**4. The proposed control charts enhanced with runs-type rules**

In this section, we carry out an extensive numerical experimentation to appraise the ability of the distribution-free monitoring schemes *Chart* 1, *Chart* 2, and *Chart* 3


#### **Table 4.**

*Comparison of the* ARLout*s with the same* ARLin *for* Chart *1.*

enhanced with runs-type rules for detecting possible shifts of the underlying distribution. The computations have been made by the aid of the simulation procedure presented in Section 3. **Tables 4** and **5** display the improved out-of-control performance of *Chart* 1, when the 2-of-3 runs-type rule is activated. We first compare the performance of the control charts by using a common *ARLin* and then evaluating the respective *ARLout* for specific shifts. Consider the case of a process with underlying in-control exponential distribution with mean equal to 2 and out-of-control exponential distribution with mean equal to 1. In **Table 4**, we present the *ARLout* values of *Chart* 1 and the proposed *Chart* 1 enhanced with 2-of-3 runs-type rule, for *ARLin* ¼ 370, 500, *m* ¼ 100, 500, and *n* ¼ 5, 11. The remaining design parameters *a*, *b*, *j*,*r*, were determined appropriately, so that *ARLin* takes on a value as close to the nominal level as possible. It is evident that the proposed monitoring scheme performs better than the one established by Balakrishnan et al. [5] for all cases considered. The fact that the *ARLout*s that exhibit *Chart* 1 with 2-of-3 runs-type rule

*Chart* **2** *Chart* **2 with 2-of-2 runs rule**

0.9239

0.9977

0.9661

0.9992

0.9736

0.9998

0.9715

0.9998

0.8627

0.9935

0.9983

0.9983

0.9618

0.9997

0.9594

0.9998

0.8627

0.9908

0.9040

0.9960

0.9479

0.9995

0.9452

0.9997

*ARout* **(***a***,** *b***) (***j***,** *k***)** *r Exact*

(20, 90) (6, 8) 1

*FAR*

0.0135

(14, 35) (4, 6) 1 0.0125 0.5527

(11, 40) (7, 9) 1 0.0110 0.9194

(23, 75) (3, 5) 1 0.0124 0.6771

(122, 350) (3, 5) 1 0.0130 0.7278

(130, 430) (8, 10) 1 0.0089 0.9551

(300, 550) (4, 6) 1 0.0110 0.6234

(27, 800) (8, 10) 1 0.0094 0.9644

(2, 45) (4, 6) 4 0.0056 0.4229

(4, 44) (8, 10) 8 0.0057 0.8633

(2, 83) (3, 5) 3 0.0055 0.7620

(5, 88) (6, 8) 4 0.0054 0.9169

(15, 420) (3, 5) 4 0.0054 0.6611

(54, 380) (8, 10) 5 0.0056 0.9288

(66, 900) (4, 6) 4 0.0047 0.5667

(87, 830) (7, 9) 7 0.0058 0.9646

(2, 49) (4, 6) 4 0.0026 0.4352

(2, 94) (3, 5) 3 0.0025 0.5758

(4, 67) (6, 8) 4 0.0029 0.8722

(12, 440) (3, 5) 4 0.0031 0.6099

(49, 445) (8, 10) 5 0.0033 0.9318

(55, 900) (4, 6) 4 0.0033 0.4673

(77, 800) (7, 9) 7 0.0032 0.9356

(6, 44) (10,

11)

*ARout*

0.9463

0.9989

0.9782

0.9563 0.9992

0.9846

0.9998

0.9715

0.9999

0.9990

0.9979

0.9983

0.9989

0.9799

0.9997

0.9596

0.9998

0.9081

0.9947

0.9670

0.9987

0.9724

0.9995

0.9529

0.9997

15 0.0031 0.7553

*FAR*

*Simulation-Based Comparative Analysis of Nonparametric Control Charts with Runs-Type Rules*

