INTRODUCTIONIn the food industry, the task of identification – thatis, determining the attribution of a food product to aparticular class in terms of condition, quality and tastecharacteristics – stands alone. For the solution of thistask there exist: a set of criteria both measurable andexpert; typical characteristics that product clusters mustmeet; and stratifying borderline values [1–5].At the same time, all the obtained relations areempirical. Besides, as discriminatory criteria areconstrued, product clusters often intersect accordingto some measured parameters, so it makes sense tointroduce a characteristic of attribution [6]. The latterwould be a unit (“the sample certainly belongs to thisproduct cluster”) in cluster centers and would decreaseat the borders (“the sample belongs to some extent toone cluster and to some extent to the neighboring one”).This would allow making product identification moretransparent and applicable to real food applications [7, 8].The method of fuzzy sets theory application tothe problems of the food industry, proposed in thispaper, will create lax regulatory restrictions on thecomposition, quality and sanitary characteristics ofthe product, taking into account the varied errorsof methods and measurements. The purpose of thisresearch was to provide food industry experts witha tool that allows building a robust multiparameteridentification criteria based on empirical product data.STUDY OBJECTS AND METHODSThe concept of fuzzy sets as applied to the foodindustry. In order to define fuzzy set A for elements𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 o∈f ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1, enter the indicator membership functionI:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0(1)I Hereafter: the indicator function and the membership function areinterchangeable concepts13Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19Concurrently, the set in the classical sense of, definedin this way, is a special case of𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11+ (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥, a∈ fℝuz𝑛𝑛z y set:𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =2(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥(2)Thus, fuzzy logic extends the Boolean one with twovalues {0,1} to the continuum of values in the intervalof [0,1]. The difference between the approaches is shownin Fig. 1. Most often, the value 𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝐴𝐴𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 _ _(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)is interpreted as asubjective assessment of x as attributed to A, for example𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝1( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116= 0.9 means that x is 90% of A [9].The example of interpretation contains the word“subjective”, which presupposes the possibility of eachsubject having an opinion concerning the relationshipof each specific set attribution on the basis of theirown indicator function. For food industry, this meansthe need of using a consensus membership functionfor each criterion; the function based on a particularfood industry experts’ consolidated opinion, as well asconfirmed experimental data [10].The subjectivity of assessment also implies theexistence of a method of translating psycholinguisticconclusions about the considered attribution to thedigital domain of the indicator function.The concept of a linguistic variable includes theobject under study, as well as a set of natural languagephrases (linguistic lexemes) that the variable can take ina fuzzy sense. The method of establishing the relevanceis individually selected for each industry and case ofstudy. Common sense is one of the primary factors,since the number of linguistic lexemes used by expertsand intended for digital transformation is extremelydiverse, for example: “true”, “false”, “almost false”,“almost true”, “unknown”, “possible”, “sometimes”,“may be”, etc. (Fig. 2).The main prerequisite for the use of fuzzy logicas applied to the food industry is the inability tobuild clear relations and criteria that link the qualityand performance of products and are not subject tomultiparameter, unamenable to expression, factors ofinfluence and measurement errors [11].In a way, the definition of a fuzzy set via theindicator function contains neither lack of focus norambiguity, so it is possible to use the fuzzy logicframework for setting standards and identificationmethods in the food industry.Basic operations with fuzzy sets. Let us examinein more detail the possible fuzzy sets manipulations andhighlight the most common operations in terms of thefood industry (in fuzzy logic it is impossible to identifya finite set of basic functions, through which all theothers could be expressed; besides, operations on setsbecome “blurred”) [12–14].Consider the sets A, B𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 𝑥𝑥𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 11.