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A Neutrosophic Repetitive Dependent-State Sampling Plan for Truncated Cycle-Life Testing of Lithium-Ion Batteries

This paper proposes a hybrid Neutrosophic Repetitive Dependent-State Sampling Plan (NH-RDSSP) for truncated cycle-life testing of Lithium-ion batteries that integrates resampling and lot-history dependency while modeling batch-to-batch uncertainty via triangular neutrosophic numbers, validated through closed-form derivations and calibration against real-world data from four independent sources.

Original authors: Vaitheeswaran Gnanaraj, Balakrishnan Vellaikannan

Published 2026-08-26

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CC BY 4.0

Original authors: Vaitheeswaran Gnanaraj, Balakrishnan Vellaikannan

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Batteries are not like lightbulbs that simply burn out at a specific moment. Instead, they are more like living things that slowly age, losing a little bit of their strength with every charge and discharge. For the lithium-ion batteries that power everything from smartphones to electric cars, engineers define the end of a battery's useful life not as a sudden stop, but as the moment its capacity fades below a specific threshold, usually eighty percent of its original power. Because testing every single battery until it dies would take years and destroy the product, manufacturers rely on a shortcut: they test a small sample of batteries for a set number of cycles and decide whether the whole batch is good enough to ship. The problem is that the old rules for making these decisions were built for parts that fail suddenly, not for things that degrade gradually. They also struggle to account for the fact that batteries from different factories, or even different batches from the same factory, age at different rates.

A team of researchers has developed a new way to make these quality checks that respects the slow, steady nature of battery aging. They created a hybrid testing plan that combines two different strategies: one that allows for re-testing a sample if the results are unclear, and another that looks at the history of previous batches to help make the current decision. Crucially, this new method does not pretend to know the exact aging rate of every battery. Instead, it embraces the uncertainty, using a mathematical approach that acknowledges what is known, what is unknown, and what is merely possible. By applying this method to real-world data from battery cycles, the researchers found that the aging patterns of these batteries are far more specific to their chemical makeup and operating conditions than previously thought. They also discovered that a popular mathematical model often used to predict battery life was actually failing to fit the data correctly, leading them to recommend a different, more reliable model for future testing.

The researchers began by rethinking the very definition of a "failure" in a battery test. In traditional quality control, a test stops at a fixed time, and any battery that hasn't failed by then is considered fine. But for a battery, failing is a process of decline. The team defined the end of life as the specific moment a battery's capacity drops below eighty percent. They then built a decision-making system that acts like a careful inspector. If a sample of batteries shows very few failures, the batch is accepted. If it shows many failures, it is rejected. But if the results fall in a gray area, the system does not just guess. It can either pull a fresh sample from the same batch to get a clearer picture, or it can look at the records of the last few batches to see if they were consistently good or bad. This hybrid approach, which the authors call a neutrosophic repetitive dependent-state plan, allows for a more nuanced decision that balances the risk of rejecting a good batch against the risk of accepting a bad one.

To make this system work, the researchers had to deal with the fact that they cannot know the exact aging speed of every battery in advance. Different batches age differently due to tiny variations in manufacturing or differences in how they are used. The team introduced a way to represent this uncertainty not as a single number, but as a range of possibilities that includes a most likely value, a lower bound, and an upper bound. They also assigned degrees of truth, uncertainty, and falsity to these values, allowing the system to calculate risks based on the worst-case scenario within that range. This ensures that the safety guarantees hold true even if the actual aging rate of a battery turns out to be at the extreme end of the expected spectrum.

The team then put this new plan to the test using real data from four different sources, including archives from NASA, MICH, and Sandia National Laboratories. In total, they analyzed data from 115 individual cells across eight distinct groups, covering different chemical compositions and testing temperatures. One of their most significant findings was that you cannot simply mix data from batteries tested at different temperatures and expect it to work. When they tried to apply a model trained on batteries tested at a warm temperature to batteries tested in the cold, the predictions failed catastrophically. The aging process is driven by temperature in a deterministic way, meaning that a model must be calibrated specifically for the conditions in which the battery will operate.

