L -fuzzy concept analysis as a decision-making tool in the development of bio-based materials

The use of L -fuzzy concept analysis for the determination of the formulations of gelatin-based ﬁlms that fulﬁll speciﬁc functional properties is reported. The requirements of water contact angle (CA), water vapour transmission rate (WVTR), L* and b* colour values, tensile strength (TS), elongation at break (EB) and gloss values were speciﬁed for both fatty food packaging and wound healing applications. Once the required formulations were estimated by L -fuzzy concepts, the experimental results showed a high accordance with the predicted values of the ﬁnal properties, as well as with the requirements. The errors were lower than 20% in all cases, except in colour (for food packaging) and gloss and elongation at break (for wound healing), in which the ﬁnal properties were even better that the estimations. This analysis allowed ﬁnding the required formulations in a highly cost-eﬀective way.


Introduction
Materials design is a common task in laboratories and industry, and many times its efficiency may be conditioned by the complexity of the formulation and/or the limitations of the time or money required for the analysis. The adjustment of a manifold of parameters with a view to achieving the desired final properties of a material can be performed on a basis of previous knowledge, but in any case, the complex interactions among these parameters make the process a challenge. The designer is commonly blind exploring a wide multidimensional space of combinations, and he/she must bear in mind that, for a given set of chemical constituent species, the properties that he/she is trying to achieve may be out of hand.
In order to address this problem, computer-aided mathematical treatment is a suitable tool that allows systematic and rational materials design. Doing so, unnecessary tasks and their attached costs are avoided, since the mathematical analysis greatly reduces experimental labor by focusing on only indispensable measurements. In this context, fuzzy sets theory [1] helps the designer when the requirements that guide design are imprecise, in the sense that either values for different magnitude may be acceptable in the proximity * Correspondence: c.alcalde@ehu.es 1 Department of Applied Mathematics, University of the Basque Country UPV/EHU, San Sebastian, Spain Full list of author information is available at the end of the article of ideal objective values; or the desirability criteria are formulated in a textual form, e.g. "it is recommended", or "it is very important that" [2,3].
In this scenario, the research on alternative solutions to conventional plastics is promoted by serious environmental problems and the rising consciousness about sustainability that population and governments are acquiring. Bio-based materials are an interesting approach to be exploited, because of their versatility and increasing functionality when applied to a wide range of requirements. In this regard, bio-based materials showed promising properties as wound dressing [4][5][6][7][8] and food packaging [9][10][11], among many others.
In reference to wound healing application, to accelerate the wound healing process, an ideal wound dressing should display adequate gas permeability, biocompatibility, provide an antibacterial environment [12,13] as well as maintain a moist environment at the wound surface [14].
Within the field of food packaging, food oxidation, that is considered a major cause of quality deterioration affecting both the nutritional and sensory quality and safety of foods, may be accelerated by moisture, light, oxygen or exposure to high temperatures [15]. Packaging for fatty food requires a particular attention, since auto-oxidation, which is the oxidation in the absence of light, can be responsible for the deterioration of olive oil flavor termed "oxidative rancidity" [16]. Unsaturated fatty acids are generally susceptible to lipid oxidation and to development of undesirable odor [17][18][19]. The prevention or inhibition of these processes during fatty food storage are thus required for maintaining their quality standard.
An efficient design of bio-based materials for both applications of wound dressing and fatty food packaging is needed. In order to tackle this problem, we propose the use of a methodology based on the analysis of L-fuzzy concepts [20][21][22] obtained from initial requirements that will provide the designer with a set of formulation parameters which will plausibly lead to the desired properties values. Or, to be precise, and attending to the fuzzy nature of the method, values which will lie near to some target values.

Materials
Fish gelatin was purchased from Healan Ingredients (East Yorkshire, UK). Glycerol and gallic acid were obtained from Panreac (Barcelona, Spain). All chemicals were used as received without further purification.

Film preparation
Fish gelatin films were prepared by mixing gelatin and gallic acid in distilled water. The acid contents employed in this work were 5, 10 or 15 wt. % on gelatin basis. Solutions were heated at 80°C for 30 min and stirred at 200 rpm. Then, 0, 5, or 10 wt. % glycerol (on gelatin basis) was added as a plasticizer and solution pH was adjusted with 1 N NaOH; the pH values used in the present work were 4.50, 7.25 and 10.00. The heating procedure was repeated and finally, solutions were poured into Petri dishes and allowed to cool for 48 h at room temperature. All films were conditioned in a controlled environment.

Mechanical behavior
Tensile tests were performed in an electromechanical testing system (MTS Insight 10) in order to determine tensile strength (TS) and elongation at break (EB). Tests were carried out according to ASTM D638-03.

