Analysis of Two Soft Computing Modeling Methodologies for Predicting Thickness Loss of Persian Hand-knotted Carpets

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1 Fibers and Polymers 2012, Vol.13, No.5, DOI /s x Analysis of Two Soft Computing Modeling Methodologies for Predicting Thickness Loss of Persian Hand-knotted Carpets A. R. Moghassem*, A. A. Gharehaghaji 1, and S. Shaikhzadeh Najar 1 Department of Textile Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran 1 Department of Textile Engineering, Amirkabir University of Technology, Tehran, Iran (Received June 18, 2011; Revised September 20, 2011; Accepted October 29, 2011) Abstract: Thickness loss caused by static and dynamic loads is one of the most important quality factors in carpets. This property is influenced by pile yarn characteristics and carpet construction parameters. Exploring the relationship between thickness loss and effective factors is highly significant to optimize the selection of the variables. Soft computing approaches, which are potent data-modeling tools in capturing complex input-output relationships, seem to be the powerful technique to decipher the above-mentioned relationship. This paper presents two different modeling methodologies for predicting thickness loss of Persian hand-knotted carpets. At first, several topologies with different architectures were used to get the best neural-network model. Since, ANN model is a black box and did not succeed in indicating inter-relationship between input and output parameters, gene expression programming (GEP) is presented here as another intelligent algorithm to predict thickness loss of the carpets. Our study showed that, GEP model has a significant priority over the ANN. The correlation coefficient (R-value) and mean square error (MSE) for GEP model were and respectively, while these parameters were and for ANN model. This indicates the desirable predictive power of GEP algorithm. Based on the proposed models, the dominant parameter on thickness loss found to be knot density and content of slipe wool. Keywords: Artificial neural network, Gene expression programming, Soft computing approach, Hand-knotted carpet, Thickness loss Introduction Hand-knotted carpet is one of the most important handicrafts and revenue creating activities in Iran [1]. Four different yarns namely, thin weft, thick weft, warp and pile from cotton, wool and cotton/polyester fibres are used to produce this kind of carpet [1,2]. Similar to other kinds of carpets, the lifetime of a handknotted carpet is determined by the deterioration in its appearance, in the course of carpet wear, during which the carpet piles are exposed to forces such as cycling loading (walking) or static loading, which may result in a reduction in the carpet thickness [3]. The loss of thickness of carpets in use is important because it affects the change in the appearance of the texture of the carpet pile [4]. Many factors such as the production methods and materials may affect this quality factor [1]. Therefore, the study on the relationship between influencing parameters and above-mentioned property is a subject of major interest to the researchers. A lot of researches have been carried out to study the effect of static and dynamic forces on the conventional carpets, but a little have been done on hand-knotted carpets [2]. In the case of conventional carpets, Ainsworth et al. [4] found a linear relation between the loss of thickness and the logarithm of the number of revolutions of the Tetrapod Walker Machine between 100 and revolutions for cut, loop and combined piles. When a carpet is used, there is *Corresponding author: armogh@yahoo.com generally a comparatively large decrease in the thickness during the first few months of wear and then a much lower, steady rate of decrease during the reminder of its life [5]. Based on the experimental study carried out by Celik et al. [6] on the thickness loss of Wilton-type carpets under dynamic loading, it was concluded that the increase in the number of impacts resulted in a decrease in the mean thickness. Acrylic-carpet structural parameters of pile height and pile density influence the carpet appearance by changing the thickness loss. Therefore, manufacturers should choose pile height and density very carefully in order to improve carpet lifetime [3]. The study carried out by Bassam [7] on Persian handknotted carpets demonstrated that, the loss of thickness under a static load were 42 %, 46 % and 56 % for the samples produced with symmetrical, asymmetrical and paired knot types. Mirjalili et al. [2] investigated