Design and Analysis of Reliability Sampling Plans Based on the Topp–Leone Generated Weibull Distribution

Abstract

As part of this study, we design a reliability acceptance sampling plan under the assumption that the lifetime of a product follows the Topp–Leone generated Weibull (TLGW) distribution, a model that exhibits structural symmetry in its hazard rate behavior and distributional form. The fundamental procedures for constructing such a plan are described. We compute and tabulate the minimum sample sizes required for given risk criteria using both binomial and Poisson models for the number of failures. We also provide the operating characteristic (OC) values for the proposed sampling plans, and determine the minimum ratios of true mean life to specified mean life needed to satisfy a given producer’s risk. The role of symmetry in the TLGW distribution is highlighted in its balanced tail properties and shape characteristics, which influence the performance of the acceptance sampling plan. Finally, we illustrate the applicability of the proposed plan with real-world data.

1. Introduction

Acceptance sampling forms an integral part of quality assurance processes, aimed at deciding whether a batch (lot) of products meets predefined quality standards. This decision is based on the analysis of a random sample extracted from the lot. The procedure entails comparing the number of defective or failed items in the sample against a predetermined threshold, enabling manufacturers to make informed decisions about lot acceptance without inspecting every unit. This, in turn, leads to significant savings in time, cost, and labor [1,2]. In reliability-focused acceptance sampling plans, the principal quality attribute under consideration is the product’s lifetime or time-to-failure. Conducting complete life tests can be costly and time-consuming, particularly because items are tested until they fail. To mitigate this, such plans typically adopt a truncated testing approach, where testing is stopped at a fixed duration t, and the number of failures observed up to this point is recorded. A test of this nature is referred to as being truncated at time t. In the present study, the lifetime distribution considered is the Topp–Leone generated Weibull (TLGW) model, which possesses structural symmetry in its distributional shape. This symmetry plays a role in balancing the tail behavior, influencing both the probability of extreme lifetimes and the design of acceptance sampling plans. By leveraging this property, the plan is designed to minimize the required sample size while maintaining a high degree of confidence in the resulting decision.

A standard single-sample reliability sampling plan is characterized by three parameters: the sample size n, the acceptance number c, and the fixed test duration t (commonly expressed as a ratio $t/{\lambda }_0$, where ${\lambda }_0$ represents the minimum acceptable mean life). According to this scheme, n units are tested for time t. The lot is accepted if the number of failures does not exceed c by time t; otherwise, it is rejected. The parameters n and c are selected to balance the producer’s risk—the probability of incorrectly rejecting a good lot ($\lambda \ge {\lambda }_0$)—and the consumer’s risk—the probability of wrongly accepting a bad lot ($\lambda <{\lambda }_0$) [3]. In practice, upper limits are set for consumer’s risk (e.g., 5% or 10%) for lots that just meet the minimum acceptable quality level, and the plan is also verified to ensure that the producer’s risk remains acceptable for lots with significantly better quality.

There exists a wide array of literature addressing lifetime-based acceptance sampling plans under various statistical distributions. For instance, Gupta and Groll [1] investigated such plans under the Gamma distribution, and Baklizi et al. [4] developed plans for the Rayleigh model. Rosaiah and Kantam [5] proposed acceptance plans using the inverse Rayleigh distribution. Generalized exponential models were addressed by Aslam et al. [6], while Rao and Kantam [7] considered percentile-based sampling using the log-logistic distribution. Other notable studies include Jose and Sivadas [8], who utilized the negative binomial Marshall–Olkin Rayleigh model, and additional plans based on the Pareto–Rayleigh distribution [2], Burr Type X distribution [9], and generalized Rayleigh–truncated negative binomial distribution [10]. Additional contributions include sampling plans for the Gumbel–Uniform distribution [11], power Lindley distribution for various lot sizes [12], and repetitive group plans based on the generalized inverted exponential distribution by Singh et al. [13]. These methodologies typically involve determining the minimum sample size and acceptance threshold that satisfy pre-defined consumer and producer risk levels.

This study explores a reliability sampling strategy where product lifetimes conform to a specific generalization of the Weibull distribution, known as the Topp–Leone generated Weibull (TLGW) distribution. The Weibull model is highly favored in reliability engineering due to its capacity to model various types of failure behavior. To enhance this flexibility, several extensions of the Weibull distribution have been developed. One such generalization is the TLGW distribution, a four-parameter model derived by applying the Topp–Leone generating mechanism to the standard two-parameter Weibull distribution [14]. According to Aryal et al. [14], this extended model offers better adaptability and frequently outperforms the traditional Weibull and several of its variants. For example, empirical analyses showed that the TLGW model provides a superior fit over the Alpha Power Law Family Weibull (ALFW) distribution [15] when modeling real-life datasets, including eruption durations of the Kiama blowhole [16] and repair times for airborne communication systems [17]. Furthermore, the log-TLGW distribution has been successfully implemented in regression analysis, such as in modeling two-arm clinical trial data [18].

