Reduction of carbon emissions under sustainable supply chain management with uncertain human learning

Richi Singh; Dharmendra Yadav; S.R. Singh; Ashok Kumar; Biswajit Sarkar; Richi Singh; Dharmendra Yadav; S.R. Singh; Ashok Kumar; Biswajit Sarkar

doi:10.3934/environsci.2023032

AIMS Environmental Science

2023, Volume 10, Issue 4: 559-592. doi: 10.3934/environsci.2023032

Previous Article Next Article

Research article Special Issues

Reduction of carbon emissions under sustainable supply chain management with uncertain human learning

1.
Department of Mathematics, Meerut College, Meerut, Uttar Pradesh 250003, India
2.
Department of Mathematics, Vardhaman College, Bijnor, Uttar Pradesh 246701, India
3.
Department of Mathematics, Chaudhary Charan Singh University, Meerut, Uttar Pradesh 250001, India
4.
Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, Seoul 03722, South Korea
5.
Center for Global Health Research, Saveetha Medical College, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai, Tamil Nadu, 600077, India

Received: 07 June 2023 Revised: 31 July 2023 Accepted: 07 August 2023 Published: 28 September 2023

Customers' growing concern for environmentally friendly goods and services has created a competitive and environmentally responsible business scenario. This global awareness of a green environment has motivated several researchers and companies to work on reducing carbon emissions and sustainable supply chain management. This study explores a sustainable supply chain system in the context of an imperfect flexible production system with a single manufacturer and multiple competitive retailers. It aims to reduce the carbon footprints of the developed system through uncertain human learning. Three carbon regulation policies are designed to control carbon emissions caused by various supply chain activities. Despite the retailers being competitive in nature, the smart production system with a sustainable supply chain and two-level screening reduces carbon emissions effectively with maximum profit. Obtained results explore the significance of uncertain human learning, and the total profit of the system increases to 0.039% and 2.23%, respectively. A comparative study of the model under different carbon regulatory policies shows a successful reduction in carbon emissions (beyond 20%), which meets the motive of this research.

Keywords:

Citation: Richi Singh, Dharmendra Yadav, S.R. Singh, Ashok Kumar, Biswajit Sarkar. Reduction of carbon emissions under sustainable supply chain management with uncertain human learning[J]. AIMS Environmental Science, 2023, 10(4): 559-592. doi: 10.3934/environsci.2023032

Related Papers:

[1]	Yuhua Ma, Ye Tao, Yuandan Gong, Wenhua Cui, Bo Wang . Driver identification and fatigue detection algorithm based on deep learning. Mathematical Biosciences and Engineering, 2023, 20(5): 8162-8189. doi: 10.3934/mbe.2023355
[2]	Zulqurnain Sabir, Hafiz Abdul Wahab, Juan L.G. Guirao . A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models. Mathematical Biosciences and Engineering, 2022, 19(1): 663-687. doi: 10.3934/mbe.2022030
[3]	Qingwei Wang, Xiaolong Zhang, Xiaofeng Li . Facial feature point recognition method for human motion image using GNN. Mathematical Biosciences and Engineering, 2022, 19(4): 3803-3819. doi: 10.3934/mbe.2022175
[4]	Rami AL-HAJJ, Mohamad M. Fouad, Mustafa Zeki . Evolutionary optimization framework to train multilayer perceptrons for engineering applications. Mathematical Biosciences and Engineering, 2024, 21(2): 2970-2990. doi: 10.3934/mbe.2024132
[5]	Biyun Hong, Yang Zhang . Research on the influence of attention and emotion of tea drinkers based on artificial neural network. Mathematical Biosciences and Engineering, 2021, 18(4): 3423-3434. doi: 10.3934/mbe.2021171
[6]	Ravi Agarwal, Snezhana Hristova, Donal O’Regan, Radoslava Terzieva . Stability properties of neural networks with non-instantaneous impulses. Mathematical Biosciences and Engineering, 2019, 16(3): 1210-1227. doi: 10.3934/mbe.2019058
[7]	Jia-Gang Qiu, Yi Li, Hao-Qi Liu, Shuang Lin, Lei Pang, Gang Sun, Ying-Zhe Song . Research on motion recognition based on multi-dimensional sensing data and deep learning algorithms. Mathematical Biosciences and Engineering, 2023, 20(8): 14578-14595. doi: 10.3934/mbe.2023652
[8]	Chun Li, Ying Chen, Zhijin Zhao . Frequency hopping signal detection based on optimized generalized S transform and ResNet. Mathematical Biosciences and Engineering, 2023, 20(7): 12843-12863. doi: 10.3934/mbe.2023573
[9]	Yongquan Zhou, Yanbiao Niu, Qifang Luo, Ming Jiang . Teaching learning-based whale optimization algorithm for multi-layer perceptron neural network training. Mathematical Biosciences and Engineering, 2020, 17(5): 5987-6025. doi: 10.3934/mbe.2020319
[10]	Jiliang Lv, Chenxi Qu, Shaofeng Du, Xinyu Zhao, Peng Yin, Ning Zhao, Shengguan Qu . Research on obstacle avoidance algorithm for unmanned ground vehicle based on multi-sensor information fusion. Mathematical Biosciences and Engineering, 2021, 18(2): 1022-1039. doi: 10.3934/mbe.2021055

Abstract

1. Introduction

According to the reports of the World Health Organization (WHO), there are many deaths occur using the tobacco epidemic and its prey most of the disabled persons during some past few years. The tobacco epidemic not only disturbs the individuals, but also become a reason for increases the health care cost, delays financial development and decreases the families' budgets ^[1]. The chain smoking is a big reason of death due to oral cavity cancer, bladder, larynx, esophagus, lung, stomach, pancreas, renal pelvis and cervix. The smoking also creates the problems of heart, chronic obstructive, lung weakness, breathing diseases, peripheral vascular and less weight of newly born children. WHO also reports that the reasoning of unproductive pregnancies, peptic ulcer disease and increase the infant mortality rate is due to smoking ^[2]. The termination of smoking is an instantaneous health support and dramatically reduces the danger of many deathly diseases and improve the respiratory system of the younger. It is the obligation of the higher authorities to teach their people and aware the communities about the drawbacks of smoking as well as develop an active policy to control this habit. Castillo et al. ^[3] discussed the mathematical model to avoid the smoking by considering the population into two kings, i.e., smokers (S) and those individuals who left smoking permanently (QP). In addition, Shoromi et al. ^[4] presented a new group temporary smoker (QT) in this mathematical model, which is defined as:

$\left\{\begin{array}{l}{P}^{\text{'}}\left(\mathit{\Omega } \right) = \mu \left(1-P\left(\mathit{\Omega } \right)\right)-\beta P\left(\mathit{\Omega } \right)S\left(\mathit{\Omega } \right), P\left(0\right) = {l}_{1}, \\ {S}^{\text{'}}\left(\mathit{\Omega } \right) = \beta P\left(\mathit{\Omega } \right)S\left(\mathit{\Omega } \right)-(\gamma +\mu )S\left(\mathit{\Omega } \right)+\alpha {Q}_{T}\left(\mathit{\Omega } \right), S\left(0\right) = {l}_{2}, \\ {Q}_{T}^{\text{'}}\left(\mathit{\Omega } \right) = \gamma (1-\sigma )S\left(\mathit{\Omega } \right)-(\mu +\alpha ){Q}_{T}\left(\mathit{\Omega } \right), {Q}_{T}\left(0\right) = {l}_{3}, \\ {Q}_{P}^{\text{'}}\left(\mathit{\Omega } \right) = -\mu {Q}_{P}\left(\mathit{\Omega } \right)+\gamma \sigma S\left(\mathit{\Omega } \right), {Q}_{P}\left(0\right) = {l}_{4}, \end{array}\right.$

(1)

where $P\left(\mathit{\Omega } \right)$ , $S\left(\mathit{\Omega } \right)$ , ${Q}_{T}\left(\mathit{\Omega } \right)$ and ${Q}_{P}\left(\mathit{\Omega } \right)$ indicate the Potential smoker (P) group, Smoker (S) group, Temporary smoker (QT) group and Permanent smoker (QS) group at time $\mathit{\Omega }$ . Whereas, $\sigma , \alpha , \gamma , \beta$ and $\mu$ represent the positive values of the constants. Furthermore, ${l}_{1}$ , ${l}_{2}$ , ${l}_{3}$ and ${l}_{4}$ designate the initial conditions (ICs) of the nonlinear smoke model (1).

The aim of this work is to investigate the nonlinear smoke model numerically to exploit a stochastic framework called Gudermannian neural works (GNNs) ^[5,6,7,8] along with the optimization procedures of global/local search terminologies based Genetic algorithm (GA) and interior-point approach (IPA), i.e., GNNs-GA-IPA. The development of the numerical solvers has been reported in various proposals for the solution of the linear/nonlinear differential models with their own applicability, stability and significance ^{[9,10,11,12,13]}, however recently artificial intelligence based numerical computing platform are introduction as a promising alternatives ^{[14,15,16,17,18]}. Whereas, GNNs-GA-IPA is never been applied before to solve the nonlinear smoke system. Some recent proposals of the stochastic solvers are functional singular system ^[19], nonlinear SIR dengue fever model ^[20], mathematical models of environmental economic systems ^[21], prey-predator nonlinear system ^[22], Thomas-Fermi system ^[23], mosquito dispersal model ^[24], transmission of heat in human head ^[25], multi-singular fractional system ^[26] and nonlinear COVID-19 model ^[27]. Some novel prominent features of the current investigations are provided as:

● The GNNs are explored efficaciously using the hybrid optimization paradigm based on GA-IPA for solving the nonlinear smoke model.

● The consistent overlapped outcomes obtained by GNNs-GA-IPA and the Runge-Kutta numerical results validate the correctness and exactness of the proposed scheme.

● The authorization of the performance is accomplished through different statistical valuations to attain the numerical outcomes of the nonlinear smoke system.

The benefits, merits and noteworthy contributions of the GNNs-GA-IPA are simply implemented to solve the nonlinear smoke model, understanding easiness, operated efficiently and inclusive with reliable applications in diversified fields.

The remaining parts of the current work are studied as: Section 2 defines the procedures of GNNs-GA-IPA along with and statistical measures. Section 3 indicates the results simulations. Section 4 provided the final remarks and future research reports.

2. Designed methodology

In this section, the proposed form of the GNNs-GA-IPA is presented in two steps to solve the nonlinear smoke model as:

● A merit function is designed using the differential system and ICs of the nonlinear smoke system.

● The necessary and essential settings are provided for the optimization procedures of GA-IPA to solve the nonlinear smoke model.

