A Swarming Meyer Wavelet Computing Approach to Solve the Transport System of Goods

Abstract

The motive of this work is to provide the numerical performances of the reactive transport model that carries trucks with goods on roads by exploiting the stochastic procedures based on the Meyer wavelet (MW) neural network. An objective function is constructed by using the differential model and its boundary conditions. The optimization of the objective function is performed through the hybridization of the global and local search procedures, i.e., swarming and interior point algorithms. Three different cases of the model have been obtained, and the exactness of the stochastic procedure is observed by using the comparison of the obtained and Adams solutions. The negligible absolute error enhances the exactness of the proposed MW neural networks along with the hybridization of the global and local search schemes. Moreover, statistical interpretations based on different operators, histograms, and boxplots are provided to validate the constancy of the designed stochastic structure.

1. Introduction

The multiscale framework in the environment represents the steady state predicated by the one-dimensional reactive transport system (RTS). The nonlinear form of the RTS is typically presented to address the problems with the soft tissue, as well as the microvascular solute transport system of fluid [1]. By performing various actions based on earth-related studies, the dynamical RTS is useful to examine biological and physical phenomena [2]. A novel hypothesis related to biology, geochemical processes, and transport, along with quantitative mass transmission, is accessible to incorporate into the field of RTS [3]. The characteristics of diffusion or convection, the transfer of heat or mass based on the phenomenon of RTS, present a dynamic form in the life of a human to examine the shifting impacts on pollutants and transport and water or air temperature [4,5]. The literature RTS form is given as [6]:

(1)$\left\{\begin{array}{l}D\frac{d^{2}x}{d{\theta }^2}-V\frac{dx}{d\theta }-r(\theta )=0, 0\le \theta \le k,\\ x(k)=x_s, \frac{dx(0)}{d\theta }=0,\end{array}\right.$

where $r(\theta )$, V, and D are the reaction process, advective velocity, and parameter of diffusivity, respectively. The system (1) shows the quantities of non-dimensional $x\left(y\right)=\frac{x(\theta )}{x_s}$, where $y=\frac{\theta }{k}$ and the Peclet number $p=\frac{Vk}{D}$ is written as:

(2)$\frac{d^{2}x}{d{\theta }^2}-p\frac{dx}{d\theta }-r(\theta )=0, 0\le \theta \le 1.$

The above system (2) is obtained by taking $k=1$ and $p=0$, i.e., without applying the advective transport form of the catalyst pellets in diffusion or reaction. Furthermore, the Michaelis–Menten reaction $r(\theta )$ is also assumed, then the system (2) takes the form of:

(3)$\left\{\begin{array}{l}\frac{d^{2}x}{d{\theta }^2}-\frac{cx(\theta )}{d+x(\theta )}=0, 0\le \theta \le 1,\\ \frac{dx}{d\theta }(0)=0, x(1)=1,\end{array}\right.$

where d > 0 shows the half-saturated concentration and c is the distinctive reaction; the reactive rate by taking $c<0$ is applied as an alternative reaction product. The RTS shown in Equation (2) without relating the transport values is considered, while the RTS presented in the system (3) is implemented in the modeling of the transport of fluid, which arises in the soft tissues and microvessels [7]. Equation (3) based on the RTS is solvable with different half-saturation concentration values and reaction rates.

The scientific community has offered comprehensive explanations depending on the RTS based on a variety of techniques. In 1995, Toride et al. [8,9] presented the analytical form of the solutions for the steady-state RTS. In 1982, RTS solutions were described by Van Genuchtenet et al. [10]. The nonlinear form of the RTS is presented by applying the Adomian decomposition and homotopy analysis approaches [11,12,13,14].

The current study performs the solutions of nonlinear RTS presented in Equation (3) based on the trucks of goods on roads by taking different c and d values. All of the aforementioned research has been presented using RTS and predictable analytical and numerical methodologies, each of which has a different degree of application, benefits, and shortcomings. The solutions of the RTS based on roads have been presented by exploiting the Meyer wavelet (MW) neural network. The RTS model, which carries trucks with goods on roads, has not been applied before by exploiting the MW neural network through the hybridization form of the global and local search procedures, i.e., swarming and interior-point algorithms. Recently, stochastic applications have been applied in biological systems [15], higher forms of singular systems [16,17], fractional models [18,19], prediction systems [20], thermal explosion models [21], and the food chain system [22]. These applications motivated the authors to present the solutions of the RTS using the MW neural network along with swarming and interior-point schemes. Some novel features of this study are highlighted as follows:

• The solutions of the RTS, which carry trucks with goods on roads, are presented successfully by using different values of c and d.

