MODELLING THE MODIFIED METHOD OF ANALYTIC HIERARCHY PROCESS BY MEANS OF CONSTRUCTIVE AND PRODUCTIVE STRUCTURES

ІНФОРМАЦІЙНО

-

КОМУНІКАЦІЙНІ

ТЕХНОЛОГІЇ

ТА

МАТЕМАТИЧНЕ

МОДЕЛЮВАННЯ

УДК

504.81:004.94

T.

М

. VASETSKA

1*

1*_{Dep. «Computer and Information Technologies», Dnipropetrovsk National University of Railway Transport} named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (098) 237 05 21, e-mail [email protected], ORCID 0000-0001-7008-2839

MODELLING THE MODIFIED METHOD OF ANALYTIC HIERARCHY

PROCESS BY MEANS OF CONSTRUCTIVE

AND PRODUCTIVE STRUCTURES

Purpose. In the study it is supposed: 1) to extend the classical method of analytic hierarchy process (AHP) for

a great number of alternatives and criteria; 2) to build a model of constructive decision making process using a modified method of analytic hierarchy process with sorting (AHPS). Methodology. To achieve this purpose the mechanism of constructive and productive structures (CPS) was used; the refining transformations of the general-ized constructive-productive structure (GCPS) were fulfilled. Findings. The developed model of the constructive process is the interaction between the three structures: the general CPS of AHPS, which allows to set criteria and alternatives and performs the decomposition of task hierarchical structure; CPS of grouping and sorting, which di-vides alternatives (criteria) into groups and implements the classic single-level AHP for each group, as well as cal-culates estimates of paired comparisons based on the input data; CPS of single-level classic AHP, which allows to fill the matrix of paired comparisons and calculates the ranks of alternatives. All three structures interact at different levels of transformations: by data conformity at the level of concretization and using of implementations. The pro-posed model allowed moving to the more abstract level in presentation of decision making problem solving for a great number of criteria and alternatives. Originality. The paper proposes to use CPS mechanism for formalizing modifications of AHP with sorting for decision making problem solving with a great number of criteria and alterna-tives. Practical value. The formalization of the presentation of the analytic hierarchy process and its modifications allows extending the range of applications of this method, as well as unifying the description of various AHP modi-fications. Such presentation provides the possibility for developing the programs to implement the method hybrid modifications. Using different interpretations presented in the article of CPS will allow for other approaches in de-termining the coherence of pairwise comparison matrices, estimate calculation and ranks of alternatives and criteria.

Keywords: modelling; constructive and productive structure; constructive process; analytic hierarchy process; modification

Introduction

Analytic Hierarchy Process (AHP) [4, 11], pro-posed by Saaty, received worldwide recognition and is used to solve the decision-making problems in different areas. There are many versions of this method, which take into account the specificity of the tasks, can reduce the existing restrictions on the

[14] presents the modification of AHP with sorting (AHPS) which may be used while ranking a large number of alternatives. The essence of this method is that all alternatives are divided into groups in threes (fours) and for each group the classical AHP is applied. If the position of alterna-tives in groups changes, the rearrangement is per-formed. Some estimates not yet identified by the expert are calculated on the basis of already deter-mined ones at each step. This greatly facilitates the work of an expert.

Purpose

The purpose of this work is to extend the clas-sical AHP for a great number of alternatives and criteria. To do this, it is proposed to present the AHPS-based constructive decision-making process by the constructive and productive structures (CPS) [6]. In [8] CPS tools formalize the alterna-tives ranking process using the classical AHP.

To represent AHPS there was developed a system of three interacting CPS: directly AHPS, grouping and sorting CPS and CPS of single-level classical AHP.

Methodology

To achieve this purpose, the mechanism of con-structive and productive structures is used. CPS is a powerful device for formalization and modelling of processes [5–8]. By performing different trans-formations of the generalized constructive and productive structure (GCPS) [6], namely, speciali-zation, interpretation, specification and implemen-tation, the different models are developed [7]. GCPS is called a triple [6]:

G , ,

C = 〈M Σ Λ〉,

where M – heterogeneous structure medium Σ – signature; consisting of sets of the binding tions, substitution and output operations; opera-tions on attributes and substitutive relaopera-tions; Λ – constructive axiomatics [6].

CPS purpose is to form the sets of structures using binding, substitution and other operations defined by axiomatic rules.

Findings

This paper presents a modified AHPS model [14] on the basis of CPS with unconstrained num-ber of criteria and alternatives.

All three CPS interact at the specification level: data coherence connection and at the implementa-tion level: CPS AHPS uses implementaimplementa-tion of grouping and sorting CPS for the criteria and for a set of alternatives for each criterion, the grouping and sorting CPS uses the implementation for each CPS group of a single-level AHP.

Constructive and productive structure of AHPS. Let us determine the GCPS specialization [6] to rep-resent the analytic hierarchy process with sorting:

, , _{S AHPS AHPS}, _AHPS, _AHPS

C= 〈M Σ Λ〉 C M Σ Λ

where C– ОКПС, M – heterogeneous medium,

Σ – signature, Λ – axiomatics, _S – specialization operation, Λ_AHPS = Λ ∪ Λ₁ ,

1 {MAHPS T1 N1,

Λ = ⊃ ∪ Σ_AHPS= Ξ Θ Φ Π

{

, , ,

}

, { ,| ,|| }

Θ = ⇒ ⇒ ⇒ ,Π = →{ },Φ = ÷{ ,*,: , , }},= < ∇

{ , , }

Ξ = ⋅ ◊ ,Ξ– binding operations, Θ – output operations, Π – substitution operations, Φ − op-eration on attributes.

Partial axiomatics Λ₁ contains the following definitions, additions and constraints that specify alphabet, medium attributes, substitutive relations, set the features of substitution and output opera-tions.

Terminal alphabet contains a set of alternatives

,

{_{name v i}x} and criteria {_{name v p,} k } with their attrib-utes:x_i − alternative identifier, name − semantics,

v − global priority (weight); k_p − criterion identi-fier.

Alternatives and criteria, valid assessment val-ues are contained in a heterogeneous medium

AHPS

M .

The following operations on attributes are in-troduced:

( ; ; )c n L

÷ − conditional n operations from the list L, if c true= , L=( , ,..., )j j_{1 2} j_n , operations are presented in the prefix form;

∗ − vector-number multiplication operation; := − assignment operation;

< − less-than comparison;

∇ − attribute value seek operation, by an ex-ternal server.

