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Development of a System-Grounded Criterion of Optimal Control

I. Lutsenko

Enterprises as structures combining great number of controlled systems are

created to meet their organizers requirements in the form of a profit at the expense

of the possibility to use a generated cash flow.

It is quite natural that enterprise organizers seek to maximize their

potentialitiesopportunities resulted from the earned profit.Accordingly, they seek

to maximize receipt sizes within the certain time period by optimizing an

enterprises controlled systems.

But at present the operation mode of most of controlled systems is rather far

from being optimal. On the one hand it is caused by the nature of design that does

not allow of optimal control; on the other hand it is due to absence of an efficiency

index that can be used as a criterion of optimal (the best for production structure

organizers) control.

Here is the way the problem of optimal control criterion was treated last

century: Engineers, researchers, economists and designers are continuously

suggesting new universal, accurate and clear objective functions. One of the

authors succeeded in gathering over 100 criteria of optimizing division processes.

in 1967. After their classification it has become clear that there is no universal

criterion and choosing a process optimization or effectiveness criterion is quite a

complicated task [1].

For example, according to the process-oriented standard ISO series9000:2000 efficiency is defined as gained results-consumed resources

relationship [2].

As is seen, the situation has not changed. Even that fundamental document

only defines efficiency at the conceptual level.

This paper is aimed at developing the estimation theory to work out the

resource efficiency index which can be used as a system-grounded criterion of

optimal control.

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Direct Estimating Technology for a Controlled System Efficiency

As regards the controlled system studied (that is defined as the executive

system ES) there can be also determined an input product feed system and an

output product consumption system (Fig.1).

Product feed system Executive system Product consumption system

Fig.1. Controlled system interaction

Three types of objects circulate among the controlled systems

technological products ( ir

- input products of ES, jp

- output products of ES),exchange products ( d - input exchange products of ES, s - output exchange

products of ES) and control signals.

By-turn control signals can be divided into exchange process control ( k,x )

and technological process control ones (z,u ).

Any ES solves two problems simultaneously providing a consumption

system with its output products and maximizing the control effect.Control effect maximization (at the expense of more effective resource

utilization) is provided due to the fact that ES output product can be formed

through different types of control; each of them has its own correlation of input

products and its own process time.

If an enterprise keeps its resources and is efficient, its financial potential

grows due to the accrual of funds. At a certain moment of time this potential is

determined as the sum of raw stock and trade stock costs, fixed assets costs

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including depreciation, the balance of funds, paid but not yet received raw and

energy products and so on after deduction ofbudgetary payments, payments for

input products (raw and energy ones), payments of wages and taxes and so on.

It is the change in an enterprises financial potential level at a certain

moment that enables to estimate direct the control process efficiency but just on

average (an enterprise, as a rule, consists of a number of controlled systems) and

with rather a great error.

As it is the maximization of the financial potential rate that is the goal of

setting up an enterprise, it is necessary to control an enterprises ES in the way that

will enable to maximize the financial potential increase within the certain period of

time. From an owners point of view this mode of an enterprise functioning is the

most efficient.

To control the financial potential level it would seem possible to use

accounting potentialities. Accounting methods enables to estimate and fix the

initial financial condition of an enterprise. Such methods provide for record of

initial funds, raw and trade stock in comparable cost values and it will also provide

their movement while the systems functioning.

But even if an enterprise consists of the only controlled system, errors in

estimating the finance potential are rather considerable.

Errors arise, for instance, at the stage of accounting cost values of fixed

assets depreciation. There are various methods of calculating depreciation to

determine depreciation value. But depreciation indices do not depend on the mode

of fixed assets exploitation. Tear and wear of instruments, especially depending on

the operation mode, is not provided for in accounting.

If equal amounts have been invested into different directions then in a

definite period of time economic analysis will make it possible to estimate resource

efficiency as a result of the control, i.e. to estimate the correctness of the chosen

direction, an enterprises dislocation, operation mode of the equipment and so on.

Besides, accounting methods only allows of estimating consequences of the

made decisions. It is impossible to use them for solving control tasks in practice as

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it is impossible to restore the initial condition of an open system and then, in a

given time, obtain another result after changing control.

A number of objective moments prevent it.

Firstly, demand for an enterprises products has probabilistic nature.