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17). The relation of inclusionof the A set B into:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)· 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴 +𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116(3)The most practical option for constructing fuzzynegation𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 _ _(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)is:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝐵𝐵 ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝐴𝐴(𝑥𝑥), 𝐵𝐵 ))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 _ _(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.5)8 ,11 + (𝑥𝑥 − 3)8 ,11 + (𝑥𝑥 − 5)12(4)There is an unlimited number of simple fuzzynegations; besides, this method is convenient forconstructing linguistic expert models, for example,the negation for “unknown” 𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥 ), 𝜒𝜒𝐵𝐵))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 )𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]1= 0.5) will also be“unknown”.The expansion of conjunction (operation “AND”) forfuzzy sets is called the t-norm (or triangular norm), andthe expansion of disjunction (operation “OR”) is calledthe s-norm. In practice, most commonly used are:The logical product of𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − (𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1and sum𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝜒𝜒𝐴𝐴 _(𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝑚𝑚𝑖𝑖=1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > :𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴() + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵()𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)(5)The algebraic product of𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝐴𝐴𝑥𝑥), 𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵() = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 _ _(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝑚𝑚and sum𝜒𝜒𝐴𝐴()∈ [0,1], 𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) 𝜒𝜒𝐴𝐴 _(𝑥𝑥) = 𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴∗) = 𝜒𝜒𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝜒𝜒𝐴𝐴|𝐴𝐴 _ _(𝑥𝑥𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑆𝑆 = (Σ 𝑖𝑖𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆Figure 1 Fuzzy and Boolean approaches to the definition of a set consisting of an element {4}Fuzzy logic Boolean logicFigure 2 An example of relation between the linguistic “attribution”variable and the intervals of the indicator functionFalseAlmost false Unknown Almost trueTrue14Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵𝑥𝑥= 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + 𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(+𝑇𝑇)0.6 (𝑡𝑡= 0.63𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑𝑒𝑒ⅇ−𝑌𝑌𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(6)The presented pairs of t- and s- norms are calleddual, since when using the above negation, de Morgan’slaws are implemented in a fuzzy form, which makestheir application practically convenient in calculations.Failure of the law of complementarity in the generalcase must be noted as an important feature of fuzzylogic. Denoting t-norm as &, s-norm as |, we have:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ () ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(7)The postulate of Boolean algebra “some criterionand its negation are simultaneously unjust” violates theintroduction of intermediate variants. In particular, thatof the lexeme “unknown”, since it and its negation areassumed to be simultaneously and equally fair. This factdemonstrates the coexistence of the property and itsnegation.With multi-criteria identification of food productsit is often necessary to assign weight numbers for eachindividual criterion while obtaining the aggregateindicator quality function. To do this, convex integrationwith𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ | 𝑋𝑋 = 𝑥𝑥)(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝1𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 30.75 )8 ,11 + (𝑥𝑥 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116− ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =−1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1coefficient (denoted as𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥= 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(+ 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝis used:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = )𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =1(𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ̅(𝑤𝑤𝑛𝑛1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1(8)This formula is easily generalized for the case ofcriteria. Supposing there are fuzzy