Minimizing driving times and greenhouse gas emissions intimber transport with a near-exact solution approach

Authors: Marco Oberscheidera,∗, Jan Zazgornika, Christian Bugge Henriksenb, Manfred Gronalta,Patrick Hirscha

a Institute of Production and Logistics, University of Natural Resources and Life Sciences,Vienna Feistmantelstraße 4, 1180 Vienna, Austria

b Department of Agriculture and Ecology, Faculty of Life Sciences, University of CopenhagenHøjbakkegard Alle 13, 2630 Taastrup, Denmark

* Corresponding author; Mail: marco.oberscheider@boku.ac.at Phone: 00431476544413 Fax:00431476544417

Abstract1

Efficient transport of timber for supplying industrial conversion and biomass power plants is a2

crucial factor for competitiveness in the forest industry. Throughout the recent years3

minimizing driving times has been the main focus of optimizations in this field. In addition to4

this aim the objective of reducing environmental impacts, represented by carbon dioxide5

equivalent (CO2e) emissions, is discussed. The underlying problem is formulated as a multi6

depot vehicle routing problem with pick-up and delivery and time windows (MDVRPPDTW)7

and a new iterative method is proposed. For the numerical studies, real life instances of8

different scale concerning the supply chain of biomass power plants are used. Small ones are9

taken to validate the optimality of the new approach. Medium and large instances are solved10

with respect to minimizing driving times and fuel consumptions separately. This paper11

analyses the trade-offs between these objectives and shows how an additional mitigation of12

CO2e emissions is achieved.13

Keywords: Green Logistics, Timber Transport, Greenhouse Gas Mitigation, Mixed Integer14

Programming, Optimization, Log-Truck Scheduling15

Introduction16

The forest based sector plays a significant role for several countries. Besides the well known17

importance for New Zealand, Sweden, Finland, Chile and Canada, it also accounts for a18

1

prominent share of the economy in other countries like Austria. In Austria more than a19

hundred thousand people are working in sectors related to forest, timber and paper industry.20

From this it follows that efficient timber transports are a main interest of freight forwarders.21

Zazgornik et al. (2012) mention approximately 500,000 log-truck trips per year that originate22

in Austrian forests, not considering transports due to timber imports. Several publications23

give estimations of the percentage of transport costs in relation to the total timber price (e.g.24

von Bodelschwingh (2001), Murphy (2003), Favreau (2006)). In all of them, a value of25

approximately 30 % of the total price of round timber is attributed to be transportation costs.26

According to Flisberg et al. (2009) log-truck scheduling is traditionally done manually. This is27

also the case in companies that the authors have worked with in Austria. Hence, finding28

efficient ways of planning the transports and thereby reducing the costs is crucial.29

The tackled problem in this paper is related to the log-truck scheduling problem (LTSP, e.g.30

Palmgren et al. (2003)) and to the pick-up and delivery problem with time windows (PDPTW,31

e.g. Ropke & Pisinger (2006)). The pick-up takes place at a wood storage location, where the32

afore empty log-truck is fully loaded. Afterwards, the delivery location, which is an industrial33

site, is visited and all transported wood is unloaded. Usually the next transport task follows34

for the empty log-truck, thus the log-truck visits another wood storage location. Therefore,35

only one wood storage location and one industrial site is visited within one trip. Due to the36

specific construction of the used log-trucks, backhauls are unusual and not considered. The37

given problem is similar to the one described in El Hachemi et al. (2010), but dependencies of38

the activity of log loaders at wood storage locations and industrial sites as well as39

consequential waiting times are neglected. Besides this difference, log-trucks have a given40

home-base or depot, as used in the LTSP, respectively. Each log-truck has to start and end its41

tour at its depot, whereas it is possible that more than one log-truck is located at one depot.42

The problem in this paper is referred to as multi depot vehicle routing problem with pick-up43

and delivery and time windows (MDVRPPDTW), whereas transport tasks are predefined, as44

used in Gronalt & Hirsch (2007) and Hirsch (2011).45

For solving the MDVRPPDTW a new method, the so-called near-exact solution46

approach (NE), is introduced. It is an iterative algorithm that solves the MDVRPPDTW in47

three stages. As a first stage an extended assignment problem (EAP) is solved to generate a48

reduced transport network. Afterwards, the given network is checked in terms of maximum49

2

tour length and time window requirements. Violations lead to the generation and addition of50

cuts to the EAP, which is thereafter solved again. If no violations occur the MDVRPPDTW is51

solved on the given network. In order to validate the NE for small problem instances, it is52

compared to a mathematical model formulation of the MDVRPPDTW implemented in the53

solver software Xpress 7.2 using standard settings.54

Sbihi & Eglese (2007) state that most research in vehicle routing and scheduling is done to55

minimize costs. Besides the economic factor, also the environmental impact of transports can56

be reduced by efficient planning and the advice of decision support systems. The objectives of57

minimum costs and minimum environmental impacts are to a certain degree not in conflict58

with each other. The European Environment Agency (EEA, 2009) states that emissions due to59

fuel consumption contain the greenhouse gases (GHGs) carbon dioxide (CO2), nitrous oxide60

(N2O) and methane (CH4), as well as particulate matter (PM ), heavy metals, toxic61

substances, carcinogenic species, ozone precursors and acidifying substances. In this paper the62

focus is on the emission of GHGs that are presented as carbon dioxide equivalents (CO2e).63

The direct proportion of CO2 emissions to fuel consumptions (e.g. ICF Consulting (2006),64

EEA (2009)) is used to estimate emissions. Additionally, a cooperation of government65

departments in the UK, namely the Department of Energy and Climate Change (DECC) and66

the Department for Environment, Food and Rural Affairs (Defra), present factors for CH4 and67

N2O emissions as CO2e that are added to the direct CO2 formation in the combustion engine68

(Defra & DECC, 2011). Furthermore, they give estimations of indirect GHG emissions per69

liter of diesel. In this paper, direct and indirect emissions are accounted for and converted in70

kilograms of CO2e. The aim is to analyze the trade-off for freight forwarders of minimizing71

CO2e emissions compared to the objective of minimizing driving times.72

Eglese & Black (2010) review some possibilities of estimating emissions. The simplest way is73

to assume an average speed or fuel consumption per kilometer traveled for the whole road74

network. This approach has flaws, due to the nonlinear speed dependency of fuel consumption.75

As a more sophisticated approach, they propose different average speeds per road class for the76

entire road network. This approach is also used in the presented numerical studies of this77

paper. Based on the computer programme to calculate emissions from road transport78

