFðS; vf ; 0; 0; 1ÞÞ. Similarly, we define the emissions

corresponding to the (reduced) model with objectivefunction F(S,80;1,0,0) as E(F(S,80;1,0,0)). This meansthat we generate a solution (S; vf ) by minimizing thetravel time and then evaluate this solution in terms ofits emissions, which is denoted by E(F(S,80;1,0,0)). Weuse the following notation.

DTT80 ¼TTðFðS; vf ; 0; 0; 1ÞÞ � FðS; 80; 1; 0; 0Þ

FðS; 80; 1; 0; 0ÞDE80 ¼

FðS; vf ; 0; 0; 1Þ � EðFðS; 80; 0; 0; 1ÞÞEðFðS; 80; 1; 0; 0ÞÞ

DTT90 ¼TTðFðS; vf ; 0; 0; 1ÞÞ � FðS; 90; 1; 0; 0Þ

FðS; 90; 1; 0; 0ÞDE90 ¼

FðS; vf ; 0; 0; 1Þ � EðFðS; 90; 0; 0; 1ÞÞEðFðS; 90; 1; 0; 0ÞÞ

In the above expressions the current situation corre-sponds to the setting with a fixed speed, that is, 80 or90 km/hour, where the minimization is on the totaltravel time. In the alternative situation the objective isto minimize emissions, that is, the objective functionFðS; vf ; 0; 0; 1Þ.Table 5 gives an overview of the potential savings

in emissions, compared to the amount of travel time

Table 3 Upper and Lower Bounds for F (S ; vf ; 0; 0; 1) (One SpeedProfile)

Set UB LB F ðS ; vf ; 0; 0; 1Þ F ðS ; vf ; 0; 0; 1Þ/LB UB/LB

32 k5 3081 2904 2976 102.5% 106.1%33 k5 2546 2443 2489 101.9% 104.2%33 k6 2838 2738 2777 101.4% 103.6%34 k5 3016 2880 2929 101.7% 104.7%36 k5 3135 2957 3088 104.4% 106.0%37 k5 2602 2479 2516 101.5% 105.0%37 k6 3684 3510 3613 102.9% 105.0%38 k5 2822 2706 2819 104.2% 104.3%39 k5 3220 3056 3175 103.9% 105.4%39 k6 3240 3071 3151 102.6% 105.5%44 k6 3634 3463 3575 103.2% 104.9%45 k6 3648 3483 3644 104.6% 104.7%45 k7 4458 4229 4466 105.6% 105.4%46 k7 3531 3386 3458 102.1% 104.3%48 k7 4172 3960 4156 104.9% 105.3%53 k7 3897 3735 3875 103.7% 104.3%54 k7 4553 4320 4563 105.6% 105.4%55 k9 4078 3961 4174 105.4% 103.0%60 k9 5238 4998 5362 107.3% 104.8%61 k9 3925 3830 4060 106.0% 102.5%62 k8 5041 4771 5104 107.0% 105.7%63 k9 6344 5980 6209 103.8% 106.1%63 k10 5042 4843 5115 105.6% 104.1%64 k9 5437 5164 5379 104.2% 105.3%65 k9 4498 4356 4666 107.1% 103.2%69 k9 4438 4298 4508 104.9% 103.3%80 k10 6900 6512 7062 108.5% 106.0%Average 104.3% 104.7%

Table 4 Results for F (S0; vf ;a; b; c) (One Speed Profile)

Cost partition

Set F ðS0; vf ; a; b; cÞ vf

CO2

(kg)Drivercost

Fuelcost

CO2

cost

32 k5 2411 84 3092 41.6% 57.0% 1.4%33 k5 1999 87 2615 40.4% 58.1% 1.4%33 k6 2267 85 2924 41.3% 57.3% 1.4%34 k5 2380 86 3061 41.4% 57.2% 1.4%36 k5 2494 85 3221 41.2% 57.4% 1.4%37 k5 2043 85 2636 41.2% 57.3% 1.4%37 k6 2934 86 3799 41.0% 57.6% 1.4%38 k5 2247 84 2937 40.5% 58.1% 1.4%39 k5 2671 85 3423 41.6% 56.9% 1.4%39 k6 2788 86 3626 40.8% 57.8% 1.4%44 k6 2952 87 3893 39.9% 58.6% 1.5%45 k6 2984 82 3750 42.8% 55.9% 1.4%45 k7 3837 87 4958 41.1% 57.4% 1.4%46 k7 2856 84 3680 41.3% 57.3% 1.4%48 k7 3341 86 4349 40.7% 57.8% 1.4%53 k7 3245 86 4208 40.9% 57.6% 1.4%54 k7 3714 82 4676 42.7% 55.9% 1.4%55 k9 3463 80 4327 43.1% 55.5% 1.4%60 k9 4534 80 5632 43.4% 55.2% 1.4%61 k9 3341 82 4247 42.1% 56.5% 1.4%62 k8 4370 86 5655 41.1% 57.5% 1.4%63 k9 5026 86 6476 41.3% 57.3% 1.4%63 k10 4414 80 5505 43.2% 55.4% 1.4%64 k9 4612 85 5916 41.6% 57.0% 1.4%65 k9 3604 83 4628 41.5% 57.1% 1.4%69 k9 3679 84 4773 40.9% 57.7% 1.4%80 k10 5631 84 7187 41.9% 56.7% 1.4%Average 41.5% 57.1% 1.4%


