International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 1

ISSN 2229-5518

A NEW PROCEDURE FOR DESIGNING SINGLE SAMPLING PLAN INDEXED THROUGH TRIGNOMETRIC RATIOS

P.R DIVYA*

*Assistant Professor in Statistics, Department of Statistics, Vimala College, Thrissur, Kerala, divyastat@gmail.com

ABSTRACT

Acceptance Sampling is the methodology which deals with procedures through which decisions to accept or reject of a lot which are based on the result of inspection of samples. The foundation of the scheme of acceptance sampling has laid by Dodge and Romig (1959). General procedures and necessary tables are provided for the selection of single sampling plan through tangent angle as proposed by Norman Bush (1953). Mandelson (1962) has explained the desirability for developing a system of sampling plans indexed through MAPD. Mayer (1967) has explained that the quality standard that the MAPD can be considered as a quality level with other conditions to specify the OC curve. Soundararajan (1975) has constructed tables for selection of single sampling plan indexed through MAPD and K (pT/p*) Suresh and Ramkumar (1996) have studied the selection of single sampling plan indexed through Maximum Allowable Average Outgoing Quality (MAAOQ) and MAPD.

This paper provides a new procedure for designing a single sampling plan indexed through trigonometric ratios, hypotenuse ratios along with decision region (d1) and probabilistic region (d2) which is more applicable in practical situations. Numerical illustrations are also provided for the construction and selection of the plan parameters using trigonometric ratios and hypotenuse ratios.

Key Words: Single Sampling Plan, Decision Region, Operating Characteristic Curve, Trigonometric ratios

Introduction

Acceptance Sampling is defined as the procedure for inspection and classification
of sample of units selected at random from a larger lot and ultimate decision about the disposition of the lot is made. Basically the “acceptance quality control” system that was developed encompasses the concept of protecting the consumer from getting unacceptable defective product, and encouraging the producer in the use of process quality control through varying the quantity and severity of acceptance inspections in direct relation to the importance of the characteristics inspected, and the inverse relation to the goodness of the quality level as indication of those inspections.
The single sampling plan is the most widely used basic sampling plan in the area of acceptance sampling. The performance of a sampling plan is identified through an OC curve. For designing a sampling inspection plan, it is the usual practice to consider the OC curve passes through any two of the quality levels. Mandelson (1962) has explained the desirability for developing such a system of sampling plans indexed through MAPD Mayer (1967) has explained that the quality standard that the MAPD can be considered as a quality level along with other conditions to specify an OC curve. Soundararajan (1975) has constructed tables for selection of single sampling plan indexed through MAPD and

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 2

ISSN 2229-5518


K= pT .Suresh and Ramkumar (1996) have studied the selection of single sampling plan

p*

indexed through Maximum Allowable Average Outgoing Quality (MAAOQ).
This paper provides a new procedure for designing attribute single sampling plan indexed through trigonometric ratios and hypotenuse ratios. Also considering the ability of the declination angles of the tangent at the inflection point on the OC curve for discrimination of the Single Sampling Plan (SSP)
0.95  L( p* )

Here, tan 1 =

d1

……………………… (1)
From (1) one can find (n,c) for a particular L(p*) and d1.So we can state that both 1 and d1 uniquely determines the SSP.

L( p* )  0.10

Similarly, tan 2 =

d 2 d1

………………… (2)
From (2) one can find (n,c) for a particular L(p*) and (d2-d1). So we can state that both 2
and (d2-d1) uniquely determines the SSP.

L( p* )


And, tan3 =

d 2

…………………………… (3)
From (3) one can find (n,c) for a particular L(p*) and d2. So we can state that both 3
d2 uniquely determines the SSP.
and
From figure1, we have  ABC represents the approximate area inscribed by the quality levels p1 and p*.  CDE represents the approximate area inscribed by the quality levels p* and p2.And the  BFG represents the approximate area inscribed by the quality levels p1
and p2. 1 is the inscribed triangle by OC with quality levels p1 and p*. 2 represent the
inscribed triangle by OC with quality levels p* and p2. And 3
OC with quality levels p1 and p2.
is the inscribed triangle by

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 3

ISSN 2229-5518


Selection of sampling plans

Table 1 is given for selected values of c. Here SSP with c=0 are not considered, since c=0 plans do not involve an inflection point on the OC curve. Tables are given for the values of L(p*) for c=1,2,……20.

