Figure 1.
Designed framework of the GNNs-GA-IPA to solve the nonlinear smoke model.
Citation: Fernanda B Haffner, Roudayna Diab, Andreea Pasc. Encapsulation of probiotics: insights into academic and industrial approaches[J]. AIMS Materials Science, 2016, 3(1): 114-136. doi: 10.3934/matersci.2016.1.114
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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\{P'(Ω)=μ(1−P(Ω))−βP(Ω)S(Ω),P(0)=l1,S'(Ω)=βP(Ω)S(Ω)−(γ+μ)S(Ω)+αQT(Ω),S(0)=l2,Q'T(Ω)=γ(1−σ)S(Ω)−(μ+α)QT(Ω),QT(0)=l3,Q'P(Ω)=−μQP(Ω)+γσS(Ω),QP(0)=l4, \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.
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.
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:
|
$\begin{array}{*{20}{c}} \left[\widehat{P}\right(\mathit{\Omega } ), \widehat{S}(\mathit{\Omega } ), {\widehat{Q}}_{T}(\mathit{\Omega } ), {\widehat{Q}}_{S}(\mathit{\Omega } \left)\right] = \left[\begin{array}{l}{\sum }_{i = 1}^{m}{k}_{P, i}{\rm T}({w}_{P, i}\mathit{\Omega } +{h}_{P, i}), {\sum }_{i = 1}^{m}{a}_{S, i}{\rm T}({w}_{S, i}\mathit{\Omega } +{h}_{S, i}), \\ {\sum }_{i = 1}^{m}{a}_{{Q}_{T}, i}{\rm T}({w}_{{Q}_{T}, i}\mathit{\Omega } +{h}_{{Q}_{T}, i}), {\sum }_{i = 1}^{m}{a}_{{Q}_{S}, i}{\rm T}({w}_{{Q}_{S}, i}\mathit{\Omega } +{h}_{{Q}_{S}, i})\end{array} \right], \\
\left[{\widehat{P}}^{\text{'}}\right(\mathit{\Omega } ), {\widehat{S}}^{\text{'}}(\mathit{\Omega } ), {\widehat{Q}}_{T}^{\text{'}}(\mathit{\Omega } ), {\widehat{Q}}_{S}^{\text{'}}(\mathit{\Omega } \left)\right] = \\
\left[∑mi=1kP,iT'(wP,iΩ+hP,i),∑mi=1kS,iT'(wS,iΩ+hS,i),∑mi=1kQT,iT'(wQT,iΩ+hQT,i),∑mi=1kQS,iT'(wQS,iΩ+hQS,i) \right].
\end{array}$
|
(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
|
$ kP=[kP,1,kP,2,...,kP,m],kS=[kS,1,kS,2,...,kS,m],kQT=[kQT,1,kQT,2,...,kQT,m],kQS=[kQS,1,kQT,2,...,kQT,m],wP=[wP,1,wP,2,...,wP,m],wS=[wS,1,wS,2,...,wS,m]wQT=[wQT,1,wQT,2,...,wQT,m],wQS=[wQS,1,wQT,2,...,wQT,m],hP=[hP,1,hP,2,...,hP,m]hS=[hS,1,hS,2,...,hS,m],hQT=[hQT,1,hQT,2,...,hQT,m],hQS=[hQS,1,hQT,2,...,hQT,m] $
|
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.
|
$\begin{array}{*{20}{c}} \left[\begin{array}{cc}\widehat{P}\left(\mathit{\Omega } \right), & \widehat{S}\left(\mathit{\Omega } \right), \\ {\widehat{Q}}_{T}\left(\mathit{\Omega } \right), & {\widehat{Q}}_{S}\left(\mathit{\Omega } \right)\end{array} \right] = \\
\left[∑mi=1kP,i(2tan−1e(wP,iΩ+hP,i)−π2),∑mi=1kS,i(2tan−1e(wS,iΩ+hS,i)−π2),∑mi=1kQT,i(2tan−1e(wQT,iΩ+hQT,i)−π2),∑mi=1kQP,i(2tan−1e(wQS,iΩ+hQS,i)−π2) \right], \\
\left[ˆP'(Ω),ˆS'(Ω),ˆQ'T(Ω),ˆQ'S(Ω) \right] = \\
\left[∑mi=12kP,iwP,i(e(wp,iΩ+np,i)1+(e(wp,iΩ+np,i))2),∑mi=12kS,iwS,i(e(wS,iΩ+nS,i)1+(e(wS,iΩ+nS,i))2),∑mi=12kQT,iwQT,i(e(wQT,iΩ+nQT,i)1+(e(wQT,iΩ+nQT,i))2),∑mi=12kQS,iwQS,i(e(wQS,iΩ+nQS,i)1+(e(wQS,iΩ+nQS,i))2) \right],
\end{array}$
|
(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).
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.
