Design and application of MDSSP using the MLL3 distribution for pandemic mortality data

1. Introduction

In the domain of quality control and reliability engineering, acceptance sampling plans are essential statistical tools used to determine whether a batch of products meets predefined quality standards. One of the most widely implemented approaches is the Single Acceptance Sampling Plan (SSP), which relies on the inspection of a random sample drawn from a production lot. If the number of defective units in the sample does not exceed a specified acceptance number, the entire lot is accepted; otherwise, it is rejected. This method became prominent during World War II for military procurement purposes and has since been adopted across various industries due to its simplicity and cost-effectiveness.

Over time, researchers have extended the SSP framework to accommodate various lifetime distributions that better reflect the behavior of real-world failure data. Notably, distributions such as the exponential Epstein (1954), log-normal Gupta (1962), log-logistic Kantam et al. (2001), Rayleigh Ayman et al. (2005), inverse Rayleigh Al-Omari (2016), Burr Type-XII Lio et al. (2010), and Birnbaum–Saunders Balakrishnan et al. (2007); Lio et al. (2009) have been utilized to design more accurate acceptance sampling plans that align with the statistical characteristics of the product’s lifetime, thereby enhancing the reliability of lot sentencing decisions. These models often incorporate truncated life tests, where the experiment is terminated at a pre-specified time, and decisions are made based on the observed number of failures within that period. To address practical constraints like limited testing time, several researchers have proposed SSP designs under truncated conditions using various flexible distributions, including the Rama distribution Al-Omari et al. (2019a), the quasi-Shanker distribution Al-Omari et al. (2021), the inverted Topp–Leone model Nassr et al. (2022), and the Zeghdoudi model AlSultan and Al-Omari (2023), which has shown promise in improving the sensitivity of acceptance decisions under truncated life testing environments.

Furthermore, studies by Aslam et al. (2019, 2021); Ahmadi Nadi and Sadeghpour Gildeh (2019); Gadde et al. (2021) demonstrated the use of SSP and MDSSP designs under time-truncated conditions in modeling lifetime data, particularly in the context of pandemic-related mortality analysis. These contributions underscore the importance of selecting appropriate life distributions when designing effective acceptance sampling plans.

Despite the operational simplicity of SSPs, they often fall short in settings where additional context, such as prior lot performance or process variability, is available. SSPs rely solely on current sample data and treat every lot in isolation, which can lead to inefficiencies, especially when dealing with continuous production environments or high-stakes reliability testing. To address these limitations, Wortham and Baker (1976) introduced the concept of the MDSSP, which allows lot acceptance decisions to depend not only on the current sample but also on the outcomes of preceding or succeeding lots. The MDSSP framework incorporates both current and historical lot outcomes, creating a multi-tiered decision structure that refines the acceptance or rejection process Aslam et al., (2020, 2019). This dependency enhances the plan’s ability to detect quality shifts while reducing the average sample number (ASN), a key performance metric in acceptance sampling.

Over time, researchers have further developed MDSSPs using more flexible statistical models and practical assumptions. For instance, Kantam et al. (2006) applied the log-logistic distribution in MDSSPs for reliability testing, while Tsai and Wu (2006) extended the design to generalized Rayleigh models. Advanced approaches such as ranked set sampling Hussain et al., (2021), Gompertz-based models Gui and Zhang, (2014), and exponential-based variations Yan et al., (2016) were also proposed to enhance plan effectiveness by improving sensitivity to process shifts and reducing the risk of misclassification in lot acceptance decisions. Additional innovations were introduced by Rao et al. (2008) using the Marshall–Olkin extended Lomax distribution, and Al-Omari et al. (2019b) who employed a three-parameter Lindley model. Furthermore, Afshari and Sadeghpour Gildeh (2017) integrated fuzzy logic into MDSSP design to address decision ambiguity in uncertain environments. Recent applications demonstrate the practicality of MDSSPs. Kolli et al. (2023) used an MDSSP based on the New Lomax Rayleigh distribution to analyze adolescent suicide data. Similarly, Adeyeye et al. (2023) proposed a Zech-based MDSSP, and Jilani et al. (2024) developed one under the Type-II generalized half-logistic model. More recently, Jeyadurga and Balamurali (2025) explored the economic design of repetitive group sampling using the two-parameter Lindley distribution.

