1 Introduction

The notion of fuzzy sets [1, 2] was introduced by Zadeh (1965) to deal with the imprecision due to uncertainty in real-world problems. Fuzzy sets [3] were generalized into Intuitionistic fuzzy sets [4], Pythagorean fuzzy sets [5], Picture fuzzy sets [6], Bi-polar fuzzy sets [7] and Neutrosophic sets [8] based on their necessity to deal with qualitative and quantitative information in various research scenarios. Theory of fuzzy numbers has given another dimension to model the various fuzzy uncertainties and vagueness in any real-time applications involving decision making, clustering and control. Various types of membership functions for fuzzy numbers such as triangular, trapezoidal, hexagonal, octagonal, nonagonal [9], decagonal, parabolic and quadratic fuzzy numbers have been proposed by many researchers to represent uncertain information. The theory of fuzzy numbers [10] has a huge role in information fusion [11], MCDM problem [12], assignment problems [13, 14] and inventory management problems [15].

There are many research papers present in the literature for applications and ranking of the interval valued fuzzy number [16, 17], fermatian fuzzy numbers [18] and intuitionistic fuzzy numbers [19]. The major area in fuzzy MCDM problems depending upon the total ordering algorithm of linear fuzzy numbers. There are many ranking methods in the literature for ‘n’-sided linear fuzzy numbers such as trapezoidal fuzzy numbers, and GTPFN  [20,21,22,23,24,25] . The ordering principle for GTPFN also satisfies the GTRFNs, since GTPFN is a generalized version of GTRFN. Some real-life applications based on pentagonal fuzzy numbers have been studied in [26]. The major subdivision of fuzzy numbers are hexagonal fuzzy numbers [15, 27,28,29] and octagonal fuzzy numbers [30,31,32,33], which are ranked based on different parameters such as centroid, value, ambiguity, rank index, score functions, upper dense sequence, etc. Some researchers work with approximation theory to deal with the fuzzy number-based multi-criteria decision making (MCDM) problems [34]. The reliability of ranking methods in MCDM problems based on the score functions on a particular class of fuzzy numbers is improved when they posses total ordering on the same class of fuzzy numbers because of the ability to differentiate the alternatives properly. So, the total ordering methods on various types of fuzzy numbers plays a huge role in MCDM problems [35,36,37].

Any fuzzy MCDM problem can be solved in two steps: (1) determining the overall effectiveness of each alternative in terms of all criteria; (2) ranking of alternatives based on its overall performances. In fuzzy MCDM problems, the cumulative performance of alternatives is evaluated as fuzzy numbers by defining appropriate operators on fuzzy numbers. Hence, to rank those alternatives, we need to differentiate them by defining total ordering on the same class of fuzzy numbers. As mentioned earlier, there are many types of fuzzy numbers available and each fuzzy numbers posses various total ordering methods in the literature [27,28,29,30,31,32,33,34]. The accuracy of experiment involving nonlinear fuzzy numbers is increased when higher-order piecewise linear fuzzy numbers are used to model nonlinear information. Hence, in this paper, a new notion of generalized ‘n’ gonal linear fuzzy numbers which generalizes all triangular, trapezoidal, hexagonal, octagonal and decagonal fuzzy numbers is introduced and the ranking of generalized ‘n’ gonal linear fuzzy numbers is studied. This study presents a novel total ordering method on the class of generalized ‘n’ gonal linear fuzzy numbers which generalizes all existing total ordering methods on triangular, trapezoidal, hexagonal, octagonal and decagonal fuzzy numbers. If a total ordering on hexagonal fuzzy numbers is required, it is enough to substitute \(n=6\) in ‘n’ gonal total ordering method, and if a total ordering on octagonal fuzzy numbers is required, it is enough to substitute \(n=8\) in ‘n’ gonal total ordering method through which we can even take any higher number based on the MCDM problem experiment. Hence, we are able to order totally any class of higher-order linear fuzzy numbers by the choice of appropriate ‘n’ based on MCDM problems which gives better solutions.

2 Preliminaries

Definition 1

(Generalized triangular fuzzy number (GTRFN)) [29] A triplet \(T=((a_1, a_2, a_3); u, \mu _{T})\), \(a_1 \le a_2 \le a_3, a_1, a_2, a_3, \in \mathbb {R}\), \(u \in [0, 1]\) with the membership function

$$\begin{aligned} \mu _T(x)=\left\{ \begin{array}{lcc} &{}u(\frac{x-a_1}{a_2-a_1}), &{} for \quad a_1 \le x \le a_2\\ &{}u, &{} for \quad x=a_2\\ &{}u(\frac{a_3-x}{a_3-a_2}), &{} for \quad a_2 \le x \le a_3\\ &{}0, &{}otherwise \end{array}\right. \end{aligned}$$

is called generalized triangular fuzzy number.

Definition 2

(Generalized trapezoidal fuzzy number (GTPFN)) [38] A quadruple \(T=((a_1, a_2, a_3, a_4); \, u, \mu _{T})\), \(a_1 \le a_2 \le a_3 \le a_4, a_1, a_2, a_3, a_4 \in \mathbb {R}\), \(u \in [0, 1]\) with the membership function

$$\begin{aligned} \mu _T(x)=\left\{ \begin{array}{lcc} &{}u(\frac{x-a_1}{a_2-a_1}), &{} for \quad a_1 \le x \le a_2\\ &{}u, &{} for \quad a_2 \le x \le a_3\\ &{}u(\frac{a_4-x}{a_4-a_3})(x), &{} for \quad a_3 \le x \le a_4\\ &{}0, &{}otherwise \end{array}\right. \end{aligned}$$

is called generalized trapezoidal fuzzy number. (See Fig. 1)

A generalized trapezoidal fuzzy number T becomes a generalized triangular fuzzy number (GTRFN) T when \(a_2=a_3\) and it is denoted by triplet \(T=((a_1, a_2, a_3); u, \mu _{T})\). In case \(u = 1\), it is a trapezoidal fuzzy number and is denoted by \(T = (a_1, a_2, a_3, a_4)\). If \(a_2 = a_3\), then T is a generalized triangular fuzzy number, denoted by triplet \(T=((a_1, a_2, a_3); u, \mu _{T})\). If \(a_2 = a_3\) and \(u = 1\), then T is a triangular fuzzy number (TRFN) which is denoted by \(T = (a_1, a_2, a_3)\). Suppose \(a_1 = a_2\) and \(a_3 = a_4\), then T is a generalized interval fuzzy number denoted by \(T = (a_1, a_4;\, u)\). If \(a_1 = a_2\) and \(a_3 = a_4\) and \(u = 1\), then T is a crisp interval number, denoted by \(T = (a_1, a_4)\). If \(a_1 = a_2 = a_3 = a_4\), then T is a generalized real number, denoted by \(T = (a_1; u)\). If \(a_1 = a_2 = a_3 = a_4\) and \(u = 1\), then T is a real number and is denoted as \(T = a_1\) .

Fig. 1
figure 1

Generalized trapezoidal number

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Definition 3

(Generalized hexagonal fuzzy number (GHXFN)) [29] A fuzzy subset H of \(\mathbb {R}\) of the form \(H = ((h_{1} ,h_{2} ,h_{3} ,h_{4} ,h_{5} ,h_{6} );{\mkern 1mu} u,u_{L} ,u_{R} ,M_{H} ),h_{1} \le h_{2} \le h_{3} \le h_{4} \le h_{5} \le h_{6} ,h_{i} \in \mathbb{R}\{ i = 1,2, \ldots 6\} ,u,u_{L} ,u_{R} \in [0,1]\) with the membership function

$$\begin{aligned} M_H(x) = \left\{{ \begin{array}{lll} u_L\left( \frac{x-h_1}{h_2-h_1}\right) , &{} if &{} h_1 \leqslant x \leqslant h_2 \\ u_L+(u-u_L)\left( \frac{x-h_2}{h_3-h_2}\right) , &{} if &{} h_2 \leqslant x \leqslant h_3 \\ u, &{} if &{} h_3 \leqslant x \leqslant h_4\\ u_R+(u-u_R)\left( \frac{h_5-x}{h_5-h_4}\right) , &{} if &{} h_4 \leqslant x \leqslant h_5 \\ u_R\left( \frac{h_6-x}{h_6-h_5}\right) , &{} if &{} h_5 \leqslant x \leqslant h_6 \\ 0, &{} &{} otherwise\\ \end{array}} \right. \end{aligned}$$

is called generalized hexagonal fuzzy number. (See Fig. 2)

A generalized hexagonal fuzzy number is in the shape of the hexagon whose vertices are \((h_1, 0), (h_2, u_L), (h_3, u),\) \((h_4, u)\), \((h_5, u_R)\) and \((h_6, 0)\) and hence it is denoted by \(((h_1, h_2, h_3, h_4, h_5, h_6); u, u_L, u_R, M_H)\). In the GHXFN H, \(u_L, u_R\) are heights of the left lower leg and the right lower leg of H,  respectively, and u is the height of H. If \(u = 1\), then H is called hexagonal fuzzy number (HFN) and is denoted by \(H=((h_1, h_2,h_3, h_4, h_5, h_6);1, u_L, u_R, M_H)\).

A GHXFN H is called generalized pentagonal fuzzy number (GPeFN) and is denoted by \(H=((h_1, h_2, h_4, h_5, h_6); u, u_L, u_R, M_H),\) when \(h_3\) and \(h_4\) are equal. When u attains the value of 1, H is called pentagonal fuzzy number (PeFN) and is denoted by \(H=((h_1, h_2, h_4, h_5, h_6);1, u_L, u_R, M_H).\)

Fig. 2
figure 2

Generalized hexagonal fuzzy number

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2.1 Motivation and Contribution

Fig. 3
figure 3

Nonlinear fuzzy number

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Fig. 4
figure 4

Nonlinear fuzzy number approximated by GPeFN

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Fig. 5
figure 5

Nonlinear fuzzy number approximated by GHXFN

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Fig. 6
figure 6

Nonlinear fuzzy number approximated by GONFN

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Fuzzy decision-making problems depend on qualitative and imprecise information which is better modeled in the form of nonlinear fuzzy numbers. The input and outputs are sometimes in the form of nonlinear fuzzy numbers [39] as shown in Fig. 3. In such cases, researchers approximate each nonlinear fuzzy number by a suitable linear fuzzy number to find the ranking between the alternatives. Then, by applying MCDM methods for linear fuzzy numbers, the objective is achieved. In Fig. 4, nonlinear fuzzy number shown in Fig. 3 is approximated by a generalized pentagonal fuzzy number which is a piecewise linear function. Similarly, in Figs. 5 and 6, nonlinear fuzzy number shown in Fig. 3 is approximated by a generalized hexagonal fuzzy number and octagonal fuzzy number, respectively. By approximating the nonlinear fuzzy number by a GPeFN (Fig. 4) in an MCDM problem, we can find the resultant ranking. But, this approximation has more data loss compabred to the approximation of the same nonlinear fuzzy number (Fig. 3) by GHXFN (Fig. 5). Similarly, approximation of nonlinear fuzzy number by GHXFN has more data loss, compared to approximating the same nonlinear fuzzy number(Fig. 3)by GONFN (Fig. 6). So, to get a better result, we can work with the approximation shown in 6. From these discussions, we obtain that when we approximate a nonlinear fuzzy number by a linear ‘n’-sided fuzzy number, the loss of data may be decreasing by the choice of appropriate ‘n’. Hence it is understood that more generalized (GTRFN is more generalized than GTFN, GPeFN is more generalized than GTRFN and GHXFN is more generalized than GPeFN ) linear fuzzy number plays a vital role in the approximation of nonlinear fuzzy number.

Therefore, the requirement of total ordering technique in the classes of higher-order generalized ‘n’-sided linear fuzzy numbers is unavoidable in this area of study. In this context, a total ordering algorithm on generalized ‘n’gonal linear fuzzy numbers is needed. Once we attain a total ordering on generalized ‘n’ gonal linear fuzzy numbers, we are able to solve any MCDM problem based on generalized ‘n’-sided linear fuzzy numbers and as well as we get a beneficiary tool to handle an MCDM problem where the ratings are in different forms of fuzzy numbers such as GTRFN, GTPN, GPeFN, GHXFN and HXFN. Thus, we need a generalized ‘n’ gonal fuzzy number and its total ordering algorithm that motivate this study.

In this paper, the notion of ‘n’ gonal linear fuzzy number is defined and some operations through which aggregations of ‘n’gonal linear fuzzy informations in MCDM have been discussed as extensions of operations on GHXFNs in Sect. 3. Further, a total ordering on ’n’ gonal linear fuzzy number is discussed in detail in Sect. 4 with applications to the MCDM in Sect. 5.1. Similarity measure on ’n’ gonal linear fuzzy number is developed with applications in Sect. 5.2. The significance of the proposed total ordering method by comparing other ranking methods in the literature is studied in Sect. 6. In literature, total ordering methods for different forms of fuzzy numbers such as GTRFN, GTPN, GPeFN, GHXFN and HXFN are discussed and it is understood that each method has its own merits and demerits. So, the proposed method is applicable in an MCDM problem with all of those forms. The limitations of the proposed research and future scope are discussed in Sect. 7 with conclusions in Sect. 8.

3 Generalized ‘n’ Gonal Linear Fuzzy Number

In this section, generalized ‘n’ gonal linear fuzzy number is defined and basic arithmetic operations on the class of generalized ‘n’ gonal linear fuzzy number for applying to MCDM problem are discussed.