Secondly, the average value of demand for finished products changes with the

lapse of time. Besides, not only automated systems operate at an enterprise but

also personnel whose knowledge, skills, motivations and so on are changeable. Tax

legislation as well as input product prices change, too; production and exchange

system processes are asynchronous. Also, results are long to expect.

The list of objective reasons can be considerably broadened but the above-

mentioned ones are enough to understand the main point.

The way out of this situation lies in developing the system of direct

determining conditional financial potential of internal (closed) controlled systems

of an enterprise (but not an enterprise as a whole). The systems should be

automated at that, where the human factor can be neglected. Within those systems

research results can be repeated many times by setting the necessary mode of feedand consumption of transformation products.

There are no exchange products in the closed systems and that is why

exchange product movement among associated closed systems is replaced by

conditional exchange product movement.

Input product cost for the studied technological process at a certain moment

of time is determined in the following way:

( )0

1

( )tI

i i

i t

re t rs rq t dt =

=

,

where i - input product identifier;

I - number of significant input products;

( )irq t - account parameter of an input product;

irs - cost estimation of an account parameter of an input product;

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( )re t - current value of a cumulative cost estimate of input products.

By-turn cost estimation of output products of the studied process at certain

moment is determined as follows:

( )01

( )tJ

j j

j t

pe t ps pq t dt

=

=

,

where j - output product identifier;J - number of significant output products;

( )j pq t - account parameter of an output product;

jps - cost estimation of an account parameter of an output product;

( )pe t - current value of cumulative cost estimates of output products.

Signals ( )re t and ( )pe t are control signals for the system of feeding

conditional exchange products of the consumption system and cumulative system

of direct control efficiency estimation. Relation between control signals and

quantitative parameters of conditional exchange products is as follows:( ) ( )de t re t = ; ( ) ( ) se t pe t =

The mathematical model of the system for determining conditional financial

potential of ES can be presented as follows

[ ]1 10 0 0

( ) ( ) ( )

f f f T T TL L

l ll lt t t

F M C de t dt se t dt z U t dt = == = =

= + + +

.

F - financial potential of the studied closed system at the moment fT ;

M - initial volume of funds in cost values;

fT - moment at which the conditional finance condition of the system is

determined;

lC - moment at which the conditional finance condition of the system l at the

moment 0t= ;

L - number of accountable mechanisms of the system studied;

l - mechanism identifier of the system studied;

de - conditional exchange product transferred by the consumption system;

se - conditional exchange product transferred by the feed system;

[ ]lz U - mechanism ldepreciation intensity depending on control.

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The structure providing implementation of the designed concept is given in Fig.2

Fig.2.

The

system

flow

block

that

provides forming of conditional financial potential

The figure shows some change of ES conditional financial potential under

heating process control change. Control change and recording of current meaningof ES conditional financial state have resulted in a diagram on fig.3.

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Fig.3. Trajectory of ES financial potential change depending on a type of control

Figure 4 shows that ES is aimed at the mode ensuring function maximum F

(U). But it is necessary to set a prolonged time period to get optimal control.

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Fig.4. Level of ES conditional financial potential at a fixed time moment t=10000

s.

It should be noted that the only way to implement the method of optimal

control search is to use closed systems.

Since the model and process duration cannot be implemented, and since

practical use of direct methods of control system efficiency estimation is not

available correct choice of control criterion should be made. The criterion should

point at the most efficient functioning mode as one that is based on comparative

estimation results of separate processes of studied control systems.

To solve the problem of the universal criterion of optimal control it isnecessary to test certain indices claiming to be efficiency indices to decide whether

they can point out the most efficient functioning mode of operated system. To

settle the task the method of direct efficiency estimation is needed.

Research based on search extremal operated system has shown that an

expense index, a conditional profit index or a conditional profitability index used

as control criteria have identical results and set control system by means of

implementing a search mode at the level of 8 units.

Fig.5. Control change in search mode using economic indices as optimization

criteria

Figure 5 shows that this control mode is not optimal one being set in the

range of 11 units. Consequently, it is necessary to develop the index of resource

use efficiency.

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3.1 AGREED AND REDUCED OPERATIONS

Any process as well as a continuous process can be presented as

consequence of processes.

Further research is based on agreed operations. It is necessary to define the

meaning of these operations.