As researchers continue to refine acceptance sampling methodologies, significant emphasis has been placed on identifying suitable lifetime distributions that offer both modeling flexibility and analytical tractability. One such promising distribution is the Modified Log- Logistic Type III (MLL3), which generalizes the traditional log-logistic family to better accommodate skewed, heavy-tailed, or bathtub-shaped failure behaviors often observed in real-world reliability data. The MLL3 distribution has proven particularly effective in modeling data under truncated life tests, making it a strong candidate for integration within advanced sampling schemes.

Recently, Afify et al. (2025) introduced a further generalization-the Generalized Kavya-Manoharan Modified Log-Logistic (GKMLL) distribution-which extends the capabilities of MLL3 by incorporating additional shape parameters to capture a wider range of hazard rate behaviors. Integrating MLL3 within the MDSSP framework presents several advantages. It allows for more precise estimation of design parameters such as sample size, acceptance number, and truncation time, while maintaining control over the producer’s and consumer’s risks. Additionally, the use of a flexible model like MLL3 enhances the discriminating power of the plan, especially when the failure mechanism deviates from exponential assumptions.

This study builds upon these developments by proposing a Multiple Dependent State Sampling Plan (MDSSP) based on the Modified Log-Logistic Type III (MLL-3) distribution under a truncated life testing environment. The objective is to construct a plan that achieves minimum ASN while satisfying the required quality constraints, thereby offering a practical and cost-effective tool for modern quality assurance systems.

2. Probability Model Based on the MLL3 Distribution

The MLL3 distribution stands out among flexible lifetime models due to its capability to accommodate right-skewed, heavy-tailed, and non-monotonic hazard behavior. This flexibility makes it a powerful candidate for modeling truncated life data, especially within reliability and quality control frameworks such as multiple dependent state sampling plans (MDSSPs). The cumulative distribution function (CDF) of the MLL3 distribution is formulated as follows:

Its corresponding probability density function (PDF) is expressed by:

To determine the lifetime corresponding to a specific cumulative probability q, the quantile function is utilized and defined as:

An important special case is the median lifetime, derived by setting $q = 0.5$, resulting in:

Let ${\nu }_q={\left(\frac{1}{1+\log \left(1-\left(\frac{e-1}{e}\right)q^{\frac{1}{b}}\right)}-1\right)}^{\frac{1}{a}},t_q=\theta {\nu }_q$ then $\theta =\frac{t_q}{{\nu }_q}$.

Let $p$ represent the probability of failure associated with a variable that follows the MLL3 distribution under a life-testing experiment. Suppose the experiment is concluded at a fixed termination time $t_0$, and the data is censored beyond a predefined truncation time $t_q$. In this context, the failure probability is given by $p=F\left(t_0\right)$, where $F(·)$ denotes the CDF of the MLL3 model. To simplify the experiment design, the termination time can be conveniently expressed as a multiple of the quantile time, i.e., $t_0= k{t}_{q0}$.

$p=F\left(t_0\right)={\left(\frac{e}{e-1}\right)}^b{\left(1-\exp \left({\left(1+{\left(\frac{k{t}_{q_0}}{\frac{t_q}{{\nu }_q}}\right)}^a\right)}^{-1}-1\right)\right)}^b$

then

In the context of quality control applications based on the MLL3 distribution, determining the probability of failure at a specified termination time $t_0$ is essential. This probability is denoted by $p=F\left(t_0\right)$, where $F(·)$ represents the CDF of the MLL3 distribution. For practical purposes, the termination time can be expressed in terms of a quantile $t_{q0}$ as $t_0= k.t_{q0}$, where $k$ is a positive constant.

The ratio $t_q/ t_{q0}$ serves as a critical measure for evaluating both producers’ and consumers’ perspectives. From the producer’s viewpoint, a value greater than one reflects acceptable quality, whereas the consumer is concerned with ensuring that the ratio does not exceed a certain threshold, typically associated with the consumer’s risk β. In this setting, the probability $p_1$ associated with $t_q/ t_{q0} > 1$ represents the Acceptable Quality Level (AQL), while $p_2$, which corresponds to the case $t_q/ t_{q0}= 1$, denotes the Limiting Quality Level (LQL).

