Genetic programming modelling for the electrical resistivity of Cu–Zn thin films
˙ISMAIL HAKKI KARAHAN1 ,∗and RASIM OZDEMIR2
1Physics Department, Faculty of Science, Mustafa Kemal University, Hatay, Turkey
2Kilis 7 Aralık University, Kilis, Turkey
∗Corresponding author. E-mail: ihkarahan@gmail.com
MS received 16 January 2018; revised 28 February 2018; accepted 1 March 2018; published online 1 August 2018 Abstract. Electrical resistivity measurement is an exact way to find defects in metals and alloys. Defects contribute to the residual resistivity, and determining their number is very important. Defining the inner electrical structure of an alloy is difficult, and especially it is unpredictable in alloys. This article offers a genetic programming formulation to learn how deposition conditions and alloy constituents affect the electrical resistivity of Cu–Zn alloy. Input parameters selected were: measurement temperature (K), Cu and Zn% content in the deposition bath and thin films, bath temperature, deposition potential, and the grain size of the samples. Electrical resistivity values were the output parameters. A total of 130 training and testing sets were selected. The comparative results prove the superior performance in predicting electrical resistivity of the films. The produced model proposes a close relationship for all the input parameters with the electrical resistivity property.
Keywords. Artificial intelligence; metal and metallic alloys; electrodeposition; electrical resistivity; genetic programming; computer simulation; Cu–Zn alloys; electrolyte conditions.
PACS Nos 73.61.Jc; 81.15.−z; 07.05.Mh
1. Introduction
Large application area of thin metal or alloy films in corrosion protection coatings, microelectronics, optics and catalysis has led to more interest in the electri- cal properties. To know and understand the details of the mechanism of electrical resistivity of the alloys is very important in the fields of physical and material sciences. The electrical resistivity is an easily accessi- ble and informative quantity to characterise a material.
Besides the experimental effort, artificial intelligence (AI) modelling is the most important approach to solve the details of the mechanism. The aim of AI is to gener- ate human-competitive results with a high AI ratio and to simulate it on the computer. The main role of this pro- gram is the use of information. Many AI techniques find applications in fields of materials science such as simu- lations, genetic algorithms, expert systems, fuzzy logic, adaptive network-based fuzzy inference systems, taboo search, artificial neural networks (NN), and genetic pro- gramming (GP) [1–3]. Artificial intelligence programs such as artificial neural network systems, fuzzy logic, and genetic programming are known as soft computing
[4]. Soft computing is useful to solve complex problems [5].
Koza is the first to propose GP [6]. GP is a computer program generalising a solution of a modelled problem.
GP uses the given data to solve specified problems, sim- ilar to GAs. Then the program tries to find a solution in a problem-independent manner [6,7]. There are some very interesting articles which describe the applications of GP [4].
The development of new materials will be of greater importance in future technological advances. Cu–Zn alloys can combine the beneficial properties of Cu and Zn. For many reasons, Cu–Zn alloys can be used in industry and science. These alloys have good adhe- sion to steel, higher corrosion protection, can be used in various actuators in electric appliances, electric machine and electronic instrument, pipe and aircraft hydraulic couplings, robotic muscles, microelectrome- chanical systems, automotive connectors and automo- bile applications, cellular phones, valves, antennae, and surgical tools and materials for biomedical applica- tions, electrical connectors, electric contact material for the microelectronic device, and for decorative purposes
[8–13]. The methods of preparation depend on the nature of the alloys to be produced [14–16]. Electrodeposition is the main technology for the deposition of metallic alloys. It is a low-cost and an alternative method to vacuum-required systems [17]. The electrodeposition depends on different deposition parameters [18–22].
Cu–Zn alloys are preferred to many other materi- als because of their superior electrical conductivity and lesser cost. The electrical resistivity of a thin film can be altered by varying the deposition conditions. Our pre- vious experimental studies have shown that Cu and Zn content in the electrolyte and thin film for Cu–Zn alloys [23–25], and Zn and Fe contents in the Zn–Fe alloys [26] are strictly affected by changing the deposition conditions.