2.1. Structure of GNN-GA-IPA

In this section, the mathematical formulations to solve the nonlinear smoke model-based groups, Potential smoker ( $\widehat{P}$ ), Temporary smoker ( ${\widehat{Q}}_{T}$ ), Permanent smoker ( ${\widehat{Q}}_{S}$ ) and Smoker ( $\widehat{S}$ ) are presented. The proposed results of these groups of the nonlinear smoke model, $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ are represented by $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ together with their derivatives are written as:

$\begin{array}{*{20}{c}} \left[\widehat{P}\right(\mathit{\Omega } ), \widehat{S}(\mathit{\Omega } ), {\widehat{Q}}_{T}(\mathit{\Omega } ), {\widehat{Q}}_{S}(\mathit{\Omega } \left)\right] = \left[\begin{array}{l}{\sum }_{i = 1}^{m}{k}_{P, i}{\rm T}({w}_{P, i}\mathit{\Omega } +{h}_{P, i}), {\sum }_{i = 1}^{m}{a}_{S, i}{\rm T}({w}_{S, i}\mathit{\Omega } +{h}_{S, i}), \\ {\sum }_{i = 1}^{m}{a}_{{Q}_{T}, i}{\rm T}({w}_{{Q}_{T}, i}\mathit{\Omega } +{h}_{{Q}_{T}, i}), {\sum }_{i = 1}^{m}{a}_{{Q}_{S}, i}{\rm T}({w}_{{Q}_{S}, i}\mathit{\Omega } +{h}_{{Q}_{S}, i})\end{array}\right], \\ \left[{\widehat{P}}^{\text{'}}\right(\mathit{\Omega } ), {\widehat{S}}^{\text{'}}(\mathit{\Omega } ), {\widehat{Q}}_{T}^{\text{'}}(\mathit{\Omega } ), {\widehat{Q}}_{S}^{\text{'}}(\mathit{\Omega } \left)\right] = \\ \left[\begin{array}{l}{\sum }_{i = 1}^{m}{k}_{P, i}{{\rm T}}^{\text{'}}({w}_{P, i}\mathit{\Omega } +{h}_{P, i}), {\sum }_{i = 1}^{m}{k}_{S, i}{{\rm T}}^{\text{'}}({w}_{S, i}\mathit{\Omega } +{h}_{S, i}), \\ {\sum }_{i = 1}^{m}{k}_{{Q}_{T}, i}{{\rm T}}^{\text{'}}({w}_{{Q}_{T}, i}\mathit{\Omega } +{h}_{{Q}_{T}, i}), {\sum }_{i = 1}^{m}{k}_{{Q}_{S}, i}{{\rm T}}^{\text{'}}({w}_{{Q}_{S}, i}\mathit{\Omega } +{h}_{{Q}_{S}, i})\end{array}\right]. \end{array}$

(2)

where W is the unknown weight vector, written as:

$W = [{W}_{P}, {W}_{S}, {W}_{{Q}_{T}}, {W}_{{Q}_{S}}] , \;{\rm{for}} \; {W}_{P} = [{k}_{P}, {\omega }_{P}, {h}_{P}] , {W}_{S} = [{k}_{S}, {\omega }_{S}, {h}_{S}] , {W}_{{Q}_{T}} = [{k}_{{Q}_{T}}, {\omega }_{{Q}_{T}}, {h}_{{Q}_{T}}]$

and $\;{W}_{{Q}_{S}} = [{k}_{{Q}_{S}}, {\omega }_{{Q}_{S}}, {h}_{{Q}_{S}}]$ , where

$\begin{array}{l} {k}_{P} = [{k}_{P, 1}, {k}_{P, 2}, ..., {k}_{P, m}], {k}_{S} = [{k}_{S, 1}, {k}_{S, 2}, ..., {k}_{S, m}], {k}_{{Q}_{T}} = [{k}_{{Q}_{T}, 1}, {k}_{{Q}_{T}, 2}, ..., {k}_{{Q}_{T}, m}], {k}_{{Q}_{S}} = \\ [{k}_{{Q}_{S}, 1}, {k}_{{Q}_{T}, 2}, ..., {k}_{{Q}_{T}, m}], {w}_{P} = [{w}_{P, 1}, {w}_{P, 2}, ..., {w}_{P, m}], {w}_{S} = [{w}_{S, 1}, {w}_{S, 2}, ..., {w}_{S, m}] {w}_{{Q}_{T}} = \\ [{w}_{{Q}_{T}, 1}, {w}_{{Q}_{T}, 2}, ..., {w}_{{Q}_{T}, m}], {w}_{{Q}_{S}} = [{w}_{{Q}_{S}, 1}, {w}_{{Q}_{T}, 2}, ..., {w}_{{Q}_{T}, m}], {h}_{P} = [{h}_{P, 1}, {h}_{P, 2}, ..., {h}_{P, m}] {h}_{S} =\\ [{h}_{S, 1}, {h}_{S, 2}, ..., {h}_{S, m}], {h}_{{Q}_{T}} = [{h}_{{Q}_{T}, 1}, {h}_{{Q}_{T}, 2}, ..., {h}_{{Q}_{T}, m}], {h}_{{Q}_{S}} = [{h}_{{Q}_{S}, 1}, {h}_{{Q}_{T}, 2}, ..., {h}_{{Q}_{T}, m}] \end{array}$

The Gudermannian function $T\left(\mathit{\Omega } \right) = 2{\mathit{tan}}^{-1}\left[\mathit{exp}(\mathit{\Omega } )\right]-0.5\pi$ is applied in the above model, this GNNs have never been implemented before the solve this model.

$\begin{array}{*{20}{c}} \left[\begin{array}{cc}\widehat{P}\left(\mathit{\Omega } \right), & \widehat{S}\left(\mathit{\Omega } \right), \\ {\widehat{Q}}_{T}\left(\mathit{\Omega } \right), & {\widehat{Q}}_{S}\left(\mathit{\Omega } \right)\end{array}\right] = \\ \left[\begin{array}{l}{\sum }_{i = 1}^{m}{k}_{P, i}\left(2{\mathit{tan}}^{-1}{e}^{({w}_{P, i}\mathit{\Omega } +{h}_{P, i})}-\frac{\pi }{2}\right), {\sum }_{i = 1}^{m}{k}_{S, i}\left(2{\mathit{tan}}^{-1}{e}^{({w}_{S, i}\mathit{\Omega } +{h}_{S, i})}-\frac{\pi }{2}\right), \\ {\sum }_{i = 1}^{m}{k}_{{Q}_{T}, i}\left(2{\mathit{tan}}^{-1}{e}^{({w}_{{Q}_{T}, i}\mathit{\Omega } +{h}_{{Q}_{T}, i})}-\frac{\pi }{2}\right), {\sum }_{i = 1}^{m}{k}_{{Q}_{P}, i}\left(2{\mathit{tan}}^{-1}{e}^{({w}_{{Q}_{S}, i}\mathit{\Omega } +{h}_{{Q}_{S}, i})}-\frac{\pi }{2}\right)\end{array}\right], \\ \left[\begin{array}{cc}{\widehat{P}}^{\text{'}}\left(\mathit{\Omega } \right), & {\widehat{S}}^{\text{'}}\left(\mathit{\Omega } \right), \\ {\widehat{Q}}_{T}^{\text{'}}\left(\mathit{\Omega } \right), & {\widehat{Q}}_{S}^{\text{'}}\left(\mathit{\Omega } \right)\end{array}\right] = \\ \left[\begin{array}{l}{\sum }_{i = 1}^{m}2{k}_{P, i}{w}_{P, i}\left(\frac{{e}^{({w}_{p, i}\mathit{\Omega } +{n}_{p, i})}}{1+{\left({e}^{({w}_{p, i}\mathit{\Omega } +{n}_{p, i})}\right)}^{2}}\right), {\sum }_{i = 1}^{m}2{k}_{S, i}{w}_{S, i}\left(\frac{{e}^{({w}_{S, i}\mathit{\Omega } +{n}_{S, i})}}{1+{\left({e}^{({w}_{S, i}\mathit{\Omega } +{n}_{S, i})}\right)}^{2}}\right), \\ {\sum }_{i = 1}^{m}2{k}_{{Q}_{T}, i}{w}_{{Q}_{T}, i}\left(\frac{{e}^{({w}_{{Q}_{T}, i}\mathit{\Omega } +{n}_{{Q}_{T}, i})}}{1+{\left({e}^{({w}_{{Q}_{T}, i}\mathit{\Omega } +{n}_{{Q}_{T}, i})}\right)}^{2}}\right), {\sum }_{i = 1}^{m}2{k}_{{Q}_{S}, i}{w}_{{Q}_{S}, i}\left(\frac{{e}^{({w}_{{Q}_{S}, i}\mathit{\Omega } +{n}_{{Q}_{S}, i})}}{1+{\left({e}^{({w}_{{Q}_{S}, i}\mathit{\Omega } +{n}_{{Q}_{S}, i})}\right)}^{2}}\right)\end{array}\right], \end{array}$

(3)

For the process of optimization, a fitness function is given as:

${\mathit{\Xi } }_{Fit} = {\mathit{\Xi } }_{1}+{\mathit{\Xi } }_{2}+{\mathit{\Xi } }_{3}+{\mathit{\Xi } }_{4}+{\mathit{\Xi } }_{5},$

(4)

${\mathit{\Xi } }_{2} = \frac{1}{N}{\sum }_{i = 1}^{N}{\left[{\widehat{S}}_{i}^{\text{'}}-\beta {\widehat{P}}_{i}{\widehat{S}}_{i}+\mu {\widehat{S}}_{i}+\gamma {\widehat{S}}_{i}-\alpha {\left({\widehat{Q}}_{T}\right)}_{i}\right]}^{2},$

(5)

${\mathit{\Xi } }_{1} = \frac{1}{N}{\sum }_{i = 1}^{N}{\left[{\widehat{P}}_{i}^{\text{'}}-\mu +\mu {\widehat{P}}_{i}+\beta {\widehat{P}}_{i}{\widehat{S}}_{i}\right]}^{2},$

(6)

${\mathit{\Xi } }_{3} = \frac{1}{N}{\sum }_{i = 1}^{N}{\left[{\left({\widehat{Q}}_{T}^{\text{'}}\right)}_{i}+\gamma \sigma {\widehat{S}}_{i}-\gamma {\widehat{S}}_{k}+\alpha {\left({\widehat{Q}}_{T}\right)}_{i}+\mu {\left({\widehat{Q}}_{T}\right)}_{i}\right]}^{2},$

(7)

${\mathit{\Xi } }_{4} = \frac{1}{N}{\sum }_{i = 1}^{N}{\left[{\left({\widehat{Q}}_{P}^{\text{'}}\right)}_{i}-\gamma \sigma {\widehat{S}}_{i}+\mu {\left({\widehat{Q}}_{P}\right)}_{i}\right]}^{2},$

(8)

${\mathit{\Xi } }_{5} = \frac{1}{4}\left[{\left({\widehat{P}}_{0}-{l}_{1}\right)}^{2}+{\left({\widehat{S}}_{0}-{l}_{2}\right)}^{2}+{\left(({\widehat{Q}}_{T}{)}_{0}-{l}_{3}\right)}^{2}++{\left(({\widehat{Q}}_{S}{)}_{0}-{l}_{4}\right)}^{2}\right],$

(9)

where ${\widehat{P}}_{i} = P\left({\mathit{\Omega } }_{i}\right), {\widehat{S}}_{i} = S\left({\mathit{\Omega } }_{i}\right), ({Q}_{T}{)}_{i} = {Q}_{T}\left({\mathit{\Omega } }_{i}\right), ({Q}_{S}{)}_{i} = {Q}_{S}\left({\mathit{\Omega } }_{i}\right), {\rm T}N = 1,$ and ${\mathit{\Omega } }_{i} = ih$ . The error functions ${\mathit{\Xi } }_{1},$ ${\mathit{\Xi } }_{2},$ ${\mathit{\Xi } }_{3}$ and ${\mathit{\Xi } }_{4}$ are related to system (1), while, ${\mathit{\Xi } }_{5}$ is based on the ICs of the nonlinear smoke model (1).

