Algorithms for Generating Neutrosophic Gamma Distributed Data

Muhammad Saleem¹ and Muhammad Aslam^2,*

¹Department of Industrial Engineering, Faculty of Engineering, King Abdulaziz University, Rabigh, 21911, Saudi Arabia
²Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
E-mail: msaleim1@kau.edu.sa; aslam_ravian@hotmail.com/magmuhammad@kau.edu.sa
*Corresponding Author

Received 26 February 2025; Accepted 28 June 2025

Abstract

The gamma distribution is well-known for its wide range of applications across various fields. Traditional gamma distributions and their associated algorithms have been used to model imprecise data; however, they face limitations in addressing uncertainty and indeterminacy. To overcome these challenges, this study introduces the neutrosophic gamma distribution, an extension of the classical gamma distribution that incorporates neutrosophic random variables to better handle imprecision and indeterminacy. Basic properties of the neutrosophic gamma distribution are presented, along with algorithms designed to generate data under different levels of indeterminacy. Simulation results reveal that as the degree of indeterminacy increases, the corresponding random variates tend to exhibit an upward trend. Comparative analysis with classical statistics highlights the significant effect of indeterminacy on data generation. Overall, the study demonstrates that the degree of indeterminacy plays a crucial role in shaping the behavior of data derived from the gamma distribution.

Keywords: Gamma distribution, random variate, indeterminacy, simulation, analysis.

1 Introduction

The gamma distribution is very important in reliability analysis, as shown in studies like [1] and [2]. In many cases, researchers use simulated data to support real data or when it’s difficult or impossible to collect real data that follows a gamma distribution. To create such data, special algorithms are used. These algorithms follow clear steps to simulate data based on specific conditions and parameters, helping ensure that the results match real-world situations as closely as possible. Simulating data using these algorithms is especially useful when collecting real data is too expensive or hard to do. Iriarte et al. [2] pointed out how useful it is to generate gamma-distributed random values in many different fields. The gamma distribution is also used in areas like Bayesian statistics and queuing theory, as shown in [3]. Other researchers have applied it in various fields – for example, Maaref and Annavajjala [4] in digital communications, Pati and Krishnan [5] in atmospheric studies, and Huber et al. [6] in economics. More details can be found in [7].

As mentioned earlier, generating random values from the gamma distribution is important in many fields. Gill [8] introduced one of the early methods for creating gamma random values. Atkinson and Pearce [9] developed algorithms for different types of statistical distributions, including the gamma distribution. Marsaglia and Tsang [10] proposed a simple and widely used method for generating gamma values. Robertson and Walls [11] provided algorithms for both the normal and gamma distributions. Tanizaki [12] created an algorithm that works for any shape parameter of the gamma distribution. Apolloni and Bassis [13] designed methods that use both parameters of the gamma distribution. Xi et al. [14] introduced a technique based on transformation to generate gamma values. Lawnik [15] explored how to simulate gamma data using an inverse chaotic transformation. Luengo [16] developed tools for generating pseudo-random gamma values.

The current algorithms for generating data from the gamma distribution are based on classical statistics and do not account for uncertainty or the degree of indeterminacy. Neutrosophic statistics, introduced by [17], is a modern approach designed to handle imprecise and uncertain data. Unlike classical methods, neutrosophic statistics includes a concept called the degree of indeterminacy, which classical statistics ignores. Neutrosophic statistics incorporates the degrees of truth, indeterminacy, and falsity, making it more informative than classical statistics. It reduces to classical statistical analysis when the degree of indeterminacy is absent. Methods for analyzing neutrosophic data have been presented by [18] and [19]. Smarandache [20] showed that neutrosophic statistics can be more effective than classical and interval-based approaches. Granados et al. [21] contributed by working on neutrosophic continuous distributions. Aslam [22] developed an algorithm to simulate neutrosophic data using the DUS-neutrosophic Weibull distribution, and later [23], introduced sine-cosine methods for generating neutrosophic data from the normal distribution. More examples of algorithms based on neutrosophic ideas can be found in [24] and [25]. Khan et al. [26] studied the neutrosophic gamma distribution and its use for complex data. Granados et al. [27] also introduced continuous distributions using neutrosophic random variables. Aslam and Alamri [28] created accept-reject-based algorithms for generating neutrosophic data from the uniform and Weibull distributions. Jdid et al. [29] proposed a method to convert uniform data into exponential data using neutrosophic statistics. In another study, Jdid et al. [30] developed algorithms to generate neutrosophic data from the gamma distribution. Aslam [31] also introduced the neutrosophic negative binomial distribution and its data generation methods, while Jdid and Smarandache [32] proposed a composition method to simulate neutrosophic data from the Poisson distribution. For more information on neutrosophic data generation algorithms, see references [22, 23, 33]. Alamri and Aslam [34] developed an algorithm for handling neutrosophic data based on the Erlang distribution.