1.1. Topp–Leone Generated Weibull Distribution

The TLGW distribution is a four-parameter extension of the Weibull model proposed in Aryal et al. [14], developed by applying the Topp–Leone generating mechanism to a baseline Weibull distribution. Let X be a non-negative random variable that follows the TLGW distribution with parameters $\alpha >0$, $\theta >0$, $\eta >0$, and $\beta >0$. Its probability density function (PDF) is defined as:

$$f_X\left(x\right)=2\alpha \theta \beta {\eta }^{\beta }{x}^{\beta -1}exp\{-{\left(\eta x\right)}^{\beta }\}{\left(1-e^{-{\left(\eta x\right)}^{\beta }}\right)}^{\theta \alpha -1}{\left[1-{\left(1-e^{-{\left(\eta x\right)}^{\beta }}\right)}^{\theta }\right]}^{\alpha -1},$$

(1)

for all $x\ge 0$. The corresponding cumulative distribution function (CDF) is given by:

$$F_X\left(x\right)={\left(1-e^{-{\left(\eta x\right)}^{\beta }}\right)}^{\alpha \theta }{\left[2-{\left(1-e^{-{\left(\eta x\right)}^{\beta }}\right)}^{\theta }\right]}^{\alpha },x\ge 0.$$

(2)

Start from the Weibull baseline $G\left(x\right)=1-exp\{-{\left(\eta x\right)}^{\beta }\}$. Apply the Topp–Leone generator to $G{\left(x\right)}^{\theta }$, which yields

$$F\left(x\right)={\left[G{\left(x\right)}^{\theta }\{2-G{\left(x\right)}^{\theta }\}\right]}^{\alpha }={\left(1-{\left[1-G{\left(x\right)}^{\theta }\right]}^2\right)}^{\alpha }.$$

Setting

$u=F\left(x\right)$ and solving for x gives a direct simulation recipe:

$$\begin{array}{cc}\hfill {(1-u)}^{1/\alpha }& ={\left[1-G{\left(x\right)}^{\theta }\right]}^2,\hfill \\ \hfill \Rightarrow G{\left(x\right)}^{\theta }& =1-{(1-u)}^{1/\left(2\alpha \right)},\hfill \\ \hfill \Rightarrow x& ={\eta }^{-1}{\left[-ln\left\{1-{\left(1-{(1-u)}^{1/\left(2\alpha \right)}\right)}^{1/\theta }\right\}\right]}^{1/\beta },\hfill \end{array}$$

(3)

which is the inverse-CDF used to generate random values.

Notably, the TLGW model can be expressed as a mixture of exponentiated Weibull distributions. The CDF admits the following binomial series expansion:

$$F_X\left(x\right)=\sum _{k=0}^{\infty }{\gamma }_k{\pi }_{(\alpha +k)\theta }\left(x\right),$$

(4)

where ${\gamma }_k={(-1)}^k{2}^{\alpha -k}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right)$ and ${\pi }_m\left(x\right)={\left(1-e^{-{\left(\eta x\right)}^{\beta }}\right)}^m$. Here, ${\pi }_m\left(x\right)$ represents the CDF of an exponentiated Weibull distribution with exponent m. Consequently, the TLGW distribution is an infinite mixture of exponentiated Weibull distributions with varying weights. Let $X_{\left(1\right)},X_{\left(2\right)},\dots ,X_{\left(n\right)}$ denote the order statistics from a sample of size n drawn from the TLGW distribution. The probability density function of the i-th order statistic $X_{\left(i\right)}$ is given by:

$$f_{X_{\left(i\right)}}\left(x\right)={\displaystyle \frac{n!}{(i-1)!(n-i)!}}{\left[F_X\left(x\right)\right]}^{i-1}{[1-F_X\left(x\right)]}^{n-i}{f}_X\left(x\right),$$

(5)

for $i=1,2,\dots ,n$. This expression can also be interpreted as a mixture of exponentiated Weibull densities using the same series expansion. For given $(\alpha ,\beta ,\theta )$, compute ${\mu }_1$ once (e.g., by a short one-dimensional numerical integral of $1-F_Z\left(x\right)$ with $\eta =1$), then obtain $\eta$ from $\lambda$ via the formula above. With $\eta$ thus fixed, $p=F_X(t;{\lambda }_0)$ in (7) is evaluated using the CDF in (2).

With the theoretical formulation in place, we now proceed to outline the development of the acceptance sampling plan based on the TLGW distribution described above. The TLGW model augments the Weibull baseline via the Topp–Leone generator, adding shape parameters $\alpha$ and $\theta$ that flexibly reallocate early- vs. late-failure mass while retaining a scale structure convenient for life-test design. Prior studies report that TLGW frequently outperforms the plain Weibull and related families in fitting diverse lifetime datasets and has proved effective in regression settings, further motivating its use for reliability planning.

Because the plan only requires the failure probability at the truncation time, $p=F_X(t;·)$, the same design carries over to other industries (mechanical/electronic components, process reliability, software, and biomedical devices) once an appropriate lifetime model is specified. The TLGW family reshapes early- vs. late-failure mass via $(\alpha ,\theta )$ while retaining a convenient scale structure, which is valuable in contexts where the plain Weibull is too rigid. The framework extends to more complex settings: (i) right/interval/progressive censoring by adapting the count model for failures by t; (ii) accelerated life testing by replacing t with the stress-specific effective time; and (iii) covariates through log-TLGW regression, linking $\eta$ to predictors while computing $p=F_X(t;\lambda )$ accordingly.

