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

Abstract

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, 13–15]. 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 [6–9]. 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

Let $\tilde{A}$ be represented by $(a, b, c, d; h_{\rm L},h_{\rm R})$ on the real line $\mathbb {R}$ such that $-\infty < a \le b \le c \le d < \infty$ 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, $h_{\rm L}$ is called the left height of the GFN $\tilde{A}$, $h_{\rm R}$ is called the right height of the GFN, $h_{\rm L}\in [0,1]$ and $h_{\rm R}\in [0,1]$ [5]. For now, let $\mathbb {F(R)}$ be the set of all GFNs with different left height and right height. If $h_{\rm L}=h_{\rm R}=1$ then the GFN reduces to a standard trapezoidal fuzzy number.

The membership function of GFN $\tilde{A}$ with different left height and right height is as given below

$$\begin{aligned} \mu _{\tilde{A}}(x)&= {\left\{ \begin{array}{ll} \mu _{1}(x), & {\text{if}}\,a \le x \le b; \\ \mu _{2}(x), & {\text{if}}\,b \le x \le c; \\ \mu _{3}(x), & {\text{if}}\,c \le x \le d; \\ 0, & {\text{otherwise}} \end{array}\right. }&\end{aligned}$$

(1)

where $\mu _{1}:[a,b]\longrightarrow [0,h_{\rm L}]$, $\mu _{2}:[b,c]\longrightarrow [h_{\rm L},h_{\rm R}] \, ({\rm or} \, [h_{\rm R},h_{\rm L}])$ and $\mu _{3}:[c,d]\longrightarrow [0,h_{\rm R}]$ are continuous. The functions $\mu _{1}(x)$ and $\mu _{3}(x)$ are strictly increasing and strictly decreasing, respectively. The function $\mu _{2}(x)$ is strictly increasing when $h_{\rm L}<h_{\rm R}$ and strictly decreasing when $h_{\rm L}>h_{\rm R}$. Then the inverse of $\mu _{\tilde{A}}(x)$ is

$$\begin{aligned} \mu _{\tilde{A}}^{-1}(y)= {\left\{ \begin{array}{ll} \mu _{1}^{-1}(y), & {\rm if }\,0 \le y \le h_{\rm L}; \\ \mu _{2}^{-1}(y), & {\rm if}\, h_{\rm L} \le y \le h_{\rm R}, {\rm or}\,(h_{\rm R} \le y \le h_{\rm L}); \\ \mu _{3}^{-1}(y), & {\text if}\,0 \le y \le h_{\rm R}; \\ 0, & {\rm otherwise} \end{array}\right. } \end{aligned}$$

(2)

where $\mu _{1}^{-1}:[0,h_{\rm L}]\longrightarrow [a,b]$, $\mu _{2}^{-1}:[h_{\rm L},h_{\rm R}] \, ({\rm or} \, [h_{\rm R},h_{\rm L}])\longrightarrow [b,c]$ and $\mu _{3}^{-1}: [0,h_{\rm R}]\longrightarrow [c,d]$ are continuous. The function $\mu _{1}^{-1}(x)$ and $\mu _{3}^{-1}(x)$ are strictly increasing and strictly decreasing, respectively. The function $\mu _{2}^{-1}(x)$ is strictly increasing when $h_{\rm L}<h_{\rm R}$ and strictly decreasing when $h_{\rm L}>h_{\rm R}$.

Let $\tilde{A}=(a, b, c, d; h_{\rm L},h_{\rm R})$ be a trapezoidal GFN with different left height and right height then the membership function is defined as

$$\begin{aligned} \mu _{\tilde{A}}(x)= {\left\{ \begin{array}{ll} \frac{h_{\rm L}(x-a)}{b-a}, & {\rm if}\,a \le x \le b; \\ \frac{h_{\rm L}(c-b)+(h_{\rm R}-h_{\rm L})(x-b)}{c-b}, & {\rm if}\,b \le x \le c; \\ \frac{h_{\rm R}(x-d)}{c-d}, & {\rm if}\,c \le x \le d; \\ 0, & {\rm otherwise.} \end{array}\right. } \end{aligned}$$

(3)

Definition 2.1.1

A GFN $\tilde{A}_{p}=(a,b,c,d;h_{\rm L}, h_{\rm R})_{p}$ is said to be a $p$-norm GFN with different left height and right height if its membership function is given by

$$\begin{aligned} \mu _{\tilde{A}_{p}}(x) = {\left\{ \begin{array}{ll} f_{\tilde{A}_{p}}^{\rm L}(x)=h_{\rm L}\left[ 1-\left( \frac{x-b}{a-b}\right) ^{p} \right] ^{\frac{1}{p}}, & {\rm if}\,a \le x \le b; \\ f_{\tilde{A}_{p}}^{\rm M}(x)=h_{\rm L}+(h_{\rm R}-h_{\rm L})\left[ 1-\left( \frac{x-c}{b-c}\right) ^{p} \right] ^{\frac{1}{p}}, & {\rm if}\,b \le x \le c; \\ f_{\tilde{A}_{p}}^{\rm R}(x)=h_{\rm R}\left[ 1-\left( \frac{x-c}{d-c}\right) ^{p} \right] ^{\frac{1}{p}}, & {\rm if}\, c \le x \le d; \\ 0, & {\rm otherwise} \end{array}\right. } \end{aligned}$$

(4)

where $p$ is a positive integer. The functions $f_{\tilde{A}_{p}}^{\rm L}:[a,b]\longrightarrow [0,h_{\rm L}]$ and $f_{\tilde{A}_{p}}^{\rm R}:[c,d]\longrightarrow [0,h_{\rm R}]$ are both continuous as well as strictly increasing and strictly decreasing functions, respectively. The function $f_{\tilde{A}_{p}}^{\rm M}:[b,c]\longrightarrow [h_{\rm L},h_{\rm R}]({\rm or} \, [h_{\rm R},h_{\rm L}])$ is strictly increasing (decreasing) when $h_{\rm L}<h_{\rm R} (h_{\rm R}<h_{\rm L}$). When $p$ is one, the $p$-norm GFN with different left height and right height reduces to trapezoidal GFN with different left height and right height as defined by Eq. (3).

