This paper presents the concept of the fat quality index (FQI), which is based and established in pursuancet of the current food regulations. It is a numerical value representing the correlated information of all parameters that provide the definition of fat according to the international guidelines. With the implementation of this index, it is possible to compare different types of fats using a single numerical value, which facilitates the elucidation of the effects of treatment processes or origins of fats. The FQI includes all the parameters considered in the regulations by incorporating a subindex for each parameter and using the minimum and maximum limit values to model and adjust an equation describing the quality of fat according to the standard. Finally, the procedure is used to obtain indices based on other experimental works that assessed the quality of fat samples produced under different operating conditions, treatment, origin or processes, allowing for better comparison and evaluation. Therefore, this index is an excellent analytical tool for assessing the quality of fats from different origins for human consumption.
Fats for human consumption have a complex composition that includes sterols, triglycerides, phospholipids, and other components that depend on the type, source, region, or processes to which they may be subjected. However, the quality parameters established in the applicable regulations are numerous, which is problematic during the comparison, analysis of results, and monitoring of processes related to the production and study of fats.
Therefore, to facilitate the interpretation of the obtained results and make them comparable, the fat quality index (FQI) has been established. It has been established in a similar manner to the water quality index (WQI) applied in Mexico (FloresJacinto
The proposed FQI integrates, via a mathematical expression, the values of different parameters and allows the use of a verbal or numeric expression or a color scale to establish the specifications and applicable uses of a specific fat under analysis: pharmaceutical applications, human consumption (Diario Oficial de la Federación,
To design the FQI, the steps outlined for the development of the WQI are followed. Initially, the selection of the parameters is established, which is performed according to the Delphi and Denius methodology based on criteria established by regulations and scholarly consensus, as well as any criteria relevant to the definition of quality. Therefore, in the case of fats, the parameters used are those established by both the current regulations in force in Latin America and the Codex Alimentarius, as well as by regulations that, although not in force, contain and provide information on different regions to the FQI, as the regulations establish quality criteria and the minimum or maximum allowed values of fats for human consumption whether they are of animal, plant, or mixed origin.
Once the selection of variables is complete, the subindex of each parameter must be established to transform the value of the dimensional parameter into a dimensionless value and homogenize each value for subsequent combination with the other subindices that make up the FQI.
To determine the subindices of each parameter, the method of curves was used, based on mathematical models in which the boundary conditions are set according to the regulations (SamboniRuiz
The selection of the parameters that determine the FQI depends on the intended use of the fat being evaluated; the most important concern is meeting the requirements for the use of fats for human consumption, either in pharmaceuticals or as food. Based on this criterion, the parameters required by the regulations for lard, cocoa butter, palm oil, butter, and margarine were analyzed.
It is known that the applicable regulations were drafted with the participation of several experts, companies, and institutions from each country. This panel of experts pooled their experience and knowledge from different countries and incorporated the idiosyncrasies of each country and region in Latin America.
The fat quality index scale ranges from 0 to 100, and represents the quality of the fat as a food product. Five intervals of fat quality are identified: 91 to 100 is classified as excellent, 71 to 90 is considered good, average quality ranges from 51 to 70, poor quality is from 26 to 50, and very poor quality ranges between 0 and 25.
The quality index value is determined by the sum of the product of the subindices and the weight that is set for each parameter.
To determine the value of the Fat Quality Index, an equation incorporating the weighed arithmetic average of each parameter was used:
Where
In the construction of the FQI, only those parameters intrinsic to fats were considered, without considering additive compounds such as colors or antioxidants, which are added during the formulation and packaging processes and, therefore, cannot be considered main components of the fats.
Fat quality parameters set by the regulations
Parameter  Symbol  Weight  Fat  Regulation applied  



