DESIGNING OF ACCEPTANCE SAMPLING PLAN BASED ON PERCENTILES FOR TOPP-LEONE GOMPERTZ DISTRIBUTION Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

DESIGNING OF ACCEPTANCE SAMPLING PLAN BASED ON PERCENTILES FOR TOPP-LEONE GOMPERTZ DISTRIBUTION

S Jayalakshmi1, Aleesha A2 •

1 Assistant Professor, Department of Statistics, Bharathiar University Coimbatore - 641 046 2 Research Scholar, Department of Statistics, Bharathiar University Coimbatore - 641 046 E-mail: 1statjayalakshmi16@gmail.com,2 aleeshaa992@gmail.com

Abstract

Acceptance sampling is a statistical technique used to inspect the quality of a batch of products. An acceptance sampling plan under which sampling inspection is performed by conducting life test upon the sampled products is termed as reliability sampling plan. In this paper, a single acceptance sampling plan based on percentile is presented for Topp-Leone Gompertz (TL-G) distribution when the life test is truncated at a pre-specified time. The minimum sample size necessary to ensure the specified life percentile is obtained under a given consumer's risk. The operating characteristic values (and curves) of the sampling plans as well as the producer's risk are presented.

Keywords: Acceptance sampling plan, Percentiles, Topp-Leone Gompertz (TL-G) distribution, Operating characteristic values, Producer's risk, Minimum sample size

1. Introduction

In statistical quality control, acceptance sampling for products is one aspect of quality assurance. If the quality characteristic is regarding the lifetime of the product, the acceptance sampling problem becomes a life test. Quality personnel would like to know whether the lifetimes of products reach the consumer's minimum standard or not. Traditionally, when the life test indicates that the mean life of products exceeds the specified one, the lot of products is accepted, otherwise it is rejected. For the purpose of reducing the test time and cost, a truncated life test may be conducted to determine the smallest sample size to ensure a certain mean life of products when the life test is terminated at a preassigned time t, and the number of failures observed does not exceed a given acceptance number 'c'.

Nzei et al. [13] developed the Topp-Leone Gompertz (TL-G) distribution. Studies regarding truncated life tests can be found in Epstein [4], Sobel and Tischendrof [20], Goode and Kao [6], Gupta and Groll [7], Gupta [8], Fertig and Mann [5], Kantam and Rosaiah [10], Baklizi [2], Wu and Tsai [22], Rosaiah et al. [19], Tsai and Wu [21], Balakrishnan et al. [3], Rao et al. [15], Aslam et al. [1], Rao et al. [17], . Mahmood et al. [12]. All these authors designed acceptance sampling plans based on the mean life time under a truncated life test using different distributions.

In contrast, Lio et al. [11] considered acceptance sampling plans for percentiles using Birnbaum-Saunders distribution. Srinivasa Rao et al. [16] studied acceptance sampling plans for percentiles based on the inverse Rayleigh distribution. Rao et al. [18] considered acceptance sampling plans for percentiles using Half Normal distribution. Pradeepa Veerakumari and Ponneeswari [14] designed acceptance sampling plan based on percentiles of exponentiated Rayleigh distribution Jayalakshmi and Vijilamery [9] studied Special Type Double Sampling Plan for truncate life test using Gompertz Frechet distribution.

S Jyalakshmi, Aleesha A RT&A, No 2 (73) DESIGNING OF ACCEPTANCE SAMPLING PLAN_Volume 18, June 2023

2. Topp-Leone Gompertz (TL-G) Distribution

The TL-G distribution was developed by Nzei et al. in 2020. The CDF and PDF of the TL-G distribution are given by

F(t; a, 5,7) = [1 - e( -1)]" (!)

f (t; a,5,7) = 2a5eYte(-MY^! - e( -25(77t-1) )](«-1); t > 0 (2)

For given 0 < q < 1 the 100qt actual percentile of the TL-G distribution can be given by

1 7 1

tq = 7ln(1 - isln(1 -q5)) (3)

The tq increase as q increases Let

n=m(1 - 25 ln(1 - q5)) (4)

Then from (3), 7 = n/tq By letting a = t/tq , F(t) becomes

F(t; a, 5,7) = [1 - e( ^Y^''} )]a (5)

Equation (5) gives the modified cdf and by partially differentiating the equation (4) w.r.t a we will get the modified pdf for percentiles of TL-G distribution where tq is the 25th percentile of the given distribution.

3. Reliability Acceptance Sampling Plan

A sampling plan in which a decision about the acceptance or rejection of a lot is based on a single sample that has been inspected is known as a Single Sampling Plan. For a single sampling plan, one sample of items is selected at random from a lot and the disposition of the lot is determined from the resulting information. Single Sampling Plans are the most common and easiest plans to use.