2.2. Optimization procedures: GA-IPA

In this section, the performance of the scheme is observed using the optimization process of GA-IPA to solve the nonlinear smoke system. The designed GNNs-GA-IPA methodology based on the nonlinear smoke system is illustrated in Figure 1.

Figure 1. Designed framework of the GNNs-GA-IPA to solve the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

GA is known as a famous global search optimization method applied to solve the solve the constrained/unconstrained models efficiently. It is commonly implemented to regulate the precise population outcomes for solving the various stiff and complex models using the optimal training process. For the best solutions of the model, GA is implemented through the process of selection, reproduction, mutation and crossover procedures. Recently, GA is applied in many famous applications that can be seen in ^{[28,29,30,31,32]} and references cited therein.

IPA is a local search, rapid and quick optimization method, implemented to solve various reputed complex and non-stiff models efficiently. IPA is implemented in various models like phase-field approach to brittle and ductile fracture ^[33], multistage nonlinear nonconvex programs ^[34], SITR model for dynamics of novel coronavirus ^[35], viscoplastic fluid flows ^[36] and security constrained optimal power flow problems ^[37]. To control the Laziness of the global search method GA, the process of hybridization with the IPA is applied for solving the nonlinear smoke model. The detailed pseudocode based on the GNNs-GA-IPA is provided in Table 1.

Table 1. The procedure of optimization using the GNNs-GA-IPA for the nonlinear smoke model.

GA process starts Inputs: The chromosomes of the same network elements are denoted as: W = [ $k$ , $w$ , $h$ ] Population: Chromosomes set is represented as: $W=[{W}_{P}, {W}_{S}, {W}_{{Q}_{T}}, {W}_{{Q}_{S}}]$ , ${W}_{P}=[{k}_{P}, {\omega }_{P}, {h}_{P}]$ , ${W}_{S}=[{k}_{S}, {\omega }_{S}, {h}_{S}]$ , ${W}_{{Q}_{T}}=\left[{k}_{{Q}_{T}}, {\omega }_{{Q}_{T}}, {h}_{{Q}_{T}}\right]$ and ${W}_{{Q}_{S}}=[{k}_{{Q}_{S}}, {\omega }_{{Q}_{S}}, {h}_{{Q}_{S}}]$ , W is the weight vector Outputs: The best global weight values are signified as: W_GA-Best Initialization: For the chromosome's assortment, adjust the W Assessment of FIT: Adjust " ${\mathit{\Xi } }_{Fit}$ " in the population (P) for each vector values using Eqs (4)–(9). ● Stopping standards: Terminate if any of the criteria is achieved [ ${\mathit{\Xi } }_{Fit}$ = 10^-19], [Generataions = 120] [TolCon = TolFun = 10^-18], [StallLimit = 150], & [Size of population = 270]. Move to [storage] Ranking: Rank precise W in the particular population for ${\mathit{\Xi } }_{Fit}$ . Storage: Save ${\mathit{\Xi } }_{Fit}$ , iterations, W_GA-Best, function counts and time. GA procedure End
Start of IPA Inputs: W_GA-Best is selected as an initial point. Output: W_GA-IPA shows the best weights of GA-IPA. Initialize: W_GA-Best, generations, assignments and other standards. Terminating criteria: Stop if [ ${\mathit{\Xi } }_{Fit}$ = 10-20], [TolFun = 10-18], [Iterations = 700], [TolCon = TolX = 10-22] and [MaxFunEvals = 260000] attained. Evaluation of Fit: Compute the values of W and E for Eqs (4)–(9). Amendments: Normalize 'fmincon' for IPA, compute ${\mathit{\Xi } }_{Fit}$ for Eqs (4)–(9). Accumulate: Transmute W_GA-IPA, time, iterations, function counts and ${\mathit{\Xi } }_{Fit}$ for the IPA trials.
IPA process End

| Show Table

DownLoad: CSV

3. Performance measures

The mathematical presentations using the statistical operators with "variance account for (VAF)", "Theil's inequality coefficient (TIC)", "mean absolute deviation (MAD)" and "semi interquartile (SI) range" together with the global operators G.VAF, G-TIC G-MAD are accessible to solve the nonlinear smoke model, written as:

$\left\{\begin{array}{l}[\text{V}\text{.}\text{A}\text{.}{\text{F}}_{P}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{S}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{T}}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{S}}] = \left[\begin{array}{l}\left(1-\frac{\mathit{var}\left({P}_{m}-{\widehat{P}}_{m}\right)}{\mathit{var}\left({P}_{m}\right)}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({S}_{m}-{\widehat{S}}_{m}\right)}{\mathit{var}\left({S}_{m}\right)}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({\left({Q}_{T}\right)}_{m}-{\left({\widehat{Q}}_{T}\right)}_{m}\right)}{{\mathit{var}\left({Q}_{T}\right)}_{m}}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({\left({Q}_{S}\right)}_{m}-{\left({\widehat{Q}}_{S}\right)}_{m}\right)}{{\mathit{var}\left({Q}_{S}\right)}_{m}}\right)\times 100\end{array}\right], \\ [\text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{P}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{S}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{T}}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{S}}] = \left[\left|\begin{array}{l}100-\text{V}\text{.}\text{A}\text{.}{\text{F}}_{P}, 100-\text{V}\text{.}\text{A}\text{.}{\text{F}}_{S}, \\ 100-\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{T}}, 100-\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{S}}\end{array}\right|\right].\end{array}\right.$

(10)

$\begin{array}{*{20}{c}} {\left[\text{T.I.}{\text{C}}_{P}, \text{T.I.}{\text{C}}_{S}, \text{T.I.}{\text{C}}_{{Q}_{T}}, \text{T.I.}{\text{C}}_{{Q}_{S}}\right] = }\\ {\left[\begin{array}{l}\frac{\sqrt{\frac{1}{n}{{\sum }_{m = 1}^{n}\left({P}_{m}-{\widehat{P}}_{m}\right)}^{2}}}{\left(\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{P}_{m}^{2}}+\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\widehat{P}}_{m}^{2}}\right)}, \frac{\sqrt{\frac{1}{n}{{\sum }_{m = 1}^{n}\left({S}_{m}-{\widehat{S}}_{m}\right)}^{2}}}{\left(\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{S}_{m}^{2}}+\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\widehat{S}}_{m}^{2}}\right)}, \\ \frac{\sqrt{\frac{1}{n}{{\sum }_{m = 1}^{n}\left({\left({Q}_{T}\right)}_{m}-{\left({\widehat{Q}}_{T}\right)}_{m}\right)}^{2}}}{\left(\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\left({Q}_{T}\right)}_{m}^{2}}+\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\left({\widehat{Q}}_{T}\right)}_{m}^{2}}\right)}, \\ \frac{\sqrt{\frac{1}{n}{{\sum }_{m = 1}^{n}\left({\left({Q}_{S}\right)}_{m}-{\left({\widehat{Q}}_{S}\right)}_{m}\right)}^{2}}}{\left(\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\left({Q}_{S}\right)}_{m}^{2}}+\sqrt{\frac{1}{n}{\sum }_{m = 1}^{n}{\left({\widehat{Q}}_{S}\right)}_{m}^{2}}\right)}\end{array}\right],} \end{array}$

(11)

$\begin{array}{l} \;\;\;\;\;[\text{M.A.}{\text{D}}_{P}, \text{M.A.}{\text{D}}_{S}\text{, M.A.}{\text{D}}_{{Q}_{T}}\text{, M.A.}{\text{D}}_{{Q}_{S}}] = \\\left[\begin{array}{l}{\sum }_{m = 1}^{n}\left|{P}_{m}-{\widehat{P}}_{m}\right|, {\sum }_{m = 1}^{n}\left|{S}_{m}-{\widehat{S}}_{m}\right|, \\ {\sum }_{m = 1}^{n}\left|{\left({Q}_{T}\right)}_{m}-{\left({\widehat{Q}}_{T}\right)}_{m}\right|, {\sum }_{m = 1}^{n}\left|{\left({Q}_{S}\right)}_{m}-{\left({\widehat{Q}}_{S}\right)}_{m}\right|\end{array}\right] \end{array}$

(12)

$\left\{\begin{array}{l}\text{S}\text{.}\text{I}\text{ }\text{R}\text{a}\text{n}\text{g}\text{e} = -0.5\times \left({Q}_{1}-{Q}_{3}\right), \\ {Q}_{1}\text{ = }{\mathrm{ }1}^{st}\text{q}\text{u}\text{a}\text{r}\text{t}\text{i}\text{l}\text{e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Q}_{3} = {3}^{rd}\text{q}\text{u}\text{a}\text{r}\text{t}\text{i}\text{l}\text{e}\text{.}\end{array}\right.$

(13)

$\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ are the approximate form of the solutions.

4. Results and simulations

The current investigations are associated to solve the nonlinear smoke model. The relative performance of the obtained solutions with the Runge-Kutta results is tested to show the exactness of the GNNs-GA-IPA. Moreover, the statistical operator's performances are used to validate the accuracy, reliability and precision of the proposed GNNs-GA-IPA. The updated form of the nonlinear smoke model given in the system (1) along with its ICs using the appropriate parameter values is shown as:

$\left\{\begin{array}{l}{P}^{\text{'}}\left(\mathit{\Omega } \right) = 20-\left(20P\left(\mathit{\Omega } \right)+0.003P\left(\mathit{\Omega } \right)S\left(\mathit{\Omega } \right)\right), P\left(0\right) = 0.3, \\ {S}^{\text{'}}\left(\mathit{\Omega } \right) = 0.003P\left(\mathit{\Omega } \right)S\left(\mathit{\Omega } \right)-20.3S\left(\mathit{\Omega } \right)+3{Q}_{T}\left(\mathit{\Omega } \right), S\left(0\right) = 0.5, \\ {Q}_{T}^{\text{'}}\left(\mathit{\Omega } \right) = 0.15S\left(\mathit{\Omega } \right)-23{Q}_{T}\left(\mathit{\Omega } \right), {Q}_{T}\left(0\right) = 0.1, \\ {Q}_{P}^{\text{'}}\left(\mathit{\Omega } \right) = 0.15S\left(\mathit{\Omega } \right)-20{Q}_{P}\left(\mathit{\Omega } \right), {Q}_{P}\left(0\right) = 0.2, \end{array}\right.$

(14)

A fitness function for the nonlinear smoke model (14) is written as:

$\begin{array}{*{20}{c}} {{\mathit{\Xi } }_{Fit} = }\\ {\frac{1}{N}{\sum }_{m = 1}^{N}\left(\begin{array}{l}{\left[{\widehat{P}}_{m}^{\text{'}}+20{\widehat{P}}_{m}-20+0.003{\widehat{P}}_{m}{\widehat{S}}_{m}\right]}^{2}+{\left[{\widehat{S}}_{m}^{\text{'}}+20.3{\widehat{S}}_{m}-0.003{\widehat{P}}_{m}{\widehat{S}}_{m}-3({Q}_{T}{)}_{m}\right]}^{2}\\ +{\left[({Q}_{T}^{\text{'}}{)}_{m}+23({Q}_{T}{)}_{m}-0.15{\widehat{S}}_{m}\right]}^{2}+{\left[({Q}_{P}^{\text{'}}{)}_{m}+20({Q}_{P}{)}_{m}-0.15{\widehat{S}}_{m}\right]}^{2}\end{array}\right) }\\ {+\frac{1}{4}\left[{\left({\widehat{P}}_{0}-0.3\right)}^{2}+{\left({\widehat{S}}_{0}-0.5\right)}^{2}+{\left(({Q}_{T}{)}_{0}-0.1\right)}^{2}+{\left(({Q}_{P}{)}_{0}-0.2\right)}^{2}\right]. } \end{array}$

(15)

The performance of the scheme is observed based on the nonlinear smoke system using the GNNs-GA-IPA for 20 independent executions using 30 numbers of variables. The proposed form of the solution based on the nonlinear smoke model is provided in the arrangement of best weights using the below equations for each group of the nonlinear smoke model and the graphical illustrations of these weights are plotted in Figure 2.