Definition 2.1.2

The inverse function of $\mu _{\tilde{A}_{p}}(x)$ as given by membership function in (4) is given by

$$\begin{aligned} \mu _{\tilde{A}_{p}}^{-1}(y) = {\left\{ \begin{array}{ll} g_{\tilde{A}_{p}}^{\rm L}(y)=b+(a-b)\left[ 1-\left( \frac{y}{h_{\rm L}}\right) ^{p} \right] ^{\frac{1}{p}}, & {\rm if}\,0 \le y \le h_{\rm L}; \\ g_{\tilde{A}_{p}}^{\rm M}(y)=c+(b-c)\left[ 1-\left( \frac{y-h_{\rm L}}{h_{\rm R}-h_{\rm L}}\right) ^{p} \right] ^{\frac{1}{p}}, & {\rm if}\,h_{\rm L} \le y \le h_{\rm R}, {\rm or}\,(h_{\rm R} \le y \le h_{\rm L}); \\ g_{\tilde{A}_{p}}^{\rm R}(y)=c+(d-c)\left[ 1-\left( \frac{y}{h_{\rm R}}\right) ^{p} \right] ^{\frac{1}{p}}, & {\rm if}\,0 \le y \le h_{\rm R}; \end{array}\right. } \end{aligned}$$

(5)

The functions $g_{\tilde{A}_{p}}^{\rm L}:[0,h_{\rm L}]\longrightarrow [a,b]$ and $g_{\tilde{A}_{p}}^{\rm R}:[0,h_{\rm R}]\longrightarrow [c,d]$ are both continuous as well as strictly increasing and strictly decreasing functions, respectively. The function $g_{\tilde{A}_{p}}^{\rm M}{:}[h_{\rm L},h_{\rm R}]({\rm or} \, [h_{\rm R},h_{\rm L}])\longrightarrow [b,c]$ is strictly increasing (decreasing) when $h_{\rm L}<h_{\rm R} (h_{\rm R}<h_{\rm L}).$

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

If $\tilde{A}$ 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 of $\tilde{A}$ is defined as

$$\begin{aligned} I_{\rm L}(\tilde{A})=\int _{0}^{h_{\rm L}}\mu _{1}^{-1}(y){\mathrm{d}}y \end{aligned}$$

(6)

Definition 3.2

If $\tilde{A}$ 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 of $\tilde{A}$ is defined as

$$\begin{aligned} I_{\rm R}(\tilde{A})=\int _{0}^{h_{\rm R}}\mu _{3}^{-1}(y){\mathrm{d}}y \end{aligned}$$

(7)

Definition 3.3

If $\tilde{A}$ 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 of $\tilde{A}$ is defined as

$$\begin{aligned} I_{\rm M}(\tilde{A})&=\int _{h_{\rm L}}^{h_{\rm R}}\mu _{2}^{-1}(y){\mathrm{d}}y \quad {\rm or} \quad I_{\rm M}(\tilde{A})= \int _{h_{\rm R}}^{h_{\rm L}}\mu _{2}^{-1}(y){\mathrm{d}}y \end{aligned}$$

(8)

Definition 3.4

If $\tilde{A}$ 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 $\alpha$ is defined as

$$\begin{aligned} I_{\rm T}^{\alpha }(\tilde{A})&=\alpha (I_{\rm R}(\tilde{A})+I_{\rm M}(\tilde{A}))+(1-\alpha )(I_{\rm L}(\tilde{A})+I_{\rm M}(\tilde{A}))&\end{aligned}$$

Proposition 3.1

Let $\tilde{A}_{p}=(a,b,c,d;h_{\rm L}, h_{\rm R})_{p}$ be a $p$ -norm GFN with different left height and right height with membership function (4), where $p$ is a positive integer. Then:

• 1. The left membership function $f_{\tilde{A}_{p}}^{\rm L}(x)$ is continuous and strictly increasing function and its left integral value is

  $$\begin{aligned} I_{\rm L}(\tilde{A}_{p})=bh_{\rm L}+\frac{a-b}{p}h_{\rm L}\frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

  (9)

  where $\Gamma (x)$ is Euler’s gamma function, defined by $\int _{0}^{\infty }y^{x-1}{\mathrm{e}}^{-y}{\mathrm{d}}y$.

• 2. The right membership function $f_{\tilde{A}_{p}}^{\rm R}(x)$ is continuous and strictly decreasing function and its right integral value is

  $$\begin{aligned} I_{\rm R}(\tilde{A}_{p})=ch_{\rm R}+\frac{d-c}{p}h_{\rm R}\frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

  (10)

• 3. The middle membership function $f_{\tilde{A}_{p}}^{\rm M}(x)$ is continuous and strictly increasing and strictly decreasing when $h_{\rm L}<h_{\rm R}$ and $h_{\rm R}<h_{\rm L}$ , respectively. The middle integral value is given by

  $$\begin{aligned} I_{\rm M}(\tilde{A}_{p})=(h_{\rm R}-h_{\rm L}) c+\frac{b-c}{p}(h_{\rm R}-h_{\rm L}) \frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

  (11)

• 4. The total integral value with optimism $\alpha$ is

  $$\begin{aligned} \nonumber I_{\rm T}^{\alpha }(\tilde{A}_{p})&=\alpha c h_{\rm R}+(1-\alpha )bh_{\rm L}+(h_{\rm R}-h_{\rm L})c\\&\quad +\left\{ \alpha h_{\rm R}(d-c)+(1-\alpha )h_{\rm L}(a-b)+(h_{\rm R}-h_{\rm L})(b-c)\right\} \frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{p\times \Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

  (12)

Proof

Continuity of the left membership function $f_{\tilde{A}_{p}}^{\rm L}(x)$ is trivial. Also, this function is strictly increasing and its integral values are inherited from [6]. Similarly, for the right membership function $f_{\tilde{A}_{p}}^{\rm L}(x)$.

