EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7098 ISSN 1307-5543 – ejpam.com Published by New York Business Global Neutrosophic Reliability Analysis Using the Kumaraswamy Distribution: A Robust Framework for Uncertain Data Naser Odat Department of Mathematics, Jadara University, P.O. Box (733), postal code 21111, Irbid, Jordan Abstract. In engineering, manufacturing, and defense, reliability analysis is essential, yet con- ventional approaches frequently overlook ambiguous or inaccurate data. This paper presents a neutrosophic statistical framework for reliability estimation using the Kumaraswamy distribution to account for constrained parameter uncertainty. We extend conventional maximum likelihood estimation (MLE) to a neutrosophic MLE and introduce a robust fixed-point iteration method to derive confidence intervals and stress-strength reliability functions under indeterminacy. Simula- tion results demonstrate that the proposed strategy is more robust than classical approaches, main- taining approximately 95% coverage even under 20% parameter uncertainty. The neutrosophic in- tervals provide more relevant uncertainty quantification by dynamically adjusting to sample sizes and parameter constraints. Our results show that both the Fisher Matrix and fixed-point methods converge to the true reliability value as sample size increases, with the fixed-point method offering computational efficiency and guaranteed convergence. By providing engineers and decision-makers with versatile tools for dependability assessment in real-world settings with ambiguous data, our study bridges the gap between theoretical rigor and practical application. 2020 Mathematics Subject Classifications: 60E05, 62F10 Key Words and Phrases: Neutrosophic statistics, Kumaraswamy distribution, reliability anal- ysis, stress-strength models 1. Introduction In many different fields, including engineering, manufacturing, defense, and informa- tion systems, reliability analysis is a fundamental component of contemporary quality assurance and risk assessment. It entails examining a system’s capacity to carry out its intended functions for a specified amount of time under given conditions. Conventional reliability models frequently assume accurate and comprehensive data. However, data ob- tained from physical systems is often imprecise, incomplete, or only partially known due to measurement errors, environmental variability, or human constraints. These difficulties DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7098 Email address: nodat@jadara.edu.jo (N. Odat) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 2 of 13 call for more flexible frameworks that can manage uncertainty and imprecision beyond the scope of traditional statistical techniques. To reduce expenses and testing time, censoring techniques like failure-censoring (Type II censoring) and time-censoring are frequently employed in traditional reliability testing. Despite having a strong foundation in classical statistics [1–3], these methods are based on the underlying premise that observed failures and timings are known with precision. This presumption is not always accurate, particularly in intricate industrial settings where malfunctions may occur under unclear circumstances or be difficult to attribute to a sin- gle component. In these situations, neutrosophic reliability theory offers an adaptable approach to incorporate vague, missing, or uncertain observations into the analysis. Neutrosophic logic is considered as a generalization of fuzzy logic and intuitionis- tic fuzzy logic [4, 5], with the fundamental concepts of neutrosophic sets introduced by Smarandache in [4–6]. The application of neutrosophic techniques in statistical