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.
| 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}}] $, W is the weight vector Outputs: The best global weight values are signified as: WGA-Best Initialization: For the chromosome's assortment, adjust the W Assessment of FIT: Adjust "$ {\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} $, iterations, WGA-Best, function counts and time. GA procedure End |
| Start of IPA Inputs: WGA-Best is selected as an initial point. Output: WGA-IPA shows the best weights of GA-IPA. Initialize: WGA-Best, generations, assignments and other standards. Terminating criteria: Stop if [$ {\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 WGA-IPA, time, iterations, function counts and $ {\mathit{\Xi } }_{Fit} $ for the IPA trials. |
| IPA process End |
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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:
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$\left\{\begin{array}{l}[\text{V}\text{.}\text{A}\text{.}{\text{F}}_{P}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{S}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{T}}, \text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{S}}] = \left[\begin{array}{l}\left(1-\frac{\mathit{var}\left({P}_{m}-{\widehat{P}}_{m}\right)}{\mathit{var}\left({P}_{m}\right)}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({S}_{m}-{\widehat{S}}_{m}\right)}{\mathit{var}\left({S}_{m}\right)}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({\left({Q}_{T}\right)}_{m}-{\left({\widehat{Q}}_{T}\right)}_{m}\right)}{{\mathit{var}\left({Q}_{T}\right)}_{m}}\right)\times 100, \\ \left(1-\frac{\mathit{var}\left({\left({Q}_{S}\right)}_{m}-{\left({\widehat{Q}}_{S}\right)}_{m}\right)}{{\mathit{var}\left({Q}_{S}\right)}_{m}}\right)\times 100\end{array} \right], \\ [\text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{P}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{S}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{T}}, \text{E}\text{-}\text{V}\text{.}\text{A}\text{.}{\text{F}}_{{Q}_{S}}] = \left[\left|100−V.A.FP,100−V.A.FS,100−V.A.FQT,100−V.A.FQS \right|\right].\end{array}\right.$
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(10) |
|
$[T.I.CP,T.I.CS,T.I.CQT,T.I.CQS]=[√1n∑nm=1(Pm−ˆPm)2(√1n∑nm=1P2m+√1n∑nm=1ˆP2m),√1n∑nm=1(Sm−ˆSm)2(√1n∑nm=1S2m+√1n∑nm=1ˆS2m),√1n∑nm=1((QT)m−(ˆQT)m)2(√1n∑nm=1(QT)2m+√1n∑nm=1(ˆQT)2m),√1n∑nm=1((QS)m−(ˆQS)m)2(√1n∑nm=1(QS)2m+√1n∑nm=1(ˆQS)2m)], $
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(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\{S.I Range=−0.5×(Q1−Q3),Q1 = 1stquartileQ3=3rdquartile. \right. $
|
(13) |
$ \widehat{P}, \widehat{S}, {\widehat{Q}}_{T} $ and $ {\widehat{Q}}_{S} $ are the approximate form of the solutions.
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\{P'(Ω)=20−(20P(Ω)+0.003P(Ω)S(Ω)),P(0)=0.3,S'(Ω)=0.003P(Ω)S(Ω)−20.3S(Ω)+3QT(Ω),S(0)=0.5,Q'T(Ω)=0.15S(Ω)−23QT(Ω),QT(0)=0.1,Q'P(Ω)=0.15S(Ω)−20QP(Ω),QP(0)=0.2, \right. $
|
(14) |
A fitness function for the nonlinear smoke model (14) is written as:
|
$ΞFit=1N∑Nm=1([ˆP'm+20ˆPm−20+0.003ˆPmˆSm]2+[ˆS'm+20.3ˆSm−0.003ˆPmˆSm−3(QT)m]2+[(Q'T)m+23(QT)m−0.15ˆSm]2+[(Q'P)m+20(QP)m−0.15ˆSm]2)+14[(ˆP0−0.3)2+(ˆS0−0.5)2+((QT)0−0.1)2+((QP)0−0.2)2]. $
|
(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.
|
$ˆP(Ω)=7.2965(2tan−1e(−5.343Ω−18.462)−0.5π)−2.7636(2tan−1e1.2606Ω+0.5134)−0.5π)+15.294(2tan−1e(1.1551Ω+2.0601)−0.5π)+8.7324(2tan−1e(19.759Ω+2.9789)−0.5π)−7.1646(2tan−1e(2.5198Ω+5.9852)−0.5π)−0.0053(2tan−1e(−0.246Ω−1.7692)−0.5π)−2.6104(2tan−1e(0.2323Ω+1.5497)−0.5π)−1.4677(2tan−1e(02.782Ω+2.0890)−0.5π)+2.4715(2tan−1e(−18516Ω−3.253)−0.5π)−2.9189(2tan−1e(10.272Ω−7.2465)−0.5π), $
|
(16) |
|
$ˆS(Ω)=7.6303(2tan−1e(−0.143Ω+0.4605)−0.5π)+9.6607(2tan−1e(0.264Ω+0.8726)−0.5π)−1.8710(2tan−1e(0.3252Ω−1.0967)−0.5π)+0.0451(2tan−1e(−1.961Ω+8.4961)−0.5π)+5.4711(2tan−1e(−9.8369Ω−1.533)−0.5π)−1.7275(2tan−1e(−9.2138Ω−2.037)−0.5π)+0.2908(2tan−1e(−2.4236Ω−0.378)−0.5π)−1.9171(2tan−1e(−9.2376Ω−0.360)−0.5π)+2.7144(2tan−1e(2.8428Ω−7.8525)−0.5π)−5.3000(2tan−1e(−4.6148Ω+4.164)−0.5π), $
|
(17) |
|
$ˆQT(Ω)=−0.563(2tan−1e(−3.245Ω+2.3616)−0.5π)−0.022(2tan−1e(−8.9859+1.2373)−0.5π)−0.5360(2tan−1e(3.2982Ω−2.4088)−0.5π)−2.1382(2tan−1e(−3.87Ω−3.100)−0.5π)−4.7812(2tan−1e(−4.1182Ω−4.994)−0.5π)+0.9505(2tan−1e(−0.038Ω−0.241)−0.5π)+17.841(2tan−1e(−14.7536Ω−4.557)−0.5π)−3.4575(2tan−1e(4.347Ω+0.7343)−0.5π)+11.8361(2tan−1e(4.5451Ω+2.013)−0.5π)+2.717(2tan−1e(3.6697Ω+7.33831)−0.5π), $
|
(18) |
|
