Generalization and ranking of fuzzy numbers by relative preference relation

Abstract

We define $2n+1$ and 2n fuzzy numbers, which generalize triangular and trapezoidal fuzzy numbers, respectively. Then, we extend the fuzzy preference relation and relative preference relation to rank $2n+1$ and 2n fuzzy numbers. When the data is representable in terms of $2n+1$ fuzzy number, we generalize the FMCDM (fuzzy multi-criteria decision making) model constructed with TOPSIS and relative preference relation. Lastly, we give an example from telecommunications to present the proposed FMCDM model and validate the results obtained.

1 Introduction

Zadeh (1965) introduced the concept of fuzzy set, and it is widely used to characterize vague or imprecise settings (conditions). Fuzzy sets have applications in automata theory, systems theory, decision theory, switching theory, pattern recognition, image thresholding, etc. [Lalotra and Singh (2020); Singh et al. (2019); Singh and Sharma (2019); Singh et al. (2020)]. Fuzzy numbers generalize real numbers and are very useful to represent data corresponding to uncertain situations. There are several methods to rank or order fuzzy numbers. Lee and Li (1988) utilized the concept of probability measure to determine the order of fuzzy numbers by considering the mean and dispersion of alternatives. Choobineh and Li (1993) proposed an indexing method to order or rank the fuzzy numbers. Dias (1993) proposed a computational approach to rank the alternatives using fuzzy numbers. Fortemps and Roubens (1996) presented a method to compare fuzzy numbers using the area compensation procedure. Cheng (1998) proposed the distance method and coefficient of variation (CV) index method to rank the fuzzy numbers. Chu and Tsao (2002) proposed a method using the area between centroid point and original point of the fuzzy numbers to facilitate ranking. Wang and Lee (2008) later revised this method. Lee (2005b) introduced the ’comparable’ property for fuzzy preference relation and showed that only O(n) comparisons of fuzzy numbers are sufficient if a fuzzy preference relation satisfies certain conditions. Asady and Zendehnam (2007) proposed a ranking method for the fuzzy numbers by obtaining the nearest point of support function with respect to fuzzy quantity. Wang (2015b) proposed a fuzzy relation with membership function representing preference degree to compare two fuzzy numbers. A relative preference relation was defined using fuzzy preference relation to compare a set of fuzzy numbers. The relative preference relation expresses preference degrees of several fuzzy numbers over average and facilitates easy and quick ranking of fuzzy numbers.

Decision-making methods often apply fuzzy sets in their computations. Jain (1976) presented a decision method that represented uncertain quantities as fuzzy sets and subsequently obtained an optimal alternative. Jain (1977) also developed a procedure for decision making using fuzzy sets by assigning quantitative numbers to qualitative terms. Wang (2014, 2015a, 2020a, 2020b) proposed various methods using relative preference relation to solve FMCDM problems. In the multi-granulation decision-theoretic rough set, Mandal and Ranadive (2019) introduced the optimistic and pessimistic fuzzy preference relation models.

In a multi-criteria decision-making problem with multiple data points, few data points, referred to as fuzzy numbers, are utilized to arrive at a decision. In this regard, different fuzzy numbers, such as triangular, trapezoidal, pentagonal and hexagonal fuzzy numbers, have been reported. These fuzzy numbers consider only a few data points to arrive at a decision. For example, the number of data points in triangular fuzzy numbers is 3, in trapezoidal fuzzy numbers it is 4, and in hexagonal fuzzy numbers it is 6. However, using a few data points to represent data leads to the loss of information. To address this situation, we generalize fuzzy numbers that encompass more data points to represent the data, thus minimizing the loss of information. Practically, when the decision problem is highly sensitive to the number of data points, it is reasonable to choose a larger value of n. The flexibility in implementing this idea is apparent in the case of data representation by $2n+1$ (or 2n) fuzzy numbers due to the choice for n. Thus, we get a natural advancement to the existing FMCDM methods.

In this paper, we define 2n and $2n+1$ fuzzy numbers. Clearly, $2n+1$ fuzzy numbers yield triangular and pentagonal cases when $n=1,2,$ and 2n fuzzy numbers coincide with trapezoidal and hexagonal fuzzy numbers when $n=2,3$, respectively. We extend the fuzzy preference relation and relative preference relation given by Wang (2015b) to rank 2n and $2n+1$ fuzzy numbers. Then, we compare the results obtained by fuzzy preference relation and relative preference relation with Wang and Lee (2008) method. Wang (2014) developed the FMCDM model with TOPSIS under fuzzy environment and relative preference relation on fuzzy numbers. We present an extension to the FMCDM model when the given data is representable in terms of $2n+1$ fuzzy numbers. We illustrate the suitability of the proposed method in solving FMCDM problems using an example. Subsequently, the proposed method results are validated and compared with VIKOR, MOORA and ELECTRE methods.

The rest of the paper is organized as follows. In Sect. 2, we present the basic definitions and related primary results. In Sect. 3, we give the definition of $2n+1$ fuzzy number and the extension of fuzzy preference and relative preference relations on $2n+1$ fuzzy numbers. In Sect. 4, we define 2n fuzzy number and the extension of fuzzy preference and relative preference relations on 2n fuzzy numbers. In Sect. 5, we present the proposed FMCDM model along with a telecommunication example. In Sect. 6, we validate the proposed method with popularly used multi-criteria decision-making methods.

2 Definitions and preliminaries

For the following definitions, we refer (Zadeh 1965; Zimmermann 1987, 1991).

Definition 2.1

A fuzzy subset A on the universe U is a set defined by a membership function $\mu _{A}$ representing a mapping $\mu _{A}:U\longrightarrow [0,1].$

Definition 2.2

$A_{\alpha }=\{x\mid \mu _{A}(x)\ge \alpha \}$ is called an $\alpha$-cut of the fuzzy set A.

Definition 2.3

Let X be a fuzzy number. Then, $X_{\alpha }^{L}$ and $X_{\alpha }^{U}$ are, respectively, defined as

$$\begin{aligned} X_{\alpha }^{L}=\inf \limits _{\mu _{X}(z)\ge \alpha }(z)~\hbox { and}~ X_{\alpha }^{U}=\sup \limits _{\mu _{X}(z)\ge \alpha }(z). \end{aligned}$$

Definition 2.4

(Lee 2005a, b; Epp 1990) A fuzzy preference relation R is a fuzzy subset of ${\mathbb {R}} \times {\mathbb {R}}$ with membership function $\mu _{R}(A, B)$ representing preference degree of fuzzy numbers A over B.

• 1. R is reciprocal if and only if $\mu _{R}(A, B)=1-\mu _{R}(B, A)$ for all fuzzy numbers A and B.

• 2. R is transitive if and only if $\mu _{R}(A, B)\ge \frac{1}{2}$ and $\mu _{R}(B, C)\ge \frac{1}{2}\Rightarrow \mu _{R}(A, C)\ge \frac{1}{2}$ for all fuzzy numbers A, B and C.

• 3. R is a fuzzy total ordering if and only if R is both reciprocal and transitive.

A is preferred to B if and only if $\mu _{R}(A, B)>\frac{1}{2}$ and A is equal to B if and only if $\mu _{R}(A, B)=\frac{1}{2}.$

Definition 2.5

(Wang 2015b) Let $\succ$ be a binary relation on fuzzy numbers defined by $A \succ B$ if and only if A is preferred to B (That is, $\mu _{R}(A, B)>\frac{1}{2}$).

Wang revised the extended fuzzy preference relation defined by Lee (2005b) as follows.

Definition 2.6

(Wang 2015b) Let A and B be two fuzzy numbers, where A is an interval $[a_{l}, a_{r}]$ and B is an interval $[b_{l}, b_{r}].$ A fuzzy preference relation P is a subset of ${\mathbb {R}}\times {\mathbb {R}}$ with membership function $\mu _{P}(A, B)$ representing preference degree of A over B.

Define

$$\begin{aligned} \mu _{P}(A, B)=\frac{1}{2}\left( \frac{\int _{0}^{1}(A-B)^{L}_{\alpha }+(A-B)^{U}_{\alpha }}{||T||}+1\right) , \end{aligned}$$

where

$$\begin{aligned} ||T||= & {} \int _{0}^{1}((T^{+}-T^{-})^{L}_{\alpha }+(T^{+}-T^{-})^{U}_{\alpha })d\alpha ~\hbox {if} ~t_{l}^{+}\ge t_{r}^{-}.\\= & {} \int _{0}^{1}((T^{+}-T^{-})^{L}_{\alpha }+(T^{+}-T^{-})^{U}_{\alpha }\\&\quad +2(t_{r}^{-}-t_{l}^{+}))d\alpha ~\hbox {if} ~t_{l}^{+}<t_{r}^{-}. \end{aligned}$$

$T^{+}$ is an interval $[t_{l}^{+},t_{r}^{+}],T^{-}$ is an interval $[t_{l}^{-},t_{r}^{-}],t^{+}_{l}=max\{a_{l},b_{l}\},$ $t^{+}_{r}=max\{a_{r},b_{r}\},t^{-}_{l}=min\{a_{l},b_{l}\}$ and $t^{-}_{r}=min\{a_{r},b_{r}\}.$

Similarly, fuzzy preference relation on triangular and trapezoidal fuzzy numbers have also been defined.

For the examples on decision-making problems, we refer (Koppula et al. 2019, 2020; Riaz et al. 2020; Chen and Huang 2021).

Definition 2.7

(Wang 2015b) Let $S=\{X_{1}, X_{2},\ldots ,X_{n}\}$ denote a set composed of n fuzzy numbers. A fuzzy number $X_{i}=[x_{il}, x_{ir}]$ belongs to the set S,  where $i=1,2,\ldots ,n.$ Assume ${\overline{X}}=\frac{\sum _{i}X_{i}}{n}$ derived by extension principle is average of the n fuzzy numbers in S. A relative preference relation $P^{*}$ with membership function $\mu _{P^{*}}(X_{i},{\overline{X}})$ represents preference degree of $X_{i}$ over ${\overline{X}}$ in S.

We define

$$\begin{aligned} \mu _{P_{*}}(X_{i}, {\overline{X}})=\frac{1}{2}\left( \frac{\int _{0}^{1}(X_{i} -{\overline{X}})^{L}_{\alpha }+(X_{i}-{\overline{X}})^{U}_{\alpha }}{||T_{s}||}+1\right) , \end{aligned}$$

where

$$\begin{aligned} \parallel T_{s} \parallel= & {} \int _{0}^{1}((T^{+}_{s}-T^{-}_{s})^{L}_{\alpha }+(T^{+}_{s}-T^{-}_{s})^{U}_ {\alpha })d\alpha ~~\hbox {if}~~ t_{sl}^{+}\ge t_{sr}^{-};\\= & {} \int _{0}^{1}((T^{+}_{s}-T^{-}_{s})^{L}_{\alpha }+(T^{+}_{s}-T^{-}_{s})^{U} _{\alpha }\\&+2(t_{sr}^{-}-t_{sl}^{+}))d\alpha ~~ \hbox {if}~~ t_{sl}^{+}<t_{sr}^{-}; \end{aligned}$$

$T^{+}_{s}$ is an interval $[t_{sl}^{+},t_{sr}^{+}],T^{-}_{s}$ is an interval $[t_{sl}^{-},t_{sr}^{-}],$ $t^{+}_{sl}=\max \nolimits _{i}\{X_{il}\},$ $t^{+}_{sr}=\max \nolimits _{i}\{X_{ir}\},$ $t^{-}_{sl}=\min \nolimits _{i}\{X_{il}\}$ and $t^{-}_{sr}=\min \nolimits _{i}\{X_{ir}\}.$

Clearly, $0< \mu _{P_{*}}(X_{i}, {\overline{X}})<1,$ where $i=1,2,\ldots ,n$. $\mu _{P_{*}}(X_{i}, {\overline{X}})<\frac{1}{2}$ expresses that ${\overline{X}}$ is preferred to $X_{i}.$ On the other hand, $\mu _{P_{*}}(X_{i}, {\overline{X}})>\frac{1}{2}$ expresses that $X_{i}$ is preferred to ${\overline{X}}.$

Similarly, relative preference relation is defined on triangular and trapezoidal fuzzy numbers.

3 Generalized $2n+1$ fuzzy number

3.1 Generalized linear $2n+1$ fuzzy number

Let $\{a_{1}, a_{2},a_{3},\ldots ,a_{2n+1}\}$ be real numbers such that $a_{1}<a_{2}<a_{3}<\cdots <a_{2n+1},~n=1,2,3,...$ and n is finite, $k\ge 2^{n-1}.$ Then, we denote

$$\begin{aligned} P(x):= & {} \dfrac{1}{k}\left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ; \\&a_{1}\le x\le a_{2} \\ S_{n}(x):= & {} \dfrac{2^{n-2}}{k}+\dfrac{2^{n-2}}{k} \left( \dfrac{x-a_{n}}{a_{n+1}-a_{n}}\right) ;\\&a_{n}\le x\le a_{n+1}\\ T_{(n,r)}(x):= & {} \dfrac{2^{n-2r}}{k}+\dfrac{2^{n-2r}}{k}\left( \dfrac{a_{n+2}-x}{a_{n+2}-a_{n+1}}\right) ; \\&a_{n+1}\le x\le a_{n+2}\\ Q_{n}(x):= & {} \dfrac{1}{k} \left( \dfrac{a_{2n+1}-x}{a_{2n+1}-a_{2n}}\right) ; \\&a_{2n}\le x\le a_{2n+1}. \end{aligned}$$

Now, $[P(x), Q_{1}(x)]$ gives fuzzy membership function of the generalized triangular fuzzy number $(a_{1}, a_{2}, a_{3};\dfrac{1}{k}).$

That is,

$$\begin{aligned} f(a_{1},a_{2},a_{3};x)= \left\{ \begin{array}{cc} \dfrac{1}{k} \left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ; &{} \quad a_{1}\le x\le a_{2}\\ \\ \dfrac{1}{k} \left( \dfrac{a_{3}-x}{a_{3}-a_{2}}\right) ; &{} \quad a_{2}\le x\le a_{3}.\\ \end{array} \right. \end{aligned}$$

If $k=1$ in the above, we get a triangular fuzzy number.

Now,

$$\begin{aligned}&f(a_{1},a_{2},a_{3},\ldots ,a_{2n+1};x)\nonumber \\&\quad = [P(x),S_{2}(x),S_{3}(x),\cdots ,S_{n-1}(x),S_{n}(x), T_{(n,1)}(x),\nonumber \\&T_{(n+1,2)}(x),T_{(n+2,3)}(x),\ldots ,T_{(2n-2,n-1)}(x),Q_{n}(x)]...(1) \end{aligned}$$

gives fuzzy membership function of the generalized fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n+1};\dfrac{2^{n-1}}{k}),$ where $n\ge 2$ and $k\ge 2^{n-1}.$

In particular, if $k=2^{n-1}$ then (1) gives fuzzy membership function of the fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n+1};1).$

For example, substitute $n=2$ in (1), then $[P(x), S_{2}(x), T_{(2,1)}(x), Q_{2}(x)]$ gives fuzzy membership function of the generalized pentagonal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5};\dfrac{2}{k}).$ That is,

$$\begin{aligned}&f(a_{1},a_{2},a_{3},a_{4},a_{5};x)\\&\quad = \left\{ \begin{array}{cc} \dfrac{1}{k} \left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ; &{}\quad a_{1}\le x\le a_{2}\\ \dfrac{1}{k}+\dfrac{1}{k} \left( \dfrac{x-a_{2}}{a_{3}-a_{2}}\right) ; &{} \quad a_{2}\le x\le a_{3} \\ \dfrac{1}{k}+\dfrac{1}{k} \left( \dfrac{a_{4}-x}{a_{4}-a_{3}}\right) ; &{} \quad a_{3}\le x\le a_{4}\\ \dfrac{1}{k}\left( \dfrac{a_{5}-x}{a_{5}-a_{4}}\right) ; &{} \quad a_{4}\le x\le a_{5}. \end{array} \right. \end{aligned}$$

If $k=2$ in the above, we get a pentagonal fuzzy number (Fig. 1).

