=Paper=
{{Paper
|id=Vol-2917/paper4
|storemode=property
|title=Solutions Outlining on the Set of Structured Technological Problems with Imposed Constraints
|pdfUrl=https://ceur-ws.org/Vol-2917/paper4.pdf
|volume=Vol-2917
|authors=Vasyl Sheketa,Roman Vovk,Volodymyr Pikh,Yulia Romanyshyn,Kostiantyn Kravtsiv,Liudmyla Poteriailo,Volodymyr Protsiuk,Mykola Pasyeka
|dblpUrl=https://dblp.org/rec/conf/momlet/SheketaVPRKPPP21
}}
==Solutions Outlining on the Set of Structured Technological Problems with Imposed Constraints==
https://ceur-ws.org/Vol-2917/paper4.pdf
Solutions Outlining on the Set of Structured Technological
Problems with Imposed Constraints
Vasyl Sheketa, Roman Vovk, Volodymyr Pikh, Yulia Romanyshyn, Kostiantyn Kravtsiv,
Liudmyla Poteriailo, Volodymyr Protsiuk and Mykola Pasyeka
Ivano-Frankivsk National Technical University of Oil and Gas, Karpatska 15 street, Ivano-Frankivsk, 76019,
Ukraine
Abstract
A method of estimating technological parameters is introduced, which allows to represent sets
of preferences and express their influence on the process of satisfaction and violation of
constraints by solving technological problems. At the final stage, a set of basic technological
parameters is allocated, which allow to fully describe the technological process of drilling, by
constructing systems of constraints and their ranking by relevance, which makes it possible to
analyze an abnormal situation as a case of violation of technological parameters with imposed
sets, systems and hierarchies of constraints. A formal structure consisting of a set of variables
(technological parameters), a set of domains (confidence intervals) and a set of constraints is
introduced, which allows to describe the technological process of drilling oil and gas wells in
terms of formal-logical constructions of representation and satisfaction of constraints possible
states.
Keywords
Constraints, comparators, preferences, weights, problems solving, reasoning, decisions-
making, intelligent decisions-making support.
1. Introduction
In general case technological process of oil and gas wells drilling, is a very complex and dynamic
process, the full formalization of which does not give the expected results in terms of completeness and
correctness. An effective method of constructing solutions to the technological problems based on
constraints [1,2] in drilling of oil and gas wells is the use of logical programming techniques in
constraints [3–5]. This implementation will consist of several parts: the first part will contain the
definition of all variables of the technological problem with their domains. Accordingly, the domains
of the variables will be reduced due to the constraints that will be set in the next steps. Therefore, the
search method in the solution space will be described by entering a label for a set of variables or by
introducing an enumeration for value generation processes for individual variable domains. In this case,
the search tree will be described based on the heuristic of the ordering of values and variables, which is
applied before the assignment of values by calling constraint propagation procedures[6,7].
_______________________________________
MoMLeT+DS 2021: 3rd International Workshop on Modern Machine Learning Technologies and Data Science, June 5, 2021, Lviv-Shatsk,
Ukraine
EMAIL: vasylsheketa@gmail.com (V. Sheketa); r.vovk@nung.edu.ua (R.Vovk); pixclg@gmail.com (V. Pikh); yulromanyshyn@gmail.com
(Y. Romanyshyn); kostya.kravtsiv@gmail.com (K. Kravtsiv); milapoteriailo@gmail.com (L. Poteriailo); v.v.protsiuk@gmail.com (V.
Protsiuk); pms.mykola@gmail.com (M. Pasyeka)
ORCID: 0000-0002-1318-4895 (V. Sheketa); 0000-0003-0681-4534 (R.Vovk); 0000-0001-9420-5522 (V. Pikh); 0000-0001-7231-8040
(Y. Romanyshyn); 0000-0002-7916-3659 (K. Kravtsiv); 0000-0002-0501-5928 (L. Poteriailo); 0000-0003-0055-2780 (V. Protsiuk) ; 0000-
0002-3058-6650 (M. Pasyeka)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
The labeling procedure can also be used to find solutions to an optimization problem with an entered
objective function. The method of binarization of constraints [8] when used as a strategy for solving
technological problems based on constraints, is expected to increase the space for solutions, so the
efficiency of the solution search procedure, in this case, will decrease. To eliminate this shortcoming in
this case, it is advisable to use special algorithms for propagating constraints to solve selected
subproblems of variables described on a certain subset by introducing global constraints [9,10].