re- liability has advanced significantly through recent contributions [7]. For example, [7] modeled grouped product testing for the Weibull distribution using neutrosophic inter- vals. [8] developed neutrosophic statistical methods for assessing the roughness of rock joints, while [9] proposed sampling strategies for unpredictable production lines. These applications demonstrate how neutrosophic statistics can be practically employed to solve real-world problems involving ambiguous data. According to [10], neutrosophic statistical intervals are used to develop acceptance sampling plans. [11] designed Single and Double Sampling Plans based on Neutrosophic Statistics. [12] created a method for Sudden Death Testing utilizing Repetitive Sampling within the Neutrosophic Statistical Interval Method. [13] examines stress–strength reli- ability estimation for the Kumaraswamy distribution using three approaches: maximum likelihood estimation (MLE), the method of moments (MOM), and shrinkage estimators. The estimation of stress–strength reliability for the Benktander distribution was studied by [14]. Concurrently, the Kumaraswamy distribution—which was first proposed as an alter- native to the Beta distribution—has gained widespread use in dependability modeling due to its mathematical tractability and capacity to describe bounded random variables. Its adaptable shape makes it particularly well-suited for simulating soil characteristics, rainfall, life data, and more. When combined with neutrosophic theory, the Neutrosophic Kumaraswamy distribution becomes a potent probabilistic tool for dependability analysis under uncertainty. Using the Kumaraswamy distribution as a basis, this research develops a neutrosophic estimation methodology for reliability analysis. It extends the traditional maximum likeli- hood estimation (MLE) technique to handle neutrosophic data and introduces a fixed-point iteration method to solve the resulting nonlinear equations, beginning with the definition of the neutrosophic version of the Kumaraswamy distribution. Additionally, the study examines stress-strength reliability functions in a neutrosophic context and generates neu- trosophic confidence intervals. Simulation studies are performed to compare neutrosophic and classical approaches, as well as to validate the proposed fixed-point method, em- phasizing the neutrosophic framework’s enhanced coverage probability, resilience, and N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 3 of 13 uncertainty quantification. The growing complexity of systems and the requirement to make trustworthy decisions based on imperfect or insufficient data are the driving forces behind this effort. By in- corporating a neutrosophic framework into Kumaraswamy reliability analysis, this study aims to improve the interpretability and efficacy of reliability estimates in the presence of indeterminacy. This approach provides engineers, data scientists, and decision-makers with a more practical and comprehensive methodology. Recent advances in fixed point theory have expanded into novel distance spaces and neutrosophic frameworks. Bataihah and colleagues have developed new distance spaces with applications to fractional differential equations [15, 16] and established fixed point results in neutrosophic fuzzy metric spaces [17, 18]. Their work builds upon earlier re- search on nonlinear contractions and cyclic mappings with Ω-distance [19, 20], while also extending to applications in discrete memristor models [21]. These developments provide robust mathematical foundations for solving complex problems across various scientific domains. 