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An NSGA-II Algorithm for the Green Vehicle
Routing Problem
Jaber Jemai, Manel Zekri, and Khaled Mellouli
1IS Department, College of Computer and Information Sciences,
Imam Mohammad Ibn Saud University. Riyadh, KSA
2Larodec Laboratory, Institut Sup´erieur de Gestion
University of Tunis. Tunis, Tunisia
Abstract. In this paper, we present and define the bi-objective Green
Vehicle Routing Problem GVRP in the context of green logistics. The
bi-objective GVRP states for the problem of finding routes for vehicles to
serve a set of customers while minimizing the total traveled distance and
the co2emissions. We review emission factors and techniques employed to
estimate co2emissions and integrate them into the GVRP definition and
model. We apply the NSGA-II evolutionary algorithm to solve GVRP
benchmarks and perform statistical analysis to evaluate and validate
the obtained results. The results show that the algorithm obtain good
results and prove the explicit interest grant to emission minimization
objective.
Keywords: Green vehicle routing, Multi-objective optimization, Evo-
lutionary algorithms, NSGA-II.
1 Introduction
A supply chain is a network [1] of suppliers, manufacturers, warehouses and
distribution channels organized to acquire materials, convert them into finished
products and distribute them to clients. The Supply Chain Management (SCM)
consists of finding best practices, policies and strategies to solve efficiently all
encountered problems. That is by employing the available resources with re-
spect to different constraints and while optimizing many different and generally
conflicting objectives. One of the most important SCM phases is the logistics
and transportation processes that allow the moving of different materials from
and to different nodes in the supply chain network. Generally, the objective of
the logistics process is to optimize transportation related costs like traveled dis-
tance, time, routes flexibility and reliability. Recently, the concept of greenness
for sustainable development has emerged to represent a human concern for the
undesirable effect of the industrial processes on the environment. This environ-
mental awareness intend to show the effect of toxic emissions on the environment
Corresponding author.
J.-K. Hao and M. Middendorf (Eds.): EvoCOP 2012, LNCS 7245, pp. 37–48, 2012.
c
Springer-Verlag Berlin Heidelberg 2012
38 J. Jemai, M. Zekri, and K. Mellouli
and to call governments and industrials to seriously consider this concern. Sev-
eral industries started enhancing their procedures to show an explicit interest
to minimize the volumes of their missions. In transportation, the aim is to con-
struct low cost routes for vehicles, trucks, planes and ships to transport goods.
However, while moving theses engines generate huge quantities of co2that affect
directly the quality of breathed air particularly in large cities. The ma jor con-
cern, for transportation firms, is the materiel benefit without reviewing vehicle
emissions and their effect on the environment. Recently, and for many reasons,
transportation companies start taking explicitly into account the emissions re-
duction objective in definition of their working plans. This trend was encouraged
by governmental regulations and customer preference to consume environment
friendly products. Then, the generated working plans must minimize costs and
co2emissions. These two objective are not necessarily positively correlated and
for some cases they are completely conflicting.
The basic transportation model generally used to represent the problem of
finding routes for vehicles to serve a set of customers is the Vehicle Routing
Problem (VRP) [27]. In the basic VRP and also in many other variants the
objective to optimize is unique and it is to minimize the overall transportation
costs in term of distance, time, number of vehicles, etc. Here, the literature is
really huge where several single objective VRP was studied and solved efficiently.
However, like other optimization problems, the objectives may be multiple and
conflicting. Then, the multi-objectiveVRPwasdefinedtorepresentaclassof
multi-objective optimization problem.
In this paper, the scope is the study and the definition of the bi-objective
Green Vehicle Routing Problem (GVRP). The bi-objective GVRP asks for
designing vehicle routes to serve set of customers while minimizing the total
traveled distance and the total co2emissions with respect to classical rout-
ing constraints mainly capacity constraints. consequently, we will implement
the NSGA-II evolutionary algorithm to solve the bi-objective GVRP model via
solving some well known benchmarks. The NSGA-II is a non-dominating sorting
genetic algorithm that solves non-convex and non-smooth multi-objective opti-
mization problems. The objective of the paper is to show the effectiveness of
explicitly considering emissions minimization as separate objective to optimize
and to prove that short routes are not necessarily less pollutant.
