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SELECTION OF SINGLE SAMPLING PLANS BY VARIABLES BASED ON GENERALIZED BETA
DISTRIBUTION
R. Vijayaraghavan3 and A. Pavithrab
Department of Statistics, Bharathiar University,
Coimbatore 641 046, Tamil Nadu, INDIA avijaystatbu@gmail.com, bpavistat95@gmail.com
Abstract
Statistical quality control (SQC) has wider applications in industries and production engineering. Product control, one of the two major categories of SQC, consists in procedures by which decisions are made on the disposition of one or more lots of finished items or materials produced by manufacturing industries. Sampling inspection by variables in product is the methodology that is employed for deciding about the disposition of a lot of individual units based on the observed measurements on a quality characteristic of randomly sampled units from the lot submitted for inspection. These procedures are defined under the assumption that the quality characteristic is measurable on a continuous scale and the functional form of the probability distribution must be known. Inspection procedures which have been developed based on the implicit assumption that the quality characteristic is distributed as normal with the related properties are found in the literature of sampling inspection procedures. The assumption of normality may not be realized often in practice and it becomes inevitable to investigate the properties of variable sampling plans based on non-normal distributions. In this paper a single sampling plan by variables is formulated and evaluated under the assumption that the quality characteristic is distributed according to a generalized beta distribution of first kind. Procedures are developed for determining the parameters of the proposed plan for specified requirements in terms of producer's and consumer's protection.
Key Words: Consumer's Quality Level, Generalized Beta Distribution, Normal Distribution, Operating Characteristic Function, Single Sampling Plan, Producer's Quality Level.
1. Introduction
Sampling inspection is an activity for taking decisions on one or more lots of finished products which have been submitted for inspection. The decision of either acceptance or rejection of the lots is usually taken by adopting suitable sampling inspection procedures, called sampling plans. Sampling plans are generally categorized into two types, namely, lot-by-lot sampling by attributes and lot-by-lot sampling by variables. In lot-by-lot inspection by attributes, one or more samples of items are drawn from a given lot of manufactured items; each item in the sample(s) is classified as conforming or nonconforming; and the decision of acceptance or rejection of the lot is made based on a specific rule. In lot-by-lot inspection by variables, one or more samples of items are drawn from a given lot; the measurement of a quality characteristic in each sampled item is recorded; and the decision of acceptance or rejection of the lot is made as a function of such measurements. The theory of inspection by variables is applicable when the quality characteristic of sampled items is measurable
on a continuous scale and the functional form of the probability distribution is assumed to be known. A variables sampling is advantageous in the sense that it generates more information from each item inspected, requires small sample and provides same protection when compared to attributes sampling. See, [1] and [2].
On the basis of the implicit assumption that the quality characteristic is distributed according to normal with mean j and standard deviation o, the concept of variables sampling inspection has been studied by many researchers. Some of the early works on variables sampling inspection are seen in [3], [4], [5], [6] and [7]. Studies relating to sampling plans when the assumption of normality of the quality characteristic fails or the functional form of the underlying distribution deviates from normal or the form of the distribution is not known are also found in the literature of acceptance sampling. [8] - [23] are few references which deal with variables inspection using non-normal distributions.
The problem of designing single sampling plans by variables, when the quality characteristic, X, follows a normal distribution with mean j and standard deviation o , has been addressed in the past. See, [24]. In the industrial situations, quite often, the assumption of normality may not be valid or the quality characteristic may be distributed according to non-normal distributions. In such cases, the selection of variable sampling plans becomes complicated. However, the literature of acceptance sampling provides procedures for the designing of variables plans when the quality characteristic follows a probability distribution other than normal. A detailed survey on various works related to variable sampling plans with emphasis on non-normality is given in [11]. A computer-aided procedure has been developed in [25] for the identification of the appropriate distribution in designing sampling inspection plans by variables when the quality characteristics are defined by compositional proportions.
A generalized probability density function, termed as double bounded probability density function has been derived in [26]. It is also called a generalized beta distribution of first kind, in which the random variable X is defined within the range (0, 1). Practical applications of variables sampling plans using a generalized beta distribution can be visualized for bulk product inspection where the quality characteristics are the compositional proportions, such as proportion of binary mixtures of pharmaceutical powder, percentage of protein in milk powder, fatty acid composition of serum lipid fractions, etc. Sampling inspection plans for compositional fractions based on the beta distribution and the procedure for designing the plans to control the proportion nonconforming levels are discussed in [27].
