Ranking p-norm generalised fuzzy numbers with different left height and right height using integral values

This paper considers ranking of generalised fuzzy numbers with different left height and right height using integral values. With the advances in new type of fuzzy number (generalised fuzzy number with different left height and right height) methods should be developed to compare them. Keeping this in view a new modified method has been proposed.


Introduction
Decision making in engineering, medical and any other real-life problems may be interpreted in terms of fuzzy. This demands ranking or ordering of fuzzy quantities to make a transparent decision. With the advances in fuzzy set theory, different ranking methods are developed. This concept was first proposed by Jain [7]. Some of the literatures that describe different approach of ranking fuzzy quantities are [1,3,4,10,[13][14][15]. Recently, ranking of trapezoidal fuzzy numbers based on the shadow length has been discussed by Pour et al. [12]. Also, ranking triangular fuzzy numbers by Pareto approach based on two dominance stages is discussed by Bahri et al. [2].
The literatures that are available on ranking fuzzy quantities based on the integral values are [6][7][8][9]. These type of methods of ranking fuzzy numbers are based on the convex combination of right and left integral values through an index of optimism found in Liou and Wang [11] and Kim and Park [8]. This concept was further generalised to rank non-normal p-norm trapezoidal fuzzy numbers [6]. However, this method was found insufficient to rank non-normal p-norm fuzzy numbers with different height; keeping this in mind, Kumar et al. [9] developed an approach to overcome those shortcoming's. With the advances of generalised fuzzy numbers (GFNs) with different left height and right height [6], Kumar's approach fails to rank them. Hence, Kumar's approach is only sufficient for ranking fuzzy numbers or non-normal p-norm fuzzy numbers with different height, but the method is insufficient for ranking GFNs with different left height and right height.
Keeping this in view, Kumar's approach has been modified in this paper to rank p-norm GFNs with different left height and right height. This modified method thus handle both normal and non-normal trapezoidal fuzzy number with different height. The modified method can also rank non-normal p-norm trapezoidal fuzzy numbers with different height.
The structure of the paper is as follows. In Sect. 2, some general concept of the GFN is put forwarded. Membership function of GFN is defined. Also the membership function of p-norm GFN with different left height and right height is defined. Section 3 starts with definitions of different integral values of p-norm GFN with different left height and right height, And finally, some properties related to them are discussed in this section. Section 4 describes the proposed modified method along with some numerical examples. Finally, in Sect. 4, conclusions are made.

Definitions and notations
In this section, brief review of some concepts of generalised fuzzy number with different left height and right height are put forwarded.

Generalised fuzzy number
LetÃ be represented by ða; b; c; d; h L ; h R Þ on the real line R such that À1\a b c d\1 is called a GFN with different left height and right height which is bounded and convex. The values a; b; c and d are real, h L is called the left height of the GFNÃ, h R is called the right height of the GFN, h L 2 ½0; 1 and h R 2 ½0; 1 [5]. For now, let FðRÞ be the set of all GFNs with different left height and right height. If h L ¼ h R ¼ 1 then the GFN reduces to a standard trapezoidal fuzzy number.
The membership function of GFNÃ with different left height and right height is as given below where l 1 : ½a; b À! ½0; h L , l 2 : ½b; c À! ½h L ; h R ðor ½h R ; h L Þ and l 3 : ½c; d À! ½0; h R are continuous. The functions l 1 ðxÞ and l 3 ðxÞ are strictly increasing and strictly decreasing, respectively. The function l 2 ðxÞ is strictly increasing when h L \h R and strictly decreasing when h L [ h R . Then the inverse of lÃðxÞ is where l À1 1 : ½0;h L À! ½a;b, l À1 2 : ½h L ;h R ðor½h R ; h L Þ À! ½b;c and l À1 3 : ½0; h R À! ½c; d are continuous. The function l À1 1 ðxÞ and l À1 3 ðxÞ are strictly increasing and strictly decreasing, respectively. The function l À1 2 ðxÞ is strictly increasing when h L \h R and strictly decreasing when h L [ h R .
LetÃ ¼ ða; b; c; d; h L ; h R Þ be a trapezoidal GFN with different left height and right height then the membership function is defined as to be a p-norm GFN with different left height and right height if its membership function is given by where p is a positive integer. The functions f L The functions g L :½h L ; h R ðor ½h R ; h L Þ À! ½b; c is strictly increasing (decreasing) when h L \h R ðh R \h L Þ:

Total integral value
Convex combination of right and left integral values through an index of optimism is called the total integral value [8,11]. The middle integral value is zero for normal and non-normal p-norm trapezoidal fuzzy numbers with different height. However, for a p-norm GFNs with different left height and right height this integral value has to be counted for a transparent decision. Keeping this in view, following definitions are put forwarded.
Definition 3.1 IfÃ is a fuzzy number with different left height and right height as defined by the membership function (1) and the inverse membership function given by (2) then the left integral value ofÃ is defined as Definition 3.2 IfÃ is a fuzzy number with different left height and right height as defined by the membership function (1) and the inverse membership function given by (2) then the right integral value ofÃ is defined as Definition 3.3 IfÃ is a fuzzy number with different left height and right height as defined by the membership function (1) and the inverse membership function given by (2) then the middle integral value ofÃ is defined as Definition 3.4 IfÃ is a fuzzy number with different left height and right height as defined by the membership function (1), then the total integral value with index of optimism a is defined as where CðxÞ is Euler's gamma function, defined by R 1 0 y xÀ1 e Ày dy. 2. The right membership function f R A p ðxÞ is continuous and strictly decreasing function and its right integral value is 3. The middle membership function f M A p ðxÞ is continuous and strictly increasing and strictly decreasing when h L \h R and h R \h L , respectively. The middle integral value is given by 4. The total integral value with optimism a is Proof Continuity of the left membership function f L A p ðxÞ is trivial. Also, this function is strictly increasing and its integral values are inherited from [6]. Similarly, for the right membership function f L A p ðxÞ.
Trivially, the function f M Now, the total integral value with optimism a is