(COPERT), developed by Ntziachristos & Samaras (2000), the EEA provides speed dependent79

formulas for estimating the fuel consumption, depending on the type of truck, its maximum80

3

weight limit, its exhaust emission standard, its load factor and the road gradient (EEA, 2009).81

In a recent comparative study of different vehicle emission models for road freight82

transportation the COPERT model provided the best estimate for heavy load vehicles with83

weights of 50 tonnes (Demir et al., 2011). Therefore, this model was used to provide input for84

calculating the fuel consumptions in this study. An alternative method for calculating CO285

emissions is used by Bektas & Laporte (2011) for the pollution-routing problem (PRP). Due to86

their given problem it is necessary to account for change in vehicle loads during a tour of a87

truck. This is not the case in the tackled problem of this paper, as there are only two states of88

a truck - fully loaded or empty. Furthermore, Eglese & Black (2010) mention congestion as an89

important factor for deviations of the average speed. To account for this factor Sbihi & Eglese90

(2007) suggest a time dependent vehicle routing and scheduling problem (TDVRSP). However,91

no data for time dependency are available for the test area used in this paper. Additionally,92

timber transports are mainly carried out on rural roads, where time dependency is less93

prominent compared to roads in urban areas. This time dependency is also assumed in the94

emission minimization vehicle routing problem (EVRP) of Figliozzi (2010). In his study he95

introduces two different problem formulations to minimize vehicle emissions in congested96

environments.97

The numerical studies of this paper are carried out with real life data - concerning the supply98

of biomass power plants - provided by the Institute of Forest Engineering of the University of99

Natural Resources and Life Sciences, Vienna. The input matrices for solving the100

MDVRPPDTW contain driving times and fuel consumptions. Both of them have been101

computed twice to account for shortest paths in terms of driving times and fuel consumptions.102

Furthermore, all instances, where fuel consumptions and driving times are compared, have103

been solved for both objectives separately.104

This paper presents a novel application for a routing problem in timber transport. With the105

NE it is possible to solve some of the real life problem instances with a size of up to 60106

transport tasks and 20 available log-trucks with their global optimum. For all other instances107

solutions close to the global optimum are obtained in a fast manner. Furthermore, a method108

for calculating and reducing CO2e emissions is introduced and applied to the presented109

problem. The objective of this paper is to give the readers an idea of how to implement the110

minimization of greenhouse gas emissions for transportation in their research or daily business111

4

and to introduce a powerful method to optimize the routing of log-trucks. Furthermore, this112

paper analyses the trade-offs between minimizing driving times and greenhouse gas emissions113

for real-life data in timber transport.114

Materials and methods115

This section presents a detailed problem description and outlines the modelling assumptions116

and data requirements. Special emphasis is given to the proposed solution approach (NE).117

Problem description118

The following features are considered in the presented problem:119

• A fleet of homogeneous log-trucks R that are located at depots H. These depots are120

typically the homes of the log-truck drivers. However, parts of the fleet can also be121

situated at a central depot.122

• A set of transport tasks T that start at wood storages W and end at industrial sites S.123

• The tour of a log-truck r ∈ R starts with an unloaded log-truck at the corresponding124

depot of r, hr ∈ H. After leaving the depot a wood storage location w ∈ W is visited,125

where the log-truck is fully loaded. In order to fulfill the task t ∈ T, the log-truck drives126

to a predefined industrial site s ∈ S to fully unload its goods. Afterwards, either another127

task is started by driving to a wood storage location or the log-truck returns to its depot128

hr.129

• Wood storage locations and industrial sites can be visited more than once throughout130

the planning period.131

• Each transport task t ∈ T has to be fulfilled.132

• As full-truck loads are assumed, transports between different wood storage locations are133

not allowed. The same holds for industrial sites. Additionally, all log-trucks have the134

same capacity. Therefore, capacity constraints are not needed.135

• A feasible tour has to fulfill constraints in terms of a maximum driving time MT and136

time windows. Time windows occur at depots [ah,bh] and industrial sites [as,bs].137

5

• For loading at the wood storage and unloading at the industrial site, service times SWw138

and SIs occur.139

With the transport tasks already defined, the objective of the log-trucks is to choose the140

optimal empty-truck rides. If this optimization is done in terms of fuel consumption, the141

driving times DT are exchanged by fuel consumptions FC of empty-truck rides. Due to the142

given problem structure, choices of following tasks occur at depots or industrial sites only,143

which - depending on the formulation - reduces the number of constraints and variables144

markedly, compared to a vehicle routing problem (VRP) without predefined transport tasks.145

Figure 1 shows an illustration of the given problem with two trucks, five wood storage146

locations and two industrial sites. In this example one log-truck is located at a depot only. At147

the beginning of the planning period the starting and end points of a transport task t ∈ T as148

well as its duration TDt and fuel consumption FCt are already known. The starting point149

equals a certain wood storage location w ∈W and the end point is a specific industrial site150

s ∈ S. The duration TDt of a transport task t equals the driving time from the wood storage151

location w to the industrial site s of t. The same concept applies to the fuel consumption FCt152

of a transport task t, which is the fuel consumption if driving fully loaded from the wood153

storage location w to the industrial site s of t.154

Figure 1: An example of the tackled problem

6

Mathematical model155

The problem is formulated as a standard MDVRPPDTW. The binary decision variable tijr is156

equal to 1, if log-truck r performs task i ∈ T ∪ H before task j ∈ T ∪ H. The first task157

i = hr, hr ∈ H starts at a depot and is followed either by a home ride j = hr, hr ∈ H -158

if log-truck r is not used within the planning period - or by a transport task j ∈ T.159

Afterwards, an arbitrary number of transport tasks may follow. As starting and end points of160

transport tasks are given, the driving time DTij of task i ∈ T to task j ∈ T equals the driving161

time from the industrial site s of i to the wood storage location w of j . If j ∈ H, DTij is the162

driving time of the last industrial site s to the depot hr, which equals a home ride.163

eir is the completion time of task i by log-truck r. str corresponds to the starting time of the164

tour of log-truck r and etr marks its end. The overall model formulation is given below:165

min∑

i∈T∪H

∑j∈T∪H

∑r∈R

DTijtijr (1)

s.t.∑

j∈T∪H

∑r∈R

tijr = 1 ∀i ∈ T ∪H (2)

∑i∈T∪H

tikr −∑

j∈T∪Htkjr = 0 ∀k ∈ T ∪H,∀r ∈ R (3)