Modeling the problem through the corresponding global constraints is one of the main ideas of logical
programming in constraints in terms of computational efficiency. In such an application, global
constraints will express a condition that must be met. In particular, when using global constraints for
constrained problems, it will be advisable to introduce additional specifiers.
In the general case, such hierarchies is built on a set of constraint labels with an additional order
relationship imposed by global constraint labels in the middle of each level of the hierarchy. Because
hierarchical labels can be thought of as variable labels in a broader sense than fuzzy labels, hierarchy-
level labels allow so far to more fully represent the semantics of labels as a whole. It should be taken
into account a more complete solution strategy obtained in this case in comparison with fuzzy labels,
which allows to show the correspondence between individual sets of labels and technological problems
based on constraints and technological problems based on fuzzy constraints in general [11-13].
The question of constructing heuristics of ordering variables in technological problems on the basis
of constraints with the choice of those variables that are most "critical" in terms of their substitution,
i.e. in considering the most "critical" sets of variables with the most important preferences, remains
unexplored.
Thus, the purpose of this research is to synthesize solutions, which can be considered as a search
heuristic that processes the search tree simultaneously. It can also be interpreted as a method of
narrowing a problem that restricts the whole set of variables, which narrows to such a level that
constraints the space of possible labels to such an extent that it will contain only tuples of solutions.
2. Solutions refinement for technological problems
The use of multilevel intelligent technologies allows to optimize the drilling process of oil and gas
wells by performing the necessary reconfigurations of equipment and applying methods of control of
the drilling process through solving technological problems and preventing emergencies [2].
Next figure presents the structuring of the drilling process in terms of available control and
automation functions.
Drilling RIG OPERATOR
Drilling SETTINGS
Dynamics of the mechanical system
of the drilling rig
weight rotary mud
on bit speed presure
The drilling process in terms of the interaction of
the bit with the rock
cutting rotative shaking pressure boost of
rate moment drilling fluid
Figure 1: Modeling of the overall structure of the drilling process as of an object of automation in
terms of controlled parameters
Thus, under the technological problem TP we do understand a certain essence of the subject area
of oil and gas wells drilling that contains a problem that requires a solution that belongs to the space of
possible solutions to the technological context and does consist in values assignment to the controlled
technological parameters.
2.1. Comparative analysis of the comparators
So far, the classes of technological problems in drilling can be reduced to the corresponding classes
of classical search problems based on constraints and there will be obviously a correspondence between
the method of constructing of an solution to a technological problem and the way of solving classical
search problems, in particular methods of working with solution space that is formed by the set of
possible assignments. Since for the intelligent system is more important the process of finding a solution
during which, by analyzing the violated and satisfied constraints, the system selects and applies some
reasonable strategy, it is advisable to analyze the process of finding of all solutions, selecting of
candidate solutions, and the process of finding of the best optimal solution accordingly to the specified
criteria. The method of narrowing of the technological problem solution space reduces the size of the
domains and possibly increases the constraints number. Other way the tightening of the constraints
helps to narrow the search space in its relevant stages as well. Accordingly, the method of narrowing of
the technological problem can be applied at any stage of the search. So far, there can be constructed
some number of strategies for combining search routines with narrowing of the technological problem
in different ways, which are effective under certain conditions presented in the form of constraints
insofar. In the case of technological problems, "domain best" comparators perform assignments that
cannot be implemented by local comparators. Assignments can be incomparable up to a certain level,
and at the next one plus level the success functions of constraints can be compared separately. Thus,
the formalization of technological problems on the basis of constraints for comparators
"lexicographically best" can be done in the form of lexicographic technological problems on the basis
of soft constraints. After introducing of weights for the comparator "best on the sum of weights" there
can be introduced the form of technological problems on the basis of soft constraints with CF (certainty
factors) and for “locally best” comparators – the form of local technological problems based on crispy
soft constraints, respectively. In particular, it can be accepted for the subject of study that the
comparators “lexicographically better” and “best in terms of weights” are so far equivalent to search
problems based on weighted constraints.
