Enhancing Transportation Efficiency with Interval-Valued Fermatean Neutrosophic Numbers: A Multi-Item Optimization Approach

Abstract

In this study, we derive a simple transportation scheme by post-optimizing the costs of a modified problem. The strategy attempts to make the original (mainly feasible) option more practicable by adjusting the building components’ costs. Next, we employ the previously mentioned cell or area cost operators to gradually restore the modified costs to their initial levels, while simultaneously implementing the necessary adjustments to the “optimal” solution. This work presents a multi-goal, multi-item substantial transportation problem with interval-valued fuzzy variables, such as transportation costs, supplies, and demands, as parameters to maintain the transportation cost. This research addresses two circumstances where task ambiguity may occur: the interval solids transportation problem and the fuzzy substantial transportation issue. In the first scenario, we express data problems as intervals instead of exact values using an interval-valued fermatean neutrosophic number; in the second case, the information is not entirely obvious. We address both models when uncertainty solely affects the constraint set. For the interval scenario, we define an additional problem to solve. Our existing efficient systems have dependable transportation, so they are also capable of handling this new problem. In the fuzzy case, a parametric technique generates a fuzzy solution to the preceding problem. Since transportation costs have a direct impact on market prices, lowering them is the primary goal. Using parametric analysis, we provide optimal parameterization solutions for complementary situations. We provide a recommended algorithm for determining the stability set. In conclusion, we offer a sensitivity analysis and a numerical example of the transportation problem involving both balanced and imbalanced loads.

1. Introduction

The complexity of today’s logistics requires sophisticated strategic optimization to balance multiple competing objectives. In this manuscript, we explore a comprehensive approach to solving complex navigation problems through the power of multi-objective optimization [1]. Multi-objective optimization is the simultaneous consideration of several objectives, which are often mutually exclusive, requiring a balance to achieve the best overall result. In navigation, the primary objectives are usually to minimize cost savings, increase reliability, and ensure on-time delivery. Traditional single-objective optimization methods do not adequately consider the trade-offs between these objectives, resulting in suboptimal solutions. By adopting multiple optimization objectives, this study aims to provide a comprehensive and effective framework for increasing transportation productivity. The transportation system refers to the movement of cargo or people between two ports or cities within a national or international network as shown in Figure 1. National and worldwide networks include carriers, matching networks, modes of transportation, network paths, itineraries, cities, depots, clients, stations, and terminals. An itinerary, a type of transportation, connects a number of nodes in the transportation system’s network. Between two nodes, the itinerary represents a single link and mode of transport. We refer to the nodes as exchange stations where we can load and unload goods, transship them, or deliver them. The main problem is the sustainability of the cost of the goods, which is directly proportional to the cost of transportation [2,3].

Two different types of costs relate to the problem of fixed-charge transportation: a fixed charge in a route, paid when a non-zero amount of products transfers between the source and the destination, and a variable cost that gradually increases with the number of items transported [4,5]. The predetermined cost could include a variety of items such as permit fees, toll taxes, subscriptions to festive occasion events, and car rentals. According to these circumstances, the transportation problem (TP) transforms into the generalized form of the fixed charge transportation problem (FCTP). The objective of those responsible for making decisions is to identify the ideal route combination that results in the lowest possible total cost of fixed and variable costs while simultaneously satisfying the supply and demand requirements of each FCTP source and destination. For example, giving a lot of different ways to deal with uncertain answers and rough assessments of fuzzy set data is a good way to solve multi-objective optimization problems [6,7]. Traditional optimization techniques fail to adequately consider the inherent fuzziness of parameters like cost, supply, demand, etc., in transportation problems, instead modeling them as fuzzy. This micro-representation causes the optimization model to more efficiently and effectively reflect real-world uncertainties, making solutions more effective. In multi-objective optimization, the conflict arises from the inability to meet all objectives with their optimal value simultaneously, a balance that the traditional single-objective optimization method cannot attain. The advantage of fuzzy sets is that they can achieve each of the goals with their own membership function [8]. For example, we can define fuzzy goals such as cost, dependability, or delivery times. We then apply this to the optimization process, aiming to find the best solution that closely aligns with the membership function of these objectives, all while implementing an efficient trade-off. This way, the possibility of achieving all of them at once is guaranteed, and the final outcome will also be rational and efficient [9,10].

Fuzzy numbers aid in flexible decision making, allowing for fuzzy constraints rather than rigid ones. These variables precisely capture specific conditions that lead to favorable results in certain domains and unfavorable outcomes in others, guaranteeing the consistent attainment of all objectives at the utmost level. For example, while the primary objective of transportation costs may not be as critical, it still allows for the consideration of slightly higher price indicators as secondary goals, such as reliability or delivery with a superior time-efficiency rating. This ensures equal resource utilization and deals appropriately with the available resources. The two main benefits of fuzzy sets include the real-world practice of binding several goals into one goal in an aggregated form through the formation of a single-objective function. Fuzzy aggregation operators, like weighted average or maximum minute, convert a group of fuzzy objectives into an aggregate of similarly fuzzy objective functions, allowing for further optimization. This method ensures that the end solution takes all objectives into account and optimizes them to the extent of their importance, or to verify their importance first. So, real-world extensions of linear programming and other types of mathematical programming, like fuzzy linear programming and fuzzy multi-objective programming, make it possible for classical techniques to deal with unknown constraints and coefficients. This lets them solve very difficult optimization problems that cannot be solved with traditional mathematical methods.

This work formulates the transportation problem as an interval-valued fermatean neutrosophic number [11] linear programming problem [12,13]. We regard the Modified Distribution (MODI) method as a standardized approach to find the best answer [14]. However, its primary drawback is that it only applies to crisp numbers, disregarding situations where determining the precise optimal value is impossible. However, experts believe Vogel serves as an approximation technique for crisp numbers [15]. Typically, it yields an initial solution that is either optimal or almost optimal. Numerous studies conducted in this area have shown that Vogel generates the best answer in over 80% of the tested cases; nevertheless, one shortcoming is that it ignores uncertainty [16,17]. Scholars in business and engineering have introduced fuzzy sets (FSs) as a vital tool to handle the uncertainties that Zadeh first advocated [18]. Indeed, scholars have described FSs as equivocal and capable of addressing hesitation in practical matters [19,20,21,22,23].

The paper arranges the remaining content as follows:

The study presents a literature review in Section 2. Section 3 discusses the mathematical basis for the preliminary work and includes annotations. Section 4 introduces the mathematical model of the multi-item transportation problem in the IVFN environment. Section 5 presents the mathematical model’s algorithm in great detail. Section 6 presents a case study of the model, accompanied by numerical examples. Section 7 provides a detailed explanation of sensitivity analysis, while Section 8 discusses the advantages of the model. Section 9 delineates the limitations imposed on the job. Section 10 presents conclusions and potential directions for further research.

2. Literature Review

Fuzzy optimization methods have attracted considerable interest in engineering and management applications because of their superior precision, effectiveness, and flexibility, resulting in high-quality and realistic outcomes [24,25]. Indeterminate optimization tactics have received thorough examination in the domains of industry, engineering, and health. Aristotle, widely regarded as the most eminent philosopher in history, is acknowledged as the originator of the concept of mathematical reasoning [26]. Furthermore, his notion of the excluded middle emerged as a fundamental instrument for substantiating assertions in mathematics. Subsequently, Cantor created the set theory [27], which is represented by a characteristic function that exclusively utilizes the numbers 0 and 1. In the literature, there are several conventional methods for addressing real-life situations using fixed data. However, as the problems become more complex, these fixed-data approaches are inadequate for accurately representing the scenario. Zadeh [18] introduced the notion of fuzzy sets (FSs), a valuable method for representing uncertainty that has been used by scholars [28,29,30,31], among others, in the domains of engineering and management. Furthermore, we observe that the FS fails to effectively address the practical issues of reluctance and skepticism. Atanassov introduced the concept of intuitionistic fuzzy sets (IFSs) [32] by incorporating a non-membership degree, thus expanding upon the concept of fuzzy sets. Iterated Function Systems (IFSs) are a contemporary and pragmatic approach to tackling the challenge of simultaneously dealing with reluctance and ambiguity. Recently, some scholars have implemented it in several technological and management domains [33,34,35,36,37]. Moreover, due to the lack of clear definitions for membership and non-membership degrees, it is evident that a single degree of membership or non-membership does not sufficiently capture the uncertainty and hesitation present in real-life problems. Consequently, we acknowledge a specific degree of extra uncertainty.

