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RELIABILITY CRITERIA FOR DESIGNING LIFE TEST SAMPLING INSPECTION PLANS BASED ON LOMAX
DISTRIBUTION
R. Vijayaraghavan1 and A. Pavithra2 •
Department of Statistics, Bharathiar University Coimbatore 641 046, Tamil Nadu, INDIA
1vijaystatbu@gmail.com, 2pavistat95@gmail.com Abstract
Acceptance sampling plays an important role in ensuring the quality of the products manufactured by the industrial production processes. Sampling inspection plans by attributes are adopted for taking decisions about the lots submitted for inspection. Such procedures are employed for sentencing individual lots or batches or lots in continuous stream. Reliability sampling is s specific inspection procedure which is used to decide whether the submitted lot or batch is acceptable or non-acceptable based on life tests. In reliability sampling, the lifetime of the items randomly drawn from the lot is considered as a random variable which follows a continuous probability distribution. In this paper, designing of single sampling plans for life tests is considered under the assumption that the lifetime random variable follows a Lomax distribution. Reliability criteria for designing life test plans when lot quality is evaluated in terms of mean life, median life, hazard rate and reliability life are proposed. Conversion factors for adapting acceptable quality levels to life and reliability testing under the assumption of Lomax distribution are determined and suitable illustrations are provided.
Keywords: Acceptable quality level, Consumer's risk, Lomax distribution, Operating characteristic function, Producer's risk, Reliability sampling, Single sampling plan.
1. Introduction
Sampling inspection is a product control strategy that decides whether a lot should be accepted or rejected based on the information obtained by the inspection of random sample(s) drawn from the submitted lot(s). Sampling inspection procedures are generally classified according to the nature of the quality characteristics, viz., measurable and non-measurable characteristics. When the quality characteristics are non-measurable, but are classified into go or no-go basis, such as good or bad, non-conforming or conforming, etc., the sampling inspection procedures are termed as attribute sampling. When the quality characteristics are measurable on a continuous scale, the corresponding sampling inspection procedures are called variables sampling, which are devised under the implicit assumption that the quality characteristic is a continuous random variable following a specific probability distribution. Reliability sampling plans, also termed as life test sampling plans, are operationally attributes sampling procedures, but involve lifetime of the components or items as a random variable which is distributed according to a specific continuous type probability distribution, such as the exponential, Weibull, lognormal, gamma distributions, etc. The lifetime of the components or items is observed by placing the sampled items under the test, called life test, which is defined as the process of evaluating the lifetime of the items through experiments. The literature in product control provides the importance of various continuous probability distributions like exponential, Weibull, lognormal and gamma distributions as well as several compound distributions for modeling lifetime data in the studies relating to the design and evaluation of reliability sampling plans.
The earlier works, which laid the foundation for the expansion of several types of sampling plans, would include the theory of reliability sampling proposed and developed by various authors. One
may refer [1] - [8] for the basic notions and terminologies of sampling inspection for life tests. The literature in statistical product control provides significant studies relating to the construction of life test sampling plans employing exponential, Weibull, lognormal and gamma distributions as well as several compound distributions for modeling lifetime data. A detailed account of the properties, methods of construction and performance of such plans is provided in [9] - [25], and the recent advances in the theory of life test sampling plans are discussed in [26] - [30].
Lomax distribution, introduced in [31], is a heavy-tailed probability distribution and is considered as Pareto Type II distribution. It has a wide range of applications in many fields which include business, economics, actuarial, medical and biological sciences. It has been proved to be much useful in reliability and life testing studies and in survival analysis. Properties of Lomax distribution and its extended form can be seen in [32] - [35]. In this paper, a specific life-test sampling plan is devised with reference to the life-time quality characteristic, which is modeled by Lomax distribution. A procedure for the selection of such plans indexed by acceptable and unacceptable mean life is evolved. Three different criteria for designing life-test plans when lot quality is evaluated in terms of mean life, hazard rate and reliability life are proposed. Factors for adapting acceptable quality level to life and reliability testing under the assumption of Lomax distribution are also illustrated.
2. Life Test Sampling Inspection Plans Based on Lomax Distribution
Sampling plans for life tests include a set of sampling procedures and rules for deciding whether to accept or reject a large number of items based on the sampled lifetime information about the items. Sampled items are tested for a set period of time under such plans. When all units are tested to failure, the standard plans can be used to compare the performance to the specified requirements, and the results can be used in an attribute sampling plan when the lifetimes are tested and the distributional assumption of the quality characteristics is fulfilled. Further, the number of failures which occur before a required time can be used with standard attributes plans in determining the disposition of the material. (See, [9]).