the effect of static loading on the characteristics of hand-knotted Persian carpets. The results indicated that, much of the change in the thickness occurred in the early stages of compression and then slowed down to a steady state condition. They found that, the thickness of the carpet with finer wool fibres as pile yarn decreases more than the thickness of the sample with coarser fibres. Thickness reduction of a carpet produced from tanned wool is more than that of the sample woven from virgin wool [2]. Also, the rate of decrease in the thickness is inversely proportional to an increase in the number of knots per unit area [2]. Exploring the relationship between thickness loss and effective factors is highly significant to optimize the 675

2 676 Fibers and Polymers 2012, Vol.13, No.5 A. R. Moghassem et al. selection of the variables. There are some research works that focus on modeling other carpet quality characteristics. Mojabi et al. [8] use statistical approach for modeling compressional creep behavior of Persian carpets during storage under different environmental conditions. Carnaby [9], Liu et al. [10] and Hearle et al. [11] developed rheological, mathematical and computational models in the mechanics of carpet wear. Dayiary et al. [12] attempted to explore a mechanistic model of cut pile carpet compression on the basis of elastic-stored bending energy. Kimura et al. [13,14] discussed the compressive deformation of pile yarns by mechanical model. Werny [15] used statistical method to model texture retention in saxony carpets. As a nonlinear problem, predicting thickness loss of a carpet can be realized by an alternative modeling method that is soft computing approach. Among various techniques in soft computing, ANN models have been shown to provide good approximations in presence of noisy data and smaller number of experimental point and the assumptions under which ANN models work are less strict than those for other methods [16]. However, ANN models are called as black box as they simply connect the inputs and outputs without understanding any physical information about the process [16,17]. Therefore, a new soft computing approach from the family of evolutionary programming that is known as Gene Expression Programming (GEP) (Ferreira, 2001) is also a promising candidate for complex prediction problems. GEP is able to provide prediction equations without requiring a cast equation, as in the case of regression analysis [18,19]. This paper makes an attempt to use ANN and GEP for predicting thickness loss of Persian hand-knotted carpets based on the three carpet constructional variables and one experimental factor which are carpet initial thickness, content of slipe wool in pile yarn, pile density and numbers of cyclic impacts. A Brief Overview of Artificial Neural Networks An artificial neural network is an information-processing system that has certain performance characteristics in common with biological neural networks. This technique is useful when there are a large number of effective factors on the specific process [17,20]. A neural network consists of a large number of simple processing elements called neurons, units, cells, nodes, or processing elements. Each neuron receives connections from other neurons and/or itself, each with an associated weight. The interconnectivity defines the topology of the ANN. The weights represent the information being used by the neural network model to solve a problem. One of the central issues in neural network design is to utilize systematic procedures to modify the weights directly from the training data without any assumptions about the data's statistical distribution [20,21]. There are different kinds of topologies and training algorithms but the feed-forward neural network with backpropagation learning algorithms is more popular. The training of a neural network by back-propagation involves three stages: the feed-forward of the input training pattern, the calculation and back-propagation of the associated error, and the adjustment of the weights [20]. ANN has been employed extensively in various textile disciplines ranging from yarn manufacturing and fabric formation to fabric properties [22-24]. In the case of carpets, Sette et al. [25] reported the objective evaluation of carpet wear using image analysis and Kohonen self-organizing network. Van Steenlandt et al. [26] applied Fourier transform image analysis and ANN to evaluate carpet wear. Further explanation of the artificial neural network has been ignored due to extensive application of the method in textile science. A