Optimization of durability of Persian hand-knotted wool carpets by using desirability functions

Abstract

Lower abrasion loss and compression, as well as higher compression recovery, are desirable for obtaining maximum carpet durability. Hand-made carpet manufacturers need to satisfy consumers on these diverse durability requirements. Therefore, it is necessary to optimize the carpet durability by considering several objectives simultaneously. In this investigation, carpet durability has been optimized by considering abrasion loss, compression and compression recovery simultaneously. The desirability function approach has been used to combine multiple objectives into a single objective. Knot density, pile height, number of plies in the pile yarn and pile yarn twist have been considered as the four independent variables. The optimum desirability of carpet durability was found to be 0.8. The validation sample also showed good agreement (error < 5%) with the optimized values of carpet durability attributes.

Keywords

abrasion resistance carpet durability compression compression recovery desirability function

Carpet is a three-dimensional home textile product, used as a covering on floors, walls and ceilings. Carpets are classified under two categories: machine-made and hand-made. Machine-made carpet manufacturing systems include tufting, weaving, knitting, braiding, needle felting (needle punching), fusion bonding and flocking. On the other hand, hand-made carpets are of three types, namely knotted, flat woven and tufted. Knotting is an extensively used method for hand-made carpet manufacturing. The texture of hand-knotted carpets is produced by independent knots. Persian or Sehna, Tibetan, Turkish or Ghiordes, Spanish and Kiwi knots are generally used in the hand-made carpet industry. Among these, the Persian knot is most popularly used in the hand-made carpet sector of India.^1–4 The purchase decision-making of hand-made carpets is primarily influenced by aesthetic appeal, such as color, design and texture. However, consumers always try to maintain a balance between aesthetic appeal and durability of hand-made carpets. Carpet durability is defined as the wear performance of a carpet in given conditions. Fiber qualities, yarn parameters, carpet constructional parameters, production of carpet, etc., have a strong influence on carpet durability.⁵ Carpet durability is a multi-criteria phenomenon; thus, it requires simultaneous fulfillment of several characteristics, such as carpet abrasion and compression behaviors. Carpet compression behaviors (compression and compression recovery) signify the wear performance of a carpet at the early stages of life. In contrast, the carpet abrasion behavior determines the wear performance of a carpet at the later stages of life.

Gupta et al.⁶ reviewed the various aspects of carpet durability. They explained the carpet durability mechanism and various techniques for measuring carpet durability attributes. Coarse wool with a low clean fleece percentage and high medullated fiber is used in carpet production.⁷ The influence of wool fiber, yarn and carpet structural parameters on durability properties was also presented. Shakyawar et al.⁸ evaluated a large number of carpet samples for Carpet Aesthetic Value (CAV) and Carpet Hand Value (CHV) subjectively by 10 different judges. They reported that carpets made from synthetic fibers were ranked poorly as compared to wool carpets, because synthetic fibers do not have the desirable attributes for carpet construction. Ishtiaque et al.^9,10 engineered the carpet yarns to purposefully position different characteristics of fibers across the yarn section using the modified SIRO spinning technique on a worsted spinning system. Carpet resiliency was found to be better for carpets manufactured from engineered yarns. Dayiary et al.^11,12 investigated the behavior of pile yarns under a compressive load. They presented the mechanism of pile deformation under compression and derived the total energy of pile deformation in terms of nonlinear bending and frictional energies.

Several researchers have developed theoretical models to predict carpet wear. Carnaby¹³ reported that the measurement of carpet thickness was the main characteristic of carpet wear. The various factors responsible for carpet thickness loss were frictional slippage, viscoelastic-plastic behavior and abrasion loss. Liu et al.^14,15 developed mathematical models for predicting the wear of cut and loop pile carpets. The life distribution function proposed by Carnaby was modified to obtain a direct measurement of model parameters. Hearle et al.¹⁶ developed a computational model of wool carpet wear. They compared this computational model with analytical models and found that the former resulted in better agreement with the experimental observations. In a recent research, the abrasion resistance of Persian hand-made wool carpets was modeled with the help of an Artificial Neural Network (ANN) by using four input parameters, namely knot density, pile height, number of plies in the pile yarn and pile yarn twist.¹⁷

