# A System Dynamics Modeling Support System Based on Computational Intelligence

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of SD Modeling Support System

#### 2.1. Inferring System Equations and Parameter Estimation

#### 2.2. Inferring CLDs, SFDs, System Equations and Parameter Estimation

## 3. Experimental Setup

#### 3.1. Setups

#### 3.1.1. Setup 1

#### 3.1.2. Setup 2

#### 3.2. Case Studies

#### 3.3. CI Algorithm Parameters

## 4. Results and Analysis

#### 4.1. Setup 1

#### 4.1.1. Case 1

#### 4.1.2. Case 2

#### 4.2. Setup 2

#### 4.2.1. Case 1

#### 4.2.2. Case 2

#### 4.3. Runtime Analysis

#### 4.4. Limitations

## 5. Conclusions

- Most pressing is the need to work towards enhancing the ability of the methods to generate models with minimal structures that can characterize the data. The methods under development should be robust and scalable with the ability to handle big data.
- It is crucial to integrate the necessary semantic domain knowledge, about the system or problem of interest, with the CI-based methods to generate valid structures. This will add a new dimension to the capability of the CI-methods for generating valuable models, especially for real-life systems where we do not have any idea about how the target model looks.
- Identification of which variables to include in the model, as part of the support system’s inference engine, is challenging because, for each variable set selected, a SD model should be built and simulated to evaluate those selected. Therefore, the efficiency of the modeling process would be greatly enhanced by finding a method which bypassed the need to build the whole SD model.
- Generating an SD model without prior knowledge of the types of variables and only limited observations is a challenging task too. However, the ability to handle these conditions since they are common in real world applications is an important feature to be added to the support system. Generating SD models under these conditions will require not only a search for the set of causal and mathematical relationships, but a search for the types of variables and for the mathematical equations for those variables that do not have any observations.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Illustration of Case 1 target behaviors, healthy people ($HP$) and sick people ($SP$), and the generated behaviors from the corresponding inferred models for Setup 1, with and without applying genetic programming depth controller.

**Figure 6.**Illustration of Case 2 target outputs, susceptible population ($SP$), infected population ($IP$) and recovery population ($RP$), and the two generated outputs from the corresponding inferred models for Setup 1, with and without applying genetic programming depth controller.

**Figure 7.**Illustration of Case 1 target outputs, healthy people ($HP$) and sick people ($SP$), and the two generated outputs from the corresponding inferred models for Setup 2 with genetic programming depth controller.

**Figure 8.**Case 1 inferred CLD by showing the correctly predicted links (

**b**), additional links (

**c**) and missing links (

**d**) compared to the target CLD (

**a**)—Setup 2.

**Figure 10.**Illustration of Case 2 target outputs, susceptible population ($SP$), infected population ($IP$) and recovery population ($RP$), and the two generated outputs from the corresponding inferred models for Setup 2 with genetic programming depth controller.

**Figure 11.**Case 2 inferred CLD by showing the correctly predicted links (

**b**); additional links (

**c**); and missing links (

**d**) compared to the target CLD (

**a**)—Setup 2.

**Table 1.**Comparison of Case 1 target system equations with inferred equations using genetic programming ensemble applied with and without depth controller parameter ${D}_{cont}$—Setup 1.

Target Equations | Inferred with ${\mathit{D}}_{\mathit{cont}}=1$ | Inferred with ${\mathit{D}}_{\mathit{cont}}=0$ |
---|---|---|

$\frac{dHP}{dt}=(2\times SP)-(\frac{5\times HP\times SP}{HP+SP})$ | $\frac{dHP}{dt}=(1.99\times SP)-(0.05\times HP\times SP)$ | $\frac{dHP}{dt}=(1.95\times SP)-(0.05\times HP\times SP-21.74)$ |

$\frac{dSP}{dt}=(\frac{5\times HP\times SP}{HP+SP})-(2\times SP)$ | $\frac{dSP}{dt}=(0.04\times HP\times SP+1.58)-(1.77\times SP)$ | $\frac{dSP}{dt}=(0.05\times HP\times SP+0.41)-(1.9\times 8SP)$ |

**Table 2.**Comparison of Case 2 target system equations with inferred equations using genetic programming ensemble applied with and without depth controller parameter ${D}_{cont}$—Setup 1.

