Теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. Частина 2.2
In this part we introduce the role of benefits, opportunities, costs and risks (BOCR) in decision- making and how to establish priorities for them. We give an example of a real life application of the US Congress acting on China joining the World Trade Organization (WTO) mailed to the US congression...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2019
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System research and information technologies| _version_ | 1867334384220110848 |
|---|---|
| author | Saaty, Thomas L. |
| author_facet | Saaty, Thomas L. |
| author_institution_txt_mv | [
{
"author": "Thomas L. Saaty",
"institution": null
}
] |
| author_sort | Saaty, Thomas L. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
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| datestamp_date | 2019-07-26T17:25:36Z |
| description | In this part we introduce the role of benefits, opportunities, costs and risks (BOCR) in decision- making and how to establish priorities for them. We give an example of a real life application of the US Congress acting on China joining the World Trade Organization (WTO) mailed to the US congressional committee before that decision. We then introduce and apply the Analytic Network Process and its concept of a supermatrix to make decisions with dependence and feedback and illustrate its application for a single "control" criterion of market share. This will be followed in Part 2.3 by a full BOCR application in the context of the ANP. |
| first_indexed | 2025-07-17T10:25:50Z |
| format | Article |
| fulltext |
© Saaty Thomas L., 2003
Системні дослідження та інформаційні технології, 2003, № 2 7
TIДC
ПРОБЛЕМИ ПРИЙНЯТТЯ РІШЕНЬ І
УПРАВЛІННЯ В ЕКОНОМІЧНИХ, ТЕХНІЧНИХ,
ЕКОЛОГІЧНИХ І СОЦІАЛЬНИХ СИСТЕМАХ
UDC 519.5
THEORY OF THE ANALYTIC HIERARCHY AND ANALYTIC
NETWORK PROCESS – EXAMPLES. PART 2.2
SAATY THOMAS L.
In this part we introduce the role of benefits, opportunities, costs and risks (BOCR)
in decision-making and how to establish priorities for them. We give an example of
a real life application of the US Congress acting on China joining the World Trade
Organization (WTO) mailed to the US congressional committee before that deci-
sion. We then introduce and apply the Analytic Network Process and its concept of a
supermatrix to make decisions with dependence and feedback and illustrate its ap-
plication for a single “control” criterion of market share. This will be followed in
Part 2.3 by a full BOCR application in the context of the ANP.
1. EVALUATING THE BOCR MERITS THROUGH STRATEGIC CRITERIA
USING RATINGS
This section was taken from an analysis carried out before the US Congress acted
favorably on China joining the WTO and was hand-delivered to many of the
members of the committee including its Chairperson [4]. Since 1986, China had
been attempting to join the multilateral trade system, the General Agreement on
Tariffs and Trade (GATT) and, its successor, the World Trade Organization
(WTO)]. According to the rules of the 135-member nation WTO, a candidate
member must reach a trade agreement with any existing member country that
wishes to trade with it. By the time this analysis was done, China signed bilateral
agreements with 30 countries — including the US (November 1999) — out of 37
members that had requested a trade deal with it.
As part of its negotiation deal with the US, China asked the US to remove its
annual review of China’s Normal Trade Relations (NTR) status, until 1998 called
Most Favored Nation (MFN) status. In March 2000, President Clinton sent a bill
to Congress requesting a Permanent Normal Trade Relations (PNTR) status for
China. The analysis was done and copies sent to leaders and some members in
both houses of Congress before the House of Representatives voted on the bill,
May 24, 2000. The decision by the US Congress on China’s trade-relations status
will have an influence on US interests, in both direct and indirect ways. Direct
impacts will include changes in economic, security and political relations between
the two countries as the trade deal is actualized. Indirect impacts will occur when
China becomes a WTO member and adheres to WTO rules and principles. China
has said that it would join the WTO only if the US gives it Permanent Normal
Trade Relations status.
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 8
It is likely that Congress will consider four options, the least likely being that
the US will deny China both PNTR and annual extension of NTR status. The
other three options are:
1. Passage of a clean PNTR bill: Congress grants China Permanent Nor-
mal Trade Relations status with no conditions attached. This option would allow
implementation of the November 1999 WTO trade deal between China and the
Clinton administration. China would also carry out other WTO principles and
trade conditions.
2. Amendment of the current NTR status bill: This option would give
China the same trade position as other countries and disassociate trade from other
issues. As a supplement, a separate bill may be enacted to address other matters,
such as human rights, labor rights, and environmental issues.
3. Annual Extension of NTR status: Congress extends China’s Normal
Trade Relations status for one more year, and, thus, maintains the status quo.
The conclusion of the study is that the best alternative is granting China
PNTR status. China now has that status.
Our analysis involves four steps. First, we prioritize the criteria in each of
the benefits, costs, opportunities and risks hierarchies. Fig. 1 shows the resulting
prioritization of these criteria. The alternatives and their priorities are shown un-
der each criterion both in the distributive and also in the ideal modes. The ideal
priorities of the alternatives were used as appropriate to synthesize their final val-
ues beneath each hierarchy.
The priorities shown in Fig. 1 were derived from judgments that compared
the elements involved in pairs. For readers to estimate the original pairwise
judgments (not shown here), one forms the ratio of the corresponding two priori-
ties shown, leave them as they are, or take the closest whole number, or its recip-
rocal if it is less than 1.0.
The idealized values are shown in parentheses after the original priorities
obtained from the eigenvector. The latter sum to 1 and are called distributive
priorities. The ideal values are obtained by dividing each of the distributive priori-
ties by the largest. For the Costs and Risks structures, the question is framed as to
which is the most costly. That is, the most costly alternative ends up with the
highest priority.
It is likely that, in a particular decision, the benefits, costs, opportunities and
risks (BOCR) are not equally important, so we must also prioritize them. This is
shown in Tabl. 1. The priorities for the economic, security and political factors
themselves were established as shown in Fig. 2 and used to rate the importance of
the benefits, costs, opportunities and risks in Tabl. 1. Finally, we used the priori-
ties of the latter to combine the synthesized priorities of the alternatives in the
four hierarchies, using the normalized reciprocal - priorities of the alternatives
under costs and risks, to obtain their final ranking, as shown in Tabl. 2.