• The design of the MW neural network is presented along with the swarming and interior-point methods for solving the RTS.

• The exactness of the stochastic MW neural network procedure is observed through the comparison of the results.

• The reducible absolute error (AE) performance validates the exactness of the stochastic procedure.

• The reliability of the MW neural network, along with the swarming and interior-point methods, is validated by using different statistical performances.

The other parts of the paper are presented as follows: Section 2 shows the MW neural network enhanced by swarming and interior-point methods along with the statistical performances. Section 3 provides the discussion of the results. Conclusions are reported in the last section.

2. Methodology

In this section, the design of the MW neural network enhanced by the swarming and interior-point schemes is provided to solve the RTS.

2.1. Modeling: MW Neural Networks

$\widehat{x}(\theta )$ and $\frac{d^{\left(n\right)}x}{d{\theta }^{(n)}}$ represent the proposed results and nth-order derivative, mathematically given as:

(4)$\begin{array}{c}\widehat{x}(\theta )={\displaystyle \sum _{f=1}^q{r}_{f}L(w_f\theta +t_f),}\\ \frac{d^{\left(n\right)}\widehat{x}}{d{\theta }^{\left(n\right)}}={\displaystyle \sum _{f=1}^q{r}_f{L}^{\left(n\right)}(w_f\theta +t_f).}\end{array}$

In the above framework, the neurons are signified as q, while $\mathit{W}=[\mathit{r},\mathit{w},\mathit{t}]$ are the unknown weights, i.e., $\mathit{r}=[r_1,r_2,...,r_q],$ $\mathit{w}=[w_1,w_2,...,w_q]$ and $\mathit{t}=[t_1,t_2,...,t_q].$ The process of the MW neural network has not been presented before for the nonlinear RTS, which carries trucks with goods on roads. The mathematical MW function form is provided as:

(5)$L(\theta )=35{\theta }^4-84{\theta }^5+70{\theta }^6-20{\theta }^7.$

An efficient form of Equation (4) based on the MW function is shown as:

(6)$\begin{array}{l}\widehat{x}(\theta )={\displaystyle \sum _{f=1}^q{r}_f\left(35{(w_f\theta +t_f)}^4-84{(w_f\theta +t_f)}^5+70{(w_f\theta +t_f)}^6-20{(w_f\theta +t_f)}^7\right)},\\ \frac{d}{d\theta }\widehat{x}(\theta )=\frac{d}{d\theta }{\displaystyle \sum _{f=1}^q{r}_f\left(35{(w_f\theta +t_f)}^4-84{(w_f\theta +t_f)}^5+70{(w_f\theta +t_f)}^6-20{(w_f\theta +t_f)}^7\right)},\end{array}$

An objective function $O_F$ using the mean square error is presented as:

(7)$O_F=O_{F.1}+O_{F.2},$

where $O_{F.1}$ is known as the unsupervised error, while $O_{F.2}$ presents the boundary conditions, given as:

(8)$O_{F.1}=\frac{1}{N}{{\displaystyle \sum _{i=1}^N\left(\frac{d^2{\widehat{x}}_i}{d{\theta }^2}-\frac{c{\widehat{x}}_i}{d+{\widehat{x}}_i}\right)}}^2,$

where $Nh=1,{\theta }_i=ih,{\widehat{x}}_i=x({\theta }_i)$ and h shows the step size.

(9)$O_{F.2}=\frac{1}{2}\left({(\frac{d{\widehat{x}}_0}{d\theta })}^2+{({\widehat{x}}_N-1)}^2\right).$

The nonlinear form of the RTS shows the unidentified weights, such as as $O_F\to 0$, and the proposed results overlap with the Adams results. The procedural steps of the scheme are presented in Figure 1. In the 1st step, the RTS is presented, which is based on the nonlinear form of the differential equations. In the 2nd step, the modeling based on the MW neural network including the unsupervised neural network and the error-based fitness function is presented. In the 3rd step, the optimization performances are presented, which are based on the global search swarming scheme, and the local search interior point is provided. In the 4th step, the data are stored based on the trained weights based on the MW neural networks, fitness valuations, time, function count, and iterations. In the last step, a comparison of the results and the statistical performances are provided.

