Prediction of Yarn Shrinkage using Neural Nets
Abstract In this study, the shrinkages of warp and weft yarn in the finished woven fabrics can be estimated by neural net. The shrinkages of warp and weft yarn in woven fabrics are affected by various factors such as loom setting, fabric type, and the properties of warp and weft yarns, which are hard to be defined and estimated. The neural nets are used to find the relationships between the shrinkage of yarns and the cover factors of yarns and fabrics. The prediction of yarn shrinkage in the off-loomed fabrics can thus be fulfilled through a prediction model constructed with neural net.
Key words yarn shrinkage, cover factor, neural network, feature parameter
(ProQuest: … denotes formula omitted.)
There is always a difference between the set weaving densities (i.e., on-the-loom) of warp and weft yarn and the actual values for the finished woven (i.e., off-the-loom) fabric due to the shrinkage of yarns. Up to now, it has always been difficult, even for an expert, to tell what the exact weaving density of warp and weft yarn for a finished woven fabric will finally be without manufacturing the textile.
The main factors [1, 2] that affect the yarn weight consumed are the length of warp beam (L), the width of the weaving machine (Width), the weaving densities of the warp (ends/inch) and weft (picks/inch) yarn (n^sub 1^ and n^sub 2^), the yarn counts (840 yd/lb) of the warp and weft yarn (N^sub 1^ and N^sub 2^), the shrinkage ratios of the warp (S^sub 1^) and weft (S^sub 2^), respectively. The weight (W) of a roll of fabric can be obtained by summing the warp (W^sub warp^) and weft (W^sub weft^) yarns as shown in equation (1).
The development of an effective way to predict the shrinkages of warp (S^sub 1^) and weft (S^sub 2^) yarn in a finished woven fabric would help the weaving mill to produce woven fabries that exactly matched the demanded specification of weaving density for a particular order. Moreover, it is essential for a woven fabric design decision support system , developed on the basis of a genetic algorithm to be equipped with a precise prediction model for the shrinkages of warp and weft yarns to attain its accuracy.
Based on Pierce’s model of a plain weave , as illustrated as Figure 1, we can obtain an equation derived by Pierce as hip = (4/3)[radical]CA, where h = (d^sub 1^ + d^sub 2^)/2 denotes crimp height, p (= 1/n) denotes thread spacing, and CR denotes crimp ratio. By taking h = (d^sub 1^ + d^sub 2^)/2 ( where d^sub 1^ and d^sub 2^ denote the diameters of the warp and weft yarns, respectively), p = 1/n (n denotes the weaving density of warp (weft) yarn (yarns/inch), and CR = S/(1 – S) [S denotes shrinkage ratio of warp (weft) yarn] into the equation h/ p = (4/3)[radical]CR, we can obtain a reformulated equation, S = 9 x n^sup 2^ x (d^sub 1^ + d^sub 2^)^sup 2^/(64 + 9 x n^sup 2^ x (d^sub 1^ + d^sub 2^)2. Using this equation, we can estimate the shrinkage of the warp and weft yarns for each combination set of weaving parameters (n^sub 1^, n^sub 2^, N^sub 1^, and N^sub 2^). For instance, in a combination set of weaving parameters (N^sub 1^ = 15.7, N^sub 2^ = 43.2, n^sub 1^ = 32.5, n^sub 2^ = 45) the shrinkages of warp and weft can be estimated as 2.99 and 5.57%, respectively, by taking n^sub 1^ = 32.5, n^sub 2^ = 45, d^sub 1^ = 1/(28[radical]N^sub 1^) = 1/(28[radical]15.7), and d^sub 2^= 1/(28[radical]N^sub 2^) = 1/(28[radical]43.2) into the reformulated equation [i.e., S = 9 x n^sup 2^ x (d^sub 1^ + d^sub 2^)^sup 2^/(64 + 9 x n2 x (d^sub 1^ + d^sub 2^)^sup 2^)], and the shrinkage of the warp yarn and weft yarn can be estimated as 2.99% (= S^sub 1^) and 5.57% (= S^sub 2^), respectively. The calculated shrinkages of warp and weft yarn by the equation are, however, different from the practical measured values of 2.53 and 7.49% for warp and weft, respectively. Differences exist between the shrinkage values measured practically from the finished woven fabrics and those calculated using equation (1). This is due to equation (1) being derived on the basis of the cross-sections of warp and weft yarn being perfect circles with a constant diameter; the bending resistances of the yarns being assumed to be negligible and the yarn assumed to be circular in cross-section, yielding a purely geometrical model, which involves no consideration of internal forces, whereas this situation never exists in the practical weaving process. The cross-sectional shape of a yarn can be deformed to an ellipse-like shape due to the impacted stress on it during picking.