How to derive the priority shown next to the goal of each of the four hierar-
chies shown in Fig. 1 is outlined in Tabl. 1. We rated each of the four merits:
benefits, costs, opportunities and risks of the dominant PNTR alternative, as it
happens to be in this case, in terms of intensities for each assessment criterion.
The intensities, Very High, High, Medium, Low, and Very Low were themselves
prioritized in the usual pairwise comparison matrix to determine their priorities.
We then assigned the appropriate intensity for each merit on all assessment crite-
ria. The outcome is as found in the bottom row of Tabl. 2.
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 9
Be
ne
fit
s S
yn
th
es
is
(I
de
al
):
P
N
TR
1
.0
0
A
m
en
d
N
TR
0
.4
8
A
nn
ua
l E
xt
en
sio
n
0.
21
C
os
ts
S
yn
th
es
is
(w
hi
ch
is
m
or
e c
os
tly
):
PN
TR
0
.3
1
A
m
en
d
N
TR
0
.5
0
A
nn
ua
l E
xt
en
sio
n
0
.8
7
C
os
ts
S
yn
th
es
is
(le
ss
co
st
ly
, a
nd
id
ea
liz
in
g)
:
P
N
TR
1
A
m
en
d
N
TR
0
.6
1
A
nn
ua
l E
xt
en
sio
n
0
.3
5
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 10
O
pp
or
tu
ni
tie
s S
yn
th
es
is
(I
de
al
):
P
N
T
R
: 1
A
m
en
d
N
T
R
: 0
.4
4
A
nn
ua
l E
xt
en
si
on
: 0
.2
0
R
is
ks
S
yn
th
es
is
(m
or
e
ri
sk
y,
Id
ea
l):
P
N
T
R
0.
51
A
m
en
d
N
T
R
0
.5
2
A
nn
ua
l E
xt
en
si
on
0
.6
1
R
is
ks
S
yn
th
es
is
(l
es
s r
is
ky
, I
de
al
):
P
N
T
R
1
A
m
en
d
N
T
R
0
.9
9
A
nn
ua
l E
xt
en
si
on
0
.8
4
Fi
g.
1
. H
ie
ra
rc
hy
fo
r R
at
in
g
B
en
ef
its
, C
os
ts
, O
pp
or
tu
ni
tie
s,
an
d
R
is
ks
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 11
T a b l e 1 . Priority Ratings for the Merits: Benefits, Costs, Opportunities, and Risks
Intensities: Very High (0.42), High (0.26), Medium (0.16), Low (0.1), Very Low (0.06)
Benefits Costs Opportunities Risks
Growth (0.19) High Very Low Medium Very
Low Economic
(0.56) Equity (0.37) Medium High Low Low
Regional (0.03) Low Medium Medium High
Non-Proliferation
(0.08) Medium Medium High High Security
(0.32)
Threat to US (0.21) High Very High High Very
High
Constituencies (0.1) High Very High Medium High Political
(0.12) American Values
(0.02) Very Low Low Low Medium
Priorities 0.25 0.31 0.20 0.24
We are now able to obtain the overall priorities of the three major decision
alternatives listed earlier, given as columns in Tabl. 2 which gives three ways of
synthesize for the ideal mode, we see in bold that PNTR is the dominant alterna-
tive any way we synthesize as in the last four columns.
T a b l e 2 . Four Methods of Synthesizing BOCR Using the Ideal Mode
A
lte
rn
at
iv
es
B
en
ef
st
s
O
pp
or
tu
ni
tie
s
C
os
ts
R
ec
ip
ro
ca
ls
o
f
C
os
ts
C
os
ts
(d
iv
id
ed
b
y
la
rg
es
t r
ec
ip
ro
ca
l)
R
is
ks
R
ec
ip
ro
ca
ls
o
f
R
is
ks
R
is
ks
(d
iv
id
ed
b
y
la
rg
es
t r
ec
ip
ro
ca
l)
B
O
/C
R
bB
+
o
O
+
c
(1
/C
) +
r(
11
-R
)
bB
+
o
O
+
c
(1
–C
) +
r
(1
–R
)
bB
+
o
O
-
cC
–
rR
(0.25) (0.20) (0.31) (0.24)
PNTR 1 1 0.31 3.23 1 0.51 1.96 1 1.65 1.01 0.78 0.23
Amend
NTR 0.48 0.44 0.50 2.00 0.62 0.52 1.92 0.98 0.22 0.64 0.51 -0.07
Annual
Exten. 0.21 0.20 0.87 1.15 0.36 0.61 1.64 0.84 0.03 0.41 0.28 -0.32
Factors for Evaluating
the Decision
Economic: 0.56
– Growth (0.33)
– Equity (0.67)
Security: 0.32
– Regional Security (0.09)
– Non-Proliferation (0.24)
– Threat to US (0.67)
Political: 0.12
– Domestic Constituencies
(0.80)
– American Values (0.20)
Fig. 2. Prioritizing the Strategic Criteria to be used in Rating the BOCR
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 12
2. THE ANALYTIC NETWORK PROCESS (ANP)
At present, in their effort to simplify and deal with complexity, people who work
in decision-making use mostly very simple hierarchic structures consisting of a
goal, criteria, and alternatives. Yet, not only are decisions obtained from a simple
hierarchy of three levels different from those obtained from a multilevel hierar-
chy, but also decisions obtained from a network can be significantly different
from those obtained from a more complex hierarchy. We cannot collapse com-
plexity artificially into a simplistic structure of two levels, criteria and alterna-
tives, and hope to capture the outcome of interactions in the form of highly con-
densed judgments that correctly reflect all that goes on in the world. We must
learn to decompose these judgments through more elaborate structures and organ-
ize our reasoning and calculations in sophisticated but simple ways to serve our
understanding of the complexity around us. Experience indicates that it is not very
difficult to do this although it takes more time and effort. Indeed, we must use feed-
back networks to arrive at the kind of decisions needed to cope with the future.
