An Innovative Ranking Method for Hexagonal Intuitionistic Fuzzy Transportation Issues ^†

Abstract

In tackling real-world transportation challenges, researchers experience various levels of uncertainty and hesitancy caused by diverse unknown circumstances. An intuitionistic fuzzy kind of information that addresses uncertainty and unwillingness has been suggested by a number of authors. Therefore, we examine transportation problems with supply, demand, expenses, ambiguity, and hesitancy in the current paper. We suggest an innovative ranking method that uses intuitionistic hexagonal fuzzy numbers to explain the problem and reflect the involvement and non-involvement elements of the fuzzy number. Three image planes are subtracted to obtain the fuzzy numbers of participation and non-participation regions, which yield the centroid of centroids of such plane images. The ranking index is explained. Additionally, a mathematical framework is employed to illustrate the effectiveness of the suggested approach.

1. Introduction

A specific kind of linear programming issue known as the “classical transportation problem” involves sending a single homogenous good from one location to another while ensuring that the overall cost of transportation is kept low. Hitchcock presented and defined the fundamental transportation problem in 1941, wherein the amounts of supply and demand, as well as the expenses associated with transport, are all discrete numbers. However, in practice, a transportation problem’s limits may be ambiguous because of a variety of unpredictable circumstances. Atanassov [1,2] proposed the concept of intuitionistic fuzzy sets in 1986 to deal with ambiguity or hesitation. The primary benefit of IFSs is that they capture each element’s degree of belonging or not belonging within the set. Bellmann and Zadeh [3] and A strategy for determining the best solution to the transportation problem with fuzzy coefficients was suggested by Chanas and Kutcha [4] in 1996. Numerous writers have explored different approaches for solving the fuzzy transportation problem (FTP). Hussain, R.J. and Senthil Kumar, P. [5] are introduced Algorithm for solving intuitionistic fuzzy transportation problem. Applying the zero-suffix approach, Gani and Abbas [6] resolved the intuitionistic fuzzy transportation problem (IFTP) with a triangular membership function in 2012. Nagoor Gani and Abbas [7], were introduced new method for solving Intuitionistic Fuzzy Transportation problem. O’Heigeartaigh [8] presented a triangle membership function FTP solution in 1982. A novel approach, the fuzzy zero-point method, was introduced by Pandian and Natarajan [9] in 2010 to discover the best solution for a particular FTP using trapezoidal fuzzy integers. Application of fuzzy logic in multi-sensor-based health service robot for condition monitoring during pandemic situations is by introduced Rout, Mahanta, Biswal, Vardhan Raj, Robot [10]. Rajarajeswari and Sahaya Sudha [11] presented Ranking of Hexagonal Fuzzy Numbers using Centroid. Zadeh [12] proposed the idea of fuzziness to address such fuzziness in decision making.

The theory of fuzzy sets can be insufficient to address the ambiguity in transportation-related issues. Thus, the introduction of intuitionistic fuzzy set (IFS) theory solves the transportation issues. IFSs have been increasingly important in decision making in fuzzy environments over the past few years.

This study presents IFTP with fuzzy supply and demand, which are hexagonal intuitionistic. We additionally arrive at the best possible outcome.

The main idea of the article is summarized as follows: A few terms are examined in Section 2. Section 3 displays the HIFN ranking. Section 4 introduces the idea of hexagonal IFTP along with its mathematical formulation. Section 5 provides an explanation of the suggested solution. In Section 6, numerical examples, differences between the suggested strategy and current practices, and an explanation are presented. Section 7 describes the conclusion of the study.

2. Preliminaries

This component comprises essential definitions (Atanassov, 1986) [1].

Fuzzy Set: Let ${\mu }_{\tilde{A}}(\mathrm{X})$ be a function from [0, 1] to a classical set $\tilde{A}$. The membership function ${\mu }_{\tilde{A}}(\mathrm{X})$ of a fuzzy set $\tilde{A}$ is defined as follows:

$$\tilde{A}=\{(X,{\mu }_{\tilde{A}}(X)):X\in \widetilde{A }\mathrm{a}\mathrm{n}\mathrm{d} {\mu }_{\tilde{A}}(X)\in [0, 1\left]\right\}$$

Fuzzy Number: In fact, a fuzzy number is an extension of an ordinary number such that it denotes an interrelated set of potential values instead of a single value, and every possible value has a value that ranges from 0 to 1. The amount of significance (membership function) represented by ${\mu }_{\tilde{A}}(x)$ satisfies the subsequent requirements:

• ${\mu }_{\tilde{A}}(x)$ has a continuous piece-wise pattern.
• ${\mu }_{\tilde{A}}(x)$ appears to be a convex fuzzy subgroup.
• ${\mu }_{\tilde{A}}(x)$ represents a fuzzy subset of a piecewise continuous pattern, indicating that the value originally filled for a minimum of one member $x_0$ must be 1, that is, ${\mu }_{\tilde{A}}\left(x_0\right)=1$.