2.2. Optimization: Swarming and Interior Point Schemes

The current section presents the optimization performances based on the hybridization of the swarming (PSO) and interior point schemes for the RTS.

The global search genetic algorithm is modified using the swarming technique PSO. It was documented in the seventh decade of the nineteenth century. For both stiff- and non-stiff-natured situations, PSO presents the optimum solution efficiency. PSO has already been used in diverse fields, such as solar energy systems [23], the cost optimization of microgrids [24], diseased plant diagnoses [25], diode photovoltaic system organization [26], feature selection in cataloging [27], big data digging of hot topics about recycled water use on micro-blog [28], reservoir operation management [29], singular functional models [30], and as a mutation operator for particle filter noise reduction in mechanical fault diagnosis [31].

Quick and efficient performances have been achieved through the combination of optimization-based swarming and local search approaches. Therefore, the interior point is applied as a local search by using the initial inputs of the PSO. It is utilized in order to produce speedy results by applying the original data of PSO. Recently, the interior-point approach has been functional in quantum key distribution rate computation [32], facility layout problems with relative-positioning constraints [33], nonsymmetric exponential-cone optimization [34], nonlinear forms of the third kind of multi-singular differential system [35], and alternating current optimal power flow [36].

2.3. Performance Procedures

The statistical procedures based on mean square error (MSE), Theil’s inequality coefficient (TIC), and semi-interquartile range (SIR) are provided to solve the mathematical RTS. The statistical performances are applied to authenticate the reliability of the proposed stochastic solver in the form of large data. The MSE, TIC and SIR performances are mathematically shown as:

(10)$MSE={\displaystyle \sum _{j=1}^n{\left(x_j-{\widehat{x}}_j\right)}^2},$

(11)$\mathrm{TIC}=\frac{\sqrt{\frac{1}{n}{\displaystyle \sum _{j=1}^n{\left(x_j-{\widehat{x}}_j\right)}^2}}}{\left(\sqrt{\frac{1}{n}{\displaystyle \sum _{j=1}^n{x}_j^2}}+\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{\widehat{x}}_j^2}}\right)}$

(12)$\mathrm{SIR}=-0.5\left(1{\mathrm{st}}^{}\mathrm{Quartile}-3\mathrm{rd}\mathrm{Quartile}\right),$

where $x(\theta )$ and $\widehat{x}(\theta )$ are the reference and proposed solutions.

3. Results Performance

In this section, numerical performances have been provided for three variations of the nonlinear RTS using the MW neural networks enhanced by the swarming and interior-point schemes. The obtained performances for RTS through the graphical and numerical forms are also presented.

Case 1: Consider the nonlinear RTS is provided for c = 0.1 and d = 1.2 in Equation (3) as:

(13)$\left\{\begin{array}{l}\frac{d^{2}x}{d{\theta }^2}-\frac{0.1x(\theta )}{1.2+x(\theta )}=0, 0\le \theta \le 1,\\ \frac{dx}{d\theta }(0)=0, x(1)=1.\end{array}\right.$

where $0\le \theta \le 1$, and the fitness $O_F$ is provided as:

(14)$O_F=\frac{1}{N}{{\displaystyle \sum _{i=1}^N\left(\frac{d^2{\widehat{x}}_i}{d{\theta }^2}-\frac{0.1{\widehat{x}}_i}{1.2+{\widehat{x}}_i}\right)}}^2+\frac{1}{2}\left({\left(\frac{d{\widehat{x}}_0}{d\theta }\right)}^2+{({\widehat{x}}_N-1)}^2\right).$

where $Nh=1,{\theta }_i=ih,{\widehat{x}}_i=x({\theta }_i)$ and h shows the step size

Case 2: Consider the nonlinear RTS is provided for c = 0.4 and d = 1.2 in Equation (3) is:

(15)$\left\{\begin{array}{l}\frac{d^{2}x}{d{\theta }^2}-\frac{0.4x(\theta )}{1.2+x(\theta )}=0, 0\le \theta \le 1,\\ \frac{dx}{d\theta }(0)=0, x(1)=1.\end{array}\right.$

The fitness $O_F$ is provided as:

(16)$O_F=\frac{1}{N}{{\displaystyle \sum _{i=1}^N\left(\frac{d^2{\widehat{x}}_i}{d{\theta }^2}-\frac{0.4{\widehat{x}}_i}{1.2+{\widehat{x}}_i}\right)}}^2+\frac{1}{2}\left({\left(\frac{d{\widehat{x}}_0}{d\theta }\right)}^2+{({\widehat{x}}_N-1)}^2\right).$

