#### Abstract

We developed a multicriteria decision-making method based on the list of novel single-valued neutrosophic hesitant fuzzy rough (SV-NHFR) weighted averaging and geometric aggregation operators to address the uncertainty and achieve the sustainability of the manufacturing business. In addition, a case study on choosing the optimum elements for a sustainable manufacturing sector was carried out. The proposed decision support method is then compared to other relevant methodologies, and a validity test is performed to show the reliability and validity of the new methodology. Sustainability is one of the most important issues the world economy is facing today. Several industrial businesses have incurred large financial losses as a result of their ignorance of sustainability issues. Manufacturers in industrialized countries have done a decent job of making sure that their businesses are sustainable over the long run. Modern companies use a lot of modern technologies. These include blockchain, artificial intelligence (AI), the Internet of Things (IoT), big data analytics (BDA), and fuzzy logic (fuzziness). These modern technologies support the continuation of life, either directly or indirectly. Therefore, it is of utmost importance to concentrate on those elements that encourage the adoption of sustainability. The goal of this study is to provide a framework for using cutting-edge technology to increase the adoption of sustainability in manufacturing firms. Under the guidance of single-valued neutrosophic hesitant fuzzy rough (SV-NHFR) aggregate information, it was advised to place a strong emphasis on addressing sustainability, waste management, environmental protection, manufacturing cost savings, and chemicals and resources. The results suggest that the proposed technique can solve the inadequacy of the existing decision method by the SV-NHFR aggregation operators in terms of decision adaptability.

#### 1. Introduction

Many researchers have used intuitionistic fuzzy sets (IFSs) to handle decision-making problems (DMPs) so far although the accuracy is not great enough at this point to handle the uncertainty. Smarandache [1] was the first to propose the neutrosophic set (NS), a philosophical discipline and mathematical tool for comprehending the origin, nature, and scope of neutralities. It is a spiritual practice that explores the origin, nature, and extent of neutralities, as well as their interactions with other ideational spectrums.

The NS generalizes the concepts of the classic set [2], fuzzy set, interval-valued fuzzy set, IFS, interval-valued IFS, paraconsistent set [3], dialetheist set, paradoxist set, and tautological set [4]. An NS is characterized by truth membership function , indeterminacy membership function and falsity membership function , where , , and are real standard or nonstandard elements from . Although an NS philosophically generalizes the notions of FS, IFS, and all the existing structures, it will be challenging to implement in real-world scientific and engineering situations.

This concept is critical in many contexts, such as information fusion, where data from several sensors is integrated. Recently, neutrosophic sets have primarily been used in engineering and other sectors to make decisions. Wang et al. [5] proposed a single-valued neutrosophic set (SV-NS), which can handle inaccurate, indeterminate, and incompatible data challenges. On the one hand, an SV-NS is an NS that allows us to convey ambiguity, imprecision, incompleteness, and inconsistency in the real world. It would be more suitable to employ uncertain information and inconsistent information matrix in decision-making [6–8]. The decision-making with the linguistic term with weakened hedge (LTWH) is very a useful tool [9]. SV-NSs, on the other hand, can be employed in scientific and technical applications since SV-NS theory is useful in modeling ambiguous, imprecise, and inconsistent data [10, 11]. The SV-NS is suitable for collecting imprecise, unclear, and inconsistent information in multicriteria decision-making analysis due to its ability to easily capture the ambiguous character of subjective judgments.

Many scholars paid close attention to SV-NS since it is a powerful universal systematic procedure. Ye [12] described the information energy and correlation of SV-NSs. The application of SV-NSs as a decision-making method was then explored by various authors [13]. The SV-NS sets are extremely useful for dealing with uncertainty challenges and improving accuracy in uncertainty challenges. The current research is inspired by this concept and concentrates on the SV-neutrosophic hesitant fuzzy rough aggregation context that is our new concept. The basic concept of SV-neutrosophic hesitant fuzzy rough sets is described in Ref. [14]. With the use of this concept, we can handle uncertainty challenges with accuracy without losing any information from the data. This contribution will be helpful for decision-makers to solve uncertainty challenges with great accuracy. There was a large gap in the literature that without defining the operators of the SV-neutrosophic hesitant fuzzy rough sets, we will not be able to solve each type of uncertainty challenges with this set as we solve a numerical example by using rough data sets and by fuzziness in it.

