Evaluating the Sustainable Human Resource Management in Manufacturing Firms Using Single-Valued Neutrosophic Distance Measure-Based RANCOM-AROMAN Model

1. Introduction

Sustainable development (SD) is one of vital topics in today's society considering several aspects of economic prosperity, social equality and environmental protection. It is defined as “the development that meets the need of the present generation without compromising the ability of the future generation to meet their own needs” (Piwowar-Sulej, 2021). The concept of SD is described as a process that incorporates the economic growth with ecological balance and environmental protection by considering social aspects (Saeidi et al., 2022). In the 2030 Plan for “sustainable development goals (SDGs)”, participant of “United Nations (UNs)” agreed on the significance of moving toward sustainable forms of consumption and creation, with the developed nations captivating the lead and with all nations profiting from the procedure (Glass & Newig, 2019; Shayan et al., 2022). Based on the UNs, corporate sustainability (CS) and triple bottom line (TBL) are guiding notions for the organizational achievement of SDGs. The concept of “CS” states to a business policy for long-term development that works in agreement with persons and the planet (de Oliveira et al., 2023), while the concept of “TBL” entails of the stability in identical harmony among the social, environmental and economic dimensions of firms (Ramalho & Martins, 2022).

Sustainable human resource management (SHRM) can be defined as “the pattern of planned or emerging human resource management (HRM) strategies and practices anticipated to enable the achievement of economic, social and ecological goals while simultaneously reproducing the human resource base over a long term” (Kramar, 2014; Macke & Genari, 2019). In actual fact, the reaction to the internal and external challenges including increasing work-related health problems (De Vos & Van der Heijden, 2017; Chams & García-Blandón, 2019), a lack of labour force (App & Büttgen, 2016), makes it essential to adapt the elegance towards the managing of people (Stankevičiūtė & Savanevičienė, 2018). SHRM has extensively applied for the HRM practices because it might lead to a “win–win” state for staffs and firms. Adoption of SHRM practices results in the better performance of the companies through improved employment relationships, long-term investment instead of just a financial assistance and employer branding (Sorribes et al., 2021; Martini et al., 2023).

The idea of SHRM is defined as the role of HRM in developing the organizational sustainability and the sustainability of HRM practices (Kramar, 2014; Macke & Genari, 2019). Tooranloo et al. (2017) identified the main indicators influencing SHRM in the manufacturing companies. Moreover, the authors have used fuzzy DEMATEL model to determine the weight of considered factors. Chams & García-Blandón (2019) presented the literature review to examine key role of SHRM and its impact on sustainability. They also investigated the relationship between SHRM and sustainable perspectives and identified the HRM processes that can contribute to attain the SDGs. Macke & Genari (2019) provided the comprehensive review on SHRM to recognize key features, developments and research challenges. In a study, Kainzbauer et al. (2021) presented a bibliometric study to cognize how SHRM contributes to CS. In addition, they highlighted that existing studies on SHRM are heavily weighted towards environmental issues. With the presence of the stakeholders in the supply chain, Ramalho & Martins (2022) presented a theoretical development of SHRM towards sustainability in the construction business. Cachon-Rodríguez et al. (2022) analyzed the result of SHRM with social capital activities on retention and loyalty of employees in higher education sector. Saeidi et al. (2022) used a survey approach to identify the main indicators of SHRM assessment and determined their weights through Pythagorean fuzzy SWARA model. In terms of the SHRM practices, the authors have ranked the manufacturing companies through Pythagorean fuzzy TOPSIS approach. Martini et al. (2023) presented the relationship between SHRM and organizational consequences. In this regard, they explored how corporate training designed at supportive employability influences the employee-employer association and worker retaining. Assessment of SHRM practices of any company depends on several factors including social, economic, environmental (SEE) dimensions of sustainability. Thus, the MCGDM models are more appropriate to deal such varieties of problems.

Due to the overwhelming vagueness and subjectivity of human’s cognizance, it is not always possible for decision experts (DEs) to express the assessment rating of an alternative in terms of crisp number. To express the uncertainty of realistic situations, Zadeh (1965) originated the idea of fuzzy sets (FSs), which has been implemented for diverse disciplines (Li & Zhang, 2023). As the FSs doctrine is described as “belongingness grade (BG)”, however, it may not continuously hold that the nonbelongningness grade (NG) of an object in FSs is equal to complement of BG. As an extension of FSs, Atanassov (1986) originated the notion of intuitionistic FSs (IFSs), which is regarded as BG, NG and indeterminacy grade (IG). Since the doctrine of IFSs can simply manage imprecise and uncertain information but it is not able to handle indeterminate and inconsistent data obtains in realistic concerns. To conquer these issues, Smarandache (1999) introduced the theory of neutrosophic sets (NSs), which is an extended version of FSs and IFSs. Further, Wang et al. (2010) developed the theory of single-valued neutrosophic sets (SVNSs), which overcomes the limitation of NSs in handling with the scientific and manufacturing problems. The SVNS theory provides a more useful and reasonable way to manage indeterminate, inconsistent and uncertain information. Due to its higher flexibility, the SVNS doctrine has been broadly explored from different perspectives.

Li et al. (2023) studied a novel MCDM approach that integrates the aspect-based sentiment analysis, TOPSIS model with SVNS doctrine and further applied for doctor selection. In the context of SVNS, Fetanat and Tayebi (2023) evaluated the hydrogen technologies using the combined CRITIC and CRADIS models. Meng et al. (2023) presented a novel SVNS and geographic information system based MCDM framework for prioritizing the sites for waste-to-energy plant development. Ye et al. (2023) proposed the SVN-score and SVN-accuracy functions for SVNSs and then developed a series of trigonometric “aggregation operators (AOs)” to combine the individual’s decision information in the context of SVNS. In addition, they presented MCGDM tool with combination of SVN-score/accuracy function and the proposed aggregation operators for solving MCGDM problems.

To derive the subjective weight of attributes, Więckowski et al. (2023) proposed the RANCOM tool. This model computes the criteria weight using DE’s knowledge. Its implementation is defined on attributes ranking order to derive the weight value. In the literature, there is no study which uses a weighting model based on SVN-RANCOM model.

To deal with the MCDM problems, several novel methods have been developed under the context of SVNSs (Mishra et al., 2021; Ridvan et al., 2021; Li et al., 2023; Luo et al., 2023; Fetanat & Tayebi, 2023). It is known that different normalization tools have different advantages and disadvantages, and different aggregation tools have different functions. Recently, Bošković et al. (2023a) developed AROMAN model to solve the decision-making problem with crisp number. In this model, two different normalization tools are included, and the gained normalized ratings are combined into averaged normalized decision-matrix. In a study, Bošković et al. (2023b) evaluated the cargo bike delivery concept selection using classical AROMAN model. Nikolic et al. (2023) presented interval type-2 fuzzy generalization of AROMAN tool and applied to enhance the sustainability of postal system in rural regions. There is no research which develops the AROMAN method from SVN information perspective.

In the following, we identify some issues in the existing studies:

• Several existing SVN-distance measures (SVN-DMs) (Ye, 2014; Xu et al., 2020; Chai et al., 2021) generate some counter-intuitive results during the computation of degree of difference between SVNSs.
• The AROMAN method has developed in the context of crisp set and interval type-2 FS (Bošković et al., 2023a,b; Nikolic et al., 2023). The methods presented by (Bošković et al., 2023a,b; Nikolic et al., 2023) ignores the significance of criteria and experts’ weights. In addition, their studies are not able to deal with SVN information.
• Some authors (Tooranloo et al., 2017; Saeidi et al., 2022) have presented different ranking techniques to rank the manufacturing firms, but these studies are not able to illustrate the indeterminate, inconsistent and uncertain data in the assessment of SHRM practices of manufacturing firms.

The key contributions of the paper are listed as follows:

• New SVN-DM is developed to avoid the limitations of extant SVN-DMs under the context of SVNSs.
• A hybrid MCGDM model is proposed to deal with the single-valued neutrosophic MCDM problems in which the information about indicators and DEs are completely unknown.
• To determine the DEs’ weight, a novel procedure is presented in the context of SVNSs.
• To find the indicators’ weights, a weight-determination model is presented with the integration of objective weights using SVN-DM-based model and subjective weights using RANCOM model with SVNSs.
• The presented SVN-DN-RANCOM-AROMAN method is applied on a case study of evaluation of SHRM of manufacturing firms in India, which illustrates its powerfulness and applicability.

The remaining part of this study is organized as follows: Section 2 confers the comprehensive literature review related to this work. Section 3 first discusses the basic concepts and then introduces a new DM for SVNSs. Section 4 develops a hybridized AROMAN method for assessing the MCDM problem from single-valued neutrosophic information perspective. Section 5 shows the introduced tool on a case study of SHRM assessment of manufacturing firms. Furthermore, this section confers the comparative and sensitivity analyses. Section 6 concludes the whole study and suggests for future researches.

2. Sustainable Human Resource Management (SHRM)

The purpose of SD is to define viable schemes considering the SEE aspects of human activity. It is the basis for today's leading global framework for international collaboration. It is a continuing process that responds to changing technology and capacity (Chams & García-Blandón, 2019). As per the SDGs, firms need to strategize about CS practices including worldwide climate change, CO₂ and waste reduction, cleaner production, poverty reduction, education improvement, quality improvement in public health, human rights promotion in their business prototypes (Maunganidze, 2022). CS states to the attainment of sustainable earnings and capacity enrichment by means of economic, social and environmental, factors (Chen, 2023). According to Jiang et al. (2018), the notion of CS is a strategic procedure for firms to search for equilibrium between the stakeholders and the financial profit, social equity and environmental concern. SHRM is perceived as an extra dimension of strategic HRM that provides an up-to-date tool to the personnel management (Stankevičiūtė & Savanevičienė, 2018). It includes TBL concepts and attempts to flourish in equilibrium between each different region (Ehnert et al., 2012).