Arithmetic operations
The arithmetic of p-norm GFNs with different left height and right height are reviewed from Chen [5]. LetÃ p ¼ ða; b; c; d; h L ; h R Þ p andB p ¼ ðq; r; s; t; h 0 L ; h 0 R Þ p be p-norm GFNs with different left height and right height. Then The following are the steps involved in this ranking method: Step Step 2. Find I L ðÃ p Þ, I R ðÃ p Þ, I M ðÃ p Þ and I L ðB p Þ, I R ðB p Þ, I M ðB p Þ, such that Step 3. Find I a T ðÃ p Þ and I a T ðB p Þ, which are given by I a T ðÃ p Þ ¼ ach 2 þ ð1 À aÞbh 1 þ ðh 2 À h 1 Þc þ ah 2 ðd À cÞ þ ð1 À aÞh 1 ða À bÞ f þ ðh 2 À h 1 Þðb À cÞg I a T ðB p Þ ¼ ash 2 þ ð1 À aÞrh 1 þ ðh 2 À h 1 Þs þ ah 2 ðt À sÞ þ ð1 À aÞh 1 ðq À rÞ f þ ðh 2 À h 1 Þðr À sÞg Step 4. Check I a T ðÃ p Þ [ I a T ðB p Þ or I a T ðÃ p Þ\I a T ðB p Þ or I a T ðÃ p Þ ¼ I a T ðB p Þ.  (vii) I a T ðÃÞ\I a T ðBÞ if ah 2 ðc À eÞ þ ð1 À aÞh 1 ðb À eÞ þ ðh 2 À h 1 Þ ðb þ c À 2eÞ\0 and (viii) I a T ðÃÞ ¼ I a T ðBÞ if ah 2 ðc À eÞ þ ð1 À aÞh 1 ðb À eÞ þ ðh 2 À h 1 Þ ðb þ c À 2eÞ ¼ 0: Proof From Eqs. (13), (14), (15) and (19), on appropriate substitutions of the variables the following could be easily obtained: ProofÃ andB 2 are GFN and 2-norm GFN with different left height and right height. Hence by the proposed method I L ðÃÞ, I R ðÃÞ, I M ðÃÞ, I a T ðÃÞ I L ðB 2 Þ, I R ðB 2 Þ, I M ðB 2 Þ and I a T ðB 2 Þ are obtained by using Eqs. (13), (14), (15) and (19) as: where h 1 ¼ minðh AL ; h BL Þ and h 2 ¼ minðh AR ; h BR Þ. Now, I L ðÃÞ À I L ðB 2 Þ ¼ 2Àp 4 h 1 ða À bÞ ! 0 and I R ðÃÞ À I R ðB 2 Þ ¼ 2Àp 4 h 2 ðd À cÞ 0. Thus the desired inequalities (i) and (ii) are obtained. ðb À cÞð2 À pÞ is always greater than or equal to zero, thus I M ðÃÞ À which prove the inequality (iii). For the inequalities (iv), (v) and (iv), we have I a M ðÃÞ À I a M ðB 2 Þ ¼ 2 À p 4 fah 2 ðd À cÞ þ ð1 À aÞh 1 ða À bÞ þ ðh 2 À h 1 Þðb À cÞg: Hence the inequalities (iv), (v) and (vi) follow immediately. h       The results of the sets A, B, C, D and E are depicted in the Table 1. Sets A and B consist of normal fuzzy numbers, hence the ranking order by the three methods is same. Sets C and D consist of non-normal fuzzy numbers, Kim and Park [8] give no option for ranking such type of fuzzy number. The method of Kumar et al. [9] and the proposed method's ranking order are same for the sets C and D. However, sets E and F which consist of p-norm generalised fuzzy numbers with different left height and right height can be ranked only by the proposed method.

Validation of the proposed modified ranking method
For the validation of the proposed ranking method, the following reasonable axioms that Wang and Kerre [13] have proposed for fuzzy numbers' ranking are considered. Let RM be an ordering method, S the set of fuzzy numbers for which the method RM can be applied, and A and A 0 finite subsets of S. The statements of two elementsÃ p andB p in A satisfy that A p has a higher ranking thanB p when RM is applied to the 1658 -fuzzy numbers in A will be written asÃ p 1B p by RM on A. A p $B p by RM on A, andÃ p #B p by RM on A are similarly interpreted. The following axioms show the reasonable properties of the ordering approach RM.
A 1 ForÃ p 2 A,Ã p "Ã p by RM on A. A 2 For ðÃ p ;B p Þ 2 A 2 ,Ã p "B p andB p "Ã p by RM on A, we should haveÃ p $B p by RM on A. A 3 For ðÃ p ;B p ;C p Þ 2 A 3 ,Ã p "B p andB p "C p by RM on A, we should haveÃ p "C p by RM on A. Therefore,Ã p þC p #B p þC p . Similarly A 0 6 also holds. h

Conclusions
In this paper, ranking of p-norm GFNs with different left height and right height is proposed. The proposed method is generalization of Kumar's approach. Kumar's approach can only deal with non-normal p-norm trapezoidal fuzzy numbers. The proposed method can handle non-normal pnorm trapezoidal fuzzy numbers as well as p-norm GFNs with different left height and right height.