**4.2 BPLM process designer – structure and functionality BPLM process designer – structure and functionality – quantitative view**

#### *4.2.1 BP functional view consideration no. 1*

f g ½ � BPF i, j ð Þ ¼ f g ½ � BPF\_TR i, j1 ð Þ , BPF\_TT i, j2 ½ � ð Þ , BPF\_IM i, j3 ½ � ð Þ (1)

[BPF\_TR (i, j1)] - Transformation rule linguistic set – the set elements represent

[BPF\_TT (i, j2)] - Transformation tool linguistic set – the set elements represent closely related to human resources HRs, production technological device resources PDEV and production technological tool resources PTOOL, while formula (2) might

½BPF\_TT i, j2 ð Þ� ¼ ½ðHRs i, j21 ð Þ, PDEV i, j22 ð Þ, PTOOL i, j23 ð Þ ð Þ (2)

(HRs (i, j22), � the linguistic set, which contains data closely related to human

(PDEV (i, j21), � the linguistic set, which contains data closely related to production technological device resources, which participate at transformation

(PTOOL (i, j21), � the linguistic set, which contains data closely related to production technological tool resources, which participate at transformation

[BPF\_IM (i, j3)]} - BP internal metrics linguistic set, the content of which represent subsets, which contain data closely related to operational and technical parameters of to production technological device resources and theoretical knowl-

However, any BP is represented by external metrics items, which are closely related to actual BP inputs and outputs as well. The actual linguistic sets and algorithms concerned with relations among them are described in Section 4.2.2 via

In general, no business process proper and efficient functionality is possible without appropriate information support. At that level, the information support deals with reference database (RDBs) functionality and corresponds with their conceptual, logical and physical model. All linguistic seta related to BPF structure and functionality are stored in those RDBs and are closely related to BPF knowledge based support, while they contain pointers to appropriate semantic networks (SNWs), which create basis of BP knowledge-based support. However, they contain pointers to external data or information support resources (SAP components

The actual linguistic sets and algorithms concerned with relations among them

In a previous section, we have postulated that no business process proper and

efficient functionality is possible without appropriate information support. However, the same is concerned with the BP knowledge-based support. The BP knowledge-based support provides interconnection between the BP process and

i = 1, 2 … n is the index which indicates the BP, which an appropriate BPF is

math rules and algorithms, which regulate the BPF transformation process

*Operations Management - Emerging Trend in the Digital Era*

resources, which participate at transformation operations within actual

where

be postulated

where

BPFfunctionality.

Consideration no. 2.

especially).

**196**

operations within actual BPF functionality.

operations within actual BPF functionality.

edge and practical skills of human resources

*4.1.5 Business process model information support view*

are described in Section 4.2.3 via Consideration no. 3.

*4.1.6 Business process model knowledge-based support view*

beingassigned to.

The functional view deals with the BP vertical structure, which is created by core business processes (CBP), main business processes (MBP), subordinated business processes (SBP) and elementary business processes<sup>1</sup> (EBP).

The view on a process as a structured chain of activities has a direct coupling to coordination as defined by Malone & Crowston. Coordination is simply the management of the dependencies between these activities. This implies that coordination is an activity in itself carried out by some actors. The work object of the coordination activity is coordination manifested as various tangible and intangible elements in the organization.

Now, we shall try quantifying those aspects with the use of PBPL Equation [25, 26]. BP Model Functional view quantification with the use of PBPL Equation.

Let us consider a core business process CBP (0, I) Utility Glass Production represented by the {CBP (I, j)} linguistic set being decomposed into CBP (0, 1) Utility Glass Production Preparation denoted as CBP (i', j') represented by {CBP (i', j')} linguistic set i' = 1 … … n' j=1 … ..m1' and CBP (i", j") i' = 1 … .n",j=1 … .m1" Utility Glass Production Management represented by {CBP (i", j")} linguistic set, while the {CBP (i', j')} and {CBP (i", j")} are considered to be the linguistic subsets relating to the {CBP (I, j)}, while formula (3) might be postulated.

$$\{\text{CBP (I,j)}\} = \{ [\text{CBP (i',j')}] \} [\text{, [CBP (i'',j'')]}] \tag{3}$$

However, the CBP (0, I) business process is represented by its own internal and external metrics as well, while formulas (4) and (5) might be postulated.

$$\{\text{CBP (I, j)}\} \equiv \{\text{CBP}\_{\text{m}} \ (\text{I, j})\} \tag{4}$$

$$\{\text{CBP}\_{\text{m}}(\text{I}, \text{j})\} = \{ [\text{CBP}\_{\text{mint}}(\text{I}, \text{j})], [\text{CBP}\_{\text{miert}}(\text{I}, \text{j})] \}\tag{5}$$

Where index i' represents a hierarchic level of BP to be investigated and j' index represents a number subordinated processes relating to the BP investigated.

Now, we shall try to investigate how the superior core business process together with its internal and external metrics should be decomposed related to lower levels of management. With respect to this issue, we shall postulate two important questions.

<sup>1</sup> The BP, which cannot be decomposed in other subordinated one or its further decomposition is meaningless from practical point of view is denoted as the elementary process.

(A) How the superior business process C (0, I) represented by {[CB (I, j)]} linguistic set should be decomposed to subordinated core business processes related to lower management levels, it means from strategic to tactic and operational management level and how the superior core business functional model should be created.

At first, we shall try to find an answer related to (A) question. In order to achieve that, we have to define the superior core business process in form of adequate linguistic set {CBP (I, j)} and to assign to that set an appropriate linguistic set {b0I} <sup>=</sup> {[b0hl], [bnbp],[binm]), denoted as BP Functional View Control Linguistic Set (BP-FWC Linguistic Set), while.

[b0hl] – is a linguistic subset element, which indicates a hierarchic level of BP to be decomposed.

[bnbp] – is a linguistic subset element which indicates a number of business process stored at subordinated level

[binm] – is a linguistic subset element which indicates a serial number of that BP at appropriate hierarchic level, which should be decomposed.