B  CF  VL  L  
Humidity%  H  4  A  A  A  A  1, 2, 3, 4, 5, 6, 11, 14, 15, 18, 21, 22, 23, 25, 26 
Melting point  MP  4  A  A  A  A  1, 5, 9, 14, 15, 17,18, 21, 22, 23, 24, 25, 26 
Solidification point of fatty acids in °C  SP  4  NA  NA  NA  A  18 
Specific gravity of 40/25 °C  ρ  4  NA  NA  A  A  17, 18, 20 
Refractive Index 313 K (40 °C)  η  4  A  NA  A  A  1, 2,3, 16,17,18, 19, 20, 22 
Saponification Index  Sap  5  A  NA  A  A  1, 2, 3, 14, 1, 5, 16,17,18, 20, 21, 22 
Unsaponifiable materials (g/kg fat)  USap  4  NA  NA  NA  A  14, 18, 20, 21, 23, 24, 25 
Iodine Index (Hanus)  I  5  A  NA  NA  A  1, 2, 3, 17,18, 19, 20, 21, 22 
Acidity expressed in lactic acid% (m/m)  A  4  A  NA  NA  NA  3 
Free acidity (%) as lauric acid%  A  NA  NA  NA  A  17, 18  
Free acidity (%) as oleic acid  A  NA  A  A  A  8,11,14,15, 16,18, 23, 24, 25  
Peroxide Index (meq. of O_{2}·kg^{−1} fat)  P  5  A  A  NA  A  4, 10, 14,18, 21, 23 
Fat% (m/m)  F  5  A  A  NA  NA  2,3,4,5,7, 11, 24 
Reichert Meisst index  Re  4  A  NA  NA  NA  1, 2 
Polenske Index  Po  A  NA  NA  NA  1, 2  
Kirchner index  Ki  A  NA  NA  NA  1  
Böemer Index  Bo  4  NA  NA  NA  A  18 
Suspended solids  Ss  4  NA  NA  NA  A  18, 19 
Rancidity  R  5  NA  NA  NA  A  18 
Sodium soap  sso  4  NA  NA  NA  A  20 
Unsolvable material  UM  4  NA  NA  NA  A  16, 18, 19, 23, 24 
Iron  Fe  5  NA  A  A  A  6, 16, 19, 20, 23, 24, 26, 27 
Copper  Cu  NA  A  A  A  5, 16, 20, 23, 24, 26, 27, 28  
Plumbum  Pb  NA  A  A  A  5, 16, 19, 20, 23, 26, 29  
Arsenic  As  NA  A  A  A  5, 16, 19, 20, 23, 24, 26, 28, 29, 30  
Nickel  Ni  NA  A  NA  A  5, 19, 23, 24, 25, 28, 29,  
Yellow  Y  4  A  A  A  A  1, 15, 19, 20, 24, 25 
Red  R  NA  A  NA  A  18, 19, 23, 25,  
Blue  B  NA  A  NA  A  18, 23, 25,  
Unsaturated/saturated 