∑i∈T∪H

(∑j∈T

(DTij + TDj)tijr +∑j∈H

DTijtijr) ≤MT ∀r ∈ R (4)

ai ≤ eir ≤ bi ∀i ∈ T, ∀r ∈ R (5)

eir + SIi +DTij + SWj + TDj −M(1− tijr) ≤ ejr ∀i, j ∈ T, ∀r ∈ R (6)

str +DThrj + SWj + TDj −M(1− thrjr) ≤ ejr ∀j ∈ T, ∀r ∈ R (7)

eir + SIi +DTihr−M(1− tihrr) ≤ etr ∀i ∈ T, ∀r ∈ R (8)

tijr ∈ {0, 1} ∀i, j ∈ T ∪H,∀r ∈ R (9)

eir ≥ 0 ∀i ∈ T ∪H,∀r ∈ R (10)

str ≥ 0 ∀r ∈ R (11)

etr ≥ 0 ∀r ∈ R (12)

7

The objective (1) seeks to minimize the sum of the driving times DTij of the empty-truck166

rides. Constraints (2) ensure that each task i ∈ T ∪ H is fulfilled. A ride from i to j, with167

i = j, is only allowed, if i, j ∈ H. This means that the log-truck does not leave the depot.168

(3) force a log-truck to leave for another task after completing task i. (4) make sure that the169

total driving time of a tour does not exceed the maximum tour length MT. Constraints (5)170

and (6) guarantee that time windows at the depots and industrial sites are met, whereas171

constant M has a large integer value, which is introduced to linearize the constraints. If j ∈ H,172

the parameters TDj , SWj and SIj are 0. (7) and (8) connect the start str and end etr of a173

tour to the end of a task eir. Constraints (9) ensure the binarity of the decision variables tijr.174

The last constraints of (10), (11) and (12) contain the non-negativity restrictions of the given175

decision variables.176

CO2e calculation177

For an estimation of the CO2 emissions, the fuel consumption of each arc within a road178

network has to be known, as the CO2 emissions are directly proportional to the fuel179

consumption. Different factors for the conversion of liters of diesel to kilograms of CO2 can be180

found in the literature. For example the EEA (2009) uses 3.14 kg CO2 per liter diesel if an181

oxidation of 100 % of the fuel carbon is reached - which is called ultimate CO2. Complete182

oxidation of all carbon components is not realistic, as also carbon monoxide (CO),183

hydrocarbons (e.g CH4) and PM are formed. Besides incomplete oxidation there are184

incompustible species present in the combustion chamber e.g. nitrogen gas (N2) or nitrogen185

oxides (NOx) out of the air (EEA, 2009). In terms of GHGs, the by-products of N2O and186

CH4 are important. Consequently, Defra & DECC (2011) give values for the formation of187

these gases as CO2e. In terms of 100 % mineral diesel 0.0012 kg CO2e of CH4 and188

0.0184 kg CO2e N2O are emitted per liter. Added to the direct formation of 2.6480 kg of CO2189

this leads to the emission of 2.6676 kg CO2e per liter diesel. Besides direct CO2e emissions190

Defra & DECC (2011) also provide information of indirect GHGs as CO2e. They originate191

from required preceding processes like the extraction and transport of primary fuels or the192

refining, distribution, storage and retail of finished fuels. As a total, indirect emissions of193

0.5085 kg CO2e per liter diesel are reported. Overall direct and indirect emissions add up to194

3.1761 kg CO2e per liter diesel. The use of a certain percentage of biofuel reduces the total195

8

emissions; e.g. a share of 3.6 vol % biofuel leads to emissions of 3.1073 kg CO2e per liter196

diesel. Worth mentioning is that the reported 3.14 kg CO2 per liter diesel for exhaust197

emissions of vehicles in European countries - which are stated in the annex of EEA (2009) -198

are also in this range.199

In EEA (2009) many formulas are available to calculate the fuel consumption for different200

types of vehicles. In this paper only heavy-duty vehicles are focused on, as they are used in201

timber transport. Different formulas are useable depending on the type of truck, its maximum202

weight, its exhaust emission standard, its load factor and the road gradient. These formulas203

provide the fuel consumption in grams per kilometer depending on the speed v. Besides the204

unknown v, up to five factors (α, β, γ, δ and ε) are used within the different formulas. These205

factors are derived from statistical analyses and are given constants that can be found in the206

annex of EEA (2009), if the aforementioned specifiers for truck and road are known.207

For example the fuel consumption fc of a half loaded truck and trailer with EURO V emission

standard and a maximum weight from 34-40 tonnes on a road with a gradient of +2 % is

calculated with equation (13).

fc = α · βv · vγ (13)

The given parameters α, β and γ have values of α = 2, 021.18, β = 1.0055 and208

γ = − 0.4261 (EEA, 2009). By inserting v = 55 km/h the formula results in a fuel209

consumption of 496.05 g/km or 49.605 kg per 100 km, respectively. The consumption of diesel210

in [g] or [kg] is not as intuitive as an indication in liters. Hence, fuel consumptions are211

transformed to liters with the mean density of diesel at a temperature of 15 ◦C from the212

European Standard (EN) 590:2009 of 0.8325 kg/l. So the above calculated use of 49.605 kg per213

100 km means a consumption of 59.59 l per 100 km.214

By the use of formula (13), it is possible to establish a network, where the specific fuel215

consumption of each arc is known. Therefore, only the length of the arc and the average speed216

on it have to be inserted. The use of a single average speed for all the arcs within a network217

would lead to a result that only differs from a weighting with driving times by a constant218

factor (Eglese & Black, 2010). This is not suitable for comparing the results of minimizing219

driving times on the one hand to the results of minimizing CO2e emissions on the other hand.220

Hence, arcs are divided into different segments depending on their road class. Each segment221

9

has a length and a corresponding average speed and by adding up the fuel consumptions of all222

the segments of an arc, the total fuel consumption from one node to another node is retrieved.223

From this information a fuel matrix is obtained by taking the minimum fuel consumption from224

each node to every other node. This matrix differs from the one with minimum driving times.225

Fuel consumptions and driving times are speed dependent, but in contrast to driving times,226

the speed dependency of fuel consumptions is not linear. This leads to different routes within227

the matrix and thereby different solutions.228

To solve the problem in a way that CO2e emissions are minimized, the objective function of

the MDVRPPDTW has to be altered to (14) accordingly.

min∑

i∈T∪H

∑j∈T∪H

∑r∈R

FCijtijr (14)