$ˆQS(Ω)=0.0762(2tan−1e(2.9990Ω+2.8605)−0.5π)+2.8390(2tan−1e(8.3177Ω+0.6804)−0.5π)+7.6702(2tan−1e(−0.0481Ω+1.9373)−0.5π)−0.4142(2tan−1e(−0.309Ω−0.401)−0.5π)−6.2827(2tan−1e(1.2291Ω+3.5918)−0.5π)+0.5077(2tan−1e(1.0574Ω+1.0166)−0.5π)+2.2061(2tan−1e(1.5133Ω+10.7487)−0.5π)−0.2776(2tan−1e(1.7337Ω+5.1928)−0.5π)−5.3623(2tan−1e(8.4564Ω+1.3100)−0.5π)+0.0684(2tan−1e(−4.1583Ω−1.6646)−0.5π), $
|
(19) |
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. Figure 3 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. Figure 4 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.
The graphic illustrations based on the statistical performances are provided in Figures 5–7 to find the convergence along with the boxplots and the histograms to solve the nonlinear smoke model. Figure 5 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 Figure 6 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 Figure 7 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.
The routines for different statistical operators, Maximum (Max), Median, Minimum (Min), standard deviation (STD) and SIR are provided in Tables 2–5 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.
| $ \mathit{\Omega } $ | P($ \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 |
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| $ \mathit{\Omega } $ | $ S\left(\mathit{\Omega } \right) $ | |||||
| 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 |
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| $ \mathit{\Omega } $ | $ {Q}_{T}\left(\mathit{\Omega } \right) $ | |||||
| 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 |
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| $ \mathit{\Omega } $ | $ {Q}_{S}\left(\mathit{\Omega } \right) $ | |||||
| 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 |
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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.
| Class | (G.MAD) | (G.TIC) | (G.EVAF) | |||
| 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 |
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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].
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.
All authors of the manuscript declare that there have no potential conflicts of interest.
| [1] | FAO W (2001) Health and nutritional properties of probiotics in food including powder milk with live lactic acid bacteria Report of a Joint FAO/WHO Expert Consultation on Evaluation of Health and Nutritional Properties of Probiotics in Food Including Powder Milk with Live Lactic Acid Bacteria: 1–34. |
| [2] | Kailasapathy K (2002) Microencapsulation of probiotic bacteria: technology and potential applications. Curr Issues Intest Microbiol 3: 39–34. |
| [3] |
Tripathi MK, Giri SK (2014) Probiotic functional foods: survival of probiotics during processing and storage. J Funct Foods 9: 225–24. doi: 10.1016/j.jff.2014.04.030
|
| [4] |
Verna EC, Lucak S (2010) Use of probiotics in gastrointestinal disorders: what to recommend? Therap Adv Gastroenterol 3: 307–319. doi: 10.1177/1756283X10373814
|
| [5] |
Fuller R (1991) Probiotics in human medicine. Gut 32: 439–442. doi: 10.1136/gut.32.4.439
|
| [6] |
Gupta S, Abu-Ghannam N (2012) Probiotic fermentation of plant based products: possibilities and opportunities. Crit Rev Food Sci Nutr 52: 183–199. doi: 10.1080/10408398.2010.499779
|
| [7] |
Lavermicocca P, Valerio F, Lonigro SL, et al. (2005) Study of adhesion and survival of lactobacilli and bifidobacteria on table olives with the aim of formulating a new probiotic food. Appl Environ Microbiol 71: 4233–4240. doi: 10.1128/AEM.71.8.4233-4240.2005
|
| [8] |
Ying D, Schwander S, Weerakkody R, et al. (2013) Microencapsulated lactobacillus rhamnosus gg in whey protein and resistant starch matrices: probiotic survival in fruit juice. J Funct Foods 5: 98–105. doi: 10.1016/j.jff.2012.08.009
|
| [9] |
Ranadheera CS, Evans CA, Adams MC, et al. (2013) Production of probiotic ice cream from goat's milk and effect of packaging materials on product quality. Small Ruminant Res 112: 174–180. doi: 10.1016/j.smallrumres.2012.12.020
|
| [10] | López de Lacey AM, Pérez-Santín E, López-Caballero ME, et al. (2014) Survival and metabolic activity of probiotic bacteria in green tea. Food Sci Technol 55: 314–322. |
| [11] | González-Sánchez F, Azaola A, Gutiérrez-López GF, et al. (2010) Viability of microencapsulated bifidobacterium animalis ssp. lactis bb12 in kefir during refrigerated storage. Int J Dairy Technol 63: 431–436. |
| [12] |