It is well-known that lower abrasion loss and compression, as well as higher compression recovery, are desirable for obtaining maximum carpet durability. However, obtaining all these desirable characteristics simultaneously is often not possible and, therefore, optimization should be done to get an acceptable solution. Problems that seek to maximize or minimize a mathematical function, that is, objective, subject to certain constraints, are called optimization problems.¹⁸ If the problem contains multiple objectives, then it is called a multi-objective optimization problem. Goal programming is a very popular method for solving multi-objective optimization problems by adding deviational variables.^19,20 The desirability function approach is also used to solve multi-objective optimization problems by combining multiple objectives into a single objective.^21–23 The use of desirability functions for predicting the global quality of knitted fabrics has been reported by some researchers. Taieb et al.²⁴ calculated the global quality index of five knitted fabrics by using a desirability function. They concluded that the desirability values depend on the objective of each property, acceptance intervals, requirement of the consumer and his or her taste. In another work, Slah et al.²⁵ developed an ANN model for predicting the global quality of knitted fabrics using 15 input parameters. Taieb and Msahli²⁶ optimized the knitted fabric quality graphically and mathematically by using the response surface methodology (RSM) and desirability function.

Although a large number of research works have been published in the area of carpet durability, there have been no quantitative attempts to optimize carpet durability by considering multiple objectives. Therefore, in this paper, an attempt has been made to optimize the durability of Persian hand-made wool carpets by using desirability functions.

Materials and methods

Materials

Four types of yarns, namely pile, warp, thick weft and thin weft, were used for the manufacturing of Persian hand-made carpets. Pile yarns of 3.9 metric counts (Nm) having three different levels of twist, that is, 3.5, 4.0 and 4.5 twists per inch (tpi), were produced from 100% wool fibers using the woolen spinning system. The mean length and length CV % of wool fiber were 75.0 mm and 28.5, respectively. The mean diameter and diameter CV % of wool fiber were 37.7 µm and 24.4, respectively. All the yarns were twisted in the ‘S’ direction. The specifications of all the yarns are given in Table 1.

Table 1.

Yarn specifications

Sl. no.	Sample	No. of plies	Yarn count (Ne)	Plying twist (inch⁻¹)	Single yarn twist (inch⁻¹)	Fiber content
1	Warp	6	0.9 (0.3)	5.5 (2.6)	11.5 (4.5)	100% cotton
2	Thin weft	2	1.6 (1.4)	5.1 (2.1)	8.5 (6.0)	60% cotton and 40% polyester
3	Thick weft	2	0.9 (0.9)	5.1 (1.6)	3.4 (14.6)	60% cotton and 40% polyester
4	Pile yarn 1	1	2.3 (8.0)	–	3.5 (9.5)	100% wool
5	Pile yarn 2	1	2.3 (8.3)	–	4.0 (9.6)
6	Pile yarn 3	1	2.3 (2.7)	–	4.5 (6.7)

Values in the parenthesis indicate coefficient of variation.

Carpet samples were manufactured by creating Persian knots, which are asymmetrical single knots. Persian knots are created by wrapping the tuft around a single warp yarn at an angle of 2π radians and then around another adjacent warp yarn at an angle of π radians, as depicted in Figure 1. A short piece of yarn is tied by the finger around two neighboring warp strands. After each row of knots is completed, two strands of weft are passed through the complete set of warp strands in alternate sheds. Then the knots and the weft threads are beaten with a comb to secure the former in place. The weaving process begins at the bottom of the loom and, as the knots and weft yarns are added, the carpet moves upwards until it is finished.

Figure 1.

Persian knot.