Colour and gloss
Colour values of films were measured using a portable colorimeter (CR-400 Minolta Croma Meter). Film specimens were placed on a white plate, and the CIELAB color scale was used to measure color: L*= 0 (black) to L*= 100 (white) and -b*(blueness) to +b*(yellowness). Standard values for the white calibration plate were L*= 97.39 and b*= 1.77. With these values and considering standard light source D65 and standard observer 2 degrees color parameters L* and b* were measured.
Gloss of the films was measured with a Minolta gloss meter (MultiGloss 268 Plus). Gloss was measured at a 60°incidence angle, according to ASTM D523-14. Results were expressed as gloss units, relative to a highly polished surface of black glass standard with a value near to 100.

Contact angle
A contact angle system (model Oca20, Dataphysics Instruments) was used to measure the contact angle of water in air on the surface of gelatin films. A film sample (20 mm × 80 mm) was put on a movable sample stage and leveled horizontally; then a drop of about 3 µL of distilled water was placed on the surface of the film using a microsyringe. The contact angle was measured in a conditioned room by recording contact angle values. Image analysis were carried out using SCA20 software.

2.6
Water vapor permeability WVTR of the films was determined according to ASTM E96-00. The sample film was cut into a circle of 7.40 cm diameter and the test area was 33 cm 2 . The setup was subjected to a temperature and relative humidity of 38°C and 90%, respectively. Water vapor transmission rate (WVTR) was calculated as where, G is the change in weigh (g), t is the time (day), and A is the test area of the film (m 2 ).
3 Description of the problem 3.1 Description of the materials designing conditions When a novel material is designed, formulation and processing conditions play a vital role in the properties of the material during service. These properties are related with different magnitudes (chemical, physical, mechanical, thermal, optical,...). For each application, we know which target output values these magnitudes should show in an optimal performance, and the question is to find the formulation that will provide us with it.
In order to achieve this goal, not only the constituent chemical species that optimize properties must be identified, but also their quantity. These quantities will be input values to be determined in an optimization study. Usually, there is a previous knowledge that helps the designer determining the approximate ranges where the ideal input values might lie in order to obtain the desired performance output values.
When finding the desired input values in their already known approximate ranges, the traditional trial and error approach with many involved variables usually require high costs of energy, time and raw materials. There are so many combinations and variation to study that this is an inefficient protocol for analyzing the influence of each parameter on the performance of the design material. In addition to this, usually the analysis is carried out via one-variable-at-atime test procedure, which does not consider the crossinteractions between the different input values of the formulation.
Often, these inputs and outputs are blurry: the designer would accept input values at around some particular input values, because he/she is willing to admit output values at around some particular output values. In this case, we can ascribe desirability fuzzy variables to them, and perform a study based on the fuzzy concepts theory.
Within this approach, and with a strict analysis of the involved variables, a set of plausible input values of the formulation can be provided in an efficient way reducing the consumption of time or resources, avoiding an inefficient protocol with unnecessary tasks and their attached costs.