Remark 1

Since the class of triangular fuzzy numbers is a subclass of trapezoidal fuzzy numbers by considering triangular fuzzy number as trapezoidal fuzzy number and any class of pentagonal fuzzy numbers is a subclass of hexagonal fuzzy number by considering pentagonal fuzzy number as hexagonal fuzzy number, it is noted that any class of odd order linear fuzzy number is a subclass of even order linear fuzzy numbers by considering odd order linear fuzzy number as an even order linear fuzzy number. Further, any generalized ‘n’gonal linear fuzzy number for any \(n < 6\) can be identified by a generalized Hexagonal fuzzy number, and we define generalized ‘n’gonal linear fuzzy number for \(n \ge 6\). Hence in this section, we define only generalized ‘n’gonal linear fuzzy number for even ‘n’ which includes odd order also.

Definition 4

(Generalized ‘n’gonal linear fuzzy number (GNLFN)) Let \(n \ge 6\) be an even number. A fuzzy subset H of \(\mathbb {R}\) denoted by \(H=((h_1, h_2, \dots h_n);\, u, u_{L_1}, u_{L_2}, \ldots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \ldots u_{R_{\frac{n-4}{2}}} ),\) where \(h_1\le h_2 \le \ldots \le h_n, h_j \in \mathbb {R}\) for \(j=1, 2, \cdots n, u, u_{L_i}, u_{R_i} \in [0, 1]\) for \(i\in 1,2,\ldots \frac{n-4}{2}\) is called generalized ‘n’gonal linear fuzzy number if the membership function is defined as

$$\begin{aligned} M_H(x) = \left\{ \begin{array}{lll} u_{L_{\frac{n-4}{2}}}\left( \frac{x-h_1}{h_2-h_1}\right) , &{} if &{} h_1 \leqslant x \leqslant h_2 \\ u_{L_{\frac{n-4}{2}}}+(u- u_{L_{\frac{n-4}{2}}})\left( \frac{x-h_2}{h_3-h_2}\right) , &{} if &{} h_2 \leqslant x \leqslant h_3 \\ u_{L_{\frac{n-6}{2}}}+(u_{L_{\frac{n-4}{2}}}-u_{L_{\frac{n-6}{2}}})\left( \frac{x-h_3}{h_4-h_4}\right) , &{} if &{} h_3 \leqslant x \leqslant h_4 \\ \vdots \\ u_{L_2}+(u_{L_2}-u_{L_1})\left( \frac{h_{\frac{n}{2}-1}-x}{h_{\frac{n}{2}-1}-h_{\frac{n}{2}-2}}\right) , &{} if &{} h_{\frac{n}{2}-2} \leqslant x \leqslant h_{\frac{n}{2}-1} \\ u_{L_1}+(u_{L_1}-u)\left( \frac{h_{\frac{n}{2}}-x}{h_{\frac{n}{2}}-h_{\frac{n}{2}-1}}\right) , &{} if &{} h_{\frac{n}{2}-1} \leqslant x \leqslant h_{\frac{n}{2}} \\ u, &{} if &{} h_{\frac{n}{2}} \leqslant x \leqslant h_{\frac{n}{2}+1}\\ u_{R_1}+(u-u_{R_1})\left( \frac{h_{\frac{n}{2}+1}-x}{h_{\frac{n}{2}+2}-h_{\frac{n}{2}+1}}\right) , &{} if &{} h_{\frac{n}{2}+1} \leqslant x \leqslant h_{\frac{n}{2}+2} \\ u_{R_2}+(u_{R_1}-u_{R_2})\left( \frac{h_{\frac{n}{2}+2}-x}{h_{\frac{n}{2}+3}-h_{\frac{n}{2}+2}}\right) , &{} if &{} h_{\frac{n}{2}+2} \leqslant x \leqslant h_{\frac{n}{2}+3} \\ \vdots \\ u_{R_{\frac{n-4}{2}}}\left( \frac{h_n-x}{h_n-h_{n-1}}\right) , &{} if &{} h_{n-1} \leqslant x \leqslant h_n \\ 0, &{} &{} otherwise\\ \end{array} \right. . \end{aligned}$$

A GNLFN \(((h_1, h_2, \dots h_n);\, u, u_{L_1}, u_{L_2}, \ldots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \ldots u_{R_{\frac{n-4}{2}}} )\) is in the shape of the ‘n’gon, whose vertices are \((h_1, 0), (h_2, u_{L_{\frac{n-4}{2}}}), (h_3, u_{L_{\frac{n-6}{2}}}),\ldots\) \((h_{\frac{n}{2}-1}, u_{L_1}),(h_{\frac{n}{2}}, u)\), \((h_{\frac{n}{2}+1}, u)\), \((h_{\frac{n}{2}+2}, u_{R_1}), (h_{\frac{n}{2}+3}, u_{R_2}),\ldots\) \((h_{n-1}, u_{R_{\frac{n-4}{2}}}),(h_n, 0)\).

In the GNLFN H, \(u_{L_i}, u_{R_i}\) are heights of the left legs and the right legs of H,  respectively, and u is the height of H (Fig. 7).

Fig. 7
figure 7

Generalized ‘N’ gonal linear fuzzy number

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Remark 2

Any ‘n’gonal linear fuzzy number for any \(n\le 4\) can be defined in the same way with \(u = u_{L_{-1}} = u_{L_{0}} = u_{R_{-1}} = u_{R_{0}}.\)

Now, we apply \(n=8\) in the above definition that results in the definition of octagonal fuzzy number defined in [28] as a particular case of GNLFN which is given below.

Definition 5

(Generalized octagonal fuzzy number)[28] A fuzzy subset H of \(\mathbb {R}\) of the form \(H=((h_1, h_2, \dots h_8); u, u_{L_1}, u_{L_2}, u_{R_1}, u_{R_2}),\) where \(h_1\le h_2 \le \ldots \le h_8, h_i, i=1, 2, \cdots 8 \in \mathbb {R}, u, u_{L_i}, u_{R_i} \in [0, 1]\), \(i\in {1,2}\) is called generalized octagonal fuzzy number whose membership function is defined by

$$\begin{aligned} M_H(x) = \left\{ \begin{array}{lll} u_{L_2}\left( \frac{x-h_1}{h_2-h_1}\right) , &{} if &{} h_1 \leqslant x \leqslant h_2 \\ u_{L_2}+(u- u_{L_2})\left( \frac{x-h_2}{h_3-h_2}\right) , &{} if &{} h_2 \leqslant x \leqslant h_3 \\ u_{L_1}+(u_{L_2}-u_{L_1})\left( \frac{x-h_3}{h_4-h_4}\right) , &{} if &{} h_3 \leqslant x \leqslant h_4 \\ u, &{} if &{} h_{4} \leqslant x \leqslant h_{5}\\ u_{R_1}+(u-u_{R_1})\left( \frac{h_{5}-x}{h_{6}-h_{5}}\right) , &{} if &{} h_{5} \leqslant x \leqslant h_{6} \\ u_{R_2}+(u_{R_1}-u_{R_2})\left( \frac{h_{6}-x}{h_{7}-h_{6}}\right) , &{} if &{} h_{6} \leqslant x \leqslant h_{7} \\ u_{R_{2}}\left( \frac{h_8-x}{h_8-h_{7}}\right) , &{} if &{} h_{7} \leqslant x \leqslant h_8 \\ 0, &{} &{} otherwise\\ \end{array} \right. . \end{aligned}$$

A generalized octagonal linear fuzzy number is in the shape of the octagon with vertices \((h_1, 0), (h_2, u_{L_2}), (h_3, u_{L_1}), (h_{4}, u)\), \((h_{5}, u)\), \((h_{6}, u_{R_1}),\ldots\) \((h_8, 0)\).

Remark 3

\(H=((h_1, h_2, \dots h_n); u, u_{L_1}, u_{L_2}, \ldots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \ldots u_{R_{\frac{n-4}{2}}} )\) and \(G=((g_1, g_2, \dots g_n);\, u', u'_{L_1}, u'_{L_2}, \ldots u'_{L_{\frac{n-4}{2}}}, u'_{R_1}, u'_{R_2}, \ldots u'_{R_{\frac{n-4}{2}}} )\) denotes generalized ‘n’gonal linear fuzzy numbers with even \(n \ge 6\) unless otherwise it is specifically mentioned.

3.1 Operations on Generalized ‘n’ Gonal Linear Fuzzy Numbers

Most of the research problems in the theory of fuzzy sets and its applications depend on operations on fuzzy numbers which make this as a crucial area of study. As a result of their uniqueness and utility, various operations have been described in literature. Since we discussed the definition of generalized ‘n’ gonal linear fuzzy numbers, for more implications and applications of GNLFN, we give some significant addition, subtraction, multiplication, and division operations on GNLFNs in this subsection.

Definition 6

(Addition of two generalized ‘n’ gonal linear fuzzy numbers) If \(H=((h_1, h_2, \dots h_n); u, u_{L_1}, u_{L_2}, \ldots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \ldots u_{R_{\frac{n-4}{2}}} )\) and \(G=((g_1, g_2, \dots g_n); u', u'_{L_1}, u'_{L_2}, \ldots u'_{L_{\frac{n-4}{2}}}, u'_{R_1}, u'_{R_2}, \ldots u'_{R_{\frac{n-4}{2}}} )\) are two generalized ‘n’gonal linear fuzzy numbers, then the addition of H and G is defined as \(H + G =(h_1+g_1, h_2+g_2, \dots h_n+g_n); w, w_{L_1}, w_{L_2}, \ldots w_{L_{\frac{n-4}{2}}}, w_{R_1}, w_{R_2}, \ldots w_{R_{\frac{n-4}{2}}} )\) whose membership function is given by \((H+G) (x)\),

$$\begin{aligned} = \left\{ \begin{array}{lll} w_{L_{\frac{n-4}{2}}}\left( \frac{x-(h_1+g_1)}{(h_2+g_2)-(h_1+g_1)}\right) , &{} if &{} x \in [h_1+g_1,h_2+g_2] \\ w_{L_{\frac{n-4}{2}}}+(w- w_{L_{\frac{n-4}{2}}})\left( \frac{x-(h_2+g_2)}{(h_3+g_3)-(h_2+g_2)}\right) , &{} if &{} x \in [h_2+g_2,h_3+g_3] \\ w_{L_{\frac{n-6}{2}}}+(w_{L_{\frac{n-4}{2}}}-w_{L_{\frac{n-6}{2}}})\left( \frac{x-(h_3+g_3)}{(h_4+g_4)-(h_4+g_4)}\right) , &{} if &{} x \in [h_3+g_3,h_4+g_4] \\ \vdots \\ w_{L_2}+(w_{L_2}-w_{L_1})\left( \frac{(h_{\frac{n}{2}-1}+g_{\frac{n}{2}-1})-x}{(h_{\frac{n}{2}-1}+g_{\frac{n}{2}-1})-(h_{\frac{n}{2}-2}+g_{\frac{n}{2}-2})}\right) , &{} if &{} x \in [h_{\frac{n}{2}-2}+g_{\frac{n}{2}-2},h_{\frac{n}{2}-1}+g_{\frac{n}{2}-1}] \\ w_{L_1}+(w_{L_1}-w)\left( \frac{(h_{\frac{n}{2}}+g_{\frac{n}{2}})-x}{(h_{\frac{n}{2}}+g_{\frac{n}{2}})-(h_{\frac{n}{2}-1}+g_{\frac{n}{2}-1})}\right) , &{} if &{} x \in [h_{\frac{n}{2}-1}+g_{\frac{n}{2}-1}, h_{\frac{n}{2}}+g_{\frac{n}{2}}] \\ w, &{} if &{} x \in [h_{\frac{n}{2}}+g_{\frac{n}{2}},h_{\frac{n}{2}+1}+g_{\frac{n}{2}+1}]\\ w_{R_1}+(w-w_{R_1})\left( \frac{(h_{\frac{n}{2}+1}+g_{\frac{n}{2}+1})-x}{(h_{\frac{n}{2}+2}+g_{\frac{n}{2}+2})-(h_{\frac{n}{2}+1}+g_{\frac{n}{2}+1})}\right) , &{} if &{} x\in [h_{\frac{n}{2}+1}+g_{\frac{n}{2}+1},h_{\frac{n}{2}+2} +g_{\frac{n}{2}+2}] \\ w_{R_2}+(w_{R_1}-w_{R_2})\left( \frac{(h_{\frac{n}{2}+2}+g_{\frac{n}{2}+2})-x}{(h_{\frac{n}{2}+3}+g_{\frac{n}{2}+3})-(h_{\frac{n}{2}+2}+g_{\frac{n}{2}+2})}\right) , &{} if &{} x\in [h_{\frac{n}{2}+2}+g_{\frac{n}{2}+2}, h_{\frac{n}{2}+3}+g_{\frac{n}{2}+3}] \\ \vdots \\ w_{R_{\frac{n-4}{2}}}\left( \frac{(h_n+g_n)-x}{(h_n+g_n)-(h_{n-1}+g_{n-1})}\right) , &{} if &{} x \in [h_{n-1}+g_{n-1},h_n+g_n] \\ 0, &{}&{} otherwise\\ \end{array} \right. , \end{aligned}$$

where \(w=u+u'-uu',w_{L_i}=u_{L_i}+u'_{L_i}-u_{L_i}u'_{L_i},w_{R_i}=u_{R_i}+u'_{R_i}-u_{R_i}u'_{R_i}\) \(\forall i \in {1,2,\ldots \frac{n-4}{2}}.\)

