1

An integrated model to predict the size of silver nanoparticles in

montmorillonite/chitosan bionanocomposites: a hybrid of data

envelopment analysis and genetic programming approach Elham Sarvestani1; Gholam Reza Khayati*2

1. Department of Materials Science and Engineering, Shahid Bahonar University of Kerman, P. O. Box No. 76135-133, Kerman, Iran;

Elham.sarvestani1993@gmail.com 2. Department of Materials Science and Engineering, Shahid Bahonar University of Kerman, P. O. Box No. 76135-133, Kerman, Iran;

khayatireza@gmail.com;098-09151903477

Abstract

Unique chemical and physical properties of silver nanoparticles (AgNPs) enhances its usages in

various categories such as medical utilities. Due to the high dependency of AgNPs properties to

the size, this study is an attempt to employ gene expression programing (GEP) for constructing a

quantitative model for estimating the size of AgNPs in montmorillonite/chitosan

bionanocomposites that prepared by chemical approach. Generalization capabilities, fault

tolerance, noise tolerance, high parallelism, nonlinearity and significant information processing

characteristics are the main advantages of GEP. Accordingly, the practical parameters including

reaction temperature, AgNO3 concentration, weight of montmorillonite in aqueous

AgNO3/chitosan solution (WMMT) and the percentage of chitosan are selected as input parameters

through GEP modeling. The accuracy of proposed models are investigated by statistical

indicators including mean absolute percentage error (MAPE), root relative squared error

(RRSE), root mean square error (RMSE) and correlation coefficient (R2). Finally, the best model

is selected by R2 = 0.987, RMSE = 0.100, RRSE = 0.146 and MAPE = 0.221. The sensitivity

analysis confirmed that the percentage of chitosan, concentration of AgNO3, WMMT and reaction

temperature are the most effecting parameters on the size of AgNPs, respectively.

Keywords

Silver nanoparticle; Gene expression programming; Montmorillonite; Chitosan;

Bionanocomposites; Sensitivity analysis

NOMENCLATURE

MAgNO3 Concentration of silver nitrate (M) MAPE Mean absolute percentage error

mailto:Elham.sarvestani1993@gmail.com

2

T Reaction temperature (C̊) C(i,j) Value returned by the individual

chromosome i for fitness case j

Cts Percentage of chitosan (w/w %) Si Sensitivity level of an input parameter

(%)

MMT montmorillonite N (=

30)

Number of datasets used for sensitivity

test

R2 Correlation coefficient ti The measured values by models

RMSE Root mean square error pi The predicted values by models

RRSE Root relative squared error

1. Introduction

Recently, nanotechnology is a hot issue for many researches in materials science due its

administrated effect on the chemical and physical properties [1]. In this regards, the

preparation of noble metal nanoparticles, e.g. Ag, Au, Pd and Pt, are the subject of various

researches [2]. Generally, there are three methods for the preparation of nanoparticles, i.e.,

chemical, physical and biological approaches. The unique potential of chemical methods for

the preparation of finer particles as well as the narrower distribution size of products

distinguishes this technique from the others. Application of metal nanoparticles play a key

rolls in the enhancement of the performance of nanocomposites with the extensive utilities in

industrials. Silver nanoparticles (AgNPs) have extensive usages in the fields of catalysis,

optics, dentistry, photography, mirrors, clothing, food industries and electronics [3].

Advanced applications of AgNPs cause to the evolution of various techniques by the

preference of the lower size as criterion. Accordingly, the preparation of AgNPs with the

smaller size as well as the lower possibility of aggregation are favorable [1].

Other challenge of the preparation of nanoparticles is their strong tendency for

agglomeration [4]. There are various approaches for decreasing the agglomeration of

nanoparticles such as the usages of layered silicates of montmorillonite (MMT) as matrix.

MMT has two-to-one layered network including one octahedral aluminum layer between the

two layers of silicon tetrahedral. The thickness of this layer is about 1 nm and the lateral

dimension of this layer is in the range of 100–1000 nm [5]. Also, MMT has outstand

3

characteristics including the ion exchange properties, swelling and intercalation. The lamellar

structure of MMT provides the possibility of its delamination to the elemental sheets without

difficulty and usages as substrate for the synthesis of nanoparticles by electroless technique.

For example the structure of MMT employs as an appropriate substrate for the preparation of

biomaterial nanoparticles through the adsorption of cationic ions and anchoring the transition

metal complexes [6].

Chitosan (Cts), i.e., a natural polymers, has unique characteristics including non-toxicity,

solubility in aqueous medium, biodegradability, excellent biocompatibility, multi-functional

groups, artificial skin and bone substitutes [7]. Cts is added to intercalate with WMMT by the

mechanism of hydrogen bonding processes as well as cationic exchange and produces bio

nanocomposites (BNCs) with functional and structural properties [8]. BNCs made of

organic/inorganic nano size filler and polymeric matrix. To the best of our knowledge,

metal/clay/polymer compounds, e.g. BNCs is a hot issues for many researches due to their

excellent properties [7, 9-11].

Gene expression programming (GEP) is used to determine the performance and

properties of engineering materials as a mathematical expression. Respect to the classical

techniques like, artificially neural network and regression analysis GEP provides a capable

environment. Compared to the classical approach, there is not any predefined function

through the GEP modeling [12, 13].

