INTRODUCTION
In the food industry, the task of identification – that
is, determining the attribution of a food product to a
particular class in terms of condition, quality and taste
characteristics – stands alone. For the solution of this
task there exist: a set of criteria both measurable and
expert; typical characteristics that product clusters must
meet; and stratifying borderline values [1–5].
At the same time, all the obtained relations are
empirical. Besides, as discriminatory criteria are
construed, product clusters often intersect according
to some measured parameters, so it makes sense to
introduce a characteristic of attribution [6]. The latter
would be a unit (“the sample certainly belongs to this
product cluster”) in cluster centers and would decrease
at the borders (“the sample belongs to some extent to
one cluster and to some extent to the neighboring one”).
This would allow making product identification more
transparent and applicable to real food applications [7, 8].
The method of fuzzy sets theory application to
the problems of the food industry, proposed in this
paper, will create lax regulatory restrictions on the
composition, quality and sanitary characteristics of
the product, taking into account the varied errors
of methods and measurements. The purpose of this
research was to provide food industry experts with
a tool that allows building a robust multiparameter
identification criteria based on empirical product data.
STUDY OBJECTS AND METHODS
The concept of fuzzy sets as applied to the food
industry. In order to define fuzzy set A for elements
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 o∈f ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
, enter the indicator membership functionI:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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 are
interchangeable concepts
13
Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19
Concurrently, the set in the classical sense of, defined
in this way, is a special case of
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1
+ (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥, a∈ fℝuz𝑛𝑛z y set:
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
(2)
Thus, fuzzy logic extends the Boolean one with two
values {0,1} to the continuum of values in the interval
of [0,1]. The difference between the approaches is shown
in Fig. 1. Most often, the value 𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
= ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)
(17)
is interpreted as a
subjective assessment of x as attributed to A, for example
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
= 0.9 means that x is 90% of A [9].
The example of interpretation contains the word
“subjective”, which presupposes the possibility of each
subject having an opinion concerning the relationship
of each specific set attribution on the basis of their
own indicator function. For food industry, this means
the need of using a consensus membership function
for each criterion; the function based on a particular
food industry experts’ consolidated opinion, as well as
confirmed experimental data [10].
The subjectivity of assessment also implies the
existence of a method of translating psycholinguistic
conclusions about the considered attribution to the
digital domain of the indicator function.
The concept of a linguistic variable includes the
object under study, as well as a set of natural language
phrases (linguistic lexemes) that the variable can take in
a fuzzy sense. The method of establishing the relevance
is individually selected for each industry and case of
study. Common sense is one of the primary factors,
since the number of linguistic lexemes used by experts
and intended for digital transformation is extremely
diverse, for example: “true”, “false”, “almost false”,
“almost true”, “unknown”, “possible”, “sometimes”,
“may be”, etc. (Fig. 2).
The main prerequisite for the use of fuzzy logic
as applied to the food industry is the inability to
build clear relations and criteria that link the quality
and performance of products and are not subject to
multiparameter, unamenable to expression, factors of
influence and measurement errors [11].
In a way, the definition of a fuzzy set via the
indicator function contains neither lack of focus nor
ambiguity, so it is possible to use the fuzzy logic
framework for setting standards and identification
methods in the food industry.
Basic operations with fuzzy sets. Let us examine
in more detail the possible fuzzy sets manipulations and
highlight the most common operations in terms of the
food industry (in fuzzy logic it is impossible to identify
a finite set of basic functions, through which all the
others could be expressed; besides, operations on sets
become “blurred”) [12–14].
Consider the sets A, B
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (
𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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 __(≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 1
1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
ℙ(𝜀𝜀 ≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)
(17)
. The relation of inclusion
of the A set B into:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
(3)
The most practical option for constructing fuzzy
negation
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝐵𝐵 ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)
𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
)
8 ,
1
1 + (𝑥𝑥 − 3
)
8 ,
1
1 + (𝑥𝑥 − 5
)
12
(4)
There is an unlimited number of simple fuzzy
negations; besides, this method is convenient for
constructing linguistic expert models, for example,
the negation for “unknown” 𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
1
= 0.5) will also be
“unknown”.