In the literature, the concept “SHRM” has been defined from various perspectives. According to Cohen et al. (2012), “SHRM highlights the role that HRM plays in supporting business sustainability by implementing practices. This concept can impact individuals and groups to develop behaviors and attitudes according to the sustainable perspective”. Kramar (2014) defined the SHRM as “the pattern of emerging human resource strategies and activities planned to enable the attainment of economic, social and environmental goals while simultaneously reproducing the human resource base over a long term”. Ehnert et al. (2016) described the SHRM as “the implementation of HRM strategies and practices that enable the attainment of economic, social and environmental goals, with impact inside and outside the organization and in the long term, while controlling unintended side effects and negative feedback”. According to Järlström et al. (2018), “SHRM encourages a more holistic view emphasizing the ability of HRM to involve with the sustainability agenda from multiple outlooks to meet the needs and interests of the organization’s numerous internal and external stakeholders”.

In the literature, several authors have considered numerous factors affecting the SHRM practices of the companies/organizations. Considering the TBL concept, Tooranloo considered the twenty six criteria to assess the SHRM practices of an organization. Stankevičiūtė & Savanevičienė (2018) identified several indicators to assess the SHRM assessment of the organizations. These indicators are objectivity and equality, amenability beyond guidelines, employee cooperation, flexibility, employee growth, external partnership, long-term coordination, care of staffs and environment, productivity and social discussion. According to Ogbeibu et al. (2020), environmental dynamic competence, product innovation, technological instability and green side creativeness are significant factors during the SHRM assessment of organizations. Further, Ramalho & Martins identified several criteria during the SHRM assessment in the supply chain. Saeidi et al. (2022) considered the three aspects of sustainability and recognized the twenty criteria including eight from social aspects, eight from environmental aspects and remaining four are from economic aspects.

On the basis of literature review and questionnaire, we are considering the TBL concept and selecting 13 criteria. Out of which, 5 of the factors are listed as social aspect, another five factors are listed as environmental aspect and the last four factors are listed as economic aspect. Table 1 and Figure 1 provide the description of chosen criteria in the assessment of SHRM for a manufacturing company in India.

3. Proposed SVN-Distance Measure (SVN-DM) for SVNSs

This section proposes SVNSs notion and SVN-DM to avoid the limitations of extant SVN-DMs.

3.1. Basic Concepts

Definition 1 (Wang et al., 2010). 

Let $\mathsf{\Omega }=\left\{x_1,x_2,...,x_n\right\}$ be a fixed set. A SVNS N on $\mathsf{\Omega }$ is described in mathematical form as

$$N=\hspace{0.17em}\left\{\langle x,\hspace{0.17em}{T}_N(x),\hspace{0.17em}{I}_N(x),F_N(x)\rangle |x\in \hspace{0.17em}\mathsf{\Omega }\right\},$$

where $T_N\left(u\right):\mathsf{\Omega }\to \left[0,1\right],$ $I_N\left(u\right):\hspace{0.17em}\mathsf{\Omega }\to \left[0,1\right]$ and $F_N\left(u\right):\mathsf{\Omega }\to \left[0,1\right]$ denote the truth BG, the indeterminacy BG and the falsity BG, respectively. For every $x\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega },$ $0\le \hspace{0.17em}{T}_N\left(x\right)+\hspace{0.17em}{I}_N\left(x\right)+F_N\left(x\right)\le 3.$ For convenience, the SVNN can be presented as $\kappa =\left(T_N,I_N,F_N\right)$.

Definition 2 (Ye 2017). 

For any two SVNNs ${\kappa }_1=\left(T_1,\hspace{0.17em}{I}_1,\hspace{0.17em}{F}_1\right)$ and ${\kappa }_2=\left(T_2,\hspace{0.17em}{I}_2,\hspace{0.17em}{F}_2\right),$ and $\lambda >0,$ the operations on SVNSs are presented as

• ${\kappa }_i^c=\left(F_i,\hspace{0.17em}\hspace{0.17em}1-\hspace{0.17em}{I}_i,\hspace{0.17em}\hspace{0.17em}{T}_i\right),\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}=1,\hspace{0.17em}2,$
• ${\kappa }_1\subseteq {\kappa }_2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{if}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{T}_1\le \hspace{0.17em}{T}_2,\hspace{0.17em}\hspace{0.17em}{I}_1\ge \hspace{0.17em}{I}_2\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}{F}_1\ge \hspace{0.17em}{F}_2,$
• ${\kappa }_1={\kappa }_2\hspace{0.17em}\iff \hspace{0.17em}\hspace{0.17em}{\kappa }_1\subseteq {\kappa }_2\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}{\kappa }_2\subseteq \hspace{0.17em}{\kappa }_1,$
• ${\kappa }_1\cup {\kappa }_2=\left(\left\{T_1\vee T_2\right\},\hspace{0.17em}\left\{I_1\wedge I_2\right\},\hspace{0.17em}\hspace{0.17em}\left\{F_1\wedge F_2\right\}\right),$
• ${\kappa }_1\cap {\kappa }_2=\left(\left\{T_1\vee T_2\right\},\hspace{0.17em}\left\{I_1\wedge I_2\right\},\hspace{0.17em}\left\{F_1\wedge F_2\right\}\right),$
• ${\kappa }_1\oplus {\kappa }_2=\left(T_1+\hspace{0.17em}{T}_2-T_1\hspace{0.17em}{T}_2,\hspace{0.17em}\hspace{0.17em}{I}_1\hspace{0.17em}{I}_2,\hspace{0.17em}\hspace{0.17em}{F}_1\hspace{0.17em}{F}_2\right),$
• ${\kappa }_1\otimes {\kappa }_2=\left(T_1\hspace{0.17em}{T}_2,\hspace{0.17em}\hspace{0.17em}{I}_1+I_2-I_1\hspace{0.17em}{I}_2,\hspace{0.17em}\hspace{0.17em}{F}_1+F_2-F_1\hspace{0.17em}{F}_2\right),$
• $\lambda \hspace{0.17em}{\kappa }_i=\left(1-{\left(1-T_i\right)}^{\lambda },\hspace{0.17em}{I}_i^{\lambda },\hspace{0.17em}{F}_i^{\lambda }\right),\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}=1,\hspace{0.17em}2,$
• ${\kappa }_i^{\lambda }=\left(T_i^{\gamma },\hspace{0.17em}\hspace{0.17em}1-{\left(1-I_i\right)}^{\gamma },\hspace{0.17em}\hspace{0.17em}1-{\left(1-F_i\right)}^{\gamma }\right),\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}=1,\hspace{0.17em}2,$
• ${\kappa }_1-{\kappa }_2=\left(\frac{T_1-T_2}{1-T_2},\hspace{0.17em}\hspace{0.17em}\frac{I_1}{I_2},\hspace{0.17em}\hspace{0.17em}\frac{F_1}{F_2}\right),$ which is valid under the conditions ${\kappa }_1\ge {\kappa }_2,\hspace{0.17em}{T}_2\ne 1,\hspace{0.17em}{I}_2\ne 0$ and $F_2\ne 0.$

Definition 3 (Smarandache, 2020). 

For a SVNN, the normalized SVN-score function is given by

$$S\left(\kappa \right)=\frac{2+\hspace{0.17em}T-\hspace{0.17em}I-F}{3},\hspace{0.17em}\hspace{0.17em}S\left(\kappa \right)\in \left[0,1\right].$$

(1)

Definition 4 (Ye, 2014). 

Let ${\kappa }_i=\left(T_i,\hspace{0.17em}{I}_i,\hspace{0.17em}{F}_i\right)$ be SVNNs and $\varpi ={\left({\varpi }_1,\hspace{0.17em}{\varpi }_2,...,{\varpi }_r\right)}^T$ be weight of ${\kappa }_i,$ with ${\varpi }_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^r{\varpi }_i}=1.$ Then the SVNWA and SVNWG operators are defined as

$$SVNWA\left({\kappa }_1,{\kappa }_2,...,{\kappa }_r\right)=\underset{i=1}{\overset{r}{\oplus }}\left({\varpi }_i\hspace{0.17em}{\kappa }_i\right)=\left(1-{\displaystyle \prod _{i=1}^r{\left(1-T_i\right)}^{{\varpi }_i}},\hspace{0.17em}{\displaystyle \prod _{i=1}^r{\left(I_i\right)}^{{\varpi }_i}},\hspace{0.17em}{\displaystyle \prod _{i=1}^r{\left(F_i\right)}^{{\varpi }_i}}\hspace{0.17em}\right),$$

(2)

$$SVNWG\left({\kappa }_1,{\kappa }_2,...,{\kappa }_r\right)=\underset{i=1}{\overset{r}{\otimes }}\left({\varpi }_i\hspace{0.17em}{\kappa }_i\right)=\left({\displaystyle \prod _{i=1}^r{\left(T_i\right)}^{{\varpi }_i}},\hspace{0.17em}1-{\displaystyle \prod _{i=1}^r{\left(1-I_i\right)}^{{\varpi }_i}},\hspace{0.17em}1-{\displaystyle \prod _{i=1}^r{\left(1-F_i\right)}^{{\varpi }_i}}\hspace{0.17em}\right).$$

(3)

Definition 5 (Majumdar & Samanta, 2014). 