Technological problem with soft constraints imposed
Local technological
Technological problem with Lexicographic technological
problem with soft
soft constraints and weights problem with soft constraints
constraints
Comparator adjustment
Lexicographically
Better on Better on better Locally Locally better
unsatisfied count weights count predicate ordered
better
Figure 2: Classification of comparators on the level of constraints hierarchy
Such kind of equivalence is based on some well defined polynomial transformation. Hierarchy of
constraints with local comparators does belong to some separate class of problems, because the formal
meta structural sets introduced in them are only partially ordered. So far, every class of local
technological problems based on soft constraints can be transformed into a class of search problems
based on constraints with weights, by applying of relevant refinement procedure with polynomial
characteristics. However, the construction of the inverse refinement procedure will be impossible due
to the nature of partial ordering of the set of assignments for the hierarchy of constraints. For all
identified classes of technological problems, it is possible to construct a relationship between these
classes and classical search problems based on constraints and weights. The related figures do present
the relationship between the basic classes of technological problems on the basis of constraints in
relation to their properties. From the practical implementation reasons variable labels can be combined
in the process of calculating global constraint labels, as well as in the process of determining the
preference for choosing a solution based on the minimum optimization routine or by introducing of
objective function.
Probabilistic Local technological problems
technological problems with soft constraints
Lexicographic
technological problem
with constraints
Retrieval Task
Technological
problem with soft with weights
Lexicographic
constraints and technological problem
weights with soft constraints
Figure 3: Relationships between basic classes of technological problems based on constraints
An alternative approach to work with labels, compared to the fuzzy approach, can be the way to
interpret them too as a kind of entities based on constraints.
Classes of simple technological problems based on soft constraints with weights and lexicographic
technological problems based on soft constraints can be considered as classes of search problems with
evaluations based on the construction of some relationship between classes of technological problems
based on soft constraints and search problems with evaluations. However, such a kind of matching
routine would be not complete insofar, cause the search tasks with evaluations do require a complete
ordering of the evaluated values of the controlled technological parameters [2]. So far , such a property
would not be satisfiable for a class of local technological problems based on soft constraints, so it is not
possible to define a class of search problems with evaluations for the hierarchy of constraints with any
of locally based comparators.
Technological problems based on soft weight constraints can be considered as the first optional level
for hierarchy of constraints with the comparator "best in the sum of weights". There should be a simple
refinement with finite characteristics from the initial hierarchy of constraints with the comparator "best
by the sum of weights" in the direction of the classical search problem with weights. Lexicographic
technological problems based on soft constraints will correspond as well to the first optional level for
hierarchy of constraints with the comparator "lexicographically better".
A local technological constraints based on soft constraints can be transformed into a problem that
is equivalent to a constraint-based search problem with weights by some finite refinement. In this case,
the optimal tuple of a local technological parameters based on soft constraints not only maximizes the
success function of an individual constraint at each level, but also minimizes the sum of the weights of
all the constraints that were violated. Thus, the local technological problem based on soft constraints
cannot be specified as a clarification of a lexicographic technological problem based on soft constraints
because of incomparable elements of their meta structures. Insofar a local technological problem based
on soft constraints must also be incomparable with the class of technological problems based on
classical search problems routines with weights for which the set of formal meta structure is completely
ordered. In classical search problems, the process for finding of optimal solution is considered as an
optimization problem. Consider the formal representation of such a process for the introduced classes
of technological problems on the basis of constraints, the formal structure of which is presented in
Figure 4. There are selected the main types of constraints that will be used to formalize technological
problems:
1) constraints with weights c weigh. ;
2) constraints with probability coefficients c probl. ;
3) constraints with possibility coefficients c posbl . ;
4) constraints with estimated values c ev . ;
5) constraints with preferences c pref. ;
6) fuzzy constraints c lv. (constraint with a linguistic label) - a linguistic label characterizes the
linguistic meaning of one of the characteristics of the constraint, such as validity .