Agreed operations are those whose integral values of output products

registered characteristics at the moment of operations completion are equal to

relevant signals of the task, and ES input products are completely transformed into

output products.

Time moment corresponding to the complete transformation of input

products of an agreed operation is defined as moment ofoperations physicalcompletion (MOPC). MOPC is marked as tf on the diagrams.

In turn, any process can be presented as a reduced vector operation (fig.6).

Consequently, it is important to determine the amplitudes of ES input and output

signals

( )1

f

s

tI

i ii t

RE rs rq t dt

=

=

; ( )1

f

s

tJ

j jj t

PE ps pq t dt

=

=

,

whereREandPEare the amplitudes of signals reflecting integral values of

registered characteristics and, therefore, of input and output products in

comparable values. An impulse signal REis correlated with the moment of

process beginning (ts) and provides re(t) function forming, and an impulse signal

PEis correlated with MOPC to form thepe (t) function.

Fig.6. Model of a reduced operation as comparable registration signals

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Using this operation model enables to gain some important data and at the

same time to develop resource use efficiency index (RUEI).

3.3 MOMENT OF OPERATIONS LOGIC COMPLETION

The two-flow model of a reduced operation based on the model on fig.6 can

be developed under consequent integration of the functions re (t) andpe (t)

0

( ) ( )t

t

ire t re t dt = ;0

( ) ( )t

t

ipe t pe t dt = .

ire(t) function is defined as resource consumption flow

ipe(t) function is defined as resource return flow

Fig.7. Model of a reduced operation as resource consumption

and resource return flows

Figure 8 shows a single flow model of a reduced operation with an

expression

0 0

( ) ( ) ( ) ( ) ( )t t

t t

ice t ire t ipe t re t dt pe t dt = + = + .

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Fig.8. Model of a reduced operation as a total flow

Input product transformation process requires certain time. For the studied

operation (Fig.6) it is 6 time periods. At the moment of physical complishing of

operation ( ft =8) the process of input products transformation is being finished.

Having studied single flow model of a reduced operation (Fig.8) it ispossible to say that its implementation is associated with losses determined by

input products time bounding. These losses, reflected by the closed flow of

involved resources, are bigger in case involved products cost estimation is higher

and the process of their transformation is longer.

Therefore, the closed flow geometry of negative half-plane reflects the loss

of control intended for goal achievement.

At the moment of ft physical complishing of operation executive system

(ES) gives input product to the consumption system. As the result resource returnflow is formed to be defined as the flow of resulted resource return ide (t).ipe (t) flows maximum value of the efficient operation is bigger than ire (t)

flow maximum value. Due to the maximum values difference flow of resulted

resource return ide (t) is formed.

Maximum value of this flow is determined by the cost values difference of

total input and output products of operation. But involved resources flow ibe (t)cannot be compensated by the cost estimation of the output product. Formal

analysis of measurement units shows it. Flow as a square has its unit of

measurement Cost x Time, and the output productp at the moment of itsintroduction into consumption system has a dimension cost. Measurement unitsof these categories are different that is why incomparable.

For example, involved resources flow ibe (t) and flow of resulted resourcereturn ide (t) can be compared. It is quite evident that operation cannot beconsidered finished till flow of resulted resource return ide (t) compensates flow ofinvolved resources ibe (t).

For the studied operation this moment is t=20. At this moment flows square

ide (t) is equal to ibe (t) flows square. This moment is called the moment of

logical complishing of operation (MOLC) and marked as lt .It was possible to determine (MOLC) in this simple way only because an

amplitude of input and output signals for the studied reduced operation were

selected to compare flows square easily.

To determine MOLC of any efficient operation it is necessary to integrate

ibe (t)and ide (t)flows. The integral function of flows module of involvedresources ibe (t) is marked as vbe (t) and the integral function of flow of resultedresource return ide (t) as vde (t). Then

0

( ) ( )

t

tvbe t ibe t dt = ;

0

( ) ( )

t

tvde t ide t dt = (Fig.9).

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Fig.9. Logical complishing moment determination of the operation by

integral functions of resource consumption and resulted resource

return flows

Integral functions vbe (t) and vde (t) unit of measurement is calculated by thecost and time component product. This unit of measurement is marked as CT.

Thereby, charts ofvbe (t) and vde (t) are intersected at the equality point ofibe (t) andide (t )flows squares.