Trivially, the function $f_{\tilde{A}_{p}}^{\rm M}(x)$ is continuous. Now,

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}x}f_{\tilde{A}_{p}}^{\rm M}(x) &=\frac{{\text {d}}}{{\text {d}}x}\left\{ h_{\rm L}+(h_{\rm R}-h_{\rm L})\left[ 1-\left( \frac{x-c}{b-c}\right) ^{p} \right] ^{\frac{1}{p}} \right\} \\ &=(h_{\rm R}-h_{\rm L})\frac{1}{p}\left[ 1-\left( \frac{x-c}{b-c}\right) ^{p}\right] ^{\frac{1}{p}-1}\left[ -p\left( \frac{x-c}{b-c}\right) ^{p-1}\left( \frac{1}{b-c}\right) \right] \end{aligned}$$

Since $0\le \left( \frac{x-c}{b-c}\right) ^{p}\le 1$, it trivially follows that $\frac{{\text {d}}}{{\text {d}}x}f_{\tilde{A}_{p}}^{\rm M}(x)\ge 0$ if $h_{\rm R}-h_{\rm L}\ge 0$ and $\frac{{\text {d}}}{{\text {d}}x}f_{\tilde{A}_{p}}^{\rm M}(x)\le 0$ if $h_{\rm R}-h_{\rm L}\le 0$. Hence the function $f_{\tilde{A}_{p}}^{\rm M}(x)$ is strictly increasing when $h_{\rm R}\ge h_{\rm L}$ and strictly decreasing when $h_{\rm R}\le h_{\rm L}$. Also,

$$\begin{aligned} I_{\rm M}(\tilde{A}_{p})&=\int _{h_{\rm L}({\rm or}\, h_{\rm R})}^{h_{\rm R}({\rm or} \, h_{\rm L})}\left[ c+(b-c)\left[ 1-\left( \frac{y-h_{\rm L}}{h_{\rm R}-h_{\rm L}}\right) ^{p} \right] ^{\frac{1}{p}}\right] {\mathrm{d}}x\\&=(h_{\rm R}-h_{\rm L}) c+\frac{b-c}{p}(h_{\rm R}-h_{\rm L}) \frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) }. \end{aligned}$$

Now, the total integral value with optimism $\alpha$ is

$$\begin{aligned} I_{\rm T}^{\alpha }(\tilde{A}_{p})&= \alpha \{I_{\rm R}(\tilde{A}_{p})+I_{\rm M}(\tilde{A}_{p})\}+(1-\alpha )\{I_{\rm L}(\tilde{A}_{p})+I_{\rm M}(\tilde{A}_{p})\}\\&= \alpha c h_{\rm R}+(1-\alpha )bh_{\rm L}+(h_{\rm R}-h_{\rm L})c\\&\quad + \left\{ \alpha h_{\rm R}(d-c)+(1-\alpha )h_{\rm L}(a-b)+(h_{\rm R}-h_{\rm L})(b-c)\right\} \frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{p\times \Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

$\square$

Remark 3.1

For a pessimistic decision maker $\alpha =0$, for an optimistic decision maker $\alpha =1$ and for a moderate decision maker $\alpha =0.5$.

Corollary 3.1

For an pessimistic decision maker $(\alpha =0)$ the total integral $I_{\rm T}^{\alpha }(\tilde{A})=I_{\rm L}(\tilde{A})+I_{\rm M}(\tilde{A})$ , for optimistic decision maker $(\alpha =1)$ the total integral value $I_{\rm T}^{\alpha }(\tilde{A})=I_{\rm R}(\tilde{A})+I_{\rm M}(\tilde{A})$ and for moderate decision maker $(\alpha =0.5)$ the total integral value $I_{\rm T}^{\alpha }(\tilde{A})=I_{\rm M}(\tilde{A})+\frac{1}{2}\{I_{\rm R}(\tilde{A})+I_{\rm L}(\tilde{A})\} .$

Note 1

The above total integral values for pessimistic decision maker, optimistic decision maker and moderate decision maker reduce to $I_{\rm T}^{\alpha }(\tilde{A})=I_{\rm L}(\tilde{A})$, $I_{\rm T}^{\alpha }(\tilde{A})=I_{\rm R}(\tilde{A})$ and $I_{\rm T}^{\alpha }(\tilde{A})= \frac{1}{2}\{I_{\rm R}(\tilde{A})+I_{\rm L}(\tilde{A})\}$, respectively, for a non-normal $p$-norm trapezoidal fuzzy number. This is because $I_{\rm M}(\tilde{A})=0$ when $h_{\rm L}=h_{\rm R}$.

Arithmetic operations

The arithmetic of $p$-norm GFNs with different left height and right height are reviewed from Chen [5]. Let $\tilde{A}_{p}=(a,b,c,d;h_{\rm L},h_{\rm R})_{p}$ and $\tilde{B}_{p}=(q,r,s,t;h'_{\rm L},h'_{\rm R})_{p}$ be $p$-norm GFNs with different left height and right height. Then

• 1. $\tilde{A}_{p}\oplus \tilde{B}_{p}=(a+q,b+r,c+s,d+t; {\rm min}\,(h_{\rm L},h'_{L}), {\rm min}\,(h_{\rm R},h'_{R}))_{p}$

• 2. $\tilde{A}_{p}\ominus \tilde{B}_{p}=(a-t,b-s,c-r,d-q; {\rm min}\,(h_{\rm L},h'_{L}), {\rm min}\,(h_{\rm R},h'_{R}))_{p}$

• 3.
  $$\begin{aligned} \lambda \tilde{A}&= {\left\{ \begin{array}{ll} (\lambda a, \lambda b, \lambda c, \lambda d; h_{\rm L}, h_{\rm R})_{p} \quad \lambda > 0 \\ (\lambda d, \lambda c, \lambda b, \lambda a; h_{\rm L}, h_{\rm R})_{p} \quad \lambda < 0 \\ \end{array}\right. }&\end{aligned}$$

The proposed method

In this section, a method of ranking $p$-norm GFNs with different left height and right height is presented. The method calculates total integral value on the basis of left integral value, right integral value and middle integral value. Ranking is done on the basis of these evaluated total integral values. Let $\tilde{A}_{p}=(a,b,c,d;h_{\rm L},h_{\rm R})_{p}$ and $\tilde{B}_{p}=(q,r,s,t;h'_{\rm L},h'_{\rm R})_{p}$ be $p$-norm GFNs with different left height and right height. Then

• 1. $\tilde{A}_{p}\succ \tilde{B}_{p}$ if $I_{\rm T}^{\alpha }(\tilde{A}_{p})>I_{\rm T}^{\alpha }(\tilde{B}_{p})$,

• 2. $\tilde{A}_{p}\prec \tilde{B}_{p}$ if $I_{\rm T}^{\alpha }(\tilde{A}_{p})<I_{\rm T}^{\alpha }(\tilde{B}_{p})$,

• 3. $\tilde{A}_{p}\sim \tilde{B}_{p}$ if $I_{\rm T}^{\alpha }(\tilde{A}_{p})=I_{\rm T}^{\alpha }(\tilde{B}_{p})$,

The following are the steps involved in this ranking method:

Step 1.:

Find $h_{1}={\rm min}\,(h_{\rm L},h'_{L})$ and $h_{2}={\rm min}\,(h_{\rm R},h'_{R})$.

