Efficiency of the RUTS systems operation and their optimization

2.2 Physical-metrological fundamentals of constructing the RUTS systemsthe RUTS systems

2.2.4 Fundamentals of constructing a RUTS system

2.2.4.4 Efficiency of the RUTS systems operation and their optimization

Let us now introduce the general concepts of “efficiency” and “quality” systems. The system efficiency (") in a general form is determined by the ratio of the effect obtained from the system to the costs of its construction:

"†E.effect/=C.costs/, (2.2.29) where the effect is determined as the extent of reaching the goal (G) of the system under consideration

EqG. (2.2.30)

The multiplierqis the system quality, i.e., the extent of system compliance to the goal achievement. This interpretation eliminates the confusion that can constantly arise when using these concepts (frequently simply identifying them) and permits them to be given a specific content.

Indeed, definition (2.2.29) becomes current and means

EfficiencyDeffect/costsDwhat does it give/what does it take (for the system with regard to some external system).

If the concept “project efficiency”, ("0), is introduced as follows:

"0Dwhat has it to give/what does it take

(for the system with regard to an external one), then"Dq="0andq(the quality) takes on the sense

qD"="0Dwhat does it give/what does it have to give (to the system).

Thus, the quality is an index of the internal properties of the system which are determined by the goal of the external system, and the efficiency is an index of its external properties caused by its quality.

The goal (G) of the system of ensuring measurement uniformity has been formulated earlier as fulfilling two conditions simultaneously for NSM:

(a) closeness of the units which various measuring instruments of a given measurand have in NSM;

(b) closeness of these units to the ideal ones (on determination).

Section 2.2 Physical-metrological fundamentals of constructing the RUTS systems 139 Let us formalize these conditions in the language of the system approach used here:

Œ' sj Œ' si q

ıi2Cıj2ıij; (2.2.31a) Œ' srŒ' 0Dırıij. (2.2.31b) Condition (goal) (2.2.31a) is provided in manufacturing and verifying working mea- suring instruments against standard measuring instruments (Sm as adopted in Sec- tion 2.2.4.3). For the most part, it is a problem of the instrument-producing industry and SEUM in the part relating to departmental metrological services.

Condition (2.2.31b) means that the error of the standard measuring instrumentSm, against which the working measuring instrumentsSi andSj were verified and which are used in solving the problemszi andzjunder conditions of measurement compati- bility (2.2.19), is negligible with regard to the errorsSiandSj, but not smaller than the difference between the unit dimension realized in this standard measuring instrument and the ideal one. This is exactly the problem that the particular RUTS system solves.

Thus,the generalized goal of a particular RUTS system(for a measurand'given in SEUM) is meeting condition (2.2.31b) for any measurement problems satisfying condition (2.2.19), or

G X

'

: ı.sm/ q

ıi2Cıj2 D Œ' smŒ' 0 q

ıi2Cıj2 1 for anyŒi ¤j in system (2.2.16.1), which satisfy the condition

'meas.zi/'meas.zj/Dq

ı2i Cıj2 and '.zi/D'.zj/. (2.2.32) Strictly speaking, it is necessary to imposeproblem situations limitson this condi- tion, as well as on condition (2.2.19) concerning SEUM, i.e., equations (2.2.19) and (2.2.32) should be applied not to any pair (zi,zj) but mainly to those pairs of problems that are solved in the systems “supplier–customer”.

In practice it is very difficult to evaluate this. However the limitation indicated also plays another role that is more essential: it allows only those pairs of measurement problems to be considered which are predominantly related toone type of applied working measuring instruments(by both the range and accuracy). This significantly reduces the dimension of problems connected with determining the efficiency of the RUTS system (and SEUM), but, to be more precise, results in dividing the variables and bringing an additive form of summation of the efficiency of working measuring instruments of different groups.

Here one more note should be added, which concerns an effect of the systems RUTS ! SEUM ! SMI ! NSM. Since the working measuring instruments are used for evaluating the product quality by both a manufacturer and a customer and as a rule tolerances for corresponding parameters being measured,'k, of a product are

similar (pm D pc D perm), in both cases, it becomes important for a customer and manufacturer to meet not only condition (2.2.19), but also theerror relationship of the working measuring instruments for both a manufacturer and a customer.

Provided, for example, ıpc > ıpm, then atperm D ıpc the customer risk (con- nected with decreasing of product quality due to such a measurement) will exceed the manufacturer risk (connected with the increasing costs of the output of the products), which creates a situation where the product quality in fulfilling tasks of a production plan decreases; and vice versa, atpermDıpc> ıpmthe quality of products obtained by the customer will increase due to the increase of manufacturer expenses (failure to execute the plan).

These important conclusions lead to the followingpractical recommendations:

(1) it is useful to equip manufacturers and customers with working measuring instru- ments of similar accuracy (ıim Dıjc);

(2) when working measuring instruments are used by manufacturers and customers it is necessary to select the relationship ı

d such, that the probability of a defect of the verification of the first or second kind would be similar:p.1/Dp.2/.