The use of GP as a powerful modelling technique related to the statistical approach seems very logi- cal for predicting electrical resistivity. Over the last decades, the interest of the GP modelling in different fields of materials science has been increased [4,27].
In our previous works, two new formulations were developed for the magnetoresistance and resistivity of electrodeposited Cu–Co–Ni thin films [28], and for the electrical resistivity of Zn–Fe thin films [29] using GP. GP and NN were used for solving various scien- tific problems in the literature. Pouraliakbar et al[27]
predicted the grain size of constrained groove-pressed aluminium by using GP. Faizabadiet alstudied NN with estimated toughness and hardness by using chemical
composition and tensile properties in microalloyed line pipe steels [30]. Eskil and Kanca [31] developed a GP-based formulation for the Fe–Mn–Si shape mem- ory alloys and investigated the relationship between alternating metal content and annealing process on the martensite start temperature. Nazari and Abdinejad tried to find a new formulation of Charpy impact energy in nanocomposites [32]. Narimaniet alstudied NN with estimated corrosion current density and potential by using chemical composition and corrosion cell char- acteristics in microalloyed line pipe steels [33]. In our latest work, we studied a new formulation for predict- ing the electrical resistivity of iron-based zinc alloys by applying the NN and GP. These formulations were com- pared with each other [34].
However, there are no studies showing the relation- ship between the electrical resistivity of Cu–Zn alloys and measurement temperature (K), Zn and Cu% con- tent in the coating bath and thin films, bath temperature, deposition potential, the grain size of the thin films to predict by applying the GP for the electrodeposited Cu–
Zn alloys.
2. Experimental details
Cu–Zn alloy was deposited in a glass beaker without stirring under nonearated conditions. Table 1 shows the conditions of the deposition bath and deposition
Table 1. Bath conditions of the Cu1−xZnxalloys.
Film No. Thin films Electrolyte in materials pH Current
(mA)
Time (min)
Temperature (◦C) CuSO4·5H2O
(mol/l) ZnSO4·7H2O
(mol/l) Na3C6H5O7
(mol/l)
1 Cu17Zn83 0.06 20
2 Cu21Zn79 0.06 30
3 Cu50Zn50 0.06 0.2 0.5 5.8 60 60 50
4 Cu36Zn63 0.08 20
5 Cu52Zn48 0.10 20
Table 2. The variables used in model construction with GP.
Code Input variable Range Code Output variable Range
d0 T 100–347 (K) D.V. ρ0 0.080–0.208 (μcm)
d1 ECu 23.1–33.3 (%)
d2 EZn 66.7–76.9 (%)
d3 FCu 17.3–52.3 (%)
d4 FZn 47.7–82.3 (%)
d5 ET 20–50 (◦C)
d6 V 2.73–3.29 (V)
d7 G 66.08–100.1 (nm)
Table3.ResultsofGPformulationsvs.experimentaltrainingresults. Film No.ThinfilmsT(K)ECu(wt.%)EZn(wt.%)FCu(wt.%)FZn(wt.%)ET(◦C)V(V)G(nm)ρ0experimental results(μcm)
