Scholarly article on topic 'Selecting Multiple Evaluator's Perception-Oriented Relevant Physical Features of Consumer Goods by Using Fuzzy Data Sensitivity and OWA Operators'

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Academic research paper on topic "Selecting Multiple Evaluator's Perception-Oriented Relevant Physical Features of Consumer Goods by Using Fuzzy Data Sensitivity and OWA Operators"

International Journal of Computational Intelligence Systems

ISSN: 1875-6891 (Print) 1875-6883 (Online) Journal homepage: http://www.tandfonline.com/loi/tcis20

Selecting Multiple Evaluator's Perception-Oriented Relevant Physical Features of Consumer Goods by Using Fuzzy Data Sensitivity and OWA Operators

I. Feki, X. Feng, A. Ghith, L. Koehl, F. Msahli & F. Sakli

To cite this article: I. Feki, X. Feng, A. Ghith, L. Koehl, F. Msahli & F. Sakli (2016) Selecting Multiple Evaluator's Perception-Oriented Relevant Physical Features of Consumer Goods by Using Fuzzy Data Sensitivity and OWA Operators, International Journal of Computational Intelligence Systems, 9:2, 213-226

To link to this article: http://dx.doi.org/10.1080/18756891.2016.1149997

Published online: 12 Feb 2016.

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Date: 13 February 2016, At: 01:33

International Journal of Computational Intelligence Systems, Vol. 9, No. 2 (2016) 213-226

Selecting Multiple Evaluator's Perception-Oriented Relevant Physical Features of Consumer Goods by Using

Fuzzy Data Sensitivity and OWA Operators

I. Feki1*, X. Zeng2, A. Ghith3, L. Koehl2, F. Msahli1 and F. Sakli4

1 Etude des Systèmes Industriels et des Energies Renouvelables (ESIER) Ecole Nationale d'Ingénieurs de Monastir University of Monastir, Tunisia.

2Laboratoire GE.M.TEX. Ecole Nationale Supérieure des Arts et Industries textiles, Roubaix Université Lille 1, France

3Département de Génie Textile Ecole Nationale d'Ingénieurs de Monastir University of Monastir, Tunisia.

4 Laboratoire de Génie Textile Institut Supérieur aux Etudes Technologiques de Ksar Hellal, Tunisia

Received 2 July 2015 Accepted 3 January 2016

Abstract

The assessment of goods quality using experts is costly task in addition to their often unavailability. In this paper, we present a new method for ranking physical features of consumer goods according to their relevancy to multiple evaluators' perception at different levels and selecting the most important ones for quality characterization. The main contribution of the paper is combining of fuzzy method and ordered weighted averaging (OWA) operators to achieve our aim. The proposed selection method, considered as a Multi-Evaluators and Multi-Criteria Decision Making (ME-MCDM) technique, has been developed using fuzzy sensitivity (FS) criterion for ranking and OWA operator to aggregate the aforementioned ranking lists. Finally, by introducing a smart percolation technique we get automatically the most relevant physical features for a given sensory descriptor. The suggested approach is applied to a selection problem of textile physical features.

Keywords: feature selection, fuzzy logic, data sensitivity, OWA operators, Multi-criteria decision making, percolation technique.

sensory evaluation and 2) physical evaluation. Sensory 1. Introduction evaluation permits to characterize human perception

In different consumer industrial sectors (food, textile

with a panel of trained evaluators who evaluates

, , f e consumer goods using a normalized procedure

and apparel, cars, furniture, cosmetics...), quality of 1

„ ^ ^ a ■ ^ . n (perceived quality) . It is very close to markets and

products is generally evaluated in two aspects: 1) n ]

consumer's behaviors but includes uncertainty and

Corresponding author. E-mail: fekiimed@yahoo.fr.

imprecision due to the involvement of human factors. Moreover, it is often difficult to quickly gather qualified evaluators to perform evaluation sessions on product quality due to their individual time availability. It is for these reasons that physical evaluation on consumer goods using measuring devices is more acceptable in industrial companies, especially for those who wish to design high quality consumer goods by engineering means2. Industrial norms on product quality are generally developed using physical evaluations3. A physical evaluation of consumer goods is performed by measuring a set of normalized physical features. It leads to precise numerical data about product quality but is relatively far from human perception on consumer goods and its interpretation with respect to human feeling should be further exploited4. For consumer goods, understanding of the relation between measured physical features and quality perceived by consumers or professional experts is very important to rely subsequently on measuring devices rather than evaluators' groups whose availableness cannot constantly be assured. It can effectively help to define and explain quality norms with respect to human perception on finished products5.

In practice, the number of measurable physical features characterizing quality of consumer goods is very large. These features are often redundant between them and not all relevant to human perception on product quality. For defining appropriate norms of consumer goods, it is necessary to reduce the quality feature space so that only the physical features the most relevant to human perception are considered in quality characterization6-8.

Selection of relevant features is an important topic which has attracted many researchers for reducing the search space and the complexity of the problem to be processed7,9. Indeed, relevant variables selection has been widely applied to pattern recognition, system modeling and data mining. The definition of the criterion of relevancy depends on the specific problems to be solved and then it has different versions in the literature. These definitions can be summarized in some sense using a general statement given by Blum and Langley10.

A variable x is relevant to a target concept c if there exists two examples A and B in the instance space such that A and B differ only in their assignment to x and c(A)^c(B).