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 4

ISSN 2229-5518

Table 1: Certain Parametric Values for SSP

c

L(p*)

np1

np2

d1

d2

1

0.735759

0.355

3.89

0.645

3.535

2

0.676676

0.818

5.322

1.182

4.504

3

0.647232

1.366

6.681

1.634

5.315

4

0.628837

1.97

7.994

2.03

6.024

5

0.615961

2.613

9.275

2.387

6.662

6

0.606303

3.286

10.532

2.714

7.246

7

0.598714

3.981

11.771

3.019

7.79

8

0.592547

4.695

12.995

3.305

8.3

9

0.587408

5.426

14.206

3.574

8.78

10

0.58304

6.169

15.407

3.831

9.238

11

0.579267

6.924

16.598

4.076

9.674

12

0.575965

7.69

17.782

4.31

10.092

13

0.573045

8.464

18.958

4.536

10.494

14

0.570437

9.246

20.128

4.754

10.882

15

0.56809

10.035

21.292

4.965

11.257

16

0.565962

10.831

22.452

5.169

11.621

17

0.564023

11.633

23.606

5.367

11.973

18

0.562245

12.442

24.756

5.558

12.314

19

0.560607

13.254

25.902

5.746

12.648

20

0.559093

14.072

27.045

5.928

12.973

Using the table it can be noted that as c increased d1, d2 increases but L(p*) decreases.

Example 1

For a given sample size n=100 and to attain an area of 0.85.Find the acceptance to be
taken for attain a better OC curve.
Using table 2 we can easily read off, for area ABC=0.8549 the corresponding acceptance number c=13.

Example 2

For a given sample size n=100 and to attain an area of 1.18, find the acceptance to be
taken for attain a better OC curve.
Using table 3 we can find that for area CDE=1.18, the corresponding acceptance number c=7.

Example3

For a given sample size n=100 and to attain an area of 3, find the acceptance to be taken
for attain a better OC curve.
Using table 4 we can easily read off, that the appropriate acceptance number is c=13.

Example 4

For an OC curve to which a tangent is drawn,it is specified the operating ratio R4=2.2394. Find the appropriate acceptance number.
Using table 8 it can be seen that the appropriate c for the operating ratio R4=2.2394 is
3.That is c=3.

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 5

ISSN 2229-5518

Construction of Tables

When the proportion of defective in the lot is small and sample size is large so that np<5 then the lot quality assumed to follow Poisson distribution. The probability of acceptance under Poisson model is given as

c

L(p) =

r 0

e np (np) r

r!

……………………………. (4)
Where p is the proportion defective of the lot, p coordinate of the inflection point will

c


obtain as p*= .

n

L(p*) represents the probability of acceptance of an utmost satisfactory quality (MAPD)

c

L(p*) =

r 0

e c (c) r

r!

………………………………..(5)
Thus L(p*)is a function of c alone, and it is constant for fixed c.
0.95  L( p* )
1  0.95  L( p* ) 

From figure, tan 1 =

d1

, the declination angle

1 = tan


  .
d1

L( p* )  0.10

1  .L( p* )  0.10 

Similarly tan 2 =

L( p* )

d 2 d1

and the declination angle is
1  .L( p* )

2 = tan



  .
d 2 d1

And tan3 =

d 2

, the declination angle is 3 = tan

  .
d 2
For different values of c=1,2,…..20, L(p*) is determined from equation (5). Substituting the appropriate values in equation (1),(2),(3) for fixed L(p*),d1,
(d2-d1), d2 and hence angle1 ,2 ,3 and (n,c) are obtained.
Table-2 provides the area of triangle ABC for a fixed n for different values of c. Table-3 provides the area of triangle CDE for a fixed n for different values of c. Table-4 provides the area of triangle BFG for a fixed n for different values of c Table-5 provides the operating ratio R1 for different values of c. Table-6 provides the operating ratioR2 for different values of c. Table-7 provides the operating ratio R3 for different values of c. Table-8 provides the operating ratio R4.

Conclusion

MAPD is the quality measure proposed for designing the sampling plan. MAPD has evolved as a world wide accepted quality measure to discriminate between good and bad lots. Many procedures for designing single sampling plan have been developing over years using MAPD as quality index. When sampling procedure fails to obtain OC curve which lies closer to the ideal one. MAPD related plans which are more efficient for achieving better quality products. Therefore Quality parameters like trigonometric ratio’s, hypotenuse ratio’s decision region (d1), probabilistic region (d2) which are more applicable in suitable situations.

