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Formation of Product Properties Determining Its Quality in a Multi-Operation Technological Process

Written By

Andrey Rostovtsev

Submitted: March 13th, 2012 Reviewed: July 25th, 2012 Published: December 19th, 2012

DOI: 10.5772/51878

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1. Introduction

Quality management of manufacture products requires knowledge of the values and interaction of all factors which form the quality. The mathematical description or the model of the process for obtaining the required product properties which correspond to the specified quality are needed for this purpose in the first place.

One of the most widespread processes in machine-building manufacture is the multi-operation technological process. As known, formation of product properties starts from receiving blank parts or raw materials to the enterprise warehouse for subsequent processing or reprocessing. After blanking operations, the main technological operations (TOs) are performed, which in most cases are concluded by final assembling. Sometimes final surface finishing and/or deposition of coating is performed after assembling.

During formation of product properties it is necessary to take into account the measurement errors which inevitably appear during quality control at each TO. In general, the technological process may be considered as a set of successive technologic states (TS) E1)[1], in which the property index (PI) or a set of PIs obtained at the completed TO have passed quality control and keep their values unchanged. This allows representing the technological process in the form of a tuple



is the symbol of ordered preference in the sense of closeness to the final TS;

r and s are the subscripts of current TS and final TS, respectively.

The question now arises: what should be regarded as parallel transformation of the properties considered here? Undoubtedly, assembling TOs should. Here this tuple is expressed in another form:


where Т is the sign of transposition of several Er in vectorial form of recording.

Initial letter of the word «Existence» – state (French)

In case of such, so to say, ‘existential’ approach to formation of product properties, TS Er must be considered as achieving of the prescribed value by property Рr at the completed TO or, in vectorial form, as achieving of the prescribed values by a set of properties (Рr), which is testified by the PIs obtained as the result of post-operation check.

For the development of mathematical model of formation of product properties (expressed by relevant PIs) during technological process, it is essential to represent each TO in the form of elementary oriented graph (fig.1), which nodes correspond to adjacent TSs (preceding TS Er-1 and subsequent TS Er), respectively [1]. Graph edge r oriented at TS Er is symbolizing a TO or, if it is principally significant, a technological step, during which the property Рr or properties (Рr) are transformed from TS Er-1 into TS Er, as shown in fig. 1 a and 1b, respectively.

Figure 1.

Mathematical model of a technological operation r of transformation of one (a) or several (b) property indices of a product from technological state Еr-1 into technological state Еr with transformation coefficientsξr,r-1 or (ξr,r-1), respectively.

For each PI achieved by TS Er, it is convenient to split the combined random error ωrΣ2into three components: inherent error ωr, extrinsic error ψr (carried from the previous TO or TOs), and check error κr, with the following equation valid for the variances of these errors [24]:

ωrΣ2= ωr2+ψr2+κr2.E3

Neglecting the infinitely small quantities of higher orders, formula (3) allows transition to the product properties transformation coefficient


However, it should be noted that in some cases, where functional connection between coefficient ξr,r-1 and PI exists in some or other form, it is not possible to neglect these infinitely small quantities of higher orders

E.g., in case of assembling fuel-regulating components of gas turbine engines.

). This coefficient is considered here as “weight” of edge r, fig. 1.

In case of several PIs, formulas (3) and (4) may be written in vectorial-matrix form:

(ωrΣ2) = (ωr2)+(ψr2)+(κr2),E5

where round brackets denote vectorial form of the relevant errors, and

(ξr,r-1Σ) = (ψr)/(ωr-1Σ),E6

where (ξr,r-1 Σ) is the matrix of transformation of PI from TS r-1 to TS r.

Passing to the nonrandom component ΔrΣof PI combined error, it is necessary to tie its center of grouping to zero reference point which corresponds to PI nominal value. Depending on accepted normalization method, such point may be either the middle of PI tolerance zone, or one of the limits (left or right) of PI tolerance zone. These limits represent the so-called functional (if related to Es) thresholds or technological (in this case) thresholds [46], left х and right х.

Hence, the requirements to PI may be represented for each of these thresholds by semi-open intervals

x  x , [x, )    and     x  x,(0, x] ,E7

respectively, and for the tolerance zone – by segment

x x x,[x,x],E8

allowing to place PI values on xnumber axis.

If TS Er contains several non-random combined errors (ΔrΣ), they may be united, similar to random errors, into the common vector of displacement of their centers of grouping. Therefore, the non-random analog of formula (5) will be:

(ΔrΣ) = (Δωr)+(Δψr)+(Δκr),E9

where Δκris set to zero because of assumed centrality of measurement errors distribution (systematic error of measurements must be close to zero due to timely certification and calibration of measuring instruments).

