Production cost optimization of enzyme washing for indigo dyed cotton denim by combining Kriging surrogate with differential evolution algorithm

Abstract

The enzyme washing process is extensively applied in the industrial production of denim garments. The process parameters of enzyme washing have significant effects on washing performances and costs. Since the relationships between the process parameters and washing performances cannot be expressed explicitly, it is impractical to determine the process parameters to obtain the optimal production cost while satisfying requirements of customers intuitively. This paper proposes an optimization methodology by combining Kriging surrogate and differential evolution (DE) algorithm to address the production cost optimization of enzyme washing for indigo dyed cotton denim. First, an experiment using Taguchi L₁₆ orthogonal array is conducted where temperature and concentration of cellulase enzyme are taken into consideration with processing time as the input parameters, while the washing performances (including color strength value, stiffness, and tensile strength in warp and weft directions of the washed denim fabrics) are the output responses. Second, the relationships between the inputs and outputs are established using the Kriging model. Third, the effects of the input parameters on the washing performances are analyzed, and the production cost optimization model is illustrated. Finally, a case study is given to depict the optimization process and a verification experiment is conducted to verify the effectiveness of the optimal values. On the whole, the proposed hybrid method, Kriging-DE, shows great capability of optimizing the production costs of the enzyme washing process for indigo dyed cotton denim.

Keywords

enzyme washing denim production cost Kriging differential evolution algorithm

Denim garments are widely accepted and popular around the world. In the evolution of denim garments, washed/vintage looks and superior hand feel, which make denim suitable for leisure wear, are key elements leading denim fashion trends, and these effects are achieved through unique washing techniques that can cause destruction of dyes or removal of surface fibers from denim fabrics. A wide range of washing methods have been created; some of the applications normally used are summarized by Paul¹ and Kan.² Among them, enzyme washing, which utilizes cellulase to weaken surface dyed fibers of denim and removes the fibers by fabric-to-fabric abrasion during washing so as to bring out partially dyed or undyed inner fibers, is extensively used in industrial production. A major reason for the wide use of enzyme washing is that it is more environmentally friendly than other methods (e.g. stone washing or bleaching washing) to give “used” or “worn” looks on denim garments,² and can be easily inactivated to ensure no harm to humans.

In the processing of enzyme washing, many parameters (e.g., washing temperature, washing time, concentration of cellulase enzyme, etc.) simultaneously affect the results of the process. However, methods to determine appropriate process parameters that can produce the items of the required quality and achieve the optimal production cost for enzyme washing is rarely reported in the denim garment industry. On the other hand, the relationship between the process parameters and washing effects cannot be expressed explicitly, which results in impossibility of figuring out the optimal solution intuitively. To grasp the relationships between the inputs of process parameters and outputs of performance values, surrogate models are frequently adopted to replace actual models that are difficult to obtain or illustrate. Surrogate models are usually created by using data-driven approaches. Polynomial response surface (PRS) methods and machine learning (ML) techniques have been studied in the textile field; for example, the response surface method (RSM) was used to investigate the effect of bleach washing parameters on denim fabrics.³ The artificial neural networks (ANN) approach, which is one of the most frequently used ML techniques, has been applied for predicting the color properties of laser-treated fabrics.^4–7 Moreover, ANN approach is also widely used for prediction of fabric hand feel,^8,9 fabric properties,^10–12 and other aspects of yarn and fabric processes.^13–15

The above literature does help to connect process parameters and the final quality indices. However, in order to ensure accuracy, ML techniques require numerous training sampling points.¹. Conversely, PRS methods usually require less, but mainly describe the trends and ignore fluctuations of detail. By contrast, the Kriging model, as one of the surrogate models, illustrates the relationships of inputs and outputs more accurately and robustly under conditions of limited samples than the above approaches, due to its interpolation and noise filtering capability.^17,18 Thus it has been frequently applied in the field of modeling surrogates.^19,20 Based on the surrogates, a further step is to seek optimal process parameters, where differential evolution (DE) is arguably one of the most popular evolutionary optimizers, and its powerful performance has been well recognized. Therefore, it has been widely applied in the optimization area in recent years.^21,22

In this paper, a hybrid approach consisting of the Kriging model and DE algorithm is proposed to tackle optimization of parameters for pursuing the optimal production cost under conditions of required enzyme washing performances. In the proposed approach, the Kriging model is adopted to construct the relationships between the inputs of enzyme washing process parameters and the outputs of performance. The DE algorithm is used to search for the best solution. Verification experiments are also conducted to prove the feasibility of the presented hybrid approach of Kriging-DE, which shows that it can be used to reduce the costs of enzyme washing in actual operation.

