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cense Attribution 4.0 International (CC BY 4.0). CybHyg-2019: International Workshop on
Cyber Hygiene, Kyiv, Ukraine, November 30, 2019.
Methods for Obtaining of Management Decisions
during Evaluating the Controlled Parameters by
Qualitative Categories
Jamil Al-Azzeh1 [0000-0002-3525-5471], Alexandr Litvinenko2 [0000-0002-8862-8032],
Dmytro Kucherov3 [0000-0002-4334-4175], Ivan-Farkhod Kashkevych4 [0000-0002-3525-5471]
and Zhenis Bagisov5 [0000-0002-8084-1234]
1Al-Balqa Applied University, Al-Salt, Jordan
2, 3, 4National Aviation University, Kyiv, Ukraine
5 M. Narikbayev KAZGUU University, Astana, Kazakhstan
azzehjamil@gmail.com, litvinen@nau.edu.ua, d_kucherov@ukr.net,
ikashkevich@gmail.com
Abstract. Expert control systems emulate the decision making ability of a hu-
man expert for solving complex problems by reasoning about knowledge. Arti-
ficial intelligence techniques are usually used for the purpose of representing
knowledge and for generating control decisions through an appropriate reason-
ing mechanism. In this paper, the generalized form of knowledge representation
models in expert control systems is represented. Furthermore, an algorithm for
deriving managerial decisions based on the method of resolving is described.
Unified control models are proposed that allow one to determine combinations
of control operations that can bring the control object to normal if it goes be-
yond the permissible ranges of several characteristics. It is proved that when as-
sessing the characteristics of the state of the control object in qualitative catego-
ries, the task of deriving a managerial decision is reduced to solving a system of
linear equations with Boolean variables or combinatorial optimization prob-
lems. Algorithms for solving such problems that implement the idea of a di-
rected enumeration of options are indicated.
Keywords: expert systems, intelligent control, logical models, decision mak-
ing, optimization.
1 Introduction
The mathematical basis of expert control systems [1-9] is formed by logical models of
knowledge representation (control models) [10-14] in conjunction with the algorithms
for logical inference of managerial decisions [15-16] corresponding to these models.
Practice shows that the main difficulties in developing such systems arise at the
stage of constructing logical models since there is no single methodology for their
formation today. The variety of forms of control models causes additional difficulties
associated with the need to develop new inference algorithms oriented to a specific
model or to adapt known methods to its structure.
The task of finding management decisions arises in those cases when one or more
characteristics of the state of the managed object go beyond the specified acceptable
ranges. In this case, the following abnormal situations are possible, each of which
requires the implementation of an appropriate algorithm for deriving a management
decision [15, 16]:
one characteristic is outside the permissible range and one control operation is
enough to normalize it;
one characteristic is outside the permissible range, but its normalization requires
the simultaneous execution of several control operations (for example, in the case
of a side effect, when the implementation of one or another control operation to
change the anomalous value of any one characteristic can lead to a change in the
values of other characteristics within acceptable ranges);
the several characteristics of the state of the managed object are out of the accepta-
ble ranges.
Characteristics of the state of the managed object can be quantified or evaluated in
qualitative categories. Assessing the characteristics of the state of the managed object
with qualitative categories requires a special approach to the development of manage-
rial decisions, which will be described below.
An analysis of the experience accumulated in publications [1-16] provides the ba-
sis for constructing a generalized logical control model that includes many predicate
expressions of four types:
1. Expressions describing possible ways of influencing the managed object (con-
trol operations), the implementation of which can lead to a change in the values of the
characteristics of its state;
2. Expressions describing the conditions for the implementation of control opera-
tions (necessary system connections and functional dependencies, availability of re-
quired resources, etc.);
3. An expression that has the following meaning: if there is a control operation that
can bring the managed object into a normal state, and the conditions for its practical
implementation are met, then it must be performed;
4. An expression reflecting the hypothesis about the possibility of implementing a
control operation that can lead to the necessary change in the value of that state char-
acteristic that is outside the allowable range.
The conclusion of the desired managerial decision based on such a model formally
reduces to proving its logical truth or inconsistency [16, 18]. For this, in expert con-
trol systems, the most commonly used resolution method is J. Robinson [16, 19]. This
method has the completeness property and guarantees to find a solution to a problem
in all cases when it objectively exists but has several significant drawbacks. The main
ones are:
1. Cumbersomeness and semantic duplication of logical expressions included in
the model;
2. Poor focus on the selection of clauses for resolution, as a result of which it is
necessary to process a large amount of information that is not used in the future;
3. The impossibility of taking into account the side effect, in which the control op-
eration performed with the aim of the required change in the value of one or another
characteristic of the managed object state can lead to unacceptable changes in the
values of other characteristics;
4. The impossibility of finding combinations of control operations in the case of
simultaneous exceeding the limits of the permissible ranges of several characteristics
of the managed object condition;
5. The inability to optimize the solution to the problem of choosing control opera-
tions according to a given criterion.
These shortcomings necessitate the development of unified management models
that adequately reflect the logic of the expert’s reasoning and allow the use of simpler
algorithms in the search for managerial decisions.
Such unified models must meet the following requirements that determine their use
in expert systems of various purposes:
1. Accounting for side effects;
2. The ability to find complex management decisions that provide for the simulta-
neous execution of combinations of control operations that can normalize the state of
the managed object as in the case of a side effect, and when going beyond the ac-
ceptable ranges of several characteristics of the states of the managed object;
3. The ability to optimize the desired management decisions according to specified
criteria.
A side effect is manifested in the fact that the control operation, which is imple-
mented to bring some characteristics into the acceptable range, can cause unaccepta-
ble changes in the values of other characteristics of the state of the managed object. In
turn, the implementation of control operations that compensate for negative changes
in the values of these other characteristics can lead to unacceptable deviations of the
next group of characteristics of the managed object condition that are not connected
with the general control operations directly with the initial characteristic, etc.
Taking into account the side effect requires the preliminary determination of the
full set of characteristics of the state of the managed object, which can change their
values as a result of the implementation of control operations aimed at bringing one or
more of the characteristics into acceptable ranges. To do this, using the special step-
by-step procedure described below.
We need to find complex management decisions that arise in two cases: when the
values of several characteristics of the state of the managed object are simultaneously
outside the acceptable ranges, as well as when there is a side effect (regardless of the
number of characteristics that have left the acceptable ranges of values).
Unified logical models of control proposed to build according to the “resource” &
“action” → “result” scheme [20], which can be considered the canonical form for
most decision-making tasks in real expert control systems. In this case, “action”
means the performance of one or more control operations, the “resource” reflects the
funds necessary for the implementation of each such operation (technological reserve
of the operation), and the required “result” determined by the current state of the
managed object.
It is possible to streamline the process of interviewing experts [21, 22] for unified
forms of logical models, automate the process of generating knowledge bases [23],
and use simple but fairly effective algorithms to derive management decisions.
The task of finding a combination of control operations is multivariant and combi-
natorial [24]. Therefore, to solve it, an algebraic model is formed that is adequate to
the logical control model and is a system of combinatorial equations of linear struc-
ture. The independent variables in the algebraic model are Boolean variables, mapped
to predicates that are part of the logical model, and describe possible control opera-
tions.
The transformation of the problem of searching for a combination of control opera-
tions from the area of mathematical logic to the field of combinatorial analysis makes
it possible to optimize the desired management solution according to a given criterion.
For this, the system of combinatorial equations is supplemented by a linear function,
the coefficients of which characterize the degree of preference for certain control
operations.
In this way, the task of finding a complex managerial solution reduced to solving a
system of combinatorial equations [25-27] or (in the presence of a criterial function)
to the problem of combinatorial optimization [26-28].
To solve this system (task), modified algorithms are used that implement the strat-
egy of directed enumeration of options [20].
2 Research method
2.1 Generalized Management Model
Summarizing the experience accumulated in publications [1–8], we can conclude that
in the vast majority of cases, the basis of logical control models is predicate expres-
sions of four types.
1. A set of predicate expressions describing possible ways of influencing the OS
(control operations), the implementation of which can lead to a change in the values
of the characteristics of its state:
ljyfDyuPyF jjj ,1)];,(),([
, (1)
where y is the conditional element of the managed object, the state characteristics of
which change as a result of the implementation of the considered control operations;
P is a predicate meaning “to be implemented”;
uj is the identifier of the ith control operation capable of changing the values of the
state characteristics of the element y;
D is a predicate describing possible changes in the characteristics of the state of the
managed object;
fj is a function that specifies changes in the characteristics of the state of the man-
aged object elements as a result of the implementation of the jth control operation;
l is the number of control operations provided by the logical control model.
2. A lot of predicate expressions describing the conditions for the implementation
of control operations (necessary system connections and functional dependencies, the
availability of required resources, etc.):
)](),([ jjlj uQcyRyF
;
lj ,1
, (2)
where R is a predicate whose values reflect the fulfillment or non-fulfillment of the
conditions for the implementation of control operations;
cj is a constant characterizing the current state of the resource necessary for the im-
plementation of the jth control operation;
Q is a predicate that reflects the possibility of practical implementation of control
operations.
3. A predicate expression that carries the following meaning: if there is a control
operation that can bring the managed object into a normal state, and the conditions for
its practical implementation are met, then it must be performed:
)]()(&)([ xPxQxExFn
;
lj ,1
(3)
where x is the identifier of the desired management decision, that is, the management
operation that must be implemented in the current situation;
E( ) is a predicate stating the existence of a control operation that can lead to the
desired result.
4. A predicate expression that displays the hypothesis about the possibility of im-
plementing a control operation that can lead to the necessary change in the character-
istics of the state of an element:
]),()([
1yfDxExyF jn
,
lj 1
(4)
Expressions (1) - (4) together form a common logical control model:
}1,1;{ njFM j
. (5)
2.2 Algorithm for deriving managerial decisions based on a generalized
model
The conclusion of the managerial decision based on model (5) is to find such a func-
tion among the set of functions that, when substituted into the formula instead of the
predicate variable, would ensure the logical following of this formula from the set of
formulas. The establishment of such a fact formally reduces to proving the truth (for
at least one interpretation) of a logical expression:
1
1
nj
n
jFF
.
This is equivalent to proving (by the principle of “the opposite”) the inconsistency of
the inverse formula:
1
1
1
1
1
1&)()(
nj
n
j
nj
n
j
nj
n
jFFFFFF
.
To this end, expert management systems often use the resolution method of J. Robin-
son [9]. It provides for the identical transformation of each of the formulas into the
corresponding set of clauses with the further formation of a complete clausal set:
1
1
n
jj
KK
.
The resolution method is based on the following theorem: a set of clauses is contra-
dictory when there is a logical conclusion from it an empty clause. It is customary to
call empty a clause that does not contain any letters and therefore is false for any in-
terpretation.
Before solving the problem of finding a managerial decision, the general manage-
ment model (5) is set up for the current situation, which consists of the following
actions:
Fixation of the managed object element in which the failure occurred: y = e.
Determination of predicate values
),( j
ceR
,
lj ,1
, reflecting the availability of
resources necessary for the implementation of control operations. If the resource nec-
essary for the implementation of the jth control operation exists, then the predicate
),( j
ceR
takes the value true, otherwise, the value false.
The inclusion in the composition of the clausal set K of the inversion of the expres-
sion that specifies the necessary result of the control action on the managed object in
the current situation:
),( efD j
.
The logical conclusion of an empty clause from the set of clauses K supplemented
by the expression
),( efD j
is carried out by sequentially resolving pairs of clauses
containing counter-letters. In the process of resolving, the predicate arguments (con-
stants, variables, and functions) of one of the clauses are substituted into the positions
of the arguments of the other clause occupied by the variables. The desired solution to
the problem is determined by the identifier of that control operation
*
j
u
,
lj *
1
,
which, at the time of receipt of the empty clause, will occupy the position of the vari-
able x.
The described method, based on the use of the logical model (5) and the disjoint
resolution algorithm, has the completeness property and guarantees the finding of a
solution to the problem in all cases when it objectively exists, but has several signifi-
cant drawbacks. The main ones include:
1. cumbersomeness and semantic duplication of logical expressions included in the
model;
2. poor focus on the selection of clauses for resolution, as a result of which it is
necessary to process a large amount of information that is not used in the future;
3. the impossibility of taking into account the side effect, in which the control opera-
tion performed with the aim of the required change in the value of one or another
characteristic of the managed object condition can lead to negative changes in the
values of other characteristics;
4. the impossibility of finding combinations of control operations in the case of sim-
ultaneous exceeding the limits of allowable ranges of several characteristics of the
managed object condition;
5. the inability to optimize the solution to the problem of choosing control operations
according to a given criterion.
These shortcomings necessitate the development of unified management models
that adequately reflect the logic of the expert’s reasoning and allow applying simpler
algorithms to finding managerial decisions.