Then formula (9) will take the form

(ΔrΣ) = (Δωr)+(Δψr),E10

Then it is necessary to reveal the inversion of PI errors, showing how the errors from the previous TSs migrate to subsequent TSs, and to perform, so to say, their mathematical convolution, uniting them into appropriate mathematical expressions [24, 6]. Let us start from consecutive transformation of errors of random PI components.

Thus, as mentioned earlier, blank parts or raw materials are received to the enterprise warehouse. Naturally, their PI has a combined error ω 1Σspecified by delivery terms (at first, let us consider the simplest case of inversion of a single PI). In this case inversion starts from TS E1 with combined technological error ω   1 Σ, and its first step is: transition from TS E1 to TS E2, with quadratic transformation of error variances corresponding to this step

ω2Σ2= ω22+ψ22+κ22 = ω22+ξ212ω1Σ2+κ22.E11

The second step performs transition from TS E2 to TS E3, which is characterized by two quadratic transformations:

ω3Σ2= ω32+ψ32+κ32=ω32+ξ322ω2Σ2+κ32 == ω32+ξ322(ω22+ξ212ω1Σ2+κ22)+κ32 == ω32+ξ322ω22+ξ322ξ212ω1Σ2+ξ322κ22+κ32.E12

Structure of formula (12) contains the forming, so to say, nucleus of inversion of manufacturing errors, or the inversion nucleus:


Using the method of mathematical induction, let us try to find out the tendencies of subsequent evolution of this nucleus in course of approaching to the final TS. For this purpose, let us perform similar quadratic transformations on the third step of inversion

ω4Σ2= ω42+ψ42+κ42=ω42+ξ432ω3Σ2+κ42 = ω42+ξ432(ω32+ξ322ω22+ξ322ξ212ω1Σ2+ξ322κ22+κ32)+κ42 = ω42+ξ432ω32+ξ432ξ322ω22++ξ432ξ322ξ212ω1Σ2+ξ432ξ322κ22+ξ432κ32)+κ42.E14

and on the fourth step of inversion

ω5Σ2= ω52+ψ52+κ52 = ω52+ξ542ω4Σ2+κ52== ω52+ξ542(ω42+ξ432ω32+ξ432ξ322ω22+ξ432ξ322ξ212ω1Σ2+ξ432ξ322κ22++ξ432κ32+κ42)+κ52 == ω52+ξ542ω42+ξ542ξ432ω32+ξ542ξ432ξ322ω22+ξ542ξ432ξ322ξ212ω1Σ2++ξ542ξ432ξ322κ22+ξ542ξ432κ32+ξ542κ42+κ52.E15

Formula (14) shows quite evidently the general tendencies of increase of inversion nucleus components and increase of the inversion structure as a whole. This allows making the first steps for generalization and more convenient perception of the results obtained.

To improve visual appearance of formula (14), let us introduce the generalizing coefficient Ξs1, denoting it as multiplicative coefficient of PI transformation. For s-1 linear transformations of PI, this coefficient is the product:

Ξs1= ξ21ξ32...ξr,r-1...ξs,s-1=Πr=2sξr,r-1.E16

Similarly, for quadratic transformation of errors characterized byξr,r-12:

Ξs12= ξ212ξ322...ξr,r-12...ξs,s-12=Πr=2sξr,r-12.E17

Now formula (14) may be rewritten in a simpler manner:

ω5Σ2= ω52+ξ542ω42+Ξ532ω32+Ξ522ω22+Ξ512ω1Σ2++Ξ522κ22+Ξ532κ32+Ξ542κ42+κ52.E18

Then let us generalize formula (17) for arbitrary number s of TSs, with parallel combining of similar terms:

ωsΣ2= ωs2+Ξs,s-12(ωs-12+κs-12)+Ξs,s-22(ωs-22+κs-22)+...++Ξs,r2(ωr2+κr2)+...Ξs,22(ω22+κ22)+Ξs,1Σ2ω1Σ2+κs2.E19

The following step for generalization of the results obtained will be introduction in formula (18) of the operator Σr=3sfor summing multiplicative coefficients Ξr,r-1of transformation for the current index r which is the number of TSs, i.e.Σr=3sΞr,r-12:


representing the mathematical convolution of combined limiting error ωsΣin the technological process containing s TOs performed consecutively.

In case of parallel execution of TOs, as mentioned above, the mathematical convolution on the basis of formula (2) will be


or in concise form


Now it is possible to consider in detail the structure of formulas (19) and (20). Formula (19) contains two inversion nuclei: the main nucleus


and additional nucleus


The additional inversion nucleus shows that the error of blank part PI or raw material PI at TS Е1 directly affects PI of the resulting TS ЕS, regardless of other TSs. Once again this demonstrates that special diligence is required for checking incoming blank parts, materials and supplies received from exterior enterprises for reprocessing. Both nuclei are circumposed by intrinsic errors ωS 2 and κS 2 of the final, S-th TO; these errors also deserve close attention.

It should be noted that the extrinsic (introduced) error ψr, is not present in formulas (19) and (20). It may be compared to a sewing needle which does not remain in the fabric sewn by it. As for the parallel transformation of PI errors given by formula (20) is concerned, the inversion of PI errors is performed here in the manner formally identical for all and every TS.

The resulting formula for the non-random component of PI error and consequently performed TOs will look like the linear analog of formula (19):


and for TOs performed in parallel – like the linear analog of formula (20):


For several PIs, according to formulas (5) and

(6), expressions (19) and (20) will become vectorial-matrix expressions, i.e.

(ω2sΣ) = (ωs2)+Σr=3s(Ξr,r-12)[(ωr-12)+(κr-12)]+(Ξs12)(ω1Σ2)+(κs2),E27


(ω2sΣ) =r=1s-1(ξs,rΣ2)(ωrΣ2),E28


The same relates to expressions (21) and (22):

sΣ) = (Δs)+Σr=3s(Ξr,r-1)(Δr-1)+(Ξs,1)(Δ1Σ)E29



In formulas (23) – (26), the round brackets indicate vectorial nature of the relevant component, excluding multiplicative transformation coefficients (Ξs1). These coefficients here are the product of matrices, either linear matrices

(Ξs1) = (ξ21)(ξ32)...(ξr,r-1)...(ξs-1) =Πr=2s(ξr,r-1)E31

or quadratic matrices

(Ξs12) = (ξ212)(ξ322)...(ξr,r-12)...(ξs,s-12) =Πr=2s(ξr,r-12)E32

If we consider the consequently performed TOs, then the combined measurement error κsΣaccumulated for one PI during the entire technological process in the resultant TS ЕS may be obtained from formula (19) in the form

(κsΣ2) = (κs2)+Σr=3s(Ξr,r-12)(κr-12)E33

In case of TOs performed in parallel, this error may be expressed according to formula (20) as:

(κsΣ2) =Σr=1s-1(ξsrΣ2)(κrΣ2).E34

When several PIs are checked, the formulas (29) and (30) will take vectorial-matrix form, i.e.

(κsΣ2) = (κs2)+Σr=3s(Ξr,r-12)(κr-12)E35


(κsΣ2) =Σr=1s-1(ξsr2)(κrΣ2)E36

Formulas (29) – (32) allow determining the share of measurement errors κsΣ2in the combined error ωsΣ2for a single PI as well as for several PIs, (κsΣ2)in (ωsΣ2), respectively, i.e. κsΣ/ωsΣin the resulting TS Еs [6]. For the current, intermediate TSs Еr this relation will have a similar form κrΣ/ωrΣ.

The described above method of mathematical convolution of errors, including measurement errors, in a multi-operational technological process has been applied to production of aggregates for shipbuilding and aerospace industry [3,4,7]. It allows not only revealing, performing mathematical convolution and determining the relationship between PI errors and measurement errors, but also creates prerequisites for comprehensive optimization of measurement errors and selection of measuring instruments at all TOs of a technological process [5].

In connection with broadening introduction of mathematically fuzzy (MF) methods in technological practice [8], it is interesting to know, at least as a first approximation, how the described above may be interpreted in MF form. In the aspect under consideration it is quite often caused by complexity or practical impossibility of actual determination of the value ξr,r-1 or values (ξr,r-1) for transformation coefficients of product PIs using analytical or, so to say, mathematically unfuzzy (MUF) methods. First of all, we are interested in MUF results of forming product PIs in a multi-operational technological process represented by formulas (2126) obtained above.

Let us regard fig. 4, which is a MF analog of fig. 1 for MUF transformation, as the first step in solving this problem. As before, TO here is represented by the oriented graph of transforming PI from TS Er-1 into Er, which edge now symbolizes MF coefficient ξr,r-1 of this transformation.

Formally this coefficient may be supposed to exist as a MF analog of formula (4) – the ratio of dividing two MF numbers in the symbolic notation

ξr,r-1= ψr/νrΣ-1,E37

where ξr,r-1, ψrand νrΣ-1are the components of formula (4) expressed in MF form, highlighted hereinafter by bold type to distinguish from MUF form.

Figure 2.

Mathematical model of a technological process with sequential transformation of one (a) or several (b) property indices from the first technological state Е1 into resulting technological state Еs with ig

Figure 3.

Mathematical model of a technological process with parallel transformation of several property indices from S-1 preceding technological states into one resulting technological state ES by performing r-1 technological operations with transformation coefficients (ξr,r-1).

Figure 4.

The outline of a component to be measured. ρ2 – total limiting technologic spread of L2 dimension, obtained in the process of elaboration of the component production technology,ρ1,3 – desired total limiting technologic spreads of L1 and L3 dimensions, required for the selection of an appropriate measuring tool.

However, here this ratio in general case is not applicable in the form of transformation coefficient, because MF operations of multiplying and dividing of MF numbers are not inverse to each other. This means that if Х and Y are MN numbers, then X ∙ Y / X ≠ Y. Regrettably, this also holds for operations of algebraic addition and deduction: (X+Y) - Y ≠ X.

Therefore, in MF case, MUF coefficient ξr,r-1 may be applied for its direct purpose only in the special case when determined relation exists between MF PIs of adjacent TSs Er and Er-1. The MF PIs obtained by some or other method shall be brought to mathematical unfuzziness (mathematically cleared)1) or defuzzied.

Then the following relationships will be true:

ξr,r-1= Βψr/ΒνrΣ-1= ψr/νrΣ-1,E38


Βψrand ΒνrΣ-1are carriers (or bases) of MF numbers ψr and νrΣ-1,

ψrand νrΣ-1are mathematically cleared (defuzzied) values of MF errors ψr and νΣr-1, respectively,

«»is the superscript of mathematical clearing of MF number.

It may be noted that Βψrand ΒνrΣ-1are analogs of ψr and νΣr-1, while ψrand νrΣ-1are analogs of Δψrand ΔνrΣ-1, respectively. This means that both MF data and MUF data are combined in one and the same MF number, allowing to present MF convolution for PI formation by one expression, rather than by two expressions, as in MUF case and in this transitional case.

For this purpose we will have to refer to MF binary relations on classical sets. The latter are a special case of MF sets defined on Cartesian product [9]. In the case under consideration, as shown in [10], for PI of TS Еr-1 and Еr, there is a fuzzy binary relation of R –order of Pr-1 and Pr, respectively

“And I saw mathematically clear…” (N.V.Gogol)


Pr-1RPr ,E39

which is a fuzzy set with membership function on unfuzzy Cartesian product of two universals Pr-1 and Pr..

Now let us determine appearance of PI quality check by measurement in MF case. For a single PI хit consists of the following [10]:

  • actual value of PI хis determined;

  • using inequalities (7) or (8), it is compared with PI value(s) specified in the act on production delivery and acceptance, i. e. with PI functional thresholds x┌ and х┐;

  • basing on these inequalities, either presence or absence of the relevant property Р хwith the product is revealed;

  • if property Рх is present, the product quality is considered as complying with the requirement imposed on it;

  • if property Рх is absent, the product quality is considered as non-complying with the requirement imposed on it

In this connection, when MF approach is used, measurement errors on the left х and right х functional thresholds and the influence of these errors on the results of product quality control are of interest.

The measurement errors here have the form of the so-called function of membership (FM)


where хmeans the PI measured, ___

θ means current (sequential) number of the term (θ = 1,Θ),

Θ means overall number of terms,

η means grade of membership (GM) of the term in respect of the measurement result (0≤η≤ 1),

+ means summation sign, considered as logical only inside angle brackets “< “ and “ >”.

A priori, when knowledge base (in the form of expert estimates, experimental data or some other precedents) is not available, it is reasonable to use the probabilistic FM composed basing on Gaussian normal differential distribution law normalized in regard of mean square deviations. For this purpose, MF unitary normalization of probabilities of this law is additionally used by means of dividing these probabilities by modal value. This value here is assumed equaling to 0.3989. Then these, now Gaussian, FM will look as follows for different Θ:

Θ =3 0,01103,0σ+1,0000+ 0,0110+3,0σ ,E41
Θ = 50,01103,0σ+0,32461,5σ+ 1,0000++0,3246+1,5σ+ 0,0110+3,0σ,E42
Θ =70,01103,0σ+0,13542,0σ+ 0,60671,0σ+ 1,0000+0,6067+1,0σ+ 0,1354+2,0σ + 0,0110+3,0σ.E43

FM (38) and (39) in graphic form are shown in fig.6 and 7, respectively.

