Prediction of Sewing Thread Consumption for Over-Edge Stitches Class 500 Using Geometrical and Multi-Linear Regression Models


 Rapid and precise methods (geometrical and statistical), which aim to predict the amount of sewing thread needed to sew a garment using different over-edge stitches of class 500 (501, 503, 504, 505, 512, 514, 515, and 516), have been provided. Using a geometrical method of different over-edge stitch shapes, sewing consumption value was determined to avoid the unused stocks for each stitch type. The prediction of the sewing thread consumption relative to each investigated over-edge stitch was proposed as a function of the studied input parameters, such as material thickness, stitch density, yarn diameter, and seam width (distance between the needle and the cutter and the distance between two needles). To improve the established models using a geometrical method, a statistical method based on multi-linear regression was studied. Geometrical and statistical results were discussed, and the coefficient R2 value was determined to evaluate the accuracy of the tested methods. By comparing the estimated thread consumption with the experimental ones, we concluded that the geometrical method is more accurate than the statistical method regarding the range of R2 (from 97.00 to 98.78%), which encourages industrialists to use geometrical models to predict thread consumption.
 All studied parameters contributing to the sewing thread consumption behavior were investigated and analyzed in the experimental design of interest. It was concluded that the most important parameter affecting thread consumption is the stitch density. The material thickness and the seam width (B1) have a little impact on thread consumption values. However, the seam thread diameter has a neglected effect on thread consumption.


Abstract:
Rapid and precise methods (geometrical and statistical), which aim to predict the amount of sewing thread needed to sew a garment using different over-edge stitches of class 500 (501, 503, 504, 505, 512, 514, 515, and 516), have been provided. Using a geometrical method of different over-edge stitch shapes, sewing consumption value was determined to avoid the unused stocks for each stitch type. The prediction of the sewing thread consumption relative to each investigated over-edge stitch was proposed as a function of the studied input parameters, such as material thickness, stitch density, yarn diameter, and seam width (distance between the needle and the cutter and the distance between two needles). To improve the established models using a geometrical method, a statistical method based on multi-linear regression was studied. Geometrical and statistical results were discussed, and the coeffi cient R2 value was determined to evaluate the accuracy of the tested methods. By comparing the estimated thread consumption with the experimental ones, we concluded that the geometrical method is more accurate than the statistical method regarding the range of R2 (from 97.00 to 98.78%), which encourages industrialists to use geometrical models to predict thread consumption.
Until now there is no work in the literature aimed to develop the geometrical relationships between the overall over-edge stitch types in the same class based on the geometrical models relative to stitch geometry consumptions (not one or two or three in the same class).
In the literature, among all stitch types of class 500, only the thread consumption of over-edge stitch type 504 has been studied. However, we investigate in the present work the most used over-edge stitch types in class 500 to sew garments. Undoubtedly, denim garments and various fabric types are seamed based mostly on these studied over-edge types.
This study aims to accurately determine the amount of sewing thread required to stitch a specific length of woven fabric using

Introduction
The sewing thread consumption has been an unsolved and a complicated problem for many years, and scientists have tried to give manufacturers of sewing threads accurate results. As the sewing thread consumption problem remains a function of several parameters, assumptions, and hypotheses, researchers used different techniques (mathematical, statistical, etc.) and industrialists are still looking for reliable, efficient, and fruitful solutions. Nevertheless, sewing thread is one of the most important components of a sewn product that contributes significantly to the useful life of a product [1]. Indeed, sewing thread consumption is very interesting not only for the garment manufacturers and sewing threads suppliers but also for researchers who try to determine scientifically the suitable method to estimate consumed thread based on different techniques.
Indeed, the sewing thread consumption topic helps manufacturers to estimate their consumed thread based on stitch types consumptions before launching their seaming process.
According to the literature, an important number of studies have analyzed some woven and knitted garment problems to find an accurate method allowing them to objectively estimate the amount of sewing thread required to prepare garments [2-5]. However, the determination of the exact required amount is still difficult. Indeed, the industrial consumptions of sewing thread depend on different input parameters, such as yarn count, mass, wastage, and rate of breaks during sewing, which makes it difficult to identify the required thread amount [6,7]. Indeed, little investigation has been done to determine the relationship 2. Materials and methods