The remaining parts of this paper are organized as follows. In the second section, the experimental works and corresponding results are presented. In the third section, the proposed hybrid approach, Kriging-DE, is introduced. In the fourth section, the main effects of various parameters on washing qualities and the optimization of parameters for production costs are demonstrated. Conclusions are given in the final section.

Experimental works

Materials and fabric preparation

Indigo dyed 100% cotton denim fabrics were desized in liquor containing soda ash (1 g/L) and desizing agent (1 g/L) with a material-to-liquor ratio of 1:30 at a temperature of 50℃ for 15 min in an industrial sample washing machine (GX-50, Jun Sheng, China). Specifications of the fabric samples are listed in Table 1.

Table 1.

Specifications of the denim fabric

Parameter name	Value
Fabric weight	358 g/m²
Composition	100% cotton
Construction	3/1 twill
Warp density	75 ends/inch
Weft density	43 picks/inch
Warp count	68 tex
Weft count	79 tex
Yarn twist	Z twist
Number of warp twist	4.4/inch
Number of weft twist	4.6/inch

Procedure of enzyme washing

The enzyme washing was carried out in liquor containing anti-back-staining agent (1.0 g/L) and cellulase with a material-to-liquor ratio of 1:30 in the washing machine. Different temperatures (abbreviated as TEMP) from 30℃ to 60℃ were used in this process with different concentrations of cellulase enzyme (abbreviated as CCE), from 1% owf to 4% owf, and various processing times (abbreviated as PT) from 10 min to 40 min.

After enzyme washing, the temperature was raised to 90℃ for 1 min to inactivate the enzyme and consequently terminate the process. Each sample was then washed thoroughly twice with clean water. A hydroextractor machine was used to dehydrate all the processed fabrics at 200 rpm for 4 min before the application of a steam dryer at 75℃ for 20 min for drying up the samples.

Measurements of washing performance

Performance of the cellulase treated denim fabrics were evaluated by color strength (K/S), stiffness, and tensile strength. K/S was used to determine the effects of discoloration caused by enzyme washing, as measured by an X-rite Color I-7 spectrophotometer (X-Rite, USA). Stiffness is considered as one of major components to evaluate fabric style, and it was measured using the bending length by conducting the cantilever test according to ASTM standard D1388-2008. Tensile strength is a normally used indicator of evaluating wear-ability. Because of the construction of denim fabric (1/3 twill), the tensile strength of the fabric is different in warp and weft directions. The strip method was performed to identify the tensile strength of each sample in warp and weft directions by a mechanical property tester (YG026H, China) on the basis of ASTM standard D 5035-1995.

According to ASTM standard D 1776-2008, all the samples were conditioned in the testing laboratory at 20℃ ± 2℃ and at 65% ± 2% relative humidity for 24 hr before conducting tests.

Experiment design

In this work, Taguchi experimental design²³ was adopted to pursue the balance between errors and efficiencies of building surrogates, since it can generate sample points with good uniform and space-filling in the design space. All the input variables were divided into four levels to cover the design space, hence, the current problem was a four-factor and five-level design problem. An L16(4³) orthogonal array could be generated as shown in Table 2, with illustrations of the performances of 16 enzyme washing experiments following the procedures and measurements described in the preceding sub-sections. An additional eight experiment points, which were sampled randomly to test the accuracy of the constructed Kriging model, are also listed in Table 2.

Table 2.