Hexagonal Fuzzy Number: Figure 1 is the membership function of a fuzzy number $\tilde{A}=(a_1,a_2,a_3,a_4,a_5,a_6)$; then, its hexagonal fuzzy number is given by the following:

$${\mu }_{\tilde{A}}\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-a_1}{a_2-a_1}}\right) , { a}_1\le x\le a_2\\ {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-a_2}{a_3-a_2}}\right) , a_2\le x\le a_3\\ 1 , { a}_3\le x\le a_4\\ 1-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-a_4}{a_5-a_4}}\right) , { a}_4\le x\le a_5\\ {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{a_6-x}{a_6-a_5}}\right), a_5\le x\le a_6\\ 0, otherwise\end{array}\right.$$

Intuitionistic Fuzzy Set (IFS):

The subsequent equation, with a similar arrangement, represents the ${\tilde{A}}^{IFS}$ IFS in $X$.

$${\tilde{A}}^{IFS}=\{(X,{\mu }_{{\tilde{A}}^{IFS}}(x),{\nu }_{{\tilde{A}}^{IFS}}(x)):x\in X\}$$

where for each x in X, the functions ${\mu }_{{\tilde{A}}^{IFS}}(x):X\to [\mathrm{0,1}]$ and ${\nu }_{{\tilde{A}}^{IFS}}(x):X\to [\mathrm{0,1}]$ denote the degree of component participation and non-participation, respectively, and $0\le {\mu }_{{\tilde{A}}^{IFS}}(x),{\nu }_{{\tilde{A}}^{IFS}}(x))\le 1$, for every $x\in X.$

Intuitionistic Fuzzy Numbers (IFNs):

A portion of the real line R in Figure 1, denoted by ${\tilde{A}}^{IFS}=\{(X,{\mu }_{{\tilde{A}}^{IFS}}(x),{\nu }_{{\tilde{A}}^{IFS}}(x)):x\in X\}$, is referred to as an IFN if it possesses the following characteristics. ${\mu }_{{\tilde{A}}^{IFS}}\left(m\right)=1 and {\nu }_{{\tilde{A}}^{IFS}}(m)=0$. ${\mu }_{{\tilde{A}}^{IFS}}:R\to [0,1]$ is continuous, and for every $x\in R,0\le {\mu }_{{\tilde{A}}^{IFS}}(x),{\nu }_{{\tilde{A}}^{IFS}}(x)\le 1$ is valid.

The role of an IFS, both involved and non-involved, is as follows:

$${\mu }_{\tilde{A}}(x)=\left\{\begin{array}{l}{f}_1(x),x\in [m-{\alpha }_1,m)\\ 1,x=m\\ h_1(x),x\in (m,m+{\beta }_1)\\ 0,otherwise\end{array}\right. and{ \nu }_{\tilde{A}}(x)=\left\{\begin{array}{l}1,x\in [-\infty ,m-{\alpha }_2)\\ f_2(x),x\in [m-{\alpha }_2,m)\\ 0,x=m,x\in [m+{\beta }_2,\infty )\\ h_2(x),x\in (m,m+{\beta }_2]\end{array}\right.$$

wherein $f_i(x)$ and $h_i(x),i=\mathrm{1,2},..$ are exclusive progressively declining functions that concurrently exist in $[m-{\alpha }_i,m)$ and $(m,m+{\beta }_i]$. Simultaneous expansions of ${\alpha }_i$ and ${\beta }_i$ appear across the top and bottom portions of ${\mu }_{\tilde{A}}(x)$ and ${\nu }_{\tilde{A}}(x)$.

Hexagonal Intuitionistic Fuzzy Number (HIFN):

An IFN ${\tilde{A}}^{HIFN}=⟨(a_1,a_2,a_3,a_4,a_5,a_6)(a_1^{\prime },a_2^{\prime },a_3^{\prime },a_4^{\prime },a_5^{\prime },a_6^{\prime })⟩$

Since its implications for involvement and non-participation are proportionally accountable, it is regarded as an HIFN shown in Figure 2.

$${\mu }_{{\tilde{A}}^{HIFN}}\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-a_1}{a_2-a_1}}\right) , { a}_1\le x\le a_2\\ {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-a_2}{a_3-a_2}}\right) , a_2\le x\le a_3\\ 1 , { a}_3\le x\le a_4\\ 1-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-a_4}{a_5-a_4}}\right) , { a}_4\le x\le a_5\\ {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{a_6-x}{a_6-a_5}}\right), a_5\le x\le a_6\\ 0, otherwise\end{array}\right. and {\upsilon }_{{\tilde{A}}^{HIFN}}\left(x\right)=\left\{\begin{array}{c}1-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-{a^{\prime }}_1}{{a^{\prime }}_2-{a^{\prime }}_1}}\right) , { a^{\prime }}_1\le x\le {a^{\prime }}_2\\ {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-{a^{\prime }}_2}{{a^{\prime }}_3-{a^{\prime }}_2}}\right) , {a^{\prime }}_2\le x\le a_3\\ 0 , { a}_3\le x\le a_4\\ {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{x-{a^{\prime }}_4}{{a^{\prime }}_5-{a^{\prime }}_4}}\right) , { a}_4\le x\le {a^{\prime }}_5\\ {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{a^{\prime }}_6-x}{{a^{\prime }}_6-{a^{\prime }}_5}}\right), {a^{\prime }}_5\le x\le {a^{\prime }}_6\\ 1, otherwise\end{array}\right.$$