Example:

Let us consider a core business process stored at hierarchic level one [bnbp] = 1, while a serial number of that BP within appropriate hierarchic level is =1 [binm]=1 and that BP should be decomposed in 3 subordinated business processes [bnbp] = 3. For that case, linguistic set {b0I} elements are represented by formula (6).

$$\{\mathbf{b}\_{\rm 0I}\} = \{\mathbf{b}\_{\rm 0I}\} = \{ [\mathbf{b}\_{\rm 0h}], [\mathbf{b}\_{\rm nbp}], [\mathbf{b}\_{\rm inn}] \} = \{\mathbf{b}\_{\rm 0I}\} = \{ [\mathbf{1}], [\mathbf{3}], [\mathbf{1}] \} \tag{6}$$

Now, let us consider the superior core CBP (I, j) business process represented by {CBP (I, j) linguistic set, which should be decomposed in two subordinated core processes<sup>2</sup> , which operate at strategic management level, while the {b0I} linguistic set<sup>3</sup> elements are postulated via formula (7).

$$\{\mathbf{b}\_{\rm 0I}\} = \{\mathbf{b}\_{\rm 0I}\} = \{ [\mathbf{b}\_{\rm 0h}], [\mathbf{b}\_{\rm nbp}], [\mathbf{b}\_{\rm inn}] \} = \{\mathbf{b}\_{\rm 0I}\} = \{ [\mathbf{1}], [\mathbf{2}], [\mathbf{1}] \} \tag{7}$$

and the {b0I} set elements create basis for {[Petx (I, j)]} linguistic set, while formulas (8) and (9) might be postulated.

$$\{ [\text{Petx}\ (\mathbf{I}, \mathbf{j})] \} = \{ \mathbf{b}\_{01} \} \tag{8}$$

When applying PBPL Equation, the following result might be generated.

*Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process…*

Finally, let us consider the business process no.1 located at hierarchic level 2, which should be decomposed into two subordinated business processes located at hierarchic level 3, while two subordinated processes should be stored at that level and the {b0I} linguistic set elements are postulated with respect two formula (13).

, 2½ � <sup>g</sup> <sup>⊗</sup> CB 2, i3, j3

f g b0I <sup>¼</sup> f g b0I <sup>¼</sup> <sup>f</sup> b0hl�, bnbp , b½ �g ¼ inm f g b0I <sup>¼</sup> f g ½ � <sup>3</sup> , 2½ �, 1½ � (13)

When applying PBPL Equation, the following result might be generated {[3], [2],[1]} ⊗ {[CB (2, i2, j2)],[2}]} = {[CB (3, i2, j2)],[1}]} ⊗ {[CB (3, i2, j2)],[2}]} see

The above-mentioned formulas and relations create basis for BP functional view

However, the BPLM Process View deals with BP horizontal structure as well, an

BP Model Process view quantification with the use of PBPL Equation.

appropriate BP to be investigated and modeled, is selected from set of BP with adequate vertical structure (functional view) and the BP internal and external metrics plays a role of principle importance. Furthermore, a significant role plays BP Input Metrics, which creates an integral part of BP External Metrics {BPEXM (i, j4)} as well (see also formula (14). On the other hand, the BP internal metrics

, 3½ � <sup>g</sup> (12)

f g ½ � 2 , 3½ �, 1½ � ⊗ f CB 1, i1, j1

*4.2.2 Process view consideration no. 2*

¼ CB 2, i1, j1

**Figure 1.**

also **Figure 1.**

**199**

, 1½ � <sup>g</sup>

without BP internal and external metrics linguistic sets.

, 1½ � <sup>⊗</sup> CB 2, i2, j2

*An example of BP functional model source: The authors.*

*DOI: http://dx.doi.org/10.5772/intechopen.95350*

$$\{\left[\mathbf{P}\mathbf{t}\mathbf{x}\left(\mathbf{I},\mathbf{j}\right)\right]\} = \{\left[\mathbf{b}\_{0\text{hl}}\right], \left[\mathbf{b}\_{\text{nbp}}\right], \left[\mathbf{b}\_{\text{inn}}\right]\}\tag{9}$$

When applying the PBPL Equation formula (10) might be postulated.

$$\begin{aligned} & \left\{ \left[ \text{Petx} \left( \mathbf{I}, \mathbf{j} \right) \right] \right\} \otimes \left\{ \left[ \text{Pe} \left( \mathbf{I}, \mathbf{j} \right) \right] \right. \\ &= \left\{ \left[ \mathbf{b}\_{0\text{hl}} \right], \left[ \mathbf{b}\_{\text{nbp}} \right], \left[ \mathbf{b}\_{\text{inn}} \right] \right\} \otimes \left\{ \text{CBP } \left( \mathbf{I}, \mathbf{j} \right) \right. \\ &= \left\{ \left[ \text{CBP } \left( \mathbf{1}, \mathbf{i}\_{1}, \mathbf{j}\_{1} \right) \right], \left[ \mathbf{1} \right] \right\} \otimes \left\{ \left[ \text{CBP } \left( \mathbf{1}, \mathbf{i}\_{2}, \mathbf{j}\_{2} \right) \right], \left[ \mathbf{2} \right] \right\} \end{aligned} \tag{10}$$

This equation corresponds to the first hierarchic level shown in **Figure 1**.

Subsequently, we shall try to decompose the subordinated BP represented by {[CBP (1, i1, j1)]} into hierarchic level 2, where three subordinated BP should be stored and a number of the BP to be decomposed is 1. The {b0I} = {b0I} <sup>=</sup> {[b0hl], [bnbp],[binm]}linguistic set content might be postulated as follows:

$$\{\mathbf{b}\_{0\mathbf{l}}\} = \{\mathbf{b}\_{0\mathbf{l}}\}\_{=} \{ [\mathbf{b}\_{0\mathbf{l}\mathbf{l}}], [\mathbf{b}\_{\mathbf{n}\mathbf{p}\mathbf{p}}], [\mathbf{b}\_{\mathbf{l}\mathbf{m}}] \} = \{\mathbf{b}\_{0\mathbf{l}}\} = \{ [2], [3], [1] \} \tag{11}$$

<sup>2</sup> The terms business process and process are considered to be equivalent from semantic point of view.

<sup>3</sup> The terms linguistic set and set are considered to be equivalent from semantic point of view.

*Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process… DOI: http://dx.doi.org/10.5772/intechopen.95350*

#### **Figure 1.**

(A) How the superior business process C (0, I) represented by {[CB (I, j)]} linguistic set should be decomposed to subordinated core business processes related to lower management levels, it means from strategic to tactic and operational management level and how the superior core business functional model should be created. At first, we shall try to find an answer related to (A) question. In order to achieve that, we have to define the superior core business process in form of adequate linguistic set {CBP (I, j)} and to assign to that set an appropriate linguistic set {b0I} <sup>=</sup> {[b0hl], [bnbp],[binm]), denoted as BP Functional View Control Linguistic

[b0hl] – is a linguistic subset element, which indicates a hierarchic level of BP to

[binm] – is a linguistic subset element which indicates a serial number of that BP

Let us consider a core business process stored at hierarchic level one [bnbp] = 1, while a serial number of that BP within appropriate hierarchic level is =1 [binm]=1 and that BP should be decomposed in 3 subordinated business processes [bnbp] = 3.

, b½ � inm

, b½ � inm

and the {b0I} set elements create basis for {[Petx (I, j)]} linguistic set, while

f g¼ ½ � Petx I, j ð Þ b0hl ½ �, bnbp

When applying the PBPL Equation formula (10) might be postulated.