Case 3: Consider the nonlinear RTS is provided for c = 0.7 and d = 1.2 in Equation (3) is:

(17)$\left\{\begin{array}{l}\frac{d^{2}x}{d{\theta }^2}-\frac{0.7x(\theta )}{1.2+x(\theta )}=0, 0\le \theta \le 1,\\ \frac{dx}{d\theta }(0)=0, x(1)=1.\end{array}\right.$

The fitness $O_F$ is provided as:

(18)$O_F=\frac{1}{N}{{\displaystyle \sum _{i=1}^N\left(\frac{d^2{\widehat{x}}_i}{d{\theta }^2}-\frac{0.7{\widehat{x}}_i}{1.2+{\widehat{x}}_i}\right)}}^2+\frac{1}{2}\left({\left(\frac{d{\widehat{x}}_0}{d\theta }\right)}^2+{({\widehat{x}}_N-1)}^2\right).$

To check the numerical observations based on the nonlinear form of the RTS for the first to third cases, the computational actions via global and local combinations are provided. The optimization performances used to find the unidentified weight vectors for 30 independent executions are presented in Equations (19)–(21), given as:

(19)$\begin{array}{ll}{\widehat{x}}_1(\theta )& =0.02\left(35{(1.92\theta -0.72)}^4-84{(1.92\theta -0.72)}^5+70{(1.92\theta -0.72)}^6-20{(1.92\theta -0.72)}^7\right)\\ & +1.23\left(35{(-0.5\theta +0.69)}^4-84{(-0.5\theta +0.69)}^5+70{(-0.5\theta +0.69)}^6-20{(-0.5\theta +0.69)}^7\right)\\ & +0.34\left(35{(-0.06\theta +0.4)}^4-84{(-0.06\theta +0.4)}^5+70{(-0.06\theta +0.4)}^6-20{(-0.06\theta +0.4)}^7\right)\\ & +...+0.26\left(35{(0.02\theta +1.2)}^4-84{(0.02\theta +1.20)}^5+70{(0.02\theta +1.2)}^6-20{(0.02\theta +1.20)}^7\right),\end{array}$

(20)$\begin{array}{ll}{\widehat{x}}_2(\theta )& =-0.87\left(35{(1.11\theta -0.2)}^4-84{(1.11\theta -0.2)}^5+70{(1.11\theta -0.2)}^6-20{(1.110\theta -0.20)}^7\right)\\ & -1.08\left(35{(-1.35\theta +1.1)}^4-84{(-1.35\theta +1.1)}^5+70{(-1.35\theta +1.1)}^6-20{(-1.35\theta +1.1)}^7\right)\\ & +4.28\left(35{(0.69\theta +0.6)}^4-84{(0.69\theta +0.67)}^5+70{(0.69\theta +0.67)}^6-20{(0.69\theta +0.67)}^7\right)\\ \hfill +...& +0.38\left(35{(-0.42\theta +1.8)}^4-84{(-0.4\theta +1.8)}^5+70{(-0.42\theta +1.8)}^6-20{(-0.42\theta +1.8)}^7\right),\hfill \end{array}$

(21)$\begin{array}{ll}{\widehat{x}}_3(\theta )& =-0.10\left(35{(2.5\theta -2.76)}^4-84{(2.5\theta -2.76)}^5+70{(2.5\theta -2.76)}^6-20{(2.5\theta -2.76)}^7\right)\\ & +1.10\left(35{(-1.1\theta +0.03)}^4-84{(-1.1\theta +0.03)}^5+70{(-1.1\theta +0.03)}^6-20{(-1.1\theta +0.03)}^7\right)\\ & -1.64\left(35{(1.46\theta -3.94)}^4-84{(1.46\theta -3.94)}^5+70{(1.46\theta -3.94)}^6-20{(1.46\theta -3.94)}^7\right)\\ \hfill +...& +2.11\left(35{(0.80\theta -0.31)}^4-84{(0.80\theta -0.31)}^5+70{(0.80\theta -0.31)}^6-20{(0.80\theta -0.31)}^7\right),\hfill \end{array}$