Reliability Single Sampling Plans are part of an inspection procedure used to determine whether to accept or reject a specific lot based on lifetime. The Reliability Single Sampling Plan can be represented as (n, c, t/tq) . Here n and c are the sample size and acceptance number for the sampling plan. Assume that a life test is conducted and will be terminated at time tq0.

3.1. Operating Procedure

The acceptance sampling plan based on truncated life tests consists of the following:

1. Take a random sample of size n from the lot and inspect them.

2. The maximum test duration time is t.

3. Count the number of defectives d in the sample of size n.

4. The benchmark of defective (d) units is c, where if d < c defectives out of n occur at the end of the test period t0q, the lot is accepted. Otherwise reject the lot.

3.2. Minimum sample size

For a fixed P* our sampling plan is characterized by (n, c, t/Here we consider sufficiently large sized lots so that the binomial distribution can be applied. The problem is to determine for given values of P* (0 < P* < 1), t0 and c, the smallest positive integer, n required to assert that tq > tq must satisfy

tf (1 - p)(n-i) < (1 - P*) (6)

i=0

where p=F(t,aq), it is the probability of failure time during time t given a specified percentile of a lifetime t0° and it depends on the a = t/(t<0) since t0° increases as q increases. Accordingly, we have

F(t, a) < F(t, S0) ^^ a < a0

Or, equivalently

F(t; tq) < F(t; t0°) ^ tq > t°° The smallest sample size n satisfying eq. (6) can be obtained for any given sampling plan

t0)

(n, c, t/10) is given in Table 1.

3.3. Operating Characteristic (OC) Function

t0)

The OC function L(p) of the acceptance sampling plan (n,c,t/10) is the probability of accepting a

lot. It is given as

L(p) = tpl(1 - p)(n-i) (7)

i=0

where p = F(t, aq) . It should be noticed that F(t, aq) can be represented as a function of aq = t/tq. Therefore, we have

t1

p = F(t, a) = F(f, d-)

tq dq

where dq=tq /t0q

Using eq. (7) the OC values can be obtained for any sampling plan (n, c, t/t^). The OC values for the proposed sampling plan is presented in Table 3.

3.4. Producer's Risk (A)

The producer's risk is defined as the probability of rejecting the lot when tq > t^. For a given value of the producer's risk, say A , we are interested in knowing the value of dq to ensure the producer's risk is less than or equal to A if a sampling plan (n, c, t/t^) is developed at a specified confidence level P*. Thus, one needs to find the smallest value dq according to eq. (7).

L( p) > 1 - A

Based on the sampling plans (n,c,t/ tq0) given in Table 2 the minimum ratios of ^0.25 at the producer's risk of A = 0.05 are presented in Table 4.

4. Illustration

Assume that the life distribution is TL-G distribution, and the experimenter is interested in showing that the true unknown 25th percentile life t0.25 is at least 1000 hrs. Let a = 1.9,^ = 0.125, Y = 1.7and A = 0.05 . It is desire to stop the experiment at time t=3500 hrs. For the acceptance number c=1 from the Table 1 one can obtain the Single Sampling Plan (n, c, t/t^) = (5,1,3.5). The optimum sample sizes needed for the given requirement is found to be as n=5. The respective OC values for the proposed acceptance sampling plan (n, c, t/1'0) with P* = 0.95

Table 1

tq/(t0) 0.75 1 1.25 1.5 1.75 2 2.25 2.5

L(p) 0.000012 0.05241 0.37108 0.67370 0.83596 0.9137 0.95184 0.97148

for TL-G distribution from the Table 2 are given in above Table 1.

This shows that if the actual 25th percentile is equal to the required 25th percentile (t0.25/ t0250 = 3.5), the producer's risk is approximately 0.94759 (1 — 0.05241). The producer's risk almost equal to 0.03 or less when the actual 25th percentile is greater than or equal to 2.5 times the specified 25th percentile.

Table 4 gives the d0.25 values for c=1 and t/t025 = 3.5 to assure that the producer's risk is less than or equal to 0.05.

In this example, the value of d0 25 is 2.229656 for c=1, t/t025 = 3.5 and A =0.05. This means the product can have a 25th percentile life of 2.229656 times the required 25th percentile lifetime. That is under the above Single Sampling Plan the product is accepted with probability of at least 0.95.

^ 1 ■n

B 0.8

w a?