Another major discovery concerned the mathematical shape of the battery's life curve. For years, researchers have debated which mathematical formula best describes how batteries age. The team tested several candidates, including a complex model called the Generalized Inverted Exponential Distribution. They found that this model consistently produced unstable and unrealistic results, with its parameters blowing up to impossible numbers. In contrast, a simpler, well-known model called the Weibull distribution fit the data reliably across all the different chemical types and conditions they tested. This suggests that the Weibull model is the correct tool for predicting battery life in these scenarios, while the more complex alternative should be discarded for this specific application.

The researchers also uncovered a hidden flaw in the data itself. In several datasets, they found a recurring error where the recording system would occasionally log the same capacity value twice or record a zero value, likely due to a glitch in how the testing equipment was programmed. This artifact, which appeared in cells from different manufacturers and with different chemistries, would have skewed the results if left uncorrected. By identifying and fixing these errors, the team was able to refine their understanding of how fast different batteries age. They found that the speed of aging varies significantly depending on the battery's chemistry; for instance, one type of battery chemistry showed a much tighter, more predictable aging pattern than another, which had a wider range of possible lifespans.

Ultimately, this work provides a more robust framework for ensuring the safety and reliability of the batteries that power our modern world. By acknowledging that uncertainty is a real part of the manufacturing process and by tailoring the testing rules to the specific conditions of the battery, the new plan offers a way to make better quality decisions with fewer tests. The researchers emphasize that their findings are not just theoretical; they are grounded in the actual behavior of real batteries across a wide variety of conditions. The study concludes that to get accurate results, manufacturers must calibrate their testing plans for specific chemical types and operating temperatures, rather than trying to use a single, universal rule for all batteries. This approach ensures that the batteries shipped to consumers are truly safe and reliable, even as the technology continues to evolve.

Technical Summary: A Neutrosophic Repetitive Dependent-State Sampling Plan for Truncated Cycle-Life Testing of Lithium-Ion Batteries

1. Problem Statement

Traditional acceptance sampling plans for truncated life tests, such as the Repetitive Acceptance Sampling Plan (RASP) and the Multiple Dependent State Sampling Plan (MDSSP), were developed for components with sharply defined failure times. These classical approaches are ill-suited for Lithium-ion (Li-ion) batteries, which degrade continuously rather than failing instantaneously. In Li-ion cells, "end of life" is conventionally defined as the point where capacity falls below a specific retention threshold (e.g., 80% of rated capacity), a process governed by gradual degradation trajectories.

Furthermore, existing literature treats RASP (resampling mechanisms) and MDSSP (lot-history dependency) as separate architectures. Additionally, classical plans often assume a fixed, known shape parameter for the lifetime distribution, failing to account for the epistemic uncertainty arising from batch-to-batch variability and manufacturing differences. Finally, previous studies have not empirically validated whether pooling data across different operating conditions (e.g., temperature) or chemistries is statistically sound for these specific components.

2. Methodology

The paper proposes a Neutrosophic Repetitive Dependent-State Sampling Plan (NH-RDSSP) designed specifically for truncated cycle-life testing of Li-ion batteries. The methodology integrates three core components:

Redefinition of Truncation: The failure event is redefined not as a fixed-time occurrence but as the crossing of a State-of-Health (SOH) threshold (e.g., capacity $\leq 80\%$ ).
Hybrid Decision Rule: The NH-RDSSP combines the resampling mechanism of RASP with the lot-history dependency of MDSSP into a single decision rule:
1. A sample of $n$ cells is tested up to a truncation point $N_0$ (cycles).
2. If the number of failures $d \leq c_1$ , the lot is accepted.
3. If $d > c_2$ , the lot is rejected.
4. If $c_1 < d \leq c_2$ , the decision depends on the history of the preceding $m$ lots. If all preceding lots were unconditionally accepted, the current lot is accepted; otherwise, a fresh sample is drawn (resampling).
Neutrosophic Uncertainty Modeling: The governing lifetime shape parameter ( $\gamma$ $γ$ ) is modeled as a single-valued triangular neutrosophic number $\tilde{\gamma} = \langle (\gamma_L, \gamma_M, \gamma_U); T_\gamma, I_\gamma, F_\gamma \rangle$ $\tilde{γ} = ⟨(γ_{L}, γ_{M}, γ_{U}); T_{γ}, I_{γ}, F_{γ} ⟩$ . This representation explicitly captures:
- Truth ( $T_\gamma$ ): Confidence in the point estimate.
- Indeterminacy ( $I_\gamma$ ): Unresolved batch-to-batch variability.
- Falsity ( $F_\gamma$ ): Evidence against the point estimate (e.g., model misfit).
- Risk constraints (Producer's and Consumer's) are enforced across the full plausible interval of the neutrosophic parameter (worst-case formulation) rather than at a single point estimate.
Base Distribution: While the framework is generic, the paper empirically evaluates the Generalized Inverted Exponential Distribution (GIED), Lognormal, and Weibull distributions. The Weibull distribution is retained as the base model due to numerical stability issues observed with GIED (divergent shape estimates) and empirical fit.