The substitution rules are written as

, , : , , , , , ,

r i j

r sr i j gr i j

substitu-tive relation, g_{r i j, ,} – a set of operations on attrib-utes, r – rule number, i, j – numbers of the first and the second pair of alternatives. Three-level indexing is used for ordering the substitution rules.

Binary partial output operation [6]

* _{( | ( , ))}

l = ⇒ Ψl (here l, _l* _{–forms before and}

after the substitution operation), consists of:

1) selecting one of the substitution rules

, , , ,

: , ,

r sr i j gr i j

ψ ∈Ψ with substitutive relations

, , r i j

s and performing the substitution operation on its basis. Availability of substitutive relation s_{r i j, ,}

is determined by the availability attribute value

, , , , r i j r i j

d ↵s : if d_{r i j, ,} ↵s_{r i j, ,} =1 the relation is avail-able, d_{r i j, ,} ↵s_{r i j, ,} =0 – not available; the availabil-ity of rules is regulated by operations on attributes or is given by axiomatics;

2) carrying out operations on attributes g_{r i j, ,} . The order of the operation on attributes in the process of performing partial output operation is given by the attribute τ_j, where τ ∈_j I,

0 1

{ }

I= τ τ, , I⊂M_AHPS, τ₀ – the operation on attribute is performed before the substitution op-eration, τ₁ – after the substitution operation.

Complete output (or output) operation is the sequential partial output operation, starting from the initial nonterminal and finishing with the con-struction that satisfies the output completion condi-tion. The result of the complete output operation is the construct containing the ordered sequence of alternatives.

The output completion condition is the absence of non-terminals in the form.

Suppose we have the following basic algo-rithmic structure (BAS) [6], which comprises the steps of performing operations by condition, ma-trix operations, as well as the launch of AHPS for criteria and alternatives:

, ,

A,AHPS A,AHPS A,AHPS A,AHPS

C = 〈M Σ Λ 〉,

where M_A,AHPS – heterogeneous medium that con-tains V_A,AHPS, Σ_A,AHPS – signature and Λ_A,AHPS – axiomatics, 0

1 , ,

{ | i j i j A A

A,AHPS A A

V ⊃ A ⋅ 0 0

22|a ba b, ,, 23| },bb A A α◊_α – a set of forming algorithms for a particular server,

and { ₂| j_{, ,} , ₃| j_, , ₄| j_, }

h q i i i

f f f

w

l l f f f

A A _Ψ A _Ψ ∪V – a set of

con-structed algorithms, 0 5 , , 6

{ |L , |P

сn L a,b

A A ,

0 0 0 0

7 | ,aa b, 8 | ,a bc, 9 | ,ba 18| ,ca b,

A A A A 0

10| ,ca b,

A 0

11| ,ca b,

A 0

12| ,ca b, A

0 0 0

13| ,, 14| ,, 15| N а c is a b a b

A A A _γ, 0 r

16| N A _γ, 0

17| ,ca

A 0

18|ca b,

A ,

0 19|ca b,

A , 0

20|ca b,

A , 0 ,

21|i, ,j p a h x x k

A }∈V_w – algorithms for operations Φ on attributes.

The above algorithms execute the following operations:

− 0

1 | , i j i j A A A A

A ⋅ – algorithm concatenation

(sequen-tial algorithm A_i after A_j);

− ₂| j_{, ,} h q i f l l f

A – substitution;

− ₃| j_, , ₄| _,

i f f

A _Ψ A Ω_{σ Ψ} – partial and complete output.

Here ,f f_{i j} –forms, σ –initial nonterminal, Ω – a set of formed constructs;

− ₅ |L_{, ,}

сn L

A – execution of n algorithms from the list L, if c true= ;

− 0

6 |ca b,

A – calculation of the product a b* , a

and b can be matrices or numbers;

− 0

7 |aa b,

A – assigning a value to a variable

a b= ;

− ₉ |b a

A – determining the value of a by an ex-ternal server;

− 0

10|сa b,

A – calculation of the quotient of a by

b;

− 0

11|сa b,

A – calculation of the remainder on di-viding a by b;

− 0

8 |ca b,

A , 0

12|ca b,

A , 0

13|ca b,

A , 0

19|ca b,

A , 0

20|ca b,

A – com-parison of numbers a and b, if the condition is satisfied (a b≤ ,a≠ b, a=b, a b> ,a b< ), then

c true= , otherwise c= false;

− 0

13|aa b,

A – assigning the value b to the vari-able a, the valuesa and b can be vectors, matri-ces, or numbers;

− 0

14|ca b,

A – logical AND of the two conditions

a and b, true, if both conditions are true;

− 0

15| N is

A _γ – calculation of the conformity

− 0 r 16|

N

A _γ – calculation of the alternative

prior-ity vector by PCM _Nγ;

− ₁₇|с

a

A – calculation of integral part of the real number a;

− 0

18|ca b,

A – calculation of the sum a b+ , a

and b can be matrices or numbers;

− 0 ,

21|i, ,j p a h x x k

A – determination of the link weight of i and j alternatives by the criterion k_p by an external server (h=1);

− 0

22|a ba b,

A –binding alternatives and criteria, where ,a b – identifiers of alternatives or criteria or links between them;

− 0

23| ,bb

A α◊_α – binding bimplementation result to non-terminal of CPS implementation use α.

Interpretation of the main CPS for the modified analytic hierarchy process:

, , ,

AHPS AHPS AHPS AHPS

C = M Σ Λ

, , , , , ,

A AHPS A AHPS A AHPS A AHPS I

C = M Σ Λ

,

, , ,

I I CAHPS = MAHPS ΣAHPS ΛI AHPS Z ,

where _I – interpretation operation; Z – a set of servers that can use all BAS algorithms; C_{A AHPS,}

, 3

I AHPS AHPS

Λ = Λ ∪Λ ,

0

3 {( 1 | , ) i j i j A A A A

A ⋅

Λ = ↵⋅ ,( |₂ j_{, ,} ),

h q i f l l f

A ↵ ⇒ ( ₃ j_, | )

i f f

A _Ψ↵ ⇒ ,

4| , || ) A Ω_{σ Ψ} ↵ ⇒ , 0

6

( |c *)

a,b

A ↵ , 0

7 ,

( |a : )

a b A ↵ = ,

0 8 ,

( |c )

a b

A ↵ < , 0 5 , ,

( |L )

c n L

A ↵÷ , ( 0

23| ,bb A α◊_α ↵◊)}. Let us represent CPS specification for the ana-lytic hierarchy process with sorting:

,

, , ,

ICAHPS = MAHPS ΣAHPS ΛI AHPS Z K

, , , , , 4

K CK AHPS = MAHPS ΣК AHPS ΛI AHPS∪Λ ∪ 5,Z

Λ ∪ , where

4 {T1 { , , ,...,x x x1 2 3 x kN, ,..., },1 kP

Λ = =

1 {P N, ,Q , ,...,1 P Q, 1,...,Q P, } N = σ λ α α φ φ χ ,

,

{_{P N} },

U = σ Ψ = ψ_K { _r: s_{r i j, ,} ,g_{r i j, ,} }, r=1,6},

r – rule number, i – number of the first alterna-tive, j – the number of the second alternative of the pair, x_i – terminal for identifier of thei-th alter-native, k_l – terminal for identifier of the l-th crite-rion, U – a set of initial non-terminals, α_l – non-terminal for processing alternatives by the l-th crite-rion, χ – non-terminal for implementation of the grouping and sorting CPS for criteria, Qλ –

non-terminal of CPS criteria (where Q={ , , }r is N – a set of attributes: r – vector of PCM priorities, is – ma-trix conformity relation, N – matrix dimension),

Q lφ – PCM alternatives by l-th criterion.

Partial axiomatic Λ₅ is as follows.

The number of criteria P and the number of al-ternatives N, as well as the semantics of alterna-tives and criteria are given at the stage of execution by an external server.

The record of the sequential concatenation of several terminals, non-terminals and sequential operations on attributes will be represented as fol-lows:

1 2

1

... _n n _i

i

x x x x

=

⋅ ⋅ ⋅ =

∏

.

The record

, , 1 1

( )

r i j

N N

d i= j i= +

α → β

∏ ∏

means that the rule consists of a sequence of substitutive rela-tions with a given availability attribute. If the sub-stitutive relation is available, then it is performed and the availability of the next relation in the se-quence is determined, otherwise this relation is omitted, and the availability of the next one in the sequence is determined.

The rules that do not change the current con-struct have void substitutive relation.

It is assumed thatd_{r i j, ,} ↵s_{r i j, ,} =1, for each r 1,5= , so this attribute in these rules is omitted. Here are the rules and their brief description.

quantity and the semantics of criteria and alterna-tives:

1,0,0 , _,0,

1

|Q ( ) ,

P P

P Q

P N _{k k} p

p

s λ

=

= σ → χ ◊ λ ⋅

∏

α

0 1,0,0

1

: ( ), : ( ), P ( :_i ( ));_i

i

g N N P P k k

τ

=

= = ∇ = ∇

∏

=∇

1

( : ( )) .

N

i i

i

x x

=

= ∇

∏

The relation s_2,0,0 uses implementation of the grouping and sorting CPS for the alternatives for each criterion, and s_3,0,0 is used to get the imple-mentation results of the sorting and grouping CPS for criteria:

2,0,0 _{, ,}

1

( |Q p )

N P P

p _{x p k} Q p

p

s φ

=

=

∏

α → χ ◊ φ .

3,0,0 | ,0, Q

Pk Q Q s = χ λ _ε◊ λ → λ .

The relation s_4,0,0 is used to obtain the imple-mentation result of the grouping and sorting CPS for alternatives:

4,0,0 _{, ,}

1

( |Q p )

N P P

p p

Q Q

x p k p

s φ

=

=

∏

χ ◊ φ → φ . The following substitutive relation is aimed to enter a set of alternatives into the construct. Opera-tions on attributes contain the calculation of global alternative priorities:

5,0,0

1 1

( ) ( )

P N

Q _{Q p v i}

p i

s x

= =

= λ ⋅

∏

φ →

∏

,

0 5,0,0 1

( : ( ) * )

N

p

i i p

i

g v x r r

τ

=

=

∏

↵ = ↵λ ↵φ . The set of substitutive relations s_6,0,0 allows ordering the alternatives into constructs according to their ranks:

6, , 1

6,0,0

1 1

( )

i j

N N

v i v j d v j v i i j i

s − x x x x

= = +

=

∏ ∏

⋅ → ⋅ ,

0

1

6,0,0 6, ,

1 1

( ( ;1; : 1))

N N

i j i j

i j i

g − v x v x d

τ

= = +

=

∏ ∏

÷ ↵ < ↵ = . The implementation of this CPS is the set of al-ternatives ordered in accordance with the calcu-lated ranks.

Constructive and productive structure of alter-natives grouping and sorting (CPS of AGS). Let us determine the GCPS specialization to represent the grouping and sorting subsystem for AHPS:

, , _{S GSA GSA GSA GSA}

C= 〈M Σ Λ〉 C M ,Σ ,Λ

where Λ_GSA = Λ ∪ Λ Λ =₆, ₆ {M_GSA ⊃T₂∪N₂ ,

{

, , ,

}

GSA

Σ = Ξ Θ Φ Π ,Π = →{ }, Θ = ⇒ ⇒ ⇒{ ,| ,|| },

{

, ,

}

Ξ = ⋅ ◊ ,Φ = ÷{ ,*, : , , ,%, , \, , ,&,= > ≤ + = ≠ , ,[]}}, Ξ– binding operation, Θ– output opera-tions, Φ − operations on attributes, Π– substitu-tion operasubstitu-tions.

Partial axiomatics Λ₆ is presented below. Terminal alphabet contains many alternatives and criteria with their attributes.

The substitution rules include a substitutive re-lation and a set of operations on attributes. The substitutive relations contain the available attribute

r

d , where r – the rule number that takes the value 1 – the relation is available and 0 – not available. For the rules with a constant availability attribute (d_r=1) this attribute is omitted for record simplic-ity.