*FAR m n* **(***a***,** *b***) (***j***,** *k***)** *r Exact*

*DOI: http://dx.doi.org/10.5772/intechopen.91040*

**0.01** 50 11 (3, 45) (4, 6) 4 0.0103 0.4575

25 (4, 45) (7, 9) 7 0.0131 0.7756

25 (6, 65) (6, 8) 4 0.0092 0.8181

25 (61, 420) (8, 10) 5 0.0092 0.9023

25 (125, 810) (8, 10) 8 0.0098 0.9125

25 (4, 44) (8, 10) 8 0.0049 0.6577

25 (5, 88) (6, 8) 4 0.0044 0.7508

25 (54, 380) (8, 10) 5 0.0046 0.8624

25 (87, 830) (7, 9) 7 0.0050 0.8832

25 (6, 44) (10, 11) 15 0.0036 0.6179

25 (4, 67) (6, 8) 4 0.0025 0.6595

25 (49, 445) (8, 10) 5 0.0026 0.8250

25 (77, 800) (7, 9) 7 0.0027 0.8435

100 11 (2, 94) (3, 5) 3 0.0028 0.3722

500 11 (12, 440) (3, 5) 4 0.0025 0.4559

1000 11 (55, 900) (4, 6) 4 0.0027 0.4673

*Comparison of the* ARout*s with the same* FAR *for* Chart *2.*

**Table 6.**

**205**

100 11 (2, 83) (3, 5) 3 0.0046 0.7508

500 11 (15, 420) (3, 5) 4 0.0050 0.5234

1000 11 (66, 900) (4, 6) 4 0.0049 0.5357

**0.0027** 50 11 (2, 49) (4, 6) 4 0.0026 0.3249

100 11 (4, 92) (3, 5) 3 0.0126 0.5832

500 11 (19, 405) (3, 5) 4 0.0099 0.5976

1000 11 (80, 894) (4, 6) 5 0.0101 0.6106

**0.005** 50 11 (2, 45) (4, 6) 4 0.0054 0.3250


**Table 5.** *Comparison of the* ARLout*s with the same* ARLin *for* Chart *1 under normal distribution (*θ*,* δ*).*

*Chart* **2** *Chart* **2 with 2-of-2 runs rule** *FAR m n* **(***a***,** *b***) (***j***,** *k***)** *r Exact FAR ARout* **(***a***,** *b***) (***j***,** *k***)** *r Exact FAR ARout* **0.01** 50 11 (3, 45) (4, 6) 4 0.0103 0.4575 0.9239 (14, 35) (4, 6) 1 0.0125 0.5527 0.9463 25 (4, 45) (7, 9) 7 0.0131 0.7756 0.9977 (11, 40) (7, 9) 1 0.0110 0.9194 0.9989 100 11 (4, 92) (3, 5) 3 0.0126 0.5832 0.9661 (23, 75) (3, 5) 1 0.0124 0.6771 0.9782 25 (6, 65) (6, 8) 4 0.0092 0.8181 0.9992 (20, 90) (6, 8) 1 0.0135 0.9563 0.9992 500 11 (19, 405) (3, 5) 4 0.0099 0.5976 0.9736 (122, 350) (3, 5) 1 0.0130 0.7278 0.9846 25 (61, 420) (8, 10) 5 0.0092 0.9023 0.9998 (130, 430) (8, 10) 1 0.0089 0.9551 0.9998 1000 11 (80, 894) (4, 6) 5 0.0101 0.6106 0.9715 (300, 550) (4, 6) 1 0.0110 0.6234 0.9715 25 (125, 810) (8, 10) 8 0.0098 0.9125 0.9998 (27, 800) (8, 10) 1 0.0094 0.9644 0.9999 **0.005** 50 11 (2, 45) (4, 6) 4 0.0054 0.3250 0.8627 (2, 45) (4, 6) 4 0.0056 0.4229 0.9990 25 (4, 44) (8, 10) 8 0.0049 0.6577 0.9935 (4, 44) (8, 10) 8 0.0057 0.8633 0.9979 100 11 (2, 83) (3, 5) 3 0.0046 0.7508 0.9983 (2, 83) (3, 5) 3 0.0055 0.7620 0.9983 25 (5, 88) (6, 8) 4 0.0044 0.7508 0.9983 (5, 88) (6, 8) 4 0.0054 0.9169 0.9989 500 11 (15, 420) (3, 5) 4 0.0050 0.5234 0.9618 (15, 420) (3, 5) 4 0.0054 0.6611 0.9799 25 (54, 380) (8, 10) 5 0.0046 0.8624 0.9997 (54, 380) (8, 10) 5 0.0056 0.9288 0.9997 1000 11 (66, 900) (4, 6) 4 0.0049 0.5357 0.9594 (66, 900) (4, 6) 4 0.0047 0.5667 0.9596 25 (87, 830) (7, 9) 7 0.0050 0.8832 0.9998 (87, 830) (7, 9) 7 0.0058 0.9646 0.9998 **0.0027** 50 11 (2, 49) (4, 6) 4 0.0026 0.3249 0.8627 (2, 49) (4, 6) 4 0.0026 0.4352 0.9081 25 (6, 44) (10, 11) 15 0.0036 0.6179 0.9908 (6, 44) (10, 11) 15 0.0031 0.7553 0.9947 100 11 (2, 94) (3, 5) 3 0.0028 0.3722 0.9040 (2, 94) (3, 5) 3 0.0025 0.5758 0.9670 25 (4, 67) (6, 8) 4 0.0025 0.6595 0.9960 (4, 67) (6, 8) 4 0.0029 0.8722 0.9987 500 11 (12, 440) (3, 5) 4 0.0025 0.4559 0.9479 (12, 440) (3, 5) 4 0.0031 0.6099 0.9724 25 (49, 445) (8, 10) 5 0.0026 0.8250 0.9995 (49, 445) (8, 10) 5 0.0033 0.9318 0.9995 1000 11 (55, 900) (4, 6) 4 0.0027 0.4673 0.9452 (55, 900) (4, 6) 4 0.0033 0.4673 0.9529 25 (77, 800) (7, 9) 7 0.0027 0.8435 0.9997 (77, 800) (7, 9) 7 0.0032 0.9356 0.9997