needed to be sacrificed to achieve this saving. Col-umns 2 and 3 exhibit the decrease in emissions,while the last two columns exhibit the increase inemissions. Considering the speed limit of 90 km/hour, Table 5 implies that an average reduction of11.4% in CO2 emissions can be achieved. However,such a reduction will result in an average increasein travel time by 17.1%. Looking to the situationwith a speed limit of 80 km/hour an average of3.2% reduction in emissions can be achieved, whichwould require an average increase of 6.6% intravel time. These results show that in the case of90 km/hour a substantial reduction of CO2 emis-sions can be achieved.

5.2. Two Speed ProfilesIn this section we consider two speed profiles. Inaddition to the profile considered in section 1, we adda similar profile but with a congestion speed equal to70 km/hour similar to (Van Woensel and Vandaele2006), instead of 50 km/hour. The two profiles wererandomly assigned to the various distances in theinstances.We ran FðS0; vf ; a; b; cÞ for the two speed profile set-

ting. Table 6 gives an overview of the performance ofthe solutions for all sets; the last three columns givethe allocation of cost among its three components.

Similar to the single speed profile case, the majority ofcosts are attributed to driving cost and fuel cost. How-ever, due to the direct relation between fuel consump-tion and CO2, speeds associated with this model arebetween 81 and 85 km/hour.Both for the single and for the two speed profile

case the resulting speeds (vf ) are within the samerange. For the single speed profile case, where conges-tion speed is 50 km/hour, the best strategy to limitfuel and emissions costs is to try to avoid congestion,and thus speeds are higher than v�ð¼ 71 km/hour).For the two speed profile case, the congestion speedsare 70 and 50 km/hour. The congestion speed of70 km/hour is rather similar to v�. For these cases,minimizing fuel and emissions costs leads to pre-fering congestion and to reduce the speed. However,this is countered by the travel time cost.As the resulting speeds for both one and two

speed profiles are within similar ranges, we examinethe case where we set vf to 85 km/hour for all sets.This value is observed in 14 sets from Table 6.Furthermore, having a single value for all sets mightbe easier from an operational point of view. Similarto the previous definitions, we define the followingrelations.

Table 5 Trade-off between Emissions and Travel Times (One SpeedProfile)

Instance DE80 DE90 DTT80 DTT90

32 k5 �0.8% � 9.2% 10.3% 20.2%33 k5 �4.2% �8.5% 4.9% 20.6%33 k6 �5.3% �12.4% 4.0% 15.1%34 k5 �2.8% �12.6% 7.3% 19.1%36 k5 �1.4% �8.2% 6.5% 20.2%37 k5 �2.1% �9.0% 9.0% 21.5%37 k6 �1.1% �11.2% 7.3% 17.5%38 k5 �1.4% �16.1% 11.1% 17.0%39 k5 �8.5% �12.0% 6.4% 21.0%39 k6 �2.0% �16.0% 8.9% 10.3%44 k6 �5.3% �9.5% 5.6% 20.4%45 k6 �3.9% �9.7% 7.6% 19.2%45 k7 �0.8% �9.3% 4.5% 12.0%46 k7 �1.4% �14.3% 8.1% 13.2%48 k7 �2.1% �12.3% 6.3% 13.5%53 k7 �5.1% �9.9% 5.8% 20.5%54 k7 �1.5% �6.7% 5.1% 17.1%55 k9 �2.1% �12.7% 7.4% 18.0%60 k9 �4.9% �11.9% 5.5% 17.0%61 k9 �2.4% �10.3% 7.7% 22.6%62 k8 �6.4% �13.9% 2.5% 11.5%63 k9 �5.3% �8.6% 3.6% 17.6%63 k10 �3.4% �11.7% 6.6% 16.7%64 k9 �4.7% �14.2% 4.4% 13.1%65 k9 �3.3% �12.0% 5.0% 15.3%69 k9 �2.5% �10.2% 6.2% 20.3%80 k10 �0.7% �14.5% 9.7% 10.5%Average �3.2% �11.4% 6.6% 17.1%