Fig. 1

Pentagonal and generalized pentagonal fuzzy number

Similarly, substitute $n=3$ in (1) then $[P(x), S_{2}(x), S_{3}(x), T_{(3,1)}(x), T_{(4,2)}(x), Q_{3}(x)]$ gives fuzzy membership function of the generalized heptagonal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7};\dfrac{4}{k}).$

That is,

$$\begin{aligned}&f(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7};x)\\&=\left\{ \begin{array}{cc} \dfrac{1}{k} \left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ; &{} a_{1}\le x\le a_{2}\\ \dfrac{1}{k}+\dfrac{1}{k} \left( \dfrac{x-a_{2}}{a_{3}-a_{2}}\right) ; &{} a_{2}\le x\le a_{3}\\ \dfrac{2}{k}+\dfrac{2}{k} \left( \dfrac{x-a_{3}}{a_{4}-a_{3}}\right) ; &{} a_{3}\le x\le a_{4}\\ \dfrac{2}{k}+\dfrac{2}{k} \left( \dfrac{a_{5}-x}{a_{5}-a_{4}}\right) ; &{} a_{4}\le x\le a_{5}\\ \dfrac{1}{k}+\dfrac{1}{k} \left( \dfrac{a_{6}-x}{a_{6}-a_{5}}\right) ; &{} a_{5}\le x\le a_{6}\\ \dfrac{1}{k} \left( \dfrac{a_{7}-x}{a_{7}-a_{6}}\right) ; &{} a_{6}\le x\le a_{7}.\\ \end{array} \right. \end{aligned}$$

If $k=4$ in the above, we get a heptagonal fuzzy number.

3.2 Generalized nonlinear $2n+1$ fuzzy number

Let $\{a_{1}, a_{2},a_{3},\ldots ,a_{2n+1}\}$ be real numbers such that $a_{1}<a_{2}<a_{3}<\cdots <a_{2n+1},~m,n=1,2,3,...$ and m, n are finite. Now, take $k\ge 2^{n-1}.$ Then, we denote

$$\begin{aligned} P^{m}(x):= & {} \dfrac{1}{k} \left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ^{m}; \\&\quad a_{1}\le x\le a_{2} \\ S_{n}^{m}(x):= & {} \dfrac{2^{n-2}}{k}+\dfrac{2^{n-2}}{k} \left( \dfrac{x-a_{n}}{a_{n+1}-a_{n}}\right) ^{m}; \\&\quad a_{n}\le x\le a_{n+1}\\ T_{(n,r)}^{m}(x):= & {} \dfrac{2^{n-2r}}{k}+\dfrac{2^{n-2r}}{k} \left( \dfrac{a_{n+2}-x}{a_{n+2}-a_{n+1}}\right) ^{m}; \\&\quad a_{n+1}\le x\le a_{n+2}\\ Q_{n}^{m}(x):= & {} \dfrac{1}{k} \left( \dfrac{a_{2n+1}-x}{a_{2n+1}-a_{2n}}\right) ^{m};\\&\quad a_{2n}\le x\le a_{2n+1}. \end{aligned}$$

Then, $[P^{m}(x), Q_{1}^{m}(x)]$ gives fuzzy membership function of the generalized nonlinear triangular fuzzy number $(a_{1}, a_{2}, a_{3};\dfrac{1}{k})$.  Now,

$$\begin{aligned}&f^{m}(a_{1},a_{2},a_{3},\ldots ,a_{2n+1};x)\nonumber \\&=[P^{m}(x),S_{2}^{m}(x),S_{3}^{m}(x),\ldots ,S_{n-1}^{m}(x),S_{n}^{m}(x),\nonumber \\&T_{(n,1)}^{m}(x), T_{(n+1,2)}^{m}(x), T_{(n+2,3)}^{m}(x),\ldots ,\nonumber \\&T_{(2n-2,n-1)}^{m}(x),Q_{n}^{m}(x)]...(3) \end{aligned}$$

gives fuzzy membership function of the generalized nonlinear fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n+1};\dfrac{2^{n-1}}{k}),$ where $n\ge 2$ and $k\ge 2^{n-1}.$

In particular, if $k=2^{n-1}$ then (3) gives fuzzy membership function of the nonlinear fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n+1};1).$

Note 3.3

If $m=1$ in the fuzzy membership function of the generalized nonlinear $2n+1$ fuzzy number, then we get a fuzzy membership function of the generalized linear $2n+1$ fuzzy number.

3.3 $\alpha$-cut of a Generalized linear $2n+1$ fuzzy number

Let $\{a_{1}, a_{2},a_{3},\ldots ,a_{2n+1}\}$ be real numbers such that $a_{1}<a_{2}<a_{3}<\cdots <a_{2n+1},~n=1,2,3,...$ and n is finite. Now, take $k\ge 2^{n-1}.$ Then, we denote

$$\begin{aligned} E(\alpha ):= & {} a_{1}+k\alpha (a_{2}-a_{1}); \\&\quad \alpha \in \left[ 0, \dfrac{1}{k}\right] \\ F_{n}(\alpha ):= & {} a_{n}+ \left( \dfrac{\alpha k}{2^{n-2}}-1\right) ({a_{n+1}-a_{n}}); \\&\quad \alpha \in \left[ \dfrac{2^{n-2}}{k}, \dfrac{2^{n-1}}{k}\right] \\ G_{(n,r)}(\alpha ):= & {} a_{n+2}- \left( \dfrac{\alpha k}{2^{n-2r}}-1\right) ({a_{n+2}-a_{n+1}}); \\&\quad \alpha \in \left[ \dfrac{2^{n-2r}}{k}, \dfrac{2^{n-2r+1}}{k}\right] \\ H_{n}(\alpha ):= & {} a_{2n+1}-k\alpha (a_{2n+1}-a_{2n}); \\&\quad \alpha \in \left[ 0, \dfrac{1}{k}\right] . \end{aligned}$$

Then, $[E(\alpha ), H_{1}(\alpha )]$ gives $\alpha$-cut of the generalized triangular fuzzy number $(a_{1}, a_{2}, a_{3};\dfrac{1}{k}).$ That is,

$$\begin{aligned}&h(a_{1},a_{2},a_{3};\alpha )\\&=\left\{ \begin{array}{cc} a_{1}+k\alpha (a_{2}-a_{1}); &{} \quad \alpha \in \left[ 0, \dfrac{1}{k}\right] \\ a_{3}-k\alpha (a_{3}-a_{2}); &{}\quad \alpha \in \left[ 0, \dfrac{1}{k}\right] .\\ \end{array} \right. \end{aligned}$$

If $k=1$ in the above, we get an $\alpha$-cut of a triangular fuzzy number.

Now,

$$\begin{aligned}&h(a_{1},a_{2},a_{3},\ldots ,a_{2n+1};\alpha )\nonumber \\&\quad =[E(\alpha ),F_{2}(\alpha ),F_{3}(\alpha ),\ldots ,F_{n-1}(\alpha ),F_{n}(\alpha ), G_{(n,1)}(\alpha ),\nonumber \\&G_{(n+1,2)}(\alpha ), G_{(n+2,3)}(\alpha ),\ldots ,G_{(2n-2,n-1)}(\alpha ),H_{n}(\alpha )]...(5) \end{aligned}$$

gives $\alpha$-cut of the generalized fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n+1};\dfrac{2^{n-1}}{k}),$ where $n\ge 2$ and $k\ge 2^{n-1}.$

For example, if $n=3$ in (5) then $[E(\alpha ), F_{2}(\alpha ), F_{3}(\alpha ), G_{(3,1)}(\alpha ), G_{(4,2)}(\alpha ), H_{3}(\alpha )]$ gives $\alpha$-cut of the generalized heptagonal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5},a_{6},a_{7};\dfrac{4}{k}).$ That is,

$$\begin{aligned}&h(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7};\alpha )\\&= \left\{ \begin{array}{cc} a_{1}+k\alpha (a_{2}-a_{1}); &{}\quad \alpha \in \left[ 0, \dfrac{1}{k}\right] \\ a_{2}+ (\alpha k-1)({a_{3}-a_{2}}); &{} \quad \alpha \in \left[ \dfrac{1}{k}, \dfrac{2}{k}\right] \\ a_{3}+ \left( \dfrac{\alpha k}{2}-1\right) ({a_{4}-a_{3}}); &{}\quad \alpha \in \left[ \dfrac{2}{k}, \dfrac{4}{k}\right] \\ a_{5}- \left( \dfrac{\alpha k}{2}-1\right) ({a_{5}-a_{4}}); &{}\quad \alpha \in \left[ \dfrac{2}{k}, \dfrac{4}{k}\right] \\ a_{6}- (\alpha k-1)({a_{6}-a_{5}}); &{}\quad \alpha \in \left[ \dfrac{1}{k}, \dfrac{2}{k}\right] \\ a_{7}-k\alpha (a_{7}-a_{6}); &{} \quad \alpha \in \left[ 0, \dfrac{1}{k}\right] .\\ \end{array} \right. \end{aligned}$$

If $k=4$ (That is $k=2^{n-1}=2^{3-1}$) in the above, we get an $\alpha$-cut of a heptagonal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5},a_{6},a_{7}).$

Infimum and Supremum of $\alpha$ -cut of a $2n+1$ fuzzy number:

Let B be any fuzzy number and $\mu _{B}(x)$ is the fuzzy membership function of B. Then $B_{\alpha }^{L}=\inf \nolimits _{\mu _{B}(x)\ge \alpha }(x)$ and $B_{\alpha }^{U}=\sup \nolimits _{\mu _{B}(x)\ge \alpha }(x)$.

For example, if $B=(a_{1}, a_{2}, a_{3};\dfrac{1}{k})$ is a triangular fuzzy number. Then, $B_{\alpha }^{L}=E(\alpha )$ and $B_{\alpha }^{U}=H_{1}(\alpha ).$

For a fuzzy number $C=(a_{1}, a_{2}, a_{3},\ldots ,a_{2n+1};\dfrac{2^{n-1}}{k}),$ $C_{\alpha }^{L}=(E(\alpha ),F_{2}(\alpha ),F_{3}(\alpha ),\ldots , F_{n-1}(\alpha ),F_{n}(\alpha ))$ and

$$\begin{aligned}&C_{\alpha }^{U}=(G_{(n,1)}(\alpha ), G_{(n+1,2)}(\alpha ), G_{(n+2,3)}(\alpha ),\ldots ,\\&G_{(2(n-1),n-1)}(\alpha ),H_{n}(\alpha )). \end{aligned}$$

3.4 Extension of fuzzy preference relation on $2n+1$ fuzzy numbers

We extend the fuzzy preference relation given by Wang (2015b) to rank 2n+1 fuzzy numbers as follows.

Definition 3.6

Let $X_{1}$ and $X_{2}$ be two fuzzy numbers, where $X_{1}=[a_{1},a_{2},\ldots ,a_{2n+1}]$ and $X_{2}=[b_{1},b_{2},\ldots ,b_{2n+1}]$. An extended fuzzy preference relation R is a subset of ${\mathbb {R}}\times {\mathbb {R}}$ with membership function $\mu _{R}(X_{1}, X_{2})$ representing preference degree of $X_{1}$ over $X_{2}$. Then,

$$\begin{aligned}&\mu _{R}(X_{1}, X_{2})\\&=\frac{1}{2}\left( \int _{0}^{\frac{2^{n-1}}{k}} \left( \dfrac{(X_{1}-X_{2})^{\alpha }_{L}+(X_{1}-X_{2})^{\alpha }_{U}}{\parallel T \parallel }\right) d\alpha +1\right) \\&\quad .....\hbox {(I)} \end{aligned}$$

where

$$\begin{aligned}&\parallel T \parallel = \int _{0}^{\frac{2^{n-1}}{k}}({(T^{+}-T^{-})^{\alpha }_{L}+(T^{+}-T^{-})^{\alpha }_{U}}) d\alpha ; ~~\\&\quad {if}~~~~~ t_{1}^{+}\ge t_{2n-1}^{-}\\&\int _{0}^{\frac{2^{n-1}}{k}}({(T^{+}-T^{-})^{\alpha }_ {L}+(T^{+}-T^{-})^{\alpha }_{U}}+2(t_{2n+1}^{-}-t_{1}^{+}))d\alpha ;~~ \quad \\&\quad {if}~~ t_{1}^{+}<t_{2n-1}^{-}, \end{aligned}$$

$T^{+}$ is an interval $[t_{1}^{+}, t_{2}^{+},\ldots ,t_{2n+1}^{+}],$ $T^{-}$ is an interval $[t_{1}^{-}, t_{2}^{-},\ldots ,t_{2n+1}^{-}]$ and $t_{1}^{+}$=max $\{a_{1}, b_{1}\}$, $t_{2}^{+}$=max $\{a_{2}, b_{2}\}, \ldots ,t_{2n+1}^{+}$=max$\{a_{2n+1}, b_{2n+1}\},$ $t_{1}^{-}$=min $\{a_{1}, b_{1}\}$, $t_{2}^{-}$=min $\{a_{2},b_{2}\},$...,$t_{2n+1}^{-}$=min $\{a_{2n+1}, b_{2n+1}\}.$

In (I), if $\int _{0}^{\frac{2^{n-1}}{k}}({(T^{+}-T^{-})^{\alpha }_{L}+(T^{+}-T^{-}) ^{\alpha }_{U}})d\alpha \ge 0,$ then $\mu _{R}(X_{1}, X_{2})\ge \frac{1}{2}$ and if $X_{1}=X_{2},$ then $\mu _{R}(X_{1}, X_{2})=\frac{1}{2}.$

Lemma 3.7

The extended fuzzy preference relation R is reciprocal. That is, $\mu _{R}(X_{1}, X_{2})=1-\mu _{R}(X_{2}, X_{1})$ for all $2n+1$ fuzzy numbers $X_{1}$ and $X_{2}.$

Lemma 3.8

The extended fuzzy preference relation R is transitive. That is, if $\mu _{R}(X_{1}, X_{2})\ge \frac{1}{2}$ and $\mu _{R}(X_{2}, X_{3})\ge \frac{1}{2}$ then $\mu _{R}(X_{1}, X_{3})\ge \frac{1}{2},$ where $X_{1},X_{2}$ and $X_{3}$ are $2n+1$ fuzzy numbers.

From Lemmas 3.7 and 3.8, the extended fuzzy preference relation R is a total ordering relation (Epp (1990) and Lee (2005b)).

Lemma 3.9

Let $X_{1}$ and $X_{2}$ be two $2n+1$ fuzzy numbers. By the extended fuzzy preference relation R, $X_{1}$ is preferred to $X_{2}$ if and only if $\mu _{R}(X_{1}, X_{2})>\frac{1}{2}.$

Lemma 3.10

$X_{1}\succ X_{2}$ if and only if $\mu _{R}(X_{1}, X_{2})>\frac{1}{2},$ where $\succ$ is a binary relation.