*Simulation-Based Comparative Analysis of Nonparametric Control Charts with Runs-Type Rules DOI: http://dx.doi.org/10.5772/intechopen.91040*

#### **Table 6.**

*Comparison of the* ARout*s with the same* FAR *for* Chart *2.*

enhanced with runs-type rules for detecting possible shifts of the underlying distribution. The computations have been made by the aid of the simulation procedure presented in Section 3. **Tables 4** and **5** display the improved out-of-control performance of *Chart* 1, when the 2-of-3 runs-type rule is activated. We first compare the performance of the control charts by using a common *ARLin* and then evaluating the respective *ARLout* for specific shifts. Consider the case of a process with underlying in-control exponential distribution with mean equal to 2 and out-of-control exponential distribution with mean equal to 1. In **Table 4**, we present the *ARLout* values of *Chart* 1 and the proposed *Chart* 1 enhanced with 2-of-3 runs-type rule, for *ARLin* ¼ 370, 500, *m* ¼ 100, 500, and *n* ¼ 5, 11. The remaining design parameters *a*, *b*, *j*,*r*, were determined appropriately, so that *ARLin* takes on a value as close to the nominal level as possible. It is evident that the proposed monitoring scheme performs better than the one established by Balakrishnan et al. [5] for all cases considered. The fact that the *ARLout*s that exhibit *Chart* 1 with 2-of-3 runs-type rule

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

*θ δ ARL ARL* 0 1 499.88 517.21 0.25 1 562.33 127.89 0.5 1 386.09 26.63 1 1 61.22 3.83 1.5 1 8.01 1.44 2 1 2.23 1.06 0.25 1.25 49.69 35.07 0.5 1.25 39.96 13.72 1 1.25 14.83 3.85 1.5 1.25 4.76 1.66 2 1.25 2.08 1.14 0.25 1.5 13.91 17.77 0.5 1.5 12.32 9.43 1 1.5 7.07 3.65 1.5 1.5 3.53 1.85 2 1.5 1.98 1.26 0.25 1.75 6.55 10.24 0.5 1.75 6.11 7.64 1 1.75 4.40 3.49 1.5 1.75 2.83 1.92 2 1.75 1.88 1.35 0.25 2 4.04 6.93 0.5 2 3.87 5.49 1 2 3.17 3.09 1.5 2 2.37 2.03 2 2 1.77 1.45

*Comparison of the* ARLout*s with the same* ARLin *for* Chart *1 under normal distribution (*θ*,* δ*).*

**Table 5.**

**204**

*Chart* **1** *Chart* **1 with 2-of-3 runs rule**

are smaller than the respective ones of *Chart* 1 indicates its efficacy to detect faster the shift of the process from the in-control distribution.

