Access to this full-text is provided by MDPI.
Content available from Energies
This content is subject to copyright.
energies
Article
Estimating the Optimal Location for the Storage of Pellet Surplus
Andrzej Bochniak 1and Monika Stoma 2, *
Citation: Bochniak, A.; Stoma, M.
Estimating the Optimal Location for
the Storage of Pellet Surplus. Energies
2021,14, 6657. https://doi.org/
10.3390/en14206657
Academic Editors: Grzegorz Karo ´n
and Fereshteh Mafakheri
Received: 26 August 2021
Accepted: 11 October 2021
Published: 14 October 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Department of Applied Mathematics and Computer Science, University of Life Sciences, Gł˛eboka 28,
20-612 Lublin, Poland; andrzej.bochniak@up.lublin.pl
2Department of Power Engineering and Transportation, University of Life Sciences, Gł˛eboka 28,
20-612 Lublin, Poland
*Correspondence: monika.stoma@up.lublin.pl
Abstract:
This paper deals with the problem of managing the surplus that arises during the seasonal
production of pellets, which will be sold in the period of increased demand. Dijkstra’s algorithm
is used in issues connected with finding a new storage place with a view of the optimisation of the
transport costs of pellets produced by a company in 18 different towns in the Lubelskie Voivodeship
in Poland. The most optimal location for the new pellet storage site has been determined, for
which the total length of the traveled routes is the shortest, taking into account the actual shares
of individual plants in the total production. The construction of the graph with the shortest paths
was made on the basis of the existing network of available transport roads, and the nodes of the
graph were their intersections. The most advantageous storage location of pellets was identified by
the calculation the total transport cost using a minimum-cost tree of shortest paths. Based on the
estimated transport assumptions, the lowest total cost of transport from all 18 plants was 3092.0 (km),
which corresponds to an average distance to production plants of 89.7 km and 61.7 km to estimated
selling distribution. The new storage point is suggested near the town of Piaski. Average cost of
travel for all trees obtained for existing plant locations and subsequent distribution to points of sale
was 4113.7 (km), while standard deviation 735.2 (km). Additionally, a relative increase in costs was
estimated in the case of selecting other locations. Using spatial interpolation and geoprocessing tools,
a map—showing the increase in pellet transport costs in relation to the most optimal solution—was
developed. The constructed map allows for a better analysis of cost increases than a single point. It
was stated that the increase in transport costs does not exceed 10% of lowest cost for 17.6% area of
studied area. It was found that the most convenient area is shifted to the south of the voivodship and
improperly adopted storage location can increase transport costs by up to 75%.
Keywords:
biomass storage; Dijkstra’s algorithm; minimizing transportation cost; geospatial analysis
1. Introduction
Biomass may be used as raw material in the forest product industry and in the chemical
industry, but also is used for biofuel production and heat/power production [
1
]. This is all
the more important as the production of electricity based on wood pellets (solid biomass)
can increasingly contribute to the achievement of climate goals. At the same time, care
should be taken to aim at low-emission in various areas of the economy while maintaining
growth economic and—with regard to electricity production—maintaining a sufficient
generation capacity [2].
Wood pellets can be produced from several types of forest raw materials, from industry
residues, forestry management residues, or pulp or timber grade pulpwood. Large-scale
production of pellets is often designed with a mix of different raw materials, depending on
local availability and cost.
Poland has a number of renewable energy resources, including energy from biomass.
It may include raw materials of plant and animal origin that are biodegradable and come
from agriculture and agro-food industry. Forest biomass, in the form of briquettes, pellets,
Energies 2021,14, 6657. https://doi.org/10.3390/en14206657 https://www.mdpi.com/journal/energies
Energies 2021,14, 6657 2 of 16
wood chips, and bales, among others, is the basic solid fuel from biomass. Biomass is
extremely varied in terms of state of matter, moisture content, calorific value, and specific
weight. This raw material has a large volume and water content, therefore transport and
storage of biomass is problematic [3].
In Poland, every region has a different energy potential. However, all regions have a
problem with the optimisation of the process of obtaining biomass and its use for energy
purposes, and hence with its transport. During transport, biomass may be damaged or its
physical and chemical properties may be changed. Pellets may serve as an example—under
the influence of moisture, their energy value is reduced, and their combustion deteriorates.
Moreover, obtaining biomass is often difficult due to the dispersed nature of its availability
Therefore, any process connected with the transport and storage has to be well thought-out
and designed due to the expenses incurred. By using appropriate methods of storage and
transport, the pellets can be prevented from uptake water by using sealed packaging or
stored in constant conditions.
Determination of an appropriate and the shortest routes to transport pellets is a serious
problem that distributors of this type of biomass face. When speed on the designated route
and quality of the pellets delivered to a customer are important, the travel route has to
be properly planned, and a storage space which is close to all users has to be designated.
It should be remembered that, on the one hand, biomass logistics includes a number of
cooperating processes, and on the other that each type of biomass requires a different
logistic process, and raises other problems and requirements related to transport, storage,
or delivery which should be taken into account when designing effective logistics systems
Various types of costs should also be taken into account, as both commercial and
service activities are a key element of sustainable development due to their contribution
to the productivity of natural and human resources although they also generate many
positive benefits for the market, society, economy, and the environment. However, they
also entail some negative problems [
4
]. Both aspects are particularly relevant to transport,
which largely contributes to the economic development of societies, but is also responsible
for negative impacts on the climate, the environment, and human health. These negative
effects can occur both at the local level, e.g., air pollution or environmental degradation
due to generated freight movements, and at the global level, e.g., climate change [
5
]. These
effects can be assessed by determining the value of externalities or external costs, as each
transport company, in connection with the cargo transport process carried out, generates
two types of costs—internal costs (own costs and infrastructure costs) and social costs of
transport, also referred to as external costs (environmental costs, e.g., pollution, noise, and
costs related to congestion and accidents).