A moment of logical complishing of the operation can be defined as the

moment of integral functions equality from functions of involved resources flowand flow of resulted resource return.

3.4. ANALYTICAL DETERMINATION OF THE MOMENT OF THE

REDUCED OPERATION LOGICAL COMPLISHING

MOLC determination based on involved resources flow ibe (t) and flow ofresulted return ide (t) gives the opportunity to realize one side of the process ofoperations result gaining. However, it is possible to calculate MOLC

mathematically basing on resource consumption ire (t) and resource return flowsipe (t) without interim transformation of these flows into ibe (t) and ide (t).

The integral function of flows module ( )ire t is marked as ( )vre t and the

integral function of flow ( )ipe t is marked as ( )vpe t (Fig. 10).

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Fig.10. Determination of logical complishing moment of operation with

integral functions of resource consumption and resource return flows

( )vre t and ( )vpe t functions are intersected at the point we have defined as

MOLC.

MOLC determination with resource consumption and resource return flows

requires less number of operations.

Thereby, MOLC can be defined as integral functions equality moment from

functions ofresource consumption and resource return flows.However, thepossibility of nonlinear functions ( )vre t and ( )vpe t replacement by linear functions

*( )vre t and *( )vpe t (Fig.11), compared to significantly nonlinear functions vbe(t)

and vde (t), is of great importance.

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Fig. 11. Determination of logical completion moment of operation with linear

functions*( )vre t and

*( )vpe t

( ) ( )

* *

svre t ire t t C C = + , 3.4.1( ) ( )* * fvpe t ipe t t C C = + . 3.4.2

Constants sC and fC are correspondingly calculated by the expressions

[ ]* s l l C ire t t = , 3.4.3

[ ]*f l l C ipe t t = . 3.4.4

Constant C determines a shift of functions ( )*vre t and ( )*vre t . For the

functions ( )*vre t and ( )*vre t (Fig.3.4.2) C=36.

onsidering (3.4.3, 3.4.4), expressions (3.4.1, 3.4.2) can be represented as

( ) [ ]* * *( ) l lvre t ire t t ire t t C = + , 3.4.5

( ) [ ]* * *( ) l lvpe t ipe t t ipe t t C = + . 3.4.6

From the expressions

( ) [ ]* *0 l lire t t ire t t C = + , 3.4.7

( ) [ ]* *

0 l lipe t t ipe t t C = + . 3.4.8time periods values are obtained when ( )*vre t and ( )*vpe t functions intersect it

[ ]( ) [ ]* */s l l r t ire t t C ire t = , 3.4.9[ ]( )* */f l l pt ipe t t C ipe t = . 3.4.10

where st - intersection moment of time axis by ( )*vre t function;

ft - the moment of time axis intersection with the function

( )*vpe t .

It can be seen that the moment st corresponds to the input products registration

moment of the given operation, the moment ft corresponds to the output products

registration moment of the given operation.

The equation system is (3.4.11)

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[ ] [ ]

[ ]

* *

* *

0

0

s s l l

f f l l

ire t t ire t t

ipe t t ipe t t

=

= 3.4.11

After solving the system relative to lt the following equation can be obtained:

[ ]

[ ]

* *

* *

f f s sl

f s

ipe t t ire t t t

ipe t ire t

=

3.4.12

Taking into consideration that*

fipe t for reduced operations is numerically

equal to fpe t or to pe , [ ]*

sire t is numerically equal to [ ]sre t or to re , the

simpler expression can be used for numerical value of determining lt .

f sl

pe t re t t

pe re

=

3.4.13

For instance , for the studied operation the following expression can be obtained:

3 8 2 220

3 2

f sl

pe t re t t

pe re

= = =

.

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The equation (1) can be generally written for the operation logic completion

moment determination of any effective operation.

[ ]0 0

0 0

( ) ( )

( ) ( )

f f

f f

t t

l t t

pe t t dt re t t dt

t

pe t dt re t dt

=

3.4.14

In discrete systems of time reading the expression 3.4.14 will be

[ ] [ ]1 1

1 1

K K

i i i ik k

l K K

i ik k

pe t re t

n

pe re

= =

= =

=

3.4.15

3.5 Operation resource capacity

The operation logic completion moment determines the time for compensation of

resulted return flow with the resource return flow. However, this compensation

takes place only in value.