HIFN arithmetic operations:

For any two HIFNs,

${\tilde{A}}^{HIFN}=\left(a_1,a_2,a_3,a_4,a_5,a_6;a_1^{\prime },a_2^{\prime },a_3,a_4,a_5^{\prime },a_6^{\prime }\right) \mathrm{a}\mathrm{n}\mathrm{d}$ ${\tilde{B}}^{HIFN}=(b_1,b_2,b_3,b_4,b_5;b_1^{\prime },b_2^{\prime },b_3^{\prime },b_4^{\prime },b_5^{\prime }),$ preceding are indeed mathematical transactions:

HIFN addition:

$${\widetilde{ A}}^{HIFN}\oplus {\tilde{B}}^{HIFN}=(a_1+b_1,a_2+b_2,a_3+b_3,a_4+b_4,a_5+b_5,a_6+b_6;{ a}_1^{\prime }+b_1^{\prime },a_2^{\prime }+b_2^{\prime },a_3^{\prime }+b_3^{\prime },a_4^{\prime }+b_4^{\prime },a_5^{\prime }+b_5^{\prime },a_6^{\prime }+b_6^{\prime })$$

HIFN subtraction:

$${\widetilde{ A}}^{HIFN}-{\tilde{B}}^{HIFN}=(a_1-b_6,a_2-b_5,a_3-b_4,a_4-b_3,a_5-b_2,a_6-b_1;{a^{\prime }}_1-{b^{\prime }}_6,{a^{\prime }}_2-{b^{\prime }}_5,{a^{\prime }}_3-{b^{\prime }}_4,{a^{\prime }}_4-{b^{\prime }}_3,{a^{\prime }}_5-{b^{\prime }}_2,{a^{\prime }}_6-{b^{\prime }}_1)$$

HIFN multiplication:

$${\tilde{A}}^{HIFN}\otimes {\tilde{B}}^{HIFN}=(a_1{b}_1,a_2{b}_2,a_3{b}_3,a_4{b}_4,a_5{b}_5,a_6{b}_6;a_1^{\prime }{b}_1^{\prime },a_2^{\prime }{b}_2^{\prime },a_3^{\prime }{b}_3^{\prime },a_4^{\prime }{b}_5^{\prime },a_5^{\prime }{b}_5^{\prime },a_6^{\prime }{b}_6^{\prime })$$

HIFN scalar multiplication:

$$k\times {\tilde{A}}^{HIFN}=\left\{\begin{array}{l}(k{a}_1,k{a}_2,k{a}_3,k{a}_4,k{a}_5,k{a}_6;k{a}_1^{\prime },k{a}_2^{\prime },k{a}_3^{\prime },k{a}_4^{\prime },k{a}_5^{\prime },k{a}_6^{\prime }),k\ge 0\\ (k{a}_6,k{a}_5,k{a}_4,k{a}_3,k{a}_2,k{a}_1;k, k{a}_5^{\prime },k{a}_4^{\prime },k{a}_3^{\prime },k{a}_2^{\prime },k{a}_1^{\prime }),k<0\end{array}\right.$$

3. The Proposed Ranking Approach for HIFNs

As an example, let HIFN ${\tilde{A}}^{HIFN}=⟨(a_1,a_2,a_3,a_4,a_5,a_6;w)(b_1,b_2,b_3,b_4,b_5,b_6;w)⟩.$ The center of a hexagon is considered as the balance point of a hexagon. The involved area of the hexagon is partitioned into three plane forms. They are a triangle CHD, a pentagonal BEFGHC, and a triangular AEB, in that order. Let $G_1,G_2, and G_3$ be the centroids of the three planar images. The center for each of these centroids, $G_1,G_2, and G_3$, acts as the guide for ranking hexagoal intuitionistic fuzzy numbers. Its centroid is located where these three plane shapes are balanced. A considerably more suitable balance point for an HIFN is the centroid of these centroid points.