From the above-mentioned discussion, we can see that fabrics are unpredictable and vary with the weaving densities of warp (n^sub 1^) and weft (n^sub 2^) yarn, the yarn count of warp (N^sub 1^) and weft (N^sub 2^) yarn. In addition, the ingredients of warp and weft yarn, and the structural class (e.g., plain weave, twill weave, and satin weave) of the fabric also affect them. In other words, the shrinkages of the yarns are influenced by loom settings, fabric type (e.g., weave), and the properties of the warp and weft yarns (e.g., relaxation).
The relating and complex influences (i.e., n^sub 1^, n^sub 2^, N^sub 1^, N^sub 2^, weave structure) on S^sub 1^ and S^sub 2^ are simplified and defined only by the cover factor of warp (weft) yarn (i.e., C^sub 1^ (C^sub 2^)), and that of fabric (i.e., C) to represent the complicated combinations of relationships between them. A prediction model for yarn shrinkage is constructed using a neural net to learn (extract) the relationships between shrinkages and cover factors in this study.
As shown in Figure 2a, the fractional factor denoted by C [1,2,5] can be defined as the fraction of the areas covered by warp and weft yarns and can be evaluated by the formula: area covered by yarns/rectangular area.
As shown in Figure 2b the total rectangular area (i.e., rectangular area ABCD) can be calculated by p^sub 1^ x p^sub 2^, and that covered by weft yarn is d^sub 2^ x p^sub 1^. The entire area covered by the yarns can be obtained by summing the areas covered by the warp and by the weft, but the summation needs to eliminate the duplication area (i.e., rectangular area DEFG = d^sub 1^ x d^sub 2^) covered by both the warp and weft yarns. Equation (2) can thus be calculated as the value of the fractional cover of the fabric:
C = (p^sub 1^d^sub 2^ + p^sub 2^d^sub 1^ – d^sub 1^d^sub 2^)/p^sub 1^p^sub 2^, (2)
where C is the fractional coverz, p^sub 1^(in), p^sub 2^(in) are the dis-tances between two adjacent yarns in the warp and weft directions, and d^sub 1^(in), d^sub 2^(in) are the diameters of the warp and weft yarns, respectively.
According to Peirce [1, 5], the derived yarn diameter and the densities of warp and weft yarns can be obtained as d = 1/(k[radical]N), p^sub 1^ = 1//n^sub 1^ and p^sub 2^ = 1/n^sub 2^, respectively, and the fractional cover of woven fabrics can be re-formulated into equation (3):
C = n^sub 1^(k[radical]N^sub 1^) n^sub 2^/(k[radical]N^sub 2^) × N^sub 2^/ k[radical]N^sub 2^) = C^sub 1^ + C^sub 2^ × C^sub 2^ (3)
where C^sub 1^, C^sub 2^, and C are the fractional covers (cover factors) for warp yarn, weft yarn, and fabric in sequence, k (= 29.28[radical]specific gravity of yarn) is the coefficient varying with the specific gravity of yarn, n^sub 1^, n^sub 2^ are the weaving densities of the warp and weft yarns (yarns/inch) respectively, and N^sub 1^, N^sub 2^ are the yarn counts of the warp and weft yarns (840 yd/lb), respectively.
The larger the value of the fabric’s fractional cover, the more compact is the fabric’s structure. The more compact the fabric’s structure, the more crimp there is in the yarn. [1, 5] Thus, there exists a relationship between the yarn shrinkage and the cover factor.
An artificial neural network (ANN) consisting of one input layer, one output layer, and one hidden layer is used in finding the relationship between shrinkage and cover factor in this study. Its construction is illustrated in Figure 3.
“Training” is equivalent to finding proper weights for all the connections of nodes between layers such that a desired output is generated for a corresponding input. The major training steps of the back propagation algorithm are as follows [6,7].
(a) Initialize all the values of connection weight (W^sub ij^s) between node j in the upper layer and node i in the layer below.
(b) Present an input for each node i in the input layer and specify the desired output for each node in the output layer.
(c) Calculate actual outputs of all the nodes using the present value of the W^sub ij^s. The output of node j, denoted by Y^sub j^, is a nonlinear function (called a sigmoid function) of its total input:
where net^sub ij^ = ∑ Y^sub i^ W^sub ij^.