In summary, lower particle size of AgNPs induced the synergetic effects on the

biological properties of prepared bio-nanocomposites. However, there is a general trend to

the agglomeration in the range of lower size of AgNPs. Accordingly, the main motivation

behind of this study are: (i) establishing the relationship among the practical variables

including the concentration of AgNO3, chitosan percentage, d-spacing of clay layers and

4

reaction temperature as input values and the size of AgNPs as output value; (ii) use of GEP

for modeling; (iii) determination the effect of each variable quantitatively on the size of

AgNPs; and (iv) employment of MMT as substrate to avoid the agglomeration.

2. Data collection and analysis

2.1. Data collection

Correct selection of practical variables that affected the output is the first step of GEP

modeling. Shabanzadeh et al. [14] investigated the size of AgNPs prepared in

montmorillonite/chitosan bionanocomposites. They showed that the size of AgNPs is

strongly depended on the concentration of AgNO3, chitosan percentage, d-spacing of clay

layers and reaction temperature. Our study uses 30 dataset from literature [14] as shown in

Table 1. The dataset are randomly divided to the training (20 data) and testing (10 data)

datasets [15].

Table 1.

2.2. Data analysis

Obviously, outlier data complicate the clear understanding of AgNPs dependency to the

practical parameters. Accordingly, determination and removing of outlier is beneficially for

providing of higher accuracy through the GEP modeling [16]. Box plot is a data analysis

procedure for determination of outlier during the practical dataset. Fig. 1 shows the box plot

of 4 practical parameters during the synthesis of AgNPs. As shown, the median of Cts and

MAgNO3 data sets are in the box center, indicating the symmetric distribution of these data.

However, the box plot of T and WMMT are skewed to the bottom and top, respectively. In

other words, the box shifted to the bottom whisker and most data are small with some minor

exceptionally large ones. The average of T and WMMT are higher than the median and is close

to the average of T (40) and WMMT (1.99). It is necessary to note that, there aren't any outlier

5

between the selected dataset for all practical parameters. Table 2 summarizes the statistical

explanation of dataset in this study.

Table 2.

Fig. 1.

Another key parameter during the GEP modeling is the independency of practical

parameters to each other. In this study, uses bivariate correlation analysis (BCA) to find the

relationships between the input parameters. BCA, at first, carried out a comprehensive

analysis to find the highly correlated pairs. Low negative/positive correlation between the

pairs is essentially during GEP modeling. In other words, significant interdependency among

the variables may be exaggerated the effect of input data during the modeling as multi

dependency [17]. Table 3 shows the amount of correlation coefficient of experimentally

variables. The main point of Table 3 is the presence of considerable dependency between

MAgNO3 with T, MAgNO3 with Cts, MAgNO3 with WMMT, T with Cts, T with WMMT, Cts with

WMMT.

Table 3.

Principal component analysis (PCA) is suitable approach for investigation the multi

dependency among the experimentally variables. This technique enables us to eliminate the

correlation between variables from a multi-dimensional space to a lower dimension space.

The variables in new space are uncorrelated [18-20]. Since, the presence of a higher

correlation coefficients between the variable is not essentially depicts the multi dependency

of dataset, to assure from possibility of PCA, at first Kaiser Mayer Olkin (KMO) [21] must be

estimated using equation 1 as criterion.

KMO =r

ij

2åår

ij

2åå + aij2åå

(1)

6

In the case that KMO is lower than 0.7, the dependency is unreal and must be neglected

[16]. Since, KMO factor of current study is equal to 0.516, there is no need to use PCA for

removing of multi dependency. In other words, high dependency between some variables in

Table 3 related to the nature of BCA. Accordingly, the selected practical parameters are

independent and as a consequence appropriate for further analysis by GEP.

3. Explanation of predictive equation

3.1. Gene expression programing (GEP)

Ferreira in 2011 proposes a novel approach by consideration of the population based on

the evolutionary strategy to compensate the disadvantageous of genetic algorithm (GA) and

genetic programing (GP) as gene expression programing (GEP) [22, 23]. GEP provides an

advanced strategy for the construction of inferring function among the input parameters with

the capability of forecasting the output by acceptable accuracy as well as minimum

estimation error. The main advantages of GEP are the possibility of finding the best

mathematical equation and its ability for checking the entire search space without trapping in

local minimum. In other words, GEP employed the advantages of GA and GP to provide

better modeling [24, 25]. Like GA, the individuals of GEP have fixed length string characters

(chromosomes) that can be connected each other by Karva language. This language enables

GEP to convert the coded programs to chromosomes. Generally, the coded solution can be

expressed as expression tree (ET) structure. GEP contains of chromosome with various genes

that constructs from 2 sections including tail and head (Fig. 2).

Fig. 2.

The head in Fig. 2 determines the functions and terminals, while the tail just determines

the terminals. It is necessary to note that, the terminals are input variable and constant value

while, the functions include the basic mathematically element including (*, /, +, -), boolean

operators (and, or, nor, not) and nonlinear functions (Exp, 3Rt, sqrt, arctan, tan, sin, -cos).