Fabric properties
Six commercial denim fabrics were chosen in our study. They have different characteristics, namely thicknesses and blend compositions (chosen to cover a wide range of thicknesses and compositions). The samples and their properties are presented in Table 1. The results presented hereby are the average of five tests for each sample.
The studied denim fabrics were prepared on a SULZER P7300 weaving loom projectile with 3/1 twill structure, whereas denim fabrics are generally made of cotton, cotton/elastane, or cotton/polyester/elastane. These specific compositions are considered in this study.

Sewing thread properties
Two commercial sewing threads, commonly used for sewing denim fabrics, were chosen. The selection of these threads was based on their linear densities, which can probably affect the used amount of sewing thread. The sewing thread linear densities were chosen according to NFG 07-117 [21]. Indeed, two "Royal" types of sewing threads were used. They are twotwisted and three-twisted polyester spun threads. The actual counts of the threads were 63.5 and 95 tex.
The seam threads' properties are shown in Table 2.

Methods used for thread consumption prediction
In this work, a geometrical method and a multi-linear regression method are considered. We tested the most useful over-edge stitch class 500 in the sewn product industry. The over-edge stitch class 500 is the class used in most types of sewn seven over-edge stitch types of class 500 (501, 503, 504, 505, 512, 514, 515, and 516). Indeed, two methods are proposed, and compared, to select the best one for the prediction of thread consumption and to help industrialists to estimate accurately the consumed amount of sewing thread. Thus, the geometrical method and a multi-linear regression method are considered.
According to the literature, many previous works used the geometrical model, regression model, fuzzy methods, and neural network model to predict the amount of sewing thread consumption. In this work, we chose geometrical and multilinear regression methods because they are the two most used methods that allow giving relevant results in the field of sewing thread consumption. In our point of view, these techniques are very useful for prediction due to their simplicity of implementation, accuracy, and relevance, especially for industrialists. Indeed, based on the literature, the same applied methods have presented their efficiency to predict and evaluate widely the sewing thread consumption for lockstitch, chain stitch, over-edge, and cover stitch types 301, 401, 504, 516, and 602, respectively. However, there is no work dealing with the geometrical and regressive relationships between overall types of stitches in the same class (based on the geometrical models relative to stitch geometry consumptions). In fact, this study underlines the originality of the present work. Referring to the published studies, among overall Over-edge Stitch types of class 500, only the thread consumption based on stitch type 516 has been studied. In addition to the originality given previously in this study, another one consists of the most used Over-edge Stitch types in class 500 to sew garments that are also investigated and predicted widely. Undoubtedly, different clothes are seamed based mostly on these studied stitch types, such as denim garments and hems of knitwear, for the laying of mesh fabric and lace underwear and lingerie.
Moreover, the effect of each input parameter has been studied and discussed to clarify to the industrialists the most significant factors to regulate before launching production. According to other parameters, such as thread tension, elongation behavior, and fabric compressibility, in this work, as in the most published ones, researchers usually do not consider the complex parameters to develop their models to facilitate their analysis. These parameters are remained constant to simplify the calculation of the consumed sewing thread based on stitch type. In the experimental analysis, all these parameters are adjusted and then, kept constant to seam overall studied fabrics. In fact, during the sewing process, the thread is fed to the sewing machine under a controlled tension, which leads to sewing thread extension. Therefore, the experimental conditions, such as thread tension and elongation behavior, are not discussed in this work despite their effect on the consumption behavior because they have been maintained constant.