The paper is organized as follow. In the next section, we present the con-
cept of green logistics, enumerate all emission factors and how co2emissions
could be estimated and then integrated into quantitative models. Section 3 is
devoted to define the vehicle routing problem with emissions, review the cor-
responding literature and propose a mathematical model for the bi-objective
GVRP. In the section 4, we present the evolutionary solving approach based
on the NSGA-II algorithm. Section 5 will report the NSGA-II implementation
details and computational results. Statistical analysis will be performed to mea-
sure the effectiveness of the model and the obtained results. Finally, we present
the conclusions of this project and state some perspectives for future work.
An NSGA-II Algorithm for the Green Vehicle Routing Problem 39
2 Green Logistics
Traditional logistics ensure the movement of materials between all actors in the
supply chain starting from raw materials locations to final customers via firms
factories. These transportation tasks should be completed efficiently to report
more benefit to the company. The efficiency is usually measured in terms of
money, time and reliability. Recently, the concept of green logistics for sustain-
able development has soared due to governmental regulations and customers
preference for green products. Consequently, transportation companies are re-
viewing their processes to take into account such concern. The revision consider
all the steps in the production process including the choice of raw materials,
factoring, packaging, alternative fuels, etc. In some cases transforming the tradi-
tional logistics systems to be environmentally friendly will give a cutting down
in costs and then it will meet classic logistics objectives. However, in many other
situations such review may cost more and come into conflict with traditional
logistics.
For transportation companies, green logistics mean transporting goods with
lower effect on the environment. Basically, the effect of transporting materials on
the environment comes from gazes emissions generated from moving engines like
trucks, planes and ships. Then, greener transportation yields to low co2emission
routes. But, those routes are generally determined using analytical model that
consider only saving money as primer objective. Then, the aim of considering the
environmental effect will be transformed into a revision of the analytical tools
used to generates routing policies and strategies. That, could be completed by
determining emission factors and quantifying trucks emissions to integrate them
into logistics systems.
2.1 Emission Factors
There are a number of factors that could affect vehicle fuel economy in real
world:
1. Vehicle weight:a vehicle carrying more weight requires more energy to run,
thus directly affect in fuel economy [4].
2. Vehicle speed and acceleration:fuel consumption and the rate of co2per mile
traveled decrease as vehicle operating speed increase up to approximately 55
to 65 mph and then begin to increase again[1]. Moreover, the co2emission
double on a per mile basis when speed drops from 30 mph to 12.5 mph or
when speed drops from 12.5 to 5 mph [3]. The relationships between these
factors and fuel economy are not simple. For example, the implication of
vehicle operating speeds on fuel consumption is not linear and depends on
vehicle type and size. It also varies on the model year and age of the vehicle.
For instance, studies of vehicle fuel economy taken during the 1990s show
less of a drop off in vehicle fuel economy above 55 miles per hours than
similar studies of vehicles during the 1970s and 1980s, due to vehicle design
changes and engine operating efficiency [14].
40 J. Jemai, M. Zekri, and K. Mellouli
3. Weather conditions: weather condition affect vehicle fuel economy. For in-
stance, head-winds reduce vehicle fuel economy as the vehicle needs addi-
tional power from the engine to combat the wind drag. Hot weather induces
the use of air conditioning, which places accessory load require on the engine.
4. Congestion level: It is commonly known that as traffic congestion increases,
co2emission (and in parallel fuel consumption) also increase. In general,
co2emission and fuel consumption are very sensitive to the type of driving
that occurs. In fact, traveling at a steady-state velocity will give much lower
emissions and fuel consumption compared to a stop-and-go movement. Thus,
by decreasing stop-and-go driving, co2emissions can be reduced [4].
2.2 Emission Estimation Techniques
To examine the environmental impact of the Vehicle Routing, it is necessary to
weigh the environmental impacts of co2emission. It is difficult to do an exact
estimation because of the uncertain effects of climate change and the setting
of a price tag on human health. The DEFRA estimated in 2007 the cost of
emitting a tone of co2at 25.5. Furthermore, the IPCC [15] published estimates
range between 5 and 25. Emissions are estimated using average grams of co2
per kilometer. The study of Mc Kinnon [25] shows that the load carried is an
important parameter to estimate emissions. Thus, we can estimates co2from
the distance traveled by vehicles and the quantity of goods carried. There are
other methods to estimate co2emission for vehicle. We can cite for example the
fuel-based approach and the distance-based method.
