*5.2.2 KPI indicator decomposition related to operational management level consideration no. 4*

#### *5.2.2.1 Determination of Production process tertiary KPI indicators*

Let us consider the {[CONTRACTD (i)]} linguistic set (see also formula (57)), which quantifies order submitted to an appropriate firm or company organization unit to produce adequate products quantified via {[MROPsel (i, j3)]} and with the use of business processes (BP), which create an integral part of a given BP group. One of those processes will be selected and demonstrated how the KPI (i, 3) indicator should be decomposed in order to describe the selected BP functionality and performance, first of all. In general any BP is represented by its own internal and external metrics, while the external metrics is concerned with BP outputs and inputs and the BP internal metrics is closely related to appropriate production human resources, production devices and production tools and those aspects are quantified via given linguistic sets.However, that decomposition will be done within several steps as well.

#### *5.2.2.2 BP external metrics KPI indicators*

#### *Step 3*

In that step, a group of selected products should be created, which is an integral part of products quantified via {[MROPsel (i, j3)]} linguistic set, while formula (66) might be postulated.

$$\{\left[\text{MROP}\_{\text{sel\\_bp}}\left(\mathbf{i}, \mathbf{j}\_3\right)\right]\} \subseteq \{\left[\text{MROP}\_{\text{sel}}\left(\mathbf{i}, \mathbf{j}\_3\right)\right]\}\tag{66}$$

Those products should be produced with the use of the selected BP (see also formula (67)).

$$\{\{\text{BP (i,j7)}\}\}\;\!\/\}\in\{\left[\text{BPG (i,j\_6)}\right]\}\tag{67}$$

Now, we have to select set input materials needed for production of the abovementioned products. We shall apply the [MROPselfincosts (i, j)]} linguistic set for those purposes, the content of consists of two subsets with respect to formula.

$$\{\text{[MROP}\_{\text{selfincots}}\left(\mathbf{i},\mathbf{j}\right)]\} = \{\left[\left[\text{MROP1}\_{\text{selfincots}}\left(\mathbf{i},\mathbf{j}\right)\right], \left[\left[\text{MROP2}\_{\text{selfmatcots}}\left(\mathbf{i},\mathbf{j}\right)\right]\right]\right\}\tag{68}$$

*Business Process Linguistic Modeling: Theory and Practice Part I: BPLM Strategy Creator DOI: http://dx.doi.org/10.5772/intechopen.95096*

Where the linguistic [[MROP1selfincosts (i, j)] subset quantifies financial costs and the [MROP2selmatcosts (i, j)] subset quantifies material costs and create basis for preparation that subset which contains material data needed for production of the above-mentioned products

$$\{\left[\text{MROP1}\_{\text{sel\\_bp}}\left(\mathbf{i}, \mathbf{j}\_3\right)\right] \} \subseteq \{\left[\text{MROP}\_{\text{sel\\_bp}}\left(\mathbf{i}, \mathbf{j}\_3\right)\right] \}\tag{69}$$

MROP2selmatcosts ½ � ð Þ i*;* j MROPsel\_bp i*;* j 3 f½ MROP1selfincosts ½ � ð Þ <sup>i</sup>*;* <sup>j</sup> , MROPselfinass ½ � ð Þ <sup>i</sup>*;* <sup>j</sup> (70)

### *5.2.2.3 Applying of PBPL equation solutions*

*Step 4*

f g ½ � CONTRACTD ið Þ ¼ Π f CUST i*;* j

*consideration no. 4*

within several steps as well.

*Step 3*

formula (67)).

**176**

might be postulated.

*5.2.2.2 BP external metrics KPI indicators*

MROPsel\_bp i*;* j

½ BP I*;* j 6 <sup>g</sup>

*Operations Management - Emerging Trend in the Digital Era*

Finally, adequate KPI indicators will be defined.

2 , <sup>f</sup> MROPsel <sup>i</sup>*;* <sup>j</sup>

j=1 …m3, j2 = 1 …..m2, j3 = 1 ….m3, j4 = 1 ….m4, j5 = 1 ….m5, j6 = 1 ….m6,

KPI (i, 3) indicator creates basis for decomposition related to operational level.

