**2. Theoretical investigation in Pareto optimization**

As far as the most general case is concerned, the system can be thought of as an ordered set of elements, relationships and their properties. The uniqueness of their assignment serves to define the system fully, notably, its structure and efficiency. The major objective of designing is to specify and define all the above-listed categories. The solution of this problem involves determining an initial set of solutions, generating a subset of pemissible solutions, assigning the criteria of the system optimality and selecting the system, which is optimal in terms of a criteria.

#### **2.1 The problem statement in optimization system**

It is assumed that the system φ = (s,β)∈Φ<sup>D</sup> is defined by the structure s (a set of elements and connections) and by the vector of parameters . β A set of input actions X <sup>а</sup>nd output results Y should be assigned for an information system. This procedure defines the system as the mapping ϕ → :X Y . The abstract determination of the system in the process of designing is considered to be exact. In particular, when formalizing the problem statement, a mathematical descripton of the working conditions (of signals, interferences) and of the functional purpose of a system (solutions obtained at the system output) are to be given, which, in fact, determine the variant of the system . ϕ∈Φ

In particular, the limitations given on conditions of work, on the structure s S ∈ D and parameters β∈Β<sup>D</sup> , as well as on values of the system quality indicators define the subset of permissible project solutions aaa Φ = ×Β S . Diverse ways of assigning a set of allowable are possible, in particular:


on a totality of given metrics describing the messages transmission quality. Thus, the formed set of the permissible design decisions is represented in the space of criteria ratings of quality indicators where, used of unconditional criteria of a preference, the subset of effective (Pareto-optimal) variants of the telecommunication network is selected. On a final stage of optimization any obtained effective variants of the network can be selected for usage. The unique variant choice of a telecommunication network with introducing some

In the present work some generalizations are made and all stages of solving multicriteria problems are analyzed with reference to telecommunication networks including the statement of a problem, finding the Pareto-optimal systems and selecting the only system variant. This chapter also considers the application particularities of multicriteria optimization methods at the operating control within telecommunication systems. The investigation results are provided on the example of solving of a particular management problem considering planning of cellular networks, optimal routing and choice of the

As far as the most general case is concerned, the system can be thought of as an ordered set of elements, relationships and their properties. The uniqueness of their assignment serves to define the system fully, notably, its structure and efficiency. The major objective of designing is to specify and define all the above-listed categories. The solution of this problem involves determining an initial set of solutions, generating a subset of pemissible solutions, assigning the criteria of the system optimality and selecting the system, which is

results Y should be assigned for an information system. This procedure defines the system as the mapping ϕ → :X Y . The abstract determination of the system in the process of designing is considered to be exact. In particular, when formalizing the problem statement, a mathematical descripton of the working conditions (of signals, interferences) and of the functional purpose of a system (solutions obtained at the system output) are to be given,

In particular, the limitations given on conditions of work, on the structure s S ∈ D and parameters β∈Β<sup>D</sup> , as well as on values of the system quality indicators define the subset of permissible project solutions aaa Φ = ×Β S . Diverse ways of assigning a set of allowable are



is defined by the structure s (a set of elements

A set of input actions X <sup>а</sup>nd output

conventional criteria of preference as some scalar goal function is also possible.

speech codec, controlling network resources, etc.

optimal in terms of a criteria.

possible, in particular:

rigorous mathematical form;

**2. Theoretical investigation in Pareto optimization** 

**2.1 The problem statement in optimization system** 

and connections) and by the vector of parameters . β

which, in fact, determine the variant of the system . ϕ∈Φ


It is assumed that the system φ = (s,β)∈Φ<sup>D</sup>

The choice of the optimal criteria is related to the formalization of the knowledge about an optimality. There exist two ways of describing the customer's preference of one variant to the other, i.e. ordinal and cardinal .

An ordinal approach is order-oriented (better-worse) and is based on introducing certain binary relations on a set of permissible alternatives. In this case the customer's preference is the binary relation R on the set ΦD which reflects the customer's knowledge that the alternative ϕ′ is better than the alternative: ϕ ϕϕ ′′ ′ ′′ : R.

Assume that a customer sticks to a certain rigorous preference , which is asymmetric and transitive, as he decides on a set of permissible alternative ΦD. The solution ϕ0 D ∈ Φ is called optimal with respect to , unless there are other solution ϕ∈ ΦD for which (0) ϕ ϕ holds true. A set of all optimal solutions in relation to is denoted by opt . ΦD A set of optimal solutions can comprise the only element, a finite or infinite number of elements as a function of the structure of a permissible set or properties of the relation . If the discernibility relation coincides with that of equality =, then the set optΦD (provided it is not empty) contains the only element.

A cardinal approach to describe the customer's preference assigns to each alternative ϕ∈ Φ<sup>D</sup> , a certain number U being interpreted as the utility of the alternative ϕ. Each utility function determines a corresponding order (or a preference) R on die set ΦD(R) ϕ ϕ ′ ′ if and only if U( ) U( ). ϕ′ ′′ ≥ ϕ In this case they say that the utility function U( )⋅ is a preference indicator R. In point of fact this approach is related to assigning a certain scalar-objective function (a conventional preference criteria) whose optimization in a general case may result in the selection of the only optimal variant of the system.