Definition 7

(Subtraction of two generalized ‘n’gonal linear fuzzy numbers) If H and G are two generalized ‘n’gonal linear fuzzy numbers, then the subtraction of H and G is defined as \(H - G =(h_1-g_n, h_2-g_{n-1}, \dots h_n-g_1); w, w_{L_1}, w_{L_2}, \ldots w_{L_{\frac{n-4}{2}}}, w_{R_1}, w_{R_2}, \ldots w_{R_{\frac{n-4}{2}}} ),\) whose membership function is given by \((H-G) (x)\)

$$\begin{aligned} = \left\{ \begin{array}{lll} w_{L_{\frac{n-4}{2}}}\left( \frac{x-(h_1-g_n)}{(h_2-g_{n-1})-(h_1-g_n)}\right) ,&{} if &{} x \in [h_1-g_n,h_2-g_{n-1}] \\ w_{L_{\frac{n-4}{2}}}+(w- w_{L_{\frac{n-4}{2}}})\left( \frac{x-(h_2-g_{n-1})}{(h_3-g_{n-2})-(h_2-g_{n-1})}\right) , &{} if &{} x \in [h_2-g_{n-1},h_3-g_{n-2}] \\ w_{L_{\frac{n-6}{2}}}+(w_{L_{\frac{n-4}{2}}}-w_{L_{\frac{n-6}{2}}})\left( \frac{x-(h_3-g_{n-2})}{(h_4-g_{n-3})-(h_4-g_{n-3})}\right) , &{} if &{} x \in [h_3-g_{n-2},h_4-g_{n-3}] \\ \vdots \\ w_{L_2}+(w_{L_2}-w_{L_1})\left( \frac{(h_{\frac{n}{2}-1}-g_{\frac{n}{2}-2})-x}{(h_{\frac{n}{2}-1}-g_{\frac{n}{2}-2})-(h_{\frac{n}{2}-2}-g_{\frac{n}{2}-3})}\right) , &{} if &{} x \in [h_{\frac{n}{2}-2}-g_{\frac{n}{2}-3},h_{\frac{n}{2}-1}-g_{\frac{n}{2}-2}] \\ w_{L_1}+(w_{L_1}-w)\left( \frac{(h_{\frac{n}{2}}-g_{\frac{n}{2}-1})-x}{(h_{\frac{n}{2}}-g_{\frac{n}{2}-1})-(h_{\frac{n}{2}-1}-g_{\frac{n}{2}-2})}\right) , &{} if &{} x \in [h_{\frac{n}{2}-1}-g_{\frac{n}{2}-2}, h_{\frac{n}{2}}-g_{\frac{n}{2}-1}] \\ w, &{} if &{} x \in [h_{\frac{n}{2}}-g_{\frac{n}{2}-1},h_{\frac{n}{2}+1}-g_{n}]\\ w_{R_1}+(w-w_{R_1})\left( \frac{(h_{\frac{n}{2}+1}-g_{n})-x}{(h_{\frac{n}{2}+2} -g_{\frac{n}{2}+1})-(h_{\frac{n}{2}+1}-g_{n})}\right) , &{} if &{} x\in [h_{\frac{n}{2}+1}-g_{n},h_{\frac{n}{2}+2} -g_{\frac{n}{2}+1}] \\ w_{R_2}+(w_{R_1}-w_{R_2})\left( \frac{(h_{\frac{n}{2}+2}-g_{\frac{n}{2}+1})-x}{(h_{\frac{n}{2}+3}-g_{\frac{n}{2}+2})-(h_{\frac{n}{2}+2}-g_{\frac{n}{2}+1})}\right) , &{} if &{} x\in [h_{\frac{n}{2}+2}-g_{\frac{n}{2}+1}, h_{\frac{n}{2}+3}-g_{\frac{n}{2}+2}] \\ \vdots \\ w_{R_{\frac{n-4}{2}}}\left( \frac{(h_n-g_1)-x}{(h_n-g_1)-(h_{n-1}-g_{2})}\right) , &{} if &{} x \in [h_{n-1}-g_{2},h_n-g_1] \\ 0, &{} &{} otherwise\\ \end{array} \right. , \end{aligned}$$

where \(w=u+u'-uu',w_{L_i}=u_{L_i}+u'_{R_i}-u_{L_i}u'_{R_i},w_{R_i}=u_{R_i}+u'_{L_i}-u_{R_i}u'_{L_i}\) \(\forall i \in {1,2,\ldots \frac{n-4}{2}}.\)

Definition 8

(Multiplication of two generalized ‘n’gonal linear fuzzy numbers) If \(H=((h_1, h_2, \dots h_n);\, u, u_{L_1}, u_{L_2}, \ldots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \ldots u_{R_{\frac{n-4}{2}}} )\) and \(G=((g_1, g_2, \dots g_n);\, u', u'_{L_1}, u'_{L_2}, \ldots u'_{L_{\frac{n-4}{2}}}, u'_{R_1}, u'_{R_2}, \ldots u'_{R_{\frac{n-4}{2}}} )\) are two generalized ‘n’gonal linear fuzzy numbers, then the multiplication of H and G is defined as follows:

if \(h_n < 0,\) then

$$\begin{gathered} H \times G = \left( {min(h_{1} g_{n} ,h_{n} g_{n} ),min(h_{2} g_{{n - 1}} ,h_{{n - 1}} g_{{n - 1}} ), \ldots } \right.\quad \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,min\left( {h_{{\frac{n}{2}}} g_{{\frac{n}{2} + 1}} ,h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2} + 1}} } \right),\quad \hfill \\ max\left( {h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2}}} ,h_{{\frac{n}{2}}} g_{{\frac{n}{2}}} } \right), \ldots max(h_{n} g_{1} ,h_{1} g_{1} );{\mkern 1mu} uv,u_{{L_{1} }} v_{{L_{1} }} , \ldots \quad \left. {u_{{L_{{\frac{{n - 4}}{2}}} }} v_{{L_{{\frac{{n - 4}}{2}}} }} ,u_{{R_{1} }} v_{{R_{1} }} , \ldots u_{{R_{{\frac{{n - 4}}{2}}} }} v_{{R_{{\frac{{n - 4}}{2}}} }} } \right); \hfill \\ \end{gathered}$$

if \(h_{n-1} \le 0, h_{n} \ge 0,\) then

$$\begin{gathered} H \times G = \left( {min(h_{1} g_{n} ,h_{n} g_{1} ),min(h_{2} g_{{n - 1}} ,h_{{n - 1}} g_{{n - 1}} ), \ldots } \right.\quad \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,min\left( {h_{{\frac{n}{2}}} g_{{\frac{n}{2} + 1}} ,h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2} + 1}} } \right),\quad \hfill \\ max\left( {h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2}}} ,h_{{\frac{n}{2}}} g_{{\frac{n}{2}}} } \right), \ldots \quad max(h_{n} g_{n} ,h_{1} g_{1} );uv,u_{{L_{1} }} v_{{L_{1} }} , \ldots u_{{L_{{\frac{{n - 4}}{2}}} }} v_{{L_{{\frac{{n - 4}}{2}}} }} ,\quad \left. {u_{{R_{1} }} v_{{R_{1} }} , \ldots u_{{R_{{\frac{{n - 4}}{2}}} }} v_{{R_{{\frac{{n - 4}}{2}}} }} } \right); \hfill \\ \end{gathered}$$

if \(h_{n-2} \le 0, h_{n-1} \ge 0,\) then

$$\begin{gathered} H \times G = \left( {min(h_{1} g_{n} ,h_{n} g_{1} ),min(h_{2} g_{{n - 1}} ,h_{{n - 1}} g_{2} ), \ldots } \right.\quad \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,min\left( {h_{{\frac{n}{2}}} g_{{\frac{n}{2} + 1}} ,h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2} + 1}} } \right),\quad \hfill \\ max\left( {h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2}}} ,h_{{\frac{n}{2}}} g_{{\frac{n}{2}}} } \right), \ldots \quad max(h_{n} g_{n} ,h_{1} g_{1} );uv,u_{{L_{1} }} v_{{L_{1} }} , \ldots u_{{L_{{\frac{{n - 4}}{2}}} }} v_{{L_{{\frac{{n - 4}}{2}}} }} ,\quad \left. {u_{{R_{1} }} v_{{R_{1} }} , \ldots u_{{R_{{\frac{{n - 4}}{2}}} }} v_{{R_{{\frac{{n - 4}}{2}}} }} } \right), \hfill \\ \end{gathered}$$

if \(h_{\frac{n}{2}} \le 0, h_{\frac{n}{2}+1} \ge 0,\) then

$$\begin{gathered} H \times G = \left( {min(h_{1} g_{n} ,h_{n} g_{1} ),min(h_{2} g_{{n - 1}} ,h_{{n - 1}} g_{2} ), \ldots } \right.\quad \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,min\left( {h_{{\frac{n}{2}}} g_{{\frac{n}{2} + 1}} ,h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2}}} } \right),\quad \hfill \\ max\left( {h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2} + 1}} ,h_{{\frac{n}{2}}} g_{{\frac{n}{2}}} } \right), \ldots \quad max(h_{n} g_{n} ,h_{1} g_{1} );uv,u_{{L_{1} }} v_{{L_{1} }} , \ldots u_{{L_{{\frac{{n - 4}}{2}}} }} v_{{L_{{\frac{{n - 4}}{2}}} }} ,\quad \left. {u_{{R_{1} }} v_{{R_{1} }} , \ldots u_{{R_{{\frac{{n - 4}}{2}}} }} v_{{R_{{\frac{{n - 4}}{2}}} }} } \right); \hfill \\ \end{gathered}$$

if \(h_{\frac{n}{2}-1} \le 0, h_{\frac{n}{2}} \ge 0,\) then

$$\begin{gathered} H \times G = \left( {min(h_{1} g_{n} ,h_{n} g_{1} ),min(h_{2} g_{{n - 1}} ,h_{{n - 1}} g_{2} ), \ldots } \right.\quad \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,min\left( {h_{{\frac{n}{2}}} g_{{\frac{n}{2}}} ,h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2}}} } \right),\quad \hfill \\ max\left( {h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2} + 1}} ,h_{{\frac{n}{2}}} g_{{\frac{n}{2} + 1}} } \right), \ldots \quad max(h_{n} g_{n} ,h_{1} g_{1} );uv,u_{{L_{1} }} v_{{L_{1} }} , \ldots u_{{L_{{\frac{{n - 4}}{2}}} }} v_{{L_{{\frac{{n - 4}}{2}}} }} ,\quad \left. {u_{{R_{1} }} v_{{R_{1} }} , \ldots u_{{R_{{\frac{{n - 4}}{2}}} }} v_{{R_{{\frac{{n - 4}}{2}}} }} } \right), \hfill \\ \end{gathered}$$

if \(h_1\ge 0,\) then

$$\begin{gathered} H \times G = \left( {min(h_{1} g_{1} ,h_{n} g_{1} ),min(h_{2} g_{2} ,h_{{n - 1}} g_{2} ), \ldots } \right.\quad \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,min\left( {h_{{\frac{n}{2}}} g_{{\frac{n}{2}}} ,h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2}}} } \right),\quad \hfill \\ max\left( {h_{{\frac{n}{2} + 1}} g_{{\frac{n}{2} + 1}} ,h_{{\frac{n}{2}}} g_{{\frac{n}{2} + 1}} } \right), \ldots \quad max(h_{n} g_{n} ,h_{1} g_{n} );uv,u_{{L_{1} }} v_{{L_{1} }} , \ldots u_{{L_{{\frac{{n - 4}}{2}}} }} v_{{L_{{\frac{{n - 4}}{2}}} }} ,\quad \left. {u_{{R_{1} }} v_{{R_{1} }} , \ldots u_{{R_{{\frac{{n - 4}}{2}}} }} v_{{R_{{\frac{{n - 4}}{2}}} }} } \right), \hfill \\ \end{gathered}$$

where \(min (a,b)=\frac{a+b}{2}\) − \(\mid \frac{a-b}{2}\mid\) and \(max (a,b)=\frac{a+b}{2}\) \(+\) \(\mid \frac{a-b}{2}\mid .\)

Remark 4

Let \(H=((h_1, h_2, \dots h_n); u, u_{L_1}, u_{L_2}, \ldots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \ldots u_{R_{\frac{n-4}{2}}} )\) be a generalized ‘n’gonal linear fuzzy number. Now the scalar multiplication of H with \(k >0\) is defined as \(kH=((kh_1, kh_2, \dots kh_n);\, u, u_{L_1}, u_{L_2}, \ldots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \ldots u_{R_{\frac{n-4}{2}}} ).\)

