Defuzzification of trapezoidal type-2 fuzzy variables and its application to solid transportation problem

2.1 Type-2 fuzzy set (T2 FS)

Type-2 fuzzy set A ˜ defined on a universe of discourse X, which is denoted as A ˜ ⊆ X, is a set of pairs { x , μ A ˜ ( x ) }, where x an element of a fuzzy set is, and its grade of membership { μ A ˜ ( x ) } in the fuzzy set A ˜ is a type-1 fuzzy set defined in the interval J _x  ⊂ [0, 1], i.e. A T2 FS A ˜ defined by Mendel and John [12] is A ˜ = { ( ( x , u ) , μ A ˜ ( x , u ) ) : ∀ x ∈ X , J x ⊂ [ 0 , 1 ] } ,

where 0 ≤ μ A ˜ ( x , u ) ≤ 1 is the type-2 membership function.

Example 1. Let us take A ˜ { ( x , μ A ˜ ( x ) ) : x ∈ X } whereX = 3, 6, 9 and the primary memberships of the points 3, 6, 9 are given by J₃ = 0.4, 0.8, 0.9, J₆ = 0.3, 0.7, 0.8, 0.9 and J₉ = 0.2, 0.7, 1.0 respectively. Then the secondary grade of the point 3 is μ A ˜ ( 3 ) = μ A ˜ ( 3 , u ) = ( 0.5 / 0.4 ) + ( 0.7 / 0.8 ) + ( 0.3 / 0.9 ) ∼( 0.4 0.8 0.9 0.5 0.7 0.3 ). i.e. μ A ˜ ( 3 , 0.4 ) = 0.5 , μ A ˜ ( 3 , 0.8 ) = 0.7 , μ A ˜ ( 3 , 0.9 ) = 0.3 .

More specifically μ A ˜ ( 3 , 0.4 ) = 0.5 means that the membership grade which is named as secondary membership grade that the point 3 has the primary membership 0.4 is 0.5. So A ˜ considers on the value 3 with membership grade ( 0.4 0.8 0.9 0.5 0.7 0.3 ), which is a RFV.

Same as, μ A ˜ ( 6 ) = μ A ˜ ( 6 , u ) = ( 0.4 / 0.3 ) + ( 0.6 / 0.7 )+ ( 0.5 / 0.8 ) + ( 0.8 / 0.9 ) ∼ ( 0.4 0.6 0.5 0.8 0.3 0.7 0.8 0.9 ) and μ A ˜ ( 9 ) = μ A ˜ ( 9 , u ) = ( 0.9 / 0.2 ) + ( 0.5 / 0.7 ) + ( 0.1 /1.0 ) ∼ ( 0.9 0.5 0.1 0.2 0.7 1.0 ).

So discrete type-2 fuzzy variable A ˜ is given by A ˜ = ( 0.5 / 0.4 ) / 3 + ( 0.7 / 0.8 ) / 3 + ( 0.3 / 0.9 ) / 3 + ( 0.3 / 0.4 ) / 6 + ( 0.7 / 0.6 ) / 6 + ( 0.8 / 0.5 ) / 6 + ( 0.9 / 0.8 ) / 0.6 + ( 0.2 / 0.9 ) / 9 + ( 0.7 / 0.5 ) / 9 + ( 1.0 / 0.1 ) 9 .

Type-2 fuzzy set A ˜ can be also express as,

A ˜ = { 3 , with possibilities ( 0.4 , 0.8 , 0.9 ) ; 6 , with possibilities ( 0.3 , 0.7 , 0.8 , 0.9 ) ; 9 , with possibilities ( 0.2 , 0.7 , 0.1 ) ;

2.2 Regular fuzzy variable (RFV) and their Critical values (CVs)

For a possibility space (φ, p, Pos), a regular fuzzy variable ξ ˜ is defined as a measurable map from φ to [0, 1] in the sense that for every t ∈ [0, 1], one has { γ ∈ φ | ξ ˜ ( γ ) ≤ t } ∈ p.