5  NA  NA  NA  A  18, 19,31 
Smell and taste  ST  5  NA  A  A  A  5, 13, 16, 18, 19, 20, 24, 28, 31, 31, 
Texture  T  4  NA  A  A  A  5, 14, 18, 19, 24, 31 
Appearance  A  4  NA  A  A  A  1, 5, 14, 19, 24, 31 
Fat  Butter (B), Composed and hydrogenated fat (CF), vegetable lard (VL) and Lard (L)  
Regulations  1) NMXF010, 
This study considered the physicochemical parameters present in all regulations and omitted parameters that only apply specifically to margarine and butter, such as pH, phosphatase, stability, and dry matter content, which are not considered in the proposed index because they only found in the regulations for a single type of fat. Among these parameters can be found nonfat solids, phosphatase, linoleic acid, casein, fat acidity, conservatives, and sodium chloride type I and II.
Fatty acid composition is complex, and the regulations consider over 15 parameters. The fatty acid profile has great importance in the quality of a fat. However, including these parameters in addition to the physicochemical ones would make the calculation of the FQI too complex and would not properly represent the interests of the fatty acid profile.
The use of the ratio of unsaturated/saturated fatty acids, which is the ratio between the sum of unsaturated fatty acids and the sum of saturated fatty acids expressed as a percentage, is proposed. This definition is a representation of the fatty acid profile that groups all the components together (Valenzuela
The maximum and minimum values of the ratio of unsaturated/saturated fatty acids are not established in the regulations, so to define these values, two limiting cases are set considering the values of each fatty acid.
Thus, to determine the minimum allowed value of the unsaturated/saturated ratio, the sum of the percentages of all unsaturated fatty acids at their minimum allowed value divided by the sum of the percentages of all saturated fatty acids at their maximum allowed value is considered. Thus, the first limiting case is obtained as a ratio that represents a fat sample with the smallest allowed amount of unsaturated fatty acids and the maximum of saturated ones. This value can be represented by the following equation:
Likewise, to obtain the maximum allowed value of the unsaturated/saturated ratio, the sum of the percentages of unsaturated fatty acids at their maximum allowed value is divided by the sum of the percentages of all saturated fatty acids at their minimum allowed value. Thus, a sample with a small content of saturated fatty acids and a large content of unsaturated ones is considered:
Finally, the organoleptic characteristics are considered, which are of great importance in the quality of the fat due to its use in foods and can significantly alter the quality of the products.
With the purpose of transforming the variables to a dimensionless scale to then integrate them into the FQI, each type of contribution is determined depending on the characteristics and trends seen in the regulations.
Organoleptic parameters (except color) and the Kreiss reaction are qualitative determinations. Thus, it is not possible to determine the value of the subindex mathematically, as it only provides nominal values such as negative or positive. Therefore, the use of the value 100 is proposed for those samples that meet the value accepted by the regulations, and a value of three is proposed for samples that do not meet the regulations.
A statistical analysis of the parameters is performed using the computer program
With the analysis and inspection of the statistical results, the most suitable mathematical model is proposed, considering the values of central tendency, kurtosis, and symmetry, in addition to the analysis of the behavior of tens of percentiles. Additionally, the lower and upper confidence limits for 95% of the population of the analyzed data are determined.
The variability and confidence intervals are determined both statistically, according to the standard deviation, and in accordance with the regulations regarding the significant values of each parameter, which are set according to the standard analytical detection method employed. This procedure allows for the use of two significant numbers in the FQI.
Furthermore, the averages were evaluated in chunks corresponding to the main types of fats: animal (e.g., butter fat, butter milk), vegetable (e.g., palm oil, cocoa butter), and mixed (margarine and hydrogenated).
Most of the parameters have subindices that are obtained by the method of curves described by mathematical equations and based on the regulations. Using these subindices, each parameter is described in an objective manner through the correlation between the actual value of the parameter and the 0–100 subindex scale of each parameter; this method is the most widely accepted (SamboniRuiz
The values of the lower and upper confidence limits or ranges of statistical acceptance of the data extracted from the regulations for each parameter are considered to determine the necessary minimum value to be labelled a fat with acceptable quality, which in this case is considered to be a value of 71.
Furthermore, depending on the type of tendency the parameter shows, the mathematical model chosen is the one that shows the best fit to the expected behavior and to the results of the statistical analysis. To determine the model, the tens of percentiles of the FQI statistical analysis are correlated with the values of 71 and 100 for a fat that meets the quality criteria.
To establish the parameters, linear, exponential, potential, and nonlinear mathematical models were used. Adjustments were made using the computer programs
However, some parameters display more complex tendencies, so it was necessary to consider multiple models that describe multiple parameters with a single index. Thus, the metals and Lovibond parameters are expressed as the algebraic sum of each parameter in an equation with multiple variables. Similarly, the ReichertMeissl, Polenske, and Kirchner values are integrated into a single parameter that represents volatile fatty acids.
In
Statistical results of fat regulations data
Parameter  Average  Standard deviation  Kurtosis  Skewness  LCI 95%  UCI 95% 