In the existing literature, the classification based feature selection using the supervised learning strategy has been frequently used in different applications. Its selection objective is to improve the classification accuracy or class label predictive accuracy of data samples. Several well-known methods include the decision-tree method11, the fuzzy TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) and fuzzy AHP (Analytic Hierarchy Process) to aid the selection decision process, have been developed using fuzzy techniques12,13. Deng et al.8 proposed a feature selection method by using the Fuzzy Combination of Data Sensitivity and Human Knowledge (FCDSHK). Compared with the classification-based feature selection methods, it is more efficient for dealing with very few numbers of experimental data and capable of integrating both experimental data and human professional knowledge into the ranking or selection procedure. Its effectiveness has been validated in many industrial applications, especially in the selection of relevant manufacturing process parameters (input variables) for a given quality feature (output variable)8.

In this paper, different from the previous applications of FCDSHK in which the output variable is a unique and normalized magnitude, we extend this method to the selection or ranking of relevant physical features of consumer goods related to quality evaluation scores provided by different evaluators14. As these evaluators consider the product quality according to different criteria, a direct aggregation of their evaluation results is not physically significant and can not completely characterize the difference between these criteria. In this context, we propose to develop an appropriate aggregation of the ranking results provided by different evaluators, which is more significant and can be more easily adjusted by production and quality experts using their professional knowledge. The aggregation of different ranking lists for a set of alternatives with single or multiple criteria has been encountered in many real scenarios and widely studied by researchers15. The earliest contributions in this area were given by Borda and Kendall16. Recently, the Ordered Weighted Averaging (OWA) operators, introduced by Yager17, have shown their good performance for aggregating ranking lists in different applications in decision making14,18,19. They can provide more choices and more flexibility than the Borda-Kendall method20. A fundamental aspect of this operator

is the re-ordering step, in particular an aggregate ai is not associated with a particular weight wi but rather a weight is associated with a particular ordered position of aggregate. When we view the OWA weights as a column vector we shall find it convenient to refer to the weights with the low indices as weights at the top and those with the higher indices with weights at the bottom21. Actually, the OWA methods have been applied in preferential voting and election systems22, parameterized estimation of fuzzy random variables23, and queuing systems of hospital's databases24.

The problem of determining weights for an OWA operator can be addressed in different ways, for example with the use of the so-called linguistic quantifiers, introduced by Zadeh25. A relative linguistic quantifier Q, such as most, few, many and all, can be represented as a fuzzy subset of the unit interval, where for a given proportion r ^ [0, 1] of the total of values to aggregate, Q(r) indicates the extent to which this proportion satisfies the semantics defined in Q. Regular Increasing Monotone (RIM) quantifiers are especially interesting for their use in OWA operators21. A quantifier Q is called RIM wherever Q(0) = 0, Q(1) = 1, if r1 > r2 then Q(r1) > Q(r2). Yager17,26 suggested to compute weights wi with the use of a RIM quantifier Q following Eq. (5).

In this paper, we aggregate the ranking lists by using the OWA operators. For this purpose, the well-known Regular Increasing Monotone (RIM) linguistic quantifier Qa(r) is applied14'27. In this quantifier, a number of linguistic principles provided by the evaluators are taken into consideration for determining the weights (Section 2.1.1).

The proposed procedure is illustrated by a real case for selecting the most relevant physical features of washed denim fabrics with respect to fabric hand feeling. The perceived quality on fabric hand has been evaluated by two different panels of evaluators. The first one is composed of 6 evaluators, selected from general public and who have received training of quantitative sensory evaluation techniques. It is used to assess and classify 41 fabric samples using a number of sensory descriptors frequently used in fabric hand evaluation. The second one, composed of 8 experts, is a panel mastering professional knowledge in textile metrology and mechanical features of fabrics. These experts have not received training on sensory evaluation. The second panel aims to provide

complementary information in order to adjust the ranking list obtained previously towards the professional knowledge given by the experts.

Therefore, this paper aims to develop a new decision making method which takes care of suitable objective and subjective criteria selection and proper evaluation of the alternatives treating it as a MCDM problem using two kinds of panel of evaluators. The proposed approach integrates experimental data, human professional knowledge, fuzzy data sensitivity and OWA operators to select automatically the set of the most relevant features from irrelevant ones. The number of features is automatically determined by the proposed percolation technique. In order to do so, the remainder of this paper is set out as follows. Each functional block of the proposed methodology is described in Section 2. In Section 3, selection of washed denim fabric physical features is applied in order to validate the effectiveness of the proposed approach. Conclusions are presented in

Section 4.

2. Proposed method for selecting relevant

features

The general structure of the proposed relevant physical features selection procedure with respect to one single sensory descriptor and multiple evaluators is described in Fig. 1. It is composed of six functional blocks. From the set of S samples, m features are measured using physical devices. Besides, all the samples are evaluated by n trained evaluators (Panel 1) using different normalized sensory descriptors.

Based on these measured and evaluated data, the fuzzy inference system (FIS1) block generates a number of ranking lists of physical features by computing the fuzzy sensitivity (FS) criterion of all the physical features to each specific sensory descriptor delivered by different evaluators. Each ranking list corresponds to one evaluator. Next, all these ranking lists are aggregated using an OWA operator (OWA1) in order to generate an aggregated ranking list for all the evaluators related to the sensory descriptor. In the same time, a number of professional experts specialized in manufacturing processes and product quality (Panel 2) give their ratings on the relevancy of each physical feature to each sensory descriptor. In practice, professional experts generally master knowledge about the relationship between physical measures and perceived quality of consumer goods but it is

incomplete and uncertain. For exploiting the complementary knowledge provided by the experts, we use another OWA operator for aggregating these different human knowledge (HK) sources (OWA2). Next, the ranking lists obtained from OWA1 will be combined with the aggregated human knowledge generated by OWA2 using fuzzy logic and another OWA respectively. We obtain then two new aggregated ranking lists of physical features for each sensory descriptor, i.e. the fuzzy data sensitivity-based ranking list (FIS2) and the OWA aggregated ranking list (OWA3). Finally, by using the percolation technique, we combine the two previous ranking lists to generate the final comprehensive ranking list for each sensory descriptor. The physical features selected from this ranking list is capable of taking into account the sensitivity of measured physical data to human perception, the difference between multiple evaluators on perceived quality and complementary knowledge provided by different professional experts. This overall procedure can be carried out for different sensory descriptors.