1. The fuel-based approach: In the fuel-based approach [11], the fuel consump-
tion is multiplied by the co2emission factor for each fuel. The emission factor
is developed based on the fuels heat content, the fraction of carbon in the
fuel that is oxidized and the carbon content coefficient. The fraction of gaso-
line oxidized depends on the transportation equipments used. Therefore, this
variability is minimal. In the US inventory, this fraction is assumed to be 99
percent. In the case of road transportation, companies and other entities have
the option to override these defaults if they have appropriate data of fuel
used. In most case, default emission factors will be used based on generic fuel
type categories( e.g., unleaded gasoline, diesel, etc) The fuel-based approach
requires essentially two main steps:
(a) Gather fuel consumption data by fuel type: Fuel use data can be obtained
from several different sources. We can cite for example fuel receipts, fi-
nancial records on fuel expenditures or direct measurements of fuel use.
When the amount of fuel is not known, it can be calculated based on dis-
tance traveled and an efficiency factor of fuel-per-distance.The distance
traveled basically come in three forms:
–distance(e.g., Kilometers)
–passenger-distance(e.g.,passenger-kilometers)
–freight-distance (e.g., ton-miles)
An NSGA-II Algorithm for the Green Vehicle Routing Problem 41
The fuel economy factors depend on the type, age and operating practice
of the vehicle in question. Thus, we obtain the following equation:
fuel consumption =distance ∗fuel economy factor
(b) Convert fuel estimate to co2emissions by multiplying results from step
1 by fuel-specific factors; The recommended approach is to first convert
fuel use data into an energy value using the heating value of the fuel.
The next step is to multiply by the emission factor of the fuel.
The fuel-based approach is the same for the different mode of transportation.
The following equation outlines the recommended approach to calculating
co2emissions based on fuel use. Thus, we obtain the following equation:
co2emissions =fuel used ∗heating value ∗emission factor
2. The distance-based method :The distance-based method [11] is another
method to estimate the carbon dioxide emissions can be calculated by using
distance-based emission factor. This method can be used when vehicle activ-
ity data is in the form of distance traveled but fuel economy data is not avail-
able. It is obvious to formulate our problem using a distance-based method
for calculating co2emissions. Calculating emissions requires two main steps:
(a) Collect data on distance traveled by vehicle type and fuel type.
(b) Convert distance estimate to co2 emissions by multiplying results from
step 1 by distance based emission factors. Thus, we obtain:
co2emissions =traveled distance ∗emission factor
The estimation of emission factor is carried out following two main steps.
The first one consists on estimate the fuel conversion factor ( 2.61kg.co2/
liter of diesel). The second step is to estimate the emission factor consists
on finding a function taking into account data related to the average fuel
consumption which depends on load factor.
3 The Vehicle Routing Problem with Emissions
3.1 Literature Review
In recent years, many research works about variants of the VRP in order to
reduce the cost and the emission of co2was conducted. The Vehicle Routing
and Scheduling Problem (VRSP) is an extension of the VRP. Its purpose is to
determine the routes and schedules for a fleet of vehicle to satisfy the demand
of a set of customers. Thus, it aims to minimize cost which is usually related to
the number of vehicles and distance. The reduction in total distance will provide
environmental benefits due to the reduction in fuel consumption.