The graphical illustrations are described in Figure 2, Figure 3, Figure 4 and Figure 5 for the mathematics RTS by taking ten as the number of neurons, an input interval of [0, 1], and a step size of 0.05. The numerical outputs are performed using the optimal weights shown in Figure 2i–iii based on Equations (19)–(21). The comparison performances are provided by taking different solutions, which are presented in Figure 2iv–vi. The plots based on the optimal and mean result performances are illustrated together with the comparison of the results. The overlapping of the optimal and mean results gives confidence to the author that the proposed scheme is correct. The illustrations based on the best AE are presented in Figure 2vii–ix, which shows that the AE is calculated at approximately 10⁻⁷–10⁻¹⁰, 10⁻⁷–10⁻⁹, and 10⁻⁶–10⁻⁹ for the first to third cases. The mean values of AE are reported as 10⁻²–10⁻⁴, 10⁻⁴–10⁻⁶, and 10⁻⁵–10⁻⁶ for the first to third cases, and even the worst form of the AE is calculated as 10⁻¹–10⁻², 10⁻³–10⁻⁴, and 10⁻⁴–10⁻⁵ for cases 1–3. The statistical performances based on the best, worse, and mean forms of the mathematical model of RTS are presented in Figure 2x–xii. For the first case, one can authenticate that the best Fitness (Fit), MSE, and TIC are performed at approximately 10⁻¹¹–10⁻¹², the mean Fit, MSE, and TIC sit at 10⁻⁴–10⁻⁶, 10⁻¹–10⁻², and 10⁻⁶–10⁻⁸, whereas the worst values of these operators are found in good measures. For the second case, the optimal Fit, MSE, and TIC are reported as 10⁻¹⁰–10⁻¹², 10⁻¹²–10⁻¹⁴, and 10⁻¹¹–10⁻¹², the mean Fit, MSE, and TIC values sit at 10⁻⁶–10⁻⁸, whereas the worst measures of these operators are found also to be satisfactory. For the third case, it is observed that the best Fit, MSE, and TIC values are calculated as 10⁻¹⁰–10⁻¹², the mean Fit, MSE, and TIC are 10⁻⁷–10⁻⁸, 10⁻⁶–10⁻⁷, and 10⁻⁸–10⁻⁹, and the worst values even show good performances. These accurate values based on the comparison of the outcomes, AE standards, and statistical operators authenticate the precision of the MW neural network along with the swarming and interior-point methods.

The statistical operator values based on the Fit, MSE, and TIC, along with the boxplots (BPs) and histograms (HTs), are presented in Figure 3, Figure 4 and Figure 5. Figure 3 presents the optimal Fit values, which are found to be 10⁻⁷–10⁻¹¹, 10⁻⁸–10⁻¹², and 10⁻⁹–10⁻¹² for each case. The optimal MSE performances are presented in Figure 4, which are 10⁻⁷–10⁻¹⁰ for the first to third cases. Likewise, Figure 5 indicates the values of TIC, which are calculated as 10⁻⁹–10⁻¹². On behalf of these calculated values, one can observe that the designed scheme is accurate. These statistical performances authenticate the reliability of the proposed scheme for solving the nonlinear mathematical RTS.

For the precision and accuracy of the MW neural network along with the optimization of swarming and interior-point schemes, the statistical presentations are tabulated in Table 1, Table 2, Table 3 and Table 4 based on the minimum (best), Mean, median, Maximum (worst), standard deviation (SD), and S.I.R for 30 executions. These operators all produce negligibly small measures for each variation of the RTS, which presents the stability of the proposed scheme.

The complexity measures to solve the mathematical form of the RTS using the MW neural network along with the optimization of swarming and interior-point schemes are provided. The deviation of parameters using the function counts, time complexity, and iterations during the decision variables of the network are also provided. Table 4 presents the computational cost investigations for the RTS in terms of numerical procedures. The iterations, used time, and function count are found to be 15.589116, 401.68888, and 1165.241312 for the respective cases of the model.

4. Concluding Remarks

In this study, the design of a novel Meyer wavelet neural network is provided for the numerical performances of the reactive transport model that carries trucks with goods on roads. This nonlinear RTS has been used to carry trucks with goods on roads by taking different c and d values. The conclusions of this study are as follows:

• 1.. When the values of d are taken as greater than zero, a half-saturated concentration is performed.

• 2.. When the values of c are taken as less than zero, which shows the distinctive reaction, the reactive rate is applied as an alternate to the reaction product.

• 3.. The solutions of the nonlinear model based on the TRS have been presented successfully by using the proposed stochastic scheme.

• 4.. An objective function has been constructed through the differential form of the RTS and its boundary conditions.