Result 3.11

Let $X_{1}=\big (a_{1}, a_{2},a_{3};\dfrac{1}{k}\big )$ and $X_{2}=(b_{1}, b_{2},b_{3};\dfrac{1}{k}\big )$ be two generalized triangular fuzzy numbers. Then, $\int _{0}^{\frac{1}{k}} [(X_{1}-X_{2})_{\alpha }^{L}+(X_{1}-X_{2})_{\alpha }^{U}]d\alpha =\dfrac{(a_{1}-b_{3})+2(a_{2}-b_{2})+(a_{3}-b_{1})}{2k}.$

Now,

$$\begin{aligned}&\mu _{R}(X_{1},X_{2})\\&=\frac{1}{2}\left( \int _{0}^{\frac{1}{k}} \left( \dfrac{\left( X_{1}-X_{2}\right) ^{\alpha }_ {L}+\left( X_{1}-X_{2}\right) ^{\alpha }_{U}}{\parallel T \parallel }\right) d\alpha +1\right) \\&=\frac{1}{2} \left( \dfrac{\left( a_{1}-b_{3}\right) +2\left( a_{2}-b_{2}\right) +\left( a_{3}-b_{1}\right) }{2k\parallel T \parallel }+1\right) . \end{aligned}$$

If $k=1$ in the above, we get

$$\begin{aligned}&\int _{0}^1 [(X_{1}-X_{2})_{\alpha }^{L}+(X_{1}-X_{2})_{\alpha }^{U}]d\alpha \\&\quad =\dfrac{(a_{1}-b_{3})+2(a_{2}-b_{2})+(a_{3}-b_{1})}{2}. \end{aligned}$$

Then, $\mu _{R}(X_{1},X_{2}) =\frac{1}{2} ( \dfrac{(a_{1}-b_{3})+2(a_{2}-b_{2})+(a_{3}-b_{1})}{2\parallel T \parallel }+1)$ (Wang (2015b)).

Lemma 3.12

Let $X_{1}=(a_{1}, a_{2},a_{3},a_{4},a_{5};\dfrac{2}{k})$ and $X_{2}=(b_{1}, b_{2}, b_{3}, b_{4}, b_{5};\dfrac{2}{k})$ be two generalized pentagonal fuzzy numbers. Then, $\mu _{R}(X_{1},X_{2}) = \frac{1}{2}\{[[(a_{1}-b_{5})+2(a_{2}-b_{4}) +2(a_{3}-b_{3}) +2(a_{4}-b_{2})+(a_{5}-b_{1})]/(2k\parallel T \parallel )]+1\}.$

Proof

Now,

$$\begin{aligned} X_{1}-X_{2}=((a_{1}-b_{5}),(a_{2}-b_{4}),(a_{3}-b_{3}), (a_{4}-b_{2}), (a_{5}-b_{1})). \end{aligned}$$

Then,

$$\begin{aligned}&\int _{0}^{\frac{2}{k}} (X_{1}-X_{2})_{\alpha }^{L}d\alpha \\&\quad =\int _{0}^{\frac{1}{k}} ((a_{1}-b_{5})+k\alpha ((a_{2}-b_{4})-(a_{1}-b_{5}))) d\alpha \\&\qquad +\int _{\frac{1}{k}}^{\frac{2}{k}} ((a_{2}-b_{4})+(k\alpha -1) ((a_{3}-b_{3})-(a_{2}-b_{4}))) d\alpha \\&\quad =\frac{(a_{1}-b_{5})}{k}+\frac{(a_{2}-b_{4})-(a_{1}-b_{5})}{2k}\\&\qquad +\frac{2(a_{2}-b _{4})}{k}+\frac{3((a_{3}-b_{3})-(a_{2}-b_{4}))}{2k}-\frac{(a_{3}-b_{3})}{k}. \end{aligned}$$

Similarly,

$$\begin{aligned}&\int _{0}^{\frac{2}{k}} (X_{1}-X_{2})_{\alpha }^{U}d\alpha \\&=\int _{0}^{\frac{1}{k}} ((a_{5}-b_{1})-k\alpha ((a_{5}-b_{1})-(a_{4}-b_{2})) d\alpha )\\&\qquad +\int _{\frac{1}{k}}^{\frac{2}{k}} ((a_{4}-b_{2})-(k\alpha -1) ((a_{4}-b_{2})-(a_{3}-b_{3})) d\alpha )\\&\quad =\frac{(a_{5}-b_{1})}{k}-\frac{(a_{5}-b_{1})-(a_{4}-b_{2})}{2k}+ \frac{2(a_{4}-b_{2})}{k}\\&\qquad -\frac{3((a_{4}-b_{2})-(a_{3}-b_{3}))}{2k} -\frac{(a_{3}-b_{3})}{k}. \end{aligned}$$

Then,

$$\begin{aligned}&\int _{0}^{\frac{2}{k}}(X_{1}-X_{2})_{\alpha }^{L}d\alpha +\int _{0}^{\frac{2}{k}} (X_{1}-X_{2})_{\alpha }^{U}d\alpha \\&\quad =\frac{1}{2k}[(a_{1}-b_{5})+2(a_{2}-b_{4})\\&\qquad +2(a_{3}-b_{3})+2(a_{4}-b_{2})\\&\qquad +(a_{5}-b_{1})]. \end{aligned}$$

Now,

$$\begin{aligned}&\mu _{R}(X_{1},X_{2})\\&\quad =\frac{1}{2}\left( \int _{0}^{\frac{2}{k}} \left( \dfrac{(X_{1}-X_{2})^{\alpha }_{L}+(X_{1}-X_{2})^{\alpha }_{U}}{\parallel T \parallel }\right) d\alpha +1\right) \\&\quad = \frac{1}{2}\{[[(a_{1}-b_{5})+2(a_{2}-b_{4})+2(a_{3}-b_{3})+2(a_{4}-b_{2})\\&\qquad +(a_{5}-b_{1})]/(2k\parallel T \parallel )]+1\}. \end{aligned}$$

$\square$

Note 3.13

If $k=2$ in Lemma 3.12, then $\mu _{R}(X_{1},X_{2})$ $= \frac{1}{2}\{[[(a_{1}-b_{5})+2(a_{2}-b_{4})+2(a_{3}-b_{3})+2(a_{4}-b_{2})+(a_{5}-b_{1})]/(4\parallel T \parallel )]+1\}.$

Lemma 3.14

Let $X_{1}=(a_{1}, a_{2},\ldots ,a_{n},a_{n+1},a_{n+2},\ldots ,a_{2n}$, $a_{2n+1};\frac{2^{n-1}}{k})$ and $X_{2}=(b_{1}, b_{2},\ldots ,b_{n},b_{n+1},\ldots ,b_{2n}$, $b_{2n+1};\frac{2^{n-1}}{k})$ be two fuzzy numbers, where $n\ge 3.$

Then,

$$\begin{aligned}&\mu _{R}(X_{1},X_{2})=\dfrac{1}{2}\{[[(a_{1}-b_{2n+1})+2(a_{2}-b_{2n})\\&\quad +3[(a_{3}-b_{2n-1})+2(a_{4}-b_{2n-2})+4(a_{5}-b_{2n-3})\\&\quad +\cdots +2^{n-3}(a_{n}-b_{n+2})]+2^{n-1}(a_{n+1}-b_{n+1})\\&\quad +3[2^{n-3}(a_{n+2}-b_{n})+2^{n-4}(a_{n+3}-b_{n-1})\\&\quad +\ldots +2(a_{2n-2}-b_{4})+(a_{2n-1}-b_{3})] +2(a_{2n}-b_{2})\\&\quad +(a_{2n+1}-b_{1})] /(2k\parallel T \parallel )]+1\}, \end{aligned}$$

where

$$\begin{aligned}&\parallel T \parallel =[(t_{1}^{+}-t_{2n+1}^{-})+2(t_{2}^{+}-t_{2n}^{-})+3[(t_{3}^{+}-t_{2n-1}^{-})\\&\quad +2(t_{4}^{+}-t_{2n-2}^{-})+4(t_{5}^{+}-t_{2n-3}^{-})+\cdots +2^{n-3}(t_{n}^{+}-t_{n+2}^{-})]\\&\quad +2^{n-1}(t_{n+1}^{+}-t_{n+1}^{-}) +3[2^{n-3}(t_{n+2}^{+}-t_{n}^{-})\\&\quad +2^{n-4}(t_{n+3}^{+}-t_{n-1}^{-})+\ldots + 2(t_{2n-2}^{+}-t_{4}^{-})+(t_{2n-1}^{+}-t_{3}^{-})]\\&\quad +2(t_{2n}^{+}-t_{2}^{-})+(t_{2n+1}^{+}-t_{1}^{-})]/2k ~~~~~~~~~\hbox {if} ~~~ (t_{1}^{+}-t_{2n+1}^{-})\ge 0.\\&\quad =\{[(t_{1}^{+}-t_{2n+1}^{-})+2(t_{2}^{+}-t_{2n}^{-})+3[(t_{3}^{+}-t_{2n-1}^{-}) \\&\quad +2(t_{4}^{+}-t_{2n-2}^{-}) +4(t_{5}^{+}-t_{2n-3}^{-})+ \ldots +2^{n-3}(t_{n}^{+}-t_{n+2}^{-})]\\&\quad +2^{n-1}(t_{n+1}^{+}-t_{n+1}^{-}) +3[2^{n-3}(t_{n+2}^{+}-t_{n}^{-})+2^{n-4}(t_{n+3}^{+}-t_{n-1}^{-})\\&\quad +\ldots +2(t_{2n-2}^{+}-t_{4}^{-})\\&\quad +(t_{2n-1}^{+}-t_{3}^{-})] +2(t_{2n}^{+}-t_ {2}^{-})+(t_{2n+1}^{+}-t_{1}^{-})] /2k\}\\&\quad + 2(t_{2n+1}^{-}-t_{1}^{+}) ~~~~~~~~~\hbox {if}~~~~~ (t_{1}^{+}-t_{2n+1}^{-})< 0, \end{aligned}$$

where $t_{1}^{+}$=max $\{a_{1}, b_{1}\}$, $t_{2}^{+}$=max $\{a_{2}, b_{2}\},$...,$t_{2n+1}^{+}$=max$\{a_{2n+1}, b_{2n+1}\},$ $t_{1}^{-}$=min $\{a_{1}, b_{1}\}$, $t_{2}^{-}$=min $\{a_{2}, b_{2}\},$...,$t_{2n+1}^{-}$=min $\{a_{2n+1}, b_{2n+1}\}.$

Proof

Now, $X_{1}-X_{2}=[(a_{1}-b_{2n+1}), (a_{2}-b_{2n}),\ldots ,(a_{n}-b_{n+2}), (a_{n+1}-b_{n+1}), (a_{n+2}-b_{n}),\ldots ,(a_{2n-1}-b_{3}),(a_{2n}-b_{2}),(a_{2n+1}-b_{1})].$

Consider

$$\begin{aligned}&\int _{0}^{\frac{2^{n-1}}{k}} (X_{1}-X_{2})_{\alpha }^{L}+(X_{1}-X_{2})_{\alpha }^{U}d\alpha \\&\quad =\int _{0}^{\frac{1}{k}}[(a_{1}-b_{2n+1})+k\alpha ((a_{2}-b_{2n})-(a_{1}-b_{2n+1}))]d\alpha \\&\quad + \int _{\frac{1}{k}}^{\frac{2}{k}}[(a_{2}-b_{2n})+(k\alpha -1)((a_{3}-b_{2n-1})-(a_{2}-b_{2n}))]d\alpha \\&\quad +\int _{\frac{2}{k}}^{\frac{4}{k}}[(a_{3}-b_{2n-1})+(\frac{k\alpha }{2}-1) ((a_{4}-b_{2n-2})\\&\quad -(a_{3}-b_{2n-1}))]d\alpha +\cdots +\int _{\frac{2^{n-3}}{k}}^ {\frac{2^{n-2}}{k}}[(a_{n-1}-b_{n+3})\\&\quad +(\frac{k\alpha }{2^{n-3}}-1)((a_{n} -b_{n+2})-(a_{n-1}-b_{n+3}))]d\alpha \\&\quad +\int _{\frac{2^{n-2}}{k}}^{\frac{2^ {n-1}}{k}}[(a_{n}-b_{n+2})+(\frac{k\alpha }{2^{n-2}}-1)((a_{n+1}-b_{n+1})\\&\quad -(a_{n}-b_{n+2}))]d\alpha +\int _{\frac{2^{n-2}}{k}}^{\frac{2^{n-1}}{k}} [(a_{n+2}-b_{n})-\left( \frac{k\alpha }{2^{n-2}}-1\right) \\&\quad ((a_{n+2}-b_{n})-(a_{n+1} -b_{n+1}))]d\alpha +\int _{\frac{2^{n-3}}{k}}^{\frac{2^{n-2}}{k}}[(a_{n+3 }-b_{n-1})\\&\quad -(\frac{k\alpha }{2^{n-3}}-1)((a_{n+3}-b_{n-1})-(a_{n+2}-b_{n}))]d\alpha \\&\quad +\cdots +\int _{\frac{2}{k}}^{\frac{4}{k}}[(a_{2n-1}-b_{3})-(\frac{k\alpha }{2}-1) ((a_{2n-1}-b_{3})\\&\quad -(a_{2n-2}-b_{4}))]d\alpha +\int _{\frac{1}{k}}^{\frac{2}{k}} [(a_{2n}-b_{2})-(k\alpha -1)\\&\quad ((a_{2n}-b_{2})-(a_{2n+1}-b_{1}))]d\alpha +\int _{0}^ {\frac{1}{k}}[(a_{2n+1}-b_{1})\\&\quad -k\alpha ((a_{2n+1}-b_{1})-(a_{2n}-b_{2}))]d\alpha \\&\quad =\dfrac{1}{2k} [(a_{1}-b_{2n+1})+2(a_{2}-b_{2n})+3[(a_{3}-b_{2n-1})\\&\quad +2(a_{4}-b_{2n-2})+4(a_{5}-b_{2n-3})+\cdots +2^{n-3}(a_{n}-b_{n+2})]\\&\quad +2^{n-1}(a_{n+1}-b_{n+1})+3[2^{n-3}(a_{n+2}-b_{n})\\&\quad +2^{n-4}(a_{n+3} -b_{n-1})+\ldots +2(a_{2n-2}-b_{4})\\&\quad +(a_{2n-1}-b_{3})] +2(a_{2n}-b_{2}) +(a_{2n+1}-b_{1}) ]. \end{aligned}$$

Then,

$$\begin{aligned}&\mu _{R}(X_{1},X_{2})\\&\quad =\frac{1}{2}\left( \int _{0}^{\frac{2^{n-1}}{k}} \left( \dfrac{(X_{1}-X_{2})^{\alpha }_{L}+(X_{1}-X_{2})^{\alpha }_{U}}{\parallel T \parallel }\right) d\alpha +1\right) \\&\quad =\frac{1}{2}\{[ [(a_{1}-b_{2n+1})+2(a_{2}-b_{2n})+3[(a_{3}-b_{2n-1})\\&\qquad +2(a_{4}-b_{2n-2})+4(a_{5}-b_{2n-3})+\cdots +2^{n-3}(a_{n}-b_{n+2})]\\&\qquad +2^{n-1}(a_{n+1}-b_{n+1})+3[2^{n-3}(a_{n+2}-b_{n})\\&\qquad +2^{n-4}(a_{n+3}-b_{n-1})+\ldots +2(a_{2n-2}-b_{4})+(a_{2n-1}-b_{3})]\\&\qquad +2(a_{2n}-b_{2})+(a_{2n+1}-b_{1})]/(2k\parallel T \parallel )]+1\} \end{aligned}$$

and clearly, we get the value of $\parallel T \parallel .$

$\square$

3.5 Extension of relative preference relation on $2n+1$ fuzzy numbers

Ranking n fuzzy numbers by fuzzy preference relation is time-consuming due to pair-wise comparisons. To reduce the time complexity, Wang (2015b) proposed relative preference relation. For example, to rank n fuzzy numbers by a preference relation, we require $n_{C_{2}} \in O(n^{2})$ fuzzy pair-wise comparisons. In contrast, it is sufficient to use relative preference relation O(n) times to rank the fuzzy numbers. We extend the relative preference relation given by Wang (2015b) to rank $2n+1$ fuzzy numbers by having the time complexity same as Wang’s method, i.e., the time complexity is O(n) for ranking n fuzzy numbers. The method is as follows:

