# A Change-Sensitive Complexity Measurement for Business Process Models Based on Control Structure

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Basic Theories

#### 3.1. BPMN

#### 3.2. Basic Control Structures

- BCS = {seq, AND, OR, XOR, cyc}
- BCS is defined as seq if t
_{0}→ t_{k}, where t ∈ Ac - BCS is defined as AND if t
_{0}has branch t_{01}, t_{02}…, t_{0S}, where t ∈ Ac - BCS is defined as OR if t
_{0}has branch t_{01}, t_{02}…, t_{0S}, where t ∈ Ac - BCS is defined as XOR if t
_{0}has branch t_{01}, t_{02}…, t_{0S}, where t ∈ Ac - BCS is defined as cyc if t
_{k}has a loop branch, where t ∈ Ac

#### 3.3. Process Model Complexity Based on the Control Structure

## 4. The Process Model Complexity Measurement

#### 4.1. Measuring the Branching Structure Complexity

#### 4.1.1. The Branching Structure Complexity

**Definition**

**1.**

_{j}represents the number of activities on the jth branch. Since intermediate events and activities are elements at the same level, we consider that they have the same impact on the process complexity. If there is an intermediate event, it is added to m

_{j}as an activity. When combined with the weight CW

_{AND}= 4 shown in Table 1, the complexity of an AND branching structure is represented by the symbol ${C}_{{AND}_{i}},$ defined as follows:

_{i}branching structure, CW

_{AND}is the weight of the AND branching structure. For example, to calculate the branching structure complexity of model P3 in Figure 1, where the AND branching structure has two branches n = 2, the number of activities on the first branch m

_{1}= 3, AND the number of activities on the second branch m

_{2}= 1, the number of activity sequences ${L}_{{AND}_{P3}}$ = 4, the weight CW

_{AND}= 4, so the complexity is ${C}_{{AND}_{P3}}$ = 16.

**Definition**

**2.**

_{XOR}= 3 shown in Table 1, the complexity of an XOR branching structure represented by the symbol ${C}_{{XOR}_{i}}$ can be calculated as:

_{XOR}is the weight of the XOR branching structure, ${L}_{{XOR}_{i}}$ is the activity sequences of the XOR

_{i}branching structure. As shown in Figure 2, the process model P4 includes an XOR branching structure with two branches n = 2, which generates two different activity sequences ${L}_{{XOR}_{P4}}$= 2. The XOR structure has the weight CW

_{XOR}= 3, so its complexity ${C}_{{XOR}_{P4}}$= 6.

**Definition**

**3.**

^{n}− 1 possible combinations of branches that can be executed simultaneously. One combination executes all the branches with a probability of $\frac{1}{{2}^{n}-1}$, which is similar to an AND structure executing all of its branches (with a probability of 1). The total probability of executing the remaining combinations is $\frac{{2}^{n}-2}{{2}^{n}-1}$, and each combination can be viewed as an XOR structure executing one of its branches (with a probability of 1/n). The number of different activity sequences generated by an OR branching structure can be represented by the symbol ${L}_{{OR}_{i}},$ defined as follows:

_{XOR}is the number of activity sequences for the corresponding XOR branching structure. When combined with the CW

_{OR}= 7 shown in Table 1, an OR branching structure complexity can be represented by the symbol ${C}_{{OR}_{i}}$ as:

_{OR}is the weight of the OR branching structure, ${L}_{{OR}_{i}}$ is the activity sequences of the OR

_{i}branching structure. As shown in Figure 2, the process model P5 has an OR branching structure. The number of different activity sequences ${L}_{{OR}_{P5}}=(\frac{1}{3}\cdot 1)\cdot 4+(\frac{2}{3}\cdot \frac{1}{2})\cdot 2=2$, the weight CW

_{XOR}= 7, so we find its complexity ${C}_{{OR}_{P5}}=14$.

#### 4.1.2. The Connection Forms between the Branching Structures

**Definition**

**4.**

_{AND}

_{1-AND2}= C

_{AND}

_{1}+ C

_{AND}

_{2}.

**Definition**

**5.**

_{AND}

_{3∗AND4}= C

_{AND}

_{3}· C

_{AND}

_{4}.