3. Key Contributions

The paper identifies and addresses three specific gaps in the literature:

Degradation Framing: It shifts the acceptance sampling paradigm from a binary "failed by time $t_0$ " event to a progressive degradation trajectory terminating at a defined SOH threshold.
Hybrid Architecture: It is the first published plan (classical or neutrosophic) to nest RASP-style resampling and MDSSP-style lot-history dependency within a single operating procedure, providing a closed-form Operating Characteristic (OC) function and Average Sample Number (ASN).
Empirical, Uncertainty-Aware Calibration: It calibrates the plan against real-world cycling data from four independent sources (NASA, HNEI, MICH, and Sandia National Laboratories), totaling 115 cells across eight physically distinct groups. This includes a rigorous diagnosis of why pooling data across temperatures fails and the correction of periodic logging artifacts in the data.

4. Results and Empirical Findings

The study utilized data from NASA Prognostics Center of Excellence, HNEI, MICH, and Sandia National Laboratories (SNL). Key findings include:

Distribution Selection: Across all eight groups, the Weibull distribution was retained as the most stable and appropriate model. The Generalized Inverted Exponential Distribution (GIED) was decisively excluded due to numerical instability (shape estimates diverging to implausible values ranging from $10^2$ to $10^{34}$ ). While Lognormal showed competitive fit in some cases (e.g., HNEI), Weibull was retained for architectural consistency.
Boundary Conditions:
- Temperature: A "Leave-One-Group-Out" analysis revealed that pooling data across different temperatures (e.g., 4°C vs. 24°C) leads to catastrophic predictive failure. The plan is valid only for a single, specified reference temperature.
- Formation Protocol: Similar generalization failures were observed when mixing different formation protocols (BLForm vs. MCForm) even at fixed temperatures.
Data Artifacts: The analysis uncovered and corrected a periodic logging artifact (paired doubled/zero-capacity rows) present in HNEI and SNL datasets, which otherwise would have biased the fitted shape parameters.
Design Efficiency: The resulting design tables show Average Sample Numbers (ASN) ranging from approximately 3 to 26 cycles. The study found that for short truncation windows ( $a=0.5$ ), the producer's risk constraint is satisfied with near certainty, making the consumer's risk the governing factor. For longer windows ( $a=1.0$ ), the producer's risk constraint becomes binding as the quality level approaches the specification.
Neutrosophic Mapping: The paper provides a reproducible recipe for mapping statistical fit diagnostics (Wald confidence levels, relative CI half-widths, and Akaike weights) to the neutrosophic triple $(T_\gamma, I_\gamma, F_\gamma)$ . This mapping was successfully instantiated across five distinct calibration scenarios.

5. Significance and Claims

The paper positions itself as a methodological and calibration-stage contribution. Its primary significance lies in:

Explicit Uncertainty Handling: By using neutrosophic statistics, the plan explicitly accounts for the epistemic uncertainty inherent in battery manufacturing batches, ensuring risk constraints hold across the full range of plausible shape parameters rather than just a point estimate.
Real-World Validation: Unlike many theoretical sampling papers that rely on simulated data, this work validates its methodology against 115 real cells from four independent sources, identifying specific boundary conditions (temperature and protocol) where the plan must not be applied without re-calibration.
Practical Recommendation: The authors conclude that shape parameters must be calibrated per chemistry and per operating condition rather than pooled. They assert that Weibull is the robust base distribution for these tests, while GIED should be avoided due to numerical instability.

The paper modestly acknowledges limitations, including small sample sizes in some calibration groups (e.g., NASA $n=4$ ) and the need for future work on formal neutrosophic inference procedures and the reconciliation of exact hybrid OC/ASN functions for specific chemistries. It does not claim to solve all battery reliability issues but offers a rigorous, uncertainty-aware framework for acceptance sampling in truncated life tests.