Definition 9

(Division of two generalized ‘n’gonal linear fuzzy numbers) If \(H=((h_1, h_2, \dots h_n);\, u, u_{L_1}, u_{L_2}, \ldots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \ldots u_{R_{\frac{n-4}{2}}} )\) and \(G=((g_1, g_2, \dots g_n);\, u', u'_{L_1}, u'_{L_2}, \ldots u'_{L_{\frac{n-4}{2}}}, u'_{R_1}, u'_{R_2}, \ldots u'_{R_{\frac{n-4}{2}}} )\) are two generalized ‘n’gonal linear fuzzy numbers such that \(H,G > 0\), then the division of H and G is defined as \(H/G =(h_1/g_n, h_2/g_{n-1}, \dots h_n/g_1); w, w_{L_1}, w_{L_2}, \ldots w_{L_{\frac{n-4}{2}}}, w_{R_1}, w_{R_2}, \ldots w_{R_{\frac{n-4}{2}}} ),\) whose membership function is given by (H/G) (x)

$$\begin{aligned} = \left\{ \begin{array}{lll} w_{L_{\frac{n-4}{2}}}\left( \frac{x-(h_1/g_n)}{(h_2/g_{n-1})-(h_1/g_n)}\right) , &{} if &{} x \in [h_1/g_n,h_2/g_{n-1}] \\ w_{L_{\frac{n-4}{2}}}+(w- w_{L_{\frac{n-4}{2}}})\left( \frac{x-(h_2/g_{n-1})}{(h_3/g_{n-2})-(h_2/g_{n-1})}\right) , &{} if &{} x \in [h_2/g_{n-1},h_3/g_{n-2}] \\ w_{L_{\frac{n-6}{2}}}+(w_{L_{\frac{n-4}{2}}}-w_{L_{\frac{n-6}{2}}})\left( \frac{x-(h_3/g_{n-2})}{(h_4/g_{n-3})-(h_4/g_{n-3})}\right) , &{} if &{} x \in [h_3/g_{n-2},h_4/g_{n-3}] \\ \vdots \\ w_{L_2}+(w_{L_2}-w_{L_1})\left( \frac{(h_{\frac{n}{2}-1}/g_{\frac{n}{2}-2})-x}{(h_{\frac{n}{2}-1}/g_{\frac{n}{2}-2})-(h_{\frac{n}{2}-2}/g_{\frac{n}{2}-3})}\right) , &{} if &{} x \in [h_{\frac{n}{2}-2}/g_{\frac{n}{2}-3},h_{\frac{n}{2}-1}/g_{\frac{n}{2}-2}] \\ w_{L_1}+(w_{L_1}-w)\left( \frac{(h_{\frac{n}{2}}/g_{\frac{n}{2}-1})-x}{(h_{\frac{n}{2}}/g_{\frac{n}{2}-1})-(h_{\frac{n}{2}-1}/g_{\frac{n}{2}-2})}\right) , &{} if &{} x \in [h_{\frac{n}{2}-1}/g_{\frac{n}{2}-2}, h_{\frac{n}{2}}/g_{\frac{n}{2}-1}] \\ w, &{} if &{} x \in [h_{\frac{n}{2}}/g_{\frac{n}{2}-1},h_{\frac{n}{2}+1}/g_{n}]\\ w_{R_1}+(w-w_{R_1})\left( \frac{(h_{\frac{n}{2}+1}/g_{n})-x}{(h_{\frac{n}{2}+2} /g_{\frac{n}{2}+1})-(h_{\frac{n}{2}+1}/g_{n})}\right) , &{} if &{} x\in [h_{\frac{n}{2}+1}/g_{n},h_{\frac{n}{2}+2} /g_{\frac{n}{2}+1}] \\ w_{R_2}+(w_{R_1}-w_{R_2})\left( \frac{(h_{\frac{n}{2}+2}/g_{\frac{n}{2}+1})-x}{(h_{\frac{n}{2}+3}/g_{\frac{n}{2}+2})-(h_{\frac{n}{2}+2}/g_{\frac{n}{2}+1})}\right) , &{} if &{} x\in [h_{\frac{n}{2}+2}/g_{\frac{n}{2}+1}, h_{\frac{n}{2}+3}/g_{\frac{n}{2}+2}] \\ \vdots \\ w_{R_{\frac{n-4}{2}}}\left( \frac{(h_n/g_1)-x}{(h_n/g_1)-(h_{n-1}/g_{2})}\right) , &{} if &{} x \in [h_{n-1}/g_{2},h_n/g_1] \\ 0, &{} &{} otherwise\\ \end{array} \right. , \end{aligned}$$

where \(w=uu',w_{L_i}=u_{L_i}u'_{R_i},w_{R_i}=u_{R_i}u'_{L_i}\) \(\forall i \in {1,2,\ldots \frac{n-4}{2}}.\)

If \(H>0\) and \(G<0,\) then H/G is defined by

\(H/G =(h_n/g_1, h_{n-1}/g_2, \dots h_1/g_n); w, w_{L_1}, w_{L_2}, \ldots w_{L_{\frac{n-4}{2}}}, w_{R_1}, w_{R_2}, \ldots w_{R_{\frac{n-4}{2}}} ),\) whose membership function is given by (H/G) (x)

$$\begin{aligned} = \left\{ \begin{array}{ccc} w_{L_{\frac{n-4}{2}}}\left( \frac{x-(h_n/g_1)}{(h_{n-1}/g_{2})-(h_n/g_1)}\right) , &{} if &{} x \in [h_n/g_1,h_{n-2}/g_2] \\ w_{L_{\frac{n-4}{2}}}+(w- w_{L_{\frac{n-4}{2}}})\left( \frac{x-(h_{n-1}/g_2)}{(h_{n-2}/g_3)-(h_{n-1}/g_2)}\right) ,&{} if &{} x \in [h_{n-1}/g_2,h_{n-2}/g_3] \\ w_{L_{\frac{n-6}{2}}}+(w_{L_{\frac{n-4}{2}}}-w_{L_{\frac{n-6}{2}}})\left( \frac{x-(h_{n-2}/g_3)}{(h_{n-3}/g_4)-(h_{n-3}/g_4)}\right) , &{} if &{} x \in [h_{n-2}/g_3,h_{n-3}/g_4] \\ \vdots \\ w_{L_2}+(w_{L_2}-w_{L_1})\left( \frac{(h_{\frac{n}{2}-2}/g_{\frac{n}{2}-1})-x}{(h_{\frac{n}{2}-2} /g_{\frac{n}{2}-1})-(h_{\frac{n}{2}-3} /g_{\frac{n}{2}-2})}\right) , &{} if &{} x \in [h_{\frac{n}{2}-3}/g_{\frac{n}{2}-2},h_{\frac{n}{2}-2}/g_{\frac{n}{2}-1}] \\ w_{L_1}+(w_{L_1}-w)\left( \frac{(h_{\frac{n}{2}-1}/g_{\frac{n}{2}})-x}{(h_{\frac{n}{2}-1}/g_{\frac{n}{2}})-(h_{\frac{n}{2}-2}/g_{\frac{n}{2}-1})}\right) , &{} if &{} x \in [h_{\frac{n}{2}-2}/g_{\frac{n}{2}-1}, h_{\frac{n}{2}-1}/g_{\frac{n}{2}}] \\ w, &{} if &{} x \in [h_{\frac{n}{2}-1}/g_{\frac{n}{2}},h_{n}/g_{\frac{n}{2}+1}]\\ w_{R_1}+(w-w_{R_1})\left( \frac{(h_{n}/g_{\frac{n}{2}+1})-x}{(h_{\frac{n}{2}+1} /g_{\frac{n}{2}+2})-(h_{n}/g_{\frac{n}{2}+1})}\right) , &{} if &{} x\in [h_{n}/g_{\frac{n}{2}+1},h_{\frac{n}{2}+1} /g_{\frac{n}{2}+2}] \\ w_{R_2}+(w_{R_1}-w_{R_2})\left( \frac{(h_{\frac{n}{2}+1}/g_{\frac{n}{2}+2})-x}{(h_{\frac{n}{2}+2}/g_{\frac{n}{2}+3})-(h_{\frac{n}{2}+1}/g_{\frac{n}{2}+2})}\right) , &{} if &{} x\in [h_{\frac{n}{2}+1}/g_{\frac{n}{2}+2}, h_{\frac{n}{2}+2}/g_{\frac{n}{2}+3}] \\ \vdots \\ w_{R_{\frac{n-4}{2}}}\left( \frac{(h_1/g_n)-x}{(h_1/g_n)-(h_{2}/g_{n-1})}\right) , &{} if &{} x \in [h_{2}/g_{n-1},h_1/g_n] \\ 0, &{} &{} otherwise\\ \end{array} \right. , \end{aligned}$$

where \(w=uu',w_{L_i}=u_{R_i}u'_{L_i},w_{R_i}=u_{L_i}u'_{R_i}\) \(\forall i \in {1,2,\ldots \frac{n-4}{2}}.\)

Remark 5

If H is an ‘n’gonal linear fuzzy number and G is a near zero ‘n’ gonal linear fuzzy number, then H/G is undefined.

Example 1

As an example of employing the above four operations, consider generalized hexagonal fuzzy numbers \(H=(2,3,5,7,8,9; 1,0.4,0.5)\) and \(G=(-1,0,2,3,4,5; 1,0.5,0.6)\). The resulting arithmetic operations of these two fuzzy numbers are

$$\begin{aligned}{} & {} (H+G) = (1,3,7,10,12,14;1,0.7,0.8),\\{} & {} (H-G) = (-3,-1,2,5,8,10; 1,0.76,0.75), \\{} & {} (HG) =(-9,0,10,21,32,45; 1,0.2,0.3). \end{aligned}$$

Since G is a generalized near zero hexagonal fuzzy number, H/G is undefined.

4 Proposed Total Ordering Algorithm

In this section, a total ordering algorithm on the generalized ‘n’ gonal linear fuzzy numbers has been derived and compared with existing ordering methods.

Let \(A = (h_1, h_2, \dots h_n; u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}})\) be a generalized ‘n’gonal linear fuzzy number. Now, u can also be viewed as \(u_0 = u_{L_0} = u_{R_0}\). We note that \(u_{L_{\frac{n-4}{2}+1}}= u_{R_{\frac{n-4}{2}+1}}= 0\). Now, we propose the following score functions which will be used in the algorithm of total ordering on the class of ‘n’gonal linear fuzzy numbers.

The score functions: Let \(\mathcal {A}\) be the set of all ’n’ gonal linear fuzzy numbers.

Definition 10

The midpoint score function \(S_1: \mathcal {A} \rightarrow \mathbb {R}\) is defined by \(S_1(A) = u \frac{(h_{\frac{n}{2}} + h_{\frac{n}{2}+1})}{2}\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n;\, u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

Definition 11

The span score function \(S_2: \mathcal {A} \rightarrow \mathbb {R}\) is defined by \(S_2 (A) = u \frac{ (h_{\frac{n}{2}+1}-h_{\frac{n}{2}} )}{2}\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n;\, u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

Definition 12

The height function \(S_3: \mathcal {A} \rightarrow [0,1]\) is defined by \(S_3 (A) = u\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n;\, u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

Definition 13

The left aggregation score function of slope \(S_{3k+1}: \mathcal {A} \rightarrow \mathbb {R}\) is defined by \(S_{3k+1} (A) = u_{L_k} (h_{\frac{n}{2}-k+1} - h_{\frac{n}{2}-k-1}) - u_{L_{k+1}} (h_{\frac{n}{2}-k+1} - h_{\frac{n}{2}-k}) + u_{L_{k-1}} (h_{\frac{n}{2}-k} - h_{\frac{n}{2}-k-1})\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n;\, u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

Definition 14

The left dissimilitude score function of slope \(S_{3k+2}: \mathcal {A} \rightarrow \mathbb {R}\) is defined by \(S_{3k+2} (A) = u_{L_k} (h_{\frac{n}{2}-k+1}-2h_{\frac{n}{2}-k} + h_{\frac{n}{2}-k-1}) - u_{L_{k+1}} (h_{\frac{n}{2}-k+1} - h_{\frac{n}{2}-k}) - u_{L_{k-1}} (h_{\frac{n}{2}-k} - h_{\frac{n}{2}-k-1})\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n; u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

Definition 15

The height function of left slope \(S_{3k+3}: \mathcal {A} \rightarrow [0,1]\) is defined by \(S_{3k+3} (A) = u_{L_k}\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n; u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

Definition 16

The right aggregation score function of slope \(S'_{3k+1}: \mathcal {A} \rightarrow \mathbb {R}\) is defined by \(S'_{3k+1} (A) = u_{R_k} (h_{\frac{n}{2}+k} - h_{\frac{n}{2}+k+2}) - u_{R_{k+1}} (h_{\frac{n}{2}+k} - h_{\frac{n}{2}+k+1}) + u_{R_{k-1}} (h_{\frac{n}{2}+k+1} - h_{\frac{n}{2}+k+2})\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n; u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

Definition 17

The right dissimilitude score function of slope \(S'_{3k+2}: \mathcal {A} \rightarrow \mathbb {R}\) is defined by \(S'_{3k+2} (A) = u_{R_k} (h_{\frac{n}{2}+k}-2h_{\frac{n}{2}+k+1}+ h_{\frac{n}{2}+k+2}) - u_{R_{k+1}} (h_{\frac{n}{2}+k} - h_{\frac{n}{2}+k+1}) - u_{R_{k-1}} (h_{\frac{n}{2}+k+1} - h_{\frac{n}{2}+k+2})\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n; u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

Definition 18

The height function of right slope \(S'_{3k+3}: \mathcal {A} \rightarrow [0,1]\) is defined by \(S'_{3k+3} (A) = u_{R_k}\) for every \(A \in \mathcal {A},\) where \(A=(h_1, h_2, \dots h_n; u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}).\)

4.1 Total Ordering Algorithm

Let \(A = (h_1, h_2, \dots h_n;\, u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}})\)

and \(B = (h'_1, h'_2, \dots h'_n;\, u', u'_{L_1}, u'_{L_2}, \dots u'_{L_{\frac{n-4}{2}}}, u'_{R_1}, u'_{R_2}, \dots u'_{R_{\frac{n-4}{2}}})\) be ‘n’gonal linear fuzzy numbers.  Using remark 1, without loss of generality assume that n is even and \(\ge 6\) such that the class of generalized ‘n’gonal linear fuzzy numbers contains both A and B.

Step 1: If \(S_1 (A) > S_1 (B),\) then we get \(A > B.\) If \(S_1 (A) < S_1 (B),\) then we get \(A < B.\) If \(S_1 (A) = S_1 (B),\) then go to step 2.