Design of experiments and corresponding performances

No.	TEMP (℃)	CCE % (owf)	PT (min)	K/S	Stiffness (cm)	Tensile strength (N)
No.	TEMP (℃)	CCE % (owf)	PT (min)	K/S	Stiffness (cm)	warp	weft
Training data
1	30	1	10	12.394	2.19	574	435
2	30	2	20	12.018	2.16	567	431
3	30	3	30	11.422	2.12	559	423
4	30	4	40	11.041	2.10	548	417
5	40	1	20	12.037	2.13	566	428
6	40	2	10	12.249	2.14	569	431
7	40	3	40	10.914	2.09	537	411
8	40	4	30	10.977	2.08	540	413
9	50	1	30	11.343	2.11	551	417
10	50	2	40	10.696	2.09	531	405
11	50	3	10	11.684	2.12	556	422
12	50	4	20	10.501	2.10	536	407
13	60	1	40	10.763	2.09	538	410
14	60	2	30	9.923	2.07	525	397
15	60	3	20	10.107	2.06	533	403
16	60	4	10	11.914	2.11	547	416
Testing data
1	35	2	25	11.962	2.15	567	431
2	30	4	20	11.672	2.13	557	427
3	55	1	15	11.8	2.11	553	425
4	40	2	30	11.755	2.12	557	421
5	55	3	40	9.79	2.05	520	395
6	45	3	30	10.656	2.06	531	406
7	50	2	30	10.874	2.08	534	404
8	40	1	10	12.096	2.16	566	434

Background and proposed optimization approach

Background of Kriging surrogate model

The Kriging surrogate model, which stems from geostatistics, is an interpolative Bayesian meta-modeling technique for approximating the relationship between sample points x and the response Y( x ). The ordinary Kriging model illustrates the response by a combination of a regression model and a non-parametric part:

Y (x) = F^{T} (x) β + Z (x)

(1)

where

F (x) = [f_{1} (x), f_{2} (x), \dots, f_{m} (x)]^{T}

is a collection of regression, m is the number of basis functions.

β = [β_{1}, β_{2} \dots, β_{m}]^{T}

is an array of the corresponding regression coefficients.

F (x)^{T} β

is a deterministic function and provides global approximation of the design space. For the non-parametric part, Z( x ), which is adopted to correct the deviation from the global model, is a stochastic process and assumed to have mean zero and covariance given by:

\begin{matrix} cov [Z (x^{(i)}), Z (x^{(j)})] = σ^{2} [R (, x^{(i)}, x^{(j)})], \\ (i . j = 1, 2, \dots, n) \end{matrix}

(2)

where n is the number of sample points;

x^{(i)}

and

x^{(j)}

denote the i-th and j-th sample points;

R (, x^{(i)}, x^{(j)})

is a correlation function with parameters, and represents the spatial correlation between training sample points. The ordinary Kriging model, it usually employs Gaussian correlation function, thus

\begin{matrix} R (, x^{(i)}, x^{(j)}) = exp [- d (x^{(i)}, x^{(j)})] \end{matrix}

(3)

where

\begin{matrix} d (x^{(i)}, x^{(j)}) = \sum_{k = 1}^{p}_{k} | x_{k}^{(i)} - x_{k}^{(j)} |^{2}; (i, j = 1, \dots, n) \end{matrix}

(4)

here, p is the number of design variables, that is, each sample point consists of p dimensions, and

x_{k}^{(i)}

is the k-th design variable value of the sample point

x^{(i)}

θ_{k}

is the correlation parameter and can be determined by using maximum likelihood estimation

\begin{matrix} {max}_{0} ϕ () = - \frac{[n ln ({\hat{σ}}^{2}) + ln | R |]}{2} \end{matrix}

(5)

where ϑ is the vector consisting of

_{k}

The Kriging model estimates the response of an arbitrary point by using the linear combination of the responses of sample points. The Kriging model further requires that the mean of prediction error is zero and the mean square error (MSE) of prediction error should be minimized. Hence, when MSE is minimized, unbiased estimate of the predictor $\hat{y} (x)$ can be expressed as an equation by introducing $r^{T} (x_{p}) = {R (x_{p}, x^{(1)}), R (x_{p}, x^{(2)}) \dots, R (x_{p}, x^{(n)})}$ which is the correlation array between unobserved point $x_{p}$ and the sample points x ²⁴.

\begin{matrix} \hat{y} (x_{p}) = \hat{β} + r^{T} (x_{p}) R^{- 1} (Y - F \hat{β}) \end{matrix}

(6)

where

\begin{matrix} \hat{β} = (F^{T} R^{- 1} F)^{- 1} F^{T} R^{- 1} Y \end{matrix}

(7)

In Equation (7), $\hat{β}$ is regarded as the scalar factor, where Y is a vector containing the responses of all sample points, F is a vector filled with ones since $F (x)$ is considered as a constant in the ordinary Kriging model.