The existing literature predominantly emphasizes classical statistical approaches to the gamma distribution, often overlooking the need for adaptable methodologies in uncertain environments. The gamma distribution under classical statistics and its associated algorithms are not equipped to effectively address indeterminacy, limiting their applicability in scenarios characterized by significant uncertainty. Although algorithms under classical statistics for generating random variates from the gamma distribution have been developed for certain environments, no existing work, to the best of our knowledge, explores algorithms using the neutrosophic gamma distribution with the accept-reject method. This study aims to address this gap and improve existing methods by introducing the neutrosophic gamma distribution and developing accept-reject-based algorithms that account for the degree of indeterminacy, thereby addressing the limitations of classical statistical approaches. The main contributions of the study are the introduction of the neutrosophic gamma distribution, the examination of its basic properties, and the development of algorithms for generating random variates using the accept-reject method under various conditions, with a particular focus on the role of indeterminacy. Simulation studies highlight the impact of increasing uncertainty on random variate generation, underscoring the need for more nuanced statistical modeling approaches. Comparative analyses with methods under classical statistics demonstrate the distinctiveness and applicability of the proposed algorithms under neutrosophic statistics in addressing uncertain scenarios effectively. Section 2 presents a brief introduction to the neutrosophic random variable. Section 3 presents the neutrosophic gamma distribution with some basic properties. Section 4 discusses the two proposed algorithms when $k < 1$ . Section 5 discusses the simulation using the algorithm when $k \geq 1$ . Section 6 discusses the comparative studies for both algorithms. Section 7 discusses the application of the proposed method. Section 8 discusses the limitations of the proposed method and Section 9 discusses the concluding remarks.

2 Neutrosophic Random Variable

In this section, we will first the neutrosophic random variable with some basic properties of expectation and variance. Let us define the neutrosophic random variable having two parts as follows

X_{N} = X_{L} + X_{L} I_{N}; I_{N} ϵ [I_{L}, I_{U}]

(1)

The first part, $X_{L}$ , is known as the determinate part and represents the random variable in classical statistics. The second part, $X_{L} I_{N}$ , is referred to as the indeterminate part, where $I_{N} ϵ [I_{L}, I_{U}]$ represents the degree of uncertainty. It is important to note that the neutrosophic random variable defined in Equation (1) is a generalization of the classical random variable, incorporating both determinate and precise values. The neutrosophic random variable reduces to $X_{L}$ when $I_{L} = 0$ . Suppose $X_{L}$ has a mean of $μ_{L}$ and a variance of $σ_{L}^{2}$ . The expectation and variance of the neutrosophic random variable’s properties are then given by [31].

$E (X_{N}) = E (X_{L} + X_{L} I_{N} t) = μ_{L} + μ_{L} I_{N}$	(2)
$V a r (X_{N}) = V a r (X_{L} + X_{L} I_{N}) = {(1 + I_{N})}^{2} σ_{L}^{2}$	(3)

3 Neutrosophic Gamma Distribution

In this section, we will introduce the neutrosophic gamma distribution using the neutrosophic random variable. Suppose that $X_{N} = X_{L} + X_{L} I_{N}; I_{N} ϵ [I_{L}, I_{U}]$ follows the neutrosophic gamma distribution, where $X_{L}$ follows the gamma distribution under classical statistics and $X_{L} I_{N}$ be the indeterminate part. The neutrosophic probability density function (npdf) of the gamma distribution, characterized by parameters $k$ (shape parameter) and $θ$ (scale parameter), is defined as follows:

f (X_{N}) = {(X_{N})}^{k - 1} {(θ)}^{k} e^{- θ X_{N}} / Γ k; I_{N} ϵ [I_{L}, I_{U}]

(4)

It is important to note that the neutrosophic gamma distribution simplifies to the classical gamma distribution when $I_{L} = 0$ .

Note that $Γ k$ refers to the gamma function and is defined as follows:

Γ k = \int_{0}^{\infty} {(X_{N})}^{k - 1} e^{- X_{N}} d X_{N}, k > 0

(5)

If $k$ is a positive integer, the neutrosophic gamma function $Γ k$ is equal to $(k - 1)!$ .

The expected value and the variance of the random variable $X_{L}$ follows the gamma distribution is expressed by

$E (X_{L}) = \frac{k}{θ}$	(6)
$Var (X_{L}) = \frac{k}{θ^{2}}$	(7)

By applying the expectation properties on a neutrosophic random variable $X_{N}$ , the expectation of a neutrosophic gamma-distributed random variable is expressed as

E (X_{N}) = \frac{k (1 + I_{N})}{θ}

(8)

The neutrosophic variance for the neutrosophic gamma distribution is defined as:

Var (X_{N}) = \frac{k {(1 + I_{N})}^{2}}{θ^{2}}

(9)

For small values of $I_{N}$ , the variance in Equation (9) can be approximated as follows

Var (X_{N}) = \frac{k (1 + I_{N})}{θ^{2}} \approx \frac{k {(1 + I_{N})}^{2}}{θ^{2}}

(10)

4 The Proposed Algorithms of Gamma Distribution

In this segment, we introduce two algorithms employing the neutrosophic gamma distribution. The behavior of these algorithms is contingent upon the parameter $k$ . The initial algorithm, as outlined by Thomopoulo [35], will be presented when the $k$ values are below one. Conversely, the second algorithm, also derived from Thomopoulo [35], will be elucidated for scenarios where the $k$ values exceed one.