Properties required for the material applications
Gelatin-based films display interesting characteristics that make them suitable for being applied to both wound healing and food packaging. Barrier properties, mechanical behaviour, and optical properties are vital properties when designing food packaging materials [23]. Regarding wound healing application, the layer is designed to cover the wound, and to give mechanical strength to the dressing. In addition, it needs to control moisture transmission to prevent fluid loss and dehydration, while allowing exudate removal [24].
Bearing this in mind, the following output properties were chosen for further analysis: contact angle (CA), water vapour transmission rate (WVTR), L* and b* (as far as colour properties are concerned), tensile strength (TS), elongation at break (EB) and gloss.
Considering the expertise in the field of the BIOMATbiopolymeric materials group, a set of target output properties for the candidate formulations for both wound healing and food packaging applications has been proposed, which takes into account the specificities of gelatin-based films. This is the first milestone in order to launch the L-fuzzy concept analysis. These target properties are listed in Table 1.
In the case of water vapor transmission rate, for fatty food packaging application no target value was selected because it is not a key property for this application. The same happened for contact angle in wound healing. It may be the case that the target properties are impossible to be achieved simultaneously for a given material, given the trade-off among them. In this case, the proposed estimation method will provide us with a compromise solution in which the estimate properties lie near the target properties. The basic concepts of the L-fuzzy contexts theory and the way of applying it to the material formulation selection are discussed in the following section.
4 Modeling of the experiment using L-fuzzy contexts 4.1 L-fuzzy concept analysis Formal concept analysis is a mathematical tool that was developed by R. Wille in 1982 [25,26] with the aim of processing conceptual knowledge and representing it in a formal way.
A formal context is a mathematical structure formed by a triple (X, Y, R), where X and Y are finite sets of objects and attributes respectively, and R ⊆ X ×Y represents the binary relation among them. Information involved in the context is provided by pairs (A, B) with A ⊆ X and B ⊆ Y , called formal concepts. The set A is the extension of the formal concept and B is its intension, and they verify that A * = B and B * = A, being (·) * the derivation operator that associates objects and attributes as follows: A * is the set of attributes common to the objects in A and B * the set of objects having all the attributes in B.
Along these lines, a formal concept can be interpreted as a group of objects A sharing all the attributes of B.
Different techniques based on formal concept analysis has been used previously to achieve predictions in in the area of chemistry [27][28][29].
In order to admit different relationship grades among objects and attributes, Burusco and Fuentes-González extended Wille's formal concepts analysis to the fuzzy case with the definition of L-fuzzy context [20][21][22]. An L-fuzzy context is defined as a tuple (L, X, Y, R) where L is a complete lattice, X and Y are respectively the set of objects and attributes, and the incidence relation R ∈ L X×Y takes values in the lattice (L, ≤).
To extract information from these L-fuzzy contexts, the derivation operators were defined for all sets A ∈ L X and B ∈ L Y by means of the following expressions [22]:  being I a fuzzy implication operator defined on L. L-fuzzy concepts constitute the tool that allows to visualize the stored information in the context. These concepts are pairs (M, M 1 ) where the set M ∈ L X is a fixed point of the constructor operator ϕ which is defined using the derivation operators as ϕ(A) = (A 1 ) 2 = A 12 for any A ∈ L X .
Equivalently, the L-fuzzy concepts can be defined from the point of view of the attributes as pairs Given a subset of objects A ∈ L X (or, a subset of attributes B ∈ L Y ), it is possible to calculate the associated L-fuzzy concept by repeatedly applying the constructor operator ϕ (or φ) until a fixed point is obtained. We can get a fixed point just applying twice the derivation operators if the used implication operator is residuated [30,31]. Thus , if I is defined from a t-norm T such that for all (a, b) ∈ [0, 1] then the pair (A 12 , A 1 ) is the L-fuzzy concept associated with A. Similarly, the L-fuzzy concept associated with B ∈ L Y will be the pair (B 2 , B 21 ).
The L-fuzzy concept associated with a subset of objects A (or attributes B) provides the "stable" situation in the context nearest to the initial conditions fixed by A (or B).
In this work, and in order to simplify the calculations, the implication operator chosen to obtain the derivated sets was Lucasiewicz implication: which is a residuated implication operator.

Experimental design for formulation optimization
The input values of formulation to be adjusted were chosen to be glycerol content, pH and gallic acid content. In order to feed the fuzzy methodology with data for the adjustment, a batch of samples was tested, in which the analyzed glycerol content values were 0%, 5% and 10%, the selected pH values 4.50, 7.25 and 10.00, whereas the gallic acid content values were 5%, 10% and 15%. The experimental values of the output properties obtained to feed the process for the construction of the L-fuzzy context are shown in Table 2. . . , y 10 } was formed by the different properties that were analyzed in each formulation: • y 1 : Glycerol content.
• y 9 : Elongation at break. • y 10 : Gloss value at 60 • incidence angle. The relation among objects and attributes R ∈ L X×Y was calculated rescaling the elements of the matrix E formed by the average experimental values in Table 2 as follows: The obtained relation is shown in Table 3.

L-fuzzy concepts associated with the initial requirements
Once defined the L-fuzzy context (L, X, Y, R), the characteristics of the material were estimated from the L-fuzzy concepts associated with the set of attributes representing the desired requirements shown in Table 1.
In order to get values in L, the required values for the properties were normalized following the rescaling method used with relation R, and the membership values of the attributes representing the unknown properties were initially supposed to be 0.
Thus, material properties for fatty food packaging application were represented by the following fuzzy set  Table 3 Relation of the L-fuzzy context.  of attributes: B ={y 1 /0, y 2 /0, y 3 /0, y 4 /0.82, y 5 /0, y 6 /0.99, y 7 /0.65, y 8 /0.11, y 9 /1, y 10 /0.20} And, after calculating the associated L-fuzzy concept, we focused on its intension: B 21 ={y 1 /0.63, y 2 /0.60, y 3 /0.63, y 4 /0.89, y 5 /0.81, y 6 /0.99, y 7 /0.65, y 8 /0.98, y 9 /1, y 10 /0.70} The intension of a concept can be interpreted as a set of attributes that are always found together in the context. Hence, the obtained set of attributes B 21 can be interpreted as the nearest set of attributes to the required set B that are given at the same time.