While examining the flow of linked resources ibe(t) and the resulted flow ide(t), it

can be seen that these flows are dispersed in time, therefore the operation logic

completion moment has irreplaceable control losses for ES.

These control losses can be determined as the integral function of mismatch at the

operation logic completion moment [3].

Thus, control losses as a closed flow of mismatch function ( )dif t can be

determined as the mismatch of the integral function of the involved resources flow( )vbe t and the integral function of the resulted return flow ( )vde t , vectoral single-

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flow model, within the interval from the beginning of the operation till the moment

of its logical completion (fig. 12).

Fig.12. Determination of the mismatch function with the usage a the single-flow

operation.

The equal approach is control losses determination by the mismatch of the integral

function for the resource consumption flow ( )vre t and the integral function for

resource return flow ( )vpe t within the interval from the beginning of the operation

till the moment of its logical completion (pic.13).

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Fig.13. Determination of the mismatch function on the basis of a two-flow

operation.

The mismatch function of the operation given as the tuple can be determined:

( )2, 2; 4, 8s fre t pe t = = = =

1. Operation logic completion moment is determined

f sl pe t re t t

pe re =

=20 .

2. The auxiliary variable is introduced and determined

[ ]0; lt t

3. The two-flow operation model within the interval [ ]0 0; l lt t = = (fig.3.5.3)is:

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0

( ) ( )ire re dv

= ,0

( ) ( )ipe pe dv

=

integral functions of resource capacity and resource return flows within the interval

[ ]0 0; l lt t = = ( fig 3.5.4) are formed.

0 0

( ) ( )vre re dv dv

=

,

0 0

( ) ( )vpe pe dv dv

=

4. Control losses as the closed flow of mismatch within the interval from the

beginning of the operation till the moment of its logical completion is

determined. (fig. 14)

0 0 0 0

( ) ( ) ( )dif re dv dv pe dv dv

=

.

Fig.14 Uncompensated control losses as the closed flow

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6. The integral function of control losses is determined (fig.15)

0 0 0 0 0

( ) ( ) ( )

v

r re dv dv pe dv dv dv

=

Fig. 15. Function of resousecapacity of operation change

Lets define the value of integral function of losses ( )r at the moment of

(MLCO), as the value of resoucecapacity of operation. Then an expression for

resoucecapacity definition will have the following form

0 0 0 0 0

( ) ( )lt

t

R re dv dv pe dv dv dv

=

, [0, ]lv t (2.29)

R the value of resoucecapasity of operation.

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The unit of measurement of resoucecapacity is .

3.6 ANALYTICAL DEFINITION OF RESOUCECAPACITY OF THEREDUCED OPERATION

As far as resoucecapacity of reduced operation, in terms of geometrical

interpretation, is the square of triangle ABC (fig.16), for reduced technological

operation it can be defined as difference of triangles ABD and CBD.

Fig. 16. Geometrical interpretation of resource capacity of operation

Substituting function ( )*ire t t and ( )*ipe t t in expressions (3.4.5, 3.4.6)

scalar values [ ]* s sire t t and*

f fipe t t , we get expressions

[ ] [ ]* *0 s s l l ire t t ire t t C = + , 3.6.1

[ ]* *0 f f l l ipe t t ipe t t C = + . 3.6.2

Forming them as the system of equations and solving relative to , we get an

expression for BD height definition.

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[ ] [ ]

[ ]

* * * *

* *.

s f s s f f

s f

ire t ipe t t ire t ipe t t BD

ire t ipe t

=

As far as resource capacity of reduced operation we defined as difference of

right triangles, an expression for its definition will look like

( ) ( )1 1

2 2l s l f R t t BD t t BD= .

Substituting BD by its value from the expression after corresponding

transformations, we get

[ ] ( )[ ]( )

2* *

* *2

f s f s

f s

ipe t ire t t t R

ipe t ire t

=

.

Taking into account that for reduced operations*

fipe t equals to the value

f pe t or to the value pe , [ ]*

sire t numerically equal [ ]sre t or just the value

re , to define the numerical value lt one can use an expression, where the

values of signals of registration and moments of their formation are used.