Definition 3.16

Let $A=\{X_{1},X_{2},\ldots ,X_{m}\}$ be m fuzzy numbers, where each $X_{i}=\{x_{i1},x_{i2},\ldots ,x_{i(2n+1)}\},i=1,2,\ldots ,m$ and ${\overline{X}}=\frac{\sum _{i} X_{i}}{m}$ is the average of the m fuzzy numbers of A. An extended relative preference relation $R_{*}$ with the fuzzy membership function $\mu _{R_{*}}(X_{i}, {\overline{X}})$ represents the preference degree of $X_{i}$ over ${\overline{X}}$ in A. Now, we define

$$\begin{aligned}&\mu _{R_{*}}(X_{i}, {\overline{X}})\\&\quad =\frac{1}{2}\left( \int _{0}^{\frac{2^{n-1}}{k}}\left( \dfrac{(X_{i}-{\overline{X}}) ^{\alpha }_{L}+(X_{i}-{\overline{X}})^{\alpha }_{U}}{\parallel T_{q} \parallel }\right) d\alpha +1\right) , \end{aligned}$$

where

$$\begin{aligned}&{\parallel T_{q} \parallel }= \int _{0}^{\frac{2^{n-1}}{k}}[{(T^{+}_{q}-T^{-}_{q})^{\alpha }_{L}+(T^{+}_{q}-T^{-}_{q})^{\alpha }_{U}}]d\alpha ; ~~ \\&\quad if~~ t_{q1}^{+}\ge t_{q(2n+1)}^{-},\\&\quad \int _{0}^{\frac{2^{n-1}}{k}}[{(T^{+}_{q}-T^{-}_{q})^{\alpha }_{L} +(T^{+}_{q}-T^{-}_{q})^{\alpha }_{U}}\\&\quad +2(t_{q(2n+1)}^{-}-t_{q1}^{+})]d\alpha ; \\&\quad ~{if}~~ t_{q1}^{+}<t_{q(2n+1)}^{-}, \end{aligned}$$

where $T^{+}_{q}$ is an interval $[t_{q1}^{+},t_{q2}^{+},\ldots ,t_{q(2n+1)}^{+}],$ $T^{-}_{q}$ is an interval $[t_{q1}^{-}, t_{q2}^{-},\ldots ,t_{q(2n+1)}^{-}]$ and $t_{q1}^{+}=\max \limits _{i}\{x_{i1}\},$

$$\begin{aligned} t_{q2}^{+}= & {} \max \limits _{i} \{x_{i2}\},\ldots ,t_{q(2n+1)}^{+}=\max \limits _{i}\{x_{i(2n+1)}\},\\ t_{q1}^{-}= & {} \min \limits _{i}\{x_{i1}\}, t_{q2}^{-}=\min \limits _{i} \{x_{i2}\},\ldots ,\\ t_{q(2n+1)}^{-}= & {} \min \limits _{i}\{x_{i(2n+1)}\},i=1,2,\ldots ,m. \end{aligned}$$

If $\mu _{R_{*}}(X_{i}, {\overline{X}}) > \frac{1}{2}$ then $X_{i}$ is preferred to ${\overline{X}}$ and

$\mu _{R_{*}}(X_{i}, {\overline{X}})< \frac{1}{2}$ then ${\overline{X}}$ is preferred to $X_{i}.$

Lemma 3.17

The extended relative preference relation $R_{*}$ is a total ordering relation.

Lemma 3.18

Let $X_{i}$ and $X_{j}$ be two fuzzy numbers in A. Then, $X_{i}$ is preferred to $X_{j}$ if and only if $\mu _{R_{*}}(X_{i}, {\overline{X}})>\mu _{R_{*}}(X_{j}, {\overline{X}}).$

Lemma 3.19

$X_{i}\succ X_{j}$ if and only if $\mu _{R_{*}}(X_{i}, {\overline{X}})>\mu _{R_{*}}(X_{j}, {\overline{X}}),$ where $\succ$ is a binary relation as defined in Definition 2.5.

Lemma 3.20

Let $A=\{X_{1},X_{2},\ldots ,X_{m}\}$ be a set of fuzzy numbers, where each $X_{i}=\{x_{i1},x_{i2},x_{i3},\ldots ,x_{i(2n+1)}\},i=1,2,\ldots ,m$ and ${\overline{X}}=({\overline{x}}_{1}, {\overline{x}}_{2},\ldots ,{\overline{x}}_{2n+1})$ be the average of m fuzzy numbers. The extended relative preference relation $R_{*}$ with membership function $\mu _{R_{*}}(X_{i}, {\overline{X}})$ represents preference degree of $X_{i}$ over ${\overline{X}}$ in A. Then,

$$\begin{aligned}&\mu _{R_{*}}(X_{i},{\overline{X}})=\dfrac{1}{2}\{[[(x_{i1}-{\overline{x}}_{2n+1}) +2(x_{i2}-{\overline{x}}_{2n})\\&\quad +3[(x_{i3}-{\overline{x}}_{2n-1})+2(x_{i4}-{\overline{x}}_{2n-2})+4(x_{i5} -{\overline{x}}_{2n-3})\\&\quad +\cdots +2^{n-3}(x_{in}-{\overline{x}}_{n+2})]+2^{n-1} (x_{i(n+1)}-{\overline{x}}_{n+1})\\&\quad +3[2^{n-3}(x_{i(n+2)}-{\overline{x}}_{n})+2^{n-4}(x_{i(n+3)} -{\overline{x}}_{n-1})\\&\quad +\cdots +2(x_{i(2n-2)}-{\overline{x}}_{4})+(x_{i(2n-1)}-{\overline{x}}_{3})] +2(x_{i2n}-{\overline{x}}_{2})\\&\quad +(x_{i(2n+1)}-{\overline{x}}_{1}) ] /(2k\parallel T_{q} \parallel )]+1\}, \end{aligned}$$

where

$$\begin{aligned}&\parallel T_{q} \parallel =[(t_{q1}^{+}-t_{q(2n+1)}^{-})+2(t_{q2}^{+}- t_{q(2n)}^{-})+3[(t_{q3}^{+}-t_{q(2n-1)}^{-})\\&\quad +2(t_{q4}^{+}-t_{q(2n-2)}^{-})+4(t_{q5}^{+}-t_{q(2n-3)}^{-})\\&\quad +\cdots +2^{n-3}(t_{qn}^{+}-t_{q(n+2)}^{-})]+2^{n-1}(t_{q(n+1)}^{+}-t_{q(n+1)}^{-})\\&\quad +3[2^{n-3}(t_{q(n+2)}^{+}-t_{qn}^{-})+2^{n-4}(t_{q(n+3)}^{+}-t_{q(n-1)}^{-})\\&\quad +\cdots +2(t_{q(2n-2)}^{+}-t_{q4}^{-})+(t_{q(2n-1)}^{+}-t_{q3}^{-})] +2(t_{q(2n)} ^{+}-t_{q2}^{-})\\&\quad +(t_{q(2n+1)}^{+}-t_{q1}^{-})]/2k ~\hbox {if} ~(t_{q1}^{+}-t_{q(2n+1)}^{-})\ge 0\\&\quad =[[(t_{q1}^{+}-t_{q(2n+1)}^{-})+2(t_{q2}^{+}-t_{q(2n)}^{-})+3[(t_{q3}^{+}-t_{q(2n-1)}^{-})\\&\quad +2(t_{q4}^{+}-t_{q(2n-2)}^{-})+4 (t_{q5}^{+}-t_{q(2n-3)}^{-})\\&\quad +\cdots +2^{n-3}(t_{qn}^{+}-t_{q(n+2)}^{-})] +2^{n-1}(t_{q(n+1)}^{+}-t_{q(n+1)}^{-})\\&\quad +3[2^{n-3}(t_{q(n+2)}^{+}-t_{qn}^{-})+2^{n-4}(t_{q(n+3)}^{+}-t_{q(n-1)}^{-})\\&\quad +\cdots +2(t_{q(2n-2)}^{+}-t_{q4}^{-})+(t_{q(2n-1)}^{+}-t_{q3}^{-})] +2(t_{q2n}^{+}-t_{q2}^{-})\\&\quad +(t_{q(2n+1)}^{+}-t_{q1}^{-}) ]/(2k)] + 2(t_{q(2n+1)}^{-}-t_{q1}^{+})~~~~\\&\quad if (t_{q1}^{+}-t_{q(2n+1)}^{-})< 0. \end{aligned}$$

where

$$\begin{aligned}&t_{q1}^{+}=\max \limits _{i}\{x_{i1}\}, {t}_{q2}^{+}=\max \limits _{i} \{x_{i2}\},\ldots ,t_{q(2n+1)}^{+}=\max \limits _{i}\{x_{i(2n+1)}\},\\&t_{q1}^{-}=\min \limits _{i}\{x_{i1}\}, {t}_{q2}^{-}=\min \limits _{i} \{x_{i2}\},\ldots ,t_{q(2n+1)}^{-}=\min \limits _{i}\{x_{i(2n+1)}\},\\&i=1,2,\ldots ,m. \end{aligned}$$

Here, we provide an example.

Example 3.21

Let $X_{1}=(2,3,5,6,8;\frac{2}{3})$ and $X_{2}=(1,2,4,7,9;\frac{2}{3})$ be two pentagonal fuzzy numbers ($K=3$).

Then,

$$\begin{aligned}X_{1}-X_{2}=(-7, -4, 1, 4, 7).\end{aligned}$$

Now,

$$\begin{aligned}t_{1}^{+}=\hbox {max}\{2,1\}=2, ~~~t_{1}^{-}=\hbox {min}\{2,1\}=1;\end{aligned}$$

Similarly,

$$\begin{aligned}&t_{2}^{+}=3, ~~~t_{2}^{-}=2;\\&t_{3}^{+}=5, ~~~t_{3}^{-}=4;\\&t_{4}^{+}=7, ~~~t_{4}^{-}=6;\\&t_{5}^{+}=9, ~~~t_{5}^{-}=8.\\&\parallel T \parallel =\frac{1}{6}(-6-6+2+10+8)+2(6)=13.3333 \end{aligned}$$

Then,

$$\begin{aligned} \mu _{R}(X_{1},X_{2})=\frac{1}{2} \left[ \frac{-7-8+2+8+7}{6(13.3333)}+1\right] =0.5125>1/2.\end{aligned}$$

This implies $X_{1} > X_{2}$.

$$\begin{aligned}&{\overline{X}}=(\overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}, \overline{x_{4}}, \overline{x_{5}})=(1.5, 2.5, 4.5, 6.5, 8.5) \end{aligned}$$

Now,

$$\begin{aligned}&t_{q1}^{+}=2 ~~~t_{q1}^{-}=1;\\&t_{q2}^{+}=3 ~~~t_{q2}^{-}=2;\\&t_{q3}^{+}=5 ~~~t_{q3}^{-}=4;\\&t_{q4}^{+}=7 ~~~t_{q4}^{-}=6;\\&t_{q5}^{+}=9 ~~~t_{q5}^{-}=8.\\&\parallel T_{q} \parallel =\frac{1}{6}[(2-8)+2(3-6)+2(5-4)\\&+2(7-2) +(9-1)]+2(6)=\frac{40}{3} \end{aligned}$$

Now,

$$\begin{aligned} \mu _{R_{*}}(X_{1},{\overline{X}})=\frac{1}{2}\left[ \frac{(2-8.5)+2(3-6.5)+2(5-4.5)+2(6-2.5)+(8-1.5)}{6\left( \frac{40}{3}\right) }+1\right] =0.5063. \end{aligned}$$

Similarly,

$$\begin{aligned} \mu _{R_{*}}(X_{2},{\overline{X}})=0.49038. \end{aligned}$$

As $\mu _{R_{*}}(X_{1},{\overline{X}})> \mu _{R_{*}}(X_{2},{\overline{X}}),$ we get $X_{1}>X_{2}.$ We compared the above result with Wang and Lee (2008)’s area method.

By Wang and Lee (2008)’s method, we get ${\overline{x}}(X_{1})=4.75$ and ${\overline{x}}(X_{2})=4.6481.$ Therefore $X_{1}>X_{2}.$

4 Generalized 2n fuzzy number

4.1 Generalized linear 2n fuzzy number

Let $\{a_{1}, a_{2},a_{3},\ldots ,a_{2n}\}$ be real numbers such that $a_{1}<a_{2}<a_{3}<\cdots <a_{2n},~n=2,3,4,...$ and n is finite, $k\ge 2^{n-2}.$ Now, we denote

$$\begin{aligned} P(x):= & {} \dfrac{1}{k}\left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ; \\&\quad a_{1}\le x\le a_{2} \\ S_{n}(x):= & {} \dfrac{2^{n-2}}{k}+\dfrac{2^{n-2}}{k} \left( \dfrac{x-a_{n-1}}{a_{n}-a_{n-1}}\right) ; \\&\quad a_{n-1}\le x\le a_{n}\\ R_{n}(x):= & {} \dfrac{2^{n-2}}{k}; \\&\quad a_{n}\le x\le a_{n+1}\\ U_{(n,r)}(x):= & {} \dfrac{2^{n-(2r+1)}}{k}+\dfrac{2^{n-(2r+1)}}{k} \left( \dfrac{a_{n+2}-x}{a_{n+2}-a_{n+1}}\right) ; \\&a_{n+1}\le x\le a_{n+2}\\ V_{n}(x):= & {} \dfrac{1}{k} \left( \dfrac{a_{2n}-x}{a_{2n}-a_{2n-1}}\right) ; \\&a_{2n-1}\le x\le a_{2n}. \end{aligned}$$

Then, $[P(x),R_{2}(x),V_{2}(x)]$ gives fuzzy membership function of the generalized trapezoidal fuzzy number $(a_{1}, a_{2}, a_{3},a_{4};\dfrac{1}{k}).$ That is,

$$\begin{aligned} g(a_{1}, a_{2}, a_{3},a_{4};x)=\left\{ \begin{array}{cc} \dfrac{1}{k} \left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ; &{} \quad a_{1}\le x\le a_{2}\\ \dfrac{1}{k} ; &{} \quad a_{2}\le x\le a_{3}\\ \dfrac{1}{k} \left( \dfrac{a_{4}-x}{a_{4}-a_{3}}\right) ; &{} \quad a_{3}\le x\le a_{4}. \end{array} \right. \end{aligned}$$

If $k=1$ in the above, we get a trapezoidal fuzzy number.