Step 2: If \(S_2 (A) > S_2 (B),\) then we get \(A > B.\) If \(S_2 (A) < S_2 (B),\) then we get \(A < B.\) If \(S_2 (A) = S_2 (B),\) then go to step 3.

Step 3: If \(S_3 (A) > S_3 (B),\) then we get \(A > B\). If \(S_3 (A) < S_3 (B),\) then we get \(A < B\). If \(S_3 (A) = S_3 (B),\) then let \(k=1\) and go to step 4.

Step 4: If \(S_{3k+1} (A) > S_{3k+1} (B),\) then we get \(A > B.\) If \(S_{3k+1} (A) < S_{3k+1} (B),\) then we get \(A < B.\) If \(S_{3k+1} (A) = S_{3k+1} (B),\) then go to step 5.

Step 5: If \(S_{3k+2} (A) > S_{3k+2} (B),\) then we get \(A > B.\) If \(S_{3k+2} (A) < S_{3k+2} (B),\) then we get \(A < B.\) If \(S_{3k+2} (A) = S_{3k+2} (B),\) then go to step 6.

Step 6: If \(S_{3k+3} (A) > S_{3k+3} (B),\) then we get \(A > B.\) If \(S_{3k+3} (A) < S_{3k+3} (B),\) then we get \(A < B.\) If \(S_{3k+3} (A) = S_{3k+3} (B),\) then go to step 7.

Step 7: Let \(k=k+1\). If \(k < \frac{n}{2} - 1\), then go to step 4. If \(k = \frac{n}{2} - 1\), then go to step 8.

Step 8: Let \(k=1\) and go to step 9.

Step 9: If \(S'_{3k+1} (A) > S'_{3k+1} (B),\) then we get \(A > B\). If \(S'_{3k+1} (A) < S'_{3k+1} (B),\) then we get \(A < B.\) If \(S'_{3k+1} (A) = S'_{3k+1} (B),\) then go to step 10.

Step 10: If \(S'_{3k+2} (A) > S'_{3k+2} (B),\) then we get \(A > B.\) If \(S'_{3k+2} (A) < S'_{3k+2} (B),\) then we get \(A < B\). If \(S'_{3k+2} (A) = S'_{3k+2} (B),\) then go to step 11.

Step 11: If \(S'_{3k+3} (A) > S'_{3k+3} (B),\) then we get \(A > B.\) If \(S'_{3k+3} (A) < S'_{3k+3} (B),\) then we get \(A < B.\) If \(S'_{3k+3} (A) = S'_{3k+3} (B),\) then go to step 12.

Step 12: Let \(k=k+1\). If \(k < \frac{n}{2} - 1\), then go to step 9. If \(k = \frac{n}{2} - 1\), then \(A=B\).

Theorem 1

The proposed total ordering algorithm derives total ordering on ‘n’ gonal linear fuzzy numbers.

Proof

By remark 1, assume that n is even and \(\ge 6\) such that the class of generalized ‘n’gonal linear fuzzy numbers. Let us assume that \(A = (h_1, h_2, \dots h_n;\, u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}})\)

and let \(B = (h'_1, h'_2, \dots h'_n;\, u', u'_{L_1}, u'_{L_2}, \dots u'_{L_{\frac{n-4}{2}}}, u'_{R_1}, u'_{R_2}, \dots u'_{R_{\frac{n-4}{2}}})\) be two ‘n’gonal linear fuzzy numbers. To prove total ordering, it is enough to show that either \(A > B\) or \(A < B\) or \(A = B.\)

Let us apply step 1. If we get \(S_1 (A)>S_1 (B)\) or \(S_1 (A)<S_1 (B)\), then we will conclude \(A > B\) or \(A < B\), which is done. Otherwise, \(S_1 (A)=S_1 (B)\) implies

$$\begin{aligned} u \left( h_{\frac{n}{2}} + h_{\frac{n}{2}+1}\right) = u' \left( h'_{\frac{n}{2}} + h'_{\frac{n}{2}+1}\right) . \end{aligned}$$
(1)

Now, by applying step 2, if we get \(S_2 (A)>S_2 (B)\) or \(S_2 (A)<S_2 (B)\), then we will conclude \(A > B\) or \(A < B\), which is done. Otherwise, \(S_2 (A)=S_2 (B)\) implies

$$\begin{aligned} u ( h_{\frac{n}{2}+1}-h_{\frac{n}{2}}) = u' ( h'_{\frac{n}{2}+1} - h'_{\frac{n}{2}} ). \end{aligned}$$
(2)

Now, by applying step 3, if we get \(S_3 (A)>S_3 (B)\) or \(S_3 (A)<S_3 (B)\), then we will conclude \(A > B\) or \(A < B\), which is done. Otherwise, \(S_3 (A)=S_3 (B)\) implies

$$\begin{aligned} u = u'. \end{aligned}$$
(3)

Substitute Eq. 3 in Eqs. 1 and 2. We get

$$\begin{aligned}{} & {} h_{\frac{n}{2}} + h_{\frac{n}{2}+1}= h'_{\frac{n}{2}} + h'_{\frac{n}{2}+1} \end{aligned}$$
(4)
$$\begin{aligned}{} & {} h_{\frac{n}{2}+1}-h_{\frac{n}{2}} = h'_{\frac{n}{2}+1} - h'_{\frac{n}{2}}. \end{aligned}$$
(5)

Now adding Eqs. 4 and 5, we get

$$\begin{aligned} h_{\frac{n}{2}+1} = h'_{\frac{n}{2}+1}. \end{aligned}$$
(6)

Substituting Eqs. 6 in 4

$$\begin{aligned} h_{\frac{n}{2}} = h'_{\frac{n}{2}}. \end{aligned}$$
(7)

At this stage, when \(S_1, S_2, S_3\) are equal, we get \(h_{\frac{n}{2}+1} = h'_{\frac{n}{2}+1}, h_{\frac{n}{2}} = h'_{\frac{n}{2}}\) and \(u=u'\).

Now, we apply score \(S_{3k}+1\) for \(k=1\). If \(S_4 (A) > S_4 (B)\), then \(A>B\). If \(S_4 (A) < S_4 (B)\), then \(A<B\). If \(S_4 (A) = S_4 (B)\), and we have

$$\begin{gathered} u_{{L_{1} }} \left( {h_{{\frac{n}{2}}} - h_{{\frac{n}{2} - 2}} } \right) - u_{{L_{2} }} \left( {h_{{\frac{n}{2}}} - h_{{\frac{n}{2} - 1}} } \right) + u_{{L_{0} }} \left( {h_{{\frac{n}{2} - 1}} - h_{{\frac{n}{2} - 2}} } \right)\quad \hfill \\ = u^{\prime}_{{L_{1} }} \left( {h^{\prime}_{{\frac{n}{2}}} - h^{\prime}_{{\frac{n}{2} - 2}} } \right)\quad \quad - u^{\prime}_{{L_{2} }} \left( {h^{\prime}_{{\frac{n}{2}}} - h^{\prime}_{{\frac{n}{2} - 1}} } \right) + u^{\prime}_{{L_{0} }} \left( {h^{\prime}_{{\frac{n}{2} - 1}} - h^{\prime}_{{\frac{n}{2} - 2}} } \right). \hfill \\ \end{gathered}$$
(8)

Then we go to the next step. Now, we apply score \(S_{3k}+2\) for \(k=1\). If \(S_5 (A) > S_5 (B)\), then \(A>B\). If \(S_5 (A) < S_5 (B)\), then \(A<B\). If \(S_5 (A) = S_5 (B)\), and we have

$$\begin{gathered} u_{{L_{1} }} \left( {h_{{\frac{n}{2}}} - 2h_{{\frac{n}{2} - 1}} + h_{{\frac{n}{2} - 2}} } \right) - u_{{L_{2} }} \left( {h_{{\frac{n}{2}}} - h_{{\frac{n}{2} - 1}} } \right) - u_{{L_{0} }} \left( {h_{{\frac{n}{2} - 1}} - h_{{\frac{n}{2} - 2}} } \right) \hfill \\ = u^{\prime}_{{L_{1} }} \left( {h^{\prime}_{{\frac{n}{2}}} - 2h^{\prime}_{{\frac{n}{2} - 1}} \qquad + h^{\prime}_{{\frac{n}{2} - 2}} } \right) - u^{\prime}_{{L_{2} }} \left( {h^{\prime}_{{\frac{n}{2}}} - h^{\prime}_{{\frac{n}{2} - 1}} } \right) - u^{\prime}_{{L_{0} }} \left( {h^{\prime}_{{\frac{n}{2} - 1}} - h^{\prime}_{{\frac{n}{2} - 2}} } \right). \hfill \\ \end{gathered}$$
(9)

Then we go to the next step. Now, we apply score \(S_{3k}+3\) for \(k=1\). If \(S_6 (A) > S_6 (B),\) then \(A>B\). If \(S_6 (A) < S_6 (B),\) then \(A<B\). If \(S_6 (A) = S_6 (B),\) and we have

$$\begin{aligned} u_{L_1} = u'_{L_1}. \end{aligned}$$
(10)

Now, by adding Eqs. 8 and 9, we get

$$\begin{aligned} (u_{L_1}-u_{L_2})(h_{\frac{n}{2}} - h_{\frac{n}{2}-1})= (u'_{L_1}-u'_{L_2})(h'_{\frac{n}{2}} - h'_{\frac{n}{2}-1}). \end{aligned}$$
(11)

By subtracting the Eqs. 9 from 8, we get

$$\begin{aligned} (u_{L_1}+u)(h_{\frac{n}{2}-1} - h_{\frac{n}{2}-2})= (u'_{L_1}+u')(h'_{\frac{n}{2}-1} - h'_{\frac{n}{2}-2}). \end{aligned}$$
(12)

By applying Eq. 10 in Eqs. 11 and 12, we get the following Eqs.:

$$\begin{aligned}{} & {} (u_{L_2})(h_{\frac{n}{2}} - h_{\frac{n}{2}-1})= (u'_{L_2})(h'_{\frac{n}{2}} - h'_{\frac{n}{2}-1}), \end{aligned}$$
(13)
$$\begin{aligned}{} & {} (h_{\frac{n}{2}-1} - h_{\frac{n}{2}-2})= (h'_{\frac{n}{2}-1} - h'_{\frac{n}{2}-2}). \end{aligned}$$
(14)

Then we go to the next step. Now we apply score \(S_{3k}+1\) for \(k=2\). If \(S_7 (A) > S_7 (B)\), then \(A>B\). If \(S_7 (A) < S_7 (B)\), then \(A<B\). If \(S_7 (A) = S_7 (B)\), then we have

$$\begin{aligned}&u_{L_2} (h_{\frac{n}{2}-1} - h_{\frac{n}{2}-3}) - u_{L_{3}} (h_{\frac{n}{2}-1} - h_{\frac{n}{2}-2}) + u_{L_{1}} (h_{\frac{n}{2}-2} - h_{\frac{n}{2}-3}) \nonumber \\&\quad = u'_{L_2} (h'_{\frac{n}{2}-1} - h'_{\frac{n}{2}-3})\nonumber \\&\quad \quad - u'_{L_{3}} (h'_{\frac{n}{2}-1} - h'_{\frac{n}{2}-2}) + u'_{L_{1}} (h'_{\frac{n}{2}-2} - h'_{\frac{n}{2}-3}) \end{aligned}$$
(15)

Now we apply score \(S_{3k}+2\) for \(k=2\). If \(S_8 (A) > S_8 (B)\), then \(A>B\). If \(S_8 (A) < S_8 (B)\), then \(A<B\). If \(S_8 (A) = S_8 (B)\), and we have

$$\begin{aligned}&u_{L_2} (h_{\frac{n}{2}-1}-2h_{\frac{n}{2}-2} + h_{\frac{n}{2}-3}) - u_{L_{3}} (h_{\frac{n}{2}-1} - h_{\frac{n}{2}-2}) \nonumber \\&\quad \quad - u_{L_{1}} (h_{\frac{n}{2}-2} - h_{\frac{n}{2}-3}) \nonumber \\&\quad =u'_{L_2} (h'_{\frac{n}{2}-1}- 2h'_{\frac{n}{2}-2} +h'_{\frac{n}{2}-3}) - u'_{L_{3}} (h'_{\frac{n}{2}-1} - h'_{\frac{n}{2}-2}) \nonumber \\&\qquad - u'_{L_{1}} (h'_{\frac{n}{2}-2} - h'_{\frac{n}{2}-3}). \end{aligned}$$
(16)

Then we go to the next step. Now, we apply score \(S_{3k}+3\) for \(k=2\). If \(S_9 (A) > S_9 (B)\), then \(A>B\). If \(S_9 (A) < S_9 (B)\), then \(A<B\). If \(S_9 (A) = S_9 (B)\), then we have

$$\begin{aligned} u_{L_2} = u'_{L_2}. \end{aligned}$$
(17)

Now adding Eqs. 15 and 16, we get

$$\begin{aligned} (u_{L_2}-u_{L_3})(h_{\frac{n}{2}-1} - h_{\frac{n}{2}-2})= (u'_{L_2}-u'_{L_3})(h'_{\frac{n}{2}-1} - h'_{\frac{n}{2}-2}). \end{aligned}$$
(18)

By subtracting the Eq. 16 from 15, we get

$$\begin{aligned} (u_{L_2}+u_{L_1})(h_{\frac{n}{2}-2} - h_{\frac{n}{2}-3})= (u'_{L_2}+u'_{L_1})(h'_{\frac{n}{2}-2} - h'_{\frac{n}{2}-3}). \end{aligned}$$
(19)