Step 2.:

Find $I_{\rm L}(\tilde{A}_{p})$, $I_{\rm R}(\tilde{A}_{p})$, $I_{\rm M}(\tilde{A}_{p})$ and $I_{\rm L}(\tilde{B}_{p})$, $I_{\rm R}(\tilde{B}_{p})$, $I_{\rm M}(\tilde{B}_{p})$, such that

$$\begin{aligned} \nonumber I_{\rm L}(\tilde{A}_{p})&=\int _{0}^{h_{1}}g_{\tilde{A}_{p}}^{\rm L}(x){\mathrm{d}}x, \quad {\rm where}\quad g_{\tilde{A}_{p}}^{\rm L}(x)=b+(a-b)\left[ 1-\left( \frac{x}{h_{1}}\right) ^{p} \right] ^{\frac{1}{p}},\\&=bh_{1}+\frac{a-b}{p}h_{1}\frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

(13)

$$\begin{aligned} \nonumber I_{\rm R}(\tilde{A}_{p})&=\int _{0}^{h_{2}}g_{\tilde{A}_{p}}^{\rm R}(x){\mathrm{d}}x, \quad {\rm where}\quad g_{\tilde{A}_{p}}^{\rm R}(x)=c+(d-c)\left[ 1-\left( \frac{x}{h_{2}}\right) ^{p} \right] ^{\frac{1}{p}},\\&=ch_{2}+\frac{d-c}{p}h_{2}\frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

(14)

$$\begin{aligned} \nonumber I_{\rm M}(\tilde{A}_{p})&=\int _{h_{1}({\rm or} \, h_{2})}^{h_{2}({\rm or} \, h_{1})}g_{\tilde{A}_{p}}^{\rm M}(x){\mathrm{d}}x, \quad {\rm where}\quad g_{\tilde{A}_{p}}^{\rm M}(x)=c+(b-c)\left[ 1-\left( \frac{x-h_{1}}{h_{2}-h_{1}}\right) ^{p} \right] ^{\frac{1}{p}},\\&=c(h_{2}-h_{1})+\frac{b-c}{p}(h_{2}-h_{1})\frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

(15)

$$\begin{aligned} \nonumber I_{\rm L}(\tilde{B}_{p})&=\int _{0}^{h_{1}}g_{\tilde{B}_{p}}^{\rm L}(x){\mathrm{d}}x, \quad {\rm where}\quad g_{\tilde{B}_{p}}^{\rm L}(x)=r+(q-r)\left[ 1-\left( \frac{x}{h_{1}}\right) ^{p} \right] ^{\frac{1}{p}},\\&=rh_{1}+\frac{q-r}{p}h_{1}\frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

(16)

$$\begin{aligned} \nonumber I_{\rm R}(\tilde{B}_{p})&=\int _{0}^{h_{2}}g_{\tilde{B}_{p}}^{\rm R}(x){\mathrm{d}}x, \quad{\rm where}\quad g_{\tilde{B}_{p}}^{\rm R}(x)=s+(t-s)\left[ 1-\left( \frac{x}{h_{2}}\right) ^{p} \right] ^{\frac{1}{p}},\\&=sh_{2}+\frac{t-s}{p}h_{2}\frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

(17)

$$\begin{aligned} I_{\rm M}(\tilde{B}_{p})&=\int _{h_{1}({\rm or} \, h_{2}}^{h_{2}({\rm or} \, h_{1})}g_{\tilde{B}_{p}}^{\rm M}(x){\mathrm{d}}x, \quad {\rm where}\quad g_{\tilde{B}_{p}}^{\rm M}(x)=s+(r-s)\left[ 1-\left( \frac{x-h_{1}}{h_{2}-h_{1}}\right) ^{p} \right] ^{\frac{1}{p}},\nonumber \\&=s(h_{2}-h_{1})+\frac{r-s}{p}(h_{2}-h_{1})\frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{\Gamma \left( \frac{2}{p}+1\right) } \end{aligned}$$

(18)

Step 3.:

Find $I_{\rm T}^{\alpha }(\tilde{A}_{p})$ and $I_{\rm T}^{\alpha }(\tilde{B}_{p})$, which are given by

$$\begin{aligned} \nonumber I_{\rm T}^{\alpha }(\tilde{A}_{p})&=\alpha c h_{2}+(1-\alpha )bh_{1}+(h_{2}-h_{1})c\\&\quad + \left\{ \alpha h_{2}(d-c)+(1-\alpha )h_{1}(a-b)+(h_{2}-h_{1})(b-c)\right\} \frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{p\times \Gamma \left( \frac{2}{p}+1\right) }, \end{aligned}$$

(19)

$$\begin{aligned} \nonumber I_{\rm T}^{\alpha }(\tilde{B}_{p})&=\alpha s h_{2}+(1-\alpha )rh_{1}+(h_{2}-h_{1})s\\&\quad + \left\{ \alpha h_{2}(t-s)+(1-\alpha )h_{1}(q-r)+(h_{2}-h_{1})(r-s)\right\} \frac{\Gamma \left( \frac{1}{p}+1\right) \Gamma \left( \frac{1}{p}\right) }{p\times \Gamma \left( \frac{2}{p}+1\right) }. \end{aligned}$$

(20)

Step 4.:

Check $I_{\rm T}^{\alpha }(\tilde{A}_{p})>I_{\rm T}^{\alpha }(\tilde{B}_{p})$ or $I_{\rm T}^{\alpha }(\tilde{A}_{p})<I_{\rm T}^{\alpha }(\tilde{B}_{p})$ or $I_{\rm T}^{\alpha }(\tilde{A}_{p})=I_{\rm T}^{\alpha }(\tilde{B}_{p})$.

Case (i):

If $I_{\rm T}^{\alpha }(\tilde{A}_{p})>I_{\rm T}^{\alpha }(\tilde{B}_{p})$ then $\tilde{A}_{p}\succ \tilde{B}_{p}$.

Case (ii):

If $I_{\rm T}^{\alpha }(\tilde{A}_{p})<I_{\rm T}^{\alpha }(\tilde{B}_{p})$ then $\tilde{A}_{p}\prec \tilde{B}_{p}$.

Case (iii):

If $I_{\rm T}^{\alpha }(\tilde{A}_{p})=I_{\rm T}^{\alpha }(\tilde{B}_{p})$ then $\tilde{A}_{p}\sim \tilde{B}_{p}$.

