Division of generalized fuzzy numbers

Introduction

The Generalized Fuzzy Number (GFN) is realized to be more flexible than the normalized fuzzy number since it considers the degrees of confidence of experts' opinions. In 1985 [1] , Chen introduced the concepts of GFNs and also formulated different arithmetic operations on GFNs by proposing function principle. Based on Chen's (1985) [1] arithmetic operations, a few researchers, of late, have provided important theoretical foundations on GFNs and their applications (e.g., Chen and Chen 2003 [3] , Chen and Hsieh 1999 [2] , Chen and Wang 2009 [8] , Chen et al. 2007 [7] , Hsieh and Chen 1999 [3] , Islam and Roy 2006 [5] , Mahapatra and Roy 2006 [6] , Wei and Chen 2009) [9] . Although function principle (Chen 1985) [1] was employed to develop arithmetic operations on GFNs, eventually some shortcomings of these operations have been observed. From mathematical point of view computation of different arithmetic operations using function principle is basically a point-wise operation. Due to this reason, it has been observed that Chen's arithmetic operations on generalized trapezoidal (and also triangular) fuzzy numbers causes loss of information, leading to inexact results. In view of this, the aim of this paper to discuss the division of generalized trapezoidal fuzzy numbers with example, which shows that the existing operations make some approximation, thereby causing loss of information.

Definition 1.1 Positive Trapezoidal Fuzzy Number

A generalized trapezoidal fuzzy number denoted as 𝐴 ̃= (𝑚 1 , 𝑚 2 ; 𝛽, 𝛾 ; 𝑤) is said to be positive if 𝑚 1 -𝛽 ≥ 0 and 𝑚 1 -𝛽, 𝑚 1 , 𝑚 2 and 𝑚 2 + 𝛾 are not identical. [1] Let us consider two positive generalized trapezoidal fuzzy numbers 𝐴 ̃1 = (𝑚 1 , 𝑚 2 ; 𝛽 1 , 𝛾 1 ; 𝑤 1 ) and 𝐴 ̃2 = (𝑚 3 , 𝑚 4 ; 𝛽 2 , 𝛾 2 ; 𝑤 2 ).

The fuzzy division operation using function principle (Chen 1985)

Then the operational laws for these two fuzzy numbers are presented as follows The division, 𝐴 ̃1is divided by 𝐴 ̃2:

Setting 𝑚 1 = 𝑚 2 and 𝑚 3 = 𝑚 4 in the above operations, the corresponding results for generalized triangular fuzzy numbers (GTFNs) may be obtained. However, in course of the study some shortcomings of Chen's operations have been observed which are better reflected through GTFNs compared to generalized trapezoidal fuzzy numbers. Therefore, in the following section the shortcomings of Chen's method have been illustrated by utilizing example of GTFNs.

Operations on GFNs

Let 𝐴 ̃1 = (𝑚 1 , 𝑚 2 ; 𝛽 1 , 𝛾 1 ; 𝑤 1 )𝑎𝑛𝑑 𝐴 ̃2 = (𝑚 3 , 𝑚 4 ; 𝛽 2 , 𝛾 2 ; 𝑤

Division of GFNs

Let 𝐴 ̃= (𝑚 1 , 𝑚 2 ; 𝛽 1 , 𝛾 1 ; 𝑤 1 ) be a positive generalized trapezoidal fuzzy number with membership function 𝜇 𝐴 ̃(𝑥).

The membership function of the inverse of 𝐴 ̃, written as 𝐴 ̃-1 , is defined as 𝜇 𝐴 ̃-1(𝑥) = 𝜇 𝐴 ̃(1/𝑥)∀𝑥 ≠ 0. Then using the concept of inverse of a fuzzy number (Dubois and Prade 1980) we can deduce the approximate formula of 𝐴 ̃-1 as 𝐴 ̃-1 ≃ (1/𝑚 2 ,/𝑚 1 ; 𝛾 1 /𝑚 2 2 𝛽 1 /𝑚 1 2 ; 𝑤 1 ). Now the division of 𝐴 ̃1(/)𝐴 ̃2 may be defined as 𝐴 ̃1(/)𝐴 ̃2 = 𝐴 ̃1(×)𝐴 ̃2 -1

where 𝐴 ̃1 = (𝑚 1 , 𝑚 2 ; 𝛽 1 , 𝛾 1 ; 𝑤 1 ) 𝑎𝑛𝑑 𝐴 ̃2 = (𝑚 3 , 𝑚 4 ; 𝛽 2 , 𝛾 2 ; 𝑤 2 ) where are two positive generalized trapezoidal fuzzy numbers. By employing the product operation on 𝐴 ̃1 𝑎𝑛𝑑 𝐴 ̃2 -1 as discussed in Theorem 1.1 the following approximate result may be obtained:

Numerical illustration Example 1.1

Let 𝐴 ̃1 = (5; 1,1; 1)𝑎𝑛𝑑 𝐴 ̃2 = (2; 1,2; 0.5) be two GTFNs. Find 𝐴 ̃1(/)𝐴 ̃2 .

Result: We obtain 𝐴 ̃1(/)𝐴 ̃2 ≃ (2.25,2.75; 2.75,1.5; 0.5)

Conclusion

In this paper, division of generalized trapezoidal and also triangular fuzzy numbers have been analyzed. For this purpose, the existing operations on GFNs have been studied and the drawbacks have been pointed out. In order to overcome the drawbacks, new arithmetic operators on generalized trapezoidal and triangular fuzzy numbers have been proposed based on extension principle. Furthermore, the proposed operators can deal with both the non-normalized and normalized fuzzy numbers and also eventually causes less loss of information than the existing ones.

References

1. Chen SH. Operations on fuzzy numbers with function principal. Tamkang J Manage Sci. 1985;6:13-25.
2. Chen SH, Hsieh CH. Graded mean integration representation of generalized fuzzy number. J Chin Fuzzy Syst Assoc. 1999;5:1-7.
3. Hsieh CH, Chen SH. Similarity of generalized fuzzy numbers with graded mean integration representation. In: Proceedings of the 8 th International Fuzzy Systems Association World Congress; 1999; Taipei, Taiwan. p. 551-555.
4. Chen SJ, Chen SM. Fuzzy risk analysis based on similarity measures of generalized fuzzy numbers. IEEE Trans Fuzzy Syst. 2003;11:45-56.
5. Islam S, Roy TK. A new fuzzy multi-objective programming: entropy based geometric programming and its application of transportation problems. Eur J Oper Res. 2006;173:387-404.