The Analytic Network Process is a generalization of the Analytic Hierarchy
Process. The basic structure is an influence network of clusters and nodes. Priori-
ties are established in the same way they are in the AHP using pairwise compari-
sons and judgment. Many decision problems cannot be structured hierarchically
because they involve the interaction and dependence of higher-level elements in a
hierarchy on lower-level elements. Not only does the importance of the criteria
determine the importance of the alternatives as in a hierarchy, but also the impor-
tance of the alternatives themselves determines the importance of the criteria.
Two bridges, both strong, but the stronger is also uglier, would lead one to choose
the strong but ugly one unless the criteria themselves are evaluated in terms of the
bridges, and strength receives a smaller value and appearance a larger value be-
cause both bridges are strong. Feedback enables us to factor the future into the
present to determine what we have to do to attain a desired future.
The feedback structure does not have the top-to-bottom form of a hierarchy
but looks more like a network, with cycles connecting its components of ele-
ments, which we can no longer call levels, and with loops that connect a compo-
nent to itself (Fig. 3). It also has sources and sinks. A source node is an origin of
paths of influence (importance) and never a destination of such paths. A sink
node is a destination of paths of influence and never an origin of such paths. A
full network can include source nodes; intermediate nodes that fall on paths from
source nodes, lie on cycles, or fall on paths to sink nodes; and finally sink nodes.
Some networks can contain only source and sink nodes. Still others can include
only source and cycle nodes or cycle and sink nodes or only cycle nodes. A deci-
sion problem involving feedback arises often in practice. It can take on the form
of any of the networks just described. The challenge is to determine the priorities
of the elements in the network and in particular the alternatives of the decision
and even more to justify the validity of the outcome. Because feedback involves
cycles, and cycling is an infinite process, the operations needed to derive the pri-
orities become more demanding than has been familiar with hierarchies.
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 13
To test for the mutual independence of elements such as the criteria, one
proceeds as follows: Construct a zero-one matrix of criteria against criteria using
the number one to signify dependence of one criterion on another, and zero oth-
erwise. A criterion need not depend on itself as an industry, for example, may not
use its own output. For each column of this matrix, construct a pairwise compari-
son matrix only for the dependent criteria, derive an eigenvector, and augment it
with zeros for the excluded criteria. If a column is all zeros, then assign a zero
vector to represent the priorities. The question in the comparison would be: For a
given criterion, which of two criteria depends more on that criterion with respect
to the goal or with respect to a higher-order controlling criterion?
In Fig. 3, a view is shown of a hierarchy and a network. A hierarchy is com-
prised of a goal, levels of elements and connections between the elements. These
connections go only to elements in lower levels. A network has clusters of ele-
ments, with the elements being connected to elements in another cluster (outer
L in e a r H ie ra rc h y
c o m p o n e n t,
c lu s te r
(L e v e l)
e le m e n t
A lo o p in d ic a te s th a t e a c h
e le m e n t d e p e n d s o n ly o n its e lf .
G o a l
S u b c rite r ia
C rite r ia
A lte rn a tiv e s
Feed back N etw ork w ith C om ponents ha vin g
Inn er an d O u ter D epend en ce am on g T heir E lem en ts
C 4
C 1
C 2
C 3
Feedback
Loop in a com ponen t ind ica tes inner dependence o f the e lem en ts in tha t com ponent
w ith respect to a com m on p rope rty.
A rc from com ponen t
C 4 to C 2 ind ica tes the
ou te r dependence o f the
e lem en ts in C 2 on the
e lem en ts in C 4 w ith respect
to a com m on p rope rty.
Fig. 3. How a Hierarchy Compares to a Network
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 14
dependence) or the same cluster (inner dependence). A hierarchy is a special case
of a network with connections going only in one direction. In a view of a hierar-
chy, such as that shown in Fig. 3, the levels in the hierarchy correspond to clusters
in a network. One example of inner dependence in a component consisting of a
father mother and baby is whom does the baby depend on more for its survival, its
mother or itself. The baby depends more on its mother than on itself. Again sup-
pose one makes advertising by newspaper and by television. It is clear that the
two influence each other because the newspaper writers watch television and need
to make their message unique in some way, and vice versa. If we think about it
carefully everything can be seen to influence everything including itself according
to many criteria. The world is far more interdependent than we know how to deal
with using our existing ways of thinking and acting. We know it but how to deal
with it. The ANP appears to be a plausible logical way to deal with dependence.
The Supermatrix of a Network
N
NnNN
N
nn eee
C
eee
C
eee
C
21
2
22221
1
11211
1 2
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
NN
N
N
NN W
W
W
W
W
W
W
W
W
W
N
Nn
N
N
n
n
N e
e
e
e
e
e
e
e
e
C
C
C
2
1
2
22
12
1
21
11
2
1
2
22
21
1
12
11
2
1
2
1
ijW Component of Supermatrix
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
)(
)(
2
)(
1
)(
)(
2
)(
1
)(
)(
2
)(
1
2
2
2
1
1
1
j
i
j
j
ii
jn
ni
jn
i
jn
i
j
ni
j
i
j
i
j
ni
j
i
j
i
ij
W
W
W
W
W
W
W
W
W
w
Fig. 4. The Supermatrix of a Network and Detail of a Component in It
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 15
The priorities derived from pairwise comparison matrices are entered as
parts of the columns of a supermatrix. The supermatrix represents the influence
priority of an element on the left of the matrix on an element at the top of the ma-
trix. A supermatrix along with an example of one of its general entry matrices is
shown in Fig. 4. The component iC in the supermatrix includes all the priority
vectors derived for nodes that are “parent” nodes in the iC cluster. Fig. 5 gives
the supermatrix of a hierarchy along with the kth power that yields the principle
of hierarchic composition in its )1,(k position.