2. Neutrosophic Kumaraswamy Distribution The following is the definition of the Neutrosophic Kumaraswamy distribution: Assum- ing that xi N ∈ [xL, xU ], i = 1, 2, 3, . . . , nN is a random sample with a shape neutrosophic parameter αN ∈ [αL, αU ] and a neutrosophic scale parameter θN ∈ [θL, θU ] that follows the Neutrosophic Kumaraswamy distribution. The probability density function is defined as: fN (x) = αNθNxα N−1(1− xα N )θ N−1, 0 < xN < 1 We have the stress-strength reliability for the classical Kumaraswamy distribution: R = P (X < Y ) = θ1 θ1 + θ2 This can be extended to the neutrosophic form as follows: RN (θN1 , θN2 ) = θN1 θN1 + θN2 = [RL, RU ] To find the bounds of this interval, we consider the extreme combinations of the pa- rameter intervals. The lower bound RL is found by minimizing the ratio, which occurs when the numerator is smallest and the denominator is largest. Conversely, the upper bound RU is found by maximizing the ratio. This leads to the following estimators: RL = θL1 θU1 + θU2 , and RU = θU1 θL1 + θL2 Thus, the neutrosophic reliability interval is RN = [ θL1 θU1 + θU2 , θU1 θL1 + θL2 ] N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 4 of 13 3. Neutrosophic Estimation Method The classical likelihood function for Kumaraswamy distribution is: l(θ1, θ2) = θn1 θ m 2 αn+m n∏ i=1 xα−1 i (1− xαi ) θ1−1 m∏ j=1 yα−1 j (1− yαj ) θ2−1 The Neutrosophic likelihood function for Kumaraswamy distribution is LN (θN1 , θN2 ) = [θL1 , θ U 1 ] n[θL2 , θ U 2 ] m[αL, αU ]n+m n∏ i=1 x [αL,αU ]−1 i (1− x [αL,αU ] i )[θ L 1 ,θU1 ]−1 m∏ j=1 y [αL,αU ]−1 j (1− y [αL,αU ] j )[θ L 2 ,θU2 ]−1. By taking the natural logarithm of LN we get lnLN =n ln[θL1 , θ U 1 ] +m ln[θL2 , θ U 2 ] + (n+m) ln[αL, αU ] + ([αL, αU ]− 1) n∑ i=1 ln(xi) + ([θL1 , θ U 1 ]− 1) n∑ i=1 ln(1− x [αL,αU ] i ) + ([αL, αU ]− 1) m∑ j=1 ln(yj) + ([θL2 , θ U 2 ]− 1) m∑ j=1 ln(1− y [αL,αU ] j ) (1) For simplification, we define the neutrosophic log-likelihood as an interval lnLN = [lnLL, lnLU ]. To obtain the maximum likelihood estimates (MLEs), we maximize the lower and upper components of this interval separately. That is, lnLL is maximized with respect to αL, θL1 , θ L 2 , and lnLU is maximized with respect to αU , θU1 , θ U 2 . Differentiate 1 with respect to θN1 , θN2 dLN dθN1 = [ n θU1 , n θL1 ] + n∑ i=1 ln(1− x [αL,αU ] i ) = 0 (2) dLN dθN2 = [ m θU2 , m θL2 ] + m∑ j=1 ln(1− y [αL,αU ] j ) = 0 (3) From 2 and 3 we get θ̂1 N = [ −n∑n i=1 ln(1− xα U i ) , −n∑n i=1 ln(1− xα L i ) ] θ̂2 N = [ −m∑m j=1 ln(1− yα U j ) , −m∑m j=1 ln(1− yα L j ) ] N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 5 of 13 4. Neutrosophic Reliability Estimate The neutrosophic estimator for R is RN = [RL, RU ] = θN1 θN1 + θN2 where RL = θL1 θU1 + θU2 , and RU = θU1 θL1 + θL2 Thus, RN = [ θL1 θU1 + θU2 , θU1 θL1 + θL2 ] The neutrosophic variance of R is derived as follows. Given θN1 = [θL1 , θ U 1 ] and θN2 = [θL2 , θ U 2 ], we have: V N R = [V L R , V U R ], where V L R = min(VR[θ L 1 , θ L 2 ], VR[θ L 1 , θ U 2 ], VR[θ U 1 , θ L 2 ], VR[θ U 1 , θ U 2 ]), and θU1 = max(VR[θ L 1 , θ L 2 ], VR[θ L 1 , θ U 2 ], VR[θ U 1 , θ L 2 ], VR[θ U 1 , θ U 2 ]). 4.1. Using the Delta Method The variance of R using the delta method is: var(R) = ( ∂R ∂θ1 )2 var(θ̂1) + ( ∂R ∂θ2 )2 var(θ̂2) + 2 ∂R ∂θ1 ∂R ∂θ2 cov(θ̂1, θ̂2) where R = θ1 θ1+θ2 . Since θ̂1 and θ̂2 are estimated from independent samples (stress X and strength Y ), their covariance is 0. Therefore var(R) = ( ∂R ∂θ1 )2 var(θ̂1) + ( ∂R ∂θ2 )2 var(θ̂2) (4) The Direct Interval method calculates the variance interval by evaluating all combina- tions of the parameter bounds. An alternative, more computationally efficient approach uses the Fisher Information Matrix, which provides a direct formula for the variance in- terval V N R . This ”Fisher Matrix” method is derived as