By applying Eq. 17 in Eqs. 18 and 19, we get the following Eqs.:

$$\begin{aligned}{} & {} (u_{L_3})(h_{\frac{n}{2}-1} - h_{\frac{n}{2}-2})= (u'_{L_3})(h'_{\frac{n}{2}-1} - h'_{\frac{n}{2}-2}), \end{aligned}$$
(20)
$$\begin{aligned}{} & {} (h_{\frac{n}{2}-2} - h_{\frac{n}{2}-3})= (h'_{\frac{n}{2}-2} - h'_{\frac{n}{2}-3}). \end{aligned}$$
(21)

Now by substituting Eq. 17 in 13 and since \(h_{\frac{n}{2}}=h'_{\frac{n}{2}}\), we get

$$\begin{aligned} h_{\frac{n}{2}-1} = h'_{\frac{n}{2}-1}. \end{aligned}$$
(22)

By substituting Eq. 22 in 14, we get

$$\begin{aligned} h_{\frac{n}{2}-2} = h'_{\frac{n}{2}-2} \end{aligned}$$
(23)

By substituting Eq. 23 in 21, we get

$$\begin{aligned} h_{\frac{n}{2}-3} = h'_{\frac{n}{2}-3}. \end{aligned}$$
(24)

From Eq. 20, we get \(u_{L_3} = u'_{L_3}\). Now we go to the next step, by induction hypothesis and continuing this process up to \(k=\frac{n}{2}-2\), we get \(u_{L_{\frac{n}{2}-2}} = u'_{L_{\frac{n}{2}-2}}, u_{L_{\frac{n}{2}-3}} = u'_{L_{\frac{n}{2}-3}},\ldots u_{L_{1}} = u'_{L_{1}}, u=u'\) and \(h_1=h'_1, h_2=h'_2,\ldots h_{\frac{n}{2}}= h'_{\frac{n}{2}},h_{\frac{n}{2}+1} = h'_{\frac{n}{2}+1}.\) Now, since we have \(k=\frac{n}{2}-1,\) we go to the next step.

Now, we apply score \(S'_{3k}+1\) for \(k=1\). If \(S'_4 (A) > S'_4 (B)\), then \(A>B\). If \(S'_4 (A) < S'_4 (B)\), then \(A<B\). If \(S'_4 (A) = S'_4 (B)\), then we have

$$\begin{aligned}&u_{R_1} (h_{\frac{n}{2}+1} - h_{\frac{n}{2}+3}) - u_{R_{2}} (h_{\frac{n}{2}+1} - h_{\frac{n}{2}+2}) + u_{L_{0}} (h_{\frac{n}{2}+2} - h_{\frac{n}{2}+3}) \nonumber \\&\quad =u'_{R_1} (h'_{\frac{n}{2}+1} - h'_{\frac{n}{2}+3}) - u'_{R_{2}} (h'_{\frac{n}{2}+1} - h'_{\frac{n}{2}+2}) + u'_{L_{0}} (h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+3}). \end{aligned}$$
(25)

Then we go to the next step. Now, we apply score \(S'_{3k}+2\) for \(k=1\). If \(S'_5 (A) > S'_5 (B)\), then \(A>B\). If \(S'_5 (A) < S'_5 (B)\), then \(A<B\). If \(S'_5 (A) = S'_5 (B)\), and we have

$$\begin{aligned}&u_{R_1} (h_{\frac{n}{2}+1}-2h_{\frac{n}{2}+2} + h_{\frac{n}{2}+3}) - u_{R_{2}} (h_{\frac{n}{2}+1} - h_{\frac{n}{2}+2}) \nonumber \\&\qquad - u_{R_{0}}(h_{\frac{n}{2}+2} - h_{\frac{n}{2}+3}) \nonumber \\&\quad =u'_{R_1} (h'_{\frac{n}{2}+1}-2h'_{\frac{n}{2}+2} + h'_{\frac{n}{2}+3}) - u'_{R_{2}} (h'_{\frac{n}{2}+1} - h'_{\frac{n}{2}+2}) \nonumber \\&\qquad - u'_{R_{0}} (h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+3}). \end{aligned}$$
(26)

Now, we go to the next step and apply score \(S'_{3k}+3\) for \(k=1\). If \(S'_6 (A) > S'_6 (B)\), then \(A>B\). If \(S'_6 (A) < S'_6 (B)\), and \(A<B\). If \(S'_6 (A) = S'_6 (B)\), then we have

$$\begin{aligned} u_{R_1} = u'_{R_1}. \end{aligned}$$
(27)

Now by adding Eqs. 25 and 26, we get

$$\begin{aligned} (u_{R_1}-u_{R_2})(h_{\frac{n}{2}+1} - h_{\frac{n}{2}+2})= (u'_{R_1}-u'_{R_2})(h'_{\frac{n}{2}+1} - h'_{\frac{n}{2}+2}). \end{aligned}$$
(28)

By subtracting Eq. 26 from 25, we get

$$\begin{aligned} (u_{R_1}+u)(h_{\frac{n}{2}+2} - h_{\frac{n}{2}+3})= (u'_{R_1}+u)(h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+3}). \end{aligned}$$
(29)

By applying Eq. 27 in Eqs. 28 and 29, we get the following Eqs.

$$\begin{aligned}{} & {} (u_{R_2})(h_{\frac{n}{2}+1} - h_{\frac{n}{2}+2})= (u'_{R_2})(h'_{\frac{n}{2}+1} - h'_{\frac{n}{2}+2}),\end{aligned}$$
(30)
$$\begin{aligned}{} & {} (h_{\frac{n}{2}+2} - h_{\frac{n}{2}+3})= (h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+3}). \end{aligned}$$
(31)

Now we go to next step and apply score \(S'_{3k}+1\) for \(k=2\). If \(S'_7 (A) > S'_7 (B)\), then \(A>B\). If \(S'_7 (A) < S'_7 (B)\), then \(A<B\). If \(S'_7 (A) = S'_7 (B)\), and we have

$$\begin{aligned}&u_{R_2} (h_{\frac{n}{2}+2} - h_{\frac{n}{2}+4}) - u_{R_{3}} (h_{\frac{n}{2}+2} - h_{\frac{n}{2}+3}) \nonumber \\&\qquad + u_{R_{1}}(h_{\frac{n}{2}+3} - h_{\frac{n}{2}+4}) \nonumber \\&\quad =u'_{R_2} (h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+4}) - u'_{R_{3}} (h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+3}) \nonumber \\&\qquad + u'_{R_{1}} (h'_{\frac{n}{2}+3} - h'_{\frac{n}{2}+4}). \end{aligned}$$
(32)

Now we apply score \(S'_{3k}+2\) for \(k=2\). If \(S'_8 (A) > S'_8 (B)\), then \(A>B\). If \(S'_8 (A) < S'_8 (B)\), and \(A<B\). If \(S'_8 (A) = S'_8 (B)\), then we have

$$\begin{aligned}&u_{R_2} (h_{\frac{n}{2}+2}-2h_{\frac{n}{2}+3} + h_{\frac{n}{2}+4}) - u_{R_{3}} (h_{\frac{n}{2}+2} - h_{\frac{n}{2}+3}) \nonumber \\&\qquad - u_{R_{1}}(h_{\frac{n}{2}+3} - h_{\frac{n}{2}+4}) \nonumber \\&\quad = u'_{R_2} (h'_{\frac{n}{2}+2}-2h'_{\frac{n}{2}+3} + h'_{\frac{n}{2}+4}) - u'_{R_{3}} (h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+3}) \nonumber \\&\qquad - u'_{R_{1}} (h'_{\frac{n}{2}+3} - h'_{\frac{n}{2}+4}). \end{aligned}$$
(33)

Now, we go to the next step and apply score \(S'_{3k}+3\) for \(k=2\). If \(S'_9 (A) > S'_9 (B)\), then \(A>B\). If \(S'_9 (A) < S'_9 (B)\), then \(A<B\). If \(S'_9 (A) = S'_9 (B)\), then we have

$$\begin{aligned} u_{R_2} = u'_{R_2}. \end{aligned}$$
(34)

Now by adding Eqs. 32 and 33, we get

$$\begin{aligned} (u_{R_2}-u_{R_3})(h_{\frac{n}{2}+2} - h_{\frac{n}{2}+3})= (u'_{R_2}-u'_{R_3})(h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+3}). \end{aligned}$$
(35)

By subtracting Eq. 33 from 32, we get

$$\begin{aligned} (u_{R_2}+u_{R_1})(h_{\frac{n}{2}+3} - h_{\frac{n}{2}+4})= (u'_{R_2}+u'_{R_1})(h'_{\frac{n}{2}+3} - h'_{\frac{n}{2}+4}). \end{aligned}$$
(36)

By applying Eq. 34 in Eqs. 35 and 19, we get the following Eqs.

$$\begin{aligned}{} & {} (u_{R_3})(h_{\frac{n}{2}+2} - h_{\frac{n}{2}+3})= (u'_{R_3})(h'_{\frac{n}{2}+2} - h'_{\frac{n}{2}+3}), \end{aligned}$$
(37)
$$\begin{aligned}{} & {} (h_{\frac{n}{2}+3} - h_{\frac{n}{2}+4})= (h'_{\frac{n}{2}+3} - h'_{\frac{n}{2}+4}). \end{aligned}$$
(38)

Now by substituting Eq. 34 in 30 and since \(h_\frac{n}{2}=h'_\frac{n}{2}\), we get

$$\begin{aligned} h_{\frac{n}{2}+2} = h'_{\frac{n}{2}+2}. \end{aligned}$$
(39)

Now by substituting Eq. 39 in 31, we get

$$\begin{aligned} h_{\frac{n}{2}+3} = h'_{\frac{n}{2}+3}. \end{aligned}$$
(40)

By substituting Eq. 40 in 38, we get

$$\begin{aligned} h_{\frac{n}{2}+4} = h'_{\frac{n}{2}+4} \end{aligned}$$
(41)

From Eq. 37, we get \(u_{R_3} = u'_{R_3}\). Now we go to the next step, by induction hypothesis, and continuing this process up to \(k=\frac{n}{2}-2\), we get \(u_{R_{\frac{n}{2}-2}} = u'_{R_{\frac{n}{2}-2}}, u_{R_{\frac{n}{2}-3}} = u'_{R_{\frac{n}{2}-3}},\ldots u_{R_{1}} = u'_{R_{1}}, u=u'\) and \(h_{\frac{n}{2}+2} = h'_{\frac{n}{2}+2},h_{\frac{n}{2}+3} = h'_{\frac{n}{2}+3},h_{\frac{n}{2}+4}= h'_{\frac{n}{2}+4},\ldots h_{n-2}=h'_{n-2},h_{n-1}=h'_{n-1}, h_n=h'_n.\) Therefore, we get \(h_i=h'_i\) for \(i \in {1,2,\ldots n}\) and \(u_{L_i}=u'_{L_i}, u_{R_i}=u'_{R_i}\) for \(i \in {1,2,\ldots \frac{n}{2}-2}\) and \(u=u',\) which implies that \(A=B\). \(\square\)

Remark 6

We have given a total ordering method for ‘n’gonal linear fuzzy numbers for any even \(n \ge 6\). To achieve total ordering for any class of ‘n’gonal linear fuzzy numbers if n is odd and \(\ge 5\), we apply the total ordering algorithm for \(n+1\) by considering ‘n’gonal linear fuzzy numbers as ‘n+1’gonal linear fuzzy numbers.

To demonstrate this easily, consider the following pentagonal fuzzy number \(A=(2,4,6,8,10; 1, 0.4,0.5)\), where ‘n’ is 5 (odd number). We rewrite this as \(A=(2,4,6,6,8,10; 1, 0.4,0.5),\) a hexagonal fuzzy number which is ‘n’gonal (\(n=6\),even). Hence, we can apply our proposed score function and ranking method.

Hence, note that our proposed total ordering method works for any \(n \in \mathbb {N},\) ‘n’gonal linear fuzzy numbers.

Remark 7

We demonstrate how our proposed algorithm attains total ordering for a particular class of octagonal fuzzy numbers. Let us assume that \(A = (h_1,h_2,h_3,h_4,h_5,h_6,h_7,h_8; u, u_{L_1}, u_{L_2}, u_{R_1}, u_{R_2})\) and \(B = (h'_1,h'_2,h'_3,h'_4,h'_5,h'_6,h'_7,h'_8; u', u'_{L_1},u'_{L_2}, u'_{R_1},u'_{R_2})\) are two octagonal fuzzy numbers. To prove total ordering on octagonal fuzzy numbers, it is enough to show that either \(A > B\) or \(A < B\) or \(A = B\). Suppose step 1, step 2 and step 3 are not able to differentiate A and B (i.e., \(A>B\) or \(A<B\)), we get \(u = u', h_{5} = h'_{5}, h_4 = h'_4\). Now, let us go to the next step by taking \(k=1\). Suppose step 4, step 5, step 6 are not able to differentiate A and B, then we have a system of Eqs. \(u_{L_2}(h_3) = u'_{L_2}(h'_3), h_3-h_2 = h'_3-h'_2, u_{L_1}=u'_{L_1}\). Now let us go to step 7 by taking \(k=2\); again we apply step 4, step 5 and step 6. If still we are not able to differentiate A and B at this stage, we have system of Eqs. \((u_{L_2})(h_3-h_2) = (u'_{L_2})(h'_3-h'_2), (u_{L_2}+u_{L_1})(h_2-h_1) = (u'_{L_2}+u'_{L_1})(h'_2-h'_1),u_{L_2}=u'_{L_2}\). By solving this system of Eqs., we conclude that \(h_1=h'_1,h_2=h'_2,h_3=h'_3,h_4=h'_4,u=u',u_{L_1}=u'_{L_1},u_{L_2}=u'_{L_2}\). Now in step 7, since \(k=3= \frac{n}{2}-1\), we go to step 8. In step 8 by taking \(k=1\), we go to step 9,10,11 if it is required. By substituting \(k=2\), in step 12, we repeat step 9,10,11 again if it is required. Now, we get \(h_5=h'_5,h_6=h'_6,h_7=h'_7,h_8=h'_8,u=u',u_{R_1}=u'_{R_1},u_{R_2}=u'_{R_2}.\) Therefore, we get \(A=B.\) Thus we have shown that our total ordering algorithm works for a particular case \(n=8.\)

5 Applications of the Proposed Method

5.1 Application to Multi-criteria Decision-Making Problems

Consider the following MCDM problem to show  the applicability of the proposed operations and total ordering method on ‘n’gonal linear fuzzy numbers. An investor has to invest his money in the following stocks A, B, C and D with respect to the criteria. Growth rate of the stock in last 5 years \((C_1)\), reliability \((C_2)\) and market value \((C_3)\) with weights as 0.4, 0.3 and 0.3,  respectively. The MCDM table attained from the expert is given in Table 1.