Remark 4.1

For any two arbitrary generalised fuzzy numbers with different left height and right height, $\tilde{A}_{p}$ and $\tilde{B}_{p}$, we have

$$\begin{aligned} I_{\rm T}^{\alpha }(\tilde{A}_{p}+\tilde{B}_{p})=I_{\rm T}^{\alpha }(\tilde{A}_{p})+I_{\rm T}^{\alpha }(\tilde{B}_{p}) \end{aligned}$$

Proposition 4.1

Let $\tilde{A}=(a,b,c,d;h_{AL},h_{AR})$ and $\tilde{B}=(a,e, d;h_{B})$ be GFN with different left height and right height and non-normal triangular fuzzy number, respectively, such that $-\infty <a\le b \le e \le c \le d < \infty$ . Then

• (i) $I_{\rm L}(\tilde{A})\le I_{\rm L}(\tilde{B}),$

• (ii) $I_{\rm R}(\tilde{A})\ge I_{\rm R}(\tilde{B}),$

• (iii) $I_{\rm M}(\tilde{A}) = I_{\rm M}(\tilde{B})$ , if $b+c=2e,$

• (iv) $I_{\rm M}(\tilde{A}) > I_{\rm M}(\tilde{B})$ , if $h_{1}<h_{2}$ and $b+c>2e$ or $h_{1}>h_{2}$ and $b+c<2e,$

• (v) $I_{\rm M}(\tilde{A}) < I_{\rm M}(\tilde{B})$ , if $h_{1}<h_{2}$ and $b+c<2e$ or $h_{1}>h_{2}$ and $b+c>2e,$

• (vi) $I_{\rm T}^{\alpha }(\tilde{A})>I_{\rm T}^{\alpha }(\tilde{B})$ if $\alpha h_{2}(c-e)+(1-\alpha )h_{1}(b-e)+(h_{2}-h_{1})(b+c-2e)>0,$

• (vii) $I_{\rm T}^{\alpha }(\tilde{A})<I_{\rm T}^{\alpha }(\tilde{B})$ if $\alpha h_{2}(c-e)+(1-\alpha )h_{1}(b-e)+(h_{2}-h_{1})(b+c-2e)<0$ and

• (viii) $I_{\rm T}^{\alpha }(\tilde{A})=I_{\rm T}^{\alpha }(\tilde{B})$ if $\alpha h_{2}(c-e)+(1-\alpha )h_{1}(b-e)+(h_{2}-h_{1})(b+c-2e)=0.$

Proof

From Eqs. (13), (14), (15) and (19), on appropriate substitutions of the variables the following could be easily obtained:

$$\begin{aligned}&I_{\rm L}(\tilde{A})=\frac{h_{1}(a+b)}{2}, \quad I_{\rm R}(\tilde{A})=\frac{h_{2}(d+c)}{2}, \quad I_{\rm M}(\tilde{A})=\frac{(h_{2}-h_{1})(b+c)}{2}, \\ & I_{\rm L}(\tilde{B})=\frac{h_{1}(a+e)}{2}, \quad I_{\rm R}(\tilde{B})=\frac{h_{2}(e+d)}{2}, \quad I_{\rm M}(\tilde{B})=e(h_{2}-h_{1}), \\ &I_{\rm T}^{\alpha }(\tilde{A})=\frac{1}{2}\{\alpha h_{2}(c+d)+(1-\alpha )h_{1}(a+b)+(h_{2}-h_{1})(c+b) \} \quad {\rm and}\\&I_{\rm T}^{\alpha }(\tilde{B})=\frac{1}{2}\{\alpha h_{2}(e+d)+(1-\alpha )h_{1}(a+e)+2e(h_{2}-h_{1}) \}\end{aligned}$$

where $h_{1}={\mathrm{min}}(h_{AL}, h_{B})$ and $h_{2}={\mathrm{min}}(h_{AR}, h_{B})$.

Now $I_{\rm L}(\tilde{A}) - I_{\rm L}(\tilde{B})=\frac{h_{1}}{2}(b-e) \le 0$ as $b\le e \le c$, hence inequality (i) is deduced. Similarly, inequality (ii) could be deduced. Again, we have $I_{\rm M}(\tilde{A})-I_{\rm M}(\tilde{B})=\frac{h_{2}-h_{1}}{2}(b+c-2e)$ hence the inequalities (iii), (iv) and (v) follow immediately. Also we have

$$\begin{aligned} I_{\rm T}^{\alpha }(\tilde{A})-I_{\rm T}^{\alpha }(\tilde{B})&=\frac{1}{2}\{ \alpha h_{2}(c-e)+(1-\alpha )h_{1}(b-e)+(h_{2}-h_{1})(c+b-2e)\}.&\end{aligned}$$

Hence the inequalities (vi), (vii) and (viii) can be deduced easily. $\square$

Corollary 4.1

[9] Let $\tilde{A}=(a,b,c,d;h_{A})$ and $\tilde{B}=(a,e, d;h_{B})$ be non-normal trapezoidal and triangular fuzzy numbers, respectively, where $-\infty <a\le b \le e \le c \le d < \infty$ . Then

• (i) $I_{\rm L}(\tilde{A})\le I_{\rm L}(\tilde{B})$,

• (ii) $I_{\rm R}(\tilde{A})\ge I_{\rm R}(\tilde{B})$,

• (iii) $I_{\rm T}^{\alpha }(\tilde{A})>I_{\rm T}^{\alpha }(\tilde{B})$ if $e<c\alpha +(1-\alpha )b$,

• (iv) $I_{\rm T}^{\alpha }(\tilde{A})=I_{\rm T}^{\alpha }(\tilde{B})$ if $e=c\alpha +(1-\alpha )b$ and

• (v) $I_{\rm T}^{\alpha }(\tilde{A})<I_{\rm T}^{\alpha }(\tilde{B})$ if $e>c\alpha +(1-\alpha )b .$

These inequalities are particular case of the inequalities in the Proposition 4.1. These can be obtained by appropriate substitutions on the inequalities in the Proposition 4.1.