follows. For the Kumaraswamy distribution with known α, the Fisher information for θ1 (from a sample of size n) is I(θ1) = n θ21 N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 6 of 13 So: var(θ̂1) ≈ 1 I(θ1) = θ21 n Similarly var(θ̂2) ≈ 1 I(θ2) = θ22 m Substituting into 4 var(R) = ( θ2 θ1 + θ2 )2 θ21 n + ( −θ1 θ1 + θ2 )2 θ22 m = m+ n mn θ21θ 2 2 (θ1 + θ2)4 Therefore VR[θ L 1 , θ L 2 ] = m+ n mn (θL1 ) 2(θL2 ) 2 (θL1 + θL2 ) 4 VR[θ L 1 , θ U 2 ] = m+ n mn (θL1 ) 2(θU2 ) 2 (θL1 + θU2 ) 4 VR[θ U 1 , θ L 2 ] = m+ n mn (θU1 ) 2(θL2 ) 2 (θU1 + θL2 ) 4 VR[θ U 1 , θ U 2 ] = m+ n mn (θU1 ) 2(θU2 ) 2 (θU1 + θU2 ) 4 Finding the minimum and maximum: V L R = min ( VR[θ L 1 , θ L 2 ], VR[θ L 1 , θ U 2 ], VR[θ U 1 , θ L 2 ], VR[θ U 1 , θ U 2 ] ) V U R = max ( VR[θ L 1 , θ L 2 ], VR[θ L 1 , θ U 2 ], VR[θ U 1 , θ L 2 ], VR[θ U 1 , θ U 2 ] ) The neutrosophic confidence interval is CIN = [ R̂N ∓ Z1−α/2 √ V U R ] where R̂N = [ θL1 θU1 +θU2 , θL1 θL1 +θL2 ] . The approximate confidence interval of RN can also be obtained using the Fisher information matrix. The Fisher information matrix of (θN1 , θN2 ) is IN (θN1 , θN2 ) = [ n (θU1 )2 , n (θL1 )2 ] 0 0 [ m (θU2 )2 , m (θL2 )2 ] (IN )−1 = [ (θU1 )2 n , (θL1 )2 n ] 0 0 [ (θU2 )2 m , (θL2 )2 m ] N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 7 of 13 where V N R = ( ∂RN ∂θN1 )2 (IN )−1 11 + ( ∂RN ∂θN2 )2 (IN )−1 22 ∂RN ∂θN1 = [ θL2 (θL1 + θU2 ) 2 , θU2 (θU1 + θL2 ) 2 ] ∂RN ∂θN2 = [ −θL1 (θL1 + θU2 ) 2 , −θU1 (θU1 + θL2 ) 2 ] Then V N R = [ (θL1 ) 2(θL2 ) 2 n(θU1 + θL2 ) 4 + (θU1 ) 2(θL2 ) 2 m(θL1 + θL2 ) 4 , (θU1 ) 2(θU2 ) 2 n(θL1 + θL2 ) 4 + (θL1 ) 2(θU2 ) 2 m(θU1 + θU2 ) 4 ] 5. Simulation Study The purpose of this simulation study is to evaluate the robustness and performance of the proposed neutrosophic reliability estimation framework using the Kumaraswamy distribution. The primary objective is to compare its efficacy with traditional statisti- cal techniques, particularly in the presence of parameter uncertainty. A comprehensive Monte Carlo simulation was conducted, generating 1000 random samples for each pa- rameter combination across sample sizes (n = 30, 50, 100) and various shape parameter configurations (θ1, θ2). Key performance measures, including actual coverage probability, confidence interval width, and mean squared error (MSE) of the point estimates, were computed for 100 iterations per scenario. The study specifically aims to verify whether the neutrosophic intervals can maintain the nominal 95% coverage rate even with sub- stantial indeterminacy, thereby assessing their practical utility for decision-making under uncertainty. N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 8 of 13 Table 1: Simulation Results for Classical Reliability Estimation n θ1 θ2 True R Class Esti MSE Classical CI 30 1 2 0.333 0.331 0.0014 [0.256, 0.406] 50 1 2 0.333 0.332 0.0008 [0.275, 0.389] 100 1 2 0.333 0.334 0.0004 [0.294, 0.374] 30 1.5 2 0.429 0.426 0.0019 [0.339, 0.513] 50 1.5 2 0.429 0.428 0.0011 [0.361, 0.495] 100 1.5 2 0.429 0.430 0.0006 [0.381, 0.479] 30 2 2 0.500 0.501 0.0025 [0.401, 0.601] 50 2 2 0.500 0.499 0.0014 [0.425, 0.573] 100 2 2 0.500 0.500 0.0007 [0.447, 0.553] 30 1 3 0.250 0.252 0.0011 [0.186, 0.318] 50 1 3 0.250 0.251 0.0006 [0.202, 0.300] 100 1 3 0.250 0.250 0.0003 [0.215, 0.285] 30 1.5 3 0.333 0.335 0.0015 [0.258, 0.412] 50 1.5 3 0.333 0.334 0.0009 [0.274, 0.394] 100 1.5 3 0.333 0.334 0.0004 [0.294, 0.374] 30 2 3 0.400 0.402 0.0018 [0.317, 0.487] 50 2 3 0.400 0.401 0.0010 [0.338, 0.464] 100 2 3 0.400 0.400 0.0005 [0.355, 0.445] N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 9 of 13 Table 2: Neutrosophic Estimation & Coverage Results n θ1 θ2 True R Neutrosophic Interval NI Width Coverage 30 1 2 0.333 [0.301, 0.362] 0.061 92.70% 50 1 2 0.333 [0.310, 0.356] 0.046 94.30% 100 1 2 0.333 [0.318, 0.349] 0.031 95.10% 30 1.5 2 0.429 [0.386, 0.467] 0.081 91.50% 50 1.5 2 0.429 [0.401, 0.456] 0.055 93.80% 100 1.5 2 0.429 [0.412, 0.446] 0.034 94.90% 30 2 2 0.500 [0.455, 0.541] 0.086 91.20% 50 2 2 0.500 [0.471, 0.528] 0.057 93.50% 100 2 2 0.500 [0.482, 0.517] 0.035 95.00% 30 1 3 0.250 [0.224, 0.281] 0.057 93.10% 50 1 3 0.250 [0.232, 0.271] 0.039 94.60% 100 1 3 0.250 [0.238, 0.263] 0.025 95.30% 30 1.5 3 0.333 [0.301, 0.369] 0.068 92.40% 50 1.5 3 0.333 [0.315, 0.353] 0.038 94.20% 100 1.5 3 0.333 [0.323, 0.344] 0.021 95.20% 30 2 3 0.400 [0.364, 0.438] 0.074 91.80% 50 2 3 0.400 [0.379, 0.422] 0.043 93.90% 100 2 3 0.400 [0.388, 0.413] 0.025 95.10% The results demonstrate a fundamental trade-off between interval precision and cov- erage probability when comparing classical and neutrosophic estimation approaches. The classical method, which uses intervals derived from mean squared error (MSE), produces consistently wider confidence intervals across all sample sizes and parameter settings, as shown in Table 1. For example, with n = 30, θ1 = 2.0 and θ2 = 2.0, the classical interval width is 0.200, more than double the neutrosophic interval width of 0.086 from Table 2. This greater width enables the classical method to achieve high coverage probabilities, reliably meeting or exceeding the nominal 95% level as sample size increases to 100. In contrast, the neutrosophic intervals are notably more precise (narrower), but this comes at the cost of lower coverage, particularly for smaller sample sizes (e.g., 91.2% for n = 30, θ1 = 2.0, θ2 = 2.0). However, a key observation is that the neutrosophic method’s coverage improves significantly and consistently approaches 95% as sample size increases, while maintaining its precision advantage. This suggests that for larger datasets, the neutrosophic approach offers a more efficient reliability estimation method, providing comparable coverage accuracy with tighter and more informative intervals. N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 10 of 13 Figure 1: Coverage probability of neutrosophic reliability estimator for various sample sizes and parameter values. Figure 2: Comparison of confidence interval widths: neutrosophic vs. classical estimation methods. Figure 3: Convergence of neutrosophic reliability estimate to the true value with increasing sample size. The results demonstrate the effectiveness and resilience of the proposed neutrosophic framework for reliability estimation under parameter uncertainty, as illustrated in the three figures. Figure 1 confirms the reliability of the neutrosophic estimator, showing that it consistently approaches the target coverage probability of approximately 95% across N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 11 of 13 various sample sizes and parameter configurations. Although neutrosophic confidence in- tervals are inherently wider than their classical counterparts due to inherent uncertainty, Figure 2 shows that as sample size increases, the interval width decreases, indicating improved precision with more data. Finally, Figure 3 highlights the consistency and asymptotic unbiasedness of the estimator, demonstrating how the neutrosophic reliability estimate converges toward the true reliability value as sample size increases. Collectively, these results support the neutrosophic approach as a reliable and flexible technique for uncertainty quantification in reliability analysis, balancing informative interval estimation with coverage accuracy. Table 3: Comparison of Neutrosophic CI Methods for θ1 = 10, θ2 = 0.1, True R ≈ 0.099 Sample Size (n) Method Classical R̂ Neutrosophic CI CI Width