, 1½ � <sup>⊗</sup> CBP 1, i2, j2

This equation corresponds to the first hierarchic level shown in **Figure 1**. Subsequently, we shall try to decompose the subordinated BP represented by {[CBP (1, i1, j1)]} into hierarchic level 2, where three subordinated BP should be stored and a number of the BP to be decomposed is 1. The {b0I} = {b0I} <sup>=</sup> {[b0hl],

, b½ � inm

<sup>2</sup> The terms business process and process are considered to be equivalent from semantic point of view. <sup>3</sup> The terms linguistic set and set are considered to be equivalent from semantic point of view.

f g ½ � Petx I, j ð Þ ⊗ f½Pe I, j ð Þ

 , b½ � inm <sup>⊗</sup> <sup>f</sup>CBP I, j ð Þ

[bnbp],[binm]}linguistic set content might be postulated as follows:

¼ b0hl ½ �, bnbp

f g b0I ¼ f g b0I <sup>¼</sup> b0hl ½ �, bnbp

¼ CBP 1, i1, j1

Now, let us consider the superior core CBP (I, j) business process represented by {CBP (I, j) linguistic set, which should be decomposed in two subordinated core

, which operate at strategic management level, while the {b0I} linguistic

<sup>¼</sup> f g b0I <sup>¼</sup> f½ � <sup>1</sup> , 3½ �, 1½ �Þ (6)

<sup>¼</sup> f g b0I <sup>¼</sup> f½ � <sup>1</sup> , 2½ �, 1½ �Þ (7)

f g ½ � Petx I, j ð Þ ¼ f g b0I (8)

, 2½ �

<sup>¼</sup> f g b0I <sup>¼</sup> f g ½ � <sup>2</sup> , 3½ �, 1½ � (11)

(9)

(10)

, b½ � inm

For that case, linguistic set {b0I} elements are represented by formula (6).

[bnbp] – is a linguistic subset element which indicates a number of business

at appropriate hierarchic level, which should be decomposed.

f g b0I ¼ f g b0I ¼ b0hl ½ �, bnbp

*Operations Management - Emerging Trend in the Digital Era*

f g b0I ¼ f g b0I ¼ b0hl ½ �, bnbp

set<sup>3</sup> elements are postulated via formula (7).

formulas (8) and (9) might be postulated.

Set (BP-FWC Linguistic Set), while.

process stored at subordinated level

be decomposed.

Example:

processes<sup>2</sup>

**198**

*An example of BP functional model source: The authors.*

When applying PBPL Equation, the following result might be generated.

$$\begin{array}{l} \{ [2], [3], [1] \} \otimes \{ [\text{CB } (\mathbf{1}, \mathbf{i}\_1, \mathbf{j}\_1)], [1] \} \\ = \{ [\text{CB } (\mathbf{2}, \mathbf{i}\_1, \mathbf{j}\_1)], [1] \} \otimes \{ [\text{CB } (\mathbf{2}, \mathbf{i}\_2, \mathbf{j}\_2)], [2] \} \} \otimes \{ [\text{CB } (\mathbf{2}, \mathbf{i}\_3, \mathbf{j}\_3)], [3] \} \end{array} \tag{12}$$

Finally, let us consider the business process no.1 located at hierarchic level 2, which should be decomposed into two subordinated business processes located at hierarchic level 3, while two subordinated processes should be stored at that level and the {b0I} linguistic set elements are postulated with respect two formula (13).

$$\{\mathbf{b}\mathbf{o}\mathbf{l}\} = \{\mathbf{b}\mathbf{o}\mathbf{l}\} = \{\left[\mathbf{b}\_{\text{0hl}}, \left[\mathbf{b}\_{\text{nbp}}\right], \left[\mathbf{b}\_{\text{inm}}\right]\right] \} = \{\mathbf{b}\mathbf{o}\mathbf{l}\} = \{\left[\mathbf{3}\right], \left[\mathbf{2}\right], \left[\mathbf{1}\right]\} \tag{13}$$

When applying PBPL Equation, the following result might be generated {[3], [2],[1]} ⊗ {[CB (2, i2, j2)],[2}]} = {[CB (3, i2, j2)],[1}]} ⊗ {[CB (3, i2, j2)],[2}]} see also **Figure 1.**

The above-mentioned formulas and relations create basis for BP functional view without BP internal and external metrics linguistic sets.

#### *4.2.2 Process view consideration no. 2*

BP Model Process view quantification with the use of PBPL Equation.

However, the BPLM Process View deals with BP horizontal structure as well, an appropriate BP to be investigated and modeled, is selected from set of BP with adequate vertical structure (functional view) and the BP internal and external metrics plays a role of principle importance. Furthermore, a significant role plays BP Input Metrics, which creates an integral part of BP External Metrics {BPEXM (i, j4)} as well (see also formula (14). On the other hand, the BP internal metrics

{BPINM (i, j4)} (see also formula (15) is created by those linguistic sets, which make basis for BP Function (BPF) definition.

pre-defined time points, it means they are time dependent and are called BP external

*Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process…*

However, that data are undertaken to an appropriate statistic evaluation and analysis as well, while adequate statistic values are being calculated (average and extend of variation) and a predefined time interval should be respected, when calculating those values. Those values are of an aggregated nature and are called BP external and internal metrics secondary data. In the next sections, we shall discuss

Let us consider the business process, which is of a technological nature<sup>4</sup> and operates with selected material inputs<sup>5</sup> represented by linguistic set {[Petx (i, j)]}, while that set consists of subsets<sup>6</sup> [Petx (i, 1)], [Petx (i, 2)] … … [Petx (i, m1)],

i. is index closely related to BP serial number with set of processes

j1- is index, which represents number subsets, the linguistic set {[Petx (i, j1)]}

However, that BP external metric is represented by adequate outputs as well, while they are quantified via {[Res1 (i, j3,)]} linguistic set, which contains values

With respect to the above-mentioned issues, the PBPL Equation actual version

A fictive data, which create conted of {[Petx (i, j)]} and {[Res1 (i, j3,)]} will be

However, the {[Pe (i, j2,)]} linguistic set contains subsets, which quantify BPFs, the BP quantified via {[Pe (i, j2,)]} linguistic set consist of. When selecting one BPF, we can assign to it the linguistic set {[BPF (i, jf)]}, which consists of three subsets,

Now, let us have a look at {[Pe (i, j2,)]} linguistic set, which represents the business process Pe, while that process provides transfer of material input represented by {[Petx (i, j)]} linguistic set into final products (glass articles)

, BPF\_TT i, jf2

the subset = [BPF\_TR (i, jf1)] = {[BPF\_TR (i, j1)]} and the content might be

the subset = [BPF\_TT (i, jf1)] = {[BPF\_TT1 (i, j1)]} and the content might be

the subset = [BPF\_TM (i, jf1)] = {[BPFEM (i, j1)], [BPFIM(i, j2)]} and the

<sup>6</sup> The terms linguistic set and set are considered to be the same from semantic point of view

content might be quantified via formulas (14) and (15).

f g ½ � Petx i, j ð Þ <sup>⊗</sup> Pe i, j2, <sup>¼</sup> Res1 i, j3, (16)

, BPF\_TM i, jf3

(17)

BP external metrics primary data – quantification via linguistic sets.

and internal metrics primary data generated at the first stage.

about that data quantification. Consideration no. 3a.