A modification of IFSs that presumes both membership function (MF) and non-membership function (NMF) are intervals instead of fixed real numbers was developed by Atanassov and Gargov [38] to enhance an IFS’s ability to handle uncertainty and reluctance. Present study activity [39,40] has focused on the activities of IVIFSs and other interesting aspects. Decision makers find it challenging to provide precise parameter values due to the increasing complexity of real-world optimization problems. Many scholars worked on multi-objective models based on internet of things [41,42], large-scale multi-objective optimization models based on a machine learning approach [43], and many more intelligent transportation models [44,45]. Consequently, several studies have focused on FS and IFS ranking research. Among the several FS generalizations, IVIFSs are a useful and interesting tool for modeling and decision making in real-world scenarios featuring hesitancy and uncertainty. Smarandache [46] presents the NS theory as an extension of the IF framework to handle the indeterminacy function (IF). To create the interval single-valued neutrosophic sets (IVSNs) in 2010, Wang [47,48] extended interval fuzzy sets. Zhang et al. [49] used interval neutrosophic sets as a concept for multi-criteria decision-making problems, and many other researchers also introduced novel models based on neutrosophic theory [50,51]. In an interval neutrosophic context, Wang, T. [52] presented a projection structure that incorporates unknown weight information [53,54] for software quality-in-use evaluation. The definition of the fermatean neutrosophic set (FNS) was given by Jansi [55], and it was motivated by the fermatean fuzzy set (FFS) [56]. In addition to deducing computations on the IVFF set class, Jeevaraj [57] introduced the concept of interval-valued fermatean fuzzy sets, or IVFFSs. We show the scoring functions of the IVFF set and analyze their properties. Liu et al. [58] addressed the aggregate functions, basic operating principles, and fermatean fuzzy linguistic term sets. Motivated by the aforementioned literature, we develop the idea of the IVFNSs for determining the best transportation option and its algorithm.

Research gaps and contributions derived from the current investigation:

• In the existing transportation models in the literature, the lowest-cost method provides a feasible solution but does not yield an optimal solution; its accuracy is only 60 percent [59]. Vogel’s Approximation Method (VAM) gives better results but its efficiency remains 70 percent, but both methods can also handle crisp values only. Frequently, these models struggle to address optimal problems. But our model is more flexible because it handles micro-level uncertainty.
• The efficiency of transport systems is an important factor in supply chain management, which directly affects operating costs and market prices. Traditional optimization methods often struggle to assess uncertainties and anomalies in transportation data, which may lead to suboptimal solutions. The goal of this study is to find ways to get around these problems and make navigation more efficient by using multi-objective optimization schemes. Transportation problems are often determined by multiple conflicting objectives, such as cost reduction, reliability enhancement, and ensuring timely delivery, and complementary factors—such as transportation cost, quantity supplied, and the amount demanded—are often uncertain and ambiguous.
• If we cannot adequately address this uncertainty, conventional methods that rely on the exact values of these parameters may not capture the true nature of the problem, leading to inefficient navigation systems. Therefore, we need a flexible and robust method that can handle these cases.
• This study addresses two main types of uncertainty: gap uncertainty and fuzzy uncertainty. In the interval condition, fermatean neutrosophic numbers with interval values express the problem parameters as intervals rather than absolute values. The existing literature has not found a method for interval values that can handle micro-level uncertainties.
• This representation allows for an accurate representation of the uncertainty in the data. In a fuzzy scenario, parametric techniques generate fuzzy solutions from fuzzy data. The goal is to design a navigation system that can handle both types of uncertainty by simultaneously optimizing multiple objectives.
• To achieve the optimality goal, researchers propose a multi-objective, multi-factor-appropriate transportation problem framework. The method initially entails adjusting feasible solutions after optimization by varying the cost components, then gradually restoring them to their initial values while maintaining optimality.
• Despite advancements in transportation optimization, a significant research gap exists in effectively addressing multi-item transportation problems under uncertainty. Specifically, there is a lack of integration of interval-valued fermatean neutrosophic numbers to manage complex and fluctuating variables such as transportation costs, supplies, and demands. Moreover, existing models often remain theoretical with limited practical application, failing to incorporate realistic constraints and dynamic environments. This gap underscores the need for comprehensive, practical optimization approaches that can handle multi-dimensional uncertainties and enhance real-world transportation efficiency.

• Motivation

Never before has transportation been as efficient as it is today, in an age of tight global supply chains. Businesses face pressure to cut costs in order to uphold high service levels of reliability and uptime. Most traditional optimization approaches are unable to handle this uncertainty well in the data-driven, rapidly changing transportation environment. These include cost fluctuations and alterations in supply and demand levels due to changes in market conditions, weather conditions, geopolitics, etc. These components form the primary inspiration for this work. IVFNNs could serve as an alternative approach to effectively address these problems. IVFNNs model uncertain areas by averaging true and false negative rates and quantifying uncertainty rates.

This technique can improve the accuracy and comprehensiveness of travel information by capturing nuances and ambiguities overlooked by traditional approaches. It needs to evolve to cope better with harsh conditions. Multi-objective optimization is another major motivation for this research. Transportation problems are inherently multi-faceted, with many conflicting objectives. For example, fare discounts can hurt reliability or delivery time. Traditional single-objective optimization methods struggle to effectively balance these competing objectives. This study aims to address this limitation by developing a balanced optimal set of objectives that satisfies multiple objectives simultaneously. This methodology is crucial to succeed a transportation system that is not only cost-effective but also reliable and well organized. Again, the benefits of improving navigation are substantial. Lower rents can have a direct impact on market trends, making infrastructure more affordable and improving industry competitiveness. Reliable and timely delivery increases customer satisfaction and loyalty, which is critical to business success. Companies and their supply chains can reap tangible benefits from offering useful tools and programs through extended incentives. It addresses gaps and ambiguities and seeks to deliver attractive solutions through a combination of sensitivity analysis and development.

3. Preliminaries

Definition 1.

We are given a set which is non-empty, say “ $\mathcal{Y}$”; then, the fuzzy set [18] will be written mathematically as we demonstrated below:

$$\tilde{Y}=[<\mathcal{Y},\mu \left({\mathtt{y}}_i\right)>|\left({\mathtt{y}}_i\right)\in \mathcal{Y}]$$

where $\mu \to [0,1]$ is defined to be the membership function of the fuzzy set “ $(\mathcal{Y},\mu \left({\mathtt{y}}_i\right))$”, “ $\mathcal{Y}$” is considered the universe of discourse, and $\mu \left({\mathtt{y}}_i\right)$ is called the degree of membership in fuzzy set $\tilde{Y}$ for every ${\mathtt{y}}_i$. In addition to this, consider “ $\mathtt{y}\in \mathcal{Y}$”, which we will call fully included in “ $\tilde{Y}$” if $\mu \left(\mathtt{y}\right)=1$, will call not included in the fuzzy set “ $\tilde{Y}$” if $\mu \left(\mathtt{y}\right)=0$, and will call partially included in the fuzzy set $\tilde{Y}$ if $0<\mu \left(\mathtt{y}\right)<1$.

Definition 2.

We have the universal set “ $\mathcal{Y}$”; then, an intuitionistic fuzzy set (IFS) [32] which we named “ $\tilde{\mathcal{I}}$” will be written mathematically as given below:

$$\tilde{\mathcal{I}}=[<{\mathtt{y}}_i,\mu \left({\mathtt{y}}_i\right),\eta \left({\mathtt{y}}_i\right)>|{\mathtt{y}}_i\in \mathcal{Y}]$$

Proceeding in this manner, if we consider any single element from the set of intuitionistic fuzzy numbers (IFNs) that is a single intuitionistic fuzzy number (IFN) which we call “ $\widetilde{{\mathcal{I}}_{*}}$”, then it can be illustrated mathematically as

$$\widetilde{{\mathcal{I}}_{*}}=[\mu \left({\mathtt{y}}_i\right),\eta \left({\mathtt{y}}_i\right)]$$

where the coordinate $\mu :\mathcal{Y}\to [0,1]$ represents the membership degree while the degree of non-membership is represented as $\eta :\mathcal{Y}\to [0,1]$ for every element ‘${\mathtt{y}}_i$’ in the set “ $\mathcal{Y}$” to the set “ ${\mathcal{I}}^F$”. We cannot consider an IFN as just an ordinary ordered pair. For the existence of IFNs, there are some limitations that exist; if any ordered pair satisfies the necessary conditions, then we can consider that an IFN. The condition we are talking about is as follows:

$$0\le \mu \left({\mathtt{y}}_i\right)+\eta \left({\mathtt{y}}_i\right)\le 1.$$

Definition 3.

We have a universal set “ $\mathcal{Y}$”; then, an interval-valued fuzzy set (IVFS) [60] which is denoted by “ $\tilde{\mathcal{S}}$” is basically a mapping defined by

$$\tilde{\mathcal{S}}:\mathcal{Y}\to \mathcal{O}[0,1]$$

where $\mathcal{O}[0,1]$ is the set consisting of all subintervals which are closed, while the membership function for every ${\mathtt{y}}_i$ can be defined as ${\mathcal{Y}}^{*}\left({\mathtt{y}}_i\right)=[{\mathcal{Y}}_W^{*}\left({\mathtt{y}}_i\right),{\mathcal{Y}}_Z^{*}\left({\mathtt{y}}_i\right)]$, where

$$0\le {\mathcal{S}}_W^{*}\left({\mathtt{y}}_i\right)\le {\mathcal{S}}_Z^{*}\left({\mathtt{y}}_i\right)\le 1$$

Moreover, ${\mathcal{S}}_W^{*}\left({\mathtt{y}}_i\right)$ and ${\mathcal{S}}_Z^{*}\left({\mathtt{y}}_i\right)$ are referred to as the membership degree of the upper bound and lower bound, respectively, for every ${\mathtt{y}}_i$ in $\mathcal{Y}$.

Definition 4.