The remainder of this paper is organized as follows. In Section 2, we formulate the reliability sampling plan for the TLGW distribution, describing how to determine the minimum sample size and other parameters given certain risk constraints. Section 3 presents the results in tabular form and includes examples illustrating how to use the tables in practice. Finally, Section 4 concludes the paper with some observations.

In time-truncated life tests, the plan depends on the failure probability by the truncation time, $p=F(t;·)$, which enters the binomial acceptance probability $L\left(p\right)$ in (8). Thus, models that flexibly shift probability mass between early and late failures can materially change p at a fixed $t/{\lambda }_0$ and reduce the minimum n that satisfies a given consumer-risk target. The TLGW CDF,

$$F\left(x\right)=G{\left(x\right)}^{\alpha \theta }{\{2-G{\left(x\right)}^{\theta }\}}^{\alpha },G\left(x\right)=1-exp\left\{-{\left(\eta x\right)}^{\beta }\right\},$$

(6)

introduces two additional shape parameters $(\alpha ,\theta )$ that control early- vs. late-failure mass while retaining simple evaluation of $p=F(t;·)$. Moreover, TLGW admits an infinite-mixture representation in exponentiated Weibull components, supplying further adaptability in the lower-tail region that dominates truncated tests. The family is scale-separable in $\eta$, so once the one-time constant $E\left[X\right]{|}_{\eta =1}$ is computed, mapping a specified mean ${\lambda }_0$ to $\eta$ is immediate.

1.2. Why the Topp–Leone Generator (vs. Other Generators Such as Weibull–G)?

Generator-based families differ in how they redistribute mass. The Topp–Leone transform $T\left(u\right)=u\{2-u\}$ for $u\in (0,1)$ reshapes the baseline $G\left(x\right)$ with two knobs $(\alpha ,\theta )$, enabling strong control of the early-failure region without sacrificing analytic tractability: $p=F(t;·)$ is closed-form and the inverse CDF is explicit. In contrast, Weibull–G [19] typically provides a single additional global shape parameter aimed at tail thickness. Because truncated plans are driven primarily by early failures, Topp–Leone’s localized control is particularly well-suited to acceptance-sampling design.

The subsequent section introduces the TLGW distribution, outlining its definition and essential mathematical properties that support the development of the proposed sampling plan. We then present the methodology for constructing the plan, followed by tables of computed results and illustrative examples. The paper concludes with a summary of findings and practical insights.

2. Reliability Sampling Plan

We now describe how to construct a reliability sampling plan under the assumption that the lifetime distribution is TLGW. Suppose the product’s lifetime X follows the TLGW distribution with known shape parameters $(\alpha ,\beta ,\theta )$ and unknown scale parameter (related to the mean life). We design a life test that will be truncated at time t, and we require that the true mean life $\lambda$ of the product be at least a specified value ${\lambda }_0$.

During the life test, if more than c failures occur before time t, the lot is rejected immediately. If the test reaches time t with at most c failures, the lot is accepted. Thus, the plan is characterized by the tuple $(n,c,t/{\lambda }_0)$. A major goal in designing such a plan is to choose the smallest sample size n that will still meet the desired reliability requirements (in terms of consumer’s and producer’s risks).

2.1. Consumer’s Risk and Minimum Sample Size

The consumer’s risk is the probability that a “bad” lot (with true mean life below the required minimum) is accepted. To control this risk, we specify a high confidence level $p^{*}$ (for example, $0.95$) so that with probability $p^{*}$ we will reject a lot that is only marginally acceptable. Equivalently, $1-p^{*}$ is the maximum consumer’s risk allowed for a lot at the minimum standard. In our context, a lot with true mean $\lambda ={\lambda }_0$ represents the borderline quality. Let $p=F_X(t;{\lambda }_0)$ be the probability that a single item fails by time t when the true mean life is ${\lambda }_0$. This value p can be obtained from the TLGW distribution’s CDF (2) for a given ratio $t/{\lambda }_0$. Now, consider the probability $L\left(p\right)$ that the lot is accepted under the plan when the true mean equals ${\lambda }_0$. This is given by the binomial distribution as:

$$L\left(p\right)=P\left(\mathrm{at}\mathrm{most}c\mathrm{failures}\mathrm{in}n\mathrm{trials}\right)=\sum _{i=0}^c\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)p^i{(1-p)}^{n-i}.$$

(7)

To guarantee the consumer’s confidence level $p^{*}$, we require

$$L\left(p\right)\le 1-p^{*},$$

(8)

so that the lot acceptance probability at the borderline quality is kept below the complement of the target confidence. In other words, (8) ensures that a bad lot will be accepted no more than $1-p^{*}$ of the time.

For given values of c, $p^{*}$ and $t/{\lambda }_0$, we must find the smallest integer n that satisfies the inequality (8). We assume the lot size is sufficiently large (effectively infinite) so that the binomial model is applicable. The resulting minimum sample sizes n are tabulated in

Table 1. Minimum sample sizes using binomial approximation for confidence interval $p^{*}$ 0.75, 0.9, 0.95, 0.99 and $t/{\lambda }_0$ = 0.628, 0.942, 1.257, 1.571, 2.356, 3.141, 3.972, 4.712.

