Approximations of Fuzzy Numbers by General Trapezoidal Fuzzy Numbers - Innovative Management in Information and Production - page 369

4.2 Example 5

− √ α ] t ,and A t 2 ,

4

Let f ( t )= 1 − λ ( t )= =[ α , 3 ,0 ≤ α ≤ 1. Find the general f-

A A

trapezoidal approximation T of

A . First, we apply Algorithm 4 to compute the general f-trapezoidal approximation
of A , as follows. By Step 1, it is easy to verify that

( ) and the general f-triangular approximation Δ ( )

1
2 ,

1
4 ,

1
6 ,

1
6 ,

1
24 ,

11
10 ,

103
180 .

a = b = c = l = l f = u = u f =

Therefore, by Step 2 we compute

⎡ ⎤ ⎡ ⎤

1
2

1
4

00 230 0

−

⎣ ⎦ ⎣ ⎦ .

1
4

1

6 00
00 2

360 0
002

φ − 1 φ = and = 4

1
4

− 3

00 4

1
6

00 − 36

Hence, we obtain

5
6 ,−

29
15 ,

8
15 )

) φ − 1

( s 1 , s 2 , s 3 , s 4 )=( l , l f , u , u f =( 1 ,

s 3 , the general f-trapezoidal approximation of A is Since s 1 ≤

5
6 − (

29
15 +

8
15 (

A

2

a 2 T ( )=[ 1 − α ) , 1 − )]

Now, we compute the general f-triangular approximation of A .ByStep3,we
compute

⎡ ⎤ ⎡ ⎤

1
4

1
4

1 46 − 6

⎣ ⎦ and

ψ − 1

⎣ ⎦ .

1
4

1
6

ψ = = 4 −

0 6159

1
4

1
6

0 − 6915

Hence, we obtain

83
60 ,−

73
40 ,

163
120 )

u f ) ψ − 1 ( t 1 , t 2 , t 3 =( l + u , l f , =(

and the general f-triangular approximation of A is

83
60 −

73
40 (

83
60 +

163
120 (

A

2 2

Δ ( )=[ 1 − α ) , 1 − α )]

as shown in the following figures: