**1. Introduction**

Quality management of manufacture products requires knowledge of the values and inter‐ action 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-opera‐ tion technological process. As known, formation of product properties starts from receiving blank parts or raw materials to the enterprise warehouse for subsequent processing or re‐ processing. After blanking operations, the main technological operations (TOs) are per‐ formed, 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 technologi‐ cal 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 con‐ trol and keep their values unchanged. This allows representing the technological process in the form of a tuple

$$\mathbf{E\_1 \prec E\_2 \prec \dots \prec E\_r \prec \dots \prec E\_{s-1} \prec E\_{s'} \ r = 1, s} \tag{1}$$

where:

≺ 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 proper‐ ties considered here? Undoubtedly, assembling TOs should. Here this tuple is expressed in another form:

$$(\mathbf{E}\_1, \mathbf{E}\_2, \dots, \mathbf{E}\_{\mathbf{r}}, \dots, \mathbf{E}\_{\mathbf{s-1}})^T \prec \mathbf{E}\_{\mathbf{s'}} \tag{2}$$

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

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

However, it should be noted that in some cases, where functional connection between coeffi‐ cient *ξ* r,r-1 and PI exists in some or other form, it is not possible to neglect these infinitely

= (4)

http://dx.doi.org/10.5772/51878

103

. This coefficient is considered here as "weight" of edge r,

= ++ (5)

= (6)

of PI combined error, it is necessary to tie its cen‐

, -1 -1 / . rr r r ξ ψω Σ Σ

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

2 2 22 ( ) ( ) ( ) ( ), r rrr ω ωψκ <sup>Σ</sup>

, -1 -1 ( ) ( ) / ( ), r r r r ξ ψω Σ Σ

) is the matrix of transformation of PI from TS r-1 to TS r.

ter 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 socalled functional (if related to Es) thresholds or technological (in this case) thresholds [4–6],

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

xx x xx x , , and , 0, , ≥∞ ≤ ) ( (7)

x x x xx , , , ≤ ≤ (8)

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

the product properties transformation coefficient

small quantities of higher orders2 )

Passing to the nonrandom component Δr<sup>Σ</sup>

respectively, and for the tolerance zone – by segment

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

allowing to place PI values on *x* number axis.

fig. 1.

where (*<sup>ξ</sup>* r,r-1 <sup>Σ</sup>

left *х* ┌ and right *х* ┐.

intervals

where Т is the sign of transposition of several Er in vectorial form of recording.<sup>1</sup>

In case of such, so to say, 'existential' approach to formation of product properties, TS E<sup>r</sup> must be considered as achieving of the prescribed value by property Р<sup>r</sup> 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 el‐ ementary 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 indi‐ ces of a product from technological state Еr-1 into technological state Еr with transformation coefficients ξr,r-1 or (ξr,r-1), respectively.

For each PI achieved by TS Er, it is convenient to split the combined random error *ωr<sup>Σ</sup>* 2 into three components: inherent error *ω* r, extrinsic error *ψ* <sup>r</sup> (carried from the previous TO or TOs), and check error *κ* <sup>r</sup>, with the following equation valid for the variances of these errors [2–4]:

$$
\omega\_{\eta\_{\Sigma}}^2 = \omega\_r^2 + \psi\_r^2 + \kappa\_r^2. \tag{3}
$$

<sup>1</sup> Initial letter of the word «Existence» – state (French)

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

$$
\xi\_{r,r-1\_{\Sigma}} = \psi\_r / \omega\_{r-1\_{\Sigma}}.\tag{4}
$$

However, it should be noted that in some cases, where functional connection between coeffi‐ cient *ξ* r,r-1 and PI exists in some or other form, it is not possible to neglect these infinitely small quantities of higher orders2 ) . 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:

$$(\omega\_{r\_\Sigma}^2) = (\omega\_r^2) + (\psi\_r^2) + (\kappa\_r^2),\tag{5}$$

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

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

Т

In case of such, so to say, 'existential' approach to formation of product properties, TS E<sup>r</sup> must be considered as achieving of the prescribed value by property Р<sup>r</sup> at the completed TO or, in vectorial form, as achieving of the prescribed values by a set of properties (Рr), which

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 el‐ ementary 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

**Figure 1.** Mathematical model of a technological operation r of transformation of one (a) or several (b) property indi‐ ces of a product from technological state Еr-1 into technological state Еr with transformation coefficients ξr,r-1 or

three components: inherent error *ω* r, extrinsic error *ψ* <sup>r</sup> (carried from the previous TO or TOs), and check error *κ* <sup>r</sup>, with the following equation valid for the variances of these errors

= ++ (3)

2 into

For each PI achieved by TS Er, it is convenient to split the combined random error *ωr<sup>Σ</sup>*

2 2 22 . r rrr ω ωψκ <sup>Σ</sup>

where Т is the sign of transposition of several Er in vectorial form of recording.<sup>1</sup>

(Рr) are transformed from TS Er-1 into TS Er, as shown in fig. 1 a and 1b, respectively.

is testified by the PIs obtained as the result of post-operation check.

1 2 r s-1 s (Е ,Е ,...,Е ,...,Е ) Е , ≺ (2)

another form:

102 Practical Concepts of Quality Control

(ξr,r-1), respectively.

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

[2–4]:

$$\left(\xi\_{\mathbf{r},\mathbf{r}\cdot\mathbf{l}\_{\Sigma}}\right) = \left(\psi\_{\mathbf{r}}\right) / \left(\omega\_{\mathbf{r}\cdot\mathbf{l}\_{\Sigma}}\right) \tag{6}$$

where (*<sup>ξ</sup>* r,r-1 <sup>Σ</sup> ) is the matrix of transformation of PI from TS r-1 to TS r.

Passing to the nonrandom component Δr<sup>Σ</sup> of PI combined error, it is necessary to tie its cen‐ ter 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 socalled functional (if related to Es) thresholds or technological (in this case) thresholds [4–6], left *х* ┌ and right *х* ┐.