Now,

$$\begin{aligned}&g(a_{1}, a_{2}, a_{3},\ldots ,a_{2n};x)\\&\quad =[P(x),S_{2}(x),S_{3}(x),\ldots ,S_{n-1}(x),R_{n}(x),U_{(n,1)}(x),\\&\quad U_{(n+1,2)}(x), U_{(n+2,3)(x)}, \ldots ,\\&\qquad U_{(2n-3,n-2)}(x),V_{n}(x)].....(2) \end{aligned}$$

gives fuzzy membership function of the generalized fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n};\dfrac{2^{n-2}}{k}),$ where $n\ge 3$ and $k\ge 2^{n-2}.$

In particular, if $k=2^{n-2}$ then (2) gives fuzzy membership function of the fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n};1),$ where $n\ge 3.$

For example, substitute $n=3$ in (2) then $[P(x), S_{2}(x), R_{3}(x), U_{(3,1)}(x),V_{3}(x)]$ gives fuzzy membership function of the generalized hexagonal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5},a_{6};\dfrac{2}{k}).$

That is,

$$\begin{aligned}&g(a_{1}, a_{2}, a_{3},a_{4},a_{5},a_{6};x)\\&\quad = \left\{ \begin{array}{cc} \dfrac{1}{k} \left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ; &{}\quad a_{1}\le x\le a_{2}\\ \dfrac{1}{k}+\dfrac{1}{k} \left( \dfrac{x-a_{2}}{a_{3}-a_{2}}\right) ; &{}\quad a_{2}\le x\le a_{3}\\ \dfrac{2}{k}; &{}\quad a_{3}\le x\le a_{4}\\ \dfrac{1}{k}+\dfrac{1}{k}\left( \dfrac{a_{5}-x}{a_{5}-a_{4}}\right) ; &{}\quad a_{4}\le x\le a_{5}\\ \dfrac{1}{k} \left( \dfrac{a_{6}-x}{a_{6}-a_{5}}\right) ; &{} \quad a_{5}\le x\le a_{6}. \end{array} \right. \end{aligned}$$

If $k=2$ in the above, we get a hexagonal fuzzy number (Fig. 2).

Fig. 2

Hexagonal and generalized hexagonal fuzzy number

Similarly, substitute $n=4$ in (2) then

$$\begin{aligned}{}[P(x), S_{2}(x),S_{3}(x), R_{4}(x), U_{(4,1)}(x),U_{(5,2)}(x),V_{4}(x)] \end{aligned}$$

gives fuzzy membership function of the generalized octagonal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5},a_{6},a_{7},a_{8};\dfrac{4}{k}).$

That is,

$$\begin{aligned}&g(a_{1}, a_{2}, a_{3},a_{4},a_{5},a_{6},a_{7},a_{8};x)\\&\quad = \left\{ \begin{array}{cc} \dfrac{1}{k}\left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ; &{} \quad a_{1}\le x\le a_{2}\\ \dfrac{1}{k}+\dfrac{1}{k} \left( \dfrac{x-a_{2}}{a_{3}-a_{2}}\right) ; &{} \quad a_{2}\le x\le a_{3}\\ \dfrac{2}{k}+\dfrac{2}{k}\left( \dfrac{x-a_{3}}{a_{4}-a_{3}}\right) ; &{} \quad a_{3}\le x\le a_{4}\\ \dfrac{4}{k}; &{} \quad a_{4}\le x\le a_{5}\\ \dfrac{2}{k}+\dfrac{2}{k}\left( \dfrac{a_{6}-x}{a_{6}-a_{5}}\right) ; &{} \quad a_{5}\le x\le a_{6}\\ \dfrac{1}{k}+\dfrac{1}{k} \left( \dfrac{a_{7}-x}{a_{7}-a_{6}}\right) ; &{} \quad a_{6}\le x\le a_{7}\\ \dfrac{1}{k} \left( \dfrac{a_{8}-x}{a_{8}-a_{7}}\right) ; &{} \quad a_{7}\le x\le a_{8}.\\ \end{array} \right. \end{aligned}$$

If $k=4$ in the above, we get an octagonal fuzzy number.

4.2 Generalized nonlinear 2n fuzzy number

Let $\{a_{1}, a_{2},a_{3},\ldots ,a_{2n}\}$ be real numbers such that $a_{1}<a_{2}<a_{3}<\cdots <a_{2n},~n=2,3,4,...$ and n is finite, $m=1,2,...$ and m is finite. Now, take $k\ge 2^{n-2}.$ Then, we denote

$$\begin{aligned}&P^{m}(x): \dfrac{1}{k} \left( \dfrac{x-a_{1}}{a_{2}-a_{1}}\right) ^{m}; \\&\quad a_{1}\le x\le a_{2} \\&S_{n}^{m}(x): \dfrac{2^{n-2}}{k}+\dfrac{2^{n-2}}{k} \left( \dfrac{x-a_{n-1}}{a_{n}-a_{n-1}}\right) ^{m}; \\&\quad a_{n-1}\le x\le a_{n}\\&R_{n}(x): \dfrac{2^{n-2}}{k}; \\&\quad a_{n}\le x\le a_{n+1}\\&U_{(n,r)}^{m}(x): \dfrac{2^{n-(2r+1)}}{k}+\dfrac{2^{n-(2r+1)}}{k} \left( \dfrac{a_{n+2}-x}{a_{n+2}-a_{n+1}}\right) ^{m}; \\&\quad a_{n+1}\le x\le a_{n+2}\\&V_{n}^{m}(x): \dfrac{1}{k} \left( \dfrac{a_{2n}-x}{a_{2n}-a_{2n-1}}\right) ^{m};\\&\quad a_{2n-1}\le x\le a_{2n}. \end{aligned}$$

Now, $[P^{m}(x),R_{2}(x),V_{2}^{m}(x)]$ gives the fuzzy membership function of the generalized nonlinear trapezoidal fuzzy number $(a_{1}, a_{2}, a_{3},a_{4};\dfrac{1}{k}).$

Now,

$$\begin{aligned}&[P^{m}(x),S_{2}^{m}(x),S_{3}^{m}(x),\ldots ,S_{n-1}^{m}(x),\\&R_{n}(x),U_{(n,1)}^{m}(x),U_{(n+1,2)} U_{(n+2,3)}^{m}(x),\ldots ,{m}(x) \\&U_{(2n-3,n-2)}^{m}(x),V_{n}^{m}(x)]....(4)U_{(n+1,2)}^{m}(x), \end{aligned}$$

gives fuzzy membership function of the generalized nonlinear fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n};\dfrac{2^{n-2}}{k}),$ where $n\ge 3$ and $k\ge 2^{n-2}.$

In particular, if $k=2^{n-2}$ then (4) gives fuzzy membership function of the nonlinear fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n};1),$ where $n\ge 3.$

Note 4.3

If $m=1$ in the fuzzy membership function of the generalized nonlinear 2n fuzzy number, then we get fuzzy membership function of the generalized linear 2n fuzzy number.

4.3 $\alpha$-cut of a generalized linear 2n fuzzy number

Let $\{a_{1}, a_{2},a_{3},\ldots ,a_{2n}\}$ be real numbers such that $a_{1}<a_{2}<a_{3}<\cdots <a_{2n},~n=1,2,3,...$ and n is finite. Now, take $k\ge 2^{n-2}.$ We denote

$$\begin{aligned} E(\alpha ):= & {} a_{1}+k\alpha (a_{2}-a_{1}); \\&\quad \alpha \in \left[ 0, \dfrac{1}{k}\right] \\ F_{n}(\alpha ):= & {} a_{n}+ \left( \dfrac{\alpha k}{2^{n-2}}-1\right) ({a_{n+1}-a_{n}}); \\&\quad \alpha \in \left[ \dfrac{2^{n-2}}{k}, \dfrac{2^{n-1}}{k}\right] \\ I_{(n,r)}(\alpha ):= & {} a_{n+2}- \left( \dfrac{\alpha k}{2^{n-(2r+1)}}-1\right) ({a_{n+2}-a_{n+1}}); \\&\quad \alpha \in \left[ \dfrac{2^{n-(2r+1)}}{k}, \dfrac{2^{n-2r}}{k}\right] \\ J_{n}(\alpha ):= & {} a_{2n}-k\alpha (a_{2n}-a_{2n-1}); \\&\quad \alpha \in \left[ 0, \dfrac{1}{k}\right] . \end{aligned}$$

Now, $[E(\alpha ), J_{1}(\alpha )]$ gives $\alpha$-cut of the generalized trapezoidal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4};\dfrac{1}{k}).$ That is,

$$\begin{aligned} l(a_{1},a_{2},a_{3},a_{4};\alpha )=\left\{ \begin{array}{cc} a_{1}+k\alpha (a_{2}-a_{1}); &{} \alpha \in \left[ 0, \dfrac{1}{k}\right] \\ a_{4}-k\alpha (a_{4}-a_{3}); &{} \alpha \in \left[ 0, \dfrac{1}{k}\right] .\\ \end{array} \right. \end{aligned}$$

If $k=1$ in the above, we get an $\alpha$-cut of a trapezoidal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4};1)$.

Now,

$$\begin{aligned}&l(a_{1},a_{2},a_{3},\ldots ,a_{2n};\alpha )\\&=[E(\alpha ),F_{2}(\alpha ),F_{3}(\alpha ),\ldots ,F_{n-1}(\alpha ),I_{(n,1)}(\alpha ),\\&I_{(n+1,2)}(\alpha ), I_{(n+2,3)}(\alpha ),\ldots ,I_{(2n-3,n-2)}(\alpha ),J_{n}(\alpha )]...(6) \end{aligned}$$

gives $\alpha$-cut of the generalized fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n};\dfrac{2^{n-2}}{k}),$ where $n\ge 3$ and $k\ge 2^{n-2}.$

For example, if $n=4$ in (6) then

$$\begin{aligned}{}[E(\alpha ), F_{2}(\alpha ), F_{3}(\alpha ), I_{(4,1)}(\alpha ),I_{(5,2)}(\alpha ),J_{4}(\alpha )] \end{aligned}$$

gives $\alpha$-cut of the generalized octagonal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5},a_{6},a_{7},a_{8};\dfrac{4}{k}).$

That is,

$l(a_{1},a_{2},a_{3},a_{4}, a_{5},a_{6},a_{7},a_{8};\alpha )$

$$\begin{aligned} =\left\{ \begin{array}{cc} a_{1}+k\alpha (a_{2}-a_{1}); &{} \quad \alpha \in \left[ 0, \dfrac{1}{k}\right] \\ a_{2}+ (\alpha k-1)({a_{3}-a_{2}}); &{} \quad \alpha \in \left[ \dfrac{1}{k}, \dfrac{2}{k}\right] \\ a_{3}+ \left( \dfrac{\alpha k}{2}-1\right) ({a_{4}-a_{3}}); &{} \quad \alpha \in \left[ \dfrac{2}{k}, \dfrac{4}{k}\right] \\ a_{6}- \left( \dfrac{\alpha k}{2}-1\right) ({a_{6}-a_{5}}); &{} \quad \alpha \in \left[ \dfrac{2}{k}, \dfrac{4}{k}\right] \\ a_{7}- (\alpha k-1)({a_{7}-a_{6}}); &{} \quad \alpha \in \left[ \dfrac{1}{k}, \dfrac{2}{k}\right] \\ a_{8}-k\alpha (a_{8}-a_{7}); &{} \quad \alpha \in \left[ 0, \dfrac{1}{k}\right] .\\ \end{array} \right. \end{aligned}$$

If $k=4$ (That is, $k=2^{n-2}=2^{4-2}$) in the above, we get an $\alpha$-cut of octagonal fuzzy number $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5},a_{6},a_{7},a_{8};1).$

Infimum and Supremum of $\alpha$-cut of a 2n fuzzy number:

Let $B=(a_{1}, a_{2}, a_{3}, a_{4};\dfrac{1}{k})$ be a trapezoidal fuzzy number and $\mu _{B}(x)$ is fuzzy membership function of B.

Then,

$$\begin{aligned}B_{\alpha }^{L}=E(\alpha )\end{aligned}$$

and

$$\begin{aligned}B_{\alpha }^{U}=J_{1}(\alpha ).\end{aligned}$$

For a fuzzy number $(a_{1}, a_{2}, a_{3},\ldots ,a_{2n};\dfrac{2^{n-2}}{k}),$ $B_{\alpha }^{L}=(E(\alpha ),F_{2}(\alpha ),F_{3}(\alpha ),\ldots ,F_{n-1}(\alpha ))$ and $B_{\alpha }^{U}=(I_{(n,1)}(\alpha ), I_{(n+1,2)}(\alpha ), I_{(n+2,3)}(\alpha ),\ldots ,I_{(2n-3,n-2)}(\alpha ),J_{n}(\alpha )).$

4.4 preference relation on 2n fuzzy numbers

We extend the fuzzy preference relation given by Wang (2015b) to rank 2n fuzzy numbers.

Definition 4.6

Let A and B be two 2n fuzzy numbers, where $A=(a_{1},a_{2},\ldots ,a_{2n})$ and $B=(b_{1},b_{2},\ldots ,b_{2n})$. An extended fuzzy preference relation $R_{2n}$ is a subset of ${\mathbb {R}}\times {\mathbb {R}}$ with membership function $\mu _{R_{2n}}(A,B)$ representing preference degree of A over B. Then,

$$\begin{aligned}&\mu _{R_{2n}}(A, B)\\&\quad =\frac{1}{2}\left( \int _{0}^{\frac{2^{n-2}}{k}} \left( \dfrac{(A-B)^{\alpha }_{L}+(A-B)^{\alpha }_{U}}{\parallel T \parallel }\right) d\alpha +1\right) \\&.....\hbox {(II)}, \end{aligned}$$

where

$$\begin{aligned}&\parallel T \parallel = \int _{0}^{\frac{2^{n-2}}{k}}({(T^{+}-T^{-})^{\alpha }_{L}+(T^{+}-T^{-})^{\alpha }_{U}})d\alpha ;~~ \\&\quad if~~~~~ t_{1}^{+}\ge t_{2n}^{-}\\&\int _{0}^{\frac{2^{n-2}}{k}}({(T^{+}-T^{-})^{\alpha }_{L}+(T^{+}-T^{-})^{\alpha }_{U}}+2(t_{2n}^{-}-t_{1}^{+}))d\alpha ;~~ \\&~~if~ t_{1}^{+}<t_{2n}^{-}, \end{aligned}$$

$T^{+}$ is an interval $[t_{1}^{+}, t_{2}^{+},\ldots ,t_{2n}^{+}],$ $T^{-}$ is an interval $[t_{1}^{-}, t_{2}^{-},\ldots ,t_{2n}^{-}]$ and $t_{1}^{+}=\max \{a_{1}, b_{1}\}$, $t_{2}^{+}=\max \{a_{2}, b_{2}\}, \ldots ,t_{2n}^{+}=\max \{a_{2n}, b_{2n}\}$, $t_{1}^{-}=\min \{a_{1}, b_{1}\}$, $t_{2}^{-}=\min \{a_{2}, b_{2}\},\ldots ,t_{2n}^{-}=\min \{a_{2n}, b_{2n}\}.$

In (II), if $\int _{0}^{\frac{2^{n-2}}{k}}({(T^{+}-T^{-})^{\alpha }_{L}+(T^{+}-T^{-})^{\alpha }_{U}})d\alpha \ge 0,$ then $\mu _{R_{2n}}(A, B)\ge \frac{1}{2}$ and if $A=B,$ then $\mu _{R_{2n}}(A, B)=\frac{1}{2}.$

Lemma 4.7

The extended fuzzy preference relation $R_{2n}$ is reciprocal. That is, $\mu _{R_{2n}}(A, B)=1-\mu _{R_{2n}}(B, A)$ for all 2n fuzzy numbers A and B.