Let us consider a universal set “ $\mathcal{Y}$”; then, a set “ $\tilde{\mathcal{P}}$” named the Pythagorean fuzzy set (PFS) [61] will be illustrated mathematically as follows:

$$\tilde{\mathcal{P}}=[<{\mathtt{y}}_i,\mu \left({\mathtt{y}}_i\right),\eta \left({\mathtt{y}}_i\right)>|{\mathtt{y}}_i\in \mathcal{Y}]$$

where the $\mu :\mathcal{Y}\to [0,1]$ is basically the membership degree while $\eta :\mathtt{Y}\to [0,1]$ is defined to be the non-membership degree for each element ‘${\mathtt{y}}_i$’ that lies in the set which we took as a universal set “ $\mathcal{Y}$” to the set “ $\tilde{\mathcal{P}}$.

If we have to define a Pythagorean fuzzy number, suppose we named it “ $\widetilde{{\mathcal{P}}_{*}}$”; then, it will be defined as follows:

$$\widetilde{{\mathcal{P}}_{*}}=[\mu \left({\mathtt{y}}_i\right),\eta \left({\mathtt{y}}_i\right)]$$

We cannot say that any random ordered pair that exists in interval [0,1] is a PFN; it will have to satisfy some important conditions for the existence of a Pythagorean fuzzy number that are as follows:

$$0\le {\left(\mu \left({\mathtt{y}}_i\right)\right)}^2+{\left(\eta \left({\mathtt{y}}_i\right)\right)}^2\le 1.$$

Definition 5.

We have a set which we consider the universal set “ $\mathcal{Y}$”; then, a fermatean fuzzy set [56] denoted by $\tilde{\mathcal{F}}$ will be defined mathematically as follows:

$$\tilde{\mathcal{F}}=[<{\mathtt{y}}_i,\mu \left({\mathtt{y}}_i\right),\eta ({\mathtt{y}}_i>|{\mathtt{y}}_i\in \mathcal{Y}]$$

where the membership degree of every element ‘${\mathtt{y}}_i$’ in “ $\mathcal{Y}$” is represented by $\mu :\mathcal{Y}\to [0,1]$ while the non-membership degree of every element ‘${\mathtt{y}}_i$’ in “ $\mathcal{Y}$” is represented by $\eta :\mathtt{Y}\to [0,1]$ to the fermatean fuzzy set.

If we define the single element of a fermatean fuzzy set, that is, the fermatean fuzzy number (FFN) denoted by $\widetilde{{\mathcal{F}}_{*}}$, it will be illustrated in mathematical form as follows:

$$\widetilde{{\mathcal{F}}_{*}}=[\mu \left({\mathtt{y}}_i\right),\eta \left({\mathtt{y}}_i\right)]$$

The essential condition for any random ordered pair to be the FFN is described as follows:

$$0\le {\left(\mu \left({\mathtt{y}}_i\right)\right)}^3+{\left(\eta \left({\mathtt{y}}_i\right)\right)}^3\le 1.$$

Definition 6.

We have a set which we consider the universal set “ $\mathcal{Y}$”; then, a neutrosophic set (NS) [46] “ $\tilde{\mathcal{N}}$” is defined to be the set which consists of ordered triples. All of the elements of an ordered triple must belong to interval [0,1] and can be defined mathematically as

$$\tilde{\mathcal{N}}=[<{\mathtt{y}}_i,(T\left({\mathtt{y}}_i\right),I\left({\mathtt{y}}_i\right),F\left({\mathtt{y}}_i\right))|{\mathtt{y}}_i\in \mathcal{Y}]$$

Proceeding in this manner, we can define a single neutrosophic number “ $\widetilde{{\mathcal{N}}_{*}}$” which will have the mathematical form as stated below:

$$\widetilde{{\mathcal{N}}_{*}}=\left[(T\left({\mathtt{y}}_i\right),I\left({\mathtt{y}}_i\right),F\left({\mathtt{y}}_i\right))\right]$$

where the degree of the membership function of truth is described by $T\left({\mathtt{y}}_i\right)\in [0,1]$, the degree of the membership function of indeterminacy is described by $F\left({\mathtt{y}}_i\right)\in [0,1]$, while the degree of the membership function of falsity is described by $F\left({\mathtt{y}}_i\right)\in [0,1]$. Additionally, we are going to define a condition that is necessary for an ordered triple to be a neutrosophic number, which is stated as follows:

$$0\le T\left({\mathtt{y}}_i\right)+I\left({\mathtt{y}}_i\right)+F\left({\mathtt{y}}_i\right)\le 3$$

Definition 7.

We have a set which we consider the universal set “ $\mathcal{Y}$”; then, we can call the set “ $\widetilde{{\mathcal{F}}^N}$” a fermatean neutrosophic set (FNS) [55] if it may have the mathematical form as follows:

$$\widetilde{{\mathcal{F}}^N}=[<{\mathtt{y}}_i,(T\left({\mathtt{y}}_i\right),I\left({\mathtt{y}}_i\right),F\left({\mathtt{y}}_i\right))|{\mathtt{y}}_i\in \mathcal{Y}]$$

As the fermatean neutrosophic set (FNS) consists of fermatean neutrosophic numbers (FNNs), we can define the single element of the fermatean neutrosophic set (FNS) as the fermatean neutrosophic number (FNN), namely “ ${\mathcal{F}}_{*}^N$”. It is an ordered triple consisting of values that belong to interval [0,1]. The FNN can be defined mathematically as follows:

$$\widetilde{{\mathcal{F}}_{*}^N}=\left[(T\left({\mathtt{y}}_i\right),I\left({\mathtt{y}}_i\right),F\left({\mathtt{y}}_i\right))\right]$$

where the degree of the membership function of truth is described by $T\left({\mathtt{y}}_i\right)\in [0,1]$, the degree of the membership function of indeterminacy is described by $F\left({\mathtt{y}}_i\right)\in [0,1]$, while the degree of the membership function of falsity is described by $F\left({\mathtt{y}}_i\right)\in [0,1]$. Additionally, we are going to define a condition that is necessary for an ordered triple to be a fermatean neutrosophic number, which is stated as follows:

$$0\le {\left[T\left({\mathtt{y}}_i\right)\right]}^3+{\left[I\left({\mathtt{y}}_i\right)\right]}^3+{\left[F\left({\mathtt{y}}_i\right)\right]}^3\le 3$$

Definition 8.

We have a set which we consider the universal set “ $\mathcal{Y}$”; then, the interval-valued fermatean neutrosophic (IVFN) set is a set “ $\tilde{\mathcal{V}}$” which can be illustrated mathematically as follows:

$$\tilde{\mathcal{V}}=[<\left({\mathtt{y}}_i\right),(T_{\mu },T_{\eta })\left({\mathtt{y}}_i\right),(I_{\mu },I_{\eta })\left({\mathtt{y}}_i\right),(F_{\mu },F_{\eta })\left({\mathtt{y}}_i\right)>|{\mathtt{y}}_i\in \mathcal{Y}]$$

Additionally, if we have to define an interval-valued fermatean neutrosophic (IVFN) number, say $\widetilde{{\mathcal{V}}_{*}}$, then it can be defined as an ordered triple of values lying between 0 and 1, and in mathematical form it can be defined as follows:

$$\widetilde{{\mathcal{V}}_{*}}=[(T_{\mu },T_{\eta })\left({\mathtt{y}}_i\right),(I_{\mu },I_{\eta })\left({\mathtt{y}}_i\right),(F_{\mu },F_{\eta })\left({\mathtt{y}}_i\right))]$$

where the pair $“(T_{\mu },T_{\eta })”\in [0,1]$ defines the degree of membership and non-membership of truth, respectively; $“(I_{\mu },I_{\eta })”\in [0,1]$ describes the membership and non-membership value of indeterminacy, respectively; and $“(F_{\mu },F_{\eta })”\in [0,1]$ defines the membership and non-membership value of falsity, respectively, for each ${\mathtt{y}}_i$ in $\mathcal{Y}$ to an interval-valued fermatean neutrosophic (IVFN) set.

In addition to this, here we define the necessary requirements for each ‘${\mathtt{y}}_i$’ ∈ “ $\mathcal{Y}$” number to be an IVFN number stated below:

$$0\le {\left[T_{\mu }\left({\mathtt{y}}_i\right)\right]}^3+{\left[F_{\mu }\left({\mathtt{y}}_i\right)\right]}^3\le 1 and 0\le {\left[I_{\mu }\left({\mathtt{y}}_i\right)\right]}^3\le 1$$

and

$$0\le {\left[T_{\eta }\left({\mathtt{y}}_i\right)\right]}^3+{\left[F_{\eta }\left({\mathtt{y}}_i\right)\right]}^3\le 1 and 0\le {\left[I_{\eta }\left({\mathtt{y}}_i\right)\right]}^3\le 1$$

If we describe the above conditions in combined form, then the limitations for the occurrence of the IVFN number will have the following form:

$$0\le {\left[T_{\mu }\left({\mathtt{y}}_i\right)\right]}^3+{\left[I_{\mu }\left({\mathtt{y}}_i\right)\right]}^3+{\left[F_{\mu }\left({\mathtt{y}}_i\right)\right]}^3\le 2$$

and

$$0\le {\left[T_{\eta }\left({\mathtt{y}}_i\right)\right]}^3+{\left[I_{\eta }\left({\mathtt{y}}_i\right)\right]}^3+{\left[F_{\eta }\left({\mathtt{y}}_i\right)\right]}^3\le 2$$

Definition 9.

