CC BY
25
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦШНО-КОМУШКАЦШШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
УДК 504.81:004.94
T. М. VASETSKA1*
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 tetyana@vasetsky.com, 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 generalized 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 divides alternatives (criteria) into groups and implements the classic single-level AHP for each group, as well as calculates 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 proposed 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 alternatives. 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 modifications. 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 determining 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], proposed 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
use of this method [1-3, 13, 14], or use AHP in combination with other decision-making methods (mathematical methods of multi-criteria analysis, statistical methods etc.) [10, 12]. There are a lot of developed software tools, which implement both the method itself and its modifications [2, 9, 14, 15].
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦШНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
[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 alternatives in groups changes, the rearrangement is performed. Some estimates not yet identified by the expert are calculated on the basis of already determined ones at each step. This greatly facilitates the work of an expert.
Purpose
The purpose of this work is to extend the classical 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 alternatives 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 constructive and productive structures is used. CPS is a powerful device for formalization and modelling of processes [5-8]. By performing different transformations of the generalized constructive and productive structure (GCPS) [6], namely, specialization, interpretation, specification and implementation, the different models are developed [7]. GCPS is called a triple [6]:
Cg ={M, E, A),
where M - heterogeneous structure medium E -signature; consisting of sets of the binding operations, substitution and output operations; operations on attributes and substitutive relations; A -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 number of criteria and alternatives.
All three CPS interact at the specification level: data coherence connection and at the implementation level: CPS AHPS uses implementation 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 represent the analytic hierarchy process with sorting:
С — {M, E, Л) s ^ СAHpS (Mahps , EAHPS , ЛАИР8
)
where С - ОКПС, M - heterogeneous medium, E - signature, Л - axiomatics, S ^ -
Л AHPS — Л^Л1 ,
Eahps — (S, ©, Ф,П},
specialization operation
A1 = {MAHPS ^ T1 ^ K
© = , n = O = {-,*,:=, <, V}},
H = {•, o, ◊}, H - binding operations, © - output operations, n - substitution operations, O - operation on attributes.
Partial axiomatics A1 contains the following definitions, additions and constraints that specify alphabet, medium attributes, substitutive relations, set the features of substitution and output operations.
Terminal alphabet contains a set of alternatives
{ name,vXi } and criteria { name,vkp } with their attributes: xi - alternative identifier, name - semantics, v - global priority (weight); kp - criterion identifier.
Alternatives and criteria, valid assessment values are contained in a heterogeneous medium
MAHPS .
The following operations on attributes are introduced:
-(c;n;L) - conditional n operations from the list L, if c = true, L = (j1, j2,..., jn), operations are presented in the prefix form;
* - vector-number multiplication operation; := - assignment operation; < - less-than comparison; V - attribute value seek operation, by an external server.
The substitution rules are written as
Vrr>i j :(Sr,i, j , gr,i, j) where Sr,i, j - substitu-
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету з^зничного транспорту, 2016, № 4 (64)
ШФОРMАЩИНО-КОMУШКАЩИШ ТЕХНОЛОГИ ТА MATEMATOTHE MОДЕЛЮВАННЯ
tive relation, gr i j - a set of operations on attributes, 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* = (¥,l)) (here l, l* -forms before and after the substitution operation), consists of:
1) selecting one of the substitution rules yr: ^sr i j, gri j} e with substitutive relations
sr i j and performing the substitution operation on
its basis. Availability of substitutive relation sriJ
is determined by the availability attribute value dri j Jsri, j: if dr i j Jsri, j = 1 the relation is available, dri j Jsri j = 0 - not available; the availability of rules is regulated by operations on attributes or is given by axiomatics;
2) carrying out operations on attributes gr i j.
The order of the operation on attributes in the process of performing partial output operation is given by the attribute t j, where t j e I,
I = {t0 , t0} , I c MAHPS , t0 - the operation on attribute is performed before the substitution operation, tj - after the substitution operation.
Complete output (or output) operation is the sequential partial output operation, starting from the initial nonterminal and finishing with the construction that satisfies the output completion condition. 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 algorithmic structure (BAS) [6], which comprises the steps of performing operations by condition, matrix operations, as well as the launch of AHPS for criteria and alternatives:
CA,AHPS = (MA,AHPS , ^ A,AHPS , ЛA,AHPS ) :
where M
A,AHPS
- heterogeneous medium that con-
tains V
A,AHPS ■
2 A ,AHPS - signature and ЛA,AHPS
A
axiomatics, VA,AHPS
^ {AGjJZ A
0 \A1 ■Aj ¿0 jaob A0 jaOb} -02 ja,b,, A23 ja,b }
a set of forming algorithms for a particular server,
and {A2 | j, f, A31 f^, A4 | f w} u Vw - a set of constructed algorithms, {A5 L, A° ,
4° ia A0 |c a° ib 4° |c 4° |c 4° |c 4° |c 7 la ,b> ^8 \a,b' ^9 la' 18 la,b' 10 la,b' 11 la,b' 12 la,b'
4° |a 4° |c 40 |is 4° |r 4° |c 4° |c
^13 |a,b ,A14\a,b ,^15 | Y ,16 | Y , 17 U, A18\a,b ,
NY N1
Aj09 |a,b, A200 |a,b, Ai |Xj } e Vw - algorithms for
operations O on attributes.
The above algorithms execute the following operations:
0 A■ • A■
- A0 |A' J. - algorithm concatenation (sequential algorithm Ai after Aj );
f
- A0j,J, f - substitution;
2 "h ,lq ^ fi
- A
if
if- ^ |CT T - partial and complete output. Here f, fj -forms, c -initial nonterminal, Q -a set of formed constructs;
- A5
- execution of n algorithms from
the list L, if c = true ;
- A
6 la,b
- calculation of the product a * b, a and b can be matrices or numbers;
- A70 a b - assigning a value to a variable
a = b
A9 \a - determining the value of a by an external server;
- A10 \a b - calculation of the quotient of a by
b ;
- Aj0j \a b - calculation of the remainder on dividing a by b ;
- A8 \a,b , A12 \a,b , A13 \a,b , A19 \a,b , A20 \a,b - comparison of numbers a and b, if the condition is satisfied ( a < b , a ^ b , a = b , a > b , a < b ), then c = irae, otherwise c = false ;
- A003 \a b - assigning the value b to the variable a , the values a and b can be vectors, matrices, or numbers;
- A004 \a b - logical AND of the two conditions a and b, true, if both conditions are true;
- A105 \!s = - calculation of the conformity rela-
N Y
tion of the pairwise comparison matrix (PCM) N Y ;
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦ1ИНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
- А0
N Y
- calculation of the alternative prior-
ity vector by PCM N Y;
- A 17|ca - calculation of integral part of the real number a ;
- A108 ia & - calculation of the sum a + b, a and b can be matrices or numbers;
- A21 !axhx. k - determination of the link weight of i and j alternatives by the criterion k p by an external server (h = 1);
- A22 at -binding alternatives and criteria, where a, b - identifiers of alternatives or criteria or links between them;
- A2310? - binding b implementation result to
non-terminal of CPS implementation use a .