The qualityq of RUTS system (') may be naturally determined by increasing the coefficient of accuracy reserve between WMI (si andsj) and MMI (sm) in condition (2.2.32), i.e.,

q X

'

D q

ıi2Cıj2

ılimm ım q

ıi2Cıj2 D ım

ılimm , (2.2.33) whereım is the really obtained in the RUTS system error of the MMI, ('), against which the given WMI group characterized byıp D q

ıipt2 Cıjpt2 D ıpt (ıi D ıj) taking into account the recommendations obtained) is verified, andılimm is the maxi- mum value of the error of these MMIs meeting condition (2.2.32).

Thus, in expression (2.2.29) for efficiency only the costs of the system, C, remain un- fixed. Considering the RUTS system as a system thatreallyallows condition (2.2.32) to be met, thetotal costs,C†, of realizing the total RUTS system (2.2.12a) in such a sense should contain: costsC1of research works on the development of its elements (MMI and methods of measurements), costsC2of manufacturing a needed park of MMI, costs C3 of their allocation, costsCex of the operation of system elements, transport costsCt rof WMIs and costsCconof realizing the control of the system:

C†D.C1CC2CC3/CEnCCexCCt rCCcon, (2.2.34) whereEnis the normative efficiency coefficient. (Here all costs should be brought to an identical interval: toT, total time of NSM at the invariability of its indices, or to Tivi 2T, the verification interval of WMIs, or toTs 2T, the selected unit of time, usually a year; then the costs will be as they are indicated.)

Section 2.2 Physical-metrological fundamentals of constructing the RUTS systems 141 The costs (C1CC2CCe) are the amount spent on pure research and engineering problems, and the costs (CeCCt rCCcon) are connected with solving and realizing or- ganizational and legal problems (including the development of normative documents concerning the RUTS system and its elements, creation of control bodies, system of metrological control and inspection intoCcon, etc.).

Since the costs of the second kind relate to elements of more generalized systems such as SEMU, SMA, and SEMQ, they will not be considered here. Then the costs relating to only to the RUTS system of the kind presented by (2.2.12a) are expressed in the form

C.RUTS/D.C1CC2/EnCex DC.'/. (2.2.34a) Then theefficiency of the RUTS system ('), satisfying goal (2.2.32), taking into account equations (2.2.33) and (2.2.34a), can be written in the form

"†.'/ E.'/

C.'/ D q.'/G.'/

C.'/ D ım ımlim at

ımlimıp 1 C.'/

ım

ıpŒ.C1CC2/EnCCe'

or

"†.',t/ ımt

ıptŒCe.'k, t /CCc.'n,t /En

(2.2.35) for each group of WMIs of the given accuracyspi;t 2.1,m/;Cc DC1C< C2. Optimization problems

A general statement of any problem on the optimization of any system†consists of founding a value of the functional:

"†DF .x,u/, (2.2.36) wherexis the uncontrollable parameters of the system anduis its controllable pa- rameters satisfying the maximum efficiency of the system"†, i.e., determination of ui.max/, corresponding to

@"

@ui D0 and uil ui.max/uih, (2.2.37) wherei 2I is the number of controllable parameters.

At the same time the following statements of the problem are possible:

(1) maximization of efficiency function at given resource limitations (and also of other types), which corresponds to finding the maximum effect in equation (2.2.29);

(2) minimization of costs (resources) at a given effect (level of a goal function and quality of the system).

In the case considered the problem of optimizing the RUTS system is formulated as follows: Find the parametersui on which the indices of the efficiency "† (RUTS) depends:

ım

ıp Dfı.ui/;CeDfe.ui/ and Cc Dfc.ui/, (2.2.38) at which the system efficiency reaches the maximum:

"0†D ı.m/

ıp.CecCCnE/ !max

ui . (2.2.39)

At the same time, the limitations can be imposed on all indices of the system in- cluding those which follow from the construction of its structure and are described in Section 2.2.4.3:

uil ui.max/uih, C CeCCcEnCmax. (2.2.40) Thus, thechoice of changeable (controllable) parametersof the system considered is determinant. This choice depends on the level of consideration, the specific problem posed when optimizing (a choice of some parameterui which is of interest), the intu- ition of the researcher, since the efficiency of optimization (the payback of costs for the optimization as a result of its effect) depends on how sensitive the system efficiency becomes due to choosing a parameter of interest).

Here various ways of posing theoptimization problemsare possible. In each of them it is necessary to choose its own set of controllable parameters which should be solved in a definite order. Let us formulate some of them.

Problem 1.Determination of an optimal accuracy relationship between the levels of RUTS system (') in one chain of the unit dimension transfer.In this case the variable (controllable) parameters are the efficiency indices ıım

p and ııi

m. Cost of the creation and operation of each element of the system (MMI) is expressed as a function of its accuracy. As a rule the dependence of the form given below is chosen:

C D C0

ırel˛ . (2.2.41)

The coefficients˛andC0are determined empirically from the available data on the cost of measuring instruments (of a given measurand) of different accuracy. Usually we have 1˛2. The parameterC0is different for the cost of creation and cost of operation of a MMI, and strongly depends on the kind of measurements (the measurand being measured).