In the illustrative example of relevant physical feature selection for washed denim fabrics, we produce 41 basic denim fabrics (5=41) using different washout treatments. These samples have been measured in the testing laboratory and evaluated by a panel of six evaluators having received sensory evaluation training (Panel 1).

The values extracted from the physical measures and sensory evaluations have been normalized into the range of [0, 1]. The total number of physical features on the 41 denim fabric samples is m=20, including direct

Initial m physical features set

Sensory parameter

Panel 1:

group of 6

sensory evaluators

Fuzzy sensitivity criterion

Rankings of m physical features corresponding to each evaluator

Aggregation of

rankings corresponding

to each evaluator of Panel 1

and indirect measured features (Table 1). The direct measures are realized directly from the related measuring devices. The indirect measures are calculated from the direct measures according to the known physical laws28'29. Also, the same samples are evaluated by the 6 trained evaluators using 9 predefined subjective sensory descriptors: smooth, fluffy, full, supple, elastic, wrinkle, soft, cold, and sliding. For each sensory descriptor, each evaluator provides an evaluation score to each fabric sample.

Table 1. Description of the physical features

Ref. Physical feature Feature description Equipment (standard)

Xl X2 X3 X5 X6 X7 X8 T2 (mm) St (mm) Bi (pN.m) B2 (pN.m) Fi (mm2) F2 (mm2) G (N/m) E100 i(%) Thickness at 2gf Surface thickness Warp bending rigidity Weft bending rigidity Warp formability Weft formability Shear rigidity Warp extensibility at 100 gf Fabric Assurance by Simple Testing: FAST system

X9 E100 2(%) Weft extensibility at 100 gf

x10 X11 Pic of tension (N) Pic of comp. (N) Peak of tension Peak of compression Stiffness tester (ASTM D4032)

x12 F Drape coefficient Drape meter (NF G07-109)

x13 x14 x15 x16 MIU 1 MIU 2 SMD 1 SMD 2 Warp friction coefficient Weft friction coefficient Warp roughness Weft roughness Kawabata Evaluating System for Fabrics 'KES-F'

x17 Mass (g/m2) Fabric weight by unit area Balance (EN 12127)

x18 P.air (l/m2/s) Air permeability Air permeability device (ISO 9237)

x19 x20 Elast1 (^m) Elast2 (p.m) Warp elastic deformation Weft elastic deformation Universal Surface Tester (UST)

FS ranking aggregation of m physical features for the sensory parameter

Panel 2:

group of 8

human knowledge experts

Rankings by rating the relevancy of m physical features to the sensory descriptor

Aggregation of

rankings corresponding to each Expert of Panel 2

Aggregation

of FS and HK

rankings Fuzzy aggregated

rank for the sensory

FIS 2 parameter FGR

Human Knowledge (HK) ranking aggregation of m physical features for the sensory parameter,

Aggregation

of FS and HK

rankings OWA aggregated '

rank for the sensory

OWA 3 parameter OWAGR

FIS : Fuzzy Inference System /

FGR : Fuzzy Global Relevancy OWAGR: OWA Global Relevancy

Fig. 1. Integrated approach for selecting the set of relevant physical features to one sensory descriptor

Percolation Selected

technique: most relevant

physical

Aggregation features for

of Fuzzy and the sensory

OWA parameter

rankings

2.1. Selection of relevant physical features using

fuzzy sensitivity (FIS1 model)

The fuzzy sensitivity criterion FS for all the physical features related to a sensory descriptor is defined in Section 1 according to the same principles given by Blum and Langley10. These principles are transformed into a set of fuzzy rules for building a fuzzy model in which the physical data variation Ax (distance between two normalized vectors in the physical feature space) and the sensory data variation Ay (distance between two normalized values of one specific sensory descriptor) are taken as two input variables, respectively, and the general sensitivity FS as output variable (model FIS1 in Fig. 1). Evidently, FS is a function of Ax and Ay, denoted as FS=FISl(Ax, Ay). This fuzzy model includes an interface of fuzzification, a base of fuzzy rules, an inference mechanism and an interface of defuzzification.

The fuzzification procedure aims to uniformly partition each of the two input variables into three fuzzy values: Small (S), Medium (M) and Big (B). The output variable is a fuzzy variable varying from 0 to 1 and composed either of three fuzzy values: Small (S), Medium (M) and Big (B) (illustrated in Fig. 2)30.

Table 2. Fuzzy rules of fuzzy sensitivity criterion

Fig. 2. Fuzzy values for each input and output data variable in FIS1 model

According to the experience of the quality experts on the relationship between physical and sensory parameters, a set of fuzzy rules are defined in Table 2. As the output variable includes fuzzy values, Mamdani's fuzzy inference method is used for aggregating these fuzzy rules and obtaining defuzzified

output values

FS And Ay

Small Medium Big

Ax Small Small Big Big

Medium Small Medium Big

Big Small Small Medium

Given a specific sensory descriptor yp, for any pair of data samples (X, y¡p) and (Xj, yjp) denoted as (i, j), the physical data variation d(Xi, Xj) is calculated as follows:

d(X,.,Xj) = |\Xt - X\[ x

and the sensory data variation d(ytp, yjp) is calculated similarly.

The corresponding sensitivity in the data pair (i, j) related to yp, denoted as FSp(i, j), can be obtained from the fuzzy model FIS1, i.e. FSp (i, j)=FIS1(d(X„ Xj), d(ytfi, yp))- FSp(i, j) can be considered as a measure of information content of all the physical features in the pair (i,j) related to the sensory descriptor yp.