The centers of these shapes on the plane are as follows:

$$G_1=\left({\displaystyle \frac{a_1+2a_2}{3}},{\displaystyle \frac{w}{6}}\right),G_2=\left({\displaystyle \frac{2a_2+a_3+a_4+2a_5}{6}},{\displaystyle \frac{w}{2}}\right) and G_3=\left({\displaystyle \frac{2a_5+a_6}{3}},{\displaystyle \frac{w}{6}}\right) respectively.$$

The centroids of $G_1,G_{2,} and G_3$ are as follows:

$$G=\left({\displaystyle \frac{\frac{a_1+2a_2}{3}+\frac{2a_2+a_3+a_4+2a_5}{6}+\frac{2a_5+a_6}{3}}{3}},{\displaystyle \frac{\frac{w}{6}+\frac{w}{2}+\frac{w}{6}}{3}}\right)$$

$$G=\left({\displaystyle \frac{2a_1+6a_2+a_3+a_4+6a_5+2a_6}{18}},{\displaystyle \frac{5w}{18}}\right)$$

The hexagonal of the non-membership function is divided into three planar shapes in a comparable way. The same method is used to evaluate the centroid of these centroids and the centroid of three plane figures. For these plane figures, a centroid is given by the following:

$$\begin{gathered} G_1^{\prime }=\left({\displaystyle \frac{b_1+2b_2}{3}},{\displaystyle \frac{w+w+\frac{w}{2}}{3}}\right), \\ {G}_2^{\prime }=\left({\displaystyle \frac{2b_2+b_3+b_4+2b_5}{6}},{\displaystyle \frac{w+\frac{w}{2}+0+0+\frac{w}{2}+w}{6}}\right), \\ and{ G}_3^{\prime }=\left({\displaystyle \frac{2b_5+b_6}{3}},{\displaystyle \frac{w+\frac{w}{2}+w}{3}}\right), respectively. \end{gathered}$$

The centroids of ${G^{\prime }}_1,{G^{\prime }}_2, and {G^{\prime }}_3$ are as follows:

$$G^{\prime }=\left({\displaystyle \frac{\frac{b_1+2b_2}{3}+\frac{2b_2+b_3+b_4+2b_5}{6}+\frac{2b_5+b_6}{3}}{3}},{\displaystyle \frac{\frac{w+w+\frac{w}{2}}{3}+\frac{w+\frac{w}{2}+0+0+\frac{w}{2}+w}{6}+\frac{w+\frac{w}{2}+w}{3}}{3}}\right)$$

$$G^{\prime }=\left({\displaystyle \frac{2a_1^{\prime }+6a_2^{\prime }+a_3+a_4+6a_5^{\prime }+2a_6^{\prime }}{18}},{\displaystyle \frac{13w}{18}}\right)$$

The average number of GHIFN centroids exploiting both non-community and community functions is given by the following:

$$\left({\bar{x}}_{\mu v},{\bar{y}}_{\mu v}\right)=\left({\displaystyle \frac{2(a_1+b_1+a_3+a_4+a_6+b_6)+6\left(a_2+b_2+a_5+b_5\right)}{36}},{\displaystyle \frac{5w+13w}{36}}\right)$$

Hexagonal intuitionistic fuzzy quantity has been specified by the ranking function.

$$R\left({\tilde{A}}^{HIFN}\right)=\left(\left({\displaystyle \frac{(a_1+b_1+a_3+a_4+a_6+b_6)+3\left(a_2+b_2+a_5+b_5\right)+2\left(a_3+b_3\right)}{18}}\right)\left({\displaystyle \frac{w}{2}}\right)\right)$$

GHIFN arithmetic operations: For any two HIFNs,

${\tilde{A}}^{GPIFN}=\left(a_1,a_2,a_3,a_4,a_5,a_6;a_1^{\prime },a_2^{\prime },a_3^{\prime },a_4^{\prime },a_5^{\prime },a_6^{\prime };{\omega }_a\right) \mathrm{a}\mathrm{n}\mathrm{d}$

${\tilde{B}}^{GPIFN}=(b_1,b_2,b_3,b_4,b_5,b_6;b_1^{\prime },b_2^{\prime },b_3^{\prime },b_4^{\prime },b_5^{\prime },b_6^{\prime };{\omega }_b)$ preceding are indeed mathematical transactions:

GHIFN addition:

$${\tilde{A}}^{GHIFN}\oplus {\tilde{B}}^{GHIFN}=\left\{\begin{array}{l}(a_1+b_1,a_2+b_2,a_3+b_3,a_4+b_4,a_5+b_5,a_6+b_6;\mathit{min}({\omega }_a,{\omega }_b))\\ (a{\prime }_1+b{\prime }_1,a_2^{\prime }+b_2^{\prime },a_3^{\prime }+b_3^{\prime },a{\prime }_4+b{\prime }_4,a{\prime }_5+b{\prime }_5,a_6^{\prime }+b_6^{\prime };\mathit{max}({\omega }_a,{\omega }_b))\end{array}\right\}$$