The expansion of conjunction (operation “AND”) for
fuzzy sets is called the t-norm (or triangular norm), and
the expansion of disjunction (operation “OR”) is called
the s-norm. In practice, most commonly used are:
The logical product of
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
and sum
𝜒𝜒𝐴𝐴(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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝑚𝑚
𝑖𝑖=1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > :
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴() + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵()
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
(5)
The algebraic product of
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 _ _(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝑚𝑚
and sum
𝜒𝜒𝐴𝐴()
∈ [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 logic
Figure 2 An example of relation between the linguistic “attribution”
variable and the intervals of the indicator function
False
Almost false Unknown Almost true
True
14
Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + 𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(+𝑇𝑇)0.6 (𝑡𝑡= 0.63
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑𝑒𝑒ⅇ−𝑌𝑌𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(6)
The presented pairs of t- and s- norms are called
dual, since when using the above negation, de Morgan’s
laws are implemented in a fuzzy form, which makes
their application practically convenient in calculations.
Failure of the law of complementarity in the general
case must be noted as an important feature of fuzzy
logic. Denoting t-norm as &, s-norm as |, we have:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ () ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)
(17)
𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥𝛷𝛷𝜀𝜀(𝑥𝑥) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖
=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(7)
The postulate of Boolean algebra “some criterion
and its negation are simultaneously unjust” violates the
introduction of intermediate variants. In particular, that
of the lexeme “unknown”, since it and its negation are
assumed to be simultaneously and equally fair. This fact
demonstrates the coexistence of the property and its
negation.
With multi-criteria identification of food products
it is often necessary to assign weight numbers for each
individual criterion while obtaining the aggregate
indicator quality function. To do this, convex integration
with
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ | 𝑋𝑋 = 𝑥𝑥)
(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 3
0.75 )
8 ,
1
1 + (𝑥𝑥 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
− ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
−1 − 𝑥𝑥)
(17)
𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)
𝛷𝛷𝜀𝜀(𝑥𝑥) =
1
2
(1 + erf (2√2))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
coefficient (denoted as
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥= 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1 ≤ 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
is used:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)
𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = )
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
(𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
̅(𝑤𝑤𝑛𝑛
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
(8)
This formula is easily generalized for the case of
criteria. Supposing there are fuzzy sets
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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)
𝑚𝑚
𝑖𝑖=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.
,
where
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖1
𝐴𝐴1, 𝐴𝐴2, … , 𝐴𝐴𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1
𝑥𝑥𝑗𝑗
′, 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 the
form:
𝜒𝜒𝑆𝑆) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 ), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
= 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) =
1
2
(1 + erf (2√2𝑥𝑥))
(9)
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + 1 − 𝜆𝜆𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (− 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
Constructing indicator functions, it is useful to
control the smoothness and speed of the transition of one
linguistic concept to another. To do this, we use a power
function that defines
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥)
≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴 +𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥)
+ (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒(𝑥𝑥 ) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
as follows:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
= ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
≤ 1 − 𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)
(17)
(10)
If
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=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)
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 for
membership to the set
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴() ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥 𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
,
1
,
1
,
1
with respect to A, at
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 standpoint
of the probability theory the indicator function can be
interpreted as conditional probability
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (− 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1
1
1
(11)
that is, the probability of membership to the set of a
random variable X, provided that it was implemented
by x value. It should be noted that this is the basic
difference between the approaches: fuzzy logic operates
by the degree of membership to a particular set. While
probability theory (and “probabilistic” logic) indicates
the probability of occurrence of mutually exclusive
events.
As an example, consider the fuzzy classification
of drinking milk by fat content (Fig. 3)II. According to
this classification, milk with a fat content of 3.75% is
both 0.5 medium-fat and 0.5 high-fat. We consciously
give no percentages here, because it is not a matter
of probability (otherwise, in a batch of milk with the
same fat content of 3.75%, half of the bottles would be
recognized as “medium-fat”, and the other half – as
“high-fat”, which makes no sense). Fuzzy sets exist in
superposition with each other, this being their main
advantage in food identification. Continuing the example
on the same classifier, 0.8 milk of average fat content
is actually the same as 0.2 of extra fat content, and this
has a direct interpretation since two linguistic postulates
describing different degrees of one measurable criterion
are associated. At the same time, it should be noted that
combining probabilistic and fuzzy methods has its own
scope; besides, probability distributions can be used as
indicator functions, as will be shown below.