Internal costs include the operational-private costs of transport and the time costs of
goods in transit [
6
]. These include, firstly, amortisation (of motor vehicles and other fixed
assets), consumption of materials and energy (fuel consumption, consumption of lubricants
and oils, and tyre wear), external services (operation of repair and service facilities), salaries,
social security, taxes, and fees and other costs [
7
]. Own costs of an enterprise providing
transport services depend on many different factors, the most important of which are: type
of cargo, type and condition of rolling stock, direction and distance of transport, degree
of use of rolling stock, condition of infrastructure, employment, weather conditions, and
topography [
8
]. According to Irannezhad and team [
4
], internal transport costs mainly
include operating costs dependent on time and distance and fixed cost of vehicles.
External costs are the monetary valuation of externalities; they are side effects of
transportation [
5
]. These include the costs of social and environmental impacts, such as
the aforementioned local and global air pollution (impact of exhaust gases on human
health and the environment—lorries that collect and distribute typically burn diesel fuel
and cause air pollution, the individual components of which can cause local damage to
surrounding buildings, green spaces, and human health), congestion costs (trucks tend to
run in densely urbanized and/or industrialized areas. They may experience congestion
and the resulting delivery delays. However, they can also cause delays for other vehicles,
Energies 2021,14, 6657 3 of 16
the costs of which are counted as externalities reflecting the loss of time of transport users
due to congestion, extension of transport time, increase in fuel costs, etc.), costs related to
accidents in transport (damage and property losses to the transport company and third
parties, in addition to the loss of life and injuries of the persons affected), costs related to
noise (generated, among others, by heavy goods vehicles involved in the collection and
distribution of loads; unacceptable limits cause irritation and, if persistent, may cause a
decrease in efficiency and have adverse health effects), and the cost of road infrastructure [
6
].
It should also be added that internalizing external costs allows pricing at the right social
cost, leading to an efficient allocation of resources [5].
The external costs of transport are quite thoroughly researched, and the results
have been presented in the extensive literature on the subject, including in the follow-
ing works: [9–13].
With regard to the analysis of wood pellet delivery costs, attention should be paid to
the studies by Miao et al. [
14
], which present an overview of the overall costs and processes
involved with biomass feedstock harvesting, processing, and delivery to biofuel plants.
Other works that include cost analysis of supply chain elements include research carried
out by Ehrig and his team [15] or Boukherroub [16].
With a view to reducing the emission of environmentally harmful substances such
as greenhouse gases and other particles and, in addition, due to the lower calorific value
of pellets in relation to fossil fuels (per mass unit), in order to make the use of pellets
profitable, the transport distances should not be too large. In the case of trucks, transport
over short distances up to 100 km is suggested [
17
]. As shown by other authors from
different countries, the route of biomass transport should be less than 1200 km [
18
]. Above
this distance, the emissions of harmful substances from transport start to exceed that from
the use of traditional energy sources.
It should be added that transport is a dynamically developing field; according to the
forecasts, by 2050 at least 25 million km of new roads are expected worldwide, which
means a 60% increase in the total length of roads compared to 2010 [
19
]. However, this is
associated with the aforementioned costs and negative aspects, especially in relation to the
environment—as global sources report, global greenhouse gas (GHG) emissions from the
transport sector have more than doubled since 1971; moreover, they are growing faster
than those of any other end-use energy sector, and more than three-quarters of this increase
comes from road vehicles [20].
Therefore, the main goal of logistics management is to minimize the total cost of
transport and proper organization of the supply chain. It depends on a given means of
transport, transported products, choice of roads and connections, transport technology,
duration of transport (the entire process), the method of loading, storage options, as
well as economic, legal, or environmental conditions. However, in the case of a specific
product, such as pellets, the costs of the supply chain are largely dependent on the specific
conditions of the supply chain, including the regional availability of the raw material,
which varies depending on the location of the pellet production plants [
2
]. Hence, it is
extremely important to properly optimize the costs associated with the broadly understood
process of transport and forwarding of goods, so that they do not exceed the profits related
to the sale or provision of services.
The logistic system of biomass supplies should therefore be designed individually
for individual solutions—at the level of communes or their associations (e.g., poviats). It
should also be remembered that sometimes distances from the sources of raw materials,
locations of energy production sites, or other conditions may cause the so-called space–
time gap. Its existence requires further improvement of logistic processes consisting in
the delivery of biomass with appropriate parameters, in the right quantity, in the right
condition, in the right place, at the right time, for the right customer, at the right cost.
Choosing an appropriate storage site of pellets may be determined using various
methods, including the construction of a minimum-cost tree with the use of Dijkstra’s
algorithm. It is used to determine the shortest distance of all connections.
Energies 2021,14, 6657 4 of 16
Therefore, the aim of the study is to find an optimal storage location of pellet surplus
in the Lubelskie Voivodeship, which in turn will lead to a reduction in transport costs of
this type of biomass. It also aims to present the essence, possibilities, and advantages that
stem from the use of Dijkstra’s algorithm in this kind of problem.
2. Materials and Methods
2.1. Data
The research was conducted in one of the voivodeships of central and eastern Poland—
the Lubelskie Voivodeship. Due to its area of 25,122 km
2
it is the 3rd largest voivodeship
of the country. However, it is a sparsely populated area characterised by a low level of
urbanisation; it also has the most sparse city network in Poland. On the other hand, com-
pared to other economic sectors, it is characterised by the highest rates in the contribution
of agricultural production in the country. The agricultural character of the voivodeship
(agricultural area constitutes 63% of the voivodeship area), diverse agricultural production,
as well as a large forest area (23% of the whole voivodeship area) result in biomass being
the most accessible source of energy, and the possibilities for its use open up in almost the
whole voivodeship.
The analysis of the determination of the biomass storage site was carried out on the
example of a certain company, which distributes pellets in the Lubelskie Voivodeship in
Poland. The company has 18 sites for obtaining chips (from which pellets are produced),
which also serve as storage facilities in the Lubelskie Voivodeship. The company plans to
build an additional storage of pellets in the Lubelskie Voivodeship, which would serve as a
storage point for seasonal surpluses for existing plants and to contribute to the reduction
in transport costs of biomass. The distances for a single transport do not exceed 220 km,
assuming transport from the most distant parts of the voivodeship or, respectively,
110 km
to the center. The conducted analyses also took into account the estimated sales of the
company in percentage terms.
