=Paper=
{{Paper
|id=Vol-2098/paper33
|storemode=property
|title=On a Problem of the Utility Network Design
|pdfUrl=https://ceur-ws.org/Vol-2098/paper33.pdf
|volume=Vol-2098
|authors=Gulzhigit Y. Toktoshov,Anastasiya N. Yurgenson,Denis A. Migov
}}
==On a Problem of the Utility Network Design==
https://ceur-ws.org/Vol-2098/paper33.pdf
On a Problem of the Utility Network Design ?
Gulzhigit Y. Toktoshov, Anastasiya N. Yurgenson, and Denis A. Migov
Institute of Computational Mathematics and Mathematical Geophysics of SB RAS,
prospekt Akademika Lavrentieva 6, 630090, Novosibirsk, Russia
tgi_tok@rambler.ru,nastya@rav.sscc.ru,mdinka@rav.sscc.ru
http://www.sscc.ru/
Abstract. An adoptable location of utility network elements for pro-
cessing and distribution of resources between consumers is a complex
technical and economic problem. For its solution, it is necessary to take
into account both the resource constraints as well as a plan of processing
and distributing the resources among consumers, thus providing an effi-
cient scheme for the functioning of a network to be designed. In this study
the problem of the utility network design is treated by the hypernet ap-
proach according to the compatibility of different types of resources with
allowance for their laying in the same track. Also, we study the reliability
aspect of the designed utility network for obtaining the optimal laying
of a reliable enough utility network on a given area with minimal costs.
The performed numerical experiments show how the method proposed
works.
Keywords: Multilayer network · Utility network · Deployment area ·
Track · Graph · Hypergraph · Hypernet · Connectivity · Network relia-
bility
1 Introduction
Utility network is a composite geographically distributed system that performs
such vital functions as providing consumers with energy and water resources,
communication facilities, information, route network, traffic, and other services.
It includes local authorities, economic objects and consumers, and, also, consid-
ers the nature of the relationship between them.
Currently, there is a significant number of published works related to de-
sign and construction of networks for various purposes [8, 2, 14, 16, 10, 9, 26, 28,
3, 13, 11, 22, 7]. In particular, in [14, 16] the authors consider the search for the
optimal routes for laying of the utility networks. The task of locating the con-
struction objects on a given territory is solved in [10, 9]. In [26, 28] the possibility
?
Supported by Russian Foundation for Basic Research under grants 17-07-00775, 18-
07-00460
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org
386 G. Y. Toktoshov et al.
of GIS-technologies applying in tracing and placing of linear objects is studied.
An application of splines in the optimization of the car roads tracing is treated
in [3]. Moreover, problems of the utility networks optimization also arise in other
adjacent areas, such as the placement of logistics facilities [13] and power sup-
ply systems [11], design and reconstruction of transport networks [22], and the
routing of service networks [7].
In all the above mentioned papers, the utility network is considered to be
a two-dimensional object located on a plane. However, such a two-dimensional
representation of networks does not completely correspond to reality, since as
the basis of the design and construction of the utility network underlies the
interaction of, at least, two interrelated objects. Unlike its predecessors, in this
work a utility network is treated as a hierarchical system, in which resources
(gas, oil, water, etc.) are transferred from producers to consumers through the
points of processing via communication (transport channels). Such systems are
characterized by the existence of the so-called fixed costs, which must be done
regardless of the volume of production and processing of products.
For the simulation of various network objects, graphs and hypergraphs are
commonly used. As was mentioned above, in many areas of applications there
is a bunch of interconnected networks, forming a hierarchical structure. In such
cases it is completely inconvenient to use graphs and hypergraphs. Instead of
them, other network models are used [8, 2, 21, 12, 19, 29, 1]. For instance, there
are such mathematical objects as nested graphs [21], layered complex networks
[12], hypernets [19, 29, 1].
For the problems of a utility network design we use the hypernet approach,
which makes it possible to operate with hierarchical, multilevel, and heteroge-
neous networks. More specifically, we study a simple case of hypernet — the
two-level network. It consists of a primary network, which represents a physical
network, and a secondary network, which represents a logical network, embed-
ded into a primary network. Our previous results on the hypernet using for the
utility network design are presented in [30, 31]. A of nodes of a secondary net-
work is a subset of nodes of a primary network, whereas each secondary network
edge is a chain in a primary network, i.e. a sequence of the adjacent edges. For
the sake of definiteness, we call edges of a primary network as branches. Note
that each branch can be included into several edges.