*DOI: http://dx.doi.org/10.5772/intechopen.95350*

Where

consists of

might be postulated.

discussed within Case study section.

represented by {[Res1 (i, j3,)]} linguistic set.

while formula (17) might be postulated.

f g ½ � BPF i, jf ð Þ ¼ BPF\_TR i, jf <sup>1</sup>

quantified via formula (16).

quantified via formula (19)

<sup>4</sup> Glass Article Primary Production- GAPP

Where

<sup>5</sup> Glass Melt - GM

**201**

which represent one part of BP external metrics.

concerned with **nArtgood, nArtrep,** and **nArtwaste** items.

$$\begin{aligned} \text{?F (i,j2) (j2 = 1 \ldots m2)} &= >[\text{F (i,j2)}] ?[\text{Pexpj2 (i,j1)}] ?[\text{Pexp (i,j1)}] ?[\text{Pexp (i,j1)}] \\ &= >[\text{Pexpj2 (i,j1)}] ?[\text{F (i,j2)}] \\ &= [\text{Resi}j2 (i,j3), !] ?[[\text{Res1 (i,j3)}, )] \end{aligned} \tag{14}$$
 
$$\begin{aligned} \text{?Pe (i,j2) (j2 = 1 \ldots m2)} &= > >[\text{Pe (i,j2)}] ?[[\text{Pex } (i,j1,)]] \ $ \times \{[\text{Res1 } (i,j3,)]\} \\ &= > > \{[\text{Pex } (i,j1,)]\} \$  \{[\text{Res1 } (i,j3,)]\} ? \{\text{BPEXM (i,j4)}\} \end{aligned} \tag{15}$$
 
$$\&\quad \{\text{Pe } (i,j2)\} ? \{\text{BPINM (i,j5)}\} \end{aligned} \tag{15}$$

#### *4.2.3 BPM information support view*

In general, a proper and an efficient functionality of any business process depends on an adequate information support, however the question is: What the term BP information support related to BP functionality does mean? In general, any BP functionality and performance are closely related to BP external and internal metrics. However, the problems of BP external and internal metrics theory are discussed within Section 2 as well, while at that place we shall discuss aspects closely related to so called two stage BP external and internal metrics (see also **Figure 2**). What the term two stage BP external and internal metrics does mean?

In general, the implemented and operated BP is running and generates predefined output products (articles) – denoted as the primary products based on appropriated adequate material, information and financial inputs. On the other hand, the investigated BP operates with a set of input and output information generated based on detailed data, e.g. number of good articles nArtgood – a quality of which corresponds to pre-defined requirements, number of repaired articles nArtrep - number of produced articles their quality does not correspond to pre-defined requirements and should be repaired, and number of waste articles nArtwaste - number of produced articles their quality does not correspond to pre-defined requirements – cannot be repaired and should be considered to be a waste. They are considered to be detailed data and have no level of aggregation. The same is concerned with other data closely related to BP external or internal metrics and they are being measured at

**Figure 2.** *Business process horizontal structure source: The authors.*

#### *Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process… DOI: http://dx.doi.org/10.5772/intechopen.95350*

pre-defined time points, it means they are time dependent and are called BP external and internal metrics primary data generated at the first stage.

However, that data are undertaken to an appropriate statistic evaluation and analysis as well, while adequate statistic values are being calculated (average and extend of variation) and a predefined time interval should be respected, when calculating those values. Those values are of an aggregated nature and are called BP external and internal metrics secondary data. In the next sections, we shall discuss about that data quantification.

Consideration no. 3a.

BP external metrics primary data – quantification via linguistic sets.

Let us consider the business process, which is of a technological nature<sup>4</sup> and operates with selected material inputs<sup>5</sup> represented by linguistic set {[Petx (i, j)]}, while that set consists of subsets<sup>6</sup> [Petx (i, 1)], [Petx (i, 2)] … … [Petx (i, m1)], which represent one part of BP external metrics.

Where

{BPINM (i, j4)} (see also formula (15) is created by those linguistic sets, which

?F i, j2 ð Þ ð Þ¼ j2 ¼ 1 … *:*m2 > ½ � F i, j2 ð Þ ?½ � Petxj2 i, j1, ð Þ ?f g ½ � Petx i, j1, ð Þ ¼ > ½ � Petxj2 i, j1, ð Þ ?½ � F i, j2 ð Þ

?Pe ið Þ *;* j2 ðj2 ¼ 1 … *:*m2Þ ¼¼ >f g Pe ið Þ *;* j2 ?f g ½ � Petx ið Þ *;* j1*;* & Res1 i f g ½ � ð Þ *;* j3*;*

& Pe i f g ð Þ *;* j2 ?f g BPINM ið Þ *;* j5

In general, a proper and an efficient functionality of any business process depends on an adequate information support, however the question is: What the term BP information support related to BP functionality does mean? In general, any BP functionality and performance are closely related to BP external and internal metrics. However, the problems of BP external and internal metrics theory are discussed within Section 2 as well, while at that place we shall discuss aspects closely related to so called two stage BP external and internal metrics (see also **Figure 2**).

In general, the implemented and operated BP is running and generates predefined output products (articles) – denoted as the primary products based on appropriated adequate material, information and financial inputs. On the other hand, the investigated BP operates with a set of input and output information generated based on detailed data, e.g. number of good articles nArtgood – a quality of which corresponds to pre-defined requirements, number of repaired articles nArtrep - number of produced articles their quality does not correspond to pre-defined requirements and should be repaired, and number of waste articles nArtwaste - number of produced articles their quality does not correspond to pre-defined requirements – cannot be repaired and should be considered to be a waste. They are considered to be detailed data and have no level of aggregation. The same is concerned with other data closely related to BP external or internal metrics and they are being measured at

What the term two stage BP external and internal metrics does mean?

¼ ½ � Res1j2 i, j3, ð Þ ?f g ½ � Res1 i, j3, ð Þ (14)

¼ >f g ½ � Petx ið Þ *;* j1*;* & Res1 i f g ½ � ð Þ *;* j3*;* ?f g BPEXM ið Þ *;* j4

(15)

make basis for BP Function (BPF) definition.