% 0.6 5

c

cd 0.4

I 02

O

0

0

Specified Lifetime

Figure 1: OC curve for the sampling plan (n = 5, c = 1, t/t0 25 = 3.5)

5. Construction of the Table

Step 1: Find the value of n for the fixed values of a = 1.9,5 = 0.125,7 = 1.7 and q=0.25 Step 2: Set the value of t/tq0 =0.7, 0.9, 0.9,1.0, 1.5, 2.0, 2.5, 3.0, 3.5

Step 3: Find the sample size n by satisfying L(p) < 1 — P* when P* = 0.99,0.95,0.90 and 0.75. Here P* is the probability of rejecting a bad lot and

L( p) = tf (1 — p)(n—l) i=0

Step 4: for the n value obtained find the d0.25 value such that L(p) > 1 — A where A = 0.05 and p = F(t/tq, 1/ dq ); dq = tq / ^

Table 2: Minimum Sample Size values necessary to assure 25th percentile for TL-G distribution

p* c t/(tq)

0.7 0.9 1.0 1.5 2.0 2.5 3 3.5

0.75 1 261 144 112 39 18 8 6 3

2 379 210 162 58 26 14 8 5

3 495 274 211 75 33 18 10 7

4 608 336 260 92 41 21 12 8

5 719 398 309 109 48 25 15 11

0.90 1 377 208 160 57 25 13 7 5

2 514 284 221 77 34 18 10 6

3 647 357 276 97 43 21 13 9

4 74 429 331 116 52 27 15 10

5 897 498 384 135 60 31 19 12

0.95 1 458 253 196 69 29 15 10 5

2 609 337 260 90 41 20 12 7

3 749 414 320 112 50 26 15 9

4 884 489 379 132 58 31 18 11

5 1017 562 435 152 68 35 21 13

0.99 1 637 353 271 94 42 21 12 7

2 803 446 343 121 53 27 15 9

3 970 534 414 144 64 32 19 12

4 1118 615 477 166 74 38 22 13

5 1263 695 539 187 83 43 25 16

Table 3: Operating characteristic values of the sampling plan (n, c = 1, t/(t0)) for a given P* under TL-G distribution

p* t/(tq 0) n tq/(tq0)

0.75 1 1.25 1.5 1.75 2 2.25 2.5

0.75 0.7 261 0.0303 0.2489 0.5162 0.7027 0.8152 0.8815 0.9214 0.9462

0.9 144 0.0255 0.2497 0.5292 0.7188 0.8293 0.8927 0.9299 0.9526

1 112 0.0225 0.2466 0.5320 0.7238 0.8342 0.8967 0.9330 0.9550

1.5 39 0.9497 0.9845 0.9938 0.9970 0.9984 0.9990 0.9994 0.9996

2 18 0.0066 0.2254 0.5666 0.7751 0.8794 0.9315 0.9588 0.9739

2.5 8 0.0104 0.2979 0.6584 0.8414 0.9216 0.9581 0.9675 0.9805

3 6 0.0008 0.1566 0.5351 0.7774 0.8905 0.9424 0.9675 0.9805

3.5 3 0.0046 0.2777 0.6755 0.8650 0.9395 0.9702 0.9840 0.9907

0.90 0.7 377 0.0038 0.0991 0.3191 0.5330 0.6879 0.7900 0.8559 0.8988

0.9 208 0.0029 0.0995 0.3321 0.5536 0.7087 0.8079 0.8704 0.9103

1 160 0.0026 0.1009 0.3407 0.5656 0.7202 0.8175 0.8779 0.9161

1.5 57 0.0010 0.0934 0.3585 0.6009 0.7568 0.8487 0.9027 0.9352

2 25 0.0005 0.0939 0.3889 0.6446 0.7963 0.8795 0.9256 0.9521

2.5 13 0.0002 0.0851 0.3998 0.6692 0.8202 0.8983 0.9395 0.9622

3 7 0.0001 0.0971 0.4461 0.7177 0.8563 0.9229 0.9560 0.9734

3.5 5 0.00001 0.0524 0.3710 0.6737 0.8359 0.9137 0.9518 0.9714

0.95 0.7 458 0.00087 0.0502 0.2214 0.4298 0.6009 0.7227 0.8054 0.8611

0.9 253 0.00062 0.0501 0.2324 0.4508 0.6246 0.7444 0.8237 0.8760

1 196 0.0005 0.0495 0.2367 0.4598 0.6348 0.7538 0.8316 0.8823

1.5 69 0.00018 0.0470 0.2583 0.5040 0.6833 0.7969 0.8172 0.9988

2 29 0.00011 0.0555 0.3082 0.5732 0.7464 0.8466 0.8667 0.9375

2.5 15 0.00003 0.0498 0.3201 0.6018 0.7757 0.8704 0.9039 0.9507

3 10 0.00001 0.0213 0.2451 0.5458 0.7456 0.8555 0.8172 0.9471

3.5 5 0.00001 0.0524 0.3710 0.6737 0.8359 0.9137 0.9518 0.9714

0.99 0.7 637 0.00003 0.0104 0.0933 0.2555 0.4306 0.5780 0.6896 0.7706

0.9 353 0.00001 0.0101 0.0993 0.2733 0.4558 0.6047 0.7143 0.7920

1 271 0.00001 0.0105 0.1049 0.2863 0.4725 0.6213 0.7290 0.8044

1.5 94 0.000004 0.0105 0.1243 0.3370 0.5379 0.6855 0.7850 0.8505

2 42 0 0.0092 0.1362 0.3750 0.5870 0.7324 0.8245 0.8822

2.5 21 0 0.0093 0.1562 0.4226 0.6409 0.7793 0.8614 0.9103

3 12 0 0.0074 0.1593 0.4450 0.6703 0.8059 0.8825 0.9262

3.5 7 0 0.0085 0.1852 0.4945 0.7177 0.8422 0.9085 0.9444

Table 4: Minimum ratio of true do.25 for the acceptability of a lot for the TL-G distribution and producer's risk of A = 0.05