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

$$\mathbf{x} \ge \mathbf{x}\neg \text{ / } \left[\mathbf{x}\neg\infty\right) \text{ and } \quad \mathbf{x} \le \mathbf{x}\neg\left(\mathbf{0}\land\neg\neg\right) \text{ .} \tag{7}$$

respectively, and for the tolerance zone – by segment

x x x xx , , , ≤ ≤ (8)

allowing to place PI values on *x* number axis.

<sup>2</sup> E.g., in case of assembling fuel-regulating components of gas turbine engines.

If TS Er contains several non-random combined errors (*Δr<sup>Σ</sup>* ), they may be united, similar to random errors, into the common vector of displacement of their centers of grouping. There‐ fore, the non-random analog of formula (5) will be:

$$(\Delta\_{r\_{\Sigma}}) = (\Delta \omega\_r) + (\Delta \psi\_r) + (\Delta \kappa\_r), \tag{9}$$

Using the method of mathematical induction, let us try to find out the tendencies of subse‐ quent evolution of this nucleus in course of approaching to the final TS. For this purpose, let

(

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

Σ

Σ

(14)

105

http://dx.doi.org/10.5772/51878

(15)

(18)

us perform similar quadratic transformations on the third step of inversion

22 2 2 2 2 2 2 2 2 22 2 32 21 1 32 2 3 4 4 43 3 43 32 2

)

2 2 2 2 2 22 2 5 5 5 5 5 54 4 5

= ++ = + +=

ω ω ψ κ ω ξω κ

Σ Σ

ξξω ξκ κ κ ω ξω ξξω

222 2 22 2 2 2 2 43 32 21 1 43 32 2 43 3 4

ξξξω ξξκ ξκ κ

+ + ++

Σ Σ

Σ

and on the fourth step of inversion

Σ

22 2 2 43 3 4 5

+ ++=

ξκ κ κ

)

+ + 22 2 2 2 2

222 2 54 43 32 2

ξξξκ

tions of PI, this coefficient is the product:

 (

2 2 2 2 2 22 2 2 2 2 22 4 4 4 4 4 43 3 4 4 43 3 32 2

+ ++=+ + +

ω ω ψ κ ω ξω κ ω ξ ω ξω

) .

2 2 2 2 2 22 2 222 2 22 2 5 54 4 43 3 43 32 2 43 32 21 1 43 32 2

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

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 transforma‐

s

s

<sup>s</sup> rr ss rr <sup>r</sup> ξξ ξ ξ ξ <sup>=</sup> Ξ = = Π (16)

<sup>s</sup> rr ss rr <sup>r</sup> ξξ ξ ξ ξ <sup>=</sup> Ξ = = Π (17)

<sup>2</sup> :

ω ξ ω ξω ξξω ξξξω ξξκ

=+ + + + + +

2 2 2 22 2 222 2 2222 2 5 54 4 54 43 3 54 43 32 2 54 43 32 21 1

ω ξω ξξω ξξξω ξξξξω

steps for generalization and more convenient perception of the results obtained.

1 21 32 , -1 , -1 , -1 <sup>2</sup> ... ... .

2 22 2 2 2 1 21 32 , -1 , -1 , -1 <sup>2</sup> ... ... .

2 2 22 2 2 2 2 22 5 5 54 4 53 3 52 2 51 1

Σ <sup>Σ</sup> = + +Ξ +Ξ +Ξ +

ω ω ξω ω ω ω

Similarly, for quadratic transformation of errors characterized by*ξr*,*r*-1

22 22 22 2 52 2 53 3 54 4 5 .

+Ξ + Ξ + Ξ +

κ κ κκ

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

54 43 3 54 4 5 ξξκ ξκ κ + + .

=+ + + + +

= ++=+ + = + + +

where Δ*κr* is 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 cali‐ bration of measuring instruments).

Then formula (9) will take the form

$$(\Delta\_{r\_\Sigma}) = (\Delta \omega\_r) + (\Delta \psi\_r)\_\prime \tag{10}$$

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 con‐ volution, uniting them into appropriate mathematical expressions [2–4, 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 ware‐ house. Naturally, their PI has a combined error *ω* 1*<sup>Σ</sup>* specified 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* <sup>1</sup> *<sup>Σ</sup>* , and its first step is: transition from TS E1 to TS E2, with quadratic transformation of error variances corresponding to this step

$$
\omega\_{\mathbf{2}\_{\Sigma}}^2 = \omega\_2^2 + \psi\_2^2 + \kappa\_2^2 = \omega\_2^2 + \xi\_{21}^2 \omega\_{\mathbf{1}\_{\Sigma}}^2 + \kappa\_2^2. \tag{11}
$$

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

$$\begin{aligned} \omega\_{3\_{\mathbb{\Sigma}}}^2 &= \omega\_3^2 + \psi\_3^2 + \kappa\_3^2 = \omega\_3^2 + \xi\_{32}^2 \omega\_{2\_{\mathbb{\Sigma}}}^2 + \kappa\_3^2 = \\ \theta &= \omega\_3^2 + \xi\_{32}^2 (\omega\_2^2 + \xi\_{21}^2 \omega\_{1\_{\mathbb{\Sigma}}}^2 + \kappa\_2^2) + \kappa\_3^2 = \\ \theta &= \omega\_3^2 + \xi\_{32}^2 \omega\_2^2 + \xi\_{32}^2 \xi\_{21}^2 \omega\_{1\_{\mathbb{\Sigma}}}^2 + \xi\_{32}^2 \kappa\_2^2 + \kappa\_3^2. \end{aligned} \tag{12}$$

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

$$
\xi\_{32}^2 \omega\_2^2 + \xi\_{32}^2 \xi\_{21}^2 \omega\_{1\_\Sigma}^2 \tag{13}
$$