Underlying distributions: exponential with mean equal to 2 (in-control) and 1 (out-of-control), respectively.

It goes without saying that the nonparametric control charts are robust in the sense that their in-control behavior remains the same for all continuous distributions. However, it is of some interest to check over their out-of-control performance for different underlying distributions. We next study the performance of the proposed *Chart* 1 enhanced with the 2-of-3 runs-type rule under normal distribution (*θ*, *δ*).

More specifically, the in-control reference sample is drawn from the standard normal distribution, while several combinations of parameters *θ*, *δ* have been examined. **Table 5** reveals that the proposed *Chart* 1 with 2-of-3 runs-type rule is superior compared to the existing *Chart* 1 for almost all shifts of the location parameter *θ* and the scale parameter *δ* considered.

We next study the out-of-control performance of *Chart* 2 presented in Section 2. In **Table 6**, three different *FAR* levels and several values of the parameters *m*, *n* have been considered. For each choice, the *AR* values under two specific Lehmann alternatives corresponding to *γ* ¼ 0*:*4 and *γ* ¼ 0*:*2 are computed via simulation for both *Chart* 2 and *Chart* 2 enhanced with 2-of-2 runs-type rule.

**Table 6** clearly indicates that, under a common *FAR*, the proposed *Chart* 2 with the 2-of-2 runs-type rule performs better than *Chart* 2, with respect to *AR* values, in all cases considered. For example, calling for a reference sample of size *m* ¼ 500, test samples of size *n* ¼ 11, and nominal *FAR* ¼ 0*:*0027, the proposed *Chart* 2 with the 2-of-2 runs-type rule achieves alarm rate of 0.6099 (0.9724) for *γ* ¼ 0*:*4 (*γ* ¼ 0*:*2), while the respective alarm rate for *Chart* 2 is 0.4559 (0.9479).

**Tables 7** and **8** shed more light on the out-of-control performance of the proposed *Chart* 2 enhanced with appropriate runs-type rules. More specifically, the schemes being under consideration are designed such as a nominal in-control *ARL* performance is attained. From the numerical comparisons carried out, it is straightforward that *Chart* 2 with 2-of-2 rule becomes substantially more efficient than *Chart* 2. Under Lehmann alternative with parameter *γ* = 0.5, the proposed chart exhibits smaller out-of-control *ARL* than *Chart* 2, and therefore it seems more capable in detecting possible shift of the process distribution.


In addition, **Table 8** depicts the out-of-control *ARL* performance of *Chart* 2 under normal distribution. More precisely, several shifts of both location and scale parameter have been considered, and *Chart* 2 with 2-of-3 rule detects the underly-

*Comparison of the* ARLout*s with the same* ARLin *for* Chart *2 under normal distribution (*θ*,* δ*).*

*θ δ Chart* **2** *Chart* **2 with 2-of-3 runs rule**

*Simulation-Based Comparative Analysis of Nonparametric Control Charts with Runs-Type Rules*

0 1 446.6 502.1 0.25 1 163.9 127.9 0.5 1 51.64 26.6 1 1 7.4 3.8 1.5 1 2.1 1.4 2 1 1.2 1.1 0.25 1.25 35.7 35.1 0.5 1.25 17.9 13.7 1 1.25 5.0 3.9 1.5 1.25 2.1 1.7 2 1.25 1.3 1.1 0.25 1.5 15.0 17.7 0.5 1.5 9.8 9.4 1 1.5 4.1 3.6 1.5 1.5 2.1 1.8 2 1.5 1.4 1.3 0.25 1.75 8.5 10.6 0.5 1.75 6.5 7.6 1 1.75 3.5 3.4 1.5 1.75 2.1 1.9 2 1.75 1.5 1.3 0.25 2 5.7 6.9 0.5 2 4.8 5.4 1 2 3.1 3.1 1.5 2 2.0 2.0 2 2 1.5 1.4

*DOI: http://dx.doi.org/10.5772/intechopen.91040*

Finally, **Tables 9** and **10** present simulation-based comparisons of the nonparametric monitoring scheme *Chart* 3 with 2-of-2 runs rule versus *Chart* 3 established by Triantafyllou [7]. For delivering the numerical results displayed in **Tables 9** and **10**, the Lehmann alternatives have been considered as the out-of-control distribution. Each cell contains the *AR*s attained for *γ* = 0.5 (upper entry) and *γ* = 0.2 (lower entry).