$\begin{array}{*{20}{c}} {\widehat{P}\left(\mathit{\Omega } \right) = 7.2965(2{\mathit{tan}}^{-1}{e}^{(-5.343\mathit{\Omega } -18.462)}-0.5\pi )-}\\ {2.7636(2{\mathit{tan}}^{-1}{e}^{1.2606\mathit{\Omega } +0.5134)}-0.5\pi )}\\ {+15.294(2{\mathit{tan}}^{-1}{e}^{(1.1551\mathit{\Omega } +2.0601)}-0.5\pi )+}\\ {8.7324(2{\mathit{tan}}^{-1}{e}^{(19.759\mathit{\Omega } +2.9789)}-0.5\pi ) }\\ {-7.1646(2{\mathit{tan}}^{-1}{e}^{(2.5198\mathit{\Omega } +5.9852)}-0.5\pi )-}\\ {0.0053(2{\mathit{tan}}^{-1}{e}^{(-0.246\mathit{\Omega } -1.7692)}-0.5\pi )}\\ {-2.6104(2{\mathit{tan}}^{-1}{e}^{(0.2323\mathit{\Omega } +1.5497)}-0.5\pi )-}\\ {1.4677(2{\mathit{tan}}^{-1}{e}^{(02.782\mathit{\Omega } +2.0890)}-0.5\pi ) }\\ {+2.4715(2{\mathit{tan}}^{-1}{e}^{(-18516\mathit{\Omega } -3.253)}-0.5\pi )-}\\ {2.9189(2{\mathit{tan}}^{-1}{e}^{(10.272\mathit{\Omega } -7.2465)}-0.5\pi ),} \end{array}$

(16)

$\begin{array}{*{20}{c}} {\widehat{S}\left(\mathit{\Omega } \right) = 7.6303(2{\mathit{tan}}^{-1}{e}^{(-0.143\mathit{\Omega } +0.4605)}-0.5\pi )+}\\ {9.6607(2{\mathit{tan}}^{-1}{e}^{(0.264\mathit{\Omega } +0.8726)}-0.5\pi )}\\ {-1.8710(2{\mathit{tan}}^{-1}{e}^{(0.3252\mathit{\Omega } -1.0967)}-0.5\pi )+}\\ {0.0451(2{\mathit{tan}}^{-1}{e}^{(-1.961\mathit{\Omega } +8.4961)}-0.5\pi )}\\ {+5.4711(2{\mathit{tan}}^{-1}{e}^{(-9.8369\mathit{\Omega } -1.533)}-0.5\pi )-}\\ {1.7275(2{\mathit{tan}}^{-1}{e}^{(-9.2138\mathit{\Omega } -2.037)}-0.5\pi ) }\\ {+0.2908(2{\mathit{tan}}^{-1}{e}^{(-2.4236\mathit{\Omega } -0.378)}-0.5\pi )-}\\ {1.9171(2{\mathit{tan}}^{-1}{e}^{(-9.2376\mathit{\Omega } -0.360)}-0.5\pi )}\\ {+2.7144(2{\mathit{tan}}^{-1}{e}^{(2.8428\mathit{\Omega } -7.8525)}-0.5\pi )-}\\ {5.3000(2{\mathit{tan}}^{-1}{e}^{(-4.6148\mathit{\Omega } +4.164)}-0.5\pi ),} \end{array}$

(17)

$\begin{array}{*{20}{c}} {{\widehat{Q}}_{T}\left(\mathit{\Omega } \right) = -0.563(2{\mathit{tan}}^{-1}{e}^{(-3.245\mathit{\Omega } +2.3616)}-0.5\pi )-}\\ {0.022(2{\mathit{tan}}^{-1}{e}^{(-8.9859+1.2373)}-0.5\pi )}\\ {-0.5360(2{\mathit{tan}}^{-1}{e}^{(3.2982\mathit{\Omega } -2.4088)}-0.5\pi )-}\\ {2.1382(2{\mathit{tan}}^{-1}{e}^{(-3.87\mathit{\Omega } -3.100)}-0.5\pi )}\\ {-4.7812(2{\mathit{tan}}^{-1}{e}^{(-4.1182\mathit{\Omega } -4.994)}-0.5\pi )+}\\ {0.9505(2{\mathit{tan}}^{-1}{e}^{(-0.038\mathit{\Omega } -0.241)}-0.5\pi ) }\\ {+17.841(2{\mathit{tan}}^{-1}{e}^{(-14.7536\mathit{\Omega } -4.557)}-0.5\pi )-}\\ {3.4575(2{\mathit{tan}}^{-1}{e}^{(4.347\mathit{\Omega } +0.7343)}-0.5\pi ) }\\ {+11.8361(2{\mathit{tan}}^{-1}{e}^{(4.5451\mathit{\Omega } +2.013)}-0.5\pi )+}\\ {2.717(2{\mathit{tan}}^{-1}{e}^{(3.6697\mathit{\Omega } +7.33831)}-0.5\pi ), } \end{array}$

(18)

$\begin{array}{*{20}{c}} {{\widehat{Q}}_{S}\left(\mathit{\Omega } \right) = 0.0762(2{\mathit{tan}}^{-1}{e}^{(2.9990\mathit{\Omega } +2.8605)}-0.5\pi )+}\\ {2.8390(2{\mathit{tan}}^{-1}{e}^{(8.3177\mathit{\Omega } +0.6804)}-0.5\pi ) }\\ {+7.6702(2{\mathit{tan}}^{-1}{e}^{(-0.0481\mathit{\Omega } +1.9373)}-0.5\pi )-}\\ {0.4142(2{\mathit{tan}}^{-1}{e}^{(-0.309\mathit{\Omega } -0.401)}-0.5\pi )}\\ {-6.2827(2{\mathit{tan}}^{-1}{e}^{(1.2291\mathit{\Omega } +3.5918)}-0.5\pi )+}\\ {0.5077(2{\mathit{tan}}^{-1}{e}^{(1.0574\mathit{\Omega } +1.0166)}-0.5\pi )}\\ {+2.2061(2{\mathit{tan}}^{-1}{e}^{(1.5133\mathit{\Omega } +10.7487)}-0.5\pi )-}\\ {0.2776(2{\mathit{tan}}^{-1}{e}^{(1.7337\mathit{\Omega } +5.1928)}-0.5\pi )}\\ {-5.3623(2{\mathit{tan}}^{-1}{e}^{(8.4564\mathit{\Omega } +1.3100)}-0.5\pi )+}\\ {0.0684(2{\mathit{tan}}^{-1}{e}^{(-4.1583\mathit{\Omega } -1.6646)}-0.5\pi ), } \end{array}$

(19)

Figure 2. Best weight vectors along with the result comparisons of the mean and best results with reference results to solve the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

A fitness function shown in the model (15) is optimized along with the hybridization of GA-IPA for the nonlinear smoke system. The proposed form of the outcomes is found using the above systems (16)–(19) for 30 variables between 0 to 1 input along with step size 0.1. The solutions of the nonlinear smoke model along with the best weight vector values are illustrated in Figures 2(a–d). The comparison of the mean and best outcomes with the reference Runge-Kutta solutions is provided in Figures 2(e–h) to solve the nonlinear smoke system. It is noticed that the mean and best results obtained by the GNNs-GA-IPA are overlapped with the reference results to solve each group of the nonlinear smoke model, which authenticate the exactness of the designed GNNs-GA-IPA. illustrates the values of the absolute error (AE) for each group of the nonlinear smoke model. It is observed that the values of the best AE for the group of potential smokers, smoker; temporary smoker and permanent smoker lie around 10-05-10-07, 10-05-10-06, 10-04-10-07 and 10-04-10-06, respectively. While, the mean AE values for these groups of the nonlinear smoke model found around 10-03-10-04, 10-03-10-05, 10-02-10-04 and 10-03-10-04, respectively. signifies the performance measures based on the operators EVAF, MAD and TIC to solve each group of the nonlinear smoke model. It is specified in the plots that the best values of the EVAF, MAD and TIC performances of each group of the nonlinear smoke model lie around 10-04-10-08, 10-03-10-05 and 10-08-10-09, respectively. The best performances of the EVAF, MAD and TIC for the $\widehat{P}\left(\mathit{\Omega } \right)$ and $\widehat{S}\left(\mathit{\Omega } \right)$ groups lie around 10-08-10-09, 10-05-10-06 and 10-09-10-10, respectively. The best performances of the EVAF, MAD and TIC for the ${\widehat{Q}}_{T}\left(\mathit{\Omega } \right)$ group of the nonlinear smoke model found around 10-05-10-06, 10-04-10-06 and 10-08-10-10 and the best performances of the EVAF, MAD and TIC for the ${\widehat{Q}}_{P}\left(\mathit{\Omega } \right)$ group of the nonlinear smoke model found around 10-07-10-08, 10-05-10-06 and 10-09-10-10. One can accomplish from the indications that the designed GNNs-GA-IPA is precise and accurate.

Figure 3. AE values for each group of the nonlinear smoke system.

DownLoad: Full-Size Img PowerPoint

Figure 4. Performances based on EVAF, MAD and TIC values to solve each group of the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

The graphic illustrations based on the statistical performances are provided in Figures 5– to find the convergence along with the boxplots and the histograms to solve the nonlinear smoke model. shows the performance of TIC for twenty runs to solve each group of the nonlinear smoke model. It is observed that most of the executions for the $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ groups lie around 10-07-10-10. The MAD performances are illustrated in that depicts most of the executions for the $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ groups lie around 10-03-10-05. The EVAF performances are illustrated in that depicts most of the executions for the $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ groups lie around 10-04-10-08. The best trial performances using the GNNs-GA-IPA are calculated suitable for the TIC, EVAF and MAD operators.

Figure 5. Convergence of the TIC values along with the Boxplots and histograms to solve each group of the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

Figure 6. Convergence of the MAD values along with the Boxplots and histograms to solve each group of the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

Figure 7. Convergence of the EVAF values along with the Boxplots and histograms to solve each group of the nonlinear smoke model.

DownLoad: Full-Size Img PowerPoint

The routines for different statistical operators, Maximum (Max), Median, Minimum (Min), standard deviation (STD) and SIR are provided in – to validate the accurateness and precision to solve the $\widehat{P}, \widehat{S}, {\widehat{Q}}_{T}$ and ${\widehat{Q}}_{S}$ groups of the nonlinear smoke system. The Max operators indicate the worst solutions, whereas the Min operators show the best results using 20 independent runs. For the group $P\left(\mathit{\Omega } \right)$ , $S\left(\mathit{\Omega } \right)$ , ${Q}_{T}\left(\mathit{\Omega } \right)$ and ${Q}_{S}\left(\mathit{\Omega } \right)$ of the nonlinear smoke model, the 'Max' and 'Min' standards lie around 10-03-10-04 and 10-06-10-08, while the Median, SIR, STD and Mean standards lie around 10-04-10-05. These small values designate the worth and values of the GNNs-GA-IPA to solve each group of the nonlinear smoke model. One can observe through these calculate measures, that the designed GNNs-GA-IPA is precise, accurate and stable.