The Time Dependent Vehicle Routing Problem (TDVRP) represents a
method which should indirectly produce less pollution and achieve environmental
benefits in congested area. The TDVRP is a variant of the VRP and has received
less attention. It was originally formulated by Malandrakiand et al. Daskin [7]
as mixed linear program. It consists of finding the solution that minimizes the
42 J. Jemai, M. Zekri, and K. Mellouli
number of tours by considering traffic conditions. The TDVRP provide environ-
mental benefits, but in an indirect way. Consequently, less pollution is created
when vehicle are traveling at the best speeds and for shorter time. The Time De-
pendent Vehicle Routing and Scheduling Problem (TDVRSP) consists of finding
the solution that minimizes the number of tours and the total traveling time. It is
motivated by the fact that traffic conditions cannot be ignored, because at peak
time, traffic congestion on popular routes will causes delays. The TDVRSP pro-
vides also environmental benefits in indirect way. There is an extensive literature
related to vehicle emission.Turkay et al. [20] and Soylu et al. [18] demonstrated a
collaborative supply chain management for mended business and for decreasing
environmentally harmful chemicals, while satisfying local regulation and Kyoto
protocol for greenhouse gas emissions. The study of Halicioglu [12] tried to em-
pirically treat the dynamic causal relationship between carbon emissions, energy
consumption, income and foreign trade in the case of Turkay [20]. Recently, Van
Woensel et al. [21] considered a vehicle routing problem with dynamic travel
time due to the traffic congestion. The approach developed introduced the traf-
fic congestion component based on queuing theory in order to determine travel
speed. A tabu search method was used to solve the model. Results showed that
the total travel time can be improved significantly when explicitly taking into
account congestion during the optimization phase. The study of Figliozzi et al.
[8] proposed a new methodology for integrating real-world network status and
travel date to TDVRP. It developed efficient algorithms TDVRP solution meth-
ods to actual road networks using historical traffic data with a limited increase
in computational time and memory. The results shows the dramatic impacts of
congestion on carriers fleet sizes and distance traveled.
Figliozzi [9] also created a new type of VRP which is denoted the Emission
Vehicle Routing Problem(EVRP). The research presented a formulation and so-
lutions approaches for the EVRP where the minimization of emission and fuel
consumption is the primary objectives or is part of a generalized cost function.
A heuristic is proposed to reduce the level of emission given a number of feasi-
ble routes for the TDVRP. Search results indicated that they may be significant
emissions saving if commercial vehicles are routed taking emissions into consider-
ation. Moreover, congestion impacts on emission levels are not uniform. Bauer et
al. [2] identified and addressed some environmental consideration in the context
of intermodal freight transportation and proposed ways to introduce environmen-
tal costs into planning model for transportation. They proposed a formulation
for scheduled service network design problem with fleet management, it is an
integer program in the form of a linear cost multi commodity and capacitated
network design formulation that minimize the amount of green house gas emis-
sion for transportation activities. The formulation has been implemented on a
real life intermodal rail network data.
3.2 The Bi-objective Green Vehicle Routing Problem
The green vehicle routing problem is an answer for the recent environmental
awareness in the field of transportation and logistics. The objective is to find
An NSGA-II Algorithm for the Green Vehicle Routing Problem 43
routing and transportation policies that give the best compromise between trav-
eling costs and co2emissions. The literature on transportation problems espe-
cially vehicle routing problems had considered this environmental interest. Later
studies show and implicit interest to handle the objective of gazes minimization
But, without viewing it as a major distinct objective like distance and time. We
can cite the TDVRP, VRSP and the emissions VRP. In this paper, we consider
the the emissions minimization as a separate major objective in addition to the
distance minimization objective. Therefor, we define a bi-objective combinatorial
optimization problem named the bi-objective green vehicle routing problem.
The bi-objective GVRP [28] could be defined as follow: Giving a set of N
customers located in a transportation network and a distance matrix Dij repre-
senting the costs of moving between customers iand jand a set of Mvehicles
hosted in a central depot. A solution of the bi-objective GVRP is composed by a
set of routes with minimum traveled distance and the minimum volume of emit-
ted co2while visiting each customer once and with respect to vehicles capacity
constraints. It is clear that the bi-ob jective GVRP is an NP-hard problem due
to the fact that it is an extension of the standard VRP which is NP-hard.
4 NSGA-II Algorithms for the Bi-ob jective GVRP
Genetic Algorithms (GA) are stochastic and evolutionary optimization algo-
rithms based on mechanisms of natural selection and genetics. GAs attempt to
solve hard non-convex single and multiobjective optimization problems. Multi-
objective GAs are based on the concept of Pareto dominance, which emphasizes
a research satisfying all objectives. They are well suited for the search of Pareto
front through their implicit parallelism to reach optimal solutions more effi-
ciently than an exhaustive method. Many multiobjective genetic algorithms can
be cited [6].