RsG GPρ0 RsG 1Cu17Zn8310023.176.917.382.3203.24100.100.0800.0791.018 2Cu17Zn8312023.176.917.382.3203.24100.100.0840.0831.011 3Cu17Zn8313023.176.917.382.3203.24100.100.0850.0851.010 4Cu17Zn8314023.176.917.382.3203.24100.100.0870.0861.005 5Cu17Zn8314923.176.917.382.3203.24100.100.0890.0881.004 6Cu17Zn8316923.176.917.382.3203.24100.100.0920.0920.995 7Cu17Zn8317923.176.917.382.3203.24100.100.0930.0940.990 8Cu17Zn8318923.176.917.382.3203.24100.100.0950.0960.987 9Cu17Zn8319823.176.917.382.3203.24100.100.0960.0980.983 10Cu17Zn8321823.176.917.382.3203.24100.100.1000.1020.977 11Cu17Zn8322823.176.917.382.3203.24100.100.1030.1040.991 12Cu17Zn8323823.176.917.382.3203.24100.100.1070.1061.005 13Cu17Zn8324823.176.917.382.3203.24100.100.1100.1081.014 14Cu17Zn8326723.176.917.382.3203.24100.100.1120.1121.000 15Cu17Zn8327723.176.917.382.3203.24100.100.1150.1141.004 16Cu17Zn8328723.176.917.382.3203.24100.100.1170.1161.001 17Cu17Zn8330723.176.917.382.3203.24100.100.1200.1210.997 18Cu17Zn8331723.176.917.382.3203.24100.100.1230.1231.001 19Cu17Zn8332723.176.917.382.3203.24100.100.1250.1250.999 20Cu17Zn8334723.176.917.382.3203.24100.100.1290.1300.997 21Cu36Zn6310028.671.436.563.5203.2966.090.1150.1111.036 22Cu36Zn6312028.671.436.563.5203.2966.090.1170.1161.014 23Cu36Zn6313028.671.436.563.5203.2966.090.1200.1181.012 24Cu36Zn6313928.671.436.563.5203.2966.090.1210.1201.007 25Cu36Zn6314928.671.436.563.5203.2966.090.1230.1231.001 26Cu36Zn6316928.671.436.563.5203.2966.090.1280.1280.998 27Cu36Zn6317928.671.436.563.5203.2966.090.1300.1310.995 28Cu36Zn6318928.671.436.563.5203.2966.090.1330.1340.997 29Cu36Zn6319828.671.436.563.5203.2966.090.1350.1360.994 30Cu36Zn6321828.671.436.563.5203.2966.090.1440.1411.016 31Cu36Zn6322828.671.436.563.5203.2966.090.1460.1441.015 32Cu36Zn6323828.671.436.563.5203.2966.090.1490.1471.014 33Cu36Zn6324828.671.436.563.5203.2966.090.1510.1501.008 34Cu36Zn6326728.671.436.563.5203.2966.090.1530.1550.985 35Cu36Zn6327728.671.436.563.5203.2966.090.1580.1581.000 36Cu36Zn6328728.671.436.563.5203.2966.090.1650.1611.020 37Cu36Zn6330728.671.436.563.5203.2966.090.1690.1671.008 38Cu36Zn6331628.671.436.563.5203.2966.090.1700.1700.997
Table3.Continued. Film No.ThinfilmsT(K)ECu(wt.%)EZn(wt.%)FCu(wt.%)FZn(wt.%)ET(◦ C)V(V)G(nm)ρ0experimental results(μcm)
RsG GPρ0 RsG 39Cu36Zn6332628.671.436.563.5203.2966.090.1720.1730.990 40Cu36Zn6334628.671.436.563.5203.2966.090.1750.1800.971 41Cu52Zn4810033.366.752.347.7203.2769.670.0790.0771.028 42Cu52Zn4812033.366.752.347.7203.2769.670.0820.0801.015 43Cu52Zn4813033.366.752.347.7203.2769.670.0820.0821.002 44Cu52Zn4814033.366.752.347.7203.2769.670.0830.0840.996 45Cu52Zn4814933.366.752.347.7203.2769.670.0850.0850.994 46Cu52Zn4816933.366.752.347.7203.2769.670.0880.0890.993 