Brief Overview of Gene Expression Programming (GEP) Genetic algorithm (GA) (Goldberg, 1989) is another component of soft computing methods. This method has different domains of applications in various engineering fields [27]. Nonetheless, offshoots of GA namely, gene expression programming (Ferreira, 2001), a natural development of Genetic algorithm (GA) and Genetic programming (GP) has not received attention by researchers. Similarly to GA, GP needs only the problem to be defined. Then the program searches for a solution in a problemindependent manner. The process begins with the random generation of the chromosomes of each of the initial population. Then the chromosomes are expressed and the fitness of each individual is assessed. The individuals are then selected on the basis of fitness to reproduce with modification, leaving progeny with new train. The individuals of the new generation are subjected to the same developmental process until a solution has been found. GEP is a genetic algorithm that consists of mainly five components; the function set, terminal set, fitness function, control parameters and stop condition. In GEP, the individuals are encoded as linear strings of fixed length (the genome or chromosomes) which are afterwards expressed as non-linear entities of different sizes and shapes (simple diagram representations or expression trees (ET)). A chromosome might be modified by one or several operators at a time or not be modified at all. The advantages of GEP are: first, the chromosomes are simple entities: linear, compact, relatively small, and easy to be genetically manipulated (replicate, mutate, recombine, transpose) and second, the expression trees are exclusively the expression of the respective chromosomes. GEP genes are composed of a head and a tail. The head contains symbols that represent both functions and terminals, whereas the tail contains only terminals. For each problem, the length of the head is chosen, whereas the length of the tail is a function of the length of the head and the number of

3 Computational Models for Predicting Thickness Loss Fibers and Polymers 2012, Vol.13, No Table 1. Specifications, mean thickness loss and relative thickness loss of the carpet samples after dynamic loading C.I.T K.D Number of impacts S. no P.S.W P.W Average CV% Average CV% (%) (%) (%) (%) (14.31) 2.18 (16.51) 2.28 (17.27) 2.43 (18.40) (5.30) 1.03 (7.80) 1.21 (9.16) 1.72 (13.03) (10.00) 1.48 (11.21) 1.73 (16.10) 2.10 (15.90) (12.28) 2.01 (19.14) 2.14 (20.38) 2.31 (22.00) (4.66) 0.76 (7.23) 1.08 (10.28) 1.24 (11.80) (11.33) 1.39 (13.23) 1.56 (14.85) 2.04 (19.42) (16.30) 1.17 (18.00) 1.43 (22.00) 1.72 (26.46) (6.15) 0.65 (10.00) 0.92 (14.15) 1.12 (17.23) (7.84) 0.79 (12.15) 0.97 (14.92) 1.21 (18.61) (8.48) 1.59 (12.04) 1.78 (13.48) 2.22 (34.15) (1.89) 0.36 (2.72) 0.71 (5.38) 0.92 (6.96) (6.66) 1.36 (12.95) 1.75 (13.25) 2.19 (16.59) (5.04) 0.76 (7.23) 1.07 (10.19) 1.27 (12.09) (2.28) 0.35 (3.33) 0.69 (6.57) 0.87 (8.28) (4.66) 0.51 (4.85) 0.95 (9.04) 1.15 (10.95) (6.76) 0.67 (10.30) 0.93 (14.30) 1.16 (17.84) (3.07) 0.28 (4.30) 0.40 (6.15) 0.66 (10.15) (5.84) 0.48 (7.38) 0.73 (11.23) 1.02 (15.69) P.S.W: percentage of slipe wool in pile yarn (%), P.W: pile weight (g/10 cm 2 ), C.I.T: carpet initial thickness (mm), K.D: knot density (knots per cm), : thickness loss (mm). arguments of the function with more arguments. For a detailed explanation of GEP, refer to reference No 19, 28. Materials and Methods Carpet Samples and Experiments Eighteen different samples of hand-knotted carpet with dimensions of cm were produced. In these samples, there were three carpet construction variables namely carpet initial thickness, content of slipe wool in pile yarn and number of knots per unit area. Two-ply woollen yarns which are used as pile yarns had been prepared in another previously-done research [29]. Hasanpour (2000) selected several skins from an Iranian sheep breed called Naini. To spin three pile yarns containing 100 % virgin wool fibres, 100 % slipe wool fibres and a blend of virgin and slipe wool fibres (70/30), all the skins were divided into two groups. Wool fibres were removed from the first group of the skins by means of a sharp blade as virgin wool fibres. Besides, slipe wool fibres were provided by applying tanning solution on the second group of the skins using the traditional method. The woollen pile yarns were dyed using acid dye in the prevalent method [29]. The general specifications of the woolen pile yarns were: 2/5.45 (N m ), 200 tpm for single ring spun yarns and 90 tpm for ply yarns. The warp ends used for