In this paper, a study on single sampling plans by variables is formulated under the assumption that the quality characteristic is assumed to have a generalized beta distribution which would be appropriate in situations where the quality characteristics are compositional fractions. A procedure for determining the parameters of the proposed plan for specified requirements in terms of producer's and consumer's protection is also developed.
2. Single Sampling Inspection Plans by Variables A single sampling inspection plan by variables is defined under the following assumptions:
(a) The quality characteristic, denoted by X, is measurable on a continuous scale and has a known form of probability distribution, represented by (x; , which is the distribution function of x with mean j and variance o2.
(b) Each individual unit in a submitted lot has a one-sided specification, say, lower specification, L or upper specification, U. If, for a unit, X > U (or X < L ), the unit is classified as a non-conforming unit.
The operating procedure of a variable sampling plan is as follows:
Step 1: Draw a random sample of n units from a lot and observe the measurements, xx, x2,..., xn of the quality characteristic, X.
Step 2: When < is known, accept the lot if x + k< < U (or x — k< > L); otherwise, reject the lot, where x is the sample mean.
When < is unknown, accept the lot, if x + ks < U (or x — ks > L); otherwise, reject the lot.
2 1 n Here, s =-
n — 1 ii
Thus, a single sampling plan by variables is designated by two parameters, namely, the sample size, n, and the acceptability constant, k. When these parameters are known, the plan could be implemented. The explicit expressions for n and k can be derived by specifying two points on the operating characteristic curve of the plan, namely, (p1, 1 — a) and (p2, JJ), where p and p are termed as producer's quality level (PQL) and the consumer's quality level (CQL), associated with the producer's risk, a and the consumer's risk, JJ, respectively. A sampling plan by variables is termed as a known < or unknown < plan according as < is known or unknown.
3. Operating Characteristic Function
An important measure of performance of a variables sampling plan is its operating characteristic function, which is a function of the proportion, p, of non-conforming units, called incoming lot quality, and provides the probability, Pa (p), of acceptance of a lot. The plot of Pa (p) against p results in a curve, called operating characteristic (OC) curve. For a given upper specification limit, U, when < is known, p and Pa (p) are defined by
p = P(X > Uu) (1)
and p (p) = P(x + ka< U|ju). (2) PQL and CQL, using (1), are defined by
pql = pi = p(x > u\ j) (3)
and CQL = p2 = P(X > U j), (4)
where j and j are the means of the underlying distribution which results in PQL and CQL, respectively.
Assume that the random variable, X, is modeled by a two-parameter generalized beta distribution. The probability density function and the cumulative distribution function of the generalized beta distribution, according to [26], are respectively given by
f (x; a, b) = abxa+!(1 - xa)b0 < x < 1 (5)
and F(x) = 1 - (1 - xa )b, (6)
where a > 0 and b > 0 are the shape parameters.
/ , (x - x) is an unbiased estimate of a2.
The moments about origin of the distribution are defined by
M-il + - V(»)
a
m =—^-, r = 1,2,3,4, •••,
<i+»+-)
from which the measures such as mean, variance, skewness and kurtosis can be derived as
2
ff = m, a2 = f2, 3i = f~j and fi2 = f\, where fi2 = m2 -m\, ju3 = m3 -3m2m12 + 2mf,
and f = m4 - 4mm + 6m2mf - 3m\.
From (1), (3) and (4), the lot quality levels, p, PQL and CL using standardized beta distribution are defined, respectively, by
p = P(T > K* ),
PQL = pi = P(T > Kpi) (7)
and CQL = p2 = P(T > Kp), (8)
wha.cT=X -l K * = U-ll P K * = U-l\ Pi mdK* =U - l 1 P2 w ere a ' p a ' Pl a an P2 a '
The producer's risk,« , and the consumer's risk, /, corresponding to AQL and LQL are, respectively, defined from (2) as
a = P(rejecting the lot | f = f ) = P(x + ka > U\ f = f ) (9)
and 1 - / = P(rejecting the lot | f = f ) = P(x + ka > U\ f = f ) . (10)
When a is unknown, the estimate s is used in the decision criterion, and hence in the evaluation of a and /.
