Designing of multiple dependent state sampling plan for Type-II generalized half logistic distribution

ABSTRACT

In this study, we propose the multiple dependent state (MDS) sampling plan for a time truncated life test when the lifetime of product follows the Type-II Generalized Half Logistic Distribution (TGHLD). The quality of the product is considered in terms of percentile lifetime. The optimal plan parameters of the our proposed MDS sampling plan, such as the sample size, the acceptance and rejection numbers, and the number of preceding lots needed for making the decision whether to accept or reject the current lot, are determined using the approach of two points on the operating characteristic curve. Tables are obtained for various values of shape parameter and results are discussed. The real data analysis is given to illustrate the applicability of the proposed in the industry. Comparison is also made with the existing plans.

KEYWORDS:

• Consumer’s risk
• producer’s risk
• percentile ratio, acceptable quality level and limiting quality level

1. Introduction

Acceptance sampling is a sampling methodology in which the decision regarding whether the submitted lot is acceptable or not is arrived at using the results of random samples drawn from the concerned lot. Acceptance sampling plays a prominent role in statistical quality control in assuring the quality of manufactured products. When implementing acceptance sampling, the decision on the disposition of the lot is immediately made by inspection of the sample items, so that the inspection cost is reduced and time is saved. Because the decision is made using the results of random samples, there may be a chance to classify s qualified lot is rejected, the producer will be affected when an unqualified lot is accepted, and the consumer will be affected. Therefore, the probability of rejecting a good lot or accepting a bad lot is usually referred to as the producer’s risk$\left(\mathit{\alpha }\right)$ and consumer’s risk$\left(\mathit{\beta }\right)$, respectively, the appropriate quality level corresponding to producer’s risk and consumer’s risk are, respectively, termed as acceptable quality level (AQL) and limiting quality level (LQL).

Kantam et al. (2014) proposed a new model called the standard Type II generalized half logistic distribution (GHLD), whose cumulative distribution function (cdf) is given by

(1) $\begin{array}{l}\mathit{F}(\mathit{t};\mathit{\theta })=1-{\left[1-\mathit{G}\left(\mathit{t}\right)\right]}^{\theta }=1-{\left[1-\left(1-\frac{2}{1+e^t}\right)\right]}^{\mathit{\theta }}\\ =1-\frac{2^{\theta }}{{\left(1+e^t\right)}^{\mathit{\theta }}}\text{ ; 0 lt t lt }\mathrm{\infty }\text{, }\mathit{\theta }>0\end{array}$(1)

where $\mathit{\theta }$ is called as the shape parameter and the corresponding probability density function (pdf) is obtained as follows:

(2) $\mathit{f}(\mathit{t};\mathit{\theta })=\frac{2^{\theta }\mathit{\theta }{e}^t}{{(1+e^t)}^{\mathit{\theta }+1}};\text{ 0 lt t lt }\mathrm{\infty }\text{, }\mathit{\theta }>0$(2)

where $\mathit{\theta }$ is shape parameter. When $\mathit{\theta }$ = 1, TGHLD reduces to half-logistic distribution.

Olapade (2014) has studied some properties of Type I GHLD and they have obtained MLEs of the parameters based on complete sample. Rosaiah et al. (2014) constructed reliability test plan for the Type II GHLD to determine the termination time of the experiment for a given sample size. Kumar et al. (2015) have established some new explicit expressions and recurrence relations for marginal and joint moment generating functions of K^th upper record values from Type I GHLD. Kumar and Farooqi (2017) have derived explicit expressions and recurrence relations for the marginal and joint moment generating function of generalized order statistics from Type I GHLD.

Awodutire et al. (2016) used three-parameter Type I GHLD as a survival model to a breast cancer data. Bello et al. (2017) have introduced five-parameter (one scale parameter, one location parameter, three shape parameters) Type I GHLD and they studied some distribution properties. Sd et al. (2018) have constructed a linear estimator of $\mathit{\sigma }$ called as linear approximate ML estimator (LAMLE) and based on a Monte Carlo simulation study.

Bello et al. (2020) have discussed ML estimation of the parameter of five-parameter Type I GHLD based on complete sample. Samuel and Olapade (2019) have introduced a new generalization of the Type I GHLD called transmuted Type I GHLD. Awodutire et al. (2020a) have derived a new distribution called Lehmann Type II GHLD and have studied statistical properties. Awodutire et al. (2020b) have introduced a new distribution called Beta Type I GHLD, which is a generalization of the Type I GHLD. Wang et al. (2023) have derived maximum likelihood estimation of the parameters of the inverse Gaussian distribution using maximum rank set sampling with unequal samples.