The choice of the optimal criteria is based on formalizing the knowledge of a die system customer (i.e. a person who makes a decision) about its optimality. However, one often fails to formalize the knowledge of a decision-making person about the system optimality rigorously. Therefore, it appears impossible to assign the implicitly of the scalar optimal criteria resulting in the choice of the only decision variant [ ] D (0) extr U( ) ϕ∈Φ ϕ = ϕ , where U( ) ϕ is

a certain objective function of the system utility (or usefulness). Therefore, at the initial design stages the system is characterized by a set of objective functions:

$$\vec{\mathbf{k}}(\boldsymbol{\Phi}) = (\mathbf{k}\_1(\boldsymbol{\Phi}), \dots, \mathbf{k}\_i(\boldsymbol{\Phi}), \dots, \mathbf{k}\_m(\boldsymbol{\Phi})),\tag{1}$$

which determines the influence of the structure *s* and the parameters β of the variant of the system ϕ= β (s, ) upon the system quality indicators. In this connection one has to deal with the newly emerged issues of optimizing approaches in terms of a collection of quality indicators, which likewise are called the problems of multicriteria or vector optimization. Basically, the statement and the solution of a multicriteria problems is related to replacing (approximation) customer's knowledge about the system optimality with a different optimality conception which can be formalized as a certain vector optimal criteria (1) and, consequently, the problem will be solved through the effective optimization procedure.

Multicriteria Optimization in Telecommunication Networks Planning, Designing and Controlling 255

When generating a set of permissible variants ΦD one has to allow for the constraints upon the structure, parameters and technical realization of elements and the system as a whole as well as for the permissible combination of elements connections and constraints up on the

Here, there exist conflicting requirements. On the one hand, it is desirable to present all conceivable variants of the system in their entirety so as not to leave out the potentially best variants. On the other hand, there are limitations specified by the permissible expenditures

After a set of permissible variant of a system has been determined in terms of a particular structure, the value of the quality indicators is estimated, a set of Pareto-optimal variants is

As a collection of objective functions is being introduced, each variant of the system ϕ is mapped from a set of permissible variants ΦD into the criteria space of estimates <sup>m</sup> V R ∈ :

<sup>m</sup> V K( ) {v R |v k( ), }. = Φ = ∈ = ϕ ϕ∈Φ D D

In this case to each approach ϕ corresponds its particular estimate of the selected quality

To the relation of the rigorous preference on the set ΦD corresponds the relation in the criteria space of estimates V. According to the Pareto axiom, for any two estimates <sup>ν</sup>, V ν ∈′′ satisfying the vector inequality ν′ ′′ ≥ ν , the relation ν′ ′′ <sup>ν</sup> is always obeyed. Besides, according to the second Pareto action for any two approaches , <sup>D</sup> ϕ ϕ′ ′′∈Φ , for which k( ) k( ) ϕ≥ ϕ ′ ′′ is true, the relation ϕ ϕ ′ ′′ always occurs. The Pareto axiom imposes

It is desirable for a customer to obtain the best possible value for each criteria. Yet in practice this case can be rarely found. Here, it should be emphasized that the quality indicators (objective function) of the system (1) may be of 3 types: neutral, consistent with one another and competing between one other. In the first two instances the system optimization can be performed separately in terms of each of indicators. In the third instance it appears impossible to arrive at a potential value of each of the individual indicators. In this case one can only attain the consistent optimum of introduced objective functions – the optimum according to the Pareto criteria which implies that each of the indicators can be further improved solly by lowering the remaining quality indicators of the system. To the Pareto optimum in the criteria space corresponds a set of Pareto-optimal estimates that satisfy the following expression:

0 m <sup>0</sup> P(V) opt V {k( ) R | k( ) V : k( ) k( )}. = = ϕ ∈ ∀ ϕ∈ ϕ≥ ϕ <sup>≥</sup>

An optimum based on the Pareto criteria can be found either directly according to (3) by the exhaustive search of all permissible variants of the system ΦD or with the use of special

procedures such as the weighting method, methods of operating characteristics.

definite limitations upon the character of the preference in multicriteria problem.

(2) and, vice versa, to each estimate corresponds an approach (in a

(2)

(3)

value of the quality indicators of the system as a whole.

distinguished and gets narrowed down to the most preferable one.

(of time and funds) on the designing of a system.

**2.3 Finding the system Pareto-optimal variants** 

general way, a single approach is not obligatory).

indicators k( ) ν = ϕ

#### **2.2 Forming a set of permissible variants of a system**

When optimizing the information systems, as their decomposition into subsystems can be assigned, it would be judicious to proceed from the morphological approach which is widely applied in designing complicated systems. In this context it is assumed that any variant of a system has a definite structure, i.e. it consists of the finite number of elements (subsystems), and the distribution of system functions amongst them can be performed by the finite number of methods.

Now consider the peculiar features of generating the structural set of permissible variants of a system. Let us assume that the functional decomposition of the system into a set of elements is

$$\{\mathfrak{op}\_{\mathbf{j}'} \colon \mathbf{j} = \overline{1, L}\_{\prime} \bigcup\_{\mathbf{j}=\mathbf{1}}^{L} \mathfrak{q}\_{\mathbf{j}} = \mathfrak{q}\}.$$

What is considered to be assigned is as follows: a finite set of elements of the system E as well as the splitting of the set E into L morphological classes σ = (l), l 1,L such as σ ∩σ =∅ (l) (l )′ at l l . ≠ ′

A concept of the morphological space 2<sup>ε</sup> Λ ⊆ is introduced, its elements being the morphological variant of the system 12 L ϕ = ( , , , ). ϕϕ ϕ Each morphological variant ϕ is a certain set of representatives of the classes ϕ ∈σ (l) (l). Here for all ϕ∈ Λ and for any l 1,L = the set ϕ∈ Λ contains a single element.