Based on the hypernet approach, we solve the task of providing the connec-
tivity of given consumers with given sources on a given area with minimization
of laying and maintenance costs and within given constraints. As a result, we
propose a new method for the selection of routes for the laying of the utility
networks taking into account the compatibility of different types of resources for
placing them in the same track. For example, in one collector it is possible to
lay in one direction the heat communications, pressure water pipes and sewerage
systems, over ten communication cables and power cables with voltage up to 10
kV. However, the joint laying of gas pipelines and pipelines with combustible
substances is not permitted due to the fact that the distance between commu-
On a Problem of the Utility Network Design 387
nications of different purposes is normalized according to building codes and
regulations [4].
Also, we introduce an algorithm for obtaining not only the cheapest solution,
but reliable enough as well. Note that reliability analysis of hypernets is not a new
task, see, for example [25, 24]. As a reliability measure, the hypernet analogue of
the 2-terminal reliability is considered, under assumption that failures occur in a
primary network, and nodes in a secondary network should be connected. In this
case we design a utility network, in which each consumer should be connected
with the necessary suppliers with probability no less than a given reliability
threshold 0 < R0 ≤ 1. For solving this problem, we introduce the ant colony
algorithm. Unlike the existing optimization techniques, the procedure proposed
makes it possible to obtain the variant of the utility network placement in a
given area, which would be appropriate in terms of reliability and minimum
total costs.
2 Definitions and Notations
We use the classical definitions from the hypernet theory [19]:
Definition: hypernet is represented by the four sets: HN = (X, V, R, F ),
where:
X = {x1 , x2 , ..., xn } – a set of nodes;
V = {v1 , v2 , ..., vg } ⊆ X × X – a set of branches;
R = {r1 , r2 , ..., rm } – a set of edges;
F : R → 2V – a mapping associating each item r ∈ R with a path F (r) ⊆ V .
These four sets determine the two graphs:
P N = (X, V ) – a graph of a primary network;
W N = (X, R) – a graph of a secondary network.
It is assumed that for the utility network laying, the following objects are
given:
D – s two-dimensional placement area;
Ysource ⊆ D – a set of point objects, which represents sources;
Yconsumer ⊆ D – a set of point objects, which represents consumers;
Y = Ysource ∪Yconsumer – a set of point objects to be connected by linear facilities
of various assignment.
Let us describe the utility network in terms of the hypernet theory.
In a primary network P N = (X, V ) we assume that:
ρ(v) – the length of v ∈ V ;
a(v) – the land cost (rent, taxes, etc.) of v ∈ V ;
b(v) – the cost of constructing (excavation) of v ∈ V ;
γ1 – a discount factor of building costs (this factor is for bringing economic
indicators of different years to comparable values).
In aP secondary network W N = (Y, R) we assume the following:
ρ(r) = v∈F (r) ρ(v) – the length of r ∈ R;
388 G. Y. Toktoshov et al.
c(r) – the cost of r ∈ R, including its installation and laying between the corre-
sponding elements of D;
γ2 – a discount factor of equipment costs;
dv (r) – maintenance costs of linear facilities on plots v ∈ F (r) for a chosen edge
(route) r;
T – a set of all types of utility resources. Thus, each edge r has the type
type(r) ∈ T .
As was said above, different resource types may be incompatible with each
other for their laying in the same track. For the description of the compatibility
of different types of resources, we introduce the following notation:
A binary relation CT ∈ T × T is defined by the rule: if (t1 , t2 ) ∈ CT , then
these types of resources can be placed in the same track, i.e. both of them can
include the same branch.
Let M inCT (t1 , . . . , th ) be the minimum number of disjoint subsets into which
a subset of types {t1 , . . . , th } can be divided.
For example, if there are types {t1 , t2 , t3 } such that (t1 , t2 ), (t2 , t3 ) ∈ CT , but
(t1 , t3 ) ∈
/ CT , then M inCT (t1 , t2 , t3 ) = 2, since these types can be divided into
the subsets {t1 , t2 } and {t3 }.
3 Problem Statement
In the general case, the task of a utility network design can be formulated as
a problem of providing the connectivity of given objects from Y = Ysource ∪
Yconsumer with minimization of the laying and maintenance costs. The obtained
configuration determines the topology of the designed utility network.
As we said above, we use the hypernet approach, so the network structure
is separated into a primary and a secondary networks. As a primary network,
we consider a discrete analogue of the placement area D for the utility network.
As a secondary network, we consider routes through branches for the laying of
various utility systems.
The problem is to find the hypernet HN , i.e. embed each path r ∈ R in W N
into the edges of the graph P N in such a way that all conditions and restrictions
imposed on the utility network be met, and the functional below would take the
minimum value:
X
Q(HN ) = a(v) + b(v) · γ1 ρ(v) · M inCT (v) + (1)
v∈V 0
X X
+ c(r) + dv (r) · γ2 ρ(r),
r∈R v∈F (r)
where v ∈ V 0 ⊆ V if ∃r ∈ R such that v ∈ F (r). Let v ∈ V 0 and v ∈ F (ri ), i =
1, h (r1 , . . . , rh ∈ R), then M inCT (v) = M inCT (type(r1 ), . . . , type(rh )).