A typical life test sampling plan can be formulated in the following manner: Suppose, n items are placed for a life test and the experiment is stopped at a predetermined time, T. The number of failures occurred until the time point T is observed, and let it be d. The lot is accepted if d is less than or equal to the acceptance number, say, c; otherwise, it is rejected. Thus, the life test sampling plan is represented by n, the number of units on test, and the maximum allowable number of failures, c, called the acceptance number. Life tests, terminated before all units have failed, may be classified into two types, namely, failure terminated and time terminated. In a failure terminated life test, a given sample of n items is tested until the specified number of failures occurs and then the test is terminated. In time terminated life test, a given sample of n items is tested until a pre-assigned termination time, t, is reached and then the test is terminated.
Generally, these tests may be defined with reference to the specifications given in terms of one of the characteristics such as (i) the mean life, that is, the expected life of the product, (ii) the median life. (iii) the hazard rate, that is, the instantaneous failure rate at some specified time, t, and (iv) the reliable life, that is, the life beyond which some specified proportion of items in the lot will survive. One of the significant features of a life test plan is that it involves a random characteristic, called lifetime or time to failure, which can be more adequately described most often by skewed distributions. Application of continuous-type of distributions such as normal, exponential, Weibull, gamma and lognormal for lifetime variables in the studies concerned with the design and evaluation of life test sampling plans has been provided in the literature of sampling inspection, demonstrating the significant contributions of those distributions in the development of life test sampling plans. The Lomax distribution, one of the lifetime distributions, is now considered as the lifetime distribution for the design and evaluation of life-test sampling plan.
3. Lomax Distribution
Let T be a random variable representing the lifetime of the components. Assume that T follows Lomax distribution. The probability density function and the cumulative distribution function of T are, respectively, defined by
f(t;9,A) = Ul+^) ,t> 0,9 > 0,A> 0 (1)
9 V 9 J
and F(t;9,A) = l - (l +|) * ,t> 0,9 > 0,A > 0, (2)
where X and 6 are the shape and scale parameters, respectively.
The mean life, the median life, the reliability function and hazard function for specified time t under Lomax distribution are. Respectively, given by
1*=^, for A>l, (3)
R(t;9,A) = (l+^) X ,t> 0,9 > 0,A> 0 (5)
and Z(t;9,A) 1 ,t > 0,9 > 0,A > 0. (6)
9 V 6/
nd = e(\r2-l), (4)
-X
e)
9)
The reliability life is the life beyond which some specified proportion of items in the lot will survive. The reliability life associated with Lomax Distribution is defined and denoted by
p(t;9,A) = 9(R-1/À -l), (7)
where R is the proportion of items surviving beyond life p. The proportion, p, of product failing before time t, is defined by the cumulative probability distribution of T and is expressed by
p = P(T <t) = F(t;9,A). (8)
4. Application of Lomax Distribution in Reliability Sampling
The techniques for determining life test sampling plans based on Weibull distribution with mean life, hazard rate, and reliability life serving as reliability criteria for the submitted lots are discussed in [36] - [38]. The dimensionless quantities, viz., t/^x100, tZ(t)x100 and t/px100, referred to as conversion factors, for determining the life test sampling plans under the reliability criteria are introduced in [39]. Analogous approaches are discussed, here, to construct the life test sampling plans using Lomax Distribution as the lifetime distribution for the lifetime quality characteristic.
The mean life criterion is determined by calculating the ratio t / E(t) = t / ju, which is associated with the proportion, p, of products that fail before reaching the termination time t. Acceptable mean life and unacceptable mean life, which are associated with the producer's risk and the consumer's risk, are the two typically stipulated requirements when dealing with a life test sampling plan in practice for providing protection to the producer and the consumer, respectively. A desired sampling plan can be determined with the specification of these indices. The quality levels, corresponding to acceptable and unacceptable mean life, are defined by p and p with associated risks, where p is the acceptable proportion of the lot failing before the specified time, t, and p is the unacceptable proportion of the lot failing before the specified time, t. Based on mean life, median life, hazard rate, and reliability life as the criteria for life test plans under the Lomax Distribution, conversion factors are obtained. These conversion factors are used for deriving the plans satisfying the requirements.