4. Designing Single Sampling Plans by Variables
In the industrial practice, the unknown standard deviation variables plans are more realistic than the known standard deviation variables plans. If the distribution is non-normal, the designing of unknown a plans is rather complicated. Such problems introducing an expansion factor in terms of measures of skewness and kurtosis are addressed in [12], which also provides a methodology for determining the parameters of sampling plans by variables under the conditions of non-normal populations using the expansion factor. The procedures for the selection of unknown standard deviation sampling plans are provided in [23] giving protection to the producer and consumer under the assumption that the quality characteristics under study follow a Pareto distribution when the measures of skewness and /or kurtosis are specified.
4.1. Case of Unknown Sigma
The methodology proposed in [12] using the expansion factor will, now, be discussed for an unknown sigma plan by variables under the assumption of generalized beta distribution for the quality characteristic, X.
In the case of unknown sigma plan, the determination of n and k is usually based on the sampling distribution of x + ks or x — ks. It is known that under the assumption of normal distribution, x and s are independent and distributed as normal. Therefore, x + ks and x — ks are normally distributed. Using these properties, formulae for finding the values of n and k can be obtained. The asymptotic distributions of x + ks and x — ks are shown to be normal having the means j = u + ka and j = ju — ka, respectively, and the common variance given by
a* =
a
n
1 + — ß-1) ± kß
(11)
where P and P2 represent the measures of skewness and kurtosis of the underlying distribution.
U — u| p
Having defined Z =- and acceptance probability function for the case of unknown
a
standard deviation as Pa (p) = Pr [x + kUs < U | p] = Pr (Y < U | p), from [12], the expressions
P , ZPQL and ZC QL (
for za , z n, zpqj and Zcgi corresponding to a, ft, PQL and CQL, respectively, are as given below:
Z„ =
U - p, + kUa)
(12)
a
U - (^\pi + kUa)
a
- Zß =
ZPQL = kU + Ka ,
(13)
(14)
7* = k — K
Z CQL kU Kß■
(15)
k 2
Here, ^ = 1 + — (P — 1) + kp is the expansion factor, which can be used to obtain the known
standard deviation plans. When the requirements are specified in terms of the points (PQL, 1—a) and (CQL, p) on the OC curve such that Pa (PQL) = 1 — a and Pa (CQL) = p, the expressions for the plan parameters n and k , derived from (14) and (15), are as given below:
nU eU
Za+ Zß
ZPQL - ZLQL
(16)
and k^ = -
Z Z + z 7
ZaZCQL + 7 ß7PQL
Za+ Zß
(17)
In a similar way, when the lower specification limit, L, is specified, the expressions for n and k can be derived.
4.2. Numerical Illustration
Suppose that a set of measurements yields P = 0.0377 and P2 = 2.0147 It is desired to determine a variables sampling plan giving protection to the producer and the consumer in terms of (PQL = 0.01, a = 0.05) and (CQL = 0.06, P = 0.10). For the given requirements, the values of a
2
e
U
n
u
2
and b are found as a = 0.65 and b = 0.80. Associated with these values are the mean and standard deviation given by M = 0.5581 and S = 0.2545, respectively. Corresponding to PQL and CQL, the values of ZPQL and ZCqL are determined from (7) and (8) as 1.6717 and 1.4619, respectively. The
normal deviates Za and Z^ are obtained as 1.645 and 1.282 by satisfying (9) and (10) for the specified sets of values of a and jj. Substituting these values in (16) and (17), the parameters of the desired plan are determined as n = 195andk = 1.554. The value of ^ is obtained as 1.67119. Thus, the
nn
parameters of a known standard deviation plan, are computed as nu =— = 117 and
eu
k'v = kv = 1.554.
4.3. Case of Known Sigma
The method of designing known sigma variables sampling plan under the assumption of Burr distribution utilizing the measures skewness and kurtosis is proposed in [15]. A similar procedure is developed here for the known sigma plan by variables when the underlying distribution is a two-parameter generalized beta distribution.
Let M and S be the mean and standard deviation of the two-parameter generalized beta distribution. Then, PQL and CQL are defined by
PQL = 1 - F(xFQL ;M, S) = (1 - xaPQL )b, a > 0, b > 0 and CQL = 1 - F(xC0t ;M, S) = (1 - xaCQL)b,a > 0, b > 0.
(18)
(19)
(20) (21)
with ZPqL and Z being the standardized values of x corresponding to PQL and CQL, respectively.
Assuming that the distribution of x is normal, a and ß are defined as area under normal curve and are expressed by
where <p(£) =^=t? - , =
and ZL_S = yfn
¡7
From equations (22) to (26), the expressions for n and k are, respectively, obtained as
(22)
(23)
(24)
(25)
(26)
(27)
If the acceptance criterion is written as x + ka < U, according to [15], the expression for k given by
is
ku =
SlZa+Zfl
.