Lemma 4.8

The extended fuzzy preference relation $R_{2n}$ is transitive. That is, if $\mu _{R_{2n}}(A, B)\ge \frac{1}{2}$ and $\mu _{R_{2n}}(B, C)\ge \frac{1}{2}$ then $\mu _{R_{2n}}(A, C)\ge \frac{1}{2},$ where A, B and C are 2n fuzzy numbers.

From Lemmas 4.7 and 4.8, the extended fuzzy preference relation $R_{2n}$ is a total ordering relation (Epp (1990) and Lee (2005b).

Lemma 4.9

Let A and B be two $2n+1$ fuzzy numbers. By extended fuzzy preference relation $R_{2n}$, A is preferred to B if and only if $\mu _{R_{2n}}(A, B)>\frac{1}{2}.$

Lemma 4.10

$A\succ B$ if and only if $\mu _{R_{2n}}(A, B)>\frac{1}{2},$ where $\succ$ is a binary relation.

Lemma 4.11

Let $A=(a_{1}, a_{2},a_{3},a_{4};\dfrac{1}{k})$ and $B=(b_{1}, b_{2},b_{3},b_{4};\dfrac{1}{k})$ be two trapezoidal fuzzy numbers. Then, $\int _{0}^{\frac{1}{k}} [(A-B)_{\alpha }^{L}+(A-B)_{\alpha }^{U}]d\alpha =\frac{1}{2k}[(a_{1}-b_{4}) +(a_{2}-b_{3})+(a_{3}-b_{2})+(a_{4}-b_{1})]$ and $\mu _{R_{4}}(A,B)=\frac{1}{2}(\int _{0}^{\frac{1}{k}} (\dfrac{(A-B)^{\alpha }_{L}+(A-B)^{\alpha }_{U}}{\parallel T \parallel })d\alpha +1){=}\frac{1}{2} ( \dfrac{(a_{1}{-}b_{4}){+}(a_{2}{-}b_{3}){+}(a_{3}{-}b_{2}){+}(a_{4}-b_{1})}{2k\parallel T \parallel }{+}1).$ where

$$\begin{aligned}&\parallel T \parallel = \frac{1}{2k}[(t_{1}^{+}-t_{4}^{-})+(t_{2}^{+}-t_{3}^{-})+(t_{3}^{+}-t_{2}^{-})+(t_{4}^{+}-t_{1}^{-})];~~ \\&\quad if~~~~~ t_{1}^{+}\ge t_{4}^{-}\\&\quad \frac{1}{2k}[(t_{1}^{+}-t_{4}^{-})+(t_{2}^{+}-t_{3}^{-})+(t_{3}^{+}-t_{2}^{-})+(t_{4}^{+}-t_{1}^{-})]\\&\quad +2(t_{4}^{-}-t_{1}^{+}); ~~ if~~~~~~~ t_{1}^{+}<t_{4}^{-}. \end{aligned}$$

$T^{+}$ is an interval $[t_{1}^{+}, t_{2}^{+},t_{3}^{+},t_{4}^{+}],$ $T^{-}$ is an interval $[t_{1}^{-}, t_{2}^{-},t_{3}^{-},t_{4}^{-}]$ and $t_{1}^{+}=\max \{a_{1}, b_{1}\}$, $t_{2}^{+}=\max \{a_{2}, b_{2}\}$, $t_{3}^{+}=\max \{a_{3}, b_{3}\}$, $t_{4}^{+}=\max \{a_{4}, b_{4}\}$, $t_{1}^{-}=\min \{a_{1}, b_{1}\}$, $t_{2}^{-}=\min \{a_{2}, b_{2}\},t_{3}^{-}=\min \{a_{3}, b_{3}\}$ and $t_{4}^{-}=\min \{a_{4}, b_{4}\}.$

Lemma 4.12

Let $A=(a_{1}, a_{2},a_{3},a_{4},a_{5},a_{6};\frac{2}{k})$ and $B=(b_{1}, b_{2},b_{3},b_{4},b_{5},b_{6};\frac{2}{k})$ be two hexagonal fuzzy numbers. Then,

$$\begin{aligned}&\int _{0}^{\frac{2}{k}} (A-B)_{\alpha }^{L}+(A-B)_{\alpha }^{U}d\alpha =\dfrac{1}{2k} [(a_{1}-b_{6})+2(a_{2}-b_{5})\\&\quad +(a_{3}-b_{4})+(a_{4}-b_{3})+2(a_{5}-b_{2})+(a_{6}-b_{1})] \end{aligned}$$

and

$$\begin{aligned}&\mu _{R_{6}}(A,B)=\frac{1}{2} \{[[(a_{1}-b_{6})+2(a_{2}-b_{5})+(a_{3}-b_{4})\\&\quad +(a_{4}-b_{3})+2(a_{5}-b_{2})+(a_{6}-b_{1})]/(2k\parallel T \parallel )]+1\}.\end{aligned}$$

where

$$\begin{aligned}&\parallel T \parallel = \frac{1}{2k}\big [(t_{1}^{+}-t_{6}^{-})+2(t_{2}^{+}-t_{5}^{-})\\&\quad +(t_{3}^{+}-t_{4}^{-})\\&\quad +(t_{4}^{+}-t_{3}^{-})+2(t_{5}^{+}-t_{2}^{-})+(t_{6}^{+}-t_{1}^{-})\big ]; ~~~ if~~~~~ t_{1}^{+}\ge t_{6}^{-}\\&\quad =\frac{1}{2k}[(t_{1}^{+}-t_{6}^{-})+2(t_{2}^{+}-t_{5}^{-})+(t_{3}^{+}-t_{4}^{-})+(t_{4}^{+}-t_{3}^{-})\\&\quad +2(t_{5}^{+}-t_{2}^{-})+(t_{6}^{+}-t_{1}^{-})]+2(t_{6}^{-}-t_{1}^{+}); ~~~~~\\&\quad if~~~~~ t_{1}^{+}<t_{6}^{-} and\end{aligned}$$

$T^{+}$ is an interval $[t_{1}^{+}, t_{2}^{+},t_{3}^{+},t_{4}^{+},t_{5}^{+},t_{6}^{+}],$ $T^{-}$ is an interval $[t_{1}^{-}, t_{2}^{-},t_{3}^{-},t_{4}^{-},t_{5}^{-},t_{6}^{-}]$ and $t_{1}^{+}$=max $\{a_{1}, b_{1}\}$, $t_{2}^{+}$=max $\{a_{2}, b_{2}\}$, $t_{3}^{+}$=max $\{a_{3}, b_{3}\}$, $t_{4}^{+}$=max $\{a_{4}, b_{4}\}$, $t_{5}^{+}$=max $\{a_{5}, b_{5}\}$, $t_{6}^{+}$=max $\{a_{6}, b_{6}\}$, $t_{1}^{-}$=min $\{a_{1}, b_{1}\}$, $t_{2}^{-}$=min $\{a_{2}, b_{2}\},t_{3}^{-}$=min $\{a_{3}, b_{3}\}$, $t_{4}^{-}$=min $\{a_{4}, b_{4}\}$, $t_{5}^{-}$=min $\{a_{5}, b_{5}\}$, $t_{6}^{-}$=min $\{a_{6}, b_{6}\}.$

Lemma 4.13

Let $A=(a_{1}, a_{2},\ldots ,a_{n},a_{n+1},\ldots ,a_{2n-1}$, $a_{2n};\frac{2^{n-2}}{k})$ and $B=(b_{1}, b_{2},\ldots ,b_{n},b_{n+1},\ldots ,b_{2n-1},b_{2n};$ $\frac{2^{n-2}}{k})$ be two fuzzy numbers, where $n>3.$

Then,

$$\begin{aligned}&\mu _{R_{2n}}(A,B)=\dfrac{1}{2}\{[((a_{1}-b_{2n})+2(a_{2}-b_{2n-1})\\&\quad +3[(a_{3}-b_{2n-2})+2(a_{4}-b_{2n-3})+4(a_{5}-b_{2n-4})\\&\quad +\cdots +2^{n-4}(a_{n-1}-b_{n+2})]+2^{n-3}(a_{n}-b_{n+1})\\&\quad +2^{n-3}(a_{n+1}-b_{n})+3[2^{n-4}(a_{n+2}-b_{n-1})\\&\quad +2^{n-5}(a_{n+3}-b_{n-2})+\cdots +2(a_{2n-3}-b_{4})\\&\quad +(a_{2n-2}-b_{3})]\\&\quad +2(a_{2n-1}-b_{2})+(a_{2n}-b_{1}))/(2k\parallel T \parallel )]+1\}, \end{aligned}$$

where

$$\begin{aligned}&\parallel T \parallel = [(t_{1}^{+}-t_{2n}^{-})+2(t_{2}^{+}-t_{2n-1}^{-})+3[(t_{3}^{+}-t_{2n-2}^{-})\\&\quad +2(t_{4}^{+}-t_{2n-3}^{-})\\&\quad +4(t_{5}^{+}-t_{2n-4}^{-})+\cdots +2^{n-4}(t_{n-1}^{+} -t_{n+2}^{-})]+2^{n-3}\\&\quad (t_{n}^{+}-t_{n+1}^{-})+2^{n-3}(t_{n+1}^{+}-t_{n}^{-}) +3[2^{n-4}(t_{n+2}^{+}-t_{n-1}^{-})\\&\quad +2^{n-5}(t_{n+3}^{+}-t_{n-2}^{-})+\cdots +2 (t_{2n-3}^{+}-t_{4}^{-})+(t_{2n-2}^{+}-t_{3}^{-})] \\&\quad +2(t_{2n-1}^{+}-t_{2}^{-}) +(t_{2n}^{+}-t_{1}^{-})]/2k ~~~~ if~~~~ (t_{1}^{+}-t_{2n}^{-})\ge 0\\&\quad =[((t_{1}^{+}-t_{2n}^{-})+2(t_{2}^{+}-t_{2n-1}^{-})+3[(t_{3}^{+}-t_{2n-2}^{-})\\&\quad +2(t_{4}^{+}-t_{2n-3}^{-})+4(t_{5}^{+}-t_{2n-4}^{-})+\cdots +2^{n-4}(t_{n-1}^{+}-t_{n+2}^{-}) ]\\&\quad +2^{n-3}(t_{n}^{+}-t_{n+1}^{-})+2^{n-3}(t_{n+1}^{+}-t_{n}^{-})+ 3[2^{n-4}(t_{n+2}^{+}-t_{n-1}^{-})\\&\quad +2^{n-5}(t_{n+3}^{+}-t_{n-2}^{-}) +\cdots +2(t_{2n-3}^{+}-t_{4}^{-})+(t_{2n-2}^{+}-t_{3}^{-})]\\&\quad +2(t_{2n-1}^{+}-t_{2}^{-})+(t_{2n}^{+}-t_{1}^{-}))/2k]+ 2(t_{2n}^{-}-t_{1}^{+}) ~~~\\&\quad if~~~~~ (t_{1}^{+}-t_{2n}^{-})< 0. \end{aligned}$$

4.5 Extension of relative preference relation on 2n fuzzy numbers

Definition 4.15

Let $A=\{X_{1},X_{2},\ldots ,X_{m}\}$ be a set of fuzzy numbers, where each $X_{i}=\{x_{i1},x_{i2},x_{i3},\ldots ,x_{i(2n)}\},i=1,2,\ldots ,m.$ and ${\overline{X}}=({\overline{x}}_{1}, {\overline{x}}_{2},\ldots ,{\overline{x}}_{2n})$ be the average of m fuzzy numbers. The relative preference relation $R_{2n}^{*}$ with membership function $\mu _{R_{2n}^{*}}(X_{i}, {\overline{X}})$ represents preference degree of $X_{i}$ over ${\overline{X}}$ in A.

Then,

$$\begin{aligned}&\mu _{R_{2n}^{*}}(X_{i},{\overline{X}}) =\dfrac{1}{2}\{[((x_{i1}-{\overline{x}}_{2n})\\&\quad +2(x_{i2}-{\overline{x}}_ {2n-1})+3[(x_{i3}-{\overline{x}}_{2n-2})\\&\quad +2(x_{i4} -{\overline{x}}_{2n-3})+4(x_{i5}-{\overline{x}}_{2n-4})+ \cdots +\\&\quad 2^{n-4}(x_{i(n-1)}-{\overline{x}}_{n+2})]+2^ {n-3}(x_{in}-{\overline{x}}_{n+1})\\&\quad +2^{n-3}(x_{i(n+1)}-{\overline{x}}_{n})+ 3[2^{n-4}(x_{i(n+2)}-{\overline{x}}_{n-1})\\&\quad +2^{n-5}(x_{i(n+3)}-{\overline{x}}_{n-2})+\cdots +2(x_{i(2n-3)}-{\overline{x}}_{4})\\&\quad +(x_{i(2n-2)}-{\overline{x}}_{3})] +2(x_{i(2n-1)}-{\overline{x}}_{2})\\&\quad +(x_{i2n}-{\overline{x}}_{1}))/(2k\parallel T_{q} \parallel )]+1\}, \end{aligned}$$

where

$$\begin{aligned}&\parallel T_{q} \parallel = [(t_{q1}^{+}-t_{q2n}^{-})+2(t_{q2}^{+}-t_{q(2n-1)}^{-})+3[(t_{q3}^{+}-t_{q(2n-2)}^{-})\\&\quad +2(t_{q4}^{+}-t_{q(2n-3)}^{-})+4(t_{q5}^{+}-t_{q(2n-4)}^{-})+\cdots +\\&\quad 2^{n-4}(t_{q(n-1)}^{+}-t_{q(n+2)}^{-})]+2^{n-3}(t_{qn}^{+}-t_{q(n+1)}^{-})\\&\quad +2^{n-3}(t_{q(n+1)}^{+}-t_{qn}^{-})+3[2^{n-4}(t_{q(n+2)}^{+}-t_{q(n-1)}^{-})\\&\quad +2^{n-5}(t_{q(n+3)}^{+}-t_{q(n-2)}^{-})+\cdots +2(t_{q(2n-3)}^{+}-t_{4}^{-})\\&\quad +(t_{q(2n-2)}^{+}-t_{q3}^{-})] +2(t_{q(2n-1)}^{+}-t_{q2}^{-})+(t_{q2n}^{+}-t_{q1}^{-})]/2k ~~~~~~\\&\quad if~~~ (t_{q1}^{+}-t_{q2n}^{-})\ge 0 \\&\quad =[((t_{q1}^{+}-t_{q2n}^{-})+2(t_{q2}^{+}-t_{q2n-1}^{-})+3[(t_{q3}^{+}-t_ {q2n-2}^{-})\\&\quad +2(t_{q4}^{+}-t_{q2n-3}^{-})+4(t_{q5}^{+}-t_{q2n-4}^{-})+\cdots +\\&\quad 2^{n-4}(t_{qn-1}^{+}-t_{qn+2}^{-})]+2^{n-3}(t_{qn}^{+}-t_{qn+1}^{-})\\&\quad +2^{n-3}(t_{qn+1}^{+}-t_{qn}^{-})+3[2^{n-4}(t_{qn+2}^{+}-t_{qn-1}^{-})\\&\quad +2^{n-5}(t_{qn+3}^{+}-t_{qn-2}^{-})+\cdots +2(t_{q2n-3}^{+}-t_{q4}^{-})\\&\quad +(t_{q2n-2}^{+}-t_{q3}^{-})] +2(t_{q2n-1}^{+}-t_{q2}^{-})+(t_{q2n}^{+}-t_{q1}^{-}))/{2k}] \\&\quad +2(t_{q2n}^{-}-t_{q1}^{+}) ~~~~~~if (t_{q1}^{+}-t_{q2n}^{-})< 0. \end{aligned}$$

Example 4.16

Let

$$\begin{aligned}&X_{1}=\{0.12, 0.2, 0.24, 0.28, 0.3, 0.35, 0.4, 0.46;1\},\\&X_{2}=\{0.1, 0.15, 0.25, 0.27, 0.32, 0.36, 0.38, 0.48;1\}\\&\hbox {and}~ X_{3}=\{0.1, 0.17, 0.2, 0.3, 0.35, 0.38, 0.41, 0.5;1\} \end{aligned}$$

be octagonal fuzzy numbers.