On a Problem of the Utility Network Design 389
4 The Two-Stage Algorithm
This Section proposes an algorithm for obtaining an approximate solution of the
problem stated. The basic idea of our algorithm is to find an initial estimation of
HN , its total cost (1), and to improve it further. The initial estimation is found
by the “greedy” algorithm or the Floyd algorithm, which is commonly used for
finding the shortest paths.
The cost of v ∈ V is (a(v) + b(v) ∗ γ1 + c(r) ∗ γ2 + dv (r)) ∗ ρ(v) (for the
sake of simplicity c(r) ∗ γ2 + dv (r) = const, ∀r) for the Floyd and the Dijkstra
algorithms in the primary network P N .
Below we outline the steps of the algorithm:
Algorithm Stage 1 (Greedy)
Divide a set of types into incompatible subsets.
repeat
Choose a subset of types T 0 with the maximum number of edges (return the
values a(v), b(v) for all v ∈ F (r) in the graph P N ).
repeat
Find all the shortest paths (xi , xj ) i, j = 1, n, i 6= j in the graph P S = (X, V )
by the Floyd algorithm (type(xi , xj ) ∈ T 0 ).
Choose the minimal value from a set of the shortest paths, where (xi , xj ) ∈ R
in the graph W N (the path (xi , xj ) is not assigned for any r ∈ R).
Assign the edge r ∈ R to the shortest path (xi , xj ) in the graph P N .
a(v) := 0, b(v) := 0 for all v ∈ F (r) in the graph P N (the costs of branches
are zero for assigning a path).
until ∀r ∈ R in the graph W N , there is an assigned path (xi , xj ) in the graph
P N.
until T 0 6= ∅.
The greedy algorithm is approximate, therefore the solutions obtained are
not always optimal. We can improve it if for an edge r ∈ R we reassign new
paths F (r). For example, a new path F (r) is cheaper if it includes a greater
number of branches v ∈ F (r) with zero cost (branches that have already been
used for other edges r ∈ R).
We can order the edges by a certain technique. Earlier we have considered
a few techniques [31], for example, ordering by the inclusion of the most rarely
used branches. However, on the average, the results obtained did not strongly
depend on the ordering method used.
We can use some iterations of the Stage 2 algorithm for finding better results.
5 Results of Numerical Experiments
For numerical experiments we have chosen the 10x10 grid as a primary network
P N . The number of edges in W N for laying in P N varied from 10 up to 4000
390 G. Y. Toktoshov et al.
Algorithm Stage 2 (Improvement)
Order ri ∈ R by decreasing the costs (renumber them) and obtain the new list {ri }.
repeat
“Delete” ri from the graph HS according to the list (i.e. ∀vk ∈ F (r), restore for
a(vk ) and b(vk ) the initial values if vk is not included into other edges).
Find a new shortest path for ri in the graph P N by the Dijkstra algorithm.
Assign it for ri .
a(v) := 0, b(v) := 0 for all v ∈ F (ri ) in the graph P N .
until All edges {ri } from the list are updated.
(the abscissae axis). For a chosen number of edges nedges , we randomly place
in the grid nedges sources and nedges consumers. As a result, a grid node can
contain more than one object: source(s) and consumer(s). The random placement
is carried out 10 times for a given nedges value. As a laying cost for nedges we
take the average among the 10 values found.
We consider the set of resource types which has been mentioned as the ex-
ample in the Section 2. Thus, we assume that there three different types of
edges.
In Figure 1 the normalized costs are shown (i.e. we find minimal cost and
divide all the obtained values by it). The results show that the Greedy algorithm
is more convenient for a small number of edges; otherwise the Floyd algorithm
gives a better solution.
In Figure 2 we show the results of the previous numerical experiments for the
case of compatibility of all resources type for laying in same track [31]. Perform-
ing Improvement stage with different orders of branches leads us to techniques
Improve1 and Improve2. The results are very similar with the numerical exper-
iment 1.
6 Utility Network Laying with Reliability Constraint
In this Section, we improve the algorithm proposed in order to obtain not only
the cheapest, but also reliable enough solution. Most of the results presented
coincide with those from our previous publication [30].