(29)
5. Determination of n and k of a Variables Sampling Plan
The parameters of a sampling plan by variables can be derived from the generalized beta distribution when the third and fourth moments of the distribution of measurements are known or specified. It is known that the measures of skewness and kurtosis, specified by /?L and of a generalized beta distribution are functions of the shape parameters a and b. Thus, for a specified values of pL and /f;, the values of a and b can be determined. In order to determine the required sampling plan by variables, the following procedure is followed:
Step 1: Specify/^ and 82.
Step 2: Specify the desired protection in terms of (pL, 1 - a) and (¡3:., fi).
Step 3: Choose the value a and b from Table 2 corresponding to the specified values of jiL and fin.
Step 4: For specified px and p2, determine xPOL and xr YJL from Fx(x), which is the cumulative
distribution function of the generalized beta distribution, satisfying the equations (18) and (19), and obtain ZpVL and Z^QL from equations (20) and (21).
Step 5: For specified a and /f, determine the normal deviates Z(.: and Zs, satisfying the equations (22) and (23).
Step 6: Determine the parameters na and ka of the plan as n^ and kv using equation (27) and (29).
Based on the procedure described, the parameters, na and ka , of the sampling plans by variables
for a wide range of values of PQL and CQL are obtained and given in Table 3 for various combination of values of a and b. The parameters provided in the table yield the maximum producer's risk of 5% and the maximum consumer's risk of 10%. To facilitate the computation of ZpQL and Z, the mean, M, and standard deviation, S, are obtained for sets of values of a and b and provided in Table 1.
5.1. Numerical Illustration
It is desired to have a single sampling plan by variables when the set of measurements drawn from a generalized beta distribution has the measure of skewness and kurtosis specified as P = 0.0654 and P = 2.1384. Suppose that the desired protection against an upper specification limit is specified in terms of (PQL = 0.01, a = 0.05) and (CQL = 0.06, P = 0.10).
Table 2 yields a = 0.750, b = 0.50,M = 0.4156 and S = 0.2363 associated with P = 0.0654 and P = 2.1384. The values of and xCQL are determined from (18) and (19) as 0.924 and
0.8458 for the specified pql = 0.01 and CQL = 0.04. The standardized deviates ZPqL and ZCqL are obtained as 2.1512 and 1.8206, respectively, from (20) and (21). The values of Za and Z^ are determined as 1.645 and 1.282. On substitution of these values in (27) and (29), the parameters of the desired plan are determined as nCT = n^ = 78, andka = 1.9654. Table 3, when entered with the
specified values of the quality levels, can be used to choose the parameters of the required plan corresponding to a = 0.75 and b = 0.50.
Table 1: Mean, M, and Standard Deviation, S of Generalized Beta Distribution
a b