2.3 Unified models and algorithms
The task of finding management decisions arises in those cases when one or more
characteristics of the state of the control object go beyond the specified allowable
ranges. The following abnormal situations are possible, each of which requires the
implementation of an appropriate algorithm for deriving a managerial decision:
one characteristic is outside the permissible range and one control operation is
enough to normalize it;
one characteristic is outside the permissible range, but its normalization requires
the simultaneous execution of several control operations (for example, in the case
of a side effect, when the implementation of one or another control operation to
change the anomalous value of any one characteristic can lead to a change in the
values of other characteristics within acceptable ranges);
out of the acceptable ranges are several characteristics of the state of the managed
object.
The characteristics of the managed object condition can be quantified or evaluated
in qualitative categories.
Unified control models and their corresponding algorithms should provide the op-
portunity to find control operations in all these cases. Similar models and algorithms
focused on the quantitative measurement of the characteristics of the managed object
state are considered in [10].
Assessing the characteristics of the state of the managed object with qualitative
categories requires a specific approach to the development of managerial decisions,
which will be described below.
In most cases, logical control models are built according to the scheme "resource"
& "action" → "result". In this case, “action” refers to the performance of one or more
control operations, the “resource” reflects the means necessary for the implementation
of each such operation (technological reserve of the operation), and the required “re-
sult” is determined depending on one or another state of the managed object.
Let
),1|( mizz i
is the vector of characteristics of the managed object state.
When assessing the characteristics of the managed object condition with qualitative
categories, the logical control model constructed according to the described scheme
can be represented by the following formula:
)],()[)(,1( ijiji azDMJjmi
, (6)
where
),(&),( jjjjj ruXsrRM
,
Ji is a lot of numbers of control operations, the implementation of which leads to a
change in the value of the ith state characteristic of the managed object;
uj is the jth identifier of the control operation;
rj is the identifier of the resource necessary for the implementation of the jth con-
trol operation;
sj is an indicator of the state of the resource rj;
aij is the indicator of changes in the characteristics zi of the managed object condi-
tion under the influence of the jth control operation;
),( jj srR
is the predicate that, after setting up the model for the situation, reflects
the fact of the presence [if
1),(
jj srR
] or absence [if
0),(
jj srR
] of the resource
necessary for the implementation of the control operation uj;
),( jj ruX
is the predicate reflecting the fact of performing [at
1),(
jj ruX
] a con-
trol operation uj using a resource rj;
),( ijiazD
is the predicate reflecting the fact of a change in the [at
1),(
ijiazD
] ith
characteristics of the managed object condition as a result of the implementation of
the control operation uj.
Depending on the specifics of the managed object, the parameters of the model (6)
can carry different semantic load. For example, an indicator sj of the state of a re-
source rj can state a certain fact (“ON”, “OFF”) or reflect a quantitative characteristic
of the resource necessary for performing the jth control operation. The indicator aij
indicates the direction of change in the characteristics zi of the managed object state
under the influence of the jth control operation (“INCREASE”, “DECREASE”).
Let
*
i
z
be a characteristic of the managed object condition, the value of which is
outside the allowable range, and let
*
i
b
be a pointer to the direction of its required
change.
The algorithm for finding a control operation capable of bringing this characteristic
into an acceptable range provides for the following actions:
1. The formation of many control operations numbers that can lead to the desired
change in this characteristic:
}:{ **** ,ijii
D
ibaJjJ
.
2. Determination of a subset
D
i
J*
of numbers of control operations that make up the
set, for the implementation of which there are necessary resources:
}1),(:{ ** jj
D
i
R
isrRJjJ
.
If it turns out that
R
i
J*
, this means that model (6) does not provide for the possi-
bility of changing the value of the characteristic
*
i
z
in the current situation.
3. The choice (at
1
*
R
i
J
) the most preferred (according to a given criterion) con-
trol operation
*
j
u
,
R
i
Jj *
*
, to be implemented.
Suppose now that the implementation of control operations requires the availability
of several types of resources. In this case, the management model takes the following
form:
)],([))(,1( iji
T
ji azDMJjmi
, (7)
where
),(&),( T
jj
T
j
T
j
TT
jruXsrRM
,
.1),(
,1),(
),(
jjtjt
Tt
jjj
T
j
T
j
TTforsrR
TforsrR
srR
j
Tj is the many types of resources necessary for the implementation of the jth con-
trol operation;
rjt is the resource identifier of tth type;
sjt is the indicator of the state of the resource rjt.