the carpets were cotton ring spun yarns with count of 9/16.60 (Ne) = 320 (Tex). Also, two different weft ends from cotton/polyester ring spun yarns were used to bind the knots in the carpets firmly. The counts of the thick and thin weft yarns were 40/15.21 (Ne) = 1552 (Tex) and 2/29.52 (Ne) = 40 (Tex), respectively. The Persian carpets used in this work had asymmetrical knots and between each carpet row there were two wefts involved. The numbers of knots per unit area were measured with a precision of 0.5 mm, based on Iranian Standards No.500 and No.456. The height of the pile and the initial thickness of the carpets were also measured based on Standards ISO2949 and ISO1765, respectively. To change knot density, in nine of the carpet samples pile yarn moved around two warp yarns and in other samples pile yarn moved around four warp yarns. Thus, all carpet samples are divided into two different groups based on the number of knots per cm that were cm and cm. Also, initial thickness of the samples consists of three mean values that were 16.5 mm, 13.5 mm and 9 mm (mean pile height of 13.5, 10.5 and 6.5 respectively). Specifications of the carpet samples have been illustrated in Table 1. Similar to other recent research works in which the thickness loss of carpets under dynamic loading has been studied [6], the loss of thickness was evaluated in accordance

4 678 Fibers and Polymers 2012, Vol.13, No.5 A. R. Moghassem et al. with Standard ISO2049 which is equivalent to Iranian Standard No.895. Shirley Dynamic Loading Machine was employed in the present work. The area of the 5 specimens used in the tests was mm. The initial thickness of the conditioned carpet specimens (65 % RH and 20 o C) were measured under a pressure of 2 kpa before applying dynamic loads. Then, the first dynamic loads representing 50 impacts are applied to the specimens as defined in the relevant Standards. The thicknesses of the specimens are immediately measured after this treatment. The specimens are then replaced on the dynamic loading machine again for further treatments. Thickness measurements are made and recorded at intervals of up to 1000 impacts. The thickness loss is calculated using variations between the initial thickness and those measured after the number of impacts stated. The results of the experiments (mean of thickness and relative thickness loss (%)) have been shown in Table 1. Model Inputs and Outputs The basic parameters of hand-knotted carpet formation namely, carpet initial thickness, content of spile wool in pile yarn and pile density accompanied by an experimental variable that is the numbers of cyclic impacts, constitute the input parameters of the models. In the developed models, X 1 is content of slipe wool; X 2 is carpet initial thickness; X 3 is knot density and X 4 is number of cyclic impacts. Thickness loss of the carpets after dynamic loading constitutes the output parameter of the models (Y). Modeling Methodologies Neural Network Parameters To predict thickness loss of the carpet samples, neural network models with four input units and one neuron in output layer (output unit) were designed. In this study, the five-fold cross-validation technique was used for evaluating the prediction error rate of neural network model. Therefore, the data set of 72 samples was divided randomly into 5 subsets each containing 14 or 15 samples, in accordance with other work [30]. The subsets were combined together and five sets of train and test data were designed. Each time, four subsets were used as training sets and one subset as the testing set. Consequently, each designed network was trained and tested five times. Since the objective of training is obtaining an effective generalization of the relationship between the inputs and the outputs, over-fitting of networks was prevented using the weight decay technique and mean square error regularization (MSEREG) performance function. This performance function causes the network to have smaller weights and biases. Data normalizing was carried out in such a way that they got zero mean and unit standard deviation. Some trials with different topologies were carried out. Hyperbolic tangent activation function for processing elements of first hidden layer, sigmoid activation function for processing elements of second hidden layer and linear activation function for processing elements of output neuron layer gave the best performance. One of the important parameters in the back-propagation learning algorithm is the learning rate. In this study we used the adaptive learning rate with momentum training algorithm to