sets𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4., ... ,𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | ), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,where𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.their convex integration will have theform:𝜒𝜒𝑆𝑆) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , +1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 ), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116= ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))(9)𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + 1 − 𝜆𝜆𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (− 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))Constructing indicator functions, it is useful tocontrol the smoothness and speed of the transition of onelinguistic concept to another. To do this, we use a powerfunction that defines𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥)≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴 +𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥)+ (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒(𝑥𝑥 ) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1as follows:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴() · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116= ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)(10)If𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 11 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)(𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4., the function reduces the requirements formembership to the set𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴() ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 _ _(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥 𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝1,1,1,1with respect to A, at𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,the function clarifies it.Linear fuzzy classification. From the standpointof the probability theory the indicator function can beinterpreted as conditional probability𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (− 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝1111(11)that is, the probability of membership to the set of arandom variable X, provided that it was implementedby x value. It should be noted that this is the basicdifference between the approaches: fuzzy logic operatesby the degree of membership to a particular set. Whileprobability theory (and “probabilistic” logic) indicatesthe probability of occurrence of mutually exclusiveevents.As an example, consider the fuzzy classificationof drinking milk by fat content (Fig. 3)II. According tothis classification, milk with a fat content of 3.75% isboth 0.5 medium-fat and 0.5 high-fat. We consciouslygive no percentages here, because it is not a matterof probability (otherwise, in a batch of milk with thesame fat content of 3.75%, half of the bottles would berecognized as “medium-fat”, and the other half – as“high-fat”, which makes no sense). Fuzzy sets exist insuperposition with each other, this being their mainadvantage in food identification. Continuing the exampleon the same classifier, 0.8 milk of average fat contentis actually the same as 0.2 of extra fat content, and thishas a direct interpretation since two linguistic postulatesdescribing different degrees of one measurable criterionare associated. At the same time, it should be noted thatcombining probabilistic and fuzzy methods has its ownscope; besides, probability distributions can be used asindicator functions, as will be shown below.In the example with milk fat linear functions areused to determine the degree of membership, being themost practically applicable for the food industry due tothe simplicity of construction and linguistic explanationof the result [15, 16]. In order to construct a linearcharacteristic function𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =(17)for some criterion A on thedomain R𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63there are three steps to follow:(1) Determination of the intervals𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖 +1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.= 1 ...𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛where 𝜒𝜒𝐴𝐴(𝑥𝑥) = (𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦); (𝑥𝑥, 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | ), 𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 () 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,where𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 ()𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.that is, belonging to such intervals ischaracterized by the lexeme “certainly Yes”;(2) Determination of the intervals𝜒𝜒𝑆𝑆(𝑥𝑥) = 𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 (𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 ± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + [–1,1] ± 0.5 [−𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊,𝜒𝜒𝑆𝑆(𝑥𝑥) = 𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 ± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + [–1,1] ± 0.5 𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓,𝜒𝜒𝑆𝑆(𝑥𝑥) 𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < (𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 ± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 [–1,1] ± 0.5 𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) = 1 ... m,where𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4., that is, their linguistic characteristics is“certainly No”;II State Standard 31450-2013. Drinking milk. Specifications. Moscow:Standartinform; 2014. 