Table 2. Statistical measures for the nonlinear smoke model of group

$P\left(\mathit{\Omega } \right)$ .

$\mathit{\Omega }$	P( $\mathit{\Omega }$ )
$\mathit{\Omega }$	Max	Min	Median	STD	SIR	Mean
0	1.9698E-03	1.5128E-07	1.3179E-05	4.3988E-04	2.8322E-05	1.5255E-04
0.1	2.2927E-03	4.0452E-05	3.7091E-04	7.5870E-04	3.5757E-04	7.1580E-04
0.2	2.8517E-04	2.5715E-06	8.0076E-05	1.0291E-04	8.7298E-05	1.0480E-04
0.3	3.7540E-04	5.0221E-06	4.6631E-05	9.6615E-05	5.7759E-05	8.0291E-05
0.4	3.1095E-04	7.2215E-07	2.5169E-05	7.9915E-05	4.8322E-05	6.4497E-05
0.5	3.7039E-04	1.0187E-07	4.0958E-05	9.9187E-05	4.0049E-05	7.4737E-05
0.6	3.4788E-04	2.7497E-06	3.6783E-05	8.6814E-05	5.3154E-05	7.3405E-05
0.7	7.2524E-04	3.8591E-06	2.6741E-05	1.6887E-04	5.3778E-05	9.7130E-05
0.8	3.1439E-04	2.3193E-06	3.0315E-05	8.9765E-05	3.8282E-05	6.8701E-05
0.9	8.2385E-04	3.6579E-06	3.6581E-05	1.8035E-04	2.2583E-05	8.9806E-05
1	3.9210E-04	8.9543E-07	1.4825E-05	1.1497E-04	3.4368E-05	6.7763E-05

| Show Table

DownLoad: CSV

Table 3. Statistical measures for the nonlinear smoke model of group

$S\left(\mathit{\Omega } \right)$ .

$\mathit{\Omega }$	$S\left(\mathit{\Omega } \right)$
$\mathit{\Omega }$	Max	Min	Median	STD	SIR	Mean
0	2.2510E-03	8.8331E-07	1.5049E-05	4.3988E-04	6.5800E-05	2.4625E-04
0.1	3.5749E-03	1.2621E-05	4.2711E-04	7.5870E-04	5.9138E-04	9.2693E-04
0.2	5.8967E-04	6.8508E-06	5.0534E-05	1.0291E-04	3.8159E-05	1.0389E-04
0.3	3.9412E-04	4.5897E-07	5.5860E-05	9.6615E-05	4.8082E-05	9.7091E-05
0.4	1.8148E-04	1.4261E-06	3.2627E-05	7.9915E-05	4.6977E-05	6.1366E-05
0.5	3.8406E-04	3.0499E-06	5.0425E-05	9.9187E-05	4.5676E-05	9.6411E-05
0.6	3.3693E-04	1.0786E-06	3.2525E-05	8.6814E-05	2.6474E-05	5.9807E-05
0.7	3.8976E-04	5.9658E-08	3.4605E-05	1.6887E-04	3.0731E-05	7.1943E-05
0.8	4.6451E-04	1.8083E-07	2.9100E-05	8.9765E-05	1.9639E-05	5.1752E-05
0.9	3.7056E-04	5.8454E-07	3.1691E-05	1.8035E-04	4.5808E-05	7.8891E-05
1	3.4425E-04	3.5833E-08	3.7245E-05	1.1497E-04	2.5924E-05	6.3636E-05

| Show Table

DownLoad: CSV

Table 4. Statistical measures for the nonlinear smoke model of group

${Q}_{T}\left(\mathit{\Omega } \right)$ .

$\mathit{\Omega }$	${Q}_{T}\left(\mathit{\Omega } \right)$
$\mathit{\Omega }$	Max	Min	Median	STD	SIR	Mean
0	6.7379E-02	2.0435E-06	7.4367E-05	4.3988E-04	1.2547E-03	6.6320E-03
0.1	8.8212E-03	1.5933E-04	6.6896E-04	7.5870E-04	6.4736E-04	1.4858E-03
0.2	1.2208E-03	1.0177E-06	1.6681E-04	1.0291E-04	1.3929E-04	2.7755E-04
0.3	1.2039E-03	6.2163E-06	3.5966E-05	9.6615E-05	7.5659E-05	2.0075E-04
0.4	5.2048E-04	1.5004E-06	9.3199E-05	7.9915E-05	1.4812E-04	1.7703E-04
0.5	7.9163E-04	4.1214E-06	7.5760E-05	9.9187E-05	9.4190E-05	1.7072E-04
0.6	7.5113E-04	3.4495E-06	5.6836E-05	8.6814E-05	1.5865E-04	1.8051E-04
0.7	6.9564E-04	2.1601E-06	8.7570E-05	1.6887E-04	1.2891E-04	1.6565E-04
0.8	8.5659E-04	2.3356E-06	5.6424E-05	8.9765E-05	6.5358E-05	1.2238E-04
0.9	8.5832E-04	1.8869E-06	8.6636E-05	1.8035E-04	1.2103E-04	1.8340E-04
1	5.9922E-04	9.8962E-07	3.9188E-05	1.1497E-04	4.8965E-05	1.1550E-04

| Show Table

DownLoad: CSV

Table 5. Statistical measures for the nonlinear smoke model of group

${Q}_{S}\left(\mathit{\Omega } \right)$ .

$\mathit{\Omega }$	${Q}_{S}\left(\mathit{\Omega } \right)$
$\mathit{\Omega }$	Max	Min	Median	STD	SIR	Mean
0	1.4249E-02	4.6038E-07	1.0193E-05	3.1752E-03	5.3007E-05	7.6325E-04
0.1	7.1456E-03	9.4140E-06	2.0718E-04	1.6018E-03	3.9882E-04	8.1152E-04
0.2	1.4449E-03	2.1615E-07	2.8536E-05	3.6835E-04	3.0447E-05	1.7480E-04
0.3	3.7091E-04	3.2754E-07	2.1791E-05	8.7324E-05	2.6186E-05	5.6609E-05
0.4	1.5545E-03	1.8846E-07	2.2389E-05	3.7612E-04	6.1143E-05	1.5810E-04
0.5	3.7972E-04	3.6651E-06	1.7656E-05	9.8654E-05	3.1268E-05	6.2521E-05
0.6	1.2224E-03	2.4032E-07	2.1050E-05	2.9478E-04	3.3053E-05	1.2732E-04
0.7	7.4101E-04	1.7539E-07	2.4276E-05	1.6625E-04	2.5398E-05	7.7130E-05
0.8	1.0726E-03	2.5213E-06	1.4100E-05	2.3771E-04	3.0541E-05	9.1329E-05
0.9	7.4407E-04	2.5770E-06	3.5282E-05	1.6258E-04	2.8865E-05	7.9482E-05
1	9.7616E-04	7.0693E-07	8.9328E-06	2.1858E-04	1.0034E-05	8.0351E-05

| Show Table

DownLoad: CSV

The global performances of the G.EVAF, G.MAD and G.TIC operators for twenty runs to solve the designed GNNs-GA-IPA are provided in Table 6 to solve each group of the nonlinear smoke model. These Min global G.MAD, G.TIC and G.EVAF performances found around 10-04-10-05, 10-08-10-09 and 10-05-10-07, whereas the SIR global values lie in the ranges of 10-04-10-05, 10-08-10-09 and 10-04-10-07 for all groups of the nonlinear smoke model. These close optimal global measures values demonstrate the correctness, accurateness and precision of the proposed GNNs-GA-IPA.

Table 6. Global measures based on the MAD, TIC and EVAF values to solve each group of the nonlinear smoke model.

Class	(G.MAD)		(G.TIC)		(G.EVAF)
Class	Min	SIR	Min	SIR	Min	SIR
$P(\mathit{\Omega } )$	1.2499E-04	9.6058E-05	7.2597E-09	7.4459E-09	2.6155E-07	7.5331E-07
$S(\mathit{\Omega } )$	9.7570E-05	1.3353E-04	8.3068E-08	1.0766E-08	1.3879E-06	5.2293E-06
${Q}_{T}\left(\mathit{\Omega } \right)$	1.4111E-04	2.5334E-04	9.1566E-09	1.8187E-08	3.9553E-05	5.1487E-04
${Q}_{S}\left(\mathit{\Omega } \right)$	6.0979E-05	4.1189E-05	3.6367E-09	4.8515E-09	1.1015E-06	9.3296E-06

| Show Table

DownLoad: CSV

5. Reference style, citation, and cross-reference

The current investigations are related to solve the nonlinear smoke model by exploiting the Gudermannian neural networks using the global and local search methodologies, i.e., GNNs-GA-IPA. The smoke model is a system of nonlinear equations contain four groups temporary smokers, potential smokers, permanent smokers and smokers. For the numerical outcomes, a fitness function is established using all groups of the nonlinear smoke model and its corresponding ICs. The optimization of the fitness function using the hybrid computing framework of GNNs-GA-IPA for solving each group of the nonlinear smoke model. The Gudermannian function is designed as a merit function along with 30 numbers of variables. The overlapping of the proposed mean and best outcomes is performed with the Runge-Kutta reference results for each group of the nonlinear smoke model. These matching and reliable results to solve the nonlinear smoke model indicate the exactness of the designed GNNs-GA-IPA. In order to show the precision and accuracy of the proposed GNNs-GA-IPA, the statistical performances based on the TIC, MAD and EVAF operators have been accessible for twenty trials using 10 numbers of neurons. To check the performance analysis, most of the runs based on the statistical TIC, MAD and EVAF performances show a higher level of accuracy to solve each group of the nonlinear smoke model. The valuations using the statistical gages of Max, Min, Mean, STD, Med and SIR further validate the value of the proposed GNNs-GA-IPA. Furthermore, global presentations through SIR and Min have been applied for the nonlinear smoke model.

In future, the designed GNNs-GA-IPA is accomplished to solve the biological nonlinear systems ^[38], singular higher order model ^[39], fluid dynamics nonlinear models ^[40] and fractional differential model ^[41].

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-12.

Conflict of interest

All authors of the manuscript declare that there have no potential conflicts of interest.

