# A Fuzzy Bivariate Poisson Control Chart

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fuzzy c Control Charts

## 3. The Proposed Chart: Fuzzy Bivariate Poisson (FBP) Control Chart

#### 3.1. Fuzzy Estimation of Parameters

**Proposition**

**1.**

**A Corollary of Proposition 1:**

#### 3.2. Design of the Fuzzy Bivariate Poisson Control Chart

#### 3.3. Average Run Length of the FBP Chart

_{0}) and an out-of-control ARL (ARL

_{1}). Simulations are carried out for several scenarios (e.g., several values of ${\alpha}^{cut}$ (0.1 to 0.9 with an increment of 0.1) and the correlation coefficients $\left(\rho \right)$ equal to 0, 0.5 and 0.8. The algorithm for ARL calculation is defined as follows:

- Step 1. Generate data $\left({X}_{1},{X}_{2}\right)$ that follows a bivariate Poisson distribution with $m=1000$ observations for each combination value of $\rho ,\text{}{\alpha}^{cut}$ as well as parameters $\left({\lambda}_{1},{\lambda}_{2}\right).$
- Step 2. Calculate the value of ${D}_{i}={X}_{1i}+{X}_{2i}$.
- Step 3. Estimate the fuzzy parameter $\tilde{\mu}=\left({\widehat{d}}_{left},\widehat{d},{\widehat{d}}_{right}\right)$ by using Proposition 1.
- Step 4. Transform each observation into $\left({d}_{jmin},{d}_{j},{d}_{jmax}.\right)$.
- Step 5. Calculate the statistics in Equation (13):
- Step 6. Calculate the control limit in Equation (14) and Equation (15).
- Step 7. Evaluate the FBP chart using the ARL, as follows:
- For each combination value of ${\alpha}^{cut}$, $\rho $ and parameter $\left({\lambda}_{1},{\lambda}_{2}\right),$ generate $m=1000$ observations that follow a bivariate Poisson distribution. Repeating and comparing statistics ${L}_{j}\left({\alpha}^{cut}\right)$ and ${L}_{R}\left({\alpha}^{cut}\right)$ with control limits obtained from step 6. If the statistics are out-of-control the first time, then the value Run Length (RL) is obtained.
- Repeat step 7(a) 1000 times and calculate the average run length for the in-control condition (ARL
_{0}). - Repeat steps 7(a) and (b) for the level shift of parameters ${\lambda}_{1}^{*}={\lambda}_{1}+\mathsf{\Delta}$ and ${\lambda}_{2}^{*}={\lambda}_{2}+\mathsf{\Delta}$, with an increment of $\mathsf{\Delta}=1$ to obtain the average run length for the out-of-control condition (ARL
_{1}). - Plot the ARL
_{0}and ARL_{1}.

_{0}) and an out-of-control ARL (ARL

_{1}). The proposed chart is simulated from a process with bivariate Poisson parameters ${\lambda}_{1}$ = 1, ${\lambda}_{2}=2$ based on several schemes for $\rho $ and ${\alpha}^{cut}$. The first scheme is selected for two independent Poisson distributions $\left(\rho =0\right)$ and the second and third schemes are selected with $\rho $ as 0.5 and 0.8. The overall scheme is simulated with ${\alpha}^{cut}$ between $0.1$ and $0.9$ with an increment of 0.1. To obtain the value of ARL

_{1}, the parameters are shifted by Δ = 1. The result shows in Table 2 and Figure 2, Figure 3 and Figure 4.

_{0}is the expected number of samples until finding the first out-of-control sample, given that the process is in control. By applying ${\alpha}^{cut}$ from 0.1 to 0.8, an ARL

_{0}of around 500 is obtained, and the value of ARL

_{0}is lower than 300 for ${\alpha}^{cut}$ equal to 0.9. In the other hand, for the highest value of ${\alpha}^{cut}=0.9,$ the value of ARL

_{0}is 220, meaning that the false alarm rate is high. With higher values of ${\alpha}^{cut}$, the value of ARL

_{0}decreases.