Suppose

${\tilde{\mathcal{V}}}_1=[(T_{\mu 1},T_{\eta 1})\left({\mathtt{y}}_i\right),(I_{\mu 1},I_{\eta 1})\left({\mathtt{y}}_i\right),(F_{\mu 1},F_{\eta 1})\left({\mathtt{y}}_i\right)]$ and

${\tilde{\mathcal{V}}}_1=[(T_{\mu 2},T_{\eta 2})\left({\mathtt{y}}_i\right),(I_{\mu 2},I_{\eta 2})\left({\mathtt{y}}_i\right),(F_{\mu 2},F_{\eta 2})\left({\mathtt{y}}_i\right)]$ are two IVFN numbers and $\kappa >0$. We can define the operation rules for different IVFN numbers as follows:

1. 

${\tilde{\mathcal{V}}}_1\oplus {\tilde{\mathcal{V}}}_2=[(\sqrt[3]{\{T_{\mu 1}{\}}^3+\{T_{\mu 2}{\}}^3-\{T_{\mu 1}{{\}}^3\{T_{\mu 2}\}}^3},\sqrt[3]{\{T_{\eta 1}{\}}^3+\{T_{\eta 2}{\}}^3-\{T_{\eta 1}{{\}}^3\{T_{\eta 2}\}}^3}),(I_{\mu 1}{I}_{\mu 2},I_{\eta 1}{I}_{\eta 2}),(F_{\mu 1}{F}_{\mu 2},F_{\eta 1}{F}_{\eta 2})]$

2. 

${\tilde{\mathcal{V}}}_1\otimes {\tilde{\mathcal{V}}}_2=[(T_{\mu 1}{T}_{\mu 2},T_{\eta 1}{T}_{\eta 2}),(\sqrt[3]{{\left\{I_{\mu 1}\right\}}^3+{\left\{I_{\mu 2}\right\}}^3-{\left\{I_{\mu 1}\right\}}^3{\left\{I_{\mu 2}\right\}}^3},\sqrt[3]{{\left\{I_{\eta 1}\right\}}^3+{\left\{I_{\eta 2}\right\}}^3-{\left\{I_{\eta 1}\right\}}^3{\left\{I_{\eta 2}\right\}}^3}),(\sqrt[3]{{\left\{F_{\mu 1}\right\}}^3+{\left\{F_{\mu 2}\right\}}^3-{\left\{F_{\mu 1}\right\}}^3{\left\{F_{\mu 2}\right\}}^3}),\sqrt[3]{{\left\{F_{\eta 1}\right\}}^3+{\left\{F_{\eta 2}\right\}}^3-{\left\{F_{\eta 1}\right\}}^3{\left\{F_{\eta 2}\right\}}^3}]$

3. 

$\kappa {\tilde{\mathcal{V}}}_1=[(\sqrt[3]{1-{[1-{\left\{T_{\mu 1}\right\}}^3]}^{\mu }},\sqrt[3]{1-{[1-{\left\{T_{\eta 1}\right\}}^3]}^{\mu }}),({\left\{I_{\mu 1}\right\}}^3,{\left\{I_{\eta 1}\right\}}^3),({\left\{F_{\mu 1}\right\}}^3,{\left\{F_{\eta 1}\right\}}^3)]$

4. 

${\left[{\tilde{\mathcal{V}}}_1\right]}^{\mu }=[({\left\{T_{\mu 1}\right\}}^3,{\left\{T_{\eta 1}\right\}}^3),(\sqrt[3]{1-{[1-{\left\{I_{\mu 1}\right\}}^3]}^{\mu }},\sqrt[3]{1-{[1-{\left\{I_{\eta 1}\right\}}^3]}^{\mu }}),(\sqrt[3]{1-{[1-{\left\{F_{\mu 1}\right\}}^3]}^{\mu }},\sqrt[3]{1-{[1-{\left\{F_{\eta 1}\right\}}^3]}^{\mu }})]$

Definition 10.

We have a score function defined by Broumi et al. [62] to rank various IVFN numbers for every ‘${\mathtt{y}}_i$’ ∈$\mathcal{Y}$ by which the degree of potential membership is provided to the IVFN numbers, $\tilde{\mathcal{V}}=[(T_{\mu },T_{\eta })\left({\mathtt{x}}_i\right),(I_{\mu },I_{\eta })\left({\mathtt{x}}_i\right),(F_{\mu },F_{\eta })\left({\mathtt{x}}_i\right)]$. This score function is defined below:

$$S\left({\mathtt{x}}_i\right)=\frac{{\left[T_{\mu }\right]}^3+{\left[T_{\eta }\right]}^3+{\left[I_{\mu }\right]}^3+{\left[I_{\eta }\right]}^3+{\left[F_{\mu }\right]}^3+{\left[F_{\eta }\right]}^3}{2}$$

4. Mathematical Model of Multi-Item Transportation Problem in IVFN Environment

First of all, we define symbols that we will use in all the processes.

Nomenclature
$\overline{O}$	Number of origins.
$\overline{D}$	Number of destinations.
$\mathfrak{i}$	The origin index for every $\overline{O}$.
$\mathfrak{j}$	The destination index for every $\overline{D}$.
${\tilde{x}}_{\mathfrak{i}\mathfrak{j}}$	The quantity of wares that we are required to deliver from one $\mathfrak{i}$th origin to the $\mathfrak{j}$th destination.
$T_{\mathfrak{i}\mathfrak{j}}$	The transportation cost per location presented in a crisp environment.
${\tilde{T}}_{\mathfrak{i}\mathfrak{j}}$	The transportation cost per location presented in an interval-valued fermatean neutrosophic environment.
$P_{\mathfrak{i}\mathfrak{j}}$	Maximum capacity of each source in a crisp environment.
${\tilde{P}}_{\mathfrak{i}\mathfrak{j}}$	Maximum capacity of each source in an interval-valued fermatean neutrosophic environment.
$Q_{\mathfrak{i}\mathfrak{j}}$	Requirement of each location in crisp values.
${\tilde{Q}}_{\mathfrak{i}\mathfrak{j}}$	Requirement of each location in interval-valued fermatean neutrosophic values.

In this section, our main purpose is to define the previously used structure for the transportation problems and the new structure we are going to use for our research work. We swap the usual parameters for the unit transportation cost, supply amount, and requirements with the new parameters defined in the IVFN environment for the better evaluation of uncertain and vague data. The previous structure in the crisp environment has the flaw that it is unable to show the vague and inaccurate data properly, which causes impreciseness in the results and a big disadvantage, costing both the supplier and buyer of the products and consequently increasing the rates of the products in the market, which is a significant issue to be dealt with. Let us take a glimpse at the structure of the solution to the transportation problem in the crisp environment where $T_{\mathfrak{i}\mathfrak{j}}$ is the variable used for the transportation cost, $P_{\mathfrak{i}}$ is the variable used for the supply of all origins, and $Q_{\mathfrak{j}}$ is the variable used for the demand of every destination:

$${\displaystyle Min=\sum _{\mathfrak{i}=0}^{\overline{O}}\sum _{\mathfrak{j}=0}^{\overline{D}}{\tilde{x}}_{\mathfrak{i}\mathfrak{j}}·T_{\mathfrak{i}\mathfrak{j}}}$$

Subject to

$$\begin{gathered} {\displaystyle \sum _{\mathfrak{j}=0}^{\overline{D}}{\tilde{x}}_{\mathfrak{i}\mathfrak{j}}=P_{\mathfrak{i}}=Supply,where\mathfrak{i}=1,2,.....,\overline{O} \\ \sum _{\mathfrak{i}=0}^{\overline{O}}{\tilde{x}}_{\mathfrak{i}\mathfrak{j}}=Q_{\mathfrak{j}}=Demand,where\mathfrak{j}=1,2,.....,\overline{D}} \end{gathered}$$

${\tilde{x}}_{\mathfrak{i}\mathfrak{j}}\ge 0\forall \mathfrak{i},\mathfrak{j}.$

Now we are going to develop a new algorithm to deal with uncertain data in transportation problems, for we will utilize the parameters in the IVFN environment instead of the parameters previously used in the crisp environment lacking the accuracy and preciseness in the numerical solutions of the real-world transportation challenges. Now the proposed parameter in the IVFN environment for the unit transportation cost is ${\tilde{T}}_{\mathfrak{i}\mathfrak{j}}$, for supply is ${\tilde{P}}_{\mathfrak{i}}$, and for demand is ${\tilde{Q}}_{\mathfrak{j}}$. Furthermore, the structure of the transportation problem in the IVFN environment is demonstrated mathematically as

$${\displaystyle Min=\sum _{\mathfrak{i}=0}^{\overline{O}}\sum _{\mathfrak{j}=0}^{\overline{D}}{\tilde{x}}_{\mathfrak{i}\mathfrak{j}}·{\tilde{T}}_{\mathfrak{i}\mathfrak{j}}}$$

Subject to

$$\begin{gathered} {\displaystyle \sum _{\mathfrak{j}=0}^{\overline{D}}{\tilde{x}}_{\mathfrak{i}\mathfrak{j}}={\tilde{P}}_{\mathfrak{i}}=Supply,where\mathfrak{i}=1,2,.....,\overline{O} \\ \sum _{\mathfrak{i}=0}^{\overline{O}}{\tilde{x}}_{\mathfrak{i}\mathfrak{j}}={\tilde{Q}}_{\mathfrak{j}}=Demand,where\mathfrak{j}=1,2,.....,\overline{D}} \end{gathered}$$

${\tilde{x}}_{\mathfrak{i}\mathfrak{j}}\ge 0\forall \mathfrak{i},\mathfrak{j}.$

The main purpose of this method is to find the most swift and efficient route for the travel and transport of goods from different sources corresponding to their production ability to various targets according to their requirements, minimizing the overall transportation cost for the financial advantage of both the seller and client and keeping in view the efficiency of the vehicle and its fuel consumption in the process. Many factors are going to affect this; for instance, the fuel consumption of the vehicles and the capacity of different vehicles according to their size and the weight they can carry. The safety of the transporting product is also a very significant area to consider regarding the privacy of the client and his product while minimizing the transportation cost in the smallest amount of time.