The algorithm for finding the control operation remains the same, except that a
subset of the numbers of control operations for the implementation of which the nec-
essary resources are available is determined by the formula:
}1),(:{ **
jtjt
Tt
D
i
R
isrRJjJ j
.
2.4 A side effect
A side effect can be manifested in the fact that the control operation
R
i
Jj *
*
, which
is carried out to bring the characteristic
*
i
z
into the acceptable range, can cause unac-
ceptable changes in the values of other characteristics zi,
)( *
1iIi E
the state of the
managed object, where
}:}{\}...,,1{{)( *
**
1 R
i
R
i
EJJimiiI
.
In turn, the implementation of control operations that compensate for negative chang-
es in the values of the characteristics zi,
)( *
1iIi E
, can lead to unacceptable devia-
tions of another group of characteristics of the state zi,
)( *
2iIi E
of the managed
object, not related to the general control operations directly with the characteristic
*
i
z
, etc.
Obviously, there is no side effect if
)( *
1iIE
. Otherwise, to normalize the state
of the managed object, it is necessary to implement a comprehensive management
solution that provides for the simultaneous execution of some combination of control
operations. To establish such a combination, it is necessary, first of all, to determine
the full set of characteristics
)( *
iI E
of the state of the managed object, which can
change their values as a result of the implementation of the control operation aimed at
bringing the characteristics
*
i
z
into the allowable range.
To do this, use the following step-by-step procedure.
Initially accepted:
}{)( **
0iiI E
,
R
i
EJiJ *
)( *
0
.
Further, at each kth step, the following sets are sequentially determined:
})(:}...,,1{{)( *
1
* iJJmiiI R
k
R
i
E
k
; (8)
)(
*
*
)( iIi
R
i
R
kE
k
JiJ
;
...,2,1k
(9)
The procedure ends if
)()( *
1
*iJiJ R
k
R
k
or, equivalently,
)()( *
1
*iIiI E
k
E
k
. Formed in
the described manner, the set
)()( ** iIiI E
k
E
contains numbers of all characteristics
of the managed object condition (including), the values of which can change as a
result of performing a control operation aimed at bringing the characteristic
*
i
z
into
an acceptable range.
Given the side effect, the logical management model takes the following form:
)],([),1( )(
*
*
*iji
iIi
T
j
Jj azDMmi E
i
. (10)
The task of determining the combination of control operations, the implementation of
which can bring the characteristic
*
i
z
of the managed object state to the acceptable
range while maintaining the acceptable values of all other characteristics zi
}{\)( ** iiIi E
, is multivariate and combinatorial in nature. Therefore, its solution is
based on the use of an algebraic model that adequate to the logical model (10).
To build such a model, you must:
To give indicators aij and bi, evaluated by qualitative categories, a quantitative
measurement;
)( *
iIi E
;
R
i
Jj
.
Let, for example, aij = 1 if the value of the characteristic zi as a result of the imple-
mentation of the jth control operation increases, and if aij = -1 it decreases. Similarly,
if bi = 1 it is necessary to increase the value of the characteristic zi; if bi = -1 it needs
to be reduced, and bi = 0 it left unchanged.
Match each predicate
);( T
jj ruX
to a Boolean variable
}1,0{
j
x
,
R
i
Jj *
.
The meaning of Boolean variables is as follows: if as a result of solving the prob-
lem it turns out that some variable
1
j
x
, this will mean that the control operation
j
u
is subject to implementation; when
0
j
x
this statement is false;
R
i
Jj *
.
The introduced assumptions allow us to reduce the procedure for determining the
desired combination of control operations to the solution of a system of linear combi-
natorial equations:
i
Jj jij bxa
R
i
,
)( *
iIi E
, (11)
where
)0(]}{\)([ ** i
EbiiIi
.
Example. Let assumes the managed object state described by five characteristics
i
z
,
5,1i
, and the numbers of control operations capable of changing their values
are given by the following sets:
}3,2,1{
1
R
J
;
}6,4,2{
2
R
J
;
}7,5{
3
R
J
;
}9,8{
4
R
J
;
}6,4{
5
R
J
.
Let the characteristic
1
z
is outside the allowable range.