Using the method of mathematical induction, let us try to find out the tendencies of subse‐ quent 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

$$\begin{aligned} \omega\_{4\_{\Sigma}}^2 &= \omega\_4^2 + \psi\_4^2 + \kappa\_4^2 = \omega\_4^2 + \xi\_{43}^2 \omega\_{3\_{\Sigma}}^2 + \kappa\_4^2 = \omega\_4^2 + \xi\_{43}^2 (\omega\_3^2 + \xi\_{32}^2 \omega\_2^2 + \xi\_{43}^2) \\ \xi\_{32}^2 \xi\_{21}^2 \omega\_{1\_{\Sigma}}^2 + \xi\_{32}^2 \kappa\_2^2 + \kappa\_3^2) + \kappa\_4^2 &= \omega\_4^2 + \xi\_{43}^2 \omega\_3^2 + \xi\_{43}^2 \xi\_{32}^2 \omega\_2^2 + \\ \xi\_{44}^2 + \xi\_{43}^2 \xi\_{32}^2 \omega\_{1\_{\Sigma}}^2 + \xi\_{43}^2 \xi\_{32}^2 \kappa\_2^2 + \xi\_{43}^2 \kappa\_3^2) + \kappa\_4^2. \end{aligned} \tag{14}$$

and on the fourth step of inversion

If TS Er contains several non-random combined errors (*Δr<sup>Σ</sup>*

fore, the non-random analog of formula (5) will be:

bration of measuring instruments).

104 Practical Concepts of Quality Control

Then formula (9) will take the form

random errors, into the common vector of displacement of their centers of grouping. There‐

where Δ*κr* is 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 cali‐

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 con‐ volution, uniting them into appropriate mathematical expressions [2–4, 6]. Let us start from

Thus, as mentioned earlier, blank parts or raw materials are received to the enterprise ware‐

us consider the simplest case of inversion of a single PI). In this case inversion starts from TS

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

Structure of formula (12) contains the forming, so to say, nucleus of inversion of manufac‐

E2, with quadratic transformation of error variances corresponding to this step

2 2 2 2 2 22 2 2 2 2 2 2 21 1 2 ω ω ψ κ ω ξω κ . <sup>Σ</sup> <sup>Σ</sup>

2 2 2 2 2 22 2 3 3 3 3 3 32 2 3 2 2 2 22 2 2 3 32 2 21 1 2 3 2 2 2 22 2 22 2 3 32 2 32 21 1 32 2 3

Σ Σ

2 2 22 2 32 2 32 21 1 ξω ξξω <sup>Σ</sup>

= ++=+ + =

ω ω ψ κ ω ξω κ ω ξ ω ξω κ κ ω ξω ξξω ξκ κ

 ( ) .

=+ + ++= =+ + + +

Σ Σ

∆ = ∆ +∆ +∆ (9)

∆ = ∆ +∆ (10)

specified by delivery terms (at first, let

(12)

, and its first step is: transition from TS E1 to TS

+ (13)

= ++ = + + (11)

( ) ( ) ( ) ( ), r rrr ωψκ <sup>Σ</sup>

( ) ( ) ( ), r rr ω ψ <sup>Σ</sup>

consecutive transformation of errors of random PI components.

house. Naturally, their PI has a combined error *ω* 1*<sup>Σ</sup>*

E1 with combined technological error *ω* <sup>1</sup> *<sup>Σ</sup>*

quadratic transformations:

turing errors, or the inversion nucleus:

), they may be united, similar to

2 2 2 2 2 22 2 5 5 5 5 5 54 4 5 2 2 2 2 2 22 2 222 2 22 2 5 54 4 43 3 43 32 2 43 32 21 1 43 32 2 22 2 2 43 3 4 5 2 2 2 22 2 222 2 2222 2 5 54 4 54 43 3 54 43 32 2 54 43 32 21 1 222 2 54 43 32 2 ( ) ω ω ψ κ ω ξω κ ω ξ ω ξω ξξω ξξξω ξξκ ξκ κ κ ω ξω ξξω ξξξω ξξξξω ξξξκ Σ Σ Σ Σ = ++ = + += =+ + + + + + + ++= =+ + + + + + + 22 2 2 2 2 54 43 3 54 4 5 ξξκ ξκ κ + + . (15)

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 transforma‐ tions of PI, this coefficient is the product:

$$
\Xi\_{s1} = \xi\_{21}\xi\_{32}...\xi\_{r,r-1}...\xi\_{s,s-1} = \prod\_{r=2}^{s} \xi\_{r,r-1}.\tag{16}
$$

Similarly, for quadratic transformation of errors characterized by*ξr*,*r*-1 <sup>2</sup> :

$$
\Xi\_{\rm s1}^2 = \xi\_{21}^2 \xi\_{32}^2 \dots \xi\_{r,r-1}^2 \dots \xi\_{s,s-1}^2 = \prod\_{r=2}^s \xi\_{r,r-1}^2. \tag{17}
$$

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

$$\begin{split} \omega\_{5\Sigma}^{2} &= \omega\_{5}^{2} + \xi\_{54}^{2}\omega\_{4}^{2} + \Xi\_{53}^{2}\omega\_{3}^{2} + \Xi\_{52}^{2}\omega\_{2}^{2} + \Xi\_{51}^{2}\omega\_{1\_{\Sigma}}^{2} + \\ &+ \Xi\_{52}^{2}\kappa\_{2}^{2} + \Xi\_{53}^{2}\kappa\_{3}^{2} + \Xi\_{54}^{2}\kappa\_{4}^{2} + \kappa\_{5}^{2}. \end{split} \tag{18}$$