GHIFN subtraction:

$${\tilde{A}}^{GHIFN}-{\tilde{B}}^{GHIFN}=\left\{\begin{array}{l}(a_1-b_6,a_2-b_5,a_3-b_4,a_4-b_3,a_5-b_2,a_6-b_1;\mathit{min}({\omega }_a,{\omega }_b))\\ ({a^{\prime }}_1-{b^{\prime }}_6,{a^{\prime }}_2-{b^{\prime }}_5,{a^{\prime }}_3-{b^{\prime }}_4,{a^{\prime }}_4-{b^{\prime }}_3,{a^{\prime }}_5-{b^{\prime }}_2,{a^{\prime }}_6-{b^{\prime }}_1;\mathit{max}({\omega }_a,{\omega }_b))\end{array}\right\}$$

GHIFN multiplication:

$${\tilde{A}}^{GHIFN}\otimes {\tilde{B}}^{GHIFN}=\left\{\begin{array}{l}(a_1{b}_1,a_2{b}_2,a_3{b}_3,a_4{b}_4,a_5{b}_5,a_6{b}_6;\mathit{min}({\omega }_a,{\omega }_b))\\ ({a^{\prime }}_1{b}_1^{\prime },a_2^{\prime }{b}_2^{\prime },a_3^{\prime }{b}_3^{\prime },a_4^{\prime }{b}_4^{\prime },a_5^{\prime }{b^{\prime }}_5,a_6^{\prime },b_6^{\prime };\mathit{max}({\omega }_a,{\omega }_b))\end{array}\right\}$$

HIFN scalar multiplication:

$$k\times {\tilde{A}}^{GTrIFN}=\left\{\begin{array}{l}(k{a}_1,k{a}_2,k{a}_3,k{a}_4,k{a}_5,k{a}_6;{\omega }_a)(k{a}_1^{\prime },k{a}_2^{\prime },k{a}_3^{\prime },k{a^{\prime }}_4,k{a}_5^{\prime },k{a}_6^{\prime };{\omega }_b) \mathit{if} k\ge 0\\ (k{a}_6,k{a}_5,k{a}_4,k{a}_3,k{a}_2,k{a}_1;{\omega }_a)(k{a}_6^{\prime },k{a}_5^{\prime },k{a}_4^{\prime },k{a}_3^{\prime },k{a}_2^{\prime },k{a}_1^{\prime };{\omega }_b) \mathit{if} k<0\end{array}\right\}$$

4. Comparison of HIFNs

In order to continuously compare HIFNs, we must rate them. To represent the ordering of all HIFNs, a ranking function is constructed that converts each HIFN into a real line, like the following: $R:F(R)\to R$. $F\left(R\right)$. Ranking functions can be used to assess HIFNs.

If ${\tilde{A}}^{HIFN}=(a_1,a_2,a_3,{\mathrm{a}}_4,{\mathrm{a}}_5,{\mathrm{a}}_6;{a^{\prime }}_1,{a^{\prime }}_2,{a^{\prime }}_3,{{\mathrm{a}}^{\prime }}_4,{{\mathrm{a}}^{\prime }}_5,{\mathrm{a}}_6^{\prime };{\mathrm{w}}_{\mathrm{a}})$ and ${\tilde{B}}^{HIFN}=\left(b_1,b_2,b_3,b_4,b_5,{\mathrm{b}}_6;{b^{\prime }}_1,{b^{\prime }}_2,{b^{\prime }}_3,{b^{\prime }}_4,{b^{\prime }}_5,{\mathrm{b}}_6^{\prime }\right)$are two HIFNs, then,

$\mathrm{R}\left({\tilde{A}}^{HIFN}\right)={\displaystyle \frac{(a_1+a_1^{\prime }+a_3+a_4+a_6+{a^{\prime }}_6)+3\left(a_2+{a^{\prime }}_2+a_5+{a^{\prime }}_5\right)+2\left(a_3+{a^{\prime }}_3\right)}{36}}$ and $\mathrm{R}\left({\tilde{B}}^{HIFN}\right)={\displaystyle \frac{(b_1+{b^{\prime }}_1+b_3+b_4+b_6+{b^{\prime }}_6)+3\left(b_2+{b^{\prime }}_2+b_5+{b^{\prime }}_5\right)+2\left(b_3+{b^{\prime }}_3\right)}{36}},\mathrm{a}\mathrm{n}\mathrm{d}$ then, orders are defined as follows:

(i)

${\tilde{A}}^{HIFN}>{\tilde{B}}^{HIFN} \mathrm{i}\mathrm{f} \mathrm{R}\left({\tilde{A}}^{HIFN}\right)>R\left({\tilde{B}}^{HIFN}\right);$

(ii)

${\tilde{A}}^{HIFN}<{\tilde{B}}^{HIFN} \mathrm{i}\mathrm{f} \mathrm{R}\left({\tilde{A}}^{HIFN}\right)<R\left({\tilde{B}}^{HIFN}\right);$

(iii)

${\tilde{A}}^{HIFN}={\tilde{B}}^{HIFN} \mathrm{i}\mathrm{f} \mathrm{R}\left({\tilde{A}}^{HIFN}\right)=\mathrm{R}\left({\tilde{B}}^{HIFN}\right).$

The following qualities of the ranking function R are also present:

(i)

$R\left({\tilde{A}}^{HIFN}\right)+R\left({\tilde{B}}^{HIFN}\right)=R\left({\tilde{A}}^{HIFN}+{\tilde{B}}^{HIFN}\right);$

(ii)

$R(k{\tilde{A}}^{HIFN})=kR({\tilde{A}}^{HIFN})\forall k\in R$.