When the test termination time is defined, the conversion factors can also be used to calculate the mean life, median life, hazard rate, and reliability life, and vice versa. The appropriate conversion factors in terms of percentages are computed and are provided in Table 1 through to 6. Numerical illustrations for demonstrating the use of tables for determining the operating characteristics of a given plan and finding the parameters of the single sampling plan satisfying the requirements in terms of acceptable quality level (acceptable mean life) and limiting quality level (unacceptable mean life) are provided in the following subsections.
4.1 Numerical Illustration
Under life testing experiments for ascertaining the reliability of components, an industrial practitioner desires to use a single sampling plan by attributes satisfying the requirements (p = 0.007, a = 0.05) and (p = 0.05, ( = 0.10). The past experimental results on the components produced by the industry have shown that the life time of the components follows Lomax distribution specified by the shape parameter A = 1.5. For the specified requirements, the parameters of an optimum sampling plan is determined as n = 105 and c = 2 using the searching algorithm given in [40]. It is assumed to employ a test termination time of 250 hours and to count the number of failures over the span of 250 hours. Under the given conditions, the operating characteristics in terms of mean life are obtained using the operating characteristic function of the single sampling plan by attributes and provided in Table 7 along with the values of k = t/jux.100 and u = t / k x 100, where t = 250 hours.
4.2 Numerical Illustration
Suppose that a single sampling plan by attributes with parameters n and c is to be defined when the requirements are specified in terms of acceptable mean life of 200 hours and unacceptable mean life of 70 hours with the associated producer's and consumer's risks of 5% and 10%, respectively. Assume that the individual items are to be tested for 3 hours and that the lifetime of the items is distributed as Lomax Distribution with the shape parameter fixed as A = 2. Then, at the specified levels, the values of k are determined as follows:
k± = t/v x 100 = (3/200) x 100 = 1.5 k2 = t/v x 100 = (3/70) x 100 = 4.286
Entering Table 1 with these values, one obtains the proportions, p = 0.03 and p = 0.08, of product failing before the specified time t corresponding to the acceptable mean life and unacceptable mean life, respectively. The operating ratio, which is the measure of discriminating good and bad lots of items, is defined by OR = 0.08/ 0.03 = 2.67, corresponding to which a single sampling plan can be chosen from [9] as (n = 159, c = 8) or from [41] as (n =157, c = 8).
In a similar manner, while Tables 2 and 3 can be used to determine conversion factors so as to obtain the life test plans and the corresponding median life and hazard rate, Tables 4 through to 6 can be utilized for obtaining reliability life for the specified values of R, viz., 0.90, 0.95 and 0.99, respectively.
4.3 Numerical Illustration
The acceptable mean life under the life test sampling plans based on Lomax distribution can be determined using the ratio k = t / ux100 for any specified value of acceptable quality level, AQL, shown in [42]. When AQL is specified as 3 percent with 95 percent acceptance probability, the test termination time is given as t = 25 hours and the shape parameter is fixed as A = 1.5, the average or
expected mean life, u, is determined as u = t/kx100 = (25/1.026)x100 = 2436.6 hours, which can be considered as an acceptable mean life. Accordingly, if a lot consisting of items which have the acceptable mean life specified at 2436.6 hours, the probability of acceptance of the lot would be 95%. Corresponding to the fixed value of AQL = 3 percent with 95 percent acceptance probability, the conversion factors for median life, hazard rate and reliability life criteria for the case in which the shape parameter is specified as A = 1.5 under Lomax Distribution are given in the following table:
Criterion Conversion Factor Value of the Factor AQL
Percent Nonconforming as per MIL - STD -105E p x 100 3 0.03
Mean Life k = t/u x 100 1.026 2436.6
Median Life k = x 100 3.492284 715.9
Hazard Rate at 25 hours tz(t) x 100 3.015203 0.001206
Reliable Life (R = 0.90) t/p x 100 28.19134 88.7
Reliable Life (R = 0.95) t/p x 100 58.96959 42.4
Reliable Life (R = 0.99) t/p x 100 305.1403 8.2
It can be noted from the above table that when the proportion, p, of products that fail before reaching the termination time, i.e., t = 25 hours is specified as the acceptable level of 3 percent, the median life of the components is 715.9 hours, 90 percent of the components will survive beyond 88.7 hours, 95 percent of the components will survive beyond 42.4 hours and 99 percent of the components will survive beyond 8.2 hours.