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

$$\begin{split} \omega\_{s\_{\Sigma}}^{2} &= \omega\_{s}^{2} + \Xi\_{s,\text{s}\cdot 1}^{2} (\omega\_{s\cdot 1}^{2} + \kappa\_{s\cdot 1}^{2}) + \Xi\_{s,\text{s}\cdot 2}^{2} (\omega\_{s\cdot 2}^{2} + \kappa\_{s\cdot 2}^{2}) + \dots + \\ &+ \Xi\_{s,\text{r}}^{2} (\omega\_{r}^{2} + \kappa\_{r}^{2}) + \dots \Xi\_{s,2}^{2} (\omega\_{2}^{2} + \kappa\_{2}^{2}) + \Xi\_{s,\text{1}\_{\Sigma}}^{2} \omega\_{\text{1}\_{\Sigma}}^{2} + \kappa\_{s}^{2} . \end{split} \tag{19}$$

2 2 1 1 ( ) <sup>s</sup> <sup>ω</sup> <sup>Σ</sup>

intrinsic errors *ω* <sup>S</sup> <sup>2</sup>

and

respectively.

and *κ* <sup>S</sup> <sup>2</sup>

For several PIs, according to formulas (5) and

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

formed TOs will look like the linear analog of formula (19):

s

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

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

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 dem‐ onstrates that special diligence is required for checking incoming blank parts, materials and supplies received from exterior enterprises for reprocessing. Both nuclei are circumposed by

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

It should be noted that the extrinsic (introduced) error *ψ* r, 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 inver‐ sion 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 per‐

, -1 -1 ,1 1 <sup>3</sup>

s s rr r s <sup>Σ</sup> <sup>r</sup> <sup>Σ</sup> <sup>=</sup> <sup>Σ</sup>


2 2 2 2 2 22 2

( ) ( ) ( ) ( ) ( ) ( )( ) ( ), <sup>s</sup> s s rr r r s <sup>s</sup> <sup>r</sup> ω ω ω κ ωκ <sup>Σ</sup> <sup>Σ</sup> =

> ( ) -1 2 22 , <sup>1</sup> ( ) ( ), <sup>s</sup> <sup>s</sup> sr r <sup>r</sup> ω ξω <sup>Σ</sup> Σ Σ <sup>=</sup>

, -1 -1 -1 1 1 <sup>3</sup>

s

, <sup>1</sup> .

Ξ (24)

http://dx.doi.org/10.5772/51878

107

of the final, S-th TO; these errors also deserve close attention.

∆ =∆ + Σ Ξ ∆ +Ξ ∆ (25)

= + Ξ + +Ξ + <sup>Σ</sup> (27)

s sr r <sup>r</sup> <sup>ξ</sup> Σ Σ <sup>=</sup> ∆ =Σ ∆ (26)

= ∑ (28)

The following step for generalization of the results obtained will be introduction in formula (18) of the operator *Σ r*=3 *s* for summing multiplicative coefficients *Ξr*,*r*-1 of transformation for the current index r which is the number of TSs, i.e.*Σ r*=3 *s Ξr*,*r*-1 <sup>2</sup> :

$$
\omega\_{s\_\Sigma}^2 = \omega\_s^2 + \sum\_{r=3}^s \Xi\_{r,r-1}^2 (\omega\_{r-1}^2 + \kappa\_{r-1}^2) + \Xi\_{s,1}^2 \omega\_{1\_\Sigma}^2 + \kappa\_s^2 \tag{20}
$$

representing the mathematical convolution of combined limiting error *ωs<sup>Σ</sup>* in the technologi‐ cal 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

$$
\omega\_{s\_\Sigma}^2 = \xi\_{s, \mathbf{1}\_\Sigma}^2 \omega\_{\mathbf{1}\_\Sigma}^2 + \xi\_{s, \mathbf{2}\_\Sigma}^2 \omega\_{\mathbf{2}\_\Sigma}^2 + \dots + \xi\_{s, r\_\Sigma}^2 \omega\_{r\_\Sigma}^2 + \dots + \xi\_{s, s-1\_\Sigma}^2 \omega\_{s-1\_\Sigma}^2 \tag{21}
$$

or in concise form

$$
\omega\_{s\_{\Sigma}}^2 = \sum\_{r=1}^{s-1} \xi\_{s,r}^2 \omega\_{r\_{\Sigma}}^2 \tag{22}
$$

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

$$\sum\_{r=3}^{s} \overleftarrow{\Xi}\_{r,r-1}^{2} (\omega\_{r-1}^{2} + \kappa\_{r-1}^{2}) \tag{23}$$

and additional nucleus

Formation of Product Properties Determining Its Quality in a Multi-Operation Technological Process http://dx.doi.org/10.5772/51878 107

$$
\Xi\_{\rm s1}^2(\omega\_{1\_{\Sigma}}^2) \tag{24}
$$

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 dem‐ onstrates 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 *ω* <sup>S</sup> <sup>2</sup> and *κ* <sup>S</sup> <sup>2</sup> of the final, S-th TO; these errors also deserve close attention.

It should be noted that the extrinsic (introduced) error *ψ* r, 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 inver‐ sion 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 per‐ formed TOs will look like the linear analog of formula (19):

$$
\Delta\_{s\_{\Sigma}} = \Delta\_s + \sum\_{r=3}^{s} \Xi\_{r,r-1} \Delta\_{r-1} + \Xi\_{s,\mathbf{1}\_{\Sigma}} \Delta\_{\mathbf{1}\_{\Sigma}} \tag{25}
$$

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

$$
\Delta\_{\mathbf{s}\_{\Sigma}} = \sum\_{r=1}^{s-1} \xi\_{\mathbf{s}\_{\Sigma},r} \Delta\_r. \tag{26}
$$