References

[1]	https://www.foodbusinessnews.net/articles/13133-sustainable-product-market-could-hit-150-billion-in-us-by-2021.
[2]	https://ecowarriorprincess.net/2018/04/carbon-intensive-industries-industry-sectors-emit-the-most-carbon
[3]	Olatunji OO, Ayo OO, Akinlabi S, et al.Competitive advantage of carbon efficient supply chain in manufacturing industry. J Clean Prod, 2019,238: 117937. https://doi.org/10.1016/j.jclepro.2019.117937 doi: 10.1016/j.jclepro.2019.117937
[4]	Parsaeifar S, Bozorgi-Amiri A, Naimi-Sadigh A, et al.A game theoretical for coordination of pricing, recycling, and green product decisions in the supply chain. J Clean Prod, 2019,226:37–49. https://doi.org/10.1016/j.jclepro.2019.03.343 doi: 10.1016/j.jclepro.2019.03.343
[5]	Ullah M, Asghar I, Zahid M, et al.Ramification of remanufacturing in a sustainable three-echelon closed-loop supply chain management for returnable products. J Clean Prod, 2021,290: 125609. https://doi.org/10.1016/j.jclepro.2020.125609 doi: 10.1016/j.jclepro.2020.125609
[6]	Xiao D, Wang J, Lu Q. Stimulating sustainability investment level of suppliers with strategic commitment to price and cost sharing in supply chain. J Clean Prod, 2020,252: 119732. https://doi.org/10.1016/j.jclepro.2019.119732 doi: 10.1016/j.jclepro.2019.119732
[7]	Indian Emission Booklet 2018. https://www.araiindia.com/pdf/Indian_Emission_Regulation_Booklet.pdf
[8]	Singh A, Raman N, Waghe U. Ecomark Scheme IN India. Int J Pharma Med Biol Sci, 2012, 1(2): 109-122.
[9]	Bai Q, Jin M, Xu X. Effects of carbon emission reduction on supply chain coordination withvendor-managed deteriorating product inventory. Int J Prod Econ, 2019,208: 83–99. https://doi.org/10.1016/j.ijpe.2018.11.008 doi: 10.1016/j.ijpe.2018.11.008
[10]	Li J, Wang L, Tan X. Sustainable design and optimization of coal supply chain network under different carbon emission policies. J Clean Prod, 2020,250: 119548. https://doi.org/10.1016/j.jclepro.2019.119548 doi: 10.1016/j.jclepro.2019.119548
[11]	Sarkar B, Guchhait R. Ramification of information asymmetry on a green supply chain management with the cap-trade, service, and vendor-managed inventory strategies. Elect Comm Res App, 2023, 60: 101274. https://doi.org/10.1016/j.elerap.2023.101274 doi: 10.1016/j.elerap.2023.101274
[12]	Gautam P, Kishore A, Khanna A. et al.Strategic defect management for a sustainable green supply chain. J Clean Prod, 2019,233: 226-241. https://doi.org/10.1016/j.jclepro.2019.06.005 doi: 10.1016/j.jclepro.2019.06.005
[13]	Ullah M. Sarkar B. Recovery-channel selection in a hybrid manufacturing-remanufacturing production model with RFID and product quality. Int J Prod Econ, 2020,219: 360–374. https://doi.org/10.1016/j.ijpe.2019.07.017 doi: 10.1016/j.ijpe.2019.07.017
[14]	Wee H, Chung C. Optimising replenishment policy for an integrated production inventory deteriorating model considering green component-value design and remanufacturing. Int J Prod Res, 2009, 47: 1343–1368. https://doi.org/10.1080/00207540701570182 doi: 10.1080/00207540701570182
[15]	Hovelaque V, Bironneau L. The carbon-constrained EOQ model with carbon emission dependent demand. Int J Prod Econ, 2015,164: 285–291. https://doi.org/10.1016/j.ijpe.2014.11.022 doi: 10.1016/j.ijpe.2014.11.022
[16]	Wu T, Kung C. Carbon emissions, technology upgradation and financing risk of the green supply chain competition. Technol For Forecast Soc, 2020,152: 119884. https://doi.org/10.1016/j.techfore.2019.119884 doi: 10.1016/j.techfore.2019.119884
[17]	Bonney M, Jaber M. Environmentally responsible inventory models: Non-classical models for a non-classical era. Int J Prod Econ, 2011,133: 43–53. https://doi.org/10.1016/j.ijpe.2009.10.033 doi: 10.1016/j.ijpe.2009.10.033
[18]	Tayyab M, Jemai J, Lim H, et al.A sustainable development framework for a cleaner multi-item multi-stage textile production system with a process improvement initiative. J Clean Prod, 2020,246: 119055. https://doi.org/10.1016/j.jclepro.2019.119055 doi: 10.1016/j.jclepro.2019.119055
[19]	https://ourworldindata.org/grapher/annual-co-emissions-by-region
[20]	Mukhopadhyay A, Goswami A. Economic production quantity (EPQ) model for three type imperfect items with rework and learning in setup. An International. J Opt Control Theor Appl, 2014, 4: 57–65. https://doi.org/10.11121/ijocta.01.2014.00170 doi: 10.11121/ijocta.01.2014.00170
[21]	Dye C, Yang C. Sustainable trade credit and replenishment decisions with credit-linked demand under carbon emission constraints. Eur J Oper Res, 2015,244: 187–200. https://doi.org/10.1016/j.ejor.2015.01.026 doi: 10.1016/j.ejor.2015.01.026
[22]	Sarkar B, Sarkar M, Ganguly B, et al.Combined effects of carbon emission and production quality improvement for fixed lifetime products in a sustainable supply chain management. Int J Prod Econ, 2021,231: 107867. https://doi.org/10.1016/j.ijpe.2020.107867 doi: 10.1016/j.ijpe.2020.107867
[23]	Mishra U. Wu Z. Sarkar B. Optimum sustainable inventory management with backorder and deterioration under controllable carbon emissions. J Clean Prod, 2021,279: 123699. https://doi.org/10.1016/j.jclepro.2020.123699 doi: 10.1016/j.jclepro.2020.123699
[24]	Tiwari S, Daryanto Y, Wee H. Sustainable inventory management with deteriorating and imperfect quality items considering carbon emission. J Clean Prod, 2018,192: 281–292. https://doi.org/10.1016/j.jclepro.2018.04.261 doi: 10.1016/j.jclepro.2018.04.261
[25]	Kundu S, Chakrabarti T. Impact of carbon emission policies on manufacturing, remanufacturing and collection of used item decisions with price dependent return rate. Opsearch, 2018, 55: 532–555. https://doi.org/10.1007/s12597-018-0336-y doi: 10.1007/s12597-018-0336-y
[26]	Jamali M, Rasti-Barzoki M. A game theoretic approach for green and non-green product pricing in chain-to-chain competitive sustainable and regular dual-channel supply chains. J Clean Prod 170: 1029–1043. https://doi.org/10.1016/j.jclepro.2017.09, 2018181 doi: 10.1016/j.jclepro.2017.09.181
[27]	Garai A, Sarkar B. Economically independent reverse logistics of customer-centric closed-loop supply chain for herbal medicines and biofuel. J Clean Prod, 2022,334: 129977. https://doi.org/10.1016/j.jclepro.2021.129977 doi: 10.1016/j.jclepro.2021.129977
[28]	Hosseini-Motlagh S, Ebrahimi S, Zirakpourdehkordi R. Coordination of dual-function acquisition price and corporate social responsibility in a sustainable closed-loop supply chain. J Clean Prod, 2020,251: 119629. https://doi.org/10.1016/j.jclepro.2019.119629 doi: 10.1016/j.jclepro.2019.119629
[29]	Yadav D, Kumari R, Kumar N, et al.Reduction of waste and carbon emission through the selection of items with cross-price elasticity of demand to form a sustainable supply chain with preservation technology. J Clean Prod, 2021,297: 126298. https://doi.org/10.1016/j.jclepro.2021.126298 doi: 10.1016/j.jclepro.2021.126298
[30]	Sarkar B, Tayyab M, Kim N, et al.Optimal production delivery policies for supplier and manufacturer in a constrained closed-loop supply chain for returnable transport packaging through metaheuristic approach. Comp Indust Eng, 2020,135: 987-1003. https://doi.org/10.1016/j.cie.2019.05.035 doi: 10.1016/j.cie.2019.05.035
[31]	Huang Y, Fang C, Lin Y. Inventory management in supply chains with consideration of logistics, green investment and different carbon emissions policies. Compt Indust Eng, 2020,139: 106207. https://doi.org/10.1016/j.cie.2019.106207 doi: 10.1016/j.cie.2019.106207
[32]	Manupati V, Jedidah S, Gupta S, et al.Optimization of a multiechelon sustainable production-distribution supply chain system with lead time consideration under carbon emission policies. Comput Ind Eng, 2019,135: 1312–1323. https://doi.org/10.1016/j.cie.2018.10.010 doi: 10.1016/j.cie.2018.10.010
[33]	Mishra U, Mashud A, Tseng M, et al.Optimizing a sustainable supply chain inventory model for controllable deterioration and emission rates in a greenhouse farm. Mathematics, 2021, 9: 495. https://doi.org/10.3390/math9050495 doi: 10.3390/math9050495
[34]	Sarkar B, Bhuniya S. A sustainable flexible manufacturing–remanufacturing model with improved service and green investment under variable demand. Exp Syst App 202, 117154. https://doi.org/10.1016/j.eswa.2022, 2022117154 doi: 10.1016/j.eswa.2022.117154