9 p.%χFigure 3 Fuzzy classification of drinking milk by fat contentχFat-freeχLow-fat%χMedium-fatχHigh-fatFat content, %χ15Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19(3) Combination of the intervals into one list with thelength of𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1… 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.and sort them in ascending orderof the left border. Since in the resulting list the intervalswith the characteristics “certainly Yes” and “certainlyNo” will alternate, it remains only to connect theboundaries by linear function.(a) For the sequence𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 11 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,;𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥= 1= 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1, 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1= 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.the function willlook like:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥 )𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵𝑥𝑥− 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 ) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116= ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =121 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥, 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − ⅇ5−𝑡𝑡−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(12)(b) For the sequence (𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1= 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝐴𝐴), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥=1𝑥𝑥𝑖𝑖+1 − ′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116+ 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63); (𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 _ _(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1− 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥− 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ − 1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 𝑚𝑚 (𝑥𝑥𝑖𝑖𝑖𝑖); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | ), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝐴𝐴)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1… 𝑚𝑚, where 𝜒𝜒𝐴𝐴𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦+1′ 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼(𝑥𝑥1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.) the function willlook like:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 () = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥 )𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2(13)Of course, in practice the most common case is thatwith one truth interval, and the function takes the formof a trapezoid, as seen in the graph of milk classification.For one truth interval the convenient approximationis:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝐵𝐵 ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝐵𝐵))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)(14)where с is the center of the interval,𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝐴𝐴() ≤ 𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴() · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11+ (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡is its range,𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥= max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥 𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,1 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0isthe smoothing fit. In the context of the example, theindicator functions for dairy products will take the form:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴· 𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴 (𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵) 𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴()𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − )𝛷𝛷𝜀𝜀() =12(1 + erf 2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)(15)for fat-free, low fat, medium fat and high fat products,respectively. The patterns of these functions, as well ascomparison of the two approaches are shown in Fig. 4.RESULTS AND DISCUSSIONTo determine the value of the indicator function𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥= 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)at a particular point, it is sometimes necessary to resortto nested fuzzy sets. This happens, for