For several PIs, according to formulas (5) and

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

$$\left(\boldsymbol{\omega}\_{\boldsymbol{s}\_{\underline{\boldsymbol{z}}}}\right) = \left(\boldsymbol{\omega}\_{\boldsymbol{s}}^{2}\right) + \sum\_{r=3}^{s} \left(\boldsymbol{\Xi}\_{r,r\cdot 1}^{2}\right) \left[\left(\boldsymbol{\omega}\_{r\cdot 1}^{2}\right) + \left(\boldsymbol{\kappa}\_{r\cdot 1}^{2}\right)\right] + \left(\boldsymbol{\Xi}\_{s1}^{2}\right) \left(\boldsymbol{\omega}\_{\mathbf{l}\_{\underline{\boldsymbol{z}}}}^{2}\right) + \left(\boldsymbol{\kappa}\_{s}^{2}\right) \tag{27}$$

and

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

, -1 -1 -1 , -2 -2 -2

The following step for generalization of the results obtained will be introduction in formula

*r*=3 *s Ξr*,*r*-1 <sup>2</sup> :

representing the mathematical convolution of combined limiting error *ωs<sup>Σ</sup>* in the technologi‐

In case of parallel execution of TOs, as mentioned above, the mathematical convolution on

2 22 2 2 2 2 2 2 ,1 1 ,2 2 , , -1 -1 ... ... s s <sup>s</sup> sr r ss s ω ξω ξ ω ξω ξ ω Σ ΣΣ Σ Σ Σ Σ Σ Σ

> -1 2 22 , <sup>1</sup>

222 , -1 -1 -1 <sup>3</sup> ( )

rr r r <sup>r</sup> ω κ <sup>=</sup>

s

Now it is possible to consider in detail the structure of formulas (19) and (20). Formula (19)

s

() ,

Σ Σ

for summing multiplicative coefficients *Ξr*,*r*-1 of transformation for the

= + Σ Ξ + +Ξ + (20)

= + ++ ++ (21)

s sr r <sup>r</sup> ω ξω Σ Σ <sup>=</sup> = Σ (22)

ΣΞ + (23)

+Ξ + + Ξ + + Ξ + (19)

( ) ( ) ...

= +Ξ + +Ξ + + +

2 22 2 2 2 2 2

s s ss s s ss s s

ω ω ωκ ωκ

22 2 22 2 22 2 , ,2 2 2 ,1 1

sr r r s s s

( ) ... ( ) .

2 2 2 2 2 22 2 , -1 -1 -1 ,1 1 <sup>3</sup>

s s rr r r s s <sup>r</sup> ωω ω κ ωκ Σ = Σ

ωκ ωκ ω κ

similar terms:

106 Practical Concepts of Quality Control

(18) of the operator *Σ*

the basis of formula (2) will be

or in concise form

and additional nucleus

Σ

*r*=3 *s*

current index r which is the number of TSs, i.e.*Σ*

s

cal process containing s TOs performed consecutively.

contains two inversion nuclei: the main nucleus

$$\{\omega^2\_{\ s\_\Sigma}\} = \sum\_{r=1}^{s-1} \{\xi\_{s,\mathfrak{z}\_\Sigma}^2\} (\omega\_{\mathfrak{z}\_\Sigma}^2)\_r \tag{28}$$

respectively.

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

$$(\Delta\_{S\_{\Sigma}}) = (\Delta\_{S}) + \sum\_{\mathbf{r}=\mathbf{3}}^{\mathbf{S}} (\Xi\_{\mathbf{r},\mathbf{r}-1})(\Delta\_{\mathbf{r}-1}) + (\Xi\_{\mathbf{S}\_{\Sigma}1})(\Delta\_{1\_{\Sigma}}) \tag{29}$$

and

$$(\Delta\_{\mathbf{S}\_{\Sigma}}) \stackrel{\text{S-1}}{=} \sum\_{\mathbf{r}=1}^{\text{S-1}} (\xi\_{\mathbf{S}\_{\Sigma},\mathbf{r}}) (\Delta\_{\mathbf{r}}).\tag{30}$$

and

error *ωs<sup>Σ</sup>*

logical process [5].

(21 – 26) obtained above.

of dividing two MF numbers in the symbolic notation

ed hereinafter by bold type to distinguish from MUF form.

formation.

*κrΣ* /*ωr<sup>Σ</sup>* .


resulting TS Еs [6]. For the current, intermediate TSs Еr this relation will have a similar form

The described above method of mathematical convolution of errors, including measure‐ ment 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 reveal‐ ing, performing mathematical convolution and determining the relationship between PI errors and measurement errors, but also creates prerequisites for comprehensive optimiza‐ tion of measurement errors and selection of measuring instruments at all TOs of a techno‐

In connection with broadening introduction of mathematically fuzzy (MF) methods in tech‐ nological practice [8], it is interesting to know, at least as a first approximation, how the de‐ scribed 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 *ξ* r,r-1 or values (*ξ* r,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

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 trans‐ forming PI from TS Er-1 into Er, which edge now symbolizes MF coefficient *ξ* r,r-1 of this trans‐

Formally this coefficient may be supposed to exist as a MF analog of formula (4) – the ratio

where *ξ* r,r-1, *ψr* and *νrΣ*-1 are the components of formula (4) expressed in MF form, highlight‐

= (37)


Formulas (29) – (32) allow determining the share of measurement errors *κs<sup>Σ</sup>*

2 for a single PI as well as for several PIs, (*κs<sup>Σ</sup>*

= Σ (36)

<sup>2</sup> ), respectively, i.e. *κs<sup>Σ</sup>*

<sup>2</sup> ) in (*ωs<sup>Σ</sup>*

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

<sup>2</sup> in the combined

http://dx.doi.org/10.5772/51878

/*ωs<sup>Σ</sup>* in the

109

In formulas (23) – (26), the round brackets indicate vectorial nature of the relevant compo‐ nent, excluding multiplicative transformation coefficients (*Ξ*s1). These coefficients here are the product of matrices, either linear matrices

$$
\langle \Xi\_{\mathfrak{s}1} \rangle = \langle \xi\_{21} \rangle \langle \xi\_{32} \rangle ... \langle \xi\_{r,r-1} \rangle ... \langle \xi\_{\mathfrak{s}-1} \rangle \\
= \prod\_{r=2}^{s} \langle \xi\_{r,r-1} \rangle \tag{31}
$$

or quadratic matrices

$$(\Xi\_{\rm s1}^2) = (\xi\_{21}^2)(\xi\_{32}^2)...(\xi\_{r,r-1}^2)...(\xi\_{s,s-1}^2) = \prod\_{r=2}^s (\xi\_{r,r-1}^2) \tag{32}$$