A discrete RFV is represented as ξ ˜ ∼ ( r 1 , r 2 ⋯ r n μ 1 , μ 2 ⋯ μ n ) where r _i  ∈ [0, 1] and μ _i  > 0, ∀ i and max i ( μ i ) = 1.

If ξ ˜ = ( r 1 , r 2 , r 3 ) with 0 ≤ r₁ < r₂ < r₃ ≤ 1, then ξ ˜ is called a triangular RFV.

Qin et al. [32] introduced three kinds of critical values (CVs). Let ξ ˜ be a RFV. Then,

The optimistic CV of ξ ˜, denoted by CV * [ ξ ˜ ] is given by,

CV * [ ξ ˜ ] = sup α ∈ [ 0 , 1 ] [ α ∧ Pos ( ξ ˜ ≥ α ) ](1)

Example 2. Let ξ ˜ be a discrete RFV define as ξ ∼ ( 0.3 0.5 0.8 0.9 0.1 0.9 0.6 0.3 )

Then we can find out that,

Pos ( ξ ˜ ≥ α ) = { 0 , if α ≤ 0.2 0.9 , if 0.2 < α ≤ 0.5 0.6 , if 0.5 < α ≤ 0.8 0.3 , if 0.8 < α ≤ 1.0

Nec ( ξ ˜ ≥ α ) = { 0.9 , if α ≤ 0.3 0.1 , if 0.3 < α ≤ 0.5 0.4 , if 0.5 < α ≤ 0.8 0.7 , if 0.8 < α ≤ 1.0

and

Cr ( ξ ˜ ≥ α ) = { 0.9 , if α ≤ 0.3 0.5 , if 0.3 < α ≤ 0.5 0.5 , if 0.5 < α ≤ 0.8 0.5 , if 0.8 < α ≤ 1.0

Then by the definitions of CVs, from (1)-(3), we have

CV * [ ξ ˜ ] = sup α ∈ [ 0 , 1 ] [ α ∧ Pos ( ξ ˜ ≥ α ) ] = sup α ∈ [ 0 , 0.2 ] [ α ∧ 0 ] ∨ sup α ∈ ( 0.2 , 0.5 ] [ α ∧ 0.9 ] ∨ sup α ∈ ( 0.5 , 0.8 ] [ α ∧ 0.6 ] ∨ sup α ∈ [ 0.8 , 1 ] ( α ∧ 0.3 ] = 0 ∨ 0.5 ∨ 0.6 ∨ 0.3 = 0.6

CV * [ ξ ˜ ] = sup α ∈ [ 0 , 1 ] [ α ∧ Nec ( ξ ˜ ≥ α ) ] = sup α ∈ [ 0 , 0.3 ] [ α ∧ 0.9 ] ∨ sup α ∈ ( 0.3 , 0.5 ] [ α ∧ 0.1 ] ∨ sup α ∈ ( 0.5 , 0.8 ] [ α ∧ 0.4 ] ∨ sup α ∈ ( 0.8 , 1 ] [ α ∧ 0.7 ] = 0 ∨ 0.1 ∨ 0.4 ∨ 0.7 = 0.7

and

CV [ ξ ˜ ] = sup α ∈ [ 0 , 1 ] [ α ∧ Cr ( ξ ˜ ≥ α ) ] = sup α ∈ [ 0 , 0.3 ] [ α ∧ 0.9 ] ∨ sup α ∈ ( 0.3 , 0.5 ] [ α ∧ 0.5 ] ∨ sup α ∈ ( 0.5 , 0.8 ] [ α ∧ 0.5 ] ∨ sup α ∈ ( 0.8 , 1 ] [ α ∧ 0.5 ] = 0 ∨ 0.5 ∨ 0.5 ∨ 0.5 = 0.5

The following theorems introduced the critical values (CVs) of trapezoidal and triangular RFVs.

Theorem 1.(Qin et al. [32]) Letξ ˜ = ( r 1 , r 2 , r 3 , r 4 )be a trapezoidal RFV. Then we have,

The optimistic CV of ξ ˜ is CV * [ ξ ˜ ] = r 4 ( 1 + r 4 - r 3 ) .