Interpretation of the main CPS for the modified analytic hierarchy process:
{CAHPS — (M.
AHPS , E AHPS ,Л AHPS
CA, AHPS — (M.
A,AHPS , EA,AHPS , ЛA,AHPs) ) I ^
>>
I ^ I CAHPS
\M
AHPS , E AHPS ,ЛI, AHPS ,
Z) ■
where j ^ - interpretation operation; Z - a set of servers that can use all BAS algorithms; CA,AHPS
Л!, AHPS — Л AHPS
и Л
3 '
Лз — {(A0 |AA J"),(
2 Wt ^^
AJ^ J|H),(A0|^b J*),(A70 J:—),
(A80 iaj<), (A50iL,n,L (A203 ia? jo )}.
Let us represent CPS specification for the analytic hierarchy process with sorting:
С
I AHPS
-\M
AHPS , E AHPS , ЛI, AHPS
■ Z)
K ^ CK, AHPS —(MAHPS , E Z, AHPS ,ЛI, AHPS
и л 4 и
ил5, z),
where
Л 4 — {T1 — ^^ x2 , x3,..., XN , k1,..., kP К
Ni ={ p, n a e ^ al,..., a P ,e e ^ P , X},
U = { p n a}, = {v r :(, gr,i, j)}, r = 1,6},
r - rule number, i - number of the first alternative, j - the number of the second alternative of the pair, xi - terminal for identifier of the i -th alternative, kl - terminal for identifier of the l -th criterion, U - a set of initial non-terminals, al - nonterminal for processing alternatives by the l -th criterion, x - non-terminal for implementation of the
grouping and sorting CPS for criteria, e ^ - nonterminal of CPS criteria (where e = {r,is,N} - a set
of attributes: r - vector of PCM priorities, is - matrix conformity relation, N - matrix dimension),
e ^ - PCM alternatives by l -th criterion. Partial axiomatic A5 is as follows. The number of criteria P and the number of alternatives N, as well as the semantics of alternatives 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 follows:
x1 " x 2 "... ' xn — xt .
■ ß) means that
The record HH (a d
i=1 j=i+1
the rule consists of a sequence of substitutive relations with a given availability attribute. If the substitutive relation is available, then it is performed and the availability of the next relation in the sequence 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 construct have void substitutive relation.
It is assumed that dr,i,j Jsr,i,j = 1, for each
r = 1,5, so this attribute in these rules is omitted. Here are the rules and their brief description.
The substitutive relation s10 0 is used to enter
the processing sequence of criteria and alternatives by criteria. Operations on attributes determine the
t—1
Наука та прогрес транспорту. Вкник Дншропетровського нащонального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦШНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
quantity and the semantics of criteria and alternatives:
4,0,0 _ \P,Nv
■(pn^XlQiLk ◊ Q^П(ap) >.
p=i
T0gi,0,0 = ( N:= V(N),P := V(P),П(k :=V(k));
i=1
N
П(x :=V(X)) >.
To ^6,0,0 H nn ( - (vJxi <vJx; ;1;d 6,i,j := 1)) ) .
\ ¿=1 j=i+1 /
The implementation of this CPS is the set of alternatives ordered in accordance with the calculated ranks.
Constructive and productive structure of alternatives grouping and sorting (CPS of AGS). Let us determine the GCPS specialization to represent the grouping and sorting subsystem for AHPS:
The relation s2 0 0 uses implementation of the
grouping and sorting CPS for the alternatives for where each criterion, and s3 0 0 is used to get the imple-
C = (M, E, Л)
S ^ CGSA(MGSA , ^GSA , AGSA)
KsA = АиЛб, Л6 = {MGSA ^ T2 ^ N2 ,
mentation results of the sorting and grouping CPS for criteria:
= <П(ap \Nxpp,Pk ◊ Qфp)>. p=i
E^ = {S,©,O,n},n = © = ,
S = {•, o, ◊}, O = {-=-, *, :=, >, < %, +, \, =, h, X,[]}}, S - binding operation, © - output operations, O - operations on attributes, n - substitution operations.
Partial axiomatics A6 is presented below. Terminal alphabet contains many alternatives and criteria with their attributes.
The substitution rules include a substitutive relation and a set of operations on attributes. The mentation result of the grouping and sorting CPS , , . ^ ,,
substitutive relations contain the available attribute
o.
'3.0.0 =< XlQ-k,0,.«Q^Q4
The relation s4 0 0 is used to obtain the imple-for alternatives:
P >ф p )).
S4,0,0 =< П (XlNxpp,Pk ◊ QФ p=1
The following substitutive relation is aimed to enter a set of alternatives into the construct. Operations on attributes contain the calculation of global alternative priorities:
/ - P N \
S5,0,0 =( Q ^-n ( Q * p ) ^n ( vxi V'
p=1 ¿=1 /
To ^5,0,0 4n (VJX- :=(rp J^)* rt J* p )Y
dr, 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 (dr=1) this attribute is omitted for record simplicity.