If we consider the consequently performed TOs, then the combined measurement error *κs<sup>Σ</sup>* accumulated for one PI during the entire technological process in the resultant TS ЕS may be obtained from formula (19) in the form

$$\left(\boldsymbol{\kappa}\_{s\_{\overline{\boldsymbol{s}}}}^2\right) = \left(\boldsymbol{\kappa}\_{\boldsymbol{s}}^2\right) + \sum\_{r=3}^{s} (\overline{\boldsymbol{\Xi}}\_{r,r-1}^2) (\boldsymbol{\kappa}\_{r-1}^2) \tag{33}$$

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

$$(\boldsymbol{\kappa}\_{s\_{\Sigma}}^2) = \mathop{\sum}\_{r=1}^{s-1} (\boldsymbol{\xi}\_{s\_{\Sigma}}^2) (\boldsymbol{\kappa}\_{r\_{\Sigma}}^2). \tag{34}$$

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

$$(\kappa\_{s\_\Sigma}^2) = (\kappa\_s^2) + \sum\_{r=3}^s (\Xi\_{r,r-1}^2)(\kappa\_{r-1}^2) \tag{35}$$

and

<sup>Σ</sup> <sup>Σ</sup> 1 11 r=3 r,r-1 r-

> Σ Σ , -1 =1 s

the product of matrices, either linear matrices

obtained from formula (19) in the form

or quadratic matrices

as:

s sr r <sup>r</sup>

1 21 32 , -1 -1 , -1 <sup>2</sup> ( ) ( )( )...( )...( ) ( )

2 22 2 2 2 1 21 32 , -1 , -1 , -1 = 2 ( ) = ( )( )...( )...( ) = ( )

2 2 22

( ) ( ) ( )( ) <sup>s</sup> s s rr r <sup>r</sup> κκ κ <sup>Σ</sup> =

> -1 2 22 1 ( ) ( )( ). <sup>s</sup> <sup>s</sup> sr r <sup>r</sup> κ ξκ <sup>Σ</sup> Σ Σ <sup>=</sup>

2 2 22

( ) ( ) ( )( ) <sup>s</sup> s s rr r <sup>r</sup> κκ κ <sup>Σ</sup> =

If we consider the consequently performed TOs, then the combined measurement error *κs<sup>Σ</sup>* accumulated for one PI during the entire technological process in the resultant TS ЕS may be

, -1 -1 <sup>3</sup>

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

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

, -1 -1 <sup>3</sup>

In formulas (23) – (26), the round brackets indicate vectorial nature of the relevant compo‐ nent, excluding multiplicative transformation coefficients (*Ξ*s1). These coefficients here are

s s s, (∆ ) = (∆ )+ ( )(∆ )+( )(∆ ) <sup>Σ</sup> Ξ Ξ (29)

s <sup>s</sup> rr s r r <sup>r</sup> ξξ ξ ξ ξ <sup>=</sup> Ξ = = Π (31)

s

<sup>s</sup> rr ss r r <sup>r</sup> Ξ Π ξξ ξ ξ ξ (32)

= +Ξ Σ (33)

= Σ (34)

= +Ξ Σ (35)

(∆ = Σ ∆ ) ( )( ). ξ (30)

s

and

108 Practical Concepts of Quality Control

$$\left(\boldsymbol{\kappa}\_{\boldsymbol{s}\_{\boldsymbol{\Sigma}}}^{2}\right) = \sum\_{r=1}^{s-1} \{\boldsymbol{\xi}\_{\boldsymbol{s}\_{\boldsymbol{r}}}^{2}\} \left(\boldsymbol{\kappa}\_{\boldsymbol{r}\_{\boldsymbol{\Sigma}}}^{2}\right) \tag{36}$$

Formulas (29) – (32) allow determining the share of measurement errors *κs<sup>Σ</sup>* <sup>2</sup> in the combined error *ωs<sup>Σ</sup>* 2 for a single PI as well as for several PIs, (*κs<sup>Σ</sup>* <sup>2</sup> ) in (*ωs<sup>Σ</sup>* <sup>2</sup> ), respectively, i.e. *κs<sup>Σ</sup>* /*ωs<sup>Σ</sup>* in the resulting TS Еs [6]. For the current, intermediate TSs Еr this relation will have a similar form *κrΣ* /*ωr<sup>Σ</sup>* .

The described above method of mathematical convolution of errors, including measure‐ ment 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 reveal‐ ing, performing mathematical convolution and determining the relationship between PI errors and measurement errors, but also creates prerequisites for comprehensive optimiza‐ tion of measurement errors and selection of measuring instruments at all TOs of a techno‐ logical process [5].

In connection with broadening introduction of mathematically fuzzy (MF) methods in tech‐ nological practice [8], it is interesting to know, at least as a first approximation, how the de‐ scribed 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 *ξ* r,r-1 or values (*ξ* r,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 (21 – 26) 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 trans‐ forming PI from TS Er-1 into Er, which edge now symbolizes MF coefficient *ξ* r,r-1 of this trans‐ formation.

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

$$
\xi\_{r,r+1} = \Psi\_r / \mathbf{v}\_{r\_{\Sigma 4}} \, \tag{37}
$$

where *ξ* r,r-1, *ψr* and *νrΣ*-1 are the components of formula (4) expressed in MF form, highlight‐ ed 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 opera‐

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

http://dx.doi.org/10.5772/51878

111

tions with transformation coefficients (ξr,r-1).