Mathematical Formulation of Hexagonal Intuitionistic Fuzzy Transportation Problem (HIFTP)

Evaluate a TP where “n” is demand and “m” suppliers. The importance of moving one result module from the supplier $ith$ to the $jth$demand is $c_{ij}$.

${\tilde{a}}_i^{HIFN}=(a_1^i,a_2^i,a_3^i,a_4^i,a_5^i,a_6^i;a_1^{i^{\prime }},a_2^{i^{\prime }},a_3^{i^{\prime }},a_4^{i^{\prime }},a_5^{i^{\prime }},a_6^{i^{\prime }})$ be IF extent at $ith$ vendor.

${\tilde{b}}_j^{HIFN}=(b_1^i,b_2^i,b_3^i,b_4^i,b_5^i,b_6^i;b_1^{i^{\prime }},b_2^{i^{\prime }},b_3^{i^{\prime }},b_4^{i^{\prime }},b_5^{i^{\prime }},b_6^{i^{\prime }})$ be IF abundant at $jth$ insistent.

${\tilde{x}}_{ij}^{HIFN}=(x_1^{ij},x_2^{ij},x_3^{ij},x_4^{ij},x_5^{ij}{,x}_6^{ij};x_1^{i^{\prime }{j}^{\prime }},x_2^{i^{\prime }{j}^{\prime }},x_3^{i^{\prime }{j}^{\prime }},x_4^{i^{\prime }{j}^{\prime }},x_5^{i^{\prime }{j}^{\prime }},x_6^{i^{\prime }{j}^{\prime }})$ be IF amount transformation of $ith$ vendor to $jth$ insistent.

Then, balanced hexagonal IFTP is given by the following:

$$\begin{array}{l}Min{\tilde{Z}}^{PIFN}={\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{c}_{ij}\times x_{ij}^{HIFN}\\ s.t.{\displaystyle \sum _{j=1}^n}{\tilde{x}}_{ij}^{HIFN}={\tilde{a}}_i^{HIFN},i=\mathrm{1,2},...,m\\ {\displaystyle \sum _{i=1}^m}{\tilde{x}}_{ij}^{HIFN}={\tilde{b}}_j^{HIFN},j=\mathrm{1,2},...,n\\ {\tilde{x}}_{ij}^{HIFN}\ge \tilde{0};i=\mathrm{1,2},...,m;j=\mathrm{1,2},...,n\end{array}$$

The transportation problem (TP) has been described as a hexagonal intuitionistic fuzzy issue, where expenses are represented by HIFNs, supplies, and demands that are real numbers. The subsequent transportation plan is being used to determine the best possible outcome.

5. Transportation Strategy

5.1. Proposed Transportation Strategy

Stage 1: First check whether the TP is balanced or not. If it is unbalanced, then introduce a dummy origin/destination with the adjusted supply/demand. Then, proceed to Stage 2. Considering the transportation issue, is it balanced? If the transportation problem fails to balance, it can be made so by adding dummy costs related to demand, as the total demand is less than the entire supply; if not, move on to Step 2.

Stage 2: Determine the mathematical average of the variances in a row or column, then write “C” for the column sum and R for the row sum. It is necessary to specify the number of columns and rows present overall. Select the column and row that vary the most.

Step 3: Choose the cell with the lowest cost from the row and column that were shown in Step 2.

Stage 4: Give the cell chosen in Stage 3 Phase 5 an equal assignment. Delete the finished column or row.

Step 5: After all identifiers are completed, go back and repeat Steps 1 through 4.

Step 6: The optimum outcome and hexagonal IFOS were attained at Step 5; thus, $\left\{x_{ij}\right\}$ is the optimal solution, and the triangle IFOS is ${\sum }_{i=1}^m{\sum }_{j=1}^n{{\tilde{c}}^{HIFN}}_{ij}\otimes x_{ij}$.

6. Numerical Example

Examine the IFTP with hexagonal intuitionistic fuzzy demands and supplies given in

Table 1. Hexagonal intuitionistic fuzzy transportation problem.