5. Algorithm for Mathematical Model

In this section of our research study, we are going to define an algorithm to find the initial feasible solution to the problem we are going to discuss in our case study. In the world of mathematics, this method is named the least-cost method (LCM), and it is very efficient for solving transportation problems of both kinds, which are balanced transportation problems and unbalanced transportation problems. In this paper, we are going to solve both of these types by using the least-cost method, and for its better illustration, we are going to define this algorithm in the following steps:

STEP 1

The values in the IVFN environment are very difficult to solve, and the evaluation of the solution becomes very complicated. So, in this step of the algorithm, we will find the score value of each IVFN number used as the transportation cost, supply of the origins, and requirements of the destinations for the better computation of the initial feasible solution to the transportation challenges.

STEP 2

There are two types of transportation problems, namely balanced transportation problems and unbalanced transportation problems. We have to deal with these types accordingly, so for the identification of the kind of transportation problem, we will sum up the total amount of supplies and check whether it is equal to the sum of the total amount of demands. Then, we will obtain two possibilities: Either the sum of the demands is equal to the sum of supplies $\sum {\tilde{P}}_{\mathfrak{i}}=\sum {\tilde{Q}}_{\mathfrak{j}}$, meaning the transportation problem is a balanced transportation problem, so we will simply go forward to STEP 3. and find the initial feasible solution to the real-life transportation problem we will be discussing at that moment. Or, the second possibility is that the sum of supplies and sum of demands is unequal to each other, meaning the transportation problem is referred to as the unbalanced transportation problem. In this case, we will add a dummy row or dummy column according to the values of the sum of supplies and demands. We formulate the table of the score values of the IVFN numbers shown as the transportation cost, supply, and demand for further steps.

STEP 3

Examine the table of the score values of each IVFN, point out the lowest transportation cost in the table, and allot the corresponding supply and demands to that specific cell. Then, subtract the allotted value from the supply and demand corresponding to the allotted cell. Extract the row or column whose value of supply or demand becomes zero. In any case, if there is an occurrence of two minimum values, allot them to the cell which can obtain the maximum of the supply and demand in the table. After canceling the row or column, allot the supply and demands to the minimum transportation cost of the remaining rows and columns. Repeat this procedure until all the requirements are met and all the supplies produced get distributed.

STEP 4

After all the supplies produced by different sources get exhausted and all the demands required by various destinations are satisfied, the table formed will give us the initial feasible solution to the transportation problem we are going to deal with. Put the values of all ${\tilde{x}}_{\mathfrak{i}\mathfrak{j}}$ in an objective function to have the net transportation cost of the complete route.

• End.

A MATLAB code of the mathematical model will be helpful for scholars to address the complexities of transportation that are given in Figure 2.

6. Case Study

This section elaborates on the main numerical work regarding transportation costs, supplies, and demands in the IVFN environment. We will deal with the real-life examples of both types of transportation problems, which are balanced transportation problems and unbalanced transportation problems, while using IVFN constraints for the accuracy and preciseness of the results. The initial feasible solution to the various transportation problems will be found out by using the algorithm defined above, and then we are going to verify whether they are the best routes we can find or if there are any other possibilities. Figure 3 illustrates the complexity of TPs, and we will address these complications in the work that we are going to conduct.

Example 1.

Four cement producers supply cement of high quality for the requirements of four cities.

Table 1. Provided data for Example 1 in IVFN environment.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	(0.2,0.41),(0.9,0.34),(0.22,0.59)	(0.13,0.79),(0.29,0.47),(0.38,0.89)	(0.24,0.39),(0.11,0.25),(0.43,0.9)	(0.19,0.11),(0.5,0.27),(0.31,0.3)	(0.29,0.11),(0.3,0.57),(0.41,0.2)
${\overline{O}}_2$	(0.9,0.12),(0.3,0.55),(0.21,0.35)	(0.11,0.22),(0.3,0.9),(0.87,0.45)	(0.23,0.1),(0.81,0.4),(0.47,0.82)	(0.29,0.28),(0.7,0.9),(0.23,0.4)	(0.19,0.28),(0.7,0.9),(0.3,0.42)
${\overline{O}}_3$	(0.9,0.2),(0.31,0.42),(0.41,0.2)	(0.89,0.21),(0.8,0.59),(0.53,0.27)	(0.4,0.25),(0.938,0.37),(0.74,0.59)	(0.17,0.47),(0.11,0.1),(0.55,0.5)	(0.27,0.47),(0.21,0.10),(0.5,0.49)
${\overline{O}}_4$	(0.12,0.19),(0.72,0.18),(0.29,0.03)	(0.37,0.11),(0.02,0.41),(0.39,0.5)	(0.25,0.31),(0.59,0.57),(0.1,0.3)	(0.13,0.24),(0.46,0.31),(0.12,0.45)	(0.44,0.82),(0.39,0.56),(0.85,0.56)
Demand	(0.57,0.11),(0.2,0.41),(0.29,0.3)	(0.42,0.19),(0.7,0.28),(0.9,0.3)	(0.5,0.21),(0.49,0.47),(0.1,0.27)	(0.56,0.82),(0.85,0.39),(0.44,0.56)	—

presents the shipping cost as interval-valued fermatean neutrosophic numbers. Additionally, every city’s demand and every company’s monthly supply are expressed in the IVFN environment to minimize the transportation cost and transport time while maintaining the safety of the products and utilizing the best route we can obtain for this purpose. The total supply produced by each company is given as

$$\begin{array}{c}Compan{y}_1=\left[\right(0.29,0.11),(0.3,0.57),(0.41,0.2\left)\right]\\ Compan{y}_2=\left[\right(0.19,0.28),(0.7,0.9),(0.3,0.42\left)\right]\\ Compan{y}_3=\left[\right(0.27,0.47),(0.21,0.10),(0.5,0.49\left)\right]\end{array}$$

and

$$Compan{y}_4=\left[\right(0.44,0.82),(0.39,0.56),(0.85,0.56\left)\right].$$

In addition, the demands of the cities are given in the IVFN environment below:

$$\begin{array}{c}Cit{y}_1=\left[\right(0.57,0.11),(0.2,0.41),(0.29,0.3\left)\right]\\ Cit{y}_2=\left[\right(0.42,0.19),(0.7,0.28),(0.9,0.3\left)\right]\\ Cit{y}_3=\left[\right(0.5,0.21),(0.49,0.47),(0.1,0.27\left)\right]\end{array}$$

and

$$Cit{y}_4=\left[\right(0.56,0.82),(0.85,0.39),(0.44,0.56\left)\right].$$

The transportation problem given in Example 1 is illustrated graphically in Figure 4.

• Step 1: In this step, we will find and substitute the score values of the IVFN numbers denoted as the transportation cost, supplies, and demands in Table 1. The calculated score values are then represented in Table 2 for the convenient calculation of the transportation cost using the algorithm defined in the above section.

Table 2. Score values.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.53	0.6916	0.449	0.1048	0.1574
${\overline{O}}_2$	0.488	0.7588	0.63189	0.5972	0.6
${\overline{O}}_3$	0.4589	0.8	0.7832	0.2012	0.1882
${\overline{O}}_4$	0.2060	0.1526	0.2319	0.1180	0.83
Demand	0.1574	0.6	0.1882	0.83	1.7756

Table 2. Score values.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.53	0.6916	0.449	0.1048	0.1574
${\overline{O}}_2$	0.488	0.7588	0.63189	0.5972	0.6
${\overline{O}}_3$	0.4589	0.8	0.7832	0.2012	0.1882
${\overline{O}}_4$	0.2060	0.1526	0.2319	0.1180	0.83
Demand	0.1574	0.6	0.1882	0.83	1.7756

• Step 2: This step examines whether the transportation problem shown in Example 1 is a balanced transportation problem or unbalanced transportation problem, and for that we will sum up all the supplies produced by various companies and all the demands required by various locations:
  $$\sum {\tilde{P}}_{\mathfrak{i}}=0.1574+0.6+0.1882+0.83=1.7756$$

  and
  $$\sum {\tilde{Q}}_{\mathfrak{j}}=0.1574+0.6+0.1882+0.83=1.7756$$

  Since the sum of all supplies is equal to the sum of all demands, we conclude that the transportation problem given in Example 1 is of the balanced type.