The spatial data processing and visualization was made in the QGIS program
ver. 3.16
[
21
].
Publicly available free vector layers of administrative borders for Poland and the Lubelskie
Voivodeship and poviats were obtained from the website of Head Office of Geodesy
and Cartography (GUGiK) [
22
]. Data on the main road network in the voivodeship was
downloaded using the Quick OSM plugin.
2.2. Dijkstra’s Algorithm
Graph theory and algorithms created for them are widely used in many areas of life.
One of the main uses is finding the shortest paths between given points of a system, as
well as control of the flow of data or goods in a studied system. Dijkstra’s algorithm,
used in this research, is a popular and well-known algorithm used for finding the shortest
paths between vertices of the graph. The literature presents its use in ICT in designing
IT networks to minimize the cost of its creation and to determine optimal locations of its
substations [
23
]. The algorithm is also used in designing a network of routes for vehicles
and the selection of the location of a distribution center [
24
,
25
]. It may be also used
in city transit in examining the accessibility of specific locations in the public transport
network [
26
]. Due to its popularity, the algorithm also has available visualizations useful
in the effective learning of its operation, e.g., in e-learning [
27
]. However, in the case of
larger data sets, when a network includes thousands of points, the use of this algorithm
may prove to be ineffective. Due to this, certain modifications are used to improve the
efficiency of the algorithm [
28
,
29
], or other heuristic or genetic algorithms are used [
30
].
In the case of uncertainties in the issue while modifying the algorithm, fuzzy set theory
may be applied [
31
]. The literature also includes the reduction in costs in the division of a
system into several subsets. It is used, among others, in the military, where the algorithm
is considered in planning military maneuvers and regrouping several facilities [
32
,
33
].
Thanks to its general character, it may also be used in the problem of finding optimal
storage locations of biomass and assessing the cost of its transport.
Energies 2021,14, 6657 5 of 16
The search of the best location of a place which may be used for storage of pellets
was based on construction of the minimum cost tree consisting of shortest paths. In graph
theory, a graph is a pair G= (V,E), where Vis a set of vertices (nodes) and Eis a set
of edges (arches). An edge is an ordered pair of vertices
vi,vj
such that
vi
,
vj∈
V. In
practice, weighted graphs are generally used. A weighted graph is defined as a structure
G= (V,E,w)
, where weights of edges are given by a weight function
w
:E
→
Rwhich
assigns a real number to each edge. Most frequently, it gives a positive real value, which
is the cost of moving directly from one vertex
u
to another
v
, i.e.,
w
:E
→
[0,
∞
). For
simplicity, the cost of an edge can be thought of as the distance between those two vertices.
A tree is an undirected graph in which any two vertices are connected by exactly one
path. For a given connected, undirected graph G, a spanning tree is in turn a subgraph
of Gwhich includes all its vertices and is a tree. For example, the vertices of the graph
can represent cities and costs of running edges represent distances between pairs of cities
connected directly by the road.
In the analyzed problem the issues connected with the shortest paths with one source
in which the graph G= (V,E) is given, and the s
∈
Vvertex is distinguished and called the
source. Moreover, the weights of all edges of the graph are nonnegative, i.e.,
wνi,νj≥
0.
The shortest path from sto vmust be found for each vertex vin the graph.
The cost of a path
p=ν0
,
ν1
,
. . .
,
νk
between two vertices is the sum of costs of the
edges in that path (Formula (1)).
w(p)=
k
∑
i=1
w(νi−1,νi)(1)
The cost δof the shortest path from vertex uto vis defined as follows (Formula (2)):
δ(u,v)=(min{w(p)}i f path p f rom u to v exists
∞elsewhere (2)
The shortest path from uto vis each path pfor which w(p)=δ(u,v).
One of the classic and most used algorithms for calculating the shortest path from
an origin to a destination is Dijkstra’s algorithm [
31
]. It was first enunciated by Edsger
Wybe Dijkstra [
34
] and is one of the most used and discussed algorithms in the literature
of graphs. Its temporal complexity is O (|E| + |V|
·
log |V|), where |E| is the number of
edges and |V| is the number of vertices of the graph. For a given source vertex (node) in
the graph, the algorithm finds the path with lowest cost (i.e., the shortest path) between that
vertex and every other vertex. If vertices denote cities, edges represent roads, and weights
mean distances between cities, then the algorithm finds the shortest routes between one
chosen city and all others [35–37].
Dijsktra’s algorithm is not efficient for searching for the shortest path in large graphs [
38
],
so various modifications to this algorithm have been proposed by several authors. Some of
these algorithms use heuristics to reduce the run time needed for searching for the shortest
path with and without data processing. Delling et al. [
39
] show an overview of routing
algorithms; all approaches show important advances in shortest path search and make
a low response time in large graphs possible using heuristics. However, in the analyzed
case the size of the graph is not large, so the time needed to complete Dijkstra’s algorithm
is short.
Dijkstra’s algorithm is based on iterations over the set of vertices. At each iteration,
the algorithm finds a vertex for which the distance from the root vertex to the selected
vertex is minimal [
40
]. In subsequent iterations, the set Sof vertices is remembered, for
which the weights of the shortest paths from the source shas been calculated, i.e., the
shortest path to the source for each vertex has been already determined. In simple terms,
Dijkstra’s algorithm involves a multiple repetition of the following operations:
1. estimation of the shortest path for each vertex u∈V–S;
2.
selection of another vertex for which the minimum weight of a shortest path has been
estimated (in other words, selecting the closest vertex);
3. addition of vertex uto the set S;
Energies 2021,14, 6657 6 of 16
4.
implementation of the so-called relaxation of edges stemming out of the vertex
u
[
27
].
The aim of relaxation is to check if after the addition of vertex
u
to set S, the distance of
each neighbor
v
of vertex
u
to root vertex sbecomes shorter than the path earlier estimated.