_{1}criterion is used to determine the sensitivity of the proposed control chart in order to detect a shift in the process. According to the first block in Table 2 and Figure 2, the trend in ARL decreases sharply for the proposed chart. The significant digits of the ARL value are varied from hundred to unit, and these indicate that the proposed chart is sensitive to detect the shifted of the parameter process. The smaller the ARL

_{1}value, the faster the proposed chart to find an out-of-control signal. Among these results, the higher the value of ${\alpha}^{cut}$, the smaller the value of ARL

_{1}. Generally, by using ${\alpha}^{cut}$ values of 0.7, 0.8 and 0.9, the proposed chart shows a faster drop than the other charts. Since the simulation study uses a significance level equal to 0.0027, the theoretical value of ARL

_{0}is around 370. Therefore, among the values of ${\alpha}^{cut}$, the values of ARL

_{0}are higher than the theoretical values.

^{th}observation (ARL

_{0}). Meanwhile, when the value of ${\alpha}^{cut}$ is equal to 0.8, the ARL

_{0}value is smaller than 500. Looking at ARL

_{1}, when the parameters ${\lambda}_{1}=1$, ${\lambda}_{2}=2$ are shifted by $\Delta $, the ARLs of the proposed chart decrease sharply. Among the overall ARLs shown in Figure 2, the greater the value ${\alpha}^{cut}$, the quicker the proposed control charts detect the shifting process. In the second scenario, despite the ability to discover values that become out-of-control more quickly, this procedure produces a low ARL

_{0}, which means that an ${\alpha}^{cut}$ equal to 0.9 has a high false alarm rate.

_{0}is around 500 for an ${\alpha}^{cut}$ equal 0.1 to 0.5. In this scheme, the proposed control chart with an ${\alpha}^{cut}$ value equal to 0.8 has a value of ARL

_{0}closer to 350. However, with an ${\alpha}^{cut}$ smaller than 0.6, the chart has a higher value of ARL

_{0}(closer to 500). However, when ${\alpha}^{cut}$ is equal to 0.9, the value of ARL

_{0}is equal to 206, which indicates a high false alarm rate. Briefly, based on Figure 4, the ARL for the proposed chart drops sharply. For the fifth level shift of the parameters, the value of ARL

_{1}is nearly 135. The proposed chart using an ${\alpha}^{cut}$ equal to 0.8 gives better performance than the other charts. In general, based on the results of the three schemes, the proposed chart provides the best performance when the correlation coefficient of the quality characteristics is higher.

## 4. Comparative Study

- Step 1. Generate data $\left({X}_{1},{X}_{2}\right)$ that follow a bivariate Poisson distribution with $m=1000$ observations, for each combination value of the correlation coefficients $\rho =0\text{},\text{}\rho =0.5\text{}$ and $\rho =0.8$ and parameters $\left({\lambda}_{1},{\lambda}_{2}\right).$
- Step 2. Calculate the statistics: ${D}_{i}={X}_{1i}+{X}_{2i}.$
- Step 3. Calculate the control limits based on the following equations:$$\begin{array}{ll}P\left(D>UCL\right)& ={\displaystyle {\displaystyle \sum}_{d=UCL}^{\infty}}exp\left\{-\left[{\displaystyle {\displaystyle \sum}_{j=1}^{J}}{\lambda}_{j}-\theta \right]\right\}{\displaystyle {\displaystyle \sum}_{i=0}^{\lfloor d/2\rfloor}}\frac{{\left({{\displaystyle \sum}}_{j=1}^{J}{\lambda}_{j}-2\theta \right)}^{d-2i}{\theta}^{i}}{\left(d-2i\right)!i!}\le \frac{\alpha}{2}\\ & =1-{\displaystyle {\displaystyle \sum}_{d=0}^{UCL-1}}exp\left\{-\left[{\displaystyle {\displaystyle \sum}_{j=1}^{J}}{\lambda}_{j}-\theta \right]\right\}{\displaystyle {\displaystyle \sum}_{i=0}^{\lfloor d/2\rfloor}}\frac{{\left({{\displaystyle \sum}}_{j=1}^{J}{\lambda}_{j}-2{\theta}_{o}\right)}^{d-2i}{\theta}^{i}}{\left(d-2i\right)!i!}\le \frac{\alpha}{2},\end{array}$$$$P\left(D<LCL\right)={\displaystyle \sum}_{d=0}^{LCL}exp\left\{-\left[{\displaystyle \sum}_{j=1}^{J}{\lambda}_{j}-\theta \right]\right\}{\displaystyle \sum}_{i=0}^{\lfloor d/2\rfloor}\frac{{\left({{\displaystyle \sum}}_{j=1}^{J}{\lambda}_{j}-2\theta \right)}^{d-2i}{\theta}^{i}}{\left(d-2i\right)!i!}\le \frac{\alpha}{2},$$
- Step 4. Evaluation of the BP chart using ARL as follows:
- For each combination value of $\rho $ and parameter $\left({\lambda}_{1},{\lambda}_{2}\right),$ generate $m=1000$ observations that follow a bivariate Poisson distribution. Repeat and compare the statistics of ${D}_{i}$ with the control limits obtained from step 3. If the statistics are out-of-control the first time, then the value for Run Length (RL) is obtained.
- Repeat step 4a 1000 times and calculate the average run length for the in-control condition (ARL
_{0}). - Repeat steps 4(a) and (b) for the level shift of parameters ${\lambda}_{1}^{*}={\lambda}_{1}+\mathsf{\Delta}$ and ${\lambda}_{2}^{*}={\lambda}_{2}+\mathsf{\Delta}$, with an increment of $\Delta =1,$ to obtain the average run length for an out-of-control condition (ARL
_{1}). - Plot the ARL
_{0}and ARL_{1}.

