Topological Structure of Single-Valued Neutrosophic Hesitant Fuzzy Sets and Data Analysis for Uncertain Supply Chains

: From production to retail, the food supply chain (FSC) encompasses all stages of food production. Food is now transmitted across continents over long ranges. People depend on supply chains for basic necessities such as food, water, drinks, etc. Any disruption in these shipment pipelines poses a serious threat to human life. Supplier selection (SS) has been identiﬁed as a crucial component of FSC, which has been contemplated as a multi-criteria decision-making (MCDM) problem in many studies. The failure of some speciﬁc MCDM problems is due to failure in contemplating the relationships between alternatives under uncertain circumstances. To address such challenges, we present a contemporary method for designating green suppliers based on single-valued neutrosophic hesitant fuzzy (SVNHF) information, in which the input assessment is taken into account using single-valued neutrosophic hesitant fuzzy numbers (SVNHFNs). The foremost purpose of this analysis is to construct a topological structure on single-valued neutrosophic hesitant fuzzy sets (SVNHFSs) as well as to validate several key properties with examples. We discuss certain properties of SVNHF topology such as the SVNHF closure, SVNHF interior, SVNHF exterior, and SVNHF frontier. We also examine the conceptualization of the SVNHF dense set and SVNHF base in SVNHF topology using comprehensive examples. Eventually, to demonstrate and validate the superiority and inferiority ranking (SIR) method and choice value (CV) method in terms of their rationality and scientiﬁc basis, a real-world example of supplier selection in a food supply chain is provided. A comparative analysis is also performed to discuss the symmetry, validity and advantage of the proposed techniques.