the structure of constraints
Complex constraints
Constraints with relevancy
constructions
degree
constraints conjunction
Constraints with weights
constraints disjunction
Constraints with probabilities
negation
Constraints with possibilities
Constraints systems Fuzzy constarints
Intersection Constraints with evaluations
Union Constraints with preferences
Ordering the systems of constraints in the form of hierarchies
Categorization of constraints hierarchies over domains set
Figure 4: Classification and structuring of constraints for technological problems
To operate with constraints in the formal representations of technological problems, there is need
for introducing of their following characteristics:
1. The degree of relevance (validity, rd relevancy degree ) – characterizes the degree of
completeness of the descriptions made by constraint for the selected technological problem. This
characteristic is considered as static profiling scheme.
2. The degree of satisfaction ( sd satisfaction degree ) - dynamic characteristics of the run time stage.
Value sd 1 corresponds to the level of full satisfaction of the constraints and value sd 0
corresponds to complete violation of the constraint. These values are marginal so far and as a rule, the
degree of satisfaction(violation) will receive values from the range [0; 1]. It is also possible to consider
the constraints in the terms of the probability of its satisfaction and of the probability of its relevance,
which will be disclosed in the following definition.
3. The weight of the constraint is characterized by the weight value (cw constraint weight) .
The system can operate both with individual constraints (with weights or estimated values,
respectively), with systems of constraints CS ( constraints system) and constraint hierarchies
CH ( constraints hierarchy ) with a given number of levels. The hierarchy distinguishes between
mandatory levels (constraints at this level must be met) and optional (satisfaction of constraints is
preferential). Constraints may be satisfied in whole or in part with a degree of satisfaction sd. Complex
constraints can be built on a set of introduced constraints on the basis of combining and intersecting of
existing sets and systems of constraints, as well as on the basis of conjunction, disjunction and negation
of individual constraints. Accordingly to the scope (activity level) constraints would be divided into
local, domain and global ones.
The degree of relevance of the constraint c to technological problem TP will be considered as an
measure of the relation of constraint c to technological problem TP in terms of completeness of its
description. The degree of relevance will be indicated by rd , rd [0;1] . Degree of relevance rd 1
means the absolute relevance of the constraint to the technological problem, and the degree of relevance
rd 0 means the absolute irrelevance of given constraint to this concrete problem.
2.2. Formal substantiation of the solutions
For each technological problem TPi from the set {TPi }i 1..n , described by its own set of constraints
ConstrSeti {cij } j 1..m , m, n N , the set of constraints with the introduced degree of relevance will
look like {ConstrSeti {cij : rd j } j 1..m }i 1..n . The weighting factor of the constraint will be considered
as an measure of the consistency for the description of the technological problem posed by this
constraint. Let’s denote the weighting of the constraint in cw, cw [0;1] . The value of the weighting
factor cw 1 means the absolute completeness of the description by constraint of the technological
problem, and the value of the weighting factor cw 0 means the complete absence of such an
description in the constraint. The weighting factor of the assignment will be considered as an
characteristic of the importance of the assignment of an certain value to the variable. The most
convenient way to represent the weight values of assignments is to use them as variable labels itself.
The set of constraints with the introduced weights will look like:
{ConstrSeti {cij : cw j } j 1..m }i 1..n . (1)
The probability coefficient of constraint will denote the measure of the probability of satisfaction or
violation of the constraint by technological problem solving routine. Let’s denote the probabilistic
coefficient for constraint as cpr,cpr [0;1] . The value of the probability coefficient cpr 1 denotes
the absolute certainty of satisfaction (violation) of the constraint by solving of technological problem,
and the value of the probability coefficient cpr 0 denotes the absolute impossibility of satisfaction
(violation) of the constraint by solving the technological problem. The set of constraints with the
probability coefficients will look like:
{ ConstrSeti { cij : cpr j }j 1..m }i 1..n . (2)
If we do denote as cpr sat. – the probability of satisfying of the constraint, then cpr viol. will define the
probability that the constraint will be violated, it is obvious that cpr sat. 1 cpr viol. will have place.