Methods

Carpet samples were manufactured by varying knot density, pile height, number of plies in the pile yarn and pile yarn twist each at three levels. A four-factor three-level Box–Behnken experimental design plan was used for the preparation of carpet samples. It is the most common quadratic RSM model based on the construction of a balanced incomplete block design.²⁷ Twenty-seven samples were produced in this case, whereas with a full factorial design 3⁴ or 81 samples would have been needed. All the samples were tested for various parameters and properties, as explained below.

Knot density

The knot density of Persian hand-made wool carpets was determined as per IS: 7877 (Part III) – 1976 (Reaffirmed 1997) by using a ruler capable of measuring to the nearest millimeter (mm). This parameter was measured at the back of the carpet in lengthwise and widthwise directions.

Pile height

The pile height of carpets was measured as per IS: 7877 (Part IV) – 1976 (Reaffirmed 1997) using flat metal gauges of known height.

Abrasion resistance

The abrasion resistance of carpets was evaluated by rubbing the samples against a standard abrader fabric for 5000 rotations. The WIRA abrasion tester was used for conducting this test based on the Schiefer principle of offset heads rotating in the same direction at the same speed. The rate of weight loss per 1000 rotations was calculated as per the IWS/TM – 283: 2000 standard. For each sample, five readings were taken and then the average was calculated.

Compression properties

The thickness of the carpet samples was measured as the distance between the reference plate on which the carpet was placed and a parallel circular presser foot. The carpet sample size for this test was 125 mm × 125 mm. The specimen was placed on the base plate so that no part of the presser foot was within 20 mm distance from the edge of the specimen or within 75 mm distance from the location of any previous measurement. The presser foot was lowered slowly on to the specimen to apply a pressure of 2 kPa and, after 30 s, the thickness was noted. Without raising the presser foot, extra weights were added to increase the pressure to 5 kPa and the thickness was noted again after 30 s. This procedure was continued for pressure levels of 10, 20, 50, 100, 150 and 200 kPa and the thickness, in each case, was noted after the specified duration of pressure application, that is, 30 s. Immediately after taking the thickness reading at the highest pressure, that is, 200 kPa, the pressure was reduced to 150 kPa. The thickness was again noted after 30 s and then the pressure was reduced to 100 kPa. The recovery cycle was continued until the original pressure of 2 kPa was reached. For each sample five readings were taken and then the average was calculated. An SDL digital carpet thickness gauge was used for conducting this test as per the BS 4098: 1975 (Reaffirmed 1982) standard. Figure 2 depicts the typical thickness–pressure curve of carpet samples.

Figure 2.

Typical thickness–pressure curve for textile floor coverings.

In Figure 2, t₂ is the initial carpet thickness at 2 kPa pressure (point A), t₂₀₀ is the carpet thickness at 200 kPa pressure (point B) and t_r is the recovered carpet thickness at 2 kPa pressure (point C) after loading up to 200 kPa pressure. Compression (t₂ − t₂₀₀) was calculated as the change in thickness of the carpet when the pressure was increased from 2 to 200 kPa. Percentage compression recovery was calculated as the change in carpet thickness when the pressure was reduced from 200 to 2 kPa, expressed as a percentage of the compression. It can be expressed as $(\frac{t_{r} - t_{200}}{t_{2} - t_{200}}) \times 100$ .

Desirability functions

Desirability is a measure of degree of fulfillment of objectives. An individual desirability d_i for each characteristic is calculated by using the desirability function. This individual desirability index varies between 0 and 1, indicating null satisfaction and total satisfaction, respectively. This index depends on the following three aspects.

the objective ascertained for each characteristic of carpet durability, such as maximize, minimize or reach a target value;

the acceptance interval [y_min, y_max] defining the lower and upper limits for each characteristic of carpet durability;

the target level of each characteristic of carpet durability.

Desirability functions d_i can be of three types: desirability functions to maximize, desirability functions to minimize and desirability functions with a target value, as depicted in Figure 3. When the desirability function is used to maximize a characteristic y_i, such as compression recovery, then d_i is calculated as follows

d_{i} = {0 if y_{i} \leq y_{min} (\frac{y_{i} - y_{min}}{y_{target} - y_{min}})^{s} if y_{min} \leq y_{i} \leq y_{target} 1 if y_{i} \geq y_{target}

(1)

Figure 3.