To obtain the total emissions in CO2e the fuel consumption is multiplied with 3.14 kg CO2e229

per liter diesel in our numerical studies.230

Solution approach231

The NE takes the structure of the given problem into account and solves the problem either232

exactly or heuristically, if computing times are unreasonable. By using this approach the233

MDVRPPDTW becomes solvable for problem sizes that cannot be reached by applying234

standard model formulations like the one presented before. Both approaches are implemented235

in the programming language Mosel and are solved with the solver software Xpress 7.2.236

The iterative approach of NE is displayed in Figure 2. In the first stage an EAP is solved to237

generate a valid network for the log-trucks. Afterwards, the feasibility of this network is238

checked by considering maximum tour length and time windows. If violations occur, cuts are239

generated - which is the second stage - and added to the EAP. Additionally to the initially240

used constraints of the EAP, the added cuts lead to a further limitation of the solution space.241

After the addition of cuts, the first stage is repeated and the resulting network gets trimmed in242

the direction of a feasible one. In the third stage the model formulation of the MDVRPPDTW243

is solved on the created constrained network. If the MDVRPPDTW still cannot be solved on244

the given network, a cut is added to the EAP that bans the actual solution. This procedure is245

repeated until the algorithm finds a network the MDVRPPDTW is solvable on. The solution246

10

is the global optimum of the problem.247

Figure 2: Activity diagram of the NE

The main advantages of this approach are the short computing times for solving the EAP and248

the reduction of the number of variables for the MDVRPPDTW. On the one hand a number249

of infeasible arcs can be excluded, because parts of the network cannot be reached by all250

log-trucks. On the other hand the values of the starting and home rides of the log-trucks -251

because they are already predetermined by the structure of the network - as well as all the arcs252

that contain the predefined transport tasks can be fixed.253

The choices for the log-trucks of where to go next occur either at their depots H or at the254

industrial sites S. Due to that structure, it is possible to solve a standard assignment problem255

that assigns all transport tasks t ∈ T either to an industrial site s or a depot h in a way that256

either the total empty-truck driving time or the fuel consumption is minimized. Additionally,257

subcycles have to be avoided and each log-truck r ∈ R needs to reach its depot hr. If this can258

be guaranteed, a valid network is obtained. However, this does not imply that the given259

11

MDVRPPDTW is feasible on it.260

The EAP can be formulated by the following system of equalities and inequalities:261

min∑

s∈H∪S

∑t∈T∪H

∑r∈R

DTstastr (15)

s.t.∑

s∈H∪S

∑r∈R

astr = 1 ∀t ∈ T ∪H (16)

∑t∈T∪H

∑r∈R

ahtr = 1 ∀h ∈ H (17)

∑r∈R

∑t∈T

ahrtr ≥ LBT (18)

∑t∈T∪H

∑r∈R

astr = |TBs| ∀s ∈ S (19)

∑s∈S

∑t∈(T\

⋃s∈S TBs)∪H

∑r∈R

astr ≥ 1 ∀S ⊂ S (20)

∑i∈H∪S:i6=s

∑t∈TBs

aitr =∑

j∈H∪{T\TBs}

asjr ∀s ∈ S, ∀r ∈ R (21)

astr ∈ {0, 1} ∀s ∈ H ∪ S, ∀t ∈ T ∪H,∀r ∈ R (22)

The binary decision variable astr is equal to 1, if transport task t ∈ T or depot t ∈ H is either262

assigned to a depot s ∈ H or an industrial site s ∈ S with a log-truck r ∈ R. If astr is 1 for263

s,t ∈ H, it follows that log-truck r is not used and stays at its associated depot hr. The starting264

ride of a log-truck is characterized by a transport task t ∈ T assigned to a depot s ∈ H,265

whereas a home ride can be identified by a depot t ∈ H assigned to an industrial site s ∈ S.266

The objective function (15) minimizes the driving times DTst for the empty-truck rides. If the267

objective is to minimize the fuel consumption, DTst is simply replaced by FCst. This leads to268

the following objective function (23), whereas the remainder of the problem stays the same:269

min∑

s∈H∪S

∑t∈T∪H

∑r∈R

FCstastr (23)

(16) ensure that all transport tasks t ∈ T are assigned to a depot s ∈ H or an industrial site270

12

s ∈ S and that all log-trucks finish at their depot t = hr. A log-truck r either leaves its271

depot hr to start with a task t ∈ T or stays at home at the depot t ∈ H (17). However, at272

least a certain number of log-trucks LBT has to leave their depots to fulfill the tasks (18).273

This minimum number of log-trucks equals LBT = dTOT/MT e, whereas TOT is the value of274

the objective function of the actual iteration plus the sum of all task durations∑t∈T TDt.275

MT is the maximum tour length of a log-truck r ∈ R. Constraints (19) guarantee that as276

much transport tasks t ∈ T or home rides t ∈ H have to be started from an industrial site s277

as belong to the industrial site s. TBs is a subset of T and contains all transport tasks t that278

deliver to s. (20) ensure that no subcycles occur. Therefore, it has to hold that there is at279

least one task t assigned to s ∈ S ⊂ S that does not deliver to industrial sites within S. (21)280

assure that each log-truck entering the transport tasks of an industrial site also has to leave281

again. Finally, (22) are the binarity constraints for the given problem.282

Solving the system of linear equations and inequations above leads to a valid network on which283

an implementation of the MDVRPPDTW can be carried out. Before solving it, some284

feasibility checks are done to avoid later violations concerning maximum tour length and time285

windows. However, as no tours are constructed yet, there is no guarantee that all violations are286

eliminated by the introduced cuts. Different types of cuts are used and added to the EAP and287

then the EAP is solved again. If no cuts need to be added at this stage, the MDVRPPDTW is288

solved. This problem is either feasible, which leads to a solution that is the global optimum, or289

infeasible. If the MDVRPPDTW is infeasible, cuts that ban the given solution are added to290

the problem and the EAP is solved again. A detailed description of the introduced cuts would291

go beyond the scope of this paper, but can be found in Oberscheider et al. (2011).292

Numerical studies293

The proposed solution methods were used to solve different problem sizes based on real-life294

data. Comparisons are done with respect to driving times and fuel consumptions, whereas295

from the latter the CO2e emissions are calculated. From August 2009 to February 2011 daily296

tours of five log-trucks were tracked and stored in a database. From this given information a297

road network consisting of 99 wood storage locations, 4 biomass power plants and 14 depots298

for the log-trucks was modelled. According to the given data, biomass power plants are used as299

industrial sites in the numerical studies. The biomass power plants are located in Austria and300

13

all the selected depots and wood storage locations are situated within a distance between 2 and301

138 km away from the power plants (Figure 3). For further information about the situation302

and potential of biomass power plants in the investigated area, see Rauch et al. (2010).

