# The Method for Risk Evaluation in Assembly Process based on the Discrete-Time SIRS Epidemic Model and Information Entropy

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## Abstract

**:**

## 1. Introduction

**Assumptions**: As risk is a common phenomenon, it occurs in all areas of manufacturing. There are various scientific disciplines that deal with risk analysis, e.g., probability calculus, statistics, econometrics, image recognition theory, reliability theory, operational research, theory of organization and management, etc. [53].

**Assumption 1**: The reliability of each assembly stations is equal and constant.

**Assumption 2**: Different production lines are independent of each other.

**Assumption 3**: Products in assembly process are divided into three compartments according to their health states. In a certain condition, three states can transform into each other.

## 2. Optimal Assembly Path Selection based on Reliability and Cost

## 3. Risk Evaluation of Assembly Process

#### 3.1. Discrete-Time SIRS Epidemic Model in Assembly Process

#### 3.2. The Calculation of Information Entropy

## 4. Case Study

#### 4.1. Case Introduction

#### 4.2. Optimal Assembly Approach Selection

#### 4.3. Risk Evaluation based on SIRS Model and Entropy

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## Notations

$p$ | The reliability of each assembly station |

${P}_{i}$ | The assembly process |

$M$ | The maximum capacity of each process |

${O}_{j}(G)$ | The assembly capacity in general line |

${O}_{j}(R)$ | The assembly capacity in rework line |

${I}_{j}$ | The input loading of line j |

${S}_{ij}$ | The sequence number of inspection station |

${t}_{ij}$ | The number of processes behind the repaired station |

${O}_{j}$ | The quantity of products assembled in production line j |

${r}_{ij}$ | The quantity of processes behind process ${P}_{ij}$ |

$k$ | The number of repaired station |

${\alpha}_{ij}$ | The indicative function |

${O}_{\mathrm{max}}$ | The maximum assembly capacity of system |

${O}_{j\mathrm{max}}$ | The maximum assembly capacity of production line j |

$D$ | The assembly task |

${d}_{j}$ | The assembly task for production line j |

${V}_{ij}$ | The sequence number of ${P}_{ij}$ in assembly line |

${Q}_{ij}(G)$ | The input quantity of each process in general line |

${Q}_{ij}(R)$ | The input quantity of each process in rework line |

${\beta}_{ij}$ | The indicative function |

${L}_{j}$ | The loading vector of assembly system |

${l}_{ij}$ | The actual input loading of each process |

${C}_{j}$ | The minimum capacity vector of production j |

${c}_{ij}$ | The minimum assembly vector |

$R$ | The reliability of whole system, that is, the capacity that assembling required products |

$\oplus $ | Take the maximum value of the corresponding element of two sets |

$\mathsf{\Lambda}$ | The products recruitment in process ${i}^{th}$ |

$\mu $ | The failure rate in process ${i}^{th}$ |

$\eta $ | Residual defect density left by the previous process |

$\epsilon $ | Product’s mortality due to this type of defect |

${\gamma}_{1},{\gamma}_{2}$ | The immune level of process ${i}^{th}$, that is, a certain degree of repairability |

$\delta $ | Loss rate of immune status |

$S/S(t)$ | The number of products in susceptible state |

$I/I(t)$ | The number of products in infectious state |

$R/R(t)$ | The number of products in recovered state |

$f(I)$ | A real local Lipschitz function |

$\beta $ | The infective rate of defects |

${S}^{\prime}$ | The derivative of $S$ with respect to time |

${I}^{\prime}$ | The derivative of $I$ with respect to time |

${R}^{\prime}$ | The derivative of $R$ with respect to time |

$\Delta t$ | A time step size |

$X$ | A random variable |

$H(X)$ | The entropy of the random variable $X$ |

${p}_{i}$ | The probability that the system is in the ${i}^{th}$ microstate |

${H}_{I}$ | The entropy of whole process at the initial time |

${H}_{c}$ | The entropy of whole process at the critical time point |

$\Delta {H}_{base}$ | The increment of entropy calculated by SIRS epidemic model in assembly process |

$\Delta H$ | The process entropy increment in optimal assembly approach |

$P$ | The overall assembly risk probability |

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**Figure 5.**The SIRS model in assembly process considering screening [51].

**Figure 11.**The variation of different states during assembly process. (

**a**) The variation of susceptive products; (

**b**) The variation of infective products; (

**c**) The variation of recovered products; and, (

**d**) The variation of total products.

**Table 1.**The comparison between defect emergence and disease propagation [52].

Definition | Infectious Source | Infectious Path | Susceptible Individuals |
---|---|---|---|

Disease propagation | Individuals with pathogens | The process that pathogens arrive and invade new susceptible individuals | Individuals susceptible to an infectious disease that lacks immunity or adaptive immunity |

Defect emergence | Process that has a positive or negative effect on product defects under process stress | A time series process in which product defects are corrected or excited under stress | Hidden risk of infection or Lack of resilience to risks |

Process | Capacity | Probability | Process | Capacity | Probability |
---|---|---|---|---|---|