In the present chapter, we investigate the in- and out-of-control performance of distribution-free control charts based on order statistics. Several runs-type rules are

ing shift sooner than *Chart* 2 does in almost all cases examined.

**5. Conclusions**

**207**

**Table 8.**

**Table 7.**

*Comparison of the* ARLout*s with the same* ARLin *for* Chart *2.*


*Simulation-Based Comparative Analysis of Nonparametric Control Charts with Runs-Type Rules DOI: http://dx.doi.org/10.5772/intechopen.91040*

#### **Table 8.**

are smaller than the respective ones of *Chart* 1 indicates its efficacy to detect faster

Underlying distributions: exponential with mean equal to 2 (in-control) and 1

It goes without saying that the nonparametric control charts are robust in the sense that their in-control behavior remains the same for all continuous distributions. However, it is of some interest to check over their out-of-control performance for different underlying distributions. We next study the performance of the proposed *Chart* 1 enhanced with the 2-of-3 runs-type rule under normal distribu-

More specifically, the in-control reference sample is drawn from the standard normal distribution, while several combinations of parameters *θ*, *δ* have been examined. **Table 5** reveals that the proposed *Chart* 1 with 2-of-3 runs-type rule is superior compared to the existing *Chart* 1 for almost all shifts of the location

We next study the out-of-control performance of *Chart* 2 presented in Section 2. In **Table 6**, three different *FAR* levels and several values of the parameters *m*, *n* have been considered. For each choice, the *AR* values under two specific Lehmann alternatives corresponding to *γ* ¼ 0*:*4 and *γ* ¼ 0*:*2 are computed via simulation for

**Table 6** clearly indicates that, under a common *FAR*, the proposed *Chart* 2 with the 2-of-2 runs-type rule performs better than *Chart* 2, with respect to *AR* values, in all cases considered. For example, calling for a reference sample of size *m* ¼ 500, test samples of size *n* ¼ 11, and nominal *FAR* ¼ 0*:*0027, the proposed *Chart* 2 with the 2-of-2 runs-type rule achieves alarm rate of 0.6099 (0.9724) for *γ* ¼ 0*:*4 (*γ* ¼ 0*:*2), while the respective alarm rate for *Chart* 2 is 0.4559 (0.9479).

**Tables 7** and **8** shed more light on the out-of-control performance of the proposed *Chart* 2 enhanced with appropriate runs-type rules. More specifically, the schemes being under consideration are designed such as a nominal in-control *ARL* performance is attained. From the numerical comparisons carried out, it is straightforward that *Chart* 2 with 2-of-2 rule becomes substantially more efficient than *Chart* 2. Under Lehmann alternative with parameter *γ* = 0.5, the proposed chart exhibits smaller out-of-control *ARL* than *Chart* 2, and therefore it seems more

*Chart* **2** *Chart* **2 with 2-of-2 runs rule**

400)

*ARLout* **(***a***,** *b***) (***j***,** *k***)** *r Exact*

4)

*ARLin*

(2, 3) 1 122.5 2.2

1 83.2 2.7

*ARLout*

the shift of the process from the in-control distribution.

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

parameter *θ* and the scale parameter *δ* considered.

both *Chart* 2 and *Chart* 2 enhanced with 2-of-2 runs-type rule.

capable in detecting possible shift of the process distribution.

*ARLin*

11 (5, 78) (3, 4) 3 88.9 2.8 (10, 83) (2,

11 (20, 410) (4, 6) 4 119.7 6.8 (50,

100 100 5 (4, 99) (2, 3) 2 115.1 4.7 (10, 80) (3, 5) 1 116.1 4.6

200 100 5 (3, 98) (2, 3) 2 185.3 6.7 (10, 85) (2, 5) 1 204.8 1.5

500 5 (15, 470) (2, 3) 2 109.8 5.2 (50, 435) (2, 3) 1 101.7 4.9

11 (4, 80) (3, 4) 3 187.7 3.5 (8, 95) (2, 3) 1 221 3.3 500 5 (10, 482) (2, 3) 2 208.9 7.2 (15, 435) (3, 5) 1 188.4 5.4 11 (18, 420) (4, 6) 4 191.1 7.5 (50, 420) (2, 3) 1 234.8 2.2

*ARL***<sup>0</sup>** *m n* **(***a***,** *b***) (***j***,** *k***)** *r Exact*

*Comparison of the* ARLout*s with the same* ARLin *for* Chart *2.*

(out-of-control), respectively.

tion (*θ*, *δ*).