To interpret the CPS of alternative grouping and sorting let us use БАС C_{A AHPS,} , described above:

, , ,

GSA GSA GSA GSA

C = M Σ Λ

, , , , , ,

A AHPS A AHPS A AHPS A AHPS I

C = M Σ Λ

,

, , ,

I I CGSA= MGSA ΣGSA ΛI GSA ZGSA ,

where

, 7

I GSA GSA

Λ = Λ ∪Λ , 0

3 {( 1 | , ), i j i j A A A A

A ⋅

Λ = ↵⋅

2 , ,

( | j ),

h q i f l l f

A ↵ ⇒ ( ₃ j_, | )

i f f

A _Ψ↵ ⇒ , A₄|Ω_{σ Ψ,} ↵ ⇒|| ),

0 6

( |c *)

a,b

A ↵ , 0

7 ,

( |a : )

a b

A ↵ = , 0

8 ,

( |c )

0 5 , ,

( |L ),

c n L

A ↵÷ 0

10

( |c /)

a,b

A ↵ , 0

11

( |c %)

a,b

A ↵ ,

0 12

( |c )

a,b

A ↵ ≠ , 0

13

( |c )

a,b

A ↵ = , 0

14

( |c &)

a,b

A ↵ ,

0 15

( | )

N is

A _γ ↵ , 0 r 16

( | )

N

A _γ ↵ , ( ₁₇|с [])

a

A ↵ ,

0 18

( |c )

a,b

A ↵+ , 0

19 ,

( |c )

a b

A ↵ > , ( 0 23| ,bb A α◊_α ↵◊)}. We concretize CPS of alternative grouping and sorting:

,

, , ,

ICGSA= MGSA ΣGSA ΛI GSA ZGSA K

, , , , , 7

K CK GSA= MGSA ΣКGSA ΛI GSA∪Λ ∪ 8,ZGSA ,

Λ ∪

where

7 {T2 T1 {qym j, } {qym j′, } {qzm j, },

Λ = = ∪ ∪ ∪

1,

m= M и j=1,4, N₂ ={_{P N,} χ,_{N p,} γ,_{p n,} β λ,_Q ,

, ,

1 ,

, ( ), ,

| n Q ,

n

m i p p n m q

i L

y p k

=

′ λ

β

υ

∏

3, ,4 ,{ }} { , , , }, k k Mα χm ∪ ch nψm ch nψ′m

1,

m= M, U={_{P N,} χ}, Ψ = ψ_K { _r: s g_r, _r }, 1,14}

r= , r – rule number,

3, ,4

k k Mα – is

responsi-ble for determining the number of groups with four and three alternatives (k k₃, ₄– number of groups with three or four alternatives in the group, respec-tively, М – total number of groups); _{ch n,} ψ_m,

,

ch nψm′ − non-terminals of m-th group of

alterna-tives, with attributes ch − flag indicating the alter-native position changes in the group (1 − alterna-tives changed their position after ranking in the group, 0 − did not change); _{N p,} γ − PCM for N

alternatives according to the criterion p, matrix elements, non-terminals _{a h i j,} γ_, with attributes: a

− evaluation of comparison of i and j alterna-tives, h − evaluation process tool, (h=1 – filled according to the evaluation by an external server, expert, 0h= – without the involvement of an ex-ternal expert on the basis of substitution rules);

, m

p nβ – PCM by criterion p for the group m, consisting of n – alternatives, this matrix elements are non-terminals _{p a h m i j, ,} β _{, ,} , where the attributes

a and h – the same as for _{a h i j,} γ_, ;

, ,

1 ,

, ( ), ,

|n Q n

m i p p n m q

i L

y p k

=

′ λ

β

υ

∏ – non-terminal of CPS

imple-mentation of the single-level classical AHP for the

group alternatives: _,

1 n

m i q i

y =

∏

– set of alternatives for AHP ranking, p – ranking criterion number,

Pk– criterion vector, Qλ – PCM of alternatives

of the group with calculated ranks and conformity relation, _nL′ – list of alternatives ordered accord-ing to the ranks, _{p n,} β_m – PCM of alternatives in the group n; χ_m – non-terminal to prepare m-th group alternatives for ranking; _allη – non-terminal to calculate the parameters of the general PCM of alternatives (missing evaluations of paired com-parisons, conformity relation and matrix comple-tion control);

{

_{q m i}y _,

}

,

{

_{q m i}y′_,

}

,

{ }

_{q m i}z _, − set of alternatives in the group m, y, y′, z − alternative identifier, q=

[

name v u r l, , , ,

]

− set of attributes where name − alternative semantics, v − global priority (weight) of an alternative, u − global number of alternative, r − alternative weight vec-tor by criteria, l − criterion number.

The first rule with the substitutive relation, which enters into the construct the sequence of alternatives, PCM by p-th criterion and non-terminal with attributes to work with groups. The operations on attributes calculate the number of groups from 3 and 4 alternatives and the total number of groups. The alternative paired compari-son evaluations are completed with default values:

3 4

1 , , , , , ,

1

|Q ( ) ,

N P N

name v u p N k k M x p k

u

s λ x

=

= χ →

∏

⋅ γ⋅ α

0g1 flag: 0, ( %4 0;3;(N k4 : N/ 4),

τ = = ÷ = ↵α =

3 4

(k ↵α =: 0),(flag: 1)), (= ÷ N=3;3;(k ↵α =: 0),

3 4

3

(k ↵α =: 2),(flag: 1)), (= ÷ flag=0;2;

4

(k ↵α =: N/ 4 (3− −N%4)),

3 4

(k ↵α =: (N k− ↵α ∗4) / 3)),

3 4

: ;

M↵α = ↵α + ↵αk k

, , ,

1 1

( ( : 0; : 0; : 0;

N N

p i j p j i p i j

i j i

a a h

= = +

↵ γ = ↵ γ = ↵ γ =

∏ ∏

, : 0;); , : 1; , : 0;) p j i p i i p i i

h↵ γ = a↵ γ = h↵ γ = . The following relation is applied for breakdown of the alternatives into the groups. The operations on attributes get the conformity between the gen-eral list of alternatives and alternatives in groups:

3 4

2 , , , ,

1 1 1

( ) ( m )

n

N M

u k k M ch n m i

q q

u m i

s x y

↵ψ ↵α

= = =

=

∏

⋅ α →

∏

ψ ⋅

∏

3 0 2 1 ( 3; k m i g n τ =

=

∏

↵ψ =

, *

1

( : ;

m n

i j i j

j

name y name x

↵ψ = ↵ = ↵

∏

3 , * 1

: )); M ( 4;

i j i j m

i k

u y u x n

= +

↵ = ↵

∏

↵ψ =

, * , *

1

( : ; : ))

m n

i j i j i j i j

j

name y name x u y u x

↵ψ

=

↵ = ↵ ↵ = ↵

∏

.

The operations on attributes of s₃ relation de-termine the attribute values for PCM elements of the alternatives:

3

3 , , ,

1 1

( m ) )

n M

ch n q m i p N

m i d

s y

↵ψ ↵α

= =

=

∏

ψ ⋅

∏

⋅ γ →

, , , ,

1 1

( m ) ) ,

m n

M

ch n q m i p n m m p N

m i y ↵ψ ↵α ↵ψ = =

→

∏

ψ ⋅

∏

⋅ β ⋅ρ ⋅ γ

0 3 , , ,, ,

1 1 1

( m( m( : ;

m i m j

n n

M

m i j u y u y

m i j

g a a

↵ψ ↵ψ

τ ↵ ↵

= = =

=

∏ ∏ ∏

↵β = ↵γ

, ,

, , : m i, m j))) m i j u y u y

h↵β = ↵γh _{↵ ↵} .