enhance the training performance. Learning rate and momentum rate were optimized at 0.07 and 0.9, respectively. The number of hidden neurons and the number of hidden layers are usually adjusted by trial and error. Neural networks with one hidden layer can approximate any function to an arbitrary degree of accuracy. However, some researchers have used more than one hidden layer to improve the performance of the network when there was a complex relationship between input and output parameters. Feed-forward multi-layer networks have been extensively and successfully applied to many function approximations and modeling problems. Therefore, feed-forward multi-layer networks with 4 inputs and one output units were developed in the present work. To achieve the best topology and to evaluate ANN in modeling, 11 topologies with one and two hidden layer(s) were assessed. The number of neurons in hidden layer(s) varied from 4 to 8. Training was done with the back-propagation based on scaled conjugate gradient learning algorithm (SCG). After training the networks with different sets of data, the mean square error of test and train data were measured by presenting them to the trained network. Then the average of mean square error of testing subsets was considered to achieve the best topology. The results showed that, the ANN with two hidden layers, seven processing elements into first and second hidden layers gave the best topology and the least MSE after 1000 epochs. Table 2 shows experimental values, predicted values and prediction error of testing samples related to the fifth set of data in a neural network with ( ) architecture. Model Construction and Analysis Using GEP Algorithm GEP offers new possibilities to solve more complex technological and scientific problems. GEP algorithms represent nature more faithfully, therefore can be used as computer models of natural evolutionary processes. In this concern, the experimental data shown in Table 1 and the mentioned variables were employed for modeling thickness loss of the same hand-knotted carpets. The same five sets used for evaluating ANN model, were used in a GEP algorithm. Then, the resulting five models were applied to the testing data sets. The major task is to define the hidden function connecting the input variables (X 1, X 2, X 3, X 4 ) and output variable (Y). This can be written in the form of Y = f (X 1, X 2, X 3, X 4 ). The function developed by GEP can be used to predict thickness loss of the carpets. The parameters used in the GEP algorithm are summarized in Table 3. There were many

5 Computational Models for Predicting Thickness Loss Fibers and Polymers 2012, Vol.13, No Table 2. The experimental and predicted value of thickness loss of testing samples in fifth class of training using a network with ( ) architecture and the best GEP structure S. no P.S.W C.I.T K.D N.D.I (experimental) (predicted) ANN model Absolute prediction errors Error (%) (predicted) GEP model Absolute prediction errors N.D.I: Number of Dynamic Loads. Error (%) Table 3. The parameters used in the GEP algorithm for different considered structures Parameters Models Inversion Constant per gene RNC mutation DC mutation DC Inversion DC IS transposition Number of generation Population size number of chromosomes head size Number of genes Linking function + - * * Mutation rate One-point recombination rate two-point recombination rate Gene recombination rate Gene transportation rate R-square on test data MSE on test data R-square on train data MSE on train Bold entries represent optimal setting.

6 680 Fibers and Polymers 2012, Vol.13, No.5 A. R. Moghassem et al. Table 4. Performance of ANN with ( ) architecture and GEP model with the best architecture on training and testing data sets ANN model GEP model Data set Training data Testing data Training data Testing data MSE R-value MSE R-value MSE R-value MSE R-value Ave Table 5. Comparison of MSE and R-value between ANN and GEP models Data MSE R-value ANN GEP Difference (%) ANN GEP Difference (%) For training data (89.41) (3.82) For testing data (57.05) (34.37) Table 6. Comparison between the prediction power of two models on independent testing data Training data Testing data Models R-value Prediction error MSE R-value MSE Min (%) Max (%) ANN (0.88) 2.15 ( ) GEP (1.95) 0.12 (75.00) Difference 0.33 (86.84%) 0.2 3(33.72%) 0.44 (89.80%) (51.72) - - different combinations of parameters which mean many GEP models. Since, running the GEP for all of them requires a long computational time therefore, a subset of these combinations was selected to investigate the performance