Figure 3: Distribution of depots, wood storage locations and biomass power plants in the test area

303

The required matrices are derived from a road network that is implemented in a geographic304

information system (GIS) according to the road classes 0-7 of Holzleitner et al. (2010). The305

average speeds per road class originate from Ganz et al. (2005), in which data were collected306

for another region of Austria. The driving times of the recorded data and the data gained by307

simulations in GIS were compared and afterwards, the average speeds have been reduced by a308

factor of 1.452. This factor is used to tackle the gap between observed and simulated values in309

order to make the scenarios more realistic. Three different matrices are extracted from the GIS310

data. The first one contains the minimum driving times from each node to every other311

required node. The second and third have the same structure, but comprise fuel consumptions.312

The second one is obtained for the fuel consumptions of empty-truck rides and the third one313

for log-trucks with full loads. The global positioning system (GPS) data for generating the314

input matrices for the algorithm, cover the rides between two locations only. It is assumed315

14

that the engine of the log-truck is turned off at the wood storage location. Fuel consumptions,316

due to reversing the log-trucks at the locations or for unloading at the industrial sites are not317

considered.318

To choose the correct formula out of the ones that are given in the annex of EEA (2009), input319

factors are determined. Besides the load of 0 % and 100 %, the type of truck is set to a truck320

and trailer with 34-40 tonnes maximum weight. EURO III is chosen as emission standard321

according to the year of manufacturing of the tracked log-trucks. As no digital elevation model322

(DEM) is used for the GIS data, an overall road gradient of 0 % is taken. Equation (24) is323

used for calculating the fuel consumptions for empty-truck rides and full-truck rides in [g/km].324

fc = ε+ α · e−β·v + γ · e−δ·v (24)

The setting of the parameters (α− ε) for the different loads can be seen in Table 1. The values325

of the parameters as well as equation (24) are taken from the annex of EEA (2009).326

Table 1: Parameter settings for the calculation of fuel consumptions

Load α β γ δ ε

0 % 547.36 0.055 1,836.32 0.43 174.44100 % 634.79 0.029 476,141.21 1.41 215.04

The received matrices for the fuel consumptions are converted from grams of diesel per path to327

liters of diesel, before used in the algorithms. This is done in the way it is described in the328

section about CO2e calculation. Figure 4 gives an overview of how much the emissions per329

kilometer vary for the valid range of driving speeds of the given formula (6 to 86 km/h)330

depending on the load of the truck.331

Three different sizes of daily datasets have been used, whereas in all three cases 20 instances332

have been generated. The smallest dataset contains scenarios with 15 tasks and 5 log-trucks.333

It is used to validate the NE approach by a comparison to the solution of the standard model334

formulation of the MDVRPPDTW. Instances with 30 tasks and 10 log-trucks were generated335

to simulate a workday of a medium-sized Austrian company in the sector of wood transport.336

Additionally, 20 instances with 60 tasks per day and 20 log-trucks were taken for the tests of337

the NE.338

15

Figure 4: CO2e emissions [g/km] on roads with a gradient of 0 % of a Euro III truck and trailer with34-40 tonnes maximum weight depending on its load and driving speed [km/h]

For the large instances the home location of log-truck 5 was taken as central depot, due to its339

geographical location. 7 log-trucks have their origin there. The wood storage locations were340

picked randomly out of the given 99, whereas the likelihood of a transport to a certain biomass341

power plant is in relation to its demand for wooden chips in bulk stacked cubic meter (BCM)342

per year (Table 2), as found in Rauch & Gronalt (2010). Due to the random choice, more than343

one transport task can have its origin at a certain wood storage location.

Table 2: Yearly demand of wooden chips of the power plants

Plant Demand in BCM Likelihood in %1 150,000 15.02 150,000 15.03 97,000 9.74 600,000 60.2

344

Besides the already mentioned input, the times for loading and unloading are needed. The345

measurements resulted in durations of 55 minutes for loading at the wood storage location and346

37 minutes for unloading at the power plant. Power plants can receive deliveries between 7 am347

and 7 pm. At the earliest, drivers may start at 5 am. The latest feasible arrival at the depot in348

the evening is at 9 pm [0, 960]. In between this time window, drivers are allowed to have a349

maximum operation time of 480 minutes. As operation time, driving times are counted only,350

16

whereas times for loading and unloading are not considered.351

Results352

The following subsections describe the results of the numerical tests. The first one contains the353

validation of the NE by a comparision to the solution of the standard model formulation that354

is generated with solver software XPress 7.2 with standard settings. This was done with small355

problem instances and the objective to minimize the driving time. The following one shows356

the results for the objective of minimizing the driving time and minimizing the fuel357

consumption or CO2e emission, respectively. Additionally, the trade-offs of using one or the358

other in terms of driving time and CO2e emissions are given.359

The NE ran till it either terminated or could not find the global optimal solution after a360

runtime of 3,600 seconds for the medium sized instances and 7,200 seconds for the large361

instances. After this duration, the driving time or fuel consumption of the current solution was362

recorded, respectively. This value serves as lower bound (LB) for the comparison with the363

actual solution, which can be gained by raising the lower bound of the number of needed364

log-trucks LBT by one. The deviation from the LB to the solution of the NE equals the365

maximum deviation from the global optimal solution. The heuristic approach of adding366

log-trucks in the NE is taken in order to get feasible solutions in reasonable computation times.367

For all the instances, for which run times are reported, the tests were performed on a single368

workstation with an Intel Core i7 with 2.8 GHz and 6.00 GB RAM, which runs on369

MS-Windows 7 as operating system.370

Comparison of NE and standard solver procedure371

The problem size for solving the model formulation of the MDVRPPDTW with the MIP372

solver XPress 7.2 using standard settings is restricted, due to computation time. Therefore,373

instances with 15 tasks and a maximum of 5 log-trucks have been chosen. 19 out of the 20374

tested instances have the same solution values for both methods. For one instance, the MIP375

solver did not finish within the maximum computation time of 43,200 seconds. In this case a376

gap of 7.64 % from the lower bound of 984.51 minutes of driving time to the best found integer377

solution of 1,066 minutes was recorded. The solution value of the NE of this instance equals378