In the example with milk fat linear functions are
used to determine the degree of membership, being the
most practically applicable for the food industry due to
the simplicity of construction and linguistic explanation
of the result [15, 16]. In order to construct a linear
characteristic function
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−1 ≤ 𝑥𝑥 + 𝜀𝜀 ≤ 1) = ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
(17)
for some criterion A on the
domain R
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
there are three steps to follow:
(1) Determination of the intervals
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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.
,
where
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 ()
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 is
characterized by the lexeme “certainly Yes”;
(2) Determination of the intervals
𝜒𝜒𝑆𝑆(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,
where
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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.
%
χ
Figure 3 Fuzzy classification of drinking milk by fat content
χ
Fat-free
χ
Low-fat
%
χ
Medium-fat
χ
High-fat
Fat content, %
χ
15
Oganesyants 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 the
length of
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝑚𝑚, 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 order
of the left border. Since in the resulting list the intervals
with the characteristics “certainly Yes” and “certainly
No” will alternate, it remains only to connect the
boundaries by linear function.
(a) For the sequence
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=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 (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)
𝑚𝑚
𝑖𝑖=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)
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)
𝑚𝑚
𝑖𝑖=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 will
look like:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 ) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
= 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) =
1
2
1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥, 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − ⅇ5−𝑡𝑡
−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(12)
(b) For the sequence (
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
≤ 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡
) = 0.63
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴𝑚𝑚, 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 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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 __(≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥=
1
𝑥𝑥𝑖𝑖+1 − ′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
+ 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
); (
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 _ _(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
− 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥− 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝑚𝑚, where 𝐴𝐴𝑖𝑖 ⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1
… 𝑚𝑚, where 𝜒𝜒𝐴𝐴𝑥𝑥) = 0 𝑘𝑘 = 𝑛𝑛 + 𝑚𝑚 𝑥𝑥𝑖𝑖, 𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖
+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 will
look like:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 () = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥 )
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(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 that
with one truth interval, and the function takes the form
of a trapezoid, as seen in the graph of milk classification.
For one truth interval the convenient approximation
is:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝐵𝐵 ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
(14)
where с is the center of the interval,
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝐴𝐴() ≤ 𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1
+ (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
≤ 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
is its range,
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥 𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)
(17)
𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)
𝛷𝛷𝜀𝜀(𝑥𝑥) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
is
the smoothing fit. In the context of the example, the
indicator functions for dairy products will take the form:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵) 𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴()𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
≤ 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() =
1
2
(1 + erf 2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑡𝑡𝑘𝑘−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 as
comparison of the two approaches are shown in Fig. 4.
RESULTS AND DISCUSSION
To determine the value of the indicator function
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥= 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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 resort
to nested fuzzy sets. This happens, for example, when
the values of the linguistic variables of the expert group
differ for the same criterion at a point. When a indicator
function of a set is realized not by a specific number, but
by another indicator function, it is called a second order
fuzzy set. In practice, it is very difficult to use such items,
and they are absolutely unsuitable for establishing legal
relations between contractors of the food industry, in
particular, producers and consumers. In this case, instead
of the nested indicator function at a point, its integral
value is considered, for example, the consensus of experts
or the probability value, if the function was represented
by the probability density [17–19].
As an example, consider a criterion of a food
product, which according to regulatory documents
should fall into the interval
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 a
fuzzy set I) with a measuring device error of ± 0.5 The
error was deliberately taken as comparable to the length
of the interval for a more visual demonstration of the
behavior of the indicator function at the boundary.
Suppose that the measurement error
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 𝑚𝑚
𝑖𝑖=1
𝑚𝑚𝑖𝑖 𝑛𝑛 𝑖𝑖 𝑖𝑖 𝐴𝐴𝑗𝑗
𝑗𝑗
𝐴𝐴𝑖𝑖𝑖𝑖 𝑖𝑖+1
′ 𝑖𝑖+1
′ 𝑟𝑟 𝑟𝑟 𝐼𝐼 𝜀𝜀𝑖𝑖𝑄𝑄 𝑇𝑇 𝑊𝑊(𝑘𝑘,𝜆𝜆𝑊𝑊(𝑘𝑘,𝜆𝜆is a normally distributed random variable with zero
expectation and dispersion, whose value can be
determined from the instrument error. If we assume
that 95.6% (which corresponds to the probability of a
normally distributed random variable falling within the
range
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 measurements
of x fall within the range x ± 0.5 (the assumption can be
strengthened or weakened depending on the conditions
and the nature of the error), it means that:
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝑡𝑡𝑘𝑘−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 the
criterion with if its measurement showed the result of?”.