The possibility coefficient of constraint will be considered as an possibility measure of satisfaction
or violation of the constraint in solving technological problem. Let’s denote the possibility coefficient
for constraint as cps,cps [0;1] . The value of the possibility coefficient cps 1 means the absolute
degree of realization of satisfaction (violation) of the constraint in solving a technological problem, and
the value of the possibility coefficient cps 0 means the absolute impossibility of satisfying (violating)
of the constraint in solving the problem. The set of constraints with the possibility coefficients will look
like:
{ ConstrSeti { cij : cps j }j 1..m }i 1..n . (3)
Constraints with evaluation – characterized by a description based on the evaluation value
evc [ 0;1 ] , which is a subjective assessment of the significance of the constraint indicated by decision
making routine. Constraints ConstrSeti with the estimated values will look like:
{ ConstrSeti { cij : evc j }j 1..m }i 1..n . (4)
Constraints with preferences – characterized by a description based on the coefficient of preference
pfc , pfc [0;1] , which is a subjective assessment of the importance (significance) of the constraint
indicated by subject domain expert. Constraints ConstrSeti with the introduced preferences will look
like:
{ ConstrSeti { cij : pfc j }j 1..m }i 1..n . (5)
Fuzzy constraints are characterized by a description based on linguistic meanings, i.e. values such
as "most likely", "in most cases", "almost never", "almost always", "always", "very often", "often",
"average", "rarely”, “very rarely”, “never”, “unknown ”, etc. The set of constraints with the linguistic
values will look like
{ ConstrSeti { cij : lv j }j 1..m }i 1..n . (6)
Thus, at each level, evaluations can be performed based on the assumption that the set of constraints
with the lowest index will be mandatory and all of its constraints will be satisfied, which will ultimately
allow building a solution at the level of the overall structure of the technological problem.
Global comparators Global constraints
Domain comparators Domain
constraints
Local Local
comparators constraints
Applying of comparators for
Refined SOLUTION
candidate SOLUTIONS
Constraints
hierarchies
Constraints systems Optimal SOLUTION for
technological problem
Full / Partial
constraints
satisfaction
Figure 5: Structuring of optimal solution refinement routine for technological problem
The objective function OF (objective function) is considered to be some function given on the
ordered set W set over a set of variables V. There is assumed that on the set W set some ordering has
been introduced W set . So far, elements of the set W set can be considered as coefficients of
preferences for over imposed constraints set. Thus, the process of finding of the optimal solution to a
technological problem TP can be considered as a process of satisfaction (violation) of the
superimposed set in form of a system or hierarchy of constraints ConstrSet ConstrSyst,ConstrHrch
with the introduced objective function OF . In the process of finding of the optimal solution Solopt. for
technological problem TP assignment we will consider more acceptable (preferential) in relation
to assignment when the value of the objective function for it is more than for the assignment ,
1 1
that is OF( ) OF( 1 ) . We will consider such an assignment as the optimal solution for the
technological problem on the basis of constraints, which is the most acceptable (preferential) one from
all possible.
To be able to evaluate assignments at constraint system levels ConstrSysti and constraint sets
level ConstrSeti in particular, it is obviously necessary to move to the level of multisets. For each
assignment and given levels of the hierarchy ConstrSeti { c1 , ,cnk } for the hierarchy of
constraints ConstrHrch value OF( ConstrSeti ) will correspond to the multiset
MultiSet lex { cw( c1 )OF( c1 ), ,cw( cnk )OF( cnk )} , (7)
that is OF( ConstrSeti ) N , which take place for every OF( c ),cw( c ) N .
This will mean that a very possible way to compare technological problems will be the method
"lexicographically better", which will add elements to the multiset OF(ConstrSeti ) either generate
a weighting of the violated constraint or assign “1” for each satisfied constraint.
Thus, the use of the constraint success function as well as the comparators itself to implement the
constraint weights at each of the levels i 1..kmax of the hierarchy would be an expectedly effective tool.