When removing xk from the whole set of physical features, the sensitivity of the remaining physical features in the data pair (i, j) related to the sensory descriptor yp can be calculated in (2)

AFSk,p (i, j)=FISl (Adk(Xv Xj), d(y,p, yJf)) (2)

where Adk (Xi ,Xj) = d (Xi )-|xki - xkj|

The general fuzzy sensitivity criterion FSkp for all the pairs of data samples when removing the physical feature xk is defined in (3)

= 1/X^Afs,

/=1 j=i+1

,(i, j)

Clearly, when FSk,p < FSp,p for all p£k, then the removed physical feature xk is considered as the least relevant among all physical features.

The FS criterion FSkp is normalized in [0,1]. In this way, the closer the value of FSkp is to 1, the more the physical feature xk is relevant to the sensory descriptor yp. The closer the value of FSkp is to 0, the less the physical feature xk is relevant to the sensory descriptor

Based on the fuzzy sensitivity criterion defined in (3), an algorithm for selecting the most relevant physical features and ranking all the physical features is proposed. The principle of this algorithm is illustrated in Fig. 3.

Fig. 3. The algorithm for ranking physical features using fuzzy sensitivity criterion

2.1.1. Aggregating FS rankings using OWA1 operator weights for Panel 1

After the application of the fuzzy model (FIS1), we obtain a number of ranking lists of all the m physical features for a given sensory descriptor. Each ranking list corresponds to the evaluation scores provided by one trained evaluator. In our approach, only the rank values of physical features are significant for further aggregation because the values of the Fuzzy Sensitivity criterion FS are less significant. In fact, their absolute values cannot be compared to each other since they are generated from different selection loops. In a ranking list of m physical features, the rank value of the most relevant one is m; that of the second one is m-1, and so on. The value of the worst ranked feature is 1.

Before computing the aggregation for different ranking lists, we test the sensitivity of each evaluator to the physical features in order to ensure their equipollency. Then, we use an OWA operator derived from the Regular Increasing Monotone (RIM) linguistic quantifier and the corresponding weight definition to set up the formula of the weights21.

Qa(r)=ra , 0< r <1 and a >0

Indeed, by applying the RIM linguistic quantifier, the weight of the ith evaluator is obtained as follows:

w,=(i/n)a-((i-1)/n)a (5)

where the coefficient a is determined using the following principle: If we wish to select the Ns most relevant physical features, their aggregated ranking scores should be from m to m-Ns+1. Also, the aggregation for n evaluators is performed according to the following principle: if k evaluators (k<n) agree to rank a physical feature as one of the p most relevant ones (i.e. its individual score is between m and m-p+1), then its aggregated score for all the n evaluators should be at least m-Ns+1.

By definition, an OWA operator of n dimensions is a mapping F : ^ ^ with an associated weight vector W= (w1,...,wn J such that ^^ Wi = 1, 0 < wi < 1, i = 1,..., n and F(a1,....,an) = ^"-iw^t where bt is the ith

largest of a1,...,a„.

We suppose that a specific physical feature is selected by the aggregation. By applying an OWA operator to the ranking scores given by all the n evaluators, denoted as a1,...,a„, and the previous aggregation principle and weight definition, we have:

Fa (a1, a2-......, an ) = S bWi ^

k " f k V

(m - p + 1)X Wi +X Wi = (m - p)|-| +1 (6)

/=1 i=k+1

where bt is the largest of a1^..,an.

This operator is a re-ordering processing. During the aggregation, the value ai is not associated with wi but with another weight obtained by comparing all the m rank values. The weights with lower indices correspond to more important features and those with higher indices to less important features. As the minimal condition for being selected corresponds to the aggregated rank m-Ns+1, the coefficient a can be calculated from the following equation:

(m - p)

— I +1 = m - Ns +1

2.1.2. Aggregating human knowledge experts (Panel 2) using OWA2 operator weights

To validate the previous ranking lists given by FIS1 and OWA1 and provide knowledge-based complementary information in order to adjust the obtained ranking lists, we integrate the results of another panel composed of q professional experts

(Panel 2). The experts of this panel have not received training on sensory evaluation but master professional knowledge on products and processes. Their role is to classify the physical features according to their relevancy to each given sensory descriptor. Each expert assigns values to each of the m physical features from the set of (1, 2, 3, 4), in which 4 stands for relevant, 3 moderately relevant, 2 little relevant and 1 no relation. For each sensory descriptor, the aggregated ranking list of the physical features for all the q experts of Panel 2 can be obtained by using another OWA operator (OWA2 model), whose principle is same to that described in Section 2.1.1.

weights issued from the FIS2 aggregation is a result of a linear (triangular) membership functions managed by fuzzy rules whereas the weights obtained from the OWA3 are extracted drawing upon Zadeh's concept of linguistic quantifiers32. These two aggregation techniques will be detailed right after. If both of them yield similar results, we consider that the obtained aggregation result is reliable. In both cases, the obtained GRs are robust and can effectively avoid the discrepancy in the ranking lists obtained from OWA1 and OWA2 due to the sensitivity of experimental data, the perceptual divergence of human evaluators and professional knowledge.

2.2. Combination between FS knowledge (HK) rankings

and human

The two previous ranking lists, i.e. the fuzzy sensitivity FS and the HK are combined in order to obtain a Global Relevancy (GR) value. At this stage, two aggregation techniques, i.e. Fuzzy Inference System (FIS2) and OWA (OWA3) (see Fig. 1) are applied in order to compare their results. In fact, the

2.2.1. Using the FIS2 fuzzy model (FGR)

The combination is based on the fuzzy rules presented in Table 3, obtained from the common qualitative knowledge. All the scores have been normalized in the range of [0, 1] to eliminate the scale effects.