Table 1 MCDM data matrix
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Note that the MCDM matrix contains data as various types (hexagonal, pentagonal, trapezoidal, triangular) of fuzzy numbers. Now, we multiply the weights of the criteria to their corresponding fuzzy numbers in the column, which is shown in Table 2.

Table 2 Weighted MCDM data matrix
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The aggregated weighted value of \(A=AC_1+AC_2+AC_3 = (1.1,1.9,3.6,5,6.7,7.5; 1,.0.99,0.99)\); similarly aggregated weighted value of \(B=(1.74,2.34,4.17,5.7,7.5,7.8;0.875,0.825,0.825), C=(1.36,1.76,3.12,3.18,4.54,4.94; 1,1,1)\) and \(D=(1.6,3.2,4.5,5.2,6.2,6.9;1,0.99,0.99).\)

By applying midpoint score function of ‘n’ gonal linear fuzzy number for \(n=6,\) we find the score value of \(A = 4.3, B=4.32, C=3.15\) and \(D=4.85,\) which implies the desired ranking as \(D>B>A>C\). Now for the same decision problem, we apply some methods in literature. Resultant ranking order is given in Table 3. Since the proposed method is the extension of the method [29], the resultant ranking of the proposed method coincides with that of [29] as expected.

Table 3 Ranking order
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5.2 Application to MCDM via Similarity Measure

In pattern recognition, fuzzy similarity measure plays a significant role. In this subsection, we try to show how to extract similarity measure for generalized ‘n’gonal linear fuzzy number using our midpoint score function \(S_1\) of the proposed total ordering algorithm as an application of score function which may be extended by aggregating all score functions. Let \(\mathbb {H}\) denote the class of all GNLFNs. We know that a function \(S: \mathbb {H} \times \mathbb {H} \rightarrow [0,1]\) is said to a be similarity measure on GNLFNs, if \(\forall H_1,H_2,H_3 \in \mathbb {H}\), S satisfies the following conditions:

  1. 1.

    \(0< S(H_1,H_2) \le 1,\)

  2. 2.

    \(S(H_1,H_2)=1 \iff H_1 = H_2,\)

  3. 3.

    \(S(H_1,H_2)=S(H_2,H_1),\)

  4. 4.

    If \(H_1 \subseteq H_2 \subseteq H_3\) then \(S(H_1,H_2) \ge S(H_1,H_3)\) and \(S(H_2,H_3) \ge S(H_1,H_3).\)

Now, we define the similarity measure on GNLFNs \(S: \mathbb {H} \times \mathbb {H} \rightarrow [0,1]\) by using our midpoint score function \(S_1\) of the proposed method by \(S(H_1,H_2)=\frac{1}{1+\mid {\frac{u(h_{\frac{n}{2}} +h_{\frac{n}{2}+1})}{2}}-{\frac{u'(h_{\frac{n}{2}} +h'_{\frac{n}{2}+1})}{2}}\mid + \sum _{i=1 \ne \frac{n}{2}}^{n} \mid h_i - h'_i \mid + \mid u - u' \mid + \sum _{i=1}^{\frac{n-4}{2}} \mid u_{L_i} - u'_{L_i} \mid + \sum _{i=1}^{\frac{n-4}{2}} \mid u_{R_i} - u'_{R_i} \mid }\) where \(H_1 = (h_1, h_2, \dots h_n; u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}})\) and \(H_2 = (h'_1, h'_2, \dots h'_n; u', u'_{L_1}, u'_{L_2}, \dots u'_{L_{\frac{n-4}{2}}}, u'_{R_1}, u'_{R_2}, \dots u'_{R_{\frac{n-4}{2}}})\) are ‘n’gonal linear fuzzy numbers. By remark 1, assume that n is even and \(n\ge 6\) such that the class of generalized ‘n’gonal linear fuzzy numbers contains both \(H_1\) and \(H_2\). We consider \(H \subseteq G\) (H is dominated by G) if \(h_i \le h'_i, u \le u' \forall i \in \{1,2, \ldots n\}\), \(u_{L_i} \le u'_{L_i}\) and \(u_{R_i} \le u'_{R_i} \forall i \in \{1,2, \ldots \frac{n-4}{2}\}\). Now, we verify that the map S defined on GNLFNs is a similarity measure. Since \(1+\mid {\frac{u(h_{\frac{n}{2}} +h_{\frac{n}{2}+1})}{2}}-{\frac{u'(h_{\frac{n}{2}} +h'_{\frac{n}{2}+1})}{2}}\mid + \sum _{i=1 \ne \frac{n}{2}}^{n} \mid h_i - h'_i \mid + \mid u - u' \mid + \sum _{i=1}^{\frac{n-4}{2}} \mid u_{L_i} - u'_{L_i} \mid + \sum _{i=1}^{\frac{n-4}{2}} \mid u_{R_i} - u'_{R_i} \mid \ge 1\), we get \(S(H_1,H_2) \le 1\). Clearly, by definition \(S(H_1,H_2) > 0\). Thus, the first condition for similarity measure is satisfied. Assume that \(S(H_1,H_2)=1\), then we get \(\mid {\frac{u(h_{\frac{n}{2}} +h_{\frac{n}{2}+1})}{2}}-{\frac{u'(h_{\frac{n}{2}} +h'_{\frac{n}{2}+1})}{2}}\mid + \sum _{i=1 \ne \frac{n}{2}}^{n} \mid h_i - h'_i \mid + \mid u - u' \mid + \sum _{i=1}^{\frac{n-4}{2}} \mid u_{L_i} - u'_{L_i} \mid + \sum _{i=1}^{\frac{n-4}{2}} \mid u_{R_i} - u'_{R_i} \mid = 0\) which implies \(H_1=H_2\). Conversely assume that \(H_1=H_2\), i.e.., \(h_i=h'_i,u=u',u_{L_i}=u'_{L_i}, u_{R_i}=u'_{R_i} \forall i \in \{1,2,\ldots n\}\) which implies \(\mid {\frac{u(h_{\frac{n}{2}} +h_{\frac{n}{2}+1})}{2}}-{\frac{u'(h_{\frac{n}{2}} +h'_{\frac{n}{2}+1})}{2}}\mid + \sum _{i=1 \ne \frac{n}{2}}^{n} \mid h_i - h'_i \mid + \mid u - u' \mid + \sum _{i=1}^{\frac{n-4}{2}} \mid u_{L_i} - u'_{L_i} \mid + \sum _{i=1}^{\frac{n-4}{2}} \mid u_{R_i} - u'_{R_i} \mid = 0\). Therefore \(S(H_1,H_2) = 1\). Thus, the second condition for similarity measure is satisfied. The third condition for similarity measure \(S(H_1,H_2)= S(H_2,H_1)\) is obvious. Let \(H_1=(h_1, h_2, \dots h_n; u, u_{L_1}, u_{L_2}, \dots u_{L_{\frac{n-4}{2}}}, u_{R_1}, u_{R_2}, \dots u_{R_{\frac{n-4}{2}}}), H_2 = (h'_1, h'_2, \dots h'_n; u', u'_{L_1}, u'_{L_2}, \dots u'_{L_{\frac{n-4}{2}}}, u'_{R_1}, u'_{R_2}, \dots u'_{R_{\frac{n-4}{2}}})\) and \(H_3= (h''_1, h''_2, \dots h''_n; u'', u''_{L_1}, u''_{L_2}, \dots u''_{L_{\frac{n-4}{2}}}, u''_{R_1}, u''_{R_2}, \dots u''_{R_{\frac{n-4}{2}}})\) be GNLFNs. Assume that \(H_1 \subseteq H_2 \subseteq H_3,\) which implies that \(\mid {\frac{u(h_{\frac{n}{2}} +h_{\frac{n}{2}+1})}{2}}-{\frac{u'(h_{\frac{n}{2}} +h'_{\frac{n}{2}+1})}{2}}\mid \le \mid {\frac{u(h_{\frac{n}{2}} +h_{\frac{n}{2}+1})}{2}}-{\frac{u''(h''_{\frac{n}{2}} +h''_{\frac{n}{2}+1})}{2}}\mid ,\) \(\sum _{i=1 \ne \frac{n}{2}}^{n} \mid h_i - h'_i \mid \le \sum _{i=1 \ne \frac{n}{2}}^{n} \mid h_i - h''_i \mid , \mid u - u' \mid \le \mid u-u'' \mid , \sum _{i=1}^{\frac{n-4}{2}} \mid u_{L_i} - u'_{L_i} \mid \le \sum _{i=1}^{\frac{n-4}{2}} \mid u_{L_i} - u''_{L_i} \mid , \sum _{i=1}^{\frac{n-4}{2}} \mid u_{R_i} - u'_{R_i} \mid \le \sum _{i=1}^{\frac{n-4}{2}} \mid u_{R_i} - u''_{R_i} \mid\). Thus, we get \(S(H_1,H_2) \ge S(H_1,H_3)\). Similarly, we can prove that \(S(H_2,H_3) \ge S(H_1,H_3)\). Thus, the fourth condition for similarity measure is satisfied. Hence, we proved that the defined function S is a similarity measure on \(\mathbb {H}\).

Now, consider the following MCDM problem. Assume that in an interview, a decision maker has to select a candidate from four candidates (\(A_1,A_2,A_3\) and \(A_4\)) with respect to the following criteria (problem solving skill \((C_1)\), communication skill \((C_2)\) and subject knowledge \((C_3)\) ). Now a GTPFN-based decision table has been formed by the experts were its range belongs to [0, 100] after examine the candidates (Tables 4, 5). Now, we are going to find the preference order by using similarity measure. We restrict the proposed similarity measure to GTPFNs \(S: \mathbb {T} \times \mathbb {T} \rightarrow [0,1]\) as \(S(T_1,T_2)=\frac{1}{1+\mid \frac{u(t_2+t_3)}{2}-\frac{u'(t'_2+t'_3)}{2}\mid +\mid t_1-t'_1\mid +\mid t_3 -t'_3 \mid +\mid t_4-t'_4\mid + \mid u - u' \mid },\) where \(T_1=(t_1,t_2,t_3,t_4;u)\) \(T_2=(t'_1,t'_2,t'_3,t'_4;u')\) and \(H \subseteq G\) (H is dominated by G) means \(t_i \le t'_i, u \le u' \forall i \in \{1,2, \ldots 4\}\). Now, the map S defined on GTPFNs is clearly a similarity measure. First we find that optimum value of this MCDM ratings is \(*=(1,1,1,1;1)\). Then we find the similarity measure between ratings of each alternatives with the optimum value \(*\) as shown in Table 6. Then we rank the alternatives from highest similarity with optimum value to lowest similarity with optimum value. Now the similarity measure value of \(A_1=\frac{0.39+0.28+0.25}{2} = 0.31\). Similarly, we find \(A_2=0.6, A_3=0.38\) and \(A_4=0.75\). Thus, our preferred order of alternatives will be \(A_4>A_2>A_3>A_1\). Likewise, we can extract a similarity measure for a particular class of fuzzy numbers by the one or aggregation of more score functions in our proposed method, which is a useful application to pattern recognition and MCDM problems.