_{AND}

_{1}= 24, and the complexity of the entire nested structure AND1*(XOR1, OR1) is calculated as C

_{AND}

_{1*(XOR1,OR1)}= C

_{AND}

_{1}(C

_{XOR}

_{1}+ C

_{OR}

_{1}) = 368. Likewise, for the nested structure AND2*(XOR1, OR1), the virtual activities I and J on the outer branching structure AND2 represent the branching structures XOR1 and OR1, respectively. The complexity of the outer branching structure AND2 is calculated as C

_{AND}

_{2}= 16, and the complexity of the entire nested structure AND2*(XOR1, OR1) is calculated as C

_{AND}

_{2*(XOR1,OR1)}= C

_{AND}

_{2}(C

_{XOR}

_{1}+ C

_{OR}

_{1}) = 736/3.

_{XOR}

_{1*AND2}= C

_{XOR}

_{1}·C

_{AND}

_{2}. We then calculate the complexity of the outer nested structure AND1*(XOR1*AND2) as C

_{AND}

_{1*(XOR1*AND2)}= C

_{AND}

_{1}C

_{XOR}

_{1*AND2}. Thus, the complexity of the entire multi-layer nested structure can be expressed as C

_{AND}

_{1*(XOR1*AND2)}=C

_{AND}

_{1}·C

_{XOR}

_{1}·C

_{AND}

_{2}.

#### 4.1.3. The Complexity Measurement for Branching Structures

**Definition**

**6.**

_{OR}

_{1}= 12, while the complexity of XOR1 is C

_{XOR}

_{1}= 9. In the nested structure AND2*(XOR2, OR2), the outer layer nested structure AND2 has a complexity of C

_{AND}

_{2}= 24, while the inner layer nested structures OR2 and XOR2 have complexities of C

_{OR}

_{2}= 12 and C

_{XOR}

_{2}= 6, respectively. Consequently, the complexity of AND1 is C

_{AND}

_{1}= C

_{AND}

_{2*(XOR2,OR2)}= C

_{AND}

_{2}·(C

_{XOR}

_{2}+ C

_{OR}

_{2}) = 432. The total complexity resulting from the branching structures in the process model is C

_{B}= C

_{AND}

_{1}+ C

_{XOR}

_{1}+ C

_{OR}

_{1}= 453.

#### 4.2. The Change-Sensitive Complexity Measurement for the Process Model Based on the Control Structure

**Definition**

**7.**

_{seq}= 1 as presented in Table 1, the formula for calculating the complexity of the sequence structure in the process model is represented by the symbol ${C}_{seq}$, defined as follows:

_{seq}is the weight of the sequence structure, the number of sequential nodes N

_{seq}includes all nodes in the sequential structure, such as the start nodes, end nodes, activity nodes, and gateway nodes.

**Definition**

**8.**

_{cyc}= 3 outlined in Table 1, the formula for calculating the complexity of the loop structure denoted by ${C}_{cyc}$ in the process model is defined as follows:

_{cyc}is the weight of the cyclic structure, N is the total number of nodes in the process model, the number of nodes in the loop N

_{cyc}includes all nodes in the loop, such as the start nodes, end nodes, activity nodes, and gateway nodes.

**Definition**

**9.**

## 5. Experiment Design and Theoretical Validation

#### 5.1. Experiment Models

Group | Purpose | Business Process Model | Figure |
---|---|---|---|

1 | This group is intended to show the effect of adding new elements on the process model complexity. | In the first case, as shown in Figure 10, process models b and c are obtained by sequentially adding the activities to the base process model a. | Figure 10 |

2 | This group is intended to show the effect of changes in the number of branches on the complexity of the process model. | In the first case, as shown in Figure 11, process models b and c are obtained by gradually adding activities to the branching structure based on process model a. | Figure 11, Figure 12 and Figure 13 |

In the second case, as shown in Figure 12, process models b and c are obtained by moving sequential activities to the branching structure based on process model a. | |||

In the third case, as shown in Figure 13, process models b and c are obtained by changing the positions of activities in the branching structure based on process model a. | |||

3 | This group is intended to show the effect of shifting branching locations on the business process complexity. | As shown in Figure 14, process models b and c are obtained by shifting the branching structure based on process model a. | Figure 14 |

4 | This group is intended to show the effect of changes in the number of activities on the branch structure on the process model complexity. | In the first case, as shown in Figure 15, process models b and c are obtained by gradually adding new activities to the branching structure based on process model a. | Figure 15 and Figure 16 |

In the second case, as shown in Figure 16, process models b and c are obtained by gradually moving the sequential activities to the branching structure based on process model a. | |||