$\mathit{t}/{\mathsf{\lambda }}_0$
${\mathit{p}}^{*}$	c	0.628	0.942	1.257	1.571	2.356	3.141	3.972	4.712
	0	9	4	2	2	1	1	1	1
	1	18	7	4	3	2	2	2	2
	2	26	10	6	5	3	3	3	3
	3	33	13	8	6	4	4	4	4
	4	41	16	10	8	6	5	5	5
0.75	5	49	19	12	9	7	6	7	6
	6	56	22	14	10	8	7	8	7
	7	64	25	16	12	9	8	9	8
	8	71	28	17	13	10	9	10	9
	9	78	31	19	15	11	10	11	10
	10	86	34	21	16	12	11	12	11
	0	15	6	3	2	1	1	1	1
	1	25	10	6	4	3	2	2	2
	2	34	13	8	6	4	3	3	3
	3	43	17	10	7	5	4	4	4
	4	52	20	12	9	6	5	5	5
0.90	5	60	23	14	10	7	6	6	6
	6	68	27	16	12	8	7	7	7
	7	76	30	18	13	10	9	8	8
	8	84	33	20	15	11	10	9	10
	9	92	36	22	16	12	11	10	11
	10	100	40	24	18	13	12	11	11
	0	19	7	4	3	2	1	1	1
	1	30	11	7	5	3	2	2	2
	2	40	15	9	6	4	3	3	3
	3	50	19	11	8	5	5	4	4
	4	59	23	13	9	6	6	5	5
0.95	5	68	26	15	12	7	7	6	6
	6	76	30	17	13	8	8	7	7
	7	85	33	20	14	9	9	8	8
	8	93	36	22	16	11	10	9	9
	9	102	40	24	17	12	11	10	10
	10	110	43	26	19	13	12	11	11
	0	29	11	6	4	2	2	1	1
	1	42	16	9	6	4	3	2	2
	2	53	20	11	8	5	4	3	3
	3	64	24	14	10	6	5	5	4
	4	74	28	16	12	7	6	6	5
0.99	5	84	32	18	13	9	7	7	6
	6	93	36	21	15	10	8	8	7
	7	102	39	23	16	11	9	9	8
	8	111	42	25	18	12	10	10	9
	9	120	46	27	20	13	12	11	10
	10	129	50	29	21	15	13	12	11

for various typical values of $p^{*}$ and $t/{\lambda }_0$ and for acceptance numbers $c=0,1,2,\dots ,10$. In Table 1, the calculations are based on the exact binomial Formula (7).

Table 2. Minimum sample sizes using Poisson probabilities with 0.75, 0.9, 0.95, 0.99 probabilities and $t/{\lambda }_0$ = 0.628, 0.942, 1.257, 1.571, 2.356, 3.141, 3.972, 4.712.

$\mathit{t}/{\mathsf{\lambda }}_0$
${\mathit{p}}^{*}$	c	0.628	0.942	1.257	1.571	2.356	3.141	3.972	4.712
	0	10	4	3	2	2	2	2	2
	1	18	8	5	4	3	3	3	3
	2	27	11	7	6	5	5	4	4
	3	35	14	9	7	6	6	6	6
	4	42	18	11	9	7	7	7	7
0.75	5	50	21	13	11	8	8	8	8
	6	57	24	15	12	10	9	9	9
	7	65	27	17	14	11	10	10	10
	8	72	30	19	15	12	12	11	11
	9	80	33	21	17	13	13	13	12
	10	87	36	23	18	14	14	14	14
	0	16	7	4	4	3	3	3	3
	1	26	11	7	6	5	4	4	4
	2	36	15	10	8	6	6	6	6
	3	45	19	12	10	8	7	7	7
	4	54	22	14	11	9	9	9	8
0.90	5	62	26	16	13	10	10	10	10
	6	71	29	19	15	12	11	11	11
	7	79	33	21	16	13	12	12	12
	8	87	36	23	18	14	14	14	14
	9	95	39	25	20	16	15	15	15
	10	103	42	27	21	17	16	16	16
	0	20	9	6	5	4	4	4	3
	1	32	13	9	7	6	5	5	5
	2	42	18	11	9	7	7	7	7
	3	52	22	14	11	9	8	8	8
	4	61	25	16	13	10	10	10	10
0.95	5	70	29	19	15	12	11	11	11
	6	79	33	21	17	13	13	12	12
	7	88	36	23	18	15	14	14	14
	8	97	40	25	20	16	15	15	15
	9	105	43	28	22	17	16	16	16
	10	113	47	30	23	19	18	18	17
	0	31	13	8	7	5	5	5	5
	1	45	19	12	9	8	7	7	7
	2	56	23	15	12	9	9	9	9
	3	67	28	18	14	11	11	11	11
	4	78	32	20	16	13	12	12	12
0.99	5	88	36	23	18	15	14	14	14
	6	97	40	26	20	16	16	15	15
	7	107	44	28	22	18	17	17	17
	8	116	48	30	24	19	18	18	18
	9	125	52	33	26	21	20	20	19
	10	135	55	35	28	22	21	21	21

provides similar results using the Poisson approximation for the number of failures (which is appropriate when n is large and/or p is small). As expected, the required sample size n increases as we demand higher confidence $p^{*}$ or as the test duration t (relative to ${\lambda }_0$) gets shorter.