Table 3. Fuzzy rules for aggregating FS and HK rankings

FS (OWA1)

(FIS2) And Very relevant Relevant Average relevant Slightly relevant Not relevant

Very relevant Very relevant Very relevant Very relevant Very relevant Not relevant

HK (OWA2) Relevant Very relevant Relevant Relevant Slightly relevant Not relevant

Average relevant Very relevant Relevant Average relevant Slightly relevant Not relevant

Slightly relevant Very relevant Slightly relevant Slightly relevant Slightly relevant Not relevant

Not relevant Not relevant Not relevant Not relevant Not relevant Not relevant

These fuzzy rules are used to build a fuzzy model in which FS and HK are taken as two input variables and the Fuzzy Global Relevancy score as output variable (FGR). The input and output variables are fuzzy variables each including five fuzzy values: Not Relevant (NR), Slightly Relevant (SR), Average Relevant (AR), Relevant (R) and Very Relevant (VR) (illustrated in Fig. 4).

Obviously, the output variable FGR varies in the interval of [0, 1]. The closer the value of FGR is to 1, the more the corresponding variable xk is relevant.

Fig. 4. Fuzzy values for each input and output data variable in the FIS2 model

2.2.2. Using the OWA3 model (OWAGR)

Similarly to OWA1 and OWA2, in order to generate the weights of OWA3 we consider that the ranking lists yielding from OWA1 and OWA2 have equal chances. We translate equal chances by the following linguistic principle: if a physical feature is ranked in the extreme case at the top of the first ranking (FS ranking) and at the bottom of the second ranking (HK ranking) then, it should be in the middle of the overall ranking (i.e. his overall score is equal to m/2). Hence, the weights of OWA3 are generated using the RIM linguistic quantifier where the coefficient a is obtained from Eq. (7) with n=2, p=1, k=1 and Ns=m/2+1, i.e.

'iY , (T

m| —j +1 - = —. The final values of the OWA

Global Relevancy (OWAGR) are also normalized in the interval of [0, 1] for the further treatment.

2.3. Distinctness between relevant and irrelevant

physical features

Until this stage, the role of the aggregation techniques used previously (Fuzzy or OWA) is restricted in building a single operation of aggregation and provide a ranking by relevance. However, for each ranking list, the scores of the physical features are usually closed each other. Therefore, we need to determine an appropriate threshold in order to separate the set of relevant physical features from irrelevant ones.

The key issue of the proposed percolation technique is to filter automatically and objectively the relevant features by creating a gap between scores of relevant and irrelevant physical features. It permits to automatically generate threshold that can effectively reduce human subjectivity and arbitrariness when manually choosing thresholds. For a specific sensory descriptor, the threshold is defined systematically by iteratively aggregating (n times) the ranking lists generated by OWA3 and FIS2.

The complete percolation technique algorithm is summarized below:

1) Initialize the percolation technique algorithm by obtaining FGR and OWAGR rankings respectively from FIS2 and OWA3 models of the previews functional blocks as shown in Fig. 1.

2) Repeat Until AE <e

a) Merge FGR and OWAGR rankings using FIS2 model to obtain a new FGR ranking

b) Merge FGR and OWAGR rankings using OWA3 model to obtain a new OWAGR ranking

c) Calculate the fuzzy and OWA Absolute Errors 'AE' for two consecutive iterations, i.e.

GRj+1 - GR]

3) Generate the final FGR and OWAGR percolated ranking lists. ■

In our experiments, £ is chosen so that the error AE is less than or equal to 1% of the maximal values of FGR. Since the latter is normalized, then we may choose s =0.01. In general, if the percolation algorithm converges very quickly, then £ can be smaller in order to obtain more precise results. However, if the convergence is slow, £ can be larger in order to reach a faster convergence.

The proposed percolation technique can effectively filter or select the relevant input variables because it's capable of making a sharp difference between the scores of the relevant and irrelevant ones.

3. The case study - Selection of stonewashed

denim fabric physical features

In this real example, we select the most relevant physical features related to hand feeling of a well known finished textile product especially the stonewashed denims, usually considered as the most important quality criteria in jeans' evaluation. The corresponding parameters, including 9 normalized sensory descriptors provided by quality evaluators, 20 physical features measured using devices, and the nature of the evaluation panels (Panel 1 and Panel 2), have been described in the previous sections.

3.1. Aggregated ranking lists generated by FIS1-OWA1 and HK-OWA2

Using OWA1, we select the six most relevant features (Ns=6) from a total of m=20 physical features. Six evaluators (n=6) perform fabric hand evaluation on the selected samples. A physical features is considered relevant if the aggregated rank score is no smaller than m - Ns +1 = 15. For a specific physical feature, if two-thirds of the evaluators (k=4) consider it as one of the p

Table 4. Aggregation weights for OWA1

w1 w2 w3 w4 w5 w6 2 wi

Smooth 0.554 0.142 0.099 0.079 0.067 0.058 1.000

Fluffy 0.589 0.151 0.106 0.084 0.071 0 1.000

Full 0.633 0.162 0.114 0.090 0 0 1.000

Supple 0.589 0.151 0.106 0.084 0.071 0 1.000

Elastic 0.589 0.151 0.106 0.084 0.071 0 1.000

Wrinkle 0.633 0.162 0.114 0.090 0 0 1.000

Soft 0.633 0.162 0.114 0.090 0 0 1.000

Cold 1.000 0 0 0 0 0 1.000

Sliding 1.000 0 0 0 0 0 1.000

Table 5. Aggregation weights for OWA2

w1 w2 w3 w4 w5 w6 w7 w8 2 wi

For each sensory parameter 0.533 0.124 0.086 0.068 0.057 0.049 0.044 0.040 1.000

most relevant features (p=4), then we obtain from Eq.(5) and Eq.(6):

Fa (17,17,17,17,1,1) = 17(w1 + w2 + w3 + w4) + w5 + w6

=14 = 0.33 16

17IIJ+1 -UJ -156,

The weights obtained from Eq.(5) are given in Table 4.