_{0}of the conventional chart is close to the theoretical ARL

_{0}for $\rho =0.5$. Furthermore, the performance of the proposed chart with ${\alpha}^{cut}$ equal to 0.8 is compared to the conventional BP chart, with $\rho =0.5.$ The comparison shows that the proposed chart is more sensitive to detecting shifts than the conventional BP chart.

#### Application FBP Control Charts

## 5. Conclusions

_{0}value of about 500. For an ${\alpha}^{cut}$ value between 0.5 to 0.8, in general, the proposed chart has an ARL

_{0}value above 370 and less than 500. The best conditions for the FBP chart are found for a condition when the $\rho =0.8$ and ${\alpha}^{cut}=0.8.$

_{0}(no shifting parameter). But, for the shifting of parameter process, the BP chart is slower to detect the shift compared to the proposed charts. In the real application, the FBP chart is more sensitive than the BP chart. For further research, the control limits of the FBP control charts could be modelled using intermediate criteria, such as “in-control”, “rather in-control”, “rather out-of-control” and “out-of-control”. Also, the proposed chart could be extended to a fuzzy multivariate Poisson control chart.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

ARL | Average run length |

FBP | Fuzzy bivariate Poisson |

BP | Bivariate Poisson |

$C{L}^{L}$ | Center line for left control chart |

$LC{L}^{L}$ | Lower control limit for left control chart |

$UC{L}^{L}$ | Upper control limit for left control chart |

$C{L}^{R}$ | Center line for right control chart |

$LC{L}^{R}$ | Lower control limit for right control chart |

$UC{L}^{R}$ | Upper control limit for right control chart |

$LCL$ | Lower control limit |

$UCL$ | Upper control limit |

$F(.)\text{}$ | Distribution function |

## Appendix A

**Proof of**

**Proposition 1.**

## Appendix B

**Proof for**

**A Corollary of Proposition 1.**

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**Figure 6.**FBP chart for monitoring the bottle production data: (

**a**) FBP left side chart, (

**b**) FBP right side chart.

$\mathit{j}$ | ${\mathit{c}}_{\mathit{j}\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{c}}_{\mathit{j}}$ | ${\mathit{c}}_{\mathit{j}\mathit{m}\mathit{a}\mathit{x}}$ |
---|---|---|---|

1 | ${c}_{1min}$ | ${c}_{1}$ | ${c}_{1max}$ |

2 | ${c}_{2min}$ | ${c}_{2}$ | ${c}_{2max}$ |

3 | ${c}_{3min}$ | ${c}_{3}$ | ${c}_{3max}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

j | ${c}_{jmin}$ | ${c}_{j}$ | ${c}_{jmax}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

m | ${c}_{mmin}$ | ${c}_{m}$ | ${c}_{mmax}$ |

$\mathit{\rho}$ | ${\mathit{\lambda}}_{1}^{*}$ | ${\mathit{\lambda}}_{2}^{*}$ | $\Delta $ | ${\mathit{\alpha}}^{\mathit{c}\mathit{u}\mathit{t}}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | ||||

0 | 1 | 2 | 0 | 500.5000 | 500.5000 | 500.5000 | 500.3762 | 499.6700 | 499.2309 | 494.9235 | 475.8822 | 220.7956 |

2 | 3 | 1 | 499.2822 | 498.5563 | 494.5448 | 445.4966 | 340.1151 | 341.7502 | 180.4167 | 73.7003 | 15.3764 | |