Assignment 1 can be considered as "better ordered" than another assignment 2 in relation to the
hierarchy of constraints, if for each of the constraints of the levels 1..k 1 , success after application
1 is equal to the success after application 2 :
k 1
c ConstrSysti ├ OF( c 1 ) OF( c 2 ) , (8)
i 1
and at the level 𝑘 the success of the constraints can be compared using their weights cw( c ) :
c ConstrSystk╞ OF([ c : cw] 1 ) cw( c ) OF([ c : cw] 2 ) . (9)
Let’s consider the hierarchy of constraints for some technological problem with weights:
kmax kmax
ConstrHrch weight ConstrSysti { ci1 : cw( ci1 ),...,cin : cw( cin )}n N . (10)
i i
i 1 i 1
The way of ordering for the formal structure ( ConstrHrch weight ,W set , W set ) allows to outline the
relationship between comparators of the type "better ordered" and "best locally". Let’s
kmax
ConstrHrch weight ConstrSysti be an hierarchy of constraints with weights and
i 1
cw : ConstrHrch weight W set - weight function. Refinement of the hierarchy ConstrHrch weight / cw
let’s present in form
kmax ni
ConstrSysti / cw,ConstrSysti / cw ConstrSetil . (11)
i 1 l 1
If the statement ConstrSystl / cw ConstrSystl takes place, then values ConstrSetil would be set for
i 1 n , l 1 ni according to the formula:
( c ConstrSysti , cl ConstrSysti :
l
( c ConstrSetil ,c ConstrSetil1 ,l1 1 ni , . (12)
1
l l1 ) ( cw( c ) W cw( c )))
Because the level ConstrSystl is mandatory and the weights have the same interpretation, it can be
assumed that:
c1 ,c2 ConstrSystl ╞ cw( c1 ) cw( c2 ) . (13)
Thus, we get the equality
ConstrSystl / cw ConstrSystl ConstrSetil . (14)
Also, since the level ConstrSyst is required, then
Sol( CH weight )╞ c, c CS . (15)
On the other hand, the refinement of the hierarchy can be seen as some new hierarchy in which the
level ConstrSeti2l2 is more important than the level ConstrSeti1l1 . Let’s the hierarchy be given as
ConstrHrch weight , weight function cw and two assignment 1 і 2 . Then here we have that if the
formal structure ( 1 , 2 , ConstrHrchweight ) is "better ordered" then we can expect that formal structure
( 1 , 2 , ConstrHrchweight / cw) would be "locally better". If the assignment 1 is a "better ordered"
solution for the hierarchy ConstrHrch weight with weight function cw , then 1 can be considered as the
"locally best" solution for refining the initial hierarchy ConstrHrch weight / cw . So far every "better
ordered" solution for the hierarchy ConstrHrch weight would be accordingly locally preferred.
Fuzzy labels for variables Hierarchical labels using
Variables labeling Using for hierarchies of
constraints
Variables ordering
Solution search for a technological problem based on constraints
Strengthening of underconstrained Weakening of
problems overconstrained problems
Domains narrowing Constraints domain
expansion
Variables adding
Variables removal
Constraint adding
Constraints removal
Adding constraints
with variables Context depending
constraints assignment
Constraints variables
labeling Preferential constraints
ordering
Values cortege matching for constraints
Figure 6: Solution control for technological problem
Accordingly the whole set of constraints imposed on the technological problem can be divided into
two subsets: a subset of absolute constraints and a subset of relative (preferential) constraints. On the
set of evaluations for assignments OF( V ) in case if some assignment does violates one of the
absolute constraints, it will accordingly be excluded from the process of finding the optimal solution.
At the same time, the violation of relative preferential constraints does not exclude the current
assignment, which violates them, but on the contrary allows to evaluate the solution (assignment) in
terms of its acceptability and accordingly to compare assignments accordingly to their acceptability.