To construct the MDSSP using the MLL3 distribution, the initial step involves estimating the distribution parameters based on relevant historical data. These estimated parameters are subsequently used to evaluate the MLL3 distribution at different combinations of $t_q$, $t_{q0}$, and $t_0$, allowing for the calculation of associated probabilities $p_1$ and $p_2$. These values are then used to derive the optimal sampling parameters, including acceptance and rejection numbers as well as the ASN.

Ultimately, this framework enables decision-makers to determine the most efficient sampling plan under specified risk levels. The approach is flexible and can be applied across various domains, utilizing appropriate real-world datasets according to the context of application.

3. Operating Procedure for MDSSP Based on MLL3 Distribution

The MDSSP designed under the Modified Log- Logistic Type III (MLL3) distribution follows a structured decision process that accounts for both producers’ and consumers’ risks. This procedure relies on key statistical measures and defined decision thresholds.

4. Designing Strategy for MDSSP Under the MLL3 Distribution

The key goal in constructing an effective sampling plan under the MLL3 distribution is to avoid complete inspection while minimizing the ASN. The MDSSP structure allows decisions about lot acceptance or rejection based on a limited sample, thus optimizing time and inspection costs.

The plan works by comparing the observed number of failures within a sample to defined thresholds. If the observed failures are within acceptable limits, the lot is accepted; otherwise, it is either rejected or evaluated using additional dependent lot information.

4.5 Case study: Designing an MDSSP based on MLL3

The COVID-19 mortality data from the United Kingdom, Table 4, spanning 76 consecutive days between April 15 and June 30, 2020, is analyzed in the context of designing multiple dependent state sampling plans using the MLL3 distribution. The estimation results of the MLL3 parameters, along with standard errors and goodness-of-fit statistics, have been presented in Table 5. The dataset used in this study was based on the COVID-19 mortality data analyzed by Amaal and Ehab (2021).

Table 4. The UK’s COVID-19 data covers 76 consecutive days between April 15 and June 30, 2020.

0.0587	0.0863	0.1165	0.1247	0.1277	0.1303
0.1652	0.2079	0.2395	0.2751	0.2845	0.2992
0.3188	0.3317	0.3446	0.3553	0.3622	0.3926
0.3926	0.4110	0.4633	0.4690	0.4954	0.5139
0.5696	0.5837	0.6197	0.6365	0.7096	0.7193
0.7444	0.8590	1.0438	1.0602	1.1305	1.1468
1.1533	1.2260	1.2707	1.3423	1.4149	1.5709
1.6017	1.6083	I.6324	I.6998	1.8164	1.8392
1.8721	I.9844	2.1360	2.3987	2.4153	2.5225
2.7087	2.7946	3.3609	3.3715	3.7840	3.9042
4.1969	4.3451	4.4627	4.6477	5.3664	5.4500
5.7522	6.4241	7.0657	7.4456	8.2307	9.6315
10.1870	11.1429	11.2019	11.4584

Table 5. MLE estimates, standard errors, and goodness-of-fit statistics for the MLL3 distribution.

Distribution	Estimates	Std error	LL	AIC	BIC	KS	P-value
MLL3	b = 1.085680 a = 1.234130 θ = 1.598351	0.7024934 0.2915602 1.1841323	284.6834	290.6834	297.6756	0.060965	0.9402

As shown in Table 5, the estimated parameters of the MLL3 distribution are $\hat{b}=1.085680,\hat{a}=1.234130$, and $\hat{\theta }= 1.598351$, with corresponding standard errors of $0.7024934, 0.2915602$, and $1.1841323$, respectively. The model demonstrates an excellent fit to the data with a log-likelihood value (LL) of $284.6834$, AIC of $290.6834$, BIC $of 297.6756$, Kolmogorov-Smirnov (KS) statistic of $0.060965$, and a high p-value of $0.9402$. These results confirm the suitability of the MLL3 distribution in modeling the given dataset and provide a strong foundation for the construction of the proposed multiple dependent state sampling plan.

Figs. 1 and 2 further support the model fit. Fig. 1 illustrates the empirical and theoretical characteristics of the MLL3 model, including the P–P plot, Q–Q plot, empirical CDF, histogram, and estimated PDF. Fig. 2 shows the T–T plot, which confirms the behavior of the underlying hazard function, a critical aspect in verifying the applicability of the MLL3 model for the proposed sampling plan.