**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 opera‐ tions with transformation coefficients (ξr,r-1).

**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

110 Practical Concepts of Quality Control

*Βψr* and *ΒνrΣ*-1

 and *νrΣ*-1 −

It may be noted that *Βψr* and *ΒνrΣ*-1

:

gle PI *х* it consists of the following [10]:

**•** actual value of PI *х* is determined;

the product is revealed;

quirement imposed on it

3 "And I saw mathematically clear…" (N.V.Gogol)

ment imposed on it;

control are of interest.

*ψr* −

ly,

of *Δψ*<sup>r</sup>

and Pr, respectively3

sals Pr-1 and Pr..

are carriers (or bases) of MF numbers *ψ* r and *νrΣ*-1

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

are mathematically cleared (defuzzied) values of MF errors *ψ* r and *νΣ<sup>r</sup>*-1

are analogs of *ψ* r and *νΣ*r-1, while *ψ<sup>r</sup>*

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

 and *Δν*rΣ-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

which is a fuzzy set with membership function on unfuzzy Cartesian product of two univer‐

Now let us determine appearance of PI quality check by measurement in MF case. For a sin‐

**•** using inequalities (7) or (8), it is compared with PI value(s) specified in the act on produc‐

**•** basing on these inequalities, either presence or absence of the relevant property Р*х* with

**•** if property Рх is present, the product quality is considered as complying with the require‐

**•** if property Рх is absent, the product quality is considered as non-complying with the re‐

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

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

tion delivery and acceptance, i. e. with PI functional thresholds x┌ and х┐;

,

−

r-1 r , P RP (39)

 and *νrΣ*-1 −

http://dx.doi.org/10.5772/51878

, respective‐

113

are analogs

**Figure 4.** The outline of a component to be measured. ρ<sup>2</sup> – 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 coef‐ ficient, 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 *ξ* r,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:

$$\dot{\hat{\xi}}\_{r,r\text{-1}} = \text{B}\psi\_r / \text{B}\nu\_{r\_{\Sigma\text{1}}} = \stackrel{\cdot}{\psi}\_r / \stackrel{\cdot}{\nu}\_{r\_{\Sigma\text{1}}} \tag{38}$$

where:

*Βψr* and *ΒνrΣ*-1 are carriers (or bases) of MF numbers *ψ* r and *νrΣ*-1 ,

*ψr* − and *νrΣ*-1 − are mathematically cleared (defuzzied) values of MF errors *ψ* r and *νΣ<sup>r</sup>*-1 , respective‐ ly,

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

It may be noted that *Βψr* and *ΒνrΣ*-1 are analogs of *ψ* r and *νΣ*r-1, while *ψ<sup>r</sup>* − and *νrΣ*-1 − are analogs of *Δψ*<sup>r</sup> and *Δν*rΣ-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, respectively3 :

$$\mathbf{P}\_{\mathbf{r}\cdot\mathbf{l}}\mathbf{R}\mathbf{P}\_{\mathbf{r}\_{\cdot}}\tag{39}$$

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

Now let us determine appearance of PI quality check by measurement in MF case. For a sin‐ gle PI *х* it consists of the following [10]:

**•** actual value of PI *х* is determined;

**Figure 4.** The outline of a component to be measured. ρ<sup>2</sup> – total limiting technologic spread of L2 dimension, obtained in the process of elaboration of the component production technology,ρ1, 3 – desired total limiting technologic

However, here this ratio in general case is not applicable in the form of transformation coef‐ ficient, 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,

Therefore, in MF case, MUF coefficient *ξ* r,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

Σ Σ

−

=Β Β = (38)

spreads of L1 and L3 dimensions, required for the selection of an appropriate measuring tool.

(mathematically cleared)1) or defuzzied.

112 Practical Concepts of Quality Control

where:

Then the following relationships will be true:

this also holds for operations of algebraic addition and deduction: (X+Y) - Y ≠ X.



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)

<sup>3 &</sup>quot;And I saw mathematically clear…" (N.V.Gogol)

$$\eta\_x \left( \theta \right) = \left\langle \eta\_1 + \eta\_2 + \dots + \eta\_\theta + \dots \eta\_\Theta \right\rangle\_{\prime} \tag{40}$$

where *х* means 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 bas‐ ing on Gaussian normal differential distribution law normalized in regard of mean square de‐ viations. 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:

$$\mathbf{S}^{\ominus} = \mathbf{3} \qquad \left\langle 0, 0110\_{-3,0\sigma} + 1, 0000 + \ 0, 0110^{+3,0\sigma} \right\rangle,\tag{41}$$

**Figure 6.** Three-term Gaussian function of membership (Θ = 3).

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

http://dx.doi.org/10.5772/51878

115

**Figure 7.** Five-term Gaussian function of membership (Θ = 5).

$$\Theta = 5 \quad \left\langle 0, 0110\_{-3,0\sigma} + 0, 3246\_{-1,5\sigma} + 1, 0000 + + 0, 3246^{+1,5\sigma} + 0, 0110^{+3,0\sigma} \right\rangle\_{\prime} \tag{42}$$

$$\Theta = 7 \qquad \left\langle 0.0110\_{-3\text{ j}\sigma} + 0.1354\_{-2,\text{0}\sigma} + 0.6067\_{-1\text{ j}\sigma} + 1,0000 + 0.6067^{+10\sigma} + 0.1354^{+2\text{ j}\sigma} + 0.0110^{+3\text{ j}\sigma} \right\rangle. \tag{43}$$

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.. **ν***<sup>r</sup>* Σ and **ν***<sup>r</sup>* Σ-1 − functions of appurtenance of property indices in the technological states of Еr and Еr-1.