Introduction
Data analysis and information aggregation techniques have been an increasing focus in various fields such as engineering, healthcare, economics, environmental concerns, and decision making. Due to uncertain information and limitations in data analysis, we cannot seek accurate and ideal evaluation in MCDM problems. To resolve such circumstances, Zadeh [1] introduced "fuzzy set theory" initially with the concept of membership function on behalf of an exact real number in [0, 1] to express the degree of belonging of objects under a criterion. The components of membership (MG) and non-membership (NMG) of objects were addressed by Atanassov [2] in terms of an "Intuitionistic fuzzy set (IFS)". "Pythagorean fuzzy set" (PFS), a new method for coping with vagueness when considering membership degree ω µ and non-membership degree ω ν was proposed by Yager [3]. It can characterize uncertain information more adequately and accurately than IFSs. Although IFSs and PFSs can effectively report the attribute values in MCDM in the vast majority of instances, there are a few instances where they are deficient. In accordance with the constraints imposed by IFSs and PFSs, the attribute value cannot be represented by both IFSs and PFSs when the square sum of ω µ and ω ν degrees exceeds unity. In order to cope with this scenario, Yager [4] established the notion of the q-rung orthopair fuzzy set (q-ROFS), which can be considered a generalization of IFS and PFS. The "picture fuzzy set" (PiFS) was developed by Cuong [5], and the "spherical fuzzy set" (SFS) initiated by Mahmood et al. [6], Ashraf et al. [7], and Gündogdu and Kahraman [8]. The idea of a neutrosophic set was suggested by Smarandache [9]. Some extensions of fuzzy sets and their constraints are expressed in Table 1. Table 1. Representations of some fuzzy sets and fuzzy numbers as well as their constraints.
Molodtsov [11] proposed "soft set" (SS) theory to cope with uncertainty of parameters and their approximate elements. Hashmi et al. [12] introduced the m-polar neutrosophic set and extended it to m-polar neutrosophic topological structure with clustering analysis and healthcare. It can be difficult to regulate an element's membership in a fixed set at times, which could be due to a misunderstanding of a collection of different values. Torra [13] suggested the notion of "hesitant fuzzy sets" (HFSs) as a generalization of FSs to better describe this situation, which permits membership degree to aid in the collection of practicable values in the closed interval [0, 1]. Experts have used HFSs to choose a variety of possible MGs to evaluate objects under suitable criteria. The theory of HFS has a diverse set of applications in a variety of disciplines, such as for computational intelligence, clustering, healthcare, and MCDM problems.
In an HFS, because of the doubts of decision makers, there is only one truth-membership hesitant function, and it is impossible to manifest this problem using only a few different values assigned by truth, indeterminacy, and falsity membership degrees. As a result, it can only represent one type of hesitancy statistics in this situation and cannot manifest three types of hesitancy statistics. To handle uncertain problems, the idea of a single-valued neutrosophic hesitant fuzzy set (SVNHFS) was introduced by Jun [14]. The truth-membership hesitancy function (TMFF), the indeterminacy-membership hesitancy function (IMHF), and the falsity-membership hesitancy function (FMHF) are three parts of the SVNHFS that can exhibit three types of hesitancy information in this state. Tanuwijaya et al. [15] proposed a novel SVNHF time series model and applied it to stock index forecasting in Indonesia and Argentina. Aggregation operators of SVNHFS were introduced by Liu et al. [16]. Wang and Li [17] proposed generalized single-valued neutrosophic hesitant fuzzy prioritized aggregation operators. The TOPSIS method for neutrosophic hesitant fuzzy multi-attribute decision making was extended by Giri et al. [18].
The CV method is a renowned and widely used MCDM basis for evaluating a set of choices using a set of criteria. Each choice is contrasted with the others by calculating a number of ratios, one per choice criterion. Every ratio is multiplied by the proportional weight of the criterion in consideration. The selection of one or more options from the set of alternatives based on the number of criteria is a fundamental task in MCDM problems.
Xu [19] proposed the SIR method, which is extension of the PROMETHEE method. The SIR method is an important MCDM approach which can grasp real data and supply the system user with six different preference structures. According to the two ranking lists, the SIR method ranks the alternatives more accurately. This method ranks alternatives using a superiority ranking list and an inferiority ranking list. The great feature of using the SIR method is that it incorporates the possessions of other MCDM techniques such as TOPSIS, SAW, and PROMETHEE. Some applications of the SIR method are given in Table 2. Certain novel concepts of neutrosophic sets, neutrosophic logic, and neutrosophic probability were explored in [31]. Seikh and Dutta [32] developed a matrix games model based on SVNSs. Saha and Paul [33] proposed generalized weighted exponential similarity measures for SVNSs. Alcantud et al. [34] developed generalized OWA aggregation operators and multi-agent decision making with N-soft sets. Sitara et al. [35] proposed the notion of q-rung picture fuzzy graph structures for decision analysis. Riaz et al. [36] studied recent trends in pharmaceutical logistics and supply chain management based on distance and similarity measures for bipolar fuzzy soft sets. Farid and Riaz [37] investigated the properties of Einstein interactive geometric information aggregation with q-ROFSs. Zararsiz and Riaz [38] introduced the idea of bipolar fuzzy metric spaces with applications in decision making. Riaz et al. [39] proposed topological data analysis with spherical fuzzy soft AHP-TOPSIS for an environmental mitigation system. Riaz et al. [40] proposed the idea of interval-valued linear Diophantine fuzzy Frank aggregation operators for computational intelligence and MCDM problems.
The foremost purpose of the paper is to construct the topological structure on a single-valued neutrosophic hesitant fuzzy set and to derive significant results. These results are explained with the help of examples. We define certain concepts of SVHF topology such as the interior of SVNHFS, the closure of SVNHFS, the exterior of SVNHFS, the frontier of SVNHFS, dense sets and the base of SVNHF topology. We establish an extension of the SIR technique towards SVNHF topology to deal with uncertain MCDM problems. Moreover, to demonstrate and validate the SIR method and the CV method, a practical example of supplier selection in a food supply chain is provided. A comparative analysis is also given to discuss the validity and advantage of the proposed techniques.
The organization of the rest of the paper is as follows. In Section 2, we examine some elementary conceptions such as NS, SVNS, HFS, SVNHFS, SF, and AF of SVNHFNs and operations on SVNHFSs. In Section 3, we introduce the topological structure of SVNHFSs. Section 4 introduces the SIR method, and the CV method is developed in Section 5 for SVNHF information. In Section 6, an application of the SIR method and the CV method for SVNHF information is illustrated for data analysis in uncertain supply chains. Section 5 concludes the article and discusses future directions.
Then, an SVNHFS ℵ Υ in K can be expressed as follows: where i varies according to alternatives, and the absolute SVNHFS is denoted as 1 ℵ Υ .
. . , n) be the collection of SVNHFEs. Then, the score function (SF) S(N i ), the accuracy function (AF) A(N i ), and the certainty function (CF) C(N i ) of N i (i = 1, 2, . . . , n) can be defined as follows: SVNHFSs are used to express the method of ranking alternatives as follows, based on the concepts of SF, AF, and CF on SVNHFSs. Let N 1 and N 2 be two SVNHFEs; the ranking method is: , l ω I k and l ω ν k are numbers of feasible membership values in N k for k = 1, 2.