Hierarchic composition yields multilinear forms which are of course nonlin-
ear and have the form
p
p
i
p
ii
ii xxx∑
,,
21
1
21 ,
Supermatrix of a Hierarchy
1
221
)1(1)1(
1
1
)2(1)2(
2
221
2
111
1
−
−
−−
−
−−
−
N
NN
nNN
NnN
NN
nNN
N
nn
ee
ee
CC
ee
С
ee
С
ee
С
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
−
−−
IW
W
W
W
e
e
e
e
e
e
C
C
C
W
nn
nn
nN
N
n
n
N
N
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1,
2,1
32
21
1
2
21
1
11
2
1
2
1
Supermatrix to nth Power Gives Hierarchical Synthesis
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
−−−−−−−−−− IWWWWWWWWWW nnnnnnnnnnnnnn
kW
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1,2,11,322,11,21322,11, ……
for 1−> nk
Fig. 5. The Supermatrix of a Hierarchy with the Resulting Limit Matrix Corresponding to
Hierarchical Composition
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 16
where ji indicates the jth level of the hierarchy and the jx is the priority of an
element in that level. The richer the structure of a hierarchy in breadth and depth,
the more elaborate are the derived multilinear forms from it. There seems to be a
good opportunity to investigate the relationship obtained by composition to co-
variant tensors and their algebraic properties.
More concretely we have the covariant tensor
iiwwww
h
h
h
hh i
NN
ii
ii
h
ii
h
i ≡=
−
−
−
−−∑
=
−
1
1
,,
1,,
21
1
11
12
1221
for the priority of the ith element in the hth level of the hierarchy. The composite
vector hW for the entire hth level is represented by the vector with covariant ten-
sorial components. Similarly, the left eigenvector approach to a hierarchy gives
rise to a vector with contravariant tensor components.
The classical problem of relating space (geometry) and time to subjective
thought can perhaps be examined by showing that the functions of mathematical
analysis (and hence also the laws of physics) are derivable as truncated series
from the above tensors by composition in an appropriate hierarchy. The foregoing
is reminiscent of the theorem in dimensional analysis that any physical variable is
proportional to the product of powers of primary variables.
Priority means dominance. If we know how capture dominance we would
know how to obtain priorities. In the ANP we look for steady state priorities from
a limit super matrix. To obtain the limit we must raise the matrix to powers. The
outcome of the ANP is nonlinear and rather complex. The limit may not converge
unless the matrix is column stochastic that is each of its columns sums to one. If
the columns sum to one then from the fact that the principal eigenvalue of a ma-
trix lies between its largest and smallest column sums, we know that the principal
eigenvalue of a stochastic matrix is equal to one. Now we know, from a theorem
due to J.J. Sylvester that when the multiplicity of each eigenvalue of a matrix W is
equal to one that an entire function )(xf (power series expansion of )(xf con-
verges for all finite values of (x) with x replaced by W , is given by
∑ ∏
∏
=
≠
≠
−
−
==
n
i
ij
ij
ij
j
iii
AI
ZZfWf
1 )(
)(
)(),()()(
λλ
λ
λλλ ,
∑
=
===
n
i
iijii ZZZZIZ
1
2 )()(,0)()(,)( λλλλλ ,
where I and 0 are the i dentity and null matrices respectively.
A similar expression is also available when some or all of the eigenvalues
have multiplicities greater than one. The matrix A itself gives the direct domi-
nance of an element on the left over another element on top. But an element can
dominate another via a third element. Dominance of an element over another
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 17
through two step transitivities is obtained by squaring the matrix. Similarly all
Nth order transitivities are obtained by raising the matrix to the Nth power which
gives the dominance of one element over another in N steps. From each matrix we
obtain the relative overall dominance of an element in steps equal to that power of
the matrix by adding the coefficients in the row of the matrix corresponding to
that element and dividing by the total. According to Cesaro summability, the limit
of the Cesaro sum ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∑∑
==∞→
N
k
Tk
N
k
Tk
N
eAeeAN
00
/1lim , )1,...,1,1(=e that repre-
sents the average of all order dominance up to N, is the same as the limit of the
sequence of the powers of the matrix i.e. TNTN
N
eAeeA /lim
∞→
and thus we need
to calculate the limiting powers of A .
How do we capture the priorities in the limit as the steady state priorities?
We see that if, as we need in our case, NWWf =)( , then N
iif λλ =)( and as
∞→N the only terms that give a finite nonzero value are those for which the
modulus of iλ is equal to one. The fact that W is stochastic ensures this. We
have:
max
11
max λ=≥∑∑
== i
j
n
j
ij
n
j
ij w
w
aa for iwmax ,
max
11
min λ=≤∑∑
== i
j
n
j
ij
n
j
ij w
w
aa for iwmin .
Thus for a row stochastic matrix we have ∑
=
≤≤=
n
j
ija
1
maxmin1 λ
∑
=
=≤
n
j
ija
1
1max , thus 1max =λ .
The same type of argument applies when a matrix that is column stochastic. For
complete treatment, see this author’s 2001 book on the ANP [1], and also the
manual for the ANP software [2].
The ANP Formulation of the Classic AHP School Example
We show in Fig. 6 below the hierarchy, the corresponding supermatrix, and its
limit supermatrix to obtain the priorities of three schools involved in a decision to
choose one for the author’s son. They are precisely what one obtains by hierarchic
composition using the AHP. The priorities of the criteria with respect to the goal
and those of the alternatives with respect to each criterion are clearly discernible
in the supermatrix itself. Note that there is an identity submatrix for the alterna-
tives with respect to the alternatives in the lower right hand part of the matrix.
The level of alternatives in a hierarchy is a sink cluster of nodes that absorbs pri-
orities but does not pass them on. This calls for using an identity submatrix for
them in the supermatrix.
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 18
The Investment Example with Criterion Weights Automatically Derived
from the Supermatrix [3]
Let us revisit the investment example that appeared in my earlier exposition in
Part 2.1 on the theory of the AHP/ANP. An individual has three alternate ways,
1A , 2A , and 3A , of investing a sum of money for the same period of time. There
are two types of returns, 1C and 2C (for example, capital appreciation and inter-
est), as shown in Tabl. 3. The question is, which is the best investment to make in
terms of actual dollars earned?