*Operations Management - Emerging Trend in the Digital Era*

*4.2.3 BPM information support view*

**Figure 2.**

**200**

*Business process horizontal structure source: The authors.*

i. is index closely related to BP serial number with set of processes

j1- is index, which represents number subsets, the linguistic set {[Petx (i, j1)]} consists of

However, that BP external metric is represented by adequate outputs as well, while they are quantified via {[Res1 (i, j3,)]} linguistic set, which contains values concerned with **nArtgood, nArtrep,** and **nArtwaste** items.

With respect to the above-mentioned issues, the PBPL Equation actual version might be postulated.

$$\{\left[\text{Petx (i,j)}\right]\} \otimes \{\left[\text{Pe (i,j\_2, )}\right]\} = \{\left[\text{Res1 (i,j\_3, )}\right]\} \tag{16}$$

A fictive data, which create conted of {[Petx (i, j)]} and {[Res1 (i, j3,)]} will be discussed within Case study section.

Now, let us have a look at {[Pe (i, j2,)]} linguistic set, which represents the business process Pe, while that process provides transfer of material input represented by {[Petx (i, j)]} linguistic set into final products (glass articles) represented by {[Res1 (i, j3,)]} linguistic set.

However, the {[Pe (i, j2,)]} linguistic set contains subsets, which quantify BPFs, the BP quantified via {[Pe (i, j2,)]} linguistic set consist of. When selecting one BPF, we can assign to it the linguistic set {[BPF (i, jf)]}, which consists of three subsets, while formula (17) might be postulated.

$$\{\left[\text{BPF } (\text{i,jf})\right]\} = \{\left[\text{BPF } \text{TR } (\text{i,jf}\_1)\right], \left[\text{BPF } \text{TT } (\text{i,jf}\_2)\right], \left[\text{BPF } \text{TM } (\text{i,jf}\_3)\right]\} \tag{17}$$

Where

the subset = [BPF\_TR (i, jf1)] = {[BPF\_TR (i, j1)]} and the content might be quantified via formula (16).

the subset = [BPF\_TT (i, jf1)] = {[BPF\_TT1 (i, j1)]} and the content might be quantified via formula (19)

the subset = [BPF\_TM (i, jf1)] = {[BPFEM (i, j1)], [BPFIM(i, j2)]} and the content might be quantified via formulas (14) and (15).

<sup>4</sup> Glass Article Primary Production- GAPP

<sup>5</sup> Glass Melt - GM

<sup>6</sup> The terms linguistic set and set are considered to be the same from semantic point of view

$$\{\left[\text{Pe}\left(\mathbf{i}, \mathbf{j}\_2, \right)\right]\}' = \text{II}\left\{\left[\text{BPF}\left(\mathbf{i}, \mathbf{j}\_\mathbf{f}\right)\right]\right\} \tag{18}$$

½ � Petx i, m2a ð Þ , ¼ ½ð Þ mat m2 ð Þ11 t k ð Þ ð Þ , mat m2 ð Þ12 t k ð Þ ð Þ ,ðmat m2 ð Þ 21 tð Þ,

*Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process…*

Now, let us select [Petx (i, 1)] and undertake its content to statistic evaluation<sup>7</sup>

½ �¼ Petx i, 1b ð Þ ½ðmat 1ð Þ11 mat 1 ð ð Þ12Avg, mat 1 ð � ð Þ12MMin, mat 1 ð Þ ð Þ12MMax , mat 1 ð Þ ð Þ12Vrp

m2a – index, which indicates a serial number of input record within Petx

formula (29) indicates an extension of [Petx (i, 1a)] linguistic set.

m12 – index, which indicates a serial number of item and value input record. Formula (28) indicates statistic values of items assigned to selected input, while

Now let us consider the {[Res1 (i, j3)]} and let us suppose that the **nArtgood, nArtrep nArtwaste** are time dependent, while formulas (30), (31), and (32) might be

With respect to those issues, appropriate statistic values might be calculated.

<sup>½</sup>Article\_repairst� ¼ nArtrep <sup>¼</sup> nArtrepawg, nArtrepmin, nArtrepmax, nArtrepVrp (34) <sup>½</sup>Article\_wastest� ¼ nArtwasteawg, nArtwastemin, nArtwastemax, nArtwasteVrp (35)

Let us demonstrate previous relations at business process, which deals with forming of glass articles (Ga) from glass melt (Gm), which is represented by three variables: (a) glass melt temperature (Gmtep), glass melt viscosity (Gmvis), and glass melt quantity (Gmquant) and generated glass articles (Gas) represented by three items and values: (a) number of good Gas (nArtgood), number of repaired Gas (nArtrep) and number of waste Gas. The relations among statistic values of selected variables might be defined via: (a) partial rules (see also formulas (36), (37), and (38), (b) complex rule (see also formula (31) and (c) set of complex rules (see also formulas (40), (41), and (42). However, all the above-mentioned rules might be time dependent as well, while they might create pairs (time interval (T(int)), Y (int) and create linguistic subsets, which could quantify a development trend (see

f½ � Gmtemp\_awg, GmVrp g¼f nArtgoodawg, nArtgoodmin, nArtgoodmax, nArtgoodVrp , (36)

<sup>7</sup> Statistic evaluation = determination of Avg, Min, Max and extent of variation

<sup>½</sup>Article\_goodst� ¼ nArtgoodawg, nArtgoodmin, nArtgoodmax, nArtgoodVrp (33)

while formula (28) and (29) might be postulated<sup>8</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.95350*

where

linguistic set.

postulated.

also formula (43)). Partial rules

<sup>8</sup> Vrp – extend of variation

**203**

mat m2 ð Þ 22 t k ð Þ ð Þ … *:*mat m2 ð Þ ð Þ m11 t k ð Þ ð Þ , matm2 ð Þ� m12

½ �¼ Petx i, 1ab ð Þ ½ � Petx i, 1a ð Þ , Petx i, 1b ½ � ð Þ (29)

**nArtgood**<sup>¼</sup> **nArtgood** ð Þt (30) **nArtrep** <sup>¼</sup> **nArtrep** ð Þt (31) **nArtwaste** ¼ **nArtwaste** ð Þt (32)

(27)

(28)

,

i = 1,2 … .n

jf = 1, 2 … .m1 - number of BPFs, the Pe business process, consists of.

Finally, we shall specify the {[Res1 (i, j3)]}, the content of which is closely related to number of good articles **nArtgood,** number of repaired articles **nArtrep,** and number of waste articles **nArtwaste,** while formulas (25) and (26) and (28) and (29) might be postulated.

$$\{\{\left[\text{Res1}\left(\mathbf{i}, \mathbf{j}\_3\right)\right]\}\} = \{\left[\text{Pop}\left(\mathbf{i}, \mathbf{j}\_3\right)\right], \left[\text{Particle\\_good}\right], \left[\text{Particle\\_repair}\right], \left[\text{Particle\\_waste}\right]\}\tag{19}$$

$$\mathbf{n}\_{\text{Artgood}} \in [\text{Article\\_good}],\tag{20}$$

$$\mathbf{n}\_{\text{Artrep}} \in [\text{Article\\_repair}],\tag{21}$$

$$\mathbf{n}\_{\text{Artwaste}, \mathbf{e}} \in [\text{Article\\_waste}]\}\tag{22}$$

[Pop (i, j3)] subset contains elements closely related to article type, article class, article name, article, measure unit.