The following substitutive relation is used for

CPS implementation of the classic single-level AHP for each alternative group:

4 , , ,

1 1

( m ))

m n

M

ch n q m i p n m m

m i s y ↵ψ ↵α ↵ψ = =

=

∏

ψ ⋅

∏

⋅ β ⋅ρ →

, 1 , , 1 ( ), , , , ( ), , 1 1 ( ( ) | ) . n m

m q m i Q

i

n m

m i p p n m m q

i n

M z

ch n q m i

y p k

m i y ↵ψ = ↵ψ ↵ψ = ↵ψ λ β = = ∏ → ψ ⋅ ⋅ υ

∏

∏

∏

The operations on attributes of the following rule supplement the general PCM with new evalua-tions: , , 1 , , 1 , ( ), 5 , , ( ), , 1 ( | ) n m

m i Q m p n m m q

i n m

m i p p n m m q

i

M z

p N y p k m s ↵ψ ↵ψ = ↵ψ ↵ψ = β λ β = ∏

= υ ⋅ γ → ∏

∏

, ,

1 1

( m , )

M n

m i Q m p N all q i m z ↵ψ = =

→

∏

∏ λ ⋅ γ⋅ η ,

1 5 ,, , , ,

1 1 1

( m( m( : ;

m i m j

n n

M

u z u z m i j

m i j

g a a

↵ψ ↵ψ

τ ↵ ↵

= = =

=

∏ ∏ ∏

↵γ = ↵β

,, , : , , ))) . m i m j

u z u z m i j

h↵γ _{↵ ↵} = ↵βh

The substitutive relation is used to calculate the general PCM elements by transitiveness and to count the uncompleted elements:

6 ,

s =

1 6 ,

1 1 1

: 0; N N ( N ( ( _{i j} 0;

i j i c

g all h

τ

= = + =

= ↵η =

∏ ∏ ∏

÷ ↵γ =

, ,

4;( ((÷ ↵γ ≠a _{c j} 0) & (a↵γ ≠_{i c} 0);2;

, ,

(sum:=sum a+ ↵γ_{i c}*a↵γ_{c j}; :q = +q 1));

, : / ; , : 0; , :

i j i j j i

a↵γ =sum q h↵γ = h↵γ =

,

,

1 0; _{j i}: ));)

i j a

a

= ↵γ = ÷ ↵γ

,

(a _{i j} 0;1;all : all 1)) .

÷ ↵γ = ↵η = ↵η +

7

7 , , , 1

1 1

( m( ) m( ), )

n

M n

m

ch n q m i q m i Q

i

m i

s y z

↵ψ _↵ψ

=

= =

=

∏

ψ ⋅

∏

⋅ ∏ λ →

, , ,

1

1 1

( m( ) m( ), );

n

M n

m

ch n q m i q m i Q

i m i y z ↵ψ _↵ψ = = =

→

∏

ψ ⋅

∏

⋅ ∏ λ

0 7

1

( : 0;

M m m g ch τ =

=

∏

↵ψ =

, ,

1

( ( ;

m n

m i m i

i

name y name z ↵ψ

=

÷ ↵ = ↵

∏

, , , ,

3;(v y↵ _{m i}:= ↵v z_{m i};name y↵ _{m i}=name z↵ _{m i};

, ,)); ( , ,;

m i m i m i m i

u y↵ = ↵u z ÷ name y↵ ≠name z↵

1;ch↵ψ =_m: 1 .

The relation s₈ is used to calculate the priority vector and the conformity relation for general ma-trix, if the position of the alternatives in the groups has not changed:

8 , , ,

1

1 1

( m( ) m( ), )

n

M n

ch n q m i q m i Q m

i

m i

s y z

↵ψ _↵ψ

=

= =

=

∏

ψ ⋅

∏

⋅ ∏ λ ×

8

, all d Q

p N

× γ⋅ η → λ ,

0 8

1

: 0; M ( ( _m 1;1; : 1));

m

g f ch f

τ

=

= =

∏

÷ ↵ψ = =

9

(f 1;1;d : 1);

÷ = =

8

(((f 0) & (all 0);1;d : 1);

÷ = ↵η = =

1g8 is : (p N, );r : (p N, ) .

τ = ↵λ = γ ↵λ = γ

The substitutive relation s₉ is used to regroup the alternatives. Operations on attributes allow set-ting the alternative attributes in the new groups:

9 , ,

1 1

( m( )

n M

ch n q m i

m i

s y

↵ψ

= =

=

∏

ψ ⋅

∏

×

, ,

1

( ), )

m n

m i Q m p N q

i z ↵ψ

=

× ∏ λ ⋅ γ ×

8 1

, ,

1 1

( m( )

n M

all d ch n m q m i

m i y ↵ψ − = = ′

× η →

∏

ψ ⋅

∏

×

, m m) , ;

p n↵ψ′ m p N

× β ⋅ρ ⋅ γ

0

1

9 1

( : [ ]; : 2; 2 M m m m n

g − l n l

τ

=

↵ψ _′ =

∏

= ↵ψ = +

, ,

1

( : ;

l

m i m l i

i

name y name z ₊

=

′

↵ = ↵

∏

, : , ; , : , ;

m i m l i m i m l i

p y↵ ′ = ↵p z ₊ v y↵ ′ = ↵v z ₊ 2

, , , 1,

1

: ); ( : ;

m i m l i m i l m i

i

u y u z ₊ name y ₊ name z ₊

=

′ ′

↵ = ↵

∏

↵ = ↵

, : 1,; , : 1,;

m i l m i m i l m i

p y↵ ′ ₊ = ↵p z ₊ v y↵ ′ ₊ = ↵v z ₊ , : 1, );

m i l m i

u y↵ ′ ₊ = ↵u z ₊

' '

1 , ,

1

9 , , ,

1 1 1

( m( m( : ;

m i m j

n n

M

m i j u y u y

m i j

g a a

↵ψ ↵ψ −

τ ↵ ↵

= = =

=

∏ ∏ ∏

↵β = ↵γ

, ,

, , : m i, m j))) . m i j u y u y

h↵β = ↵γh _{↵ ↵}

The substitutive relation s₁₀ is for implementa-tion of classic AHP for new alternative groups:

1

10 , , ,

1 1

( m( ) )

m n

M

ch n m q m i p n m m

m i s y ↵ψ − ′ ↵ψ = = ′ ′

=

∏

ψ ⋅

∏

⋅ β ⋅ρ →

1 1 M m − = →

∏

× , 1 , , 1 ( ), , , , ( ), , 1 ( ) | . n m

m q m i Q

i

n m

m i p p n m m q

i

n z

ch n q m i

y p k i y ′ ↵ψ = ′ ↵ψ ′ ↵ψ = ↵ψ λ ′ β = ⎛ _∏ ⎞ ⎜ _{′ ′} ⎟

×_⎜ ψ ⋅ ⋅ υ _⎟

⎜ ∏ ⎟

⎝ ⎠

∏

The following rule contains the substitutive re-lation to get the AHP ranking result in each group and to save the evaluation entered into the general PCM by the expert:

, , 1 , , 1

1 , ( ),

11 ,

, ( ), , 1

( | )

n m

m i Q m p n m m q

i n m

m i p p n m m q

i

M z

p N y p k m s ′ ↵ψ ′ ↵ψ = ′ ↵ψ ′ ↵ψ =

− β λ

′ β

=

∏

= υ ⋅ γ → ∏

∏

1 , , 1 1

( m( ), ) ,

M n

m

m i Q p N all q

i m

z − ↵ψ′

= =

' '

1 , ,

1

11 , , ,

1 1 1

( m( m( : ;

m i m j

n n

M

u z u z m i j

m i j

g a a

↵ψ ↵ψ −

τ ↵ ↵

= = =

=

∏ ∏ ∏

↵γ = ↵β

,, , : , , ))); 6: 1; 12: 1 . m i m j

u z u z m i j

h↵γ _{↵ ↵} = ↵βh d = d =

Operations on attributes of the following rule allow determining the changes in the positions of alternatives in the groups after AHP application:

12 ,

s = _{0 12} 1

1

( : 0;

M

m m

g − ch

τ

=

′ =

∏

↵ψ =

, , , ,

1

( ( ;1; : ),

m n

m i m i m i m i

i

name y name z v y v z

′ ↵ψ

=

′ ′

÷ ↵ = ↵ ↵ = ↵

∏

, ,

(name y_{m i}′ name z_{m i};1;ch _m′ : 1

÷ ↵ ≠ ↵ ↵ψ = . The substitutive relation s₁₃ is used to calculate the priority vector and the conformity relation for general matrix, if the position of the alternatives in the groups has not changed:

' 1

13 , , ,

1

1 1

( m( ) m( ), )

n

M n

ch n q m i q m i Q m

i

m i

s y z

↵ψ

− ↵ψ′

=

= =

′ ′

=

∏

ψ ⋅

∏

⋅ ∏ λ ×

13

, all d Q ,

p N

× γ⋅ η → λ

0

1

13

1

: 0;M ( ( _m 1;1; : 1));

m

g f − ch f

τ

=

′

= =

∏

÷ ↵ψ = =

13

(f 1;1;d : 1);

÷ = =

14

(((f 0) & (all 0);1;d : 1) .

÷ = ↵η = =

1g13 is : (p N, );r : (p N, ) .

τ = ↵λ = γ ↵λ = γ

The following substitutive relation is used to restore alternatives in the groups, if their position has changed:

1

14 , , ,

1 1 1

( m( )) (

n

M M

-ch n q m i ch n m

m i m

s y (

↵ψ

= = =

′ =

∏

ψ ⋅

∏

⋅

∏

ψ ×

'

14

, ,

1 1

( ) ( ), )

m _m

n _n

m i m i Q m d

q q

i i

y z

↵ψ _↵ψ′

= =

′

×

∏

⋅ ∏ λ →

, ,

1 1

( m ) ,

n M

ch n q m i

m i

y ↵ψ

= =

→

∏

ψ ⋅

∏

0

1

14 1

( : [ ]; 2

M

m

m

n

g − l

τ

=

↵ψ =

∏

=

, ,

1

( : ;

l

m l i m i

i

name y ₊ name z =

↵ = ↵

∏

, : ,; , : ,;

m l i m i m l i m i

p y↵ ₊ = ↵p z v y↵ ₊ = ↵v z 2

, , 1, ,

1

: ); ( : ;

m l i m i m i m i l

i

u y ₊ u z name y ₊ name z ₊

=

↵ = ↵

∏

↵ = ↵

1, : , ; 1, : .

m i m i l m i

p y↵ ₊ = ↵p z ₊ v y↵ ₊ =

, ; 1, : , )

m i l m i m i l

v z ₊ u y ₊ u z ₊

= ↵ ↵ = ↵

1g14 d3: 1;d14: 0 .

τ = = =

Implementation of CPS of alternative grouping and sorting is the non-terminal with calculated al-ternative rank attributes and the conformity rela-tion for PCM of the alternatives.

Constructive and productive structure for clas-sical single-level AHP. CPS of classical single-level AHP implements completing by an external expert of some paired comparison evaluations, finding the proper number of the matrix, confor-mity relation of PCM and alternative ranks.

Let us determine GCPC specialization to repre-sent classical single-level AHP:

, , _{S AHP AHP AHP AHP}

C= 〈M Σ Λ〉 C M ,Σ ,Λ ,

where Λ_AHP = Λ ∪ Λ Λ =₉, ₉ {M_AHP ⊃T₃∪N₃,

{

, ,

}

,

AHP

Σ = Ξ Θ Φ Θ = ⇒ ⇒ ⇒ →{ ,| ,|| , },Ξ = ⋅

{ }

, , { , : , /, , , , , , }}

Φ = ÷ = ≤ > = .

The operation ( , , )x x k_{i j p} allows setting a val-ue of the link weight between i and j alternatives (criteria) byk_p criterion, ( , , )x x_{i j} ε allows setting a value of the link weight between criteria. These operations are executed by an external server.

Terminal alphabet contains many alternatives and criteria with their attributes.

to the assessment by an external server expert, 0

h= – without the involvement of an external expert on the basis of substitution rules);

, ( )

a h x xi j ε – link of i and j alternative

(crite-rion) if a comparison criterion is not given; Qλ –

pairwise comparison matrix for alternatives;

1

( )

N

r i i

x =

∏

– sorted alternative sequence.