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Fig. 1.

Goodness-of-fit plots for the MLL3 distribution: (a) P–P Plot, (b) Q–Q Plot, (c) Empirical vs Theoretical CDF, (d) Histogram with theoretical PDF.

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Fig. 2.

Further model diagnostics for the MLL3 distribution: (a) T–T Plot, (b) Estimated hazard rate function (EHRF).

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Table 6 presents the optimal parameters for the proposed MDSSP constructed using the MLL3D distribution (MLL3D). Each block in the table corresponds to a fixed consumer’s risk level β (ranging from 0.25 to 0.01). Within each block, different values of the quantile ratio $t_q/t{ }_{q0}$ represent how far the testing quantile deviates from a baseline or reference level. For each scenario, the table provides:

• The sample size $n$ to be used in each stage.

• The acceptance number $c_1$and rejection number $c_2$.

• The number of inspection stages m in the MDSSP plan.

• The acceptance probability $p_a\left(p_1\right)$ at the consumer’s quality level $p_1$.

Table 6. Optimal parameters of the proposed MDSSP for MLL3D with â = 1.234130, $\hat{b}$ =1.085680.

β	tq/tq0	k=0.5					k=0.7					k=1.0
		n	c1	c2	m	Pa(P1)	n	c1	c2	m	Pa(P1)	n	c1	c2	m	Pa(P1)
0.25	2	25	5	7	2	0.954251	20	5	9	1	0.954507	18	7	9	3	0.951420
	4	9	1	2	2	0.967791	7	1	2	1	0.963807	5	1	2	1	0.958658
	6	5	0	1	1	0.959784	4	0	2	1	0.956000	5	1	2	1	0.990272
	8	5	0	1	1	0.979734	3	0	1	3	0.962231	3	0	1	1	0.960526
	10	5	0	1	1	0.988303	3	0	1	3	0.977544	3	0	1	1	0.976582
0.1	2	41	7	12	1	0.951325	33	8	13	1	0.951042	28	10	15	2	0.952543
	4	14	1	4	1	0.953590	12	2	3	3	0.960730	9	2	3	2	0.950435
	6	12	1	2	2	0.982761	9	1	2	1	0.981294	7	1	2	1	0.969473
	8	8	0	1	1	0.951420	6	0	2	1	0.953037	7	1	2	1	0.989335
	10	7	0	1	2	0.964465	5	0	1	2	0.957564	4	0	1	1	0.959433
0.05	2	53	9	14	1	0.952399	44	11	14	1	0.951677	38	13	17	1	0.956198
	4	20	2	3	1	0.956849	14	2	4	2	0.960465	11	2	4	1	0.952019
	6	15	1	2	1	0.974114	11	1	2	1	0.965406	8	1	2	1	0.953900
	8	15	1	2	1	0.991224	11	1	2	1	0.987972	8	1	2	1	0.983435
	10	9	0	2	2	0.952666	7	0	1	1	0.950377	5	0	2	1	0.953900
0.01	2	83	14	19	1	0.956144	64	15	1122	1	0.950369	57	19	24	1	0.954989
	4	31	3	5	2	0.963652	23	3	5	1	0.960113	17	3	6	1	0.950782
	6	21	1	3	1	0.961268	15	1	4	1	0.955838	14	2	3	1	0.960087
	8	20	1	2	2	0.972490	15	1	2	1	0.969113	11	1	2	1	0.955970
	10	20	1	2	2	0.988419	15	1	2	1	0.986465	11	1	2	1	0.980182

The values are optimized under three test truncation levels: $k = 0.5, k = 0.7$, and $k = 1.0$. This allows the user to assess how different truncation points affect sampling plan efficiency and strictness.

Illustrative Example:

Assume β = 0.05, $t_q/t{ }_{q0} = 6$, and $k = 0.7$. From Table 6, the corresponding optimal sampling plan is:

• $n=11,c_1=1,c_2=2,m=1$

• $p_a\left(p_1\right)=0.965406$

This indicates that a one-stage sampling plan is used, where 11 items are tested. If the number of failures is 0 or 1, the lot is accepted. If it exceeds 2, the lot is rejected. An intermediate value of 2 would invoke the dependent-state decision rule depending on prior outcomes.