Therefore, the estimated variance of the output model (i.e., the ${\hat{σ}}^{2}$ in Equation (5)) can be calculated as follows

{\hat{σ}}^{2} = \frac{(Y - F \hat{β})^{T} R^{- 1} (Y - F \hat{β})}{n}

(8)

DE algorithm

Evolutionary computing techniques are powerful stochastic search algorithms, they imitate the natural selection process based on Darwin’s theory of evolution to find near optimal results, and have been widely applied in various optimization areas.^25–27 One of the most famous evolutionary computational techniques is the genetic algorithm (GA),²⁸ inspired by the science of genetics and natural selection phenomena. Thus, a simple GA contains three main operators: mutation, crossover, and selection. The standard GA has two main shortcomings, however: its local search ability is not good enough, and convergence is premature.²⁹ To overcome these disadvantages, the DE algorithm was proposed by Storn and Price.³⁰

Similar to GA, DE algorithm is also a population-based algorithm with three operators: mutation, crossover, and selection, however, the details of each operator are different. The flowchart of the basic DE algorithm is given in Figure 1.

Figure 1.

Flowchart of the basic DE algorithm.

Initialization

At generation G = 0, an initial population is assigned a randomly chosen value from the feasible solution space

\begin{matrix} x_{i, j, 0} = X_{j}^{min} + u_{j} (X_{j}^{max} - X_{j}^{min}), i = 1, 2, \dots, N_{p}, \\ j = 1, 2, \dots, D \end{matrix}

(9)

where

N_{p}

is the population size, and each solution vector in a population has D parameters.

X_{j}^{max}

and

X_{j}^{min}

are the upper and lower bounds of the j-th parameter. u_j indicates a uniformly distributed random number within the range [0, 1] for the j-th parameter.

Mutation

At each generation G, DE creates a mutant vector $v_{i, G + 1}$ according to the following expression:

v_{i, G + 1} = x_{r 1, G} + F (x_{r 2, G} - x_{r 3, G})

(10)

where i, r1, r2, r3 ∈ {1, 2,…,

N_{p}

} are distinct integers and selected randomly. The parameter

F \in [0, 2]

is termed the scaling factor which affects the amplification of the differential variation

(x_{r 2, G} - x_{r 3, G})

Crossover

DE also performs a binomial crossover to increase the diversity of the parameter vectors. A trial vector $w_{i, G + 1}$ is created as below:

w_{i, j, G + 1} = {\begin{matrix} v_{i, j, G + 1} & if rand (0, 1) \leq C_{r} or j = j_{rand} \\ x_{i, j, G} & otherwise \end{matrix}

(11)

where

C_{r} \in [0, 1]

is the crossover constant, and

j_{rand}

is a randomly chosen integer from the set {1, 2,…, D} which helps ensure at least one parameter from

v_{i, j, G + 1}

Selection

Selection is conducted to select the better one from trial vector $w_{i, G + 1}$ and its predecessor $x_{i, G}$ to the next generation, as below

_{i}, G + 1 = \begin{matrix} v_{i, G + 1} & if f (v_{i, G + 1}) < f (x_{i, G}) \\ x_{i, G} & otherwise \end{matrix}

(12)

where

f (\cdot)

is the fitness function.

Proposed hybrid Kriging-DE method

In this section, a hybrid method consisting of the Kriging model and DE algorithm (Kriging-DE) is proposed to seek optimal enzyme washing cost. In this method Kriging is applied to construct the relationships between the inputs of washing process parameters and outputs of effects, and the relationships constructed are considered as constraints in the enzyme washing cost model, while the DE algorithm is used to facilitate the global optimum search. The flowchart of the optimization procedure is shown in Figure 2 and depicted as follows:

Define the enzyme washing optimization problem including objective function, constraints, design variables, and ranges. In this paper, the objective is to search for the minimum enzyme washing cost under constraints of washing effects required by consumers. The three variables are: TEMP, CCE, and PT;