Figure 5.

Mathematically fuzzy model of a technological operation r of transformation of one of property indices of a product from technological state Еr-1 into technological state Еr..νrΣandνrΣ-1− functions of appurtenance of property indices in the technological states of Еr and Еr-1.

Figure 6.

Three-term Gaussian function of membership (Θ = 3).

Figure 7.

Five-term Gaussian function of membership (Θ = 5).

It is important to note that though fig. 6 in appearance resembles the so-called MF triangular number, but in no case should be confused with it, because of “eine grosse Kleinigkeit” (German) – zero GM value at its left and right edges.

Logical summands of FM (37) – (39) are the GM of terms provided with subscripts or superscripts, except the modal term, which GM always equals to 1. These subscripts and superscripts indicate the number of root-mean-square deviations σ along PI xaxis of current terms from the modal term, with relevant sign. Positive deviations are contained in superscripts, negative deviations – in subscripts.

For the majority of practical measurements, it is quite sufficient to evaluate the combined limiting measurement error κr using three-term FM (37). Combined limiting spread of PI хis most conveniently represented by five-term FM (38) and by seven-term FM (39).

Let us assume that the dimension of the component is checked by a checking measurement system employing a double-limit electric contact sensor, and has FM (37) for the limiting spread of sensor contacts triggering.

νsensor=0.011+ 1.00+ 0.01+1,E44

figure 8 а, where values −1,0 μm of subscript and +1,0 μm of superscript of GM 0,01 for two utmost terms correspond to combined limiting error ±1 μm of sensor contacts triggering.

Let us assume a priori, in the first approximation, that the spread of the dimension of a component corresponds to FM (39) in the form

νcomp=0.013+0.142+ 0.611+ 1.00 + 0.61+1+ 0.14+2+ 0.01+3E45

graphically presented in fig. 8 b.

As seen from FM (39), the width of its carrier in the units of measurement of subscripts and superscripts equals to 6 μm. GM values in formulas (40) and (41) are given with accuracy of two digits after decimal point, which is practically sufficient for performing logical operations (algebraic operations using GM values will not be given here at all).

As a result of this, FM (41) is “fuzzified”, creating the combined FM determined by MF summing shown in figure 8.

Figure 8.

Mathematically fuzzy relationships during check by measurement of component dimensions using electrical contact sensor at the lower limit of tolerance zone.аib– functions of appurtenance of electric contact check errors and controlled component dimensions, respectively.

Then let us proceed with check by measurement. From MF point of view, check operation means alignment of the left (х) or, as the case may be, right (х) thresholds – limits of tolerance zone of component dimension, i.e. FM carrier (38), with the appropriate position of sensor contacts triggering adjusted for each of these thresholds. This alignment causes triggering of sensor contacts, in this case – at the low limit of sensor adjustment, introducing into FM (38) the check error characterized by FM (39). As the result, FM (38) is “fuzzified”,

creating the combined FM determined by MF summing shown in figure 8.

Eventually, we get the required sum

0,014+0,013+ 0,142+ 0,611+ 1,00+0,61+1+ 0,14+2  +0,01+3+0,01+4,E46

which is the seven-term FM (39),“fuzzified” by two terms up to nine-term FM.

This leads to the following conclusions related to quality check by measurement:

  1. Adjustment of triggering of any threshold checking device to one of the limits of the specified tolerance zone of PI хof the product causes additional error ωsensor, located symmetrically to the left and to the right of this zone as ωsensor/2 with MF normalized GM η, which is not over 0,01 (more precisely, 0,0110) for a priori assumed Gaussian FM;

  2. If PI хof a product is given as a functional or technological threshold, then the error ωsensor introduced by threshold checking device is located symmetrically to the left and to the right from this threshold, with the same MF indices of precision as for the tolerance zone mentioned above;

  3. Manufacturing of a product which quality corresponds to PI хspecified by some or other method may be guaranteed by symmetrical respective narrowing of its tolerance zone.

  4. In order to increase the accuracy of the results of checking PI of the product ωsensor/2 at the left side and at the right side, or by the same displacement to the right and to the left of the left threshold х or right threshold х specified instead of it, it is necessary to reduce the error ωsensor to reasonable technical-economic limits, while MF normalized GM shall be not over 0.01.


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  • Initial letter of the word «Existence» – state (French)
  • E.g., in case of assembling fuel-regulating components of gas turbine engines.
  • “And I saw mathematically clear…” (N.V.Gogol)

Written By

Andrey Rostovtsev

Submitted: March 13th, 2012 Reviewed: July 25th, 2012 Published: December 19th, 2012