When the expected number of failures is small—practically, when $np\lesssim 3$ (and $p\le 0.10$)—the Poisson approximation $X\sim \mathrm{Pois}\left(np\right)$ yields acceptance probabilities that are within a percent or two of the exact binomial for $c\le 2$, and the resulting minimum n typically differs by at most $0–2$ across our grids; see the side-by-side comparison in Table 1 and Table 2.

2.2. Operating Characteristic Function

Once a plan $(n,c,t/{\lambda }_0)$ is determined, it is important to understand its discrimination power over a range of true mean lives. This is described by the operating characteristic (OC) function of the plan, which gives the probability of lot acceptance as a function of the actual quality of the lot. If the true distribution has failure probability $p=F_X(t;\lambda )$ by time t (where $\lambda$ is the actual mean life), then the acceptance probability is

$$L\left(p\right)=\sum _{i=0}^c\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)p^i{(1-p)}^{n-i},$$

similar to (7) but now with $p=F_X(t;\lambda )$ corresponding to the actual mean $\lambda$. We can express this dependence in terms of the ratio $\lambda /{\lambda }_0$. For a given plan, one can compute $L\left(p\right)$ for various multiples of the specified mean life. The OC curve is then a plot of $L\left(p\right)$ versus $\lambda /{\lambda }_0$.

Table 3. OC values for the plan $(n,c,t/{\lambda }_0)$ for given confidence level $p^{*}$, c = 2, $\gamma ,\beta ,\theta =2$.

${\mathit{p}}^{*}$	n	$\frac{\mathit{t}}{{\mathsf{\lambda }}_0}$	$\mathsf{\lambda }/{\mathsf{\lambda }}_0$
			2	4	6	8	10	12
	26	0.628	0.9860	0.9999	1	1	1	1
	10	0.942	0.9729	0.9999	0.9999	1	1	1
	6	1.257	0.9521	0.9998	0.9999	1	1	1
	5	1.571	0.8917	0.9994	0.9999	0.9999	1	1
0.75	3	2.356	0.8504	0.9979	0.9999	0.9999	1	1
	3	3.141	0.5922	0.9836	0.9992	0.9999	0.9999	0.9999
	3	3.972	0.3333	0.9340	0.9949	0.9994	0.9999	0.9999
	3	4.712	0.1804	0.8504	0.9835	0.9979	0.9996	0.9999
	34	0.628	0.9713	0.9999	1	1	1	1
	13	0.942	0.9446	0.9998	0.9999	1	1	1
	8	1.257	0.8938	0.9996	0.9999	1	1	1
	6	1.571	0.8234	0.9988	0.9999	0.9999	1	1
0.90	4	2.356	0.6399	0.9925	0.9997	0.9999	0.9999	1
	3	3.141	0.5922	0.9835	0.9992	0.9999	0.9999	0.9999
	3	3.972	0.3333	0.9341	0.9949	0.9994	0.9999	0.9999
	3	4.712	0.1804	0.8504	0.9835	0.9979	0.9996	0.9999
	40	0.628	0.9565	0.9999	0.9999	1	1	1
	15	0.942	0.9204	0.9998	0.9999	1	1	1
	9	1.257	0.8579	0.9994	0.9999	1	1	1
0.95	6	1.571	0.8234	0.9988	0.9999	0.9999	1	1
	4	2.356	0.6399	0.9925	0.9997	0.9999	0.9999	1
	3	3.141	0.5922	0.9935	0.9992	0.9999	0.9999	0.9999
	3	3.972	0.3333	0.9940	0.9949	0.9994	0.9999	0.9999
	3	4.712	0.1804	0.8504	0.9835	0.9979	0.9996	0.9999
	53	0.628	0.9143	0.9998	0.9999	1	1	1
	20	0.942	0.8448	0.9995	0.9999	1	1	1
	11	1.257	0.7770	0.9988	0.9999	0.9999	1	1
	8	1.571	0.6673	0.9969	0.9999	0.9999	1	1
0.99	5	2.356	0.4423	0.9830	0.9994	0.9999	0.9999	0.9999
	4	3.141	0.2760	0.9467	0.9970	0.9997	0.9999	0.9999
	3	3.972	0.3333	0.9340	0.9949	0.9994	0.9999	0.9999
	3	4.712	0.1804	0.8504	0.9835	0.9979	0.9996	0.9999

provides representative OC values for several scenarios. In particular, for each combination of $p^{*}$ and $t/{\lambda }_0$ considered in Table 1, we first obtain the corresponding minimum n (for $c=2$) and then calculate the acceptance probability $L\left(p\right)$ at $\lambda /{\lambda }_0=2,4,6,8,10,$ and 12. These calculations assume $\alpha =\beta =\theta =2$ for the TLGW distribution. From the table, one can observe that as the true mean life increases relative to the specified mean, the acceptance probability approaches 1. For example, when $\lambda /{\lambda }_0=2$ (true mean is twice the specified value), the OC value is around 0.94 for the case $p^{*}=0.90$, $t/{\lambda }_0=0.942$; when $\lambda /{\lambda }_0$ reaches 4 or higher, the acceptance probability is essentially 1 for all listed cases, indicating a very high chance of accepting a clearly superior lot.