4.1 Algorithm when $k < 1$

This section outlines the development of a procedure for generating random number $X_{N}$ when $k < 1$ , as per Thomopoulo [35] and Ahrens and Dieter [36]. The proposed algorithm employs the accept-reject method. It is important to note that ${\overset{´}{X}}_{N}$ follows the neutrosophic gamma distribution with $θ = 1$ , while $X_{N}$ adheres to the gamma distribution with any positive value of $θ$ . The algorithm is executed as follows:

Step-1: Pre-fix the values of $I_{N}$ and set $b = \frac{(e + k)}{e}$ , where $e = 2.71828$

Step-2: Generate two uniform random numbers $u_{1}$ and $u_{2}$ from the uniform distribution $u \sim U (0, 1)$ , calculate $p_{N} = b u_{1}$

if $p_{N} > 1$ ; then go to step-4

if $p_{N} \leq 1$ ; then go to step-3

Step-3: $y_{N} = p_{N}^{k}$

if $u_{2} \leq e^{- y_{N}}$ then set ${\overset{´}{X}}_{N} = y_{N}$ , go to step-5

if $u_{2} > e^{- y_{N}}$ then set ${\overset{´}{X}}_{N} = y_{N}$ , go to step-2

Step-4: $y_{N} = {- \ln [(b - p_{N}) / k]}$

if $u_{2} \leq y_{N}^{k - 1}$ then set ${\overset{´}{X}}_{N} = y_{N}$ , go to step-5

if $u_{2} > y_{N}^{k - 1}$ then set ${\overset{´}{X}}_{N} = y_{N}$ , go to step-2

Step-5: Return $\frac{{\overset{´}{X}}_{N}}{θ}$

The algorithm to generate random variates when $k < 1$ is shown in Figure 1.

Figure 1 The algorithm to generate random variates when $k < 1$ .

It is important to note that the proposed algorithm for the neutrosophic gamma distribution when $k < 1$ extends the classical statistical algorithm. As illustrated in Figure 1, the proposed algorithm simplifies to the one presented by Thomopoulos [35] when no uncertainty is present in the data. Note that the neutrosophic mean given in Equation (8) and the approximate variance in Equation (10) were used to calculate the values of $k$ and $θ$ , following the method outlined by Thomopoulos [35].

4.2 Algorithm when $k \geq 1$

This section outlines the development of a procedure for generating random number $X_{N}$ when $k \geq 1$ , as per Thomopoulos [35] and Cheng [37]. The proposed algorithm employs the accept-reject method. It is important to note that ${\overset{´}{X}}_{N}$ follows the neutrosophic gamma distribution with $θ = 1$ , while $X_{N}$ adheres to the gamma distribution with any positive value of $θ$ The algorithm is executed as follows:

Step-1: Pre-fix the values of $I_{N}$ and set $a_{N} = \frac{1}{\sqrt{2 k - 1}}$ ; $b_{N} = (k - \ln 4)$ ; $q_{N} = (k + \frac{1}{a_{N}})$ ; $c = 4.5$ and $d = 1 + \ln (4.5)$ .

Step-2: Generate two uniform random numbers $u_{1}$ and $u_{2}$ from the uniform distribution $u \sim U (0, 1)$ , calculate $v_{N} = (a_{N} \times \ln [u_{1} / (1 - u_{1})])$ ; $y_{N} = (k e^{v_{N}})$ ; $z_{N} = u_{1}^{2} u_{2}$ and $w_{N} = (b + q_{N} v_{N} - y_{N})$ .

Step-3: if $w_{N} + d - c z_{N} \geq 0$ , then set ${\overset{´}{X}}_{N} = y_{N}$ , go to step-5

if $w_{N} + d - c z_{N} < 0$ , go to step-4

Step-4: if $w_{N} \geq \ln (z_{N})$ , then set ${\overset{´}{X}}_{N} = y_{N}$ , go to step-5

if $w_{N} < \ln (z_{N})$ , go to step-2.

Step-5: Return $\frac{{\overset{´}{X}}_{N}}{θ}$

The algorithm to generate random variates when $k \geq 1$ is shown in Figure 2.

Figure 2 The algorithm to generate random variates when $k \geq 1$ .

It is important to note that the proposed algorithm for the neutrosophic gamma distribution when $k \geq 1$ extends the classical statistical algorithm. As illustrated in Figure 2, the proposed algorithm simplifies to the one presented by Thomopoulos [35] when no uncertainty is present in the data. Note that the neutrosophic mean given in Equation (8) and the approximate variance in Equation (10) were used to calculate the values of $k$ and $θ$ , following the method outlined by Thomopoulos [35].