If the path from sto
v
via
u
is shorter, then it is changed (detailed information on the
operation of the algorithm can be found in, e.g., [41]).
2.3. Transport Cost Calculation
After determining the shortest paths by Dijkstra’s algorithm for a given graph Gand
a selected root vertex s, a tree Tof shortest paths for the root vertex sto any other vertex
v
can be constructed. The cost of a tree is the sum of costs of all paths from optimal location
to any of the current pellet production sites. In case of different shares, weights
ui
of
individual points can be considered, which is associated with more or less frequent need
to cover a given route. The weighted average length of the path from the selected central
vertex to the nother vertices was determined using the Formula (3):
ds(s)=
ns
∑
i=1
ui·w(s, . . . , νi)/
ns
∑
i=1
ui(3)
The above formula relates to the average distance travelled from the pellet production
sites to the new surplus storage site;
ns
denotes number of production sites. Similarly, the
weighted average distance between the designated storage point and individual poviats
pi
(exactly to their centroids) can be expressed according to the expected sales shares
ri
by
Formula (4):
dp(s)=
np
∑
i=1
ri·w(s, . . . , pi)/
np
∑
i=1
ri(4)
Finally, the total cost of transport can be determined using Formula (5):
c(s)=
n
∑
i=1
Nsi·fi·dsi(s)+
n
∑
i=1
Npi·fi·dpi(s)(5)
where nrefers to number of vehicle types,
Ns
and
Np
to the numbers of paths travelled to
new storage site and to poviats, respectively, using vehicle of type iand
fi
to its relative
fuel consumption. In the following research,
Ns
= 18 and
Np=
24 was adopted, which cor-
respond to the number of production plants and number of poviats, respectively.
One type
of vehicle was considered.
2.4. Research Methodology
The methodology adopted in the research is presented in Figure 1. It considers
the choice of a suitable storage of pellet surplus, which can be used in the period of
increased demand.
The construction of the graph, and the implementation of Dijkstra’s algorithm and
the tree with the minimum cost of transport, were made in MATLAB based on spatial
data determined in the QGIS program (locations of pellet production points, junctions
determined by the intersection of main roads, and distances between nodes as measures
of transport costs). Based on the obtained pellet transport cost estimates, a map was
developed showing their increase in relation to the optimal solution. Spatial interpolation
was performed with the use of spline functions.
Energies 2021,14, 6657 7 of 16
Figure 1. The order of procedures in the research methodology.
3. Results
As mentioned before, the research was carried out for the Lubelskie Voivodeship
(central-eastern Poland). The analysis aimed to find a location for the storage of pellet
surpluses with the shortest distances from individual production points and to the later
distribution to points of sale in the period of increased demand for energy. A total of
18 localities in which pellets are stored were the vertices of the graph. The localities
in the graph were connected with edges indicating actual road connections, including
express, national, voivodeship, poviat, and municipality roads. The map of the studied
road network was presented in Figure 2. The markings of locations used in Figures 2–7
and on the map are characterized in Table 1.
Table 1. Symbols of places on the map and graphs.
Symbol City Symbol City
BP Biała Podlaska MP Mi˛edzyrzec Podlaski
Bi Biłgoraj Pi Piszczac
Ch Chełm Pu Puławy
JL Janów Lubelski RP Radzy´n Podlaski
Jo Józefów So Sosnowica
Ks Krasnystaw Sw ´
Swidnik
Kr Kra´snik TL Tomaszów Lubelski
Lb Lubartów Wl Włodawa
Lk Łuków Zw Zwierzyniec
Energies 2021,14, 6657 8 of 16
Figure 2. Locations of biomass depots and the network of main roads in Lubelskie Voivodeship.
Figure 3. Graph presenting locations of pellets production and road connections.
Energies 2021,14, 6657 9 of 16
Figure 4. Estimated sale shares of stored by poviat.
Figure 5. The minimum cost tree rooted in optimal location.
Energies 2021,14, 6657 10 of 16
Figure 6. Relative cost increase for original locations in comparison with the optimal one.
Figure 7. Map with contours for relative cost increase in comparison with the optimal one.
During the next stage, the road network was simplified. The exact topography and the
actual shape of roads is not needed for graph algorithms; it was approximated with the use
of straight lines. The graph constructed in this way is presented in Figure 3. The vertices of
the graph show the current locations of the pellet production points (vertices filled with
black color) and the points of intersection of main roads in the Lubelskie Voivodeship as
approximate locations of a potential pellet storage point (filled with white color). In total,
the graph contains 122 vertices.
Each edge connecting vertices was assigned with a cost reflecting the distance between
chosen localities. For each vertex, the shortest path to each pellet production locality was
determined with the use of Dijkstra’s algorithm.
As already mentioned, the large number of small farms, which is characteristic of
rural, poorly industrialized areas with a lower level of economic development, does not
have a positive effect on the conclusion of contracts for the supply of biomass. Therefore,
when designing a logistics system, models should be developed that take into account the
companies that purchase and process biomass already on the market, as well as, on the
Energies 2021,14, 6657 11 of 16
one hand, the availability of a given type of biomass on the market and, on the other hand,
the demand for it. Therefore, when estimating the costs of pellet logistics, the estimated
demand for pellets in individual poviats of the voivodship was additionally taken into
account (Figure 4).
As shown in Figure 4, in the analyzed Lubelskie Voivodeship, several areas with
different estimated demand for pellets can be observed, with the greatest demand in the
south of the voivodeship, due to the large number of farms and the large potential of
bio-mass sources. In turn, the lowest demand is estimated in the central and western part
of the voivodeship, due to the higher population density and industrialization, resulting in
a greater number of households connected to the central heating network.
According to the next step of the research procedure, after determining the shortest
paths, a tree with minimum transport cost was constructed. Total costs for individual
trees rooted in the current locality were presented in Table 2(if a new location is near to
an existing one). The columns present the production share of a given plant, the average
distance to other pellet production sites, the total cost including all 18 locations, and the
absolute and relative increase in transport costs in relation to the optimal location.