Table 1. Values of t/jx 100Based on Lomax Distribution for Specified Values of A
p% Shape Parameter, A
1.25 1.50 1.75 2.00 2.50 3.00
1 0.201817 0.336136 0.431968 0.503781 0.604234 0.671146
2 0.407337 0.677979 0.870847 1.015254 1.217073 1.351392
3 0.616667 1.025686 1.316821 1.534616 1.838731 2.040957
4 0.829918 1.379418 1.770079 2.062073 2.469426 2.740067
5 1.047205 1.739346 2.230818 2.597835 3.109387 3.448954
6 1.268648 2.105644 2.699241 3.142125 3.758848 4.167861
7 1.494373 2.478495 3.175561 3.695169 4.418056 4.897038
8 1.72451 2.858088 3.659998 4.257207 5.087262 5.636744
9 1.959193 3.244622 4.152782 4.828484 5.76673 6.38725
10 2.198566 3.638299 4.654149 5.409256 6.456732 7.148834
11 2.442774 4.039335 5.164347 5.999788 7.157552 7.921784
12 2.691971 4.447953 5.683634 6.600358 7.869484 8.706402
13 2.946319 4.864383 6.212277 7.211253 8.592833 9.502999
14 3.205984 5.288869 6.750557 7.832773 9.327917 10.3119
15 3.471141 5.721661 7.298763 8.465229 10.07507 11.13344
16 3.741973 6.163023 7.857198 9.108945 10.83462 11.96797
17 4.018672 6.613231 8.426179 9.76426 11.60695 12.81585
18 4.301438 7.072571 9.006036 10.43153 12.39241 13.67746
19 4.590479 7.541343 9.597112 11.11111 13.19139 14.5532
p% Shape Parameter, A
1.25 1.50 1.75 2.00 2.50 3.00
20 4.886016 8.01986 10.19977 11.8034 14.00431 15.44347
21 5.188277 8.508452 10.81438 12.50879 14.83158 16.3487
22 5.497506 9.007462 11.44134 13.2277 15.67364 17.26935
23 5.813954 9.51725 12.08106 13.96058 16.53095 18.20587
24 6.137887 10.03819 12.73397 14.70787 17.404 19.15874
25 6.469584 10.57069 13.40052 15.47005 18.29327 20.12848
50 18.52753 29.37005 36.44957 41.42136 47.92619 51.98421
60 27.03458 42.10079 51.60637 58.11389 66.40499 71.44177
70 40.50026 61.57216 74.22759 82.57418 92.79668 98.76032
80 65.59747 96.20089 113.1363 123.6068 135.5481 141.9952
90 132.7393 182.0794 204.5695 216.2277 226.7829 230.8869
Table 2. Values of t / /Jd x 100Based on Lomax Distribution for Specified Values of A
p% Shape Parameter, A
1.25 1.50 1.75 2.00 2.50 3.00
1 1.089282 1.144486 1.185111 1.216236 1.260759 1.291057
2 2.19855 2.308402 2.389184 2.451041 2.539475 2.599621
3 3.328382 3.492284 3.61272 3.704892 3.836589 3.92611
4 4.479376 4.696683 4.856241 4.978283 5.152561 5.270959
5 5.652156 5.922175 6.120285 6.271729 6.487866 6.634618
6 6.847369 7.169357 7.405412 7.58576 7.842995 8.01755
7 8.065691 8.438851 8.712204 8.920928 9.218458 9.42024
8 9.307824 9.731301 10.04127 10.27781 10.61479 10.84319
9 10.5745 11.04738 11.39323 11.65699 12.03252 12.28691
10 11.86648 12.38779 12.76873 13.0591 13.47224 13.75193
11 13.18456 13.75325 14.16847 14.48477 14.93453 15.23883
12 14.52958 15.14452 15.59314 15.93467 16.42001 16.74817
13 15.90238 16.56239 17.04349 17.40951 17.92931 18.28055
14 17.30389 18.00769 18.52026 18.90999 19.46309 19.8366
15 18.73505 19.48128 20.02428 20.43687 21.02205 21.41696
16 20.19683 20.98404 21.55635 21.99094 22.60689 23.02231
17 21.69028 22.51692 23.11736 23.57301 24.21838 24.65335
18 23.21647 24.08089 24.70821 25.18393 25.85728 26.3108
19 24.77653 25.67698 26.32984 26.82459 27.5244 27.99542
20 26.37166 27.30625 27.98323 28.49593 29.22058 29.708
21 28.00308 28.96982 29.66943 30.19889 30.94671 31.44936