Target Equations | Inferred with ${\mathit{D}}_{\mathit{cont}}=1$ | Inferred with ${\mathit{D}}_{\mathbf{cont}}=0$ |
---|---|---|

$\frac{dSP}{dt}=-\frac{1.5\times SP\times IP}{N}$ | $\frac{dSP}{dt}=-\frac{1.5\times SP\times IP}{N}$ | $\frac{dSP}{dt}=-\frac{1.5\times SP\times IP}{N}$ |

$\frac{dIP}{dt}=\frac{1.5\times SP\times IP}{N}-0.5\times IP$ | $\frac{dIP}{dt}=\frac{1.501\times SP\times IP}{N}-0.5\times IP$ | $\frac{dIP}{dt}=\frac{1.5\times SP\times IP}{N}-0.501\times IP$ |

$\frac{dRP}{dt}=0.5\times IP$ | $\frac{dRP}{dt}=0.5\times IP$ | $\frac{dRP}{dt}=0.5\times IP$ |

**Table 3.**Comparison of Case 1 target system equations with inferred equations using genetic programming ensemble applied with depth controller parameter ${D}_{cont}$—Setup 2.

Target Equations | Inferred with ${\mathit{D}}_{\mathit{cont}}=1$ |
---|---|

$\frac{dHP}{dt}=(2\times SP)-(\frac{5\times HP\times SP}{HP+SP})$ | $\frac{dHP}{dt}=(-0.05\times SP\times HP)-(HP-SP-99.968)$ |

$\frac{dSP}{dt}=(\frac{5\times HP\times SP}{HP+SP})-(2\times SP)$ | $\frac{dSP}{dt}=-(0.05\times HP\times SP)+(2.001\times SP)$ |

**Table 4.**Comparison of Case 1 target system equations with inferred equations using genetic programming ensemble applied with depth controller parameter ${D}_{cont}$—Setup 2.

Target Equations | Inferred with ${\mathit{D}}_{\mathit{cont}}=1$ |
---|---|

$\frac{dSP}{dt}=-\frac{1.5\times SP\times IP}{N}$ | $\frac{dSP}{dt}=-\frac{1.5\times SP\times IP}{N}$ |

$\frac{dIP}{dt}=\frac{1.5\times SP\times IP}{N}-0.5\times IP$ | $\frac{dIP}{dt}=\frac{SP\times IP}{N}-0.00005\times IP\times (N-SP)$ |

$\frac{dRP}{dt}=0.5\times IP$ | $\frac{dRP}{dt}=\frac{83.96\times IP\times SP+N}{167.91\times SP}$ |

**Table 5.**Runtime for proposed computational intelligence methods in generating system dynamics models for each case with different setups.

Case Studies | Setup 1 | Setup 2 |
---|---|---|

Case 1 | 6.2 min | 4 h |

Case 2 | 6.9 min | 5 h |

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**MDPI and ACS Style**

Abdelbari, H.; Shafi, K.
A System Dynamics Modeling Support System Based on Computational Intelligence. *Systems* **2019**, *7*, 47.
https://doi.org/10.3390/systems7040047

**AMA Style**

Abdelbari H, Shafi K.
A System Dynamics Modeling Support System Based on Computational Intelligence. *Systems*. 2019; 7(4):47.
https://doi.org/10.3390/systems7040047

**Chicago/Turabian Style**

Abdelbari, Hassan, and Kamran Shafi.
2019. "A System Dynamics Modeling Support System Based on Computational Intelligence" *Systems* 7, no. 4: 47.
https://doi.org/10.3390/systems7040047