The lifetime is 100q-th percentile of TGHLD given as follows:

(3) $t_q=\mathit{\sigma }{\eta }_q,\text{where}{\eta }_q=\mathit{ln}\left(2{(1-\mathit{q})}^{-1/\theta }-1\right)$(3)

The lifetime is 50^th percentile of TGHLD given as follows:

$t_{0.5}=\mathit{\sigma }\left[\mathit{log}\left(2{(1-0.5)}^{-1/\theta }-1\right)\right]$

If the lifetime of the product t follows TGHLD, then $\mathit{p}=\mathit{F}(t_0;{\sigma }_0,{\theta }_0)$, in a convenient approach, to determine the experiment test termination time $t_0$, as $t_0={\delta }_q^0{t}_q^0$ for a constant ${\delta }_q^0$and the targeted 100qth lifetime percentile, $t_q^0$, suppose $t_q$ be the true 100qth lifetime percentile. Then, p can be written as follows:

(4) $\begin{array}{l}\mathit{p}=1-\frac{2^{\theta }}{{\left(1+\exp \left(\frac{t_0}{\mathit{\sigma }}\right)\right)}^{\mathit{\theta }}}=1-\frac{2^{\theta }}{{\left(1+\exp \left(\frac{{\eta }_q{\delta }_q^0}{\left(t_q/t_q^0\right)}\right)\right)}^{\mathit{\theta }}}.\end{array}$(4)

The concept of multiple dependent state (MDS) sampling plan was first introduced by Wortham and Baker (1976). The chain sampling plan proposed by Dodge (1977) Later, many authors have studied MDS sampling plan under various situations Govindaraju and Subramani (1993), Balamurali and Jun (2007) and Balamurali et al. (2016). For more details on designing an attribute MDS sampling plans, please refer to Soundararajan and Vijayaraghavan (1990), Subramani and Haridoss (2012), Aslam et al. (2013), Aslam et al. (2014), Wu et al. (2015), Wu et al. (2016), Balamurali et al. (2017a), Balamurali et al. (2017b) and Aslam et al. (2019). Later, the concept of multiple dependent state sampling has been used in control chart design. For example, Aslam, Nazir, et al. (2015) and Aslam, Azam, et al. (2015) studied a control chart for an exponential distribution using MDS sampling and Aslam et al. (2017) studied a control chart for gamma distribution using MDS sampling.

In this paper, the main aim of this study is to develop a MDS sampling plan to ensure the percentile ratio of the product based on time truncated life test assuming the lifetime distribution of the product follows a TGHLD. The operating procedure, the operating characteristic (OC) function of the multiple dependent state sampling plans, and the proposed methodology of the MDS sampling plan under TGHLD are given. Tables are obtained for various values of shape parameter and results are discussed. A comparison between the performances of the proposed plan and single sampling plan (SSP) and repetitive sampling plan (RSP) are discussed.

2. Multiple dependent state sampling plan based on TGHLD

The operating procedure proposed and procedure of the MDS sampling plans under TGHLD is discussed. The following step by step procedure for MDS sampling plan is suggested by Balamurali and Jun (2007).

Operating procedure is as follows:

Step-1: For each lot, select a random sample of n items from the current lot. Then put them on life test for specified time $t_0$.

Step-2: The observed number of failures before the experiment time $t_0$, it is denoted as D.

Step-3: If $\mathit{D}\le c_1$ accept the lot, if $\mathit{D}>c_2$, reject the current lot. If $c_1<\mathit{D}\le c_2$, accept the current lot if the number of failures before the experiment time $t_0$ is less than or equal to $c_1$ in the previous $\mathit{m}$lots.

Thus, our proposed MDS sampling plan is specified by four parameters are $c_1,c_2,\mathit{m}\text{and}\mathit{n}$, where $c_1$ is the maximum number of allowable failure items for unconditional acceptance, $c_2$ is the maximum number of additional failure items for conditional acceptance and $\mathit{m}$ is the number of previous lots needed to make a decision and $\mathit{n}$ is the sample number.