of the GEP in predicting thickness loss of the carpets. After training the GEP models with different sets of data, the mean square error of test and train data were measured by presenting them to the trained model. Then the average of mean square error of testing subsets was considered to achieve the best structure. Table 3 shows the structure of the best and some of the other GEP algorithms that were used for predicting thickness loss of the samples. Also, Table 2 illustrates experimental values, predicted values and prediction error of testing samples related to fifth set of data in the best GEP structure. Results and Discussion Correlation coefficient (R-value) and mean square error (MSE) were used for evaluating and comparing the prediction performance of ANN and GEP models. The results of two algorithms for predicting training and testing data sets are shown in Table 4. The obtained results of average MSE and R-value of five subsets of testing data indicated that the performance of GEP model was better than ANN model. The differences between the MSE values of two models for predicting thickness loss of testing and training data were (57.05 %) and (89.41 %), respectively. Also, differences between the R-values of two models on testing and training data were (34.37 %) and (3.82 %), respectively. In relation to thickness loss, the maximum and minimum MSE in the ANN model for predicting testing data belonged to the second and fifth data sets, respectively, as they were 1.290, 0,025. They were and for the second and fifth data sets in the GEP model. Therefore, it is important to consider the lowest MSE in the prediction of testing data occurring in the GEP model. Table 5 shows the difference between MSE and R-value of the two models. Prediction Performance of Models The prediction performance of the models was assessed using an independent test set. A set of fourteen number of randomly selected carpet samples constituted the test set. From the prediction results, statistical performance indicators (MSE and R-value) were calculated. The results are shown in Table 6, for both training and testing data sets. Table 6 shows that, the difference between the MSE values for predicting thickness loss of testing data was 0.44 that is %. Also, the difference between the R-values for predicting thickness loss of testing data was that is

7 Computational Models for Predicting Thickness Loss Fibers and Polymers 2012, Vol.13, No Table 7. Prediction performance of independent test set carpets P.S.W C.I.T K.D N.D.I ANN model GEP model (experimental) A.P.E E% A.P.E E% (predicted) (predicted) A.P.E: Absolute Prediction Error, E%: Error %. Details of testing data have been presented in Table 7. The ability of the best ANN model to predict the testing data has been shown in Figure 1 schematically. Results reveal that, minimum and maximum prediction errors in the best ANN model are respectively 0.88 % and %. The performance of the best gene expression programming architecture on the same testing data has been shown in Figure 2. The equation (1) represents the mathematical function generated by the best structure of GEP approach. Y = log( log( (( d( 4) d( 2) ) ) + ( d( 1) d( 2) ))) Figure 1. Evaluation of the ANN model to predict the thickness loss of the carpet samples. + ( exp( ((( ) d( 1) )) + ( log( d2) ))) d( 2))) + (( d( ( 2) (( cos( (( sqrt( d( 3) ) d( 1) )) + d( 2) )) 3 )))) + ( ((( log( d( 4) ) ) atan( d( 1) )) ( d( 4) )) + (( d( 2) cos( atan( d( 1) )))) ezp( atan( log( d( 3) ))) (1) In this equation, d(1), d(2), d(3) and d(4) are equivalent to X 1, X 2, X 3 and X 4 that are inputs of the models. As it is clear in Figure 2, there was a closer match between the actual and predicted thickness loss values than ANN model. Based on the obtained results, minimum and maximum prediction error in the best GEP model were 1.95 % and %, respectively. Therefore, the GEP function was able to closely follow the trend of the actual data. All the data in Table 6 and Figures 1, 2 confirmed the excellent capability of GEP model in predicting thickness loss of the carpet samples, compared with ANN model. The better performance of GEP algorithm could be explained based on the method of optimization of its parameter which Figure 2. Evaluation of the GEP model to predict the thickness loss of the carpet samples. was based on genetic algorithm. Although applying genetic algorithm to optimize the ANN model parameters is a useful method to improve the predictive performance of ANN model, this increases the complexity of modeling process in comparison with GEP algorithm.