The NSGA-II is more efficient than its previous version NSGA [5]. This algo-
rithm tends to spread quickly and appropriately when a certain non dominated
region is found. The main advantage is that the strategy of preserving of diver-
sity used in NSGA-II requires no parameters to fix. For these reasons, we choose
to resolve our problem using this approach. In NSGA-II, the child population
Q(t) is first created from the parent population P(t)(randomly filled). They are
then met into a set R(t)=P(t)Q(t) that is sorted according to the princi-
ple of dominance: all non-dominated solutions of the population are assigned
a fitness value 1 (first front), then they are removed from the population. All
non-dominated solutions of the population are assigned a fitness value 2 (second
front), then they are removed from the population. And so on. This process is
iterated until all solutions whose fitness value is upon to evaluate is empty [6].
To select subsets that will be placed in the population, a measure of the density
of solutions in the space of criteria called crowding distance is used.
5 Implementation and Computational Results
In order to evaluate the effectiveness of the proposed model and to prove the ef-
fect of considering explicitly the emissions minimization objective, the NSGA-II
44 J. Jemai, M. Zekri, and K. Mellouli
algorithm was implemented to solve bi-objective GVRP instances. The proposed
algorithm was implemented using the ParadisEO-MOEO library [26]. The per-
formance of the metaheuristic has been tested on different instances taken from
the VRPLIB [23]. These instances involve between 16 and 500 nodes. The num-
ber at the end of an instances name represents the number of vehicles while
the number at the first is the number of customers. The stopping condition of
all tests is based on the number of generation (100 generation). Computational
runs were performed on an Intel Core 2 Duo CPU (2.00 GHz) machine with 2G∅
RAM. The results presented below are based on the following GA parameters:
–Chromosome encoding: a solution chromosome is represented by an integer
string. A gene is a customer number, while a sequence of genes dictates
a group of customers assigned to a vehicle. For instance, the chromosome
(0,3,6,1,0,2,4,0,5,0) contains three routes (0 :: 3 :: 6 :: 1 :: 0), (0 :: 2 :: 4 :: 0)
and (0 :: 5 :: 0). The population size is set to 100 chromosomes.
–Crossover: We utilized the standard crossover operator Partially-Mapped-
Crossover (PMX). The first step is to Select a substring uniformly in two
parents at random. The next step is to exchange these two substrings to pro-
duce proto-offspring. The third step is to determine the mapping relationship
according to these two substrings. The last step is to legalize proto-offspring
with the mapping relationship. The crossover probability is 0.25.
–Mutation: In the mutation stage, two customers are selected from different
routes randomly. They are going to be swapped only if constraints are met
after this operation. After swap, insertion is done in which we select randomly
a customer from a route and try to insert rest of any one route if it satisfies
all the constraints.The mutation probability is fixed to 0.35.
5.1 Computational Results
To demonstrate the efficiency of the metaheuristic implementation, measures
related the computation time are computed and reported in Table.1. We can
remark that the computation time of the implemented algorithm increases pro-
portionally to the size of the instance due to algorithm complexity and especially
the complexity of the computation of the crowding distance O(MNlogN). It is
important to observe that the cardinality of the pareto fronts is small. This fact
can be explained by the correlation between our two objectives; for instance the
emissions objectives was written as a function of the distance objective. From
another side, we can see for four instances, that obtaining solutions with minimal
distance does not imply minimal emissions.
5.2 Statistical Analysis
To evaluate the quality of the obtained solutions and measure the performance of
the algorithm, metric measurements have been selected and calculated. We use
three metrics: the first is the Generational Distance (GD) which measures how far
from the Front Pareto is located a set of solutions, the second is the Spacing (S)
metric which measures the distribution uniformity of points of the set of solution
An NSGA-II Algorithm for the Green Vehicle Routing Problem 45
Table 1 . The obtained Pareto fronts and the needed computation time
Instance Pareto front CTime (s)
Obj1(km) Obj2(kg.co2)
E101-08e 83.413
1946 1411
1961 1398
1977 1349
E301-28k 99.621
2352 1598
2298 1643
2357 1597
2277 1683
2349 1602
2360 1592
2303 1631
2302 1641
2302 1596
E421-41k 126.547
4163 2982
4153 2998
4168 2971
4071 3094
E484-19k 135.330
2307 1365
2306 1361
2325 1348
in the plan (obj1,obj2), the third indicator of performance is the Entropy (E)
metric that uses the concept of niche to evaluate the distribution of solutions
on the front. The NSGA-II algorithm give different approximations for each
execution. Thus, to empirically analyze the performance of our algorithms, we
first run the same algorithm several times on the same instance of the problem.