5 | This group is intended to show the effect of changing the location of activities on the branches on the business process complexity. | As shown in Figure 17, process models b and c are obtained by changing the positions of activities in the branching structure based on process model a. | Figure 17 |

6 | This group is intended to show the effect of the change of branch logic type on the process model complexity. | As shown in Figure 18, process models b and c are obtained by changing the branch logic type based on process model a. | Figure 18 |

7 | This group is intended to show the effect of exchanging the branch logic type on the process model complexity. | As illustrated in Figure 19, process models b and c are obtained by exchanging the branch logic types based on process model a. | Figure 19 |

8 | This group is intended to show the effect of changing the connection forms between the branching structures on the process model complexity. | As shown in Figure 20, process models b and c are obtained by changing the connection form between branching structures based on process model a. | Figure 20 |

9 | This group is intended to show the effect of changing the number of cycles and activities in the loop on the process model complexity. | As shown in Figure 21, process models b, c, and d are obtained by changing the number of activities in the loop and the number of loops based on process model a. | Figure 21 |

#### 5.2. Experiment Data

#### 5.3. Theoretical Validation

- Property 1: (∃P) (∃Q) (|P|≠|Q|). There exist two distinct processes of P and Q, which are not of the same complexity. This property requires that the measure can distinguish at least two different complexity process models, meaning that all process models cannot be considered the same complexity. The CP can differentiate the complexity of the different process models. As shown in Table 3, the results of the complexity measurements of CPs are not the same. Therefore, the CP satisfies property 1.
- Property 2: Let c be a non-negative number. Then there are only finite processes for which |P| = c. This property requires that the result of the complexity measurement must be non-negative. As shown in Table 3, the measurement values of CPs are positive for different process models. Therefore, the CP satisfies Property 2.
- Property 3: There are distinct processes P and Q such that |P| = |Q|. There exist distinct process models P and Q, and their complexity measurements are equal. This property requires that different process models can have the same complexity. As shown in Table 3, the CP measure yields the same complexity for different process models Figure 13a–c. Therefore, the CP satisfies property 3.
- Property 4: (∃P) (∃Q) (P ≡ Q & |P| ≠ |Q|). There exist two process models P and Q with the same function but different structural designs, such that their complexity measurements are not equal. This property requires that the complexity measurement method can distinguish between two process models with the same functionality but different structural designs. The CP describes a process model complexity by analyzing the complexity of its structural design. Thus, the CP can distinguish between two process models with the same functionality but different structural designs. Therefore, the CP satisfies this property.
- Property 5: (∀P) (∀Q); (|P| ≤ |P; Q| &|Q| ≤ |P; Q|). For any two process models P and Q, if they are combined to form a new process model, then the complexity of the combined process model is not less than the complexity of each process model. When two process models are combined in sequence to form a new process model, the CP describes the complexity of the combined process model as the sum of the complexities of the two individual process models. Therefore, the CP satisfies property 5.
- Property 6:a. (∃P) (∃Q) (∃R) (|P| = |Q| & |P; R| ≠ |Q; R|);b. (∃P) (∃Q) (∃R) (|P| = |Q| & | R; P| ≠ |R; Q|)
- Two processes P and Q with the same complexity are combined with process R in the same way to form a new process. The measure can distinguish the complexity of the two process models obtained after composition. When P and Q are sequentially connected to R to form a new process, the CP measures that the complexity of the two resulting process models is the same, meaning that it is impossible to distinguish between the complexity of the two combined process models. Therefore, the CP does not satisfy this property.
- Property 7: If Q is formed by permuting the order of the activities of P, then the complexity of P and Q may be different, |P|≠|Q|. Changes to the position of elements in the process (such as activities) may affect the complexity of the process model. As shown in Table 3, the complexity values of the process models Figure 15a,b change as the positions of the activities are altered. Therefore, the CP satisfies this property.
- Property 8: If P is a renaming of Q, then |P| = |Q|. Changing the names of the components of a process model does not affect its complexity. This property requires that renaming the activity or any other structural components of a model should not change the complexity. The CP describes the complexity of a process model based on its structure, and the measurement results are not affected by changes in the structure’s names. Therefore, the CP satisfies the property.
- Property 9: (∃P) (∃Q); (|P| + |Q| < |P; Q|). There are two processes P and Q, whose complexity when combined into a process model is greater than the sum of the complexities of each process model. This means that the complexity of a whole process model is at least equal to the sum of the complexities of all its local components. When two processes are combined by nesting, the CP describes their complexity to be greater than the sum of their complexities. Therefore, the CP satisfies property 9.