**Table 7.**

**206**

*Comparison of the* ARLout*s with the same* ARLin *for* Chart *2 under normal distribution (*θ*,* δ*).*

In addition, **Table 8** depicts the out-of-control *ARL* performance of *Chart* 2 under normal distribution. More precisely, several shifts of both location and scale parameter have been considered, and *Chart* 2 with 2-of-3 rule detects the underlying shift sooner than *Chart* 2 does in almost all cases examined.

Finally, **Tables 9** and **10** present simulation-based comparisons of the nonparametric monitoring scheme *Chart* 3 with 2-of-2 runs rule versus *Chart* 3 established by Triantafyllou [7]. For delivering the numerical results displayed in **Tables 9** and **10**, the Lehmann alternatives have been considered as the out-of-control distribution.

Each cell contains the *AR*s attained for *γ* = 0.5 (upper entry) and *γ* = 0.2 (lower entry).

### **5. Conclusions**

In the present chapter, we investigate the in- and out-of-control performance of distribution-free control charts based on order statistics. Several runs-type rules are


#### **Table 9.**

*Comparison of the* ARout*s with the same* FAR *for* Chart *3.*

employed in order to enhance the ability of the aforementioned nonparametric monitoring schemes to detect possible shifts in distribution process. The *AR* and the *ARL* behavior of the underlying control charts is studied under several out-ofcontrol situations, such as the so-called Lehmann alternatives and the exponential or the normal distribution model. The numerical experimentation carried out depicts the melioration of the proposed schemes with the runs-type rules. It is of some research interest to branch out the incorporation of such runs rules (or even more complicated) to additional nonparametric control charts based on well-known test statistics.

**Author details**

**Table 10.**

Thessaly, Greece

**209**

Ioannis S. Triantafyllou

Department of Computer Science and Biomedical Informatics, University of

*Chart* **3** *Chart* **3 with 2-of-2 runs rule**

(3, 40)

(13, 78)

(12, 81)

(3, 45)

(10, 80)

(10, 82)

1.02

1.22

1.02

1.02

1.29

1.04

*ARLout* **(***a***,** *b***) (***i***,** *c***,** *j***,** *d***)** *r Exact*

*ARLin*

(2, 1, 2, 4) 1 111.4 2.3

(4, 1, 2, 4) 1 102.4 4.72

(5, 5, 3, 5) 1 114.0 2.28

(2, 1, 2, 4) 1 187.5 2.23

(3, 1, 2, 4) 1 230.9 5.62

(5, 5, 3, 5) 1 192.2 3.10

*ARLout*

1.02

1.14

1.02

1.02

1.24

1.04

*ARLin*

*Simulation-Based Comparative Analysis of Nonparametric Control Charts with Runs-Type Rules*

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: itriantafyllou@uth.gr

*ARL***<sup>0</sup>** *m n* **(***a***,** *b***) (***i***,** *c***,** *j***,** *d***)** *r Exact*

*DOI: http://dx.doi.org/10.5772/intechopen.91040*

100 50 10 (1, 45) (2, 1, 2, 4) 4 101.06 2.3

200 50 10 (1, 48) (3, 1, 2, 4) 3 213.49 2.30

*Comparison of the* ARLout*s with the same* ARLin *for* Chart *3.*

100 5 (3, 100) (4, 1, 2, 4) 3 123.51 4.73

100 10 (2, 89) (3, 1, 2, 4) 3 101.82 2.31

100 5 (1, 100) (3, 1, 2, 4) 2 244.88 6.59

100 10 (4, 94) (5, 5, 3, 5) 4 222.69 3.20

*Each cell contains the AR's attained for Y = 0.5 (upper entry) and Y = 0.2 (lower entry).*

provided the original work is properly cited.

*Simulation-Based Comparative Analysis of Nonparametric Control Charts with Runs-Type Rules DOI: http://dx.doi.org/10.5772/intechopen.91040*


#### **Table 10.**

*Comparison of the* ARLout*s with the same* ARLin *for* Chart *3.*