Leaming Frends School
Life
Vocational
Training
College
Prep.
Music
Classes
School
A
School
B
School
C
Goal
Satisfaction with School
Goal Learning Friends School life Vocational trainingCollege preparation Music classes A B C
Goal 0 0 0 0 0 0 0 0 0 0
Learning 0 0 0 0 0 0 0 0 0 0
Friends 0 0 0 0 0 0 0 0 0 0
School life 0 0 0 0 0 0 0 0 0 0
Vocational training 0 0 0 0 0 0 0 0 0 0
College preparation 0 0 0 0 0 0 0 0 0 0
Music classes 0 0 0 0 0 0 0 0 0 0
Alternative A 0.3676 0.16 0.33 0.45 0.77 0.25 0.69 1 0 0
Alternative B 0.3781 0.59 0.33 0.09 0.06 0.5 0.09 0 1 0
Alternative C 0.2543 0.25 0.34 0.46 0.17 0.25 0.22 0 0 1
Goal Learning Friends School life Vocational trainingCollege preparation Music classes A B C
Goal 0 0 0 0 0 0 0 0 0 0
Learning 0.32 0 0 0 0 0 0 0 0 0
Friends 0.14 0 0 0 0 0 0 0 0 0
School life 0.03 0 0 0 0 0 0 0 0 0
Vocational training 0.13 0 0 0 0 0 0 0 0 0
College preparation 0.24 0 0 0 0 0 0 0 0 0
Music classes 0.14 0 0 0 0 0 0 0 0 0
Alternative A 0 0.16 0.33 0.45 0.77 0.25 0.69 1 0 0
Alternative B 0 0.59 0.33 0.09 0.06 0.5 0.09 0 1 0
Alternative C 0 0.25 0.34 0.46 0.17 0.25 0.22 0 0 1
The School Hierarchy as Supermatrix
Limiting Supermatrix & Hierarchic Composition
Fig. 6. Supermatrix of School Choice Hierarchy gives same Result as Hierarchic Com-
position
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 19
It is easy to calculate the actual total cost for each alternative by simply add-
ing the two numbers; the relative cost is then obtained by normalizing as shown in
the table.
T a b l e 3 . Calculating Returns Arithmetically
Alternatives
Criterion C1
Unnormalized
weight = 1.0
Criterion C2
Unnormalized
weight = 1.0
Weighted Sum
Unnormalized
Normalized or
Relative values
A1 200 150 350 350/1300=0.269
A2 300 50 350 350/1300=0.269
A3 500 100 600 600/1300=0.462
Column Totals 1000 300 1300 1
Since we are dealing with tangibles we normalize each column to obtain the
priorities for the alternatives under each criterion. We also normalize each row to
obtain the priorities of the criteria with respect to each alternative. We enter these
in a supermatrix as shown in Tabl. 4; there is no need to weight the supermatrix
because it is already column stochastic, so we can raise it to limiting powers and
obtain the limit supermatrix in Tabl. 5 in which all the columns are identical. Be-
cause the supermatrix is column stochastic the priorities for the alternatives and
the criteria each add to 50% of the value in a column. We see that the supermatrix
saves us the arithmetic of determining criteria weights based on the values of the
alternatives under each criterion.
T a b l e 4 . The Unweighted Supermatrix of the Investment Example
Alternatives Criteria
A1 A2 A3 C1 C2
A1 0.000 0.000 0.000 0.200 0.500
A2 0.000 0.000 0.000 0.300 0.167 Alternatives
A3 0.000 0.000 0.000 0.500 0.333
C1 0.571 0.857 0.833 0.000 0.000
Criteria
C2 0.429 0.143 0.167 0.000 0.000
T a b l e 5 . The Limit Supermatrix of the Investment Example
Alternatives Criteria
A1 A2 A3 C1 C2
Alternatives A1 0.135 0.135 0.135 0.135 0.135
A2 0.135 0.135 0.135 0.135 0.135
A3 0.231 0.231 0.231 0.231 0.231
Criteria C1 0.385 0.385 0.385 0.385 0.385
C2 0.115 0.115 0.115 0.115 0.115
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 20
Normalizing the results for the alternatives, that is, dividing by the sum of
their values in Tabl. 5, which is .5, gives the same ratios we obtained for the over-
all return for each investment in Tabl. 3.
3. TWO EXAMPLES OF ESTIMATING MARKET SHARE — THE ANP WITH A
SINGLE BENEFITS CONTROL CRITERION
A market share estimation model is structured as a network of clusters and nodes.
The object is to try to determine the relative market share of competitors in a par-
ticular business, or endeavor, by considering what affects market share in that
business and introducing them as clusters, nodes and influence links in a network.
The decision alternatives are the competitors and the synthesized results are their
relative dominance. The relative dominance results can then be compared against
some outside measure such as dollars. If dollar income is the measure being used,
the incomes of the competitors must be normalized to get it in terms of relative
market share.
The clusters might include customers, service, economics, advertising, and
quality of goods. The customers cluster might then include nodes for the age
groups of the people that buy from the business: teenagers, 20–33 year olds, 34–
55 year olds, 55–70 year olds, and over 70. The advertising cluster might include
newspapers, TV, Radio, and Fliers. After all the nodes are created start by picking
a node and linking it to the other nodes in the model that influence it. The “chil-
dren” nodes will then be pairwise compared with respect to that node as a “par-
ent” node. An arrow will automatically appear going from the cluster the parent
node is in to the cluster with its children nodes. When a node is linked to nodes in
its own cluster, the arrow becomes a loop on that cluster and we say there is inter-
dependence.
The linked nodes in a given cluster are pairwise compared for their influence
on the node they are linked from (the parent node) to determine the priority of
their influence on the parent node. Comparisons are made as to which is more
important to the parent node in capturing “market share”. These priorities are then
entered in the supermatrix for the network.