example, whenthe values of the linguistic variables of the expert groupdiffer for the same criterion at a point. When a indicatorfunction of a set is realized not by a specific number, butby another indicator function, it is called a second orderfuzzy set. In practice, it is very difficult to use such items,and they are absolutely unsuitable for establishing legalrelations between contractors of the food industry, inparticular, producers and consumers. In this case, insteadof the nested indicator function at a point, its integralvalue is considered, for example, the consensus of expertsor the probability value, if the function was representedby the probability density [17–19].As an example, consider a criterion of a foodproduct, which according to regulatory documentsshould fall into the interval𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where (𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) ± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀([–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = (we consider it as afuzzy set I) with a measuring device error of ± 0.5 Theerror was deliberately taken as comparable to the lengthof the interval for a more visual demonstration of thebehavior of the indicator function at the boundary.Suppose that the measurement error𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴() 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.,𝑆𝑆𝑖𝑖𝐴𝐴𝑖𝑖 𝑚𝑚𝑖𝑖=1𝑚𝑚𝑖𝑖 𝑛𝑛 𝑖𝑖 𝑖𝑖 𝐴𝐴𝑗𝑗𝑗𝑗𝐴𝐴𝑖𝑖𝑖𝑖 𝑖𝑖+1′ 𝑖𝑖+1′ 𝑟𝑟 𝑟𝑟 𝐼𝐼 𝜀𝜀𝑖𝑖𝑄𝑄 𝑇𝑇 𝑊𝑊(𝑘𝑘,𝜆𝜆𝑊𝑊(𝑘𝑘,𝜆𝜆is a normally distributed random variable with zeroexpectation and dispersion, whose value can bedetermined from the instrument error. If we assumethat 95.6% (which corresponds to the probability of anormally distributed random variable falling within therange𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = (𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, ± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.relative to the mean value) of measurementsof x fall within the range x ± 0.5 (the assumption can bestrengthened or weakened depending on the conditionsand the nature of the error), it means that:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)(16)To construct the indicator function of the criterion,let us ask: “what probability does the product satisfy thecriterion with if its measurement showed the result of?”.Obviously, a second order fuzzy set emerges: for each xthere is an error probability density that can serve (aftersome manipulations) as a nested indicator function. Aspreviously stated, it is more convenient to assume theintegral value as the value at the point, that is, from aprobabilistic point of view, to calculate the conditionalprobability𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4., where𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 < 𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 is the real valueof the indicator. In this case,𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖′ 𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 1.25 0.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝis the possible realvalue𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 𝜒𝜒𝐴𝐴 _(𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝜒𝜒𝐴𝐴&𝐴𝐴 _ _(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 𝑥𝑥= 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑚𝑚𝑖𝑖=1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 0.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥2𝑥𝑥on the interval𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where (𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) ± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 probability densityFigure 4 Approximation of fuzzy classification of drinking milk by fat contentFat content, % Fat co%ntent, %χFat-free%χLow-fat%χMedium-fatχHigh-fat16Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19integral (i.e., the probability of𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63falling into thespecified interval):𝐴𝐴 𝐴𝐴 _𝑛𝑛𝑖𝑖𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥 𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥 ) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63𝜒𝜒𝐴𝐴&𝐴𝐴 _(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(17)The last expression is nothing but the differencebetween the distribution functions𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1… , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1= 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = + 𝜀𝜀 𝜒𝜒𝐼𝐼 𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.of the randomvariable ε. The formula is:𝐴𝐴 _𝐴𝐴 _𝜆𝜆 𝛼𝛼 𝑥𝑥 2𝑌𝑌(𝑥𝑥) 𝑖𝑖𝑖𝑖𝑛𝑛𝑖𝑖=15−𝑡𝑡𝑘𝑘𝑘𝑘0.6 5−𝑡𝑡 20.6 𝐴𝐴⋃𝐵𝐵𝐴𝐴𝐵𝐵 𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(18)𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63erf x is the error function, included for convenience ofcalculations in many packages of mathematical dataprocessing, in particular, MS Excel.In the classical approach to identification, anymeasurement