p* t/(t0) n

0.75 0.7 261 2.5451 2.5464 2.5454 2.5462 2.5467 2.5470 2.5499 2.5496

0.9 144 2.4588 2.4600 2.4599 2.4611 2.4614 2.4622 2.4578 2.4750

1 112 2.4264 2.4276 2.4277 2.4290 2.4293 2.4301 2.4273 2.4281

1.5 39 2.2498 2.2507 2.2517 2.2433 2.2472 2.2521 2.2500 2.2600

2 18 2.1487 2.1495 2.1506 2.1457 2.1494 2.1478 2.1416 2.1542

2.5 8 1.9254 1.9191 1.9240 1.9192 1.9183 1.9201 1.9305 1.9229

3 6 2.0570 2.0575 2.0514 2.0556 2.0526 2.0559 2.0600 2.0635

3.5 3 1.8133 1.8095 1.8127 1.8122 1.8089 1.8143 1.8122 1.8124

0.90 0.7 377 3.0392 3.0515 3.0508 3.0483 3.0461 3.0442 3.0389 3.0529

0.9 208 2.9272 2.9362 2.9356 2.934 2.9337 2.9324 2.9304 2.9324

1 160 2.8693 2.8768 2.8763 2.8754 2.8753 2.8742 2.8734 2.8761

1.5 57 2.6634 2.6585 2.6691 2.6692 2.6573 2.6571 2.6620 2.6675

2 25 2.4689 2.4683 2.4662 2.4693 2.4698 2.4717 2.4658 2.4750

2.5 13 2.3440 2.3433 2.3431 2.3452 2.3463 2.3419 2.3398 2.3515

3 7 2.1876 2.1875 2.1887 2.1896 2.1861 2.1826 2.1875 2.1911

3.5 5 2.2309 2.2296 2.2298 2.2313 2.2249 2.2319 2.2275 2.2250

0.95 0.7 458 3.3592 3.3560 3.3529 3.3445 3.3597 3.3578 3.3548 3.3483

0.9 253 3.2281 3.2253 3.2241 3.2174 3.2289 3.2283 3.2264 3.2238

1 196 3.1704 3.1678 3.1667 3.1612 3.1563 3.1710 3.1695 3.1681

1.5 69 2.9000 2.9104 2.9096 2.9082 2.9081 2.9065 2.9054 2.9086

2 29 2.6336 2.6372 2.6368 2.6370 2.6371 2.6260 2.6323 2.6365

2.5 15 2.4881 2.4836 2.4807 2.4846 2.4853 2.4879 2.4804 2.5

3 10 2.5268 2.5294 2.5290 2.5197 2.5206 2.5241 2.5288 2.5250

3.5 5 2.2309 2.2296 2.2298 2.2313 2.2249 2.2319 2.2275 2.2250

0.99 0.7 637 3.9519 3.9484 3.9580 3.9540 3.9405 3.9577 3.9512 3.9411

0.9 353 3.7928 3.7890 3.7985 3.7958 3.7860 3.7990 3.7958 3.7899

1 271 3.7035 3.6996 3.7089 3.7065 3.6989 3.7096 3.7076 3.7032

1.5 94 3.3554 3.3517 3.3462 3.3578 3.3558 3.3537 3.3496 3.3429

2 42 3.1019 3.0989 3.0974 3.0906 3.1031 3.1027 3.1017 3.1013

2.5 21 2.8693 2.8672 2.8660 2.8642 2.8648 2.8632 2.8635 2.8679

3 12 2.7162 2.7239 2.7230 2.7227 2.7231 2.7232 2.7245 2.7183

3.5 7 2.5509 2.5545 2.5539 2.5544 2.5546 2.5469 2.5534 2.5537

6. CONCLUSION

In this paper we have derived the acceptance sampling plans based on percentiles for the Topp-Leone Gompertz (TL-G) distribution when the life test is truncated at a pre-fixed time. The minimum sample size required to decide upon accepting or rejecting a lot based on its specified 25th percentile, the operating characteristic function values and corresponding producer's risk are obtained. Tables provided are helpful for the industrial use to save the cost and time of the experiment.

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