Desirability functions to maximize, minimize and with a target value.

The degree of desirability requirement is represented in this function by the s value. If s = 1, then the desirability is fairly required. On the other hand, if s ≫ 1, then the desirability is very much required and, finally, when s ≪ 1, the desirability is very little required. For example, if y_target and y_min are 20 and 10, respectively, and y_i is 15, then d_i becomes 0.5 when s = 1. However, if s = 2, implying more stringent desirability, then d_i becomes 0.25, indicating a lower level of attainment of objective.

When the desirability function is used to minimize a characteristic y_i, such as compression or abrasion loss, then d_i is calculated as follows

d_{i} = {1 if y_{i} \leq y_{target} (\frac{y_{i} - y_{max}}{y_{target} - y_{max}})^{t} if y_{target} \leq y_{i} \leq y_{max} 0 if y_{i} \geq y_{max}

(2)

In this case, the degree of desirability requirement is represented by the t value. When the desirability function is used to reach a target value of the characteristic, then d_i is calculated as follows

d_{i} = {0 if y_{i} \leq y_{min} or y_{i} \geq y_{max} (\frac{y_{i} - y_{min}}{y_{target} - y_{min}})^{s} if y_{min} \leq y_{i} \leq y_{target} 1 if y_{i} = y_{target} (\frac{y_{i} - y_{max}}{y_{target} - y_{max}})^{t} if y_{target} \leq y_{i} \leq y_{max}

(3)

When the characteristic has a target value, as shown in Figure 3, there could be two different degrees of desirability requirement. They are represented by s and t when the measured values are lower and higher, respectively, than the target value. In this study, s = t was considered, which implies the same degrees of desirability requirement in the case of over-attainment or under-attainment of the target value of a characteristic.

Every transformation d_i presents a satisfaction degree calculated by the desirability function of a characteristic while considering the fixed target, acceptance intervals and degree of desirability requirement. All the individual desirability indexes are amalgamated into a single term of global desirability (d_G) by using the Derringen and Suich desirability function,²⁸ defined as follows

d_{G} = \sqrt[w]{d_{1}^{w_{1}} \times d_{2}^{w_{2}} \times \dots \times d_{m}^{w_{m}}}

(4)

where d_i is the individual desirability index of the ith characteristic, i ∈ [1, … , m], w_i the weight of the ith characteristic, w is the sum of w_i and m is the number of characteristics.

The compromise between the characteristics is better when d_G is higher and vice versa. The global desirability becomes perfect when d_G is equal to 1, that is, all the individual desirability (d_i) has been fulfilled. If the individual desirability (d_i) of any characteristic is equal to 0, then it implies a value beyond the acceptance interval. Thus, the function d_G yields 0 and so the compromise is rejected.

Results and discussion

Regression models for durability attributes of carpet

Carpet specifications (knot density, pile height, number of plies in the pile yarn and pile yarn twist level) and the measured values of various carpet durability attributes (abrasion loss, compression and compression recovery) are shown in Table 2.

Table 2.