General observations:

• 1.

  Lower consumer risk (β) leads to a higher sample size and possibly more stages to ensure tighter control.

• 2.

  Higher quantile ratios $t_q/t{ }_{q0}$ generally result in higher acceptance probabilities, indicating more lenient conditions.

• 3.

  As the truncation point k increases (from 0.5 to 1.0), the sample size tends to reduce, but careful tuning of $c_1$, $c_2$, and m keeps plan performance stable.

Overall, Table 6 provides an effective reference for constructing cost-efficient and reliable sampling plans tailored to varying operational risks and product quality expectations using the MLL3D model.

5. Evaluating Sampling Efficiency of MDSSP versus SSP Using the MLL3 Model

To evaluate the performance of the proposed MDSSP under the MLL3 distribution, a comparative analysis was carried out against the conventional Single Sampling Plan (SSP). The assessment employed fixed distributional parameters $a = 1.99$ and $b = 1.99$, across various consumer risk levels $\beta = 0.25, 0.10, 0.05, 0.01$, and quantile ratios $t_q/t{ }_{q0}= 2, 4, 6, 8, 10$. For each configuration, the sampling parameters were optimized to satisfy both the producer’s and consumer’s risk constraints.

Conceptually, the MDSSP functions as a multi-tiered decision mechanism, incorporating conditional zones that extend beyond the binary structure of SSP. This layered approach allows for a more refined evaluation of lot quality, thereby enhancing flexibility while maintaining statistical rigor. In contrast, SSP imposes a rigid threshold, potentially leading to premature rejection of borderline-acceptable lots.

The results summarized in Table 7 reveal a consistent pattern: MDSSP achieves comparable or better acceptance probabilities using smaller sample sizes. For example, at$\beta = 0.01$ and a quantile ratio of 2, MDSSP attains $p_a\left(p_1\right)= 0.963092$with $n = 46$, while SSP requires $n = 69$ to reach a similar performance. Such reductions in sample size directly translate to lower inspection costs and faster decision cycles, both vital in time-sensitive production environments.

Additionally, the operational characteristics of the MDSSP are visually reflected in the OC curves Fig. 3. These curves exhibit sharper declines as failure probabilities increase, signifying higher sensitivity to deviations from acceptable quality. This visual advantage reinforces the operational superiority of the MDSSP across varying quality conditions. Altogether, the MDSSP structured under the MLL3 model stands out as more than just a viable alternative to traditional SSP; it offers a strategically superior approach that blends statistical accuracy with practical efficiency, making it particularly well-suited for modern quality control environments where resource optimization and reliable decision-making are critical.

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Fig. 3.

OC Curves for MDSSP and SSP under MLL3.

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The proposed MDSSP design offers practical benefits over the SSP, such as reduced sample size and improved decision efficiency. These gains imply cost and time savings in real-world applications, particularly relevant in resource-constrained contexts such as during the COVID-19 pandemic.

Fig. 3 illustrates that the MDSSP provides a steeper OC curve compared to the SSP. This indicates a sharper transition between acceptance and rejection regions, which enhances the discriminating power of the sampling plan. The MDSSP also maintains comparable or lower acceptance probabilities across varying failure rates, reflecting higher reliability in detection. While this design assumes that MLL3 parameters are accurately estimated from preliminary or historical data, the potential impact of estimation uncertainty on the plan’s efficiency and robustness should be acknowledged. Future studies may incorporate such uncertainty into the MDSSP framework.

6. Conclusions

This study proposed and evaluated an MDSSP based on the MLL3 distribution within a truncated life testing framework. The proposed design incorporates key performance metrics such as the OC function and the ASN, while satisfying both producers’ and consumers’ risk constraints. The application of the plan to real COVID-19 mortality data demonstrates its effectiveness in modeling skewed lifetime data and minimizing sampling efforts. Compared to classical plans, the MLL3-based MDSSP provides enhanced flexibility and efficiency, making it highly suitable for industrial and public health applications. Future research may explore the integration of Bayesian approaches, fuzzy environments, or hybrid censoring schemes within the MDSSP framework using MLL3 and related distributions.