For interpretation of this CPS we use the BAS

A,AHPS

C described above:

, , ,

AHP AHP AHP AHP A,AHPS

C = M Σ Λ C =

, ,

A,AHPS A,AHPS A,AHPS I M

= Σ Λ

,

, АAHPS , , , ,

I I C CAHP = MAHP ΣAHP ΛI AHP Z ,

where

, 11

I AHP AHP

Λ = Λ ∪Λ , 0

11 {( 1 | , ), i j i j A A A A

A ⋅

Λ = ↵⋅

2 , ,

( | j ),

h q i f l l f

A ↵ ⇒ ₃|i_, | )

i f f

A _Ψ ↵ ⇒ , A₄|Ω_{σ Ψ,} ↵ ⇒|| ),

0 7 ,

( |a : )

a b

A ↵ = , 0

8 ,

( |c )

a b

A ↵ ≤ , 0

5 , ,

( |L )

c n L

A ↵÷ ,

0 10

( |c /)

a,b

A ↵ , 0

12

( |c )

a,b

A ↵ ≠ , 0

13

( |c )

a,b

A ↵ =

0 19 ,

( |c )

a b A ↵ > ,

0 15

( | )

N is

A _γ ↵ , 0 r

16

( | )

N

A _γ ↵ , 0

21 ,

( |c )

a b

A ↵ ,

0 22 ,

( |a b )

a b

A ↵ }.

Let us perform specification of the interpreted CPS for a single-level AHP:

,

, АAHP , , , ,

I C CAHP = MAHP ΣAHP ΛI AHP Z K

, , , , , 9 10,

K AHP AHP К AHP I AHP

C = M Σ Λ ∪Λ ∪Λ z ,

where

4 {T3 T N1, 3 {N ,P , , ,P N, ,Q ,p N, }

Λ = = = ξ ρ δ µ υ λ β ,

,

{_{P N} }

U= υ ,Ψ = ψ_K { _r: s_{r i j, ,} ,g_{r i j, ,} }, r=1,12},

r – rule number, i and j – number of the first and second pairs of alternatives, U – set of initial non-terminals, _Nξ – non-terminal to indicate

al-ternative links, _Pρ– non-terminal to indicate crite-ria links, β_{p i j, ,} – non-terminals to indicate links between alternatives ,i j by p-th criterion (for

simplicity indicated as matrix _{p N,} β), Qλ –

pairwise comparison matrix for alternatives (where [ , , ]

Q= v is N – vector of attributes: v – vector of PCM priorities, is – matrix conformity relation,

N – the number of alternatives).

The substitutive relation s_1,0,0 serves to change PCM completion and ranking of alternatives x_i by criterion k_p. All axiom input parameters are added to the medium.

, 1 ,

, ( )

1,0,0 | _{, , ,} , .

N i p N q

i N P p N

x

p N x p k

s =

β

β

∏

= υ → β ⋅ δ ⋅µ

The following substitutive relation is used, if a ranking criterion is specified:

2,0,0

2,0,0 , ,

1 1

( )

p N N

d p i j k

p N

i j s

= =

= β ⋅ δ →

∏∏

β ⋅ ρ ,

0g2,0,0 (p 0;1;d2,0,0: 1), τ = ÷ ↵β > =

3,0,0

(p 0;1;d : 1)

÷ ↵β = = .

The substitutive relation s_3,0,0 is used to form PCM alternatives, if a criterion is not specified:

3,0,0

3,0,0 , ,

1 1

( )

N N

d p i j

p N

i j s

= =

= β ⋅ δ →

∏∏

β . The following rule contains the substitutive re-lation to determine a connection between the alter-natives. Operations on attributes determine the evaluation of alternatives links:

4,0,0 , ,

1 1

( ( )

N N

i j i j i j

i j

s x x

= =

=

∏∏

β → ε ⋅β ,

1g4,0,0 a x x( i j ) : a i j, , τ = ↵ ε = ↵β

,

( _{i j} ) : _{i j}

h x x↵ ε = ↵βh .

and j= +1 i N, :

5, ,i j

s = ,

1g5, ,i j a x x( i j ) : ( , , ),x xi j

τ = ↵ ε = ε

, : ( ); ( ) : 1,

i j i j j i

a↵β = ↵h x x ε ↵h x x ε =

( _{j i} ) : 1 ( ( _{i j} )),

a x x↵ ε = a x x↵ ε

, : ( ); , : ( )

j i j i i j j i

a↵β = ↵a x x ε ↵β = ↵h h x x ε . The relation for PCM completion check. If the matrix is completed, then based on the operations on attributes we calculate the conformity relation and fill the alternative priority vector:

6,0,0 1 1

6,0,0

1 1

( )

( ) )

N N

i j d

i j N N

Q i j i j

x x s

x x = =

= =

ε → =

→ ε ⋅ λ

∏∏

∏∏

0g6,0,0 d6,0,0: 1,full: 1,

τ = = =

1 1

( ( ( ) 0;1;

N N

i j i j i

a x x = = +

÷ ↵ ε ≤

∏ ∏

6,0,0

: 0)), ( 0;1; : 0)

full = ÷ full= d = ,

1g6,0,0 is : (p N, );r : (p N, ) .

τ = ↵λ = β ↵λ = β

The relation s_7,0,0 enters the sequence of alter-natives with the weights into the construct, if PCM conformity relation is valid:

7 1 1 7,0,0

1

( )

( )

N N

i j i j

N

Q _{d q i} Q

i x x s

x = =

=

ε × =

× λ ⋅µ → ⋅ λ

∏∏

∏

0g7,0,0 ((is ) 0,011; 1;d7,0,0: 1)

τ = ÷ ↵λ ≤ = ,

1 7,0,0 7,0,0

1

( : ); : 0

N

i i

i

g v x r d

τ

=

=

∏

↵ = ↵λ = . The following set of rules (i=1,N−1,

1,

j i= + N) defines the relations for descending ordering of alternatives according to their weights:

8, ,

8, ,i j (v i v j) d i j (v j v i) ,

s = x ⋅ x → x ⋅ x

0g8, ,i j (v xj v xi;1;d8, ,i j: 1) τ = ÷ ↵ > ↵ = .

The substitutive relation s_9,0,0 is used to estab-lish the link between the alternatives by the given criteria:

9,0,0 , ,

1 1

( ( ) ,

p N N

i j k i j p i j

i j

s x x k

= =

=

∏∏

β ⋅ ρ → ⋅β

1g9,0,0 a x x( i j kp) : a i j, ,

τ = ↵ = ↵β

,

( _{i j p}): _{i j}

h x x↵ k = ↵βh .