Properties of Persian hand-knotted wool carpets

Sl. no.	Knot density (inch⁻¹)	Pile height (mm)	Number of plies (pile yarn)	Pile yarn twist (inch⁻¹)	Abrasion loss (mg)	Compression (mm)	Compression recovery (%)
1	5	10	3	4.0	37.3 (5.5)	10.0 (2.2)	62.1 (3.4)
2	5	16	3	4.0	59.2 (1.8)	12.9 (2.7)	58.6 (5.3)
3	7	10	3	4.0	52.7 (3.9)	11.3 (2.8)	72.3 (4.8)
4	7	16	3	4.0	56.2 (3.0)	14.4 (3.1)	71.7 (2.8)
5	5	13	2	4.0	64.5 (2.9)	10.8 (2.2)	43.5 (7.8)
6	5	13	4	4.0	68.2 (1.6)	11.4 (7.1)	60.9 (7.9)
7	7	13	2	4.0	69.0 (3.3)	12.5 (5.6)	62.3 (4.4)
8	7	13	4	4.0	57.8 (2.6)	13.1 (3.3)	76.1 (4.5)
9	5	13	3	3.5	48.3 (1.3)	9.5 (5.8)	50.9 (6.0)
10	5	13	3	4.5	51.8 (7.3)	9.8 (7.7)	63.4 (5.4)
11	7	13	3	3.5	66.1 (3.4)	15.2 (2.1)	72.7 (3.4)
12	7	13	3	4.5	56.0 (1.7)	12.4 (2.8)	68.8 (2.6)
13	6	10	2	4.0	38.4 (4.8)	10.6 (5.9)	64.4 (2.0)
14	6	10	4	4.0	43.2 (5.3)	11.1 (5.2)	78.0 (3.7)
15	6	16	2	4.0	67.3 (2.3)	11.5 (5.3)	51.7 (6.6)
16	6	16	4	4.0	56.4 (1.9)	14.3 (5.6)	71.0 (8.0)
17	6	10	3	3.5	33.7 (7.4)	11.5 (5.7)	68.6 (4.1)
18	6	10	3	4.5	41.1 (2.3)	13.0 (3.7)	64.5 (4.1)
19	6	16	3	3.5	56.4 (4.8)	13.7 (6.2)	66.1 (7.1)
20	6	16	3	4.5	41.0 (1.9)	11.8 (5.4)	59.8 (6.9)
21	6	13	2	3.5	61.1 (4.4)	12.3 (1.7)	51.2 (8.2)
22	6	13	2	4.5	61.5 (0.8)	9.8 (8.9)	59.8 (5.0)
23	6	13	4	3.5	63.6 (2.7)	12.9 (4.3)	71.0 (2.6)
24	6	13	4	4.5	59.7 (0.8)	14.3 (3.6)	68.3 (5.0)
25	6	13	3	4.0	49.1 (2.2)	14.1 (9.0)	62.5 (1.3)
26	6	13	3	4.0	43.7 (2.2)	13.9 (7.2)	63.9 (2.4)
27	6	13	3	4.0	39.9 (1.0)	12.6 (3.7)	67.4 (4.4)

Values in the parenthesis indicate CV%.

Table 3 shows a comparative analysis of three types of regression models (linear, linear with interaction and quadratic) of carpet durability attributes based on the experimental data shown in Table 2. Linear regression and linear with interaction models, which comprise five and 11 terms, respectively, have lower values of coefficients of determination (R²). Besides, from the p-values shown in Table 3, it can be inferred that linear and linear with interaction models are not statistically significant for abrasion loss at the 95% confidence level (p = 0.05). However, the quadratic model comprises 15 terms and it yields higher values of coefficients of determination (R²). This indicates that the quadratic model is the best among the three types of models.

Table 3.

Comparative analysis of various regression models

Carpet attributes	Model	No. of parameters	R ²	p-value
Abrasion loss (y₁)	Linear	5	0.27	0.1240
	Linear with interaction	11	0.40	0.4318
	Quadratic	15	0.91	0.0002
Compression (y₂)	Linear	5	0.55	0.0011
	Linear with interaction	11	0.71	0.0074
	Quadratic	15	0.81	0.0160
Compression recovery (y₃)	Linear	5	0.81	<0.0001
	Linear with interaction	11	0.88	<0.0001
	Quadratic	15	0.92	0.0002