Obviously, a second order fuzzy set emerges: for each x
there is an error probability density that can serve (after
some manipulations) as a nested indicator function. As
previously stated, it is more convenient to assume the
integral value as the value at the point, that is, from a
probabilistic point of view, to calculate the conditional
probability
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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.
, where
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 value
of the indicator. In this case,
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖′ 𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 1.25 0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
is the possible real
value
𝜒𝜒𝐴𝐴(𝑥𝑥) ∈ [0,1], 𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 𝑥𝑥= 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑚𝑚
𝑖𝑖=1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥2
𝑥𝑥
on the interval
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 density
Figure 4 Approximation of fuzzy classification of drinking milk by fat content
Fat content, % Fat co%ntent, %
χ
Fat-free
%
χ
Low-fat
%
χ
Medium-fat
χ
High-fat
16
Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19
integral (i.e., the probability of
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=
1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
falling into the
specified interval):
𝐴𝐴 𝐴𝐴 _
𝑛𝑛𝑖𝑖
𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)
𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥 𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(𝑥𝑥 + 𝜀𝜀 ≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
𝑥𝑥) − ℙ(𝜀𝜀 ≤ −1 − 𝑥𝑥)
(17)
𝜒𝜒𝐼𝐼 (𝑥𝑥 ) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)
𝛷𝛷𝜀𝜀(𝑥𝑥) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
𝜒𝜒𝐴𝐴&𝐴𝐴 _(𝑥𝑥) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(17)
The last expression is nothing but the difference
between the distribution functions
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=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 random
variable ε. The formula is:
𝐴𝐴 _
𝐴𝐴 _
𝜆𝜆 𝛼𝛼 𝑥𝑥 2𝑌𝑌(𝑥𝑥) 𝑖𝑖
𝑖𝑖
𝑛𝑛𝑖𝑖
=1
5−𝑡𝑡
𝑘𝑘𝑘𝑘0.6 5−𝑡𝑡 2
0.6 𝐴𝐴⋃𝐵𝐵𝐴𝐴𝐵𝐵 𝜒𝜒𝐴𝐴∗𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)
𝜒𝜒𝐴𝐴+𝐵𝐵(𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥) + 𝜒𝜒𝐵𝐵(𝑥𝑥) − 𝜒𝜒𝐴𝐴(𝑥𝑥) · 𝜒𝜒𝐵𝐵 (𝑥𝑥)
𝜒𝜒𝐴𝐴&𝐴𝐴 __(𝑥𝑥) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
erf x is the error function, included for convenience of
calculations in many packages of mathematical data
processing, in particular, MS Excel.
In the classical approach to identification, any
measurement that falls within the interval [–1,1] ± 0.5
will be recognized as corresponding to the criterion (to
simplify the example, the questions of additional and
outlier measurements are omitted here), while already
at the values –1 and 1 the level of belonging to the
criterion in the fuzzy approach will be equal to only
0.5 (Fig. 5), and when approaching the boundaries of
a large interval –1.5 and 1.5, there is no chance for the
criterion. Moreover, to provide the characteristic “most
likely 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 account
specifically at the boundaries of the interval
identification. For example, when establishing a
boundary for foods of mixed composition with milk
fat content the following definition is proposed: if
milk fat content exceeds 51% of the total fat phase, the
product is called milk-based. If it makes less than 50%
– milk-containing, respectively, with a measurement
error of ± 0.5%. In this case, products containing milk
fat in the range of [50.25%, 50.75%] will not belong to
any specified class with a sufficiently high level of
confidence.
Despite measurement errors, the boundary of
identification classes should be set without taking them
into account. Regardless of the nature (except for the
assumption of distribution symmetry) and the type of
error at the point of the boundary, the indicator function
of both classes will be equal to 0.5. This is a logical
assumption to refer the product to a particular class
if the measurement gave a boundary indicator. In the
above example, this boundary will be the point 50%.
However, if indicator functions of two identification
classes, being adjacent linguistic characteristics of the
same criterion, take the same value of 0.5 at a point, it
makes sense setting a boundary between these classes at
this point. For the multidimensional case, the boundary
will be represented by a hyperplane, but in practice the
dimension exceeding two is rarely considered.