Random graphs are commonly used for modeling the networks whose el-
ements are subject to random failures [5, 15, 17, 18, 27, 23, 20]. As a rule, the
network reliability is defined as some connectivity measure. The most common
reliability measure of such networks is the probability that all terminal nodes
in a network can keep being connected together, given the reliability of each
network node and edge. The problem of calculation of the network probabilistic
connectivity is known to be NP-hard [5]. Nevertheless, it is possible to do the
exact calculation of reliability for networks with a dimension of practical interest
by taking into consideration some special features of real network structures and
based on modern supercomputers [15, 17, 18].
Another well-accepted network reliability measure is the 2-terminal proba-
bilistic connectivity [27]. The average pairwise connectivity [23, 20] is the aver-
On a Problem of the Utility Network Design 391
Fig. 1. Numerical results: the normalized costs for the 10x10 grid, the number of edges
from 10 up to 800.
Fig. 2. Numerical results: the normalized costs for the 10x10 grid, the number of edges
from 10 up to 1000.
392 G. Y. Toktoshov et al.
age between all 2-terminal probabilistic connectivities of the network. The later
parameter demonstrates the network reliability from the point of connectivity
between a pair of nodes even if a network is disconnected.
Let us assume that branches (edges of the primary network) are subject to
random failures that occur statistically independently with known probabilities
pi , 1 ≤ i ≤ g.
The reliability of an edge r ∈ R we define as
Y
Rr (HN ) = p(v). (2)
v∈F (r)
If for an edge r ∈ R the path F (r) has the endpoints a and b, we use the
notation Rab (HN ) instead Rr (HN ). If for the nodes a and b there are more
than one of such edges, we choose one with a maximum reliability value.
We consider the following reliability measure R(HN ), taking into account
the fact that failures occur in the primary network, and all consumers should be
connected with the corresponding sources:
R(HN ) = min{Rab (HN )}, a ∈ Ysource , b ∈ Yconsumer . (3)
Therefore, R(HN ) is the minimum among all 2-terminal probabilistic connec-
tivities Rab (HN ), where b is a consumer and a is a corresponding source.
Based on the two-stage algorithm, we propose a new algorithm with a reliabil-
ity constraint. It is assumed that we are given a reliability threshold 0 < R0 ≤ 1.
The objective is to find a sufficiently reliable solution of problem (1). In other
words, we find a solution HS of problem (1) such that R(HS) ≥ R0 .
7 Ant Colony Algorithm for the Reliable Utility Network
Design
In the case of the reliability constraint, we use the Ant Colony Algorithm [6]
instead of the Floyd Algorithm in the two-stage algorithm. For each edge, we
create l ants acting according to the following rules:
– The probability of an ant moving to the next vertex at the iteration of t is
defined as:
α β
τij νij
Pij (t) = P
ανβ
τij ij
– Every ant li (0 < i ≤ l) has TimeToLive label (T T L(li ))
Y
T T L(li ) = p(v),
v∈P ath(li )
where P ath(li ) is the path of the ant li . If T T L(li ) < R0 , then ant li dies.
– After finding an edge, each ant lays the pheromone on a branch. ∆τij (t) =
Q/length(t)
– In a set of the most frequent routes we find the minimum and fix it, so we
decrease the cost of branches of the primary network.
On a Problem of the Utility Network Design 393
8 Numerical Results for the Reliable Utility Network
Design
Below in Figure 3 we present the numerical results of the algorithm proposed
for pi = 0.99, 1 ≤ i ≤ g, R0 = 0.9. For the primary network P N we consider the
10x10 grid (|X| = 100), α := 1; β := 3; τ0 := 1; Q := 50.
The number of edges was from 10 up to 100 in W N (the abscissae axis). Axis
of ordinate is cost Q(HN ). The first and the second algorithms are the Floyd
and the Greedy respectively, without constraint (2). Therefore, the found value
Q(HN ) is less than the result of FloydProb and AntColony (FloydProb is the
Floyd algorithm with constraint (2)). Figure 2 shows that the AntColony results
are better than the FloydProb results only for a small number of edges.
Fig. 3. Numerical results: costs for 10x10 grid, the number of edges from 10 up to 100
9 Conclusion
The new approach to the utility network design has been studied. Based on the
hypernet theory, we consider a natural-technical system of a land plot and a
utility network within the framework of the unified mathematical model. As a
result, we propose the new method to provide the connectivity between given
consumers and corresponding sources with minimization of laying and main-
tenance costs and within given constraints. This method gives an appropriate
solution with allowance for the compatibility of different types of resources for
placing them in the same track.
For ensuring the reliability of a designed utility network, we, also, introduce
the method for obtaining not only a cheap solution, but a reliable enough one
394 G. Y. Toktoshov et al.
as well assuming that failures occur in a primary (physical) network. As a re-
liability measure we consider the minimum among all 2-terminal probabilistic
connectivities between each consumer and corresponding sources.
The conducted numerical experiments which show the applicability of the
methods proposed are presented.
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