Let $M,\hspace{0.17em}N,\hspace{0.17em}P\hspace{0.17em}\in \hspace{0.17em}SVNSs\left(\mathsf{\Omega }\right).$ A real-valued mapping $d\hspace{0.17em}:SVNSs\left(\mathsf{\Omega }\right)\times SVNSs\left(\mathsf{\Omega }\right)\to [0,\hspace{0.17em}\hspace{0.17em}1]$ is called SVN-DM which holds the following axioms:

(a₁)

$0\le d\left(M,\hspace{0.17em}N\right)\le 1,$

(a₂)

$d\left(M,\hspace{0.17em}N\right)=d\left(N,\hspace{0.17em}M\right),$

(a₃)

$d\left(M,\hspace{0.17em}N\right)=0\iff M=\hspace{0.17em}N,$

(a₄)

If $M\hspace{0.17em}\subseteq \hspace{0.17em}N\subseteq \hspace{0.17em}P,$ then $d\left(M,\hspace{0.17em}P\right)\ge d\left(M,\hspace{0.17em}N\right)$ and $d\left(M,\hspace{0.17em}P\right)\ge d\left(N,\hspace{0.17em}P\right),$ $\forall \hspace{0.17em}\hspace{0.17em}M,\hspace{0.17em}N,\hspace{0.17em}P\hspace{0.17em}\hspace{0.17em}\in SVNSs\left(\mathsf{\Omega }\right).$

3.2. SVN-DM and Its Effectiveness Over the Existing DMs

In the context of SVNSs, the concept of DM has widely used for medical diagnosis, cluster analysis etc. Ye (2014) proposed Hamming, Euclidean and generalized DMs for SVNSs and presented a clustering algorithm to cluster SVN data. Chai et al. (2021) suggested some SVN-DM and SVN-similarity measure (SVN-SM) with their application in medical diagnosis and pattern recognition. Based on matrix norm, Luo et al. (2022) presented a novel DM for SVNSs, which satisfies the axiomatic definition of the metric. Ali et al. (2023) introduced two schemes to compute the Hausdorff DM and its equivalent SVN-SM.

For $M,\hspace{0.17em}N\hspace{0.17em}\in \hspace{0.17em}SVNSs\left(\mathsf{\Omega }\right),$ we propose new SVN-DM, given as

$$d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}\hspace{0.17em}=\frac{1}{9\hspace{0.17em}r}{\displaystyle \sum _{i\hspace{0.17em}=1}^r\left[\begin{array}{l}\left|T_M(x_i)-\hspace{0.17em}{T}_N(x_i)\right|+\left|I_M(x_i)-\hspace{0.17em}{I}_N(x_i)\right|+\hspace{0.17em}\left|F_M(x_i)-\hspace{0.17em}{F}_N(x_i)\right|+\left|T_M(x_i)\hspace{0.17em}{I}_N(x_i)-T_N(x_i)\hspace{0.17em}{I}_M(x_i)\right|\\ +\hspace{0.17em}\left|T_M(x_i)\hspace{0.17em}{F}_N(x_i)-T_N(x_i)\hspace{0.17em}{F}_M(x_i)\right|+\hspace{0.17em}\left|I_M(x_i)\hspace{0.17em}{F}_N(x_i)-I_N(x_i)\hspace{0.17em}{F}_M(x_i)\right|\\ +(\left|\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|+\left|\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\\ +(\left|\min \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_N(x_i)\right\}-\hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|+\left|\max \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|)\\ +(\left|\min \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|+\left|\max \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\end{array}\right]}.$$

(4)

The properties of $d\left(M,\hspace{0.17em}N\right)$ are discussed as follows:

Property 3.1: 

For $M,\hspace{0.17em}N\hspace{0.17em}\in \hspace{0.17em}SVNSs\left(\mathsf{\Omega }\right),$ $0\le \hspace{0.17em}d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}\hspace{0.17em}\le 1.$

Proof: 

Let M and N be SVNSs, then $0\hspace{0.17em}\le \hspace{0.17em}{T}_M(x_i)\le \hspace{0.17em}1$ $0\hspace{0.17em}\le \hspace{0.17em}{I}_M(x_i)\le \hspace{0.17em}1$ and $0\hspace{0.17em}\le \hspace{0.17em}{F}_M(x_i)\le \hspace{0.17em}1,$ and $0\hspace{0.17em}\le \hspace{0.17em}{T}_M(x_i),\hspace{0.17em}{I}_M(x_i),\hspace{0.17em}{F}_M(x_i)\hspace{0.17em}\le \hspace{0.17em}3.$ Thus, it is obvious that $d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}\ge \hspace{0.17em}0.$ Also, $d\left(M,\hspace{0.17em}N\right)$ is defined with its normalization, which implies $d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}\le \hspace{0.17em}1.$ Hence, $0\hspace{0.17em}\le d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}\le \hspace{0.17em}1.$

Property 3.2: 

$d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=0$ iff M = N.

Proof: 

For given two IVIFSs M and N, if M = N, then

$$d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}\frac{1}{9\hspace{0.17em}r}{\displaystyle \sum _{i\hspace{0.17em}=1}^r\left[\begin{array}{l}\left|T_M(x_i)-\hspace{0.17em}{T}_M(x_i)\right|+\left|I_M(x_i)-\hspace{0.17em}{I}_M(x_i)\right|+\hspace{0.17em}\left|F_M(x_i)-\hspace{0.17em}{F}_M(x_i)\right|+\left|T_M(x_i)\hspace{0.17em}{I}_M(x_i)-T_M(x_i)\hspace{0.17em}{I}_M(x_i)\right|\\ +\hspace{0.17em}\left|T_M(x_i)\hspace{0.17em}{F}_M(x_i)-T_M(x_i)\hspace{0.17em}{F}_M(x_i)\right|+\hspace{0.17em}\left|I_M(x_i)\hspace{0.17em}{F}_M(x_i)-I_M(x_i)\hspace{0.17em}{F}_M(x_i)\right|\\ +(\left|\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|+\left|\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\\ +(\left|\min \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_M(x_i)\right\}-\hspace{0.17em}\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|+\left|\max \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|)\\ +(\left|\min \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|+\left|\max \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\end{array}\right]}\hspace{0.17em}=\hspace{0.17em}0.$$

Conversely, if $d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=0,$ then

$$d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}\hspace{0.17em}=\frac{1}{9\hspace{0.17em}r}{\displaystyle \sum _{i\hspace{0.17em}=1}^r\left[\begin{array}{l}\left|T_M(x_i)-\hspace{0.17em}{T}_N(x_i)\right|+\left|I_M(x_i)-\hspace{0.17em}{I}_N(x_i)\right|+\hspace{0.17em}\left|F_M(x_i)-\hspace{0.17em}{F}_N(x_i)\right|+\left|T_M(x_i)\hspace{0.17em}{I}_N(x_i)-T_N(x_i)\hspace{0.17em}{I}_M(x_i)\right|\\ +\hspace{0.17em}\left|T_M(x_i)\hspace{0.17em}{F}_N(x_i)-T_N(x_i)\hspace{0.17em}{F}_M(x_i)\right|+\hspace{0.17em}\left|I_M(x_i)\hspace{0.17em}{F}_N(x_i)-I_N(x_i)\hspace{0.17em}{F}_M(x_i)\right|\\ +(\left|\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|+\left|\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\\ +(\left|\min \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_N(x_i)\right\}-\hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|+\left|\max \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|)\\ +(\left|\min \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|+\left|\max \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\end{array}\right]}=\hspace{0.17em}0.$$

(5)

Since every term of Eq. (5) is non-negative, then by $d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}0,$ we can observe that every term of Eq. (5) should be equal to zero, it implies that we have $T_M\left(x_i\right)=\hspace{0.17em}{T}_N\left(x_i\right),$ $I_M\left(x_i\right)=\hspace{0.17em}{I}_N\left(x_i\right)$ and $F_M\left(x_i\right)=\hspace{0.17em}{F}_N\left(x_i\right).$ Thus, $d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=0$ iff $M=N.$

Property 3.3: 

$d\left(M,\hspace{0.17em}\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}d\left(N,\hspace{0.17em}M\right).$

Proof: 

For given SVNSs M and N, we get

$$d\left(N,\hspace{0.17em}M\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{1}{9\hspace{0.17em}r}{\displaystyle \sum _{i\hspace{0.17em}=1}^r\left[\begin{array}{l}\left|T_N(x_i)-\hspace{0.17em}{T}_M(x_i)\right|+\left|I_N(x_i)-\hspace{0.17em}{I}_M(x_i)\right|+\hspace{0.17em}\left|F_N(x_i)-\hspace{0.17em}{F}_M(x_i)\right|+\left|T_N(x_i)\hspace{0.17em}{I}_M(x_i)-T_M(x_i)\hspace{0.17em}{I}_N(x_i)\right|\\ +\hspace{0.17em}\left|T_N(x_i)\hspace{0.17em}{F}_M(x_i)-T_M(x_i)\hspace{0.17em}{F}_N(x_i)\right|+\hspace{0.17em}\left|I_N(x_i)\hspace{0.17em}{F}_M(x_i)-I_M(x_i)\hspace{0.17em}{F}_N(x_i)\right|\\ +(\left|\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\right\}\right|+\left|\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\right\}\right|)\\ +(\left|\min \left\{T_N(x_i)\hspace{0.17em},\hspace{0.17em}{F}_M(x_i)\right\}-\hspace{0.17em}\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{F}_N(x_i)\right\}\right|+\left|\max \left\{T_N(x_i)\hspace{0.17em},\hspace{0.17em}{F}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{F}_N(x_i)\right\}\right|)\\ +(\left|\min \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\right\}\right|+\left|\max \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\right\}\right|)\end{array}\right]}$$

$$=\hspace{0.17em}\hspace{0.17em}\frac{1}{9\hspace{0.17em}r}{\displaystyle \sum _{i\hspace{0.17em}=1}^r\left[\begin{array}{l}\left|T_M(x_i)-\hspace{0.17em}{T}_N(x_i)\right|+\left|I_M(x_i)-\hspace{0.17em}{I}_N(x_i)\right|+\hspace{0.17em}\left|F_M(x_i)-\hspace{0.17em}{F}_N(x_i)\right|+\left|T_M(x_i)\hspace{0.17em}{I}_N(x_i)-T_N(x_i)\hspace{0.17em}{I}_M(x_i)\right|\\ +\hspace{0.17em}\left|T_M(x_i)\hspace{0.17em}{F}_N(x_i)-T_N(x_i)\hspace{0.17em}{F}_M(x_i)\right|+\hspace{0.17em}\left|I_M(x_i)\hspace{0.17em}{F}_N(x_i)-I_N(x_i)\hspace{0.17em}{F}_M(x_i)\right|\\ +(\left|\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|+\left|\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\\ +(\left|\min \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_N(x_i)\right\}-\hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|+\left|\max \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|)\\ +(\left|\min \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|+\left|\max \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\end{array}\right]}\hspace{0.17em}=\hspace{0.17em}d\left(M,\hspace{0.17em}N\right).$$