Proposition 4.2

Let $\tilde{A}=(a,b,c,d;h_{AL},h_{AR})$ and $\tilde{B}_{2}=(a,b,c,d;h_{BL},h_{BR})_{2}$ be GFN and $2$ -norm GFN with different left height and right height, respectively. Then

• (i) $I_{\rm L}(\tilde{A})\ge I_{\rm L}(\tilde{B}_{2}),$

• (ii) $I_{\rm L}(\tilde{A})\le I_{\rm L}(\tilde{B}_{2}),$

• (iii) $I_{\rm M}(\tilde{A})\ge (\le ) I_{\rm M}(\tilde{B}_{2})$ if $h_{1}< (>)h_{2},$

• (iv) $I_{\rm M}^{\alpha }(\tilde{A}) > I_{\rm M}^{\alpha }(\tilde{B}_{2})$ if $\alpha h_{2}(d-c)+h_{1}(1-\alpha )(a-b)+(h_{2}-h_{1})(b-c)<0 ,$

• (v) $I_{\rm M}^{\alpha }(\tilde{A})< I_{\rm M}^{\alpha }(\tilde{B}_{2})$ if $\alpha h_{2}(d-c)+h_{1}(1-\alpha )(a-b)+(h_{2}-h_{1})(b-c)>0$ and

• (vi) $I_{\rm M}^{\alpha }(\tilde{A}) = I_{\rm M}^{\alpha }(\tilde{B}_{2})$ if $\alpha h_{2}(d-c)+h_{1}(1-\alpha )(a-b)+(h_{2}-h_{1})(b-c)=0 .$

Proof

$\tilde{A}$ and $\tilde{B}_{2}$ are GFN and $2$-norm GFN with different left height and right height. Hence by the proposed method $I_{\rm L}(\tilde{A})$, $I_{\rm R}(\tilde{A})$, $I_{\rm M}(\tilde{A})$, $I_{\rm T}^{\alpha }(\tilde{A})$ $I_{\rm L}(\tilde{B}_{2})$, $I_{\rm R}(\tilde{B}_{2})$, $I_{\rm M}(\tilde{B}_{2})$ and $I_{\rm T}^{\alpha }(\tilde{B}_{2})$ are obtained by using Eqs. (13), (14), (15) and (19) as:

$$\begin{aligned}&I_{\rm L}(\tilde{A})=\frac{h_{1}(a+b)}{2}, \quad I_{\rm R}(\tilde{A})=\frac{h_{2}(d+c)}{2}, \quad I_{\rm M}(\tilde{A})=\frac{(h_{2}-h_{1})(b+c)}{2}, \\ &I_{\rm L}(\tilde{B}_{2})=bh_{1}+\frac{a-b}{4}h_{1}\pi , \quad I_{\rm R}(\tilde{B}_{2})=ch_{2}+\frac{d-c}{4}h_{2}\pi , \quad I_{\rm M}(\tilde{B}_{2})= \frac{b+3c}{4}(h_{2}-h_{1})\pi , \\ &I_{\rm T}^{\alpha }(\tilde{A})=\frac{1}{2}\{\alpha h_{2}(c+d)+(1-\alpha )h_{1}(a+b)+(h_{2}-h_{1})(c+b) \} \quad {\rm and} \\ &I_{\rm T}^{\alpha }(\tilde{B}_{2})=\frac{1}{4}\left[ (h_{2}-h_{1})\{ 4c+\pi (b-c)\}+(1-\alpha )h_{1}\{4b+\pi (a-b) \}+\alpha h_{2}\{ 4c+\pi (d-c)\}\right] \end{aligned}$$

where $h_{1}={\mathrm{min}}(h_{AL},h_{BL})$ and $h_{2}={\mathrm{min}}(h_{AR},h_{BR})$.

Now, $I_{\rm L}(\tilde{A}) - I_{\rm L}(\tilde{B}_{2})=\frac{2-\pi }{4}h_{1}(a-b)\ge 0$ and $I_{\rm R}(\tilde{A}) - I_{\rm R}(\tilde{B}_{2})=\frac{2-\pi }{4}h_{2}(d-c)\le 0$. Thus the desired inequalities (i) and (ii) are obtained. $(b-c)(2-\pi )$ is always greater than or equal to zero, thus $I_{\rm M}(\tilde{A}) - I_{\rm M}(\tilde{B}_{2})= (h_{2}-h_{1})\frac{(b-c)(2-\pi )}{4}\ge (\le ) 0$ if $h_{1}< (>)h_{2}$, which prove the inequality (iii). For the inequalities (iv), (v) and (iv), we have

$$\begin{aligned} I_{\rm M}^{\alpha }(\tilde{A}) - I_{\rm M}^{\alpha }(\tilde{B}_{2})&=\frac{2-\pi }{4}\{\alpha h_{2}(d-c)+(1-\alpha )h_{1}(a-b)+(h_{2}-h_{1})(b-c) \}.&\end{aligned}$$

Hence the inequalities (iv), (v) and (vi) follow immediately. $\square$

Corollary 4.2

[9] Let $\tilde{A}=(a,b,c,d;h_{A})$ and $\tilde{B}_{2}=(a,b,c,d;h_{B})$ be non-normal trapezoidal fuzzy number and non-normal $2$ -norm trapezoidal fuzzy number, respectively, then

• (i) $I_{\rm L}(\tilde{A})\ge I_{\rm L}(\tilde{B}_{2})$,

• (ii) $I_{\rm L}(\tilde{A})\le I_{\rm L}(\tilde{B}_{2})$,

• (iii) $I_{\rm T}^{\alpha }<I_{\rm T}^{\alpha }(\tilde{B})$ if $\alpha (d-c)+(1-\alpha )(a-b)>0$,

• (iv) $I_{\rm T}^{\alpha }=I_{\rm T}^{\alpha }(\tilde{B})$ if $\alpha (d-c)+(1-\alpha )(a-b)=0$ and

• (v) $I_{\rm T}^{\alpha }>I_{\rm T}^{\alpha }(\tilde{B})$ if $\alpha (d-c)+(1-\alpha )(a-b)<0$.

These inequalities are particular case of the inequalities in the Proposition 4.2. These can be obtained by appropriate substitutions on the inequalities in the Proposition 4.2.

Proposition 4.3

Let $\tilde{A}=(a,a,a,a;1,1)$ and $\tilde{B}=(b,b,b,b;1,1)$ be GFNs with height 1. Then

• (i) $I_{\rm L}(\tilde{A})\ge (\le ) I_{\rm L}(\tilde{B})$ , if $a\ge (\le ) b$,

• (ii) $I_{\rm M}(\tilde{A})\ge (\le ) I_{\rm M}(\tilde{B}_{2})$ if $a\ge (\le )b$ and

• (iii) $I_{\rm T}^{1}(\tilde{A}) >(<) I_{\rm T}^{1}(\tilde{B})$ if $a>(<)b .$

The Proposition 4.3 validates that the proposed method can also be applied for real numbers.