Any single-sample truncated-life plan computes n by evaluating $p=F(t;·)$ for the chosen lifetime and applying (6). Thus, an apples-to-apples comparison across distributions (Gamma, Rayleigh, log-logistic, generalized exponential, etc.) fixes $(c,t/{\lambda }_0,p^{*})$ and compares the resulting n or OC. Because TLGW re-shapes early-failure mass through $(\alpha ,\theta )$, it can produce smaller p at a given $t/{\lambda }_0$ when data exhibit stronger early-failure behavior than the plain Weibull captures, often reducing n. See [1,4,6,7,20] for alternative baselines and [14] for evidence on TLGW’s adaptability.

2.3. Producer’s Risk

The producer’s risk $\gamma$ is the probability that a “good” lot (with true mean life higher than ${\lambda }_0$) is rejected by the sampling plan. Producers typically want this risk to be quite low (e.g., 5%). To assess the producer’s risk for our plan, we consider a lot with true mean life $\lambda$ greater than ${\lambda }_0$. Using the OC function notation as above, the producer’s risk for a given $\lambda$ is $1-L\left(p\right)$, where $p=F_X(t;\lambda )$. To ensure the producer’s risk is at most $\gamma$, we need $L\left(p\right)\ge 1-\gamma$. Often, one determines a particular quality level (say, $\lambda ={\lambda }_1>{\lambda }_0$) such that $L\left(p\right)$ at ${\lambda }_1$ is $1-\gamma$. Equivalently, we can find the minimum ratio $\lambda /{\lambda }_0$ that satisfies

$$L\left(F_X(t;\lambda )\right)\ge 1-\gamma .$$

(9)

Given a plan $(n,c,t/{\lambda }_0)$ and a specified producer’s risk (for example $\gamma =0.05$), we solve (9) for $\lambda /{\lambda }_0$.

Table 4. Minimum ratio of true mean life to specified mean life ($\lambda /{\lambda }_0$) for the acceptability of a lot with $\alpha =$ 0.05.

$\mathit{t}/{\mathsf{\lambda }}_0$
${\mathit{p}}^{*}$	c	0.628	0.942	1.257	1.571	2.356	3.141	3.972	4.712
	0	2.985	3.476	3.761	4.7	5.573	7.43	9.395	11.145
	1	1.987	2.185	2.363	2.647	3.202	4.269	5.398	6.404
	2	1.717	1.842	2.003	2.325	2.528	3.37	4.262	5.056
	3	1.598	1.691	1.819	1.946	2.167	2.89	3.654	4.335
	4	1.495	1.587	1.708	1.913	2.386	2.634	3.331	3.952
0.75	5	1.448	1.513	1.628	1.726	2.144	2.435	3.615	3.653
	6	1.397	1.467	1.576	1.607	2.005	2.274	3.38	3.412
	7	1.374	1.427	1.536	1.614	1.9	2.164	3.203	3.247
	8	1.346	1.398	1.456	1.532	1.826	2.077	3.079	3.116
	9	1.319	1.375	1.433	1.543	1.743	2.002	2.939	3.003
	10	1.319	1.354	1.416	1.483	1.683	1.941	2.837	2.912
	0	3.445	3.913	4.179	4.641	5.573	7.43	9.395	11.145
	1	2.213	2.48	2.766	2.953	3.912	4.243	5.366	6.366
	2	1.879	2.029	2.256	2.484	3.035	3.34	4.224	5.011
	3	1.717	1.865	2.003	2.114	2.605	2.886	3.649	4.329
	4	1.625	1.732	1.856	2.02	2.326	2.607	3.297	3.911
0.90	5	1.561	1.636	1.758	1.841	2.142	2.416	3.055	3.624
	6	1.495	1.596	1.691	1.806	2.005	2.274	2.876	3.412
	7	1.457	1.542	1.634	1.706	2.102	2.533	2.737	3.247
	8	1.422	1.501	1.59	1.688	2.013	2.419	2.626	3.116
	9	1.397	1.463	1.554	1.611	1.922	2.324	2.531	3.487
	10	1.382	1.456	1.525	1.611	1.853	2.244	2.455	2.912
	0	3.584	4.067	4.571	5.32	6.96	7.43	9.395	11.145
	1	2.377	2.575	2.973	3.222	3.912	4.243	5.366	6.366
	2	1.987	2.158	2.372	2.484	3.035	3.34	4.224	5.011
	3	1.811	1.941	2.087	2.264	2.597	3.462	3.649	4.329
	4	1.7	1.842	1.928	2.013	2.601	3.128	3.297	3.911
0.95	5	1.611	1.72	1.819	2.035	2.386	2.859	3.055	3.624
	6	1.549	1.668	1.74	1.89	2.227	2.673	2.876	3.412
	7	1.516	1.606	1.719	1.778	2.102	2.533	2.737	3.247
	8	1.476	1.555	1.669	1.758	2.005	2.419	2.626	3.116
	9	2.985	3.476	3.761	4.7	5.573	7.43	9.395	11.145
	10	1.422	1.501	1.59	1.666	1.853	2.244	2.455	2.912
	0	4.029	4.584	5.131	5.7	6.935	9.245	9.395	11.145
	1	2.563	2.875	3.174	3.453	4.429	5.216	5.366	6.366
	2	2.147	2.36	2.554	2.81	3.417	4.076	4.224	5.011
	3	1.941	2.108	2.312	2.503	2.913	3.462	4.378	4.329
	4	1.813	1.959	2.106	2.32	2.601	3.102	3.922	3.906
0.99	5	1.734	1.865	1.971	2.119	2.601	2.859	3.615	3.624
	6	1.667	1.798	1.933	2.05	2.42	2.673	3.38	3.412
	7	1.611	1.714	1.835	1.919	2.273	2.533	3.203	3.247
	8	1.563	1.657	1.773	1.878	2.162	2.419	3.059	3.116
	9	1.538	1.626	1.736	1.678	2.072	2.57	2.947	3.003
	10	1.505	1.597	1.678	1.769	2.116	2.47	2.837	2.912