7

The head length, is predefined by GEP user while, the tail length, is dependent to h. nmax (the

maximum number of argument) is correlated to the function as equation 2.

t = h(n

max-1) +1

(2)

The chromosome of GEP is made of random; illustrate by a subset of sub-ET that

correlates to the other using linking function (*, /, +, -) [26-29]. As shown in Fig. 3, GEP

starts by generation of chromosomes in a random way and provide the first population as

Karva language (K-Expression). At following, each chromosomes depicted is ET and

mathematical equation, respectively. By consideration of fitness function as criterion,

evaluating the cost of individuals. The best ones selected for the next generation. If f (Si (t))

defined as fitness of individual Si in the population at t generation, by consideration of the

fitness-proportionate selection, the probability of individual Si will be copied to the next

population generation as a result of any one reproduction operation (equation 3) [30]:

f (Si(t))

f (Si(t))

j=1

M

å (3)

After the estimation of probability values of each chromosomes, the selection process

continues by selection as shown in Fig. 3. In this step, by consideration of fitness value as

criterion, the individuals select as candidate for selection. At the next phase, genetic operator

(transposition, recombination and mutation inversion) utilizes to the chromosomes to

generate the better individuals at future generation. The same process is keep on till and

suitable solution is found [31-34]. At following the genetic operators explains in summary:

I. Mutation

The main characteristics of mutation is its intrinsic modification power which are able to

perform anywhere in the chromosome length. Conversion a terminal to the other terminal is

8

performed in the tail by mutation operator while, in the head mutation provide the possibility

of the changes in functions or terminals each other [22, 30, 35].

II. Inversion

This operator is active through the chromosomes head and enables the user to reverse the

length of 1 to 3 in head.

III. Transposition

This operator has three types including: (i) insertion sequence transposition; i.e.,

responsible for the transportation of a fragment to the head of its own or other genes; (ii) root

insertion sequence transposition; i.e., responsible for the transposition of a segment with a

function in the first position of the root of the genes; (iii) gene transposition; i.e. responsible

to transpose all of genes at first chromosomes [22, 30, 35].

Fig. 3.

3.2. Training GEP as an intelligent approach for modeling and obtaining the

mathematical equation

The aim of this study is to propose a mathematical equation as GEP model (GEP-1 to

GEP-10) for prediction of the size of AgNPs. Table 4 summarizes the inferring equation of

GEP-1 to GEP-10 that have the better performance. The number of genes is supposed to eight

(sub-ETs), the multiplication and addition randomly, selects as correlating function. During

the testing and training phases of GEP models uses 10 various sub-ETs MAgNO3, T, Cts and

WMMT as input parameters and AgNPs as output value. Between 30 collected data sets, 20

trails are selected as training data set and 10 trails are chosen for testing phase. The fitness

function, fi, of an individual program, estimates from equation 4:

fi=

1

N

ti- p

i

ti

i=1

i=N

å ´100 (4)

9

In the case of the precision |C(ij)-Tj| ≤ 0.01, the precision = 0 and i = fmax = MCt. While,

for fmax = 1000, M = 100. Such approach provide the possibility of finding the optimum

condition [36, 37].

Afterwards, the set of terminals (T) and the set of functions (F) for generation of

chromosomes are preferred. The former including MAgNO3, T, Cts, and WMMT while, the later

including four basic arithmetic operators (-, +, /, *) and some basic mathematical functions (-,

+, /, *, 3Rt, Inv, Tanh, Arctan, Cos, Sin, Log, Ln, Exp, X2) are utilized. Simultaneously, the

performance of proposed GEP models are monitored during the testing and training phases.

Selection of the chromosomal tree is another step of GEP. In this study at first, initially

considered the single gene and two lengths of heads as first approximation and then the

amount of each ones are increased during each runs. Table 5 abbreviates the investigated

characteristics during the GEP modeling in current study and the size of AgNPs is a function

of MAgNO3, T, Cts and WMMT [36].

Table 4.

Table 5.

3.3. Evaluating the training performance of the model

To find the approximation performance of GEP equations, the real output and predicted

output compare by consideration of statistical performance indices including mean absolute

percentage error (MAPE), root relative squared error (RRSE), root mean square error (RMSE)

and correlation coefficient (R2) as criteria (equations 5-8) [38, 39].

MAPE =1

N

ti- p

i

tii=1

i=N

å ´100 (5)

RRSE =

(ti- p

i)2

i=1

i=N

å

ti- (

1

N) t

ii=1

i=N

å )2å

èç

å

å÷i=1

i=N

å

(6)

10

RMSE =(t

i- p

i)2

Ni=1

i=N

å (7)

R2 = 1-

(ti- p

i)2

i=1

N

å

pi

2

i=1

N

å (8)

Table 6 summarizes the values of these criteria for GEP models.

Table 6.

3.4. Sensitivity analysis

The significance of each practical parameters and relative effects on the amount of

AgNPs size may be investigated by sensitivity analysis (equation 9). Accordingly, each

parameter varies between the maximum and minimum of its level while, the other practical

variable suppose to be constant at their mean values. Then, the changes of output are

measured and uses as threshold for comparing the effect of each parameter. Typically, for

determination of WMMT changes on the size of AgNPs, all practical variables (except WMMT)

are supposed to be fixed in their mean values while, the WMMT changes [39]. To do the

sensitivity analysis a step-by-step methodology is carried out on most appropriate GEP model

by changing of input parameters, one at a time in a constant rates [15].