Harrington’s function as an example of indicator
function. One of the applied tools in the qualitative
assessment of the developed food industry identification
methods is Harrington’s desirability function [20].
The idea of Harrington’s function is to transform
the values of the criteria into a dimensionless
desirability scale that allows comparing and combining
the characteristics of products of different nature.
It establishes compliance between experts’ psycholinguistic
assessments and natural indicators of criteria.
In addition, it has all the necessary practical properties
of the indicator function, which allows using it actively
in 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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(19)
where
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 relation
between the values of the experimental variable
and the dimensionless scale [21]. In practice, it is
almost always linear, being accountable for the
shift and steepness of Harrington’s function curve
in accordance with application needs. It is so as
to correspond to the well-established mapping
of the function value intervals to the linguistic
variable 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 desirability
functions
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴∗ (𝑥𝑥) ∈ {0,1}, 𝑥𝑥 ∈ ℝ𝑛𝑛
𝐴𝐴 ⊂ 𝐵𝐵 ⟺ 𝜒𝜒𝐴𝐴(𝑥𝑥) ≤ 𝜒𝜒𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ ℝ𝑛𝑛
𝜒𝜒𝐴𝐴 _ (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
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(20)
The useful property of the function is insensitivity
within the range of 0 to 1 values (estimates “very bad”
and “very good”, respectively). It can be used in the
construction of criteria linked to the product’s shelf life.
Figure 5 Indicator functions of the interval criterion in strict
and fuzzy approaches
Criterion interval Specified interval
17
Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19
In food products ‒ complex biological systems ‒
quality deterioration is often subject to the exponential
law, and one of the key factors is the change in
microbiological parameters. So, Harrington’s function
can be considered as
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 health
regulations. Here storage time
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 us
first consider its validity indicator function without the
use of fuzzy logic (Fig. 6).
Identification areas I “the product meets the
standards and is ready for consumption” and III “the
product must be disposed of” are shown, respectively.
Within strict logic, at point {4}, it is expected that the
function has a gap of the first kind. As an applicable
rule, the function reflects rather Cinderella’s carriage
qualitative characteristics before and after midnight than
those 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) ≥ 0
𝜒𝜒𝐴𝐴|𝐴𝐴 __(𝑥𝑥) ≤ 1
𝜒𝜒(𝐴𝐴+𝐵𝐵)𝜆𝜆 (𝑥𝑥) = 𝜆𝜆𝜒𝜒𝐴𝐴(𝑥𝑥) + (1 − 𝜆𝜆)𝜒𝜒𝐵𝐵(𝑥𝑥)
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(21)
The function is a fuzzy negation of Harrington’s
function, but this is natural when smaller values are
assumed to have a larger desirability value. Its graph is
shown in Fig. 7.
In addition to the identification areas I and III,
whose linguistic characteristics remain the same, there
appears area II – “the product is safe for use, but already
undergoing degrading qualitative changes”. In this
sense, products from identification area II are no longer
suitable for end-users and must be sent for extending
shelf life processing (sterilization, canning, etc.). At
the same time, it should be noted that, for example,
when canning, a fuzzy logic device must also be used
to establish the final shelf life of the product from raw
materials within the boundaries of identification area II.
As it was mentioned above, changes in
microbiological parameters have a direct impact
on the quality of the product. Logically the phases
of microbiological cultures’ development can be
compared with Harrington’s function identification
areas; in particular, area I corresponds to the lag phase,
area II – to acceleration and exponential growth of
microorganisms phase, and area III – to deceleration
and stationarity phase. At the same time, substrate and
other biotechnological characteristics of change in the
population of microorganisms are calculated for each
specific product, which may lead to diversities in the
general matches given.
Combination of the product’s quality
characteristics. The construction of indicator functions
is inextricably linked with decision-making systems.
In the case of one criterion (for example, safety, as
described above), the fuzzy logic apparatus gives no
clear advantage over a strict approach. In the end,
all the contractors of the food industry (consumers,
manufacturers, law enforcement agencies, etc.) make
a binary decision whether a particular product sample
complies with a criterion [22, 23]. Due to the fact that
the criterion is unique (for example, expiration date)
they identify the above decision with the function of the
ultimate goal (“buying” vs. “not buying”, “recalling”
vs. “not recalling”, “fining” vs. “not fining”).