Definition 9. Let us consider two SNVHFEs N
Complement: The complements of SVNHFEs N 1 and N 1 can be expressed as follows: The union of two SVNHFEs is defined as follows:

4.
The intersection of two SVNHFEs is defined as follows:

SVNHF Topology
Ye [14] proposed the idea of SVNHFS as an efficient model for modeling uncertainties. Biswas et al. [41] suggested the notions of SF, AF, and CF for SVNHFEs. In this section, the notion of SVNHF topology is introduced using fundamental characteristics of SVNHFSs. Definition 10. Let K be a set and τ be the collection of SVNHFSs in K. Then, τ is called an SVNHF topology if it satisfies following properties:  Tables 3 and 4 show the union and intersection, respectively, of the SVNHFSs ℵ Υ 1 and ℵ Υ 2 . Table 3. Union of SVNHFSs.
Proof. The proof is obvious.

Example 5. From Example 4, let us consider
Thus, τ 2 is SVNHF coarser than τ 4 and τ 4 is SVNHF finer than τ 2 . Definition 14. Assume a universal set K, and the assemblage of all SVNHFSs τ is defined over K. Then, τ is an SVNHF discrete topology on K, and (K, τ) is known as SVNHF discrete topological space over K.
Proof. The proof is obvious.
Theorem 4. Assume that (K, τ) is an SVNHF topological space over K, ℵ Υ 1 and ℵ Υ 2 are SVNHFSs over K. Then, 1. ( This is obvious from the definition of the SVNHF interior. 2. Since (ℵ Υ 1 ) • is an SVNHF open set and it is also the biggest SVNHF open subset of itself, From part (4), Theorem 5. Suppose that K is a universal set, that (K, τ) is an SVNHF topological space over K, and that ℵ Υ 1 and ℵ Υ 2 are SVNHFSs over K. Then, 1.
The proof is obvious.
As we know that ℵ Υ ( Proof. Straight forward.

Definition 20. Consider an SVNHF topological space
Example 10. Let us consider SVNHF topological space given in Example 4 an SVNHFS ℵ Υ defined in Example 6. We see that ℵ Υ = 1 ℵ Υ . This shows that ℵ Υ is dense in K.

Extension of SIR Method for SVNHF Information
In this section, a new MCDM method is developed based on Algorithm 1, which is an extension of the SIR method to SVNHFSs.
Attain the performance function f ij : where Attain the preference intensity P I j (ω x i , ω x t ): We define P I j (ω x i , ω x t )(i, t = 1, 2, · · · , l, i = t; j = 1, 2, · · · , m) as the preference intensity of alternatives ω x i to the corresponding attribute ω C j , which is given as follows: where φ j (d) is a non-decreasing function from the real number to [0, 1]. Normally, φ j (d) is from a set of six generalized threshold functions [42], or interpreted by the experts themselves.

3.
Acquire superiority matrix and inferiority matrix: Inferiority index (I-index): I = (I ij ) l×m Step 5: Calculate the superiority flow and inferiority flow as follows: S-flow By using Equation (1), we calculate the score function of the corresponding S-flow Ψ > (ω x i ) and I-flow Ψ < (ω x i ), respectively. Hence, we obtain the S-flow and I-flow of alternative . It seems that if the S-flow Ψ > (ω x i ) is larger and the I-flow Ψ < (ω x i ) is smaller, the alternative ω x i is preferable.

Algorithm 1 Cont.
Inferiority ranking rule (I R-Rule): Step 7: By incorporating the SR-Rule and the IR-Rule, we can achieve the best alternative ω x i (i = 1, 2, · · · , l). A flow chart of the SIR method for supplier selection is shown in Figure 1.

Extension of CV Method for SVNHF Information
In this section, a new MCDM method is developed in Algorithm 2, which is extension of the CV method to SVNHFSs. Table 5 includes various types of cases where different systems were used to help with operations or decision-making in the food supply chain.