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 6

ISSN 2229-5518

Table 2: The area of triangle ABC for a fixed n

c

d1

L(p*)

.95-L(p*)

AC

areaABC

1

0.645

0.7358

0.2142

0.6797

0.0691

2

1.182

0.6767

0.2733

1.2132

0.1615

3

1.634

0.6472

0.3028

1.6618

0.2474

4

2.03

0.6288

0.3212

2.0552

0.3260

5

2.387

0.6160

0.3340

2.4103

0.3987

6

2.714

0.6063

0.3437

2.7357

0.4664

7

3.019

0.5987

0.3513

3.0394

0.5303

8

3.305

0.5925

0.3575

3.3243

0.5907

9

3.574

0.5874

0.3626

3.5923

0.6480

10

3.831

0.5830

0.3670

3.8485

0.7029

11

4.076

0.5793

0.3707

4.0928

0.7556

12

4.31

0.5760

0.3740

4.3262

0.8060

13

4.536

0.5730

0.3770

4.5516

0.8549

14

4.754

0.5704

0.3796

4.7691

0.9022

15

4.965

0.5681

0.3819

4.9797

0.9481

16

5.169

0.5660

0.3840

5.1832

0.9925

17

5.367

0.5640

0.3860

5.3809

1.0358

18

5.558

0.5622

0.3878

5.5715

1.0776

19

5.746

0.5606

0.3894

5.7592

1.1187

20

5.928

0.5591

0.3909

5.9409

1.1586

Table 3: The area of triangle CDE for a fixed n

c

d2-d1

L(p*)

L(p*)-.10

CE

area CDE

1

2.89

0.73576

0.6358

2.9591

0.9187

2

3.322

0.67668

0.5767

3.3717

0.9579

3

3.681

0.64723

0.5472

3.7215

1.0072

4

3.994

0.62884

0.5288

4.0289

1.0561

5

4.275

0.61596

0.5160

4.3060

1.1029

6

4.532

0.6063

0.5063

4.5602

1.1473

7

4.771

0.59871

0.4987

4.7970

1.1897

8

4.995

0.59255

0.4925

5.0192

1.2301

9

5.206

0.58741

0.4874

5.2288

1.2687

10

5.407

0.58304

0.4830

5.4285

1.3059

11

5.598

0.57927

0.4793

5.6185

1.3415

12

5.782

0.57597

0.4760

5.8016

1.3760

13

5.958

0.57304

0.4730

5.9767

1.4092

14

6.128

0.57044

0.4704

6.1460

1.4414

15

6.292

0.56809

0.4681

6.3094

1.4726

16

6.452

0.56596

0.4660

6.4688

1.5032

17

6.606

0.56402

0.4640

6.6223

1.5327

18

6.756

0.56224

0.4622

6.7718

1.5615

19

6.902

0.56061

0.4606

6.9174

1.5896

20

7.045

0.55909

0.4591

7.0599

1.6172

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 7

ISSN 2229-5518

Table 4: The area of triangle BFG for a fixed n

c

L(p*)