The procedure for determining the full set
)( *
iI E
of characteristics of the state of
the managed object, which can change their values as a result of the implementation
of control operations
)( *
iI E
designed to bring the characteristics
*
i
z
into the accepta-
ble range, consists in sequentially performing the following actions:
1. The fixation of the initial conditions:
1
*i
;
}1{)1(
0
E
I
;
}3,2,1{)1(
0
R
J
;
2. The definition of the set
)1(
1
E
I
of numbers of characteristics of the state of the
managed object, which can change their values as a result of the implementation of
control operations
)1(
0
R
Jj
, following by (8):
}3,2,1{}3,2,1{}3,2,1{)1(
01 RR JJ
;
}2{}3,2,1{}6,4,2{)1(
02 RR JJ
;
}3,2,1{}7,5{)1(
03 RR JJ
;
}3,2,1{}9,8{)1(
04 RR JJ
;
}3,2,1{}6,4{)1(
05 RR JJ
.
Then
}2,1{)1(
1
E
I
.
3. The definition of the set
)1(
1
R
J
of numbers of control operations designed to
change the values of the characteristics
1
z
and
2
z
in according to (9):
}6,4,3,2,1{}6,4,2{}3,2,1{)1(
1
R
J
.
4. The definition of the set
)1(
2
E
I
of numbers of characteristics of the state of the
managed object, which can change their values as a result of the implementation of
control operations
)1(
1
R
Jj
:
}3,2,1{}6,4,3,2,1{}3,2,1{)1(
11 RR JJ
;
}6,4,2{}6,4,3,2,1{}6,4,2{)1(
12 RR JJ
;
}6,4,3,2,1{}7,5{)1(
13 RR JJ
;
}6,4,3,2,1{}9,8{)1(
14 RR JJ
;
}6,4{}6,4,3,2,1{}6,4{)1(
15 RR JJ
.
Then
}5,2,1{)1(
2
E
I
.
5. The definition of the set
)1(
2
R
J
of numbers of control operations designed to
change the values of the characteristics
1
z
,
2
z
, and
5
z
:
}6,4,3,2,1{}6,4{}6,4,2{}3,2,1{)1(
2
R
J
.
Since
)1()1( 12 RR JJ
, the procedure ends.
The full set of characteristics
)1(
E
I
of the managed object state, which can change
their values as a result of the control operation aimed at bringing the characteristics
1
z
into the acceptable range, is determined by the formula:
}5,2,1{)1()1( 2 EE II
.
Conclusion: to compensate for the side effect, in the system of equations (11), it is
necessary to include expressions corresponding to the characteristics of the managed
object condition
1
z
,
2
z
, and
5
z
:
1313212111 bxaxaxa
,
0
626424222 xaxaxa
,
0
656454 xaxa
,
where
}1,1{
1b
, depending on the direction of the required change in characteristics
1
z
.
To solve this system, a modified algorithm can be used that implements a strategy
of directed enumeration of variants [10, 11].
The numbers of variables
j
x
,
}6,4,3,2,1{)1()1( 2 RR JJj
that, as a result of
solving this system of equations take a value of unity, determine the set of control
operations
j
u
that must be performed to bring the characteristics
1
z
into an accepta-
ble range under the side effect.
2.5 Definition of combinations of control operations
Let I* be the set of numbers of characteristics of the state of the managed object, the
values of which are outside the allowable ranges, and
}1,1{
i
b
,
*
Ii
let be the
direction indicators of the required changes in their values.
In the absence of a side effect, the logical control model is presented in the form
(7), and in its presence, in the form (10).
The corresponding systems of combinatorial equations are formed from expres-
sions of the form (11). In the first case, they are formed separately for each
*
Ii
. In
the second - for everyone
)( *
IIi E
, where
)( *
IIE
is the set of numbers of all char-
acteristics of the managed object condition, the values of which can change as a result
of control operations implemented to bring the totality of characteristics
i
z
,
*
Ii
into an acceptable range. Wherein
)0(]\)([ ** i
EbIIIi
.
To determine the set
)( *
IIE
, the step-by-step procedure described above is used,
during the execution of which it is initially accepted:
**
0)( IIIE
;
R
i
Ii
EJIJ *
*
)( *
0
.
Further, at each kth step, the following sets are sequentially determined:
})(:}...,,1{{)( *
1
* IJJmiII R
k
R
i
E
k
; (12)
)(
*
*
)( IIi
R
i
R
kE
k
JIJ
;
...,2,1k
(13)
The procedure ends with finding the set
)()()( **
1
*IIIIII EE
k
E
k
.