Formation of Product Properties Determining Its Quality in a Multi-Operation Technological Process http://dx.doi.org/10.5772/51878 115

**Figure 6.** Three-term Gaussian function of membership (Θ = 3).

( ) 1 2 , <sup>х</sup> <sup>θ</sup> ηθ η η η η = + +…+ +… <sup>Θ</sup> (40)

3,0

+ + <sup>Θ</sup> − − <sup>+</sup> + ++ + (42)

<sup>+</sup> <sup>Θ</sup> <sup>−</sup> + + (41)

1,5 3,0

1,0 2,0 3,0

− functions of appurtenance of property

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

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 bas‐ ing on Gaussian normal differential distribution law normalized in regard of mean square de‐ viations. 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

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

equaling to 0.3989. Then these, now Gaussian, FM will look as follows for different Θ:

3,0 = 3 0,0110 1,0000 0,0110 , <sup>σ</sup> σ

3,0 1,5 = 5 0,0110 0, 3246 1,0000 0, 3246 0,0110 , σ σ

3,0 2,0 1,0 =7 0,0110 0,1354 0,6067 1,0000 0,6067 0,1354 0,0110 . σσσ

++ + <sup>Θ</sup> −− − + + ++ + + (43)

**Figure 5.** Mathematically fuzzy model of a technological operation r of transformation of one of property indices of a

Σ and **ν***<sup>r</sup>* Σ-1

σ σ

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

σσ σ

product from technological state Еr-1 into technological state Еr.. **ν***<sup>r</sup>*

indices in the technological states of Еr and Еr-1.

where *х* means the PI measured, \_\_\_

114 Practical Concepts of Quality Control

Θ means overall number of terms,

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

**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 super‐ scripts, except the modal term, which GM always equals to 1. These subscripts and super‐ scripts indicate the number of root-mean-square deviations σ along PI *x* axis of current terms from the modal term, with relevant sign. Positive deviations are contained in super‐ scripts, negative deviations – in subscripts.

For the majority of practical measurements, it is quite sufficient to evaluate the combined limiting measurement error *κ* r using three-term FM (37). Combined limiting spread of PI *х* is 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.

$$\mathbf{v}\_{\text{sensor}} = \left\langle 0.01\_{-1} + 1.00 + 0.01^{+1} \right\rangle\_{\text{'}} \tag{44}$$

**Figure 8.** Mathematically fuzzy relationships during check by measurement of component dimensions using electrical contact sensor at the lower limit of tolerance zone. а i *b* – functions of appurtenance of electric contact check errors

**1.** Adjustment of triggering of any threshold checking device to one of the limits of the specified tolerance zone of PI *х* of the product causes additional error *ω* sensor, located symmetrically to the left and to the right of this zone as *ω* sensor/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 *х* of a product is given as a functional or technological threshold, then the error *ω* sensor 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

**3.** Manufacturing of a product which quality corresponds to PI *х* specified by some or oth‐ er method may be guaranteed by symmetrical respective narrowing of its tolerance

−− − − + + + ++ + + + (46)

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

http://dx.doi.org/10.5772/51878

117

1 2 3 4

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

43 2 1 0,01 0,01 0,14 0,61 1,00 0,61 0,14 0,01 0,01 , + ++ +

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:

and controlled component dimensions, respectively.

Eventually, we get the required sum

zone mentioned above;

zone.

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 com‐ ponent corresponds to FM (39) in the form

$$\mathbf{v}\_{\rm comp} = \left\langle 0.01\_{-3} + 0.14\_{-2} + 0.61\_{-1} + 1.00 \right\rangle + \left\langle 0.61^{+1} + 0.14^{+2} + 0.01^{+3} \right\rangle \tag{45}$$

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 opera‐ tions (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 sum‐ ming shown in figure 8.

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 trig‐ gering 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",

Formation of Product Properties Determining Its Quality in a Multi-Operation Technological Process http://dx.doi.org/10.5772/51878 117

**Figure 8.** Mathematically fuzzy relationships during check by measurement of component dimensions using electrical contact sensor at the lower limit of tolerance zone. а i *b* – functions of appurtenance of electric contact check errors and controlled component dimensions, respectively.

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

Eventually, we get the required sum

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"

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

For the majority of practical measurements, it is quite sufficient to evaluate the combined limiting measurement error *κ* r using three-term FM (37). Combined limiting spread of PI *х*

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

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 com‐

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 opera‐

As a result of this, FM (41) is "fuzzified", creating the combined FM determined by MF sum‐

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 trig‐ gering 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",

tions (algebraic operations using GM values will not be given here at all).

32 1 0.01 0.14 0.61 1.00 0.61 0.14 0.01 comp <sup>ν</sup> +++ <sup>=</sup> −− − ++ + + + <sup>+</sup> (45)

1

<sup>1</sup> 0.01 1.00 0.01 , sensor <sup>ν</sup> <sup>+</sup> <sup>=</sup> <sup>−</sup> + + (44)

123

is most conveniently represented by five-term FM (38) and by seven-term FM (39).

(German) – zero GM value at its left and right edges.

scripts, negative deviations – in subscripts.

116 Practical Concepts of Quality Control

spread of sensor contacts triggering.

ponent corresponds to FM (39) in the form

graphically presented in fig. 8 b.

ming shown in figure 8.

$$\left\langle 0, 01\_{-4} + 0, 01\_{-3} + 0, 14\_{-2} + 0, 61\_{-1} + 1, 00 + 0, 61^{+1} + 0, 14^{+2} + 0, 01^{+3} + 0, 01^{+4} \right\rangle\_{\prime} \tag{46}$$

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:


**4.** In order to increase the accuracy of the results of checking PI of the product *ω* sensor/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 *ω* sensor to reasonable technical-economic limits, while MF normalized GM shall be not over 0.01.

[10] RostovtsevA.M.,(2011). Quality Check by Measurement from the Point of View of Mathematical Fuzziness. [in Russian] Proceedings of the 11th scientific and technical conference «State and Problems of Measurements». Moscow State Technical Univer‐

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

http://dx.doi.org/10.5772/51878

119

sity n.a. N.E. Bauman, 26-28 of April, , 112-114.