## 6. Results and Discussion

- Group 1: Adding new elements to the process model

- Group 2: Adding branches to the process model

- Group 3: Shifting the location of the branching structure

- Group 4, 5, 7: Changing the number of activities, activity position, and exchanging branch logic type

- Group 6: Changing the logic type of the branching structure

- Group 8: Changing the connection form between the branching structures

- Group 9: Changing the number of loop nodes and loop in the process model

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Control Structure | Weight | Source |
---|---|---|

Sequence | 1 | [11] |

Parallel Split (AND) | 4 | |

Exclusive Choice (XOR) | 3 | |

Multiple Choice (OR) | 7 | |

Looping (Cyclic) | 3 | [19] |

Process Models | CP | YC | CFC | CADAC | Scale | CW |
---|---|---|---|---|---|---|

AND/XOR/OR | AND/XOR/OR | AND/XOR/OR | AND/XOR/OR | AND/XOR/OR | AND/XOR/OR | |

Figure 10a | 3.00 | 19.00 | 0 | 21 | 0 | 1 |

Figure 10b | 4.00 | 21.00 | 0 | 23 | 0 | 1 |

Figure 10c | 5.00 | 23.00 | 0 | 25 | 0 | 1 |

Figure 11a | 12.00/10.00/13.33 | 45.00/43.00/51.00 | 1/2/3 | 48/46/51 | 16/32/48 | 5/4/8 |

Figure 11b | 29.00/14.00/17.00 | 65.40/50.40/104.40 | 1/3/7 | 51/49/54 | 19/57/133 | 5/4/8 |

Figure 11c | 102.00/18.00/23.73 | 141.33/57.33/325.33 | 1/4/15 | 54/52/57 | 22/88/130 | 5/4/8 |

Figure 12a | 14.00/12.00/15.33 | 46.67/44.67/52.67 | 1/2/3 | 52/50/55 | 20/40/60 | 5/4/8 |

Figure 12b | 29.00/14.00/17.00 | 66.00/51.00/105.00 | 1/3/7 | 48/46/51 | 21/63/147 | 5/4/8 |

Figure 12c | 100.00/16.00/21.73 | 141.33/57.33/325.33 | 1/4/15 | 51/49/54 | 22/88/330 | 5/4/8 |

Figure 13a | 24.00/10.00/20.33 | 54.00/52.00/60.00 | 1/2/3 | 54/52/57 | 22/44/66 | 5/4/8 |

Figure 13b | 84.00/13.00/30.00 | 71.00/56.00/110.00 | 1/3/7 | 55/53/58 | 23/69/161 | 5/4/8 |

Figure 13c | 244.00/16.00/38.53 | 144.00/60.00/328.00 | 1/4/15 | 56/54/59 | 24/96/360 | 5/4/8 |

Figure 14a | 30.00/15.00/18.00 | 46.67/31.67/85.67 | 1/3/7 | 52/50/55 | 20/60/140 | 5/4/8 |

Figure 14b | 30.00/15.00/18.00 | 46.67/31.67/85.67 | 1/3/7 | 52/50/55 | 20/60/140 | 5/4/8 |

Figure 14c | 30.00/15.00/18.00 | 46.67/31.67/85.67 | 1/3/7 | 52/50/55 | 20/60/140 | 5/4/8 |

Figure 15a | 12.00/10.00/13.33 | 45.00/43.00/51.00 | 1/2/3 | 48/46/51 | 16/32/48 | 5/4/8 |

Figure 15b | 17.00/11.00/16.67 | 49.40/47.40/55.40 | 1/2/3 | 50/48/53 | 18/36/54 | 5/4/8 |

Figure 15c | 30.00/12.00/24.67 | 51.33/49.33/57.33 | 1/2/3 | 52/50/55 | 20/40/60 | 5/4/8 |

Figure 16a | 14.00/12.00/15.33 | 46.67/44.67/52.67 | 1/2/3 | 52/50/55 | 20/40/60 | 5/4/8 |

Figure 16b | 17.00/11.00/16.67 | 46.67/44.67/52.67 | 1/2/3 | 52/50/55 | 20/40/60 | 5/4/8 |