Hence, $d\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}d\left(N,\hspace{0.17em}M\right).$

Property 3.4: 

If $M\hspace{0.17em}\subseteq \hspace{0.17em}N\subseteq \hspace{0.17em}P,$ then $d\left(M,\hspace{0.17em}P\right)\ge d\left(M,\hspace{0.17em}N\right)$ and $d\left(M,\hspace{0.17em}P\right)\ge d\left(N,\hspace{0.17em}P\right),$ $\forall \hspace{0.17em}\hspace{0.17em}M,\hspace{0.17em}N,\hspace{0.17em}P\hspace{0.17em}\hspace{0.17em}\in SVNSs\left(\mathsf{\Omega }\right).$

Proof: 

For given three SVNSs $M\hspace{0.17em}=\hspace{0.17em}\left\{\langle x_i,\hspace{0.17em}{T}_M(x_i),\hspace{0.17em}{I}_M(x_i),\hspace{0.17em}{F}_M(x_i)\rangle \hspace{0.17em}:\hspace{0.17em}{x}_i\in \hspace{0.17em}\mathsf{\Omega }\right\},$ $N\hspace{0.17em}=\hspace{0.17em}\left\{\langle x_i,\hspace{0.17em}{T}_N(x_i),\hspace{0.17em}{I}_N(x_i),\hspace{0.17em}{F}_N(x_i)\rangle \hspace{0.17em}:\hspace{0.17em}{x}_i\in \hspace{0.17em}\mathsf{\Omega }\right\}$ and $P\hspace{0.17em}=\hspace{0.17em}\left\{\langle x_i,\hspace{0.17em}{T}_P(x_i),\hspace{0.17em}{I}_P(x_i),\hspace{0.17em}{F}_P(x_i)\rangle \hspace{0.17em}:\hspace{0.17em}{x}_i\in \hspace{0.17em}\mathsf{\Omega }\right\},$ if $M\subseteq \hspace{0.17em}N\subseteq \hspace{0.17em}P,$then $T_M\left(x_i\right)\le \hspace{0.17em}{T}_N\left(x_i\right)\le \hspace{0.17em}{T}_P\left(x_i\right),$ $I_P\left(x_i\right)\le \hspace{0.17em}{I}_N\left(x_i\right)\le \hspace{0.17em}{I}_M\left(x_i\right)$ and $F_P\left(x_i\right)\le \hspace{0.17em}{F}_N\left(x_i\right)\le \hspace{0.17em}{F}_M\left(x_i\right),\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}{x}_i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega }.$ Therefore, we have $\left|T_M(x_i)-\hspace{0.17em}{T}_P(x_i)\right|\hspace{0.17em}\ge \hspace{0.17em}\left|T_M(x_i)-\hspace{0.17em}{T}_N(x_i)\right|,$ $\left|I_M(x_i)-\hspace{0.17em}{I}_P(x_i)\right|\hspace{0.17em}\ge \hspace{0.17em}\left|I_M(x_i)-\hspace{0.17em}{I}_N(x_i)\right|$ and $\left|F_M(x_i)-\hspace{0.17em}{F}_P(x_i)\right|\ge$ $\left|F_M(x_i)-\hspace{0.17em}{F}_N(x_i)\right|,\hspace{0.17em}\forall \hspace{0.17em}{x}_i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega }.$ It will also imply that $0\hspace{0.17em}\le \hspace{0.17em}\min \left\{T_M(x_i),\hspace{0.17em}{I}_P(x_i)\right\}$ $\le \hspace{0.17em}\min \left\{T_M(x_i),\hspace{0.17em}{I}_N(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}{I}_M(x_i)\right\}\le \hspace{0.17em}\min \left\{T_P(x_i),\hspace{0.17em}{I}_M(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}1,$ $0\hspace{0.17em}\le \hspace{0.17em}\min \left\{T_M(x_i),\hspace{0.17em}{F}_P(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}$ $\min \left\{T_M(x_i),\hspace{0.17em}{F}_N(x_i)\right\}\le \hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}{F}_M(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}\min \left\{T_P(x_i),\hspace{0.17em}{I}_M(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}1,$ $0\hspace{0.17em}\le \hspace{0.17em}\max \left\{T_M(x_i),\hspace{0.17em}{I}_P(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}$ $\max \left\{T_M(x_i),\hspace{0.17em}{I}_N(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}{I}_M(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}\max \left\{T_P(x_i),\hspace{0.17em}{I}_M(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}1,$and $0\hspace{0.17em}\le \hspace{0.17em}\max \left\{T_M(x_i),\hspace{0.17em}{F}_P(x_i)\right\}$ $\le \hspace{0.17em}\max \left\{T_M(x_i),\hspace{0.17em}{F}_N(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}{F}_M(x_i)\right\}\hspace{0.17em}$ $\le \hspace{0.17em}\min \left\{T_P(x_i),\hspace{0.17em}{I}_M(x_i)\right\}\hspace{0.17em}\le \hspace{0.17em}1,\hspace{0.17em}\forall \hspace{0.17em}{x}_i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega }$

Thus, it follows that

$$\begin{array}{l}(\left|\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_P(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{T_P(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|\ge \hspace{0.17em}(\left|\min \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|\\ +\left|\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_P(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_P(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)+\left|\max \left\{T_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|),\end{array}$$

$$\begin{array}{l}(\left|\min \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_P(x_i)\right\}-\hspace{0.17em}\min \left\{T_P(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|\ge \hspace{0.17em}(\left|\min \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_N(x_i)\right\}-\hspace{0.17em}\min \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|\\ +\left|\max \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_P(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_P(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|)+\left|\max \left\{T_M(x_i)\hspace{0.17em},\hspace{0.17em}{F}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{T_N(x_i),\hspace{0.17em}\hspace{0.17em}{F}_M(x_i)\right\}\right|),\end{array}$$

$$\begin{array}{l}(\left|\min \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_P(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{F_P(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|\\ +\left|\max \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_P(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{F_P(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|)\end{array}$$

$$\begin{array}{l}\ge \hspace{0.17em}(\left|\min \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\min \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|\\ +\left|\max \left\{F_M(x_i),\hspace{0.17em}\hspace{0.17em}{I}_N(x_i)\hspace{0.17em}\right\}-\hspace{0.17em}\max \left\{F_N(x_i),\hspace{0.17em}\hspace{0.17em}{I}_M(x_i)\right\}\right|),\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}{x}_i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega }.\end{array}$$

In addition, we have

$$\left|T_M(x_i)\hspace{0.17em}{I}_P(x_i)-T_P(x_i)\hspace{0.17em}{I}_M(x_i)\right|\ge \left|T_M(x_i)\hspace{0.17em}{I}_N(x_i)-T_N(x_i)\hspace{0.17em}{I}_M(x_i)\right|,$$

$$\left|T_M(x_i)\hspace{0.17em}{F}_P(x_i)-T_P(x_i)\hspace{0.17em}{F}_M(x_i)\right|\ge \left|T_M(x_i)\hspace{0.17em}{F}_N(x_i)-T_N(x_i)\hspace{0.17em}{F}_M(x_i)\right|,$$

$$\left|I_M(x_i)\hspace{0.17em}{F}_P(x_i)-I_P(x_i)\hspace{0.17em}{F}_M(x_i)\right|\ge \left|I_M(x_i)\hspace{0.17em}{F}_N(x_i)-I_N(x_i)\hspace{0.17em}{F}_M(x_i)\right|,\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}{x}_i\hspace{0.17em}\in \mathsf{\Omega }.$$

It implies that $d\left(M,\hspace{0.17em}P\right)\ge d\left(M,\hspace{0.17em}N\right),$ $\forall \hspace{0.17em}\hspace{0.17em}M,\hspace{0.17em}N,\hspace{0.17em}P\hspace{0.17em}\hspace{0.17em}\in SVNSs\left(\mathsf{\Omega }\right).$ Similarly, if $M\subseteq \hspace{0.17em}N\subseteq \hspace{0.17em}P,$then $T_M\left(x_i\right)\le \hspace{0.17em}{T}_N\left(x_i\right)\le \hspace{0.17em}{T}_P\left(x_i\right),$ $I_P\left(x_i\right)\le \hspace{0.17em}{I}_N\left(x_i\right)\le \hspace{0.17em}{I}_M\left(x_i\right)$ and $F_P\left(x_i\right)\le \hspace{0.17em}{F}_N\left(x_i\right)\le \hspace{0.17em}{F}_M\left(x_i\right),\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}{x}_i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega },$ therefore, we can prove that $d\left(M,\hspace{0.17em}P\right)\hspace{0.17em}\le \hspace{0.17em}d\left(N,\hspace{0.17em}P\right).$

Proposition 3.1: 

From the Properties 3.1-3.4, Eq. (4) holds all the necessary conditions of Definition

3.5. Hence, Eq. (4) is valid SVN-distance measure on $SVNSs\left(\mathsf{\Omega }\right).$

To demonstrate the efficacy of developed SVN-DM, we compare it with some of the extant SVN-DMs given by (Ye, 2014; Xu et al., 2020; Chai et al., 2021), which are presented by Eqs. (6)-(10). The required results are presented in Table 2.