0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550
0.500 0.3694 0.4063 0.4382 0.4661 0.4909 0.5132 0.5333 0.5517 M
0.1738 0.1868 0.1969 0.205 0.2116 0.2168 0.2211 0.2246 S
0.550 0.3387 0.3759 0.4082 0.4367 0.4622 0.4852 0.506 0.5251 M
0.1727 0.1871 0.1985 0.2076 0.2151 0.2212 0.2262 0.2303 S
0.600 0.3111 0.3484 0.381 0.4099 0.4359 0.4594 0.4808 0.5004 M
0.1707 0.1863 0.1989 0.2091 0.2175 0.2244 0.2301 0.2349 S
0.650 0.2862 0.3234 0.3561 0.3853 0.4116 0.4356 0.4574 0.4775 M
0.168 0.1848 0.1985 0.2097 0.2189 0.2266 0.233 0.2384 S
0.700 0.2638 0.3007 0.3334 0.3627 0.3893 0.4135 0.4357 0.4562 M
0.1648 0.1827 0.1973 0.2094 0.2195 0.2279 0.235 0.241 S
0.750 0.2435 0.28 0.3126 0.3419 0.3687 0.3931 0.4156 0.4364 M
0.1612 0.1801 0.1957 0.2086 0.2195 0.2286 0.2363 0.2429 S
0.800 0.2251 0.2611 0.2934 0.3228 0.3495 0.3741 0.3968 0.4178 M
0.1573 0.1771 0.1936 0.2073 0.2189 0.2287 0.2371 0.2442 S
0.850 0.2084 0.2438 0.2758 0.305 0.3318 0.3565 0.3793 0.4005 M
0.1533 0.1739 0.1911 0.2056 0.2179 0.2284 0.2373 0.245 S
0.900 0.1932 0.2279 0.2596 0.2886 0.3153 0.34 0.363 0.3843 M
0.1492 0.1705 0.1884 0.2036 0.2165 0.2276 0.2371 0.2453 S
0.950 0.1793 0.2134 0.2446 0.2734 0.3 0.3247 0.3477 0.3691 M
0.145 0.1669 0.1855 0.2013 0.2149 0.2265 0.2366 0.2452 S
1.000 0.1667 0.2 0.2308 0.2593 0.2857 0.3103 0.3333 0.3548 M
0.1409 0.1633 0.1824 0.1988 0.213 0.2251 0.2357 0.2449 S
1.500 0.0852 0.1108 0.136 0.1604 0.1841 0.2068 0.2286 0.2494 M
0.1029 0.1276 0.15 0.1701 0.1881 0.2042 0.2185 0.2313 S
2.000 0.0476 0.0667 0.0865 0.1068 0.127 0.147 0.1667 0.1859 M
0.0753 0.0992 0.1219 0.1432 0.1628 0.1808 0.1972 0.2122 S
2.500 0.0284 0.0426 0.0583 0.0749 0.092 0.1095 0.127 0.1444 M
0.0562 0.0782 0.1001 0.1212 0.1412 0.16 0.1775 0.1936 S
3.000 0.0179 0.0286 0.041 0.0547 0.0693 0.0844 0.1 0.1157 M
0.043 0.0628 0.0833 0.1037 0.1235 0.1425 0.1604 0.1772 S
Table 1 (Continued)
a b
0.600 0.650 0.700 0.750 0.800 0.850 0.900
0.500 0.5686 0.5841 0.5985 0.6119 0.6243 0.6359 0.6468 M
0.2274 0.2296 0.2314 0.2328 0.2339 0.2347 0.2352 S
0.550 0.5426 0.5588 0.5738 0.5877 0.6007 0.6129 0.6243 M
0.2337 0.2365 0.2387 0.2405 0.2419 0.243 0.2439 S
0.600 0.5185 0.5352 0.5507 0.5652 0.5787 0.5914 0.6032 M
0.2388 0.2421 0.2448 0.247 0.2488 0.2502 0.2513 S
0.650 0.496 0.5132 0.5292 0.5441 0.5581 0.5712 0.5835 M
0.2429 0.2466 0.2498 0.2524 0.2545 0.2563 0.2577 S
0.700 0.4751 0.4927 0.5091 0.5244 0.5388 0.5523 0.565 M
0.246 0.2503 0.2539 0.2569 0.2594 0.2615 0.2633 S
0.750 0.4556 0.4736 0.4903 0.506 0.5207 0.5345 0.5475 M
0.2485 0.2532 0.2572 0.2607 0.2635 0.266 0.268 S
0.800 0.4374 0.4556 0.4726 0.4886 0.5036 0.5177 0.531 M
0.2503 0.2555 0.2599 0.2637 0.267 0.2697 0.2721 S
0.850 0.4203 0.4387 0.456 0.4723 0.4875 0.5019 0.5155 M
0.2515 0.2572 0.2621 0.2663 0.2698 0.2729 0.2755 S
0.900 0.4043 0.4229 0.4404 0.4569 0.4723 0.487 0.5008 M
0.2524 0.2584 0.2637 0.2683 0.2722 0.2756 0.2785 S
0.950 0.3892 0.408 0.4257 0.4423 0.458 0.4728 0.4869 M
0.2528 0.2593 0.2649 0.2698 0.2741 0.2778 0.281 S