Conduct the experiment at selected sample points by adopting the Taguchi method to obtain the corresponding responses;

Construct relationships by the Kriging surrogate model;

Check whether the desired accuracy of the constructed Kriging surrogates is achieved or not. If yes, the obtained surrogates can be used as constraints for the enzyme washing cost optimization, otherwise, go back to Step 2;

Conduct DE to obtain the optimum input parameters. During optimization, the fitness values are calculated by the enzyme washing cost model with several constraints (illustrated below), and the values of constraints are predicted by the constructed Kriging surrogates;

Determin the termination criteria for the optimization;

Output the optimum process parameters and conduct verifications.

Figure 2.

Flowchart of optimization procedure.

Results and analysis

Kriging surrogate modeling

A Kriging surrogate model can be considered as consisting of a regression model and a non-parametric part (as described above). In this paper, a zero-order polynomial regression function and a Gaussian correlation function illustrated as Equation (3) were employed for the two portions, respectively. All of the training data in Table 2 were used to construct the Kriging model, while another five testing points were chosen randomly and corresponding performances were gained (as shown in Table 3) to verify the accuracy of the constructed Kriging models.

Table 3.

ANOVA results and comparison of accuracies of the Kriging surrogates and square polynomial functions

		Kriging surrogate			Square polynomial functions			P-value
No.	OV	PV_kr	error		PV_pf	error		P-value
K/S
1	11.962	11.766	1.64%	MSE	11.67	2.44%	MSE	TEMP
2	11.672	11.876	−1.75%	0.06818	11.775	−0.88%	0.07302	0.0148
3	11.8	11.312	4.14%	RMSE	11.395	3.43%	RMSE	CCE
4	11.755	11.437	2.71%	0.26111	11.377	3.22%	0.27022	0.3667
5	9.79	9.966	−1.80%	MAE	9.478	3.19%	MAE	PT
6	10.656	10.703	−0.44%	0.23225	10.544	1.05%	0.24725	0.0129
7	10.874	10.681	1.77%	MRAE	10.666	1.91%	MRAE
8	12.096	12.332	−1.95%	0.02024	12.264	−1.39%	0.02189
Stiffness
1	2.15	2.13	0.93%	MSE	2.13	0.93%	MSE	TEMP
2	2.13	2.12	0.47%	0.00036	2.14	−0.47%	0.0002	0.0001
3	2.11	2.1	0.47%	RMSE	2.09	0.95%	RMSE	CCE
4	2.12	2.12	0.00%	0.01904	2.11	0.47%	0.01414	0.0037
5	2.05	2.08	−1.46%	MAE	2.06	−0.49%	MAE	PT
6	2.06	2.09	−1.46%	0.01625	2.08	−0.97%	0.01250	0.0003
7	2.08	2.09	−0.48%	MRAE	2.08	0.00%	MRAE
8	2.16	2.14	0.93%	0.00775	2.15	0.46%	0.00593
Tensile strength (warp direction)
1	567	561	1.06%	MSE	560	1.23%	MSE	TEMP
2	557	562	−0.90%	32.5000	567	−1.80%	55.6250	0.0005
3	553	560	−1.27%	RMSE	561	−1.45%	RMSE	CCE
4	557	553	0.72%	5.70088	551	1.08%	7.45822	0.0566
5	520	512	1.54%	MAE	509	2.12%	MAE	PT
6	531	536	−0.94%	5.50000	536	−0.94%	7.12500	0.0031
7	534	537	−0.56%	MRAE	539	−0.94%	MRAE
8	566	572	−1.06%	0.01005	571	−0.88%	0.01304
Tensile strength (weft direction)
1	431	426	1.16%	MSE	424	1.62%	MSE	TEMP
2	427	426	0.23%	14.2500	431	−0.94%	18.5000	0.0002
3	425	418	1.65%	RMSE	421	0.94%	RMSE	CCE
4	421	420	0.24%	3.77492	418	0.71%	4.30116	0.0651
5	395	391	1.01%	MAE	390	1.27%	MAE	PT
6	406	408	−0.49%	3.25000	408	−0.49%	4.00000	0.0021
7	404	407	−0.74%	MRAE	409	−1.24%	MRAE
8	434	431	0.69%	0.00777	432	0.46%	0.00959

The key in the construction of Kriging models is to gain the unknown correlation parameter θ in Equation (3) and the scalar factor $\hat{β}$ in Equation (7). Take Kriging surrogate for K/S as an example, the correlation parameter θ = [0.1563, 0.0515, 0.2062] was calculated by solving Equation (5), and the $\hat{β}$ = 0.7782 was obtained using Equation (7). Based on the known θ and $\hat{β}$ , the response of any unobserved point could be predicted by conducting Equation (6).