8 682 Fibers and Polymers 2012, Vol.13, No.5 A. R. Moghassem et al. Finally, presenting a specific mathematical equation describing the relation between dependent and independent parameters was a time consuming process, especially when there was not any clear relationship between input and output parameters. But, this case is easily obtainable by GEP algorithm. The GEP algorithm results in an equation that can be easily programmed even into a pocket calculator to use in future predictions. All of the obtained results accompanied with this benefit, demonstrate the advantages of GEP model when compared with ANN model. Analysis of the Impact of Input Parameters on Thickness Loss of the Carpet Samples The relative contribution of each of the carpet construction parameters and number of the cyclic impacts has been evaluated using two different modeling methodologies. An input significance test was conducted by eliminating one designated input from the model at each time. The model was trained again and the prediction was made from the testing data. The increase in the mean square error of prediction as compared to that parent model was considered as the indicator of importance of the eliminated input. Study showed that, knot density and content of slipe wool are the major contributing parameters to the thickness loss of the carpets in the order of descending importance. The effects of the input parameters on thickness loss have been predicted as well by ANN and GEP models. Based on the experimental data, increase in content of slipe wool fibers leads to more reduction in carpet thickness after applying compressive load. Dense carpets give less compression and thickness loss after loading. An increase in carpet thickness (pile height) improves the compressibility and decreases its elastic recovery. Consequently, thickness loss of the carpet samples increases by increase in the number of cyclic impacts. Conclusion In this study, Gene Expression Programming (GEP) algorithm, as a new intelligent methodology, was applied to obtain a predictive model of thickness loss of Persian handknotted carpets based on the three carpet constructional parameters and one experimental variable. Artificial neural network (ANN) model was also developed as a criterion to evaluate the predictive power of GEP algorithm. Our Study showed that, ANN model, with two hidden layers and seven processing elements in each of them, was the best model. The obtained results from extensive computational tests indicated the better prediction performance of the GEP model in comparison with ANN model. The difference between the MSE and R-value of two proposed models in predicting testing data was (57 %) and (34 %). GEP was found as a powerful programming algorithm in predicting thickness loss of hand-knotted carpets. Also, another advantage of GEP is its ability to explore mathematical equation that can be used during the carpet producing process. References 1. M. Kamali Dolatabadi, M. Montazer, and M. Latifi, J. Text. Inst., 96, 1 (2005). 2. S. A. Mirjalili and M. Sharzehee, J. Text. Inst., 96, 287 (2005). 3. Y. Korkmaz and S. Dalci Kocer, J. Text. Inst., 101, 236 (2010). 4. E. A. Ainsworth and G. E. Cusick, J. Text. Inst., 56, T25 (1965). 5. G. Dorothy and Clegg, L. Anderson, J. Text. Inst., 53, T347 (1962). 6. N. Celik and E. Koc, Fibers Text. East. Eur., 18, 54 (2010). 7. S. J. Bassam, A Study on the Physical and mechanical properties of symmetrical and Asymmetric knots, pp.21-22, 40-42, Research Center for the Persian Carpets (PCRC), Iran, S. A. Mojabi, S. Shaikhzadeh Najar, S. H. Hashemi, A. Rashidi, and S. J. Bassam, Fibers Text. East. Eur., 16, 57 (2008). 9. G. A. Carnaby, Text. Res. J, 51, 514 (1981). 10. H. Liu, S. K. Tandon, and E. J. Wood, Text. Res. J, 72, 954 (2002). 11. J. W. S. Hearle, H. Liu, S. K. Tandon, and E. J. Wood, J. Text. Inst., 96, 137 (2005). 12. M. Dayiary, S. Shaikhzadeh Najar, and M. Shamsi, J. Text. Inst., 100, 688 (2009). 13. K. Kimura, S. Kawabata, and H. Kawai, J. Text. Mach. Soc. Jap., 23, 67 (1970). 14. K. Kimura and S. Kawabata, J. Text. Mach. Soc. Jap., 24, 207 (1971). 15. F. Werny, Text. Res. J., 63, 194 (1993). 16. T. Chen, C. Zhang, X. Chen, and L. Li, Res. J. Text. Appl., 11, 80 (2007). 17. R. Rajamanickam, S. Hansen, and S. Jayaraman, Text. Res. J., 67, 39 (1997). 18. A. Baykasoglu, A. Oztas, and E. Ozbay, Exp. Sys. Appl., 36, 6145 (2009). 19. M. Dayik, Text. Res. J., 79, 963 (2009). 20. L. Fausett, Fundamentals of Neural Networks, Prentice Hall, New Jersey, J. C. Principe, N. R. Euliano, and W. C. Lefebvre, Neural and Adaptive Systems, John Wiley & Sons, USA, New York, R. Beltran, L. Wang, and X. Wang, Text. Res. J., 74, 757 (2004). 23. A. Majumdar, P. K. Majumdar, and B. Sarkar, Ind. J. Fiber. Text. Res., 30, 19 (2005). 24. R. Beltran, L. Wang, and X. Wang, J. Text. Inst., 97, 129 (2005).

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