Si(%) =

1

N

%change inoutput

%change in input

å

èçå

å÷j=1

N

åj

(9)

4. Results and discussion

In current study, R2 (which closer to 1 is better), RMSE (which closer to 0 is better),

RRSE (which closer to 0 is better) and MAPE (which closer to 0 is better) criteria are utilized

to evaluate the accuracy of proposed GEP models. As shown in Table 6, 0.945 < R2 < 0.995;

0.073 < RMSE < 0.319; 0.039 < RRSE < 0.219 and 0.221 < MAPE < 3.039. Fig. 4 shows

the changes of selected statistical indices for 10 most appropriate models. Accordingly, the

high value of R2 belong to GEP-5 model and equal to the 0.995 and 0.985 for the training and

11

testing phases, respectively. If uses RMSE as metric for GEP validation, the lower RMSE

belongs to GEP-3 due to the closer value to zero equal to 0.067 and 0.100 during the training

and testing phases. Similar selection GEP model by utilization of RRSE reveals the GEP-3

with RRSE equal to 0.125 for training and 0.146 for testing phases has the best performances

and in the case of MAPE uses as criteria, GEP-3 selected as the most appropriate model.

Consequently, using of one statistical approach can't be satisfied simultaneously the lower

values of errors (RMSE, RRSE, MAPE) and higher values of R2. Since, to contribute all

statistical indices for the selection of most appropriate GEP models defined Fitness value as

equation 10.

Fitness - value =1/ R2 + RMSE + RRSE + MAPE (10)

Fig. 4.

In this approach, the most appropriate model has the lower fitness value (Eq. 10). Fig. 5

shows the change of fitness value for various GEP models. Accordingly, the best structure

belongs to GEP-3 with 3 genes, 8 head and 30 chromosomes. In which, the genes linked each

other using the addition function.

Fig. 5.

Figure 6 shows the expression tree of GEP-3. The inferring equation between the

practical parameters and Ag particles size of GEP-3 model in Table 6 reveals the complexity

of practical parameters on output.

Fig. 6.

Figure 7 compares the practical and the predicted values of the size of AgNPs. In which,

the experiment number is defined by the number in the circle and hexagonal and the number

in vertical axis are the Ag nanoparticles size. The values of statistical indices as well as

fitness value are added to Fig. 7. As shown, GEP-3 provides the acceptable accuracy during

the testing and training set, respectively.

12

Fig. 7.

Figure 8 shows the effect of experimentally parameters on outputs as 3D diagrams. Fig.

8(a) in which the variation of AgNPs size drawn as a function of MAgNO3 and Cts confirm the

significant effect of these parameters. Fig. 8(b), in which MAgNO3 and T select as variables,

shows the relatively higher effect of MAgNO3 respect to T on the output. The yellow and red

points inside of Fig. 8(c) shows MAgNO3 is higher effective respect to WMMT. Also, the output

value changes directly by increasing of MAgNO3 and WMMT. Analysis of Fig. 8(d) reveals that

the effect of Cts on the output is higher than T. Also, WMMT factor has higher effect on

AgNPs size respect to the T (Fig. 8(e)). Simultaneously, increasing of WMMT and Cts could

be increased the size of output (Fig. 8(f)).

Fig. 8.

Generally, the prediction of AgNPs by consideration of the effect of four variables,

including Cts, MAgNO3, WMMT and T are studied. As expected, the size of AgNPs increases by

increasing of AgNO3 concentration, temperature of reaction and amount of MMT. However,

the influence of temperature of reaction is very low on the enhancement of the size of AgNPs

and with the increases of chitosan percentages, the AgNPs size decreases. However, with the

simultaneous increases of chitosan with other parameters, the size of the AgNPs increases. As

shown in Fig. 9, the important factors in the size of AgNPs are the Cts, MAgNO3, WMMT and T

of the reaction, respectively.

Fig. 9.

5. Conclusion

In the present study, a gene expression programming is utilized to forecast the size of AgNPs

by consideration of the effect of MAgNO3, T, Cts and the WMMT as input variables. The results

confirm the GEP structure with 3 genes, 8 head and 30 chromosomes has the better behavior

during the training algorithm with acceptable performance during GEP molding of prepared

13

bionanocomposites. As a results, GEP-3 model with R2=0.977, RRSE=0.146, RMSE=0.100

and MAPE=0.221 has been suggested for the estimation of the size of AgNPs prepare at

various practical parameter. The results confirm that by increasing of the amount of Ag+ in

concentration and temperature, the size of AgNPs would be increased. While, at a higher

chitosan percentage, the size of output decreased. Also, GEP models provide a suitable

approach for the prediction of the size of AgNPs.

6. References

1. Shabanzadeh, P., Yusof, R. and Shameli, K. "Artificial neural network for modeling the

size of silver nanoparticles’ prepared in montmorillonite/starch bionanocomposites",

Journal of Industrial and Engineering Chemistry, 24, pp. 42-50 (2013).

2. Veisi, H., Azizi, S. and Mohammadi, P., "Green synthesis of the silver nanoparticles

mediated by Thymbra spicata extract and its application as a heterogeneous and

recyclable nanocatalyst for catalytic reduction of a variety of dyes in water", Journal of

Cleaner Production, 170:, pp. 1536-1543 (2016).

3. Rai, M., Yadav, A. and Gade, A., "Silver nanoparticles as a new generation of

antimicrobials". Biotechnology advances, 27(1): pp. 76-83 (2009).

4. Shameli, K., et al., "Green biosynthesis of silver nanoparticles using Callicarpa maingayi

stem bark extraction", Molecules, 17(7), pp. 8506-8517 (2012).

5. Shameli, K., et al., "Synthesis of silver/montmorillonite nanocomposites using γ-

irradiation", International journal of nanomedicine, 5, pp. 1067 (2010).