To interpret the CPS of alternative grouping and sorting let us use БАС CAAHPS, described
above:
{CGSA = {MGSA , EGSA , ЛGSA) ,
CA, AHPS = (M
A,AHPS , EA,AHPS , ЛA,AHPs)) I ^
I i =1
The set of substitutive relations s6 0 0 allows
ordering the alternatives into constructs according to their ranks:
I ^ I CGSA = {MGSA , E
(N-1 N
П П ( vXi • Vх; d 6
i=1 j=i+1
where
Л I,,
fj
GSA' ЛI ,GSA , ZGSA) ,
= Л(Ш U Л7 , Лз = {(A10 \AJ-),
Vх; • vX )
( A2,l„, f
0 c 1*4 t л0 a , ч / A0 c
(А0Ць J*), (AX J :=), (AX
i=1
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦШНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
(A50t,L J-), (Ai00j /), (A^ J%),
(Ai02ia,b j^), ( a" j=), ( ao4 j&),
(A« Г = JÄ), (Ai°6 |r = JX), (A i7ia j[]),
Ш N1
(A« ia,b j+), ( ax j>), ( a203 jo )}.
We concretize CPS of alternative grouping and sorting:
CGSA = {MGSA , EGSA ,ЛI ,GSA , ZGSA )
,GSA = (MGSA , EZ,GSA ,ЛI,GSA U Л7 U UAs, ZGSÄ) ,
where
Л 7 = {To = T U {-ym,} U {-ylj} U {^, j},
m = 1,M и j = 1,4, No = {pnX, n,pY, p,„ß, gЯ
U i = n ,
p,n ßm П(qymi )• P,kp i=1
k3 ,k4 ,M a,{Xm }} U { ahn V m , ей n V'm },
m
= 1, M , U = { p,N X}, ^ = {V r :(Sr , gr )},
pn Pm - PCM by criterion p for the group m,
consisting of n - alternatives, this matrix elements are non-terminals pahPmiJ, where the attributes
a and h - the same as for ah Yi j;
- non-terminal of CPS imple-
U yL''Q"a
r = 1,14}, r - rule number, a - is responsible for determining the number of groups with four and three alternatives (k3, k4- number of groups with three or four alternatives in the group, respectively, M - total number of groups); ch,n vm,
chn v'm - non-terminals of m -th group of alternatives, with attributes ch - flag indicating the alternative position changes in the group (1 - alternatives changed their position after ranking in the
group, 0 - did not change); Np y - PCM for N alternatives according to the criterion p , matrix elements, non-terminals ahyiJ with attributes: a - evaluation of comparison of i and j alternatives, h - evaluation process tool, (h = 1 - filled according to the evaluation by an external server, expert, h = 0 - without the involvement of an external expert on the basis of substitution rules);
pn Pm n(qymi P,kp i=1
mentation of the single-level classical AHP for the
n
group alternatives: n ~ymi - set of alternatives
i=1
for AHP ranking, p - ranking criterion number, Pk - criterion vector, e^ - PCM of alternatives
of the group with calculated ranks and conformity relation, nL' - list of alternatives ordered according to the ranks, pn Pm - PCM of alternatives in the group n; xm - non-terminal to prepare m -th group alternatives for ranking; all n - non-terminal to calculate the parameters of the general PCM of alternatives (missing evaluations of paired comparisons, conformity relation and matrix completion control); { q yma } , { q ym i } , { q ^ ) - set of alternatives in the group m, y, y', z - alternative identifier, q = [name,v,u,7,l] - set of attributes
where name - alternative semantics, v - global priority (weight) of an alternative, u - global number of alternative, r - alternative weight vector 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 nonterminal 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 comparison evaluations are completed with default values:
S = X i- r-
1 \ Л 1 NX,p, pk
П ( name,v Xu ) p,N У
k3,k4,Ma /,
т0 g1 = ( flag := 0, * (N%4 = 0;3;(k4 Ja := N /4), k Ja := 0),(flag := 1)),*(N = 3;3;(k4Ja := 0), k Ja := 1),(flag := 1)), * (N = 5;3;(k4Ja := 0),
I
u=1
Наука та прогрес транспорту. Вкник Дншропетровського нащонального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦ1ИНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
(кз Ja := 2),(flag := 1)), + (flag = 0;2; (k4 Ja := N /4 - (3 - N%4)), (k3 Ja := ( N - k4 Ja* 4)/3)),
M Ja := k3 Ja + k4 Ja;
n (n (aJ pYi,j := 0aJ pY„ := 0;hJ pyu := 0;
i=1 j=i+1
hJpyhl := 0;);aJ^yUi := 1;hJ^yUi := 0;) ).
The following relation is applied for breakdown of the alternatives into the groups. The operations on attributes get the conformity between the general list of alternatives and alternatives in groups:
/ N MJa "Jym
s2 ^n ( ) • k3,k4, M n ( cM y- n ~qym; )
i u =1
m=1
i=1
"3
,g2 = ( П (nJ^m = 3;
i=1
nJVm
П (nameJyi j := nameJx* j ;
j=1
uJy,j := uJx*j)); П (nJvm = 4;
=k3+1
= ( П ( Oh, n V-П ~qym,, ) • p, N Y)
M Ja nJ^m = _
П ( oh,n V-П qym,i ' p,nJ^ïï ß m 'P m ) - p,N y) ),
m=1 i =1
M nJVm nJVm
g =< П(П (П (aJßm,i,j :=aJY.
m=1 i =1 j =1
uJVi ,uJ-V j'
h Jßm,i,j := ^YuJ^uJ^))) ).
The following substitutive relation is used for
CPS implementation of the classic single-level AHP for each alternative group:
M Ja nJ^m
= ( П (oh,nУ-П qy>
m=1
i=1
q У
ßm -PJ)-
M nJVi J} ( -qZi i ),Q X
'П ( oh,n У'Ц ( qymi )\ _ ^
m=1 i=1 F,nJvm ßm , П ( qymi )'p'kF
The operations on attributes of the following rule supplement the general PCM with new evaluations:
S5 = ( (U \ _ nJVm > " p,N ,
=1 p,nJym ßm , П ( qym,i ),p,kp i=1
= nJVm
M ßm • П ( qzm,i ),Q Xm
i=1
) • p,N Y-
m=
nJyVm
m =1
=1 i =1
M nJ^m nJ^m
tg5 = < П(П (П (aJY
П ( П qZm,i ,Q X m ) - p, N 4' all n )■
:= aJßmi,, ;
m=1 i =1 j =1
hJYu
:= hJßm,i,j))) >.