In the example with the fety function, three clear
identification areas can be introduced. For them, for
instance, the seller will have a system of specific
actions (I – “selling”, II – “reselling for recycling”,
III – “recycling”).
However, even when the second criterion in the
decision-making system is engaged, it is much harder to
establish the precise boundaries of identification classes.
Consider the instance with the safety criterion
with an additional indicator “consumer quality” – a
characteristic that demonstrates the taste and overall
satisfaction from the consumption of the product –
added. In the fuzzy logic the unction of this indicator
decreases faster than the safety function. For example,
for baking and confectionery products, taste profiles
degrade much earlier than the products become unfit
for use. The taste of “fresh bread” is of great value to
the consumer and has its impact on their purchase
preferences, but it is not unique or decisive, as shelf life
is also taken into account.
To construct an example of the consumer quality
function
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 fresh
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 makes
sense to put an experiment to determine the distribution
histogram of the moment of fresh taste degradation.
However, for the purposes of exemplification, it will be
simulated with the help of Weibull distribution, used in
survival analysis, giving a good approximation in the
study of products’ storage stability [25]. The density of
this distribution
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 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) =
1
2
erf 𝑥𝑥 =
𝑑𝑑(𝑥𝑥) = 𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = 𝜒𝜒𝑄𝑄 (𝑡𝑡) 𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 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, classically
Q (quality)
t, days
18
Oganesyants L.A. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 12–19
As a quality function we take the survival function
+ ( 𝑥𝑥
0.75)
8 1 + (𝑥𝑥 − 1.5
0.75 )
8 1 + (𝑥𝑥 − 3
0.75 )
8 1 (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
𝜀𝜀 ≤ −1 − 𝑥𝑥)
(17)
𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)
𝛷𝛷𝜀𝜀(𝑥𝑥) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
(22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
,
1 + ( 𝑥𝑥
0.75)
8 1 + (𝑥𝑥 − 1.5
0.75 )
8 1 (𝑥𝑥 − 3
0.75 )
8 1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
≤ 1) − ℙ(𝑥𝑥 + 𝜀𝜀 ≤ −1) =
𝜀𝜀 ≤ −1 − 𝑥𝑥)
(17)
𝜒𝜒𝐼𝐼 (𝑥𝑥) = 𝛷𝛷𝜀𝜀(1 − 𝑥𝑥) − 𝛷𝛷𝜀𝜀(−1 − 𝑥𝑥)
𝛷𝛷𝜀𝜀(𝑥𝑥) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
for the given distribution. It is equal to the
probability that the value of the random variable under
study will exceed t, in this case, the probability that
the taste has not yet been lost by t. For the Weibull
distribution, it has a convenient expression:
𝜒𝜒𝐴𝐴𝑥𝑥=
𝑥𝑥𝑖𝑖+ 1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑥𝑥𝑖𝑖+ 1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(23)
Supposing that for a product with the safety function
described by formula (23) the average taste profile is
lost on the second day, an approximation of distribution
parameters with
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
⊂ ℝ𝑛𝑛 𝛼𝛼 < 1𝛼𝛼 > 1 (𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1 … 𝑛𝑛, where 𝜒𝜒𝐴𝐴(𝑥𝑥) = 1
𝜒𝜒𝐴𝐴(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 ε
1,1] ± 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 𝑚𝑚
𝑖𝑖=1
𝑛𝑛 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 establish
the time when the product should already be sold at the
residual price (the time of entering area III) taking into
account the safety and consumer quality criteria.
