International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 1
ISSN 2229-5518
*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
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
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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, tan 3 =
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
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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.
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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.
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.
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.
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.
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.
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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 tan 3 =
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.
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.
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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 |
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 |
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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 |
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 |
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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 |
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 |
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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 |
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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.
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