Specific gravity  0.9157  0.033  1.60  1.1  0.8432  0.9745 
Melting point  37.7  8.0  −0.15  0.026  30.5  54.6 
Solidification point  38.0  6.9  −0.24  −2.1  29.7  50.5 
Refractive index  1.4560  0.0051  0.057  −0.52  1.4483  1.4635 
Saponification index  207.3  19. 4  0.78  −0.36  185.1  243.1 
Iodine index  45.62  17.5  0.074  −0.92  36.1  88.5 
Unsaponifiable  10.1  5.1  −0.051  −0.92  5.0  20.2 
Acidity index  1.602  1.9  2.3  7.04  –  5.1 
Humidity  6.69  8.1  0.51  −1.8  –  23.9 
Peroxide  5.19  3.5  0.41  −1.4  13.6  13.6 
Böemer  72.3  1.2  −1.7  –  75.1  75.1 
Reichertmeissl  27.8  5.4  0.29  −1.7  36.4  36.4 
Polenske  2.9  1.9  0.79  −0.12  6.6  6.6 
Kirchner  22.5  5.0  –  –  32.4  32.4 
Fat percent  78.3  12.4  0.20  0.12  67  – 
Sodium soap  0.0043  0.0010  −0.97  –  0.0068  
Unsolvable material  0.0463  0.011  −2.8  8  –  0.07 
Iron  1.5938  1.0  2.5  8.4  –  8.3 
Copper  0.1667  0.13  1.5  0.14  –  3.6 
Plumbum  0.0946  0.019  −3.6  13  –  0.38 
Arsenic  0.1  0.017  −4.6  5  –  0.14 
Nickel  0.3667  0.56  2.3  5.6  –  0.1 
Yellow  36  13.7  −1.0  1.6  22  62.6 
Blue  5  1.8  2.3  5.9  3  8.7 
Red  4  4.2  –  –  0  12.5 
Unsaturated/saturated  3.5  –  –  –  0.2827  6.3240 
By analyzing the measures of central tendency, it is possible to establish the value of each parameter of an average fat. Based on the standard deviation, the data variability is established, which also allows the determination of the values of the lower and upper limits of confidence for 95% of the population.
Furthermore, the analyses of the dispersion values allow us to analyze the data distribution, in addition to the distribution of percentiles, as seen in (
Graphs showing the correlation function of specific gravity. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of melting point. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of solidification point. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of refractive index. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of saponification index. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of iodine index. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of unsaponifiable materials. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of acidity index. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of humidity. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of suspended solids. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of peroxide index. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of Böemer index. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of fat percent. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of sodium soap. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of unsolvable material. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of metals. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of parameters comprising the Lovibond color. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of the parameters comprising the volatile fatty acids. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
Graphs showing the correlation function of the coefficient unsaturated/saturated fatty acids. Black dots showing percentiles and dotted lines with circle adjustments are proposed equations to describe the parameter.
In
Statistical values and subindex of the quality parameters of different types of fats and the overall average
Parameter  Lard  Vegetable lard  Composed and hydrogenated fat  Average  