Table 2. The cost of trees rooted at given cities by Dijkstra’s algorithm in relation to optimal place.
City
Prod.
Share
u
(%)
Average
Distance
ds(km)
Average
Distance
dp(km)
Tree
Cost c
(km)
Cost
Increase
(km)
Relative
Cost
Increase
(%)
Krasnystaw (Ks) 4.8 89.7 61.5 3092.0 24.9 0.8
´
Swidnik (Sw) 1.9 97.4 65.4 3321.6 254.5 8.3
Chełm (Ch) 4 96.1 66.7 3331.8 264.6 8.6
Zwierzyniec (Zw) 4.9 102.2 73.6 3605.8 538.6 17.6
Sosnowica (So) 5.2 103.3 73.6 3625.5 558.4 18.2
Lubartów (Lu) 5.1 104.4 74.4 3665.2 598.0 19.5
Biłgoraj (Bi) 15 96.7 82.3 3715.6 648.5 21.1
Kra´snik (Kr) 1 103.2 81.2 3805.7 738.6 24.1
Janów Lubelski (JL) 8.1 103.4 81.4 3813.3 746.1 24.3
Józefów (Jo) 12.6 102.2 82.3 3816.1 749.0 24.4
Włodawa (Wl) 7.2 107.5 83.7 3944.0 876.8 28.6
Tomaszów Lubelski (TL)
5.4 110.9 88.2 4113.0 1045.9 34.1
Radzy´n Podlaski (RP) 1.1 127.9 92.4 4519.4 1452.2 47.3
Puławy (Pu) 5.7 128.4 95.9 4613.0 1545.9 50.4
Mi˛edzyrzec Podlaski
(MP) 5.5 143.1 107.0 5143.6 2076.5 67.7
Piszczac (Pi) 4.5 146.1 109.4 5255.4 2188.3 71.3
Biała Podlaska (BP) 7 148.6 110.5 5326.4 2259.3 73.7
Łuków (Lk) 1 150.0 109.9 5338.4 2271.3 74.1
On the basis of the calculated costs, it was found that the lowest total cost (
3067.2 km
)
was indicated for the trees rooted in the vertex number 101, located near the town of Piaski
(GPS coordinates 51.1594 N, 22.9459 E). For this root vertex, the average distance to all the
production plants was 88.1 km, and average distance to all poviats 61.7 km. The graph
presenting a tree with the lowest cost is shown in Figure 5. The location for which the
presented tree has a minimum cost, obtained with the use of Dijkstra’s algorithm, is marked
with a red triangle.
Considering only existing production plant localities, the smallest difference with the
optimal one occurred in the case of Krasnystaw—Ks (total tree cost equals 3092.0 km). This
location is closest to the optimal location. The average distance to other plants is 89.7 km,
and this is only a 0.8% increase in cost in relation to the optimal one. On the other hand,
Łuków (Lk) and Biała Podlaska (BP) are the worst sites for potential storage locations of
biomass in the form of pellets in the Lubelskie Voivodeship, due to the costs obtained for
these localities; the average distance to all production sites is 150.0 km, and the total cost is
5338.4 km for Łuków and 148.4 km and 5326.4 km, respectively, for Biała Podlaska.
The average cost of travel for all trees obtained for individual locations amounted
to 4113.7 (km), while standard deviation amounted to 735.2 (km). A visual comparison
Energies 2021,14, 6657 12 of 16
regarding the relative difference of costs for towns with storage locations of biomass was
presented in Figure 6.
The relative increase in costs in comparison to the optimal one is the smallest in in
the case of Krasnystaw (Ks) at 0.8%; slightly higher, 8.3%, for ´
Swidnik (Sw); and 8.6%
for Chelm (Ch). On the other hand, Łuków (Lu) and Biała Podlaska (BP) had the largest
relative cost increase, which is, respectively, 74.1% and 73.7% higher than the optimum. It
shows that these towns do not provide an optimal storage location of biomass and had to
be rejected. If the relative difference of costs is large, a given locality does not qualify as a
place to create a storage location of biomass.
For visualization purposes, based on the obtained pellet transport cost estimates
determined for trees for all possible graph nodes, a map was developed showing their
increase in relation to the optimal solution. Spatial interpolation was performed with
the use of spline functions. The constructed map is shown in Figure 7. The map shows
contours specifying the equal cost of pellet transport in the case of the location of a new
storage place.
It can be seen on the map that, despite the fact that the location of the optimal storage
site is determined approximately in the center of gravity of the entire voivodeship, the
area where the relative cost increase does not exceed 10% is quite stretched. This area
has 42,122.36 km
2
, which is approximately 17.6% of the entire area of the voivodship
(
25,122.35 km2
). This area is shifted to the south of the voivodeship, where two larger
production plants are located and an increased demand for energy from pellets is present.
4. Discussion
The transport, storage, and reloading of pellets are very important issues for all market
participants—producers, distributors, consumers, and the entire society. This is due to
both economic reasons and the quality of the final product [
42
–
44
]. Inadequate storage
and transport conditions may reduce the calorific value and other quality features of wood
pellets, such as mechanical strength, degree of crushing, or ash content, which are key
determinants of the selection of the appropriate type of fuel [45].
It should be added that a large part of the current use of pellets is facilitated by the
intense activities of global trade, and the growing use of bioenergy in combination with
the spatial distribution of biomass demand and supply will further increase trade in solid
biomass [
46
]. In the literature on the subject, one can find the results of research that
allow us to assess the economic feasibility of transporting biomass and biofuel; however,
this transport is carried out on a large scale and over long distances. Large-scale and
long-distance transport of biomass is a cost-effective transport option for biofuel plants
and for coal plants which consider biomass co-firing [
47
–
49
]. On the other hand, it should
be remembered that the large distance between the production site and the locations of
end users may cause pellets to be subject to variable storage conditions, which in turn may
cause their physical and chemical degradation [50].
Therefore, in the research and analysis carried out, the focus was on the issues of
biomass transport on a local scale. In order to indicate the optimal pellet storage location,
Dijkstra’s algorithm was used. Dijkstra’s algorithm is a well-known algorithm used to find
the shortest paths between vertices of a graph. Shortest path search has been widely studied.