If the weight of safety indicator is set at 0.6, and
the weight of consumer quality indicator is set at 0.4,
respectively, a convex criteria combination (8) will take
the form:
𝑆𝑆 = (Σ 𝐴𝐴𝑖𝑖
𝑚𝑚
𝑖𝑖=1
)
𝛬𝛬
𝛬𝛬 = {𝜆𝜆1, 𝜆𝜆2, … , 𝜆𝜆𝑚𝑚}, Σ 𝜆𝜆𝑖𝑖
𝑚𝑚
𝑖𝑖=1
= 1
𝜒𝜒𝐴𝐴𝛼𝛼 (𝑥𝑥) = 𝜒𝜒𝐴𝐴(𝑥𝑥)𝛼𝛼 , 𝛼𝛼 > 0
𝜒𝜒𝐴𝐴(𝑥𝑥) = 𝑃𝑃(𝑋𝑋 ∈ 𝐴𝐴 | 𝑋𝑋 = 𝑥𝑥)
𝜒𝜒𝐴𝐴(𝑥𝑥) = −
1
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
𝑥𝑥 +
𝑦𝑦𝑖𝑖
𝑥𝑥𝑖𝑖+1
′ − 𝑦𝑦𝑖𝑖
+ 1, 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖; 𝑥𝑥𝑖𝑖+1
′ ]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ 𝑥𝑥 −
𝑦𝑦𝑖𝑖
′
𝑥𝑥𝑖𝑖+1 − 𝑦𝑦𝑖𝑖
′ , 𝑥𝑥 ∈ [𝑦𝑦𝑖𝑖
′; 𝑥𝑥𝑖𝑖+1]
𝜒𝜒𝐴𝐴(𝑥𝑥) =
1
1 + (𝑥𝑥 − 𝑐𝑐
𝑙𝑙 )
𝑝𝑝
1
1 + ( 𝑥𝑥
0.75)
8 ,
1
1 + (𝑥𝑥 − 1.5
0.75 )
8 ,
1
1 + (𝑥𝑥 − 3
0.75 )
8 ,
1
1 + (𝑥𝑥 − 5
1.25 )
12
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
1) = 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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63
(24)
Solving the equation (0.63 being Harrington’s
function upper exponent for “satisfactory”):
0.5 = 2𝜎𝜎, 𝜎𝜎2 =
1
16
ℙ(−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) =
1
2
(1 + erf (2√2𝑥𝑥))
erf 𝑥𝑥 =
2
√𝜋𝜋
∫ 𝑒𝑒−𝑡𝑡2𝑑𝑑𝑑𝑑
𝑥𝑥
0
𝑑𝑑(𝑥𝑥) = 𝑒𝑒ⅇ−𝑌𝑌(𝑥𝑥) , 𝑥𝑥 ∈ ℝ
𝑑𝑑̅(𝑥𝑥1, … , 𝑥𝑥𝑛𝑛) = (Π𝑑𝑑(𝑥𝑥𝑖𝑖)𝑖𝑖
𝑤𝑤𝑖𝑖
𝑛𝑛
𝑖𝑖=1
)
1
Σ 𝑤𝑤𝑖𝑖
𝑛𝑛𝑖𝑖
=1
𝜒𝜒𝑄𝑄 (𝑡𝑡) = 1 − 𝑒𝑒ⅇ5−𝑡𝑡
𝑓𝑓𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) =
𝑘𝑘
𝜆𝜆𝑘𝑘 𝑡𝑡𝑘𝑘−1𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0 (22)
𝑆𝑆𝑊𝑊(𝑘𝑘,𝜆𝜆)(𝑡𝑡) = 𝑒𝑒−(𝑡𝑡⁄𝜆𝜆)𝑘𝑘, 𝑡𝑡> 0
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.6(1 − 𝑒𝑒ⅇ5−𝑡𝑡 ) + 0.4 𝑒𝑒−(𝑡𝑡⁄2)2
𝜒𝜒(𝑄𝑄+𝑇𝑇)0.6 (𝑡𝑡) = 0.63 (25)
we obtain
𝜒𝜒𝑆𝑆(𝑥𝑥) = Σ 𝜆𝜆𝑖𝑖𝜒𝜒𝐴𝐴𝑖𝑖 (𝑥𝑥)
𝑚𝑚
𝑖𝑖=1
𝐴𝐴1, 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 the
product must be sold in traditional or alternative ways.
Guided by rate expiry date only, the seller would get
the value of 5, thus having no time left for operational
maneuvers. The type of function graphs and their
convex combination is shown in Fig. 8.
CONCLUSION
Thus, the apparatus of fuzzy logic allows building
multi-criteria decision-making systems in the food
industry. They help effectively make decisions about
products’ quality and safety and, in the case of violations
and arbitral bodies’ involvement, differentiate the
administrative impact on the contractors of the food
industry.
CONFLICT OF INTEREST
The authors declare that there is no conflict of
interest related to this article.

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