Geometrical model
The stitch modeling was realized using a mathematical calculation based on geometrical shapes of over-edge stitch geometries and some assumptions to facilitate the calculation of the consumed thread. However, to represent the overall shape geometries of the over-edge stitch class 500, we used two software: 3ds Max 2014 and keyshot 5. In our case, we must indicate that we assume that the cross-section of threads is circular and incompressible in nature. Also, in all cases, the needle threads and looper threads are the same.
3.1.1. Case of over-edge stitch type 504 Figure 1A indicates the actual seam configuration and the positions of interlacing points formed by needle(s) and looper(s) threads. Figure 1B indicates a stitch configuration (a rectangular shape) of an over-edge stitch seam with a repeating unit. A unit cell of a seam has spacing. The thread length of a unit cell is assumed to be the same throughout the seam length for all the repeated units. Further elaboration of Figure 1 concluded that the stitch class 504 consists of three parts, i.e., needle thread, upper lopper thread, and lower lopper thread. Therefore, the sewing thread consumption of stitch class 504 depends on these three threads. Total thread consumed in a stitch will be the sum of needle thread consumed, upper lopper thread consumed, and lower lopper thread consumed. From the geometrical representation of each studied over-edge stitch, the length of the sewing thread was calculated for each stitch based on simple mathematical principles (sum, shape calculation using basic mathematical relationships, circular section of thread, incompressible threads, incompressible fabrics, etc.). It was assumed that the yarn is circular and incompressible in nature. The diameters of needle and looper(s) thread are equal; these are also equal to yarn present in a given fabric. The fabric is also incompressible in nature. The model is two dimensional in nature.
Equation (1) represents the total thread consumed by a stitch.
products. Its thread consumption is as high as class 400. Both stitch classes are usually used for sewing denim garments (such as trousers, shirts, jackets, knitwear, mesh fabric, and lingerie). For this reason, eight types of stitch are chosen. In this work, we try to establish models to explain consumed thread based on over-edge stitch class 500, thus making one of the originalities of the present study, because until now there is no work dealing with over all types of stitch in the same class.
Nevertheless, for the multi-linear regression, a statistical method based on an experimental design using MINITAB-17 software was applied to conduct objectively this modeling technique. Moreover, based on these multi-linear regression models, thanks to MINITAB-17 software, individual effects of input parameters were determined, analyzed, and classified. For both geometric and statistical methods, five input parameters with their levels were taken into account [material thickness (six levels), stitch density (three levels), yarn diameter (two levels), and stitch width (distance between the needle and the cutter (three levels) and the distance between two needles (two levels)]. It is highly complicated to optimize all parameter effects during the sewing process. The predicted thread consumption is still related to the stitched fabric plies or layers. In this work, all samples were sewn using two fabrics plies.
Also, he said that X is linear but here, we observe that the thread form is rather rounded (Figures 4-6). Then, With With (10) Regarding that X represents 0.8 times of SL (according to Abher) and Y is equal to SW, then Then, The amount of sewing thread 504 C needed for the 504 overedge stitch, whose geometry is represented in Figure 1, is estimated by the following formula: (1) where n504 C is the consumption of needle thread (n1), ul504 C is the consumption of upper looper thread, and ll504 C is the consumption of lower looper thread.     The amount of sewing thread 505 C needed for the 505 overedge stitch, whose geometry is represented previously in Figure 2, is estimated by the following formula: (23) where n505 C is the consumption of needle thread (n1), ul504 C is the consumption of upper looper thread, and ll504 C is the consumption of lower looper thread (Figure 9).
Where (24) Regarding that X represents 0.9 times of SL (according to Abher) and Y is equal to SW, then 3.1.2. Case of over-edge stitch type 503 The amount of sewing thread 503 C needed for the 503 overedge stitch, whose geometry is represented previously in Figure 2, is estimated by the following formula: where n505 C is the consumption of needle thread (n1) and ll514 C is the consumption of lower looper thread (Figure 7).
3.1.3. Case of over-edge stitch type 501 The amount of sewing thread 501 C needed for the 501 overedge stitch, whose geometry is represented previously in Figure 2, is estimated by the following formula: where 1 C is the consumption of looper thread (n 1 ) ( Figure 8).
Where    The amount of sewing thread 514 C needed for the 514 overedge stitch, whose geometry is represented previously in Figure 2, is estimated by the following formula: C is the consumption of needle thread (n1), n 2 C is the consumption of needle thread (n2), ul514 C is the consumption of upper looper thread, and ll514 C is the consumption of lower looper thread (Figure 11).
Regarding that ul505 C is equal to ul504 C with SW is equal to B , then: 3.1.4. Case of over-edge stitch type 512 The amount of sewing thread 512 C needed for the 512 overedge stitch, whose geometry is represented previously in Figure 2, is estimated by the following formula: where n1 C is the consumption of needle thread (n1), n 2 C is the consumption of needle thread (n 2 ), ul512   ) C was determined to improve the obtained results. The limit of the mean absolute error values used in the modeling is estimated to be 6% [23].
(40) Table 5 presents the average error for each type of over-edge stitch class 500.
The average absolute error does not exceed 3.45%. Thus, the result of testing thread consumption is widely verifi ed. Regarding the fi ndings, the error values are much lower than 6%. Thus, the studied models are well-justifi ed and the fi ndings are highly signifi cant.
The experimental and theoretical values comparison showed a good agreement between geometrical and experimental consumption values. Indeed, the coeffi cient of regression ranged from 97.00 to 98.78% ( Figure 13) refl ects signifi cant and effi cient relationships between experimental and theoretical fi ndings.
Comparisons between the theoretical and experimental consumption values for over-edge stitch types 503, 504, 516, and 514 are shown in Figure 13A-C. Experimental thread consumption (x-axis) is plotted against theoretical thread consumption (y-axis). Linear regression is applied and a line is fi tted to the data. The R 2 value of the fi tted line is 98.46, 98.42, 97, and 98.78%, which means that the difference between the two data sets is very small. The high value of R 2 verifi es the model accuracy and depicts that the model has taken all the variables into account on which the thread consumption depends. This result is in a good agreement with Abher et al.
[1] who concluded, using geometrical models, that the difference between the actual and predicted thread consumption, for 504 3.1.7. Case of over-edge stitch type 516 Regarding that the over-edge stitch type 516 is composed of the chain stitch type 401 and the over-edge stitch type 504 (Figure 4), then the amount of sewing thread 516 C needed for the 516 over-edge stitch is estimated by the following formula: To verify these models, 108 samples were prepared by varying the different factors for 503, 504, 516, and 514 over-edge stitches.
For both over-edge stitch types 503, 504, 514, and 516, the sewing machines type Siruba 503 M2-05, Siruba 504 M2-04, Siruba 514 M2-24, and Siruba 516 M2-35 over-edge stitch (speed: 2,000 rpm), respectively, =were used. Table 3 presents the different factors and levels for each type of over-edge stitch point. Table 4 presents the levels of each factor for preparing the samples of each type of over-edge stitch class 500.
The samples were prepared and a 10-cm seam length of each sample was investigated. Then, the seam was unstitched to get the needle(s) and looper(s) thread consumed in 5-cm length. After unstitching, the sewed thread length was determined to measure the value of the consumed thread per centimeter.    Regarding that, the error rate is very low and the regression coeffi cient values are very high (from 97.00 to 98.78%), it was concluded that the obtained geometrical models are fruitful and could be recommended for the prediction of thread consumption in the experimental design of interest.
As the results are well-verifi ed and validated for the consumption of the stitch 503, 504, 514, and 516, we can validate the other over-edge stitches in this class, which are 501, 505, 512, and 515.