P_{1} | 0 | 0.010 | P_{5} | 0 | 0.001 |

100 | 0.010 | 60 | 0.002 | ||

150 | 0.010 | 120 | 0.002 | ||

200 | 0.010 | 180 | 0.005 | ||

250 | 0.010 | 200 | 0.010 | ||

300 | 0.020 | 240 | 0.005 | ||

350 | 0.020 | 260 | 0.005 | ||

400 | 0.910 | 280 | 0.970 | ||

P_{2} | 0 | 0.005 | P_{6} | 0 | 0.010 |

150 | 0.010 | 50 | 0.010 | ||

200 | 0.010 | 100 | 0.010 | ||

250 | 0.015 | 150 | 0.020 | ||

300 | 0.010 | 200 | 0.020 | ||

350 | 0.010 | 220 | 0.005 | ||

450 | 0.020 | 240 | 0.003 | ||

600 | 0.920 | 260 | 0.001 | ||

P_{3} | 0 | 0.002 | P_{7} | 280 | 0.001 |

50 | 0.003 | 300 | 0.920 | ||

100 | 0.005 | 0 | 0.005 | ||

150 | 0.010 | 150 | 0.005 | ||

200 | 0.010 | 170 | 0.015 | ||

250 | 0.015 | 190 | 0.015 | ||

300 | 0.955 | 210 | 0.015 | ||

P_{4} | 0 | 0.001 | P_{8} | 230 | 0.025 |

50 | 0.001 | 250 | 0.010 | ||

100 | 0.001 | 270 | 0.910 | ||

150 | 0.002 | 0 | 0.001 | ||

200 | 0.002 | 50 | 0.002 | ||

250 | 0.003 | 100 | 0.002 | ||

280 | 0.010 | 150 | 0.005 | ||

300 | 0.010 | 200 | 0.010 | ||

320 | 0.010 | 250 | 0.015 | ||

350 | 0.960 | 300 | 0.965 |

Process | ${\mathbf{S}}_{\mathbf{i}\mathbf{j}}$ | Following Processes | ${\mathbf{t}}_{\mathbf{i}\mathbf{j}}$ | ${\mathsf{\alpha}}_{\mathbf{i}\mathbf{j}}$ | ${\mathsf{\beta}}_{\mathbf{i}\mathbf{j}}$ |
---|---|---|---|---|---|

P_{1} | 1 | P_{2}, P_{3}, P_{4}, P_{5}, P_{6}, P_{7}, P_{8} | 7 | 1 | 0 |

P_{2} | 2 | P_{3}, P_{4}, P_{5}, P_{6}, P_{7}, P_{8} | 6 | 1 | 0 |

P_{3} | 3 | P_{4}, P_{5}, P_{6}, P_{7}, P_{8} | 5 | 0 | 1 |

P_{4} | 4 | P_{5}, P_{6}, P_{7}, P_{8} | 4 | 0 | 1 |

P_{5} | 5 | P_{6}, P_{7}, P_{8} | 3 | 0 | 1 |

P_{6} | 6 | P_{7}, P_{8} | 2 | 0 | 1 |

P_{7} | 7 | P_{8} | 1 | 0 | 1 |

P_{8} | 8 | - | 0 | 0 | 1 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|

${Q}_{i1}(G)$ | 302.874 | 287.73 | 273.344 | 259.677 | 246.693 | 234.358 | 222.64 | 211.508 |

${Q}_{i1}(R)$ | 0 | 0 | 12.9838 | 12.3346 | 11.7179 | 11.132 | 10.5754 | 10.0466 |

${l}_{i1}$ | 302.874 | 287.73 | 286.328 | 272.011 | 258.411 | 245.49 | 233.216 | 221.555 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|

${Q}_{i2}(G)$ | 216.339 | 205.522 | 195.246 | 185.483 | 176.209 | 167.399 | 159.029 | 151.077 |

${Q}_{i2}(R)$ | 0 | 0 | 9.2747 | 8.81046 | 8.36994 | 7.95144 | 7.55387 | 7.17618 |

${l}_{i2}$ | 216.339 | 205.522 | 204.52 | 194.294 | 184.579 | 175.35 | 166.583 | 158.254 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|

${Q}_{i1}(G)$ | 259.606 | 246.626 | 234.295 | 222.58 | 211.451 | 200.878 | 190.835 | 181.293 |

${Q}_{i1}(R)$ | 0 | 0 | 11.129 | 10.5726 | 10.0439 | 9.54173 | 9.06464 | 8.61141 |

${l}_{i1}$ | 259.606 | 246.626 | 245.424 | 233.153 | 221.495 | 210.42 | 199.899 | 189.904 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|

${Q}_{i2}(G)$ | 259.606 | 246.626 | 234.295 | 222.58 | 211.451 | 200.878 | 190.835 | 181.293 |

${Q}_{i2}(R)$ | 0 | 0 | 11.129 | 10.5726 | 10.0439 | 9.54173 | 9.06464 | 8.61141 |

${l}_{i2}$ | 259.606 | 246.626 | 245.424 | 233.153 | 221.495 | 210.42 | 199.899 | 189.904 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|