Normalized hamming distance (Ye, 2014)

$$d_1\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{1}{3}\hspace{0.17em}\left(\left|T_M(x_i)-\hspace{0.17em}{T}_N(x_i)\right|+\hspace{0.17em}\left|I_M(x_i)-\hspace{0.17em}{I}_N(x_i)\right|+\hspace{0.17em}\left|F_M(x_i)-\hspace{0.17em}{F}_N(x_i)\right|\right).$$

(6)

Hausdorff distance (Xu et al., 2020)

$$d_2\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\max \hspace{0.17em}\left(\left|T_M(x_i)-\hspace{0.17em}{T}_N(x_i)\right|,\hspace{0.17em}\hspace{0.17em}\left|I_M(x_i)-\hspace{0.17em}{I}_N(x_i)\right|,\hspace{0.17em}\hspace{0.17em}\left|F_M(x_i)-\hspace{0.17em}{F}_N(x_i)\right|\right).$$

(7)

SVN-distance measure by Chai et al. (2021)

$$d_3\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{1}{3\hspace{0.17em}\left|\mathsf{\Omega }\right|}\hspace{0.17em}{\displaystyle \sum _{x_i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega }}\left(\left|T_M^2(x_i)-\hspace{0.17em}{T}_N^2(x_i)\right|+\hspace{0.17em}\left|I_M^2(x_i)-\hspace{0.17em}{I}_N^2(x_i)\right|+\hspace{0.17em}\left|F_M^2(x_i)-\hspace{0.17em}{F}_N^2(x_i)\right|\right)}.$$

(8)

$$d_4\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{1}{3\hspace{0.17em}\left|\mathsf{\Omega }\right|}\hspace{0.17em}{\displaystyle \sum _{x_i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega }}\left|\left(T_M^2(x_i)-\hspace{0.17em}{T}_N^2(x_i)\right)-\hspace{0.17em}\left(I_M^2(x_i)-\hspace{0.17em}{I}_N^2(x_i)\right)-\hspace{0.17em}\left(F_M^2(x_i)-\hspace{0.17em}{F}_M^2(x_i)\right)\right|}.$$

(9)

$$d_5\left(M,\hspace{0.17em}N\right)\hspace{0.17em}=\hspace{0.17em}1-\hspace{0.17em}\hspace{0.17em}\frac{1}{\left|\mathsf{\Omega }\right|}\hspace{0.17em}{\displaystyle \sum _{x_i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Omega }}\left(\frac{\left(T_M^2(x_i)\wedge T_N^2(x_i)\right)\hspace{0.17em}+\hspace{0.17em}\left(I_M^2(x_i)\wedge \hspace{0.17em}{I}_N^2(x_i)\right)+\hspace{0.17em}\left(F_M^2(x_i)\wedge \hspace{0.17em}{F}_N^2(x_i)\right)}{\left(T_M^2(x_i)\vee T_N^2(x_i)\right)\hspace{0.17em}+\hspace{0.17em}\left(I_M^2(x_i)\vee \hspace{0.17em}{I}_N^2(x_i)\right)+\hspace{0.17em}\left(F_M^2(x_i)\vee \hspace{0.17em}{F}_N^2(x_i)\right)}\right)}.$$

(10)

Based on the computational outcomes obtained in Table 2, we present the following results:

• For two different sets (set-I and set-II), the measures d₂(M,N) and d₄(M,N) present the same results, which are 1 and 0.3333, respectively.
• For two different sets (set-II and set-III), the Hamming measure d₁(M,N) obtains the same value, which is 0.3333. For these two sets, d₄(M,N) obtains the value, which dissatisfies the property (a₁) of Definition 5.
• For the set-IV and set-V, the Hausdroff distance measure d₂(M,N) obtains the same value 0.1.
• For all the sets, the proposed measure successfully describes the difference between SVNSs, which prove its effectiveness.

4. A hybrid SVN-RANCOM-AROMAN approach

This section firstly proposes hybrid SVN-DM-RANCOM-AROMAN approach to solve the MCDM problems under SVNSs setting. To determine the solution of MCGDM problems, the developed model involves the following steps:

• Step 1: Let $F=\hspace{0.17em}\left\{F_1,\hspace{0.17em}{F}_2,\hspace{0.17em}...,\hspace{0.17em}{F}_m\right\}$ and $H\hspace{0.17em}=\hspace{0.17em}\left\{H_1,\hspace{0.17em}{H}_2,\hspace{0.17em}...,\hspace{0.17em}{H}_n\right\}$ be the set of options and criteria, respectively. Let $\hspace{0.17em}E=\left\{e_1,\hspace{0.17em}{e}_2,\hspace{0.17em}...,\hspace{0.17em}{e}_l\right\}$ be DEs who offers their opinion for each option F_i over H_j in term of “linguistic ratings (LRs)”. Let $R={\left({\xi }_{ij}^{\left(k\right)}\right)}_{m\hspace{0.17em}\times \hspace{0.17em}n}$ be “linguistic assessment matrix (LAM)” presented by DEs, where ${\xi }_{ij}^{\left(k\right)}$ denotes assessment information of an option F_i over a criterion H_j in the form of LRs and further changed into SVNNs.
• Step 2: Compute the DEs’ significance values.
  Assume that $e_k=\left(T_k,\hspace{0.17em}{I}_k,\hspace{0.17em}{F}_k\right),\hspace{0.17em}k=1,2,...,l$ be the performance of k^th DE. Then the procedure for estimating the numeric significance value of k^th DE is as follows:
• Step 2a: Determine the matrix using score function.
  Each SVN $e_k$ is normalized and computed using SVN-score function as

  $${\overline{e}}_k=\frac{\left(2+\hspace{0.17em}{T}_{\kappa }-\hspace{0.17em}{I}_{\kappa }-F_{\kappa }\right)}{{\displaystyle \sum _{k=1}^l\left(2+T_{\kappa }-\hspace{0.17em}{I}_{\kappa }-F_{\kappa }\right)}},\hspace{0.17em}\forall \hspace{0.17em}k.$$

  (11)
• Step 2b: Determine the rank of DEs’ performances and compute the DE’s significance value$l-{\rho }_k+1$, wherein ${\rho }_k$ denotes the priority of k^th expert. The normalization process is used to normalize each significance value:

  $${\overline{e}}_k^r=\frac{l-{\rho }_k+1}{{\displaystyle \sum _{k=1}^l\left(l-{\rho }_k+1\right)}},\forall \hspace{0.17em}k.$$

  (12)
• Step 2c: Compute the numeric weights.

In accordance with the combination of Eq. (11) and Eq. (12), DE’s weighting formula is given as

$${\vartheta }_k=\frac{1}{2}\left(\left({\overline{e}}_k\right)+\left({\overline{e}}_k^r\right)\right),\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}k, \mathrm{wherein} {\vartheta }_k\ge \hspace{0.17em}0 \mathrm{and} {\displaystyle \sum _{k=1}^l{\vartheta }_k=\hspace{0.17em}1.}$$

(13)

• Step 3: Create the “aggregated SVN decision matrix (A-SVNDM)”.

In the procedure of MCGDM, there is a need to aggregate the individual DEs’ opinions. We create the A-SVNDM $Z={\left({\eta }_{ij}\right)}_{m\hspace{0.17em}\times \hspace{0.17em}n},$ using SVNWA operator, where

$${\eta }_{ij}=\left(T_{ij},\hspace{0.17em}{I}_{ij},\hspace{0.17em}{F}_{ij}\right)=\left(1-{\displaystyle \prod _{k=1}^l{\left(1-T_{ij}^{\left(k\right)}\right)}^{{\vartheta }_k}},\hspace{0.17em}{\displaystyle \prod _{k=1}^l{\left(I_{ij}^{\left(k\right)}\right)}^{{\vartheta }_k}},{\displaystyle \prod _{k=1}^l{\left(F_{ij}^{\left(k\right)}\right)}^{{\vartheta }_k}}\hspace{0.17em}\right).$$

(14)

• Step 4: Estimation of weight of attributes

Let $w={\left(w_1,w_2,...,w_n\right)}^T$ be the weight of attributes satisfying ${\displaystyle \sum _{j=1}^n{w}_j}\hspace{0.17em}=1$ and$\hspace{0.17em}{w}_j\in \left[0,\hspace{0.17em}\hspace{0.17em}1\right].$ Next, we develop the SVN-DM-RANCOM tool to estimate the integrated weight of attribute.

• Case I: Determine the objective weights using SVN-DM-based model given by Eq. (15).

  $$w_j^o=\frac{\frac{1}{m-1}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{k=1}^{m}d\left({\eta }_{ij},\hspace{0.17em}{\eta }_{kj}\right)}}}{{\displaystyle \sum _{j=1}^n\left(\frac{1}{m-1}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{k=1}^{m}d\left({\eta }_{ij},\hspace{0.17em}{\eta }_{kj}\right)}}\right)}},\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}j.$$

  (15)
• Case II: Determining of subjective weight through SVN-RANCOM model.

This method considers the following steps for computing the subjective criteria weighs:

• Step 4a: Estimate the A-SVNDM based on the linguistic assessment degrees provided by the DEs through introduced SVNWA operator and obtained given as

  $$N={\left({\xi }_j\right)}_{1\times \hspace{0.17em}n}=SVNW{A}_{{\vartheta }_k}\left({\xi }_j^{\left(1\right)},{\xi }_j^{\left(2\right)},...,{\xi }_j^{\left(l\right)}\right),$$

  (16)

  or

  $$N={\left({\xi }_j\right)}_{1\times \hspace{0.17em}n}=SVNW{G}_{{\vartheta }_k}\left({\xi }_j^{\left(1\right)},{\xi }_j^{\left(2\right)},...,{\xi }_j^{\left(l\right)}\right).$$
• Step 4b: Find the SVN-SM.