The clusters are also pairwise compared to establish their importance with
respect to each cluster they are linked from, and the resulting matrix of numbers is
used to weight the components of the original unweighted supermatrix to give the
weighted supermatrix. This matrix is then raised to powers until it converges to
give the limit supermatrix. The relative values for the companies are obtained
from the columns of the limit supermatrix that are all the same. Normalizing these
numbers yields the relative market share.
If comparison data in terms of sales in dollars, or number of members, or
some other known measures are available, one can use these relative values to
validate the outcome. The AHP/ANP has a compatibility metric to determine how
close the ANP result is to the known measure. It involves taking the Hadamard
product of the matrix of ratios of the ANP outcome and the transform of the ma-
trix of ratios of the actual outcome summing all the coefficients and dividing by
n2. The requirement is that the value should be close to 1 and certainly not much
more than 1.1.
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 21
We will give three examples of market share estimation showing details of
the process in the first example and showing only the models and results in the
second and third examples.
Example 1. Estimating the relative market share of Walmart, Kmart and
Target
The network for the ANP model shown in Fig. 7 well describes the influences that
determine the market share of these companies. We will not use space in this pa-
per to describe the clusters and their nodes in greater detail.
The Unweighted Supermatrix
The unweighted supermatrix is constructed from the priorities derived from the
different pairwise comparisons. The nodes, grouped by the clusters they belong
to, are the labels of the rows and columns of the supermatrix. The column for a
node a contains the priorities of the nodes that have been pairwise compared with
respect to a. The supermatrix for the network in Fig. 7 is shown in Tabl. 6.
The Cluster Matrix
The cluster themselves must be compared to establish their relative importance
and use it to weight the supermatrix to make it column stochastic. A cluster im-
pacts another cluster when it is linked from it, that is, when at least one node in
the source cluster is linked to nodes in the target cluster. The clusters linked from
the source cluster are pairwise compared for the importance of their impact on it
with respect to market share, resulting in the column of priorities for that cluster
in the cluster matrix. The process is repeated for each cluster in the network to
Fig. 7. The Clusters and Nodes of a Model to Estimate the Relative Market Share of
Walmart, Kmart and Target
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 22
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Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 23
Te
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Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 24
obtain the matrix shown in Tabl. 7. An interpretation of the priorities in the first
column is that Merchandise (0.442) and Locations (0.276) have the most impact
on Alternatives, the three competitors.
T a b l e 7 . The Cluster Matrix
1. Alter-
natives
2. Adver-
tising
3. Loca-
tions
4. Customer
Groups
5. Mer-
chandise
6. Characteris-
tics of Store
1. Alternatives 0.137 0.174 0.094 0.057 0.049 0.037
2. Advertising 0.091 0.220 0.280 0.234 0.000 0.000
3. Locations 0.276 0.176 0.000 0.169 0.102 0.112
4. Customer Groups 0.054 0.429 0.627 0.540 0.252 0.441
5. Merchandise 0.442 0.000 0.000 0.000 0.596 0.316
6. Characteristics of Store 0.000 0.000 0.000 0.000 0.000 0.094
Weighted Supermatrix
The weighted supermatrix shown in Tabl. 8 is obtained by multiplying each entry
in a block of the component at the top of the supermatrix by the priority of influ-
ence of the component on the left from the cluster matrix in Tabl. 7. For example,
the first entry, 0.137, in Tabl. 7 is used to multiply each of the nine entries in the
block (Alternatives, Alternatives) in the unweighted supermatrix shown in
Tabl. 6. This gives the entries for the (Alternatives, Alternatives) component in
the weighted supermatrix of Tabl. 8. Each column in the weighted supermatrix
has a sum of 1, and thus the matrix is stochastic.
The limit supermatrix shown in Tabl. 9 is obtained from the weighted su-
permatrix by raising it to powers until it converges so that all columns are
identical.
Synthesized Results
The relative market shares of the alternatives, 0.599, 0.248 and 0.154 are dis-
played as synthesized results in the Super Decisions Program, shown in the mid-
dle column of Tabl. 10. They are obtained by normalizing the values for Walmart,
Kmart and Target: 0.057, 0.024 and 0.015, taken from the limit supermatrix. The
idealized values are obtained from the normalized values by dividing each value
by the largest value in that column.
Actual Relative Market Share Based on Sales
The object was to estimate the market share of Walmart, Kmart, and Target. The
normalized results from the model were compared with sales as reported in the
Discount Store News of July 13, 1998, p.77, of $58, $27.5 and $20.3 billions of
dollars respectively. Normalizing the dollar amounts shows their actual relative
market shares to be 54.8, 25.9 and 19.2. The relative market share from the model
was compared with the sales values by computing the compatibility index using
the Hadamard multiplication method below; it was equal to 1.016. Since that
value is less than 1.1 it is acceptable.
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 25
T
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Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 26
Te
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Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 27
T
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Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 28
Te
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Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 29
Competitor ANP Results Dollar Sales Actual Market Share as
Dollar Sales Normalized
Walmart 59.8 $58.0 billion 54.8
Kmart 24.8 $27.5 billion 35.9
Target 15.4 $20.3 billion 19.2
Compatibility Index 1.016
T a b l e 10. The Synthesized Results for the Alternative
Alternatives Ideal
Values
Normalized
Values
Values from Limit Super-
matrix
Walmart 1.000 0.594 0.057
KMart 0.414 0.250 0.024
Target 0.271 0.156 0.015
Example 2. Estimating Relative Market Share of Airlines
An ANP model to estimate the relative market share of 8 American Airlines is
shown in Fig. 8. The results from the model are shown in Tabl. 10 below and the
comparison with the relative actual market share is shown in Tabl. 11.
Fig. 8. ANP Network to Estimate Relative Market Share of 8 US Airlines
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 30
T a b l e 11. Comparing Model Results with Actual Market Share Data
Model Results Actual Market Share (yr 2000)
American 23.9 24.0
United 18.7 19.7
Delta 18.0 18.0
Northwest 11.4 12.4
Continental 9.3 10.0
US Airways 7.5 7.1
Southwest 5.9 6.4
American West 4.4 2.9
Compatibility Index 1.0247
We summarize by giving the reader a list of the steps we have followed in
applying the ANP.