Example 4.1

Let $\tilde{A}=(5,7,8,9;0.5,0.6)$ and $\tilde{B}_{2}=(5,7,8,9;0.7,0.6)_{2}$ be GFN and $2$-norm GFN with different left height and right height, which are depicted in Fig. 1. But, according to the proposed modified method $h_{1}={\mathrm{min}}(0.5,0.6)$ and $h_{2}={\mathrm{min}}(0.7, 0.6)$. Also, $I_{\rm T}^{0.5}(\tilde{A})=4.8000$ and $I_{\rm T}^{0.5}(\tilde{B}_{2})=4.7144$. Thus, $I_{\rm T}^{0.5}(\tilde{A})>I_{\rm T}^{0.5}(\tilde{B}_{2})$.

Fig. 1

Fuzzy numbers $\tilde{A}=(5,7,8,9;0.5,0.6)$ and $\tilde{B}_{2}=(5,7,8,9;0.7,0.6)_{2}$

Example 4.2

Let $\tilde{A}=(0.1659, 0.2803, 0.7463, 1.154; 0.5, 0.6)$, $\tilde{B}=(0.1611, 0.2475, 0.5696, 0.8187; 0.4, 0.5)$ and $\tilde{C}=(0.1645, 0.2445, 0.5869, 0.8894 ; 0.5, 0.6)$, are GFNs with different left height and right height. Figure 2 depicts the membership function of the above fuzzy numbers.

Fig. 2

Fuzzy numbers $\tilde{A}=(0.1659, 0.2803, 0.7463, 1.154; 0.5, 0.6)$, $\tilde{B}=(0.1611, 0.2475, 0.5696, 0.8187; 0.4, 0.5)$ and $\tilde{C}=(0.1645, 0.2445, 0.5869, 0.8894 ; 0.5, 0.6)$

Here $I_{\rm T}^{0.5}(\tilde{A})=0.3335$, $I_{\rm T}^{0.5}(\tilde{B})=0.2553$ and $I_{\rm T}^{0.5}(\tilde{C})=0.2670$, and $I_{\rm T}^{1}(\tilde{A})=0.1406$, $I_{\rm T}^{1}(\tilde{B})=0.1226$ and $I_{\rm T}^{1}(\tilde{C})=0.1234$. Thus a moderate decision and a pessimistic decision maker rank them as $\tilde{A}>\tilde{C}>\tilde{B}$.

Example 4.3

Let $\tilde{A}_{2}=(-2, -1, 0, 1; 0.5, 0.5)_{2}$, $\tilde{B}_{2}=(-1.5, -0.5, 0.5, 1.5; 0.5, 0.6)_{2}$ and $\tilde{C}_{2}=(1, 1.5, 2, 2.5; 0.6, 0.5)_{2}$, are GFNs with different left height and right height. The membership functions of the fuzzy numbers are depicted in Fig. 3. Here, $I_{\rm T}^{0.5}(\tilde{A}_{2})=-0.2500$, $I_{\rm T}^{0.5}(\tilde{B}_{2})=0.000$ and $I_{\rm T}^{0.5}(\tilde{C}_{2})=0.6787$, and $I_{\rm T}^{0}(\tilde{A}_{2})=-0.8927$, $I_{\rm T}^{0}(\tilde{B}_{2})=-0.6427$ and $I_{\rm T}^{0}(\tilde{C}_{2})=0.5537$. A moderate decision maker ($\alpha =0.5$) ranks $\tilde{A}$, $\tilde{B}$ and $\tilde{C}$ as $\tilde{C}>\tilde{B}>\tilde{A}$ and also a pessimistic decision maker ranks them in the same order.

Fig. 3

Fuzzy numbers $\tilde{A}_{2}=(-2, -1, 0, 1; 0.5, 0.5)_{2}$, $\tilde{B}_{2}=(-1.5, -0.5, 0.5, 1.5; 0.5, 0.6)_{2}$ and $\tilde{C}_{2}=(1, 1.5, 2, 2.5; 0.6, 0.5)_{2}$

Example 4.4

Let $\tilde{A}=(1,1,1,1;1,1)$ and $\tilde{B}=(2,2,2,2; 1, 1)$ be GFNs which are actually real numbers. Now by Proposition 4.3 $\tilde{B} > \tilde{A}$ trivially.

Example 4.5

Consider the following set of fuzzy numbers,

Set A:

$\tilde{A}=(0.4,0.5,0.5,0.6;1,1)$, $\tilde{B}=(0.2,0.4,0.6,0.8;1,1)$.

Set B:

$\tilde{A}=(0.2,0.3,0.3,0.6;1,1)$, $\tilde{B}=(0.2,0.4,0.4,0.6;1,1)$, $\tilde{C}=(0.2,0.5,0.5,0.6;1,1)$.

Set C:

$\tilde{A}=(0.2,0.4,0.4,0.6;0.9,0.9)$, $\tilde{B}=(0.2,0.4,0.4,0.6;1,1)$, $\tilde{C}=(0.2,0.4,0.4,0.6;0.5,0.5)$.

Set D:

$\tilde{A}=(0.4,0.5,0.5,0.6;0.5,0.5)$, $\tilde{B}=(0.2,0.4,0.6,0.8;0.6,0.6)$.

Set E:

$\tilde{A}=(0.4,0.5,0.5,0.6;0.6,0.7)$, $\tilde{B}=(0.2,0.4,0.6,0.8;0.5,0.6)$.

Set F:

$\tilde{A}=(0.4,0.5,0.5,0.6;0.6,0.7)$, $\tilde{B}=(0.2,0.4,0.6,0.8;0.5,0.6)_{2}$.

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.

Table 1 Comparison of the proposed method with Kim and Park [8] and Kumar et al. [9] methods

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 $\mathcal {A}$ and $\mathcal {A'}$ finite subsets of $S$. The statements of two elements $\tilde{A}_{p}$ and $\tilde{B}_{p}$ in $\mathcal {A}$ satisfy that $\tilde{A}_{p}$ has a higher ranking than $\tilde{B}_{p}$ when $RM$ is applied to the fuzzy numbers in $\mathcal {A}$ will be written as $\tilde{A}_{p}\succ \tilde{B}_{p}$ by $RM$ on $\mathcal {A}$. $\tilde{A}_{p}\sim \tilde{B}_{p}$ by $RM$ on $\mathcal {A}$, and $\tilde{A}_{p} \succeq \tilde{B}_{p}$ by $RM$ on $\mathcal {A}$ are similarly interpreted. The following axioms show the reasonable properties of the ordering approach $RM$.