Next, compute the SVN-SM of A-SVN-DM based on Eq. (17).

$${\overline{\eta }}_j=\frac{1}{2}\left(S\left({\xi }_j\right)\hspace{0.17em}+\hspace{0.17em}1\right).$$

(17)

Step 4c: Determine the prioritization of attributes.

The minimum score value represents the most significant rank of the attribute. Some attributes may have equal score values, which reveals that ties are offered through the DEs’ result. Though, the changes that happened in prioritization would not influence the computed weight unless they comprise diverse attributes hierarchy.

Step 4d: Found the “matrix of ranking comparison (MRC)”.

The MRC is computed based on the pairwise comparison of prioritization of attributes by DEs ratings. The MRC is represented by Eq. (18), where ${\phi }_{ij}=\hspace{0.17em}1,$ 0.5 and 0 for $\rho \left({\overline{\eta }}_j\right)<\rho \left({\overline{\eta }}_t\right),$ $\rho \left({\overline{\eta }}_j\right)=\rho \left({\overline{\eta }}_t\right)$ and $\rho \left({\overline{\eta }}_j\right)>\rho \left({\overline{\eta }}_t\right),$ respectively and $\rho \left({\overline{\eta }}_j\right)$ is the rank of j^th criterion.

$$\begin{array}{l}\begin{array}{cccc}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_1& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_2& \hspace{0.17em}\hspace{0.17em}\cdots & \hspace{0.17em}\hspace{0.17em}{r}_n\end{array}\\ {\alpha }_{tj}=\begin{array}{c}{r}_1\\ r_2\\ ⋮\\ r_n\end{array}\left[\begin{array}{cccc}{\phi }_{11}& {\phi }_{11}& \cdots & {\phi }_{1n}\\ {\phi }_{21}& {\phi }_{22}& \cdots & {\phi }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\phi }_{n1}& {\phi }_{n2}& \cdots & {\phi }_{nn}\end{array}\right].\end{array}$$

(18)

Step 4e: Compute the summed criteria weights (SCWs).

In accordance with previous step, the horizontal vector of the SCW is computed using Eq. (19).

$$SC{W}_j={\displaystyle \sum _{t=1}^n{\alpha }_{tj}},\hspace{0.17em}j=1,2,...,n.$$

(19)

Step 4f: Derive the subjective weight of attribute.

Thus, the subjective weight of attributes is defined as

$$w_j^s=\frac{SC{W}_j}{{\displaystyle {\sum }_{j=1}^{n}SC{W}_j}},\hspace{0.17em}j=1,2,...,n.$$

(20)

Case III: Combined SVN-DM-RANCOM weight-determining tool.

In order to consider the benefits of objective and subjective weights of criteria through SVN-distance measure and SVN-RANCOM models, respectively, we present an integrated weighting model, given as

$$w_j=\hspace{0.17em}\zeta w_j^o+\left(1-\zeta \right)w_j^s,$$

(21)

where $\zeta \hspace{0.17em}\in \hspace{0.17em}\left[0,\hspace{0.17em}1\right]$ represents the weighting precision parameter.

• Step 5: Normalize the A-SVNDM.

In this step, the A-SVNDM $Z={\left({\eta }_{ij}\right)}_{m\hspace{0.17em}\times \hspace{0.17em}n}$ is normalized A-SVNDM (NA-SVNDM) into $\mathbb{N}={\left({\epsilon }_{ij}\right)}_{m\hspace{0.17em}\times \hspace{0.17em}n}$ using the linear and vector normalization tools.

• Step 5.1 (Linear normalization). It eliminates the dimensions of criteria using the doctrine of max-min operator. A linear NA-SVN-DM ${\mathbb{N}}^{\left(1\right)}\hspace{0.17em}=\hspace{0.17em}{\left({\epsilon }_{ij}^{\left(1\right)}\right)}_{m\times n}$ is created using Eq. (22), where

  $${\epsilon }_{ij}^{\left(1\right)}=\left(T_{ij}^{\left(1\right)},\hspace{0.17em}{I}_{ij}^{\left(1\right)},\hspace{0.17em}{F}_{ij}^{\left(1\right)}\right)=\frac{{\eta }_{ij}}{{\max }_{i}S\left({\eta }_{ij}\right)}.$$

  (22)
• Step 5.2 (Vector normalization). A vector NA-SVN-DM ${\mathbb{N}}^{\left(2\right)}\hspace{0.17em}=\hspace{0.17em}{\left({\epsilon }_{ij}^{\left(2\right)}\right)}_{m\times n}$ is constructed using Eq. (23), where

  $${\epsilon }_{ij}^{\left(2\right)}=\left(T_{ij}^{\left(2\right)},\hspace{0.17em}{I}_{ij}^{\left(2\right)},\hspace{0.17em}{F}_{ij}^{\left(2\right)}\right)=\frac{{\eta }_{ij}}{\sqrt{{\displaystyle {\sum }_{i=1}^m{\left(S\left({\eta }_{ij}\right)\right)}^2}}}.$$

  (23)
• Step 5.3. Find the averaged NA-SVN-DM
  The averaged NA-SVN-DM $\mathbb{N}={\left({\epsilon }_{ij}\right)}_{m\hspace{0.17em}\times \hspace{0.17em}n},$ where ${\epsilon }_{ij}=\left({\overline{T}}_{ij},\hspace{0.17em}{\overline{I}}_{ij},\hspace{0.17em}{\overline{F}}_{ij}\right)$ is done by applying the following expression as follows:

  $${\epsilon }_{ij}=\beta {\epsilon }_{ij}^{\left(1\right)}+\left(1-\beta \right){\epsilon }_{ij}^{\left(2\right)},$$

  (24)

  where ${\epsilon }_{ij}$ denotes the averaged NA-SVN-DM and β represents the normalization parameter changing from 0 to 1. Here, we take β= 0.5.
• Step 6: Calculate the weighted averaged NA-SVN-DM.
  Corresponding to Eq. (25), the weighted NA-SVM-DM ${\mathbb{N}}_w={\left({\stackrel{⌢}{\tau }}_{ij}\right)}_{m\hspace{0.17em}\times \hspace{0.17em}n},$ where ${\stackrel{⌢}{\tau }}_{ij}=\left({\stackrel{⌢}{T}}_{ij},\hspace{0.17em}{\stackrel{⌢}{I}}_{ij},\hspace{0.17em}{\stackrel{⌢}{F}}_{ij}\right)$ is constructed, where

  $${\stackrel{⌢}{\tau }}_{ij}=\left({\stackrel{⌢}{T}}_{ij},\hspace{0.17em}{\stackrel{⌢}{I}}_{ij},\hspace{0.17em}{\stackrel{⌢}{F}}_{ij}\right)=\left(1-{\displaystyle \prod _{k=1}^l{\left(1-{\overline{T}}_{ij}\right)}^{w_j}},\hspace{0.17em}{\displaystyle \prod _{k=1}^l{\left({\overline{I}}_{ij}\right)}^{w_j}},{\displaystyle \prod _{k=1}^l{\left({\overline{F}}_{ij}\right)}^{w_j}}\hspace{0.17em}\right).$$

  (25)
• Step 7: Evaluate the SVN-score ratings of weighted NA-SVN-DM of each option.
  From weighted NA-SVN-DM, we find the weighted normalized rating (L_i) for benefit-type attribute and the weighted normalized rating (M_i) for cost-type attribute as

  $$L_i\hspace{0.17em}=\hspace{0.17em}{\displaystyle {\sum }_{j\in C_b}S\left({\stackrel{⌢}{\tau }}_{ij}\right),}$$

  (26)

  $$M_i\hspace{0.17em}=\hspace{0.17em}{\displaystyle {\sum }_{j\in {\mathrm{C}}_c}S\left({\stackrel{⌢}{\tau }}_{ij}\right),}$$

  (27)

  where $S\left({\stackrel{⌢}{\tau }}_{ij}\right)$ symbolizes the SVN-score function of each rating of weighted averaged NA-SVN-DM.
• Step 8: Calculate the “final utility degree (FUD)” of each option.
  The FUD (g_i) of each option is obtained using Eq. (28) as

  $$s_i\hspace{0.17em}=\hspace{0.17em}\lambda \hspace{0.17em}{L}_i\hspace{0.17em}+\hspace{0.17em}\left(1-\hspace{0.17em}\lambda \right)\hspace{0.17em}{M}_i,$$

  (28)

  wherein λ exemplifies the parameter of changing the attribute type. If we involve both types of criteria, then we take λ= 0.5. Though, there is a choice to change the parameter λ by taking the different criterion type.
• Step 9: Based on the FUD (s_i), where i = 1, 2, …, m, prioritize the options.

5. Case Study: SHRM Assessment of Manufacturing Firms

HRM contributes considerably to manufacturing firms in India as each participant has an essential portion of firms, and their distinct features will reinforce the manufacturing procedures, attaining an growing in production (Saeidi et al., 2022). Thus, human resources are the main aspect for the affordability, efficacy, and expansion of manufacturing firms in India. This region being inside very competitive and dynamic market that needs continuous progress in all corporate regions, particularly paying better consideration to essential human resources, since they are persons that offer awareness, and capabilities to the firms and involve to the advance of innovative policies to attain continuous development and attainment of manufacturing firms. Additionally, the firms must distinguish the significance of the HRM to assurance its better management (Yánez Sarmiento et al., 2018). As whole, HRMs are vital for the appropriate working of a business, irrespective of its firm and size. Furthermore, the competitiveness of manufacturing firms is huge because of the feature of their HRM, so as to the abilities of staffs and their constant growth, in which training subjugates a vital place, is to become a stable aspect of competitive returns. Hence, committing time and resources to attain, sustain, and progress the capabilities of the firm's HRM becomes an essential strategic goal ( Saeidi et al., 2022).