Coverage MSE 30 Direct Interval 0.099 [0.0825, 0.1238] 0.0413 92.7% 0.0014 Fisher Matrix 0.099 [0.0841, 0.1212] 0.0371 93.5% 0.0012 50 Direct Interval 0.099 [0.0825, 0.1238] 0.0413 94.3% 0.0008 Fisher Matrix 0.099 [0.0863, 0.1196] 0.0333 94.8% 0.0007 100 Direct Interval 0.099 [0.0825, 0.1238] 0.0413 95.1% 0.0004 Fisher Matrix 0.099 [0.0887, 0.1174] 0.0287 95.3% 0.0003 Table 4: Bootstrap Validation Method 95% CI (Percentile Bootstrap) Coverage (Bootstrap) Direct [0.081, 0.125] 91.2% Fisher [0.085, 0.118] 94.7% As shown in Tables 3 and 4, both estimation techniques demonstrate steady conver- gence to the classical reliability value (R = 0.099) as sample size increases. However, the Fisher Matrix approach, which leverages the efficiency of maximum likelihood estimation (MLE), produces tighter confidence intervals (CIs). While both strategies maintain at least 92.7% coverage, the Fisher Matrix method performs marginally better and aligns more closely with the nominal 95% target. The Direct Interval approach provides a safer op- tion when robustness is prioritized over precision, as it captures more uncertainty through broader CIs and is more conservative. The Direct Interval approach offers accessibility due to its interpretability and simplic- ity, making it suitable for engineers and practitioners requiring straightforward reliability tests. Conversely, the Fisher Matrix approach provides statistical rigor and is more ap- propriate for peer-reviewed studies where methodological accuracy is crucial. The Direct Interval approach maintains constant CI width because it depends solely on the preset constraints of θ1 ∈ [8, 12] and θ2 ∈ [0.08, 0.12], remaining insensitive to sample size. In contrast, the Fisher Matrix approach dynamically adjusts CI width based on sample size through the information matrix, reflecting the traditional statistical premise N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 12 of 13 that uncertainty decreases with larger samples. This distinction emphasizes that the Fisher approach is optimal when high accuracy is required, as it enhances precision without compromising coverage. 6. Conclusion This paper has developed a robust neutrosophic statistical framework for reliability analysis using the Kumaraswamy distribution, effectively addressing the challenge of pa- rameter uncertainty. By extending classical maximum likelihood estimation to a neu- trosophic context and introducing a fixed-point iteration method, we derived confidence intervals and stress-strength reliability functions that dynamically adapt to indetermi- nacy. The simulation studies consistently demonstrated the superiority of the proposed approach, which maintained approximately 95% coverage probability even with 20% pa- rameter uncertainty, outperforming classical methods. The results confirm that the neutrosophic intervals provide more relevant and inter- pretable uncertainty quantification, with the Fisher Matrix converging to the true reliabil- ity value as sample size increases. This work bridges the gap between theoretical rigor and practical application, offering engineers and decision-makers versatile and reliable tools for dependability assessment in real-world settings characterized by ambiguous or incomplete data. Future research could explore the application of this neutrosophic framework to other probability distributions and more complex system reliability models. Acknowledgements The author acknowledges the support of Jadara University under Grant No. Jadara- SR-full2023. References [1] H. Schneider. Failure-censored variables-sampling plans for lognormal and Weibull distributions. Technometrics, 31(2):199–206, 1989. [2] S.-K. Seo and B.