22 29.6721 30.66887 31.3895 31.9345 32.70371 33.22037
23 31.38009 32.40461 33.14459 33.70382 34.49253 35.02192
24 33.12847 34.17833 34.93586 35.50793 36.31417 36.85492
25 34.91876 35.99138 36.76455 37.34802 38.16968 38.72038
50 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000
60 145.9157 143.346 141.5829 140.2993 138.5568 137.4297
70 218.595 209.6426 203.6446 199.3517 193.6242 189.9814
80 354.054 327.5476 310.3914 298.4132 282.8268 273.1506
90 716.4437 619.9492 561.2397 522.0199 473.1921 444.1481
Table 3. Values of tz(t) x 100Based on Lomax Distribution for Specified Values of A
p% Shape Parameter, A
1.25 1.50 1.75 2.00 2.50 3.00
1 1.001004 1.001674 1.002153 1.002513 1.003016 1.003352
2 2.004032 2.006727 2.008654 2.010101 2.01213 2.013483
3 3.00911 3.015203 3.019566 3.022844 3.027441 3.03051
4 4.016262 4.027152 4.034955 4.040821 4.049051 4.054551
5 5.025514 5.04262 5.054887 5.064113 5.077068 5.085729
6 6.036893 6.061658 6.079431 6.092805 6.111597 6.124167
7 7.050427 7.084316 7.108656 7.126985 7.152751 7.169998
8 8.066143 8.110646 8.142635 8.166739 8.200644 8.223352
9 9.084069 9.140701 9.181444 9.21216 9.255394 9.284369
10 10.10424 10.17454 10.22516 10.26334 10.31712 10.35319
11 11.12667 11.21221 11.27385 11.32038 11.38595 11.42995
12 12.15141 12.25377 12.3276 12.38337 12.46201 12.51481
13 13.17847 13.29929 13.3865 13.45242 13.54543 13.60792
14 14.20791 14.34883 14.45064 14.52763 14.63635 14.70944
15 15.23973 15.40243 15.52008 15.60911 15.73491 15.81953
16 16.27399 16.46018 16.59493 16.69697 16.84125 16.93836
17 17.31072 17.52214 17.67528 17.79133 17.95552 18.06611
18 18.34994 18.58836 18.76123 18.89230 19.07787 19.20295
19 19.39171 19.65893 19.85286 20.00000 20.20847 20.34908
20 20.43605 20.73392 20.95029 21.11456 21.34748 21.50467
21 21.483 21.81339 22.0536 22.23611 22.49505 22.66994
22 22.5326 22.89743 23.16293 23.36478 23.65138 23.84508
23 23.58491 23.98612 24.27836 24.50071 24.81664 25.03031
24 24.63995 25.07952 25.40002 25.64404 25.99101 26.22584
25 25.69776 26.17773 26.52803 26.79492 27.17469 27.43191
50 53.20635 55.50592 57.23373 58.57864 60.53543 61.88984
60 64.94378 68.56747 71.33223 73.5089 76.71379 78.95812
70 77.29026 82.77893 87.04709 90.45549 95.54978 99.17011
80 90.50676 98.70072 105.2368 110.5573 118.6736 124.5589
90 105.1888 117.6835 128.0528 136.7544 150.4732 160.7523
Table 4. Values of t/px 100 when R = 0.90 Based on Lomax Distribution
for Specified Values of A
p% Shape Parameter, A
1.25 1.50 1.75 2.00 2.50 3.00
1 9.179481 9.238821 9.281348 9.313323 9.358197 9.388186
2 18.52739 18.6345 18.7112 18.76883 18.84968 18.90367
3 28.04859 28.19134 28.29348 28.37019 28.47773 28.54951
4 37.74813 37.91381 38.03227 38.12118 38.24575 38.32885
5 47.63126 47.80655 47.93179 48.02573 48.15727 48.24497
6 57.70344 57.87439 57.99643 58.08791 58.21594 58.30125
7 67.97035 68.12234 68.23074 68.31197 68.42555 68.50119
8 78.43793 78.5556 78.63946 78.70226 78.79002 78.84843
9 89.11233 89.1796 89.2275 89.26335 89.31342 89.34673
10 99.99997 99.99998 99.99998 99.99997 99.99998 99.99997
11 111.1076 111.0226 110.9622 110.917 110.8541 110.8122
12 122.4421 122.2536 122.1197 122.0197 121.8803 121.7877