Our everyday lives have become increasingly concerned with environmental issues in recent years. Sustainable manufacturing is only one of several subcategories that fall under the larger umbrella of sustainable development. Throughout the manufacturing process, many environmental and social challenges also arise. Sustainable manufacturing practices may be used to overcome these obstacles in the production process. Ecologically friendly and resource-efficient manufacturing is the aim of sustainable manufacturing. Because these firms are financially sound, they are also safe for workers, communities, and consumers [15]. The three components of a sustainable manufacturing strategy are the selection of acceptable indicators for monitoring the sustainability of production, an assessment tool for identifying weak areas, and system improvements to strengthen the sustainable manufacturing process [16]. Sustainable manufacturing strategies are crucial for long-term success in the manufacturing industry for both large, small, and medium-sized firms (SMSF). Sustainability practices in manufacturing SMSF differ substantially from those in huge companies due to qualities such as customized management, a lack of finance, insufficient resources, increased flexibility, a lateral framework, a small number of customers, access to a limited market, and a lack of knowledge. Sustainable manufacturing in SMSF cannot be viewed as a scaled-down version of larger organizations based on these characteristics. The bulk of sustainable manufacturing strategies are built on indicators and assessment models that have been established and tested in large manufacturing companies [6, 17].

Indicators for technology assessment can be used in two ways: to evaluate a technology system’s overall performance or to compare at least two technology systems. Rather than creating a generic collection of indicators suitable for all applications, Dewulf and Van Langenhove [18] recommended utilizing a “fit for purpose” method to apply indicators. Indicators are classified as descriptive, performance, or efficiency indicators and can be quantitative or qualitative [19]. When choosing acceptable indicators, a UN report [20] establishes a number of guidelines. In summary, they should be straightforward and instructive, and approaches should be simple and devoid of a huge number of subsets. Changes in the environment and accompanying human activities should be reflected in indicators. They should be precise, should be unambiguous, and should provide a comparison point. Environmental indicators include greenhouse gas emissions, energy consumption, resource renewability, emission toxicity, material reuse, waste material recoverability, and efficiency. Some of the recommended economic matrixes for industrial ovens are net sales, operational production costs, gross margin, and overhead costs [21].

Finally, societal factors are typically linked to toxicity and safety [22]. It is also critical to examine the indicator set using a multicriteria analysis method appropriate for that particular application in addition to establishing a relevant set of sustainability indicators. Tokos et al. [23], for example, were able to establish a framework for assessing integrated sustainability performance in processing sectors. Multicriteria analysis is a decision-making tool that gathers data on a variety of criteria, or indicators, to see how several objectives might be met most effectively. It allows for the evaluation of indicators with different units beside one another. Fuzzy set theory is a well-established topic within multicriteria analysis that offers a solution to problems that standard multicriteria analysis had previously been unable to solve. It is concerned with estimation rather than exact argumentation [24], enabling uncertainty to be logically addressed by assigning an acceptability grade to quantitative and qualitative data. Fuzzy indicator sets, which include both qualitative and quantitative data, have recently been shown to be a tool for assessing sustainability indicators [25] by allowing objective decision-making of indicators that are often subjective. In the case of qualitative indicators, uncertainty can be caused by imprecise measurements, average or outdated data, proxies and incomplete data, approximations in modeling, normalization and weighting [26], assessment and linguistic descriptors by experts, and their assigned values. When utilizing traditional multicriteria analysis to solve problems, uncertainty in the assessment of sustainable development presents complications. Fuzzy theory, on the other hand, is based on multivalued logic and deals with events that have no clear meaning, allowing fuzziness to characterize the degree to which an event occurs (and soft thresholds) [24].

The study established a mechanism for systematically assessing and analyzing social sustainability goals. The study’s scientific worth is the establishment of a theoretical model for evaluating social sustainability projects, computation of the fuzzy social sustainability index, and the identification of weaker features. A typical crisp methodology was used to confirm the fuzzy technique’s results. Fuzzy sets are assigned a degree of membership rather than being in or out. Fuzzy approaches are effective for appraising complex or ill-defined problems, making them ideal for sustainability indicators. The uncertainty of fuzzy indicators is attributed to generality, ambiguity, or vagueness rather than error or randomness.

##### 1.1. Motivation

By acknowledging the global market, manufacturers can easily extend trade abroad and even operate their businesses in low-cost nations. Moreover, they have taken the initiative to reduce industrial emissions into the atmosphere in the current scenario of climate change and damage to the human ecological environment. Intriguingly, the government has also taken steps for the protection of the environment and laws for their implication and regulation, which force producers to design eco-friendly products. Sustainable production of goods can be defined as minimal environmental impact, social security of employment, and the welfare of the community and consumers during the whole span of the product’s life.