47Cu52Zn4817933.366.752.347.7203.2769.670.0900.0900.993 48Cu52Zn4818933.366.752.347.7203.2769.670.0910.0920.993 49Cu52Zn4819833.366.752.347.7203.2769.670.0930.0940.994 50Cu52Zn4821833.366.752.347.7203.2769.670.0960.0970.992 51Cu52Zn4822833.366.752.347.7203.2769.670.0980.0990.991 52Cu52Zn4823833.366.752.347.7203.2769.670.1000.1010.995 53Cu52Zn4824833.366.752.347.7203.2769.670.1020.1030.998 54Cu52Zn4826733.366.752.347.7203.2769.670.1060.1060.997 55Cu52Zn4827733.366.752.347.7203.2769.670.1080.1081.001 56Cu52Zn4828733.366.752.347.7203.2769.670.1100.1100.999 57Cu52Zn4830733.366.752.347.7203.2769.670.1130.1130.994 58Cu52Zn4831733.366.752.347.7203.2769.670.1140.1150.992 59Cu52Zn4832633.366.752.347.7203.2769.670.1170.1170.996 60Cu52Zn4834633.366.752.347.7203.2769.670.1210.1211.002 61Cu21Zn7910023.176.921.478.6303.0295.910.1490.1540.964 62Cu21Zn7912023.176.921.478.6303.0295.910.1530.1580.970 63Cu21Zn7913023.176.921.478.6303.0295.910.1560.1600.975 64Cu21Zn7914023.176.921.478.6303.0295.910.1580.1620.978 65Cu21Zn7914923.176.921.478.6303.0295.910.1610.1640.982 66Cu21Zn7917023.176.921.478.6303.0295.910.1650.1680.985 67Cu21Zn7918023.176.921.478.6303.0295.910.1670.1700.987 68Cu21Zn7919023.176.921.478.6303.0295.910.1700.1720.988 69Cu21Zn7919923.176.921.478.6303.0295.910.1720.1730.991 70Cu21Zn7921923.176.921.478.6303.0295.910.1770.1770.995 71Cu21Zn7922923.176.921.478.6303.0295.910.1810.1801.007 72Cu21Zn7923923.176.921.478.6303.0295.910.1820.1821.004 73Cu21Zn7924923.176.921.478.6303.0295.910.1850.1841.007 74Cu21Zn7926823.176.921.478.6303.0295.910.1900.1881.013 75Cu21Zn7927823.176.921.478.6303.0295.910.1920.1901.014 76Cu21Zn7928823.176.921.478.6303.0295.910.1940.1921.012 77Cu21Zn7930723.176.921.478.6303.0295.910.1990.1961.015
Table3.Continued. Film No.ThinfilmsT(K)ECu(wt.%)EZn(wt.%)FCu(wt.%)FZn(wt.%)ET(◦C)V(V)G(nm)ρ0experimental results(μcm)
RsG GPρ0 RsG 78Cu21Zn7931723.176.921.478.6303.0295.910.1990.1981.007 79Cu21Zn7932723.176.921.478.6303.0295.910.2050.2001.024 80Cu21Zn7934623.176.921.478.6303.0295.910.2080.2041.017 81Cu50Zn5010023.176.949.950.1502.7369.720.0950.0921.029 82Cu50Zn5012023.176.949.950.1502.7369.720.0980.0961.026 83Cu50Zn5013023.176.949.950.1502.7369.720.1000.0981.019 84Cu50Zn5014023.176.949.950.1502.7369.720.1010.0991.013 85Cu50Zn5015023.176.949.950.1502.7369.720.1020.1011.009 86Cu50Zn5017023.176.949.950.1502.7369.720.1050.1051.002 87Cu50Zn5018023.176.949.950.1502.7369.720.1070.1070.999 88Cu50Zn5019023.176.949.950.1502.7369.720.1080.1090.996 89Cu50Zn5020023.176.949.950.1502.7369.720.1100.1110.998 90Cu50Zn5021923.176.949.950.1502.7369.720.1150.1141.002 91Cu50Zn5022923.176.949.950.1502.7369.720.1150.1160.987 92Cu50Zn5023823.176.949.950.1502.7369.720.1170.1180.991 93Cu50Zn5024823.176.949.950.1502.7369.720.1200.1201.002 94Cu50Zn5026723.176.949.950.1502.7369.720.1250.1241.010 95Cu50Zn5027723.176.949.950.1502.7369.720.1260.1261.005 96Cu50Zn5028723.176.949.950.1502.7369.720.1270.1280.999 97Cu50Zn5030723.176.949.950.1502.7369.720.1300.1320.988 98Cu50Zn5031623.176.949.950.1502.7369.720.1320.1330.987 99Cu50Zn5032623.176.949.950.1502.7369.720.1340.1350.987 100Cu50Zn5034623.176.949.950.1502.7369.720.1380.1400.991