13 134.0109 133.6993 133.4782 133.3132 133.0833 132.9307
14 145.8215 145.3665 145.0438 144.8031 144.468 144.2459
15 157.882 157.2619 156.8227 156.4952 156.0397 155.7378
16 170.2006 169.3929 168.8213 168.3955 167.8035 167.4114
17 182.786 181.7671 181.0466 180.5102 179.765 179.2718
18 195.6474 194.3922 193.5055 192.8458 191.93 191.3243
19 208.7942 207.2766 206.2055 205.4092 204.3044 203.5744
20 222.2365 220.4288 219.1543 218.2074 216.8947 216.0278
21 235.9846 233.8579 232.3599 231.2478 229.7072 228.6904
22 250.0496 247.5734 245.8309 244.5383 242.7488 241.5687
23 264.443 261.5851 259.5761 258.0868 256.0266 254.669
24 279.1768 275.9034 273.6046 271.9018 269.548 267.9981
25 294.2638 290.5392 287.9263 285.9922 283.3208 281.5631
50 842.7096 807.2465 783.1628 765.7495 742.2667 727.1703
60 1229.646 1157.156 1108.825 1074.341 1028.461 999.3483
70 1842.121 1692.333 1594.869 1526.535 1437.208 1381.488
80 2983.647 2644.116 2430.87 2285.098 2099.329 1986.27
90 6037.54 5004.519 4395.421 3997.365 3512.348 3229.714
Table 5. Values of t / px 100when R = 0.95 Based on Lomax Distribution
for Specified Values of A
p% Shape Parameter, A
1.25 1.50 1.75 2.00 2.50 3.00
1 19.27196 19.32542 19.36365 19.39236 19.43257 19.4594
2 38.89754 38.97895 39.03713 39.08078 39.1419 39.18266
3 58.88693 58.96959 59.02862 59.07288 59.13483 59.17612
4 79.25073 79.3067 79.34663 79.37656 79.4184 79.4463
5 99.99998 99.99997 99.99998 99.99998 99.99998 99.99999
6 121.1461 121.0595 120.9978 120.9516 120.8871 120.8442
7 142.7011 142.4958 142.3496 142.2403 142.0876 141.9861
8 164.6774 164.3197 164.0653 163.8751 163.6098 163.4334
9 187.0879 186.5426 186.1551 185.8656 185.4619 185.1938
10 209.946 209.1762 208.6297 208.2216 207.6529 207.2754
11 233.266 232.233 231.5002 230.9533 230.1917 229.6865
12 257.0625 255.7255 254.778 254.0714 253.0879 252.436
13 281.3507 279.6673 278.4753 277.5869 276.3513 275.5327
14 306.1467 304.0722 302.6046 301.5115 299.9921 298.9862
15 331.4672 328.9547 327.1788 325.857 324.0209 322.8062
16 357.3296 354.3299 352.2115 350.6359 348.4488 347.0027
17 383.7522 380.2136 377.717 375.8613 373.2872 371.5865
18 410.7541 406.6223 403.71 401.5468 398.5482 396.5683
19 438.3553 433.5734 430.206 427.7065 424.2441 421.9597
20 466.5768 461.0847 457.221 454.3551 450.3881 447.7725
21 495.4405 489.1753 484.7719 481.5081 476.9936 474.0191
22 524.9694 517.8648 512.8765 509.1816 504.0749 500.7125
23 555.1877 547.174 541.5529 537.3925 531.6466 527.8662
24 586.1208 577.1245 570.8206 566.1584 559.7242 555.4942
25 617.7953 607.739 600.6999 595.4977 588.324 583.6111
50 1769.236 1688.568 1633.911 1594.456 1541.338 1507.245
60 2581.594 2420.495 2313.338 2237.011 2135.629 2071.404
70 3867.462 3539.96 3327.371 3178.576 2984.404 2863.486
80 6264.051 5530.865 5071.519 4758.068 4359.318 4117.05
90 12675.58 10468.27 9170.156 8323.379 7293.492 6694.403
Table 6. Values of t / p x 100when R = 0.99 Based on Lomax Distribution
for Specified Values of A
p% Shape Parameter, A
1.25 1.50 1.75 2.00 2.50 3.00
1 100.0001 100.0001 100.0001 100.0001 100.0001 100.0001
2 201.8351 201.698 201.6002 201.5269 201.4244 201.3561
3 305.5578 305.1403 304.8427 304.6198 304.3081 304.1006
4 411.2234 410.3754 409.7714 409.3192 408.6875 408.2672