In particular, the attributes of MDS sampling plan are the ordinary case of a single sampling plan (SSP) and it’s concurrence to SSP when either $\mathit{m}\to \mathrm{\infty }$ and/or $c_2=c_1=\mathit{c}.$ The OC function of the MDS sampling plan under TGHLD for a lifetime truncated test is given by the following:

(5) $\begin{array}{l}{P}_a\left(\mathit{p}\right)=\mathit{p}\left(\mathit{D}\le c_1\right)+\mathit{p}\left(c_1<\mathit{D}\le c_2\right){\left(p\left(D\le c_1\right)\right)}^{\mathit{m}}.]\end{array}$(5)

Under the binomial distribution, Equation 5(5) $\begin{array}{l}{P}_a\left(\mathit{p}\right)=\mathit{p}\left(\mathit{D}\le c_1\right)+\mathit{p}\left(c_1<\mathit{D}\le c_2\right){\left(p\left(D\le c_1\right)\right)}^{\mathit{m}}.]\end{array}$(5) can be rewritten as follows:

(6) $\begin{array}{l}{P}_a\left(\mathit{p}\right)=\sum _{\mathit{D}=0}^{c_1}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right)p^D{\left(1-\mathit{p}\right)}^{\mathit{n}-\mathit{D}}+\sum _{D=c_1+1}^{c_2}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right)\\ p^D{\left(1-\mathit{p}\right)}^{\mathit{n}-\mathit{D}}{\left[\sum _{\mathit{D}=0}^{c_1}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right)p^D{\left(1-\mathit{p}\right)}^{\mathit{n}-\mathit{D}}\right]}^{\mathit{m}}.\end{array}$(6)

Mostly, the sampling plans are designed to minimize the average sample number (ASN). Commonly, any sampling plan with minimum average sample number would always be desirable because when the average sample number is minimum, the corresponding inspection time and inspection cost will be reduced. The ASN of the MDS sampling plans is the sample size n only. Hence, the design parameters for the proposed plan with minimum sample size will be obtained by solving the below optimization technique.

(7) $\begin{array}{l}\text{Minimum }\mathit{ASN}\left(\mathit{p}\right)=\mathit{n}\\ \text{subject to }{p}_a\left(p_1\right)\ge 1-\mathit{\alpha }\\ p_a\left(p_2\right)\le \mathit{\beta }\\ \mathit{n}>1,\mathit{m}\ge 1,\\ c_2>c_1\ge 0\end{array}$(7)

(8) $\begin{array}{l}{P}_a\left(p_1\right)=\sum _{\mathit{D}=0}^{c_1}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right){p_1}^{\mathit{D}}{\left(1-p_1\right)}^{\mathit{n}-\mathit{D}}+\sum _{D=c_1+1}^{c_2}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right)\\ {p_1}^{\mathit{D}}{\left(1-p_1\right)}^{\mathit{n}-\mathit{D}}{\left[\sum _{\mathit{D}=0}^{c_1}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right){p_1}^{\mathit{D}}{\left(1-p_1\right)}^{\mathit{n}-\mathit{D}}\right]}^{\mathit{m}}\end{array}$(8)

(9) $\begin{array}{l}{P}_a\left(p_2\right)=\sum _{\mathit{D}=0}^{c_1}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right){p_2}^{\mathit{D}}{\left(1-p_2\right)}^{\mathit{n}-\mathit{D}}+\sum _{D=c_1+1}^{c_2}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right)\\ {p_2}^{\mathit{D}}{\left(1-p_2\right)}^{\mathit{n}-\mathit{D}}{\left[\sum _{\mathit{D}=0}^{c_1}\left(\begin{array}{l}\mathit{n}\\ \mathit{D}\end{array}\right){p_2}^{\mathit{D}}{\left(1-p_2\right)}^{\mathit{n}-\mathit{D}}\right]}^{\mathit{m}}\end{array}$(9)

where $p_1$ be probability of a failure corresponding to the producer’s risk and $p_2$ be the probability of a failure corresponding to the consumer’s risk. In this approach, the quality level is the ratio of its true percentile lifetime to the true lifetime, $t_q/t_q^0$. The percentile ratio concept helps the producer to enhance the product quality. The percentile ratio is $t_q/t_q^0=1$ at the consumer’s risk and $t_q/t_q^0=2,4,6,8\text{and}10$ at producer’s risk.

Tables 1–3 present best parameters of the proposed MDS sampling plan for TGHLD with known shape parameters $\mathit{\theta }=1.5,2.0\text{and}2.5$ under truncated life tests. Given the producer’s risk $\mathit{\alpha }=0.05$ and consumer’s risk $\mathit{\beta }=0.25,0.10,0.05\text{and}0.01$ and the termination ratio is taken as $\mathit{a}=0.5,0.7\text{and}1.0$ at 50^th percentile value. From these tables, it is to be noticed that for fixed values of $\mathit{\beta },t_q/t_q^0\text{and}\mathit{\theta }$ when the value of $\mathit{a}$ is increased from 0.5 to 1.0, the sample size $\mathit{n}$is decreased. If $\mathit{\beta }$ value decreases from 0.25 to 0.01, the sample size $\mathit{n}$ increases when all other parametric combinations are same. Also, if parameter $\mathit{\theta }$ increases, sample size $\mathit{n}$ decreases when all other parametric combinations are fixed.