Algorithm 2 (CV method for SVNHFSs)
Step 1: Obtain the decision matrices from the decision makers, with alternative ω x j evaluated on the basis of criterion ω C i , given in Table 6. The aggregated decision matrix p ij is obtained using step 1, step 2 and step 3(2).
Step 2: Decision makers also give weight w to the criteria with the condition that the sum of weights must be equal to 1. Then, the multiplication of decision matrix is computed with criteria weights, to obtain the matrix q ij .
Step 3: Find the score function of each SVNHFN.
Step 4: Compute the ranking of the alternatives according to their score function values.

Case Study
The food industry is critical for delivering the essentials for a variety of uses and tendencies [43]. Food must be stored, supplied, and marketed as soon as it is cultivated or produced so that it can reach the ultimate customers on time. Every year, worldwide food loss would supply more than enough to nourish the world's nearly 1 billion starving people. In Pakistan, it is anticipated that 40% of food is wasted. Food is produced in sufficient quantities to feed the overall population of Pakistan, but due to food waste, an expected 6 out of 10 inhabitants go to bed without dinner. Pakistan stands 107th out of 118 developing countries on the International Poverty Index. Approximately one-third of all produced food is destroyed or wasted each year (approximately 1.3 billion tonnes) [44]. Two-thirds of all waste in food (roughly 1 billion tonnes) occurs at the supply chain stage, which encompasses cultivation, shipments, and storage [45].
The term FSCM has been utilized to depict the activities or procedures occurring during the yield, dispersion, and the use of various foods in order to preserve their quality and safety in effective and efficient ways [46,47]. The relevance of factors such as safety, food quality, and freshness within a specified time frame distinguishes FSCM from many other supply chains including furniture logistics and supply chain management, making the underpinning supply chain more convoluted and unmanageable [48]. As the challenges of global coordination have increased, the attention turned from a single echelon, such as food production, to the effectiveness and efficiency of the comprehensive supply chain. That is, the food supply chain resources such as vehicles, storage areas, transport services, and laborers will be used proficiently to ensure the quality and safety of food through effective efforts including optimization decisions [49]. The relevance of value chains in FSCM is that they benefit both consumers and producers. The traceability of food has become increasingly popular in recent decades, with a wide range of applications. Because of the emergence of food, FSCM is becoming more dynamic and complex in order to meet the diversifying and globalized industries.

FSCM IT Systems
There is no doubt that IT systems are crucial in FSCM because so much can go wrong, such as vehicles, food suppliers, data entry, and so on. The decision-making systems and traceability for FSCM are used as examples of current state-of-the-art situations that professionals can use when instituting IT-based solutions. A food's traceability consists in a data trail that follows the physiological trail of the food through different phases [50]. Some systems track food all the way from the retailer to the farm, while others only examine key stages of the supply chain. Some traceability systems gather information only for tracking food products to the minute of manufacturing or logistics path, while others track only superficial data, such as in a vast geographical area [51]. Aside from FSCM's traceability systems, other decision-making processes in the food industry include implementation, strategy, vehicle maintenance, and WMS.
The internationalization of food production, logistics, and utilization has resulted in an integrated world for FSCM, whose models are critical in ensuring consistently high standards and security in food products [52]. Quality of food, high delivery performance, and food security appear to be more important in these models. Multi-objective considerations are also common; for example, food quality assurance is incorporated into decision models. Recently, supply chain effectiveness and value chain evaluation have received special attention, since the international FSCM is becoming increasingly important.

Instance Firm Network Enhancement
Aramyan et al. [53] Tomato firm Performance Efficiency and flexibility measurement system have both improved, food quality has improved, and there is a faster response time.
Bevilacqua et al. [54] Tronto Valley ARIS Three types of costs are being reduced; improved traceability.
Tuncel and Alpan [56] A medium size Risk management The percentage of orders completed on time has increased to 90.6%, with risk reduction rising by 9.9%.
Zhu et al. [57] A food manufacturer Customer cooperation Customer cooperation has improved; system internal environment management has been improved.
Jacxsens et al. [58] A fresh producer Food safety Food of higher quality; management system improved risk management ability.
Friel et al. [59] Agri-food supply chain H&S food A more nutritious diet, decision-making with improved environmental system sustainability. Choosing a supplier is a critical element of any business's operations. Reputation, reliability, service, cost, and value for money are all important considerations. The aim of supplier selection is to identify the best supplier who delivers the best value for money in terms of product or service. Suitable supplier selection yields good profit and quality in the end. The supplier is treated as an integral part of the organization in this strategic alliance. All purchasers must choose a supplier, and it is a critical step in the acquisition process. Purchasers should go through several stages of decision making and develop their own selection criteria for selecting appropriate suppliers.