The Kriging surrogates for K/S, stiffness (bending length), tensile strength (warp direction), and tensile strength (weft direction) are demonstrated in Figure 3. In order to evaluate the performance of the Kriging surrogates, four statistical criteria were employed, including MSE, root mean square error (RMSE), mean absolute error (MAE), and mean relative absolute error (MRAE).

\begin{matrix} MSE = \frac{1}{n} \sum_{i = 1}^{n} (e_{i} - p_{i})^{2} \end{matrix}

(13)

\begin{matrix} RMSE = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} (e_{i} - p_{i})^{2}} \end{matrix}

(14)

\begin{matrix} MAE = \frac{1}{n} \sum_{i = 1}^{n} | e_{i} - p_{i} | \end{matrix}

(15)

MRAE = \frac{1}{n} \sum_{i = 1}^{n} \frac{| e_{i} - p_{i} |}{e_{i}}

(16)

where e_i is the observed values (OV) from the real experiments and p_i is the corresponding predicted values from the constructed Kriging surrogates (PV_kr).

Figure 3.

Kriging surrogates for (a) K/S, (b) stiffness, (c) tensile strength in warp direction, and (d) tensile strength in weft direction.

Additionally, the square polynomial functions based on the training data in Table 2 were built for the accuracy comparison. The predicted values of the constructed polynomial functions (PV_pf) together with OV and PV_kr are illustrated in Table 3 for comparison. MSE, RMSE, MAS, and MRAE for all trained Kriging surrogates and square polynomial functions are also listed in Table 3 for performance evaluation. In order to reflect the prediction performance directly, the predicted results for K/S value, stiffness (bending length), tensile strength (warp direction) and tensile strength (weft direction) versus observed experimental data are illustrated in Figure 3(a), (b), (c), and (d), respectively.

As can be seen in Table 3 and Figure 4, both Kriging surrogates and square polynomial functions have very high prediction accuracy, which may be for two reasons. On the one hand, in the defined ranges of the process parameters, relatively small variation was observed in the washing performances of the samples with the variation of any input parameter, and seldom was dramatic fluctuation observed. Hence, it is not difficult to predict the unobserved values of unobserved experimental points. On the other hand, the relative changes of washing performances resulting from the variations of parameters are small and can be evaluated by Equation (17).

ratio = \frac{x_{max} - x_{min}}{\frac{1}{n} \sum_{i = 1}^{i = n} x_{i}} \times 100 %

(17)

where x is a data point, and x_max and x_min indicate the maximum and minimum values in all data. According to the above equation, based on the 16 training samples in Table 2, the relative changes of K/S, stiffness, and tensile strength in warp and weft directions can be calculated as 21.96%, 6.16%, 8.93%, and 7.68%, respectively. Usually, the prediction deviation would be much less than the above relative changes in a monotonically increasing/decreasing function, and we can find that all surrogates owning monotonicity from Figure 3. Therefore, the prediction error looks quite small.

Figure 4.

Predicted data output by the Kriging surrogates and square polynomial functions versus observed experimental data: (a) K/S, (b) stiffness, (c) tensile strength in warp direction, and (d) tensile strength in weft direction.

Table 3 also presents a comparison of the prediction accuracies of the Kriging surrogates and the square polynomial functions. It is found that the Kriging surrogate performs slightly worse on the stiffness prediction (MSE = 0.00036, RMSE = 0.01904, MAE = 0.01625, MRAE = 0.00775) compared with the square polynomial function (MSE = 0.0002, RMSE = 0.01414, MAE = 0.01250, MRAE = 0.00593). However, for other indicators including K/S and tensile strength in warp and weft, the Kriging surrogates perform better in prediction. In general, the Kriging surrogates can be considered as an appropriate model for prediction of enzyme washing performance.