• 5.. The optimization of the merit function has been performed by using the hybridization of global swarming and local search interior-point algorithms.

• 6.. The correctness of the scheme has been observed by performing a comparison of the results and reducible AE for three cases of the RTS.

• 7.. For the stability of the scheme, the statistical performances based on different operators have been provided using 30 trials.

The designed structure can be tested in the future for solving quantum models [37], lonngren-wave systems [38], singular models [39], nonlinear differential models [40], fractional types of systems [41,42,43,44], pricing economy networks [45], Gemini virus models [46], and other related systems [47,48,49,50].

Author Contributions

Conceptualization and Methodology: Z.S. and T.S., Solution of the model: J.L.G.G., A.V. and J.M.S., writing the manuscript: A.V. and J.M.S. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

This paper does not contain any data not stated in the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1. Procedural steps for solving the mathematical nonlinear RTS.

Figure 2. Optimal weight vectors, AE, comparison, and statistical values for each case of RTS.

Figure 3. Convergence performances for the RTS based on the FIT performances.

Figure 4. Convergence measures for the RTS through the MSE operators.

Figure 5. TIC performances for the mathematical RTS.

Table 1

Statistical observations based on stochastic procedure for the first case of the RTS.

$\mathit{\theta }$	Minimum	M × 10 an	M × 10 Dian	Maximum	SD	S.I.R
0	1.4527 × 10−8	1.1993 × 10−3	1.9190 × 10−6	3.5750 × 10−2	6.5257 × 10−3	1.2439 × 10−6
0.05	2.0419 × 10−8	1.1502 × 10−3	1.9322 × 10−6	3.4274 × 10−2	6.2561 × 10−3	1.2728 × 10−6
0.1	3.8890 × 10−9	1.1002 × 10−3	1.9125 × 10−6	3.2774 × 10−2	5.9823 × 10−3	1.3728 × 10−6
0.15	2.8178 × 10−8	1.0487 × 10−3	1.8718 × 10−6	3.1235 × 10−2	5.7013 × 10−3	1.4261 × 10−6
0.2	1.9187 × 10−8	9.9533 × 10−4	1.6947 × 10−6	2.9648 × 10−2	5.4116 × 10−3	1.5783 × 10−6
0.25	1.2690 × 10−9	9.4028 × 10−4	1.4543 × 10−6	2.8014 × 10−2	5.1134 × 10−3	1.6929 × 10−6
0.3	2.6290 × 10−8	8.8380 × 10−4	1.2071 × 10−6	2.6340 × 10−2	4.8079 × 10−3	1.4238 × 10−6
0.35	5.6202 × 10−8	8.2619 × 10−4	9.0459 × 10−7	2.4632 × 10−2	4.4963 × 10−3	1.2307 × 10−6
0.4	3.9326 × 10−8	7.6774 × 10−4	6.3965 × 10−7	2.2898 × 10−2	4.1798 × 10−3	1.0408 × 10−6
0.45	6.7536 × 10−8	7.0861 × 10−4	5.3210 × 10−7	2.1140 × 10−2	3.8588 × 10−3	8.1470 × 10−7
0.5	8.6172 × 10−8	6.4879 × 10−4	4.1044 × 10−7	1.9358 × 10−2	3.5335 × 10−3	8.0040 × 10−7
0.55	1.2342 × 10−7	5.8831 × 10−4	5.0963 × 10−7	1.7549 × 10−2	3.2034 × 10−3	5.3053 × 10−7
0.6	7.3231 × 10−8	5.2692 × 10−4	5.6119 × 10−7	1.5710 × 10−2	2.8676 × 10−3	5.8202 × 10−7
0.65	1.1715 × 10−7	4.6448 × 10−4	7.2612 × 10−7	1.3833 × 10−2	2.5250 × 10−3	6.6468 × 10−7
0.7	1.0742 × 10−7	4.0064 × 10−4	8.3261 × 10−7	1.1913 × 10−2	2.1744 × 10−3	8.7671 × 10−7
0.75	9.8509 × 10−8	3.3513 × 10−4	8.2785 × 10−7	9.9417 × 10−3	1.8144 × 10−3	8.2716 × 10−7
0.8	8.0672 × 10−8	2.6764 × 10−4	8.6879 × 10−7	7.9116 × 10−3	1.4438 × 10−3	8.2038 × 10−7
0.85	3.7794 × 10−8	1.9791 × 10−4	7.6176 × 10−7	5.8157 × 10−3	1.0611 × 10−3	8.2012 × 10−7
0.9	4.8047 × 10−8	1.2575 × 10−4	5.1176 × 10−7	3.6475 × 10−3	6.6533 × 10−4	6.6801 × 10−7
0.95	1.1990 × 10−8	5.0879 × 10−5	2.2844 × 10−7	1.4007 × 10−3	2.5545 × 10−4	9.9951 × 10−7
1	8.1075 × 10−8	3.5194 × 10−5	1.7002 × 10−7	9.3014 × 10−4	1.6981 × 10−4	1.0300 × 10−6