3 | 4 | 2 | 420.4424 | 427.5666 | 312.8545 | 82.0889 | 36.2809 | 34.4991 | 19.3954 | 9.9167 | 3.5195 | |

4 | 5 | 3 | 90.3181 | 80.7588 | 46.8710 | 14.0316 | 8.0395 | 7.7880 | 5.1136 | 3.6686 | 1.8113 | |

5 | 6 | 4 | 17.0719 | 18.5737 | 10.4823 | 4.2816 | 3.3956 | 3.1855 | 2.3477 | 1.8376 | 1.2132 | |

6 | 7 | 5 | 6.1447 | 5.8631 | 4.2326 | 2.4632 | 1.9245 | 1.7552 | 1.5059 | 1.3105 | 1.0865 | |

7 | 8 | 6 | 2.9205 | 2.9935 | 2.2925 | 1.6491 | 1.2869 | 1.3602 | 1.2232 | 1.0714 | 1.0769 | |

8 | 9 | 7 | 1.7298 | 1.9136 | 1.6197 | 1.2092 | 1.1747 | 1.1357 | 1.0686 | 1.0227 | 1.0000 | |

9 | 10 | 8 | 1.3452 | 1.4130 | 1.4331 | 1.1140 | 1.0606 | 1.0706 | 1.0000 | 1.0000 | 1.0000 | |

10 | 11 | 9 | 1.2727 | 1.2072 | 1.0826 | 1.1304 | 1.0435 | 1.0833 | 1.0000 | 1.0000 | 1.0000 | |

11 | 12 | 10 | 1.0976 | 1.0632 | 1.0270 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

12 | 13 | 11 | 1.0444 | 1.0303 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

13 | 14 | 12 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

0.5 | 1 | 2 | 0 | 500.5000 | 500.4438 | 500.2137 | 500.2686 | 491.5374 | 491.5374 | 497.9363 | 440.4627 | 339.3639 |

2 | 3 | 1 | 496.3325 | 487.9893 | 470.2235 | 488.5853 | 239.3342 | 239.3342 | 352.3361 | 56.3104 | 29.7720 | |

3 | 4 | 2 | 393.0880 | 301.4079 | 178.3637 | 294.5800 | 32.8209 | 32.8209 | 54.1805 | 11.6062 | 7.4959 | |

4 | 5 | 3 | 95.5691 | 55.1628 | 34.1244 | 57.5133 | 8.6225 | 8.6225 | 14.4826 | 4.2045 | 3.2586 | |

5 | 6 | 4 | 24.3012 | 15.9264 | 12.0475 | 15.6603 | 3.7615 | 3.7615 | 4.9493 | 2.4232 | 1.9244 | |

6 | 7 | 5 | 8.9751 | 6.5880 | 4.7337 | 6.3533 | 2.4211 | 2.4211 | 2.8989 | 1.6003 | 1.3769 | |

7 | 8 | 6 | 4.4603 | 3.4896 | 2.7533 | 3.1933 | 1.6544 | 1.6544 | 1.9817 | 1.3096 | 1.2010 | |

8 | 9 | 7 | 2.8164 | 2.2916 | 1.9074 | 2.3437 | 1.2638 | 1.2638 | 1.5943 | 1.1623 | 1.0909 | |

9 | 10 | 8 | 1.8460 | 1.6450 | 1.4289 | 1.6855 | 1.2512 | 1.2512 | 1.2698 | 1.0667 | 1.0714 | |

10 | 11 | 9 | 1.5119 | 1.3694 | 1.2569 | 1.4331 | 1.1096 | 1.1096 | 1.1500 | 1.0600 | 1.0400 | |

11 | 12 | 10 | 1.2474 | 1.2258 | 1.1973 | 1.2424 | 1.0233 | 1.0233 | 1.1600 | 1.0435 | 1.0000 | |

12 | 13 | 11 | 1.1356 | 1.1622 | 1.0735 | 1.0784 | 1.0000 | 1.0000 | 1.0789 | 1.0000 | 1.0000 | |

13 | 14 | 12 | 1.0500 | 1.0274 | 1.0600 | 1.1250 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

14 | 15 | 13 | 1.0238 | 1.0588 | 1.0000 | 1.0345 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

0.8 | 1 | 2 | 0 | 500.1596 | 500.1514 | 500.2378 | 499.6190 | 496.4722 | 473.6515 | 434.2774 | 354.6419 | 206.9041 |