provides these solutions for $\gamma =0.05$ (5% producer’s risk) across various values of c, $p^{*}$, and $t/{\lambda }_0$. For instance, looking at the row for $c=2$ in Table 4, one can find the factor by which the true mean must exceed ${\lambda }_0$ so that the lot will be accepted at least 95% of the time. Generally, as c (the allowed number of failures) increases, the required $\lambda /{\lambda }_0$ for a given producer’s risk tends to decrease, reflecting a less stringent plan from the producer’s perspective.

3. Numerical Results and Discussion

We have computed a variety of plans and their characteristics for different parameter settings, and present the results in Table 1, Table 2, Table 3 and Table 4. These tables serve as a practical reference for implementing the sampling plan under the TLGW distribution.

Table 1 displays the minimum sample size n required for a range of acceptance numbers ($c=0$ to 10), confidence levels ($p^{*}=0.75,0.90,0.95,0.99$), and normalized test times ($t/{\lambda }_0=0.628,0.942,1.257,1.571,2.356,3.141,3.972,$ and $4.712$). Each entry in the table is the smallest n that satisfies the consumer’s risk requirement (8) for the corresponding c and $t/{\lambda }_0$. Table 2 provides the analogous results when using the Poisson approximation to the binomial. Comparing the two tables, one can see that the Poisson-based sample sizes are very close to the exact values for most cases, especially when n is large. Minor differences occur in some cases (for example, at smaller n or larger p values), but the Poisson approximation is quite accurate here.

Table 3 presents selected operating characteristic values for an acceptance number $c=2$. For each combination of $p^{*}$ and $t/{\lambda }_0$ (from Table 1), the corresponding minimum sample size n is used to calculate the OC at several values of $\lambda /{\lambda }_0$ (specifically, $2,4,6,8,10,$ and 12). As mentioned earlier, these calculations assume $\alpha =\beta =\theta =2$ for the TLGW distribution. The table clearly shows that when the true mean is well above the specified mean, the probability of accepting the lot is extremely high (often effectively 1.0 for $\lambda /{\lambda }_0\ge 8$). Even at $\lambda /{\lambda }_0=2$, the acceptance probability is typically above 0.85 for the plans considered. This illustrates that the sampling plan will protect the producer (low risk of rejection) if the product quality is substantially better than the specification.

Table 4 contains the minimum ratios $\lambda /{\lambda }_0$ required to achieve a producer’s risk of $\gamma =0.05$ for various scenarios. The entries are given for $p^{*}=0.75,0.90,0.95,0.99$ and $c=0,1,\dots ,10$ over the same range of $t/{\lambda }_0$ values as before. For example, if $p^{*}=0.90$, $t/{\lambda }_0=0.942$, and $c=2$, Table 4 indicates that $\lambda /{\lambda }_0$ must be at least about $2.03$ to ensure a 95% chance of lot acceptance (i.e., producer’s risk 5%). We observe from the table that stricter plans (with smaller c or higher $p^{*}$ or shorter test times) demand a larger multiple of the specified mean to achieve the same producer’s risk. This is intuitive; if we allow very few failures or have a very short test time, only a much higher actual mean life will guarantee that the lot consistently passes.

Illustrative Examples

To demonstrate the use of these tables, we provide two illustrative examples:

• Designing a Sampling Plan. Consider a product whose lifetime follows the TLGW distribution. Suppose the specified average life is ${\lambda }_0=1000$ h. We choose a consumer confidence level $p^{*}=0.90$ and plan to truncate the life test at $t=942$ h (so that $t/{\lambda }_0=0.942$).
  If we select an acceptance number $c=2$, Table 1 (binomial model) indicates that the minimum required sample size is $n=13$. The Poisson approximation in Table 2 gives a similar suggestion of $n=15$. Let us use the plan $(n=13,c=2,t/{\lambda }_0=0.942)$. For this plan, the OC values can be obtained from Table 3. In particular, when the true mean life is twice the specified value ($\lambda /{\lambda }_0=2$), the acceptance probability is about $0.9446$. If the true mean is four times the specified value ($\lambda /{\lambda }_0=4$), the acceptance probability rises to $0.9998$, and it is essentially $1.0$ for $\lambda /{\lambda }_0\ge 8$. Thus, the lot has a very high chance of being accepted as long as the product’s actual mean life is significantly better than the requirement.
  From Table 4, for $c=2$ and $p^{*}=0.90$ at $t/{\lambda }_0=0.942$, the minimum ratio $\lambda /{\lambda }_0$ that guarantees a producer’s risk of $\gamma =0.05$ is approximately $2.029$. In other words, if the true mean life is about $2.029\times 1000\approx 2029$ h or more, the probability of rejecting such a good lot is 5% or less under this plan.