Table 2

Statistical observations based on stochastic procedure for the second case of the RTS.

$\mathit{\theta }$	Minimum	M × 10 an	M × 10 Dian	Maximum	SD	S.I.R
0	1.9638 × 10−8	1.4758 × 10−5	2.2665 × 10−6	2.7830 × 10−4	5.0733 × 10−5	2.2574 × 10−6
0.05	2.3255 × 10−8	1.4359 × 10−5	2.3012 × 10−6	2.6391 × 10−4	4.8142 × 10−5	2.2600 × 10−6
0.1	3.3426 × 10−8	1.3988 × 10−5	2.4301 × 10−6	2.4823 × 10−4	4.5328 × 10−5	2.3297 × 10−6
0.15	3.1179 × 10−8	1.3481 × 10−5	2.3980 × 10−6	2.3151 × 10−4	4.2342 × 10−5	2.3230 × 10−6
0.2	5.6182 × 10−9	1.2801 × 10−5	2.2211 × 10−6	2.1455 × 10−4	3.9319 × 10−5	2.3689 × 10−6
0.25	2.3805 × 10−8	1.1973 × 10−5	1.9095 × 10−6	1.9810 × 10−4	3.6381 × 10−5	2.0592 × 10−6
0.3	1.4197 × 10−8	1.1043 × 10−5	1.5541 × 10−6	1.8266 × 10−4	3.3609 × 10−5	2.1100 × 10−6
0.35	2.3342 × 10−9	1.0096 × 10−5	1.3575 × 10−6	1.6841 × 10−4	3.1025 × 10−5	2.0331 × 10−6
0.4	1.9974 × 10−8	9.1717 × 10−6	1.0868 × 10−6	1.5526 × 10−4	2.8626 × 10−5	1.8203 × 10−6
0.45	8.9137 × 10−9	8.2857 × 10−6	1.1243 × 10−6	1.4296 × 10−4	2.6388 × 10−5	1.5645 × 10−6
0.5	5.4568 × 10−9	7.4630 × 10−6	8.7409 × 10−7	1.3112 × 10−4	2.4253 × 10−5	1.3701 × 10−6
0.55	1.8057 × 10−8	6.7222 × 10−6	6.9428 × 10−7	1.1931 × 10−4	2.2156 × 10−5	1.2701 × 10−6
0.6	4.9173 × 10−8	6.2622 × 10−6	6.2897 × 10−7	1.0713 × 10−4	1.9971 × 10−5	1.3442 × 10−6
0.65	4.0434 × 10−8	5.7940 × 10−6	6.0506 × 10−7	9.4260 × 10−5	1.7726 × 10−5	1.4266 × 10−6
0.7	2.3268 × 10−8	5.2882 × 10−6	5.0749 × 10−7	8.0491 × 10−5	1.5387 × 10−5	7.6547 × 10−7
0.75	3.1087 × 10−9	4.9108 × 10−6	7.8657 × 10−7	6.5769 × 10−5	1.2875 × 10−5	1.2075 × 10−6
0.8	3.5008 × 10−9	4.3955 × 10−6	7.2060 × 10−7	5.0222 × 10−5	1.0321 × 10−5	1.5987 × 10−6
0.85	1.0835 × 10−8	3.7330 × 10−6	6.4426 × 10−7	3.4182 × 10−5	7.8691 × 10−6	1.4233 × 10−6
0.9	1.6673 × 10−8	2.9877 × 10−6	5.0698 × 10−7	2.4524 × 10−5	5.8277 × 10−6	1.0828 × 10−6
0.95	1.2772 × 10−8	2.2740 × 10−6	4.2338 × 10−7	2.2550 × 10−5	4.7879 × 10−6	1.0993 × 10−6
1	1.8408 × 10−9	2.4009 × 10−6	4.5024 × 10−7	2.0584 × 10−5	4.8422 × 10−6	8.9874 × 10−7

Table 3

Statistical observations based on stochastic procedure for the third case of the RTS.