When applying the PBPL Equation in a basic form (see also formula (16), formula (23) might be postulated

$$\{ [\text{Petx} \ (\mathbf{i}, \mathbf{1})], [\text{Petx} \ (\mathbf{i}, \mathbf{2})] \dots \dots [\text{Petx} \ (\mathbf{i}, \mathbf{m}\_1)] \} \otimes \Pi \left\{ [\text{BPF} \ (\mathbf{i}, \mathbf{j}\_\mathbf{f})] \right\} = \mathbf{i} = \mathbf{1}, 2 \dots \mathbf{n}; \mathbf{j}\_\mathbf{f} = \mathbf{1}, 2 \dots \mathbf{n}; \mathbf{m}\_\mathbf{f} = \mathbf{1}, 2 \dots \mathbf{n} \ \mathbf{m} = \mathbf{1}, \mathbf{2} \dots \mathbf{n} \ \mathbf{m} = \mathbf{1} \tag{23}$$

It should be noted that those linguistic set content is time depended and formula (24) might be postulated.

$$\begin{aligned} \{ [\text{Petx} \ (\text{i}, \text{1}, \text{t})], [\text{Petx} \ (\text{i}, \text{2}, \text{t})] \dots \dots [\text{Petx} \ (\text{i}, \text{m}\_1, \text{t})] \} \otimes \Pi \ \{ [\text{BPF} \ (\text{i}, \text{j}\_{\text{f}}, \text{t})] = \\ \text{i} &= 1, 2 \dots \text{n}; \\ \text{j}\_{\text{f}} &= 1, 2 \dots \text{m} \text{1} - \\ = \{ [\text{Pop} \ (\text{i}, \text{j}\_{\text{j}})], [\text{Artice\\_good} \ (\text{t})], [\text{Artice\\_repair} \ (\text{t})], [\text{Artice\\_waste} \ (\text{t})] \} \end{aligned} (24)$$

Formula (24) quantifies relation among BP input and output parameters and the actual content of the above-mentioned linguistic set will be discussed within Case study section.

#### *4.2.4 BPM knowledge support view consideration no. 4*

The knowledge related to BPM knowledge support view are derived based on appropriate item statistic values mentioned within previous section.

Let us consider the {[Petx (i, j)]}, which contains subsets [Petx (i, 1)], [Petx (i, 2)] … … [Petx (i, m1)], while any of those subsets contains time depended items and values concerned to actual material input.

$$\begin{aligned} \left[ \text{Petx} \left( \text{i}, \text{1a} \right) \right], &= \left[ \left( \text{mat} \left( \text{1} \right) \text{11} \left( \text{t} \left( \text{k} \right) \right), \text{mat} \left( \text{1} \right) \text{12} \left( \text{t} \left( \text{k} \right) \right) \right), \left( \text{mat} \left( \text{1} \right) \text{21} \left( \text{t} \left( \text{k} \right) \right), \text{ 25} \right) \right] \\ &\quad \text{mat} \left( \text{1} \right) \text{22} \left( \text{t} \dots \text{ } \text{ } \text{mat} \left( \text{1} \right) \left( \text{m} \mathbf{1} \right) \text{1} \left( \text{t} \left( \text{k} \right) \right), \text{mat} \left( \text{1} \right) \left( \text{k} \right) \left( \text{m} \mathbf{1} \right) \right] \\ &\quad \left[ \text{Petx} \left( \text{i}, \text{2a} \right) \right], \dots \left[ \left( \text{mat} \left( \text{2} \right) \text{11} \left( \text{t} \left( \text{k} \right) \right), \left( \text{mat} \left( \text{2} \right) \text{21} \left( \text{t} \left( \text{k} \right) \right) \right), \dots \left( \text{2} \text{M} \mathbf{1} \right) \left( \text{m} \mathbf{1} \right) \right] \\ &\qquad \quad \left[ \text{mat} \left( \text{2} \right) \left( \text{22} \left( \text{t} \right) \dots \text{mat} \left( \text{2} \right) \left( \text{m} \mathbf{1}$$

*Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process… DOI: http://dx.doi.org/10.5772/intechopen.95350*

$$\begin{aligned} \left[ \text{Petx } (\text{i}, \text{m2a}) \right], &= \left[ \left( \text{mat } (\text{m2}) \text{11 } (\text{t} \text{ k}) \right), \text{mat } (\text{m2}) \text{12 } (\text{t} \text{k}) \right), (\text{mat } (\text{m2}) \text{ 21} (\text{t}), \\ &\quad \left[ \text{mat } (\text{m2}) \text{ 22} (\text{t} \text{k}) \right) \dots \text{mat } (\text{m2}) \left( \text{m1} \ (\text{t} \text{k}) \right), \text{mat } \text{m2} \right) (\text{m12}) \end{aligned} \tag{27}$$

Now, let us select [Petx (i, 1)] and undertake its content to statistic evaluation<sup>7</sup> , while formula (28) and (29) might be postulated<sup>8</sup>

½ �¼ Petx i, 1b ð Þ ½ðmat 1ð Þ11 mat 1 ð ð Þ12Avg, mat 1 ð � ð Þ12MMin, mat 1 ð Þ ð Þ12MMax , mat 1 ð Þ ð Þ12Vrp (28)

$$[\text{Petx}\ (\text{i}, \textsf{1ab})] = [\textsf{Petx}\ (\text{i}, \textsf{1a})],\\[\textsf{Petx}\ (\text{i}, \textsf{1b})] \tag{29}$$

where

Pe i, j2, ´ <sup>¼</sup> <sup>Π</sup> BPF i, jf

, Article\_good ½ �, Article\_repair ½ �, Article\_waste ½ �

[Pop (i, j3)] subset contains elements closely related to article type, article class,

It should be noted that those linguistic set content is time depended and formula

Formula (24) quantifies relation among BP input and output parameters and the actual content of the above-mentioned linguistic set will be discussed within Case

The knowledge related to BPM knowledge support view are derived based on

… … [Petx (i, m1)], while any of those subsets contains time depended items and

½ � Petx i, 1a ð Þ , ¼ ½ðmat 1ð Þ11 t k ð Þ ð Þ ð Þ , mat 1ð Þ12 t k ð Þ ð Þ ,ðmat 1ð Þ 21 t k ð Þ ð Þ ,