Here, we use the proposed approach on an empirical study of manufacturing companies’ assessment from SVN information perspective. In this section, a panel of four DEs is created to identify the criteria and assess the alternatives based on considered criteria. Based on the literature review and online questionnaire, we have considered four companies alternatives and thirteen criteria. Details description of indicators is given in Table 1.

The required implementation procedure is presented in the following steps:

• Steps 1-2: Table 3 presents the linguistic variables and the consequent SVNNs (Hezam et al., 2023). Using Table 3 and Eqs (11)-(13), the significance degree of each DE is derived and depicted in Table 4.
• Step 3: Considering the LRs into mind, the DEs provide their opinions for each alternative with respect over indicators and required LAM is given in Table 5. To create the A-SVNDM, the SVNWA operator (14) is applied on Table 5 and required results are presented in Table 6.
• Step 4: From Eq. (5) and Eq. (15), the objective weight of indicators for SHRM assessment in manufacturing firms are estimated and presented as $w_j^o$ = {0.1003, 0.1015, 0.0654, 0.0883, 0.0855, 0.0926, 0.0807, 0.0307, 0.0852, 0.0613, 0.0296, 0.0867, 0.0922}.

Next, to determine indicators ranking for SHRM assessment in manufacturing firms, we compute the A-SVNDM and SVN-score value of indicators using Eq. (16)-Eq. (17) and given in Table 7. Based on the comparisons made by DEs, the MRC is determined using Eq. (18) and presented in Table 8. Based on the MRC, the SCWs are calculated using Eq. (19) and given in Table 8. From Eq. (20), we have calculated the subjective weights using the SVN-RANCOM tool of each indicator for SHRM assessment in manufacturing firms. The resultant values are shown in Figure 1 and presented as follows:

$w_j^s$ = {0.1006, 0.0769, 0.0355, 0.1124, 0.0533, 0.1243, 0.0651, 0.0059, 0.0888, 0.1361, 0.1479, 0.0355, 0.0178}.

From Eq. (21), we integrate the SVN-DM-based tool and the SVN-RANCOM models. The integrated weight with the combination of the SVN-distance-RANCOM tool for $\tau =0.5$ for SHRM assessment in manufacturing firms is depicted in the Figure 3 and is given by w_j = (0.1005, 0.0892, 0.0504, 0.1004, 0.0694, 0.1084, 0.0729, 0.0183, 0.087, 0.0987, 0.0887, 0.0611, 0.055).

Here, Figure 3 shows the variation of weight of diverse criteria for SHRM assessment in manufacturing firms. Green management and leadership (0.1084) is the most important indicators for SHRM assessment in manufacturing firms. Social investment (0.1005) is the second most criteria for SHRM assessment in manufacturing firms. Social integrity (diversity management) (0.1004) is third, Green employee relations (0.0987) is fourth, Equal employment opportunities (0.0892) is fifth most important criterion for SHRM assessment in manufacturing firms and remaining are measured crucial indicators for SHRM evaluation in manufacturing firms.

• Step 5: From Table 5 and Eq. (22)-Eq. (23), linear normalization matrix and vector normalization matrix are constructed and presented in Table 9 and Table 10. The next step is to combine the linear and vector normalized A-SVN-DM using Eq. (24) to determine the averaged NA-SVN-DM and depicted in Table 11.
• Step 6: From Eq. (25) and Table 11, we determine the weighted averaged normalized A-SVN-DM using SVNWAO or SVNWGO and criteria weights obtained in Eq. (21) and given in Table 12.
• Step 7-9: The L_i and M_i ratings are estimated using Eq. (26)-Eq. (27). The FUDs (s_i) value is calculated using Eq. (28). Here, we take λ = 0.5 as S₁, S₂, S₃ and S₄, are considered as cost-type and rest are benefit-type. Table 13 shows the SVN-score values of weighted averaged NA-SVN-DM to estimate the L_i, M_i and OADs (g_i). In this way, the following ranking order of manufacturing firms for SHRM assessment in manufacturing firms is obtained: F₁ (0.4241) > F₂ (0.3843) > F₃(0.3742) > F₄ (0.3672). Thus, the manufacturing firm-I (F₁) is most suitable one with highest FUD (0.4241) for SHRM assessment in manufacturing firms.

5.1. Sensitivity Analysis

The variation of indicators’ weight to objective and subjective for different parameter (ζ) degrees in the proposed SVN-distance measure-RANCOM tool, changes in type of S-HRM indicators from benefit to cost with different parameter (λ) and varying the parameter β from linear to vector normalization to illustrate the FUDs of manufacturing firms for SHRM assessment in manufacturing firms. The investigations are performed by considering three cases.

• Case I: When employing the normalization tool over diverse parameter (β) values. The changing in β from linear to vector normalization type is helping us to assess the sensitivity of the developed SVN-distance measure-RANCOM-AROMAN model to the prominence of normalization types. Table 14 and Figure 4 exemplify sensitivity of manufacturing firms for SHRM assessment in manufacturing firms over different normalization parameter β. According to the results, we find the same prioritizations F₁$\succ$F₂$\succ$F₃$\succ$F₄ for β = 0.0 to β = 1.0, which provides firm-I (S₁) is the best choice, while firm-IV (F₄) has the last rank for SHRM assessment in manufacturing firms. Thus, it is observed that the developed model holds suitable solidity with diverse parameter degrees.
• Case II: When considering weight-determining tool, objective (for ζ = 1.0) and subjective (for ζ = 0.0) weights are registered to offer enhanced weights for SHRM assessment in manufacturing firms. In this context, the weights of indicators are estimated by considering the objective and subjective weight separately in place of integrated weight of drivers. The prioritizations have been determined by changing the drivers’ weights from SVN-distance measure to SVN-RANCOM instead of SVN-distance measure-RANCOM weighting tool and depicted in Table 15 and Figure 5. Using SVN-distance measure (for ζ = 1.0), the FUD and preference of manufacturing firms are presented as follows: the FUD of option as F₁= 0.4179, F₂ = 0.3792, F₃ = 0.3749 and F₄ = 0.3609 and prioritization of firms is obtained as F₁$\succ$F₂$\succ$F₃$\succ$F₄. Applying the SVN-RANCOM (for ζ = 0.0) tool, the FUD and prioritization of manufacturing firm are estimated as follows: the FUD of options as F₁ = 0.4283, F₂ = 0.3877, F₃ =0.3718, and F₄ = 0.3717 and prioritization of manufacturing firms for S-HRM assessment in manufacturing firms is obtained as F₁$\succ$F₂$\succ$F₃$\succ$F₄. Based on aforesaid investigation, it is determined that by changing diverse parameter degrees will enhance the performance of developed SVN-distance measure-RANCOM-AROMAN method.
• Case III: When considering attribute changing parameter, benefit (for λ= 1.0) and cost-type (for λ = 0.0) indicators for SHRM assessment in manufacturing firms are taken separately to show changes of prioritization of manufacturing firms for SHRM assessment in manufacturing firms. In this context, the FUD of firms are computed by considering the benefit and cost indicators separately in place of integrated FUD of manufacturing firms for SHRM assessment in manufacturing firms. The prioritizations have been determined by changing the indicators from benefit to cost for SHRM assessment in manufacturing firms and depicted in Table 16 and Figure 6. Considering only benefit indicators (for λ = 1.0), the FUD and preference of manufacturing firms are presented as follows: the FUD of option as F1=0.7398, F2 =0.6707, F3 =0.6461 and F4 =0.6216 and prioritization of manufacturing firms is obtained as F1$\succ$F2$\succ$F3$\succ$F4. Applying the cost indicators (for λ = 0.0) tool, the FUD and prioritization of manufacturing firms are estimated as follows: the FUD of manufacturing firms as F1 = 0.1085, F2 = 0.0980, F3 =0.1022, and F4 = 0.1128 and prioritization of firms for SHRM assessment in manufacturing firms is obtained as F4 $\succ$F1$\succ$F3$\succ$F2. Based on aforesaid investigation, it is determined that by changing indicators type will enhance the performance of developed SVN-DM-RANCOM-AROMAN method.

5.2. Comparative Investigation

Here, we compare the results of proposed method and some of extant methods such as SVN-COPRAS (Hezam et al., 2022), SVN-WASPAS (Zavadskas et al., 2016), SVN-TOPSIS (Nancy and Garg, 2019) and SVN-CoCoSo (Rani and Mishra, 2020) tools.

5.2.1. SVN-COPRAS

The SVN-COPRAS model is applied on SHRM assessment problem of manufacturing firms. The sum of maximization criteria is obtained as ${\wp }_1=$ (0.489, 0.443, 0.483), ${\wp }_2=$ (0.466, 0.488, 0.519), ${\wp }_3=$ (0.446, 0.497, 0.527) and ${\wp }_4=$(0.434, 0.511, 0.543) and the sum of minimization criteria is estimated as ${\Im }_1=$ (0.127, 0.877, 0.864), ${\Im }_2=$ (0.118, 0.891, 0.877), ${\Im }_3=$ (0.120, 0.884, 0.869) and ${\Im }_4=$ (0.133, 0.871, 0.861). The relative degree of each alternative is ${\ell }_1=$ 0.3212, ${\ell }_2=$0.3103, ${\ell }_3=$0.3011 and ${\ell }_4=$ 0.2885. Finally, the degree of utility of each option is obtained as ${\alpha }_1=$ 100, ${\alpha }_2=$96.61, ${\alpha }_3=$93.73 and ${\alpha }_4=$ 89.82. Then the ranking order of the firms is $F_1\succ F_2\succ F_3\succ F_4$ and the “firm-1 (F₁)” is considered to be the best choice among the others for SHRM assessment in manufacturing firms.