Under the assumption that there exist a multitude of alternative model of implementing each subsystem l k ϕ= = , k 1,L, l 1,L , the following morphological table can be specified:


Table 1. Morphological table.

As an example (see table 1), a q -th morphological variant of the system 2 <sup>q</sup> <sup>φ</sup> <sup>=</sup> <sup>φ</sup>12 2K l3 L1 ,<sup>φ</sup> ,…,<sup>φ</sup> ,…,φ that determines the system structure is distinguished. The total number of all possible morphological variants of the system is generally determined as L l l 1 Q K = <sup>=</sup> ∏ .

When generating a set of permissible variants ΦD one has to allow for the constraints upon the structure, parameters and technical realization of elements and the system as a whole as well as for the permissible combination of elements connections and constraints up on the value of the quality indicators of the system as a whole.

Here, there exist conflicting requirements. On the one hand, it is desirable to present all conceivable variants of the system in their entirety so as not to leave out the potentially best variants. On the other hand, there are limitations specified by the permissible expenditures (of time and funds) on the designing of a system.

After a set of permissible variant of a system has been determined in terms of a particular structure, the value of the quality indicators is estimated, a set of Pareto-optimal variants is distinguished and gets narrowed down to the most preferable one.

#### **2.3 Finding the system Pareto-optimal variants**

254 Telecommunications Networks – Current Status and Future Trends

When optimizing the information systems, as their decomposition into subsystems can be assigned, it would be judicious to proceed from the morphological approach which is widely applied in designing complicated systems. In this context it is assumed that any variant of a system has a definite structure, i.e. it consists of the finite number of elements (subsystems), and the distribution of system functions amongst them can be performed by

Now consider the peculiar features of generating the structural set of permissible variants of a system. Let us assume that the functional decomposition of the system into a set of

L

j j j 1 { , j 1,L, }. = <sup>ϕ</sup> <sup>=</sup> <sup>ϕ</sup> <sup>=</sup> <sup>ϕ</sup>

What is considered to be assigned is as follows: a finite set of elements of the system E as well as the splitting of the set E into L morphological classes σ = (l), l 1,L such as

A concept of the morphological space 2<sup>ε</sup> Λ ⊆ is introduced, its elements being the morphological variant of the system 12 L ϕ = ( , , , ). ϕϕ ϕ Each morphological variant ϕ is a certain set of representatives of the classes ϕ ∈σ (l) (l). Here for all ϕ∈ Λ and for any l 1,L =

Under the assumption that there exist a multitude of alternative model of implementing each subsystem l k ϕ= = , k 1,L, l 1,L , the following morphological table can be specified:

> Number of modes of implementing the system

Possible models of implementing the system elements

σ(1) <sup>1</sup> 11 12 13 1K ϕϕϕ ϕ [ ] K1 σ(2) <sup>2</sup> 21 22 13 2K ϕϕϕ ϕ [ ] K2 ……… ………………………… ………

σ(l) <sup>l</sup> l1 l2 l3 lK ϕϕ ϕ ϕ [ ] Kl ……… ………………………… ……… σ(L) L1 L2 L3 LKL [ ] ϕ ϕϕ ϕ KL

As an example (see table 1), a q -th morphological variant of the system

<sup>q</sup> <sup>φ</sup> <sup>=</sup> <sup>φ</sup>12 2K l3 L1 ,<sup>φ</sup> ,…,<sup>φ</sup> ,…,φ that determines the system structure is distinguished. The total number of all possible morphological variants of the system is generally determined as

**2.2 Forming a set of permissible variants of a system** 

the finite number of methods.

σ ∩σ =∅ (l) (l )′ at l l . ≠ ′

Morphological classes

Table 1. Morphological table.

2

L l l 1 Q K = <sup>=</sup> ∏ .

the set ϕ∈ Λ contains a single element.

elements is

As a collection of objective functions is being introduced, each variant of the system ϕ is mapped from a set of permissible variants ΦD into the criteria space of estimates <sup>m</sup> V R ∈ :

$$\mathbf{V} = \vec{\mathbf{K}}(\Phi\_{\rm D}) = \{ \vec{\mathbf{v}} \in \mathbb{R}^{m} \mid \vec{\mathbf{v}} = \vec{\mathbf{k}}(\boldsymbol{\upphi}), \,\boldsymbol{\upphi} \in \Phi\_{\rm D} \}. \tag{2}$$

In this case to each approach ϕ corresponds its particular estimate of the selected quality indicators k( ) ν = ϕ (2) and, vice versa, to each estimate corresponds an approach (in a general way, a single approach is not obligatory).

To the relation of the rigorous preference on the set ΦD corresponds the relation in the criteria space of estimates V. According to the Pareto axiom, for any two estimates <sup>ν</sup>, V ν ∈′′ satisfying the vector inequality ν′ ′′ ≥ ν , the relation ν′ ′′ <sup>ν</sup> is always obeyed. Besides, according to the second Pareto action for any two approaches , <sup>D</sup> ϕ ϕ′ ′′∈Φ , for which k( ) k( ) ϕ≥ ϕ ′ ′′ is true, the relation ϕ ϕ ′ ′′ always occurs. The Pareto axiom imposes definite limitations upon the character of the preference in multicriteria problem.