Table 4 Decision matrix from the experts
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Table 5 Normalized decision matrix
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Table 6 Similarity measure values of the alternatives with respect to optimum value
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6 Significance of the Proposed Method and Comparative Analysis

Our proposed total algorithm will be a useful and beneficial tool to deal with the MCDM problem and similarity measure-based MCDM problems as we have shown in the last Sect. 5. Furthermore, it can be extended to various situations of MCDM problems. For example, consider a real-life MCDM situation where data can be collected in various forms of fuzzy numbers. One decision maker might deal with the problem with trapezoidal fuzzy information, another decision maker might deal with the same problem with hexagonal fuzzy number, and another decision maker might approach the same problem with another ‘n’ sided fuzzy number. Then each decision maker would apply appropriate ordering algorithm in literature corresponding to the fuzzy number with which they are dealing. Since each ordering algorithms need not to be coincided and each method has its own merits and demerits, we might have a different ordering preference for each decision maker for the same real-life MCDM situation, which is considered as a major drawback in this area of study. Now when each decision maker follows our proposed algorithm, their preference order will be the same and more reliable because of our method’s uniformity, which makes us to claim that our proposed method is superior to other extension-based fuzzy numbers in dealing with MCDM problems. To demonstrate this, consider the same MCDM problem assigned to two decision makers. Both of them are going to apply hexagonal and octagonal fuzzy number, respectively, as the ratings of the alternatives with respect to the criteria. Suppose that a decision maker, who applies hexagonal fuzzy number in his MCDM table, gets the ranking order as \(A<B\) for the following two GHXFNs: \(A= (5,10,15,20,25,30; 1,0.5,0.5)\) and \(B=(2,4,18,20,22,24; 1,0.5,0.5)\) by [29]. Another decision maker, who applies octagonal fuzzy number in his MCDM table, gets the ranking order as \(A>B\) for the same hexagonal fuzzy numbers \(A= (5,10,15,20,25,30; 1,0.5,0.5)\) and \(B=(2,4,18,20,22,24; 1,0.5,0.5)\) as GONFNs by [30]. Therefore for the same MCDM problem, we get different preferences in the alternatives. Here, we are not able to use the method [29] for GONFNs, since [29] is dealing up to GHXFNs. Now for the same MCDM problem, when both decision makers use our proposed algorithm for their GHXFNs, GONFNs, we get the ranking as \(A<B\) from both the decision makers, which is more sensible, and we will get the same preference order even if we use different extensions of fuzzy numbers-based MCDM table for the same scenario. Now, we compare our proposed method with existing methods in the literature as follows.

6.1 Comparison with Existing Ordering Methods for Generalized Trapezoidal Fuzzy Number

In [23], we have two GTPFNs \(A=(0.1,0.2,0.4,0.5;1)\) and \(B=(0.1,0.3,0.3,0.5;1),\) which are ranked as \(A>B\). Our method also suggests the same ranking for these two GTPFNs. In [23], we have two GTPNs \(A=(3, 5, 5,7; 1)\) and \(B=(4,5,5,\frac{51}{8}; 1),\) which are ranked as \(A<B\) that is against our intuition and our method ranks \(A>B\). The ordering method given in [23] is claimed to be better than Yager, Fortemps Roubens, Liou and Wang, Chen methods by getting results as \(D>B>C>A\) for the following four GTPFNs \(A=(0.1.0.2,0.2,0.3;1), B=(0.2,0.5,0.5,0.8;1), C=(0.3,0.4,0.4,0.9;1)\) and \(D=(0.6,0.7,0.7,0.8;1)\). Our method also gets the same ranking. For the following GTPFNs \(A=(4,5,5,6;1), B=(1.75,2,2,3;1), C=(7,9,9,10;1)\) and (1, 1, 2, 2; 1), the ordering method in [24] ranks \(C>A>D>B\), whereas our method ranks \(C>A>B>D\). For the following GTPFNs \(A=(0.1,0.3,0.3,0.5;1)\) and \(B=(0.2,0.3,0.3,0.4;1)\), the ordering method in [25] ranks \(A<B\), whereas our method ranks \(A>B.\)

6.2 Comparison with Existing Ordering Methods for Generalized HXFN

Consider the following generalized HXFNs. Let \(A= (5,10,15,20,25,30; 1,0.5,0.5)\) and \(B=(2,4,18,20,22,24; 1,0.5,0.5)\). By Haar wavelet ranking method [27], we get the ordering as \(A>B\). From our method, we get \(A<B\) in first step itself, which is more acceptable. Further, by using Haar wavelet method, the ranking is made without considering the heights of the hexagonal fuzzy numbers, which leads to a misunderstanding. For example, consider two generalized hexagonal fuzzy numbers \(C = (3,6,9,12,15,18; 1, 0.8,0.4)\) and \(D = (3,6,9,12,15,18; 1,0.6,0.3)\), by using Haar wavelet method, the ranking would be given as \(C=D\) which is absurd. But when we use our method, we would get the ranking as \(C>D\) which is more reliable. We also note that Haar wavelet method does not derive a total ordering between GHXFNs, since it ranks the HXFN \((h_1,h_2,h_3,h_4,h_5,h_6;\, 1,0.5,0.5)\) and the set of all HXFNs \(\{(h_1,h_2,h_3,h_4,h_5,h_6; u,u_L,u_R)/ u, u_L, u_R \in \mathbb {R}\}\) as same which contradicts the total ordering. But we have shown that our method holds total ordering.

Consider the following two generalized hexagonal fuzzy numbers \(A= (0.02,0.03,0.05,0.06,0.08,0.09; 0.7,0.35,0.35)\) and \(B=(0.04,0.06,0.1,0.12,0.16,0.18; 0.35,0.175,0.175)\). By ordering of generalized hexagonal fuzzy number using rank, mode, divergence and spread method [28], the ordering is \(A>B\) after four steps. From our method, we get \(A>B\) in the third step itself. In the same method for HXFN numbers, \(C= (0.17,0.2,0.25,0.38,0.4,0.45; 0.4,0.2,0.2)\) and \(D= (0.34,0.4,0.5,0.76,0.8,0.9; 0.8,0.4,0.4)\), \(C>D\) which is illogical, but our method gives \(C<D\) which is more acceptable.

Consider the following two generalized hexagonal fuzzy numbers \(A= (1,2,3,4,5,6; 1,0.5,0.5)\) and \(B=(3,3,3,4,6,7;1,0.5,0.5 )\). By [29], we get the ordering as \(A>B\) after five steps. From our method, we get \(A>B\) in the fourth step itself. Consider the following GHXFNs: \(A=(1,2,3,4,5,6;1,0.5,0.5)\) and \(B=(2,2.4,2.5,4,5,6;1,0.5,0.5)\); by applying the ordering method in [15] we get \(A=B,\) whereas our method give the preference as \(A>B\). Our method works better because it inherits total ordering. Further, our proposed algorithm works for any ‘n’gonal linear fuzzy numbers which will be the main tool for approximation any nonlinear fuuzy numbers without much loss of data/information.

6.3 Comparison with Existing Ordering Methods for Generalized Octagonal Fuzzy Numbers

Consider the following generalized octagonal fuzzy numbers \(A= (5,10,12,15,20,25,30,35; 1,0.5,0.25,0.5,0.25)\) and \(B=(2,4,12,12,15,15,20,40; 1,0.5,0.25,0.5,0.25)\). By Haar wavelet ranking method [30], we get the ordering as \(A<B.\) From our method, we get \(A>B\) which is more acceptable. For the same GNOFN numbers, we get the ranking as \(A>B\) by using the score function which is given in [32]. It coincides with our ranking for these two generalized octagonal fuzzy numbers. But, when we use the same method for the following generalized octagonal fuzzy numbers \(A'= (12,14,16,24,26,30,32,34; 1,0.5,0.25,0.5,0.25)\) and \(B'=(18,18.5,19,20,22,30,31,40; 1,0.5,0.25,0.5,0.25)\), we get the preference order as \(A'<B'\) by [32], but our method gives the ranking order as \(A'>B'\) which shows that our method and study on solving octagonal fuzzy numbers using modified vogel’s approximation method need not to give same ranking always.

Moreover, in Haar wavelet method [30] and [32], the ranking is made without considering the heights of the generalized octagonal fuzzy numbers, which leads to a misunderstanding. For example, consider two generalized octagonal fuzzy numbers \(C = (2,4,6,9,12,15,18,20; 0.8,0.4,0.2,0.4,0.2)\) and \(D = (2,4,6,9,12,15,18,20; 0.6,0.3,0.2,0.3,0.2)\). By using Haar wavelet method [30] and [32] the ranking would be given as \(C=D\) which is absurd. But when we use our method, we would get the ranking as \(C>D\) which is more acceptable. We also note that these two methods do not derive a total ordering between generalized octagonal fuzzy numbers, since both of them rank the generalized octagonal fuzzy number \((h_1,h_2,h_3,h_4,h_5,h_6; 1,0.5,0.25,0.5,0.25)\) and the set of all generalized octagonal fuzzy numbers \(\{(h_1,h_2,h_3,h_4,h_5,h_6; u,u_{L_1},u_{L_2},u_{R_1},u_{R_2})/ u, u_{L_1},u_{L_2},u_{R_1},u_{R_2} \in \mathbb {R}\}\) as same, which contradicts the total ordering. But we have shown that our method holds total ordering.

7 Limitations and Future Scope of the Proposed Method

To derive total ordering in any generalized ‘n’gonal linear fuzzy number for any \(n < 6\), it can be identified or can be written as a generalized hexagonal fuzzy number and then the ordering algorithm can be applied as we mentioned earlier in remarks 1, 2, and 6. However, we have a minor limitation in this methodology which can be rectified as follows. To demonstrate the limitation of our proposed method, we consider the class of GTPFNs. There are four ways to view any GTPFN \(T = (t_1,t_2,t_3,t_4; u)\) as GHXFN as follows:

  • \(T=(t_1, t_2, t_2, t_3, t_3, t_4; u,u,u),\)

  • \(T=(t_1,t_1,t_2,t_3,t_3,t_4; u,0,u),\)

  • \(T=(t_1,t_2,t_2,t_3,t_4,t_4;u,u,0),\)

  • \(T=(t_1,t_1,t_2,t_3,t_4,t_4;u,0,0).\)

Here, when we write \(T=(t_1,t_2,t_3,t_4;\,u)\) as \(T =(t_1, t_2, t_2, t_3, t_3, t_4;\, u,u,u)\), from our proposed algorithm we get score functions as \(S_1 = \frac{u(t_2 + t_3)}{2}, S_2=\frac{u(t_3-t_2)}{2}, S_3=u, S_4=\frac{u(t_2-t_1)}{2}, S_5=\frac{u(t_1-t_2)}{2}, S_6= u, S_7=\frac{u(t_4-t_3)}{2}, S_8 =0, S_9=u\). Clearly, these score functions will determine a total ordering on GTPFNs. Hence, we suggest this way to view any GTPFN \(T = (t_1,t_2,t_3,t_4;\, u)\) as GHXFN, when applying our proposed ordering method.

Now, if we want to represent any GTPFN as GHXFN in the other three ways, we have the following limitations. While we write \(T=(t_1,t_2,t_3,t_4;u)\) as \(T=(t_1,t_1,t_2,t_3,t_3,t_4; u,0,u)\), from our proposed algorithm we get score functions as \(S_1 = \frac{u(t_2 + t_3)}{2}, S_2=\frac{u(t_3-t_2)}{2}, S_3=u, S_4,S_4,S_6=0, S_7=\frac{u(t_4-t_3)}{2}, S_8 =\frac{u(t_3-t_4)}{4}\)and \(S_9=u,\) which does not inherit total ordering. To overcome this limitation and to derive total ordering, we need to add another function \(S_{10} = \frac{u(t_2-t_1)}{2}\) with the condition that‘ if \(S_{10} (A) > S_{10} (B),\) then \(A>B\)’.

Similarly, while we represent a GTPFN \(T=(t_1,t_2,t_3,t_4;\,u)\) as a GHXFN \(T=(t_1,t_2,t_2,t_3,t_4,t_4;\,u,u,0)\), from our proposed algorithm we get a set of score functions which does not inherit total ordering. To overcome this limitation and to derive total ordering, we need to add another function \(S_{10} = \frac{u(t_4-t_3)}{2}\) with the condition that‘ if \(S_{10} (A) > S_{10} (B),\) then \(A>B\)’.

Similarly, while we represent a GTPFN \(T=(t_1,t_2,t_3,t_4;u)\) as a GHXFN \(T=(t_1,t_2,t_2,t_3,t_4,t_4;\,u,u,0)\), from our proposed algorithm we get a set of score functions which does not gives total ordering. To overcome this limitation and to derive total ordering, we need to add another two function \(S_{10} = \frac{u(t_2-t_1)}{2}\), \(S_{11} = \frac{u(t_4-t_3)}{2}\) with the conditions that‘ if \(S_{10} (A) > S_{10} (B),\) then \(A>B\) and if \(S_{10} (A) > S_{10} (B),\) and \(A>B\)’.

In future, this work can be extended to find similarity measure for ’n’gonal linear fuzzy numbers by using all the score function of our proposed method. Moreover, total ordering on nonlinear ‘n’gonal fuzzy number can be analyzed in future research. An approximation of convex nonlinear fuzzy number and nonconvex nonlinear fuzzy number by ‘n’gonal fuzzy number can also be discussed in the future.

8 Conclusion

In this work, we have proposed a total ordering method for ‘n’gonal linear fuzzy numbers. We have also shown that how the proposed total algorithm works for some of its particular cases (when \(n=8\)) like hexagonal and octagonal fuzzy numbers and we have shown a comparative analysis with existing ranking methods in the literature. There are many ranking methods exist for various fuzzy numbers like triangular(\(n=3\)), trapezoidal(\(n=4\)), hexagonal(\(n=6\)), octagonal(\(n=8\)) fuzzy numbers which are quit useful in MCDM problem. Since each of those ranking methods are different and some ranking methods could be better than another method in particular situations, the decision maker has to use an appropriate ranking method for a particular problem. Moreover, some research work presents a ranking method which works very well for hexagonal fuzzy number and it may have some shortfalls when it is extended for further fuzzy numbers like octagonal fuzzy numbers. Similarly, some research work presents a ranking method which works very well for octagonal fuzzy number and it may have some shortfalls when it is used for trapezoidal and triangular fuzzy numbers. So, a decision maker may require to deal with each kind of particular fuzzy number by choosing different ranking methods which makes him unable to use a same structured ranking principles to various fuzzy numbers. Since different ranking methods for various types of linear fuzzy numbers are used, the reliability of such MCDM is questioned which can be solved by the proposed uniformed ranking method for any ‘n’.