Table 1 Random numbers when $k = 0.5$ and $θ = 5$

$I_{N} = 0$	$I_{N} = 0$ .1	$I_{N} = 0.2$	$I_{N} = 0.3$	$I_{N} = 0.4$	$I_{N} = 0.5$	$I_{N} = 0.6$	$I_{N} = 0.7$	$I_{N} = 0.8$	$I_{N} = 0.9$
0.141	0.150	0.158	0.164	0.170	0.175	0.180	0.184	0.188	0.191
0.189	0.195	0.200	0.195	0.193	0.192	0.192	0.193	0.194	0.197
0.124	0.133	0.141	0.149	0.155	0.161	0.166	0.171	0.175	0.179
0.185	0.181	0.179	0.178	0.179	0.181	0.183	0.186	0.190	0.195
0.097	0.106	0.115	0.123	0.130	0.136	0.142	0.147	0.152	0.157
0.087	0.097	0.105	0.113	0.120	0.127	0.133	0.139	0.144	0.148
0.075	0.084	0.093	0.101	0.108	0.115	0.121	0.126	0.132	0.137
0.103	0.113	0.121	0.129	0.136	0.142	0.148	0.153	0.158	0.162
0.121	0.130	0.138	0.145	0.152	0.158	0.163	0.168	0.172	0.176
0.127	0.136	0.144	0.151	0.158	0.163	0.168	0.173	0.177	0.181
0.054	0.062	0.070	0.078	0.085	0.092	0.098	0.104	0.110	0.115
0.158	0.166	0.173	0.179	0.184	0.189	0.193	0.196	0.200	0.200
0.048	0.056	0.064	0.071	0.078	0.085	0.091	0.097	0.103	0.108
0.156	0.164	0.171	0.177	0.183	0.187	0.192	0.195	0.199	0.202
0.179	0.186	0.192	0.197	0.199	0.196	0.195	0.195	0.196	0.198

5 Simulation Using Algorithm when $k < 1$

In this section, we will showcase simulation studies conducted with the algorithm under the condition $k < 1$ . We generated random variates from the gamma distribution, varying the mean, variance, $k$ , $I_{N}$ and $θ$ values to analyze their impact on the data generated using the algorithm is presented in Figure 1. Table 1 displays results for a mean of 0.1, the variance of 0.02, $k$ of 0.5, and $θ$ of 5. Likewise, Tables 2 and 3 present outcomes for different sets of parameters. For instance, Table 2 presents values corresponding to a mean of 0.05, the variance of 0.03, $k$ of 0.08, and $θ$ of 1.67. Table 3 showcases results for a mean of 0.06, the variance of 0.04, $k$ of 0.09, and $θ$ of 1.5. Analyzing Tables 1–3 reveals that as the degree of indeterminacy increases from 0.1 to 0.9, there is a general upward trend in the random variates. For instance, in Table 1, when $I_{N} = 0.1$ , the random variate is 0.150, while for $I_{N} = 0.9$ , it increases to 0.191. Figure 3 illustrates the trend in random variates when $k = 0.5$ and $θ = 0.5$ , while Figures 4 and 5 depict the trends for $k = 0.08$ with $θ = 1.67$ and $k = 0.09$ with $θ = 1.5$ , respectively. Notably, Figures 3–5 show that as the values of $I_{N}$ increase, the random variates from the gamma distribution also increase. The curves corresponding to $I_{N} = 0.9$ consistently occupy higher positions compared to other $I_{N}$ values. In conclusion, based on this simulation study, it is inferred that when $k < 1$ , increasing the degree of uncertainty significantly influences the generation of random variates.

Table 2 Random numbers when $k = 0.08$ and $θ = 1.67$

$I_{N} = 0$	$I_{N} = 0$ .1	$I_{N} = 0.2$	$I_{N} = 0.3$	$I_{N} = 0.4$	$I_{N} = 0.5$	$I_{N} = 0.6$	$I_{N} = 0.7$	$I_{N} = 0.8$	$I_{N} = 0.9$
0.014	0.021	0.028	0.036	0.046	0.056	0.066	0.077	0.088	0.099
0.080	0.099	0.118	0.138	0.157	0.176	0.194	0.212	0.229	0.245
0.006	0.010	0.015	0.020	0.026	0.033	0.040	0.048	0.057	0.065
0.161	0.187	0.212	0.236	0.259	0.280	0.301	0.320	0.338	0.354
0.001	0.003	0.004	0.006	0.009	0.012	0.016	0.020	0.025	0.030
0.001	0.001	0.002	0.004	0.006	0.008	0.011	0.014	0.017	0.021
0.000	0.001	0.001	0.002	0.003	0.004	0.006	0.008	0.010	0.013
0.002	0.004	0.006	0.009	0.012	0.016	0.020	0.025	0.031	0.037
0.005	0.009	0.013	0.018	0.023	0.029	0.036	0.044	0.052	0.060
0.007	0.012	0.017	0.022	0.029	0.036	0.044	0.053	0.062	0.071
0.000	0.000	0.000	0.000	0.001	0.001	0.002	0.003	0.003	0.005
0.027	0.037	0.048	0.060	0.073	0.086	0.099	0.112	0.126	0.139
0.000	0.000	0.000	0.000	0.000	0.001	0.001	0.002	0.002	0.003
0.026	0.036	0.046	0.058	0.070	0.083	0.096	0.109	0.122	0.135
0.059	0.076	0.092	0.110	0.127	0.144	0.161	0.178	0.194	0.210