It is desirable for a customer to obtain the best possible value for each criteria. Yet in practice this case can be rarely found. Here, it should be emphasized that the quality indicators (objective function) of the system (1) may be of 3 types: neutral, consistent with one another and competing between one other. In the first two instances the system optimization can be performed separately in terms of each of indicators. In the third instance it appears impossible to arrive at a potential value of each of the individual indicators. In this case one can only attain the consistent optimum of introduced objective functions – the optimum according to the Pareto criteria which implies that each of the indicators can be further improved solly by lowering the remaining quality indicators of the system. To the Pareto optimum in the criteria space corresponds a set of Pareto-optimal estimates that satisfy the following expression:

$$\mathbf{P}(\mathbf{V}) = \mathbf{opt}\_{\geq} \mathbf{V} = \{ \vec{\mathbf{k}}(\boldsymbol{\upphi}^{0}) \in \mathbf{R}^{m} \mid \forall \vec{\mathbf{k}}(\boldsymbol{\upphi}) \in \mathbf{V} \colon \vec{\mathbf{k}}(\boldsymbol{\upphi}) \geq \vec{\mathbf{k}}(\boldsymbol{\upphi}^{0}) \}. \tag{3}$$

An optimum based on the Pareto criteria can be found either directly according to (3) by the exhaustive search of all permissible variants of the system ΦD or with the use of special procedures such as the weighting method, methods of operating characteristics.

Multicriteria Optimization in Telecommunication Networks Planning, Designing and Controlling 257

the Pareto-optimal surface in the criteria space being obtained, the multidimensional potential characteristics of the system and related multidimensional exchange diagram are

It should be noted that they are different types of optimization problems depending upon

*Discrete selection.* The initial set ΦD is specified by a finite number of variants of constructing the system { , l 1,L , }. ϕ = ϕ∈Φ l D D It is required that set of Pareto-optimal

*Parametric optimization.* The structure of the system SD is specified. It is necessary to find the

*Structural-parametric optimization.* It is necessary to synthesize the structure s S ∈ D and to

The first two types of problems have been adequately developed in the theoiy of multicriteria optimization. The solution of the third-type problems is most complicated. To synthesize the Pareto-optimal structure and find the optimal parameters a set of functionals

case appears to be a rather challenging task from both the mathematical and some no less importants standpoints. In the case of a vector the solution to these types of problems becomes still more complicated. Therefore, in designing the systems with regard to a set of the quality indicators one has to simplify the optimization problem by decomposing the system into simpler subsystems, to reduce the number of quality indicators as the system

If the set of Pareto-optimal systems variants, which has been found following the optimization procedure, turned out to be a narrow one, then any of them can be made use of as an optimal one. In this case the rigorous preference relation may be thought of as

However, in practice the set P(V) proves to be sufficiently wide. This implies that the relations and ≥ (although they are connected through the Pareto axiom) do not show a close agreement. Here, the inclusions opt V P(V) ⊂ and D D <sup>k</sup> opt P ( ) Φ⊂ Φ are valid. Therefore, we will have to deal with an emerging problem of narrowing the found Pareto-optimal solutions involving additional information about the relation of the customer's rigorous preference. Yet the ultimate selection of optimal approaches should

is to be optimized. Yet optimizing functionals even in a scalar

at which ϕ= β ∈ Φ (s, ) opt . <sup>D</sup>

at which ϕ= β ∈ Φ (s, ) opt . <sup>D</sup>

found.

the problem statement.

magnitude of the vectors <sup>0</sup> β ∈BD

k (s, ), k (s, ),..., k (s, ) 12 m ββ β

structure is being synthesized.

**system** 

respect to a precise variant.

variants of the system opt . ΦD should be selected.

find the magnitude of the vector of the parameters β∈BD

coinciding with the relation ≥ and, therefore, opt V P(V). =

only be made within the limits of the found set of Pareto-optimal solution.

**2.4 Narrowing of the set of Pareto-optimal solutions down to the only variant of a** 

The formal model of the Pareto optimization problem does not contain any information to select the only alternative. In this particular instance a set of permissible variants gets narrowed only to a set of Pareto-optimal solution by eliminating the worse variants with

With the Pareto *weighting method* being employed. The optimal decisions are found by optimizing the weighted sum of objective functions

$$\mathop{\rm extr}\_{\mathfrak{q}\mathfrak{e}\mathfrak{e}\mathfrak{G}\_{\text{D}}} \{ \mathbf{k}\_{\text{p}}(\mathfrak{q}) = \lambda\_{1}\mathbf{k}\_{1}(\mathfrak{q}) + \lambda\_{2}\mathbf{k}\_{2}(\mathfrak{q}) + \dots + \lambda\_{m}\mathbf{k}\_{m}(\mathfrak{q}) \} \, \tag{4}$$

in which the weighting coefficients λλ λ 12 m ,,, are selected from the condition m i i i 1 0, 1 = λ> λ= . The Pareto-optimal decisions are the system variants that satisfy eq. (4) with different permissible combination of the weighting coefficients λλ λ 12 m , ,, . When solving this problem one can observe the variation in the alternative systems ϕ = (s, ) β ∈ Φ<sup>D</sup> within the limits of specified.

*The method of operating characteristics* consists all the objective functions, except for a single one, say, the first one, are transferred into a category of limitations of an inequality type, and its optimum is sought on a set of permissible alternatives

$$\mathop{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\cdot}}}}}}}}}}}}}}}}}}}}}}}}}}} \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\text{\text{\text{\mathrm{\text{\mathrm{\text{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\bullet}}}}}}}}}}}}}{}}} }} \mathrm{\mathrm{\cdot}} \mathrm{\mathrm{\text{\text{\text{\text{\text{\text{\text{\cdot}}}}}}}}} }} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} } \mathrm{\cdot} } \mathrm{\cdot} \mathrm{\cdot} } \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} } \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} } \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} } \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} } \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} } \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} \mathrm{\cdot} } \mathrm{\$$

Here K ,K ,...,K 23 m ϕϕ ϕ are the certain fixed, but arbitrary quality indicators values.