Example. Suppose the managed object state is described by seven characteristics
i
z
,
7,1i
, and the numbers of control operations capable of changing their values
are given by the following sets:
}3,1{
1
R
J
;
}4,2{
2
R
J
;
}5,1{
3
R
J
;
}9,7{
4
R
J
;
}6,4{
5
R
J
;
}11,10{
6
R
J
;
}6,5{
7
R
J
.
Suppose that the characteristics
1
z
and
2
z
are outside the allowable range.
The procedure for determining the full set
)( *
IIE
of characteristics of the state of
the managed object, which can change their values as a result of the implementation
of control operations designed to bring the characteristics
i
z
,
*
Ii
into the accepta-
ble range, consists in sequentially performing the following actions:
1. The fixation of the initial conditions:
}2,1{)( **
0 III E
;
}4,3,2,1{}4,2{}3,1{)( *
0 IJE
.
2. The definition of the set
)( *
1IIE
of numbers of characteristics of the state of the
managed object, which can change their values as a result of the implementation of
control operations
)( *
0IJj R
following by (12):
}4,3,2,1{}4,3,2,1{}3,1{)( *
01 IJJ RR
;
}4,3,2,1{}4,3,2,1{}4,2{)( *
02 IJJ RR
;
}1{}4,3,2,1{}5,1{)( *
03 IJJ RR
;
}4,3,2,1{}9,7{)( *
04 IJJ RR
;
}4{}4,3,2,1{}6,4{)( *
05 IJJ RR
;
}4,3,2,1{}11,10{)( *
06 IJJ RR
;
}4,3,2,1{}6,5{)( *
07 IJJ RR
.
Therefore
}5,3,2,1{)( *
1IIE
.
3. The definition of the set
)( *
1IJ R
of numbers of control operations designed to
change the values of the characteristics
1
z
,
2
z
,
3
z
and
5
z
according to (13):
}6,5,4,3,2,1{}6,5{}5,1{}4,2{}3,1{)( *
1 IJ R
.
4. The definition of the set
)( *
2IIE
of numbers of characteristics of the state of the
managed object, which can change their values as a result of the implementation of
control operations
)( *
1IJj R
:
}6,5,4,3,2,1{}6,5,4,3,2,1{}3,1{)( *
11 IJJ RR
;
}6,5,4,3,2,1{}6,5,4,3,2,1{}4,2{)( *
12 IJJ RR
;
}5,1{}6,5,4,3,2,1{}5,1{)( *
13 IJJ RR
;
}6,5,4,3,2,1{}9,7{)( *
14 IJJ RR
;
}6,4{}6,5,4,3,2,1{}6,4{)( *
15 IJJ RR
;
}6,5,4,3,2,1{}11,10{)( *
16 IJJ RR
;
}6,5{}6,5,4,3,2,1{}6,5{)( *
17 IJJ RR
.
Thus
}7,5,3,2,1{)( *
2IIE
.
5. The definition of the set
)( *
2IJ R
of numbers of control operations designed to
change the values of the characteristics
1
z
,
2
z
,
3
z
and
5
z
:
}6,5,4,3,2,1{}6,5{}5,1{}4,2{}3,1{)( *
2 IJ R
.
Since
)()( *
1
*
2IJIJ RR
, the procedure ends.
The complete set
)( *
IIE
of characteristics of the state of the managed object,
which can change their values as a result of the implementation of the control opera-
tion aimed at bringing the characteristics
1
z
into the acceptable range, is determined
by the formula:
}7,5,3,2,1{)()( *
2
* IIII EE
.
Conclusion: to compensate for the side effect, in the system of equations (11), it is
necessary to include expressions corresponding to the characteristics of the managed
object condition
1
z
,
2
z
,
3
z
and
5
z
:
1313111 bxaxa
,
2424222 bxaxa
,
0
535131 xaxa
,
0
656454 xaxa
,
0
676575 xaxa
,
where
}1,1{
i
b
,
}2,1{i
, depending on the directions of the required change in
characteristics
1
z
and
2
z
.
The numbers of variables
j
x
,
}6,5,4,3,2,1{)()( *
2
* IJIJj RR
that, as a re-
sult of solving this system of equations take the value of unity, determine the set of
control operations
j
u
that must be performed to bring the characteristics
1
z
and
2
z
into the acceptable range under the side effect.
The task of finding a combination of control operations that can bring the managed
object into a normal state is of a multivariate and, therefore, optimization nature.