Noorbakhsh R, Yaghmaee P, Durance T (2013) Radiant energy under vacuum (rev) technology: a novel approach for producing probiotic enriched apple snacks. J Funct Foods 5: 1049–1056. doi: 10.1016/j.jff.2013.02.011
|
| [13] |
Sheehan VM, Ross P, Fitzgerald GF (2007) Assessing the acid tolerance and the technological robustness of probiotic cultures for fortification in fruit juices. Innov Food Sci Emerg Technol 8: 279–284. doi: 10.1016/j.ifset.2007.01.007
|
| [14] |
Nualkaekul S, Cook MT, Khutoryanskiy VV, et al. (2013) Influence of encapsulation and coating materials on the survival of lactobacillus plantarum and bifidobacterium longum in fruit juices. Food Res Int 53: 304–311. doi: 10.1016/j.foodres.2013.04.019
|
| [15] |
Mestry AP, Mujumdar AS, Thorat BN (2011) Optimization of spray drying of an innovative functional food: fermented mixed juice of carrot and watermelon. Dry Technol 29: 1121–1131. doi: 10.1080/07373937.2011.566968
|
| [16] |
Alegre I, Viñas I, Usall J, et al. (2011) Microbiological and physicochemical quality of fresh-cut apple enriched with the probiotic strain lactobacillus rhamnosus gg. Food Microbiol 28: 59–66. doi: 10.1016/j.fm.2010.08.006
|
| [17] |
Rößle C, Auty MA, Brunton N, et al. (2010) Evaluation of fresh-cut apple slices enriched with probiotic bacteria. Innov Food Sci Emerg Technol 11: 203–209. doi: 10.1016/j.ifset.2009.08.016
|
| [18] |
Possemiers S, Marzorati M, Verstraete W, et al. (2010) Bacteria and chocolate: a successful combination for probiotic delivery. Int J Food Microbiol 141: 97–103. doi: 10.1016/j.ijfoodmicro.2010.03.008
|
| [19] |
Roopashri AN, Varadaraj MC (2014) Hydrolysis of flatulence causing oligosaccharides by α-d-galactosidase of a probiotic lactobacillus plantarum mtcc 5422 in selected legume flours and elaboration of probiotic attributes in soy-based fermented product. Eur Food Res Technol 239: 99–115. doi: 10.1007/s00217-014-2207-y
|
| [20] |
Hugo AA, Pérez PF, Añón MC, et al. (2014) Incorporation of lactobacillus delbrueckii subsp lactis (cidca 133) in cold-set gels made from high pressure-treated soybean proteins. Food Hydrocolloid 37: 34–39. doi: 10.1016/j.foodhyd.2013.10.025
|
| [21] | Agheyisi R (2014) The probiotics market: ingredients, supplements, foods. BCC Research Food, Beverage Report: 1–25. |
| [22] | Del Piano M, Carmagnola S, Andorno S, et al. (2010) Evaluation of the intestinal colonization by microencapsulated probiotic bacteria in comparison with the same uncoated strains. J Clin Gastroenterol 44 Suppl 1: S42–6. |
| [23] | Piano MD, Carmagnola S, Ballarè M, et al. (2012) Comparison of the kinetics of intestinal colonization by associating 5 probiotic bacteria assumed either in a microencapsulated or in a traditional, uncoated form. J Clin Gastroenterol 46 Suppl: S85–92. |
| [24] |
Heidebach T, Först P, Kulozik U (2012) Microencapsulation of probiotic cells for food applications. Crit Rev Food Sci Nutr 52: 291–311. doi: 10.1080/10408398.2010.499801
|
| [25] |
Cook MT, Tzortzis G, Charalampopoulos D, et al. (2012) Microencapsulation of probiotics for gastrointestinal delivery. J Control Release 162: 56–67. doi: 10.1016/j.jconrel.2012.06.003
|
| [26] |
Rodrigues D, Sousa S, Rocha-Santos T, et al. (2011) Influence of l-cysteine, oxygen and relative humidity upon survival throughout storage of probiotic bacteria in whey protein-based microcapsules. Int Dairy J 21: 869–876. doi: 10.1016/j.idairyj.2011.05.005
|
| [27] | Teixeira PC, Castro MH, Malcata FX, et al. (1995) Survival of lactobacillus-delbrueckii ssp. bulgaricus following spray-drying. J Dairy Sci 78: 1025–1031. |
| [28] | Oxley J (2014) Overview of microencapsulation process technologies, In: Gaonkar AG, Vasisht N, Khare AR, Sobel R (Eds.), Microencapsulation in the food industry, 1 Eds., San Diego: Elsevier, 35–46. |
| [29] |
Burgain J, Gaiani C, Linder M, et al. (2011) Encapsulation of probiotic living cells: from laboratory scale to industrial applications. J Food Eng 104: 467–483. doi: 10.1016/j.jfoodeng.2010.12.031
|
| [30] |
Gbassi GK, Vandamme T (2012) Probiotic encapsulation technology: from microencapsulation to release into the gut. Pharmaceutics 4: 149–163. doi: 10.3390/pharmaceutics4010149
|
| [31] |
Riaz QUA, Masud T (2013) Recent trends and applications of encapsulating materials for probiotic stability. Crit Rev Food Sci Nutr 53: 231–244. doi: 10.1080/10408398.2010.524953
|
| [32] | Krasaekoopt W, Bhandari B, Deeth H (2003) Evaluation of encapsulation techniques of probiotics for yoghurt. Int Dairy J 13: 3–13. |