Table 6 Results for FðS;vf; a;b; cÞ (Two Speed Profiles)

Cost partition

Set F ðS0; vf ; a;b; cÞ vf

CO2

(kg)Drivercost

Fuelcost

CO2

cost

32 k5 2487 85 3231 40.8% 57.7% 1.4%33 k5 1968 85 2564 40.7% 57.9% 1.4%33 k6 2234 83 2887 41.1% 57.4% 1.4%34 k5 2332 85 3039 40.7% 57.9% 1.4%36 k5 2450 84 3161 41.2% 57.3% 1.4%37 k5 2004 85 2625 40.4% 58.2% 1.4%37 k6 2925 85 3823 40.5% 58.1% 1.4%38 k5 2199 85 2874 40.5% 58.1% 1.4%39 k5 2632 85 2874 40.8% 57.7% 1.4%39 k6 2763 84 3571 41.1% 57.4% 1.4%44 k6 2875 84 3728 40.9% 57.6% 1.4%45 k6 2850 85 3729 40.4% 58.2% 1.4%45 k7 3633 85 4744 40.5% 58.0% 1.4%46 k7 2781 85 3641 40.4% 58.2% 1.4%48 k7 3393 83 4371 41.3% 57.2% 1.4%53 k7 3221 85 4198 40.6% 57.9% 1.4%54 k7 3736 84 4851 40.9% 57.7% 1.4%55 k9 3319 81 4243 41.8% 56.8% 1.4%60 k9 4371 82 5593 41.7% 56.9% 1.4%61 k9 3189 82 4114 41.2% 57.3% 1.4%62 k8 4279 85 5588 40.5% 58.0% 1.4%63 k9 4976 83 6394 41.5% 57.1% 1.4%63 k10 4069 84 5300 40.7% 57.9% 1.4%64 k9 4554 85 5951 40.5% 58.1% 1.4%65 k9 3826 85 5027 40.2% 58.4% 1.4%69 k9 3639 85 4804 39.9% 58.7% 1.5%80 k10 5673 82 7250 41.8% 56.8% 1.4%average 40.8% 57.7% 1.4%


DðTTÞ85 ¼TTðFðS; vf ; a; b; cÞÞ � TTðFðS; 85; a; b; cÞÞ

TTðFðS; 85; a; b; cÞÞDðEÞ85 ¼

EðFðS; vf ; a; b; cÞÞ � EðFðS; 85; a; b; cÞÞEðFðS; 85; a; b; cÞÞ

DðFÞ85 ¼FðS; vf ; a; b; cÞ � FðS; 85; a; b; cÞ

FðS; 85; a; b; cÞKðFÞ85 ¼ FðS; vf ; a; b; cÞ � FðS; 85; a; b; cÞ

Table 7 exhibits the results of setting vf to 85 km/hour.We note that the differences are not substantial. Thecost for FðS; vf ; a; b; cÞÞ is on average only 1.1% lowerthan the cost for F(S,85;a,b,c). However, in absolutevalues, as depicted by KðFÞ85, differences of up to144.6 € are observed. For the analyzed sets, we con-clude that setting the vehicle speed to 85 km/hourresults in a good compromise between travel timeminimization and emissions minimization.

6. Conclusions and Future Research

Carrier companies need to consider employinggreener practices in the future. This article establishesa framework for modeling CO2 emissions in a time-dependent VRP context, by the E-TDVRP model. Itproposes an efficient solution method for theE-TDVRP. We show the potential reduction in emis-

sions when weighed against travel time. Furthermore,we show that reducing emissions may have a positiveimpact with respect to reducing cost.We analyzed the effect of limiting vehicle speed in a

time-dependent VRP setting on the generated amountof CO2 emissions. Time-dependency was establishedby subjecting distances to variable speed profiles. CO2

emissions were modeled as a function of speed. Thespeed, which optimizes CO2 emissions per km, wasnot observed in any of our experimental settings. Thisis explained by the relation between limiting free flowspeed and the likelihood of running into congestion.In congestion, the vehicle is forced to drive moreslowly and emits more CO2. Thus, increasing the freeflow speed decreases the amount of time spent in con-gestion, and thereby may result in less emissions.Based on the optimal VRP solutions in the litera-

ture, we constructed an upper and lower bound forCO2 emissions. We showed that these bounds aretight, on average the gap was 4.7%. From a practicalpoint of view these bounds are simple and fast to cal-culate. They require standard VRP solutions that arecommonly used in practice. The bounds enable theassessment of the potential reduction in CO2 emis-sions.Considering CO2 emissions, the results of the E-