6. Makwana, D., et al., "Characterization of Agar-CMC/Ag-MMT nanocomposite and

evaluation of antibacterial and mechanical properties for packaging applications",

Arabian Journal of Chemistry, 13(1), pp. 3092-3099 (2018).

7. Lavorgna, M., et al., "MMT-supported Ag nanoparticles for chitosan nanocomposites:

structural properties and antibacterial activity", Carbohydrate polymers, 102: pp. 385-392

(2014).

8. Usman, M.S., et al., "Copper nanoparticles mediated by chitosan: synthesis and

characterization via chemical methods", Molecules, 17(12): pp. 14928-14936 (2012).

9. Shameli, K., et al., "Synthesis and characterization of silver/montmorillonite/chitosan

bionanocomposites by chemical reduction method and their antibacterial activity".

International journal of nanomedicine, 6: pp. 271 (2011).

10. Ahmad, M.B., et al., "Antibacterial effect of silver nanoparticles prepared in bipolymers

at moderate temperature", Research on Chemical Intermediates, 40(2): pp. 817-832

(2014).

https://www.sciencedirect.com/science/journal/18785352/13/1

14

11. Shameli, K., et al., "Green synthesis of silver/montmorillonite/chitosan

bionanocomposites using the UV irradiation method and evaluation of antibacterial

activity", International journal of nanomedicine, 5, pp. 5:875 (2010).

12. Jafari, M.M. and Khayati, G.R., "Prediction of hydroxyapatite crystallite size prepared by

sol–gel route: gene expression programming approach", Journal of Sol-Gel Science and

Technology, 86(1), pp. 112-125 (2018).

Ebrahimzade, H., Khayati, G.R. and Schaffie, M., "A novel predictive model for

estimation of cobalt leaching from waste Li-ion batteries: Application of genetic

programming for design", Journal of Environmental Chemical Engineering, 6(4), pp. 3999-4007 (2018)

13. Shabanzadeh, P., et al., "Prediction of silver nanoparticles’ diameter in

montmorillonite/chitosan bionanocomposites by using artificial neural networks",

Research on Chemical Intermediates, 41(5), pp. 3275-3287 (2015).

14. Khayati, G.R., "Adaptive neuro-fuzzy inference system and neural network in predicting

the size of monodisperse silica and process optimization via simulated annealing

algorithm", Journal of Ultrafine Grained and Nanostructured Materials, 51(1): pp. 43-52

(2018).

15. Faradonbeh, R.S. and Monjezi, M., "Prediction and minimization of blast-induced ground

vibration using two robust meta-heuristic algorithms", Engineering with Computers,

33(4): pp. 835-851 (2017).

16. Sun, M., "Prediction of viscosity of branched alkanes using gene expression

programing."Petroleum Science and Technology, 36(23): pp. 2049-2056 (2018).

17. Tiryaki, B., "Predicting intact rock strength for mechanical excavation using multivariate

statistics, artificial neural networks, and regression trees", Engineering Geology, 99(1-2):

pp. 51-60 (2008).

18. Sayadi, A.R., Khalesi, M.R. and Borji, M.K., "A parametric cost model for mineral

grinding mills". Minerals Engineering, 55, pp. 96-102 (2014).

19. Sayadi, A.R., Lashgari, A. and Paraszczak, J.J., "Hard-rock LHD cost estimation using

single and multiple regressions based on principal component analysis", Tunnelling and

underground space technology, 27(1), pp. 133-141 (2012).

20. Kaiser, H.F., "An index of factorial simplicity", Psychometrika, 39(1), pp. 31-36 (1974).

21. Alanni, R., et al., "New gene selection method using gene expression programing

approach on microarray data sets." International Conference on Computer and

Information Science. Springer, Cham, (2018).

22. Dikmen, E., "Gene expression programming strategy for estimation performance of LiBr–

H 2 O absorption cooling system." Neural Computing and Applications, 26(2), pp.409-

415 (2015).

23. Steeb, W.-H., "The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural

Networks, Genetic Algorithms, Fuzzy Logic with C++, Java, Symbolicc++ and Reduce

Programs". World Scientific, (2001).

https://www.sciencedirect.com/science/journal/22133437/6/4

15

24. Brownlee, J., "Clever algorithms: nature-inspired programming recipes", Jason Brownlee,

2011.

25. Monjezi, M., et al., "Modification and prediction of blast-induced ground vibrations

based on both empirical and computational techniques". Engineering with Computers,

32(4), pp. 717-728 (2016).

26. Faradonbeh, R.S., et al., "Prediction of ground vibration due to quarry blasting based on

gene expression programming: a new model for peak particle velocity prediction".

International journal of environmental science and technology, 16(6), pp. 1453-1464

(2016).

27. Khandelwal, M., et al., "A new model based on gene expression programming to estimate

air flow in a single rock joint", Environmental Earth Sciences, 75(9), pp. 739 (2016).

28. Aval, S. B., Ketabdari, H. and Gharebaghi, S. A., "Estimating shear strength of short

rectangular reinforced concrete columns using nonlinear regression and gene expression

programming", Structures, Elsevier, 12: (2017).

29. Ferreira, C., "Gene expression programming: mathematical modeling by an artificial

intelligence", Springer, 21, (2006).

30. Jafari, M. M., and Khayati, G. R., "Prediction of hydroxyapatite crystallite size prepared

by sol–gel route: gene expression programming approach." Journal of Sol-Gel Science

and Technology, 86(1), pp. 112-125 (2018).

