Let us consider the set of some kind of physical values p ∈ [ P m i n , P m a x ] .

To normalize these values of p we utilize the following expression

Hence, in our research we always consider to operate on a universe of discourse U , i.e. on a set of integers rather, than on original physical scale of p ∈ [ P m i n , P m a x ] . For this purpose, we use the following mapping

Let P be correspondent to p n o r m from (2.1) type-1FS and let the relevant membership function(MF) of u ∈ U in P be μ p ( u ) , which is a crisp number in [ 0 , 1 ] .

Thus, the fuzzy set is presented as

On the other hand, to determine the estimates of the MF in terms of singletons from (2.3) in the form μ p ( u i ) / u i | ∀ i ∈ [ 1 , C a r d U ] , given (2.1) and (2.2) we propose the following procedure. ∀ i ∈ [ 1 , C a r d U ]

where α i is a weight coefficient of the i-th input cluster [9] for ∀ i ∈ [ 1 , C a r d U ] .

In more details from (2.4) we have

In type-1 fuzzy set P is represented as a triplet of the form

where P ˜ is normal FS with correspondent MF μ p : U → [ 0 , 1 ] . T ( u ) could be the term from Table 1.

For each fuzzy set P one could define a linguistic term from the following set

The following Table 1 depicts linguistic terms from (2.7).

Table 1. Linguistic terms.

To determine the estimates of the MF in terms of singletons from (2.5) in the form μ p ( u i ) / u i | ∀ i ∈ [ 1 , C a r d U ] , for each k-th term from Table 1. we use the following procedure.

Note, that MF from (2.8) has triangular form, which is defined on entire universe of discourse U .

A type-2 fuzzy set in X is P ˜ and the membership grade of x ∈ X in P ˜ is μ P ˜ ( x ) , which is a type-1FS in [ 0 , 1 ] . The elements of the domain of μ P ˜ ( x ) are called primary memberships of x in P ˜ and the memberships of the primary memberships in μ P ˜ ( x ) are called secondary memberships of x in P ˜ ; the latter defines the possibilities for the primary membership, e.g., the type-2FS, P ˜ ∈ X has fuzzy grades (FS in J X ⊆ [ 0 , 1 ] ), μ P ˜ ( x ) that can be denoted for each x ∈ X as μ P ˜ : X → [ 0 , 1 ] J X

And to define v i ∈ J X in (2.7) we propose the following expression

Thus, we presume that membership grade μ P ˜ ( x ) is normal, therefore ∃ ! l * ∈ [ 1 , C a r d J X ] | f ( v l * ) = max ∀ i ∈ [ 1 , C a r d J X ] { f ( v i ) } =   = 1 and the MF of fuzzy grades f ( v i ) we define as

Example 1.

Suppose the C a r d J X = 6 ,then from (2.10) we obtain J X = { 0 , 0.2 , 0.4 , 0.6 , 0.8 , 1 } ⊆ [ 0 , 1 ] . And also suppose the X = { 1 , 2 , 3 , 4 , 5 , 6 } is the set of numbers u from (2.2), which is correspondent to the set of normalized values p n o r m from (2.1) and that P is FS of type-2 of Normalized Variable(Figure 1).

P = NormalizedVariable = low/1 + middle/3 + high/5

Figure 1. Type-2 MF of “Normalized Variable” from Ex. 1.

From (2.9) and (2.11) by using terms from Table 1. we find the following membership grades( μ P ˜ ( 1 ) , μ P ˜ ( 3 ) and μ P ˜ ( 5 ) are shown on Figure 1).

μ P ˜ ( 1 ) = low = 1/0 + 0.8/0.2 + 0.6/0.4 + 0.4/0.6 + 0.2/0.8 + 0/1

μ P ˜ ( 3 ) = middle = 0.6/0 + 0.8/0.2 + 1/0.4 + 0.8/0.6 + 0.6/0.8 + 0.4/1

μ P ˜ ( 5 ) = high = 0/0 + 0.2/0.2 + 0.4/0.4 + 0.6/0.6 + 0.8/0.8 + 1/1

Note, that in Ex.1 J 1 = J 3 = J 5 = J X , whereas J 2 = J 4 = J 6 = ∅ .