${Q}_{i1}(G)$ | 216.339 | 205.522 | 195.246 | 185.483 | 176.209 | 167.399 | 159.029 | 151.077 |

${Q}_{i1}(R)$ | 0 | 0 | 9.2747 | 8.81046 | 8.36994 | 7.95144 | 7.55387 | 7.17618 |

${l}_{i1}$ | 216.339 | 205.522 | 204.52 | 194.294 | 184.579 | 175.35 | 166.583 | 158.254 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|

${Q}_{i2}(G)$ | 302.874 | 287.73 | 273.344 | 259.677 | 246.693 | 234.358 | 222.64 | 211.508 |

${Q}_{i2}(R)$ | 0 | 0 | 12.9838 | 12.3346 | 11.7179 | 11.132 | 10..5754 | 10.0466 |

${l}_{i2}$ | 302.874 | 287.73 | 286.328 | 272.011 | 258.411 | 245.49 | 233.216 | 221.555 |

Position | Path | D | Input loading | Minimum capacity vector | Reliability |
---|---|---|---|---|---|

Position1 | Rework 1 | (210,150) | 302.874 | [350,300,300,280,260,260,250,250, 250,250,250,200,200,200,170,200] | 0.68757846320055 |

216.339 | |||||

(180,180) | 259.606 | [300,250,250,250,240,220,210,200, 300,250,250,250,240,220,210,200] | |||

259.606 | |||||

(150,210) | 216.339 | [250,250,250,200,200,200,170,200, 350,300,300,280,260,260,250,250] | |||

302.874 | |||||

Rework 2 | (210,150) | 302.187 | [350,300,300,280,260,260,250,250, 250,250,200,200,200,200,170,200] | 0.69951463495432 | |

215.848 | |||||

(180,180) | 259.018 | [300,250,250,250,240,220,210,200, 300,250,250,250,240,220,210,200] | |||

259.018 | |||||

(150,210) | 215.848 | [250,250,200,200,200,200,170,200, 350,300,300,280,260,260,250,250] | |||

302.187 | |||||

Rework 3 | (210,150) | 301.468 | [350,300,300,280,260,260,250,250, 250,250,200,200,200,200,170,200] | 0.69951463495432 | |

215.334 | |||||

(180,180) | 258.401 | [300,250,250,250,240,220,210,200, 300,250,250,250,240,220,210,200] | |||

258.401 | |||||

(150,210) | 215.334 | [250,250,200,200,200,200,170,200, 350,300,300,280,260,260,250,250] | |||

301.468 | |||||

Position2 | Rework 1 | (210,150) | 303.529 | [350,300,300,280,260,260,250,250, 250,250,250,200,200,200,170,200] | 0.68757846320055 |

216.807 | |||||

(180,180) | 260.168 | [300,250,250,250,240,220,210,200, 300,250,250,250,240,220,210,200] | |||

260.168 | |||||

(150,210) | 216.807 | [250,250,250,200,200,200,170,200, 350,300,300,280,260,260,250,250] | |||

303.529 | |||||

Rework 2 | (210,150) | 302.187 | [350,300,200,280,260,260,250,250, 250,250,200,200,200,200,170,200] | 0.71474526994618 | |

215.848 | |||||

(180,180) | 259.018 | [300,250,250,250,240,220,210,200, 300,250,250,250,240,220,210,200] | |||

259.018 | |||||

(150,210) | 215.848 | [250,250,200,200,200,200,170,200, 350,300,200,280,260,260,250,250] | |||

302.187 |

Path | Reliability | Rank | |
---|---|---|---|

Position 1 | Rework 1 | 0.687578 | 4 |

Rework 2 | 0.699514 | 3 | |

Rework 3 | 0.699514 | 2 | |

Position 2 | Rework 1 | 0.687578 | 5 |

Rework 2 | 0.714745 | 1 |

Position | Rework path | $\Delta \mathit{H}$ |
---|---|---|

1 | 1 | 1.099 |

2 | 1.095 | |

3 | 1.090 | |

2 | 1 | 1.103 |

2 | 1.095 |

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## Share and Cite

**MDPI and ACS Style**

Wu, M.; Dai, W.; Lu, Z.; Zhao, Y.; Wang, M.
The Method for Risk Evaluation in Assembly Process based on the Discrete-Time SIRS Epidemic Model and Information Entropy. *Entropy* **2019**, *21*, 1029.
https://doi.org/10.3390/e21111029

**AMA Style**

Wu M, Dai W, Lu Z, Zhao Y, Wang M.
The Method for Risk Evaluation in Assembly Process based on the Discrete-Time SIRS Epidemic Model and Information Entropy. *Entropy*. 2019; 21(11):1029.
https://doi.org/10.3390/e21111029

**Chicago/Turabian Style**

Wu, Mengyao, Wei Dai, Zhiyuan Lu, Yu Zhao, and Meiqing Wang.
2019. "The Method for Risk Evaluation in Assembly Process based on the Discrete-Time SIRS Epidemic Model and Information Entropy" *Entropy* 21, no. 11: 1029.
https://doi.org/10.3390/e21111029