31. Zhong, J., Feng, L., and Ong, Y. S., "Gene expression programming: A survey." IEEE

Computational Intelligence Magazine, 12(3), pp. 54-72 (2017).

32. Faradonbeh, R.S., "Monjezi, M. and Armaghani, D.J., Genetic programing and non-linear

multiple regression techniques to predict backbreak in blasting operation", Engineering

with Computers, 32(1), pp. 123-133 (2016).

33. Baykasoğlu, A., et al., "Prediction of compressive and tensile strength of limestone via

genetic programming". Expert Systems with Applications, 35(1-2), pp. 111-123 (2008).

34. Ferreira, C. and Gepsoft, U., "What is gene expression programming". (2008).

35. Sarıdemir, M., "Effect of specimen size and shape on compressive strength of concrete

containing fly ash: Application of genetic programming for design", Materials & Design

(1980-2015), 56, pp. 297-304 (2014).

36. Jafari, S., and Mahini, S. S., "Lightweight concrete design using gene expression

programing", Construction and Building Materials 139, pp. 93-100 (2017).

37. Fallahpour, A., et al., "An integrated model for green supplier selection under fuzzy

environment: application of data envelopment analysis and genetic programming

approach", Neural Computing and Applications, 27(3), pp. 707-725 (2016).

38. Khozani, Z. S., Bonakdari, S., and Ebtehaj, I., "An analysis of shear stress distribution in

circular channels with sediment deposition based on Gene Expression

Programming." International Journal of Sediment Research, 32(4), pp. 575-584 (2017).

16

39. Hoseinian, F.S., et al., "Semi-autogenous mill power model development using gene

expression programming", Powder Technology, 308, pp. 61-69 (2017).

40.

41. Elham Sarvestani is received his M.Sc. in Materials Science Engineering from Shahid Bahonar

University of Kerman (Iran) in 2020 and B.Sc. Materials Science Engineering from Shahid

Bahonar University of Kerman (Iran) in 2017.

42. Gholam Reza Khayati is currently a faculty of Materials Science and Engineering at Shahid

Bahonar University of Kerman. He received his Ph.D. in materials science from Shiraz University

of Shiraz (Iran) and M.Sc. (honor student) in Materials Science and Engineering at Department of

Materials Science & Engineering, Tehran University (Iran). He received his B.Sc. in Materials

Science from Shahid Bahonar University of Kerman, Iran in 2004. He has published more than 40

scientific papers in the field of synthesis and characterization of nanostructures materials.

Figures caption:

Fig. 1. Boxplot for MAgNO3, T, Cts and WMMT variables during the green synthesis of AgNPs.

Fig. 2. Typically, illustration of Karva language and expression tree for a chromosome with

two genes.

Fig. 3. Flowchart of GEP strategy.

Fig. 4. Comparison of validation criteria for GEP models (a) R2, (b) RMSE, (c) RRSE and (d)

MAPE for GEP-3 structure.

Fig. 5. The changes of fitness value for 10 GEP models.

Fig. 6. Expression tree of GEP-3 with 3 genes to forecast the average particle size of AgNPs.

Fig. 7. Comparison of predicted values by GEP-3 and actual data, (a) training data sets and

(b) testing data sets.

Fig. 8. Three-dimensional surfaces plots: (a) the effects of MAgNO3 and Cts (w/w%), (b) the

effects of MAgNO3 and T, (c) the effects of MAgNO3 and WMMT, (d) the effects of Cts (w/w%)

and T, (e) the effects of T and WMMT, (f) the effects of Cts (w/w%) and WMMT on size of

AgNPs.

Fig. 9. Sensitivity analysis of the practical parameter during the preparation of AgNPs.

17

Tables caption:

Table 1. Illustration of practical data for the preparation of AgNPs by green synthesis.

Table 2. Statistical distribution of MAgNO3, T, Cts and WMMT in development of GEP models.

Table 3. Correlation coefficients among MAgNO3, T, Cts and WMMT for preparation of AgNPs

by green synthesis.

Table 4. Representation of inferring equations for different GEP models.

Table 5. The most appropriate characteristics of GEP algorithm in current study.

Table 6. Various statistical criteria for most 10 appropriate GEP models.

18

Table 1. Illustration of practical data for the preparation of AgNPs by green synthesis

No. Factors Response

M AgNO3

(M)

T

(̊C)

Cts

(w/w %)

WMMT

(gr) AgNPs size (nm)

1 0.5 25 0.25 1.34 3.51

2 0.5 30 0.25 1.35 3.54

3 0.5 40 0.25 1.39 3.58

4 0.5 50 0.25 1.40 3.62

5 0.5 60 0.25 1.41 3.63

6 1.0 25 0.5 1.55 3.85

7 1.0 30 0.5 1.56 3.87

8 1.0 40 0.5 1.60 3.94

9 1.0 60 0.5 1.69 4.08

10 1.5 30 1 1.93 4.14

11 1.5 35 1 1.97 4.15

12 1.5 40 1 2.11 4.18

13 1.5 50 1 2.16 4.21

14 2.0 25 1 2.29 4.25

15 2.0 30 1.5 2.08 4.52

16 2.0 60 1.5 2.33 5.14

17 5.0 30 2 2.17 4.87

18 5.0 40 2 2.25 5.32

19 5.0 50 2 2.35 5.59

20 5.0 60 2 2.38 5.632

21 1.5 60 1 2.26 4.22

22 2 35 1.5 2.12 4.60

23 2 40 1.5 2.18 4.64

24 2 50 1.5 2.29 4.69

25 5 35 2 2.20 4.99

26 0.5 35 0.25 1.37 3.55

27 1 35 0.5 1.58 3.89

28 1 50 0.5 1.64 3.97

29 1.5 25 1 1.88 4.11

30 5 25 2 2.14 4.76

19

Table 2. Statistical distribution of MAgNO3, T, Cts and WMMT in development of GEP models

Variable Min. Max. Mean Median Std.