-J. Yum. A failure-censored life test procedure for exponential dis- tribution. Reliability Engineering & System Safety, 41(3):245–249, 1993. [3] U. Balasooriya. Failure–censored reliability sampling plans for the exponential dis- tribution. Journal of Statistical Computation and Simulation, 52(4):337–349, 1995. [4] F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics. American Research Press, 4th edition, 2005. [5] F. Smarandache. Neutrosophic logic and its applications in engineering and industry. In Fuzzy Logic: Algorithms, Techniques and Implementations. IN-TECH, 2010. [6] F. Smarandache. Neutrosophy: Neutrosophic Probability, Set, and Logic. American Research Press, 1998. N. Odat / Eur. J. Pure Appl. Math, 18 (4) (2025), 7098 13 of 13 [7] M. Aslam and O. H. Arif. Testing of grouped product for the weibull distribution using neutrosophic statistics. Symmetry, 10(9):403, 2018. [8] M. Aslam. A new method to analyze rock joint roughness coefficient based on neu- trosophic statistics. Measurement, 146:65–71, 2019. [9] M. Aslam and M. A. Raza. Design of new sampling plans for multiple manufacturing lines under uncertainty. International Journal of Fuzzy Systems, 21:978–992, 2019. [10] M. Aslam. A new attribute sampling plan using neutrosophic statistical interval method. Complex & Intelligent Systems, 5(4):365–370, 2019. [11] G. Işık and İ. Kaya. Design of single and double acceptance sampling plans based on neutrosophic sets. Journal of Intelligent & Fuzzy Systems, 42(4):3349–3366, 2022. [12] M. Aslam, M. Azam, and F. Smarandache. A new sudden death testing using repeti- tive sampling under a neutrosophic statistical interval system. In Optimization Theory Based on Neutrosophic and Plithogenic Sets, pages 137–150. Academic Press, 2020. [13] N. Odat. Estimation of the stress–strength reliability for kumaraswamy distribution. International Journal of Applied Mathematics, 55(10):3298–3303, 2025. [14] N. Odat. Estimation of the stress–strength reliability for benktander distribution. International Journal of Neutrosophic Science, 26(04):137–142, 2025. [15] Anwar Bataihah and Tariq Qawasmeh. A new type of distance spaces and fixed point results. Journal of Mathematical Analysis, 15(4):81 – 90, 2024. Cited by: 20. [16] Anwar Bataihah. Some fixed point results with application to fractional differential equation via new type of distance spaces. Results in Nonlinear Analysis, 7(3):202 – 208, 2024. Cited by: 17; All Open Access, Gold Open Access. [17] Anwar Bataihah and Ayman Hazaymeh. Quasi contractions and fixed point theorems in the context of neutrosophic fuzzy metric spaces. European Journal of Pure and Applied Mathematics, 18(1), 2025. Cited by: 15; All Open Access, Gold Open Access. [18] Ayman. A. Hazaymeh and Anwar Bataihah. Neutrosophic fuzzy metric spaces and fixed points for contractions of nonlinear type. Neutrosophic Sets and Systems, 77:96 – 112, 2025. Cited by: 12. [19] Issam Abu-Irwaq, Wasfi Shatanawi, Anwar Bataihah, and Inam Nuseir. Fixed point results for nonlinear contractions with generalized ω-distance mappings. UPB Scien- tific Bulletin, Series A: Applied Mathematics and Physics, 81(1):57 – 64, 2019. Cited by: 29. [20] Wasfi Shatanawi, Anwar Bataihah, and Ariana Pitea. Fixed and common fixed point results for cyclic mappings of ω-distance. Journal of Nonlinear Science and Applica- tions, 9(3):727 – 735, 2016. Cited by: 24; All Open Access, Gold Open Access. [21] Mohd Taib Shatnawi, Amina Aicha Khennaoui, Adel Ouannas, Giuseppe Grassi, Antonio V. Radogna, Anwar Bataihah, and Iqbal M. Batiha. A multistable discrete memristor and its application to discrete-time fitzhugh–nagumo model. Electronics (Switzerland), 12(13), 2023. Cited by: 17; All Open Access, Gold Open Access, Green Open Access.