Table 3 Random numbers when $k = 0.09$ and $θ = 1.5$

$I_{N} = 0$	$I_{N} = 0$ .1	$I_{N} = 0.2$	$I_{N} = 0.3$	$I_{N} = 0.4$	$I_{N} = 0.5$	$I_{N} = 0.6$	$I_{N} = 0.7$	$I_{N} = 0.8$	$I_{N} = 0.9$
0.021	0.030	0.040	0.051	0.063	0.076	0.088	0.102	0.115	0.129
0.106	0.129	0.152	0.175	0.198	0.219	0.240	0.261	0.280	0.298
0.010	0.016	0.022	0.029	0.038	0.047	0.056	0.066	0.077	0.088
0.202	0.233	0.262	0.289	0.314	0.338	0.361	0.382	0.402	0.420
0.003	0.004	0.007	0.010	0.014	0.019	0.024	0.029	0.036	0.042
0.001	0.003	0.004	0.006	0.009	0.013	0.017	0.021	0.026	0.031
0.001	0.001	0.002	0.003	0.005	0.007	0.010	0.013	0.016	0.020
0.004	0.006	0.009	0.013	0.018	0.024	0.030	0.036	0.044	0.051
0.009	0.014	0.019	0.026	0.034	0.042	0.051	0.061	0.071	0.081
0.012	0.018	0.025	0.033	0.041	0.051	0.061	0.072	0.083	0.094
0.000	0.000	0.000	0.001	0.001	0.002	0.003	0.004	0.006	0.008
0.039	0.052	0.066	0.082	0.097	0.113	0.129	0.145	0.161	0.177
0.000	0.000	0.000	0.000	0.001	0.001	0.002	0.003	0.004	0.005
0.037	0.050	0.064	0.079	0.094	0.110	0.125	0.141	0.157	0.172
0.080	0.100	0.121	0.142	0.163	0.183	0.203	0.222	0.240	0.258

Figure 3 Random variates when $k = 0.5$ and $θ = 0.5$ .

Figure 4 Random variates when $k = 0.08$ and $θ = 1.67$ .

Figure 5 Random variates when $k = 0.09$ and $θ = 1.50$ .

Table 4 Random numbers when $k = 1.5$ and $θ = 0.15$

$I_{N} = 0$	$I_{N} = 0$ .1	$I_{N} = 0.2$	$I_{N} = 0.3$	$I_{N} = 0.4$	$I_{N} = 0.5$	$I_{N} = 0.6$	$I_{N} = 0.7$	$I_{N} = 0.8$	$I_{N} = 0.9$
2.936	3.508	4.096	4.698	5.312	5.939	6.575	7.222	7.877	8.540
2.790	3.345	3.917	4.503	5.103	5.714	6.337	6.970	7.611	8.262
16.243	17.292	18.363	19.449	20.544	21.645	22.749	23.856	24.964	26.073
3.396	4.018	4.654	5.302	5.961	6.630	7.308	7.995	8.690	9.392
5.126	5.898	6.677	7.463	8.254	9.050	9.852	10.659	11.470	12.285
12.830	13.878	14.931	15.989	17.049	18.110	19.171	20.232	21.293	22.354
5.211	5.990	6.775	7.566	8.362	9.164	9.971	10.783	11.599	12.419
10.464	11.475	12.487	13.499	14.511	15.523	16.535	17.547	18.559	19.571
9.325	10.306	11.287	12.267	13.248	14.228	15.209	16.190	17.172	18.154
22.719	23.645	24.647	25.700	26.785	27.895	29.020	30.157	31.302	32.454
16.676	17.721	18.792	19.879	20.976	22.080	23.188	24.299	25.411	26.525
8.386	9.335	10.283	11.232	12.181	13.131	14.082	15.033	15.985	16.939
21.712	22.666	23.687	24.750	25.843	26.955	28.081	29.217	30.361	31.508
10.000	11.000	12.000	13.000	14.000	15.000	16.000	17.000	18.000	19.000
1.579	1.967	2.377	2.806	3.253	3.716	4.192	4.682	5.185	5.698