MAgNO3 0.50 5.00 2.00 1.50 1.61

T 25.00 60.00 40.00 37.50 12.11

Cts 0.25 2.00 1.04 1.00 0.65

WMMT 1.34 2.38 1.99 21.02 0.36

Table 3. Correlation coefficients among MAgNO3, T, Cts and WMMT for preparation of AgNPs

by green synthesis

Variable MAgNO3 T Cts WMMT

MAgNO3 1 0 0.921 0.723

T 0 1 0.033 0.193

Cts 0.921 0.033 1 0.891

WMMT 0.723 0.193 0.891 1

20

Table 4. Representation of inferring equations for different GEP models

Models Predicted equation

GEP-1

Y = ((((WMMT* MAgNO3)* T)*( 0.034*0.034))+(( 0.034*WMMT)*(

MAgNO3+Cts)))+((((Cts -WMMT )*(WMMT* 0.074))*((WMMT*WMMT )-

MAgNO3))- T)+((MAgNO3/(((-2.33)+Cts)* T))-(((-2.33)- WMMT )- T))

GEP-2

Y = ((((MAgNO3-Cts)*(0.006*T))+((0.006/Cts )+ WMMT))/WMMT

)*((((WMMT +WMMT )*Cts)/((-0.446)-WMMT))+((WMMT+MAgNO3)/(WMMT*(-

0.446))))*((((0.587-Cts)*(0.587/T))/((MAgNO3/0.587)-WMMT))+0.587) *

((WMMT + WMMT)/(((WMMT /T)-(-0.220))-(MAgNO3+WMMT)))

GEP-3

Y = (((((((((((-0.126)-Cts)+WMMT)*MAgNO3)*

MAgNO3)))^(1/3))))^(1/3))+(1.0/((((WMMT+(-0.830))-

tanh(MAgNO3))+(WMMT+Cts)/(((T))^(1/3))))))+(d(4)+((((-0.834)*Cts)-

tanh((-0.834)))*(tanh(MAgNO3)-WMMT)))

GEP-4

Y = ((Atan((T*(-0.432)))+Atan(WMMT))*((MAgNO3*(-0.432))+ (-

0.432)))+exp(Atan(((exp(MAgNO3)+(Cts/WMMT))+(0.150/Cts))))+(((-

0.007)+ MAgNO3)/(((T+WMMT)*(-0.007))*(WMMT+T)))

GEP-5

Y = (Log((sin(sin(sin(MAgNO3)))+(Log(((MAgNO3*(-

1.020))+T))/Log(10))))/Log(10))+sin((((WMMT*Cts)-((-

0.304)/WMMT))*(((-0.304)+WMMT)*sin((-

0.304)))))+(T+((sin(WMMT)+(WMMT-T))+((0.891*Cts)+0.891)))

GEP-6

Y = ((cos((WMMT+WMMT))^2)-((Cts*WMMT)-(0.006+0.006))) +

((WMMT^2)+(((MAgNO3/0.321)*(Cts-MAgNO3))/(T*Cts)))+(cos((cos(WMMT)-

((-0.530)^2)))-((Cts^2)*((-0.530)+(-0.530)))) +

(((cos(WMMT)/Cts)+(Cts/T))*(( WMMT-MAgNO3)*(0.200*WMMT)))

GEP-7

Y = ((((((((T))^(1/3))+(1.176-WMMT))*((1.176+1.176)+(Cts*

MAgNO3)))))^(1/3))+((((((0.607+Cts)+(MAgNO3+MAgNO3))+((0.607+

MAgNO3)+(0.607+MAgNO3)))))^(1/3))+((((MAgNO3-

WMMT)*Cts)*MAgNO3)/(((((-0.682)))^(1/3))*(T+MAgNO3)))

GEP-8

Y = ((0.520*Cts)+((((0.520-MAgNO3*Cts))))^(1/3)))+((((((((-

1.110)))^(1/3))^3)+(((MAgNO3))^(1/3)))*((((((T))^(1/3))))^(1/3)))^3)+((((((

Cts))^(1/3))/T)+( 0.333*Cts))+((((0.333))^(1/3))+(Cts+WMMT)))+(0.391-

((( MAgNO3^3)/(WMMT*T))/((Cts^3)- MAgNO3)))

GEP-9

Y = ((1.0/((((Cts-WMMT)*(T/(-0.333)))+((-0.333)-

T))))+WMMT)+(1.0/((WMMT+(((WMMT+(-0.970))/((-

0.970)+Cts))*(WMMT/T)))))+((tanh((WMMT*MAgNO3))-((Cts*(-0.317))/(

MAgNO3+(-0.317))))-(-0.317))+(((-0.185)/((T/ MAgNO3)-WMMT))-((MAgNO3*(-

0.185))-(T-T)))