| [33] | Vidhyalakshmi R, Bhakyaraj R, Subhasree RS (2009) Encapsulation "the future of probiotics"-a review. Adv Biol Res 3: 96–103. |
| [34] | Poncelet D, Dulieu C, Jacquot M (2001) Description of the immobilisation procedures, In: Wijffels RH (Ed.), Immobilized cells, 1 Eds., Heidelbert: Springer, 15–30 |
| [35] | Okuro PK, Junior FE, Favaro-Trindade CS (2013) Technological challenges for spray chilling encapsulation of functional food ingredients. Food Technol Biotechnol : 1–12. |
| [36] | Pinto SS, Fritzen-Freire CB, Muñoz IB, et al. (2012) Effects of the addition of microencapsulated bifidobacterium bb-12 on the properties of frozen yogurt. J Food Eng 111: 563–569. |
| [37] |
Ying D, Sun J, Sanguansri L, et al. (2012) Enhanced survival of spray-dried microencapsulated lactobacillus rhamnosus gg in the presence of glucose. J Food Eng 109: 597–602. doi: 10.1016/j.jfoodeng.2011.10.017
|
| [38] |
Chávez BE, Ledeboer AM (2007) Drying of probiotics: optimization of formulation and process to enhance storage survival. Dry Technol 25: 1193–1201. doi: 10.1080/07373930701438576
|
| [39] |
Cheow WS, Kiew TY, Hadinoto K (2014) Controlled release of lactobacillus rhamnosus biofilm probiotics from alginate-locust bean gum microcapsules. Carbohydr Polym 103: 587–595. doi: 10.1016/j.carbpol.2014.01.036
|
| [40] |
Behboudi-Jobbehdar S, Soukoulis C, Yonekura L, et al. (2013) Optimization of spray-drying process conditions for the production of maximally viable microencapsulated l. acidophilusncimb 701748 Dry Technol 31: 1274–1283. doi: 10.1080/07373937.2013.788509
|
| [41] |
De Castro-Cislaghi FP, Silva CD, Fritzen-Freire CB, et al. (2012) Bifidobacterium bb-12 microencapsulated by spray drying with whey: survival under simulated gastrointestinal conditions, tolerance to nacl, and viability during storage. J Food Eng 113: 186–193. doi: 10.1016/j.jfoodeng.2012.06.006
|
| [42] |
Estevinho BN, Rocha F, Santos L, et al. (2013) Microencapsulation with chitosan by spray drying for industry applications – a review. Trends Food Sci Tech 31: 138–155. doi: 10.1016/j.tifs.2013.04.001
|
| [43] |
Yonekura L, Sun H, Soukoulis C, et al. (2014) Microencapsulation of lactobacillus acidophilus ncimb 701748 in matrices containing soluble fibre by spray drying: technological characterization, storage stability and survival after in vitro digestion. J Funct Foods 6: 205–214. doi: 10.1016/j.jff.2013.10.008
|
| [44] | Pedroso DL, Dogenski M, Thomazini M, et al. (2014) Microencapsulation of bifidobacterium animalis subsp. lactis and lactobacillus acidophilus in cocoa butter using spray chilling technology. Braz J Microbiol 15: 1–7. |
| [45] |
Okuro PK, Thomazini M, Balieiro JC, et al. (2013) Co- encapsulation of lactobacillus acidophilus with inulin or polydextrose in solid lipid microparticles provides protection and improves stability. Food Res Int 53: 96–103. doi: 10.1016/j.foodres.2013.03.042
|
| [46] |
Gouin S (2004) Micro-encapsulation: industrial appraisal of existing technologies and trends. Trends Food Sci Technol 15: 330–347. doi: 10.1016/j.tifs.2003.10.005
|
| [47] |
Lahtinen SJ, Ouwehand AC, Salminen SJ, et al. (2007) Effect of starch- and lipid-based encapsulation on the culturability of two bifidobacterium longum strains. Lett Appl Microbiol 44: 500–505. doi: 10.1111/j.1472-765X.2007.02110.x
|
| [48] |
Dianawati D, Mishra V, Shah NP (2013) Survival of bifidobacterium longum 1941 microencapsulated with proteins and sugars after freezing and freeze drying. Food Res Int 51: 503–509. doi: 10.1016/j.foodres.2013.01.022
|
| [49] | Shoji AS, Oliveira AC, Balieiro JC, et al. (2013) Viability of l. acidophilus microcapsules and their application to buffalo milk yoghurt. Food Bioprod Process 91: 83–88. |
| [50] |
Mantzouridou F, Spanou A, Kiosseoglou V (2012) An inulin-based dressing emulsion as a potential probiotic food carrier. Food Res Int 46: 260–269. doi: 10.1016/j.foodres.2011.12.016
|
| [51] | Muthukumarasamy P, Allan-Wojtas P, Holley RA (2006) Stability of lactobacillus reuteri in different types of microcapsules. J Food Sci : 1–5. |
| [52] |
Amine KM, Champagne C, Raymond Y, et al. (2014) Survival of microencapsulated bifidobacterium longum in cheddar cheese during production and storage. Food Control 37: 193–199. doi: 10.1016/j.foodcont.2013.09.030
|
| [53] |
López de Lacey AM, López-Caballero ME, Gómez-Estaca J, et al. (2012) Functionality of lactobacillus acidophilus and bifidobacterium bifidum incorporated to edible coatings and films. Innov Food Sci Emerg Technol 16: 277–282. doi: 10.1016/j.ifset.2012.07.001