6 Simulation Using Algorithm when $k \geq 1$

In this section, we will showcase simulation studies conducted with the algorithm under the condition $k \geq 1$ . We generated random variates from the gamma distribution, varying the mean, variance, $k$ , $I_{N}$ and $θ$ values to analyze their impact on the data generated using the algorithm is presented in Figure 2. Table 4 displays results for a mean of 10, the variance of 66.6, $k$ of 1.5, and $θ$ of 0.15. Likewise, Tables 5 and 6 present outcomes for different sets of parameters. For instance, Table 5 presents values corresponding to a mean of 12, the variance of 72, $k$ of 2.00, and $θ$ of 0.17. Table 6 showcases results for a mean of 14, the variance of 65.4, $k$ of 3.00, and $θ$ of 0.21. Analyzing Tables 4–6 reveals that as the degree of indeterminacy increases from 0.1 to 0.9, there is a general upward trend in the random variates. For instance, in Table 4, when $I_{N} = 0.1$ , the random variate is 3.508, while for $I_{N} = 0.9$ , it increases to 8.540. Figure 6 illustrates the trend in random variates when $k = 1.5$ and $θ = 0.15$ , while Figures 7 and 8 depict the trends for $k = 2.0$ with $θ = 0.17$ and $k = 3.0$ with $θ = 0.21$ , respectively. Notably, Figures 6–8 show that as the values of $I_{N}$ increase, the random variates from the gamma distribution also increase. The curves corresponding to $I_{N} = 0.9$ consistently occupy higher positions compared to other $I_{N}$ values. In conclusion, based on this simulation study, it is inferred that when $k \geq 1$ , increasing the degree of uncertainty significantly influences the generation of random variates.

Table 5 Random numbers when $k = 2$ and $θ = 0.17$

$I_{N} = 0$	$I_{N} = 0$ .1	$I_{N} = 0.2$	$I_{N} = 0.3$	$I_{N} = 0.4$	$I_{N} = 0.5$	$I_{N} = 0.6$	$I_{N} = 0.7$	$I_{N} = 0.8$	$I_{N} = 0.9$
4.408	5.153	5.914	6.692	7.483	8.287	9.102	9.927	10.762	11.607
4.229	4.955	5.700	6.461	7.236	8.024	8.824	9.635	10.456	11.286
17.837	19.154	20.479	21.807	23.137	24.468	25.799	27.128	28.457	29.784
4.965	5.762	6.574	7.400	8.238	9.087	9.946	10.814	11.691	12.576
6.950	7.903	8.864	9.833	10.809	11.791	12.780	13.774	14.773	15.777
14.710	15.982	17.256	18.529	19.803	21.075	22.347	23.617	24.887	26.155
7.045	8.004	8.971	9.946	10.927	11.915	12.909	13.909	14.913	15.922
12.453	13.668	14.882	16.096	17.310	18.524	19.738	20.951	22.164	23.377
11.334	12.511	13.688	14.865	16.043	17.222	18.400	19.580	20.759	21.939
23.463	24.781	26.128	27.494	28.872	30.258	31.648	33.042	34.436	35.831
18.224	19.545	20.874	22.207	23.543	24.879	26.216	27.551	28.886	30.219
10.392	11.532	12.672	13.814	14.958	16.102	17.248	18.395	19.543	20.693
22.610	23.934	25.282	26.647	28.022	29.402	30.787	32.173	33.561	34.948
12.000	13.200	14.400	15.600	16.800	18.000	19.200	20.400	21.600	22.800
2.655	3.201	3.770	4.360	4.970	5.596	6.238	6.895	7.565	8.247

Table 6 Random numbers when $k = 3$ and $θ = 0.21$

$I_{N} = 0$	$I_{N} = 0$ .1	$I_{N} = 0.2$	$I_{N} = 0.3$	$I_{N} = 0.4$	$I_{N} = 0.5$	$I_{N} = 0.6$	$I_{N} = 0.7$	$I_{N} = 0.8$	$I_{N} = 0.9$
6.442	7.396	8.367	9.354	10.355	11.369	12.394	13.430	14.475	15.529
6.238	7.174	8.129	9.099	10.085	11.083	12.093	13.115	14.146	15.186
19.034	20.586	22.137	23.685	25.230	26.772	28.312	29.849	31.383	32.914
7.065	8.069	9.090	10.124	11.171	12.229	13.297	14.375	15.461	16.555
9.169	10.324	11.488	12.661	13.841	15.028	16.222	17.421	18.625	19.835
16.393	17.876	19.358	20.837	22.315	23.790	25.264	26.736	28.207	29.676
9.265	10.426	11.596	12.775	13.960	15.153	16.352	17.556	18.765	19.980
14.408	15.824	17.239	18.654	20.068	21.482	22.896	24.309	25.722	27.135
13.394	14.770	16.146	17.523	18.900	20.278	21.657	23.036	24.416	25.796
23.541	25.164	26.791	28.419	30.045	31.670	33.292	34.911	36.527	38.139
19.354	20.913	22.471	24.026	25.578	27.128	28.674	30.218	31.759	33.297
12.523	13.860	15.200	16.541	17.884	19.228	20.575	21.922	23.271	24.621
22.875	24.491	26.110	27.728	29.345	30.960	32.571	34.180	35.786	37.388
14.000	15.400	16.800	18.200	19.600	21.000	22.400	23.800	25.200	26.600
4.349	5.103	5.880	6.679	7.498	8.334	9.186	10.053	10.934	11.827

Figure 6 Random variates when $k = 1.5$ , and $θ = 0.15$ .