5 518.8889 517.4535 516.4318 515.6675 514.6003 513.8907
6 628.6139 626.4268 624.8713 623.7084 622.0856 621.0071
7 740.4604 737.3495 735.1389 733.4872 731.1837 729.6538
8 854.4928 850.2781 847.2856 845.051 841.9366 839.8694
9 970.7784 965.2712 961.3644 958.4489 954.3879 951.6942
10 1089.387 1082.390 1077.43 1073.731 1068.583 1065.169
11 1210.392 1201.698 1195.541 1190.952 1184.568 1180.338
12 1333.869 1323.261 1315.755 1310.164 1302.392 1297.245
13 1459.898 1447.149 1438.135 1431.426 1422.105 1415.938
14 1588.561 1573.433 1562.747 1554.797 1543.761 1536.463
15 1719.947 1702.188 1689.656 1680.339 1667.413 1658.872
16 1854.144 1833.493 1818.933 1808.116 1793.119 1783.215
17 1991.248 1967.429 1950.651 1938.195 1920.938 1909.549
18 2131.358 2104.082 2084.887 2070.647 2050.931 2037.928
19 2274.578 2243.541 2221.721 2205.544 2183.162 2168.412
20 2421.016 2385.900 2361.235 2342.962 2317.699 2301.062
21 2570.786 2531.255 2503.517 2482.981 2454.611 2435.941
22 2724.009 2679.710 2648.658 2625.685 2593.972 2573.116
23 2880.808 2831.371 2796.752 2771.159 2735.856 2712.656
24 3041.316 2986.351 2947.900 2919.496 2880.344 2854.634
25 3205.672 3144.767 3102.206 3070.789 3027.518 2999.124
50 9180.371 8737.558 8438.036 8222.095 7931.735 7745.597
60 13395.61 12524.94 11946.82 11535.54 10989.96 10644.75
70 20067.84 18317.65 17183.60 16390.89 15357.76 14715.19
80 32503.47 28619.66 26190.94 24535.82 22433.07 21157.14
90 65772.19 54168.42 47357.61 42920.97 37532.35 34401.93
Table 7. Operating Characteristics of the Life Test Sampling Plan
Based on Mean Life Criterion (n =105, c = 2 and A = 1.5)
p P ( P) X = 1.5
k = t / jux100 j = 250/ k x100
0.010 0.911201 0.336136 74374.66
0.015 0.790632 0.506334 49374.48
0.020 0.649366 0.677979 36874.30
0.025 0.510198 0.851089 29374.12
0.030 0.386710 1.025686 24373.94
0.035 0.284587 1.201788 20802.33
0.040 0.204328 1.379418 18123.58
0.045 0.143657 1.558597 16040.07
0.050 0.099187 1.739346 14373.22
0.055 0.067404 1.921687 13009.40
0.060 0.045163 2.105644 11872.85
0.070 0.019543 2.478495 10086.77
0.080 0.008106 2.858088 8747.105
0.090 0.003242 3.244622 7705.059
0.100 0.001256 3.638299 6871.342
5. Conclusion
A procedure for deriving single sampling plans for life tests is described under the condition that the lifetime quality characteristic is modeled by a Lomax distribution. The tables for the determining the sampling plans when lot quality is evaluated using four criteria, namely, mean life, median life, hazard rate and reliability life are constructed fixing a set of values of the shape parameter. Practitioners can generate the necessary sampling plans for other values of the shape parameter as per their requirements using the procedure described in this paper. Conversion factors are also included to assist in the calculation of the mean life, median life, and hazard rate at a certain test termination time and vice versa. The factors for adapting acceptable quality level as the index to mean life, median life, hazard rate and reliability life are also provided.
6. Acknowledgment
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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