	${\mathbf{D}}_1$	${\mathbf{D}}_2$	${\mathbf{D}}_3$	${\mathbf{D}}_4$	$\mathbf{S}\mathbf{U}\mathbf{P}\mathbf{P}\mathbf{L}\mathbf{Y}$
${\mathbf{S}}_1$	5	6	12	9	(7,9,11,13,16,20) (5,7,11,13,19,23)
${\mathbf{S}}_2$	3	2	8	4	(6,8,11,14,19,25) (4,7,11,14,21,27)
${\mathbf{S}}_3$	7	11	20	9	(9,11,13,15,18,20) (8,10,13,15,19,22)
$\mathbf{D}\mathbf{E}\mathbf{M}\mathbf{A}\mathbf{N}\mathbf{D}$	(3,4,5,6,8,10) (2,4,5,6,10,12)	(3,5,7,9,12,15) (2,4,7,9,13,17)	(6,7,9,11,13,16) (5,6,9,11,16,18)	(10,12,14,16,20,24) (8,10,14,16,20,25)

.

Solution: Here, supply is not equal to demand, i.e., given transportation problem is balanced. Select the maximum and minimum HPIFN in each row and column and calculate the difference as given in

Table 2. Row and column difference table.

	${\mathbf{D}}_1$	${\mathbf{D}}_2$	${\mathbf{D}}_3$	${\mathbf{D}}_4$	$\mathbf{S}\mathbf{U}\mathbf{P}\mathbf{P}\mathbf{L}\mathbf{Y}$	Row Diff.
${\mathbf{S}}_1$	5	6	12	9	(7,9,11,13,16,20) (5,7,11,13,19,23)	1
${\mathbf{S}}_2$	3	2	8	4	(6,8,11,14,19,25) (4,7,11,14,21,27)	1
${\mathbf{S}}_3$	7	11	20	9	(9,11,13,15,18,20) (8,10,13,15,19,22)	4
$\mathbf{D}\mathbf{E}\mathbf{M}\mathbf{A}\mathbf{N}\mathbf{D}$	(3,4,5,6,8,10) (2,4,5,6,10,12)	(3,5,7,9,12,15) (2,4,7,9,13,17)	(6,7,9,11,13,16) (5,6,9,11,16,18)	(10,12,14,16,20,24) (8,10,14,16,20,25)		6
Col. Diff	2	4	4	5	C = 15	R = 12

.

The problem in Table 2 was turned into

Table 3. Initial allotment table.

	${\mathbf{D}}_1$	${\mathbf{D}}_2$	${\mathbf{D}}_3$	${\mathbf{D}}_4$	$\mathbf{S}\mathbf{U}\mathbf{P}\mathbf{P}\mathbf{L}\mathbf{Y}$
${\mathbf{S}}_1$	5	6	12	9	(7,9,11,13,16,20) (5,7,11,13,19,23)
${\mathbf{S}}_2$	3	2	8	4 (6,8,11,14,19,25) (4,7,11,14,21,27)
${\mathbf{S}}_3$	7	11	20	9	(9,11,13,15,18,20) (8,10,13,15,19,22)
$\mathbf{D}\mathbf{E}\mathbf{M}\mathbf{A}\mathbf{N}\mathbf{D}$	(3,4,5,6,8,10) (2,4,5,6,10,12)	(3,5,7,9,12,15) (2,4,7,9,13,17)	(6,7,9,11,13,16) (5,6,9,11,16,18)	(−15,−7,0,5,12,18) (−19,−11,0,5,13,21)

, utilizing Stage 3 as well as the proposed method, and Section 5.1’s Step 5 was used to assign the initial allocation.

Using Stage 5 of Section 5.1, remove $D_2$ from Table 3. The new reductions are shown in

Table 4. Updated reduction table.

	${\mathbf{D}}_1$	${\mathbf{D}}_2$	${\mathbf{D}}_3$	${\mathbf{D}}_4$	$\mathbf{S}\mathbf{U}\mathbf{P}\mathbf{P}\mathbf{L}\mathbf{Y}$
${\mathbf{S}}_1$	5	6	12	9	(7,9,11,13,16,20) (5,7,11,13,19,23)
${\mathbf{S}}_3$	7	11	20	9	(9,11,13,15,18,20) (8,10,13,15,19,22)
$\mathbf{D}\mathbf{E}\mathbf{M}\mathbf{A}\mathbf{N}\mathbf{D}$	(3,4,5,6,8,10) (2,4,5,6,10,12)	(3,5,7,9,12,15) (2,4,7,9,13,17)	(6,7,9,11,13,16) (5,6,9,11,16,18)	(−15,−7,0,5,12,18) (−19,−11,0,5,13,21)

, and procedure in Section 5.1 is again applied.

In similar way, the second allocation is shown in

Table 5. Second allocation table.