In the simplest case the parametersmandniof the RUTS system (') are given, i.e., are evaluated in advance on the basis of algorithms (see Section 2.2.4.3). Moreover,

Section 2.2 Physical-metrological fundamentals of constructing the RUTS systems 143 the efficiency functional is constructed:

"0†.'/D ım ıp

Xm iD0

niC0e

ıi˛ CEn

Xm iD0

niC0c

ıi˛

!1

D ım

ıp .C0eCEnC0c/ Xm iD0

ni ıi˛

!1

(2.2.42) and its maximum is found under the condition (limitation) resulting from the law of error accumulation at the time of transferring the unit dimension:

ıp D vu utXm

iD0

ıi2. (2.2.42.1)

Further, the problem can be complicated for the case of all transfer chains.

Problem 2. Determination of the optimal parameters of the RUTS system structure, (mandnk), and MMI errors at given WMI parameters,.Np,Tp,ıp/, as well as the means of reproducing the unit,.n0D1,T0,ı0/, i.e., of a reference MMI,.S0/.In this case the controllable parameters arem,nk andık .k D 1,: : :,m1/. At the same time a limitation is imposed:

mY1 kD0

ık ıkC1 ı0

ıp and ıp D vu

utı02C Xm

iDkC1

ıi2, (2.2.43) and a form of empirical dependence connecting the transfer capacity of MMI and their cost with the accuracy are chosen.

Problem 3. Determination of the optimal parameters of the RUTS system structure, (mand nk), and the MMI errors, taking into account the probability of the defects of verification both when transferring the unit dimension within the system limits and when verifying WMIs.The problem is similar to the preceding one, but is more com- plicated and requires additional initial data:

dependence on the accuracy of a cost of creating and operating both suitable and unsuitable MMIs;

cost of losses due to operating unsuitable WMIs;

laws of error distribution of all MMIs and WMIs (this is the main difficulty).

Problem 4.Optimization of the parameters of the RUTS system on the structure of in- teractions among its elements (dependence of methods of transferring unit dimensions on a MMI type).This problem may be formulated on the language of graph theory but is difficult for practical realization.

Problems of the types indicated have already been considered in the literature: the statement of the problem of the RUTS system optimization on theextent of unit repro- duction centralization(see Section 2.2.5.1). Actually, the problem is reduced to deter- mining such a value forn0(the number of MMI at a top level of the RUTS system) at which the corresponding functional (2.2.39) becomes the maximum. For that it is nec- essary to somehow expressn0through other parameters of the system. For example, among them there can be recurrent relationships of type (2.2.28), to which the empir- ical dependence of type (2.2.41) of one of controllable parameters.i,Ti,ti1,qi1,li/ on the error of the corresponding MMI is added, as well as taking into account the probability of verification defects, etc.

It is clear that atn0D1 we havecomplete centralizationof the unit reproduction, and atn0!Npthetotal decentralization.

In this connection it is convenient to describe the extent of centralizationby the coefficient which is invers ton0:

n01; 1 Np1. (2.2.44)

Problem 5.Determination of the optimal extent of centralization () of reproducing the unit in the RUTS system (') while varying the remaining parameters of the system.

ni DniC1tiC1

TiC1 1CqiC1

ili , iD0,m;

0niniC1tiC1

TiC1 1CqiC1

ili <1;

i Df0.tiC1, TiC1, i, qiC1, li/;

pi Dfp i

i ,i1

, i Dfp.i/;

CCi DfC.i/, CEi DfE.i/;

p D vu utXm

iD0

i2, (2.2.45)

where is the mean square (standard) deviation of the unit transfer;is the permis- sible error of MMI;pis the probability of a verification defect;is the distribution of the mean square deviation of the given MMI; the remaining designations have been introduced earlier.

The solution in the general form of such a complicated problem is practically impos- sible. Therefore, it is necessary to make some allowance based on the earlier obtained empirical relationships that are sufficiently general. Some of them can be

(1) the choice of empirical dependencies of costs on errors in the form (2.2.41) at 1˛2;

Section 2.2 Physical-metrological fundamentals of constructing the RUTS systems 145 (2) the limitation of the interval of a relationship of adjacent levels with the values obtained from the previously solved optimization problems: 2< .ıiC1=ıi/ <3.5.

It is known that the influence of errors of the top elements of the system.0,1,: : :/

will be less, the more the number of levels separating them from WMI. Consequently, possible and useful are

decreasing the relationship of errors of the adjacent categories transfer in moving towards the top levels;

not taking into account the verification defects at the upper levels on the verification defects at the bottom level, and the probability of revealing unsuitable WMIs (at m1), and others.

Certainly, these issues require additional independent investigations.

Here it is important to stress that the known solution for problem 5 (byn0a number of working standards was meant) has shown that there exists an optimal relationship betweenn0,m, andi atn0 ¤ 1, i.e., at a definite extent of decentralization of the reproduction. In other words, decentralization can be economically substantiated.