TDVRP were rather close to the lower bound, with adifference of 4.3% on average. This indicates that thechosen solution strategy was adequate. We comparedtwo extreme cases of the E-TDVRP model, consider-ing only travel times and considering only CO2 emis-sions. For the first case, two possible speed limitswere considered, 80 and 90 km/hour. For a speedlimit of 90 km/hour our results showed that achiev-ing an average reduction of 11.4% in CO2 emissionsrequired an average increase in travel time of 17.1%.However, for a speed limit of 80 km/hour, an averageincrease of 6.6% in travel times achieves a reductionof 3.2% in CO2 emissions.Considering one and two congestion speeds, the

experiments showed that from a realistic cost per-spective it is beneficial for vehicles to go below90 km/hour. We also showed that adopting a speedlimit of 85 km/hour yields good results in terms oftotal cost over all sets. Finally, we conclude that mini-mizing CO2 emissions by limiting vehicle speed canbe costly in terms of travel time. Nevertheless, limit-ing vehicle speed to a certain extent might be bothcost and emission effective.Additional enhancement of the algorithm may

facilitate the solution of larger instances. Futureresearch can also focus upon more accurate emissionsmodeling. For example, the weight of the vehicle hasan impact on the amount of emissions, and in theVRP context this can be incorporated as a function ofsatisfied demand. In addition, more sophisticated

Table 7. Comparison of F(S ; vf ; a;b; c) and F(S,85;a,b,c) (Two SpeedProfiles)

Set F ðS0; 85; a; b; cÞ DðTT Þ85 DðEÞ85 DðF Þ85 KðF Þ8532 k5 2489 0.0% 0% 0% 033 k5 1968 0.0% 0% 0% 0.033 k6 2234 �2.9% �5% �4% �100.134 k5 2384 0.0% 0% 0% 0.036 k5 2450 �0.7% �5% �4% �89.037 k5 2040 0.0% 0% 0% 0.037 k6 2960 0.0% 0% 0% 0.038 k5 2229 0.0% 0% 0% 0.039 k5 2632 �0.9% �2% �2% �47.939 k6 2763 �1.7% �2% �2% �54.844 k6 2875 �3.8% �5% �4% �129.045 k6 2899 0.0% 0% 0% 0.045 k7 3643 0.0% 0% 0% 0.046 k7 2908 0.0% 0% 0% 0.048 k7 3393 2.0% �2% 0% �8.553 k7 3258 0.0% 0% 0% 0.054 k7 3736 0.1% �1% �1% �25.955 k9 3319 3.9% �4% �1% �25.460 k9 4371 1.6% �2% �1% �35.861 k9 3189 0.2% �5% �3% �104.762 k8 4282 0.0% 0% 0% 0.063 k9 4976 0.5% �3% �2% �78.663 k10 4069 �2.9% �4% �4% �144.664 k9 4738 0.0% 0% 0% 0.065 k9 3891 0.0% 0% 0% 0.069 k9 3665 0.0% 0% 0% 0.080 k10 5673 0.9% �4% �2% �124.5average �0.1% �1.7% �1.1% �35.9


travel time profiles can be constructed to encompassacceleration and deceleration, which in return willenable more accurate emissions modeling. Further-more, the proposed model can be employed to inves-tigate the costs and benefits of using alternative fuelsin the context of VRP.We have shown that the current market cost of CO2

has no tangible economic impact on the results. How-ever, exploring this value is yet another possibleextension to this work. Examining the impact of thisvalue on transportation-related strategies is a relevantresearch question.

Acknowledgments

The research of Ola Jabali has been funded by the Nether-lands Organization for Scientific Research (NWO), projectnumber 400-05-185. Thanks are due to the referees for theirvaluable comments.

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Supporting InformationAdditional Supporting Information may be found in theonline version of this article:

Appendix S1. Validation of the solution method

Please note: Wiley-Blackwell is not responsible for thecontent or functionality of any supporting materials sup-plied by the authors. Any queries (other than missing mate-rial) should be directed to the corresponding author for thearticle.


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