After interviewing the experts of Panel 2, we found that all of them agreed on the following principle: if a physical parameter has all but two excellent assigned values then his overall score should be about 3.75, which is a cut-off value for the relevant physical feature to a sensory one27. In other words, by applying this principle to OWA2, a physical feature is considered as relevant if at least six (k=6) of the eight experts (n=8) in Panel 2 take it as relevant (the assigned value is 4). Then, the final score should be m - Ns +1 = 3.75. In this case (application of OWA2), we have m=4 and p=1.

Fa (4,4,4,4,4,4,1,1) = 4(w1 + w2 + w3 + w4 + w5 + w6) + w7 + w8 = 3.75

6 Y +1 -( 63.75 of* = ^

= 0.303

According to Eq. (5), we calculate the weights for all the sensory descriptors, shown in Table 5.

By aggregating the ranking lists given by different evaluators in Panel 1 and Panel 2, we obtain the scores

and ranking lists for three sensory descriptors, presented in Table 6.

For instance, when considering the sensory descriptor "Smooth", the physical parameters (x13, x16, x19, x12, x2) have been classified on top of relevant parameters by OWA1. When using OWA2 the same parameters have been considered as most relevant but in addition to the physical parameter x15 which was classified on bottom of the list using OWA1.

In order to make more reliable decisions in selection of relevant physical features, it is necessary to further combine these two ranking lists. The combination is performed using two different approaches, i.e. FIS2 and

3.2. Aggregated ranking lists generated by FIS2 and OWA3

Having obtained the aggregated ranking lists with OWA1 and OWA2 models, we further aggregate them to form the final ranking list of physical features for each specific sensory feature. This combination will permit to integrate both sources of information (perception of sensory evaluators and professional knowledge of experts) and overcome the sensitivity of experimental data and perceptual divergence of human evaluators.

Table 6. Aggregation results obtained from OWA1 (FS) and OWA2 (HK) for three sensory descriptors

Smooth

Supple

Elastic

FS HK FS HK FS HK

Ref. Score Ref. Score Ref. Score Ref. Score Ref. Score Ref. Score

x13 19.508 x16 | 4.000 x13 19.774 X5 4.000 x13 19.041 x8 4.000

x16 19.371 x15 4.000 x16 18.881 x6 4.000 x19 18.926 x9 4.000

x19 18.105 x13 3.960 x19 18.691 x12 4.000 x16 18.620 x19 3.960

x14 17.888 x14 3.960 x12 17.951 x3 3.960 x12 17.488 x20 3.960

x12 17.134 x19 3.471 x2 17.922 x4 3.960 x2 17.161 x5 3.917

X2 17.097 x20 3.471 x18 16.427 x7 3.867 x20 16.647 x6 3.917

x18 16.363 x2 3.429 x14 15.328 x1 3.728 x14 16.280 x7 3.771

x17 14.390 x3 3.385 x10 14.611 x19 3.728 x17 16.087 x3 3.743

x10 13.904 x4 3.385 x17 13.847 x20 3.728 x10 15.807 x4 3.743

x6 12.292 x12 3.361 x20 12.328 x8 3.660 x18 15.130 x10 3.571

x20 12.050 x7 3.002 x7 12.311 x9 3.660 x5 13.267 x11 3.361

x1 11.552 x17 2.934 x6 11.186 x11 3.620 x6 13.028 x1 3.237

x7 11.428 x5 2.660 x5 8.774 x2 3.611 x15 10.785 x12 3.144

xs 7.758 x6 2.660 x15 8.058 x17 3.571 x4 9.735 x17 3.058

x4 7.589 x11 2.611 x1 7.881 x10 3.535 x7 8.800 x2 2.618

x3 6.747 x1 2.574 x4 7.736 x13 2.660 x1 6.669 x13 2.468

x15 6.367 x18 2.574 x3 6.877 x14 2.660 x3 5.169 x14 2.468

x9 4.546 x10 2.525 x9 4.514 x15 2.468 x9 4.845 x15 1.811

x8 3.225 x8 2.344 x11 2.766 x16 2.468 x8 2.774 x16 1.811

x11 1.796 x9 2.344 x8 2.518 x18 2.468 x11 1.589 x18 1.743

Table 7. Aggregation of the ranking lists by using FIS2 (FGR) and OWA3 (OWAGR) for three sensory descriptors

