Introduction
Analytic Hierarchy Process (AHP) [6, 7], 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, 2, 3, 8, 9], or use AHP in combination with other decision-making methods (mathematical methods of multi-criteria analysis, statistical methods etc.) [4, 5]. There are a lot of developed software tools, which implement both the method itself and its modifications [2, 9, 14, 15].
[9] 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) [12]. In [10] 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 [10 13]. By performing different transformations of the generalized constructive and productive structure (GCPS) [12], namely, specialization, interpretation, specification and implementation, the different models are developed [13]. GCPS is called a triple [12]:
,
where
–
heterogeneous structure medium,
–signature,
consisting of sets of the binding operations,
substitution and output operations, operations on attributes and
substitutive relations,
–
constructive axiomatics [12].
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 [9] 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 [12] to represent the analytic hierarchy process with sorting:
where
–
ОКПС,
– heterogeneous medium,
–signature,
–axiomatics,
–specialization operation,
,
,
,
,
– binding
operations,
– output
operations,
– substitution
operations,
operation on attributes.
Partial
axiomatics
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
and criteria
with their attributes:
alternative identifier,
semantics,
global priority (weight);
criterion identifier.
Alternatives
and criteria, valid assessment values are contained in a
heterogeneous medium
.
The following operations on attributes are introduced:
conditional
operations from the list
,
if
,
,
operations are presented in the prefix form;
vector-number
multiplication operation;
:= assignment operation;
< less-than comparison;
attribute
value seek operation, by an external server.
The
substitution rules are written as
where
– substitutive
relation,
– a
set of operations on attributes,
–rule number,
,
– numbers of the first and the second pair of alternatives.
Three-level indexing is used for ordering the substitution rules.
Binary
partial output operation [12]
(here
,
–forms before and after the substitution operation), consists of:
1)
selecting one of the substitution rules
with substitutive relations
and performing the substitution operation on its basis. Availability
of substitutive relation
is determined by the availability attribute value
:
if
the relation is available,
–not available; the availability of rules is regulated by
operations on attributes or is given by axiomatics;
2)
carrying
out operations on attributes
.
The
order of the operation on attributes in the process of performing
partial output operation is given by the attribute
,
where
,
,
,
– the
operation on attribute is performed before the substitution
operation,
– 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) [12], which comprises the steps of performing operations by condition, matrix operations, as well as the launch of AHPS for criteria and alternatives:
,
where
– heterogeneous
medium
that
contains
,
– signature
and
– axiomatics,
– a
set
of
forming
algorithms
for
a
particular
server,
and
–
a
set
of
constructed algorithms,
,
,
,
,
,
,
– algorithms
for
operations
on
attributes.
The above algorithms execute the following operations:
– algorithm concatenation (sequential algorithm
after
);
– substitution;
– partial and complete output. Here
–forms,
–initial nonterminal,
– a set of formed constructs;
– execution of
algorithms from the list
, if
;
– calculation of the product
,
and
can be matrices or numbers;
– assigning a value to a variable
;
– determining the value of
– calculation of the quotient of
– calculation of the remainder on dividing
,
,
,
,
– comparison of numbers
,
,
), then
;
– assigning the value
– logical AND of the two conditions
– calculation of the conformity relation of the pairwise comparison matrix (PCM)
;
– calculation of the alternative priority vector by PCM
.
– calculation of integral part of the real number
– calculation of the sum
,
– determination of the link weight of
and
alternatives by the criterion
by an external server (
);
–binding alternatives and criteria, where
– identifiers of alternatives or criteria or links between them.
– binding
.
Interpretation of the main CPS for the modified analytic hierarchy process:
,
where
–interpretation
operation;
–a
set
of
servers
that
can
use
all
BAS
algorithms;
,
,
,
,
,
,
,
,
(
)}.
Let us represent CPS specification for the analytic hierarchy process with sorting:
,
where ,
,
,
–rule number,
–number of the first alternative,
– the
number of the second alternative of the pair,
– terminal
for identifier
of
the
-th
alternative,
– terminal
for identifier of the
-th
criterion,
–a
set of initial non-terminals,
– non-terminal
for processing alternatives by the
-th
criterion,
–non-terminal
for
implementation
of the grouping and sorting
CPS
for
criteria,
– non-terminal
of
CPS criteria (where
– a
set of attributes:
– vector
of PCM priorities,
– matrix
conformity relation,
– matrix
dimension),
–
PCM
alternatives by
-th
criterion.
Partial
axiomatic
is
as
follows.
The
number of criteria P and the number of alternatives
,
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:
.
The
record
means that 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,
for each
,
so this attribute in these rules is omitted. Here are the rules and
their brief description.
The
substitutive
relation
is
used
to
enter
the
processing
sequence
of
criteria
and
alternatives
by
criteria.
Operations
on
attributes
determine
the
quantity
and
the semantics
of criteria
and
alternatives:
The
relation
uses implementation of the grouping and sorting CPS for the
alternatives for each criterion, and
is
used to get the implementation results of the sorting and grouping
CPS for criteria:
.
.
The
relation
is
used
to
obtain
the
implementation
result
of
the
grouping
and
sorting
CPS
for
alternatives:
.
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:
,
.
The
set of substitutive relations
allows
ordering the alternatives into constructs according to their ranks:
,
.
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:
where
,
,
,
,
,
,
,
–
binding
operation,
–
output operations,
operations
on
attributes,
–
substitution
operations.
Partial
axiomatics
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 substitutive relations contain the
available attribute,
where
– the rule number that takes the value 1 – the relation is
available and 0 – not available. For the rules with a constant
availability attribute (
=1)
this attribute is omitted for record simplicity.
To
interpret the CPS of alternative grouping and sorting let us use БАС
,
described
above:
,
where ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(
)}.