Therefore, in addition to the system of combinatorial equations (11), in the mathemat-
ical model of this problem, it is necessary to include the criterion function:
X
Jj jj xvxf )(
, (14)
where
X
J
is the set of identifiers of the considered control operations:
X
Ii
R
i
XJJ
X
I
is the many numbers of characteristics of the state of the managed object, such
that
*
II X
in the absence of a side effect and
)( *
III EX
if there is one;
j
v
is the coefficients reflecting the degree of preference for certain control opera-
tions (for example, the costs of their implementation, technological advantages, etc.);
X
Jj
;
x
is the vector of independent Boolean variables:
)( X
jJjxx
;
}1,0{
j
x
;
X
Jj
.
In a formal statement, the problem is reduced to finding a vector of values of Boolean
variables
)( X
jJjxx
that turn into an optimum (maximum or minimum) criteri-
on function (14) subject to a system of constraints (11).
To solve it, an algorithm can be used that implements the idea of a directed enu-
meration of options in adaptation to the structure of the constraint system [20].
3 Discussion of results
The following research results obtained:
A generalized logical control model proposed, which can be considered the canon-
ical form of knowledge representation in intelligent control systems using a resolu-
tion algorithm or any other deductive inference algorithms to find management so-
lutions.
Unified forms of logical control models have been developed that allow one to find
optimal (according to a given criterion) integrated management decisions in condi-
tions of a side effect and simultaneously out beyond the acceptable ranges of sev-
eral characteristics of the managed object.
A new approach to finding managerial solutions proposed, providing for the con-
struction of an algebraic model adequate to the logical control model. It is proved
that when assessing controlled parameters by qualitative categories, the basis of
such an algebraic model is a system of linear combinatorial equations with a uni-
modular coefficient matrix. This makes it possible to use effective algorithms that
implement a strategy of directed enumeration of options to find managerial deci-
sions.
The transition from logical control models to algebraic forms allows you to:
set complex management decisions (combinations of control operations) in cases
when several characteristics of the state of a managed object that go beyond the ac-
ceptable ranges simultaneously;
to compensate for the side effect when the control operation performed with the
aim of the required change in the value of one or another characteristic of the man-
aged object condition leads to unacceptable changes in the values of other charac-
teristics;
to optimize integrated management decisions according to specified criteria.
Formalized procedures:
determining the set of control operations that can bring the managed object into a
normal state and possess the resources necessary for their practical implementa-
tion;
determination of the characteristics of the state of the managed object, to which it
is necessary to apply control operations in the condition a side effect;
construction of algebraic models adequate to unified logical control models.
The following research results have scientific novelty:
a generalized logical management model, creating the basis for standardizing the
methodology of interviewing experts in the subject area under consideration;
unified forms of logical control models that allow using simple and effective algo-
rithms to find optimal integrated management solutions;
a formalized procedure for determining the characteristics of the state of the man-
aged object, for which it is necessary to apply control operations in the face of a
side effect;
a formalized procedure for determining the set of control operations that can bring
the managed object into a normal state and have the resources necessary for their
practical implementation;
a formalized procedure for constructing algebraic models adequate to unified logi-
cal control models.
The practical value of the research results is that the application of the proposed
methods allows you to:
standardize the methodology of interviewing experts in the subject area;
automate the process of formalizing knowledge gained from experts;
reduce the design time of software of intelligent control systems;
reduce the cost of computing time to find managerial decisions during the opera-
tion of such systems;
increase the effectiveness of management decisions by using optimization algo-
rithms.
4 Conclusions
The contribution of this study lies in the proposal of unified logical control models,
as well as search algorithms based on them for complex management decisions.
The given logical models are the result of the generalization of various forms of
knowledge representation in intelligent control systems for various purposes. This
proves their adequacy to the real tasks of making managerial decisions based on logi-
cal models that reflect the logic of the reasoning of a person - an expert in the subject
area under consideration.
The developed models allow us to determine the optimal (according to the given
criteria) integrated management decisions that can bring the managed object to a
normal state, taking into account the available resources necessary for the practical
implementation of control operations, and a side effect. Also, they make it possible to
standardize the procedure for interviewing experts, to automate the process of formal-
izing the knowledge gained from them, as well as the technology for designing rele-
vant knowledge bases.
Converting the presented logical control models to algebraic form allows us to
transform the search for managerial solutions to solving a system of linear equations
with Boolean variables or (if there is a criterion function) to solving a combinatorial
optimization problem. For this, simple, but quite effective algorithms for directional
enumeration of options are used.
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