The substitutive relation is used to calculate the general PCM elements by transitiveness and to count the uncompleted elements:
nJ¥»
n (nameJyi j := nameJx*j ; uJyi,j := uJx*j )) ) .
j=i
The operations on attributes of s3 relation determine the attribute values for PCM elements of the alternatives:
M Ja nJ^m
= ( ),
N N N
tjg6 = ( allJn:= 0;nn (nWhJYi, = 0;
i=1 j=i+1 C=1
4;(-((aJYCj * 0)&(aJy,.c * 0);2;
(sum := sum + aJYi c * aJYc j; q := q +1)); aJYi j := sum / q; hJYi j := 0; hJjj i :=
= 0;aJYj,i := 0 ));) -
aJY i, j
-(aJYi,j = 0;1; allJn := allJn +1)) ).
s7 - substitutive relation for comparison of the alternatives in groups after AHP application. If the order of the alternatives in the group is changed, the corresponding attribute is set to one:
i=1
T
s
6
d
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦ1ИНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
nJVm
«JV,
nJVm
— ( П ( ch,n V-П ( qym,t ) • П (-Zm,t ),Q ^m ) ^ X fl„ Ц ^ ^П ( ch,n V'm П ( ~аУт,г У'
t—1
M
"JVm nJVm -
П ( ch,n У'Ц ( ~аУт,г ) " П ( qzm,t ),Q ^m );
m—1
t—1
Т0 g7 —
M
П (chJVm :— 0;
m —1
nJVm
m—1
t—1
t—1
X N Y" ail П d8 ^ Q 4
M
— ( f :— 0;П ("KchJvm —1;1; f :— 1));
m—1
H-(f —1;1; d.9 :— 1);
н-(((f — 0) & (ail Jn — 0);1;d8 :— 1);)
— (— h( p, N Y); rJX:— X( N Y )).
4 p,nJ^m ßm P m ) p,N Y /;
n H^^m, = namejzm,i;
i=1
3;( vJym,i := vJzm,i; nameJym, = nameJzma;
ujym,i = ujzm,i ));-(nameJym,1 * nameJzm,r ;
1;chJvm := 1 )•
The relation s8 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:
M nJVm nJvm -
S8 = ( n ( ch,n V^ ( ~qym,i ) • n ( qzm,i ),e X m ) X
g — Щ(l :— [nJTm];nJVm :— l + 2;
\m—1 2
П (nameJy'm,t :— nameJzm,1+t ; t—1
pjym,t:— pjzm,i+i ; vjym,t:— ^Jz,
i,i+t '
uJym, := uJzm,l+,■); n (nameJym,!+l := nameJzm+u; i=1
pJym,i+l := pJzm+1,i; vjym,i+l := vjzm+1,i;
uJy'm,i +l := uJzm+1,i ); M-1 nJVm nJVm
X! g9 Hn ( n ( n (X JPm,i,; :=a JYujymj,, ;
m=1 i=1 j=1
hJPm,, j := hJYuJymt uJymJ ))) ).
The substitutive relation s10 is for implementation of classic AHP for new alternative groups:
M-1
nJVm
S10 — ( ^П ( ch,n Vm ' ^П ( qym,t ) " p,nJvm ßm 'P m ) ^
m—1 t—1
M-1
^Пх
m—1
f «jvm ^
nJVm П (qZmt ),QЯ.
ch,nV'" П (qym,t)"u| t—1
t —1
V
_ "JVm
,,nJVm ßm . П ( qy'mt )'P,kp
J
The substitutive relation S9 is used to regroup The following rule contains the substitutive re-the alternatives. Operations on attributes allow set- lation to get the AHP ranking result in each group
ting the alternative attributes in the new groups:
nJVm
— (П(ch,nV" П (qym,t);
nJVm
X П ( qzm,t ),Q ^ m ) • p,N YX
t —1
and to save the evaluation entered into the general PCM by the expert:
= nJV'm
M-1 p,nJV'm Pm , J! ( ~qzmi )-e "Am =
s11 = \ H(u 1 _ n ' )• p,NY^
m=1 p,nJv'm Pm , n ( qym* -1'p'kp i=1
nJvm _ =
^n ( n (qzmi ),e Xm ) • p,N Y^ all n
m =1 i =1
M -1
T
T
0
T
Наука та прогрес транспорту. Вкник Дншропетровського нащонального ушверситету зашзничного транспорту, 2016, № 4 (64)
ШФОРМАЦ1ИНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
M-1 KJVm nJ\Vm
1 g11 = ( П ( П ( П (aJYuJZm„uJZmJ '= aJßm,i,j- ;
m=1 i =1 j=1
hJYu
:= hJßm,i,j)));d6:= 1;du := 1).
Operations on attributes of the following rule allow determining the changes in the positions of alternatives in the groups after AHP application:
m-1
si2 =( ), T0 gl2 = ( H (chJVm := 0;
, gH = /f[<' := [,,
m=1
П (nameJym,l+i := name Jzm,i ;
i=1
pJym,l+i := pJZm,i ;V Jym,l+i := VjZm,i ;
2
u Jy m ,l+i := uJZm,i ); П (name Jy m+1,i := »ameJZm,i+l ;
i=1
m=1
П (-(nameJym ,,■ = nameJzm i ;1; vJym ,,■ := vJzm,i),
p Jym+1,i := pJZm,i+l ; Vjy,
m+1,i "
i=1
+(nameJy'mф nameJzmi ;1; ohJvm := ^. The substitutive relation s13 is used to calculate
= V Jzm,i+l ;u Jy«+1,i := u Jzm,i+l )
T1 g14 = (d3:= 1; d14 := 0).
the priority vector and the conformity relation for general matrix, if the position of the alternatives in the groups has not changed:
Ш »Д»
s13 НП ( ch,n У'Ц ( -qy^ ) • П ( -qzm,t ),Q X m )
i m=1
i=1
,g13 = ( f := 0; П (HohJVm = 1;1; f := 1));
Implementation of CPS of alternative grouping and sorting is the non-terminal with calculated alternative rank attributes and the conformity relation for PCM of the alternatives.
Constructive and productive structure for classical single-level AHP. CPS of classical singlelevel AHP implements completing by an external expert of some paired comparison evaluations, x P,N Y' aii n dl3 ^ Q M, finding the proper number of the matrix, conformity relation of PCM and alternative ranks.