MCDM Process
The RH Flour Mills in Lahore wants to find the best supplier for one of its key components in the manufacturing process. Four suppliers were left as alternatives. The four criteria considered were: quality and safety, delivery, social responsibility, and service. The suppliers are evaluated using the recommended methodology by a group of decision makers. In multi-criteria decision making with a fuzzy environment, four decision makers were chosen, consisting of supplier experts and expert academics. The steps in the procedure for selecting the best green supplier are as follows.
The decision-making process using Algorithm 1 is illustrated as follows: , ω x 4 } is the set of alternatives and C = {ω C 1 , ω C 2 , ω C 3 } is the set of attributes. Assume that E = {ω e 1 , ω e 2 , ω e 3 } is the set of experts. Then, the single-valued neutrosophic hesitant fuzzy decision matrices are expressed in Table 6, the weights of the experts are given in Table 7, and the weights of the attributes are shown in Table 8.  Step 1: Compute the individual measure degree ρ k (k = 1, 2, 3) using Equation (6), given by ρ = (0.432, 0.466, 0.573) µ Step 2: Acquire the normalized vector using Equation (7), given as = (0.294, 0.317, 0.390) µ Step 3: The attribute weights can be obtained using Equation (8), which are expressed as follows:ẇ The aggregated decision matrices can be obtained using Equation (9), and can be written as follows: Step 4: Acquire the performance function f ij using Equation (10): The threshold attribute function was set to Acquire the superiority matrix (S-matrix) using Equation (12) Step 5: Measure the flow of superiority and inferiority using Equations (14) and (15), which are exhibited in Tables 9 and 10. Table 9. SVNHF superiority flow.  Step 6: Integrate Table 9 with the SR-Rule, and the following seems to be accessible: Table 10 with the IR-Rule, and the following seems to be accessible: Step 7: The most desirable alternative, according to the results of the SR-Rule and the IR-Rule, is ω x 1 . A representation of SVNHF superiority and inferiority flow is shown in Figure 2.  The decision-making process using Algorithm 2 is illustrated as follows: Step 1: Consider SVNHF decision matrices given in Table 6. Obtain aggregated decision matrix (p ij ) 4×3 using Step 1, Step 2, and Step 3.
Step 2: The decision makers provide weights to three criteria as w 1 = 0.235, w 2 = 0.312, and w 3 = 0.453, with ∑ w i = 1 Step 3: Compute the score values of each alternative. The score values of the alternatives are given in Table 11. Table 11. Score values.

Alternatives
Score Values Step 4: Rank the alternatives according to their score values. As a result, ω x 1 is the best supplier among the four alternatives according to the qualities of all criteria.

Comparative Analysis
This paper develops new techniques for modeling uncertainties using SVNHF information. We compare the ranking of alternatives using proposed SIR method and the CV method for SVNHFSs. If we use the SVNHF SIR approach to assemble the alternatives, they are ranked for superiority flow as Based on these findings, it is clear that the ranking of the alternatives is not same. However, the optimal alternative ω x 1 remains identical in both MCDM methods. The ranking of alternatives using the SIR method and the CV method is also shown in Figure 3.

Conclusions
This paper was designed to introduce the concept of single-valued neutrosophic hesitant fuzzy (SVNHF) topology and its applications in data analysis for uncertain supply chains. An SVNHFS is a hybrid structure of a hesitant fuzzy set (HFS) and a single-valued neutrosophic set (SVNS), which is a novel concept for modeling uncertainties in real-life circumstances with key features of three membership functions: truth-hesitancy membership function, indeterminacy-hesitancy membership function and falsity-hesitancy membership function. Using the characteristics of SVNHFSs, we defined the notion of SVNHF topology. We investigated the fundamental properties of SVNHF topology, such as the SVNHF closure, the SVNHF interior, the SVNHF exterior, and the SVNHF frontier, as well as the SVNHF dense set and the SVNHF base, with the help illustrative examples. Eventually, to demonstrate and validate the SIR method and the CV method in terms of rationality and scientific basis, a real-life example of supplier selection in a food supply chain was provided. A comparative analysis was also given to discuss the validity and advantage of proposed techniques. The proposed methods can be further extended to investigate the dynamics of human decision analysis, humanized computing, data analysis, computational intelligence, and healthcare.