The rules for setting the alternative pairwise comparison values by an expert by p-th criterion, where i=1,N and j= +1 i N, :

10, ,i j

s = ,

1g10,i,j a x x( i j kp) : ( , ,x x ki j p),

τ = ↵ =

, : ( ); ( ) : 1,

i j i j p i j p

a↵β = ↵a x x k h x x↵ k =

, : ( ); , : ( );

i j i j p j i j i p

h↵β = ↵h x x k a↵β = ↵a x x k

( _{j i p}) : 1 ( _{i j p})

a x x k↵ = a x x↵ k ,

( _{j i p}) : 1; _{j i}: ( _{j i p})

h x x k↵ = h↵β = ↵h x x k . The operations on attributes of the next rule check the PCM completion of alternatives by p-th criterion. If everything is completed, then the con-formity is calculated and the alternative priority vector is filled, otherwise the rule does not apply:

11,0,0 11,0,0

1 1

( )

N N

i j p d

i j

s x x k

= =

=

∏∏

→

1 1

( ) ,

N N

i j p Q

i j

x x k

= =

→

∏∏

⋅ λ

0g11 full: 1,d11,0,0: 1,

τ = = =

1 1

( ( ( ) 0

N N

i j p

i j

a x x k

= =

÷ ↵ ≤

∏∏

;

11,0,0

1g11,0,0 is : (p N, );r : (p N, )

τ = ↵λ = β ↵λ = β .

The relation s_12,0,0 is used to generate a se-quence of alternatives, if the PCM of alternatives has a valid conformity level. After the substitution on the basis of operations on attributes, the ranks of alternatives are determined:

12,0,0

12,0,0 ,

1 1

( )

N N

i j p p N Q d

i j

s x x k

= =

=

∏∏

⋅ β ⋅ λ ⋅µ →

, 1

( )

N

i Q

p N q

i x =

→ β ⋅

∏

⋅ λ ,

0g12,0,0 ((is ) 0,011; 1;d12,0,0: 1)

τ = ÷ ↵λ ≤ = ,

1 12,0,0 12,0,0

1

( : ); : 0 .

N

i i

i

g v x r d

τ

=

=

∏

↵ = ↵λ =

Implementation of CPS for a single-level AHP is the ranked list of alternatives, the completed PCM and the calculated conformity relation values for the formed matrix.

Originality and practical value

The developed model of constructive process for alternative ranking by modified AHPS can solve the problem with a large number of criteria and alternatives (more than ten), and can also be used under conditions of incomplete information, as part of evaluations is entered by an expert, and the part is calculated based on the input. This method can improve the conformity of expert judgments. CPS-modeling opens wide possibilities for automated hybridization of AHP modifications taking into account the specifics of the tasks.

Conclusions

The developed modeling system for construc-tive alternaconstruc-tives ranking process consists of three CPS, interacting at different levels of refinement transformations. Disaggregation of process com-ponents makes it possible to independently change some models, change their interpretation, which allows applying this approach to solve more spe-cific tasks.

CPS-formalization allows moving to a higher level of abstraction when describing a method for decision making problem solving, which in turn

provides an opportunity for the development of programs that implement the hybrid modification of the decision-making methods, in particular the various modifications of AHP.

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Т

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М

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ВАСЕЦЬКА

1*

1*_{Каф. «Комп’ютерніінформаційнітехнології», Дніпропетровськийнаціональнийуніверситетзалізничного}

транспортуіменіакадемікаВ. Лазаряна, вул. Лазаряна, 2, Дніпро, Україна, 49010, тел./факс +38 (098) 237 05 21,

ел. пошта [email protected], ORCID 0000-0001-7008-2839

МОДЕЛЮВАННЯ

МОДИФІКОВАНОГО

МЕТОДУ

АНАЛІЗУ

ІЄРАРХІЙ

ЗАСОБАМИ

КОНСТРУКТИВНО

-

ПРОДУКЦІЙНИХ

СТРУКТУР

Мета. У дослідженні передбачається: 1) розширити можливості класичного методу аналізу ієрархій

(МАІ) длявеликоїкількостіальтернативтакритеріїв; 2) побудуватимодельконструктивногопроцесупри

-йняттярішеньізвикористанняммодифікованогометодуаналізуієрархійізсортуванням (МАІС). Методика. Длядосягненняпоставленоїметивикористовуєтьсямеханізмконструктивно-продукційнихструктур (КПС).

Виконано уточнюючі перетворення узагальнюючої конструктивно-продукційної структури (УКПС).

Результати. Розроблена модель конструктивного процесу представляє собою взаємодію трьох структур: 1) загальноїструктуриКПСМАІС, якадозволяєзадатиальтернативитакритерії, виконуючидекомпозицію ієрархічної структури задачі; 2) КПС групування та сортування, яка розбиває альтернативи (критерії) на групитареалізуєкласичнийоднорівневийМАІдлякожноїгрупи, атакожрозраховуєоцінкипарнихпорів

-няньнаоснові введенихданих; 3) КПС однорівневого класичногоМАІ, якадозволяє заповнити матрицю парнихпорівнянь та розрахувати ранги альтернатив. Всі три структуривзаємодіють міжсобою нарізних рівняхуточнюючихперетворень: черезузгодженняподанимнарівніконкретизаціїтавикористанняреалі

-зацій. Запропонованамодельдозволилаперейтинабільшабстрактнийрівеньпредставленнярозв’язкузадач прийняттярішеньдлявеликоїкількостікритеріївтаальтернатив. Наукова новизна. Зарезультатамироботи пропонуєтьсявикористовуватимеханізмКПСдляформалізаціїмодифікаційМАІізсортуваннямдлярозв’я

-зкузадачприйняттярішеньізвеликоюкількістюкритеріївтаальтернатив. Практична значимість.Форма

-лізаціяпредставленняяк самогометодуаналізу ієрархій, такі йогомодифікаційдозволяє розширитиколо застосуванняданого методу, впорядкуватиописирізнихмодифікаційМАІ. Такепредставленнязабезпечує можливістьрозробкипрограмдляреалізаціїгібриднихмодифікаційметоду. Використаннярізнихінтерпре

-таційзапропонованихвстаттіКПСдозволитьвикористатиіншіпідходипривизначенніузгодженостімат

-рицьпарнихпорівнянь, розрахункуоціноктарангівальтернативікритеріїв.

Ключові слова: моделювання; конструктивно-продукційні структури; конструктивний процес; метод