Let us determine GCPC specialization to represent classical single-level AHP:
i=1
-(f = 1;1;dj3 := 1); -(((f = 0) & (allJn = 0);1;dj4 := 1) ).
T0 gi3 = (is JX: = h( p,N Y); rJX := *( p,N Y^.
The following substitutive relation is used to restore alternatives in the groups, if their position has changed:
M nJym M-0
si4 = ( n ( ch,n n ( qymi )) - (n (ch,n y'm X
m=1
i =1
m=1
nJVm »JVm
ХП ( ~qy'mi ) • П ( q^i ),Q X m ) d14 ^
i=1
i=1
M nJVm
^П ( oh,n У П qym;) ),
C = <M, ^ A> S ^ CAHp(M
AHP > ^AHP > AAHP / ,
where AAHP = AuA9, A9 = {MAHP 373, UN3, Eahp = {S,©,O}, © = , S = {•, o},
O = {-,:=,/, <, >, = >}}.
The operation > (xi, x. , kp) allows setting a value of the link weight between i and j alternatives (criteria) by kp criterion, > (xi, x., s) 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.
The output process forms the construct that will include the following forms: ah(xi ° x. ° kp) - link
of i and j alternatives by criterion kp, a - link
weight, h -attribute, responsible for the weight value derivation process (h = 1 - filled according
T
m =1
Наука та прогрес транспорту. Вкник Дншропетровського нащонального ушверситету залiзничного транспорту, 2016, № 4 (64)
IНФOРМАЦIИНO-КOМУНIКАЦIИНI ТЕХНOЛOПÏ ТА МАТЕМАТИЧНЕ МOДЕЛЮВАННЯ
to the assessment by an external server expert, ternative links, Pp - non-terminal to indicate crite-
h = 0 - without the involvement of an external ria links, ppiJ- non-terminals to indicate links
expert on the basis of substitution rules); ' '
a,h (x o x} o s) - link of i and j alternative (crite- between alternatives 1,j by p-th criterion_(for rion) if a comparison criterion is not given; q X -
simplicity indicated as matrix pN ß ), q à -
pairwise comparison matrix for alternatives; pairwise comparison matrix for altematives (where
e = [v, is, N] - vector of attributes: v - vector of
N
П ( rxi ) - sorted alternative sequence.
PCM priorities, is - matrix conformity relation,
For interpretation of this CPS we use the BAS N - the number of alternatives).
The substitutive relation sj 0 0 serves to change
PCM completion and ranking of alternatives xi by
criterion kp . All axiom input parameters are added
to the medium.
i=1
CA,AHPS described above:
{CAHP = {MAHP , EAHP ,ЛAHP ), CA,AHPS =
= (M A, AHPS , E A, AHPS ,Л A, AHPS ) ) I ^
Ca
M
AHP , E AHP ,ЛI, AHP ,
Z>.
si,o,o = (
p,n ß :п( qxi )
where
U| - '=i - ^ nN
'ß . NX,p, Pk p-N
ß ^Ц )■
ЛI,AHP = ЛAHP U Л11 , Л11 = {(Л1« \Л ,Aj
^if^fJ^), Лз lf,T JN, Л4 J\H), (ЛХ J :=), (Л80 ia,b J<): (Л50 \L,n,L J-), ( Alo \cab j /),( aî2 ia,b л,0, ia,b J=)
( Ai9 a ,bJ >).
(A" |is Y Jft),( A106 |r Y JX),( A201 ia,b J »),
N Y N Y
(A202 ix;?' j°)}.
Let us perform specification of the interpreted CPS for a single-level AHP:
The following substitutive relation is used, if a ranking criterion is specified:
= NN
S2,0,0 = ( pN P -S d2 0 0 ^nn( p Pi,, • kp P )),
i=1 j=1
T0 g2,0,0 = (pJP > 0;1;d2,0,0 := 1),
-(pjp = 0;1; ¿3,0,0 := 1)).
The substitutive relation s3,0,0 is used to form PCM alternatives, if a criterion is not specified:
3,0,0
= ( p,N ß •S
N N
ПП( p ßi.j )>■
i=1 j=1
I ,Ca
a ahp CAHP = {MAHP , EAHP , ЛI, AHP , Z) CK, AHP ={¡MAHP , EК, AHP ,ЛI, AHP U Л9 U Л10, Z) :
where
Л4 = {T3 = Ti, N3 = { n Ç, p P, S, Ц, p,n U, Q À, p,n ß},
U = {p,nU},^K = {Vr :(sri,j:gr,i,j>}: r = TT2},
The following rule contains the substitutive relation to determine a connection between the alternatives. Operations on attributes determine the evaluation of alternatives links:
(N N \
n№ i.j^(Xi0Xj°s)-Pj,
T g4,0,0 = ( xJ( Xi ° Xj ° S):= ajPi, j ,
h J(xi °Xj o s) := hjpt .
The rules for setting the alternative pairwise
r - rule number, i and j - number of the first
and second pairs of alternatives, U - set of initial
• , e . • , . • j. . , comparison values by an expert, where i = 1,N
-terminals, N ç - non-terminal to indicate al- ^ J *
non
Наука та прогрес транспорту. Вкник Дншропетровського нащонального ушверситету залiзничного транспорту, 2016, № 4 (64)
1НФОРМАЩИНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
and j = 1 + i,N :
s5,i, j = ( ) , T0 g5,i, j = (a J(X i °Xj ° S):= > (Xi, Xj , s) ,
aJ$t j := hJ (xi oXj ° s);hJ (x j °xi ° s) := 1,
a J( x j ° xi o s) := y (a J( x i ° Xj ° s)),
aJpj i := aJ(x j °xi ° s); hJ$i j := hJ (x j °xi ° s) ) .
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:
N N
'nn(Xi ° Xj ° s) d6,0,0 -1 j=1
NN _
^nn(Xi ° Xj ° s) ' Q k);
i=1 j=1
T0 g6,0,0 = (d6,0,0 := 1, M := 1,
N N
nn(-(aJ(Xi°Xj os) < 0;1;
i=1 j=i+1
full := 0)), -(full = 0;1; d^,0 := 0)),
Ti g6,0,0 = (isJk: = h( p,np); rJk := X( p,NP )).