Let us deliberate some important feature of fuzzy grades, which could be utilize later in this study.

Lemma 1.

If fuzzy grade μ P ( x ) = ∑ i f ( v i ) v i , v i ∈ J X

from (2.9) is defined as (2.10), then it is a convex for any integers i , k ∈ [ 1 , C a r d J X ] , ∃ j ∈ [ 1 , C a r d J X ] | i ≤ j ≤ k , i.e. f ( v j ) ≥ min { f ( v i ) , f ( v k ) } | i ≤ j ≤ k .

Proof:

Let us consider f ( v n − 1 ) , f ( v n ) , f ( v n + 1 ) from (2.11) ∀ n ∈ [ 1 , C a r d   J X ] . We have the following cases

1) n + 1 ≤ l * ;

2) n ≤ l * , n + 1 > l * ;

3) n − 1 ≤ l * , n > l * ;

4) n − 1 > l * .

For case 1. from (2.11)

From where we have

For case 2. from (2.11)

Given that Δ n + 1 − Δ n = 2 ⋅ l * − 2 ⋅ n − 1 | n ≤ l * ⇒ Δ n + 1 − Δ n < 0 we have

For case 3. from (2.11)

Given that Δ n − Δ n − 1 = 2 ⋅ n − 2 ⋅ l * − 1 | n > l * ⇒ Δ n − Δ n − 1 < 0 we have

For case 4. from (2.11)

From where we have

⇒ f ( v n ) > min { f ( v n − 1 ) , f ( v n + 1 ) } . ■

Let μ P ( x ) and μ Q ( x ) be two fuzzy grades that is, FSs in J X ⊆ [ 0 , 1 ] of FSs of type 2, P and Q , respectively, represented as

where the functions f and g are MFs of fuzzy grades (FSs in J X ⊆ [ 0 , 1 ] ) μ P ( x ) and μ Q ( x ) , respectively, and the values f ( v i ) and g ( w j ) in [0, 1] denote the grades for v i and w j in J X , respectively. Thus, the operations for FS of type 2 are expressed by the following.

Union.

where ∧ represent min and ∨ denote max, whereas ⊔ is fuzzy grades operation join.

Intersection.

where ∧ represent min, whereas ⊓ is fuzzy grades operation meet.

Remark.

In general, if μ P ( x ) and μ Q ( x ) are two fuzzy grades that is, FSs in J X ⊆ [ 0 , 1 ] of FS of type 2, P and Q in X respectively, denoted as (2.12) and (2.13), and ⋇ is a binary operation defined in X , then the operation ⋇ can be extended to FS P and Q by the definition relation (extension principle).

Example 2.

Let J X = { 0 , 0.2 , 0.4 , 0.6 , 0.8 , 1 } and let fuzzy grades μ P ( x ) and μ Q ( x ) be given as

μ P ( x ) = 0.8/0+ 1/0.2+ 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1

μ Q ( x ) = 0.2/0 + 0.4/0.2 + 0.6/0.4 + 0.8/0.6 + 1/0.8 + 0.8/1

Then for Join operation from (2.14) we have

μ p ( x ) ⊔ μ Q ( x ) = (0.8/0 + 1/0.2+ 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1) ⊔ (0.2/0 + 0.4/0.2 + 0.6/0.4 + 0.8/0.6 + 1/0.8 + 0.8/1)

=

=0.2/0 + (0.2∨0.4∨0.4)/0.2 + (0.2∨0.4∨0.6∨0.6∨0.6)/0.4 + (0.2∨0.4∨0.6∨0.6∨0.8∨0.8∨0.8)/0.6 + (0.2∨0.4∨0.4∨0.4∨0.4∨ 0.6∨0.8∨1∨0.8)/0.8 + (0.2∨0.2∨0.2∨0.2∨0.2∨0.2∨0.4∨0.6∨0.8∨0.8∨0.8)/1 = 0.2/0 + 0.4/0.2 + 0.6/0.4 + 0.8/0.6 + 1/0.8 + 0.8/1

