**Author details**

Ioannis S. Triantafyllou Department of Computer Science and Biomedical Informatics, University of Thessaly, Greece

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

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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

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

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

*FAR*

(15, 75) (7, 2, 4, 7) 1 0.0086 0.7064

(10, 79) (6, 2, 6, 12) 1 0.0108 0.9103

(80, 300) (33, 2, 4, 7) 1 0.0088 0.7326

(75, 300) (49, 2, 7, 9) 1 0.0089 0.9585

(155, 690) (65, 2, 4, 7) 1 0.0096 0.7161

(165, 600) (99, 2, 7, 9) 1 0.0129 0.9778

(14, 75) (5, 2, 4, 7) 1 0.0054 0.6312

(10, 81) (5, 2, 6, 12) 1 0.0051 0.9227

(70, 300) (31, 2, 4, 7) 1 0.0049 0.6653

(70, 314) (43, 2, 7, 9) 1 0.0046 0.9485

(150, 700) (65, 2, 4, 7) 1 0.0047 0.7051

(150, 600) (87, 2, 7, 9) 1 0.0056 0.9645

(11, 75) (4, 2, 4, 7) 1 0.0032 0.5283

(9, 80) (6, 2, 6, 9) 1 0.0024 0.8803

(70, 345) (28, 2, 4, 7) 1 0.0024 0.6624

(70, 320) (38, 2, 7, 4) 1 0.0030 0.9497

(135, 705) (65, 2, 4, 7) 1 0.0023 0.6445

(145, 600) (78, 2, 7, 9) 1 0.0029 0.9587

*ARout*

0.9825

0.9992

0.9876

0.9999

0.9873

0.9999

0.9787

0.9989

0.9827

0.9998

0.9870

0.9998

0.9660

0.9988

0.9834

0.9996

0.9823

0.9997

*FAR*

0.9537

0.9992

0.9583

0.9999

0.9584

0.9999

0.9198

0.9983

0.9536

0.9998

0.9583

0.9998

0.8905

0.9971

0.9453

0.9996

0.9452

0.9997

*FAR m n* **(***a***,** *b***) (***i***,** *c***,** *j***,** *d***)** *r Exact*

**0.01** 100 11 (6, 93) (7, 2, 4, 7) 4 0.0104 0.5363

25 (5, 86) (6, 2, 6, 12) 10 0.0102 0.8181

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

25 (30, 450) (49, 2, 7, 9) 8 0.0101 0.9122

25 (30, 960) (99, 2, 7, 9) 8 0.0098 0.9181

25 (4, 88) (5, 2, 6, 12) 10 0.0051 0.7508

25 (30, 450) (43, 2, 7, 9) 8 0.0052 0.8755

25 (30, 960) (87, 2, 7, 9) 8 0.0050 0.8832

25 (5, 86) (6, 2, 7, 9) 7 0.0025 0.6951

25 (30, 450) (38, 2, 7, 9) 8 0.0027 0.8338

25 (30, 960) (78, 2, 7, 9) 8 0.0028 0.8479

500 11 (24, 480) (28, 2, 4, 7) 4 0.0028 0.4715

1000 11 (53, 990) (55, 2, 4, 7) 6 0.0027 0.4673

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

500 11 (26, 478) (31, 2, 4, 7) 5 0.0052 0.5094

1000 11 (60, 985) (65, 2, 4, 7) 6 0.0050 0.5299

**0.0027** 100 11 (3, 94) (4, 2, 4, 7) 4 0.0029 0.3410

500 11 (31, 484) (33, 2, 4, 7) 6 0.0108 0.5335

1000 11 (60, 967) (65, 2, 4, 7) 6 0.0099 0.5311

**0.005** 100 11 (4, 93) (5, 2, 4, 7) 4 0.0051 0.4134

test statistics.

**208**

**Table 9.**