Table 3 furthermore illustrates p-values obtained by conducting significance tests. Here, the main effects of all inputs were taken into consideration but ignoring effects caused by interactions of the inputs on account of limited data. The p-values indicate that the effectiveness of TEMP and PT is significant at a high confidence level (p-values < 0.05). However, the p-values of CCE on most outputs are relatively high (p-values > 0.05), especially on K/S value (p-value = 0.3667). The main reason for this is that the effectiveness of cellulase is strongly affected by temperature, so increasing CCE at unsuitable temperatures will produce limited effects.

Discussion of multi-parameter effects on the washing performances

The main effects of the process parameters on washing performances are depicted by conducting global sensitivity analysis³¹ on the constructed model, which can quantify the relative importance of each input in influencing the value of the output, as demonstrated in Figure 5. The magnitude of the bars in Figure 5 illustrates its degree of influence to the washing performances. Figure 5(a) and Figure 5(b) demonstrate the influences of all parameters on K/S and stiffness, respectively, in which the impact derived from TEMP is the most obvious. At the same time, the TEMP also has the most significantly effect on the tensile strength in warp and weft directions as shown in Figure 5(c) and Figure 5(d). The reason for this phenomenon probably is that the cellulase activity is improved nonlinearly with the increase of TEMP in the selected ranges, and the effects from CCE / PT tend to vary linearly. In other words, the washing performances caused by the growth of TEMP have more intensive change than the effects caused by the variations of CCE/PT.

Figure 5.

Contribution rates of the process parameters to washing performances: (a) K/S, (b) stiffness, (c) tensile strength in warp direction, and (d) tensile strength in weft direction.

Optimization of the process parameters

The aim of the optimization is to pursue optimal cost in the production, and the proposed optimization model is described in Equation (18). The object function in the model is to minimize the total cost caused by TEMP x_TEMP, CCE x_CCE, and PT x_PT, but it is subject to the constraints of washing performances.

\begin{matrix} minimize f_{cost} = \frac{x_{TEMP} - l_{fTEMP}}{l_{uTEMP} - l_{fTEMP}} \cdot c_{TEMP} + \frac{x_{CCE} - l_{fCCE}}{l_{uCCE} - l_{fCCE}} \cdot c_{CCE} + \frac{x_{PT} - l_{fPT}}{l_{uPT} - l_{fPT}} \cdot c_{PT} \\ subject to f_{KS} (x_{TEMP}, x_{CCE}, x_{PT}) \geq t_{KS} or \\ f_{KS} (x_{TEMP}, x_{CCE}, x_{PT}) \leq t_{KS} \\ f_{BL} (x_{TEMP}, x_{CCE}, x_{PT}) \geq t_{BL} or \\ f_{BL} (x_{TEMP}, x_{CCE}, x_{PT}) \leq t_{BL} \\ f_{TS - warp} (x_{TEMP}, x_{CCE}, x_{PT}) \geq t_{TS - warp} \\ f_{TS - weft} (x_{TEMP}, x_{CCE}, x_{PT}) \geq t_{TS - weft} \end{matrix}

(18)

where the

l_{fTEMP}

l_{fCCE}

, and

l_{fPT}

are the floor limit of x_TEMP, x_CCE, and x_PT;

l_{uTEMP}

l_{uCCE}

, and

l_{uPT}

are the upper limit of x_TEMP, x_CCE, and x_PT;

c_{TEMP}

c_{CCE}

, and

c_{PT}

are the coefficients, which are related to the costs raised by changes of concentration of cellulase enzyme, temperature, and processing time, respectively;

f_{KS} (\cdot)

f_{BL} (\cdot), f_{TS - warp} (\cdot)

, and

f_{TS - weft} (\cdot)

represent the surrogates constructed by the Kriging method that illustrates the relationships between input parameters (x_TEMP, x_CCE, x_PT) and K/S, stiffness(bending length), and tensile strength in warp and weft directions, respectively.

t_{KS}

t_{BL}

t_{TS - warp}

, and

t_{TS - weft}

indicate the technical specifications depending on the requirements of customers.