d2

FG

area BFG

1

0.7358

3.535

3.611

1.300

2

0.6767

4.504

4.555

1.524

3

0.6472

5.315

5.354

1.720

4

0.6288

6.024

6.057

1.894

5

0.6160

6.662

6.690

2.052

6

0.6063

7.246

7.271

2.197

7

0.5987

7.79

7.813

2.332

8

0.5925

8.3

8.321

2.459

9

0.5874

8.78

8.800

2.579

10

0.5830

9.238

9.256

2.693

11

0.5793

9.674

9.691

2.802

12

0.5760

10.092

10.108

2.906

13

0.5730

10.494

10.510

3.007

14

0.5704

10.882

10.897

3.104

15

0.5681

11.257

11.271

3.197

16

0.5660

11.621

11.635

3.289

17

0.5640

11.973

11.986

3.377

18

0.5622

12.314

12.327

3.462

19

0.5606

12.648

12.660

3.545

20

0.5591

12.973

12.985

3.627

Table 5: The ratio of area of triangle ABC and CDE

c

area ABC

area CDE

R1=CDE/ABC

1

0.0691

0.9187

13.2962

2

0.1615

0.9579

5.9297

3

0.2474

1.0072

4.0717

4

0.3260

1.0561

3.2397

5

0.3987

1.1029

2.7663

6

0.4664

1.1473

2.4599

7

0.5303

1.1897

2.2436

8

0.5907

1.2301

2.0825

9

0.6480

1.2687

1.9581

10

0.7029

1.3059

1.8578

11

0.7556

1.3415

1.7755

12

0.8060

1.3760

1.7071

13

0.8549

1.4092

1.6483

14

0.9022

1.4414

1.5976

15

0.9481

1.4726

1.5532

16

0.9925

1.5032

1.5145

17

1.0358

1.5327

1.4797

18

1.0776

1.5615

1.4491

19

1.1187

1.5896

1.4209

20

1.1586

1.6172

1.3957

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 8

ISSN 2229-5518

Table 6: The ratio of area of triangle ABC and BFG

c

area ABC

area BFG

R2=BFG/ABC

1

0.069093

1.300454

18.8219

2

0.161534

1.523875

9.4338

3

0.247362

1.720019

6.9535

4

0.325981

1.894057

5.8103

5

0.398676

2.051765

5.1464

6

0.466397

2.196635

4.7098

7

0.530266

2.33199

4.3978

8

0.590691

2.459071

4.1630

9

0.647951

2.578722

3.9798

10

0.702912

2.693061

3.8313

11

0.755554

2.801913

3.7084

12

0.806045

2.906321

3.6057

13

0.854935

3.006765

3.5170

14

0.902222

3.103746

3.4401

15

0.948093

3.197492

3.3726

16

0.992545

3.288525

3.3132

17

1.03577

3.376523

3.2599

18

1.077571

3.461742

3.2125

19

1.118725

3.545281

3.1690

20

1.15865

3.626554

3.1299

Table 7: The ratio of area of triangle CDE and BFG

c

area CDE

area BFG

R3=BFG/CDE

1

0.9187

1.3005

1.4156

2

0.9579

1.5239

1.5909

3

1.0072

1.7200

1.7078

4

1.0561

1.8941

1.7935

5

1.1029

2.0518

1.8604

6

1.1473

2.1966

1.9146

7

1.1897

2.3320

1.9602

8

1.2301

2.4591

1.9990

9

1.2687

2.5787

2.0325

10

1.3059

2.6931

2.0622

11

1.3415

2.8019

2.0887

12

1.3760

2.9063

2.1121

13

1.4092

3.0068

2.1337

14

1.4414

3.1037

2.1533

15

1.4726

3.1975

2.1713

16

1.5032

3.2885

2.1877

17

1.5327

3.3765

2.2030

18

1.5615

3.4617

2.2170

19

1.5896

3.5453

2.2304

20

1.6172

3.6266

2.2426

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 9

ISSN 2229-5518

Table 8: The hypotenuse values and their ratios

c

AC

CE

R4=CE/AC

1

0.6797

2.9591

4.3539

2

1.2132

3.3717

2.7792

3

1.6618

3.7215

2.2394

4

2.0552

4.0289

1.9603

5

2.4103

4.3060

1.7865

6

2.7357

4.5602

1.6669

7

3.0394

4.7970

1.5783

8

3.3243

5.0192

1.5099

9

3.5923

5.2288

1.4555

10

3.8485

5.4285

1.4105

11

4.0928

5.6185

1.3728

12

4.3262

5.8016

1.3410

13

4.5516

5.9767

1.3131

14

4.7691

6.1460

1.2887

15

4.9797

6.3094

1.2670

16

5.1832

6.4688

1.2480

17

5.3809

6.6223

1.2307

18

5.5715

6.7718

1.2154

19

5.7592

6.9174

1.2011

20

5.9409

7.0599

1.1884

IJSER © 2012 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 10

ISSN 2229-5518

REFERENCES

1. Cameron.J.M (1952): Tables for constructing and for computing characteristics of single sampling plan, Industrial Quality Control.9, 37-39.
2. Carroll W. M. J (1963): Application of an Inspection Scheme for attributes, Ph.D Thesis.
3. Douglas C Montgomery (2001): Introduction to Statistical Quality Control, 4th
Edition, Arizona State University.
4. Lilly Christina. A (1989): Sampling Inspection Plans indexed by Inflection point a review, M.phil contributed to Bharathiar University, Coimbatore, Tamilnadu, India.
5. Soundararajan. V (1975): Maximum Allowable Percent Defective (MAPD) Single Sampling Inspection by Attributes Plan, Journal of Quality Technology, Vol.7 No.4, 173-182
6. Suresh .K.K and Ramkumar .T.B (1996): selection of sampling Plans indexed with Maximum Allowable Average Outgoing Quality.
7. Suresh K.K (1993): A study on Acceptance Sampling using Acceptable Quality
Levels. Ph.D Thesis Bharathiar University, Coimbatore, Tamilnadu, India.

IJSER © 2012 http://www.ijser.org