17

the best found integer solution of the MIP solver.379

The computation times for the NE as well as for the MIP solver vary markedly. The range for380

the NE goes from 0.1 seconds to 27,072.1 seconds. The latter shows an exceptional higher381

computation time than the rest of the instances, due to a large number of iterations that had382

to be performed. However, also the computation times for the instances that were solvable to383

optimality by the MIP solver vary from 4.1 seconds to 15,322.8 seconds. The median of the384

computation time required to solve the instances by NE is 7.5 seconds, whereas for the MIP385

solver it is 20.3 seconds. It follows that for the given instances the NE is the faster method. It386

provides proven global optimum solutions in 19 out of 20 instances.387

Minimizing driving time versus minimizing CO2e emissions388

In Table 3 a summary of the results, which can be found in detail in the Appendix, is given.389

The means of the presented fuel consumptions and driving times are given for the empty-truck390

rides only, as they are minimized by the NE. For medium-sized instances of both objectives391

the NE found the global optimal solution for 16 out of 20 instances. For the remaining392

instances one log-truck was added after 3,600 seconds of runtime. They also have similar393

values in terms of runtimes as well as maximum deviations from the LB.394

From the large instances with the scenario of minimizing driving times, Instance 1 could be395

solved in a provable exact way only. Out of the remaining 19 instances, 14 have been solved by396

adding one log-truck, while for five of them the number of log-trucks had to be increased by397

two. In terms of minimizing fuel consumption, 4 globally optimal solutions could be obtained.398

For 12 instances it was sufficient to add one truck, whereas for 4 instances two trucks had to399

be added to the given problem. Runtimes show higher variations as for medium-sized400

instances and maximum deviations from the LB are slightly below 3 %.401

Table 3: Summary of the results of medium (M) and large (L) instances for the objectives ofminimizing driving time (DT) and minimizing fuel consumption (FC)

Mean Mean Global Max. DEV Mean Trucks Range of MeanFC DT opt. found from LB Trucks added runtimes [s] runtime

Scenario [l] [min] [no.] [%] [no.] 1 2 Min Max [s]

DT M 310.8 1,791.2 16 3.10 8.3 4 0 2.7 3,603.4 924.7FC M 308.7 1,830.3 16 3.57 8.4 4 0 2.2 3,607.0 935.7DT L 585.2 3,365.0 1 2.94 16.4 14 5 594.0 15,183.6 9,052.6FC L 581.2 3,445.3 4 2.97 16.5 12 4 149.4 16,853.9 7,883.6

For a comparison of the objectives of minimizing fuel consumption and minimizing driving402

18

time, different input matrices for driving times and fuel consumptions have to be computed.403

This follows the fact that also the shortest paths between two points within the network may404

change, due to different objectives. Therefore, the fastest way does not have to be the most405

efficient one in relation to fuel consumption. Hence, the comparison leads to even higher406

deviations than an optimization that uses the same input matrices for both scenarios. This is407

not true for empty-truck rides only, but also for transport tasks with fully loaded log-trucks.408

Therefore, empty-truck rides and full-truck rides have to be aggregated before the solutions of409

the two scenarios are compared to each other. The main focus of the comparison is on reducing410

total CO2e emissions by changing objectives. Additionally, it is important to report the change411

of total driving times, as this is the main interest of drivers and freight forwarders. A summary412

of these results is presented in Table 4, whereas detailed results can be found in the Appendix.413

Table 4: Total CO2e emissions versus total driving times of medium (M) and large (L) instances forthe objectives of minimizing driving time (DT) and minimizing fuel consumption (FC)

Mean total CO2e [kg] CO2e reduction [kg] Mean total DT [min] DT extension [min]Scenario min FC min DT Mean SD min FC min DT Mean SD

M 2,748.1 2,781.1 33.0 14.4 3,745.1 3,663.6 81.6 36.1L 5,334.2 5,396.5 62.3 24.7 7,229.6 7,059.4 170.2 50.7

For the medium sized instances, the exchange of the objective function leads on the one hand414

to reduced CO2e emissions, but on the other hand to higher driving times for all tested415

instances. The average reduction of CO2e is 33 kg and the extension of the driving time has a416

mean value of 81.6 minutes. As stated in Table 3 approximately 8 log-trucks have to be used.417

Hence, changing the objective to minimizing CO2e emissions leads to an average increase of418

approximately 10 minutes per driver and a mean reduction of roughly 4 kg CO2e per tour.419

According to the tests with large instances, Instance 9 is an outlier (see Appendix), due to a420

reduced driving time, when minimizing fuel consumption. The reason for this is that two421

log-trucks had to be added for Instance 9 to solve it within the maximum computation time422

regarding the minimization of driving times. The remaining instances showed the expected423

behavior of reduced total CO2e emissions and increased total driving time. By using the424

objective of minimizing CO2e emissions, a mean reduction of 62.3 kg is obtained. The average425

extension of driving times is 170.2 minutes, which is shared by 16 to 17 log-trucks. Similar to426

the medium sized instances the CO2e emissions decrease by approximately 4 kg CO2e per427

driver, whereas the tour duration of a driver rises by around 10 minutes.428

19

Discussion429

Based on the comparison of the results of the NE and the MIP solver for small instances it can430

be concluded that the NE is a promising method for solving problems with the given structure.431

For the tested instances it was enough to raise the lower bound of trucks by a maximum of 10432

% of the available log-trucks. In order to save computation time it is possible to raise this433

lower bound already at the beginning of the computation. This is an option for medium and434

large instances where a solution could not be found within the given maximum time. Hence, if435

used in praxis, the algorithm should be started parallel with additional log-trucks on one hand436

and the attempt to solve the problem exactly on the other hand. Thereby, fast solutions can be437

obtained. In our test case this would lead to a maximum computation time of 2,453.9 seconds438

for Instance 20 of the large instances with the objective of minimizing fuel consumption.439

It is likely that some of the presented solutions, for which the global optimal solution cannot440

be guaranteed, are solved with their global optimum. This arises from the fact that a rather441

straight forward approach is used to set the lower bound of log-trucks. For example, for solving442

Instance 7 of the medium sized instances in terms of minimizing driving time, the lower bound443

of log-trucks is set to eight. The division of the lower bound of the solution by the maximum444

tour length leads to a minimum number of log-trucks of 7.98. Thus, in the opinion of the445

authors, it is unlikely that eight log-trucks are sufficient to solve the given problem, because446

the average driving time of a log-truck would be 478.9 minutes. As single routes are indivisible,447

an assignment of 30 tasks to eight log-trucks within the maximum tour length is improbable.448