The optimization problem (5) is solved sequentially for all permissible combinations of the values K K , K K ,..., K K . 2 2D 3 3D m mD ϕϕ ϕ ≤≤ ≤ In each instance an optimal value of the indicator k is sought by variations 1opt ϕ∈ ΦD. As a result a certain multidimensional working space in the criteria space is sought

$$\mathbf{k}\_{1\text{opt}} = \mathbf{f}\_{\text{p}}(\mathbf{K}\_{2\text{p}}, \mathbf{K}\_{3\text{p}}, \dots, \mathbf{K}\_{\text{m}\text{p}}).\tag{6}$$

If the found relation (6) is monotonously decreasing in nature for each of the arguments, the working surface coincides with a Pareto-optimal surface. This surface can be connected, nonconnected and just a set of isolated points.

It should be pointed out that each point of the pareto-optimal surface offers the property of a m -fold optimum, i.e. this point checks with a potentially attainable (with variation ϕ∈ Φ<sup>D</sup> ) value of one of the indicators k at the fixed (corresponding to this point) value 1opt of other ( m 1 − ) quality indicators. The Pareto-optimal surface can be described by any of the following relationships

$$\mathbf{k}\_{1\text{opt}} = \mathbf{f}\_{\text{no}}^1(\mathbf{k}\_2, \mathbf{k}\_3, \dots, \mathbf{k}\_{\text{m}}), \dots, \mathbf{k}\_{\text{mopt}} = \mathbf{f}\_{\text{no}}^{\text{m}}(\mathbf{k}\_1, \mathbf{k}\_2, \dots, \mathbf{k}\_{\text{m}-1}), \tag{7}$$

which represent the multidimensional diagram of the exchange between the quality indicators showing the way in which the potentially attainable value of the corresponding indicator depends upon the values of other indicators.

Thus, the Pareto-optimal surface connects the potentially attainable values of index is Paretooptimum consistent, generally dependent and competing quality indicators Therefore, with

With the Pareto *weighting method* being employed. The optimal decisions are found by

extr[k ( ) k ( ) k ( ) ... k ( )], p 11 22 m m ϕ∈Φ

in which the weighting coefficients λλ λ 12 m ,,, are selected from the condition

λ> λ= . The Pareto-optimal decisions are the system variants that satisfy eq. (4)

with different permissible combination of the weighting coefficients λλ λ 12 m , ,, . When solving this problem one can observe the variation in the alternative systems ϕ = (s, ) β ∈ Φ<sup>D</sup>

*The method of operating characteristics* consists all the objective functions, except for a single one, say, the first one, are transferred into a category of limitations of an inequality type, and

extr[k ( )], k ( ) K ; k ( ) K ,..., k ( ) K . 1 2 23 3 m m ϕϕ ϕ ϕ∈Φ

The optimization problem (5) is solved sequentially for all permissible combinations of the values K K , K K ,..., K K . 2 2D 3 3D m mD ϕϕ ϕ ≤≤ ≤ In each instance an optimal value of the indicator k is sought by variations 1opt ϕ∈ ΦD. As a result a certain multidimensional

If the found relation (6) is monotonously decreasing in nature for each of the arguments, the working surface coincides with a Pareto-optimal surface. This surface can be connected,

It should be pointed out that each point of the pareto-optimal surface offers the property of a m -fold optimum, i.e. this point checks with a potentially attainable (with variation ϕ∈ Φ<sup>D</sup> ) value of one of the indicators k at the fixed (corresponding to this point) value 1opt of other ( m 1 − ) quality indicators. The Pareto-optimal surface can be described by any of

which represent the multidimensional diagram of the exchange between the quality indicators showing the way in which the potentially attainable value of the corresponding

Thus, the Pareto-optimal surface connects the potentially attainable values of index is Paretooptimum consistent, generally dependent and competing quality indicators Therefore, with

Here K ,K ,...,K 23 m ϕϕ ϕ are the certain fixed, but arbitrary quality indicators values.

ϕ = λ ϕ + λ ϕ + +λ ϕ (4)

ϕ ϕ= ϕ= ϕ= (5)

k f (K ,K ,...,K ). 1opt p 2 3 m = ϕϕ ϕ (6)

1 m k f (k ,k ,...,k ),...,k f (k ,k ,...,k ), 1opt no 2 3 m mopt no 1 2 m 1 = = <sup>−</sup> (7)

optimizing the weighted sum of objective functions

D

its optimum is sought on a set of permissible alternatives

D

working space in the criteria space is sought

nonconnected and just a set of isolated points.

indicator depends upon the values of other indicators.

the following relationships

m i i i 1 0, 1 =

within the limits of specified.

the Pareto-optimal surface in the criteria space being obtained, the multidimensional potential characteristics of the system and related multidimensional exchange diagram are found.

It should be noted that they are different types of optimization problems depending upon the problem statement.

*Discrete selection.* The initial set ΦD is specified by a finite number of variants of constructing the system { , l 1,L , }. ϕ = ϕ∈Φ l D D It is required that set of Pareto-optimal variants of the system opt . ΦD should be selected.