Then, $\parallel T \parallel$ of $X_{1}$ and $X_{2}$ is

$$\begin{aligned}&\frac{1}{8}[(0.12-0.46)+2(0.2-0.38)+3(0.25-0.35)\\&\qquad +2(0.28-0.3) +2(0.32-0.27)+3(0.36-0.24)\\&\qquad +2(0.4-0.15) +(0.48-0.1)]+2(0.46-0.12)\\&\quad =0.7175, \end{aligned}$$

where

$$\begin{aligned} \begin{array}{cc} t_{1}^{+}= \hbox {max}\{0.12, 0.1\}=0.12; &{} t_{1}^{-}= \hbox {min}\{0.12, 0.1\}=0.1\\ t_{2}^{+}= \hbox {max}\{0.2, 0.15\}=0.2; &{} t_{2}^{-}= \hbox {min}\{0.2, 0.15\}=0.15\\ t_{3}^{+}= \hbox {max}\{0.24, 0.25\}=0.25; &{} t_{3}^{-}= \hbox {min}\{0.24, 0.25\}=0.24\\ t_{4}^{+}= \hbox {max}\{0.28, 0.27\}=0.28; &{} t_{4}^{-}= \hbox {min}\{0.28, 0.27\}=0.27\\ t_{5}^{+}= \hbox {max}\{0.3, 0.32\}=0.32; &{} t_{5}^{-}= \hbox {min}\{0.3, 0.32\}=0.3\\ t_{6}^{+}= \hbox {max}\{0.35, 0.36\}=0.36; &{} t_{6}^{-}= \hbox {min}\{0.35, 0.36\}=0.35\\ t_{7}^{+}= \hbox {max}\{0.4, 0.38\}=0.4 &{} t_{7}^{-}= \hbox {min}\{0.4, 0.38\}=0.38\\ t_{8}^{+}= \hbox {max}\{0.46, 0.48\}=0.48 &{} t_{8}^{-}= \hbox {min}\{0.46, 0.48\}=0.46.\\ \end{array} \end{aligned}$$

Then,

$$\begin{aligned}&\mu _{R_{8}}(X_{1},X_{2})=\frac{1}{2}\{[(0.12-0.48)+2(0.2-0.38)\\&\quad +3(0.24-0.36)+ 2(0.28-0.32)+2(0.3-0.27)\\&\quad +3(0.35-0.25)+2(0.4-0.15)+(0.46-0.1)]\\&\quad /(8\times 0.7175)]+1\}=0.5052>\frac{1}{2}. \end{aligned}$$

Therefore $X_{1}$ is preferred to $X_{2}.$

Similarly, $\mu _{R_{8}}(X_{1},X_{3})=0.4924<\frac{1}{2}$.

This implies $X_{3}$ is preferred to $X_{1}$ and $\mu _{R_{8}}(X_{2},X_{3})=0.4885<\frac{1}{2}.$ This implies $X_{3}$ is preferred to $X_{2}.$ Therefore $X_{3} \succ X_{1} \succ X_{2}.$

By extended relative preference relation,

$$\begin{aligned} {\overline{X}}= & {} (0.1067, 0.1733, 0.23, 0.2833, 0.3233, 0.3633, \\&0.3967, 0.48).\\ t_{q1}^{+}= & {} \hbox {max}\{0.12, 0.1,0.1\}=0.12\\ t_{q1}^{-}= & {} \hbox {min}\{0.12, 0.1,0.1\}=0.1\\ t_{q2}^{+}= & {} \hbox {max}\{0.2, 0.15,0.17\}=0.2\\ t_{q2}^{-}= & {} \hbox {min}\{0.2, 0.15,0.17\}=0.15 \end{aligned}$$

Similarly,

$$\begin{aligned}&t_{q3}^{+}=0.25;~~~~t_{q3}^{-}=0.2\\&t_{q4}^{+}=0.3;~~~~~~~t_{q4}^{-}=0.27\\&t_{q5}^{+}=0.35;~~~~~~~t_{q5}^{-}=0.3\\&t_{q6}^{+}=0.38;~~~~~~~t_{q6}^{-}=0.35\\&t_{q7}^{+}=0.41;~~~~~~~t_{q7}^{-}=0.38\\&t_{q8}^{+}=0.5;~~~~~~~t_{q8}^{-}=0.46 \end{aligned}$$

and

$$\begin{aligned}&\parallel T_{q} \parallel =[[(0.12-0.46)+2(0.2-0.38)+3(0.25-0.35)\\&\quad +2(0.3-0.3) +2(0.35-0.27) +3(0.38-0.2)\\&\quad +2(0.41-0.15)+(0.5-0.1)]/(2*4)]\\&\quad +2(0.46-0.12)=0.7175. \end{aligned}$$

Now,

$$\begin{aligned}&\mu _{R_{8}}^{*}(X_{1}, {\overline{X}})=\dfrac{1}{2}\{[[(0.12-0.48)+2(0.2-0.3967)\\&\quad +3(0.24-0.3633)\\&\quad +2(0.28-0.3233)+2(0.3-0.2833)\\&\quad +3(0.35-0.23) +2(0.4-0.1733)\\&\quad +(0.46-0.1067)]/(8*0.7175)]+1\}=0.4991. \end{aligned}$$

Similarly, we get $\mu _{R_{8}}^{*}(X_{2}, {\overline{X}})=0.4939$ and $\mu _{R_{8}}^{*}(X_{3}, {\overline{X}})=0.5070.$

As $\mu _{R_{8}}^{*}(X_{3}, {\overline{X}})>\mu _{R_{8}}^{*}(X_{1}, {\overline{X}})>\mu _{R_{8}}^{*}(X_{2}, {\overline{X}}),$ we get $X_{3}\succ X_{1} \succ X_{2}.$

The above result is compared with Wang and Lee (2008)’s method and according to this method the values are ${\overline{x}}(X_{1})=0.2954$, ${\overline{x}}(X_{2})=0.2853$ and ${\overline{x}}(X_{3})=0.2976.$

Hence we get $X_{3}\succ X_{1} \succ X_{2}.$

5 Fuzzy multi-criteria decision-making (FMCDM) model

In this section, we extend FMCDM model given by Wang (2014) using the relative preference relation on $2n+1$ fuzzy numbers.

In this algorithm, we consider the generalized $2n+1$ fuzzy numbers with $k=2^{n-1}$ and we denote them as $(a_{1}, a_{2},\ldots ,a_{2n+1})$ instead of $(a_{1}, a_{2},\ldots ,a_{2n+1};1).$

We take $E_{1},E_{2},\ldots ,E_{r}$ as the experts who provide their opinion on criteria $C_{1}, C_{2},\ldots ,C_{t}$ of the alternatives $A_{1},A_{2},\ldots ,A_{p}$. Let $B_{ijl}{=}(b_{ijl1},b_{ijl2},b_{ijl3},\ldots ,b_{ijl(2n{+}1)})$ be the evaluation rating given by the expert $E_{l}$ for alternative $A_{i}$ on criterion $C_{j},$ where $i=1,2,\ldots ,p$, $j=1,2,\ldots ,t$, $l=1,2,\ldots ,r.$

Then,

$$\begin{aligned}B_{ij}=(b_{ij1}, b_{ij2}, b_{ij3},\ldots ,b_{ij(2n+1)}),\end{aligned}$$

where

$$\begin{aligned}&b_{ij1}= \frac{1}{r}\sum \limits _{l=1}^{r}b_{ijl1}\\&b_{ij2}= \frac{1}{r}\sum \limits _{l=1}^{r}b_{ijl2}\\&b_{ij3}= \frac{1}{r}\sum \limits _{l=1}^{r}b_{ijl3}\\&\vdots \\&b_{ij(2n+1)}=\frac{1}{r}\sum \limits _{l=1}^{r}b_{ijl(2n+1)},~i=1,2,\ldots ,p,\\&j=1,2,\ldots ,t, n=1,2,3,...~ \hbox {and}~ n~\hbox { is finite.} \end{aligned}$$

The normalized value of $B_{ij}$ is denoted by ${\widetilde{B}}_{ij}$ and it is classified as follows.

If $B_{ij}$ belongs to cost criteria then

$$\begin{aligned} {\widetilde{B}}_{ij}=\left( \frac{b_{j1}^{-}}{b_{ij(2n+1)}}, \frac{b_{j1}^{-}}{b_{ij(2n)}}, \frac{b_{j1}^{-}}{b_{ij(2n-1)}},\ldots ,\frac{b_{j1}^{-}}{b_{ij2}}, \frac{b_{j1}^{-}}{b_{ij1}}\right) , \end{aligned}$$

where

$$\begin{aligned} b_{j1}^{-}=\min \limits _{i}\{b_{ij1}\},\forall ~j \end{aligned}$$

If $B_{ij}$ belongs to benefit criteria then

$$\begin{aligned} {\widetilde{B}}_{ij}\!=\!\!\left( \!\frac{b_{ij1}}{b_{j(2n+1)}^{+}}, \frac{b_{ij2}}{b_{j(2n+1)}^{+}}, \frac{b_{ij3}}{b_{j(2n+1)}^{+}},\ldots ,\frac{b_{ij(2n)}}{b_{j(2n+1)}^{+}}, \frac{b_{ij(2n+1)}}{b_{j(2n+1)}^{+}}\!\right) , \end{aligned}$$

where

$$\begin{aligned} b_{j(2n+1)}^{+}=\max \limits _{i}\{b_{ij(2n+1)}\},\forall ~j. \end{aligned}$$

Let $W_{jl}=(w_{jl1}, w_{jl2},\ldots ,w_{jl(2n+1)})$ be the weight of the criterion $C_{j}$ given by the expert $E_{l},$ where

$$\begin{aligned} j=1,2,\ldots ,t;~~ l=1,2,\ldots ,r. \end{aligned}$$

Then,

$$\begin{aligned} W_{j}=(w_{j1},w_{j2},\ldots ,w_{j(2n+1)}),\end{aligned}$$

where

$$\begin{aligned}&w_{j1}=\frac{1}{r}\sum \limits _{l=1}^{r}w_{jl1}\\&w_{j2}=\frac{1}{r}\sum \limits _{l=1}^{r}w_{jl2}\\&\vdots \\&w_{j(2n+1)}=\frac{1}{r}\sum \limits _{l=1}^{r}w_{jl(2n+1)},~j=1,2,\ldots ,t. \end{aligned}$$

The normalized values of $i^{th}$ alternative on t criteria are presented as follows.

$$\begin{aligned} A_{i}=[{\widetilde{B}}_{i1}, {\widetilde{B}}_{i2},\ldots ,{\widetilde{B}}_{it}],~~ i=1,2,\ldots ,p. \end{aligned}$$

Then, we find ideal and anti-ideal solutions according to as Wang et al. (2003).

The ideal solution is $A^{+}=[{\widetilde{B}}_{1}^{+}, {\widetilde{B}}_{2}^{+},\ldots ,{\widetilde{B}}_{t}^{+}],$ where ${\widetilde{B}}_{j}^{+}=(t_{qj1}^{+}, t_{qj2}^{+},\ldots ,t_{qj(2n+1)}^{+})$ is the best rating on jth criterion for all normalized values,  $j=1,2,\ldots ,t.$

The anti-ideal solution is $A^{-}=[{\widetilde{B}}_{1}^{-}, {\widetilde{B}}_{2}^{-},\ldots ,{\widetilde{B}}_{t}^{-}],$ where $B_{j}^{-}=(t_{qj1}^{-},t_{qj2}^{-},\ldots ,t_{qj(2n+1)}^{-})$ is the worst rating on jth criterion for all normalized values, $j=1,2,\ldots ,t.$

Now, $\mu _{R_{*}}({\widetilde{B}}_{j}^{+},{\widetilde{B}}_{ij})$ indicates the relative preference degree of $A^{+}$ over the alternative $A_{i}$ to the jth criterion,

$$\begin{aligned} ~i=1,2,\ldots ,p,~j=1,2,\ldots ,t. \end{aligned}$$

Table 1 Data given by experts about various criteria with respect to alternatives

Then,

$$\begin{aligned}&\mu _{R_{*}}({\widetilde{B}}_{j}^{+}, {\widetilde{B}}_{ij})=\frac{1}{2}\{[[(t_{qj1}^{+}-{\widetilde{b}}_{ij(2n+1)}) +2(t_{qj2}^{+}-{\widetilde{b}}_{ij(2n)})\\&\quad +3[(t_{qj3}^{+}-{\widetilde{b}}_{ij(2n-1)})+2(t_{qj4}^{+}-{\widetilde{b}}_{ij(2n-2)})\\&\quad +4(t_{qj5}^{+} {\widetilde{b}}_{ij(2n-3)})+\ldots +2^{n-3}(t_{qjn}^{+}-{\widetilde{b}}_{ij(n+2)})]\\&\quad +2^{n-1}(t_{qj(n+1)}^{+}- {\widetilde{b}}_{ij(n+1)})+3[2^{n-3}(t_{qj(n+2)}^{+}-{\widetilde{b}}_{ijn})\\&\quad +2^{n-4}(t_{qj(n+3)}^{+}-{\widetilde{b}}_{ij(n-1)})+\ldots +2(t_{qj(2n-2)}^{+} -{\widetilde{b}}_{ij4})\\&\quad +(t_{qj(2n-1)}^{+}-{\widetilde{b}}_{ij3})] +2(t_{qj(2n)}^{+} -{\widetilde{b}}_{ij2})\\&\quad +(t_{qj(2n+1)}^{+}-{\widetilde{b}}_{ij1})]/(2\times 2^{n-1} \parallel T_{q_{j}}\parallel )]+1\} \end{aligned}$$

and

$$\begin{aligned}&\mu _{R_{*}}({\widetilde{B}}_{ij}, {\widetilde{B}}_{j}^{-})=\frac{1}{2}\{[[({\widetilde{b}}_{ij1}-t_{qj(2n+1)}^{-}) +2({\widetilde{b}}_{ij2}-t_{qj(2n)}^{-})\\&\quad +3[({\widetilde{b}}_{ij3}-t_{qj(2n-1)}^{-}) +2({\widetilde{b}}_{ij4}-t_{qj(2n-2)}^{-})\\&\quad +4({\widetilde{b}}_{ij5}-t_{qj(2n-3)}^{-}) +\cdots +2^{n-3}({\widetilde{b}}_{ijn}-t_{qj(n+2)}^{-})]\\&\quad +2^{n-1}({\widetilde{b}}_{ij(n+1)} -t_{qj(n+1)}^{-})+3[2^{n-3}({\widetilde{b}}_{ij(n+2)}-t_{qjn}^{-})\\&\quad + 2^{n-4}({\widetilde{b}}_{ij(n+3)} -t_{qj(n-1)}^{-})+\cdots +2 ({\widetilde{b}}_{ij(2n-2)}-t_{qjj4}^{-})\\&\quad +({\widetilde{b}}_{ij(2n-1)}-t_{qj3}^{-})] +2({\widetilde{b}}_{ij(2n)}-t_{qj2}^{-})\\&\quad +({\widetilde{b}}_{ij(2n+1)}-t_{qj1}^{-})] /(2\times 2^{n-1}\parallel T_{q_{j}}\parallel )]+1\} \end{aligned}$$