Example 3. Let ξ ˜ = ( 0.3 , 0.4 , 0.8 , 0.9 ) be a trapezoidal RFV. Then according to the Theorem 1 we have, CV * [ ξ ˜ ] = 9 11 , CV * [ ξ ˜ ] = 4 11 , CV [ ξ ˜ ] = 1 2 .

Theorem 2.(Qin et al. [32]) Letξ ˜ = ( r 1 , r 2 , r 3 )be a triangular RFV. Then we have,

The optimistic CV of ξ ˜ is CV * [ ξ ˜ ] = r 3 ( 1 + r 3 - r 2 ) .

Example 4. Let ξ ˜ = ( 0.6 , 0.7 , 0.9 ) be a triangular RFV. Then according to the Theorem 2 we have, CV * [ ξ ˜ ] = 3 4 , CV * [ ξ ˜ ] = 7 11 , CV [ ξ ˜ ] = 2 3 .

3.1 Defuzzification using CV

According to Chen et al. [45] for a trapezoidal type-2 fuzzy fuzzy variable ξ ˜ = ( r 1 , r 2 , r 3 , r 4 ; θ l , θ r ), where r _i  ∈ R, ∀ i and θ _l , θ _r  ∈ [0, 1] are the two parameters that characterize the degree of uncertainty that ξ ˜ takes a value say x and the corresponding secondary possibility distribution function μ ˜ ξ ˜ ( x ) is given by,

μ ˜ ξ ˜ ( x ) = ( x - r 1 r 2 - r 1 - θ l min { x - r 1 r 2 - r 1 , r 2 - x r 2 - r 1 } , x - r 1 r 2 - r 1 , x - r 1 r 2 - r 1 + θ r min { x - r 1 r 2 - r 1 , r 2 - x r 2 - r 1 } )

For any x ∈ [r₁, r₂], μ ˜ ξ ˜ ( x ) = 1 ˜ for x ∈ (r₂, r₃) and μ ˜ ξ ˜ ( x ) = ( r 4 - x r 4 - r 3 - θ l min { r 4 - x r 4 - r 3 , x - r 3 r 4 - r 3 } , r 4 - x r 4 - r 3 ,r 4 - x r 4 - r 3 + θ r min { r 4 - x r 4 - r 3 , x - r 3 r 4 - r 3 } )

For any x ∈ [r₃, r₄].

Theorem 3.(Qin et al. [32]) Letξ ˜ = ( r 1 , r 2 , r 3 , r 4 ; θ l , θ r )be a type-2 trapezoidal fuzzy variable. Then we have,

Using the optimistic CV reduction method, the reduction ξ₁ of ξ ˜ has the following possibility distribution,

μ ξ 1 ( x ) = { ( 1 + θ r ) ( x - r 1 ) r 2 - r 1 + θ r ( x - r 1 ) , if x ∈ [ r 1 , r 1 + r 2 2 ] ( 1 - θ r ) x - r 1 + θ r r 2 r 2 - r 1 + θ r ( r 2 - x ) , if x ∈ ( r 1 + r 2 2 , r 2 ] 1 ˜ , if x ∈ ( r 3 , r 4 ) ( 1 + θ r ) ( r 4 - x ) r 4 - r 3 + θ r ( r 4 - x ) , if x ∈ ( r 3 , r 3 + r 4 2 ] ( - 1 + θ r ) x - θ r r 3 + r 4 r 4 - r 3 + θ r ( x - r 4 ) , if x ∈ ( r 3 + r 4 2 , r 4 ] .

Theorem 4.(Qin. et al. [32]) Supposeξ _i be the reduction of type-2 fuzzy variable ξ ˜ = ( r ˜ 1 i , r ˜ 2 i , r ˜ 3 i , r ˜ 4 i ; θ l , i , θ r , i ) obtained by the CV reduction method for i = 1, 2, 3, ⋯ , n and ξ₁, ξ₂, ⋯ , ξ _n are mutually independent, and k _i  ≥ 0 for i = 1, 2, 3, ⋯ , n .