We get then a sample of approximation. We run the algorithm ten times for each
instances. Table 2 presents averages of metrics GD, S and E over ten runs of the
four instances.
Table 2 . Averages of metrics GD, S and E for the algorithm NSGA-II
Instances GD S E
E101-08e 3.931 4.632 0.227
E301-28k 4.063 6.878 0.422
E421-41k 3.009 6.515 0.095
E484-19k 5.396 2.570 0.371
46 J. Jemai, M. Zekri, and K. Mellouli
The values obtained by the GD metric are small and vary between 0.6282 and
6.250 so are not large enough. We can then conclude that the set of solutions are
near of the Pareto front. For the S metric, the results obtained for variants E101-
08e and E421-41k and E484-19k are close to 0, so the points are well distributed
in the set Parto front. For the instance E301-28k, the mean value is equal to
6.878, therefore the worse. By exploiting the solutions obtained by the Entropy
metric, we note that the value found in the instance E421-41k is the closer to 1,
thus the distribution of solutions for this instance on the front is better than the
three other instances. To evaluate the metaheuristic rigorously and to estimate
the confidence of the results to be scientifically valid, statistical tests are per-
formed on the indicators of performance. Experiments are performed on the four
instances E101-08e, E301-28k, E421-41k and E484-19k. The three algorithms are
executed ten times for each instance and calculations of metrics GD, S and E are
made. In order to determine whether the mean of the experiments are different
or not at a statically significant level, an analysis of variance is done. By applying
a Shapiro-test on the distribution, we found that this one follow a normal low.
Consequently, we used a one factor analysis of variance (ANOVA) test which is
based on the central assumption of normally data distribution to check whether
a factor has a significant effect on the performance of the algorithm. In our case,
the experiments are taken as factor and the metrics are taken as dependents
variables. The hypothesis is:
H0:µ1=µ2=µ3=µ4Versu s H1:µi=µj
with i, j =1,2,3,4andi=j
Table.3 shows the ANOVA for metrics GD, S and E. The first ANOVA for met-
ric GD don’t found significant differences for the different experiments. Hence,
the effect of the factor experiment does not influence the variables of measures
of performance. The second ANOVA for metric S found significant differences.
Consequently, there is an effect of the factor on the variable of measures of
performance. The third ANOVA for metric E found also significant differences.
Table 3 . ANOVA table for metrics GD, S and E
Sum sq DF Mean sq F-value Prob >F
GD
Factor 7.352 3 2.4508 0.7742 0.517
Residual 3.1728 36
S
Factor 137.09 3 45.696 10.977 2.9 exp−5
Residual 149.86 36 4.163
E
Factor 0.3981 3 0.1327 10.759 3.428 exp−5
Residual 0.4441 36 0.0123
An NSGA-II Algorithm for the Green Vehicle Routing Problem 47
6 Conclusions
The green vehicle routing problem consists of designing a set of routes for a set of
vehicles to serve customers over a transportation network. We model the GVRP
as bi-objective optimization problem where the first objective is to minimize the
overall traveled distance and the second ob jective is to minimize the volume
of emitted co2. Many solving approaches and algorithms are envisaged. In this
paper, we choose evolutionary algorithms to find better pareto fronts for the
GVRP. This choice is explained by the performance of evolutionary algorithms
especially elitist algorithm like NSGA-II, SPEA-II and the IBEA algorithms
for solving multi-objective combinatorial optimization problems. Hence, we im-
plement the NSGA-II algorithm for solving GVRP benchmarks. The obtained
results show and prove the effectiveness of considering the emissions minimiza-
tion as a separate objective. Performed statistical tests confirm the quality of the
generated pareto fronts and then the performance of the NSGA-II algorithm.
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