[35]	Alamri O, Jayaswal M, Khan F, et al.An EOQ model with carbon emissions and inflation for deteriorating imperfect quality items under learning effect. Sustainability, 2022, 14: 1365. https://doi.org/10.3390/su14031365 doi: 10.3390/su14031365
[36]	Wang S, Wang X, Chen S. Global value chains and carbon emission reduction in developing countries: does industrial upgrading matter? Environ Impact Assess, 2022, 97: 106895. https://doi.org/10.1016/j.eiar.2022.106895 doi: 10.1016/j.eiar.2022.106895
[37]	Sun H, Zhong Y. Carbon emission reduction and green marketing decisions in a two-echelon low-carbon supply chain considering fairness concern. J Bus Ind Mark, 2023, 38: 905–29. https://doi.org/10.1108/JBIM-02-2021-0090 doi: 10.1108/JBIM-02-2021-0090
[38]	Kang K, Tan BQ. Carbon emission reduction investment in sustainable supply chains under cap-and-trade regulation: An evolutionary game-theoretical perspective. Expert Syst Appl, 2023,227: 120335. ttps://doi.org/10.1016/j.eswa.2023.120335 doi: 10.1016/j.eswa.2023.120335
[39]	Sarkar B, Ullah M, Sarkar M. Environmental and economic sustainability through innovative green products by remanufacturing. J Clean Prod, 2022,332: 129813. https://doi.org/10.1016/j.jclepro.2021.129813 doi: 10.1016/j.jclepro.2021.129813
[40]	Glock C. Batch sizing with controllable production rates. Int J Prod Res, 2010, 48: 5925–5942. https://doi.org/10.1080/00207540903170906 doi: 10.1080/00207540903170906
[41]	Glock C. Batch sizing with controllable production rates in a multi-stage production system. Int J Prod Res, 2011, 49: 6017–6039. https://doi.org/10.1080/00207543.2010.528058 doi: 10.1080/00207543.2010.528058
[42]	Singhal S, Singh S. Volume flexible multi-items inventory system with imprecise environment. Int J Ind Eng Comp, 2013, 4: 457–468. https://doi.org/10.5267/j.ijiec.2013.07.002 doi: 10.5267/j.ijiec.2013.07.002
[43]	Singhal S, Singh S. Modeling of an inventory system with multi variate demand under volume flexibility and learning. Uncertain Supply Chain Manag, 2015, 3: 147–158. https://doi.org/10.5267/j.uscm.2014.12.006 doi: 10.5267/j.uscm.2014.12.006
[44]	Tayal S, Singh S, Sharma R. An integrated production inventory model for perishable products with trade credit period and investment in preservation technology. Int J Mathematics Oper Res, 2016, 8: 137–163. https://doi.org/10.1504/IJMOR.2016.074852 doi: 10.1504/IJMOR.2016.074852
[45]	Manna A, Dey J, Mondal S. Imperfect production inventory model with production rate dependent defective rate and advertisement dependent demand. Comput Ind Eng, 2017,104: 9–22. https://doi.org/10.1016/j.cie.2016.11.027 doi: 10.1016/j.cie.2016.11.027
[46]	Sarkar M, Chung B. Flexible work-in-process production system in supply chain management under quality improvement. Int J Prod Res, 2020, 58: 3821–3838. https://doi.org/10.1080/00207543.2019.1634851 doi: 10.1080/00207543.2019.1634851
[47]	Dey B, Pareek S, Tayyab M, et al.Autonomation policy to control work-in-process inventory in a smart production system. Int J Prod Res, 2021, 59: 1258–1280. https://doi.org/10.1080/00207543.2020.1722325 doi: 10.1080/00207543.2020.1722325
[48]	Sarkar M, Sarkar B. How does an industry reduce waste and consumed energy within a multi-stage smart sustainable biofuel production system? J Clean Prod, 2020,262: 121200. https://doi.org/10.1016/j.jclepro.2020.121200 doi: 10.1016/j.jclepro.2020.121200
[49]	Mridha B, Pareek S, Goswami A, et al.Joint effects of production quality improvement of biofuel and carbon emissions towards a smart sustainable supply chain management. J Clean Prod, 2023,386: 135629. https://doi.org/10.1016/j.jclepro.2022.135629 doi: 10.1016/j.jclepro.2022.135629
[50]	Bera U, Mahapatra N, Maiti M. An imperfect fuzzy production-inventory model over a finite time horizon under the effect of learning. Int J Mathematics Oper Res, 2009, 1: 351–371. https://doi.org/10.1504/IJMOR.2009.024290 doi: 10.1504/IJMOR.2009.024290
[51]	Glock C, Schwindl K, Jaber M. An EOQ model with fuzzy demand and learning in fuzziness. Int J Serv Op Manag, 2012, 12: 90–100. https://doi.org/10.1504/IJSOM.2012.046675 doi: 10.1504/IJSOM.2012.046675
[52]	Pathak S, Kar S, Sarkar S. Fuzzy production inventory model for deteriorating items with shortages under the effect of time dependent learning and forgetting: A possibility/necessity approach. Opsearch, 2013, 50: 149–181. https://doi.org/10.1007/s12597-012-0102-5 doi: 10.1007/s12597-012-0102-5
[53]	Yadav D, Singh S, Kumari R. Inventory model with learning effect and imprecise market demand under screening error. Opsearch, 2013, 50: 418–432. https://doi.org/10.1007/s12597-012-0118-x doi: 10.1007/s12597-012-0118-x
[54]	Kumar R, Goswami A. EPQ model with learning consideration, imperfect production and partial backlogging in fuzzy random environment. Int J Syst Sci, 2015, 46: 1486–1497.
[55]	Kazemi N, Shekarian E, Cárdenas-Barrón L E, et al.Incorporating human learning into a fuzzy EOQ inventory model with backorders. Comput Indust Eng, 2015, 87: 540–542. https://doi.org/10.1016/j.cie.2015.05.014 doi: 10.1016/j.cie.2015.05.014
[56]	Shekarian E, Olugu E, Abdul-Rashid S, et al.An economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning. J Intell Fuzzy Syst, 2016, 30: 2985–2997. https://doi.org/10.3233/IFS-151907 doi: 10.3233/IFS-151907
[57]	Sarkar B, Omair M, Kim N. A cooperative advertising collaboration policy in supply chain management under uncertain conditions. App Soft Comput, 2020, 88: 105948. https://doi.org/10.1016/j.asoc.2019.105948 doi: 10.1016/j.asoc.2019.105948
[58]	Giri B, Masanta M. Developing a closed-loop supply chain model with price and quality dependent demand and learning in production in a stochastic environment. Int J Syst Sci Oper, 2020, 7: 147–163. https://doi.org/10.1080/23302674.2018.1542042 doi: 10.1080/23302674.2018.1542042
[59]	Saha S, Chakrabarti T. A supply chain model under return policy considering refurbishment, learning effect and inspection error. Croat Oper Res Rev, 2020, 11: 53–66. https://doi.org/10.17535/crorr.2020.0005 doi: 10.17535/crorr.2020.0005
[60]	Dey BK, Bhuniya S, Sarkar B. Involvement of controllable lead time and variable demand for a smart manufacturing system under a supply chain management. Exp Syst App, 2021,184: 115464. https://doi.org/10.1016/j.eswa.2021.115464 doi: 10.1016/j.eswa.2021.115464
[61]	Jayaswal M, Mittal M, Sangal I, et al.Fuzzy-based EOQ model with credit financing and backorders under human learning. Int J Fuzzy Syst Appl, 2021, 10: 14–36. https://doi.org/10.4018/IJFSA.2021100102 doi: 10.4018/IJFSA.2021100102
[62]	Poursoltan L, Mohammad Seyedhosseini S, Jabbarzadeh A. A two-level closed-loop supply chain under the constract of vendor managed inventory with learning: A novel hybrid algorithm. J Ind Prod Eng, 2021, 38: 254–270. https://doi.org/10.1080/21681015.2021.1878301 doi: 10.1080/21681015.2021.1878301
[63]	Alsaedi BS, Alamri OA, Jayaswal MK, et al.A sustainable green supply chain model with carbon emissions for defective items under learning in a fuzzy environment. Mathematics, 2023, 11: 301. https://doi.org/10.3390/math11020301 doi: 10.3390/math11020301
[64]	Habib MS, Asghar O, Hussain A, et al.A robust possibilistic programming approach toward animal fat-based biodiesel supply chain network design under uncertain environment. J Clean Prod, 2021,278: 122403. https://doi.org/10.1016/j.jclepro.2020.122403 doi: 10.1016/j.jclepro.2020.122403
[65]	Lee S, Kim D (2104) An optimal policy for a single-vendor single-buyer integrated production–distribution model with both deteriorating and defective items. Int J Prod Econ 147: 161–170. https://doi.org/10.1016/j.ijpe.2013.09.011 doi: 10.1016/j.ijpe.2013.09.011
[66]	Singh SK, Chauhan A, Sarkar B. Sustainable biodiesel supply chain model based on waste animal fat with subsidy and advertisement. J Clean Prod, 2023,382: 134806. https://doi.org/10.1016/j.jclepro.2022.134806 doi: 10.1016/j.jclepro.2022.134806
[67]	Wright T. Factors affecting the cost of airplanes. J Aeronaut Sci, 1936, 3: 122–128. https://doi.org/10.2514/8.155 doi: 10.2514/8.155
[68]	Saxena N, Sarkar B, Wee HM, et al.A reverse logistic model with eco-design under the Stackelberg-Nash equilibrium and centralized framework. J Clean Prod, 2023,387: 135789. https://doi.org/10.1016/j.jclepro.2022.135789 doi: 10.1016/j.jclepro.2022.135789