We concretize CPS of alternative grouping and sorting:
where
и
,
,
,
,
– rule
number,
– is
responsible
for
determining
the
number
of
groups
with
four
and
three
alternatives
(
–
number
of
groups
with three
or
four
alternatives
in
the
group,
respectively,
М
–
total
number
of
groups);
,
non-terminals
of
-th
group
of
alternatives,
with
attributes
flag
indicating
the
alternative
position
changes
in
the
group
(1
alternatives
changed
their
position
after
ranking
in
the
group,
0
did not
change);
PCM
for
alternatives
according
to
the
criterion
,
matrix elements,
non-terminals
with
attributes:
evaluation
of comparison
of
and
alternatives,
evaluation process tool, (
– filled
according
to
the
evaluation by
an
external
server,
expert,
– without
the
involvement
of
an
external
expert
on
the
basis
of
substitution
rules);
– PCM
by criterion
for
the
group
,
consisting
of
– alternatives,
this matrix
elements
are
non-terminals
,
where
the attributes
and
– the
same
as for
;
–
non-terminal of CPS
implementation
of
the
single-level
classical
AHP
for
the
group alternatives:
– set
of
alternatives
for
AHP ranking,
p
–
ranking
criterion
number,
–
criterion
vector,
– PCM
of
alternatives
of the group
with
calculated
ranks
and
conformity
relation,
– list
of
alternatives
ordered
according
to the ranks,
– PCM
of
alternatives
in
the
group
n;
– non-terminal
to prepare
- th
group
alternatives for ranking;
–
non-terminal
to
calculate the parameters of the general PCM of alternatives (missing
evaluations of paired comparisons, conformity relation and matrix
completion control);
,
,
set
of alternatives in the group
m,
,
,
alternative
identifier,
set of attributes where
alternative
semantics,
global priority (weight) of an alternative,
global number of alternative,
alternative weight vector by criteria,
criterion number.
The first rule with the substitutive relation, which enters into the construct the sequence of alternatives, PCM by p-th criterion and non-terminal with attributes to work with groups. The operations on attributes calculate the number of groups from 3 and 4 alternatives and the total number of groups. The alternative paired comparison evaluations are completed with default values:
.
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:
.
.
The
operations
on
attributes
of
relation
determine
the attribute
values
for PCM elements
of the
alternatives:
.
The following substitutive relation is used for CPS implementation of the classic single-level AHP for each alternative group:
The operations on attributes of the following rule supplement the general PCM with new evaluations:
,
The substitutive relation is used to calculate the general PCM elements by transitiveness and to count the uncompleted elements:
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:
The
relation
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:
,
The
substitutive
relation
is
used
to
regroup
the alternatives.
Operations
on attributes allow setting the alternative attributes in the new
groups:
The
substitutive
relation
is
for
implementation
of
classic
AHP
for
new
alternative
groups:
.
The following rule contains the substitutive relation to get the AHP ranking result in each group and to save the evaluation entered into the general PCM by the expert:
Operations on attributes of the following rule allow determining the changes in the positions of alternatives in the groups after AHP application:
.
The
substitutive
relation
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:
The following substitutive relation is used to restore alternatives in the groups, if their position has changed:
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 single-level AHP implements completing by an external expert of some paired comparison evaluations, 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:
,
where
,
,
.
The
operation
allows
setting a value of the link weight between
i
and
j
alternatives
(criteria) by
criterion,
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:
– link of
and
alternatives by criterion
,
–link weight,
–attribute, responsible for the weight value derivation process (
– filled according to the assessment by an external server expert,
– without the involvement of an external expert on the basis of
substitution rules);
– link of
and
alternative (criterion) if a comparison criterion is not given;
–pairwise comparison matrix for alternatives;
– sorted alternative sequence.
For
interpretation of this CPS we use the BAS
described above:
,
where
,
,
,
,
,
,
,
,
,
,
,
,
,
}.
Let us perform specification of the interpreted CPS for a single-level AHP:
,
where ,
,
,
,
–
rule number,
and
– number of the
first and second pairs of alternatives,
– set of initial non-terminals,
– non-terminal
to indicate alternative links,
–
non-terminal to indicate criteria links,
–
non-terminals to indicate links between alternatives
by
-th
criterion (for simplicity indicated as matrix
),
–pairwise comparison matrix for alternatives (where
– vector of attributes:
– vector of PCM priorities,
– matrix conformity relation,
– the number of alternatives).
The
substitutive
relation
serves
to change PCM completion and ranking of alternatives
by
criterion
.
All
axiom input parameters are added to the medium.
The following substitutive relation is used, if a ranking criterion is specified:
,
.
The
substitutive relation
is
used to form PCM alternatives, if a criterion is not specified:
.
The following rule contains the substitutive relation to determine a connection between the alternatives. Operations on attributes determine the evaluation of alternatives links:
,
The
rules for setting the alternative pairwise comparison values by an
expert, where
and
:
,
.
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:
,
The
relation
enters
the
sequence
of
alternatives
with
the
weights
into
the
construct,
if
PCM
conformity
relation
is
valid:
,
.
The
following set of rules (,
)
defines the relations for descending ordering of alternatives
according to their weights:
.
The
substitutive relation
is used to establish the link between the alternatives by the given
criteria:
.
The
rules for setting the alternative pairwise comparison values by an
expert by
-th
criterion, where
and
:
,
.
The
operations
on
attributes
of
the
next
rule
check
the
PCM
completion
of
alternatives
by
-th
criterion.
If
everything
is completed,
then
the
conformity is calculated
and
the
alternative
priority vector
is
filled,
otherwise
the
rule
does
not
apply:
,
.
The
relation
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:
,
,
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.
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