Multi-linear regression models
To help industrialists to use the best method to predict the consumed thread to sew denim garments, the multi-linear regression method was applied to compare its results with those obtained using the geometrical method.  SST is the sum of squared distances represents the total variation in the experimental data according to these relationships: However, it is important to analyze the statistical analysis variance. Based on the p-value statistical coeffi cient, it is possible to evaluate the importance of different parameters. Indeed, three conditions are presented: • p-value is null: very signifi cant parameter.
• p-value is ranged from 0 to 0.05: signifi cant parameter.
• p-value is higher than 0.05: negligible parameter. Each parameter having a p-value higher than 0.05 is neglected. As a consequence, referring to Table 8, all parameters are not neglected except the d parameter. Then, the multi-linear regression models presented in Table 8 are considered.
Considering the range of R 2 values obtained (from 96.47 to 98.46%), it may be recognized as reliable and signifi cant. Therefore, this range of R 2 shows the effectiveness of the statistical method to determine the suitable consumption value. Moreover, to predict consumed thread using multi-linear regression models, industrialists can use statistical results for the estimation of the consumed amount. Nevertheless, compared to the geometrical modeling technique, the statistical method remained less accurate, less powerful, and less predictable regarding its range of R 2 (from 97.00 to 98.78% and from 96.47 to 98.46%, for geometrical and statistical models, respectively). In fact, from a point of view of mathematical modeling, the geometrical method takes into account some hypotheses and assumptions, such as considering sewing thread section circular, fabric incompressible, the width of stitch constant, and threads incompressible, and makes the calculation of consumed thread more precise and independent of parameters considered in the statistical technique variables and interacting with each other. Regarding regressive and geometrical methods, the geometrical technique is found more effective than the regressive method and fi tted the experimental results widely. This difference can be explained by the hypothesis used to simplify the determination of an approximate consumption value. The developed geometrical models allow a fruitful prediction of the consumed thread amount for the experimental design of interest. Indeed, the calculation of the sewing thread consumption based on different input parameters generates some errors during stitching and unstitching thread, explaining the range of the coeffi cient of regression from 96.47 to 98.46%. Hence, all measures can be subject to error taking into account wastage, reproducibility of unstitching steps, etc. For these reasons, the geometric modeling technique gives more accurate results than the statistical one. These fi ndings seem in good agreement with those in the literature [1,9].