Let us consider the {[CONTRACTD (i)]} linguistic set (see also formula (57)), which quantifies order submitted to an appropriate firm or company organization unit to produce adequate products quantified via {[MROPsel (i, j3)]} and with the use of business processes (BP), which create an integral part of a given BP group. One of those processes will be selected and demonstrated how the KPI (i, 3) indicator should be decomposed in order to describe the selected BP functionality and performance, first of all. In general any BP is represented by its own internal and external metrics, while the external metrics is concerned with BP outputs and inputs and the BP internal metrics is closely related to appropriate production human resources, production devices and production tools and those aspects are quantified via given linguistic sets.However, that decomposition will be done

In that step, a group of selected products should be created, which is an integral part of products quantified via {[MROPsel (i, j3)]} linguistic set, while formula (66)

> 3 (66)

<sup>g</sup> (67)

6

3 ⊆ MROPsel i*;* j

Those products should be produced with the use of the selected BP (see also

f½BP ið Þgg *;* j7 g∈ ½ BPG i*;* j

Now, we have to select set input materials needed for production of the abovementioned products. We shall apply the [MROPselfincosts (i, j)]} linguistic set for those purposes, the content of consists of two subsets with respect to formula.

MROPselfincosts ½ ð Þ i, j �g ¼ f½ MROP1selfincosts ½ � ð Þ i, j , ½ MROP2selmatcosts ½ ð Þ i, j �g (68)

*5.2.2 KPI indicator decomposition related to operational management level -*

*5.2.2.1 Determination of Production process tertiary KPI indicators*

3 , MROPselfinass i*;* j

KPI ið Þ¼ *;* 1 f g ½ � CONTRACTB ið Þ (63) KPI ið Þ¼ *;* 2 f g ½ � CONTRACTC ið Þ (64) KPI ið Þ¼ *;* 3 f g ½ � CONTRACTD ið Þ (65)

4

(62)

,

When applying the PBPL Equation, KPI indicator for the selected BP functionality and performance might be derived, while the modified PBPL Equation is postulated with respect to formula (61).

f MROP2selmatcosts ½ � ð Þ i*;* j , MROP1selfincosts ½ � ð Þ i*;* j , MROPselfinass ½ � ð Þ i*;* j , MROPselmatass ½ ð Þ i*;* j �g ⊗ MROPsel\_bp i*;* j 3 <sup>¼</sup> Tbex i*;* <sup>j</sup> 8 ⊗ Retx i*;* j 9 

(71)

*5.2.2.4 PBPL equation solution results*

*Step 5*

$$\{\left[\text{Tbx}\left(\mathbf{i}, \mathbf{j}\_{\text{8}}\right)\right]\} = \left(\left[\text{MROP}\_{\text{selmattas}}\left(\mathbf{i}, \mathbf{j}\right)\right], \left[\text{MROP}2\_{\text{selmattosts}}\left(\mathbf{i}, \mathbf{j}\right)\right]\right) \tag{72}$$

f Retx i*;* j 8 g¼f MROP2selmatcosts ½ � ð Þ <sup>i</sup>*;* <sup>j</sup> , MROP1selfincosts ½ � ð Þ <sup>i</sup>*;* <sup>j</sup> , MROPselfinass ½ � ð Þ <sup>i</sup>*;* <sup>j</sup> , MROPselmatass ½ ð Þ i*;* j �g

$$\sigma\_{\beta}$$

*5.2.2.5 BP external metrics KPI indicators*

*Step 6*

When dealing with BP External metrics KPI Indicators, so called basic and external KPI indicators will be defined.

$$\text{KPIemb } (\mathbf{i}, \mathbf{3}) = \left\{ \left[ \mathbf{Tbes \left( \mathbf{i}, \mathbf{j}\_8 \right)} \right] \right\} = \left( \left[ \mathbf{MROP}\_{\text{selmatas}} \left( \mathbf{i}, \mathbf{j} \right) \right], \left[ \mathbf{MROP}2\_{\text{selmatas} \cup \text{ts}} \left( \mathbf{i}, \mathbf{j} \right) \right] \right) \tag{74}$$

$$\begin{aligned} \text{KPIement } (\mathbf{i}, \mathbf{3}) &= \{ \left[ \text{Rectx } (\mathbf{i}, \mathbf{j\_8}) \right] \} = \{ \left[ \text{MROP2}\_{\text{selmatcots}} \left( \mathbf{i}, \mathbf{j} \right) \right], \left[ \text{MROP1}\_{\text{seldimcots}} \left( \mathbf{i}, \mathbf{j} \right) \right], \\ &\quad \left[ \text{MROP}\_{\text{selfinas}} \left( \mathbf{i}, \mathbf{j} \right) \right] \text{[MROP}\_{\text{selmatcats}} \left( \mathbf{i}, \mathbf{j} \right)] \} \end{aligned}$$

(75)

$$\text{KPIem} = \text{KPIemb} \ (\text{i}, \text{3}) \otimes \text{KPIemb} \ (\text{i}, \text{3}) \tag{76}$$

#### *5.2.2.6 BP Internal metrics KPI Indicators*

#### *Step 7*

The similar algorithm might be applied, when deriving BP Internal metrics KPI Indicators.