The relation s7 0 0 enters the sequence of alternatives with the weights into the construct, if PCM conformity relation is valid:
N N
nn( x- ° xj °s) x i=1 j=1
__N _
x Q d7 — qx,) ' Q k< 0 g7,0,0 = (-((isJk) < 0,011; 1;d^ := 1)),
N
T g7,0,0 = ( n (vJXi := ri Jk); d7,0,0 := 0) .
i=1
s8,i, j = ^( vXi ' vXj ) d8ij — ( vXj ' vXi
T0 gs,i,j- =(-(vJxj > vJx d8,i,j :=0)).
The substitutive relation s9 0 0 is used to establish the link between the alternatives by the given
criteria:
(N N \
nil n ^ i, j ' kp p—( x i ° xj ° kp) 'p i, j /,
t0 g9,0,0 = (aJ(xi °Xj ° kp ):= aJPi, j ,
hJ(XioXj okp):= hJpj .
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:
si0,i, j = ( ) , T0 gl0,ij = (a J(x i °xj ° kp ):= » (xi, xj , kp ),
aJPi, j := aJ(^ Ox. O kp );hJ(^ Ox. O kp ) := 0
hJPi,j :=hJ(xi °xj ° kp);aJpj,i := aJ(x. °x ° kp);
aJ(x j o^ ° kp ):= 1/aJ(x, °x. ° kp) hJ(x j °x ° kp ):= 1;hJp. t := hJ(x. °x ° kp )).
The operations on attributes of the next rule check the PCM completion of alternatives by p -th
criterion. If everything is completed, then the conformity is calculated and the alternative priority vector is filled, otherwise the rule does not apply:
J11,0,0
The following set of rules ( i = 1, N -1, j = i +1, N ) defines the relations for descending ordering of alternatives according to their weights:
= ( ПП(Х 0 xj 0 kp ) dn,0,0 ^ i=1 j =1
N N -
^ПП(х °Xj оkp)• QXX
i=1 j=1
t0 g11 =(M := d11,0,0 := 1,
N N
ПП(^(aJ(Xi°Xj °kp) < 0;
i=1 j=1
1; full : = 0)),<full = 0;1;d„,0,0 := 0^,
Наука та прогрес транспорту. Вкник Дншропетровського нащонального ушверситету залiзничного транспорту, 2016, № 4 (64)
1НФОРМАЩИНО-КОМУШКАЩИШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
x, g11,0,0 = (iS JX: = h( p,N P ); rjX := *( p,NP )) .
The relation s12 0 0 is used to generate a sequence 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:
NN — _
S12,0,0 =( nn(Xi ° Xj ° kp ) - p, N P-e ^ ¿12 0 0 ^ i=1 j=1
= N
^ pnP-n(qXi)• eV,
i=1
x0g12,0,0 = (-((isjI) < 0,011; 1;¿12,0,0 := 1)> ,
x, g12,0,0 = ^TI^ := ri J^); d12,0,0 :=
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 constructive alternatives ranking process consists of three CPS, interacting at different levels of refinement transformations. Disaggregation of process components makes it possible to independently change some models, change their interpretation, which allows applying this approach to solve more specific 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.
LIST OF REFERENCE LINKS
1. Колесникова, С. И. Модификация метода анализа иерархий для динамических наборов альтернатив / С. И. Колесникова // Прикладная дискретная математика. - 2009. - № 4 (6). -С. 102-109.
2. Миронова, Н. А. Интеграция модификаций метода анализа иерархий для систем поддержки принятия групповых решений / Н. А. Миронова // Радиоэлектроника, информатика, управление. - 2011. - № 2 (25) - С. 47-54.
3. Саати, Т. Л. Об измерении неосязаемого. Подход к относительным измерениям на основе главного собственного вектора матрицы парных сравнений [Electronic resource] / Т. Л. Саати // Cloud оf Science. - 2015. - Т. 2, № 1. - C. 5-40. - Available at: https://cloud-ofscience.ru/sites/default/files/pdf/CoS_2_1 .pdf. -Title from the screen. - Accessed : 17.06.2016.
4. Саати, Т. Л. Принятие решений при зависимостях и обратных связях: Аналитические сети / Т. Л. Саати. - Москва : Изд-во ЛКИ, 2008. -360 с.
5. Шинкаренко, В. И. Конструктивная модель адаптации структур данных в оперативной памяти: Часть I. Конструирование текстов программ / В. И. Шинкаренко, Г. В. Забула // Наука та прогрес транспорту. - 2016. - № 1 (61). - С. 109-121. doi: 10.15802/stp2016/60998.
6. Шинкаренко, В. И. Конструктивно-продукционные структуры и их грамматические интерпретации. I. Обобщенная формальная конструктивно-продукционная структура / В. И. Шинкарен-ко, В. М. Ильман // Кибернетика и системный анализ. - 2014. - № 5. - С. 8-16.
7. Шинкаренко, В. И. Конструктивно-продукционные структуры и их грамматические интер претации. II. Уточняющие преобразования /
B. И. Шинкаренко, В. М. Ильман // Кибернетика и системный анализ. - 2014. - № 6. - С. 15-28.
8. Шинкаренко, В. I. Моделювання процесу ран-жування альтернатив методом анатзу ieрархiй засобами конструкцшно-продукцшних структур / В. I. Шинкаренко, Т. М. Васецька // Мате-матичш машини i системи. - 2016. - № 1. -
C. 39-47.
9. Expert Choice for Collaborative Decision Making [Electronic resource]. - Available at: http://expertchoice.com/. - Title from the screen. -Accessed : 17.06.2016.
Наука та прогрес транспорту. Вкник Дншропетровського нащонального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦШНО-КОМУШКАЦШШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
10. Saaty, T. L. Models, Methods, Concepts & Ap- 13. plications of the Analytic Hierarchy Process /
T. L Saaty, L. G. Vargas. - New York : Springer Science & Business Media, 0010. - 345 p.
11. Saaty, T. L. Relative Measurement and Its Generalization in Decision Making Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intangible Factors The Analytic Hier- 14. archy/Network Process / T. L. Saaty // Revista de
la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A: Matematicas. - 2008. -Vol. 102. - Iss. 2. - P. 251-318. doi: 10.1007/B-F03191825.