Table4.ResultsofGPformulationsvs.experimentaltestingresults. Film No.ThinfilmsT(K)ECu(wt.%)EZn(wt.%)FCu(wt.%)FZn(wt.%)ET(◦ C)V(V)G(nm)ρ0experimental results(Mcm)
RsG GPρ0 RsG 1Cu17Zn8311023.176.917.382.3203.24100.100.0810.0811.000 2Cu17Zn8315923.176.917.382.3203.24100.100.0900.0901.000 3Cu17Zn8320823.176.917.382.3203.24100.100.0980.1000.980 4Cu17Zn8325723.176.917.382.3203.24100.100.1120.1101.018 5Cu17Zn8329723.176.917.382.3203.24100.100.1200.1191.008 6Cu17Zn8333623.176.917.382.3203.24100.100.1280.1271.008 7Cu36Zn6311028.671.436.563.5203.2966.090.1160.1131.027 8Cu36Zn6315928.671.436.563.5203.2966.090.1260.1261.000 9Cu36Zn6320828.671.436.563.5203.2966.090.1370.1390.986 10Cu36Zn6325728.671.436.563.5203.2966.090.1530.1531.000 11Cu36Zn6329728.671.436.563.5203.2966.090.1680.1641.024 12Cu36Zn6333628.671.436.563.5203.2966.090.1730.1770.977 13Cu52Zn4811033.366.752.347.7203.2769.670.0810.0791.025 14Cu52Zn4815933.366.752.347.7203.2769.670.0860.0870.989 15Cu52Zn4820833.366.752.347.7203.2769.670.0950.0951.000 16Cu52Zn4825733.366.752.347.7203.2769.670.1040.1041.000 17Cu52Zn4829733.366.752.347.7203.2769.670.1110.1120.991 18Cu52Zn4833633.366.752.347.7203.2769.670.1190.1191.000 19Cu21Zn7911023.176.921.478.6303.0295.910.1510.1560.968 20Cu21Zn7915923.176.921.478.6303.0295.910.1630.1660.982 21Cu21Zn7920823.176.921.478.6303.0295.910.1740.1750.994 22Cu21Zn7925923.176.921.478.6303.0295.910.1870.1861.005 23Cu21Zn7929823.176.921.478.6303.0295.910.1960.1941.010 24Cu21Zn7933623.176.921.478.6303.0295.910.2030.2021.005 25Cu50Zn5011023.176.949.950.1502.7369.720.0970.0941.032 26Cu50Zn5016023.176.949.950.1502.7369.720.1040.1031.010 27Cu50Zn5020823.176.949.950.1502.7369.720.1130.1121.009 28Cu50Zn5025723.176.949.950.1502.7369.720.1230.1221.008 29Cu50Zn5029723.176.949.950.1502.7369.720.1290.1300.992 30Cu50Zn5033623.176.949.950.1502.7369.720.1370.1380.993
Table 5. List of function set.
Code Function set (p1)
S1 +, –, *, /,√ x,x2
S2 +, –, *, /,√
x,x2, 10x
S3 +, –, *, /,√
x,x2, exp(x)
S4 +, –, *, /,√
x,x2, x3,10x, sin(x) S5 +, –, *, /,√
x,x2,x3,10x, ln(x), sin(x), cos(x) S6 +, –, *, /,√
x,x2, pow(x,y), ln(x) S7 +, –, *, /,√
x,x2,x3, exp(x), ln(x), sin(x), cos(x) S8 +, –, *, /,x2,x3, exp(x), ln(x),√3
x, sin(x)
parameters for the Cu–Zn films. Our previously published article gives detailed procedures for preparing Cu–Zn films [24,25]. Properties of genetic program- ming are explained in detail in our previous papers [34].