Effects of input parameters
After determining the regression models, we determined the effect of each input parameter. Figure 14 presents the infl uence of fabric thickness, stitch density, yarn linear density, and the

Material thickness (MT) effect
Based on Figure 14A-D, it is notable that the increase of fabric thickness from 0.150 to 0.240 mm increases thread consumption by 6.63, 7.27, 9.01, and 8.16% for 503, 504, 514 and 516 over-edge stitches, respectively. Thus, fabric thickness does not have a very important effect, but it remains an infl uential factor infl uencing thread consumption. In fact, the increase in fabric thickness increases thread consumption for the three studied over-edge stitches. In addition, the increase in fabric thickness increases seam thickness as well as thread consumption. Thus, less thick fabric requires less thread, but thicker fabric requires more thread. This result is confi rmed by Jaouachi and Abher's fi nding that consumption values increase with the increase of fabric thickness. This means that the thickness of fabric samples affects the consumed thread [4,9]. This fi nding is also in accordance with the fi ndings of other researchers [24,25]. Besides, according to Jaouachi, a heavy fabric consumes more sewing thread than an average fabric. This consumption is estimated for heavy fabrics, which are thicker than the average fabrics [24]. Sharma et al. [13] proved that fabric thickness has the most important infl uence in seam consumption because the fabric thickness has a very large range (from 1.64 to 6.56 mm). However, Abher et al. [1] concluded that the contribution ratio of material thickness was around just 38%.

Yarn diameter effect
According to Figure 14A-C, Yarn linear density is not an important impact on thread consumption. Indeed, it has no impact on thread consumption of 516 over-edge stitch. Also, the increase of yarn diameter from 0.024 to 0.036 cm (relative to increase of yarn linear density from 63.5 to 95 tex) decreases thread consumption by 2.5 and 1.19% for 503 and 504 over-edge stitches, respectively, and increase by 3.73% for 514 over-edge stitch. This result is not in a good agreement with Jaouachi results [22]. In addition, if 100% PES thread composition is used instead of 100% cotton yarn, thread consumption becomes much less signifi cant. In fact, in the case of polyester thread composition, the increase in its linear density does not have an important effect on thread consumption. This may be related to the close values of linear densities used by Jaouachi (equal to 16.5 and 16.67 tex). So, this result is in accordance with the properties of each thread.