12. Saaty, T. L. The possibility of group choice: pair- 15. wise comparisons and merging functions /
T. L. Saaty, L. G. Vargas // Social Choice and Welfare. - 2012. - Vol. 38. - Iss. 3. - P. 481-496. doi: 10.1007/s00355-011-0541-6.
T. M. BACEUUKKA1*
The Group Decision Support System to Evaluate the ICT Project Performance Using the Hybrid Method of AHP, TOPSIS and Copeland Score / H. Setiawan, J. E. Istiyanto, R. Wardoyo, P. Santoso // Intern. J. of Advanced Computer Science and Applications. - 2016. - № 7 (4). -P. 334-341. doi: 10.14569/IJACSA.2016.070444. Shynkarenko, V. I. Reducing the number of expert judgments in analytic hierarchy process by sorting and survey management / V. I. Shynkarenko, T. M. Vasetska, E. Y. Boiko // Системш технологи: perioH. мгжвуз. зб. наук. пр. - Дншропет-ровськ, 2016. - Вип. 2 (103). - С. 16-31. Super Decisions Software [Electronic resource]. -Available at: http://www.super-decisions.com/. -Title from the screen. - Accessed : 17.06.2016.
1 Каф. «Комп'ютерт шформацшт технологи», Дтпропетровський нацюнальний утверситет затзничного транспорту ÍMem академжа В. Лазаряна, вул. Лазаряна, 2, Дшпро, Украша, 49010, тел./факс +38 (098) 237 05 21, ел. пошта tetyana@vasetsky.com, ORCID 0000-0001-7008-2839
МОДЕЛЮВАННЯ МОДИФ1КОВАНОГО МЕТОДУ АНАЛ1ЗУ 1СРАРХШ ЗАСОБАМИ КОНСТРУКТИВНО-ПРОДУКЦ1ЙНИХ СТРУКТУР
Мета. У дослвдженш передбачаеться: 1) розширити можливостi класичного методу аналiзу ieрархiй (МА1) для велико! шлькосп альтернатив га критерив; 2) побудуваги модель конструктивного процесу при-йняггя рiшень iз використанням модифiкованого методу аналiзу ieрархiй iз сортуванням (МА1С). Методика. Для досягнення поставлено! мети використовуеться механiзм конструктивно-продукцшних структур (КПС). Виконано уточнюючi перетворення узагальнюючо! конструктивно-продукцiйно! структури (УКПС). Результата. Розроблена модель конструктивного процесу представляе собою взаемодш трьох структур: 1) загально! структури КПС МА1С, яка дозволяе задати альтернативи та критери, виконуючи декомпозицш iерархiчно! структури задачц 2) КПС групування та сортування, яка розбивае альтернативи (критерi!) на групи та реалiзуе класичний однорiвневий МА1 для кожно! групи, а також розраховуе оцiнки парних порiв-нянь на основi введених даних; 3) КПС однорiвневого класичного МА1, яка дозволяе заповнити матрицю парних порiвнянь та розрахувати ранги альтернатив. Ва три структури взаемодшть мгж собою на рiзних рiвнях уточнюючих перетворень: через узгодження по даним на рiвнi конкретизацi!' та використання реаль зацiй. Запропонована модель дозволила перейти на бшьш абстрактний рiвень представлення розв'язку задач прийняття ршень для велико! кiлькостi критерив та альтернатив. Наукова новизна. За результатами роботи пропонуеться використовувати мехашзм КПС для форматзаци модифiкацiй МА1 iз сортуванням для розв'язку задач прийняття ршень iз великою шльшстю критерi!в та альтернатив. Практична значимкть. Форма-лiзацiя представлення як самого методу анатзу iерархiй, так i його модифiкацiй дозволяе розширити коло застосування даного методу, впорядкувати описи рiзних модифiкацiй МА1. Таке представлення забезпечуе можливiсть розробки програм для реатзаци гiбридних модифiкацiй методу. Використання рiзних штерпре-тацiй запропонованих в статп КПС дозволить використати iншi подходи при визначеннi узгодженостi мат-риць парних порiвнянь, розрахунку оцшок та рангiв альтернатив i критерив.
Ключовi слова: моделювання; конструктивно-продукцiйнi структури; конструктивний процес; метод аналiзу iерархiй; модифiкацiя
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦШНО-КОМУШКАЩЙШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
Т. Н. ВАСЕЦКАЯ1*
1 Каф. «Компьютерные информационные технологии», Днепропетровский национальный университет железнодорожного транспорта имени академика В. Лазаряна, ул. Лазаряна, 2, Днипро, Украина, 49010, тел./факс +38 (098) 237 05 21, эл. почта tetyana@vasetsky.com, ORCID 0000-0001-7008-2839
МОДЕЛИРОВАНИЕ МОДИФИЦИРОВАННОГО МЕТОДА АНАЛИЗА ИЕРАРХИЙ СРЕДСТВАМИ КОНСТРУКТИВНО-ПРОДУКЦИОННЫХ СТРУКТУР
Цель. В исследовании предполагается: 1) расширить возможности классического метода анализа иерархий (МАИ) для большого количества альтернатив и критериев; 2) построить модель конструктивного процесса принятия решений с использованием модифицированного метода анализа иерархий с сортировкой (МАИС). Методика. Для достижения поставленной цели используется механизм конструктивно-продукционных структур (КПС). Выполнены уточняющие преобразования обобщенной конструктивно-продукционной структуры. Результаты. Разработанная модель конструктивного процесса представляет собой взаимодействие трех структур: 1) общей КПС МАИС, которая позволяет определить альтернативы и критерии, выполняя декомпозицию иерархической структуры задачи; 2) КПС группировки и сортировки, которая разбивает на группы альтернативы и критерии, реализуя для каждой из групп классический одноуровневый МАИ, а также рассчитывая оценки парных сравнений на основании введенных данных; 3) КПС одноуровневого классического МАИ, которая позволяет заполнить матрицу парных сравнений и рассчитать ранги альтернатив. Все три структуры взаимодействуют между собой на разных уровнях уточняющих преобразований: посредством согласования по данным на уровне конкретизации и использования реализаций. Предложенная модель позволила перейти на более абстрактный уровень представления разрешения задач принятия решений для большого количества критериев и альтернатив. Научная новизна. По результатам работы предлагается использовать механизм КПС для формализации модификаций МАИ с сортировкой для разрешения задач принятия решений с большим количеством критериев и альтернатив. Практическая значимость. Формализация представления как самого метода анализа иерархий, так и его модификаций позволяет расширить круг применения данного метода; унифицировать описания различных модификаций МАИ. Такое представление обеспечивает возможность разработки программ для реализации гибридных модификаций данного метода. Использование разных интерпретаций представленных в статье КПС позволит использовать другие подходы при определении согласованности матриц парных сравнений, расчета оценок и весов альтернатив и критериев.