${\mathrm{A}}_{1}$ :

For $\tilde{A}_{p}\in \mathcal {A}$, $\tilde{A}_{p}\preceq \tilde{A}_{p}$ by $RM$ on $\mathcal {A}$.

${\mathrm{A}}_{2}$ :

For $(\tilde{A}_{p}, \tilde{B}_{p})\in \mathcal {A}^{2}$, $\tilde{A}_{p}\preceq \tilde{B}_{p}$ and $\tilde{B}_{p}\preceq \tilde{A}_{p}$ by $RM$ on $\mathcal {A}$, we should have $\tilde{A}_{p}\sim \tilde{B}_{p}$ by $RM$ on $\mathcal {A}$.

${\mathrm{A}}_{3}$ :

For $(\tilde{A}_{p}, \tilde{B}_{p}, \tilde{C}_{p})\in \mathcal {A}^{3}$, $\tilde{A}_{p}\preceq \tilde{B}_{p}$ and $\tilde{B}_{p}\preceq \tilde{C}_{p}$ by $RM$ on $\mathcal {A}$, we should have $\tilde{A}_{p}\preceq \tilde{C}_{p}$ by $RM$ on $\mathcal {A}$.

${\mathrm{A}}_{4}$ :

For $(\tilde{A}_{p}, \tilde{B}_{p})\in \mathcal {A}^{2}$, ${\rm inf} \, {\mathrm{supp}}(\tilde{B}_{p})>{\rm sup}\, {\mathrm{supp}}(\tilde{A}_{p})$, we should have $\tilde{A}_{p} \preceq \tilde{B}_{p}$ by $RM$ on $\mathcal {A}$.

${\mathrm{A}}_{4}'$ :

For $(\tilde{A}_{p}, \tilde{B}_{p})\in \mathcal {A}^{2}$, ${\rm inf}\, {\mathrm{supp}}(\tilde{B}_{p})>{\rm sup} \, {\mathrm{supp}}(\tilde{A}_{p})$, we should have $\tilde{A}_{p} \prec \tilde{B}_{p}$ by $RM$ on $\mathcal {A}$.

${\mathrm{A}}_{5}$ :

Let $(\tilde{A}_{p}, \tilde{B}_{p})\in (\mathcal {A}\cap \mathcal {A'})^{2}$. We obtain the ranking order $\tilde{A}_{p} \preceq \tilde{B}_{p}$ by $RM$ on $\mathcal {A'}$ if and only if $\tilde{A}_{p} \preceq \tilde{B}_{p}$ by $RM$ on $\mathcal {A}$.

${\mathrm{A}}_{6}$ :

Let $\tilde{A}_{p}$, $\tilde{B}_{p}$, $\tilde{A}_{p}+\tilde{C}_{p}$ and $\tilde{B}_{p}+\tilde{C}_{p}$ be elements of $S$. If $\tilde{A}_{p}\succeq \tilde{B}_{p}$ by $RM$ on $\{\tilde{A}_{p}, \tilde{B}_{p}\}$, then $\tilde{A}_{p}+\tilde{C}_{p}\succeq \tilde{B}_{p}+\tilde{C}_{p}$ by $RM$ on $\{\tilde{A}_{p}+\tilde{C}_{p}, \tilde{B}_{p}+\tilde{C}_{p}\}$.

${\mathrm{A}}_{6}'$ :

Let $\tilde{A}_{p}$, $\tilde{B}_{p}$, $\tilde{A}_{p}+\tilde{C}_{p}$ and $\tilde{B}_{p}+\tilde{C}_{p}$ be elements of $S$. If $\tilde{A}_{p}\succ \tilde{B}_{p}$ by $RM$ on $\{\tilde{A}_{p}, \tilde{B}_{p}\}$, then $\tilde{A}_{p}+\tilde{C}_{p}\succ \tilde{B}_{p}+\tilde{C}_{p}$ by $RM$ on $\{\tilde{A}_{p}+\tilde{C}_{p}, \tilde{B}_{p}+\tilde{C}_{p}\}$ when $\tilde{C}_{p}\ne \phi$.

Proposition 4.4

The proposed ranking method $RM$ has the properties ${\mathrm{A}}_{1}$, ${\mathrm{A}}_{2}$, ${\mathrm{A}}_{3}$, ${\mathrm{A}}_{4}$, ${\mathrm{A}}_{4}'$, ${\mathrm{A}}_{5}$, ${\mathrm{A}}_{6}$ and ${\mathrm{A}}_{6}' .$

Proof

It is easy to verify that properties ${\mathrm{A}}_{1}$–${\mathrm{A}}_{5}$ are hold. For the proof of ${\mathrm{A}}_{6}$, consider the generalised fuzzy numbers with different left height and right height as $\tilde{A}_{p}=(a,b,c,d;h_{\rm L},h_{\rm R})_{p}$, $\tilde{B}_{p}=(q,r,s,t;h'_{\rm L},h'_{\rm R})_{p}$ and $\tilde{C}_{p}=(l, m,n,o,;h''_{\rm L},h''_{\rm R})_{p}$. Let $\tilde{A}_{p}\succeq \tilde{B}_{p}$ by $RM$, hence

$$\begin{aligned}\nonumber I_{\rm T}^{\alpha }(\tilde{A}_{p})&\ge I_{\rm T}^{\alpha }(\tilde{B}_{p}),&\end{aligned}$$

by adding $I_{\rm T}^{\alpha }(\tilde{C}_{p})$

$$\begin{aligned} I_{\rm T}^{\alpha }(\tilde{A}_{p})+I_{\rm T}^{\alpha }(\tilde{C}_{p})&\ge I_{\rm T}^{\alpha }(\tilde{B}_{p})+I_{\rm T}^{\alpha }(\tilde{C}_{p}),&\end{aligned}$$

and by Remark 4.1

$$\begin{aligned} I_{\rm T}^{\alpha }(\tilde{A}_{p}+\tilde{C}_{p})&\ge I_{\rm T}^{\alpha }(\tilde{B}_{p}+\tilde{C}_{p}). \end{aligned}$$

Therefore, $\tilde{A}_{p}+\tilde{C}_{p}\succeq \tilde{B}_{p}+\tilde{C}_{p}$. Similarly ${\mathrm{A}}_{6}'$ also holds. $\square$

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 $p$-norm trapezoidal fuzzy numbers as well as $p$-norm GFNs with different left height and right height.