Many applications can be found in different fields of science, especially in Geographic
Information Systems. Additionally, various modifications of this algorithm with the use of
heuristics are proposed in order to reduce its execution time [40].
The selection of the optimal location for a new storage point for pellet surplus—
presented in this paper—concerns the problem in which this point becomes the central
point of distribution of goods. Additionally, actual shares of pellet production in each
plant have been taken into account, so each edge has the different weight. The weights
take into account, e.g., the intensity of following specific routes. The construction of the
minimum-cost tree also depends on the way the goods are distributed to selling points.
One can consider a model in which distribution is also possible between individual (closest)
Energies 2021,14, 6657 13 of 16
locations, as well as the demand for the product in the vicinity of production and storage
points. However, when it comes to the place of production of pellets, one of the most
important factors when choosing the right place is the availability of the raw material. This
aspect affects not only the price of the raw material, but also the costs of transporting the
raw material to the pellet plant [2].
Similar studies were carried out in, for example, the USA, where attempts were
made to develop mathematical logistic models allowing for the estimation of distribution
channels, transport, and quantities for the domestic demand for wood pellets. Lacoa and
his team [
51
] investigated two cases: distribution to power plants and distribution to retail
stores. Models developed using Dijkstra’s algorithm can be used as tools to minimize the
costs of wood pellet distribution in a selected area.
It should be added that more and more often in computer-aided systems of choosing
the optimal distribution routes for goods, multi-criteria mathematical models are used,
which take into account many qualitative and quantitative parameters; this requires their
additional standardization and determination of their priority level [
52
]. An example is
the work of Aghalari et al. [
53
], in which they presented the problem of optimizing the
entire biomass-to-pellet supply system. They have developed a two-stage stochastic model
that takes into account various elements such as: harvesting, storage, transportation, or
quality inspection. However, they focused largely on the impact of biomass quality and its
transport to processing points.
With regard to transport processes, it can therefore be concluded that biomass can
be transported in many ways. When choosing the optimal form of transport in terms of
economic and ecological aspects, one should take into account many factors, including, first
of all: the type and form of biomass, its quantity, but also the distance to the destination,
and the requirements and needs of customers. Taking into account the typical locations of
biomass sources and the considered distances in the analyzed area, it seems that the most
beneficial in many respects will be the use of road infrastructure and road transport. Then,
with regard to storage processes, it should be mentioned that many types of biomass are
characterized by seasonal availability, as they are collected and then processed at a specific
time of the year, hence they must be stored. Therefore, building small, local depots seems
to be a good solution. Storage points (warehouses) may be located in specially dedicated
places for this purpose.
5. Conclusions
After analyzing the literature, and on the basis of this research, it was possible to draw
the following conclusions:
1. Dijkstra’s algorithm allowed finding the shortest path between vertices (localities).
2.
The optimal location for a new storage place for pellets was indicated near the city of
Piaski. The lowest total cost of tree rooted in this place was 3067.2 km, which relates
to the average distance 88.1 km to a single plant and 61.7 km to poviat.
3.
From 18 current localities of the company’s sites in the Lubelskie Voivodeship, the
closest optimal costs were obtained for Krasnystaw (0.8% higher), ´
Swidnik (8.3%
higher), and Chełm (8.6% higher).
4.
Average cost of travel for all trees obtained for individual locations amounted to
4113.7 (km), while standard deviation amounted to 735.2 (km).
5.
The area with the most convenient locations is shifted from the center towards the
southern part of the voivodship, and the increase in transport costs does not exceed
10% of lowest cost for 17.6% area of the Lubelskie voivodship.
6. Improperly adopted storage location can increase transport costs by up to 75%.
The results obtained in the analysis in order to estimate the potential storage location of
biomass in the form of pellets on a technical level may serve as the basis for further research.
It may be carried out for different territorial units with the use of tools of technical and
economic optimization of various undertakings of production and processing of biomass
into useful forms of energy. Dijkstra’s algorithm allows us to find the shortest path between
Energies 2021,14, 6657 14 of 16
vertices (localities). Individual edges of a graph may have different weights that define the
cost of going from one vertex to another on a given edge, which allows the determination
of the potential of the optimization procedure on the economic level. This is all the more
important as reducing supply chain costs can contribute to increased use of pellets. This
manuscript concerns mainly one of the components of the entire pellet production chain,
which is the cost of transport. In the next step, the authors intend to take into account other
factors in order to model and optimize the problem as a whole. Free GIS software will also
be used for implementation so that the whole problem is solved in one analytical program.
Author Contributions:
Conceptualization, A.B. and M.S.; methodology, A.B. and M.S.; software,
A.B.; validation, A.B. and M.S.; formal analysis, A.B. and M.S.; investigation, A.B. and M.S.; resources,
A.B. and M.S.; data curation, M.S.; writing—original draft preparation, A.B. and M.S.; writing—
review and editing, A.B. and M.S.; visualization, A.B.; supervision, M.S.; project administration, A.B.
and M.S.; and funding acquisition, M.S. All authors have read and agreed to the published version of
the manuscript.
Funding:
Funded from the ‘Excellent science’ program of the Ministry of Science and Higher Educa-
tion as a part of the contract no. DNK/SP/465641/2020 “The role of the agricultural engineering and
environmental engineering in the sustainable agriculture development”.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or
in the decision to publish the results.
References
1.
Borjesson, M.; Ahlgren, E.O.; Lundmark, R.; Athanassiadis, D. Biofuel futures in road transport—A modeling analysis for Sweden.
Transp. Res. Part D-Transp. Environ. 2014,32, 239–252. [CrossRef]
2.
Visser, L.; Hoefnagels, R.; Junginger, M. Wood pellet supply chain costs—A review and cost optimization analysis. Renew. Sustain.
Energy Rev. 2020,118, 109506. [CrossRef]
3. Lewandowski, W.M. Proekologiczne Odnawialne ´
Zródła Energii; Wydawnictwa Naukowo-Techniczne: Warszawa, Poland, 2012.