indicates relative preference degree of alternative $A_{i}$ over $A^{-}$ to the jth criterion, where

$$\begin{aligned}&\parallel T_{q_{j}}\parallel =\frac{1}{2}[(t_{qj1}^{+}-t_{qj(2n+1)}^{-}) +2(t_{qj2}^{+}-t_{qj2n}^{-})\\&\quad +3[(t_{qj3}^{+}-t_{qj(2n-1)}^{-})+2(t_{qj4}^{+}- t_{qj(2n-2)}^{-})\\&\quad +4(t_{qj5}^{+}-t_{qj(2n-3)}^{-})+\ldots +2^{n-3}(t_{qjn}^{+} -t_{qj(n+2)}^{-})]\\&\quad +2^{n-1}(t_{qj(n+1)}^{+}-t_{qj(n+1)}^{-})+3[2^{n-3} (t_{qj(n+2)}^{+}-t_{qjn}^{-})\\&\quad +2^{n-4}(t_{qj(n+3)}^{+}-t_{qj(n-1)}^{-}) +\cdots +2(t_{qj(2n-2)}^{+}-t_{qj4}^{-})\\&\quad +(t_{qj(2n-1)}^{+}-t_{qj3}^{-})] +2(t_{qj(2n)}^{+}-t_{qj2}^{-})\\&\quad +(t_{qj(2n+1)}^{+}-t_{qj1}^{-})]/2k ~\hbox {if}~ (t_{qj1}^{+}-t_{qj(2n+1)}^{-})\ge 0. \end{aligned}$$

$$\begin{aligned}&\frac{1}{2}[[(t_{qj1}^{+}-t_{qj(2n+1)}^{-})+2(t_{qj2}^{+}-t_{qj2n}^{-}) +3[(t_{qj3}^{+}-t_{qj(2n-1)}^{-})\\&\quad +2(t_{qj4}^{+}-t_{qj(2n-2)}^{-})+4(t_{qj5}^{+} -t_{qj(2n-3)}^{-})\\&\quad +\cdots +2^{n-3}(t_{qjn}^{+}-t_{qj(n+2)}^{-})]+2^{n-1}(t_{qj(n+1)} ^{+}-t_{qj(n+1)}^{-})\\&\quad +3[2^{n-3}(t_{qj(n+2)}^{+}-t_{qjn}^{-})+2^{n-4}(t_{qj(n+3)} ^{+}-t_{qj(n-1)}^{-})\\&\quad +\cdots +2(t_{qj(2n-2)}^{+}-t_{qj4}^{-})\\&\quad +(t_{qj(2n-1)}^{+}-t_{qj3}^{-})] +2(t_{qj(2n)}^{+} -t_{qj2}^{-})\\&\quad +(t_{qj(2n+1)}^{+}-t_{qj1}^{-})]/2k] +2(t_{qj(2n+1)}^{-}-t_{qj1}^{+})~\\&\quad \hbox { if}~ (t_{qj1}^{+}-t_{qj(2n+1)}^{-})< 0, \end{aligned}$$

Table 2 Linguistic terms of feedback and corresponding fuzzy numbers

Table 3 Linguistic terms of weights of criteria and corresponding fuzzy numbers

Table 4 represents linguistic terms of weights of criteria given by experts Tables 5, 6 and 7 provide related calculations.

Table 4 Linguistic terms of weights of criteria

Table 5 Fuzzy group decision matrix

Table 6 Normalized group decision matrix

where

$$\begin{aligned}&t_{qj1}^{+}=\max \limits _{i}\{{\widetilde{b}}_{ij1}\}\\&t_{qj2}^{+}=\max \limits _{i}\{{\widetilde{b}}_{ij2}\}\\&\vdots \\&t_{qj(2n+1)}^{+}=\max \limits _{i}\{{\widetilde{b}}_{ij(2n+1)}\} \end{aligned}$$

and

$$\begin{aligned}&t_{qj1}^{-}=\min \limits _{i}\{{\widetilde{b}}_{ij1}\}\\&t_{qj2}^{-}=\min \limits _{i}\{{\widetilde{b}}_{ij2}\}\\&\vdots \\&t_{qj(2n+1)}^{-}=\min \limits _{i}\{{\widetilde{b}}_{ij(2n+1)}\},~~~i=1,2,\ldots ,p. \end{aligned}$$

$D_{i}^{+}$ is derived by using $\mu _{R_{*}}({\widetilde{B}}_{j}^{+}, {\widetilde{B}}_{ij})$ and $W_{j},~j=1,2,\ldots ,t$ and indicates the weighted preference degree of $A^{+}$ over the alternative $A_{i}$ and $D_{i}^{-}$ is derived by using $\mu _{R_{*}}({\widetilde{B}}_{ij},{\widetilde{B}} _{j}^{-})$ and $W_{j},~j=1,2,\ldots ,t$ and indicates the weighted preference degree of $A_{i}$ over $A^{-}$ for $i=1,2,\ldots ,p.$

Now,

$$\begin{aligned} D_{i}^{+}&=(W_{1}\mu _{R_{*}}({\widetilde{B}}_{1}^{+}, {\widetilde{B}}_{i1}))+(W_{2}\mu _{R_{*}}({\widetilde{B}}_{2}^{+}, {\widetilde{B}}_{i2}))+\cdots \\&\quad +(W_{t}\mu _{R_{*}}({\widetilde{B}}_{t}^{+}, {\widetilde{B}}_{it})) \end{aligned}$$

and $D_{i}^{-}=(W_{1}\mu _{R_{*}}({\widetilde{B}}_{i1}, {\widetilde{B}}_{1}^{-}))+(W_{2}\mu _{R_{*}}({\widetilde{B}}_{i2}, {\widetilde{B}}_{2}^{-}))+\cdots +(W_{t}\mu _{R_{*}}({\widetilde{B}}_{it}, {\widetilde{B}}_{t}^{-})),~~i=1,2,\ldots ,p.$

Then, we get $D_{i}^{+}$ and $D_{i}^{-}$ as $2n+1$ fuzzy numbers and are denoted by

$$\begin{aligned}&D_{i}^{+}=(d_{i1}^{+}, d_{i2}^{+},\ldots ,d_{i(2n+1)}^{+}) ~\hbox {and}~\\&D_{i}^{-}=(d_{i1}^{-}, d_{i2}^{-},\ldots ,d_{i(2n+1)}^{-}), i=1,2,\ldots ,p. \end{aligned}$$

Then,

$$\begin{aligned}&{{\overline{D}}}^{+}=({\overline{d}}_{1}^{+},\\&{\overline{d}}_{2}^{+},\ldots ,{\overline{d}}_{2n+1}^{+})~\hbox { and}\\&{\overline{D}}^{-}=({\overline{d}}_{1}^{-},\\&{\overline{d}}_{2}^{-},\ldots ,{\overline{d}}_{2n+1}^{-}), \end{aligned}$$

Table 7 Fuzzy weights of criteria

Table 8 Ideal and anti-ideal solutions of internet service providers

Table 9 Relative preference degree of ideal solution over internet providers on criteria

Table 10 Relative preference degree of internet service providers over anti-ideal solution on criteria

Table 11 Weighted preference degree of ideal solution over internet service providers on criteria

Table 12 Weighted preference degree of internet service providers over anti-ideal solution on criteria

where

$$\begin{aligned}&{\overline{d}}_{1}^{+}=\frac{1}{p}\sum \limits _{i=1}^{p}d_{i1}^{+}\\&{\overline{d}}_{2}^{+}=\frac{1}{p}\sum \limits _{i=1}^{p}d_{i2}^{+}\\&{\overline{d}}_{3}^{+}=\frac{1}{p}\sum \limits _{i=1}^{p}d_{i3}^{+}\\&\vdots \\&{\overline{d}}_{2n+1}^{+}=\frac{1}{p}\sum \limits _{i=1}^{p}d_{i(2n+1)}^{+} \end{aligned}$$

and

$$\begin{aligned}&{\overline{d}}_{1}^{-}=\frac{1}{p}\sum \limits _{i=1}^{p}d_{i1}^{-}\\&{\overline{d}}_{2}^{-}=\frac{1}{p}\sum \limits _{i=1}^{p}d_{i2}^{-}\\&\vdots \\&{\overline{d}}_{2n+1}^{-}=\frac{1}{p}\sum \limits _{i=1}^{p}d_{i(2n+1)}^{-}. \end{aligned}$$

Now, the relative closeness coefficient $D_{i}$ of the alternative $A_{i}$ is defined as

$$\begin{aligned} D_{i}=\frac{\mu _{R_{*}}(D_{i}^{-}, {\overline{D}}^{-})}{\mu _{R_{*}}(D_{i}^{-}, {\overline{D}}^{-})+\mu _{R_{*}}(D_{i}^{+}, {\overline{D}}^{+})},~i=1,2,...p,\end{aligned}$$

where $\mu _{R_{*}}(D_{i}^{-}, {\overline{D}}^{-})$ is the relative preference degree of $D_{i}^{-}$ over ${\overline{D}}^{-}$ and $\mu _{R_{*}}(D_{i}^{+}, {\overline{D}}^{+})$ is the relative preference degree of $D_{i}^{+}$ over $D^{+}.$

Clearly, $D_{i},~i=1,2,\ldots ,p$ is in the interval [0, 1]. The bigger the value $D_{i},~i=1,2,\ldots ,p$ is closer the ideal solution is. Alternatively, smaller the value $D_{i},~i=1,2,\ldots ,p$ is closer the anti-ideal solution is. Thus, the P alternatives are ranked according to their relative closeness coefficients $D_{1}, D_{2},\ldots ,D_{p}.$

Analysis of computational complexity

The computational complexity of the proposed algorithm is O(mn), where m is the number of alternatives with n criteria. Like in the other FMCDM methods, it is assumed that the number of experts is a constant for the purpose of computational complexity. If $m=O(n)$, then the algorithm takes quadratic $(O(n^{2}))$ time. In general, FMCDM methods have O(mn) computational complexity. This coincides with the complexity of the algorithm given in this paper.

Example 5.1

In real-world conditions, selection of an internet service provider is often based on various criteria such as Monthly cost ($C_{5}$) as Cost criteria and Volume of data ($C_{1}$), Speed of internet ($C_{2}$), Subscription to OTT platforms ($C_{3}$), dependable customer service ($C_{4}$) as benefit criteria. If the available internet service providers ($A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$) are assessed on this criterion by four different experts ($E_{1}$, $E_{2}$, $E_{3}$, $E_{4}$) with each expert considering one of the benefit criteria to be more important than others, then there will be a dilemma in the selection as each service provider may have at least one parameter in which they are superior to other internet service providers. In such situations, we provide a solution to arrive at a decision as follows.

Table 1 represents the consolidated information about the performance of various internet service providers ($A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$) with respect to various selection criteria ($C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$, $C_{5}$) by various experts ($E_{1}$, $E_{2}$, $E_{3}$, $E_{4}$) wherein the feedback on benefit criteria is expressed as Very Good (VG), Good (G), Average (A), Poor (P) or Very Poor (VP). The feedback on cost criteria, on the other hand, is expressed as numerical values (for example, here it represents price of the service).

Linguistic terms used for expressing the feedback on benefit criteria are assigned with fuzzy numbers as shown in Table 2.

Table 3 represents weights of criteria and their corresponding fuzzy numbers.

Ideal and anti-ideal solutions of 4 internet service providers are presented in Table 8. Subsequent calculations are shown in Tables 9, 10, 11, 12, 13, 14, 15, 16.

From Table 17, the relative closeness coefficients of 4 internet service providers are $A_{1}: 0.4993$, $A_{2}: 0.5767,$ $A_{3}: 0.3607$ and $A_{4}: 0.5624$ and hence the rank order of internet service providers is $A_{2}>A_{4}>A_{1}>A_{3}.$ This indicates that the internet service provider $A_{2}$ is better compared to other internet service providers.

6 Validation with existing methods

Wang (2014) proposed an algorithm to give decision in a triangular FMCDM using relative preference relation along with TOPSIS method. In the present study, we extended Wang’s method to arrive at a decision in FMCDM problems involving $2n +1$ fuzzy numbers. Similar extension is possible for FMCDM problems involving 2n fuzzy numbers.

The suitability of the proposed method was verified by comparing it with the popularly used multi-criteria decision-making methods such as VIKOR, MOORA and ELECTRE. Prior to the application of these methods, alternative values of various criteria given by experts and weights of criteria from the above example are converted to crisp values by centroid defuzzification method.

1.VIKOR: Following table of values is obtained by applying VIKOR method (Opricovic (1998, 2002)).

	$A_{1}$	$A_{2}$	$A_{3}$	$A_{4}$
S	1.5384	0.3488	2.3193	0.5928
R	0.6950	0.2182	0.6950	0.4504
Q	0.8019	0	1	0.3054

From the above table, rank order of the alternatives is $A_{2}>A_{4}>A_{1}>A_{3}.$

2. Multi-objective Optimization by Ratio analysis (MOORA): By applying MOORA reference point method ( Brauers and Zavadskas (2006, 2010)), deviations from the reference points and ranking of the alternatives are presented in the following table.

	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$	max. value	Ranking
$A_{1}$	0	0.0386	0.11	0.0682	0	0.11	3
$A_{2}$	0.0527	0	0	0	0.0404	0.0527	1
$A_{3}$	0.1329	0.0793	0.0535	0.0211	0.1111	0.1329	4
$A_{4}$	0.0775	0.0203	0.0253	0.0235	0.005	0.0775	2

3. ELECTRE: By using ELECTRE method (Roy (1991)), following global matrix is obtained.

From the global matrix, it is clear that $A_{2}>A_{3}>A_{1}$ and $A_{4}>A_{3}>A_{1}$ which can also be expressed as $A_{2}\ge A_{4}>A_{3}>A_{1}$.

The rank order of alternatives obtained from all three methods indicates the proposed method’s suitability in FMCDM problems. The proposed method involves less operational complexity, it is easy to compute, and it minimizes the loss of information.

7 Conclusions

We have defined $2n+1$ and 2n fuzzy numbers as generalizations of triangular and trapezoidal fuzzy numbers, respectively. Fuzzy preference relation and relative preference relation of Wang (2015b) are extended to rank 2n and $2n+1$ fuzzy numbers and the results are compared with Wang and Lee (2008) method. Wang (2014)’s method was extended to arrive at a decision in FMCDM problems when the given data is in terms of $2n+1$ fuzzy numbers. An illustrative example was provided to explain the suitability of the proposed method, and the results were validated using VIKOR, MOORA and ELECTRE methods. The proposed method can also be extended to 2n fuzzy numbers.

Table 13 Weighted preference degrees of ideal solution over internet service providers and average

Table 14 Weighted preference degrees of internet service providers over anti-ideal solution and average

Table 15 Relative preference degrees $\mu _{P}^{*}(D_{i}^{+}, {\overline{D}}^{+})$

Table 16 Relative preference degrees $\mu _{P}^{*}(D_{i}^{-}, {\overline{D}}^{-})$

Table 17 Relative closeness coefficients of internet service providers