Table 1. Best parameters of the proposed MDS plan for TGHLD with $\mathit{\theta }$ = 1.5

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=0.5$					$\mathit{a}=0.7$					$\mathit{a}=1.0$
		n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$
0.25	2	31	6	11	2	0.9633	21	5	9	1	0.9510	14	5	13	2	0.9543
	4	10	1	9	2	0.9538	7	1	6	2	0.9584	5	1	2	1	0.9583
	6	9	1	3	4	0.9835	7	1	6	2	0.9890	5	1	2	1	0.9875
	8	6	0	2	1	0.9599	4	0	1	1	0.9533	3	0	2	1	0.9595
	10	5	0	4	2	0.9651	4	0	1	1	0.9689	3	0	2	1	0.9732
0.10	2	50	9	19	2	0.9517	36	9	19	2	0.9578	25	8	15	1	0.9500
	4	18	2	6	2	0.9658	13	2	3	1	0.9524	9	2	8	2	0.9714
	6	13	1	5	2	0.9697	9	1	2	2	0.9590	7	1	3	1	0.9789
	8	13	1	5	2	0.9882	9	1	2	2	0.9824	7	1	3	1	0.9920
	10	13	1	5	2	0.9945	6	0	2	1	0.9508	7	1	3	1	0.9964
0.05	2	-	-	-	-	-	-	-	-	-	-	31	10	14	1	0.9501
	4	26	3	8	2	0.9759	16	2	5	1	0.9529	11	2	5	1	0.9599
	6	17	1	4	1	0.9607	12	1	4	1	0.9631	8	1	3	1	0.9665
	8	16	1	2	1	0.9708	11	1	10	2	0.9796	8	1	3	1	0.9870
	10	16	1	2	1	0.9846	11	1	10	2	0.9903	8	1	3	1	0.9940
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	39	4	7	1	0.9738	28	4	8	1	0.9787	17	3	8	1	0.9546
	6	28	2	5	1	0.9758	20	2	5	1	0.9770	14	2	6	1	0.9795
	8	22	1	3	1	0.9585	16	1	5	1	0.9629	11	1	5	1	0.9658
	10	22	1	3	1	0.9795	15	1	2	1	0.9668	11	1	5	1	0.9832

Table 2. Best parameters of the proposed MDS plan for TGHLD with $\mathit{\theta }$ = 2

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=0.5$					$\mathit{a}=0.7$					$\mathit{a}=1.0$
		n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$
0.25	2	30	6	16	3	0.9533	22	6	8	2	0.9529	16	6	8	2	0.9588
	4	11	1	4	1	0.9621	7	1	6	2	0.9549	5	1	2	1	0.9554
	6	9	1	4	3	0.9862	7	1	6	2	0.9879	5	1	2	1	0.9864
	8	6	0	2	1	0.9575	4	0	1	1	0.9506	3	0	2	1	0.9573
	10	6	0	2	1	0.9718	3	0	2	4	0.9518	3	0	2	1	0.9716
0.10	2	49	9	16	2	0.9505	36	9	19	2	0.9500	26	9	19	2	0.9547
	4	18	2	12	2	0.9616	13	2	12	2	0.9625	9	2	8	2	0.9683
	6	13	1	11	2	0.96	9	1	3	2	0.9693	7	1	3	1	0.9769
	8	13	1	11	2	0.9868	9	1	3	2	0.9881	7	1	3	1	0.9912
	10	13	1	11	2	0.9939	9	1	3	2	0.9945	7	1	3	1	0.9960
0.05	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	26	3	4	1	0.9505	16	2	6	1	0.9500	11	2	5	1	0.9558
	6	17	1	5	1	0.9577	12	1	4	1	0.9596	8	1	3	1	0.9635
	8	16	1	2	1	0.9681	11	1	2	1	0.9729	8	1	3	1	0.9856
	10	16	1	2	1	0.9830	11	1	2	1	0.9858	8	1	3	1	0.9933
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	38	4	14	2	0.9601	27	4	9	2	0.9646	19	4	14	2	0.9687
	6	27	2	12	2	0.9593	20	2	6	1	0.9752	14	2	6	1	0.9769
	8	22	1	4	1	0.9600	16	1	6	1	0.9594	11	1	5	1	0.9626
	10	21	1	5	2	0.9686	15	1	2	1	0.9638	11	1	5	1	0.9815

Table 3. Best parameters of the proposed MDS plan for TGHLD with $\mathit{\theta }$ = 2.5