4.
Irannezhad, E.; Prato, C.G.; Hickman, M. The effect of cooperation among shipping lines on transport costs and pollutant
emissions. Transp. Res. Part D-Transp. Environ. 2018,65, 312–323. [CrossRef]
5.
Merchan, A.L.; Leonard, A.; Limbourg, S.; Mostert, M. Life cycle externalities versus external costs: The case of inland freight
transport in Belgium. Transp. Res. Part D-Transp. Environ. 2019,67, 576–595. [CrossRef]
6.
Janic, M. Modelling the full costs of an intermodal and road freight transport network. Transp. Res. Part D-Transp. Environ.
2007
,
12, 33–44. [CrossRef]
7.
Zimon, G. Cost Analysis of Transport Car Companies. Zesz. Nauk. Uniw. Szczeci ´nskiego. Finans. Rynk. Finans. Ubezpieczenia
2015
,
77, 349–354. [CrossRef]
8.
Gil, L.; Ignaciuk, P. Influence of transportation distance on transportation costs. Autobusy. Tech. Eksploat. Syst. Transp.
2014
,5,
50–52.
9.
Musso, A.; Rothengatter, W. Internalisation of external costs of transport-A target driven approach with a focus on climate change.
Transp. Policy 2013,29, 303–314. [CrossRef]
10.
Moliner, E.; Vidal, R.; Franco, V. A fair method for the calculation of the external costs of road traffic noise according to the
Eurovignette Directive. Transp. Res. Part D-Transp. Environ. 2013,24, 52–61. [CrossRef]
11.
Mostert, M.; Limbourg, S. External Costs as Competitiveness Factors for Freight Transport A State of the Art. Transp. Rev.
2016
,
36, 692–712. [CrossRef]
12.
Mostert, M.; Caris, A.; Limbourg, S. Road and intermodal transport performance: The impact of operational costs and air
pollution external costs. Res. Transp. Bus. Manag. 2017,23, 75–85. [CrossRef]
13.
Dente, S.M.R.; Tavasszy, L.A. Impacts of trade related sustainability strategies on freight transportation: Modelling framework
and application for France. Transp. Res. Part D-Transp. Environ. 2018,58, 308–319. [CrossRef]
14.
Miao, Z.W.; Grift, T.E.; Hansen, A.C.; Ting, K.C. An overview of lignocellulosic biomass feedstock harvest, processing and supply
for biofuel production. Biofuels-Uk 2013,4, 5–8. [CrossRef]
15.
Ehrig, R.; Behrendt, F.; Worgetter, M.; Strasser, C. Economics and Price Risks in International Pellet Supply Chains; Springer:
Berlin/Heidelberg, Germany, 2014.
Energies 2021,14, 6657 15 of 16
16.
Boukherroub, T.; Lebel, L.; Lemieux, S. An integrated wood pellet supply chain development: Selecting among feedstock sources
and a range of operating scales. Appl. Energy 2017,198, 385–400. [CrossRef]
17.
Interantional Reanewable Energy Agency, Solid Biomass Supply for Heat and Power. Technology Brief. Abu Dhabi. 2018.
Available online: https://irena.org (accessed on 28 September 2021).
18.
Forsberg, G. Biomass energy transport: Analysis of bioenergy transport chains using life cycle inventory method. Biomass
Bioenergy 2000,19, 17–30. [CrossRef]
19.
Laurance, W.F.; Clements, G.R.; Sloan, S.; O’Connell, C.S.; Mueller, N.D.; Goosem, M.; Venter, O.; Edwards, D.P.; Phalan, B.;
Balmford, A.; et al. A global strategy for road building. Nature 2014,513, 229–232. [CrossRef]
20.
Broin, E.O.; Guivarch, C. Transport infrastructure costs in low-carbon pathways. Transp. Res. Part D-Transp. Environ.
2017
,55,
389–403. [CrossRef]
21.
QGIS Documentation QGIS Desktop User Guide/Manual (QGIS 3.16). Available online: https://docs.qgis.org/3.16/en/docs/
user_manual/ (accessed on 23 August 2021).
22.
Główny Urz ˛ad Geodezji i Kartografii Dane PgiK. Available online: http://www.gugik.gov.pl/pzgik (accessed on
22 August 2021
).
23.
Wong, Y.K.; Shi, K.L.; Chan, T.F. Effective algorithms for designing power distribution networks. Microprocess. Microsyst.
1996
,20,
251–258. [CrossRef]
24.
Graba´nski, S. Setting the Location of a Distribution Center. Modeling with the Use of Geoinformation Systems; Pozna ´n University of
Economics and Business: Pozna ´n, Poland, 2015.
25.
Liu, S.; Chan, F.T.S.; Chung, S.H. A study of distribution center location based on the rough sets and interactive multi-objective
fuzzy decision theory. Robot. Comput. Integr. Manuf. 2011,27, 426–433. [CrossRef]
26.
Cichoci´nski, P.; D˛ebi´nska, E. Badanie dost ˛epno´sci komunikacyjnej wybranej lokalizacji z wykorzystaniem funkcji analiz
sieciowych. Rocz. Geomatyki 2012,10, 41–48.
27.
Sanchez-Torrubia, M.G.; Torres-Blanc, C.; Lopez-Martinez, M.A. PathFinder: A Visualization eMathTeacher for Actively Learning
Dijkstra’s Algorithm. Electron. Notes Theor. Comput. Sci. 2009,224, 151–158. [CrossRef]
28.
Peyer, S.; Rautenbach, D.; Vygen, J. A generalization of Dijkstra’s shortest path algorithm with applications to VLSI routing. J.
Discret. Algorithms 2009,7, 377–390. [CrossRef]
29. Wang, S.X. The Improved Dijkstra’s Shortest Path Algorithm and Its Application. Procedia Eng. 2012,29, 1186–1190. [CrossRef]
30.
Karadimas, N.V.; Doukas, N.; Kolokathi, M.; Defteraiou, G. Routing optimization heuristics algorithms for urban solid waste
transportation management. WSEAS Trans. Comput. 2008,7, 2022–2031.