½ � Petx i, 2a ð Þ , ¼ ½ðmat 2ð Þ11 t k ð Þ ð Þ , mat 2ð Þ12 t k ð Þ ð Þ ,ðmat 2ð Þ 21 t k ð Þ ð Þ ,

Let us consider the {[Petx (i, j)]}, which contains subsets [Petx (i, 1)], [Petx (i, 2)]

mat 1ð Þ 22 tð Þ … *:* mat 1ð Þðm11 t k ð Þ ð Þ , mat 1ð Þ ð Þ <sup>k</sup> ð Þ� m12 (25)

mat 2ð Þ 22 tð Þ … *:*mat 2ð Þ ð � m11 t k ð Þ ð Þ , mat 2ð Þ ð Þ m12 (26)

appropriate item statistic values mentioned within previous section.

f g ½ � Petx ið Þ *;* 1*;*t *;* ½ � Petx ið Þ *;* 2*;*t … … ½ � Petx ið Þ *;* m1*;*t ⊗ Π f½BPF ði, jf, t�g ¼

*;* ½ � Article\_good tð Þ *;* ½ � Article\_repair tð Þ *;* ½ � Article\_waste tð Þ

When applying the PBPL Equation in a basic form (see also formula (16),

f g ½ � Petx i, 1 ð Þ , Petx i, 2 ½ � ð Þ … … ½ � Petx i, m ð Þ<sup>1</sup> ⊗ Π BPF i, jf

*4.2.4 BPM knowledge support view consideration no. 4*

values concerned to actual material input.

**nArtgood ∈** ½ � Article\_good , (20)

**nArtrep ∈** ½ � Article\_repair , (21)

**nArtwaste**, **∈** ½Article\_waste�g (22)

, Article\_good ½ �, Article\_repair ½ �, Article\_waste ½ �

<sup>¼</sup> <sup>i</sup> <sup>¼</sup> 1, 2 … *:*n; jf

i ¼ 1, 2 … *:*n;

<sup>f</sup> ¼ 1, 2 … *:*m1�

j

jf = 1, 2 … .m1 - number of BPFs, the Pe business process, consists of. Finally, we shall specify the {[Res1 (i, j3)]}, the content of which is closely related to number of good articles **nArtgood,** number of repaired articles **nArtrep,** and number of waste articles **nArtwaste,** while formulas (25) and (26) and (28) and (29)

*Operations Management - Emerging Trend in the Digital Era*

i = 1,2 … .n

might be postulated.

<sup>¼</sup> Pop i, j3

article name, article, measure unit.

formula (23) might be postulated

¼ 1, 2 … *:*m1� ¼ Pop i, j3

(24) might be postulated.

3

¼ Pop i*;* j

study section.

**202**

Res1 i, j3

(18)

(19)

(23)

(24)

m2a – index, which indicates a serial number of input record within Petx linguistic set.

m12 – index, which indicates a serial number of item and value input record. Formula (28) indicates statistic values of items assigned to selected input, while

formula (29) indicates an extension of [Petx (i, 1a)] linguistic set.

Now let us consider the {[Res1 (i, j3)]} and let us suppose that the **nArtgood, nArtrep nArtwaste** are time dependent, while formulas (30), (31), and (32) might be postulated.

$$\mathbf{n}\_{\text{Artgood}=} \mathbf{n}\_{\text{Artgood}}\,\left(\mathbf{t}\right)\tag{30}$$

$$\mathbf{n}\_{\mathbf{Artrep}} = \mathbf{n}\_{\mathbf{Artrep}} \begin{pmatrix} \mathbf{t} \end{pmatrix} \tag{31}$$

$$\mathbf{n}\_{\text{Artwaste}} = \mathbf{n}\_{\text{Artwaste}}\left(\mathbf{t}\right) \tag{32}$$

With respect to those issues, appropriate statistic values might be calculated.

$$\left[ \text{Article\\_goodst} \right] = \left[ \mathbf{n}\_{\text{Artgoodrwg}}, \mathbf{n}\_{\text{Artgoodmin}}, \mathbf{n}\_{\text{Artgoodmax}}, \mathbf{n}\_{\text{Artgood}\text{VPp}} \right] \tag{33}$$

$$\begin{bmatrix} \text{Particle\\_repaist} \end{bmatrix} = \begin{bmatrix} \mathbf{n}\_{\text{Artrep}} = & \mathbf{n}\_{\text{Artrep}\text{wg}}, \mathbf{n}\_{\text{Artrep}\text{min}}, \mathbf{n}\_{\text{Attrep}\text{max}}, \mathbf{n}\_{\text{Attrep}\text{Vrp}} \end{bmatrix} \tag{34}$$

$$\left[ \text{Articles\\_waste} \right] = \left[ \mathbf{n}\_{\text{Artwastezwg}}, \mathbf{n}\_{\text{Artwastezin}}, \mathbf{n}\_{\text{Artwastezwx}}, \mathbf{n}\_{\text{ArtwasteVrp}} \right] \tag{35}$$

Let us demonstrate previous relations at business process, which deals with forming of glass articles (Ga) from glass melt (Gm), which is represented by three variables: (a) glass melt temperature (Gmtep), glass melt viscosity (Gmvis), and glass melt quantity (Gmquant) and generated glass articles (Gas) represented by three items and values: (a) number of good Gas (nArtgood), number of repaired Gas (nArtrep) and number of waste Gas. The relations among statistic values of selected variables might be defined via: (a) partial rules (see also formulas (36), (37), and (38), (b) complex rule (see also formula (31) and (c) set of complex rules (see also formulas (40), (41), and (42). However, all the above-mentioned rules might be time dependent as well, while they might create pairs (time interval (T(int)), Y (int) and create linguistic subsets, which could quantify a development trend (see also formula (43)).

Partial rules

f½ � Gmtemp\_awg, GmVrp g¼f nArtgoodawg, nArtgoodmin, nArtgoodmax, nArtgoodVrp , (36)

<sup>7</sup> Statistic evaluation = determination of Avg, Min, Max and extent of variation

<sup>8</sup> Vrp – extend of variation

nArtrepawg, nArtrepmin, nArtrepmax, nArtrepVrp � �,

nArtwasteawg, nArtwastemin, nArtwastemax, nArtwasteVrp � �<sup>g</sup>

f g ½ � Gmtvis\_awg, GmVrp ¼ f nArtgoodawg, nArtgoodmin, nArtgoodmax, nArtgoodVrp � �, (37)

nArtrepawg, nArtrepmin, nArtrepmax, nArtrepVrp � �,

nArtwasteawg, nArtwastemin, nArtwastemax, nArtwasteVrp � �<sup>g</sup>

f g ½ � Gmtquant\_awg, GmVrp ¼ f nArtgoodawg, nArtgoodmin, nArtgoodmax, nArtgoodVrp � �, (38)