4. OUTLINE OF THE STEPS OF THE ANP
1. Describe the decision problem in detail including its objectives, criteria
and subcriteria, actors and their objectives and the possible outcomes of that deci-
sion. Give details of influences that determine how that decision may come out.
2. Determine the control criteria and subcriteria in the four control hierar-
chies one each for the benefits, opportunities, costs and risks of that decision and
obtain their priorities from paired comparisons matrices. If a control criterion or
subcriterion has a global priority of 3% or less, you may consider carefully elimi-
nating it from further consideration. The software automatically deals only with
those criteria or subcriteria that have subnets under them. For benefits and oppor-
tunities, ask what gives the most benefits or presents the greatest opportunity to
influence fulfillment of that control criterion. For costs and risks, ask what incurs
the most cost or faces the greatest risk. Sometimes (very rarely), the comparisons
are made simply in terms of benefits, opportunities, costs, and risks in the aggre-
gate without using control criteria and subcriteria.
3. Determine the most general network of clusters (or components) and
their elements that applies to all the control criteria. To better organize the devel-
opment of the model as well as you can, number and arrange the clusters and their
elements in a convenient way (perhaps in a column). Use the identical label to
represent the same cluster and the same elements for all the control criteria.
4. For each control criterion or subcriterion, determine the clusters of the
general feedback system with their elements and connect them according to their
outer and inner dependence influences. An arrow is drawn from a cluster to any
cluster whose elements influence it.
5. Determine the approach you want to follow in the analysis of each cluster
or element, influencing (the preferred approach) other clusters and elements with
respect to a criterion, or being influenced by other clusters and elements. The
sense (being influenced or influencing) must apply to all the criteria for the four
control hierarchies for the entire decision.
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 31
6. For each control criterion, construct the supermatrix by laying out the
clusters in the order they are numbered and all the elements in each cluster both
vertically on the left and horizontally at the top. Enter in the appropriate position
the priorities derived from the paired comparisons as subcolumns of the corre-
sponding column of the supermatrix.
7. Perform paired comparisons on the elements within the clusters them-
selves according to their influence on each element in another cluster they are
connected to (outer dependence) or on elements in their own cluster (inner de-
pendence). In making comparisons, you must always have a criterion in mind.
Comparisons of elements according to which element influences a given element
more and how strongly more than another element it is compared with are made
with a control criterion or subcriterion of the control hierarchy in mind.
8. Perform paired comparisons on the clusters as they influence each cluster
to which they are connected with respect to the given control criterion. The de-
rived weights are used to weight the elements of the corresponding column blocks
of the supermatrix. Assign a zero when there is no influence. Thus obtain the
weighted column stochastic supermatrix.
9. Compute the limit priorities of the stochastic supermatrix according to
whether it is irreducible (primitive or imprimitive [cyclic]) or it is reducible with
one being a simple or a multiple root and whether the system is cyclic or not. Two
kinds of outcomes are possible. In the first all the columns of the matrix are iden-
tical and each gives the relative priorities of the elements from which the priori-
ties of the elements in each cluster are normalized to one. In the second the limit
cycles in blocks and the different limits are summed and averaged and again nor-
malized to one for each cluster. Although the priority vectors are entered in the
supermatrix in normalized form, the limit priorities are put in idealized form be-
cause the control criteria do not depend on the alternatives.
10. Synthesize the limiting priorities by weighting each idealized limit vector
by the weight of its control criterion and adding the resulting vectors for each of
the four merits: Benefits (B), Opportunities (O), Costs (C) and Risks (R). There
are now four vectors, one for each of the four merits. An answer involving mar-
ginal values of the merits is obtained by forming the ratio BO/CR for each alter-
native from the four vectors. The alternative with the largest ratio is chosen for
some decisions. Companies and individuals with limited resources often prefer
this type of synthesis.
11. Governments prefer this type of outcome. Determine strategic criteria
and their priorities to rate the four merits one at a time. Normalize the four ratings
thus obtained and use them to calculate the overall synthesis of the four vectors.
For each alternative, subtract the costs and risks from the sum of the benefits and
opportunities. At other times one may add the weighted reciprocals of the costs
and risks. Still at other times one may subtract the costs from one and risks from
one and then weight and add them to the weighted benefits and opportunities. In
all, we have four different formulas for synthesis.
12. Perform sensitivity analysis on the final outcome and interpret the results
of sensitivity observing how large or small these ratios are. Can another outcome
that is close also serve as a best outcome? Why? By noting how stable this out-
come is. Compare it with the other outcomes by taking ratios. Can another out-
come that is close also serve as a best outcome? Why?
Saaty Thomas L.
ISSN 1681–6048 System Research & Information Technologies, 2003, № 2 32
5. CONCLUSIONS
Complete examples applying the AHP to a decision involving BOCR can be
found in references 1 and 2. Numerous other examples along with the software
Super Decisions for the ANP can be obtained from rozann@creativedecisions.net.
The reader now should have a good idea as to how to use the process in a com-
plex decision. The AHP and ANP have found application in practice by many
companies and governments. My book Decision Making for Leaders is in more
than a half a dozen languages. What is happening now is the wide interest shown
in the ANP and its applicability to the long discussed project of building a Na-
tional Missile Defense This application was done in September 2000 and pre-
sented at the National Defense University in Washington, DC, in February 2002.
Its conclusions were affirmed by President Bush’s decision of late December
2002 to construct such a system. Another recent policy study was done regarding
whether the US should challenge Iraq directly or go through the UN. The admini-
stration decided to go through the UN. There is also the hopeless Middle East
conflict. An ANP analysis showed that the best option is for Israel and the US to
help the Palestinians both set up a state and in particular achieve a viable econ-
omy. There are two things to tell the reader about it in this regard. The ANP book
is now translated to Russian and will soon appear and the manual for the Super
Decisions sofware can be obtained by sending an email to ro-
zann@creativedecisions.net. My forthcoming book The Encyclicon will have
nearly 100 examples of such applications and will be out in the near future.