By using two different objective functions, the potential savings of CO2e emissions compared449

to an optimization in terms of driving times were presented. In general, it is the opinion of the450

authors that the biggest part of possible savings is achieved by the use of a decision support451

system per se, even if it optimizes driving times instead of fuel consumptions. If optimized in452

terms of fuel consumption, the potential reduction of approximately 4 kg CO2e emissions per453

driver and day is bought by an extension of the driving time of approximately 10 minutes per454

driver and day. Specifications in terms of road gradients would increase the accuracy of the455

results, but these data were not available. Additionally, factors as the driving behavior or456

maintenance of the trucks play a crucial role for actual emissions, but are hard to indicate.457

In Figure 4 the impact of driving speeds on CO2e emissions is shown. In the given range the458

emissions per kilometer are decreasing with increasing driving speeds. Therefore, it can be459

20

advantageous for the vehicle to choose a road with a road class that has a higher average460

speed, even if this leads to rising driving times of the truck due to longer distances. Especially461

at driving speeds below 25 km/h the slope becomes steep, which leads to comparatively high462

increases of CO2e emissions per kilometer. Additionally, to the choice of roads with higher463

average speeds, congested areas need to be avoided. Hence, information on time-dependent464

travel speeds would be another interesting input factor for increasing the accuracy of the given465

results even though they are less relevant in timber transport, since it takes place in rural466

areas mainly (see e.g. Piecyk et al. (2010)).467

The given problem implies full-truck loads, whereas the model formulation of the LTSP of468

Palmgren et al. (2003) allows more than one pick-up and/or delivery per route. Therefore, it469

would be interesting to use the findings of minimizing emissions of Bektas & Laporte (2011) on470

that problem formulation. By using the objective of minimizing greenhouse gas emissions, the471

sequence of nodes in a route is also depending on how much is loaded or unloaded at the472

corresponding node, respectively. As shown in Figure 4 a higher load factor leads to higher473

emissions. Therefore, it could be advantageous in case of multiple delivery locations to474

perform deliveries with higher weights first, even if this leads to longer driven distances. Vice475

versa, the same is true for multiple pick-up locations.476

In conclusion, the selection of the objective of minimizing fuel consumptions leads to a477

significant reduction of CO2e emissions compared to the use of the objective of minimizing478

driving times. Therefore, the used approach gives an idea of how to implement the479

minimization of greenhouse gas emissions in timber transport in research or daily business and480

introduces a powerful method to optimize the routing of log-trucks. Further tests will be481

performed on available instances (e.g. Hirsch (2011) and Zazgornik et al. (2012)) to validate482

the range of applications of the NE.483

Acknowledgments484

Thanks go to Andrea Trautsamwieser for helping with mathematical formulations. For485

providing very important input in terms of data, the authors want to thank Franz Holzleitner486

of the Institute of Forest Engineering of the University of Natural Resources and Life Sciences,487

Vienna.488

21

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24

Appendix560

Notations561

H Set of depots

R Set of log-trucks

S Set of industrial sites

S Subset of S

T Set of transport tasks

TBs Set that contains the tasks that end in/belong to industrial site s

W Set of wood storage locations

M Constant that has a large integer value

MT Maximum driving time

α, β, γ, δ, ε Parameters derived from statistics to calculate the fuel consumption fc of a

segment of an arc

ai Begin of time window of i

bi End of time window of j

fc Fuel consumption of a segment of an arc

DTij Driving time of an empty-truck ride from node i to node j

FCij Fuel consumption from node i to node j

FCt Fuel consumption of transport task t : The fuel consumption from the wood

storage location w to the industrial site s of transport task t

hr The corresponding depot of log-truck r

LBT Lower bound of the number of log-trucks

SIs The service time at the industrial site s

SWw The service time at the wood storage w

TDt Task duration of transport task t : The driving time from the wood storage

location w to the industrial site s of transport task t

TOT Equals the total driving time (empty-truck and full-truck rides) of a solution

v Average speed on a segment of an arc

astr Binary decision variable, equals 1 if transport task or depot t is assigned to

depot or industrial site s with a log-truck r, 0 otherwise

eir Completion time of task i by log-truck r

etr End time of the tour of log-truck r

str Starting time of the tour of log-truck r

tijr Binary decision variable, equals 1 if log-truck r performs task i before task j,

0 otherwise

562

25

List of abbreviations563

BCM Bulk stacked cubic meter

CH4 Methane

CO Carbon monoxide

CO2 Carbon dioxide

CO2e Carbon dioxide equivalent

COPERT Computer programme to calculate emissions from road transport

DECC Department of Energy and Climate Change of the UK

Defra Department for Environment, Food and Rural Affairs of the UK

DEM Digital elevation model

EAP Extended assignment problem

EEA European Environment Agency

EN European standard

EVRP Emission minimization vehicle routing problem

GHG Greenhouse gas

GIS Geographic information system

GPS Global positioning system

LB Lower bound

LTSP Log-truck scheduling problem

MDVRPPDTW Multi depot vehicle routing problem with pick-up and delivery and time

windows

NE Near-exact solution approach

N2 Nitrogen gas

N2O Nitrous oxide

NOx Nitrogen oxides

PDPTW Pick-up and delivery problem with time windows

PM Particulate matter

PRP Pollution-routing problem

TDVRSP Time dependent vehicle routing and scheduling problem

VRP Vehicle routing problem

564

26

Resu

lts

MIN

IMIZ

EDRIV

ING

TIM

E(D

T)

MIN

IMIZ

EFUEL

CONSUM

PTIO

N(F

C)

DT

versusFC

Instance

LB

DT

DEV

FC

Tru

cks

Run-

LB

FC

DEV

DT

Tru

cks

Run-

DT:Tota

lFC:Tota

lDEV

DT:Tota

lFC:Tota

lDEV

used

time

used

time

CO

2e

CO

2e

DT

DT

[min]

[min]

[%]

[l]

[no.]

[s]

[l]

[l]

[%]

[min]

[no.]