|
| [54] |
Sathyabama S, Ranjith kumar M, Bruntha devi P, et al. (2014) Co-encapsulation of probiotics with prebiotics on alginate matrix and its effect on viability in simulated gastric environment. LWT - Food Science and Technology 57: 419–425. doi: 10.1016/j.lwt.2013.12.024
|
| [55] |
Doherty SB, Auty MA, Stanton C, et al. (2012) Survival of entrapped lactobacillus rhamnosus gg in whey protein micro-beads during simulated ex vivo gastro-intestinal transit. Int Dairy J 22: 31–43. doi: 10.1016/j.idairyj.2011.06.009
|
| [56] |
Jiménez-Pranteda ML, Poncelet D, Náder-Macías ME, et al. (2012) Stability of lactobacilli encapsulated in various microbial polymers. J Biosci Bioeng 113: 179–184. doi: 10.1016/j.jbiosc.2011.10.010
|
| [57] |
Brachkova MI, Duarte MA, Pinto JF (2010) Preservation of viability and antibacterial activity of lactobacillus spp. in calcium alginate beads. Eur J Pharm Sci 41: 589–596. doi: 10.1016/j.ejps.2010.08.008
|
| [58] |
Khan NH, Korber DR, Low NH, et al. (2013) Development of extrusion-based legume protein isolate–alginate capsules for the protection and delivery of the acid sensitive probiotic, bifidobacterium adolescentis. Food Res Int 54: 730–737. doi: 10.1016/j.foodres.2013.08.017
|
| [59] | Chavarri M, Maranon I, Carmen M (2012) Encapsulation technology to protect probiotic bacteria, In: Rigobelo E (Ed.), Probiotics, 1 Eds, Brazil: Intech, 501–539. |
| [60] |
Baker CG, McKenzie KA (2005) Energy consumption of industrial spray dryers. Dry Technol 23: 365–386. doi: 10.1081/DRT-200047665
|
| [61] |
Rodríguez-Huezo ME, Durán-Lugo R, Prado-Barragán LA, et al. (2007) Pre-selection of protective colloids for enhanced viability of bifidobacterium bifidum following spray-drying and storage, and evaluation of aguamiel as thermoprotective prebiotic. Food Res Int 40: 1299–1306. doi: 10.1016/j.foodres.2007.09.001
|
| [62] |
Corcoran BM, Ross RP, Fitzgerald GF, et al. (2004) Comparative survival of probiotic lactobacilli spray-dried in the presence of prebiotic substances. J Appl Microbiol 96: 1024–1039. doi: 10.1111/j.1365-2672.2004.02219.x
|
| [63] |
Conrad PB, Miller DP, Cielenski PR, et al. (2000) Stabilization and preservation of lactobacillus acidophilus in saccharide matrices. Cryobiology 41: 17–24. doi: 10.1006/cryo.2000.2260
|
| [64] | Selmer-Olsen E, Sorhaug T, Birkeland S, et al. (1999) Survival of lactobacillus helveticus entrapped in ca-alginate inrelation to water content, storage and rehydration. J Ind Microbiol Biot 23: 1–7. |
| [65] |
Dong Q, Chen M, Xin Y, et al. (2013) Alginate-based and protein-based materials for probiotics encapsulation: a review. Int J Food Sci Technol 48: 1339–1351. doi: 10.1111/ijfs.12078
|
| [66] |
Ananta E, Birkeland S, Corcoran B, et al. (2004) Processing effects on the nutritional advancement of probiotics and prebiotics. Microb Ecol Health D 16: 113–124. doi: 10.1080/08910600410032277
|
| [67] |
Simpson PJ, Stanton C, Fitzgerald GF, et al. (2005) Intrinsic tolerance of bifidobacterium species to heat and oxygen and survival following spray drying and storage. J Appl Microbiol 99: 493–501. doi: 10.1111/j.1365-2672.2005.02648.x
|
| [68] |
Ananta E, Volkert M, Knorr D (2005) Cellular injuries and storage stability of spray-dried lactobacillus rhamnosus gg. Int Dairy J 15: 399–409. doi: 10.1016/j.idairyj.2004.08.004
|
| [69] |
Wang Y, Yu R, Chou C (2004) Viability of lactic acid bacteria and bifidobacteria in fermented soymilk after drying, subsequent rehydration and storage. Int J Food Microbiol 93: 209–217. doi: 10.1016/j.ijfoodmicro.2003.12.001
|
| [70] | Champagne C, Gardner N, Brochu E, et al. (1991) The freeze-drying of lactic acid bacteria. a review. Canadian Institute of Food Science and Technology : 1–11. |
| [71] |
De Vos P, Faas MM, Spasojevic M, et al. (2010) Encapsulation for preservation of functionality and targeted delivery of bioactive food components. Int Dairy J 20: 292–302. doi: 10.1016/j.idairyj.2009.11.008
|
| [72] | Rutherford WM, Allen JE, Schlameus HW, et al. (1994) Process for preparing rotary disc fatty acid microspheres of microorganisms. United States Patent 5,292,657. |
| [73] |
Seo JK, Kim S, Kim M, et al. (2010) Direct-fed microbials for ruminant animals. Asian-Aust J Anim Sci 23: 1657–67. doi: 10.5713/ajas.2010.r.08
|
| [74] |
European Food Safety Authority (EFSA) (2012) Scientific opinion on the efficacy of bactocell (pediococcus acidilactici) when used as a feed additive for fish. EFSA Journal 10: 2886. doi: 10.2903/j.efsa.2012.2886