As a final word, the AHP has developed a group of critics who think it can-
not be used for multicriteria methods because it is simple. They say that it is lin-
ear. I say it is non-linear. Hierarchic composition yields nonlinear forms that are
dense in the space of polynomials (multinomials). According to the theorem of
Weierstrasse the latter can be used to approximate a continuous function arbitrar-
ily close. Multilinear forms and polynomials are intimately related, particularly
when we see that we can identify all the variables into a single or into several
variables raised to powers. Thus depending on how rich a hierarchic structure is,
one can use hierarchic composition to come close to the real answer underlying a
decision. This author, a mathematician, with a Ph.D. in mathematics from Yale
University, and postdoctoral work at the Sorbonne in Paris, has written many first
works in the field of Operations Research. Among them is Mathematical Methods
of Operations Research, translated to many languages. Also, he has spent the first
half of his career working at the Pentagon, the Navy Department, and the De-
partment of State, all the time searching for and applying, when feasible, mathe-
matical models of operations research. The biggest weakness of these methods is
that they could only be understood by the experts and occasionally used by the
practicing layman, with doubt and hesitation. As an adult I took it upon myself
since my first involvement in research on negotiations between the US and the
Soviet Union in Geneva in the 1960’s and after having written a book on the
Mathematical Models of Arms Control and Disarmament in 1968, translated to
Russian in 1977, to always try to simplify the AHP so that even a child can use it.
And children have used it without the need to explain the mathematics but even if
one has to, it is possible to do that with some patience but without a great deal of
Theory of the analytic hierarchy and analytic network process – examples. Part 2.2
Системні дослідження та інформаційні технології, 2003, № 2 33
prolixity and confusion. This is why the AHP and ANP are intentionally simple.
More full blown decision examples will be given in Part 2.3.
REFERENCES
1. Saaty, Thomas L. The Analytic Network Process, RWS Publications, 4922 Ellsworth
Avenue, Pittsburgh, Pa. 15213, 2001.
2. Saaty, Rozann W. Decision Making in Complex Environments: The Analytic Network
Process (ANP) for Dependence and Feedback; a manual for the ANP Software;
Creative Decisions Foundation, 4922 Ellsworth Avenue, Pittsburgh, PA 15213,
2002.
3. Saaty, Thomas L. Fundamentals of the Analytic Hierarchy Process, RWS Publica-
tions, 4922 Ellsworth Avenue, Pittsburgh, Pa. 15413, 2000.
4. Saaty, Thomas L and Yeonmin Cho “The Decision by the US Congress on China’s
Trade Status: A Multicriteria Analysis,” Socio-Economic Planning Sciences,
35(2001) 243-252.
Received 29.04.2003
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| id | journaliasakpiua-article-174060 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:25:50Z |
| publishDate | 2019 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
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| resource_txt_mv | journaliasakpiua/e5/54da3fc083a556252cf0731d628d34e5.pdf |
| spelling | journaliasakpiua-article-1740602019-07-26T17:25:36Z Theory of the analytic hierarchy and analytic network processes – examples. Part 2.2 Теория аналитических иерархических процессов и аналитические сетевые процессы — примеры. Часть 2.2 Теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. Частина 2.2 Saaty, Thomas L. In this part we introduce the role of benefits, opportunities, costs and risks (BOCR) in decision- making and how to establish priorities for them. We give an example of a real life application of the US Congress acting on China joining the World Trade Organization (WTO) mailed to the US congressional committee before that decision. We then introduce and apply the Analytic Network Process and its concept of a supermatrix to make decisions with dependence and feedback and illustrate its application for a single "control" criterion of market share. This will be followed in Part 2.3 by a full BOCR application in the context of the ANP. В этой части вводится роль прибылей, возможностей, стоимостей и рисков (BOCR) в выработке решений. Показано, как установить для них приоритеты. Приводится направленный в комитет Конгресса США пример применения в реальной жизни методов в деятельности Конгресса США по поводу присоединения Китая ко Всемирной Организации Торговли (WTO) до принятия решения. Затем вводится и применяется Аналитический Сетевой Процесс (ANP) и его концепция суперматрицы для выработки решений в условиях зависимости и обратной связи. Иллюстрируется его применение для единственного "управляющего" критерия раздела рынка. Эта статья будет продолжена в части 2.3 рассмотрением применения BOCR в контексте ANP. У цій частині введено роль прибутків, можливостей, вартостей та ризиків (BOCR) у виробленні рішень. Показано, як визначити для них пріоритети. Наводиться поданий до комітету Конгресу США ще до прийняття ним рішення приклад застосування у реальному житті методів діяльності Конгресу США щодо приєднання Китаю до Всесвітньої Організації Торгівлі (WTO). Далі введено і застосовано Аналітичний Мережний Процес (ANP) і його концепцію суперматриці для вироблення рішень в умовах залежності і зворотнього зв’язку та проілюстровано його застосування для єдиного "керуючого" критерію розподілу ринку. Цю статтю буде продовжено у частині 2.3, де буде розглянуте застосуванням BOCR у контексті ANP. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2019-07-26 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/174060 System research and information technologies; No. 2 (2003); 7-33 Системные исследования и информационные технологии; № 2 (2003); 7-33 Системні дослідження та інформаційні технології; № 2 (2003); 7-33 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/174060/173971 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Saaty, Thomas L. Теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. Частина 2.2 |
| title | Теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. Частина 2.2 |
| title_alt | Theory of the analytic hierarchy and analytic network processes – examples. Part 2.2 Теория аналитических иерархических процессов и аналитические сетевые процессы — примеры. Часть 2.2 |
| title_full | Теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. Частина 2.2 |
| title_fullStr | Теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. Частина 2.2 |
| title_full_unstemmed | Теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. Частина 2.2 |
| title_short | Теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. Частина 2.2 |
| title_sort | теорія аналітичинх ієрархічних процесів та аналітичні мережеві процеси — приклади. частина 2.2 |
| url | https://journal.iasa.kpi.ua/article/view/174060 |
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