(d) Find an error term for each output node and hidden node. If d^sub j^ and Y^sub j^ stand for desired and actual values of a node, respectively, for an output node,
δ^sub j^ = (d^sub j^-Y^sub j^)Y^sub j^(1 – Y^sub j^) (5)
and for a hidden layer node,
δ^sub j^ = Y^sub j^(1-Y^sub j^)∑δ^sub j^W^sub ij^ (6)
where k is over all nodes in the layer above node j. (e) Update weights by
W^sub ij^(t + 1) = W^sub ij^(t) + αδY^sub i^ + γ (W^sub ij^(t) – W^sub ij^(t -1)) (7)
where W^sub ij^(t) stands for connection weight value between node j in the upper layer and node i in the layer below, α is a learning rate, and the momentum factor γ is a constant between O and 1.
(f) Return to step (b) to present another new input for each node until all the training sets have been learned and the weights have stabilized.
Using known data series for the input and output parameters, a training rule is applied to calculate the weight factors in each neuron to obtain the best correspondence between the known values for the output parameters and those calculated with the neural network. Training a neural net is a time-consuming job. Once the final parameters have been selected, however, the output values for given input values can be calculated within seconds.
Samples Acquiring and Yarn Shrinkage Measuring
As mentioned above, the shrinkage of yarns is affected by the related and complex influences among the n^sub 1^, n^sub 2^, N^sub 1^, N^sub 2^, parameters of the weave structure, and the properties of the warp and weft yarns. For simplification, the weaving machine, the weave structure, and material yarn were set as air jet loom, plain weave, and T/C warp (weft) yarn, respectively, in this study.
There were 26 sample woven fabrics manufactured by air jet loom in this study. The characteristics of the material yarns used for weaving are illustrated in Table 1. There was only one kind of material yarn, i.e., T/C, used for the experiment. The material warp yarn was chosen to be one kind of yarn count 15.7’S. There are various kinds of yarn counts for material weft yarns, e.g., 10.8’S, 19.3’S, 29.9’S, and 43.2’S. The weaving density (on-the-loom) for warp and weft yarn is set at the range of 45-95 and 20-45, respectively, during weaving.
In order to proceed with sampling in a more objective way, each sample was selected from a different area (i.e., the left, the middle, and the right) of a manufactured woven fabric. The selected sample fabric was cut to a size of 20 cm × 20 cm.
The shrinkage (off-the-loom) (i.e., S^sub 1^) of warp yarn and that (i.e., S^sub 2^) of weft yarn were calculated by using equation (8). The measuring of yarn shrinkage is performed according to Chinese National Standard. During measuring the length of the yarn unraveled from sample fabric (i.e., with a size of 20 cm × 20 cm), each yarn was hung with a loading of 346/N (g), where N is the yarn count (840 yds/lb) of the yarn for testing.
S = (L-L’)/L (8)
where L denotes the measured length of the warp (weft) yarn, L’ denotes the length of the fabric in the warp (weft) direction.
Selecting Feature Parameters
Woven fabric consists of warp and weft yarn interlaced with each other. In the unit cell geometry model developed by Peirce , he established seven related equations with eleven parameters. The solution of Peirce’s equations was simplified using graphical techniques by Painter , and later by Adams et al. . The simplified monograph  represents the seven relationships. As can be imagined, it is difficult to find the exact relationship between the yarn shrinkage and the yarn material ingredients, the weaving densities of warp and weft yarn, and the yarn counts of warp and weft.
The status of a piece of fabric is effectively balanced among various directions. Regarding the shrinkage of warp and weft, we simplify the problem with a strong-weak relationship between warp and weft directions instead of a complex problem of stress analysis and deformation mechanism. It can be expected that the stronger the warp (or the weft), the weaker the weft (or the warp). It is the only way that a fabric can keep balanced. Using only the cover factor of the fabric (i.e., C) as the input layer for the neural net to learn, we can expect this to come up with bad testing results. For instance, two pieces of fabrics are of the same cover factor value without necessary being of the same cover factor value of warp or weft. If the value of warp is larger than that of weft, there exists bigger crimp for weft than warp. This happens probably due to different properties (e.g., strength, elongation, relaxation, etc.) between the warp and weft yarn directions in the woven fabric. By only using single cover factor of fabric it is not possible to discriminate the strong-weak relationship between warp and weft direction in the woven fabric. The degree of success of a prediction model heavily depends on how adequate, general, and compact is the representation of each sample.
To promote the correct predicting rate in yarn shrinkage, apart from the cover factor (i.e., C) of the fabric, there are two more feature parameters the cover factor (i.e., C^sub 1^) of the warp yarn, and the one (i.e., C^sub 2^) of the weft yarn are included as a feature vector to be learned by the ANN. In this study, we included an additional two cover factors of the warp (i.e., C^sub 1^) and weft (i.e., C^sub 2^) yarns as the input of the input layer for the net to learn by ANN. These three factors (i.e., C^sub 1^, C^sub 2^, and C), which has a statistically significant effect  on the shrinkages (i.e., S^sub 1^ and S^sub 2^) of the warp and weft yarn under a 95% confidence level. With the cover factor of warp (i.e., C^sub 1^) and that of weft (i.e., C^sub 2^), the strong-weak relationship between warp and weft direction in a specific woven fabric can therefore be effectively expressed.