1.000 0.375 0.3939 0.4118 0.4286 0.4444 0.4595 0.4737 M
0.2528 0.2598 0.2658 0.2711 0.2756 0.2796 0.2831 S
1.500 0.2693 0.2884 0.3066 0.3239 0.3405 0.3564 0.3716 M
0.2428 0.253 0.2622 0.2704 0.2777 0.2842 0.2901 S
2.000 0.2045 0.2227 0.2402 0.2571 0.2735 0.2893 0.3045 M
0.2258 0.2382 0.2494 0.2596 0.2689 0.2773 0.2849 S
2.500 0.1616 0.1785 0.1951 0.2113 0.227 0.2423 0.2572 M
0.2086 0.2223 0.2349 0.2465 0.2572 0.2669 0.2759 S
3.000 0.1315 0.1472 0.1627 0.178 0.1931 0.2078 0.2222 M
0.1928 0.2074 0.221 0.2336 0.2452 0.2559 0.2658 S
Table 2: Measures, p1 and p2, of Skewness and Kurtosis Generalized Beta Distribution
a b
0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550
0.500 0.0676 0.0342 0.0133 0.0025 0.0001 0.0049 0.0157 0.0316 px
2.4664 2.3723 2.3018 2.2505 2.2149 2.192 2.18 2.1771 p2
0.550 0.1318 0.0804 0.0441 0.02 0.0059 0.0003 0.0017 0.0091 P1
2.5366 2.4147 2.3212 2.2503 2.1977 2.1601 2.135 2.1204 P^
0.600 0.2138 0.1431 0.0904 0.0522 0.026 0.0096 0.0015 0.0003 P1
2.6341 2.4822 2.3641 2.2727 2.2026 2.1499 2.1115 2.085 P^
0.650 0.3122 0.2205 0.1503 0.0974 0.0585 0.031 0.0131 0.0032 P1
2.7563 2.572 2.4276 2.3145 2.2262 2.1579 2.1059 2.0675 P^
0.700 0.426 0.3114 0.2226 0.154 0.1018 0.0628 0.0348 0.016 P1
2.9013 2.6818 2.5094 2.3734 2.266 2.1815 2.1156 2.0651 P^
0.750 0.5542 0.415 0.306 0.2209 0.1547 0.1037 0.0654 0.0374 P1
3.0679 2.8101 2.6076 2.4473 2.3199 2.2186 2.1384 2.0755 P^
0.800 0.6965 0.5304 0.4 0.2972 0.2162 0.1528 0.1037 0.0664 P1
3.255 2.9558 2.721 2.5349 2.3864 2.2677 2.1726 2.0969 P2
0.850 0.8523 0.6572 0.5036 0.3821 0.2857 0.2092 0.149 0.102 P1
3.4621 3.1178 2.8484 2.6349 2.4643 2.3273 2.2169 2.1282 P2
0.900 1.0216 0.7949 0.6166 0.4751 0.3624 0.2723 0.2005 0.1436 P1
3.6887 3.2956 2.9889 2.7464 2.5525 2.3964 2.2702 2.1679 P2
0.950 1.2042 0.9432 0.7384 0.5758 0.4458 0.3415 0.2577 0.1905 P1
3.9347 3.4886 3.142 2.8686 2.6502 2.4741 2.3314 2.2154 P2
1.000 1.4 1.102 0.8687 0.6836 0.5355 0.4163 0.32 0.2422 P1
4.2 3.6964 3.3072 3.0009 2.7566 2.5598 2.4 2.2696 P2
1.500 4.1297 3.2597 2.6111 2.1141 1.7247 1.4143 1.1633 0.9582 P1
7.9786 6.5824 5.5687 4.8073 4.2201 3.7575 3.3868 3.0856 P2
2.000 8.531 6.5435 5.1512 4.1346 3.3675 2.7735 2.3038 1.9259 P1
14.2288 11.0692 8.9317 7.412 6.2892 5.4341 4.7668 4.2355 P2
2.500 15.1716 11.1897 8.5727 6.7539 5.4343 4.4441 3.6804 3.0784 P1
23.8586 17.5151 13.5161 10.8234 8.9175 7.515 6.4505 5.622 P2
3.000 24.8095 17.5104 13.0029 10.0171 7.9316 6.4136 5.2717 4.3895 P1
38.094 26.3968 19.5076 15.1041 12.1123 9.9818 8.4073 7.2085 P2
Table 2 (Continued)
a b
0.600 0.650 0.700 0.750 0.800 0.850 0.900
0.500 0.052 0.0762 0.1037 0.1342 0.1672 0.2026 0.24 ß1
2.182 2.1934 2.2107 2.2329 2.2595 2.2899 2.3237 ß2
0.550 0.0216 0.0386 0.0594 0.0835 0.1106 0.1403 0.1723 ß1
2.1145 2.1162 2.1244 2.1382 2.1569 2.1798 2.2065 ß2
0.600 0.005 0.0148 0.029 0.0469 0.0682 0.0925 0.1193 ß1
2.0686 2.0606 2.06 2.0656 2.0766 2.0924 2.1124 ß2
0.650 0 0.0026 0.0102 0.0221 0.0377 0.0566 0.0784 ß1