Figure 7 Random variates when $k = 2.00$ , and $θ = 0.17$ .

Figure 8 Random variates when $k = 3.00$ , and $θ = 0.21$ .

7 Comparative Studies

As previously mentioned, neutrosophic statistics serve as an extension of classical statistics. Consequently, the algorithms proposed within neutrosophic statistics converge to their classical counterparts when $I_{N}$ equal to zero. In order to assess the outcomes of the proposed algorithms against those introduced by Thomopoulos [35], random variates sourced from the gamma distribution are generated. The results are organized in Tables 1–3 for scenarios where $k$ is less than 1. For cases where $k$ is greater than or equal to 1, the random variates derived using classical statistics are presented in Tables 4–6. Our analysis will initially focus on comparing the two algorithms when $k$ is less than 1, followed by a comparison when $k$ is greater than or equal to 1.

7.1 Comparative Study when $k < 1$

We now examine the random number generation using both the proposed and existing algorithms within the framework of classical statistics as presented in Thomopoulos [35], with specific values assigned to $k = 0.09$ and $θ = 1.50$ . The results are compared by illustrating the random variates curves generated by the two algorithms in Figure 9. The upper curve depicts the random variate curve for $I_{N} = 0.50$ , whereas the lower curve represents values of random variates for $I_{N} = 0$ as described in Thomopoulos [35]. Analysis of Figure 9 reveals an increase in random variates as the degree of indeterminacy rises. Notably, the existing algorithm, operating under classical statistics, produces smaller values of random variates, as depicted in the figure. This observation underscores the influence of the degree of indeterminacy on random number generation from the gamma distribution in uncertain environments. Consequently, it is recommended to account for the degree of uncertainty when utilizing random numbers in practical applications.

Figure 9 Random variates when $k = 0.09$ and $θ = 1.50$ .

Figure 10 Random variates when $k = 3.00$ , and $θ = 0.21$ .

7.2 Comparative Study when $k \geq 1$

Presently, we investigate the random number generation using both the proposed and existing algorithms within the framework of classical statistics as presented in Thomopoulos [35], with specific values assigned to $k = 3$ and $θ = 0.21$ . The obtained results are compared by depicting the random variates curves generated by the two algorithms in Figure 10. The upper curve illustrates the random variate curve for $I_{N} = 0.50$ , while the lower curve represents values of random variates for $I_{N} = 0$ as mentioned in Thomopoulos [35]. An examination of Figure 10 reveals an increase in random variates as the degree of indeterminacy rises. Significantly, the existing algorithm, functioning under classical statistics, yields smaller values of random variates, as portrayed in the figure. To further understand the behavior of random variates with varying $k$ values, Figure 11 is presented, showcasing the random variates’ response as $k$ transitions from 1.5 to 3. Figure 11 clearly demonstrates an ascending trend in random variates as the values of $k$ increase. This finding emphasizes the impact of the degree of indeterminacy on random number generation from the gamma distribution in uncertain environments. Consequently, it is advisable to consider the degree of uncertainty when incorporating random numbers into practical applications.

Figure 11 Random variates curves for various $k$ .

8 Application in Acceptance Sampling Plan

This section presents the application of neutrosophic data following the neutrosophic gamma distribution in the area of acceptance sampling plans. Acceptance sampling is widely used for product testing and inspection, particularly when failure times are modeled by a gamma distribution. Previous studies, such as those by [38–40], demonstrate the design of acceptance sampling plans using gamma-distributed failure times. However, failure time data is often imprecise, and neutrosophic statistics can capture this inherent uncertainty. Here, we compare acceptance sampling plans under two scenarios: with uncertainty (neutrosophic parameters) and without it. In acceptance sampling, the acceptance number $c$ and sample size $n$ are key parameters. Testing is conducted over a fixed duration $t$ , and the lot is accepted if the number of failures is less than $c$ within that time; otherwise, it is rejected. For illustration, we assume $c = 2$ , $n = 15$ , and $t = 5$ hours, with product failure times following a gamma distribution where $k = 2$ and $θ = 0.17$ . Table 5 shows failure times under two uncertainty levels: when $I_{N} = 0$ (no uncertainty) and $I_{N} = 0.1$ (some uncertainty), enabling comparison of plan effectiveness in handling imprecise data.

$I_{N} =$ 0: 2.655, 4.229, 4.408, 4.965, 6.95, 7.045, 10.392, 11.334, 12, 12.453, 14.71, 17.837, 18.224, 22.61, 23.463

$I_{N} =$ 0.1: 3.201, 4.955, 5.153, 5.762, 7.903, 8.004, 11.532, 12.511, 13.2, 13.668, 15.982, 19.154, 19.545, 23.934, 24.781

Using a gamma distribution under classical statistics, a product lot will be accepted if fewer than 2 failures occur within 5 hours. In this case, the analysis shows 4 failures within this period, leading to a lot of rejection. However, applying the proposed neutrosophic gamma distribution to the same data reveals only 2 failures within 5 hours, resulting in lot acceptance. This comparison highlights how assuming precise versus imprecise failure times can impact decisions about the product lot. Consequently, incorporating degrees of uncertainty is recommended when using acceptance sampling plans with a gamma distribution, as this approach accommodates indeterminacy in failure times, providing a more flexible and potentially accurate decision-making process.