Finally two technological problems TP (V ,D,C ) і TP (V ,D ,C ) can be considered as
equivalent when they have the same set of variables and the same set of solutions:
TP(V ,D,C ) ~ TP(V ' ,D' ,C' )╞ V V Sol set (V ,D,C ) Sol set (V ,D ,C ) . (16)
Technological problem TP (V ,D ,C ) can be considered as narrowed in relation to the initial
technological problem TP (V ,D,C ) if problems TP and TP are equivalent; the domain of each
variable Di is a subset of the corresponding domain Di , Di Di ; set of constraints C ' does more
strictly constrain the set of all possible variables assignments as the initial set C .
Since every of imposed constraint can to be understood finally as some subset of all possible
assignments, the narrowing of the constraint satisfaction problem can be understood as the removal
from the constraint of some assignments that do not participate in any of the relevant solution tuples.
Satisfaction Constraint
degree preference Possibility
Weight Evaluation
Const-
raints Probability
based Single Constraints Constraints
ERROR constraints systems hierarchies Linguistic
value
Relevancy level Satisfaction degree Evaluations
Violation degree
Figure 7: The structure of levels and attributes of constraints
Excessive assignments in the constraint will be marked such a constraint as one that is not a
projection of any of the possible solution tuples. That is
if ( s1 ,...sni ) Soli ,i N then , p Soli ├ p . (17)
An redundant domain value is a value that is not part of any of the solution tuples:
redundant( di1 Dj )╞ di Soli ,i,i1 , j N . (18)
Assignments and values that are interpreted as "redundant" can be removed from the problem.
3. Discussion. Assumptions and limitations of the research.
If the domain of any variable or of any constraint can be reduced to an empty set of constraints, it
can be concluded that the problem has no solutions in the general case. Accordingly, narrowing of the
problem will reduce the number of potential solutions and, consequently, the problem will be simpler
in terms of finding possible solutions. Such methods are based on assigning values to certain variables
with subsequent verification of assignments for compatibility with the constraints imposed on the
technological problem. The choice of such values should not be random, but should be compared with
the assignments that were made before. With the primary substitution of more preferential variables,
the task of assigning them more acceptable values becomes simpler. Variable preferences (labels) allow
to express the user's preferences along with his expectations about the complexity of assigning relevant
values. The application of this technique eliminates the difficulties that arise when solving
superimposed problems or problems with a large solution space. The most complex variables are the
source for the constraint propagation procedure, and their initial initialization can substantially narrow
the solution space. For a given constraint system with input labels for variables, the ordering of variables
is calculated based on global variable labels. This way of ordering variables belongs to the class of static
ordering. The final ordering calculation depends on the choice of annotation triplets. When choosing a
substitute for an annotation triplet, it is necessary to consider all the properties of a given problem. If it
is desirable to clearly distinguish between single variables, then the best solution is to consider
assignments at each level. If the assignment does violates certain constraints, the next value for this
variable is selected, if it exists. If there is no relevant value for a variable that does not violate any of
the constraints, a step back is performed and the variable to which the value was assigned before the
current variable would be reassigned as well. This process continues until a solution is found, or until
all combinations of variable assignments are proved to be erroneous. In this case, it can be concluded
that the technological problem is inconsistent due to imposed constraints.
4. Conclusions
Formal methods of choosing a solution from the set of all assignments that will reflect the possible
semantics of annotation of variables and assignment of variables to labels are presented. Labels will
determine how variables are arranged within process problems based on constraints with preferences as
in the case of optimization problems. The task of displaying labels on a set of technological problems
with imposed constraints and displaying in the case of a hierarchy of constraints, allows to specify the
process of finding solutions to superimposed problems, based on the specifications of classical formal
structures. An assessment of possible domain comparators for technological problems arising in the
drilling process allows to form a structure for building a solution for the introduced hierarchy of
constraints. A partial assignment is extended by including of new variables until a solution is found, or
until all partial assignments would be checked. The main idea of the approach is to collect the sets of
all partial assignments that do not violate the constraint for growing set of variables. The future research
should ensure the correctness of the approach, when all partial assignments that violate certain
constraints at a certain step of the routine would be removed, and to ensure the completeness of the
approach, when the set of all partial assignments that do not violate any of the constraints must be
controlled.
5. References
[1] M.Chesanovskyy, K.Kravtsiv, V.Protsiuk, L.Poteriailo, Software outlines for decisions making
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