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=0.5$					$\mathit{a}=0.7$					$\mathit{a}=1.0$
		n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$
0.25	2	30	6	9	2	0.9564	22	6	9	2	0.9596	16	6	8	2	0.9559
	4	11	1	4	1	0.9597	7	1	6	2	0.9523	5	1	2	1	0.9533
	6	9	1	8	3	0.9852	7	1	2	1	0.9842	5	1	2	1	0.9856
	8	6	0	2	1	0.9559	7	1	2	1	0.9933	3	0	2	1	0.9557
	10	5	0	4	2	0.9616	4	0	1	1	0.9657	3	0	2	1	0.9706
0.10	2	51	9	14	1	0.9517	37	9	13	1	0.9523	26	9	19	2	0.9507
	4	18	2	12	2	0.9584	13	2	12	2	0.9596	9	2	8	2	0.9661
	6	13	1	2	1	0.9590	9	1	4	2	0.9690	7	1	3	1	0.9755
	8	13	1	2	1	0.9817	9	1	4	2	0.9879	7	1	3	1	0.9906
	10	13	1	2	1	0.9905	9	1	4	2	0.9944	7	1	3	1	0.9957
0.05	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	26	3	5	1	0.9736	19	3	5	1	0.9739	11	2	5	1	0.9529
	6	16	1	3	1	0.9568	12	1	5	1	0.9577	8	1	3	1	0.9613
	8	16	1	3	1	0.9826	11	1	2	1	0.9713	8	1	3	1	0.9846
	10	15	1	5	3	0.9842	11	1	2	1	0.9849	8	1	3	1	0.9928
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	38	4	7	1	0.9703	27	4	14	2	0.9609	19	4	14	2	0.9656
	6	27	2	4	1	0.9671	19	2	12	2	0.9604	14	2	6	1	0.9751
	8	22	1	5	1	0.9582	15	1	3	1	0.9591	11	1	5	1	0.9603
	10	21	1	2	1	0.9602	15	1	3	1	0.9798	11	1	5	1	0.9802

3. Application of the proposed MDS sampling plan with real data set

In this section, we illustrate that the proposed MDS sampling plans with the help of real data set are given by Lawlees (2003). The following data represent the number of 1000s of cycles to failure for electrical appliances in a life test.

Table

0.014	0.034	0.059	0.061	0.069	0.080	0.123	0.142	0.165	0.210
0.381	0.464	0.479	0.556	0.574	0.839	0.917	0.969	0.991	1.064
1.088	1.091	1.174	1.270	1.355	1.397	1.477	1.578	1.649	1.702
1.893	1.932	2.001	2.161	2.292	2.326	2.337	2.628	2.785	2.811
2.993	3.122	3.248	3.715	3.790	3.857	3.912	4.100	4.106	4.116
4.315	4.510	4.580	5.267	5.299	5.583	6.065	0.9701

The maximum likelihood estimate of TGHLD with known $\mathit{\sigma }$ for the above data is $\hat{\theta }=\text{0}\text{.5749}\text{.}$ To test the goodness of fit, we apply the Kolmogorov-Smirnov test, and it is observed that the data set Kolmogorov-Smirnov statistic is 0.10511 with p-value 0.4887. To emphasize the goodness of fit, we also plot histogram and superimposed empirical density. Further, goodness of fit is stressed with Q-Q plot, and both plots are displayed in Figure 1.

Figure 1. Histogram superimposed empirical density plots and Q-Q plot of the fitted TGHLD for cycles to failure for electrical appliances data.

Suppose that lifetime of the products follows TGHLD with parameter $\hat{\theta }=\text{0}\text{.5749}\text{.}$ Assuming that $\mathit{\alpha }=0.05$,$\mathit{\beta }=0.25$ $\mathit{a}=0.5$ and $t_q/t_q^0=2,$ from Table 4, we note that the optimal design parameters are $\mathit{n}=30,c_1=5,c_2=8\text{and}\mathit{m}=2.$ Thus, the design can be implemented as follows: Take a random sample of size 30 from the current lot. Accepted the current lot if five failures occur before 0.069 cycles to failures for electrical appliances. If more than eight failures, the current lot will be rejected. If the number of failures are from five to eight, the disposition of the current lot will be dependent until the next preceding lot is tested. If the next lot is accepted, then accept the current lot. Otherwise, reject current lot and next lot.