31.
Deng, Y.; Chen, Y.X.; Zhang, Y.J.; Mahadevan, S. Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment.
Appl. Soft Comput. 2012,12, 1231–1237. [CrossRef]
32.
Tarapata, Z. Models and methods of movement planning and simulation in simulation aided system for operational training.
In Proceedings of the 6th NATO Regional Conference on Military Communication and Information Systems, Zegrze, Poland,
6–8 October 2004; pp. 152–161.
33.
Tarapata, Z. Zastosowanie metod wyznaczania przepływu w sieciach do planowania manewru wojsk. Biul. Inst. Syst. Inform.
2008,2, 31–44.
34. Dijkstra, E.W. A note on two problems in connexion with graphs. Numer. Math. 1959,1, 269–271. [CrossRef]
35.
Fagerholt, K.; Heimdal, S.I.; Loktu, A. Shortest path in the presence of obstacles: An application to ocean shipping. J. Oper. Res.
Soc. 2000,51, 683–688. [CrossRef]
36.
Romero, G.; Duran, G.; Marenco, J.; Weintraub, A. An approach for efficient ship routing. Int. Trans. Oper. Res.
2013
,20, 767–794.
[CrossRef]
37.
Neumann, T. Method of Path Selection in the Graph—Case Study. Transnav-Int. J. Mar. Navig. Saf. Sea Transp.
2014
,8, 557–562.
[CrossRef]
38.
Fuhao, Z.; Jiping, L. An Algorithm of Shortest Path Based on Dijkstra for Huge Data. In Proceedings of the Sixth International
Conference on Fuzzy Systems and Knowledge Discovery, Tianjin, China, 14–16 August 2009; pp. 244–247.
39.
Delling, D.; Sanders, P.; Schultes, D.; Wagner, D. Engineering Route Planning Algorithms. In Algorithmics of Large and Complex
Networks. Lecture Notes in Computer Science; Lerner, J., Wagner, D., Zweig, K., Eds.; Springer: Berlin/Heidelberg, Germany, 2009;
Volume 5515, pp. 117–139, ISBN 0302-9743 1611-3349.
40.
Rodriguez-Puente, R.; Lazo-Cortes, M.S. Algorithm for shortest path search in Geographic Information Systems by using reduced
graphs. Springerplus 2013,2. [CrossRef] [PubMed]
41. Cormen, T.H. Introduction to Algorithms, 3rd ed.; MIT Press: Cambridge, MA, USA, 2009; ISBN 9780262033848.
42.
Rynkiewicz, M.; Dobek, T.K. Selected physical and mechanical properties of pellets depending on the composition and tempera-
ture of their storage. Agric. Eng. 2013,3, 321–330.
43.
Jose-Vicente, O.V.; Enrique, G.A.; Patricia, G.G. Analysis of Durability and Dimensional Stability of Hydrothermal Carbonized
Wooden Pellets. Wood Res. 2016,61, 321–330.
44.
Whittaker, C.; Shield, I. Factors affecting wood, energy grass and straw pellet durability—A review. Renew. Sustain. Energy Rev.
2017,71, 1–11. [CrossRef]
45.
Kuranc, A.; Stoma, M.; Rydzak, L.; Pilipiuk, M. Durability Assessment of Wooden Pellets in Relation with Vibrations Occurring
in a Logistic Process of the Final Product. Energies 2020,13, 5890. [CrossRef]
Energies 2021,14, 6657 16 of 16
46.
Matzenberger, J.; Kranzl, L.; Tromborg, E.; Junginger, M.; Daioglou, V.; Goh, C.S.; Keramidas, K. Future perspectives of
international bioenergy trade. Renew. Sustain. Energy Rev. 2015,43, 926–941. [CrossRef]
47.
Roni, M.S.; Eksioglu, S.D.; Searcy, E.; Jacobson, J.J. Estimating the variable cost for high-volume and long-haul transportation of
densified biomass and biofuel. Transp. Res. Part D-Transp. Environ. 2014,29, 40–55. [CrossRef]
48.
Argo, A.M.; Tan, E.C.D.; Inman, D.; Langholtz, M.H.; Eaton, L.M.; Jacobson, J.J.; Wright, C.T.; Muth, D.J.; Wu, M.M.;
Chiu, Y.W.; et al.
Investigation of biochemical biorefinery sizing and environmental sustainability impacts for conventional
bale system and advanced uniform biomass logistics designs. Biofuels Bioprod. Biorefining-Biofpr 2013,7, 282–302. [CrossRef]
49.
Bai, Y.; Hwang, T.S.; Kang, S.; Ouyang, Y.F. Biofuel refinery location and supply chain planning under traffic congestion. Transp.
Res. Part B-Methodol. 2011,45, 162–175. [CrossRef]
50.
Gilvari, H.; Cutz, L.; Tiringer, U.; Mol, A.; de Jong, W.; Schott, D.L. The Effect of Environmental Conditions on the Degradation
Behavior of Biomass Pellets. Polymers 2020,12, 970. [CrossRef]
51.
Lacoa, U.; Velarde, G.; Kay, M.; Blanco, E.; Saloni, D. Design and Development of Logistics Models for Residential and Commercial
Biomass Pellets for Heat and Power Generation in the US. Bioresources 2017,12, 1506–1531. [CrossRef]
52.
Rosita, Y.D.; Rosyida, E.E.; Rudiyanto, M.A. Implementation of Dijkstra Algorithm and Multi-Criteria Decision-Making for
Optimal Route Distribution. Fifth Inf. Syst. Int. Conf. 2019,161, 378–385. [CrossRef]
53.
Aghalari, A.; Aladwan, B.S.; Marufuzzaman, M.; Tanger, S.; Da Silva, B.K.; Gude, V.G. Optimizing a pellet supply system:
Market-specific pellet production with biomass quality considerations. Comput. Chem. Eng. 2021,153. [CrossRef]
Available via license: CC BY 4.0
Content may be subject to copyright.