Figure 16c | 28.00/10.00/22.67 | 46.67/44.67/52.67 | 1/2/3 | 52/50/55 | 20/40/60 | 5/4/8 |

Figure 17a | 124.00/13.00/40.00 | 73.50/58.50/112.50 | 1/3/7 | 57/55/60 | 25/75/175 | 5/4/8 |

Figure 17b | 244.00/13.00/70.00 | 73.50/58.50/112.50 | 1/3/7 | 57/55/60 | 25/75/175 | 5/4/8 |

Figure 17c | 364.00/13.00/100.00 | 73.50/58.50/112.50 | 1/3/7 | 57/55/60 | 25/75/175 | 5/4/8 |

Figure 18a | 28.00 | 65.40 | 1 | 51 | 19 | 5 |

Figure 18b | 13.00 | 40.50 | 3 | 49 | 57 | 4 |

Figure 18c | 4.00 | 104.50 | 7 | 54 | 133 | 8 |

Figure 19a | 372.00 | 93.50 | 6 | 85 | 204 | 15 |

Figure 19b | 108.00 | 93.50 | 6 | 85 | 204 | 15 |

Figure 19c | 4.00 | 93.50 | 6 | 85 | 204 | 15 |

Figure 20a | 21.00/17.00/23.67 | 65.00/61.00/77.00 | 2/4/6 | 61/57/67 | 54/108/162 | 9/7/15 |

Figure 20b | 101.00/41.00/113.89 | 69.00/65.00/81.00 | 2/4/6 | 75/71/81 | 54/108/162 | 9/7/15 |

Figure 20c | 196.00/40.00/178.22 | 71.00/67.00/83.00 | 2/4/6 | 75/71/81 | 54/108/162 | 9/7/15 |

Figure 21a | 7.20/7.20/7.20 | 39.42/39.42/39.42 | 1/2/3 | 48/46/51 | 20/40/60 | 4/4/4 |

Figure 21b | 6.50/6.50/6.50 | 42.13/42.13/42.13 | 1/2/3 | 48/46/51 | 20/40/60 | 4/4/4 |

Figure 21c | 5.80/5.80/5.80 | 44.83/44.83/44.83 | 1/2/3 | 48/46/51 | 20/40/60 | 4/4/4 |

Figure 21d | 6.75/6.75/6.75 | 47.50/47.50/47.50 | 2/4/6 | 51/47/57 | 50/100/150 | 7/7/7 |

Properties | CP | YC | Scale | CADAC | CFC | CW |
---|---|---|---|---|---|---|

1 | √ | √ | √ | √ | √ | √ |

2 | √ | √ | √ | √ | × | √ |

3 | √ | √ | √ | √ | √ | √ |

4 | √ | √ | √ | √ | √ | √ |

5 | √ | √ | √ | √ | √ | √ |

6 | × | × | × | × | × | × |

7 | √ | √ | × | × | √ | √ |

8 | √ | √ | √ | √ | √ | √ |

9 | √ | √ | √ | √ | √ | √ |

Groups | CP | YC | Scale | CADAC | CFC | CW |
---|---|---|---|---|---|---|

1 | √ | √ | × | √ | × | × |

2 | √ | √ | √ | √ | × | × |

3 | × | × | × | × | × | × |

4 | √ | √ | × | × | × | × |

5 | √ | × | × | × | × | × |

6 | √ | √ | √ | √ | √ | √ |

7 | √ | × | × | × | × | × |

8 | √ | √ | × | √ | × | × |

9 | √ | √ | × | × | × | × |

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## Share and Cite

**MDPI and ACS Style**

Zhou, C.; Zhang, D.; Chen, D.; Liu, C.
A Change-Sensitive Complexity Measurement for Business Process Models Based on Control Structure. *Systems* **2023**, *11*, 250.
https://doi.org/10.3390/systems11050250

**AMA Style**

Zhou C, Zhang D, Chen D, Liu C.
A Change-Sensitive Complexity Measurement for Business Process Models Based on Control Structure. *Systems*. 2023; 11(5):250.
https://doi.org/10.3390/systems11050250

**Chicago/Turabian Style**

Zhou, Changhong, Dengliang Zhang, Deyan Chen, and Cong Liu.
2023. "A Change-Sensitive Complexity Measurement for Business Process Models Based on Control Structure" *Systems* 11, no. 5: 250.
https://doi.org/10.3390/systems11050250