Given the generalized credibility level α ∈ (0, 0.5], if α ∈ (0, 0.25], then Cr ˜ { ∑ i = 1 n k i ξ i ≤ t } ≥ α is equivalent to, ∑ i = 1 n ( 1 - 2 α + ( 1 - 4 α ) θ r , i ) k i r 1 i + 2 α k i r 2 i 1 + ( 1 - 4 α ) θ r , i ≤ t , and if α ∈ (0.25, 0.5], then Cr ˜ { ∑ i = 1 n k i ξ i ≤ t } ≥ α is equivalent to ∑ i = 1 n ( 1 - 2 α ) k i r 1 i + ( 2 α + ( 4 α - 1 ) θ l , i ) k i r 2 i 1 + ( 4 α - 1 ) θ l , i ≤ t .

Proof: We are proof here only case 2, the other can be proof in similar way.

For a trapezoidal type-2 fuzzy variable the possibility distribution obtained by CV reduction method is given by, μ ξ 3 ( x ) = { ( 1 + θ r ) ( x - r 1 ) r 2 - r 1 + 2 θ r ( x - r 1 ) , if x ∈ [ r 1 , r 1 + r 2 2 ] ( 1 - θ l ) x + θ l r 2 - r 1 r 2 - r 1 + 2 θ l ( r 2 - x ) , if x ∈ ( r 1 + r 2 2 , r 2 ] 1 ˜ , if x ∈ ( r 3 , r 4 ) ( - 1 + θ l ) x - θ l r 3 + r 4 r 4 - r 3 + 2 θ l ( x - r 3 ) , if x ∈ ( r 3 , r 3 + r 4 2 ] ( 1 + θ r ) ( r 4 - x ) r 4 - r 3 + 2 θ r ( r 4 - x ) , if x ∈ ( r 3 + r 4 2 , r 4 ] .

Let ξ^′ = ∑kξ

Now if α > 0.5 then we have,

Cr ˜ ( ξ ′ ≤ t ) = 1 2 ( 1 + sup x ≤ t μ ξ ′ ( x ) - sup x > t μ ξ ′ ( x ) )

= 1 2 ( 1 + 1 - sup x > t μ ξ ′ ( x ) )

Therefore Cr ˜ ( ξ ≤ t ) ≥ α is equivalent to 

If we define ξ s u p ′ ( α ) = s u p { r | sup x > t μ ξ ′ ( x ) ≥ α } for α ∈ (0, 1] then we have, ξ s u p ′ ( 2 − 2 α ) ≤ t

So we can write, using the above, ξ s u p ′ ( 2 − 2 α ) ≤ t = ( ∑ k ξ ) s u p ( 2 − 2 α ) = ∑ k ξ s u p ( 2 − 2 α ) ≤ t

Now if we put x = r 3 + r 4 2 in the possibility distribution function then it become 0.5 i.e. μ ξ ( r 3 + r 4 2 ) = 0.5

If (2 -2α) ≥0.5 i.e. α ∈ (0.5, 0.75] then ξ _sup  (2 -2α) is the solution of the following equation.

( - 1 + θ l ) x - θ l r 3 + r 4 r 4 - r 3 + 2 θ l ( x - r 3 ) = 2 - 2 α(4) Solving the equation (4) we get, x = ( 2 α - 1 ) k i r 4 i + ( 2 ( 1 - α ) + ( 3 - 4 α ) θ l , i ) k i r 3 i 1 + ( 3 - 4 α ) θ l , i ,

A trapezoidal type-2 fuzzy fuzzy variable is defined as ξ ˜ = ( r 1 , r 2 , r 3 , r 4 ; θ l , θ r ), then by optimistic CV, pessimistic CV and CV reduction method we have

Using the optimistic CV reduction method, the reduction ξ₁ of ξ ˜ has the following possibility distribution, μ ξ 1 ( x ) = { ( 1 + θ r ) ( x - r 1 ) r 2 - r 1 + θ r ( x - r 1 ) , if x ∈ [ r 1 , r 1 + r 2 2 ] ( 1 - θ r ) x - r 1 + θ r r 2 r 2 - r 1 + θ r ( r 2 - x ) , if x ∈ ( r 1 + r 2 2 , r 2 ] 1 ˜ , if x ∈ ( r 3 , r 4 ) ( 1 + θ r ) ( r 4 - x ) r 4 - r 3 + θ r ( r 4 - x ) , if x ∈ ( r 3 , r 3 + r 4 2 ] ( - 1 + θ r ) x - θ r r 3 + r 4 r 4 - r 3 + θ r ( x - r 4 ) , if x ∈ ( r 3 + r 4 2 , r 4 ] .