Table 4. Best parameters of the proposed MDS plan for TGHLD with $\mathit{\theta }$ = 0.5749

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=0.5$					$\mathit{a}=0.7$					$\mathit{a}=1.0$
		n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$	n	$c_1$	$c_2$	m	$P_a\left(p_1\right)$
0.25	2	28	5	8	2	0.9565	20	5	15	2	0.9630	14	5	13	2	0.9676
	4	10	1	3	2	0.9666	7	1	6	2	0.9708	5	1	2	1	0.9684
	6	10	1	3	2	0.9916	7	1	6	2	0.9929	5	1	2	1	0.9911
	8	5	0	1	2	0.9526	4	0	1	1	0.9632	3	0	2	1	0.9677
	10	5	0	1	2	0.9684	4	0	1	1	0.9758	3	0	2	1	0.9790
0.10	2	48	8	18	2	0.9531	32	7	12	1	0.9517	21	7	11	2	0.9529
	4	19	2	6	2	0.9743	13	2	3	2	0.9601	9	2	8	2	0.9813
	6	14	1	11	2	0.9747	10	1	2	1	0.9705	7	1	3	1	0.9855
	8	14	1	11	2	0.9904	10	1	2	1	0.9874	7	1	3	1	0.9948
	10	9	0	2	1	0.9570	7	0	6	1	0.9504	4	0	1	1	0.9529
0.05	2	63	10	14	1	0.9557	41	9	13	1	0.9511	29	9	15	1	0.9581
	4	22	2	5	2	0.9505	16	2	4	1	0.9627	11	2	5	1	0.9732
	6	17	1	11	2	0.9531	12	1	3	1	0.9719	8	1	3	1	0.9767
	8	17	1	2	1	0.9764	12	1	3	1	0.9894	8	1	3	1	0.9914
	10	17	1	2	1	0.9877	12	1	3	1	0.9952	8	1	3	1	0.9962
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	4	36	3	7	1	0.9610	24	3	13	2	0.9510	17	3	8	1	0.9717
	6	29	2	3	1	0.9567	20	2	4	1	0.9826	14	2	6	1	0.9876
	8	23	1	3	1	0.9689	16	1	4	1	0.9751	11	1	5	1	0.9768
	10	23	1	3	1	0.9850	16	1	4	1	0.9881	11	1	5	1	0.9891

In order to comparative study, the efficiency of the proposed MDS sampling plan with the SSP and RSP. Tables 5 and 6 present the ASN of the proposed MDS sampling plan, SSP and RSP. From Tables 5 and 6, we observed that the ASN of the proposed MDS sampling plan is smaller than the RSP for many combinations and is also smaller than SSP for all combinations. For example, when $\mathit{\alpha }=0.05$,$\mathit{\beta }=0.25$, $\mathit{\theta }=1.5$, $\mathit{a}=0.5$ and $t_q/t_q^0=2$, the ASN of the proposed MDS sampling plan is 31, whereas the ASN of SSP is 50 and ASN of the RSP is 43.53. Similarly, when $\mathit{\alpha }=0.05$, $\mathit{\beta }=0.25$, $\mathit{\theta }=1.5$, $\mathit{a}=1.0$ and $t_q/t_q^0=2$, the ASN of the proposed MDS sampling plan is 14, whereas the ASN of SSP is 27 and ASN of the RSP is 23.05. Hence, we conclude that the proposed MDS sampling plan under TGHLD life time model is more economical compared to SSP and RSP.

Table 5. ASN values of the proposed MDS plan, repetitive sampling plan (RSP) and single sampling plan (SSP) for the TGHLD when $\mathit{a}=0.5$

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=1.5$			$\mathit{a}=2.0$			$\mathit{a}=2.5$
		MDS	RSP	SSP	MDS	RSP	SSP	MDS	RSP	SSP
0.25	2	31	43.53	50	30	39.92	54	30	42.96	49
	4	10	14.32	18	11	17.02	18	11	13.43	18
	6	9	8.98	14	9	8.93	14	9	8.89	14
	8	6	8.98	9	6	8.93	9	6	8.89	9
	10	5	8.98	9	6	8.93	9	5	8.89	9
0.10	2	50	58.32	79	49	59.94	82	51	57.74	81
	4	18	21.58	28	18	21.93	27	18	16.86	27
	6	13	16.42	23	13	16.32	23	13	16.26	22
	8	13	16.42	18	13	10.37	18	13	10.33	18
	10	13	11.13	18	13	10.37	18	13	10.33	18
0.05	2	-	66.39	–	-	65.86	–	-	16.85	–
	4	26	22.46	36	26	22.21	36	26	23.73	35
	6	17	17.06	26	17	16.83	26	16	16.68	26
	8	16	17.06	21	16	16.83	21	16	16.68	21
	10	16	11.82	21	16	16.83	21	15	11.71	21
0.01	2	-	79.17	–	-	78.53	–	-	81.91	–
	4	39	28.69	54	38	28.25	53	38	27.99	53
	6	28	23.32	39	27	22.99	38	27	22.78	38
	8	22	18.08	34	22	17.92	33	22	17.82	33
	10	22	18.08	28	21	17.92	27	21	17.82	27