Keeping in view the mentioned factors, manufacturing engineers should be more accountable and aware of environmental, economic, and social concerns because manufacturing emissions are a genius problem. Hence, a comprehensive assessment of the available alternatives, multicriteria decision-making (MCDM), is a strategy for tackling the real issues of the world that is the best solution. MCDM-based approaches are gaining popularity due to their wide use in various fields, including medicine, architecture, economics, and a lot more scientific and technological fields.

The MCDM technique has become complicated because of the complexity and uncertainty in data, making it difficult for decision-makers to get the best outcomes. Consequently, SV-NSs provide a better approach to handling such issues. Therefore, relying on the sensitivity of the problem, the simple techniques of SV-NSs are no more useful to get accuracy. So to seek better results, SV-neutrosophic hesitant fuzzy rough sets (SV-NHFRSs) have been discussed, and the analysis depends upon SV-NHFR weighted averaging, and weighted geometric operators.

The goal of the study and the task are mentioned in the following part.

The purpose of this research is to increase the sustainability of manufacturing work cells by using multicriteria decision-making. Two activities have been identified to help achieve this goal:(a)Define and quantify matrix, determine and implement an appropriate weighting mechanism, and determine and execute a suitable ranking system as part of a decision-making approach(b)Demonstrate the process by identifying and describing a representative work cell and utilizing the integrated sustainability assessment method

Some consequential endowment of the current study are as follows:(1)Firstly, we recall the concept of SV-neutrosophic hesitant fuzzy rough sets from literature(2)We proposed novel fundamental operational laws for SV-NHFRSs(3)Design a decision-making strategy that employs proposed aggregation operators to aggregate uncertain data for decision-making difficulties in the part of best option for manufacturing industry sustainability that is based on the Internet of things (IoT)

The remainder of this study is structured as follows: Section 1 presents some basic concepts of SV-FSs, HFSs, and rough set theory briefly. Basic notations and concepts are described in Section 2. A novel notion of SV-neutrosophic hesitant fuzzy sets (SV-NHFSs) are presented in Sections 3 and 4, respectively. Section 5 presents a list of algebraic SV-hesitant fuzzy aggregation operators for combining uncertain data in decision-making. The validity and reliability tests are presented in Section 6 to ensure that the suggested approach is effective. This manuscript comes to a close with Section 7.

#### 2. Preliminaries

In this constituent, we study the elementary concepts for hesitant fuzzy sets (HFS), neutrosophic sets (NS), single-valued neutrosophic sets (SV-NS), SV-neutrosophic hesitant fuzzy set (SV-NHFS), rough sets (RSs), SV-neutrosophic RS (SV-NRS), and SV-neutrosophic hesitant fuzzy RSs (SV-NHFRSs).

*Definition 1. *Let be a fixed set. The representation of HFS which is explained in Ref. [27] and is mathematically denoted aswhere is a set of values in [0, 1], which indicate the grade of membership of the element in .

*Definition 2. *Assume is a set and . A neutrosophic set [1], in is denoted as membership , an indeterminacy , and a falsity membership values. , , and are real standard and nonstandard subset of andThe representation of neutrosophic set (NS) is mathematically defined aswhere

*Definition 3. *Let be a set and . A single valued neutrosophic set (SV-NS) [5], in is characterized by truth-membership function , an indeterminacy-membership function and a falsity-membership function . and are real standard and nonstandard subsets of and thenThe representation of SV-NS is mathematically defined aswhere

*Definition 4. *Suppose be a fixed set. The representation of SV-NHFS [28], then is mathematically defined aswhere , indicates the hesitant grade of membership, indeterminacy, and falsity of the element to the set .

*Definition 5. *For a fixed set , the (see [29]) is represented mathematically as follows:where and are in the range and show the membership, indeterminacy, and nonmembership values sequentially. It has the following characteristics:andandFor simplicity, we will use a pair to mean .

*Definition 6. *Assume is a universal set and is mapping on A set valued relation (see the relation in [30]) is defined asfor , where is called a beneficiary neighborhood of with relation . A pair is called (crisp) surmise space. For any set, , the lower and upper (*L* ∼ *H*) surmise of to surmise space is defined asThe pair is called the fuzzy rough set and both are *L* ∼ *H* surmise operators.