  $\lambda /{\lambda }_0$	2	4	6	8	10	12
  OC	0.9446	0.9998	0.9999	1	1	1

  Figure 1 illustrates the OC curves for this sampling plan under various acceptance numbers. The plot shows the probability of acceptance versus $\lambda /{\lambda }_0$ for different values of c. Such a graph helps in visualizing the stringency of the plan; a smaller acceptance number c leads to a lower curve (meaning the plan is more stringent and less likely to accept lots unless $\lambda$ is much larger than ${\lambda }_0$), whereas a larger c makes the plan more lenient (higher acceptance probability for a given $\lambda /{\lambda }_0$). By examining these curves, a practitioner can choose an acceptance number that provides an appropriate balance between consumer and producer protection for their specific situation.
• Application to Real Data. We now consider a real dataset of failure times to illustrate how the sampling plan can be applied. The dataset, reported by Wood [21], consists of 16 ordered failure times (in hours) of a software system:

  $$519,968,1430,1893,2490,3058,3625,4422,5218,$$

  $$5823,6539,7083,7487,7846,8205,8564.$$
  Assume that the lifetime of this software follows the TLGW distribution. Let the specified mean life be ${\lambda }_0=1000$ h, and set the consumer’s risk at $10\%$ (so $p^{*}=0.90$). We choose a test truncation time of $t=942$ h (as before, $t/{\lambda }_0=0.942$). Based on the earlier results, a reasonable plan is $(n=16,c=2,t/{\lambda }_0=0.942)$ since our sample size is 16 and allowing two failures meets the desired confidence level (indeed, from Table 1, the minimum n for $c=2$ and $p^{*}=0.90$ at $t/{\lambda }_0=0.942$ was 13, so $n=16$ is adequate).
  By the time the test reached $t=942$ h, only one failure had occurred in the sample (the first failure was at 519 h; the second failure was at 968 h, which is just beyond t). Because the observed number of failures (1) does not exceed the acceptance number $c=2$, the lot (software) would be accepted. In summary, using this reliability sampling plan, the software passes the test, providing at least 90% confidence that its mean lifetime is not below the specified 1000 h.

Because the family is scaled in $\eta$, changes in $(\alpha ,\beta ,\theta )$ primarily reshape early- vs. late-failure mass and thus alter $p=F_X(t;\lambda )$. Larger $\beta$ (steeper Weibull tail) and larger $\theta$ push mass to the right, lowering p at fixed $t/{\lambda }_0$ and reducing the required n; the OC correspondingly shifts upward (higher acceptance at a given $\lambda /{\lambda }_0$). Increasing $\alpha$ (stronger Topp–Leone effect) similarly reduces early failures. These trends are monotone through the mapping $p\mapsto L\left(p\right)$ in (6).

To complement the software example, Appendix A reports a second real-world application on mechanical component lifetimes, showing how $p=F_X(t;{\lambda }_0)$, the minimum n, and the OC are computed step by step. The conclusions mirror the software case:; when the empirical data exhibit heavier early failures than the plain Weibull captures, TLGW yields smaller p at a fixed $t/{\lambda }_0$ and hence smaller required n.

The design presumes i.i.d. lifetimes from a correctly specified TLGW with known shape parameters; substantial mis-specification of $(\alpha ,\beta ,\theta )$ distorts $p=F_X(t;\lambda )$ and thus the computed n and OC. The plan is calibrated at a single truncation time t; if t is too short relative to ${\lambda }_0$, p may be large and the required n correspondingly high. The Poisson shortcut is appropriate only when $np$ is small; otherwise, the exact binomial in (6) should be preferred. Finally, when $(\alpha ,\beta ,\theta )$ are estimated from small pilot studies, uncertainty in p can be material; sensitivity checks in ± plausible ranges of $(\alpha ,\beta ,\theta )$ are recommended alongside the reported plan.

Across Table 1 and Table 2, the minimum n decreases as the normalized test time $t/{\lambda }_0$ increases, and increases with higher confidence $p^{*}$; larger acceptance numbers c reduce n. Poisson- and binomial-based n values agree closely when $np$ is small, consistent with the well-known approximation. Table 3 shows OC curves that increase rapidly with $\lambda /{\lambda }_0$, approaching one for clearly superior lots, while Table 4 maps producer’s risk targets directly to actionable thresholds in $\lambda /{\lambda }_0$ for planning.

4. Conclusions

In this paper, we developed an acceptance sampling plan for truncated life tests when the product lifetime follows the Topp–Leone generated Weibull distribution. We outlined the methodology to determine the plan parameters and presented comprehensive tables for minimum sample sizes under various scenarios. We also tabulated operating characteristic values and the minimum true-to-specified mean life ratios required to achieve a desired producer’s risk. These results, along with the illustrative examples provided, demonstrate how the proposed sampling plans can be employed in practice to make informed accept-or-reject decisions for lots of products. The flexibility of the TLGW distribution in modeling diverse lifetime data makes the plan broadly applicable in reliability engineering contexts.