Table 6. ASN values of the proposed MDS plan, repetitive sampling plan (RSP) and single sampling plan (SSP) for the TGHLD when $\mathit{a}=1.0$

$\mathit{\beta }$	$t_q/t_q^0$	$\mathit{a}=1.5$			$\mathit{a}=2.0$			$\mathit{a}=2.5$
		MDS	RSP	SSP	MDS	RSP	SSP	MDS	RSP	SSP
0.25	2	14	23.05	27	16	23.05	27	16	23.05	27
	4	5	7.84	10	5	10.24	10	5	10.00	10
	6	5	4.80	7	5	4.80	7	5	4.80	7
	8	3	4.80	5	3	4.80	5	3	4.80	5
	10	3	4.80	5	3	4.80	5	3	4.80	5
0.10	2	25	29.81	39	26	30.97	44	26	30.97	44
	4	9	9.41	14	9	11.76	14	9	11.76	14
	6	7	8.37	9	7	8.37	12	7	8.37	12
	8	7	5.33	9	7	5.33	9	7	5.23	9
	10	7	5.33	9	7	5.33	9	7	5.23	9
0.05	2	31	32.41	51	-	32.41	51	-	34.25	53
	4	11	11.96	18	11	11.75	18	11	11.76	18
	6	8	8.93	13	8	8.93	13	8	8.93	13
	8	8	8.93	11	8	8.93	11	8	8.93	11
	10	8	5.93	11	8	5.93	11	8	5.93	11
0.01	2	-	38.86	–	-	43.23	–	-	43.23	–
	4	17	14.83	27	19	14.83	27	19	14.83	27
	6	14	12.03	19	14	12.03	19	14	12.03	19
	8	11	9.31	17	11	9.31	17	11	9.31	17
	10	11	9.31	14	11	9.31	14	11	9.31	14

In addition, Figure 2 OC curves for comparison between MDS sampling plan and SSP under TGHLD. The MDS and SSP plans are selected to have the same acceptance number and sample size under the time truncated life test. The parameters of the proposed MDS sampling plan considered in Figure 2 are $\mathit{n}=20,c_1=1,c_2=2,\text{and}\mathit{m}=2$ single sampling $\mathit{n}=20,\mathit{c}=1.$ From this Figure 2, it is to be notice that for small values of failure probability, the probability of acceptance of the lot under the MDS sampling plan is higher than that of SSP and the OC curve of the MDS plan moves towards the OC curve of the SSP when the failure probability is increased. For the higher values of the failure probability, the two plans give almost the same probability of acceptance. From this, we can see that the MDS plan provides more protection for the producer than SSP when the quality level is better.

Figure 2. OC curves for comparison between MDS and SSP.

With another comparison, Figure 3 provides the OC curve of the proposed MDS sampling plans $\mathit{n}=20,c_1=1,c_2=2,\text{and}\mathit{m}=2$, RSP $\mathit{n}=20,c_1=1,c_2=2$ and SSP $\mathit{n}=20,\mathit{c}=1$based on the time truncated life test under TGHLD with known shape parameter. From this Figure, we see that when the mean ratio is small, the probability of acceptance of a submitted lot under the MDS plan is smaller than that of RSP, whereas if the mean ratio is high, the MDS plan provides more probability of acceptance than that of SSP. From this, we conclude that the proposed MDS sampling plan protects the producer when quality level is good, as well as safeguard the consumer when quality is bad.

Figure 3. OC curves for comparison between MDS, RSP and SSP.

The extension of these classical MDS sampling plans will be extended for neutrosophic statistics MDS sampling plans under uncertain environment. Neutrosophic statistics is the extension of classical statistics and is applied when the data come from a complex process or from an uncertain environment. In recent years, Aslam and Arif (2018) developed sudden death sampling plan using neutrosophic statistics, Aslam (2018) worked on a variable sampling plan based on exponential distribution under neutrosophic statistics; Aslam (2019) proposed a plan for testing of reliability under neutrosophic statistics; Aslam (2021) studied testing average wind speed using sampling plan for Weibull distribution under indeterminacy; and Rao and Aslam (2021) developed a inspection plan for COVID-19 patients for Weibull distribution using repetitive sampling under indeterminacy.