Value  Sub  Value  Sub  Value  Sub  Value  Sub  
Specific gravity  0.9  98  0.9262  99  0.9157  100  0.9157  100 
Melting point  37.3  100  34.6  99  43.0  97  37.7  100 
Solidification point  37.9  99  38.5  98  38.0  99  38.0  99 
Refractive index  1.4553  99  1.4574  100  1.4560  99  1.4560  99 
Saponification index  209.68  100  202.70  100  207.28  100  207.28  100 
Iodine index  47.83  100  40.63  98  45.62  100  45.62  100 
Unsaponifiable  9.25  88  10.67  86  13.00  82  10.25  87 
Acidity index  1.289  91  2.217  84  0.700  95  1.602  89 
Humidity  6.29  99  0.23  100  14.25  94  6.69  99 
Suspended Solids  0.1333  90  0.1333  90  0.1333  90  0.1333  90 
Rancidity  +  100  +  100  +  100  +  100 
Peroxide  5.64  84  5.13  85  3.00  91  5.19  85 
Böemer  72.3  73  72.3  73  72.3  73  72.3  73 
ReichertMeissl  27.8  78  27.8  78  27.8  78  27.8  78 
Polenske  2.9  86  2.9  86  2.9  86  2.9  86 
Kirchner  22.5  75  22.5  75  22.5  75  22.5  75 
Volatile fatty acids  79  79  79  79  
Fat percent  83.2  90  74.0  83  74.0  83  78.3  86 
Sodium soap  0.0050  71  0.0037  79  0.0043  75  0.0043  75 
Unsolvable material  0.0500  71  0.0425  75  0.0463  73  0.0463  73 
Iron  1.3750  92  2.6250  85  1.0000  94  1.5938  91 
Copper  0.19  86  0.1750  87  0.1  93  0.1667  88 
Lead  0.1  71  0.1000  71  0.0860  75  0.0946  73 
Arsenic  0.1  71  0.1000  71  0.1  71  0.1  71 
Nickel  0.1  98  0.6333  88  0.6333  88  0.3667  93 
Metals  79  77  80  79  
Yellow  27  100  43  83  36  95  36  95 
Blue  6  81  5  99  5  95  5  95 
Red  4  95  6  98  4  95  4  95 
Lovibond  94  92  95  95  
Unsaturated/saturated  1.7  88  6.0  100  3.5  95  3.5  95 
Smell and taste  +  100  +  100  +  100  +  100 
Texture  +  100  +  100  +  100  +  100 
Appearance  +  100  +  100  +  100  +  100 
FQI  93.05  93.28  93.61  93.51 
In
Parameter's subscript equations and range of validity
Specific gravity  



Melting point  


Solidification point  


Refractive Index  


Saponification Index  


Iodine Index  


Unsaponifiable materials  


Acidity Index  


Humidity  


Suspended solids  


Peroxide  


Böemer Index  


Index Volatile Fatty Acids  


Fat percent  


Sodium soap  


Unsolvable material  


Metals  


Lovibond  


Unsaturated/Saturated  

Finally, the quality index for each type of fat according to its origin and the overall average shows values above 93, which corresponds to fats of excellent quality within all international guidelines.
Employment of the fat Quality Index on five experimental works, demonstrating its usefulness in interpreting experimental fat samples.
RoseMonde
Thus, applying the FQI model, the obtained values are 84.65 for the fat with a beige color (A), 83.85 for the fat with a yellow color (B), and 86.26 for the fat with a grey color (C), which facilitates choosing the raw material of the highest quality.
Yellow shea butter is consumed most frequently and can be processed to improve its physicochemical properties; a comparison of these properties before and after treatment is shown in
In 2012, Dhurvey studied the effect of hydrogenation on ghee (III), a traditional butter from India that is prepared at home, and he evaluated its hydrogenation (Dhurvey
Sağdiç (
In 2010, Okullo evaluated shea butter from different regions of Uganda (V), Pader, Lira, Katakwi, and Arua, and he showed the variation in the quality parameters in each of these regions (Okullo
To use the proposed quality index, the FQI calculator developed in MS EXCEL is made available at the following address:
The proposed FQI achieves the grouping of all standard quality parameters in a single weighed numerical value, making it a viable option for the interpretation of the values of the physicochemical and organoleptic variables inherent in the quality of fats, as the different variables are combined to result in a value that can be easily interpreted by the general population.
Each quality parameter has a different behavior regarding its statistical distribution, based on the results obtained in accordance with the regulations; therefore, different types of mathematical models were implemented to adequately describe these behaviors, obtaining a subindex for each.
Finally, the applications of the fat quality index shown in this paper demonstrate its usefulness for the discussion of results and the elucidation of conclusions with appropriate grouping and interpretation of all parameters that make up the quality of the fat under study.
The authors thank the Academic groups UVERCA226: “Estudio integral de ingenieríaaplicada” and UVERCA160: “Química de compuestos con posible actividad biológica”, Faculty of Chemistry of the University of Veracruz Orizaba, for providing facilities for the development of this work. JACM acknowledges support from the program “Cátedras Conacyt”.