	${\mathbf{D}}_1$	${\mathbf{D}}_2$	${\mathbf{D}}_3$	${\mathbf{D}}_4$	$\mathbf{S}\mathbf{U}\mathbf{P}\mathbf{P}\mathbf{L}\mathbf{Y}$	Row Diff
${\mathbf{S}}_1$	5	6	12	9	(7,9,11,13,16,20) (5,7,11,13,19,23)	1
${\mathbf{S}}_3$	7	11	20	9	(9,11,13,15,18,20) (8,10,13,15,19,22)	2
$\mathbf{D}\mathbf{E}\mathbf{M}\mathbf{A}\mathbf{N}\mathbf{D}$	(3,4,5,6,8,10) (2,4,5,6,10,12)	(3,5,7,9,12,15) (2,4,7,9,13,17)	(6,7,9,11,13,16) (5,6,9,11,16,18)	(−15,−7,0,5,12,18) (−19,−11,0,5,13,21)		C = 3
Col. Diff	2	5	8	0	R = 15

.

The allotment is determined as described in

Table 6. Final allocation table.

	${\mathbf{D}}_1$	${\mathbf{D}}_2$	${\mathbf{D}}_3$	${\mathbf{D}}_4$
${\mathbf{S}}_1$	5	6 (−9,−4,0,4,9,14) (−13,−9,0,4,13,18)	12 (6,7,9,11,13,16) (5,6,9,11,16,18)	9
${\mathbf{S}}_2$	3	2	8	4 (6,8,11,14,19,25) (4,7,11,14,21,27)
${\mathbf{S}}_3$	7 (3,4,5,6,8,10) (2,4,5,6,10,12)	11 (−11,−4,3,9,16,24) (−16,−9,3,9,22,30)	20 (−15,−7,0,5,12,18) (−19,−11,0,5,13,21)	9

, using Stage 6 of Section 5.1 model.

Stage 7: Optimum solution and hexagonal intuitionistic fuzzy optimum value

The optimum solution, obtained in Stage 5, is as follows: $x_{13}=35,x_{14}=15,x_{24}=10,x_{25}=30,x_{31}=30,x_{32}=25,x_{34}=15$.

The IF optimum value of hexagonal intuitionistic fuzzy transportation problem, given in Table 1, is as follows:

$$\begin{array}{l}6\otimes ((-9,-4,0,4,9,14) (-13,-9,0,4,13,18))\oplus 12\otimes ((6,7,9,11,13,16) (5,6,9,11,16,18)) \oplus \\ 4\otimes ((6,8,11,14,19,25) (4,7,11,14,21,27))\oplus 7\otimes ((3,4,5,6,8,10) (2,4,5,6,10,12))\oplus \\ 11\otimes (-11,-4,3,9,16,24) (-16,-9,3,9,22,30))\oplus 20\otimes ((-15,-7,0,5,12,18) (-19,-1,0,5,13,21))\\ =(\left(-358,-64,220,453,778,1070\right) (-544,-45,220,453,926,1266))\end{array}$$

Statistical comparison with the proposed methods are shown in

Table 7. Statistical comparison with the proposed methods.

Name of the Method	North–West Corner	Row Minima	Column Minima	Matrix Minima	VAM	Penalty Difference Algorithm	Optimum Cost
Problem 1	90	92	92	92	76	76	76
Deviation %	0.18	0.21	0.21	0.21	0.00	0.00	0.00
Problem 2	82	88	77	77	73	77	73
Deviation %	0.12	0.21	0.05	0.05	0.00	0.05	0.00
Problem 3	167	61	56	56	56	56	56
Deviation %	1.98	0.09	0.00	0.00	0.00	0.00	0.00
Problem 4	730	555	555	555	555	555	555
Deviation %	0.32	0.00	0.00	0.00	0.00	0.00	0.00
Problem 5	1207	916	1102	896	782	782	782
Deviation %	0.54	0.17	0.41	0.15	0.00	0.00	0.00
Problem 6	128	152	121	156	114	123	114
Deviation %	0.12	0.33	0.06	0.37	0.00	0.08	0.00
Problem 7	1615	1620	1465	1600	1510	1540	1465
Deviation %	0.10	0.11	0.00	0.09	0.03	0.05	0.00
Problem 8	974	1014	931	984	936	783	768
Deviation %	0.27	0.32	0.21	0.28	0.22	0.02	0.00
Problem 9	1195	355	380	355	355	385	355
Deviation %	2.37	0.00	0.07	0.00	0.00	0.08	0.00
Problem 10	845	590	555	745	555	590	555
Deviation %	0.52	0.06	0.00	0.34	0.00	0.06	0.00
Number of best solution	0	2	3	3	8	5
Mean deviation (%)	0.65	0.15	0.10	0.15	0.03	0.03

and graphical representations of existing slotions shown in Figure 3 and Deviation from the best solution Figure 4.

7. Conclusions

In general, trapezoidal or triangular intuitionistic fuzzy numbers are utilized for addressing transportation-related issues. In order to deal with IFTP, a new hexagonal intuitionistic fuzzy number has been implemented in this study. The results are derived using mathematical operations on hexagonal intuitionistic fuzzy integers. There are six parameter intuitionistic fuzzy problems that HIFNs can solve. In order to address issues in intuitionistic fuzzy environments, we suggest using generalized hexagonal intuitionistic fuzzy numbers in future research.