Now according to the definition of α-cut (Wu [10]), we have the following α-cuts of the reductions of ξ ˜

Using the optimistic CV reduction method, ξ 1 L ( α ) = { ( 1 + θ r ) r 1 + ( r 2 - r 1 - θ r r 1 ) α ( 1 + θ r ) - θ r α , if 0 ≤ α ≤ 0.5 ( r 1 - θ r r 1 ) + ( r 2 - r 1 - θ r r 2 ) α ( 1 - θ r ) + θ r α , if 0.5 < α ≤ 1

ξ 1 R ( α ) = { ( r 4 - θ r r 3 ) - ( r 4 - r 3 - θ r r 3 ) α ( 1 - θ r ) - θ r α , if 0.5 ≤ α ≤ 1 ( 1 + θ r ) r 4 - ( r 4 - r 3 + θ r r 4 ) α ( 1 + θ r ) - θ r α , if 0 ≤ α < 0.5

With this method for the α-cuts of optimistic CV, pessimistic CV or CV reduction of the trapezoidal type-2 fuzzy number ξ ˜ we can find the nearest interval approximation of ξ ˜.

5.1 Chance-constrained programming using generalized credibility

Chance-constrained programming with fuzzy number was developed by Liu et al. [4], Yang et al. [17], Kundu et al. [28, 29] using credibility measure. Since the parameters of the proposed problem are taken as trapezoidal type-2 fuzzy variable, so we defuzzify them using the proposed defuzzification methods (given in Section 3).

Let c ˜ ijk ′ , t ˜ ijk ′ , a ˜ i ′ , b ˜ j ′ , e ˜ k ′ are type reduced form for the type-2 fuzzy variables c ˜ ijk , t ˜ ijk , a ˜ i , b ˜ j , e ˜ k respectively based on critical value reduction method [32]. With this reduced form, we formulate a chance constrained programming model to find solution of the proposed problem. Basically in chance constrained programming, the uncertain constraints are allowed to be violated such that constraints constraints must be satisfied at some chance or confidence level specified by the DM. Also we can not use the usual credibility measure to the type reduced variables as they may not be normalized and hence we use the generalized credibility. With this, we formulate the following chance constrained programming model for the proposed problem given by (5)–(10). Min ︸ x Min ︸ f 1 ¯ f 1 ¯ Min ︸ x Min ︸ f 2 ¯ f 2 ¯ s . t .(11) Cr ˜ { ∑ i = 1 m ∑ j = 1 n ∑ k = 1 l c ˜ ijk ′ x ijk ≤ f 1 ¯ } ≥ α c Cr ˜ { ∑ i = 1 m ∑ j = 1 n ∑ k = 1 l t ˜ ijk ′ y ijk ≤ f 2 ¯ } ≥ α t Cr ˜ { ∑ j = 1 n ∑ k = 1 l x ijk ≤ a ˜ i ′ } ≥ α i , i = 1 , 2 , ⋯ , m Cr ˜ { ∑ i = 1 m ∑ k = 1 l x ijk ≥ b ˜ j ′ } ≥ α j , j = 1 , 2 , ⋯ , n Cr ˜ { ∑ i = 1 m ∑ j = 1 n x ijk ≤ e k ′ } ≥ α k , k = 1 , 2 , ⋯ , l with non negativity restriction given by constraint (10). Here α ^c , α ^t , α _i , α _j , and α _k , ∀ i, j, k are the predetermined different generalized credibility levels for the chance constraints respectively.