( )( )

2

2

r p pe re t t R

pe re

=

. 3.4.13

For example, for the operation under consideration (fig.2.18) we

get, (. 2.18)

( )( )

2

3 2 36108

2 2

r ppe re t t R

pe re

= = =

T.

The value of resoucecapacity of resulted operation equals to the ratio of

product of cost estimation of value of the input product module and the output

product value and a squared difference value of the moments of their registration

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to double value of difference of cost estimation of module value of input and

output products.

3.7 ABSOLUTE POTENTIAL EFFECT OF OPERATION

As it was mentioned, ES starts to get control effect from the moment of

logical conclusion of operation. Till this moment, resulted flow compensates losses

connected with the operation execution.

As the resulted flow is open, the potential effect of the operation within the

single interval can be determined.

Expression structure for the potential effect of the operation must correspond to

the expression structure for resource capacity.

Its necessary to gain the effect to maintain activity ES (ineffective systems die

in straight and figurative sense). The essence of CS (functioning for synthetic

CS) is in new operations formation. It means that formation of consequent

operations is based on the using in time the results of previous operations.

Therefore the potential effect of the operation (A) can be determined as the

potential resulted resource return within the single interval of time t from theoperation logic completion moment.

Thus, d lt t t= +

The given operation shows the steps, which are necessary to determine absolute

potential effect of the operation ES

1. Operation logic completion moment can be determined:

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f sl

pe t re t t

pe re

=

=20 .

2. The position of the right boundary point of the range of function definition is

20 1 21d lt t t= + = + = .

3. Auxiliary variables , can be introduced and determined

[ ]0; dt t ; [ ];l dt t .

4. A single flow model of the operation can be formed

0 0

( ) ( ) ( )ice pe s ds re s ds

= + , . .c e

5. The function ( )vde within the interval [ ];l dt t (fig. 17) can be determined

0 0

( ) ( ) ( )lt t t

vde pe s ds re s ds dv

= +

.

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Fig.17 Formation of the function ( )vde

6. The function ( )wde within the interval [ ];l dt t (fig.18)

0 0

( ) ( ) ( )

l l

t

t t t t

wde t pe s ds re s ds dv d

= +

.

Fig. 18 Figure of function construction ( )wde

Absolute potential effect of Executive system (ES) operation equals the

value of function ( )wde at a time moment dt . So, an expression for absolute

potential effect of the function determination, in the case of vectorial presentation

of initial model, will have the following appearance

0 0

( ) ( )d

l l

t

t t t t

A pe s ds re s ds dv d

= +

.

On the assumption of geometrical interpretation of absolute potential effect

factor (APEF) (pic.18), effective factor value of the given operation, can be

defined from the following expression

[ ] [ ]( ) ( ) / 2l l i l d l A ipe t ire t t t = ,

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Using the signals of registration we will get

( ) ( ) / 2d l A pe re t t = .

RESOURCE USE EFFICIENCY FACTOR

Definition of absolute potential effect and resource capacity of technological

operation allows to obtain resource use efficiency factor as a ratio AE R= .

Substituting A and R for corresponding integral expressions we get

0 0 0

0 0 0 0 0

( ) ( )

( ) ( )

d

l

l

t

t

t

t

pe d re dv d d

E

re dv dv pe dv dv dv

+

=

,

[ ]0 0 0 0 0; ; ;l d dt t t t = = = = ; 0[ , ]dv t t ; [ , ]l dt t ; 0[ , ]lt t .

Resource use efficiency factor for the given operation we get in the form of

analytical expression

( ) ( ) ( ) ( )

( )

( )

( )

2

2 2

d d d l

s sf f

ipe t ire t pe re t t pe reE

pe re t t pe re t t

= =

.

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Fig.19 Change of efficiency of control depending on types of control

Picture 19 shows the process of optimum of extreme system of control

search, used criterion is resource use efficiency factor. As its seen on the pic.20,

given control corresponds to the control where ES provides maximization of

conditioned financial potential.

Fig. 20 Change of control in the process of optimum search

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Conclusion. Resource use efficiency factor which can be used as a criterion of

optimization, showing control that guarantees maximization of financial potential

of controlling system has been elaborated.

On the basis of produced estimation theory the grounds of economic theory can

be generated as well.

Literature:

1. .., ..

. .:, , 1978, .25

2. 9001

3. ..

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