2 | 3 | 1 | 484.9163 | 484.0479 | 459.7577 | 422.8506 | 337.2877 | 136.9043 | 68.0762 | 38.9847 | 22.0416 | |

3 | 4 | 2 | 290.3477 | 282.2958 | 186.3935 | 102.5621 | 58.8145 | 22.6175 | 14.6593 | 9.4666 | 6.4562 | |

4 | 5 | 3 | 59.2883 | 63.3660 | 37.6905 | 24.0107 | 15.9634 | 7.6190 | 5.4739 | 3.8403 | 3.0144 | |

5 | 6 | 4 | 19.1030 | 17.6349 | 11.9913 | 9.4512 | 6.7731 | 3.7473 | 2.8958 | 2.2732 | 1.9483 | |

6 | 7 | 5 | 6.9379 | 7.8796 | 5.4919 | 4.2981 | 3.7087 | 2.2773 | 1.9121 | 1.5686 | 1.3836 | |

7 | 8 | 6 | 3.9895 | 4.2529 | 3.2495 | 2.7780 | 2.2431 | 1.6430 | 1.4298 | 1.3440 | 1.3223 | |

8 | 9 | 7 | 2.6617 | 2.6448 | 2.2773 | 1.9415 | 1.6178 | 1.3566 | 1.2623 | 1.1795 | 1.1327 | |

9 | 10 | 8 | 1.9186 | 1.8547 | 1.7790 | 1.5094 | 1.3815 | 1.1759 | 1.1348 | 1.0598 | 1.1129 | |

10 | 11 | 9 | 1.6305 | 1.6224 | 1.3197 | 1.3099 | 1.1855 | 1.1085 | 1.0625 | 1.0448 | 1.0345 | |

11 | 12 | 10 | 1.3713 | 1.3718 | 1.2817 | 1.1304 | 1.1216 | 1.0694 | 1.0250 | 1.0455 | 1.0000 | |

12 | 13 | 11 | 1.1865 | 1.2045 | 1.1260 | 1.0777 | 1.1026 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

13 | 14 | 12 | 1.0847 | 1.1322 | 1.0667 | 1.0725 | 1.1111 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

14 | 15 | 13 | 1.1127 | 1.0200 | 1.0435 | 1.0476 | 1.0400 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

${\mathit{\lambda}}_{1}^{*}$ | ${\mathit{\lambda}}_{2}^{*}$ | $\Delta $ | $\mathit{\rho}$ | ||
---|---|---|---|---|---|

0 | 0.5 | 0.8 | |||

1 | 2 | 0 | 761.3600 | 381.5000 | 586.7870 |

2 | 3 | 1 | 487.9755 | 274.1520 | 370.0240 |

3 | 4 | 2 | 338.6280 | 197.5360 | 261.0067 |

4 | 5 | 3 | 256.9445 | 152.0770 | 199.9752 |

5 | 6 | 4 | 206.5924 | 123.0724 | 161.6114 |

6 | 7 | 5 | 172.6343 | 103.2250 | 135.4560 |

7 | 8 | 6 | 148.2440 | 88.8520 | 116.5456 |

8 | 9 | 7 | 129.8954 | 77.9851 | 102.2551 |

9 | 10 | 8 | 115.5977 | 69.4892 | 91.0911 |

10 | 11 | 9 | 104.1505 | 62.6713 | 82.1280 |

11 | 12 | 10 | 94.7784 | 57.0829 | 74.7811 |

12 | 13 | 11 | 86.9658 | 52.4195 | 68.6478 |

13 | 14 | 12 | 80.3536 | 48.4702 | 63.4546 |

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**MDPI and ACS Style**

Wibawati; Mashuri, M.; Purhadi; Irhamah.
A Fuzzy Bivariate Poisson Control Chart. *Symmetry* **2020**, *12*, 573.
https://doi.org/10.3390/sym12040573

**AMA Style**

Wibawati, Mashuri M, Purhadi, Irhamah.
A Fuzzy Bivariate Poisson Control Chart. *Symmetry*. 2020; 12(4):573.
https://doi.org/10.3390/sym12040573

**Chicago/Turabian Style**

Wibawati, Muhammad Mashuri, Purhadi, and Irhamah.
2020. "A Fuzzy Bivariate Poisson Control Chart" *Symmetry* 12, no. 4: 573.
https://doi.org/10.3390/sym12040573