GEP-10 Y = ((((((Cts*WMMT)/(((0.369))^(1/3)))+(

MAgNO3*Ln(T)))))^(1/3))+Ln(Ln((T+(((1.685^2)^2)-(

MAgNO3^2)))))+(((((((((Cts^2)-(WMMT-(-

21

0.209)))+(MAgNO3/Cts))))^(1/3))))^(1/3))

Table 5. The most appropriate characteristics of GEP algorithm in current study

GEP parameters GEP-1 to GEP-10

Number of runs 100

Parameters Value

Used functions *, /, +, -, 3Rt, Inv, Tanh, Arctan, Cos,

Sin, Log, Ln, Exp, X2

chromosomes Number 34 (GEP-2), 32 (GEP-8), 30 for other

models

size of Head 8

genes Number 4 (GEP-2), (GEP-8) and (GEP-9), 3

for other models

Linking function Addition , Multiplication

Fitness function error type Root relative squared error

Constant per gene 1

Mutation rate 0.0044

Inversion rate 0.1

One-point recombination rate 0.2

Two-point recombination rate 0.2

Gene recombination rate 0.1

Gene transportation rate 0.1

22

Table 6. Various statistical criteria for most 10 appropriate GEP models

Models Performance

indices R2 RMSE RRSE MAPE

GEP-1 Training set 0.992 0.046 0.086 0.989

Testing set 0.954 0.153 0.222 2.134

GEP-2 Training set 0.995 0.039 0.073 0.628

Testing set 0.945 0.160 0.233 2.215

GEP-3 Training set 0.985 0.067 0.125 1.238

Testing set 0.987 0.100 0.146 0.221

GEP-4 Training set 0.980 0.076 0.141 1.212

Testing set 0.953 0.168 0.244 2.092

GEP-5 Training set 0.995 0.040 0.074 0.564

Testing set 0.985 0.102 0.149 1.274

GEP-6 Training set 0.991 0.050 0.094 1.034

Testing set 0.969 0.129 0.188 2.089

GEP-7 Training set 0.990 0.052 0.097 0.916

Testing set 0.963 0.143 0.208 1.827

GEP-8 Training set 0.988 0.058 0.107 1.095

Testing set 0.982 0.113 0.164 1.554

GEP-9 Training set 0.987 0.062 0.114 1.090

Testing set 0.973 0.219 0.319 3.039

GEP-10 Training set 0.986 0.064 0.119 0.957

Testing set 0.980 0.129 0.188 1.714

23

Fig. 1. Boxplot for MAgNO3, T, Cts and WMMT variables during the green synthesis of AgNPs.

WMMT (gr)Cts (w/w%)T (̊C)MAgNO3 (M)

24

Fig. 2. Typically, illustration of Karva language and expression tree for a chromosome with

two genes.

Fig. 3. Flowchart of GEP strategy.

Sub-ET1: Sub-ET2:

(𝑎 + 𝑏) × 𝑎

𝑏

(𝑎 − 𝑏)

Gene1Gene2

Head Tail

Position

number

25

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1 2 3 4 5 6 7 8 9 10

R2

GEP Model

Train Testa)

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7 8 9 10

RM

SE

GEP Model

Train Testb)

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 10

RR

SE

GEP Model

Train Testc)

26

Fig. 4. Comparison of validation criteria for GEP models (a) R2, (b) RMSE, (c) RRSE and (d)

MAPE for GEP-3 structure.

Fig. 5. The changes of fitness value for 10 GEP models.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 2 3 4 5 6 7 8 9 10

MA

PE

GEP Model

Train Testd)

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10

Fit

nes

s V

alue

GEP Model

Train Test

27

Fig. 6. Expression tree of GEP-3 with 3 genes to forecast the average particle size of AgNPs.

Fig. 7. Comparison of predicted values by GEP-3 and actual data, (a) training data sets and

(b) testing data sets.

Sub-ET 1: Sub-ET 2: Sub-ET 3:

a) GEP- Training data

0123456

12

3

4

5

6

7

8

910

1112

13

14

15

16

17

18

1920

Target

Model

R2=0.985

RRSE=0.125

RMSE=0.067

MAPE=1.238

Fitness Value=2.445

b) GEP- Testing data

0123456

1

2

3

4

5

6

7

8

9

10

Target

Model

R2=0.987

RRSE=0.146

RMSE=0.100

MAPE=0.221

Fitness Value=1.48

28

Cts (w/w %)MAgNO3

T (̊C)MAgNO3

WMMT (gr) MAgNO3

29

Fig. 8. Three-dimensional surfaces plots: (a) the effects of MAgNO3 and Cts (w/w%), (b) the

effects of MAgNO3 and T, (c) the effects of MAgNO3 and WMMT, (d) the effects of Cts (w/w%)

T (̊C)Cts (w/w %)

WMMT (gr) T (̊C)

WMMT (gr) Cts (w/w %)

30

and T, (e) the effects of T and WMMT, (f) the effects of Cts (w/w%) and WMMT on size of

AgNPs.

Fig. 9. Sensitivity analysis of the practical parameter during the preparation of AgNPs.

0

20

40

60

80

100

120

MAgNO3 (M) T ( ̊C) Cts (w/w%) WMMT (gr)

Sen

siti

vit

y le

vel

(%

)

MAgNO3 (M)

5%-5%10%

-10%15%

-15%

WMMT (gr)