2.0404 2.0229 2.0135 2.0111 2.0147 2.0236 2.0371 ß2
0.700 0.0048 0.0002 0.0013 0.0071 0.0171 0.0309 0.0478 ß1
2.0272 2.0001 1.982 1.9717 1.9681 1.9702 1.9775 ß2
0.750 0.0182 0.0063 0.0007 0.0005 0.005 0.0136 0.0258 ß1
2.0267 1.9899 1.9632 1.945 1.9342 1.9298 1.9309 ß2
0.800 0.0389 0.0196 0.0074 0.0012 0.0002 0.0037 0.0111 ß1
2.0371 1.9906 1.9552 1.9292 1.9113 1.9003 1.8954 ß2
0.850 0.066 0.0393 0.0204 0.0081 0.0016 0.0001 0.0028 ß1
2.0571 2.0006 1.9564 1.9226 1.8976 1.8802 1.8694 ß2
0.900 0.0989 0.0646 0.0389 0.0205 0.0085 0.0019 0 ß1
2.0853 2.0188 1.9657 1.924 1.892 1.8682 1.8515 ß2
0.950 0.137 0.0948 0.0622 0.0377 0.0201 0.0085 0.002 ß1
2.1209 2.0441 1.9821 1.9325 1.8934 1.8632 1.8407 ß2
1.000 0.1796 0.1295 0.0899 0.0592 0.036 0.0193 0.0082 ß1
2.163 2.0758 2.0048 1.9471 1.9008 1.8642 1.836 ß2
1.500 0.7889 0.6482 0.5307 0.4322 0.3495 0.28 0.2217 ß1
2.8381 2.6328 2.4614 2.3174 2.1959 2.0932 2.0062 ß2
2.000 1.6175 1.3629 1.1506 0.9721 0.8211 0.6927 0.5829 ß1
3.8057 3.4531 3.1605 2.9155 2.7086 2.5329 2.3827 ß2
2.500 2.5949 2.2007 1.8751 1.6033 1.3743 1.1798 1.0136 ß1
4.9637 4.4316 3.9953 3.633 3.3292 3.0721 2.8529 ß2
3.000 3.6928 3.1325 2.6749 2.2963 1.9795 1.7119 1.484 ß1
6.273 5.528 4.9247 4.4289 4.0164 3.6697 3.3756 ß2
Table 3: Sample Size, n, and Acceptability Constant, k, of Single Sampling Plans by Variables
Based on Generalized Beta Distribution Having a Maximum Producer's Risk of 5 Percent (a = 0.05) and a Maximum Consumer's Risk of 10 Percent (P = 0.10)
PQL CQL b a
0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850
0.005 0.01 0.200 244 209 180 155 135 118 104 91 n
2.4205 2.5076 2.5939 2.6792 2.7637 2.8473 2.9298 3.0114 k
0.005 0.01 0.250 67 58 50 43 38 34 30 26 n
2.2105 2.2836 2.3556 2.4264 2.4962 2.5648 2.6324 2.6988 k
0.005 0.02 0.300 89 77 67 59 51 45 40 36 n
2.1311 2.2 2.2679 2.3346 2.4004 2.4651 2.5289 2.5916 k
0.006 0.03 0.350 70 60 53 47 41 36 33 29 n
1.9813 2.041 2.0996 2.157 2.2134 2.2686 2.3228 2.376 k
0.007 0.04 0.400 63 55 49 43 38 34 30 27 n
1.863 1.916 1.9678 2.0185 2.068 2.1165 2.1638 2.2102 k
0.008 0.05 0.450 61 54 48 42 38 34 30 27 n
1.7649 1.8127 1.8593 1.9047 1.949 1.9922 2.0344 2.0756 k
0.01 0.04 0.500 142 125 111 98 88 78 70 64 n
1.7406 1.7878 1.8338 1.8787 1.9226 1.9654 2.0073 2.0483 k
0.01 0.07 0.550 142 125 111 98 88 78 70 64 n
1.7406 1.7878 1.8338 1.8787 1.9226 1.9654 2.0073 2.0483 k
0.02 0.09 0.600 75 67 60 54 48 44 40 36 n
1.4919 1.5258 1.5584 1.5899 1.6203 1.6497 1.678 1.7054 k
0.02 0.03 0.650 2922 2585 2298 2052 1840 1656 1496 1356 n
1.6059 1.6474 1.6878 1.7272 1.7657 1.8033 1.84 1.876 k
0.03 0.04 0.700 4520 4014 3582 3211 2889 2610 2366 2152 n
1.5208 1.5578 1.5938 1.6288 1.6629 1.696 1.7283 1.7598 k
0.03 0.05 0.750 4520 4014 3582 3211 2889 2610 2366 2152 n
1.5208 1.5578 1.5938 1.6288 1.6629 1.696 1.7283 1.7598 k
0.04 0.06 0.800 2002 1787 1603 1444 1305 1185 1079 986 n
1.4037 1.4354 1.466 1.4957 1.5244 1.5523 1.5793 1.6056 k