that falls within the interval [–1,1] ± 0.5will be recognized as corresponding to the criterion (tosimplify the example, the questions of additional andoutlier measurements are omitted here), while alreadyat the values –1 and 1 the level of belonging to thecriterion in the fuzzy approach will be equal to only0.5 (Fig. 5), and when approaching the boundaries ofa large interval –1.5 and 1.5, there is no chance for thecriterion. Moreover, to provide the characteristic “mostlikely the product has a criterion” (function value 0.8),the measurement value must fall within the range[–0.79, 0.79].This approach should be taken into accountspecifically at the boundaries of the intervalidentification. For example, when establishing aboundary for foods of mixed composition with milkfat content the following definition is proposed: ifmilk fat content exceeds 51% of the total fat phase, theproduct is called milk-based. If it makes less than 50%– milk-containing, respectively, with a measurementerror of ± 0.5%. In this case, products containing milkfat in the range of [50.25%, 50.75%] will not belong toany specified class with a sufficiently high level ofconfidence.Despite measurement errors, the boundary ofidentification classes should be set without taking theminto account. Regardless of the nature (except for theassumption of distribution symmetry) and the type oferror at the point of the boundary, the indicator functionof both classes will be equal to 0.5. This is a logicalassumption to refer the product to a particular classif the measurement gave a boundary indicator. In theabove example, this boundary will be the point 50%.However, if indicator functions of two identificationclasses, being adjacent linguistic characteristics of thesame criterion, take the same value of 0.5 at a point, itmakes sense setting a boundary between these classes atthis point. For the multidimensional case, the boundarywill be represented by a hyperplane, but in practice thedimension exceeding two is rarely considered.Harrington’s function as an example of indicatorfunction. One of the applied tools in the qualitativeassessment of the developed food industry identificationmethods is Harrington’s desirability function [20].The idea of Harrington’s function is to transformthe values of the criteria into a dimensionlessdesirability scale that allows comparing and combiningthe characteristics of products of different nature.It establishes compliance between experts’ psycholinguisticassessments and natural indicators of criteria.In addition, it has all the necessary practical propertiesof the indicator function, which allows using it activelyin fuzzy logic applications.Generally, Harrington’s function is of the form of:𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(19)where𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.is a function that establishes a relationbetween the values of the experimental variableand the dimensionless scale [21]. In practice, it isalmost always linear, being accountable for theshift and steepness of Harrington’s function curvein accordance with application needs. It is so asto correspond to the well-established mappingof the function value intervals to the linguisticvariable of desirability: “very good” – 0.8, 1;“good” – 0.63, 0.8; “satisfactory” – 0.37, 0.63; “bad” –0.2, 0.37; “very bad” – 0.0.37.If there are n criteria with corresponding desirabilityfunctions𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where (𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) [–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥),, 𝜒𝜒th𝑄𝑄e( 𝑡𝑡c)o 𝑡𝑡n>so0li d𝜒𝜒a𝑇𝑇t(e𝑡𝑡d), e (s𝑡𝑡ti=m0a)t e 𝑓𝑓 i𝑊𝑊s( 𝑘𝑘e,x𝜆𝜆)p(r𝑡𝑡e) s s𝑆𝑆e𝑊𝑊d(𝑘𝑘 a,𝜆𝜆s)( 𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = a weighted geometric mean:𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛𝜒𝜒𝐴𝐴 _ (𝑥𝑥) = 1 − 𝜒𝜒𝐴𝐴(𝑥𝑥)𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵(𝑥𝑥))𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴(𝑥𝑥), 𝜒𝜒𝐵𝐵 (𝑥𝑥))𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(20)The useful property of the function is insensitivitywithin the range of 0 to 1 values (estimates “very bad”and “very good”, respectively). It can be used in theconstruction of criteria linked to the product’s shelf life.Figure 5 Indicator functions of the interval criterion in strictand fuzzy approachesCriterion interval Specified interval17Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19In food products ‒ complex biological systems ‒quality deterioration is often subject to the exponentiallaw, and one of the key factors is the change inmicrobiological parameters. So, Harrington’s functioncan be considered as𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇 (𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘indicator function of Q fuzzy ,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.set – i.e., the products corresponding to public healthregulations. Here storage time𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄(𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒is𝑇𝑇 (u𝑡𝑡s)e, d ( 𝑡𝑡a=s a0 )p r𝑓𝑓o𝑊𝑊d(u𝑘𝑘,c𝜆𝜆)t( 𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.characteristic.Supposing a product has 4 days’ shelf-life. Let usfirst consider its validity indicator function without theuse of fuzzy logic (Fig. 6).Identification areas I “the product meets thestandards and is ready for consumption” and III “theproduct must be disposed of” are shown, respectively.Within strict logic, at point {4}, it is expected that thefunction has a gap of the first kind. As an applicablerule, the function reflects rather Cinderella’s carriagequalitative characteristics before and after midnight thanthose of the actual food product.Since shelf life usually has a margin of 20–25%,consider the following indicator function:𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(21)The function is a fuzzy negation of Harrington’sfunction, but this is natural when smaller values areassumed to have a larger desirability value. Its graph isshown in Fig. 7.In addition to the identification areas I and III,whose linguistic characteristics remain the same, thereappears area II – “the product is safe for use, but alreadyundergoing degrading qualitative changes”. In thissense, products from identification area II are no longersuitable for end-users and must be sent for extendingshelf life processing (sterilization, canning, etc.). Atthe same time, it should be noted that, for example,when canning, a fuzzy logic device must also be usedto establish the final shelf life of the product from rawmaterials within the boundaries of identification area II.As it was mentioned above, changes inmicrobiological parameters have a direct impacton the quality of the product. Logically the phasesof microbiological cultures’ development can becompared with Harrington’s function identificationareas; in particular, area I corresponds to the lag phase,area II – to acceleration and exponential growth ofmicroorganisms phase, and area III – to decelerationand stationarity phase. At the same time, substrate andother biotechnological characteristics of change in thepopulation of microorganisms are calculated for eachspecific product, which may lead to diversities in thegeneral matches given.Combination of the product’s qualitycharacteristics. The construction of indicator functionsis inextricably linked with decision-making systems.In the case of one criterion (for example, safety, asdescribed above), the fuzzy logic apparatus gives noclear advantage over a strict approach. In the end,all the contractors of the food industry (consumers,manufacturers, law enforcement agencies, etc.) makea binary decision whether a particular product samplecomplies with a criterion [22, 23]. Due to the fact thatthe criterion is unique (for example, expiration date)they identify the above decision with the function of theultimate goal (“buying” vs. “not buying”, “recalling”vs. “not recalling”, “fining” vs. “not fining”).In the example with the fety function, three clearidentification areas can be introduced. For them, forinstance, the seller will have a system of specificactions (I – “selling”, II – “reselling for recycling”,III – “recycling”).However, even when the second criterion in thedecision-making system is engaged, it is much harder toestablish the precise boundaries of identification classes.Consider the instance with the safety criterionwith an additional indicator “consumer quality” – acharacteristic that demonstrates the taste and overallsatisfaction from the consumption of the product –added. In the fuzzy logic the unction of this indicatordecreases faster than the safety function. For example,for baking and confectionery products, taste profilesdegrade much earlier than the products become unfitfor use. The taste of “fresh bread” is of great value tothe consumer and has its impact on their purchasepreferences, but it is not unique or decisive, as shelf lifeis also taken into account.To construct an example of the consumer qualityfunction𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄(𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇(𝑡𝑡),, (𝑡𝑡le=t 0u) s 𝑓𝑓𝑊𝑊u(𝑘𝑘s,e𝜆𝜆) (𝑡𝑡t)h e𝑆𝑆 𝑊𝑊(p𝑘𝑘,r𝜆𝜆o)(b𝑡𝑡a)b i l𝜆𝜆ity= 2t,h𝑘𝑘e=or2y 𝑡𝑡 ≅ 4.apparatus, assuming that fresh𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ ± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o [–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄(𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇(𝑡𝑡), (𝑡𝑡 = 0) p𝑓𝑓r𝑊𝑊o(d𝑘𝑘,u𝜆𝜆)c(t𝑡𝑡 )h a𝑆𝑆s𝑊𝑊 (s𝑘𝑘o,𝜆𝜆m)(e𝑡𝑡 ) 𝜆𝜆 = 2, 𝑘𝑘 = 2 taste profile lost on expiry [24]. In practice, it makessense to put an experiment to determine the distributionhistogram of the moment of fresh taste degradation.However, for the purposes of exemplification, it will besimulated with the help of Weibull distribution, used insurvival analysis, giving a good approximation in thestudy of products’ storage stability [25]. The density ofthis