|
| [75] | Harel M, Kohari-Beck K (2007) A delivery vehicle for probiotic bacteria comprising a dry matrix of polysaccharides, saccharides and polyols in a glass form and methods of making same. WO 2007079147 A2. |
| [76] |
Nguyen HT, Razafindralambo H, Blecker C, et al. (2014) Stochastic exposure to sub-lethal high temperature enhances exopolysaccharides (eps) excretion and improves bifidobacterium bifidum cell survival to freeze–drying. Biochem Eng J 88: 85–94. doi: 10.1016/j.bej.2014.04.005
|
| [77] | Basholli-Salihu M, Mueller M, Salar-Behzadi S, et al. (2014) Effect of lyoprotectants on β-glucosidase activity and viability of bifidobacterium infantis after freeze-drying and storage in milk and low ph juices. Food Sci Technol 57: 276–282. |
| [78] |
Champagne CP, Fustier P (2007) Microencapsulation for the improved delivery of bioactive compounds into foods. Curr Opin Biotechnol 18: 184–190. doi: 10.1016/j.copbio.2007.03.001
|
| [79] |
Schell D, Beermann C (2014) Fluidized bed microencapsulation of lactobacillus reuteri with sweet whey and shellac for improved acid resistance and in-vitro gastro-intestinal surviva. Food Res Int 62: 308–314. doi: 10.1016/j.foodres.2014.03.016
|
| [80] |
Semyonov D, Ramon O, Kovacs A, et al. (2012) Air-suspension fluidized-bed microencapsulation of probiotics. Dry Technol 30: 1918–1930. doi: 10.1080/07373937.2012.708692
|
| [81] |
Bensch G, Rüger M, Wassermann M, et al. (2014) Flow cytometric viability assessment of lactic acid bacteria starter cultures produced by fluidized bed drying. Appl Microbiol Biot 98: 4897–4909. doi: 10.1007/s00253-014-5592-z
|
| [82] |
Champagne C, Raymond Y, Tompkins TA (2010) The determination of viable counts in probiotic cultures microencapsulated by spray-coating. Food Microbiol 27: 1104–1111. doi: 10.1016/j.fm.2010.07.017
|
| [83] | Wu WH, Roe WS, Gimino VG, et al. (2000) Low melt encapsulation with high laurate canola oil. United States Patent 61,532,36 A. |
| [84] | Ubbink JB, Schaer-Zammaretti P, Cavadini C (2012) Probiotic delivery system. United States Application 2005/0,153,018 A1. |
| [85] | Durand H, Panes J (2001) Particles coated with a homogeneous, hydrophobic protective layer for use in pharmaceuticals, dietic or feed compositions, comprise agglomerates of microorganisms. WO200168808-A1. |
| [86] | Beck NT, Franch G, Geneau DL (2002) Edible emulsion comprising live micro-organisms and dressings or side sauces comprising said edible emulsion. WO 2002030211 A1. |
| [87] | Vos H, Poortinga AT (2010) Double emulsion and method to produce such. WO 2010039036 A1. |
| [88] | Mazer T, Kessler T (2014) Methods for extruding powered nutritional products using a high shear element. WO 2014093832 A1. |
| [89] | Frenken LG, Hammarstroem LG, Ledeboer AM (2007) Food products comprising probiotic micro-organisms and antibodies. WO 2007019901 A1. |
| [90] | Gregoriadis G, Antimisiaris SG, Gursel I (2001) Liposomes containing particulate materials. US 6451338 B1. |
| [91] |
Gerez CL, Font de Valdez G, Gigante ML, et al. (2012) Whey protein coating bead improves the survival of the probiotic lactobacillus rhamnosus crl 1505 to low ph. Lett Appl Microbiol 54: 552–556. doi: 10.1111/j.1472-765X.2012.03247.x
|
| [92] | Agüeros BM, Esparza CI, Gamazo DL, et al. (2014) Microparticles for encapsulating probiotics, production and uses thereof. WO 2014006261 A2. |
| [93] | Bhushani JA, Anandharamakrishnan C (2014) Electrospinning and electrospraying techniques: potential food based applications. Trends Food Sci Technol 38 : 21–33. |
| [94] | Borges S, Barbosa J, Camilo R, et al. (2011) Effects of encapsulation on the viability of probiotic strains exposed to lethal conditions. Int J Food Sci Technol 47: 416–421. |
| [95] |
Paques J, van der Linden E, van Rijn CJM, et al. (2013) Alginate submicron beads prepared through w/o emulsion and gelation with cacl2 nanoparticles. Food Hydrocolloid 31: 428–434. doi: 10.1016/j.foodhyd.2012.11.012
|
| [96] |
Laelorspoen N, Wongsasulak S, Yoovidhya T, et al. (2014) Microencapsulation of lactobacillus acidophilus in zein–alginate core–shell microcapsules via electrospraying. J Funct Foods 7: 342–349. doi: 10.1016/j.jff.2014.01.026
|
| [97] | De Prisco A, Maresca D, Ongeng D, et al. (2015) Microencapsulation by vibrating technology of the probiotic strain lactobacillus reuteri dsm 17938 to enhance its survival in foods and in gastrointestinal environment. LWT-Food Sci Technol 61: 452–62. |