*Definition 7. *Let universal set and let SV-NHFRS be relation [31], then(i)is reflexive if (ii)is symmetric if (iii)is transitive if , and

*Definition 8. *Let universal set and let SV-NHFRS be SV-NF relation [32]. The pair represents a V-NF approximation space. Let be any subset of SV-NS , i.e., SV-NS . Then on the basis of SV-NF approximation space , then the *L* ∼ *H* surmises of are represented as and that is given as follows:wheresuch thatandAs and are , so :SV-NFS SV-NFS are *L* ∼ *H* surmise operators. So the pairis called the SV-NF rough set. For simplicity, it can be denoted as and are known as the SV-NF rough number (SV-NFRN).

*Example 1. *Suppose is an arbitrary set and is the approximation space with which is the mapping, as given in Table 1. Now a decision professional presents the optimum normal decision object which is a .

andThen, it follows thatIn a similar way, we obtain the other values:Similarly,In a similar way, we obtain the other values:Similarly,In a similar way, we obtain the other values:For lower approximation,In a similar way, we obtain the other values:In a similar way, we obtain the other values:In a similar way, we obtain the other values:Thus, *L* ∼ *H* SV-NHFR sets are as follows:where

*Definition 9 (see [33]). *Let and be two SV-NHFNs.

The following are the basic appropriate measures actions:

*Definition 10 (see [30]). * Assume a universal set with be a (crisp) mapping. Then(1) is reflexive if , for every (2) is symmetric if , , then (3) is transitive if and

#### 3. Construction of Single-Valued Neutrosophic Hesitant Fuzzy Rough Sets

In this dissertation, we introduce the concept of SV-NF rough set (SV-NHFRS), which is a hybrid rough set structure.

We also introduce the SV-NHFRS’s scoring and accuracy features, as well as its basic operational regulations.

*Definition 11. *Assume universal set and let be relation. The pair () represents a approximation space. Let be any subset of , i.e., . Then on the basis of approximation space , the *L* ∼ *H* approximations of are represented as and and given as follows:wheresuch thatandAs and are , so are *L* ∼ *H* approximation operators. So the pairis called the rough set. For simplicity, it can be denoted asis known as the rough number .

*Definition 12. *Let be the universal set, then any subset is known an SV-neutrosophic hesitant fuzzy mapping. The pair is called SV-NHFRS approximation space. If for any then the *L* ∼ *H* operators of to SV-NHFRS approximation space are two SV-NHFRSs, which are given by and and are defined aswheresuch thatandAs are , then are operators. The pairwill be called the -neutrosophic hesitant fuzzy rough set. For simplicity,is described asand is called SV-NHFRSs.

*Definition 13. *Let and be two SV-NHFRSs. Then(i)(ii)

*Definition 14. *Let and be two SV-NHFRSs. Then(i)(ii)(iii) and (iv), for (v), for (vi), , where and shows the aggregate of SV-neutrosophic fuzzy rough operators and , that is, (vii) iff and The score function is used to compare/rank two or more SV-NHFRNs. The SV-NHFRNs has the greater score value are said to be superior SV-NHFRNs. When the score values are equal, we will use the accuracy function.

*Definition 15. *The score function for SV-NHFRNsis given asThe accuracy function for SV-NHFRNsis given aswhere , and represent the number of elements in and , respectively.

*Definition 16. *Suppose and are two SV-NHFNs. Then(i)If , then (ii)If , then (iii)If , then(a)If then (b)If then (c)If then

#### 4. SV-Neutrosophic Hesitant Fuzzy Rough Aggregation Operators

We introduce a novel concept of SV-NHF rough aggregation operators in this article by combining rough sets and SV-NHF aggregation operators to produce the aggregation concepts SV-NHFRWA, SV-NHFROWA, and SV-NHFRHWA. These ideas’ fundamental features are addressed in this article.

##### 4.1. Single-Valued Neutrosophic Hesitant Fuzzy Rough Weighted Averaging Operator

*Definition 17. *Consider the set of values of SV-NHFRNs with weight vector such that and . The SV-NHFRWA operator is determined as

Theorem 1. *Let be the set of values of NHFRNs with weight vector . Then the NHFRWA operator is defined as*

*Proof. *Applying mathematical induction to proof. Applying the operational law, it follows thatandIf , thenHence, the result is correct for . Let it be correct for , that is,