Results and Discussion
Model Construction and ANN Training
There are three factors, i.e., warp yarn cover factor C^sub 1^, weft yarn cover factor C^sub 2^, and fabric cover factor C included as input parameters for the neural network training. From the 26 sample fabrics manufactured, we gathered experimental data. They were randomly partitioned into two sets as shown in Table 2. One half (i.e., 13 samples) was used for training, the other half (i.e., 13 samples) for testing.
In this study, we used a back propagation neural net [6, 7], which consisted of three input nodes, one hidden layers of 16 nodes, and two output nodes that gave the calculated evaluations of the yarn shrinkage for warp and weft. Learning rate and momentum factor are 0.9 and 0.7, respectively. We chose a sigmoid function given by equation (4) as the transfer function for all nodes. The neural net is built on a personal computer with a P4 3.0 G processor using the Neural Works software package.
Table 3 shows the overall mean errors for the ANN during training and testing. There was a continuous decrease of the deviation of known and calculated outputs in accordance with the increase in the number of learning cycles. The number of learning cycles, during which the weights were adapted, denotes the number of times that the data points passed through the network. The obtained networks were tested with the test set. There were minimal mean errors of 0.0090 (i.e., 0.90%) and 0.0059 (i.e., 0.59%) for the shrinkage ratios of warp (i.e., S^sub 1^) and weft (i.e., S^sub 2^) yarns, respectively, at 160000 learning cycles. Figure 4 shows the simulation results for the first 20 000 iteration process during ANN training. As can be seen from Figure 4, there is a steadily convergent trend proceeding alongside the iterations.
Accuracy and Efficiency of Prediction
The way to measure the regression ability of an equation to predict is to compare the standard error of estimate, S^sub e^ (= ∑ (y- Y’)2, where Y denotes a predicted value of Y), to the standard deviation of the response variable, S^sub Y^ (= ∑ (Y – …)2, where y is the mean of variable Y). The idea is that S^sub e^ is (essentially) of the residuals that we would obtain from a horizontal regression line at height Y, the response variable’s mean. Therefore, if S^sub e^ is small compared to SY (i.e., if S^sub e^/S^sub Y^ is small), then the regression line has evidently done a good job in explaining the variation of the response variable. The ^sup R^2 measure  can be obtained as (1 – S^sub e^/S^sub Y^), which indicates that when the residuals are small, then the R^sup 2^ will be close to 1, but when they are large, R^sup 2^ will be close to 0.
Figures 5 and 6 show the test results for warp yarn and weft yarn. There was a close match between the actual and predicted shrinkage of both warp and weft yarn. The test result yielded R^sup 2^ values of 0.8509 and 0.8723 one for the shrinkage of warp (i.e., S^sub 1^) and weft (i.e., S^sub 2^) respectively. According to the large R^sup 2^ values obtained for both the shrinkage of warp (= 0.8509) and weft (= 0.8723) yarn in the test results, the ANN prediction model was shown able to closely follow the trend of the actual data.
In this study, we have presented a neural net model for predicting shrinkage of warp and weft yarn using the results of cover factors of warp, weft, and fabric as input. The neural net was trained with 13 experimental data points. A test on 13 data points showed that the mean errors between the known output values and the output values calculated using the neural net were only 0.0090 and 0.0059 for the shrinkage ratio of warp (S^sub 1^) and weft (S^sub 2^) yarn, respectively. Furthermore, there was a close match between the actual and predicted shrinkage of the warp (weft) yarn. The test results gave R^sup 2^ values of 0.85 and.87 for the shrinkage of the warp (i.e., S^sub 1^) and weft (i.e., S^sup 2^), respectively. This showed that the neural net produced good results for predicting the shrinkage of yarns in woven fabrics. We are trying to experiment with woven fabric manufactured on other different looms (e.g., rapier, gripper, etc.), raw material yarn ingredients (e.g., T/C × T/R, T/R × T/R, T/C × C, etc.), and fabric structural class (e.g., twill, satin, etc.) to measure the shrinkage ratio of warp and weft yarns. Then, using the developed neural net model to train the obtained data as demonstrated in this study, it is expected that the ANN can show good results for predicting each specific combination of weaving parameters.
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Department of Information Management, Vanung University, Chung-Li, Tao-Yuan, Taiwan, R.O.C.
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