| [98] |
Graff S, Hussain S, Chaumeil J, et al. (2008) Increased intestinal delivery of viable saccharomyces boulardii by encapsulation in microspheres. Pharm Res 25: 1290–1296. doi: 10.1007/s11095-007-9528-5
|
| [99] | Alisch G, Brauneis E, Pirstadt B, et al. (1995) Process and plant for the production of spherical alginate pellets. US 5,472,648. |
| [100] | Van LBH (2001) Encapsulation of sensitive components into a matrix to obtain discrete shelf-stable particles. WO 2001025414. |
| [101] | Asada M, Hatano Y, Kamaguchi R, et al. (2003) Capsules containing vital cells or tissues. WO 2003001927 A1. |
| [102] | Doherty S, Brodkorb A (2010) Production of microbeads e.g. used in vehicle for the delivering active agent to lower intestine of subject by providing suspension of denatured whey protein and active component, and treating suspension to generate microbeads. WO2010119041 A2. |
| [103] | Shah NP (1999) Probiotic bacteria: selective enumeration and survival in dairy foods. J Dairy Sci : 894–907. |
| [104] |
Rokka S, Rantamäki P (2010) Protecting probiotic bacteria by microencapsulation: challenges for industrial applications. Eur Food Res Technol 231: 1–12. doi: 10.1007/s00217-010-1246-2
|
| [105] |
Foster JA, McVey Neufeld K (2013) Gut-brain axis: how the microbiome influences anxiety and depression. Trends Neurosci 36: 305–312. doi: 10.1016/j.tins.2013.01.005
|
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| 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 |
Figures(2) / Tables(2)
Fernanda B Haffner, Roudayna Diab, Andreea Pasc. Encapsulation of probiotics: insights into academic and industrial approaches[J]. AIMS Materials Science, 2016, 3(1): 114-136. doi: 10.3934/matersci.2016.1.114
| 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}}] $, W is the weight vector Outputs: The best global weight values are signified as: WGA-Best Initialization: For the chromosome's assortment, adjust the W Assessment of FIT: Adjust "$ {\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} $, iterations, WGA-Best, function counts and time. GA procedure End |
| Start of IPA Inputs: WGA-Best is selected as an initial point. Output: WGA-IPA shows the best weights of GA-IPA. Initialize: WGA-Best, generations, assignments and other standards. Terminating criteria: Stop if [$ {\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 WGA-IPA, time, iterations, function counts and $ {\mathit{\Xi } }_{Fit} $ for the IPA trials. |
| IPA process End |
DownLoad:
CSV
| $ \mathit{\Omega } $ | P($ \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 |
DownLoad:
CSV
| $ \mathit{\Omega } $ | $ S\left(\mathit{\Omega } \right) $ | |||||
| 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 |
DownLoad:
CSV
| $ \mathit{\Omega } $ | $ {Q}_{T}\left(\mathit{\Omega } \right) $ | |||||
| 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 |
DownLoad:
CSV
| $ \mathit{\Omega } $ | $ {Q}_{S}\left(\mathit{\Omega } \right) $ | |||||
| 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 |
DownLoad:
CSV
| Class | (G.MAD) | (G.TIC) | (G.EVAF) | |||
| 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 |
DownLoad:
CSV
| 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}}] $, W is the weight vector Outputs: The best global weight values are signified as: WGA-Best Initialization: For the chromosome's assortment, adjust the W Assessment of FIT: Adjust "$ {\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} $, iterations, WGA-Best, function counts and time. GA procedure End |
| Start of IPA Inputs: WGA-Best is selected as an initial point. Output: WGA-IPA shows the best weights of GA-IPA. Initialize: WGA-Best, generations, assignments and other standards. Terminating criteria: Stop if [$ {\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 WGA-IPA, time, iterations, function counts and $ {\mathit{\Xi } }_{Fit} $ for the IPA trials. |
| IPA process End |
| $ \mathit{\Omega } $ | P($ \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 |
| $ \mathit{\Omega } $ | $ S\left(\mathit{\Omega } \right) $ | |||||
| 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 |
| $ \mathit{\Omega } $ | $ {Q}_{T}\left(\mathit{\Omega } \right) $ | |||||
| 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 |
| $ \mathit{\Omega } $ | $ {Q}_{S}\left(\mathit{\Omega } \right) $ | |||||
| 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 |
| Class | (G.MAD) | (G.TIC) | (G.EVAF) | |||
| 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 |