5. Conclusions

In this study, we proposed an MDS sampling plan for a Type-II Generalized half distribution. In this approach, the quality level as the ratio of its true percentile lifetime to the product follows a Type-II Generalized half logistic distribution. The optimal parameters of our proposed MDS sampling plan using the approach of two points on the operating characteristic curve by considering the producer’s and consumer’s risk simultaneously. The proposed MDS sampling plan is effective in terms of ASN. The ASN of the proposed MDS sampling plan is smaller than the single sampling plan for all combinations of AQL and LQL. From this study, it is concluded that the proposed MDS sampling plan under TGHLD life time model is more economical compared to SSP and RSP. This study has some limitations that indicate possible areas for future research. Our proposed MDS plans were developed under a quality characteristic with assumptions of TGHLD. However, if the quality characteristics are connected with uncertain environment, we could not adopt the proposed MDS sampling plans. The execution of our developed MDS sampling plans is a precious area for future research to develop an MDS plans under neutrosophic statistics. Hence, our work can be carried out using neutrosophic statistics as future research.

Public interest statement

The main aim of this study is to develop a multiple dependent state (MDS) sampling plan to ensure the percentile ratio of the product based on time truncated life test assuming the lifetime distribution of the product follows a Type-II Generalized Half Logistic Distribution (TGHLD). The operating procedure, the operating characteristic (OC) function of the MDS sampling plans, and the proposed methodology of the MDS sampling plan under TGHLD are given. Tables are obtained for various values of shape parameter and results are discussed. A comparison between the performances of the proposed plan and single sampling plan (SSP) and repetitive sampling plan (RSP) are discussed. A real life example is also provided to study the performance of the proposed sampling design.

Authors’ contributions

Methodology and Computations carried out by GS Rao and writing and data collection done by Jilani and AV Rao. All authors read and approved the final manuscript.

Acknowledgements

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of the paper. The first author would like to acknowledge the technical support received from the Office of the Deputy Vice Chancellor Academic, Research and Consultancy, The University of Dodoma, Tanzania.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Sd. Jilani

Sd. Jilani received his MSc in Statistics (2015) and PhD in Statistics (2021) from the Acharya Nagarjuna University, GUNTUR, A.P., INDIA. He is presently working as a Guest Faculty, Department of Statistics, Acharya Nagarjuna University, GUNTUR, A.P. He published 15 publications in different peer-reviewed journals in national and international well-reputed journals. His research interests include topics related to statistical quality control and Statistical Inference.

Gadde Srinivasa Rao

Gadde Srinivasa Rao received his MSc in Statistics (1988), MPhil in Statistics (1994) and PhD in Statistics (2002) from the Acharya Nagarjuna University, Guntur, India. He is presently working as a Professor of Statistics at the Department of Mathematics and Statistics, The University of Dodoma, Tanzania. He boasts more than 170 publications in different peer-reviewed journals in national and international well-reputed journals including, for example including the Journal of Applied Statistics, International Journal of Advanced Manufacturer Technology, IEEE Access, Communications in Statistics Theory and methods, Communications in Statistics Simulation and Computation, Journal of Testing and Evaluation, Arabian Journal for Science and Engineering, International Journal of Quality & Reliability Management, Economic Quality Control Quality and Reliability Engineering International, Quality and Reliability Engineering International and Journal of Statistical Computation and Simulation. His papers have been cited more than 2382 times with h-index 23 and i-10 index 56 (Google Coalitions). He is a reviewer of more than 60 well-reputed international journals. He is the author of three book/book chapters published in Springer Nature and Elsevier. His research interests include statistical inference, statistical process control, applied Statistics, acceptance sampling plans, reliability estimation and Neutrosophic statistics. He got world scientists’ recognition in AD Scientific Index- 2023. He got the first rank among the faculty member at the University of Dodoma, Tanzania. He is the President of the NSIA branch of Tanzania, Neutrosophic Science International Association (NSIA).

A. Vasudeva Rao

Vasudeva Rao Ananthasetty is a Honorary Professor of Statistics in the Department of Statistics, Acharya Nagarjuna University, GUNTUR, A.P., INDIA. He has been working as teaching faculty member in the same institution since 1985. He received MSc in Statistics (1982), MPhil in Statistics (1986) and PhD in Statistics (1990) from Acharya Nagarjuna University. His research areas are Statistical Inference (Estimation Theory), reliability estimation and about 35 publications in reputed National and International Journals are to his credit. So far one PhD and 12 MPhils are awarded under his guidance and a number of students are pursuing Ph.D. under him. Attended about 25 National and International Conferences. He is life member for Indian Society for Probability and Statistics (ISPS), Society for Development of Statistics, A.P. and Kerala Statistical Association. Prof. A V Rao is an Academic Senate Member of Acharya Nagarjuna University and in 2018 and he has received University Best Teacher Award from the Government of A.P.