• Step 3: In this step, the cell having the minimum transportation cost is going to be allotted the maximum value that can be allotted according to the given supplies and demands. These allocations are shown in Table 3.

Table 3. Indicated minimum cells and allotments assigned.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.53	0.6916	0.449	0.1048|(0.1574)	0.1574
${\overline{O}}_2$	0.488	0.7588|(0.4118)	0.63189|(0.1882)	0.5972	0.6
${\overline{O}}_3$	0.4589|(0.1574)	0.8|(0.0308)	0.7832	0.2012	0.1882
${\overline{O}}_4$	0.2060	0.1526|(0.1574)	0.2319	0.1180|(0.6726)	0.83
Demand	0.1574	0.6	0.1882	0.83	1.7756

Table 3. Indicated minimum cells and allotments assigned.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.53	0.6916	0.449	0.1048|(0.1574)	0.1574
${\overline{O}}_2$	0.488	0.7588|(0.4118)	0.63189|(0.1882)	0.5972	0.6
${\overline{O}}_3$	0.4589|(0.1574)	0.8|(0.0308)	0.7832	0.2012	0.1882
${\overline{O}}_4$	0.2060	0.1526|(0.1574)	0.2319	0.1180|(0.6726)	0.83
Demand	0.1574	0.6	0.1882	0.83	1.7756

• After assigning all the allocations, we found the following initial feasible solution to our transportation problem defined in Example 1:

$$\begin{array}{c}({\overline{O}}_1,{\overline{D}}_4)={\tilde{x}}_{14}=0.1574,({\overline{O}}_2,{\overline{D}}_2)={\tilde{x}}_{22}=0.4118,\\ ({\overline{O}}_2,{\overline{D}}_3)={\tilde{x}}_{23}=0.1882,({\overline{O}}_3,{\overline{D}}_1)={\tilde{x}}_{31}=0.1574,\\ ({\overline{O}}_3,{\overline{D}}_2)={\tilde{x}}_{32}=0.0308,({\overline{O}}_4,{\overline{D}}_2)={\tilde{x}}_{42}=0.1574,\\ ({\overline{O}}_4,{\overline{D}}_4)={\tilde{x}}_{44}=0.6726\end{array}$$

• Step 4: Putting the values of each ${\tilde{x}}_{\mathfrak{i}\mathfrak{j}}$ in the objective function.
  $$\begin{array}{c}Min=0.1574\times 0.1048+0.4118\times 0.7588+0.1882\times 0.63189+0.1574\times 0.4589+0.0.0308\times 0.8\\ +0.1574\times 0.1526+0.6726\times 0.1180\end{array}$$
  $$\begin{array}{c}Min=0.01649552+0.31247384+0.118921698+0.07223086+0.02464+0.02401924+0.0793668\\ Min=0.648147958\end{array}$$

  Hence, from the above calculations using the least-cost method (LCM) to find the initial feasible solution and then apply the perposed formulation, we obtain the net transportation cost for the problem defined in Example 1.

Example 2.

Alexander’s primary source of income comes from his three ranches, where he uses the animals to produce a variety of animal products such as meat, milk, and so on. He sells these products in three separate cities to meet their needs.

Table 4. Provided data for Example 2 in IVFN environment.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	(0.13,0.24),(0.46,0.31),(0.12,0.45)	(0.09,0.36),(0.08,0.07),(0.09,0.389)	(0.59,0.83),(0.47,0.45),(0.31,0.3)	(0.31,0.1),(0.42,0.3),(0.52,0.8)	(0.19,0.11),(0.5,0.27),(0.31,0.3)
${\overline{O}}_2$	(0.42,0.67),(0.6,0.21),(0.7,0.43)	(0.95,0.01),(0.84,0.32),(0.66,0.59)	(0.49,0.23),(0.05,0.55),(0.11,0.43)	(0.48,0.26),(0.57,0.78),(0.48,0.72)	(0.29,0.28),(0.7,0.9),(0.23,0.4)
${\overline{O}}_3$	(0.14,0.32),(0.41,0.02),(0.57,0.04)	(0.09,0.57),(0.041,0.2),(0.9,0.07)	(0.28,0.43),(0.33,0.47),(0.39,0.26)	(0.15,0.26),(0.71,0.01),(0.09,0.37)	(0.17,0.47),(0.11,0.1),(0.55,0.5)
Demand	(0.12,0.19),(0.72,0.18),(0.29,0.03)	(0.37,0.11),(0.02,0.41),(0.39,0.5)	(0.25,0.31),(0.59,0.57),(0.1,0.3)	(0.364,0.97),(0.24,0.2),(0.3,0.5)	—

presents the shipping costs for these support products, indicating intermediate IVFN number values. Additionally, the IVFN scenario, which accounts for supply and demand uncertainty, expresses each municipality’s need for livestock resources and each farm’s annual supply:

$$\begin{array}{c}Far{m}_1=\left[\right(0.19,0.11),(0.5,0.27),(0.31,0.3\left)\right]\\ Far{m}_2=\left[\right(0.29,0.28),(0.7,0.9),(0.23,0.4\left)\right]\end{array}$$

and

$$Far{m}_3=\left[\right(0.17,0.47),(0.11,0.1),(0.55,0.5\left)\right].$$

In addition, the demands of the towns are given in the IVFN environment below:

$$\begin{array}{c}Tow{n}_1=\left[\right(0.12,0.19),(0.72,0.18),(0.29,0.03\left)\right]\\ Tow{n}_2=\left[\right(0.37,0.11),(0.02,0.41),(0.39,0.5\left)\right]\\ Tow{n}_3=\left[\right(0.25,0.31),(0.59,0.57),(0.1,0.3\left)\right]\end{array}$$

and

$$Tow{n}_4=\left[\right(0.364,0.97),(0.24,0.2),(0.3,0.5\left)\right].$$

The transportation problem given in Example 2 is illustrated graphically in Figure 5.

• Step 1: This step involves locating and substituting the score values of the IVFN numbers in Table 4, which represent the transportation cost, supplies, and demands. The computed score values are then shown in Table 5 so that the net transportation cost may be easily estimated using the algorithm described in the previous section.

Table 5. Score values.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.1180	0.0297	0.5242	0.3263	0.1048
${\overline{O}}_2$	0.5113	0.9878	0.1885	0.6358	0.5972
${\overline{O}}_3$	0.1448	0.4616	0.1590	0.215	0.2012
Demand	0.2060	0.1526	0.2319	0.5673	—

Table 5. Score values.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.1180	0.0297	0.5242	0.3263	0.1048
${\overline{O}}_2$	0.5113	0.9878	0.1885	0.6358	0.5972
${\overline{O}}_3$	0.1448	0.4616	0.1590	0.215	0.2012
Demand	0.2060	0.1526	0.2319	0.5673	—

• Step 2: In order to determine if the transportation issue in Example 1 is an unbalanced or balanced issue, we must add up all of the supplies made by numerous companies and all of the demands required by different locations:
  $$\sum {\tilde{P}}_{\mathfrak{i}}=0.1048+0.5972+0.2012=0.9032$$

  and
  $$\sum {\tilde{Q}}_{\mathfrak{j}}=0.2060+0.1526+0.2319=0.5905$$

  Since $\sum {\tilde{P}}_{\mathfrak{i}}\ne \sum {\tilde{Q}}_{\mathfrak{j}}$. In this case, we are going to add a dummy row to balance the unbalanced transportation problem and further proceed to our next steps to solve this transportation problem.
  We may conclude that the transportation problem in Table 6 has been balanced by the addition of a dummy row.

Table 6. Addition of dummy row to balance the supplies and demands.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.1180	0.0297	0.5242	0.3263	0.1048
${\overline{O}}_2$	0.5113	0.9878	0.1885	0.6358	0.5972
${\overline{O}}_3$	0.1448	0.4616	0.1590	0.215	0.2012
${\overline{O}}_4$	0	0	0	0	0.2546
Demand	0.2060	0.1526	0.2319	0.5673	—

Table 6. Addition of dummy row to balance the supplies and demands.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.1180	0.0297	0.5242	0.3263	0.1048
${\overline{O}}_2$	0.5113	0.9878	0.1885	0.6358	0.5972
${\overline{O}}_3$	0.1448	0.4616	0.1590	0.215	0.2012
${\overline{O}}_4$	0	0	0	0	0.2546
Demand	0.2060	0.1526	0.2319	0.5673	—

• Step 3: Identify the cell having the least amount of transportation cost and allot the maximum supply or demand that can be allocated according to the corresponding row and column until all supplies have been used and demands have been met. These allocations are shown in Table 7.