Smooth

Supple

Elastic

Ref. Score Ref. Score Ref. Score Ref. Score Ref. Score Ref. Score

x16 0.996 x16 0.920 x12 0.950 x6 0.920 x19 0.988 x13 0.914

x13 0.989 x15 0.920 x19 0.883 x12 0.909 x20 0.926 x19 0.905

x14 0.944 x13 0.919 x2 0.823 x5 0.908 x5 0.824 x20 0.867

x19 0.807 x14 0.862 x6 0.764 x19 0.825 x6 0.817 x5 0.848

x2 0.765 x19 0.812 x7 0.749 x7 0.806 x10 0.812 x6 0.844

x12 0.747 x12 0.778 x20 0.702 x2 0.792 x12 0.773 x12 0.794

x15 0.648 x2 0.777 x10 0.699 x20 0.761 x17 0.713 x17 0.764

x20 0.632 x20 0.670 x5 0.698 x3 0.731 x4 0.687 x10 0.759

x17 0.543 x17 0.475 x17 0.690 x4 0.715 x13 0.678 x4 0.668

x18 0.498 x4 0.456 x4 0.655 x17 0.706 x7 0.668 x9 0.657

x4 0.486 x7 0.450 x3 0.632 x10 0.690 x2 0.653 x7 0.574

x7 0.474 x18 0.370 x13 0.585 x13 0.501 x9 0.614 x2 0.550

x3 0.463 x3 0.365 x1 0.580 x1 0.436 x14 0.595 x3 0.483

x10 0.411 x5 0.242 x16 0.499 x9 0.287 x3 0.563 x14 0.467

x6 0.402 x6 0.241 x9 0.464 x14 0.218 x8 0.558 x1 0.381

x1 0.355 x1 0.224 x14 0.450 x11 0.119 x16 0.527 x8 0.356

x5 0.267 x10 0.209 x18 0.424 x15 0.086 x1 0.486 x16 0.248

x11 0.085 x9 0.089 x8 0.409 x18 0.084 x18 0.408 x15 0.135

x9 0.082 x11 0.088 x11 0.402 x16 0.083 x11 0.377 x11 0.082

x8 0.042 x8 0.087 x15 0.169 x8 0.081 x15 0.291 x18 0.081

The combination using fuzzy techniques (FIS3) takes FS and HK as fuzzy input variables and fuzzy global relevancy FGR as output variable. The five membership functions for these fuzzy variables are presented in Fig. 4.

The combination using OWA3 is based on the following rule: if a physical feature has, in the most divergent condition, a good score in one ranking list and bad score in another one, then its overall ranking score should be around 10. Therefore, by applying Eq. (6) and (7), we have: Fa (20,1) = 20w1 + w2

= 0.928

, ,1] -10H

v2) U) U) 19

According to the Eq. (5), we deduce the following weights (0.526 0.474) for aggregating the FS and HK ranking lists respectively, to obtain the OWA global relevancy OWAGR ranking list.

By using the two previous combination methods (FIS2 and OWA3), we obtain the values of FGR and OWAGR respectively and then can easily rank the 20 physical features from the most to the least relevant one. Indeed, in Table 7 and for "Smooth" sensory descriptor, FGR and OWAGR agreed that the eight most relevant physical features are (x2, x12, x13, x14, x15, x16, x19, x20) although with different classifications. However, we cannot sharply distinguish the relevant features from irrelevant ones due to very close aggregated scores between them (See Table 7). At the next step, we use the percolation technique to cause an abrupt variation in the scores of the relevant and irrelevant features.

3.3. Separation of the relevant physical features from irrelevant ones

Having applied the percolation method described in Section 2.3 with e = 10-4, we can observe that the set of relevant physical features is separated as shown in Table 8. The number of iterations needed to achieve the convergence is around 26. When we change the percolation threshold to s = 0.01, similar results are obtained within about 15 iterations.

Having applied the percolation technique, we separate the relevant physical features from irrelevant ones for each sensory descriptor. The results of three descriptors are shown in Table 8. From the final scores, we can clearly note that the relevant features are very

different from the irrelevant ones. Besides, we can easily observe that the most relevant features have not changed when comparing Table 7 and Table 8. Nevertheless, apparently the order of the physical features changes after applying percolation, yet the scores several physical features are identical therefore, the order is indeed not different. For the presented descriptors, the numbers of relevant physical features vary: 8 selected relevant features for "Smooth" and 12 for "Supple" and "Elastic". The originality and performance of the proposed relevant feature selection method can be shown from these various results. Instead of selecting identical numbers of features with a predefined threshold, the proposed method can be adapted to the specific natures of the complex relations between sensory descriptors and physical features, in order to propose lists of relevant features of different sizes for different descriptors.

In the real application of stone washed denim fabrics, the final physical feature selection results can be validated by the professional textile knowledge or results in textile research. Some analysis is given below. The relevant physical features for the descriptor "Smooth" include: surface thickness "St" (x2), coefficient of friction "MIU" (x13 and x14), geometric roughness "SMD" (x15 and x16) and the elastic deformation of the surface texture (x19 and x20). Obviously, all the surface features are selected except the drape coefficient "F" (x12). "F" is a parameter that describes the way in which fabric flows or falls with gravity. From the point of physical view, this feature is less related to surface smoothness. Nevertheless, the relevancy of "F" to smoothness can be validated by the study given by Hu and Chan33, who found that the mean deviation of friction coefficient is highly correlated with the fabric drape coefficient.

The selected relevant features for the descriptor "Supple" are mainly physical parameters related to the suppleness, apart from (x13) which describes fabric surface (see Table 8). However, the relevancy of x13 to "Supple" can also be validated by the study of Hu and Chan33.

According to the Australian Wool Textile Objective Measurement Executive Committee AWTOMEC34, the descriptor "Elastic" is related to tensile, compression and shear. These three properties are represented by the set of physical features (x5, x6, x9, x10) for tensile, (x2, x19, x20) for compression and (x7) for shear. Indeed,

all the stated physical features have been taken as the most relevant ones to the descriptor "Elastic" in Table 8. Also, conform to the conclusion obtained by Kim and Vaughn35, the other selected relevant physical features of "Elastic" have a good correlation with the extensibility property.

The previous discussion on fabric mechanical properties show that the proposed relevant feature selection method is effectively conform to the conclusions obtained in the other textile research work and general textile standards.