Stitch density (SPC) effect
Referring to Figure 14A-D, stitch density seems an important parameter. In fact, the increase of stitch density from 3 to 5 stitches/cm increases the thread consumption percentage of 53.71, 35.93, 48.77, and 30.86% for 503, 504, 514, and 516 over-edge stitches, respectively. Thus, the increase in stitch density increases thread consumption. Besides, the increase in the consumed amount of sewing thread results from the increase in interlacing zone numbers inside assembled fabric layers. This result is in accordance with the fi ndings of Jaouachi, Kennon, Hayes, and Abher [9,24,26,27] and confi rms our fi nding that with the fabric feed in motion, the thread is required to form the upper and lower lengths during the validity of these models because for the knitted fabric, for example, having extensibility and deformability character, other parameters must be integrated, such as thread tension, compressibility, and elongation behavior, which also has a notable infl uence on the sewing thread consumption. Hence, these input parameters must be included such infl uential factors.
Notwithstanding, to select the input parameters that are most infl uential on the consumption behavior, a classifi cation of the signifi cance using Minitab Software is applied.
The increase in each parameter increases the seam thread consumption, except in the case of the seam thread diameter that has a very low impact. The parameter that is most infl uential on the increase in thread consumption is the stitch density. The material thickness and seam width, in the range has been studied, have the lowest impact on thread consumption values. Stitch density has more impact because stitches of class 500 are made by using 1, 2, 3, 4, or 5 different threads. Each thread has its own path but the path of lopper thread is very curvy; therefore, a little increase in stitch density contributes signifi cantly toward the total thread consumption. Also, the increase in stitch width infl uences the seam thread consumption because it has the same impact of increasing stitch density.
To conclude, the proposed models can accurately predict the thread consumption based on the over-edge stitch class 500 thanks to developed geometrical relationships. Therefore, it can be used to better estimate the required thread. Further work follows. [7] Amirbayat, J., Alagha, M. J. (1993). Further studies on balance and thread consumptions of lock-stitch seams. International Journal of Clothing Science and Technology, stitch length modifi cation. Moreover, this result is in good agreement with the fi ndings of Lauriol's study [12] which proves that if the sewing length decreases from 2.5 to 2 mm (4-5 stitches/cm), thread consumption increases, approximately, by 10%. According to Sharma et al.'s [13] study, the stitch density presents the second most infl uential parameter on-seam thread consumption because the stitch density has a small range (from 2 to 3.5 stitches/cm).

References
Abher calculated the contribution ratio percentage and concluded that the contribution ratio of stitch density was around 62%. In his case, stitch density has more impact because stitch class 504 is made up of three different threads forming chain shape. Each thread has its own path but the path of lopper thread is very curvy, therefore, a little increase in stitch density contributes signifi cantly toward the total thread consumption [1].

Seam width (B 1 ) effect
According to Figure 14A-D for 503, 504, 514, and 516 overedge stitches, stitch width is an important parameter. The increase in the seam width value edge stitches, stitch width is an important parameter. The increases thread consumption. In fact, increasing 1 B from 4 to 5 mm, increased thread consumption by 9.61, 12.07, 5.67, and 10.10% for 503, 504, 514, and 516 Over edge stitches, respectively. In our case, the increase percentage value is not very important because the difference between two-width levels is not very important, which can explain the low impact of this parameter. This is because the increase in sewing width increases the stitch length. This result is confi rmed by Goldnfi ber who reported that a higher width of seam requires more thread to sew the garments [28].

Conclusion
This work deals with the prediction of the required amount of sewing thread using denim samples and over-edge stitch class 500 using more than three signifi cant input parameters and studying their effects through two methods: mathematical (geometrical technique) and statistical (multi-linear regressive technique) methods. Comparing the results showed the accuracy of the geometrical modeling method to explain the consumption behavior in each over-edge stitch shape for class 500.
In fact, using statistical analysis, multi-linear regression models were obtained and investigated. Their corresponding R 2 values ranged from 96.47 to 98.46% have shown their accuracy. Nevertheless, using the geometrical modeling technique, the R 2 values (R 2 ranged from 97.00 to 98.78%) seem more signifi cant than those given by the statistical method. This superiority encourages industrialists to apply the geometrical models for thread consumption prediction based on the overedge stitch geometry class 500.
This proposed methodology remains valid for denim fabrics and other types of fabric with characteristics similar to those of denim. For other fabric types, it was necessary to ensure