Ключевые слова: моделирование; конструктивно-продукционные структуры; конструктивный процесс; метод анализа иерархий; модификация
REFERENCES
1. Kolesnikova S.I. Modifikatsiya metoda analiza ieyrarkhiy dlya dinamicheskikh naborov alternativ [Modification of hierarchies analysis method for dynamic set of alternatives]. Prikladnaya diskretnaya matematika -Applied Discrete Mathemathics, 2009, no. 4 (6), pp. 102-109.
2. Mironova N.A. Integratsiya modifikatsiy metoda analiza iyerarkhiy dlya sistem podderzhki prinyatiya grup-povykh resheniy [Integration of modifications of hierarchy analysis method for support systems of group decision making]. Radioelektronika, informatika, upravleniye - Electronics, Computer Science, Management, 2011, no. 2 (25), pp. 47-54.
3. Saati T.L. Ob izmerenii neosyazayemogo. Podkhod k otnositelnym izmereniyam na osnove glavnogo sobstvennogo vektora matritsy parnykh sravneniy [On the measurement of the intangible. The approach to the relative measurements based on the main eigenvector of pairwise comparisons matrix]. Cloud of Science, 2015, volume 2, issue 1, pp. 5-40. Available at: https://cloudofscience.ru/sites/default/files/pdf/CoS_2_1.pdf (Accessed 17 June 2016).
4. Saati T.L. Prinyatiye resheniy pri zavisimostyakh i obratnykh svyazyakh: Analiticheskiye seti [Decision making at dependences and feedbacks: Analytic networks]. Moscow, LKI Publ., 2008. 360 p.
5. Shinkarenko V.I., Zabula H.V. Konstruktivnaya model adaptatsii struktur dannykh v operativnoy pamyati: Chast I. Konstruirovaniye tekstov programm. [Constructive model of data structures adaptation in ram: Part I.
Наука та прогрес транспорту. Вкник Дншропетровського нащонального ушверситету залiзничного транспорту, 2016, № 4 (64)
ШФОРМАЦШНО-КОМУШКАЦШШ ТЕХНОЛОГИ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ
Program text constructing]. Nauka ta prohres transportu - Science and Transport Progress, 2016, no. 1 (61), pp. 109-121. doi 10.15802/stp2016/60998.
6. Shinkarenko V.I, Ilman V.M. Konstruktivno-produktsionnyye struktury i ikh grammaticheskiye interpretatsii. Obobshchennaya formalnaya konstruktivno-produktsionnaya struktura. [Constructive-synthesizing Structures and their Grammatical Interpretations. I. Generalized Formal Constructive-synthesizing Structure]. Kiber-netika i sistemnyy analiz - Cybernetics and Systems Analysis, 2014, vol. 49, issue 5, pp. 1-15.
7. Shinkarenko V.I, Ilman V.M. Konstruktivno-produktsionnyye struktury i ikh grammaticheskiye interpretatsii. Obobshchennaya formalnaya konstruktivno-produktsionnaya struktura. [Constructive-synthesizing Structures and their Grammatical Interpretations. I. Generalized Formal Constructive-synthesizing Structure]. Kiber-netika i sistemnyy analiz - Cybernetics and Systems Analysis, 2014, no. 6, pp. 15-28.
8. Shynkarenko V.I., Vasetska T.M. Modeliuvannia protsesu ranzhuvannia alternatyv metodom analizu iierarkhii zasobamy konstruktsiino-produktsiinykh struktur [Simulation of ranking process of alternatives using the method of analytic hierarchy by means of structurally-productive structures]. Matematychni mashyny i systemy - Mathematical Machines and Systems, 2016, issue 1, pp. 39-47.
9. Expert Choice for Collaborative Decision Making. Available at: http://expertchoice.com/ (Accessed 17 June 2016).
10. Saati T.L., Vargas L.G. Models, Methods, Concepts & Applications of the Analytic Hierarchy Process. New York, Springer Science & Business Media Publ., 2012. 345 p.
11. Saaty T.L. Relative Measurement and Its Generalization in Decision Making Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intangible Factors The Analytic Hierarchy/Network Process. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2008, vol. 102, issue 2, pp. 251-318. doi: 10.1007/BF03191825.
12. Saaty T.L., Vargas L.G. The possibility of group choice: pairwise comparisons and merging functions. Social Choice and Welfare, 2012, vol. 38, issue 3, pp. 481-496. doi: 10.1007/s00355-011-0541-6.
13. Setiawan H., Istiyanto J.E., Wardoyo R., Santoso P. The Group Decision Support System to Evaluate the ICT Project Performance Using the Hybrid Method of AHP, TOPSIS and Copeland Score. International Journal of Advanced Computer Science and Applications, 2016, no. 7 (4), pp. 334-341. doi: 10.14569/IJACSA.2016.070444.
14. Shynkarenko V.I., Vasetska T.M., Boiko E.Y. Reducing the number of expert judgments in analytic hierarchy process by sorting and survey management. Systemni tekhnolohii. Rehionalnyi mizhvuzivskyi zbirnyk nau-kovykh prats [System technologies. Regional Interuniversity collection of scientific papers], 2016, vol. 2 (103), pp. 16-31.
15. Super Decisions Software. Available at: http://www.superdecisions.com/ (Accessed 17 June 2016).
Prof. V. Ye. Belozerov, Dr. Sc. (Phys. and Math.) (Ukraine); Prof. V. I. Shinkarenko, Dr. Sc. (Tech.)
(Ukraine) recommended this article to be published
Received: March 22, 2016
Accepted: July 20, 2016