Three response surface equations (5)–(7) were developed for relating abrasion loss (y₁), compression (y₂) and compression recovery (y₃), respectively, with the independent variables (knot density or x₁, pile height or x₂, number of plies in pile yarn or x₃ and pile yarn twist level or x₄) by using the quadratic model. The coefficients of determination (R²) of all three equations were higher than 0.8, implying that they can explain at least 80% of the total variability of the dependent variable

y_{1} = - 31.852 - 49.348 x_{1} + 41.946 x_{2} - 25.972 x_{3} - 12.673 x_{4} - 1.531 x_{1} x_{2} - 3.710 x_{1} x_{3} - 6.810 x_{1} x_{4} - 1.306 x_{2} x_{3} - 3.805 x_{2} x_{4} - 2.150 x_{3} x_{4} + 9.167 x_{1}^{2} - 0.428 x_{2}^{2} + 12.122 x_{3}^{2} + 13.307 x_{4}^{2}

(5)

[R² = 0.91]

y_{2} = - 120.021 + 18.374 x_{1} + 3.576 x_{2} - 5.103 x_{3} + 28.702 x_{4} + 0.016 x_{1} x_{2} + 0.015 x_{1} x_{3} - 1.525 x_{1} x_{4} + 0.201 x_{2} x_{3} - 0.568 x_{2} x_{4} + 1.975 x_{3} x_{4} - 0.945 x_{1}^{2} - 0.065 x_{2}^{2} - 0.785 x_{3}^{2} - 2.343 x_{4}^{2}

(6)

[R² = 0.81]

y_{3} = - 308.010 + 50.978 x_{1} - 8.722 x_{2} + 38.992 x_{3} + 94.705 x_{4} + 0.241 x_{1} x_{2} - 0.914 x_{1} x_{3} - 8.215 x_{1} x_{4} + 0.478 x_{2} x_{3} - 0.355 x_{2} x_{4} - 5.640 x_{3} x_{4} - 0.956 x_{1}^{2} + 0.246 x_{2}^{2} - 1.577 x_{3}^{2} - 2.898 x_{4}^{2}

(7)

[R² = 0.92]

It was found during the experiments that very low and very high knot density (x₁) is detrimental for abrasion. Knot density of 6 inch⁻¹ gives the minimum abrasion loss. Equation (5) has been used for the modeling of abrasion loss. Figure 4 shows the change in abrasion loss with the change in knot density keeping remaining independent variables (x₂, x₃ and x₄) constant at their mid-values. Here, the lowest abrasion loss is obtained when the value of knot density is close to 6 inch⁻¹. Thus, the coefficients of linear and quadratic terms ensure proper representation of the actual influence of knot density on abrasion loss. It was also found during the experiments that abrasion loss increases with an increase in pile height (x₂) due to more fiber loss. Figure 5 depicts the change in abrasion loss with the change in pile height keeping the remaining independent variables (x₁, x₃ and x₄) constant at their mid-values. In this case, also the coefficient of linear and quadratic terms ensures proper representation of the actual influence of pile height on abrasion loss. Similar observations were also obtained for other durability properties of carpets. This gives interpretability of the regression models developed in this work.

Figure 4.

Influence of knot density on abrasion loss.

Figure 5.

Influence of pile height on abrasion loss.

Desirability functions for the durability of carpet

Table 4 presents the range of experimental values of various attributes of carpet durability. The maximum values of carpet pile abrasion loss, compression and compression recovery are 69.0 mg, 15.2 mm and 78.0%, respectively. On the other hand, the minimum values of carpet pile abrasion loss, compression and compression recovery are 33.7 mg, 9.5 mm and 43.5%, respectively. The objectives and target values for various durability attributes are also shown in Table 4.

Table 4.

Objectives and target values of carpet durability attributes

Carpet attributes	Objective	Minimum value	Target value	Maximum value
Abrasion loss (mg)	Minimize	33.7	33.7	69.0
Compression (mm)	Minimize	9.5	9.5	15.2
Compression recovery (%)	Maximize	43.5	78.0	78.0

