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.


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].[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][6][7][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]: where M -heterogeneous structure medium Σsignature; consisting of sets of the binding operations, substitution and output operations; operations on attributes and substitutive relations; Λconstructive 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: , , , , where C -ОКПС, M -heterogeneous medium, Σ -signature, Λ -axiomatics, S specialization operation,  The order of the operation on attributes in the process of performing partial output operation is given by the attribute j τ , where τ -the operation on attribute is performed before the substitution operation, 1 τ -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: , , where A,AHPS M -heterogeneous medium that con- a set of forming algorithms for a particular server, -algorithms for operations Φ on attributes.
The above algorithms execute the following operations: -partial and complete output.
Here , where I -interpretation operation; Z -a set of servers that can use all BAS algorithms; Let us represent CPS specification for the analytic hierarchy process with sorting: , , , , , , , where φ -PCM alternatives by l -th criterion.
Partial axiomatic 5 Λ is as follows.
The number of criteria P and the number of alternatives N , as well as the semantics of alterna- tives and criteria are given at the stage of execution by an external server.
The record of the sequential concatenation of several terminals, non-terminals and sequential operations on attributes will be represented as follows: 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 , , , , , so this attribute in these rules is omitted.Here are the rules and their brief description.
The relation 2,0,0 s uses implementation of the grouping and sorting CPS for the alternatives for each criterion, and 3,0,0 s is used to get the implementation results of the sorting and grouping CPS for criteria: 2,0,0 The relation 4,0,0 s 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 6,0,0 s 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: , , ,[]}} , Ξ -binding operation, Θ -output operations, Φ − operations on attributes, Π -substitution operations.
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 r d , where r -the rule number that takes the value 1 -the relation is available and 0 -not available.For the rules with a constant availability attribute ( r d =1) this attribute is omitted for record simplicity.
To interpret the CPS of alternative grouping and sorting let us use БАС , , We concretize CPS of alternative grouping and sorting: , , , , , , , , k k M α -is responsible for determining the number of groups with four and three alternatives ( 3 4 , k k -number of groups with three or four alternatives in the group, respectively, М -total number of groups); , : 0; : 0; ); : 1; : 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: , ; ) 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: : 0; ( ( ( 0; ) ; ; )); ( ; The relation 8 s 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 9 s is used to regroup the alternatives.Operations on attributes allow setting the alternative attributes in the new groups: ) ; ( : ; : ; ) ; ; .
The substitutive relation 10 s 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: ) ))); : 1; : 1 .
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 13 s 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 singlelevel 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: , , , { , x x k allows setting a value of the link weight between i and j alternatives (criteria) by p k criterion, ( , , ) i j x x ε allows setting a value of the link weight between criteria.These operations are executed by an external server.
Terminal alphabet contains many alternatives and criteria with their attributes.
The output process forms the construct that will include the following forms: , ( ) to the assessment by an external server expert, 0 h = -without the involvement of an external expert on the basis of substitution rules); , ( ) x x ε -link of i and j alternative (criterion) if a comparison criterion is not given; Q λpairwise comparison matrix for alternatives; , , , , ) 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: The substitutive relation 3,0,0 s 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 relation for PCM completion check.If the matrix is completed, then based on the operations on attributes we calculate the conformity relation and fill the alternative priority vector: 6,0,0 1 1 6,0,0 ) ,0,0 7,0,0 1 ( : ); : 0 The following set of rules ( 1, ) defines the relations for descending ordering of alternatives according to their weights: , : 1 ) The substitutive relation 9,0,0 s is used to establish the link between the alternatives by the given criteria: 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: 11,0,0 11,0,0 1 1 ( ) ( )

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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forms, σ -initial nonterminal, Ωa set of formed constructs; a and b , if the condition is satisfied ( a b ≤ , a ≠ b , a = b , a b > , value b to the variable a , the values a and b can be vectors, matrices, or numbers; of the two conditions a and b , true, if both conditions are true; − 0 15 | N is A γ -calculation of the conformity relation of the pairwise comparison matrix (PCM) N γ ; and criteria, where , a b -identifiers of alternatives or criteria or links between them; binding b implementation result to non-terminal of CPS implementation use α .Interpretation of the main CPS for the modified analytic hierarchy process: , , , non-terminals of m -th group of alterna- tives, 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); , N p γ − PCM for N alternatives according to the criterion p , matrix elements, non-terminals , , a h i j γ with attributes: a − evaluation of comparison of i and j alternatives, h − evaluation process tool, ( 1 h = -filled according to the evaluation by an external server, expert, 0 h = -without the involvement of an external expert on the basis of substitution rules); , m p n β -PCM by criterion p for the group m , consisting of n -alternatives, this matrix elements are non-terminals , , , , p a h m i j β , where the attributes a and h -the same as for terminal of CPS implementation of the single-level classical AHP for the group alternatives: alternatives for AHP ranking, p -ranking criterion number, P k -criterion vector, Q λ -PCM of alternatives of the group with calculated ranks and conformity relation, n L′ -list of alternatives ordered accord- ing to the ranks, , m p n β -PCM of alternatives in the group n; m χ -non-terminal to prepare m -th group alternatives for ranking; all η -non-terminal to calculate the parameters of the general PCM of alternatives (missing evaluations of paired comparisons, conformity relation and matrix completion control); { } , alternatives in the group m, y , y′ , z − alternative identifier, 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: Наука та прогрес транспорту.Вісник Дніпропетровського національного університету залізничного транспорту, 2016, № 4 (64) ІНФОРМАЦІЙНО-КОМУНІКАЦІЙНІ ТЕХНОЛОГІЇ ТА МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ doi 10.15802/stp2016/77926 © T. M. Vasetska, attributes of 3 s relation determine the attribute values for PCM elements of the alternatives:

3
The relation 7,0,0 s enters the sequence of alternatives with the weights into the construct, if PCM conformity relation is valid:

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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.