Table 8. Extraction of the set of the relevant physical features

Smooth

Supple

Elastic

OWAGR FGR OWAGR FGR OWAGR FGR

Ref. Score Ref. Score Ref. Score Ref. Score Ref. Score Ref. Score

x16 1.000 x16 0.920 x12 1.000 x12 0.920 x19 1.000 x19 0.920

x13 0.807 x13 0.757 x6 0.806 x6 0.757 x13 0.806 x13 0.757

x15 0.806 x15 0.757 x5 0.806 x5 0.757 x20 0.806 x20 0.757

x14 0.806 x14 0.757 x19 0.806 x19 0.757 x5 0.806 x5 0.757

x19 0.806 x19 0.757 x2 0.806 x2 0.757 x6 0.806 x6 0.757

x2 0.806 x2 0.757 x7 0.806 x7 0.757 x12 0.806 x12 0.757

x12 0.806 x12 0.757 x20 0.806 x17 0.757 x10 0.806 x10 0.757

x20 0.806 x20 0.757 x3 0.806 x20 0.757 x17 0.806 x17 0.757

x17 0.194 x17 0.243 x4 0.806 x3 0.757 x4 0.806 x4 0.757

x4 0.194 x4 0.243 x17 0.806 x4 0.757 x9 0.806 x9 0.757

x7 0.194 x7 0.243 x10 0.806 x10 0.757 x7 0.806 x7 0.757

x18 0.194 x1 0.243 x13 0.806 x13 0.757 x2 0.806 x2 0.757

x3 0.194 x3 0.243 x1 0.194 x1 0.243 x14 0.194 x14 0.243

xg 0.194 x5 0.243 xg 0.194 xg 0.243 x3 0.194 x3 0.243

x10 0.194 xg 0.243 x14 0.194 x14 0.243 x8 0.194 x8 0.243

x1 0.194 x10 0.243 x11 0.194 x11 0.243 x1 0.194 x1 0.243

x5 0.194 x18 0.243 x18 0.194 x18 0.243 x1g 0.194 x1g 0.243

xg 0.194 xg 0.243 x1g 0.194 x1g 0.243 x11 0.194 x11 0.243

x11 0.194 x11 0.243 x8 0.194 x8 0.243 x18 0.194 x18 0.243

x8 0.000 x8 0.080 x15 0.000 x15 0.080 x15 0.000 x15 0.080

8 Relevant physical features n =30 iterations

12 Relevant physical features n =26 iterations

12 Relevant physical features n =26 iterations

3.4. Correlations between the selected physical parameters

In practice, there are often dependencies between input variables. It is more reasonable to choose the physical parameters that are more sensitive to the sensory descriptor, but less correlated with other input variables. For this reason, we propose to calculate the Bravais-Pearson coefficient R obtained from Eq. (9) for each pair of the relevant parameters (xk, xp) as:

R( xk > xp ) =

X (xik - xk )-(xip - xp) (9)

X (xk - xk )2 JX (xip ~ xp )

After selecting the set of relevant parameters, the objective of the next step is to keep one of two highly correlated variables with other variables in the list of the relevant physical parameters according to the following algorithm:

1) For each pair of the relevant parameters (xk, xp), calculate the coefficient (R);

2) If xk and xp e Xr; k ± p; |R(xk, xp)| > t (correlation threshold) and FSt,i> FSp?l, xp then must be removed from the list of the relevant parameters Xr.

3) Xr = Xr \{xp} ■

Undergone this algorithm, the input variables obtained in the XR final list are the most relevant variables to a specific sensory descriptor. The threshold is defined by the experts. The higher the threshold t, the less variable in the final list are correlated. The chosen

threshold t is 0.75. Following this threshold value, we obtain the sets of most physical parameters as presented in Table 9.

Table 9. Sets of the most relevant physical features

Smooth x16 x13 X15 x14 X19 X12

x20 X2

Fluffy x13 x16 X15 X2 X14 X19

x17 x20

Full Xl x13 x16 X12 X6 X19

x10 X7 X5 x14 x15

Supple x12 X6 X5 X19 X2 X7

x20 x10 x17 x13

Elastic x19 x13 X20 X5 X6 X12

x17 x10 X4 X9 X2

Wrinkle x19 X5 x6 x16 X13 X20

x12 X2 X7 x14 x17 x10

Soft x16 x15 X13 X2 X14 X12

X7 x19 x6 x18

Cold x13 x16 X15 X2 x18 X14

x12 x19

x13 x16 x14 X12 X15 X19

Sliding x20 x17 x6

After checking the correlations between physical parameters, the number of inputs was reduced according to the number of possible correlations between them (see Table 9).

4. Conclusion

This paper presents an integrated method for selecting relevant physical features related to a specific sensory descriptor of a given consumer product. This proposed method integrates the data sensitivity criterion and human professional knowledge by using the techniques of fuzzy logic and OWA operators. Considered as an approach in Multi-Evaluators and Multi-Criteria Decision Making (MCDM), we originally integrate evaluator perception on products and professional knowledge of experts. The proposed selection method is applicable to any consumer good, as far as its quality is objectively quantified by physical measurements and subjectively evaluated by persons. Our approach combines the degree of relevancy obtained from the experimental data-based Fuzzy Sensitivity (FS) and the Human Knowledge (HK) using both the fuzzy model and OWA operators.

In the proposed aggregation procedures for ranking lists obtained by different sources, the OWA operators

associated with some linguistic rules have been applied. They have been proved to realize aggregations between different objectives, by allowing a good compensation between scores. The effectiveness of the OWA operators as well as the data sensitivity-based fuzzy model has been validated through the real example of stonewashed denim fabric. In fact, the ranking lists of the physical features corresponding to FGR and OWAGR are very close each other and both of them are robust, insensitive to measured data noises and conflicts. Moreover, they are more sensitive to significant data and more efficient for physical interpretation. In order to obtain more reliable results for selection of relevant physical features, a percolation technique has been applied for combining the fuzzy global relevancy and OWA global relevancy criteria in order to clearly distinguish scores of the relevant physical features from those of irrelevant ones.

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