[s]

[kg]

[kg]

[%]

[min]

[min]

[%]

M1

1,760

1,760

0.00

298.2

8840.0

297.6

297.6

0.00

1,791

83.5

3,670

3,628

1.16

2,681

2,697

0.60

M2

1,817

1,817

0.00

313.1

82.7

310.5

310.5

0.00

1,854

85.4

3,768

3,694

2.00

2,730

2,778

1.75

M3

1,533

1,533

0.00

265.2

733.8

261.0

261.0

0.00

1,560

71,564.9

3,248

3,199

1.53

2,421

2,450

1.19

M4

1,805

1,805

0.00

317.6

84.2

312.7

312.7

0.00

1,851

82.2

3,754

3,653

2.76

2,766

2,803

1.34

M5

1,929

1,929

0.00

336.7

914.2

333.4

333.4

0.00

1,961

94.2

3,962

3,892

1.80

2,881

2,940

2.06

M6

1,895

1,895

0.00

326.0

910.6

324.1

324.1

0.00

1,923

9627.0

3,975

3,901

1.90

2,926

2,967

1.43

M7

1,847

1,894

2.54

331.8

93,603.4

328.3

328.3

0.00

1,938

910.3

3,972

3,878

2.42

2,937

2,982

1.54

M8

1,943

1,943

0.00

335.7

919.4

334.0

334.0

0.00

1,977

95.6

3,988

3,925

1.61

2,934

2,962

0.94

M9

1,588

1,637

3.09

278.8

83,603.2

269.0

278.6

3.57

1,655

83,603.1

3,381

3,322

1.78

2,447

2,463

0.65

M10

1,794

1,794

0.00

313.8

85.2

312.1

312.1

0.00

1,830

8124.3

3,610

3,537

2.06

2,650

2,676

0.98

M11

1,767

1,767

0.00

302.8

8120.7

299.4

308.7

3.11

1,845

93,602.6

3,880

3,743

3.66

2,806

2,825

0.67

M12

2,003

2,003

0.00

347.4

9183.1

344.1

344.1

0.00

2,053

9389.4

4,160

4,080

1.96

3,058

3,084

0.86

M13

1,792

1,843

2.85

323.8

93,603.3

312.6

312.6

0.00

1,846

8558.0

3,767

3,709

1.56

2,749

2,814

2.36

M14

1,756

1,756

0.00

298.8

84.0

295.1

295.1

0.00

1,807

82.6

3,653

3,547

2.99

2,598

2,641

1.67

M15

1,668

1,668

0.00

291.2

843.4

289.6

289.6

0.00

1,693

82.9

3,540

3,494

1.32

2,635

2,657

0.82

M16

1,939

1,939

0.00

342.6

928.9

339.9

339.9

0.00

1,987

9110.0

4,069

3,951

2.99

3,009

3,056

1.54

M17

1,548

1,596

3.10

284.3

83,603.4

273.0

282.2

3.37

1,626

83,603.1

3,268

3,204

2.00

2,463

2,483

0.82

M18

1,690

1,690

0.00

286.9

8680.0

283.5

292.8

3.28

1,799

93,607.0

3,832

3,638

5.33

2,765

2,786

0.77

M19

1,805

1,805

0.00

310.0

82,085.7

307.3

307.3

0.00

1,828

8881.7

3,824

3,763

1.62

2,818

2,840

0.81

M20

1,749

1,749

0.00

311.6

85.0

308.8

308.8

0.00

1,782

86.1

3,581

3,513

1.94

2,688

2,716

1.06

L1

3,225

3,225

0.00

553.0

16

594.0

549.5

556.8

1.33

3,337

16

7,583.3

6,767

6,920

2.26

5,080

5,059

0.42

L2

3,300

3,335

1.06

576.7

16

7,370.3

568.1

571.8

0.65

3,393

16

8,183.1

6,976

7,093

1.68

5,262

5,192

1.35

L3

3,189

3,219

0.94

557.7

16

7,388.0

548.2

551.5

0.60

3,302

16

7,393.6

6,825

6,997

2.52

5,244

5,173

1.39

L4

3,195

3,236

1.28

564.9

16

15,183.6

556.5

559.8

0.59

3,301

16

7,305.2

6,743

6,906

2.42

5,140

5,083

1.12

L5

3,207

3,232

0.78

557.9

16

7,341.5

550.6

561.0

1.89

3,327

17

14,460.1

6,999

7,196

2.81

5,411

5,376

0.65

L6

3,366

3,438

2.14

601.2

17

14,542.4

590.3

596.8

1.10

3,529

17

7,425.6

7,268

7,485

2.99

5,640

5,553

1.57

L7

3,451

3,523

2.09

618.3

17

15,099.1

606.3

612.8

1.07

3,604

17

7,439.5

7,232

7,423

2.64

5,548

5,471

1.40

L8

3,278

3,325

1.43

584.3

16

7,286.3

570.1

577.4

1.28

3,403

16

7,308.2

6,968

7,122

2.21

5,358

5,294

1.21

L9

3,229

3,324

2.94

566.1

17

14,508.7

548.7

548.7

0.00

3,270

15

1,195.6

6,862

6,858

-0.06

5,109

5,036

1.47

L10

3,322

3,370

1.44

586.1

16

8,930.9

576.1

593.2

2.97

3,496

17

14,557.7

6,949

7,153

2.94

5,309

5,294

0.28

L11

3,263

3,310

1.44

569.5

16

7,387.3

563.8

572.2

1.49

3,429

17

8,136.2

7,121

7,340

3.08

5,475

5,423

0.95

L12

3,612

3,660

1.33

643.2

17

7,602.4

634.9

634.9

0.00

3,776

17

914.4

7,712

7,915

2.63

5,976

5,907

1.17

L13

3,229

3,276

1.46

571.8

16

8,784.8

559.5

566.8

1.30

3,336

16

7,992.0

6,838

7,043

3.00

5,205

5,142

1.22

L14

3,488

3,537

1.40

600.5

17

7,444.0

588.4

588.4

0.00

3,577

16

1,643.6

7,348

7,504

2.12

5,500

5,411

1.64

L15

3,364

3,459

2.82

608.7

17

14,452.0

594.2

603.3

1.53

3,549

17

7,307.9

7,218

7,344

1.75

5,522

5,473

0.89

L16

3,234

3,264

0.93

571.5

16

7,301.1

563.7

568.5

0.85

3,325

16

9,129.2

6,940

7,121

2.61

5,354

5,258

1.82

L17

3,253

3,299

1.41

572.5

16

7,519.2

556.0

563.3

1.31

3,374

16

7,357.0

6,829

7,021

2.81

5,184

5,097

1.71

L18

3,574

3,623

1.37

634.9

17

7,347.7

627.3

627.3

0.00

3,691

17

149.4

7,585

7,746

2.12

5,826

5,725

1.77

L19

3,238

3,285

1.45

568.3

16

7,604.4

555.8

572.2

2.95

3,383

17

15,336.2

6,966

7,138

2.47

5,313

5,276

0.70

L20

3,312

3,359

1.42

596.3

16

7,364.4

582.3

598.0

2.70

3,504

17

16,853.9

7,042

7,267

3.20

5,473

5,440

0.59

27