The constraints in Equation (18) are nonlinear, and the values of input parameters are taken as integers or infinite decimals in actual productions, hence, the proposed optimization model can be treated as an integer nonlinear programming problem. The DE algorithm is an effective means to solve the problem. In order to verify the effectiveness and feasibility of the Kriging-DE model for optimizing the cost of denim enzyme washing, a case study is given in the following section.

Case study

In this case, the ranges of x_TEMP, x_CCE, and x_PT were still set as 30℃–60℃, 1% owf–4% owf and 10 min–40 min, respectively. $c_{TEMP}$ , and in Equation (18) are defined as 0.08, 0.4, and 0.52, which were derived from actual situations in a factory. The requirement of washing performance was proposed as below.

The K/S value of the processed denim fabric should achieve 11;

The bending length of the fabric should be less than 2.1;

The minimum tensile strength of the fabric in warp and weft directions should be 540 N and 410 N, respectively.

Therefore, Equation (18) can be written as below

\begin{matrix} minimize f_{cost} = \frac{x_{TEMP} - 30}{60 - 30} \cdot 0.08 + \frac{x_{CCE} - 1}{4 - 1} \cdot 0.4 \\ + \frac{x_{PT} - 10}{40 - 10} \cdot 0.52 subject to f_{KS} (x_{TEMP}, x_{CCE}, x_{PT}) = 11 \\ f_{BL} (x_{TEMP}, x_{CCE}, x_{PT}) \leq 2.1 \\ f_{TS - warp} (x_{TEMP}, x_{CCE}, x_{PT}) \geq 540 \\ f_{TS - weft} (x_{TEMP}, x_{CCE}, x_{PT}) \geq 410 \end{matrix}

(19)

The optimization of the process parameters was completed by DE, which was coded in MATLAB 2014a software. In the solving process, the iteration curve of object function is demonstrated in Figure 6. The calculated minimum value is 0.18133, and the corresponding values of x_TEMP, x_CCE, and x_PT are 59℃, 1.0% owf, and 16 min. The calculation time is 4.66 s under computer configurations: CPU: i7-6560U@ 2.2 GHz, RAM: 8 G.

Figure 6.

Iteration curve of object function using Kriging-DE method.

An experiment was conducted to verify the effectiveness of the proposed method. A denim sample was washed with x_TEMP, x_CCE, x_PT adopting 59℃, 1.0% owf, and 16 min under the procedures and requirements mentioned above. After washing, K/S, stiffness, and tensile strength in warp and weft directions were tested as 11.172, 2.09 cm, 553 N and 419 N, respectively. The error of the K/S was 1.56% which was within the allowable range, and the other three indices satisfied the requirements. The washed sample is shown in Figure 7, alongside an unwashed sample. The optimal cost was achieved under the model given by Equation (19), while the washed sample met the needs. It demonstrates that Kriging-DE is feasible and can be applied to guide actual enzyme washing production.

Figure 7.

Denim samples from the verification experiment.

Conclusion

In this paper, a hybrid optimization methodology combining Kriging surrogate and DE algorithm is proposed to address the optimization of parameters for pursuing the optimal cost under conditions of required enzyme washing performances. The following conclusions can be drawn from the experiment and optimization results:

The prediction capability of the Kriging surrogate for enzyme washing performances outperforms the second-order RSM. The RMSE of K/S, stiffness (bending length), and tensile strength in warp and weft directions using Kriging surrogate are 0.2697, 0.0173, 6.1644, and 4.8166, and the maximum errors are −4.14%, 1.46%, −1.57% and −1.64%, respectively, which are within the acceptable ranges.

By the constructed Kriging surrogates, it is found that temperature makes the most significant contribution to the washing performances in the defined ranges, which are temperature (30℃–60℃), concentration of cellulase enzyme (1% owf–4% owf), and processing time (10 min–40 min).

A production cost optimization model is offered and can be effectively solved by using the proposed Kriging-DE approach. A case study and a verification experiment were conducted to prove the feasibility of the methodology. Overall, it is demonstrated that optimization of the enzyme washing cost by the Kriging-DE method is effective and the method can be applied to guide actual enzyme washing production.

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 a grant from the scientific research project of Hubei provincial department of education, China (Project: Q20191707).

ORCID iDs

Jie Xu

Zhenglei He

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