5.2.2. SVN-WASPAS

The SVN-WASPAS model (Zavadskas et al., 2016) is implemented on aforesaid SHRM assessment problem. Using this model, the measures obtained through WSM are $U_1^{(1)}\hspace{0.17em}=$(0.633, 0.345, 0.391), $U_2^{(1)}\hspace{0.17em}=$ (0.625, 0.353, 0.401), $U_3^{(1)}\hspace{0.17em}=$ (0.605, 0.363, 0.418) and $U_4^{(1)}\hspace{0.17em}=$(0.590, 0.383, 0.418). In addition, the score values of WSM measures are $S\left(U_1^{(1)}\right)$= 0.632, $S\left(U_2^{(1)}\right)=$0.624, $S\left(U_3^{(1)}\right)=$0.608 and $S\left(U_4^{(1)}\right)=$ 0.597. The measures obtained through WPM are $U_1^{(2)}=$ (0.621, 0.322, 0.400), $U_2^{(2)}\hspace{0.17em}=$(0.616, 0.342, 0.416), $U_3^{(2)}\hspace{0.17em}=$ (0.599, 0.346, 0.414) and $U_4^{(2)}\hspace{0.17em}=$ (0.583, 0.368, 0.441). The score values of WPM measures are $S\left(U_1^{(2)}\right)=$0.633, $S\left(U_2^{(2)}\right)=$0.619, $S\left(U_3^{(2)}\right)=$0.613 and $S\left(U_4^{(2)}\right)=$ 0.591. Next, the UD of each firm is estimated as $U_1=$ 0.6327, $U_2=$0.6216, $U_3=$0.6106 and $U_4=$ 0.5939. Thus, the prioritization of manufacturing firm is $F_1\succ F_2\succ F_3\succ F_4$ and the most appropriate firm for SHRM assessment in manufacturing firms is manufacturing firm-1 (F₁).

5.2.3. SVN-TOPSIS

The SVN-TOPSIS model (Nancy & Garg, 2019) is implemented on aforesaid SHRM assessment problem. Using this model, the SVN-IS and SVN-AIS of options are calculated as ${\varphi }_j^{+}=${(0.294, 0.740, 0.690), (0.609, 0.357, 0.389), (0.666, 0.262, 0.315), (0.574, 0.356, 0.394), (0.778, 0.171, 0.201), (0.642, 0.286, 0.331), (0.628, 0.312, 0.356), (0.319, 0.697, 0.671), (0.660, 0.271, 0.320), (0.649, 0.273, 0.325), (0.304, 0.713, 0.673), (0.288, 0.752, 0.702), (0.249, 0.761, 0.734)} and ${\varphi }_j^{-}=${(0.419, 0.569, 0.568), (0.456, 0.471, 0.483), (0.583, 0.360, 0.399), (0.461, 0.528, 0.522), (0.644, 0.296, 0.340), (0.525, 0.438, 0.455), (0.526, 0.428, 0.454), (0.358, 0.662, 0.621), (0.548, 0.396, 0.426), (0.572, 0.384, 0.409), (0.337, 0.672, 0.641), (0.396, 0.607, 0.577), (0.363, 0.597, 0.566)}. Next, the degrees of discrimination of each option from SVN-IS are 0.006, 0.010, 0.013 and 0.018 and the degrees of discrimination of each option from SVN-AIS are presented as 0.020, 0.012, 0.011 and 0.006. The “relative closeness coefficient (RCC)” of each firm over the SVN-IS is estimated as 0.7735, 0.5600, 0.4698 and 0.2415. Thus, prioritization firms is $F_1\succ \hspace{0.17em}{F}_2\succ \hspace{0.17em}{F}_3\succ \hspace{0.17em}{F}_4$ and the most appropriate firm for SHRM assessment in manufacturing firms is manufacturing firm-1 (F₁).

5.2.4. SVN-CoCoSo

The SVN-CoCoSo model is applied on the SHRM assessment problem of manufacturing firms. In this model, the balanced compromise degrees of alternatives are determined as $Q_1^{\left(1\right)}=$0.2573, $Q_2^{\left(1\right)}=$0.2528, $Q_3^{\left(1\right)}=$0.2483 and $Q_4^{\left(1\right)}=$ 0.2415, $Q_1^{\left(2\right)}=$2.1309, $Q_2^{\left(2\right)}=$2.0932, $Q_3^{\left(2\right)}=$2.0563 and $Q_4^{\left(2\right)}=$ 2.0000, $Q_1^{\left(3\right)}=$1.0000, $Q_2^{\left(3\right)}=$0.9823, $Q_3^{\left(3\right)}=$0.9650 and $Q_4^{\left(3\right)}=$ 0.9386. Next, the OCDs of manufacturing firms are estimated as Q₁ = 1.9479, Q₂ = 1.9135, Q₃ = 1.8797 and Q₄ = 1.8283. Then the ranking order of manufacturing firms is $F_1\succ \hspace{0.17em}{F}_2\succ \hspace{0.17em}{F}_3\succ \hspace{0.17em}{F}_4$ and the most appropriate firm for SHRM assessment in manufacturing firms is manufacturing firm-1 (F₁).

Figure 7. Comparison of assessment degree of firms for SHRM assessment with various methods.

Figure 7. Comparison of assessment degree of firms for SHRM assessment with various methods.

5.3. Discussion

At the present time, sustainability is the major apprehensions of administrations and society. Along with all sectors of each country’s business, organizations are considerably trying to achieve SDGs. In this way and to promote firm’s sustainability, various firms are trying to establish relation with sustainability to HRM, which is formed an innovative standard named as “sustainability human resource management (SHRM)”. Corresponding to Ehnert et al. (2016), SHRM is associated to those actions which lead to attaining the SEE sustainability dimensions objectives inside and outside of the manufacturing firms.

From Figure 7, it can effortlessly be seen that most suitable “manufacturing firm (F₁)” is same for all the MCGDM approaches, namely SVN-COPRAS, SVN-WASPAS, SVN-TOPSIS and SVN-CoCoSo. The main advantages of the developed SVN-DM-RANCOM-AROMAN methodology are as follows:

• The SVN-DM proposed in this study overcomes the limitations of existing measures in order to enumerate the amount of distance between SVNSs.
• The developed tool uses weighting model using the combination of SVN-DM-based tool for objective weight and SVN-RANCOM model for subjective weight, which results in more accurate, optimal weights.
• Existing SVN-WASPAS and SVN-TOPSIS models consider direct weight of criteria, while the proposed model has used a rank sum-based procedure to compute the DEs weights. Thus, the proposed model has good effectiveness through the evaluation of manufacturing firms for SHRM assessment.
• The SVN-DM-RANCOM-AROMAN model uses the linear and vector normalization models to aggregate the indeterminate, inconsistent and uncertain information, therefore it provides more accurate decision than existing methods.

Figure 7. Depiction of the Sustainability aspects weight of SEE analysis by SVN-DM-RANCOM.

Figure 7. Depiction of the Sustainability aspects weight of SEE analysis by SVN-DM-RANCOM.

Based on the previous studies, organizations can do various activities in order to reach SHRM. In this study, in order to consider all earlier studies providing a systematic review and planning interviews, we have characterized aspects of SHRM into three SEE sustainability dimensions. This work has been documented five indicators to the social dimension: social investment, equal employment opportunities, poverty reduction and social equity, social integrity and corporate social responsibility, considered five indicators with the environmental dimension as green management and leadership, green culture strategy, training and development, green workspace, green employee relationship, and finally three indicators comprising of green performance appraisal, green profitability, employee efficiency, creating the economic dimension.

Moreover, in Figure 7, assessment outcomes of manufacturing firms for S-HRM assessment are prioritized as: Social (0.4099) ≻ Environmental (0.3853) ≻ Economic (0.2048), which means social dimension have highest impact on manufacturing firms for S-HRM assessment, followed by Environmental and Economic and risks. Aforementioned assessments, four manufacturing firms for S-HRM assessment continuously preserve the prioritization in spite of how indicators weight varies. It can be observed the firmness and practicality of SVN-DM-RANCOM-AROMAN model.

6. Conclusions

SHRM plays a crucial role in SEE sustainability dimensions management. In this work, we have introduced hybrid MCGDM model called “SVN-DM-RANCOM-AROMAN” to evaluate SHRM indicators in the manufacturing firms. For this purpose, new SVN-distance measure has been presented for SVNSs, which overcomes the limitations of extant SVN-DMs. In the developed model, the objective weight of indicators has been computed with the SVN-DM-based model, while the subjective weights of indicators have been derived through novel SVN-RANCOM model. Further, the integrated weight of indicators for SHRM assessment in manufacturing firms has been determined with SVN-DM-RANCOM tool. To rank the firms, the AROMAN method has been combined with the SVN-DM-RANCOM model from SVNNs perspectives. To validate the proposed method, we have executed it on a case study of SHRM assessment in manufacturing firms in India with multiple indicators and DEs. In this study, four manufacturing firms have been extensively assessed by means of 3 SEE dimensions and 13 indicators in the presence of four DEs. Furthermore, comparison with extant models has been presented to confirm the robustness of developed model. Moreover, sensitivity investigation has been made over diverse normalization and weighting parameters. The advantages of the developed approach include the development of new SVN-distance measure, linear and vector normalization tools and criteria weighting through SVN-distance measure-RANCOM model. Thus, the developed model is more capable and accurate while making decisions under SVNSs environment.

This study has some limitations such as it does not consider the interrelationships among the criteria. In addition, more decision experts should be considered during the assessment of SHRM factors in manufacturing firms. In future, we will try to evade the shortcomings of presented work by developing new model based on Shapley function, prospect theory and evidential reasoning under SVNSs environment. Moreover, we can extend the present study under the context of Plithogenic sets, linear Diophantine sets, hyperbolic sets and others.