Environ-10-04-032-s001.pdf

This article has been cited by:

1.	Wajaree Weera, Thongchai Botmart, Samina Zuhra, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Salem Ben Said, A Neural Study of the Fractional Heroin Epidemic Model, 2023, 74, 1546-2226, 4453, 10.32604/cmc.2023.033232
2.	Nabeela Anwar, Iftikhar Ahmad, Adiqa Kausar Kiani, Shafaq Naz, Muhammad Shoaib, Muhammad Asif Zahoor Raja, Intelligent solution predictive control strategy for nonlinear hepatitis B epidemic model with delay, 2023, 1745-5030, 1, 10.1080/17455030.2023.2178827
3.	Yue Li, Jianyou Zhao, Zenghua Chen, Gang Xiong, Sheng Liu, A Robot Path Planning Method Based on Improved Genetic Algorithm and Improved Dynamic Window Approach, 2023, 15, 2071-1050, 4656, 10.3390/su15054656
4.	Muneerah AL Nuwairan, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Anwar Aldhafeeri, An advance artificial neural network scheme to examine the waste plastic management in the ocean, 2022, 12, 2158-3226, 045211, 10.1063/5.0085737
5.	Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Nadia Mumtaz, Irwan Fathurrochman, R. Sadat, Mohamed R. Ali, An Investigation Through Stochastic Procedures for Solving the Fractional Order Computer Virus Propagation Mathematical Model with Kill Signals, 2022, 1370-4621, 10.1007/s11063-022-10963-x
6.	Wajaree Weera, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Salem Ben Said, Maria Emilia Camargo, Chantapish Zamart, Thongchai Botmart, An Artificial Approach for the Fractional Order Rape and Its Control Model, 2023, 74, 1546-2226, 3421, 10.32604/cmc.2023.030996
7.	Zulqurnain Sabir, Dumitru Baleanu, Muhammad Asif Zahoor Raja, Evren Hincal, A hybrid computing approach to design the novel second order singular perturbed delay differential Lane-Emden model, 2022, 97, 0031-8949, 085002, 10.1088/1402-4896/ac7a6a
8.	Thongchai Botmart, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Wajaree weera, Mohamed R. Ali, R. Sadat, Ayman A. Aly, Ali Saad, A hybrid swarming computing approach to solve the biological nonlinear Leptospirosis system, 2022, 77, 17468094, 103789, 10.1016/j.bspc.2022.103789
9.	J. F. Gómez-Aguilar, Zulqurnain Sabir, Manal Alqhtani, Muhammad Umar, Khaled M. Saad, Neuro-Evolutionary Computing Paradigm for the SIR Model Based on Infection Spread and Treatment, 2022, 1370-4621, 10.1007/s11063-022-11045-8
10.	Watcharaporn Cholamjiak, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Manuel Sánchez-Chero, Dulio Oseda Gago, José Antonio Sánchez-Chero, María-Verónica Seminario-Morales, Marco Antonio Oseda Gago, Cesar Augusto Agurto Cherre, Gilder Cieza Altamirano, Mohamed R. Ali, Artificial intelligent investigations for the dynamics of the bone transformation mathematical model, 2022, 34, 23529148, 101105, 10.1016/j.imu.2022.101105
11.	Zulqurnain Sabir, Salem Ben Said, Qasem Al-Mdallal, Mohamed R. Ali, A neuro swarm procedure to solve the novel second order perturbed delay Lane-Emden model arising in astrophysics, 2022, 12, 2045-2322, 10.1038/s41598-022-26566-4
12.	Tianqi Yu, Lei Liu, Yan-Jun Liu, Observer-based adaptive fuzzy output feedback control for functional constraint systems with dead-zone input, 2022, 20, 1551-0018, 2628, 10.3934/mbe.2023123
13.	Prem Junsawang, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Soheil Salahshour, Thongchai Botmart, Wajaree Weera, Novel Computing for the Delay Differential Two-Prey and One-Predator System, 2022, 73, 1546-2226, 249, 10.32604/cmc.2022.028513
14.	Chenyang Hu, Yuelin Gao, Fuping Tian, Suxia Ma, A Relaxed and Bound Algorithm Based on Auxiliary Variables for Quadratically Constrained Quadratic Programming Problem, 2022, 10, 2227-7390, 270, 10.3390/math10020270
15.	Rahman Ullah, Qasem Al Mdallal, Tahir Khan, Roman Ullah, Basem Al Alwan, Faizullah Faiz, Quanxin Zhu, The dynamics of novel corona virus disease via stochastic epidemiological model with vaccination, 2023, 13, 2045-2322, 10.1038/s41598-023-30647-3
16.	Thanasak Mouktonglang, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Saira Bhatti, Thongchai Botmart, Wajaree Weera, Chantapish Zamart, Designing Meyer wavelet neural networks for the three-species food chain model, 2023, 8, 2473-6988, 61, 10.3934/math.2023003
17.	Lihe Hu, Yi Zhang, Yang Wang, Qin Jiang, Gengyu Ge, Wei Wang, A simple information fusion method provides the obstacle with saliency labeling as a landmark in robotic mapping, 2022, 61, 11100168, 12061, 10.1016/j.aej.2022.06.002
18.	Sakda Noinang, Zulqurnain Sabir, Shumaila Javeed, Muhammad Asif Zahoor Raja, Dostdar Ali, Wajaree Weera, Thongchai Botmart, A Novel Stochastic Framework for the MHD Generator in Ocean, 2022, 73, 1546-2226, 3383, 10.32604/cmc.2022.029166
19.	Suthep Suantai, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Watcharaporn Cholamjiak, Numerical Computation of SEIR Model for the Zika Virus Spreading, 2023, 75, 1546-2226, 2155, 10.32604/cmc.2023.034699
20.	Zulqurnain Sabir, Juan L. G. Guirao, A Soft Computing Scaled Conjugate Gradient Procedure for the Fractional Order Majnun and Layla Romantic Story, 2023, 11, 2227-7390, 835, 10.3390/math11040835
21.	Kanit Mukdasai, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Peerapongpat Singkibud, R. Sadat, Mohamed R. Ali, A computational supervised neural network procedure for the fractional SIQ mathematical model, 2023, 1951-6355, 10.1140/epjs/s11734-022-00738-9
22.	Jingzhi Tu, Gang Mei, Francesco Piccialli, An improved Nyström spectral graph clustering using k-core decomposition as a sampling strategy for large networks, 2022, 34, 13191578, 3673, 10.1016/j.jksuci.2022.04.009
23.	Thongchai Botmart, Qusain Hiader, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Wajaree Weera, Stochastic Investigations for the Fractional Vector-Host Diseased Based Saturated Function of Treatment Model, 2023, 74, 1546-2226, 559, 10.32604/cmc.2023.031871
24.	Zulqurnain Sabir, Neuron analysis through the swarming procedures for the singular two-point boundary value problems arising in the theory of thermal explosion, 2022, 137, 2190-5444, 10.1140/epjp/s13360-022-02869-3
25.	Muhammad Umar, Fazli Amin, Soheil Salahshour, Thongchai Botmart, Wajaree Weera, Prem Junswang, Zulqurnain Sabir, Swarming Computational Approach for the Heartbeat Van Der Pol Nonlinear System, 2022, 72, 1546-2226, 6185, 10.32604/cmc.2022.027970
26.	Thongchai Botmart, Zulqurnain Sabir, Afaf S. Alwabli, Salem Ben Said, Qasem Al-Mdallal, Maria Emilia Camargo, Wajaree Weera, Computational Stochastic Investigations for the Socio-Ecological Dynamics with Reef Ecosystems, 2022, 73, 1546-2226, 5589, 10.32604/cmc.2022.032087
27.	Qusain Haider, Ali Hassan, Sayed M. Eldin, Artificial neural network scheme to solve the hepatitis B virus model, 2023, 9, 2297-4687, 10.3389/fams.2023.1072447
28.	Farhad Muhammad Riaz, Shafiq Ahmad, Juniad Ali Khan, Saud Altaf, Zia Ur Rehman, Shuaib Karim Memon, Numerical Treatment of Non-Linear System for Latently Infected CD4+T Cells: A Swarm- Optimized Neural Network Approach, 2024, 12, 2169-3536, 103119, 10.1109/ACCESS.2024.3429279
29.	Zulqurnain Sabir, Shahid Ahmad Bhat, Muhammad Asif Zahoor Raja, Sharifah E. Alhazmi, A swarming neural network computing approach to solve the Zika virus model, 2023, 126, 09521976, 106924, 10.1016/j.engappai.2023.106924
30.	Chenxu Wang, Mengqin Wang, Xiaoguang Wang, Luyue Zhang, Yi Long, Multi-Relational Graph Representation Learning for Financial Statement Fraud Detection, 2024, 7, 2096-0654, 920, 10.26599/BDMA.2024.9020013
31.	ZULQURNAIN SABIR, DUMITRU BALEANU, MUHAMMAD ASIF ZAHOOR RAJA, ALI S. ALSHOMRANI, EVREN HINCAL, COMPUTATIONAL PERFORMANCES OF MORLET WAVELET NEURAL NETWORK FOR SOLVING A NONLINEAR DYNAMIC BASED ON THE MATHEMATICAL MODEL OF THE AFFECTION OF LAYLA AND MAJNUN, 2023, 31, 0218-348X, 10.1142/S0218348X23400169
32.	Zulqurnain Sabir, Mohamed R. Ali, R. Sadat, Simulation of fractional order mathematical model of robots for detection of coronavirus using Levenberg–Marquardt backpropagation neural network, 2024, 0941-0643, 10.1007/s00521-024-10361-5
33.	MUHAMMAD SHOAIB, RAFIA TABASSUM, KOTTAKKARAN SOOPPY NISAR, MUHAMMAD ASIF ZAHOOR RAJA, FAROOQ AHMED SHAH, MOHAMMED S. ALQAHTANI, C. AHAMED SALEEL, H. M. ALMOHIY, DESIGN OF BIO-INSPIRED HEURISTIC TECHNIQUE INTEGRATED WITH SEQUENTIAL QUADRATIC PROGRAMMING FOR NONLINEAR MODEL OF PINE WILT DISEASE, 2023, 31, 0218-348X, 10.1142/S0218348X23401485
34.	Manal Alqhtani, J.F. Gómez-Aguilar, Khaled M. Saad, Zulqurnain Sabir, Eduardo Pérez-Careta, A scale conjugate neural network learning process for the nonlinear malaria disease model, 2023, 8, 2473-6988, 21106, 10.3934/math.20231075
35.	Soumen Pal, Manojit Bhattacharya, Snehasish Dash, Sang-Soo Lee, Chiranjib Chakraborty, A next-generation dynamic programming language Julia: Its features and applications in biological science, 2024, 64, 20901232, 143, 10.1016/j.jare.2023.11.015
36.	Syed T. R. Rizvi, Aly R. Seadawy, Samia Ahmed, Homoclinic breather, periodic wave, lump solution, and M-shaped rational solutions for cold bosonic atoms in a zig-zag optical lattice, 2023, 12, 2192-8029, 10.1515/nleng-2022-0337
37.	Juan Luis García Guirao, On the stochastic observation for the nonlinear system of the emigration and migration effects via artificial neural networks, 2023, 1, 2956-7068, 177, 10.2478/ijmce-2023-0014
38.	Honghua Liu, Chang She, Zhiliang Huang, Lei Wei, Qian Li, Han Peng, Mailan Liu, Uncertainty analysis and optimization of laser thermal pain treatment, 2023, 13, 2045-2322, 10.1038/s41598-023-38672-y
39.	Zulqurnain Sabir, Salem Ben Said, Qasem Al-Mdallal, An artificial neural network approach for the language learning model, 2023, 13, 2045-2322, 10.1038/s41598-023-50219-9
40.	Nabeela Anwar, Iftikhar Ahmad, Adiqa Kausar Kiani, Muhammad Shoaib, Muhammad Asif Zahoor Raja, Intelligent solution predictive networks for non-linear tumor-immune delayed model, 2024, 27, 1025-5842, 1091, 10.1080/10255842.2023.2227751
41.	Manoj Gupta, Achyuth Sarkar, Stochastic intelligent computing solvers for the SIR dynamical prototype epidemic model using the impacts of the hospital bed, 2024, 1025-5842, 1, 10.1080/10255842.2023.2300684
42.	Zulqurnain Sabir, Atef Hashem, Adnène Arbi, Mohamed Abdelkawy, Designing a Bayesian Regularization Approach to Solve the Fractional Layla and Majnun System, 2023, 11, 2227-7390, 3792, 10.3390/math11173792
43.	Juan Luis García Guirao, On the stochastic observation for the nonlinear system of the emigration and migration effects via artificial neural networks, 2023, 0, 2956-7068, 10.2478/ijmce-2023-00014
44.	A. A. Alderremy, J. F. Gómez-Aguilar, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Shaban Aly, Numerical investigations of the fractional order derivative-based accelerating universe in the modified gravity, 2024, 39, 0217-7323, 10.1142/S0217732323501808
45.	A. A. Alderremy, J. F. Gómez-Aguilar, Zulqurnain Sabir, Shaban Aly, J. E. Lavín-Delgado, José R. Razo-Hernández, Numerical performances based artificial neural networks to deal with the computer viruses spread on the complex networks, 2024, 101, 0020-7160, 314, 10.1080/00207160.2024.2326926
46.	Abdulgafor Alfares, Yusuf Abubakar Sha'aban, Ahmed Alhumoud, Machine learning -driven predictions of lattice constants in ABX3 Perovskite Materials, 2025, 141, 09521976, 109747, 10.1016/j.engappai.2024.109747
47.	Kamel Guedri, Rahat Zarin, Mowffaq Oreijah, Samaher Khalaf Alharbi, Hamiden Abd El-Wahed Khalifa, Artificial neural network-driven modeling of Ebola transmission dynamics with delays and disability outcomes, 2025, 115, 14769271, 108350, 10.1016/j.compbiolchem.2025.108350
48.	Farhad Muhammad Riaz, Junaid Ali Khan, A Swarm Optimized ANN-based Numerical Treatment of Nonlinear SEIR System based on Zika Virus, 2025, 2147-9429, 1, 10.2339/politeknik.1543179
49.	Kamel Guedri, Rahat Zarin, Mowffaq Oreijah, Samaher Khalaf Alharbi, Hamiden Abd El-Wahed Khalifa, Modeling syphilis progression and disability risk with neural networks, 2025, 314, 09507051, 113220, 10.1016/j.knosys.2025.113220
50.	Zulqurnain Sabir, Basma Souayeh, Zahraa Zaiour, Alyn Nazal, Mir Waqas Alam, Huda Alfannakh, A machine learning neural network architecture for the accelerating universe based modified gravity, 2025, 29, 11108665, 100635, 10.1016/j.eij.2025.100635
51.	Zhou Yang, Zhi Li, Xikun Jin, Yimin Zhang, Sensitivity analysis of subjective and objective reliability of disc brakes based on the parallel efficient global optimization algorithm, 2025, 0305-215X, 1, 10.1080/0305215X.2025.2475000
52.	Hanadi Alkhudhayr, Internet of things based parking slot detection and occupancy classification for smart city traffic management, 2025, 152, 09521976, 110802, 10.1016/j.engappai.2025.110802
53.	Rahat Zarin, Artificial neural network-based approach for simulating influenza dynamics: A nonlinear SVEIR model with spatial diffusion, 2025, 176, 09557997, 106230, 10.1016/j.enganabound.2025.106230
54.	Zulqurnain Sabir, Abdallah Srayeldine, A machine learning radial basis Bayesian regularization neural network procedure for the cholera disease model, 2025, 14, 2192-6670, 10.1007/s13721-025-00539-9

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)