It was aimed to be in the best fitness zone, to minimise errors in training and testing steps, to reduce complexity, and to ensure that the predicted result is the best [27].
3. Application of genetic programming
Data used for this modelling were obtained from our own experimental results. The most important task in this stage is to ascertain the relationship between the hidden function connecting the input variables (d0,d1, d2,d3,d4,d5,d6, andd7) and output D.V. The follow- ing equations show the empirical models as a function of the experimental conditions:
D.V.= f(d0,d1,d2,d3,d4,d5,d6,d7), (1) ρ0 = f(T,ECu,EZn,FCu,FZn,ET,V,G), (2) where T is the temperature in the film (K), ECuis the electrolyte Cu content (%), EZn is the electrolyte Zn content (%),FCuis the % Cu content in the film,FZnis the % Zn content in the film,ETis the temperature in the electrolyte (◦),V is the applied circuit voltage between the anode and the cathode,G is the grain size (nm),ρ0
is the resistivity.
The formulas obtained by GP will be used for esti- mating the relationship between film components and electrical resistivity properties of Cu1−xZnxalloys. The variables used in the GP models are presented in table2.
The testing and training databases are obtained as two sets. The testing and training sets are selected without prior planning and created from 30 and 100 mixtures of all 130 alloys, respectively. The results obtained from GP are valid between the ranges given in tables3and4, respectively.
4. GP formulations
Previously obtained experimental results were used to create tool parameters as function list and all stud- ied combinations. Used toolbox for implementing the formulate the given data were mentioned in our previ- ous study [5]. Table5presents the GP parameters and table6gives the function list and all studied alternatives.
Innately selected sets are used for the performance of the program. The best equations for the prediction of elec- trical resistivity are as follows:
U1=(−7)+
√T EZn+FZn
V2+ FGCu (3)
U2=(−7)+ FZn G−T
FCu−V + E7Zn (4)
U3=(−4)+ V
V2−ET2 +4FZn+ECu
(5)
U4= T
(FCu+G−√
T)2 +9 (6)
U5=9+
FCu G·9 FCu−9−FZn
FZn (7)
G=U1+U2+U3+U4+U5 (8) G =
√T EZn+FZn
V2+ FGCu + FZn G−T
FCu−V + E7Zn
+ V
V2−ET2
+4FZn+ECu
+ T
(FCu+G−√ T)2 +
FCu
G·9 FCu−9−FZn
FZn . (9) Equation (9) presents the final formula to estimate the electrical resistivity predictions of electrodeposited alloys for the best solutions by GP algorithm.
Table 6. The best results obtained from the GP tests.