*Parametric optimization.* The structure of the system SD is specified. It is necessary to find the magnitude of the vectors <sup>0</sup> β ∈BD at which ϕ= β ∈ Φ (s, ) opt . <sup>D</sup> 

*Structural-parametric optimization.* It is necessary to synthesize the structure s S ∈ D and to find the magnitude of the vector of the parameters β∈BD at which ϕ= β ∈ Φ (s, ) opt . <sup>D</sup> 

The first two types of problems have been adequately developed in the theoiy of multicriteria optimization. The solution of the third-type problems is most complicated. To synthesize the Pareto-optimal structure and find the optimal parameters a set of functionals k (s, ), k (s, ),..., k (s, ) 12 m ββ β is to be optimized. Yet optimizing functionals even in a scalar case appears to be a rather challenging task from both the mathematical and some no less importants standpoints. In the case of a vector the solution to these types of problems becomes still more complicated. Therefore, in designing the systems with regard to a set of the quality indicators one has to simplify the optimization problem by decomposing the system into simpler subsystems, to reduce the number of quality indicators as the system structure is being synthesized.

If the set of Pareto-optimal systems variants, which has been found following the optimization procedure, turned out to be a narrow one, then any of them can be made use of as an optimal one. In this case the rigorous preference relation may be thought of as coinciding with the relation ≥ and, therefore, opt V P(V). =

However, in practice the set P(V) proves to be sufficiently wide. This implies that the relations and ≥ (although they are connected through the Pareto axiom) do not show a close agreement. Here, the inclusions opt V P(V) ⊂ and D D <sup>k</sup> opt P ( ) Φ⊂ Φ are valid. Therefore, we will have to deal with an emerging problem of narrowing the found Pareto-optimal solutions involving additional information about the relation of the customer's rigorous preference. Yet the ultimate selection of optimal approaches should only be made within the limits of the found set of Pareto-optimal solution.

#### **2.4 Narrowing of the set of Pareto-optimal solutions down to the only variant of a system**

The formal model of the Pareto optimization problem does not contain any information to select the only alternative. In this particular instance a set of permissible variants gets narrowed only to a set of Pareto-optimal solution by eliminating the worse variants with respect to a precise variant.

Multicriteria Optimization in Telecommunication Networks Planning, Designing and Controlling 259

Let X be a certain set of possible magnitudes of a particular quality indicator of a system. The fuzzy set G on the set X is assigned by the membership function ξ<sup>G</sup> : X [0,1] → which brings the real number ξG over the interval [0,1] in line with each element of the set X . The value ξG defined the degree of membership of the set X elements to the fuzzy set G . The nearer is the value ξG(x) to unity, the higher is the membership degree. The membership function ξG(x) is the generalization of the characteristic function of sets, which takes two values only : 1 – at x G∈ ; 0 – at x G∉ . For discrete sets X the fuzzy set G is

Thus, according to the theory of fuzzy sets each of the quality indicators can be assigned in

<sup>j</sup> k {k , (k )}, j j = <sup>k</sup> <sup>j</sup> ξ

where kj ξ ( ) is the membership function of the specific value of the j -th index to the

This type of writing is highly informative, since it gives an insight into its physical meaning and "worth" in relation to the optimal (extreme) value which is characterized by the

The main difficulty over the practical implementation of the considered approach consists in choosing the type of a membership function. In some sense the universal form of the membership function being interpreted in terms of the theory of fuzzy sets with regard to

k 12 m k j

ξ =ξ

<sup>1</sup> (k ,k ,...,k ) [ (k )] . m

The advantage of this form is that depending upon the parameter β a wide class of functions is implemented. These functions range from the linear additive form at 1 β = to

It should be pointed out that with this particular approach it is essential that the information obtained from a customer by an expert estimates method be used to pick out a membership

*Selecting optimal approaches at quality indicators strictly ordered in terms of the level of their importance.* Occasionally it appears desirable for a customer to obtain die maximum magnitude of one of the indicators, say, k1 even at the expense of the "lasses" for the remaining indicators. This means that the indicator k1 is found to be more important than

In addition, there may be the case where the whole set of indicators k ,k , ,k 12 m is strictly ordered in terms of their importance such k1 is more important that other indicators k ,k , ,k 12 m ; k2 is more essential than all the indicators k ,k , ,k 12 m , etc. This corresponds the instance where the lexico-graphical relation lex is employed when a comparison is made

between the estimates of approaches. Now we give the definition of the above relation.

j

m

l 1

=

1

<sup>β</sup> <sup>β</sup>

(9)

written as the set of pairs G x, (x) = ξ { <sup>G</sup> } .

the form of a fuzzy set

optimal magnitude.

membership function kj ξ ( ) .

the collection of indicators is written as:

function and a variety of coefficients.

other indicators.

the particularly nonlinear relationships at β→∞ .

However, the only variant of a system is normally to be chosen to ensure the subsequent designing stages. It is just for this reason why one feels it necessary to narrow the set of Pareto-optimal solutions down to the only variant of a system and to make use of some additional information about a customer's preference. This type of information is produced following the comprehensive analysis of Pareto-optimal variants of a system, particularly, of a structure, parameters, operating characteristics of the obtained variants of a system, a relative importance of input quality indicators, etc. Some additional information thus obtained concerning the customer's preferences is employed to construct choice function (an objective scalar function) whose optimization tends to select the sole variants of a system.

In order to solve the problem of narrowing a set of Pareto-optimal solution a diversity of approaches, especially those based on the theory of utility, the theory of fuzzy sets, etc. Now let us take a brief look at some of them.

*The selection of optimal approaches using the scalar value function.* One of the commonly used methods of narrowing a set of Pareto-optimal solution is constructing the scalar value function, which, if applied, gives rise to selecting one of the optimal variants of a system.