Table 7. Indicated minimum cells and allotments assigned.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.1180|(0.1048)	0.0297	0.5242	0.3263	0.1048
${\overline{O}}_2$	0.5113|(0.0048)	0.9878|(0.0478)	0.1885|(0.2319)	0.6358|(0.3127)	0.5972
${\overline{O}}_3$	0.1448|(0.2012)	0.4616	0.1590	0.215	0.2012
${\overline{O}}_4$	0	0	0	0|(0.2546)	0.2546
Demand	0.2060	0.1526	0.2319	0.5673	—

Table 7. Indicated minimum cells and allotments assigned.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.1180|(0.1048)	0.0297	0.5242	0.3263	0.1048
${\overline{O}}_2$	0.5113|(0.0048)	0.9878|(0.0478)	0.1885|(0.2319)	0.6358|(0.3127)	0.5972
${\overline{O}}_3$	0.1448|(0.2012)	0.4616	0.1590	0.215	0.2012
${\overline{O}}_4$	0	0	0	0|(0.2546)	0.2546
Demand	0.2060	0.1526	0.2319	0.5673	—

• We discovered the following initial feasible solution after allocating every allocation for the transportation problem shown in Example 2:

$$\begin{array}{c}({\overline{O}}_1,{\overline{D}}_1)={\tilde{x}}_{11}=0.1048,({\overline{O}}_2,{\overline{D}}_1)={\tilde{x}}_{21}=0.0048,\\ ({\overline{O}}_2,{\overline{D}}_2)={\tilde{x}}_{22}=0.0478,({\overline{O}}_2,{\overline{D}}_3)={\tilde{x}}_{23}=0.2319,\\ ({\overline{O}}_2,{\overline{D}}_4)={\tilde{x}}_{24}=0.3127,({\overline{O}}_3,{\overline{D}}_1)={\tilde{x}}_{31}=0.2012,\\ ({\overline{O}}_4,{\overline{D}}_4)={\tilde{x}}_{44}=0.2546\end{array}$$

.

• Step 4: To obtain the basic feasible solution given in Example 2, we are going to insert the values of all ${\tilde{x}}_{\mathfrak{i}\mathfrak{j}}$ in the objective function:

$$\begin{array}{c}Min=0.1048\times 0.0297+0.0048\times 0.5113+0.0478\times 0.9878+0.2319\times 0.1885+0.3127\times 0.6358\\ +0.2012\times 0.1448+0.2546\times 0.\end{array}$$

$$Min=0.00311256+0.00245424+0.04721684+0.04371315+0.19881466+0.02913376+0.$$

$$Min=0.32444521.$$

From the above method known as the least-cost method (LCM) for IVFNNs, we obtained our initial feasible solution and then applied the perposed formulation; we found that the transportation cost acquired from it is 0.32444521.

The main key contribution of this study is as below:

In this study, we handled the transportation problems by applying the interval-valued fermatean neutrosophic numbers (IVFNNs). There are a lot of unknowns when it involves transportation-related issues, like erratic visitor patterns and variable demand. Traditional methods typically produce less-than-ideal results because they are unable to regulate this unpredictability to the necessary extent. However, we significantly improve the anomaly tolerance by incorporating the IVFNNs into the problem-solving process.

• This study aims to explore how IVFNNs adapt and function in various domains when faced with unpredictability in transportation challenges.
• We described the fundamental functions that algorithms perform in the context of IVFNNs, so make sure you have a solid grasp of the concepts that lie beneath the surface.
• A new algorithm that is used at the micro-level is tailored to the derived set and aimed at improving the effectiveness of the transportation problem-solving process.
• To obtain relevant data, it is necessary to provide a detailed description and analysis of the method in IVFNNs, taking into account the most significant features. We also provide this information, which enhances the validity of our model.
• We can apply the algorithm in mathematical models, demonstrating its practical efficacy and potential for real-world implementation.
• We are committed to providing comprehensive answers to problems with balanced and unbalanced transportation, as well as comprehensive tactics for dealing with a wide range of circumstances.

7. Discussion and Sensitivity Analysis

In this section, we discuss the model’s validity and also explain its validity through the numerical process. We verify that our model gives an optimal solution and is applicable in a real-life context.

The suggested model uses interval-valued fermatean neutrosophic numbers (IVFNNs) in a multi-objective optimization framework to show that it is better by analyzing the sensitivity of both balanced and unbalanced transport problems with a number of costs, suppliers, and middle men. Representing uncertain parameters such as demand, the model captures inherent uncertainties more accurately than traditional methods. This micro-method allows for a robust and reliable optimization process, ensuring that the solution remains even more efficient in changing real-world conditions. Since the method simultaneously accomplishes several conflicting objectives, such as cost reduction, greater reliability, and ensuring timely delivery, this holistic approach ensures a balanced and responsive transportation plan. It goes a long way toward effectively solving the multi-faceted transportation challenges. Many researchers propose various models for optimality and multi-item transmission, but upon micro-level analysis, we find that existing models are unable to handle the same information, thereby preventing us from comparing them with our own data. For its validity, we tested our model’s validity using the Stepping Stone method. After aggregating the interval values, we can use this method to test our results.

7.1. Sensitivity Analysis for Balanced TP

The initial feasible solution from Table 3 may or may not give us the minimum value of the net transportation cost of the balanced or unbalanced transportation problem. For this confirmation, we will use the Stepping Stone method to see whether this initial solution is the best solution we can get out of the transportation problem stated in Example 1 or if there is any other route which can provide the minimum value of the total transportation cost. Simply said, either this is the most optimal way to allot the supplies and demands or we can have another way to make allocations. We have found that our initial feasible solution is not the optimal one, and we will obtain the most efficient allocations in

Table 8. Updated allocations for transportation problem in Example 1.

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.53	0.6916	0.449	0.1048|(0.1574)	0.1574
${\overline{O}}_2$	0.488|(0.1574)	0.7588	0.63189|(0.1882)	0.5972|(0.2544)	0.6
${\overline{O}}_3$	0.4589	0.8	0.7832	0.2012|(0.1882)	0.1882
${\overline{O}}_4$	0.2060	0.1526|(0.6)	0.2319	0.1180|(0.23)	0.83
Demand	0.1574	0.6	0.1882	0.83	1.7756

by using the Stepping Stone method.

Formulate the solution from Table 8, which is the optimal solution to Example 1:

$$\begin{array}{c}({\overline{O}}_1,{\overline{D}}_4)={\tilde{x}}_{14}=0.1574,({\overline{O}}_2,{\overline{D}}_1)={\tilde{x}}_{21}=0.1574,\\ ({\overline{O}}_2,{\overline{D}}_3)={\tilde{x}}_{23}=0.1882,({\overline{O}}_2,{\overline{D}}_4)={\tilde{x}}_{24}=0.2544,\\ ({\overline{O}}_3,{\overline{D}}_4)={\tilde{x}}_{34}=0.1882,({\overline{O}}_4,{\overline{D}}_2)={\tilde{x}}_{42}=0.6,\\ ({\overline{O}}_4,{\overline{D}}_4)={\tilde{x}}_{44}=0.23\end{array}$$

Substitute all the values of ${\tilde{x}}_{\mathfrak{i}\mathfrak{j}}$ in the objective function to obtain the total transportation cost of the transportation problem defined in Example 1:

$$\begin{array}{c}Min=0.1574\times 0.1048+0.1574\times 0.488+0.1882\times 0.63189+0.2544\times 0.5972+0.1882\times 0.2012\\ +0.6\times 0.1526+0.23\times 0.1180\end{array}$$

$$Min=0.01649552+0.0768112+0.118921698+0.15192768+0.03786584+0.09156+0.02714$$

$$Min=0.520721938$$

After applying the Stepping Stone method with the proposed formulation and utilizing the IVFN numbers for the better results, we obtain the minimum value 0.520721938 for the provided data in Table 1.

7.2. Sensitivity Analysis for Unbalanced TP

Let us investigate if the first feasible solution to Example 2 that we arrived at in Table 7 is the best option available or if there is another configuration of allocations that would solve the transportation problem more effectively. For this, the Stepping Stone method was employed. Using the Stepping Stone method, we discovered that our first feasible answer is not the most effective one, and we will obtain the optimal approach, which is shown in

Table 9. Updated allocations for transportation problem in Example 2:

$\mathfrak{S}$	${\overline{\mathit{D}}}_1$	${\overline{\mathit{D}}}_2$	${\overline{\mathit{D}}}_3$	${\overline{\mathit{D}}}_4$	Supply
${\overline{O}}_1$	0.1180	0.0297|(0.1048)	0.5242	0.3263	0.1048
${\overline{O}}_2$	0.5113|(0.2060)	0.9878	0.1885|(0.2319)	0.6358|(0.1593)	0.5972
${\overline{O}}_3$	0.1448	0.4616	0.1590	0.215|(0.2012)	0.2012
${\overline{O}}_4$	0	0|(0.0478)	0	0|(0.2068)	0.2546
Demand	0.2060	0.1526	0.2319	0.5673	—

.