For Meet operation from (2.15)

μ p ( x ) ⊓ μ Q ( x ) = (0.8/0 + 1/0.2+ 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1) ⊓ (0.2/0 + 0.4/0.2 + 0.6/0.4 + 0.8/0.6 + 1/0.8 + 0.8/1)

=

=(0.2∨0.2∨0.2∨0.2∨0.2∨0.2∨0.4∨0.6∨0.8∨0.8∨0.8)/0 + (0.2∨0.4∨0.4∨0.4∨0.4∨0.6∨0.8∨1∨0.8)/0.2 + (0.2∨0.4∨0.6∨ 0.6∨0.8∨0.8∨0.8)/0.4 + (0.2∨0.4∨0.6∨0.6∨0.6)/0.6 + (0.2∨0.4∨0.4)/0.8 + 0.2/1 = 0.8/0 +1/0.2+ 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1

Note. For both Joinand Meet operations the number N 1 of calculations needed was N 1 = 3 × C a r d J X 2 , In our case for C a r d J X = 6 we obtain N 1 = 108 ,

Now we introduce significantly more effective way to operate with type-2 FS.

Lemma 2.

If fuzzy grades

and

are both normal, i.e.

then the following features are taking place

Proof:

1) v k * , w l * ∈ J | v k * ≤ w l * ⇒ b * = v k * ∧ w l * = v k *

2) v k * , w l * ∈ J | v k * > w l * ⇒ b * = v k * ∧ w l * = w l *

3) v k * , w l * ∈ J | v k * ≤ w l * ⇒ b * = v k * ∨ w l * = w l *

4) v k * , w l * ∈ J | v k * > w l * ⇒ b * = v k * ∨ w l * = v k *

μ p ( x ) ⊔ μ Q ( x ) = 1 + ∑ v i ∨ w j ≠ b *   f ( v i ) ∧ g ( w j ) = μ p ( x ) . ■

Corollary 1.

From the conditions (2.17) of Lemma 2 we can formulate the following fast and simple implementations of Meet and Join operations for type-2 FS

Note. For both Joinand Meet operations the number N 2 of calculations needed was N 2 = ( 2 × C a r d J X ) + 1 , In our case for C a r d J X = 6 we obtain N 2 = 13 , Therefore, this algorithm is more than 8.3 times faster than the original one.

In this chapter we revisit and reconsider some important features of type-2FLS, in which the antecedent or consequent membership functions are type-2 FS.

Figure 2. The structure of a type-2 FLS.

Figure 2 shows the structure of a type-2FLS[6]. It is very similar to the structure of a type-1FLS. The fuzzifier maps the crisp input into a FS. This fuzzy set can, in general, be a type-2 set. The inference engine combines rules and gives a mapping from input type-2FS to output type-2FS. To do this one needs to find unions and intersections of type-2FS sets, as well as compositions of type-2 relations. In the type-2 case, the output of the inference engine is a type-2 set. Usually, in type-2FLSextended defuzzification is utilized, which gives a type-1 fuzzy set. Since this operation transforms type-2 output sets to the FLS to a type-1 set, this operation is called type reduction and the type-reduced set so obtained a type-reduced set. In [6]-[8] it was mentioned that in order to get a crisp output from a type-2 FLS, the type-reduced set has to be de-fuzzified.

Hence, in order to develop a type-2 FLS, one needs to be able to:

1) perform the set theoretic operations of union, intersection, and complement on type-2 sets;

2) know properties (e.g., commutativity, associativity, identity laws) of membership grades of type-2 sets;

3) deal with type-2 fuzzy relations and their compositions;

4) perform type reduction and defuzzification to obtain a set-valued or crisp output from the FLS [7][8].