The numerical function F(v ,v ,...,v ) 12 m of m variables is referred to as the value (utility) function for the relation if for the arbitrary estimates ' '' v,v V<sup>∈</sup> the inequality ' '' F(v ) F(v ) <sup>&</sup>gt; occurs if and only if ' '' v v. If there exists the function of utility F(v) for the relation , then it is obvious that

$$\text{opt}\_{\succ} \mathbf{V} = \{ \vec{\mathbf{v}}^0 \in \mathbf{V} \colon \mathbf{F}(\vec{\mathbf{v}}^0) = \max\_{\vec{\mathbf{v}} \in \mathbf{V}} \mathbf{F}(\vec{\mathbf{v}}) \}$$

and finding an optimal estimate boils down to solving the single-criteria problem of optimizing the function F(v) on the set V. The value function of the type

$$F(\mathbf{v}\_1, \mathbf{v}\_2, \dots, \mathbf{v}\_m) = \sum\_{j=1}^m \mathbf{c}\_j \mathbf{f}\_j(\mathbf{v}\_j)\_\prime \tag{8}$$

where j c is the scaling factor, j j f( ) ν are the certain unidimensional value function which are the estimates of usefulnen of the system variant ϕ in terms of the index k( ) <sup>j</sup> ϕ .

The construction of the value function (8) consists in estimating the scale factors, forming unidimensional utility function j j f( ) ν as well as in validating their independence and consistency. Here, use is made of the data obtained from interrogating a customer. Special interrogation procedures and program packages intended to acquire some additional information about the customer's preferences have been worked out.

*The selection of optimal approaches based upon the theory of fuzzy sets.* This procedure is based on the fact that due to the apriori uncertainty with regard to the customer's preference, the concept such as "the best variant of a system" cannot be accurately defined. This concept may be thought of as constituting a fuzzy set and in order to make an estimate of the system, the basic postulates of the fuzzy- set theory can be employed.

However, the only variant of a system is normally to be chosen to ensure the subsequent designing stages. It is just for this reason why one feels it necessary to narrow the set of Pareto-optimal solutions down to the only variant of a system and to make use of some additional information about a customer's preference. This type of information is produced following the comprehensive analysis of Pareto-optimal variants of a system, particularly, of a structure, parameters, operating characteristics of the obtained variants of a system, a relative importance of input quality indicators, etc. Some additional information thus obtained concerning the customer's preferences is employed to construct choice function (an objective scalar function) whose optimization tends to select the sole

In order to solve the problem of narrowing a set of Pareto-optimal solution a diversity of approaches, especially those based on the theory of utility, the theory of fuzzy sets, etc. Now

*The selection of optimal approaches using the scalar value function.* One of the commonly used methods of narrowing a set of Pareto-optimal solution is constructing the scalar value function, which, if applied, gives rise to selecting one of the optimal variants of a system.

The numerical function F(v ,v ,...,v ) 12 m of m variables is referred to as the value (utility) function for the relation if for the arbitrary estimates ' '' v,v V<sup>∈</sup> the inequality ' '' F(v ) F(v ) <sup>&</sup>gt; occurs if and only if ' '' v v.

0 0

opt V {v V : F(v ) maxF(v)} <sup>∈</sup> =∈ = 

and finding an optimal estimate boils down to solving the single-criteria problem of

12 m jj j j 1

F(v ,v ,...,v ) c f (v ),

where j c is the scaling factor, j j f( ) ν are the certain unidimensional value function which are

The construction of the value function (8) consists in estimating the scale factors, forming unidimensional utility function j j f( ) ν as well as in validating their independence and consistency. Here, use is made of the data obtained from interrogating a customer. Special interrogation procedures and program packages intended to acquire some additional

*The selection of optimal approaches based upon the theory of fuzzy sets.* This procedure is based on the fact that due to the apriori uncertainty with regard to the customer's preference, the concept such as "the best variant of a system" cannot be accurately defined. This concept may be thought of as constituting a fuzzy set and in order to make an estimate of the

m

=

optimizing the function F(v) on the set V. The value function of the type

the estimates of usefulnen of the system variant ϕ in terms of the index k( ) <sup>j</sup> ϕ .

information about the customer's preferences have been worked out.

system, the basic postulates of the fuzzy- set theory can be employed.

If there exists the function of utility F(v) for the

<sup>=</sup> (8)

v V

variants of a system.

let us take a brief look at some of them.

relation , then it is obvious that

Let X be a certain set of possible magnitudes of a particular quality indicator of a system. The fuzzy set G on the set X is assigned by the membership function ξ<sup>G</sup> : X [0,1] → which brings the real number ξG over the interval [0,1] in line with each element of the set X . The value ξG defined the degree of membership of the set X elements to the fuzzy set G . The nearer is the value ξG(x) to unity, the higher is the membership degree. The membership function ξG(x) is the generalization of the characteristic function of sets, which takes two values only : 1 – at x G∈ ; 0 – at x G∉ . For discrete sets X the fuzzy set G is written as the set of pairs G x, (x) = ξ { <sup>G</sup> } .

Thus, according to the theory of fuzzy sets each of the quality indicators can be assigned in the form of a fuzzy set

$$\mathbf{k}\_{\mathbf{j}} = \{ \mathbf{k}\_{\mathbf{j}}, \mathbf{\tilde{\xi}}\_{\mathbf{k}\_{\mathbf{j}}} (\mathbf{k}\_{\mathbf{j}}) \}\_{\mathbf{\tilde{\xi}}}$$

where kj ξ ( ) is the membership function of the specific value of the j -th index to the optimal magnitude.

This type of writing is highly informative, since it gives an insight into its physical meaning and "worth" in relation to the optimal (extreme) value which is characterized by the membership function kj ξ ( ) .