Formulate the solution based on Table 9 for Example 2.

$$\begin{array}{c}({\overline{O}}_1,{\overline{D}}_2)={\tilde{x}}_{12}=0.1048,({\overline{O}}_2,{\overline{D}}_1)={\tilde{x}}_{21}=0.2060\\ ({\overline{O}}_2,{\overline{D}}_3)={\tilde{x}}_{23}=0.2319,({\overline{O}}_2,{\overline{D}}_4)={\tilde{x}}_{24}=0.1593,\\ ({\overline{O}}_3,{\overline{D}}_4)={\tilde{x}}_{34}=0.2012,({\overline{O}}_4,{\overline{D}}_2)={\tilde{x}}_{42}=0.0478,\\ ({\overline{O}}_4,{\overline{D}}_4)={\tilde{x}}_{44}=0.2068.\end{array}$$

In order to obtain the total cost of transportation for the transportation problem outlined in Example 2, replace all of the values of ${\tilde{x}}_{\mathfrak{i}\mathfrak{j}}$ in the objective function:

$$\begin{array}{c}Min=0.1048\times 0.0297+0.2060\times 0.5113+0.2319\times 0.1885+0.1593\times 0.6358+0.2012\times 0.215\\ +0.0478\times 0+0.2068\times 0.\end{array}$$

$$Min=0.00311256+0.1053278+0.04371315+0.10128294+0.043258+0+0.$$

$$Min=0.29669445.$$

For the data supplied in Table 4, we obtain the minimal value of 0.29669445 by applying various strategies (such as the least-cost method and Stepping Stone method, combined with existing formulations) while using something new acknowledged as the IVFN approach for the most effective results.

A sensitivity analysis reveals the model’s flexibility and robustness, demonstrating that it can maintain optimal solutions in a variety of situations, including balanced and unbalanced transportation problems. This dynamic environment alters costs as they progress, slowly returning to their original values. They also ensure that transport systems in the Netherlands remain profitable and efficient. Furthermore, the model offers unique and superior solutions by addressing gaps, ambiguities, and uncertainties that traditional optimization methods cannot achieve. It makes the model a unique and applicable technique for implementation, ultimately contributing to efficiency and cost-effectiveness.

8. Comparative Analysis

Here, we conducted a comprehensive comparison between our transmission model and various existing techniques and found that our model outperforms them in terms of accuracy, as demonstrated numerically in Table 1 and graphically in Figure 1. Traditional models, such as least cost, Vogel’s approximation, and north–west corner techniques, primarily focus on identifying initial feasible solutions. These approaches struggle to effectively regulate and manage uncertainties and flat costs. Our purposeful approach can handle both interval and fuzzy uncertainty, making it adaptable to a wide range of real-world situations. Its ability to handle both intervals and fuzzy uncertainty makes it adaptable to a wide range of real-world situations. We compare our findings to the optimal transportation cost (OTC) of both models, which we calculate using a sensitive analysis. We compared our results with the least-cost method (LCM) [59], north–west corner method (NWCM) [63], Stepping Stone method (SSM) [64], Zero-point Allocation by Vogel’s Approximation Method (ZP-VAM) [65], Modified Distribution method (MODI) [66], Total Opportunity Cost Matrix Approach (TOCMA) [67], Transportation Algorithm by Dantzig (DTA) [68], Hungarian Method (HM) [69], Russells Approximation Method (RAM) [70], Dhouib-Matrix-Tp1 (DMTp1) [71], and neutrosophic basic feasible solution method (NFSM) [72] with the propose transportation model (PTM).

The following

Table 10. Comparison analysis for balanced problem.

Method	Example 1 OTC	OTC	Validity %
LCM	0.3154	0.5207	60%
NWCM	0.3446	0.5207	66%
SSM	0.4381	0.5207	75%
ZP-VAM	No Result	0.5207	0%
MODI	0.2894	0.5207	55%
TOCMA	No Result	0.5207	0%
DTA	No Result	0.5207	0%
HM	No Result	0.5207	0%
RAM	1.2765	0.5207	41%
DMTp1	0.7869	0.5207	66%
NFSM	0.6877	0.5207	75%
PTM	0.6481	0.5207	81%

Note: the proposed method gives validity of 100% ± 0.024 numerical error.

and

Table 11. Comparison analysis for unbalanced problem.

Method	Example 2 OTC	OTC	Accuracy %
LCM	0.2146	0.2967	72%
NWCM	0.2102	0.2967	70%
SSM	0.2486	0.2967	83%
ZP-VAM	No Result	0.2967	0%
MODI	0.2299	0.2967	77%
TOCMA	No Result	0.2967	0%
DTA	No Result	0.2967	0%
HM	No Result	0.2967	0%
RAM	0.5644	0.2967	52%
DMTp1	0.4962	0.2967	60%
NFSM	0.3932	0.2967	75%
PTM	0.3244	0.2967	91%

Note: the proposed method gives validity of 100% ± 0.09 numerical error.

show the numerical values of the comparison analysis for balanced and unbalanced problems, respectively. Similarly, Figure 6 and Figure 7 illustrate this contrast graphically, and the trend line shows the good accuracy of our model. Please keep in mind that we used MATLAB software R2020a to find both the ideal solution and the solution to our current approaches.

9. Advantages of the Model

The proposed model, which uses interval-valued fermatean neutrosophic numbers (IVFNNs) in a multi-objective optimization scheme, outperforms traditional navigation optimization methods in finding solutions independently and has the following advantages:

• The model utilized interpolated interval-valued fermatean neutrosophic numbers (IVFNNs) to illustrate and assess errors and unknowns in navigation data. This gives strong and reliable answers.
• It can simultaneously optimize multiple conflicting goals, such as reducing costs, increasing reliability, and ensuring on-time delivery, while providing a balanced and comprehensive solution.
• Many people use the flawed line-of-best-fit method to estimate varying membership levels with changing transporters, supply, and demand, but this method eliminates it and provides a different perspective on how variable uncertainty actually is compared to using only the small, higher-performing data points.
• The technique of optimization post-adjustment alters the cost values to display the initially feasible solutions and then restores the cost values back to their original values, keeping all the parameters flexible enough such that the transport system is profitable as well as working efficiently.
• The numerical examples include parametric analysis and the stability set, which serve as a reminder to understand the impact of the parameters on the transport process and ensure that the solution converged under various conditions.
• Cost adjustment and redesign: it follows an iterative approach which always guarantees that the transport system is gradually improving, making it able to be used and adapt in the dynamic environment.
• Our model, by handling multiple objectives and uncertainties, yields a much more comprehensive and realistic optimization algorithm as opposed to single-objective methods.
• A balance among several objectives yielding such an outcome leads to an optimally cost-effective, reliable, and efficient transportation system. Multi-dimensional transportation challenges constrain most solutions to this era-defining problem.
• The numerical and sensitivity analysis demonstrate that the model can be successfully used for the balanced as well as unbalanced loading conditions, typically encountered in real life.
• These elements found within the model provide a clear structure for defining stability sets, sensitivity analysis, and application tools that businesses can use to squeeze more transportation efficiency out of the resources they have today.

10. Limitations

Although the proposed model is highly effective, it has some limitations. This can be computationally intensive, requiring advanced algorithms and significant resources. Accurate and high-quality data entry is essential to our productivity. Initial model calibration can be difficult without specialized knowledge. Scaling large transport networks can be challenging and can lead to long computation times. The iterative process should be sensitive to initial conditions and assumptions. Applications can vary from industry to industry, benefiting greatly from a dynamic environment. Maintaining accuracy requires the continuous updating of data and parameters, which can be resource-consuming. Despite these limitations, the model remains widely applicable and effective for horizontal navigation under uncertain conditions.

11. Conclusions

This study used fermatean neutrosophic numbers with interval values to solve the transportation problem, which had three variables. In the context of issue solving, optimization is synonymous with selecting the best possible solution from among the available options. Prior investigative studies demonstrate that a wide variety of domains can utilize this method. On the other hand, however, this work’s MODI approach, which completely eliminates uncertainty and employs crisp values, only achieves an accuracy of 80% when applied to such problems. By generating a simple and easy-to-understand solution, it reduces complications, allowing for its application in other areas to optimize other problems. The advantages of utilizing these approaches are addressed in greater depth below:

1.

We start by discussing the key aspects of transportation issues and the decline in merchandise costs.

2.

Second, we demonstrated that fuzzy set theory is the only method capable of handling difficulties and reducing uncertainty. Consequently, our proposed operators for fermatean neutrosophic numbers, which utilize interval values, exhibit greater flexibility compared to conventional methods.

3.

Third, we developed and applied the algorithm, and we found that the results provided by the algorithm are reliable and accurate when compared to other strategies that are already in use, indicating the usefulness of these algorithms in real-world scenarios.

4.

We think this method could help us learn more about things like personalized individual uniformity control agreement, acceptance reaching when dealing with problems related to non-cooperative behavior control, and symmetrical corresponding making choices when criteria are incomplete and have many levels of detail. The study of the constraints imposed by the proposed algorithm disregards the degrees of involvement, abstinence, and non-membership. The planned operations are currently implementing a new hybrid structure consisting of interactive and prioritized techniques.

5.

We will use state-of-the-art methods in our upcoming work to explore the theoretical underpinnings of transportation problems in various fuzzy environments. We will also discuss the diverse fields that employ these techniques, such as the social sciences, finance, robotics, horticulture, intelligent systems, soft computing, and human resource management.

6.

Future research should improve the accuracy and flexibility of navigation solutions by combining real-time data and machine learning algorithms to dynamically adjust model parameters based on current conditions.

7.

There is a need to develop flexible and efficient optimization algorithms to handle large transport networks, reduce computation time, and manage increased complexity for broader applications and impacts.