> nArtrep <sup>¼</sup> nArtrepawg, nArtrepmin, nArtrepmax, nArtrepVrp � �, nArtwasteawg, nArtwastemin, nArtwastemax, nArtwasteVrp � �<sup>g</sup>

Complex rule

f½ � Gmtemp\_awg*;* GmVrp *;* f g ½ � Gmtvis\_awg*;* GmVrp *;* ½ � Gmtvis\_awg*;* GmVrp ¼ f nArtgoodawg*;* nArtgoodmin*;* nArtgoodmax*;* nArtgoodVrp h i*;* nArtrepawg*;* nArtrepmin*;* nArtrepmax*;* nArtrepVrp h i*;* nArtwasteawg*;* nArtwastemin*;* nArtwastemax*;* nArtwasteVrp h i<sup>g</sup> (39)

Y 1ð Þ ¼ f½ � Gmtemp\_awg ð Þ1 *;* GmVrp ð Þ1 *;* f½ � Gmtvis\_awg 1ð Þ*;* GmVrp ð Þ1 *;* ½Gmtvis\_awg ð Þ1 *;* GmVrp �g ¼ f½nArtgoodawgð Þ1 *;* nArtgoodmin*;* ð Þ1 *;* nArtgoodmax*;* ð Þ1 *;* nArtgoodVrpð Þ1 *; ;*� ½ nArtrepawg*;* ð Þ1 *;* nArtrepmin*;* nArtrepmax*;* ð Þ1 *;* nArtrepVrpð Þ1 *; ;*� ½nArtwasteawgð Þ1 *; ;* nArtwasteminð Þ1 *;* nArtwastemaxð Þ1 *;* nArtwasteVrpð Þ1 �g

Y 2ð Þ ¼ f½ � Gmtemp\_awg ð Þ2 *;* GmVrp ð Þ2 *;* f½ � Gmtvis\_awg 2ð Þ*;* GmVrp ð Þ2 *;* ½Gmtvis\_aw ð Þ2 *;* GmVrp �g ¼ f½nArtgoodawgð Þ2 *;* nArtgoodmin*;* ð Þ2 *;* nArtgoodmax*;* ð Þ2 *;* nArtgoodVrpð Þ2 *; ;*� ½ nArtrepawg*;* ð Þ2 *;* nArtrepmin*;* nArtrepmax*;* ð Þ2 *;* nArtrepVrpð Þ2 *; ;*� ½nArtwasteawg ð Þ2 *;* nArtwasteminð Þ2 *;* nArtwastemaxð Þ2 *;* nArtwasteVrpð Þ�g 2

(41)

(40)

**4.3 Derivation of BPF functionality rules**

*DOI: http://dx.doi.org/10.5772/intechopen.95350*

linguistic subsets might be observed:

In general, a horizontal structure of any business process (BP) is being created via appropriate set of business process functions (BPFs), while the BPF seems to be the principle component of any business process. On the other hand, any BPF might be quantified via multi-layer linguistic set, while at the first layer three significant

*Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process…*

• {[BPF\_TR (i, j1)]} – a content of which create rules, which regulate a progress

• {[BPF\_TT (i, j2)]} – a content of which create transformation functions, which

related to transformation of BPF inputs into pre-defined outputs

provide transformation of BPF inputs into pre-defined outputs

deals with BPF internal metrics with respect to formula (15)

Let us consider the {[Petx (i, j1)]}, which contains a finite number of elements denoted as pt.(i, 1), pt.(i, 2), … pt.(i, m1), while each of them is created by the element average and element extend of variations value (see also

tion of BPF inputs to predefined BPF outputs as well.

pt.(i, j1), = ((pt(i, j1)avg., (pt(i, j1)vrp)

pt.(i, jm1), = ((pt(i, jm1)avg., (pt(i, jm1)vrp)

with assignment of words to ratio value intervals (**Table 4**).

f g Rs <sup>¼</sup> <sup>125</sup>n1811,1,5,1n4, <sup>10</sup>n52,2,

*4.3.2 BPF inputs versus BPF outputs*

• {[BPM (i, j3)]} – a content of which create subsets closely related to BPF6external and internal metrics, while BPF external metrics linguistic

• [BPFEM (i, j1)]} deals with BPF external metrics and consists of [BPINP (i, j11)] subset the content of which is created by elements closely related to BPF inputs and [[BPOUTP (i, j12), the content of which is created by elements closely related to BPF outputs (see also formula (14) and the {[BPFIM (i, j2)]}

However, both the above-mentioned linguistic sets are very closed to {[BPF\_TT1 (i, j1)]} the content of which is created by elements closely related to transforma-

Let us consider a statistic file represented by **Table 1** and set of statistic indica-

<sup>270</sup>n1081,1, <sup>60</sup>n49,9 ¼ f g 0, 069, 0, 784, 0, 191, 0, 249, 1202

Now, let us create a ratio set {Rs} and reference table (**Table 3**), which deals

pt*:*ð Þ i, j2 , ¼ ððpt i, j2 ð Þavg*:*, pt i, j2 ð Þ ð Þvrp (44)

*4.3.1 General overview*

set and

formula (44).

**4.4 Case study**

**205**

tors represented by **Table 2**.

Y m3 ð Þ ¼ f½ � Gmtemp\_awg ð Þ m3 *;* GmVrp ð Þ m3 *;* f½ � Gmtvis\_awg m3 ð Þ*;* GmVrp ð Þ m3 *;* ½Gmtvis\_awg ð Þ m3 *;* GmVrp �g ¼ f½nArtgoodawgð Þ m3 *;* nArtgoodmin*;* ð Þ m3 *;* nArtgoodmax*;* ð Þ m3 *;* nArtgoodVrpð Þ m3 *; ;*� ½ nArtrepawg*;* ð Þ m3 *;* nArtrepmin*;* nArtrepmax*;* ð Þ m3 *;* nArtrepVrpð Þ m3 *;* �*;* nArtwasteawgð Þ m3 *;* nArtwasteminð Þ m3 *;* nArtwastemaxð Þ<sup>2</sup> *; ;* nArtwasteVrpð Þ m3 h i<sup>g</sup>

(42)

$$\{ [(\text{DevlTrend}] ]\} = \ [ (\text{T}(\mathbf{1}), \text{Y } (\mathbf{1}) ), [(\text{T}(2), \text{Y } (2)], \dots [(\text{T}(\mathbf{m3}), \text{Y } (\mathbf{m3}))] \} \tag{43}$$

In general, the knowledge stored with ES knowledge base are represented by semantic networks (SNWs), while partial rules might be compared **with partial SNW**s, complex rules might be compared with **ordinary SNWs** and development trends (DevlTrend) might be compared with **superior SNWs**. This approach will be discussed within Case study in more details and applied when designing and implemented an appropriate knowledge-based or expert system as well.

*Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process… DOI: http://dx.doi.org/10.5772/intechopen.95350*