We will concentrate our attention on 1., 2. and 3.

Let us consider a type-2 FLS having K inputs, x 1 ∈ X 1 , x 2 ∈ X 2 , ⋯ , x K ∈ X K , and one output y ∈ Y . We assume that this FLShas Mrules where the m-th rule has the form

This rule represents a type-2 fuzzy relation between the input space X 1 × X 2 × ⋯ × X K and the output space Y of the FLS. We denote the membership function of this type-2 relation as,

∀ m ∈ [ 1 , M ] | μ P 1 m ˜ × P 2 m ˜ × ⋯ × P K m ˜ → Q ˜ m ( x , y ) , where P 1 m ˜ × P 2 m ˜ × ⋯ × P K m ˜ denotes the Cartesian product of P 1 m ˜ , P 2 m ˜ , ⋯ , P K m ˜ and x = { x 1 , x 2 , ⋯ , x K } . When an input x ′ is applied, the composition of the fuzzy set P ′ ˜ to which x ′ have its place and the rule R m is found by using the extended sup-star composition [6]-[8]

Let us take a look at the implication membership function from (3.2) for each m-th rule and denote it in the following form

We have to underline the fact that the implication operation in (3.3) is an essential element of a decision-making mechanism, known as Fuzzy Conditional Inference Rule (FCIR) of the (3.1) form.

Thus, we use the following type of implication [10]

For practical purposes, described down below, we will use Fuzzy Conditional Rule (FCR) [11] of the following type

From (3.3) the correspondent fuzzy binary relationship would look like that ∀ m ∈ [ 1 , M ]

Given (3.5) from (3.6) we are getting ∀ m ∈ [ 1 , M ]

To further investigate the structure and possible implementation of [ ∏ ∀ i ∈ [ 1 , K ] μ P i m ˜ ( x ) ] in (3.7) we introduce the following.

Definition1.

The set of values J X = { v i } = { v 1 , v 2 , ⋯ , v N } ⊆ [ 0 , 1 ] | is a lattice ( J X , ≤ ), i.e. v 1 < v 2 < ⋯ < v N and ∀ a , b ∈ J X | a ≤ b ⇒ a ∧ b = a ; a ∨ b = b .

Definition2.

For a lattice J X ⊆ [ 0 , 1 ] and N various normal fuzzy grades in J X , i.e.

∃ ! k j * ∈ [ 1 , C a r d J X ] | f j ( v k j * ) = max ∀ i ∈ [ 1 , C a r d J ] { f j ( v i j ) } =   = 1 , we define

Therefore, the relevant indexes from (2.8) are

From (3.8) and (3.10) we define the upper fuzzy grade

And from (3.9) and (3.11) we define its lower counterpart

Based on Definition1 and Definition2 we shall prove the following.

Lemma 3.

If J X ⊆ [ 0 , 1 ] is a lattice, then for N various normal fuzzy grades in J X , i.e.

the following is taking place

Proof:

From Definition 2 we have the set V ⊆ J X = { v k * } = { v k 1 * , v k 2 * , ⋯ , v k M * } | V ⊆ [ v l o w e r , v u p p e r ] .

Since v l o w e r , v u p p e r ∈ J X | v l o w e r < v u p p e r , from (3.12) and (3.13) we obtain the following

∐ j = 1 , N ¯   μ P j ( x ) = μ P l o w e r ( x ) ⊔ μ P u p p e r ( x ) = μ P u p p e r ( x ) . ■

It was mentioned in [6]-[8] that “the distinction between type-1 and type-2 is associated with the nature of the MFs, which is not important while forming ‘IF-THEN’ rules”. In opposite, from our point of view, this aspect needs additional attention. To further elaborate on type-2 FLS inference specificity and on correspondent “IF-THEN” rules formulation, we have to mention that in contrast with type-1 case, these rules are describing not a knowledge about input/output system behavior, but rather between our semantic interpretation/understanding of a nature of things under consideration. For this particular reason we are formulating the following.