0.04 0.08 0.850 634 567 510 460 417 379 346 317 n
1.3458 1.3749 1.4029 1.4301 1.4563 1.4816 1.5061 1.5299 k
0.05 0.10 0.900 501 450 406 367 334 304 279 256 n
1.2823 1.3085 1.3337 1.3579 1.3813 1.4038 1.4255 1.4464 k
Table 3 (Continued)
PQL CQL b a
0.900 0.950 1.000 1.500 2.000 2.500 3.000
0.005 0.01 0.200 81 72 64 24 12 7 4 n
3.092 3.1716 3.2501 3.9699 4.5555 4.993 5.2813 k
0.005 0.01 0.250 23 21 19 8 4 2 1 n
2.7642 2.8284 2.8915 3.4603 3.9127 4.2494 4.4768 K
0.005 0.02 0.300 32 29 26 10 5 3 2 N
2.6534 2.7141 2.7739 3.3179 3.765 4.1177 4.3816 K
0.006 0.03 0.350 26 24 21 9 5 3 2 n
2.4281 2.4792 2.5294 2.9786 3.3394 3.62 3.8287 k
0.007 0.04 0.400 25 22 20 9 5 3 2 n
2.2555 2.2999 2.3433 2.7277 3.0319 3.2667 3.4418 k
0.008 0.05 0.450 25 22 20 9 5 3 2 n
2.1157 2.155 2.1933 2.5298 2.793 2.9954 3.1469 k
0.01 0.04 0.500 58 52 48 21 12 7 5 n
2.0883 2.1275 2.1657 2.5055 2.7778 2.9938 3.162 k
0.01 0.07 0.550 58 52 48 21 12 7 5 n
2.0883 2.1275 2.1657 2.5055 2.7778 2.9938 3.162 k
0.02 0.09 0.600 33 30 28 14 8 5 4 n
1.7319 1.7575 1.7822 1.9885 2.1343 2.2343 2.2987 k
0.02 0.03 0.650 1232 1124 1028 477 261 160 106 n
1.9111 1.9455 1.9791 2.2801 2.5277 2.7323 2.9013 k
0.03 0.04 0.700 1963 1796 1648 788 443 278 189 n
1.7904 1.8203 1.8494 2.1056 2.3092 2.4715 2.6002 k
0.03 0.05 0.750 1963 1796 1648 788 443 278 189 n
1.7904 1.8203 1.8494 2.1056 2.3092 2.4715 2.6002 k
0.04 0.06 0.800 903 830 765 379 220 142 98 n
1.631 1.6558 1.6798 1.8871 2.0463 2.169 2.2627 k
0.04 0.08 0.850 291 268 247 125 74 48 34 n
1.5529 1.5751 1.5967 1.7808 1.919 2.023 2.1004 k
0.05 0.10 0.900 235 217 201 104 62 41 30 n
1.4666 1.4861 1.5049 1.6621 1.7752 1.856 1.9121 k
6. Conclusion
The literature in statistical quality control provides various sampling inspection procedures which been developed based on the assumption that the quality characteristic under study follows a normal distribution. While such procedures are widely used in the industries, the departure from the assumption of normality or the violation of distributional assumptions are the major concern for the industrial practitioners as the decision that is made on the lot disposition in such situations would be inappropriate. Focusing on this vital aspect, in this paper, procedures for designing single sampling plans by variables are devised under the assumption that the quality characteristic is distributed according to a generalized beta distribution of first kind. The procedures and tables presented are appropriate for bulk inspection procedures where the quality characteristics are defined by compositional proportions.
7. Acknowledgments
The authors are grateful to Bharathiar University, Coimbatore for providing necessary facilities to carry out this research work. The second author is indebted to the Department of Science and Technology, India for awarding the DST-INSPIRE Fellowship under which the present research has been carried out.
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