Results and discussion
Given the experimental data for the L-fuzzy concept analysis, the estimated formulation values for an op-timal performance are shown in Table 4. For a fatty food packaging application, this methodology suggests values of around 6.3% in glycerol content, 11% of gallic acid content and a pH of 8; whereas for wound healing use, the predicted values are approximately 1% of glycerol, 9% of gallic acid and a pH of 8.
In reference to fatty food packaging application, as can be observed in Table 5, the estimated achievable values were very similar to the target functional properties values, apart from gloss values and tensile strength, and the proposed values by the L-fuzzy concept analysis were higher than the selected objective values. As far as gloss values are concerned, according to the obtained estimation, obtaining values near to 50 is not possible taking into account the constraints in the rest of the properties, suggesting that gloss values must be greater. Since gloss values are inversely related to the roughness of the film surface [32], greater gloss values would indicate smoother surfaces. Something similar happened with tensile strength values, estimated values are greater than the target. It is worth mentioning that the mechanical behavior of the material proposed by the estimation method is even better than that of the specification, having greater tensile strength values without decreasing elongation at break values. In the case of the water vapor transmission rate, as for fatty food packaging application no target value was selected due to the fact that it is not a key property for this application, for the proposed formulation the L-fuzzy concept estimates a value of 1243 g m 2 ·day for this property. To verify the employed estimation method, a biobased material was developed with the proposed formulation and the experimental responses of material functional properties were measured. As shown in Table 5, for the majority of the properties, the values given through the L-fuzzy context are reasonably similar to the experimental values. The obtained errors are less than 21% for all cases, apart from L* colour value in which the error is higher and the obtained experimental value is lower than the estimation. This gives rise to a higher luminosity of the film, greater than the specified in the requirement, which is adequate for the food packaging application. Moreover, it could be confirmed that, as the L-fuzzy concept predicted, gloss values were higher than the objective values and the mechanical performance of the material is even better than required for the estimated formulation.
Concerning wound healing application, the formulation proposed by our estimation method was approximately 1% glycerol content, 9% gallic acid content and a pH of 8. It is worthy of note that values of the functional properties of that formulation were very close to the defined objective values for the application. Gloss is again the principal magnitude that is not so close to the required value, and it was slightly higher than the estimation. In the case of the material application for wound healing, no target for contact angle was selected as it is not considered a key factor, and the estimation given by the L-fuzzy concept was 71.7°for this functional property.
As in the case of fatty food packaging application, for the validation of the estimation method, a material with the proposed formulation was developed and its functional properties were measured. Obtained material performance was very similar to the predicted by the L-fuzzy concept, as the measured values for most of the properties were very close to the values obtained by the calculated estimation. Obtained errors were lesser than 18% except in gloss values and elongation at break. For these properties, the errors were 33.5% and 26.4% respectively. Experimental values of elongation at break were higher than the estimated values provided by the L-fuzzy concept, giving rise to a slight improvement in the mechanical performance of the material, although tensile strength values were slightly lower than expected. Gloss values are related with the surface of the material. Having greater values in the bio-based material for wound healing application could lead to a decrease of the film roughness. Although the values were greater than the specification, the properties obtained lead to a better performance of the material.

Conclusions
In the present work, L-fuzzy concept analysis was employed as decision-making tool to support the material formulation decision for obtaining required functional properties for fatty food packaging and wound healing applications. Results obtained after the implementation of the proposed methodology revealed that values predicted by the L-fuzzy concepts were very similar to the target values for the majority of the properties for both applications. Moreover, after analyzing the material performance with the estimated formulations, the experimental values were very close to the proposed approximate ones. The error between the approximate values estimated by the L-fuzzy concepts and experimental values carried out with the proposed formulation were lesser than the 20% in most of the functional properties. In addition, in the cases in which the variation between predicted value and experimental value was greater than the 20%, this difference causes an improvement of the required properties, obtaining a better performance of the material.
Definitely, the formulations proposed by L-fuzzy concept analysis would be the best starting point for studying the effect of each parameter in the functional Table 5 Material requirements, fuzzy estimation, experimental values and obtained error for fatty food packaging and wound healing applications. Parameters of material formulation are glycerol content (GLY), gallic acid (GA) and solution pH. Studied material properties are water contact angle (CA), water vapour transmission rate (WVTR), L* and b* colour values, tensile strength (TS), elongation at break (EB) and gloss values at 60°incidence angle.

Formulation
Bio-based material properties properties of the material. Instead of having a vast array of formulations to study, using this estimation method helps to identify approximately the values of the parameters in the formulation to obtain the desired properties, and to know if it is possible to obtain those functional properties for a specific system.