distribution𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄(𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇(𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) h 𝑆𝑆a𝑊𝑊s( 𝑘𝑘th,𝜆𝜆)e( 𝑡𝑡f)o r m 𝜆𝜆:= 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ 𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ 𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝜒𝜒𝐴𝐴 _(𝑥𝑥) 𝜒𝜒𝐴𝐴⋂𝐵𝐵(𝑥𝑥) = min(𝜒𝜒𝐴𝐴⋃𝐵𝐵(𝑥𝑥) = max(𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) 𝜒𝜒𝐴𝐴&𝜒𝜒𝐴𝐴𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝑆𝑆 = 𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … 𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 0.5 = ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 𝛷𝛷𝜀𝜀(𝑥𝑥) =12erf 𝑥𝑥 =𝑑𝑑(𝑥𝑥) = 𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(𝜒𝜒(𝑄𝑄+𝑇𝑇)(22)Figure 6 Product quality indicator function, classicallyQ (quality)t, days18Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19As a quality function we take the survival function+ ( 𝑥𝑥0.75)8 1 + (𝑥𝑥 − 1.50.75 )8 1 + (𝑥𝑥 − 30.75 )8 1 (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡(22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63,1 + ( 𝑥𝑥0.75)8 1 + (𝑥𝑥 − 1.50.75 )8 1 (𝑥𝑥 − 30.75 )8 1 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =116≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63for the given distribution. It is equal to theprobability that the value of the random variable understudy will exceed t, in this case, the probability thatthe taste has not yet been lost by t. For the Weibulldistribution, it has a convenient expression:𝜒𝜒𝐴𝐴𝑥𝑥=𝑥𝑥𝑖𝑖+ 1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑥𝑥𝑖𝑖+ 1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(23)Supposing that for a product with the safety functiondescribed by formula (23) the average taste profile islost on the second day, an approximation of distributionparameters with𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2,, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.𝑆𝑆𝑖𝑖𝐴𝐴𝑖𝑖 𝑚𝑚𝑖𝑖=1𝑛𝑛 𝑖𝑖 𝑖𝑖 𝐴𝐴𝐴𝐴𝑖𝑖𝑖𝑖 𝑖𝑖+1′ 𝑖𝑖+1′ 𝑟𝑟 𝐼𝐼 𝜀𝜀[–𝑡𝑡𝑊𝑊(𝑘𝑘,𝜆𝜆𝑊𝑊(𝑘𝑘,𝜆𝜆2, c a≅n b4e. derived.The problem of food products’ seller is to establishthe time when the product should already be sold at theresidual price (the time of entering area III) taking intoaccount the safety and consumer quality criteria.If the weight of safety indicator is set at 0.6, andthe weight of consumer quality indicator is set at 0.4,respectively, a convex criteria combination (8) will takethe form:𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖𝑚𝑚𝑖𝑖=1)𝛬𝛬𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖𝑚𝑚𝑖𝑖=1= 1𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)𝜒𝜒𝐴𝐴(𝑥𝑥) = −1𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖𝑥𝑥 +𝑦𝑦𝑖𝑖𝑥𝑥𝑖𝑖+1′ − 𝑦𝑦𝑖𝑖+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1′ ]𝜒𝜒𝐴𝐴(𝑥𝑥) =1𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ 𝑥𝑥 −𝑦𝑦𝑖𝑖′𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖′; 𝑥𝑥𝑖𝑖+1]𝜒𝜒𝐴𝐴(𝑥𝑥) =11 + (𝑥𝑥 − 𝑐𝑐𝑙𝑙 )𝑝𝑝11 + ( 𝑥𝑥0.75)8 ,11 + (𝑥𝑥 − 1.50.75 )8 ,11 + (𝑥𝑥 − 30.75 )8 ,11 + (𝑥𝑥 − 51.25 )120.5 = 2𝜎𝜎, 𝜎𝜎2 =1161) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63(24)Solving the equation (0.63 being Harrington’sfunction upper exponent for “satisfactory”):0.5 = 2𝜎𝜎, 𝜎𝜎2 =116ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) == ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)(17)𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)𝛷𝛷𝜀𝜀(𝑥𝑥) =12(1 + erf (2√2𝑥𝑥))erf 𝑥𝑥 =2√𝜋𝜋∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑𝑥𝑥0𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1)1Σ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖=1𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =𝑘𝑘𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63 (25)we obtain𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)𝑚𝑚𝑖𝑖=1𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1(𝑥𝑥𝑗𝑗′, 𝑦𝑦𝑗𝑗′), 𝑗𝑗 = 1 … 𝑚𝑚, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 (𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖 ); (𝑥𝑥𝑖𝑖+1′ , 𝑦𝑦𝑖𝑖+1′ ) 𝜀𝜀 ~ 𝛮𝛮(0, 𝜎𝜎2)± 2𝜎𝜎 ℙ(𝑥𝑥𝑟𝑟 ∈ 𝐼𝐼 | 𝑥𝑥), 𝑥𝑥𝑟𝑟 = 𝑥𝑥 + 𝜀𝜀 𝜒𝜒𝐼𝐼 (𝑥𝑥) 𝑥𝑥 + 𝜀𝜀 [−1, 1] 𝛷𝛷𝜀𝜀(·) o ε[–1,1] ± 0.5 [−0.79, 0.79] 𝑌𝑌(𝑥𝑥)𝑑𝑑𝑖𝑖(𝑥𝑥), 𝜒𝜒𝑄𝑄(𝑡𝑡) 𝑡𝑡 > 0 𝜒𝜒𝑇𝑇(𝑡𝑡), (𝑡𝑡 = 0) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) 𝜆𝜆 = 2, 𝑘𝑘 = 2 𝑡𝑡 ≅ 4.. This means that after four days theproduct must be sold in traditional or alternative ways.Guided by rate expiry date only, the seller would getthe value of 5, thus having no time left for operationalmaneuvers. The type of function graphs and theirconvex combination is shown in Fig. 8.CONCLUSIONThus, the apparatus of fuzzy logic allows buildingmulti-criteria decision-making systems in the foodindustry. They help effectively make decisions aboutproducts’ quality and safety and, in the case of violationsand arbitral bodies’ involvement, differentiate theadministrative impact on the contractors of the foodindustry.CONFLICT OF INTERESTThe authors declare that there is no conflict ofinterest related to this article.