Individual desirability indexes were calculated based on the maximum and minimum values of each of the durability attributes. Lower abrasion loss and compression, as well as higher compression recovery, are desirable for obtaining maximum carpet durability. Therefore, the respective minimum values were taken as the targets for abrasion loss and compression. In contrast, the maximum value was considered as the target for compression recovery. Individual desirability indexes for abrasion loss (d₁), compression (d₂) and compression recovery (d₃) were calculated with the help of equations (8)–(10), respectively, using the limits mentioned in Table 4

d_{1} = {1 if y_{3} \leq 33.7 (\frac{y_{1} - 33.7}{69.0 - 33.7}) if 33.7 \leq y_{3} \leq 69.0 0 if y_{3} \geq 69.0

(8)

d_{2} = {1 if y_{2} \leq 9.5 (\frac{y_{2} - 15.2}{15.2 - 9.5}) if 9.5 \leq y_{2} \leq 15.2 0 if y_{2} \geq 15.2

(9)

d_{3} = {0 if y_{3} \leq 43.5 (\frac{y_{3} - 78.0}{43.5 - 78.0}) if 43.5 \leq y_{3} \leq 78.0 1 if y_{3} \geq 78.0

(10)

Formulation of the optimization problem and its solution

The optimization problem of carpet durability was formulated as shown below.

Objective: $Maximize global desirability = d_{G} = (d_{1} \times d_{2} \times d_{3})^{\frac{1}{3}}$

Subject to:

The optimization problem was solved using the solver add-in of Microsoft Excel. The values of x₁, x₂, x₃ and x₄ were found to be 6 inch⁻¹, 10 mm, 3 plies and 3.5 inch⁻¹, respectively. The corresponding optimized values of abrasion loss, compression and compression recovery were 32.0 mg, 10.9 mm and 67.8%. The global desirability index was found to be 0.8, whereas the individual desirability indexes for abrasion loss, compression and compression recovery were 1.0, 0.7 and 0.7, respectively. This implies that the fulfillment of abrasion loss was better than those of compression and compression recovery.

Validation of optimization model

One validation sample was prepared using the solution set of input variables. The validation sample, shown in Figure 6, was tested for the three attributes of durability, and the results are shown in Table 5. The abrasion loss, compression and compression recovery of the validation carpet sample were found to be 33.3 mg, 11.5 mm and 67.0%, respectively. The durability attributes of the validation sample showed reasonably good agreement with the optimized values. The deviation between the optimized and achieved values for abrasion loss, compression and compression recovery were 4.1%, 4.7% and 1.2%, respectively. Although Persian wool carpets have been used in this research to demonstrate the efficacy of the desirability function approach, this method can also be used for optimization of durability properties of other carpets having Tibetan or Turkish knots.

Figure 6.

Carpet sample prepared for validation.

Table 5.

Optimized and achieved values of durability attributes

Parameter	Abrasion loss (mg)	Compression (mm)	Compression recovery (%)
Optimized value	32.0	10.9	67.8
Achieved value	33.3	11.5	67.0
Deviation (%)	4.1	4.7	1.2

Conclusions

Persian hand-made wool carpets were manufactured by varying the pile yarn (number of plies and pile yarn twist) and carpet construction parameters (knot density and pile height) and following the four-factor three-level Box–Behnken experimental design plan. The quadratic regression model of the RSM was applied for the analysis of experimental data of carpet durability attributes. The desirability function approach was used to optimize the durability (abrasion loss, compression and compression recovery) of carpets. The global desirability index was found to be 0.8, whereas individual desirability indexes for abrasion loss, compression and compression recovery were 1.0, 0.7 and 0.7, respectively. The optimized values of input variables were 6 inch⁻¹, 10 mm, 3 plies and 3.5 inch⁻¹ for knot density, pile height, number of plies in the pile yarn and pile yarn twist, respectively. A validation sample was also prepared using the solution set of input variables. The achieved values of durability attributes in the validation sample were in good agreement (error < 5%) with the optimized values. The proposed approach can also be applied for carpets having different types of fibers and knots.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the office of the Development Commissioner (Handicrafts), Ministry of Textile, Government of India, for the promotion of hand-made carpet through improved and predictable wear performance of hand-made woolen carpets (research and development scheme No. K-12012/4/14/2014-15/R&D dated 11.09.2014).

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