P1 P2 P3 P4 P5 P6 P7 R2error
Training data Test data
S1 50 10 5 322984 Addition RRSE 0.989 0.989
S1 50 10 5 55479 Addition RRSE 0.989 0.987
S1 50 10 5 165543 Addition RRSE 0.997 0.996
S1 60 10 6 236871 Addition RRSE 0.988 0.988
S1 50 12 5 705778 Addition RRSE 0.991 0.991
S1 60 10 5 145688 Multiplication RRSE 0.927 0.928
S1 50 10 4 116281 Addition RRSE 0.988 0.985
S1 50 10 5 99862 Division RRSE 0.968 0.965
S1 50 10 5 126275 Multiplication RRSE 0.991 0.988
S1 40 8 4 176924 Addition RRSE 0.990 0.989
S1 50 8 5 172384 Subtraction RRSE 0.987 0.986
S1 50 8 3 223971 Addition RRSE 0.981 0.980
S1 40 8 5 191805 Multiplication RRSE 0.957 0.956
S1 50 8 5 226128 Addition MSE 0.975 0.972
S1 50 8 4 231599 Addition MSE 0.969 0.969
S1 50 8 5 215271 Multiplication MSE 0.920 0.911
S1 50 10 5 169168 Addition RMSE 0.992 0.990
S1 50 8 5 233665 Addition MSE 0.985 0.985
S1 50 8 5 208894 Addition RMSE 0.960 0.957
S1 50 10 5 185560 Addition rRRSE 0.985 0.985
S1 50 8 5 71015 Addition rRMSE 0.981 0.980
S1 50 10 5 185560 Addition rRRSE 0.985 0.985
S1 50 10 5 290720 Multiplication rMSE 0.990 0.987
S2 60 10 5 218621 Addition RRSE 0.992 0.992
S2 60 12 5 829560 Addition RRSE 0.982 0.980
S2 70 10 5 93838 Addition RRSE 0.990 0.989
S2 30 8 4 184405 Addition RRSE 0.992 0.991
S2 40 8 4 273363 Addition MSE 0.974 0.973
S2 40 8 4 188671 Multiplication MSE 0.973 0.971
S2 40 8 4 158959 Addition MSE 0.954 0.951
S2 40 10 4 312344 Addition MSE 0.898 0.886
S2 50 10 4 310338 Subtraction MSE 0.984 0.983
S3 50 10 5 171482 Addition RRSE 0.972 0.970
S4 50 10 5 279665 Addition RRSE 0.986 0.987
S5 50 8 5 220458 Multiplication RRSE 0.986 0.985 S5 50 9 5 326890 Multiplication RMSE 0.940 0.937 S6 50 8 5 324289 Multiplication RMSE 0.977 0.973
S7 50 10 5 212102 Addition RRSE 0.975 0.730
S7 50 8 4 109248 Multiplication MSE 0.972 0.971
S7 40 10 5 310142 Multiplication RRSE 0.978 0.976
S7 60 10 5 239051 Addition RRSE 0.982 0.981
S7 50 9 5 280598 Multiplication RMSE 0.991 0.991
S7 50 8 5 138275 Subtraction rRRSE 0.994 0.993
S8 50 8 5 340819 Multiplication MAE 0.943 0.937
S8 50 8 5 115477 Subtraction rRRSE 0.985 0.983
5. Results and discussion
Figures 1–4 show the generalisation performance of GP for the electrical resistivity of Cu–Zn alloys by comparing the experimental and the predicted values.
It is clearly observed that the formulation proposed by GP for electrical resistivity is very successful. The
experimental and predicted values of testing and training dataset are shown in figures 1 and 3, respec- tively. As shown, the predictability of GP is very successful. However, generalisation ability is the main success gauge of genetic programming. This program can accurately predict the output of unseen data and this was achieved by validating the dataset as shown in
0 5 10 15 20 25 30 35
Electrical Resisvity (μΩcm)
Experimental No Target GP Model
0.25
0.20
0.15
0.10
0.05
Figure 1. Testing evaluation of the GP method for the electrical resistivity prediction as a function of experiment number.
Electrical Resisvity (μΩcm)
Electrical Resisvity (μΩcm)
Target GP
0.25
0.20
0.15
0.10
0.05
0.05 0.10 0.15 0.20 0.25
Figure 2. Testing evaluation of the GP methods for the electrical resistivity prediction.
0 20 40 60 80 100
Electrical Resisvity (μΩcm)
Experimental No Model Target 0.25
0.20
0.15
0.10
0.05
GP
Figure 3. Training evaluation of the GP method for the electrical resistivity prediction as a function of experiment number.
figures2and4. In fact, we can say that GP has an explicit approach to the electrical resistivity problem.
This model was in accordance with the set of speci- fications for the experimental values in the training set.
However, there was a significant difference between the
GP values and the experimental values when the formula was applied to the testing set.
Statistical parameters of the formulations were given in table7, whereRis the coefficient of correlation; RAE is the relative absolute error, RSE is the relative squared