The main difficulty over the practical implementation of the considered approach consists in choosing the type of a membership function. In some sense the universal form of the membership function being interpreted in terms of the theory of fuzzy sets with regard to the collection of indicators is written as:

$$\mathfrak{S}\_{\vec{\mathbf{k}}}(\mathbf{k}\_1, \mathbf{k}\_2, \dots, \mathbf{k}\_m) = \frac{1}{\mathbf{m}} \left\{ \sum\_{l=1}^{\mathbf{m}} \|\mathfrak{S}\_{\mathbf{k}\_j}(\mathbf{k}\_j)\|^{\beta} \right\}^{\frac{1}{\beta}}.\tag{9}$$

The advantage of this form is that depending upon the parameter β a wide class of functions is implemented. These functions range from the linear additive form at 1 β = to the particularly nonlinear relationships at β→∞ .

It should be pointed out that with this particular approach it is essential that the information obtained from a customer by an expert estimates method be used to pick out a membership function and a variety of coefficients.

*Selecting optimal approaches at quality indicators strictly ordered in terms of the level of their importance.* Occasionally it appears desirable for a customer to obtain die maximum magnitude of one of the indicators, say, k1 even at the expense of the "lasses" for the remaining indicators. This means that the indicator k1 is found to be more important than other indicators.

In addition, there may be the case where the whole set of indicators k ,k , ,k 12 m is strictly ordered in terms of their importance such k1 is more important that other indicators k ,k , ,k 12 m ; k2 is more essential than all the indicators k ,k , ,k 12 m , etc. This corresponds the instance where the lexico-graphical relation lex is employed when a comparison is made between the estimates of approaches. Now we give the definition of the above relation.

Multicriteria Optimization in Telecommunication Networks Planning, Designing and Controlling 261

transmission through the communication channels. The procedures of the messages packing

have simulated a batch data transmission with a mode of the window load control.

Fig. 1. Choice of Pareto-optimal variants of the telecommunication network.

channel.

(equal 8).

terminals, alarm installations, etc).

The procedures of a packet transmission were simulated by the processes of transfer using duplex communication channels with errors. The simulation analysis of the transfer delays was stipulated at a packet transmission in the communication lines connected with final velocity of signals propagation in communication channels, fixed transmission channel capacity and packets arrival time in the queue for their transfer trough the communication

Different variants of the telecommunication network functioning were realized at the simulation analysis, they differed in disciplines of service in the queues, ways of routing in a packet transmission and size of the window of the transport junction. In the considered example thirty six variants of the network functioning were obtained. Network functioning variants were estimated by the following quality indicators: average time of deliveries k T <sup>1</sup> = and average probability of message loss k P <sup>2</sup> = . These quality indicators had contradictory character of interconnection. The obtained permissible set of network variants is presented in a criteria space (fig. 1). The subset of the Pareto-optimal network operation is selected by the exclusion of the inferior variants. The left low bound set of the valid variants corresponds to Pareto-optimal variants. Among Pareto-optimal variants of the network Ф<sup>0</sup> was selected a single variant from the condition of a minimum of the introduced resulting quality indicator k Ck Ck pn 1 1 2 2 = + . For the case C 0,4 <sup>1</sup> = , C 0,6 <sup>2</sup> = the single variant 11 was selected; the discipline service of the requests (in the random order) was established for it as well as the way of routing (weight method) and size of the "transmission window"

The given task is urgent for practical applications being critical to the delivery time (in telecommunication systems of video and voice intelligences, systems of the banking

Let there be two vectors of estimates <sup>m</sup> ν ν∈ ⊂ , VR ′ . The lexico-graphical relation lex is determined in the following way: the relation lex occurs if and only if one the following conditions is satisfied.

$$\begin{aligned} \text{1) } & \mathbf{v}\_{1}^{\cdot} > \mathbf{v}\_{1}^{\cdot\cdot}; \\ \text{2) } & \mathbf{v}\_{1}^{\cdot} = \mathbf{v}\_{1}^{\cdot\cdot}; \mathbf{v}\_{2}^{\cdot} > \mathbf{v}\_{2}^{\cdot} \\ \text{3) } & \mathbf{v}\_{j}^{\cdot} = \mathbf{v}\_{j}^{\cdot\cdot}; \mathbf{j} = \mathbf{1}\_{\prime} \mathbf{2}\_{\prime}, \dots, \mathbf{m} - \mathbf{1}\_{\prime} \mathbf{v}\_{m}^{\cdot} > \mathbf{v}\_{m}^{\cdot\cdot} \\ \text{v}^{\cdot} = (\mathbf{v}\_{1}^{\cdot}, \mathbf{v}\_{2}^{\cdot}, \dots, \mathbf{v}\_{m}^{\cdot}); \quad \mathbf{v}^{\cdot} = (\mathbf{v}\_{1}^{\cdot\cdot}, \mathbf{v}\_{2}^{\cdot\cdot}, \dots, \mathbf{v}\_{m}^{\cdot\cdot}). \end{aligned}$$

In this case the components v ,v , ,v 12 m , i.e. the estimates of the system quality indicators k ( ),k ( ),...,k ( ) 12 m ϕϕ ϕ are said to be strictly order in terms of their importance. As the relation ' '' v lexv is satisfied they say that from the lexico-graphical stand point the vector ν′ is greater than the vector . ν′ At m 1 <sup>−</sup> the lexico-graphical relation coincides with the relation on the subset of real numbers.

In determining the lexico-graphical relation a major role is played by the order of enumerating quality indicators. The change in the numeration of quality indicators give rise to a different lexico-graphical relation.