$\mathit{\theta }$	Minimum	M × 10 an	M × 10 Dian	Maximum	SD	S.I.R
0	1.2692 × 10−7	1.0850 × 10−5	3.5253 × 10−6	1.3120 × 10−4	2.4262 × 10−5	4.5906 × 10−6
0.05	8.0136 × 10−9	1.0724 × 10−5	3.6738 × 10−6	1.2795 × 10−4	2.3635 × 10−5	4.6424 × 10−6
0.1	8.3734 × 10−8	1.0465 × 10−5	3.7088 × 10−6	1.2305 × 10−4	2.2713 × 10−5	4.7483 × 10−6
0.15	5.5947 × 10−8	9.7839 × 10−6	3.3621 × 10−6	1.1363 × 10−4	2.1035 × 10−5	4.1961 × 10−6
0.2	1.7855 × 10−8	8.6888 × 10−6	2.8874 × 10−6	9.9654 × 10−5	1.8657 × 10−5	3.0460 × 10−6
0.25	2.7752 × 10−8	7.6144 × 10−6	2.7272 × 10−6	8.2601 × 10−5	1.5766 × 10−5	2.6415 × 10−6
0.3	6.6981 × 10−8	6.5909 × 10−6	2.1657 × 10−6	6.4539 × 10−5	1.2858 × 10−5	3.1148 × 10−6
0.35	8.3396 × 10−8	5.7635 × 10−6	2.1566 × 10−6	4.7516 × 10−5	1.0340 × 10−5	2.7891 × 10−6
0.4	1.2963 × 10−7	5.0440 × 10−6	1.6800 × 10−6	3.3704 × 10−5	8.5866 × 10−6	2.0396 × 10−6
0.45	5.2079 × 10−8	4.4612 × 10−6	1.4913 × 10−6	3.3523 × 10−5	7.6228 × 10−6	1.4828 × 10−6
0.5	9.3220 × 10−8	4.1111 × 10−6	1.5086 × 10−6	3.3048 × 10−5	7.1512 × 10−6	1.0982 × 10−6
0.55	1.7663 × 10−7	4.0022 × 10−6	1.4248 × 10−6	3.2201 × 10−5	6.8970 × 10−6	9.8590 × 10−7
0.6	9.3265 × 10−8	4.0334 × 10−6	1.4263 × 10−6	3.0951 × 10−5	6.7883 × 10−6	9.4095 × 10−7
0.65	2.9198 × 10−8	4.1475 × 10−6	1.5320 × 10−6	2.9326 × 10−5	6.8164 × 10−6	1.0183 × 10−6
0.7	1.2750 × 10−8	4.3450 × 10−6	1.2376 × 10−6	2.7398 × 10−5	6.8970 × 10−6	1.2565 × 10−6
0.75	2.6347 × 10−8	4.5013 × 10−6	1.4200 × 10−6	2.5276 × 10−5	6.9542 × 10−6	1.6041 × 10−6
0.8	2.0621 × 10−8	4.5455 × 10−6	1.3296 × 10−6	2.5618 × 10−5	6.9195 × 10−6	2.0174 × 10−6
0.85	1.7131 × 10−8	4.4486 × 10−6	1.3861 × 10−6	2.8828 × 10−5	6.9066 × 10−6	2.4376 × 10−6
0.9	6.6650 × 10−10	4.3191 × 10−6	1.3299 × 10−6	3.2659 × 10−5	7.2203 × 10−6	1.8780 × 10−6
0.95	2.8647 × 10−9	4.7253 × 10−6	1.2215 × 10−6	3.7108 × 10−5	7.8977 × 10−6	3.7568 × 10−6
1	7.1753 × 10−10	5.1313 × 10−6	7.7154 × 10−7	4.1628 × 10−5	9.0769 × 10−6	3.8721 × 10−6

Table 4

Complexity measures for the mathematical form of the RTS.

Case	Iterations		Implemented Time		Fun. Counts
	Minimum	SD	Minimum	SD	Minimum	SD
1	15.45159273	2.390238385	395.0666667	54.40710738	24,763.46667	3326.960576
2	15.77388434	0.944774262	405	0	25,409.63333	94.49593435
3	15.54187202	0.691946317	405	0	25,400.93333	74.26742607

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