9 Utilization, Assumptions and Limitations

The comparative analysis of simulation studies reveals distinguishable differences in the random variates generated from classical statistics and those under neutrosophic statistics. The findings underscore the significant impact of the degree of indeterminacy on random variate generation. Consequently, it is recommended to update computer software using the proposed algorithm for generating random variates from the gamma distribution. To account for the degree of indeterminacy, the existing algorithms in computer systems should be revised accordingly. However, it is important to note that the proposed algorithm has limitations and makes certain assumptions. Specifically, it is designed solely for generating gamma variates, and it is not applicable for obtaining normal variates. Moreover, the proposed algorithm is not suitable for scenarios where there is no uncertainty present in the data.

10 Concluding Remarks

In conclusion, this study introduced the neutrosophic gamma distribution as a valuable extension of classical statistics, enabling the handling of uncertainty and indeterminacy. Additionally, we proposed two algorithms for generating neutrosophic data from the gamma distribution. These algorithms effectively produced random variates under varying conditions, highlighting distinct trends influenced by the degree of indeterminacy. Simulation studies illustrated how increasing uncertainty impacts random variate generation. For example, analyzing Tables 1–3 shows a general upward trend in random variates as the degree of indeterminacy ( $I_{N}$ ) increases from 0.1 to 0.9. Specifically, in Table 1, when $I_{N} = 0.1$ , the random variate is 0.150, while at $I_{N} = 0.9$ , it rises to 0.191. Furthermore, the proposed method was applied to an acceptance sampling plan. Under the classical gamma distribution, a lot was rejected with four failures in five hours. However, using the neutrosophic gamma distribution, the lot was accepted with only two failures, showcasing the practical advantages of the proposed approach in decision-making under uncertainty. The comparative analyses with classical statistics underscored the significance of accounting for indeterminacy, highlighting the limitations of traditional methods in uncertain environments. This study is significant as it addresses the increasing need to update existing statistical methods to effectively handle the degree of uncertainty commonly encountered in practice. Recommendations included updating computer software using the proposed algorithm and acknowledging the degree of uncertainty in practical applications. While the proposed algorithms showed promise, it was crucial to recognize their limitations, particularly in their applicability only to gamma variates and not normal variates. These findings contributed to the evolving landscape of statistical methodologies, encouraging further research into adapting statistical tools to better accommodate uncertainty. Overall, this study paves the way for future research into the practical applications and broader implications of the neutrosophic gamma distribution across various fields, such as statistical process control and reliability analysis. Developing specialized software and adapting existing modules to incorporate the proposed algorithms represents a valuable direction for future work. Moreover, an in-depth exploration of the statistical properties of the neutrosophic gamma distribution offers a promising avenue for further investigation and theoretical advancement.

Declarations

Ethics Approval and Consent to Participate

Not applicable.

Consent for Publication

Not applicable.

Availability of Data and Materials

The data is given in the paper.

Competing Interests

The authors declare no conflict of interest.

Authors’ Contributions

M.S and M.A wrote the paper.

Funding

No funds for the paper.

Acknowledgements

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality and personation of the paper. We acknowledge the use of ChatGPT to improve the clarity and grammar of the manuscript’s English language.

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Biographies

Muhammad Saleem received the master’s degree in computer science & communications engineering from the University of Duisburg-Essen, Germany, and the Ph.D. degree in engineering from the University of Federal Armed Forces, Munich, Germany. He has more than 15 years of teaching, research, and administrative experience with the Department of Industrial Engineering, University of Duisburg-Essen, and King Abdulaziz University, Saudi Arabia. He is currently an Associate Professor with King Abdulaziz University. He is actively involved in curriculum development and accreditation processes of engineering programs. His research interests include industrial quality control, artificial intelligence, and engineering management.

Muhammad Aslam was the first to introduce the field of neutrosophic statistical quality control (NSQC) and is the founder of several key areas within neutrosophic statistics, including neutrosophic inferential statistics, advanced neutrosophic distribution theory, neutrosophic survey sampling, neutrosophic design of experiments, neutrosophic reliability analysis, and neutrosophic index numbers. His major contribution lies in the original development of a comprehensive neutrosophic statistical theory for inspection, inference, and process control. Prof. Aslam extended the foundational concepts of classical statistics to neutrosophic statistics, formally initiating this advancement in 2018.

Journal of Reliability and Statistical Studies, Vol. 18, Issue 2 (2025), 289–312.
doi: 10.13052/jrss0974-8024.1822
© 2025 River Publishers